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+{"text":"\\section{Introduction}\n    Phase retrieval aims to reconstruct a complex-valued signal from its intensity-only measurements. It is a crucial problem in crystallography, optical imaging, astronomy, X-ray, electronic imaging, etc., because most existing measurement systems can only detect the relative low-frequency magnitude information of the wave fields. The study of the phase retrieval problems originated in the $1970$s. The researchers of the optics community developed many reconstruction algorithms \\cite{Gerchberg1972APA, Fienup:82, rodenburg2008ptychography}. \n    \n    Mathematically, the phase retrieval problem can be described by the following equation:\n    \\begin{equation}\n    \t\\label{CDP}\n    \t\\begin{aligned}\n    \t\t\\text { Find } \\mathbf{x} \\in \\mathbb{C}^{N} \\quad \\text { s.t. } \\ \\mathbf{y}_{i}=\\left|\\mathcal{F}\\left(\\mathbf{I}_{i} \\circ \\mathbf{x}\\right)\\right|^{2}, i=1, \\ldots, M,\n    \t\\end{aligned}\n    \\end{equation}\n    where $\\mathbf{x}\\in \\mathbb{C}^N$ is the complex-valued signal of interest, and $ \\mathbf{y}$ is its Fourier intensity measurements; $\\circ $ and $\\mathcal{F}$ refer to elementwise multiplication and Fourier transform operator, respectively. The pre-defined optical masks $ \\mathbf{I} $ provide the constraints to reduce the ill-posedness of the problem. It is implemented in many different ways. For instance, early-stage phase retrieval algorithms treat the support of the signal as the optical mask \\cite{Fienup:82}. Specifically, $|\\mathbf{x}_i| \\neq 0$ when $i \\in \\mathcal{S}$, where $\\mathcal{S}$ denotes the support of the signal. However, these early-stage algorithms cannot guarantee the convergence of the optimization process nor provide globally optimal solutions. Recently, pre-defined random optical masks have been used as the constraints to improve the reconstruction performance \\cite{Anand07, Horisaki2014SingleshotPI, candes2015phase, YE2022106808}. The random masks can be implemented using a spatial light modulator (SLM) or digital micromirror device (DMD) \\cite{Horisaki2014SingleshotPI, Zheng2017DigitalMD}. Although the usage of the random masks can lead to better reconstruction performance, there are several disadvantages. The cost of DMD and SLM is one concern, the error due to the global gray-scale phase mismatch and spatial non-uniformity from the devices is another. Besides, it was empirically shown that about $ 4 - 6 $ measurements are required for exact recovery with random binary masks \\cite{candes2015code, candes2015phase}. It increases the measurement time and can affect the quality of the reconstructed image, particularly for dynamic objects.\n    \n    Recently, data-driven methods, like deep neural networks, have been widely applied to explore the data distributions through extensive training samples \\cite{LeCun_2015}. Among them, the so-called Convolutional Neural Network (CNN) that exploits the features based on convolution kernels has been successfully adopted in image processing tasks, like image restoration \\cite{He_2016_CVPR, zhang2017beyond, wang2020lightening}. Besides, an important technique named attention mechanism that mimics cognitive attention became popular in the CNN structure to enhance some parts of the feature maps and thus improve the performance \\cite{attention_17}. Since CNN can learn discriminative representations from large training datasets, researchers have spent effort in using CNN for solving phase retrieval problems \\cite{deng_li_goy_kang_barbastathis_2020, White:19, Shi:19, Sinha17}. For instance, Chen et al. reconstructed signals from Fresnel diffraction patterns with CNN and the contrast transfer function \\cite{Bai:19}. Kumar et al. proposed the deep unrolling networks that solved the iterative phase retrieval problem with CNN \\cite{Kumar:21}. Besides, the conditional generative adversarial network (GAN), a kind of data-driven generative method based on specific conditions \\cite{cond_gan_14}, was adopted for reconstructing the phase information from Fourier intensity measurements \\cite{uelwer2021phase}. Wu et al. also proposed a CNN to reconstruct the crystal structures with coherent X-ray diffraction patterns \\cite{Wu_cw5029, Wu_2021}. The above  methods adopt supervised learning that trains the networks with the paired ground-truth data. In recent years, training with unpaired or unsupervised (no ground-truth data) datasets has been a popular topic. For example, Fus et al. proposed the unsupervised learning for in-line holography phase retrieval \\cite{Fus_18}, while Zhang et al. developed the GAN-based phase retrieval with unpaired datasets \\cite{Zhang_21}. While the above approaches have achieved some success, phase retrieval with Fraunhofer diffraction patterns (i.e., Fourier intensity) for general applications is still a challenge. It is partly because of the large domain discrepancy between the Fourier plane and image plane; the large variation in data distributions of general applications also introduces much difficulty in designing a model structure that is optimal for different applications.  In this paper, we propose a single-shot maskless phase retrieval method named \\textit{SiPRNet} that is benefited from the advanced deep learning technology. Compared with the traditional algorithms that need multiple intensity measurements, the proposed method only collects one Fourier intensity measurement for reconstruction. Besides, no optical mask is needed in the proposed method; thus, the cost  for expensive SLM\/DMD devices is saved. The proposed \\textit{SiPRNet} is designed to work with general image datasets without assuming a specific application domain. It has demonstrated its generality by achieving state-of-the-art performance on three different datasets, as shown in our experimental results. It significantly outperforms the existing optimization-based and learning-based phase retrieval methods.\n    \n    The contributions of our method are: (1) We propose a novel CNN structure - \\textit{SiPRNet} that utilizes advanced deep learning techniques for single-shot maskless phase retrieval. It can reliably reconstruct the original signal based on only one Fourier intensity measurement without using optical masks.  (2) We propose a Multi-Layer Perception (MLP) block to extract global representations from the intensity measurement. In addition, we propose to embed a dropout layer into the MLP block to constrain the behavior of the fully connected layers, which promotes the MLP to extract global representation. (3) We propose to use residual learning and self-attention mechanism in the Up-sampling and Reconstruction blocks, which improves the quality of the reconstructed signals. (4) We verify the proposed method on an optical platform to demonstrate its practicality. The source codes of the proposed method have been released on Github: \\url{https:\/\/github.com\/Qiustander\/SiPRNet}.\n    \n    The rest of this paper is organized as follows. Section \\ref{Sec:Method} introduces the defocused phase retrieval system we developed in this study and the proposed end-to-end deep-learning phase retrieval network \\textit{SiPRNet}. Section \\ref{Sec:simulation} presents the model analysis and simulation results. Section \\ref{Sec:experiment} provides the experimental results on an optical system for validating the performance of the proposed method. \n      \n\\section{Proposed Method} \n\\label{Sec:Method}\n\n\\subsection{Phase Retrieval System} \\label{Sec:PRsystem}\n\nAs indicated in Eq. \\eqref{CDP}, a phase retrieval system aims to reconstruct a complex-valued signal $ \\mathbf{x} $ from its Fourier intensity measurements. The proposed method does not require multiple measurements or optical masks, so $ M=1 $ and all the values of the mask $ \\mathbf{I} $ are equal to one in Eq. \\eqref{CDP}. In this case, the original complex-valued signal $ \\mathbf{x} $ is reconstructed only from its Fourier intensity. This seemingly impossible task is known to have a solution, although with ambiguities. From \\cite{hayes1982reconstruction}, it is known that $ \\mathbf{x} $ can be uniquely defined, except for trivial ambiguities, by its Fourier intensity, with an oversampling factor over $ 2 $ in each dimension, if $ \\mathbf{x} $ has a finite support and is non-symmetric. The above has an important implication to the optical system required for the proposed phase retrieval method. Fig. \\ref{fig:opticalpath} shows the system we have constructed to implement the proposed method. In the system, we increase the resolution of the CMOS camera to make sure that the captured Fourier intensity image has a sampling rate at least two times in each dimension higher than that of the object image $ \\mathbf{x} $. As a result, the number of samples in the zero diffraction order of the Fourier intensity image is $ 762\\times762 $, while the number of samples of $ \\mathbf{x} $ is $ 128\\times128 $. We then extract the central $ 128\\times128 $ pixels of the Fourier intensity measurements for feeding to the proposed CNN model for phase retrieval. \n\n\\begin{figure}[htb]               \n\t\\centering\\includegraphics[width=0.8\\linewidth]{Figures\/Fig_Optical_Path.pdf}\n\t\\caption{Optical path of the defocus phase retrieval system. $\\mathbf{x}$ represents the complex-valued object, and $\\mathbf{y}$ is its Fourier intensity measurement; $\\mathbf{X} = \\mathcal{F}\\left(\\mathbf{I}_{i} \\circ \\mathbf{x}\\right)$,  $\\mathbf{H}$ and $\\ast$ denote the Fourier plane in complex-valued form, the defocus function generated via Fresnel diffraction and the convolution operation, respectively. }\n\t\\label{fig:opticalpath}\n\t\n\\end{figure}\n\nWhen capturing Fourier intensity images, it is common to have the saturation problem. It is because most structured signals have the energy concentrated in the low-frequency areas, in particular the zero frequency. Due to the limited analog-to-digital resolution of standard scientific cameras ($14$ - $16$ bits), the central regions of the Fourier intensity measurement are often seriously saturated. To lessen the effect of the saturation problem, we convolve the Fourier intensity measurement with a defocus function $ \\mathbf{H} $ to reduce the dynamic range \\cite{goodman2017introduction}. It can be implemented by moving the camera beyond the Fourier plane. Fig. \\ref{fig:opticalpath} shows a typical optical path for implementing the defocus phase retrieval system. Fig. \\ref{fig:compvisual} shows some examples of the Fourier intensity measurements obtained in our experiments. \n\nBesides the lens-based system that we have implemented in this work, we can also find lensless phase retrieval systems \\cite{Sinha17}. Different from the lens-based system that reconstructs images from their Fraunhofer diffraction patterns, the lensless system reconstructs images based on their Fresnel diffraction patterns \\cite{Sinha17}. Compared with the lens-based system, the lensless system requires a much longer distance between the sensors and objects. For instance, in \\cite{Sinha17}, the distance between the sensor and object is reported as $37.5 cm$  or longer. It would lead to severe aberrations and energy loss. By using a lens, the lens-based system can generate high-quality Fraunhofer diffraction patterns at a short propagation distance between the lens and sensor ($3 cm$  or less), thus avoiding the drawbacks stated above.\n\n\n\\subsection{Deep Learning Model: \\textit{SiPRNet}}\n\nWith the oversampling constraint as mentioned above, it is possible to reconstruct a complex-valued signal (with trivial ambiguities) from only its Fourier intensity without other constraints (such as optical masks). However, such a reconstruction problem is highly non-convex that is difficult to solve with traditional optimization methods. Due to their non-linear characteristic, deep neural networks have been successfully applied to many non-convex optimization problems. In this paper, we propose to regard the single-shot maskless phase retrieval challenge as an end-to-end learning task, and formulate the retrieval process directly by a deep neural network model (denoted as $P(\\mathbf{y};\\theta)$) as:\n\n\\begin{dmath}\n\t\\hat{\\mathbf{x}} = P(\\mathbf{y};\\theta),\n\\end{dmath}\nwhere $\\theta$ represents the learnable parameters of the deep learning model. The training target is to minimize the error between the predicted $P(\\mathbf{y};\\theta)$ and the ground-truth complex-valued image $ \\mathbf{x} $ regularized with the total variation (TV) norm $ TV(\\hat{\\mathbf{x}}) $. The training process can be written as:\n\n\\begin{equation}\n\tP(y;\\theta) = \\mathop{argmin}\\limits_{\\hat{\\mathbf{x}}} \\left( \\left\\| \\mathbf{x}-\\hat{\\mathbf{x}} \\right\\|_1 +\\lambda TV\\left(\\hat{\\mathbf{x}}\\right)\\right), \n\\end{equation}\nwhere $\\left\\| \\cdot \\right\\|_1$ denotes the $\\ell_1$-norm distance  and $ \\lambda $ is a constant determined empirically.\n\nFig. \\ref{fig:structure} shows the overall structure of the proposed phase retrieval network: \\textit{SiPRNet}. It mainly consists of two parts: feature extraction and phase reconstruction. It takes the Fourier intensity measurement $ \\mathbf{y} $ as input and reconstruct the complex-valued image $ \\mathbf{x} $ as its output.\n\n\\begin{figure} [htb]\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{Figures\/Fig_Network_Structure.pdf}\n\t\\caption{Structure of the proposed network that mainly consists of two parts: Feature Extraction and Phase Reconstruction.}\n\t\\label{fig:structure}\n\\end{figure}\n\n\\subsubsection{\\textbf{Feature Extraction}}\nTraditional learning-based approaches often use a CNN as the front end to extract the features of the Fourier intensity measurement. However, the convolution operation of CNN shares the parameters (e.g., weights and bias) for all data of the measurement but ignores their different characteristics. Therefore, we propose to use a new Multi-Layer Perceptron (MLP) block to fuse the information of different frequency data as shown in Fig. \\ref{fig:structure}(a). The MLP block consists of three Fully Connected (FC) and one Dropout layer. The input Fourier measurement is firstly reshaped from the size of $128\\times 128\\times 1$ to $1\\times1\\times16,384$, and then fed to the first FC layer of the MLP block. Each FC layer has $1,024$ neurons for extracting the features required for reconstructing the original complex-valued images. We propose to embed a Dropout layer after the first FC layer. The Dropout layer randomly deactivates a small number (empirically $20\\%$) of neurons at the training stage. At the inference stage, all neurons are activated that forms an ensemble to reduce the risk of overfitting \\cite{srivastava2014dropout}. Our design with embedding a Dropout layer after the first FC layer constrains the behavior of the FC layers so that the features extracted by the MLP block do not depend on a particular input frequency measurement. The effectiveness of the dropout layer has been evaluated and shown in Section \\ref{Sec:ablationstudy}. Fig. \\ref{fig:structure} shows an example of the original complex-valued image (real and imaginary), the Fourier intensity measurement, and the features extracted by the MLP block. It can be seen that the MLP block has performed the domain conversion. The extracted feature image contains the features of the original complex-valued image. A decoder is needed to reconstruct the real and imaginary images from the extracted features. \n\n\n\\subsubsection{\\textbf{Phase Reconstruction}}\nThe phase reconstruction part contains two Up-sampling and Reconstruction (UR) blocks and one Post-processing block, as shown in Fig. \\ref{fig:structure}(b) and (c). The UR block is responsible for reconstructing the complex-valued image from the extracted features and progressively upsampling it back to the original size. The basic processing unit of the UR block is the convolution block that has three components: convolutional, instance normalization, and Parametric Rectified Linear Unit (PReLU) layers. The convolutional layer contains $3\\times 3$ learnable filters that aim to implicitly approximate the nonlinear backward mapping from the extracted features to the desired complex-valued image $ \\mathbf{x} $. The instance normalization and PReLU layers are non-linear functions that increase the non-linear capacity and learning ability of the deep learning model. \n\nA self-attention unit is followed after each convolutional block. Recently, the self-attention technique has been popularly used in image reconstruction tasks, e.g., the scattering problem \\cite{non-local, Wang:21}. Traditional CNN using small convolution kernels can only extract the local correlation of an image. However, natural images are often structural that exhibit global correlation. Self-attention is an effective way to exploit the global correlation in images. The structure of the self-attention layer used in the UR block is similar to that in \\cite{non-local}, as shown in Fig. \\ref{fig:attn}. At the first stage, the input features are fed to three $1\\times1$ convolutional layers ($\\theta, \\phi$ and $g$). The outputs are denoted as key, query, and value, respectively, all with the same dimensions. Next, the cross-correlation matrix is computed between the key and query. It represents the dependency between any two pixels of the input features. The softmax function is then performed to normalize the correlation scores and build an attention map. Large values in the attention map denote strong correlations between the features concerned. Since the correlation is computed among all features, the attention is made not only to those with local correlations but also the global ones. The result is finally multiplied with the value to generate the self-attention feature maps. They are then added back to the original features to strengthen the features of interest. Such an attention mechanism is important to the training of the UR block. It allows the UR block to reconstruct images with correlated features. For instance, for the extracted features as shown in Fig. \\ref{fig:structure}, the UR block will try to reconstruct both the left and right eyes together since they have a high correlation as indicated in the training face images in the dataset. Since the correlation of the features may change at different resolutions, another UR block is used after the first one on the features of higher resolution.\n\n\\begin{figure} [htb]\n\t\\centering\n\t\\includegraphics[width=0.5\\linewidth]{Figures\/Fig_Atten_Structure.pdf}\n\t\\caption{Structure of the attention mechanism.}\n\t\\label{fig:attn}\n\\end{figure}\n\nDirect stacking covolutional and attention layers would lead to the obvious performance drop because of weak information flow and gradient vanishing \\cite{He_2016_CVPR}. Therefore, we add a shortcut connection after the convolution block and attention mechanism that forms a residual structure (shown in Fig. \\ref{fig:structure}(b)), which is widely adopted in image processing tasks \\cite{Wang_2017_CVPR, zhang2021explainable, park2021dynamic}. The residual setting avoids the gradient vanishing problem and therefore reduces the learning difficulty. As shown in Fig. \\ref{fig:structure}, the self-attention layers effectively estimate the residue between the input feature and desired one. Finally, we utilize the unshuffle layer to reshape the feature map to enlarge the spatial sizes. The effectiveness of the residual self-attention structure is further discussed in Section \\ref{Sec:ablationstudy}.\n\n\n\\subsubsection{\\textbf{Post-processing block}}\nThe Post-processing block contains two convolutional layers (shown in Fig. \\ref{fig:structure}(c)). It estimates the real $\\hat{\\mathbf{x}}_{Re}$ and imaginary $\\hat{\\mathbf{x}}_{Im}$ parts of the image from the up-sampled features. Then, the phase part $\\hat{\\mathbf{x}}$ of the image can be calculated by:\n\\begin{equation}\n\t\\label{Eq:converttophase}\n\t\\hat{\\mathbf{x}}=tan^{-1}\\left( \\frac{\\hat{\\mathbf{x}}_{Im}}{\\hat{\\mathbf{x}}_{Re}} \\right).\n\\end{equation}\n\n\n\\subsubsection{\\textbf{Loss Function}}\nWe regard the phase retrieval problem as a supervised learning task. At the training stage, we prepare a set of paired training data. Each Fourier intensity measurement has a paired complex-valued image as the ground-truth target. Then, the deep learning model can be optimized by minimizing the $\\ell_1$-norm distance between the estimation and the ground-truth images plus the TV-norm to improve the image edges. The loss function $\\mathcal{L}$ is defined as: \n\\begin{equation}\n\t\\mathcal{L} = \\left\\| \\mathbf{x}_{Re}-\\hat{\\mathbf{x}}_{Re} \\right\\|_1 + \\left\\| \\mathbf{x}_{Im}-\\hat{\\mathbf{x}}_{Im} \\right\\|_1 + \\lambda TV\\left(\\hat{\\mathbf{x}}\\right).\n\\end{equation}\nThe symbols $\\mathbf{x}_{Re}$ and $\\mathbf{x}_{Im}$ represent the ground-truth real and imaginary parts of the image, respectively. \n\n    \\section{Simulation Results} \\label{Sec:simulation}\n\n\\subsection{Simulation Setup}\n\nWe conducted the simulation on two different datasets: the Real-world Affective Faces (RAF) dataset \\cite{li2017reliable} and Fashion-MNIST dateset \\cite{xiao2017\/online}. Specifically, the training data was converted to gray-scale image and resized to $128 \\times 128$. Then the pre-processed images $\\mathbf{x} \\in [0, 1]$ were used to generate complex-valued images of two forms: phase-only and magnitude-phase. For generating the phase-only images, we mapped $\\mathbf{x}$ to the $2\\pi$ phase domain via an exponential function: $\\exp(2\\pi i \\mathbf{x})$, with the magnitude kept as $1$. On the other hand, we used the images of Fashion-MNIST dataset to generate the magnitude-phase images. To be specific, we directly used the pre-processed images $\\mathbf{x} \\in [0, 1]$ as the magnitudes of the images, i.e., $\\mathbf{x}_{mag} = \\mathbf{x} \\in [0, 1]$. Then we set the phase parts through $\\mathbf{x}_{phase} = \\exp(2\\pi i \\mathbf{x}_{mag})$ and combined them by $\\mathbf{x} = \\mathbf{x}_{mag} \\circ \\mathbf{x}_{phase}$ as the magnitude-phase training and testing images, where $\\circ$ denotes the elementwise multiplication.\n\nAs mentioned in Section \\ref{Sec:Method}, we suggest to reduce the saturation problem by convolving the Fourier intensity measurement with a defocus function. Based on the convolution property of the Fourier transform, we can simulate it by multiplying the testing image with a defocus kernel $\\mathbf{h}$ in the spatial domain. It can be generated through the Fresnel propagation \\cite{goodman2017introduction}. Denote the original image as $x(p, q)$ and $X(u, v)$ as its  Fourier transform, then the optical field $X_L(u', v')$ in the defocus plane is expressed as:\n\\begin{dmath}\n\t\\label{Eq:defocuskernel}\n\tX_L(u', v') \n\t=  C \\iint X(u, v) H(u - \\frac{u'}{\\lambda L}, v - \\frac{v'}{\\lambda L}) du\\ dv,\n\\end{dmath}\nwhere $\\lambda$, $L$ and $H$ denotes the wavelength, distance between the lens after the object and the defocus plane, and the Fourier transform of the defocus kernel $\\mathbf{h}$, respectively. $C$ is the constant and $h(p, q)$ is given as $e^{\\frac{j\\pi}{\\lambda L}c(p^{2}+q^{2})}$. Based on the convolution theorem, Eq. \\eqref{Eq:defocuskernel} is equivalent to the elementwise multiplication between $h$ and $x$ in the spatial domain with the scaling factor $\\lambda L$. \n\nThen, oversampled discrete Fourier transform (DFT) with size $ 762\\times762 $ (the same size as in real experiments) was performed to generate the intensity measurements. For each image, the Fourier intensity measurement was capped at the value $ 4095 $ ($12$-bit) to simulate the dynamic range of the practical scientific cameras. The central $128\\times 128$ patch was cropped as the input of the \\textit{SiPRNet} (the performance of different input sizes is discussed in Section \\ref{Sec:ablainputsize}). The total epoch and mini-batch size were set as $1,000$ and $32$. The Adam optimizer \\cite{kingma2017adam} with the learning rate of $10^{-4}$ was implemented to update the gradients in each training step. Note that the learning rate was updated through \\textit{StepLR} scheduler in PyTorch every $100$ epoches with $\\gamma = 0.9$.\n\nWe adopted Peak Signal to Noise Ratio (PSNR, higher the better), Structural Similarity (SSIM, higher the better) and Mean Absolute Error (MAE, lower the better) as the performance metrics to measure the difference between the reconstructed images $\\hat{\\mathbf{x}}$ and ground truth images $\\mathbf{x}$. For each dataset, we selected the first $1,000$ images in the test set as the testing samples for evaluation. We converted the reconstructed real and imaginary parts to phase-only images via Eq. \\eqref{Eq:converttophase}. The phase values of both images were shifted by $\\pi$ to ensure non-negative values for the computation of PSNR. The average PSNR, SSIM and MAE were acquired among all $1,000$ testing images for evaluation. \n\n\\subsection{Ablation Study} \\label{Sec:ablationstudy}\n\nTo gain a deeper insight into the improvements obtained by \\textit{SiPRNet}, we conducted additional ablation studies on different components of our framework and analyzed their impact on the speed and accuracy of the network.\n\n\n\\subsubsection{\\textbf{Impact of Input Size}} \\label{Sec:ablainputsize}\n\nFor the two-dimensional Fourier spectrum, the central regions represent the low-frequency information while outer areas contain the high-frequency components. Structured signals (e.g., natural images) usually have their energy concentrated in the low-frequency regions; hence the high-frequency components have relatively lower intensity and are dominated by noise. An ablation analysis was performed to determine the size of the central regions to be used. In our experiments, the intensity measurements have the size of $762 \\times 762$. We fed the central $[32 \\times 32, 64 \\times 64, 128 \\times 128, 256 \\times 256, 512 \\times 512]$ of the intensity measurement to the same network and study the performance, as shown in Fig. \\ref{fig:Inputsize}. The average testing PSNR from $30000th$ to $35000th$ training steps was chosen for comparison. The box-plot of the performance, the floating point operations per second (FLOPs) and the numbers of trainable parameters with respect to the sizes are shown in Fig. \\ref{fig:Inputsize}. Although the model performance keeps increasing along with the increase of input size, the number of the trainable parameters also boosts quickly, which brings a huge computational burden. For example, it can be seen that the number of parameters dramatically increases from $ 19.3M $ to $ 69.6M $ when the input size changes from $128 \\times 128$ to $256 \\times 256$. While the growth speed of the performance is linear when the input size is smaller than $256 \\times 256$, it saturates as the input size increases to about $128 \\times 128$. Therefore, we adopt $128 \\times 128$ as the input size to balance performance and efficiency. \n\n\\begin{figure}[htb]\t\t\n\t\\centering\n\t\\includegraphics[width=0.7\\linewidth]{Figures\/Fig_Ablation_Input_Size.pdf}\n\t\\caption{Ablation study on the input sizes. The results are obtained with the RAF dataset \\cite{li2017reliable}.}\n\t\\label{fig:Inputsize}\n\\end{figure}\n\n\\subsubsection{\\textbf{Impact of Defocus Distance}} \\label{Sec:abladefocusdist}\n\nThe effect of the defocus distance $L$ is discussed in this section. We investigated $6$ situations of the defocus distance: $0mm$ (no defocusing), $15mm$, $20mm$, $30mm$, $45mm$, and $75mm$. The defocus kernel is a pure-phase object generated by Holoeye built-in function.  The default defocus distance is $30mm$ in our proposed \\textit{SiPRNet}. The size of the input intensity measurements is $128 \\times 128$. The box-plot for average PSNR is presented in Fig. \\ref{fig:Defocusdist}. As shown in the figure, the reconstruction performance significantly degrades when there is no defocusing ($L = 0 mm$). It is because some pixels are severely saturated and give wrong information to the network. On the other hand, the performances are similar when $L = 15$ to $45 mm$ although the PSNR is slightly higher when $L = 30 mm$. It shows that the performance of the system is not sensitive to the choice of $L$ as long as it is not too small or too big.  When the defocus distance is large, i.e. $75 mm$, the PSNR significantly drops. It is because a portion of the Fourier intensity measurements (high-frequency areas) become very small due to the decrease in light intensity with the long defocus distance.  To balance the illumination efficiency and the reconstruction performances, the defocus distance $L$ was selected as $30 mm$ for all experiments.\n\n\\begin{figure}[htb]\t\t\n\t\\centering\t\\includegraphics[width=0.7\\linewidth]{Figures\/Fig_Ablation_DefocusDist.pdf}\n\t\\caption{Ablation study on the defocus distance. The results are obtained with the RAF dataset \\cite{li2017reliable}. }\n\t\\label{fig:Defocusdist}\n\\end{figure}\n\n\n\\subsubsection{\\textbf{Impact of Dropout Layer}} \\label{Sec:abladropout}\n\nWe investigated the effect of the number of Dropout layers in feature extraction part. We set the dropout rate as $0.2$ in every Dropout layer. Since $3$ FC layers are used in the MLP blocks, we investigated $4$ situations when introducing the Dropout layers: $0$ Dropout layer, $1$ Dropout layer after the first FC layer, $2$ Dropout layers after each of the first two FC layers, and $3$ Dropout layers after each of the FC layers. One Dropout layer is the default setting in our proposed \\textit{SiPRNet}. The size of the input intensity measurements is $128 \\times 128$. The box-plot for average PSNR is presented in Fig. \\ref{fig:Dropout}. As shown in the figure, if there is no Dropout layer, the average PSNR of the reconstructed images is the lowest because of the over-fitting problem. The network is lazy to investigate the hidden frequency representations, and the reconstruction relies on a small number of inputs. The Dropout layer randomly deactivates a part of the inputs at the training stage, which inspires the network to find more global and robust representations. At the testing stage, all input signals are used to form an ensemble learning structure that increases the generalization power of the whole network. After implementing the Dropout layer, the model performance boosts as expected. Interestingly, with more dropout layers applied, there is too much information ignored in the training stage that hurts the learning progress of the network and decreases the performance. Therefore, we adopt $30 mm$ as the defocus distance to balance performance and illumination efficiency. \n\n\\begin{figure}[htb]\t\t\n\t\\centering\t\\includegraphics[width=0.7\\linewidth]{Figures\/Fig_Ablation_Dropout.pdf}\n\t\\caption{Ablation study on the number of Dropout layers in feature extraction. The results are obtained with the RAF dataset \\cite{li2017reliable}. }\n\t\\label{fig:Dropout}\n\\end{figure}\n\n\n\\subsubsection{\\textbf{Impact of Residual Units}} \\label{Sec:ablaresidual}\n\nWe also investigated the effects of the number of residual units in the phase reconstruction part. As shown in Fig. \\ref{fig:structure}, each residual unit contains two layers of Conv Blocks interlaced with two self-attention layers plus the PReLU at the end. The size of the input images to the network is still $128 \\times 128$. With everything remaining the same but just changing the number of residual units in each UR block, the average PSNRs of the estimated images are obtained and shown in Fig. \\ref{fig:ResLayer}. It can be seen that using the residual units brings in better PSNR performance. The model with one residual unit has similar or even better performance than the models with more residual units. To pursue better PSNR and efficiency, we choose to use one residual unit in the UR block.\n\n\\begin{figure}[htb]\t\t\n\t\\centering\t\\includegraphics[width=0.7\\linewidth]{Figures\/Fig_Ablation_ResLayer.pdf}\n\t\\caption{Ablation study on the number of residual units. The results are obtained with the RAF dataset \\cite{li2017reliable}. }\n\t\\label{fig:ResLayer}\n\\end{figure}\n\n\\subsubsection{\\textbf{Model Robustness}} \\label{Sec:CrossValidation}\nBesides the generalization ability, the robustness is also a crucial metric to evaluate neural networks. The more robust the network, the less probable the overfitting problem would happen in the network. In order to demonstrate the robustness of the proposed \\textit{SiPRNet}, we utilized the K-Fold cross validation, a statistical method based on the resampling procedure \\cite{Stone_1974}. The procedure of the K-Fold cross validation can be briefly described as follows: \\textit{1.} Randomly shuffle the dataset. \\textit{2.} Split the dataset into $K$ groups.  \\textit{3.} In each fold, take a group as the testing dataset and the remaining groups as the training dataset. Then, fit the deep learning model on the training dataset and do the evaluation on the testing dataset. \\textit{4.} Reset the model parameters and repeat step \\textit{3} until $K$ folds are accomplished. \\textit{5.} Summarize the model evaluation scores obtained in each fold. To be specific, we adopted $K=5$ folds for evaluation and we ran $100$ epochs in each fold. In each fold, $80\\%$ of the samples were treated as the training data and the rest $20\\%$ were utilized for testing. We used the training (for the training dataset) and testing (for the testing dataset) losses as the evaluation scores. The training loss was recorded every $20$ training batches and the testing loss was collected every epoch. The simulated results on the RAF and Fashion-MNIST datasets are shown in Fig.\\ref{Fig:KFold}. As can be seen, both losses decrease quickly with the growth of the training epoch for both datasets. The curves of the losses are similar in each fold. Although there are small variations at the beginning of the training stage ($5-15$ epochs in Fig.\\ref{Fig:KFold}(a) and  $10-25$ epochs in Fig.\\ref{Fig:KFold}(b)), all folds converge to the optimal direction and thus are close to the average fold curve. In conclusion, the K-fold cross validation results demonstrate the robustness and effectiveness of \\textit{SiPRNet}. \n\n\\begin{figure}[htb]\n\t\\centering\n\t\\begin{subfigure}[t]{\\textwidth}\n\t\t\\begin{tabular}{*{2}{c@{\\extracolsep{0em}}} }\n\t\t\t\\includegraphics[width=0.5\\textwidth]{Figures\/Fig_Ablation_KFold_RAF.pdf}\n\t\t\t\\includegraphics[width=0.5\\textwidth]{Figures\/Fig_Ablation_KFold_RAF_test.pdf}\n\t\t\\end{tabular}\n\t\t\\caption{}\n\t\\end{subfigure}\n\t\n\t\\begin{subfigure}[t]{\\textwidth}\n\t\t\\begin{tabular}{*{2}{c@{\\extracolsep{0em}}} }\n\t\t\t\\includegraphics[width=0.5\\textwidth]{Figures\/Fig_Ablation_KFold_Fashion.pdf}\n\t\t\t\\includegraphics[width=0.5\\textwidth]{Figures\/Fig_Ablation_KFold_Fashion_test.pdf}\n\t\t\\end{tabular}\n\t\t\\caption{}\n\t\\end{subfigure}\n\t\n\t\\caption{Training \\& Testing errors of $5$-Fold cross validation on (a) RAF dataset, and (b) Fashion-MNIST dataset, respectively. Fold Avg. (colored in red) denotes the mean of $5$ folds.}\n\t\\label{Fig:KFold}\n\\end{figure}\n\n\n\\subsection{Simulation Results}\n\nIn this paper, we compare our \\textit{SiPRNet} with the traditional algorithms and learning-based methods. Compared methods include the traditional optimization-based algorithms such as Gerchberg\u2013Saxton (GS) \\cite{Gerchberg1972APA}, Hybrid Input-Output (HIO) \\cite{Fienup:82}, and total variation ADMM (ADMM-TV) \\cite{chang2018total}. In addition, we compare the proposed \\textit{SiPRNet} with other CNN-based methods including: Plain Network (denoted as PlainNet) \\cite{He_2016_CVPR}: the basic CNN structure that stacks convolutional layers one by one; Residual Network (denoted as ResNet) \\cite{He_2016_CVPR, Nishizaki_2020}: a popular deep-learning structure and has several skip connections among the convolutional layers; residual dense network (denoted as ResDenseNet) \\cite{Zhang_2018_CVPR}: a network that adds more skip connections to achieve better efficiency; Lensless imaging network (denoted as LenlessNet) \\cite{Sinha17}: a residual UNet to recover the phase-only images in a lensless imaging system; Conditional Generative Adversarial Phase Retrieval Network (denoted as PRCGAN) \\cite{uelwer2021phase}: a network that utilizes the conditional adversarial learning strategy  for phase retrieval task; and NNPhase \\cite{Wu_cw5029}: a network to recover the complex-valued images from Fraunhofer diffraction patterns. \n\n\\textbf{\\textit{Phase-only datasets:}} The phase retrieval results with the phase-only datasets are shown in Table \\ref{table:RAFresult}, Table \\ref{table:Fashionresult}, and Fig. \\ref{fig:compvisual}. To save space, the qualitative results of PlainNet and ResDenseNet are not included in Fig. \\ref{fig:compvisual} due to their inferior performances. All the CNN-based methods were trained with the same settings on the RAF and Fashion-MNIST datasets. Note that the  hyper-parameters of all traditional methods were optimally fine-tuned and fixed when the methods were evaluated with all testing images. For each traditional optimization-based method, we did three different trials of evaluation for each testing image, where each trial had $ 400 $ iterations. We manually selected the reconstructed image with the highest PSNR as the final result for comparison. For the CNN-based methods, we compared the average PSNR of reconstructed images given by the trained models at the last epoch.\n\nAs can be seen in Table \\ref{table:RAFresult} and Table \\ref{table:Fashionresult}, the proposed \\textit{SiPRNet} achieves much better performance than all compared methods. Compared with the traditional optimization-based methods, \\textit{SiPRNet} has average PSNR and SSIM gains of at least $6.544dB$ and $0.457$, respectively, on the RAF dataset (Table \\ref{table:RAFresult}). The PSNR and SSIM gains can even reach $19.58dB$ and $0.816$, respectively, on the Fashion-MNIST dataset (Table \\ref{table:Fashionresult}). This is because traditional optimization-based methods cannot converge to satisfactory results with only one intensity measurement. As shown in Fig. \\ref{fig:compvisual}, only ADMM-TV method can recover the contour of the facial images ($7$th column). In addition, \\textit{SiPRNet} can achieve PSNR and SSIM gains of at least $1.631dB$ and $0.065$, respectively, compared with other state-of-the-art CNN-based methods on the two datasets. It can be seen in Fig. \\ref{fig:compvisual} that the reconstructed images of the proposed \\textit{SiPRNet} outperform other CNN-based methods qualitatively. In addition, the error maps ($4$th column) also show that the estimated images are close to the ground-truth images. As can be seen, the reconstructed images only differ from the original ones in some local areas with minor errors, while most regions are close to $0$.\n\nNote that we train all the networks with our quantized defocused Fraunhofer diffraction patterns datasets. Although the reconstructed images via PRCGAN \\cite{uelwer2021phase} seem to have better high-frequency information (facial details), the reconstructed image does not share similarities with the original image (last second column).  In fact, the performance of PRCGAN in our simulation is poorer than that reported in the original paper of PRCGAN. It is because the dataset used in the original simulation of PRCGAN does not consider the saturation problem, which is commonly found when imaging the Fourier intensity. Besides, the authors assume the object images are magnitude-only (no phase information), which is far from real situations.  As for NNPhase \\cite{Wu_2021}, it aims to deal with the crystal data that the application is quite specific. Besides, LenslessNet \\cite{Sinha17} is initially designed for the phase retrieval task based on Fresnel diffraction patterns with lensless imaging systems. When testing with the more general and realistic datasets used in our simulations, the performances of these networks drop significantly compared with those reported in their original papers. We will show in the next section that these networks also suffer from the same problems in the real experiments. The above simulation results show that the proposed \\textit{SiPRNet} outperforms the traditional optimization-based and CNN-based methods both quantitatively and qualitatively. More simulation results can be found in \\nameref{Sec:Moreresults}. \n\nWe also evaluate the inference speed of different methods by collecting the mean execution time of one hundred samples. The results are presented in Table \\ref{table:RAFresult} (last column). Since our approach is designed in an end-to-end manner that can effectively utilize the huge computational power of the current advanced GPU devices, it can be seen in Table \\ref{table:RAFresult} that the proposed the \\textit{SiPRNet} is very fast and can realize real-time performance ($3.569$ ms\/img). It is comparable with other learning-based approaches. In contrast, the inference speed of the traditional algorithms is quite slow (in the order of seconds) since lots of iterations are required, and they can only be implemented with the CPU.\n\n\\textbf{\\textit{Magnitude-phase dataset:}} The simulation results are shown in Fig. \\ref{fig:ampphase}. We select HIO, NNPhase, and PRCGAN for comparison since they all perform Fourier phase retrieval. It can be seen that the proposed \\textit{SiPRNet} outperforms all compared approaches. In particular, the results show that PRCGAN cannot deal with the complex-valued phase retrieval tasks, and it is difficult for HIO to reconstruct complex-valued images with single-shot intensity measurement. The magnitude and phase images given by these approaches have poor quality. As is presented in Fig. \\ref{fig:ampphase}(b), the proposed \\textit{SiPRNet} achieves much better performance than all compared methods evaluated by different metrics. For the magnitude part, the proposed \\textit{SiPRNet} achieves average PSNR and SSIM gains of at least $2.145dB$ and $0.156$, respectively. For the phase part, the increases in PSNR and SSIM gains can reach $3.214 dB$ and $0.181$, respectively. The above simulation results show that the proposed \\textit{SiPRNet} can be generalized to complex-valued image datasets (containing both magnitude and phase parts) and outperform the existing methods.\n\n\\begin{table}[ht]\n\t\\centering\n\n\n\t\\caption{Quantitative comparison (average MAE\/PSNR\/SSIM\/Inference Time) with the state-of-the-art methods for phase retrieval on the RAF dataset. Best and second best performances are in \\textcolor{red}{red} and \\textcolor{blue}{blue} colors, respectively.}\n\t\\begin{adjustbox}{width=0.8\\columnwidth, center}\n\t\t\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\t\t\t\\hline \\multirow{2}{*}{ Methods } & \\multicolumn{2}{c|}{ MAE $\\downarrow$} & \\multicolumn{2}{c|}{ PSNR $\\uparrow$} & \\multicolumn{2}{c|}{ SSIM $\\uparrow$} & \\multirow{2}{*}{\\parbox{1.2cm}{\\centering Inference Time (ms\/img) }}\\\\\n\t\t\t\\cline { 2 - 7 } & Sim. & Exp. & Sim. & Exp. & Sim. & Exp. & \\\\\n\t\t\t\\hline\n\t\t\t\\hline GS \\cite{Gerchberg1972APA} & $0.969$ & $1.147$ & $15.099$ & $12.770$ & $0.250$ & $0.129$ & $1.233\\times 10^{3}$\\\\\n\t\t\tHIO \\cite{Fienup:82} & $1.007$ & $1.185$ & $14.854$ & $12.554$ & $0.239$ & $0.127$ & $1.226\\times 10^{3}$\\\\\n\t\t\tADMM-TV \\cite{chang2018total} &  $1.119$ & $1.600$ & $12.660$ & $9.902$ & $0.206$ & $0.025$ & $1.324\\times 10^{4}$\\\\\n\t\t\tPlainNet \\cite{He_2016_CVPR} & $0.975$ & $1.180$ & $13.841$ & $12.479$ & $0.425$ & $0.377$ & $1.583$\\\\\n\t\t\tResNet \\cite{He_2016_CVPR, Nishizaki_2020} & $0.894$ & $0.812$ & $15.178$ & $15.831$ & $0.456$ & $0.471$ & $2.438$\\\\\n\t\t\tResDenseNet \\cite{Zhang_2018_CVPR} & $1.084$ & $1.010$ & $13.344$ & $13.800$ & $0.380$ & $0.387$ & $1.732$\\\\\n\t\t\tLenslessNet \\cite{Sinha17} & $\\textcolor{blue}{0.671}$ & $\\textcolor{blue}{0.704}$ & $\\textcolor{blue}{17.573}$ & $\\textcolor{blue}{17.132}$ & $\\textcolor{blue}{0.598}$ & $\\textcolor{blue}{0.596}$ & $7.396$\\\\\n\t\t\tPRCGAN \\cite{uelwer2021phase} & $0.951$ & $0.945$ & $14.738$ & $14.833$ & $0.511$ & $0.508$ & $0.983$\\\\\n\t\t\tNNPhase \\cite{Wu_cw5029} & $0.755$ & $0.931$ & $16.753$ & $15.416$ & $0.558$ & $0.544$ & $5.834$\\\\\n\t\t\t\\textbf{\\textit{SiPRNet} (Ours)} & $\\textcolor{red}{0.570}$ & $\\textcolor{red}{0.582}$ & $\\textcolor{red}{19.204}$ & $\\textcolor{red}{18.905}$ & $\\textcolor{red}{0.663}$ & $\\textcolor{red}{0.663}$ & $3.569$\\\\\n\t\t\t\\hline\n\t\t\\end{tabular}\n\t\\end{adjustbox}\n\t\\label{table:RAFresult}\n\\end{table}\n\n\\begin{table}[ht]\n\t\\centering\n\n\n\n\t\\caption{Quantitative comparison (average MAE\/PSNR\/SSIM) with the state-of-the-art methods for phase retrieval on the Fashion-MNIST dataset. Best and second best performances are in \\textcolor{red}{red} and \\textcolor{blue}{blue} colors, respectively.}\n\t\\begin{adjustbox}{width=0.7\\columnwidth,center}\n\t\t\\begin{tabular}{|c|c|c|c|c|c|c|}\n\t\t\t\\hline \\multirow{2}{*}{ Methods } & \\multicolumn{2}{c|}{ MAE $\\downarrow$} & \\multicolumn{2}{c|}{ PSNR $\\uparrow$} & \\multicolumn{2}{c|}{ SSIM $\\uparrow$} \\\\\n\t\t\t\\cline { 2 - 7 } & Sim. & Exp. & Sim. & Exp. & Sim. & Exp. \\\\\n\t\t\t\\hline\n\t\t\t\\hline GS \\cite{Gerchberg1972APA} & $1.571$ & $1.681$ & $9.852$ & $9.176$ & $0.088$ & $0.053$ \\\\\n\t\t\tHIO \\cite{Fienup:82} & $1.594$ & $1.709$ & $9.629$ & $9.082$ & $0.068$ & $0.052$ \\\\\n\t\t\tADMM-TV \\cite{chang2018total} & $0.935$ & $1.366$ & $13.772$ & $9.973$ & $0.226$ & $0.132$ \\\\\n\t\t\tPlainNet \\cite{He_2016_CVPR} & $0.589$ & $0.601$ & $16.916$ & $16.672$ & $0.629$ & $0.623$  \\\\\n\t\t\tResNet \\cite{He_2016_CVPR, Nishizaki_2020} & $0.289$ & $0.304$ & $21.394$ & $20.806$ & $0.739$ & $0.726$  \\\\\n\t\t\tResDenseNet \\cite{Zhang_2018_CVPR} & $0.381$ & $0.419$ & $19.406$ & $18.252$ & $0.675$ & $0.657$  \\\\\n\t\t\tLenslessNet \\cite{Sinha17} & $\\textcolor{blue}{0.179}$ & $\\textcolor{blue}{0.221}$ & $\\textcolor{blue}{26.784}$ & $\\textcolor{blue}{23.833}$ & $\\textcolor{blue}{0.842}$ & $\\textcolor{blue}{0.802}$\\\\\n\t\t\tPRCGAN \\cite{uelwer2021phase} & $1.598$ & $1.602$ &  $10.102$ & $10.126$ & $0.422$ & $0.417$ \\\\\n\t\t\tNNPhase \\cite{Wu_cw5029} & $0.404$ & $0426$ & $23.956$ & $22.778$ & $0.739$ & $0.443$ \\\\\n\t\t\t\\textbf{\\textit{SiPRNet} (Ours)} & $\\textcolor{red}{0.127}$ & $\\textcolor{red}{0.144}$ & $\\textcolor{red}{29.209}$ & $\\textcolor{red}{28.132}$ & $\\textcolor{red}{0.884}$ & $\\textcolor{red}{0.872}$   \\\\\n\t\t\t\\hline\n\t\t\\end{tabular}\n\t\\end{adjustbox}\n\t\\label{table:Fashionresult}\n\\end{table}\n\n   \n\n\\begin{figure}[htbp]\n\n\n\n\t\\begin{adjustbox}{width=1\\textwidth,center}\n\t\t\\begin{tabular}{c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}}\n\t\t\t\n\t\t\t\\includegraphics[height=0.09\\textwidth, valign=t]{Figures\/main_result\/measurement_12_sim.jpg}~\n\t\t\t&\\includegraphics[height=0.09\\textwidth, valign=t]{Figures\/main_result\/SiPRNet_img_0011_angle_gt.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/SiPRNet_img_0011_angle.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/img_0011_residual.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/\/GS_RAF_simulation_12.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/\/HIO_RAF_simulation_12.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/\/ADMM_sim.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/\/Res18_simulation.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/lensless_sim_img_0011_angle.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/PRCGAN_img_0011_angle.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/NNPhase_img_0011_angle_sim.png}\n\t\t\t\\\\\n\t\t\t\n\t\t\t\\includegraphics[height=0.09\\textwidth, valign=t]{Figures\/main_result\/measurement_12_exp.jpg}~\n\t\t\t&\\includegraphics[height=0.09\\textwidth, valign=t]{Figures\/main_result\/SiPRNet_img_0011_angle_gt_exp.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/SiPRNet_img_0011_angle_exp.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/img_0011_residual_exp.png}~\t\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/GS_RAF_exp_12.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/HIO_RAF_exp_12.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/ADMM_exp.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/Res18_exp.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/lensless_exp_img_0011_angle.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/PRCGAN_img_0011_angle_exp.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/NNPhase_img_0011_angle.png}\n\t\t\t\n\t\t\t\\\\\n\t\t\t\n\t\t\t\\includegraphics[height=0.09\\textwidth, valign=t]{Figures\/main_result\/measurement_304_exp.jpg}~\n\t\t\t&\\includegraphics[height=0.09\\textwidth, valign=t]{Figures\/main_result\/SiPRNet_img_0303_angle_gt.png}~\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/SiPRNet_img_0303_angle.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/img_0303_residual.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/GS_RAF_exp_304.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/HIO_RAF_exp_304.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/ADMM_exp.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/Res18_exp2.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/lensless_exp_img_0303_angle.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/PRCGAN_img_0303_angle.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/NNPhase_img_0303_angle.png}\n\t\t\t\\\\\n\t\t\t\n\t\t\t\\includegraphics[height=0.09\\textwidth, valign=t]{Figures\/main_result\/measurement_0847.png}\n\t\t\t&\\includegraphics[height=0.09\\textwidth, valign=t]{Figures\/main_result\/SiPRNet_img_0847_angle_gt.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t, valign=t]{Figures\/main_result\/SiPRNet_img_0847_angle.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t, valign=t]{Figures\/main_result\/img_0847_residual.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t, valign=t]{Figures\/main_result\/GS_img_0847_angle.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t, valign=t]{Figures\/main_result\/HIO_img_0847_angle.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t, valign=t]{Figures\/main_result\/ADMM_RAF_0847.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t, valign=t]{Figures\/main_result\/ResNet_img_0847_angle.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t, valign=t]{Figures\/main_result\/LenslessNet_img_0847_angle.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t, valign=t]{Figures\/main_result\/PRCGAN_img_0847_angle.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t, valign=t]{Figures\/main_result\/NNPhase_img_0847_angle.png}\n\t\t\t\\\\    \t\t\t\n\t\t\t\n\t\t\t\\includegraphics[height=0.09\\textwidth, valign=t]{Figures\/main_result\/measurement_646_exp_fashion.jpg}\n\t\t\t&\\includegraphics[height=0.09\\textwidth, valign=t]{Figures\/main_result\/SiPRNet_img_0645_angle_gt.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/SiPRNet_img_0645_angle.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/img_0645_residual.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/GS_Fashion_exp_646.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/HIO_Fashion_exp_646.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/ADMM_exp_fashion.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/ResNet_exp_fashion.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/lensless_exp_img_0645_angle.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/PRCGAN_img_0645_angle.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/NNPhase_img_0645_angle.png}\n\t\t\t\\\\\n\t\t\t\n\t\t\t\\includegraphics[height=0.09\\textwidth, valign=t]{Figures\/main_result\/measurement_0007.png}\n\t\t\t&\\includegraphics[height=0.105\\textwidth, width=0.09\\textwidth, valign=t]{Figures\/main_result\/SiPRNet_img_0007_angle_gt.pdf}~         &\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/SiPRNet_img_0007_angle.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/img_0007_residual.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/GS_img_0007_angle.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/HIO_img_0007_angle.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/ADMM_Fashion_0007.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/ResNet_img_0007_angle.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/LenslessNet_img_0007_angle.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/PRCGAN_img_0007_angle.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/NNPhase_img_0007_angle.png}\n\t\t\t\\\\    \n\t\t\t\n\t\t\t{\\scriptsize \\makecell[c]{Fourier\\\\measurement}} & {\\scriptsize \\makecell[c]{Ground-truth \\\\ (phase)}} & {\\scriptsize \\makecell[c]{\\textbf{\\textit{SiPRNet}}\\\\(ours)} }& {\\scriptsize \\makecell[c]{\\textbf{\\textit{SiPRNet}} \\\\(Error Map)} }  & {\\scriptsize GS } & {\\scriptsize HIO} & {\\scriptsize ADMM-TV} & {\\scriptsize ResNet}\n\t\t\t& {\\scriptsize LenslessNet}  & {\\scriptsize PRCGAN}  & {\\scriptsize NNPhase}  \n\t\t\t\\\\\n\t\t\\end{tabular}\n\t\\end{adjustbox}\n\t\\vspace{-0.2cm}\n\t\\caption{Simulation and experimental results of different phase retrieval methods on the RAF \\cite{li2017reliable} and the Fashion-MNIST datasets \\cite{xiao2017\/online}. The first \\textbf{column} denotes the Fourier intensity measurements (pixel values: $0 \\rightarrow 4095$). The second \\textbf{column} shows the corresponding phase parts of the complex-valued images with scale bars (for experimental results) at the bottom left corners. The fourth \\textbf{column} denotes the error maps of \\textit{SiPRNet}. The other \\textbf{columns} present the reconstructed images through different methods. Except for the Fourier intensity measurements (the first \\textbf{column}), the colormap of the rest \\textbf{columns} ranges from $0$ to $2\\pi$. The first and second \\textbf{rows} are the simulation and experimental results of the same image from the RAF dataset. The third and the fourth \\textbf{rows} show the experimental performances of other images in the RAF dataset. The fifth and the sixth \\textbf{rows} indicate the experimental results of images from the Fashion-MNIST dataset. Please zoom in for better view. More results can be found in \\nameref{Sec:Moreresults}. }\n\t\\label{fig:compvisual}\n\n\\end{figure}\n\n\n\\begin{figure}[htb]\n\t\n\t\\begin{subfigure}{\\textwidth}\t\n\t\t\\centering\n\t\t\\begin{adjustbox}{width=1\\textwidth, center}        \\begin{tabular}{c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}}\n\t\t\t\t\n\t\t\t\t\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/measurement_0093.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/SiPRNet_img_0093_magnitude_gt.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/SiPRNet_img_0093_magnitude.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/HIO_img_0093_magnitude.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/NNPhase_img_0093_magnitude.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/PRCGAN_img_0093_magnitude.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/SiPRNet_img_0093_angle_gt.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/SiPRNet_img_0093_angle.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/HIO_img_0093_angle.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/NNPhase_img_0093_angle.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/PRCGAN_img_0093_angle.png}~\n\t\t\t\t\\\\\n\t\t\t\t\n\t\t\t\t\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/measurement_0550.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/SiPRNet_img_0550_magnitude_gt.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/SiPRNet_img_0550_magnitude.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/HIO_img_0550_magnitude.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/NNPhase_img_0550_magnitude.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/PRCGAN_img_0550_magnitude.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/SiPRNet_img_0550_angle_gt.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/SiPRNet_img_0550_angle.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/HIO_img_0550_angle.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/NNPhase_img_0550_angle.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/PRCGAN_img_0550_angle.png}~\n\t\t\t\t\\\\\n\t\t\t\t\n\t\t\t\t\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/measurement_0756.png}~\n\t\t\t\t&\\includegraphics[height = 0.118\\textwidth,width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/SiPRNet_img_0756_magnitude_gt.pdf}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/SiPRNet_img_0756_magnitude.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/HIO_img_0756_magnitude.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/NNPhase_img_0756_magnitude.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/PRCGAN_img_0756_magnitude.png}~\n\t\t\t\t&\\includegraphics[height = 0.118\\textwidth,width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/SiPRNet_img_0756_angle_gt.pdf}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/SiPRNet_img_0756_angle.png}~\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/HIO_img_0756_angle.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/NNPhase_img_0756_angle.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/PRCGAN_img_0756_angle.png}~\n\t\t\t\t\\\\\n\t\t\t\t\n\t\t\t\t{\\scriptsize \\makecell[c]{Fourier\\\\measurement}} & {\\scriptsize \\makecell[c]{Ground-truth \\\\ (mag.)}} & {\\scriptsize \\makecell[c]{\\textbf{\\textit{SiPRNet} }  \\\\ (mag.)}} & {\\scriptsize \\makecell[c]{HIO \\\\ (mag.)}} & {\\scriptsize \\makecell[c]{NNPhase \\\\ (mag.)}} & {\\scriptsize \\makecell[c]{PRCGAN \\\\ (mag.)}} & {\\scriptsize \\makecell[c]{Ground-truth \\\\ (phase)}} & {\\scriptsize \\makecell[c]{\\textbf{\\textit{SiPRNet}  } \\\\ (phase)}} & {\\scriptsize \\makecell[c]{HIO \\\\ (phase)}} & {\\scriptsize \\makecell[c]{NNPhase \\\\ (phase)}} & {\\scriptsize \\makecell[c]{PRCGAN \\\\ (phase)}} \n\t\t\t\t\\\\\n\t\t\t\\end{tabular}\n\t\t\\end{adjustbox}\n\t\t\\vspace*{-2mm}\n\t\t\\caption{}\n\t\\end{subfigure}\n\t\n\t\\begin{subfigure}{\\textwidth}\t\n\t\t\\centering\n\t\t\\begin{adjustbox}{width=0.7\\columnwidth, center}\n\t\t\t\\begin{tabular}{|c|c|c|c|c|c|c|}\n\t\t\t\t\\hline \\multirow{2}{*}{ Methods } & \\multicolumn{2}{c|}{ MAE $\\downarrow$} & \\multicolumn{2}{c|}{ PSNR $\\uparrow$} & \\multicolumn{2}{c|}{ SSIM $\\uparrow$} \\\\\n\t\t\t\t\\cline { 2 - 7 } & Mag. & Phase & Mag. & Phase & Mag. & Phase \\\\\n\t\t\t\t\\hline\n\t\t\t\t\\hline HIO \\cite{Fienup:82} & $0.298$ & $1.898$ &  $7.709$ & $8.317$ & $0.330$ & $0.043$ \\\\\n\t\t\t\tPRCGAN \\cite{uelwer2021phase} & $0.073$ & $1.654$ &  $18.672$ & $9.924$ & $0.588$ & $0.387$ \\\\\n\t\t\t\tNNPhase \\cite{Wu_cw5029} & $0.109$ & $0.687$ & $18.169$ & $18.752$ & $0.395$ & $0.46$ \\\\\n\t\t\t\t\\textbf{\\textit{SiPRNet} (Ours)} & $\\mathbf{0.053}$ & $\\mathbf{0.359}$ & $\\mathbf{20.817}$ & $\\mathbf{21.966}$ & $\\mathbf{0.744}$ & $\\mathbf{0.594}$   \\\\\n\t\t\t\t\\hline\n\t\t\t\\end{tabular}\n\t\t\\end{adjustbox}\n\t\t\\caption{}\n\t\\end{subfigure}\n\t\n\t\\vspace{-0.2cm}\n\t\\caption{(a) Qualitative simulation results of different phase retrieval methods on the magnitude-phase Fashion-MNIST datasets \\cite{xiao2017\/online}. The pixel values of the Fourier intensity measurements (first column) range from $0$ to $4095$. The colormap of the magnitude parts (second to the sixth columns) ranges from $0$ to $1$, while the colormap of the phase parts (seventh to the last columns) ranges from $0$ to $2\\pi$. Please zoom in for better view. (b) Quantitative comparisons (average MAE\/PSNR\/SSIM) on the same dataset. Best performances are denoted in \\textbf{bold} font.}\n\t\\label{fig:ampphase}\n\t\n\\end{figure}\n\n    \\section{Experimental Results} \\label{Sec:experiment}\n\n\\subsection{Experimental Setup}\n\nWe also evaluated our proposed \\textit{SiPRNet} with a practical optical system. A picture of the experimental setup is shown in Fig. \\ref{fig:setup}.  It comprises a Thorlabs $ 10 mW$ HeNe laser with wavelength $ \\lambda = 632.8 nm$ ; and a $ 12 $-bit $ 1920\\times1200 $ Kiralux CMOS camera with pixel pitch $ 5.86 \\mu m$ . Besides, the optical system includes one $75 mm$, three $100 mm$, and one $150 mm$ lenses to form a standard imaging system. The choice of the lenses is for generating Fourier intensity images of appropriate size to match the size of the CMOS sensor. In practice, these lenses can be compactly arranged such that the whole imaging system can be made very small. However, the lens design is out of the scope of this study. A $ 1920\\times1080 $ Holoeye Pluto phase-only SLM with pixel pitch $ \\delta_{SLM}=8\\mu m $ was used to generate the testing objects in the experiments. Specifically, the images in the two datasets mentioned above were loaded to the SLM to impose multiple-level phase changes. The images were pre-multiplied with a defocus kernel; thus, there was no need to move the camera beyond the focal plane. It is to ensure that the defocusing process, which is not our main focus, will be perfectly implemented so that we only need to focus on the reconstruction performances of different methods. The defocus kernel is a pure-phase object generated by Holoeye built-in function corresponding to $L = 30 mm$. The reason for choosing $L = 30mm$ has been explained in Section \\ref{Sec:abladefocusdist}. We utilized the central $ 128\\times 128 $ SLM pixels in all experiments. The resulting Fourier intensity measurements were captured by the camera and formed the training and testing datasets for \\textit{SiPRNet}. The size of the acquired measurement is $\\frac{\\lambda L}{d}$ where $d$ denotes the pixel pitch of the SLM. \n\n\\begin{figure}[ht]               \n\t\\centering\\includegraphics[width=0.8\\linewidth]{Figures\/Fig_Setup.pdf}\n\t\\caption{Experimental setup of the defocus phase retrieval system.}\n\t\\label{fig:setup}\n\t\n\\end{figure}\n\n\\subsection{Experimental Results}\n\nWith the training dataset, we fine-tuned all the pre-trained models acquired in the simulation with the experimental measurements. That is, we re-trained the models with the trained parameters from the simulation and used the experimental measurements as the new training inputs. The quantitative and qualitative experimental results are presented in Table \\ref{table:RAFresult}, Table \\ref{table:Fashionresult}, and Fig. \\ref{fig:compvisual}, respectively. \n\nIt is clearly shown in Table \\ref{table:RAFresult} and Table \\ref{table:Fashionresult} that the experimental results of \\textit{SiPRNet} only decline a little compared with the simulation results. It proves the robustness of the proposed method. As shown in Fig. \\ref{fig:compvisual}, all traditional optimization-based methods fail to reconstruct the original images because it is insufficient to converge to satisfactory solutions with only one intensity measurement for those methods. Compared with other state-of-the-art CNN-based methods, \\textit{SiPRNet} can achieve at least $1.773dB$ and $0.067$ gains in PSNR and SSIM, respectively, on the two datasets. As can be clearly seen in Fig. \\ref{fig:compvisual}, the proposed \\textit{SiPRNet} can reconstruct the most similar contours as the ground truth and preserve more high-frequency details compared with other methods.  The above experimental results show that the proposed \\textit{SiPRNet} outperforms all traditional optimization-based and CNN-based methods quantitatively and qualitatively when using in practical phase retrieval systems. However, there is always room for improvement. It is shown in the results of all learning-based approaches that the reconstructed images are a bit blurry compared with the ground truths. We note that for the captured intensity images, the values of the high-frequency components are much smaller than those in the low-frequency areas. The noise in the images further makes the high-frequency components difficult to identify. It introduces much difficulty to the neural networks to extract the features of the high-frequency components. Further investigation is needed to improve the sharpness of the reconstructed images. Extra denoising and attention modules may be applied to better recover the high-frequency components.\n\n\\section{Conclusion} \n\tIn conclusion, this paper proposed an end-to-end CNN structure for single-shot maskless phase retrieval. We have shown how the optical system should be constructed to ensure the feasibility of the solution and reduce the saturation problem due to the high dynamic range of Fourier intensity measurements. We proposed a novel CNN named \\textit{SiPRNet} to reconstruct phase-only images with the central areas of the intensity measurements. To fully extract the representative features, we proposed an MLP block embedded with a Dropout layer to fuse the information of the frequency components. A self-attention structure was included to enhance the global correlation in the reconstructed phase images. The simulation and experimental results demonstrate that the proposed \\textit{SiPRNet} performs much better than the traditional phase retrieval algorithms and other CNN structures, making it a promising solution to practical phase retrieval applications. At the moment, we only conduct simulations and experiments on synthesized objects. Further research is underway to enhance the network structure for working with more complex image datasets and realistic objects. \n\n    \\section*{Appendix}  \\label{Sec:Moreresults}\n\n\tIn this Appendix, we show additional qualitative comparison results.  Fig. \\ref{fig:firstimage}, \\ref{fig:secondimage}, \\ref{fig:thirdimage} present the simulation and experimental results. The detailed explanations are attached with each figure.\n\t\n\t   \n\t\\begin{figure}[htb]\n\t\t\\begin{adjustbox}{width=1\\textwidth,center}        \\begin{tabular}{c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}}\n\t\t\t\t\n\t\t\t\t\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/measurement_288.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/SiPRNet_img_0287_angle_gt.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/SiPRNet_img_0287_angle.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/img_0287_residual.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/GS_RAF_simulation_288.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/HIO_RAF_simulation_288.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/ADMM_sim_288.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/Res18_simulation_288.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/lensless_sim_img_0287_angle.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/CGAN_simulation_288.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/NNPhase_img_0287_angle.png}\n\t\t\t\t\n\t\t\t\t\\\\\n\t\t\t\t\n\t\t\t\t\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/measurement_545.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/SiPRNet_img_0544_angle_gt.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/SiPRNet_img_0544_angle.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/img_0544_residual.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/GS_RAF_simulation_545.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/HIO_RAF_simulation_545.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/ADMM_sim_545.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/Res18_simulation_545.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/lensless_sim_img_0544_angle.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/CGAN_simulation_545.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/NNPhase_img_0544_angle.png}\n\t\t\t\t\\\\\n\t\t\t\t\n\t\t\t\t\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/measurement_722.png}~\n\t\t\t\t&\\includegraphics[height=0.118\\textwidth,width=0.1\\textwidth, valign=t]{Figures\/first\/SiPRNet_img_0721_angle_gt.pdf}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/SiPRNet_img_0721_angle.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/img_0721_residual.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/GS_RAF_simulation_722.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/HIO_RAF_simulation_722.