diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzbidm" "b/data_all_eng_slimpj/shuffled/split2/finalzzbidm" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzbidm" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nThe brain is organized into a hierarchy of functionally specialized regions, which selectively coordinate during behavior \\cite{hipp_oscillatory_2011,vezoli_brain_2021,engel_dynamic_2001,salinas_correlated_2001} and rest \\cite{fox_spontaneous_2007,brookes_investigating_2011,de_pasquale_cortical_2012}. Effective function relies on dynamic coordination between brain regions, in response to a changing environment, on an essentially fixed and limited anatomical substrate \\cite{kopell_beyond_2014,quiroga_closing_2020,tononi_information_2004,kohn_principles_2020}. Through these anatomical connections multiplexing occurs: multiple signals that combine for transmission through a single communication channel must then be differentiated at a downstream target location \\cite{akam_oscillatory_2014,becker_resolving_2020}. How information \u2013 communicated via coordinated transmission of spiking activity \\cite{hahn_portraits_2019} \u2013 dynamically routes through the brain's complex, distributed, hierarchical network remains unknown \\cite{palmigiano_flexible_2017}. \n\n\\noindent Brain rhythms \u2013 approximately periodic fluctuations in neural population activity \u2013 have been proposed to control the flow of information within the brain network \\cite{akam_oscillatory_2014, fries_mechanism_2005,gonzalez_communication_2020,canolty_functional_2010,bonnefond_communication_2017,buzsaki_brain_2012} and proposed as the core of cognition \\cite{palva_discovering_2012,siegel_spectral_2012,siebenhuhner_genuine_2020,williams_modules_2021}. Through periodic modulations in neuronal excitability, rhythms may support flexible and selective communication, allowing exchange of information through coordination of phase at rhythms of the same frequency (e.g., coherence \\cite{fries_mechanism_2005,bonnefond_communication_2017,bastos_communication_2015,fries_rhythms_2015,schroeder_low-frequency_2009}) and different frequencies (e.g., phase-amplitude coupling \\cite{canolty_functional_2010,lisman_theta-gamma_2013,hyafil_neural_2015} or n:m phase locking \\cite{palva_phase_2005,belluscio_cross-frequency_2012,tass_detection_1998}). Recent evidence shows that neural oscillations appear as transient, isolated events \\cite{lundqvist_gamma_2016,sherman_neural_2016}; how such transient oscillations route information through neural networks remains unclear \\cite{van_ede_neural_2018}. \n\n\\noindent Significant evidence supports the organization of brain rhythms into a small set of discrete frequency bands (e.g., theta [4-8 Hz], alpha [8-12 Hz], beta [12-30 Hz], gamma [30-80 Hz]) \\cite{buzsaki_rhythms_2011,buzsaki_neuronal_2004}. Consistent frequency bands appear across mammalian species (mouse, rat, cat, macaque, and humans \\cite{buzsaki_scaling_2013}) and in some cases the biological mechanisms that pace a rhythm are well-established (e.g., the decay time of inhibitory post-synaptic potentials sets the timescale for the gamma rhythm \\cite{whittington_inhibition-based_2000}). Why brain rhythms organize into discrete bands, and whether these rhythms are fixed by the brain's biology or organized to optimally support brain communication, remains unclear. For example, an alternative organization of the brain's rhythms (e.g., into a larger set of different frequency bands) may better support communication but remain inaccessible given the biological mechanisms available to pace brain rhythms.\n\n\\noindent While much evidence supports the existence of brain rhythms and their importance to brain function, few theories explain their arrangement. Different factors have been proposed for the spacing between the center frequencies of neighboring bands: Euler's number ($e\\approx2.718$) \\cite{penttonen_natural_2003}, the integer 2 \\cite{klimesch_algorithm_2013}, or the golden ratio ($\\phi\\approx1.618$) \\cite{roopun_temporal_2008}. Existing theory shows that irrational factors (e.g., $e$ and $\\phi$) minimize interference between frequency bands, in support of separate rhythmic communication channels for multiplexing information in the brain \\cite{izhikevich_weakly_1997,hoppensteadt_thalamo-cortical_1998,izhikevich_weakly_1999}. However, if separate rhythmic channels communicate different information, and the organization of brain rhythms prevents interference, how a target location coordinates information across these rhythms is unclear. For example, how in theory a neural population integrates top-down and bottom-up input communicated in separate rhythmic channels (lower [$<$40 Hz] and higher [$>$40 Hz] frequency ranges, respectively \\cite{bastos_communication_2015,fries_rhythms_2015,michalareas_alpha-beta_2016,fontolan_contribution_2014,bastos_layer_2020}) remains unclear. We propose a solution to this problem: addition of a third rhythm. Motivated by an existing mathematical theory \\cite{izhikevich_weakly_1997,hoppensteadt_thalamo-cortical_1998,izhikevich_weakly_1999}, we show that effective communication among three rhythms is optimal for rhythms arranged according to the golden ratio.\n\n\\noindent In what follows, we show that golden rhythms \u2013 rhythms organized by the golden ratio \u2013 are the optimal choice to integrate information among separate rhythmic communication channels. We propose that brain rhythms organize in the discrete frequency bands observed, with the specific spacing observed, to optimize segregation and integration of information transmission in the brain.\n\n\\section{Methods}\n\nAll simulations and analysis methods to reproduce the manuscript results and figures are available at \\url{https:\/\/github.com\/Mark-Kramer\/Golden-Framework}.\n\n\\subsection{Damped harmonic oscillator model} \\label{meth:dsho}\nAs a simple model of rhythmic neural population activity (e.g., observed in the local field potential (LFP) or magneto\/electroencephalogram (M\/EEG)) we implement a network of coupled damped harmonic oscillators \\cite{moorman_golden_2007}. We choose the damped harmonic oscillator for three reasons. First, a harmonic oscillator (e.g., a spring) mimics the restorative mechanisms governing displacements about a stable equilibrium in neural dynamics (e.g., excitation followed by inhibition in the gamma rhythm \\cite{whittington_inhibition-based_2000,fries_gamma_2007}, depolarization followed by hyperpolarization \u2013 and vice versa \u2013 in bursting rhythms \\cite{izhikevich_synchronization_2001}). Second, brain rhythms are transient \\cite{lundqvist_gamma_2016,sherman_neural_2016}. In the model, damping (e.g., friction) produces transient oscillations that decay to a stable equilibrium. Third, the damped harmonic oscillator driven by noise is equivalent to an autoregressive model of order two (AR(2), see Appendix \\ref{Appendix1}). The AR(2) model simulates stochastic brain oscillations \\cite{spyropoulos_spontaneous_2020}, consistent with the concept of a neural population with resonant frequency driven by random inputs.\n\\\\\n\n\\noindent We simulate an 8-node network of damped, driven harmonic oscillators. We model the activity $x_k$ at node $k$ as,\n\\begin{equation}\\label{eq:dsho}\n \\ddot{x}_k + 2 \\beta \\dot{x}_k + \\omega_k^2 x_k = \\left( \\bar{g}_C + \\bar{g}_S\\cos{\\omega_S t} \\right) \\sum_{j \\neq k} x_j \\ ,\n\\end{equation}\nwhere $\\beta$ is the damping constant, and $\\omega_k=2 \\pi f_k$ is the natural frequency of node $k$. The activity $x_j$ summed from all other nodes ($j \\neq k$) drives node $k$. We modulate this drive by a gain function with two terms: a constant gain $\\bar{g}_C$ and a sinusoidal gain with amplitude $\\bar{g}_S$ and frequency $\\omega_S=2 \\pi f_S$. To include noise in the dynamics, we represent the second order differential equation in Equation (\\ref{eq:dsho}) as two first order differential equations for the position and velocity of the oscillator. We add to the position dynamics a noise term, normally distributed with mean zero and standard deviation equal to the average standard deviation of the evoked response at all oscillators simulated without noise, excluding the perturbed oscillator from the average. In this way, we add meaningful noise of the same magnitude to all oscillators. We numerically simulate the model with noise using the Euler-Maruyama method. To examine the impact of different noise levels, we multiply the noise term by factors $\\{0, 0.5, 1.0, 1.5, 2.0\\}$. For each noise level, we repeat the simulation 100 times with random noise instantiations.\n\n\\section{Results}\nIn what follows, we propose that brain rhythms organized according to the golden ratio produce triplets of rhythms that establish a hierarchy of cross-frequency coupling. We conclude with four hypotheses deduced from this framework and testable in experiments.\n\n\\subsection{Rhythms organized by the golden ratio support selective cross-frequency coupling}\n\nIn the case of weakly-connected oscillatory populations, whether the populations interact or not depends on their frequency ratios \\cite{izhikevich_weakly_1997,hoppensteadt_thalamo-cortical_1998,izhikevich_weakly_1999}; rational frequency ratios support interactions, while irrational frequency ratios do not. Motivated by this theory, we consider a network of interacting, rhythmic neural populations (Figure \\ref{fig:schematic}). We model each population as a damped harmonic oscillator, with each oscillator assigned a natural frequency $f_k$. To couple the populations, we drive each oscillator with the summed activity of all other oscillators (i.e., the connectivity is all-to-all). We modulate this drive by a gain function ($g$) with constant ($\\bar{g}_C$) and sinusoidal (amplitude $\\bar{g}_S$, frequency $f_S$) terms: $g=\\bar{g}_C+\\bar{g}_S \\cos{\\left(2\\pi f_S t \\right)}$; see {\\it Methods}. Analysis of this coupled oscillator system reveals resonance (i.e., a large amplitude response) at a target oscillator in two cases. To describe these cases, we denote the frequency of a target oscillator as $f_T$ and the frequency of a driver oscillator as $f_D$. A large amplitude (resonant) response occurs at the target oscillator in the following cases,\n\\\\\n\nconstant gain modulation:\n\\vspace{-0.15in}\n\\begin{eqnarray}\n 0 = f_T - f_D \\label{eq:const}\n\\end{eqnarray}\n\nsinusoidal gain modulation:\n\\vspace{-0.15in}\n\\begin{subequations} \\label{eq:sin}\n\\begin{eqnarray}\n f_S = f_T - f_D \\label{eq:sin_a} \\\\\n f_S = f_D - f_T \\label{eq:sin_b} \\\\\n f_S = f_D + f_T \\label{eq:sin_c}\n\\end{eqnarray}\n\\end{subequations}\n\n\\noindent The first case (Equation \\ref{eq:const}) corresponds to the standard result for a damped target oscillator driven by sinusoidal input; when the sinusoidal driver frequency $f_D$ matches the natural frequency of the target $f_T$, the response amplitude at the target is largest (e.g., see Chapter 5 of \\cite{taylor_classical_2005}). The next three cases (Equation \\ref{eq:sin}) correspond to a damped target oscillator driven by sinusoidal input modulated by sinusoidal gain. If the gain frequency $f_S$ equals the sum or difference of the target and driver frequencies, then the response amplitude at the target is largest (see Appendix \\ref{Appendix2}). We note that the first case corresponds to within-frequency coupling (i.e., the driver and target have the same frequency) while the next three cases correspond to cross-frequency coupling (i.e., the driver and target have different frequencies). We also note that, in this model, we assume an oscillator responds to an input by exhibiting a large amplitude response; in this way, we consider the oscillation amplitude as encoding information, consistent with notion of information encoded in firing rate modulations \\cite{akam_oscillations_2010}.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=6cm]{Figure-1.pdf}\n\\caption{\\textbf{Illustration of the coupled oscillator network.} Oscillators with frequency $f_k$ receive input from all other oscillators. Input from one oscillator (frequency $f_1$) to all other oscillators $(f_2,f_3,\\ldots,f_8)$ is shown (black curves); similar connectivities exist from all other oscillators (not shown). Constant ($\\bar{g}_C$) and sinusoidal ($\\bar{g}_S$) gain modulates each input (gray lines).} \\label{fig:schematic}\n\\end{figure}\n\n\\noindent The results in Equations (\\ref{eq:const}, \\ref{eq:sin}) hold for any choice of driver, target, and gain frequencies without additional restrictions. We now apply an additional restriction, and consider the damped harmonic oscillator network with oscillator and gain frequencies $f_k$ satisfying,\n\\begin{eqnarray} \\label{eq:fk}\n f_k = f_0 \\, c^k \\, ,\n\\end{eqnarray}\nwhere $f_0 > 0$ determines the frequency at $k=0$. As discussed above, candidate values for $c$ deduced from {\\it in vivo} observations include Euler's number ($e\\approx2.718$) \\cite{penttonen_natural_2003}, the integer 2 \\cite{klimesch_algorithm_2013}, or the golden ratio ($\\phi\\approx1.618$) \\cite{roopun_temporal_2008}. Then, given the set of three neighboring frequencies $\\{f_k,f_{k+1},f_{k+2}\\}$, what choice of $c$ supports cross-frequency coupling in the network? To answer this, we choose $f_S = f_{k+2}$, $f_D = f_{k+1}$, and $f_T = f_k$ so that Equation (\\ref{eq:sin_c}) becomes\n\\begin{equation*}\n f_{k+2} = f_{k+1}+f_k \\, .\n\\end{equation*}\nSubstituting Equation (\\ref{eq:fk}) into this expression and solving for $c$, we find\n\\begin{equation*}\n c^2 - c - 1 = 0\n\\end{equation*}\nwith solution\n\\begin{equation*}\n c = \\dfrac{1 + \\sqrt{5}}{2} = \\phi \\, ,\n\\end{equation*}\nthe golden ratio. The same solution holds for all Equations (\\ref{eq:sin}) with appropriate selection of $\\{f_S,f_D,f_T\\}$ from $\\{f_k,f_{k+1},f_{k+2}\\}$. We conclude that, for a system of damped coupled oscillators with oscillator and gain frequencies spaced by the multiplicative factor $c$, cross-frequency coupling between three neighboring rhythms requires $c = \\phi$, the golden ratio. In other words, we propose that frequencies organized according to the golden ratio are particularly suited to support these cross-frequency interactions.\n\\\\\n\n\\noindent To illustrate this result, we consider a network of 8 damped, coupled oscillators each with a different natural frequency determined by the golden ratio ($f_k=\\phi^k$, where $\\phi= \\frac{(1+\\sqrt{5})}{2}$; Figure \\ref{fig:8nodes_phi}); we label these rhythms \u2013 scaled by factors of the golden ratio \u2013 as {\\it golden rhythms}. Starting all nodes in a resting state, we perturb one oscillator ($f_D = \\phi^6\\approx17.9$~Hz) to produce a transient oscillation at that node. With only a constant gain $(\\bar{g}_C=50, \\bar{g}_S=0)$, the impact of the perturbation on the other oscillators is small (Figure \\ref{fig:8nodes_phi}B); because $f_T \\neq f_D$ for any oscillator pair, the network impact of the perturbation is small, despite the constant coupling.\n\\\\\n\n\\noindent Including the sinusoidal gain modulation ($\\bar{g}_C=50, \\bar{g}_S=50$) results in selective communication between the oscillators. For example, choosing $f_S= \\phi^7 \\approx 29.0$ Hz, we observe an evoked response at two oscillators (Figure \\ref{fig:8nodes_phi}C): $f_T= \\phi^8 \\approx 47.0$ Hz (consistent with Equation (\\ref{eq:sin_a})) and $f_T= \\phi^5 \\approx 11.1$ Hz (consistent with Equation (\\ref{eq:sin_c})). We note that the frequency of evoked responses matches the natural frequency of each oscillator. We also note that no solution exists for Equation (\\ref{eq:sin_b}) because $f_T>0$. Different choices of gain frequency $f_S$ result in different pairs of cross-frequency coupling between the driver ($f_D$) and response oscillators (Figure \\ref{fig:8nodes_phi}D). Cross-frequency coupling occurs when Equations (\\ref{eq:sin}) are satisfied with $f_D \\approx17.9$~Hz. The coupling is selective; for example, choosing a gain modulation of $f_S=11.1$ Hz results in cross-frequency coupling between the driver ($f_D=17.9$ Hz) and faster ($29$ Hz) and slower ($6.9$ Hz) golden rhythms. In this case, sinusoidal gain frequencies $f_S$ exist that support cross-frequency coupling and occur at factors of the golden ratio: i.e., $f_S = \\phi^k$ (Figure \\ref{fig:8nodes_phi}D, circles). We note that evoked responses also occur when $f_S \\neq \\phi^k$ (Figure \\ref{fig:8nodes_phi}D, X's); in these cases, frequencies outside the original rhythm sequence $f_k = \\phi^k$ must exist to support cross-frequency coupling. We conclude that if brain rhythmic activity \u2013 both oscillator and gain frequencies \u2013 organizes according to the golden ratio, then cross-frequency coupling is possible between a subset of separate rhythmic communication channels.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=14.5cm]{Figure-2.pdf}\n\\caption{\\it{\\textbf{Rhythms organized by the golden ratio support selective cross-frequency coupling.} \\textbf{(A)} We perturb one oscillator (natural frequency 17.9 Hz, red), with connectivity to all other oscillators; $\\phi$ is the golden ratio. \\textbf{(B)} With only constant gain modulation, the perturbation ($t=0$, red) has little impact on other nodes. \\textbf{(C)} With sinusoidal gain modulation at 29 Hz, two oscillators (natural frequencies 11.1 Hz and 47.0 Hz) selectively respond to the perturbation. \\textbf{(D)} Average response amplitude (logarithm base 10) from $t=0$ to $t=1.5$ s at each oscillator versus gain frequency $f_S$. Different choices of gain frequency support selective coupling between the perturbed oscillator (natural frequency 17.9 Hz) and other oscillators. Peaks in response amplitude occur at golden rhythms (yellow circles, vertical lines) or other frequencies (red X's). Minimum response set to 0 for each curve, and vertical scale bar indicates 1. \\textbf{(E)} Average amplitude (black curve) and range ($2.5\\%$ to $97.5\\%$ from 100 simulations, shaded region) of evoked responses versus noise level. The oscillator with frequency $17.9$ Hz is directly perturbed, and sinusoidal gain modulation occurs with $f_S \\approx 29.0$ Hz. Oscillators at golden rhythms exhibit different behavior with perturbation (red) versus without perturbation (gray). Code to simulate this network and create this figure is available \\href{https:\/\/github.com\/Mark-Kramer\/Golden-Framework\/blob\/main\/Figure-2.ipynb}{here}}.} \\label{fig:8nodes_phi}\n\\end{figure}\n\n\\noindent We now consider the impact of noise on this cross-frequency communication. With sinusoidal gain modulation ($\\bar{g}_C=50, \\bar{g}_S=50$, and $f_S= \\phi^7 \\approx 29.0$ Hz) and including noise in the oscillator dynamics (see {\\it Methods}), we show the results for two cases: with perturbation and without perturbation to one oscillator ($f_D \\approx 17.9$ Hz, as above). Without perturbation (gray in Figure \\ref{fig:8nodes_phi}E), we find no evidence of an evoked response at any node, as expected; the amplitude remains small at all nodes, with a small gradual increase as the noise increases. With the perturbation (red in Figure \\ref{fig:8nodes_phi}E), we find an evoked response at the perturbed oscillator ($f_D \\approx 17.9$ Hz) and two other oscillators: $f_T \\approx 47.0$ Hz (consistent with Equation (\\ref{eq:sin_a})) and $f_T \\approx 11.1$ Hz (consistent with Equation (\\ref{eq:sin_c})). As the noise increases, so does the variability in the evoked response. For the lower frequency $f_T \\approx 11.1$ Hz oscillator, the evoked response remains evident as the noise increases; in Figure \\ref{fig:8nodes_phi}E, the perturbed (red) and unperturbed (gray) responses remain separate. For the higher frequency $f_T \\approx 47.0$ Hz oscillator, the evoked response becomes more difficult to distinguish from the unperturbed case as the noise increases; in Figure \\ref{fig:8nodes_phi}E, the perturbed (red) and unperturbed (gray) responses begin to overlap with increasing noise. We note that the amplitude of evoked responses decreases with frequency. Therefore, the same amount of noise impacts the higher frequency ($f_T \\approx 47.0$ Hz) oscillator more than the lower frequency ($f_T \\approx 11.1$ Hz) oscillator, making an evoked response more difficult to distinguish from background noise in the higher frequency case. We also note that oscillators not satisfying Equation (\\ref{eq:sin}) (i.e., $f_T \\approx \\{6.9, 29.0, 76.0\\}$ Hz when $f_D \\approx 17.9$ Hz and $f_S \\approx 29.0$ Hz) exhibit little evidence of an evoked response at any noise level.\n\\\\\n\n\\vspace{-0.075in}\n\\noindent To illustrate the utility of the golden ratio, we consider an alternative network of oscillators with frequencies organized by a factor of 2 (Figure \\ref{fig:8nodes_2}A); such integer relationships have been proposed as important to neural communication \\cite{palva_phase_2005,marin_garcia_genuine_2013,klimesch_algorithm_2013}. As expected, with only constant gain ($\\bar{g}_C = 50$) a perturbation to one node ($f_D=16$ Hz) does not impact the rest of the network (Figure \\ref{fig:8nodes_2}B). Including sinusoidal gain with frequency $f_S$ can produce cross-frequency coupling. For example, choosing $f_S=8$ Hz results in cross-frequency coupling between the $f_D=16$ Hz and $f_T=8$ Hz rhythms (Figure \\ref{fig:8nodes_2}C). Similarly, choosing $f_S=16$ Hz results in cross-frequency coupling between the $f_D=16$ Hz and $f_T=32$ Hz rhythms; however, this choice of $f_S$ also results in strong cross-frequency coupling between $f_D=16$ Hz and lower frequency rhythms ($f_T=8,4,2,1$ Hz; Figure \\ref{fig:8nodes_2}D). Importantly, we note that cross-frequency coupling typically occurs at sinusoidal gain frequencies that differ from the set of oscillator frequencies at $2^k$ Hz (vertical lines in Figure \\ref{fig:8nodes_2}D); a new set of rhythms (and rhythm generators) must exist to support cross-frequency coupling in this network.\n\\\\\n\n\\vspace{-0.075in}\n\\noindent To summarize, in a network of damped coupled oscillators (Equation \\ref{eq:dsho}), sinusoidal gain modulation supports cross-frequency coupling (Equation \\ref{eq:sin}). If oscillator and gain frequencies organize according to a multiplicative factor (Equation \\ref{eq:fk}), then cross-frequency coupling between neighboring frequencies requires a multiplicative factor of $\\phi$, the golden ratio (e.g., Figure \\ref{fig:8nodes_phi}D). While oscillators organized with a different multiplicative factor can still produce cross-frequency coupling, the frequencies of effective gain modulation are not part of the original rhythmic sequence (e.g., Figure \\ref{fig:8nodes_2}D), thus requiring the brain devote more resources to implementing a larger set of rhythms in support of cross-frequency interactions.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=14.5cm]{Figure-3.pdf}\n\\caption{\\it{\\textbf{An integer scaling between oscillators limits cross-frequency interactions.} \\textbf{(A)} In a network of oscillators with frequencies organized by a factor of 2, we perturb one oscillator (natural frequency 16 Hz, red). \\textbf{(B)} With constant gain, the impact of the perturbation is limited. \\textbf{(C)} With sinusoidal modulation at $f_S=8$ Hz, a response appears at another oscillator (natural frequency 8 Hz). \\textbf{(D)} The average response amplitude at each oscillator versus gain frequency $f_S$. Many oscillators respond when the gain frequency is 8 Hz, and responses tend not to occur at the oscillator frequencies; see Figure \\ref{fig:8nodes_phi}D for plot details. Code to simulate this network and create this figure is available \\href{https:\/\/github.com\/Mark-Kramer\/Golden-Framework\/blob\/main\/Figure-3.ipynb}{here}}.} \\label{fig:8nodes_2}\n\\end{figure}\n\n\\subsection{Rhythms organized by the golden ratio support ensembles of cross-frequency coupling}\n\nIn the previous section, we considered a network of nodes oscillating at different natural frequencies. As an alternative example, we now consider a network with two ensembles of nodes oscillating at different frequencies. The two ensembles consist of nodes oscillating at frequencies $\\phi^k$ or $\\phi^{k+2}$, where $\\phi$ is the golden ratio. With only constant gain, a perturbation to any node impacts only nodes of the same ensemble (i.e., with the same frequency). Including sinusoidal gain modulation with (intermediate) frequency $f_S= \\phi^{k+1}$, a perturbation to any node impacts nodes in both ensembles. We illustrate this in the 8-node network with 4 nodes in each ensemble oscillating at natural frequencies $\\phi^4 \\approx 6.85$ Hz or $\\phi^6 \\approx 17.9$ Hz (Figure \\ref{fig:ensemble_phi}A). With only constant gain ($\\bar{g}_C=50, \\bar{g}_S=0$), a perturbation to one $\\phi^6 \\approx 17.9$ Hz (driver) node impacts the amplitude of all other nodes in the same ensemble (Figure \\ref{fig:ensemble_phi}B). Including sinusoidal gain modulation ($\\bar{g}_C=50, \\bar{g}_S=50$) with frequency $\\phi^5 \\approx 11.1$ Hz, the same perturbation now impacts all nodes in both ensembles (Figure \\ref{fig:ensemble_phi}C). From Equation \\ref{eq:sin} we determine that two sinusoidal gain frequencies support cross-frequency coupling between the driver ($f_D= \\phi^6 \\approx 17.9$ Hz) and target ($f_T=\\phi^4 \\approx 6.85$ Hz) ensembles,\n\\begin{subequations}\n\\begin{eqnarray*}\n f_S=f_T-f_D=6.85-17.9<0.00 \\, , \\\\\n f_S=f_D-f_T=17.9-6.85=11.1 \\, \\mathrm{Hz} \\, , \\\\\n f_S=f_D+f_T=17.9+6.85=24.6 \\, \\mathrm{Hz} \\, .\n\\end{eqnarray*}\n\\end{subequations}\n\\noindent However, of these two frequencies, only the former ($f_S=11.1$ Hz) is also a golden rhythm (Figure \\ref{fig:ensemble_phi}D, box). In this case, cross-frequency coupling occurs when ensemble and gain rhythms organize in a \"golden triplet\" $(f_T, f_S, f_D) =(\\phi^k,\\phi^{k+1},\\phi^{k+2}) \\approx (6.85, 11.1, 17.9)$ Hz, where $\\phi^k+\\phi^{k+1} = \\phi^{k+2}$.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=14.5cm]{Figure-4.pdf}\n\\caption{\\it{\\textbf{Golden rhythms support coupling among an ensemble of nodes.} \\textbf{(A)} The network consists of two ensembles with frequencies: $\\phi^4$ and $\\phi^6$. \\textbf{(B)} With constant gain, perturbing a node in one ensemble impacts (red) other nodes in the same ensemble (black). \\textbf{(C)} With sinusoidal gain at frequency $f_S=11.1$ Hz, the same perturbation impacts both ensembles. \\textbf{(D)} Average amplitude response versus gain frequency for all nodes in both ensembles. The response at the unperturbed ensemble (blue) increases when the gain frequency is a golden rhythm (yellow box); see Figure \\ref{fig:8nodes_phi}D for additional plot details. Code to simulate this network and create this figure is available \\href{https:\/\/github.com\/Mark-Kramer\/Golden-Framework\/blob\/main\/Figure-4.ipynb}{here}}.} \\label{fig:ensemble_phi}\n\\end{figure}\n\n\\noindent An alternative choice of irrational frequency ratio between the brain's rhythms is Euler's number ($e$) \\cite{penttonen_natural_2003}. Repeating the simulation with two ensembles of frequency $e^k$ or $e^{k+2}$ results in cross-frequency coupling between ensembles only when $f_S=e^{k+2} \\pm e^k$ (see Figure \\ref{fig:ensemble_e} for an example with $k=2$). We therefore find similar results for the \"Euler triplet\" $(f_D, f_T,f_S)=(e^{k+2},e^k,e^{k+2} \\pm e^k)$ or specifically for $k=2$, $(f_D, f_T, f_S)=(e^4,e^2,e^4 \\pm e^2)$. However, this Euler triplet is not consistent with the ratio of $e$ observed {\\it in vivo}, where three neighboring frequency bands appear at multiplicative factors of $e$ (e.g., $(f,ef,e^2f)$) and the two slower rhythms do not sum to equal the faster rhythm (e.g., $f+ef \\neq e^2f)$. Only for three neighboring frequency bands related by the golden ratio $(f,\\phi f, \\phi^2f)$ do the frequencies of the slower rhythms sum to the faster rhythm (i.e., $f+\\phi f=\\phi^2f$).\n\\\\\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=14.5cm]{Figure-5.pdf}\n\\caption{\\it{\\textbf{Rhythms organized by Euler's number do not support coupling between ensembles of nodes.} \\textbf{(A)} The network consists of two ensembles with frequencies: $e^4$ and $e^2$. \\textbf{(B)} With constant gain, perturbing a node in one ensemble impacts (red) other nodes in the same ensemble (black). \\textbf{(C)} With sinusoidal gain at frequency $f_S=47.2$ Hz, the same perturbation impacts both ensembles. \\textbf{(D)} Average amplitude response versus gain frequency for all nodes in both ensembles. The response at the unperturbed ensemble (blue) does not increase when the gain frequency is a factor of the Euler number (black vertical lines); see Figure \\ref{fig:8nodes_phi}D for additional plot details. Code to simulate this network and create this figure is available \\href{https:\/\/github.com\/Mark-Kramer\/Golden-Framework\/blob\/main\/Figure-5.ipynb}{here}}.} \\label{fig:ensemble_e}\n\\end{figure}\n\n\\subsection{Golden rhythms establish a hierarchy of cross-frequency interactions}\nWe now consider results derived for weakly coupled oscillators, which motivated the study of (strongly) coupled damped harmonic oscillators presented above. In \\cite{izhikevich_weakly_1997,hoppensteadt_thalamo-cortical_1998}, Hoppensteadt and Izhikevich consider the general case of intrinsically oscillating neural populations with weak synaptic connections. When uncoupled, each neural population exhibits periodic activity (i.e., a stable limit cycle attractor) described by the phase of oscillation. We note that, in our study of coupled damped harmonic oscillators, we instead consider the amplitude of each oscillator. When Hoppensteadt and Izhikevich include weak synaptic connections between the neural populations, the phases of the neural populations interact only when a resonance relation exists between frequencies, i.e., \n\\begin{equation*}\n \\sum_{i}{k_i f_i}=0 ,\n\\end{equation*}\nwhere $k_i$ is an integer and not all 0, and $f_i$ is the frequency of neural population $i$. The resonance order is then defined as the summed magnitudes of the integers $k_i$,\n\\begin{equation*}\n \\mathrm{resonance\\ order}= \\sum_i ||{k_i}|| .\n\\end{equation*}\nFor the case of two neural populations, if\n\\begin{equation*}\n k_1 f_1+k_2 f_2 = 0\n\\end{equation*}\nfor integers $k_1$ and $k_2$, then\n\\begin{equation*}\n \\frac{f_2}{f_1}=-\\frac{k_1}{k_2} = \\mathrm{rational} .\n\\end{equation*}\nIn other words, if the frequency ratio $f_2\/f_1$ of the two neural populations is rational (i.e., the ratio of two integers), then the neural populations may interact, with the strength of interaction decreasing as either $k_1$ or $k_2$ increases (i.e., stronger interactions correspond to smaller resonance orders)\\footnote{See Proposition 9.14 of \\cite{izhikevich_weakly_1997}}. Alternatively, if this frequency ratio is irrational,\n\\begin{equation*}\n \\frac{f_2}{f_1} = \\mathrm{irrational} ,\n\\end{equation*}\nthen the two neural populations behave as if uncoupled.\n\\\\\n\n\\noindent Consistent with the results presented here, Hoppensteadt and Izhikevich show that golden triplets possess the lowest resonance order, and therefore the strongest cross-frequency coupling \\cite{izhikevich_weakly_1997,hoppensteadt_thalamo-cortical_1998,izhikevich_weakly_1999}. However, other resonances exist due to the recursive nature of rhythms organized by the golden ratio. To illustrate these relationships, we consider a set of golden rhythms $\\{f^k\\}$ \u2013 rhythms organized by the golden ratio so that,\n\\begin{equation} \\label{eq:hierarchy1}\n f_{k-1}+f_k=f_{k+1} \\, ,\n\\end{equation}\nwhere $k$ is an integer. Because\n\\begin{equation*}\n f_{k-1}+f_k-f_{k+1}=0\n\\end{equation*}\nthe resonance order is $3$; this golden triplet supports strong cross-rhythm communication. Replacing $k$ with $k-1$ in Equation \\ref{eq:hierarchy1}, we find\n\\begin{equation}\\label{eq:hierarchy2}\n f_{k-2}+f_{k-1}=f_k \\, .\n\\end{equation}\nThen, replacing $f_k$ in Equation \\ref{eq:hierarchy1} with the expression in Equation \\ref{eq:hierarchy2}, we find\n\\begin{equation*}\n f_{k-1}+\\big(f_{k-2}+f_{k-1}\\big)=f_{k+1}\n\\end{equation*}\nor\n\\begin{equation} \\label{eq:heirarchy3}\n f_{k-2}+2f_{k-1}-f_{k+1}=0 \\, ,\n\\end{equation}\nwhich has resonance order $4$. Continuing this procedure to replace $f_{k-2}$ in the equation above, we find\n\\begin{equation} \\label{eq:heirarchy4}\n {-f}_{k-3}+{3\\ f}_{k-1}{-\\ f}_{k+1}=0 \\, ,\n\\end{equation}\nwhich has resonance order $5$. In this way, golden rhythms support specific patterns of preferred coupling between rhythmic triplets, with the strongest coupling (lowest resonance order) between golden triplets.\n\n\\noindent As a specific example, we fix $f_{k+1}=40$ Hz and list in Table \\ref{tab:ex} the sequence of golden rhythms beginning with this generating frequency. We expect strong coupling between $(f_{k-1},f_k,f_{k+1}) = (15.3, 25, 40)$ Hz, a golden triplet, which has resonance order $3$. Using Equations \\ref{eq:heirarchy3}, \\ref{eq:heirarchy4}, and Table \\ref{tab:ex}, we compute additional triplets with higher resonance orders: $(9.4, 15.3, 40)$ Hz with resonance order $4$, and $(5.8,15.3,40)$ Hz with resonance order $5$. Continuing this procedure organizes golden rhythms into triplets with different resonance orders (Figure \\ref{fig:resonance_order}). Triplets with low resonance order appear near the target frequency of $f_{k+1}=40$ Hz (see gold, silver, and bronze circles in Figure \\ref{fig:resonance_order}), and resonance orders tend to increase for frequencies further from $f_{k+1}=40$ Hz, with exceptions (e.g., $( f_{k-1},f_k, f_{k+1})=(2.2, 9.4, 40)$ Hz has resonance order $6$). We conclude that - based on theory developed for weakly coupled oscillators - golden rhythms support both separate communication channels and a hierarchy of cross-frequency interactions between rhythmic triplets with varying coupling strengths. While here we consider three interacting rhythms, we note that the theory also applies to four (or more) interacting rhythms. The implications of these results for networks of (strongly) coupled (damped) oscillators remains unclear.\n\n\\begin{table}[tbh]\n\\caption{{\\it {\\bf Example sequence of golden rhythms.} Beginning with $f_{k+1}=40$ Hz we compute the sequence of golden rhythms by multiplying or dividing by the golden ratio.}}\n\\label{tab:ex}\n\\centering\n\\begin{threeparttable}\n\\begin{tabular}{cccccc||c||cccc}\n\\headrow\n\\thead{$f_{k-5}$} & \\thead{$f_{k-4}$} & \\thead{$f_{k-3}$} & \\thead{$f_{k-2}$} & \\thead{$f_{k-1}$} & \\thead{$f_{k}$} & \\thead{$f_{k+1}$} & \\thead{$f_{k+2}$} & \\thead{$f_{k+3}$} & \\thead{$f_{k+4}$} & \\thead{$f_{k+5}$} \\\\\n2.2 & 3.6 & 5.8 & 9.4 & 15.3 & 24.7 & 40 & 64.7 & 104.7 & 169.4 & 274.2 \\\\\n\\hline \n\\end{tabular}\n\\end{threeparttable}\n\\end{table}\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=5.5cm]{Figure-6.pdf}\n\\caption{\\it{\\textbf{Golden rhythms establish triplets with a discrete set of resonance orders.} The resonance order (numerical value, marker size) for triplets generated from 40 Hz. Lower resonance orders (3,4,5) indicated in color (gold, silver, bronze, respectively). Code to simulate this network and create this figure is available \\href{https:\/\/github.com\/Mark-Kramer\/Golden-Framework\/blob\/main\/Figure-6.ipynb}{here}}.} \\label{fig:resonance_order}\n\\end{figure}\n\n\\subsection{Four experimental hypotheses}\nWe propose that golden rhythms optimally support separate and integrated communication channels between oscillatory neural populations. We now describe four hypotheses deduced from this theory. First, if the organization of brain rhythms follows the golden ratio, then we expect a discrete sequence of three frequency bands subdivides the existing gamma frequency band, broadly defined from 30-100 Hz \\cite{buzsaki_neuronal_2004,fries_gamma_2007}, with peak frequencies separated by a factor of $\\phi$. For example, using the sequence of golden rhythms with generating frequency 40 Hz (Table \\ref{tab:ex}), we identify multiple distinct rhythms (at 40 Hz, 65 Hz, 105 Hz) corresponding to this gamma band. Consistent with this hypothesis, multiple distinct rhythms have been identified within the gamma band (e.g., \\cite{lopes-dos-santos_parsing_2018,zhang_sub-second_2019,fernandez-ruiz_entorhinal-ca3_2017,edwards_high_2005,crone_functional_1998,vidal_visual_2006,colgin_frequency_2009,colgin_slow_2015,zhou_methodological_2019}). While different choices of generating frequency produce quantitatively different results, the qualitative result is the same: organized according to the golden ratio, multiple distinct rhythms exist within the gamma frequency range, each capable of supporting a separate communication channel.\n\\\\\n\n\\noindent Second, if rhythms organize according to the golden ratio, then evidence for this relationship should exist {\\it in vivo}. To that end, we consider examples of two or more frequency bands reported in the literature (predominately in rodent hippocampus; Figure \\ref{fig:empirical}). These preliminary observations suggest that, in these cases, frequency bands separated by a factor of $\\phi$ or $\\phi^2$ commonly occur. \n\\\\\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=14.5cm]{Figure-7.pdf}\n\\caption{\\it{\\textbf{Empirical observations of rhythms organized by the golden ratio in vivo.} \\textbf{(A)} Pairs of frequencies reported in the literature; see legend. When only a frequency band is reported, we select the mean frequency of the band. \\textbf{(B)} Histogram of the frequency ratio for each point in (A). Lines (golden) indicate frequency bands organized by $\\phi \\approx 1.6$ or $\\phi^2 \\approx 2.6$. Code to create this figure is available \\href{https:\/\/github.com\/Mark-Kramer\/Golden-Framework\/blob\/main\/Figure-7.ipynb}{here}}.} \\label{fig:empirical}\n\\end{figure}\n\n\\noindent Third, if rhythms organize according to the golden ratio, then we propose that rhythmic triplets support cross-frequency communication. Nearly all existing research in cross-frequency coupling focuses on interactions between two rhythms (e.g., theta-gamma \\cite{canolty_functional_2010,lisman_theta-gamma_2013,hyafil_neural_2015}), and many measures exist to assess and interpret bivariate coupling between rhythms \\cite{hyafil_neural_2015,siebenhuhner_genuine_2020,tort_measuring_2010,aru_untangling_2015}. Yet, brain rhythms coordinate beyond pairwise interactions; trivariate interactions between three brain rhythms include coordination of beta, low gamma, and high gamma activity by theta phase \\cite{belluscio_cross-frequency_2012,lopes-dos-santos_parsing_2018,zhang_sub-second_2019,fernandez-ruiz_entorhinal-ca3_2017,colgin_frequency_2009,jiang_distinct_2020}; coordination between ripples (140-200 Hz), sleep spindles (12-16 Hz), and slow oscillations (0.5-1.5 Hz) \\cite{buzsaki_brain_2012}; and coordination between (top-down) beta, (bottom-up) gamma, and theta rhythms \\cite{fries_rhythms_2015}. To assess trivariate coupling, an obvious initial choice is the bicoherence, which assesses the phase relationship between three rhythms: $f_1$, $f_2$, and $f_1+f_2$ \\cite{barnett_bispectrum_1971,kramer_sharp_2008,shahbazi_avarvand_localizing_2018}. However, the bicoherence may be too restrictive (requiring a constant phase relationship between the three rhythms), and estimation of alternative interactions (e.g., between amplitudes and phases) will require application and development of alternative methods \\cite{haufler_detection_2019}.\n\\\\\n\n\\noindent Fourth, why brain rhythms occur at the specific frequency bands observed, and not different bands, remains unknown. To address this, we combine the golden ratio scaling proposed here with a fundamental timescale for life on Earth: the time required for Earth to complete one rotation (i.e., the sidereal period) of 23 hr, 56 min. Beginning from this fundamental frequency ($1\/86160$ Hz), we compute higher frequency bands by repeated multiplication of the golden ratio (Table \\ref{tab:earth}). Doing so, we identify frequencies consistent with the canonical frequency bands (i.e., delta, theta, alpha, beta, low gamma, middle gamma, high gamma, ripples, fast ripples; see last column of Table \\ref{tab:earth}). We note that broad frequency ranges define the canonical frequency bands, for example the gamma band from (30, 100) Hz. Therefore, model predictions that identify rhythms within a band is not surprising. We propose instead that the relevant model prediction is the subdivision of the canonical frequency bands (e.g., the 10-30 Hz beta band into two sub-bands, the 30-100 Hz gamma band into three sub-bands), not the specific frequency values identified. We note that the lower frequencies may include \"body oscillations\", such as heart rate and breathing frequency \\cite{tort_respiration-entrained_2018}, as proposed for a harmonic frequency relationship (factor of 2) in \\cite{klimesch_algorithm_2013,klimesch_frequency_2018}. We hypothesize that if intelligent life were to evolve on a planet like Earth, in a star system like our own, with neural physiology like our own, then rhythmic bands would exist with center frequencies that depend on the planet's circadian cycle. We acknowledge that this hypothesis, and the proposed association between neural rhythms and the sidereal period in Table \\ref{tab:earth}, remain speculation, without robust supporting evidence.\n\n\\begin{table}[h]\n\\caption{{\\it {\\bf Golden rhythms, beginning with the sidereal period, align with the brain's rhythms.