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+{"text":"\n\\section{Platform and Evaluation}\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.96\\linewidth]{Images\/tasks_simple_to_complex.png} \\vspace{-3mm}\n\t\\caption[FishGym Tasks]{The control results for benchmark tasks using our trained policies and two-way coupled fluid-structure interaction solver. Top row: cruising in a shallow fluid; second row: cruising in a deep fluid (with buoyancy control); third row: pose control with a U-turn; fourth row: two-fish schooling; bottom row: path following with an arbitrarily specified path.}\n\t\\label{fig:tasksequence} \n\t\\vspace{-15pt}\n\\end{figure}\n\nIn the following, we first describe our setup on simulation and evaluate our fluid simulation platform with our learning algorithms for robot fish swimming in multiple aspects.\n\n\\subsection{Platform setup}\n\\paragraph{Simulation}\nIn most our training tasks, all the fish robots have a density of 1080$kg\/m^3$, and we used local non-inertial fluid-structure interaction solver for simulating robot fish dynamics, with a grid resolution of $100\\times100\\times100$ and a physical time step of 0.004s.\nThe solver costs around 3.5 seconds for simulating one physical second on an NVidia TitanXp GPU with 12G memory.\nFor two-fish schooling, we extended the local domain horizontally to simultaneously include two fishes, with a grid resolution of $150\\times50\\times100$ and the same physical time step, which costs around 4 seconds for simulating one physical second on the same GPU.\n\n\\paragraph{Training}\nThe policy network consists of two layers, each containing 256 units, with an ReLU activation function. \nFor each task, the policy is trained using SAC \\cite{Haarnoja2018}.\nWe train each policy for a total of 2000 simulation rollouts, each of which contains 50 time steps, where a candidate policy executes a new action at each time step. \nWe train the policy network with a batch size of 256.\nThe model parameters are updated for each step, and one gradient step is performed after each rollout.\nWe train all policies on a machine with an NVidia TitanXP GPU, and the training process usually starts to converge after 6 hours (1000 episodes), and have a smooth convergence within 10 hours. \nThe training could be several times faster if we use the most state-of-the-art GPUs, such as NVidia GeForce RTX3090.\n\n\\subsection{Comparison for different types of robot fishes}\nOur platform can support robot fishes designed with different skeleton connectivity and skin shapes.\nFig.~\\ref{fig:different_fish_type_result} shows the training results for three types of robot fishes swimming along an arbitrarily given path.\nThe average distances from the path (around 7 meters long) are as close as 0.03m, 0.05m, 0.08m, respectively, indicating the capability of our platform in supporting a variety of robot fishes.\n\\subsection{Comparison for different simulation models}\nIn the literature, a simple empirical model was proposed as the simulator for robot fish swimming~\\cite{Terzopoulos1994}, which has been used until now~\\cite{grzeszczuk1998neuroanimator,si2014realistic,min2019softcon}.\nIt models the instantaneous force on the surface of the robot due to viscous fluid as:\n\\vspace{-5pt}\n\\begin{equation}\n\tF = -k\\int_{S}(\\mathbf{n} \\cdot \\mathbf{v}) \\mathbf{n} d s ,\n\t\\label{empirical_model_eq}\n\t\\vspace{-5pt}\n\\end{equation}\nwhere $\\mathbf{n}$ is the unit outward normal; $\\mathbf{v}$ is the relative velocity between the surface and the fluid (since there is no fluid simulation, the fluid velocity is assumed to be zero), and $k$ is a constant manually tuned for different robot fishes and the surrounding fluids.\nNote that for a specific system, $k$ can only be determined either by real measurement data or from other physically more accurate simulators.\n\nTo examine the similarity and difference between the empirical model and our physical simulator for robot fish swimming, we conduct two test cases with analyses below.\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.96\\linewidth]{\"Images\/different_robot_fishes\"} \\vspace{-3mm}\n\t\\caption[Fish Gym Tasks]{Different types of robot fishes trained using local policy learning for arbitrary path following. Top: koi robot fish; middle: flatfish robot fish with a different skeleton topology (with branching during modeling); bottom: eel robot fish with a long concatenated skeleton.}\\vspace{-3mm}\n\t\\label{fig:different_fish_type_result}\n\\end{figure}\n\\paragraph{Path following}\nPath following is a primitive task for robot control.\nHere, we compare the similarity and difference of path following using an empirical model and our physical simulator.\nSince there is no clue on how to tune the parameter $k$ in an empirical model, we arbitrarily choose one and learn a policy. \n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.96\\linewidth]{\"Images\/Parameter_need_tune\"} \\vspace{-3mm}\n\t\\caption{Comparison for path following using (left) empirical model with an arbitrarily chosen parameter and (right) our physical simulator. Note that due to improper boundary force, serious drifting happens for empirical model when turning. The numbers indicate the order during path following.} \\vspace{-3mm}\n\t\\label{fig:empirical_failed_path}\n\\end{figure}\n\nFig.~\\ref{fig:empirical_failed_path} (left) shows the control result, indicating serious drifting when turning, while our physical simulator does not require any parameter turning (we directly specify the physical parameter for the fluid as $\\rho$=1000$kg\/m^3$ and $\\nu$=0.00089$m^2\/s$ with a zero-velocity initialization to match that used in the empirical model), leading to a reasonable path following result as shown in Fig.~\\ref{fig:empirical_failed_path} (right).\nNote that the mean path deviation for the empirical model with an arbitrary parameter is as large as 0.5m, while our physical simulator only has a mean path deviation of 0.03m.\nThe empirical model can be much improved if we collect data from our simulator and fit the parameter $k$, leading to a very similar result in a static fluid environment but runs much faster.\nHowever, in some cases where we cannot assume static fluid background, empirical model can completely fail no matter how we fit the parameter $k$, and we demonstrate this case in the following two-fish schooling task.\n\\paragraph{Two-fish schooling}\nFish schooling describes a common phenomenon where fishes tend to swim in a group and one fish follows the other.\nIt has been revealed by scientists that due to the vortex ring generated behind the leader fish, the follower fish tries to utilize the vortex ring to reduce drag and pass through it in order to catch up with the leader fish more efficiently \\cite{Novati2017,Verma2018}.\nWe demonstrate this behavior in Fig.~\\ref{fig:teaser} (bottom row).\nDue to more accurate modeling to capture complex fluid flows, the training can successfully obtain a policy that utilizes the vortex ring, see Fig.~\\ref{fig:teaser} (bottom), while empirical model, on the other hand, fails to learn such a policy due to lack of a real fluid-structure interaction.\n\n\n\n\n\n\n\n\n\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.98\\linewidth]{Images\/density_compare} \\vspace{-3mm}\n\t\\caption{Control behaviors of a koi-like robot fish for different fluid densities. Top: $\\rho=1000$; bottom: $\\rho=10$.}\n\t\\label{fig:densitycompare}\n\t\\vspace{-18pt}\n\\end{figure} \n\n\\subsection{Comparison for different fluid settings}\nIn contrast to the empirical model, our simulator can freely set physical parameters, leading to different control behaviors for a trained robot fish.\nHere, we change the fluid density (with a higher density $\\rho=1000$ and a lower density $\\rho=10$, see Fig.~\\ref{fig:densitycompare}) for simulation and train with the same learning parameters to achieve a cruising task.\nIt is observed that the robot fish in a higher density fluid seems to move with less swing amplitude, while the one in a lower density fluid has a larger swing amplitude to generate greater propulsion force, which is consistent with the physical expectations.\n\n\\subsection{Effects of reward weighting}\n\nIn this experiment we investigate effects of different reward weighting. Table~\\ref{energy_table} summarizes the total energy cost by varying weights $w_v$ and  $w_e$ in the cruising task, where $w_v$ encourages fish to reach the target position while $w_e$ encourages fish to save its energy.\nA good balance between energy preservation and the time to accomplish the task can be made, e.g., $w_v=1.0$ and $w_e=0.5$.\n\n\\begin{table}[h]\n\\vspace{-3mm}\n\t\\caption{Impact of of weights in the reward} \\vspace{-3mm}\n\t\\label{energy_table}\n\t\\begin{center}\n\t\\begin{tabular}{c|c|c|c}\n\t\\hline\n\t$w_v$ & $w_e$ & Total Energy Cost & Total Time (sec.)\\\\\n\t\\hline\n \t0.00 & 1.00 & 0.0413 & 10.0 \\\\\n \t0.20 & 1.00 & 12.354 & 5.2 \\\\\n \t1.00 & 1.00 & 15.046 & 4.6 \\\\\n \t1.00 & 0.50 & 17.705 & 4.2 \\\\\n \t1.00 & 0.00 & 40.969 & 4.2 \\\\\n\t\\hline\n\t\\end{tabular}\n\t\\end{center}\n\\vspace{-15pt}\n\\end{table}\n\n\n\n\\section{Conclusion}\nIn this paper, we propose a new open-to-use simulation platform for training underwater fish-like robots.\nThe whole platform consists of a new modeling for fish-like underwater robot, a GPU-based non-inertial high-performance fluid-structure interaction solver (as a training environment), and reinforcement learning algorithms with both global and local policy learning.\nFour different benchmark tasks were proposed and trained with our platform, with expected results.\nWe compared and analyzed the new training platform in terms of different results in multiple aspects to evaluate the advantages.\n\nThere are also some limitations.\nFirst, the fish model is a reduced model that may deviate from the real robot design, and is now hence difficult to directly transfer to a real robot once learned.\nSecond, since we use local simulation, the platform is unable to training fish robot with a more complex external environment, e.g., a large vortex.\nFinally, the grid resolution around the fish is not fine enough (otherwise, it will become very slow for training), and the accuracy is not sufficiently high.\nSupporting simulation in more complex fluid environment and developing more efficient training method with higher accuracy deserve our future work.\n\\section{Introduction}\n\\label{sec:intro}\n\nBio-inspired underwater robots often demonstrate strong maneuverability, propulsion efficiency, and deceptive visual appearance.\nThese advantages have motivated a set of academic studies on bio-inspired soft robots and biomimetric fish-like robots in the past years~\\cite{Du2015,Paley2021,Duraisamy2019}.\nIt also opens up some important applications, such as marine education, navigation and rescue, seabed exploration, scientific surveying, etc.~\\cite{Kopman2012,Picardi2020,Berlinger2021,Li2021,Katzschmann2018}.\nHowever, due to lack of sufficient datasets and high physical cost, training their intelligent behaviors in real environments that at least mimic the bionic creatures or even exceed their capabilities in accomplishing complex tasks is still quite challenging.\n\nAlternatively, simulation has been considered as a viable and important tool for acquiring a large number of datasets in different scenarios~\\cite{AvilaBelbutePeres2018,Lee2019,James2019,Bergamin2019}.\nMost of the currently available simulators for robot training mainly target rigid and soft body systems~\\cite{Coumans2015,Lee2018,Todorov2012}.\nExisting simulators for fluid environment are either highly inaccurate (e.g., based on an empirical model~\\cite{Terzopoulos1994}), too restrictive to support different agents or  environments~\\cite{Song2017,Song2020,Verma2018}, or expensive to generate a large amount of training data \\cite{Verma2018,Novati2017}.\nThere is currently a dearth of simulation platform which is able to provide a versatile, efficient yet accurate results that could be used for training control policies of underwater robots. \n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.98\\linewidth]{\"Images\/school_comparision_teaser_2fig.png\"} \\vspace{-3mm}\n\t\\caption{\\textbf{Two-fish schooling behaviors.}  With the empirical model (1st row), the follower fish always stays in the same line as the leader fish. However, with our physical simulator (2nd row), the follower fish can utilize the wake vortex ring (blue) and gradually pass through the vortex ring to preserve energy. This motion behavior cannot be acquired through a simple empirical model, highlighting the advantages of the proposed FishGym platform. }  \\vspace{-3mm}\n\t\\label{fig:teaser}\n\t\\vspace{-10pt}\n\\end{figure}\nWe propose \\textbf{FishGym}\\footnote{https:\/\/github.com\/fish-gym\/gym-fish}, a high-performance simulation platform targeting the two-way interaction dynamics between fish-like underwater robots and the surrounding fluid environment. The robots are modeled by the skeletons of arbitrary topology with surface skinning, whose motion is driven by the articulated rigid body dynamics~\\cite{Weinstein2006}, while the fluid-structure interaction is achieved using a recently proposed GPU-optimized lattice Boltzmann solver~\\cite{Li2020,Chen2021gpu}, where immersed boundary method~\\cite{li2016immersed,wu2010improved} is employed for efficient two-way coupling.\nTo support simulations in a local fluid domain around the robot to enable training in an infinitely large physical domain, we propose a modification of the original lattice Boltzmann solver that enables simulations in a local frame of reference with acceleration, with higher flexibility in acquiring various training environments. \nThe whole simulation module is then coupled with a reinforcement learning module implemented using PyTorch~\\cite{Raffin2019,Paszke2019}.   \n\nTo demonstrate the capability of the proposed simulator, we evaluated existing reinforcement learning algorithm with reward functions and training procedures tailored for underwater robots.\nWe compare the learned control policies with that from the empirical model on several underwater planning and control tasks to assess the feasibility and advantages of our framework. \nAnalyses on the emerged behaviors also indicate consistency with previous studies on fish motion in nature.   \nIn summary, we have made the following contributions to training bionic underwater robots: \n\\begin{itemize}\n\\item A GPU-accelerated lattice Boltzmann solver that enables high-performance fluid-structure interaction in a local moving frame of reference to allow robot swimming in an infinitely large domain;\n\\item A high-performance simulation platform to help explore training bionic underwater robots;\n\\item A learning algorithm tailored for bionic under-water robots that is able to acquire natural and efficient control policies for swimming;\n\\item A collection of benchmark tasks for underwater robot to evaluate and compare different learning methods and control policies. \n   \n\n\n\n\n\\end{itemize}\n\t \n\n\n\n\n\n\n\n\n\n\\section{FishGym Framework}\nOur physics-based robot learning framework  consists of three components:\n1) the robot model for which we focus on fish-like robots, 2) \nthe simulation method for predicting fluid-robot interaction, \nand 3) the robot learning method that leverages our simulation method.\nWe now present their details.\n\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.48\\textwidth]{\"Images\/two_fish_skeletons\"}\n\t\\caption[Fish Skeletons]{Illustration of modeled fishes with different skeletons and skins, where the number of joints, the length of each skeleton edge as well as the topology of the skeleton can vary for different types of fishes.}\n\t\\label{fig:fish_model} \n\t\\vspace{-20pt}\n\\end{figure}\n\n\\subsection{Robot model}\nMotivated by the anatomy of the fish structure,\nwe model a fish robot's locomotion by its skeletal structure, that is, bones connected by joints.\nCovering the skeleton are flesh and skin. Provided a skeleton configuration (e.g., with a certain set of joint angles), we use linear blend skinning~\\cite{magnenat1988joint} to determine \nthe fish's surface shape. In our fluid-robot simulation, the fish surface is assumed to be inelastic, and the flesh is treated as a rigid body under the current skeleton configuration. We make this assumption for the sake of computational efficiency.\nBy changing its joint angles, a fish robot can adjust its skeleton pose,\nwhich in turn determines its surface shape.\nThree examples of fish robots with different skeletal structures are shown in Fig.~\\ref{fig:fish_model}.\n\n\n\n\n\nThe bone skeleton is driven by the articulated rigid body dynamics \\cite{Weinstein2006}:\n\\vspace{-8pt}\n\\begin{equation}\n\\mathbf{M}(\\mathbf{q})\\ddot{\\mathbf{q}}+\\mathbf{C}(\\mathbf{q}, \\dot{\\mathbf{q}})=\\boldsymbol{\\tau}_{int}+\\boldsymbol{\\tau}_{ext} ,\n\\vspace{-8pt}\n\\end{equation}\nwhere $\\mathbf{q}, \\dot{\\mathbf{q}}$ and $\\ddot{\\mathbf{q}}$ are respectively the vectors of generalized positions, velocities and accelerations of all joints. $\\mathbf{M}(\\mathbf{q})$ is the mass matrix, and $\\mathbf{C}(\\mathbf{q}, \\dot{\\mathbf{q}})$ accounts for the Coriolis and centrifugal forces;\ndetails of these forces will be described shortly.\n$\\boldsymbol{\\tau}_{ {int }}$ and $\\boldsymbol{\\tau}_{ {ext }}$ are the vectors representing the generalized internal forces (including the spring forces on joints to enable elasticity, damping forces due to velocities, friction forces, as well as the actuation given by the controller) and the generalized external forces (caused by gravity, possible collisions and the surrounding fluids) exerted on the multi-body system.\nWe employ DART \\cite{Lee2018} to solve the above multi-body system, and for each time step, we control the bone shape by applying generalized actuation forces on joints.\nThe skin surface is achieved by employing linear blending method proposed by \\cite{magnenat1988joint}.\n\n\\subsection{GPU-accelerated localized fluid-structure interaction}\nThere exist many simulation methods that may \npredict fluid-robot interaction~\\cite{klingner2006fluid,lv2010novel,dai2005adaptive}. \nThese methods, however, require a fixed simulation domain, inside which the \nunderwater robot moves. \nWhen the simulation domain is large,\nsimulation is costly.\nTo reduce the cost,\nwe assume that the fluid \nfurther away from the robot by a certain distance will not influence the robot motion. \nThereby, we can fix the size of the simulation domain centered around the fish robot and allow the domain to move along with the fish. \nThis setup allows the fish robot to move in an \ninfinite spatial domain while keeping the simulation domain limited. \nBut then, to capture fluid dynamics correctly, we need to simulate fluid-structure interaction in a moving frame of reference.\nBeing able to swim in an infinitely large domain is very important for training fish-like underwater robots; Also crucial is the simulation performance, as\ntraining the learning algorithm will often run fluid simulations\nmany times (see Section~\\ref{sec:fish_learning}).\nWe tackle both problems next.\n\n\\subsubsection{Formulation}\nFluid in a fixed frame of reference is often governed by the following NS equation:\n\\vspace{-6pt}\n\\begin{equation}\n\\frac{\\partial \\mathbf{u}}{\\partial t}+(\\mathbf{u} \\cdot \\nabla) \\mathbf{u}=-\\frac{1}{\\rho} \\nabla p+\\nu \\nabla^{2} \\mathbf{u} + \\mathbf{F},\n\\vspace{-6pt}\n\\end{equation}\nwhere $\\rho$, $\\mathbf{u}$, $p$ and $\\mathbf{F}$ represent the density, velocity, pressure and external force fields, and $\\nu$ is the kinematic viscosity.\nThis equation cannot be used to solve flows in a moving frame of reference around the fish, which should be reformulated in a frame of reference with acceleration.\nAccording to \\cite{Asmuth2016}, by transforming with time-dependent relative translation $\\mathbf{p}$ and rotation $\\mathbf{r}$ (represented as Euler angles) between consecutive frames, the NS equation in an accelerating frame of reference results in an additional virtual force added to the system:\n\\vspace{-6pt}\n\\begin{equation}\n\t\\vspace{-6pt}\n\\mathbf{F}_{ni}=-\\ddot{\\mathbf{p}}-\\ddot{\\mathbf{r}} \\times \\mathbf{x}^{\\prime}-\\dot{\\mathbf{r}} \\times\\left(\\dot{\\mathbf{r}} \\times \\mathbf{x}^{\\prime}\\right)-2 \\dot{\\mathbf{r}} \\times \\mathbf{u}^{\\prime} ,\n\\end{equation}\nwhere all the physical quantities are measured in a moving frame of reference.\nIn case of any immersed solid, e.g., the swimming robot, we apply Neumann boundary condition (i.e., slipping) as an approximation.\n\n\\subsubsection{Simulation}\nTo simulate the above dynamics in an efficient manner, we discard the traditional NS solver; instead, we employ a GPU-optimized LBM solver with immersed boundary (IB) method \\cite{Li-2020} for fluid-structure interaction, whose high efficiency has been demonstrated.\nThe fish-like robot surface is uniformly sampled before simulation, and the external virtual force due to acceleration can be added into the system very easily.\nThe difficulty we need to address is the domain boundary, which in theory should be set according the large-domain simulation.\nHowever, in practice, we do not know the exact values of the domain boundary, and when fish moves with different velocities and accelerations, the flow can go into and outside from any portion of the domain boundary.\nThus, we need a domain boundary treatment which can adapt to this situation, and through a set of experiments, we found that the method described in \\cite{ZhaoLi2002} satisfy this requirement and has been employed in our IB-LB simulation.\nFig.~\\ref{fig:complocalglobal} compares the accuracy between a full global fluid domain simulation (left) where we directly use IB-LB method in \\cite{Li-2020} and the proposed localized fluid-structure interaction (right). \nBoth methods produce nearly identical robot motion trajectories.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.98\\linewidth]{Images\/comp_local_global} \\vspace{-3mm}\n\t\\caption{Comparison between fluid-structure interaction in a global static fluid domain (left) and the non-inertial counterpart in a moving local fluid domain (right), indicating the closeness of the paths with a known policy.} \\vspace{-5mm}\n\t\\label{fig:complocalglobal}\n\t\\vspace{-3pt}\n\\end{figure}\n\n\\subsection{Reinforcement learning for fish-like robot control}\n\\label{sec:fish_learning}\nDue to complex fish dynamic model, the control of fish-like robot swimming cannot be simply achieved by a model-based controller.\nIn addition, since the control input is usually high-dimensional and cannot be fully decoupled, the PID controller cannot be used either.\nThus, reinforcement learning (RL) is a common choice to train complex control policies for swimming.\nIn this paper, we propose four benchmark tasks that will be trained using RL in order to demonstrate the power of the new simulator together with the learning algorithms.\nThese benchmark tasks are:\n\\begin{itemize}\n\t\\item \\textbf{Cruising.}\n\tThe robot fish tries to swim to reach a given target location that is a distant away from the robot.\n\t\\item \\textbf{Pose control.}\n\tA robot fish tries to control its pose in order to make a U-turn.\n\t\\item \\textbf{Two-fish schooling.}\n\tA robot fish follows a leader fish as closely as possible, where the leader fish is controlled to swim in a straight path.\n\t\\item \\textbf{Path following.}\n\tA robot fish follows a given arbitrary path as closely and efficiently as possible.\n\\end{itemize}\n\nIn RL, an agent learns a policy for a specific task through repeated interaction with the environment.\nGiven a state $\\mathbf{s}_i$, the RL tries to learn a parametric policy $\\pi_{\\boldsymbol{\\theta}}$, which is usually represented by a fully-connected neural network, to produce an action $\\mathbf{a}_i$;\nThe action can be taken by the agent to transit to the next state $\\mathbf{s}_{i+1}$, where a reward $r_i$ is evaluated.\nThe agent iterates transitions until it satisfies one of the exit conditions, e.g., finite time horizon, or success\/failure of a given task.\nA parametric policy $\\pi_{\\boldsymbol{\\theta}}$ is learned by finding the optimal parameters $\\boldsymbol{\\theta}^{*}$ that maximize the expected return:\n\\vspace{-8pt}\n\\begin{equation}\n\t\\vspace{-6pt}\nJ(\\theta)=\\mathbb{E}_{\\tau \\sim p_{\\theta}(\\tau)}\\left[\\sum_{t=0}^{T} \\gamma^{t} r_{t}\\right] ,\n\\end{equation}\nwhere $T$ is the maximum number of control time steps, $\\gamma$ is the discounting factor, and $\\tau$ is the sampled trajectory containing a sequence of states and actions, i.e., $\\tau = (s_0, a_0 , s_1, a_1, ..., s_i, a_i, ..., s_t, a_t)$.\nThe policy learning can be achieved in two different ways.\nDepending on the available resources, we can sample the trajectories from one single task, or from multiple tasks to learn a \\textit{global policy}.\nHowever, if the variety of tasks is large, it is time consuming especially when the simulator is not fast enough.\nOn the other hand, if the tasks can be subdivided into small and simple sub-tasks, we can gather all these sub-tasks together and sample from them to learn a \\textit{local policy}, which is expected to be much more generalizable given a relatively small training set, and could be affordable for limited resources.\nWe adopt both approaches to train different tasks.\n\n\nIn the following, we specify in detail the specific designs on how we train these benchmark tasks.\n\\paragraph{State} \nFor all benchmark tasks, we consider the following state variable:\n\\vspace{-8pt}\n$$\n\\mathbf{s}=(\\mathbf{s}_d, \\mathbf{s}_p, \\mathbf{s}_r, \\mathbf{s}_{task}) ,\n\\vspace{-6pt}\n$$\nwhere $\\mathbf{s}_d = (\\mathbf{q},\\dot{\\mathbf{q}})$ contains the dynamic states, with $\\mathbf{q}$ the vector of generalized joint positions, and $\\dot{\\mathbf{q}}$ is the vector of generalized joint velocities;\n$\\mathbf{s}_p=(\\mathbf{p},\\dot{\\mathbf{p}})$ contains translation, where $\\mathbf{p}$ is the relative translation vector between simulation time steps, and $\\dot{\\mathbf{p}}$ is the relative velocity vector;\n$\\mathbf{s}_r = \\mathbf{r} $ contains rotation (Euler angles);\nand $\\mathbf{s}_{task}$ contains task-specific states. \nIn practice, to reduce input dimension, all state variables are expressed in a local coordinate system. \n\n\\paragraph{Action}\nFor all benchmark tasks, the action can be generally defined as:\n\\vspace{-10pt}\n$$\n\\mathbf{a} = (\\boldsymbol{\\sigma},\\Delta v) ,\n\\vspace{-6pt}\n$$\nwhere $\\boldsymbol{\\sigma}$ is the vector containing the actuation forces applied to the joints, and\n$\\Delta v$ is the change of the bladder's volume inside the robot fish, which controls buoyancy to enable going up and down in a fluid.\n\\paragraph{Reward}\nThe reward for training all benchmark tasks can also be written in a general mathematical form as:\n\\vspace{-4pt}\n$$\nr = w_pr_p + w_vr_v + w_rr_r + w_er_e + w_{task}r_{task} ,\n\\vspace{-4pt}\n$$\nwhere $r_p= \\exp( -\\|\\mathbf{p}\\|_2)$ and $r_v = \\|\\dot{\\mathbf{p}}\\|_2$ drive the robot towards its target position and velocity as fast as possible;\n$r_r= 1-\\|\\mathbf{r}\\|_2$ drives the robot towards its target rotation (pose);\n$r_e =\\|\\mathbf{\\tau}\\|^2$ measures the effort exhausted during the swimming;\n$r_{task}$ is a task-specific reward, which will be specified later;\nand $w_p,w_v,w_r,w_e,w_{task}$ are the corresponding weights for different components in the reward.\nThe weights for each benchmark task are listed in Table.~\\ref{weight_table}.\n\\begin{table}[t]\t\n\t\\caption{Weights used in the reward of each task} \n\t\\vspace{-10pt}\n\t\\label{weight_table}\n\t\\begin{center}\n\t\\begin{tabular}{c|c|c|c|c|c}\n\t\t& $w_v$ & $w_p$ & $w_r$ & $w_e$ & $w_{task}$ \\\\ \\hline\n\t\tCruising           & 1  &0   & 0.2   & 0.5   & 0          \\\\\n\t\tPose Control       & 0  &0   & 1     & 0     & 0          \\\\\n\t\tTwo Fish Schooling & 0  &1   & 0     & 0.1   & 0          \\\\\n\t\tPath Following     & 1  &0   & 0     & 0.5   & 1       \\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{center}\n\\vspace{-25pt}\n\\end{table}\nOur proposed four benchmark tasks are trained using either global or local policy learning approaches we described.\n\\paragraph{Global policy learning}\nFor cruising, pose control and two-fish schooling tasks, we use global policy learning with a single input task, meaning that we train robot fish separately for each task, where $r_{task}=0$. Fig.~\\ref{fig:tasksequence} shows the snapshots of the swimming results, where the first two rows show cruising inside a shallow and a deep fluid; the third row shows the pose control for U-turn, and the fourth row shows the two-fish schooling result, which has not been demonstrated in previous works.\n\n\\paragraph{Local policy learning}\n\\begin{figure}[htb]\n\t\\vspace{-6pt}\n\t\\centering\n\t\\includegraphics[width=0.98\\linewidth]{Images\/local_policy.png}\n\t\\vspace{-12pt}\n\t\\caption{Illustration of local policy training for arbitrary path following. Left: a local target is sampled given a random $d$ and $\\theta$; right: when applying learned policy for path following, we always select a local target ahead on the desired path, which changes for each time step.}\n\t\\label{fig:local_policy}\n\t\\vspace{-10pt}\n\\end{figure}\nTraining robot fish following an arbitrarily long path is more difficult, and global policy learning could be resource demanding and time consuming.\nTo make the training easier while also retaining generalizability, we use local policy learning instead.\nIn fact, following an arbitrarily long path can be viewed as following a sequence of short and straight local paths along nearly the tangent direction of the global path given a robot location, greatly simplifying the training process. \nDuring training, we randomly sample local paths (parameterized by $(d,\\theta)$, see Fig.~\\ref{fig:local_policy} (left), where $d$ is the distance to the local path and $\\theta$ is the angle to the target; note that we restrict the robot fish to swim in a horizontal 3D plane) and form a set of trajectories that are representative of the local conditions of a global path; then we can train the local policy once and apply it to any specified path at any time step, similar in idea to \\cite{Peng2017}. \nIn our local policy learning, the task specific state is defined as $s_{task} = \\mathbf{d}$, and the task specific reward $r_{task}$ is:\n\\vspace{-4pt}\n\\begin{equation}\nr_{task} = \\|\\dot{\\mathbf{d}}\\|_2+\\exp( -\\|\\mathbf{d}\\|_2) ,\n\\vspace{-4pt}\n\\end{equation}\nwhich encourages fast and stable convergence to the local path.\nHere, $\\mathbf{d}$ is a vector containing relative distance to the path.\nThe training is initialized randomly by the technique proposed in \\cite{Peng2018}, and on each trial, random initial velocity is enforced on the robot fish and random angles and velocities are set on the joints.\nOnce learned, we apply the policy every some time steps based on the input state and a local target location that is 0.5m ahead on the local path, see Fig.~\\ref{fig:local_policy} (right), and Fig.~\\ref{fig:tasksequence} (bottom) shows a path following result.\n\\section{Related Work}\nWe herein review the relevant work in the literature for both simulation and learning algorithms before we dig into the details of our whole framework.\n\n\\subsection{Simulation environments in robotics}\n\nRobot training can be achieved following OpenAI Gym~\\cite{Brockman2016}, which is an open-source robot learning framework with general definitions, and can be implemented for training a variety of robots with different environments.\nHowever, both the simulator and learning framework should be provided separately.\nAt present, the most commonly used physics simulators in robot are based on rigid-body, soft-body and cloth dynamics~\\cite{Coumans2015,Todorov2012}. \nIn particular, for articulated rigid body systems, DART~\\cite{Lee2018} can be a good choice.\n\nFluid environment was traditionally provided by solving N\\\"avier-Stoke (NS) equation coupled with a rigid body simulator~\\cite{Tan2011}; but efficiency limited their application especially for vortical flows. Recently, a new simulation environment for underwater soft-body creatures appeared relying on the finite-element method and projection dynamics~\\cite{min2019softcon}; however, its choice of empirical formula~\\cite{Terzopoulos1994} on hydrodynamics makes it impossible to create complex flow environment involving vortices and turbulence. The same issue also applies to some marine vehicle simulators~\\cite{cieslak2019stonefish,manhaes2016uuv} based on Fossen model\\cite{fossen2011handbook}.\nVery recently, Gan et al.~\\cite{Gan2020} proposed a fluid environment, but only for limited tasks and accuracy.\n\nThere is lack of versatile and efficient yet accurate fluid simulation environment upon which more general underwater robot training can be performed, and our proposed ``FishGym'' tries to fill the gap by providing highly efficient GPU-based simulator for two-way coupled fluid-structure interaction.\nThere are also some learning frameworks that can be used based on OpenAI Gym, e.g., rllib~\\cite{Liang2018}, Coach~\\cite{Caspi2017} and stable-baselines3~\\cite{Raffin2019}, and we adopted ``stable-baselines3'' for training our fish-like underwater robots. \n\n\n\n\n\\iffalse\n\\subsection{Fish dynamics and actuation}\\label{sec:fish_model_work}\nIn this paper, we focus on the fish-like underwater robots, and we briefly discuss the dynamics and actuation of fishes in order to motivate our later design.\n\n\\paragraph{Fish modeling}\nFish modeling targets the description of fish structure in a proper way for later simulation, and various models have been proposed with different complexity and accuracy.\nOne of the simplest models is to regard fish as a surface mesh with a mass-spring system to approximate the functionality of muscles \\cite{Terzopoulos1994}.\nAnother type of simple fish model is to employ non-deformable rigid\/flexible foil \\cite{Lu2013,Kaya2007,Hover2004,Wang2018}.\nHowever, these simple models cannot produce accurate fish dynamics. \nTo improve accuracy for simulation, articulated rigid body systems involving connected links with joint constraints were considered for underwater creatures \\cite{Tan2011, Liu2015}, but it is inaccurate for fishes.\nA better deformable fish model which is commonly used employs triangle meshes with center-line curvature driven deformation~\\cite{Ming2018,Song2017,Song2020,Novati2017,Verma2018}, but it is less flexible to allow different types of fishes and control behaviors.\nA more accurate physically-based approach is to model fish as a general volumetric, elastically deformable body \\cite{Pan2018}, but it is very costly during fish dynamics simulation.\n\nIn contrast, our fish is modeled by the skeletons of arbitrary topology with surface skinning, whose motion is driven by the articulated rigid body dynamics. This intuitive approach is easy to extend to other skeleton-based bionic robot shape. Secondly, compared to deformable approaches, the utilization of a mature articulated solver can be expected to step into real robotics. Third, our later coupling method which treat the skin as point samples which is physically more accurate and adaptive, compared to surface voxelization\\cite{Tan2011}.\n\nFish dynamics aims to describe the dynamic interaction between fishes and the surrounding fluid.\nLighthill proposed a large-amplitude elongated-body theory \\cite{Lighthill1975}, which first explains mathematically how fish moves in a fluid. \nStarting from that, researchers either used empirical force model \\cite{Terzopoulos1994} or more accurate fluid-structure interaction simulation ~\\cite{Song2017,Song2020,Verma2018,Novati2017} for fish dynamics description. \nWhile empirical model is fast, it is usually inaccurate; in addition, hyper-parameters inside the model are also difficult to tune in order to match the real physical system.\nDirect numerical simulation ~\\cite{dai2012dynamic,song2014three,Novati2017}, on the other hand, is more accurate and reliable, but it is usually very slow, leading to a very long training period. \nWhen fish is immersed inside fluid, the fish surface is often marked as the solid boundary condition for fluid simulation \\cite{Ming2018,Song2017,Song2020,Tan2011, Liu2015}, the fluid forces calculated at the surface are applied to drive the fish motion.\n\nThere are several models could be used for defining a fish's actuation. \nFor example, the torso of the fish is deformed through controlling the contraction ratio of springs in \\cite{Terzopoulos1994}.\nAn alternative way is to manipulate the centerline curvatures using mathematical functions ranging from interpolation \\cite{Song2017}, sinusoidal \\cite{Song2020}, exponential and polynomial \\cite{Ming2018} to weighted sum of cubic splines \\cite{Novati2017,Verma2018}. \nIn \\cite{Tan2011}, a robot is actuated by specifying expected joint poses for a PD controller to output torque signals. \n\\fi \n\n\n\n\\subsection{Fluid-structure interaction}\n\nFluid simulation has been studied for decades.\nTwo different fields have intensively progressed its development.\nIn computational fluid dynamics (CFD), fluid simulation mostly targets accuracy, and a set of fluid solvers are available, from finite difference \\cite{godunov1959finite,Rai-1991,smolarkiewicz1998mpdata}, to finite volume \\cite{eymard2000finite,versteeg2007introduction,pinelli2010immersed}, as well as to finite elements \\cite{wilson1983finite,girault2012finite,elman2014finite}.\nThese algorithms are typically very expensive, which are difficult for training underwater robots.\nIn computer graphics (CG), fluid simulation concerns efficiency more than accuracy, and a large number of more efficient yet less accurate solvers were proposed~\\cite{Stam2001,Kim2005FlowFixer,Becker2007Weakly,Ihmsen2014,Jiang2015Affine,Zehnder2018,Qu2019}.\nHowever, even though GPU acceleration has been used in some of these solvers, efficiency is still not high enough.