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/ADMM_sim_722.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/Res18_simulation_722.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/lensless_sim_img_0721_angle.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/CGAN_simulation_722.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/NNPhase_img_0721_angle.png}\n\t\t\t\t\\\\\n\t\t\t\t\n\t\t\t\t{\\scriptsize \\makecell[c]{Fourier\\\\measurement}} & {\\scriptsize \\makecell[c]{Ground-truth \\\\ (phase)}} & {\\scriptsize \\makecell[c]{\\textbf{\\textit{SiPRNet}}\\\\(ours)} }& {\\scriptsize \\makecell[c]{\\textbf{\\textit{SiPRNet}} \\\\(Error Map)} }  & {\\scriptsize GS } & {\\scriptsize HIO} & {\\scriptsize ADMM-TV} & {\\scriptsize ResNet}\n\t\t\t\t& {\\scriptsize LenslessNet}  & {\\scriptsize PRCGAN}  & {\\scriptsize NNPhase}  \n\t\t\t\t\\\\\n\t\t\t\\end{tabular}\n\t\t\\end{adjustbox}\n\t\t\\vspace{-0.2cm}\n\t\t\\caption{Simulation results of different phase retrieval methods on the RAF dataset \\cite{li2017reliable}. The first \\textbf{column} denotes the Fourier intensity measurements (pixel values: $0 \\rightarrow 4095$). The second \\textbf{column} shows the corresponding phase parts of the complex-valued images. The fourth \\textbf{column} denotes the error maps of the \\textit{SiPRNet}. The other \\textbf{columns} present the reconstructed images through different methods.  Except for the Fourier intensity measurements (the first \\textbf{column}), the colormap of the rest \\textbf{columns} ranges from $0$ to $2\\pi$. Each \\textbf{row} presents the simulation results of a testing image.  Please zoom in for better view.}\n\t\t\\label{fig:firstimage}\n\t\t\n\t\\end{figure}\n\t\n\t\n\t\n\n\t\\begin{figure}[htb]\n\t\t\\begin{adjustbox}{width=1\\textwidth,center}          \\begin{tabular}{c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}}\n\t\t\t\t\n\t\t\t\t\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/measurement_178.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/SiPRNet_img_0177_angle_gt.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/SiPRNet_img_0177_angle.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/img_0177_residual.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/GS_RAF_exp_178.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/HIO_RAF_exp_178.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/ADMM_exp_178.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/Res18_exp_178.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/lensless_exp_img_0177_angle.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/CGAN_exp_178.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/NNPhase_img_0177_angle.png}\n\t\t\t\t\\\\\n\t\t\t\t\n\t\t\t\t\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/measurement_288.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/SiPRNet_img_0287_angle_gt.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/SiPRNet_img_0287_angle.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/img_0287_residual.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/GS_RAF_exp_288.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/HIO_RAF_exp_288.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/ADMM_exp_288.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/Res18_exp_288.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/lensless_exp_img_0287_angle.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/CGAN_exp_288.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/NNPhase_img_0287_angle.png}\n\t\t\t\t\\\\\n\t\t\t\t\n\t\t\t\t\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/measurement_545.png}~\n\t\t\t\t&\\includegraphics[height=0.118\\textwidth, width=0.1\\textwidth, valign=t]{Figures\/second\/SiPRNet_img_0544_angle_gt.pdf}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/SiPRNet_img_0544_angle.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/img_0544_residual.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/GS_RAF_exp_545.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/HIO_RAF_exp_545.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/ADMM_exp_545.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/Res18_exp_545.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/lensless_exp_img_0544_angle.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/CGAN_exp_545.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/NNPhase_img_0544_angle.png}\n\t\t\t\t\\\\\n\t\t\t\t\n\t\t\t\t{\\scriptsize \\makecell[c]{Fourier\\\\measurement}} & {\\scriptsize \\makecell[c]{Ground-truth \\\\ (phase)}} & {\\scriptsize \\makecell[c]{\\textbf{\\textit{SiPRNet}}\\\\(ours)} }& {\\scriptsize \\makecell[c]{\\textbf{\\textit{SiPRNet}} \\\\(Error Map)} }  & {\\scriptsize GS } & {\\scriptsize HIO} & {\\scriptsize ADMM-TV} & {\\scriptsize ResNet}\n\t\t\t\t& {\\scriptsize LenslessNet}  & {\\scriptsize PRCGAN}  & {\\scriptsize NNPhase}  \n\t\t\t\t\\\\\n\t\t\t\\end{tabular}\n\t\t\\end{adjustbox}\n\t\t\\vspace{-0.2cm}\n\t\t\\caption{Experimental results of different phase retrieval methods on the RAF dataset \\cite{li2017reliable}. The first \\textbf{column} denotes the Fourier intensity measurements (pixel values: $0 \\rightarrow 4095$). The second \\textbf{column} shows the corresponding phase parts of the complex-valued images with scale bars at the bottom left corners.  The fourth \\textbf{column} denotes the error maps of \\textit{SiPRNet}. The other \\textbf{columns} present the reconstructed images through different methods. Except for the Fourier intensity measurements (the first \\textbf{column}), the colormap of the rest \\textbf{columns} ranges from $0$ to $2\\pi$. Each \\textbf{row} presents the experimental results of a testing image. Please zoom in for better view.}\n\t\t\\label{fig:secondimage}\n\t\t\n\t\\end{figure}     \n\t\n\t\n\n\t\\begin{figure}[H]\n\t\t\\begin{adjustbox}{width=1\\textwidth, center}          \\begin{tabular}{c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}}\n\t\t\t\t\n\t\t\t\t\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/measurement_sim_107.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/SiPRNet_img_0106_angle_gt.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/SiPRNet_img_0106_angle.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/img_0106_residual.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/GS_Fashion_simulation_107.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/HIO_Fashion_simulation_107.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/ADMM_sim_107.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/ResNet_simulation_107.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/lensless_sim_img_0106_angle.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/CGAN_simulation_107.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/NNPhase_img_0106_angle_sim.png}\n\t\t\t\t\\\\\n\t\t\t\t\n\t\t\t\t\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/measurement_exp_107.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/SiPRNet_img_0106_angle_gt_exp.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/SiPRNet_img_0106_angle_gt_exp.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/img_0106_residual_exp.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/GS_Fashion_exp_107.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/HIO_Fashion_exp_107.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/ADMM_exp_107.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/ResNet_exp_107.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/lensless_exp_img_0106_angle.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/CGAN_exp_107.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/NNPhase_img_0106_angle.png}\n\t\t\t\t\\\\\n\t\t\t\t\n\t\t\t\t\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/measurement_sim_927.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/SiPRNet_img_0926_angle_gt.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/SiPRNet_img_0926_angle.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/img_0926_residual.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/GS_Fashion_simulation_927.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/HIO_Fashion_simulation_927.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/ADMM_sim_927.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/ResNet_simulation_927.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/lensless_sim_img_0926_angle.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/CGAN_simulation_927.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/NNPhase_img_0926_angle_sim.png}\n\t\t\t\t\\\\\n\t\t\t\t\n\t\t\t\t\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/measurement_exp_927.png}~\n\t\t\t\t&\\includegraphics[height=0.118\\textwidth,width=0.1\\textwidth, valign=t]{Figures\/third\/SiPRNet_img_0926_angle_gt_exp.pdf}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/SiPRNet_img_0926_angle_exp.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/img_0926_residual_exp.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/GS_Fashion_exp_927.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/HIO_Fashion_exp_927.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/ADMM_exp_927.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/ResNet_exp_927.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/lensless_exp_img_0926_angle.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/CGAN_exp_927.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/NNPhase_img_0926_angle.png}\n\t\t\t\t\\\\\n\t\t\t\t\n\t\t\t\t{\\scriptsize \\makecell[c]{Fourier\\\\measurement}} & {\\scriptsize \\makecell[c]{Ground-truth \\\\ (phase)}} & {\\scriptsize \\makecell[c]{\\textbf{\\textit{SiPRNet}}\\\\(ours)} }& {\\scriptsize \\makecell[c]{\\textbf{\\textit{SiPRNet}} \\\\(Error Map)} }  & {\\scriptsize GS } & {\\scriptsize HIO} & {\\scriptsize ADMM-TV} & {\\scriptsize ResNet}\n\t\t\t\t& {\\scriptsize LenslessNet}  & {\\scriptsize PRCGAN}  & {\\scriptsize NNPhase}  \n\t\t\t\t\\\\\n\t\t\t\t\n\t\t\t\\end{tabular}\n\t\t\\end{adjustbox}\n\t\t\\vspace{-0.2cm}\n\t\t\\caption{Simulation and experimental results of different phase retrieval methods on the Fashion-MNIST dataset \\cite{xiao2017\/online}. The first \\textbf{column} denotes the Fourier intensity measurements (pixel values: $0 \\rightarrow 4095$). The second \\textbf{column} shows the corresponding phase parts of the complex-valued images with scale bars (for experimental results) at the bottom left corners. The fourth \\textbf{column} denotes the error maps of \\textit{SiPRNet}. The other \\textbf{columns} present the reconstruction images through different methods. Except for the Fourier intensity measurements (the first \\textbf{column}), the colormap of the rest \\textbf{columns} ranges from $0$ to $2\\pi$. The first and second \\textbf{rows} present the simulation and experimental results of the same image. The third and the fourth \\textbf{rows} show the simulation and experimental results of another image. Please zoom in for better view.}\n\t\t\\label{fig:thirdimage}\n\t\t\n\t\\end{figure}   \n\n    \\section*{Data availability} The datasets used in this paper are available in  \\cite{li2017reliable, xiao2017\/online}. Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.\n   \t\n   \t\\section*{Funding} General Research Fund (PolyU 15225321), Centre for Advances in Reliability and Safety (CAiRS)\n    \n    \\section*{Acknowledgments} This work was supported by the Hong Kong Research Grant Council under General Research Fund no. PolyU 15225321, and the Centre for Advances in Reliability and Safety (CAiRS) admitted under AIR@InnoHK Research Cluster.\n    \n    \\section*{Disclosures} The authors declare no conflicts of interest.\n\n\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nCertainly the existence of black holes was one of the most interesting predictions of General Relativity. However, the discovery that these objects, due to quantum fluctuations, emit as black bodies with temperatures dictated by the surface gravity \\cite{Bekenstein:1972tm,Bekenstein:1973ur,Hawking:1974rv,Hawking:1974sw}, a purely geometric magnitude, simultaneously exposed  black holes as  thermodynamic objects and showed that black holes are scenarios where geometry and thermodynamics intertwine.\n\nThe first law of the black hole thermodynamics (see for instance Ref.\\cite{Wald:1984rg}) is given by\n\\begin{equation}\\label{1aleySinPdV}\ndM=TdS+\\Omega dJ+ \\phi dQ,\n\\end{equation}\nand represents the balance of energy through the modification of the macroscopic parameters of the black hole. Here $M$ corresponds to the mass parameter and was considered the internal energy of the system, namely the (ADM) mass. Finally, in Eq.(\\ref{1aleySinPdV}) $T$ is the temperature computed as the $(4\\pi)^{-1} \\kappa$ with $\\kappa$ is the surface gravity of the black hole horizon, $S$ is the entropy, $\\Omega$ is the angular velocity, $J$ is the angular momentum, $\\phi$ is the electrostatic potential and $Q$ is the electric charge.\n\nBy comparing Eq.(\\ref{1aleySinPdV}) with the first law of thermodynamics one can notice the absence of the pressure\/volume term, namely $-pdV$, which would stand for \\textit{the macroscopic work} done by the system. In a matter of speaking, this was due to the lack of the concepts of volume and pressure for a black hole in the original derivation, based on accretion processes. Several works have studied this issue in the last 20 years \\cite{Padmanabhan:2002sha,Tian:2010gn,Cvetic:2010jb,Dolan:2011xt}. To address this problem let us consider the first law of thermodynamics (see for example \\cite{Dolan:2011xt}):\n\\begin{equation}\\label{1aley1}\ndU = TdS - pdV +\\Omega dJ+ \\phi dQ,\n\\end{equation}\nwhere $U$ stands for the internal energy. To extend Eq.(\\ref{1aleySinPdV}) to match Eq.(\\ref{1aley1}), one can think of assigning a volume to the black hole by considering the \\textit{volume}  defined its radius, for example $V=\\frac{4}{3}\\pi r_+^3$ for $d=4$ case. Unfortunately, since entropy is a function of the horizon radius as well, then Eq.(\\ref{1aley1}) would be inconsistent due to $dS$ and $dV$ would not be independent directions. To address this problem, in Ref.\\cite{Kastor1}, was proposed to reinterprete the mass parameter as the Enthalpy of the black hole, instead of internal energy $U$. Moreover, the cosmological constant was connected with the thermodynamic pressure. The law obtained is called the first law of (black hole) thermodynamics in the {\\it extended phase space}. In Ref.\\cite{MotorTermico1}, on the other hand, the promotion of the mass parameter to the enthalpy is based on the fact that to form a black hole would require to cut off a region of the space, and therefore an initial energy equal to $E_0 = -\\rho V$, with $\\rho$ the energy density of the system, is needed. In four dimensions the thermodynamics volume corresponds to $V=\\frac{4}{3}\\pi r_+^3$. Moreover, the presence of the cosmological constant defines $\\rho = -P$ and therefore the mass parameter can be considered equivalent to\n\\begin{equation}\nM=U-\\rho V = U+pV,\n\\end{equation}\nHere $M$ is to be recognized as the enthalpy $H$ of the system. With this in mind, the first law, in this  extended phase space, yields\n\\begin{equation} \\label{1aley2}\ndM=dH=TdS+Vdp+\\phi dQ+\\Omega dJ.\n\\end{equation}\nThis extended first law (\\ref{1aley2}) also was derived in reference \\cite{Mann1} by using Hamiltonian formalism. It is worth to mention that the definition of the extended phase space have allowed to construct a {\\it heat engine} in terms of a black hole, see some examples in references \\cite{MotorTermico1,MotorTermico5,MotorTermico7,MotorTermico13,MotorTermico14,MotorTermico15}, adding a new layer to our understanding of the black hole thermodynamics. Another interesting applications is the Joule Thompson expansion for black holes studied in \\cite{JT1,JT2,JT3,JT4}.\n\n\\subsection{Phase Transitions}\nThe study of phase transitions in black hole physics has called a renewed attention in the last years due to the AdS\/CFT conjecture. For instance, it is well known that the {\\it Hawking-Page phase transition} \\cite{Hawking:1982dh}, in the context of the AdS\/CFT correspondence, has been re-interpreted as the plasma gluon confinement\/deconfinement phase transition in the would-be dual (conformal) field theory. Similarly, the AdS Reissner Nordstr\\\"om's transitions in the $(\\phi-q)$ diagram have been interpreted as liquid\/gas phase transitions of Van der Waals fluids \\cite{Chamblin:1999tk,Chamblin:1999hg}.\n\nRecently the analysis of the $p-V$ critical behaviors (in the extended phase space) have been under studied extensively\\cite{RegularBHquimica, PV1,PV2,PV3,PV4,PV5,PV6,PV7,PV8,PV9,PV10,PV11,PV12,Hendi:2017fxp,Hendi:2015cqz,Hendi:2015hoa,Hendi:2014kha}. For instance, in \\cite{Mann2} was studied in the context of the charged $4D$ AdS black holes how the phase transitions between small\/large black hole are analogous to liquid\/gas transitions in a Van der Waals fluid. Moreover, it was also shown that the critical exponents, near the critical points, recovers those of Van der Waals fluid with the same \\textit{compressibility factor} $Z=3\/8$. In reference \\cite{Mann1} was introduced a \\textit{new interpretation} of the Hawking Page phase transition \\cite{Hawking:1982dh} mentioned above, but in the context of $p-V$ critical behavior. \n\n\\subsection{Higher Dimensions, Lovelock and thermodynamics}\n\nDuring the last years 50 years several branches of theoretical physics have noticed that considering higher dimensions is plausible. Now, considering higher dimension in gravity opens up a range of new possibilities that retain the core of the Einstein gravity in four dimensions. Lovelock is a one of these possibilities as, although includes higher powers of curvature corrections, its equations of motion are of second order and thus causality is still insured. Generic Lovelock theory is the sum of the Euler densities $L_n$ in lower dimensions ($2n \\leq d$) multiplied by coupling constants $\\alpha_n$ \\footnote{In even dimensions the maximum order, $n=d\/2$, is in turn the corresponding Euler density of the dimension and therefore does not contribute to the equations of motion,}. The Lagrangian is\n\\begin{equation}\n \\displaystyle L  =  \\sum_{n=0}^{[d\/2]} \\alpha_n L_n,\n\\end{equation}\nwhere $L_n=\\frac{1}{2^{n}}\\,\\sqrt{-g}\\ \\delta_{\\nu_{1}...\\nu_{2n}}^{\\mu_{1}...\\mu_{2n}}\\ R_{\\mu_{1}\\mu_{2}}^{\\nu_{1}\\nu_{2}}\\cdots R_{\\mu_{2n-1}\\mu_{2n}}^{\\nu_{2n-1}\\nu_{2n}}$. The $L _0 \\propto 1$ term is related by the cosmological constant, $L_1 \\propto R $ is related by the Ricci scalar, and $L_2 \\propto R^{\\alpha\\beta}_{\\hspace{2ex}\\mu\\nu}R^{\\mu\\nu}_{\\hspace{2ex}\\alpha\\beta}-4 R^{\\alpha\\nu}_{\\hspace{2ex}\\beta\\nu} R^{\\beta\\mu}_{\\hspace{2ex}\\alpha\\mu} + R^{\\alpha\\beta}_{\\hspace{2ex}\\alpha\\beta}R^{\\mu\\nu}_{\\hspace{2ex}\\mu\\nu}$ is the Gauss Bonnet density. The EOM are:\n\\begin{align}\\label{LovelockGenerico}\n\\frac{1}{\\sqrt{g}}\\frac{\\delta }{\\delta g_{\\mu\\nu}} (L\\sqrt{g}) &= G^{\\mu}_{ \\hspace{1ex} \\nu} \\nonumber \\\\\n&= \\sum_{n} \\alpha_{n} \\frac{1}{2^n} \\delta^{\\mu_1 \\ldots \\mu_{2n} \\mu}_{\\nu_1 \\ldots \\nu_{2n} \\nu} R^{\\nu_{1} \\nu_{2}}_{\\hspace{2ex}\\mu_{1} \\mu_{2}}\\ldots R^{\\nu_{2n-1} \\nu_{2n}}_{\\hspace{2ex}\\mu_{2n-1} \\mu_{2n}}\n\\end{align}\nwhere $G^{\\mu \\nu}_{(LL)}$   which, by definition, satisfies $\\nabla_\\mu G^{\\mu\\nu}_{(LL)} \\equiv 0$.\n\nIn higher curvature the $p-V$ criticality have been studied for example in reference \\cite{Mann3} for generic Lovelock black holes with conformal scalar hair arising from coupling of a real scalar field to the dimensionally extended Euler densities and in reference \\cite{Nam:2019clw} for a regular charged AdS black hole in Einstein Gauss Bonnet gravity. See also \\cite{Hendi:2015soe,Xu:2014tja}.\n\nOne drawback of a generic Lovelock gravity is the existence of more than single ground state, namely more than a single constant curvature spaces solution, or equivalently more than a single potential \\textit{effective cosmological constants} \\cite{Camanho:2011rj}. However, there are two families of Lovelock gravities that have indeed a single ground state. The first family has been originally studied in \\cite{Banados:1994ur} and has a unique but $k$ degenerated ground state. \n\nThe second case is usually called {\\it Pure Lovelock gravity}. In this case the Lagrangian is a just the $n$-single term of Lagrangian plus a cosmological constant, \\textit{i.e.}, $L = \\alpha_n L_n + \\alpha_0 L_0$. This has a single and non-degenerated ground state of negative curvature $-1\/l^2$ for $n$ odd. Several solutions of Pure Lovelock theories are known. For instance vacuum black hole solutions can be found in references \\cite{Cai:2006pq,DadhichBH,Dadhich3,PureLovelock1,PureLovelock2,Aranguiz:2015voa} and regular black holes in references \\cite{milko1,milko2}. See other applications in references \\cite{Dadhich1,Dadhich2,Dadhich4,Dadhich5}.\n\nIn the next sections the thermodynamic of the Pure Lovelock solutions will be carried out in details. For this, the cosmological constant will be promoted to the intensive thermal pressure of the system. Under this assumption, it will be shown that the system behaves as Van der Waals fluid. Finally, the critical coefficient near the critical points will be computed.\n\n\\subsubsection{{\\bf Vacuum black hole in Pure Lovelock gravity: Uncharged Asymptotically AdS Case}}\n\nLet us consider a static geometry, represented in Schwarzschild coordinates by\n\\begin{equation}\\label{linelement}\nds^2 =-f(r) dt^2+ \\frac{dr^2}{f(r)} + r^2 d \\Omega^2_{d-2}.\n\\end{equation}\nFor simplicity the cosmological constant is fixed such that\n\\begin{equation} \\label{Lambda1}\n    \\Lambda = - \\frac{(d-1)(d-2)}{2 l^{2n}}.\n\\end{equation}\nThis is similar to the definition in \\cite{Aranguiz:2015voa} but a numerical factor. The equations of motion reduce to merely\n\\begin{equation}\n\\frac{d}{dr} \\left ( r^{d-2n-1}\\big (1-f(r) \\big)^n +  \\dfrac{ r^{d-1}}{{l}^{2n}} \\right ) =0.\n\\end{equation}\nSolutions of these equations have been studied in references \\cite{Cai:2006pq,Dadhich1}. In terms of Eq.(\\ref{linelement}) the solution is defined by\n\\begin{equation} \\label{funcionPLvacio}\nf(r) =1 - \\left (\\frac{2M}{r^{d-2n-1}} - \\frac{r^{2n}}{l^{2n}} \\right )^{1\/n} .\n\\end{equation}\nOne can notice that $f(r)$ can take complex values for even $n$. Because of this, in this work will be considered only odd $n$. The thermodynamic pressure can be read off this definition as\n\\begin{equation} \\label{p1}\n    p=-\\frac{\\Lambda}{(d-2)\\Omega_{d-2}},\n\\end{equation}\nwhere $\\Omega_{d-2}$ is the unitary area of a $d-2$ sphere. It is worth to stress that Eqs.(\\ref{Lambda1},\\ref{p1}) coincide with the definitions in Ref.\\cite{Mann1,Mann2} for $n=1$ and $d=4$. Now, replacing Eqs.(\\ref{Lambda1},\\ref{p1}) into Eq.(\\ref{funcionPLvacio}) yields\n\\begin{equation} \\label{funcionPLvacio1}\n    f(r)=1 - \\left (\\frac{2M}{r^{d-2n-1}} - \\frac{2 \\Omega_{d-2}r^{2n}}{d-1}p  \\right )^{1\/n},\n\\end{equation}\nwhere the dependence of $f(r)$ on $p,M$ have been made explicit.\n\n\\subsubsection{{\\bf Charged Pure Lovelock solution}}\nThe charged case is slightly difference since it is necessary to solve the Maxwell equations and to include an energy momentum tensor into the gravitational equations. Let us start by defining $A_\\mu = A_t(r) \\delta_\\mu^t$, which defines only the non-vanishing component of the Maxwell tensor \n\\begin{equation} \\label{PotencialVectorial}\n    F_{tr}=-\\partial_rA_t(r), \n\\end{equation}\nand the single non vanishing component of the Maxwell equations, \n\\begin{equation} \\label{TensorElectromagnetico}\n  \\nabla_{\\mu}F^{\\mu\\nu} =0 \\rightarrow  \\frac{d}{dr} \\left ( r^{d-2} F_{tr} \\right) =0.\n\\end{equation}\nLikewise, the gravitational equations are reduced to the single independent component\n\\begin{equation}\\label{GravitationalEMEq}\n \\frac{d}{dr} \\left ( r^{d-2n-1}\\big (1-f(r) \\big)^n +  \\dfrac{ r^{d-1}}{{l}^{2n}} \\right ) = (F_{tr})^2 r^{d-2}.\n\\end{equation}\nIt is straightforward to integrate Eqs.(\\ref{TensorElectromagnetico},\\ref{GravitationalEMEq}) yielding \\cite{Aranguiz:2015voa}\n\\begin{equation} \\label{funcionPLvacioQ}\nf(r) =1 - \\left (\\frac{2M}{r^{d-2n-1}} - \\frac{r^{2n}}{l^{2n}} - \\frac{Q^2}{(d-3) r^{2d-2n-4}} \\right )^{1\/n},\n\\end{equation}\nand\n\\begin{equation}\n    F_{tr}=\\frac{Q}{r^{d-2}} \\rightarrow A_t(r)= \\phi_\\infty + \\frac{Q}{(d-3)r^{d-3}},\n\\end{equation}\nAs usual $\\phi_{\\infty}=0$ is fixed such that $\\displaystyle \\lim_{r\\rightarrow \\infty} A_t(r) =0$. \n\nNow,  by direct observation, one can notice that for even $n$ the existence of certain ranges of $r$ where $f(r)$ can take complex values. To avoid this only odd $n$ will be considered from now on.\n\nAs done previously, by replacing Eqs(\\ref{Lambda1},\\ref{p1}) into Eq.(\\ref{funcionPLvacioQ}), $f(r)$ can be written in terms of the thermodynamics variables as\n\\begin{equation} \\label{funcionPLvacio1Q}\n    f(r)=1 - \\left (\\frac{2M}{r^{d-2n-1}} - \\frac{2 \\Omega_{d-2}r^{2n}}{d-1}p  - \\frac{Q^2}{(d-3) r^{2d-2n-4}}   \\right )^{1\/n}.\n\\end{equation}\n \n\n\\section{Extended phase space in Vacuum Pure Lovelock Gravity}\n\nIn this section the extended phase space formalism, mentioned above, will be introduced to analyze the solutions. One can notice that the fist law of the thermodynamics, Eq.(\\ref{1aley2}), for the non rotating case takes the form\n\\begin{equation} \\label{1aleyconcarga}\n   dH= dM=TdS+Vdp + \\phi dQ.\n\\end{equation}\nNow, in order to construct a thermodynamic interpretation one must notice that under any transformation of the parameters the function $f(r_+,M,Q,l)$  must still vanishes, otherwise the transformation would not be mapping black holes into black holes in the space of solutions. Indeed, $\\delta f(r_+,M,Q,l) = 0$ and $f(r_+,M,Q,l)=0$ are to be understood as constraints on the evolution along the space of parameters. However, there is another approach by recalling that the mass parameter, $M$, is also to be understood as a function of the parameters $M(r_+,l,Q)$ as well.\n\nSince the thermodynamic parameters are $S,p$ and $Q$, therefore it is convenient to reshape $M=M(S,p,Q)$ in order to explicitly obtain \n\\begin{equation} \\label{1aley3}\n    dM= \\left ( \\frac{\\partial M}{\\partial S}    \\right)_{p,Q}dS + \\left ( \\frac{\\partial M}{\\partial p}    \\right)_{S,Q}dp + \\left ( \\frac{\\partial M}{\\partial Q}    \\right)_{p,S}dQ .\n\\end{equation}\nThis corresponds to the definitions of the component of the tangent vector in the space of parameters, but also correspond to the definitions of the temperature, thermodynamic volume and electric potential in the form of\n\\begin{eqnarray}\nT &=&\\left ( \\frac{\\partial M}{\\partial S}    \\right)_{p,Q},\\label{temperatura} \\\\\nV &=& \\left ( \\frac{\\partial M}{\\partial p}    \\right)_{S,Q} \\textrm{ and }\\label{volumen} \\\\\n\\phi &=& \\left ( \\frac{\\partial M}{\\partial Q}    \\right)_{S,p}. \\label{potencial}\n\\end{eqnarray}\n\nOn the other hand, the variation along the space of parameters of the condition defined by $f(r_+,M,p,Q)=0$,\n\\begin{eqnarray}\n  df(r_+,M,p,Q) &=& 0\\label{variacion}\\\\\n   &=& \\frac{\\partial f}{\\partial r_+}  dr_+ + \\frac{\\partial f}{\\partial M} dM + \\frac{\\partial f}{\\partial p} dp + \\frac{\\partial f}{\\partial Q} dQ,\\nonumber\n\\end{eqnarray}\nyields a second expression for $dM$ given by\n\\begin{align} \\label{variacion1}\n    dM &=  \\left (\\frac{1}{4\\pi} \\frac{\\partial f}{\\partial r_+} \\right ) \\left( -\\frac{1}{4 \\pi} \\frac{\\partial f}{\\partial M} \\right)^{-1} dr_+ \\nonumber \\\\\n    &+ \\left( - \\frac{\\partial f}{\\partial M} \\right)^{-1} \\left ( \\frac{\\partial f}{\\partial p} \\right ) dp +\\left( - \\frac{\\partial f}{\\partial M} \\right)^{-1} \\left ( \\frac{\\partial f}{\\partial Q} \\right ) dQ,\n\\end{align}\nwhich must coincide with equation (\\ref{1aley3}). In Eq.(\\ref{variacion1}) one can recognize presence of the temperature, which geometrically is defined as\n\\begin{equation} \\label{temperaturaconocida}\n    T= \\frac{1}{4\\pi} \\frac{\\partial f}{\\partial r_+},\n\\end{equation}\n yielding \n\\begin{equation} \\label{dS1}\n\\left( -\\frac{1}{4 \\pi} \\frac{\\partial f}{\\partial M} \\right)^{-1} dr_+= dS,\n\\end{equation}\nwhich is a very known result. Now, by the same token, the thermodynamic volume and electric potential are given by\n\\begin{equation} \\label{volumen1}\n    V=\\left ( \\frac{\\partial M}{\\partial p}    \\right)_{S,Q}= \\left( - \\frac{\\partial f}{\\partial M} \\right)^{-1} \\left ( \\frac{\\partial f}{\\partial p} \\right ) \n\\end{equation}\nand\n\\begin{equation} \\label{potencial1}\n    \\phi = \\left ( \\frac{\\partial M}{\\partial Q}    \\right)_{S,p}= \\left( - \\frac{\\partial f}{\\partial M} \\right)^{-1} \\left ( \\frac{\\partial f}{\\partial Q} \\right ),\n\\end{equation}\nrespectively. These expressions will be discussed in the next sections.\n\n\\subsection{Smarr Expression}\nConsidering Plank units, one can notice that $p$, the pressure, has units of $[p]=\\ell^{-2n}$. See Eqs.(\\ref{Lambda1},\\ref{p1}, \\ref{presionPL1}). Likewise, one can check that $[M] = \\ell^{d-2n-1}$, $[Q] = \\ell^{d-n-2}$ and $[S] = \\ell^{d-2n}$. \n\nFollowing {\\it Euler's theorem} \\cite{Altamirano:2014tva}, with $M(S,p,Q)$, one can construct the {\\it Smarr formula} for Pure Lovelock gravity given by\n\\begin{equation}\n    \\frac{d-2n-1}{d-2n}M=TS+\\frac{d-n-2}{d-2n} \\phi Q - \\frac{2n}{d-2n}VP.\n\\end{equation}\nThis coincides with the definitions discussed in \\cite{Altamirano:2014tva,Mann2,Mann1} for $n=1$ . Derivations of the Smarr formula for generic Lovelock theory are discussed in \\cite{Dolan:2014vba,Kastor:2010gq}. \n\n\\subsection{ Uncharged asymptotically AdS}\nReplacing the solution of Eq.(\\ref{funcionPLvacio1}) into Eq.(\\ref{dS1}) yields\n\\begin{align}\n\\left( -\\frac{1}{4 \\pi} \\frac{\\partial f}{\\partial M} \\right)^{-1} dr_+&=2\\pi n r_+^{d-2n-1} \\nonumber \\\\\n&=d \\left ( \\frac{2}{d-2n}n \\pi r_+^{d-2n}  \\right ) = dS,\n\\end{align}\nor equivalently\n\\begin{equation} \\label{entropia1}\n    S= \\frac{2}{d-2n}n \\pi r_+^{d-2n},\n\\end{equation}\nwhich coincides with reference \\cite{Cai:2006pq}. Eq.(\\ref{volumen1}), on the other hand,  yields\n\\begin{equation} \\label{volumen11}\n    \\left( - \\frac{\\partial f}{\\partial M} \\right)^{-1} \\left ( \\frac{\\partial f}{\\partial p} \\right )  = V = \\frac{\\Omega_{d-2}}{d-1}r_+^{d-1},\n\\end{equation}\nwhich corresponds to the volume of a $(d-2)$ sphere of radius $r_+$ . This coincides with the definition in Refs.\\cite{Mann1,Mann2}.\n\n\\subsubsection{{ \\bf Fluid equation of state}}\nIt is worth to notice at this point that the temperature, see Ref.\\cite{milko1}, can be expressed as \n\\begin{equation} \\label{TemperaturaPL1}\n    4\\pi n T= \\frac{d-2n-1}{r_+} + (d-1)\\frac{ r_+^{2n-1}}{l^{2n}},\n\\end{equation}\nwhere, the temperature has units of $\\ell^{-1}$. One can make explicit the dependence on the pressure, by inserting Eqs.(\\ref{Lambda1},\\ref{p1}) into Eq.(\\ref{TemperaturaPL1}). Therefore, \n\\begin{equation} \\label{presionPL1}\n    2 \\Omega_{d-2}p = 4 \\pi n \\frac{T}{r_+^{2n-1}} - \\frac{d-2n-1}{r_+^{2n}}.\n\\end{equation}\n\n\\subsubsection{\\bf Physical Pressure}\nBefore to proceed a digression is necessary. As mentioned above $[p] = \\ell^{-2n}$, however the physical pressure, $p_G$, must be satisfied [Force\/Area]$=\\ell_p^{-2}$, since the area has units of [area]$=\\ell_p^{d-2}$. In the literature $p_g$ is called the geometrical pressure. Since $p$ and $p_G$ must be connected by $p_G \\sim \\alpha_n p$, namely $[p_{G}] = \\left [\\alpha_n p \\right]=\\ell_p^{-d}$, therefore\n \\begin{itemize}\n    \\item the {\\it coupling constants} must satisfy $[\\alpha_n]=\\ell^{2n-d}$ \\cite{Oliva1}. For $n=1$ this coincides with the inverse of the higher dimensional Newton constant $G_d^{-1}$ \\cite{Maartens:2003tw} and\n    \\item  there is still room for a dimensionless constant which can be used to adjust the definition. \n\\end{itemize}\nWith this in mind,  $p_G$ can be taken as \n\\begin{equation}\n    p_{G} =2 \\Omega_{d-2} \\ell_p^{2n-d} p = 4 \\pi n \\frac{T}{\\ell_p^{d-2n}r_+^{2n-1}} + \\ldots,\n\\end{equation}\nwhich, in turns, defines the \\textit{specific volume} of the system as\n\\begin{equation} \\label{volumenespecificoPL}\n    v=\\frac{\\Omega_{d-2}}{2\\pi} \\ell_p^{d-2n}r_+^{2n-1}.\n\\end{equation}\n\nNotice that $v$ indeed has units of volume, namely $[v]= \\ell_p^{d-1}$ \\cite{Hennigar:2018cnh}. Finally, after some replacements, \n\\begin{equation} \\label{presionPL2}\n   p = \\frac{n T}{v} - \\frac{1}{2\\Omega_{d-2}(2\\pi\/\\Omega_{d-2})^{2n\/(2n-1)}}\\frac{d-2n-1}{v^{2n\/(2n-1)}},\n\\end{equation}\nwhich can be recognized as the Van der Walls equation for $n=1$, namely $P=T\/(v-b)-a\/v^2$, with $b=0$ \\cite{Mann1}.  \n\n\\subsubsection{{\\bf Hawking-Page phase transition}}\n\nIn reference \\cite{Hawking:1982dh} was analyzed the thermodynamic behavior of Schwarzschild AdS space, which differs from the Schwarzschild case due to the presence of the AdS gravitational potential,  namely the presence of $\\sim r^2\/l^2$ in $f(r)$ \\cite{Czinner:2015eyk}. In this case there is a phase transition between black hole and AdS radiation at a critical temperature $T_{HP}$ where the Gibbs free energy, $G=M-TS$ vanishes \\cite{Mann2,Czinner:2015eyk,Wang:2019vgz}. \n\nFig.\\ref{HP} displays the numerical behavior of the Gibbs free energy v\/s temperature for $n=3$ and $d=10$ for Pure Lovelock. The upper curve represents the unstable small black hole (namely with negative heat capacity) and the lower curve represents the stable large black hole. It is direct to show that this behavior is similar for any other set of values of $n$ and $d$. \n\nThe generic behavior is displayed in Fig.\\ref{HPgenerico}. The upper green curve represents the unstable small black hole, the lower blue curve represents the stable large black hole, and the orange curve represents the thermal AdS radiation whose Gibbs free energy vanishes \\cite{Mann1,Czinner:2015eyk,Wang:2019vgz,Costa:2015gol}. One can notice that for $[T_{min},T_{HP}[$ the large stable black hole has positive Gibbs free energy,  therefore, the preferred state corresponds to the thermal AdS radiation. On the other hand, for $T>T_{HP}$, the preferred state is the large stable black hole whose Gibbs free energy is negative. This hints the existence of a Hawking Page phase transition between radiation and the large black hole states at $T=T_{HP}$.  \n\nFinally, it is worth to notice, from Fig.\\ref{HP}, that the value of $T_{HP}$ increases as the pressure increases. This behavior is similar to the HP phase transition for the Schwarzschild AdS black holes \\cite{Mann1,Belhaj:2015hha} or the polarized AdS black holes \\cite{Costa:2015gol}.\n\n\\begin{figure}\n\\centering\n{\\includegraphics[width=3in]{HP1.eps}}\n\\caption{HP phase transitions for $n=3$, $d=10$ and $Q=1$, $p=0,0000045$(blue), $p=0,0000050$(red), $p=0,0000055$ (green). T (horizontal axis) v\/s G (vertical axis). }\n\\label{HP}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n{\\includegraphics[width=3in]{HPgenerico.eps}}\n\\caption{Generic behavior of Gibbs free energy (vertical axis) v\/s T (horizontal axis). HP phase transition at $T_{HP}$}\n\\label{HPgenerico}\n\\end{figure}\n\n\\subsection{ Charged Pure Lovelock solution}\nBy replacing Eq.(\\ref{funcionPLvacio1Q}) into Eqs.(\\ref{dS1},\\ref{volumen1}) the expressions for the entropy (\\ref{entropia1}) and the volume (\\ref{volumen11}) are obtained . On the other hand, replacing Eq.(\\ref{funcionPLvacio1Q}) into Eq.(\\ref{potencial1}) yields \n\\begin{equation} \\label{potencial2}\n    \\phi= \\frac{Q}{(d-3)r^{d-3}}.\n\\end{equation}\nThis thermodynamics quantities, obtained by mean of variation of parameters, are consistent with the values presented in the literature. \n\n\\subsubsection{\\bf Fluid equation of state}\nIn this case the temperature can be written as\n\\begin{equation} \\label{TemperaturaPL1Q}\n    4\\pi n T= \\frac{d-2n-1}{r_+} + (d-1)\\frac{ r_+^{2n-1}}{l^{2n}}- \\frac{Q^2}{r_+^{2d-2n-3}} .\n\\end{equation}\nBy inserting equations (\\ref{Lambda1},\\ref{p1}) into equation (\\ref{TemperaturaPL1Q}) is obtained\n\\begin{equation} \\label{presionPL2Q}\n    2 \\Omega_{d-2}p = 4 \\pi n \\frac{T}{r_+^{2n-1}} - \\frac{d-2n-1}{r_+^{2n}} + \\frac{Q^2}{r_+^{2d-4}}.\n\\end{equation}\nNow, by using the definition of the specific volume, defined in Eq.(\\ref{volumenespecificoPL}) with $\\ell_p=1$,\n\\begin{eqnarray}\n p &=& \\frac{n T}{v} - \\frac{1}{2\\Omega_{d-2}(2\\pi\/\\Omega_{d-2})^{2n\/(2n-1)}}\\frac{d-2n-1}{v^{2n\/(2n-1)}} \\nonumber \\\\\n   &+& \\frac{1}{2\\Omega_{d-2}(2\\pi\/\\Omega_{d-2})^{(2d-4)\/(2n-1)}}\\frac{Q^2}{v^{(2d-4)\/(2n-1)}}. \\label{presionPL3}\n\\end{eqnarray}\nOne can notice the similarity of this relation with the corresponding one for a Van der Waals fluid for $n=1$ but with the presence of an addition third term.  \n\n\\subsubsection{\\bf Critical points and compressibility factor}\nTo compare Eq.(\\ref{presionPL3}) with the behavior of a Van der Waals fluid it is necessary to determine the critical points of the system. The second order critical points are defined by the conditions,  \n\\begin{equation}\n    \\frac{\\partial p}{\\partial v}=0 \\textrm{ and } \\frac{\\partial^2 p}{\\partial v^2}=0.\n\\end{equation}\nThese determine the critical values\n\\begin{equation}\n    v_c = \\frac{\\Omega_{d-2}}{2\\pi}\\left ( \\frac{2d^2-2dn-7d+4n+6}{n(d-2n-1)}Q^2   \\right )^{(2n-1)\/(2d-2n-4)} ,\n\\end{equation}\nand\n\\begin{align}\n    T_c &= \\frac{n(d-2n-1)}{2\\pi n (2n-1)(2\\pi\/\\Omega_{d-2})^{1\/(2n-1)}v^{1\/(2n-1)}} \\nonumber \\\\\n        &-\\frac{(d-2)Q^2(2\\pi\/\\Omega_{d-2})^{(2n-2d+4)\/(2n-1)}v^{(2n-2d+4)\/(2n-1)}}{2\\pi n (2n-1)(2\\pi\/\\Omega_{d-2})^{1\/(2n-1)}v^{1\/(2n-1)}}\n\\end{align}\nNotice that $p_c=p(v_c,T_c)$ can be determined from equation (\\ref{presionPL3}) by evaluation on the critical values $T_c$ and $v_c$. \n\nIn the table (\\ref{tabla1}) critical values $v_c$, $T_c$ and  $p_c$ and the {\\it compressibility factor} $Z=p_cv_v\/T_c$ are displayed for different values of $n$ and $d$ . For $n=1$ and $d=4$ the compressibility factor has the exact value $Z=3\/8$ which coincides with the value of the compressibility for a Van der Waals fluid. In general, compressibility factor can be written as\n\\begin{equation} \\label{factorcompresibilidad}\n    Z=\\frac{2d-2n-3}{4(d-2)},\n\\end{equation}\nwhich implies that $Z<1$ extrictly. This implies that this can be interpreted as a low pressure gas. \n\\begin{table*} \n\\caption{Critical values and Compressibility Factor ($Z$) .}\n\\begin{tabular*}{\\textwidth}{@{\\extracolsep{\\fill}}lccccc@{}}\n\\hline\n\\multicolumn{1}{c}{$n$   } & \\multicolumn{1}{c}{$d$ } & \\multicolumn{1}{c}{$v_c$}& \\multicolumn{1}{c}{$T_c$}&\\multicolumn{1}{c}{$p_c$}&\\multicolumn{1}{c}{$Z=\\dfrac{p_cv_c}{T_c}$} \\\\\n \\hline\n$1$ & $4$  & $\\Omega_{d-2}\/(2\\pi)Q\\sqrt{6}$ & $1\/(18\\pi Q) \\sqrt{6}$ &$1\/(24\\Omega_{d-2}Q^2)$ & $3\/8$   \\\\\n\\hline \n$1$ & $5$  & $\\Omega_{d-2}\/(4\\pi)(120Q^2)^{1\/4}$ & $4\/(75\\pi)(6750\/Q^2)^{1\/4}$ &$2\/(45\\Omega_{d-2}Q)\\sqrt{30}$ & $5\/12$   \\\\\n\\hline\n$1$ & $6$  & $\\Omega_{d-2}\/(6\\pi)(6804Q^2)^{1\/6}$ & $9\/(98\\pi)(201.684\/Q^2)^{1\/6}$ &$9\/(112\\Omega_{d-2})(294\/Q^2)^{1\/3}$ & $7\/16$   \\\\\n\\hline\n$3$ & $8$  & $\\Omega_{d-2}\/(2\\pi)(14Q^2)^{5\/6}$ & $3\/(490\\pi) (14^5\/Q^2)^{1\/6}$ &$1\/(280\\Omega_{d-2}Q^2)$ & $7\/24$   \\\\\n\\hline\n$3$ & $9$  & $\\Omega_{d-2}\/(4\\pi)(21^5\\cdot 2^3Q^{10})^{1\/8}$ & $8\/(945\\pi) (21^7\\cdot 2\/Q^2)^{1\/8}$ &$4\/(735\\Omega_{d-2})(21\\cdot 2^3\/Q^6)^{1\/4}$ & $9\/28$   \\\\\n\\hline\n$3$ & $10$  & $\\Omega_{d-2}\/(18\\pi)(792)^{1\/2}Q$ & $3\/(242\\pi) (2^7\\cdot 11^9 \\cdot 9\/Q^2)^{1\/10}$ &$9\/(704\\Omega_{d-2})(726\/Q^6)^{1\/5}$ & $11\/32$   \\\\\n\\hline\n$5$ & $12$  & $\\Omega_{d-2}\/(2\\pi)(22Q^2)^{9\/10}$ & $5\/(2178\\pi)(22^9\/Q^2)^{1\/10}$ &$1\/(792\\Omega_{d-2}Q^2)$ & $11\/40$   \\\\\n\\hline\n\\end{tabular*} \\label{tabla1}\n\\end{table*}\n\n\\subsubsection{\\bf $P-v$ curve.}\n\nIn Fig.\\ref{p-v} is displayed for $n=3$ and $d=10$ the behavior of the curve $p-v$ defined by equation (\\ref{presionPL3}). Although this is an example still this behavior is generic for any values of $n$ and $d$. \n\nFor values of temperature $T>T_{c}$, the second and third factors of Eq.(\\ref{presionPL3}) are negligible in comparison with the first one, and therefore the curve approximates the form  $p \\cdot v \\propto T$ and thus mimicking the behavior of an ideal gas. Conversely, for $T<T_{c}$  as $v$ increases the $p$, which diverges for $v=0$, decreases until reach a local minimum. Next, $p$ increases until reaching a local maximum. Finally $p$ decreases asymptotically until reaching $p=0$. Thus for $T<T_{c}$ ( or for $p<p_{c}$ due that $T \\propto p$) the behavior is analogue to the Van der Waals fluid.  \n\nIn standard vapor-liquid theory is well established that an increase in pressure must be correlated with a decrease in volume and vice versa. Therefore, see Fig.\\ref{p-v}, one can notice the existence of a range of the specific volume, says $]v_{min},v_{max}[$, which must be considered nonphysical due to both pressure and volume increase simultaneously. On the other hand, it can be also noticed that for a single value of the specific pressure, there might exist up to three possible values of specific volume $v$ with one of them always within the nonphysical region. Therefore, for analysis one must only consider the two proper solutions that satisfy either $v_1<v_{min}$ or $v_2>v_{max}$ with $p(v_1)=p(v_2)$. In fluid theory these two solutions are known as the {\\it Van der Waals loop} and physically this corresponds to a {\\it vapor liquid equilibrium}, highlighting that phase transitions take place. \n\n\\begin{figure}\n\\centering\n{\\includegraphics[width=3in]{p-v.eps}}\n\\caption{$p-v$ curve for $T_1<T_2<T_3<T_{c}<T_4<T_5$}\n\\label{p-v}\n\\end{figure}\n\n\\subsubsection{\\bf Temperature}\nThe behavior of the temperature is displayed in figure \\ref{FigTemperatura}(a) for $p>p_{c}$, in figure \\ref{FigTemperatura}(b) for $p=p_{c}$ and in figure \\ref{FigTemperatura}(c) for $p<p_{c}$ . Although for these figures $n=3$ and $d=10$ it is straightforward to show that this enfolds the generic behavior for any $n$ and $d$. One can notice the presence of an extreme black hole case for small $r_+$ where the temperature vanishes. For $p>p_{c}$ the temperature is an increasing function of $r_+$. For $p=p_{c}$ the temperature has one inflexion point at $r_+=r_{infl}$. More relevant for this discussion is the case for $p<p_{c}$, where the fluid is analogous to the Van der Waals, and where the temperature has a local minimum and a local maximum at $r_+=r_{min}$ and $r_+=r_{max}$, respectively.\n\n\\begin{figure}\n\\centering\n\\subfigure[Temperature for $p=1>p_{c}$ .]{\\includegraphics[width=75mm]{TemperaturaMayor.eps}} \n\\subfigure[Temperature for $p=p_{c} \\approx 0,001608$ ]{\\includegraphics[width=75mm]{TemperaturaCritica.eps}}\n\\subfigure[Temperature for $p=0,00001<p_{c}$ ]{\\includegraphics[width=75mm]{TemperaturaMenor.eps}}\n\\caption{Temperature behavior for $n=3$ and $d=10$ with $Q=1$ .}\n\\label{FigTemperatura}\n\\end{figure}\n\n\\subsubsection{\\bf Heat Capacity}\nThe heat capacity is displayed in Fig.\\ref{CalorEspecificoMayoroigual}(a) for $p>p_c$, in figure \\ref{CalorEspecificoMayoroigual}(b) for $p=p_c$ and in figure \\ref{CalorEspecificoMenor} for $p<p_c$. As previously, although $n=3$ and $d=10$ it is straightforward to show that Fig.\\ref{CalorEspecificoMayoroigual} enfold the generic behavior for any $n$ and $d$. One can notice that \n\n\\begin{itemize}\n    \\item For $p>p_c$ the heat capacity is a positive increase function of $r_+$, and there is no phase transition, thus black hole is always stable.\n    \\item For $p=p_c$ small and large black hole coexist at the inflexion point $r_+=r_{infl}$, where the heat capacity $C \\to \\infty$.\n    \\item For the $p<p_c$ case, the derivative $(dT\/dr_+)$ can vanish for two values of  $r_+=r_{min}$ and $r_+=r_{max}$, as observed in the figure \\ref{FigTemperatura}(c). This implies, due to $C= (dS\/dr_+)\/(dT\/dr_+)|_{p,Q}$, that the heat capacity becomes ill-defined at those values of $r_+$. In this way $r_{min}$ and $r_{max}$ define three regions. For $r_+<r_{max}$ one can notice that $C>0$ defining a small stable black hole. Next, there is small unstable ($C<0$) region for $r \\in ]r_{max},r_{min}[$. Finally there is a third region $r_+>r_{min}$ where the system is a large stable black hole $(C>0)$. This hints the existence of phase transitions but, by means of the following analysis of the Gibbs free energy, one can check that only the \\textit{small stable bh\/large stable bh} transition is allowed. \n\\end{itemize}\n\n\\begin{figure}\n\\centering\n\\subfigure[Heat capacity for $p=1>p_c$.]{\\includegraphics[width=75mm]{Cmayor.eps}}\n\\subfigure[Heat capacity for $p=p_c \\approx 0,001608 $]{\\includegraphics[width=75mm]{Ccritica.eps}}\n\\caption{Behavior of Heat Capacity for $n=3$ and $d=10$ with $Q=1$  .}\n\\label{CalorEspecificoMayoroigual}\n\\end{figure}\n\n\n\\begin{figure}\n\\centering\n\\subfigure[Heat capacity for $r_+ \\in (0.896,1.6)$.]{\\includegraphics[width=75mm]{CMenor1.eps}}\n\\subfigure[Heat capacity for $r_+ \\in (2.7,3.5)$]{\\includegraphics[width=75mm]{CMenor2.eps}}\n\\caption{Behavior of Heat Capacity for $n=3$ and $d=10$ with $Q=1$, $p=0,00001<p_c$ .}\n\\label{CalorEspecificoMenor}\n\\end{figure}\n\n\\subsubsection{\\bf Gibbs Free energy}\nThe Gibbs free energy, defined as $G=M-TS$ \\cite{Mann1,Hennigar:2018cnh,Czinner:2015eyk,Wang:2019vgz}, is displayed for $n=3$ and $d=10$ in figure \\ref{GibbsT}(a) for $p>p_c$, in figure \\ref{GibbsT}(b) for $p=p_c$ and in \\ref{GibbsT}(c) for $p<p_c$. It is direct to check that this behavior is similar for other values of $n$ and $d$. As well known \\cite{RegularBHquimica} the Gibbs free energy describes the global stability of the system. It is worth to recall that the global minimum represents the most likely state, while the preferred state at fixed temperature corresponds to the minimal value of the Gibbs free energy. \n\\begin{itemize}\n    \\item For pressure larger than the critical pressure, the Gibbs free energy is a single valued function.\n    \\item For $p=p_c$ there is a cusp at $T=T_c$, which coincides with the radius $r_+=r_{infl}$, thus, since at this point $C=T dS\/dT=-T (\\partial^2G\/\\partial T^2) \\to \\infty$, the discontinuity on the second derivative of Gibbs function implies the presence of second order phase transition between small stable\/large stable black holes.\n    \\item The generic behavior of the Gibbs free energy for $p<p_c$ is displayed in figure \\ref{GibbsT1}. We see three possibles black hole states: small stable, small unstable and large stable. The preferred state is such that the Gibbs free energy has the minimum value. Thus, for $]T_0,T_{max}]$ the preferred state correspond to stable large black hole. However, for $T<T_0$ the preferred state corresponds to the small stable black hole. Thus, at $T=T_0$ there is a first order phase transition between large\/small stable black hole.\n\\end{itemize}\n\n\\begin{figure}\n\\centering\n\\subfigure[Gibbs for $p=1>p_{c}$ .]{\\includegraphics[width=63mm]{GibbsMayor1.eps}} \\hspace{15mm}\n\\subfigure[ Gibbs for $p=p_{c} \\approx 0,001608$ ]{\\includegraphics[width=63mm]{GibbsCritico1.eps}}\n\\subfigure[Gibbs for $p=0,00001<p_{c}$ ]{\\includegraphics[width=63mm]{GibbsMenor1.eps}}\n\\caption{Gibbs free energy (vertical axis) v\/s T (horizontal axis) for $n=3$ and $d=10$ with $Q=1$.}\n\\label{GibbsT}\n\\end{figure}\n\n\n\\begin{figure}\n\\centering\n{\\includegraphics[width=3in]{GibbsT1.eps}}\n\\caption{Generic behavior of Gibbs free energy (vertical axis) v\/s T (horizontal axis)}\n\\label{GibbsT1}\n\\end{figure}\n\n\\subsubsection{\\bf Behavior near critical points}\nLet's define the following dimensionless variables \n\\begin{equation} \\label{omega}\n    \\omega =\\left ( \\frac{V}{V_c} \\right)^{2n-1}-1=\\left ( \\frac{v}{v_c} \\right)^{d-1}-1\n\\end{equation}\nand \n\\begin{equation} \\label{t}\n    t = \\frac{T}{T_c}-1,\n\\end{equation}\nto analyze the behavior nearby the critical points. It is direct to check that near the critical points, \\textit{i.e.} $V \\to V_c$ (or $v \\to v_c$) and $T \\to T_c$, the variables $\\omega \\to 0$, and $t \\to 0$, respectively. The pressure in Eq.(\\ref{presionPL3}) is displayed in table \\ref{tabla2} at $O(t\\omega^2, \\omega^4)$. The truncation will be justified below . \n\\begin{table*} \n\\caption{Behavior of pressure near critical points .}\n\\begin{tabular*}{\\textwidth}{@{\\extracolsep{\\fill}}llc@{}}\n\\hline\n\\multicolumn{1}{l}{$n$   } & \\multicolumn{1}{l}{$d$ } & \\multicolumn{1}{c}{$p$} \\\\\n \\hline\n$1$ & $4$  & $p \\approx 1 + 8\/3 t - 8\/9t \\omega -4\/81 \\omega^3$    \\\\\n\\hline \n$1$ & $5$  & $p \\approx 1 + 12\/5t - 3\/5t \\omega -1\/32 \\omega^3$  \\\\\n\\hline\n$1$ & $6$  & $p \\approx 1 + 16\/7t - 16\/35t \\omega -8\/375 \\omega^3$   \\\\\n\\hline\n$3$ & $8$  & $p \\approx 1 + 72\/7t - 72\/49t \\omega -12\/8575 \\omega^3$   \\\\\n\\hline\n$3$ & $9$  & $p \\approx 1 + 28\/3t - 7\/6t \\omega -7\/6400 \\omega^3$   \\\\\n\\hline\n$3$ & $10$  & $p \\approx 1 + 96\/11t - 32\/33t \\omega -16\/18225 \\omega^3$   \\\\\n\\hline\n$5$ & $12$  & $p \\approx 1 + 200\/11t - 200\/121t \\omega -100\/323433 \\omega^3$    \\\\\n\\hline\n\\end{tabular*} \\label{tabla2}\n\\end{table*}\n\nIn general it is possible to approximate the pressure as\n\\begin{equation} \\label{presiongenerica}\n    p \\approx 1 + \\frac{n}{Z}t - \\frac{n t}{(d-1)Z} \\omega - \\frac{(2d-2n-3)n}{6Z(d-1)^3(2n-1)^2} \\omega^3,\n\\end{equation}\nwhereas \n\\begin{equation}\n    dP= -\\frac{P_c n}{Z (d-1)} \\left ( t+ \\frac{(2d-2n-3)}{2(d-1)^2(2n-1)^2} \\omega^2 \\right ) d \\omega.\n\\end{equation}\nTo compute the \\textit{ critical exponents} one can follow \\cite{Mann2}. \n\\begin{itemize}\n    \\item The {\\it $\\alpha$ exponent} describes the behavior of the heat capacity at constant volume defined as\n    \\begin{equation}\n        C_v = T \\frac{\\partial S}{\\partial T} \\propto |t|^{-\\alpha}.\n    \\end{equation}\n    In the case at hand, since entropy and volume are both functions of horizon radius, see Eqs.(\\ref{entropia1},\\ref{volumen11}), then a constant volume implies a constant entropy as well. Therefore it is satisfied that $C_v=0$ which implies that there is no dependence on $|t|$. In turn this implies that\n    \\begin{equation}\n        \\alpha=0.\n    \\end{equation}\n  \\item The \\textit{exponent} $\\beta$ describes the behavior of the \\textit{order parameter} defined as\n  \\begin{equation} \\label{eta}\n      \\eta = V_l-V_s \\propto |t|^\\beta.\n  \\end{equation}\n  To compute this order parameter one can use the \\textit{Maxwell's area law}\n\\begin{equation} \\label{integral1}\n    \\oint VdP=0,\n\\end{equation}\nwhere, the volume (\\ref{omega}) is approximated such that $VdP$ is truncated to $O(t\\omega^3, \\omega^5)$, in other words, \n\\begin{equation} \\label{aproxVolumen}\n    V \\approx V_c \\left (  1 + \\frac{1}{2n-1} \\omega   \\right ).\n\\end{equation}\nTherefore, closed integral (\\ref{integral1}) becomes\n\\begin{equation} \\label{integral2}\n  V_c  \\oint  dP + \\frac{V_c}{2n-1} \\oint \\omega dP =0.\n\\end{equation}\nThe second integral of the left side of (\\ref{integral2}) yields\n\\begin{equation}\n \\int^{\\omega_s}_{\\omega_l}\\omega dP= \\int^{\\omega_s}_{\\omega_l} \\omega   \\left ( t+ \\frac{(2d-2n-3)}{2(d-1)^2(2n-1)^2} \\omega^2 \\right ) d \\omega =0.\n\\end{equation}\nwhich has the non trivial solution given by\n\\begin{equation} \\label{condicion}\n    \\omega_s = -\\omega_l.\n\\end{equation}\nOn the other hand, the first integral of the left side of (\\ref{integral2}) yields\n\\begin{align}\n    &\\int^{\\omega_s}_{\\omega_l}dP = 0 \\nonumber \\\\\n&1 + \\frac{n}{Z}t - \\frac{n t}{(d-1)Z} \\omega_l - \\frac{(2d-2n-3)n}{6Z(d-1)^3(2n-1)^2} \\omega^3_l = \\nonumber \\\\\n&1 + \\frac{n}{Z}t - \\frac{n t}{(d-1)Z} \\omega_s - \\frac{(2d-2n-3)n}{6Z(d-1)^3(2n-1)^2} \\omega^3_s,\n\\end{align}  \nThis, by condition (\\ref{condicion}), yields\n\\begin{equation} \\label{omegal}\n    \\omega_l=(d-1)(2n-1) \\sqrt{- \\frac{6}{2d-2n-3}t},\n\\end{equation}\n with $t<0$. Replacing Eqs. (\\ref{aproxVolumen}) and (\\ref{omegal}) into Eq.(\\ref{eta}) one can obtain \n \\begin{equation}\n     \\eta = \\frac{2V_c}{2n-1} \\omega_l = 2(d-1)V_c\\sqrt{- \\frac{6}{2d-2n-3}t},\n \\end{equation}\n Comparing this result with Eq.(\\ref{eta}) one can uncover that \n \\begin{equation}\n     \\beta = \\frac{1}{2}.\n \\end{equation}\n\\item Now one can compute the \\textit{exponent} $\\gamma$ which describes the behavior  under \\textit{isothermal compressibility}, $\\kappa_T$, defined by\n\\begin{equation}\n\\kappa_T= - \\frac{1}{V}\\frac{\\partial V}{\\partial P} \\bigg |_T \\propto |t|^{-\\gamma}.\n\\end{equation}\n\nBy using Eqs.(\\ref{presiongenerica},\\ref{aproxVolumen}) one can prove that\n\\begin{equation}\n    \\frac{\\partial P}{\\partial V}=P_c \\frac{\\partial p}{\\partial \\omega} \\frac{\\partial \\omega}{\\partial V} \\propto -(2n-1)\\frac{P_c}{V_c} \\frac{n}{(d-1)Z}t,\n\\end{equation}\nand therefore\n\\begin{equation}\n    \\kappa_T \\propto \\frac{(d-1) Z}{P_c n(2n-1)t},\n\\end{equation}\nfrom which is direct to read that\n\\begin{equation}\n    \\gamma=1.\n\\end{equation}\n\n\\item Finally, one can compute \\textit{exponent} $\\delta$ which describes the behavior on the critical isotherm $T=T_c$, and therefore for $t = 0$. In this case this is defined by\n\\begin{equation}\n    |P-P_c| \\propto |V-V_c|^\\delta.\n\\end{equation}\nFrom Eq.(\\ref{presiongenerica}), at $t=0$, it is possible to notice that\n\\begin{equation} \\label{presiongenerica1}\n    p -1 \\approx - \\frac{(2d-2n-3)n}{6Z(d-1)^3(2n-1)^2} \\omega^3.\n\\end{equation}\nUsing the approximation of Eq.(\\ref{aproxVolumen}) one can show that\n\\begin{equation}\n    \\frac{P-P_c}{P_c} \\approx - \\frac{(2d-2n-3)n}{6Z(d-1)^3(2n-1)^2} \\left ( (2n-1) \\frac{V-V_c}{V_c}   \\right )^3,\n\\end{equation}\nand therefore\n\\begin{equation}\n    \\delta=3. \n\\end{equation}\n\\end{itemize}\n\nThis critical exponents just computed are similar to those of Van der Waals gas. Although the presence of extra dimensions and the value of $n$ modify the value of the compressibility factor respect to the well known value $Z=3\/8$, they do not affect the value of the critical exponents, and thus, the behavior is still similar to that of Van der Waals fluid near the critical exponents.\n\n\\section{Conclusion and discussion} \nIn this article it has been analyzed the thermodynamics of the Pure Lovelock solutions in $d$ dimensions, in an extended phase space including the introduction of pressure and volume as dual thermodynamic variables and the mass parameter standing for the enthalphy of the system. A linear relation between the cosmological constant and the thermodynamics pressure, valid for all value of $n$ (odd) and $d$, has been established. The thermodynamic volume obtained corresponds to a generalization of the black hole \\textit{volume}. The Smarr formula for Pure Lovelock gravity was constructed as well.\n\nIn addition, it was shown that the variation of parameters are constrained to follow a first law of thermodynamics in the extended phase space. The temperature is defined by surface gravity, as usual, the entropy coincides with the value computed {\\it a la Wald} \\cite{Cai:2006pq, milko1} and the electric potential matches the usual known definition. \n\nThe equation state for the uncharged and charged cases was obtained. For the uncharged case, a Hawking-Page-Like phase transitions between thermal radiation and large stable black hole has been found. For the charged case, it was found that the compressibility factor, $Z$, is a generic function of $d$ and $n$ given by equation (\\ref{factorcompresibilidad}). Remarkably, $Z<1$ estrictly and therefore the behavior always corresponds to a real gas. \n\nIt was found the existence of a critical temperature, $T_c$, where phase transitions occur. The mapping of the $p-v$ curves indicate that for Pure Lovelock gravity the behavior is similar to an ideal gas for $T>T_c$. For $T<T_c$ the behavior is analogous to a Van der Walls fluid. Furthermore, there are a first order phase transition between small stable\/large stable black hole, which are analogous to liquid\/gas phase transitions.\n\nFinally, we have computed the critical exponents whose values are similar to those of the Van der Waals gas. Furthermore, it is possible to write the pressure as a generic function of $n$ and $d$ given by equation (\\ref{presiongenerica}). Thus, the presence of extra dimensions and the value of $n$ does not modify the known values of the critical points, therefore the general behavior can be considered similar to a  Van der Waals fluid near the critical exponents.\n\n\\section*{Acknowledgments}\nThis work was partially funded by grant DI-08-19 UNAB.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{References}}\n\\bibliographystyle{elsarticle-num}\n\n\\begin{document}\n\t\\begin{frontmatter}\n\t\t\\title{Minimum $2$-vertex strongly biconnected spanning directed subgraph problem}\n\t\t\\author{Raed Jaberi}\n\t\t\n\t\t\\begin{abstract}  \n\t    A directed graph $G=(V,E)$ is strongly biconnected if $G$ is strongly connected and its underlying graph is biconnected. A strongly biconnected directed graph $G=(V,E)$ is called $2$-vertex-strongly biconnected if $|V|\\geq 3$ and the induced subgraph on $V\\setminus\\left\\lbrace w\\right\\rbrace $ is strongly biconnected for every vertex $w\\in V$. In this paper we study the following problem.\t\t\n\t\tGiven a $2$-vertex-strongly biconnected directed graph $G=(V,E)$, compute an edge subset $E^{2sb} \\subseteq E$ of minimum size such that the subgraph $(V,E^{2sb})$ is $2$-vertex-strongly biconnected.\n\t\t\\end{abstract} \n\t\t\\begin{keyword}\n\t\t\tDirected graphs \\sep Approximation algorithms  \\sep Graph algorithms \\sep strongly connected graphs \\sep Strongly biconnected directed graphs\n\t\t\\end{keyword}\n\t\\end{frontmatter}\n\t\\section{Introduction}\n\t  A directed graph $G=(V,E)$ is strongly biconnected if $G$ is strongly connected and its underlying graph is biconnected. A strongly biconnected directed graph $G=(V,E)$ is called $k$-vertex-strongly biconnected if $|V|> k$ and for each $L\\subset V$ with $|L|<k$, the induced subgraph on $V\\setminus L$ is strongly biconnected. The minimum $k$-vertex-strongly biconnected spanning subgraph problem (denoted by MKVSBSS) is formulated as follows.\t\t\n\tGiven a $k$-vertex-strongly biconnected directed graph $G=(V,E)$, compute an edge subset $E^{ksb} \\subseteq E$ of minimum size such that the subgraph $(V,E^{ksb})$ is $k$-vertex-strongly biconnected. In this paper we consider the MKVSBSS problem for $k=2$.\n Each $2$-vertex-strongly-biconnected directed graph is $2$-vertex-connected, but the converse is not necessarily true. Thus, optimal solutions for minimum $2$-vertex-connected spanning subgraph (M2VCSS) problem are not necessarily feasible solutions for the $2$-vertex strongly biconnnected spanning subgraph problem, as shown in Figure \\ref{figure:sbiconnectedexampleoptimalsolutions}.\n\n\\begin{figure}[htp]\n\t\\centering\n\t\n\t\\subfigure[]{\n\t\n\\scalebox{0.79}{\n\n\t\t\\begin{tikzpicture}[xscale=2]\n\t\t\\tikzstyle{every node}=[color=black,draw,circle,minimum size=0.9cm]\n\t\t\\node (v1) at (-1.6,3.1) {$1$};\n\t\t\\node (v2) at (-2.4,0) {$2$};\n\t\t\\node (v3) at (-0.5, -2.5) {$3$};\n\t\t\\node (v4) at (3.6,-2.1) {$4$};\n\t\t\\node (v5) at (0.5,1.6) {$5$};\n\t\t\\node (v6) at (3.2,0.5) {$6$};\n\t\t\\node (v7) at (3.7,3.2) {$7$};\n\t\n\t\t\\begin{scope}   \n\t\t\\tikzstyle{every node}=[auto=right]   \n\t\t\\draw [-triangle 45] (v2) to (v5);\n\t\t\t\\draw [-triangle 45] (v2) to (v6);\n\t\t\t\t\\draw [-triangle 45] (v3) to (v5);\n\t\\draw [-triangle 45] (v2) to (v1);\n\t\\draw [-triangle 45] (v1) to[bend right] (v2);\n\t\\draw [-triangle 45] (v1) to (v5);\n\t\\draw [-triangle 45] (v5) to[bend right] (v1);\n\t\\draw [-triangle 45] (v5) to (v7);\n\t\\draw [-triangle 45] (v7) to[bend right] (v5);\n\t\\draw [-triangle 45] (v7) to (v6);\n\t\\draw [-triangle 45] (v6) to[bend right] (v7);\n\t\\draw [-triangle 45] (v6) to (v4);\n\t\\draw [-triangle 45] (v4) to[bend right] (v6);\n\t\\draw [-triangle 45] (v4) to (v3);\n\t\\draw [-triangle 45] (v3) to[bend right] (v4);\n\t\\draw [-triangle 45] (v3) to (v2);\n\t\\draw [-triangle 45] (v2) to[bend right] (v3);\n\t\\draw [-triangle 45] (v1) to[bend left] (v7);\n\t\\draw [-triangle 45] (v5) to (v6);\n\t\\draw [-triangle 45] (v6) to (v3);\n\t\\draw [-triangle 45] (v4) to (v2);\n\t\\draw [-triangle 45] (v3) to (v1);\n\t\\draw [-triangle 45] (v4) to (v5);\n\t\\draw [-triangle 45] (v2) to (v5);\n\t\\draw [-triangle 45] (v7) to [bend left](v4);\n\t\t\\end{scope}\n\t\t\\end{tikzpicture}\n\t}\n\t}\n\t\\subfigure[]{\n\t\t\\scalebox{0.79}{\n\t\t\\begin{tikzpicture}[xscale=2]\n\t\t\\tikzstyle{every node}=[color=black,draw,circle,minimum size=0.9cm]\n\t\\node (v1) at (-1.6,3.1) {$1$};\n\\node (v2) at (-2.4,0) {$2$};\n\\node (v3) at (-0.5, -2.5) {$3$};\n\\node (v4) at (3.6,-2.1) {$4$};\n\\node (v5) at (0.5,1.6) {$5$};\n\\node (v6) at (3.2,0.5) {$6$};\n\\node (v7) at (3.7,3.2) {$7$};\n\t\t\n\t\t\\begin{scope}   \n\t\t\\tikzstyle{every node}=[auto=right]   \n\t\t\n\t\t\\draw [-triangle 45] (v2) to (v1);\n\t\t\\draw [-triangle 45] (v1) to[bend right] (v2);\n\t\t\\draw [-triangle 45] (v1) to (v5);\n\t\t\\draw [-triangle 45] (v5) to[bend right] (v1);\n\t\t\\draw [-triangle 45] (v5) to (v7);\n\t\t\\draw [-triangle 45] (v7) to[bend right] (v5);\n\t\t\\draw [-triangle 45] (v7) to (v6);\n\t\t\\draw [-triangle 45] (v6) to[bend right] (v7);\n\t\t\\draw [-triangle 45] (v6) to (v4);\n\t\t\\draw [-triangle 45] (v4) to[bend right] (v6);\n\t\t\\draw [-triangle 45] (v4) to (v3);\n\t\t\\draw [-triangle 45] (v3) to[bend right] (v4);\n\t\t\\draw [-triangle 45] (v3) to (v2);\n\t\t\\draw [-triangle 45] (v2) to[bend right] (v3);\n\t\n\t\t\\end{scope}\n\t\t\\end{tikzpicture}}}\n\\subfigure[]{\n\t\\scalebox{0.79}{\n\t\\begin{tikzpicture}[xscale=2]\n\t\\tikzstyle{every node}=[color=black,draw,circle,minimum size=0.9cm]\n\t\\node (v1) at (-1.6,3.1) {$1$};\n\\node (v2) at (-2.4,0) {$2$};\n\\node (v3) at (-0.5, -2.5) {$3$};\n\\node (v4) at (3.6,-2.1) {$4$};\n\\node (v5) at (0.5,1.6) {$5$};\n\\node (v6) at (3.2,0.5) {$6$};\n\\node (v7) at (3.7,3.2) {$7$};\n\t\n\t\\begin{scope}   \n\t\\tikzstyle{every node}=[auto=right]   \n\t\\draw [-triangle 45] (v2) to (v5);\n\t\\draw [-triangle 45] (v2) to (v1);\n\n\t\\draw [-triangle 45] (v1) to (v5);\n\t\n\t\\draw [-triangle 45] (v5) to (v7);\n\t\n\t\\draw [-triangle 45] (v7) to (v6);\n\n\t\\draw [-triangle 45] (v6) to (v4);\n\n\t\\draw [-triangle 45] (v4) to (v3);\n\n\t\\draw [-triangle 45] (v3) to (v2);\n\t\n\t\\draw [-triangle 45] (v1) to[bend left] (v7);\n\t\\draw [-triangle 45] (v5) to (v6);\n\t\\draw [-triangle 45] (v6) to (v3);\n\t\\draw [-triangle 45] (v4) to (v2);\n\t\\draw [-triangle 45] (v3) to (v1);\n\t\\draw [-triangle 45] (v2) to (v5);\n\t\\draw [-triangle 45] (v7) to [bend left](v4);\n\t\\end{scope}\n\t\t\\end{tikzpicture}}}\n\\caption{(a) A $2$-vertex strongly biconnected graph. (b) An optimal solution for the minimum $2$-vertex-connected spanning subgraph problem. But note that this subgraph is not $2$-vertex strongly biconnected.(c) An optimal solution for the minimum $2$-vertex strongly biconnected spanning subgraph problem}\n\\label{figure:sbiconnectedexampleoptimalsolutions}\n\\end{figure}\n\nThe problem of finding a $k$-vertex-connected spanning subgraph of a $k$-vertex-connected directed graph is NP-hard for $k\\geq 1$ \\cite{G79}. Results of Edmonds \\cite{Edmonds72} and Mader \\cite{Mader85} imply that the number of edges in each minimal $k$-vertex-connected directed graph is at most $2kn$ \\cite{CT00}. Cheriyan and Thurimella \\cite{CT00} gave a $(1+1\/k)$-approximation algorithm for the minimum $k$-vertex-connected spanning subgraph problem. Georgiadis \\cite{Georgiadis11} improved the running time of this algorithm for the M2VCSS problem and presented a linear time approximation algorithm that achieves an approximation factor of $3$ for the M2VCSS problem. Georgiadis et al. \\cite{GIK20} provided linear time $3$-and $2$- approximation algorithms based on the results of \\cite{GT16,G10,ILS12,FILOS12} for the M2VCSS problem. Furthermore, Georgiadis et al. \\cite{GIK20} improved the algorithm of Cheriyan and Thurimella when $k=2$. Strongly connected components of a directed graph and blocks of an undirected graphs can be found in linear time using Tarjan's algorithm \\cite{TAARJAN72}.\n Wu and Grumbach \\cite{WG2010} introduced the concept of strongly biconnected directed graph and strongly connected components.  The MKVSBSS problem is NP-hard for $k\\geq 1$. In this paper we study the MKVSBSS problem when $k=2$ (denoted by M2VSBSS). \n\t\n\\section{Approximation algorithm for the M2VSBSS problem} \nIn this section we present an approximation algorithm (Algorithm \\ref{algo:approximationalgorithmfor2sb}) for the M2VSBSS Problem. This algorithm is based on b-articulation points, minimal 2-vertex-connected subgraphs, and Lemma \\ref{def:addingedgestoreducesbcs}. A vertex $w$ in a strongly biconnected directed graph $G$ is a b-articulation points if $G\\backslash\\left\\lbrace w \\right\\rbrace $ is not strongly biconnected \\cite{Jaberi20}. \n\n\n\\begin{lemma} \\label{def:addingedgestoreducesbcs}\n\tLet $G_s=(V,E_s)$ be a subgraph of a strongly biconnected directed graph $G=(V,E)$ such that $G_s$ is strongly connected and $G_s$ has $t>0$ strongly biconnected components. Let $(u,w)$ be an edge in $E\\setminus E_s$  such that $u,w $ are in not in the same strongly\n\tbiconnected component of $G_s$. Then the directed subgraph $(V,E\\cup \\left\\lbrace (u,w) \\right\\rbrace )$ contains at most $t-1$ strongly biconnected components.\n\\end{lemma}\n\\begin{proof}\n\tSince $G_s$ is strongly connected, there exists a simple path $p$ from $w$ to $u$ in $G_s$. Path $p$ and edge $(u,w)$ form a simple cycle. Consequently, the vertices $u,w$ are in the same strongly biconnected component of the subgraph $(V,E\\cup \\left\\lbrace (u,w) \\right\\rbrace )$.\n\\end{proof}\n\n\\begin{figure}[htbp]\n\t\\begin{myalgorithm}\\label{algo:approximationalgorithmfor2sb}\\rm\\quad\\\\[-5ex]\n\t\t\\begin{tabbing}\n\t\t\t\\quad\\quad\\=\\quad\\=\\quad\\=\\quad\\=\\quad\\=\\quad\\=\\quad\\=\\quad\\=\\quad\\=\\kill\n\t\t\t\\textbf{Input:} A $2$-vertex strongly biconnected directed graph $G=(V,E)$ \\\\\n\t\t\t\n\t\t\t\\textbf{Output:} a $2$-vertex strongly biconnected subgraph $G_{2s}=(V,E_{2s})$\\\\\n\t\t\t{\\small 1}\\> find a minimal $2$-vertex-connected subgraph $G_{1}=(V,E_{1})$ of $G$.\\\\\n\t\t\t{\\small 2}\\> \\textbf{if} $G_{1}$ is $2$-vertex strongly biconnected \\textbf{then} \\\\\n\t\t\t{\\small 3}\\>\\>output  $G_{1}$\\\\\n\t\t\t{\\small 4}\\> \\textbf{else}\\\\\n\t\t\t{\\small 5}\\>\\>$E_{2s} \\leftarrow E_{1}$\\\\\n\t\t\t{\\small 6}\\>\\>  $G_{2s}\\leftarrow (V,E_{2s})$ \\\\\n\t\t\t{\\small 7}\\>\\>  identify the b-articulation points of $G_{1}$.\\\\\n\t\t\t{\\small 8}\\>\\>  \\textbf{for} evry b-articulation point $b\\in V$ \\textbf{do} \\\\\n\t\t\t{\\small 9}\\>\\>\\>  \\textbf{while} $G_{2s}\\setminus\\left\\lbrace b \\right\\rbrace $ is not  strongly biconnected \\textbf{do}\\\\ \n\t\t{\\small 10}\\>\\>\\>\\>  calculate the  strongly biconnected components of $G_{2s}\\setminus\\left\\lbrace b \\right\\rbrace $\\\\\n\t\t{\\small 11}\\>\\>\\>\\> find an edge $(u,w) \\in E\\setminus E_{2s}$ such that $u,w $ are not in \\\\\n\t\t{\\small 12}\\>\\>\\>\\>the same strongly biconnected components of $G_{2s}\\setminus\\left\\lbrace b \\right\\rbrace $.\\\\\n\t\t{\\small 13}\\>\\>\\>\\> $E_{2s} \\leftarrow E_{2s} \\cup\\left\\lbrace  (u,w)\\right\\rbrace $ \\\\\n\t\t{\\small 14}\\>\\>\\>output $G_{2s}$\n\t\n\t\t\\end{tabbing}\n\t\\end{myalgorithm}\n\\end{figure}\n\n\\begin{lemma} \nAlgorithm \\ref{algo:approximationalgorithmfor2sb} returns a $2$-vertex strongly biconnected directed subgraph.