} The period ($T$) and frequency ($f$) of rhythms beginning with the sidereal period ($T=86160$ s) and multiplying the frequency by the golden ratio (number of multiplications indicated by the value in column {\\bf Power}). Traditional frequency band labels (from \\cite{buzsaki_neuronal_2004, buzsaki_scaling_2013, roopun_temporal_2008}) indicated in the last column.}}\n\\label{tab:earth}\n\\centering\n\\begin{threeparttable}\n\\begin{tabular}{ccc||ccc||ccc r}\n\\headrow\n\\thead{Power} & \\thead{$T$ [s]} & \\thead{$f$ [Hz]} & \n\\thead{Power} & \\thead{$T$ [s]} & \\thead{$f$ [Hz]} &\n\\thead{Power} & \\thead{$T$ [s]} & \\thead{$f$ [Hz]} & Label \\\\\n0&\t86160&\t1.16E-05&\t\t12&\t268&\t0.004&\t24&\t0.83&\t1.20&\tSlow 1 \\\\\n1&\t53250&\t1.88E-05&\t\t13&\t165&\t0.006&\t25&\t0.51&\t2&\tDelta \\\\\n2&\t32910&\t3.04E-05&\t\t14&\t102&\t0.010&\t26&\t0.32&\t3&\tDelta \\\\\n3&\t20340&\t4.92E-05&\t\t15&\t63.2&\t0.016&\t27&\t0.20&\t5&\tTheta \\\\\n4&\t12571&\t7.96E-05&\t\t16&\t39.0&\t0.026&\t28&\t0.12&\t8&\tAlpha \\\\\n5&\t7769&\t1.29E-04&\t\t17&\t24.1&\t0.041&\t29&\t0.07&\t13&\tBeta1 \\\\\n6&\t4802&\t2.08E-04&\t\t18&\t14.9&\t0.067&\t30&\t0.05&\t22&\tBeta2 \\\\\n7&\t2968&\t3.37E-04&\t\t19&\t9.2&\t0.11&\t31&\t0.03&\t35&\tLow Gamma \\\\\n8&\t1834&\t5.45E-04&\t\t20&\t5.7&\t0.18&\t32&\t0.02&\t57&\tMid Gamma \\\\\n9&\t1133&\t8.82E-04&\t\t21&\t3.5&\t0.28&\t33&\t0.01&\t91&\tHigh Gamma \\\\\n10&\t701&\t0.001&\t\t 22&\t2.2&\t0.46&\t34&\t0.01&\t148&\tRipple \\\\\n11&\t433&\t0.002&\t\t 23&\t1.3&\t0.74&\t35&\t0.004&\t239&\tFast Ripples \\\\\n\\hline \n\\end{tabular}\n\\end{threeparttable}\n\\end{table}\n\n\\section{Discussion}\nWhy do brain rhythms organize into the small subset of discrete frequencies observed? Why does the alpha rhythm peak at 8-12 Hz and the (low) gamma rhythm peak at 35-55 Hz, across species \\cite{buzsaki_scaling_2013}? Why does the brain not instead exhibit a continuum of rhythms, or a denser set of frequency bands, or different frequency bands? Here we provide a theoretical explanation for the organization of brain rhythms. Imposing a ratio of $\\phi$ (the golden ratio) between the peaks of neighboring frequency bands, we constrain activity to a small subset of discrete brain rhythms, consistent with those observed {\\it in vivo}. Organized in this way, brain rhythms optimally support the separation and integration of information in distinct rhythmic communication channels.\n\\\\\n\n\\noindent The framework proposed here combines insights developed in existing works. Mathematical analysis of weakly coupled oscillators established the importance of resonance order for effective communication between neural populations oscillating at different frequencies \\cite{izhikevich_weakly_1997,hoppensteadt_thalamo-cortical_1998,izhikevich_weakly_1999,nunez_brain_2010}. Experimental observations and computational models have established the importance of brain rhythms \\cite{buzsaki_rhythms_2011,whittington_inhibition-based_2000,wang_neurophysiological_2010}, their interactions \\cite{canolty_functional_2010,fries_rhythms_2015}, and their organization according to the golden ratio \\cite{roopun_temporal_2008,kramer_rhythm_2008,roopun_period_2008}. Here, we combine these previous results with simulations and analysis of a network of damped, coupled oscillators in support of the proposed theory.\n\\\\\n\n\\noindent The framework proposed here is consistent with existing theories for the role of brain rhythms. Like the communication-through-coherence (CTC) hypothesis \\cite{fries_mechanism_2005,fries_rhythms_2015} and the frequency-division multiplexing hypothesis \\cite{akam_oscillatory_2014,akam_efficient_2012}, in the framework proposed here neural populations communicate dynamically along anatomical connections via coordinated rhythms. Organization by the golden ratio complements these existing theories in two ways. First, by proposing which rhythms participate \u2013 namely, rhythms spaced by factors of the golden ratio. Second, by proposing the importance of three rhythms to establish cross-frequency interactions and proposing a hierarchical organization to these interactions.\n\\\\\n\n\\noindent We considered a network of damped, coupled oscillators with sinusoidal gain modulation. In that network, cross-frequency coupling occurs when the gain frequency equals the sum or difference of the oscillator frequencies (Equation \\ref{eq:sin}). This result holds without additional restrictions on the oscillator or gain frequencies. However, golden rhythms are unique in that oscillator and gain frequencies chosen from this set support cross-frequency coupling; no rhythms beyond this set are required. Alternative irrational scaling factors (e.g., Euler's number $e$) establish different sets of oscillator frequencies (e.g., $\\ldots e^1, e^2, e^3, \\ldots$) and separate communication channels, but require gain frequencies beyond this set to support cross-frequency coupling. In this alternative scenario, two distinct sets of rhythms exist: one reflecting local population activity, and another the cross-frequency coupling between populations. Rhythms organized by the golden ratio support a simpler framework: one set of frequencies (oscillator and gain) that reflect both local oscillations and their cross-frequency coupling. Golden rhythms are the smallest set of rhythms that support both separate communication channels and their cross-frequency interactions. Requiring fewer rhythms simplifies implementation, reducing the number of mechanisms required to produce these rhythms.\n\\\\\n\n\\noindent These results are consistent with existing proposals that the golden ratio organizes brain rhythms and minimizes cross-frequency interference \\cite{roopun_period_2008,weiss_golden_2003,pletzer_when_2010}. We extend these proposals by showing how triplets of golden rhythms facilitate cross-frequency coupling. An integer ratio of $2$ between frequency bands (with bandwidth determined by the golden ratio) provides an alternative organization to support cross-frequency coupling \\cite{klimesch_algorithm_2013,klimesch_frequency_2018}. In this scenario, cross-frequency interactions have been proposed to occur via a shift in frequency. For example, two regions - with an irrational frequency ratio - remain decoupled until the center frequencies shift to establish $1:2$ phase coupling \\cite{rodriguez-larios_mindfulness_2020,rodriguez-larios_thoughtless_2020}. We instead propose both regions maintain their original frequencies and couple when an appropriate third rhythm appears (e.g., a golden triplet). Our simulation results suggest more widespread coupling between populations oscillating at a $1:2$ frequency ratio compared to a golden ratio (Figure \\ref{fig:8nodes_2}D). Interpreted another way, integer ratios between frequency bands may facilitate a \"coupling superhighway\"; a target region shifts frequency to enter the coupling superhighway and receive strong inputs from all upstream regions oscillating at integer multiples (or factors) of the target frequency. Rhythms organized by a golden ratio require coordination with a third input to establish cross-frequency coupling. Investigating these proposals requires analysis of larger networks with multiple rhythms, and perhaps multiple organizing frequency ratios.\n\\\\\n\n\\noindent While we do not propose the specific mechanisms that support golden rhythms, proposals do exist. A biologically motivated sequence exists to create golden rhythms from the beta1 (15 Hz), beta2 (25 Hz), and gamma (40 Hz) bands. Through {\\it in vitro} experiments and computational models, a process of period concatenation \u2013 in which the mechanisms producing the faster beta2 and gamma rhythms concatenate to create the slower beta1 rhythm \u2013 was proposed \\cite{roopun_period_2008,roopun_temporal_2008,kramer_rhythm_2008}. Alternatively, golden rhythms may emerge when two input rhythms undergo a nonlinear transformation \\cite{haufler_detection_2019,ahrens_spectral_2002}. The framework proposed here suggests that the emergent rhythms \u2013 appearing at the sum and difference of the two rhythms \u2013 may support local coordination of the input rhythms. \n\\\\\n\n\\noindent The simplicity of the proposed framework (compared to the complexity of brain dynamics) results in at least four limitations. First, brain rhythms appear as broad spectral bands, not sharply defined spectral peaks. Therefore, the meaning of a precise frequency ratio, or the practical difference between an irrational frequency ratio ($\\phi$) and a rational frequency ratio (e.g., $1.6$) is unclear. Second, rhythm frequencies may vary systematically and continuously with respect to stimulus or behavioral parameters \\cite{ray_differences_2010,kropff_frequency_2021}. Whether the brain maintains a constant frequency ratio between varying frequency rhythms, and what mechanisms could support this coordination, is unclear. Third, no evidence suggests the tuning of brain rhythms specifically to support separate communication channels. Instead, brain rhythms may occur at the frequencies observed due to the biological mechanisms available for coordination of neural activity (e.g., due to the decay time of inhibitory postsynaptic potentials that coordinate excitatory cell activity). Fourth, identification of rhythms in noisy brain signals remains a practical challenge, with numerous opportunities for confounds \\cite{zhou_methodological_2019,cole_brain_2017,donoghue_methodological_2021}. Therefore, the best approach to compare this theory with data remains unclear.\n\\\\\n\n\\noindent However, the simplicity of the golden framework is also an advantage. The framework consists of only one parameter (the golden ratio) compared to the many \u2013 typically poorly constrained \u2013 parameters of biologically detailed models of neural rhythms. In this way, the golden framework is broadly applicable and requires no specific biological mechanisms or rhythm frequencies; instead, only the relationship between frequencies is constrained.\n\\\\\n\n\\noindent No theoretical framework exists to explain the discrete set of brain rhythms observed in nature. Here, we propose a candidate framework, simply stated: brain rhythms are spaced according to the golden ratio. This simple statement implies brain rhythms establish communication channels optimal for separate and integrated information flow. While the specific purpose of brain rhythms remains unknown, perhaps the brain evolved to these rhythms in support of efficient multiplexing on a limited anatomical network.\n\n\\section*{Acknowledgements}\nThe author would like to acknowledge Dr.\\ Catherine Chu for writing assistance and tolerating many conversations about the golden ratio. \n\n\\printendnotes\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\nCredit reports have played a central role in consumer lending in the United States since the 1950's and are widely used around the globe. Today credit agencies collect hundreds of variables on each individual who is active in the consumer credit market. The central idea is to capture a consumer's credit worthiness through their past history of borrowing and payments. Most consumers are familiar with the aggregation of this information into a credit score which they often encounter when they apply for a mortgage or a car loan. By capturing the detailed credit history of a consumer, the report provides a unique observational window of the behavior of individuals across a number of core economic activities such as housing, credit card spending, car purchases etc. Credit reports thus capture a multi-modal view of individual behavior. Importantly, consumers retain some private information concerning their unobservable traits, including their underlying health status, those traits, unobservable to the researcher, are however central to the consumer decision process on how to behave in the credit market: i.e. how much to borrow, when, for what purpose, whether to repay, etc. The relevance of such asymmetric information, between parties of a contract, is a centerpiece of contract theory (see \\cite{StiglitzWeiss1981}, and \\cite{BoltonDewatripont2005}).\nAt the same time, various major shocks experienced by the consumer will have both short and long term implications on consumers' behavior which will be indirectly captured in the credit data. We use recent developments in machine learning to show that it is possible to predict a seemingly unrelated outcome, mortality, from the data recorded in a consumer report. \nThere are three channels through which we believe our approach has a large potential to predict outcomes: i. asymmetric or private information; ii. economic shocks leading the deterioration in health (access to treatment and directly linked to lower resources and the so call death of despair); iii. as a tangible signal of an occurring major health shock which would not be recorded otherwise in easily available data.\n\nWe will not try to pinpoint which one is the most likely channel in this project, they could all be at play. We see our work as pioneering the use of large available data in a specific domain to inform decision-making in seemingly unrelated domains.\n\nWe also note that the ability to use available Big Data generated in a specific domain to predict outcomes in seemingly unrelated domains is both an opportunity for innovation and also a potential privacy concern (\\cite{harding2014future}). While not specifically addressing the current COVID-19 crisis, this study highlights the potential of such data to capture the extent to which complex factors such as economic circumstances and choices can be predictive of health outcomes including mortality.\n\nTo the best of our knowledge, this is the first paper to study the relationship between credit and mortality directly. However, there is a rich literature that links mortality and health to other economic variables and economic activity. For example, it has been shown that heath is counter-cyclical in the sense that during economic booms health tends to deteriorate \\cite{ruhm2000recessions, ruhm2003good, ruhm2005healthy}. Economic growth is associated with increased levels of obesity and a decrease in physical activities, diet quality and leisure time. Also, a reduction of unemployment is associated with a fast increase in coronary heart disease \\cite{ruhm2007healthy}. An experimental study showed that higher income is associated with less risk of cardiovascular diseases \\cite{wang2019longitudinal}. The relationship however appears to be heterogeneous and it has been documented that recessions are correlated with increases in infant mortality \\cite{alexander2011quantifying} and differential across age, race, and education groups \\cite{HoynesMillerSchaller2012}.\n\nResearch also attempts to identify the underlying mechanisms for the previously mentioned counter-cyclical evidence \\cite{stevens2015best}. The evidence is that during economic growth mortality increases mostly amongst elderly women, which suggests that this relation may be linked to factors other than labor. This assumption is supported by the fact that health care may also be counter-cyclical in the sense that staff hiring in nursing facilities increases during recessions. Another theory states that higher mortality is actually related to worst economic conditions during early life \\cite{van2006economic}.\n\nOur work contributes this literature of mortality and economics by looking at how credit operations and individual death probabilities are related. We use a very rich data set of microdata from the Experian credit bureau, which is one of the agencies responsible for the credit scores in the United States. We follow a large pool of individuals credit activities through the years and to estimate a modified actuarial life table\\footnote{The actuarial life table shows the probability of death within the next year by age and gender. Examples can be found in the Social Security Administration page: www.ssa.gov.} that uses detailed credit variables to get more accurate results. We use machine learning models to deal with the complexity of the data in a pool of more than 2 million individuals (a random 1\\% sample of the US population with credit score) and more than a thousand variables in the models. \nTo reiterate, our analysis provides an individual level forecasting exercise on mortality making use of available data collected by credit reporting agencies. The ability to predict mortality using routinely performed credit operations opens up to possibility for policy interventions as well as for tailor-made contracts, but at the same time raises privacy concerns over and beyond the classic sharing of sensitive health information\n\n\n\\section{Data}\n\n\nOur data comes from the Experian Credit Bureau. \nIt consists of a yearly sample of more than two million anonymous individuals and 429 variables covering the period between 2004 and 2016. The data is representative for the population of individuals in the US with access to credit.\\footnote{Overall over 80\\% of the US adult population is represented in the data, with a coverage increasing by age up to age 65 where below 5\\% of people do not have a credit score. See for example \\cite{LeevanderKlaauw2010}.} The variables are divided in categories such as mortgage loans, car loans, credit cards, installments, etc. The full list of categories is presented in table \\ref{tab:groups}. Within the categories there are variables such as number of trades, trade balance, delinquency, etc. A ``deceased flag\" in the data will be used to create the mortality variable used in the estimation process. The state of residency is also recorded in the data and will be used separately as discussed below due to potentially confounding effects of geography on mortality given the health disparities in the US. Similar data sets were used to estimate improved credit scores and consumer credit risk using machine learning methods \\cite{albanesi2019predicting,khandani2010consumer} but have not been evaluated for their potential to predict outcomes recorded in the data such as mortality. \n\nThe deceased flag is 1 if the individual is dead or died within the reference year. We use this variable to define a mortality variable that is 1 if the individual died in the reference year and 0 otherwise. This is going to be our outcome variable in the machine learning models. \n\n\n\n\\begin{table}[htb]\n\\caption{Experian Groups of Variables}\n\\label{tab:groups}\n\\begin{tabular}{cc}\n\\hline\nGroup & Description \\\\ \\hline\nALJ & Joint Trades \\\\\nALL & ALL Trade Types \\\\\nAUA & Auto Loan or Lease trades \\\\\nAUL & Auto Lease trades \\\\\nAUT & Auto Loan trades \\\\\nBCA & Bankcard Revolving and charge trades \\\\\nBCC & Bankcard Revolving trades \\\\\nBRC & Credit Card Trades \\\\\nBUS & Personal Liable Business Loan Line of Credit \\\\\nCOL & Collection trades \\\\\nCRU & Credit Union \\\\\nFIP & Personal Finance \\\\\nILJ & Joint Installment Trades \\\\\nILN & Installment trades \\\\\nIQ & Inquiries \\\\\nMTA & Mortgage type trades \\\\\nMTF & First mortgage trades \\\\\nMTJ & Joint mortgage trades \\\\\nMTS & second mortgage trades \\\\\nPIL & Personal installment trades \\\\\nREC & Recreational Merchendise trades \\\\\nREJ & Joint Revolving trades \\\\\nREV & Revolving trades \\\\\nRPM & Real-estate property managment trades \\\\\nRTA & Retail trades \\\\\nRTI & Installment retail trades \\\\\nRTR & Revolving retail trades \\\\\nSTU & Student trades \\\\\nUSE & Authorized user trades \\\\\nUTI & Utility trades \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\\subsection{Estimation Details}\n\n\nOur estimation uses the full data set available to us consisting of 429 credit variables for 2.2 million individuals covering the period from 2004 to 2016. Besides credit, we also have age and state of residency and we build a variable that counts how many times an individual moved to a different state in the training sample. Our goal is to estimate the probability of death for each individual within the next year using all these variables in the same fashion as the life table published by the Social Security Administration (SSA) conditional only on age and gender. Since our data does not have gender, gender is a protected category and cannot be used in credit decisions and therefore it is not available to us, our benchmark is the probability of death within the next year conditional only on age. Our estimates are not directly comparable with SSA numbers because our sample comprises people with a valid credit score, i.e. they are likely to be wealthier on average than a random sample taken from the entire population, as mentioned earlier for the age groups above 25 the sample covers over 90\\% of the US population.\n\nLet $y_{i,t}$ be a binary variable that is 1 if individual $i$ died in period $t$, $X_{i,t}$ contains all credit variables and $Z_{i,t}$ contains variables that are not related to credit, such as age and state of residency. Our estimation framework can be defined by the following equation:\n\n\n\\begin{equation}\n\\label{eq:1}\n P(y_{i,t+1} = 1 | Z_{i,t}, X_{i,t}, X_{i,t-1},\\dots,X_{i,t-k}) = f(Z_{i,t}, X_{i,t}, X_{i,t-1},\\dots,X_{i,t-k}), \n\\end{equation}\nwhere $f(\\cdot)$ is a general function that will depend on the chosen model and $k$ is the number of lags. Our out-of-sample period is from 2012 to 2016. For each out-of-sample year we estimated several models in the framework of equation \\ref{eq:1} using as many lags as possible given the timing of the observation in relation to the length of the available data. The model used to predict mortality in 2012, for example, was estimated with data from 2004 to 2011 (7 lags of the credit variables), which accounts for 3055 variables. The model used to predict mortality in 2016 uses all data from 2004 to 2015, which results in 11 lags and 4771 variables. \n\nThe models we used for the function $f(\\cdot)$ are the Random Forest \\cite{breiman2001random} and the Gradient Boosting\\footnote{Details on the algorithms, implementation and tuning of both models are available in the Supplementary Information.} \\cite{friedman2001greedy} in its stochastic version following \\cite{friedman2002stochastic} and with early stopping \\cite{zhang2005boosting}. The choice of these models is due to several facts. First, they can easily deal with a large number of variables and observations and the algorithms are computationally efficient in very large data sets such as the one we used here, where the outcome variable is an event of very low probability \\cite{pike2019bottom}; second, these models are suited for complex data sets where one expects to have many interactions between variables; third, these models are known to be consistent \\cite{zhang2005boosting, scornet2015consistency}; finally, both Gradient Boosting and Random Forest are well established machine learning with several successful applications in many different fields \\cite{ehrentraut2018detecting, touzani2018gradient, li2018machine}. Our aim is to show the feasibility of the mortality prediction using established techniques which are widely available.\n\n\n\n\\subsection{Predicting Mortality}\n\n\nTable \\ref{tab:probs} shows the estimated probabilities of dying conditional to the true outcomes from a model trained at $t$ and tested in $t+1$. The values were averaged across all out-of-sample years (2012-2016). The first row is the unconditional probability of death and the remaining rows are the probability of death when the true outcome was death ($y = 1$) and when it was not death ($y = 0$) for the models described in table \\ref{tab:abbrev}. The best way to understand this table is to compare for a given model the difference between the probabilities assigned to death when $y=0$ and to survival when $y = 1$. A big gap between these two means that the model is able to allocate higher probabilities to individuals that actually died in year $t+1$.\nWe notice that conditional on age we have a very small improvement compared to the unconditional model given that the model is assigning bigger probabilities to individuals where the true outcome was death. The inclusion of state of residency does not bring significant improvements however. This is likely due to the fact that while mortality rates differ by geography, in the absence of more detailed location information for the individual, the state of residence by itself carries little predictive power. However, when we evaluate the models with credit data variables, the improvement is significant, especially for older people. The Gradient Boosting with credit assign twice the probability of death in cases where the individual actually died for most age groups. The Age model, on the other hand assigns probabilities of death less than 10\\% larger to people that died in $t+1$ and the models with State dummies do not provide a sizeable improvement. Finally, the Random Forest model performs better, with assigned probability of dying at least 60\\% larger for people who actually died at $t+1$. \n\nWe present a more formal comparison between models in table \\ref{tab:auc}, which shows the Area Under the Curve (AUC) for all models in all years and the DeLong \\cite{delong1988comparing} test to compare their differences. The AUC is the area under the curve of the false positive rate (1-Specificity) against the true positive rate (Sensitivity) for all possible cuts between 0 and 1. If the AUC is larger than 0.5 it means that the model has improved predictive ability compared to the unconditional probabilities. In other words, the AUC tells how much a model is able to distinguish between classes. The values in parentheses in table \\ref{tab:auc} show the p-value of the test that compares the Credit Random Forest and the Credit Gradient Boosting to the Age only model. The null hypothesis is that the difference between the AUC of the two tested models is 0. We omitted the models using the state of residence from this table to given that table \\ref{tab:probs} already shows that these models do not significantly improve on the models conditional on age alone. The results show that for individuals older than 55 the models with credit data have significantly higher AUCs than traditional actuarial calculations based on age. As we start looking at younger individuals the Random Forest and the Gradient Boosting get closer to the age model to the point where there is no significant difference between them. However, this only happens in general for groups of people younger than 50 is we consider the Gradient Boosting and table \\ref{tab:probs} shows that even for younger groups the Gradient Boosting and the Random Forest separate better between the classes. The main explanation to this result is that the number of deaths in the sample for younger groups is very small, as expected, which makes it harder for the models to understand the relation between credit and mortality in a way that can be generalized. Lastly, in table \\ref{tab:auc}, the Gradient Boosting algorithm produces slightly higher AUCs than the Random Forest in most cases for older individuals, where the credit models outperform the simpler model based on age. At younger ages, our improved models, while still performing much better than simpler models, suffer from the scarcity of events in the data and somewhat shorter credit histories for younger individuals.\n\n\n\n\n\n\n\\begin{table\n\\caption{Model Abbreviations}\n\\label{tab:abbrev}\n\\begin{tabular}{lcc}\n\\hline\n\\textbf{Abbreviation} & \\textbf{Model} & \\textbf{Variables} \\\\ \\hline\nUncond. & Unconditional Probabilities & \\\\\nAge & Conditional Probabilities & Age \\\\\nState Lin. & Linear - Logistic & Age, State \\\\\nState GB & Gradient Boosting & Age, State \\\\\nState RF & Random Forest & Age, State \\\\\nCredit GB & Gradient Boosting & Age, State, Credit \\\\\nCredit RF & Random Forest & Age, State, Credit \\\\ \\hline\n\\end{tabular}%\n\\end{table}\n\n\n\\begin{table\n\\begin{threeparttable}\n\\caption{Average Estimated Probability of dying conditional to the true outcomes in $t+1$}\n\\label{tab:probs}\n\\resizebox{\\textwidth}{!}{%\n\\begin{tabular}{lcccccccccc}\n\\hline\n & & \\multicolumn{9}{c}{\\textbf{Age Cohorts}} \\\\\n & & 81-100 & 76-80 & 71-75 & 66-70 & 61-65 & 56-60 & 51-55 & 46-50 & 41-45 \\\\ \\hline\nUncond. & & 2.99 & 1.52 & 0.98 & 0.61 & 0.41 & 0.25 & 0.16 & 0.10 & 0.07 \\\\ \\hline\nAge & y=1 & 3.27 & 1.54 & 1.00 & 0.63 & 0.42 & 0.25 & 0.16 & 0.10 & 0.07 \\\\\n & y=0 & 3.03 & 1.52 & 0.99 & 0.62 & 0.41 & 0.25 & 0.16 & 0.10 & 0.07 \\\\ \\hline\nState Lin. & y=1 & 3.30 & 1.55 & 1.01 & 0.64 & 0.43 & 0.25 & 0.16 & 0.10 & 0.07 \\\\\n & y=0 & 3.02 & 1.52 & 0.99 & 0.62 & 0.41 & 0.25 & 0.16 & 0.10 & 0.07 \\\\ \\hline\nState GB & y=1 & 3.31 & 1.56 & 1.00 & 0.64 & 0.43 & 0.26 & 0.17 & 0.11 & 0.06 \\\\\n & y=0 & 3.00 & 1.52 & 0.99 & 0.62 & 0.41 & 0.25 & 0.16 & 0.10 & 0.07 \\\\ \\hline\nState RF & y=1 & 3.29 & 1.55 & 1.00 & 0.63 & 0.42 & 0.26 & 0.16 & 0.10 & 0.06 \\\\\n & y=0 & 3.01 & 1.52 & 0.99 & 0.62 & 0.41 & 0.25 & 0.16 & 0.10 & 0.07 \\\\ \\hline\nCredit GB & y=1 & 4.06 & 0.76 & 0.43 & 0.29 & 0.19 & 0.07 & 0.04 & 0.03 & 0.01 \\\\\n & y=0 & 1.93 & 0.42 & 0.25 & 0.15 & 0.09 & 0.04 & 0.02 & 0.01 & 0.01 \\\\ \\hline\nCredit RF & y=1 & 5.56 & 2.68 & 1.82 & 1.28 & 0.87 & 0.51 & 0.34 & 0.26 & 0.15 \\\\\n & y=0 & 3.63 & 1.99 & 1.33 & 0.88 & 0.60 & 0.35 & 0.23 & 0.17 & 0.10 \\\\ \\hline\n\\end{tabular}\n}%\n\\begin{tablenotes}\n\\small\n\\item The table shows the estimated probabilities of dying conditional to the true outcomes from a model trained in $t$ and tested in $t+1$. For example, under Prob by Age we have the probability of dying given that the true outcome was death ($y=1$) and the probability of dying given that the true outcome was not death ($y = 0$). The results were averaged across all estimation windows.\n\\end{tablenotes}\n\\end{threeparttable}\n\\end{table}\n\n\\begin{landscape}\n\\begin{table}\n\\begin{threeparttable}\n\\caption{Out-of-Sample Area Under the Curve for all Test Years}\n\\label{tab:auc}\n\\begin{adjustbox}{max width=\\textwidth}\n\\begin{tabular}{lccccccccccccccccccc}\n\\hline\n{ \\textbf{}} & \\multicolumn{17}{c}{{\\textbf{Out-of-Sample Area Under the Curve}}} & { \\textbf{}} & { \\textbf{}} \\\\\n{ } & \\multicolumn{3}{c}{{ Test year = 2012}} & { } & \\multicolumn{3}{c}{{ Test year = 2013}} & { } & \\multicolumn{3}{c}{{ Test year = 2014}} & { } & \\multicolumn{3}{c}{{ Test year = 2015}} & { } & \\multicolumn{3}{c}{{ Test year = 2016}} \\\\\nCohort & Age & Credit GB & Credit RF & & Age & Credit GB & Credit RF & & Age & Credit GB & Credit RF & & Age & Credit GB &Credit RF & & Age &Credit GB & Credit RF \\\\ \\cline{1-4} \\cline{6-8} \\cline{10-12} \\cline{14-16} \\cline{18-20} \n81-100 & 0.587 & 0.659 & 0.657 & & 0.576 & 0.673 & 0.678 & & 0.578 & 0.693 & 0.684 & & 0.576 & 0.682 & 0.663 & & 0.562 & 0.703 & 0.683 \\\\\n & & (0.000) & (0.000) & & & (0.000) & (0.000) & & & (0.000) & (0.000) & & & (0.000) & (0.000) & & & (0.000) & (0.000) \\\\\n76-80 & 0.523 & 0.604 & 0.619 & & 0.525 & 0.607 & 0.607 & & 0.526 & 0.608 & 0.610 & & 0.542 & 0.608 & 0.603 & & 0.542 & 0.612 & 0.609 \\\\\n & & (0.000) & (0.000) & & & (0.000) & (0.000) & & & (0.000) & (0.000) & & & (0.000) & (0.000) & & & (0.000) & (0.000) \\\\\n71-75 & 0.518 & 0.596 & 0.620 & & 0.521 & 0.595 & 0.619 & & 0.522 & 0.592 & 0.615 & & 0.534 & 0.606 & 0.595 & & 0.546 & 0.587 & 0.594 \\\\\n & & (0.000) & (0.000) & & & (0.000) & (0.000) & & & (0.000) & (0.000) & & & (0.000) & (0.000) & & & (0.000) & (0.000) \\\\\n66-70 & 0.545 & 0.603 & 0.603 & & 0.543 & 0.619 & 0.611 & & 0.528 & 0.612 & 0.592 & & 0.531 & 0.612 & 0.607 & & 0.535 & 0.617 & 0.617 \\\\\n & & (0.000) & (0.000) & & & (0.000) & (0.000) & & & (0.000) & (0.000) & & & (0.000) & (0.000) & & & (0.000) & (0.000) \\\\\n61-65 & 0.491 & 0.583 & 0.592 & & 0.519 & 0.598 & 0.594 & & 0.531 & 0.602 & 0.604 & & 0.546 & 0.594 & 0.605 & & 0.528 & 0.599 & 0.592 \\\\\n & & (0.000) & (0.000) & & & (0.000) & (0.000) & & & (0.000) & (0.000) & & & (0.000) & (0.000) & & & (0.000) & (0.000) \\\\\n56-60 & 0.511 & 0.619 & 0.568 & & 0.550 & 0.610 & 0.574 & & 0.539 & 0.595 & 0.595 & & 0.545 & 0.591 & 0.582 & & 0.537 & 0.594 & 0.590 \\\\\n & & (0.000) & (0.001) & & & (0.000) & (0.152) & & & (0.000) & (0.001) & & & (0.002) & (0.015) & & & (0.000) & (0.000) \\\\\n51-55 & 0.530 & 0.616 & 0.601 & & 0.502 & 0.566 & 0.580 & & 0.545 & 0.591 & 0.579 & & 0.527 & 0.591 & 0.573 & & 0.541 & 0.578 & 0.562 \\\\\n & & (0.000) & (0.001) & & & (0.002) & (0.000) & & & (0.018) & (0.083) & & & (0.000) & (0.013) & & & (0.040) & (0.228) \\\\\n46-50 & 0.559 & 0.650 & 0.595 & & 0.515 & 0.617 & 0.558 & & 0.545 & 0.599 & 0.581 & & 0.553 & 0.603 & 0.583 & & 0.510 & 0.610 & 0.577 \\\\\n & & (0.001) & (0.184) & & & (0.000) & (0.085) & & & (0.042) & (0.148) & & & (0.028) & (0.216) & & & (0.000) & (0.003) \\\\\n41-45 & 0.539 & 0.557 & 0.551 & & 0.536 & 0.574 & 0.597 & & 0.546 & 0.594 & 0.537 & & 0.549 & 0.549 & 0.527 & & 0.536 & 0.592 & 0.575 \\\\\n & & (0.582) & (0.706) & & & (0.213) & (0.043) & & & (0.108) & (0.746) & & & (0.973) & (0.438) & & & (0.083) & (0.187) \\\\ \\hline\n\\end{tabular}\n\\end{adjustbox}\n\\begin{tablenotes}\n\\small\n\\item The table shows the out-of-sample Area Under the Curve (AUC) for all test years, cohorts and models. Values in parenthesis are \\\\ p-values from \\citet{delong1988comparing} test for the Random Forest and the Gradient Boosting against the probabilities conditional on \\\\ age only. The null hypothesis is that the difference between the AUC of both models is 0.\n\\end{tablenotes}\n\\end{threeparttable}\n\\end{table}\n\n\\end{landscape}\n\nThe same results presented in table \\ref{tab:auc} can be observed in more details in figures (\\ref{f:auc2012} and \\ref{f:auc2016}). These figures show the AUC plots for years 2012 and 2016 and all age cohorts. It is very clear that all curves deviate more from the $45^{\\circ}$ line for cohorts with older people. The small deviations in younger people are likely due to the low mortality rate for these cohorts resulting in a number of deaths in each year in the sample that is insufficient to successfully apply these algorithms. The same behavior persists through all out-of-sample years. The number of lags does not seem to increase the performance of the models. The 2012 model had 7 lags and the 2016 model had 11 lags. \nWe go back to the discussion on predictive value of distant lags in the next section.\n\n\n\\begin{figure}[htb]\n\\caption{Out-of-Sample Area Under The Curve - 2012}\n\\label{f:auc2012}\n\\includegraphics[width=0.95\\textwidth]{2012.pdf}\n\\end{figure}\n\n\\begin{figure}[htb]\n\\caption{Out-of-Sample Area Under The Curve - 2016}\n\\label{f:auc2016}\n\\includegraphics[width=0.95\\textwidth]{2016.pdf}\n\\end{figure}\n\n\\subsection{Variable Importance}\n\nIn this subsection we address which variables are more relevant to reduce the prediction errors in our models. We present results for the Gradient Boosting, which was slightly more accurate in general than the Random Forest. Figure \\ref{f:importance_group} shows the relative importance between the 10 most relevant groups of variables by age cohorts. The list of all groups is presented in table \\ref{tab:groups}. The four most important groups (BCA, BCC, BRC and REV) are related to credit and bank cards and revolving trades, unsecured credit. Mortgage and joint trades variables (MTA, MTF, ALJ) are also of some importance. Auto loan (AUA) variables also appear in the most important groups for all cohorts. As for the difference between age cohorts, mortgage and auto related loans are less relevant for older people than for younger ones. Of less importance but still worth noting are Installment trades. These results seem intuitive given that credit cards and other revolving trades are classes of credit that can change very fast with individual circumstances. Furthermore, mortgage decisions are very central to many younger individuals and are driven by long run expectations. Likewise, one would expect mortgage and auto loans to be less important for older individuals. Overall, though the pattern of variable importance is remarkably consistent across age cohorts. The only exception appears to be the cohort of consumers over 80 years old. It is possible that many of these consumers ``simplify\" their financial life. For example mortgages are paid off, the elderly tend to reduce consumption in many areas and thus need to rely less on credit cards to smooth out consumption over time etc.\n\n\nFigure \\ref{f:importance_lags} shows the relative importance between the 10 most relevant groups of variables by lags. What is quite interesting is that for the class of variables that are the most responsive in the short run, they are also very central for the long lags. For example, the 5-years lags overall appears to have about a 1\/3 importance with respect to the 1-year lag, yet that rules out some simple story of running into default because of death or health shocks only. It seems that part of the predictive power, especially in long-lags, is due to individual revelation of their types rather than reaction to shocks. This is consistent with the hypothesized mechanism where credit behavior reveals the particular individual type, a story of private information rather than economic shocks. \n\nNote that the most relevant variables are the same as the ones in figure \\ref{f:importance_group}. However, we can see a bigger change in their relative importance as we move from lag 1 to lag 7. Inquiries (IQ) appear only for bigger lags (5 and 7) and in lag 1 we have collection trades (COL) as a relevant group. Finally, figure \\ref{f:importance_lag_only} shows the relative importance between lags. Lag 1 clearly dominates the remaining lags but the overall importance of lags 2 to 7 combined is bigger than lag 1 alone. This is evidence that the relationship between credit and mortality has a long term component and financial positions taken several years in the past may be predictive of mortality (and potentially other health outcomes) in the present. We take this as suggestive of sizeable private information retained by consumers on their health status, i.e. the predictive power of distant past behavior could be reflecting private information on one's general health and life-expectancy rather than short term shocks. It is less likely that these results are consistent with a simple story of a health shock leading to bankruptcy or an income shock reducing the individual ability to access health care. It is interesting to note that while there is a sharp reduction in the importance of the variables from lag 1 to lag 2, the importance after does not decline perhaps as rapidly as we might expect. We think that this is consistent with a process where credit reports capture both immediate shocks and more persistent effects. For example mortality is driven both by sudden health events such as a stroke and long run uncontrolled blood pressure which is related to lifestyle choices such as a sedentary life and bad nutrition.\n\n\n\\begin{figure}[htb]\n\\caption{Variable Importance by Groups and Age Cohorts}\n\\label{f:importance_group}\n\\includegraphics[width=0.95\\textwidth]{importance_groups.pdf}\\\\\n\\floatfoot{The figure shows the relative importance of the 10 most relevant groups of variables based on the Experian classification for several age cohorts. Values were adjusted so that the sum of all bars is equal 1. The importance was measured as the contribution of each variable to reduce the model prediction error.}\n\\end{figure}\n\n\\begin{figure}[htb]\n\\caption{Variable Importance by Groups and Lags}\n\\label{f:importance_lags}\n\\includegraphics[width=0.95\\textwidth]{importance_lags.pdf}\\\\\n\\floatfoot{The figure shows the relative importance of the 10 most relevant groups of variables based on the Experian classification for lags 1-7. Values were adjusted so that the sum of all bars is equal 1. The importance was measured as the contribution of each variable to reduce the model prediction error.}\n\\end{figure}\n\n\n\\begin{figure}[htb]\n\\caption{Importance by lags}\n\\label{f:importance_lag_only}\n\\includegraphics[width=0.5\\textwidth]{importance_lag_only.pdf}\\\\\n\\floatfoot{The figure shows the relative importance between lags 1 to 7 averaged across all models. Values were adjusted so that the sum of all bars is equal 1. The importance was measured as the contribution of each variable to reduce the model prediction error.}\n\\end{figure}\n\n\\section{Conclusion}\n\n\nWe have shown that data routinely collected by credit bureaus such as Experian in the US have substantial power in predicting mortality at the individual level. We employed data on over 2 million individuals and 429 credit related variables to estimate Gradient Boosting and Random Forest models for the probability of an individual dying within the next year. Our models significantly outperform actuarial tables conditional on age in terms of AUC, which shows the model ability to distinguish between classes. Moreover, the Gradient Boosting assigns probabilities of death twice as big to individuals that actually died at $t+1$ against an increase of less than 10\\% in the actuarial age model. A limitation of our study is that we do not observe gender in our data which is commonly reported in actuarial tables, but it is unavailable in credit data. The predictive performance of our algorithms improves with the age of the individual. It is not clear whether this is due to the increased information content of credit data for older Americans or whether this is an artefact of the estimation strategy and the relatively low number of deaths in younger cohorts. The measured variable importance seems to suggest that the results are driven by the measured changes in credit and bank cards (such as balances and payment amounts). Additional variable groups related to mortgage activity and other loans are also predictive but to a lesser extent.