\nWhen rigid body dynamics is coupled for fluid-structure interaction \\cite{klingner2006fluid,lv2010novel,dai2005adaptive}, the efficiency can be even lower.\n\nIn recent years, lattice Boltzmann method (LBM) has been considered as a very promising alternative to traditional fluid solvers~\\cite{Liu-2012,Daniel-2014,Rosis-2017,Li-2018,Li-2020}, exhibiting excellent efficiency and accuracy (usually an order of magnitude faster than the NS counterpart with comparable accuracy on GPU).\nIts pure local dynamics without solving global equations greatly benefits the highly parallel implementation~\\cite{Li-2018,Li-2020,Chen2021gpu}.\nWhen LBM is coupled with immersed boundary (IB) method~\\cite{Li-2020}, it can be easily used to simulate two-way coupled fluid-structure interaction.\nIn particular, Chen et al.~\\cite{Chen2021gpu} proposed a GPU-optimized implementation of IB-LBM, which provides a super-efficient solver for fluid-structure interaction, making the originally expensive fluid simulation now affordable for robot training purposes.\nOur proposed platform is based on such a solver, with modifications to allow it for simulating fish dynamics in a local moving domain for higher flexibility, which is not supported in any previous works. \n\n\n\n\n\\subsection{Reinforcement learning for robot control}\nReinfocement learning (RL) is a branch of machine learning which aims to train agents using data collected through interaction with the surrounding environment. \nFor real world problems in robotics, model-free RL algorithms are often used~\\cite{Nian2020}. \nThere are two main approaches of model-free RL: policy optimization and Q-learning. \nPolicy optimization algorithms, like PPO~\\cite{Schulman2017a} and A2C\/A3C~\\cite{Mnih2016}, are stable but sample-inefficient. \nQ-learning methods, like DQN~\\cite{Mnih2013} and C51~\\cite{Bellemare2017}, are more sample-efficient but less stable.\nBoth of them have wide applications. \nFor example, PPO was used in multi-robot collision avoidance task~\\cite{long2018towards}, bipedal robot locomotion~\\cite{li2021reinforcement} etc. \nDQN also proves to work well on a certain type of tasks in real robots~\\cite{kato2017autonomous,xin2017application,chen2021non}.\nRecently, SAC~\\cite{Haarnoja2018} emerges to combine the strengths of the above two main approaches and has proved its capability in real robot problems like Dexterous manipulation~\\cite{haarnoja2018soft}, mobile robot navigation~\\cite{de2021soft}, robot arm control~\\cite{wong2021motion}, multi-legged robot~\\cite{haarnoja2018learning}, etc.\nDue to its sample efficiency and wide applications, we adopt SAC in this paper.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nAlthough the minimal Standard Model (SM) agrees well with \ncurrent electroweak experiments~\\cite{lep_slc_97}, \nit is important to examine consequences of new physics \nmodels beyond the SM at current or future collider experiments. \nOne of the simplest extensions of the SM is to introduce an \nadditional U(1) gauge symmetry, ${\\rm U(1)'}$, \nwhose breaking scale is close to the electroweak scale. \nThe ${\\rm U(1)'}$ symmetry is predicted in a certain class \nof grand unified theories (GUTs) with gauge group whose rank \nis higher than that of the SM. \nIn general, the additional ${\\rm U(1)'}$ gauge boson \n$Z'$ can mix with the hypercharge ${\\rm U(1)}_Y$ gauge boson \nthrough the kinetic term at above the electroweak scale, \nand also it can mix with the SM $Z$ boson after the \nelectroweak symmetry is spontaneously broken.  \nThrough those mixings, the $Z'$ boson can affect the electroweak \nobservables at the $Z$-pole and the $W$ boson mass $m_W$. \nBoth the $Z$-$Z'$ mixing and the direct $Z'$ contribution \ncan affect neutral current experiments off the $Z$-pole. \nThe presence of an additional $Z'$ boson can be explored \ndirectly at $p \\bar{p}$ collider experiments. \n\nThe supersymmetric (SUSY) $E_6$ models are the promising candidates \nwhich predict an additional $Z'$ boson at the weak scale (for \na review, see~\\cite{hewett_rizzo}). \nThe gauge group $E_6$ can arise from the perturbative heterotic \nstring theory as a consequence of its compactification.  \nIn the $E_6$ models, the SM matter fields in each generation \nare embedded into its fundamental representation ${\\bf 27}$ \nthat also contains several exotic matter fields -- two SM singlets, \na pair of weak doublets and color triplets. \nBecause $E_6$ is a rank-six group, it can have two extra \n${\\rm U(1)}$ factors besides the SM gauge group. \nA superposition of the two extra ${\\rm U(1)}$ groups may \nsurvive as the ${\\rm U(1)'}$ gauge symmetry at the GUT scale. \nThe ${\\rm U(1)'}$ symmetry may break spontaneously at the weak \nscale through the radiative corrections to the mass term of the \nSM singlet scalar field~\\cite{radiative_u1prime}. \n\nIn this paper, we study constraints on the $Z'$ bosons predicted \nin the SUSY $E_6$ models. \nAlthough there are several previous works~\\cite{altarelli,\nlangacker_luo,chm,gherghetta,zprime_lep}, we would like to \nupdate their studies by using the recent results of \nelectroweak experiments, and by allowing for an arbitrary \nkinetic mixing~\\cite{holdom, eta_model,general_zzmixing} \nbetween the $Z'$ boson and the hypercharge $B$ boson. \nIn our study, we use the results of $Z$-pole experiments \nat LEP1 and SLC, and the $m_W$ measurements at Tevatron and LEP2 \nwhich were reported at the summer conferences in \n1997~\\cite{lep_slc_97}. \nWe also study the constraints from low-energy neutral current \n(LENC) experiments: lepton-quark, lepton-lepton scattering \nexperiments and atomic parity violation measurements. \n\nWe find that the lower mass limit of the heavier mass \neigenstate $Z_2$ is obtained as a function of the effective $Z$-$Z'$ \nmixing term $\\zeta$, which is a combination of the mass and kinetic \nmixings. \nIn principle, $\\zeta$ is calculable, together with the gauge \ncoupling $g_E$, once the particle spectrum of the $E_6$ model \nis specified. \nWe show the theoretical prediction for $\\zeta$ and $g_E$ \nin the SUSY $E_6$ models by assuming the minimal particle \ncontent which satisfies the anomaly free condition and the \ngauge coupling unification.  \nFor those models, the electroweak data give stringent \nlower mass bound on the $Z_2$ boson. \n\nThis paper is organized as follows. \nIn the next section, we briefly review the additional $Z'$ \nboson in the SUSY $E_6$ models and the generic feature of \n$Z$-$Z'$ mixing in order to fix our notation. \nWe show that the effects of $Z$-$Z'$ mixing and direct \n$Z'$ boson contribution are parametrized \nby the following three terms: \n(i) a tree-level contribution to the $T$ parameter~\\cite{peskin_takeuchi}, \n$T_{\\rm new}$, \n(ii) the effective $Z$-$Z'$ mass mixing angle $\\bar{\\xi}$ \nand \n(iii) a contact term $g_E^2\/c^2_\\chi m_{Z_2}^2$ which appears in the low-energy \nprocesses. \nIn Sec.~3, we collect the latest results of electroweak experiments. \nThere, the theoretical predictions for the electroweak \nobservables are shown together with the SM radiative corrections. \nIn Sec.~4, we show constraints on the $Z'$ bosons from the \nelectroweak data. \nThe presence of non-zero kinetic mixing between the \n${\\rm U(1)}_Y$ and ${\\rm U(1)'}$ gauge bosons \nmodifies the couplings between the $Z'$ boson and the SM \nfermions. \nWe discuss impacts of the kinetic mixing term \non the $\\chi^2$-analysis. \nThe 95\\% CL lower mass limit of the heavier mass eigenstate $Z_2$ \nis given as a function of the effective $Z$-$Z'$ mixing parameter \n$\\zeta$. \nThe $\\zeta$-independent constraints from the low-energy \nexperiments and those from the direct search experiments \nat Tevatron are also discussed. \nIn Sec.~5, \nwe find the theoretical prediction for $\\zeta$ in some \nSUSY $E_6$ models ($\\chi, \\psi, \\eta, \\nu$) \nby assuming the minimal particle content.  \nStringent $Z_2$ boson mass bounds are found for most models.\nSec.~6 summarizes our findings. \n\\section{$Z$-$Z'$ mixing in supersymmetric $E_6$ model}\n\\subsection{$Z'$ boson in supersymmetric $E_6$ model}\n\\setcounter{equation}{0} \nSince the rank of $E_6$ is six, \nit has two ${\\rm U(1)}$ factors besides the SM gauge \ngroup which arise from the following decompositions: \n\\begin{eqnarray}\n\t\\begin{array}{rl}\n\tE_6 &\\supset {\\rm SO(10)} \\times {\\rm U(1)}_\\psi  \n\\\\\n\t&\\supset {\\rm SU(5)} \\times {\\rm U(1)}_\\chi \\times {\\rm U(1)}_\\psi. \n\t\\end{array}\n\\end{eqnarray}\n\\label{eq:e6_breaking}\n\\hsp{-0.3}\nAn additional $Z'$ boson in the electroweak scale \ncan be parametrized as \na linear combination of the ${\\rm U(1)}_\\psi$ gauge boson \n$Z_\\psi$ and the ${\\rm U(1)}_\\chi$ gauge boson \n$Z_\\chi$ as~\\cite{PDG} \n\\begin{equation}\nZ' = Z_\\chi \\cos \\beta_E + Z_\\psi \\sin \\beta_E. \n\\end{equation}\nIn this paper, we study the following four $Z'$ models \nin some detail: \n\\begin{eqnarray}\n\\begin{array}{|c||c|c|c|c|} \\hline \n~~~\\beta_E~~~ & ~~~0~~~ &~~~\\pi\/2~~~ & \\tan^{-1}(-\\sqrt{5\/3}) \n& \\tan^{-1}(\\sqrt{15}) \\\\ \\hline\n{\\rm model} & \\chi      & \\psi         & \\eta      & \\nu      \n\\\\  \\hline \n\\end{array}\n\\end{eqnarray}\nIn the SUSY-$E_6$ models, each generation of the SM quarks \nand leptons is embedded into a {\\bf 27} representation. \nIn Table~1, we show all the matter fields \ncontained in a {\\bf 27} and their classification in SO(10) \nand SU(5). \nThe ${\\rm U(1)'}$ charge assignment on the matter fields \nfor each model is also given in the same table. \nThe normalization of the ${\\rm U(1)'}$ charge follows \nthat of the hypercharge. \n\\esix_contents\nBesides the SM quarks and leptons, there are two SM singlets \n$\\nu^c$ and $S$, a pair of weak doublets $H_u$ and $H_d$, \na pair of color triplets $D$ and $\\overline{D}$ in each generation. \nThe $\\eta$-model arises when $E_6$ breaks into a rank-5 group \ndirectly in a specific compactification of the heterotic \nstring theory~\\cite{eta_ellis}. \nIn the $\\nu$-model, the right-handed neutrinos $\\nu^c$ are \ngauge singlet~\\cite{nu_model} and can have large Majorana \nmasses to realize the see-saw mechanism~\\cite{see-saw}. \n\nThe ${\\rm U(1)'}$ symmetry breaking occurs if the scalar \ncomponent of the SM singlet field develops the vacuum \nexpectation value (VEV). \nIt can be achieved at near the weak scale via radiative \ncorrections to the mass term of the SM singlet scalar \nfield. \nFor example, the terms $SD\\overline{D}$ and $S H_u H_d$ appear \nin the ${\\rm SU(3)}_C \\times {\\rm SU(2)}_L \\times \n{\\rm U(1)}_Y \\times {\\rm U(1)'}$ invariant superpotential. \nIf the Yukawa couplings of the $SD\\overline{D}$ term and\/or \n$S H_u H_d$ term are $O(1)$, the squared mass of the \nscalar component of $S$ can become negative at the weak \nscale through the renormalization group equations (RGEs) \nwith an appropriate boundary condition at the GUT scale. \nRecent studies of the radiative ${\\rm U(1)'}$ symmetry \nbreaking can be found, {\\it e.g.}, in ref.~\\cite{radiative_u1prime}. \n\nSeveral problems may arise in the $E_6$ models from view \nof low-energy phenomenology~\\cite{hewett_rizzo}.  \nFor example, the scalar components of extra colored triplets \n$D, \\overline{D}$ in ${\\bf 27}$ could mediate an instant proton \ndecay. \nIt should be forbidden by imposing a certain discrete symmetry \non the general \n${\\rm SU(3)}_C \\times {\\rm SU(2)}_L \\times {\\rm U(1)}_Y \\times\n{\\rm U(1)'}$ invariant superpotential. \nExcept for the $\\nu$-model~\\cite{nu_model}, the large Majorana \nmass of $\\nu^c$ is forbidden by the ${\\rm U(1)'}$ gauge \nsymmetry, and the fine-tuning is needed to make the Dirac neutrino \nmass consistent with the observation. \nFurther discussions can be found in ref.~\\cite{hewett_rizzo}.  \nIn the following, we \nassume that these requirements are satisfied by an unknown \nmechanism. \nMoreover we assume that all the super-partners of the SM \nparticles and the exotic matters do not affect the radiative \ncorrections to the electroweak observables significantly,  \n{\\it i.e.}, they are assumed to be heavy enough to decouple from \nthe weak boson mass scale. \n\n\\subsection{Phenomenological consequences of $Z$-$Z'$ mixing}\nIf the SM Higgs field carries a non-trivial ${\\rm U(1)'}$ \ncharge, its VEV induces the $Z$-$Z'$ mass mixing. \nOn the other hand, the kinetic mixing between the hypercharge \ngauge boson $B$ and the ${\\rm U(1)'}$ gauge boson $Z'$ \ncan occur through the quantum effects below the GUT scale. \nAfter the electroweak symmetry is broken, the effective \nLagrangian for the neutral gauge bosons in the \n${\\rm SU(2)}_L \\times {\\rm U(1)}_Y \\times {\\rm U(1)'}$ \ntheory is given by~\\cite{eta_model} \n\\begin{eqnarray}\n{\\cal L}_{gauge} \n\t&=&  -\\frac{1}{4}Z^{\\mu\\nu}Z_{\\mu\\nu}\n            -\\frac{1}{4}Z'^{\\mu\\nu}Z'_{\\mu\\nu} \n\t    -\\frac{\\sin \\chi}{2}B^{\\mu\\nu}Z'_{\\mu\\nu}\n\t    -\\frac{1}{4}A^{0\\mu\\nu}A^{0}_{\\mu\\nu} \n\\nonumber \\\\ \n\t& & + m^2_{ZZ'} Z^{\\mu}Z'_{\\mu}\n\t    +\\frac{1}{2} m^2_Z Z^{\\mu}Z_{\\mu}\n\t    +\\frac{1}{2} m^2_{Z'} Z'^\\mu Z'_{\\mu}, \n\\label{eq:l_gauge}\n\\end{eqnarray} \nwhere $F^{\\mu\\nu} (F=Z,Z',A^0,B)$ represents the gauge field strength. \nThe $Z$-$Z'$ mass mixing and the kinetic mixing are characterized \nby $m^2_{ZZ'}$ and $\\sin \\chi$, respectively. \nIn this basis, the interaction Lagrangian for the neutral current \nprocess is given as \n\\begin{eqnarray}\n{\\cal L}_{NC} &=& -\\sum_{f,\\, \\alpha}  \\left\\{ \n\t\\; e \\, Q_{f^{}_{\\alpha}} \\overline{f^{}_{\\alpha}}\n\t\\gamma^{\\mu}f^{}_{\\alpha} A^0_{\\mu} +\n\tg^{}_Z \\overline{f^{}_{\\alpha}} \\gamma^{\\mu}\n\t\\left( I^3_{f_L} - Q_{f^{}_{\\alpha}} \\sin^2\\theta_W \\right)\n\tf^{}_{\\alpha} Z^{}_{\\mu} \\right. \\nonumber \\\\ \n\t& & \\left. + g^{}_E Q^{f^{}_{\\alpha}}_E \n\t\\overline{f^{}_{\\alpha}}\\gamma^{\\mu}f^{}_{\\alpha} \n\tZ'_{\\mu} \\right\\}, \n\\label{eq:neutraC}\n\\end{eqnarray}\nwhere $g_Z = g\/\\cos\\theta_W = g_Y\/\\sin\\theta_W$. \nThe ${\\rm U(1)'}$ gauge coupling constant is denoted by $g_E$ \nin the hypercharge normalization. \nThe symbol $f_\\alpha$ denotes the quarks or leptons with \nthe chirality $\\alpha$ ($\\alpha = L$ or $R$). \nThe third component of the weak isospin, the electric charge \nand the ${\\rm U(1)'}$ charge of $f_\\alpha$ are given by \n$I^3_{f_\\alpha}$, $Q_{f_\\alpha}$ and $Q_E^{f_\\alpha}$, respectively. \nThe ${\\rm U(1)'}$ charge of the quarks and leptons listed in Table~1 \nshould be read as \n\\begin{eqnarray}\n\t\\left.\n\t\\begin{array}{l}\nQ_E^Q = Q_E^{u_L} = Q_E^{d_L},~~~Q_E^L = Q_E^{\\nu_L} = Q_E^{e_L}, \n\\\\ \nQ_E^{f^c} = -Q_E^{f_R} ~~~(f=e,u,d),\n\t\\end{array}\n\t\\right \\}. \n\\label{eq:charge_rule}\n\\end{eqnarray}\nThe mass eigenstates $(Z_1,Z_2,A)$ is obtained by \nthe following transformation; \n\\begin{equation}\n\\left( \\begin{array}{c}  Z \\\\  Z' \\\\  A^0  \n\\end{array}\\right) \n= \n\\left(\n\t\\begin{array}{ccc}\n\\cos \\xi + \\sin \\xi \\sin \\theta_W \\tan \\chi &\n-\\sin \\xi + \\cos \\xi \\sin\\theta_W \\tan \\chi & 0 \\\\\n\\sin \\xi \/ \\cos \\chi & \\cos \\xi \/ \\cos \\chi & 0 \\\\\n-\\sin\\xi \\cos \\theta_W \\tan \\chi & \n- \\cos \\xi \\cos \\theta_W \\tan \\chi & 1   \n\t\\end{array}\n\\right) \n\\left( \\begin{array}{c} {Z_1} \\\\ \n{Z_2} \\\\ {A} \\end{array}\\right). \n\\end{equation}\nHere the mixing angle $\\xi$ is given by \n\\begin{equation}\n\\tan 2\\xi = \\frac{-2c^{}_{\\chi}(m^2_{ZZ'}+s^{}_W s^{}_{\\chi}\n            m^2_Z)}{m^2_{Z'} - (c^2_{\\chi}-s^2_W s^2_{\\chi})m^2_Z+\n            2s^{}_W s^{}_{\\chi} m^2_{ZZ'}}~, \n\\label{eq:angle_xi}\n\\end{equation}\nwith the short-hand notation, \n$c_\\chi =  \\cos\\chi$, $s_\\chi = \\sin\\chi$ and $s_W = \\sin\\theta_W$. \nThe physical masses $m_{Z_1}$ and $m_{Z_2}$ ($m_{Z_1} < m_{Z_2}$) \nare given as follows; \n\\begin{subequations}\n\\begin{eqnarray}\nm_{Z_{1}}^2 \n\t &=& m_Z^2 (c_\\xi + s_\\xi s_W t_\\chi)^2 \n\t+ m_{Z'}^2 \\biggl( \\frac{s_\\xi}{c_\\chi} \\biggr)^2\n\t+ 2 m^2_{ZZ'} \\frac{s_\\xi}{c_\\chi} (c_\\xi + s_\\xi s_W t_\\chi),  \n\\label{eq:light_Z1}\n\\\\\nm_{Z_{2}}^2 \n\t &=& m_Z^2 (c_\\xi s_W t_\\chi - s_\\xi)^2 \n\t+ m_{Z'}^2 \\biggl( \\frac{c_\\xi}{c_\\chi} \\biggr)^2\n\t+ 2 m^2_{ZZ'} \\frac{c_\\xi}{c_\\chi} (c_\\xi s_W t_\\chi -s_\\xi), \n\\end{eqnarray}\n\\end{subequations}\nwhere $c_\\xi = \\cos\\xi$, $s_\\xi = \\sin\\xi$ and $t_\\chi = \\tan\\chi$. \nThe lighter mass eigenstate $Z_1$ should be identified with \nthe observed $Z$ boson at LEP1 or SLC. \nThe excellent agreement between the current experimental results \nand the SM predictions at the quantum level implies that the \nmixing angle $\\xi$ have to be small. \nIn the limit of small $\\xi$, the interaction Lagrangians \nfor the processes \n$Z_{1,2} \\rightarrow f_\\alpha \\overline{f_\\alpha}$ are expressed as \n\\begin{subequations}\n\\begin{eqnarray}\n{\\cal L}_{Z_1} &=& -\\sum_{f,\\, \\alpha} g^{}_Z \n\t\\overline{f_{\\alpha}} \\gamma^{\\mu} \\left[\n\t\\left( I^{3}_{f_{L}}-Q_{f_{\\alpha}}\\sin^2\\theta_W \n\t\\right) + \\tilde{Q}^{f_{\\alpha}}_E \\bar{\\xi} \\right]\n\tf_{\\alpha}  Z_{1 \\mu}, \n\\label{eq:neutral1}\\\\ \n{\\cal L}_{Z_2} &=& -\\sum_{f,\\,\\alpha}\\frac{g^{}_E}{c^{}_{\\chi}}\n\t\\overline{f_{\\alpha}}\\gamma^{\\mu} \\left[ \\tilde{Q}^{f_{\\alpha}}_E\n\t-\\left( I^3_{f_{\\alpha}} - Q_{f_{\\alpha}}\n\t\\sin^2\\theta_W \\right) \\frac{g^{}_Z c^{}_{\\chi}}{g^{}_E}\n\t\\xi \\right]f_{\\alpha} Z_{2 \\mu}, \n\\label{eq:neutral2}\n\\end{eqnarray}\n\\label{eq:neutralboth}\n\\end{subequations}\n\\hsp{-0.3} \nwhere the effective mixing angle $\\bar{\\xi}$ \nin eq.~(\\ref{eq:neutral1}) is given as \n\\begin{equation}\n\\bar{\\xi} = \\frac{g_E}{g_Z\\cos \\chi } \\xi. \n\\end{equation}\nIn eq.~(\\ref{eq:neutralboth}), the effective ${\\rm U(1)'}$ \ncharge $\\tilde{Q}_E^{f_\\alpha}$ is introduced as \na combination of $Q_E^{f_\\alpha}$ and the hypercharge $Y_{f_\\alpha}$: \n\\begin{subequations}\n\\begin{eqnarray}\n\\tilde{Q}^{f_\\alpha}_E &\\equiv& Q^{f_\\alpha}_E + Y_{f_\\alpha} \\delta, \n\\label{eq:effective_charge}\\\\\n\\delta &\\equiv&  -\\frac{g^{}_Z}{g^{}_E}s^{}_W s^{}_{\\chi}, \n\\end{eqnarray}\n\\label{eq:u1_charge}\n\\end{subequations}\n\\hsp{-0.3}\nwhere the hypercharge $Y_{f_\\alpha}$ should be read from Table~1 \nin the same manner with $Q_E^{f_\\alpha}$ \n(see, eq.~(\\ref{eq:charge_rule})).  \nAs a notable example, one can see from Table~1 \nthat the effective charge $\\tilde{Q}_E^{f_\\alpha}$ of \nthe leptons ($L$ and $e^c$) disappears in the \n$\\eta$-model if $\\delta$ is taken to be $1\/3$~\\cite{eta_model}. \n\nNow, due to the $Z$-$Z'$ mixing, the observed \n$Z$ boson mass $m_{Z_1}$ at LEP1 or SLC is shifted from \nthe SM $Z$ boson mass $m_Z$: \n\\begin{equation}\n\\Delta m^2 \\equiv m_{Z_1}^2 - m_Z^2 \\le 0. \n\\label{eq:mass_shift}\n\\end{equation}\nThe presence of the mass shift affects the \n$T$-parameter~\\cite{peskin_takeuchi} at tree level. \nFollowing the notation of ref.~\\cite{hhkm}, the $T$-parameter \nis expressed in terms of the effective form factors $\\bar{g}_Z^2(0), \n\\bar{g}_W^2(0)$ and the fine structure constant $\\alpha$:\n\\begin{subequations}\n\\begin{eqnarray}\n\\alpha T &\\equiv& 1 - \\frac{\\bar{g}^2_{W}(0)}{m^2_W}\n\t\\frac{m^2_{Z_1}}{\\bar{g}^2_Z(0)}  \\\\ \n\t&=&\n\t\\alpha \\left(T_{\\rm SM}^{}+  T_{\\rm new}^{}\\right), \n\\end{eqnarray}\n\\end{subequations}\nwhere \n$T_{\\rm SM}^{}$ and the new physics contribution \n$T_{\\rm new}$ are given by: \n\\begin{subequations}\n\\begin{eqnarray}\n\\alpha T_{\\rm SM} \n\t&=& 1 - \\frac{\\bar{g}^2_{W}(0)}{m^2_W}\n\t\t\\frac{m^2_{Z}}{\\bar{g}^2_Z(0)}, \\\\\n\\alpha T_{\\rm new}\n\t& = & -\\frac{ \\Delta m^2}{m^2_{Z_1}} \\geq 0. \n\\end{eqnarray}\n\\end{subequations}\nIt is worth noting that the sign of $T_{\\rm new}$ is \nalways positive. \nThe effects of the $Z$-$Z'$ mixing in the $Z$-pole experiments \nhave hence been parametrized by the effective mixing angle \n$\\bar{\\xi}$ and the positive parameter $T_{\\rm new}$. \n\nWe note here that we retain \nthe kinetic mixing term $\\delta$ as a part of the effective $Z_1$ \ncoupling $\\tilde{Q}_E^{f_\\alpha}$ in eq.~(\\ref{eq:effective_charge}). \nAs shown in refs.~\\cite{eta_model,general_zzmixing,holdom2}, \nthe kinetic mixing term $\\delta$ can be absorbed into a further \nredefinition of $S$ and $T$. \nSuch re-parametrization may be useful \nif the term $Y_{f_\\alpha} \\delta$ in eq.~(\\ref{eq:effective_charge}) \nis much larger than the $Z'$ charge $Q_E^{f_\\alpha}$. \nIn the $E_6$ models studied in this paper, we find no merit in \nabsorbing the $Y_f \\delta$ term because, the remaining \n$Q_E^{f_\\alpha}$ term is always significant. \nWe therefore adopt $\\tilde{Q}_E^{f_\\alpha}$ as the effective \n$Z_1$ couplings and $T_{\\rm new}$ accounts only for the \nmass shift (\\ref{eq:mass_shift}). \nAll physical consequences such as the bounds on $\\bar{\\xi}$ and \n$m_{Z_2}$ are of course independent of our choice of the \nparametrization. \n\nThe two parameters $T_{\\rm new}$ and $\\bar{\\xi}$ \nare complicated functions of the parameters of the effective \nLagrangian (\\ref{eq:l_gauge}). \nIn the small mixing limit, \nwe find the following useful expressions \n\\begin{subequations}\n\\begin{eqnarray}\n\\bar{\\xi} &=& -\\biggl( \\frac{g_E}{g_Z}\\frac{m_Z}{m_{Z'}} \n\t\\biggr)^2 \\zeta \\biggl[ 1+ O(\\frac{m_Z^2}{m_{Z'}^2})\\biggr], \n\\\\\n\\alpha T_{\\rm new} &=& \\hphantom{-} \\biggl( \\frac{g_E}{g_Z}\n\t\t\\frac{m_Z}{m_{Z'}} \\biggr)^2 \\zeta^2\n\t\\biggl[ 1+ O(\\frac{m_Z^2}{m_{Z'}^2})\\biggr], \n\\end{eqnarray}\n\\label{eq:tnew_xibar}\n\\end{subequations}\n\\hsp{-0.3}\nwhere we introduced an effective mixing parameter $\\zeta$ \n\\begin{equation}\n\\zeta = \\frac{g_Z}{g_E}\\frac{m_{ZZ'}^2}{m_Z^2} - \\delta.  \n\\label{eq:zeta}\n\\end{equation}\nThe $Z$-$Z'$ mixing effect disappears at $\\zeta = 0$. \nStringent limits on $m_{Z'}$ and hence on $m_{Z_2}$ \ncan be obtained through the mixing effect if $\\zeta$ is \n$O(1)$. \nWe will show in Sec.~5 that $\\zeta$ is calculable \nonce the particle spectrum of the model is specified. \nThe parameter $\\zeta$ plays an essential role in \nthe analysis of $Z'$ models. \n\nIn the low-energy neutral current processes, effects of \nthe exchange of the heavier mass eigenstate $Z_2$ can be \ndetected. \nIn the small $\\bar{\\xi}$ limit, they constrain the contact \nterm $g_E^2\/c_\\chi^2 m_{Z_2}^2$. \n\\section{Electroweak observables in the $Z'$ model}\n\\setcounter{equation}{0}\nIn this section, we give the theoretical predictions for the \nelectroweak observables which are used in our analysis. \nThe experimental data of the $Z$-pole experiments and the $W$ \nboson mass measurement~\\cite{lep_slc_97} \nare summarized in Table~2. \nThose for the low-energy experiments~\\cite{chm} are listed in \nTable~\\ref{table:low_energy}. \n\\lep_table \nL.E.N.C.\\ _table \n\\subsection{Observables in $Z$-pole experiments}\nThe decay amplitude for the process \n$Z^{}_1 \\rightarrow f_\\alpha \\overline{f_\\alpha}$ is written as\n\\begin{equation}\nT(Z_1 \\rightarrow f_{\\alpha} \\overline{f_{\\alpha}}) \n\t= M^f_{\\alpha}~\\epsilon_{Z_1} \\cdot J_{f_{\\alpha}}, \n\\label{eq:decay_amp}\n\\end{equation}\nwhere $\\epsilon_{Z_1}^\\mu$ is the polarization vector of the \n$Z^{}_1$ boson and \n$J^{\\mu}_{f_{\\alpha}} = \\overline{f_\\alpha}\\gamma^\\mu f_\\alpha$ is \nthe fermion current without the coupling factors. \nThe pseudo-observables of the $Z$-pole experiments are \nexpressed in terms of the real scalar amplitudes $M_\\alpha^f$ \nwith the following normalization~\\cite{lep_slc_97} \n\\begin{equation}\ng^f_{\\alpha} = \\frac{M^f_{\\alpha}}{\\sqrt{4\\sqrt{2}G_Fm^2_{Z_1}}} \n\t\\approx \\frac{M^f_{\\alpha}}{0.74070}. \n\\end{equation}\n\nFollowing our parametrization of the $Z$-$Z'$ mixing in \neq.~(\\ref{eq:neutral1}), the effective coupling $g^f_{\\alpha}$ \nin the $Z'$ models can be expressed as \n\\begin{eqnarray}\ng_\\alpha^f = (g_\\alpha^f)_{\\rm SM} + \\tilde{Q}_E^{f_\\alpha} \\bar{\\xi}. \n\\end{eqnarray}\nThe SM predictions~\\cite{hhkm, hhm} for the effective \ncouplings $(g_\\alpha^f)_{\\rm SM}$ can be parametrized as \n\\begin{subequations}\n\\begin{eqnarray}\n(g^{\\nu}_L)_{\\rm SM} &=& \\makebox[3.3mm]{ } 0.50214 \n\t+ 0.453\\,{\\Delta \\bar{g}^2_Z},\n\\label{eq:amp_nu} \\\\\n(g^e_L)_{\\rm SM} &=& - 0.26941 - 0.244\\,{\\Delta \\bar{g}^2_Z} \n\t+ 1.001\\,{\\Delta \\bar{s}^2},  \n\\label{eq:amp_el}\\\\\n(g^e_R)_{\\rm SM} &=& \\makebox[3.3mm]{ } 0.23201 + 0.208\\,\n\t{\\Delta \\bar{g}^2_Z} + 1.001\\,{\\Delta \\bar{s}^2}, \n\\label{eq:amp_er}\\\\\n(g^u_L)_{\\rm SM} &=&  \\makebox[3.3mm]{ } 0.34694 + 0.314\\,\n\t{\\Delta \\bar{g}^2_Z} - 0.668\\,{\\Delta \\bar{s}^2}, \n\\label{eq:amp_ul}\\\\\n(g^u_R)_{\\rm SM} &=& - 0.15466 - 0.139\\,{ \\Delta \\bar{g}^2_Z}  \n\t- 0.668 \\,{\\Delta \\bar{s}^2},  \n\\label{eq:amp_ur}\\\\\n(g^d_L)_{\\rm SM} &=& - 0.42451 - 0.383\\,{\\Delta \\bar{g}^2_Z} \n\t+ 0.334\\,{\\Delta \\bar{s}^2}, \n\\label{eq:amp_dl}\\\\\n(g^d_R)_{\\rm SM} &=& \\makebox[3.3mm]{ }  0.07732\n\t+ 0.069\\,{ \\Delta \\bar{g}^2_Z} + 0.334\\,{\\Delta \\bar{s}^2},  \n\\label{eq:amp_dr}\\\\\n(g^b_L)_{\\rm SM} &=& - 0.42109 - 0.383\\,{ \\Delta \\bar{g}^2_Z} \n\t+ 0.334\\,{\\Delta \\bar{s}^2} \n\t+ 0.00043 x_t, \n\\label{eq:amp_bl}\n\\end{eqnarray}\n\\label{eq:amp_sm}\n\\end{subequations}\n\\hsp{-0.3}\nwhere the SM radiative corrections are expressed in terms of \nthe effective couplings $\\Delta \\gzbar$ and $\\Delta \\sbar$~\\cite{hhkm, hhm} \nand the top-quark mass dependence of the $Zb_L^{} b_L^{}$ vertex \ncorrection in $(g^b_L)_{\\rm SM}$ is parametrized by \nthe parameter $x_t^{}$ \n\\begin{equation}\nx_t^{} \\equiv \\frac{m_t - 175~{\\rm GeV}}{10~{\\rm GeV}}. \n\\end{equation}\nThe gauge boson propagator corrections, $\\Delta \\gzbar$ and $\\Delta \\sbar$, \nare defined as the shift in the effective couplings \n$\\bar{g}_Z^2(m_{Z_1}^2)$ and $\\bar{s}^2(m_{Z_1}^2)$~\\cite{hhkm} \nfrom their SM reference values at $m_t^{} = 175~{\\rm GeV}$ and \n$m_H^{} = 100~{\\rm GeV}$. \nThey can be expressed in terms of the $S$ and $T$ parameters as \n\\begin{subequations}\n\\begin{eqnarray}\n\\!\\!\\!\\!\\!\\!\\!\\! \\Delta \\bar{g}^2_Z &=&  \\bar{g}_Z^2(m_{Z_1}^2) - 0.55635 \n\t= 0.00412 \\Delta T + 0.00005[1-(100~{\\rm GeV}\/m^{}_H)^2], \\\\  \n\\!\\!\\!\\!\\!\\!\\!\\! \\Delta \\bar{s}^2 &=& \\bar{s}^2(m_{Z_1}^2) - 0.23035 \n\t= 0.00360 \\Delta S - 0.00241 \\Delta T - 0.00023 x_\\alpha^{}, \n\\end{eqnarray}\n\\end{subequations}\nwhere the expansion parameter $x_\\alpha^{}$ is introduced to estimate \nthe uncertainty of the hadronic contribution to the QED coupling \n$1\/\\overline{\\alpha}(m_{Z_1}^2) = 128.75 \\pm 0.09$~\\cite{eidelman}:\n\\begin{equation}\nx_\\alpha^{} \\equiv \\frac{1\/\\overline{\\alpha}(m_{Z_1}^2) - 128.75}{0.09}. \n\\label{eq:xa_qed}\n\\end{equation}\nHere, $\\Delta S,\\Delta T,\\Delta U$ parameters are also measured from their \nSM reference values and they are given as the sum of the SM \nand the new physics contributions \n\\begin{equation}\n\\Delta S = \\Delta S^{}_{\\rm SM}+S^{}_{\\rm new}, \\;\\;\n\\Delta T = \\Delta T^{}_{\\rm SM}+T^{}_{\\rm new}, \\;\\; \n\\Delta U = \\Delta U^{}_{\\rm SM}+U^{}_{\\rm new}. \n\\label{eq:stu_delta}\n\\end{equation}\nThe SM contributions can be parametrized as~\\cite{hhm}\n\\begin{subequations}\n\\begin{eqnarray}\n\\Delta S^{}_{\\rm SM} &=& - 0.007 x^{}_t +0.091 x^{}_H -0.010 x^2_H , \\\\\n\\Delta T^{}_{\\rm SM} &=& (0.130 - 0.003 x^{}_H) x^{}_t +0.003 x^{}_t \n- 0.079 x^{}_H -0.028 x^2_H \\nonumber \\\\ & & +0.0026 x^3_H, \\\\\n\\Delta U^{}_{\\rm SM} &=& 0.022 x^{}_t -0.002x^{}_H, \n\\end{eqnarray}\n\\end{subequations}\nwhere $x_H^{}$ is defined by\n\\begin{eqnarray}\nx_H^{} &\\equiv& \\log (m_H^{}\/100~{\\rm GeV}). \n\\label{eq:higgs_xh}\n\\end{eqnarray}\n\nThe pseudo-observables of the $Z$-pole experiments \nare given by using the above eight effective couplings $g_\\alpha^f$ \nas follows. \nThe partial width of $Z_1$ boson is given by \n\\begin{eqnarray}\n\\Gamma_f &=& \\frac{G_Fm_{Z_1}^{3}}{3\\sqrt{2} \\pi} \\left\\{ \n\t\\left| g^f_L + g^f_R \\right|^2\\frac{C_{fV}}{2} \n\t+ \\left| g^f_L - g^f_R \\right|^2 \\frac{C_{fA}}{2}  \\right\\}\n\t\\left( 1+\\frac{3}{4}Q^2_f\\frac{\\bar{\\alpha}(m^2_{Z_1})}{\\pi}\\right)\n        \\makebox[10mm][l]{,}\n\\label{eq:partial_width}\n\\end{eqnarray}\nwhere the factors $C_{fV}$ and $C_{fA}$ account for the \nfinite mass corrections and the final state QCD corrections \nfor quarks. \nTheir numerical values are listed in Table~\\ref{tab:cvca}. \nThe $\\alpha_s$-dependence in $C_{qV}, C_{qA}$ \nis parametrized in terms of the parameter $x_s^{}$\n\\begin{equation}\nx_s^{} \\equiv \\frac{\\alpha_s(m_{Z_1})-0.118}{0.003}. \n\\end{equation}\nThe last term proportional to $\\bar{\\alpha}(m^2_{Z_1})\/\\pi$ \nin eq.~(\\ref{eq:partial_width}) accounts for the \nfinal state QED correction. \n\\cvca_tab\nThe total decay width $\\Gamma_{Z_1}$ and the hadronic decay \nwidth $\\Gamma_h$ are given in terms of $\\Gamma_f$: \n\\begin{subequations}\n\\begin{eqnarray}\n\\Gamma_{Z_1} &=& 3\\Gamma_{\\nu} + \\Gamma_e \n\t+ \\Gamma_{\\mu} + \\Gamma_{\\tau} + \\Gamma_h, \n\\label{eq:total_width}\\\\\n\\Gamma_h &=& \\Gamma_u + \\Gamma_c + \\Gamma_d + \\Gamma_s + \\Gamma_b. \n\\label{eq:hadron_width}\n\\end{eqnarray}\n\\end{subequations}\nThe ratios $R_\\ell^{}, R_c^{}, R_b^{}$ and the hadronic peak \ncross section $\\sigma_h^0$ are given by: \n\\begin{equation}\nR_{\\ell} = \\frac{\\Gamma_h}{\\Gamma_e},\\;\nR_c      = \\frac{\\Gamma_c}{\\Gamma_h},\\;\nR_b      = \\frac{\\Gamma_b}{\\Gamma_h},\\;\n\\sigma^0_h = \\frac{12\\pi}{m^2_{Z_1}}\n\t\\frac{\\Gamma_e\\Gamma_h}{\\Gamma_{Z_1}^2}. \n\\end{equation}\n\nThe left-right asymmetry parameter $A^{f}$ is also \ngiven in terms of the effective couplings $g_\\alpha^f$ as  \n\\begin{equation}\nA^f \t= \\frac{(g^{f}_L)^2-(g^{f}_R)^2}{(g^{f}_L)^2+(g^{f}_R)^2}. \n\\end{equation}\nThe forward-backward (FB) asymmetry $A^{0,f}_{FB}$ \nand the left-right (LR) asymmetry $A^{0,f}_{LR}$ \nare then given as follows: \n\\begin{subequations}\n\\begin{eqnarray}\nA^{0,f}_{FB} &=& \\frac{3}{4}A^{e}A^{f}, \\\\\nA^{0,f}_{LR} &=&  A^{f}. \n\\end{eqnarray}\n\\end{subequations}\n\n\\subsection{$W$ boson mass}\nThe theoretical prediction of $m^{}_W$\ncan be parametrized as~\\cite{hhkm,hhm}\n\\begin{equation}\nm_W^{}{\\rm (GeV)} =80.402-0.288\\,{\\it \\Delta S}+0.418\\,{\\it \\Delta T} \n+0.337\\,{\\it \\Delta U}+0.012\\,{\\it x_{\\alpha}}, \n\\end{equation}\nby using the same parameters, $\\Delta S, \\Delta T, \\Delta U$ (\\ref{eq:stu_delta}) \nand $x_\\alpha^{}$ (\\ref{eq:xa_qed}). \n\n\\subsection{Observables in low-energy experiments} \n\\setcounter{equation}{0}\nIn this subsection, we show the theoretical predictions for the \nelectroweak observables in the low-energy neutral current \nexperiments (LENC) --- \n(i) polarization asymmetry of the charged lepton scattering off \nnucleus target (\\ref{section_slac}--\\ref{section_mainz}), \n(ii) parity violation in cesium atom (\\ref{section_apv}),  \n(iii) inelastic $\\nu_\\mu$-scattering off nucleus target \n(\\ref{section_nq}) and \n(iv) neutrino-electron scattering (\\ref{section_ne}). \nThe experimental data are summarized in Table~\\ref{table:low_energy}. \nTheoretical expressions for the observables of \n(i) and (ii) are conveniently given in terms of \nthe model-independent parameters $C_{1q}, C_{2q}$~\\cite{jekim} \nand $C_{3q}$~\\cite{chm}.  \nThe $\\nu_\\mu$-scattering data (iii) and (iv) are expressed \nin terms of the parameters $g_{L\\alpha}^{\\nu_\\mu f}$. \nAll the model-independent parameters can be expressed \ncompactly in terms of the reduced helicity amplitudes \n$M_{\\alpha\\beta}^{f f'}$~\\cite{chm,hhkm} of the \nprocess $f_\\alpha f'_\\beta \\rightarrow f_\\alpha f'_\\beta$: \n\\begin{subequations}\n\\begin{eqnarray}\nC_{1q} &=& \\frac{1}{2 \\sqrt{2} G_F} ( \\hphantom{-} M_{LL}^{\\ell q}\n\t+ M_{LR}^{\\ell q} - M_{RL}^{\\ell q} - M_{RR}^{\\ell q} ), \n\\\\\nC_{2q} &=& \\frac{1}{2 \\sqrt{2} G_F} ( \\hphantom{-} M_{LL}^{\\ell q}\n\t- M_{LR}^{\\ell q} + M_{RL}^{\\ell q} - M_{RR}^{\\ell q} ), \n\\\\\nC_{3q} &=& \\frac{1}{2 \\sqrt{2} G_F} ( \t-M_{LL}^{\\ell q}\n\t+ M_{LR}^{\\ell q} + M_{RL}^{\\ell q} - M_{RR}^{\\ell q} ), \n\\\\\ng_{L\\alpha}^{\\nu_\\mu f} &=& \\frac{1}{2 \\sqrt{2} G_F} \n\t(-M_{L\\alpha}^{\\nu_\\mu f} ). \n\\label{eq:nutrino_amplitude}\n\\end{eqnarray}\n\\end{subequations}\nBelow, we divide these model-independent parameters into two pieces as \n\\begin{subequations}\n\\begin{eqnarray}\nC_{iq} &=& (C_{iq})_{\\rm SM} + \\Delta C_{iq}, \\\\\ng_{L\\alpha}^{\\nu_\\mu f} &=& (g_{L\\alpha}^{\\nu_\\mu f})_{\\rm SM} + \n\t\\Delta g_{L\\alpha}^{\\nu_\\mu f}, \n\\end{eqnarray}\n\\end{subequations}\nwhere the first term in each equation is the SM contribution\nwhich is parametrized conveniently by $\\Delta S$ and $\\Delta T$ in \nref.~\\cite{chm}. \nThe terms $\\Delta C_{iq}$ and $\\Delta g_{L\\alpha}^{\\nu_\\mu f}$ \nrepresent the additional contributions from the $Z$-$Z'$ mixing \nand the $Z_2$ exchange:\n\\begin{subequations}\n\\begin{eqnarray}\n\\Delta C^{}_{1u} &=& \n\t(-0.19s^{}_\\beta-0.15c^{}_\\beta+0.65\\delta )\\bar{\\xi} \n\t-\\frac{g^2_E}{c^2_\\chi}\n\t\\frac{(\\tilde{Q}^L_E-\\tilde{Q}^E_E)\n\t(\\tilde{Q}^Q_E+\\tilde{Q}^U_E)}\t{2\\sqrt{2}G^{}_Fm^2_{Z_2}}, \\\\\t\n\\Delta C^{}_{1d} &=& \n\t(0.36s^{}_\\beta-0.54c^{}_\\beta+0.17\\delta )\\bar{\\xi}\n\t-\\frac{g^2_E}{c^2_\\chi}\n\t\\frac{(\\tilde{Q}^L_E-\\tilde{Q}^E_E)(\\tilde{Q}^Q_E+\\tilde{Q}^D_E)}\n\t{2\\sqrt{2}G^{}_Fm^2_{Z_2}}, \\\\\n\\Delta C^{}_{2u} &=& \n\t(0.02s^{}_\\beta-0.84c^{}_\\beta+1.48\\delta )\\bar{\\xi} \n\t-\\frac{g^2_E}{c^2_\\chi}\\frac{(\\tilde{Q}^L_E+\\tilde{Q}^E_E)\n\t(\\tilde{Q}^Q_E-\\tilde{Q}^U_E)}{2\\sqrt{2}G^{}_Fm^2_{Z_2}}, \\\\\n\\Delta C^{}_{2d} &=&\n\t(0.02s^{}_\\beta+0.84c^{}_\\beta-1.48\\delta )\\bar{\\xi}\n\t-\\frac{g^2_E}{c^2_\\chi}\\frac{(\\tilde{Q}^L_E+\\tilde{Q}^E_E)\n\t(\\tilde{Q}^Q_E-\\tilde{Q}^D_E)}{2\\sqrt{2}G^{}_Fm^2_{Z_2}}, \\\\\n\\Delta C^{}_{3u} &=&\n\t(-0.82c^{}_\\beta+1.00\\delta )\\bar{\\xi}\n\t-\\frac{g^2_E}{c^2_\\chi}\\frac{(\\tilde{Q}^L_E-\\tilde{Q}^E_E)\n\t(\\tilde{Q}^U_E-\\tilde{Q}^Q_E)}{2\\sqrt{2}G^{}_Fm^2_{Z_2}}, \\\\\n\\Delta C^{}_{3d} &=&\n\t(1.06s^{}_\\beta-0.82c^{}_\\beta-1.00\\delta )\\bar{\\xi}\n\t-\\frac{g^2_E}{c^2_\\chi}\\frac{(\\tilde{Q}^L_E-\\tilde{Q}^E_E)\n\t(\\tilde{Q}^D_E-\\tilde{Q}^Q_E)}{2\\sqrt{2}G^{}_Fm^2_{Z_2}}, \\\\\n\\Delta g_{LL}^{\\nu u} &=& \n\t(0.44s^{}_\\beta+0.22c^{}_\\beta-0.18\\delta )\\bar{\\xi}\n\t+\\frac{g^2_E}{c^2_\\chi}\\frac{\\tilde{Q}^L_E \\tilde{Q}^Q_E}\n\t{2\\sqrt{2}G^{}_Fm^2_{Z_2}}, \\\\\n\\Delta g_{LR}^{\\nu u} &=& \n\t(-0.35s^{}_\\beta+0.01c^{}_\\beta+0.82\\delta )\\bar{\\xi}\n\t+\\frac{g^2_E}{c^2_\\chi}\\frac{\\tilde{Q}^L_E \\tilde{Q}^U_E}\n\t{2\\sqrt{2}G^{}_Fm^2_{Z_2}}, \\\\\n\\Delta g_{LL}^{\\nu d} &=& \n\t(0.04s^{}_\\beta-0.72c^{}_\\beta+0.59\\delta )\\bar{\\xi}\n\t+\\frac{g^2_E}{c^2_\\chi}\\frac{\\tilde{Q}^L_E \\tilde{Q}^Q_E}\n\t{2\\sqrt{2}G^{}_Fm^2_{Z_2}}, \\\\\n\\Delta g_{LR}^{\\nu d} &=& \n\t(-0.22s^{}_\\beta-0.52c^{}_\\beta-0.41\\delta )\\bar{\\xi}\n\t+\\frac{g^2_E}{c^2_\\chi}\\frac{\\tilde{Q}^L_E \\tilde{Q}^D_E}\n\t{2\\sqrt{2}G^{}_Fm^2_{Z_2}},\\\\\n\\Delta g_{LL}^{\\nu e} &=& (0.12 s^{}_\\beta + 0.28 c^{}_\\beta \n\t- 0.23 \\delta) \\bar{\\xi} \n\t+\\frac{g^2_E}{c^2_\\chi}\\frac{\\tilde{Q}^L_E \\tilde{Q}^L_E}\n\t{2\\sqrt{2}G^{}_Fm^2_{Z_2}},\n\\\\\n\\Delta g_{LR}^{\\nu e} &=& (-0.14 s^{}_\\beta + 0.49 c^{}_\\beta \n\t- 1.23 \\delta) \\bar{\\xi}\n\t+\\frac{g^2_E}{c^2_\\chi}\\frac{\\tilde{Q}^L_E \\tilde{Q}^e_E}\n\t{2\\sqrt{2}G^{}_Fm^2_{Z_2}}. \n\\end{eqnarray}\n\\label{eq:lenc_extra}\n\\end{subequations}\nwhere $c^{}_\\beta = \\cos\\beta_E$ and $s^{}_\\beta = \\sin\\beta_E$. \n\n\\subsubsection{SLAC $e$D experiment}\n\\label{section_slac}\nThe parity asymmetry in the inelastic scattering of polarized \nelectrons from the deuterium target was measured at SLAC~\\cite{slac}. \nThe experiment constrains the parameters \n$2C_{1u}-C_{1d}$ and $2C_{2u}-C_{2d}$. \nThe most stringent constraint shown in Table~\\ref{table:low_energy} \nis found for the following combination\n\\begin{subequations}\n\\begin{eqnarray}\nA_{\\rm SLAC} &=& 2C_{1u}-C_{1d} +0.206(2C_{2u}-C_{2d}) \\\\\n\t&=& 0.745 - 0.016\\,{\\it \\Delta S} + 0.016\\,{\\it \\Delta T} \n\t\\nonumber \\\\\n\t&&~~~~\n\t+ 2\\Delta C_{1u}- \\Delta C_{1d} \n\t+ 0.206(2\\Delta C_{2u}- \\Delta C_{2d}), \n\\end{eqnarray}\n\\label{eq:slac_sm1}\n\\end{subequations}\nwhere the theoretical prediction~\\cite{chm} is evaluated \nat the mean momentum transfer $\\langle Q^2 \\rangle = $ 1.5 GeV$^2$. \n\n\\subsubsection{CERN $\\mu^\\pm$C experiment}\nThe CERN $\\mu^\\pm$C experiment~\\cite{cern} measured \nthe charge and polarization asymmetry of deep-inelastic \nmuon scattering off the ${}^{12}$C target.\nThe mean momentum transfer of the experiment may be estimated \nat $\\langle Q^2 \\rangle = $ 50 GeV$^2$~\\cite{souder}.\nThe experiment constrains the parameters \n$2C_{2u}-C_{2d}$ and $2C_{3u}-C_{3d}$. \nThe most stringent constraint is found for the following \ncombination~\\cite{chm}\n\\begin{subequations}\n\\begin{eqnarray}\nA_{\\rm CERN} &=& 2C_{3u}-C_{3d}+0.777(2C_{2u}-C_{2d}) \\\\\n\t &=& -1.42-0.016\\,{\\it \\Delta S}+0.0006\\,{\\it \\Delta T}  \n\t\\nonumber \\\\\n\t&&~~~+ 2\\Delta C_{3u}- \\Delta C_{3d}\n\t+ 0.777(2\\Delta C_{2u}- \\Delta C_{2d}). \n\\end{eqnarray}\n\\label{eq:cern_sm1}\n\\end{subequations}\n\\subsubsection{Bates $e$C experiment} \nThe polarization asymmetry of the electron elastic scattering\noff the ${}^{12}$C target was measured at Bates \\cite{bates}.\nThe experiment constrains the combination\n\\begin{subequations}\n\\begin{eqnarray}\nA_{\\rm Bates} &=& C_{1u}+C_{1d} \\\\\n\t&=& - 0.1520 - 0.0023\\,{\\it \\Delta S} \n\t+ 0.0004\\,{\\it \\Delta T} \n\t+ \\Delta C_{1u} + \\Delta C_{1d}, \n\\end{eqnarray}\n\\end{subequations}\nwhere the theoretical prediction~\\cite{chm} is evaluated \nat $\\langle Q^2 \\rangle = $ 0.0225 GeV$^2$.\n\\subsubsection{Mainz $e$Be experiment }\n\\label{section_mainz}\nThe polarization asymmetry of electron quasi-elastic scattering\noff the ${}^9$Be target was measured at Mainz \\cite{mainz}.