\n\\end{lemma}\t\n\\begin{proof}\nIt follows from Lemma \\ref{def:addingedgestoreducesbcs}.\n\\end{proof}\n\nThe following lemma shows that each optimal solution for the M22VSBSS problem has at least $2n$ edges.\n\\begin{lemma} \\label{def:optsolutionatlesttwon}\n\tLet $G=(V,E)$ be a $2$-vertex-strongly biconnected directed graph. Let $O\\subseteq E$ be an optimal solution for the M2VSBSS problem. Then $|O|\\geq 2n$. \n\t\n\\end{lemma}\n\\begin{proof}\n\tfor any vertex $x\\in V$, the removal of $x$ from the subgraph $(V,O)$ leaves a strongly biconnected directed subgraph. Since each strongly biconnected directed graph is stronly connected,\n\tthe subgraph $(V,O)$ has no strong articulation points. Therefore, the directed subgraph $(V,O)$ is $2$-vertex-connected.\n\\end{proof}\nLet $l$ be the number of b-articulation points in  $G_{1}$. The following lemma shows that Algorithm \\ref{algo:approximationalgorithmfor2sb} has an approximation factor of $(2+l\/2)$.\n\\begin{theorem}\n\tLet $l$ be the number of b-articulation points in  $G_{1}$. Then, $|E_{2s}|\\leq l(n-1)+4n$.\n\\end{theorem}\n\\begin{proof}\n\tResults of Edmonds \\cite{Edmonds72} and Mader \\cite{Mader85} imply that $|E_1|\\leq 4n$  \\cite{CT00,Georgiadis11}. Moreover, by Lemma \\ref{def:optsolutionatlesttwon}, every optimal solution for the M22VSBSS problem has size at least $2n$. For every b-articulation point in line $8$,  Algorithm \\ref{algo:approximationalgorithmfor2sb} adds at most $n-1$ edge to $E_{2s}$ in while loop. Therefore, $|E_{2s}|\\leq l(n-1)+4n$\n\\end{proof}\n\\begin{Theorem}\n\tThe running time of Algorithm \\ref{algo:approximationalgorithmfor2sb} is $O(n^{2}m)$.\n\\end{Theorem}\n\\begin{proof}\n\tA minimal $2$-vertex-connected subgraph can be found in time $O(n^2)$ \\cite{Georgiadis11,GIK20}. B-articulation points can be computed in $O(nm)$ time.\n\tThe strongly biconnected components of a directed graph can be identified in linear time \\cite{WG2010}. \n\tFurthermore, by Lemma \\ref{def:addingedgestoreducesbcs}, lines $9$--$13$ take $O(nm)$ time.\n\\end{proof}\n\n\\section{Open Problems}\n Results of Mader \\cite{Mader71,Mader72} imply that the number of edges in each minimal $k$-vertex-connected undirected graph is at most $kn$ \\cite{CT00}. Results of Edmonds \\cite{Edmonds72} and Mader \\cite{Mader85} imply that the number of edges in each minimal $k$-vertex-connected directed graph is at most $2kn$ \\cite{CT00}. These results imply a $2$-approximation algorithm \\cite{CT00} for minimum $k$-vertex-connected spanning subgraph problem for undirected and directed graphs \\cite{CT00} because every vertex in a $k$-vertex-connected undirected graphs has degree at least $k$ and every vertex in a $k$-vertex-connected directed graph has outdegree at least $k$ \\cite{CT00}. Note that these results imply a $7\/2$ approximation algorithm for the M2VSBSS problem by calculating a minimal $2$-vertex-connected directed subgraph of a $2$-vertex strongly biconnected directed graph $G=(V,E)$ and a minimal $3$-vertex connected undirected subgraph of the underlying graph of $G$.\n \n \\begin{lemma}\nLet $G=(V,E)$ be a $2$-vertex strongly biconnected directed graph. Let $G_1=(V,L)$ be a minimal $2$-vertex-connected subgraph of $G$ and let $G_2=(V,U)$ be a minimal $3$-vertex-connected subgraph of the underlying graph of $G$. Then the directed subgraph $G_s=(V,L\\cup A )$ is $2$-vertex strongly connected, where $A=\\left\\lbrace (v,w) \\in E \\text{ and } (v,w) \\in U \\right\\rbrace $. Moreover, $|L\\cup A|\\leq 7n$\n \\end{lemma}\n\\begin{proof}\n\tLet $w$ be any vertex of the subgraph $G_s$. Since the $G_1=(V,L)$ is $2$-vertex-connected, subgraph $G_s$ has no strong articulation points. Therefore, $G_s\\setminus\\left\\lbrace w\\right\\rbrace $ is strongly connected. Moreover, the underlying graph of  $G_s\\setminus\\left\\lbrace w\\right\\rbrace $ is biconnected because the underlying graph of $G_s$ is $3$-vertex-connected. Results of Edmonds \\cite{Edmonds72} and Mader \\cite{Mader85} imply that $|L|<4n$. Results of Mader \\cite{Mader71,Mader72} imply that $|U|\\leq 3n$.\n\\end{proof}\n \n An open problem is whether each minimal $2$-vertex strongly biconnected directed graph has at most $4n$ edges.\n\n Cheriyan and Thurimella \\cite{CT00} presented a $(1+1\/k)$-approximation algorithm for the minimum $k$-vertex-connected spanning subgraph problem for directed and undirected graphs. The algorithm of Cheriyan and Thurimella \\cite{CT00} has an approximation factor of $3\/2$ for the minimum $2$-vertex-connected directed subgraph problem. Let $G=(V,E)$ be a $2$-vertex strongly biconnected directed graph and let $E^{CT}$ be the output of the algorithm of Cheriyan and Thurimella \\cite{CT00}. The directed subgraph $(V,E^{CT})$ is not necessarily $2$-vertex strongly biconnected. But a $2$-vertex strongly biconnected subgraph can be obtained by performing the following third phase. For each edge $e\\in E\\setminus E^{CT}$, if the underlying graph of $G\\setminus \\left\\lbrace e\\right\\rbrace $ is $3$-vertex-connected, delete $e$ from $G$. We leave as open problem whether this algorithm has an approximation factor of $3\/2$ for the M2VSBSS problem.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\chapter{Acknowledgement}\\label{Acknowledgement}\n\n\nIt seems incredible but the Ph.D. really comes to the end! \n\nFor sure this has been the most intense period of my life both by the scientific point of view, with {\\em my} Di-Bosoni, the Higgs, tons of new things learned and discovered, and also by the personal point of view,\nwith thousands of miles across the World and several houses and cities where I spent my time. \n\nThe real risk was to get lost along the way but, luckily, I have been continuously surrounded, helped and supported by so so many nice people that it is really hard to thank everybody in one page.\n\nFor sure I couldn't had made it without the support of Prof. Giorgio Chiarelli (who now knows me by the far 2007); the work wouldn't had been possible without him but neither without the continuous exchange of ideas with Dr. Sandra Leone and within all the CDF Pisa Group.\n\nFermilab has been the other indispensable ingredient: the best place in the Chicago suburbs to do Physics and also all the rest. I met there a great research team (all the WHAM'ers), wonderful scientists and researchers from whom I learned so much.\n\nHowever there have not just been Physics in these years\\footnote{Although often friendship, study, work, spare time and Physics all overlaps in a indistinguishable way...}, my family has been always with me also from the other side of the ocean or farther away. No possibility to get anywhere without their help. Then there is Celine, ``la Piccolina'', that I met once, lost, and luckily I met again and now we are together!\n\nAnd finally there are all the friends from {\\em Sancasciani}, {\\em Cottonwood}, {\\em Siena}, {\\em Chicago}, {\\em Pisa}, the {\\em old} and the {\\em new Summer Students}, the {\\em ControTV} group and others I am surely missing. I spend with all of you a very good time and first or later I will meet again everybody scattered all around the World.\n\nThis travel lasted three years (well almost four), a pretty long period in the lifespan of somebody that is not even thirty but it flowed rapidly like a mountain torrent.\n\n\n\n\\chapter[First Evidence of Diboson in $\\ell\\nu+ HF$ ($7.5~fb^{-1}$)]{First Evidence of Diboson in Lepton plus Heavy Flavor Channel ($7.5~\\textrm{fb}^{-1}$)}\\label{App:7.5}\n\nThe first evidence of diboson production in the  $\\ell \\nu + HF$ final state was already obtained in the Summer of 2011 with a preliminary version of this analysis performed on a smaller dataset, $7.5$~fb$^{-1}$ of CDF data. \n\nThe differences with respect to the work described in the main parts of the thesis are the followings:\n\\begin{itemize}\n\\item a dataset of $7.5$~fb$^{-1}$ of CDF data.\n\\item The forward electrons category (PHX) was not used.\n\\item A previous version~\\cite{CHEP_svm} of the SVM multi-jet rejection algorithm was employed. The algorithm was optimized only for binary classification and it was not possible to use the shape of the output distribution. Therefore the \\mbox{${\\not}{E_T}$}  distribution was used to in the normalization of the $W + $ jets and the multi-jet backgrounds.\n\\item  No flavor separator ($KIT-NN$) information was used for single-tag events. Therefore only the di-jet invariant mass distribution was used as the final signal to background discriminator both for the single and double tagged signal regions.\n\\item The statistical analysis of the significance of the observation was performed in a different way. The likelihood ratio~\\cite{mclimit} of the signal and {\\em test} hypothesis, after the fit over the nuisance parameter, was used\\footnote{This method can not be easily applied to the significance estimate of two signals (for example $WW$ {\\em vs} $WZ\/ZZ$) therefore, in the main part of the thesis, we moved to the method described in Section~\\ref{sec:sigma_eval} for the statistical analysis}. \n\\item We also estimated $95$\\% Confidence Level (CL) limits, both in the case of diboson signal and no diboson signal. The first case can be used to constrain new physics models which produces an increase of the TGC couplings.\n\\end{itemize}\nA summary of the event selection, the background estimate and the statistical analysis is reported in the following.\n\n\n\n\\section{Event Selection and Background Estimate}\\label{sec:Data}\n\nWe select events consistent with the $\\ell\\nu+HF$ signature. \n\nThe charged lepton candidate online and offline identification is described in Chapter~\\ref{chap:sel} but we consider only the tight central lepton candidates (CEM, CMUP, CMX), the loose lepton candidates (BMU, CMU, CMP, CMIO, SCMIO, CMXNT) and the isolated track candidates (ISOTRK). Loose leptons and ISOTRK are classified together in the EMC category. The $W\\to \\ell \\nu$ selection is completed by a cut on the \\mbox{${\\not}{E_T}$}  variable, corrected for the presence of muons and jets. We require a \\mbox{\\mbox{${\\not}{E_T}$} $>20$~GeV} for CEM and EMC leptons while we relax the cut down to \\mbox{\\mbox{${\\not}{E_T}$} $>10$~GeV} for the tight muons categories. \n\nThe $HF$ jets selection is also identical to the one reported in Chapter~\\ref{chap:sel}: two central ($|\\eta_{Det}| < 2.0$) jets with E$_T^{cor} > 20$~GeV (energy corrected for detector effects) \non which we require the identification of one or two secondary vertices with the \\texttt{SecVtx} $b$-tagging algorithm. The $b$-tagging requirement divides the selected sample in a {\\em pretag} control region (no $b$-tag requirement) and two signal regions characterized by exactly one or two $b$-tagged jets.\n\nA relevant difference in the event selection comes from the multi-jet background rejection strategy. We used a previous version of the SVM algorithm~\\cite{CHEP_svm} based on the following six variables (described in Section~\\ref{sec:var_sel}):\n\\begin{itemize}\n\\item Lepton $p_T$, \\mbox{${\\not}{E_T}$} , $MetSig$, $\\Delta\\phi(lep,\\mbox{${\\not}{E_T^{raw}}$})$,  $E_T^{raw,jet2}$ and $E_T^{cor,jet2}$.\n\\end{itemize}\nThe discriminant was optimized to work in the central region of the detector and, as binary classifier, the performances were similar to the present algorithm described in Appendix~\\ref{chap:AppSvm}.\n\nThe background estimate was done with the same methodology described in Chapter~\\ref{chap:bkg}. The only difference is that the fits, used to estimate $W + $ jets and multi-jet normalizations in the different lepton categories and tag regions, were performed on the \\mbox{${\\not}{E_T}$}  distributions. Figure~\\ref{fig:qcd_fit_met} shows the result of the pretag maximum likelihood fit on the \\mbox{${\\not}{E_T}$} , pretag sample, distribution of the different lepton categories: CEM, CMUP, CMX, EMC.\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.495\\textwidth]{App7.5\/Figs\/PretafFit_CEM}\n\\includegraphics[width=0.495\\textwidth]{App7.5\/Figs\/PretafFit_CMUP}\n\\includegraphics[width=0.495\\textwidth]{App7.5\/Figs\/PretafFit_CMX}\n\\includegraphics[width=0.495\\textwidth]{App7.5\/Figs\/PretafFit_ISOTRK}\n\\caption[\\mbox{${\\not}{E_T}$}  Fit Used for $W +$ Jets Estimates in the Pretag Sample]{$W + $ jets  and non-$W$ fraction estimates, on the pretag sample, with a maximum likelihood fit on the \\mbox{${\\not}{E_T}$}  distributions. The non-$W$ background is shown in pink while the $W + $ jets component is in green. The dashed line is the sum of all the backgrounds and the points are the data. The figures show (left to right and top to bottom) the CEM, CMUP, CMX and EMC \n  charged lepton categories.} \\label{fig:qcd_fit_met}\n\\end{center}\n\\end{figure}\n\nThe total background estimate is reported in Tables~\\ref{tbl:1tag} and~\\ref{tbl:2tags}. The final statistical analysis of the selected events is performed on the $M_{Inv}(jet1, jet2)$ distribution of four channels: single and double tagged candidates for the central tight leptons (CEM+CMUP+CMX) and EMC leptons. Figure~\\ref{fig:mjj_1tag} shows the high-statistics single-tag  $M_{Inv}(jet1, jet2)$ distribution of central tight leptons and of the EMC leptons.\n\n\\begin{table}[htb]\n  \\begin{center}\n    \\begin{tabular}{lcccc}\n      \\toprule\n      \\multicolumn{5}{c}{\\bf Single-tag Event Selection}\\\\\n      Lepton ID & CEM                       &             CMUP             & CMX    & EMC    \\\\  \n      \\midrule    \n\n     Pretag Data       & 61596  & 29036 & 18878  &  27946 \\\\\\midrule\n      $Z+$jets            &27.9 $\\pm$ 3.5     & 43.0 $\\pm$ 5.5    & 27.3 $\\pm$ 4.2     & 65.0 $\\pm$ 8.4     \\\\  \n      $t\\bar{t}$        &201.3 $\\pm$ 19.6   &109.8 $\\pm$ 10.7   & 55.0 $\\pm$ 7.1     & 171.9 $\\pm$ 16.9    \\\\  \n      Single Top $s$    &52.9 $\\pm$ 4.8     &28.2 $\\pm$ 2.6     & 14.0 $\\pm$ 1.8     & 38.0 $\\pm$ 3.5       \\\\  \n      Single Top $t$    &71.4 $\\pm$ 8.4     &37.4 $\\pm$ 4.4     & 19.8 $\\pm$ 2.9     & 49.5 $\\pm$ 5.8       \\\\  \n      $WW$                &68.0 $\\pm$ 9.4     &33.3 $\\pm$ 4.6     & 20.3 $\\pm$ 3.3     & 38.4 $\\pm$ 5.3       \\\\  \n      $WZ$                &21.8 $\\pm$ 2.3     & 11.5 $\\pm$ 1.25   & 7.4 $\\pm$ 1.0      & 14.1 $\\pm$ 1.6     \\\\ \n      $ZZ$                &0.44 $\\pm$ 0.04    & 0.65 $\\pm$ 0.06   & 0.42 $\\pm$ 0.05    & 0.86 $\\pm$ 0.08    \\\\    \n      $W + b\\bar{b}$    &632.9 $\\pm$ 254.2  & 309.8$\\pm$ 124.1  & 192.3 $\\pm$ 77.1   & 308.9 $\\pm$ 124.3  \\\\\n      $W + c\\bar{c}$    &331.0 $\\pm$ 133.7  & 155.1 $\\pm$ 62.5  & 96.2 $\\pm$ 38.8    & 164.2 $\\pm$ 66.4   \\\\\n      $W + cj$            &259.9 $\\pm$ 105.0  & 127.8 $\\pm$ 51.5  & 75.3 $\\pm$ 30.4    & 106.4 $\\pm$ 43.0   \\\\\n      $W+LF$            &605.2 $\\pm$ 71.3   & 283.8 $\\pm$ 31.7   & 181.0 $\\pm$ 20.6  & 346.2 $\\pm$ 39.2   \\\\   \n      Non-$W$             &173.9 $\\pm$ 69.6   & 45.8 $\\pm$ 18.3  & 2.8 $\\pm$ 1.1       & 100.9 $\\pm$ 40.4   \\\\   \n      \\midrule                                                                                               \n       {\\bf Prediction}        &{\\bf 2446 $\\pm$ 503} & {\\bf 1186 $\\pm$ 242 } &{\\bf  692 $\\pm$ 149}  & {\\bf 1404 $\\pm$ 243} \\\\\n       {\\bf Observed}          &{\\bf 2332}               & {\\bf 1137}              & {\\bf 699}                & {\\bf 1318}               \\\\\n      \\midrule                                                                                            \n      {\\bf Dibosons}             &{\\bf 89.7 $\\pm$ 10.2}    & {\\bf 44.8 $\\pm$ 5.05}   & {\\bf  27.7 $\\pm$ 3.9}   &  {\\bf 52.5 $\\pm$ 5.9}     \\\\\n      \\bottomrule                                                   \n    \\end{tabular}\n\n    \\caption[Observed and Expected Events with One \\texttt{SecVtx} Tags]{Summary of  observed and expected events with one \\texttt{SecVtx} tag, in the $W+2$ jets sample, in \n      7.5 fb$^{-1}$ of data.}\\label{tbl:1tag}\n  \\end{center}\n\\end{table}\n\n\\begin{table}[htb]\n\\begin{center}\n    \\begin{tabular}{lcccc}\n    \\toprule\n    \\multicolumn{5}{c}{\\bf Double-tag Event Selection}\\\\\n    Lepton ID & CEM                       &             CMUP             & CMX    & EMC    \\\\  \n    \\midrule    \n      \n       Pretag Data       & 61596   &  29036  &  18878  &  27946 \\\\\\midrule\n       $Z+$jets            & 0.9 $\\pm$ 0.1   &2.0 $\\pm$ 0.3    & 1.2 $\\pm$ 0.2   & 3.1 $\\pm$ 0.4  \\\\    \n       $t\\bar{t}$        & 42.2 $\\pm$ 6.1  &22.2 $\\pm$ 3.2   & 11.1 $\\pm$ 1.9  & 34.4 $\\pm$ 5.0  \\\\\n       Single Top $s$    & 14.1 $\\pm$ 2.0  &7.6 $\\pm$ 1.1    & 3.7 $\\pm$ 0.6   & 10.2 $\\pm$ 1.4  \\\\ \n       Single Top $t$    & 4.2 $\\pm$ 0.7   &2.3 $\\pm$ 0.4    & 1.2 $\\pm$ 0.2   & 3.1 $\\pm$ 0.5   \\\\ \n       $WW$                & 0.6 $\\pm$ 0.1   & 0.26 $\\pm$ 0.07 & 0.16 $\\pm$ 0.04 & 0.33 $\\pm$ 0.08\\\\  \n       $WZ$                & 4.0 $\\pm$ 0.6   & 1.9 $\\pm$ 0.3   & 1.4 $\\pm$ 0.2   & 2.4 $\\pm$ 0.4  \\\\  \n       $ZZ$                & 0.06 $\\pm$ 0.01 & 0.12 $\\pm$ 0.02 & 0.09 $\\pm$ 0.01 & 0.16 $\\pm$ 0.02  \\\\\n       $W + b\\bar{b}$    & 81.9 $\\pm$ 33.2 &42.2 $\\pm$ 17.1  & 23.4 $\\pm$ 9.5  & 44.9 $\\pm$ 18.2\\\\\n       $W + c\\bar{c}$    & 4.7 $\\pm$ 1.9   &2.3 $\\pm$ 1.0    & 1.3 $\\pm$ 0.5   & 2.8 $\\pm$ 1.1   \\\\ \n       $W + cj$            & 3.7 $\\pm$ 1.5   &1.9 $\\pm$ 0.8    & 1.0 $\\pm$ 0.4   & 1.8 $\\pm$ 0.7  \\\\    \n       $W+LF$            & 3.2 $\\pm$ 0.7   &1.6 $\\pm$ 0.3    & 0.9 $\\pm$ 0.2   & 2.2 $\\pm$ 0.4  \\\\  \n       Non-$W$             & 7.9 $\\pm$ 3.2   &4.8 $\\pm$ 1.9    & 0.1 $\\pm$ 0.5   & 0.0 $\\pm$ 0.5   \\\\   \n       \\midrule                                                                               \n       {\\bf Prediction}        & {\\bf 167.3 $\\pm$ 38.0} &{\\bf 88.9 $\\pm$ 19.6}  & {\\bf 45.4 $\\pm$ 10.9} & {\\bf 105.3 $\\pm$ 21.5}\\\\\n       {\\bf Observed}          & {\\bf 147}             &{\\bf 74}               &{\\bf  39}              & {\\bf 106}             \\\\  \n       \\midrule\n       {\\bf Dibosons}             &{\\bf  4.6 $\\pm$ 0.6}   &{\\bf  2.1 $\\pm$ 0.3}   & {\\bf 1.5 $\\pm$ 0.2}   &{\\bf  2.7 $\\pm$ 0.4}   \\\\\n       \\bottomrule\n       \\end{tabular}  \n\n       \\caption[Observed and Expected Events with Two \\texttt{SecVtx} Tags]{Summary of observed and expected events with two \\texttt{SecVtx} tags, in the $W+2$ jets sample, in $7.5$~fb$^{-1}$ of data.}\\label{tbl:2tags}\n\\end{center}\n\\end{table}\n\n\n\\begin{figure} [!ht]\n\\centering\n\\includegraphics[width=0.495\\textwidth]{App7.5\/Figs\/1svt_cen}\n\\includegraphics[width=0.495\\textwidth]{App7.5\/Figs\/1svt_emc} \n\\caption[$M_{Inv}(jet1,jet2)$ Distribution for Single \\texttt{SecVtx} Tag Events]{$M_{Inv}(jet1,jet2)$ distribution for the single \\texttt{SecVtx} tagged events. Tight leptons (CEM+CMUP+CMX combined) on the left and EMC leptons on the right. The best fit values for the rate and shape of the backgrounds are used in the figures.}  \n\\label{fig:mjj_1tag}\n\\end{figure}\n\n\n\n\\section{Statistical Analysis}\n\nAt the time of this analysis, the process $WZ\/WW \\rightarrow \\ell \\nu $ + Heavy Flavors was not yet observed at hadron colliders. Therefore we started by evaluating a 95\\% exclusion CL on a potential signal and then we evaluated the cross section of the process and its significance. \n\nMost of the statistical analysis procedure has been described in Chapter~\\ref{chap:StatRes}. We build a likelihood function (Equation~\\ref{eq:likelihood_full}) with templates derived from the selected data and backgrounds, both shape and rate systematics are taken into account as described in Section~\\ref{sec:sys_desc}. The only relevant change concerns the $b$-tag SF for $c$-marched jets where we applied the same prescriptions of the $WH$ CDF search, it does not double the uncertainty for $c$-matched quarks. \n\nThe last relevant difference is the evaluation of the 95\\% CLs for the diboson signal. The CLs are evaluated by integrating the likelihood distribution over the unknown parameter $\\alpha$ (i.e. the diboson cross section) up to cover 95\\% of the total possible outcomes. In formulas:\n\\begin{equation}\n\\int^{\\bar{\\alpha}}_{0} \\mathscr{L}(\\alpha)\\mathrm{d}\\alpha.  \n\\end{equation}\nwhere $\\mathscr{L}(\\alpha)$ is derived from Equation~\\ref{eq:likelihood_full} after the integration over the nuisance parameters, $\\vec{\\alpha}$ is reduced to just one dimension because we perform CLs on only one signal and the limit on the integration, $\\bar{\\alpha}$, is given by the condition of 95\\% coverage.\n\nA first set of expected CLs are obtained assuming no SM diboson production and generating Pseudo Experiments (PEs) on the base of the expected background yields varied within the assigned systematics. \nCombining single--tagged and double--tagged results for all lepton categories, we find an expected limit of:\n\\begin{equation}\n\\left( 0.575^{+0.33}_{-0.31} \\right) \\times \\textrm{SM prediction.}\n\\end{equation}\n\nWe also calculated a second set of expected CLs assuming the predicted SM diboson yield in the PEs generation. In this case an excess in the observed CLs would indicate the presence of {\\em new physics}.\nCombining single--tagged and double--tagged results for all lepton categories, we find an expected limit of:\n\\begin{equation}\n\\left( 1.505 ^{+0.46}_{-0.61}\\right) \\times \\textrm{SM prediction.}\n\\end{equation}\n\nThe observed limit of $1.46$ times the SM prediction is thereby consistent with the existence of a signal and no presence of new physics.\n\nTable~\\ref{tab:limits} summarizes the expected (and observed) 95\\% production limits in units of the SM prediction. \n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{lccc}\n\\toprule\n Category     & Expected 95\\% CL & Expected 95\\% CL & Observed \\\\ \n&  (No Diboson) &  (With Diboson)  & 95\\% CL \\\\\\midrule\nSingle Tag  &&&\\\\\nCEM+CMUP+CMX          & $0.715 ^{+0.42}_{-0.38}$ & $1.625 ^{+0.54}_{-0.69}$ & 2.07 \\\\\nEMC                   & $1.215 ^{+0.57}_{-0.46}$ & $2.125 ^{+0.76}_{-0.6}$ & 1.70 \\\\ \\midrule\nDouble Tag &&&\\\\\nCEM+CMUP+CMX         & $4.015 ^{+1.94}_{-1.50}$ & $4.875 ^{+2.15}_{-1.7}$ & 3.03 \\\\ \nEMC                  & $6.075 ^{+2.92}_{-2.20}$ & $7.015 ^{+3.15}_{-2.42}$ & 8.59 \\\\ \\midrule\nAll combined                & $0.575 ^{+0.33}_{-0.31}$ & $1.505 ^{+0.46}_{-0.61}$ & 1.46 \\\\\n\\bottomrule                                                                                                       \n\\end{tabular}\n\\caption[Expected and Observed 95\\% CL for Diboson Production]{Expected and observed 95\\% exclusion confidence levels for each lepton category, single and double tagged events, in units of the SM cross section for diboson production. Expected CLs are produced also including the diboson signal generated PEs, thus probing the contribution of new physics processes.}\\label{tab:limits}\n\\end{center}\n\\end{table}\n\n\n\n\\subsection{Significance and Cross Section Measurement}\n\nTo compute the significance of the signal, we performed a hypothesis test comparing the data to the likelihood ratio of the {\\em null} and {\\em test} hypotheses. \n\nThe {\\em null} hypothesis, $H_0$, assumes all the predicted processes except diboson production. The {\\em test} hypothesis, $H_1$, assumes that the diboson production cross section and the branching ratio into $HF$ are the ones predicted by the SM. The likelihood ratio is defined as:\n\\begin{equation}\\label{eq:l_ratio}\n-2 \\ln Q = -2 \\ln \\frac{ \\mathscr{L}(data|H_1,\\hat{\\theta})}{ \\mathscr{L}(data|H_0,\\hat{\\hat{\\theta}})}\n\\end{equation}\nwhere $\\mathscr{L}$ is defined by Equation~\\ref{eq:likelihood_full}, $\\theta$ represents the nuisance parameters describing the uncertain values of the quantities studied for systematic error, $\\hat{\\theta}$ the best fit values of $\\theta$  under $H_1$ and $\\hat{\\hat{\\theta}}$ are the best fit values of the nuisance parameters under $H_0$.\n\nTo perform the hypothesis test we generated two sets of PEs, one assuming $H_0$ and a second one assuming $H_1$ and we evaluated Equation~\\ref{eq:l_ratio} for each pseudo-data outcome. The distributions of the  $- 2 \\ln Q$ values are shown in Figure~\\ref{fig:gaussians}. The integral of the $H_0$ $- 2 \\ln Q$ distribution below the $- 2 \\ln Q$ value of the real data gives the observed $p-$value of the signal.\n\nWe obtained an observed $p-$value of 0.00120, corresponding to a 3.03$\\sigma$ excess and producing the {\\em evidence} of the diboson signal.  The result is compatible with the expectation as the {\\em test} hypothesis $p-$value, obtained from the median of the $H_1$ distribution of $ - 2 \\ln Q$, is $0.00126$.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.95\\textwidth]{App7.5\/Figs\/combined_significance}\n\\caption[Significance Estimate of Diboson Signal with $ - 2 \\ln Q$]{Distributions  of $ - 2 \\ln Q$  for the {\\em test} hypothesis $H_1$, which assumes the estimated backgrounds plus SM diboson production and decay into $HF$ (blue histogram), and for the {\\em null} hypothesis, $H_0$, which assumes no diboson  (red histogram). The observed value of $ - 2 \\ln Q$  is indicated with a solid, vertical line and the $p$-value is the fraction of the $H_0$ curve integral to the left of the data.}\\label{fig:gaussians}\n\\end{figure}\n\nIn order to measure the diboson production cross section, a Bayesian marginalization technique is applied as described in Section~\\ref{sec:likelihood_meas}. The nuisance parameters are integrated, the maximum of the posterior distribution returns the cross section measurement while the smallest interval containing 68\\% of the integral gives the 1-$\\sigma$ confidence interval. The resulting cross section measurement is:\n\\begin{equation}\n \\sigma_{Diboson}^{Obs}=\\left(1.08^{+0.26}_{-0.40} \\right)\\times \\sigma_{Diboson}^{SM} =  \\left(19.9^{+4.8}_{-7.4}\\right)\\textrm{~pb,}\n\\end{equation}\nwhere the errors include statistical and systematic uncertainties and $\\sigma^{SM}_{Diboson}$ is the SM predicted cross section derived from Table~\\ref{tab:mc_dib}\n\n\\chapter[SVM Multi-Jet Rejection]{Support Vector Machines Multi-Jet Rejection}\\label{chap:AppSvm}\n\nAn innovative multivariate method, based on the Support Vector Machines algorithm (SVM), is used in this thesis to drastically reduce the multi-jet background. \n\nOne of the crucial points in the search for diboson production in the $\\ell\\nu + HF$ final state is the maximization of the signal acceptance while keeping the background under control. This is a challenge because, in hadronic collider environment, jets are produced with a rate several order of magnitude larger than $W\\to \\ell \\nu$ events, therefore, as a jet can fake the lepton identification with not negligible probability (especially for electrons identification algorithms), multi-jet events are introduced in the sample.\n \nThe multi-jet background, a mixture of detector and physics processes, is challenging to parametrize and, usually, approximate data-driven models are obtained by appropriate fake-enriched selections (see Section~\\ref{sec:qcd}). These models are often statistically limited and the use of a different selection can produce unexpected biases in the simulated variables. It is obviously not trivial the use of multivariate techniques to tackle such a problem.\n\nThe SVM algorithm, described in Section~\\ref{sec:SVM}, is considered to perform well in this case as it offers good non-linear separation and stable solutions also on low statistical training samples~\\cite{svmbook, BShop_machine_learning}. It was never used before and we had to develop original solutions to address the major challenges: evaluate the robustness of the SVM against biases in the training set and establish the best, minimal set of input variables providing optimal performances. Section~\\ref{sec:method} describes how we solved the first problem while Section~\\ref{sec:var_sel} describes the input variable selection criteria.\n\nThe results, reported in Section~\\ref{sec:svm_res}, in terms of signal efficiency and background rejection, are superior to any other cut based or multi-variate method previously applied at CDF.\n\n\n\n\\section{Support Vector Machines}\\label{sec:SVM}\n\nThe SVM is a supervised learning binary classifier whose basic concept is the identification of the {\\em best separating hyper-plane} between two classes of $n$-dimension vectors. \n\nIn the case of linear separation the algorithm produces, given a training set of the vectors of the two classes, an {\\em unique} solution where the plane is defined by the minimum amount of vectors, called {\\em support vectors}, at the boundary of the two classes. In the case of non-linear separation, the plane is found in an abstract space, defined by a transformation of the input vectors. However it is not necessary to know the exact transformation, but just its effect on the scalar product between the vectors, named {\\em Kernel}, thus allowing a feasible solution. Finally the cases of not perfect separability of the two samples are included by introducing a penalty parameter accounting for the contamination. \n\nThe main advantages of the SVM with respect to other machine learning algorithms are the unique convergence of the problem, a small number of free tunable parameters (usually related to the Kernel choice) and good performances for low statistics training sets because only a small number of training vectors (the support vectors) are important  for the final solution.\n\nIt is possible to find more details in~\\cite{svmbook, BShop_machine_learning}, but a short overview of the algorithm is also given in the following. For the actual, numerical, implementation of the SVM algorithm we relied on the \\texttt{LIBSVM} open source library~\\cite{libsvm}.\n\n\n\n\\subsection{The Linear Case}\nFigure~\\ref{fig:svm_linear} shows a basic example of the SVM linear classification separating two classes of bi-dimensional training vectors with a maximum margin hyperplane (a line for this simple case)\n\n\\begin{figure}\n  \\begin{center}  \n    \\includegraphics[width=0.8\\textwidth]{AppSVM\/Figs\/vgv4}\n    \\caption[Example of SVM Linear Separation]{An example of SVM: two linearly separable classes of data are represented with red and blue dots. The hyperplane (in this case a simple line) leading to a maximum margin separation is defined by the weight vector $w$ and the bias vector $b$.}\\label{fig:svm_linear}\n  \\end{center}\n\\end{figure}\n\nThe problem can be formalized in a general way as the minimization of $|\\vec{w}|^2$ (with $\\vec{w}$ = vector normal to the plane) with the constraint:\n\\begin{equation}\\label{eq:constraint}\n  y_i(\\vec{x_i}\\cdot\\vec{w} + b)-1\\geq 0\\qquad\n  \\left\\{\\begin{array}{ll}\n  y_i=+1; & i \\in \\textrm{signal} \\\\\n  y_i=-1; & i \\in \\textrm{background}\\\\\n  \\end{array} \\right.\n\\end{equation}\nThe problem has an unique solution obtained by the maximization of:\n\\begin{equation}\\label{eq:lagrange_1}\n  L=\\sum_i \\alpha_i - \\frac{1}{2}\\sum_{i,j}\\alpha_i\\alpha_j y_i y_j\\vec{x_i}\\cdot\\vec{x_j} ,\n\\end{equation}\nobtained with the application of the Lagrange multipliers to Equation~\\ref{eq:constraint}.\nThe solution identifies, for some $i$, $\\alpha_i>0$. The associate vectors are the \\textit{support vectors}, i.e. a subset of the training sample that define the best hyper-plane (see Figure~\\ref{fig:svm_linear}).\n\nTo solve the case of not completely separable classes of vectors, a {\\em penalty parameter}, $C$, is added into the target function to account for the contamination. So that we have a new minimization condition:\n\\begin{equation}\n  |\\vec{w}|^{2} + C \\sum_{i}{\\xi_{i}};\n\\end {equation}\nand a new constraint (derived again from Equation~\\ref{eq:constraint}):\n\\begin{equation}\\label{eq:lagrange_2}\n  y_i(\\vec{x_i}\\cdot\\vec{w} + b) \\geq 1-\\xi_{i}\\quad\\textrm{with}\\quad \\xi \\geq 0\\textrm{.}\n\\end{equation}\n\nThe parameter $C$ defines, before the training, the SVM implementation therefore it represents as one of the {\\em hyper parameter} of the SVM.\n\nA newly seen vector, $\\vec{X}$, is classified according the position with respect to the plane defined by the support vectors $\\vec{x_i}$ and the parameters $\\alpha_i$:\n\\begin{equation}\\label{eq:distance}\n  D(\\vec{X})= \\sum_i\\alpha_i y_i \\vec{x_i}\\cdot \\vec{X_i} - b,\n\\end{equation}\nwhere $b$ is a bias term of the solution. The sign of $D(\\vec{X})$ defines the classification but the value itself can be seen as the distance of a test vector $\\vec{X}$ from the classification plane. However, as we will see in the next paragraph, a non-linear classification is possible only thanks to a not-explicit transformation in a different vector space, where $D$ looses it immediate geometrical meaning. \n\nCommonly SVMs are used as binary classifiers but, here, we add a large degree of flexibility by exploiting the full information of the variable $D$. We see it as a dimensionality reducer that summarizes all the information obtained during the training and classification process. \n\n\n\n\\subsection{Kernel Methods}\n\nNon-linearly separable classes of vectors can be classified by transforming them into linearly separable\nclasses. An opportune function, $\\Phi(\\vec{x})$, can be used to map the elements into another\nspace, usually with higher dimension where the separation is possible. \n\nHowever the  identification of $\\Phi(\\vec{x})$ is {\\em non trivial} and the, so called, {\\em Kernel trick} is often used: a Kernel function, $\\mathbf{K}(x_i,x_j)$, generalizes the scalar product appearing in Equation~\\ref{eq:lagrange_1} (or Equation~\\ref{eq:lagrange_2}) without the need of explicitly know $\\Phi(\\vec{x})$. Or in equations, we compose the mapping $\\Phi(\\vec{x})$ with the inner product:\n\\begin{equation}\n  \\mathbf{K}(x_i,x_j)= \\Phi(x_i)\\cdot\\Phi(x_j)\\quad\\mathrm{with}\\quad \\Phi : \\Re^n\\mapsto \\mathcal{H}.\n\\end{equation}\n\nThe function $\\mathbf{K}$ should satisfy to a general set of roles to be a Kernel, but we want only to briefly describe the {\\em Gaussian} Kernel we used in this work. It is expressed as:\n\\begin{equation}\\label{eq:gauss_k}\nK(x_{i},x_{j}) = e^{-\\gamma |\\vec{x}_{i}-\\vec{x}_{j}|^2}\n\\end{equation}\nThe corresponding $\\Phi(x)$ maps to an infinite dimension space and it is not known. The Kernel is defined only by one hyper-parameter, $\\gamma$, that should be defined before the training.\n\n\n\n\n\n\\section{SVM Training in a Partially Biased Sample}\\label{sec:method}\n\nThe assumption behind the supervised learning is that the labelled samples, used for the classifier training, are drawn from the same probability distribution of the unclassified events. \nHowever in our case of study, where only an approximate and statistically limited model of the background processes is available (see Sections~\\ref{sec:cdf_set} and~\\ref{sec:qcd} for the multi-jet background description), we do not expected the previous assumption to hold for every region of the phase space. To cope with this problem, we developed an original methodology to evaluate the SVM training performances.\n\nSection~\\ref{sec:SVM} shows that, for each choice of hyper-parameters and training vectors, only one optimal SVM solution exists and, for it, we need to evaluate the performances. \n\nAs a performances estimator we use the {\\em confusion matrix} of the classifier: the element $(i,j)$ of the matrix is the fraction of the class $i$ classified as member of class $j$. Figure~\\ref{fig:confusion} shows a representation of it in the two classes case, where one class is considered the {\\em background} and the other the {\\em signal}. We obtain a reliable estimate of the classifier quality by filling the confusion matrix in two independent ways and combining all the available information.\n\n\\begin{figure}[h!]\n  \\begin{center}   \n    \\begin{tabular}{|c|c|}\n      \\toprule\n      {\\em Sgn} classified as {\\em Sgn} & {\\em Bkg} classified as {\\em Sgn}\\\\\n      \\midrule\n      {\\em Sgn} classified as {\\em Bkg} & {\\em Bkg} classified as {\\em Bkg}\\\\\n      \\bottomrule\n    \\end{tabular}   \n    \\caption[Definition of Confusion Matrix]{Definition of confusion matrix for a two classes ($Sgn$ and $Bkg$) classification problem. This reproduces the case of an algorithm used to discriminate signal {\\em vs} background: the elements of the matrix are the signal and background classification performances and the cross contamination.}\\label{fig:confusion}\n  \\end{center}\n\\end{figure}\n\nThe first performance evaluation method is the {\\em $k$-fold cross-validation}: the training set is divided into $k$ sub-samples of which one is used as a validation set and the remaining $k - 1$ are used in the training; the confusion matrix is then evaluated applying the trained discriminant to the validation set. The cross-validation process is repeated $k$ times, the {\\em folds}, and the final performance is given by the average on all the folds. This method is solid against over fitting but it has no protection against biases on the complete training sample.\n\nThe second method, a {\\em key feature} of this work, is based on a bi-component fit that uses signal and background templates and it is performed on a significant distribution of the unclassified events, the {\\em data}. While the signal and background templates are derived in the same way of the training samples, the unclassified data events are, by definition composed by an unknown mixture of the {\\em true} signal and background events. The fit is performed by maximizing a binned likelihood function, $\\lambda$, where the Poisson statistic of the templates is used and the fractions of the signal and of background templates, from which we can derive the elements of the confusion matrix, are free parameters. The fitting function is implemented in the ROOT~\\cite{rootr} analysis package and it is derived from~\\cite{Barlow_1993}. \nFigure~\\ref{fig:toysamplefit} shows an example of the fit used on the toy model described in the next Section.\n\nIf the variable considered in the fit is not well reproduced in the simulation then we expect that the fitted fractions will differ greatly from the results obtained with the $k$-fold cross-validation. At the same time we can evaluate quantitatively the agreement between the data shape and the fitted templates because the quantity:\n\\begin{equation}\n  \\chi^2 = -2\\ln(\\lambda),\n\\end{equation}\nfollows a $\\chi^2$ probability distribution (under general assumptions).\n\nThe last critical point is the identification of a {\\em sensitive} variable to be used in the fit. In a previous work~\\cite{CHEP_svm} we exploited the \\mbox{${\\not}{E_T}$}  distribution as it is sensitive to the multi-jet contamination. A much more general approach, by the machine learning point of view, is the use of the SVM distance value, $D$, defined in Equation~\\ref{eq:distance}. If the SVM training performances are optimal, also the variable $d$ offers an optimal discrimination, furthermore the cross check on the $\\chi^2$ of the template fit ensure a good shape agreement between the data and signal and background templates. We verified the validity of the fit procedure with a toy example reported in the following.\n\n\\begin{figure}[!h]\n  \\begin{center}\n  \\includegraphics[width=0.9\\textwidth]{AppSVM\/Figs\/toysample}\n  \\caption[Bi-component Template Fit on Toy Data]{A bi-component fit is performed on the SVM distance, $D$ (Equation~\\ref{eq:distance}), of a signal (blue) and background (red) templates toy data.}\\label{fig:toysamplefit}    \n  \\end{center}\n\\end{figure}\n  \n\n\\subsection{A Toy Example}\n\nWe built a toy example in order to verify the robustness of the proposed method for an SVM performances evaluation when partially biased samples are available.\n\nThe toy is composed by three data-sets generated with known probability distributions:\n\\begin{description}\n\\item[signal model:] $10^5$ vectors generated from a $2-$Dim Gaussian distribution with the following mean, $\\vec{\\mu}_{Sgn}$ and standard deviation, $\\tilde{\\sigma}_{Sgn}$:\n  \\begin{equation}\\label{eq:sgn_gaus}\n    \\vec{\\mu}_{Sgn} = \\left[\n      \\begin{array}{c}\n        - 3  \\\\\n        0  \\end{array} \\right]\n    ,\\qquad \n     \\tilde{\\sigma}_{Sgn} = \\left[\n       \\begin{array}{cc}\n         8 & 0 \\\\\n       0 & 8  \\end{array} \\right].\n  \\end{equation}\n  \n \\item[Background model:] $10^5$ vectors generated from a $2-$Dim Gaussian distribution with the following mean, $\\vec{\\mu}_{Bkg}$ and standard deviation, $\\tilde{\\sigma}_{Bkg}$:\n   \n   \\begin{equation}\\label{eq:bkg_gaus}\n     \\vec{\\mu}_{Bkg} = \\left[ \n     \\begin{array}{c}\n       3  \\\\\n       0  \\end{array} \\right]\n ,\\qquad \n     \\tilde{\\sigma}_{Bkg} = \\left[\n     \\begin{array}{cc}\n       8 & 0 \\\\\n       0 & 8  \\end{array} \\right].\n   \\end{equation}\n \\item[Data:] a mixture of $5 \\cdot 10^4$ vectors generated from the same distribution of the signal model (Equation~\\ref{eq:sgn_gaus}) and $5\\cdot 10^4$ vectors generated from a {\\em true} background distribution similar to the background model (Equation~\\ref{eq:bkg_gaus}) but with $\\tilde{\\sigma}_{Bkg}$ increased by 20\\% in one direction to simulate a mismatch between the real background and the model.\n \\end{description}\n %\n We tested several combinations of the hyper-parameters $C$ and $\\gamma$ (over a grid) using the signal and background model in the training. For obtained SVM we evaluated the $k$-fold cross validation and we performed the template fit on the SVM distance, $D$, evaluated on the data sample. The result is reported in Figure~\\ref{fig:ToyResult} with the real performances reported on the $x$ axis of the diagram (we know the true label of the data vectors). The evaluation of the performances obtained with the fit is on the diagonal of the plane, therefore it gives a much more realistic estimate of the true performances of the classifier.\n\n \\begin{figure}[!h]\n   \\begin{center}\n   \\includegraphics[width=0.75\\textwidth,height=0.5\\textwidth]{AppSVM\/Figs\/ToyResult}\n   \\caption[Toy Model SVM Performances Estimate with Fit and Cross-Validation]{SVM performances estimate with a $k$-fold cross-validation (green circles) and with a bi-component signal and background template fit on the SVM distance (red triangle) of toy data of known composition. The {\\em true} performances of the SVM classifier are reported on the $x$ axis. The fit evaluation appears on the diagonal of the plane, signaling a more realistic estimate of the true performances of the classifier. }\\label{fig:ToyResult}     \n   \\end{center}\n \\end{figure}\n\n\n\n\\section{Performances on the CDF Dataset}\\label{sec:cdf_set}\n\nThe final goal of the SVM discriminant we discussed is the realization of a tool able to reject the multi-jet background in a wide range of searches performed in the lepton plus jets channel in a hadron collider environment. \n\nWe performed the SVM training on $W\\to e\\nu + $jets candidate, as the electron identification is more tamed by the multi-jet background. Furthermore  we performed the training process two times, one in the central ($|\\eta|<1.1$) and one in the forward ($1.2<|\\eta|<2.0$) region of the detector, as both the electron identification algorithm (see Section~\\ref{sec:lep_id}) and the kinematic of the events are different. \n\n\n\\subsection{Training Sample Description}\\label{sec:training}\n\nWe defined both a central and a forward training set using $7000$ $W\\to e\\nu +$jets signal events and $3500$ multi-jet background events:\n\\begin{description}\n  \\item[Signal:] $W + 2,3$ partons ALPGEN~\\cite{alpgen} MC, where the $W$ is forced to decay into electron and neutrino. We have about $10^5$ generated events and we keep approximately  $9\\times 10^4$ events as a control sample (i.e. not used for training). The CEM and PHX electron identification algorithms are used for the central and forward sample selection. \n\\item[Background:] we obtain a suitable background sample with a data-driven approach. The anti-electrons selection described in Section~\\ref{sec:qcd} is used for the central background training while for the forward training we had study a improved multi-jet model. \n\nIn particular, we noticed that the anti-PHX sample produced large over estimates in the detector region of $1.2<|\\eta|<1.4$ where two different calorimeter sub-systems are connected. These events are clear fake leptons and they are usually rejected by very loose kinematic requirements, nevertheless the quantity of them produced a too large bias in the SVM training efficiency estimate. Non-Isolated PHX electrons (with $IsoRel>0.1$) were found to give a reliable training set after a correction to the lepton $E_T$ equal to the amount of energy in the outer isolation cone ($E_T^{0.1<\\Delta R<0.4}$).\n\\item[Data:] The data sample used for the bi-component fit validation corresponds to the data periods (see Table~\\ref{tab:datasets}) from p18 to p24. They represent only a small fraction of the total CDF data with intermediate luminosity profile and run conditions.\n\\end{description}\nA close to final lepton plus jet selection with loose kinematic cuts is applied to all the samples. We require two or more jets reconstructed (see Section~\\ref{sec:jetCone}) in the central region of the detector ($\\eta<2.0$), with energy corrected at L5 and $E_T> 18$~GeV, also a minimal amount of \\mbox{${\\not}{E_T}$} $>15$~GeV is used as signature of the escaping neutrino. We also removed events with not understood behaviour like electron $E_T>300$~GeV, $M_T^W>200$~~GeV$\/c^{2}$ and, only for the forward sample, \\mbox{${\\not}{E_T^{raw}}$}$<20$~GeV (this last cut was needed to avoid trigger turn-on effects).\n\n\n\n\\subsection{Variable Selection}\\label{sec:var_sel}\n\nA multivariate algorithm relies on a given set of input variables. The {\\em feature selection} problem is fundamental in machine-learning and, if possible, even more in the present case where the background sample does not guaranteed a perfectly model of all the variables. \n\nWe started from a large set of twenty-four variables chosen according to two basic criteria: no correlation with respect to the lepton identification variables and exploit of the kinematic difference between real $W+$ jets events and multi-jet fakes. These requirements allowed the development of a flexible multi-jet rejection algorithm, applied with very good performances also to muon and isolated track lepton selections.\n\nTable~\\ref{tab:input_vars} shows all the input variables that we used in the optimization process. Many of them were introduced in Chapter~\\ref{chap:objects} but the following are new:\n\\begin{itemize}\n\\item \\mbox{${\\not}{p_T}$} is the missing momentum defined as the momentum imbalance on the transverse plane. It is computed adding all the reconstructed charged tracks transverse momenta, $\\vec{p_i}$:\n  \\begin{equation}\n    \\vec{{\\not}{p_T}}\\equiv - \\sum_i{\\vec{p^i}_T}\\quad\\textrm{with }|\\vec{p^i}_T|>0.5\\textrm{GeV}\/c\\textrm{;}\n  \\end{equation}\n\\item $M_T^W$ is the {\\em transverse mass} of the reconstructed $W$ boson:\n  \\begin{equation}\n    M_T^W=\\sqrt{2(E_T^{lep}\\cancel{E}_T - E_x^{lep}\\cancel{E}_x - E_y^{lep}\\cancel{E}_y)}\\mathrm{.}\n  \\end{equation}\n\\item $MetSig$ is the \\mbox{${\\not}{E_T}$}  {\\em significance}, a variable that relates the reconstructed \\mbox{${\\not}{E_T}$}  with the detector activity (jets and unclustered energy):\n  \\begin{equation}\n    MetSig =\\frac{\\cancel{E}_T}{\\sqrt{ \\Delta E^{jets}+ \\Delta E^{uncl}}}\\mathrm{,}\n  \\end{equation}\n  where:\n  \\begin{eqnarray}\n    \\Delta E^{jets} =\\sum_{j}^{jets}(cor_{j}^2\\cos^2\\Big(\\Delta\\phi\\big(\\vec{p_j},\\cancel{\\vec{E}}_T\\big)\\Big)E_{T}^{raw, j}\\mathrm{,}\\\\\n    \\Delta E^{uncl} = \\cos^2\\Big(\\Delta\\phi\\big(\\vec{E}_T^{uncl}\\cancel{\\vec{E}}_T\\big)\\Big)E_T^{uncl}\\mathrm{,}\n  \\end{eqnarray}\n  $uncl$ refers to the calorimeter energy not clustered into electrons or jets and $cor_j$ is the total correction applied to each jet.\n\\item $\\nu^{Min},\\quad \\nu^{Max}$ are the two possible reconstruction of the neutrino momenta. As the $p_z^{\\nu}$ component is not directly measurable we infer it from the $W$ boson mass and the lepton momentum. The constraints lead to a quadratic equation which may have two real solutions, one real solution, or two complex solutions\\footnote{The real part is chosen in this case}. The reconstructed $\\nu^{Min}$, $\\nu^{Max}$ derive from the distinction of $p_z^{\\nu,Max}$ and $p_z^{\\nu,Min}$.\n\\end{itemize}\n\n\\begin{table}\n  \\begin{tabular}{cl| cl|  cl| cl }\n    \\toprule\n    \\multicolumn{8}{c}{Possible Input Variables }\\\\\n    \\midrule\n    1 & $p_T^{lep}$           &  7 & $E_T^{raw, jet1} $   & 13 &  $\\Delta\\phi( {\\not}{p_T},$ $lep)$ &   19 & $\\Delta R( lep,$ $jet 2)$ \\\\\n    2 & \\mbox{${\\not}{E_T}$}                  &  8 & $E_T^{raw, jet2} $   & 14 &  $\\Delta\\phi( {\\not}{p_T},$ \\mbox{${\\not}{E_T}$} $)$ & 20 & $\\Delta R( \\nu^{min},$ $jet 1)$ \\\\ \n    3 & \\mbox{${\\not}{E_T^{raw}}$}                &  9 & $E_T^{cor,jet1} $       & 15 & $\\Delta\\phi( {\\not}{p_T},$ \\mbox{${\\not}{E_T^{raw}}$}$)$ &  21 & $\\Delta R( \\nu^{min},$ $jet 2)$ \\\\ \n    4 & \\mbox{${\\not}{p_T}$} & 10 & $E_T^{cor,jet2} $       &  16 & $\\Delta\\phi( lep,$ \\mbox{${\\not}{E_T}$} $)$ &    22 & $\\Delta R( \\nu^{min},$ $lep)$ \\\\ \n    5 & $M_T^W$              & 11 & $\\Delta\\phi( jet 1,$ \\mbox{${\\not}{E_T}$} $)$ & 17 & $\\Delta\\phi( lep,$ \\mbox{${\\not}{E_T^{raw}}$}$)$ &  23  & $\\Delta R( \\nu^{max},$ $jet 1)$ \\\\ \n    6 & $MetSig$             & 12 & $\\Delta\\phi( jet 2,$ \\mbox{${\\not}{E_T}$} $)$ &  18 & $\\Delta R( lep,$ $jet 1)$ &  24 & $\\Delta R( \\nu^{max},$ $jet 1)$ \\\\ \n\\bottomrule\n  \\end{tabular}\n  \\caption[Possible SVM Input Variables]{All the possible input variables used for the SVM training and optimization. See Section~\\ref{sec:var_sel} for a detailed description.}\\label{tab:input_vars}  \n\\end{table}\n\nUnluckily the extensive research over all the possible combinations of variables across all the $C, \\gamma$ phase space of a given SVM training, is computationally unfeasible. To scan the most relevant sectors of the phase space we applied factorized and incremental optimization:\n\\begin{itemize}\n\\item for all the configuration of {\\em three} variables and the given training set, we evaluate a grid of  $C, \\gamma$ values in the intervals\\footnote{The use of a logarithmic scale allows to scan the parameters across different orders of magnitude.}:\n  \\begin{equation}\n   \\log_2 C\\in [-3,8] \\quad\\textrm{and}\\quad \\log_2\\gamma\\in[-4,5].\n  \\end{equation}\nWe select only the best training configuration according to the confusion matrix evaluation.\n\\item For each {\\em best} SVM of a given variable configuration we perform a bi-component fit on the SVM distance $D$. We evaluate the  $\\chi^2$ of the fit, reduced by the Number of Degrees of Freedom ($NDoF$), and we compare the fitted background contamination, $f_{Bkg}^{Fit}$, against the one obtained from the $n$-fold cross-validation, $f_{Bkg}^{n-fold}$. The SVM under exam is rejected if:\n  \\begin{equation}\n    \\frac{\\chi^2}{NDoF}>3\\quad  or \\quad\\frac{f_{Bkg}^{Fit}}{f_{Bkg}^{n-fold}}>2\n  \\end{equation}\n\n\\item The remaining SVMs are displayed on a {\\em signal-efficiency vs background-contamination} scatter plot like the one in Figure~\\ref{fig:discriminants}. The 5 best variable combinations are \nselected for further processing.\n\\item We add other 2 or 3 variables to the best variables combinations obtained in the previous step and we iterate the chain.\n\\end{itemize}\nAfter a couple of iterations the best variable combination and $C,\\gamma$ hyper-parameters choice remains stable within $1\\div 2\\%$.\n\n\\begin{figure}[!h]\n  \\begin{center}    \n    \\includegraphics[width=0.9\\textwidth]{AppSVM\/Figs\/discriminants_bests_trainCEM3}\n  \\caption[Signal Efficiency {\\em vs} Background Contamination for Multiple SVM]{Different SVM configurations (i.e. with different input variables), obtained for the central region training, are displayed on a {\\em signal-efficiency vs background-contamination} scatter plot. The signal efficiency is directly estimated from simulation while the background contamination is obtained from the bi-component template fit of the SVM distance, $D$, described in Section~\\ref{sec:method}. The blue starts represent the performances obtained from a three (out of twenty-four) input variables training. The best five configurations w.r.t. the Euclidean distance from the optimal point ($\\epsilon_{Sig}=1$, $f_{Bkg}=0$) are circled in yellow. The Euclidean distance of the fifth best SVM is represented by a dotted line. Three iterations, with {\\em three, six and nine} input variables are represented by dotted lines, after that no more appreciable improvement occurs.}\\label{fig:discriminants}\n  \\end{center}\n\\end{figure}\n\n\n\\section{Final SVM Results}\\label{sec:svm_res}\nThe SVM configurations obtained by the process described in this appendix are finally used in the multi-jet rejection phase of the analysis (Section~\\ref{sec:svm_sel}) and in the background normalization estimates\\footnote{By construction a fit on the SVM distance, $D$, offers a reliable estimate of the multi-jet background normalization.} (see Sections~\\ref{sec:wjets_pretag} and~\\ref{sec:qcd}).\n\nThe optimal hyper-parameter configurations that we obtain for the central (superscript $c$) and forward (superscript $f$) SVMs are:\n\\begin{equation}\nC^c = 7,\\quad  \\gamma^c = \u2212- 1;\n\\end{equation}\n\\begin{equation}\nC^f = 8,\\quad  \\gamma^f = \u2212- 1.\n\\end{equation}\nTable~\\ref{tab:svm_fin_vars} reports the final eight input variables used for the central SVM and the six ones used for the forward SVM.\n\\begin{table}\n  \\begin{center}   \n  \\begin{tabular}{lccc}\n    \\toprule\n    \\multicolumn{4}{c}{ Final SVM Input Variables}\\\\\n    \\midrule\n    Central SVM: &  $M^W_T$ & \\mbox{${\\not}{E_T^{raw}}$} &  ${\\not}{p_T}$\\\\ \n                 & $MetSig$ & $\\Delta \\phi({\\not}{p_T},$ \\mbox{${\\not}{E_T}$} ) & $\\Delta \\phi(lep,$ \\mbox{${\\not}{E_T}$} $)$ \\\\\n                 & $\\Delta R(\\nu^{Min},$ $lep)$  & $\\Delta \\phi(Jet1,$ \\mbox{${\\not}{E_T}$} )  & \\\\\n    \\midrule\n    Forward SVM: & $M^W_T$ & \\mbox{${\\not}{E_T^{raw}}$} & ${\\not}{p_T}$ \\\\\n                 &  $MetSig$ &  $\\Delta \\phi({\\not}{p_T},$ \\mbox{${\\not}{E_T}$} ) & $\\Delta \\phi({\\not}{p_T},$ \\mbox{${\\not}{E_T^{raw}}$})  \\\\\n    \\bottomrule\n  \\end{tabular}\n  \\caption[Final SVM Input Variables]{Input variables used for the configuration of the {\\em central} and {\\em forward} SVM multi-jet discriminants.}\\label{tab:svm_fin_vars}\n  \\end{center}\n\\end{table}\nFigure~\\ref{fig:final_svm} shows the complete shape of the two discriminants for the multi-jet background models and the $W+2$ partons signal. A quantitative measurement of the performances can be seen in Table~\\ref{tab:svm_eff}.\n\n\\begin{figure}[!h]\n  \\begin{center}    \n    \\includegraphics[width=0.495\\textwidth]{AppSVM\/Figs\/central_svm}\n    \\includegraphics[width=0.495\\textwidth]{AppSVM\/Figs\/forward_svm}\n  \\caption[Final SVM Discriminants for Central and Forward Detector Regions]{Distribution of the SVM distance, $D$, described in Equation~\\ref{eq:distance} for the central (left) and the forward (right) SVM discriminants obtained from the optimization process. Multi-jet background models are shown in red, $W+2$ partons MC signal is shown in blue.}\\label{fig:final_svm}\n  \\end{center}\n\\end{figure}\n\nWe can conclude that we successfully built a multi-jet rejection tool based of the SVM algorithm, nevertheless the challenges of a multi-variate approach to this problem. The CDF II dataset was a perfect test-bench for this problem with very good performances, however the procedure can be exported also to any of the LHC experiments.\n\n\n\\chapter{Rate Systematics Summary}\\label{app:rate_sys}\n\nTables from~\\ref{tab:1tag_cem} to~\\ref{tab:2tag_emc}: rate systematics variations for each channel entering in the statistical analysis (see Chapter~\\ref{chap:StatRes}), same name systematics are fully correlated.\n\n\\renewcommand{\\arraystretch}{0.97}\n\\renewcommand{\\tabcolsep}{1.5pt}\n\n\\begin{table}[!h]\n  \\begin{scriptsize}\n    \\begin{tabular}{ c  c c c c c c c c c c c } \\toprule\n      & nonW & $t\\bar{t}$ & s-top $s$ & s-top $t$ & Zjets & W+LF & Wbb & Wcc & Wcj & WW & WZ\/ZZ \\\\\\midrule\n$\\texttt{ XS\\_ttbar}$ & $0\/0$  & $10\/-10$  & $10\/-10$  & $10\/-10$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFBTAGSF}$ & $0\/0$  & $3\/-3$  & $3\/-3$  & $5\/-5$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $11\/-11$  & $5\/-5$  \\\\\n$\\texttt{ CDFLUMI}$ & $0\/0$  & $6\/-6$  & $6\/-6$  & $6\/-6$  & $6\/-6$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $6\/-6$  & $6\/-6$  \\\\\n$\\texttt{ LEPACC\\_CEM}$ & $0\/0$  & $2\/-2$  & $2\/-2$  & $2\/-2$  & $2\/-2$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $2\/-2$  & $2\/-2$  \\\\\n$\\texttt{ QCD\\_CEM}$ & $40\/-40$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFVHF}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $30\/-30$  & $30\/-30$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFWC}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $30\/-30$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFMISTAG}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $11\/-11$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ ZJETS}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $45\/-45$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ ISRFSRPDF}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $4\/-4$  & $4\/-4$  \\\\\n$\\texttt{ CDFJES}$ & $0\/0$  & $-12\/12$  & $-3\/2$  & $-2\/1$  & $1\/6$  & $1\/-1$  & $-6\/6$  & $-3\/6$  & $-5\/6$  & $-3\/-4$  & $2\/-2$  \\\\\n$\\texttt{ CDFQ2}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $-2\/-1$  & $4\/11$  & $5\/8$  & $3\/9$  & $0\/0$  & $0\/0$  \\\\\n\\hline\n$\\texttt{TOT}$ & $40\/40$  & $17\/17$  & $12\/12$  & $13\/13$  & $45\/46$  & $11\/11$  & $31\/33$  & $31\/32$  & $30\/32$  & $14\/14$  & $9\/11$  \\\\\n\\hline\n    \\end{tabular}\n  \\end{scriptsize}\n  \\caption[Rate Uncertainties: Single-Tag, CEM Channel]{Per-sample rate uncertainty (\\% up\/down). Single-tag, CEM  channel.}\\label{tab:1tag_cem}\n  \\end{table}\n\n  \\begin{table}[!h]\n    \\begin{scriptsize}\n  \\begin{tabular}{ c c c c c c c c c c c c} \\toprule\n    & nonW & $t\\bar{t}$ & s-top $s$ & s-top $t$ & Zjets & W+LF & Wbb & Wcc & Wcj & WW & WZ\/ZZ \\\\\\midrule\n$\\texttt{ XS\\_ttbar}$ & $0\/0$  & $10\/-10$  & $10\/-10$  & $10\/-10$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFBTAGSF}$ & $0\/0$  & $3\/-3$  & $3\/-3$  & $5\/-5$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $11\/-11$  & $5\/-5$  \\\\\n$\\texttt{ CDFLUMI}$ & $0\/0$  & $6\/-6$  & $6\/-6$  & $6\/-6$  & $6\/-6$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $6\/-6$  & $6\/-6$  \\\\\n$\\texttt{ LEPACC\\_PHX}$ & $0\/0$  & $2\/-2$  & $2\/-2$  & $2\/-2$  & $2\/-2$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $2\/-2$  & $2\/-2$  \\\\\n$\\texttt{ QCD\\_PHX}$ & $40\/-40$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFVHF}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $30\/-30$  & $30\/-30$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFWC}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $30\/-30$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFMISTAG}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $11\/-11$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ ZJETS}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $45\/-45$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ ISRFSRPDF}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $4\/-4$  & $4\/-4$  \\\\\n$\\texttt{ CDFJES}$ & $0\/0$  & $-8\/12$  & $-1\/2$  & $-3\/3$  & $2\/-6$  & $-1\/-1$  & $-4\/4$  & $-5\/4$  & $-8\/6$  & $-7\/-5$  & $-10\/-3$  \\\\\n$\\texttt{ CDFQ2}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $-1\/-2$  & $0\/15$  & $0\/18$  & $2\/10$  & $0\/0$  & $0\/0$  \\\\\n\\hline\n$\\texttt{TOT}$ & $40\/40$  & $14\/17$  & $12\/12$  & $13\/13$  & $45\/46$  & $11\/11$  & $30\/34$  & $30\/35$  & $31\/32$  & $15\/14$  & $14\/10$  \\\\\n\\hline\n\\end{tabular}\n    \\end{scriptsize}\n  \\caption[Rate Uncertainties: Single-Tag, PHX Channel]{Per-sample rate uncertainty (\\% up\/down). Single-tag, PHX channel.}\\label{tab:1tag_phx}\n  \\end{table}\n\n  \\begin{table}\n    \\begin{scriptsize}\n\n\\begin{tabular}{ c c c c c c c c c c c c } \\toprule\n    & nonW & $t\\bar{t}$ & s-top $s$ & s-top $t$ & Zjets & W+LF & Wbb & Wcc & Wcj & WW & WZ\/ZZ \\\\\\midrule\n$\\texttt{ XS\\_ttbar}$ & $0\/0$  & $10\/-10$  & $10\/-10$  & $10\/-10$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFBTAGSF}$ & $0\/0$  & $3\/-3$  & $3\/-3$  & $5\/-5$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $11\/-11$  & $5\/-5$  \\\\\n$\\texttt{ CDFLUMI}$ & $0\/0$  & $6\/-6$  & $6\/-6$  & $6\/-6$  & $6\/-6$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $6\/-6$  & $6\/-6$  \\\\\n$\\texttt{ LEPACC\\_MU}$ & $0\/0$  & $1\/-1$  & $1\/-1$  & $1\/-1$  & $1\/-1$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $1\/-1$  & $1\/-1$  \\\\\n$\\texttt{ QCD\\_MU}$ & $40\/-40$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFVHF}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $30\/-30$  & $30\/-30$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFWC}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $30\/-30$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFMISTAG}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $11\/-11$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ ZJETS}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $45\/-45$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ ISRFSRPDF}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $4\/-4$  & $4\/-4$  \\\\\n$\\texttt{ CDFJES}$ & $0\/0$  & $-11\/12$  & $-4\/3$  & $-1\/1$  & $9\/-6$  & $1\/1$  & $-8\/6$  & $-5\/5$  & $-6\/7$  & $-1\/-4$  & $1\/-2$  \\\\\n$\\texttt{ CDFQ2}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $-3\/-4$  & $2\/16$  & $6\/12$  & $2\/15$  & $0\/0$  & $0\/0$  \\\\\n\\hline\n$\\texttt{TOT}$ & $40\/40$  & $17\/17$  & $13\/12$  & $13\/13$  & $46\/46$  & $11\/12$  & $31\/35$  & $31\/33$  & $31\/34$  & $13\/14$  & $9\/16$  \\\\\n\\hline\n\\end{tabular}\n    \\end{scriptsize}\n\\caption[Rate Uncertainties: Single-Tag, CMUP $+$ CMX Channel]{Per-sample rate uncertainty (\\% up\/down). Single-tag, CMUP $+$ CMX channel.}\\label{tab:1tag_emc}\n  \\end{table}\n\n\n  \\begin{table}\n    \\begin{scriptsize}\n\\begin{tabular}{ c c c c c c c c c c c c } \\toprule\n    & nonW & $t\\bar{t}$ & s-top $s$ & s-top $t$ & Zjets & W+LF & Wbb & Wcc & Wcj & WW & WZ\/ZZ \\\\\\midrule\n$\\texttt{ XS\\_ttbar}$ & $0\/0$  & $10\/-10$  & $10\/-10$  & $10\/-10$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFBTAGSF}$ & $0\/0$  & $3\/-3$  & $3\/-3$  & $5\/-5$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $11\/-11$  & $5\/-5$  \\\\\n$\\texttt{ CDFLUMI}$ & $0\/0$  & $6\/-6$  & $6\/-6$  & $6\/-6$  & $6\/-6$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $6\/-6$  & $6\/-6$  \\\\\n$\\texttt{ LEPACC\\_EMC}$ & $0\/0$  & $5\/-5$  & $5\/-5$  & $5\/-5$  & $5\/-5$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $5\/-5$  & $5\/-5$  \\\\\n$\\texttt{ QCD\\_EMC}$ & $40\/-40$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFVHF}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $30\/-30$  & $30\/-30$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFWC}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $30\/-30$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFMISTAG}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $11\/-11$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ ZJETS}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $45\/-45$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ ISRFSRPDF}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $4\/-4$  & $4\/-4$  \\\\\n$\\texttt{ CDFJES}$ & $0\/0$  & $-10\/9$  & $0\/-2$  & $0\/-3$  & $11\/-8$  & $0\/-0$  & $-5\/6$  & $-8\/4$  & $-3\/2$  & $9\/-6$  & $8\/-6$  \\\\\n$\\texttt{ CDFQ2}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $-3\/-2$  & $11\/13$  & $12\/11$  & $11\/16$  & $0\/0$  & $0\/0$  \\\\\n\\hline\n$\\texttt{TOT}$ & $40\/40$  & $17\/16$  & $13\/13$  & $14\/14$  & $47\/46$  & $11\/11$  & $32\/33$  & $33\/32$  & $32\/34$  & $17\/15$  & $14\/19$  \\\\\n\\hline\n\\end{tabular}\n    \\end{scriptsize}\n\\caption[Rate Uncertainties: Single-Tag, EMC Channel]{Per-sample rate uncertainty (\\% up\/down). Single-tag, EMC channel.}\\label{tab:1tag_mu}\n  \\end{table}\n\n\n  \\begin{table}\n    \\begin{scriptsize}\n\\begin{tabular}{ c c c c c c c c c c c c } \\toprule\n    & nonW & $t\\bar{t}$ & s-top $s$ & s-top $t$ & Zjets & W+LF & Wbb & Wcc & Wcj & WW & WZ\/ZZ \\\\\\midrule\n$\\texttt{ XS\\_ttbar}$ & $0\/0$  & $10\/-10$  & $10\/-10$  & $10\/-10$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFBTAGSF}$ & $0\/0$  & $11\/-11$  & $11\/-11$  & $12\/-12$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $24\/-24$  & $11\/-11$  \\\\\n$\\texttt{ CDFLUMI}$ & $0\/0$  & $6\/-6$  & $6\/-6$  & $6\/-6$  & $6\/-6$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $6\/-6$  & $6\/-6$  \\\\\n$\\texttt{ LEPACC\\_CEM}$ & $0\/0$  & $2\/-2$  & $2\/-2$  & $2\/-2$  & $2\/-2$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $2\/-2$  & $2\/-2$  \\\\\n$\\texttt{ QCD\\_CEM}$ & $40\/-40$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFVHF}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $30\/-30$  & $30\/-30$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFWC}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $30\/-30$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFMISTAG}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $21\/-21$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ ZJETS}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $45\/-45$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ ISRFSRPDF}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $4\/-4$  & $4\/-4$  \\\\\n$\\texttt{ CDFJES}$ & $0\/0$  & $-4\/10$  & $-5\/3$  & $-4\/2$  & $-9\/-1$  & $3\/-3$  & $-7\/8$  & $-15\/2$  & $-11\/9$  & $-18\/6$  & $-4\/-5$  \\\\\n$\\texttt{ CDFQ2}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/1$  & $18\/9$  & $6\/17$  & $5\/-9$  & $0\/0$  & $0\/0$  \\\\\n\\hline\n$\\texttt{TOT}$ & $40\/40$  & $17\/19$  & $17\/16$  & $17\/17$  & $46\/45$  & $21\/21$  & $35\/32$  & $34\/35$  & $32\/32$  & $31\/26$  & $14\/14$  \\\\\n\\hline\n\\end{tabular}\n    \\end{scriptsize}\n\\caption[Rate Uncertainties: Double-Tag, CEM Channel]{Per-sample rate uncertainty (\\% up\/down). Double-tag, CEM channel.}\\label{tab:2tag_cem}\n  \\end{table}\n\n  \\begin{table}\n    \\begin{scriptsize}\n\\begin{tabular}{ c c c c c c c c c c c c } \\toprule\n    & nonW & $t\\bar{t}$ & s-top $s$ & s-top $t$ & Zjets & W+LF & Wbb & Wcc & Wcj & WW & WZ\/ZZ \\\\\\midrule\n$\\texttt{ XS\\_ttbar}$ & $0\/0$  & $10\/-10$  & $10\/-10$  & $10\/-10$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFBTAGSF}$ & $0\/0$  & $11\/-11$  & $11\/-11$  & $12\/-12$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $24\/-24$  & $11\/-11$  \\\\\n$\\texttt{ CDFLUMI}$ & $0\/0$  & $6\/-6$  & $6\/-6$  & $6\/-6$  & $6\/-6$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $6\/-6$  & $6\/-6$  \\\\\n$\\texttt{ LEPACC\\_PHX}$ & $0\/0$  & $2\/-2$  & $2\/-2$  & $2\/-2$  & $2\/-2$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $2\/-2$  & $2\/-2$  \\\\\n$\\texttt{ QCD\\_PHX}$ & $40\/-40$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFVHF}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $30\/-30$  & $30\/-30$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFWC}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $30\/-30$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFMISTAG}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $21\/-21$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ ZJETS}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $45\/-45$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ ISRFSRPDF}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $4\/-4$  & $4\/-4$  \\\\\n$\\texttt{ CDFJES}$ & $0\/0$  & $-7\/11$  & $-5\/-2$  & $-9\/0$  & $1\/-5$  & $3\/-2$  & $-6\/5$  & $-17\/8$  & $-35\/32$  & $-50\/-50$  & $-4\/-4$  \\\\\n$\\texttt{ CDFQ2}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $5\/0$  & $6\/-7$  & $-0\/35$  & $-41\/9$  & $0\/0$  & $0\/0$  \\\\\n\\hline\n$\\texttt{TOT}$ & $40\/40$  & $17\/20$  & $17\/16$  & $19\/17$  & $45\/46$  & $22\/21$  & $31\/31$  & $34\/47$  & $62\/45$  & $56\/56$  & $14\/14$  \\\\\n\\hline\n\\end{tabular}\n    \\end{scriptsize}\n\\caption[Rate Uncertainties: Double-Tag, PHX Channel]{Per-sample rate uncertainty (\\% up\/down). Double-tag, PHX channel.