\n\nThe current study is not meant to fully uncover the underlying mechanism which are likely to be complex and potentially subject to many feedback cycles between credit behavior and health. Our insights are however consistent with the current state of knowledge which documents correlations between economic shocks and health outcomes, and potentially with individuals retaining a substantial amounts of private information on their health status. These correlations appear to be highly predictive of mortality outcomes. It is important to note that lags of the credit variables are also predictive which is indicative of both a short and long run component of the relationships between health and consumer finance. \n\nThe documented predictive power of credit variables for individual mortality has a number of implications. From an economic perspective, mortality predictions play an important role in a number of markets such those for life insurance and reverse mortgages. Life expectancy calculations are also key in legal proceedings which rely on evaluations of the expected value of life. Our study shows that actuarial tables that are usually relied upon can be significantly improved upon at the individual level using relatively common if proprietary data collected routinely for most Americans. Thus even without access to any sort of health information on the individual, health outcomes such as mortality can in fact be inferred from credit data.\n\n\n\\clearpage \n\n\n\\bibliographystyle{agsm}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\setcounter{equation}{0}\n\nThe construction and study of discrete Painlev\\'e equations has been a topic of research interest \nfor almost two decades, \\cite{NP,FIK,RGH,JimSakai}. Reviews of the subject may be found in \\cite{GNR,GR}.\nThe subject has culminated in the classification by H. Sakai of discrete as well continuous \nPainlev\\'e equations based on the algebraic geometry of the corresponding rational \nsurfaces associated with the spaces of initial conditions \\cite{Sakai}. \nAs a byproduct of the latter treatment, \na ``mistress'' discrete Painlev\\'e equation with elliptic dependence on the independent variable\nwas discovered. \n\nIn the history of the Painlev\\'e program, after the classification results for second-order first-degree equations, Painlev\\'e's students, Chazy and Garnier, \\cite{Chazy9,Chazy11,Garn}, investigated the classification of second-order second-degree equations and third-order first-degree equations. The classification of the second-degree class was completed by Cosgrove in recent years, \\cite{Cosg,CosgS}. A partial classification for the third-order case was also obtained by the aforementioned authors.\nThe work of Bureau, \\cite{Bur72, Bur87}, is also important in this respect. \nNo classification results exist for the analogous discrete case and hardly any examples of \nsecond-order second-degree difference equations exist to date, with the notable exception of an (additive difference) equation given by Est\\'evez and Clarkson \\cite{Clarkson}.\n\nA key result of this letter is a second-order second-degree equation, \nwhich can be considered as a $q$-analogue of an equation in the Chazy-Cosgrove class, \ntogether with its Lax pair \n(i.e., isomonodromic $q$-difference problem). \nThis new equation contains four free parameters, which suggests that it \ncould be a $q$-difference analogue of the second-order second-degree \ndifferential equation that is a counterpart of the sixth Painlev\\'e \nequation. \nThere are several forms of a second-order second-degree equation\nrelated to the sixth Painlev\\'e \nequation that have appeared in the literature, notably\none derived by Fokas and Ablowitz \\cite{FA} and another appearing in the\nwork of Okamoto \\cite{Oka}.\nDifference analogues of the Fokas-Ablowitz equation have been \nprovided by Grammaticos and Ramani, \\cite{RG}, but these \ndifference equations were all of first-degree. It has \nbeen argued by these authors that equations that are second-degree in the highest iterate cannot be viewed as \\lq\\lq integrable\\rq\\rq , however, for the new equation we establish integrability through a Lax pair in the form of an isomonodromic deformation system. \nFurthermore we show that the equation arises as a similarity reduction from an integrable partial $q$-difference equation. Through the same procedure we also construct \nhigher-order second-degree equations, which form a hierarchy associated with the new equation.\n\n\\section{$q$-Difference Similarity Reduction}\\label{lKdVsaac}\n\\setcounter{equation}{0}\n\nLattice equations of KdV-type were introduced and studied over the last three decades \\cite{hirota:77,NQC}, \nsee \\cite{KDV} for a review. These lattice equations can be formulated as partial difference equations on a lattice with step sizes that enter as parameters of the equation. \nConventionally we think of these parameters as fixed constants. However, in agreement with the integrability of these equations, there exists the freedom to take the parameters as functions of the local lattice coordinate in each corresponding direction. \nIn this paper we consider the case when the parameters depend exponentially with base $q$ on the lattice coordinates.\n\nWe work in a space ${\\mathcal F}$ of functions $f$ of arbitrarily many variables $a_i$ ($i=1,\\dots,M$ for any $M$) on which we define the $q$-shift operations\n\\[\n\\,_q\\!T_i\\,f(a_1,\\ldots, a_N):=f(a_1,\\ldots, q\\,a_i,\\ldots, a_M)\\ . \n\\]\nFor $u, v, z \\in {\\mathcal F}$, we consider the following systems of nonlinear\npartial $q$-difference equations:\n\\begin{equation}\\label{eq:qKdV}\n\\left(u-\\,_q\\!T_i\\,_q\\!T_ju\\right)\\left(\\,_q\\!T_ju-\\,_q\\!T_iu\\right)\n= (a_i^2-a_j^2) q^2 \\ , \n\\end{equation} \n\\begin{equation}\\label{eq:qmKdV}\na_j(_q\\!T_jv)\\,_q\\!T_i\\,_q\\!T_jv + a_i(_q\\!T_jv)v = \na_i(_q\\!T_iv)\\,_q\\!T_j\\,_q\\!T_iv + a_j(_q\\!T_iv)v\\ \n\\end{equation}\nand\n\\begin{eqnarray}\\nonumber\n&&a_i^2\\left(z-\\,_q\\!T_iz\\right)\\left(\\,_q\\!T_jz-\\,_q\\!T_i\\,_q\\!T_jz\\right)\\\\\n\\label{eq:qSKdV}&&\\qquad\\qquad =\na_j^2\\left(z-\\,_q\\!T_jz\\right)\\left(\\,_q\\!T_iz-\\,_q\\!T_i\\,_q\\!T_jz\\right) , \n\\end{eqnarray}\nwhere $i,j=1,\\dots,M$. \nEach of these systems, (\\ref{eq:qKdV}) to (\\ref{eq:qSKdV}), \nrepresents a multi-dimensionally consistent family of partial difference equations, in the sense of \\cite{NW,BS}, which implies that they constitute holonomic systems of nonlinear partial $q$-difference equations. \nAnother way to formulate this property is through an underlying linear system which takes the form\n\\begin{equation}\\label{eq:Tjphi}\n\\,_q\\!T_i^{-1}\\boldsymbol{\\phi}=\\boldsymbol{M}_i(k)\\boldsymbol{\\phi} , \n\\end{equation}\nwhere $\\boldsymbol{\\phi}=\\boldsymbol{\\phi}(k;\\{a_j\\})$ is a two-component vector-valued function and by consistency, $\\,_qT_i^{-1}\\,_qT_j^{-1}\\boldsymbol{\\phi}=\\,_qT_j^{-1}\\,_qT_i^{-1}\\boldsymbol{\\phi}$, leads to the \nset of Lax equations (for each pair of indices $i,j$)\n\\begin{equation}\\label{eq:Laxeqs}\n(\\,_qT_i^{-1}\\boldsymbol{M}_j)\\boldsymbol{M}_i=(\\,_qT_j^{-1}\\boldsymbol{M}_i)\\boldsymbol{M}_j\\ . \n\\end{equation} \nWe will consider three different cases, associated respectively with equations (\\ref{eq:qKdV})--(\\ref{eq:qSKdV}).\nTo avoid proliferation of symbols we use the same symbol $\\boldsymbol{M}_i(k)$ for each of the respective Lax matrices. For specific choices of the matrices $\\boldsymbol{M}_i$ the Lax equations (\\ref{eq:Laxeqs}) lead to the nonlinear equations given above. \nIn the case of the $q$-lattice KdV, (\\ref{eq:qKdV}), the Lax matrices $\\boldsymbol{M}_i$ are given by\n\\begin{equation}\\label{eq:MjKdV}\n\\boldsymbol{M}_i(k;\\{a_j\\})=\\frac{1}{a_i-k}\\left(\\begin{array}{ccc} a_i-\\,_q\\!T_i^{-1}u &,& 1 \\\\ \nk^2-a_i^2+(a_i+u)(a_i-\\,_q\\!T_i^{-1}u) &,& a_i+u\\end{array}\\right) \\ . \n\\end{equation} \nIn the case of the $q$-lattice mKdV, (\\ref{eq:qmKdV}), the Lax matrices $\\boldsymbol{M}_i$ are given by\n\\begin{equation}\\label{eq:Mj}\n\\boldsymbol{M}_i(k;\\{a_j\\})=\\frac{1}{a_i-k}\\left(\\begin{array}{ccc} a_i(_q\\!T_i^{-1}v)\/v &,& k^2\/v \\\\ \n\\,_q\\!T_i^{-1}v &,& a_i\\end{array}\\right) \\ . \n\\end{equation} \nFinally, in the case of the $q$-lattice SKdV, (\\ref{eq:qSKdV}), the Lax matrices $\\boldsymbol{M}_i$ are given by \n\\begin{equation}\\label{eq:MjSKdV}\n\\boldsymbol{M}_i(k;\\{a_j\\})=\\frac{a_i}{a_i-k}\\left(\\begin{array}{ccc} 1 &,& (k^2\/a_i^2)\\left(z-\\,_q\\!T_i^{-1}z\\right)^{-1} \\\\ \nz-\\,_q\\!T_i^{-1}z &,& 1\\end{array}\\right) \\ . \n\\end{equation} \nThese Lax matrices are straightforward generalizations of those \nwith constant lattice parameters given in e.g. \\cite{NRGO}. \n\nWe mention that the solutions of the equations (\\ref{eq:qKdV}) to (\\ref{eq:qSKdV}) \nare related through discrete Miura type relations, namely\n\\numparts\\label{eq:Miuras}\n\\begin{eqnarray}\n&& a_i\\left(z-\\,_q\\!T_i^{-1}z\\right)=v\\left(\\,_q\\!T_i^{-1}v\\right)\\quad, \\label{eq:CH} \\\\ \n&& s=\\left(a_i-\\,_q\\!T_i^{-1}u\\right)v-a_i\\,_q\\!T_i^{-1}v\\quad,\\label{eq:Miura} \\\\ \n&& \\,_q\\!T_i^{-1}s=a_iv-(a_i+u)\\,_q\\!T_i^{-1}v\\ , \\label{eq:Miura2} \n\\end{eqnarray}\n\\endnumparts\nwhere $s\\in{\\mathcal F}$ is an auxiliary dependent variable. From these relations, the partial $q$-difference equations (\\ref{eq:qKdV}) to (\\ref{eq:qSKdV}) can be derived by eliminating $s$. \n\nSimilarity reductions of lattice equations have been considered in \\cite{NP,Nijh:Dorf,NW,DIGP,NRGO} where it was shown that scaling invariance of the solution can be implemented through additional compatible constraints on the lattice equations. In the present case of (\\ref{eq:qKdV})\nto (\\ref{eq:qSKdV}) these constraints adopt the following form \\cite{FJN}\n\\numparts\n\\begin{eqnarray}\n&& u(\\{q^{-N}a_i\\})=q^{-N}\\frac{1-\\lambda(q^N-1)(-1)^{\\sum_i\\,^{q}\\!\\log\\,a_i}}\n{1+\\lambda(q^N-1)(-1)^{\\sum_i\\,^{q}\\!\\log\\,a_i}}\\,u(\\{ a_j\\}) , \n \\label{eq:uconstr} \\\\\n&& v(\\{q^{-N}a_i\\})=\\frac{1-\\lambda(q^N-1)(-1)^{\\sum_i\\,^{q}\\!\\log\\,a_i}}{1+\\mu(q^N-1)}\\,v(\\{ a_j\\}) ,\n\\label{eq:vconstr} \\\\ \n&& z(\\{q^{-N}a_i\\})=q^N\\frac{1-\\mu(q^N-1)}{1+\\mu(q^N-1)}\\,z(\\{ a_j\\})\\quad,\\label{eq:zconstr} \n\\end{eqnarray}\n\\endnumparts\nwhere $\\lambda$ and $\\mu$ are constant parameters of the reduction and $N \\in \\mathbb{N}$ \nrepresents a \\lq\\lq periodicity freedom\\rq\\rq. The notation $^{q}\\!\\log\\,x$ refers to the logarithm of $x$\nwith base $q$.\n\nIn order to compute the corresponding isomonodromic deformation problems associated with the \nsimilarity reductions we have the following constraints on the vector function of the Lax pairs. \nIn the case of (\\ref{eq:qKdV}) we have\n\\begin{eqnarray}\\label{eq:phiforqKdV}\n\\fl\\boldsymbol{\\phi}(q^N\\,k;\\{a_j\\})= \\\\ \n\\fl\\nonumber\\left(\\begin{array}{lcl}\n\\qquad\\left(1+\\lambda(q^N-1)(-1)^{\\sum_i\\,^{q}\\!\\log\\,a_i}\\right)&,&0\\\\\n-2\\lambda q \\frac{q^N-1}{q-1}\\left(\\sum_i\\,a_i\\right)(-1)^{\\sum_i\\,^{q}\\!\\log\\,a_i}\n &,&q^N \\left(1-\\lambda(q^N-1)(-1)^{\\sum_i\\,^{q}\\!\\log\\,a_i}\\right) \\\\\n\\end{array}\\right)\\boldsymbol{\\phi}(k;\\{q^{-N}\\,a_j\\}) .\n\\end{eqnarray}\nIn the case of (\\ref{eq:qmKdV})\n\\begin{eqnarray}\\label{eq:phiforqmKdV}\n\\fl\\boldsymbol{\\phi}(q^N\\,k;\\{a_j\\})=\\\\\n\\fl\\nonumber\\qquad\\left(\\begin{array}{ll}\n\\left(1+\\lambda(q^N-1)(-1)^{\\sum_i\\,^{q}\\!\\log\\,a_i}\\right)&0\\\\\n0&q^{-N}(1+\\mu(q^N-1))\\\\\n\\end{array}\\right)\\boldsymbol{\\phi}(k;\\{q^{-N}\\,a_j\\}) .\n\\end{eqnarray}\nIn the case of (\\ref{eq:qSKdV})\n\\begin{eqnarray}\\label{eq:phiforqSKdV}\n\\fl\\boldsymbol{\\phi}(q^N\\,k;\\{a_j\\})=\\\\\n\\fl\\nonumber\\qquad\\left(\\begin{array}{ll}\n\\left(1-\\mu(q^N-1)\\right)&0\\\\\n0&q^{-N}(1+\\mu(q^N-1))\\\\\n\\end{array}\\right)\\boldsymbol{\\phi}(k;\\{q^{-N}\\,a_j\\}) .\n\\end{eqnarray}\nThe similarity constraints, (\\ref{eq:phiforqKdV}) to (\\ref{eq:phiforqSKdV}), in conjunction with the discrete linear equations (\\ref{eq:MjKdV}) to (\\ref{eq:MjSKdV}) can be used to derive corresponding $q$-isomonodromic deformation problems. \nThat is, (\\ref{eq:phiforqKdV}) to (\\ref{eq:phiforqSKdV}) lead to\n$q$-difference equations in the spectral variable $k$, hence together\nwith the lattice equation Lax pairs we obtain $q$-isomonodromic deformation\nproblems for the corresponding reductions.\n\n\\paragraph{Remarks:} \n\\begin{enumerate}\n\\item The similarity constraints above were obtained through an approach based on Jackson-type integrals, the details of which will be presented elsewhere \\cite{FJN}. By construction, these constraints are compatible with the underlying lattice equations, which can be checked \\textit{a posteriori} \nby an explicit calculation, presented in the appendix. \n\\item In this approach, the dynamics in terms of the variables $a_i$ appear through appropriately chosen $q$-analogues of exponential functions, whereas the relevant Jackson integrals exhibit an invariance through scaling by factors $q^N$.\n\\item The parameters $\\lambda$ and $\\mu$ arise in this setting\nthrough boundary contributions in a manner analogous to the derivation in \\cite{NRGO}.\n\\end{enumerate}\n\nIn the remainder of this letter our aim is to implement the similarity constraint \nto obtain explicit reductions to ordinary $q$-difference equations. \nFor simplicity we consider only the reduction of the $q$-mKdV equation (\\ref{eq:qmKdV}), leaving\nconsiderations of the $q$-KdV and $q$-SKdV to a future publication \\cite{FJN}. \nThere are two possible scenarios to derive similarity reductions of the lattice equations using the constraint (\\ref{eq:vconstr}).\n\\paragraph{\\textit{``Periodic'' similarity reduction:}} \nBy fixing $M=2$ and allowing $N$ to vary, we select two lattice directions, say the variables $a_1$ and $a_2$, and consider similarity reductions with different values of $N$. This is a $q$-variant of the periodic staircase type reduction of partial difference equations on the two-dimensional lattice. \nFor instance, with $N=2$ the reduction is a second-order first-degree $q$-Painlev\\'e equation.\nIncreasing $N$ leads to $q$-difference Painlev\\'e type equations of increasing order.\nHowever, we will not pursue this route here but leave it to a subsequent publication \\cite{FJN}. These reductions are reminiscent of the work \\cite{Hay,SahadevanCapel,Carstea}. \n\\paragraph{\\textit{Multi-variable similarity reduction:}} \nThe similarity constraints provide the mechanism to couple together two or more lattice directions. \nBy considering the case $N=1$ we implement the similarity constraints on an extended lattice of three or more dimensions in order to obtain coupled ordinary $q$-difference equations, \nin a way that is reminiscent of the approach of \\cite{NW}. \nThis is considered in the next section.\n\n\\paragraph{{}} \nWe have not considered the more general case of arbitrary $M, N \\in \\mathbb{N}$, \nwhich we will postpone to a future publication \\cite{FJN}. \n\n\\section{Multi-variable similarity reduction} \\label{multivarsec}\n\\setcounter{equation}{0}\nIn this section we consider explicitly the $M=1$, 2 and 3 cases for $N=1$.\n\nFor simplicity we shall in what follows denote the coefficient in (\\ref{eq:vconstr}) as $\\gamma$, i.e. \n\\begin{equation}\\label{eq:gamma}\n\\gamma=\\frac{1-\\lambda(q-1)(-1)^{\\sum_i\\,^{q}\\!\\log\\,a_i}}{1+\\mu(q-1)}\\quad\\Rightarrow\\quad\n v(\\{q^{-1} a_i\\})=\\gamma \\, v(\\{a_i\\})\\ , \n\\end{equation}\nwhere $\\gamma$ alternates between two values, i.e., $\\,_q\\!T_i^2\\gamma=\\gamma$. \n\nIn contrast to the usual difference case which was explored in \\cite{NW} where \nin the case of two variables we obtain a nontrivial O$\\Delta$E as a reduction, \nin the $q$-difference case we have to consider at least three independent variables\nto obtain a nontrivial system of O$\\Delta$Es as a \nreduction.\n\nIn \\cite{NW} the compatibility between the similarity constraint and the lattice \nsystem was established and led \nto a system of higher order difference equations in the reduction, namely equations which were on \nthe level of the first Garnier system. \nIn contrast to the $q=1$ work, the 3D similarity constraint here is somewhat \nsimpler and leads to a second-order equation (which is of second-degree, and \nis a principal result of this letter). \n\n\n\n\n\\subsection*{Two-variable case}\nLet us now select among the collection of variables $\\{a_j\\}$ two specific ones\nwhich for simplicity we will call $a$ and $b$. \nDenote the $q$-shifts in these variables by an over-tilde $\\widetilde{\\phantom{C}}$ \nand an over-hat $\\widehat{\\phantom{C}}$ respectively. Equation\n(\\ref{eq:qmKdV}) may now be written\n\\begin{equation}\\label{qmKdVab}\nb \\widehat{v} \\widehat{\\widetilde{v}} + a \\widehat{v} v = a \\widetilde{v} \\widehat{\\widetilde{v}} + b v \\widetilde{v},\n\\end{equation}\nwhere the over-tilde $\\widetilde{\\phantom{v}}$ refers to the\n$q$-translation $a \\mapsto q a$ and the over-hat $\\widehat{\\phantom{v}}$ refers to the $q$-translation $b \\mapsto q b$\n(so if $v \\equiv v(a,b)$, $\\widetilde{v} \\equiv v(q a,b)$, $\\undertilde{v} \\equiv v( q^{-1}a ,b)$, \n$\\widehat{v} \\equiv v(a,bq)$, \\ldots).\n\nEquation (\\ref{eq:gamma})\ngives the constraint $v = \\gamma \\widehat{\\widetilde{v}}$ to impose on (\\ref{qmKdVab})\n(where $\\widehat{\\widetilde{\\gamma}} = \\widetilde{\\widetilde{\\gamma}} = \\gamma$).\nThis leads to the linear first-order (in that it is a two point) ordinary\ndifference equation\n\\begin{equation}\\label{Neq1}\n\\undertilde{v} = C \\,\\tilde{v},\n\\end{equation}\nwhere $C=\\widetilde{\\gamma} (a\\gamma + b)\/(a+b\\gamma)$.\nIn the appendix the\nconsistency between the lattice equation (\\ref{qmKdVab}) and the constraint\n(\\ref{Neq1}) is shown by direct computation.\n\n\n\n\n\\subsection*{Three-variable case}\n\n\nTake three copies of the lattice mKdV equation with $a_1=a$, $a_2=b$, $a_3=c$, \n\\numparts\n\\begin{eqnarray}\\label{qmKdVabc}\nb \\widehat{v} \\widehat{\\widetilde{v}} + a \\widehat{v} v = a \\widetilde{v} \\widehat{\\widetilde{v}} + b v \\widetilde{v}, \\label{eq:qmKdVab} \\\\ \nc \\ol{v} \\widetilde{\\ol{v}} + a \\ol{v} v = a \\widetilde{v} \\widetilde{\\ol{v}} + c v \\widetilde{v}, \\label{eq:qmKdVac}\\\\ \nc \\ol{v} \\widehat{\\ol{v}} + b \\ol{v} v = b \\widehat{v} \\widehat{\\ol{v}} + c v \\widehat{v}, \\label{eq:qmKdVbc}\n\\end{eqnarray}\n\\endnumparts\nwhere $\\ol{c} = q c$, together with the constraint \n\\begin{equation}\\label{eq:abcconstr}\nv(q^{-1}a,q^{-1}b,q^{-1}c)=\\gamma v(a,b,c).\n\\end{equation} \n(The similarity constraint is shown by a direct computation to be compatible with the multidimensionally \nconsistent system of mKdV lattice equations in the appendix.)\nWe now proceed to derive the \nreduced system which leads to a (higher-degree) ordinary $q$-difference \nequation in terms of one selected independent\nvariable, say the variable $a$. The remaining variables $b$ and $c$ \nwill play the role of parameters \nin the reduced equation. Thus, \nwe can derive the following system of two coupled O$\\Delta$Es for $v(a,b,c)$ and \n$w(a,b,c)\\equiv v(a,b,q^{-1}c)$:\n\\numparts\\label{eq:vw}\n\\begin{eqnarray}\n\\gamma v&=&w \\frac{a\\widetilde{\\gamma}\\widetilde{v}-b\\undertilde{w}}{a\\undertilde{w}-b\\widetilde{\\gamma}\\widetilde{v}}\\ , \\label{eq:vwa} \\\\ \n\\undertilde{w}&=&v \\frac{aw-c\\undertilde{v}}{a\\undertilde{v}-cw}\\ , \\label{eq:vwb} \n\\end{eqnarray}\n\\endnumparts\nwhere $\\widetilde{\\widetilde{\\gamma}}=\\gamma$. We consider the system (\\ref{eq:vwa}) and (\\ref{eq:vwb}) \nto constitute a $q$-Painlev\\'e system with four free \nparameters. \n\nThe system (\\ref{eq:vwa}) and (\\ref{eq:vwb}) \ncan be reduced to a second-order second-degree ordinary difference equation \nas follows. Introduce the variables\n\\begin{equation}\\label{eq:XVW} \nX=\\frac{v}{w}\\quad,\\quad V=\\frac{\\widetilde{v}}{v}\\quad,\\quad W=\\frac{\\widetilde{w}}{w}\\ , \n\\end{equation} \nthen from (\\ref{eq:vwa}) we obtain\n\\begin{equation}\\label{eq:VW}\n\\widetilde{\\gamma}\\frac{\\widetilde{v}}{\\undertilde{w}}=\\widetilde{\\gamma}VX\\undertilde{W}=\\frac{a\\gamma X+b}{b\\gamma X+a}\\ , \n\\end{equation} \nwhereas from (\\ref{eq:vwb}) we get \n\\begin{equation}\\label{eq:VV}\nW=\\frac{VX}{\\widetilde{X}}=\\frac{a+q^{-1}cXV}{q^{-1}c\/X+aV}\\ , \n\\end{equation} \nusing also the definitions (\\ref{eq:XVW}). Thus, we obtain a quadratic equation for $V$ in terms \nof $X$ and $\\widetilde{X}$ and hence also we have $W$ in terms of $X$ and $\\widetilde{X}$. Inserting these into \n(\\ref{eq:VW}) we obtain a second-order algebraic equation for $X$. Alternatively, avoiding \nthe emergence of square roots, the following second-order second-degree equation for $X$ may be \nderived\n\\begin{eqnarray}\\label{eq:Xeq}\n\\fl\n\\left[ \\widetilde{\\gamma}^2\\widetilde{X}\\undertilde{X}-\\left(\\frac{a\\gamma X+b}{b\\gamma\nX+a}\\right)^2\\right]^2 \\nonumber \\\\\n = \\widetilde{\\gamma}\\frac{c^2}{a^2}\\frac{1}{X}\n\\left(\\frac{a\\gamma X+b}{b\\gamma X+a}\\right)\\left[\\widetilde{\\gamma}\\widetilde{X}(1-X\\undertilde{X})+\nq^{-1}(1-X\\widetilde{X})\\frac{a\\gamma X+b}{b\\gamma X+a}\\right] \\nonumber \\\\\n\\qquad\\quad\\times \\left[ q^{-1}\\widetilde{\\gamma}\\undertilde{X}(1-X\\widetilde{X})+ (1-X\\undertilde{X})\n\\frac{a\\gamma X+b}{b\\gamma X+a}\\right] \\ .\n\\end{eqnarray}\nWe consider this second-degree equation to be one of the main results of this letter.\n\n\n\nWe now proceed to present the Lax pair for the $q$-Painlev\\'e system (\\ref{eq:vwa}) and (\\ref{eq:vwb}) and the second-order \nsecond-degree equation (\\ref{eq:Xeq}).\nThe Lax pair is formed by considering the compatibility of two paths on the lattice:\nalong a `period' then in the $a$ direction and evolving in the $a$ direction\nthen along a `period'.\nUsing (\\ref{eq:phiforqmKdV}) the evolution along a period is converted into a\ndilation of the spectral parameter, $k$, by $q$. \nThe result is the following isomonodromic $q$-difference system for the vector\n$\\phi(k;a)$ \nwhich using the results of section \\ref{lKdVsaac} yields \n\\numparts\n\\begin{eqnarray}\n\\phi(k;q^{-1}a)&=&\\boldsymbol{M}(k;a)\\phi(k;a)\\ , \\\\ \n\\phi(qk;a)&=&\\boldsymbol{L}(k;a)\\phi(k;a)\\ , \n\\end{eqnarray}\n\\endnumparts\nwhere \n\\numparts\n\\begin{equation}\\label{eq:M}\n\\boldsymbol{M}(k;a)=\\frac{1}{a-k}\\left(\\begin{array}{cc} a\\undertilde{v}\/v & k^2\/v \\\\ \n\\undertilde{v} & a\\end{array}\\right) \\ , \n\\end{equation} \nand \n\\begin{equation}\\label{eq:L}\n\\boldsymbol{L}(k;a)= \n\\frac{1}{a-k}\\left(\\begin{array}{cc} a\\gamma v\/\\widetilde{v} & k^2\/\\widetilde{v} \\\\ \nq^{-1}\\gamma v & q^{-1}a \\end{array}\\right)\\, \n\\left(\\begin{array}{cc} b\\widetilde{\\gamma}\\widetilde{v}\/w & k^2\/w \\\\ \n\\widetilde{\\gamma}\\widetilde{v} & b \\end{array}\\right)\\, \n\\left(\\begin{array}{cc} cw\/v & k^2\/v \\\\ \nw & c \\end{array}\\right)\\, \n\\end{equation}\n\\endnumparts\nwhere we have suppressed the dependence on the variables $b$ and $c$ (which now \nplay the role of parameters) and omitted the unnecessary prefactors $(b-k)^{-1}$ \nand $(c-k)^{-1}$, as well as an over factor $q^{-1}(1+\\mu(q-1))$. \n\n\nThe consistency condition \nobtained from the two ways of expressing $\\phi(qk;q^{-1}a)$ in terms of $\\phi(k;a)$ is formed by the \nLax equation \n\\begin{equation}\\label{eq:Laxeq}\n\\boldsymbol{L}(k;q^{-1}a)\\boldsymbol{M}(k;a)=\\boldsymbol{M}(qk;a)\\boldsymbol{L}(k;a)\\ . \n\\end{equation} \nA gauge transformation can be obtained expressing the \nLax matrices in terms of the variables introduced in (\\ref{eq:XVW}). Setting \n\\numparts\n\\begin{equation} \\label{eq:LMa}\n\\mathcal{M}(k;a) = \\frac{1}{a-k}\\left(\\begin{array}{cc} \na\/\\undertilde{V} & k^2 \\\\ 1 & a\\undertilde{V} \\end{array}\\right) ,\n\\end{equation}\n\\begin{equation}\\label{eq:LMb}\n\\mathcal{L}(k;a) = \\frac{1}{a-k}\\left(\\begin{array}{ccc} \n\\widetilde{\\gamma}(ab\\gamma X+k^2) & & k^2(a\\gamma X+b)\/V \\\\ \nq^{-1}\\widetilde{\\gamma} V(a+b\\gamma X) & & q^{-1}(ab+k^2\\gamma X)\\end{array}\\right) \n\\left(\\begin{array}{cc} c\/X & k^2 \\\\ 1\/X & c\\end{array}\\right) \\ , \n\\end{equation}\n\\endnumparts\nthe Lax equations (\\ref{eq:Laxeq}) (replacing $\\boldsymbol{L}$ and $\\boldsymbol{M}$ by $\\mathcal{L}$ and $\\mathcal{M}$ respectively) \nyield a set of relations equivalent to the following two equations: \n\\begin{eqnarray}\\label{eq:laxconds}\n&& \\widetilde{\\gamma}V\\undertilde{V}\\undertilde{X}=\\frac{a\\gamma X+b}{a+b\\gamma X} \\ , \\label{eq:laxcondsa} \\\\ \n&& aV^2+q^{-1}c\\left( \\frac{1}{X}-\\widetilde{X}\\right) V-a\\frac{\\widetilde{X}}{X}=0\\ , \\label{eq:Laxcondsb} \n\\end{eqnarray} \nusing also $\\widetilde{\\gamma}=\\undertilde{\\gamma}$. This set follows directly from (\\ref{eq:VW}) and (\\ref{eq:VV}). Thus \n(\\ref{eq:LMa}) and (\\ref{eq:LMb}) \nform a $q$-isomonodromic Lax pair for the second-degree equation (\\ref{eq:Xeq}). \n\n\n\\subsection*{Four-variable case:}\n\n\n\n\nSuppose we have $4$ variables $a_i$, $i=1,\\dots, 4$. Select $a=a_1$ to be the \nindependent variable after reduction. \nIntroduce the dependent variables $w_{j-2}=\\,_q\\!T_j^{-1}v$, \n$j=3,4$. Then directly from the $q$-lattice mKdV equation (\\ref{eq:qmKdV}) we have the set \nof equations\n\\begin{equation}\\label{eq:multi-qmKdV}\n\\underaccent{\\wtilde}{w}_j\n=v\\frac{aw_j-a_{j+2}\\undertilde{v}}{a\\undertilde{v}-a_{j+2}w_j}\\quad,\\quad j=1,2\\ , \n\\end{equation}\nwhere as before the tilde denotes a $q$-shift in the variable $a$. \nAt the same time the multiply shifted\nobject $_q\\!T_3^{-1} \\,_q\\!T_4^{-1}\\undertilde{v}$ can be expressed in a unique way (due to the CAC property) \nin terms of $\\undertilde{v}$ and $\\,_q\\!T_j^{-1}v=w_{j-2}$, $j=3,4$, by iterating the relevant copies of the \n$q$-lattice mKdV equation in the variables $a_j$, $j\\neq 2$, leading to an expression of the form\n$_q\\!T_3^{-1} \\,_q\\!T_4^{-1}\\undertilde{v}=:F(\\undertilde{v},w_1,w_{2})$, where $F$\nis easily obtained explicitly. \nImposing the similarity constraint (\\ref{eq:gamma}) \nwe obtain $\\widetilde{\\gamma}\\,_q\\!T_2v=F(\\undertilde{v},w_1,w_{2})$\nand inserting this expression into the $q$-lattice mKdV (\\ref{eq:qmKdV}) with $i=1,j=2$ we obtain \n\\begin{equation}\\label{eq:vvF}\n\\left( a+\\frac{a_2}{\\gamma} \\frac{a_3 w_2-a_4 w_1}{a_3 w_1 -a_4 w_2}\\right)\n\\left(a_2\\widetilde{\\gamma}^{-1}F(\\undertilde{v},w_1,w_{2})-a\\widetilde{v}\\right)=\n(a_2^2-a^2)v\\widetilde{v}\\ . \n\\end{equation} \nWith the explicit form of $F(\\undertilde{v},w_1,w_{2})$ equation (\\ref{eq:vvF}) reads\n\\numparts\n\\begin{eqnarray}\\label{eq:4dsysta}\n\\fl\n(a_2^2-a^2)\\gamma\\widetilde{\\gamma}\\widetilde{v}(a_3w_1-a_4w_2)\n\\left[ a(a_3^2-a_4^2)\\undertilde{v}+a_3(a_4^2-a^2)w_1+a_4(a^2-a_3^2)w_2\\right] \\nonumber \\\\\n=[(a_2a_3-\\gamma aa_4)w_2+(\\gamma aa_3-a_2a_4)w_1]\\,\\left[ a(a_3^2-a_4^2)(a_2 w_1w_2-\\widetilde{\\gamma}a\\widetilde{v}\\undertilde{v}) \\right. \n\\nonumber \\\\\n\\left. +\\left(a_2a_4(a^2-a_3^2)\\undertilde{v}-aa_3(a_4^2-a^2)\\widetilde{\\gamma}\\widetilde{v}\\right)w_1 +\n\\left(a_2a_3(a_4^2-a^2)\\undertilde{v}-\\widetilde{\\gamma}aa_4(a^2-a_3^2)\\widetilde{v}\\right)w_2 \\right] \\ , \\nonumber \\\\ \n\\end{eqnarray} \nand this is supplemented by the two equations\n\\begin{eqnarray}\n&& avw_1+a_3w_1\\underaccent{\\wtilde}{w}_1=a_3v\\undertilde{v}+a\\undertilde{v}\\underaccent{\\wtilde}{w}_1\\ , \\label{eq:4dsystb} \\\\ \n&& avw_2+a_4w_2\\underaccent{\\wtilde}{w}_2=a_4v\\undertilde{v}+a\\undertilde{v}\\underaccent{\\wtilde}{w}_2\\ , \\label{eq:4dsystc} \n\\end{eqnarray}\n\\endnumparts\nwhich is equivalent to a five-point \n(fourth-order) $q$-difference equation in terms of \n$v$ alone, containing five free parameters: $a_2$, $a_3$, $a_4$, $\\lambda$ and $\\mu$ (inside $\\gamma$ and $\\widetilde{\\gamma}$). This would be an algebraic equation, \nso we proceed as follows in order to derive a higher-degree \n$q$-difference system. Introduce the variables\n\\begin{equation}\\label{eq:XW}\nX_i=\\frac{v}{w_i}\\quad,\\quad W_i=\\frac{\\widetilde{w}_i}{w_i}\\quad,\\quad i=1,2\\ , \n\\end{equation} \nwhile retaining the variable $V=\\widetilde{v}\/v$ as before. By definition we have\n\\begin{equation}\\label{eq:XWV}\n\\frac{V}{W_i}=\\frac{\\widetilde{X}_i}{X_i}\\quad,\\quad i=1,2 , \n\\end{equation}\nand from (\\ref{eq:4dsystb}), (\\ref{eq:4dsystc}) we obtain\n\\begin{equation}\\label{eq:WXV}\nW_i=\\frac{qa+a_{i+2}VX_i}{qaV+a_{i+2}\/X_i}=\\frac{VX_i}{\\widetilde{X}_i}\\quad,\\quad i=1,2\\ , \n\\end{equation} \nwhilst from (\\ref{eq:4dsysta}) we get\n\\begin{eqnarray}\\label{eq:WVX} \n\\fl\n(a_2^2-a^2)\\gamma\\widetilde{\\gamma}V\\left(\\frac{a_3}{X_1}-\\frac{a_4}{X_2}\\right)\\left( a\\frac{a_3^2-a_4^2}{\\undertilde{V}}+\na_3\\frac{a_4^2-a^2}{X_1}+a_4\\frac{a^2-a_3^2}{X_2}\\right) \\nonumber \\\\\n =\\left(\\frac{a_2a_3-\\gamma aa_4}{X_2}+\\frac{\\gamma aa_3-a_2a_4}{X_1}\\right)\\,\\left[ a(a_3^2-a_4^2)(\\frac{a_2}{X_1X_2}\n-\\widetilde{\\gamma}a\\frac{V}{\\undertilde{V}}) \\right. \\nonumber \\\\\n\\left. +\\left(a_2a_4\\frac{a^2-a_3^2}{\\undertilde{V}}-aa_3(a_4^2-a^2)\\widetilde{\\gamma}V\\right)\\frac{1}{X_1} +\n\\left(a_2a_3\\frac{a_4^2-a^2}{\\undertilde{V}}-\\widetilde{\\gamma}aa_4(a^2-a_3^2)V\\right)\\frac{1}{X_2} \\right] \\ . \\nonumber \\\\ \n\\end{eqnarray}\n{}From (\\ref{eq:WXV}) we obtain the set of quadratic equations for $V$\n\\begin{equation}\nqa\\frac{X_i}{\\widetilde{X}_i}V^2+a_{i+2}\\left(\\frac{1}{\\widetilde{X}_i}-X_i\\right)V - qa=0\\quad,\\quad i=1,2\\ , \n\\end{equation} \nfrom which by eliminating $V$ we obtain\n\\begin{eqnarray}\\label{eq:YX}\n\\fl\n\\left[a_3(1-X_1\\widetilde{X}_1)X_2-a_4(1-X_2\\widetilde{X}_2)X_1\\right]\\left[a_3(1-X_1\\widetilde{X}_1)\\widetilde{X}_2-a_4(1-X_2\\widetilde{X}_2)\\widetilde{X}_1\\right] \\nonumber \\\\ \n= q^2a^2 (X_1\\widetilde{X}_2-X_2\\widetilde{X}_1)^2\\ . \n\\end{eqnarray} \nFurthermore, solving $V$ from the quadratic system as\n\\begin{equation}\\label{eq:V}\nV=qa\\frac{X_2\\widetilde{X}_1-X_1\\widetilde{X}_2}{a_3(1-X_1\\widetilde{X}_1)X_2-a_4(1-X_2\\widetilde{X}_2)X_1}\\ , \n\\end{equation} \nand inserting this into (\\ref{eq:WVX}) \nwe obtain a second-order equation in both $X_1$, $X_2$ coupled to the equation \n(\\ref{eq:YX}) which is first order in both $X_1$, $X_2$. It is this coupled system of two equations in $X_1$, $X_2$ which \nforms our higher order generalisation of (\\ref{eq:Xeq}). \nThe system of (\\ref{eq:WVX}) and (\\ref{eq:YX}) with (\\ref{eq:V})\nconstitutes a third-order system with five parameters.\n\nThe derivation of the Lax pair follows the same approach as that for the\nthree-variable case\n(with an extra factor in $\\mathcal{L}$ due to the additional lattice direction).\nWe omit details here, which we intend to publish in the future \\cite{FJN}.\n\n\n\\subsection*{Beyond the four-variable case:} \n\nIt is straight-forward to give the form of the full hierarchy, however due to lack of space we postpone this until a later publication \\cite{FJN}.\n \n\n\\section{Conclusion and discussion}\\label{limsanddeg}\n\\setcounter{equation}{0}\n\nIn this letter we have presented the results of \na scheme to derive partial $q$-difference equations \nof KdV type and consistent symmetries of the equations \nand demonstrated how it can be implemented. \nLax matrices follow from this approach. A notable result is the \nderivation of the higher-degree equation (\\ref{eq:Xeq}), showing that \nthe scheme presented here allows for the derivation of new results\nwithin the field of discrete integrable systems. \n\nThe first-, second- and third-order members of the $N=1$ hierarchy have been presented.\nThe scheme continues to give successively higher-order\nequations by considering successively higher dimensions of the original lattice equation. \nOne may ask the natural question\nas to whether this gives an `interpolating' hierarchy which,\ncontrary to the usual cases, increases the order and number of parameters\nof the equations\nby one in each step, rather than a two step increase. A further natural\nquestion connected with this hierarchy is its\nrelation to the $q$-Garnier systems of Sakai \\cite{GarnSakai}.\n\nWe will present full details of the \nscheme from which the lattice equations (\\ref{eq:qKdV})\nto (\\ref{eq:qSKdV}) and their associated constraints follow in a future publication \\cite{FJN}.\nThere we will consider the most general case of symmetry reductions \n(arbitrary $N \\in \\mathbb{N}$) of all three lattice equations.\n\nWe also intend \nto return in a future publication to the question of limits and degeneracies of the \nequations presented in this paper. These include the \n$q\\rightarrow 1$ continuum limit,\nthe $q\\rightarrow 1$ discrete limit \nand the $q\\rightarrow 0$ crystal or ultradiscrete limit. \n\n\\section*{Acknowledgements} \n\nDuring the writing of this letter C.M. Field has been supported by\nthe Australian Research Council Discovery Project Grant $\\#$DP0664624\nand by the Netherlands Organization\nfor Scientific Research (NWO) in the VIDI-project ``Symmetry and\nmodularity in exactly solvable models''.\nN. Joshi is also supported by \nthe Australian Research Council Discovery Project Grant $\\#$DP0664624.\nPart of the work presented here was performed whilst N. Joshi \nwas visiting the University of Leeds.\n\n\\section{Appendix}\n\nIn this appendix we present the result of explicit calculations showing the consistency\nof the lattice equations and constraints.\n\n\n\\subsection*{Two-variable consistency}\n\nWe shall check the consistency between the lattice equation (\\ref{qmKdVab}) and the \nconstraint $v= \\gamma \\widehat{\\widetilde{v}}$ by direct computation. \nThis computation is illustrated in the following diagram: \n\\vspace{.2cm}\n\\begin{center}\n\n\n\\setlength{\\unitlength}{0.00043489in}\n\\begingroup\\makeatletter\\ifx\\SetFigFont\\undefined%\n\\gdef\\SetFigFont#1#2#3#4#5{%\n \\reset@font\\fontsize{#1}{#2pt}%\n \\fontfamily{#3}\\fontseries{#4}\\fontshape{#5}%\n \\selectfont}%\n\\fi\\endgroup%\n{\\renewcommand{\\dashlinestretch}{30}\n\\begin{picture}(5245,4530)(0,-10)\n\\put(-1010,2205){.}\n\\put(-1010,3985){.}\n\\put(-1010,380){.}\n\n\\put(6235,2205){.}\n\\put(6235,3985){.}\n\\put(6235,380){.}\n\n\\put(765,3985){.}\n\\put(4410,380){.}\n\n\\put(2565,2205){\\circle*{303}}\n\\put(4410,2205){\\circle*{303}}\n\\put(765,2205){\\circle{303}}\n\\put(2365,3885){{\\Large$\\times$}\n\n\\put(4410,4005){\\circle{303}}\n\\put(615,325){{\\large$\\otimes$}}\n\\put(2610,439){\\circle{303}}\n\\thicklines\n\\dashline{90.000}(4410,4005)(2565,4005)(765,2205)\n\t(765,405)(2610,405)(4410,2205)(4410,4005)\n\\drawline(2565,2205)(4410,2205)\n\\put(2385,1725){\\makebox(0,0)[lb]{\\Large $v_0$}}\n\\put(4410,1770){\\makebox(0,0)[lb]{\\Large $v_1$}}\n\\put(4815,3980){\\makebox(0,0)[lb]{\\Large $v_{12}$}}\n\\put(2340,4305){\\makebox(0,0)[lb]{\\Large $v_2$}}\n\\put(0,2205){\\makebox(0,0)[lb]{\\Large $v_{-1}$}}\n\\put(295,-15){\\makebox(0,0)[lb]{\\Large $v_{-1,-2}$}}\n\\put(2475,-75){\\makebox(0,0)[lb]{\\Large $v_{-2}$}}\n\\end{picture}\n}\n\nFig 1. Consistency on the 2D lattice. \n\\end{center}\n\\noindent\nAssuming the values $v_0$, $v_1$ as indicated in Fig 1 are given, \nwe compute successively $v_{12}$,$v_2$ etc., \nwhere the subscripts refer to the shifts in the lattice variables $a$,$b$ respectively, as is evident \nfrom Fig 1.\nPoints other than $v_0$ and $v_1$ are computed using either \nthe lattice equation (indicated by $\\times$) or by using the similarity constraint (indicated by \n$\\bigcirc$). The value $v_{-1,-2}$ is the first point which can be calculated in two different ways \n(hence indicated in the diagram by $\\otimes$). Without making any particular assumptions on how \n$\\gamma$ depends on $a$ and $b$, a straightforward calculation shows that the two ways of computing \n$v_{-1,-2}$ are indeed the same, for any choice of initial data $v_0$ and $v_1$, provided that $\\gamma$ \nobeys the relation\n\\begin{equation}\\label{eq:gammarel}\n\\left(\\frac{a+b\\gamma}{b+a\\gamma}\\right)^{\\!\\widehat{\\widetilde{\\phantom{a}}}}\n\\left(\\frac{a+b\\gamma}{b+a\\gamma}\\right)^{-1}=\\frac{\\widetilde{\\gamma}}{\\widehat{\\gamma}}\\ . \n\\end{equation} \nA particular solution of this relation is\n\\begin{equation}\\label{eq:gammasol} \n\\widehat{\\widetilde{\\gamma}}=\\gamma\\quad\\Leftrightarrow\\quad \\widehat{\\gamma}=\\widetilde{\\gamma}\\ , \n\\end{equation} \nand hence $\\widetilde{\\widetilde{\\gamma}}=\\gamma$ implying that $\\gamma$ is an alternating ``constant'' which is in \naccordance with the value given in (\\ref{eq:vconstr}). The reduced equation \nin this case is (\\ref{Neq1}),\nwhich can be readily integrated. \n\nMore generally, equation (\\ref{eq:gammarel}) can be resolved by setting \n\\begin{equation}\\label{eq:nurels} \\frac{a+b\\gamma}{b+a\\gamma}=\\frac{\\widetilde{\\nu}}{\\widehat{\\nu}}\\quad,\\quad \n\\gamma=\\frac{\\widehat{\\widetilde{\\nu}}}{\\nu}\\ , \\end{equation} \nleading to the consequence that $\\nu$ has to solve the $q$-lattice mKdV (\\ref{qmKdVab}). \nIn principle we could take for $\\nu$ any solution of the reduced equation (\\ref{Neq1}) \nand use this to parametrise the reduced equation for $v$ via the relations (\\ref{eq:nurels}). \nIn any event, we see that the two-variable case does not lead to interesting nonlinear equations. \n\n\n\n\\subsection*{Three-variable consistency}\nIn this \ncase the consistency diagram is as follows: \n\\vspace{.2cm} \n\n\\begin{center}\n\n\n\\setlength{\\unitlength}{0.00057489in}\n\\begingroup\\makeatletter\\ifx\\SetFigFont\\undefined%\n\\gdef\\SetFigFont#1#2#3#4#5{%\n \\reset@font\\fontsize{#1}{#2pt}%\n \\fontfamily{#3}\\fontseries{#4}\\fontshape{#5}%\n \\selectfont}%\n\\fi\\endgroup%\n{\\renewcommand{\\dashlinestretch}{30}\n\\begin{picture}(6666,3630)(0,-10)\n\\put(3465,405){\\circle{180}}\n\\put(5265,1800){\\circle{180}}\n\\put(2430,1760){$\\otimes$\n\\put(1800,1665){\\circle*{180}}\n\\put(675,405){\\circle{180}}\n\\put(4200,1580){$\\times$\n\\put(6210,3060){\\circle*{180}}\n\\put(3510,3105){\\circle*{180}}\n\\thicklines\n\\drawline(720,405)(3465,405)\n\\drawline(3510,405)(5220,1800)\n\\drawline(3465,3105)(6210,3105)\n\\drawline(1778,1682)(3488,3077)\n\\dashline{90.000}(3465,3105)(3465,405)\n\\dashline{90.000}(2610,1800)(5175,1800)\n\\dashline{90.000}(2565,1800)(720,450)\n\\dashline{90.000}(1800,1665)(4365,1665)\n\\dashline{90.000}(6187,3017)(4342,1667)\n\\put(3240,3465){\\makebox(0,0)[lb]{\\Large $v_0$}}\n\\put(6390,3375){\\makebox(0,0)[lb]{\\Large $v_1$}}\n\\put(1080,1710){\\makebox(0,0)[lb]{\\Large $v_2$}}\n\\put(3375,0){\\makebox(0,0)[lb]{\\Large $v_{-3}$}}\n\\put(4000,1300){\\makebox(0,0)[lb]{\\Large $v_{1,2}$}}\n\\put(5500,1750){\\makebox(0,0)[lb]{\\Large $v_{-2,-3}$}}\n\\put(0,0){\\makebox(0,0)[lb]{\\Large $v_{-1,-3}$}}\n\\put(2565,1925){\\makebox(0,0)[lb]{\\Large $v_{-1,-2,-3}$}}\n\\end{picture}\n}\n\nFig 2. Consistency on the 3D lattice.\n\\end{center}\n\\vspace{.2cm}\n\n\\noindent \nA similar notation as the previous case is used as is evident from Fig 2.\nThe initial data $v_0$, $v_1$ and $v_2$ are given, and the indicated values on the vertices are \ncomputed either by using one of the lattice equations (\\ref{eq:qmKdVab}) to (\\ref{eq:qmKdVbc}) \nor the similarity \nconstraint (\\ref{eq:abcconstr}) over the diagonal. Thus, $v_{1,2}$ is obtained from (\\ref{eq:qmKdVab}) \nyielding \n$$ v_{1,2}=v_0\\,\\frac{av_2-bv_1}{av_1-bv_2}\\ , $$\nwhilst from the similarity constraint we obtain \n$$v_{-1,-3}=\\gamma_2 v_2 \\quad,\\quad v_{-3}=\\gamma_{1,2}v_{1,2}\\quad,\\quad \nv_{-2,-3}= \\gamma_1 v_1\\ , $$\nassuming that $\\gamma$ shifts along the lattice, indicated by the indices, \nand finally the value of $v_{-1,-2,-3}$ can be computed in \ntwo different ways, leading to\n$$v_{-1,-2,-3}=\\gamma_0 v_0=\\frac{av_{-2,-3}-bv_{-1,-3}}{av_{-1,-3}-bv_{-2,-3}}v_{-3}=\n\\frac{a\\gamma_1v_1-b\\gamma_2v_2}{a\\gamma_2v_2-b\\gamma_1v_1}\\gamma_{1,2}v_0 \\frac{av_2-bv_1}{av_1-bv_2}\\ , $$ \nleading quadratic identity in $v_1$ and $v_2$. Assuming that the latter must hold identically, \nand thus setting all coefficients equal to zero, we obtain the following conditions on $\\gamma$:\n$$ \\gamma_{1,2,3}=\\gamma_1=\\gamma_2=\\gamma_3\\quad , $$ \nfrom which we conclude that $\\gamma$ is an alternating ``constant'', for instance \n\\begin{equation}\\label{form:gamma}\n\\gamma=\\alpha\\,\\beta^{(-1)^{n+m+\\dots}}\\quad \\quad (\\alpha,\\beta\\ \\ {\\rm constants}) \\ \n\\end{equation} \nand this leads to the conditions\nfrom which it is easily deduced that the form (\\ref{eq:gamma}) of $\\gamma$ satisfies these \nconditions. \n\n\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{s-intro}\n\nOver the last 10 years, in particular since the installation of the Cosmic Origins Spectrograph (COS) on the {\\it Hubble Space Telescope} ({\\em HST}), we have made significant leaps in empirically characterizing the circumgalactic medium (CGM) of galaxies at low redshift where a wide range of galaxy masses can be studied (see recent review by \\citealt{tumlinson17}). We appreciate now that the CGM of typical star-forming or quiescent galaxies have a large share of galactic baryons and metals in relatively cool gas-phases ($10^4$--$10^{5.5}$ K) \\citep[e.g.,][]{stocke13,bordoloi14,liang14,peeples14,werk14,johnson15,burchett16,prochaska17,chen19,poisson19}. We have come to understand that the CGM of galaxies at $z \\la 1$ is not just filled with metal-enriched gas ejected by successive galaxy outflows, but has also a large amount of metal poor gas ($<1$--2\\% solar) in which little net chemical enrichment has occurred over several billions of years \\citep[e.g.,][]{ribaudo11,thom11,lehner13,lehner18,lehner19,wotta16,wotta19,prochaska17,kacprzak19,poisson19,zahedy19}. The photoionized gas around $z\\la 1$ galaxies is very chemically inhomogeneous, as shown by large metallicity ranges and the large metallicity variations among kinematically distinct components in a single halo (\\citealt{wotta19,lehner19}, and see also \\citealt{crighton13a,muzahid15,muzahid16, rosenwasser18}). Such a large metallicity variation is not only observed in the CGM of star-forming galaxies, but also in the CGM of passive and massive galaxies where there appears to be as much cold, bound \\ion{H}{1}\\ gas as in their star-forming counterparts \\citep[e.g.,][]{thom12,tumlinson13,berg19,zahedy19}.