\nThe data shown in Table~\\ref{table:low_energy} is for \nthe combination\n\\begin{subequations}\n\\begin{eqnarray}\nA_{\\rm Mainz} &=& -2.73 C_{1u} + 0.65 C_{1d} - 2.19 C_{2u} \n\t+ 2.03 C_{2d} \n\\\\\n\t&=&  -0.876 + 0.043\\Delta S - 0.035\\Delta T\n\t\\nonumber \\\\\n\t&&~~~\n\t-2.73 \\Delta C_{1u} + 0.65 \\Delta C_{1d} \n\t- 2.19 \\Delta C_{2u} + 2.03 \\Delta C_{2d}, \n\\end{eqnarray}\n\\end{subequations}\nwhere the theoretical prediction~\\cite{chm} is evaluated \nat $\\langle Q^2 \\rangle = $ 0.2025 GeV$^2$. \n\\subsubsection{Atomic Parity Violation}\n\\label{section_apv}\nThe experimental results of parity violation in the atom \nare often given in terms of the weak charge $Q^{}_W(A,Z)$\nof nuclei. By using the model-independent parameter $C_{1q}^{}$,\nthe weak charge of a nuclei can be expressed as \n\\begin{equation}\nQ^{}_W(A,Z)=2ZC_{1p}^{}+2(A-Z)C_{1n}^{}. \n\\end{equation}\nBy taking account of the long-distance photonic \ncorrection~\\cite{apv_photonic}, \nwe find $C_{1p}$ and $C_{1n}$ as \n\\begin{subequations}\n\\begin{eqnarray}\nC_{1p} &=& \\hphantom{-} 0.03601  -0.00681 \\Delta S + 0.00477\\,\\Delta T \n\t+ 2\\Delta C_{1u} + \\Delta C_{1d}, \\\\\nC_{1n} &=& -0.49376 - 0.00366\\,{\\it \\Delta T} \n\t+ \\Delta C_{1u} + 2\\Delta C_{1d}. \n\\end{eqnarray}\n\\end{subequations}\nThe data for cesium atom $^{133}_{55}Cs$~\\cite{noecker,wood} \nis given in Table~\\ref{table:low_energy} and \nthe theoretical prediction of the weak charge \nis found to be~\\cite{chm} \n\\begin{equation}\nQ_{W}(^{133}_{55}Cs) =  -73.07 -0.749\\,{\\it \\Delta S}\n\t- 0.046\\,{\\it \\Delta T} \n\t+ 376 \\Delta C_{1u} + 422 \\Delta C_{1d}. \n\\end{equation}\n\\subsubsection{Neutrino-quark scattering} \n\\label{section_nq}\nFor the $\\nu_\\mu$-quark scattering, the experimental results \nup to the year 1988 were summarized in ref.~\\cite{fh} in terms \nof the model-independent parameters $g_L^2, g_R^2, \\delta_L^2, \\delta_R^2$. \nThe most stringent constraint on the result in ref.~\\cite{fh} \nis found for the \nfollowing combination: \n\\begin{eqnarray}\nK_{\\rm FH} &=& g_L^2 + 0.879 g_R^2 -0.010 \\delta_L^2 -0.043 \\delta_R^2.\n\\end{eqnarray}\nMore recent CCFR experiment at Tevatron measured the following \ncombination~\\cite{ccfr}\n\\begin{eqnarray}\nK_{\\rm CCFR} &=&  1.7897 g_L^2 + 1.1479 g_R^2 - 0.0916 \\delta_L^2 \n\t- 0.0782 \\delta_R^2. \n\\end{eqnarray}\nThe data are shown in Table~\\ref{table:low_energy} and the \nSM predictions are calculated from our reduced \namplitudes (\\ref{eq:nutrino_amplitude}) as \nfollows~\\cite{chm,hhkm}\n\\begin{eqnarray}\ng_\\alpha^2 = (g_{L\\alpha}^{\\nu_\\mu u})^2 \n\t+ (g_{L\\alpha}^{\\nu_\\mu d})^2, ~~~ \n\\delta_\\alpha^2 = (g_{L\\alpha}^{\\nu_\\mu u})^2 \n\t- (g_{L\\alpha}^{\\nu_\\mu d})^2,  \n\\end{eqnarray}\nfor $\\alpha = L$ and $R$, respectively, where \n\\begin{subequations}\n\\begin{eqnarray}\ng_{LL}^{\\nu_\\mu u} &=& \n\t\\hphantom{-} 0.3468 - 0.0023 \\Delta S + 0.0041 \\Delta T, \n\\\\\ng_{LR}^{\\nu_\\mu u} &=& \n\t-0.1549 - 0.0023 \\Delta S + 0.0004 \\Delta T, \n\\\\\ng_{LL}^{\\nu_\\mu d} &=& \n\t-0.4299 + 0.0012 \\Delta S - 0.0039 \\Delta T, \n\\\\\ng_{LR}^{\\nu_\\mu d} &=& \n\t\\hphantom{-} 0.0775 + 0.0012 \\Delta S - 0.0002 \\Delta T.\n\\end{eqnarray}\n\\end{subequations}\nThe above predictions are obtained at the momentum transfer \n$\\langle Q^2 \\rangle = 35~{\\rm GeV}^2$ relevant for the \nCCFR experiment~\\cite{ccfr}. \nThe estimations are found to be valid~\\cite{chm} also \nfor the data of ref.~\\cite{fh}, whose typical scale is \n$\\langle Q^2 \\rangle = 20~{\\rm GeV}^2$. \n\n\\subsubsection{Neutrino-electron scattering} \n\\label{section_ne}\nThe $\\nu_\\mu$-$e$ scattering experiments measure the neutral \ncurrents in a purely leptonic channel. \nThe combined results~\\cite{chm,charm-II} are given in \nTable~\\ref{table:low_energy}. \nThe theoretical predictions \n\\begin{subequations}\n\\begin{eqnarray}\ng_{LL}^{\\nu_\\mu e} &=& -0.273 + 0.0033 \\Delta S - 0.0042 \\Delta T \n\t+ \\Delta g_{LL}^{\\nu_\\mu e}, \n\\\\\ng_{LR}^{\\nu_\\mu e} &=& \\hphantom{-} 0.233 + 0.0033 \\Delta S - 0.0006 \\Delta T \n\t+ \\Delta g_{LR}^{\\nu_\\mu e}, \n\\end{eqnarray}\n\\end{subequations}\nare evaluated at $\\langle Q^2 \\rangle = 2m_e E_\\nu$ \nwith $E_\\nu = 25.7~{\\rm GeV}$ for the CHARM-II \nexperiment~\\cite{charm-II}. \n\\section{Constraints on $Z'$ bosons from electroweak experiments}\n\\setcounter{equation}{0} \nFollowing the parametrization presented in Sec.~3, \nwe can immediately obtain the constraints on $T_{\\rm new}, \\bar{\\xi}$ \nand $g_E^2\/c^2_\\chi m_{Z_2}^2$ from the data listed in Table~2 \nand Table~\\ref{table:low_energy}. \nSetting $S_{\\rm new} = U_{\\rm new} = 0$, we find that \nthe $Z$-pole measurements constrains $T_{\\rm new}$ and $\\bar{\\xi}$ \nwhile $m_W$ data constrains $T_{\\rm new}$. \nThe contact term $g_E^2\/c^2_\\chi m_{Z_2}^2$ is constrained from the LENC data. \nThe number of the free parameters is, therefore, six: \nthe above three parameters and the SM parameters, \n$m_t^{}, \\alpha_s(m_{Z_1})$ and $\\bar{\\alpha}(m_{Z_1}^2)$. \nThroughout our analysis, we use \n\\begin{subequations}\n\\begin{eqnarray}\nm_t^{} &=& 175.6 \\pm 5.5~{\\rm GeV}~\\cite{mt96}, \\\\\n\\alpha_s (m_{Z_1}^{}) &=& 0.118 \\pm 0.003~\\cite{PDG}, \\\\\n1\/\\bar{\\alpha}(m_{Z_1}^2) &=& 128.75 \\pm 0.09~\\cite{eidelman}, \n\\end{eqnarray}\n\\end{subequations}\nas constraints on the SM parameters. \nThe Higgs mass dependence of the results are parametrized \nby $x_H^{}$ (\\ref{eq:higgs_xh}) \nin the range $77~{\\rm GeV} <  m_H^{} ~{\\rlap{\\lower 3.5pt\\hbox{$\\mathchar\\sim$}}\\raise 1pt\\hbox{$<$}}\\, 150~{\\rm GeV}$. \nThe lower bound is obtained at the LEP experiment~\\cite{higgs_direct}. \nThe upper bound is the theoretical limit on the lightest Higgs \nboson mass in any supersymmetric models that accommodate \nperturbative unification of the gauge couplings~\\cite{kane}. \nWe first obtain the constraints from the $Z$-pole experiments \nand $W$ boson mass measurement only, and then obtain \nthose by including the LENC experiments.  \n\n\\subsection{Constraints from $Z$-pole and $m_W$ data}\nLet us examine first the constraints from the $Z$-pole and \n$m_W$ data by performing the five-parameter fit for $T_{\\rm new}, \n\\bar{\\xi}, m_t^{}, \\alpha_s(m_{Z_1})$ and $\\bar{\\alpha}(m_{Z_1}^2)$. \nThe results for the $\\chi, \\psi, \\eta$ and $\\nu$ models at \n$\\delta = 0$ are summarized as follows: \n\\def(\\roman{enumi}){(\\roman{enumi})}\n\\def\\roman{enumi}{\\roman{enumi}}\n\\begin{enumerate}\n\\item $\\chi$-model ($\\delta = 0$)\n\\begin{eqnarray}\n\\begin{array}{l}\n\t\\left.\n\t\\begin{array}{lcl}\n\tT_{\\rm new} &=& -0.040 + 0.15x_H^{} \\pm 0.12 \\\\\n\t\\bar{\\xi} &=& \\hphantom{-} 0.00017 - 0.00005x_H^{} \\pm 0.00046\n\t\\end{array}\n\t\\right \\} \\rho_{\\rm corr} = 0.28,  \\\\\n\\chi^2_{\\rm min}\/({\\rm d.o.f.}) = (16.5 + 0.7 x_H^{})\/(12), \n\\end{array}\n\\label{eq:const_chi}\n\\end{eqnarray}\n\\item $\\psi$-model ($\\delta = 0$)\n\\begin{eqnarray}\n\\begin{array}{l}\n\t\\left.\n\t\\begin{array}{lcl}\n\tT_{\\rm new} &=& -0.043 + 0.16x_H^{} \\pm 0.11 \\\\\n\t\\bar{\\xi} &=& \\hphantom{-} 0.00019 + 0.00012x_H^{} \\pm 0.00050\n\t\\end{array}\n\t\\right \\} \\rho_{\\rm corr} = 0.20, \\\\\n\\chi^2_{\\rm min}\/({\\rm d.o.f.}) = (16.5 + 0.4 x_H^{})\/(12), \n\\end{array}\n\\label{eq:const_psi}\n\\end{eqnarray}\n\\item $\\eta$-model ($\\delta = 0$)\n\\begin{eqnarray}\n\\begin{array}{l}\n\t\\left.\n\t\\begin{array}{lcl}\n\tT_{\\rm new} &=& -0.053 + 0.14x_H^{} \\pm 0.11 \\\\\n\t\\bar{\\xi} &=& -0.00014 - 0.00062x_H^{} \\pm 0.00108 \n\t\\end{array}\n\t\\right \\} \\rho_{\\rm corr} = 0.09, \\\\\n\\chi^2_{\\rm min}\/({\\rm d.o.f.}) = (16.6 + 0.4 x_H^{})\/(12), \n\\end{array}\n\\label{eq:const_eta}\n\\end{eqnarray}\n\\item $\\nu$-model ($\\delta = 0$)\n\\begin{eqnarray}\n\\begin{array}{l}\n\t\\left.\n\t\\begin{array}{lcl}\n\tT_{\\rm new} &=& -0.042 + 0.15x_H^{} \\pm 0.11 \\\\\n\t\\bar{\\xi} &=& \\hphantom{-} 0.00016 +  0.00007x_H^{} \\pm 0.00042 \n\t\\end{array}\n\t\\right \\} \\rho_{\\rm corr} = 0.23, \\\\\n\\chi^2_{\\rm min}\/({\\rm d.o.f.}) = (16.5 + 0.5 x_H^{})\/(12). \n\\end{array}\n\\label{eq:const_nu}\n\\end{eqnarray}\n\\end{enumerate}\nIn the above four $Z'$ models, the results for $T_{\\rm new}$ \nand $\\bar{\\xi}$ are consistent with zero for $x_H^{} = 0$. \nMoreover, the best fits of $T_{\\rm new}$ in all the $Z'$ models \nare in the unphysical \nregion, $T_{\\rm new} < 0$. \nThe parameter $T_{\\rm new}$ could be positive for the large $x_H^{}$: \nFor example, $x_H^{} = 0.41$ ($m_H^{} = 150~{\\rm GeV}$) makes $T_{\\rm new}$ \nin all the four $Z'$ models positive. \nThe allowed range of the effective mixing angle $\\bar{\\xi}$ is \norder of $10^{-3}$ for the $\\eta$-model and $10^{-4}$ for \nthe other three models in 1-$\\sigma$ level. \nThe $x_H^{}$-dependence of $\\bar{\\xi}$ in the $\\eta$-model is \nlarger than the other three models. \nFor comparison, we show the result for the leptophobic \n$\\eta$-model ($\\delta=1\/3$) \n\\begin{enumerate}\n\\addtocounter{enumi}{4}\n\\item leptophobic $\\eta$-model ($\\delta = 1\/3$)\n\\begin{eqnarray}\n\\begin{array}{l}\n\t\\left.\n\t\\begin{array}{lcl}\n\tT_{\\rm new} &=& -0.049 + 0.15x_H^{} \\pm 0.11 \\\\\n\t\\bar{\\xi} &=& \\hphantom{-} 0.00269 +  0.00026x_H^{} \\pm 0.00309 \n\t\\end{array}\n\t\\right \\} \\rho_{\\rm corr} = 0.03, \\\\\n\\chi^2_{\\rm min}\/({\\rm d.o.f.}) = (15.9 + 0.5 x_H^{})\/(12).\n\\end{array}\n\\label{eq:zpole_leptophobic}\n\\end{eqnarray}\n\\end{enumerate}\nBy comparing the $\\eta$-model with no kinetic mixing ($\\delta = 0$) \nin eq.~(\\ref{eq:const_eta}), we find significantly \nweaker constraint on $\\bar{\\xi}$. \n\\tnew_xi \nIn Fig.~\\ref{allowed_eta}, \nwe show the 1-$\\sigma$ and 90\\% CL allowed region on \nthe $(\\bar{\\xi}, T_{\\rm new})$ plane \nin the $\\eta$-model with $\\delta = 0$ and $1\/3$ for \n$m_H^{} = 100~{\\rm GeV}$. \n\nThe best fit results at $m_H^{} = 100~{\\rm GeV}$ under the \nconstraint $T_{\\rm new} \\geq 0$ are shown in \nTable~2. \nWe can see from Table~2 that \nthere is no noticeable improvement of the fit \nfor the $\\chi,\\psi,\\eta$ and $\\nu$ models at $\\delta = 0$. \nThe $\\chi^2_{\\rm min}$ remains almost the same as that of the SM, \neven though each model has two new free parameters, \n$T_{\\rm new}$ and $\\bar{\\xi}$. \nThe fit slightly improves for the leptophobic $\\eta$-model \n($\\delta = 1\/3$) because of the smaller pull factor \nfor the $R_b$ data. \nThe probability of the fit, 18.7\\% CL, is still less \nthan that of the SM, 26.2\\% CL, because the $\\chi^2_{\\rm min}$ \nreduces only 0.8 despite two additional free parameters. \n\nWe explore the whole range of the parameters, $\\beta_E$ and $\\delta$. \nIn Fig.~\\ref{chisq_distribution}, we show the improvement in \n$\\chi^2_{\\rm min}$ over the SM value, $\\chi^2_{\\rm min}({\\rm SM})\n= 16.9$ (see Table~2): \n\\begin{eqnarray}\n\\Delta \\chi^2 \\equiv \\chi_{\\rm min}^2(\\beta_E, \\delta) \n\t- \\chi^2_{\\rm min}({\\rm SM}), \n\\end{eqnarray}\nwhere $\\chi^2_{\\rm min}(\\beta_E, \\delta)$ is evaluated \nat the specific value of $\\beta_E$ and $\\delta$ for \n$m_H^{} = 100~{\\rm GeV}$.\n\\fig_beta_delta \nAs we seen from Fig.~\\ref{chisq_distribution}, the $\\chi^2_{\\rm min}$ \ndepends very mildly in the whole range of the $\\beta_E$ and $\\delta$ \nplane, except near the leptophobic $\\eta$-model \n($\\beta_E = \\tan^{-1}(\\sqrt{5\/3})$ and $\\delta = 1\/3$)~\\cite{eta_model}. \nEven for the best choice of $\\beta_E$ and $\\delta$, the improvement in \n$\\chi^2_{\\rm min}$ is only 1.5 over the SM. \nBecause each model has two additional parameters $T_{\\rm new}$ and \n$\\bar{\\xi}$, we can conclude that no $Z'$ model in this framework \nimproves the fit over the SM. \nThe ``$\\times$'' marks plotted in Fig.~\\ref{chisq_distribution} show \nthe specific models which we will discuss in the next section. \n\n\\subsection{Constraints from $Z$-pole + $m_W$ + LENC data}\nNext we find constraints on the contact term $g_E^2\/c^2_\\chi m_{Z_2}^2$ \nby including the low-energy data in addition to the \n$Z$-pole and $m_W$ data.\nBecause $T_{\\rm new}$ and $\\bar{\\xi}$ are already constrained \nseverely by the $Z$-pole and $m_W$ data, only the contact \nterms proportional to $g_E^2\/c_\\chi^2 m_{Z_2}^2$ \ncontribute to \nthe low-energy observables, except for the special case \nof the leptophobic $\\eta$-model ($\\delta = 1\/3$). \n\nWe summarize the results of the six-parameter \nfit for the $\\psi, \\chi, \\eta$ and $\\nu$ models:\n\\begin{flushleft}\n(i) $\\chi$-model\n\\begin{subequations}\n\\begin{eqnarray}\n& & \\!\\!\\!\\!\\left. \\!\\!\n\\begin{array}{r c l c l c l }\nT_{\\rm new} &\\!\\!=\\!\\!&\\!\\! -0.063\\!\\!&\\!\\!+\\!\\!&\\!\\! 0.14  x^{}_H\\!\\!\n&\\!\\!\\pm\\!\\!&\\!\\!0.11\\!\\!\\!\\!  \\\\\n\\bar{\\xi}   &\\!\\!=\\!\\!&\\!\\! -0.00005\\!\\!&\\!\\!-\\!\\!&\\!\\!0.00006x^{}_H\\!\\!\n&\\!\\!\\pm\\!\\!&\\!\\! 0.00044\\!\\!\\! \\\\\ng^2_E\/c^2_{\\chi}m^2_{Z_2} &\\!\\!=\\!\\!& \n\\makebox[3.3mm]{}\\!\\!0.26\\!\\!&\\!\\!+\\!\\!&\n\\!\\!0.01 x^{}_H\\!\\! &\\!\\!\\pm\\!\\!&\\!\\!0.21\\!\\!\n\\end{array} \\right\\} \n\\rho_{\\rm corr} \\!=\\! \\left(\n\\begin{array}{rrr}\n\\!\\!1.00\\!&\\!0.25\\!& \\!0.09\\!\\! \\\\\n        &\\!1.00\\!& \\!0.15\\! \\\\\n        &        & \\!1.00\\!\n\\end{array} \\right)\\!\\!,\\makebox[6mm]{ }  \n \\\\ \\!\\!\\!\\!\\!&&\\chi^2_{\\rm min}\/ ({\\rm d.o.f.}) \n= (19.9 + 0.9 x^{}_H)\/(20), \n\\end{eqnarray}\n\\end{subequations}\n\\end{flushleft}\n\\begin{flushleft}\n(ii) $\\psi$-model\n\\begin{subequations}\n\\begin{eqnarray}\n& & \\!\\!\\!\\!\\left. \\!\\!\n\\begin{array}{r c l c l c l }\nT_{\\rm new} &\\!\\!=\\!\\!&\\!\\! -0.065 \\!\\!  &\n\\!\\!+\\!\\!&\\!\\! 0.15  x^{}_H\\!\\!&\\!\\!\\pm\\!\\!&\\!\\!0.11\\!\\! \\\\\n\\bar{\\xi}   &\\!\\!=\\!\\!&\\!\\! -0.00014\\!\\! &\n\\!\\!+\\!\\!&\\!\\! 0.00012x^{}_H\\!\\!&\\!\\!\\pm\\!\\!&\\!\\!0.00050\\!\\!\\! \\\\\ng^2_E\/c^2_{\\chi}m^2_{Z_2} &\\!\\!=\\!\\!&\\!\\! \\makebox[3.3mm]{}1.66\\!\\!&\n\\!\\!+\\!\\!&\\!\\!0.19 x^{}_H\\!\\!&\\!\\!\\pm\\!\\!&\\!\\! 2.90\\!\\!\n\\end{array} \\right\\} \n\\rho^{}_{\\rm corr} \\!=\\! \\left( \n\\begin{array}{rrr}\n\\!\\!1.00\\! & \\!0.19\\! & \\!0.07\\!\\! \\\\\n         & \\!1.00\\! & \\!0.03\\! \\\\\n         &          & \\!1.00\\! \n\\end{array} \\right)\\!\\!,\\makebox[6mm]{ }  \n \\\\ \\!\\!\\!\\!\\!&&\\chi^2_{\\rm min}\/ ({\\rm d.o.f.}) \n\t= (21.1+0.8 x^{}_H)\/(20), \n\\end{eqnarray}\n\\end{subequations}\n\\end{flushleft}\n\\begin{flushleft}\n(iii) $\\eta$-model\n\\begin{subequations}\n\\begin{eqnarray}\n\\hsp{-2.5}& & \\!\\!\\!\\!\\left. \\!\\!\n\\begin{array}{r c l c l c l }\nT_{\\rm new}\\! &\\!\\!\\!=\\!\\!&\\!\\! -0.074\\!\\! &\\!\\!+\\!\\!&\n\\!\\! 0.14  x^{}_H\\!\\!&\\!\\!\\pm\\!\\!&\\!\\!0.11\\!\\!\\! \\\\\n\\bar{\\xi}\\!   &\\!\\!\\!=\\!\\!&\\!\\! -0.00038\\!\\! &\\!\\!-\\!\\!& \n\\!\\!0.00063x^{}_H\\!\\!&\\!\\!\\pm\\!\\!&\\!\\!0.00106\\!\\!\\!\\! \\\\\ng^2_E\/c^2_{\\chi}m^2_{Z_2}\\! &\\!\\!=\\!\\!&\\!\\!-0.62\\!\\!&\\!\\!+\\!\\!&\n\\!\\!0.08 x^{}_H\\!\\!&\\!\\!\\pm\\!\\!&\\!\\!0.87\\!\\!\\!\n\\end{array} \\right\\}\\! \n\\rho^{}_{\\rm corr} \\!=\\! \\left( \n\\begin{array}{rrr}\n\\!\\!1.00\\! & \\!0.06\\! & \\!-0.05\\!\\! \\\\\n         & \\!\\!1.00\\! & \\!-0.22\\!\\! \\\\\n         &          & \\!1.00\\!  \n\\end{array} \\right)\\!\\!,\\makebox[6mm]{ }  \n \\\\ \\!\\!\\!\\!\\!&&\\chi^2_{\\rm min}\/ ({\\rm d.o.f.}) \n\t= (20.8+0.5 x^{}_H)\/(20), \n\\end{eqnarray}\n\\end{subequations}\n\\end{flushleft}\n\\begin{flushleft}\n(iv) $\\nu$-model\n\\begin{subequations}\n\\begin{eqnarray}\n\\hsp{-1.5}& & \\!\\!\\!\\!\\left. \\!\\!\n\\begin{array}{r c l c l c l }\nT_{\\rm new} &\\!\\!=\\!\\!&\\!\\! -0.061\\!\\! &\\!\\!+\\!\\!&\n\\!\\! 0.15  x^{}_H\\!\\!&\\!\\!\\pm\\!\\!&\\!\\!0.11\\!\\! \\\\\n\\bar{\\xi}   &\\!\\!=\\!\\!&\\!\\! \\makebox[3.3mm]{} 0.00010\\!\\! &\\!\\!+\\!\\!&\n\\!\\! 0.00006x^{}_H\\!\\!&\\!\\!\\pm\\!\\!&\\!\\!0.00041\\!\\!\\!\\! \\\\\ng^2_E\/c^2_{\\chi}m^2_{Z_2}\\! &\\!\\!=\\!\\!&\\!\\!-0.65\\!\\!&\\!\\!+\\!\\!&\n\\!\\!0.04 x^{}_H\\!\\!&\\!\\!\\pm\\!\\!&\\!\\! 0.54\\!\\!\n\\end{array} \\right\\} \n\\rho^{}_{\\rm corr} \\!=\\! \\left( \n\\begin{array}{rrr}\n\\!\\!1.00\\! & \\!0.21\\! & \\!0.07\\! \\\\\n         & \\!1.00\\! & \\!0.03\\! \\\\\n         &          & \\!1.00\\!\n\\end{array} \\right)\\!\\!,\\makebox[6mm]{ }  \n \\\\ \\!\\!\\!\\!\\!&&\\chi^2_{\\rm min}\/ ({\\rm d.o.f.}) \n\t= (20.1+0.8 x^{}_H)\/(20).\n\\end{eqnarray}\n\\end{subequations}\n\\end{flushleft}\nThe contact term $g_E^2\/c^2_\\chi m_{Z_2}^2$ in the $\\psi$ and $\\eta$ models \nis consistent with zero in 1-$\\sigma$ level. \nBoth the best fit and the 1-$\\sigma$ error of \nthe parameters $T_{\\rm new}$ and $\\bar{\\xi}$ in all the \n$Z'$ models are slightly affected \nby including the LENC data: The best fit value of $T_{\\rm new}$ \nin all the $Z'$ models cannot be positive \neven for the $m_H^{} = 150~{\\rm GeV}$ $(x_H^{} = 0.41)$. \nSince the leptophobic $\\eta$-model does not have the contact term, \nthe low-energy data constrain the same parameters $T_{\\rm new}$ \nand $\\bar{\\xi}$. \nAfter taking into account both the high-energy and low-energy \ndata, we find \n\\begin{enumerate}\n\\addtocounter{enumi}{4}\n\\item leptophobic $\\eta$ model ($\\delta = 1\/3$)\n\\begin{eqnarray}\n\\begin{array}{l}\n\\left.\n\\begin{array}{lrl}\nT_{\\rm new} &=&  -0.074 + 0.148 x_H^{} \\pm 0.110\\\\ \n\\bar{\\xi}      &=&  \\hphantom{-} 0.00157 + 0.00019 x_H^{} \\pm 0.00279\n\\end{array}\n\t\\right \\}\\rho_{\\rm corr} = 0.02, \\\\\n\\chi^2_{\\rm min}\/({\\rm d.o.f.}) \n\t= (21.2 + 1.0 x_H^{})\/(21). \n\\end{array}\n\\end{eqnarray}\n\\end{enumerate}\nThe allowed range of $\\bar{\\xi}$ is slightly severe as \ncompared to eq.~(\\ref{eq:zpole_leptophobic}). \n\nThe best fit results for $m_H^{} = 100~{\\rm GeV}$ under the \ncondition $T_{\\rm new} \\geq 0$ are shown in Table~\\ref{table:low_energy}. \nIt is noticed that the best fit values for the weak charge of \ncesium atom $^{133}_{55}Cs$ in the $\\chi, \\eta$ and $\\nu$ models \nare quite close to the experimental data. \nThese models lead to $\\Delta \\chi^2 = -1.8$ ($\\chi$), \n$-0.8$ $(\\eta)$ and $-1.6$ $(\\nu)$. \nNo other noticeable point is found in the table. \n\\mass_lenc \n\nThe above constraints on $g_E^2\/c^2_\\chi m_{Z_2}^2$ from the LENC data give \nthe lower mass bound of the heavier mass eigenstate $Z_2$ \nin the $Z'$ models except for the leptophobic $\\eta$-model. \nIn Fig.~\\ref{mass_distribution}, the contour plot of \nthe 95\\% CL lower mass limit of $Z_2$ boson from the \nLENC experiments \nare shown on the $(\\beta_E, \\delta)$ plane by setting \n$g_E = g_Y$ and $m_H^{} = 100~{\\rm GeV}$ \nunder the condition $m_{Z_2} \\geq 0$. \nIn practice, we obtain the 95\\% CL lower limit of \nthe $Z_2$ boson mass $m_{95}$ in the following way: \n\\begin{eqnarray}\n0.05 =\n \\frac{\n\\int^\\infty_{m_{95}} dm_{Z_2} P(m_{Z_2})}\n{\\int^\\infty_{0} dm_{Z_2} P(m_{Z_2})}, \n\\end{eqnarray}\nwhere we assume that the probability density function \n$P(m_{Z_2})$ is proportional to ${\\rm exp}(-\\chi^2(m_{Z_2})\/2)$. \n\nWe can read off from Fig.~\\ref{mass_distribution} that \nthe lower mass bound of the $Z_2$ boson in the $\\psi$ model \nat $\\delta = 0$ is much weaker than those of the other $Z'$ \nmodels. \nIt has been pointed out that the most stringent \nconstraint on the contact term is the APV measurement \nof cesium atom~\\cite{chm}. \nSince  all the SM matter fields in the $\\psi$ model \nhave the same ${\\rm U(1)'}$ charge (see Table~1), \nthe couplings of contact interactions are Parity conserving, \nwhich makes constraint from the APV measurement useless. \nWe also find in Fig.~\\ref{mass_distribution} that the lower mass \nbound of the $Z_2$ boson disappears near the leptophobic $\\eta$-model \n($\\beta_E = \\tan^{-1}(\\sqrt{5\/3})$ and $\\delta = 1\/3$)~\\cite{eta_model}. \n\nWe summarize the 95\\% CL lower bound on $m_{Z_2}$ \nfor the $\\chi,\\psi,\\eta$ and $\\nu$ models ($\\delta = 0$) \nin Table~\\ref{mzelenc}. \nFor comparison, we also show the lower bound of $m_{Z_2}$  \nin the previous study~\\cite{cvetic_langacker_review} \nin the same table. \nThe bounds on the $Z_\\chi$ and $Z_\\nu$ masses \nare more severely constrained \nas compared to ref.~\\cite{cvetic_langacker_review}.  \nAlthough we used the latest electroweak data, our result \nfor the $Z_\\psi$ boson mass is somewhat weaker than that of \nref.~\\cite{cvetic_langacker_review}.  \nIn the analysis of ref.~\\cite{cvetic_langacker_review}, \nthe $e^+e^- \\rightarrow \\mu^+\\mu^-, \n\\tau^+\\tau^-$ data below the $Z$ pole \nare also used besides the $Z$-pole, $m_W$ and the LENC data. \nAs we mentioned before, the lower mass bound of the $Z'$ boson \nis obtained from the LENC data, not from the $Z$-pole data.  \nBecause the APV measurement which is most stringent constraint \nin the LENC data does not well constrain the $\\psi$ model, \nit is expected that the $e^+ e^-$ annihilation data below \nthe $Z$-pole play an important role to obtain the bound of \n$Z_\\psi$ boson mass. \n\\bound_lenc \n\nOur results in Table~\\ref{mzelenc} are also slightly weaker than \nthose in ref.~\\cite{chm}. \nThe results in ref.~\\cite{chm} have been obtained \nwithout including the $Z$-$Z'$ mixing effects and by setting \n$m_t^{} = 175~{\\rm GeV}, m_H^{} = 100~{\\rm GeV}, x_\\alpha^{} = 0$ \nand $T_{\\rm new} = 0$. \n\n\\subsection{Lower mass bound of $Z_2$ boson}\nWe have found that the $Z$-pole, $m_W$ and the LENC data \nconstrain ($T_{\\rm new}, \\bar{\\xi}$), $T_{\\rm new}$ and $g_E^2\/c^2_\\chi m_{Z_2}^2$, \nrespectively. \nWe can see from eq.~(\\ref{eq:zeta}) that, for a given $\\zeta$, \nconstraints on $T_{\\rm new}, \\bar{\\xi}$ and $g_E^2\/c^2_\\chi m_{Z_2}^2$ can be \ninterpreted as the bound on $m_{Z_2}$. \nWe show the 95\\% CL lower mass bound of the $Z_2$ boson \nfor $m_H^{} = 100~{\\rm GeV}$ \nin four $Z'$ models as a function of $\\zeta$. \nThe bound is again found under the condition $m_{Z_2} \\geq 0$. \nResults are shown in Fig.~\\ref{mass_95cl}.(a) $\\sim$  \n\\ref{mass_95cl}.(d) for the $\\chi,\\psi,\\eta,\\nu$ models, respectively. \nThe lower bound from the $Z$-pole and $m_W$ data, and \nthat from the LENC data are separately plotted in the same figure. \nIn order to see the $g_E$-dependence of the $m_{Z_2}$ bound \nexplicitly, we show the lower mass bound for the combination \n$m_{Z_2}g_Y\/g_E$. \nWe can read off from Fig.~\\ref{mass_95cl} that the bound on \n$m_{Z_2}g_Y\/g_E$ is approximately independent of $g_E$ for \n$g_E\/g_Y = 0.5 \\sim 2.0$ in each model. \n\\zprime_mass \nAs we expected from the formulae for \n$T_{\\rm new}$ and $\\bar{\\xi}$ in the small \nmixing limit (eq.~(\\ref{eq:tnew_xibar})), \nthe $Z_2$ mass is unbounded from the $Z$-pole data at \n$\\zeta = 0$. \nFor models with very small $\\zeta$, the lower bound on \n$m_{Z_2}$, therefore, comes from the LENC experiments \nand the direct search experiment at Tevatron. \nFor comparison, we plot the 95\\% CL lower bound on \n$m_{Z_2}$ obtained from the direct search experiment~\\cite{direct_search} \nin Fig.~\\ref{mass_95cl}. \nIn the direct search experiment, the $Z'$ decays into the exotic \nparticles, {\\it e.g.}, the decays into the light right-handed neutrinos \nwhich are expected for some models, are not taken into account. \nWe summarize the 95\\% CL lower bound on $m_{Z_2}$ \nfor the $\\chi,\\psi,\\eta$ and $\\nu$ models ($\\delta = 0$) \nobtained from the low-energy data and \nthe direct search experiment~\\cite{direct_search} \nin Table~\\ref{mzelenc}. \n\nThe lower bound of $m_{Z_2}$ is affected by the Higgs boson mass \nthrough the $T$ parameter. \nAs we seen from eqs.~(\\ref{eq:const_chi}) $\\sim$ (\\ref{eq:const_nu}),  \n$T_{\\rm new}$ tends to be in the physical region ($T_{\\rm new} \\geq 0$) \nfor large $m_H^{}$ $(x_H^{})$. \nThen, we find that the large Higgs boson mass decreases the lower \nbound of $m_{Z_2}$. \nFor $\\zeta = 1$, \nthe lower $m_{Z_2}$ bound in the $\\chi,\\psi,\\nu$ ($\\eta$) models \nfor $m_H^{} = 150~{\\rm GeV}$ is weaker than that for $m_H^{} = 100~{\\rm GeV}$ \nabout 7\\% (11\\%). \nOn the other hand, the Higgs boson with $m_H^{} = 80~{\\rm GeV}$ \nmakes the lower $m_{Z_2}$ bound in all the $Z'$ models \nsevere about 5\\% as compared to the case for $m_H^{} = 100~{\\rm GeV}$.  \nBecause $T_{\\rm new}$ and $\\bar{\\xi}$ are proportional to \n$\\zeta^2$ and $\\zeta$, respectively (see eq.~(\\ref{eq:tnew_xibar})), \nand it is unbounded at $|\\zeta| \\simeq 0$, \nthe lower bound of $m_{Z_2}$ may be independent of $m_H^{}$ \nin the small $|\\zeta|$ region. \nThe $m_H^{}$-dependence of the lower mass bound obtained \nfrom the LENC data is safely negligible. \n\n\nIt should be noted that,  at $\\zeta = 0$, \nonly the leptophobic $\\eta$-model ($\\delta = 1\/3$) is \nnot constrained from both the $Z$-pole and the low-energy \ndata. \nThe precise analysis and discussion for the lower mass bound \nof the $Z_2$ boson in the leptophobic $\\eta$-model can be \nfound in ref.~\\cite{uch}. \nIt is shown in ref.~\\cite{uch} that \nthe $\\zeta$-dependence of the lower mass bound is slightly \nmilder than that of the $\\eta$-model with $\\delta = 0$ \nin Fig.~\\ref{mass_95cl}.(c). \n\nIt has been discussed that the presence of $Z_2$ boson \nwhose mass is much heavier than the SM $Z$ boson mass,  \nsay 1 TeV, may lead to a find-tuning problem to stabilize \nthe electroweak scale against the ${\\rm U(1)'}$ scale~\\cite{drees}. \nThe $Z_2$ boson with $m_{Z_2} \\leq 1~{\\rm TeV}$ for $g_E = g_Y$ \nis allowed by the electroweak data only if $\\zeta$ satisfies \nthe following condition: \n\\begin{eqnarray} \n\\begin{array}{ll}\n-0.6 ~{\\rlap{\\lower 3.5pt\\hbox{$\\mathchar\\sim$}}\\raise 1pt\\hbox{$<$}}\\, \\zeta ~{\\rlap{\\lower 3.5pt\\hbox{$\\mathchar\\sim$}}\\raise 1pt\\hbox{$<$}}\\, +0.3 & ~~{\\rm for~the}~\\chi,\\psi,\\nu~{\\rm models}, \n\\\\\n-0.7 ~{\\rlap{\\lower 3.5pt\\hbox{$\\mathchar\\sim$}}\\raise 1pt\\hbox{$<$}}\\, \\zeta ~{\\rlap{\\lower 3.5pt\\hbox{$\\mathchar\\sim$}}\\raise 1pt\\hbox{$<$}}\\, +0.6 & ~~{\\rm for~the}~\\eta~{\\rm model}. \n\\end{array}\n\\label{eq:zeta_condition}\n\\end{eqnarray}\nIn principle, the parameter $\\zeta$ is calculable, together \nwith the gauge coupling $g_E$, once the particle spectrum \nof the $E_6$ model is specified. \nIn the next section, we calculate the $\\zeta$ parameter in \nseveral $E_6$ $Z'$ models. \n\n\n\\section{Light $Z'$ boson in minimal SUSY $E_6$-models}\n\\setcounter{equation}{0}\nIt is known that the gauge couplings are not unified \nin the $E_6$ models with three generations of {\\bf 27}. \nIn order to guarantee the gauge coupling unification, \na pair of weak-doublets, $H'$ and $\\overline{H'}$, \nshould be added into  \nthe particle spectrum at the electroweak scale~\\cite{dienes}. \nThey could be taken from ${\\bf 27} + {\\bf \\overline{27}}$ \nor the adjoint representation {\\bf 78}. \nThe ${\\rm U(1)'}$ charges of the additional weak doublets \nshould have the same magnitude and opposite sign, $a$ and $-a$,  \nto cancel the ${\\rm U(1)'}$ anomaly. \nIn addition, a pair of the complete SU(5) multiplet such as \n${\\bf 5 + \\overline{5}}$ can be added without spoiling the unification \nof the gauge couplings~\\cite{eta_model, dienes}. \n\nThe minimal $E_6$ model which have three generations of {\\bf 27} \nand a pair ${\\bf 2} + {\\bf \\overline{2}}$ depends in principle on the \nthree cases; $H'$ has the same quantum number as $L$ or $H_d$ of \n{\\bf 27}, or $\\overline{H_u}$ of ${\\bf\\overline{27}}$. \nAll three cases will be studied below. \n\\e6_extra \nThe hypercharge and ${\\rm U(1)'}$ charge of \nthe extra weak doublets for the $\\chi,\\psi,\\eta,\\nu$ models \nare listed in Table~\\ref{table:extra_higgs}.  \nFor comparison, we also show those in the model of Babu \n{\\it et al.~}~\\cite{eta_model}, where two pairs of \n${\\bf 2 + \\overline{2}}$ from {\\bf 78} and a pair of ${\\bf 3 + \\overline{3}}$ \nfrom ${\\bf 27 + \\overline{27}}$ are introduced \nto achieve the quasi-leptophobity at the weak scale. \n\nLet us recall the definition of $\\zeta$; \n\\begin{equation}\n\\zeta = \\frac{g_Z}{g_E}\\frac{m_{ZZ'}^2}{m_Z^2} - \\delta.  \n\\label{eq:zetadef}\n\\end{equation}\nIn the minimal model, \nthe following eight scalar-doublets can develop VEV to \ngive the mass terms $m_Z^2$ and $m_{ZZ'}^2$ in eq.~(\\ref{eq:l_gauge}): \nthree generations of $H_u,  H_d$, \nand an extra pair, $H'$ and $\\overline{H'}$. \nThen, $m_Z^2$ and $m_{ZZ'}^2$ are written in terms of \ntheir VEVs as \n\\begin{subequations}\n\\begin{eqnarray}\nm_Z^2 &=& \\frac{1}{2} g_Z^2\\biggl[ \n\t \\sum_{i=1}^3 \\biggl\\{ \n\t\\langle H_u^i \\rangle^2 + \\langle H_d^i \\rangle^2 \n\t\\biggr\\} + \\langle H' \\rangle^2 \n\t+ \\langle \\overline{H'} \\rangle^2 \\biggr], \n\\\\\nm_{ZZ'}^2 &=& \\! g_Z g_E \\biggl[ \n\t\\sum_{i=1}^{3} \\biggl\\{ \n\t-Q_E^{H_u} \\langle H_u^i \\rangle^2 \n\t+ Q_E^{H_d} \\langle H_d^i \\rangle^2 \n\t\\biggr \\}\n\t+ Q_E^{H'} \\langle H' \\rangle^2 \n\t- Q_E^{\\overline{H'}} \\langle \\overline{H'} \\rangle^2 \n\t\\biggr]\\!, \n\\end{eqnarray}\n\\end{subequations}\nwhere $i$ is the generation index. \nThe third component of the weak isospin $I_3$ for the Higgs \nfields are\n\\begin{equation}\nI_3(H_d) = I_3(H') = -I_3(H_u) = -I_3(\\overline{H'}) = 1\/2. \n\\end{equation}\nTaking account of the ${\\rm U(1)'}$ charges of \nthe extra Higgs doublets, $Q_E^{H'} = - Q_E^{\\overline{H'}}$, \nwe find from eq.~(\\ref{eq:zetadef}) \n\\begin{eqnarray}\n\\zeta &=& 2 \\frac{\\displaystyle{ \\sum_{i=1}^3 }\\biggl\\{  \n\t-Q_E^{H_u} \\langle H_u^i \\rangle^2 \n\t+Q_E^{H_d} \\langle H_d^i \\rangle^2 \n\t\\biggr \\}\n\t+ Q_E^{H'} \\biggl( \\langle H' \\rangle^2 \n\t+  \\langle \\overline{H'}\\rangle^2  \\biggr) }\n\t{\\displaystyle{ \\sum_{i=1}^3 }\\biggl\\{ \n\t\\langle H_u^i \\rangle^2 + \\langle H_d^i \\rangle^2 \n\t\\biggr\\}\n\t+ \\langle H'\\rangle^2 + \\langle \\overline{H'} \\rangle^2 } \n\t- \\delta.  \n\\label{eq:zeta_final}\n\\end{eqnarray}\nWe note here that the observed $\\mu$-decay constant \nleads to the following sum rule \n\\begin{eqnarray}\nv_u^2 + v_d^2 + v_{H'}^2 + v_{\\overline{H'}}^2 \\equiv \nv^2 = \\frac{1}{\\sqrt{2}G_F} \\approx (246~{\\rm GeV})^2,  \n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\left.\n\\begin{array}{rr}\n\t\\displaystyle{\\sum_{i=1}^3\\langle H_u^i \\rangle^2 = \\frac{v_u^2}{2},}\n\t&\n\t\\displaystyle{\\sum_{i=1}^3\\langle H_d^i \\rangle^2 = \\frac{v_d^2}{2}, }\n\t\\\\\n\t\\displaystyle{\\langle H' \\rangle^2 = \\frac{v_{H'}^2}{2}, }\n\t&\n\t\\displaystyle{\\langle \\overline{H'} \\rangle^2 = \\frac{v_{\\overline{H'}}^2}{2}}. \n\\end{array}\n\\right.\n\\end{eqnarray}\nBy further introducing the notation\n\\begin{subequations}\n\\begin{eqnarray}\n\\tan \\beta &=& \\frac{v_u}{v_d}, \n\\\\\nx^2 &=& \\frac{v_{H'}^2 + v_{\\overline{H'}}^2}{v^2}, \n\\end{eqnarray}\n\\end{subequations}\nwe can express eq.~(\\ref{eq:zeta_final}) as \n\\begin{eqnarray}\n\\zeta &=& 2 \\biggl\\{\n\t-Q_E^{H_u}(1-x^2)\\sin^2\\beta +Q_E^{H_d}(1-x^2)\\cos^2\\beta \n\t+Q_E^{H'}x^2\n\t\\biggr\\} \n\t- \\delta. \n\\end{eqnarray}\nBecause $H'$ and $\\overline{H'}$ are taken from {\\bf 27} + \n{$\\bf\\overline{27}$}, the ${\\rm U(1)'}$ charge of \n$H'$, $Q_E^{H'}$, is identified with that of $L$, \n$H_d$ or $\\overline{H_u}$. \n\nAmong all the models, only in the $\\chi$-model one can have \nsmaller number of matter particles. \nIn the $\\chi$-model, three generations of the matter fields \n{\\bf 16} and  a pair of Higgs doublets make the model \nanomaly free. \nIn this case, $\\zeta$ is found to be independent of $\\tan\\beta$: \n\\begin{subequations}\n\\begin{eqnarray}\n\\zeta &=& 2 \\frac{ Q_E^{H_d} - Q_E^{H_u} \\tan^2 \\beta  }\n\t{1 + \\tan^2 \\beta}\n\t- \\delta \n\\\\\n\t&=& 2 Q_E^{H_d} - \\delta. \n\\end{eqnarray}\n\\end{subequations}\n\n\\coeff_rge\nLet us now examine the kinetic mixing parameter $\\delta$ in each model. \nThe boundary condition of $\\delta$ at the GUT scale is $\\delta = 0$. \nThe non-zero kinetic mixing term can arise at low-energy \nscale through the following RGEs: \n\\begin{subequations}\n\\begin{eqnarray}\n\\frac{d}{dt} \\alpha_i &=& \\frac{1}{2\\pi}b_i \\alpha_i^2,  \n\\label{eq:rgea}\n\\\\\n\\frac{d}{dt} \\alpha_4 &=& \\frac{1}{2\\pi}\n( b_E + 2 b_{1E} \\delta + b_1 \\delta^2 ) \\alpha_4^2 ,\n\\label{eq:rgeb}\\\\\n\\frac{d}{dt} \\delta &=& \\frac{1}{2\\pi}\n( b_{1E} + b_1 \\delta ) \\alpha_1, \n\\label{eq:rgec}\n\\end{eqnarray}\n\\label{eq:rge}\n\\end{subequations}\nwhere $i=1,2,3$ and $t=\\ln \\mu$. \nWe define $\\alpha_1$ and $\\alpha_4$ as \n\\begin{eqnarray}\n\\alpha_1 \\equiv \\frac{5}{3}\\frac{g_Y^2}{4\\pi}, \n~~~~~~~\n\\alpha_4 \\equiv \\frac{5}{3}\\frac{g_E^2}{4\\pi}. \n\\end{eqnarray}\nThe coefficients of the $\\beta$-functions for $\\alpha_1, \n\\alpha_4$ and $\\delta$ are: \n\\begin{eqnarray}\nb_1 = \\frac{3}{5} {\\rm Tr} (Y^2), \n~~~~\nb_E = \\frac{3}{5} {\\rm Tr} (Q_E^2), \n~~~~\nb_{1E} = \\frac{3}{5} {\\rm Tr} (Y Q_E). \n\\end{eqnarray}\nFrom eq.~(\\ref{eq:rgec}), we can clearly see that the non-zero $\\delta$ \nis generated at the weak scale if $b_{1E} \\neq 0$ holds. \nIn Table~\\ref{table:coeff_rge}, we list $b_1, b_E$ and $b_{1E}$ \nin the minimal $\\chi,\\psi,\\eta,\\nu$ models and the $\\eta_{\\rm BKM}$ \nmodel~\\cite{eta_model}. \nAs explained above, the $\\chi(16)$ model has three generations \nof {\\bf 16}, and the $\\chi(27)$ model has three generations of \n{\\bf 27}. \nWe can see from Table~\\ref{table:coeff_rge} that the \nmagnitudes of the differences $b_1 - b_2$ and $b_2 - b_3$ are \ncommon among all the models including the minimal supersymmetric \nSM. \nThis guarantees the gauge coupling unification \nat $\\mu = m_{GUT} \\simeq 10^{16}~{\\rm GeV}$. \n\n\\ge_value \nIt is straightforward to obtain $g_E(m_{Z_1})$ and \n$\\delta(m_{Z_1})$ for each model. \nThe analytical solutions of eqs.~(\\ref{eq:rgea})$\\sim$\n(\\ref{eq:rgec}) are as follows:\n\\begin{subequations}\n\\begin{eqnarray}\n\\frac{1}{\\alpha_i(m_{Z_1})} &=& \\frac{1}{\\alpha_{GUT}} \n\t+ \\frac{1}{2\\pi}b_i \\ln \\frac{m_{GUT}}{m_{Z_1}}, \n\\\\\n\\delta(m_{Z_1}) &=& -\\frac{b_{1E}}{b_1}\\biggl( \n\t1 - \\frac{\\alpha_1(m_{Z_1})}{\\alpha_{GUT}} \\biggr), \n\\\\\n\\frac{1}{\\alpha_4(m_{Z_1})} &=& \\frac{1}{\\alpha_{GUT}} \n\t+ \\biggl\\{ \\frac{b_E}{b_1} - \n\t\\biggl(\\frac{b_{1E}}{b_1} \\biggr)^2 \n\t\\biggr \\} \n\t  \\biggl\\{ \\frac{1}{\\alpha_1(m_{Z_1})} - \\frac{1}{\\alpha_{GUT}}\n\t\\biggr \\} \n\\nonumber \\\\\n\t&& \n\t- \\biggl( \\frac{b_{1E}}{b_1} \\biggr)^2 \n\t\\frac{ \\alpha_1(m_{Z_1}) - \\alpha_{GUT}}{\\alpha_{GUT}^2}, \n\\end{eqnarray}\n\\end{subequations}\nwhere $\\alpha_{GUT}$ denotes the unified gauge coupling at \n$\\mu = m_{GUT}$. \nIn our calculation, \n$\\alpha_3(m_{Z_1}) = 0.118$ and \n$\\alpha(m_{Z_1}) = e^2(m_{Z_1})\/4\\pi = 1\/128$ \nare used as example. \nThese numbers give $g_Y(m_{Z_1}) = 0.357$. \nWe summarize the predictions for $g_E$ \nand $\\delta$ at $\\mu = m_{Z_1}$ in the \nminimal $E_6$ models and the $\\eta_{\\rm BKM}$ model \nin Table~\\ref{table:ge_delta}. \nIn all the minimal models, the ratio $g_E\/g_Y$ is approximately \nunity and $|\\delta|$ is smaller than about 0.07. \nOn the other hand, the $\\eta_{\\rm BKM}$ model \npredicts \n\\tnb_zeta\n$g_E\/g_Y \\sim 0.86$ and $\\delta \\sim 0.29$, which is close \nto the leptophobic-$\\eta$ model at $\\delta = 1\/3$. \nIn Figs.~\\ref{chisq_distribution} and \\ref{mass_distribution}, \nwe show the predictions of all the models by ``$\\times$'' symbol.\n\nNext, we estimate the parameter $\\zeta$ for several \nsets of $\\tan\\beta$ and $x$. \nIn Table~\\ref{table:zetasummary}, we show the predictions for \n$\\zeta$ in the minimal $\\chi,\\psi,\\eta,\\nu$ models and the \n$\\eta_{\\rm BKM}$ model. \nThe results are shown for $\\tan\\beta = 2$ and $30$, and \n$x^2 = 0$ and $0.5$. \nWe find from the table that the parameter $\\zeta$ is \nin the range $|\\zeta| ~{\\rlap{\\lower 3.5pt\\hbox{$\\mathchar\\sim$}}\\raise 1pt\\hbox{$<$}}\\, 1.35$ for all the models \nexcept for the $\\eta_{\\rm BKM}$ model, where the \npredicted $\\zeta$ lies between $-2.0$ and $-1.2$.  \nIt is shown in Fig.~\\ref{mass_95cl}\nthat $m_{Z_2}g_Y\/g_E$ is approximately \nindependent of $g_E\/g_Y$.  \nActually, we find in Table~\\ref{table:ge_delta} and \nTable~\\ref{table:zetasummary} that \nthe predicted $|\\delta|$ is smaller than about 0.1 \nand $g_E\/g_Y$ is quite close to unity in all the minimal models.  \nWe can, therefore, read off from Fig.~\\ref{mass_95cl} \nthe lower bound of $m_{Z_2}$ in the minimal models at $g_E = g_Y$. \nIn Table~\\ref{table:mass95_zeta}, we summarize the 95\\% CL lower \n$m_{Z_2}$ bound for the minimal $\\chi,\\psi,\\eta,\\nu$ models and \nthe $\\eta_{\\rm BKM}$ model which correspond to the predicted $\\zeta$ \nin Table~\\ref{table:zetasummary}. \n\\masszeta\nMost of the lower mass bounds in Table~\\ref{table:mass95_zeta} \nexceed 1 TeV. \nThe $Z_2$ boson with $m_{Z_2} \\sim O(1~{\\rm TeV})$ should be \nexplored at the future collider such as LHC. \nThe discovery limit of the $Z'$ boson in the $E_6$ models at LHC \nis expected as~\\cite{cvetic_bound}\n\\begin{eqnarray}\n\\begin{array}{cccc}\\hline \n\\chi & \\psi & \\eta & \\nu \\\\ \\hline \n3040 & 2910 & 2980 & *** \\\\ \\hline \n\\end{array}\n\\end{eqnarray}\nAll the lower bounds of $m_{Z_2}$ listed in \nTable~\\ref{table:mass95_zeta} are smaller than 2 TeV \nand they are, therefore, in the detectable range of LHC. \nBut, it should be noticed that most of them \n($1~{\\rm TeV} ~{\\rlap{\\lower 3.5pt\\hbox{$\\mathchar\\sim$}}\\raise 1pt\\hbox{$<$}}\\, m_{Z_2}$) may require the fine-tuning to stabilize \nthe electroweak scale against the ${\\rm U(1)'}$ scale~\\cite{drees}. \n\nThe lower bound of the $Z_2$ boson mass in the $\\eta_{\\rm BKM}$ \nmodel for the predicted $\\zeta$ can be read off from Fig.~2 in \nref.