}\\label{tab:2tag_phx}\n  \\end{table}\n\n  \\begin{table}\n    \\begin{scriptsize}\n\\begin{tabular}{ c c c c c c c c c c c c } \\toprule\n    & nonW & $t\\bar{t}$ & s-top $s$ & s-top $t$ & Zjets & W+LF & Wbb & Wcc & Wcj & WW & WZ\/ZZ \\\\\\midrule\n$\\texttt{ XS\\_ttbar}$ & $0\/0$  & $10\/-10$  & $10\/-10$  & $10\/-10$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFBTAGSF}$ & $0\/0$  & $11\/-11$  & $11\/-11$  & $12\/-12$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $24\/-24$  & $11\/-11$  \\\\\n$\\texttt{ CDFLUMI}$ & $0\/0$  & $6\/-6$  & $6\/-6$  & $6\/-6$  & $6\/-6$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $6\/-6$  & $6\/-6$  \\\\\n$\\texttt{ LEPACC\\_MU}$ & $0\/0$  & $1\/-1$  & $1\/-1$  & $1\/-1$  & $1\/-1$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $1\/-1$  & $1\/-1$  \\\\\n$\\texttt{ QCD\\_MU}$ & $40\/-40$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFVHF}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $30\/-30$  & $30\/-30$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFWC}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $30\/-30$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFMISTAG}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $21\/-21$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ ZJETS}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $45\/-45$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ ISRFSRPDF}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $4\/-4$  & $4\/-4$  \\\\\n$\\texttt{ CDFJES}$ & $0\/0$  & $-7\/9$  & $-3\/2$  & $-2\/0$  & $7\/2$  & $-0\/0$  & $-10\/9$  & $-6\/10$  & $-14\/7$  & $-1\/-6$  & $-2\/-3$  \\\\\n$\\texttt{ CDFQ2}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $-2\/-5$  & $14\/10$  & $-6\/-8$  & $-8\/-11$  & $0\/0$  & $0\/0$  \\\\\n\\hline\n$\\texttt{TOT}$ & $40\/40$  & $18\/19$  & $16\/16$  & $17\/17$  & $46\/45$  & $21\/22$  & $35\/33$  & $31\/33$  & $34\/33$  & $25\/26$  & $13\/13$  \\\\\n\\hline\n\\end{tabular}\n    \\end{scriptsize}\n\\caption[Rate Uncertainties: Double-Tag, CMUP $+$ CMX Channel]{Per-sample rate uncertainty (\\% up\/down). Double-tag, CMUP $+$ CMX channel.}\\label{tab:2tag_mu}\n  \\end{table}\n\n\n  \\begin{table}\n    \\begin{scriptsize}\n\\begin{tabular}{ c  c c c c c c c c c c c  } \\toprule\n    & nonW & $t\\bar{t}$ & s-top $s$ & s-top $t$ & Zjets & W+LF & Wbb & Wcc & Wcj & WW & WZ\/ZZ \\\\\\midrule\n$\\texttt{ XS\\_ttbar}$ & $0\/0$  & $10\/-10$  & $10\/-10$  & $10\/-10$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFBTAGSF}$ & $0\/0$  & $11\/-11$  & $11\/-11$  & $12\/-12$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $24\/-24$  & $11\/-11$  \\\\\n$\\texttt{ CDFLUMI}$ & $0\/0$  & $6\/-6$  & $6\/-6$  & $6\/-6$  & $6\/-6$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $6\/-6$  & $6\/-6$  \\\\\n$\\texttt{ LEPACC\\_EMC}$ & $0\/0$  & $5\/-5$  & $5\/-5$  & $5\/-5$  & $5\/-5$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $5\/-5$  & $5\/-5$  \\\\\n$\\texttt{ QCD\\_EMC}$ & $40\/-40$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFVHF}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $30\/-30$  & $30\/-30$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFWC}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $30\/-30$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFMISTAG}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $21\/-21$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ ZJETS}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $45\/-45$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ ISRFSRPDF}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $4\/-4$  & $4\/-4$  \\\\\n$\\texttt{ CDFJES}$ & $0\/0$  & $-7\/9$  & $-0\/-0$  & $1\/-5$  & $5\/-3$  & $-1\/1$  & $-10\/11$  & $-5\/19$  & $-7\/6$  & $-10\/-14$  & $8\/-8$  \\\\\n$\\texttt{ CDFQ2}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $-3\/-1$  & $17\/5$  & $15\/0$  & $27\/20$  & $0\/0$  & $0\/0$  \\\\\n\\hline\n$\\texttt{TOT}$ & $40\/40$  & $18\/19$  & $17\/17$  & $18\/18$  & $46\/46$  & $21\/21$  & $36\/32$  & $34\/36$  & $41\/37$  & $27\/29$  & $16\/16$  \\\\\n\\hline\n\\end{tabular}\n    \\end{scriptsize}\n\\caption[Rate Uncertainties: Double-Tag, EMC Channel]{Per-sample rate uncertainty (\\% up\/down). Double-tag, EMC channel.}\\label{tab:2tag_emc}\n  \\end{table}\n\n\\renewcommand{\\tabcolsep}{6pt}\n\n\n\\chapter{Kinematic Distribution of the Signal Regions}\n\nFigures from~\\ref{fig:1svt_jet} to~\\ref{fig:2svt_dphi} show several kinematic variables for the single-tag and double-tag signal regions after the composition of all the background and all the lepton categories. The normalization of each background has been changed to match the {\\em best fit} values returned by statistical analysis of the likelihood described in by Equation~\\ref{eq:likelihood_full}, in Chapter~\\ref{chap:StatRes}. For each variable we show:\n\\begin{itemize}\n\\item the total background prediction overlaid to the selected data.\n\\item The reduced $\\chi^2$ of the prediction against the data distribution. As we use the best-fit normalization is used for the backgrounds, no rate uncertainty is applied in the $\\chi^2$ evaluation, however MC statistical uncertainty is accounted.\n\\item The probability of the Kolmogorov-Smirnov test derived from the predicted and observed shapes.\n\\item Background subtracted data histogram.\n\\item JES and $Q^2$ shape variations added in quadrature.\n\\end{itemize}\n\n\\begin{sidewaysfigure}\n  \\includegraphics[width=0.49\\textwidth]{AppTagKin\/Figs\/WZ_2JET_SVT_CEMCMUPCMXISOTRKPHX_J1Et}\n  \\includegraphics[width=0.49\\textwidth]{AppTagKin\/Figs\/WZ_2JET_SVT_CEMCMUPCMXISOTRKPHX_J2Et}\\\\\n  \\includegraphics[width=0.49\\textwidth]{AppTagKin\/Figs\/WZ_2JET_SVT_CEMCMUPCMXISOTRKPHX_J1Eta}\n  \\includegraphics[width=0.49\\textwidth]{AppTagKin\/Figs\/WZ_2JET_SVT_CEMCMUPCMXISOTRKPHX_J2Eta}\n  \\caption[Single Tag Region Jet Kinematic Variables]{Variables relative to the jet kinematic for all the lepton categories combined in the single-tag signal region. Jet 1 $E_T$ (top left), jet 2 $E_T$ (top right), jet 1 $\\eta$ (bottom left), jet 2 $\\eta$ (bottom left).}\\label{fig:1svt_jet}\n\\end{sidewaysfigure}\n\n\\begin{sidewaysfigure}\n  \\includegraphics[width=0.49\\textwidth]{AppTagKin\/Figs\/WZ_2JET_SVT_CEMCMUPCMXISOTRKPHX_LepPt}\n  \\includegraphics[width=0.49\\textwidth]{AppTagKin\/Figs\/WZ_2JET_SVT_CEMCMUPCMXISOTRKPHX_LepEta}\\\\\n  \\includegraphics[width=0.49\\textwidth]{AppTagKin\/Figs\/WZ_2JET_SVT_CEMCMUPCMXISOTRKPHX_met}\n  \\includegraphics[width=0.49\\textwidth]{AppTagKin\/Figs\/WZ_2JET_SVT_CEMCMUPCMXISOTRKPHX_WMt}\n  \\caption[Single Tag Region Lepton Kinematic Variables]{Variables relative to the lepton kinematic for all the lepton categories combined in the single-tag signal region. Lepton $P_T$ (top left), lepton $\\eta$ (top right), \\mbox{${\\not}{E_T}$}  (bottom left), $M_T^W$ (bottom left).}\\label{fig:1svt_lep}\n\\end{sidewaysfigure}\n\n\\begin{sidewaysfigure}\n\\includegraphics[width=0.49\\textwidth]{AppTagKin\/Figs\/WZ_2JET_SVT_CEMCMUPCMXISOTRKPHX_dPhiMetJet1}\n\\includegraphics[width=0.49\\textwidth]{AppTagKin\/Figs\/WZ_2JET_SVT_CEMCMUPCMXISOTRKPHX_dRLepJ1}\n\\includegraphics[width=0.49\\textwidth]{AppTagKin\/Figs\/WZ_2JET_SVT_CEMCMUPCMXISOTRKPHX_dRJ1J2}\n\\includegraphics[width=0.49\\textwidth]{AppTagKin\/Figs\/WZ_2JET_SVT_CEMCMUPCMXISOTRKPHX_wPt}\\\\\n\\caption[Single Tag Region Angular Kinematic Variables]{Angular variables for all the lepton categories combined in the single-tag signal region. $\\Delta\\phi($\\mbox{${\\not}{E_T}$} $,jet1)$ (top left), $\\Delta R(Lep,jet1)$ (top right), $\\Delta R(jet1,jet2)$, (bottom left), $P_T^W$ (bottom left).}\\label{fig:1svt_dphi}\n\\end{sidewaysfigure}\n\n\n\\begin{sidewaysfigure}\n  \\includegraphics[width=0.49\\textwidth]{AppTagKin\/Figs\/WZ_2JET_SVTSVT_CEMCMUPCMXISOTRKPHX_J1Et}\n  \\includegraphics[width=0.49\\textwidth]{AppTagKin\/Figs\/WZ_2JET_SVTSVT_CEMCMUPCMXISOTRKPHX_J2Et}\\\\\n  \\includegraphics[width=0.49\\textwidth]{AppTagKin\/Figs\/WZ_2JET_SVTSVT_CEMCMUPCMXISOTRKPHX_J1Eta}\n  \\includegraphics[width=0.49\\textwidth]{AppTagKin\/Figs\/WZ_2JET_SVTSVT_CEMCMUPCMXISOTRKPHX_J2Eta}\n  \\caption[Double Tag Region Jet Kinematic Variables]{Variables relative to the jet kinematic for all the lepton categories combined in the double tag-signal region. Jet 1 $E_T$ (top left), jet 2 $E_T$ (top right), jet 1 $\\eta$ (bottom left), jet 2 $\\eta$ (bottom left).}\\label{fig:2svt_jet}\n\\end{sidewaysfigure}\n\n\\begin{sidewaysfigure}\n  \\includegraphics[width=0.49\\textwidth]{AppTagKin\/Figs\/WZ_2JET_SVTSVT_CEMCMUPCMXISOTRKPHX_LepPt}\n  \\includegraphics[width=0.49\\textwidth]{AppTagKin\/Figs\/WZ_2JET_SVTSVT_CEMCMUPCMXISOTRKPHX_LepEta}\\\\\n  \\includegraphics[width=0.49\\textwidth]{AppTagKin\/Figs\/WZ_2JET_SVTSVT_CEMCMUPCMXISOTRKPHX_met}\n  \\includegraphics[width=0.49\\textwidth]{AppTagKin\/Figs\/WZ_2JET_SVTSVT_CEMCMUPCMXISOTRKPHX_WMt}\n  \\caption[Double Tag Region Lepton Kinematic Variables]{Variables relative to the lepton kinematic for all the lepton categories combined in the double tag-signal region. Lepton $P_T$ (top left), lepton $\\eta$ (top right), \\mbox{${\\not}{E_T}$}  (bottom left), $M_T^W$ (bottom left).}\\label{fig:2svt_lep}\n\\end{sidewaysfigure}\n\n\\begin{sidewaysfigure}\n\\includegraphics[width=0.49\\textwidth]{AppTagKin\/Figs\/WZ_2JET_SVTSVT_CEMCMUPCMXISOTRKPHX_dPhiMetJet1}\n\\includegraphics[width=0.49\\textwidth]{AppTagKin\/Figs\/WZ_2JET_SVTSVT_CEMCMUPCMXISOTRKPHX_dRLepJ1}\n\\includegraphics[width=0.49\\textwidth]{AppTagKin\/Figs\/WZ_2JET_SVTSVT_CEMCMUPCMXISOTRKPHX_dRJ1J2}\n\\includegraphics[width=0.49\\textwidth]{AppTagKin\/Figs\/WZ_2JET_SVTSVT_CEMCMUPCMXISOTRKPHX_wPt}\\\\\n\\caption[Double Tag Region Angular Kinematic Variables]{Angular variables for all the lepton categories combined in the double-tag signal region. $\\Delta\\phi($\\mbox{${\\not}{E_T}$} $,jet1)$ (top left), $\\Delta R(Lep,jet1)$ (top right), $\\Delta R(jet1,jet2)$, (bottom left), $P_T^W$ (bottom left).}\\label{fig:2svt_dphi}\n\\end{sidewaysfigure}\n\n\n\\chapter{WHAM: WH Analysis Modules}\\label{chap:AppWHAM}\n\nHigh energy physics data analysis is done with the help of complex software frameworks which allow to manage the huge amount of information collected by the detector. \nThe analysis framework which was used for this analysis is named $WH$ {\\em Analysis Modules} or WHAM and I was one of the main developers of the package.  \n\nThe aim of the software is a reliable event selection and background estimate in the $\\ell\\nu+HF$ channel, with the possibility to easily implement new features and studies that can improve the Higgs search.\nWHAM plays a relevant role in the CDF low-mass Higgs boson search, especially in the $WH\\to \\ell\\nu+b\\bar{b}$ channel but also in other contexts, like $t\\bar{t}H$ and lately also $ZH$ searches. \n\nThe analysis package tries to incorporate the CDF knowledge about the $\\ell\\nu+HF$ channel, most of it coming from the top~\\cite{top_lj2005} and single-top~\\cite{single_top} analyses. An effort was also made in the direction of code modularization and analysis customization with option loading at run-time.\n\nA detailed explanation of the package is beyond the goal of this thesis, however a general overview of the package structure and functionality is given. More information is available in the CDF internal pages~\\cite{wham_twiki} although a comprehensive documentation is not yet available.\n\n\n\\section{Package Structure}\n\nThe package is organized in a folder structure organized according to the purpose of each of the sub-elements. The {\\em first level} directories are:\n\\begin{description}\n\\item[Setup:] contains the scripts needed to setup the analysis environment, both first installation and every day use, the references to all the external tools and any patche that needs to be applied.\n\\item[Documentation:] contains all the internal and public documentation. It is easily accessible and customizable by all the analysis group collaborators.\n\\item[Inputs:] contains the database files of the MC samples, the parametrization of the triggers and $SF$s, the re-weighting templates, the option configuration files. Basically every input to the analysis elaboration is here, except the data and the MC ntuples themselves.\n\\item[Commands:] this is more an utilities repository, it contains scripts to run the analysis on the CDF Grid for parallel computing (named CAF~\\cite{caf_Sfiligoi2007}), plus a wide set of macros and scripts used for single-sample studies, text file processing or small data-handling tasks. \n\\item[Results:] contains all the information elaborated by the rest of the analysis packages. This includes pre-processed data and MC samples as well as the final templates obtained after the complete elaboration. Several commands expect to find the input files here.\n\\item[Modules:] this is the core of the analysis package. It contains the C++ code used for the selection, the background estimate and the final production of the templates used for the statistical analysis and the validation of the kinematic distributions. Next section will describe it in more details.\n\\end{description}\n\n\\subsection{WHAM Modules}\n\nThe core of the WHAM analysis framework is the Modules directory. Here each functional step of the analysis is classified in a {\\em module}, i.e. a self consistent C++ class built with standardized structure to allow straightforward compilation and testing. \n\nThe modules are of two kinds: {\\em functional modules} and {\\em construction modules}. The formers are in charge of actually perform an operation, for example the event selection or the drawing of stacked histograms. The second kind of modules are the sub-components used by the first, for example the selection code needs to known the format of the input and output data as well as the definition of the lepton-object or jet-object.\n\nThe level of abstraction offered by the building modules proved to be extremely powerful. Two minimal examples (on which I contributed) that revealed to be extremely useful are: the handling of the configuration options and of the $b$-tag efficiency estimate. \nFor the first, I implemented, using the \\verb'libconfig' library~\\cite{libconfig}, a text file reading utility that allows a single location definition and the run-time loading of all the options needed by the functional modules. For the second, it was necessary to identify the minimal amount of information needed to define a $b$-tagging algorithm. The only two values needed\\footnote{Functional dependencies and correlations should be already taken into account.} are: $SF_{Tag}$ and $p_{Mistag}^j$ (see Section~\\ref{sec:tag-sel}). With this information, it was possible to develop a single algorithm for the combination of any number of different $b$-tagging algorithms, for any required tag and jet multiplicity. The code works iteratively on the jets of the event requiring the definition of a $b$-tag in priority order, defined by the user.\n\n\nThe functional modules are four and, in the directory structure, are identified by the \\verb'process' prefix:\n\\begin{description}\n\\item[Sample Selection:] access to the production ntuples (see Section~\\ref{sec:data-struc}), as well as lepton and jet selections, are performed here. The result is a small size ntuple, the \\texttt{EvTree}, containing the 4-vectors of the identified particles plus all the relevant information needed in the next steps. The \\texttt{EvTree} is defined by a class with complex {\\em methods} working on the simple stored variables: this allows a huge saving of disk space and computing time at selection stage. Furthermore the portability of the ntuple class ease the reproducibility of the same algorithms and allows faster checks.\n\\item[Sample Pre-processing:] here the complete selection is applied to the \\texttt{EvTree}'s and the pre-processing with more analysis-specific algorithms is performed. For example, MC samples are scaled to the expected yield with the application of the latest available $SF$'s and trigger efficiencies. Also, the multivariate discriminants are evaluated here. The pre-processing has the possibility to be interfaced to other ntuples than the \\texttt{EvTree}.\n\\item[Background Estimate:] this is the last step in the analysis of the $\\ell\\nu+HF$ channel. The background estimate described in Chapter~\\ref{chap:bkg} is applied here and the templates of the different signals, backgrounds and data samples are stored in a ROOT file with the derived normalizations. In general any other background estimate method can be plugged at the end of the analysis chain but for the moment only the one described in Chapter~\\ref{chap:bkg} is available.\n\\item[Result Display:] this step completes the analysis in the sense that it allows the comparison of the final estimate with the observed data distributions. Histograms are produced together with statistical indicators of the shape agreement: $\\chi^2$, Kolmogorov-Smirnov tests, systematic overlay, background subtracted plots. Tables and histograms can appear in several formats, from html pages to simple eps files.\n\\end{description}\nThe very last step in each analysis, the statistical interpretation of the results, is implemented in a different software tool~\\cite{mclimit} that is interfaced with the templates produced by the WHAM background estimate.\n\n\n\\section{Relevant Results}\n\nBeyond the analysis presented here, a wide range of other analyses exploit the WHAM package. Between the most relevant: the new single-top cross section measuremens~\\cite{single_top_benwu_10793}, the $t\\bar{t}H$~\\cite{tth_10801} search in the lepton plus jets channel and, lately, also top-properties~\\cite{topR_10887} and SUSY searches are exploiting the package.\n\nFigure~\\ref{fig:WH_improve}, probably, shows the most striking result: the improvement of the $WH$ search sensitivity (7.5~fb$^{-1}$ and 9.4~fb$^{-1}$ versions~\\cite{wh75_note10596, wh94_note10796}). The several improvements produced for the Higgs search were readily implemented and tested thanks to the backbone of a reilable framework. \n\nNevertheless the shrinking of the CDF collaboration, the final sensitivity to the Higgs boson exceeded the best expectations.\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.85\\textwidth,height=0.75\\textwidth]{AppWHAM\/Figs\/WH_Projection_new}\n\\caption[Improvements to the CDF $WH$ Search Sensitivity]{Improvements to the CDF $WH\\to \\ell\\nu+ b\\bar{b}$ search sensitivity for a Higgs boson of $m_H=115$~GeV$\/c^{2}$. Both the 2011~\\cite{wh75_note10596} and 2012~\\cite{wh94_note10796} results are mainly obtained within the WHAM analysis framework. The sensitivity reached improves more than 125\\% over the first analysis of the same channel.}\\label{fig:WH_improve}\n\\end{center}\n\\end{figure}\n\n\n\n\\chapter{Background Estimate}\\label{chap:bkg}\n\nIn this analysis an accurate background evaluation is crucial since a low signal over background ratio is expected and several components contribute to the final $M_{Inv}$ line shape used for the signal extraction. \n\nBackground estimate should be as independent as possible from the signal regions, the single and double-tagged $\\ell\\nu+2 $~jets samples. One possible approach is the simulation of background events with Monte Carlo generators, another approach is the extrapolation of background components from data control regions. Both methods present advantages and disadvantages, for example a MC simulation is completely under control but can be limited by the knowledge of the underlying physics phenomena parametrization. Data-driven techniques, on the other hand, are based on the assumption that a background in the control region can be simply extrapolated to the signal region. This brings some approximations, eventually covered by systematic uncertainties. \n\nIn our case we use a mixture of MC and data-driven estimates depending on the different background component being addressed.\n\nThe complete machinery used for the $\\ell\\nu+HF$ jets background estimate is named {\\em Method II}\\footnote{The name derives from a methodology developed for the lepton plus jets $t\\bar{t}$ decay channel. Method II appeared after the {\\em Method I}, the data-driven background estimate used in the top discovery~\\cite{cdf_top_evidence}.} and it was extensively studied at CDF, first on $t\\bar{t}$~\\cite{top_lj2005} analyses, and later in single-top~\\cite{single_top} and $WH$~\\cite{wh75_note10596, wh94_note10796} analyses. Together with the MC, two data control samples play a major role in this kind of background estimate: the {\\em pretag} control sample, without any $b$-tag requirement, and the fake-$W$ enriched sample, without any multi-jet rejection requirement.\n\nThe background evaluation can be divided in four categories: \n\\begin{description}\n\\item[Electroweak and Top:] sometimes dubbed $EWK$, these are backgrounds due to contributions of well known physical processes:  $t\\bar{t}$, single-top and $Z+ $~jets  production. For these processes, described in detail in Section~\\ref{sec:mc_bkg}, we completely rely on MC simulation for the line-shape and theoretical prediction or previous measurements for the normalization. \n\\item[$W+$ Heavy Flavors:] also named $W+HF$, is the part of the $W+$~jets sample where the $W$ is produced in association with $HF$ quarks. It is classified as {\\em irreducible} because it has the same final state ($W+c\/b$ quarks) of the $diboson\\to \\ell\\nu+HF$ signal. The line shape is derived by a LO $W+n$~partons MC while the normalization is obtained from the {\\em pretag} data control region and from MC derived $W+HF$ fractions. Section~\\ref{sec:whf} describes in detail the procedure.\n\\item[$W+$ Light Flavors:] also named {\\em mistags} or $W+LF$. These are events where the $W$ is produced in association with one or more Light Flavor ($LF$) jets. This component is drastically reduced by the requirement of one or more $\\mathtt{SecVtx}$ tagged jets. Remaining events are due to mis-measured jet tracks and long living $LF$ hadrons. The background evaluation, explained in Section~\\ref{sec:mistag}, is obtained using LO $W+ n$~partons MC where appropriate reweighting and normalization are derived from a data control sample of multi-jet events.\n\\item[Fake-$W$:] sometimes also named multi-jet or non-$W$ background, is composed by events without a real $W$ passing the selection. Those are due to multi-jet QCD production where the jets fake the charged lepton and the \\mbox{${\\not}{E_T}$}  of the neutrino. Section~\\ref{sec:qcd} describes the normalization and line-shape evaluation procedure, both completely data-driven.\n\\end{description}\nThe normalization of the four categories proceeds through sequential steps: the $EWK$ contribution is evaluated first, then $W+$~jets normalization is extracted from the pretag control region so that the $W+HF$ fraction can be scaled to the single and double tag signal regions.\n\nNext step is the estimate, from data, of $W+LF$ contribution and, finally, the {\\em tagged} non-$W$ contribution is evaluated before the application of the SVM requirement.\n\n\n\\section{Electroweak and Top Processes}\\label{sec:mc_bkg}\n\nThis category contains all the processes whose shape is evaluated directly from MC samples and whose normalization is obtained from theory prediction or experimental measurements. Figure~\\ref{fig:fey_ewk} shows some of the tree-level Feynman diagrams contributing to this background set. \n\nThe components of $EWK$ backgrounds are: \n\\begin{description}\n\\item[$Z\/\\gamma$ + jets:] Drell-Yan and $Z\\to \\ell\\ell$ events can be produced in association with jets emitted by QCD radiation. If one lepton is misidentified, they enter as a background in our selection. The simulation of  $Z\/\\gamma+n$ partons is available only at LO level (with \\texttt{ALPGEN}~\\cite{alpgen} Matrix Element generator) therefore the process is evaluated with the same procedure of $W+$~jets background (see Section~\\ref{sec:whf}). Luckily, the small acceptance and the possibility to use a cross section measured at CDF~\\cite{z_jets_cx} allows us to evaluate the $Z\/\\gamma+$jets background directly from MC. The $Z+$jets cross section measurement refers only to the on-shell $Z$ contribution, therefore a further scaling factor is derived by comparing the \\texttt{ALPGEN} LO predictions for the on-shell and the inclusive $Z\/\\gamma+$ jets MC sample. \n\n\\item[Top pair:] $t\\bar{t}$ production and decay in the lepton plus jets ($\\ell\\nu+b\\bar{b}+2$ jets) and di-lepton ($2\\ell+2\\nu+b\\bar{b}$) channels can enter in our selection if one or more of the final state objects is misidentified. Although the probability is small, the background becomes sizable in the $HF$-tagged sample because of the presence of two $b$-quarks.  \n\\item[Single-top:] $t$ and $s$ channel production have cross sections comparable to the diboson one and the semi-leptonic decay channel produces the same final state we are interested in. On the other hand, the signal to background discrimination is performed by the $M_{Inv}(jet1,jet2)$ where the single-top has no peak.\n\\end{description}\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.25\\textwidth]{Bkg\/Figs\/zjet}\n\\includegraphics[width=0.7\\textwidth]{Bkg\/Figs\/ttbar}\\\\\n\\includegraphics[width=0.7\\textwidth]{Bkg\/Figs\/single-top_s_t}\n\\caption[Some Tree-Level Diagrams Contributing to the $EWK$ Backgrounds]{Examples of tree-level Feynman diagrams contributing to the $EWK$ background set. They are $Z+$jets production (top left in association with a $b\\bar{b}$ quark pair), $t\\bar{t}$ production and decay in semi-leptonic (top center) and di-leptonic channels (top right), single-top production in the $s$ (bottom left) and $t$ (bottom right) channels and semi-leptonic decay.}\\label{fig:fey_ewk}\n\\end{center}\n\\end{figure}\n\nThe normalization, $N_{l,k}^{EKW}$, of each $EWK$ process is obtained for each lepton category, $l$, in the pretag and $1,2$-Tag regions, $k$, by using Equation~\\ref{eq:mc_estimate}.  A summary of the production cross sections and the MC generator information are reported in Table~\\ref{tab:mc_ewk}.\n\n  \\begin{table}[h]\n    \\begin{center}\n    \\begin{tabular}{cccc}\n      \\toprule\n      Sample & $\\sigma$ (pb)& Source & Initial Events \\\\\n      \\midrule\n      $Z\/\\gamma+$jets    & 787.4 $\\pm$ 85    & Measured~\\cite{z_jets_cx} & Several MCs \\\\ \n      $t\\bar{t}$         & 7.04  $\\pm$ 0.49  & NNLO~\\cite{ttbar_cx} & 6.7 M\\\\  \n       single-top, $s$   & 1.017 $\\pm$ 0.065 & NNLO~\\cite{stops_kidonakis} & 3.5 M\\\\ \n       single-top, $t$   & 2.04  $\\pm$ 0.18  & NNLO~\\cite{stopt_kidonakis} & 3.5 M\\\\ \n      \\bottomrule\n    \\end{tabular}\n    \\caption[MC Information for the $EWK$ Backgrounds]{MC information for backgrounds of the $EWK$ category, i.e. the ones estimated from simulation. $Z+$ jets cross section~\\cite{z_jets_cx} refers to the on-shell $Z$ contribution and \\texttt{ALPGEN} LO MC prediction is used to derive a correction factor. \\texttt{HERWIG} plus \\texttt{PYTHIA} PS are used for $t\\bar{t}$, \\texttt{POWHEG} plus \\texttt{PYTHIA} PS are used for single-top $s$ and $t$ channels. The generated top mass is $m_t=172.5$~GeV$\/c^{2}$.}\\label{tab:mc_ewk}\n    \\end{center}\n  \\end{table}\n\nAs the three MC derived background processes and the diboson signal are estimated in the same way, the label $EWK$ will be used to indicate generically all of them:\n\\begin{equation}\n  \\sum\\limits_{EWK} N_{l,k}^{EWK} = N_{l,k}^{Zjets} + N_{l,k}^{t\\bar{t}} + N_{l,k}^{s-top,s} + N_{l,k}^{s-top,t} +  \\sum\\limits_{diboson} N_{l,k}^{diboson}\\mathrm{.}\n\\end{equation}\n\n\n\n\\section{$W$ plus Heavy Flavor Background}\\label{sec:whf}\n\nThe associated $W+HF$ production (Figure~\\ref{fig:fey_wjets} shows few examples of tree-level diagram contributions) is the most significant contribution to the total background in the single and double tag signal samples. For its estimate we rely partially on MC and partially on data. \n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth]{Bkg\/Figs\/wjet}\n\\caption[Some Tree Level Diagrams Contributing to $W+HF$ and $W+LF$ Backgrounds]{Examples of tree-level Feynman diagrams contributing to the $W+HF$ and the $W+LF$ backgrounds. NLO diagrams also play a relevant role in the $W+$~QCD radiation emission.}\\label{fig:fey_wjets}\n\\end{center}\n\\end{figure}\n\nNormalization and kinematic distributions of the $W+$~jets sample, and of the $W+HF$ sub-sample, present several theoretical difficulties.  NLO MC simulations became available lately for $W+b\\bar{b}$ processes~\\cite{wbb_powheg} but large higher order contributions enhance the theoretical uncertainty. Furthermore the prediction of the complete $HF$ spectrum ($W+c\\bar{c}$, $W+b\\bar{b}$, $W+c$) is beyond the current capabilities. \n\nThe two aspects of the problem can be solved by factorizing the evaluation of the kinematic properties from the estimate of the normalization:\n\\begin{itemize}\n\\item a solution to estimate the kinematic properties and the composition of $W+$~jets QCD emission was proposed in~\\cite{first_lo_match} in 2001: the appropriate matching scheme of a LO calculation with a properly tuned Parton Shower (PS) simulation can reproduce the experimental results. Section~\\ref{sec:alpgen_comp} describes in detail our use of the \\texttt{ALPGEN} LO generator, interfaced with \\texttt{PHYTIA} PS. $W+HF$ relative components or $HF$ fractions, $f^{HF}$, can be extracted with this method.\n\\item the only free parameter remaining is, now, the $W+$~jets normalization. We derive it completely from data using a maximum likelihood fit on the SVM output distribution: this variable, described in Appendix~\\ref{chap:AppSvm}, has a high discriminating power between non-$W$ and real $W$ events. Therefore, as all the other $EWK$ components are known, both $N^{nonW}_{pretag}$ and $N^{Wjets}_{pretag}$  can be extracted from the fit. Section~\\ref{sec:wjets_pretag} describes the procedure.\n\\end{itemize}\n\n\n\n\\subsection{$W +$ Jets Spectrum Composition}\\label{sec:alpgen_comp}\n\n$W+$~jets composition and kinematic properties are evaluated from MC. We employ a large set of $W+n$~partons \\texttt{ALPGEN}~\\cite{alpgen} LO MC that includes $HF$ production in the ME calculation and is interfaced with \\verb'PYTHIA'~\\cite{pythia} PS:\n\\begin{itemize}\n\\item[-]$W$ plus $0$, $1$, $2$, $3$, $\\ge 4$ partons (for $W+ n$~generic jets description);\n\\item[-]$W+b\\bar{b}$ plus $0$, $1$, $2$ partons;\n\\item[-]$W+c\\bar{c}$ plus $0$, $1$, $2$ partons;\n\\item[-]$W+c$ (or $W+\\bar{c}$) plus $0$, $1$, $2$, $3$ partons.\n\\end{itemize}\nIn total we used ninety samples: fifteen to be multiplied by the three leptonic $W$ decay modes ($W\\to e,\\mu,\\tau +\\nu_e,\\nu_{\\mu}\\nu_{\\tau}$) and by another factor two for {\\em low luminosity} and {\\em high luminosity} running periods. Table~\\ref{tab:samples} summarizes the total of ninety samples used along with the respective LO cross sections and the approximate number of events initially generated.\n\\begin{table}\\begin{center}\n\\begin{tabular}{ccc}\n\\toprule\nSample& $\\sigma$ (pb)& Initial Events\\\\\n\\midrule\n$W+0p$& 1810& 7 M\\\\\n$W+1p$& 225 & 7 M\\\\\n$W+2p$& 35.3& 1.4 M\\\\\n$W+3p$& 5.59& 1.4 M\\\\\n$W+4p$& 1.03& 0.5 M\\\\\n$W+b\\bar{b}+0p$& 2.98& 2.1 M\\\\\n$W+b\\bar{b}+1p$& 0.888& 2.1 M\\\\\n$W+b\\bar{b}+2p$& 0.287& 2.1 M\\\\\n$W+c\\bar{c}+0p$& 5& 2.8 M\\\\\n$W+c\\bar{c}+1p$& 1.79& 2.8 M\\\\\n$W+c\\bar{c}+2p$& 0.628& 2.8 M\\\\\n$W+c+0p$& 17.1& 2.8 M\\\\\n$W+c+1p$& 3.39& 2.8 M\\\\\n$W+c+2p$& 0.507& 2.8 M\\\\\n$W+c+3p$& 0.083& 2.8 M\\\\\n\\bottomrule\n\\end{tabular}\n\\caption[ALPGEN $W + $ Partons MC Samples]{\\texttt{ALPGEN} LO plus \\texttt{PYTHIA} (PS) MC samples used to estimate the various contributions to the $W+HF$ and $W+LF$ backgrounds. The second column lists the LO cross section of the samples, the third column shows the approximate number of generated events. Different MC samples are used for the three lepton flavors ($e, \\mu, \\tau$) of the $W$ decay.}\\label{tab:samples}\n\\end{center}\\end{table}\n\nThe basic composition algorithm is simple: the acceptance of each sample, for a given jet multiplicity and for each lepton category, is weighted by its own LO production cross section, then a normalization factor equal to the sum of all the samples is applied. The key idea is that, if the PS matching is well tuned, the higher order corrections to the LO $W+n$~partons simulation, should simplify when the normalization factor is applied. \n\nThe procedure is more complex in our case because we want an accurate estimate of the $W+HF$ spectrum: we need to derive the $HF$ fractions, $f^{HF}$, from the $W+n$~partons samples and relate them to the physical observables of $W+b\\bar{b}\/c\\bar{c}\/c$ jets with one or two \\texttt{SecVtx} $HF$-tags.\n\nThe $W+HF$ production is evaluated explicitly at ME level in some of the $W+n$ partons samples, however a certain amount of $HF$ quarks is also produced by the PS algorithm in all the samples. Some care must be used when combining all the samples because the double counting of a process will degrades the relative weights.\n\nThe {\\em overlap removal} technique used in this work was previously developed by the CDF collaboration and it is documented in~\\cite{hf_8765}. It consists in the selection of $HF$ production from the PS {\\em or} from the ME on the base of detector level reconstructed jets. We explicitly veto events where the ME heavy flavor quarks wind up in the same jet, and we also remove events where the $HF$ quark pair from the shower is divided in two jets. In both cases, the distinction is made with simple kinematic cuts that define a parton to be {\\em in a jet} if:\n\\begin{equation}\n  \\Delta R(parton, jet)<0.4\\quad\\textrm{and}\\quad E_T^{jet,cor}> 12~\\textrm{GeV}\n\\end{equation}\nThe reason for this choice is that PS and ME predict different $\\Delta R$ (separation) between heavy quarks: PS model is tuned on the more collinear gluon splitting quark pairs but fails in the limit of large opening angles, while ME generation works better in latter regime. Figure~\\ref{fig:overlap_rem_spectrum} shows the generator level $\\Delta R$ distribution of $b\\bar{b}$ and $c\\bar{c}$ pair production after the composition of the $W+n$~partons samples and the overlap removal procedure. The $\\Delta R $ from PS-only is also reported in Figure~\\ref{fig:overlap_rem_spectrum} showing disagreement at large $\\Delta R $ with respect to the $HF$ generated with ME. The smooth transition between the different  $W+n$~partons contributions indicates that the overlap removal scheme is working properly.\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/cc_dr}\n\\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/bb_dr}\n\\caption[Generator Level $\\Delta R$ Distributions of $b\\bar{b}$ and $c\\bar{c}$ ]{Generator level $\\Delta R$ distributions of $b\\bar{b}$ (left) and $c\\bar{c}$ (right) pair production after the composition of the $W+n$~partons samples and the overlap removal procedure~\\cite{hf_8765}. The $\\Delta R $ from PS-only is also reported showing disagreement at large $\\Delta R $ w.r.t. the $HF$ generated with ME.}\\label{fig:overlap_rem_spectrum}\n\\end{center}\n\\end{figure}\n\nThe complete procedure for the determination of the $f^{HF}$'s is the following:\n\\begin{itemize}\n\\item each MC sample is assigned a weight:\n  \\begin{equation}\\label{eq:weight}\n    w^{m}=\\frac{\\sigma^m}{N_{z_0}^m} \\mathrm{,}\n  \\end{equation}\n  where $m$ varies on all the different samples of Table~\\ref{tab:samples}, $\\sigma^m$ is the cross section and $N_{z_0}^m$ is the total number of generated events with $|z_0^{vtx}|<60$~cm.\n\\item After pretag selection, the denominators of the $HF$ fractions are calculated as:\n  \\begin{equation}\n    Den=\\sum_{m=1}^{samples} w^{m}N^m\\mathrm{,}\n  \\end{equation}\n  where $N^m$ are the {\\em unique} events (i.e. not vetoed by the overlap removal) falling in each jet-bin for sample $m$.\n\\item Each selected jet is classified as $HF$ jet or not: $HF$ jets have a $b$ or $c$ parton (at generation level) {\\em match}, i.e. laying inside the cone (\\mbox{$\\Delta R(jet,parton)<0.4$}). We distinguish four $HF$ categories depending on the number of matched jets and on the kind of $HF$ parton ($b$ or $c$). The categories are: $1B$, $2B$, $1C$ and $2C$ depending if we have 1 or $\\ge 2$, $b$ or $c$ quarks matched; no $b$ quarks should be matched in the case of $1,2C$ categories.\n\\item The numerators are defined by the sum of the events in each $HF$ category (with weights given by Equation~\\ref{eq:weight}) over all the samples, in each jet-bin:\n  \\begin{equation}\n    Num^{HF}=\\sum_{m=1}^{samples}w^m N^{m,HF}\\mathrm{.}\n  \\end{equation}\n\\end{itemize}\nThe computation of the $f^{HF}$'s is now straightforward. Table~\\ref{tab:hf_frac} summarizes the final $HF$ fractions averaged on all the lepton categories.\n\\begin{table}\\begin{center}\n\\begin{tabular}{cccc}\n\\toprule\n\\multicolumn{4}{c}{Heavy Flavor Fractions}\\\\\n\\midrule\n$f^{1B}$ & $f^{2B}$ & $f^{1C}$ & $f^{2C}$ \\\\\n$0.024$ & $ 0.014$  & $ 0.114$  & $ 0.022$ \\\\ \n\\bottomrule\n\\end{tabular}\n\\caption[$HF$ fractions, $f^{HF}$, for Two Jets Selection]{$HF$ fractions, $f^{HF}$, for the two jets selection derived from \\texttt{ALPGEN} MC composition. They are classified in the different categories. $1B$: events with $1$ $b$-jet. $2B$: events with $\\ge 2$ $b$-jet. $1C$: events with $1$ $c$-jet and no $b$-jet. $2C$: events with $\\ge 2$ $c$-jet and no $b$-jet.}\\label{tab:hf_frac}\n\\end{center}\\end{table}\n\nHowever $f^{HF}$'s  are not immediately usable to predict $W+HF$ background. They need an additional correction, $K$, to account for different $HF$ composition in data and in the LO ME simulation:\n\\begin{equation}\\label{eq:k}\nK^{HF}=\\frac{f^{HF}_{data}}{f^{HF}_{MC}}\\mathrm{,}\n\\end{equation}\nThe $HF$ calibration is carried out on a data sample not used in this measurement: we base our correction on the studies performed in~\\cite{single_top}. There, the $W+1$~jet data sample is used to derive a correction of \\mbox{$K^{c\\bar{c}\/b\\bar{b}}=1.4$} to the $W+c\\bar{c}\/b\\bar{b}$ processes, while no correction ($K^{c}=1.0$) is quoted for the $W+c$ process\\footnote{$W+c$ production is an electroweak production process measured at CDF in~\\cite{cdf_wc}.}. A large uncertainty of $30$\\% is associated to all these $K$ factors as the correction is extrapolated from the $W+1$~jet data sample to the $W\\ge 2$ jets data sample. A previous study~\\cite{k_8768}, on a multi-jet control sample, obtained a $K$ closer to $1$ and it is consistent within the uncertainty. Table~\\ref{tab:k-fac} shows a summary of the applied $K$ corrections.\n\\begin{table}\\begin{center}\n\\begin{tabular}{ccc}\n\\toprule\n\\multicolumn{3}{c}{Heavy Flavor $K-$Factor Corrections}\\\\\n\\midrule\n$K^{W+b\\bar{b}}$ & $K^{W+c\\bar{c}}$ & $K^{W+c}$ \\\\\n$1.4\\pm 0.42$ & $1.4\\pm 0.42$ & $1.0\\pm 0.33$ \\\\ \n\\bottomrule\n\\end{tabular}\n\\caption[$HF$ $K-$Factor Correction]{$K-$Factor corrections needed for the calibration of the $HF$ production in data. Studies performed in~\\cite{single_top}.}\\label{tab:k-fac}\n\\end{center}\\end{table}\n\nThe final estimate of $f^{HF}$ is:\n\\begin{equation}\nf^{HF}=K\\frac{Num^{HF}}{Den}\\mathrm{.}\n\\end{equation}\n\nThe $HF$ fractions can be used in the construction of the pretag sample shapes but are of marginal importance because of the overwhelming $LF$ contribution.\nOn the other hand $HF$ are fundamental in the tagged sample composition. To evaluate it we need the tagging efficiency of each $HF$ category ($\\epsilon^{HF}$): they are derived from the MC samples requiring a \\verb'SecVtx' tag, then we apply the $\\omega_{k-Tag}$ correction as described in Section~\\ref{sec:jetSel}. Tagging efficiencies for each $HF$ category are summarized in Table~\\ref{tab:hf_tag_ef}.\n\\begin{table}\\begin{center}\n\\begin{tabular}{ccccc}\n\\toprule\n\\multicolumn{5}{c}{Heavy Flavor Tagging Efficiency}\\\\\n\\midrule\n& $\\epsilon^{1B}$ & $\\epsilon^{2B}$ & $\\epsilon^{1C}$ & $\\epsilon^{2C}$ \\\\\n1-Tag & $ 22$\\%  & $ 30$\\% & $6.7$\\%  & $ 9.1$\\% \\\\ \n2-Tag & $ 0.4$\\%  & $8.6$ & $ 0.06$\\%  & $ 0.4$\\% \\\\ \n\\bottomrule\n\\end{tabular}\n\\caption[Heavy Flavor Tagging Efficiency $\\epsilon^{HF}$ ]{Heavy Flavor tagging efficiency $\\epsilon^{HF}$ in each category. $1B$: events with $1$ $b$-jet. $2B$: events with $\\ge 2$ $b$-jet. $1C$: events with $1$ $c$-jet. $2C$: events with $\\ge 2$ $c$-jet.}\\label{tab:hf_tag_ef}\n  \\end{center}\n\\end{table}\n\n\nFinally we can combine all the pieces to obtain the {\\em final} $HF$ tagged contribution. For each lepton $l$ and tag category $k$, it is:\n\\begin{equation}\\label{eq:tag_hf}\nN^{HF}_{l,k}=N^{Wjets}_{l,k}\\cdot f^{HF} K^{HF}\\epsilon^{HF}_{k}\\mathrm{,}\n\\end{equation}\nWhere the $HF$ label distinguishes the number of jet-matched heavy flavor hadrons: $1,2B$, $1,2C$. The $1,2B$ fractions represent $W+b\\bar{b}$, $2C$ represents $W+c\\bar{c}$ prediction and $1C$ is related to $W+c$ production. The extraction of the $W+$~jets pretag normalization ($N^{Wjets}$) is discussed in the next Section.\n\n\n\n\\subsection{Pretag $W +$ Jets Normalization Estimate}\\label{sec:wjets_pretag}\n\nAccording to Equation~\\ref{eq:tag_hf}, the $W+$~jet normalization, $N^{Wjets}$,  is the only missing piece in the complete $W+HF$ estimate.  $N^{Wjets}$ is obtained solving:\n\\begin{equation}\\label{eq:pretag_comp}\nN^{Data}_{Pretag} = F^{nonW}_{Pretag} \\cdot N^{Data}_{Pretag} + F^{Wjets}_{Pretag} \\cdot N^{Data}_{Pretag} + \\sum_{EWK}N^{EWK}_{Pretag} \\mathrm{,}\n\\end{equation}\nwhere:\n\\begin{itemize}\n\\item $N^{EWK}$ is derived from Section~\\ref{sec:mc_bkg};\n\\item the fractions $F^{nonW}_{Pretag}$ and $F^{Wjets}_{Pretag}$ are unknown parameters obtained by a maximum likelihood fit in the pretag region. This provides a $W+$~jets estimate unbiased by the $b$-tag requirement;\n\\item the SVM output distribution is used in the fit before the application of the multi-jet rejection cut (Section~\\ref{sec:svm_sel}) so that the fake-$W$ fraction of the sample can be extrapolated from the sideband. \n\\end{itemize}\nWe build a binned likelihood function on the SVM distribution on the base of Equation~\\ref{eq:pretag_comp}. Events are Poisson distributed and a Gaussian constraint is imposed to the  $N^{EWK}$ normalizations while $F^{nonW}_{Pretag}$ and $F^{Wjets}_{Pretag}$ are left free to vary.\n\nTemplates used in the fit are derived in different ways according to the background source: $EWK$ and $diboson$ SVM distributions are derived from MC, $W+$~jets template is derived from the $W+n$~partons MC composition described in previous Section. Non-$W$ template is obtained in data by reversing appropriate lepton identification cuts, a detailed description is given in Section~\\ref{sec:qcd}.\nTemplates are built for each lepton category but we decided to combine all the EMC algorithms in one template to reduce statistical fluctuations exploiting the homogeneous behaviour of the different components of EMC.\n\nThe validation of the SVM output distribution is another important step of the fit procedure. Most of the previous analyses~\\cite{single_top, wh94_note10796} used a fit to the \\mbox{${\\not}{E_T}$}  distribution to derive the $W+$ jets fraction. Here a slightly different strategy is used as the SVM was trained to discriminate the $W+$~jets sample against multi-jet events. This physics interpretation was checked during the training process (see Appendix~\\ref{chap:AppSvm}). A simplified version of the fit (slightly different selection and composed only by non-$W$ and $W+$~jets samples) was performed at each training step to reject unreliable configurations. We based the rejection on two figures of merit: \n\\begin{itemize}\n\\item the $\\chi^2$ evaluation of post fit template\/data shapes; \n\\item the comparison of the non-$W$ and $W+$~jets expectation and fit estimate.\n\\end{itemize}\n\nThe final step is the actual fit evaluation for the CEM, PHX, CMUP, CMX and EMC lepton categories. Figure~\\ref{fig:pretag_fit} shows the result: the template composition with the fractions returned by the maximization likelihood fit. Table~\\ref{tab:wjet_pretag_qcd_fit} summarizes $F^{Wjets}_{Pretag}$ and $F^{nonW}_{Pretag}$ extracted from the fit. \n\nThe pretag non-$W$ fraction, $F^{nonW}_{Pretag}$, is not used any more in the background estimate but the total non-$W$ pretag normalization, $N^{nonW}_{Pretag}$, is needed to cross check the kinematic of the pretag control sample. We derive it with the following equation:\n\\begin{equation}\n  N^{nonW}_{Pretag} = F^{nonW}_{Pretag} \\cdot N^{Data}_{Pretag} + \\sigma^{EWK}\\textrm{;}\n\\end{equation}\nwhere $\\sigma^{EWK}$ is the change in the normalization of the $EWK$ backgrounds as returned by the Likelihood fit.\n\n\n\\begin{table}\\begin{center}\n\\begin{tabular}{cccccc}\n\\toprule\n Lepton & CEM & PHX & CMUP & CMX &  EMC \\\\\\midrule\n$F^{nonW}_{Pretag}$ & $ 8.5 \\pm 0.2$\\% &  $ 13.6\\pm 0.1 $\\% &  $ 1.9\\pm 0.2 $\\% &  $2.2 \\pm 0.2 $\\% & $5.9\\pm 0.2$\\%\\\\\n$F^{Wjets}_{Pretag}$ &  $ 84.0 \\pm 0.4$\\% &  $ 82.5 \\pm 0.6$\\% &  $86.7 \\pm 0.7 $\\% &  $ 87.9 \\pm 0.9$\\% &  $ 78.7\\pm 0.9 $\\%\\\\\n\\bottomrule\n\\end{tabular}\n\\caption[$W +$Jets and Non-$W$ Pretag Sample Compositions]{$W+$ jets and non-$W$ fractional compositions of the pretag selection sample estimated by a maximum likelihood fit on the SVM output distributions reported in Figure~\\ref{fig:pretag_fit}. The statistical error of the fit is reported.}\\label{tab:wjet_pretag_qcd_fit}\n\\end{center}\\end{table}\n\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/NonWFit_2JET_PretagSVTSVT_CEM}\n\\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/NonWFit_2JET_PretagSVTSVT_PHX}\\\\\n\\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/NonWFit_2JET_PretagSVTSVT_CMUP}\n\\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/NonWFit_2JET_PretagSVTSVT_CMX}\\\\\n\\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/NonWFit_2JET_PretagSVTSVT_ISOTRK}\n\\caption[$W +$ Jets and Non-$W$ Pretag Sample Estimate]{Background templates composition with proportions returned by the maximization likelihood fit on the SVM pretag distribution for the different lepton categories: CEM (top left), PHX (top right), CMUP (center left), CMX (center right), EMC (bottom). An arrow indicates the selection cut value in each lepton category.}\\label{fig:pretag_fit}\n\\end{center}\n\\end{figure}\n\n\\subsection{$W +$ Jets Scale Uncertainty}\nThe simulation of the $W+$~jets sample should take into account one more effect, the uncertainty on the factorization and renormalization scale, $Q^2$, of the ALPGEN event generator. \n\nThe $Q^2$ of the generator can be seen as the momentum scale of the hard interaction. Technically this value is used in two contexts:\n\\begin{itemize}\n\\item in the evaluation of the PDFs of the hard interaction, the {\\em factorization scale};\n\\item in the calculation of the ME perturbative expansion of a given QCD process, the {\\em renormalization scale};\n\\end{itemize}\nA reasonable choice for it, in the $W+n$~partons MC, is given by the following equation: \n\\begin{equation}\\label{eq:scale}\n  Q^2 = M^2_W + \\sum^{partons} p_T^2  \n\\end{equation}\nwhere $M_W$ is the $W$ boson mass, the sum extends over all the partons and $p_T$ is their transverse energy.\n\nHowever the $Q^2$ is not a physical observable, as it is an artifact of the perturbative approximation needed to solve QCD problems, therefore an appropriate uncertainty should be taken into account. \nTable~\\ref{tab:q2samples} shows the characteristics of two new sets of $W+n$~partons MC samples used for systematic variation and generated with the $Q^2$ parameter doubled and halved.\n\\begin{table}\\begin{center}\n\\begin{tabular}{lcccc}\n\\toprule\n&\\multicolumn{2}{c}{\\bf $\\mathbf{Q^2=2.0}$ (Up)}& \\multicolumn{2}{c}{\\bf $\\mathbf{Q^2=0.5}$ Down}\\\\\nSample& $\\sigma$ (pb)& Initial Events &  $\\sigma$ (pb)& Initial Events\\\\\n\\midrule\n$W+0p$        & 1767  & 3.5 M & 1912  & 3.5 M\\\\\n$W+1p$        & 182.9 & 3.5 M & 303   & 3.5 M\\\\\n$W+2p$        & 24.6  & 0.7 M & 57.6  & 0.7 M\\\\\n$W+3p$        & 3.36  & 0.7 M & 10.8  & 0.7 M\\\\\n$W+4p$        & 0.54  & 0.7 M & 2.24  & 0.7 M\\\\\n$W+b\\bar{b}+0p$& 2.30  & 0.7 M & 4.14  & 0.7 M\\\\\n$W+b\\bar{b}+1p$& 0.550 & 0.7 M & 1.64  & 0.7 M\\\\\n$W+b\\bar{b}+2p$& 0.152 & 0.7 M & 0.615 & 0.7 M\\\\\n$W+c\\bar{c}+0p$& 3.89  & 1.4 M & 6.88  & 1.4 M\\\\\n$W+c\\bar{c}+1p$& 1.08  & 1.4 M & 3.18  & 1.4 M\\\\\n$W+c\\bar{c}+2p$& 0.323 & 1.4 M & 1.30  & 1.4 M\\\\\n$W+c+0p$       & 13.8  & 1.4 M & 23.3  & 1.4 M\\\\\n$W+c+1p$       & 2.30  & 1.4 M & 5.72  & 1.4 M\\\\\n$W+c+2p$       & 0.294 & 1.4 M & 1.03  & 1.4 M\\\\\n$W+c+3p$       & 0.042 & 1.4 M & 0.189 & 1.4 M\\\\\n\\bottomrule\n\\end{tabular}\n\\caption[ALPGEN $W + $Partons MC Samples for $Q^2$ Systematics]{\\texttt{ALPGEN} LO plus \\texttt{PYTHIA} (PS) MC samples used to estimate the $Q^2$ systematic variation of the $W+HF$ and $W+LF$ backgrounds. The $Q^2$ is halved and doubled with respect to the default value given by Equation~\\ref{eq:scale}. Different MC samples are used for the three lepton flavors ($e, \\mu, \\tau$) of the $W$ decay.}\\label{tab:q2samples}\n\\end{center}\\end{table}\n\nThe $Q^2$ variation affects both the kinematic and the parton composition of the $W+$~jets sample, therefore, for each $Q^2$ systematic variation, the background estimate was redone as described Section~\\ref{sec:wjets_pretag}. The results are different shapes and rates for the $W+$~jets components. Figure~\\ref{fig:wjets_mjj_q2} shows the effect on the single-tag $M_{Inv}(jet1,jet2)$ distributions.\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/mjj_wlf_q2sys}\n\\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/mjj_wc_q2sys}\\\\\n\\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/mjj_wbb_q2sys}\n\\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/mjj_wcc_q2sys}\n\\caption[$M_{Inv}(jet1,jet2)$ Distribution and $Q^2$ Variations for $W + ~LF\/c\/c\\bar{c}\/b\\bar{b}$ Samples]{$M_{Inv}(jet1,jet2)$ distribution for single \\texttt{SecVtx} tagged events, after combination of all the lepton categories: the default estimate of the four $W+$~jets samples is marked by the continuous line while the $Q^2$ systematic variations are marked by the dashed lines. Top left: $W+LF$. Top right: $W+c$. Bottom left: $W+c\\bar{c}$. Bottom right: $W+b\\bar{b}$.}\\label{fig:wjets_mjj_q2}\n\\end{center}\n\\end{figure}\n\n\n\n\\section{$W$ plus Light Flavors}\\label{sec:mistag}\n\nThe $W+$~jets events originating from a $LF$ quark can produce a secondary vertex for several reasons. Long living $LF$ hadrons produce a small amount of real \\verb'SecVtx' tags while false \\verb'SecVtx' tags are due to track reconstuction errors or interaction with the detector material and the beam pipe.\n\nThe $LF$ pretag fraction ($f_j^{LF}$) is derived from the $W+n$~partons MC composition, in the same way as the $HF$ fraction. However after requiring a $b$-tag, only a very small fraction of $LF$ jets remains in the sample. As the effects that generate mistags are not adequately simulated, appropriate parametrization is obtained studying the {\\em mistag} behaviour of the \\verb'SecVtx' algorithm. \n\nAs explained in Section~\\ref{sec:btag_sf}, a multi-jet control sample is used to parametrize a per-jet mistag probability,  $p^j_{Mistag}$, function of six variables specific of the event and the jet. The $p^j_{Mistag}$ evaluation can be applied to any data sample or to MC. This gives the possibility to divide the normalization and the shape evaluation issues:\n\\begin{itemize}\n\\item the normalization is derived directly from the selected data sample. For each event we calculate an {\\em event-mistag} estimate, $w_{Mistag}^{ev}$, by using  Equations~\\ref{eq:1tag} and~\\ref{eq:2tag} with the $p^j_{Mistag}$ of the jets as inputs. The sum of all the event-mistag gives a raw normalization of the total $LF$ contribution:\n  \\begin{equation}\n    N^{rawLF}=\\sum_{ev} w_{Mistag}^{ev}\\textrm{,}\n  \\end{equation}\nthat needs to be corrected for the contribution of the other backgrounds. The $W+LF$ only part is:\n\\begin{equation}\n  N^{W+LF} = N^{rawLF} \\frac{N^{Wjets} - N^{W+HF}- \\sum\\limits_{EWK}{N^{EWK}}}{N^{Wjets}+ \\sum\\limits_{EWK}N^{EWK}}\\textrm{.}\n\\end{equation}\n\n\\item Shape is derived from the $W+LF$ composition of ALPGEN MC (no $HF$ matched jets are used) where each simulated event is weighted by its own $w_{Mistag}$ to reproduce the kinematic beaviour of the mistagged jets. Figure~\\ref{fig:wlf_weight} shows the normalized ratios between the single-tagged $W+LF$ weighted MC and the original pretag $W+LF$ distribution: $LF$ jets have a higher mistag probability in the central $\\eta$ region of the detector (where also \\texttt{SecVtx} $b$-tagging efficiency is higher) and for large $E_T$. The effect on the $W+LF$ $M_{Inv}(jet1, jet2)$ spectrum after the $b$-tagging requirement is also shown.\n\n\\end{itemize}\n\nA rate uncertainty of $11$\\% and $21$\\% is included in the single and double-tagged mistag estimates respectively. It is derived from the mistag matrix parametrization and takes into account the statistical uncertainties and correlations between jets which fall in the same jet-bin of the mistag matrix. \n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/wlf_eta}\n\\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/wlf_et}\\\\\n\\includegraphics[width=0.7\\textwidth]{Bkg\/Figs\/wlf_mjj}\n\\caption[Normalized Ratios of $W+LF$ Pretag and Tag Distributions]{Normalized ratios between the single-tagged $W+LF$ weighted MC and the original pretag $W+LF$ MC distribution: jet 1 $\\eta$ distribution (top left), jet 1 $E_T$ distribution (top right) and $M_{Inv}(jet1, jet2)$ (bottom).}\\label{fig:wlf_weight}\n\\end{center}\n\\end{figure}\n\n\n\n\n\\section{Multi-jet Background}\\label{sec:qcd}\n\nAnother source of background comes from fake-$W$ events, where a QCD multi-jet event fakes the charged lepton and produces enough \\mbox{${\\not}{E_T}$}  to pass the event selection (\\mbox{${\\not}{E_T}$} $>15$~GeV). \n\nThe probability to fake the lepton identification both at trigger and at offline level selection is very small but, at hadron colliders, multi-jet events are produced at such a high rate that this background becomes important. The selection of a fake lepton can happen for a mixture of physics and detector effects: for example a jet with large EM fraction and low track multiplicity can easily fake an electron or a high $p_T$ hadron can reach the muon chambers leaving little energy in not well instrumented sections of the calorimeter. A simulation of these effects would need an extremely high statistics, a perfect simulation of the detector and an accurate QCD prediction at high orders. This is not feasible therefore we rely on data to model the multi-jet sample (both shape and normalization). \n\nA fake-$W$ sample is obtained by reversing one or more $W\\to \\ell \\nu$ identification criteria so that the final sample will be enriched in multi-jet events and orthogonal to the signal selection. However the correlation between the reversed identification variables and the related kinematic quantities are lost in the process. The fake-$W$ sample may present discrepancies with respect to the truly multi-jet events selected in the signal region. \n\nThe production mechanism is strictly related to the lepton identification algorithm, therefore we employ three different models:\n\\begin{description}\n\\item[Not isolated muons:] the $IsoRel <0.1$ cut is one of the most important $W$ identification requirements therefore we can employ real muons but in the sideband with $IsoRel>0.2$ to select fake-$W$ candidates for CMX, CMUP and EMC categories\\footnote{EMC is composed by 7 lepton identification algorithms but the non-isolated EMC category is build only by four of them: CMU, CMP, CMXNT and BMU. The multi-jet sample properties are reproduced well enough also with this simpler model.}.\n For a better simulation, the jet associated with the non-isolated muon is removed from the jet-multiplicity count and its energy (without the muon) accounted for the \\mbox{${\\not}{E_T}$}  correction (Section~\\ref{sec:metObj}).\n\\item[Fake-CEM:] CEM electron case is different because the isolation cut is correlated with the EM cluster energy, used both at trigger level selection and in the calorimeter \\mbox{${\\not}{E_T}$}  calculation. A fake-$W$ model that maintains $IsoRel<0.1$ is obtained reversing at least two out of five shower identification requirements: Table~\\ref{tab:antiEle} shows the reversed cuts. As in the muon case, jet multiplicity should be corrected for the jet associated with the lepton, however it is not straightforward to decide if any correction needs to be applied to the \\mbox{${\\not}{E_T}$}  calculation. \n\nWe studied the behaviour of the fake model on data, after full selection, but with no SVM cut and for \\mbox{\\mbox{${\\not}{E_T}$} $<15$~GeV}. The region is supposed to be dominated by multi-jet events (except for a small $Z$ contamination) therefore we compared data to the fake-$W$ model.  Figure~\\ref{fig:chi2} shows how the reduced $\\chi^2$ test changes when comparing $M^W_T$ shapes for data and fake-$W$ model with different \\mbox{${\\not}{E_T}$}  corrections. The best $\\chi^2$ value is obtained when the \\mbox{${\\not}{E_T}$}  is corrected as if the jet corresponding to the anti-CEM is present in the event but with an energy of $0.45\\cdot E_T^{jet,raw}$. We apply this correction.\n\n\\item[Fake-PHX:] also the fake-$W$ PHX model is obtained with an anti-PHX selection where 2 out of 5 shower identification cuts are reversed (listed in Table~\\ref{tab:antiEle}). In this case no special prescription to the \\mbox{${\\not}{E_T}$}  correction was found to drastically improve the kinematic model. Although some properties of the multi-jet sample are not perfectly reproduced, the high rejection power of the SVM discriminant reduces this to a minor problem.\n\\end{description}\nThe accurate description of the multi-jet background and its rejection by using the SVM discriminant played a major role in this analysis and in the latest $WH$ search results~\\cite{wh75_note10596,wh94_note10796}.\n\n\\begin{table}\\begin{center}\n\\begin{tabular}{c|c}\n\\toprule\n\\multicolumn{2}{c}{\\bf Anti-Electron: $\\ge 2$ Failed Cuts }\\\\\nCEM  & PHX \\\\\\midrule\n$E^{Had}\/E^{EM}<0.055+0.0045 E^{EM}$& $E^{Had}\/E^{EM}<0.05$ \\\\        \n$L_{shr}<0.2$                      & PEM$3\\times 3$Fit $\\ne 0$ \\\\    \n$\\chi^2_{\\mathrm{{CES strip}}}<10$ & $\\chi^2_{PEM3\\times 3} <10$ \\\\  \n$\\Delta z(CES,trk)<3$~cm           & $PES5\\mathrm{by}9U>0.65$ \\\\     \n$-3.0<q \\Delta x(CES,trk)<1.5$~cm  & $PES5\\mathrm{by}9V>0.65$ \\\\     \n\\bottomrule\n\\end{tabular}\n\\caption[CEM and PHX Multi-jet Model Selection Cuts]{Multi-jet models for CEM and PHX electrons should fail at least 2 of the shower identification cuts listed here. The models built in this way are named {\\em anti-CEM} and {\\em anti-PHX} leptons.}\\label{tab:antiEle}\n\\end{center}\\end{table}\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.7\\textwidth]{Bkg\/Figs\/cem_scan100}\n\\caption[Reduced $\\chi^2$ Used to Tune \\mbox{${\\not}{E_T}$}  Correction on Fake-$W$ Model]{Reduced $\\chi^2$ obtained comparing the normalized $M^W_T$ shapes of data and anti-CEM fake-$W$ model in the multi-jet enriched region  \\mbox{${\\not}{E_T}$} $<15$~GeV. The test changes as we correct the \\mbox{${\\not}{E_T}$}  for fractions of the energy associated with the jet corresponding to the anti-CEM.}\\label{fig:chi2}\n\\end{center}\n\\end{figure}\n\nThe fake-$W$ models can now be used in the background evaluation. Section~\\ref{sec:wjets_pretag} already gives the formula for the pretag $N^{nonW}$ estimate, however this result is not used in this analysis except for control purposes and to plot the background composition in the pretag region. On the other hand, the tagged region normalization of the fake-$W$ sample is the only missing piece in the full background evaluation. To estimate it we use again the multi-jet enriched sideband of the SVM output distribution.\n\nWe perform another maximum likelihood fit similar to the one described in Section~\\ref{sec:wjets_pretag} with two differences. The first is a modification of the template: we require events with at least $1$ ($2$) {\\em taggable} jets when evaluating the single (double) tag fake-$W$ templates, this allows an approximate simulation of the $b$-tag requirement retaining most of the statistics of the samples. The second difference is in the fit strategy: all the real-$W$ samples are added into one template whose normalization is left free to float in the fit together with $N^{nonW}_{k-Tag}$. This procedure and the use of the sidebands make the fake-$W$ determination as much uncorrelated as possible from the other backgrounds and from a possible unknown signal. Figures~\\ref{fig:nonw_tag1} and~\\ref{fig:nonw_tag2} show the results of the $N^{nonW}_{k-Tag}$ fit on the single and double tagged SVM distributions for the CEM, PHX, CMUP, CMX and EMC lepton categories.\nA conservative rate uncertainty of 40\\% is applied uniformly on all the lepton categories to account for low statistics in some of the fits and for the extrapolation from the sideband region. Fake-$W$ contamination is summarized in Table~\\ref{tab:wjet_tag_qcd_fit}, confirming the excellent rejection power of the SVM discriminant.\n\\begin{table}\\begin{center}\n\\begin{tabular}{cccccc}\n\\toprule\n Lepton & CEM & PHX & CMUP & CMX &  EMC \\\\\\midrule\n$F^{nonW}_{1-Tag}$ & $ 9.7 \\pm 0.5$\\% &  $ 19.1 \\pm 0.7  $\\% &  $ 3.7 \\pm 0.7  $\\% &  $ 3.3\\pm 1.0 $\\% & $ 6.2 \\pm 0.5$\\%\\\\\n $F^{nonW}_{2-Tag}$ &  $ 3.2 \\pm 1.8$\\% &  $ 10.8 \\pm 2.4$\\% &  $ 4.7 \\pm 3.6$\\% &  $ 0.0 \\pm 2.2$\\% &  $ 0.0\\pm 0.3$\\%\\\\\n\\bottomrule\n\\end{tabular}\n\\caption[Non-$W$ Background Contamination in Single and Double Tag Samples]{Estimate of the non-$W$ contamination in the single and double-tagged selection samples. Only the statistical error of the fit is reported, a systematic uncertainty of $40$\\% is used in the final estimate.}\\label{tab:wjet_tag_qcd_fit}\n\\end{center}\\end{table}\n\n\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/NonWFit_2JET_SVT_CEM}\n\\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/NonWFit_2JET_SVT_PHX}\\\\\n\\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/NonWFit_2JET_SVT_CMUP}\n\\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/NonWFit_2JET_SVT_CMX}\\\\\n\\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/NonWFit_2JET_SVT_ISOTRK}\n\\caption[Single Tag SVM Distribution for Non-$W$ Background Estimate]{Result of the maximum likelihood fit performed on the single-tagged SVM distribution to estimate the non-$W$ background normalization: $N^{nonW}_{1-Tag} = F^{nonW}_{1-Tag}\\cdot N^{Data}_{k-Tag}$. Results are reported for the different lepton categories: CEM (top left), PHX (top right), CMUP (center left), CMX (center right), EMC (bottom). An arrow indicates the used selection cut value in each lepton category.}\\label{fig:nonw_tag1}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/NonWFit_2JET_SVTSVT_CEM}\n\\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/NonWFit_2JET_SVTSVT_PHX}\\\\\n\\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/NonWFit_2JET_SVTSVT_CMUP}\n\\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/NonWFit_2JET_SVTSVT_CMX}\\\\\n\\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/NonWFit_2JET_SVTSVT_ISOTRK}\n\\caption[Double Tag SVM Distribution for Non-$W$ Background Estimate]{Result of the maximum likelihood fit performed on the double-tagged SVM distribution to estimate the non-$W$ background normalization: $N^{nonW}_{1-Tag} = F^{nonW}_{1-Tag}\\cdot N^{Data}_{k-Tag}$. Results are reported for the different lepton categories: CEM (top left), PHX (top right), CMUP (center left), CMX (center right), EMC (bottom). An arrow indicates the used selection cut value in each lepton category.}\\label{fig:nonw_tag2}\n\\end{center}\n\\end{figure}\n\n\\section{Correction of Luminosity Effects}\n\nThe effect of instantaneous luminosity should be corrected in the available MC samples before completing of the background estimate. \n\nThe instantaneous luminosity of the collisions influences several parameters like the occupancy of the detector, the number of vertices in the interactions and so on. Most of these effects are parametrized within the CDF simulation (see Section~\\ref{sec:mcGen}) and they are simulated in a run-dependent way (i.e. each MC section corresponds to a period of data taking with appropriate tuning). Unluckily the available MC samples reproduce only a first section (up to 2008) of the CDF dataset while the highest luminosities (easily above $\\mathscr{L} = 2\\times 10^{32}$~cm$^{-2}$s$^{-1}$) were delivered by the Tevatron in the latest years. \n\nAn approximate solution to the problem is the {\\em reweighting} of the MCs as a function of the observed number of good quality primary vertices\\footnote{Quality$\\ge 12$, see Table~\\ref{tab:pVtx}.} (a variable highly correlated with the instantaneous luminosity). The reweighting function is evaluated comparing, in the pretag control region, the number of vertices estimated in MC against the one observed in data. Figure~\\ref{fig:nvtx_rew} shows the distribution of the number of primary vertices withouth any correction and after correcting the background estimate with the reweighting function that we apply to the final evaluation.\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/WZ_2JET_PretagSVT_CEMCMUPCMXISOTRKPHX_nQ12Vtx_noW}\n\\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/WZ_2JET_PretagSVT_CEMCMUPCMXISOTRKPHX_nQ12Vtx}\n\\caption[Number of \\mbox{{\\em Quality} $\\ge 12$} Before and After Reweighting]{Distribution (for all the lepton categories) of the number of \\mbox{{\\em Quality} $\\ge 12$} primary vertices for the events selected in the pretag control region before (left) and after (right) the application of a reweighting function to the MC events.}\\label{fig:nvtx_rew}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\n\\section{Final Background Evaluation}\n\nAs explained in the previous part of this Chapter, we derive the shape and normalization contributions of each background in an independent way. The composition of them should reproduce, within uncertainties, the observed number of selected events and the kinematic distributions of data in control and signal regions. If the background estimate is proven to be solid, then it is possible to investigate the presence of a signal with adequate tools. \n\nWe present here the final background estimate. Statistical analysis and the measured properties of the $diboson\\to \\ell \\nu + HF$ signal will be described in the next Chapter.\n\nThe first important validation criteria is the agreement of the kinematic distributions in the pretag control region. As the normalization of the region is constrained by data the most important effect comes from the evaluation of the shapes. Figures from~\\ref{fig:pret_jetEt} to~\\ref{fig:pret_mjj} show several kinematic variables after the composition of all the background for all the lepton categories. For each variable, we estimated a central pretag background value and four systematic variations: \n\\begin{itemize}\n\\item jet energy scale plus and minus one sigma (JES $+1\\sigma$, JES $-1\\sigma$);\n\\item $W+n$~partons renormalization scale variation: $Q^2$ is doubled ($Q^2_{Up}= 2.0 Q^2 $) and halved ($Q^2_{Down}= 0.5 Q^2$).