\n\nThese empirical results have revealed both expected and unexpected properties of the CGM of galaxies and they all provide new means to understand the complex relationship between galaxies and their CGM. Prior to these empirical results, the theory of galaxy formation and evolution was mostly left constraining the CGM properties indirectly by their outcomes, such as galaxy stellar mass and ISM properties. Thus the balance between outflows, inflows, recycling, and ambient gas--and the free parameters controlling them--were tuned to match the optical properties of galaxies rather than implemented directly as physically-rigorous and self-consistent models. These indirect constraints suffer from problems of model uniqueness: it is possible to match stellar masses and metallicities with very different treatments of feedback physics \\citep[e.g.,][]{hummels13,liang16}. Recent empirical and theoretical advances offer a way out of this model degeneracy. New high-resolution, zoom-in simulations employ explicit treatments of the multiple gas-phase nature and feedback from stellar population models \\citep[e.g.,][]{hopkins14,hopkins18}. It is also becoming clear that not only high resolution inside the galaxies but also in their CGM is required to capture more accurately the complex processes in the cool CGM, such as metal mixing \\citep{hummels19,peeples19,suresh19,vandevoort18,corlies19}.\n\nA significant limitation in interpreting the new empirical results in the context of new high-resolution zoom simulations is that only average properties of the CGM are robustly derived from traditional QSO absorption-line techniques for examining halo gas. In the rare cases where there is a UV-bright QSO behind a given galaxy, the CGM is typically probed along a single ``core sample\" through the halo of each galaxy. These measurements are then aggregated into a statistical map, where galaxies with different inclinations, sizes, and environments are blended together and the radial-azimuthal dependence of the CGM is essentially lost. All sorts of biases can result: phenomena that occur strongly in only a subset of galaxies can be misinterpreted as being weaker but more common, and genuine trends with mass or star formation rate can be misinterpreted as simply scatter with no real physical meaning (see also \\citealt{bowen16}). Simulations also suggest that time-variable winds, accretion flows, and satellite halos can induce strong halo-to-halo variability, further complicating interpretation \\citep[e.g.][]{hafen17,oppenheimer18a}. Observational studies of single galaxy CGM with multiple sightlines are therefore required to gain spatial information on the properties of the CGM. \n\nMulti-sightline information on the CGM of single galaxies has been obtained in a few cases from binary or multiple (2--4) grouped QSOs behind foreground galaxies \\citep[e.g.,][]{bechtold94,martin10,keeney13,bowen16}, gravitationally-lensed quasars \\citep[e.g.,][]{smette92,rauch01,ellison04,lopez05,zahedy16,rubin18,kulkarni19}, giant gravitational arcs \\citep[e.g.,][]{lopez19}, or extended bright background objects observed with integral field units \\citep[e.g.,][]{peroux18}. These observations provide better constraints on the kinematic relationship between the CGM gas and the galaxy and on the size of CGM structures. However, they yield limited information on the gas-phase structure owing to a narrow range of ionization diagnostics or poor quality spectral data. Thus, it remains unclear how tracers of different gas phases vary with projected distance $R$ or azimuth $\\Phi$ around the galaxy. \n\nThe CGM that has been pierced the most is that of the Milky Way (MW), with several hundred QSO sightlines \\citep{wakker03,shull09,lehner12,putman12,richter17} through the Galactic halo. However, our position as observers within the MW disk severely limits the interpretation of these data (especially for the extended CGM, see \\citealt{zheng15, zheng20}) and makes it difficult to compare with observations of other galaxies. \n\nWith a virial radius that spans over $30\\degr$ on the sky, M31 is the only $L^*$ galaxy where we can access more than 5 sightlines without awaiting the next generation of UV space-based telescope (e.g., \\citealt{luvoir19}). With current UV capabilities, it is the only single galaxy where we can study the global distribution and properties of metals and baryons in some detail.\n\nIn our pilot study (\\citealt{lehner15}, hereafter \\citetalias{lehner15}), we mined the {\\em HST}\/COS G130M\/G160M archive available at the \\textit{Barbara A. Mikulski Archive for Space Telescopes} (MAST) for sightlines piercing the M31 halo within a projected distance of $\\sim 2 \\ensuremath{R_{\\rm vir}}$ (where $R_{\\rm vir}=300$ kpc for M31, see below). There were 18 sightlines, but only 7 at projected distance $R \\la \\ensuremath{R_{\\rm vir}}$. Despite the small sample, the results of this study were quite revealing, demonstrating a high covering factor (6\/7) of M31 CGM absorption by \\ion{Si}{3}\\ (and other ions including, e.g., \\ion{C}{4}, \\ion{Si}{2}) within $1.1\\ensuremath{R_{\\rm vir}}$ and a covering factor near zero (1\/11) between $1.1 \\ensuremath{R_{\\rm vir}}$ and $2 \\ensuremath{R_{\\rm vir}}$. We found also a drastic change in the ionization properties, as the gas is more highly ionized at $R \\sim \\ensuremath{R_{\\rm vir}} $ than at $R<0.2\\ensuremath{R_{\\rm vir}}$. The \\citetalias{lehner15} results strongly suggest that the CGM of M31 as seen in absorption of low ions (\\ion{C}{2}, \\ion{Si}{2}) through intermediate (\\ion{Si}{3}, \\ion{Si}{4}) and high ions (\\ion{C}{4}, \\ion{O}{6}) is very extended out to at least the virial radius. However, owing to the small sample within \\ensuremath{R_{\\rm vir}}, the variation of the column densities ($N$) and covering factors (\\ensuremath{f_{\\rm c}}) with projected distances and azimuthal angle remain poorly constrained.\n\nOur Project AMIGA (Absorption Maps In the Gas of Andromeda) is a large {\\em HST}\\ program (PID: 14268, PI: Lehner) that aims to fill the CGM with 18 additional sightlines at various $R$ and $\\Phi$ within $1.1 \\ensuremath{R_{\\rm vir}}$ of M31 using high-quality COS G130M and G160M observations, yielding a sample of 25 background QSOs probing the CGM of M31. We have also searched MAST for additional QSOs beyond $1.1 \\ensuremath{R_{\\rm vir}}$ up to $R=569$ kpc from M31 ($\\sim 1.9 \\ensuremath{R_{\\rm vir}}$) to characterize the extended gas around M31 beyond its virial radius. This archival search yielded 18 suitable QSOs. Our total sample of 43 QSOs probing the CGM of a single galaxy from 25 to 569 kpc is the first to explore simultaneously the azimuthal and radial dependence of the kinematics, ionization level, surface-densities, and mass of the CGM of a galaxy over its entire virial radius and beyond. With these observations, we can also test how the CGM properties derived from one galaxy using multiple sightlines compares with a sample of galaxies with single sightline information and we can directly compare the results with cosmological zoom-in simulations.\n\nWith the COS G130M and G160M wavelength coverage, the key ions in our study are \\ion{C}{2}, \\ion{C}{4}, \\ion{Si}{2}, \\ion{Si}{3}, \\ion{Si}{4}\\ (other ions and atoms include \\ion{Fe}{2}, \\ion{S}{2}, \\ion{O}{1}, \\ion{N}{1}, \\ion{N}{5}, but are typically not detected, although the limit on \\ion{O}{1}\\ constrains the level of ionization). These species span ionization potentials from $<1$ to $\\sim$4 Rydberg and thus trace neutral to highly ionized gas at a wide range of temperatures and densities. We have also searched the {\\it Far Ultraviolet Spectroscopic Explorer}\\ ({\\em FUSE}) to have coverage of \\ion{O}{6}, which resulted in 11 QSOs in our sample having both COS and {\\em FUSE}\\ observations. The \\ion{H}{1}\\ Ly$\\alpha$\\ absorption can unfortunately not be used because the MW dominates the entire Ly$\\alpha$\\ absorption. Instead we have obtained deep \\ion{H}{1}\\ 21-cm observations with the Robert C. Byrd Green Bank Telescope (GBT) toward all the targets in our sample and several additional ones (\\citealt{howk17}, hereafter \\citetalias{howk17}), showing no detection of any \\ion{H}{1}\\ down to a level $N_{\\rm H\\,I} \\simeq 4\\times 10^{17}$ cm$^{-2}$\\ ($5\\sigma$; averaged over an area that is 2 kpc at the distance of M31). Our non-detections place a limit on the covering factor of such optically thick \\ion{H}{1}\\ gas around M31 to $\\ensuremath{f_{\\rm c}}\\ < 0.051$ (at 90\\% confidence level) for $R\\la \\ensuremath{R_{\\rm vir}}$.\n\nThis paper is organized as follows. In \\S\\ref{s-data}, we provide more information about the criteria used to assemble our sample of QSOs and explain the various steps to derive the properties (velocities and column densities) of the absorption. In that section, we also present the line identification for each QSO spectrum, which resulted in the identification of 5,642 lines. In \\S\\ref{s-ms}, we explain in detail how we remove the foreground contamination from the Magellanic Stream (MS, e.g., \\citealt{putman03,nidever08,fox14}), which extends to the M31 CGM region of the sky with radial velocities that overlap with those expected from the CGM of M31. For this work, we have developed a more systematic and automated methodology than in \\citetalias{lehner15} to deal with this contamination. In \\S\\ref{s-dwarfs}, we present the sample of the M31 dwarf satellite galaxies to which we compare the halo gas measurements. In \\S\\ref{s-properties}, we derive the empirical properties of the CGM of M31 including how the column densities and velocities vary with $R$ and $\\Phi$, the covering factors of the ions and how they change with $R$, and the metal and baryon masses of the CGM of M31. In \\S\\ref{s-disc}, we discuss the results derived in \\S\\ref{s-properties} and compare them to observations from the COS-Halos survey \\citep{tumlinson13,werk14} and to state-of-the-art cosmological zoom-ins from in particular the Feedback in Realistic Environments (FIRE, \\citealt{hopkins19}) and and Figuring Out Gas \\& Galaxies In Enzo (FOGGIE, \\citealt{peeples19}) simulations projects. In \\S\\ref{s-sum}, we summarize our main conclusions.\n\nTo properly compare to other work, and to simulations, we must estimate a characteristic radius for M31. We use the radius $R_{200}$ enclosing a mean overdensity of $200$ times the critical density: $R_{200} = (3 M_{\\rm 200} \/ 4 \\pi \\, \\Delta\\; \\rho_{\\rm crit})^{1\/3}$, where $\\Delta = 200$ and $\\rho_{\\rm crit}$ is the critical density. For M31, we adopt $M_{200} = 1.26 \\times 10^{12}$ M$_\\sun$ (e.g., \\citealt{watkins10,vandermarel12}), implying $R_{200} \\simeq 230$ kpc. For the virial mass and radius (\\ensuremath{M_{\\rm vir}}\\ and \\ensuremath{R_{\\rm vir}}), we use the definition that follows from the top-hat model in an expanding universe with a cosmological constant where $\\ensuremath{M_{\\rm vir}} = 4\\pi\/3 \\; \\rho_{\\rm vir} \\ensuremath{R_{\\rm vir}}^3$ where the virial density $ \\rho_{\\rm vir} = \\Delta_{\\rm vir} \\Omega_{\\rm m} \\rho_{\\rm crit}$ \\citep{klypin11,vandermarel12}. The average virial overdensity is $\\Delta_{\\rm vir} = 360$ assuming a cosmology with $h=0.7$ and $\\Omega_{\\rm m} = 0.27$ \\citep{klypin11}. Following, e.g, \\citet{vandermarel12}, $\\ensuremath{M_{\\rm vir}} \\simeq 1.2 M_{200} \\simeq 1.5 \\times 10^{12}$ M$_\\sun$ and $\\ensuremath{R_{\\rm vir}} \\simeq 1.3 R_{200} \\simeq 300$ kpc. The escape velocity at $R_{200}$ for M31 is then $v_{200}\\simeq 212$ ${\\rm km\\,s}^{-1}$. A distance of M31 of $d_{\\rm M31} = 752 \\pm 27$ kpc based on the measurements of Cepheid variables \\citep{riess12} is assumed throughout. We note that this distance is somewhat smaller than the other often adopted distance of M31 of 783 kpc \\citep[e.g.,][]{brown04,mcconnachie05}, but for consistency with our previous survey as well as the original design of Project AMIGA, we have adopted $d_{\\rm M31} = 752$ kpc. All the projected distances were computed using the three dimensional separation (coordinates of the target and distance of M31).\n\n\\begin{figure*}[tbp]\n\\epsscale{0.9}\n\\plotone{f1.pdf}\n\\caption{Locations of the Project AMIGA pointings relative to the M31--M33 system. The axes show the angular separations converted into physical coordinates relative to the center of M31. North is up and east to the left. The 18 sightlines from our large {\\em HST}\\ program are in red filled circles; the 25 archival COS targets are in open red circles. Crosses show the GBT \\ion{H}{1}\\ 21-cm observations described in \\citetalias{howk17}. Dotted circles show impact parameters $R = 100$, 200, 300, 400, 500 kpc. $\\ensuremath{R_{\\rm vir}} = 300$ kpc is marked with a heavy dashed line. The sizes and orientations of the two galaxies are taken from RC3 \\citep{devaucouleurs91} and correspond to the optical $R_{25}$ values. The light blue dashed line shows the plane of the Magellanic Stream ($b_{\\rm MS} =0\\degr$) as defined by \\citet{nidever08}. The shaded region within $b_{\\rm MS} \\pm 20\\degr$ of the MS midplane is the approximate region where we identify most of the MS absorption components contaminating the M31 CGM absorption (see \\S\\ref{s-ms}).\\label{f-map}}\n\\end{figure*}\n\n\\section{Data and Analysis}\\label{s-data}\n\\subsection{The Sample}\\label{s-sample}\n\nThe science goals of our {\\em HST}\\ large program require estimating the spatial distributions of the kinematics and metal column densities of the M31 CGM gas within about $1.1\\ensuremath{R_{\\rm vir}}$ as a function of azimuthal angle and impact parameter. The search radius was selected based on our pilot study where we detected M31 CGM gas up to $\\sim 1.1\\ensuremath{R_{\\rm vir}}$, but essentially not beyond \\citepalias{lehner15} (a finding that we revisit in this paper with a larger archival sample, see below). With our {\\em HST}\\ program, we observed 18 QSOs at $R\\la 1.1\\ensuremath{R_{\\rm vir}}$ that were selected to span the M31 projected major axis, minor axis, and intermediate orientations. The sightlines do not sample the impact parameter space or azimuthal distribution at random. Instead, the sightlines were selected to probe the azimuthal variations systematically. The sample was also limited by a general lack of identified UV-bright AGNs behind the northern half of M31's CGM owing to higher foreground MW dust extinction near the plane of the Milky Way disk. Combined with 7 archival QSOs, these sightlines probe the CGM of M31 in azimuthal sectors spanning the major and minor axes with a radial sample of 7--10 QSOs in each $\\sim$100 kpc bin in $R$.\n\nIn addition to target locations, the 18 QSOs were optimized to be the brightest available QSOs (to minimize exposure time) and to have the lowest available redshifts (in order to minimize the contamination from unrelated absorption from high redshift absorbers). For targets with no existing UV spectra prior to our observations, we also required that the GALEX NUV and FUV flux magnitudes are about the same to minimize the likelihood of an intervening Lyman limit system (LLS) with optical depth at the Lyman limit $\\tau_{\\rm LL}>2$. An intervening LLS could absorb more or all of the QSO flux we would need to measure foreground absorption in M31. This strategy successfully kept QSOs with intervening LLS out of the sample. \n\nAs we discuss below and as detailed by \\citetalias{lehner15}, the MS crosses through the M31 region of the sky at radial velocities that can overlap with those of M31 (see also \\citealt{nidever08,fox14}). To understand the extent of MS contamination and the extended gas around M31 beyond the virial radius, we also searched for targets beyond $1.1\\ensuremath{R_{\\rm vir}}$ with COS G130M and\/or G160M data. This search identified another 18 QSOs at $1.1\\la R\/\\ensuremath{R_{\\rm vir}} <1.9$ that met the data quality criteria for inclusion in the sample\\footnote{This search found eight additional targets at $R>1.6 \\ensuremath{R_{\\rm vir}}$ that are not included in our sample. SDSSJ021348.53+125951.4, 4C10.08, LBQS0052-0038 were excluded because of low S\/N in the COS data. NGC7714 has smeared absorption lines. LBQS0107-0232\/3\/5 lie at $z_{\\rm em}\\simeq 0.7$--1 and have extremely complex spectra. HS2154+2228 at $z_{\\rm em} = 1.29$ has no G130M wavelength coverage making the line identification highly uncertain. \\label{foot-reason}}. Our final sample consists of 43 sightlines probing the CGM of M31 from 25 to 569 kpc; 25 of these probe the M31 CGM from 25 to 342 kpc, corresponding to $0.08 - 1.1 \\ensuremath{R_{\\rm vir}}$. Fig.~\\ref{f-map} shows the locations of each QSO in the M31--M33 system (the filled circles being targets obtained as part of our {\\em HST}\\ program PID: 14268 and the open circles being QSOs with archival {\\em HST}\\ COS G130M\/G160M data), and Table~\\ref{t-sum} lists the properties of our sample QSOs ordered by increasing projected distances from M31. In this table, we list the redshift of the QSOs (\\ensuremath{z_{\\rm em}}), the J2000 right ascension (RA) and declination (Dec.), the MS coordinate (\\ensuremath{l_{\\rm MS}}, \\ensuremath{b_{\\rm MS}}, see \\citealt{nidever08} for the definition of this coordinate system), the radially ($R$) and cartesian ($X,Y$) projected distances, the program identification of the {\\em HST}\\ program (PID), the COS grating used for the observations of the targets, and the signal-to-noise ratio (SNR) per COS resolution element of the COS spectra near the \\ion{Si}{3}\\ transition (except otherwise stated in the footnote of this table). \n\n\\subsection{UV Spectroscopic Calibration}\\label{s-calib}\n\nTo search for M31 CGM absorption and to determine the properties of the CGM gas, we use ions and atoms that have their wavelengths in the UV (see \\S\\ref{s-prop}). Any transitions with $\\lambda>1144$ \\AA\\ are in the {\\em HST}\\ COS bandpass. All the targets in our sample were observed with {\\em HST}\\ using the COS G130M grating ($R_\\lambda \\approx 17,000$). All the targets observed as part of our new {\\em HST}\\ program were also observed with COS G160M, and all the targets but one within $R<1.1 \\ensuremath{R_{\\rm vir}}$ have both G130M and G160M wavelength coverage. \n\nWe also searched for additional archival UV spectra in MAST, including the {\\em FUSE}\\ ($R_\\lambda \\approx 15,000$) archive to complement the gas-phase diagnostics from the COS spectra with information from the \\ion{O}{6}\\ absorption. We use the {\\em FUSE}\\ observations for 11 targets with adequate SNR near \\ion{O}{6}\\ (i.e., $\\ga 5$): RX\\_J0048.3+3941, IRAS\\_F00040+4325, MRK352, PG0052+251, MRK335, UGC12163, PG0026+129, MRK1502, NGC7469, MRK304, PG2349-014 (only the first 6 targets in this list are at $R\\la 1.1 \\ensuremath{R_{\\rm vir}}$). We did not consider {\\em FUSE}\\ data for quasars without COS observations because the available UV transitions in the far-UV spectrum (\\ion{O}{6}, \\ion{C}{2}, \\ion{C}{3}, \\ion{Si}{2}, \\ion{Fe}{2}) are either too weak or too contaminated to allow for a reliable identification of the individual velocity components in their absorption profiles. \n\nThere are also 3 targets (MRK335, UGC12163, and NGC7469) with {\\em HST}\\ STIS E140M ($R_\\lambda \\simeq 46,500$) observations that provide higher resolution information.\\footnote{For 2 targets, we also use COS G225M (3C454.3) and FOS NUV (3C454.3, PG0044+030) observations to help with the line identification (see \\S\\ref{s-lineid}). The data processing follows the same procedure as the other data.}\n\nInformation on the design and performance of COS, STIS, {\\em FUSE}\\ can be found in \\citet{green12}, \\citet{woodgate98}, and \\citet{moos00}, respectively. For the {\\em HST}\\ data, we use the pipeline-calibrated final data products available in MAST. The {\\em HST}\\ STIS E140M data have an accurate wavelength calibration and the various exposure and echelle orders are combined into a single spectrum by interpolating the photon counts and errors onto a common grid, adding the photon counts and converting back to a flux.\n\nThe processing of the {\\em FUSE}\\ data is described in detail by \\citet{wakker03} and \\citet{wakker06}. In short, the spectra are calibrated using version 2.1 or version 2.4 of the {\\em FUSE}\\ calibration pipeline. The wavelength calibration of {\\em FUSE}\\ can suffer from stretches and misalignments. To correct for residual wavelength shifts, the central velocities of the MW interstellar lines are determined for each detector segment of each individual observation. The {\\em FUSE}\\ segments are then aligned with the interstellar velocities implied by the STIS E140M spectra or with the velocity of the strongest component seen in the 21-cm \\ion{H}{1}\\ spectrum. Since the \\ion{O}{6}\\ absorption can be contaminated by H$_2$ absorption, we remove this contamination following the method described in \\citet{wakker06}. This contamination can be removed fairly accurately with an uncertainty of about $\\pm 0.1$ dex on the \\ion{O}{6}\\ column density \\citep{wakker03}.\n\nFor the COS G130M and G160M spectra, the spectral lines in separate observations of the same target are not always aligned, with misalignments of up to $\\pm 40$ ${\\rm km\\,s}^{-1}$\\ that varying as function of wavelength. This is a known issue that has been reported previously \\citep[e.g.,][]{savage14,wakker15}. While the COS team has improved the wavelength solution, we find that this problem can still be present sometimes. \nSince accurate alignment is critical for studying multiple gas-phases probed by different ions and since there is no way to determine {\\it a priori} which targets are affected, we uniformly apply the \\cite{wakker15} methodology to coadd the different exposures of the COS data to ensure proper alignment of the absorption lines. In short, we identify the various strong ISM and IGM weak lines and record the component structures and identify possible contamination of the ISM lines by IGM lines. We cross-correlate each line in each exposure, using a $\\sim$3 \\AA\\ wide region, and apply a shift as a function of wavelength to each spectrum. To determine the absolute wavelength calibration, we compare the velocity centroids of the Gaussian fits to the interstellar UV absorption lines (higher velocity absorption features being Gaussian fitted separately) and the \\ion{H}{1}\\ emission observed from our 9$\\arcmin$ GBT \\ion{H}{1}\\ survey \\citepalias{howk17} or otherwise from 21-cm data from the Leiden\/Argentine\/Bonn (LAB) survey \\citep{kalberla05} or the Parkes Galactic All Sky Survey (GASS) \\citep{kalberla10}. The alignment is coupled with the line identification into an iterative process to simultaneously determine the most accurate alignment and line identification (see \\S\\ref{s-lineid}). To combine the aligned spectra, we add the total counts in each pixel and then convert back to flux, using the average flux\/count ratio at each wavelength (see also \\citealt{tumlinson11a,tripp11}); the flux error is estimated from the Poisson noise implied by the total count rate.\n\n\\subsection{Line Identification}\\label{s-lineid}\nWe are interested in the velocity range $-700 \\le \\ensuremath{v_{\\rm LSR}}\\ \\le -150$ ${\\rm km\\,s}^{-1}$\\ where absorption from the M31 CGM may occur (see \\S\\ref{s-prop} for the motivation of this velocity range). It is straightforward to identify M31 absorption or its absence in this pre-defined velocity range, but we must ensure that there is either no contamination from higher redshift absorbers, or if there is, that we can correct for it.\n\nFor ions with multiple transitions, it is relatively simple to determine whether contamination is at play by comparing the column densities and the shapes of the velocity profiles of the available transitions. The profiles of atoms or ions with a single transition can be compared to other detected ions to check if there is some obvious contamination in the single transition absorption. However, some contamination may still remain undetected if it directly coincides with the absorption under consideration. Furthermore, when only a single ion with a single transition is detected (\\ion{Si}{3}\\ $\\lambda$1206 being the prime example), the only method that determines if it is contaminated or not is to undertake a complete line identification of all absorption features in each QSO spectrum.\n\nFor the 18 targets in our large {\\em HST}\\ program, our instrument setup ensures that we have the complete wavelength coverage with no gap between 1140 and 1800 \\AA. As part of our target selection, we also favor QSOs at low redshift (44\\% are at $\\ensuremath{z_{\\rm em}} \\le 0.1$, 89\\% at $\\ensuremath{z_{\\rm em}} \\le 0.3$). This assures that Ly$\\alpha$\\ remains in the observed wavelength range out to the redshift of the QSO (Ly$\\alpha$\\ redshifts out the long end of the COS band at $z = 0.48$) and greatly reduces the contamination from EUV transitions in the COS bandpass. The combination of wavelength coverage and low QSO redshift ensures the most accurate line identification. At $R<351$ kpc (i.e., $\\la 1.2 \\ensuremath{R_{\\rm vir}}$), 93\\% have Ly$\\alpha$\\ coverage down to $z = \\ensuremath{z_{\\rm em}}$ that remains in the observed wavelength range (one target has only observation of G130M and another QSO is at $z=0.5$, see Table~\\ref{t-sum}). On the other hand, for the targets at $R>351$ kpc, the wavelength coverage is not as complete over 1140--1800 \\AA\\ (55\\% of the QSOs have only 1 COS grating---all but one have G130M, and 4 QSOs have $\\ensuremath{z_{\\rm em}} \\ga 0.48$). We note that the QSOs of 6\/10 G130M observations have $\\ensuremath{z_{\\rm em}} <0.17$, setting all the Ly$\\alpha$\\ transitions within the COS G130M bandpass. \n\nThe overall line identification process is as follows. First, we mark all the ISM absorption features (i.e., any absorption that could arise from the MW or M31) and the velocity components (which is done as part of the overall alignment of the spectra, see \\S\\ref{s-calib}). Local (approximate) continua are fitted near the absorption lines to estimate the equivalent widths ($W_\\lambda$) and their ratios for ions with several transitions are checked to determine if any are potentially contaminated. We then search for any absorption features at $z = \\ensuremath{z_{\\rm em}}$, again identifying any velocity component structures in the absorption. We then identify possible Ly$\\alpha$\\ absorption and any other associated lines (other \\ion{H}{1}\\ transitions and metal transitions) from the redshift of QSO down to $z=0$. In each case, if there are simultaneous detections of Ly$\\alpha$, Ly$\\beta$, and\/or Ly$\\gamma$\\ (and weaker transitions), we check that the equivalent width ratios are consistent. If there are any transitions left unidentified, we check whether it could be \\ion{O}{6}\\ $\\lambda\\lambda$1031, 1037 as this doublet can be sometimes detected without any accompanying \\ion{H}{1}\\ \\citep{tripp08}. Finally we check that the alignment in each absorber with multiple detected absorption lines is correct or whether it needs some additional adjustment.\n\nIn the region $R\\la 1.1\\ensuremath{R_{\\rm vir}}$ and for 84\\% of the sample at any $R$, we believe the line identifications are reliable and accurate at the 98\\% confidence level. In the Appendix, we provide some additional information regarding the line identification, in particular for the troublesome cases. We also make available in a machine-readable format the full line identification for all the targets listed in Table~\\ref{t-sum} (see Appendix~\\ref{a-lineid}).\n\n\\subsection{Determination of the Properties of the Absorption at $-700 \\le v_{\\rm LSR} \\le -150$ ${\\rm km\\,s}^{-1}$ }\\label{s-prop}\n\nOur systematic search window for absorption that may be associated with the CGM of M31 is $-700 \\le \\ensuremath{v_{\\rm LSR}}\\ \\le -150$ ${\\rm km\\,s}^{-1}$\\ \\citepalias{lehner15}. The $-700$ ${\\rm km\\,s}^{-1}$\\ cutoff corresponds to about $-100$ ${\\rm km\\,s}^{-1}$\\ less than the most negative velocities from the rotation curve of M31 ($\\sim -600$ ${\\rm km\\,s}^{-1}$, see \\citealt{chemin09}). The $-150$ ${\\rm km\\,s}^{-1}$\\ cutoff is set by the MW lines that dominate the absorption in the velocity range $-150 \\la \\ensuremath{v_{\\rm LSR}}\\ \\la +50$ ${\\rm km\\,s}^{-1}$. The $-100 \\la \\ensuremath{v_{\\rm LSR}} \\la -50$ ${\\rm km\\,s}^{-1}$\\ range is dominated by low and intermediate-velocity clouds that are observed in and near the Milky Way disk. Galactic high-velocity clouds (HVCs) down to velocities $\\ensuremath{v_{\\rm LSR}} \\sim -150$ ${\\rm km\\,s}^{-1}$\\ further above the MW disk have also been observed toward distant Galactic halo stars in the general direction of M31 \\citep{lehner15,lehner12,lehner11a}. Since the M31 disk rotation velocities extend to about $-150$ ${\\rm km\\,s}^{-1}$\\ in the northern tip of M31, there is a small window that is inaccessible for studying the CGM of M31 (see also \\citealt{lehner15} and \\S\\ref{s-dwarfs-vel}).\n\nTo search for M31 CGM gas and determine its properties, we use the following atomic and ionic transitions: \\ion{O}{1}\\ $\\lambda$1302, \\ion{C}{2}\\ $\\lambda\\lambda$1036, 1334, \\ion{C}{4}\\ $\\lambda\\lambda$1548, 1550, \\ion{Si}{2}\\ $\\lambda\\lambda$1190, 1193, 1260, 1304, 1526 \\ion{Si}{3}\\ $\\lambda$1206, \\ion{Si}{4}\\ $\\lambda\\lambda$1393, 1402, \\ion{O}{6}\\ $\\lambda$1031, \\ion{Fe}{2}\\ $\\lambda\\lambda$1144, 1608, \\ion{Al}{2}\\ $\\lambda$1670. We also report results (mostly upper limits on column densities) for \\ion{N}{5}\\ $\\lambda\\lambda$1238, 1242, \\ion{N}{1}\\ $\\lambda$1199 (\\ion{N}{1}\\ $\\lambda\\lambda$1200, 1201 being typically blended in the velocity range of interest $-700 \\le \\ensuremath{v_{\\rm LSR}}\\ \\le -150$ ${\\rm km\\,s}^{-1}$), \\ion{P}{2}\\ $\\lambda$1301, \\ion{S}{3}\\ $\\lambda$1190, and \\ion{S}{2}\\ $\\lambda\\lambda$1250, 1253, 1259. \n\nTo determine the column densities and velocities of the absorption, we use the apparent optical depth (AOD) method (see \\S\\ref{s-aod}), but in the Appendix~\\ref{s-pf} we confront the AOD results with measurements from Voigt profile fitting (see also \\S\\ref{s-gen-comments}). As much as possible at COS resolution, we derive the properties of the absorption in individual components. Especially toward M31, this is important since along the same line of sight in the velocity window $-700 \\le \\ensuremath{v_{\\rm LSR}}\\ \\le -150$ ${\\rm km\\,s}^{-1}$, there can be multiple origins of the gas (including the CGM of M31 or MS, see Fig.~\\ref{f-map} and \\citetalias{lehner15}) as we detail in \\S\\ref{s-ms}. However, the first step to any analysis of the absorption imprinted on the QSO spectra is to model the QSO's continuum.\n\n\\subsubsection{Continuum Placement}\\label{s-continuum}\nTo fit the continuum near the ions of interest, we generally use the automated continuum fitting method developed for the COS CGM Compendium (CCC, \\citealt{lehner18}). Fig. 3 in \\citet{lehner18} shows an example of an automatic continuum fit. In short, the continuum is fitted near the absorption features using Legendre polynomials. A velocity region of about $\\pm$1000--2000 ${\\rm km\\,s}^{-1}$\\ around the relevant absorption transition is initially considered for the continuum fit, but could be changed depending on the complexity of the continuum placement in this region. In all cases the interval for continuum fitting is never larger than $\\pm$2000 ${\\rm km\\,s}^{-1}$\\ or smaller than $\\pm$250 ${\\rm km\\,s}^{-1}$. Within this pre-defined region, the spectrum is broken into smaller sub-sections and then rebinned. The continuum is fitted to all pixels that did not deviate by more than $2\\sigma$ from the median flux, masking pixels from the fitting process that may be associated with small-scale absorption or emission lines. Legendre polynomials of orders between 1 and 5 are fitted to the unmasked pixels, with the goodness of the fit determining the adopted polynomial order. Typically the adopted polynomials are of orders between 1 and 3 owing to the relative simplicity of the QSO continua when examined over velocity regions of 500--4000 ${\\rm km\\,s}^{-1}$. The only systematic exception is \\ion{Si}{3}\\ where the polynomial order is always between 2--3 and 5 owing to this line being in the wing of the broad local Ly$\\alpha$\\ absorption profile.\n\nThis procedure is applied to our pre-defined set of transitions, with the continuum defined locally for each. Each continuum model is visually inspected for quality control. In a few cases, the automatic continuum fitting fails owing to a complex continuum (e.g., near the peak of an emission line or where many absorption lines were present within the pre-defined continuum window). In these cases, we first try to adjust the velocity interval of the spectrum to provide better-constrained fits; if that still fails, we manually select the continuum region to be fitted. \n\n\\begin{figure*}[tbp]\n\\epsscale{1}\n\\plotone{f2.pdf}\n\\caption{Example of normalized absorption lines as a function of the LSR velocity toward RX\\_J0043.6+3725 showing the typical atoms and ions probed in our survey. High negative velocity components likely associated with M31 are shown in colors, and each color represents a different component identified at the COS G130M-G160M resolution. In this case, significant absorption is observed in the two identified components in \\ion{C}{2}, \\ion{Si}{2}, and \\ion{Si}{3}. Higher ions (\\ion{Si}{4}, \\ion{C}{4}) are observed in only one of the components, showing a change in the ionization properties with velocity. Some species are not detected, but their limits can still be useful in assessing the physical properties of the gas. The MW absorption is indicated between the two vertical dotted lines and is observed in all the species but \\ion{N}{5}. At $\\ensuremath{v_{\\rm LSR}} \\ga -100$ ${\\rm km\\,s}^{-1}$, airglow emission lines can contaminate \\ion{O}{1}, and hence the MW absorption is contaminated, but typically that is not an issue for the surveyed velocity range $-700 \\le \\ensuremath{v_{\\rm LSR}}\\ \\le -150$ ${\\rm km\\,s}^{-1}$. \n\\label{f-example-spectrum}}\n\\end{figure*}\n\n\n\\subsubsection{Velocity Components and AOD Analysis}\\label{s-aod}\n\nThe next step of the analysis is to determine the velocity components and integrate them to determine the average central velocities and column densities for each absorption feature. In Fig.~\\ref{f-example-spectrum}, we show an example of the normalized velocity profiles. In the supplemental material, we provide a similar figure for each QSO in our sample. Although we systematically search for absorption in the full velocity range $-700 \\le \\ensuremath{v_{\\rm LSR}}\\ \\le -150$ ${\\rm km\\,s}^{-1}$, the most negative velocity of detected absorption in our sample is $\\ensuremath{v_{\\rm LSR}}= -508$ ${\\rm km\\,s}^{-1}$; that is, we do not detect any M31 absorption in the range $-700 \\la \\ensuremath{v_{\\rm LSR}} \\la -510 $ ${\\rm km\\,s}^{-1}$. In Fig.~\\ref{f-example-spectrum}, MW absorption at $-100 \\la \\ensuremath{v_{\\rm LSR}} \\la 100$ ${\\rm km\\,s}^{-1}$\\ is clearly seen in all species but \\ion{N}{5}. Absorption observed in the $-510 \\le \\ensuremath{v_{\\rm LSR}}\\ \\le -150$ ${\\rm km\\,s}^{-1}$\\ that is not color-coded is produced by higher-redshift absorbers or other MW lines.\n\nTo estimate the column density in each observed component, we use the AOD method \\citep{savage91}. In this method, the absorption profiles are converted into apparent optical depth per unit velocity, $\\tau_a(v) = \\ln[F_{\\rm c}(v)\/F_{\\rm obs}(v)]$, where $F_c(v)$ and $F_{\\rm obs}(v)$ are the modeled continuum and observed fluxes as a function of velocity. The AOD, $\\tau_a(v)$, is related to the apparent column density per unit velocity, $N_a(v)$, through the relation $N_a(v) = 3.768 \\times 10^{14} \\tau_a(v)\/(f \\lambda(\\mbox{\\AA})$) ${\\rm cm}^{-2}\\,({\\rm km\\,s^{-1}})^{-1}$, where $f$ is the oscillator strength of the transition and $\\lambda$ is the wavelength in \\AA. The total column density is obtained by integrating the profile over the pre-defined velocity interval, $N = \\int_{v_1}^{v_2} N_a(v) dv $, where $[v_1,v_2]$ are the boundaries of the absorption. We estimate the line centroids with the first moment of the AOD $v_a = \\int v \\tau_a(v) dv\/\\int \\tau_a(v)dv$ ${\\rm km\\,s}^{-1}$. As part of this process, we also estimate the equivalent widths, which we use mainly to determine if the absorption is detected at the $\\ge 2\\sigma$ level. In cases where the line is not detected at $\\ge 2\\sigma$ significance, we quote a 2$\\sigma$ upper limit on the column density, which is defined as twice the 1$\\sigma$ error derived for the column density assuming the absorption line lies on the linear part of the curve of growth.\n\nFor features that are detected above the $2\\sigma$ level, the estimated column densities are stored for further analysis. Since we have undertaken a full identification of the absorption features in each spectrum (see \\S\\ref{s-lineid}, Appendix \\ref{a-lineid}), we can reliably assess if a given transition is contaminated using in particular the conflict plots described in the Appendix (see Appendix \\ref{a-conflicplot}). If there is evidence of some line contamination and several transitions are available for this ion (e.g., \\ion{Si}{2}, \\ion{Si}{4}, \\ion{C}{4}), we exclude it from our list. \n\nWe find contamination affects the \\ion{Si}{3}\\ and \\ion{C}{2}\\ in the velocity range $-700 \\le \\ensuremath{v_{\\rm LSR}}\\ \\le -150$ ${\\rm km\\,s}^{-1}$\\ in a few rare cases (6 components of \\ion{Si}{3}\\ and 3 components of \\ion{C}{2}\\ $\\lambda$1334).\\footnote{Toward RX\\_J0048.3+3941, \\ion{C}{2}\\ $\\lambda$1334 is contaminated in the third component, but \\ion{C}{2}\\ $\\lambda$1036 is available to correct for it in this case.}. For all but one of these contaminated \\ion{Si}{3}\\ components, we can correct the contamination because the interfering line is a Lyman series line from a higher redshift and the other \\ion{H}{1}\\ transitions constrain the equivalent width of the contamination. The one case we cannot correct this way is the $-340$ ${\\rm km\\,s}^{-1}$\\ component toward PHL1226 (see also Appendix~\\ref{a-lineid}), which is associated with the MS. In the footnote of Table~\\ref{t-results}, we list the ions that are found to be contaminated at some level. For any column density that is corrected for contamination, the typical correction error is about 0.05--0.10 dex depending on the level of contamination as well as the SNRs of the spectrum in that region.\n\nThe last step is to check for any unresolved saturation. When the absorption is clearly saturated (i.e., the flux level reaches zero-flux in the core of the absorption), the line is automatically marked as saturated and a lower limit is assigned to the column density. In \\S\\ref{s-ms}, we will show how we separate the MS from the M31 CGM absorption, but we note that only the \\ion{Si}{3}\\ components associated with the MS and the MW have their absorption reaching zero-flux level, not the components associated with the CGM of M31.