~\\cite{uch}. \nBecause somewhat large $\\zeta$ is predicted in the $\\eta_{\\rm BKM}$ \nmodel, $1 ~{\\rlap{\\lower 3.5pt\\hbox{$\\mathchar\\sim$}}\\raise 1pt\\hbox{$<$}}\\, |\\zeta|$, the lower mass bound is also large \nas compared to the minimal models. \n\n\\section{Summary}\nWe have studied constraints on $Z'$ bosons in the SUSY \n$E_6$ models. \nFour $Z'$ models --- the $\\chi,\\psi,\\eta$ and $\\nu$ models \nare studied in detail. \nThe presence of the $Z'$ boson affects the electroweak processes \nthrough the effective $Z$-$Z'$ mass mixing angle $\\bar{\\xi}$, \na tree level contribution $T_{\\rm new}$ which is a positive \ndefinite quantity, and the contact term $g_E^2\/c^2_\\chi m_{Z_2}^2$. \nThe $Z$-pole, $m_W$ and LENC data constrain ($T_{\\rm new}, \n\\bar{\\xi}$), $T_{\\rm new}$ and $g_E^2\/c^2_\\chi m_{Z_2}^2$, respectively. \nThe convenient parametrization of the electroweak \nobservables in the SM and the $Z'$ models are presented. \nFrom the updated electroweak data, we find that the $Z'$ models \nnever give the significant improvement of the $\\chi^2$-fit \neven if the kinetic mixing is taken into accounted. \nThe 95\\% CL lower mass bound of the heavier mass eigenstate $Z_2$ \nis given as a function of the effective $Z$-$Z'$ mixing \nparameter $\\zeta$. \nThe approximate scaling low is found for the $g_E\/g_Y$-dependence \nof the lower limit of $m_{Z_2}$. \nBy assuming the minimal particle content of the $E_6$ model,  \nwe have found the theoretical predictions for $\\zeta$. \nWe have shown that the $E_6$ models \nwith minimal particle content which is consistent with \nthe gauge coupling unification predict the non-zero kinetic mixing \nterm $\\delta$ and the effective mixing parameter $\\zeta$ of \norder one. \nThe present electroweak experiments lead to the lower \nmass bound of order 1 TeV or larger for those models. \n\n\\section*{Acknowledgment}\nThis work is supported in part by Grant-in-Aid for Scientific \nResearch from the Ministry of Education, Science and Culture of Japan.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{s1}\n\nIn this paper, we will study the loop quantum cosmology (LQC)\n\\cite{aa-rev,mbrev} of the Bianchi type II model. These models are\nof special interest to the issue of singularity resolution\nbecause of the intuition derived from the body of results related to\nthe Belinksii, Khalatnikov, Lifshitz (BKL) conjecture\n\\cite{bkl1,bkl2} on the nature of generic, spacelike singularities\nin general relativity (see, e.g., \\cite{bb}). Specifically, as the\nsystem enters the Planck regime, dynamics at any fixed spatial point\nis expected to be well described by the Bianchi I evolution.\nHowever, there are transitions in which the parameters\ncharacterizing the specific Bianchi I space-time change and the\ndynamics of these transitions mimics the Bianchi II time evolution.\nIn a recent paper \\cite{awe2}, we studied the Bianchi I model in the\ncontext of LQC. In this paper we will extend that analysis to the\nBianchi II model. We will follow the same general approach and use\nthe same notation, emphasizing only those points at which the\npresent analysis differs from that of \\cite{awe2}.\n\nBianchi I and II models are special cases of type A Bianchi models\nwhich were analyzed already in the early days of LQC (see in\nparticular \\cite{mb-hom, bdv}). However, as is often the case with\npioneering early works, these papers overlooked some important\nconceptual and technical issues. At the classical level,\ndifficulties faced by the Hamiltonian (and Lagrangian) frameworks in\nnon-compact, homogeneous space-times went unnoticed. In these cases,\nto avoid infinities, it is necessary to introduce an elementary cell\nand restrict all integrals to it \\cite{as,abl}. The Hamiltonian\nframeworks in the early works did not carry out this step. Rather,\nthey were constructed simply by dropping an infinite volume integral\n(a procedure that introduces subtle inconsistencies). In the quantum\ntheory, the kinematical quantum states were assumed to be periodic\n---rather than almost-periodic--- in the connection, and the quantum\nHamiltonian constraint was constructed using a ``pre-$\\mu_o$''\nscheme. Developments over the intervening years have shown that\nthese strategies have severe limitations (see, e.g.,\n\\cite{aps3,acs,cs1,aa-badhonef,cs2}). In this paper, they will be\novercome using ideas and techniques that have been introduced in the\nisotropic and Bianchi I models in these intervening years. Thus, as\nin \\cite{awe2} the classical Hamiltonian framework will be based on\na fiducial cell, quantum kinematics will be constructed using almost\nperiodic functions of connections and quantum dynamics will use the\n``$\\bar\\mu$ scheme.'' Nonetheless, the space-time description of\nBianchi II models in \\cite{mb-hom, bdv}, tailored to LQC, will\nprovide the point of departure of our analysis.\n\nNew elements required in this extension from the Bianchi I model\ncan be summarized as follows. Recall first that the spatially\nhomogeneous slices $M$ in Bianchi models are isomorphic to\n3-dimensional group manifolds. The Bianchi I group is the\n3-dimensional group of translations. Hence the the three Killing\nvectors ${}^o\\xi^a_i$ on $M$ ---the left invariant vector fields on\nthe group manifold--- commute and coincide with the right\ninvariant vector fields ${}^o\\!e^a_i$ which constitute the fiducial\northonormal triads on $M$. In LQC one mimics the strategy used in\nLQG and spin foams and defines the curvature operator in terms of\nholonomies around plaquettes whose edges are tangential to these\nvector fields. The Bianchi II group, on the other hand, is\ngenerated by the two translations and the rotation on a null\n2-plane. Now the Killing vectors ${}^o\\xi^a_i$ no longer commute and\nneither do the fiducial triads ${}^o\\!e^a_i$. Therefore we have to\nfollow another strategy to build the elementary plaquettes.\nHowever, this situation was already encountered in the\nk=$1$, isotropic models \\cite{warsaw,apsv}. There, the desired\nplaquettes can be obtained by alternating between the integral\ncurves of right and left invariant vector fields which do commute.\nHowever, in the isotropic case, the gravitational connection is\ngiven by $A_a^i = c\\,\\, {}^o\\!\\omega_a^i$, where ${}^o\\!\\omega_a^i$ are the covectors\ndual to ${}^o\\!e^a_i$ and the holonomies around these plaquettes turned\nout to be almost periodic functions of the connection component\n$c$ \\cite{warsaw,apsv}. By contrast, in the Bianchi II model we\nhave three connection components $c^i$ because of the presence of\nanisotropies, and, unfortunately, the holonomies around our\nplaquettes are no longer almost periodic functions of $c^i$. (This\nis also the case in more complicated Bianchi models.) Since the\nstandard kinematical Hilbert space of LQC consists of almost\nperiodic functions of $c^i$, these holonomy operators are not\nwell-defined on this Hilbert space. Thus, the strategy \\cite{abl}\nused so far in LQC to define the curvature operator is no longer\nviable.\n\nOne could simply enlarge the kinematical Hilbert space to\naccommodate the new holonomy functions of connections. But then the\nproblem quickly becomes as complicated as full LQG. To solve the\nproblem within the standard, symmetry reduced kinematical framework\nof LQC, one needs to generalize the strategy to define the curvature\noperator. Of course, the generalization must be such that, when\napplied to all previous models, it is compatible with the procedure\nof computing holonomies around suitable plaquettes used there. We\nwill carry out this task by suitably modifying ideas that have\nalready appeared in the literature. This generalization will enable\none to incorporate \\emph{all} class A Bianchi models in the LQC\nframework.\n\nOnce this step is taken, one can readily construct the quantum\nHamiltonian constraint and the physical Hilbert space, following\nsteps that were introduced in the analysis \\cite{awe2} of the\nBianchi I model. However, because Bianchi II space-times have\nspatial curvature, the spin connection compatible with the\northonormal triad is now non-trivial. It leads to two new terms in\nthe Hamiltonian constraint that did not appear in the Bianchi I\nHamiltonian. We will analyze these new terms in some detail. In\nspite of these differences, the big bang singularity is resolved in\nthe same precise sense as in the Bianchi I model \\cite{awe2}: If a\nquantum state is initially supported only on classically\nnon-singular configurations, it continues to be supported on\nnon-singular configurations throughout its evolution.\n\nThe paper is organized as follows. Section \\ref{s2} summarizes the\nclassical Hamiltonian theory describing Bianchi II models. Section\n\\ref{s3} discusses the quantum theory. We first define a non-local\nconnection operator $\\hat{A}_a^i$ and use it to obtain the\nHamiltonian constraint. We then show that the singularity is\nresolved and the Bianchi I quantum dynamics is recovered in the\nappropriate limit. In Section \\ref{s4}, we introduce effective\nequations for the model (with the same caveats as in the Bianchi I\ncase \\cite{awe2})\nFinally, in section V we summarize our results and discuss the new\nelements that appear in the Bianchi II model. In Appendix A we\nimprove on the discussion of discrete symmetries presented in\n\\cite{awe2}. The results on the Bianchi I model obtained in\n\\cite{awe2} carry over without any change. But the change of\nviewpoint is important to the LQC treatment of the Bianchi II model\nand more general situations.\n\n\n\\section{Classical Theory}\n\\label{s2}\n\nThis section is divided into two parts. In the first we recall the\nstructure of Bianchi II space-times and in the second we summarize\nthe phase space formulation, adapted to LQC.\n\n\\subsection{Diagonal Bianchi II Space-times}\n\\label{s2.1}\n\nBecause the issue of discrete symmetries is subtle in background\nindependent contexts, and because it plays a conceptually important\nrole in the quantum theory of Bianchi II models, we will begin with\na brief summary of how various fields are defined\n\\cite{alrev,aa-dis}. This stream-lined discussion brings out the\nassumptions which are often only implicit, making the discussion of\ndiscrete symmetries clearer.\n\nIn the Hamiltonian framework underlying loop quantum gravity (LQG),\none fixes an \\emph{oriented} 3-manifold $M$ and a 3-dimensional\n`internal' vector space $I$ equipped with a positive definite metric\n$q_{ij}$. The internal indices $i,j,k,\\ldots$ are then freely\nlowered and raised by $q_{ij}$ and its inverse. A spatial triad\n$e^a_i$ is an isomorphism from $I$ to tangent space at each point of\n$M$ which associates a vector field $v^a:= e^a_i v^i$ on $M$ to each\nvector $v^i$ in $I$.%\n\\footnote{Thus, in LQG one begins with non-degenerate triads and\nmetrics, passes to the Hamiltonian framework and then, at the end,\nextends the framework to allow degenerate geometries.}\nThe dual co-triads are denoted by $\\omega_a^i$. Given a triad, we\nacquire a positive definite metric $q_{ab}:= q_{ij} \\omega_a^i\n\\omega_b^j$ on $M$. The metric $q_{ab}$ in turn singles out a 3-form\n$\\epsilon_{abc}$ on $M$ which has \\emph{positive orientation} and\nsatisfies $ \\epsilon_{abc}\\epsilon_{def}\\, q^{ad} q^{be}q^{cf}= 3!$.\nOne can then define a 3-form $\\epsilon_{ijk}$ on $I$ via\n$\\epsilon_{ijk} = \\epsilon_{abc} e^a_i e^b_j e^c_k$. Note that\n$\\epsilon_{ijk}$ is automatically compatible with $q_{ij}$, i.e.,\n$\\epsilon_{ijk}\\epsilon_{lmn}\\, q^{il} q^{jm} q^{kn}= 3!$. If a\ntriad $\\bar{e}^a_i$ is obtained by flipping an odd number of the\nvectors in the triad $e^a_i$, then $\\bar{e}^a_i$ and $e^a_i$ have\nopposite orientations and the fields they define satisfy\n$\\bar{q}_{ab} = {q}_{ab},\\, \\bar\\epsilon_{abc} = \\epsilon_{abc}$ but\n$\\bar\\epsilon_{ijk} = - \\epsilon_{ijk}$. Had we fixed\n$\\epsilon_{ijk}$ once and for all on $I$, then $\\epsilon_{abc}$\nwould have flipped sign under this operation and volume integrals on\n$M$ computed with the unbarred and barred triads would have had\nopposite signs. With our conventions, these volume integrals will\nnot change and the parity flips will be symmetries of the symplectic\nstructure and the Hamiltonian constraint.\n\nThe triad also determines an unique spin connection $\\Gamma_a^i$ via\n\\begin{equation} \\label{sc} D_{[a} \\omega_{b]}^i\\, \\equiv \\,\n\\partial_{[a}\\omega_{b]}^i + \\epsilon^{i}{}_{jk} \\Gamma_{[a}^j\n\\omega_{b]}^k \\, =\\, 0\\, .\\end{equation}\nThe gravitational configuration variable $A_a^i$ is then given by\n$A_a^i = \\Gamma_a^i + \\gamma K_a^i$ where $K_{ab} := K_a^i\n\\omega_{bi}$ is the extrinsic curvature of $M$ and $\\gamma$ is the\nBarbero-Immirzi parameter, representing a quantization ambiguity.\n(The numerical value of $\\gamma$ is fixed by the black hole entropy\ncalculation.) The momenta $E^a_i$ carry, as usual, density weight 1\nand are given by: $E^a_i = \\sqrt{q} e^a_i$. The fundamental Poisson\nbracket is:\n\\begin{equation} \\{A_a^i(x), \\, E^b_j(y)\\} = 8\\pi G\\gamma\\,\\, \\delta_a^b\\,\n\\delta^i_j\\, \\delta^3(x,y)\\, .\\end{equation}\n\nIn Bianchi models \\cite{taub,bianchi,atu}, one restricts oneself to\nthose phase space variables admitting a 3-dimensional group of\nsymmetries which act simply and transitively on $M$. Thus, the\n3-metrics $q_{ab}$ under consideration admit a 3-parameter group of\nisometries and $M$ is diffeomorphic to a 3-dimensional Lie group\n$G$. (However, there is no canonical diffeomorphism, so that there\nis no preferred point on $M$ corresponding to the identity element\nof $G$.) To avoid a proliferation of spaces and types of indices, it\nis convenient to identify the internal space $I$ and the Lie-algebra\n$\\mathcal{L} G$ of $G$ via a fixed isomorphism. Then, there is a natural\nisomorphism ${}^o\\xi^a_i$ between $\\mathcal{L} G \\equiv I$ and Killing vector\nfields on $M$: for each internal vector $v^i$, ${}^o\\xi^a_i v^i$ is a\nKilling field on $M$. For brevity we will refer to ${}^o\\xi^a_i$ as\n(left invariant) vector fields on $M$. There is a canonical triad\n${}^o\\!e^a_i$\n---the right invariant vector fields--- which is Lie dragged by the\n${}^o\\xi^a_i$. This triad and the dual co-triad ${}^o\\!\\omega_a^i$ satisfy:\n\\begin{eqnarray} [ {}^o\\xi_i,\\, {}^o\\!e_j ] &=&0, \\quad\\quad  [{}^o\\!e_i,\\, {}^o\\!e_j] =\n- {}^o C_{ij}^k\\, {}^o\\!e_k,\\nonumber\\\\\n\\mathcal{L}_{{}^o\\xi_i}\\,( {}^o\\!\\omega^j) &=&0, \\quad\\quad {\\rm d}\\,{}^o\\!\\omega^k = \\frac{1}{2}\\,\n{}^o C_{ij}^k {}^o\\!\\omega^i\\wedge{}^o\\!\\omega^j,\\end{eqnarray}\nwhere ${}^o C_{ij}^k$ denotes the structure constants of $\\mathcal{L} G$. It is\nconvenient to use the fixed fields ${}^o\\!e^a_i$ and ${}^o\\!\\omega_a^i$ as\n\\emph{fiducial} triads and co-triads.\n\nIn the case when $G$ is the Bianchi II group, we have ${}^o C_{ik}^k =0$\nas in all class A Bianchi models and, furthermore, the symmetric\ntensor $k^{kl}:={}^o C_{ij}^k \\, \\epsilon^{ijl}$ has signature +,0,0.\nTherefore, we can fix, once and for all an orthonormal basis\n${}^o b_1^i, {}^o b_2^i, {}^o b_3^i$ in $I$ such that the only non-zero\ncomponents of ${}^o C_{ij}^k$ are\n\\begin{equation} {}^o C_{23}^1 = - {}^o C_{32}^1 = \\tilde{\\alpha}\\, ,\\end{equation}\nwhere $\\tilde{\\alpha}$ is a non-zero real number.%\n\\footnote{Without loss of generality $\\tilde{\\alpha}$ can be chosen to be 1. We\nkeep it general because we will rescale it later (see Eq.\n(\\ref{tilde})) and because we want to be able to pass to the Bianchi\nI case by taking the limit $\\tilde{\\alpha}\\to0$.}\nWe will assume that this basis is so oriented that\n\\begin{equation} \\label{ve1} \\epsilon_{123}\\, :=\\, \\epsilon_{ijk} \\, {}^o b^i_1\\,\n{}^o b^j_2\\, {}^o b^k_3\\, \\, =\\, \\varepsilon\\end{equation}\nwhere $\\varepsilon = \\pm 1$ depending on whether the frame $e^a_i$ (which\ndetermines the sign of $\\epsilon_{ijk}$) is right or left handed.\nThroughout this paper we will set ${}^o\\xi^a_1 = {}^o\\xi^a_i {}^o b^i_1,\\,\n{}^o\\!e^a_1 = {}^o\\!e^a_i{}^o b^i_1,\\, {}^o\\!\\omega_a^1 = {}^o\\!\\omega_a^i{}^o b^1_i$, etc.\n\nThe form of the components of ${}^o C^k_{ij}$ in this basis implies that\n$M$ admits global coordinates $x,y,z$ such that the Bianchi II\nKilling vectors have the fixed form\n\\begin{equation} {}^o\\xi_1^a = \\left(\\frac{\\partial}{\\partial x}\\right)^a, \\qquad\n {}^o\\xi^a_2 = \\left(\\frac{\\partial}{\\partial y}\\right)^a, \\qquad\n {}^o\\xi^a_3 = \\tilde{\\alpha} y\\left(\\frac{\\partial} {\\partial x}\\right)^a+\n \\left(\\frac{\\partial}{\\partial z}\\right)^a. \\end{equation}\nThese expressions bring out the fact that, if we were to attempt to\ncompactify the spatial slices to pass to a $\\mathbb{T}^3$ topology\n---as one can in the Bianchi I model--- we will no longer have\nglobally well-defined Killing fields. Thus, in the Bianchi II model,\nwe are forced to deal with the subtleties associated with\nnon-compactness of the spatially homogeneous slices.\n\nIn the $x,y,z$ chart, the right invariant triad is given by\n\\begin{equation} {}^o\\!e^a_1 = \\left(\\frac{\\partial}{\\partial x}\\right)^a, \\qquad {}^o\\!e^a_2\n= \\tilde{\\alpha} z \\left(\\frac{\\partial}{\\partial\nx}\\right)^a+\\left(\\frac{\\partial}{\\partial y} \\right)^a, \\qquad {}^o\\!e^a_3\n= \\left(\\frac{\\partial}{\\partial z}\\right)^a, \\end{equation}\nand the dual co-triad by\n\\begin{equation} {}^o\\!\\omega_a^1=({\\rm d} x)_a-\\tilde{\\alpha} z({\\rm d} y)_a, \\qquad{}^o\\!\\omega_a^2=({\\rm d} y)_a,\n\\qquad{}^o\\!\\omega_a^3=(dz)_a. \\end{equation}\nThey determine a fiducial 3-metric ${}^o\\!q_{ab}:= q_{ij}{}^o\\!\\omega_a^i{}^o\\!\\omega_b^j$\nwith Bianchi II symmetries:\n\\begin{equation} {}^o\\!q_{ab} {\\rm d} x^a {\\rm d} x^b =  ({\\rm d} x-\\tilde{\\alpha} z\\:{\\rm d} y)^2\\,+\\,{\\rm d}\ny^2\\,+\\, {\\rm d} z^2. \\end{equation}\n\nIn the diagonal models, the physical triads $e^a_i$ are related to\nthe fiducial ones by%\n\\footnote{There is no sum if repeated indices are both covariant or\ncontravariant.  As usual, the Einstein summation convention holds if a\ncovariant index is contracted with a contravariant index.}\n\\begin{equation} \\label{edef} \\omega_a^i = a^i(\\tau){}^o\\!\\omega_a^i, \\qquad \\mathrm{and}\n\\qquad a_i(\\tau) e^a_i = {}^o\\!e^a_i\n \\end{equation}\nwhere the $a_i$ are the three directional scale factors. Since the\nphysical spatial metric is given by  $q_{ab} =\n\\omega_a^i\\omega^{}_{bi}$, the space-time metric can be expressed as\n\\begin{equation} \\label{metric} {\\rm d} s^2= -N {\\rm d}\\tau^2 + a_1(\\tau)^2\\:({\\rm d} x-\\tilde{\\alpha}\nz\\:{\\rm d} y)^2+a_2(\\tau)^2\\:{\\rm d} y^2+a_3(\\tau)^2\\:{\\rm d} z^2 \\end{equation}\nwhere $N$ is the lapse function adapted to the time coordinate\n$\\tau$.\n\nFor later use, let us calculate the spin connection (\\ref{sc})\ndetermined by triads $e^a_i$. From the definition of $\\Gamma_a^i$ it\nfollows that\n\\begin{equation} \\Gamma_a^i =\n-\\epsilon^{ijk}\\,e^b_j\\,\\left(\\partial_{[a}\\omega_{b]k}+\\frac{1}{2}e^c_k\n\\omega^l_a\\partial_{[c}\\omega_{b]l}\\right)\\, . \\end{equation}\nUsing (\\ref{ve1}), the components of $\\Gamma_a^i$ in the internal\nbasis ${}^o b^i_1, {}^o b^i_2, {}^o b^i_3$ can be expressed as\n\\begin{equation} \\Gamma_a^1 = \\frac{\\tilde{\\alpha}\\varepsilon a_1^2}{2a_2a_3}\\:{}^o\\!\\omega_a^1; \\qquad \\Gamma_a^2\n=-\\frac{\\tilde{\\alpha}\\varepsilon a_1}{2a_3}\\:{}^o\\!\\omega_a^2; \\qquad \\Gamma_a^3 = -\\frac{\\tilde{\\alpha}\\varepsilon\na_1}{2a_2}\\:{}^o\\!\\omega_a^3. \\end{equation}\n\nBefore studying the dynamics of the model, let us examine the action\nof internal parity transformation $\\Pi_k$ which flips the $k$th\ntriad vector and leaves the orthogonal vectors alone. (For details\nsee Appendix and \\cite{aa-dis}). Under the parity transformation\n$\\Pi_1$, for example, we have: $e^a_1\\,\\to \\, -e^a_1,\\, e^a_2\\, \\to\ne^a_2,\\, e^a_ 3\\, \\to \\, e^a_3$ and $a_1\\to -a_1,\\, a_2\\to a_2, a_3\n\\to a_3$ whence $\\Gamma_a^1 \\to -\\Gamma^a_1,\\, \\Gamma_a^2 \\to\n\\Gamma^a_2,\\, \\Gamma_a^3 \\to \\Gamma^a_3$. Thus, both $e^a_i$ and\n$\\Gamma_a^i$ are \\emph{proper} internal vectors. $\\varepsilon$ on the other\nhand is a pseudo internal scalar, $\\varepsilon \\to -\\varepsilon$ under every\n$\\Pi_k$. Note that the fiducial quantities carrying a label $o$ do\nnot change under this transformation; it affects only the physical\nquantities.\n\n\\subsection{The Bianchi II Phase space}\n\\label{s2.2}\n\nAs is usual in LQC, we will now use the fiducial triads and\nco-triads to introduce a convenient parametrization of the phase\nspace variables, $E^a_i, A_a^i$. Because we have restricted\nourselves to the diagonal model and these fields are symmetric under\nthe Bianchi II group, from each equivalence class of gauge related\nphase space variables we can choose a pair of the form\n\\begin{equation} \\label{var} E^a_i = \\tilde{p}_i\\sqrt{|{}^o\\!q|}\\,{}^o\\!e^a_i \\qquad \\mathrm{and}\n\\qquad A_a^i = \\tilde{c}^i \\,{}^o\\!\\omega_a^i, \\end{equation}\nwhere, as spelled out in footnote 3, there is no sum over $i$. Thus,\na point in the phase space is now coordinatized by six real numbers\n$\\tilde{p}_i,\\tilde{c}^i$. One would now like to use the symplectic structure in\nfull general relativity to induce a symplectic structure on our\nsix-dimensional phase space. However, because of spatial homogeneity\nand the ${\\mathbb{R}}^3$ spatial topology, the integrals defining the\nsymplectic structure, the Hamiltonian (and the action) all diverge.\nTherefore we have to introduce a fiducial cell $\\mathcal{V}$ and\nrestrict integrals to it \\cite{as,abl}. We will take the fiducial\ncell to be rectangular with edges along the coordinate axes and\nlengths of $L_1, L_2$ and $L_3$ with respect to the \\emph{fiducial}\nmetric ${}^o\\!q_{ab}$. It then follows that the volume of the fiducial\ncell with respect to ${}^o\\!q_{ab}$ is $V_o=L_1L_2L_3$. Then the non-zero\nPoisson brackets are given by:\n\\begin{equation} \\label{pb1} \\{\\tilde{c}^i,\\, \\tilde{p}_j\\} \\, = \\, \\frac{8\\pi G \\gamma}{V_o}\\,\n\\delta^i_j \\end{equation}\nwhere $\\gamma$ is the Barbero-Immirzi parameter. As in the Bianchi I\ncase, we have a 1-parameter ambiguity in the symplectic structure\nbecause of the explicit dependence on $V_o$ and we have to make sure\nthat the final physical results are either independent of $V_o$ or\nremain well-defined as we remove the `regulator' and take the limit\n$V_o \\to \\infty$.\n\nIt is convenient to rescale variables to absorb this dependence in\nthe phase space coordinates (as was done in the treatment of Bianchi\nI model in \\cite{awe2}). Let us set\n\\begin{equation} p_1= L_2L_3\\tilde{p}_1, \\qquad p_2=L_3L_1\\tilde{p}_2, \\qquad p_3= L_1L_2\\tilde{p}_3,\n\\end{equation}\n\\begin{equation} \\label{tilde} c_1=L_1\\tilde{c}_1, \\qquad c_2=L_2\\tilde{c}_2, \\qquad\nc_3=L_3\\tilde{c}_3 \\qquad \\mathrm{and} \\qquad \\alpha =\n\\frac{L_2L_3}{L_1}\\tilde{\\alpha}\\, , \\end{equation}\nwhere the last rescaling has been introduced to absorb factors of\n$L_i$ which would otherwise unnecessarily obscure the expression of\nthe Hamiltonian constraint. The Poisson brackets between these new\nphase space coordinates is given by%\n\\begin{equation} \\label{pb2}\\{c^i,\\, p_j\\} \\, = \\, 8\\pi G \\gamma \\,\\delta^i_j \\,\n. \\end{equation}\nThese variables have direct physical interpretation. For example,\n$p_1$ is the (oriented) area of the 2-3 face of the elementary cell\nwith respect to the \\emph{physical} metric $q_{ab}$ and $h^{(1)} =\n\\exp c_1\\tau_1$ is the holonomy of the physical connection $A_a^i$\nalong the first edge of the elementary cell.\n\n\nOur choice (\\ref{var}) of physical triads and connections has fixed\nthe internal gauge as well as the diffeomorphism freedom.\nFurthermore, it is easy to explicitly verify that, thanks to\n(\\ref{var}), the Gauss and the diffeomorphism constraints are\nautomatically satisfied. Thus, as in \\cite{awe2}, we are left just\nwith the Hamiltonian constraint\n\\begin{equation} \\label{Hgen} \\mathcal{C}_H = \\int_\\mathcal{V}\n\\Big[\\frac{NE^a_iE^b_j}{16\\pi G\\sqrt{|q|}}\n\\big(\\epsilon^{ij}{}_kF_{ab}{}^k-2(1+\\gamma^2)K_{[a}^iK_{b]}^j \\Big)\n+ N \\mathcal{H}_{{\\rm matt}}\\big]\\, {\\rm d}^3x\\, , \\end{equation}\nwhere\n\\begin{equation} F_{ab}{}^k=2\\partial_{[a}A_{b]}^k+\\epsilon_{ij}{}^kA_a^iA_b^j \\end{equation}\nis the curvature of $A_a^i$ and $\\mathcal{H}_{\\rm matt}$ is the matter\nHamiltonian density. As in \\cite{awe2}, our matter field will\nconsist only of a massless scalar field $T$ which will later serve\nas a relational time variable a la Liebniz. (Additional matter\nfields can be incorporated in a straightforward manner, modulo\npossible intricacies of essential self-adjointness.) Thus,\n\\begin{equation} \\mathcal{H}_{{\\rm matt}} = \\frac{1}{2}\\frac{p_T^2}{\\sqrt{|q|}}. \\end{equation}\n\nSince we want to use the massless scalar field as relational time,\nit is convenient to use a harmonic-time gauge, i.e., assume that the\ntime coordinate $\\tau$ in (\\ref{metric}) satisfies $\\Box \\tau=0$.\nThe corresponding lapse function is $N=\\sqrt{|p_1p_2p_3|}$. With\nthis choice, the Hamiltonian constraint simplifies considerably.\nNote first that the basic canonical variables can be expanded as\n\\begin{equation} E^a_i = \\frac{p_i}{V_o}L_i\\sqrt{|{}^o\\!q|}{}^o\\!e^a_i \\qquad {\\rm and} \\qquad\nA_a^i = \\frac{c^i}{L^i}{}^o\\!\\omega_a^i, \\end{equation}\nand the extrinsic curvature is given by\n $$\\qquad  K_a^i = \\gamma^{-1} (A_a^i-\\Gamma_a^i).$$\nNext, using $p_1 = ({\\rm sgn}a_1)\\, |a_2a_3|\\,L_2L_3$ etc, the\ncomponents of the spin connection become:\n\\begin{equation} \\Gamma_a^1 = \\frac{\\alpha\\varepsilon p_2p_3}{2p_1^2}\\frac{{}^o\\!\\omega_a^1}{L_1}, \\qquad\n\\Gamma_a^2= -\\frac{\\alpha\\varepsilon p_3}{2p_1}\\frac{{}^o\\!\\omega_a^2}{L_2}, \\qquad\n\\Gamma_a^3=-\\frac{\\alpha\\varepsilon p_2}{2p_1} \\frac{{}^o\\!\\omega_a^3}{L_3} \\, .\\end{equation}\nCollecting terms, the Hamiltonian constraint (\\ref{Hgen}) becomes\n\\begin{align} \\label{Hcl} \\mathcal{C}_H&=-\\frac{1}{8\\pi G\\gamma^2}\n\\Big[p_1p_2c_1c_2+p_2p_3 c_2c_3+p_3p_1c_3c_1+\\alpha\\varepsilon p_2p_3c_1\n\\nonumber \\\\\n&\\qquad\\qquad-(1+ \\gamma^2)\\,\\big(\\frac{\\alpha p_2p_3}{2p_1}\\big)^2\\Big]\n + \\frac{1}{2}p_T^2 \\\\\n& = \\mathcal{C}_H^{\\rm (BI)} - \\frac{1}{8\\pi G\\gamma^2}\\Big[\\alpha\\varepsilon\np_2p_3c_1-(1+ \\gamma^2)\\,\\big(\\frac{\\alpha p_2p_3}{2p_1}\\big)^2\\Big],\n\\end{align}\nwhere $\\mathcal{C}_H^{\\rm (BI)}$ is the Hamiltonian constraint\n(including the matter term) for Bianchi I space-times which has\nalready been studied in \\cite{awe2}.  Note that this constraint is\nrecovered in the limit $\\alpha\\to 0$, as it must be.\n\n\nKnowing the form of the Hamiltonian constraint, it is now possible to derive\nthe time evolution of any classical observable $\\mathcal{O}$ by taking its\nPoisson bracket with $\\mathcal{C}_H$:\n\\begin{equation} \\dot{\\mathcal{O}} = \\{\\mathcal{O},\\mathcal{C}_H\\}\\, , \\end{equation}\nwhere the `dot' stands for derivative with respect to harmonic time\n$\\tau$. This gives\n\\begin{equation} \\label{ceom1} \\dot{p_1}=\\gamma^{-1}(p_1p_2c_2+p_1p_3c_3+\\alpha\\varepsilon p_2p_3), \\end{equation}\n\\begin{equation} \\dot{p_2}=\\gamma^{-1}(p_2p_1c_1+p_2p_3c_3), \\end{equation}\n\\begin{equation} \\dot{p_3}=\\gamma^{-1}(p_3p_1c_1+p_3p_2c_2), \\end{equation}\n\\begin{equation}\n\\dot{c_1}=-\\frac{1}{\\gamma}\\Big(p_2c_1c_2+p_3c_1c_3+\\frac{1}{2p_1}(1+\\gamma^2)\n\\big(\\frac{\\alpha p_2p_3}{p_1}\\big)^2\\Big), \\end{equation}\n\\begin{equation} \\dot{c_2}=-\\frac{1}{\\gamma}\\Big(p_1c_2c_1+p_3c_2c_3+\\alpha\\varepsilon\np_3c_1-\\frac{1}{2p_2} (1+\\gamma^2)\\big(\\frac{\\alpha\np_2p_3}{p_1}\\big)^2\\Big), \\end{equation}\n\\begin{equation} \\label{ceom2}\n\\dot{c_3}=-\\frac{1}{\\gamma}\\Big(p_1c_3c_1+p_2c_3c_2+\\alpha\\varepsilon p_2\nc_1-\\frac{1}{2p_3}(1+\\gamma^2)\\big(\\frac{\\alpha p_2p_3}{p_1}\\big)^2\\Big).\n\\end{equation}\nAny initial data satisfying the Hamiltonian constraint can be\nevolved by using the six equations above. It is straightforward to\nextend these results if there are additional matter fields.\n\nFinally, let us consider the parity transformation $\\Pi_k$ which\nflips the $k$th \\emph{physical} triad vector $e^a_k$. (As noted\nbefore, this transformation does not act on any of the fiducial\nquantities which carry a label $o$.) Under this map, we have:\n$q_{ab} \\to q_{ab}, \\, \\epsilon_{abc} \\to \\epsilon_{abc}\\,$ but\n$\\epsilon_{ijk} \\to -\\epsilon_{ijk}, \\, \\varepsilon \\to -\\varepsilon$. The canonical\nvariables $c^i, p_i$ transform as proper internal vectors and\nco-vectors: For example\n\\begin{equation} \\Pi_1(c_1,c_2,c_3) \\rightarrow (-c_1, c_2, c_3) \\qquad {\\rm and}\n\\qquad \\Pi_1(p_1,p_2,p_3) \\rightarrow (-p_1, p_2, p_3)\\, . \\end{equation}\nConsequently, both the symplectic structure and the Hamiltonian\nconstraint are left invariant under any of the parity maps $\\Pi_k$.\n\nThis Hamiltonian description will serve as the point of departure\nfor loop quantization in the next section.\n\n\n\\section{Quantum Theory}\n\\label{s3}\n\nThis section is divided into three parts.  In the first, we discuss\nthe kinematics of the model, in the second we define an operator\ncorresponding to the connection $A_a^i$ using holonomies and in the\nthird we introduce the Hamiltonian constraint operator and describe\nits action on states.\n\n\n\\subsection{LQC Kinematics}\n\nThe kinematics for the LQC of Bianchi II models is almost identical\nto that for Bianchi I models. Therefore, in the sub-section we\nclosely follow \\cite{awe2}.\n\nLet us begin by specifying the elementary functions on the\nclassical phase space which will have unambiguous analogs in the\nquantum theory. As in the Bianchi I model, the elementary\nvariables are the momenta $p_i$ and holonomies of the\ngravitational connection $A_a^i$ along the integral curves of the\nright invariant vector fields ${}^o\\!e^a_i$. Let $\\tau_i$ be a basis of\nthe Lie algebra of SU(2), satisfying $\\tau_i \\tau_j =\n\\frac{1}{2}\\epsilon_{ij}{}^k \\tau_k- \\frac{1}{4} \\delta_{ij}\\mathbb{I}$\nwhere $\\mathbb{I}$ is the unit $2\\times2$ matrix. Consider an edge\nof length $\\ell L_k$ with respect to the fiducial metric\n${}^o\\!q_{ab}$, parallel to ${}^o\\!e^a_k$. The holonomy $h_k^{(\\ell)}$ along\nit is given by\n\\begin{equation} \\label{hol} h_k^{(\\ell)}(c_1,c_2,c_3) = \\exp\\left(\\ell\nc_k\\tau_k\\right) = \\cos\\frac{\\ell c_k}{2} \\mathbb{I} + 2\\sin\\frac{\\ell\nc_k}{2}\\tau_k. \\end{equation}\n(Note that $\\ell$ depends of the fiducial cell but not on the\nfiducial metric.) This family of holonomies is completely\ndetermined by the almost periodic functions $\\exp(i\\ell c_k)$ of\nthe connection. These almost periodic functions will be our\nelementary configuration variables which will be promoted\nunambiguously to operators in the quantum theory.\n\nIt is simplest to use the $p$-representation to  specify the\ngravitational sector $\\mathcal{H}_{\\rm kin}^{\\rm grav}$ of the kinematic Hilbert space. The\northonormal basis states $|p_1,p_2,p_3\\rangle$ are eigenstates of\nquantum geometry. For example, in the state $|p_1,p_2,p_3\\rangle$\nthe face $S_{23}$  of the fiducial cell $\\mathcal{V}$ (given by $x$\n={\\rm const}) has area $|p_1|$.\nThe basis is orthonormal in the sense\n\\begin{equation} \\langle p_1,p_2,p_3|p_1',p_2',p_3'\\rangle = \\delta_{p_1^{}p_1'}\n\\delta_{p_2^{}p_2'}\\delta_{p_3^{}p_3'}\\, , \\end{equation}\nwhere the right side features Kronecker symbols rather than the\nDirac delta distributions. Hence kinematical states can consist only\nof \\emph{countable} linear combinations\n\\begin{equation} |\\Psi\\rangle \\,=\\,\n\\sum_{p_1,p_2,p_3}\\Psi(p_1,p_2,p_3)|p_1,p_2,p_3\\rangle\\ \\end{equation}\nof these basis states for which the norm\n\\begin{equation} \\label{norm} ||\\Psi ||^2\\, =\\, \\sum_{p_1,p_2,p_3}\\,\n|\\Psi(p_1,p_2,p_3)|^2 \\end{equation}\nis finite. Because the right side features a sum over a countable\nnumber of points on ${\\mathbb{R}}^3$, rather than a 3-dimensional integral,\nLQC kinematics are inequivalent to those of the Schr\\\"odinger\napproach used in Wheeler-DeWitt quantum cosmology.\n\nNext, recall that on the classical phase space the three reflections\n$\\Pi_i:\\,\\,e^a_i\\,\\to\\, -e^a_i$ are large gauge transformations\nunder which physics does not change (since both the metric and the\nextrinsic curvature are left invariant). These large gauge\ntransformations have a natural induced action, denoted by\n$\\hat\\Pi_i$, on the space of wave functions $\\Psi(p_1,p_2,p_3)$. For\nexample,\n\\begin{equation} \\hat\\Pi_1\\Psi(p_1,p_2,p_3)=\\Psi(-p_1,p_2,p_3). \\end{equation}\nSince $\\hat\\Pi_i^2$ is the identity, for each $i$, the group of\nthese large gauge transformations is simply $\\mathbb{Z}_2$. As in Yang-Mills\ntheory, physical states belong to its irreducible representation.\nFor definiteness, as in the isotropic and Bianchi I models, we will\nwork with the symmetric representation. It then follows that\n$\\mathcal{H}_{\\mathrm{kin}}^{\\mathrm{grav}}$ is spanned by wave\nfunctions $\\Psi(p_1,p_2,p_3)$ which satisfy\n\\begin{equation} \\label{parity} \\Psi(p_1,p_2,p_3)=\\Psi(|p_1|,|p_2|,|p_3|) \\end{equation}\nand have a finite norm (\\ref{norm}).\n\nThe action of the elementary operators on\n$\\mathcal{H}_{\\mathrm{kin}}^{\\mathrm{grav}}$ is as follows: the\nmomenta act by multiplication whereas the almost periodic\nfunctions in $c_i$ shift the $i$th argument. For example,\n\\begin{equation} [\\hat p_1 \\Psi](p_1,p_2,p_3) = p_1\\, \\Psi(p_1,p_2,p_3) \\,\\quad\n\\mathrm{and} \\,\\quad \\Big[\\widehat{\\exp(i\\ell c_1)}\\Psi\\Big](p_1,\np_2, p_3) = \\Psi(p_1-8\\pi\\gamma G\\hbar \\ell, p_2, p_3)\\, . \\end{equation}\nThe expressions for $\\hat p_2, \\widehat{\\exp(i\\ell c_2)}, \\hat\np_3$ and $\\widehat{\\exp(i\\ell c_3)}$ are analogous.  Finally, we\nneed to define the operator $\\hat{\\varepsilon}$ since $\\varepsilon$ features in\nthe expression of the Hamiltonian constraint. In the classical\ntheory, $\\varepsilon$ is unambiguously defined on non-degenerate triads,\ni.e., when $p_1p_2p_3 \\not= 0$. In quantum theory, wave functions\ncan have support also on degenerate configurations. We will extend\nthe definition to degenerate triads using the basis\n$|p_1,p_2,p_3\\rangle$:\n\\begin{equation} \\label{ve2} \\hat{\\varepsilon}\\,|p_1,p_2,p_3\\rangle := \\left\\{\n\\rlap{\\raise2ex\\hbox{\\,\\,$\\quad|p_1,p_2,p_3 \\rangle$ if $p_1p_2p_3\n\\ge 0$,}}{\\lower2ex\\hbox{\\,\\,$ -\\,|p_1,p_2,p_3 \\rangle$ if\n$p_1p_2p_3<0$.}} \\right. \\end{equation}\nFinally, the full kinematical Hilbert space\n$\\mathcal{H}_{\\mathrm{kin}}$ will be the tensor product\n$\\mathcal{H}_{\\mathrm{kin}}=\\mathcal{H}_{\\mathrm{kin}}^\n{\\mathrm{grav}}\\otimes\\mathcal{H}_{\\mathrm{kin}}^{\\mathrm{matt}}$,\nwhere $\\mathcal{H}_{\\mathrm{kin}}^{\\mathrm{matt}}=L^2({\\mathbb{R}},dT)$ is\nthe matter kinematical Hilbert space for the homogeneous scalar\nfield.  On $\\mathcal{H}_ {\\mathrm{kin}}^{\\mathrm{matt}}$, $\\hat T$\nwill act by multiplication and $\\hat p_T:=-i\\hbar \\mathrm{d}_T$\nwill act by differentiation. As in isotropic and Bianchi I models,\nour final results would remain unaffected if we use a ``polymer\nrepresentation'' also for the scalar field.\n\n\n\\subsection{The connection operator $\\hat{A}_a^i$}\n\\label{s3.2}\n\nTo define the quantum Hamiltonian constraint, we cannot directly use\nthe symmetry reduced classical constraint (\\ref{Hcl}) because it\ncontains connection components $c_k$ themselves and in LQC only\nalmost periodic functions of $c_k$ have unambiguous operator\nanalogs. Indeed, in all LQC models considered so far\n\\cite{abl,aps3,warsaw,apsv,kv1,ls,bp,awe2}, we were led to return to\nthe expression (\\ref{Hgen}) in the full theory and mimic the\nprocedure used in LQG \\cite{tt}. More precisely, the key strategy\nwas to follow full LQG (and spin foams) and define a ``field\nstrength operator'' using holonomies around suitable closed loops.\nIn the Bianchi I model, these closed loops were formed by following\nintegral curves of right invariant vector fields (which are also\nleft invariant). As mentioned in section \\ref{s2}, in the Bianchi II\nmodel the right invariant vector fields define the fiducial triads\n${}^o\\!e^a_i$, the left invariant vector fields, the Killing fields\n${}^o\\xi^i$. Neither constitutes a commuting set, whence their integral\ncurves cannot be used to form closed loops. However, as in the k=1\ncase \\cite{warsaw,apsv}, one can hope to exploit the fact that the\nright invariant vector fields do commute with the left invariant\nones and construct the closed loops by alternately following right\nand left invariant vector fields. But, as mentioned in section\n\\ref{s1}, a new problem arises: unlike in the k=1 (or Bianchi I)\nmodel the resulting holonomies are no longer almost periodic\nfunctions of $c_k$, whence the Hilbert space $\\mathcal{H}_{\\rm kin}^{\\rm\ngrav}$ does not support these holonomy operators. For completeness\nwe will first show this fact explicitly and then introduce a new\navenue to bypass this difficulty.\n\nThe problematic curvature component turns out to be $F_{yz}{}^1$. To\nconstruct the corresponding operator, following the strategy used in\nthe k=1 case \\cite{warsaw,apsv}, we will construct a closed loop\n$\\Box_{yz}$ as follows. In the coordinates $(x,y,z)$,\\,\\, i) Move\nfrom $(0,0,0)$ to $(0,\\bar\\mu_2L_2,0)$ following $\\xi^a_2$;\\,\\, ii)\nthen move from $(0,\\bar\\mu_2L_2,0)$ to\n$(0,\\bar\\mu_2L_2,\\bar\\mu_3L_3)$ following ${}^o\\!e^a_3$;\\,\\, iii) then\nmove from $(0,\\bar\\mu_2L_2,\\bar\\mu_3L_3)$ to $(0,0,\\bar\\mu_3L_3)$\nfollowing $-\\xi^a_2$;\\,\\, and, finally, iv) move from\n$(0,0,\\bar\\mu_3L_3)$ to $(0,0,0)$ following $-{}^o\\!e^a_3$. The\nparameters $\\bar\\mu_i$ which determine the `lengths' of these edges\ncan be fixed by the semi-heuristic correspondence between LQC and\nLQG exactly as in the Bianchi I model \\cite{awe2} because the\ngeometric considerations used in that analysis continue to hold\nwithout any modification in all Bianchi models with $\\mathbb{R}^3$ spatial\ntopology:\n\\begin{equation} \\label{mubar} \\bar\\mu_1 =\n\\sqrt\\frac{|p_1|\\Delta\\,\\ell_{\\mathrm{Pl}}^2}{|p_2p_3|}, \\qquad  \\bar\\mu_2 =\n\\sqrt\\frac{|p_2|\\Delta\\,\\ell_{\\mathrm{Pl}}^2}{|p_1p_3|}, \\qquad \\bar\\mu_3 =\n\\sqrt\\frac{|p_3| \\Delta\\,\\ell_{\\mathrm{Pl}}^2}{|p_1p_2|} \\end{equation}\nwhere $\\Delta\\,\\ell_{\\mathrm{Pl}}^2 = 4\\sqrt{3}\\pi\\gamma\\,\\ell_{\\mathrm{Pl}}^2$ is the `area gap'.\nThe holonomy around this closed loop $\\Box_{yz}$ is given by\n\\begin{equation} {h}_{\\Box_{yz}} = \\frac{2}{c\\,\\,\\bar\\mu_2\\bar\\mu_3L_2L_3}\\cos\\left(\n\\frac{\\bar\\mu_2c_2}{2}\\right)\\sin\\left(\\frac{\\bar\\mu_2 c}{2}\\right)\n\\Big(c_2\\sin(\\bar\\mu_3c_3)+\n\\alpha\\bar\\mu_3c_1\\cos(\\bar\\mu_3c_3)\\Big) \\end{equation}\nwhere\n\\begin{equation} \\label{c12} c = \\sqrt{\\alpha^2\\bar\\mu_3^2c_1^2+c_2^2}. \\end{equation}\nIf we were to shrink the loop so that the area it encloses goes to\nzero, we do indeed recover the classical expression of $F_{yz}{}^1$.