\n\\end{itemize}\nThe positive (negative) systematic variations are added in quadrature and the result is shown together with the central background evaluation. The agreement is excellent within uncertainties. Few distributions that show mis-modeling, like the $\\eta$ of the second jet shown in Figure~\\ref{fig:pret_jetEta}, become irrelevant after the tagging requirement that enforces a more central selection of the jets.\n\n  \\begin{figure}\n    \\begin{center}    \n  \\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/WZ_2JET_PretagSVT_CEMCMUPCMXISOTRKPHX_J1Et}\n  \\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/WZ_2JET_PretagSVT_CEMCMUPCMXISOTRKPHX_J2Et}\n    \\end{center}\n  \\caption[Jets $E_T$ Distributions in Pretag Control Region]{Jets corrected $E_T$ distributions in the pretag control region, for all the lepton categories combined: jet 1 $E_T$ (left) and jet 2 $E_T$ (right) for all the lepton categories combined, in the pretag control region.}\\label{fig:pret_jetEt}\n  \\end{figure}\n\n  \\begin{figure}\n    \\begin{center}    \n    \\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/WZ_2JET_PretagSVT_CEMCMUPCMXISOTRKPHX_J1Eta}\n    \\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/WZ_2JET_PretagSVT_CEMCMUPCMXISOTRKPHX_J2Eta}\n    \\end{center}\n    \\caption[Jets $\\eta$ Distributions in Pretag Control Region]{Jets $\\eta$ distributions in the pretag control region, for all the lepton categories combined: jet 1 $\\eta$ (left) and jet 2 $\\eta$ (right).}\\label{fig:pret_jetEta}\n  \\end{figure}\n  \n  \\begin{figure}\n    \\begin{center}    \n    \\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/WZ_2JET_PretagSVT_CEMCMUPCMXISOTRKPHX_LepPt}\n    \\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/WZ_2JET_PretagSVT_CEMCMUPCMXISOTRKPHX_LepEta}\n    \\end{center}\n    \\caption[Lepton $p_T$ and $\\eta$ Distributions in Pretag Control Region]{Two lepton related kinematic distributions in the pretag control region, for all the lepton categories combined: $p_T$ (left) and $\\eta$ (right).}\\label{fig:pret_lepKin}\n  \\end{figure}\n\n  \\begin{figure}\n    \\begin{center}\n  \\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/WZ_2JET_PretagSVT_CEMCMUPCMXISOTRKPHX_met}\n  \\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/WZ_2JET_PretagSVT_CEMCMUPCMXISOTRKPHX_dPhiMetJet1}\n    \\end{center}\n  \\caption[$\\nu$ Kinematic Distributions in Pretag Control Region]{Two $\\nu$ related kinematic distributions in the pretag control region, for all the lepton categories combined: \\mbox{${\\not}{E_T}$}  (left) and $\\Delta\\phi(jet1, $\\mbox{${\\not}{E_T}$} $)$ (right).}\\label{fig:pret_metKin}\n  \\end{figure}\n\n  \\begin{figure}\n    \\begin{center}      \n      \\includegraphics[width=0.49\\textwidth]{Bkg\/Figs\/WZ_2JET_PretagSVT_CEMCMUPCMXISOTRKPHX_wPt}\n      \\includegraphics[width=0.49\\textwidth]{Bkg\/Figs\/WZ_2JET_PretagSVT_CEMCMUPCMXISOTRKPHX_WMt}\n      \\caption[$W$ Kinematic Distributions in Pretag Control Region]{Two $W$ related kinematic distributions in the pretag control region, for all the lepton categories combined: $p_T^W$ (left) and $M_T^W$ (right).}\\label{fig:pret_wkin}\n    \\end{center}\n  \\end{figure}\n\n  \\begin{figure}\n    \\begin{center}\n      \\includegraphics[width=0.49\\textwidth]{Bkg\/Figs\/WZ_2JET_PretagSVT_CEMCMUPCMXISOTRKPHX_dRLepJ1}\n      \\includegraphics[width=0.49\\textwidth]{Bkg\/Figs\/WZ_2JET_PretagSVT_CEMCMUPCMXISOTRKPHX_dRJ1J2}     \n    \\end{center}\n    \\caption[Angular Separation Distributions in Pretag Control Region]{Two angular separation distributions in the pretag control region, for all the lepton categories combined: $\\Delta R(lep,jet1)$ (left), $\\Delta R(jet1,jet2)$, (right).}\\label{fig:pret_dphi}\n  \\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.98\\textwidth]{Bkg\/Figs\/WZ_2JET_PretagSVT_CEMCMUPCMXISOTRKPHX_Mj1j2}\n\\caption[$M_{Inv}(jet1, jet2)$ Distribution in Pretag Control Region]{Invariant mass distribution of the selected di-jet pair in the pretag control region, for all the lepton categories combined.}\\label{fig:pret_mjj}\n\\end{center}\n\\end{figure}\n\nThe other validation method is the actual comparison of the expected backgrounds in the signal region against the observed selection. Tables~\\ref{tab:1tag_final} and~\\ref{tab:2tag_final} report the complete background composition in each of the separate lepton categories for the single and double \\texttt{SecVtx}-tagged selections, neither JES nor $Q^2$ variations are reported in the tables. All the estimates are within the uncertainties, that are, however, rather large because of the $K^{HF}$ factor uncertainties listed in Table~\\ref{tab:k-fac}. \n\n\\begin{sidewaystable}\n  \\begin{center}\n    \\begin{tabular}{lccccc}\n  \\toprule\n        \\multicolumn{6}{c}{\\bf Single-tag Event Selection}\\\\\n        Lepton ID & CEM                  &  PHX         &             CMUP             & CMX    & EMC    \\\\  \n\\midrule                                                         \n   $Z+$jets           & 55.53 $\\pm$ 4.73     & 7.76 $\\pm$ 0.67       & 65.3 $\\pm$ 5.73        & 37.14 $\\pm$ 3.28    & 104.18 $\\pm$ 10.8  \\\\    \n   $t\\bar{t}$      & 237.55 $\\pm$ 23.3    & 46.93 $\\pm$ 4.59      & 139.68 $\\pm$ 13.8      & 62.22 $\\pm$ 6.14    & 228.58 $\\pm$ 25.6   \\\\   \n   Single-top $s$  & 64.23 $\\pm$ 5.88     & 11.08 $\\pm$ 1.02      & 36.42 $\\pm$ 3.35       & 16.11 $\\pm$ 1.49    & 51.68 $\\pm$ 5.48    \\\\   \n   Single-top $t$  & 84.95 $\\pm$ 9.99     & 16.86 $\\pm$ 1.98      & 47.75 $\\pm$ 5.64       & 22.5 $\\pm$ 2.66     & 66.22 $\\pm$ 8.56    \\\\   \n   $WW$              & 84.35 $\\pm$ 11.8          & 25.05 $\\pm$ 3.54       & 43.7 $\\pm$ 6.17        & 23.68 $\\pm$ 3.35      & 53.11 $\\pm$ 8     \\\\\n   $ZZ$              & 1.85 $\\pm$ 0.19           & 0.21 $\\pm$ 0.02        & 2.45 $\\pm$ 0.25        & 1.37 $\\pm$ 0.14       & 3.45 $\\pm$ 0.39  \\\\ \n   $WZ$              & 29.2 $\\pm$ 2.95           & 12.32 $\\pm$ 1.22       & 16 $\\pm$ 1.66          & 9.21 $\\pm$ 0.94       & 20.54 $\\pm$ 2.39  \\\\\n   $W+b\\bar{b}$    & 858.71 $\\pm$ 258        & 263.08 $\\pm$ 79.1       & 428.58 $\\pm$ 129       & 238.99 $\\pm$ 71.9    & 409.97 $\\pm$ 123   \\\\\n   $W+c\\bar{c}$    & 441.77 $\\pm$ 134        & 145.08 $\\pm$ 44.1       & 212.98 $\\pm$ 64.8      & 120.26 $\\pm$ 36.6    & 214 $\\pm$ 65       \\\\\n   $W+cj$            & 342.86 $\\pm$ 104        & 96.42 $\\pm$ 29.3       & 171.77 $\\pm$ 52.2      & 93.01 $\\pm$ 28.3     & 143.47 $\\pm$ 43.6  \\\\ \n   $W+LF$         & 809.01 $\\pm$ 87.1       & 302.78 $\\pm$ 32.2       & 408.74 $\\pm$ 43.3      & 230.75 $\\pm$ 24.7    & 463.65 $\\pm$ 53.5  \\\\\n   Non-$W$           & 302.69 $\\pm$ 121        & 205.06 $\\pm$ 82       & 58.75 $\\pm$ 23.5       & 27.96 $\\pm$ 11.2     & 106.35 $\\pm$ 42.5    \\\\\n\\midrule                                                                                                                                    \n   {\\bf Prediction}      & {\\bf 3312.68 $\\pm$ 521}       & {\\bf 1132.63 $\\pm$ 176 }      & {\\bf 1632.11 $\\pm$ 253 }     & {\\bf 883.21 $\\pm$ 140}     & {\\bf 1865.2 $\\pm$ 248}   \\\\\n   {\\bf Observed }       & {\\bf 3115}                    & {\\bf 1073}                  & {\\bf 1577 }                  & {\\bf 830}                   & {\\bf 1705}               \\\\\n\\midrule                                                                                                                                \n   {\\bf Dibosons}     & {\\bf 115.39 $\\pm$ 13}           & {\\bf 37.57 $\\pm$ 4.05}      & {\\bf 62.15 $\\pm$ 6.87}       & {\\bf 34.26 $\\pm$ 3.75}    & {\\bf 77.09 $\\pm$ 9.35}    \\\\\n\\bottomrule\n    \\end{tabular}  \n  \\end{center}\n    \\caption[Observed and Expected Events with One \\texttt{SecVtx} Tags]{Summary of observed and expected events with one secondary vertex tag (\\texttt{SecVtx}), in the $W+2$~jets sample, in \n      $9.4$~fb$^{-1}$ of data in the different lepton categories. Statistical and systematics rate uncertainties are included in the table except for the contribution of the JES and the $Q^2$  variations. Diboson expected yield is added in the prediction.}\\label{tab:1tag_final}\n\\end{sidewaystable}\n\n\\begin{sidewaystable}\n  \\begin{center}\n\n  \\begin{tabular}{lccccc}\n        \\toprule \n        \\multicolumn{6}{c}{\\bf Double-tag Event Selection}\\\\\n        Lepton ID         & CEM                     & PHX                    &      CMUP                   & CMX                     & EMC               \\\\ \n        \\midrule                                                                                                                             \n      \n      \n       $Z+$jets                  & 1.43 $\\pm$ 0.13      & 0.27 $\\pm$ 0.02      & 2.84 $\\pm$ 0.27        & 1.41 $\\pm$ 0.13          & 4.6 $\\pm$ 0.5           \\\\\n       $t\\bar{t}$             & 48.21 $\\pm$ 6.99     & 9.91 $\\pm$ 1.44      & 27.31 $\\pm$ 3.98       & 12.37 $\\pm$ 1.8          & 44.89 $\\pm$ 6.95        \\\\\n       Single-top $s$         & 16.89 $\\pm$ 2.36     & 2.87 $\\pm$ 0.4       & 9.68 $\\pm$ 1.36        & 4.17 $\\pm$ 0.58          & 13.72 $\\pm$ 2.05        \\\\\n       Single-top $t$         & 5.07 $\\pm$ 0.81      & 1.13 $\\pm$ 0.18      & 2.85 $\\pm$ 0.46        & 1.34 $\\pm$ 0.22          & 4.16 $\\pm$ 0.71         \\\\\n       $WW$                     & 0.72 $\\pm$ 0.19      & 0.18 $\\pm$ 0.05      & 0.35 $\\pm$ 0.09        & 0.2 $\\pm$ 0.05           & 0.49 $\\pm$ 0.13         \\\\\n       $ZZ$                     & 0.26 $\\pm$ 0.04      & 0.03 $\\pm$ 0         & 0.46 $\\pm$ 0.06        & 0.29 $\\pm$ 0.04          & 0.63 $\\pm$ 0.1          \\\\  \n      $WZ$                     & 5.28 $\\pm$ 0.75      & 2.6 $\\pm$ 0.37       & 2.52 $\\pm$ 0.36        & 1.67 $\\pm$ 0.24          & 3.52 $\\pm$ 0.54         \\\\\n       $W+b\\bar{b}$           & 114.7 $\\pm$ 35.1     & 33.92 $\\pm$ 10.4     & 59.06 $\\pm$ 18.1       & 29.49 $\\pm$ 9.04         & 60.71 $\\pm$ 18.6        \\\\\n       $W+c\\bar{c}$           & 6.68 $\\pm$ 2.1       & 2.16 $\\pm$ 0.68      & 3.41 $\\pm$ 1.08        & 1.63 $\\pm$ 0.51          & 4 $\\pm$ 1.25            \\\\     \n       $W+cj$                   & 5.18 $\\pm$ 1.63      & 1.43 $\\pm$ 0.45      & 2.75 $\\pm$ 0.87        & 1.26 $\\pm$ 0.4           & 2.69 $\\pm$ 0.84         \\\\   \n       $W+LF$           & 4.53 $\\pm$ 0.94      & 1.7 $\\pm$ 0.36       & 2.35 $\\pm$ 0.48        & 1.28 $\\pm$ 0.27          & 2.98 $\\pm$ 0.66         \\\\    \n       Non-$W$                  & 5.58 $\\pm$ 2.23      & 6.79 $\\pm$ 2.72      & 4.31 $\\pm$ 1.73        & 0 $\\pm$ 0.5              & 0 $\\pm$ 0.5             \\\\ \n       \\midrule                                                          \n      {\\bf  Prediction}            & {\\bf 214.53 $\\pm$ 40.5}    & {\\bf 62.99 $\\pm$ 12.1}     & {\\bf 117.92 $\\pm$ 21.1}      & {\\bf 55.11 $\\pm$ 10.4}         & {\\bf 142.38 $\\pm$ 23.3}       \\\\\n       {\\bf Observed}                 & {\\bf 175}                 & {\\bf 62 }                   & {\\bf 92}                        & {\\bf 49 }                        & {\\bf 126}              \\\\     \n       \\midrule                                                          \n       {\\bf Dibosons }          & {\\bf 6.26 $\\pm$ 0.79}           & {\\bf 2.8 $\\pm$ 0.38}       & {\\bf 3.34 $\\pm$ 0.4}         & {\\bf 2.15 $\\pm$ 0.26}          & {\\bf 4.64 $\\pm$ 0.61 }        \\\\\n       \\bottomrule\n      \\end{tabular}        \n  \\end{center}\n  \\caption[Observed and Expected Events with Two \\texttt{SecVtx} Tags]{Summary of observed and expected events with two secondary vertex tags (\\texttt{SecVtx}), in the $W+2$~jets sample, in \n      $9.4$~fb$^{-1}$ of data in the different lepton categories. Statistical and systematics rate uncertainties are included in the table except for the contribution of the JES and the $Q^2$  variations. Diboson expected yield is added in the prediction.}\\label{tab:2tag_final}\n\\end{sidewaystable}\n\nBackground is still large although the diboson signal is sizable. An analysis of the $diboson\\to \\ell\\nu+HF$ properties with a counting experiment is not feasible. Therefore, in the next Chapter, we exploit the separation power of two variables: \n\\begin{itemize}\n\\item the di-jet invariant mass, $M_{Inv}(jet1,jet2)$, improves the separation between the non-resonant $W+HF$ production and the diboson.\n\\item KIT-NN improves the $b$-jets $vs$ $c$-jets separation.\n\\end{itemize}\nThe shape analysis and the fitting procedure also constraint the large normalization uncertainties, thereby increasing the significance of the measurement.\n\n\n\n\n\\chapter{Conclusions}\\label{chap:conclusions}\n\nThe result of this thesis is the evidence, at $3.08\\sigma$, for the associate production of massive vector bosons ($W$ and $Z$) detected at the CDF II experiment in a final state with one lepton, \\mbox{${\\not}{E_T}$}  and $HF$-tagged jets. \n\nSuch a result, obtained on the complete CDF II dataset ($9.4$~fb$^{-1}$ of data), was possible thanks to a simultaneous effort in several directions. The signal acceptance was extended both at online and offline selection level; the multi-jet background was strongly suppressed using a SVM-based multi-variate algorithm; second vertex $HF$-tagging was used in conjunction with a flavor-separator NN (KIT-NN). \n\nThe signals, both inclusive diboson production and $WW$ {\\em vs} $WZ\/ZZ$ separately, were extracted from the invariant mass distribution, $M_{Inv}(jet1,jet2)$, of single and double $HF$-tagged jet pairs. For single-tagged  events the $b$ quark {\\em vs} $c$ quark discrimination was obtained by using a bi-dimensional distribution $M_{Inv}(jet1,jet2)$ {\\em vs} KIT-NN.\n\nWe measure the total diboson production cross section, fixing the $WW$ and $WZ\/ZZ$ relative contribution to the SM prediction. We obtain:\n\\begin{equation}\n \\sigma_{Diboson}^{Obs}=\\left(0.79\\pm 0.28\\right)\\times \\sigma_{Diboson}^{SM} = \\left(14.6\\pm 5.2\\right)\\textrm{~pb,}\n\\end{equation}\nwhere the errors include statistical and systematic uncertainty and $\\sigma^{SM}_{Diboson}$ is the SM predicted cross section.\n\nThen, after removing the constraint on the $WW$ and $WZ\/ZZ$ relative contribution, we leave them free to float independently. We perform a simultaneous measurement of $\\sigma_{WW}$ and $\\sigma_{WZ\/ZZ}$ where all correlations are included. We can also perform a separate measurement of each contribution one at the time by considering the other as a background. In this case we obtain:\n\\begin{equation}\n\\sigma^{Obs}_{WW}=\\left(0.45^{ +0.35}_{ -0.32}\\right)\\times\\sigma^{SM}_{WW} = \\left(5.1^{+4.0}_{-3.6}\\right)\\textrm{~pb}\n\\end{equation}\nand\n\\begin{equation}\n  \\sigma^{Obs}_{WZ\/ZZ}=\\left(1.64^{+0.83}_{-0.78}\\right)\\times\\sigma^{SM}_{WZ\/ZZ} = \\left(11.6^{+5.9}_{-5.5}\\right)\\mathrm{~pb,}    \n\\end{equation}\nwhere the errors include statistical and systematic uncertainties.\nThe significance of the measurements is $1.78\\sigma$ for $WW$ signal and $2.54\\sigma$ for $WZ\/ZZ$ signal.\n\nAll the results are consistent with the SM prediction and they confirm the CDF capability of identify a small signal in this challenging final state. In particular the previous version of this analysis, performed with a dataset of $7.5$~fb$^{-1}$ and reported in Appendix~\\ref{App:7.5}, was the first evidence of diboson production in $\\ell\\nu+HF$ final state at a hadron collider.\n\nBeyond the pure testing of SM predicted processes, several of the techniques developed for this thesis were also applied to the $WH$ search at CDF, with a relevant improvement of the sensitivity to this process.\n\n\\chapter{The CDF II Experiment}\\label{chap:detector}\nIn this Chapter the accelerator facility and the detector apparatus are described in their main features. \n\nThe analyzed dataset presented in this thesis corresponds to $9.4$~fb$^{-1}$ of data collected by the CDF II (Collider Detector at Fermilab for Run II) experiment along its entire operation time\\footnote{Most of the presented work was performed during the data taking period therefore some experimental features are presented as if the operations are not yet concluded.}, from February 2002 to September $30^{th}$, 2011.\nThe CDF multi-purpose detector was located at one of the two instrumented interaction points along the Tevatron accelerator ring where $p\\bar{p}$ beams collided at an energy of $\\sqrt{s}=1.96$~TeV. \n\n\\section{The Tevatron}\n\nThe Tevatron collider was a proton-antiproton storage ring and circular accelerator located at the Fermi National Accelerator Laboratory (FNAL or Fermilab), $50$~Km west from Chicago (Illinois, U.S.A.). With a center-of-mass energy of $\\sqrt{s}=1.96$~TeV, it was the world highest energy accelerator~\\cite{tevatronweb} before the beginning of the Large Hadron Collider (LHC) era\\footnote{The Tevatron collision energy record was exceed by the LHC on March $30^{th}$, 2010, when the first $pp$ collisions at $\\sqrt{s}=7$~TeV took place.} and the largest anti-matter source in the world. The decommissioning of the accelerator started at the end of 2011 with the last collision and the stop of the operations on September $30^{th}$, 2011.\n\nThe history of the Tevatron is marked by impressive technology achievements and physics results. For example, starting the operations in 1983, it was the fist super-conducting magnet accelerator ring, in 1995, the {\\em top} quark was discovery here~\\cite{CDF_top_obs, d0_top_obs} or in 2006, $B_s$ mixing was observed~\\cite{cdf_bsmix_obs}.\n\nOne of most striking achievement of the Tevatron was the whole process of the proton-antiproton production and acceleration that involved the simultaneous operation of a chain of accelerator machines. Figure~\\ref{accel_sec} shows a view of Tevatron complex and of its sections~\\cite{tevatronweb,rookie_book}, next paragraphs summarize their operation and performances.\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.99\\textwidth]{Detector\/Figs\/accel_sec}\n\\caption[CDF Detector Schema]{Schematic view of Tevatron accelerator complex at Fermilab, different colors mark different accelerator sections.}\\label{accel_sec}\n\\end{center}\n\\end{figure}\n\n\\subsection{Proton and Antiproton Production}\n\nThe first stage, proton extraction and initial acceleration, takes place in the Pre-Accelerator (\\emph{PreAc}). Hot hydrogen gas molecules ($H_{2}$) are \nsplit by an intense local electrostatic field and charged with two electrons; $H^{-}$ ions are accelerated up to $750$~KeV by a Cockcroft-Walton accelerator every $66$~ms.\n\nPreAc ion source\\footnote{The ion sources are actually two, named H- and \nI-, and working alternatively.} constantly produces beams at $15$~Hz rate and \nsend them to the \\emph{Linac}: a linear accelerator that increases the ions \nenergy $750$~KeV to $400$~MeV. It is made of two sections: a low energy \ndrift tube and a high energy coupled cavity at the end.\n\nNext acceleration stage is performed by a circular accelerator (synchrotron)\nof $75$~m radius called \\emph{Booster}. The insertion of a thin carbon foil \nstrips off electrons from the $400$~MeV ions and a sweep, from $\\sim38$ to \n$\\sim53$~MHz in radio-frequency (RF), carries resulting protons to an energy \nof $8$~GeV. The use of negative ions permits injection of more particles from \nthe Linac, otherwise the magnetic field needed to catch the protons would also\nkick away protons already inside the Booster. Bunches are extracted when about\n$8\\cdot 10^{12}$ protons are collected.\n\nResulting bunches have the correct energy to be sent to the \\emph{Main Injector}, a larger synchrotron with radius of $\\approx 0.5$~Km): here conventional magnets and 18 accelerating cavities are employed to accelerate protons up to $120$~GeV or $150$~GeV, depending upon their use. The $120$~GeV protons are used to produce the antiprotons, while $150$~GeV protons are further accelerated into the Tevatron main ring. \n\nAntiproton production takes place in the \\emph{Antiproton Source}. This machine is composed by several parts (see Figure~\\ref{accel_sec}): first \nthere is a target station where the $120$~GeV protons, extracted from Main Injector, collide with a Nickel target and $8$~GeV $\\bar{p}$ are selected from all the resulting particles. Typically, $10\\div 20$ $\\bar{p}$ are collected for each $10^{6}$ protons on target. After production, antiprotons have a large spatial and momentum spread while acceleration into Main Injector requires narrow $8$~GeV packets. Therefore they are sent to the \\emph{Debuncher}: a triangular shape synchrotron, with a mean radius of $90$~m. The bunch signal is picked up and analyzed at one side of the ring and then it is corrected on the other side in a process named \\emph{stochastic cooling} and \\emph{bunch rotation}~\\cite{Marriner:2003mn_cooling}.\nThe final step of the $\\bar{p}$ production is the accumulation: the beam is sent to a smaller synchrotron (with a mean radius of $75$~m) inside Debuncher ring called \\emph{Accumulator}. Other cooling methods are applied here.\n\nFrom the Accumulator, $8$~GeV antiprotons can be transferred either to the Main\nInjector or to the Recycler ring. The latter is a $3.3$~Km long ring of permanent magnets, located in the Main Injector enclosure, which is used to gather antiprotons as a final storage\nbefore the injection into the Tevatron, thus allowing the Accumulator to operate at its optimal efficiency.\n\nDuring the last year of running, 2011, a store could start with up to $3.5\\times 10^{12}$ antiprotons, collected in $10\\div 20$ hours of production.\n\n \n\\subsection{Collision and Performance}\n\nLast acceleration stage takes place into the \\emph{Tevatron} Main Ring: with a radius of one kilometer this is the largest of the Fermilab accelerators, and, thanks to superconducting magnets, it can store and \naccelerate beams from an energy of $150$~GeV (Main Injector result) to $980$~GeV.\nTable~\\ref{tab:FermilabAcceleratorComplex} summarizes the acceleration characteristics of the different stages of the Fermilab $p\\bar{p}$ Accelerator Complex.\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{small}\n\\begin{tabular}{cccccccc}\\toprule\nAcc. & H & $H^{-}$ & C-W & L & B & M & T\\\\\n\\midrule\nE &0.04 eV & 25 KeV &750 KeV & 400 MeV&8 GeV& 150 GeV&0.98 TeV\\\\\n$\\beta$ & $9.1\\cdot10^{-8}$ & $0.01$& $0.04$ & $0.71$ & $0.99$ & $1$ &$1$\\\\\n$\\gamma$ & $1$ & $1$ & $1$ & $1.43$ & $9.53$ & $161$ & $1067$\\\\\n\\bottomrule\n\\end{tabular}\n\\end{small}\n\\caption[Performance of Fermilab Accelerator Complex]\n{Performances of the Fermilab Accelerator Complex. The steps along the accelerator chain (Acc.), with the corresponding labelling, are: Cockwroft-Walton (C-W), Linac (L), Booster (B), Debuncher and Recycler, Main Injector (M), Tevatron (T). The energy reached at the end of the step is E, $\\beta=\\frac{v}{c}$ expresses\nthe speed of the particle as a fraction of the speed of light in vacuum and \\mbox{$\\gamma=\\frac{E}{pc}=\\frac{1}{\\sqrt{1-(\\frac{v}{c})^{2}}}$} is the relativistic factor.}\\label{tab:FermilabAcceleratorComplex}\n\\end{center}\n\\end{table}\n\n\nWhen beams production and acceleration is complete, a Tevatron \\emph{store} is started: 36 protons and 36 antiprotons bunches, containing respectively \n$\\sim 10^{13}$ and $\\sim 10^{12}$ particles, are injected into the Main Ring at location\\footnote{The Tevatron is divided into six sections (see \nFigure~\\ref{accel_sec}) and each junction zone, named form A to F, has a different function: most important areas are B0, D0 and F0, the first two are experimental areas where CDF and DO~detectors are placed, while F0 contains RF cavity for beam acceleration and switch areas to connect Main Injector and the Tevatron.} F0 to be collided. The $0.5$~mm thin proton and antiproton bunches share the same beam pipe, magnets and vacuum system and they follows two non intersecting orbits kept $5$~mm away from each other. Beam control is obtained through nearly 1000 superconducting magnets $6$~m long, cooled to $4.3$~K and capable of $4.2$~T fields. \n\\\\\n\nBeside energy, the other fundamental parameter of an accelerator is the \ninstantaneous luminosity ($\\mathscr{L}$), as the rate of a physical process\nwith cross section $sigma$ is:\n\\begin{equation}\\label{ist_ev}\n\\frac{dN}{dt}[\\textrm{events s}^{-1}]=\\mathscr{L}[\\textrm{cm}^{-2}\\textrm{s}^{-1}]\\sigma[\\textrm{cm}^{2}]\\mathrm{.}\n\\end{equation}\nHigh energy permits an insight to incredibly small scale physics but only very \nhigh instantaneous luminosity and very large integrated (in time) luminosity\nallow to see rare events. Figure~\\ref{processes} shows the production\ncross section of different physical processes\\footnote{Due to the tiny cross sections we deal with, through most of this work we will be using {\\it picobarns} (pb) where $1$~pb$= 10^{-36}\\mathrm{~cm}^{2}$.}. \n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.75\\textwidth]{Detector\/Figs\/processes_new}\n\\caption[Production Cross Section of Physical Processes at CDF]{Predicted production cross section of physical processes at CDF along with their measured values. Higgs cross section varies with the Higgs mass.}\\label{processes}\n\\end{center}\n\\end{figure}\n\nThe instantaneous luminosity of an accelerator, usually measured in cm$^{-2}$s$^{-1}$, is given by:\n\\begin{equation}\\label{ist_lum}\n\\mathscr{L}=\\frac{N_{p}N_{\\bar{p}}B f}{2\\pi(\\sigma^2_{p}+\\sigma^2_{\\bar{p}})} F\\Big(\\frac{\\sigma_l}{\\beta^\\ast}\\Big),\n\\end{equation}\nwhere $N_{p}$ ($N_{\\bar{p}}$) are the number of protons (antiprotons) per bunch, $B$ is the number of bunches inside accelerator, $f$ is the bunch \ncrossing frequency, $\\sigma_{p(\\bar{p})}$ is the r.m.s. of the proton (antiproton) beam at the interaction point and $F$ is a beam shape form factor depending on the ratio between the the longitudinal r.m.s. of the bunch, $\\sigma_l$,  and the beta function, $\\beta^\\ast$, a measure of the beam extension in the $x,y$ phase space.\nSeveral of these parameters are related to the accelerator structure so they are (almost) fixed inside the Tevatron:\n36 $p\\bar{p}$ bunches crossed with frequency of $396$~ns and, at interaction points, quadrupole magnets focus beams in $\\approx 30$~$\\mu$m spots with $\\beta^\\ast\\simeq 35$~cm and $\\sigma_l \\simeq 60$~cm.\n\nHigher luminosities were achieved in Run II thanks to the increased antiproton stack rate. Figures~\\ref{initLum} and~\\ref{lumi} show the delivered luminosity and the constant progress in the performances of the machine with an instantaneous luminosity record of $4.3107\\times 10^{32}$~cm$^{-2}$s$^{-1}$ on May, 3$^{rd}$ 2011~\\cite{acc_records}. \n\nThe total integrated luminosity produced by the Tevatron is more than $12$~fb$^{-1}$ and CDF wrote on tape, on average, about $85\\%$ of it, with small inefficiencies due\nto detector calibration, stores not used to collect data for physics or dead time during the start-up of the data taking. This analysis uses all the data collected by CDF in Run II, corresponding, after data quality requirements, to a integrated luminosity of about $9.4$~fb$^{-1}$.\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.99\\textwidth]{Detector\/Figs\/initLum}\n\\caption[Initial Instantaneous Luminosity Delivered by the Tevatron to CDF]{Initial instantaneous luminosity delivered by the Tevatron accelerator to the CDF II detector. Performances grown by two order of magnitude from the beginning of the operation (2002) to the final collisions (2011)~\\cite{lum}.}\\label{initLum}\n\\end{center}\n\\end{figure}\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.99\\textwidth]{Detector\/Figs\/lum}\n\\caption[Integrated Luminosity Delivered by the Tevatron for each Run]{Integrated luminosity delivered by the Tevatron for each physics run (blue) and averaged on a week by week basis (green bars)~\\cite{lum}.}\\label{lumi}\n\\end{center}\n\\end{figure}\n\n\n\n\\section{The CDF Detector}\n\nCDF II is a multi-purpose solenoidal detector situated at the B0 interaction point along the Tevatron main accelerator ring. Thanks to accurate charged \nparticle tracking, fast projective calorimetry and fine grained muon detection, the CDF II detector can measure energy, momentum and charge of most particles resulting from $\\sqrt{s}=1.96$~TeV $p\\bar{p}$ collisions. \n\nThe first original design goes back to 1981 but CDF underwent many upgrades \nduring the past twenty years. The last and most extensive one began in \n1996 and ended in 2001 when Tevatron \\emph{Run II} started. At present the \nCDF II experiment is operated by an international collaboration\nthat embraces more than 60 institutions from 13 different countries, for a total\nof about 600 researchers.\n\n\n\\subsection{Overview and Coordinate system}\n\nCDF is composed by many parts (sub-detectors) for a total of about \n5000 tons of metal and electronics, a length of $\\sim 16$~m and a diameter of \n$\\sim 12$~m. It is approximately cylindrical in shape with axial and \nforward-backward symmetry about the B0 interaction point. Before going further\nwe describe the coordinate system used at CDF and through this thesis.\n\nB0 is taken as the origin of CDF right-handed coordinate system: $x$-axis\nis horizontal pointing North\\footnote{Outward with respect to the center of \nTevatron.}, $y$-axis is vertical pointing upward and $z$-axis is along beam \nline pointing along proton direction, it identifies {\\em forward} and \n{\\em backward} regions, respectively at $z>0$, East, and $z<0$, West. Sometimes \nit is convenient to work in cylindrical ($r$, $z$, $\\phi$) coordinates where \nthe azimuthal angle $\\phi$ is on the $xy$-plane and is measured from the \n$x$-axis. \nThe $xy$-plane is called \\emph{transverse}, quantities projected on it are \nnoted with a T subscript. Two useful variables are the transverse momentum, $p_{T}$, and energy, $E_{T}$, of a particle:\n\\begin{equation}\n\\vec{p}_{T}\\equiv p\\sin(\\theta)\\mathrm{,}\\qquad E_{T}\\equiv E \\sin(\\theta)\\mathrm{,}\n\\end{equation}\nin collider physics another widely used variable, used in place of $\\theta$, is the \\emph{pseudorapidity}: \n\\begin{equation}\n\\eta\\equiv-\\ln\\big(\\tan (\\theta\/2)\\big).\n\\end{equation}\nIf $(E,\\vec{p})$ is the 4-momentum of a particle, the  pseudorapidity is the \nhigh energy approximation ($p\\gg m$) of the \\emph{rapidity}:\n\\begin{equation}\ny=\\frac{1}{2}\\ln\\frac{E+p\\cos(\\theta)}{E-p\\cos(\\theta)}\\stackrel{p \\gg m}{\\rightarrow} \\frac{1}{2}\\ln\\frac{p+p\\cos(\\theta)}{p-p\\cos(\\theta)}= -\\ln\\big( \\tan (\\theta\/2)\\big) \\equiv \\eta.\n\\end{equation}\nA Lorentz boost along the $\\hat{z}$ direction adds a constant $\\ln(\\gamma + \\gamma\\beta)$ to $y$, therefore rapidity differences are invariant.\nThe statistical distribution of final state particles is roughly flat in $y$ because, in hadronic colliders, the interactions between the (anti)proton \nconstituents, which carry only a fraction of the nucleon energy, may have large momentum imbalances along $\\hat{z}$.\n\nFigure~\\ref{CDF_schema} shows an isometric view of the CDF detector and of \nits various sub-detectors. The part inside the $1.4$~T superconducting \nsolenoid contains the integrated tracking system: three silicon sub-detectors \n(the \\emph{Layer00}, the \\emph{Silicon Vertex detector II} and the \n\\emph{Intermediate Silicon Layers}) are the inner core of CDF II. The high \nresolution capability of silicon microstrips is necessary to have good track \nresolution near the interaction point, where particle density is higher. Afterward \nan open cell drift chamber (the \\emph{Central Outer Tracker}) covers until \n$r\\simeq 130$~cm, in the region $|\\eta|<1.0$, the extended lever arm provides \nvery good momentum measurement (\\mbox{$\\Delta p_T\/p^2_T \\simeq 10^{-3}~$GeV$\/c^{-1}$}).\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.99\\textwidth]{Detector\/Figs\/CDF_schema2}\n\\caption[Isometric View of the CDF II Detector]{Isometric view of the CDF II detector, the various sub-detectors are highlighted in different colors and listed.}\\label{CDF_schema}\n\\end{center}\n\\end{figure}\n\nCalorimeter systems are located outside the superconducting solenoid. They are\nbased on \\emph{shower sampling calorimeters} made of sequential layers of \nhigh-Z passive absorbers and active signal generator plastic scintillators. \nThe system is composed by towers with $\\eta - \\phi$ segmentation, each one \ndivided in electromagnetic and hadronic part, they cover the region up to \n$|\\eta|\\simeq 3.6$ ($\\theta\\simeq3^{\\circ}$) and are organized in two main\nsections: the \\emph{Central Calorimeter} covering the region \n$|\\eta|\\lesssim 1.1$ and the \\emph{Plug Calorimeter} \nextending the coverage up to $|\\eta|\\simeq 3.6$. While the central calorimeter\nis unchanged since 1985, the plug calorimeter active part was completely \nrebuilt  for Run II, replacing gas chambers with plastic scintillator tiles to\nbetter cope with the higher luminosity.\n\nThe outermost part of CDF detector, outside calorimeters, is occupied by the\nmuon detectors. They are multiple layers of drift chambers arranged in various \nsubsections which cover the region $|\\eta| \\lesssim 1.5$. Only high penetrating\ncharged particles, such as muons, can go across the entire detector.\n\nOther detectors are used for a better particle identification, calibration or \nmonitoring. However a detailed description of the entire CDF detector is far \nfrom the scope of this work. The next paragraphs will focus on tracking and \ncalorimeter systems which play a significant role in the analysis. A complete \ndescription of CDF II detector can be found in~\\cite{TDR}.\n\n\n\n\n\\subsection{Integrated Tracking System}\\label{track_par}\n\nThe trajectory of a charged particle in a uniform \nmagnetic field in vacuum is a helix. A tracking detector identifies some points along particle path so \nthat it is possible to obtain momentum measurements by reconstructing the \nhelix parameters\\footnote{See Section~\\ref{sec:track} for track reconstruction \ndetails.}. A schematic view of CDF tracking volume can be seen in \nFigure~\\ref{CDF_track}: the three main components are the superconducting \nmagnet, the silicon sub-detectors and the central drift chamber.\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.75\\textwidth]{Detector\/Figs\/CDF_track}\n\\caption[CDF II Tracking Volume and Calorimeter]{View of CDF II tracking volume and calorimeter location.}\\label{CDF_track}\n\\end{center}\n\\end{figure}\n\nThe solenoidal magnet, made  by NbTi\/Cu superconducting coils, maintains a \nbending magnetic field with a central value of $1.4116$~Tesla, oriented along \nthe positive $\\hat{z}$ direction and nearly uniform in all the tracking volume \n($r\\lesssim 150$~cm and $|z|\\lesssim 250$~cm). \nThe momentum threshold for a particle to radially escape the magnetic field is \n$p_{T}\\gtrsim 0.3$~GeV\/$c$ and the radial thickness of the coil is $0.85$ \nradiation lengths ($X_{0}$).\n\n\n\n\\subsection*{Silicon System}\n\nThe silicon system is the first tracking sub-detector encountered by particles\nexiting from the primary interaction vertex. Semiconductor detectors offer\nexcellent spatial resolution and fast response time.\nTherefore it permits the reconstruction of secondary vertices displaced from the\nprimary, produced in the decay of long lived $b$-hadrons\\footnote{Correct \nidentification of $b$-hadrons is fundamental in many analyses e.g. $b$-hadrons \nare one of the decay products of $top$ quark and also Higgs boson has a high \nbranching ratio to $b$ quarks for $m_H\\lesssim 140$~GeV$\/c^{2}$.}.\n\nCDF employs $\\sim7$~m$^2$ silicon active-surface for a total \nof 722,432 different channels read by about 5500 integrated custom chips. The\ncomplete silicon tracking detector is displayed in Figure~\\ref{silicons}.\nOf the three subsystems composing the core of CDF, the \n\\emph{Layer00 }~\\cite{l00} (L00 ) is the innermost.\nIt consists of a single layer of single-sided silicon sensors directly mounted \non the beam pipe at radii, alternating in $\\phi$, of $1.35$~cm or $1.62$~cm, \ncovering the region $|z|\\lesssim 47$~cm. During the construction of the SVX II \nmicrovertex (see below) CDF realized that the multiple scattering due to the \npresence of read-out electronics and cooling systems installed \ninside tracking volume was going to degrade the impact parameter resolution.\nL\\O\\O~ was designed to recover it thanks to its proximity to the beam. \nFurthermore, being made of state-of-the-art radiation-tolerant sensors, it will\nensure a longer operating lifetime to the entire system.\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{Detector\/Figs\/cdfiisi_rz}\n\\includegraphics[width=0.49\\textwidth]{Detector\/Figs\/cdf_silicon_endview}\n\\caption[Side and Front View of Silicon System]{Side and front view of silicon tracking system at CDF.}\\label{silicons}\n\\end{center}\n\\end{figure}\n\nThe main component of the silicon system is SVX II~\\cite{svx}, the \n\\emph{Silicon VerteX detector}\nis made of three cylindrical barrels for a total length of about $96$~cm along\n$z$, covering the luminosity region until $\\simeq 2.5~\\sigma_{l}$, and with a \npseudo-rapidity\nrange $|\\eta|\\lesssim 2$. Each barrel is divided in twelve identical wedges in\n$\\phi$, arranged in five concentric layers between radii \n$2.4$~cm and $10.7$~cm. Each layer is divided into independent longitudinal \nread-out units, called {\\em ladders}. Each ladder consists of a low-mass support\nfor a double-sided silicon microstrip detector. Three out of five layers \ncombine an $r-\\phi$ measurement on one side with $90^{\\circ}$ stereo measurement\non the other, the remaining two layers combine an $r-\\phi$ measure with a small\nangle $r-z$ stereo measurement (with tilt angle of $1.2^{\\circ}$). \nThe highly parallel fiber based data acquisition \nsystem reads out the entire sub-detector in approximately $10~\\mu$s.\n\nThe \\emph{Intermediate Silicon Layers} detector~\\cite{isl} (ISL) is the outermost of the three silicon sub-detectors, radially\nlocated between SVX II and the drift chamber covering the region $|\\eta|\\lesssim 2$. It is divided in three barrels segmented into $\\phi$ \nwedges. The central barrel ($|\\eta|\\lesssim 1$) is made of one layer of silicon sensors at radius of $22$~cm, instead the two outer barrels \n($1\\lesssim|\\eta|\\lesssim 2$) are made of two layers at radii of $20$~cm and \n$28$~cm. Its purpose is to strengthen the CDF tracking in the central region and to add precision hits in a region not fully covered by the \ndrift chamber. Track reconstruction can be extended to the whole region $|\\eta|<2$ using the silicon detector.\n\nThe complete silicon sub-detector (L00, SVX II and ISL) has an asymptotic \nresolution of $40~\\mu$m in impact parameter and of $70~\\mu$m along $z$ \ndirection. The total amount of material varies roughly as:\n\\begin{equation}\n\\frac{0.1X_{0}}{\\sin(\\theta)}\n\\end{equation}\nin the central region and doubles in the forward region because of the presence of read-out electronics, cooling system and support frames~\\cite{sil_performance}.\n\n\n\n\\subsection{Central Outer Tracker}\n\nThe \\emph{Central Outer Tracker}~\\cite{cot} (COT) is an open-cell drift \nchamber used for particles tracking at large radii. It has an \nhollow-cylindrical geometry and covers $43.3<r<132.3$~cm, $|z|\\lesssim 155$~cm.\nFigure~\\ref{CDF_track} shows that COT fully covers the central region \n($|\\eta|\\lesssim 1$) with some residual capability up to $|\\eta| \\approx 1.8$\n\nThe COT (see Figure~\\ref{cot1}) is structured into eight {\\em super-layers} each \ndivided into $\\phi$ cells; each cell contains twelve sampling wires, spaced \n$0.583$~cm, to collect the ions produced by passing charged particles. The \narrangement of the cells has a $\\chi=35^{\\circ}$ tilt with respect to the \nchamber radius to partially compensate the Lorentz angle of the electrons \ndrifting in the magnetic field and obtain the best \nresolution\\footnote{Electrons drifting in a gas within an electromagnetic \nfield $(\\vec{E},\\vec{B})$ move with an angle \n$\\chi\\simeq\\arctan\\big(\\frac{v(E,B=0)B}{kE}\\big)$, where $k$ is empirical \nparameter of gas and electric field and $v(E,B=0)$ is the velocity without the\nmagnetic field. The angle $\\chi$ is also known as Lorentz angle.}.\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.98\\textwidth]{Detector\/Figs\/cot1}\n\\caption[A $1\/6$ Section of the COT End-plate]{A $1\/6$ section of the COT end-plate with the eight super-layers structure and the location of cell slots.}\\label{cot1}\n\\end{center}\n\\end{figure}\n\nThe final structure has $8 \\times 12$ sampling planes alternated with \nplanes of potential wires (see Figure~\\ref{cot2}), $96$ hits are measured for a \nparticle crossing the entire COT ($|\\eta|<1$). Four super-layers employ \nsense-wires parallel\nto the beam axis for measurements in $r-\\phi$ plane, the other four interspacing\nsuper-layers are named {\\em stereo} super-layers because their wires are \nalternately canted at angles of $+2^{\\circ}$ and $-2^{\\circ}$ with respect to \nthe beam line and are used to measure $r-z$ coordinates. The electric \ndrift field (see Figure~\\ref{cot2}) is $1.9$~kV\/cm. A $50:50$ gas admixture of \nargon and ethane bubbled through isopropyl alcohol ($1.7$\\%) constantly flows \nin the chamber volume. The drift velocity is about $100~\\mu$m\/cm for a\nmaximum drift space of $0.88$~cm. The material of the COT is about \n$0.017X_{0}$, mostly concentrated in the inner and outer shell.\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.98\\textwidth]{Detector\/Figs\/cot2}\n\\caption[Equipotential Line and Wire Layout Inside one of the COT Cells]{Left: equipotential line inside one of the COT super-layer cell. \nRight: layout of sense-wires, field-wires and shaper-wires inside one COT cell.}\\label{cot2}\n\\end{center}\n\\end{figure}\n\n\n\n\\subsection{Calorimeter System}\n\nLocated immediately outside the solenoid, the calorimeter system covers\na solid angle of nearly $4\\pi$ around $p\\bar{p}$ interaction point and has the\nfundamental role to measure energies of electrons, photons, particle clusters \n(\\emph{jets}) and the imbalance in transverse energy flow (signature of \n\\emph{neutrinos}). \nThe location of calorimeter sections is visible in Figure~\\ref{CDF_track}.\nBoth plug and central are sampling calorimeters divided into an \nelectromagnetic section (lead\/scintillator), optimized to collect all the \nenergy of electrons and photons, and a subsequent hadronic section \n(iron\/scintillator), thicker and optimized for hadron energy measurement. \nCalorimeters have an in-depth segmentation, finer near the collision point and\ncoarser outward. The $\\eta-\\phi$ plane is mapped in rectangular cells, each one\ncorresponding to the independent read-out of a projective electromagnetic or \nhadronic tower. Thanks to the fast response of scintillators, signals from\ncalorimeters are quickly processed and used at various trigger levels. \nFollowing paragraphs explains in more detail the composition \nof the different subsections and Table~\\ref{cal_tab} summarizes their main \ncharacteristics.\n\\begin{table}\n  \\begin{center}\n    \\begin{small}\n\n      \\begin{tabular}{ccccc}\n        \\toprule\n        & En. Resolution & $\\eta$ Coverage & Absorber & Longitudinal Depth\\\\\n        \\midrule\n        CEM & $13.5\\%\/\\sqrt{E} \\oplus 2\\% $ &  $|\\eta|<1.1$  & lead  & $19X_{0}$,  $1\\lambda$\\\\\n        CHA & $50\\%\/\\sqrt{E}\\oplus 3\\%$ &  $|\\eta|<0.9$      & iron  & $4.5\\lambda$\\\\\n        WHA & $75\\%\/\\sqrt{E}\\oplus 4\\%$ &  $0.7<|\\eta|<1.3$  & iron  & $4.5\\lambda$\\\\\n        PEM & $16\\%\/\\sqrt{E}\\oplus 1\\%$ &  $1.1<|\\eta|<3.6$  & lead  & $21X_{0}$, $1\\lambda$ \\\\\n        PHA & $74\\%\/\\sqrt{E}\\oplus 4\\%$ & $1.3<|\\eta|<3.6$   & iron  & $7\\lambda$ \\\\\n        \\bottomrule\n      \\end{tabular}      \n    \\end{small}\n    \\caption[CDF II Calorimeter System]{Main characteristics of CDF II calorimeter system.}\\label{cal_tab}\n  \\end{center}\n  \\end{table}\n\n\n\\subsection*{Central Calorimeter}\n\nThe central region of the detector is covered by the \\emph{Central Electromagnetic} (CEM) and the \\emph{Central HAdronic} (CHA) calorimeters~\\cite{cen_cal}, corresponding to the pseudo-rapidity region $|\\eta|<1.1$ and $|\\eta|<0.9$ respectively. \n\nThe CEM is a hollow cylinder located at $173<r<208$~cm, divided in four $180^{\\circ}$ arches each composed by 12 azimuthal sections ($\\Delta\\phi=15^{\\circ}$) and 10 pseudo-rapidity sections ($\\Delta\\eta\\simeq0.11$) for a total of 478 instrumented towers\\footnote{Two towers are missing to permit access to the solenoid, the\nso-called {\\em chimney}.}. \nThe CHA covers region $|\\eta|<0.9$ and it is divided into 9x12 $\\eta-\\phi$ \ntowers corresponding to CEM segmentation for a total of 384 towers. Central \nhadronic calorimeter covering is extended up to $|\\eta|\\simeq 1.3$ thanks to \nthe \\emph{Wall HAdron Calorimeter}~\\cite{wha} (WHA). It has same $\\phi$ \nsegmentation and six additional $\\eta$ towers: the first three overlap\nCHA and the last three extend $\\eta$ coverage.\n\nFigure~\\ref{cen_sector} shows a wedge of the central calorimeter system. Each\nCEM sector is a sampling device made of 31 layers of polystyrene scintillator \n($5$~mm thick) radially alternated with layers of aluminum-clad lead \n($3.18$~mm thick). Some of the 30 lead layers are replaced by acrylic \n(Plexiglas) as a function of $\\theta$ to maintain a uniform thickness in \n$X_{0}$.\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.75\\textwidth]{Detector\/Figs\/cen_sector}\n\\caption{Structure of a wedge of CDF central calorimeter.}\\label{cen_sector}\n\\end{center}\n\\end{figure}\nAs particles loose energy into the absorber medium, the blue light emitted by \nactive scintillator medium is collected by thin bars of blue-to-green \nwave-length shifter acrylic material placed on the sides of each tower that \nbring the light to two photomultiplier tubes (PMT) outside CHA.\nCEM contains also the \\emph{Central Electron Strip chambers} (CES) and the \n\\emph{Central PReshower detector} (CPR). CES is a multi-wire proportional \nchamber placed at a radial depth of $\\sim6X_{0}$ and is used to determine shower \nposition and transverse shower development with an accuracy of $\\sim 0.2$~cm. \nCPR is a layer of finely segmented scintillators located immediately outside \nthe solenoid and is used to monitor photon conversion started in the tracking \nregion.\n\nThe structure of hadronic calorimeters is similar to electromagnetic ones but\nabsorber materials are 32 steel, $2.5$~cm-thick, layers in CHA and 15 steel, \n$5.1$~cm thick, layers in WHA both alternated with acrylic scintillator, \n$1.0$~cm thick.\n\nThe total thickness of electromagnetic section is nearly uniform and \ncorresponds to $19$ radiation lengths ($X_{0}$) or $1$ interaction length \n($\\lambda_{int}$). Based on test beam data, the CEM energy resolution \nfor an electron going through the center of a tower is found to be:\n\\begin{equation}\\label{cem_res}\n\\frac{\\sigma_{E}}{E}=\\frac{13.5\\%}{\\sqrt{E(\\mathrm{GeV})}}\\oplus 2\\%.\n\\end{equation}\nThe total thickness of hadronic section is $\\sim4.5\\lambda_{int}$ and the \nenergy resolution is:\n\\begin{equation}\\label{cha_res}\n\\frac{\\sigma_{E}}{E}=\\frac{50\\%}{\\sqrt{E(\\mathrm{GeV})}}\\oplus 3\\%, \\qquad \\frac{\\sigma_{E}}{E}=\\frac{75\\%}{\\sqrt{E(\\mathrm{GeV})}}\\oplus 4\\%.\n\\end{equation}\nrespectively for CHA and WHA.\n\n\n\\subsection*{Forward Calorimeter}\\label{forward_cal_sec}\n\nPlug calorimeters~\\cite{plug} are two identical structures, East and \nWest, covering region $1.1\\lesssim|\\eta|\\lesssim 3.6$. Figure~\\ref{plug1} \nshows the structure\nof plug calorimeters, in a way similar to the central device: there is a \n\\emph{Plug ElectroMagnetic calorimeter} section (PEM), a \n\\emph{Plug PReshower} (PPR) detector before the calorimeter, a \n\\emph{Plug Electromagnetic Shower-maximum} detector (PES) embedded (at \n$6X_{0}$)  and a subsequent \\emph{Plug HAdronic calorimeter} section (PHA).\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.75\\textwidth]{Detector\/Figs\/plug1}\n\\caption{Elevation view of one quarter of the CDF plug calorimeter.}\\label{plug1}\n\\end{center}\n\\end{figure}\n\nElectromagnetic section is $21X_{0}$ thick and is composed by 23 annular \nplates, of $2.77$~m outer diameter and an inner hole for the beam pipe made of\n$4.5$~mm thick lead absorber. Towers have \na segmentation with varying $\\Delta\\eta$ and $\\Delta\\phi$ as \nTable~\\ref{cal_segment} shows, with an azimuthal-angle covering of \n$7.5^{\\circ}$ down to $\\eta=2.11$ and of $15^{\\circ}$ further. Active elements\nare $4$~mm thick scintillator tiles read-out by embedded wavelength shifters \nconnected to PMT. All is assembled in triangular shape \\emph{pizza-pans} that\nenclose a slice of a $\\Delta\\phi=15^{\\circ}$ sector. Two layers are different:\nthe first scintillator layer is $10$~mm thick and is used as a preshower detector,\nand another layer, at about $5$~cm from the surface, is used as the Plug Electron Shower-max (PES) detector, it is made by two strips of scintillators that provide shower maximum \nposition measurement with $\\sim1$~mm accuracy. \n\n\\begin{table}\\begin{center}\n\\begin{tabular}{ccc}\n\\toprule\n$|\\eta|$ Range & $\\Delta\\phi$ & $\\Delta\\eta$ \\\\\n\\midrule\n$0.-1.1 (1.3 H)$ & $15^{\\circ}$ & $\\sim 0.1$ \\\\\n$1.1(1.3 H)-1.8$ & $7.5^{\\circ}$ & $\\sim 0.1$ \\\\\n$1.8-2.1$ & $7.5^{\\circ}$ & $\\sim 0.16$ \\\\\n$2.1-3.64$ & $15^{\\circ}$ & $0.2 - 0.6$ \\\\\n\\bottomrule\n\\end{tabular}\n\\caption[CDF II Calorimeter Segmentation]{CDF II calorimeter segmentation, $H$ stands for the hadronic section.}\\label{cal_segment}\n\\end{center}\n\\end{table}\n\nHadronic section is about $7\\lambda_{int}$ thick and segmented in \n$\\Delta\\phi=30^{\\circ}$ for a total of 12 sections of 23 iron $5$~cm-thick \nlayers alternated with $6$~mm scintillator active material layers. The \ncharacteristic {\\em plug} shape is due to the growing radii of the layers far \nfrom interaction point to match WHA coverage. Energy resolution is:\n\\begin{equation}\\label{pem_res}\n\\frac{\\sigma_{E}}{E}=\\frac{16\\%}{\\sqrt{E(\\mathrm{GeV})}}\\oplus 1\\%, \\qquad \\frac{\\sigma_{E}}{E}=\\frac{74\\%}{\\sqrt{E(\\mathrm{GeV})}}\\oplus 4\\%.\n\\end{equation}\nrespectively for PEM and PHA. Figure~\\ref{plug2} shows the segmentation of a\n$\\Delta\\phi=15^{\\circ}$ sector and describes the distribution of trigger \ntowers.\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.75\\textwidth]{Detector\/Figs\/plug2}\n\\caption[Plug Calorimeter Segmentation]{Segmentation of the plug calorimeter and tower location inside one wedge.}\\label{plug2}\n\\end{center}\n\\end{figure}\n\n\n\\subsection{Muon Detectors}\\label{sec:mu_cham}\nAlthough nearly all particles are absorbed by the calorimeter system, muons pass through the calorimeters as minimum ionizing particles and can exit the calorimeter system\\footnote{Muons from $Z^0$ decays, for instance, deposit on average about $0.4$~GeV in the electromagnetic portion of the calorimeter and $4$~GeV in the hadronic one.}, therefore the outermost sub-detector of CDF is the muon detection system~\\cite{muon}. It is made out of single wire drift chambers and scintillator counters for fast timing, located radially just outside the calorimeter system.\n\nThere are various muon subsystems with slightly different characteristics and named according to their locations: the Central Muon Detector (CMU), the Central Muon uPgrade Detector (CMP), the Central Scintillator uPgrade (CSP), the Central Muon eXtension Detector (CMX), the Central Scintillator eXtension (CSX), the Toroid Scintillator Upgrade (TSU), the Barrel Muon Upgrade (BMU) and the Barrel Scintillator Upgrade (BSU). \nThe CMU, CMP and CSP systems cover an $\\eta$ range of $|\\eta|<0.6$, the CMX and CSX systems cover an $\\eta$ range of $0.6<|\\eta|<1.0$ and the TSU, BMU and BSU subsystems cover an $\\eta$ range of $1.0<|\\eta|<2.0$. A diagram of the muon subsystems coverage can be seen in Figure~\\ref{figure:MuonCoverage} .\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[angle=0,width=0.75\\textwidth,clip=]{.\/Detector\/Figs\/mu_coverage}\n\\caption[CDF II Muon System $\\eta$-$\\phi$ Diagram]\n{Diagram in the $\\eta$-$\\phi$ plane of the muon systems at CDF: CMU, CMP, CMX and BMU muon detectors. The BMU detector is referred in this diagram as IMU.}\\label{figure:MuonCoverage}\n\\end{center}\n\\end{figure} \n\nThe innermost muon system is CMU, it was built for CDF I and is located just outside the CHA calorimeter, at a radius of $350$~cm and arranged in $12.6$$^\\circ$ wedges in $\\phi$. Each wedge is made of three layers ({\\em stacks}) composed by four rectangular drift tubes. Each drift tube operates in proportional mode, with an Argon-Ethane mixture gas and a single 50~$\\mu$m sense wire in the middle of the cell, parallel to the $z$ axis: absolute differences of drift arrival time between two tubes provide a measurement of the azimuthal coordinate, while the charge division at each end of a wire can be used to determine the $z$ coordinate. The CMU is followed by another muon system of similar structure, the CMP, installed beyond a $60$~cm thick layer of steel. The minimal $p_T$ threshold for the CMU (CMP) is $1.4$ ($2.2$)~GeV$\/c$. \n\nOutside the CMP we find the CSP: a fast response detector used for triggering and formed by a single scintillator layer connected to a light guide and a PMT. \n\nThe CMX muon system is located at each edge between the CDF barrel and forward regions. It has a conical geometry with drift chambers similar to the CMP. Also, it has a scintillating system called the CSX, similar to the CSP. The CMX system covers $360$$^\\circ$ with 15 wedges in $\\phi$. Each wedge is formed of eight layers of drift chambers in the radial direction. \nVarious properties of the CMU\/CMP\/CMX subsystems are summarized in Table~\\ref{table:MuonSubsystems}.\n\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabular}{cccc}\\toprule\nGeneral Parameters & CMU & CMP & CMX\\\\\n\\midrule\n$\\eta$ coverage & 0-0.6 & 0-0.6   & 0.6-1.0\\\\\n$p_T$ Threshold [GeV$\/c$] & 1.4 & 2.2 & 1.4\\\\\n\\midrule \nDrift Tubes & CMU &  CMP & CMX\\\\\n\\midrule \nThickness [cm] & 2.68 & 2.5 & 2.5 \\\\\nWidth [cm] & 6.35 & 15 & 15 \\\\\nLength [cm] & 226 & 640 & 180 \\\\\nMax. drift time [$\\mu$s] & 0.8 & 1.4 & 1.4 \\\\\n\\midrule \nScintillators & N\/A & CSP & CSX\\\\\n\\midrule \nThickness [cm] & N\/A & 2.5 & 1.5 \\\\\nWidth [cm] & N\/A & 30 & 30-40 \\\\\nLength [cm] & N\/A & 320 & 180 \\\\\n\\bottomrule\n\\end{tabular}\n\\caption[CDF II Summary of Muon Subsystem Properties]{Summary of the properties of the muon subsystems at CDF. \\label{table:MuonSubsystems}}\n\\end{center}\n\\end{table}\n\n\nMuon identification proceed on the base of short ionization tracks left in the drift chambers (called {\\em stubs}) and reconstructed thanks to the timing information provided by the individual drift chambers. Then a COT track is matched to the stubs to confirm the muon candidate providing an accurate measurement of the the muon momentum.\n\n\\subsection{Cherenkov Luminosity Counters}\\label{clc_par}\n\nThe \\emph{Cherenkov Luminosity Counters}~\\cite{clc_performances} (CLC) are two symmetrical detector modules designed to measure the instantaneous luminosity through the rate of $p\\bar{p}$ interactions in the forward region.\n\nEach counter is made of 48 conical, isobuthane gas filled, Cherenkov counters pointing to the nominal interaction region and located inside \neach plug calorimeter in a forward pseudo-rapidity region (\\mbox{$3.7<|\\eta|<4.7$}). Cones are disposed in a concentric way, with smaller counters at the center (length $110$~cm, initial diameter $2$~cm) and larger ones outward (length $180$~cm, initial diameter $6$~cm). The narrow shape and orientation is optimal to collect particles outgoing from the \ninteraction point that produce an important Cherenkov light yield. On the other hand particles from beam halo or from secondary interactions have larger crossing angle, hence they produce a much smaller signal. The excellent time resolution (less than $100$~ps) allows the analysis of the coincidence between the two modules (East and West) and it is an additional tool to remove \nbackground interactions. Figure~\\ref{clc_coincidence} shows the time distribution of the hits on the two modules. \n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.75\\textwidth]{Detector\/Figs\/clc_coincidence}\n\\caption[Time Distribution of CLC Modules Signal]{Time distribution of East and West CLC modules signals. A $p\\bar{p}$ collision deposits a coincidence signal in the two modules.}\\label{clc_coincidence}\n\\end{center}\n\\end{figure}\n\nThe CLC signal shape is used to measure the average number of inelastic interactions per bunch crossing ($\\bar{N}$), the instantaneous luminosity $\\mathscr{L}$ is inferred from the relation:\n\\begin{equation}\\label{lum1}\n\\bar{N} f_{b.c.}=\\mathscr{L} \\sigma_{in} \\epsilon ,\n\\end{equation}\nwhere the bunch crossing frequency ($f_{b.c.}$) is precisely known from the Tevatron RF, $\\epsilon$ is the CLC acceptance for inelastic scattering and $\\sigma_{in}$ is the inelastic $p\\bar{p}$ cross section. The last parameter comes from the averaged CDF and E811 luminosity independent measurements at $\\sqrt{s}=1.80$~TeV~\\cite{cx_ppbar_cdf, cx_ppbar_e811}, extrapolated to $\\sqrt{s}=1.96$~TeV:\n\\begin{equation}\\label{lum2}\n\\sigma_{in}(1.80\\mathrm{~TeV})=60.4\\pm 2.3\\mathrm{~mb}\\rightarrow\\sigma_{in}(1.96\\mathrm{~TeV})=61.7\\pm 2.4\\mathrm{~mb}.\n\\end{equation}\nThe combined systematic uncertainty on the luminosity measurement~\\cite{clc_performance} is $6$\\%: a\n4\\% due to the extrapolation applied in Equation~\\ref{lum2} and about 4\\% due to the uncertainty of the CLC acceptance. \n\n\n\n\n\n\n\\section{Trigger and Data Handling}\\label{sec:trigger}\n\nThe purpose  of the trigger system is the on-line selection of useful physics events from the background of uninteresting processes produced at much higher rate. The online selection step is rather important because only a fraction of data can be stored for offline physics analysis.\n\nAt the B0 interaction point, with a bunch crossing frequency of $2.5$~MHz, an inelastic $p\\bar{p}$ cross section of $\\sigma_{in}\\simeq 60$~mb and an instantaneous luminosity of \n\\mbox{$\\mathscr{L}\\simeq 10^{32}$~cm$^{-2}$s$^{-1}$}, there are about $1\\div 2$ inelastic collision in each bunch crossing. It is clearly impossible to store the entire detector information for each collision, as the maximum recording rate is $50\\div 100$~Hz, and it would also be useless, because interesting processes have much smaller cross section than generic inelastic interactions (diboson production cross sections are $\\mathscr{O}(10^{-9})$ w.r.t. generic jet production, see Figure~\\ref{processes}). The CDF trigger system is designed for the efficient selection of the interesting events.\n\nThe system is composed by three levels, L1, L2 and L3 (see \nFigure~\\ref{trigger_lev}); each one provides a sufficient rate reduction to allow\nthe feeding and processing by the next level with, virtually, no dead-time\\footnote{\\emph{Dead-time} occurs when events must be rejected because trigger system is occupied processing a preceding event.}.\n\nEach level filters the events using a set of programmable conditions, step by \nstep more complex as the detector read-out completes and more elaboration time becomes available.\nA, so-called, \\emph{trigger path} is the logic combination of criteria from different levels. \n\nA peculiar requirement that needs to be described is the {\\em PreScale} (PS) condition: a known fraction of the events selected by L1 or L2 are immediately discarded before the elaboration of the following trigger level. PS can be fixed or {\\em Dynamic} (DPS): the first is applied to auxiliary trigger paths, used for efficiency estimates or data quality control, the second is used for physics trigger paths that would require excessive computing time, thus their scaling factor is optimized during the data taking according to the available bandwidth at each trigger level.\n\nThe several trigger paths used this analysis are described in Section~\\ref{sec:onlineSel}, however the CDF experiment collects about 150 trigger paths. They are arranged in a \\emph{trigger table} aimed to maximise the acceptance of interesting events allowing a maximum acquisition dead time of $5\\%$.\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.525\\textwidth]{Detector\/Figs\/trigger_lev}\n\\includegraphics[width=0.465\\textwidth]{Detector\/Figs\/trigger}\n\\caption[CDF II Trigger System]{The CDF II trigger system. Left: block diagram of three-levels trigger\n and DAQ system. Right: L1 and L2 trigger streams.}\\label{trigger_lev}\n\\end{center}\n\\end{figure}\n\n\n\\subsection{Level-1}\n\nThe first trigger level is a custom designed hardware system composed by three\nparallel streams which feed inputs to Global Level-1 decision unit (see Figure~\\ref{trigger_lev}). One stream, \\verb'L1_CAL', collects prompt colorimetric response: it is divided into {\\em object triggers}, i.e. single hadronic or electromagnetic deposits above threshold, and {\\em global triggers}, i.e. total event transverse energy ($\\sum{E_{T}}$) or the presence of raw missing transverse energy\\footnote{Missing transverse energy is defined as $\\vec{\\mathrm{\\cancel{E}}}_{T}\\equiv - \\sum_i{\\vec{E}_T^i}$, with $i$ index of the calorimeter towers. See Section~\\ref{sec:metObj} for more details.} (MET or \\cancel{E}$_T$ ).\nA second stream, \\verb'L1_MUON', collects information from muon chambers thus identifying muon stubs. The last stream, \\verb'L1_TRAK', comes from the eXtreme Fast Tracker (XFT), a powerful\nparallel pattern recognition algorithm used to feed COT raw tracking information to L1 trigger. \nThe collected information already allows a coarse but efficient reconstruction of candidate physics objects: a track plus matched EM deposit is an electron candidate, a track plus a matched stub is a muon candidate, MET can identify a neutrino, a hadronic cluster in the calorimeter can be a jet and so on. \n\nWhile the L1 trigger takes a decision, the events are stored in forty-two pipelined buffers synchronized with Tevatron clock cycles ($132$~ns). In a latency time of $132\\cdot42\\simeq 5.5~\\mu$s, L1 drastically reduces the rate of accepted events from $2.5$~MHz to about $50$~KHz, the feed rate of L2.\n\n\n\\subsection{Level-2}\n\nAt the second trigger level there is enough time to readout the information of slower sub-detectors and perform more complex physic object identification algorithms. The main components of L2 are:\nthe readout of the shower-max trigger boards, the 3D reconstruction of the XFT tracks~\\cite{xft_3d}, the \\verb'L2CAL' hardware~\\cite{l2cal_up} clustering of calorimeter towers and the Silicon Vertex Trigger~\\cite{svt_up} (SVT) for the reconstruction of displaced secondary vertices. \n\nThe shower-maximum detector information ensures a better electron or photon recognition with azimuthal information and better spatial resolution. Electron identification is also improved by the XFT 3D track matching. The requirement of a track matched to a muon chamber stub improves also the muon fake rejection.\n\n\\verb'L2CAL' is a custom hardware setup able to perform raw energy clustering. Adjacent towers above a predefined threshold (usually few GeV), are merged iteratively to build refined trigger objects like jets or EM clusters. The use of physics objects allows to define advanced selection criteria on the base of  detector $\\eta-\\phi$ position, multiplicity or $E_T$ threshold for the full objects.\n\nThe last fundamental piece of the L2 is SVT: the SVX II detector $r-\\phi$ side is readout and the silicon hits, together with XFT information, are compared on on the fly with a large batch of Associative Memories (AM) where all the possible track configurations (of a certain resolution) are simulate and stored. The comparison with pre-processed simulation is the key for an extremely fast reconstruction of the track parameters and it allows the identification of displaced secondary vertices, a possible signature of beauty hadrons present in the event.\n\nStarting from the L1 input rate of $\\sim 50$~KHz, L2 must bring the accept rate to $300$~Hz exploiting four memory buffers and a short latency time of \\mbox{$5.5\\times 4 = 22~\\mu$s}. The high luminosity delivered by the Tevatron after 2005 produced events with lager detector occupancy that required more time for L2 elaboration. To cope with this, an extensive upgrade of the L2 hardware  took place between 2005 and 2006. My first experience with CDF happened in this context and, in particular, I was part of the SVT~\\cite{svt_up} upgrade team.\n\n\n\\subsection{Level-3} \n\nThe last stage of the trigger system, L3, is composed by a farm \nof several hundreds processors exploiting LINUX OS and C++ based software for the full event reconstruction. The complete detector information is assembled by the \\emph{EVent Builder}~\\cite{evb}\n(EVB) and processed by a simplified version of the offline reconstruction code.\n\nIf the L3 trigger requirements are satisfied, the {\\em event record} corresponding to the given bunch crossing is transferred to the Consumer Server\/Data Logger (CSL) that streams the data to disk, while a fraction of the output is also used for real time monitoring. The L3 accept rate suitable for disk storage is $100$~Hz.\n\n\n\n\\subsection{Data Structure}\\label{sec:data-struc}\n\nWhen an event record is saved on disk, it is labelled with a progressive number, grouped and classified.\n\nAll the events belonging to a continuous data taking period  are expected to have  very similar detector conditions (i.e. active sub-detecors, calibration parameters, trigger table, etc) so they define a {\\em run}. A large set of runs is grouped into a {\\em data period}, that usually corresponds to an integrated luminosity of a few hundreds of pb$^{-\u22121}$. The thirty-eight data periods that compose the complete CDF dataset are summarized in Table~\\ref{tab:datasets} together with the corresponding integrated luminosities and run ranges.\n\nEvents are also classified into {\\em data-streams}, a four-character label, describing similar trigger properties or common physical interest. This analysis uses four of the several data-streams available at CDF: \n\\begin{description}\n  \\item[\\texttt{bhel}:] high $E_T$ central electron stream;\n  \\item[\\texttt{bpel}:] high $E_T$ forward electron stream;\n  \\item[\\texttt{bhmu}:] high $p_T$ muon stream;\n  \\item[\\texttt{emet}:] high \\mbox{${\\not}{E_T}$}  stream.\n\\end{description}\n\n\\renewcommand{\\arraystretch}{0.95}\n\\begin{table}\n  \\begin{center}\n    \\begin{small}\n    \\begin{tabular}{cccc}\\toprule\n      Data Period & Run range & Period $\\int\\mathscr{L}dt$ (pb$^{-1}$) & Total $\\int\\mathscr{L}dt$ (pb$^{-1}$)\\\\\n\n\\midrule\np0\t&138425-186598\t&550\t&550   \\\\\np1\t&190697-195408\t&130\t&680   \\\\\np2\t&195409-198379\t&130\t&810   \\\\\np3\t&198380-201349\t&100\t&910   \\\\\np4\t&201350-203799\t& 95\t&1005  \\\\\np5\t&203819-206989\t&135 \t&1140 \\\\\np6\t&206990-210011\t& 110 \t&1250 \\\\\np7\t&210012-212133\t& 50 \t&1300 \\\\\np8\t&217990-222426\t& 210 \t&1510 \\\\ \np9\t&222529-228596\t&180 \t&1690 \\\\\np10\t&228664-233111\t&280 \t&1970  \\\\\np11\t&233133-237795\t&264 \t&2234  \\\\\np12\t&237845-241664\t&185 \t&2419  \\\\\np13\t&241665-246231\t&317 \t&2736  \\\\\np14\t&252836-254683\t&44.5 \t&2780  \\\\\np15\t&254800-256824\t&159 \t&2939  \\\\\np16\t&256840-258787\t&142 \t&3081  \\\\\np17\t&258880-261005\t&188 \t&3269  \\\\\np18\t&261119-264071\t&407 \t&3676  \\\\\np19\t&264101-266513\t&287 \t&3963  \\\\\np20\t&266528-267718\t&256 \t&4219  \\\\\np21\t&268155-271047\t&520 \t&4739  \\\\\np22\t&271072-272214\t&292 \t&5031  \\\\\np23\t&272470-274055\t&232 \t&5263  \\\\\np24\t&274123-275848\t&283 \t&5546  \\\\\np25\t&275873-277511\t&236 \t&5782  \\\\\np26\t&282976-284843\t&189 \t&5971  \\\\\np27\t&284858-287261\t&422 \t&6393  \\\\\np28\t&287294-289197\t&333  \t&6726  \\\\\np29\t&289273-291025\t&360  \t&7086  \\\\\np30\t&291294-293800\t&460 \t&7546  \\\\\np31\t&293826-294777\t&172 \t&7718  \\\\\np32\t&294778-299367\t&435 \t&8153  \\\\\np33\t&299368-301303\t&357 \t&8510  \\\\\np34\t&301952-303854\t&359 \t&8869  \\\\\np35\t&304266-306762\t&364 \t&9233  \\\\\np36\t&306791-308554\t&462 \t&9695 \\\\\np37\t&308570-310441\t&174 \t&9869 \\\\\np38\t&310472-312510\t&252 \t&10121 \\\\\n\\bottomrule\n    \\end{tabular}\n    \\end{small}\n    \\caption[Data Period and Luminosity]{Summary of data taking periods for the complete CDF dataset. Corresponding run range and collected integrated luminosity are reported. \\label{tab:datasets}}\n  \\end{center}\n\\end{table}  \n\\renewcommand{\\arraystretch}{1.