\n\nWhen the flux does not reach a zero-flux level, the procedure for checking saturation depends on the number of transitions for a given ion or atom. We first consider ions with several transitions (\\ion{Si}{2}, \\ion{C}{4}, \\ion{Si}{4}, sometimes \\ion{C}{2}) since they can provide information about the level of saturation for a given peak optical depth. For ions with several transitions, we compare the column densities with different $f\\lambda$-values to determine whether there is a systematic decrease in the column density as $f\\lambda$ increases. If there is not, we estimate the average column density using all the available measurements and propagate the errors using a weighted mean. For the \\ion{Si}{2}\\ transitions, \\ion{Si}{2}\\ $\\lambda$1526 shows no evidence for saturation when detected based on the comparison with stronger transitions while \\ion{Si}{2}\\ $\\lambda$1260 or $\\lambda$1193 can be saturated if the peak optical $\\tau_a \\ga 0.9$. For doublets (e.g., \\ion{C}{4}, \\ion{Si}{4}), we systematically check if the column densities of each transition agree within $1\\sigma$ error; if they do not and the weak transition gives a higher value (and there is no contamination in the weaker transition), we correct for saturation following the procedure discussed in \\citet{lehner18} (and see also \\citealt{savage91}). For \\ion{C}{4}\\ and \\ion{Si}{4}, there is rarely any evidence for saturation (we only correct once for saturation of \\ion{C}{4}\\ in the third component observed in the MRK352 spectrum; in that component the peak optical $\\tau_a \\sim 0.9$). For single strong transitions (in particular \\ion{Si}{3}\\ and often \\ion{C}{2}), if the peak optical depth is $\\tau_a >0.9$, we conservatively flag the component as saturated and adopt a lower limit for that component. We adopt $\\tau_a >0.9$ as the threshold for saturation based on other ions with multiple transitions (in particular \\ion{Si}{2}) where the absorption starts to show some saturation at this peak optical depth.\n\nTo estimate how the column density of silicon varies with $R$ (which has a direct consequence for the CGM mass estimates derived from silicon in \\S\\S\\ref{s-nsi-vs-r} and \\ref{s-mass}), it is useful to assess the level of saturation of \\ion{Si}{3}, which is the only silicon ion that cannot be directly corrected for saturation\\footnote{Some of the \\ion{Si}{2}\\ transitions (especially, \\ion{Si}{2}\\ $\\lambda\\lambda$1193, 1260) have evidence for saturation, but weaker transitions are always available (e.g., \\ion{Si}{2}\\ $\\lambda$1526), and therefore we can determine a robust value of the column density of \\ion{Si}{2}.}. The lower limits of the \\ion{Si}{3}\\ components associated with the CGM of M31 are mostly observed at $R\\la 140$ kpc (only 2 are observed at $R>140$ kpc), but they do not reach zero-flux level; these components are conservatively marked as saturated because their peak apparent optical depth is $\\tau_a > 0.9$ (not because $\\tau_a \\gg 2$) and because the comparison between the different \\ion{Si}{2}\\ transitions show in some cases evidence for saturation (see above). Hence the true values of the column densities of these saturated components is most likely higher than the adopted lower-limit values but are very unlikely to be overestimated by a factor $\\gg 3$--4. We can estimate how large the saturation correction for \\ion{Si}{3}\\ might be using the strong \\ion{Si}{2}\\ lines (e.g., \\ion{Si}{2}\\ $\\lambda$1193 or \\ion{Si}{2}\\ $\\lambda$1260) compared to the weaker ones (e.g., \\ion{Si}{2}\\ $\\lambda$1526). Going through the 8 sightlines showing some saturation in the components of \\ion{Si}{3}\\ associated with the CGM of M31 (see Table~\\ref{t-results}), for all the targets beyond 50 kpc, the saturation correction is likely to be small $<0.10$--0.15 dex based on the fact that many show no evidence of saturation in \\ion{Si}{2}\\ $\\lambda$1260 (when there is no contamination for this transition) or \\ion{Si}{2}\\ $\\lambda$1193. On the other hand, for the two most inner targets, the saturation correction is at least 0.3 dex and possibly as large as 0.6 dex based on the column density comparison between saturated \\ion{Si}{2}\\ and weaker, unsaturated transitions. The latter would put $N_{\\rm Si}\\simeq 14.5$ close to the maximum values derived with photoionization modeling in the COS-Halos sample (see \\S\\ref{s-coshalos}). Therefore for the components associated with the CGM of M31 at $R>50$ kpc when we estimate the functional form of $N_{\\rm Si}$ with $R$, we adopt an increase of 0.1 dex of the lower limits. For the two inner targets at $R<50$ kpc, we explore how an increase of 0.3 and 0.6 dex affects the estimation of $N_{\\rm Si}(R)$.\n\n\\subsubsection{High Resolution Spectra and Profile Fitting Analysis}\\label{s-gen-comments}\n\nIn the Appendices~\\ref{s-comp-fit-aod} and \\ref{s-pf} we explore the robustness of the AOD results by comparing high- and low-resolution spectra and by comparing to a Voigt profile fitting analysis. There is good overall agreement in the column densities derived from the STIS and COS data and our conservative choice of $\\tau_a \\sim 0.9$ as the threshold for saturation in the COS data is adequate (see Appendix~\\ref{s-comp-fit-aod}). For the profile fitting analysis, we consider the most complicated blending of components in our sample and demonstrate there are some small systematic differences between the AOD and PF derived column densities (see Appendix~\\ref{s-pf}). However these difference are small and a majority of our sample is not affected by heavy blending. Hence the AOD results are robust and are adopted for the remaining of the paper.\n\n\\subsection{Correcting for Magellanic Stream Contamination}\\label{s-ms}\nPrior to determining the properties of the gas associated with the CGM of M31, we need to identify that gas and distinguish it from the MW and the MS. We have already removed from our analysis any contamination from higher redshift intervening absorbers and any contamination from the MW (defined as $-150 \\la \\ensuremath{v_{\\rm LSR}}\\ \\la 100$ ${\\rm km\\,s}^{-1}$). However, as shown in Fig.~\\ref{f-map} and discussed in \\citetalias{lehner15}, the MS is another potentially large source of contamination: in the direction of M31, the velocities of the MS can overlap with those expected from the CGM of M31. The targets in our sample have MS longitudes and latitudes in the range $-132\\degr \\le \\ensuremath{l_{\\rm MS}} \\le -86\\degr$ and $-14\\degr \\le \\ensuremath{b_{\\rm MS}} \\le +41\\degr$. The \\ion{H}{1}\\ 21-cm emission GBT survey by \\citet{nidever10} finds that the MS extends to about $\\ensuremath{l_{\\rm MS}} \\simeq -140\\degr$. Based on this and previous \\ion{H}{1}\\ emission surveys, \\citet{nidever08,nidever10} found a relation between the observed LSR velocities of the MS and \\ensuremath{l_{\\rm MS}}\\ that can be used to assess contamination in our targeted sightlines based on their MS coordinates. Using Fig.~7 of \\citet{nidever10}, we estimate the upper and lower boundaries of the \\ion{H}{1}\\ velocity range as a function of \\ensuremath{l_{\\rm MS}}, which we show in Fig.~\\ref{f-nidever} by the curve colored area. The MS velocity decreases with decreasing \\ensuremath{l_{\\rm MS}}\\ up to $\\ensuremath{l_{\\rm MS}} \\simeq -120\\degr$ where there is an inflection point where the MS LSR velocity increases. We note that the region beyond $\\ensuremath{l_{\\rm MS}} \\la -135\\degr$ is uncertain but cannot be larger than shown in Fig.~\\ref{f-nidever} (see also \\citealt{nidever10})---however, this does not affect our survey since all our data are at $\\ensuremath{l_{\\rm MS}} \\ga -132\\degr$.\n\n\n\\begin{figure*}[tbp]\n\\epsscale{1.}\n\\plotone{f3.pdf}\n\\caption{The LSR velocity of the \\ion{Si}{3}\\ components (circles) observed in our sample as a function of the MS longitude \\ensuremath{l_{\\rm MS}}, color-coded according to the absolute MS latitude. Shaded regions show the velocities that can be contaminated by the MS and MW (by definition of our search velocity window, any absorption at $\\ensuremath{v_{\\rm LSR}} >-150$ ${\\rm km\\,s}^{-1}$\\ was excluded from our sample). We also show the data (squares) from the MS survey from \\citet{fox14} and the radial velocities of the M31 dwarf galaxies (stars). \n\\label{f-nidever}}\n\\end{figure*}\n\nWe take a systematic approach to removing the MS contamination that does not reject entire sightlines based on their MS coordinates. Not all velocity components may be contaminated even on sightlines close to the MS. In Fig.~\\ref{f-nidever}, we show LSR velocity of the \\ion{Si}{3}\\ components as a function of the MS longitude. We choose \\ion{Si}{3}\\ as this ion is the most sensitive to detect both weak and strong absorption and is readily observed the physical conditions of the MS and M31 CGM \\citep{fox14,lehner15}. We consider the individual components as for a given sightline, several components can be observed falling in or outside the boundary region associated with the MS as illustrated in Fig.~\\ref{f-nidever}. We find that $28\/74\\simeq 38\\%$ of the detected \\ion{Si}{3}\\ components are within MS boundary region shown in Fig.~\\ref{f-nidever}. We note that changing the upper boundary by $\\pm 5$ ${\\rm km\\,s}^{-1}$\\ would change this number by about $\\pm 3\\%$.\n\nTo our own sample, we also add data from two different surveys: the {\\em HST}\/COS MS survey by \\citet{fox14} and the M31 dwarfs (\\citealt{mcconnachie12} and see \\S\\ref{s-dwarfs}). For the MS survey, we restrict the sample $-150\\degr \\le \\ensuremath{l_{\\rm MS}} \\le -20\\degr $, i.e., overlapping with our sample but also including higher \\ensuremath{l_{\\rm MS}}\\ value while still avoiding the Magellanic Clouds region where conditions may be different. The origin of the sample for the M31 dwarf galaxies is fully discussed in \\S\\ref{s-dwarfs} . The larger galaxies M33, M32, and NGC\\,205 are excluded here from that sample as their large masses are not characteristic. The LSR velocities of the M31 dwarfs as a function of \\ensuremath{l_{\\rm MS}}\\ are plotted with a star symbol in Fig.~\\ref{f-nidever}. For the MS survey, we select the LSR velocities of \\ion{Si}{3}\\ for the MS survey (note these are average velocities that can include multiple components), which are shown with squares in Fig.~\\ref{f-nidever}. Most ($\\sim90\\%$) of the squares fall between the two curves in Fig.~\\ref{f-nidever}, confirming the likelihood that these sightlines probe the MS (although we emphasize that this test was not initially used by \\citealt{fox14} to determine the association with the MS).\n\nThe M31 dwarf galaxies are of course not affected by the MS, but can help us to determine how frequently they fall within the velocity range where MS contamination is likely. For $\\ensuremath{l_{\\rm MS}} \\ga -132\\degr$ (where all the QSOs are and to avoid the uncertain region), only 9\\% (2\/22) of the dwarfs are within the velocity region where MS contamination occurs. If the velocity distributions of the M31 dwarfs and M31 CGM gas are similar, this would strongly suggest that velocity components with the expected MS velocities are indeed more likely associated with the MS. We, however, note two additional dwarfs are close to the upper boundary, which would change the frequency of the dwarfs in the MS velocity-boundary region to 18\\%.\n\nObservations of \\ion{H}{1}\\ 21-cm emission toward the QSOs observed with COS in MS survey \\citep{fox14} and Project AMIGA \\citep{howk17} show only \\ion{H}{1}\\ detections within $|\\ensuremath{b_{\\rm MS}}|\\la 11\\degr$. In the region defined by $-150\\degr \\le \\ensuremath{l_{\\rm MS}} \\le -20\\degr $, the bulk of the \\ion{H}{1}\\ 21-cm emission is observed within $|\\ensuremath{b_{\\rm MS}}|\\la 5\\degr$ \\citep{nidever10}. We therefore expect the metal ionic column densities to have a strong absorption when $|\\ensuremath{b_{\\rm MS}}|\\la 10\\degr$ and a weaker absorption as $|\\ensuremath{b_{\\rm MS}}|$ increases. In Fig.~\\ref{f-col-cont}, we show the total column densities of \\ion{Si}{3}\\ for the velocity components from the Project AMIGA sample found within the MS boundary region shown in Fig.~\\ref{f-nidever}, i.e., we added the column densities of the components that are likely associated with the MS. We also show in the same figure the results from the \\citet{fox14} survey. Both datasets show the same behavior of the total \\ion{Si}{3}\\ column densities with $|\\ensuremath{b_{\\rm MS}}|$, an overall decrease in $N_{\\rm Si\\,III}$ as $|\\ensuremath{b_{\\rm MS}}|$ increases. Treating the limits as values, combining the two samples, and using the Spearman rank order, the test confirms the visual impression that there is a strong monotonic anti-correlation between $N_{\\rm Si\\,III}$ and $|\\ensuremath{b_{\\rm MS}}|$ with a correlation coefficient $r_{\\rm S} = -0.72$ and a p-value $\\ll 0.1\\%$.\\footnote{We note that if we increase the lower limits by 0.15 dex or more and similarly decrease the upper limits, the significance of the anti-correlation would be similar.} There is a large scatter (about $\\pm 0.4$ dex around the dotted line) at any \\ensuremath{b_{\\rm MS}}, making it difficult to determine if any data points may not be associated with the MS (as, e.g., the three very low $N_{\\rm Si\\,III}$ at $12\\degr<|\\ensuremath{b_{\\rm MS}}|<18\\degr$ from our sample or the very high value at $|\\ensuremath{b_{\\rm MS}}|\\sim 27\\degr$ from the \\citealt{fox14} sample).\n\n\\begin{figure}[tbp]\n\\epsscale{1.2}\n\\plotone{f4.pdf}\n\\caption{The total column density of \\ion{Si}{3}\\ that are associated with the MS as a function of the absolute MS latitude. We also show the MS survey by \\citet{fox14} restricted to data with $-150\\degr \\le \\ensuremath{l_{\\rm MS}} \\le -20\\degr $. The lighter gray squares with downward arrows are non-detections in the \\citeauthor{fox14} sample. The dashed line is a linear fit to the data treating the limits as values. A Spearman ranking correlation test implies a strong anti-correlation with a correlation coefficient $r_{\\rm S} = -0.72$ and $p\\ll 0.1\\%$.\n\\label{f-col-cont}}\n\\end{figure}\n\nIn Fig.~\\ref{f-col-vs-rho-ex}, we show the individual column densities of \\ion{Si}{3}\\ as a function of the impact parameter from M31 for the Project AMIGA sightlines where we separate components associated with the MS from those that are not. Looking at Figs.~\\ref{f-map} and \\ref{f-col-cont}, we expect the strongest column densities associated with the MS to be at $|\\ensuremath{b_{\\rm MS}}|\\la 10\\degr$ and $R\\ga 300$ kpc, which is where they are located on Fig.~\\ref{f-col-vs-rho-ex}. We also expect a positive correlation between $N_{\\rm Si\\,III}$ and $R$ for the MS contaminated components while for uncontaminated components, we expect the opposite (see \\citetalias{lehner15}). Treating again limits as values, the Spearman rank order test demonstrates a strong monotonic correlation between $N_{\\rm Si\\,III}$ and $R$ ($r_{\\rm S} = 0.68$ with $p \\ll 0.1\\%$) while for uncontaminated components there is a strong monotonic anti-correlation ($r_{\\rm S} = -0.57$ with $p \\ll 0.1\\%$), in agreement with the expectations. Based on these results, it is therefore reasonable to consider any absorption components observed in the COS spectra within the MS boundary region defined in Fig.~\\ref{f-nidever} as most likely associated with the MS. We therefore flag any of these components (28 out 74 components for \\ion{Si}{3}) as contaminated by the MS and those are not included further in our sample.\n\n\n\\begin{figure}[tbp]\n\\epsscale{1.2}\n\\plotone{f5.pdf}\n\\caption{Logarithm of the column densities of the individual components for \\ion{Si}{3}\\ as a function of the projected distances from M31 of the background QSOs where the separation is made for the components associated or not with the MS.\n\\label{f-col-vs-rho-ex}}\n\\end{figure}\n\nFinally, we noted above that only a small fraction of the dwarfs are found in the MS contaminated region. While that fraction is small (9\\%), this could still suggest that in the MS contaminated region, some of the absorption could be a blend between of both MS and M31 CGM components. However, considering the uncontaminated velocities along sightlines in (29 components) and outside (17 components) the contaminated regions, with $p$-value of 0.74 the Kolmogorov-Smirnov (KS) comparison of the two samples cannot reject the null-hypothesis that the distributions are the same. This strongly suggests that the correction from the MS contamination does not bias much the velocity distribution associated with the CGM of M31 (assuming that there is no strong change of the velocity with the azimuth $\\Phi$; as we explore this further in \\S\\S\\ref{s-dwarfs-vel} and \\ref{s-map-vel}, there is, however, no strong evidence a velocity dependence with $\\Phi$).\n\n\\section{M31 Dwarf Galaxy Satellites}\\label{s-dwarfs}\nWhile Project AMIGA is dedicated to understanding the CGM of M31, our survey also provides a unique probe of the dwarf galaxies found in the halo of M31. In particular, we have the opportunity to assess if the CGM of dwarf satellites plays an important role in the CGM of the host galaxy, as studied by cosmological and idealized simulations~\\citep[e.g.][]{angles-alcazar17, hafen19a, hafen19b, bustard18}. When considering the dwarf galaxies in our analysis we have two main goals: 1) to determine if the velocity distribution of the dwarfs and the absorbers are similar, and 2) assess if some of absorption observed toward the QSOs could be associated directly with the dwarfs, either as gas that is gravitationally bound or recently stripped. \n\nThe sample for the M31 dwarf galaxies is mostly drawn from the \\citet{mcconnachie12} study of Local Group dwarfs, in which the properties of 29 M31 dwarf satellites were summarized. Four additional dwarfs (Cas\\,II, Cas\\,III, Lac\\,I, Per\\,I) are added from recent discoveries \\citep{collins13,martin14,martin16,martin17}. M33 is excluded from that sample as its large mass is not characteristic of satellites.\\footnote{In Appendix~\\ref{a-m33}, we further discuss and present some evidence that the CGM of M33 is unlikely to contribute much to the observed absorption in our sample.} Table~\\ref{t-dwarf} summarizes our adopted sample of M31 dwarf galaxies (sorted by increasing projected distance from M31), listing some of their key properties. As listed in this table, most of the M31 satellite galaxies are dwarf spheroidal (dSph) galaxies, which have been shown to have been stripped of most of their gas most likely via ram-pressure stripping \\citep{grebel03}, a caveat that we keep in mind as we associate these galaxies with absorbers. \n\n\\subsection{Velocity Transformation}\\label{s-vel-trans}\nSo far we have used LSR velocity to characterize MW and MS contamination of gas in the M31 halo. However, as we now consider relative motions over $30\\degr$ on the sky, we cannot simply subtract M31's systemic radius velocity to place these relative motions in the correct reference frame. Over such large sky areas, tangential motion must be accounted for because the ``systemic'' sightline velocity of the M31 system changes with sightline. To eliminate the effects of ``perspective motion\", we follow \\citet{gilbert18} (and see also \\citealt{veljanoski14}) by first transforming the heliocentric velocity ($v_\\sun$) into the Galactocentric frame, $v_{\\rm Gal}$, which removes any effects the solar motion could have on the kinematic analysis. We converted our measured radial velocities from the heliocentric to the Galactocentric frame using the relation from \\citet{courteau99} with updated solar motions from \\citet{mcmillan11} and \\citet{schonrich10}:\n\\begin{equation}\\label{e-gal}\n\\begin{aligned}\nv_{\\rm Gal} = & v_\\sun + 251.24\\, \\sin(l)\\cos(b) +\\\\\n& 11.1\\, \\cos(l)\\cos(b) + 7.25\\, \\sin(b)\\,,\n\\end{aligned}\n\\end{equation}\nwhere $(l,b)$ are the Galactic longitude and latitude of the object. To remove the bulk motion of M31 along the sightline to each object, we use the heliocentric systemic radial velocity for M31 of $-301$ ${\\rm km\\,s}^{-1}$\\ \\citep{vandermarel08,chemin09}, which is $v_{\\rm M31,r}=-109$ ${\\rm km\\,s}^{-1}$\\ in the Galactocentric velocity frame. The systemic transverse velocity of M31 is $v_{\\rm M31,t}=-17$ ${\\rm km\\,s}^{-1}$\\ in the direction on the sky given by the position angle $\\theta_t = 287\\degr$ \\citep{vandermarel12}. The removal of M31's motion from the sightline velocities resulting in peculiar line-of-sight velocities for each absorber or dwarf, $v_{\\rm M31}$, is then given by \\citep{vandermarel08}:\n\\begin{equation}\\label{e-vm31}\n\\begin{aligned}\nv_{\\rm M31} = & v_{\\rm Gal} - v_{\\rm M31,r}\\, \\cos(\\rho) + \\\\\n & v_{\\rm M31,t}\\, \\sin(\\rho)\\cos(\\phi - \\theta_t),\n\\end{aligned}\n\\end{equation}\nwhere $\\rho$ is the angular separation between the center of M31 to the QSO or dwarf position, $\\phi$ the position angle of the QSO or dwarf with respect to M31's center. We note that the transverse term in Eqn.~\\ref{e-vm31} is more uncertain \\citep{vandermarel08,veljanoski14}, but its effect is also much smaller, and indeed including it or not would not quantitatively change the results; we opted to include that term in the velocity transformation. We apply these transformations to change the LSR velocities to heliocentric velocities to Galactocentric velocities to peculiar velocities for each component observed in absorption toward the QSOs and for each dwarf. With this transformation, an absorber or dwarf with no peculiar velocity relative to M31's bulk motion has $v_{\\rm M31}=0$ ${\\rm km\\,s}^{-1}$, regardless of its position on the sky \\citep{gilbert18}.\n\n\n\\subsection{Velocity Distribution}\\label{s-dwarfs-vel}\n\n\\begin{figure}[tbp]\n\\epsscale{1.2}\n\\plotone{f6.pdf}\n\\caption{{\\it Left}: The M31 peculiar velocity (as defined by Eqn.~\\ref{e-vm31}) against the projected distances for the observed absorption components associated with M31 (using \\ion{Si}{3}) and M31 dwarf galaxies. The dotted curves show the escape velocity divided by $\\sqrt{3}$ to account for the unknown tangential motions of the absorbers and galaxies. {\\it Right}: The M31 velocity distributions with the same color-coding definition. \n\\label{f-v_vs_r_dwarfs}}\n\\end{figure}\n\nIn Fig.~\\ref{f-v_vs_r_dwarfs}, we compare the M31 peculiar velocities of the absorbers using \\ion{Si}{3}\\ and dwarfs against the projected distance (see \\S\\ref{s-vel-trans}). In Fig.~\\ref{f-v_vs_r_dwarfs}, we also show the expected escape velocity, $v_{\\rm esc}$, as a function of $R$ for a $1.3\\times 10^{12}$ M$_\\sun$ point mass. We conservatively divide $v_{\\rm esc}$ by $\\sqrt{3}$ in that figure to account for remaining unconstrained projection effects. Nearly all the CGM gas traced by \\ion{Si}{3}\\ within $\\ensuremath{R_{\\rm vir}}$ is found at velocities consistent with being gravitationally bound, and this is true even at larger $R$ for most of the absorbers. This finding also holds for most of the dwarf galaxies, and, as demonstrated by \\citet{mcconnachie12}, it holds when the galaxies' 3D distances are used (i.e., using the actual distance of the dwarf galaxies, instead of the projected distances used in this work). Therefore, both the CGM gas and galaxies probed in our sample at both small and large $R$ are consistent with being gravitationally bound to M31.\n\nFig.~\\ref{f-v_vs_r_dwarfs} also informs us that the dwarf satellite and CGM gas velocities overlap to a high degree but do not follow identical distributions. The mean and standard deviation of the M31 velocities for the dwarfs are $+34.2 \\pm 110.0$ ${\\rm km\\,s}^{-1}$\\ and $+36.6 \\pm 68.0$ ${\\rm km\\,s}^{-1}$\\ for the CGM gas. There is therefore a slight asymmetry favoring more positive peculiar motions. A simple two-sided KS test of the two samples rejects the null hypothesis that the distributions are the same at 95\\% level confidence ($p=0.04$). And indeed while the two distributions overlap and the means are similar, the velocity dispersion of the dwarfs is larger than that of the QSO absorbers. For the QSO absorbers, all the components but one have their M31 velocities in the interval $-80 \\le v_{\\rm M31} \\le +160$ ${\\rm km\\,s}^{-1}$, but 9\/32 (28\\%) of the dwarfs are outside that range. Four of the dwarfs are in the range $+160 < v_{\\rm M31} \\le +210$ ${\\rm km\\,s}^{-1}$, a velocity interval that cannot be probed in absorption owing to foreground MW contamination. The other five dwarfs have $v_{\\rm M31}< -80$ ${\\rm km\\,s}^{-1}$, while only one out of 46 \\ion{Si}{3}\\ components (2\\%) have $v_{\\rm M31}< -80$ ${\\rm km\\,s}^{-1}$. Both the small fraction of dwarfs at $v_{\\rm M31}> +160$ ${\\rm km\\,s}^{-1}$\\ and $v_{\\rm M31}< -80$ ${\\rm km\\,s}^{-1}$\\ and the even smaller fraction of absorbers at $v_{\\rm M31}< -80$ ${\\rm km\\,s}^{-1}$\\ suggest that there is no important population of absorbers at the inaccessible velocities $v_{\\rm M31}> +160$ ${\\rm km\\,s}^{-1}$\\ (see also \\S\\ref{s-ms}).\n\n\\begin{figure*}[tbp]\n\\epsscale{1}\n\\plotone{f7.pdf}\n\\caption{Locations of the QSOs ({\\it squares}) and dwarfs ({\\it circles}) relative to M31 (see Fig.~\\ref{f-map}). The data are color-coded according to the relative velocities of the detected \\ion{Si}{3}\\ (multiple colors in a symbol indicate multiple detected components) or the dwarfs. The black circles centered on the dwarfs indicate their individual $R_{200}$. \n\\label{f-velmap-dwarfs}}\n\\end{figure*}\n\n\\subsection{The Associations of Absorbers with Dwarf Satellites}\\label{s-dwarfs-cgm}\n\nUsing the information from Table~\\ref{t-dwarf}, we cross-match the sample of dwarf galaxies and QSOs to determine the QSO sightlines that are passing within a dwarf's $R_{200}$ radius. There are 11 QSOs (with 58 \\ion{Si}{3}\\ components) within $R_{200}$ of 16 dwarfs. In Table~\\ref{t-xmatch-dwarf}, we summarize the results of this cross-match. Fig.~\\ref{f-velmap-dwarfs}, we show the map of the QSOs and dwarf locations in our survey where the M31 velocities of the \\ion{Si}{3}\\ components and dwarfs are color coded on the same scale and the circles around each dwarf represent their $R_{200}$ radius. \n\nTable~\\ref{t-xmatch-dwarf} and Fig.~\\ref{f-velmap-dwarfs} show that several absorbers can be found within $R_{200}$ of several dwarfs when \\ion{Si}{3}\\ is used as the gas tracer. For example, the two components observed in \\ion{Si}{3}\\ toward Zw535.012 are found within $R_{200}$ of 6 dwarf galaxies. In Table~\\ref{t-xmatch-dwarf}, we also list the escape velocity ($v_{\\rm esc}$) at the observed projected distance of the QSO relative to the dwarf as well as the velocity separation between the QSO absorber and the dwarf ($\\delta v \\equiv |v_{\\rm M31, Si\\,III} - v_{\\rm M31,dwarf}|$). So far we have not considered the velocity separation $\\delta v$ between the dwarf and the absorber, but it is likely that if $\\delta v \\gg v_{\\rm esc}$ then the observed gas traced by the absorber is unlikely to be bound to the dwarf galaxy even if $\\Delta_{\\rm sep} = R\/R_{200}<1$. \n\nIf we set $\\delta v 3.9\\times 10^{10}$ M$_\\sun$ are removed from the sample. Applying a cross-match where $\\delta v < v_{\\rm esc}$ and $\\Delta_{\\rm sep}<1$ can reduce the degeneracy between different galaxies, especially if one excludes the four most massive galaxies. For example, RXS\\_J0118.8+3836 is located at $0.40 R_{200}$ and $0.72 R_{200}$ from Andromeda\\,XV and Andromeda\\,XXIII, but only in the latter case $\\delta v \\ll v_{\\rm esc}$ (and in the former case $\\delta v > v_{\\rm esc}$), making the two components observed toward RXS\\_J0118.8+3836 more likely associated with Andromeda\\,XXIII.\n\nSeveral sightlines therefore pass within $\\Delta_{\\rm sep}<1$ of a dwarf galaxy and show a velocity absorption within the escape velocity. This gas could be gravitationally bound to the dwarf. However, there are also 5 absorbers where $\\delta v 89\\%$ level. In case 2, the mean and dispersion is $[$\\ion{O}{1}\/Si$]<-0.74 \\pm 0.51$, so that the ionization fraction is still $>81\\%$ on average. These are upper limits because typically \\ion{O}{1}\\ is not detected. However, even in the 5 cases where \\ion{O}{1}\\ is detected, 4\/5 are upper limits too because \\ion{Si}{3}\\ is saturated and hence only a lower limit on the column density of Si can be derived. In that case, $[$\\ion{O}{1}\/Si$]$ ranges from $<-1.78$ to $-0.43$ (or to $<-0.85$ if we remove the absorber where the \\ion{O}{1}\\ absorption is just detected at the $2\\sigma$ level), i.e., even when \\ion{O}{1}\\ is detected to more than $3\\sigma$, the gas is still ionized at levels $>86\\%$--$98\\%$.\n\nThe combination of \\ion{Si}{2}, \\ion{Si}{3}, and \\ion{Si}{4}\\ allows us to probe gas within the ionization energies 8--45 eV, i.e., the bulk of the photoionized CGM of M31. The high ions, \\ion{C}{4}\\ and \\ion{O}{6}, have ionization energies 48--85 eV and 114--138 eV, respectively, and are not included in the above calculation. The column density of H can be directly estimated in the ionization energy 8--45 eV range from the observations via $\\log N_{\\rm H} = \\log N_{\\rm Si} - \\log Z\/Z_\\sun$. As we show below, Si varies strongly with $R$ with values $\\log N_{\\rm Si}\\ga 13.7$ at $R\\la 100$ kpc and $\\log N_{\\rm Si} \\la 13.3$ at $R\\ga 100$ kpc, which implies $N_{\\rm H} \\ga 1.5\\times 10^{18} (Z\/Z_\\sun)^{-1}$ cm$^{-2}$\\ and $\\la 0.6\\times 10^{18} (Z\/Z_\\sun)^{-1}$ cm$^{-2}$, respectively. For the high ions, a ionization correction needs to be added, and, e.g., for \\ion{O}{6}, $\\log N_{\\rm H} = \\log N_{\\rm O\\,VI} - \\log Z\/Z_\\sun - \\log f^i_{\\rm O\\,VI }$ where $f^i_{\\rm O\\,VI }\\la 0.2$ is the ionization fraction of \\ion{O}{6}\\ that peaks around 20\\% for any ionizing models \\citep[e.g.][]{oppenheimer13,gnat07,lehner14}. As discussed below, there is little variation of $N_{\\rm O\\,VI}$\\ with $R$ and is always such that $\\log N_{\\rm O\\,VI} \\ga 14.4$--$14.9$ within 300 kpc from M31, which implies $N_{\\rm H} \\ga (2.5$--$8.1)\\times 10^{18} (Z\/Z_\\sun)^{-1}$ cm$^{-2}$. Therefore the CGM of M31 is not only mostly ionized (often at levels close to 100\\%), but it also contains a substantial fraction of highly ionized gas with higher column densities than the weakly photoionized gas.\n\n\n\\subsection{Ion Column Densities versus $R$}\\label{s-n-vs-r}\n\n\\begin{figure*}[tbp]\n\\epsscale{1}\n\\plotone{f8.pdf}\n\\caption{Total column densities of the ions as a function of $R$ with ionization potential increasing from top to bottom panels. The column densities are shown in logarithm values (with the same relative vertical scale of about 3 dex in each panel) on the left and in linear units on the right. Blue circles are detections, while gray circles with downward arrows are non-detections. A blue circle with an upward arrow denotes that the absorption is saturated, resulting into a lower limit. The components associated with the MS have been removed. The dashed vertical line marks $R_{200}$. Note how \\ion{Si}{3}\\ and \\ion{O}{6}\\ are detected at high frequency well beyond $R_{200}$.\n\\label{f-coltot-vs-rho}}\n\\end{figure*}\n\nIn Fig.~\\ref{f-coltot-vs-rho}, we show the logarithmic (left) and linear (right) values of the total column densities of the components associated with M31 for \\ion{C}{2}, \\ion{Si}{2}, \\ion{Si}{3}, \\ion{Si}{4}, \\ion{C}{4}, and \\ion{O}{6}\\ as a function of the projected distances from M31. Gray data points are upper limits, while blue data with upward arrows are lower limits owing to saturated absorption. Overall, the column densities decrease at higher impact parameter. As the ionization potentials of the ions increase, the decrease in the column densities becomes shallower; \\ion{O}{6}\\ is almost flat. These conclusions were already noted in \\citetalias{lehner15}, but now that the region from 50 to 350 kpc is filled with data, these trends are even more striking. However, our new sample shows also an additional feature with a remarkable change around $R_{200} \\simeq 230$ kpc of M31 notable especially for the low and intermediate ions whereby high \\ion{C}{2}, \\ion{Si}{2}, \\ion{Si}{3}, and \\ion{Si}{4}\\ column densities are observed solely at $R\\la R_{200}$. Low column densities \\ion{C}{2}, \\ion{Si}{2}, \\ion{Si}{3}, and \\ion{Si}{4}\\ are observed at all $R$, but strong absorption is observed only at $R\\la R_{200}$. The frequency of strong absorption is also larger at $R\\la 0.6 R_{200}$ than at larger $R$ for all ions. A similar pattern is observed for \\ion{C}{4}\\ and \\ion{O}{6}, but the difference between the low and high column densities is smaller: for \\ion{C}{2}, \\ion{Si}{2}, \\ion{Si}{3}, and \\ion{Si}{4}, the difference between low and high column densities is a factor $\\ga 5$--10, while it drops to a factor 2--4 for \\ion{C}{4}, and possibly even less for \\ion{O}{6}.\n\n\nIn Fig.~\\ref{f-col-vs-rho}, we show the logarithmic values of the column densities derived from the individual components for \\ion{C}{2}, \\ion{Si}{2}, \\ion{Si}{3}, \\ion{Si}{4}, \\ion{C}{4}, and \\ion{O}{6}\\ as a function of the projected distances from M31. Similar trends are observed in Fig.~\\ref{f-coltot-vs-rho}, but Fig.~\\ref{f-col-vs-rho} additionally shows that 1) more complex velocity structures (i.e., multiple velocity components) are predominantly observed at $R\\la R_{200}$, and 2) factor $\\ga 2$--10 changes in the column densities are observed across multiple velocity components along a given sightline.\n\n\n\\begin{figure}[tbp]\n\\epsscale{1.2}\n\\plotone{f9.pdf}\n\\caption{Logarithm of the column densities for the individual components of various ions (low to high ions from top to bottom) as a function of the projected distances from M31 of the background QSOs. Blue circles are detections, while gray circles with downward arrows are non-detections. A blue circle with an upward arrow denotes that the absorption is saturated, resulting into a lower limit. The components associated with the MS have been removed. The dashed vertical lines shows the $R_{200}$ location. The same relative vertical scale of about 3 dex is used in each panel for comparison between the different ions. \n\\label{f-col-vs-rho}}\n\\end{figure}\n\n\n\\subsection{Silicon Column Densities versus $R$}\\label{s-nsi-vs-r}\n\nWith \\ion{Si}{2}, \\ion{Si}{3}, and \\ion{Si}{4}, we can estimate the total column density of Si within the ionization energy range 8--45 eV without any ionization modeling. Gas in this range should constitute the bulk of the cool photoionized CGM of M31 (see \\S\\ref{s-ionization}). In Fig.~\\ref{f-coltotsi-vs-rho}, we show the total column density of Si (estimated following \\S\\ref{s-ionization}) against the projected distance $R$ from M31. The vertical-ticked bar in Fig.~\\ref{f-coltotsi-vs-rho} indicate data with some upper limits, and the length of the vertical bar represents the range of $N_{\\rm Si}$ values allowed between cases 1 and 2 (see \\S\\ref{s-ionization}). \n\n\n\\begin{figure}[tbp]\n\\epsscale{1.2}\n\\plotone{f10.pdf}\n\\caption{Total column densities of Si (i.e., $N_{\\rm Si} = N_{\\rm Si\\,II} + N_{\\rm Si\\,III} + N_{\\rm Si\\,IV}$) as a function of the projected distances from M31 of the background QSOs. The vertical-ticked bars show the range of values allowed if the upper limit of a given Si ion is negligible or not. The lower limits have upward arrows, and the upper limits are flagged using downward arrows. The orange, green, and blue curves are the H, SPL, GP-derived models to the data, respectively (see text for details regarding how censoring is treated in each model). The dotted and dashed curves correspond to model where the lower limits at $R<50$ kpc are increased by 0.3 or 0.6 dex. The blue areas correspond to the dispersion derived from the GP models (see Appendix~\\ref{a-model} for more detail).\n\\label{f-coltotsi-vs-rho}}\n\\end{figure}\n\nFig.~\\ref{f-coltotsi-vs-rho} reinforces the conclusions observed from the individual low ions in Figs.~\\ref{f-coltot-vs-rho} and \\ref{f-col-vs-rho}. Overall there is a decrease of the column density of Si at larger $R$. This decrease has a much stronger gradient in the inner region of the M31 CGM between ($R\\la 25$ kpc) and about $R\\sim 100$--$150$ kpc than at $R\\ga 150$ kpc. $N_{\\rm Si}$ changes by a factor $>5$--$10$ between about 25 kpc and 150 kpc while it changes by a factor $\\la 2$ between 150 kpc and 300 kpc. The scatter in $N_{\\rm Si}$ is also larger in the inner regions of the CGM than beyond $\\ga 120$--150 kpc.\n\nTo model this overall trend (which is also useful to determine the baryon and metal content of the CGM, see \\S\\ref{s-mass}), we consider three models, a hyperbolic (H) model, single-power law (SPL) model, and a Gaussian Process (GP) model. We refer the refer to Appendix~\\ref{a-model} where we fully explain the modeling process and how lower and upper limits are accounted for in the modeling. Fig.~\\ref{f-coltotsi-vs-rho} shows these 3 models greatly overlap. The non-parametric GP model overlaps more with the SPL model than with the H fit in the range $250\\la R\\la 400$ kpc and at $R<90$ kpc (especially for the high H fit, see Fig.~\\ref{f-coltotsi-vs-rho}). While there are some differences between these models (and we will explore in \\S\\ref{s-mass} how these affect the mass estimates of the CGM), they all further confirm the strong evolution of the column density of Si with $R$ between $\\la 25$ and $90$--150 kpc and a much shallower evolution with $R$ beyond 200 kpc. In \\S\\ref{s-mass}, we use these models to constrain the metal and baryon masses of the cool CGM gas probed by \\ion{Si}{2}, \\ion{Si}{3}, \\ion{Si}{4}.\n\n\\subsection{Covering Factors}\\label{s-fc}\nAs noted in \\S\\ref{s-n-vs-r}, the diagnostic ions behave differently with $R$ in a way that probably reflects the underlying physical conditions. For example, \\ion{Si}{2}\\ has a high detection rate within $R<100$ kpc, a sharp drop beyond $R>100$ kpc, and a total absence at $R\\ga 240$ kpc (see Figs.~\\ref{f-coltot-vs-rho} and \\ref{f-col-vs-rho}). On the other hand, \\ion{Si}{3}\\ and \\ion{O}{6}\\ are mostly detected at all $R$, but the column densities of \\ion{Si}{3}\\ fall significantly with $R$ while \\ion{O}{6}\\ remains relatively flat. In this section, we quantity further the detection rates, or the covering factors, for each ion.\n\n\\begin{figure*}[tbp]\n\\epsscale{1}\n\\plotone{f11.pdf}\n\\caption{Cumulative covering factors for impact parameters less than $R$ without ({\\it left}) and with ({\\it right}) some threshold cut on the column densities (for Si ions, $\\log N_{\\rm th} = 13$; for C ions, $\\log N_{\\rm th} = 13.8$; for \\ion{O}{6}, $\\log N_{\\rm th} = 14.5$, and for \\ion{H}{1}, $\\log N_{\\rm th} = 17.6$, see text for more detail and \\citetalias{howk17}). Confidence intervals (vertical bars) are at the 68\\% level and data points are the median values. On the left panel, the solid lines are polynomial fits to median values of \\ensuremath{f_{\\rm c}}\\ for \\ion{Si}{3}, \\ion{O}{6}, \\ion{C}{2}--\\ion{C}{4}, \\ion{Si}{2}--\\ion{Si}{4}\\ (i.e., taking the mean value of \\ensuremath{f_{\\rm c}}\\ between these two ions at a given $R$). On the right panel, the orange line is a polynomial fit to the mean values of \\ensuremath{f_{\\rm c}}\\ for \\ion{C}{2}, \\ion{C}{4}, \\ion{Si}{2}, \\ion{Si}{3}, and \\ion{Si}{4}\\ while the gray line is polynomial fit to the median values of \\ensuremath{f_{\\rm c}}\\ for \\ion{O}{6}.\n\\label{f-fccum-vs-rho}}\n\\end{figure*}\n\nTo calculate the covering factors of the low and high ions, we follow the methodology described in \\citetalias{howk17} for \\ion{H}{1}\\ by assuming a binomial distribution. We assess the likelihood function for values of the covering factor given the number of detections against the total sample, i.e., the number of targets within a given impact parameter range (see \\citealt{cameron11}). As demonstrated by \\citet{cameron11}, the normalized likelihood function for calculating the Bayesian confidence intervals on a binomial distribution with a non-informative (uniform) prior follows a $\\beta$-distribution.\n\nIn Fig.~\\ref{f-fccum-vs-rho}, we show the {\\em cumulative} covering factors (\\ensuremath{f_{\\rm c}}) for the various ions, where each point represents the covering factor for all impact parameters less than the given value of $R$. The vertical bars are 68\\% confidence intervals. As discussed in \\S\\S\\ref{s-n-vs-r}, \\ref{s-nsi-vs-r}, for all the ions but \\ion{O}{6}, the highest column densities are only observed at $R\\la 100$--150 kpc, with a sharp decrease beyond that. For the covering factors, we therefore consider 1) the entire sample (most of the upper limits---non-detections---are at the level of lowest column densities of a detected absorption, so it is adequate to do that), and 2) the sample where we set a threshold column density ($N_{\\rm th}$) to be included in the sample. In the left panel of Fig.~\\ref{f-fccum-vs-rho}, we show the first case, while in the right panel, we focus on the strong absorbers only. For the Si ions, we use $\\log N_{\\rm th} = 13$; for the C ions, $\\log N_{\\rm th} = 13.8$; for \\ion{O}{6}, $\\log N_{\\rm th} = 14.5$. These threshold column densities are chosen based on Fig.~\\ref{f-coltot-vs-rho}. We also show in the right panel of Fig.~\\ref{f-fccum-vs-rho}, the results for the \\ion{H}{1}\\ emission from \\citetalias{howk17}.\n\nThese results must be interpreted in light of the fact that the intrinsic strength of the diagnostic lines varies by ion. The oscillator strength, $f \\lambda$, of these different ions are listed in Table~\\ref{t-strength} along with the solar abundances of these elements. The optical depth is such that $\\tau \\propto f\\lambda N $ (see \\S\\ref{s-aod}) and $f\\lambda $ is a good representation of the strength of a given transition. For the Si ions, \\ion{Si}{3}\\ has the strongest transition, a factor 2.7--5.5 stronger than \\ion{Si}{4}\\ and a factor 1.3--5.7 stronger than \\ion{Si}{2}\\ (the weaker \\ion{Si}{2}\\ $\\lambda$1526 is sometimes used but mostly to better constrain the column density of \\ion{Si}{2}\\ if the absorption is strong). \\ion{Si}{2}\\ and \\ion{Si}{4}\\ have more comparable strength, which is also the case between \\ion{C}{2}\\ and \\ion{C}{4}. Comparing between different species, while $(f\\lambda)_{\\rm Si\\,III} \\simeq 14.4 (f\\lambda)_{\\rm O\\,VI}$, but this is counter-balanced by oxygen being 15 times more abundant than silicon (and a similar conclusion applies comparing \\ion{Si}{3}\\ with \\ion{C}{2}\\ or \\ion{C}{4}).