\nHowever, because of presence of the term $c$, if $\\alpha\\not=0$ the\nright side fails to be almost periodic in $c_1$ and $c_2$. Hence\nthis holonomy operator fails to exist on $\\mathcal{H}_{\\rm kin}$. It is clear\nfrom the expression (\\ref{c12}) of $c$ that the problem is\nindependent of the specific way $\\bar\\mu_i$ are fixed.\n\nWe will bypass this difficulty by mimicking another strategy used in\nfull LQG \\cite{tt}: We will use holonomies along segments parallel\nto ${}^o\\!e^a_i$ to define an operator corresponding to the connection\nitself. This is a natural strategy because holonomies along these\nsegments suffice to separate the Bianchi II connections (\\ref{var}).\nLet us set $A_a := A_a^k\\tau_k$. Then we have the identity:\n\\begin{equation} \\label{classA} A_a =  \\lim_{\\ell_k \\to 0}\\, \\sum_k\n\\,\\frac{1}{2\\ell_kL_k}\\,\\, \\Big(h_k^{(\\ell_k)} -\n(h_k^{(\\ell_k)})^{-1}\\Big) \\end{equation}\nwhere $h_k^{(\\ell_k)}$ is given by (\\ref{hol}). In the expressions\nof physically interesting operators such as the Hamiltonian\nconstraint of full LQG, one often replaces $A_a$ with the (analog of\nthe) right side of (\\ref{classA}). But because of the specific forms\nof these operators, the limit trivializes on diffeomorphism\ninvariant states of LQG. In LQC, we have gauge fixed the system and\ntherefore cannot appeal to diffeomorphism invariance. Indeed, while\nthe holonomies are well-defined for each non-zero $\\ell_k$, the\nlimit fails to exist on $\\mathcal{H}_{\\rm kin}^{\\rm grav}$. A natural\nstrategy is to shrink $\\ell_k$ to a judiciously chosen non-zero\nvalue. But what would this value be? In the case of plaquettes, we\ncould use the interplay between LQG and LQC directly because the\nargument $p_i$ of LQC quantum states refers to \\emph{quantum} areas\nof faces of the elementary cell $\\mathcal{V}$ \\cite{awe2}. For edges\nwe do not have such direct guidance. There is, nonetheless a natural\nprinciple one can adopt: Normalize $\\ell_k$ such that the numerical\ncoefficient in front of the curvature operator constructed from the\nresulting connection agrees with that in the expression of the\ncurvature operator constructed from holonomies around closed loops,\nin all cases where the second construction is available. We will use\nthis strategy. Let us apply it to the Bianchi I model where\n$F_{ab}{}^k = \\epsilon_{ij}{}^k\\, A_a^i A_b^j$. Using holonomies\naround closed loops one obtains the field strength operator\n\\begin{equation} \\hat{F}_{ab}{}^k = \\epsilon_{ij}{}^k\\,\n\\big(\\frac{\\sin\\bar{\\mu}c}{\\bar{\\mu}L}\\, {}^o\\!\\omega_a\\big)^i\\,\n\\big(\\frac{\\sin\\bar{\\mu}c}{\\bar{\\mu}L}\\, {}^o\\!\\omega_b\\big)^j \\end{equation}\nwhere\n\\begin{equation} \\big(\\frac{\\sin\\bar{\\mu}c}{\\bar{\\mu}L}\\, {}^o\\!\\omega_a\\big)^i =\n\\big(\\frac{\\sin\\bar{\\mu_i}c_i}{\\bar{\\mu_i}L_i}\\, {}^o\\!\\omega_a^i\\big) \\quad\\quad\n\\hbox{\\rm (no sum over i)} \\nonumber \\end{equation}\n(see Eqs (3.12) and (3.13) in \\cite{awe2}). Therefore, our strategy\nyields $\\ell_k = 2\\bar\\mu_k$, that is,\n\\begin{equation} \\label{Aop} \\hat{A}_a^k =\n\\frac{\\widehat{\\sin(\\bar\\mu^kc^k)}}{\\bar\\mu^kL_k}\\,\\,{}^o\\!\\omega_a^k, \\end{equation}\nwhere there is no sum over $k$. Note that the principle stated above\nleads us unambiguously to the factor $2$ in $\\ell_k = 2\\bar\\mu_k$;\nwithout recourse to a systematic strategy, one may have naively set\n$\\ell_k =\\bar\\mu_k$.\n\nIf we compare the expression (\\ref{Aop}) of the connection operator\nwith the expression (\\ref{var}) of the classical connection, we have\neffectively defined an operator $\\hat{c}$ via\n\\begin{equation} \\hat{c}_k = \\frac{\\widehat{\\sin(\\bar\\mu^kc^k)}}{\\bar\\mu^k} \\end{equation}\nwhere there is again no sum over $k$. In the literature such a\nquantization of $c$ is often called ``polymerization.'' Our approach\nis an improvement over such strategies in two respects. First, we\ndid not just make the substitution $c \\rightarrow \\sin \\ell c\/\\ell$\nby hand; a priori one could have used another substitution such as\n$c \\rightarrow \\tan \\ell c\/\\ell$. Rather, as in full LQG, we used\nthe strategy of expressing the connection in term of holonomies,\n`the elementary variables'. But this still leaves open the question\nof what $\\ell$ one should use. We determined this by requiring that\nthe overall normalization of $\\hat{F}_{ab}{}^k$ constructed from\n$\\hat{A}_a^i = c^i (L^i)^{-1}\\,{}^o\\!\\omega_a^i$ should agree with that of\n$\\hat{F}_{ab}{}^k$ constructed from holonomies around appropriate\nclosed loops, when the second construction is possible. Therefore,\nour construction is a bona-fide generalization of the previous\nconstructions used successfully in LQC.\n\nThis strategy has some applications beyond the Bianchi II model\nstudied in this paper. First, the k=$-1$ isotropic case has been\nstudied in detail in \\cite{kv1,ls}. The analysis uses the $\\bar\\mu$\nscheme, carries out a numerical simulation using exact LQC equations\nand shows that the effective equations of the ``embedding approach\"\n\\cite{jw,vt} (discussed in section \\ref{s4}) provide an excellent\napproximation to the quantum evolution. While this is an essentially\nexhaustive treatment, as \\cite{kv1,ls} itself points out, the\ntreatment has a conceptual limitation in that it builds holonomies\naround the closed loops using the extrinsic curvature $K_a^i$\n---rather than $A_a^i$--- as a ``connection''. This limitation can\nbe overcome in a straightforward fashion using our current strategy.\nMore importantly, this strategy is applicable to all class A Bianchi\nmodels, including type IX. Thus, it opens the door to the LQC\ntreatment of all these models in one go.\n\n\n\\subsection{The quantum Hamiltonian constraint}\n\\label{s3.3}\n\nWith the connection operator at hand, one can construct the\nHamiltonian constraint operator starting either from the general LQG\nexpression (\\ref{Hgen}) or the symmetry reduced expression\n(\\ref{Hcl}). We will begin by a small change in the representation\nof kinematical states which will facilitate this task.\n\n\\subsubsection{A more convenient representation}\n\\label{s3.3.1}\n\nIgnoring factor ordering ambiguities for the moment, the\nconstraint operator $\\hat{\\mathcal{C}}_H$ is given by\n\\begin{align} \\label{qHam1} \\hat{\\mathcal{C}}_H = -\\frac{1}{8\\pi\nG\\gamma^2\\Delta\\ell_{\\mathrm{Pl}}^2}&\\Big[p_1 p_2|p_3|\\sin\\bar\\mu_1c_1\n\\sin\\bar\\mu_2c_2+|p_1|p_2p_3\\sin\\bar\\mu_2c_2\\sin\\bar\\mu_3c_3 \\nonumber \\\\\n&+p_1|p_2|p_3\\sin\\bar\\mu_3c_3\\sin\\bar\\mu_1c_1\\Big]-\n\\frac{1}{8\\pi G\\gamma^2}\\Big[\\alpha\\hat{\\varepsilon}p_2p_3\\sqrt\\frac{|p_2p_3|}{|p_1|\\Delta\n\\ell_{\\mathrm{Pl}}^2}\\sin\\bar\\mu_1c_1\\nonumber \\\\ & -(1+\\gamma^2)\\left(\\frac{\\alpha\np_2p_3}{2p_1}\\right)^2\\Big]+\\frac{1}{2}\\hat{p}_T^2 \\end{align}\nwhere for simplicity of notation here and in what follows we have\ndropped the hats on the $p_i$ and $\\sin\\bar\\mu_ic_i$ operators.\nRecall that, classically, the Bianchi II symmetry group reduces to\nthe Bianchi I symmetry group if we set $\\alpha=0$. If one sets\n$\\alpha=0$ in (\\ref{qHam1}), the last two terms disappear and the\noperator $\\hat{\\mathcal{C}}_H$ reduces to that of the Bianchi I\nmodel \\cite{awe2} showing explicitly that our construction is a\nnatural generalization of the strategy used there.\n\nTo obtain the action of operators corresponding to terms of the form\n$\\sin\\bar\\mu_ic_i$ we use the same strategy as in \\cite{awe2}. As\nshown there, it is simplest to introduce dimensionless variables\n\\begin{equation}\n\\lambda_i=\\frac{\\mathrm{sgn}(p_i)\\sqrt{|p_i|}}{(4\\pi\\gamma\\sqrt\\Delta\\ell_{\\mathrm{Pl}}^3)^{1\/3}}\\,\n. \\end{equation}\nThen the kets $|\\lambda_1,\\lambda_2,\\lambda_3\\rangle$ constitute an orthonormal\nbasis in which the operators $p_k$ are diagonal\n\\begin{equation} p_k|\\lambda_1,\\lambda_2,\\lambda_3\\rangle\\, =\\,\n[\\mathrm{sgn}(\\lambda_k)(4\\pi\\gamma\\sqrt\\Delta\\ell_{\\mathrm{Pl}}^3)^{2\/3}\n\\lambda_k^2]\\,\\,|\\lambda_1,\\lambda_2,\\lambda_3\\rangle\\, . \\end{equation}\nQuantum states will now be represented by functions\n$\\Psi(\\lambda_1,\\lambda_2,\\lambda_3)$. The operator $e^{i\\bar\\mu_1c_1}$ acts on\nthem as follows\n\\begin{align} \\big[e^{i\\bar\\mu_1c_1}\\,\\Psi\\big] (\\lambda_1,\\lambda_2,\\lambda_3)\n&= \\Psi(\\lambda_1- \\frac{1}{|\\lambda_2\\lambda_3|},\\lambda_2,\\lambda_3) \\nonumber \\\\\n&= \\Psi(\\frac{v-2\\mathrm{sgn}(\\lambda_2\\lambda_3)}{v}\\cdot\\, \\lambda_1,\\lambda_2,\\lambda_3),\n\\end{align}\nwhere we have introduced the variable $v=2\\lambda_1\\lambda_2\\lambda_3$ which is\nproportional to the volume of the fiducial cell:\n\\begin{equation} \\hat{V}\\,\\Psi(\\lambda_1,\\lambda_2,\\lambda_3)\\, =\\,\n[2\\pi\\gamma\\sqrt\\Delta\\,|v|\\,\\ell_{\\mathrm{Pl}}^3]\\, \\Psi(\\lambda_1, \\lambda_2,\\lambda_3). \\end{equation}\n(The $e^{i\\bar\\mu_1c_1}$ operator is well-defined in spite of the\nappearance of $|\\lambda_2\\lambda_3|$ in the denominator; see \\cite{awe2}.)\nThe operators $e^{i\\bar\\mu_2 c_2}$ and $e^{i\\bar\\mu_3c_3}$ have\nanalogous action.\n\nWe are now ready to write the Hamiltonian constraint explicitly in\nthe $\\lambda_i$-representation. As noted above, the three terms in the\nfirst square bracket on the right hand side of Eq. (\\ref{qHam1})\nconstitute the gravitational part of $\\hat{\\mathcal{C}}_H$ for the\nLQC of Bianchi I model%\n\\footnote{There are some minor changes in the action of these three\nterms since $\\gamma$ is no longer treated as a pseudoscalar (see\nAppendix \\ref{a1}), but these do not affect the discussion.}\nand have been discussed in \\cite{awe2}. In the next two\nsub-sections we will now discuss the last two terms, which are\nspecific to the Bianchi II model.\n\n\\subsubsection{The Fourth term in $\\hat{\\mathcal{C}}_H$}\n\\label{s3.3.2}\n\nUsing a symmetric factor ordering, the fourth term becomes\n\\begin{equation} \\label{hc4} \\hat{\\mathcal{C}}_H^{(4)} = -\\frac{\\alpha\np_2p_3\\sqrt{|p_2p_3|}} {16\\pi\nG\\gamma^2\\sqrt\\Delta\\ell_{\\mathrm{Pl}}}\\,\\,\\widehat{|p_1|^{-1\/4}}\\,(\\hat{\\varepsilon}\\,\n\\sin\\bar\\mu_1c_1+\\sin\\bar\\mu_1c_1\\,\\hat{\\varepsilon})\\,\\widehat{|p_1|^{-1\/4}}\n\\, . \\end{equation}\n(Note that $p_2$ and $p_3$ commute with the other terms in\n$\\hat{\\mathcal{C}}_H^{(4)}$). The operator $p_1$ is self-adjoint on\n$\\mathcal{H}_{\\rm kin}^{\\rm grav}$ whence any measurable function of $p_1$ is\nalso a well-defined self-adjoint operator. However, since kets\n$|\\lambda_1=0, \\lambda_2,\\lambda_3\\rangle$ are normalizable in $\\mathcal{H}_{\\rm kin}^{\\rm\ngrav}$, the naive inverse powers of $\\hat{p}_1$ fail to be densely\ndefined and cannot be self-adjoint. To define inverse powers, as is\nusual in LQG, we will use a variation on the Thiemann inverse triad\nidentities \\cite{tt}. Classically, we have the identity\n\\begin{equation} \\label{class} |p_1|^{-1\/4} = -\\frac{i\\,\\mathrm{sgn}(p_1)}{2\\pi G\\gamma}\n\\sqrt\\frac{|p_2p_3|}{\\Delta\\ell_{\\mathrm{Pl}}^2}\\,\\,\ne^{-i\\bar\\mu_1c_1}\\,\\{e^{i\\bar\\mu_1c_1},|p_1|^{1\/4}\\}\\, . \\end{equation}\nwhich holds for any choice of $\\bar\\mu_1$. Since it is most natural\nto use the same $\\bar\\mu_1$ that featured in the definition of the\nconnection operator, we will make this choice. Eq (\\ref{class})\nsuggests a natural quantization strategy for $|p_1|^{-1\/4}$. Using\nit and the parity considerations, we are led to the following factor\nordering:%\n\\footnote{In the classical theory, $(L_2L_3)^{1\/4}\\,|p_1|^{-1\/4}$ is\nindependent of the choice of the elementary cell. As pointed out in\n\\cite{kv1} the inverse triad operators, by contrast, depend on the\nchoice of the cell. However, one can verify that as we remove the\nregulator, i.e., take the limit $\\mathcal{V} \\to \\mathbb{R}^3$, as in the\nclassical theory, the limiting\n$(L_2L_3)^{1\/4}\\,\\widehat{|p_1|^{-1\/4}}$ has a well defined limit.}\n\\begin{equation} \\widehat{|p_1|^{-1\/4}} = - \\frac{i\\,\\mathrm{sgn}(p_1)}{2\\pi\nG\\gamma}\\sqrt\\frac{|p_2p_3|}{\\Delta\\ell_{\\mathrm{Pl}}^2}\\,\\,e^{-i\\bar\\mu_1c_1\/2}\\,\\,\n\\frac{1}{i\\hbar}[e^{i\\bar\\mu_1c_1},|p_1|^{1\/4}]\\,\\,\ne^{-i\\bar\\mu_1c_1\/2}\\, , \\end{equation}\nwhere, as is common in LQC, $\\mathrm{sgn}(p_1)$ is defined as\n\\begin{equation} \\mathrm{sgn}(p_1) = \\left\\{\\rlap{\\rlap{\\raise4ex\\hbox{\\,\\,$+1$ if $p_1>0$,}}\n{\\raise0ex\\hbox{\\,\\,$0$ if $p_1=0$,}}}\n{\\lower4ex\\hbox{\\,\\,$-1$ if $p_1<0$.}} \\right. \\end{equation}\n\nAt first it may seem surprising that the expression of\n$\\widehat{|p_1|^{-1\/4}}$ involves operators other than ${p_1}$. It\nis therefore important to verify that it has the standard desirable\nproperties. First, as one would hope, it is indeed diagonal in the\neigenbasis of the operators $\\hat{p}_k$:\n\\begin{equation} \\label{inv} \\widehat{|p_1|^{-1\/4}}\\, |\\lambda_1,\\lambda_2,\\lambda_3\\rangle =\n\\frac{\\sqrt2 \\mathrm{sgn}(\\lambda_1)\\,\\sqrt{|\\lambda_2\\lambda_3|}}\n{(4\\pi\\gamma\\sqrt\\Delta\\ell_{\\mathrm{Pl}}^3)^{1\/6}}\n\\left(\\sqrt{|v+\\mathrm{sgn}(\\lambda_2\\lambda_3)|}-\\sqrt{|v-\\mathrm{sgn}(\\lambda_2\\lambda_3)|}\\right)\\,\n|\\lambda_1,\\lambda_2,\\lambda_3\\rangle. \\end{equation}\nSecond, on eigenkets with large volume, the eigenvalue is indeed\nwell-approximated by $p_1^{-1\/4}$, whence on semi-classical states\nit behaves as the inverse of $\\hat{p}^{1\/4}$, just as one would\nhope. Thus, (\\ref{inv}) is a viable candidate for\n$\\widehat{|p_1|^{-1\/4}}$. But there are interesting\nnon-trivialities in the Planck regime. In particular, although\ncounter-intuitive, as is common in LQC the operator annihilates\nstates $|\\lambda_1,\\lambda_2,\\lambda_3\\rangle$ with $v = 2\\lambda_1\\lambda_2\\lambda_3 =0$\n\nFinally, note that the operator $\\hat{\\varepsilon}$ appearing in the\nexpression (\\ref{hc4}) of $\\hat{\\mathcal{C}}_H^{(4)}$ either\noperates immediately before or after $\\widehat{|p_1|^{-1\/4}}$. Since\n$\\widehat{|p_1|^{-1\/4}}$ annihilates all zero volume states and\n$\\hat{\\varepsilon}$ acts on such states as the identity operator, we only\nneed to consider the action of $\\hat{\\varepsilon}$ on states with nonzero\nvolume. In this case, $\\hat{\\varepsilon}$ acts as $\\mathrm{sgn}(v)$. Therefore the\naction of $\\hat{\\mathcal{C}}_H^{(4)}$ can be written as:\n\\begin{align}\n\\Big[\\hat{\\mathcal{C}}_H^{(4)}\\,\\Psi\\Big](\\lambda_1,\\lambda_2,\\lambda_3) =&\n-\\frac{i\\alpha\\pi\\sqrt\\Delta\\hbar\\ell_{\\mathrm{Pl}}^2}{(4\\pi\\gamma\\sqrt\\Delta)^{1\/3}}\\,\\,\n\\mathrm{sgn}(v)\\,\\, (\\lambda_2\\lambda_3)^4\\nonumber\\\\\n\\left(\\sqrt{|v+\\mathrm{sgn}(\\lambda_2\\lambda_3)|}-\\sqrt{|v-\\mathrm{sgn}(\\lambda_2\\lambda_3)|} \\right)\n& \\quad \\Big[\\Phi^+(\\lambda_1,\\lambda_2,\\lambda_3) -\\Phi^-(\\lambda_1,\\lambda_2,\\lambda_3)\\Big]\n\\label{c4}\n\\end{align}\nwhere\n\\begin{align} \\Phi^\\pm(\\lambda_1,\\lambda_2,\\lambda_3) =& \\Big(\\sqrt{\\left|v\\pm2\\mathrm{sgn}(\\lambda_2\n\\lambda_3)+\\mathrm{sgn}(\\lambda_2\\lambda_3))\\right|}\n-\\sqrt{\\left|v\\pm2\\mathrm{sgn}(\\lambda_2\\lambda_3)-\\mathrm{sgn}(\\lambda_2\\lambda_3)\\right|}\\, \\Big)\n\\nonumber \\\\  & \\quad\\: \\times\n\\big(\\mathrm{sgn}(v)+\\mathrm{sgn}(v\\pm2 \\mathrm{sgn}(\\lambda_2\\lambda_3))\\big)\\,\\,\n\\Psi(\\frac{v\\pm2\\mathrm{sgn}(\\lambda_2\\lambda_3)}{v}\\lambda_1,\\lambda_2,\\lambda_3).\\label{phi} \\end{align}\n\nRecall that in the classical theory the singularity corresponds\nprecisely to the phase space points at which the volume vanishes.\nTherefore, as in the Bianchi I model, states with support only on\npoints with $v=0$ will be called `singular' and those which vanish\nat points with $v=0$ will be called regular. The total Hilbert space\n$\\mathcal{H}_{\\rm kin}^{\\rm grav}$ is naturally decomposed as a direct sum $\\mathcal{H}_{\\rm kin}^{\\rm grav} = \\mathcal{H}^{\\rm\ngrav}_{\\rm sing}\\oplus \\mathcal{H}^{\\rm grav}_{\\rm reg}$ of singular and\nregular sub-spaces. We will conclude this discussion by examining\nthe action of $\\hat{\\mathcal{C}}_H^{(4)}$ on these sub-spaces. Note\nfirst that in the action (\\ref{hc4}) of $\\hat{\\mathcal{C}}_H^{(4)}$,\nthe state is first acted upon by the operator\n$\\widehat{|p_1|^{-1\/4}}$. Since this operator annihilates states\n$|\\lambda_1 \\lambda_2,\\lambda_3\\rangle$ with $v = 2\\lambda_1\\lambda_2\\lambda_3 =0$, singular\nstates are simply annihilated by $\\hat{\\mathcal{C}}_H^{(4)}$. In\nparticular this implies that the singular sub-space is mapped to\nitself under this action. It is clear from (\\ref{phi}) that if\n$\\Psi$ is regular, i.e. vanishes on all points with $v =0$,\n$\\Phi^\\pm$ also vanish at these points. Thus the regular sub-space\nis also preserved by this action. This fact will be used in the\ndiscussion of singularity resolution in section \\ref{s3.3.4}.\n\n\n\\emph{Remark:}\\, Our definition of the operator\n$\\widehat{|p|^{-1\/4}}$ is not unique; as is common with non-trivial\nfunctions of elementary variables, there are factor ordering\nambiguities. For example, for $0<n<1\/2$, we have the classical\nidentity\n\\begin{equation}\n|p_1|^{n-1\/2}=\\frac{-i\\mathrm{sgn}(p_1)\\sqrt{|p_2p_3|}}{8\\pi\\gamma\\sqrt\\Delta\nG\\ell_{\\mathrm{Pl}} n} e^{-i\\bar\\mu_1c_1}\\left\\{e^{i\\bar\\mu_1c_1},|p_1|^n\\right\\}\\,\n. \\nonumber \\end{equation}\nHence, it is possible to instead define $\\widehat{p_1^{-1\/4}}$ as\n\\begin{equation} \\widehat{p_1^{-1\/4}} =\n\\left(\\widehat{|p_1|^{n-1\/2}}\\right)^{-1\/(4n-2)} \\nonumber \\end{equation}\nwhere\n\\begin{equation} \\widehat{|p_1|^{n-1\/2}} =\n-\\frac{(4\\pi\\gamma\\sqrt\\Delta\\ell_{\\mathrm{Pl}}^3)^{(2+2n)\/3}} {4^n(8\\pi\\gamma\nG\\sqrt\\Delta\\ell_{\\mathrm{Pl}})^3n}\\mathrm{sgn}(\\lambda_1)|\\lambda_2\\lambda_3|^{1-2n}\\Big[|v+\n\\mathrm{sgn}(\\lambda_2\\lambda_3)|^{2n}-|v-\\mathrm{sgn}(\\lambda_2\\lambda_3)|^{2n}\\Big]. \\nonumber \\end{equation}\nFor $n\\ne1\/4$, this choice for the operator $\\widehat{p_1^{-1\/4}}$\nis not equivalent to the one we chose.  These two choices are both\nwell-defined and admit the same classical limit but they differ in\nthe Planck regime. It is also possible to construct other such\ninequivalent $\\widehat{p_1^{-1\/4}}$ candidate operators. For\ndefiniteness we have made the `simplest' choice.\n\n\n\n\\subsubsection{The fifth term in $\\hat{\\mathcal{C}}_H$}\n\\label{s3.3.3}\n\nLet us now consider the last term in the expression of the\ngravitational part of the Hamiltonian constraint\n\\begin{equation} \\hat{\\mathcal{C}}_H^{(5)} = \\frac{\\alpha^2}{32\\pi\nG\\gamma^2}(1+\\gamma^2)\\,(p_2 p_3)^2\\,\\,\\widehat{p_1^{-2}}. \\end{equation}\nThis term is simpler since it only involves powers of $p_k$ and we\nare working in a representation where $p_k$ are diagonal. From our\ndiscussion of the last section, it is natural to set\n\\begin{equation} \\widehat{p_1^{-2}}:=\\left(\\widehat{p_1^{-1\/4}}\\right)^8\\, , \\end{equation}\nthen we have\n\\begin{align} \\label{c5} \\hat{\\mathcal{C}}_H^{(5)}\\,\\Psi(\\lambda_1,\\lambda_2,\n\\lambda_3)\\, =\\, & \\frac{8\\pi\\alpha^2\\Delta(1+\\gamma^2)\\hbar\\ell_{\\mathrm{Pl}}^2}{(4\\pi\\gamma\\sqrt\n\\Delta)^{2\/3}}\\,\\mathrm{sgn}(\\lambda_1)^8\\lambda_2^8\\lambda_3^8 \\nonumber \\\\ & \\times \\left(\n\\sqrt{|v+\\mathrm{sgn}(\\lambda_2\\lambda_3)|}-\\sqrt{|v-\\mathrm{sgn}(\\lambda_2\\lambda_3)|}\\right)^8 \\Psi(\\lambda_1,\\lambda_2,\n\\lambda_3). \\end{align}\nAgain, it is clear that if $v=0$, the wave function is annihilated\nby this part of the constraint.  Also, it follows by inspection that\nthe singular and regular subspaces are both mapped to themselves by\nthe action of $\\hat{\\mathcal{C}}_H^{(5)}$.\n\n\\subsubsection{Singularity resolution}\n\\label{s3.3.4}\n\nWe can now determine the gravitational part $\\hat{\\mathcal{C}}_{\\rm grav}$\nof the Hamiltonian constraint by combining the results of\n\\cite{awe2} and Eqs. (\\ref{c4}) and (\\ref{c5}). We have:\n\\begin{equation} \\label{qHam2} \\hat{\\mathcal{C}}_{\\rm grav} = \\hat{\\mathcal{C}}_{\\rm grav}^{\\rm\n(BI)} + \\hat{\\mathcal{C}}_H^{(4)} + \\hat{\\mathcal{C}}_H^{(5)} \\end{equation}\nwhere $\\hat{\\mathcal{C}}_{\\rm grav}^{\\rm (BI)}$ is the gravitational part of\nthe Hamiltonian constraint in the Bianchi I model \\cite{awe2}. There\nis however, a conceptual subtlety. In the classical theory the\nHamiltonian density $\\mathcal{C}_{\\rm grav}\/(L_1L_2L_3)^2$ is independent of\nthe choice of the elementary cell (where we have to divide by\n$(L_1L_2L_3)^2$ because the lapse corresponding to harmonic time\nscales as $(L_1L_2L_3)$ and the Hamiltonian constraint is obtained\nby integration over the elementary cell $\\mathcal{V}$). As shown in\nthe section V of \\cite{awe2}, $\\hat{\\mathcal{C}}_{\\rm grav}^{\\rm\n(BI)}\/(L_1L_2L_3)^2$ is again independent of the choice of the\nelementary cell $\\mathcal{V}$. However, the two additional terms\nthat are special to the Bianchi II model are not independent of\n$\\mathcal{V}$ because they involve the inverse-triad operators\n\\cite{kv1}. Nonetheless, in the limit as we take the regulator away,\ni.e., $\\mathcal{V} \\to \\mathbb{R}^3$, the operator\n$\\hat{\\mathcal{C}}_{\\rm grav}\/(L_1L_2L_3)^2$ has a well-defined limit (see\nfootnote 5). Strictly speaking, in the discussion of Bianchi II\nquantum dynamics, we have to work with this limit, rather than with\noperators defined using a fixed cell.\n\nAs in the Bianchi I model, the action simplifies if we replace one\nof the $\\lambda_i$ by $v$.  In the Bianchi I model, it does not matter\nwhich of the $\\lambda_i$ is replaced because of the additional symmetry\nof that model. In the Bianchi II case, while it remains possible to\nreplace any of the $\\lambda_i$, it is simplest to replace $\\lambda_1$ by $v$\nand represent quantum states as $\\Psi=\\Psi(\\lambda_2,\\lambda_3,v;T)$. This\nchange of variables would be nontrivial if, as in the Wheeler-DeWitt\ntheory, we had used the Lesbegue measure in the gravitational\nsector. However, it is quite tame here because the norms are defined\nusing a discrete measure. The inner product on $\\mathcal{H}_{\\rm kin}^{\\rm grav}$ is now given\nby\n\\begin{equation} \\langle\\Psi_1|\\Psi_2\\rangle_{\\rm kin} = \\sum_{\\lambda_2,\\lambda_3,v}\n\\,\\,\\bar{\\Psi}_1(\\lambda_2,\\lambda_3,v)\\,\\Psi_2(\\lambda_2,\\lambda_3,v) \\end{equation}\nand states are symmetric under the action of $\\hat\\Pi_k$. In\nAppendix \\ref{a1}, we show that, under the action of reflections\n$\\hat\\Pi_i$, the operators $\\sin\\bar\\mu_ic_i$ have the same\ntransformation properties that $c_i$ have under reflections $\\Pi_i$\nin the classical theory. As a consequence, $\\hat{\\mathcal{C}}_{\\rm grav}$ is\nalso reflection symmetric. Therefore, its action is well defined on\n$\\mathcal{H}_{\\rm kin}^{\\rm grav}$: $\\hat{\\mathcal{C}}_{\\rm grav}$ is a densely defined, symmetric\noperator on this Hilbert space. In the isotropic case, its analog\nhas been shown to be essentially self-adjoint \\cite{warsaw2}. In\nwhat follows we will assume that (\\ref{qHam2}) is essentially\nself-adjoint on $\\mathcal{H}_{\\rm kin}^{\\rm grav}$ and work with its self-adjoint extension.\n\nIt is now straightforward to write down the full Hamiltonian\nconstraint on $\\mathcal{H}_{\\rm kin}^{\\rm grav}$:\n\\begin{equation} \\label{qHam3} -\\hbar^2\\, \\partial^2_T \\, \\Psi(\\lambda_2,\\lambda_3,v; T) =\n\\Theta\\, \\Psi(\\lambda_2,\\lambda_3,v; T)\\quad {\\rm where}\\quad \\Theta = -2\n\\hat{\\mathcal{C}}_{\\rm grav} \\end{equation}\nAs in the isotropic case \\cite{aps2}, one can obtain the physical\nHilbert space $\\mathcal{H}_{\\rm phy}$ by a group averaging procedure and the\nfinal result is completely analogous. Elements of $\\mathcal{H}_{\\rm phy}$\nconsist of `positive frequency' solutions to (\\ref{qHam3}), i.e.,\nsolutions to\n\\begin{equation} \\label{qHam4} -i\\hbar \\partial_T \\Psi(\\lambda_2,\\lambda_3,v; T)\\, = \\,\n\\sqrt{|\\Theta|}\\, \\Psi(\\lambda_2,\\lambda_3,v; T)\\, ,\\end{equation}\nwhich are symmetric under the three reflection maps $\\hat\\Pi_i$,\ni.e. satisfy\n\\begin{equation} \\label{sym} \\Psi(\\lambda_2,\\lambda_3,v;\\, T) = \\Psi(|\\lambda_2|,|\\lambda_3|,|v|;\\,\nT)\\, . \\end{equation}\nThe scalar product is given simply by:\n\\begin{eqnarray} \\label{ip1} \\langle \\Psi_1|\\Psi_2\\rangle_{\\rm phys} &=& \\langle\n\\Psi_1(\\lambda_2,\\lambda_3, v; T_o)|\\Psi_2(\\lambda_2,\\lambda_3,v; T_o) \\rangle_{\\rm\nkin} \\nonumber\\\\\n&=& \\sum_{\\lambda_1,\\lambda_2,\\lambda_3} \\bar\\Psi_1(\\vec{\\lambda}, T_o)\\,\n\\Psi_2(\\vec{\\lambda}, T_o) \\end{eqnarray}\nwhere $T_o$ is any ``instant'' of internal time $T$.\n\nWe can now address the issue of singularity resolution using general\nproperties of various operators. Recall that the gravitational part\nof the Hamiltonian constraint operator in the Bianchi I model shares\ntwo properties with the fourth and the fifth terms studied above\nwhich are specific to the Bianchi II model. First, it annihilates\nsingular states and, second, singular states decouple from the\nregular states under its action. Therefore the full Bianchi II\nHamiltonian constraint also has these two properties. Since the\nsingular states decouple from regular states%\n\\footnote{Singular states are in the kernel of $\\Theta$ and regular\nstates are orthogonal to the singular ones. From spectral\ndecomposition one expects $\\sqrt{\\Theta}$ to have the same property.\nHowever, to complete this argument, one would have to establish that\n$\\hat{\\mathcal{C}}_{\\rm grav}$ is essentially self-adjoint and its self\nadjoint extension also shares this property.},\nan initial state in the regular sub-space cannot become singular\nduring evolution. It is in this precise sense that the classical\nsingularity is resolved. Sometimes one considers weaker forms of\nsingularity resolution. For example, it could happen that the\nevolution of the wave function is always well defined but a regular\nstate can evolve to the singular sub-space. For the Bianchi I and II\nmodels, the singularity is resolved in a stronger sense: \\emph{Not\nonly is the evolution well defined at all times, but the singular\nstates (are stationary and) decouple entirely from the regular\nones.}\n\n\\subsubsection{The explicit form of the Hamiltonian constraint}\n\\label{s.3.5}\n\nWe will conclude by providing an explicit form of the full quantum\nconstraint equation that will be needed in numerical simulations.\n\nRecall that in the Bianchi I model \\cite{awe2} symmetries enabled us\nto restrict our attention to the positive octant of the\n3-dimensional space spanned by $(\\lambda_1,\\lambda_2,\\lambda_3)$. This is again the\ncase for the Bianchi II model. More precisely, elements of $\\mathcal{H}_{\\rm kin}^{\\rm grav}$\nare invariant under the three parity maps $\\hat{\\Pi}_k$ and, as\nshown in the Appendix \\ref{a1}, the Hamiltonian constraint satisfies\n$\\hat{\\Pi}_k\\, \\hat{\\mathcal{C}}_{\\rm grav} \\hat{\\Pi}_k =\n\\hat{\\mathcal{C}}_{\\rm grav}$. Therefore, knowledge of the restriction of\nthe image $\\hat{ \\mathcal{C}}_{\\rm grav}\\Psi$ of $\\Psi$ to the positive\noctant suffices to determine $\\hat{\\mathcal{C}}_{\\rm grav}\\Psi$ completely.\nIn the positive octant, $\\mathrm{sgn}(\\lambda_k)$ can only be 0 or 1 which\nsimplifies the action of operators. Therefore, in the remainder of\nthis section we will restrict the argument of $\\hat{\n\\mathcal{C}}_H\\Psi$ to the positive octant. The full action is given\nsimply by\n\\begin{equation} \\big(\\hat{\\mathcal{C}}_{\\rm grav}\\Psi\\big)(\\lambda_2,\\lambda_3, v)=\\big(\\hat{\n\\mathcal{C}}_{\\rm grav}\\Psi\\big)(|\\lambda_2|,|\\lambda_3|,|v|). \\end{equation}\n\nSince the singular states are annihilated by\n$\\hat{\\mathcal{C}}_{\\rm grav}$, their evolution is trivial:\n\\begin{equation} \\partial_T^2\\, \\Psi(\\lambda_2,\\lambda_3,v=0;T) = 0\\, . \\end{equation}\nNon-singular states are physically more relevant. On them, the\nexplicit form of the full constraint is given by:\n\\begin{align} \\partial_T^2\\, \\Psi(\\lambda_2,\\lambda_3,v;T) =& \\frac{\\pi G}{2}\\Bigg[\\sqrt{v}\n\\bigg((v+2)\\sqrt{v+4}\\,\\Psi^+_4(\\lambda_2,\\lambda_3,v;T) - (v+2)\\sqrt v\\,\n\\Psi^+_0( \\lambda_2, \\lambda_3,v;T)\\nonumber \\\\& -(v-2)\\sqrt\nv\\,\\Psi^-_0(\\lambda_2,\\lambda_3,v;T)+(v-2)\\sqrt{|v-4|}\n\\,\\Psi^-_4(\\lambda_2,\\lambda_3,v;T)\\bigg)\\nonumber \\\\ &\n+\\frac{2i\\alpha\\sqrt\\Delta}{(4\\pi\n\\gamma\\sqrt\\Delta)^{1\/3}}(\\lambda_2\\lambda_3)^4\\left(\\sqrt{v+1}-\\sqrt{|v-1|}\\right)\\bigg(\n\\Phi^--\\Phi^+\\bigg)(\\lambda_2,\\lambda_3,v;T) \\nonumber\\\\& \\label{qHamfin} +\n\\frac{16\n\\alpha^2\\Delta(1+\\gamma^2)}{(4\\pi\\gamma\\sqrt\\Delta)^{2\/3}}(\\lambda_2\\lambda_3)^8\\:\n(\\sqrt{v+1}-\\sqrt{|v-1|})^8\\:\\Psi(\\lambda_2,\\lambda_3,v;T)\\Bigg] \\end{align}\nwhere $\\Psi^\\pm_{0,4}$ are defined as follows:\n\\begin{align} \\Psi^\\pm_4(\\lambda_2,\\lambda_3,v;T)=& \\:\\Psi\\left(\\frac{v\\pm4}{v\\pm2}\\cdot\n\\lambda_2,\\frac{v\\pm2}{v}\\cdot\\lambda_3,v\\pm4;T\\right)+\\Psi\\left(\\frac{v\\pm4}{v\\pm2}\\cdot\\lambda_2,\n\\lambda_3,v\\pm4;T\\right)\\nonumber\\\\& +\\Psi\\left(\\frac{v\\pm2}{v}\\cdot\\lambda_2,\\frac{v\\pm4}{v\n\\pm2}\\cdot\\lambda_3,v\\pm4;T\\right)+\\Psi\\left(\\frac{v\\pm2}{v}\\cdot\\lambda_2, \\lambda_3,v\\pm4;T\n\\right) \\nonumber \\\\ & +\\Psi\\left(\\lambda_2,\\frac{v\\pm2}{v}\\cdot\\lambda_3,v\\pm4;T\\right)+\n\\Psi\\left(\\lambda_2,\\frac{v\\pm4}{v\\pm2}\\cdot\\lambda_3,v\\pm4;T\\right), \\end{align}\nand\n\\begin{align} \\Psi^\\pm_0(\\lambda_2,\\lambda_3,v;T)= & \\:\\Psi\\left(\\frac{v\\pm2}{v}\\cdot\\lambda_2,\n\\frac{v}{v\\pm2}\\cdot\\lambda_3,v;T\\right)+\\Psi\\left(\\frac{v\\pm2}{v}\\cdot\\lambda_2,\\lambda_3,v;T\n\\right) \\nonumber \\\\ & +\\Psi\\left(\\frac{v}{v\\pm2}\\cdot\\lambda_2,\\frac{v\\pm2}{v}\\cdot\\lambda_3,\nv;T\\right)+\\Psi\\left(\\frac{v}{v\\pm2}\\cdot\\lambda_2,\\lambda_3,v;T\\right) \\nonumber \\\\ & +\n\\Psi\\left(\\lambda_2,\\frac{v}{v\\pm2}\\cdot\\lambda_3,v;T\\right)+\\Psi\\left(\\lambda_2,\\frac{v\\pm2}{v}\n\\cdot\\lambda_3,v;T\\right)\\, ,\\end{align}\nwhile $(\\Phi^--\\Phi^+)$ is given by\n\\begin{align} \\big(\\Phi^--\\Phi^+\\big)(\\lambda_2,\\lambda_3,v;T)\\, =& \\,(\\sqrt{|v-2+\n\\mathrm{sgn}(v-2)|}-\\sqrt{|v-2-\\mathrm{sgn}(v-2)|}) \\nonumber \\\\ & \\qquad \\times (1+\\mathrm{sgn}(v-2))\n\\Psi(\\lambda_2,\\lambda_3,v-2;T) \\nonumber \\\\ & \\: -2(\\sqrt{v+3}-\\sqrt{v+1})\n\\Psi(\\lambda_2,\\lambda_3,v+2;T). \\end{align}\n(The imaginary coefficients in (\\ref{qHamfin}) come from the\naction of single $\\sin\\bar\\mu_ic_i$ terms.)\n\nEq. (\\ref{qHamfin}) immediately implies that, as in the Bianchi I\nmodel, the steps in $v$ are uniform: the argument of the wave\nfunction only involves $v-4, v-2, v, v+2$ and $v+4$. Thus, there is\na superselection in $v$. For each $\\epsilon\\in[0,2),$ let us\nintroduce a lattice $\\mathcal{L}_\\epsilon$ of points $v=2n+\\epsilon$\nif $\\epsilon$ is 0 or 1 or $v=n+\\epsilon$ otherwise.\n\\footnote{The lattice for $\\epsilon \\not= 0,1$ is twice as large as\nthat for $\\epsilon=0$ or $\\epsilon=1$ due to the symmetry properties\nof the wave function.}\nThen the quantum evolution ---and the action of the Dirac\nobservables $\\hat{p}_T$ and $\\hat{V}|_{T}$ commonly used in\nLQC--- preserves the subspaces $\\mathcal{H}^\\epsilon_{\\mathrm{phy}}$\nconsisting of states with support in $v$ on $\\mathcal{L}_\\epsilon$.\nThe most interesting lattice is the one corresponding to\n$\\epsilon=0$\nsince it includes the classically singular points $v=0$.\n\nFinally, it is obvious from (\\ref{qHamfin}) that in the limit\n$\\alpha\\to0$ quantum dynamics of the Bianchi II model reduces to\nthat of the Bianchi I model discussed in \\cite{awe2}. In particular,\nit is possible to obtain the LQC dynamics for the $k$=0 FRW\ncosmology from this model by first taking $\\alpha\\to0$ and then\nfollowing the projection map defined in section IVA in \\cite{awe2}.\n\n\n\n\\section{Effective Equations}\n\\label{s4}\n\nIn the isotropic models, effective equations have been introduced\nvia two different approaches ---the embedding and the truncation\nmethods. Both start by regarding the space of quantum states as an\ninfinite dimensional symplectic manifold ---the quantum phase\nspace--- which is also equipped with a K\\\"ahler structure that\ndescends from the Hermitian inner product on the Hilbert space. In\nthe first method, one finds a judicious embedding of the classical\nphase space into the quantum phase space which is approximately\npreserved by the quantum evolution vector field \\cite{jw,vt}. By\nprojecting this vector field into the image of the embedding one\nobtains quantum corrected effective equations. In the isotropic case\nthese effective equations provide an excellent approximation to the\nfull quantum evolution of states which are Gaussians at late times,\neven in the $\\Lambda\\not=0$ as well as k=$\\pm 1$ cases where the\nmodels are not exactly soluble. In the second method one uses\nexpectation values, uncertainties, and higher moments to define a\nconvenient system of coordinates on the infinite dimensional phase\nspace. The exact quantum evolution equations are then a set of\ncoupled non-linear ordinary differential equations for these\ncoordinates. By a judicious truncation of this system one obtains\neffective equations containing quantum corrections \\cite{bs}. In its\nspirit the first method is analogous to the `variational principle\ntechnique' used in perturbation theory, in that it requires a\njudicious combination of art (of selecting the embedding) and\nscience. It is often simpler to use and can be surprisingly\naccurate. The second method is more systematic, similar in our\nanalogy to the standard, order by order perturbation theory. It is\nalso more general in the sense that it is applicable to a wide\nvariety of states. In this section we will use the first method to\ngain qualitative insights into leading order quantum effects.\n\nTo obtain the effective equations, without loss of generality we can\nrestrict our attention to the positive octant of the classical phase\nspace (where $\\varepsilon=1$). Then the quantum corrected Hamiltonian\nconstraint is given by the classical analogue of (\\ref{qHam1}):\n\\begin{equation} \\label{Heff}\n\\frac{p_T^2}{2}+\\mathcal{C}^{\\mathrm{eff}}_{\\mathrm{grav}} = 0, \\end{equation}\nwhere\n\\begin{align} \\mathcal{C}^{\\mathrm{eff}}_{\\mathrm{grav}} =&\n-\\frac{p_1p_2p_3}{8\\pi G\\gamma^2\\Delta\\ell_{\\mathrm{Pl}}^2}\n\\Bigg[\\sin\\bar\\mu_1c_1\\sin\\bar\\mu_2c_2+\\sin\\bar\\mu_2c_2\n\\sin\\bar\\mu_3c_3+\\sin\\bar\\mu_3c_3\\sin\\bar\\mu_1c_1\\Bigg] \\nonumber\n\\\\ & \\quad - \\frac{1}{8\\pi\nG\\gamma^2}\\Bigg[\\frac{\\alpha(p_2p_3)^{3\/2}}{\\sqrt\\Delta\\ell_{\\mathrm{Pl}}\n\\sqrt{p_1}}\\sin\\bar\\mu_1c_1 -(1+\\gamma^2)\\left(\\frac{\\alpha\np_2p_3}{2p_1}\\right)^2 \\Bigg]. \\end{align}\nUsing the expressions (\\ref{mubar}) of $\\bar\\mu_k$, it is easy to\nverify that far away from the classical singularity ---more\nprecisely in the regime in which the (gauge fixed) spin connection\nand the extrinsic curvature are sufficiently small so that\n$c_k\\bar\\mu_k \\ll 1$---  the effective Hamiltonian constraint\n(\\ref{Heff}) is well-approximated by the classical one (\\ref{Hcl}).\n\nSince $\\sin\\theta$ is bounded by 1 for all $\\theta$, these equations\nimply that the matter density $\\rho_{\\mathrm{matt}}=\np_T^2\/2V^2=p_T^2\/2p_1p_2p_3$ satisfies\n\\begin{equation} \\rho_{\\mathrm{matt}} \\le \\frac{3}{8\\pi\\gamma^2\\Delta G\\ell_{\\mathrm{Pl}}^2}+\n\\frac{1}{8\\pi \\gamma^2 G} \\left[\\frac{x}{\\sqrt\\Delta\\ell_{\\mathrm{Pl}}} -\n\\frac{(1+\\gamma^2)x^2}{4} \\right] \\end{equation}\nwhere we have introduced $x:=\\alpha\\sqrt{p_2p_3\/p_1^3}$. The maximum\nof the expression in square brackets is attained at\n$x=2\/(1+\\gamma^2)\\sqrt\\Delta\\ell_{\\mathrm{Pl}}$, whence\n\\begin{equation} \\rho_{\\mathrm{matt}} \\le \\frac{3+(1+\\gamma^2)^{-1}}\n{8\\pi\\gamma^2\\Delta G \\ell_{\\mathrm{Pl}}^2} \\approx 0.54 \\rho_{\\mathrm{Pl}}. \\end{equation}\nThus, on the constraint surface in the phase space defined by\n(\\ref{Heff}), the matter energy density is bounded by $0.54\n\\rho_{\\mathrm{Pl}}$. But this bound may be far from being optimal.\nIn all isotropic models, the optimal bound on matter density was\nfound to be $0.41 \\rho_{\\rm Pl}$. In the Bianchi I model, available\nsimulations by Vandersloot (private communication) show that the\n`volume bounce' occurs when matter density is \\emph{lower} than\n$0.41 \\rho_{\\rm Pl}$ because there is also energy density in\ngravitational waves. It would be interesting to use numerical\nsimulations to find out what happens for generic solutions to the\nBianchi II effective equations.\n\nFinally, to obtain the effective equations for each variable, one\nsimply takes its Poisson bracket with the effective Hamiltonian\nconstraint.  This gives the effective equations\n\\begin{equation} \\dot{p_1} = \\gamma^{-1}\\left(\\frac{p_1^2}{\\bar\\mu_1}(\\sin\\bar\\mu_2c_2+\n\\sin\\bar\\mu_3c_3)+\\alpha p_2p_3\\right)\\cos\\bar\\mu_1c_1, \\end{equation}\n\\begin{equation} \\dot{p_2} = \\frac{p_2^2}{\\gamma\\bar\\mu_2}(\\sin\\bar\\mu_1c_1+\\sin\\bar\\mu_3c_3)\n\\cos\\bar\\mu_2c_2, \\end{equation}\n\\begin{equation} \\dot{p_3} = \\frac{p_3^2}{\\gamma\\bar\\mu_3}(\\sin\\bar\\mu_1c_1+\\sin\\bar\\mu_2c_2)\n\\cos\\bar\\mu_3c_3, \\end{equation}\n\\begin{align} \\dot{c_1} &= -\\frac{1}{\\gamma}\\Big[\\frac{p_2p_3}{\\Delta\\ell_{\\mathrm{Pl}}^2}\\big(\n\\sin\\bar\\mu_1c_1\\sin\\bar\\mu_2c_2+\\sin\\bar\\mu_1c_1\\sin\\bar\\mu_3c_3+\\sin\\bar\\mu_2\nc_2\\sin\\bar\\mu_3c_3 \\nonumber \\\\ & \\qquad +\\frac{\\bar\\mu_1c_1}{2}\\cos\\bar\\mu_1c_1(\n\\sin\\bar\\mu_2c_2+\\sin\\bar\\mu_3c_3)-\\frac{\\bar\\mu_2c_2}{2}\\cos\\bar\\mu_2c_2(\\sin\\bar\n\\mu_1c_1+\\sin\\bar\\mu_3c_3) \\nonumber \\\\ & \\qquad -\\frac{\\bar\\mu_3c_3}{2}\\cos\\bar\n\\mu_3c_3(\\sin\\bar\\mu_1c_1+\\sin\\bar\\mu_2c_2)\\big)+(1+\\gamma^2)\\alpha^2\\frac{(p_2\np_3)^2}{2p_1^3} \\nonumber \\\\ & \\qquad +\\frac{\\alpha}{2\\sqrt\\Delta\\ell_{\\mathrm{Pl}}}\\left(\n\\frac{p_2p_3}{p_1}\\right)^{3\/2}\\!\\!