}\n\nAfter that the event information is saved on disk and properly classified, it is possible to start the offline processing, also named {\\em production}. At this stage, the low-level detector data is extracted, corrected with the calibration constants and appropriate algorithms are used to reconstruct high-level physics objects (tracks, electrons, muons and jets described in Chapter~\\ref{chap:objects}).\n\nThe software is an object oriented framework where all the algorithms are defined by a self consistent \\texttt{C++} module~\\cite{data_proc}; this allows an independent testing and development of the separate algorithms and improves flexibility. For example the same {\\em track module} can be used both to reconstruct a single track information or as an element of a more complex algorithm, like the electron identification module. The final format of the physics data is a large array ({\\em n-tuple}) that can be analyzed with commercial software\\footnote{The open source \\texttt{ROOT}~\\cite{rootr} analysis software is widely used in the high energy physics community.}.  \n\nAlthough each of the \\texttt{AC++} module can be improved or modified, the analyzed data should be as much stable and uniform as possible therefore there is always a {\\em recommended} analysis prescription for the production code version to use in each data period: Table~\\ref{tab:prod_prescription} reports the one used in this thesis and approved by the Higgs Discovery  Group (HDG).\n\n\\begin{table}\n  \\begin{center}\n    \\begin{tabular}{cccc}\\toprule\n      Data Period & p0 & p1-p17 & p18-p38 \\\\\\midrule\n      Production Version & 5.3.1 & 6.1.1 & 6.1.6p+ \\\\      \n      \\bottomrule \n    \\end{tabular}\n    \\caption[Data Code Production Version]{Production code versions used the different data periods.}\\label{tab:prod_prescription}\n  \\end{center}\n\\end{table}\n\n\n\n\n\\section{Monte Carlo Simulation}\\label{sec:mcGen}\n\nAn accurate Monte Carlo (MC) simulation of the physics processes of interest and of the detector response are a fundamental tool for most of the high energy physics experiments. A wide variety of MC samples is available to the CDF users and, as explained in Chapters~\\ref{chap:sel} and~\\ref{chap:bkg}, this analysis relies on MC both for signal evaluation and for part of the background estimate. \n\nThe simulation of an high energy hadron collision event proceeds through four independent phases:\n\\begin{description}\n\\item[Parton Density Function Application:] quarks and gluons are the initial states of any interaction. Unluckily they are {\\em confined} by the strong interaction within the $p$ and the $\\bar{p}$ making impossible a perfect knowledge of the initial parameters. The Particle Density Functions (PDFs) overcome this problem by giving a parametrization of the interaction probability as a function of the momentum transfer ($Q^2$) during the $p\\bar{p}$ collision. When a MC event generator starts the evaluation of a process, the first step is the extraction of two partons of given $Q^2$ from the PDFs. Several parametrization of the PDFs exist but all the MCs used in this analysis employ the CTEQ5L~\\cite{cteq5l} PDFs. Systematic variations are obtained by using different  PDFs sets and varying the prediction within the theoretical uncertainties.\n\n\\item[Event Generation:] once that two initial partons are extracted from the PDFs functions, a hard interaction between them is calculated to obtain the simulation of the desired final state.\nWithin the perturbative approximation~\\cite{qftbook_weinberg}, the Matrix Element (ME) equation is derived with numerical integration of Leading-Order (LO) or Next-Leading Order (NLO) equations. The softwares performing such calculations are named {\\em generators}. Here we exploit a variety of them depending of the different final states: \\verb'ALPGEN'~\\cite{alpgen}, v2.1, for $W+$jets and $Z+$Jets prediction, \\verb'PYHTIA'~\\cite{pythia} v6.216 for $WW$, $WZ$, $ZZ$ and $t\\bar{t}$ prediction and, finally, \\verb'POWHEG'~\\cite{powheg1,powheg2, powheg3} v6.510, for single-top $s$ and $t$ channel NLO prediction.\n\n\\item[Parton Shower:] away from the hard interaction, higher order QCD processes are needed. The perturbative approach breaks and simulation is based on analytical parametrization and QCD models. Parton Shower (PS) programs like \\verb'PYHTIA' and \\verb'HERWIG'~\\cite{herwig} are used to simulate quark hadronization, soft gluon emission or underlying-event processes (i.e. secondary soft interactions and spectator quark interactions).\nPartons are evolved until they form real final state particles that can be undergo physical interaction in the detector. \n\n\\item[Detector Simulation:] the MC receives, as input, the positions, the four-momenta, and the identities of all particles and it reproduces the response of the different sub-detectors, including resolution effects, passage through passive material (such as cables or support structures) and secondary decays. CDF uses the \\verb'GEANT3'~\\cite{geant}, V.3.15, program to model the tracking volume of the detector. A mathematical model is used with full  simulation of charged particles passage, showering and secondary or tertiary particle production. The calorimeter section is not completely simulated because it would be too much time and CPU consuming. A much faster parametric response program, called \\verb'GFLASH'~\\cite{geant4}, tuned on test beam data, is used.\n\\end{description}\nWhen all the simulation is completed, the MC events are saved into a data structure identical to the one used for collision data, thus allowing reconstruction algorithms to work in the same way on data and MC events.\n\n\\chapter{Introduction}\n\nThe topic of this thesis is the measurement of $W$ and $Z$ bosons associate production in the lepton plus neutrino plus Heavy Flavor ($HF$) quarks final state:\n\\begin{equation}\n  p\\bar{p}\\to WW\/WZ \\to \\ell\\nu + HF,\n\\end{equation}\nidentified by the CDF II experiment at the Tevatron collider at $\\sqrt{s}=1.96$~TeV.\n\nThe associate production of the massive vector bosons $W$ and $Z$, as well as the different final states, are predicted by the Standard Model of the elementary particles (SM). The SM, briefly described in Chapter~\\ref{chap:physics}, is an extremely successful theory in which a minimal set of equations explains most of the known interactions. However the mechanism responsible for the mass of the particles still needs to be fully proved by the discovery of the (predicted) Higgs boson.\n\nThe CDF II experiment, described in Chapter~\\ref{chap:detector}, set tight constraints on the existence of the Higgs boson. Most of the sensitivity for the low mass Higgs boson production comes from:\n\\begin{equation}\n  p\\bar{p}\\to WH\\to \\ell\\nu+b\\bar{b}.\n\\end{equation}\nThis happens to be extremely similar to the process studied in this thesis\\footnote{I personally contributed to the $p\\bar{p}\\to WH\\to \\ell\\nu+b\\bar{b}$ CDF result as a developer of the common analysis framework described in Appendix~\\ref{chap:AppWHAM}.}, thus it allows to test, on a well known physics process, the correctness of the analysis procedure used in the Higgs search. \n\nHowever, the observation of diboson production in the  $\\ell\\nu + HF$ final state is not a simple task and, before this work (in particular its preliminary version described in Appendix~\\ref{App:7.5} and presented in 2011), no evidence was observed at a hadron collider experiment.\n\nThe actual analysis procedure is divided in four steps: object identification, event selection, background estimate and statistical analysis. \n\nFirst, a set of advanced identification algorithms, described in Chapter~\\ref{chap:objects}, is exploited for the recognition of the final state objects: one charged lepton ($\\ell$), a neutrino ($\\nu$), and two high energy jets, of which at least one tagged by the identification of the secondary decay vertex produced by a $HF$ hadron. Key element of the selection, the $HF$-tagging allows an efficient identification of \\mbox{$Z\\to c\\bar{c}\/b\\bar{b}$} signal candidates together with $W\\to c\\bar{s}$ candidates.\n\n\nFurther specific event selection criteria, described in Chapter~\\ref{chap:sel}, are imposed to maximise the accepted signal events while keeping the background under control. An original technique, based on the support vector machine algorithm and described in Appendix~\\ref{chap:AppSvm}, was developed to suppress the multi-jet events, a background, difficult to model, due to events in which no real $W\\to \\ell\\nu$ decay is present.\n\n\nSuccessively, as described in Chapter~\\ref{chap:bkg}, the total background is estimated. A variety of methods are used for the different background sources. We exploit Monte Carlo information for several backgrounds (e.g. for top quark production), a completely data-driven approach for the multi-jet contamination and a combination of data and Monte Carlo information for the $W+$ jets background estimate.\n\nAfter the full selection a large irreducible background fraction (i.e. with the same final state signature) remains. In particular, the total $W+HF$ non-resonant production is estimated to be more than a factor twenty larger than the expected signal. The shape analysis of the di-jet invariant mass distribution, $M_{Inv}(jet1, jet2)$, allows the extraction of the combined diboson signal, but the separation between $WW$ and $WZ$ contributions is still not feasible due to the close mass of the $W$ and the $Z$ bosons. To overcome this last issue, we exploited the discriminative power of a flavor separation neural network (KIT-NN described in Section~\\ref{sec:kitnn}) to classify the single-tagged events according to their $c$ quark or $b$ quark origin. This was not necessary for the double-tagged selection as it is dominated only by events with two $b$ quarks in the final state.\n\nChapter~\\ref{chap:StatRes} describes the statistical analysis of the bi-dimensional distribution $M_{Inv}(jet1, jet2)$ $vs$ KIT-NN, of the single-tagged events, together with the simple $M_{Inv}(jet1, jet2)$ distribution, of the double-tagged events. This allowed both the measurement of the cross sections for the total diboson associate production processes ($WW+ WZ\/ZZ$) and to separate the $WW$ and $WZ\/ZZ$ contributions.\n\nA summary of the results and the conclusions are reported in Chapter~\\ref{chap:conclusions}.\n\n\\chapter{Physics Objects Identification}\\label{chap:objects}\n\nThe final state topology of this analysis presents four different high-$p_T$ physics objects reconstructed combining the data of several sub-detectors. The identification and reconstruction methodologies are introduced here and discussed more in depth in the rest of the Chapter.\n\nEvents are selected online by the three-level trigger system described in Section~\\ref{sec:trigger} with two trigger strategies (described more in detail in next Chapter): {\\em single high-$p_T$ lepton} triggers and {\\em multiple-objects} triggers. After this, the digitalized electrical pulses recorded by the CDF sub-detectors are analyzed to reconstruct the physics objects of interest: the primary vertex of the interaction, one charged lepton, missing transverse energy (signaling the escape of an undetected neutrino) and two high-$E_T$ jets containing a reconstructed secondary vertex that {\\em tags} the presence of a Heavy Flavor ($HF$) hadron decay.\n\nGiven the small expected signal yield, the efficient identification of charged lepton candidates is a key feature of the analysis. We identify three main lepton categories: electrons, muons and isolated tracks. Electrons are defined as electromagnetic energy clusters matched to a charged track, muons are defined by ionization deposits in the muon chambers ({\\em stubs}) matched to a charged track and, finally, isolated tracks are just high-quality charged tracks isolated from other detector activity. The variety of identification criteria, with a total of {\\em eleven} different selection algorithms, allows to identify a large fraction of the $W\\to\\ell\\nu$ decays: factorizing the trigger efficiency and the jet and neutrino selection, we estimate an approximate lepton acceptance of $35$\\% for electrons, $45$\\% for muons and $4$\\% for taus\\footnote{$\\tau$ identification is not enforced but leptonic and one prong $\\tau$ decays contributes to the selected lepton categories.}.\n\n$W$ selection is completed by the neutrino identification (Section~\\ref{sec:metObj}). A neutrinos is the only particle that leaves the experimental apparatus completely undetected, however its presence is revealed by a large imbalance in the total transverse energy of the event (MET or \\mbox{${\\not}{E_T}$} ), since the total transverse energy of the $p\\bar{p}$ interaction is expected to be zero.\n\nAnother key feature of the analysis is the identification of $HF$ jets (Section~\\ref{sec:jetObj}). Jets are the experimental signature of high momentum quark and gluon production that hadronizes in a narrow shower of particles. They appear as a energy deposit (clustered) in both electromagnetic and hadronic calorimeters.\nHeavy flavor quarks hadronize to meta-stable particles that can travel a distance away from the primary interaction vertex before they decay in other particles. The identification of a secondary decay vertex, displaced from the primary, is used to {\\em tag} the jet as coming from a $HF$, $b$ or $c$ quark, hadronization. The selection efficiency for an event containing a $b\\bar{b}$ quark pair is approximately $50$\\%, going down to about $12$\\% for $c\\bar{c}$ pairs; a residual contamination of $1\\div 2$\\% Light-Flavor ($LF$) quarks selection is also present. A Neural-Network flavor separator (named KIT-NN and described in Section~\\ref{sec:kitnn}) is also applied on each tagged jet to separate $b$ quarks from $c$ and $LF$ quark components, thus giving further discrimination between $W\\to cs$ and \\mbox{$Z\\to b\\bar{b}$} signals.\n\n\n\\section{Charged Tracks Reconstruction}\\label{sec:track}\n\nThe ability to detect and reconstruct charged particle trajectories is essential for particle identification and momentum reconstruction. Precise, high efficiency tracking is the first step in the lepton identification, moreover track reconstruction allows the measurement of the track impact parameter, thus the identification of secondary vertices.\n\nA charged particle moving in a uniform magnetic field ($\\vec{B}$ with $|\\vec{B}|=1.4$~T for CDF) produces an helicoidal trajectory that can be uniquely described by five parameters. At CDF we use (see Figure~\\ref{coordinate}):\n\\begin{itemize}\n\\item $C$: the half-curvature of the trajectory, $C\\equiv 1\/2qr$ with $r$ equal to the helix radius and $q$ the measured charge of the particle. It has\n  the same sign of the particle charge and it is related to the transverse momentum of the track:\n  \\begin{equation}\n    p_T=\\frac{cB}{2 |C|}\n  \\end{equation}\n  %\n\\item $d_0$: the impact parameter, i.e. the distance of closest approach in the transverse plane between the helix and the origin. It is defined as:\n  \\begin{equation}\n    d_0=q\\Big(\\sqrt{x_0^2+y_0^2}-r\\Big)\\mathrm{,}\n  \\end{equation}\n  where $x_0$ and $y_0$ are the coordinates of the center obtained by the projection of the helix on the transverse plane and $r=1\/2C$. The quality of $d_0$ measurement is often parametrized by the {\\em impact parameter significance} defined as $|d_0\/\\sigma_{d_0}|$.\n  %\n\\item $\\lambda$: the helix pitch, i.e. the cotangent of the polar angle between the track and the $z$-axis ($\\cot \\theta_0$). The longitudinal component of the momentum is given by:\n \\begin{equation}\n   p_z=p_T\\cot \\theta_0\\mathrm{.}\n \\end{equation}\n  %\n\\item $z_0$: the $z$ position of the track vertex.\n  %\n\\item $\\phi_0$: the azimuthal angle of the track at its vertex. \n\\end{itemize}\nThe helix is completely described by these five parameters. Indeed every point along the trajectory satisfies the following equations~\\cite{track_eq}:\n\\begin{eqnarray}\nx&=& r\\sin \\phi -(r-d_0)\\sin \\phi_0\\mathrm{,}\\\\\ny&=& -r\\cos \\phi +(r-d_0)\\cos \\phi_0\\mathrm{,}\\\\\nz&=& z_0 +s\\lambda\\mathrm{,}\n\\end{eqnarray}\nwhere $s$ is the length projected along the track, and $\\phi= 2Cs+\\phi_0$.\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.7\\textwidth]{PhysObj\/Figs\/coordinate}\n\\caption[Track Reconstruction Parameters]{CDF track parameters and coordinate system.}\\label{coordinate}\n\\end{center}\n\\end{figure}\n\nCDF exploits several tracking algorithms~\\cite{cdf_trk}, optimized for different detector regions, to derive the previously defined parameters.\nThe main ones, described in the following paragraphs, are: the Outside-In algorithm (OI), the Silicon-Stand-Alone (SiSA) algorithm and the Inside-Out (IO) algorithm.\n\n\\paragraph{Outside-In Algorithm}\n\nThe Outside-In is the most reliable of CDF tracking algorithms as it is based on COT coverage, efficient up to $|\\eta|\\simeq 1$.\n\nTrack pattern recognition starts in the COT outer layers (lower hit density) and proceeds through four steps: first\neach super-layer is searched for groups of three aligned hits that are fit to a straight line with the least squares method. Then the \ntracks are reconstructed from the information of the {\\em axial} super-layers that are linked by two different algorithms ({\\em segment linking} and \n{\\em histogram linking} algorithms~\\cite{hist_alg}). During the third step, the information of the stereo layers is added and the algorithm searches for the vertex of the track. As final step, a global refit of the track is performed taking into account corrections for the non-uniformity of the magnetic field and for the modeling of the electrons drift.\n\nAt second stage, the track found in the COT is propagated into the silicon system. A road around a track is defined using the uncertainties on the COT track parameters and silicon hits are added if they lie inside this predefined road. When a hit is added, the track parameters are recalculated and the search is performed again. The impact parameter resolution of COT + SVX tracks is found to be $\\sigma_{d_0}\\simeq 20~\\mu$m.\n\n\\paragraph{Silicon-Stand-Alone Algorithm}\n\nThe hits in silicon sub-detectors not used by OI tracking are available to the Silicon-Stand-Alone algorithm~\\cite{sisa}, it covers the region $|\\eta|<2$ with a small residual capability up to $|\\eta|\\simeq 2.8$. \n\nThe SiSA algorithm starts from a collection of at least four hits in the SVX II detector in the $r-\\phi$ plane (SVX has five axial layers, three $90^{\\circ}$ layers and two small angle layers) and fits the $C$, $d_0$ and $\\lambda$ parameters to obtain a projection of the helix on the transverse plane. Then the algorithm creates a 3-D seed track adding small angle hits and the primary vertex information. At this point the $90^{\\circ}$ stereo hits are added and a global refit is performed. \n\nSiSA tracks reconstructed using only SVX II have a poor resolution for high $p_T$ tracks so hits are searched in LOO and ISL with the SVX II track as seed. The track is refit if other layers can be added. However, the performances on momentum and impact parameter resolution are limited and indeed SiSA tracks are not used for secondary vertexing.\n\n\\paragraph{Inside-Out Algorithm}\n\nThe third tracking algorithm, the Inside-Out~\\cite{io}, tries to recover efficiency and $p_T$ resolution in the region $1.2<|\\eta|<1.8$ where the COT coverage is limited. SiSA tracks are used as seeds which are extrapolated to the COT inner cylinder. Matching hits in the COT are added, the track is refitted and all duplicates are removed.\n\n\n\n\\section{Primary Vertex Identification}\\label{sec:primVtx}\n\nPrecise identification of the primary interaction vertex (PV) is the very first step in the event reconstruction process. Due to the relatively long $\\sigma_Z$ of the beam ($\\sigma_z\\simeq 28$~cm), important correction to $E_T$ and \\cancel{E}$_T$ may be needed. Furthermore, PV position allows the individuation of displaced secondary vertices in an event, the signature of long living $HF$ hadrons.\n\nThe algorithm used to reconstruct primary vertices is \\verb'PrimVtx'~\\cite{primVtx}: a seed vertex is calculated as the average $z$ position measured during \ncollisions and is provided as input, then all tracks with $|z_{0}-z_{PV}|< 1$~cm, $|d_0|<1$~cm and $|d_0\/\\sigma_{d_0}|<3$ are collected and ordered in decreasing\n$p_T$. They are fitted to a new $3D$ vertex and the tracks with $\\chi^2 > 10$ are removed. The procedure is iterated until all accepted tracks have $\\chi^2< 10$.\nA quality index (see Table~\\ref{tab:pVtx}) is assigned to the primary vertex depending on track multiplicity and type: a quality $\\ge 12$ is required for primary interaction vertex reconstruction. \n\nThe PV position is defined by $\\left(x_{PV},y_{PV},z_{PV}\\right)$. Typical $x_{PV}$ and $y_{PV}$ are of the order of tens of microns while a cut of $|z_{PV}|\\leq 60$~cm (luminosity region fiducial selection) is applied to constrain the collisions in the geometrical region where the detector provides optimal coverage.\n\n\n  \\begin{table}[h]\n\\begin{center}\n    \\begin{tabular}{cc}\\toprule\n       Criterion & Quality Value\\\\\\midrule\n       Number Si -tracks$\\ge$3 & 1\\\\\n       Number Si -tracks$\\ge$6 & 3\\\\\n       Number COT-tracks$\\ge$1 & 4\\\\\n       Number COT-tracks$\\ge$2 & 12\\\\\n       Number COT-tracks$\\ge$4 & 28\\\\\n       Number COT-tracks$\\ge$6 & 60\\\\\n      \\bottomrule\n    \\end{tabular}\n    \\caption[Primary Vertex Quality Criteria]{Primary Vertex quality criteria: a quality $\\ge 12$ is required for primary interaction vertex reconstruction.}\\label{tab:pVtx}\n\\end{center}\n  \\end{table}\n\n\n\n\n\\section{Lepton Identification Algorithms}\\label{sec:lep_id}\n\nIn this section the specific lepton identification algorithms\\footnote{A lepton is, by the experimental point of view, an electron or a $\\mu$ with no distinction between a particle and its anti-particle. Also $\\tau$ leptonic decays can enter in the lepton sample but the algorithms are not optimized for them.} are discussed. We distinguish two {\\em tight electron} identification algorithms (CEM, PHX), two {\\em tight muon} identification algorithms (CMUP, CMX), six {\\em loose muon} identification algorithms (BMU, CMU, CMP, SCMIO, CMIO, CMX-NotTrig) and one isolated track identification algorithm (ISOTRK). \nFigure~\\ref{fig:split_alg} shows, for a $WZ$ MC, the detector $\\eta-\\phi$ coverage separately for each lepton identification category. Figure~\\ref{fig:all_alg} shows them for all the categories together. The eleven lepton identification algorithms are described in the following sections.\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=0.495\\textwidth]{PhysObj\/Figs\/lepSet1}\n\\includegraphics[width=0.495\\textwidth]{PhysObj\/Figs\/lepSet2}\\\\\n\\includegraphics[width=0.495\\textwidth]{PhysObj\/Figs\/lepSet3}\n\\includegraphics[width=0.495\\textwidth]{PhysObj\/Figs\/lepSet4}\n\\caption[Coverage, in the $\\eta - \\phi$ Plane, for the Lepton Identification Algorithms]\n        {Coverage of the detector $\\eta - \\phi$ plane for the different lepton identification algorithms evaluated on a $WZ$ Monte Carlo: CEM and PHX tight electrons (top left); CMUP and CMX tight muons (top right); BMU, CMU, CMP, SCMIO, CMIO, CMX-NotTrig loose muons (bottom left); ISOTRK isolated tracks (bottom right).}\\label{fig:split_alg}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=0.99\\textwidth]{PhysObj\/Figs\/lepSet0}\n\\caption[Combined coverage, in the $\\eta - \\phi$ Plane, for the Lepton Identification Algorithms Combined]\n{Coverage of the detector $\\eta - \\phi$ plane for all the combined lepton identification algorithms evaluated on a $WZ$ Monte Carlo.}\\label{fig:all_alg}\n\\end{center}\n\\end{figure}\n\n\n\n\\subsection{Electron Identification}\\label{sec:eleSel}\n\nA candidate electron is ideally defined as an energy deposit ({\\em cluster}) in the EM section of the calorimeters and a charged track matched to it. The track requirements removes most of the ambiguity due to photon EM showers. \n\nThe two electron identification algorithms used in this analysis are divided according to the CDF calorimeter segmentation in CEM (Central ElectroMagnetic object) and PHX (Plug electromagnetic object with pHoeniX tracking), for $|\\eta|<1.1$ and $1.2<|\\eta|<3.6$ respectively.\n\nThe CDF EM clustering algorithm~\\cite{ele_cal} works in a simple but efficient way. The physical space corresponding to the calorimeter towers is mapped in the $\\eta-\\phi$ plane and the algorithm creates two lists of the calorimeter towers ordered by decreasing energy measurement: the {\\em usable list} (working towers with energy $>100$~MeV) and the {\\em seed list} (towers with energy $>2$~GeV). Then, it takes the first seed tower and creates an $\\eta-\\phi$ cluster by adding the neighboring towers to form a $2\\times 2$ or $3\\times 3$ $\\eta-\\phi$ area. An EM cluster is found if:\n\\begin{equation}\n  E^{Had}\/E^{EM}<0.125\\textrm{,}\n\\end{equation}\nwhere $E^{Had}$ is the energy deposited in the hadronic calorimeter section and $E^{EM}$ is the corresponding quantity for the EM section. As final step, the $\\eta-\\phi$ centroid of the cluster is calculated and the used towers are removed from the list. The algorithm selects the next seed tower and iterates the process until there are no more seed towers available. \n\nThe $3\\times 3$ clustering is used for the CEM algorithm while the $2\\times 2$ clusters are used in the Plug region. A cluster is not allowed to cross the boundary between different sub-detectors. \n\nSeveral corrections are applied to reconstruct the final energy: lateral leakage, location inside the physical tower, on-line calibration and response curve drawn by the test beam data. Also the energy measured in the shower-max (CES, PES) and preshower (CPR, PPR) detectors is added to the final reconstructed energy. The shower-max profile is also compared to the calibration profiles of electrons or photons and, last but not least, it is used to measure the position of the EM shower centroid. \n\nBeyond the EM energy measurement, the calorimeter information is further exploited for a better particle identification. The following variables are used:\n\\begin{itemize}\n\\item $E^{Had}\/E^{EM}$ ratio: studies performed with candidate $Z^0\\to e^+e^-$ events~\\cite{w_z_prl} show that electrons detected in the central or in the plug region have a little deposit in the hadronic part of the calorimeter (Figure~\\ref{ele_deposit}). \n\\item {\\em Lateral shower sharing} variable, $L_{shr}$, compares the sharing of energy deposition between the towers in the CEM to the expected in true electromagnetic showers taken with test beam data:\n\\begin{equation}\n  L_{shr}=0.14\\sum_i \\frac{E_i^{adj}- E_i^{expect}}{\\sqrt{(0.14\\sqrt{E_i^{adj}})^2+(\\Delta E_i^{expect})^2}}\\mathrm{,}\n\\end{equation}\nwhere the sum is over the towers adjacent (\\emph{adj}) to the seed tower of the cluster, $0.14\\sqrt{E_i^{adj}}$ is the error on the energy measure and $\\Delta E_i^{expect}$ is the error on the energy estimate.\n\\item The $\\chi^2$ of the fit between the energy deposit and the one obtained from test beam data ($\\chi^2_{strip}$ for CEM and $\\chi^2_{towers}$ for PHX).\n\\end{itemize}\n\\begin{figure}[!h]\n\\begin{center}\n\\includegraphics[width=0.92\\textwidth]{PhysObj\/Figs\/ele_deposit_cen}\\\\\n\\includegraphics[width=0.445\\textwidth]{PhysObj\/Figs\/ele_hadem_plug}\n\\includegraphics[width=0.445\\textwidth]{PhysObj\/Figs\/ele_iso_plug}\n\\caption[Electron Identification Criteria for $Z^0\\to e^+e^-$ Candidate Events]{$E^{Had}\/E^{EM}$ (left) and isolation (right) distribution of central (top) and plug (bottom) calorimeter electron selection from unbiased, second legs of $Z^0\\to e^+e^-$ candidate events in data~\\cite{w_z_prl}.}\\label{ele_deposit}\n\\end{center}\n\\end{figure}\n\nFinally, a reconstructed track matched to the EM cluster is used to suppress photons in the central region. Due to poor tracking in the forward region, PHX electrons are defined by a different strategy that relies on the {\\em Phoenix} matching scheme~\\cite{phx_alg}. \n\nTracks candidate electrons in the central region must satisfy the following requirements:\n\\begin{itemize}\n\\item track quality: $\\ge 3$ ($\\ge 2$) COT Axial (Stereo) segments with at least five hits each associated with the track.\n\\item $E^{cluster}\/p^{trk}<2$: bremsstrahlung photons are emitted colinearly to the electron and energy is radiated in the same EM cluster, therefore the measured track momentum is lower than the original $p_T$ of the electron.\n\\item Track-shower matching: \n  \\begin{equation}\n   3.0<q \\Delta x<1.5\\textrm{~cm}\\quad\\textrm{and}\\quad\\Delta z<3\\textrm{~cm,}\n  \\end{equation}\n  where $\\Delta x=|x^{CES}-x^{trk}|$, $\\Delta z=|z^{CES}-z^{trk}|$ and $q$ is the charge. \n\\end{itemize}\n\nElectron candidates in the PEM region do not have any track matching requirement due to the limited coverage of the tracking system for $|\\eta|>1.1$. However to provide some fake-electron rejection a road is built starting from the $x, y$ position in the PES detector, the PV position and the curvature provided by the $E_T$ cluster. The Phoenix matching succeeds if at at last three hits in the silicon detectors are found, thus allowing the selection of a PHX electron candidate. This is not considered a {\\em real} track matching because only four points are used in the fit and it is impossible to reconstruct a five-parameter helix, moreover some events have a high density of silicon hits, increasing the ambiguity of the matching. \n\nThe other important identification requirement is the {\\em Isolation} or $IsoRel$, a variable describing how much calorimeter activity surrounds the lepton. It is defined as: \n\\begin{equation}\n  IsoRel\\equiv E_T^{iso}\/E_T^{cluster}<0.1,\\quad E_T^{iso}=E_T^{0.4}-E_T^{cluster},\n\\end{equation}\n$E_T^{0.4}$ is the energy collected by the calorimeters within a radius $\\Delta R=0.4$ from the centroid of the EM cluster. Isolation is used in analyses involving a $W^{\\pm}$ or $Z^0$ boson because the kinematic region allowed to leptons coming from the bosons decay is usually far from jets or other particles (see Figure~\\ref{ele_deposit}).\n\nTable~\\ref{tab:ele_alg} summarizes all the CEM and PHX identification criteria while the top left part of Figure~\\ref{fig:split_alg} shows the detector coverage of the two algorithms in the $\\eta - \\phi$ plane for a $WZ$ Monte Carlo.\n\\begin{table}\\begin{center}\n\\begin{tabular}{cc}\n\\toprule\nElectrons & Identification Cuts\\\\\n\\midrule\n& EM fiduciality \\\\\n& $E_T^{EM}>20$~GeV \\\\\n& $E^{Had}\/E^{EM}<0.055+0.0045 E^{EM}$ \\\\\n& $L_{shr}<0.2$ \\\\\n& $\\chi^2_{\\mathrm{{CES strip}}}<10$ \\\\\nCEM &$\\ge 3$ axial $\\ge 2$ stereo COT Layers \\\\\n& $\\Delta z(CES,trk)<3$~cm \\\\\n& $-3.0<q \\Delta x(CES,trk)<1.5$~cm \\\\\n&$E\/p^{trk}<2.0$ or $E_T^{\\mathrm{cluster}}>100$~GeV \\\\\n&$|z_0^{trk}-z_0^{PV}|<5$~cm \\\\\n& $|z_0^{trk}|<60$~cm \\\\\n&$IsoRel<0.1$ \\\\\n\\midrule\n& $1.2<|\\eta^{PES}|<2.8$ \\\\\n& $E_T^{EM}>20$~GeV \\\\\n& $E^{Had}\/E^{EM}<0.05$ \\\\\n& PEM$3\\times 3$Fit $\\ne 0$ \\\\\n& $\\chi^2_{PEM3\\times 3} <10$ \\\\\nPHX & $PES5\\mathrm{by}9U>0.65$ \\\\\n& $PES5\\mathrm{by}9V>0.65$ \\\\\n& $\\Delta R(PES,\\mathrm{cluster})<3$~cm \\\\\n& N Si Hits $\\ge 3$ \\\\ \n& $\\mathrm{Phoenix~Track}$\\\\\n&$|z_0^{PHX}-z_0^{PV}|<5$~cm \\\\\n& $|z_0^{PHX}|<60$~cm \\\\\n&$IsoRel<0.1$ \\\\\n\\bottomrule\n\\end{tabular}\n\\caption[CEM and PHX Electron Selection Requirements]{Summary of CEM and PHX \\emph{tight electron} selection requirements.}\\label{tab:ele_alg}\n\\end{center}\\end{table}\n\n\n\n\\subsection{Muon Identification}\n\nA muon behaves like a minimum ionizing particle due to its rest mass, which is about 200 times larger than the electron one. Therefore muons deposit very little energy in the calorimeter systems and can leave a signal in the outer layer of the CDF detector which is instrumented with arrays of gas detector and scintillators (muon chambers). Ionization deposits from a muon candidate in a given muon detector constitute a {\\em stub} and the candidate naming depends by the coverage of the muon detector that records them (see Section~\\ref{sec:mu_cham}). \n\nThe basic selection for a muon candidate~\\cite{muon_id} is a high quality COT track pointing to a Minimum Ionizing Particle (MIP) energy deposit in the EM and HAD calorimeters and matched to a stub in the muon chambers. The precise requirements are the following:\n\\begin{itemize}\n\\item track quality: the reconstructed COT track must have a minimal amount of axial ($\\ge 3$) and stereo ($\\ge 2$) COT super-layers;\n\\item the $\\chi^2_{trk}$ returned by the track fitting algorithm should be less than $2.3$;\n\\item track\/stub matching in the $\\phi$-plane is required by an appropriate $\\Delta x(trk,\\mathrm{stub}) = | x^{trk} - x^{\\mathrm{stub}}|$ with cut values different for each specific muon sub-detector\n\\end{itemize}\n\nIf we are interested to a muon coming from $Z$ or $W$ decay, we expect the muon to be isolated from other detector activity (as in the electron case). The isolation is defined exploiting the muon candidate momentum and MIP energy deposit in the calorimeter:\n\\begin{equation}\n  IsoRel\\equiv E_T^{iso}\/p_T<0.1;\\quad E_T^{iso}=E_T^{0.4}-E_T^{MIP},\n\\end{equation}\nwhere the $p_T$ is the COT track momentum and $E_T^{MIP}$ is the transverse energy deposited in the towers crossed by the track.\n\nThe two {\\em tight} muon reconstruction algorithms are the CMUP and CMX. The first covers the region $|\\eta|<0.6$ where a track is required to match stubs in both CMU and CMP muon detectors. The second covers the region $0.65<|\\eta|<1.0$: it requires a stub in the CMX muon detector and a minimum curvature of the COT track, $\\rho_{COT}$, of $140$~cm\\footnote{This last requirement  ensures appropriate efficiency of the CMX trigger.}. Table~\\ref{tab:central_mu_alg} summarizes all the CMUP and CMX identification criteria while the top right section of Figure~\\ref{fig:split_alg} shows the detector coverage of the two algorithms in the $\\eta - \\phi$ plane for a $WZ$ Monte Carlo.\n\\begin{table}\\begin{center}\n\\begin{tabular}{cc}\n\\toprule\nMuons & Identification Cuts\\\\\n\\midrule\n&$p_T^{trk}>20$~GeV$\/c$ \\\\\n&$E_T^{EM}<2+\\mathrm{max}\\big(0, ((p^{trk}-100)0.0115)\\big)$~GeV \\\\\n&$E_T^{Had}<6+\\mathrm{max}\\big(0, ((p^{trk}-100)0.028)\\big)$~GeV \\\\\n& $\\ge 3$ axial $\\ge2$ stereo COT Layers \\\\\nCMUP & $|d_0^{trk}|<0.2$~cm (track w\/o silicon) \\\\\nCMX & $|d_0^{trk}|<0.02$~cm (track with silicon) \\\\\n& $|z_0^{trk}-z_0^{PV}|<5$~cm \\\\\n& $|z_0^{trk}|<60$~cm \\\\\n& $\\chi^2_{trk}<2.3$\\\\\n&$IsoRel<0.1$ \\\\ \\midrule\n& CMU Fiduciality \\\\ \nCMUP & CMP Fiduciality \\\\\n& $\\Delta X_{CMU}(trk, \\mathrm{stub})<7$~cm \\\\\n& $\\Delta X_{CMP}(trk, \\mathrm{stub})<5$~cm \\\\\\hline\n& CMX Fiduciality \\\\ \nCMX & $\\rho_{COT}>140$~cm \\\\\n& $\\Delta X_{CMX}(trk, \\mathrm{stub})<6$~cm \\\\\n\\bottomrule\n\\end{tabular}\n\\caption[CMUP and CMX Muon Selection Requirements]{Summary of CMUP and CMX \\emph{tight muons} selection requirements.}\\label{tab:central_mu_alg}\n\\end{center}\\end{table}\n\n\n\n\n\\subsection{Loose Muons Identification}\\label{sec:LooseMuId}\n\nMuons can be faked by cosmic rays or hadrons showering deep inside the calorimeters or not showering at all, however the muon candidates usually have a very clean signature because of the many detector layers used to identify them. On the other hand the signal acceptance for the tight muon categories is geometrically limited, therefore a set of lower quality muon identification criteria, {\\em loose muons}, was developed. Loose muons were used in other analysis~\\cite{single_top} to increase the signal yield. These are the main requirements of the loose muons algorithms:\n\\begin{itemize}\n\\item BMU: forward isolated muons ($1.0<|\\eta|<1.5$) with hits in {\\em Barrel} muon chambers.\n\\item CMU: central isolated muons with hits {\\em only} in the CMU chambers and not in CMP.\n\\item CMP: central isolated muons with hits {\\em only} in the CMP chambers and not in CMU.\n\\item SCMIO: a good quality track matched to an isolated MIP deposit and a {\\em non-fiducial} stub in the muon detector (Stubbed Central Minimum Ionizing particle).\n\\item CMIO: a good quality track {\\em only} matched to an isolated MIP deposit and failing any other muon identification criteria (Central Minimum Ionizing particle).\n\\item CMX-NotTrigger: also named CMXNT, are isolated muon detected in the CMX chamber but which are {\\em not} triggered by the CMX specific triggers because of the geometrical limits of the COT ($\\rho_{COT}<140$~cm).\n\\end{itemize}\nTable~\\ref{tab:loose_alg} summarizes all the Loose muons identification criteria while the bottom left section of Figure~\\ref{fig:split_alg} shows the detector coverage of the six algorithms in the $\\eta - \\phi$ plane for a $WZ$ Monte Carlo.\n\n\n\\begin{table}\\begin{center}\n\\begin{tabular}{cc}\n\\toprule\nMuons & Identification Cuts \\\\\n\\midrule\n&$p_T^{trk}>20$~GeV$\/c$ \\\\\n&$E_T^{EM}<2+\\mathrm{max}\\big(0, ((p^{trk}-100)0.0115)\\big)$~GeV \\\\\nBMU &$E_T^{Had}<6+\\mathrm{max}\\big(0, ((p^{trk}-100)0.028)\\big)$~GeV \\\\\nCMU & $|d_0^{trk}|<0.2$~cm (track w\/o silicon) \\\\\nCMP & $|d_0^{trk}|<0.02$~cm (track with silicon) \\\\\nSCMIO& $|z_0^{trk}-z_0^{PV}|<5$~cm \\\\\nCMIO& $|z_0^{trk}|<60$~cm \\\\\nCMXNT& $\\chi^2_{trk}<2.3$\\\\\n&$IsoRel<0.1$ \\\\ \\midrule\n\n& BMU Fiduciality \\\\ \nBMU & $\\Delta X_{BMU}(trk, \\mathrm{stub})<9$~cm \\\\\n& COT Hits Frac. $>0.6$ \\\\\\midrule\n\n& CMU Fiduciality \\\\ \nCMU & $\\Delta X_{CMU}(trk, \\mathrm{stub})<7$~cm \\\\\n& $\\ge 3$ axial $\\ge 2$ stereo COT Layers \\\\\\midrule\n\n& CMP Fiduciality \\\\ \nCMP & $\\Delta X_{CMP}(trk, \\mathrm{stub})<7$~cm \\\\\n& $\\ge 3$ axial $\\ge 2$ stereo COT Layers \\\\\\midrule\n\n& Stub Not Fiducial  \\\\ \nSCMIO & $\\ge 3$ axial $\\ge 2$ stereo COT Layers \\\\\n&$E_T^{EM} + E_T^{Had} > 0.1$  \\\\\\midrule\n\n& No Stub  \\\\ \nCMIO & $\\ge 3$ axial $\\ge 2$ stereo COT Layers \\\\\n&$E_T^{EM} + E_T^{Had} > 0.1$  \\\\\\midrule\n\n\n& CMX Fiduciality \\\\ \nCMXNT & $\\Delta X_{CMX}(trk, \\mathrm{stub})<6$~cm \\\\\n& $\\ge 3$ axial $\\ge 2$ stereo COT Layers \\\\\n& $\\rho_{COT}<140$~cm, Not trigger CMX \\\\\n\\bottomrule\n\\end{tabular}\n\\caption[Loose Muons Selection Requirements]{Summary of the BMU, CMU, CMP, SCMIO, CMIO, CMX-Not-Trigger (CMXNT) {\\em Loose muons} selection requirements.}\\label{tab:loose_alg}\n\\end{center}\\end{table}\n\n\n\n\n\n\n\\subsection{Isolated Track Identification}\\label{sec:IsoTrk}\n\nThe isolated tracks are last lepton category used in this analysis. They are defined to be high-$p_T$ good quality tracks isolated from energy deposits in the tracking systems. The track isolation is defined as:\n\\begin{equation}\\label{TrackIsolation}\n\\rm{TrkIso}=\\frac{p_T({\\rm{trk\\ candidate}})}{p_T({\\rm{trk\\ candidate}})+\\sum p_T({\\rm{other\\ trk)}}}\\,\\rm{,}\n\\end{equation}\nwhere $p_T({\\rm{trk\\ candidate}}$ is the transverse momentum of the specific track we analyze (candidate) and $\\sum p_T({\\rm{other\\ trk)}}$ is the sum of the transverse momenta of all good quality tracks within a cone radius of 0.4 of the candidate track. The isolation requirement is necessary in order to ensure that the track corresponds to a charged lepton produced in a decay of a $W$ boson and it is not part of an hadronic jet. A track is fully isolated if $TrkIso = 1.0$, thus a cut value of $0.9$ is used in the analysis.\nA selection requirement of $\\Delta R>0.4$ between the track and any tight jet\\footnote{Central, high $E_T$ jets are classified as {\\em tight}: $|\\eta|<2.0$, $E_T^{cor}>20$ (see Section~\\ref{sec:jetObj}).} is also applied to remove jets with low track multiplicity.\n\nThe fact that the isolated track is not required to match a calorimeter cluster or a muon stub allows to recover real charged leptons that arrive in non-instrumented regions of the calorimeter or muon detectors, as seen in Figure~\\ref{fig:split_alg}. \n\nTable~\\ref{iso_tab} summarizes the criteria used to select good quality isolated tracks while the bottom right section of Figure~\\ref{fig:split_alg} shows the detector coverage of the algorithms in the $\\eta - \\phi$ plane for a $WZ$ Monte Carlo.\n\n\\begin{table}\\begin{center}\n\\begin{tabular}{cc}\n\\toprule\nTrack Leptons & Identification Cuts \\\\\n\\midrule\n&$p_T^{trk}>20$~GeV$\/c$ \\\\\n& $\\ge 24$ axial $\\ge 20$ stereo COT Hits \\\\\n& $|d_0^{trk}|<0.2$~cm (track w\/o silicon) \\\\\n& $|d_0^{trk}|<0.02$~cm (track with silicon)\\\\\nIsolated & $|z_0^{trk}-z_0^{PV}|<5$~cm \\\\\nTracks & $|z_0^{trk}|<60$~cm \\\\\n& $\\chi^2_{trk}$ probability $<10^{-8}$\\\\\n& N Si Hits $\\ge 3$ \\\\ \n&$TrkIso>0.9$ \\\\\n& $\\Delta R(trk, \\mathrm{tight~jets})>0.4$\\\\\n& $\\Delta \\phi(trk, \\mathrm{jet 1})>0.4$\\\\\n\\bottomrule\n\\end{tabular}\n\\caption[Isolated Tracks Selection Requirements]{Summary of ISOTRK \\emph{isolated track} leptons selection requirements.}\\label{iso_tab}\n\\end{center}\\end{table}\n\n\n\n\\section{Jet Identification}\\label{sec:jetObj}\n\nThe QCD theory tells us that the partons composing the (anti)proton can be treated perturbatively as free particles if they are stuck by an external \nprobe\\footnote{I.e. a lepton, a photon or a parton from a different hadron.} with sufficient high energy (so called {\\em hard scattering}). However partons resulting from the \ninteraction can not exist as free particles because at longer distances (i.e. lower energies) the \\emph{strong potential} can not be treated perturbatively and \npartons must form colorless hadrons. This process, called {\\em hadronization} or {\\em showering}, produces a {\\em jet}, i.e. a narrow spray of stable particles that retains the information of the initial parton (for a pictorial representation see Figure~\\ref{jet1}). \n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.6\\textwidth]{PhysObj\/Figs\/jet1}\n\\caption[Schematic Representation of a Hadronic Jet]{A parton originating from a hard scattering hadronizes and generates a\na narrow spray of particles identified as a {\\em jet}.}\\label{jet1}\n\\end{center}\n\\end{figure}\n\nFrom an experimenter's point of view a jet is defined as a large energy deposit in a localized area of the detector (see Figure~\\ref{lego_jet}). The challenge of a physics analysis\nis to recover from detector information the initial energy, momentum and, possibly, the kind of the parton produced in the original interaction. \n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.85\\textwidth]{PhysObj\/Figs\/lego_jet}\n\\caption[Event Display with Jet Deposits]{Calorimetric deposit in the $\\eta-\\phi$ plane as represented in the CDF event display. EM deposits are red while HAD deposits are blue.}\\label{lego_jet}\n\\end{center}\n\\end{figure}\nA {\\em jet identification algorithm} is a tool to reconstruct such information and it should satisfy at best the following requirements~\\cite{jet_req}:\n\\begin{itemize}\n\\item\\emph{Infrared safety}: the presence of soft radiation between two jets may cause a merging of the two jets. This should not occur to avoid an incorrect parton attribution.\n\\item\\emph{Collinear safety}: the jet reconstruction should be independent from any collinear radiation in the event, i.e. different energy distribution of particles inside calorimetric towers.\n\\item\\emph{Invariance under boost}: the same jets should be found independently from boosts in longitudinal direction.\n\\item\\emph{Boundary stability}: kinematic variables should be independent from the details of the final state.\n\\item\\emph{Order independence}: the same reconstructed quantities should appear looking at parton, particle and detector levels.\n\\item\\emph{Straightforward implementation}: algorithm should be easy to implement in perturbative calculations.\n\\end{itemize}\nBeyond this theoretical aspects a jet algorithm should be experimentally efficient with a high reconstruction efficiency, good resolution and robust at high instantaneous luminosity.\n\n\n\\subsection{CDF Cone Algorithm}\\label{sec:jetCone}\n\nCDF uses several algorithms, none of them completely satisfying all the above requirements. The most common one, that is also the one used in this ana\\-lysis,\nis \\verb'JETCLU'~\\cite{jetclu}, an iterative fixed cone jet reconstruction algorithm based only on calorimetric information.\n\nThe algorithm starts by creating a list of the seed towers from all the calorimeter towers with transverse energy above the threshold of $1$~GeV. \nStarting with the highest-$E_T$ seed tower, a precluster is formed by combining\ntogether all adjacent seed towers within a cone of given radius $R$ in the $\\eta-\\phi$ space\\footnote{In this analysis we use $R=0.4$.}. This procedure is repeated, starting with the next unused seed tower, until the list is exhausted. The $E_T$-weighted centroid is then formed from the towers in the precluster and a new cone of radius $R$ is formed around this centroid. All towers with energy above the lower threshold of $100$~MeV within this new cone are added to the cluster. Then, a new centroid is calculated from the set of towers within the cluster and a new cone drawn. This process is iterated until the centroid of the energy deposition within the cone is aligned with the geometric axis of the cone (stable solution).\n\nThe initial clusters found can overlap so the next step is to merge or separate overlapping clusters, since each tower may belong only to one jet, each particle should not be assigned to more than one jet. Two clusters are merged if the total energy of the overlapping towers is greater than $75\\%$ of the energy of the smallest cluster. If the shared energy is below this cut, the shared towers are assigned to the cluster that is closer in $\\eta-\\phi$ space. The process is iterated again until the list of clusters remains fixed. \n\nThe final step of the jet identification happens on a second stage, after electron candidate identification and it is called {\\em reclustering}. If an EM calorimeter cluster is found to be compatible with a tight electron identification (CEM, PHX both isolated or not), the EM calorimeter towers are removed and the jet clustering algorithm is iterated.\n\nMassless four-vector momenta are assigned to the towers in the clusters for EM and HAD components with a magnitude equal to the energy deposited in the tower and the direction defined by a unit vector pointing from the event vertex to the center of the calorimeter tower at the depth that corresponds to the shower maximum. A cluster four-vector is then defined summing over the towers in the cluster:\n\\begin{eqnarray}\n&& E=\\sum_{i=1}^N(E_i^{EM}+E_i^{HAD})\\mathrm{,}\\\\\n&& p_x=\\sum_{i=1}^N(E_i^{EM}\\sin\\theta_i^{EM}\\cos\\phi_i^{EM}+E_i^{HAD}\\sin\\theta_i^{HAD}\\cos\\phi_i^{HAD})\\mathrm{,}\\\\\n&& p_y=\\sum_{i=1}^N(E_i^{EM}\\sin\\theta_i^{EM}\\sin\\phi_i^{EM}+E_i^{HAD}\\sin\\theta_i^{HAD}\\sin\\phi_i^{HAD})\\mathrm{,}\\\\\n&& p_z=\\sum_{i=1}^N(E_i^{EM}\\cos\\theta_i^{EM}+E_i^{HAD}\\cos\\theta_i^{HAD})\\mathrm{.}\n\\end{eqnarray}\nwhere the index $i$ runs over the towers in the cluster.\n\nIn order to study jet characteristics, other variables (number of tracks, energy deposited in the HAD and EM calorimeters, etc.) are reconstructed and associated to the final jet analysis-object.\n\n\n\n\\subsection{Jet Energy Corrections}\n\nThe ultimate goal of the jet reconstruction algorithm is the determination of the exact energy of the outgoing partons coming from the hard interaction, i.e. the Jet Energy Scale (JES). Clearly many factors produce a mismatch between the raw energy measured by the algorithm and the one of the parton before the hadronization. \n\nCDF developed a set of jet energy corrections depending of $\\eta$, $E_T^{raw}$ and $R$ of the jet reconstructed by \\verb'JETCLU' algorithm. The corrections are divided into five levels\\footnote{The actual naming skips $L2$, because it is absorbed in $L1$, and $L3$, as it was introduced as a temporary MC calibration in Run II.} ($L$-levels) that can be applied in a standard way to different analyses~\\cite{jet_corr1, jet_corr2}: $\\eta$-dependent response($L1$), effect of multiple interactions ($L4$), absolute energy scale ($L5$), underlying event ($L6$) and out-of-cone ($L7$) corrections. The correction $L1$ and $L5$ are multiplicative factors ($f_{L1}$ and $f_{L5}$) on the raw $E_T$ of the jet, the others are additive constants ($A_{L4}$, $A_{L6}$ and $A_{L7}$). The equation for the complete correction is:\n\\begin{equation}\\label{eq:fullCor}\nE_T^{FullCor}(\\eta, E_T^{raw}, R)=(E_T^{raw} f_{L1}-A_{L4})f_{L5}-A_{L6}+A_{L7}.\n\\end{equation}\nA description of each term is given in the following paragraphs while Figure~\\ref{totsys} shows the separate systematic uncertainties ($\\sigma_{JES}$) associated to each term of Equation~\\ref{eq:fullCor}.\n\nRecent studies~\\cite{gluonJES_10829} have shown that the simulated detector response to jets originating from high-$p_T$ gluons is improved by lowering the reconstructed $E_T$ by two times the uncertainty used in Equation~\\ref{eq:fullCor} ($-2\\sigma_{JES}$). More accurate studies are ongoing within the CDF collaboration to fully understand the effect of a {\\em parton dependent} jet energy correction. The $-2\\sigma_{JES}$ gluon-jets prescription  has been applied also to this analysis, however the result is  negligible due to the small fraction of gluon-jets present in the $HF$-enriched signal region.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.99\\textwidth]{PhysObj\/Figs\/totsys2}\n\\caption[Jet Energy Scale Systematic Uncertainty]{Jet $p_T$ dependence of the systematic uncertainties ($\\sigma_{JES}$) relative to the different terms of Equation~\\ref{eq:fullCor} for the \\texttt{JETCLU} algorithm with $R=0.4$ cone size.}\\label{totsys}\n\\end{center}\n\\end{figure}\n\nIn this analysis, like in many others~\\cite{single_top, wh94_note10796}, we choose to use jet corrections only up to L5. Therefore we define the Level-5 only correction:\n\\begin{equation}\nE_T^{cor}(\\eta, E_T^{raw}, R)=(E_T^{raw} f_{L1}-A_{L4})f_{L5}.\n\\end{equation}\n\nL6 and L7 corrections are fundamental for the measurement of the unknown mass of a particle in the hadronic final state (for example top quark mass measurements) but they are much less relevant in a cross section measurement where only the relative data\/MC energy scale matters  and not its absolute value. Furthermore, Figure~\\ref{totsys} shows that a large uncertainty is associated to the out-of-cone correction for $R=0.4$ jets, which would decrease the di-jet invariant mass resolution, lowering the sensitivity of the present measurement.\n\nDepending on, L5 corrected, $E_T^{cor}$ and the jet centroid position in the detector, jets are classified as {\\em tight} or {\\em loose}:\n\\begin{itemize}\n\\item{\\bf Tight:} $|\\eta^{det}|<2.0$, $E_T^{cor}>20$~GeV.\n\\item{\\bf Loose:} $|\\eta^{det}|<2.4$, $E_T^{cor}>12$~GeV and the jet is not tight.\n\\end{itemize}\nEvent selection (see Section~\\ref{sec:jetSel}) is based on the number of tight jets while both tight and loose jets are used in the \\mbox{${\\not}{E_T}$}  correction. Jet clusters of lower energy are not analyzed, it is assumed that they produce a negligible noise on the top of the unclustered energy of the event.\n\n\n\n\n\n\\paragraph{Level-1: $\\eta$ Dependent Corrections}\n\n$L1$ correction is applied to raw jet energy measured in the calorimeters to \nmake the detector response uniform in $\\eta$, it takes into account aging of\nthe sub-detectors\\footnote{This was the $L2$ correction during Run I} and other \nhardware non-uniformities (for example the presence of cracks).\nThis correction is obtained using a large di-jet sample:\nevents with one jet (\\emph{trigger jet}) in the central region of the \ncalorimeter ($0.2<|\\eta|<0.6$), where the detector response is well known and \nflat in $\\eta$, and a second jet (\\emph{probe jet}), allowed to range anywhere\nin the calorimeter ($|\\eta|<3.6$). In a perfect detector the jets should be\nbalanced in $p_T$, a balancing fraction is formed:\n\\begin{equation}\nf_b\\equiv \\frac{\\Delta p_T}{p_T^{ave}}=\\frac{p_T^{probe}-p_T^{trigger}}{(p_T^{probe}+p_T^{trigger})\/2}\\mathrm{,}\n\\end{equation}\nthe average of $f_b$ in the analyzed $\\eta$ bin is used to define the $\\beta$\nfactor\\footnote{The definition of Equation~\\ref{beta_eq} has a average value equal\nto $p_T^{probe}\/p_T^{trig}$ but is less sensitive to presence of non-Gaussian \ntails in the usual $p_T^{probe}\/p_T^{trig}$ ratio.} (Figure~\\ref{eta_correction} \nshows the $\\beta$ distribution for different cone radii):\n\\begin{equation}\\label{beta_eq}\n\\beta\\equiv\\frac{2+<f_b>}{2-<f_b>}\\mathrm{.}\n\\end{equation}\nThe final $L1$ correction is defined as $f_{L1}(\\eta, E_T^{raw}, R)=1\/\\beta$ \nand reproduces an approximately flat response in $\\eta$ with an error varying \nfrom $0.5\\%$ to $7.5\\%$.\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.85\\textwidth]{PhysObj\/Figs\/eta_correction}\n\\caption[Jet Energy Correction as a Function of $\\eta$]{$\\eta$-dependence of $\\beta$ factor for cone radii $R=0.4$, $0.7$ and \n$1.0$, measured in the di-jet component of $\\mathtt{jet20}$\nsample.}\\label{eta_correction}\n\\end{center}\n\\end{figure}\n\n\\paragraph{Level-4: Multiple Interactions Corrections}\n\nJet energy measurement is also degraded by the presence of minimum-bias events\nthat come from multiple $p\\bar{p}$ interactions. This correction becomes more \nrelevant at high luminosity, indeed the number of $p\\bar{p}$ interactions is \nPoisson distributed with mean value approximately linear with instantaneous luminosity:\n\\begin{equation} \n\\langle N(\\mathscr{L}\\simeq 10^{32}~\\mathrm{cm}^{-2}\\mathrm{s}^{-1})\\rangle\\simeq3\\mathrm{,}\\quad \\langle N(\\mathscr{L}\\simeq 3\\cdot 10^{32}~\\mathrm{cm}^{-2}\\mathrm{s}^{-1})\\rangle\\simeq 8\\mathrm{.}\n\\end{equation}\nThe energy of particles coming from those processes is estimated from \nminimum-bias events drawing a cone in a random\nposition in the region $0.1<\\eta<0.7$. Figure~\\ref{mp_correction} shows that\nthe measured minimum-bias $E_T$ grows linearly with the number of primary \nvertices\\footnote{Good quality primary vertices are reconstructed through at \nleast 2 COT tracks.}, such quantity, $A_{L4}$, must be subtracted by jet raw \nenergy. The total uncertainty is about $15\\%$, it mostly depends on luminosity \nand event topology.\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.99\\textwidth]{PhysObj\/Figs\/mp_correction}\n\\caption[$E_T$ Correction due to Multiple Interactions]{$E_T$ correction deriving from multiple interactions as a function of primary vertex number for cones with $R=0.4$ (right) and $R=0.7$ (left).}\\label{mp_correction}\n\\end{center}\n\\end{figure}\n\n\\paragraph{Level-5: Absolute Energy Scale Corrections}\n\nWhile $L1$ and $L4$ make jet reconstruction uniform over the whole detector and\nover the global behavior of $p\\bar{p}$ beam interaction, $L5$ correction \n($f_{L5}$) aims to derive, from the detector jet energy measurement, the $E_T$ of particles originating the jet.\n\nThe study is MC driven: first jet events are generated with full CDF detector \nsimulation, then jets are reconstructed both at calorimeter and hadron\ngeneration levels (HEPG) with the use of same clustering algorithm. A \ncalorimeter jet (C) is associated to the corresponding hadron jet (H) if \n$\\Delta R <0.1$. For both HEPG and detector jets the transverse momentum, \n$p_T^C$ and $p_T^H$, is calculated. \nThe absolute jet energy is defined as $\\mathcal{P}(p_T^C|p_T^H)$, the probability\\footnote{Different \n$p_T^H$ can give the same $p_T^C$, in this case the maximum is taken.} to measure $p_T^C$ with a given $p_T^{H}$.\n\nFigure~\\ref{abs_correction} shows the correction factor $f_{L5}$ for different \ncone sizes as function of the different jet transverse energies. The total \nuncertainty is about $3\\%$ and it mainly arises from the determination of\ncalorimetric response to single particles and MC fragmentation modeling.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.85\\textwidth]{PhysObj\/Figs\/abs_correction2}\n\\caption[Absolute Jet Energy Correction as a Function of $p_T$]{Absolute jet energy scale correction and systematic uncertainty ($f_{L5}$) for \\texttt{JETCLU} algorithm with $R=0.4$ cone size.}\\label{abs_correction}\n\\end{center}\n\\end{figure}\n\n\\paragraph{Level-6 \\& Level-7: Underlying Event and Out-of-cone Corrections}\n\nAlthough we do not use L6 and L7 corrections, their description is reported for completeness. They are the last two corrections needed to infer the initial energy of the parton originating the jet. \n\nThe underlying event correction ($L6$) takes into account the interaction \nprocesses which can occur between spectator partons or that originates from initial \nstate radiation (usually soft gluon radiation) while the out-of-cone correction\n($L7$) considers the fraction of particles coming from the original parton that fall outside the jet cone. \n\nThe underlying event energy ($A_{L6}$) must be subtracted to the total jet \nenergy. It was measured studying minimum-bias events during Run I and is \nparametrized with a constant value that scale with the cone radius. Out of cone\nenergy ($A_{L7}$) must be added to the total jet energy, studies are carried \nout with the same jet-parton matching method of $L5$. \n\n\n\n\\section{Neutrino Reconstruction}\\label{sec:metObj}\n\nNeutrinos are the only subatomic particles that leaves the detector completely undetected, therefore their signature is {\\em missing energy}. \n\nSince the longitudinal energies of the colliding partons are unknown and not necessarily equal, we can only say that the total transverse energy of the $p \\bar{p}$ collision is zero. Therefore the total amount of missing transverse energy \\cancel{E}$_T$ (or MET) gives a measurement of the neutrino transverse momentum\\footnote{For a massless neutrino $p_{T}=E_{T}$.} and it is defined as:\n\\begin{equation}\\label{rawmet}\n\\vec{\\mathrm{\\cancel{E}}}_T\\equiv - \\sum_i{\\vec{E}_T^i}\n\\end{equation}\nwhere $\\vec{E}_T^i$ is a vector with magnitude equal to the transverse energy \ncollected by the i-th calorimeter tower and pointing from the interaction\nvertex to the center of the tower. The sum involves all the towers with total \nenergy above $0.1$~GeV in the region $|\\eta|<3.6$.\n\nThe \\mbox{${\\not}{E_T}$}  obtained from Equation~\\ref{rawmet}, close to the online reconstructed missing energy, is often referred to as {\\em raw} \\cancel{E}$_T$ (or \\mbox{${\\not}{E_T^{raw}}$}). The fully reconstructed \\mbox{${\\not}{E_T}$}  is corrected for the true $z$ vertex position, for the difference between the raw and corrected $E_T$ of the tight and loose jets, for the presence of muons in the event, by subtracting the momenta of minimum ionizing high-$p_T$ muons and adding back the transverse energy of the MIP deposit in the calorimeter towers. These corrections can be summarized in the following equation:\n\\begin{equation}\n\\mbox{${\\not}{E_T}$}  = \\mbox{${\\not}{E_T^{raw}}$} - \\sum_{\\text{muon}} p_T + \\sum_{\\text{muon}} (E_T^{EM} + E_T^{HAD}) - \\sum_{jet}(E_T - E_T^{cor}).\n\\end{equation}\n\n\n\n\\section{Secondary Vertex Tagging}\\label{sec:secvtx}\n\nThe algorithms able to select a jet coming from a Heavy-Flavor ($HF$) quark hadronization process are called {\\em $b$-taggers} or {\\em heavy-flavor taggers} and they are of fundamental importance in this and in many other analyses. For example both the top quark and the SM Higgs boson (for $m_H\\lesssim 135$~GeV$\/c^{2}$) have large branching fraction in $b$-quark, therefore an efficient $b$-tagging can dramatically reduce the background of uninteresting physical processes which contain only light-flavor ($LF$) hadrons in their final state.\n\nWe employ the Secondary Vertex Tagger\\footnote{Historically it was the most important \ncomponent in top discovery in 1995.} algorithm (\\verb'SecVtx') to select a $HF$ enriched sample by requiring a $b$-tag on one or both the selected jets. In this way, it is possible to discriminate the $W\\to cs$ and $Z\\to c\\bar{c}\/b\\bar{b}$ decays against the generic $W\/Z\\to u,d, s$ decays. \n\nIn a successive phase of the analysis a flavor separator Neural Network~\\cite{Richter_stop2007,kit_note7816}, developed by the Karsrhue Institute of Technology (also named KIT Flavor Separator), is used to separate jets originating from $c$ and $b$ quarks, allowing a separate measurement of $W\\to cs$ and $Z\\to c\\bar{c}\/b\\bar{b}$.\n\n\n\\subsection{The SecVtx Algorithm}\\label{sec:secvtx_alg}\n\nThe \\verb'SecVtx' algorithm takes advantage of the long life time of $b$-hadrons: a $c\\tau$ value of about $450~\\mu$m together with a \nrelativistic boost due to a momentum of several GeV$\/c$ permits to a $b$-hadron to fly several millimeters\\footnote{The average transverse momentum of a $b$-hadron coming from a $WH$ events is about $40$~GeV$\/c$ for a Higgs boson mass of $120$~GeV$\/c^2$; in that condition a neutral $B^0$ meson of mass $5.28$~GeV$\/c^2$ undergoes a boost $\\beta\\gamma=7.6$ and the average decay length is $3.5$~mm.} away from the primary interaction \nvertex. The relevant quantity is the $c\\tau$ which is approximately the average\nimpact parameter of the outgoing debris of $b$-hadron decays. \nThe decay produces small sub-jets composed by tracks with large impact \nparameter ($d_0$). The silicon detectors (see \nsection~\\ref{track_par}) are able to reconstruct $d_0$ with adequate \nprecision to separate displaced tracks from the prompt tracks coming from the \nprimary interaction. Figure~\\ref{svx_example} shows as a $W+$ jets candidate\nevent with two displaced secondary vertices is identified by \\verb'SecVtx' and \nreconstructed by the CDF event display.\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.7\\textwidth]{PhysObj\/Figs\/svx_example}\n\\caption[$W +$ Jets Candidate Event with Two Reconstructed Secondary Vertices]{$W+$ jets candidate event with two secondary vertices tagged by \n$\\mathtt{SecVtx}$ (run 166063, event 279746). The \\cancel{E}$_T$ direction, a \nmuon track, a prompt track and tracks from the secondary vertices are shown.}\\label{svx_example}\n\\end{center}\n\\end{figure}\n\nTagging is performed for all the jets with $|\\eta|<2.4$ in an event. The algorithm searches for secondary vertices using the tracks within the jet cone of radius \\mbox{$\\Delta R=0.4$} . The usable tracks must satisfy the following requirements:\n\\begin{itemize}\n\\item $p_T>0.5$~GeV$\/c$;\n\\item $|d_0|<0.15$~cm and $|d_0\/\\sigma_0|>2.0$;\n\\item $|z_0-z_{PV}|<2.0$~cm;\n\\item have a minimum number (depending on track reconstruction quality and position) of hits in the silicon detector;\n\\item be seeded or confirmed in the COT;\n\\end{itemize}\na {\\em taggable} jet is defined as a jet containing at least two usable tracks.\n\nThe algorithm works on a two step basis and has two main operation modes\\footnote{An {\\em ultra-tight} operation mode exists but it is rarely used.}, {\\em tight} (the standard one) and {\\em loose}. The operating modes are defined by track and vertex quality criteria~\\cite{secvtx_opt7578} but the two-step selection algorithm remains identical.\n\n\nIn the \\emph{Pass 1} at least three tracks are required to pass loose selection\ncriteria. At least one of the tracks used is required to have $p_T > 1.0$~GeV$\/c$. The selected tracks are \ncombined two by two until a seed secondary vertex is reconstructed, then all \nthe others are added one by one and a quality $\\chi^2$ is computed. Tracks are \nadded or removed depending of their contribute to the $\\chi^2$.\n\nThe \\emph{Pass 2} begins if \\emph{Pass 1} does not find a secondary vertex. Now only \ntwo tracks are required to form a secondary vertex but they must pass tighter requirements: $p_t>1.0$~GeV$\/c$, $|d_0\/\\sigma_0|>3.5$ and one of the tracks must have $p_T > 1.5$~GeV$\/c$.\n\nIf a secondary vertex is identified in a jet, the jet is {\\em tagged}. The bi-dimensional decay length $L_{xy}$ is calculated as the projection on the jet axis, in the $r-\\phi$ \nplane, of the \\verb'SecVtx' vector, i.e. the one pointing from the primary vertex \nto the secondary. The sign of $L_{xy}$ is defined by the angle $\\alpha$ between\nthe jet axis and the \\verb'SecVtx' vector. Figure~\\ref{fig:svx} explains the \ngeometry.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.99\\textwidth]{PhysObj\/Figs\/svx}\n\\caption[Reconstructed Secondary Vertex: True and Fake $\\mathtt{SecVtx}$]{Schematic representation of $\\mathtt{SecVtx}$ variables and geometry. Left: true reconstructed secondary vertex. Right: negative $\\mathtt{SecVtx}$ tag, falsely reconstructed secondary vertex.}\\label{fig:svx}\n\\end{center}\n\\end{figure}\n\nA secondary vertex coming from a $HF$ hadron is expected to have \nlarge $L_{xy}$. To reduce background due to mismeasured tracks \n\\mbox{$|L_{xy}\/\\sigma_{L_{xy}}|>7.5$} is required\\footnote{Negative $L_{xy}$ has\nno physical meaning but it is important to estimate the mistag probability due \nto resolution effects.}. Other requirements are applied on the \ninvariant mass of the pair of tracks, to avoid $K$ and $\\lambda$ decays, and on\nvertex multiplicity and impact parameter to reject secondary vertices due to \ninteraction with material inside the tracking volume.\n\n\n\\subsection{Tagging Performances and Scale Factors}\\label{sec:btag_sf}\n\nThe performances of a $b$-tagger are evaluated on its efficiency, i.e the rate \nof correctly identified $b$-hadrons over all the produced $b$-hadrons, and on \nits purity, i.e the rate of falsely identified $b$-hadrons in a sample with no \ntrue $b$-hadrons. CDF uses $b\\bar{b}$ QCD MC to evaluate \\verb'SecVtx' efficiency \nrelying on detector and physical processes simulation. Figure~\\ref{fig:tag_eff} \nshows the $b$-tagging efficiency as a function of jet $\\eta$ and $E_T$ for the \ntight and loose \\verb'SecVtx' operating modes. Tagging efficiency drops at large $|\\eta|$ because of tracking acceptance.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.99\\textwidth]{PhysObj\/Figs\/tag_eff}\n\\caption[\\texttt{SecVtx} Efficiency]{Efficiency of the  \\texttt{SecVtx} $b$-tagging algorithm for the tight and loose operating modes.}\\label{fig:tag_eff}\n\\end{center}\n\\end{figure}\n\nAs MC does not reproduce the exact $b$-tagging efficiency of \\verb'SecVtx' a {\\em Scale Factor} ($SF_{Tag}$) is introduced to account for data\/MC difference in the form:\n\\begin{equation}\nSF_{Tag}\\equiv\\frac{\\epsilon_{data}}{\\epsilon_{MC}}.\n\\end{equation}\n\n\nCDF uses two methods to calculate $SF_{Tag}$: both use di-jet samples where one of the two jets contains a low-$p_T$ lepton, electron or muon, which increases the presence of a $HF$ hadron decay, while the other jet can be $b$-tagged by \\verb'SecVtx' or not. \n\nThe method exploiting the electrons~\\cite{eleSF_10178} finds algebraically the relative number of  $b$-tagged $HF$-jets with respect to the number of $b$-jets with a semileptonic electron decay. To extract the fraction, it also needs the number of jets with a low-$p_T$ electron but not containing $HF$ hadrons: this is obtained from a jet sample where the $\\gamma\\to e^+e^-$ conversion process is identified, and therefore, with depleted  $HF$ content. \n\nThe second method, used as a validation of the first one, exploits the muon semileptonic decay of the $HF$ hadrons~\\cite{muSF_8640}. The algorithm is similar to the electron based one, but the fraction of  low-$p_T$ muons not coming from $HF$ hadrons is extracted from a fit of the muon $p_T$ relative to the jet axis (named $p_T^{rel}$) that has a peculiar distribution for $HF$ hadron decays. \n\nThe efficiency on data are therefore compared to the MC results and the $SF_{Tag}$ is extracted.