\n\nWith that in mind, we first consider the left panel of Fig.~\\ref{f-fccum-vs-rho}, i.e., where all the absorbers irrespective of their absorption strengths are taken into account to estimate the cumulative covering factors. We fitted 4 low-degree polynomials to the data: \\ion{Si}{3}, \\ion{O}{6}, and treating in pair \\ion{C}{2}-\\ion{C}{4}\\ and \\ion{Si}{2}-\\ion{Si}{4}\\ as they appear to follow each other reasonably well, respectively. For \\ion{C}{2}-\\ion{C}{4}\\ and \\ion{Si}{2}-\\ion{Si}{4}, we fit the mean covering factors of each ionic pair. For \\ion{O}{6}, we only fitted data beyond 200 kpc owing to the smaller size sample (there are only 3 data points within 200 kpc and 11 in total, see Fig.~\\ref{f-coltot-vs-rho}). It is striking how the cumulative covering factors of \\ion{Si}{3}\\ and \\ion{O}{6}\\ vary with $R$ quite differently from each other and from the other ions. The cumulative covering factor of \\ion{Si}{3}\\ increases with $R$, reaches a maximum somewhere between 250 and 300 kpc, and then decreases, but still remains much higher than \\ensuremath{f_{\\rm c}}\\ of \\ion{C}{2}-\\ion{C}{4}\\ or \\ion{Si}{2}-\\ion{Si}{4}. The cumulative covering factor of \\ion{O}{6}\\ monotonically increases with $R$ up to $R\\sim 569$ kpc. In contrast, while the cumulative covering factors of \\ion{C}{2}-\\ion{C}{4}\\ or \\ion{Si}{2}-\\ion{Si}{4}\\ are offset from each other, they both monotonically decrease with $R$. There seems to be a plateau in \\ion{C}{2}-\\ion{C}{4}\\ covering factor beyond 400 kpc, which is not observed for \\ion{Si}{2}\\ or \\ion{Si}{4}.\n\nTurning to the right panel of Fig.~\\ref{f-fccum-vs-rho} where we show \\ensuremath{f_{\\rm c}}\\ for a given column density threshold that changes with species (see above), the relation between \\ensuremath{f_{\\rm c}}\\ and $R$ is quite different. For all the ions, the cumulative covering factors monotonically decrease with increasing $R$. For \\ion{C}{2}, \\ion{C}{4}, \\ion{Si}{2}, \\ion{Si}{3}, and \\ion{Si}{4}, the covering factors are essentially the same within $1 \\sigma$, and the orange line in Fig.~\\ref{f-fccum-vs-rho} shows a second-degree polynomial fit to the mean values of \\ensuremath{f_{\\rm c}}\\ between these different ions. Ignoring data at $R<200$ kpc owing to the small sample size, \\ion{O}{6}\\ has a similar evolution of \\ensuremath{f_{\\rm c}}\\ with $R$, but overall \\ensuremath{f_{\\rm c}}\\ is tentatively a factor $\\sim$1.5 times larger than for the other ions at any $R$.\n\nThe contrast between the two panels of Fig.~\\ref{f-fccum-vs-rho} strongly suggests that the CGM of M31 has three main populations of absorbers: 1) the strong absorbers that are found mostly at $R \\la 100$--$150$ kpc ($0.3$--$0.5 \\ensuremath{R_{\\rm vir}}$) probing the denser regions and multiple gas-phase (singly to highly ionized gas) of the CGM, 2) weak absorbers probing the diffuse CGM traced principally by \\ion{Si}{3}\\ (but also observed in higher ions and more rarely in \\ion{C}{2}) that are found at any surveyed $R$ but more frequent at $R\\la \\ensuremath{R_{\\rm vir}}$, and 3) hotter, more diffuse CGM probed by \\ion{O}{6}, \\ion{O}{6}\\ having the unique property compared to the ions that its column density remains largely invariant with the radius of the M31 CGM.\n\n\\subsection{Ion ratios and their Relation with $R$}\\label{s-ratio-vs-r}\n\nIn \\S\\ref{s-ionization}, we show that the ratio of \\ion{O}{1}\\ to Si ions provides a direct estimate of the ionization fraction of the CGM gas of M31. Using ratios of the main ions studied here (\\ion{C}{2}, \\ion{C}{4}, \\ion{Si}{2}, \\ion{Si}{3}, \\ion{Si}{4}, \\ion{O}{6}), we can further constrain the ionization and physical conditions in the CGM of M31 and how they may change with $R$. To estimate the ionic ratios, we consider the component analysis of the absorption profiles, i.e., we compare the column densities estimated over the same velocity range. However, coincident velocities do not necessarily mean that they probe the same gas, especially if their ioniziation potentials are quite different (such as for \\ion{C}{2}\\ and \\ion{C}{4}). In Fig.~\\ref{f-ratio-vs-rho}, we show the results for several ion ratios as a function of $R$.\n\n\n\\begin{figure}[tbp]\n\\epsscale{1.2}\n\\plotone{f12.pdf}\n\\caption{Logarithmic column density ratios of different ions as a function of the projected distances from M31 of the background QSOs. The column densities in individual components are compared to estimate the ionic ratios. Blue symbols indicate that both ions in the ratio are detected. Blue down or up arrows indicate that the absorption is saturated for the ion in the denominator or numerator of the ratio. Gray symbols indicate that one of the ions in the ratio is not detected at $>2\\sigma$. The components associated with the MS have again been removed. The dashed vertical lines mark $R_{200}$.\n\\label{f-ratio-vs-rho}}\n\\end{figure}\n\n\n\\subsubsection{The \\ion{Si}{2}\/\\ion{Si}{3}\\ and \\ion{Si}{4}\/\\ion{Si}{3}\\ ratios}\\label{s-si2-si3-si4}\nThe \\ion{Si}{2}\/\\ion{Si}{3}\\ and \\ion{Si}{4}\/\\ion{Si}{3}\\ ratios are particularly useful because they trace different ionization levels independently of relative elemental abundances. The ionization potentials for these ions are 8.1--16.3 eV for \\ion{Si}{2}, 16.3--33.5 eV for \\ion{Si}{3}, and 33.5--45.1 eV for \\ion{Si}{4}. The top two panels of Fig.~\\ref{f-ratio-vs-rho} show the ratios \\ion{Si}{2}\/\\ion{Si}{3}\\ and \\ion{Si}{4}\/\\ion{Si}{3}\\ as a function of $R$. In both panels, there are many upper limits and there is no evidence of any correlation with $R$, except perhaps for the for the \\ion{Si}{2}\/\\ion{Si}{3}\\ ratio where the only detections of \\ion{Si}{2}\\ are at $R\\la R_{200}$ (see also Fig.~\\ref{f-col-vs-rho}).\n\nWith so many upper limits, we use the Kaplan-Meier estimator (see \\S\\ref{s-abund}) to estimate the mean of these ratios: $\\langle \\log N_{\\rm Si\\,II}\/N_{\\rm Si\\,III}\\rangle = (-0.50 \\pm 0.04) \\pm 0.23$ (mean, error on the mean from the Kaplan-Meier estimator, and standard deviation for 44 data points with 38 upper limits) and $\\langle \\log N_{\\rm Si\\,IV}\/N_{\\rm Si\\,III}\\rangle = (-0.49 \\pm 0.07) \\pm 0.20$ (43 data points with 32 upper limits). There are only 4\/44 components where $ \\log N_{\\rm Si\\,II}\/N_{\\rm Si\\,III} \\simeq 0$ and 8\/43 where $ \\log N_{\\rm Si\\,IV}\/N_{\\rm Si\\,III} \\ga 0$. In the latter cases, \\ion{Si}{4}\\ could be produced by another mechanism such as collisional ionization. Among the three Si ions in our survey, \\ion{Si}{3}\\ is the dominant ion at any $R$ from M31 in the ionizing energy range 8.1--45.1 eV. Ions (of any element) with ionizing energies in the range 16.3--33.5 eV are therefore expected to be dominant ions at least for processes that are dominated by photoionization. \n\nThe \\ion{Si}{2}\/\\ion{Si}{3}\\ ratio has previously been used to constrain the properties of the photoionized gas. According to photoionization modeling produced by \\citet{oppenheimer18a}, an ionic ratio of $\\langle \\log N_{\\rm Si\\,II}\/N_{\\rm Si\\,III}\\rangle = (-0.50 \\pm 0.04) \\pm 0.23$ would imply gas density in the range $-3 \\la \\log n_{\\rm H} \\la -2.5 $ and a temperature of the gas around $10^4$ K (see Fig.~16 in \\citealt{oppenheimer18a}). \n\n\\subsubsection{The \\ion{C}{2}\/\\ion{C}{4}\\ ratio}\\label{s-c2-c4}\nFor the \\ion{C}{2}\/\\ion{C}{4}\\ ratio the ionizing energy ranges are well separated with 11.3--24.3 eV for \\ion{C}{2}\\ and 47.9--64.4 eV for \\ion{C}{4}. In fact with an ionization potential above the \\ion{He}{2}\\ ionization edge at 54.4 eV, \\ion{C}{4}\\ can be also produced not just photoionization but also by collisional ionization. Therefore \\ion{C}{2}\\ and \\ion{C}{4}\\ are unlikely to probe the same ionization mechanisms or be in a gas with the same density. We note that \\ion{C}{2}\\ has ionization energies that overlap with \\ion{Si}{3}\\ and larger than those of \\ion{Si}{2}, which certainly explain the presence of \\ion{C}{2}\\ beyond $R_{200}$ where \\ion{Si}{2}\\ is systematically not detected.\n\nThe third panel of Fig.~\\ref{f-ratio-vs-rho} shows the \\ion{C}{2}\/\\ion{C}{4}\\ ratios. There is again no strong relationship between \\ion{C}{2}\/\\ion{C}{4}\\ and $R$, but $ \\log N_{\\rm C\\,II}\/N_{\\rm C\\,IV} \\ga 0$ is more frequently observed at $RR_{200}$ (2\/9), consistent with the observation made in \\S\\ref{s-n-vs-r} that the gas becomes more highly ionized as $R$ increases. With the survival analysis (considering the only lower limit as a detection), we find $\\langle \\log N_{\\rm C\\,II}\/N_{\\rm C\\,IV}\\rangle = (-0.21 \\pm 0.11) \\pm 0.40$ (21 data points with 8 upper limits). Considering data at $RR_{200}$.\n\n\\subsubsection{The \\ion{C}{4}\/\\ion{Si}{4}\\ ratio}\\label{s-c4-si4}\nFor the \\ion{C}{4}\/\\ion{Si}{4}\\ ratio, different species are compared, but as we discuss in \\S\\ref{s-abund}, the relative abundances of C and Si are consistent with the solar ratio owing to little evidence of any strong dust depletion or nucleosynthesis effects, i.e., these effects should not affect the observed ratio of \\ion{C}{4}\/\\ion{Si}{4}. \\ion{Si}{4}\\ and \\ion{C}{4}\\ have near adjacent ionization energies 33.5--45.1 eV to 47.9--64.4 eV, respectively. Both photoionization and collisional ionization processes can be important at these ionizing energies, but if $\\log N_{\\rm C\\,IV}\/N_{\\rm Si\\,IV}>0$, then the ionization from hot stars is unimportant (see Fig.~13 in \\citealt{lehner11b}), which is nearly always the case, as illustrated in Fig.~\\ref{f-ratio-vs-rho}. A harder photoionizing spectrum or collisional ionization must be at play to explain the origin of these ions.\n\nFig.~\\ref{f-ratio-vs-rho} suggest a moderate correlation between $\\log N_{\\rm C\\,IV}\/N_{\\rm Si\\,IV}$ and $R$. If the two data points beyond 400 kpc are removed (and treating the limits as actual values), a Spearman rank order implies a monotonic correlation between $\\log N_{\\rm C\\,IV}\/N_{\\rm Si\\,IV}$ and $R$ with a correlation coefficient $r_{\\rm S} = +0.45$ and $p=0.019$ for the gas at $R<1.2\\ensuremath{R_{\\rm vir}}$. Considering the entire sample, the Spearman rank test yield $r_{\\rm S} = 0.34$ and $p=0.07$. This is again consistent with our earlier conclusion that the gas becomes more highly ionized as $R$ increases. With the survival analysis (considering the 3 upper limits as detections),\\footnote{If these 3 upper limits are included or excluded from the sample, the means are essentially the same. } we find $\\langle \\log N_{\\rm C\\,IV}\/N_{\\rm Si\\,IV}\\rangle = (+0.87 \\pm 0.07) \\pm 0.24$ (29 data points with 9 lower limits). This is about a factor 1.9 larger than the mean derived for the broad \\ion{C}{4}\\ and \\ion{Si}{4}\\ components in the Milky Way disk and low-halo \\citep{lehner11b}, which is about $1\\sigma$ larger.\n\n\\subsubsection{The \\ion{C}{4}\/\\ion{O}{6}\\ ratio}\\label{s-c4-o6}\nFinally, in the last panel of Fig.~\\ref{f-ratio-vs-rho}, we show the \\ion{C}{4}\/\\ion{O}{6}\\ ratio as a function of $R$. As for the \\ion{C}{4}\/\\ion{Si}{4}\\ ratio, different species are compared, and for the same reasons, the relative dust depletion or nucleosynthesis effects should be negligible. With 113.9--138.1 eV ionizing energies needed to produce \\ion{O}{6}, this is the highest high ion in the sample and as we demonstrated in the previous section the \\ion{O}{6}\\ properties (covering factor and column density as a function of $R$) are quite unique. Not surprisingly Fig.~\\ref{f-ratio-vs-rho} does not reveal any relation between $\\log N_{\\rm C\\,IV}\/N_{\\rm O\\,VI}$ and $R$.\n\nIf we treat the two lower limits as detections, then the survival analysis yields $\\langle \\log N_{\\rm C\\,IV}\/N_{\\rm O\\,VI}\\rangle = (-0.93 \\pm 0.11) \\pm 0.32$ (16 data points with 6 upper limits). The mean and range of $\\log N_{\\rm C\\,IV}\/N_{\\rm O\\,VI}$ are smaller than observed in the Milky Way disk and low halo where the full range varies from $-1$ to $+1$ dex (see, e.g., Fig.~14 of \\citealt{lehner11b}). This demonstrates that the highly ionized gas in the 113.9--138.1 eV range is much more important than in the 47.9--64.4 eV range at any $R$ of the M31 CGM.\n\n\\subsection{Metal and Baryon Mass of the M31 CGM}\\label{s-mass}\n\nWith a better understanding of the column density variation with $R$, we can estimate with more confidence the metal and baryon mass of the M31 CGM than in our original survey where we had very little information between 50 and 300 kpc \\citepalias{lehner15}. The metal mass can be directly estimated from the column densities of the metal ions. With the silicon ions, we have information on its three dominant ionization stages in the $T<7\\times 10^4$ K ionized gas (ionizing energies in the range 8--45 eV, see \\S\\ref{s-nsi-vs-r}), so we can obtain a direct measured metal mass without any major ionization corrections. Following \\citetalias{lehner15} (and see also \\citealt{peeples14}), the metal mass of the cool photoionized CGM is\n$$\nM^{\\rm cool}_{\\rm Z} = 2\\pi\\, \\mu_{\\rm Si}^{-1}\\, m_{\\rm Si}\\, \\int R \\,N_{\\rm Si}(R) \\,dR\\,,\n$$\nwhere $\\mu_{\\rm Si}=0.064$ is the solar mass fraction of metals in silicon (i.e., $12+\\log ({\\rm Si\/H})_\\sun = 7.51$ and $Z_\\sun = 0.0142$ from \\citealt{asplund09}), $m_{\\rm Si} = 28 m_{\\rm p}$, and for $N_{\\rm Si}(R)$ we use the hyperbola (``H model'', Eqn.~\\ref{e-colsi-vs-r}), single power-law (``SPL model\", Eqn.~\\ref{e-colsi-vs-r1}), and GP models that we determine in \\S\\ref{s-nsi-vs-r} and Appendix~\\ref{a-model} (see Fig.~\\ref{f-coltotsi-vs-rho}).\n\nA direct method to estimate the total mass is to convert the total observed column density of Si to total hydrogen column density via $N_{\\rm H} = N_{\\rm H\\,I} + N_{\\rm H\\,II} = N_{\\rm Si}\\, ({\\rm Si}\/{\\rm H})_\\sun^{-1}\\, ({\\rm Z\/Z}_\\sun)^{-1}$. The baryonic mass of the CGM of M31 is then:\n$$\nM^{\\rm cool}_{\\rm g} = 2\\pi\\, m_{\\rm H}\\, \\mu\\, \\ensuremath{f_{\\rm c}} \\, \\Big(\\frac{\\rm Si}{\\rm H}\\Big)_\\sun^{-1} \\Big(\\frac{Z}{Z_\\sun}\\Big)^{-1}\\, \\int R\\, N_{\\rm Si}(R)\\, dR\\,,\n$$\nwhere $\\mu \\simeq 1.4$ (to correct for the presence of He), $m_{\\rm H} = 1.67\\times 10^{-24}$ g is the hydrogen mass, \\ensuremath{f_{\\rm c}}\\ is covering fraction (that is 1 over the considered radii), and $\\log ({\\rm Si\/H})_\\sun = -4.49$ is the solar abundance of Si. Inserting the values for each parameter, $M^{\\rm cool}_{\\rm g}$ can be simply written in terms of $M^{\\rm cool}_{\\rm Z}$: $M^{\\rm cool}_{\\rm g} \\simeq 10^2 (Z\/Z_\\sun)^{-1} M^{\\rm cool}_{\\rm Z}$.\n\nIn Table~\\ref{t-mass}, we summarize the estimated metal mass over different regions of the CGM for the three models of $N_{\\rm Si}(R)$, within $R_{\\rm 200}$ (first entry), within \\ensuremath{R_{\\rm vir}}\\ (second entry), within $1\/2 \\ensuremath{R_{\\rm vir}}$ (third entry), between $1\/2 \\ensuremath{R_{\\rm vir}}$ and \\ensuremath{R_{\\rm vir}}\\ (fourth entry), and within 360 kpc (fifth entry), which corresponds to the radius where at least one of the Si ions is always detected (beyond that, the number of detections drastically plummets). A key difference between the H\/SPL models and the GP model is that the range of values for the H\/SPL models is derived using the low (dotted) and high (dashed) curves in Fig.~\\ref{f-coltotsi-vs-rho} while for the GP models we actually use the standard deviations from the low and high models (i.e., the top and bottom of the shaded blue curve in Fig.~\\ref{f-coltotsi-vs-rho}). Hence it is not surprising that the mass ranges for the GP model are larger. Nevertheless there is a large overlap between the three models. As the GP results overlap with the other models and provide empirical confidence intervals, we adopt them for the remaining of the paper. At \\ensuremath{R_{\\rm vir}}, the metal and cool gas masses are therefore $(2.0 \\pm 0.5) \\times 10^7$ and $2\\times 10^9\\, (Z\/Z_\\sun)^{-1}$ M$_\\sun$, respectively. Owing to the new functional form of $N_{\\rm Si}(R)$ and how the lower limits are treated, this explains the factor 1.4 times increase in the metal mass compared to that derived in \\citetalias{lehner15}.\n\nThese masses do not include the more highly ionized gas traced by \\ion{O}{6}\\ or \\ion{C}{4}. Even though the sample with \\ion{O}{6}\\ is smaller than \\ion{C}{4}, we use \\ion{O}{6}\\ to probe the higher ionization gas phase because as we show above the properties of \\ion{O}{6}\\ (column density and covering fraction as a function of $R$) are quite different from all the other ions, including \\ion{C}{4}, which behaves more like the other, lower ions. Furthermore, \\citet{lehner11b} using 1.5--3 ${\\rm km\\,s}^{-1}$\\ resolution UV spectra show that \\ion{C}{4}\\ can probe cool and hotter gas while the profiles of \\ion{N}{5}\\ and \\ion{O}{6}\\ are typically broad and more consistent with hotter gas. Since \\ion{O}{6}\\ is always detected and there is little evidence for variation with $R$ (see Fig.~\\ref{f-coltot-vs-rho}), we can simply use the mean column density $\\log N_{\\rm O\\,VI} = 14.46 \\pm 0.10$ (error on the mean using the survival method for censoring) to estimate the baryon mass assuming a spherical distribution:\n$$\nM^{\\rm warm}_{\\rm g} = \\pi r^2 \\, m_{\\rm H}\\, \\mu\\, \\ensuremath{f_{\\rm c}} \\, \\frac{N_{\\rm O\\,VI}}{f^i_{\\rm O\\,VI}}\\, \\Big(\\frac{\\rm O}{\\rm H}\\Big)_\\sun^{-1}\\, \\Big(\\frac{Z}{Z_\\sun}\\Big)^{-1},\n$$\nwhere the \\ion{O}{6}\\ ionization fraction is $f^i_{\\rm O\\,VI} \\la 0.2$ (see \\S\\ref{s-ionization}), $\\ensuremath{f_{\\rm c}} =1$ for \\ion{O}{6}\\ at any $R$ (see Fig.~\\ref{f-coltot-vs-rho}). At \\ensuremath{R_{\\rm vir}}, we find $M^{\\rm warm}_{\\rm g}\\ga 9.3\\times 10^9\\,(Z\/Z_\\sun)^{-1}$ M$_\\sun$ or $M^{\\rm warm}_{\\rm g} \\ga 4.4 \\,M^{\\rm cool}_{\\rm g}$ (assuming the metallicity is about similar in the cooler and hotter gas-phases). At $R_{200}$, we find $M^{\\rm warm}_{\\rm g}\\ga 5.5\\times 10^9\\,(Z\/Z_\\sun)^{-1}$ M$_\\sun$ (assuming the metallicity is similar in the cooler and hotter gas-phases). These are lower limits because the fraction of \\ion{O}{6}\\ could be much smaller than 20\\% and the metallicity of the cool or warm ionized gas is also likely to less than solar (see below). In terms of metal mass in the highly ionized gas-phase, we have $M^{\\rm warm}_{\\rm g} \\simeq 10^2 (Z\/Z_\\sun)^{-1} M^{\\rm warm}_{\\rm Z}$ and hence also $ M^{\\rm warm}_{\\rm Z} \\ga 4.4 \\, M^{\\rm cool}_{\\rm Z}$. Since \\ion{O}{6}\\ is detected out to the maximum surveyed radius of 569 kpc, and at that radius (i.e., $1.9 \\ensuremath{R_{\\rm vir}}$), $M^{\\rm warm}_{\\rm g}\\ga 34 \\times 10^9\\,(Z\/Z_\\sun)^{-1}$ M$_\\sun$.\n\n\nBy combining both the cool and hot gas-phase masses, we can find the baryon mass for gas in the temperature range $\\sim 10^3$--$10^{5.5}$ K at \\ensuremath{R_{\\rm vir}}:\n$$\n\\begin{aligned}\nM_{\\rm g} & = M^{\\rm cool}_{\\rm g} + M^{\\rm warm}_{\\rm g}\\\\\n& \\ga 1.1\\times 10^{10}\\, \\Big(\\frac{Z}{Z_\\sun}\\Big)^{-1} \\; {\\rm M_\\sun}\\,.\n\\end{aligned}\n$$\nWithin $R_{200}$, the total mass $M_{\\rm g} \\ga 7.2\\times 10^9$ M$_\\sun$. As the stellar mass of M31 is about $10^{11}$ M$_{\\sun}$ \\citep[e.g.,][]{geehan06,tamm12}, the mass of the diffuse weakly and highly ionized CGM of M31 within $1\\ensuremath{R_{\\rm vir}}$ is therefore at least 10\\% of the stellar mass of M31 and could be significantly larger than 10\\%.\n\nThis estimate does not take into account the hot ($T\\ga 10^6$ K) coronal gas. The diffuse X-ray emission is observed to extend to about 30--70 kpc around a handful of massive, non-starbursting galaxies \\citep{anderson11,bregman18} or in stacked images of galaxies \\citep{anderson13,bregman18}, but beyond $50$ kpc, the CGM is too diffuse to be traced with X-ray imaging, even though a large mass could be present. Using the results summarized recently by \\citet{bregman18}, the hot gas mass of spiral galaxy halos is in the range $M^{\\rm hot}_{\\rm g}\\simeq 1$--$10 \\times 10^9$ M$_\\sun$ within 50 kpc. For M31, $M_{\\rm g} = M^{\\rm cool}_{\\rm g} + M^{\\rm warm}_{\\rm g} \\ga 0.4\\times 10^9$ M$_\\sun$ within 50 kpc. Extrapolating the X-ray results to \\ensuremath{R_{\\rm vir}}, \\citet{bregman18} find masses of the hot X-ray gas similar to the stellar masses of these galaxies in the range $M^{\\rm hot}_{\\rm g}\\simeq1$--$10\\times 10^{11}$ M$_\\sun$. For the MW hot halo within $1\\ensuremath{R_{\\rm vir}}$, \\citet{gupta17} (but see also \\citealt{gupta12,gupta14,wang12,henley14}) derive $3$--$10\\times 10^{10}$ M$_\\sun$, i.e., on the low side of the mass range listed in \\citet{bregman18}. The hot gas could therefore dominate the mass of the CGM of M31. There are, however, two caveats to that latter conclusion. First, if $f_{\\rm O\\,VI}\\ll 0.2$, then $M^{\\rm warm}_{\\rm g}$ could be become much larger. Second, the metallicity of the hot X-ray gas ranges from 0.1 to $0.5 Z_\\sun$ with a mean metallicity of $0.3 Z_\\sun$ \\citep{bregman18,gupta17}, while for the cooler gas we have conservatively adopted a solar abundance. If instead we adopt a $0.3 Z_\\sun$ metallicity (consistent with the rough limits set in \\S\\S\\ref{s-metallicity}, \\ref{s-abund}), then $M_{\\rm g} \\simeq 3.7\\times 10^{10}$ M$_\\sun$ at \\ensuremath{R_{\\rm vir}}, which is now comparable to the hot halo mass of the MW. If we adopt the average metallicity derived for the X-ray gas, then $M^{\\rm cool}_{\\rm g} + M^{\\rm warm}_{\\rm g}$ would be comparable to the hot gas mass if $M^{\\rm hot}_{\\rm g} \\sim 5\\times 10^{10}$ M$_\\sun$ at \\ensuremath{R_{\\rm vir}}\\ for M31. Depending on the true metallicities and the actual state of ionization, the cool and warm gas in the M31 halo could therefore contribute to a substantial enhancement of the total baryonic mass compared to our conservative assumptions. \n\n\n\\subsection{Mapping the Metal Surface Densities in the CGM of M31}\\label{s-map-metal}\n\n\n\\begin{figure*}[tbp]\n\\epsscale{1.}\n\\plotone{f13.pdf}\n\\caption{Positions of the Project AMIGA targets relative to the M31, where the axes show the physical impact parameter from the center of M31 (north is up, east to the left). Dotted circles are centered on M31 to mark 100 kpc intervals. The dashes lines represent the projected minor and major axes of M31 and the thin dotted lines are $\\pm 45\\degr$ from the major\/minor axes (which by definition of the coordinate systems also correspond to the vertical and horizontal zero-axis). Each panel corresponds to a different ion. In each panel, the column densities of each velocity component are shown and color coded according the vertical color bar. Circles represent detections while triangles are non-detections. Circles with several color indicate the observed absorption along the sightlines have more than one component. \n\\label{f-colmap}}\n\\end{figure*}\n\nThus far, we have ignored the distribution of the targets in azimuthal angle ($\\Phi$) relative to the projected minor and major axes of M31, where different physical processes may occur. In Fig.~\\ref{f-colmap}, we show the distribution of the column densities of each ion in the X--Y plane near M31 where the circles represent detections and downward triangles are non-detections. Multiple colors in a given circle indicate several components along that sightline for that ion. In that figure, we also show the projected minor and major axes of M31 (dashed lines). The overall trends that are readily apparent from Fig.~\\ref{f-colmap} are the ones already described in the previous sections: 1) overall the column density decreases with increasing $R$, 2) the decrease in $N$ is much stronger for low ions than high ions, 3) \\ion{Si}{3}\\ and \\ion{O}{6}\\ are observed at any $R$ while singly ionized species tend to be more frequently observed at small impact parameters. This figure (and Fig.~\\ref{f-col-vs-rho}) also reveals that absorption with two or more components is observed more frequently at $R<200$ kpc: using \\ion{Si}{3}, 64\\%--86\\% of the sightlines have at least 2 velocity components at $R<200$ kpc, while this drops to 14\\%--31\\% at $R>200$ kpc (68\\% confidence intervals using the Wilson score interval); similar results are found using the other ions. However, the complexity of the velocity profiles does not change with $\\Phi$.\n\nConsidering various radius ranges (e.g., 25--50 kpc, 50--100 kpc, etc.) up to $1\\ensuremath{R_{\\rm vir}}$, there is no indication that the column densities strongly depend on $\\Phi$. Considering \\ion{Si}{3}\\ first, it is equally detected along the projected major and minor axes and in-between (wherever there is a sigthline) and overall the strength of the absorption mostly depends on $R$, not $\\Phi$. Considering the other ions, they all show a mixture of detections and non-detections, and the non-detections (that are mostly beyond 50 kpc) are not preferentially observed along a certain axis or one of the regions shown in Fig.~\\ref{f-colmap}. We therefore find no strong evidence of an azimuthal dependence in the column densities. \n\nBeyond $\\ga 1.1 \\ensuremath{R_{\\rm vir}}$, the situation is different with all but one detection (in \\ion{C}{4}\\ and \\ion{O}{6}\\ only) being near the southern projected major axis and about $52\\degr$ east off near the $X=0$ kpc axis. There is detection in this region of \\ion{Si}{3}, \\ion{C}{4}, \\ion{Si}{4}, \\ion{O}{6}, and also \\ion{C}{2}. That is the main region where \\ion{C}{2}\\ is detected beyond 200 kpc. In contrast, between the $X=0$ kpc axis and southern projected minor axis, the only region where there are several QSOs beyond \\ensuremath{R_{\\rm vir}}, there is no detection in any of the ions (excluding \\ion{O}{6}\\ because there is no {\\em FUSE}\\ observations in these directions). Although that direction is suspiciously in the direction of the MS, it is very unlikely to be additional contamination from the MS because 1) the velocities would be off from those expected of the MS in these directions (see Fig.~\\ref{f-nidever} and also \\S\\ref{s-map-vel}), and 2) there is no overall decrease of the column densities as $|b_{\\rm MS}|$ increases, a trend observed for the components identified as the MS components (see Fig.~\\ref{f-col-cont}). In fact, regarding the second point, the opposite trend is observed with the highest column densities being more frequently at $|b_{\\rm MS}|\\ga 15 \\degr$ than near the MS main axis ($b_{\\rm MS}\\sim 0\\degr$). Therefore while at $R<\\ensuremath{R_{\\rm vir}}$ there is no apparent trend between $N$ and $\\Phi$ for any ions (although we keep in mind that the azimuthal information for \\ion{O}{6}\\ is minimal), most of the detections at $R>\\ensuremath{R_{\\rm vir}}$ are near the southern projected major axis and $52\\degr$ east off of that axis. \n\nThe fact that the gas is observed mainly in a specific region of the CGM beyond \\ensuremath{R_{\\rm vir}}\\ suggests an IGM filament feeding the CGM of M31, as is observed in some cosmological simulations. In particular, \\citet{nuza14} study the gas distribution in simulated recreations of MW and M31 using a constrained cosmological simulation of the Local Group from the Constrained Local UniversE Simulations (CLUES) project. In their Figures 3 and 6, they show different velocity and density projection maps where the central galaxy (M31 or MW) is edge-on. They find that some of the gas in the CGM can flow in a filament-like structure, coming from outside the virial radius all the way down to the galactic disk. \n\n\n\\begin{figure*}[tbp]\n\\epsscale{1.}\n\\plotone{f14.pdf}\n\\caption{Similar to Fig.~\\ref{f-colmap}, but we now show the distribution of the M31 velocities for each component observed for each ion. Circles with several color indicate the observed absorption along the sightlines have more than one components. By definition, in the M31 velocity frame, an absorber with no peculiar velocity relative to M31's bulk motion has $v_{\\rm M31}=0$ ${\\rm km\\,s}^{-1}$.\n\\label{f-velmap}}\n\\end{figure*}\n\n\\subsection{Mapping the Velocities in the CGM of M31}\\label{s-map-vel}\n\n\\begin{figure*}[tbp]\n\\epsscale{1.}\n\\plotone{f15.pdf}\n\\caption{Same as Fig.~\\ref{f-velmap}, but for the average velocities.\n\\label{f-velavgmap}}\n\\end{figure*}\n\nHow the velocity field of the gas is distributed in $R$ and $\\Phi$ beyond 25--50 kpc is a key diagnostic of accretion and feedback. However, a statistical survey using one sightline per galaxy (such as COS-Halos) cannot address this problem because it observes many galaxies in an essential random mix of orientations and inclinations, which necessarily washes out any coherent velocity structures. An experiment like Project AMIGA is needed to access information about large-scale flows in a sizable sample of lines of sight for a single galaxy. The velocity information remains limited because we have only the (projected) radial velocity along pencil beams piercing the CGM at various $R$ and $\\Phi$. Nevertheless as we show below some trends are apparent thanks to the large size of the sample. We use here the $v_{\\rm M31}$ peculiar velocities as defined by Eqn.~\\ref{e-vm31}. By definition, in the M31 velocity frame, an absorber with no peculiar velocity relative to M31's bulk motion has $v_{\\rm M31}=0$ ${\\rm km\\,s}^{-1}$. In \\S\\ref{s-dwarfs-vel}, we show that the M31 peculiar velocities of the absorbers seen toward the QSOs and the velocities of the M31 dwarf satellites largely overlap. We now review how the velocities of the absorbers are distributed in the CGM of M31 over the entire surveyed range of $R$.\n\nIn Figs.~\\ref{f-velmap} and \\ref{f-velavgmap}, we show the distribution of the M31 peculiar velocities of the individual components identified for each ion and column-density-weighted average velocities of each ion, respectively. Circles with several colors indicate that the observed absorption appears in more than one component. Both Figs.~\\ref{f-velmap} and \\ref{f-velavgmap} demonstrate that in many cases there is some overlap in the velocities between low ions (\\ion{Si}{2}, \\ion{C}{2}, \\ion{Si}{3}) and higher ions (\\ion{Si}{4}, \\ion{C}{4}, \\ion{O}{6}). This strongly implies that the CGM of M31 has multiple gas-phases with overlapping kinematics when they are observed in projection (a property also readily observed from the normalized profiles shown in Fig.~\\ref{f-example-spectrum} and as supplemental material in Appendix~\\ref{a-supp-fig}). There are also some rarer cases where there is no velocity correspondence in the velocities between \\ion{Si}{3}\\ and higher ions (see, e.g., near $X\\simeq -335$ $Y\\simeq -95$ kpc), indicating that the observed absorption in each ion is dominated by a single phase--that is, the components are likely to be distinct single-phase objects.\n\nThe full range of velocities associated with the CGM of M31 are between $-249 \\le v_{\\rm M31}\\le +175$ ${\\rm km\\,s}^{-1}$\\ for \\ion{Si}{3}, but for all the other ions it is $-53 \\la v_{\\rm M31}\\le +175$ ${\\rm km\\,s}^{-1}$. Furthermore there is only one absorber\/component of \\ion{Si}{3}\\ that has $ v_{\\rm M31}=-249$ ${\\rm km\\,s}^{-1}$. We emphasize that the rarity of velocity $v_{\\rm M31}<-249$ ${\\rm km\\,s}^{-1}$\\ (corresponding to $\\ensuremath{v_{\\rm LSR}} <-510$ ${\\rm km\\,s}^{-1}$\\ in the direction of this sightline) is not an artifact since velocities below these values are not contaminated by any foreground gaseous features. \n\nWe show in \\S\\ref{s-dwarfs-vel} that the velocity dispersion of the M31 dwarf satellites have a velocity dispersion that is larger (110 ${\\rm km\\,s}^{-1}$\\ for the dwarfs vs. 68 ${\\rm km\\,s}^{-1}$\\ for the \\ion{Si}{3}\\ absorbers) and the M31 dwarfs have some velocities in the velocity range contaminated by the MW and MS. While the CGM gas velocity field distribution may not follow that of the dwarf satellites, it remains plausible that some of the absorption from the extended region of the M31 CGM could be lost owing to contamination from the MW or MS. Therefore we may not be fully probing the entire velocity distribution of the M31 CGM. However, as discussed in \\S\\ref{s-ms}, there is no evidence that the velocity distributions of the \\ion{Si}{3}\\ component in and outside the MS contamination zone are different (see also Figs.~\\ref{f-map} and \\ref{f-velmap}), and hence it is quite possible that at least the MS contamination does not affect much the velocity distribution of the M31 CGM. With these caveats, we now proceed describing the apparent trends of the velocity distribution in the CGM of M31.\n\nFrom Fig.~\\ref{f-velmap}, the first apparent property was already noted in the previous section: the velocity complexity (and hence full-width) of the absorption profiles increases with decreasing $R$ (see \\S\\ref{s-map-metal}). Within $R\\la 100$ kpc or $\\la 200$ kpc, about 75\\% of the \\ion{Si}{3}\\ absorbers have at least two components (at the COS G130M-G160M resolution). This drops to about 33\\% at $200100$ kpc. From this table and for all the ions besides \\ion{O}{6}, $\\langle v_{\\rm M31} \\rangle=90$ ${\\rm km\\,s}^{-1}$\\ at $R\\le 100$ kpc, while at $R>100$ kpc, $\\langle v_{\\rm M31} \\rangle=20$ ${\\rm km\\,s}^{-1}$, a factor 4.5 times smaller. There are only two data points for \\ion{O}{6}, at $R\\le 100$ kpc, but the average at $R>100$ kpc is $\\langle v_{\\rm M31} \\rangle=22$ ${\\rm km\\,s}^{-1}$, following a similar pattern as observed for the other ions. For all the ions but \\ion{C}{2}, the velocity dispersions or IQRs are smaller at $R\\le 100$ kpc than at $R>100$ kpc.\n\nThe third property observed in Fig.~\\ref{f-velmap} or Fig.~\\ref{f-velavgmap} is that at $R\\le 100$ kpc, there is no evidence for negative M31 velocities, while at $R>100$ kpc, about 40\\% of the \\ion{Si}{3}\\ sample has blueshifted $v_{\\rm M31}$ velocities. This partially explains the previous result, but even if we consider the absolute velocities, $\\langle|v_{\\rm M31}|\\rangle=40$ ${\\rm km\\,s}^{-1}$\\ at $R>100$ kpc, implying $\\langle |v_{\\rm M31}(R>100)|\\rangle = 0.44\\langle |v_{\\rm M31}|(R\\le 100) \\rangle$, i.e., in absolute terms or not, $v_{\\rm M31}$ is smaller at $R>100$ kpc than at $R\\le 100$ kpc. Therefore at $R>100$ kpc, not only are the peculiar velocities of the CGM gas less extreme, but they are also more uniformly distributed around the bulk motion of M31. At $R<100$ kpc, the peculiar velocities of the CGM gas are more extreme and systematically redshifted relative to the bulk motion of M31.\n\nThe fourth property appears in Fig.~\\ref{f-velmap-dwarfs} where we compare $v_{\\rm M31}$ velocities of the M31 dwarfs and \\ion{Si}{3}\\ absorbers, which shows that overall the velocities of the satellites and the CGM gas do not follow each other. As noted in \\S\\ref{s-dwarfs-cgm} (see also Table~\\ref{t-xmatch-dwarf}), some velocity components seen in absorption toward the QSOs are found with $\\Delta_{\\rm sep}<1$ and have $\\delta v100$ kpc or $R\\le 100$ kpc, $\\langle v_{\\rm M31,dwarf} \\rangle \\simeq 34$ ${\\rm km\\,s}^{-1}$ ($\\langle |v_{\\rm M31,dwarf}|\\rangle \\simeq 102$ ${\\rm km\\,s}^{-1}$), remarkably contrasting with the properties of the CGM gas described in the previous two paragraphs. These findings strongly suggests that the velocity fields of the dwarfs and CGM gas are decoupled. We infer from this decoupling that (1) gas bound to satellites does not make a significant contribution to CGM gas observed in this way, and (2) the velocities of gas removed from satellites via tidal or ram-pressure interactions, if it is present, becomes decoupled from the dwarf that brought it in (as one might expect from its definition as unbound to the satellites). \n\nThe fifth property is more readily apparent considering the average velocities shown in Fig.~\\ref{f-velavgmap} where considering the CGM gas in different annuli, there is an apparent change in the sign of the average $v_{\\rm M31}$ velocities with on average a positive velocity in at $R<200$ kpc, negative velocity in $2000$ (see \\S\\ref{s-ms}). Absorption occurring in the velocity range $-50 \\la v_{\\rm M31}\\la +150$ ${\\rm km\\,s}^{-1}$\\ is also not contaminated by the MS.\n\n\\section{Discussion}\\label{s-disc}\n\nThe major goal of Project AMIGA is to determine the global distribution of the gas phases and metals through the entire CGM of a representative galaxy. With a large sample of QSOs accumulated over many surveys, and newly observed by {\\em HST}\/COS, we are able to probe multiple sightlines that pierce M31 at different radii and azimuthal angles. Undertaking this study in the UV has been critical since only in this wavelength band there are the diagnostics and spectral resolution to constrain the physical properties of the multiple gas-phases existing in the CGM over $10^{4-5.5}$ K (for $ z = 0$, the hottest phase can only be probed with X-ray observations). With 25 sightlines within about $1.1\\ensuremath{R_{\\rm vir}}$ and 43 within 569 kpc ($\\la 1.9\\ensuremath{R_{\\rm vir}}$) of M31, the size of the sample and the information as a function of $R$ and $\\Phi$ are unparalleled. We will now consider the broad patterns and conclusions we can draw from this unique dataset. \n\n\\subsection{Pervasive Metals in the CGM of M31}\\label{s-disc-metal-mass}\n\nA key finding of Project AMIGA is the ubiquitous presence of metals in the CGM of M31. While the search for \\ion{H}{1}\\ with $\\log N_{\\rm H\\,I} \\ga 17.5$ in the CGM of M31 toward pointed radio observations has been unsuccessful in the current sample (\\citetalias{howk17} and see Fig.~\\ref{f-map}), the covering factor of \\ion{Si}{3}\\ (29 sightlines) is essentially 100\\% out to $1.2 \\ensuremath{R_{\\rm vir}}$, while \\ion{O}{6}\\ associated with M31 is detected toward all 11 sightlines with FUSE data, all the way out to $1.9 \\ensuremath{R_{\\rm vir}}$, the maximum radius of our survey (see \\S\\S\\ref{s-n-vs-r}, \\ref{s-fc}). From the ionization range probed by Project AMIGA, we further show that \\ion{Si}{3}\\ and \\ion{O}{6}\\ are key probes of the diffuse gas (see \\S\\ref{s-ratio-vs-r}). With information from \\ion{Si}{2}, \\ion{Si}{3}, and \\ion{Si}{4}, we demonstrate that \\ion{Si}{3}\\ is the dominant ion in the ionizing energy range 8--45 eV (see \\S\\ref{s-si2-si3-si4}). The fact that \\ion{Si}{3}\\ and \\ion{O}{6}\\ have such high covering factors suggests that these ions are not produced in small clumps within a hotter medium; instead it must be more pervasively distributed. \n\nThe finding of pervasive metals in the CGM of M31 is a strong indication of ongoing and past gas outflows that ejected metals well beyond their formation site. Based on a specific star-formation rate of ${\\rm SFR\/M_\\star} = (5\\pm 1) \\times 10^{-12}$ yr$^{-1}$ \\citep[using the stellar mass M$_\\star$ and SFR from][]{geehan06,kang09}, M31 is not currently in an active star-forming episode. In fact, \\citet{williams17} show that the bulk of star formation occurred in the first $\\sim$6 billion years and the last strong episode happened over $\\sim$2 billion years ago (see also Fig.~6 in \\citealt{telford19} for a metal production model of M31). Hence most of the metals seen in the CGM of M31 have most likely been ejected by previous star-forming episodes and\/or stripped from its dwarfs and more massive companions. However, the fact that metals are detected beyond \\ensuremath{R_{\\rm vir}}, and, that beyond \\ensuremath{R_{\\rm vir}}\\ they are found predominantly in a certain direction, also suggests that some of the metals may be coming from the Local group medium, possibly recycling metals from the MW or M31 (see \\S\\ref{s-map-metal}), or from an IGM filament in that particular direction. \n\nIn \\S\\ref{s-mass}, we estimate that the mass of metals $M^{\\rm cool}_{\\rm Z} = (2.0 \\pm 0.5) \\times 10^7$ M$_\\sun$ within \\ensuremath{R_{\\rm vir}}\\ for the predominantly photoionized gas probed by \\ion{Si}{2}, \\ion{Si}{3}, and \\ion{Si}{4}. For the gas probed by \\ion{O}{6}, we find that $M^{\\rm warm}_{\\rm Z} > 4.4 M^{\\rm cool}_{\\rm Z} \\ga 9\\times 10^7$ M$_\\sun$ at \\ensuremath{R_{\\rm vir}}\\ (this is a lower limit because the fractional amount of \\ion{O}{6}\\ is an upper limit, see \\S\\ref{s-mass}). The sum of these two phases yields a lower limit to the CGM metal mass because the hotter phase probed by the X-ray and metals bound in dust are not included. If the hot baryon mass of M31 is not too different from that estimated for the MW (see \\S\\ref{s-mass}), then we expect $M^{\\rm hot}_{\\rm Z} \\approx M^{\\rm warm}_{\\rm Z}$. The CLUES simulation of the Local group estimates that the mass of the hot ($>10^5$ K) gas is a factor 3 larger than the cooler ($<10^5$ K) gas \\citep{nuza14}. The dust CGM mass remains quite uncertain, but could be at the level of $5\\times 10^7$ M$_\\sun$ according to estimates around $0.1$--$1L^*$ galaxies \\citep{menard10,peeples14,peek15}. Hence the total metal mass of the CGM of M31 out to \\ensuremath{R_{\\rm vir}}\\ could be as large as $M^{\\rm CGM}_{\\rm Z}\\ga 2.5\\times 10^8$ M$_\\sun$.