(\\bar\\mu_1c_1\\cos\\bar\\mu_1c_1-\\sin\\bar\\mu_1c_1)\n\\Big], \\end{align}\n\\begin{align} \\dot{c_2} &= -\\frac{1}{\\gamma}\\Big[\\frac{p_1p_3}{\\Delta\\ell_{\\mathrm{Pl}}^2}\\big(\n\\sin\\bar\\mu_1c_1\\sin\\bar\\mu_2c_2+\\sin\\bar\\mu_1c_1\\sin\\bar\\mu_3c_3+\\sin\\bar\\mu_2\nc_2\\sin\\bar\\mu_3c_3 \\nonumber \\\\ & \\qquad -\\frac{\\bar\\mu_1c_1}{2}\\cos\\bar\\mu_1c_1(\n\\sin\\bar\\mu_2c_2+\\sin\\bar\\mu_3c_3)+\\frac{\\bar\\mu_2c_2}{2}\\cos\\bar\\mu_2c_2(\\sin\\bar\n\\mu_1c_1+\\sin\\bar\\mu_3c_3) \\nonumber \\\\ & \\qquad -\\frac{\\bar\\mu_3c_3}{2}\\cos\\bar\n\\mu_3c_3(\\sin\\bar\\mu_1c_1+\\sin\\bar\\mu_2c_2)\\big)-(1+\\gamma^2)\\alpha^2\\frac{p_2\np_3^2}{2p_1^2} \\nonumber \\\\ & \\qquad +\\frac{\\alpha p_3}{2\\bar\\mu_1}(3\\sin\\bar\n\\mu_1c_1-\\bar\\mu_1c_1\\cos\\bar\\mu_1c_1) \\Big], \\end{align}\n\\begin{align} \\dot{c_3} &= -\\frac{1}{\\gamma}\\Big[\\frac{p_1p_2}{\\Delta\\ell_{\\mathrm{Pl}}^2}\\big(\n\\sin\\bar\\mu_1c_1\\sin\\bar\\mu_2c_2+\\sin\\bar\\mu_1c_1\\sin\\bar\\mu_3c_3+\\sin\\bar\\mu_2\nc_2\\sin\\bar\\mu_3c_3 \\nonumber \\\\ & \\qquad -\\frac{\\bar\\mu_1c_1}{2}\\cos\\bar\\mu_1c_1(\n\\sin\\bar\\mu_2c_2+\\sin\\bar\\mu_3c_3)-\\frac{\\bar\\mu_2c_2}{2}\\cos\\bar\\mu_2c_2(\\sin\\bar\n\\mu_1c_1+\\sin\\bar\\mu_3c_3) \\nonumber \\\\ & \\qquad +\\frac{\\bar\\mu_3c_3}{2}\\cos\\bar\n\\mu_3c_3(\\sin\\bar\\mu_1c_1+\\sin\\bar\\mu_2c_2)\\big)-(1+\\gamma^2)\\alpha^2\\frac{p_2^2\np_3}{2p_1^2} \\nonumber \\\\ & \\qquad +\\frac{\\alpha p_2}{2\\bar\\mu_1}(3\\sin\\bar\n\\mu_1c_1-\\bar\\mu_1c_1\\cos\\bar\\mu_1c_1) \\Big]. \\end{align}\nIn the ``embedding approach'' these effective equations provide the\nleading-order quantum corrections to the classical equations of\nmotion Eqs.~(\\ref{ceom1})~--~(\\ref{ceom2}). It would be very\ninteresting to numerically test if the accuracy they display in the\nisotropic case for states which are Gaussians at late times carries\nover to the Bianchi II case.\n\n\n\\section{Discussion}\n\\label{s5}\n\nIn this paper, we analyzed the ``improved'' LQC dynamics of the\nBianchi II model.  As in the isotropic and Bianchi I cases, we chose\nthe matter source to be a massless scalar field since it continues\nto serve as a viable relational time parameter in the classical as\nwell as the quantum theory. It is again rather straightforward to\naccommodate additional matter fields in this framework.\n\nOur broad strategy is the same as that used in the Bianchi I model\n\\cite{awe2}. However, because Bianchi II models have anisotropies\n\\emph{as well as} spatial curvature, holonomies around closed curves\nare no longer guaranteed to be almost periodic functions of the\nconnection. Hence, one cannot use them to construct the field\nstrength operator on the LQC Hilbert space; a new conceptual and\ntechnical input is necessary to define the quantum Hamiltonian\nconstraint operator. We overcame this difficulty by generalizing the\nstrategy used so far \\cite{abl,aps3,warsaw,apsv,kv1,ls,bp,awe2}.\nSpecifically, we used holonomies around open segments parallel to\nthe fiducial triads ${}^o\\!e^a_i$ to define a connection operator. This\nstrategy is also inspired by methods introduced by Thiemann in the\nfull theory \\cite{tt}. However, because of gauge fixing LQC does not\nenjoy the manifest diffeomorphism invariance of full LQG. As a\nconsequence, in LQC one needs a principle to fix the `length' of the\nopen segment along which holonomy is evaluated. We required that the\n`length' be so chosen that the field strength operator constructed\nfrom the resulting connection should agree with that constructed\nfrom holonomies around closed loops whenever the second construction\nis available. This guarantees that (apart from `tame' factor\nordering ambiguities) the new procedure reduces to the one used in\nthe LQC literature before. Moreover, the strategy of defining the\nHamiltonian constraint through this connection operator can be used\nalso in more general contexts. In particular, it enables one to\novercome a conceptual limitation of the otherwise complete treatment\nof the isotropic, k=$-1$ model given in \\cite{kv1,ls}. More\nimportantly, it extends to more general class A Bianchi models. A\nsystematic treatment of the Bianchi IX model along the lines of this\npaper would be especially interesting.\n\nThere is a second ---but primarily technical--- difference from the\nBianchi I case: The Hamiltonian operator now contains inverse powers\nof $p_1$. This was handled following a general method introduced by\nThiemann to define inverse triad operators in LQG \\cite{tt}. As\nusual, there is a factor ordering ambiguity. In the main discussion\nwe used the simplest operator which has the same symmetries with\nrespect to parity as its classical counterpart.\n\nAfter addressing these two issues, we obtained a well defined\nquantum Hamiltonian constraint and showed that the singularity in\nBianchi II models is resolved in the same precise sense as in the\nFRW and Bianchi I models. The Kinematical Hilbert space $\\mathcal{H}_{\\rm kin}^{\\rm grav}$ can\nbe decomposed as $\\mathcal{H}_{\\rm kin}^{\\rm grav} = \\mathcal{H}_{\\rm sing}^{\\rm grav} \\oplus \\mathcal{H}^{\\rm\ngrav}_{\\rm reg}$ where states in the singular subspace have support\nonly on configurations with zero volume, while those in the regular\nsub-space have no support on these singular configurations.\n\\emph{The Hamiltonian constraint annihilates states in $\\mathcal{H}_{\\rm\nsing}^{\\rm grav}$ and maps $\\mathcal{H}^{\\rm grav}_{\\rm reg}$ to itself.} We\nalso provided an explicit form of the Hamiltonian constraint which\nshould be helpful in performing numerical simulations.\n\nFinally, we obtained effective equations using the ``embedding\nmethod'' introduced by Willis \\cite{jw} and further developed by\nTaveras \\cite{vt} in the isotropic case. There, although the\nassumptions made in the derivation fail in the deep Planck regime,\nthe final equations provide a surprisingly accurate approximation to\nthe full quantum evolution of states which are Gaussians at late\ntimes. This holds not only for the exactly soluble k=0, $\\Lambda=0$\nmodel but also for the much more complicated $\\Lambda\\not=0$ and\nk=$\\pm 1$ models. It would be interesting to see if this phenomenon\nextends also to Bianchi II models. Furthermore, numerical solutions\nof these effective equations themselves may be of considerable\ninterest because the simplest upper bound on matter density they\nlead to is higher than that in all other models studied so far,\nincluding Bianchi I. Numerical simulations of effective equations\nwill answer several questions within this approximation. Is the\nupper bound optimal, i.e., do generic solutions to effective\nequations come close to saturating it? In the Bianchi I case,\nnumerical simulations by Vandersloot (private communication)\nrevealed that, unlike in the isotropic model, there are several\ndistinct kinds of `bounces.' Roughly, anytime a shear ---or a Weyl\ncurvature--- scalar enters the Planck regime, quantum geometry\nrepulsion comes into pay in a dominant manner and `dilutes' that\nscalar, preventing a blow up. How do additional terms in the Bianchi\nII effective equations affect this scenario? Qualitative lessons\nfrom numerical simulations would be valuable in developing further\nintuition for various quantum geometry effects.\n\n\n\\section*{Acknowledgements:}\n\nWe would like to thank Gianluca Calcagni, Alex Corichi, Jerzy\nLewandowski, Simone Mercuri, Tomasz Pawlowski and Hanno Sahlmann for\nhelpful discussions. This research was supported in part by NSF\ngrant PHY0854743, the George A. and Margaret M. Downsbrough\nEndowment, the Eberly research funds of Penn State, Le Fonds\nqu\\'eb\\'ecois de la recherche sur la nature et les technologies and\nthe Edward A. and Rosemary A. Mebus Fellowship.\n\n\n\\begin{appendix}\n\n\n\\section{Parity Symmetries}\n\\label{a1}\n\nIn non-gravitational physics, parity transformations are normally\ntaken to be discrete diffeomorphisms $x_i \\rightarrow -x_i$ in the\nphysical space which are isometries of the flat 3-metric thereon. In\nthe phase space formulation of general relativity,  we do not have a\nflat metric ---or indeed, any fixed metric. Therefore these discrete\nsymmetries are no longer meaningful (except in the weak field\nlimit). However, if the dynamical variables have internal indices\n---such as the triads and connections used in LQG--- we can use the\nfact that the internal space $I$ is a vector space equipped with a\nflat metric $q_{ij}$ to define parity operations on the internal\nindices. Associated with any unit internal vector $v^i$, there is a\nparity operator $\\Pi_v$ which reflects the internal vectors across\nthe 2-plane orthogonal to $v^i$. This operation induces a natural\naction on triads $e^a_i$, the connections $A_a^i$ and the conjugate\nmomenta $P^a_i =: (1\/8\\pi G\\gamma)\\, E^a_i$ (since they are internal\nvectors or co-vectors).\n\nThe triads $e^a_i$ are proper internal co-vectors. In previous\nreferences \\cite{bd,awe2}, conventions were such that the spin\nconnection $\\Gamma^a_i$ turned out to be an internal \\emph{pseudo}\nvector. It was then natural to regard the Barbero-Immirzi parameter\n$\\gamma$ to be a pseudo quantity so that the connection $A_a^i$ has\ndefinite parity namely, it transforms as an internal pseudo-vector.\nThis in turn led to the conclusion that $P^a_i$ is also an internal\npseudo-vector (as one would expect because it is canonically\nconjugate to $A_a^i$) \\cite{awe2}. While this is all\nself-consistent, these conventions lead to two undesirable\nconsequences. First, in the classical theory, it is not possible to\nreconstruct the triads $e^a_i$ unambiguously starting from the\nmomenta $P^a_i$. Therefore, one cannot recover the space-time\ngeometry unambiguously starting from the Hamiltonian theory. Second,\nthe momenta $P^a_i$ are subject to a non-holonomic constraint which\nobstructs the passage to quantum theory a la LQG. However, if one\nsets conventions as in section \\ref{s2.1}, then $\\Gamma_a^i, \\gamma,\nA_a^i$ and $P_a^i$ are all \\emph{proper} quantities and the two\ndifficulties disappear \\cite{aa-dis}. In the main text we have used\nthis strategy. We now summarize the differences from the Appendix of\n\\cite{awe2} that it leads to.\n\nIn diagonal Bianchi models, we can restrict ourselves just to three\nparity operations $\\Pi_i$. Under their action, the canonical\nvariables $c_i,p_i$ transform as follows:\n\\begin{equation} \\label{P1} \\Pi_1 (c_1,c_2, c_3) = (-c_1, c_2, c_3), \\quad \\quad\n\\Pi_1 (p_1, p_2, p_3) = (-p_1, p_2, p_3)\\, , \\end{equation}\nand the action of $\\Pi_2, \\Pi_3$ is given by cyclic permutations.\nThus, $c^i$ and $p_i$ are \\emph{proper} internal vectors and\nco-vectors. Under any of these maps $\\Pi_i$, the symplectic\nstructure (\\ref{pb2}), the Hamiltonian (\\ref{Hcl}), and hence also\nthe Hamiltonian vector field, are left invariant. This is just as\none would expect because $\\Pi_i$ are simply large gauge\ntransformations of the theory under which the physical metric\n$q_{ab}$ and the extrinsic curvature $K_{ab}$ do not change. Also,\nit is clear from the action of (\\ref{P1}) that if one knows the\ndynamical trajectories on the octant $p_i\\ge 0$ of the phase space,\nthen dynamical trajectories on any other octant can be obtained just\nby applying a suitable (combination of) $\\Pi_i$. Therefore, in the\nclassical theory one can restrict one's attention just to the\npositive octant.\n\nLet us now turn to the quantum theory. We now have three operators\n$\\hat\\Pi_i$. Their action on states is given by\n\\begin{equation} \\hat\\Pi_1 \\Psi(\\lambda_1,\\lambda_2,\\lambda_3) = \\Psi(-\\lambda_1,\\lambda_2\\lambda_3)\\, , \\end{equation}\netc. What is the induced action on operators? Since\n\\begin{align} \\label{P2}\n\\hat{\\Pi}_1\\hat{\\lambda}_1\\hat{\\Pi}_1\\Psi(\\lambda_1,\\lambda_2,\\lambda_3)\n&= \\hat{\\Pi}_1 \\Big({\\lambda}_1\\,\\Psi(-\\lambda_1,\\lambda_2,\\lambda_3)\\Big) \\nonumber \\\\\n&= -\\lambda_1\\Psi(\\lambda_1,\\lambda_2,\\lambda_3), \\end{align}\nwe have\n\\begin{equation} \\label{P3} \\hat{\\Pi}_1\\hat{\\lambda}_1\\hat{\\Pi}_1  = -\\hat{\\lambda}_1. \\end{equation}\nThe Hamiltonian constraint operator, modulo factor ordering which is\nnot important here, is given by Eq. (\\ref{qHam1}). To calculate its\ntransformation property under parity maps, in addition to\n(\\ref{P3}), we also need the transformation property of the\noperators $\\sin \\bar\\mu_ic_i$ and $\\hat{\\varepsilon}$ and operators\ncorresponding to inverse powers of $p_1$. Due to the symmetries of\ntype A Bianchi models, to know the properties of $\\sin \\bar\\mu_ic_i$\nunder parity transformations, it is sufficient to calculate\n$\\hat{\\Pi}_i \\sin \\bar\\mu_1c_1\\hat{\\Pi}_i$.  We have:\n\\begin{align} \\hat{\\Pi}_1\\sin\\bar\\mu_1c_1\\hat{\\Pi}_1\\Psi(\\lambda_1,\\lambda_2,\\lambda_3)\n&= \\frac{1}{2i}\\, \\hat\\Pi_1\\,\\Big[\\Psi(-\\lambda_1+\\frac{1}{|\\lambda_2\\lambda_3|},\\lambda_2,\\lambda_3)-\n\\Psi(-\\lambda_1-\\frac{1} {|\\lambda_2\\lambda_3|},\\lambda_2,\\lambda_3)\\Big] \\nonumber \\\\\n&= \\frac{1}{2i}\\Big[\\Psi(\\lambda_1+\\frac{1}{|\\lambda_2\\lambda_3|},\\lambda_2,\\lambda_3)-\\Psi(\\lambda_1-\n\\frac{1}{|\\lambda_2\\lambda_3|},\\lambda_2,\\lambda_3)\\Big] \\nonumber \\\\ &=\n-\\sin\\bar\\mu_1c_1\\Psi(\\lambda_1,\\lambda_2,\\lambda_3), \\end{align}\nwhence\n\\begin{equation} \\hat{\\Pi}_1 \\sin\\bar\\mu_1c_1\\hat\\Pi_1 = -\\sin\\bar\\mu_1c_1. \\end{equation}\nAn identical calculation shows that\n\\begin{align} \\hat{\\Pi}_2\\sin\\bar\\mu_1c_1\\hat{\\Pi}_2\\,\\Psi(\\lambda_1,\\lambda_2,\\lambda_3)\n&= \\frac{1}{2i}\\, \\hat\\Pi_2\\, \\Big[ \\Psi(\\lambda_1-\\frac{1}{|\\lambda_2\\lambda_3|},\n-\\lambda_2,\\lambda_3)-\\Psi(\\lambda_1+\\frac{1}{|\\lambda_2\\lambda_3|},-\\lambda_2,\\lambda_3)\\Big] \\nonumber\\\\\n&= \\frac{1}{2i}\\Big[\\Psi(\\lambda_1-\\frac{1}{|\\lambda_2\\lambda_3|},\\lambda_2,\\lambda_3)-\n\\Psi(\\lambda_1+\\frac{1}{|\\lambda_2\\lambda_3|},\\lambda_2,\\lambda_3)\\Big] \\nonumber \\\\\n&= \\sin\\bar\\mu_1c_1\\Psi(\\lambda_1,\\lambda_2,\\lambda_3)\\, , \\end{align}\nand similarly for $\\hat\\Pi_3$. Therefore, we have:\n\\begin{equation} \\hat{\\Pi}_2 \\sin\\bar\\mu_1c_1\\hat\\Pi_2 = \\sin\\bar\\mu_1c_1, \\quad\n{\\rm and} \\quad \\hat{\\Pi}_3 \\sin\\bar\\mu_1c_1\\hat{\\Pi}_3 =\n\\sin\\bar\\mu_1c_1.\\end{equation}\nAs expected, these transformation properties of $\\sin\\bar\\mu_1c_1$\nunder $\\hat\\Pi_i$ mirror those of $c_1$ under the three parity\noperations $\\Pi_i$ in the classical theory. (Note that, because of\nthe absolute value signs in the expressions (\\ref{mubar}),\n$\\bar\\mu_i$ do not change under any of the parity maps.)  Finally,\nit is clear from Eq. (\\ref{ve2}) that\n\\begin{equation}  \\hat\\Pi_i\\,\\hat{\\varepsilon}\\,\\hat\\Pi_i = \\left\\{ \\rlap{\\raise2ex\\hbox{\n$\\hat{\\varepsilon}$ if $v=0$,}}{\\lower2ex\\hbox{$-\\hat{\\varepsilon}$ otherwise,}}\n\\right. \\end{equation}\nand from Eq. (\\ref{inv}) that\n\\begin{equation} \\hat\\Pi_i\\,\\widehat{|p_1|^{-1\/4}}\\,\\hat\\Pi_i =\n\\widehat{|p_1|^{-1\/4}}. \\end{equation}\n(Note incidentally that this need not be the case for different\nfactor ordering choices in Eq. (\\ref{inv}).)\n\nWe can now collect these results to study the transformation\nproperty of the Hamiltonian constraint. Consider first the regular\nsubspace $\\mathcal{H}_{\\rm reg}^{\\rm grav}$ of $\\mathcal{H}_{\\rm kin}^{\\rm grav}$ spanned by states which\nhave no support on points with $v=0$. From Eq. (\\ref{qHam1}) it\nfollows that the restriction to $\\mathcal{H}_{\\rm reg}^{\\rm grav}$ of the\ngravitational part of the Hamiltonian constraint is left invariant\nunder $\\hat\\Pi_i$. Since $\\hat{p}_T^2$ is manifestly invariant, on\nthe regular sub-space we have\n\\begin{equation} \\label{Pham} \\hat{\\Pi}_i\\,\\, \\hat{\\mathcal{C}}_H \\,\\,\\hat{\\Pi}_i\n= \\hat{\\mathcal{C}}_H \\end{equation}\nNext, since the gravitational part of the Hamiltonian constraint\nannihilates the states in the singular sub-space (i.e. those with\nsupport only on those points at which $v=0$), we have\n\\begin{equation} \\hat{\\mathcal{C}}_H\\Psi=-\\hbar^2\\partial_T^2\\Psi=\\hat{\\Pi}_i\\,\\,\n\\hat{\\mathcal{C}}_H \\,\\, \\hat{\\Pi}_i\\Psi. \\end{equation}\nThus, the Hamiltonian constraint operator is left invariant by all\nthe parity operators, mirroring the behavior of its classical\ncounterpart.\n\nThis invariance implies that, given any state $\\Psi \\in\n\\mathcal{H}_{\\rm kin}^{\\rm grav}$, the restriction to the positive\noctant of its image under $\\hat{\\mathcal{C}}_{\\rm grav}$ determines\nits image everywhere on $\\mathcal{H}_{\\rm kin}^{\\rm grav}$. This\nproperty simplifies calculations and was used to arrive at the form\nof the Hamiltonian constraint given in (\\ref{qHamfin}).\n\n\n\\end{appendix}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and Statement of Results}\n\nA level $p$ modular function $f(\\tau)$ is a holomorphic function\non the complex upper half-plane which satisfies \\[\nf\\left(\\frac{a\\tau+b}{c\\tau+d}\\right)=f(\\tau)\\text{ for all }\\left(\\begin{smallmatrix}a & b\\\\\nc & d\\end{smallmatrix}\\right)\\in\\Gamma_{0}(p)\\]\n and is meromorphic at the cusps of $\\Gamma_{0}(p)$. Equivalently,\n$f(\\tau)$ is a weakly holomorphic modular form of weight $0$ on\n$\\Gamma_{0}(p)$. Such a function will necessarily have a $q$-expansion\nof the form $f(\\tau)=\\sum_{n=n_{0}}^{\\infty}a(n)q^{n}$, where $q=e^{2\\pi i\\tau}$.\n\nOf particular interest in the study of modular forms is the\nclassical $j$-invariant, $j(\\tau)=q^{-1}+744 + \\sum_{n=1}^{\\infty}\nc(n)q^{n}$, which is a modular function of level $1$. The\ncoefficients $c(n)$ of the $j$-function, like the Fourier\ncoefficients of many other modular forms, are of independent\narithmetic interest; for instance, they appear as dimensions of a\nspecial graded representation of the Monster group.\n\nIn 1949 Lehner showed \\cite{Lehner:1,Lehner:2} that \\[\nc(2^{a}3^{b}5^{c}7^{d}n)\\equiv0\\pmod{2^{3a+8}3^{2b+3}5^{c+1}7^{d}},\\]\n proving that the coefficients $c(n)$ are often highly divisible\nby small primes. Similar results have recently been proven for other\nmodular functions in~\\cite{Griffin}, and for modular forms of level\n1 and small weight in~\\cite{Doud:padic}, \\cite{DoudJenkinsLopez}. It\nis natural to ask whether such congruences hold for the Fourier\ncoefficients of modular functions of higher level, such as those\nstudied by Ahlgren~\\cite{Ahlgren:theta} in his work on Ramanujan's\n$\\theta$-operator.\n\nLehner's results for $j(\\tau)$ are in fact more general; in~\\cite{Lehner:2}\nhe pointed out that for $p=2,3,5,7$, similar congruences hold for\nthe coefficients of level $p$ modular functions which have integral\ncoefficients at both cusps and have poles of order less than $p$\nat the cusp at infinity.\n\nIn this paper, for $p\\in\\{2,3,5,7\\}$, we examine canonical bases for\nspaces of level $p$ modular functions which are holomorphic at the\ncusp $0$. To construct these bases, we introduce the level $p$\nmodular function $\\psi^{(p)}(\\tau)$, defined as \\[\n\\psi^{(p)}(\\tau)=\\left(\\frac{\\eta(\\tau)}{\\eta(p\\tau)}\\right)^{\\frac{24}{p-1}}\n\\text{ where }\\eta(\\tau) =\nq^{\\frac{1}{24}}\\prod_{n=1}^{\\infty}(1-q^{n}).\\] The integer\n$\\frac{24}{p-1}$ for $p=2,3,5,7$ will appear frequently, so we will\ndenote it $\\lambda^{(p)}$, or simply $\\lambda$ where no confusion\narises. The function $\\psi^{(p)}(\\tau)$ is a modular function of\nlevel $p$ with a simple pole at $\\infty$ and a simple zero at 0.  We\nwill also use the modular function \\[\\phi^{(p)}(\\tau) =\n(\\psi^{(p)}(\\tau))^{-1}.\\]\n\nFollowing Ahlgren \\cite{Ahlgren:theta}, and using the notation of\nDuke and Jenkins \\cite{Duke:zeros}, for $p=2,3,5,7$ we construct a\nbasis $\\{f_{0,m}^{(p)}(\\tau)\\}_{m=0}^{\\infty}$ for the space of\nlevel $p$ modular functions which are holomorphic at 0 as follows:\n\\[ f_{0,0}^{(p)}(\\tau)=1,\\]\n\\[f_{0,m}^{(p)}(\\tau)=q^{-m}+O(1) =\n\\psi^{(p)}(\\tau)^{m}-Q(\\psi^{(p)}(\\tau)),\\] where $Q(x)$ is a\npolynomial of degree $m-1$ with no constant term, chosen to\neliminate all negative powers of $q$ in $\\psi^{(p)}(\\tau)^{m}$\nexcept for $q^{-m}$. Since $\\psi^{(p)}(\\tau)$ vanishes at $0$ and\nthe polynomial $Q$ has no constant term, we see that the functions\n$f_{0,m}^{(p)}$ also vanish at $0$ when $m>0$. We write\\[\nf_{0,m}^{(p)}=q^{-m}+\\sum_{n=0}^{\\infty}a_{0}^{(p)}(m,n)q^{n}\\]\n so that for $n\\geq0$, the symbol $a_{0}^{(p)}(m,n)$ denotes the coefficient\nof $q^{n}$ in the $m^{th}$ basis element of level $p$.  Note that\nthe function $f_{0, m}^{(p)}$ corresponds to Ahlgren's $j_m^{(p)}$.\n\nFor an example of some of these functions, consider the case $p=2$:\n\\begin{align*}\nf_{0,1}^{(2)}(\\tau) & =\\psi^{(2)}(\\tau)\\\\\n & =q^{-1}-24+276q-2048q^{2}+11202q^{3}-49152q^{4}+\\ldots\\\\\nf_{0,2}^{(2)}(\\tau) & =\\psi^{(2)}(\\tau)^{2}+48\\psi^{(2)}(\\tau)\\\\\n & =q^{-2}-24-4096q+98580q^{2}-1228800q^{3}+10745856q^{4}+\\ldots\\\\\nf_{0,3}^{(2)}(\\tau) & =\\psi^{(2)}(\\tau)^{3}+72\\psi^{(2)}(\\tau)^{2}+900\\psi^{(2)}(\\tau)\\\\\n & =q^{-3}-96+33606q-1843200q^{2}+43434816q^{3}-648216576q^{4}+\\ldots\\end{align*}\nThe function $f_{0,m}^{(p)}$ is a level $p$ modular function that\nvanishes at $0$ (if $m\\neq0$) and has a pole of order $m$ at\n$\\infty$. The conditions at the cusps determine this function\nuniquely; if two such functions exist, their difference is a\nholomorphic modular function, which must be a constant. Since both\nfunctions vanish at 0, this constant must be $0$.\n\nThe functions comprising these bases for $p=2,3,5,7$ have\ndivisibility properties which bear a striking resemblance to the\ndivisibility properties of $j(\\tau)$; in many cases they are\nidentical. As an example of some of the divisibility properties we\nencounter with this basis, we experimentally examine the $2$-adic\nvaluation of the even indexed coefficients of $f_{0,m}^{(2)}(\\tau)$\nfor $m=1,3,5,7$ in Table \\ref{tab:2-Adic-Table}.  As the data in the\ntable suggest, the $2$-divisibility which $j(\\tau)$ exhibits gives\nus a lower bound on the $2$-divisibility of the odd-indexed $p=2$\nbasis elements.\n\n\n\\begin{table}[h]\n\\noindent \\begin{centering}\n\\label{Flo:2-Adic-Float}\n\\par\\end{centering}\n\\noindent \\begin{centering}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\\hline\n\\multicolumn{1}{|c}{} &  & \\multicolumn{1}{c}{$a_{0}^{(2)}(m,2)$} & \\multicolumn{1}{c}{$a_{0}^{(2)}(m,4)$} & \\multicolumn{1}{c}{$a_{0}^{(2)}(m,6)$} & \\multicolumn{1}{c}{$a_{0}^{(2)}(m,8)$} & \\multicolumn{1}{c}{$a_{0}^{(2)}(m,10)$} & $a_{0}^{(2)}(m,12)$\\tabularnewline\n\\hline\n & $1$ & $11$ & $14$ & $13$ & $17$ & $12$ & $16$\\tabularnewline\n\\cline{2-8}\n & $3$ & $13$ & $16$ & $15$ & $19$ & $14$ & $18$\\tabularnewline\n\\cline{2-8}\n$m$ & $5$ & $12$ & $15$ & $14$ & $18$ & $13$ & $17$\\tabularnewline\n\\cline{2-8}\n & $7$ & $14$ & $17$ & $16$ & $20$ & $15$ & $19$\\tabularnewline\n\\cline{2-8}\n & min & $11$ & $14$ & $13$ & $17$ & $12$ & $16$\\tabularnewline\n\\hline\n\\multicolumn{8}{|c|}{}\\tabularnewline\n\\hline\n$j(\\tau)$ &  & $11$ & $14$ & $13$ & $17$ & $12$ & $16$\\tabularnewline\n\\hline\n\\end{tabular}\n\\par\\end{centering}\n\\noindent \\centering{}\\caption{\\label{tab:2-Adic-Table}$2$-adic\nvaluation of $a_{0}^{(2)}(m,n)$ compared to corresponding\ncoefficients in $j(\\tau)$}\n\\end{table}\n\nNote that these functions form a basis for $M_0^\\infty(p)$, the\nspace of modular forms of weight 0 and level $p$ with poles allowed\nonly at the cusp at $\\infty$.  A full basis for the space $M_0^!(p)$\nof weakly holomorphic modular forms of weight $0$ and level $p$ is\ngenerated by the $f_{0, m}^{(p)}(\\tau)$ and the functions\n$(\\phi^{(p)}(\\tau))^n$ for integers $n\\geq 1$.\n\nRecall that the concluding remarks of Lehner's second paper\n\\cite{Lehner:2} state that the coefficients of certain level $p$\nmodular functions having a pole of order less than $p$ at $\\infty$\nhave the same $p$-divisibility properties as the coefficients $c(n)$\nof $j(\\tau)$. More precisely, we have the following theorem.\n\\begin{thm}[Lehner]\n\\label{thm:Lehner-Main} Let $p\\in\\{2,3,5,7\\}$ and let $f(\\tau)$ be a\nmodular function on $\\Gamma_{0}(p)$ having a pole at $\\infty$ of\norder $<p$ and $q$-series of the form\n\\[f(\\tau)=\\sum_{n=n_{0}}^{\\infty}a(n)q^{n},\\]\n\\[ f(-1\/p\\tau) = \\sum_{m=m_0}^\\infty b(n) q^n,\\]\nwhere $a(n), b(n) \\in\\mathbb{Z}$. Then the coefficients $a(n)$\nsatisfy the following congruence properties:\n\\[\n\\begin{array}{rll}\na(2^{a}n)\\equiv0 & \\pmod{2^{3a+8}} & \\text{if }p=2\\\\\na(3^{a}n)\\equiv0 & \\pmod{3^{2a+3}} & \\text{if }p=3\\\\\na(5^{a}n)\\equiv0 & \\pmod{5^{a+1}} & \\text{if }p=5\\\\\na(7^{a}n)\\equiv0 & \\pmod{7^{a}} & \\text{if }p=7.\\end{array}\\]\n\\end{thm}\nNote that Lehner's original statement of this theorem mistakenly\nstates that a function on $\\Gamma_{0}(p)$ inherits the\n$p$-divisibility property for \\emph{every }prime in $\\{2,3,5,7\\}$,\nnot just the prime matching the level.\n\nA necessary condition in the statement of Lehner's theorem is that\nthe function must have an integral $q$-expansion at $0$. This\ncondition is quite strong; in fact, neither the function\n$\\phi^{(p)}(\\tau)$ nor any of its powers satisfy it, although the\nfunctions $f_{0, m}^{(p)}(\\tau)$ do.\n\nFurther, Lehner's theorem assumes that the order of the pole at\n$\\infty$ must be less than $p$. In this paper, we remove this\nrestriction on the order of the pole to show that every function in\nthe $f_{0,m}^{(p)}$ basis has divisibility properties similar to\nthose in Theorem \\ref{thm:Lehner-Main}.  Specifically, we prove the\nfollowing theorem.\n\\begin{thm}\n\\label{thm:Andersen-Main}Let $p\\in\\{2,3,5,7\\}$, and let \\[\nf_{0,m}^{(p)}(\\tau)=q^{-m}+\\sum\\limits _{n=0}^{\\infty}a_{0}^{(p)}(m,n)q^{n}\\]\n be an element of the basis described above, with $m=p^{\\alpha}m'$\nand $(m',p)=1$. Then, for $\\beta > \\alpha$,\n\\[\n\\begin{array}{rll}\na_{0}^{(2)}(2^{\\alpha}m',2^{\\beta}n)\\equiv0 & \\pmod{2^{3(\\beta-\\alpha)+8}} & \\text{if }p=2\\\\\na_{0}^{(3)}(3^{\\alpha}m',3^{\\beta}n)\\equiv0 & \\pmod{3^{2(\\beta-\\alpha)+3}} & \\text{if }p=3\\\\\na_{0}^{(5)}(5^{\\alpha}m',5^{\\beta}n)\\equiv0 & \\pmod{5^{(\\beta-\\alpha)+1}} & \\text{if }p=5\\\\\na_{0}^{(7)}(7^{\\alpha}m',7^{\\beta}n)\\equiv0 & \\pmod{7^{(\\beta-\\alpha)}} & \\text{if }p=7.\\end{array}\\]\n\\end{thm}\n\nNote that for basis elements $f_{0,m}^{(p)}$ with $(m,p)=1$, these\ndivisibility properties match those in Theorem\n\\ref{thm:Lehner-Main}; in fact, Lehner's proof is easily extended to\nprove the congruences in these cases. For basis elements with\n$m=p^{\\alpha}m'$ and $\\alpha\\ge1$, the divisibility is ``shifted.''\nThis shifting occurs in the $(\\beta-\\alpha)$ factor in the exponent\nof the modulus.\n\nFor the coefficients $a_0^{(p)}(p^\\alpha m', p^\\beta n)$ with\n$\\alpha > \\beta$, computations suggest that similar congruences\nshould hold.  Additionally, it appears that powers of the function\n$\\phi^{(p)}(\\tau)$ have Fourier coefficients with slightly weaker\ndivisibility properties, despite the fact that their Fourier\ncoefficients at $0$ are not integral. It would be interesting to\nmore fully understand these congruences.\n\n\\section{Preliminary Lemmas and Definitions}\n\nIn this section, we provide the necessary definitions and background\nfor the proof of the main theorem.\n\nFor a prime $p$ we define the level $p$ Hecke operator $U_{p}$ by\\[\nU_{p}f(\\tau)=\\frac{1}{p}\\sum_{\\ell=0}^{p-1}f\\left(\\frac{\\tau+\\ell}{p}\\right),\\]\nusing the notation $U_{p}^{n}f=U_{p}U_{p}\\cdots U_{p}f$ for repeated\napplication of $U_{p}$. When $f$ has the Fourier expansion\n$f(\\tau)=\\sum_{n=n_{0}}^{\\infty}a(n)q^{n}$, this operator takes the\nform\\[ U_{p}f(\\tau)=\\sum_{n=n_{0}}^{\\infty}a(pn)q^{n},\\] essentially\n{}``pulling out'' all of the coefficients of $f$ whose index is\ndivisible by $p$. This operator preserves modularity:\nif $f$ is a level $p$ modular function, then $U_{p}f$ is also a level\n$p$ modular function.\n\nFor the primes $p=2,3,5,7$ the topological genus of\n$\\Gamma_{0}(p)\\backslash\\mathcal{H}$ is zero, so the field of level\n$p$ modular functions is generated by a single modular function\ncalled a Hauptmodul.  For the primes in consideration, one such\nfunction is $\\psi^{(p)}(\\tau)$. Note that the modular function\n$\\phi^{(p)}(\\tau)=\\psi^{(p)}(\\tau)^{-1}=q+O(q^{2})$ is also a\nHauptmodul.\n\nFurther, for these primes, the fundamental domain for\n$\\Gamma_{0}(p)$ has precisely two cusps, which may be taken to be at\n$\\infty$ and at $0$. Hence, we are most concerned with the behavior\nof these functions at $\\infty$ and at $0$. To switch between cusps,\nwe make the substitution $\\tau\\mapsto-1\/p\\tau$. The following lemma\ngives relations for $\\psi^{(p)}(\\tau)$ and $\\phi^{(p)}(\\tau)$ at\n$0$, and makes clear that powers of $\\phi^{(p)}$ do not satisfy\nLehner's integrality condition.\n\\begin{lem}\n\\label{lem:Psi-At-0}The functions $\\psi^{(p)}(\\tau)$ and $\\phi^{(p)}(\\tau)$\nsatisfy the relations \\begin{align}\n\\psi^{(p)}(-1\/p\\tau) & =p^{\\lambda\/2}\\phi^{(p)}(\\tau)\\label{eq:Psi-At-0}\\\\\n\\phi^{(p)}(-1\/p\\tau) & =p^{-\\lambda\/2}\\psi^{(p)}(\\tau)\\label{eq:Phi-At-0}\\end{align}\n\\end{lem}\n\\begin{proof}\nThe functional equation for $\\eta(\\tau)$ is\n$\\eta(-1\/\\tau)=\\sqrt{-i\\tau}\\eta(\\tau)$. Using this, we find that \\[\n\\psi^{(p)}\\left(\\frac{-1}{p\\tau}\\right) =\n\\left(\\frac{\\eta(-1\/p\\tau)} {\\eta(-1\/\\tau)}\\right)^{\\lambda} =\n\\left(\\frac{\\sqrt{-ip\\tau}\n \\eta(p\\tau)}{\\sqrt{-i\\tau} \\eta(\\tau)}\\right)^{\\lambda} = (\\sqrt{p})^{\\lambda}\n\\left(\\frac{\\eta(p\\tau)}{\\eta(\\tau)}\\right)^{\\lambda} =\np^{\\lambda\/2}\\phi^{(p)}(\\tau).\\] The second statement follows after\nreplacing $\\tau$ by $-1\/p\\tau$ in the first statement.\n\\end{proof}\nWe next state a well-known lemma which gives a formula for\ndetermining the behavior of a modular function at $0$ after $U_{p}$\nhas been applied. A proof can be found in \\cite{Apostol:modular}.\n\\begin{lem}\n\\label{lem:Main-Up-Formula}Let $p$ be prime and let $f(\\tau)$ be\na level $p$ modular function. Then\\begin{equation}\np(U_{p}f)(-1\/p\\tau)=p(U_{p}f)(p\\tau)+f(-1\/p^{2}\\tau)-f(\\tau).\\label{eq:Main-Up-Formula}\\end{equation}\n\n\\end{lem}\nLehner's original papers included the following lemma and its proof, which gives\ntwo important equations. The first gives a formula for $U_{p}\\phi^{(p)}$\nas a polynomial with integral coefficients in $\\phi^{(p)}$; the second\ngives an algebraic relation which is satisfied by $\\phi^{(p)}(\\tau\/p)$.\n\\begin{lem}\n\\label{lem:Modular-Eq-Phi}Let $p\\in\\{2,3,5,7\\}$. Then there exist\nintegers $b_{j}^{(p)}$ such that\n\n\\[\n\\begin{array}{cc}\n\\text{(a)} & U_{p}\\phi^{(p)}(\\tau)=p\\sum\\limits _{j=1}^{p}b_{j}^{(p)}\\phi^{(p)}(\\tau)^{j}.\\end{array}\\]\n\n\nFurther, let $h^{(p)}(\\tau)=p^{\\lambda\/2}\\phi^{(p)}(\\tau\/p).$ Then\\[\n\\begin{array}{cc}\n\\text{(b)} & \\big(h^{(p)}(\\tau)\\big)^{p}+\\sum\\limits _{j=1}^{p}(-1)^{j}g_{j}(\\tau)\\big(h^{(p)}(\\tau)\\big)^{p-j}=0\\end{array}\\]\n\n\nwhere $g_{j}(\\tau)=(-1)^{j+1}p^{\\lambda\/2+2}\\sum\\limits _{\\ell=j}^{p}b_{\\ell}\\phi^{(p)}(\\tau)^{\\ell-j+1}$.\\end{lem}\n\\begin{proof}\n\n(a) Since $\\phi$ vanishes at $\\infty$, $U_{p}\\phi$ also vanishes\nat $\\infty$; we will now consider its behavior at $0$. Using (\\ref{eq:Main-Up-Formula})\nand replacing $\\tau$ by $p\\tau$ in (\\ref{eq:Psi-At-0}) we obtain\\begin{align*}\nU_{p}\\phi(-1\/p\\tau) & =U_{p}\\phi(p\\tau)+p^{-1}\\phi(-1\/p^{2}\\tau)-p^{-1}\\phi(\\tau)\\\\\n & =U_{p}\\phi(p\\tau)+p^{-1}\\psi(p\\tau)-p^{-1}\\phi(\\tau)\\\\\n & =O(q^{p})+p^{-\\lambda\/2-1}q^{-p}+O(1)-p^{-1}q+O(q^{2})\\\\\np^{\\lambda\/2+1}U_{p}\\phi(-1\/p\\tau) & =q^{-p}+O(1)\\end{align*}\n\n\nThe right side of this equation is a level $p$ modular function with\ninteger coefficients, so we may write it as a polynomial in\n$\\psi(\\tau)$ with integer coefficients. The polynomial will not have\na constant term since the left side vanishes at $0$. Therefore,\\[\np^{\\lambda\/2+1}U_{p}\\phi(-1\/p\\tau)=\\sum_{j=1}^{p}c_{j}\\psi(\\tau){}^{j}.\\]\n\n\nNow, replacing $\\tau$ by $-1\/p\\tau$, we find\\[\np^{\\lambda\/2+1}U_{p}\\phi(\\tau)=\\sum_{j=1}^{p}c_{j}p^{\\lambda j\/2}\\phi(\\tau){}^{j}.\\]\n\n\nAfter cancelling the $p^{\\lambda\/2+1}$, we find that\n$U_{p}\\phi(\\tau)=\\sum\\limits _{j=1}^{p}c_{j}'\\phi(\\tau)^{j}$ and we\ncompute the coefficients $c_{j}'$ (the authors used\n\\noun{mathematica}). The computation is finite, and we find that\neach coefficient $c_{j}'$ has a factor of $p$, so the coefficients\n$b_{j}^{(p)}$ are integral. A complete table of values of the\n$b_{j}^{(p)}$ is found in Table \\ref{tab:b_j-Table}.\n\n\\begin{table}[h]\n\\noindent \\begin{centering}\n\\label{Flo:b_j-Float}\n\\par\\end{centering}\n\n\\noindent \\begin{centering}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\cline{3-6}\n\\multicolumn{1}{c}{} &  & \\multicolumn{4}{c|}{$p$}\\tabularnewline\n\\cline{3-6}\n\\multicolumn{1}{c}{} &  & 2 & 3 & 5 & 7\\tabularnewline\n\\hline\n & 1 & $3\\cdot2^{2}$ & $10\\cdot3^{1}$ & $63\\cdot5^{0}$ & $82\\cdot7^{0}$\\tabularnewline\n\\cline{2-6}\n & 2 & $2^{10}$ & $4\\cdot3^{6}$ & $52\\cdot5^{3}$ & $176\\cdot7^{2}$\\tabularnewline\n\\cline{2-6}\n & 3 &  & $3^{10}$ & $63\\cdot5^{5}$ & $845\\cdot7^{3}$\\tabularnewline\n\\cline{2-6}\n$j$ & 4 &  &  & $6\\cdot5^{8}$ & $272\\cdot7^{5}$\\tabularnewline\n\\cline{2-6}\n & 5 &  &  & $5^{10}$ & $46\\cdot7^{7}$\\tabularnewline\n\\cline{2-6}\n & 6 &  &  &  & $4\\cdot7^{9}$\\tabularnewline\n\\cline{2-6}\n & 7 &  &  &  & $7^{10}$\\tabularnewline\n\\hline\n\\end{tabular}\n\\par\\end{centering}\n\n\\noindent \\centering{}\\caption{\\label{tab:b_j-Table}Values of $b_{j}^{(p)}$}\n\n\\end{table}\n\n\n(b) We again apply (\\ref{eq:Main-Up-Formula}) to $\\phi(\\tau)$, this\ntime using what we know from $(a)$.\\[\npU_{p}\\phi(-1\/p\\tau)=pU_{p}\\phi(p\\tau)+\\phi(-1\/p^{2}\\tau)-\\phi(\\tau)\\]\n\\[\np^{2}\\sum_{j=1}^{p}b_{j}\\phi(-1\/p\\tau)^{j}=p^{2}\\sum_{j=1}^{p}b_{j}\\phi(p\\tau)^{j}+\\phi(-1\/p^{2}\\tau)-\\phi(\\tau).\\]\nWe now use Lemma \\ref{lem:Psi-At-0} with the knowledge that $\\psi(\\tau)=\\phi(\\tau)^{-1}$\nto obtain\n\\[p^{2}\\sum_{j=1}^{p}b_{j}p^{-\\lambda j\/2}\\phi(\\tau)^{-j}-p^{2}\\sum_{j=1}^{p}b_{j}\\phi(p\\tau)^{j}+\\phi(\\tau)-p^{-\\lambda\/2}\\phi(p\\tau)^{-1}=0.\\]\nAfter replacing $\\tau$ by $\\tau\/p$ and multiplying by $p^{\\lambda\/2}$,\nthis becomes\\begin{equation}\np^{\\lambda\/2+2}\\sum_{j=1}^{p}b_{j}\\big(h(\\tau)^{-j}-\\phi(\\tau)^{j}\\big)+h(\\tau)-\\phi(\\tau)^{-1}=0.\\label{eq:Intermediate-Modular-Eq-Phi}\\end{equation}\n\n\nWe now divide by $h^{-1}-\\phi$. Note two facts: \\[\nh^{-j}-\\phi^{j}=(h^{-1}-\\phi)\\sum_{\\ell=0}^{j-1}h^{-\\ell}\\phi^{j-\\ell-1}\\]\n\\[\n\\frac{h-\\phi^{-1}}{h^{-1}-\\phi}=\\frac{h(h\\phi-1)}{\\phi(1-h\\phi)}=-\\frac{h}{\\phi}.\\]\nSo (\\ref{eq:Intermediate-Modular-Eq-Phi}) becomes\\[\np^{\\lambda\/2+2}\\sum_{j=1}^{p}b_{j}\\sum_{\\ell=0}^{j-1}h^{-\\ell}\\phi^{j-\\ell-1}-\\phi^{-1}h=0\\]\nwhich, after multiplying by $\\phi h^{p-1}$, becomes\\[\np^{\\lambda\/2+2}\\sum_{j=1}^{p}b_{j}\\sum_{\\ell=0}^{j-1}h^{p-\\ell-1}\\phi^{j-\\ell}-h^{p}=0.\\]\nWe now change the order of summation and rearrange terms to obtain\nthe desired formula:\\[\nh(\\tau)^{p}=\\sum_{j=1}^{p}\\big(p^{\\lambda\/2+2}\\sum_{\\ell=j}^{p}b_{\\ell}\\phi(\\tau)^{\\ell-j+1}\\big)h(\\tau){}^{p-j}.\\]\n\n\\end{proof}\nThe next lemma states that when you apply $U_{p}$ to a certain type\nof polynomial in $\\phi_{p}$, you get a similar polynomial back which\nhas picked up a power of $p$. The details of this lemma are found in\nboth \\cite{Lehner:1} and \\cite{Lehner:2}, scattered throughout the\nproofs of the main theorems. For our purposes, it will be more\nuseful in the following form.\n\\begin{lem}\n\\label{lem:Phi-Polynomials}Let $p\\in\\{2,3,5,7\\}$ and let $R^{(p)}$\ndenote the set of polynomials in $\\phi^{(p)}$ of the form\\[\nd_{1}\\phi^{(p)}(\\tau)+\\sum_{n=2}^{N}d_{n}p^{\\gamma}\\phi^{(p)}(\\tau)^{n}\\]\n\\[\n\\begin{array}{ll}\n\\text{where }\\gamma= & \\begin{cases}\n8(n-1) & \\text{if }p=2\\\\\n4(n-1) & \\text{if }p=3\\\\\nn & \\text{if }p=5\\\\\nn & \\text{if }p=7.\\end{cases}\\end{array}\\] Then $U_{p}$ maps\n$R^{(p)}$ to $p^{\\delta}R^{(p)}$ where $\\delta=3,2,1,1$ for\n$p=2,3,5,7$, respectively. That is, applying $U_{p}$ to a polynomial\nof the above form yields a polynomial of the same form with an extra\nfactor of $p^{\\delta}$.\\end{lem}\n\\begin{proof}\nConsider the function\\[\nd_{1}U_{p}\\phi(\\tau)+\\sum_{n=2}^{r}d_{n}p^{\\gamma}U_{p}\\phi(\\tau)^{n}.\\]\nFor the first term, Lemma \\ref{lem:Modular-Eq-Phi}(a) shows that\n$U_{p}\\phi(\\tau)\\in p^{\\delta}R_{p}$ since, by inspection, the $b_{j}^{(p)}$\nintegers are divisible by sufficiently high powers of $p$.\n\nFor the remaining terms, we will prove\\begin{equation}\np^{\\gamma}U_{p}\\phi^{n}=p^{\\delta}r\\label{eq:Up-Phi^t}\\end{equation}\nwhere $r\\in R_{p}$. The result will immediately follow.\n\nBy the definition of $U_{p}$ we have\\begin{equation}\nU_{p}\\phi^{n}=p^{-1}\\sum_{\\ell=0}^{p-1}\\phi\\left(\\frac{\\tau+\\ell}{p}\\right)^{n}=p^{-1-\\lambda t\/2}\\sum_{\\ell=0}^{p-1}h_{\\ell}(\\tau)^{n}\\label{eq:Up-Def-W-Sum}\\end{equation}\nwhere $h_{\\ell}(\\tau)=p^{\\lambda\/2}\\phi\\left(\\frac{\\tau+\\ell}{p}\\right)$\nis related to $h$ from Lemma \\ref{lem:Modular-Eq-Phi}(b). Let $S_{n}$\nbe the sum of the $n^{th}$ powers of the $h_{\\ell}$ so that \\[\nS_{n}=\\sum_{\\ell=0}^{p-1}h_{\\ell}^{n}.\\]\n\n\nDefine the polynomial\n$F(x)=\\sum_{j=0}^{p}(-1)^{j}g_{j}(\\tau)x^{p-j}$ where the\n$g_{j}(\\tau)$ are as in Lemma \\ref{lem:Modular-Eq-Phi}. In the same\nlemma, if we replace $\\tau$ with $\\tau+\\ell$, the $g_{j}(\\tau)$ are\nunaffected since $\\phi(\\tau+1)=\\phi(\\tau)$. Therefore, that lemma\ntells us that the $p$ roots of the polynomial $F(x)$ are precisely\nthe $h_{\\ell}$. Using Newton's formula for the $n^{th}$ power sum of\nthe roots of a polynomial, we obtain\\begin{equation}\nS_{n}=\\sum_{\\ell=0}^{p-1}h_{\\ell}^{n}=\\sum_{j=1}^{n}(-1)^{j+1}g_{j}S_{n-j}\\label{eq:Newtons-Formula}\\end{equation}\nwhere $g_{j}=0$ for $j>p$ and $S_{0}=n$.\n\nWe now proceed case-by-case. The $p=2$ case illustrates the method,\nso we will only include the intermediate steps in the $p=3,5,7$ cases.\n\\begin{caseenv}\n\\item $p=2$. Then, using (\\ref{eq:Up-Def-W-Sum}), equation (\\ref{eq:Up-Phi^t}) is\nequivalent to\\[\n2^{8(n-1)}\\big(2^{-1-12n}S_{n}\\big)=2^{3}r,\\text{ or}\\]\n\\begin{equation}\nS_{n}=2^{4n+12}r.