\nFigure~\\ref{fig:tag_sf} shows the $SF_{Tag}$ dependency with respect to the jet $E_T$ as determined by the electron method.\nTable~\\ref{tab:tag_sf} reports the $SF_{Tag}$ for the loose and tight \\verb'SecVtx' operation modes integrated over the variables of interest for the $SF_{Tag}$ parametrization (no strong dependency is seen in any of them).\n The total per-jet tagging efficiency, deconvoluted from tracking effects is about $40$\\% for $b$-jets and $6$\\% for $c$-jets.\n\\begin{table}\\begin{center}\n\\begin{tabular}{cccc}\n\\toprule\n mode & $SF_{Tag}$ & stat. err. & sys. err.\\\\\n\\midrule\nTight &  $0.96$ & $0.01$ & $0.05$ \\\\\nLoose & $0.98$ & $0.01$ & $0.05$ \\\\\n\\bottomrule\n\\end{tabular}\n\\caption[\\texttt{SecVtx} Efficiency Scale Factors]{$\\mathtt{SecVtx}$ Scale Factors ($SF_{Tag}$) for the tight and loose $\\mathtt{SecVtx}$ operation modes.}\\label{tab:tag_sf}\n\\end{center}\\end{table}\n\\begin{figure}[!h]\n\\begin{center}\n\\includegraphics[width=0.99\\textwidth]{PhysObj\/Figs\/tag_sf}\n\\caption[\\texttt{SecVtx} Scale Factor]{Evaluation of the \\texttt{SecVtx} $SF_{Tag}$ with the electron method~\\cite{eleSF_10178} and its dependence over the $E_T$ of the jet. The linear fit shows almost no dependency from the $E_T$ of the jet. Table~\\ref{tab:tag_sf} reports the evaluation of $SF_{Tag}$ integrated over the variables of interest for the $HF-$tagger parametrization.}\\label{fig:tag_sf}\n\\end{center}\n\\end{figure}\n\nThe number of falsely \\verb'SecVtx' tagged jets is dubbed \\emph{mistags}.\nMistags can be due to track resolution, long living $LF$ hadrons or \nsecondary interactions with detector material.\n\nThe rate of $W$ plus mistag jets is derived from a sample of events collected with \nan inclusive jet-based trigger with no $HF$ requirement\\footnote{The presence of a \nsmall $HF$ contamination in the sample is a source of systematic uncertainty.}. \nThe mistag parametrization~\\cite{mistag} is obtained from an inclusive jet sample using {\\em negative tags} (see Figure~\\ref{fig:svx}), i.e. $b$-jets which appear to travel back toward the \nprimary vertex. Resolution and material effects are expected to produce false \ntags in a symmetric pattern around the primary interaction vertex. The mistag \nrate is then corrected for the effects of long-lived $LF$ hadrons to take\ninto account the LF contamination giving real secondary vertices. \nA {\\em per-jet} mistag probability, $p^j_{Mistag}$ is parametrized in bins of:\n\\begin{itemize}\n\\item total number of jets in the event, \n\\item jet $E_T$, \n\\item jet $\\eta$, \n\\item track multiplicity within the jet, \n\\item total $E_T$ of the event, \n\\item number of interaction vertices,\n\\item the $z$ vertex position.\n\\end{itemize}\nThis defines the, so called, {\\em Mistag Matrix}. Figure~\\ref{fig:tag_mistag} shows the mistag rate as function of $E_T$ and $\\eta$ of the tagged jets for the tight and loose \\verb'SecVtx'\noperation modes. The approximate per-jet fake rate is about $1$\\% for the tight operation mode and $2$\\% for the loose one.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.99\\textwidth]{PhysObj\/Figs\/tag_mistag}\n\\caption[$E_T$ and $\\eta$ Mistag Rate]{Rate of wrongly $\\mathtt{SecVtx}$ tagged jets (\\emph{mistags}) as a function of $E_T$ and $\\eta$ of the jets for the tight and loose $\\mathtt{SecVtx}$ operation modes. The rate is derived from an inclusive jet sample.}\\label{fig:tag_mistag}\n\\end{center}\n\\end{figure}\n\n\\section{Neural Network Flavor Separator (KIT-NN)}\\label{sec:kitnn}\n\nThe $b$-tag requirement drastically reduces the contamination of $LF$ jets, however also a per-jet fake rate of $1$\\% can produce enough mistag background to decrease the sensitivity to rare processes. On the other hand a more stringent $b$-tagging requirement would reduce the signal yield. The KIT Flavor Separator Neural Network~\\cite{Richter_stop2007,kit_note7816} (KIT-NN) offers a possible solution to these problems: after a \\verb'SevVtx' tag, the Neural-Network (NN) exploits a broad range of $b$ quark discriminative variables to obtain a continuous distribution with a good separation power between real $b$-jets and $LF$ jets (see Figure~\\ref{fig::kit_nn_out}). \n\n\\begin{figure}[!h]\n\\begin{center}\n\\includegraphics[width=0.99\\textwidth]{PhysObj\/Figs\/kit_yield}\n\\caption[KIT-NN Output Distribution and $b$-jet Efficiency\/Purity Curve]{Left: Output value of the KIT-NN function for $b$, $c$ and $LF$ jets. Right: curve of the $b$-jet efficiency {\\em vs} $b$-jet purity for different selection thresholds of the KIT-NN output.}\\label{fig::kit_nn_out}\n\\end{center}\n\\end{figure}\nBeyond the $LF$ separation power, in this analysis we exploit the KIT-NN distribution in an original way. Thanks to the substantial shape difference between $c$-jets and $b$-jets KIT-NN values, we managed to measure separately $W\\to cs$ and $Z\\to c\\bar{c}\/b\\bar{b}$ contributions (as shown in Chapeter~\\ref{chap:StatRes}).\n\nA short description of the KIT-NN implementation, developed for the single-top search at CDF~\\cite{single_top}, can be useful to understand the physical meaning of such variable. \n\nA Bayesian NN is a supervised learning algorithm~\\cite{BShop_machine_learning} that associates a score to an {\\em event}, according to its likeliness to be {\\em signal} or {\\em background}. An event is defined by an array of $K$ input variables, $\\alpha_i$, and the ouput score, $o \\in [-1,1]$, is obtained from the following function:\n\\begin{equation}\n  o =S  \\Big( \\sum^H_{j=0} \\omega_j S \\Big( \\sum_{i=0}^{K}\\omega_{i,j} \\alpha_i +\\mu_{0,j} \\Big) \\Big),\n\\end{equation}\nwhere $\\omega_i$, $\\omega_{i,j}$ and $\\mu_{0,j}$ are tunable parameters and $S$ is a sigmoid, or activation, function:\n\\begin{equation}\n  S(x) =\\frac{2}{1+e^{-x}} -1,\n\\end{equation}\nThe parameters are derived from a target function optimized over the training set of signal and background labeled events. \n\nThe final result is a per-event signal-or-background {\\em posterior probability} that takes into account non-linear correlations between the input variables. \n\nIn our case, after that a jet has been tagged by \\verb'SevVtx', twenty-five input variables, relative to the secondary vertex and to the jet, are fed into a NeuroBayes\\textsuperscript{\\textregistered} NN~\\cite{neurobayes}. The training signal sample is composed by $t\\bar{t}$, single-top and $W+b\\bar{b}$ MC samples, while the background is built with $W+c\\bar{c}$ and $W+LF$ MC samples. The variables exploit lifetime, mass, and decay multiplicity of the $b$ hadrons using several characteristics of the identified secondary vertex, the properties of the tracks inside the jet and information from the $b$-tag algorithm, for example if {\\em Pass 1} or {\\em Pass 2} reconstruction is used (see Section~\\ref{sec:secvtx_alg}).\nFigure~\\ref{fig::kit_nn_out} shows the obtained classification values for  $b$, $c$ and $LF$ jets.\n\nAlthough a careful validation of the input variables was performed, the use of a MC-driven $LF$ training sample introduced a relevant discrepancy in the KIT-NN evaluation of {\\em real} mistagged jets. Therefore a  $LF$ correction function was derived from a fake-enriched data sample selected by negative $\\mathtt{SecVtx}$ tags. The final $LF$ KIT-NN distribution is obtained by assigning, to each $LF$-jet, a random value extracted from the corrected $LF$ distribution. The uncorrected template distribution is used as an {\\em optimistic} systematic variation. A similar effect has been hypothized also for $c$-jets but it appears in a less relevan way. For $c$-jets the KIT-NN output is evaluated per-jet each jet and the $LF$-like variation is used as a {\\em pessimistic} systematic variation. Figure~\\ref{fig:kit_sys}\nshows the default (central) KIT-NN distribution and systematic variations for $LF$ and $c$ jets.\n\n\\begin{figure}[!h]\n\\begin{center}\n\\includegraphics[width=0.99\\textwidth]{PhysObj\/Figs\/kit_sys}\n\\caption[KIT-NN Distribution and Systematic Variation for $LF$ and $c$ Jets]{KIT-NN default distribution (continuous lines) and systematic variations (dashed lines) for $LF$-jets (left) and $c$-jets (right).}\\label{fig:kit_sys}\n\\end{center}\n\\end{figure}\n\n\\chapter{Events Selection}\\label{chap:sel}\n\nIn the following we describe the selection requirements applied to produce samples of data enriched in $diboson\\to \\ell\\nu+ HF$ candidates.\n\nEvery physics analysis starts with some kind of candidate signal selection. In the case of collider experiments, it is possible to see three ingredients in it: online (trigger) selection (Section~\\ref{sec:onlineSel}), offline selection (Sections from~\\ref{sec:w_sel} to~\\ref{sec:svm_sel}) and the efficiency evaluation of each of the steps.\n\nThe variety of different physics object that compose the diboson decay channel under investigation is a characteristic of this analysis: charged leptons (electrons, muons and isolated tracks), missing energy, jets and $HF$ jets. These represent a good fraction of all the objects that can be reconstructed at a hadron collider experiment. Furthermore the need to relax the selection requirements in order to increase the statistics and the background-rich $W+2$ jets sample increase the challenges. It worths noticing that the same issues arise also in other primary analyses at CDF: for example single-top or $WH$ searchs.\n\nIn order to deal with these difficulties, one of the major task of this thesis work was the development of a robust analysis framework that could exploit the $\\ell\\nu+HF$ channel in an efficient and reliable way. The package, named $WH$ Analysis Modules (WHAM), is described in Appendix~\\ref{chap:AppWHAM}. Its strengths are modularity, a wide set of configuration options and the use, with improvements, of CDF most advanced tools. The modular and object-oriented approach of the software allowed the handling of the complex selection formed by eleven different lepton types and four data-streams. At the same time it was also possible to develop a set of tools useful also in other contexts: for example a new multivariate multi-jet background rejection tool (see Appendix~\\ref{chap:AppSvm}) or a versatile $b$-tag efficiency evaluation algorithm.\n\nThe actual selection process is divided into two main steps. First we select candidates that have the $\\ell\\nu+2$ jets signature, this defines the {\\em pretag control} sample. Successively we require one or both the jets to be $b$-tagged, obtaining two signal samples ({\\em single} and {\\em double} tagged) enriched in $HF$. Section~\\ref{sec:data_sig} shows the final result of the selection on the CDF dataset and on the diboson signal.\n\n\n\\section{Online Event Selection}\\label{sec:onlineSel}\nThe trigger system, described in Section~\\ref{sec:trigger}, is in charge of the online selection.\n\nIn order to maximize the signal acceptance we use a complex trigger strategy based on several trigger paths collected in four data-streams:\n\\begin{itemize}\n\\item high energy central electrons or \\verb'bhel';\n\\item high energy forward electrons or \\verb'bpel';\n\\item high momentum central muons or \\verb'bhmu';\n\\item high missing transverse energy or \\verb'emet'.\n\\end{itemize}\nAfter online trigger selection but before offline analysis, the quality of the recorded data is crosschecked and the final luminosity is calculated (see Section~\\ref{sec:data-struc}).\n\nA detailed description of the trigger selection, the trigger efficiency ($\\epsilon_{trig}$) evaluation and of the data quality requirements follows in the Section. \n\n\n\\subsection{High Energy Central Electron Trigger}\n\nThe trigger path used for the selection of tight central electron candidate events (CEM) is named \\verb'ELECTRON_CENTRAL_18' and it is stored in the \\verb'bhel' data-stream together with other auxiliary triggers. Single high energy electron identification is based on a calorimeter electromagnetic cluster matched with a reconstructed track.\n\nThe selection proceeds through the three trigger levels, first with minimal requirements and then with more and more sophisticated object reconstruction. L1 requirements are a track with $p_T > 8$~GeV\/c, a central\n($|\\eta| < 1.2$) calorimeter tower with $E_T > 8$~GeV and the ratio between the\nenergy deposited in the hadronic calorimeter to that in the electromagnetic\ncalorimeter ($E^{HAD}\/E^{EM}$) less than $0.125$. At L2, it is required a calorimeter cluster with\n$E_T > 16$~GeV matched to a track of $p_T > 8$~GeV$\/c$. At L3, it requires an\nelectron candidate with $E_T > 18$~GeV matched to a track of $p_T > 9$~GeV$\/c$.\n\nThe \\verb'ELECTRON_CENTRAL_18' trigger path was extensively studied in precision CDF measurements involving $W$ and $Z$ bosons~\\cite{w_z_prl} and now, after each data taking period, the efficiency is evaluated with a standard set of tools~\\cite{perfidia}. A data sample of $W$ boson collected with a \\verb'NO_TRACK' trigger is used to measure the tracking efficiency while a backup trigger is used to measure the efficiency of the calorimeter clustering. The efficiency is \\mbox{$\\epsilon^{CEM}_{trig} =0.982\\pm 0.003$} across the complete dataset with a small $E_T$ and $\\eta$ dependency (see Figure~\\ref{fig:cem_trig}). The trigger turn-on $E_T$ dependency is parametrized with the following function:\n\\begin{equation}\\label{eq:cem_et}\n  \\epsilon_{trig}(x) = A - Be^{-c x};\n\\end{equation}\nwhere $x \\equiv E_T$ and $A$, $B$, $c$ are the parameters of the fit. The following equation is used to model the small effect across $\\eta$:\n\\begin{equation}\\label{eq:cem_eta}\n  \\epsilon_{trig}(x) = A - \\frac{C}{2\\pi\\sigma} e^{-\\frac{x^2}{2\\sigma^2}};\n\\end{equation}\nwhere $x \\equiv  \\eta$ and $A, C, \\sigma$ are the parameters of the fit. Figure~\\ref{fig:cem_trig} shows the parametrization, derived from the full CDF dataset, and applied to MC events.\n\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.495\\textwidth, height=0.36\\textwidth]{Sel\/Figs\/cem_et_trig}\n\\includegraphics[width=0.495\\textwidth, height=0.36\\textwidth]{Sel\/Figs\/cem_eta_trig}\n\\caption[\\texttt{ELECTRON\\_CENTRAL\\_18} Trigger Efficiency Parametrization]{\\texttt{ELECTRON\\_CENTRAL\\_18} trigger efficiency parametrization as a function of $E_T$ (left) and $\\eta$ (right) of the candidate electrons. $E_T$ dependence is parametrized by Equation~\\ref{eq:cem_et}, $\\eta$ dependence by Equation~\\ref{eq:cem_eta}. The fit parameters are reported in the legends.}\\label{fig:cem_trig}\n\\end{center}\n\\end{figure}\n\n\n\n\\subsection{High Energy Forward Electron Trigger}\n\nThe candidate high energy forward electrons (in $1.2 < |\\eta_{Det}| < 2.0$ or Plug region) are stored in the \\verb'bpel' data-stream. \n\nElectron triggering in the plug region is particularly challenging because it lacks of the COT coverage and because a large number of soft interactions are boosted in the region closer to the\nbeam-line. A partial solution to these problems is the use of a multiple-object trigger, named \\verb'MET_PEM', dedicated to $W\\to e\\nu$ selection instead of an inclusive electron trigger: there is no track selection but large \\mbox{${\\not}{E_T}$}  is required together with a high energy cluster in the forward electromagnetic calorimeter. The L1 requirements are a EM tower of $E_T > 8$~GeV in the forward region and \\mbox{${\\not}{E_T}$} $ > 15$~GeV. L2 confirms \\mbox{${\\not}{E_T}$}  requirements but a reconstructed EM cluster of $E_T > 20$~GeV is required and, finally, L3 confirms again the same quantities (\\mbox{\\mbox{${\\not}{E_T}$} $> 15$}~GeV and EM Cluster \\mbox{$E_T > 20$}~GeV) with L3 reconstruction algorithms. \n\nThe trigger shows a turn-on for both the $E_T$ of the cluster and the \\mbox{${\\not}{E_T^{raw}}$}, i.e. corrected only for the position of the primary vertex. The following equation is used for the parametrization of the turn-on:\n\\begin{equation}\\label{eq:phx_trig}\n  \\epsilon_{trig}(x) = \\frac{1}{1+ e^{-A(x-B)}};\n\\end{equation}\nwhere $x\\equiv E_T$ or $x\\equiv$\\mbox{${\\not}{E_T^{raw}}$} and $A$ and $B$ are the parameters of the fit, reported in Figure~\\ref{fig:phx_trig}.\n\nThe \\verb'MET_PEM' trigger parametrization was part of my initial work in\nthe CDF collaboration~\\cite{met_pem_trig9493} and is now included in the CDF analysis\ntools~\\cite{perfidia}.\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.495\\textwidth]{Sel\/Figs\/phx_et_trig}\n\\includegraphics[width=0.495\\textwidth]{Sel\/Figs\/phx_rawmet_trig}\n\\caption[\\texttt{MET\\_PEM} Trigger Efficiency Parametrization]{\\texttt{MET\\_PEM} trigger efficiency parametrization as a function of $E_T$ (left) of the electron candidate and \\mbox{${\\not}{E_T^{raw}}$} (right) of the event. The \\mbox{${\\not}{E_T^{raw}}$} is corrected only for the position of the primary vertex and therefore it is closer to the trigger level \\mbox{${\\not}{E_T}$} . The $E_T$ and \\mbox{${\\not}{E_T^{raw}}$} dependence is parametrized by Equation~\\ref{eq:phx_trig} and the fit parameters are reported in the legends.}\\label{fig:phx_trig}\n\\end{center}\n\\end{figure}\n\n\n\n\n\\subsection{High Momentum Central Muon Trigger}\n\nCentral tight muon candidate events (CMUP and CMX) are collected by the \\verb'MUON_CMUP18' and the \\verb'MUON_CMX18' triggers and saved into the \\verb'bhmu' data-stream. \n\nFor \\verb'MUON_CMUP18' the main trigger requirements are: at L1, a track (\\mbox{$p_T > 4$}~GeV$\/c$) matched to a stub in both the CMU and CMP detectors. At L2, a confirmed track ($p_T > 15$~GeV$\/c$) and a calorimeter deposit consistent with a MIP in the direction of the track and muon stubs. Finally at L3, a fully reconstructed COT track ($p_T > 18$~GeV$\/c$) whose extrapolation matches the CMU (CMP) hits within $\\Delta x_{CMU} < 10$~cm ($\\Delta x_{CMP} < 20$~cm). \nSimilar requirements are made also for \\verb'MUON_CMX18' but the sub-detector of interest is the CMX (covering $0.6 < |\\eta| < 1.1$) and the L1 requirement is a track with $p_T > 8$~GeV$\/c$ matched to a stub in the CMX chamber.  \n\nAs for tight electron candidates, also the central muon triggers were extensively studied~\\cite{w_z_prl} and the efficiency is\nmeasured with a standard procedure~\\cite{perfidia}. Using a data sample of $Z\\to \\mu^+\\mu^-$, a muon passes the trigger requirements and the second muon is checked to pass the trigger selection or not. \n\nAs the CMUP sample represents a relevant fraction of the good quality muon data, an additional effort was done, in the context of the $WH$ analysis~\\cite{hao_trig10747}, to improve the CMUP trigger collection efficiency. Basically we exploited all the CMUP trigger candidates present in the \\verb'bhmu' stream and collected by secondary and auxiliary triggers. The purity of those events was found to be the same of the \\verb'MUON_CMUP18' selected, and we evaluated, with the bootstrapping method, a recover of about $12$\\% efficiency on CMUP online selection. \n\nThe trigger efficiencies are \\mbox{$\\epsilon^{CMUP}_{trig} = 0.976\\pm 0.001$} and \\mbox{$\\epsilon^{CMX}_{trig}=0.873\\pm0.001$}, flat in the kinematic variables. \n\n\n\n\\subsection{High Missing Transverse Energy Triggers}\n\nIn order to recover lepton acceptance lost because of the limited geometrical coverage of the single-lepton triggers, we rely on data collected by the \\texttt{emet} data-stream. \n\nThe \\texttt{emet} stream contains samples with large online \\mbox{${\\not}{E_T}$}  or \\mbox{${\\not}{E_T}$}  plus reconstructed jets. No charged lepton information is used, therefore the acceptance of $\\ell\\nu + 2$~jets events can be recovered. In particular we use the \\verb'emet' stream to collect Loose Muons (Section~\\ref{sec:LooseMuId}) and Isolated Tracks (Section~\\ref{sec:IsoTrk}) candidates, altogether dubbed Extended Muon Categories (EMC).\n\nThe trigger optimization~\\cite{higgs_ttf8875} was performed with the aim to increase the sensitivity to Higgs boson production, but, as diboson and $WH$ production share the same $HF$ final state, an identical strategy was employed for this analysis. The three main triggers used are: \\verb'MET45', \\verb'MET2J' and \\verb'METDI'. The exact naming and requirements of the trigger paths changed many times along the data taking period due to different instantaneous luminosity conditions but the main classification and characteristics are the following:\n\\begin{itemize}\n\\item \\verb'MET45' requires large \\mbox{${\\not}{E_T}$}  in the event, first at L1 and, with increased quality and cut levels, at L2 and L3. The trigger was present along all the data taking with slightly different specifications, however in the last 2\/3 of the data taking the L3 \\mbox{${\\not}{E_T}$}  cut was $40$~GeV.\n\\item \\verb'MET2J' requires high \\mbox{${\\not}{E_T}$}  in the event at L1 (\\mbox{${\\not}{E_T}$} $ > 28$~GeV), two jets at\nL2 with one of them reconstructed in the central region of the detector ($|\\eta| < 0.9$) and, finally, L3 object confirmation and \\mbox{${\\not}{E_T}$} $ > 35$~GeV. The trigger was Dynamically Pre-Scaled (DPS) to cope with the increasing luminosity in the second half of the data taking.\n\\item \\verb'METDI' requires two jets and \\mbox{${\\not}{E_T}$}  in the event, very similarly to \\verb'MET2J', but the L1 requirement, with \\mbox{${\\not}{E_T}$} $ > 28$~GeV and at least one jet, is different.\nThis trigger was introduced after the first $2.4$~fb$^{-\u22121}$ of integrated luminosity.\n\\end{itemize}\n\nThe overlap and composite structure of the \\verb'emet' triggers makes their use less straightforward than the inclusive lepton ones. Furthermore the use of multiple-objects requires to model, on signal and background MC, the possible selection efficiency correlations (e.g. \\mbox{${\\not}{E_T}$}  and jets correlations).\nEfficiency studies were performed in~\\cite{Nagai:2010zza, Buzatu:2011zz}. The main results are the definition of appropriate kinematic regions where $\\epsilon_{trig}$ is flat (plateau region) with respect to the jet variables and can be parametrized as a function of the \\mbox{${\\not}{E_T}$}  only. \\mbox{${\\not}{E_T}$}  is corrected for primary vertex and jet energy but not for the presence of muons. In this way, tight muon candidates, collected from\nother triggers (e.g. \\verb'MUON_CMUP18'), can be used to evaluate the turn-on curves as their contribution to the \\mbox{${\\not}{E_T}$}  is not accounted at trigger level. \n\nAn optimal trigger combination strategy for \\texttt{emet} triggers was evaluated in~\\cite{Buzatu:2011zz}. The trigger selection maximizes the $WH$ acceptance after that the \\mbox{$\\epsilon_{trig}(\\mbox{${\\not}{E_T}$} )$} function is evaluated for each data period and kinematic configuration of the three triggers. \n\nThe final combined trigger efficiency reaches $\\epsilon_{trig}\\simeq 0.5$ for $WZ$ simulated events.\n\n\n\n\n\n\\subsection{Data Quality Requirements and Luminosity Estimate}\\label{sec:lumi}\n\nThe last step before offline event selection is the application of strict data quality requirements. These are ensured by the Good Run List (GRL) selection: i.e. it is possible to use only the run periods where the performances of the sub-detectors are optimal and well parametrized.\n\nA conservative approach, used in many of the CDF analyses, is to consider a run as good only if all the sub-detectors used to define the physics objects of the analysis are fully functional. However here, as in the latest $WH$ search results~\\cite{wh94_note10796}, an ad-hoc approach is used~\\cite{ext_grl}. We require a full functionality of the silicon detector (used for jet $b$-tagging) and calorimeters, while, as we are interested in the one-lepton final state, we de-correlate inefficiencies due to shower-max detectors functionality (CES and PES used for CEM and PHX identification) from muon chambers (CMU, CMP, CMX problematic runs). In this way it was possible to recover 8\\% of the collected luminosity.\n\nOnce that the trigger paths and the appropriate GRLs are defined, it is possible to calculate the final integrated luminosity used for the various lepton types:\n\\begin{itemize}\n\\item CEM and PHX electrons use an integrated luminosity of $\\int \\mathcal{L}\\mathrm{dt} = 9.446$~fb$^{\u2212-1}$.\n\\item CMUP muons use an integrated luminosity of $\\int \\mathcal{L}\\mathrm{dt} = 9.494$~fb$^{-\u22121}$. \n\\item CMX muons use an integrated luminosity of $\\int \\mathcal{L} \\mathrm{dt}= 9.396$~fb$^{-1}$.\n\\item EMC charged leptons use an integrated luminosity of $\\int \\mathcal{L}\\mathrm{dt} = 9.288$~fb$^{-\u22121}$.\n\\end{itemize}\nThe calculated luminosity has systematic uncertainty of $6$\\% (see Section~\\ref{clc_par}).\n\n\n\\section{Offline Event Selection}\n\nNow, on good data, we apply the object identification algorithms described in Chapter~\\ref{chap:objects} to select the desired final state of $\\ell\\nu + HF$ jets. \n\nTable~\\ref{tab:cuts} summarizes the complete selection {\\em cut flow} and the following Sections (from \\ref{sec:w_sel} to~\\ref{sec:svm_sel}) describe in detail the selection criteria. Part of the cut flow is common to many other CDF analyses while the applied multi-jet rejection algorithm, as well as some other specific choices, are completely original.\n\nBecause we rely on simulation to evaluate the signal acceptance, the data\/MC agreement is cross checked for each one of the employed algorithms with data driven methods. Well known physics processes are selected in control samples and the efficiency on data ($\\epsilon^{Data}_{alg}$) and MC ($\\epsilon^{MC}_{alg}$) is estimated. The ratio between them, with appropriate correlations and systematic errors evaluation, is a correction Scale Factor ($SF_{alg}$) to be applied on the simulation:\n\\begin{equation}\n  SF_{alg} =\\frac{ \\epsilon^{MC}_{alg}}{\\epsilon^{Data}_{alg}}\n\\end{equation}\nThe $SF_{alg}$ associated to the lepton identification ($SF_{l}$) and to the $b$-tagging ($SF_{Tag}$) are particularly important in this analysis because of the large number of different lepton reconstruction algorithms and because of the $HF$ final state of the signal. The $SF$s are discussed along with the selection criteria in the following Sections\n\n\n\\begin{table}[h]\n  \\begin{center}\n   \\begin{tabular}{c}\n        \\toprule\n  {\\bf Selection Cuts}\\\\\\midrule\n\n   Trigger Fired \\\\\n   Good Run Requirement\\\\\n  $|z_0^{PV}| \\le 60$~cm \\\\\n  Charged Lepton Selection \\\\\n  $|z_0^{lep} \u2212- z_0^{PV}| < 5$~cm\\\\\n  Di-Lepton Veto\\\\\n  $Z\\to \\ell\\ell$ Veto\\\\\n  Cosmic Ray Veto \\\\\n  $\\gamma\\to e^+e^-$ Conversion Veto\\\\\n  \\mbox{${\\not}{E_T}$} $ > 15$~GeV\\\\\n  2 Tight Jets Selection \\\\\n  $M_{Inv}(jet1, jet2)>20$~GeV$\/c^2$\\\\\n  1 or 2 \\texttt{SecVtx} Tags\\\\\n  {\\bf Multi-jet Rejection:} \\\\\n {\\em SVM}$_{CEM, EMC}>0$, {\\em SVM}$_{CMUP, CMX}>-0.5$, {\\em SVM}$_{PHX}>1$\\\\\n      \\bottomrule\n    \\end{tabular}\n    \\caption[Selection Summary]{Summary of all the selection requirements applied in the analysis.}\\label{tab:cuts}\n  \\end{center}\n\\end{table}\n\n\n\n\n\\subsection{$W\\to\\ell\\nu$ Offline Selection}\\label{sec:w_sel}\n\nThe leptonic decay of the $W$ boson is selected by requiring exactly one high energy charged lepton and missing transverse energy signaling the presence of the neutrino.\n\nThe correct application of these two basic requirements exploits several of the identification algorithms described in the previous Chapter and other selection criteria in order to purify the lepton sample.\nThe summary of all the selection steps, applied on data and MC, is the following:\n\\begin{itemize}\n\\item identification of the primary interaction vertex (Section~\\ref{sec:primVtx}) within the interaction fiducial region of 60 cm from the detector center ($|z_0^{PV}| \\le 60$~cm). Minimum bias events are used to evaluate on data the efficiency of this cut. The result, averaged over all the data taking periods, is\\footnote{The initial number of MC events, used to normalize all the simulated acceptances, is derived after this cut so that $\\epsilon_{z_0}$ should be applied to obtain the final yields.}:\n  \\begin{equation}\n    \\epsilon_{z_0} = 0.9712\\pm 0.0006.\n  \\end{equation}\n\\item Identification of one charged and isolated lepton candidate originating\nfrom the primary vertex ($|z_0^{lep} \u2212- z_0^{PV}| < 5$~cm). A charged lepton candidate is one of the eleven lepton identification algorithms described in Sections from~\\ref{sec:eleSel} to~\\ref{sec:IsoTrk}: CEM, PHX (tight electrons), CMUP, CMX (tight muons), BMU, CMU, CMP, CMIO, SCMIO, CMXNT (loose muons) and ISOTRK (isolated tracks). This requirement does not produce any appreciable signal loss.\n\\item Veto of di-leptonic candidate events: if an event contains two tight or loose electrons or muons both isolated or not, it is rejected as a Drell-Yan event or $t\\bar{t}$ dileptonic candidate. Events with an ISOTRK candidate are vetoed if reconstructed together with another ISOTRK or a tight lepton. An efficiency of approximately 90\\% is estimated from signal MC.\n\\item Further rejection of $Z\\to\\ell\\ell$ candidates is obtained for CEM, PHX, CMUP and CMX leptons by vetoing events in which the tight lepton can be paired with a track or a EM calorimeter cluster that form an invariant mass within the range $[76, 116]$~GeV$\/c^2$.  An efficiency of approximately 90\\% is estimated from signal MC.\n\\item A cosmic tagger was implemented within the CDF offline code~\\cite{cosmic_tag}\nto reject high-$p_T$ cosmic muons which interact with the detector simultaneously with a bunch crossing. The algorithm exploits muon chambers, COT and TOF, calorimeter energy and timing information, providing\nalmost 100\\% rejection of cosmic ray events, with a negligible loss of signal efficiency.\n\\item Rejection of electrons originating from photon conversions, $\\gamma\\to e^+e^{-\u2212}$,\nis obtained by applying the CDF conversion tagger~\\cite{conversion_tag}. The algorithm looks for two opposite-sign tracks (one of them belonging to the identified electron) and requires $\\Delta \\cot(\\theta) \\le 0.02$ and the distance at the closest approach between them $D_{xy} \\le 0.1$~ cm. The rejection efficiency of the conversion veto is about $65$\\% with a $1\\div 2$\\% signal loss.\n\\item Finally, \\mbox{${\\not}{E_T}$} $ > 15$~GeV is required to signal the presence of a neutrino. The \\mbox{${\\not}{E_T}$}  is fully corrected for position of the primary interaction vertex, presence of jets and muons in the event (see Section~\\ref{sec:metObj}). Approximatly 7\\% of the signal is lost with this requirement.\n\\end{itemize}\n\nFor each run period and lepton identification algorithm, the agreement\nbetween data and MC is measured comparing the $Z\\to\\ell\\ell$ data sample against the simulation. Table~\\ref{tab:lepId} reports, averaged on all the run range, the yield correction, $SF_{l}$ that covers the\ndata\/MC differences. The main differences arise in the simulation of the isolation and of the muon chamber response.\n\n   \\begin{table}[h]\n \\begin{center}\n     \\begin{tabular}{cccc}\n       \\toprule\n\n       CEM & PHX & CMUP & CMX \\\\\n       $0.973 \\pm 0.005$ &   $0.908 \\pm 0.009$ & $0.868\\pm 0.008$ & $ 0.940\\pm 0.009$\\\\\n     \\end{tabular}\n  \n      \\begin{tabular}{cccccc}\n        \\midrule\n        BMU & CMU & CMP & SCMIO & CMIO & ISOTRK \\\\\n\n        $ 1.06\\pm 0.02$ & $ 0.88\\pm 0.02$ & $ 0.86\\pm0.01$ & $ 1.02\\pm 0.01$ & $ 0.97 \\pm 0.02$  &  $ 0.94 \\pm 0.04$ \\\\\n        \\bottomrule \n      \\end{tabular}\n\n     \\caption[Lepton Algorithm Data\/MC Scale Factors ($SF_l$)]{Data\/MC Scale Factors ($SF_l$) for each of the single lepton identification algorithms used in the analysis, tight charged leptons in the first row and EMC categories in the lower row.}\\label{tab:lepId}\n \\end{center}\n   \\end{table}\n\nIf several lepton identification algorithms cover the same detector region they can not be considered independent and a further correction\\footnote{This correction was introduced for the first time in this analysis and in the $WH\\to \\ell\\nu+ b\\bar{b}$ search~\\cite{wh94_note10796}.} to the simulation is needed. As shown in Figure~\\ref{fig:split_alg}, loose muon and ISOTRK largely overlap with the tight lepton selection and, often, a MC lepton candidate is reconstructed by more than one lepton selection algorithm. Because of this and to avoid ambiguity, we apply a prioritized selection, i.e. we label a lepton candidate with the highest priority algorithm according to:\n  \\begin{equation}\n\\begin{gathered}\n CEM > PHX > CMUP > CMX > BMU > CMU >\\\\\n CMP > SCMIO > CMIO > CMXNT >  ISOTRK\n\\end{gathered}\n  \\end{equation}  \n\n\nIn most of the cases, selection cuts are orthogonal and only one lepton reconstruction is possible, however, if this is not true {\\em and} $SF_{l}\\ne 1$, the same simulated lepton has a probability greater than zero of not being identified by the highest priority algorithm\\footnote{Quality criteria and historical reasons are at the base of the prioritization of the algorithms. Prioritization is indirecly present also in the $SF_{l}$ measurements: for example in the ISOTRK $SF_{l}$ measurement the lepton candidates are checked not to be reconstructed as tight leptons, while the check is not performed in the opposite case, i.e. no ISOTRK identification is performed during the tight $SF_{l}$ measurement.}. In this case the MC event under consideration is classified under more lepton categories with different weights, according to:\n\\begin{equation}\\label{eq:composite_sf}\n  \\omega_{l} = SF_{l} + SF_{l}\\prod_{l'>l} \\big(1- SF_{l'} \\big)\n\\end{equation}\nwhere $\\omega_{l}$ is a composite $SF_{l}$. Equation~\\ref{eq:composite_sf} reduces to $\\omega_{l} = SF_{l}$ when a lepton candidate is reconstructed only by one identification algorithm, otherwise it takes into account the probability of {\\em not} identification from algorithms with higher priority ($l'>l$).\n\nFigure~\\ref{fig:eta_multisf} shows the corrected  $\\eta$  distribution for the EMC lepton category (combination of loose muon and ISOTRK algorithms) while Figure~\\ref{fig:eta_pre_post} shows the complete background estimate distribution for the combined $\\eta$ of the EMC category before and after the correction of Equation~\\ref{eq:composite_sf}.\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.8\\textwidth]{Sel\/Figs\/emc_eta_sf}\n\\caption[$\\eta$ Distribution of the $\\omega_{l}$ Correction on EMC]{Effect of the $\\omega_{l}$ correction described in Equation~\\ref{eq:composite_sf} across the $\\eta$ distribution of the EMC lepton category selected on $WZ$ MC. The correction appears as a factor $f_{\\omega_l}\\ne 1$ for each event where the reconstructed lepton is identified by more than one algorithm.}\\label{fig:eta_multisf}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.495\\textwidth]{Sel\/Figs\/WZ_Pretag_ISOTRK_LepEta_noSFCorr}\n\\includegraphics[width=0.495\\textwidth]{Sel\/Figs\/WZ_Pretag_ISOTRK_LepEta_wSFCorr}\n\\caption[Effect of $\\omega_{l}$ on EMC $\\eta$ Distribution after Background Estimate]{$\\eta$ distribution of the EMC lepton category with full background evaluation before (left) and after (right) the application of the multiple $SF_{l}$ correction defined by Equation~\\ref{eq:composite_sf}.}\\label{fig:eta_pre_post}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\\subsection{Offline Jets Selection}\\label{sec:jetSel}\n\nThe final aim of this analysis is the study of the invariant mass spectrum of a two jet, $HF$ enriched, final state. \n\nJet selection is based on the \\verb'JETCLU' $R = 0.4$ algorithm, described in Section~\\ref{sec:jetObj}, and the \\texttt{SecVtx} $HF$-tagger algorithm, described in Section~\\ref{sec:secvtx}.\nOne of the goals of this analysis is to demonstrate the ability of the CDF experiment to identify a resonance in the background rich sample of $W+HF$ jets. Therefore the jet selection was kept simple with the aim to understand the sample itself. The specific requirements are the following:\n\\begin{itemize}\n\\item  selection of exactly two tight jets. Jets central in the detector and with large $E_T$ (after L5 energy correction) are named tight jets if:\n\n\\begin{equation}\n  E^{jet,cor}_T > 20\\mathrm{~GeV}\\quad\\textrm{and}\\quad |\\eta^{jet}_{Det}| < 2.0;\n\\end{equation}\n\n\\item a minimal lower bound on the invariant mass distribution of the jet pair: $M_{Inv}(jet1, jet2) > 20$~GeV$\/c^2$;\n\\item one or both of the selected jets $HF$-tagged by the \\texttt{SecVtx} algorithm;\n\\item jets are ordered according to their $E_T^{jet,cor}$ and classified as: $jet1$ and $jet2$. \n\\end{itemize}\nNo other requirements are applied on the jets and, in particular, we do not apply any criteria to the loose jets present in the event. \n\nThe $HF$-tag requirement divides the $\\ell\\nu+ 2$ jets sample in three analysis regions:\n\\begin{description}\n\\item[Pretag:] no $HF$-tag requirement is applied and, due to the small $HF$ contamination, the sample is used as a control region (see Section~\\ref{sec:wjets_pretag}).\n\\item[Single-tag:] events with exactly one jet tagged as $HF$ compose this signal region. The statistics of the selected sample is quite large therefore it is important for the sensitivity of this analysis to  $WW\\to\\ell\\nu + c\\bar{s}$ decay.\n\\item[Double-tag:] events with both jets tagged as $HF$. The statistics of this sample is small because of the strict double-tag requirement, however a large fraction of the sensitivity for the $WZ\\to\\ell\\nu + b\\bar{b}$ signal comes from this region as the background contamination is also low.\n\\end{description}\n\nAccording to Section~\\ref{sec:btag_sf}, the $HF$ selection efficiency of the \\texttt{SecVtx} algorithm must be corrected with an appropriate data\/MC $SF_{Tag}$. Furthermore a contamination due to $LF$ quarks producing secondary vertices will contribute to the signal sample. The $HF$ and $LF$ components for $1$\u2212-Tag and $2$--Tag selection are accounted in the signal MC with a per-event tagging probability, $\\omega_{k-\u2212Tag}$ obtained by the combination of per-jet tagging probabilities:\n\\begin{equation}\\label{eq:1tag}\n  \\omega_{1-Tag}=\\sum_{i=1}^{n}p^i_{Tag}\\cdot\\Big(\\prod^n_{j=1,j\\ne i}\\big( 1 - p^j_{Tag}\\big)\\Big),\n\\end{equation}\n\\begin{equation}\\label{eq:2tag}\n  \\omega_{2-Tag}=\\sum_{i=1}^{n-1}p^i_{Tag}\\cdot\\Big( \\sum_{j>i}^{n}p^i_{Tag} \\cdot \\Big( \\prod^n_{h=1,h\\ne i,h\\ne j}\\big( 1 - p^k_{Tag}\\big)\\Big)\\Big),\n\\end{equation}\nwhere the per-jet tagging probability is defined by $p^j_{Tag} \\equiv SF_{Tag}$, if the jet is matched\\footnote{The matching is satisfied if $\\Delta R(jet, had) < 0.4$ where we test the fully reconstructed jet 4-vector against all the final state hadrons 4-vectors.}  to a $HF$ hadron, or $p^j_{Tag} \\equiv p^j_{Mistag}$, if the event is matched to a $LF$ hadron.\n\nEquations~\\ref{eq:1tag} and~\\ref{eq:2tag} are applied to obtain \\texttt{SecVtx} 1-Tag or 2-Tags probabilities.  A further generalization of the previous equations was implemented in the WHAM analysis framework (see Appendix~\\ref{chap:AppWHAM}) so that efficiencies can be calculated for any $b$-tagging algorithm (and combinations of them), any tag and jet multiplicity.\n\n\n\\subsection{Multi-Jet Background Rejection}\\label{sec:svm_sel}\n\nMulti-jet events can fake the $W$ boson signature when one jet passes the high $p_T$ lepton selection criteria and fake \\mbox{${\\not}{E_T}$}  is generated through jets mis-measurement. Although the probability for such an event is small, the high rate of multi-jet events, combined with the small cross section of the processes of interest, make this an important background. Additional difficulties arise in the simulation of the  mixture of physics and detector processes contributing to this background. Therefore it is desirable to reduce the multi-jet events as much as possible.\n\nFor this purpose, an original and efficient multi-variate method based on the Support Vector Machine (SVM) algorithm was developed for this analysis.\n\nThe basic concept behind a SVM classifier is quite simple: given two sets of $n$-dimensional vectors (the training sets) the algorithm finds the {\\em best separating hyper-plane} between the two. The plane is then used again to classify a newly presented vector in one of the two sets. Non linear separation can be achieved with the use of appropriate transformations called {\\em Kernels}. SVM classification performs better than other multivariate algorithms, like Neural Networks, in the case of statistically limited training sets and shows more stability with respect to over-training~\\cite{svmbook}.\n\nThe work described here improves a previously developed software package\\footnote{The previously created SVM based multi-jet rejection~\\cite{CHEP_svm} was already tested and employed in previous analyses~\\cite{wh75_note10596, wh94_note10796, cdfDiblvHF_Mjj2011}.}, based on the \\texttt{LIBSVM}~\\cite{libsvm} library, able to perform algorithm training, variable ranking, signal discrimination and robustness test. Here we report only a summary of the results while a detailed description of optimization, training procedure and variable definition is given in Appendix~\\ref{chap:AppSvm}. \n\nWe trained two specific SVMs, one for the {\\em central} and one for the {\\em forward} region of the detector. \nThe SVMs aimed to improve the purity of the $W\\to e\\nu+2$ jets sample, however we avoided the use of input variables related to the electron identification (i.e. cluster or track related) while we focused on the kinematic variables. This gave an additional result as the multi-jet rejection proved to be optimal also for other lepton identification algorithms. For example the central SVM was used on all the lepton identification algorithms in the region $|\\eta|<1.2$. \n\nAppendix A describes in details all the input variables used in the trainings.\nThe SVM for the {\\em central} region exploits the following eight input variables:\n\\begin{itemize}\n\\item $W$ related variables: $M^W_T$, \\mbox{${\\not}{E_T^{raw}}$}, ${\\not}{p_T}$, $\\Delta \\phi(lep,$ \\mbox{${\\not}{E_T}$} $)$, $\\Delta R(\\nu^{min},$ $lep)$;\n\\item global variables:  $MetSig$, $\\Delta \\phi(jet1,$ \\mbox{${\\not}{E_T}$} ), $\\Delta \\phi({\\not}{p_T},$ \\mbox{${\\not}{E_T}$} ).\n\\end{itemize} \nThe specific SVM for the {\\em forward} region of the detector (used only for the PHX electron selection) exploits six input variables:\n\\begin{itemize}\n\\item $W$ related variables: $M^W_T$, \\mbox{${\\not}{E_T^{raw}}$}, ${\\not}{p_T}$; \n\\item global variables:  $MetSig$, $\\Delta \\phi({\\not}{p_T},$ \\mbox{${\\not}{E_T^{raw}}$}), $\\Delta \\phi({\\not}{p_T},$ \\mbox{${\\not}{E_T}$} ).\n\\end{itemize}\n\nTable~\\ref{tab:cuts} shows different SVM selection values for the different lepton types:\n\\begin{equation}\n\\textrm{{\\em SVM}}_{CEM, EMC}>0\\textrm{;}\n\\end{equation}\n\\begin{equation}\n  \\textrm{{\\em SVM}}_{CMUP, CMX}>-0.5\\textrm{;}\n\\end{equation}\n\\begin{equation}\n  \\textrm{{\\em SVM}}_{PHX}>1\\textrm{.}\n\\end{equation}\nWe choose different thresholds due to the different probabilities to fake a $W$ signature: lower for the tight muon selection (CMUP, CMX), higher for CEM electrons and EMC lepton category, and way higher for PHX electrons, identified in a detector region with low track reconstruction efficiency and large calorimeter occupancies. \n\nFigure~\\ref{fig:svm_templates} shows the SVM output distribution for CEM and PHX leptons for the signal and background MC samples as well as for the multi-jet background models (see Section~\\ref{sec:qcd} for their description).\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.495\\textwidth]{Sel\/Figs\/svm_cem} \\includegraphics[width=0.495\\textwidth]{Sel\/Figs\/svm_phx}\n\\caption[SVM Output Distribution for CEM and PHX Leptons]{SVM output distribution for CEM (left) and PHX (right) leptons for the signal and background MC samples as well as for the multi-jet background model (see Section~\\ref{sec:qcd} for the description).}\\label{fig:svm_templates}\n\\end{center}\n\\end{figure}\n\nTable~\\ref{tab:svm_eff} summarizes the signal selection efficiencies for the SVM cuts in the different lepton categories; all the analysis selection cuts are applied except for the tagging requirement. The evaluation of the final multi-jet background contamination is described in Section~\\ref{sec:qcd} and a summary of the results is reported in Tables~\\ref{tab:wjet_pretag_qcd_fit} and~\\ref{tab:wjet_tag_qcd_fit}.\n\nThe training procedure requires the SVM to pass several quality criteria about data\/MC agreement. We also checked the stability of the selection efficiency. We found it stable against a variation of up to 5\\% of the SVM input variables. Therefore we did not find necessary to add a specific SF for the SVM selection.\n\n  \\begin{table}\n\\begin{center}\n    \\begin{tabular}{cccccc}\n      \\toprule\n      Lepton & CEM & PHX & CMUP & CMX &  EMC \\\\\\midrule\n      $\\epsilon_{SVM}^{WW}$ & $0.95 \\pm 0.02$ & $0.82 \\pm 0.03$ & $0.97 \\pm 0.03$ & $0.97 \\pm 0.04$ &  $0.94 \\pm 0.03$ \\\\\n      $\\epsilon_{SVM}^{WZ}$ & $0.96 \\pm 0.03$ & $0.83 \\pm 0.04$ & $0.98 \\pm 0.04$  &  $0.99 \\pm 0.05$ & $0.95 \\pm 0.03$ \\\\\n      \\bottomrule\n    \\end{tabular}\n    \\caption[$WW$ and $WZ$ SVM Selection Efficiency]{$WW$ and $WZ$ signal selection efficiencies for the SVM cuts used in the lepton selection. Reported errors are statistical only.}\\label{tab:svm_eff}\n\\end{center}\n  \\end{table}\n\n\n\n\\section{Final Signal Estimate and Data Selection}\\label{sec:data_sig}\n\nThe complete event selection can now be  applied to both the full data sample and the signal MC. \n\nThe final signal yield is derived from the following equation, for each signal process ($Sig=WW$, $WZ$, $ZZ$), each lepton identification algorithm ($l$) and for single and double tagged ($k=1$, $2$) selections:\n\\begin{equation}\\label{eq:mc_estimate}\n  N^{Sig} = \\epsilon_{l, k}^{Sig}\\cdot\\sigma_{p\\bar{p}\\to Sig}\\cdot\\int\\mathcal{L}_{l}\\mathrm{dt};\n\\end{equation}\nthe factors are:\n\\begin{itemize}\n\\item the cross sections, $\\sigma_{p\\bar{p}\\to Sig}$, for inclusive diboson production, reported in Table~\\ref{tab:mc_dib}, are evaluated at NLO~\\cite{mcfm_prd,mcfm_url};\n  \\begin{table}\n    \\begin{center}\n    \\begin{tabular}{ccc}\n      \\toprule\n      Signal & $\\sigma$ (pb), NLO &  Initial Events \\\\\n      \\midrule\n      $WW$    & 11.34 $\\pm$ 0.66  & 4.6 M\\\\ \n      $WZ$    & 3.47  $\\pm$ 0.21  & 4.6 M\\\\  \n      $ZZ$    & 3.62  $\\pm$ 0.22  & 4.8 M\\\\ \n      \\bottomrule\n    \\end{tabular}\n    \\caption[Diboson MC Information and NLO Cross Section]{MC information about the diboson signal. The reported cross sections are evaluated at NLO precision~\\cite{mcfm_prd,mcfm_url}.  \\texttt{PYTHIA}~\\cite{pythia} v6.216 generator and parton-showering is used for the MC simulation.}\\label{tab:mc_dib}\n    \\end{center}\n  \\end{table}\n\n\\item the integrated luminosity, $\\int\\mathcal{L}_{l}dt$, is calculated in Section~\\ref{sec:lumi} for the different leptons\/data-streams;\n\\item $\\epsilon_{l, k}^{Sig}$ is the simulated selection acceptance accounting for trigger and $|z_0|$ cut efficiencies ($\\epsilon_{trig}$, $\\epsilon_{z_0}$), lepton and $b$-tag efficiency corrections ($\\omega_l$, $\\omega_{k-Tag}$):\n  \\begin{equation}\\label{eq:all_w_tag}\n    \\epsilon_{l, k}^{Sig} = \\epsilon_{trig}\\cdot\\epsilon_{z_0}\\cdot\\omega_{l}\\cdot\\omega_{k-Tag}\\cdot\\frac{N^{Sig}_{l,k}}{N^{Sig}_{z_0}}\n  \\end{equation}\nwhere $N^{Sig}_{l,k}$ is the number of MC events passing the lepton and tag selection and $N^{Sig}_{z_0}$ is the number of events generated in the fiduciality region of the detector. Equation~\\ref{eq:all_w_tag} simplifies slightly in case we are evaluating the pretag region yield:\n\\begin{equation}\\label{eq:all_w_pretag}\n    \\epsilon_{l, pretag}^{Sig} = \\epsilon_{trig}\\cdot\\epsilon_{z_0}\\cdot\\omega_{l}\\cdot\\frac{N^{Sig}_{l,pretag}}{N^{Sig}_{z_0}}\n\\end{equation}\nwhere $N^{Sig}_{l,pretag}$ is the number of MC events passing the lepton selection at pretag level. \n\\end{itemize}\nThe final result of the event selection is reported in Table~\\ref{tab:data_sig} for the CDF dataset and the signal expectation; also the pretag control region is shown for comparison. \nThe selection suppresses the $Z\\to\\ell\\ell$ events making the total $ZZ$ contribution almost negligible ($\\approx 3$\\% of the total diboson yield), however this component is included in the signal because no distinction is possible between the $Z\\to HF$ decay originating from $WZ$ or $ZZ$ production.\n\n\\begin{table}[h]\n  \\begin{small}\n    \\begin{center}\n    \\begin{tabular}{cccccc}\n      \\toprule\n \\multicolumn{6}{c}{\\bf Pretag Selection}\\\\\n Lepton & CEM & PHX & CMUP & CMX &  EMC \\\\\\midrule\n    {\\bf Data} & {\\bf 80263}  &{\\bf 27759}  &    {\\bf 39045}  & {\\bf 22465}      & {\\bf 35810} \\\\                      \n  $WW$            & 2012.4 $\\pm$ 168.7    &  705.5 $\\pm$ 59.3     & 1047.7 $\\pm$ 88.5   & 537.0 $\\pm$ 45.5  &1157.7 $\\pm$ 115.3 \\\\\n  $ZZ$            & 22.4 $\\pm$ 1.9        &  6.1 $\\pm$ 0.5        & 25.2 $\\pm$ 2.2      & 13.2 $\\pm$ 1.1    &37.5 $\\pm$ 3.8\\\\\n  $WZ$            &  307.0 $\\pm$ 26.2     & 135.7 $\\pm$ 11.6      & 168.6 $\\pm$ 14.5    & 95.3 $\\pm$ 8.2    &225.7 $\\pm$ 22.8\\\\\n    \\midrule\n   \\multicolumn{6}{c}{\\bf Single-Tag Selection}\\\\\n Lepton & CEM & PHX & CMUP & CMX &  EMC \\\\ \\midrule\n    {\\bf Data}        & {\\bf 3115}           &{\\bf 1073 }     & {\\bf 1577}          & {\\bf 830}          & {\\bf 1705}          \\\\                           \n\n   $WW$            & 84.35 $\\pm$ 11.8 & 25.05 $\\pm$ 3.54          & 43.7 $\\pm$ 6.17        & 23.68 $\\pm$ 3.35      & 53.11 $\\pm$ 8           \\\\\n   $ZZ$            & 1.85 $\\pm$ 0.19  & 0.21 $\\pm$ 0.02           & 2.45 $\\pm$ 0.25        & 1.37 $\\pm$ 0.14       & 3.45 $\\pm$ 0.39        \\\\ \n   $WZ$            & 29.2 $\\pm$ 2.95  & 12.32 $\\pm$ 1.22          & 16 $\\pm$ 1.66          & 9.21 $\\pm$ 0.94       & 20.54 $\\pm$ 2.39        \\\\\n\n\\midrule                                                                                        \n  \\multicolumn{6}{c}{\\bf Double-Tag Selection}\\\\\n Lepton & CEM & PHX & CMUP & CMX &  EMC \\\\\\midrule\n  {\\bf Data}               &{\\bf 175 }       & {\\bf 62 }     & {\\bf 92 }            & {\\bf 49 }             & {\\bf 126}              \\\\     \n       $WW$           & 0.72 $\\pm$ 0.19   & 0.18 $\\pm$ 0.05        & 0.35 $\\pm$ 0.09        & 0.2 $\\pm$ 0.05           & 0.49 $\\pm$ 0.13      \\\\\n       $ZZ$           & 0.26 $\\pm$ 0.04   & 0.03 $\\pm$ 0.01           & 0.46 $\\pm$ 0.06        & 0.29 $\\pm$ 0.04          & 0.63 $\\pm$ 0.10     \\\\  \n       $WZ$           & 5.28 $\\pm$ 0.75   & 2.6 $\\pm$ 0.37         & 2.52 $\\pm$ 0.36        & 1.67 $\\pm$ 0.24          & 3.52 $\\pm$ 0.54      \\\\\n       \\bottomrule\n\n    \\end{tabular}\n    \\caption[Final Signal Estimate and Data Selection]{Final result of the event selection for the $W+2$ jets dataset and signal expectation in the different lepton categories. The pretag control region is shown together with single and double-tagged signal regions. The uncertainties from MC statistics, productionn cross section, luminosity, lepton identification, trigger and $HF$-tagging are considered in the table.}\\label{tab:data_sig}\n    \\end{center}\n  \\end{small}\n  \\end{table}\n\n\\chapter{Statistical Analysis and Results}\\label{chap:StatRes}\n\nThe measurement of a potential signal (and its properties) over a predicted background requires a statistical analysis of the selected events.\n\nThe detection of diboson events in the $\\ell\\nu+HF$ final state is a challenging problem as it combines a small expected signal yield, a sizable irreducible background and large systematic uncertainties typical of the hadronic environment.\n\nAfter the full event selection (Chapter~\\ref{chap:sel}), the remaining background processes are mainly of irreducible nature ($W+c\/c\\bar{c}\/b\\bar{b}$) and still overwhelm the signal by more than a factor twenty. \n\nAs the diboson production is a resonant process while $W+HF$ is not, a feasible and optimal signal extraction strategy is the shape analysis of the di-jet invariant mass distribution, $M_{Inv}(jet1,jet2)$.\n\nThe use of the di-jet mass has two effects. Firstly, we can not distinguish $WZ\\to\\ell\\nu+b\\bar{b}\/c\\bar{c}$ decay from $ZZ\\to\\ell\\not{\\ell}+b\\bar{b}\/c\\bar{c}$ decay where one lepton is lost. Therefore, from now on, the $WZ$ diboson production is considered together with the small amount of selected $ZZ$ events\\footnote{Table~\\ref{tab:data_sig} shows that $ZZ$ is about $1\/10$ of the $WZ$ expected events and 1\/30 of the total diboson sample.}. A more important effect is that, due to the low mass resolution, typical of the hadronic final states, we can not separate contribution from $WW\\to \\ell\\nu+c\\bar{s}$ from the $WZ\\to \\ell\\nu+b\\bar{b}\/c\\bar{c}$, in the single-tagged event selection\\footnote{In the double-tagged event selection the only signal contribution comes from $WZ\\to \\ell\\nu+b\\bar{b}$.}. We solve this problem with the use of a second variable, the NN flavor-separator (KIT-NN) described in Section~\\ref{sec:kitnn}. We use it on single-tagged events to build a bi-dimensional distribution:\n\\begin{equation}\n  M_{Inv}(jet1,jet2)\\textrm{ {\\em vs }KIT-NN}  \\textrm{.}\n\\end{equation}\nThis, together with the simple $M_{Inv}(jet1,jet2)$ distribution of the double-tagged events, allows to measure $WW$ and $WZ\/ZZ$ contributions separately.\n\nThe statistical analysis is performed with a methodology and a software tool, named {\\em mclimit}~\\cite{mclimit, single_top}, also used in the CDF and Tevatron Higgs searches~\\cite{cdf_cmb, tev_cmb} and adapted to the present analysis.\n\nThe technique, described in Section~\\ref{sec:likelihood}, is based on the integration, over data, of a binned likelihood function where the diboson production cross section is a free parameter. The systematic uncertainties, listed in Section~\\ref{sec:sys_desc}, are included with a Bayesian approach and marginalized to increase the sensitivity. \n\nThe likelihood is built by dividing the selected data in eight orthogonal channels depending on the kinematic properties and on the background composition:\n\\begin{itemize}\n\\item {\\bf 4 Lepton categories} are derived from the data-streams used to collect the events (see Section~\\ref{sec:onlineSel}): central electrons (CEM), forward electrons (PHX), central muons (CMUP+CMX) and extended muons (EMC). The kinematic is homogeneous within each sample.\n\\item {\\bf 2 Tag categories} are derived from the single or double \\texttt{SecVtx} tagged event selection for each lepton category. The sample composition drastically changes in the two regions. Section~\\ref{sec:mjj_kit_distrib} shows the $M_{Inv}(jet1,jet2)$ distribution in the double-tagged categories and slices of the bi-dimensional distribution, $M_{Inv}(jet1,jet2)$ {\\em vs} KIT-NN, used for the single-tagged categories. Figure~\\ref{fig:2d_templates} shows four examples (for the CEM lepton selection) of the bi-dimensional templates used for the single-tagged channels evaluation: $WW$, $WZ\/ZZ$ signals and $W+c\\bar{c}$, $W+b\\bar{b}$ backgrounds.\n\\end{itemize}\n\\begin{figure}\n  \\includegraphics[width=0.495\\textwidth]{StatRes\/Figs\/ww_templ}\n  \\includegraphics[width=0.495\\textwidth]{StatRes\/Figs\/wzzz_templ}\\\\\n  \\includegraphics[width=0.495\\textwidth]{StatRes\/Figs\/wcc_templ}\n  \\includegraphics[width=0.495\\textwidth]{StatRes\/Figs\/wbb_templ}\n  \\caption[Example of $M_{Inv}(jet1,jet2)$ {\\em vs } KIT-NN Templates]{Four examples of the bi-dimensional templates, $M_{Inv}(jet1,jet2)$ {\\em vs } KIT-NN, used for the statistical analysis of the  single-tagged channels in the CEM lepton selection: $WW$ signal (top left), $WZ\/ZZ$ signals (top right), $W+c\\bar{c}$ background (bottom left), $W+b\\bar{b}$ background (bottom right). The contribution of $c$ and $b$ quarks cluster at opposite sides of the KIT-NN distribution, the diboson production cluster around an invariant mass resonant peak while the generic $HF$ production spreads along the $M_{Inv}(jet1,jet2)$ direction.}\\label{fig:2d_templates}\n\\end{figure}\n\nThe measured diboson cross sections, $\\sigma_{WW}$ and  $\\sigma_{WZ\/ZZ}$, are obtained in Section~\\ref{sec:cx_measure} from the evaluation of the Bayesian posterior distribution of the combined likelihood. \n\nAs a first result, we measure the total diboson production cross section, fixing the $WW$ and the $WZ\/ZZ$ relative contributions to the SM prediction. Then we remove the constraint on the $WW$ and the $WZ\/ZZ$ relative fractions. We let them free to float independently so to obtain a measurement where the full correlation between $\\sigma_{WW}$ and  $\\sigma_{WZ\/ZZ}$ is accounted. Finally we also obtain two individual measurements integrating out $\\sigma_{WW}$ or  $\\sigma_{WZ\/ZZ}$ one at the time from the Bayesian posterior distribution.\n\nIn Section~\\ref{sec:sigma_eval}, we discuss the significance of the different measurements and we show the evidence of the total diboson production. The previous version of this analysis~\\cite{cdfDiblvHF_Mjj2011}, performed with a dataset of $7.5$~fb$^{-1}$ and reported in Appendix~\\ref{App:7.5}, was the first evidence of this process at a hadron collider.\n\nIn general the result confirms the SM prediction for diboson production also when they are detected in the $HF$ final state. This provides additional confidence on the capability and the methods used by the CDF collaboration for the Higgs search in the $H\\to b\\bar{b}$ channel. \n\n\n\\section{Construction of the Likelihood Function}\\label{sec:likelihood}\n\nThe comparison of observed data with the templates derived from signal and background is possible with the use of a likelihood function in which we use a Bayesian approach to include the prior probabilities of the background and systematic effects.\n\nWe assume that the outcome of the data observation $n_{i,j}$, in the $j^{th}$ bin of the $i^{th}$ input histogram follows the Poisson statistics and the expectation value, $\\mu_{i,j}$, depends by the estimated backgrounds and signals. Therefore the complete likelihood function, $\\mathscr{L}$, of the bin by bin outcome is:\n\\begin{equation}\\label{eq:likelihood}\n  \\mathscr{L}(\\vec{\\alpha}, \\vec{s}, \\vec{b}| \\vec{n}, \\vec{\\beta}) =\\prod^{Hists}_{i} \\prod^{Bins}_{j} \\frac{e^{-\\mu_{i,j}}\\mu_{i,j}^{n_{i,j}}}{n_{i,j}!}\n\\end{equation}\nwhere the first product runs over the total number of input channels, $Hist$ (i.e. the different histograms provided for them), and the second runs on the bins of each distribution. The bin expectation value contains the signal and background dependence: \n\\begin{equation}\\label{eq:mu_likelihood}\n  \\mu_{i,j}(\\vec{\\alpha}, \\vec{s}, \\vec{b}| \\vec{\\beta}) = \\sum_p^{Sgn} \\alpha_{p} s_{i,j,p}(\\vec{\\beta}) + \\sum_q^{Bkg} b_{i,j,q}(\\vec{\\beta})\\textrm{,}\n\\end{equation}\nwhere $s_{i,j,p}$ represents the $p^{th}$ unknown signal, $b_{i,j,q}$ is the $q^{th}$ estimated background process and $\\alpha_{p}$ is the {\\em scaling parameter} used to measure the amount of the different signals. The unknown expectation of the signal is parametrized by flat, uniform, positive prior distribution in $\\vec{\\alpha}$. The last element of Equations~\\ref{eq:likelihood} and~\\ref{eq:mu_likelihood} is $\\vec{\\beta}$, the vector of the nuisance parameters: it incorporates in the likelihood the systematic uncertainties. They are parametrized as fractional variations on the $s_{i,j,p}$ and $b_{i,j,q}$ rates: \n\\begin{equation}\n   s_{i,j,p}(\\vec{\\beta})= s_{i,j,p}^{central}\\prod_{k}^{Sys}(1+\\frac{\\sigma_{i,j,k}}{s_{i,j,p}^{central}}\\beta_k)\n\\end{equation}\n\\begin{equation}\n   b_{i,j,p}(\\vec{\\beta})= b_{i,j,p}^{central}\\prod_{k}^{Sys}(1+\\frac{\\sigma_{i,j,k}}{b_{i,j,p}^{central}}\\beta_k)  \n\\end{equation}\nwhere $\\frac{\\sigma_{i,j,k}}{s_{i,j,p}^{central}}$, and $\\frac{\\sigma_{i,j,k}}{b_{i,j,p}^{central}}$ are the relative uncertainties related to the systematic effect $k$ and the $\\beta_k$ variables. It is important to notice that we distinguish:\n\\begin{itemize}\n\\item {\\em total rate uncertainties}, like the luminosity dependence of the signal, where there is no bin by bin variation of the systematic effect: $\\sigma_{i,j,k}\\equiv \\sigma_{i,k}$.\n\\item {\\em shape uncertainties} where the rate variation can change on a bin by bin basis, evaluated by the ratios between the central and the varied histograms.\n\\end{itemize}\nTo account for the nuisance parameters effect and correlate them across different channels we introduce in Equation~\\ref{eq:likelihood}, for each systematic, a Gaussian probability constraint centered in zero and with unitary variance. The final result is:\n\\begin{equation}\\label{eq:likelihood_full}\n  \\mathscr{L}(\\vec{\\alpha}, \\vec{s}, \\vec{b}| \\vec{n}, \\vec{\\beta}) =\\prod^{Hists}_{i} \\prod^{Bins}_{j} \\frac{e^{-\\mu_{i,j}}\\mu_{i,j}^{n_{i,j}}}{n_{i,j}!}\\prod_{k}^{Sys}e^{-\\beta_k^{2}\/2}\\textrm{,}\n\\end{equation}\nthe previous equation is the full likelihood used in the analysis of the results.\n\n\\subsection{Parameter Measurement and Likelihood Integration}\\label{sec:likelihood_meas}\n\nBy maximizing the likelihood, we obtain the {\\em measurement} of the unknown $\\vec{\\alpha}$.\n\nThe maximum of Equation~\\ref{eq:likelihood_full} can be found in different ways: with a fit in the multidimensional space of $\\vec{\\alpha}$ and $\\vec{\\beta}$ or with the integration ({\\em marginalization}) of the nuisance parameters over their prior probability distributions. \n\nThe second technique, exploited here for the measurements, returns a {\\em Bayesian posterior distribution} of $\\vec{\\alpha}$: the minimal extension that covers $68$\\% of the distribution around the maximum gives the one-standard-deviation confidence band and the uncertainty on the measurement.\n\nSection~\\ref{sec:cx_measure} reports the results of the marginalization, where the integration of the nuisance parameters is performed numerically with a Markov-chain adaptive integration~\\cite{mk-chain_book}.\n\nHowever we perform also a fit of the likelihood function because it gives the {\\em best} outcome for all the nuisance parameters. Those values, the result of the fit and the reduced $\\chi^2$, should be consistent with the results obtained by the marginalization.\n\n\n\n\\section{Sources of Systematic Uncertainties}\\label{sec:sys_desc}\n\nAlong the preceding Chapters, we discussed several sources of rate and shape systematics and we need to include them in the likelihood Equation~\\ref{eq:likelihood_full}. \n\nThe {\\em rate-only} systematic effects that we include are summarized in the following points:\n\\begin{description}\n %\n\\item{\\em Initial and Final State Radiation} uncertainties are estimated by changing (halving and doubling) the parameters related to ISR and FSR emission on the signal MC. Half of difference between the two shifted samples is taken as the systematic uncertainty on the signal samples. The total effect on the signal acceptance is about $4\\%$.\n\\item{\\em Parton Distribution Functions} uncertainties are evaluated by reweighting each event of the signal MC according to several PDFs parametrisation and to the generator level information of the event. Then the acceptance is evaluated again giving a rate variation with respect to the original PDFs. The exact procedure is described in~\\cite{pdf_note7051}. The effect on the signal acceptance was evaluated in previous similar analyses to be around $1\\div 2$\\%. It is added in quadrature with the ISR\/FSR systematic as they both influence only the signal acceptance.\n\\item{\\em $b$-tag Scale Factor} uncertainty  comes from the measured \\texttt{SecVtx} $b$-tag efficiency variation. The $SF_{b-Tag}$ is $0.96\\pm0.5$ for a single $b$-matched and \\texttt{SecVtx} tagged jet. The uncertainty, propagated through the per-event tagging probability, is assumed to be double in the case of $c$-jets. The uncertainty is applied on the signal and on all the EWK backgrounds described in Section~\\ref{sec:mc_bkg}.\n\\item{\\em Luminosity} measurement uncertainty contributes for an overall 6\\% rate uncertainty on the signal and on all the EWK backgrounds.\n\\item{ \\em Lepton Acceptance} uncertainty derives from the quadrature sum of trigger efficiency measurements and lepton ID Scale Factors. They range from $1 \\div 2$\\% for tight lepton categories (CEM, CMUP+CMX, PHX) to $6$\\% for the EMC leptons collected by the \\mbox{${\\not}{E_T}$}  plus jets triggers. These uncertainties are applied on signal and EWK samples.\n\\item {\\em Top Production} uncertainty  is a $10$\\% rate uncertainty applied on all the top-related processes ($t\\bar{t}$, single-top $s$ and $t$ channels). It covers the cross section theoretical prediction uncertainty and the acceptance differences due to systematic top quark mass variation. \n\\item {\\em QCD} normalization, independent for each lepton channel, is constrained to be within 40\\% of the value extracted in Section~\\ref{sec:qcd}.\n\\item{\\em Mistag} uncertainty is derived from the mistag matrix $\\pm 1\\sigma$ variation and propagated to the final $W+LF$ sample. The rate variations are $11$\\% and 21\\% for single and double-tagged channels respectively.\n\\item{\\em K-Factor} uncertainty is of 30\\% on $W+HF$ rate estimate. It also includes the $b$-tag SF uncertainty used to derive this correction. We consider an uncorrelated uncertainty for $W+c\\bar{c}\/W+b\\bar{b}$  and  $W+c$ backgrounds as the first process is produced by strong interaction while the latter is of electroweak nature.\n\\item {\\em $Z+$Jets} normalization uncertainty is, conservatively, set to $45$\\% as it includes the uncertainty on $Z+HF$ production.\n\\end{description}\nAppendix~\\ref{app:rate_sys} summarizes the systematic rate variation on the different templates.\n\nWe also account for {\\em shape} systematic variations from three sources: JES, $Q^2$ and KIT modeling.\n\\begin{itemize}\n\\item {\\em Jet Energy Scale} shape uncertainty (JES) is estimated  by shifting the JES of the input templates by $\\pm 1\\sigma$ from the nominal value. The acceptance of the process is allowed to change, therefore a new background estimate is performed with the JES varied templates. This produces a simultaneous rate and shape uncertainty. All the templates are affected except the fake-$W$ sample that is derived from data.\n\\item {\\em $W+$ Jets $Q^2$ Scale} uncertainty is obtained by halving and doubling the nominal generation $Q^{2}$ (defined in Equation~\\ref{eq:scale}) of all the $W+$ jets samples ($W+b\\bar{b}$, $W+c\\bar{c}$, $W+c$, $W+LF$). Again, we have a shape and rate uncertainty as a new background estimate is performed with the  varied templates.\n\\item {\\em KIT-NN} shape uncertainty is the last systematic that we take into account. We apply two kind of independent variations: one on the fake-$W$ templates and the other on $c$-quarks and mistag templates. \nFor the multi-jet template, a variation of the flavor composition is applied allowing more or less $b$-quark like events in the fake-$W$ KIT-NN template. For templates with relevant $c$-quark component  ($WW$, $WZ$, $ZZ$, $W+c$ and $W+c\\bar{c}$) and $W+LF$ a correction to the KIT-NN output is derived from negative tags in data. This systematic variation influences only the shape and not the rate of the final discriminant because it leaves untouched the background composition.\n\n\\end{itemize}\n\nIn the case of templates with few events passing all requirements\\footnote{The MC events can be spread across a large number of bins, especially in the bi-dimensional case where each template is composed by $50\\times 4$ bins.