\n\nThe stellar mass of M31 is $(1.5\\pm 0.2) \\times 10^{11}$ M$_\\sun$ \\citep[e.g.,][]{williams17}. Using this result, \\citet{telford19} estimated that the current metal mass in stars is $3.9\\times 10^8$ M$_\\sun$, i.e., about the same amount that is found in the entire CGM of M31 up to \\ensuremath{R_{\\rm vir}}. \\citet{telford19} also estimated the metal mass of the gas in the disk of M31 to be around $(0.8$--$3.2)\\times 10^7$ M$_\\sun$, while \\citet{draine14} estimated the dust mass in the disk to be around $5.4\\times 10^7$ M$_\\sun$, yielding a total metal mass in the disk of M31 of about $M^{\\rm disk}_{\\rm Z}\\simeq 5\\times 10^8$ M$_\\sun$. Therefore M31 has in its CGM within \\ensuremath{R_{\\rm vir}}\\ at least 50\\% of the present-day metal mass in its disk. As we show in \\S\\ref{s-n-vs-r} and \\S\\ref{s-mass} and discuss above, metals are also found beyond \\ensuremath{R_{\\rm vir}}, especially in the more highly ionized phase traced by \\ion{O}{6}\\ (and even higher ions). These metals could come from M31 or being recycled in the Local group from the MW or dwarf galaxies. \n\n\\subsection{Comparison with COS-Halos Galaxies}\\label{s-coshalos}\nThe Project AMIGA experiment is quite different from most of the surveys of the CGM of galaxies done so far. Outside the local universe, surveys of the CGM of galaxies involve assembling samples of CGM gas in aggregate by using one sightline per galaxy (see \\S\\ref{s-intro}), and in some nearby cases up to 3--4 sightlines (e.g., \\citealt{bowen16,keeney17}). By assembling a sizable sample of absorbers associated with galaxies in a particular sub-population (e.g. $L^*$, sub-$L^*$, passive or star-forming galaxies), one can then assess how the column densities change with radii around that kind of galaxy, and from this estimate average surface densities, mass budgets, etc. can then be evaluated. By contrast, Project AMIGA has assembled almost as many sightlines surrounding M31 as COS-Halos had for its full sample of 44 galaxies. We can now make a direct comparison between these two types of experiments. For this comparison, we use the COS-Halos survey of $0.30.9R_{200}$, there is no COS-Halos observation (owing to the design of the survey). The extrapolated model to the COS-Halos observations shown in gray in Fig.~\\ref{f-cos-halos} is a factor 2--4 higher than the models of the Project AMIGA data shown in blue depending on $R\/R_{200}$. \n\nA likely explanation for the higher column density absorbers is that some of these COS-Halos absorbers could be fully or partly associated with a closer galaxy than the initially targeted COS-Halos galaxies where the gas can contain more neutral and weakly ionized gas. Indeed, while the COS-Halos galaxies were selected to have no bright companion, that selection did not preclude fainter nearby companions such as dwarf satellites (see \\citealt{tumlinson13}). Galaxy observation follow-up by \\citet{werk12} found several $L > 0.1 L^*$ galaxies within 160 kpc of the targeted COS-Halos galaxy. Comparing the results from other surveys of galaxies\/absorbers at low redshift \\citep{stocke13,bowen02}, \\citet{bregman18} also noted a higher preponderance of high \\ion{H}{1}\\ column density absorbers in the COS-Halos survey. However, the higher COS-Halos column densities at large radii could also be an effect of evolution in the typical CGM, as COS-Halos probed a slightly higher cosmological redshift. It is also possible that the M31 CGM is less rich in ionized gas at these radii than the typical $L^*$ galaxy at $z \\sim 0.2$, because of its star formation history or environment. \n\n\\subsubsection{CGM Mass Comparison}\\label{s-cos-halos-mass}\nAmong a key physical parameter of the CGM is its mass, which is obtained from the column density distribution of the gas and assuming a certain geometry of the gas. For M31, we cannot derive the baryonic mass of CGM gas without assuming a metallicity since the \\ion{H}{1}\\ column density remains unknown toward all the targets in our sample (but see \\S\\ref{s-metallicity}, \\ref{s-mass}). However, the metal mass of the cool gas probed by \\ion{Si}{2}, \\ion{Si}{3}, and \\ion{Si}{4}\\ can be straightforwardly estimated directly from the observations without any ionization modeling (see top panel of Fig.~\\ref{f-cos-halos}). \n\nEven though both \\citet{peeples14} and \\citet{werk14} use the Si column densities derived from phoionization models, as illustrated in Fig.~\\ref{f-cos-halos}, this would not change the outcome that the metal mass of the cool CGM gas derived from the COS-Halos survey is about a factor 2--3 higher than the metal mass derived in Project AMIGA. This is because there are 7 COS-Halos Si column densities at $R\/R_{200}<0.3$ that are much higher owing to saturation in the weak \\ion{Si}{2}\\ transitions (see above), driving the overall model of $N_{\\rm Si}(R)$ substantially higher. The fact that these high $N_{\\rm Si}$ are not found in the CGM of M31 or lower redshift galaxies at similar impact parameters \\citep[e.g.,][]{bowen02,stocke13} suggests a source of high-column \\ion{H}{1}\\ and \\ion{Si}{2}\\ absorbers in the COS-Halos sample that could be recent outflows, strong accretion\/recycling, or gas associated with closer satellites to the sightline. With only 5 targets within $R\/R_{200}<0.3$ and none below $R\/R_{200}<0.1$ for M31, it would be quite useful to target more QSOs in the inner region of the CGM of M31 to better determine how $N_{\\rm Si}(R)$ varies with $R$ at small impact parameters.\n\nFor the warm-hot gas probed by \\ion{O}{6}, the COS-Halos star-forming galaxies have $\\langle N_{\\rm O\\,VI}\\rangle = 10^{14.5}$ cm$^{-2}$, a detection rate close to 100\\%, and no large variation of $N_{\\rm O\\,VI}$\\ with $R$ \\citep{tumlinson11a}. For M31, we have a similar average \\ion{O}{6}\\ column density, hit rate, and little evidence for any large variation of $N_{\\rm O\\,VI}$\\ with $R$ (see \\S\\ref{s-mass}). This implies that the masses of the warm-hot CGM of M31 and COS-Halos star-forming galaxies are similar. M31 has a specific SFR that is a factor $\\ga 10$ lower than the COS-Halos star-forming galaxies, but its halo mass is on the higher side of the COS-Halos star-forming galaxies (but lower than the COS-Halos quiescent galaxies). As discussed in \\S\\ref{s-disc-pers-comp} in more detail, M31 and the COS-Halos star-forming galaxies have halo masses in the range $M_{200} \\simeq 10^{11.7}$--$10^{12.3}$ M$_\\sun$, corresponding to a virial temperature range that overlaps with the temperature at which the ionization fraction of \\ion{O}{6}\\ peaks, which may naturally explaining some of the properties of the \\ion{O}{6}\\ in the CGM of ``$L^*$\" galaxies \\citep{oppenheimer18}. It is also possible that some \\ion{O}{6}\\ arises in photoionized gas or combinations of different phases (see \\S\\ref{s-disc-pers-comp}). \n\nBased on the comparison above, we find that the \\ion{O}{6}\\ is less subject to the uncertainty in the association of the absorber to the correct galaxy owing to its column density being less dependent on $R$ (see also \\S\\ref{s-disc-change}). Therefore this leads to similar metal masses of the CGM of the $z\\sim 0.2$ COS-Halos galaxies and M31 for the \\ion{O}{6}\\ gas-phase. For the lower ions, their column densities are more dependent on $R$ (see also \\S\\ref{s-disc-change}). Therefore the association of the absorber to the correct galaxy is more critical to derive an accurate column density profile with $R$ and hence derive an accurate CGM metal mass. However, we note that despite these uncertainties the metal mass of the cool CGM of the COS-Halos galaxies is only a factor 2--3 higher than that derived for M31. \n\n\n\\subsection{A Changing CGM with Radius}\\label{s-disc-change}\nA key discovery from Project AMIGA is that the properties of the CGM of M31 change with $R$. This is reminiscent of our earlier survey \\citepalias{lehner15}, but the increase in the size sample has transformed some of the tentative results of our earlier survey into robust findings. In particular the radius around $R\\sim 100$--150 kpc appears critical in view of several properties changing near this threshold radius:\n\\begin{enumerate}[wide, labelwidth=!, labelindent=0pt]\n \\item For any ions, the frequency of strong absorption is larger at $R\\la 100$--150 kpc than at larger $R$.\n \\item The column densities of Si and C ions change by a factor $>5$--$10$ between about 25 kpc and 150 kpc, while they change only by a factor $\\la 2$ between 150 kpc and 300 kpc.\n \\item The detection rate of singly ionized species (\\ion{C}{2}, \\ion{Si}{2}) is close to 100\\% at $R<150$ kpc, but sharply decreases beyond (see Fig.~\\ref{f-col-vs-rho}), and therefore the gas has a more complex gas-phase structure at $R<150$ kpc.\n \\item The peculiar velocities of the CGM gas are more extreme and systematically redshifted relative to the bulk motion of M31 at $R\\la 100$ kpc, while at $R\\ga 100$ kpc, the peculiar velocities of the CGM gas are less extreme and more uniformly distributed around the bulk motion of M31.\n\\end{enumerate}\nThere are also two other significant regions: 1) beyond $R_{200}\\simeq 230$ kpc the gas is becoming more ionized and more highly ionized than at lower $R$ (e.g., there is a near total absence of \\ion{Si}{2}\\ absorption beyond $R_{200}$---see Fig.~\\ref{f-col-vs-rho}, or, a higher \\ion{C}{2}\/\\ion{C}{4}\\ ratio on average at $R\\ga R_{200}$ than at lower $R$---see \\S\\ref{s-c2-c4}); and 2) beyond $1.1\\ensuremath{R_{\\rm vir}}$ the gas is not detected in all the directions away from M31, as it is at smaller radii, but only in a cone near the southern projected major axis and about $52\\degr$ east off the $X=0$ kpc axis (see \\S\\ref{s-map-metal}). \n\nThe overall picture that can be drawn out from these properties is that the inner regions of the CGM of M31 are more dynamic and complex, while the more diffuse regions at $R\\ga 0.5\\ensuremath{R_{\\rm vir}}$ are more static and simpler. Zoom-in cosmological simulations capture in more detail and more accurately the structures of the CGM than large-scale cosmological simulations thanks to their higher mass and spatial resolution. Below we use several results from zoom simulations to gain some insights on these observed changes with $R$. However, the results laid out in \\S\\ref{s-properties} also now provide a new testbed for zoom simulations, so that not only qualitative but also quantitative comparison can be undertaken. We note that most of the zoom simulations discussed here have only a single massive halo. However, according to the ELVIS simulations of Local group analogs \\citep{garrison-kimmel14}, there should be no major difference at least within about \\ensuremath{R_{\\rm vir}}\\ for the distribution of the gas between isolated and paired galaxies.\n\n\\subsubsection{Visualization and Origins of the CGM Variation}\\label{s-disc-qual-comp}\nTo help visualize the properties described above and gain some insights into the possible origins of these trends, we begin by qualitatively examining two zoom simulations. First, we consider the Local group zoom simulations from the CLUES project \\citep{nuza14} where the gas distribution around MW and M31-like galaxies is studied. This paper does not show the distribution of the individual ions, but examines the two main gas-phases above and below $10^5$ K in an environment that is a constrained analog to the Local group. Interestingly, considering Fig.~3 (simulated M31) or Fig.~6 (simulated MW) in \\citeauthor{nuza14}, the region within 100--150 kpc appears more complex, with a large covering factor for both cool and hot gas phases and higher velocities than at larger radii. In these simulations, this is a result of the combined effects of cooling and supernova heating affecting the closer regions of the CGM of M31. This simulation also provides an explanation for the gas observed beyond $1.1\\ensuremath{R_{\\rm vir}}$ that is preferentially observed in a limited region of the CGM of M31 (see Fig.~\\ref{f-colmap} and see middle right panel of their Fig.~3) whereby the $\\la 10^5$ K gas might be accreting onto the CGM of M31. We also note that \\citet{nuza14} find a mass for the $\\la 10^5$ K CGM gas of $1.7\\times 10^{10}$ M$_\\sun$, broadly consistent with our findings (see \\S\\ref{s-mass}). More quantitative comparisons between the CLUES (or Local group analog simulations like ELVIS-FIRE simulations, \\citealt{garrison-kimmel14,garrison-kimmel19}) and Project AMIGA results are beyond the scope of this paper, but they would be valuable to undertake in the future. \n\nSecond, we consider the zoom Eris2 simulation of a massive, star-forming galaxy at $z = 2.8$ presented in \\citet{shen13}. The Eris2 galaxy being $z = 2.8$ and with a star-formation rate of 20 M$_\\sun$\\,yr$^{\u22121}$ is nothing like M31, but this paper shows the distribution of the gas around the central galaxy using some of the same ions that are studied in Project AMIGA, specifically \\ion{Si}{2}, \\ion{Si}{4}, \\ion{C}{2}, \\ion{C}{4}, and \\ion{O}{6}\\ (see their Figs.~3a and 4a, b). Because Eris2 is so different from M31, we would naively expect their CGM properties to be different, and yet: 1) Eris2 is surrounded by a large diffuse \\ion{O}{6}\\ halo with a near unity covering factor all the way out to about $3\\ensuremath{R_{\\rm vir}}$; 2) the covering factor of absorbing material in the CGM of Eris2 declines less rapidly with impact parameter for \\ion{C}{4}\\ or \\ion{O}{6}\\ compared to \\ion{C}{2}, \\ion{Si}{2}, or \\ion{Si}{4}; 3) beyond \\ensuremath{R_{\\rm vir}}, the covering factor of \\ion{Si}{2}\\ drops more sharply than \\ion{C}{2}. There are also key differences, like the strongest absorption in any of these ions being observed in the bipolar outflows perpendicular to the plane of the disk, which is unsurprisingly not observed in M31 since it currently has a low star-formation rate \\citep[e.g.,][]{williams17}. However, the broad picture of the CGM of M31 and the simulated Eris2 galaxy are remarkably similar. This implies that some of the properties of the CGM may depend more on the micro-physics producing the various gas-phases than the large-scale physical processes (outflow, accretion) that vary substantially over time. In fact, the Eris2 simulation shows that inflows and outflows coexist and are both traced by diffuse \\ion{O}{6}; In Eris2, a high covering factor of strong \\ion{O}{6}\\ absorbers seems to be the least unambiguous tracer of large-scale outflows. \n\n\n\\subsubsection{Quantitative Comparison in the CGM Variation between Observations and Simulations}\\label{s-disc-quant-comp}\n\nTwo simulations of M31-like galaxies in different environments at widely separated epochs show some similarity with some of the observed trends in the CGM of M31. We now take one step further by quantitatively comparing the column density variation of the different ions as a function of $R$ in three different zoom-in cosmological simulations, two being led by members of the Project AMIGA team (FIRE and FOGGIE collaborations), and a zoom-in simulation from the Evolution and Assembly of GaLaxies and their Environments (EAGLE) simulation project \\citep{oppenheimer18a,schaye15,crain15}. \n\n\\noindent\n{\\it $\\bullet$ Comparison with FIRE-2 Zoom Simulations}\n\nWe first compare our observations with column densities modeled using cosmological zoom-in simulations from the FIRE project\\footnote{FIRE project website: \\url{http:\/\/fire.northwestern.edu}}. Details of the simulation setup and CGM modeling methods are presented in \\citet{ji19}. Briefly, the outputs analyzed here are FIRE-2 simulations evolved with the {\\small GIZMO} code using the meshless finite mass (MFM) solver \\citep{hopkins15}. The simulations include a detailed model for stellar feedback including core-collapse and Type Ia SNe, stellar winds from OB and AGB stars, photoionization, and radiation pressure \\citep[for details, see][]{hopkins18}. We focus on the ``m12i'' FIRE halo, which has a mass $M_{\\rm vir} \\approx 1.2 M_{200} \\approx 1.2\\times10^{12}\\,M_\\odot$ at $z=0$, which is comparable to the halo mass of M31. However, neither the SFR history nor the present-day SFR are similar. The ``m12i'' FIRE halo has a factor 10--12 higher SFR (see Fig.~3 in \\citealt{hopkins19}) than the present-day SFR of M31 of 0.5\\,M$_\\sun$\\,yr$^{-1}$ \\citep[e.g.][]{kang09}. We compare Project AMIGA to FIRE-2 simulations with two different sets of physical ingredients. The ``MHD'' run includes magnetic fields, anisotropic thermal conduction and viscosity, and the ``CR'' run includes all these processes plus the ``full physics'' treatment of stellar cosmic rays. The CR simulation assumes a diffusion coefficient $\\kappa_{||}=3\\times10^{29}$ cm$^{-2}$\\,s$^{-1}$, which was calibrated to be consistent with observational constraints from $\\gamma-$ray emission of the MW and some other nearby galaxies \\citep{hopkins19,chan19}. \\citet{ji19} showed cosmic rays can potentially provide a large or even dominant non-thermal fraction of the total pressure support in the CGM of low-redshift $\\sim L^*$ galaxies. As a result, in the fiducial CR run analyzed here, the volume-filling CGM is much cooler ($\\sim 10^{4}-10^{5}$ K) and is thus photoionized in regions where in the run without CRs prefers by hot gas that is more collisionally ionized.\n\nThe column densities are generated as discussed in \\citet{ji19}. For the ionization modeling, a hybrid treatment combining the FG09 \\citep{faucher-giguere09} and HM12 \\citep{haardt12} UV background models is used.\\footnote{We use this mixture because, based on the recent UV background analysis of \\citet{faucher-giguere19}, the FG09 model is in better agreement with the most up-to-date low-redshift empirical constraints at energies relevant for low and intermediate ions (\\ion{C}{2}, \\ion{Si}{2}, \\ion{Si}{3}, \\ion{Si}{4}, and \\ion{C}{4}). However, the HM12 model is likely more accurate for high ions such as \\ion{O}{6}\\ because the FG09 model used a crude AGN spectral model which under-predicted the higher-energy part of the UV\/X-ray background. \\citet{ji19} shows how some ion columns depend on the assumed UV background model.}\n\n\n\\begin{figure}[tbp]\n\\epsscale{1.2}\n\\plotone{f17.pdf}\n\\caption{Comparison of ion column density profiles between Project AMIGA (total column densities) and FIRE-2 simulations, where ``MHD'' and ``CR'' runs. Thick curves show median values of an ensemble of sightlines produced from simulations, and shaded regions show the full range across all model sightlines. \n\\label{f-col-vs-rho-fire}}\n\\end{figure}\n\nIn Fig.~\\ref{f-col-vs-rho-fire}, we compare the ion column densities from FIRE-2 simulations with observationally-derived total column densities around M31 as a function of $R\/R_{200}$. The green and orange curves show the median simulated column densities for the MHD and CR runs, respectively, while the shaded regions show the full range of columns for all sightlines at a given impact parameter (the lowest values are truncated to match the scales that are adequate for the observations, see \\citealt{ji19} for the full range of values). The CR run produces higher column densities and better agreement with observations than the MHD run for all ions presented. The much higher column densities of low\/intermediate ions (\\ion{C}{2}, \\ion{Si}{2}, \\ion{Si}{3}, and \\ion{Si}{4}) in the CR run owing to the more volume-filling and uniform cool phase, which produces higher median values of ion column densities and smaller variations across different sightlines. In contrast, in the MHD run the cool phase is pressure confined by the hot phase to compact and dense regions, leading to smaller median columns but larger scatter for the low and intermediate ions. We note, however, that even in the CR runs the predicted column densities are lower than observations at the larger impact parameters $R\\ga 0.5 R_{200}$. This might be due to insufficient resolution to resolve fine-scale structure in outer halos, or it may indicate that feedback effects are more important at large radii than in the present simulations. This difference is quite notable owing to the fact that the star formation of the ``m12i\" galaxy has been continuous with a SFR in the range 5--20 M$_\\sun$\\,yr$^{-1}$ \\citep{hopkins19} over the last $\\sim$8 billion years while M31 had only a continuous SFR around 6--8 M$_\\sun$\\,yr$^{-1}$ over its first 5 billion years while over the last 8 billion years it had only two short bursts of star formation about 4 and 2 billion years ago \\citep{williams17}. While there are some discrepancies, the simulations also follow some similar trends: 1) the simulated column densities of the low ions decrease more rapidly with $R$ than the high ions, 2) \\ion{O}{6}\\ is observed beyond $1.7R_{200}$ where there is no substantial amount of low\/intermediate ions, and 3) a larger scatter is observed in the column densities of the low and intermediate ions than \\ion{O}{6}. \n\nIn the FIRE-2 simulations, both collisional ionization and photoionization can contribute significantly to the simulated \\ion{O}{6}\\ columns, typically with an increasing contribution from photoionization with increasing impact parameter, driven by decreasing gas densities. In the MHD run, most of the \\ion{O}{6}\\ in the inner halo ($R\\la 0.5 R_{200}$) is produced by collisional ionization, but photoionization can dominate at larger impact parameters. In the CR run, collisional ionization and photionization contribute comparably to the \\ion{O}{6}\\ mass at radii $50 < R < 200$ kpc \\citep{ji19}. The actual origins of the CGM in terms of gas flows in FIRE-2 simulations without magnetic fields or cosmic rays were analyzed in \\cite{hafen19a}, although the results are expected to be similar for simulations with MHD only. In these simulations, \\ion{O}{6}\\ exists as part of a well-mixed hot halo, with contributions from all the primary channels of CGM mass growth: IGM accretion, wind, and contributions from satellite halos (reminiscent of the Eris2 simulations, see above and \\citealt{shen13}). The metals responsible for \\ion{O}{6}\\ absorption originate primarily in winds, but IGM accretion may contribute a large fraction of total gas mass traced by \\ion{O}{6}\\ since the halo is well-mixed and IGM accretion contributes $\\ga 60\\%$ of the total CGM mass. In the simulations, the hot halo gas persists in the CGM for billions of years, and the gas that leaves the CGM does so primarily by accreting onto the central galaxy \\citep{hafen19b}.\n\n\\noindent\n{\\it $\\bullet$ Comparison with FOGGIE Simulations}\n\n\nWe also compare the observed total column densities to the Milky-Way like-mass ``Tempest'' ($M_{\\rm 200} \\approx 4.2 \\times 10^{11}$ M$_{\\odot}$) halo from the FOGGIE simulations,\\footnote{FOGGIE project website: \\url{http:\/\/foggie.science}} which has a halo mass of $M_{\\rm 200} \\approx 4.2 \\times 10^{11}$ M$_{\\odot}$ \\citep{peeples19}. We use the $z=0$ output (see \\citealt{zheng20} for simulation details), but because of the size difference between M31 and the Tempest galaxy, we again scale all distances by $R_{200}$ ($R_{200} = 159$ kpc for the simulated halo compared to 230 kpc for M31). The only ``feedback'' included in this FOGGIE run is thermal explosion-driven SNe outflows. While this feedback is limited in scope compared to FIRE, FOGGIE achieves higher mass resolution than FIRE-2 by using a ``forced refinement'' scheme that applies a fixed computational cell size of $\\sim 381 h^{-1}$ pc within a moving cube centered on the galaxy that is $\\sim 200 h^{-1}$ ckpc on a side. This refinement scheme enforces constant {\\it spatial resolution} on the CGM, resulting in a variable and very small mass resolution in the low density gas, with typical cell masses of ($\\la 1$--100 M$_\\sun$). The individual small-scale structures that contribute to the observed absorption profiles can therefore be resolved. These small-scale structures that become only apparent in high-resolution simulations are hosts to a significant amount of cool gas, enhancing the column densities in especially the low ionization state of the gas (\\citealt{peeples19,corlies19}, and see also \\citealt{vandevoort18,hummels19,rhodin19}). \n\n\n\\begin{figure}[tbp]\n\\epsscale{1.2}\n\\plotone{f18.png}\n\\caption{Comparison of ion column density profiles between Project AMIGA (total column densities) and the ``Tempest\" halo from the FOGGIE simulations. The pink and green shaded areas are projected total column densities from the simulated halo with and without galaxy\/satellite contributions, respectively, while the rest of the figure is analogous to Fig.~\\ref{f-coltot-vs-rho}. The vertical line shows the extent of the forced resolution cube in the FOGGIE simulation. \n\\label{f-col-vs-rho-foggie}}\n\\end{figure}\n\n\nAs for the FIRE-2 simulations, we compare the total Project AMIGA column densities to FOGGIE because in the simulation we do not (yet) separate individual components, but look at the projected column densities through the halo. We note that the CGM is not necessarily self-similar, so some differences between the simulation predictions and M31 observations at rescaled impact parameter could be due to the halo mass difference. This is especially so since the halo mass range $M_{\\rm h}\\approx 3\\times10^{11}$--$10^{12}$ M$_{\\odot}$ corresponds to the expected transition between cold and hot accretion \\citep[e.g.,][]{birnboim03,keres03,faucher-giguere11,stern19}.\n\nIn Fig.~\\ref{f-col-vs-rho-foggie}, we compare the simulated and observed column densities for each ion probed by our survey. The pink and green shaded areas are the data points from the simulation (with and without satellite contribution, respectively) and show the total column density in projection through the halo. The scatter in the simulated data points comes from variation in the structures along the mock sightline and most of the scatter is in fact below $10^{11}$ cm$^{-2}$. The peaks in the column densities are due to small satellites in the halo, which enhance primarily the low-ion column densities. We show the green points to highlight the difference between the mock column densities with and without satellites. For the high ionization lines the difference is negligible, while the difference in the low ions is significant. \n\nOverall, the metal line column densities are systematically lower than in the observations at any $R$. Only at $R\\la 0.3R_{200}$, there is some overlap for the singly ionized species between the FOGGIE simulation and observations. However, the discrepancy is particularly striking for \\ion{Si}{3}\\ and the high ions. This can be understood by the current feedback implementation in FOGGIE, which does not expel enough metals from the stellar disk into the CGM \\citep{hamilton20} to be consistent with known galactic metal budgets\\citep{peeples14}. This effect is expected to be stronger for the high ions than the low ions, due to the additional heating and ionization of the CGM that would be expected from stronger feedback, and indeed the discrepancy between the simulation and observations is larger for the high ions (and \\ion{Si}{3}) than for the singly ionized species. However, while the absolute scale of the column densities is off, there are also some similarities between the simulation and observations in the behavior of the relative scale of the column density profiles with $R$: 1) the column densities of the low ions drop more rapidly with $R$ than the high ions; 2) despite the inadequate feedback in the current simulations, the \\ion{O}{6}-bearing gas (and \\ion{C}{4}\\ to a lesser extent) is observed well-beyond $R_{200}$; 3) a large scatter is observed in the column densities of the low and intermediate ions than \\ion{O}{6}. It is striking that the overall slope of the \\ion{O}{6}\\ profile resembles the observations but at significantly lower absolute column density. In the FOGGIE simulation, the low ions tracing mainly dense, cool gas are preferentially found in the disk or satellites, while the hotter gas traced by the higher ions is more homogeneously distributed in the halo.\n\n\\noindent\n{\\it $\\bullet$ Comparison with EAGLE Simulations}\n\n\nFinally, we compare our results with the EAGLE zoom-in simulations (EAGLE \\emph{Recal-L025N0752} high-resolution volume) discussed in length in \\citet{oppenheimer18a}. The EAGLE simulations have successfully reproduced a variety of galaxy observables \\citep[e.g.,][]{crain15,schaye15} and achieved ``broad but imperfect\" agreement with some of the extant CGM observations (e.g., \\citealt{turner16,rahmati18,oppenheimer18a,lehner19,wotta19}). \n\n \\citet{oppenheimer18a} aimed to directly study the multiphase CGM traced by low metal ions and to compare with the COS-Halos survey (see \\S\\ref{s-coshalos}). As such, they explored the circumgalactic metal content traced by the same ions explored in Project AMIGA in the CGM galaxies with masses that comprise that of M31. Overall \\citeauthor{oppenheimer18a} find agreement between the simulated and COS-Halos samples for \\ion{Si}{2}, \\ion{Si}{3}, \\ion{Si}{4}, and \\ion{C}{2}\\ within a factor two or so and larger disagreement with \\ion{O}{6}, where the column density is systematically lower. With Project AMIGA, we can directly compare the results with one of the EAGLE galaxies that has a mass very close to M31 and also compare the column densities beyond 160 kpc, the maximum radius of the COS-Halos survey \\citep{tumlinson13,werk13}. We refer the reader to \\citet{oppenheimer16}, \\citet{rahmati18}, and \\citet{oppenheimer18a} for more detail on the EAGLE zoom-in simulations. We also refer the reader to Fig.~1 in \\citet{oppenheimer18a} where in the middle column they show the column density map for galaxy halo mass of $\\log M_{200} = 12.2$ at $z \\simeq 0.2$, which qualitatively shows similar trends described in \\S\\ref{s-disc-qual-comp}. \n\n\n\\begin{figure}[tbp]\n\\epsscale{1.2}\n\\plotone{f19.pdf}\n\\caption{Comparison of ion column density profiles between Project AMIGA and EAGLE zoom-in simulation of a galaxy with $\\log M_{200}\\simeq 12.1 $ at $z=0$ (from the models presented in \\citealt{oppenheimer18a}). For the EAGLE simulation, the mean column densities are shown. Note that here we only plot the column density profiles out to about \\ensuremath{R_{\\rm vir}}.\n\\label{f-col-vs-rho-eagle}}\n\\end{figure}\n\nIn Fig.~\\ref{f-col-vs-rho-eagle}, we compare the EAGLE and observed column densities as a function of the impact parameter out to \\ensuremath{R_{\\rm vir}}. As in the previous two figures, the blue and gray circles are detections and non-detections in the halo of M31. The green curve in each panel represents the mean column density for each ion as a function of the impact parameter for the EAGLE galaxy with $\\log M_{200} = 12.1$ at $z=0$. In contrast to FIRE-2 or FOGGIE simulations, the EAGLE simulations appear to produce a better agreement between $N$ and $R$ for low and intermediate ions (\\ion{Si}{2}, \\ion{Si}{3}, \\ion{Si}{4}), and \\ion{C}{4}\\ out to larger impact parameters. However, as already noted in \\citet{oppenheimer18a}, this agreement is offset by producing too much column density for the low and intermediate ions at small impact parameters (see, e.g., \\ion{Si}{2}, which is not affected by lower limits, is clearly overproduced at $R\\la 80$ kpc). The flat profile of \\ion{O}{6}, with very little dependence on $R$, is similar to the observations and other models, but overall the EAGLE \\ion{O}{6}\\ column densities are a factor 0.2--0.6 dex smaller than observed. \\citet{oppenheimer18a} (and also \\citealt{oppenheimer16}) already noted that issue from their comparison with the COS-Halos galaxies (see also \\S\\ref{s-coshalos}), requiring additional source(s) of ionization for the \\ion{O}{6}\\ such as AGN flickering \\citep{oppenheimer13a,oppenheimer18} or possibly CRs as shown for the FIRE-2 simulations (see \\citealt{ji19} and above). While the results are shown only to \\ensuremath{R_{\\rm vir}}, as in the other simulations and M31, \\ion{O}{6}\\ is also observed well beyond \\ensuremath{R_{\\rm vir}}\\ in the EAGLE simulations (see Fig.~1 in \\citealt{oppenheimer18a}). \n\n\\subsubsection{Insights from the Observation\/Simulation Comparison}\\label{s-disc-pers-comp}\n\nThe comparison with the simulations shows that the CGM is changing in zoom-in simulations on length scales roughly similar to those observed in M31. The low ions and high ions follow substantially different profiles with radius, in both data and simulations. In the zoom-in simulations described above, the inner regions of the CGM of galaxies are more directly affected by large-scale feedback and recycling processes between the disk and CGM of galaxies. Therefore it is not surprising that the M31 CGM within 100--150 kpc shows a large variation in column density profiles with $R$, a more complex gas-phase structure, and larger peculiar velocities even though the current star formation rate is low. While both accretion and large-scale outflow coexist in the CGM and are responsible for the gas flow properties, stellar feedback is required to produce substantial amount of metals in the CGM at large impact parameters (see Figs.~\\ref{f-col-vs-rho-fire}, \\ref{f-col-vs-rho-foggie}, \\ref{f-col-vs-rho-eagle}). M31 has currently a low SFR, but it had several episodes of bursting star formation in the past \\citep[e.g.,][]{williams17}, likely ejecting a large portion of its metals in the CGM during these episodes.\n\nVarious models simulating different galaxy masses at different epochs, with distinct SFRs or feedback processes can reproduce at some level the diffuse \\ion{O}{6}\\ observed beyond \\ensuremath{R_{\\rm vir}}. All the simulations we have reviewed produce \\ion{O}{6}\\ profiles that are flatter than the low ions and which extend to beyond \\ensuremath{R_{\\rm vir}}\\ with significant column density. While the galaxy halo masses are different, they are all roughly in the range of about $10^{11.5}$--$10^{12.3}$ M$_\\sun$, which is a mass range where their virial temperatures overlap with the temperature at which the ionization fraction of \\ion{O}{6}\\ peaks \\citep{oppenheimer16}. Using the EAGLE simulations, \\citet{oppenheimer16} show that the virial temperature of the galaxy halos can explain the presence of strong \\ion{O}{6}\\ in the CGM of star-forming galaxies with $M_{200} \\simeq 10^{11.5}$--$10^{12.3}$ M$_\\sun$ and the absence of strong \\ion{O}{6}\\ in the CGM of quiescent galaxies that have overall higher halo masses ($M_{200} 10^{12.5}$--$10^{13.5}$ M$_\\sun$) and hence higher virial temperatures, i.e., halo mass, not SFR largely drives the presence of strong \\ion{O}{6}\\ in the CGM of galaxies (cf. \\S\\ref{s-coshalos}). Production the \\ion{O}{6}\\ in volume-filling virialized gas could explain why \\ion{O}{6}\\ is widely spread in the CGM of simulated galaxies and the real M31. Additional ionization mechanisms from cosmic rays (\\citealt{ji19} and see Fig.~\\ref{f-col-vs-rho-fire}) or fluctuating AGNs \\citep{oppenheimer13a,oppenheimer18} can further boost the \\ion{O}{6}\\ production, but halo masses with their virial temperatures close to the temperature at which the ionization fraction of \\ion{O}{6}\\ peaks appear to provide a natural source for the diffuse, extended \\ion{O}{6}\\ in the CGM of $L^*$ galaxies. Conversely, a number of studies have shown that significant \\ion{O}{6}\\ can arise in active outflows, with the outflow column densities varying strongly with the degree of feedback \\citep{hummels13, hafen19a}. Right now, no clear observational test can distinguish \\ion{O}{6}\\ in warm virialized gas and direct outflows. However, any model that attempts to distinguish them will be constrained by the flat profile and low scatter seen by Project AMIGA. \n\nOn the other hand, the cooler, diffuse ionized gas probed predominantly by \\ion{Si}{3}, and also low ions (\\ion{C}{2}, \\ion{Si}{2}) at smaller impact parameters, is not well-reproduced in the simulations. In the FIRE-2 and FOGGIE simulations, the column densities of \\ion{Si}{3}\\ and low ions within $\\la 0.3 R_{200}$ are reasonably matched, but their covering fractions drop sharply and much more rapidly than observed for M31 when $ R > 0.3 R_{200}$. Only near satellite galaxies within $0.3 R_{200}$ do the column densities of these ions increases. This is, however, not a fair comparison as M31 lacks gas-rich satellites within this radius. Furthermore the near unity covering factor of \\ion{Si}{3}\\ out to $1.65 R_{200}$ in the CGM of M31 could not be explained by dwarf satellites anyway. For the EAGLE simulation, this problem is not as extreme as in the other simulations, but EAGLE does overproduce low and intermediate ions in the inner regions ($\\la 0.3 R_{200}$) of the CGM. Possibly maintaining a high resolution out to \\ensuremath{R_{\\rm vir}}\\ would be needed to accurately model the small-scale structures of the cool gas content and preserve it over longer periods of times \\citep{hummels19,peeples19,vandevoort18}. \n\nWhile the observations of M31 and simulations discussed above show some discrepancy, there is an overall trend that is universally observed: when the ionization energies increase from the singly-ionized species (\\ion{Si}{2}, \\ion{C}{2}) to intermediate ions (\\ion{Si}{3}, \\ion{Si}{4}) to \\ion{C}{4}\\ to \\ion{O}{6}, the column density dispersions and dependence on $R$ decrease. While the larger scatter in the low and intermediate column densities compared to \\ion{O}{6}\\ was observed previously \\cite[e.g.,][]{werk13,liang16}, that trend with $R$ was not as obvious owing to a larger scatter at any $R$, in part caused by neighboring galaxies or different galaxy masses \\citep{oppenheimer18a}. This general trend is the primary point of agreement between the observations and simulations, especially considering that the simulations were not tuned to match the CGM properties. This trend most likely arises from the physical conditions of the gas: in the inner regions of the CGM the gas takes on a density that favors the production of the low and intermediate ions. At these densities \\ion{O}{6}\\ would need to be collisionally ionized or distributed in pockets of low-density photoionized gas. In the outer regions of the CGM, the overall gas must have a much lower density where \\ion{O}{6}\\ and weak \\ion{Si}{3}\\ and nearly no singly ionized species can be produced predominantly by photoionization processes. This basic structure of the CGM appears in broad agreement between Project AMIGA, statistical sampling of many galaxies like COS-Halos, and three different suites of simulations. \n\n\\subsection{Implications for the MW CGM}\n\nBased the findings from Project AMIGA, it is likely that the MW has not only an extended hot CGM \\citep{gupta14,gupta17}, but also an extended CGM of cool (\\ion{Si}{2}, \\ion{Si}{3}, \\ion{Si}{4}) and warm-hot (\\ion{C}{4}, \\ion{O}{6}) gas that extends all the way to about 300 kpc (\\ensuremath{R_{\\rm vir}}), and even farther away for the \\ion{O}{6}. In fact, the MW and M31 \\ion{O}{6}\\ CGMs most likely already overlap as it can be seen, e.g., in the CLUES simulations of the Local group \\citep{nuza14} since the distance between M31 and MW is only 752 kpc. \n\nA large covering factor of the CGM of M31 is not detected at high peculiar velocities (see Fig.~\\ref{f-velavgmap}), and in fact beyond 100 kpc, the velocities $v_{\\rm M31}$ are scattered around 20 ${\\rm km\\,s}^{-1}$. Even within 100 kpc, the average velocity is about 90 ${\\rm km\\,s}^{-1}$, which would barely constitute a HVC studied in the MW. In the MW most of the absorption within $\\pm 90$ ${\\rm km\\,s}^{-1}$\\ relative to the systemic velocity of the MW in a given direction is dominated by the disk, i.e., material within a few hundreds of pc from the galactic plane. Because the HVC velocities are high enough to separate them from the disk absorption, HVCs in the MW have been studied for many years to determine the ``halo\" properties of the MW \\citep[e.g.][]{wakker97,putman12,richter17}. However, we know now that the distances of these HVCs, including the predominantly ionized HVCs, are not at 100s of kpc from the MW, but most of them are within 15--20 kpc from the sun \\citep[e.g.][]{wakker01,wakker08,thom08,lehner11a,lehner12}, i.e., in a radius not even explored by Project AMIGA and many other surveys of the galaxy CGM at higher redshifts \\citep[e.g.,][]{werk13,liang14,borthakur16,burchett16}. Only the MS allows us to probe the interaction between the MW and the Magellanic clouds in the CGM of the MW out to about 50--100 kpc \\citep[e.g.,][]{donghia16}. The results from Project AMIGA strongly suggest that the CGM of the MW is hidden in the low velocity absorption arising from its disk (see also \\citealt{zheng15}. To complicate the matter, the column densities of the low, intermediate ions, and \\ion{C}{4}\\ drop substantially beyond 100-150 kpc (see, e.g., Figs.