\\label{eq:St-Powers-of-2}\\end{equation}\nWe now use (\\ref{eq:Newtons-Formula}) to calculate $S_{1}$ and $S_{2}$:\\[\nS_{1}=g(1)\\]\n\\[\nS_{2}=g_{1}S_{1}-2g_{2}=g_{1}^{2}-2g_{2}.\\]\nFrom Lemma \\ref{lem:Modular-Eq-Phi} we can compute the values of\nthe $g_{j}$. Using the $b_{j}$ values from the table in that lemma,\nwe have\\[\ng_{1}=2^{14}(b_{1}\\phi_{2}+b_{2}\\phi_{2}^{2})=2^{16}(3\\phi_{2}+2^{8}\\phi_{2}^{2})\\]\n\\[\ng_{2}=-2^{14}b_{2}\\phi_{2}=-2^{24}\\phi_{2}.\\]\nWe can now see that \\[\nS_{1}=g_{1}=2^{16}(3\\phi_{2}+2^{8}\\phi_{2}^{2})\\]\n\\[\nS_{2}=2^{32}(3\\phi_{2}+2^{8}\\phi_{2}^{2})^{2}+2^{25}\\phi_{2}=2^{20}(2^{5}\\phi_{2}+2^{12}3^{2}\\phi_{2}^{2}+2^{21}3\\phi_{2}^{3}+2^{28}\\phi_{2}^{4}).\\]\nThus (\\ref{eq:St-Powers-of-2}) is satisfied for $n=1,2$. We proceed\nby induction. Assume (\\ref{eq:St-Powers-of-2}) is satisfied for all\nintegers $<n$. We show that it is satisfied for $n$. For ease of\ncomputation, we introduce the set \\[\nR^{*}=2^{8}R^{(2)}=\\left\\{ \\sum_{i=1}^{m}d_{i}2^{8i}\\phi_{2}^{i}\\big|d_{i}\\in\\mathbb{Z},m\\in\\mathbb{Z}^{+}\\right\\} \\]\nwhich, the reader will notice, is a ring. From (\\ref{eq:Newtons-Formula})\nwe obtain\\[\nS_{n}=g_{1}S_{n-1}-g_{2}S_{n-2}=2^{8}r_{1}^{*}\\cdot2^{4n}r_{2}^{*}+2^{16}r_{3}^{*}\\cdot2^{4(n-1)}r_{4}^{*}=2^{4n+8}r_{5}^{*}=2^{4n+16}r\\]\nwhere $r_{i}^{*}\\in R^{*}$ and $r\\in R^{(2)}$.\n\\item $p=3$. We want to show \\begin{equation}\nS_{n}=3^{2n+7}r\\label{eq:St-Powers-of-3}\\end{equation}\n where $r\\in R^{(3)}$. We compute the $g_{j}$ and $S_{n}$ as follows,\nusing the $b_{j}$ from the table:\\[\n\\begin{array}{ccc}\ng_{1}=3^{9}(3^{9}\\phi_{3}^{3}+3^{5}4\\phi_{3}^{2}+10\\phi_{3}), &\ng_{2}=3^{14}(-3^{4}\\phi_{3}^{2}-4\\phi_{3}), &\ng_{3}=3^{18}\\phi_{3},\\end{array}\\]\n\\[\n\\begin{array}{ccc}\nS_{1}=g_{1}, & S_{2}=g_{1}^{2}-2g_{2}, &\nS_{3}=g_{1}{}^{3}-3g_{1}g_{2}+3g_{3}.\\end{array}\\] From this, we\nobtain\\[\nS_{1}=3^{9}(3^{9}\\phi_{3}^{3}+3^{5}4\\phi_{3}^{2}+10\\phi_{3})\\]\n\\[\nS_{2}=3^{14}(8\\phi_{3}+3^{5}34\\phi_{3}^{2}+3^{9}80\\phi_{3}^{3}+3^{13}68\\phi_{3}^{4}+3^{18}8\\phi_{3}^{5}+3^{25}\\phi_{3}^{6})\\]\n\\[\nS_{3}=3^{19}(\\phi_{3}+3^{5}40\\phi_{3}^{2}+3^{8}1174\\phi_{3}^{3}+3^{15}136\\phi_{3}^{4}+3^{18}581\\phi_{3}^{5}+3^{25}16\\phi_{3}^{6}+3^{27}58\\phi_{3}^{7}+3^{32}4\\phi_{3}^{8}+3^{35}\\phi_{3}^{9})\\]\nwhich proves (\\ref{eq:St-Powers-of-3}) for $n=1,2,3$. For the inductive\nstep, let $R^{*}$ be the ring $3^{4}R^{(3)}$ so that\\begin{align*}\nS_{n} & =g_{1}S_{n-1}-g_{2}S_{n-2}+g_{3}S_{n-3}\\\\\n & =3^{5}r_{1}^{*}3^{2n+1}r_{2}^{*}+3^{10}r_{3}^{*}3^{2n-1}r_{4}^{*}+3^{14}r_{5}^{*}3^{2n-3}r_{6}^{*}\\\\\n & =3^{2n+6}r_{7}^{*}\\\\\n & =3^{2n+10}r\\end{align*}\nwhere $r_{i}^{*}\\in R^{*}$ and $r\\in R^{(3)}$.\n\\item $p=5$. We want\\begin{equation}\nS_{n}=5^{2n+2}r\\label{eq:St-Powers-of-5}\\end{equation}\nwhere $r\\in R^{(5)}$. Computing the $S_{n}$ we find\\[\n\\begin{array}{ccc}\nS_{1}=5^{5}r_{1} & S_{2}=5^{8}r_{2} & S_{3}=5^{10}r_{3}\\end{array}\\]\n\\[\n\\begin{array}{cc}\nS_{4}=5^{13}r_{4} & S_{5}=5^{16}r_{5}\\end{array}\\]\nfor some $r_{1},\\ldots,r_{5}\\in R^{(5)}$. This proves (\\ref{eq:St-Powers-of-5})\nfor $n=1,\\ldots,5$. For the inductive step, let $R^{*}$ be the ring\n$5R^{(5)}$ so that\\begin{align*}\nS_{n} & =g_{1}S_{n-1}-g_{2}S_{n-2}+g_{3}S_{n-3}-g_{4}S_{n-4}+g_{5}S_{n-5}\\\\\n & =5^{4}r_{1}^{*}5^{2n-1}r_{2}^{*}-\\ldots+5^{14}r_{9}^{*}5^{2n-9}r_{10}^{*}\\\\\n & =5^{2n+3}r_{11}^{*}\\\\\n & =5^{2n+4}r\\end{align*}\nwhere $r_{i}^{*}\\in R^{*}$ and $r\\in R^{(5)}$ .\n\\item $p=7$. We want\\begin{equation}\nS_{n}=7^{n+2}r\\label{eq:St-Powers-of-7}\\end{equation}\nwhere $r\\in R^{(7)}$. Computing the $S_{n}$ we find\\[\n\\begin{array}{cccc}\nS_{1}=7^{4}r_{1} & S_{2}=7^{6}r_{2} & S_{3}=7^{7}r_{3} & S_{4}=7^{9}r_{4}\\end{array}\\]\n\\[\n\\begin{array}{ccc}\nS_{5}=7^{11}r_{5} & S_{6}=7^{13}r_{6} & S_{7}=7^{15}r_{7}\\end{array}\\]\nfor some $r_{1},\\ldots,r_{7}\\in R^{(7)}$. This proves (\\ref{eq:St-Powers-of-7})\nfor $n=1,\\ldots,7$. For the inductive step, let $R^{*}$ be the ring\n$7R^{(7)}$ so that\\begin{align*}\nS_{n} & =\\sum\\limits _{i=1}^{7}(-1)^{i+1}g_{i}S_{n-i}\\\\\n & =7^{3}r_{1}^{*}7^{n}r_{2}^{*}-\\ldots+7^{13}r_{13}^{*}7^{n-6}r_{14}^{*}\\\\\n & =7^{n+3}r_{15}^{*}\\\\\n & =7^{n+4}r\\end{align*}\nwhere $r_{i}^{*}\\in R^{*}$ and $r\\in R^{(7)}$ .\n\\end{caseenv}\n\\end{proof}\n\n\\section{Proof of the Theorem}\n\nTo remind the reader of the main result of the paper, we include it\nhere.\n\\begin{thm*}\nLet $p\\in\\{2,3,5,7\\}$, and let $f_{0,m}^{(p)}(\\tau)=q^{-m}+\\sum\na_{0}^{(p)}(m,n)q^{n}$ be an element of the basis described above,\nwith $m=p^{\\alpha}m'$ and $(m',p)=1$. Then, for $\\beta > \\alpha$,\n\\[\n\\begin{array}{rll}\na_{0}^{(2)}(2^{\\alpha}m',2^{\\beta}n)\\equiv0 & \\pmod{2^{3(\\beta-\\alpha)+8}} & \\text{if }p=2\\\\\na_{0}^{(3)}(3^{\\alpha}m',3^{\\beta}n)\\equiv0 & \\pmod{3^{2(\\beta-\\alpha)+3}} & \\text{if }p=3\\\\\na_{0}^{(5)}(5^{\\alpha}m',5^{\\beta}n)\\equiv0 & \\pmod{5^{(\\beta-\\alpha)+1}} & \\text{if }p=5\\\\\na_{0}^{(7)}(7^{\\alpha}m',7^{\\beta}n)\\equiv0 & \\pmod{7^{(\\beta-\\alpha)}} & \\text{if }p=7.\\end{array}\\]\n\\end{thm*}\nThe proof is in three cases. The first illustrates the method for\nthe simplest basis elements, namely those with $(m,p)=1$. The second\ndemonstrates the ``shifting'' property at its first occurence,\n$f_{0,p}^{(p)}$. The third is the general case; it builds\ninductively upon the methods of the first two cases.\n\n\n\\subsection{Case 1: $(m,p)=1$}\n\\begin{proof}\nThis proof is almost identical to Lehner's proof of Theorem 3 in\n\\cite{Lehner:2}, however it applies not only to functions which\nhave poles of order bounded by $p$, but to all basis elements with\n$(m,p)=1$. For ease of notation, let $f(\\tau)=f_{0,m}^{(p)}(\\tau)$.\n\nWe will demonstrate the method with $m=1$, then generalize it to\nall $m$ relatively prime to $p$. First, we will write $U_{p}f(\\tau)$\nas a polynomial in $\\phi(\\tau)$ with integral coefficients, all of\nwhich are divisible by the desired power of $p$. Since $U_{p}$ isolates\nthe coefficients whose index is divisible by $p$, we will have proven\nthe theorem for $\\beta=1$. We will then apply $U_{p}$ repeatedly\nto the polynomial, showing that the result is always another polynomial\nin $\\phi$ with integral coefficients, all of which are divisible\nby the desired higher power of $p$.\n\nConsider the level $p$ modular function $g(\\tau)=pU_{p}f(\\tau)+p^{\\lambda\/2}\\phi(\\tau)$.\nNotice that $g(\\tau)$ is holomorphic at $\\infty$ since both $U_{p}f(\\tau)$\nand $\\phi(\\tau)$ are holomorphic there. The $q$-expansion at $0$\nfor $g(\\tau)$ is given by\\[\ng(-1\/p\\tau)=p(U_{p}f)(-1\/p\\tau)+p^{\\lambda\/2}\\phi(-1\/p\\tau)\\]\nwhich, by Lemmas \\ref{lem:Psi-At-0} and \\ref{lem:Main-Up-Formula} becomes\\[\ng(-1\/p\\tau)=p(U_{p}f)(p\\tau)+f(-1\/p^{2}\\tau)-f(\\tau)+\\psi(\\tau).\\]\nWhen we notice that $f(\\tau)=\\psi(\\tau)$ in this $m=1$ case, we\nobtain\\begin{align*}\ng(-1\/p\\tau) & =p(U_{p}f)(p\\tau)+\\psi(-1\/p^{2}\\tau)-\\psi(\\tau)+\\psi(\\tau)\\\\\n & =p(U_{p}f)(p\\tau)+p^{\\lambda\/2}\\phi(p\\tau),\\end{align*}\nwhich is holomorphic at $\\infty$. Hence, $g(\\tau)$ is a holomorphic\nmodular function on $\\Gamma_{0}(p)$, so it must be constant. Therefore,\n\\begin{equation}\nU_{p}f(\\tau)=c_{0}-p^{\\lambda\/2-1}\\phi(\\tau)\\label{eq:Up-f1=00003DPhi}\\end{equation}\nfor some constant $c_{0}$. The proof is complete for $\\beta=1$.\n\nNote: the prime $13$, having genus zero, would work in this construction;\nhowever, in that case $\\lambda=\\frac{24}{13-1}=2$, so $13^{\\lambda\/2-1}=1$,\nand we gain no new information.\n\nWe now iterate the above process to prove the theorem for $\\beta>1$.\nNotice that \\[\nU_{p}(U_{p}f(\\tau))=c^{(p)}-p^{\\lambda\/2-1}U_{p}\\phi(\\tau).\\] We\nknow from Lemma \\ref{lem:Modular-Eq-Phi} that $U_{p}\\phi$ is a\npolynomial in $\\phi$; in fact, by inspection of the $b_{j}^{(p)}$\nvalues we see that we may write\\begin{align*}\nU_{2}\\phi^{(2)}(\\tau) & =2^{3}\\big(d_{1}^{(2)}\\phi^{(2)}(\\tau)+\\sum_{n=2}^{2}d_{n}^{(2)}2^{8(n-1)}\\phi^{(2)}(\\tau)^{n}\\big)\\\\\nU_{3}\\phi^{(3)}(\\tau) & =3^{2}\\big(d_{1}^{(3)}\\phi^{(3)}(\\tau)+\\sum_{n=2}^{3}d_{n}^{(3)}3^{4(n-1)}\\phi^{(3)}(\\tau)^{n}\\big)\\\\\nU_{5}\\phi^{(5)}(\\tau) & =5\\big(d_{1}^{(5)}\\phi^{(5)}(\\tau)+\\sum_{n=2}^{5}d_{n}^{(5)}5^{n}\\phi^{(5)}(\\tau)^{n}\\big)\\\\\nU_{7}\\phi^{(7)}(\\tau) &\n=7\\big(d_{1}^{(7)}\\phi^{(7)}(\\tau)+\\sum_{n=2}^{7}d_{n}^{(7)}7^{n}\\phi^{(7)}(\\tau)^{n}\\big)\\end{align*}\nfor some integers $d_{n}^{(p)}$. This shows that the second $U_{p}$\niteration is divisible by the correct power of $p$. Further, it\ngives us a polynomial of a suitable form to iterate the process\nusing Lemma \\ref{lem:Phi-Polynomials}. In each of the polynomials\nabove, notice that $U_{p}\\phi(\\tau)=p^{\\delta}r$ for some $r\\in\nR^{(p)}$. Using Lemma \\ref{lem:Phi-Polynomials}, we find that\\[\nU_{p}(U_{p}\\phi)(\\tau)=p^{2\\delta}r'\\] for some $r'\\in R^{(p)}$, and\nfurther\\[ U_{p}^{\\beta}\\phi(\\tau)=p^{\\beta\\delta}r^{(\\beta)}\\] for\nsome $r_{\\beta}\\in R^{(p)}$. This completes the proof for $m=1$.\n\nNow, if $(m,p)=1$, then $U_{p}f(\\tau)$ is holomorphic at $\\infty$,\njust as it was with $m=1$. Moving to the cusp at $0$ we find\nthat $(U_{p}f)(-1\/p\\tau)$ can be written as a polynomial in $\\psi(\\tau)$\nwhich appears as a polynomial in $\\phi(\\tau)$ when we return to $\\infty$.\nSimilar to (\\ref{eq:Up-f1=00003DPhi}), we obain the equality\\[\nU_{p}f(\\tau)=c_{0}+\\sum_{i=1}^{M}p^{\\lambda i\/2-1}c_{i}\\phi(\\tau)^{i}\\]\nfor some $c_{i}\\in\\mathbb{Z}$ and $M\\in\\mathbb{Z}^{+}$. The only\ndifference between this equation and (\\ref{eq:Up-f1=00003DPhi}) is\nthat in this more general case, we find that $U_{p}f$ is a higher\ndegree polynomial in $\\phi$. This formula can easily be iterated\nas before to obtain the desired result.\n\\end{proof}\n\n\\subsection{Case 2: $m=p$}\n\\begin{proof}\nAgain, for ease of notation, denote $f_{0,p}^{(p)}(\\tau)$ by\n$f(\\tau)$. For the $m=p$ case, we will proceed as before; however,\nwe will find that $U_{p}f(\\tau)$ has poles at both $\\infty$ and $0$,\nand that $U_{p}f(\\tau)$ does not possess any interesting\ndivisibility properties, but $U_{p}^{2}f(\\tau)$ does. This property\nwill manifest itself as the {}``shifting'' previously mentioned.\n\nNotice first that $U_{p}f(\\tau)=q^{-1}+O(1)$ has a simple pole at\n$\\infty$. Therefore, we shall deal primarily with the function $U_{p}f(\\tau)-\\psi(\\tau)$,\nwhich is holomorphic at $\\infty$. We can use Lemmas \\ref{lem:Psi-At-0}\nand \\ref{lem:Main-Up-Formula} to view this function at $0$:\n\n\\begin{align*}\np(U_{p}f)(-1\/p\\tau)-p\\psi(-1\/p\\tau) & = p(U_{p}f)(p\\tau)+f(-1\/p^{2}\\tau)-f(\\tau)-p^{\\lambda\/2+1}\\phi(\\tau) \\\\\n & =pq^{-p}+O(1)+O(1)-q^{-p}+O(1)+O(q)\\\\\n & =c_{0}+\\sum_{i=1}^{p}c_{i}\\psi(\\tau)^{i}\n\\end{align*}\nfor some integers $c_{i}$. Replacing $\\tau$ by $-1\/p\\tau$, we obtain\n\\begin{equation}\n(U_{p}f)(\\tau)=\\frac{c_{0}}{p}+\\psi(\\tau)+\\sum_{i=1}^{p}c_{i}p^{\\lambda\ni\/2-1}\\phi(\\tau)^{i}.\\label{eq:Case-2-Upf}\n\\end{equation}\n\n\nThe $\\psi(\\tau)$ term in the equation makes any attempt at\n$p$-divisibility fail; for example, computation shows that the\n$7^{th}$ coefficient of $\\psi^{(2)}(\\tau)$ is odd. However,\n$\\psi(\\tau)$ satisfies Lehner's divisibility properties, so $U_{p}f$\ninherits its $p$-divisibility from $\\psi(\\tau)$. So the function\\[\nU_{p}^{2}f(\\tau)=c_{0}+U_{p}\\psi(\\tau)+\\sum_{i=1}^{p}c_{i}p^{\\lambda\ni\/2-1}U_{p}\\phi(\\tau)^{i}\\] has the same $p$-divisibility as\n$f_{0,1}^{(p)}$; hence, the shift.\n\\end{proof}\n\n\\subsection{Case 3: $m=p^{\\alpha}m'$}\n\\begin{proof}\nWe prove this case using induction on $\\alpha$. Case 1 showed that the theorem is true for\nall $m'$ relatively prime to $p$, so the $\\alpha=0$ base case is complete. Assume Theorem\n\\ref{thm:Andersen-Main}\nholds for all $m$ of the form $m=p^{\\ell}m'$ with $\\ell<\\alpha$.\nWe will show it holds for $m=p^{\\alpha}m'$.\nTo simplify notation,\nlet $f_{\\alpha}(\\tau)=f_{0,p^{\\alpha}m'}^{(p)}(\\tau)$.\n\nSince $f_{\\alpha}(\\tau)=q^{-p^{\\alpha}m'}+O(1)$, we find that\n$U_{p}f_{\\alpha}(\\tau)=q^{-p^{\\alpha-1}m'}+O(1)$ has a pole of order\n$p^{\\alpha-1}m'$ at $\\infty$. So we focus our attention on\n$U_{p}f_{\\alpha}(\\tau)-f_{\\alpha-1}(\\tau)$, which is holomorphic at\n$\\infty$. Using (\\ref{eq:Main-Up-Formula}) we examine this function\nat $0$:\\begin{align*}\np(U_{p}f_{\\alpha})\\left(\\frac{-1}{p\\tau}\\right) -\npf_{\\alpha-1}\\left(\\frac{-1}{p\\tau}\\right) &\n=p(U_{p}f_{\\alpha})(p\\tau)+f_{\\alpha}\\left(\\frac{-1}{p^{2}\\tau}\\right)\n-\nf_{\\alpha}(\\tau)-pf_{\\alpha-1}\\left(\\frac{-1}{p\\tau}\\right)\\\\\n & =pq^{-p^{\\alpha}m'}+O(1)+O(1)-q^{-p^{\\alpha}m'}-O(1)-O(1)\\\\\n & =(p-1)q^{-p^{\\alpha}m'}+O(1).\\end{align*}\n\n\nAs before, we write this function as a polynomial in $\\psi(\\tau)$\nwith integral coefficients $c_{i}$:\\[\np(U_{p}f_{\\alpha})(-1\/p\\tau)-pf_{\\alpha-1}(-1\/p\\tau)=c_{0}+\\sum_{i=1}^{p^{\\alpha}m'}c_{i}\\psi(\\tau)^{i},\\]\nwhich, after switching back to the $q$-expansion at $\\infty$, becomes\\begin{equation}\nU_{p}f_{\\alpha}(\\tau)=\\frac{c_{0}}{p}+f_{\\alpha-1}(\\tau)+\\frac{1}{p}\\sum_{i=1}^{p^{\\alpha}m'}c_{i}p^{\\lambda i\/2}\\phi(\\tau)^{i}.\\label{eq:Case-3-Upf}\\end{equation}\n\n\nNotice that (\\ref{eq:Case-3-Upf}) looks very similar to\n(\\ref{eq:Case-2-Upf}), where $\\psi(\\tau)$ is replaced by\n$f_{\\alpha-1}(\\tau)$, so $U_{p}f_{\\alpha}(\\tau)$ inherits whatever\ndivisibility properties $f_{\\alpha-1}(\\tau)$ has. Our inductive\nhypothesis states that $f_{\\alpha-1}(\\tau)$ exhibits Lehner's\ndivisibility properties only after $U_{p}$ is applied $\\alpha-1$\ntimes. Therefore, applying $U_{p}$ to (\\ref{eq:Case-3-Upf})\n$\\alpha-1$ times, we obtain\\[\nU_{p}^{\\alpha}f_{\\alpha}(\\tau)=\\frac{c_{0}}{p}+U_{p}^{\\alpha-1}f_{\\alpha-1}(\\tau)\n+\\frac{1}{p}\\sum_{i=1}^{p^{\\alpha}m'}c_{i}p^{\\lambda\ni\/2}U_{p}^{\\alpha-1}\\phi(\\tau)^{i}\\] showing that\n$U_{p}^{\\alpha}f_{\\alpha}(\\tau)$ exhibits Lehner's divisibility\nproperties.\n\\end{proof}\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction\\label{Introduction}}\n\nThe radar head echo is a signal that reflects from the plasma surrounding the\nfast-descending meteoroid and is doppler-shifted by approximately the\nmeteoroid velocity. Only a small volume of the dense plasma sufficiently close\nto the meteoroid contributes to the corresponding radar wave reflection.\nQuantitative knowledge of the spatial structure of the near-meteoroid plasma\nis crucial for accurate modeling the head echo radar reflections\n\\citep{Bronshten:Physics83,Ceplecha:Meteor1998,Close:NewMethod2005,Campbell-Brown:Meteoroid2007}.\n\nIn the companion paper \\citep{Dimant:Formation1_17}, hereinafter referred to as\nPaper~1, we developed a first-principle kinetic theory of the plasma formed\naround a small meteoroid as it moves through the atmosphere at hypersonic\nspeeds. Using a number of easily justified assumptions, we obtained\napproximate analytic expressions describing velocity distributions of meteoric\nions and neutrals. In this paper, we calculate the spatial structure of the\nplasma density that follows from the kinetic theory developed in Paper~1. This\ncalculation demonstrates that this spatial structure differs dramatically from\na simple Gaussian or exponential distribution currently employed for modeling\nradar wave scattering from the meteor plasma \\citep{Close:NewMethod2005,Marshall:FDTD15}. \nThis research does not describe the distribution of\nplasma or neutrals in the meteoroid tail where particles lag well behind the\nmeteoroid after having collided more than once.\n\nSimple analysis of individual collisions between particles indicates that\nheavy meteoric particles in the near-meteor sheath consist predominantly of\nthe `primary' and `secondary' particles. By a primary particle we mean an\nablated meteoroid particle that moves freely with a ballistic trajectory until\nit collides with an atmospheric molecule. These primary particles are\npredominantly neutral. A secondary particle is a former primary particle that\nexperienced exactly one collision, either scattering or ionizing. Most of the\nnear-meteoroid ions responsible for head echoes belong to the group of\nsecondary particles. The vast majority of ions that experienced multiple\ncollisions since the original ablation lag behind the fast-moving meteoroid\nand form a long-lived extended column of plasma visible to radars through\nspecular or non-specular echoes.\n\nGiven the velocity distributions developed in Paper 1 as a function of spatial\ncoordinates, one can integrate over velocity variables to find the\ncorresponding particle density. However, the complexity of these analytic\nexpressions makes this non-trivial. This paper makes an additional simplifying\nassumption about the collision model (the isotropic differential cross-section\nof ionization) and then integrates over the velocities to obtain the meteor\ngas and plasma density as a function of distance from the meteoroid.\n\nThe paper is organized as follows.\nSection~\\ref{Summary of the ion distribution function} summarizes the results\nof Paper~1 on the ion distribution function.\nSection~\\ref{Plasma density calculations} performs the calculations of the\nnear-meteoroid plasma density. Section~\\ref{Discussion} discusses implications\nof our theory and some caveats. Section~\\ref{Conclusions} lists the major\nunderlying assumptions and discusses the paper results.\n\n\\section{Summary of the ion distribution\nfunction\\label{Summary of the ion distribution function}}\n\nPaper~1 does all our calculations in the rest frame of a meteoroid moving\nthrough the atmosphere with the local velocity $-\\vec{U}$, so that in this\nframe the impinging atmospheric particles move with the opposite velocity,\n$\\vec{U}$. We define the coordinate system with the major axis passing through\nthe meteoroid center and parallel to $\\vec{U}$. Due to the axial symmetry\nabout $\\vec{U}$, we characterize the real space by two spherical coordinates:\nthe radial distance from the meteoroid center, $R$, and the polar angle,\n$\\theta$, measured from the major axis ($\\theta=0$ corresponds to the major\nsemi-axis behind the meteoroid, while $\\theta=\\pi$ corresponds to the opposite\nsemi-axis in front of it). Figure~\\ref{Fig:Cartoon_reproduced}, reproduced\nfrom Paper~1, explains all relevant notations.\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=30pc]{Cartoon_5_20_2015.pdf}\n\\caption{Nomenclature of spatial coordinates and velocity variables. The\nspatial variables $R=|\\vec{R}|$, $\\theta$, $\\varphi$ denote the radius and two\nangles of the spherical coordinate system with the origin at the meteoroid\ncenter and the major axis anti-parallel to the meteoroid velocity (shown on\nthe left). All other variables pertain to the particle velocity space:\n$V=|\\vec{V}|$ is the particle speed, $\\vartheta$ is the polar angle of\n$\\vec{V}$ with respect to the local axis parallel to $\\vec{U}$, $\\Phi$ is the\naxial angle measured from the common $\\vec{U}$-$\\vec{R}$ plane; $\\Theta$ is\nthe polar angle of $\\vec{V}$ with respect to the local radial distance\n$\\vec{R}$.}\n\\label{Fig:Cartoon_reproduced}\n\\end{figure}\n\n\nThe velocity distribution of secondary ions, $f^{(2)}$, is expressed as a\nfunction of three velocity variables that are invariants of the ion\ncollisionless motion. These variables include the ion speed, $V$, the cosine\nof the angle between the ion velocity vector $\\vec{V}$ and $\\vec{U}$,\n$\\mu=\\cos\\vartheta$, and a normalized angular momentum variable, $R_{0}$,\nwhich to the minimum distance between the ion trajectory and the meteoroid\ncenter, $R_{0}=R\\sin\\Theta$, where $\\Theta$ is the polar angle of $\\vec{V}$\nwith respect to the local radius vector $\\vec{R}$. The entire set of\nvelocity-space variables also includes a discrete variable $\\sigma_{R}$ which\ntakes two values, $\\pm1$, depending on the sign of the particle radial\nvelocity,%\n\\begin{equation}\nV_{R}=\\frac{dR}{dt}\\equiv V\\cos\\Theta=\\sigma_{R}\\sqrt{1-\\frac{R_{0}^{2}}%\n{R^{2}}}\\ V. \\label{V_r_snova}%\n\\end{equation}\nThe value of $\\sigma_{R}=+1$ corresponds to the outgoing particles, $V_{R}>0$,\nwhile $\\sigma_{R}=-1$ corresponds to the incoming particles, $V_{R}<0$. At any\nlocation, the entire distribution function is given by a sum of the two\ncorresponding functions,%\n\\begin{equation}\nf^{(2)}(V,\\mu,R_{0};R,\\theta)=\\sum_{\\sigma_{R}=\\pm1}f_{\\sigma_{R}}^{(2)}%\n(V,\\mu,R_{0};R,\\theta). \\label{sum_f}%\n\\end{equation}\nThe functions $f_{\\sigma_{R}}^{(2)}$ are non-zero provided $\\mu=\\cos\n\\vartheta>0$; otherwise $f_{\\sigma_{R}}^{(2)}=0$,%\n\\begin{align}\n&  \\left.  f_{\\sigma_{R}}^{(2)}(V,R,\\theta)\\right\\vert _{\\mu>0}=L_{\\sigma_{R}%\n}\\ \\delta\\left(  V-\\frac{2m_{\\beta}\\mu U}{m+m_{\\beta}}\\right)  ,\\nonumber\\\\\nL_{\\sigma_{R}}  &  =\\frac{G_{\\mathrm{ion}}(U,1-2\\mu^{2})n_{0}n_{\\mathrm{A}}%\n}{\\sqrt{3}\\ \\mu U^{2}}\\left(  1+\\frac{m}{m_{\\beta}}\\right)  ^{3}%\nI(R,R_{0}),\\label{f^(2)_via_I}\\\\\n&  \\left.  f_{\\sigma_{R}}^{\\left(  2\\right)  }(V,R,\\theta)\\right\\vert _{\\mu\n<0}=0.\\nonumber\n\\end{align}\nThe quantities $n_{0}$ and $n_{\\mathrm{A}}$ are the densities of the ablated\nparticles at the meteoroid surface and of the atmospheric particles at a given\naltitude, respectively. The quantity $G_{\\mathrm{ion}}(U,1-2\\mu^{2})$\noriginates from the differential cross-section of ionizing collisions,\n$G_{\\mathrm{ion}}$, expressed as a function of the relative speed between the\ntwo colliding particles, $u=\\left\\vert \\vec{u}\\right\\vert $, and the cosine of\nthe scattering angle, $\\Theta_{\\mathrm{sc}}$. In this paper, we simplify our\ntreatment by assuming $G_{\\mathrm{ion}}$ to be a function of only $u\\approx\nU$. The corresponding angular dependence in the relevant energy range is\ngenerally unknown, but the assumption of isotropic $G_{\\mathrm{ion}}(U)$ is reasonable.\n\nThe condition $\\mu>0$ is fulfilled if either\n\\begin{subequations}\n\\label{sgn}%\n\\begin{align}\n&  \\sigma_{R}=\\mathrm{sgn}(\\cos\\theta)\\qquad\\text{and}\\qquad0<R_{0}%\n<R_{c}(\\theta,\\Phi)\\label{sigma=sgn}\\\\\n\\text{or}  & \\nonumber\\\\\n&  \\sigma_{R}=-\\mathrm{sgn}(\\cos\\theta)\\qquad\\text{and}\\qquad R_{c}%\n(\\theta,\\Phi)<R_{0}<R, \\label{sigma=-sgn}%\n\\end{align}\nwhere $\\mathrm{sgn}(x)$ means the sign of $x$ and%\n\\end{subequations}\n\\begin{equation}\nR_{c}(\\theta,\\Phi)=\\frac{R\\left\\vert \\cos\\theta\\right\\vert }{\\sqrt{1-\\sin\n^{2}\\theta\\sin^{2}\\Phi}}. \\label{R_c(Phi)}%\n\\end{equation}\nHere $\\Phi$ is the axial angle of the particle velocity $\\vec{V}$ around the\ndirection of the local radius-vector $\\vec{R}$ (see\nFigure~\\ref{Fig:Cartoon_reproduced}) and we set the origin $\\Phi=0$ where\n$\\vec{V}$ lies in the common $\\vec{R}$-$\\vec{U}$ plane.\n\nIn this paper, we consider the meteor plasma located not too close to the\nmeteoroid, $R\\gg r_{\\mathrm{M}}$. The corresponding multiplier $I(R,R_{0})$ in\nequation~(\\ref{f^(2)_via_I}) has the following piecewise definition:%\n\\begin{equation}\nI(R,R_{0})=\\left\\{\n\\begin{array}\n[c]{ccc}%\nJ_{R}^{\\infty} & \\text{for} & \\sigma_{R}=-1,\\\\\n&  & \\\\\nJ_{R_{0}}^{\\infty}+J_{R_{0}}^{R}=2J_{R_{0}}^{\\infty}-J_{R}^{\\infty} &\n\\text{for} & \\sigma_{R}=+1,\n\\end{array}\n\\right.  \\label{III}%\n\\end{equation}\nwhere, under constraints of $R,R_{0}>3r_{\\mathrm{M}}$, the well-convergent\nintegral $J_{a}^{b}$, taken as a function of its integration limits, $b>a\\geq\nR_{0}$, is given by%\n\\begin{equation}\nJ_{a}^{b}\\approx r_{\\mathrm{M}}^{2}\\int_{a}^{b}\\left[  1+\\left(\n\\frac{R^{\\prime}}{\\lambda_{T}^{(1)}}\\right)  ^{2\/3}\\right]  \\exp\\left[\n-\\ \\frac{3}{2}\\left(  \\frac{R^{\\prime}}{\\lambda_{T}^{(1)}}\\right)\n^{2\/3}\\right]  \\frac{dR^{\\prime}}{R^{\\prime}\\sqrt{\\left(  R^{\\prime}\\right)\n^{2}-R_{0}^{2}}}. \\label{J_a^b_R>3epsilon}%\n\\end{equation}\nHere $\\lambda_{T}^{(1)}$ is the mean free path of the primary (ablated)\nparticles,%\n\\begin{equation}\n\\lambda_{T}^{(1)}=\\frac{V_{T}}{\\nu_{T}^{(1)}},\\qquad V_{T}=\\left(  \\frac\n{T_{M}}{m_{\\mathrm{M}}}\\right)  ^{1\/2},\\qquad\\nu_{T}^{(1)}\\approx2\\pi\nn_{\\mathrm{A}}U\\int_{-1}^{1}G^{\\left(  1\\right)  }(U,\\Lambda)d\\Lambda,\n\\label{lambda_T^(1)}%\n\\end{equation}\nwhere $T_{M}$ and $m_{\\mathrm{M}}$ are the temperature and mass of the primary\nmeteor particles. The quantity $G^{\\left(  1\\right)  }(U,\\Lambda)$ includes\nall collisions that result in scattering of the primary neutral particles. The\nexpression for $G^{\\left(  1\\right)  }$, as that for $G_{\\mathrm{ion}}$, takes\ninto account that $V_{T}\\ll U$, so that the collision frequency $\\nu_{T}%\n^{(1)}$ depends only on the meteoroid speed, $U$, and hence is the same for\nall particles. This reduces $\\lambda_{T}^{(1)}$ to a constant value which\nbecomes the characteristic length-scale of the near-meteoroid plasma.\n\nThe general integral $J_{a}^{b}$ cannot be calculated exactly, but the\nparticular integral $J_{R_{0}}^{\\infty}$ has an almost perfect analytic\napproximation,%\n\\begin{equation}\nJ_{R_{0}}^{\\infty}\\approx\\frac{\\pi r_{\\mathrm{M}}^{2}}{2R_{0}}\\sqrt{1+\\frac\n{2}{\\pi}\\left(  \\frac{R_{0}}{\\lambda_{T}^{(1)}}\\right)  ^{2\/3}}\\ \\exp\\left[\n-\\ \\frac{3}{2}\\left(  \\frac{R_{0}}{\\lambda_{T}^{(1)}}\\right)  ^{2\/3}\\right]  ,\n\\label{J_R_0_interpo}%\n\\end{equation}\naccurate within $1\\%$ for all $R_{0}$. As we demonstrate below, depending on\nthe specific calculation, it may become beneficial to use either the exact\noriginal integral expression for $J_{a}^{b}$ given by\nequation~(\\ref{J_a^b_R>3epsilon}) or (only for $J_{R_{0}}^{\\infty}$) its\napproximation given by equation~(\\ref{J_R_0_interpo}).\n\nFor local calculations of the ion density it is more convenient to pass from\nthe invariant velocity variables $V,R_{0},\\mu$ to local variables\n$V,R_{0},\\Phi$, where $\\Theta$ and $\\Phi$ are the polar and axial angles of\nthe ion velocity about the direction of the local radius-vector $\\vec{R}$, as\ndepicted by Figure~\\ref{Fig:Cartoon_reproduced}.\n\n\\section{Plasma density calculations\\label{Plasma density calculations}}\n\nRadar head echo is determined by the spatial distribution of the electron\ndensity around the meteoroid. The near-meteoroid plasma is quasi-neutral, so\nthat the electron density almost equals that of ions, $n_{e}\\approx n_{i}=n$.\nWe calculate the spatial distribution of the ion density based on the\ndistribution function explained in\nsection~\\ref{Summary of the ion distribution function}.\n\nThe ion density can be easily calculated in the far region of $R\\gg\\lambda\n_{T}^{(1)}$. Albeit less simple, but $n^{(2)}(R,\\theta)$ can also be\nexplicitly calculated in the opposite limit of $R\\ll\\lambda_{T}^{(1)}$. In the\nentire space of arbitrary $R$, we were unable to find a unified purely\nalgebraic expression for $n^{(2)}(R,\\theta)$. However, we have reduced the\ngeneral 3D velocity-space integral to a much simpler expression for $n^{(2)}$\nin terms of normalized variables and parameters, as explained below. This\nuniversal expression involves only treatable analytical functions and two 1D\nintegral functions suitable for simple numerical integration and tabulation.\nThe resultant universal expression for $n^{(2)}$ makes the future analysis and\ncomputer modeling of the radar signal much easier.\n\n\\subsection{Preliminary remarks}\n\nAt a given location determined by $R$ and $\\theta$, the ion density is given\nby $n^{\\left(  2\\right)  }=\\sum_{\\sigma_{R}=\\pm1}\\int f_{\\sigma_{R}}%\n^{(2)}V^{2}dVd\\Omega$, where $d\\Omega=d\\left(  \\cos\\Theta\\right)  d\\Phi$\ndenotes the elementary volume of the local solid angle. Choosing instead of\n$\\Theta$ the new variable $R_{0}=R\\sin\\Theta$, we obtain%\n\\begin{equation}\nn^{\\left(  2\\right)  }=\\frac{1}{R}\\sum_{\\sigma_{R}=\\pm1}\\int_{0}^{R}%\n\\frac{R_{0}dR_{0}}{\\sqrt{R^{2}-R_{0}^{2}}}\\int_{0}^{2\\pi}d\\Phi\\int_{0}%\n^{\\infty}f_{\\sigma_{R}}^{(2)}V^{2}dV. \\label{n^(2)_snova}%\n\\end{equation}\n\n\nFirst, we integrate over $V$ to eliminate the $\\delta$-function in equation\n(\\ref{f^(2)_via_I}),%\n\\begin{equation}\n\\int_{0}^{\\infty}f_{\\sigma_{R}}^{(2)}V^{2}dV=\\frac{4\\mu G_{\\mathrm{ion}}%\nn_{0}n_{\\mathrm{A}}}{\\sqrt{3}}\\left(  1+\\frac{m}{m_{\\beta}}\\right)\nI(R,R_{0}). \\label{yields}%\n\\end{equation}\nAs a result of this simple integration, the previously singular factor $\\mu$\nin the expression for $L_{\\sigma_{R}}$ has moved from its denominator to the\nnumerator, reducing dramatically the relative contribution of the\n`small-angle' ($\\Theta_{\\mathrm{sc}}=1-2\\mu^{2}\\approx1$, $\\mu\\ll1$)\nionization where some assumptions of our general theory are invalid\n\\citep{Dimant:Formation1_17}. Since we assumed above isotropic $G_{\\mathrm{ion}%\n}=G_{\\mathrm{ion}}(U)$, the only $\\Phi$-dependent quantity in the right-hand\nside (RHS) of equation (\\ref{yields}) is $\\mu$ in the numerator. This variable\nis expressed in terms of $R_{0}$ and $\\Phi$ as%\n\\begin{equation}\n\\mu=\\sigma_{R}\\sqrt{1-\\frac{R_{0}^{2}}{R^{2}}}\\cos\\theta+\\frac{R_{0}\\sin\n\\theta}{R}\\ \\cos\\Phi. \\label{mumumu}%\n\\end{equation}\nThe function $I(R,R_{0})$, along with the corresponding integral expressions\nfor $J_{a}^{b}$, are described by equations~(\\ref{III}) to\n(\\ref{J_R_0_interpo}).\n\n\\subsection{Long-distance asymptotics, $R\\gg\\lambda_{T}^{(1)}$, behind the\nmeteoroid\\label{Long-distance asymptotics}}\n\nWe start by calculating the ion density at the simplest limit of long radial\ndistances, $R\\gg\\lambda_{T}^{(1)}$, behind the meteoroid. Ignoring\nexponentially small densities (as explained below), we will consider only the\nspace behind the meteoroid, $\\cos\\theta>0$. Outgoing particles within the\ndominant beam-like (along $\\vec{R}$) velocity distribution make the major\ncontribution to $n^{\\left(  2\\right)  }$. In equations (\\ref{n^(2)_snova}) and\n(\\ref{yields}), setting $R_{0}\\ll R$, $\\vartheta\\approx\\theta$, $\\mu\n\\approx\\cos\\theta$, while neglecting the exponentially small quantity\n$J_{R}^{\\infty}$ in equation~(\\ref{III}), we obtain $I(R,R_{0})\\approx\n2J_{R_{0}}^{\\infty}$. This allows us to easily integrate the RHS of\nequation~(\\ref{n^(2)_snova}) over $\\Phi$. In the exact integral expression\ngiven by (\\ref{J_a^b_R>3epsilon}), the primary contribution to $J_{R_{0}%\n}^{\\infty}$ arises from components of $f_{+}^{(2)}$ near the meteoroid,\n$R^{\\prime}\\lesssim\\lambda_{T}^{(1)}\\ll R$. This allows us to extend the upper\nintegration limit to infinity. This yields%\n\\begin{align}\n&  \\left.  n^{\\left(  2\\right)  }\\right\\vert _{R\\gg\\lambda_{T}^{(1)}}%\n\\approx\\frac{16\\pi r_{\\mathrm{M}}^{2}G_{\\mathrm{ion}}n_{0}n_{\\mathrm{A}}%\n\\cos\\theta}{R^{2}\\sqrt{3}}\\left(  1+\\frac{m}{m_{\\beta}}\\right) \\nonumber\\\\\n&  \\times\\int_{0}^{\\infty}R_{0}dR_{0}\\int_{R_{0}}^{\\infty}\\left[  1+\\left(\n\\frac{R^{\\prime}}{\\lambda_{T}^{(1)}}\\right)  ^{2\/3}\\right]  \\exp\\left[\n-\\ \\frac{3}{2}\\left(  \\frac{R^{\\prime}}{\\lambda_{T}^{(1)}}\\right)\n^{2\/3}\\right]  \\frac{dR^{\\prime}}{R^{\\prime}\\sqrt{\\left(  R^{\\prime}\\right)\n^{2}-R_{0}^{2}}}. \\label{n^(2)_long_start}%\n\\end{align}\nChanging here the order of integration with the corresponding adjustment of\nthe integration limits, $\\int_{0}^{\\infty}dR_{0}\\int_{R_{0}}^{\\infty}\\left(\n\\cdots\\right)  dR^{\\prime}=\\int_{0}^{\\infty}dR^{\\prime}\\int_{0}^{R^{\\prime}%\n}\\left(  \\cdots\\right)  dR_{0}$, and using the simple identities\n\\begin{align*}\n\\int_{0}^{R^{\\prime}}\\frac{R_{0}dR_{0}}{\\sqrt{\\left(  R^{\\prime}\\right)\n^{2}-R_{0}^{2}}}  &  =R^{\\prime},\\\\\n\\int_{0}^{\\infty}\\left[  1+\\left(  \\frac{R^{\\prime}}{\\lambda_{T}^{(1)}%\n}\\right)  ^{2\/3}\\right]  \\exp\\left[  -\\ \\frac{3}{2}\\left(  \\frac{R^{\\prime}%\n}{\\lambda_{T}^{(1)}}\\right)  ^{2\/3}\\right]  dR^{\\prime}  &  =\\sqrt{\\frac{2\\pi\n}{3}}\\ \\lambda_{T}^{(1)},\n\\end{align*}\nwe obtain for $\\cos\\theta>0$:\n\\begin{equation}\n\\left.  n^{\\left(  2\\right)  }\\right\\vert _{R\\gg\\lambda_{T}^{(1)}}\\approx\n\\frac{kr_{\\mathrm{M}}^{2}\\lambda_{T}^{(1)}G_{\\mathrm{ion}}n_{0}n_{\\mathrm{A}}%\n}{R^{2}}\\left(  1+\\frac{m}{m_{\\beta}}\\right)  \\cos\\theta, \\label{n_R>>lambda}%\n\\end{equation}\nwhere $k=16\\sqrt{2}\\ \\pi^{\\frac{3}{2}}\/3$. Equation~(\\ref{n_R>>lambda}) shows\nthat at $R\\gg\\lambda_{T}^{(1)}$ the density of the major ion population fall\noff as $(\\cos\\theta)\/R^{2}$.\n\nIf, however, instead of using the exact integral expression for $J_{R_{0}%\n}^{\\infty}$ given by equation (\\ref{J_a^b_R>3epsilon}), we employed its\napproximation given by equation~(\\ref{J_R_0_interpo}) then, applying the\nidentity\n\\begin{equation}\n\\int_{0}^{\\infty}\\sqrt{\\left(  1+\\frac{4x}{3\\pi}\\right)  x}\\ \\exp\\left(\n-\\ x\\right)  dx=\\frac{\\sqrt{3\\pi}}{4}\\exp\\left(  \\frac{3\\pi}{8}\\right)\nK_{1}\\left(  \\frac{3\\pi}{8}\\right)  , \\label{K_1}%\n\\end{equation}\ndeduced from \\citep[][equation~3.372]{Gradshteyn:Table94}, with the modified\nBessel function of the second kind $K_{\\nu}(x)$, we would obtain\nequation~(\\ref{n_R>>lambda}) with $k$ replaced by a formally different\ncoefficient, $k_{1}=(2^{\\frac{3}{2}}\\pi^{\\frac{5}{2}}\/\\sqrt{3})\\exp\\left(\n3\\pi\/8\\right)  K_{1}\\left(  3\\pi\/8\\right)  $. However, the numerical values of\n$k\\approx42$ and $k_{1}\\approx41.74$ are so close to each other that can be\nconsidered as essentially the same. This confirms that the approximate\nexpression $J_{R_{0}}^{\\infty}$ given by equation~(\\ref{J_R_0_interpo}) is\nreasonably accurate and can be successfully used in other occasions, as done below.\n\n\\subsection{General distances \\label{Moderate distances}}\n\nFor all but largest meteors the radar head echo is formed within moderate\nradial distances of $R\\sim\\lambda_{T}^{(1)}$, the most difficult domain to\ntreat analytically. Below, we reduce the general expression for $n^{\\left(\n2\\right)  }$ to a simpler form, more suitable for a further analytic or\nnumerical treatment, and then obtain the explicit spatial distribution of\n$n^{\\left(  2\\right)  }$ for $R\\ll\\lambda_{T}^{(1)}$, the limit opposite to\nthat considered in section~\\ref{Long-distance asymptotics}. After that, we\nwill discuss the general case, using numerical integrations.\n\n\\subsubsection{Reduction of the general ion density \\label{Reduction}}\n\nUnder assumption of the isotropic differential cross-section, $G_{\\mathrm{ion}%\n}(U)$, equation~(\\ref{yields}) involves $\\mu$ only as a linear multiplier. For\nthe further analysis, equation~(\\ref{n^(2)_snova}) with the integration over\n$\\Phi$ is no longer convenient. More advantageous is integrating over $\\mu$,\nwhere $\\mu$ is given by equation~(\\ref{mumumu}). Introducing a dimensionless\nvariable%\n\\begin{equation}\n\\xi_{0}=\\frac{R_{0}}{R}=\\sin\\Theta\\leq1 \\label{xi_0}%\n\\end{equation}\nand changing variables $R_{0},\\Phi$ to $\\xi_{0},\\mu$, we arrive at\n\\begin{subequations}\n\\label{n^(2),I_muha}%\n\\begin{align}\n&  n^{\\left(  2\\right)  }=2\\times\\frac{4n_{0}n_{\\mathrm{A}}}{\\sqrt{3}}\\left(\n1+\\frac{m}{m_{\\beta}}\\right)  G_{\\mathrm{ion}}M\\label{n^(2)_prome_2}\\\\\n&  M\\equiv\\sum_{\\sigma_{R}=\\pm1}\\int_{0}^{1}\\frac{I(R,\\xi_{0}R)\\xi_{0}d\\xi\n_{0}}{\\sqrt{1-\\xi_{0}^{2}}}\\ I_{\\mu}(\\xi_{0}),\\label{M}\\\\\nI_{\\mu}(\\xi_{0})  &  =\\int\\frac{\\mathrm{H}\\left(  \\mu\\right)  \\mu d\\mu}%\n{\\sqrt{\\xi_{0}^{2}\\sin^{2}\\theta-\\left(  \\mu-\\sigma_{R}\\sqrt{1-\\xi_{0}^{2}%\n}\\cos\\theta\\right)  ^{2}}}, \\label{I_muha}%\n\\end{align}\nwhere the function $I(R,R_{0})$ is given by equation~(\\ref{III}) and\n$\\mathrm{H}\\left(  x\\right)  $ is the Heaviside step-function ($\\mathrm{H}%\n\\left(  x\\right)  =1$ for $x\\geq0$ and $\\mathrm{H}\\left(  x\\right)  =0$ for\n$x<0$). The latter takes into account the fact that the distribution function\nof secondary particles is non-zero only for positive $\\mu$, as discussed in\nPaper~1. The integration in $I_{\\mu}(\\xi_{0})$ is performed over the entire\n$\\mu$-range where the expression under the square root is non-negative. The\nfactor `$2$' in front of the RHS of equation~(\\ref{n^(2)_prome_2}) takes into\naccount the fact that each value of $\\mu$ corresponds to two symmetric values\nof $\\Phi$ with the same $\\cos\\Phi$ but opposite $\\sin\\Phi$. Introducing%\n\\end{subequations}\n\\begin{equation}\n\\mu_{1}=\\sigma_{R}\\sqrt{1-\\xi_{0}^{2}}\\cos\\theta-\\xi_{0}\\sin\\theta,\\qquad\n\\mu_{2}=\\sigma_{R}\\sqrt{1-\\xi_{0}^{2}}\\cos\\theta+\\xi_{0}\\sin\\theta,\n\\label{mu_1,2}%\n\\end{equation}\nand eliminating the step-function, we can rewrite $I_{\\mu}(\\xi_{0})$ in\nequation~(\\ref{I_muha}) as%\n\\begin{equation}\nI_{\\mu}(\\xi_{0})=\\int_{\\max(\\mu_{1},0)}^{\\max(\\mu_{2},0)}\\frac{\\mu d\\mu}%\n{\\sqrt{\\left(  \\mu_{2}-\\mu\\right)  \\left(  \\mu-\\mu_{1}\\right)  }},\n\\label{I_mu}%\n\\end{equation}\nwhere the integration limits take into account that in general case $\\mu\n_{1,2}$ can be negative. For all signs of $\\cos\\theta$, the relations between\n$\\mu_{1,2}$ and $0$ are listed in this table:%\n\\begin{equation}%\n\\begin{tabular}\n[c]{|l|l|l|}\\hline\n& $\\xi_{0}<\\left\\vert \\cos\\theta\\right\\vert $ & $\\xi_{0}>\\left\\vert \\cos\n\\theta\\right\\vert $\\\\\\hline\n$\\sigma_{R}\\cos\\theta<0$ & $\\mu_{1}<\\mu_{2}<0$ & $\\mu_{1}<0<\\mu_{2}$\\\\\\hline\n$\\sigma_{R}\\cos\\theta>0$ & $\\mu_{2}>\\mu_{1}>0$ & $\\mu_{1}<0<\\mu_{2}$\\\\\\hline\n\\end{tabular}\n\\ \\ \\ \\ \\ . \\label{Table:mu>>}%\n\\end{equation}\nAll this yields for $I_{\\mu}$, defined by equation~(\\ref{I_muha}), a\npiece-wise expression:%\n\\begin{equation}\nI_{\\mu}(\\xi_{0})=\\left\\{\n\\begin{array}\n[c]{ccc}%\n\\int_{\\mu_{1}}^{\\mu_{2}}\\frac{\\mu d\\mu}{\\sqrt{\\left(  \\mu_{2}-\\mu\\right)\n\\left(  \\mu-\\mu_{1}\\right)  }} & \\text{if} & \\xi_{0}<\\left\\vert \\cos\n\\theta\\right\\vert \\text{ and }\\sigma_{R}\\cos\\theta>0,\\\\\n\\int_{0}^{\\mu_{2}}\\frac{\\mu d\\mu}{\\sqrt{\\left(  \\mu_{2}-\\mu\\right)  \\left(\n\\mu-\\mu_{1}\\right)  }} & \\text{if} & \\xi_{0}>\\left\\vert \\cos\\theta\\right\\vert\n\\text{ for all }\\sigma_{R}\\cos\\theta,\\\\\n0 & \\text{if} & \\xi_{0}<\\left\\vert \\cos\\theta\\right\\vert \\text{ and }%\n\\sigma_{R}\\cos\\theta<0.