} large statistical fluctuations can introduce a bias in the evaluation of the shape systematics. Therefore, shape variations are filtered to reduce the statistical noise. Filters are widely employed in modern image processing for different purposes. We choose a {\\em median filter smoothing}~\\cite{median_filter} because it maintains long range correlations among the histogram bins. We use a 5-bin filter along the $M_{Inv}(jet1,jet2)$ direction for the bi-dimensional, single-tagged, templates while a 3-bin filter is used for double-tagged templates\\footnote{A double bin width is used for double-tagged $M_{Inv}(jet1,jet2)$ templates w.r.t. single-tagged ones as a 5-bin long range correlation would spoil the systematic variation.}. We also apply a low and high boundary to the possible template variations to reduce them in $[0.5,2.0]$ range.\nFigure~\\ref{fig:smoothing} shows an example of the double-tagged CEM lepton $WZ$ template before and after the smoothing filtering.\n\n\n\\begin{figure} [!h]\n\\centering\n\\includegraphics[width=0.4\\textwidth,height=0.35\\textwidth]{StatRes\/Figs\/noSmooth_SVTSVT_CEM}\\hspace{0.7cm}\n\\includegraphics[width=0.4\\textwidth,height=0.35\\textwidth]{StatRes\/Figs\/smooth_SVTSVT_CEM}\n\\caption[Example of $M_{Inv}(jet1,jet2)$ Template Before and After Smoothing]{Example of the double-tagged CEM lepton $WZ$ $M_{Inv}(jet1,jet2)$ template rate variation due to JES systematic before (left) and after (right) a 3-bin median filter smoothing~\\cite{median_filter}. The red (blue) line indicates the one sigma down (up) JES variation.}\\label{fig:smoothing}\n\\end{figure}\n\n\n\n\n\n\\section{$M_{Inv}(jet1,jet2)$ and KIT-NN Distributions}\\label{sec:mjj_kit_distrib}\n\nThe final templates, including all the systematic effects, are then used to build the likelihood of Equation~\\ref{eq:likelihood_full}. \n\nWe combine a total of eight different channels: {\\em four} lepton sub-samples (CEM, PHX, Tight Muons, EMC) times {\\em two} $b$-tag prescriptions (single and double \\texttt{SecVtx} tags). For the double \\texttt{SecVtx} tagged events, the signal discrimination is based only on the di-jet invariant mass, $M_{Inv}(jet1,jet2)$, while for the single-tagged events we exploit the bi-dimensional distribution of $M_{Inv}(jet1,jet2)$ {\\em vs} KIT-NN flavor separator. The KIT-NN output ranges from $-1$ to $1$ and it is divided in four equal size bins: the rightmost is highly enriched in $b$-like jets, while the others have variable composition of $b$-like, $c$-like and $LF$-like jets. \n\nAs noted in Section~\\ref{sec:likelihood_meas}, we first obtain the maximum value of the likelihood with a fit. The  best fit result allows to check the agreement of the predicted distribution with data. The reduced $\\chi^2$ of the fit is:\n \\begin{equation}\n   \\frac{\\chi^2}{ NDoF}=\\frac{744.4}{664} = 1.12\\textrm{,}\n \\end{equation}\ncorresponding to a probability $P=0.984$. To further investigate the agreement of data and prediction, several post-fit distributions are shown from Figure~\\ref{fig:mjj_2tag_nokit} to~\\ref{fig:mjj_1tag_allkit}. We evaluate on them both the $\\chi^2$ the Kolmogorov-Smirnov tests.\n\\begin{itemize}\n\\item Figure~\\ref{fig:mjj_2tag_nokit} shows the  $M_{Inv}(jet1,jet2)$ distribution for double-tagged events;\n\\item Figure~\\ref{fig:kit_1tag} shows the KIT-NN distribution for single-tagged events after the integration of all the $M_{Inv}(jet1,jet2)$ values used in the bi-dimensional distribution;\n\\item Figure~\\ref{fig:mjj_1tag_allkit} shows the $M_{Inv}(jet1,jet2)$ distribution  for the single-tagged channel, added for all the lepton categories and integrated across all the KIT-NN values as well as the two most interesting KIT-NN regions: KIT-NN$>0.5$, $b$-enriched, and KIT-NN$<0.5$ with contribution from $c$ and $LF$ quarks.\n\\end{itemize}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.99\\textwidth]{StatRes\/Figs\/AddSelected_DT_TIG_EMC_PHX}\n\\caption[$M_{Inv}(jet1,jet2)$ Distribution for 2 \\texttt{SecVtx} Tag Events]{$M_{Inv}(jet1,jet2)$ distribution for candidate events with two \\texttt{SecVtx} tags, all the lepton categories have been added together. The best fit values for the rate and shape of the backgrounds are used in the figure. }  \n\\label{fig:mjj_2tag_nokit}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.99\\textwidth]{StatRes\/Figs\/AddSelected_ST_MjjKitProj_TIG_EMC_PHX}\n\\caption[KIT-NN Distribution for 1 \\texttt{SecVtx} Tag Events]{KIT-NN distribution for candidate events with a single \\texttt{SecVtx} tag, all the lepton categories have been added together. The best fit values for the rate and shape of the backgrounds are used in the figure.}  \n\\label{fig:kit_1tag}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.99\\textwidth]{StatRes\/Figs\/AddSelected_ST_KitAll_TIG_EMC_PHX}\\\\\\vspace{0.5cm}\n\\includegraphics[width=0.495\\textwidth]{StatRes\/Figs\/AddSelected_ST_KitLow_TIG_EMC_PHX}\n\\includegraphics[width=0.495\\textwidth]{StatRes\/Figs\/AddSelected_ST_KitHigh_TIG_EMC_PHX}\n\\caption[$M_{Inv}(jet1,jet2)$ Distribution for 1 \\texttt{SecVtx} Tag Events]{$M_{Inv}(jet1,jet2)$ distribution for candidate events with a single \\texttt{SecVtx} tag, all the lepton category have been added together. Top distribution shows the results for all KIT-NN values while the bottom distributions are separated for KIT-NN$<0.5$ (bottom left) and KIT-NN$>0.5$ (bottom right). The best fit values for the rate and shape of the backgrounds are used in the figures.}\n\\label{fig:mjj_1tag_allkit}\n\\end{figure}\n\n\n\n\n\n\n\\section{Cross Section Measurement}\\label{sec:cx_measure}\n\nThe actual cross section measurement is performed by marginalizing the likelihood with respect to the nuisance parameters and studying the resulting Bayesian posterior as a function of the diboson signal cross section.\n\nFirst we measure the total diboson cross section, $\\sigma_{Diboson}^{Obs}$, constraining the relative  $WW$ and $WZ\/ZZ$ cross sections to the SM prediction. The resulting Bayesian posterior distribution is shown in Figure~\\ref{fig:posterior_dib} together with the 68\\% and 95\\% confidence intervals. The measured cross section is:\n\\begin{equation}\n \\sigma_{Diboson}^{Obs}=\\left(0.79\\pm 0.28\\right)\\times \\sigma_{Diboson}^{SM} = \\left(14.6\\pm 5.2\\right)\\textrm{~pb,}\n\\end{equation}\nwhere the errors include statistical and systematic uncertainties and $\\sigma^{SM}_{Diboson}$ is the SM predicted cross section derived from Table~\\ref{tab:mc_dib}:\n\\begin{equation}\n  \\sigma_{Diboson}^{SM}= \\sigma^{SM}_{WW}+ \\sigma^{SM}_{WZ\/ZZ} = \\left( 18.43\\pm0.73\\right)\\textrm{~pb.}\n\\end{equation}\n\nIn order to separate the different components, we exploit the $c$ versus $b$ classification power of KIT-NN and the different sample composition of single and double-tagged events to obtain a separate measurement of $WW$ and $WZ\/ZZ$. \n\nWe iterate the cross section measurement procedure but, this time, $\\sigma_{WW}$ and $\\sigma_{WZ\/ZZ}$ are left free to float independently (i.e. not constrained to the SM ratio). Figure~\\ref{fig:posterior_2d} shows the resulting Bayesian posterior distribution. The maximum of the posterior distribution gives the value of the measured cross sections:\n\\begin{equation}\n\\sigma^{Obs,2D}_{WW}=\\left(0.50^{+0.51}_{-0.46}\\right)\\times\\sigma^{SM}_{WW} = \\left(5.7^{+5.8}_{-5.2}\\right)\\textrm{~pb}\n\\end{equation}\nand\n\\begin{equation}\n  \\sigma^{Obs,2D}_{WZ\/ZZ}=\\left(1.56^{+1.22}_{-0.73}\\right)\\times\\sigma^{SM}_{WZ\/ZZ}= \\left(11.1^{+8.7}_{-6.3}\\right)\\textrm{~pb,}\n\\end{equation}\nwhere the SM predictions are $\\sigma^{SM}_{WW} = 11.34\\pm0.66$~pb, $\\sigma^{SM}_{WZ\/ZZ} =7.09\\pm0.30$~pb and the errors are evaluated by the intersection of the $x$ and $y$ position of the maximum with the boundary of the smallest area enclosing the 68\\% of the posterior distribution. The smallest areas enclosing 68\\%, 95\\% and 99\\% of the posterior integrals give the contours of one, two and three standard deviations and are explicitly shown in Figure~\\ref{fig:posterior_2d} with correlation between $\\sigma^{Obs,2D}_{WW}$ and  $\\sigma^{Obs,2D}_{WZ\/ZZ}$.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.9\\textwidth]{StatRes\/Figs\/1d_posterior_blessSyle}\n\\caption[Bayesian Posterior Distribution of $\\sigma_{Diboson}$ Measurement]{The Bayesian posterior, marginalized over nuisance parameters, is shown. The maximum value is the central value of the cross-section, the blue and azure areas represent the smallest intervals enclosing 68\\%, 95\\% of the posterior integrals, respectively. The final cross section measurement is\n\\mbox{$\\sigma_{Diboson}^{Obs}=\\left(0.79\\pm 0.28\\right)\\times \\sigma_{Diboson}^{SM} = \\left(14.6\\pm 5.2\\right)\\textrm{~pb}$}.}\n\\label{fig:posterior_dib}\n\\end{figure}\n\nThe $WW$ and $WZ\/ZZ$ channels are also analyzed separately by projecting the two-dimensional Bayesian posterior on the $\\sigma_{WW}$ and the $\\sigma_{WZ\/ZZ}$ axes. In this way, the two processes are considered as background one at the time. For both $WW$ and $WZ\/ZZ$ we re-computed the maximum values and confidence intervals. Figure~\\ref{fig:posterior_2d_proj} shows the results, the measured cross sections are:\n\\begin{equation}\n\\sigma^{Obs}_{WW}=\\left(0.45^{ +0.35}_{ -0.32}\\right)\\times\\sigma^{SM}_{WW} = \\left(5.1^{+4.0}_{-3.6}\\right)\\textrm{~pb}\n\\end{equation}\nand\n\\begin{equation}\n  \\quad \\sigma^{Obs}_{WZ\/ZZ}=\\left(1.64^{+0.83}_{-0.78}\\right)\\times\\sigma^{SM}_{WZ\/ZZ} = \\left(11.6^{+5.9}_{-5.5}\\right)\\mathrm{~pb,}    \n\\end{equation}\nwhere the errors include statistical and systematic uncertainties.\n\n\\begin{figure}[h!]\n  \\centering\n\\includegraphics[width=0.99\\textwidth]{StatRes\/Figs\/2d_posterior}\n\\caption[2-Dim Bayesian Posterior Distribution of $\\sigma_{WW}$ and $\\sigma_{WZ\/ZZ}$ Combined Measurement]{The Bayesian posterior, marginalized over nuisance parameters (scaled to SM expectation), is shown in the plane $\\sigma_{WW}$ {\\em vs} $\\sigma_{WZ\/ZZ}$. The measured cross sections correspond to the maximum value of $\\sigma^{Obs,2D}_{WW}=0.50\\times\\sigma^{SM}_{WW} = 5.7\\textrm{~pb}$ and $\\sigma^{Obs,2D}_{WZ\/ZZ}=1.56\\times\\sigma^{SM}_{WZ\/ZZ}= 11.1\\textrm{~pb}$. The red, blue and azure areas represent smallest areas enclosing 68\\%, 95\\% and 99\\% of the posterior integrals, respectively.}\n\\label{fig:posterior_2d}\n\\end{figure}\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.495\\textwidth]{StatRes\/Figs\/1d_WW_posterior_blessSyle}\n\\includegraphics[width=0.495\\textwidth]{StatRes\/Figs\/1d_WZZZ_posterior_blessSyle}\n\\caption[Bayesian Posterior Distributions of $\\sigma_{WW}$ and $\\sigma_{WZ\/ZZ}$ Separate Measurements]{The Bayesian posterior, function of $\\sigma_{WW}$ and $\\sigma_{WZ\/ZZ}$ marginalized over nuisance parameters, is shown after projection on the $\\sigma_{WW}$ (left) and $\\sigma_{WZ\/ZZ}$ (right) axes.\nThe maximum value is the central value of the cross-section, the blue and azure areas represent the smallest intervals enclosing 68\\%, 95\\% of the posterior integrals, respectively. The final cross section measurements are: \\mbox{$\\sigma^{Obs}_{WW}=\\left(0.45^{ +0.35}_{ -0.32}\\right)\\times\\sigma^{SM}_{WW} = \\left(5.1^{+4.0}_{-3.6}\\right)\\textrm{~pb}$} and \\mbox{$\\quad \\sigma^{Obs}_{WZ\/ZZ}=\\left(1.64^{+0.83}_{-0.78}\\right)\\times\\sigma^{SM}_{WZ\/ZZ} = \\left(11.6^{+5.9}_{-5.5}\\right)\\mathrm{~pb}$}.}\\label{fig:posterior_2d_proj}\n\\end{figure}\n\n\n\\section{Evaluation of the Statistical Significance}\\label{sec:sigma_eval}\n\nTo compute the significance of the measurements we perform a {\\em hypothesis test} comparing data observation to the {\\em null hypothesis} ($H_0$).\n\nRandom generated Pseudo Experiments (PEs) are extracted from the predicted background processes distribution in the $H_0$ hypothesis (i.e. excluding  the diboson production): this is straightforward once we know the probability distribution of the background and of the nuisance parameters. Then we repeat the cross section measurements with the complete marginalization of the likelihood. We expect a distribution peaking at $\\sigma_{Diboson}^{PEs}\/\\sigma_{Diboson}^{SM} =0$ and we compare it to the measured cross section.\n\nFigure~\\ref{fig:significance_dib} shows the possible outcomes of many cross section PEs in a back\\-ground-only and in a background-plus-signal hypothesis. The number of times that a background fluctuation produces a cross section measurement greater than \\mbox{$\\sigma^{Obs}_{Diboson}=0.79\\times\\sigma^{SM}_{Diboson}$} has a $p-$value of $0.00209$. \n\nThe result is an evidence for the diboson production in $\\ell\\nu +HF$ final state with a significance\\footnote{A two sided significance estimate is used because both upper and lower fluctuation of the cross section measurement are considered in the integral of the {\\em null} hypothesis cross section distribution above the measured value.} of $3.08\\sigma$.\n\nThen, we evaluate the single $WW$ and $WZ\/ZZ$ significances in a similar way: PEs are generated with null hypothesis for both $WW$ and $WZ\/ZZ$ signals. Then, the cross section PEs measurements are projected along the $\\sigma_{WW}$ vs $\\sigma_{WZ\/ZZ}$ axes and compared with $\\sigma^{Obs}_{WZ}$ and $\\sigma^{Obs}_{WZ\/ZZ}$. The result of the $p-$value estimates are shown in Figure~\\ref{fig:significance_2d_proj}. \n\nWe obtain: $p-$value$_{WW}=0.074565$ and $p-$value$_{WZ\/ZZ}=0.011145$. They correspond to a significance of $1.78\\sigma$ and $2.54\\sigma$ for $WW$ and $WZ\/ZZ$ respectively.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.9\\textwidth]{StatRes\/Figs\/significance_diboson}\n\\caption[Diboson Significance Evaluation]{Possible outcomes of many diboson cross section measurements from Pseudo Experiments (PEs) generated in a background-only and in a background-plus-signal hypothesis. The $p-$value for $\\sigma^{Obs}_{Diboson}=0.79\\times\\sigma^{SM}_{Diboson}$ is $0.00209$, corresponding to a significance of $3.08\\sigma$.}\n\\label{fig:significance_dib}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.495\\textwidth]{StatRes\/Figs\/significance_WW}\n\\includegraphics[width=0.495\\textwidth]{StatRes\/Figs\/significance_WZZZ}\n\\caption[$WW$ and $WZ$ Separate Significance Evaluation]{Possible outcomes of many diboson cross section measurements from Pseudo Experiments (PEs) generated in a background-only and in a background-plus-signal hypothesis in the $\\sigma_{WW}$ vs $\\sigma_{WZ\/ZZ}$ plane and then projected on the  $\\sigma_{WW}$ (left) and $\\sigma_{WZ\/ZZ}$ (axis). The $p-$values of $0.074565$ and $0.011145$ correspond to a significance of $1.78\\sigma$ and $2.54\\sigma$ for $WW$ and $WZ\/ZZ$ respectively.}\n\\label{fig:significance_2d_proj}\n\\end{figure}\n\n\n\n\\chapter{Theoretical and Experimental Overview}\\label{chap:physics}\n\n\nThe goal of particle physics is the understanding of the principles of Nature.\n\nThe quest is pursued through the scientific method: the observation of a phenomenon is explained by a hypothesis that must be, successively, verified or rejected by experimental evidences.\n\nIn this prospect, the observed phenomenon is the existence itself of the atomic and sub-atomic structure of matter, the hypothesis is the Standard Model theory of Elementary Particles and Fundamental Interaction (SM) while the experimental tools are the high energy physics colliders and detectors, available nowadays.  \n\nThe main infrastructure of the SM~\\cite{pdg2010, glasgow, weinberg_lep} was developed in the 70's and, since then, it showed to be a very successful theory. One of its main success was the prediction of new elementary particles, later observed at hadron collider experiments. The discovery of the $W$ and $Z$ force carrier vector bosons~\\cite{ua1_w_obs, ua2_w_obs, ua1_z_obs,ua2_z_obs} and of the $top$ quark~\\cite{CDF_top_obs,d0_top_obs} shed light on the fundamental structure of the matter. \n\nStrengthened by these results, SM describes the {\\em electromagnetic}, {\\em weak} and {\\em strong} interactions, three of the four fundamental forces that compose the physics description of Nature. The fourth force, gravitational interaction, is left out but it is negligible at atomic and subatomic scale. \n\nDespite the great success of the SM, one predicted particle has not yet been observed: the \\emph{Higgs boson}, an essential element for the inclusion of the mass of the particles in the equation of motion~\\cite{higgs1964,englert1964}. Because of this, a considerable effort is ongoing to prove or disprove the existence of the Higgs boson\\footnote{Some hints of its existence are confirmed by the present experiments~\\cite{tev_cmb, atlas_cmb, cms_cmb}.} and the analysis of the data collected by the CDF II experiment, situated at the Tevatron $p\\bar{p}$ collider, plays a relevant role in it~\\cite{cdf_cmb}.\n\nIn the scenario~\\cite{Tev4LHC} of a low-mass Higgs ($m_H\\lesssim 135$~GeV$\/c^{2}$), one of the most relevant CDF search channels is the $WH$ associate production with a $\\ell\\nu+b\\bar{b}$ final state. In light of this, the  diboson decay channel considered for the presented analysis becomes a perfect benchmark for the Higgs boson search. The accurate SM prediction for the diboson production and decay can be used as a standard comparison for an unknown process. \n\nThis Chapter introduces the relevant aspects of the SM theory (Section\\ref{sec:sm_teo}), some of the latest results of the Higgs boson search (Section\\ref{sec:higgs_search}) and several, diboson related, experimental confirmation of the SM validity (Section\\ref{sec:dib_res}).\n\n\n\n\n\\section{The Standard Model of Elementary Particles}\\label{sec:sm_teo}\n\nThe SM is defined by the language of mathematics and theoretical physics so that it can be used to produce accurate predictions that have to be verified by the experiments. \nIn this language a particle is defined by a local quantum field. If no interaction is present, the free field is described by only two quantum numbers, the spin and the mass; if interactions are presents, the {\\em Gauge symmetries} can elegantly describe them: new quantum numbers classify the type and the strength of force while new particles, force-mediator vector bosons, are used to propagate the interaction.\n\nThe fundamental building blocks of matter, observed up to now, are the spin-$1\/2$ fields (\\emph{fermionic}), named \\emph{quarks} and \\emph{leptons}, and the spin-$1$ ({\\em vector}) fields, named \\emph{gauge bosons}. The leptons are divided into three \\emph{generations}, or \\emph{families}, and are\ngrouped in a left weak isospin doublet\\footnote{See Section~\\ref{sec:SM} for \nthe explanation of the weak isospin quantum number.} and a right weak isospin singlet. Also quarks are divided into three \\emph{flavor families} but weak \nisospin classification mixes quark doublets of different families. Quarks are also subject to the strong interaction, described by the \\emph{color} quantum number. Finally the {\\em charge} quantum number is used, for both quarks and leptons, to describe the electromagnetic interaction. \n\nThe force mediators are $W^\\pm$, $Z^0$, $\\gamma$, that carry electroweak \nforce, and $g$ (\\emph{gluons}), which mediate strong interaction. A short \nsummary of the SM fundamental particles is reported in Figure~\\ref{fig:particles}.\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.7\\textwidth]{Theory\/Figs\/SM_Particles}\n\\caption[SM Particles and Their Properties]{Quarks, leptons and gauge bosons in Standard Model and some of their \ncharacteristics~\\cite{pdg2010}, for each particle the corresponding antiparticle exists.}\\label{fig:particles}\n\\end{center}\n\\end{figure}\n\n\n\\subsection{Gauge Theory Example: QED}\n\nThe Quantum Electro-Dynamic (QED) is a perfect example to explain the importance of gauge invariance.\n\nThe equation that describes free fermionic fields is the {\\em Dirac} Lagrangian:\n\\begin{equation}\\label{eq:dirac}\n\\mathscr{L}(x)=\\bar{\\psi}(x)(i\\gamma^{\\mu}\\partial_{\\mu}- m)\\psi(x),\n\\end{equation}\nwhere $\\psi$ is the Dirac field of mass $m$ and $\\gamma^{\\mu}$ are the Dirac's \nmatrices. Equation~\\ref{eq:dirac} satisfies the \\emph{global} $U(1)$ symmetry\ntransformation: \n\\begin{equation}\n\\psi(x)\\to e^{iQ\\alpha}\\psi(x),\n\\end{equation}\nwith the electric charge $Q$ and the space independent parameter $\\alpha$ ($x$ is a space-time 4-vector).\nThe \\emph{Noether theorem}~\\cite{qftbook_weinberg} states that when a symmetry \nappears in a Lagrangian there is a corresponding conserved current. In the case of the Dirac field:\n\\begin{equation}\n\\partial_{\\mu}j^{\\mu} = 0\\quad\\mathrm{with}\\quad j^{\\mu} = -Q\\bar{\\psi}\\gamma^{\\mu}\\psi,\n\\end{equation}\ndescribes the conservation of charge, i.e the time component of the current 4-vector $j^{\\mu}$, integrated over the space, is a constant.\n\nAn elegant way to introduce interaction in the free Lagrangian is to shift from\nthe global, i.e. space independent, $U(1)$ transformation to a \\emph{local} \n$U(1)$ transformation, i.e. with a space dependent parameter $\\alpha(x)$:\n\\begin{equation}\n\\psi(x)\\to e^{iQ\\alpha(x)}\\psi(x).\n\\end{equation}\nTo maintain the gauge invariance condition in the Lagrangian~\\ref{eq:dirac}, a \ncovariant derivative $D_{\\mu}$ is introduced:\n\\begin{equation}\\label{eq:em_derivative}\n\\partial_{\\mu}\\to D_{\\mu}=\\partial_{\\mu}+iQA_{\\mu}, \\qquad\nD_{\\mu}\\psi(x)\\to e^{iQ\\alpha(x)}D_{\\mu}\\psi(x),\n\\end{equation}\nwhere the new vector field $A_{\\mu}$ is defined to transform in the following manner:\n\\begin{equation}\nA_{\\mu}\\to A_{\\mu}-\\frac{1}{Q}\\partial_{\\mu}\\alpha(x)\\mathrm{.}\n\\end{equation}\nEquations~\\ref{eq:em_derivative} and~\\ref{eq:dirac} can be composed to give the final QED Lagrangian:\n\\begin{equation}\\label{eq:L_qed}\n\\mathscr{L}_{QED}=\\bar{\\psi}(x)(i\\gamma^{\\mu}D_{\\mu}- m)\\psi(x) - \\frac{1}{4}F_{\\mu\\nu}F^{\\mu\\nu},\n\\end{equation}\nwhere $F_{\\mu\\nu}\\equiv\\partial_{\\mu}A_{\\nu}-\\partial_{\\nu}A_{\\mu}$ is the covariant kinetic term of $A_{\\mu}$.\nThe Dirac equation of motion for a field $\\psi$ undergoing electromagnetic interaction is obtained by applying the Euler-Lagrange equation~\\cite{qftbook_weinberg} to the QED Lagrangian:\n\\begin{equation}\n(i\\gamma^{\\mu}\\partial_{\\mu}- m)\\psi(x) = Q\\gamma^{\\mu}A_{\\mu}\\psi(x)\\mathrm{,}\n\\end{equation}\nthe force is mediated by the massless vector field $A_{\\mu}$. A mass term in \nthe form $\\frac{1}{2}m^{2}A_{\\mu}A^{\\mu}$ would break apart gauge invariance of\nEquation~\\ref{eq:L_qed}, indeed this is consistent with zero mass of the photon.\n\n\n\n\\subsection{Standard Model Theory}\\label{sec:SM}\n\nThe leptonic sector of the SM\\footnote{Only electroweak interaction on the leptons is considered here to simplify the discussion.} is based on the gauge group:\n\\begin{equation}\n  SU(2)\\otimes U(1)\\textrm{,}  \n\\end{equation}\nwhere $SU(2)$ is the non-Abelian group used in the spin algebra, and $U(1)$ is the Abelian group equivalent to the one used in QED. \nThe quantum number arising from $SU(2)$ is the \\emph{weak isospin}, $\\vec{T}$, and the one arising from $U(1)$ is {\\em hypercharge}, $Y$. They are related to the observed charge of real particles, $Q$, by the \nthe Gell-Mann-Nishijima equation:\n\\begin{equation}\nQ = T_{3}+\\frac{Y}{2}\\mathrm{.}\n\\end{equation}\nwhere $T_{3}$ is the third component of weak isospin.\n\nElectroweak interaction can be explained with a simplified model containing only two spin $1\/2$, elementary, massless, fermions, $f$ and $f'$, such that \n$Q_{f}=Q_{f'}+1$ ($Q$ is the electric charge). Weak interaction is built from V-A currents, i.e. left and right components are \ndefined and collected into a left doublet field and into two right singlet fields:\n\\begin{equation}\\label{eq:smL}\n\\psi_{1} \\equiv \\left( \\begin{array}{c} f_{L}(x) \\\\ f'_{L}(x) \\end{array}\\right), \\qquad \\psi_{2} \\equiv f_{R}(x) \n\\qquad \\psi_{3} \\equiv f'_{R}(x),\n\\end{equation}\nwith:\n\\begin{equation}\nf_{L,R}(x)=\\frac{1}{2}(1\\pm\\gamma_{5})f(x), \\qquad \n\\bar{f}_{L,R}(x)=\\frac{1}{2}\\bar{f}(x)(1\\pm\\gamma_{5}),\n\\end{equation}\n\\begin{equation}\nf'_{L,R}(x)=\\frac{1}{2}(1 \\pm \\gamma_{5})f'(x), \\qquad \n\\bar{f'}_{L,R}(x)=\\frac{1}{2}\\bar{f'}(x)(1 \\pm \\gamma_{5}).\n\\end{equation}\nAll the leptonic sector of the SM is explained by such pattern: the left doublet with $T_{3}=\\pm 1\/2$, $Y = 1$ is the charged lepton $f$\nplus the corresponding neutrino $f'$, while the right singlet with $T_{3}=0$, $Y = -2$ is only the charged lepton.\n\nThe electroweak interaction is introduced through $SU(2)\\otimes U(1)$ gauge transformation:\n\\begin{equation}\n\\psi_{j}(x)\\to\\psi'_{j}(x)= e^{i\\frac{\\tau}{2}\\cdot\\vec{\\alpha}(x) + iY_{j}\\beta(x)}\\psi_{j}(x),\n\\end{equation}\nof the free field Lagrangian:\n\\begin{equation}\\label{eq:freeL}\n\\mathscr{L_{0}}(x)=\\sum_{j=1}^{3}i\\bar{\\psi}_{j}(x)\\gamma^{\\mu}\\partial_{\\mu}\\psi_{j}(x)\\textrm{,}\n\\end{equation}\nwhere a covariant derivative is also introduced to maintain gauge invariance. The result is:\n\\begin{equation}\\label{eq:SML}\n\\mathscr{L}_{I}(x)=\\sum_{j=1}^{3}i\\bar{\\psi}_{j}(x)\\gamma^{\\mu}D_{\\mu}^{j}\\psi(x)_{j} - \\frac{1}{4}\\vec{W}_{\\mu,\\nu} \\vec{W}^{\\mu,\\nu} - \\frac{1}{4} B_{\\mu,\\nu}B^{\\mu,\\nu},\n\\end{equation}\n\\begin{equation}\\label{eq:bosons}\n\\mathrm{with}\\qquad D_{\\mu}^{j}= \\partial_{\\mu}-ig\\frac{\\tau}{2}\\cdot\\vec{W}_{\\mu}(x)-ig'Y_{j}B_{\\mu}(x)\\mathrm{,}\n\\end{equation}\n\\begin{equation}\\label{eq:tensor}\n\\mathrm{and}\\quad \\vec{W}_{\\mu,\\nu}=\\partial_{\\mu}\\vec{W}_{\\nu} - \\partial_{\\nu} \\vec{W}_{\\mu} + g \\vec{W}_{\\mu}\\times\\vec{W}_{\\nu}\\textrm{,}\\quad B_{\\mu,\\nu}= \\partial_{\\mu}B_{\\nu} - \\partial_{\\nu} B_{\\mu}.\n\\end{equation}\nEquation~\\ref{eq:bosons} contains three vector bosons ($\\vec{W}_{\\mu}$) from the $SU(2)$ \ngenerators, one vector boson ($B_{\\mu}$) from the $U(1)$ generator and four coupling constants: \n\\begin{equation}\n  g,\\quad g'Y_{j} \\quad \\mathrm{with}\\quad j=1,2,3\\mathrm{.}\n\\end{equation}\nAfter some algebra the Lagrangian~\\ref{eq:SML} can be written in the form:\n\\begin{equation}\n\\mathscr{L}_{I}(x)= \\mathscr{L}_{CC}(x) + \\mathscr{L}_{NC}(x)\\mathrm{,}\n\\end{equation}\nwith a {\\em charged current} contribution ($\\mathscr{L}_{CC}$) and a {\\em neutral\ncurrent contribution} ($\\mathscr{L}_{NC}$). The charged current contribution is seen only by left doublet fields:\n\\begin{equation}\\label{eq:W}\n\\mathscr{L}_{CC}(x)=\\frac{g}{2\\sqrt{2}}\\Big\\{\\bar{f}(x)\\gamma^{\\mu}(1-\\gamma_{5})f'(x)\\frac{1}{\\sqrt{2}}W_{\\mu}^{+}(x) + h.c.\\Big\\},\n\\end{equation}\nwith $W_{\\mu}^{+}(x)$ defined by a linear combination of $W_{\\mu}^{1}(x)$ and $W_{\\mu}^{2}(x)$. Equation~\\ref{eq:W} defines the Lagrangian for charged current \ninteractions mediated by the $W$ boson.\n\nThe fermion coupling to $Z^{0}$ field and photon ($A$) field is produced in a similar way, by an appropriate orthogonal linear combination of neutral vector fields $B_{\\mu}(x)$\nand $W_{\\mu}^{0}(x)$:\n\\begin{equation}\n\\mathscr{L}_{NC}(x)= \\mathscr{L}_{NC}^{A}(x) + \\mathscr{L}_{NC}^{Z}(x)\\textrm{,}\n\\end{equation}\nwhere:\n\\begin{eqnarray}\n\\mathscr{L}_{NC}^{A}(x)=\\sum^{3}_{j=1}\\bar{\\psi}_{j}(x)\\gamma^{\\mu}\\big[g\\frac{\\tau_{3}}{2}\\sin \\theta_{W}\n+ g'Y_{j}\\cos \\theta_{W}\\big]\\psi_{j}(x)A_{\\mu}(x),\\\\\n\\mathscr{L}_{NC}^{Z}(x)=\\sum^{3}_{j=1}\\bar{\\psi}_{j}(x)\\gamma^{\\mu}\\big[g\\frac{\\tau_{3}}{2}\\cos \\theta_{W}\n+ g'Y_{j}\\sin \\theta_{W}\\big]\\psi_{j}(x)Z_{\\mu}(x),\n\\end{eqnarray}\nthe parameter $\\theta_{W}$ is named Weinberg angle and the generic four coupling constants, arising from the SM group structure, have now a physical meaning:\n\\begin{equation}\ng\\sin \\theta_{W} = e,\n\\end{equation}\n\\begin{equation}\ng'\\cos \\theta_{W}Y_{1}=e(Q_{f}-1\/2), \\qquad g'\\cos \\theta_{W}Y_{2}=eQ_{f},\n\\end{equation}\n\\begin{equation}\ng'\\cos \\theta_{W}Y_{3}=eQ_{f'}\\textrm{.}\n\\end{equation}\nPrevious equations are the core of the Standard Model. However one problem remains as no mass term appears for any of the fields: the spontaneous symmetry breaking and Higgs mechanism can generate the mass term without breaking the gauge invariance.\n\n\n\\subsection{Spontaneous Symmetry Breaking}\n\nSpontaneous symmetry breaking can be applied to Equation~\\ref{eq:smL} to give mass to $W^{\\pm}$ and $Z^{0}$ bosons. The actual application procedure is named {\\em Higgs mechanism}: two complex scalar fields are introduced such that they form an iso-doublet with respect to $SU(2)$:\n\\begin{equation}\\label{eq:phi_doublet}\n\\phi(x)\\equiv\\left(\\begin{array}{c} \\phi^{+}(x) \\\\\\phi^{0}(x) \\end{array}\\right),\n\\end{equation}\nthe field $\\phi^{+}(x)$ is the charged component of the doublet and $\\phi^{0}(x)$ is neutral component. The Higgs potential, $ V_{H}(x)$, is then defined as:\n\\begin{equation}\\label{eq:h_pot}\n V_{H}(x)\\equiv -\\mu^{2}\\phi^{\\dagger}(x)\\phi(x)-h\\big[\\phi^{\\dagger}(x)\\phi(x)\\big] ^{2},\n\\end{equation}\nwith $h>0$ and $\\mu^{2}<0$. The neutral scalar field $\\phi^{0}(x)$ has an unconstrained (i.e. to be obtained from measurements) vacuum expectation value of $\\frac{\\lambda}{\\sqrt{2}}$, so that (at first order)\nthe field  $\\phi(x)$ is:\n\\begin{equation}\\label{eq:doublet}\n\\phi(x)=e^{\\frac{i}{\\lambda}\\vec{\\tau}\\cdot\\vec{\\theta(x)}}\\left(\\begin{array}{c} 0 \\\\\\frac{1}{\\sqrt{2}}\\big(\\lambda + \\chi(x)\\big) \\end{array}\\right),\n\\end{equation}\nwhere the $SU(2)$ gauge freedom is explicit. This permits to \\emph{gauge away} three of the four components of field $\\phi(x)$ leaving only one real scalar field: \n\\begin{equation}\\label{eq:}\n  \\phi^{0}(x)= \\frac{1}{\\sqrt{2}}\\big(\\lambda + \\chi(x)\\big).\n\\end{equation}\nThe explicit evaluation of Equation~\\ref{eq:h_pot} and the coupling of $\\phi^{0}(x)$ with the electroweak force carriers ($W^{\\pm}$, $Z^0$) gives the last piece of the SM Lagrangian:\n\\begin{eqnarray}\\label{eq:finalL}\n\\mathscr{L}(x)&=& \\frac{1}{4}g^{2}\\lambda^{2}W_{\\mu}^{\\dagger}(x)W^{\\mu}(x) + \\frac{1}{1}(g^{2}+g'^{2})\\lambda^{2}Z_{\\mu}(x)Z^{\\mu}\\\\\n&&+\\frac{1}{2}g^{2}\\lambda W^{\\dagger}_{\\mu}(x)W^{\\mu}(x)\\chi(x)+\\frac{1}{4}g^{2}W^{\\dagger}_{\\mu}W^{\\mu}\\chi^{2}(x)\\nonumber\\\\\n&&+\\frac{1}{4}(g^{2}+g'^{2})\\lambda Z_{\\mu}(x)Z^{\\mu}(x)\\chi(x)+\\frac{1}{8}g^{2}Z_{\\mu}(x)Z^{\\mu}(x)\\chi^{2}(x)\\nonumber\\\\\n&&+\\frac{1}{2}\\big[\\partial^{\\mu}\\chi(x)\\partial_{\\mu}\\chi(x) + 2\\mu^{2}\\chi^{2}(x)\\big]\\nonumber\\\\\n&&+\\frac{\\mu^{2}}{\\lambda}\\chi^{3}(x)+\\frac{\\mu^{2}}{4\\lambda^{2}}\\chi^{4}(x)-\\frac{1}{4}\\lambda^{2}\\mu^{2}\\nonumber\\textrm{.}\n\\end{eqnarray}\nWe conclude that the $Z^{0}$ and $W^{\\pm}$ bosons have acquired mass:\n\\begin{equation}\n  M_{W}=\\frac{1}{2}\\lambda g,  \n\\end{equation}\n\\begin{equation}\n  M_{Z}=\\frac{1}{2}\\lambda \\sqrt{g^{x}+g'^{x}}=\\frac{1}{2}\\frac{\\lambda g}{\\cos \\theta_{w} },  \n\\end{equation}\nsome parameters are now constrained, for example:\n\\begin{equation}\n  M_{Z}=\\frac{M_{W}}{cos\\theta{w}}\\geqslant M_{W},\n\\end{equation}\n\\begin{equation}\n  \\frac{G_{F}}{\\sqrt{2}}=\\frac{g^{2}}{8M^{2}_{W}},\n\\end{equation}\nwhile the Higgs mass, $M_{\\chi}=\\sqrt{-2\\mu^{2}}$ ($m_{H}$ is also used), remains a free parameter to be measured by the experiments. \nThe Higgs mechanism can generate also fermion masses if a Yukawa coupling is added:\n\\begin{eqnarray}\\label{eq:h_yukawa}\n\\mathscr{L_{f}}(x)&=&c_{f'}\\Bigg[(\\bar{f}(x),\\bar{f}'(x))_{L} \\left(\\begin{array}{c} \\phi^{+}(x) \\\\\\phi^{0}(x) \\end{array}\\right)\\Bigg]f_{R}'(x)\\\\\n&&+ c_{f}\\Bigg[(\\bar{f}(x),\\bar{f}'(x))_{L} \\left(\\begin{array}{c} -\\bar{\\phi}^{0}(x) \\\\\\phi^{-}(x) \\end{array}\\right)\\Bigg]f_{R}(x) + h.c.\\mathrm{,}\\nonumber\n\\end{eqnarray}\ntherefore, after symmetry breaking, fermion masses have the form:\n\\begin{equation}\nm_{f}=-c_{f}\\frac{\\lambda}{\\sqrt{2}}\\mathrm{,}\\qquad m_{f'}=-c_{f'}\\frac{\\lambda}{\\sqrt{2}}\\mathrm{,}\n\\end{equation}\nwhere the constants $c_{f}$ and $c_{f'}$ can be derived by the measurements of the fermion masses.\n\n\n\n\n\\section{Higgs Boson Search and Results}\\label{sec:higgs_search}\n\nThe mechanism that generates the mass of all the SM particles is a key element for the understanding of Nature, therefore it is not a surprise that the Higgs boson search is considered, by the High Energy Physics community, one of the most interesting research topics. \n\nAlthough the existence of the Higgs particle is unknown, its hypothetical couplings and decay properties are important for the interpretation of the experimental results: Figure~\\ref{fig:higgs_cx} shows the Higgs production cross section~\\cite{Tev4LHC}, at $\\sqrt{s}=1.96$~TeV and $\\sqrt{s}=7$~TeV, and Figure~\\ref{fig:higgs_br} shows the Higgs decay Branching Ratios~\\cite{LHCHiggs_cx_wg:2011ti} (BR) for a mass range $100< m_H<200$~GeV$\/c^{2}$. \n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[angle=-90,width=0.495\\textwidth]{Theory\/Figs\/tev_higgs_cx}\n\\includegraphics[angle=-90,width=0.495\\textwidth]{Theory\/Figs\/lhc_higgs_cx}\n\\caption[Higgs Boson Production Cross Sections]{Higgs boson production cross sections in different modes~\\cite{Tev4LHC}. At the Tevatron $p\\bar{p}$, $\\sqrt{s}=1.96$~TeV (left) and at the LHC $pp$, $\\sqrt{s}=7$~TeV (right). In 2012 the LHC raised the collision energy to  $\\sqrt{s}=8$~TeV increasing still more the Higgs production cross section.}\\label{fig:higgs_cx}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.75\\textwidth]{Theory\/Figs\/YRHXS2_BR_Fig1-2}\n\\caption[Higgs Boson Decay Branching Ratios]{Higgs boson decay branching ratios~\\cite{LHCHiggs_cx_wg:2011ti}. The Higgs boson couples to the mass of the particles therefore the decay to the $b\\bar{b}$ quark pair is favored for $m_H\\lesssim 135$~GeV$\/c^{2}$ while $H\\to W^+W^-$ decay dominates for larger masses.}\\label{fig:higgs_br}\n\\end{center}\n\\end{figure}\n\nThe LEP experiments were the first to test the existence of  Higgs boson for masses larger that $100$~GeV$\/c^{2}$, but, as no signal evidence was found~\\cite{lep}, all the searches were combined to provide a lower mass limit of $m_H>114.4$~GeV$\/c^{2}$, at 95\\% Confidence Level (CL). In the latest years also the experiments situated at the Tevatron and LHC colliders provided several mass exclusion limits~\\cite{tev_cmb, cms_cmb, atlas_cmb}. Figure~\\ref{fig:tev_cmb} gives a summary of the 95\\% CLs of all the three colliders, overlaid to the Tevatron result in the mass range $100< m_H<200$~GeV$\/c^{2}$, only a tiny fraction of the phase space is still available to the Higgs presence and, interestingly, a broad excess appears in mass range $110\\lesssim m_H \\lesssim 140$~GeV$\/c^{2}$. \n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth]{Theory\/Figs\/tev_lhx_excl}\n\\caption[Tevatron Combined 95\\% Exclusion CL]{Mass exclusion limits (95\\% CL) obtained from the combination of all the Tevatron searches for a SM Higgs boson~\\cite{tev_cmb}. Exclusion limits obtained  from the CMS~\\cite{cms_cmb} and Atlas~\\cite{atlas_cmb} Collaboration are overlaid.}\\label{fig:tev_cmb}\n\\end{center}\n\\end{figure}\n\nIn a short time, as the LHC continues the data taking, a conclusive statement about the Higgs existence will be possible.\n\nHowever, the Tevatron and LHC results are also complementary because they investigate different couplings of the Higgs boson.\nThe LHC experiments base most of the low-mass ($m_H\\lesssim 135$~GeV$\/c^{2}$) sensitivity on the $H\\to\\gamma\\gamma$ final state. This channel offers an excellent mass resolution and background rejection although at the price of a very low BR (see Figure~\\ref{fig:higgs_br}). This is optimal for the higher background rate and Higgs production cross sections availables at a the LHC $pp$ collisions of energy $\\sqrt{s}=7$~TeV and $\\sqrt{s}=8$~TeV.\n\nThe Tevatron experiments rely more on the $H\\to b\\bar{b}$ final state, where the Higgs is produced in association with a vector boson ($WH$ and $ZH$ production). The lower production cross section is compensated by the larger BR (see Figures~\\ref{fig:higgs_cx} and~\\ref{fig:higgs_br}) while the presence a leptonic decay of the $W$ or $Z$ boson allows to keep the background under control. Figure~\\ref{fig:tev_bb} shows the  $H\\to b\\bar{b}$ only Tevatron combined search result. Furthermore the investigation of the $H\\to b\\bar{b}$ BR is important to understand the coupling of the Higgs with the fermion masses and for the confirmation of the SM assumption coming from Equation~\\ref{eq:h_yukawa}.\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth]{Theory\/Figs\/tev_bb_cmb}\n\\caption[Tevatron $H\\to b\\bar{b}$ Combined 95\\% Exclusion CLs]{Mass exclusion limits (95\\% CL) obtained from the combination of the Tevatron searches~\\cite{tev_cmb} exploiting the $H\\to b\\bar{b}$ final state.}\\label{fig:tev_bb}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\\section{Status of the Diboson Measurements}\\label{sec:dib_res}\n\nIn the context of the Higgs searches at the Tevatron, the diboson observation in  $\\ell\\nu + b\\bar{b}$  final state is particularly relevant as it is a direct check of the  $p\\bar{p}\\to WH\\to \\ell\\nu + b\\bar{b}$ analyses.\n\nThe relevant tree-level diagrams involved in dibosons production are shown in Figure~\\ref{fig:wz_prod}. \n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.75\\textwidth]{Theory\/Figs\/wz_prod}\n\\caption[$WW$ and $WZ$ Tree-Level Diagrams]{$WW$ and $WZ$ production Feynman diagrams at tree-level, $t$-channel\n(left) and $s$-channel (right).}\\label{fig:wz_prod}\n\\end{center}\n\\end{figure}\nThe simultaneous emission of a $W$ and the $Z$ vector bosons can happen in the $t$-channel (left of Figure~\\ref{fig:wz_prod}), with the exchange of a virtual quark, or in the $s$-channel (right of Figure~\\ref{fig:wz_prod}) with the exchange of a virtual force carrier. The second case is due to the {\\em non-Abelian} characterisctics of the $SU(2)$ group that origins a {\\em Triple Gauge Coupling} (TGC) in the kinetic term (Equation~\\ref{eq:tensor}) of the SM Lagragnian. The cross sections for the $WW$ and $WZ$ production\\footnote{In this analysis we consider also $ZZ\\to \\ell\\ell + HF$ as a signal when a lepton is misidentified, however this contributes to less than 3\\% of the total diboson signal yield.} calculated at NLO~\\cite{mcfm_prd,mcfm_url}, for $p\\bar{p}$ collision at $\\sqrt{s}=1.96$~TeV, are:\n\n\\begin{equation}\n  \\sigma_{p\\bar{p}\\to WW} = 11.34 \\pm 0.66\\mathrm{~pb};\\quad\\sigma_{p\\bar{p}\\to WZ} = 3.47 \\pm 0.21\\mathrm{~pb};  \n\\end{equation}\nAn increase in the TGC, $s$-channel, production cross section would point to a possible contribution from New Physics (NP) processes. However, the precision that we can obtain in the $\\ell\\nu+HF$ final state is not comparable to the one achievable in other channels with higher leptonic multiplicity (see Table~\\ref{tab:dib_measurements}).\n\nFigure~\\ref{fig:torte} shows the small dibosons BR in $b$ or $c$ quarks\\footnote{The experimental identification of $HF$ quarks, described in Section~\\ref{sec:secvtx} has also a low efficiency.}, furthermore \nthe hadronic final state is background rich and has a low invariant mass resolution.  The search is challenging but it is a perfect standard candle to confirm the understanding of the $\\ell\\nu+HF$ dataset on a well known SM process.\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.75\\textwidth]{Theory\/Figs\/wzpie}\\vspace{0.2cm}\n\\caption[Diboson Final States BR]{Branching ratios into the different final states allowed to $WW$ and $WZ$ production. $\\ell\\nu+ HF$ final state is highlighted in red.}\\label{fig:torte}\n\\end{center}\n\\end{figure}\n\n\nDiboson related experimental results are widely present in literature. LEP~\\cite{lep_diboson} performed the first measurements exploiting all the decay channels of the $WW$ and $ZZ$ processes: their cross sections were measured with good precision. The use of an $e^+e^-$ machine allowed also the observation of hadronically decaying $W$'s, in addition to the semi-leptonic $WW$ decays. However $WZ$ production was not allowed at LEP, since it is forbidden by charge conservation.\n\nHadron colliders, both Tevatron and LHC, observed $WW$, $WZ$ and $ZZ$ production in their fully leptonic decay modes, obtaining excellent agreement with the SM prediction~\\cite{cdfWZlllv_2012, d0WZZZlep_2012, cmsDib_2011, cdfZZllll_2012, atlasZZ_2012, cdfWWllvv_2012, d0WWllvv_2009, cmsWW_2012, atlasWW_2012, cdfDibMetJet_2009, cdfDiblvJet_Mjj2010, cdfDiblvJet_ME2010, d0DiblvJet_2011, cdfDiblvHF_Mjj2011, cdfWZZZCombHF_2012, d0WZZZCombHF_2011,  tevWZZZCombHF_2012}.\n\nThe semi-leptonic final states, more difficoult to isolate due to the background rich hadronic environment, were observed at the Tevatron, both at CDF and D0 experiments. In this thesis we present the measurement in the channel $p\\bar{p}\\to WW\/WZ\\to \\ell\\nu +HF$ with an update and improvement of the analysis described in Appendix~\\ref{App:7.5}~\\cite{cdfDiblvHF_Mjj2011}, performed on a smaller dataset in 2011 ($7.5$~fb$^{-1}$). \n\nRecently~\\cite{cdfWZZZCombHF_2012, d0WZZZCombHF_2011, tevWZZZCombHF_2012}, CDF and D0 produced the evidence for $WZ\/ZZ$ production with $HF$ jets in the final state and $WW$ production considered as background. Both experiments produced single results in the three semileptonic diboson decay modes:\n\\begin{equation}\n  WZ\/ZZ \\to \\ell\\ell+ HF\\textrm{;}\n\\end{equation}\n\\begin{equation}\n  WZ\/ZZ \\to \\ell\\nu + HF\\textrm{;}\n\\end{equation}\n\\begin{equation}\n  WZ\/ZZ \\to \\nu\\nu+ HF\\textrm{;}\n\\end{equation}\nwhere the $\\nu$ may indicate a lepton failing the identification. The analyses were performed as an exact replica of corresponding Higgs searches in those channels, with the final signal discriminants re-optimized for $WZ\/ZZ$ extraction.  Heavy usage of multivariate tecniques, as for example in the $HF$ selection strategy~\\cite{HOBIT, BDT_D0} and in the final signal--background discrimination, is the key of the impressive sensitivity of these analyses. The final combined cross section measurement, with a significance of $4.6\\sigma$, is: \n\\begin{equation}\n  \\sigma_{WZ+ZZ}  =  4.47\\pm 0.67^{+0.73}_{-0.72}\\textrm{~pb,}\n\\end{equation}\nwhere the SM ratio between $WZ$ and $ZZ$ is imposed. This confirms the SM production prediction\\footnote{Both $\\gamma$ and $Z$ components are assumed in the neutral current exchange and corresponding production of dilepton final states for $75\\le m_{\\ell^+\\ell^{-}} \\le 105$~~GeV$\/c^{2}$.} of  $\\sigma_{WZ+ZZ}=4.4\\pm0.3$~pb~\\cite{mcfm_prd,mcfm_url}. The evidence of the $WZ\/ZZ$ signal, obtained independently by each experiment and their combination, strongly supports the Tevatron $H\\to b\\bar{b}$ search results.\n\nA summary of all the present diboson measuremens is reported in Table~\\ref{tab:dib_measurements}.\n\n\\renewcommand{\\tabcolsep}{3pt}\n\\begin{table}[h]\n\\begin{center}\n  \\begin{small}\n\n\\begin{tabular}{ccccc}\\toprule\nChannel & Experiment &  $\\mathscr{L}$ (fb$^{-1}$) & Measured $\\sigma$(pb) & Theory $\\sigma$(pb) \\\\\\midrule\n\n$WZ\\to \\ell\\ell\\ell\\nu$ & CDF II\\cite{cdfWZlllv_2012} & $7.1$ & $3.9\\pm 0.8$    &  $3.47 \\pm 0.21$ \\\\\n                                    & D0\\cite{d0WZZZlep_2012} & $8.6$ & $4.5^{+0.6}_{-0.7} $  &  $3.47 \\pm 0.21$ \\\\\n                                    & CMS\\cite{cmsDib_2011} & $1.1$ & $17.0 \\pm 2.4\\pm 1.5$    &  $ 17.3^{+1.3}_{-0.8}$ \\\\\n                                    & Atlas\\cite{cdfWZlllv_2012} & $$ & $20.5 \\pm ^{+3.2+1.7}_{-2.8-1.5}$ & $ 17.3^{+1.3}_{-0.8}$ \\\\\n\\midrule\n\n$ ZZ \\to \\ell\\ell\n+\\ell\\ell\/\\nu\\nu$ \n & CDF II\\cite{cdfZZllll_2012} & $6.1$ & $ 1.64^{+0.44}_ {-0.38}$ & $1.4\\pm 0.1$\\\\\n&  D0\\cite{d0WZZZlep_2012} & $8.6$ & $ 1.44^{+0.35}_{-0.34}$ & $1.4\\pm 0.1$\\\\\n\\midrule\n\n  $ZZ \\to \\ell\\ell\\ell\\ell$ & Atlas\\cite{atlasZZ_2012} & $1.02$ & $ 8.5^{+2.7+0.5}_ {-2.3-0.4}$ & $6.5^{+0.3}_{-0.2}$\\\\\n                            & CMS\\cite{cmsDib_2011} & $1.1$ & $ 3.8^{+1.5}_{-1.2}\\pm 0.3 $ & $6.5^{+0.3}_{-0.2}$\\\\\n\n\\midrule\n\n$WW\\to \\ell\\ell\\nu\\nu$ & CDF II\\cite{cdfWWllvv_2012} & $3.6$ & $12.1\\pm 0.9 ^{+1.6}_{-1.4}$  &  $11.3 \\pm 0.7$ \\\\\n                                    & D0\\cite{d0WWllvv_2009} & $1.0$ & $11.5\\pm 2.2$   &  $11.3 \\pm 0.7$ \\\\\n                                    & CMS\\cite{cmsWW_2012} & $4.92$ & $52.4\\pm 2.0\\pm 4.7$    &  $ 47.0\\pm 2.0 $ \\\\\n                                    & Atlas\\cite{atlasWW_2012} & $1.02$ & $54.4\\pm 4.0 \\pm 4.4$  &  $ 47.0\\pm 2.0$ \\\\\n\n\\midrule\n$ Diboson\\to \\nu\\nu + jets$ & CDF II\\cite{cdfDibMetJet_2009} & $3.5$ & $18.0\\pm 2.8 \\pm 2.6$  &  $16.8 \\pm 0.5$ \\\\\n\n\\midrule\n$ WW\/WZ\\to \\ell\\nu + jets$ & CDF II\\cite{cdfDiblvJet_Mjj2010} & $4.3$ & $18.1\\pm 3.3\\pm 2.5$  &  $16.8 \\pm 0.5$ \\\\\n                                      & CDF II\\cite{cdfDiblvJet_ME2010} & $4.6$ & $16.5^{+3.3}_{-3.0}$  & \\\\\n\n  & D0\\cite{d0DiblvJet_2011} & $4.3$ & $19.6^{+3.2}_{-3.0}$  &  $16.8 \\pm 0.5$ \\\\\n \n\\midrule\n\n$ WW\/WZ\\to \\ell\\nu + HF$ & CDF II\\cite{cdfDiblvHF_Mjj2011} & $7.5$ & $18.1^{+3.3}_{-6.7}$  &  $16.8 \\pm 0.5$ \\\\\n\n\\midrule\n\n & CDF II\\cite{cdfWZZZCombHF_2012} & $9.45$ & $4.1^{+1.8}_{-1.3}$  &  $4.4 \\pm 0.3$ \\\\\n$  WZ\/ZZ \\to \\ell\\ell\/\\ell\\nu\/\\nu\\nu+ HF$ & D0\\cite{d0WZZZCombHF_2011} & $8.4$ & $5.0 \\pm 1.0 ^{+1.3}_{-1.2}$  &  $4.4 \\pm 0.3$ \\\\\n & D0,CDF II\\cite{ tevWZZZCombHF_2012} & $7.5 - 9.5$ & $4.47\\pm 0.67^{+0.73}_{-0.72}$  &  $4.4 \\pm 0.3$ \\\\\n\n\\bottomrule\n\\end{tabular}\n\\caption[Recent Diboson Measurements]{Summary of the recent measurements of the diboson production cross section in leptonic and semi-leptonic final states. The reference to the individual measurements are reported in the table as well as the used integrated luminosity, the statistical uncertainty appears before the systematic uncertainties (when both are available), the theoretical predictions are calculated with NLO precision~\\cite{mcfm_prd,mcfm_url}.}\\label{tab:dib_measurements}\n    \n  \\end{small}\n\\end{center}\n\\end{table}\n\\renewcommand{\\tabcolsep}{6pt}\n\n\\chapter*{\\centering \\begin{large}Abstract\\end{large}}\n\n\n\\clearpage\n\\include{Introduction\/Introduction}\n\n\n\\mainmatter\n\\clearpage\n\\include{Theory\/Theory}\n\n\\clearpage\n\\include{Detector\/Detector}\n\n\\clearpage\n\\include{PhysObj\/PhysObj}\n\n\\clearpage\n\\include{Sel\/Sel}\n\n\\clearpage\n\\include{Bkg\/Bkg}\n\n\\clearpage\n\\include{StatRes\/StatRes}\n\n\\clearpage\n\\include{Conclusions\/Conclusions}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe current generation of short pulse lasers reach intensities beyond $\\mathrm{I} > 10^{20}\\ \\mathrm{W\/cm^2}$, and hold great potential for many applications in high energy density science such as high energy x-ray backlighters and particle beams\\cite{Note1}\\cite{Note2}. While lower intensity lasers couple their energy into the transverse motion of electrons, relativistic electrons can couple energy into longitudinal motion via the laser's magnetic field. However, as intensity increases on relativistic laser platforms, the long ns pre-pulse pedestal that arrives prior to the main interaction intensifies as well. This pre-pulse, generated by amplified spontaneous emission processes in the laser, typically has an intensity contrast ratio $\\mathrm{(I_{Main Beam}\/I_{Pre-Pulse})}$ of $10^{6}$. In high intensity interactions this results in a pre-pulse of high enough intensity to ionize material, creating an underdense pre-plasma that extends for hundreds of\\SI{}{\\ \\micro\\meter}. \n\nWhile pre-plasma has been seen as detrimental to electron acceleration for applications such as fast ignition due to filamentation and self focusing instabilities\\cite{MacPhee}\\cite{Sunahara}, it actually can be beneficial in other ways, enabling a number of mechanisms that can generate energetic electrons. Accelerating electrons is the primary way to couple the energy in a laser to a plasma and provides the basis for a wide range of phenomena and applications. The generation of large quantities of x-ray and energetic secondary particles, such as ions \\cite{Pomerantz}, neutrons\\cite{Schollmeier} and positrons\\cite{Chen} highly depend on relativistic electrons. The electron acceleration in these experiments occur in a regime where the plasma response time is short compared to the laser pulse. In these experiments the density profile evolves slowly relative to the laser and develops into a quasi-steady state, opposite of the regime necessary for processes such as wakefield acceleration. Therefore, understanding the parameters that control the quantity, energy and trajectory of these electrons is critical to properly understanding the production of other energetic particles from secondary interactions.\n\nA common element for electron acceleration mechanisms is the important role played by quasi-static transverse and longitudinal pre-plasma electric fields in enhancing the energy transfer from the laser pulse to pre-plasma electrons \\cite{Booshan1}-\\cite{Robinson}. These fields are relatively weak compared to the field of the laser pulse and are unable to directly transfer considerable energy to the electrons. However, they do change the phase between the oscillating electric field of the laser and the electron velocity, which can result in a net energy gain with each laser period. This is the essence of the mechanism called direct laser acceleration (DLA) that leads to acceleration of the so-called super-ponderomotive electrons in an extended pre-plasma. These electrons have a corresponding relativistic $\\gamma$-factor greatly exceeding the conventional estimate of $\\gamma \\approx a_0^2$, where $a_0$ is the normalized laser amplitude. Indeed it has been shown that these super-ponderomotive electrons can enhance the energy of target normal sheath accelerated (TNSA) protons for radiography purposes but requires carefully controlled conditions for the rise time and trajectory of the electron beam \\cite{Kim}. \n\nPrior experiments using pulses with durations of 150 fs \\cite{Peebles},400 fs \\cite{Tanaka}, 500 fs \\cite{Toshi} and 700 fs \\cite{Ma} suggest that appreciable quantities of super-ponderomotive electrons should be accelerated for pulse lengths 400 fs or longer. The goal of the experiment presented here was to fill a gap between theory and experiment by accelerating large quantities of super-ponderomotive electrons using a very high intensity Texas Petawatt (TPW) laser while varying the interaction pulse length. These super-ponderomotive electrons were observed with energy up to $150$ MeV or more. However, contrary to prior experiments and simulations they were only detected at the longest pulse length tested (600 fs). In this paper we report the experimental measurement of super-ponderomotive electrons accelerated by DLA and the significance that a quasi static magnetic field has on their trajectory.\n\n\\section{Experimental Setup and Initial Simulations}\n\n\\begin{figure}[b]\n\t\\includegraphics[width=\\columnwidth]{Exp-Setup3}\n\t\\caption{Setup for the experiment on the TPW laser. A long pulse beam (green) generates an underdense plasma on the surface of the target while the high intensity short pulse (red) accelerates electrons measured by the primary EPPS diagnostics.}\n\\end{figure}\n\nSimilar to previous experiments \\cite{Peebles}-\\cite{Willingale} a low intensity, 2 ns beam with a large focal spot was used to generate a controllable, uniform, underdense plasma. This beam was injected 4 ns prior to the arrival of a short pulse beam, which had a variable pulse length (150, 450, 600 fs) and energy (30-105 J). Intensity was kept nominally to $2-3 \\times 10^{20}\\ \\mathrm{W\/cm^2}$ for all pulse lengths, and the beam was focused to a \\SI{10}{\\ \\micro\\meter} diameter spot, incident on the target at a $21.8^{\\circ}$ angle. The intrinsic pre-pulse from amplified spontaneous emission of the high intensity beam was nearly negligible after recent upgrades to the TPW, which improved the pre-pulse intensity contrast to over $3 \\times 10^{10}$ \\cite{GaulContrast}. Both beams had a wavelength of 1057 nm. The high-contrast laser was incident on a 1 $\\mathrm{mm^2}$ \\SI{100}{\\ \\micro\\meter} thick planar foil target composed primarily of aluminum, with a buried \\SI{20}{\\ \\micro\\meter} thick copper layer for diagnostic purposes. This foil was attached to a bulk plastic block used to inhibit electron refluxing, preventing double counting of electrons in our diagnostics. The experimental setup with the positions of diagnostics are shown in Figure 1.\n\n\\begin{figure}[b]\n\t\\includegraphics[width=\\columnwidth]{Sims1-a-revised3}\n\t\\caption{Simulation results of a 150, 450 and 600 fs beam (traveling left to right) incident on a target with underdense plasma. Electrons with energy $>$ 100 MeV traveling in the forward direction are color coded by angle relative to the target normal. The deep purple and green sections represent areas with significant electromotive force in the transverse direction on a 100 MeV electron.}\n\\end{figure}\n\nDiagnostics were fielded to characterize hot electron temperature and trajectory. While bremsstrahlung spectrometers (BMXS) and copper K$\\alpha$ imaging (SCI) were used, their signals are more sensitive to lower temperature electrons. High energy electrons were measured with EPPSes, a set of  calibrated magnetic electron spectrometers \\cite{EPPS}. The spectrometers had 1 x 2 mm entrance pinholes, a strong magnet to deflect charged particles and differentiate them by energy and charge and two image plate detectors. The energy range was 5-100 MeV with energy resolution scaling with energy, from 5 keV at low energies to 1 MeV at higher energies. Two spectrometers were placed in the chamber, the first 43 cm away from the interaction facing the rear surface normal of the target and the second was placed 55.5 cm away from the interaction, 3 degrees off the front surface normal of the target. Spectra taken from the EPPS were characterized by the Half Maximum Integrated Energy (HMIE) value, which represents the energy value where 50 percent of the total energy in the spectrum is contained, e.g. a HMIE of 10 MeV means 50\\% of the energy in the spectrum is contained in electrons with less than 10 MeV energy. Hence a higher HMIE value will result from a spectrum with a greater proportion of high energy electrons.\n\nInitial hydrodynamic simulations with 2D FLASH \\cite{FLASH} were conducted to estimate the density profile of the underdense plasma using the parameters of the long pulse beam in the experiment. The resulting density profile was approximated as a sum of exponential decays with a density of around $10^{19}\\ \\mathrm{cm}^{-3}$ \\SI{150}{\\ \\micro\\meter} away from the solid surface. This density profile was input into the 2D particle-in-cell (PIC) code EPOCH \\cite{EPOCH} and truncated \\SI{160}{\\ \\micro\\meter} away from the target surface. The simulation box size was \\SI{250}{\\ \\micro\\meter} long, \\SI{60}{\\ \\micro\\meter} wide, with 10 electron and 5 proton macro particles per cell. The resolution was 30 cells\/\\SI{}{\\\\\\micro\\meter} in the x direction and 15 cells\/\\SI{}{\\\\\\micro\\meter} in the y direction. A laser was injected into the box with an intensity of $3 \\times 10^{20}\\ \\mathrm{W\/cm^2}$, a 70 fs rise time and a \\SI{10}{\\ \\micro\\meter} diameter spot.\n\nIn Figure 2 we present initial snapshots simulations for pulse lengths of 150, 450 and 600 fs taken just after the center of the pulse impacts the target in time. With the 150 fs pulse no significant super-ponderomotive electrons were accelerated. For 450 fs super-ponderomotive electrons are accelerated and travel primarily along the laser trajectory, while at 600 fs these electrons are dispersed by fields that develop near the critical density. Based on these simulations we anticipated measurable quantities of super-ponderomotive electrons ($\\mathcal{E} >$  60 MeV, $a_0$ = 15.6) using pulse lengths greater than 400 fs.\n\n\\section{Results and Discussion}\n\nOver 35 experimental shots were performed over the course of the campaign with time split evenly between the 3 pulse length settings (150, 450 and 600 fs); half of the total shots did not use the long pulse beam to serve as the no pre-plasma cases. We verified that no super-ponderomotive electrons were measured with the shortest pulse length. When comparing the spectra for the 150 fs pulse length case there was no difference in the HMIE values when the long pulse beam was included. Upon increasing the pulse length to 450 fs the results were largely the same and no significant departure was seen between shots with and without pre-plasma. This was a surprising result since prior experiments \\cite{Tanaka}\\cite{Toshi} and our simulations with similar pulse lengths exhibited super-ponderomotive electrons and a change in spectrum shape.\n\n\\begin{figure}[b]\n\t\\includegraphics[width=\\columnwidth]{Exp-EPPS1-d2}\n\t\\caption{(a): Raw data from EPPS1 comparing cases with and without underdense plasma. Background subtracted lineouts are taken of the data and convolved with the spectrometer's calibrated dispersion to produce electron spectra. \\\\\\hspace{\\textwidth} (b): Electron spectra leaving the rear of the target measured by EPPS1 in the calibrated region of the diagnostic. Two 600 fs shots with pre-plasma shown in black and red produced substantial quantities of super-ponderomotive electrons $>$ 60 MeV resulting in a dramatic change to the electron spectrum compared to the no pre-plasma case (blue). \\\\\\hspace{\\textwidth}(c): HMIE values taken for all shots at 600 fs. Shots from the above spectra are circled with their respective color. At 600 fs, 3 out of 9 shots with underdense plasma measured a significant quantity of super-ponderomotive electrons}\n\\end{figure}\n\nExtending the pulse length further to 600 fs yielded significant changes to the electron spectrum shown in Figure 3; 3 out of the 9 shots with the long pulse beam had differently shaped spectra than their counterparts. These spectra contained electrons so energetic that they were detected at the end of the spectrometer (beyond its known calibration) and had energy exceeding 150 MeV. These results contradict the initial expectations from simulations shown in Figure 2. At 450 fs in simulations copious quantities of high energy electrons are accelerated nearly \\SI{50}{\\ \\micro\\meter} from the target surface and move forward into the target, while in the experiment no super-ponderomotive electrons were measured. At 600 fs super-ponderomotive electrons were measured on experiment while in simulations most electrons were severely deflected and few super-ponderomotive electrons propagated directly forward toward the diagnostic.\n\n\\begin{figure}[t]\n\t\\includegraphics[width=\\columnwidth]{Sims2-revised2}\n\t\\caption{Identical simulation setup to Figure 2 with the laser at a 20 degree incidence angle on target. The electrons are color coded by angle bin with respect to the EPPS diagnostic placed in the target normal direction. At 450 fs electrons travel in the direction of the laser similar to those in Figure 2 (b), while at 600 fs the electrons are deflected upwards in the target normal direction. This is due to the development of a large asymmetry in the electromotive force.}\n\\end{figure}\n\nWhat could cause this large discrepancy between the simulations and experimental data? The cause can be partly attributed to the EPPS diagnostic itself. The EPPS collects electrons only near the target rear normal and the solid angle ($\\sim 10^{-5}$ steradian) is an extremely small fraction of the solid angle where electrons are seen to travel in the simulations. Another difference between simulation and experiment lies in the inclusion of the laser incidence angle. In our experiment the laser was incident at roughly a $\\mathrm{20^o}$ angle and the EPPS diagnostic was placed facing the target normal, while our simulations used a normal incidence beam. When conducting another simulation with the same parameters but with the laser incidence angle and diagnostic angles included, the trajectories of super-ponderomotive electrons were altered significantly.\n\nFigure 4 shows the results from the angled simulation taken at the same times as those in Figure 2. Initially the $>$ 100 MeV electrons propagate primarily in the laser direction, not in the direction of the diagnostic. Measuring the angle of the electrons with respect to the viewing angle of the EPPS shows that the EPPS would miss the majority of super-ponderomotive electrons at 450 fs. However, later in time, electrons were scattered weakly toward the target normal rather than in all directions. At 600 fs the bulk of high energy electrons were deflected in the direction of the target normal and EPPS diagnostic, resulting in a greater chance of detection, which is directly supported by the experimental results.\n\n\\begin{figure}[t]\n\t\\includegraphics[width=\\columnwidth]{Exp-EPPS2-b}\n\t\\caption{HMIE values for electrons measured from the front surface of the target. Higher energy electrons were measured when an underdense plasma was present, regardless of pulse length suggesting that electrons in the pre-plasma were accelerated by the reflected beam.}\n\\end{figure}\n\\begin{figure}[]\n\t\\includegraphics[width=\\columnwidth]{Sims3-4-revised2}\n\t\\caption{(a): A current density plot for the 20 degree laser incidence angle case taken at the same time as Figure 4 (c). The effect of the asymmetric magnetic field on incident hot electrons is gradual causing a smaller deflection. Arrows indicate the location and direction of the majority of current flow.\\\\\\hspace{\\textwidth} (b): A plot of a simulation with a 10 degree laser incidence angle taken earlier in time. \\\\\\hspace{\\textwidth} (c): Plot of the 10 degree incidence simulation at the same time as the top 20 degree image. The current accelerated by the reflected beam expands and encounters the incident current more directly than in the 20 degree case. The super-ponderomotive electrons see a sudden, larger increase in the deflecting magnetic field as the fields join together.}\n\\end{figure}\n\nUpon closer examination of electron deflection at these later times it is found that the electrons encounter a counter-propagating current of colder electrons. The effect of this reverse current can be seen by mapping the electromagnetic force on a forward going electron (seen in purple and green in Figs 2 and 4). The forward going electrons generate a self confining magnetic field based on their current. Extending from the target surface, a counter-propagating current generates an opposite field that causes the electrons to split upwards or downwards. As pulse length increases this opposing field gains strength and causes more severe deflections. This effect has the largest impact on super-ponderomotive electrons, as they are accelerated via DLA far from the target surface and travel through this field. Lower energy, ponderomotive scaling electrons are less affected since they are accelerated near the critical density, closer to the target surface and do not interact with these fields over an extended distance.\n\nThe development of counter-propagating current can be traced to electrons in the pre-plasma near the target surface that are accelerated by reflected laser light away from the critical surface. This effect has been characterized in 3D PIC simulations conducted by F. Perez \\textit{et al.}\\cite{Perez}, which show the development of strong magnetic fields due to counter-propagating current after 473 fs. In the normal incidence case the two currents directly oppose each other and incoming electrons are deflected in all directions; in the angled case the opposing fields develop at an angle with respect to the target surface. This angle causes the fields from the counter-propagating current to build upon the fields from the incident current rather than oppose them, leading to a large asymmetry and significant deflection toward target normal.\n\nExperimental evidence of pre-plasma electrons accelerated by reflected light was found in the data of the spectrometer facing the front surface of the target (EPPS2). Measuring the HMIE values for electrons traveling in this direction shows distinctly higher energies for nearly all shots with the long pulse beam, regardless of pulse length (Fig 5). However, as our simulations show, the field from this reflected current requires time to develop and only has enough strength to deflect significantly electrons after 500 fs.\n\n\\section{Summary and Future Considerations}\n\nIn summary, we found that in experiments no super-ponderomotive electrons were detected in the forward direction for $\\tau_L$ = 150 fs, nor for $\\tau_L$ = 450 fs, in contradiction to earlier normal-incidence simulations, and only sporadically for $\\tau_L=$ 600 fs. Simulations including the experimental incidence angle showed that the super-ponderomotive electrons in fact were generated as early as $450$ fs, but were likely traveling in the direction of the laser direction and not towards the diagnostic placed in the target normal direction. As fields from an opposing current increased later in time these electrons were eventually deflected preferentially in the direction of the diagnostic. These results raise the question, how sensitive are super-ponderomotive electron trajectories to changes in laser incidence angle? To address this, another simulation with identical parameters was conducted, which used a 10 degree incidence angle rather than 20. As seen in figure 6 (c), electrons in the small angle case were deflected \\textit{more} significantly than in either of the previous two cases (0 and 20 degree), with a large portion deflected away from the target entirely late in time. Due to the smaller incidence angle, the region where laser light reflects off the critical density surface becomes narrower causing the opposing current to become more concentrated. More importantly, halving the incidence angle causes the relative angle between the reflected and incident light to be quartered. The opposing current therefore has a more head on trajectory with the incident electrons leading to greater deflection angle unlike the ``glancing blow'' seen in the large angle case.\n\nThese results point to a few considerations for future experiments and simulations. Understanding the effect of laser incidence angle is crucial for setting up an experiment. Many high intensity laser facilities do not allow for direct normal laser incidence due to the potential damage caused by reflected light, which means that incidence angle effects are a factor that must be considered for experiments that have been or will be performed at these facilities. Even for experiments that use perfectly normal incidence, the effect of counter-propagating current can result in deflecting electrons away from the laser direction as shown in Figure 2. Electron trajectories change significantly when incidence angle is varied and must be taken into consideration when placing diagnostics and when accelerating protons via TNSA using non-relativistic beams \\cite{Kim}. Due to the nature of high energy electrons, diagnostics capable of measuring super-ponderomotive electrons use very small solid angle apertures. Measuring these directional, high energy electrons poses a challenge to the diagnostics community to develop new tools capable of providing spatial information for higher energy electrons.\n\n\n\\section*{Acknowledgments}\n\nExperimental time and resources were supported by the United States Department of Energy under contract DE-NA0001858 (DOE). Simulations were performed using the EPOCH code (developed under UK EPSRC Grants No. EP\/G054940\/1, No. EP\/G055165\/1, and No. EP\/G056803\/1) using HPC resources provided by the TACC at the University of Texas. The work by A. Arefiev ,CHEDS researchers, and the TPW itself were supported by NNSA Cooperative Agreement DE-NA0002008.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}