\n\\end{array}\n\\right.  . \\label{I_mu_proma}%\n\\end{equation}\nUsing the definitions given by equation~(\\ref{mu_1,2}), we obtain:%\n\\begin{equation}\n\\int_{\\mu_{1}}^{\\mu_{2}}\\frac{\\mu d\\mu}{\\sqrt{\\left(  \\mu_{2}-\\mu\\right)\n\\left(  \\mu-\\mu_{1}\\right)  }}=\\pi\\sigma_{R}\\sqrt{1-\\xi_{0}^{2}}\\cos\\theta,\n\\label{Int_mu_full_again}%\n\\end{equation}\nand (for $\\xi_{0}>\\left\\vert \\cos\\theta\\right\\vert $):%\n\\begin{align}\n&  \\int_{0}^{\\mu_{2}}\\frac{\\mu d\\mu}{\\sqrt{\\left(  \\mu_{2}-\\mu\\right)  \\left(\n\\mu-\\mu_{1}\\right)  }}\\nonumber\\\\\n&  =\\sigma_{R}\\left(  \\frac{\\pi}{2}+\\arcsin\\frac{\\sigma_{R}\\sqrt{1-\\xi_{0}%\n^{2}}\\cos\\theta}{\\xi_{0}\\sin\\theta}\\right)  \\sqrt{1-\\xi_{0}^{2}}\\cos\n\\theta+\\sqrt{\\xi_{0}^{2}-\\cos^{2}\\theta}. \\label{Int_mu_partial_again}%\n\\end{align}\nRecalling equation~(\\ref{III}), for the quantity $M$, defined by\nequation~(\\ref{M}), we obtain%\n\\begin{align}\nM  &  =\\int_{0}^{1}\\frac{\\left(  2J_{R_{0}}^{\\infty}-J_{R}^{\\infty}\\right)\n_{\\sigma_{R>0}}\\xi_{0}d\\xi_{0}}{\\sqrt{1-\\xi_{0}^{2}}}\\int_{\\max(\\mu_{1}%\n,0)}^{\\max(\\mu_{2},0)}\\frac{\\mu d\\mu}{\\sqrt{\\left(  \\mu_{2}-\\mu\\right)\n\\left(  \\mu-\\mu_{1}\\right)  _{\\sigma_{R=+1}}}}\\label{K}\\\\\n&  +\\int_{0}^{1}\\frac{\\left(  J_{R}^{\\infty}\\right)  _{\\sigma_{R<0}}\\xi\n_{0}d\\xi_{0}}{\\sqrt{1-\\xi_{0}^{2}}}\\int_{\\max(\\mu_{1},0)}^{\\max(\\mu_{2}%\n,0)}\\frac{\\mu d\\mu}{\\sqrt{\\left(  \\mu_{2}-\\mu\\right)  \\left(  \\mu-\\mu\n_{1}\\right)  _{\\sigma_{R=-1}}}}\\nonumber\n\\end{align}\nRegrouping the terms, and using equations~(\\ref{I_mu_proma}%\n)--(\\ref{Int_mu_partial_again}), for all $\\theta$ we obtain%\n\\begin{align}\n&  M=\\pi\\left\\vert \\cos\\theta\\right\\vert \\int_{0}^{\\left\\vert \\cos\n\\theta\\right\\vert }J_{R_{0}}^{\\infty}\\xi_{0}d\\xi_{0}+\\left(  \\pi\\cos\n\\theta\\right)  \\int_{0}^{1}J_{R_{0}}^{R}\\xi_{0}d\\xi_{0}\\nonumber\\\\\n&  +2\\int_{\\left\\vert \\cos\\theta\\right\\vert }^{1}J_{R_{0}}^{\\infty}\\xi\n_{0}\\sqrt{\\frac{\\xi_{0}^{2}-\\cos^{2}\\theta}{1-\\xi_{0}^{2}}}\\ d\\xi\n_{0}\\label{K_snova}\\\\\n&  +2\\left\\vert \\cos\\theta\\right\\vert \\int_{\\left\\vert \\cos\\theta\\right\\vert\n}^{1}J_{R_{0}}^{\\infty}\\xi_{0}\\arcsin\\frac{\\sqrt{1-\\xi_{0}^{2}}\\left\\vert\n\\cos\\theta\\right\\vert }{\\xi_{0}\\sin\\theta}\\ d\\xi_{0}\\ ,\\nonumber\n\\end{align}\nwhere $J_{R_{0}}^{R}=J_{R_{0}}^{\\infty}-J_{R}^{\\infty}$. All terms in the RHS\nof equation~(\\ref{K_snova}) are symmetric with respect to the sign of\n$(\\cos\\theta)$, except the second term which is antisymmetric. This term is\nresponsible for the entire asymmetry between the locations in front of the\nmeteoroid ($\\cos\\theta<0$) and behind it ($\\cos\\theta>0$).\n\nTo simplify further, we introduce other variables and parameters,%\n\\begin{equation}\n\\eta=\\frac{R^{\\prime}}{R},\\qquad\\beta=\\left(  \\frac{R}{\\lambda_{T}^{(1)}%\n}\\right)  ^{\\frac{2}{3}},\\qquad q=\\frac{r_{\\mathrm{M}}^{2}}{R},\n\\label{beta_snova}%\n\\end{equation}\nwhere $R^{\\prime}$ is the integration variable in $J_{a}^{b}$, defined by\nequation~(\\ref{J_a^b_R>3epsilon}). With use of these dimensionless quantities,\nequation~(\\ref{J_a^b_R>3epsilon}) yields\n\\begin{subequations}\n\\label{J_R_0^infty,1}%\n\\begin{align}\nJ_{R_{0}}^{\\infty}  &  =q\\int_{\\xi_{0}}^{\\infty}\\left(  1+\\beta\\eta^{\\frac\n{2}{3}}\\right)  \\exp\\left(  -\\ \\frac{3\\beta\\eta^{\\frac{2}{3}}}{2}\\right)\n\\frac{d\\eta}{\\eta\\sqrt{\\eta^{2}-\\xi_{0}^{2}}},\\label{J_R_0^infty}\\\\\nJ_{R_{0}}^{R}  &  =q\\int_{\\xi_{0}}^{1}\\left(  1+\\beta\\eta^{\\frac{2}{3}%\n}\\right)  \\exp\\left(  -\\ \\frac{3\\beta\\eta^{\\frac{2}{3}}}{2}\\right)\n\\frac{d\\eta}{\\eta\\sqrt{\\eta^{2}-\\xi_{0}^{2}}}. \\label{J_R_0^1}%\n\\end{align}\nFor some calculations, we will also need approximate\nequation~(\\ref{J_R_0_interpo}) for $J_{R_{0}}^{\\infty}$,%\n\\end{subequations}\n\\begin{equation}\nJ_{R_{0}}^{\\infty}\\approx\\frac{\\pi q}{2\\xi_{0}}\\sqrt{1+\\frac{2}{\\pi}\\ \\beta\n\\xi_{0}^{\\frac{2}{3}}}\\exp\\left(  -\\ \\frac{3}{2}\\ \\beta\\xi_{0}^{\\frac{2}{3}%\n}\\right)  . \\label{J_R_)_approx_dimensionless}%\n\\end{equation}\nFor $J_{R_{0}}^{R}$ we need no approximations, as will become clear soon.\n\nBefore proceeding, we check that the long-distance limit of $R\\gg\\lambda\n_{T}^{(1)}$ (i.e., $\\beta\\gg1$) for the above equations provides a smooth\ntransition to the range of long distances considered in\nsection~\\ref{Long-distance asymptotics}. Beyond a narrow vicinity around\n$\\theta=\\pi\/2$, given by $\\left\\vert \\cos\\theta\\right\\vert \\lesssim\n\\beta^{-3\/2}$, we can easily see that in the RHS of equation~(\\ref{K_snova})\nthe third and fourth terms are exponentially small. Neglecting them and\nextending the same accuracy to the upper integration limit in the first and\nsecond terms, $J_{R_{0}}^{R}\\approx J_{R_{0}}^{\\infty}$, we obtain\n\\begin{equation}\n\\left.  M\\right\\vert _{\\beta\\gg1}\\approx\\left\\{\n\\begin{array}\n[c]{ccc}%\n2\\pi\\left(  \\cos\\theta\\right)  \\int_{0}^{\\infty}J_{R_{0}}^{\\infty}\\xi_{0}%\nd\\xi_{0} & \\text{if} & \\cos\\theta>0,\\\\\n&  & \\\\\n0 & \\text{if} & \\cos\\theta<0.\n\\end{array}\n\\right.  \\label{K_beta>>1}%\n\\end{equation}\nUsing for $J_{R_{0}}^{\\infty}$ the exact equation~(\\ref{J_R_0^infty}) and\napplying for the double integration the same approach as in\nsection~\\ref{Long-distance asymptotics}, we obtain\n\\[\n\\left.  M\\right\\vert _{\\beta\\gg1}\\approx\\frac{2\\pi q\\cos\\theta}{\\beta\n^{\\frac{3}{2}}}\\sqrt{\\frac{2\\pi}{3}}.\n\\]\nReturning from the temporary dimensionless parameters $\\beta,q$ to the\noriginal coordinate $R$ and inserting the corresponding $M$ to\nequation~(\\ref{n^(2)_prome_2}), for $\\cos\\theta>0$ we recreate\nequation~(\\ref{n_R>>lambda}).\n\n\\subsubsection{Short distances, $R\\ll\\lambda_{T}^{(1)}$%\n\\label{Very short distances}}\n\nNow we consider the short-distance limit of $R\\ll\\lambda_{T}^{(1)}$ ($\\beta\n\\ll1$) which is opposite to that discussed in\nsection~\\ref{Long-distance asymptotics}. For not too large integration\nvariables $R^{\\prime}$, $\\eta=R^{\\prime}\/R\\ll\\beta^{-3\/2}$, all factors with\n$\\beta\\eta^{\\frac{2}{3}}$ in equation~(\\ref{J_R_0^infty,1}) can be neglected.\nSince this range of $R^{\\prime}$ makes the dominant contribution to all\nintegrals, we extend this approximation to the entire range of $\\eta$, so\nthat\n\\begin{subequations}\n\\label{J_R,R_0_reduced}%\n\\begin{align}\nJ_{R_{0}}^{\\infty} &  \\approx q\\int_{\\xi_{0}}^{\\infty}\\frac{d\\eta}{\\eta\n\\sqrt{\\eta^{2}-\\xi_{0}^{2}}}=\\frac{\\pi q}{2\\xi_{0}},\\label{J_R_0_reduced}\\\\\nJ_{R_{0}}^{R} &  \\approx q\\int_{\\xi_{0}}^{1}\\frac{d\\eta}{\\eta\\sqrt{\\eta\n^{2}-\\xi_{0}^{2}}}=\\frac{q}{\\xi_{0}}\\arccos\\xi_{0}.\\label{J_R_reduced}%\n\\end{align}\nIn this limit, the first two terms in the RHS\\ of equation~(\\ref{K_snova}) can\nbe easily integrated, yielding $\\pi q[\\pi(\\cos^{2}\\theta)\/2+\\cos\\theta]$. The\ntwo remaining terms can be expressed in terms of the complete elliptic\nintegrals of the 1st and 2nd kind,%\n\\end{subequations}\n\\begin{equation}\n\\mathrm{K}\\left(  k\\right)  =\\int_{0}^{1}\\frac{dt}{\\sqrt{\\left(\n1-t^{2}\\right)  \\left(  1-k^{2}t^{2}\\right)  }},\\qquad\\mathrm{E}\\left(\nk\\right)  =\\int_{0}^{1}\\sqrt{\\frac{1-k^{2}t^{2}}{1-t^{2}}}%\n\\ dt,\\label{F,E_complete}%\n\\end{equation}\nrespectively, where $0\\leq k<1$. Indeed, for constant $\\xi_{0}J_{R_{0}%\n}^{\\infty}$, as in equation~(\\ref{J_R_0_reduced}), the third term in\nequation~(\\ref{K_snova}) becomes proportional to%\n\\[\nI_{1}=\\int_{\\left\\vert \\cos\\theta\\right\\vert }^{1}\\sqrt{\\frac{\\xi_{0}^{2}%\n-\\cos^{2}\\theta}{1-\\xi_{0}^{2}}}\\ d\\xi_{0}.\n\\]\nThis integral already resembles an elliptic integral, but reducing $I_{1}$ to\nthose with the real arguments requires additional efforts. Substituting\n$\\xi_{0}=\\sqrt{1-z^{2}\\sin^{2}\\theta}$, we reduce $I_{1}$ to $\\mathrm{E}%\n\\left(  \\sin\\theta\\right)  -\\left(  \\cos^{2}\\theta\\right)  \\mathrm{K}\\left(\n\\sin\\theta\\right)  $. In a similar way, we can also calculate the fourth term\nin equation~(\\ref{K_snova}). With constant $J_{R_{0}}^{\\infty}\\xi_{0}$, this\nterm becomes proportional to%\n\\[\nI_{2}=\\int_{\\left\\vert \\cos\\theta\\right\\vert }^{1}\\arcsin\\frac{\\sqrt{1-\\xi\n_{0}^{2}}\\left\\vert \\cos\\theta\\right\\vert }{\\xi_{0}\\sin\\theta}\\ d\\xi_{0}.\n\\]\nIntegration of the corresponding indefinite integral by parts gives%\n\\begin{align*}\n&  \\int\\arcsin\\frac{\\sqrt{1-\\xi_{0}^{2}}\\left\\vert \\cos\\theta\\right\\vert }%\n{\\xi_{0}\\sin\\theta}\\ d\\xi_{0}\\\\\n&  =\\xi_{0}\\arcsin\\frac{\\sqrt{1-\\xi_{0}^{2}}\\left\\vert \\cos\\theta\\right\\vert\n}{\\xi_{0}\\sin\\theta}+\\left\\vert \\cos\\theta\\right\\vert \\int\\frac{d\\xi_{0}%\n}{\\sqrt{\\left(  1-\\xi_{0}^{2}\\right)  \\left(  \\xi_{0}^{2}-\\cos^{2}%\n\\theta\\right)  }}.\n\\end{align*}\nMaking\\ the same substitution for the remaining integral in the RHS as done\nfor $I_{1}$ and evaluating everything over the proper integration limits, we\nobtain $I_{2}=(\\mathrm{K}\\left(  \\sin\\theta\\right)  -\\pi\/2)|\\cos\\theta|$. When\nadding all terms in the RHS of equation~(\\ref{K_snova}), the $\\mathrm{K}%\n\\left(  \\sin\\theta\\right)  $-terms in $I_{1,2}$ cancel and equation~(\\ref{M})\nreduces to a simple expression: $M\\approx\\pi q[\\cos\\theta+\\mathrm{E}\\left(\n\\sin\\theta\\right)  ]$. As a result, the ion density in the short-distance\nlimit reduces to%\n\\begin{equation}\n\\left.  n^{\\left(  2\\right)  }\\right\\vert _{R\\ll\\lambda_{T}^{(1)}}\\approx\n\\frac{8\\pi r_{\\mathrm{M}}^{2}G_{\\mathrm{ion}}n_{0}n_{\\mathrm{A}}}{\\sqrt{3}%\n\\,R}\\left(  1+\\frac{m}{m_{\\beta}}\\right)  \\left[  \\cos\\theta+\\mathrm{E}\\left(\n\\sin\\theta\\right)  \\right]  .\\label{n_R<<lambda}%\n\\end{equation}\n\n\nComparing equation~(\\ref{n_R<<lambda}) with the opposite limit given by\nequation~(\\ref{n_R>>lambda}) shows that the $1\/R^{2}$-dependency of\n$n^{\\left(  2\\right)  }|_{R\\gg\\lambda_{T}^{(1)}}$ transforms to the\n$1\/R$-dependency for $n^{\\left(  2\\right)  }|_{R\\ll\\lambda_{T}^{(1)}}$. The\nangular $\\theta$-dependency also changes significantly. While in the\nlong-distance limit of $R\\gg\\lambda_{T}^{(1)}$ ions occupy almost entirely the\nhalf-space behind the meteoroid ($0\\leq\\theta<\\pi\/2$), in the short-distance\nlimit of $R\\ll\\lambda_{T}^{(1)}$ ions show a noticeable presence in front of\nthe meteoroid ($\\pi\/2\\leq\\theta\\leq\\pi$) as well. Red dashed curves in\nFigure~\\ref{Fig:M_versus_theta} show the corresponding angular dependencies\nnormalized to their maximum values at $\\theta=0$.\n\n\\subsubsection{Arbitrary distances \\label{General case}}\n\nThe case of moderate distances $R\\sim\\lambda_{T}^{(1)}$ is covered by general\nequations (\\ref{n^(2),I_muha}) and (\\ref{K_snova}) with $J_{R_{0}}^{\\infty}$\nand $J_{R_{0}}^{R}$ expressed in the original integral form by\nequation~(\\ref{J_R_0^infty,1}), or in an approximate, but explicit, form for\n$J_{R_{0}}^{\\infty}$ by equation~(\\ref{J_R_)_approx_dimensionless}). Unlike\n$J_{R_{0}}^{\\infty}$, the integral $J_{R_{0}}^{R}$ is involved only in the\nsecond term in the RHS of equation~(\\ref{K_snova}). As we show below, this\nterm can be calculated exactly by using the integral form of\nequation~(\\ref{J_R_0^1}). Below we obtain the explicit analytic expressions\nfor the first and second terms in the RHS of equation~(\\ref{K_snova}). Being\nunable to obtain a general analytic approximation for the two last integral\nterms, we will integrate them numerically.\n\n\\paragraph{First term in\\ equation~(\\ref{K_snova}).}\n\nUsing~equation~(\\ref{J_R_0^infty}), for the integral in the first term of the\nexpression for $M$ in~(\\ref{K_snova}), we have%\n\\begin{align}\n&  Q_{1}\\equiv\\frac{1}{q}\\int_{0}^{\\left\\vert \\cos\\theta\\right\\vert }J_{R_{0}%\n}^{\\infty}\\xi_{0}d\\xi_{0}\\nonumber\\\\\n&  =\\int_{0}^{\\left\\vert \\cos\\theta\\right\\vert }\\left[  \\int_{\\xi_{0}}%\n^{\\infty}\\left(  1+\\beta\\eta^{\\frac{2}{3}}\\right)  \\exp\\left(  -\\ \\frac\n{3\\beta\\eta^{\\frac{2}{3}}}{2}\\right)  \\frac{d\\eta}{\\eta}\\right]  \\frac{\\xi\n_{0}d\\xi_{0}}{\\sqrt{\\eta^{2}-\\xi_{0}^{2}}}, \\label{Q_1}%\n\\end{align}\nwhere the dimensionless variables $\\eta$, $\\beta$, and $q$ are defined by\nequation~(\\ref{beta_snova}). Changing the order of integration, we obtain%\n\\begin{align*}\n&  Q_{1}=\\int_{0}^{\\left\\vert \\cos\\theta\\right\\vert }\\left(  1+\\beta\n\\eta^{\\frac{2}{3}}\\right)  \\exp\\left(  -\\ \\frac{3\\beta\\eta^{\\frac{2}{3}}}%\n{2}\\right)  \\frac{d\\eta}{\\eta}\\ \\int_{0}^{\\eta}\\frac{\\xi_{0}d\\xi_{0}}%\n{\\sqrt{\\eta^{2}-\\xi_{0}^{2}}}\\\\\n&  +\\int_{\\left\\vert \\cos\\theta\\right\\vert }^{\\infty}\\left(  1+\\beta\n\\eta^{\\frac{2}{3}}\\right)  \\exp\\left(  -\\ \\frac{3\\beta\\eta^{\\frac{2}{3}}}%\n{2}\\right)  \\frac{d\\eta}{\\eta}\\ \\int_{0}^{\\left\\vert \\cos\\theta\\right\\vert\n}\\frac{\\xi_{0}d\\xi_{0}}{\\sqrt{\\eta^{2}-\\xi_{0}^{2}}}.\n\\end{align*}\nThese integrations yield%\n\\begin{align}\n&  Q_{1}=\\sqrt{\\frac{2\\pi}{3\\beta^{3}}}\\operatorname{erf}\\left(  \\sqrt\n{\\frac{3\\beta}{2}}\\left\\vert \\cos\\theta\\right\\vert ^{\\frac{1}{3}}\\right)\n+J_{1}\\nonumber\\\\\n&  -\\left(  \\left\\vert \\cos\\theta\\right\\vert ^{\\frac{2}{3}}+\\frac{2}{\\beta\n}\\right)  \\left\\vert \\cos\\theta\\right\\vert ^{\\frac{1}{3}}\\exp\\left(\n-\\ \\frac{3\\beta\\left\\vert \\cos\\theta\\right\\vert ^{\\frac{2}{3}}}{2}\\right)  ,\n\\label{First_term}%\n\\end{align}\nwhere $\\operatorname{erf}(x)=(2\/\\sqrt{\\pi})\\int_{0}^{x}e^{-x^{2}}dx$ is the\nstandard error-function and%\n\\begin{equation}\nJ_{1}=\\int_{\\left\\vert \\cos\\theta\\right\\vert }^{\\infty}\\left(  1+\\beta\n\\eta^{\\frac{2}{3}}\\right)  \\exp\\left(  -\\ \\frac{3\\beta\\eta^{\\frac{2}{3}}}%\n{2}\\right)  \\left(  1-\\sqrt{1-\\frac{\\left\\vert \\cos\\theta\\right\\vert ^{2}%\n}{\\eta^{2}}}\\right)  d\\eta. \\label{J_1_snova}%\n\\end{equation}\nThe integral $J_{1}$ cannot be taken exactly, but directly below we obtain its\napproximate value. However, even without doing this, one can easily verify\nthat equations~(\\ref{First_term}) and (\\ref{J_1_snova}) provide both the\ncorrect limit of short distances, $\\lim_{\\beta\\rightarrow0}Q_{1}=\\pi\\left\\vert\n\\cos\\theta\\right\\vert \/2$, and the large-distance asymptotics of $\\beta\\gg1$,\n$Q_{1}\\approx(2\\pi\/3)^{1\/2}\\beta^{-3\/2}$.\n\nNow we find an approximate expression for $J_{1}$ by constructing a proper\nanalytic interpolation between two limiting cases. For small $\\beta$, we have%\n\\begin{equation}\n\\left.  J_{1}\\right\\vert _{\\beta\\rightarrow0}=\\left(  \\frac{\\pi}{2}-1\\right)\n\\left\\vert \\cos\\theta\\right\\vert . \\label{J_1_beta->0}%\n\\end{equation}\nIn the opposite limit of large $\\beta$, the major contribution to the integral\n$J_{1}$ is made in the small vicinity of the lower integration limit. This\nyields the following asymptotics,%\n\\begin{equation}\nJ_{1}\\approx\\left(  1-\\sqrt{\\frac{\\pi}{2\\beta\\left\\vert \\cos\\theta\\right\\vert\n^{\\frac{2}{3}}}}\\right)  \\exp\\left(  -\\ \\frac{3\\beta\\left\\vert \\cos\n\\theta\\right\\vert ^{\\frac{2}{3}}}{2}\\right)  \\left\\vert \\cos\\theta\\right\\vert\n. \\label{J_1_beta>>1}%\n\\end{equation}\nInterpolating between equations ~(\\ref{J_1_beta->0}) and (\\ref{J_1_beta>>1})\nas%\n\\begin{equation}\nJ_{1}\\approx\\left(  1-\\frac{\\left(  4-\\pi\\right)  \\sqrt{2\\pi}}{2\\sqrt\n{2\\pi+\\left(  4-\\pi\\right)  ^{2}\\beta\\left\\vert \\cos\\theta\\right\\vert\n^{\\frac{2}{3}}}}\\right)  \\exp\\left(  -\\ \\frac{3\\beta\\left\\vert \\cos\n\\theta\\right\\vert ^{\\frac{2}{3}}}{2}\\right)  \\left\\vert \\cos\\theta\\right\\vert\n, \\label{J_1_approxy}%\n\\end{equation}\nwe obtain a reasonably good approximation for $J_{1}$, valid in the entire\nrange of $\\beta$. Even in the worst case of $\\beta\\left\\vert \\cos\n\\theta\\right\\vert ^{\\frac{2}{3}}\\sim6$, the mismatch between the actual\nintegral value and this approximation is only $\\simeq6\\%$.\n\nCombining equations~(\\ref{First_term}) with (\\ref{J_1_approxy}), for the\ndouble integral $Q_{1}$ defined by equation~(\\ref{Q_1}), to a good accuracy we\nobtain%\n\\begin{align}\n&  Q_{1}\\approx\\sqrt{\\frac{2\\pi}{3\\beta^{3}}}\\ \\operatorname{erf}\\left(\n\\sqrt{\\frac{3\\beta}{2}}\\left\\vert \\cos\\theta\\right\\vert ^{\\frac{1}{3}}\\right)\n\\nonumber\\\\\n&  -\\left[  \\frac{\\left(  4-\\pi\\right)  \\sqrt{2\\pi}\\left\\vert \\cos\n\\theta\\right\\vert }{2\\sqrt{2\\pi+\\beta\\left(  4-\\pi\\right)  ^{2}\\left\\vert\n\\cos\\theta\\right\\vert ^{\\frac{2}{3}}}}+\\frac{2\\left\\vert \\cos\\theta\\right\\vert\n^{\\frac{1}{3}}}{\\beta}\\right]  \\exp\\left(  -\\ \\frac{3\\beta\\left\\vert\n\\cos\\theta\\right\\vert ^{\\frac{2}{3}}}{2}\\right)  . \\label{First_term_final}%\n\\end{align}\n\n\n\\paragraph{Second term in\\ equation~(\\ref{K_snova}).}\n\nNow we calculate the integral%\n\\begin{align}\n&  Q_{2}\\equiv\\frac{1}{q}\\int_{0}^{1}J_{R_{0}}^{R}\\xi_{0}d\\xi_{0}\\nonumber\\\\\n&  =\\int_{0}^{\\left\\vert \\cos\\theta\\right\\vert }\\left[  \\int_{\\xi_{0}}%\n^{1}\\left(  1+\\beta\\eta^{\\frac{2}{3}}\\right)  \\exp\\left(  -\\ \\frac{3\\beta\n\\eta^{\\frac{2}{3}}}{2}\\right)  \\frac{d\\eta}{\\eta}\\right]  \\frac{\\xi_{0}%\nd\\xi_{0}}{\\sqrt{\\eta^{2}-\\xi_{0}^{2}}}. \\label{Q_2}%\n\\end{align}\nUnlike $Q_{1}$, this integral can be calculated exactly. Indeed, changing the\norder of integration, we obtain%\n\\begin{align}\n&  Q_{2}=\\int_{0}^{1}\\left(  1+\\beta\\eta^{\\frac{2}{3}}\\right)  \\exp\\left(\n-\\ \\frac{3\\beta\\eta^{\\frac{2}{3}}}{2}\\right)  \\frac{d\\eta}{\\eta}\\ \\int%\n_{0}^{\\eta}\\frac{\\xi_{0}d\\xi_{0}}{\\sqrt{\\eta^{2}-\\xi_{0}^{2}}}\\nonumber\\\\\n&  =\\int_{0}^{1}\\left(  1+\\beta\\eta^{\\frac{2}{3}}\\right)  \\exp\\left(\n-\\ \\frac{3\\beta\\eta^{\\frac{2}{3}}}{2}\\right)  d\\eta\\nonumber\\\\\n&  =\\sqrt{\\frac{2\\pi}{3\\beta^{3}}}\\ \\operatorname{erf}\\left(  \\sqrt\n{\\frac{3\\beta}{2}}\\right)  -\\left(  1+\\frac{2}{\\beta}\\right)  \\exp\\left(\n-\\ \\frac{3\\beta}{2}\\right)  . \\label{Second_term_final}%\n\\end{align}\n\n\n\\paragraph{Density along the major axis.}\n\nNow we consider two particular positions along the major axis: strictly behind\nthe meteoroid ($\\theta=0$) and strictly in front of it ($\\theta=\\pi$). In both\nthese positions, we have $\\left\\vert \\cos0\\right\\vert =1$, so that the third\nand fourth terms in the RHS\\ of equation~(\\ref{K_snova}) become zero. The\ncombination of the two first terms there is given by%\n\\begin{equation}\n\\pi\\left\\vert \\cos\\theta\\right\\vert \\int_{0}^{\\left\\vert \\cos\\theta\\right\\vert\n}J_{R_{0}}^{\\infty}\\xi_{0}d\\xi_{0}+\\pi\\left(  \\cos\\theta\\right)  \\int_{0}%\n^{1}J_{R_{0}}^{R}\\xi_{0}d\\xi_{0}=\\pi q\\left(  \\left\\vert \\cos\\theta\\right\\vert\nQ_{1}+\\left(  \\cos\\theta\\right)  Q_{2}\\right)  \\label{combination}%\n\\end{equation}\nAs a result, at the major axis behind the meteoroid we obtain\n\\begin{align}\n&  \\left.  n^{\\left(  2\\right)  }\\right\\vert _{\\theta=0}=\\frac{8\\pi\nr_{\\mathrm{M}}^{2}n_{0}n_{\\mathrm{A}}}{R\\sqrt{3}}\\left(  1+\\frac{m}{m_{\\beta}%\n}\\right)  G_{\\mathrm{ion}}\\left\\{  2\\sqrt{\\frac{2\\pi}{3}}\\frac{\\lambda\n_{T}^{(1)}}{R}\\operatorname{erf}\\left[  \\sqrt{\\frac{3}{2}}\\left(  \\frac\n{R}{\\lambda_{T}^{(1)}}\\right)  ^{\\frac{2}{3}}\\right]  \\right. \\nonumber\\\\\n&  \\left.  -\\left[  \\frac{\\left(  4-\\pi\\right)  \\sqrt{2\\pi}}{2\\sqrt\n{2\\pi+\\left(  4-\\pi\\right)  ^{2}(R\/\\lambda_{T}^{(1)})^{\\frac{2}{3}}}%\n}+1+4\\left(  \\frac{\\lambda_{T}^{(1)}}{R}\\right)  ^{\\frac{2}{3}}\\right]\n\\exp\\left[  -\\ \\frac{3}{2}\\left(  \\frac{R}{\\lambda_{T}^{(1)}}\\right)\n^{\\frac{2}{3}}\\right]  \\right\\}  . \\label{n^(2)_theta=0}%\n\\end{align}\nSimilarly, at the major axis in front of of the meteoroid we obtain\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=30pc]{M_versus_theta.pdf}\n\\caption{Density versus the polar angle $\\theta$ for different distances\n(black solid curves from the top to the bottom: $R\/\\lambda_{T}^{(1)}%\n=0.1;0.3;1;3;10$). Red dashed curves show asymptotic solutions given by\nequation~(\\ref{K_beta>>1}) (the top curve) and by equation~(\\ref{n_R<<lambda}%\n). All density distributions are normalized to the maximum values strictly\nbehind the meteoroid, $\\theta=0$ (see\nFig.~\\ref{Fig:Density_vs_Radius_logarithm}).}\n\\label{Fig:M_versus_theta}%\n\\end{figure}\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=30pc]{Density_vs_Radius_logarithm.pdf}\n\\caption{Normalized density versus the relative radial distance,\n$R\/\\lambda_{T}^{(1)}$, for several equally separated angles $\\theta$. The\nnormalization corresponds to the factor $M$ with $q\\rightarrow\\lambda\n_{T}^{(1)}\/R$. Red dashed curves for $\\theta=0$ (the top curve) and for\n$\\theta=\\pi$ (the bottom curve) are described analytically by\nequations~(\\ref{n^(2)_theta=0}) and (\\ref{n^(2)_theta=pi}), respectively.\nBlack solid curves show $M(R,\\theta)$ obtained by combining analytic\nequations~(\\ref{Q_1}) and (\\ref{Q_2}) for the two first terms of\nequation~(\\ref{K_snova}) with numerically calculated two remaining integral\nterms. The black solid curves, from the top to the bottom, correspond to\n$\\theta=\\pi\/8$; $\\pi\/4$; $3\\pi\/8$; $\\pi\/2$; $5\\pi\/8$; $3\\pi\/4;$ and $7\\pi\/8$,\nrespectively. The black solid curves with $\\theta=\\pi\/8$ and $\\theta=7\\pi\/8$\nare almost indistinguishable from the top and bottom red dashed curves,\n$\\theta=0,\\pi$.}%\n\\label{Fig:Density_vs_Radius_logarithm}\n\\end{figure}\n\\begin{align}\n&  \\left.  n^{\\left(  2\\right)  }\\right\\vert _{\\theta=\\pi}=\\frac{8\\pi\nr_{\\mathrm{M}}^{2}n_{0}n_{\\mathrm{A}}}{R\\sqrt{3}}\\left(  1+\\frac{m}{m_{\\beta}%\n}\\right)  G_{\\mathrm{ion}}\\nonumber\\\\\n&  \\times\\left[  1-\\frac{\\left(  4-\\pi\\right)  \\sqrt{2\\pi}}{2\\sqrt\n{2\\pi+\\left(  4-\\pi\\right)  ^{2}(R\/\\lambda_{T}^{(1)})^{\\frac{2}{3}}}}\\right]\n\\exp\\left[  -\\ \\frac{3}{2}\\left(  \\frac{R}{\\lambda_{T}^{(1)}}\\right)\n^{\\frac{2}{3}}\\right]  . \\label{n^(2)_theta=pi}%\n\\end{align}\n\n\nFigure~\\ref{Fig:Density_vs_Radius_logarithm} shows these two radial\ndependencies with $\\cos\\theta=1$ ($\\theta=0$) and $\\cos\\theta=-1$ ($\\theta\n=\\pi$) are actually representative for slightly more general $\\cos\\theta$. The\nradial-distance dependencies for positive and negative $\\cos\\theta$ look\nqualitatively different, as we discuss below.\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=30pc]\n{Density_Contours_3D.pdf}\n\\caption{Spatial distribution of plasma density around a small meteoroid. The\nfigure shows a cross-section of an axially symmetric distribution by a plane\nthat includes the symmetry axis (blue dashed line). The meteoroid is shown by\na small white circle at the symmetry axis. The color coding signifies the\nplasma density in relative units; $r$ is the distance to the symmetry axis.}\n\\label{Fig:3Ddensity}\n\\end{figure}\n\nBehind the meteoroid, $\\cos\\theta>0$, the radial dependence of the density\ngradually changes from $n^{\\left(  2\\right)  }\\varpropto1\/R$ for $R\\ll\n\\lambda_{T}^{(1)}$, as described by equation~(\\ref{n_R<<lambda}), to\n$n^{\\left(  2\\right)  }\\varpropto1\/R^{2}$ for $R\\gg\\lambda_{T}^{(1)}$, as\ndescribed by equation~(\\ref{n_R>>lambda}). This change in the power-law radial\ndependence of $n^{\\left(  2\\right)  }$ occurs for the following reason. The\nsource for the secondary ions are the primary neutral particles, whose density\nfalls off near the meteoroid roughly as $1\/R^{2}$. At a given location $R$,\nthe number of ions moving in a certain direction is determined by the total\ncollisional ionization over the preceding segment of the straight-line\ntrajectory aligned with that direction. For $R\\ll\\lambda_{T}^{(1)}$, the total\nintegration over the entire ionization path acquires an additional factor\n$\\varpropto R$ which gradually transforms $1\/R^{2}$ to $1\/R$. On the other\nhand, for $R\\gg\\lambda_{T}^{(1)}$ only localized ionization within $R^{\\prime\n}\\lesssim\\lambda_{T}^{(1)}$ plays a role, resulting in `saturation' of the\nprevious additional factor $R$ at a constant value $\\sim\\lambda_{T}^{(1)}$.\nThis leaves the $\\varpropto1\/R^{2}$ dependence of $n^{\\left(  2\\right)  }$\nessentially untouched. This transition works only for locations behind the\nmeteoroid, $\\cos\\theta>0$, because almost all freshly born ions have\nvelocities $\\vec{V}$ with positive $\\mu=\\cos\\vartheta$, where $\\vartheta$ is\nthe angle between $\\vec{V}$ and $-\\vec{U}$, as shown in\nFigure~\\ref{Fig:Cartoon_reproduced}. Regardless of how far away from the\nmeteoroid this $R,\\theta$-point is located, all preceding straight-line\ntrajectory segments with $\\mu>0$ always cross the near-meteoroid volume\n$R^{\\prime}\\lesssim\\lambda_{T}^{(1)}$,\n\nA significantly different situation takes place in front of the meteoroid,\n$\\cos\\theta<0$. At $R\\ll\\lambda_{T}^{(1)}$, straight-line trajectory segments\nwith $\\mu>0$ also cross a part of the near-meteoroid volume $R^{\\prime\n}\\lesssim\\lambda_{T}^{(1)}$. That is why here $\\left.  n^{\\left(  2\\right)\n}\\right\\vert _{\\theta=\\pi}$ is quite noticeable, although a few times smaller\nthan $\\left.  n^{\\left(  2\\right)  }\\right\\vert _{\\theta=0}$. On the other\nhand, at larger distances, $R\\gtrsim\\lambda_{T}^{(1)}$, there are almost no\npreceding trajectory segments with positive $\\mu$ that would cross the near\nregion of $R^{\\prime}\\lesssim\\lambda_{T}^{(1)}$. These trajectories cross the\nregions where the number of the primary particles is itself exponentially\nsmall, so that $\\left.  n^{\\left(  2\\right)  }\\right\\vert _{\\theta=\\pi\n}\\varpropto\\exp[-\\ (3\/2)(R\/\\lambda_{T}^{(1)})^{2\/3}]$. This exponentially\ndecreasing density remains much less than that behind the meteoroid where\n$n^{\\left(  2\\right)  }$ decreases largely by a power law.\n\n\\paragraph{General case.\\label{General case_again}}\n\nFor the general case of $R\\sim\\lambda_{T}^{(1)}$ with $\\cos\\theta\\neq\\pm1$, we\nwere unable to find acceptable analytic approximations for the two last\n(integral) terms in the RHS\\ of equation~(\\ref{K_snova}). Therefore, we will\nintegrate those 1D integrals numerically.\n\nNow we summarize the entire expression for $n^{\\left(  2\\right)  }$ by\ncombining equation~(\\ref{n^(2)_prome_2}) with (\\ref{K_snova}), where in the\nfirst two terms the integrals $\\int_{0}^{\\left\\vert \\cos\\theta\\right\\vert\n}J_{R_{0}}^{\\infty}\\xi_{0}d\\xi_{0}=qQ_{1}$ and $\\int_{0}^{1}J_{R_{0}}^{R}%\n\\xi_{0}d\\xi_{0}=qQ_{2}$ were calculated with $Q_{1,2}$ explicitly given by\nequations~(\\ref{First_term_final}) and (\\ref{Second_term_final}) (as it was\ndone above when calculating the ion density at the major axis). For the two\nremaining integral terms in equation~(\\ref{K_snova}) we use the approximation\nfor $J_{R_{0}}^{\\infty}$ given by equation~(\\ref{J_R_)_approx_dimensionless}).\nThis gives%\n\\begin{align}\nn^{\\left(  2\\right)  }  &  =\\frac{8\\pi r_{\\mathrm{M}}^{2}n_{0}n_{\\mathrm{A}}%\n}{\\sqrt{3}R}\\left(  1+\\frac{m}{m_{\\beta}}\\right)  G_{\\mathrm{ion}%\n}(U)\\nonumber\\\\\n&  \\times\\left[  f_{1}\\left(  R,\\cos\\theta\\right)  \\left\\vert \\cos\n\\theta\\right\\vert +f_{2}\\left(  R\\right)  \\cos\\theta+f_{3}\\left(  R,\\cos\n\\theta\\right)  \\right]  , \\label{n^(2)_general}%\n\\end{align}\nwhere%\n\\begin{align}\n&  f_{1}\\left(  R,\\cos\\theta\\right)  =\\frac{\\lambda_{T}^{(1)}}{R}\\sqrt\n{\\frac{2\\pi}{3}}\\operatorname{erf}\\left[  \\sqrt{\\frac{3}{2}}\\left(  \\frac\n{R}{\\lambda_{T}^{(1)}}\\right)  ^{\\frac{1}{3}}\\left\\vert \\cos\\theta\\right\\vert\n^{\\frac{1}{3}}\\right] \\nonumber\\\\\n&  -\\left[  \\frac{\\left(  4-\\pi\\right)  \\left\\vert \\cos\\theta\\right\\vert\n}{2\\sqrt{1+\\left.  \\left(  4-\\pi\\right)  ^{2}(R\/\\lambda_{T}^{(1)})^{\\frac\n{2}{3}}\\left\\vert \\cos\\theta\\right\\vert ^{\\frac{2}{3}}\\right\/  (2\\pi)}%\n}+2\\left(  \\frac{\\lambda_{T}^{(1)}}{R}\\right)  ^{\\frac{2}{3}}\\left\\vert\n\\cos\\theta\\right\\vert ^{\\frac{1}{3}}\\right] \\nonumber\\\\\n&  \\times\\exp\\left[  -\\ \\frac{3\\left\\vert \\cos\\theta\\right\\vert ^{\\frac{2}{3}%\n}}{2}\\left(  \\frac{R}{\\lambda_{T}^{(1)}}\\right)  ^{\\frac{2}{3}}\\right]  ,\n\\label{f_1}%\n\\end{align}\n\n\\begin{align}\nf_{2}\\left(  R\\right)   &  =\\frac{\\lambda_{T}^{(1)}}{R}\\sqrt{\\frac{2\\pi}{3}%\n}\\operatorname{erf}\\left[  \\sqrt{\\frac{3}{2}}\\left(  \\frac{R}{\\lambda\n_{T}^{(1)}}\\right)  ^{\\frac{1}{3}}\\right] \\nonumber\\\\\n&  -\\left[  1+2\\left(  \\frac{\\lambda_{T}^{(1)}}{R}\\right)  ^{\\frac{2}{3}%\n}\\right]  \\exp\\left[  -\\ \\frac{3}{2}\\left(  \\frac{R}{\\lambda_{T}^{(1)}%\n}\\right)  ^{\\frac{2}{3}}\\right]  , \\label{f_2}%\n\\end{align}\n\n\\begin{align}\nf_{3}(R,\\cos\\theta)  &  =\\int_{\\left\\vert \\cos\\theta\\right\\vert }^{1}%\n\\sqrt{1+\\frac{2\\xi_{0}^{\\frac{2}{3}}}{\\pi}\\left(  \\frac{R}{\\lambda_{T}^{(1)}%\n}\\right)  ^{\\frac{2}{3}}}\\exp\\left[  -\\ \\frac{3\\xi_{0}^{\\frac{2}{3}}}%\n{2}\\left(  \\frac{R}{\\lambda_{T}^{(1)}}\\right)  ^{\\frac{2}{3}}\\right]\n\\nonumber\\\\\n&  \\times\\sqrt{\\frac{\\xi_{0}^{2}-\\cos^{2}\\theta}{1-\\xi_{0}^{2}}}\\ d\\xi\n_{0}\\nonumber\\\\\n&  +\\left\\vert \\cos\\theta\\right\\vert \\int_{\\left\\vert \\cos\\theta\\right\\vert\n}^{1}\\sqrt{1+\\frac{2\\xi_{0}^{\\frac{2}{3}}}{\\pi}\\left(  \\frac{R}{\\lambda\n_{T}^{(1)}}\\right)  ^{\\frac{2}{3}}}\\exp\\left[  -\\ \\frac{3\\xi_{0}^{\\frac{2}{3}%\n}}{2}\\left(  \\frac{R}{\\lambda_{T}^{(1)}}\\right)  ^{\\frac{2}{3}}\\right]\n\\nonumber\\\\\n&  \\times\\arcsin\\frac{\\sqrt{1-\\xi_{0}^{2}}\\left\\vert \\cos\\theta\\right\\vert\n}{\\xi_{0}\\sqrt{1-\\cos^{2}\\theta}}\\ d\\xi_{0}. \\label{f_3}%\n\\end{align}\nFor the isotropic differential cross-section the mean free path defined\nequation by equation~(\\ref{lambda_T^(1)}) reduces to\n\\begin{equation}\n\\lambda_{T}^{(1)}=\\frac{1}{4\\pi Un_{\\mathrm{A}}G(U)}\\left(  \\frac\n{T_{\\mathrm{M}}}{m_{\\mathrm{M}}}\\right)  ^{1\/2}. \\label{lambda_new}%\n\\end{equation}\n\n\nFigures~\\ref{Fig:M_versus_theta}, \\ref{Fig:Density_vs_Radius_logarithm}, and\n\\ref{Fig:3Ddensity}\\ illustrate the general $\\theta,R$-dependences of\n$n^{\\left(  2\\right)  }$.\n\nAs might be expected, in Figure~\\ref{Fig:M_versus_theta} the normalized curves\nwith intermediate values of $R\/\\lambda_{T}^{(1)}$ smoothly and uniformly fill\nthe gap between the two asymptotic solutions corresponding to the long,\n$R\\gg\\lambda_{T}^{(1)}$, and short, $R\\ll\\lambda_{T}^{(1)}$, distances, as\ndescribed by equations~(\\ref{n_R>>lambda}) and (\\ref{n_R<<lambda}). Similarly,\nin Figure~\\ref{Fig:Density_vs_Radius_logarithm} the curves with intermediate\nvalues of $\\cos\\theta$ fill the gap between the solutions strictly in front of\nthe meteoroid ($\\cos\\theta=-1$) and behind it ($\\cos\\theta=1$), as described\nby equations~(\\ref{n^(2)_theta=pi}) and (\\ref{n^(2)_theta=0}).\n\nFigure~\\ref{Fig:3Ddensity} shows the entire 3D structure of the ion density in\ncolor coding. Since the spatial distribution of the plasma density is axially\nsymmetric, this figure shows a meridional cross-section that includes the\nmajor axis. Behind the meteoroid, at $z\\equiv R\\cos\\theta\\gg\\lambda_{T}^{(1)}$\nthe presented distribution may be inaccurate because it does not properly\ndescribe the extended trail formed lagging behind multiply scattered ions.\nHowever, around the meteoroid this distribution should be reasonably accurate.\nFigure~\\ref{Fig:3Ddensity} shows a significant span in the density values\n(seven orders of magnitude). Notice visible bulges in constant density\ncontours in front of the meteoroid. This feature has not been predicted in\nprevious PIC simulations \\citep{Dyrud:Plasma2008EMaP,Dyrud:Plasma2008AdSpR}.\n\n\\section{Discussion\\label{Discussion}}\n\nThe spatial structure of the near-meteoroid plasma shown in\nFigure~\\ref{Fig:3Ddensity} scales with the collisional mean free path of the\nprimary (ablated) particles, $\\lambda_{T}^{(1)}$. The latter, in turn, scales\nwith the altitude($h$)-dependent atmospheric density as $\\lambda_{T}%\n^{(1)}\\propto1\/n_{\\mathrm{A}}(h)$. The frontal edge of the meteoroid plasma\nhas a bulge located around the major axis at the distance $\\sim\\lambda\n_{T}^{(1)}$ from the meteoroid.\n\nThe most striking feature of the meteor-plasma spatial structure is not the\nangular dependence of $n^{(2)}$ but rather the fact that near the meteoroid,\n$R\\lesssim\\lambda_{T}^{(1)}$, the plasma density behaves as $n^{\\left(\n2\\right)  }\\simeq F(\\theta)\/R$. The singular $1\/R$ dependence holds almost to\nthe meteoroid surface, $R\\sim r_{\\mathrm{M}}$. This fact has serious\nimplications for head-echo observations and for the corresponding modeling of\nthe radar wave propagation.\n\nTo analyze radio wave propagation through the dense meteoroid plasma with\n$n_{e}\\approx n_{i}\\approx n^{(2)}\\gg n_{\\mathrm{A}}$, we apply the simplest\ncriterion of geometrical optics applicable (the WKB approximation):\n\\begin{equation}\n\\frac{\\lambda_{0}}{2\\pi}\\ \\frac{\\left\\vert \\nabla\\tilde{n}\\right\\vert\n}{|\\tilde{n}|^{2}}\\ll1 \\label{WKB}%\n\\end{equation}\n\\citep{Ginzburg:Propagation71}. Here $\\lambda_{0}=c\/f$ is the vacuum wavelength\nwith the frequency $f$, and index of refraction (neglecting the geomagnetic\nfield) given by $\\tilde{n}=\\left(  1-n_{e}\/n_{\\mathrm{cr}}\\right)  ^{1\/2}$,\nwhere%\n\\begin{equation}\nn_{\\mathrm{cr}}=\\frac{4\\pi^{2}\\epsilon_{0}m_{e}f^{2}}{e^{2}}\\approx\n1.24\\times10^{4}%\n\\operatorname{cm}%\n^{-3}\\left(  \\frac{f}{1%\n\\operatorname{MHz}%\n}\\right)  ^{2} \\label{n_cr}%\n\\end{equation}\nis the critical plasma density corresponding to $\\tilde{n}=0$.\n\nMost of small meteoroids have a radius within the hundreds-of-microns range,\ni.e., many orders of magnitude less than the mean free path, $\\lambda\n_{T}^{(1)}$. This means that the radar beam with the wave frequency in the\ntens-of-MHz range (e.g., Jicamarca with $f=50~$MHz) will almost certainly\ncross the boundary $R=R_{\\mathrm{cr}}(\\theta)$ where $n_{\\mathrm{cr}}%\n=n_{e}\\approx n^{\\left(  2\\right)  }(R_{\\mathrm{cr}},\\theta)$. Under the\ngeometric-optics approximation, the boundary $R\\approx R_{\\mathrm{cr}}%\n(\\theta)$ represents the wave reflection level.\n\nAccording to equation~(\\ref{WKB}), geometric optics applies beyond the wave\nreflection level of $\\tilde{n}=0$ and, for $n_{e}\\propto1\/R$, where\n\\begin{equation}\nR\\gg\\sqrt{\\frac{\\lambda_{0}R_{\\mathrm{cr}}}{4\\pi|\\tilde{n}|^{3}}}.\n\\label{proximity}%\n\\end{equation}\nFor small radii, $R\\ll R_{\\mathrm{cr}}$, for which $|\\tilde{n}|\\approx\n(R_{\\mathrm{cr}}\/R)^{1\/2}\\gg1$, this requires $R\\gg\\lambda_{0}^{2}\/(16\\pi\n^{2}R_{\\mathrm{cr}})$, meaning that geometric optics applicability will\nnecessarily fail in the close proximity to the meteoroid.\n\nIf $R_{\\mathrm{cr}}\\gg\\lambda_{0}\/(4\\pi)$ then the radio wave propagating from\nlarger to smaller $R$ will reach the reflection level of the plasma density,\n$R=R_{\\mathrm{cr}}$, well before reaching the volume where the WKB becomes\ninvalid. This corresponds to normal overdense reflection of the radar wave, so\nthat the $1\/R$ dependence of $n^{\\left(  2\\right)  }$ will not prevent using\nthe regular geometric-optics technique. If, however, $R_{\\mathrm{cr}}%\n\\lesssim\\lambda_{0}$ then the applicability of the geometric-optics\npropagation breaks down before the radio wave reaches its reflection level.\nThis requires modeling the radar wave propagation using full Maxwell's\nequations \\citep{Marshall:FDTD15}. The characteristics of the radar signal\nobtained in this way may differ from those obtained in the framework of the\nregular ray tracing.\n\nTo conclude this discussion, we note that in the underlying kinetic theory we\nhave neglected the effect of fields on the ion collisionless motion. Directly\nnear the meteoroid, where the calculated density sharply increases with\ndecreasing $R$ this assumption may become invalid. This is especially true\nwithin the Debye length from the meteoroid surface, where one may expect a\nsignificant net positive charge formed by slow ions after fast electrons moved\naway from that region. This charge separation can create a significant\npotential barrier for electrons that, at the same time, will accelerate ions\naway from the meteoroid. This may modify the $1\/R$ rate near the meteoroid\nsurface. However, the relevant distances are typically much smaller than the\nradar wavelength, so that this localized modification should not affect the\noverall radio wave propagation and formation of the radar head echo.\n\n\\section{Conclusions\\label{Conclusions}}\n\nBased on the kinetic theory developed in the companion paper\n\\citep{Dimant:Formation1_17}, we have calculated the spatial distribution of\nthe meteor plasma responsible for the radar head echo.\n\nThe underlying theory assumes that (1) plasma electrons obey the Boltzmann\ndistribution; (2) most plasma ions originate from the ablated meteoroid\nmaterial after exactly one ionizing collision, while further collisions can be\nneglected; (3) the ion motion between collisions is largely unaffected by\nfields. In order to make plasma density calculations easier, in this paper we\nhave additionally assumed the isotropic differential cross-section of the\nmeteor-particle ionization.\n\nEquations~(\\ref{n^(2)_general}) to (\\ref{f_3}) describe the entire spatial\ndistribution of the near-meteoroid plasma density.\nFigures~\\ref{Fig:M_versus_theta}, \\ref{Fig:Density_vs_Radius_logarithm}, and\n\\ref{Fig:3Ddensity}\\ illustrate the solution. The major feature of the\nnear-meteoroid plasma density spatial distribution is its dominant $1\/R$\ndependence in most of the plasma. This quasi-singular behavior makes the radar\npenetrated near-meteoroid plasma to be overdense at locations sufficiently\nclose to the meteoroid. This and other features are important for modeling the\nradar head echo. In near future, we are planning to employ the obtained plasma\ndensity distribution for such modeling.\n\n\\section*{Acknowledgments}\n\nThis work was supported by NSF Grant AGS-1244842.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}