~\\ref{f-coltot-vs-rho}, \\ref{f-coltotsi-vs-rho}). Owing to its strength and little dependence on $R$, \\ion{O}{6}\\ is among the best ultraviolet diagnostic of the extended CGM (see also the recent FOGGIE simulation results by \\citealt{zheng20}). \n\n\\section{Summary}\\label{s-sum}\nWith Project AMIGA, we have surveyed the CGM of a galaxy with an unprecedented number of background targets (43) piercing it at various azimuths and impact parameters, 25 from $0.08 \\ensuremath{R_{\\rm vir}}$ to about $1.1\\ensuremath{R_{\\rm vir}}$ and the additional 18 between $1.18 \\times 10^7$ M$_\\sun$. The total metal mass could be as large as $\\ga 2.5 \\times 10^8$ M$_\\sun$ if the dust and hot X-ray gas are accounted for. Since the total metal mass in the disk of M31 is about $M^{\\rm disk}_{\\rm Z}\\simeq 5\\times 10^8$ M$_\\sun$, the CGM of M31 has at least 50\\% of the present-day metal mass of its disk and possibly much more. \n\\item We estimate the baryon mass of the $\\sim 10^4$--$10^{5.5}$ K gas is $\\ga 3.7 \\times 10^{10}\\,(Z\/0.3\\,Z_\\sun)^{-1}$ M$_\\sun$ at \\ensuremath{R_{\\rm vir}}. The dependence on the largely unknown metallicity of the CGM makes the baryon mass estimate uncertain, but it is broadly comparable to other recent observational results or estimates in zoom-in simulations. \n\\item We study if any of the M31 dwarf satellites could give rise to some of the observed absorption associated with the CGM of M31. We find it is plausible that few absorbers within close spatial and velocity proximity of the dwarfs could be associated with the CGM of dwarfs if they have a gaseous CGM. However, these are Sph galaxies, which have had their gas stripped via ram-pressure and unlikely to have much gas left in their CGM. And, indeed, none of the properties of the absorbers in close proximity to these dwarf galaxies show any peculiarity that would associate them to the CGM of the satellites rather than the CGM of M31.\n\\end{enumerate}\n\n\n\\section*{Acknowledgements}\nWe thank David Nidever for sharing his original fits of the MS \\ion{H}{1}\\ emission and Ben Oppenheimer for sharing the EAGLE simulations shown in Fig.~\\ref{f-col-vs-rho-eagle}. Support for this research was provided by NASA through grant HST-GO-14268 from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-26555. CAFG and ZH were also supported by NSF through grants AST-1517491, AST-1715216, and CAREER award AST-1652522; by NASA through grants NNX15AB22G and 17-ATP17-0067; by STScI through grants HST-GO-14681.011 and HST-AR-14293.001-A; and by a Cottrell Scholar Award from the Research Corporation for Science Advancement. Based on observations made with the NASA-CNES-CSA Far Ultraviolet Spectroscopic Explorer, which was operated for NASA by the Johns Hopkins University under NASA contract NAS5-32985.\n\n\\software{Astropy \\citep{price-whelan18}, emcee \\citep{foreman-mackey13}, Matplotlib \\citep{hunter07}, PyIGM \\citep{prochaska17a}}\n\n\\facilities{HST(COS); HST(STIS); FUSE}\n\n\\bibliographystyle{aasjournal}\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{ Introduction}\n\n\n\n The process $e^-e^+\\to W^-W^+$ has been studied theoretically\n and experimentally since a long time, as it provides sensitive\n tests of the gauge structure of the\n electroweak interactions \\cite{Boehm, Beenakker,Hahn, Denner}, and checks the possible\n presence of non standard new physics (NP) contributions.\n A detail history of the subject and a list of references may be seen in \\cite{Denner}.\n\n First experimental studies of this process have been done at LEP2 \\cite{LEP2WW}.\n No signal of departures from SM has been found, but the accuracy is not sufficient\n to eliminate the possibility of NP effects at a high scale.\n\n LHC studies involving production of W pairs also exist; but their detailed studies\n require a difficult event analysis, because of various sources of background\n \\cite{LHCWW}.\n\n Future high energy $e^-e^+$ colliders are therefore deeply desired,\n in order to provide fruitful information about this subject \\cite{ILC,CLIC}.\n\n From the theoretical side, the present situation of an 1loop electroweak (EW)\n order analysis, aiming e.g. at searching\n for any non standard effects, is quite complex.\n This is already true at the SM level, and if one includes\n SM extensions, like e.g. SUSY, the sensitivity to any benchmark\n choice has to be considered. Particularly for\n amplitudes involving longitudinal $W$'s, the numerical situation is more difficult,\n because of the huge cancelations taking place. In both the SM and SUSY cases,\n very lengthy numerical codes are required to describe\n the complete 1loop EW contribution; see e.g. \\cite{Hahn, Denner}.\n\n The aim of the present paper is to call attention to the fact that at high energies,\n the 1loop electroweak (EW) corrections to the helicity amplitudes for $e^-e^+\\to W^-W^+$,\n acquire very simple forms, in both, the SM and MSSM cases. To establish them we have\n done a complete calculation of the 1loop diagrams and then taken the high energy,\n fixed angle, limit using \\cite{asPV}. The soft photon bremsstrahlung can then\n be added as usual \\cite{Boehm, Beenakker,Hahn, Denner}.\n\n\n Our procedure is the same as the one used previously for other 2-to-2 processes,\n leading to the \"supersimple\" (sim) 1loop EW expressions for the dominant high energy\n helicity conserving (HC) amplitudes; the helicity violating (HV)\n ones are quickly vanishing\\footnote{The notations HC and HV are fully\n defined in the next section.}\n \\cite{super, ttbar}. We find very simple and quite accurate\n expressions for the high energy HC amplitudes, in both the SM and MSSM frameworks, which\n nicely show their relevant dynamical contents.\n\n The use of this description, which clearly indicates the relevant physical parameters,\n should very much simplify the analysis of the experimental results. Particularly because,\n its accuracy turns out to be\n sufficient for distinguishing 1loop SM (or MSSM) effects,\n from e.g. various types of additional New Physics contributions, like AGC couplings\n or $Z'$ exchange; see for example \\cite{Andreev}.\n\n The content of the paper is the following. In Section 2 we present\n the various properties of the high energy $e^-e^+\\to W^-W^+$ amplitudes,\n with special attention to their helicity conservation\n (HCns) property \\cite{heli1, heli2}. The explicit supersimple expressions\n are discussed in the later part of Section 2 and in Appendix A.\n In Section 3 we present the energy and angular dependencies\n of the cross sections, for polarized and unpolarized electron beams,\n in either SM or MSSM. And subsequently, we compare these SM or MSSM contributions\n to those due to anomalous gauge couplings (AGC) or $Z'$ effects; both given\n in Appendix B. We find that the accuracy of the supersimple\n expressions is sufficient for distinguishing these various types\n of contributions. Thus, they may be used instead of the complete 1loop results.\n The conclusions summarize these results.\\\\\n\n\n\n\n\n\n\\section{Supersimplicity in $e^-e^+\\to W^-W^+$}\n\nThe process studied, to the 1loop Electroweak (EW) order, is\n\\bq\ne^-_\\lambda (l) ~ e^+_{\\lambda'} (l') \\to W^-_\\mu (p)~ W^+_{\\mu'}(p')~~, \\label{process}\n\\eq\nwhere $(\\lambda, \\lambda')$ denote the helicities of the incoming $(e^-, e^+)$ states,\nand $(\\mu, \\mu')$ the helicities of the outgoing $(W^-, W^+)$. The corresponding\nmomenta are denoted as $(l,l',p,p')$. Kinematics are defined through\n\\bqa\n && s=(l+l')^2=(p+p')^2 ~~,~~ t=(l-p)^2=(l'-p')^2 ~~,~~ \\nonumber \\\\\n && p_W=\\sqrt{{s\\over4}-m^2_W}~~,~~ \\beta_W=\\sqrt{1-\\frac{4 m_W^2}{s}} ~~, \\label{kinematics}\n\\eqa\nwhere $p_W, \\beta_W$ denote respectively the $W^\\mp$ three-momentum and velocity in the\n$W^-W^+$-rest frame. Finally, the angle between the incoming $e^-$\nmomentum $l$ and the outgoing $W^-$ momentum $p$, in the center of mass frame,\nis denoted as $\\theta$.\n\nDue to the smallness of the electron mass,\nnon-negligible amplitudes at high energies only appear for $\\lambda=-\\lambda'=\\mp 1\/2$.\nThe helicity amplitudes for\nthis process are therefore determined by three helicity indices and denoted\nas $F_{\\lambda,\\mu,\\mu'}(\\theta)$, where $(e^-, W^-)$\nare treated as particles No.1, and $(e^+,W^+)$ as particles No.2, in the standard\nJacob-Wick notation \\cite{JW}.\n\nAssuming CP invariance, we obtain the constraint\n\\bq\nF_{\\lambda, \\mu,\\mu'}(\\theta)=F_{\\lambda, -\\mu',-\\mu}(\\theta) ~~, \\label{CP-cons}\n\\eq\nwhich means that the process is described by just 12 independent helicity\namplitudes.\n\nAt high energy, the helicity conservation (HCns) rule implies\nthat only the amplitudes satisfying\n\\bq\n\\lambda +\\lambda'=0= \\mu + \\mu' ~~, \\label{heli-cons}\n\\eq\ncan dominate \\cite{heli1,heli2}. These are the\n helicity conserving (HC) amplitudes, which explicitly are\n\\bq\n F_{\\mp-+}~~,~~ F_{\\mp+-}~~,~~ F_{\\mp 00} ~~.~~ \\label{6HC-amp-list}\n\\eq\nThe purely left-handed $W$ couplings though, forces the HC amplitudes\n\\bq\nF_{++-}~~,~~F_{+-+}~~, \\label{R-HC-amp-list}\n\\eq\nto vanish at Born-level and be very small at 1loop.\nThus, only four leading HC helicity amplitudes remain at high energy,\nnamely\n\\bq\nF_{--+}~~,~~ F_{-+-}~~,~~ F_{\\pm00}~~.~~ \\label{4HC-amp-list}\n\\eq\nThe remaining amplitudes, which violate (\\ref{heli-cons}), are termed as\nhelicity violating (HV) ones. Explicitly these are\n\\bq\nF_{-0+}~~,~~F_{---}~~,~~F_{-+0}~~,~~F_{+0+}~~,~~F_{+--}~~,~~F_{++0}~~,~~ \\label{HV-amp-list}\n\\eq\n and are expected to be suppressed like $m_W\/\\sqrt{s}$ ~ or $m^2_W\/ s$, at\n high energy.\\\\\n\n\n\\subsection{Born contribution to the helicity amplitudes}\n\n We next turn to the Born contribution to the HC and HV amplitudes in (\\ref{4HC-amp-list})\n and (\\ref{HV-amp-list}) respectively. The relevant diagrams involve\n neutrino exchange in the t-channel and photon+Z exchange in the s-channel. The resulting\n amplitudes satisfy the HCns constraints \\cite{heli1, heli2}.\n In the usual Jacob and Wick convention \\cite{JW}, their exact expressions are:\\\\\n\n\n\\noindent\n{\\bf Transverse-Transverse (TT) amplitudes ($\\mu,\\mu'=\\pm1$)}\n\nUsing (\\ref{kinematics}), we find\n\\bqa\nF^{\\rm Born}_{\\lambda \\mu \\mu' }&=& {se^2\\sin\\theta\\over16ts^2_W}\n\\delta_{\\lambda,-}\n\\left \\{ \\mu+\\mu'+\\beta_W(1+\\mu\\mu')-2\\mu(1+\\mu'\\cos\\theta) \\right \\}\n\\nonumber\\\\\n&&+{se^2\\over4}\\left [{Q_e\\over s}+{a_{eL}\\delta_{\\lambda,-}+a_{eR}\\delta_{\\lambda,+}\\over\n2s^2_W(s-m^2_Z)}\\right ](1+\\mu\\mu')(2\\lambda)\\beta_W\\sin\\theta ~~,\n\\label{FBorn-TT}\n\\eqa\nwith\n\\bq\nQ_e=-1~~,~~a_{eL}=1-2s^2_W~~,~~a_{eR}=2s^2_W~~, \\label{e-couplings}\n\\eq\ndetermining the electron charge, and the $Z$ left- and right-couplings.\nBecause of the purely left-handed $W$ coupling, Eqs.(\\ref{FBorn-TT}) leads to\n\\bq\nF^{\\rm Born}_{+{1\\over2},\\mu, -\\mu}=0 ~~, \\label{R-HC-TT-Born}\n\\eq\nas already said just after (\\ref{R-HC-amp-list}).\nIn addition, (\\ref{FBorn-TT}) leads at high energy to\n\\bq\nF^{\\rm Born}_{\\lambda\\mu\\mu} \\to 0 ~~, \\label{Born-asym-TT-HV}\n\\eq\nin agreement with HCns \\cite{heli1, heli2}, and\n\\bq\nF^{\\rm Born}_{-{1\\over2}\\mu-\\mu}\\to - {e^2 \\sin\\theta (\\mu-\\cos\\theta)\n\\over 4s^2_W (\\cos\\theta-1)}~~. \\label{Born-asym-TT-HC}\n\\eq\nThis confirms that the first two HC Born amplitudes in (\\ref{4HC-amp-list}),\ngo to constants, asymptotically.\\\\\n\n\n\\noindent\n{\\bf Transverse-Longitudinal (TL) and Longitudinal-Transverse (LT)\\\\\namplitudes ($\\mu=\\pm1,\\mu'=0$, $\\mu=0,\\mu'=\\pm1$)}\n\nUsing again (\\ref{kinematics}), we obtain\n\\bqa\nF^{\\rm Born}_{\\lambda\\mu 0}&=& {s\\sqrt{s}e^2\\over8\\sqrt{2}m_Wts^2_W}\\delta_{\\lambda,-}\n\\left \\{ (\\beta_W-\\cos\\theta)(1-\\mu\\cos\\theta)- {2m^2_W\\over s}(\\mu-\\cos\\theta)\\right \\}\n\\nonumber\\\\\n&&-{s\\sqrt{s}e^2\\over2\\sqrt{2}m_W}\\left [{Q_e\\over s}+{a_{eL}\\delta_{\\lambda,-}\n+a_{eR}\\delta_{\\lambda,+}\\over\n2s^2_W(s-m^2_Z)} \\right ]\\beta_W(1+2\\lambda\\mu\\cos\\theta)~~, \\label{FBorn-TL} \\\\\nF^{\\rm Born}_{\\lambda 0 \\mu'}&=& {s\\sqrt{s}e^2\\over8\\sqrt{2}m_Wts^2_W}\\delta_{\\lambda,-}\n\\left \\{(\\beta_W-\\cos\\theta)(1+\\mu'\\cos\\theta)- {2m^2_W\\over s}(\\mu'+\\cos\\theta) \\right \\}\n\\nonumber\\\\\n&&-{s\\sqrt{s}e^2\\over2\\sqrt{2}m_W}\n\\left [{Q_e\\over s}+{a_{eL}\\delta_{\\lambda,-}+a_{eR}\\delta_{\\lambda,+}\\over\n2s^2_W(s-m^2_Z)} \\right ]\\beta_W(1-2\\lambda\\mu'\\cos\\theta)~~. \\label{FBorn-LT}\n\\eqa\n\nThe amplitudes in (\\ref{FBorn-TL}, \\ref{FBorn-LT}) are both HV, and at high energies\nthey are quickly suppressed like $m_W\/ \\sqrt{s}$.\\\\\n\n\\noindent\nThe {\\bf Longitudinal-Longitudinal (LL) amplitudes ($\\mu=0,\\mu'=0$)}\nare\n\\bqa\nF^{\\rm Born}_{\\lambda 00}&=& {se^2\\sin\\theta\\over16ts^2_W}\\delta_{\\lambda,-}\n\\left \\{ {s\\over m^2_W}(\\beta_W-\\cos\\theta)+2\\beta_W \\right\\}\n\\nonumber\\\\\n&&+{(2\\lambda)s^2e^2\\over8m^2_W}\n\\left [{Q_e\\over s}+{a_{eL}\\delta_{\\lambda,-}+a_{eR}\\delta_{\\lambda,+}\\over\n2s^2_W(s-m^2_Z)} \\right ]\\beta_W(3-\\beta_W^2)\\sin\\theta~~, \\label{FBorn-LL}\n\\eqa\nwhere (\\ref{kinematics}) have again been used.\nAt high energy, keeping terms to order $m^2_Z\/ s$ and $m^2_W\/ s$, one gets\n\\bqa\nF^{\\rm Born}_{-{1\\over2}00} & \\to & - {e^2\\over8s^2_Wc^2_W} \\sin\\theta ~~,\n\\nonumber \\\\\nF^{\\rm Born}_{+{1\\over2}00} & \\to & {e^2\\over4c^2_W} \\sin\\theta ~~, \\label{Born-asym-LL-HC}\n\\eqa\nwhich together with (\\ref{Born-asym-TT-HC}) confirm that all Born HC amplitudes in\n(\\ref{4HC-amp-list}), go to constants, asymptotically.\nOn the contrary, all six HV amplitudes listed in (\\ref{HV-amp-list}) vanish,\n in this limit.\\\\\n\n\nThe Born level properties of the helicity amplitudes are illustrated\nin Figs.\\ref{HV-HC-Born-amp}. The two HC amplitudes listed in (\\ref{R-HC-amp-list}),\nare not shown, since they vanish, when coefficients proportional to the\nelectron-mass are neglected.\\\\\n\n\n\n\n\n\n\n\\subsection{ Helicity amplitudes to the 1loop electroweak (EW) order.}\n\n\nThe relevant contributions come from up and down triangle diagrams in the t-channel;\ninitial and\nfinal triangle diagrams in the s-channel; direct, crossed and\ntwisted box diagrams; specific triangles involving a 4-leg gauge boson\ncouplings; and finally neutrino, photon and Z self-energies.\nCounter terms in the Born contributions, which help canceling the\ndivergences induced by self-energy and triangle diagrams, are also included, leading to the so-called\non-shell renormalization scheme \\cite{OS}.\n\n\n\nSuch type of computations have already been done; see for example \\cite{Hahn, Denner}.\nBut our aim here is to look at the specific properties of each helicity\namplitudes, and to derive simple high energy expressions for the\nHC ones. For this reason we repeated the complete calculation of the 1loop EW corrections\nand then computed their high energy expressions that we call supersimple (sim),\nusing the expansion of \\cite{asPV}. A special attention is paid to the\nvirtual photon exchange diagrams leading to infrared singularities (when $m_\\gamma\\to 0$)\nwhich are then cancelled by the addition of the soft photon bremsstrahlung contribution. The sim\nexpressions are given (in Appendix A) in the two possible choicess , arbitrary small $m_\\gamma$\nvalue, or $m_\\gamma=m_Z$ which can be considered as \"small\" at high energies. This second\nchoice, also used in previous studies \\cite{super, ttbar}, has the advantage of\nleading to even simpler expressions as we can see in Appendix A.\\\\\n\nAs already said and numerically shown below, the HV amplitudes amplitudes in (\\ref{HV-amp-list})\n are negligible at high energies. Only the four HC amplitudes appearing in (\\ref{4HC-amp-list})\n are relevant there. Turning to them, we present in Appendix A.1\n the very simple sim expressions\n for the TT amplitudes $F_{--+}$, ~$F_{-+-}$; while the corresponding expressions for the LL\namplitudes $F_{-00}$,~ $F_{+00}$ appear in Appendix A.2.\nThe results (\\ref{simSM--+}, \\ref{simSM-+-}, \\ref{simSM+00}, \\ref{simSM-00})\ngive the SM predictions,\nwhile (\\ref{simMSSM--+}, \\ref{simMSSM-+-}, \\ref{simMSSM+00}, \\ref{simMSSM-00})\ngive the MSSM ones, always corresponding to the $m_\\gamma=m_Z$ choice.\nThe corrections to be done to them in order to obtain the general result for any $m_\\gamma$,\nappear in (\\ref{deltaFTT}, \\ref{deltaF00}).\n\n\nFor deriving these, we start from the complete 1loop EW results\nin terms of Passarino-Veltman (PV) functions \\cite{PV},\nand then use their high energy expansions given in \\cite{asPV}.\nFor the TT amplitudes $F_{--+}$, ~$F_{-+-}$, the derivation is quite\nstraightforward.\n\nFor the two LL amplitudes $F_{-00}$, $F_{+00}$ though, the derivation is very delicate,\nbecause of huge gauge cancelations\namong contributions exploding like\\footnote{ Particularly for neutralinos, this demands\na very accurate determination of their mixing matrices, like the one supplied e.g. by \\cite{LeMouel}.}\n $s\/ m^2_W$. Such cancelations also occur at Born level,\nbetween t- and s-channel terms. But at 1loop level, the situation is\nmuch more spectacular, because more diagrams are involved.\nTechnically, the derivation of the limiting expressions can be facilitated by using\nthe equivalence theorem and looking at the Goldstone process $e^-e^+\\to G^-G^+$\n\\cite{equivalence}.\\\\\n\n\n\n\n\n\n\nWe next turn to the infrared divergencies implied by the presence of\n $m_\\gamma$ in the $e^-e^+\\to W^-W^+$ amplitudes. As usual, these are canceled at the cross section\nlevel by adding to the 1loop EW results for $d\\sigma (e^-e^+\\to W^-W^+)\/d\\Omega$, the Born-level\n cross section describing the soft photon bremsstrahlung, given by\n\\bq\n\\frac{d\\sigma_{\\rm brems}(e^-e^+\\to W^-W^+\\gamma)}{d\\Omega}=\n \\frac{d\\sigma^{\\rm Born} (e^-e^+\\to W^-W^+)}{d\\Omega} \\delta_{\\rm brems}(m_\\gamma,\\Delta E)~~,\n \\label{brems-sigma}\n\\eq\nwhere $\\delta_{\\rm brems}(m_\\gamma,~\\Delta E)$ is given by\\footnote{Parameter $\\lambda$ in \\cite{Boehm} corresponds to our $m_\\gamma$} Eqs. (5.18) in \\cite{Boehm}, while\n $\\Delta E$ describes the highest energy of the emitted unobservable\nsoft photon, satisfying\n\\bq\nm_\\gamma \\leq \\Delta E \\ll \\sqrt{s} ~~. \\label{brems-kin}\n\\eq\nThe only requirement for this cancelation to happen is that\n $m_\\gamma$ is {\\it small}; i.e. that terms proportional to a power of $m_\\gamma$\n (not inside a high energy logarithm) are always negligible. Under these condition, any\n $m_\\gamma$-dependence cancels out in the sum $d\\sigma (e^-e^+\\to W^-W^+)\/d\\Omega$ plus\n $d\\sigma_{\\rm brems}\/d\\Omega$.\n\n But, at the high energies of $\\sqrt{s} \\gg m_Z$ we are interested in,\n the $Z$ mass is also {\\it small}; since\n any such $m_Z$ coefficient is necessarily suppressed by an energy denominator. In other words,\n since the infrared $m_\\gamma$ effects cancel out in the cross section including bremsstrahlung (\\ref{brems-sigma}) contribution, they will also cancel in the special case $m_\\gamma=m_Z$.\n As already said we made this choice because it leads to the simplest expressions.\n The illustrations given below correspond to it.\n\n In order to obtain the (infrared sensitive) unpolarized cross section\n $d\\sigma (e^-e^+\\to W^-W^+)\/d\\Omega$ from the experimental data, one has obviously to subtract the\n bremsstrahlung contribution. Consequently, the difference between the values of this cross section\n regularized at an arbitrary $m_\\gamma$ or at $m_\\gamma=m_Z$, for the same $\\Delta E$,\n is given by\n \\bqa\n && \\frac{d\\sigma (e^-e^+\\to W^-W^+) }{d\\Omega}\\Big |_{m_\\gamma} -\n \\frac{d\\sigma (e^-e^+\\to W^-W^+) }{d\\Omega}\\Big |_{m_\\gamma \\to m_Z} \\nonumber \\\\\n && = \\frac{d\\sigma^{\\rm Born}}{d\\Omega}\n {\\alpha\\over\\pi}\\left ( \\ln {m_Z\\over m_\\gamma} \\right )\n \\left ( 4 -2\\ln{s\\over m^2_e} +4\\ln{m^2_W-u \\over m^2_W-t}\n +2{s-2m^2_W\\over s\\beta_W}\\ln{1-\\beta_W\\over 1+\\beta_W} \\right ) ; \\label{mgamma-mZ-effect}\n \\eqa\n see our eqs.(\\ref{kinematics}, \\ref{brems-sigma}) and eq.(5.18) of \\cite{Boehm}.\n If one wants to keep the usual choice of an arbitrary small $m_\\gamma$ in the bremsstrahlung cross section,\n one would have to use our extended sim expressions given in (\\ref{deltaFTT}, \\ref{deltaF00})\n of Appendix A.\\\\\n\n\nTurning now to the numerical illustrations, we first check\nthat all HV amplitudes quickly vanish at high energy, in both MSSM and SM \\cite{heli1, heli2}.\nFor the MSSM case, we use benchmark S1 of \\cite{bench}, where the EW\n scale values of all squark masses are at the 2 TeV level, $A_t=2.3$ TeV,\n the slepton masses are at $0.5$ TeV, and the remaining mass parameters (in TeV) are\n \\bq\n \\mu =0.4~~,~~ M_1=0.25 ~~,~~ M_2=0.5 ~~,~~ M_3=2 ~~, \\label{bench-param}\n \\eq\nwhile $\\tan\\beta=20$.\nSuch a benchmark is consistent with present LHC constraints \\cite{bench}.\nAll MSSM results shown in this paper, are using this benchmark.\nSimilar results are also obtained for other LHC-consistent MSSM benchmarks, like those\nlisted e.g. in the Snowmass suggestion \\cite{Snowmass},\nor the very encouraging cMSSM ones given in \\cite{Konishi}.\n\n\nComparing the SM and MSSM results in Figs.\\ref{HV-full-amp},\nwe see that for all HV amplitudes, the purely supersymmetric\ncontribution mostly cancel the (already suppressed)\npure SM ones; this is more spectacular for energies above the SUSY\nscale. Thus, Figs.\\ref{HV-full-amp} indeed show that the six HV amplitudes listed\nin (\\ref{HV-amp-list}), are quickly suppressed in MSSM, as well as in SM.\\\\\n\n\n\nWe next turn to the high energy description of the four\nleading (HC) amplitudes listed in (\\ref{4HC-amp-list}).\n As it is shown in Figs.\\ref{HC-full-amp}, the supersimple (sim) approximations to them,\n follow very closely the complete expressions for the 1loop electroweakly corrected\n amplitudes, in both SM and MSSM.\nFor the TT amplitudes $F_{--+}$, $F_{-+-}$, this appears in the upper panels\nof Figs.\\ref{HC-full-amp}, for SM and the MSSM benchmark mentioned above.\nThe corresponding numerical illustrations for the LL HC amplitudes are shown in the lower\npanels. These results indicate that all four 1loop predictions; i.e. the complete SM and\nMSSM results, as well their sim SM and sim MSSM approximations,\nare very close to each other at high energies. Moreover, a comparison of\nFigs.\\ref{HV-full-amp} and \\ref{HC-full-amp} immediately shows that soon above 0.5TeV\nthe HC amplitudes in (\\ref{4HC-amp-list}) are much larger than all other ones.\n\nThere are two main conclusions we draw from this, for energies up to a TeV or so:\nThe first is that the process\n $e^-e^+\\to W^-W^+$ is rather insensitive to MSSM contributions, for benchmarks consistent\nwith the present SUSY constraints \\cite{bench, Snowmass, Konishi}. And the second conclusion\nis that (\\ref{simSM--+}, \\ref{simSM-+-}, \\ref{simSM+00}, \\ref{simSM-00})\nprovide a true description of the sources of the relevant dynamics.\\\\\n\n\n\n\\section{Application to the $e^-e^+\\to W^-W^+$ observables}\n\n The observables we study here are the unpolarized differential cross sections\n\\bq\n{d\\sigma\\over d\\cos\\theta}={\\beta_W\\over 128\\pi s}\n\\Sigma_{\\lambda \\mu \\mu'}|F_{\\lambda \\mu \\mu'}(\\theta)|^2 ~~, \\label{dsigma-unpol}\n\\eq\nas well as the polarized differential cross sections using right-handedly polarized\n electron beams $e^-_R$,\n\\bq\n{d\\sigma^R\\over d\\cos\\theta}={\\beta_W\\over 64\\pi s}\n\\Sigma_{\\mu \\mu'}|F_{+{1\\over2},\\mu \\mu'}(\\theta)|^2 ~~, \\label{dsigma-pol}\n\\eq\nwhere (\\ref{kinematics}) is used.\n\nThese cross sections are shown in Figs.\\ref{sigmas}, where the complete\n1loop EW order SM results are compared to the corresponding\nsupersimple (sim) ones. The later are constructed\nby using the expressions of Appendix A for the HC amplitudes, while the HV amplitudes\nare approximated by the quickly vanishing Born contributions\\footnote{If instead\nwe had completely ignored the HV amplitudes in the sim cross sections, then\nappreciable differences would only appear for energies below 1TeV,\nparticularly for the $e^-_R$ cross sections.} in\n(\\ref{FBorn-TT}, \\ref{FBorn-TL}, \\ref{FBorn-LT}). As shown in\nFigs.\\ref{sigmas}, the sim results very closely follow the SM ones.\n\n In addition, we show in the same figures, how the complete 1loop SM results are changed,\n when an anomalous contribution is added like e.g. AGC1 or AGC2, respectively\ndefined by (\\ref{AGC1-choice}) or (\\ref{AGC2-gLgR}, \\ref{AGC2-choice}) of Appendix B.1; or\na $Z'$-effect defined Appendix B.2.\n\nLeft panels in Figs.\\ref{sigmas} present results for\nthe unpolarized $e^-e^+$ cross sections; while right panels show results for the\n$e^-_Re^+$ cross sections involving a right-handedly polarized electron.\n\nThe upper panels present the energy dependencies at $\\theta =30^o$;\nwhile the middle (lower) panels indicate the angular dependencies at\n$\\sqrt{s}=1$TeV ($\\sqrt{s}=5$TeV).\n\nIn all cases, the supersimple (sim) description is very good.\nNo MSSM or sim MSSM illustrations are given, since they are very close to the corresponding\n SM ones; at the 1-2\\% level, for benchmarks consistent with the current LHC\n constraints \\cite{bench, Snowmass, Konishi}.\n\n In other words, at the scale of Figs.\\ref{sigmas},\n the SM and MSSM results for \\cite{bench}, would coincide.\nSuch a weakness of the pure supersymmetric\n contributions, has been already noticed in previous analyses,\n \\cite{Hahn}. Because of the different\nmass scales of the supersymmetric partners, at a given energy,\nthe absolute magnitudes of the SUSY 1loop effects\nmay differ notably. But relative to the SM contributions (Born + 1 loop),\nthey always remain very small.\n\nConcerning the relevant dynamics for the unpolarized $e^-e^+$ cross sections,\nwe note that, at forward angles, they are dominated by the left-handed-$e^-$ TT\namplitudes.\\\\\n\nFor specific experimental studies of the LL amplitudes,\none can either make a final polarization analysis of the $W^\\mp$-decays;\nor use a right-handedly polarized $e^-$-beam, so that\nthe usual TT amplitudes do not contribute.\nIn the right panels in Figs.\\ref{sigmas}, we show the energy and angular dependencies\nof these $e^-_R e^+$ cross sections.\n\n These LL studies are probably the best place to search for\nanomalous contributions, like those from the AGC effects presented in Section B.1.\nAs seen in (\\ref{FAGC-TT}-\\ref{FAGC-LL}), such\nAGC contributions do not appear in the HC TT amplitudes;\nbut they do appear in the HC LL amplitudes, as well as\nin all the HV ones (TT, TL and LT).\nThis is a remarkable property that should be checked by a careful\nanalysis of experimental signals.\n\n The most simple-minded implication of AGC physics is presented\n by the AGC1 model in Figs.\\ref{HV-full-amp}, \\ref{sigmas}, \\ref{HC-full-NPamp},\n where the parameters in Appendix B.1 are fixed as in (\\ref{AGC1-choice}).\n In this case, the anomalous contributions to the LL\namplitudes increase like $ s\/ m^2_W$,\ncausing a strong increase of the cross sections\nwith the energy.\n\nSuch a strong increase may be tamed though, by the existence of\n scales $M$ in the various anomalous couplings, which transforms them to\n form factors decreasing like $M^2\/(s+ M^2)$.\n\nAnother way of taming the above strong AGC increase, is by the addition\nof new exchanges in the\nt-channel, such that one gets cancelations between s- and t-channel contributions,\n like in the Born SM case.\nA purely ad-hoc phenomenological solution of this kind is given by AGC2, presented\nin Appendix B.1, and determined by\n(\\ref{AGC2-gLgR}, \\ref{AGC2-choice}). In the effective lagrangian framework\nmany such possibilities exist; see e.g. \\cite{ef-Lagrangian}.\n\n\n The AGC1, AGC2 results of in Figs.\\ref{HV-full-amp}, \\ref{HC-full-NPamp}, \\ref{sigmas},\nshow various amplitudes and cross-sections where such anomalous\nbehaviours may be seen and compared to the SM and MSSM results.\n\nPresent experimental constraints on fixed AGC couplings,\n from LEP2 \\cite{LEP2WW} are of the order of\n $\\pm 0.04$.\n From LHC \\cite{LHCWW}, they are of the order of $\\pm 0.1$;\n compare with (\\ref{AGC1-choice}, \\ref{AGC2-choice}).\n These values are much larger than the uncertainties of\n our description.\\\\\n\nAnother type of anomalous contribution is a $Z'$ exchange in the s-channel;\n see \\cite{Andreev} and Appendix B2.\n Here also one can impose a good high energy behaviour to the\nLL and LT amplitudes. A simple solution is a $Z-Z'$ mixing\nsuch that, the total s-channel contribution at high energy, cancels the standard\nt-channel exchange at Born-level.\n Figs.\\ref{HV-full-amp}, \\ref{sigmas}, \\ref{HC-full-NPamp} show the\n behaviours of the various amplitudes\n and cross-sections under the presence of such $Z'$ contributions,\n and compare them to the corresponding SM and MSSM ones.\\\\\n\n>From the above illustrations one sees that our supersimple expressions\nare sufficiently accurate to distinguish 1loop SM or MSSM corrections\nfrom such New Physics. But these are examples. More elaborated\nanalyses could of course be done,\n for example in the spirit of \\cite{Andreev}; still remaining in a\n sensitivity region where supersimple expressions sufficiently describe\nSM physics. The existence of this possibility constitutes an important\n motivation for supersimplicity. \\\\\n\n\n\n\\section{Conclusions}\n\nIn this paper we have analyzed the high energy behaviour\nof the 1loop EW corrections to the\n$e^-e^+\\to W^- W^+$ helicity amplitudes. And we have\nverified that soon above threshold,\nthe four helicity conserving amplitudes in (\\ref{4HC-amp-list}) are much larger than\nall other ones, in both SM and MSSM.\n\nWe have then established the so-called supersimple (sim) expressions for\nthe HC amplitudes in (\\ref{4HC-amp-list}), both in SM and in MSSM.\nThese expressions (explicitly\nwritten in Appendix A) are really simple and provide a panoramic view of\nthe dynamics; i.e., of the fermion, gauge and higgs exchanges,\nand (in the supersymmetric part) of the sfermion, additional higgses,\ncharginos and neutralinos exchanges.\n\nMoreover, the accuracy of these sim expressions\nis sufficient to allow their use in order to search\nfor possible new physics contributing in addition to SM or MSSM.\nIn other words, sim expressions may be used to avoid the enormous codes\nneeded when using the complete 1loop expressions.\nThus, analyses done by only using Born terms, can be easily upgraded\nto the 1loop EW order.\n\nIn previous work \\cite{super, ttbar},\nwe have emphasized the peculiar simplicity arising in the MSSM case.\nHowever in the process $e^-e^+\\to W^- W^+$, the purely\nsupersymmetric contributions are rather small. So even in the purely\nSM case, we get simple accurate expressions, that are valid at LHC energies.\n\nAt present there is no signal of supersymmetry at LHC. The discovery of the Higgs\nboson at 125 GeV is nevertheless a source of questions about the\npossibility of various kinds of New Physics effects \\cite{Altarelli}.\nThe process $e^-e^+\\to W^- W^+$ is a typical place where such\neffects can be looked for. For our illustrations,\nwe have taken the cases of AGC or $Z'$ contributions,\nwhich have been often discussed. Other possibilities may of course\nbe tried \\cite{Andreev}.\n\nOur supersimple\nexpressions are intended to help differentiating such New Physics effects from\nstandard or supersymmetric corrections, in a way which is as simple as possible,\nwhile at the same time allowing us to directly see the responsible dynamics. \\\\\n\n\n\n\n\\vspace*{1cm}\n\n\n\\renewcommand{\\thesection}{A}\n\\renewcommand{\\theequation}{A.\\arabic{equation}}\n\\setcounter{equation}{0}\n\n\n\n\n\\section{Appendix: Supersimple expressions for the 4 HC amplitudes}\n\n\nThe purpose of this Appendix is to present the {\\it supersimple} (sim) expressions\nfor the four leading HC amplitudes listed in (\\ref{4HC-amp-list}).\nThe procedure is valid for of any 2-to-2 process at 1loop EW order,\nin either MSSM or SM, provided the Born contribution is non-negligible.\nAnd it is based on the fact that the\n helicity conserving (HC) amplitudes, are the only relevant\n ones at high energy \\cite{heli1,heli2}.\n\n To derive these sim expressions, we start from a complete 1loop EW order calculation,\n and then take the high energy\nlimit using \\cite{asPV}. As in the analogous cases studied in \\cite{super, ttbar},\nthese expressions\nconstitute a very good high energy approximation, to the HC amplitudes, renormalized\non-shell \\cite{OS}.\n\n\nApart from possible additive constants,\nthese sim expressions consist of linear combinations of just four forms \\cite{super, ttbar}.\nFor $e^-e^+\\to W^-W^+$, the structure of these forms simplifies as\n\\bqa\n&& \\overline{\\ln^2x_{Vi}} = \\ln^2x_{V}+4L_{aVi} ~~,~~\nx_V \\equiv \\left (\\frac{-x-i\\epsilon}{m_V^2} \\right )~~, \\label{Sud-ln2-form} \\\\\n&& \\overline{\\ln x_{ij}} = \\ln x_{ij}+b^{ij}_0(m_a^2)-2 ~~ , ~~\n\\ln x_{ij}\\equiv \\ln \\frac{-x-i\\epsilon}{m_im_j} ~~, \\label{Sud-ln-form} \\\\\n&& \\overline{\\ln^2r_{xy}}=\\ln^2r_{xy}+ \\pi^2 ~~~,~~~\nr_{xy} \\equiv \\frac{-x-i\\epsilon}{-y-i\\epsilon} ~~~~, \\label{ln2r-form} \\\\\n&& \\ln r_{xy} ~~~,~~ \\label{lnr-form}\n\\eqa\nwhere $(x,y)$ denotes any two of the Mandelstam variables $(s,t,u)$.\n\nThe indices $(i,j,V)$ in the first two forms (\\ref{Sud-ln2-form}, \\ref{Sud-ln-form}),\ncalled Sudakov augmented forms \\cite{super}, denote internally exchanged particles,\nin the various 1loop diagrams; while $V$ always\nrefers to a gauge exchange. The index \"$a$\" always refers to a particle such that\nthe tree-level vertices $aVi$ or $aij$ are non-vanishing. This particle $a$, could\neither be an external particle (i.e. $e^\\mp$ or $W^\\mp$ for the process studied here),\nor a particle contributing at tree level through an exchange in the $s,~t$ or $u$\nchannel (i.e. $\\nu_e$, or\\footnote{As always, for an internal photon we use a mass $m_\\gamma$,\nin order to regularize possible infrared singularities.} $\\gamma, Z$ in our case).\nUsing these, the energy-independent\nexpressions in (\\ref{Sud-ln2-form}, \\ref{Sud-ln-form}) may be written as\n\\cite{super, ttbar, asPV}\n\\bqa\n L_{aVi}& = & \\rm Li_2 \\left ( \\frac{2m_a^2+i\\epsilon}{m_V^2-m_i^2+m_a^2+i\\epsilon +\n\\sqrt{\\lambda (m_a^2+i\\epsilon, m_V^2, m_i^2)}} \\right )\n\\nonumber \\\\\n&& + \\rm Li_2 \\left ( \\frac{2m_a^2+i\\epsilon }{m_V^2-m_i^2+m_a^2+i\\epsilon -\n\\sqrt{\\lambda (m_a^2+i\\epsilon, m_V^2, m_i^2)}} \\right )~~,\\label{LaVi-term} \\\\[0.5cm]\nb_0^{ij}(m_a^2)& \\equiv& b_0(m_a^2; m_i,m_j) =\n2 + \\frac{1}{m_a^2} \\Big [ (m_j^2 -m_i^2)\\ln\\frac{m_i}{m_j}\\nonumber\\\\\n&& + \\sqrt{\\lambda(m_a^2+i\\epsilon, m_i^2, m_j^2)} {\\rm ArcCosh} \\Big\n(\\frac{m_i^2+m_j^2-m_a^2-i\\epsilon}{2 m_i m_j} \\Big ) \\Big ] ~~, \\label{b0ij}\n\\eqa\nwhere\n\\bq\n\\lambda(a,b,c)=a^2+b^2+c^2-2ab-2ac-2bc~~. \\label{lambda-function}\n\\eq\n\n\nThe other two forms (\\ref{ln2r-form}, \\ref{lnr-form}) are solely induced\n by box contributions to the asymptotic amplitudes \\cite{asPV}.\n The forms (\\ref{lnr-form}) in particular, have no dependence on mass scales and never arise\n from differences of the augmented Sudakov linear-log contributions,\n of the type (\\ref{Sud-ln-form}).\\\\\n\nAs already said, apart from possible additive constants,\n the sim expressions consist of linear combinations of the\n four forms (\\ref{Sud-ln2-form}-\\ref{lnr-form}).\nThe coefficients of these forms may\n involve ratios of Mandelstam variables, as well as constants.\nParticularly for the Sudakov augmented forms\n(\\ref{Sud-ln2-form}, \\ref{Sud-ln-form}) though, their coefficients should be\nsuch that, when differences in the scales of masses and Mandelstam variables are disregarded,\n then, the complete coefficients in the implied e.g. $\\ln s $ and $\\ln^2 s$ terms\nbecome the constants given by general rules\n\\cite{MSSMrules1,MSSMrules2,MSSMrules3,MSSMrules4}.\n\n\nGenerally, these \\emph{supersimple} HC helicity amplitudes, differ from the\non-shell renormalized ones \\cite{OS}, by small additive constant terms,\nin both, the MSSM and SM cases.\nWe have checked numerically that\nfor the process studied here, these are indeed negligible.\n\nIn the next two subsections we give the \\emph{supersimple} expressions\nfor the $e^-e^+\\to W^-_TW^+_T$ and $e^-e^+\\to W^-_LW^+_L$ HC amplitudes respectively.\nIn these, we first give the results for the case where infrared singularities are\nregularized by using $m_\\gamma=m_Z$ \\cite{super, ttbar};\nand subsequently quote the corrections\nfor the $m_\\gamma \\neq m_Z$ case.\nIn each case, we give separately the SM and the MSSM predictions.\n\n\n\n\n\\subsection{The $e^-e^+\\to W^-_TW^+_T$ HC amplitudes}\n\n\nThere are two such HC amplitudes listed in the left part of\n(\\ref{4HC-amp-list}); namely $F_{-{1\\over2}-+}$ and $F_{-{1\\over2}+-}$.\nIn the $m_\\gamma=m_Z$ case, using the Born results\nin (\\ref{Born-asym-TT-HC}),\n\n\\vspace*{0.1cm}\n\n\\noindent\nthe asymptotic supersimple {\\bf sim SM } expressions are\n\\bqa\nF_{-{1\\over2}-+}&=&F^{\\rm Born}_{-{1\\over2}-+} \\left ({\\alpha\\over16\\pi s^2_W}\\right )\n\\Bigg \\{\\overline{\\ln t_{Ze}}\\left ({3+2c^2_W\\over c^2_W}-{4t\\over s}+{4s\\over u}\\right )\n+\\overline{\\ln t_{W\\nu}}\\left ({-1+10c^2_W\\over c^2_W}-{8t\\over s} \\right )\\nonumber\\\\\n&&\n+{\\overline{\\ln t_{Z\\nu}}\\over c^2_W} +2\\overline{\\ln t_{We}}\n+\\overline{\\ln u_{Ze}}\\left ({4t\\over u}-{4t\\over s} \\right )\n+{8t\\over s}(\\overline{\\ln s_{W\\nu}}+\\overline{\\ln s_{Ze}})-4\\overline{\\ln u_{W\\nu}}\n\\nonumber\\\\\n&&-3\\overline{\\ln^2t_{Ze}}-\\overline{\\ln^2t_{ZW}}\n-3\\overline{\\ln^2t_{W\\nu}}-\\overline{\\ln^2t_{WZ}}\\nonumber\\\\\n&&- {1\\over c^2_W}(\\overline{\\ln^2s_{Ze}}\n+4c^2_W\\overline{\\ln^2s_{ZW}})\n-2\\overline{\\ln^2s_{WZ}}+2\\overline{\\ln^2u_{Ze}}+2\\overline{\\ln^2u_{ZW}}\n\\nonumber\\\\\n&&-{2t\\over u}(\\overline{\\ln^2s_{W\\nu}}+\\overline{\\ln^2s_{WZ}}-\\overline{\\ln^2t_{W\\nu}}\n-\\overline{\\ln^2t_{WZ}})\\nonumber\\\\\n&&+\\overline{\\ln^2r_{ts}}\\left [{2u^3+2t^3+6ut^2+2tu^2)\\over 2u^3c^2_W}\n+{6u^3-6t^3)\\over u^3} \\right ]\\nonumber\\\\\n&&+{4s\\over u}\\overline{\\ln^2r_{ut}}+{4(t-u)\\over u}\\overline{\\ln^2r_{us}}\n+\\left [{t(2t+5u)\\over u^2c^2_W}+{t(12t^2+6u^2+6tu)\\over su^2}\\right ]\\ln r_{ts}\n\\nonumber\\\\\n&&\n+~{t(16u+12t)\\over su}\\ln r_{us}-\\left ({8t\\over u}+4 \\right)\\ln r_{tu}\n+~{t(1-6c^2_W)\\over uc^2_W} \\Bigg\\} ~~,\n\\label{simSM--+} \\\\[0.5cm]\nF_{-{1\\over2}+-}&=&F^{\\rm Born}_{-{1\\over2}+-}\\left ({\\alpha\\over16\\pi s^2_W} \\right )\n\\Bigg\\{\\overline{\\ln t_{Ze}}\\left ({3+2c^2_W\\over c^2_W}- {4t\\over s}+{4s\\over u}\\right )\n+\\overline{\\ln t_{W\\nu}}\\left ({-1+10c^2_W\\over c^2_W}-{8t\\over s}\\right )\\nonumber\\\\\n&&\n+{1\\over c^2_W}\\overline{\\ln t_{Z\\nu}} +2\\overline{\\ln t_{We}}\n+\\overline{\\ln u_{Ze}}\\left ({4t\\over u}-{4t\\over s} \\right )\n+ {8t\\over s} \\left (\\overline{\\ln s_{W\\nu}}+\\overline{\\ln s_{Ze}}\\right )\n-4\\overline{\\ln u_{W\\nu}}\n\\nonumber\\\\\n&&-3\\overline{\\ln^2t_{Ze}}-\\overline{\\ln^2t_{ZW}}\n-3\\overline{\\ln^2t_{W\\nu}}-\\overline{\\ln^2t_{WZ}}\\nonumber\\\\\n&&- {1\\over c^2_W} (\\overline{\\ln^2s_{Ze}} +4c^2_W\\overline{\\ln^2s_{ZW}})\n-2\\overline{\\ln^2s_{WZ}}+2\\overline{\\ln^2u_{Ze}}+2\\overline{\\ln^2u_{ZW}}\n\\nonumber\\\\\n&&- {2t\\over u}\\left (\\overline{\\ln^2s_{W\\nu}}+\\overline{\\ln^2s_{WZ}}\n-\\overline{\\ln^2t_{W\\nu}}-\\overline{\\ln^2t_{WZ}} \\right )\n+\\overline{\\ln^2r_{ts}}\\left [{u-t\\over uc^2_W}+{6(u-t)\\over u} \\right ]\\nonumber\\\\\n&&\n+{4s\\over u}\\overline{\\ln^2r_{ut}}\n+({4t^2+2ut+6u^2\\over ut})\\overline{\\ln^2r_{us}}\n+\\left [{-3\\over c^2_W}+{18u^2+30ut\\over su} \\right ]\n\\ln r_{ts}\\nonumber\\\\\n&&\n+({4t\\over u}+8)\\ln r_{tu}+ ({4t\\over s}+12)\\ln r_{us}\n- {1-6c^2_W\\over c^2_W}\\Bigg\\} ~~, \\label{simSM-+-}\n\\eqa\n\n\n\n\\vspace*{0.5cm}\n\n\\noindent\nwhile the {\\bf sim MSSM} results, always assuming CP conservation,\nare\n\\bqa\nF_{-{1\\over2}-+}&=&F^{\\rm Born}_{-{1\\over2}-+}\\left ({\\alpha\\over16\\pi s^2_W} \\right )\n\\Bigg\\{{1\\over c^2_W}\\left ( 3\\overline{\\ln t_{Ze}}\n-\\overline{\\ln t_{W\\nu}}+\\overline{\\ln t_{Z\\nu}} \\right ) -2\\overline{\\ln t_{Ze}}\n\\nonumber\\\\\n&& +6\\overline{\\ln t_{W\\nu}}\n+2\\overline{\\ln t_{We}}-3\\overline{\\ln^2t_{Ze}}-\\overline{\\ln^2t_{ZW}}\n-3\\overline{\\ln^2t_{W\\nu}}-\\overline{\\ln^2t_{WZ}}\\nonumber\\\\\n&&- {1\\over c^2_W}(\\overline{\\ln^2s_{Ze}} +4c^2_W\\overline{\\ln^2s_{ZW}})\n-2\\overline{\\ln^2s_{WZ}}+2\\overline{\\ln^2u_{Ze}}+2\\overline{\\ln^2u_{ZW}}\n\\nonumber\\\\\n&&-~{2t\\over u}(\\overline{\\ln^2s_{W\\nu}}+\\overline{\\ln^2s_{WZ}}\n-\\overline{\\ln^2t_{W\\nu}}-\\overline{\\ln^2t_{WZ}})\n\\nonumber\\\\\n&&+{4s\\over u}\\ln r_{tu}-{12t^2\\over su}\\ln r_{ts}+({4t\\over s}-{8t\\over u})\\ln r_{us}\n+{2t\\over uc^2_W}\\ln r_{ts}\n\\nonumber\\\\\n&&-~{1\\over c^2_W}\n\\Big \\{\\sum_j|Z^N_{1j}s_W+Z^N_{2j}c_W|^2\\overline{\\ln t_{\\chi^0_j\\tilde{e_L}}}\n+2c^2_W\\sum_j|Z^+_{1j}|^2\\overline{\\ln t_{\\chi^+_j\\tilde{\\nu}}} \\Big \\}\n\\nonumber\\\\\n&&+\\overline{\\ln^2r_{ts}}\\left [{t^2+u^2\\over u^2c^2_W}+{6t^2+6u^2)\\over u^2}\\right ]\n+~{4s\\over u}\\overline{\\ln^2r_{ut}}+{4(t-u)\\over u}\\overline{\\ln^2r_{us}}\\Bigg\\} ~~,\n\\label{simMSSM--+} \\\\[0.5cm]\nF_{-{1\\over2}+-}&=&F^{\\rm Born}_{-{1\\over2}+-}\\left ({\\alpha\\over16\\pi s^2_W} \\right )\n\\Bigg\\{{1\\over c^2_W}[3\\overline{\\ln t_{Ze}}\n-\\overline{\\ln t_{W\\nu}}+\\overline{\\ln t_{Z\\nu}}] -2\\overline{\\ln t_{Ze}}\n\\nonumber\\\\\n&& +6\\overline{\\ln t_{W\\nu}} +2\\overline{\\ln t_{We}}\n-3\\overline{\\ln^2t_{Ze}}-\\overline{\\ln^2t_{ZW}}\n-3\\overline{\\ln^2t_{W\\nu}}-\\overline{\\ln^2t_{WZ}}\n\\nonumber\\\\\n&&-~{1\\over c^2_W}(\\overline{\\ln^2s_{Ze}}\n+4c^2_W\\overline{\\ln^2s_{ZW}})-2\\overline{\\ln^2s_{WZ}}\n+2\\overline{\\ln^2u_{Ze}}+2\\overline{\\ln^2u_{ZW}}\n\\nonumber\\\\\n&&-~{2t\\over u}(\\overline{\\ln^2s_{W\\nu}}+\\overline{\\ln^2s_{WZ}}\n-\\overline{\\ln^2t_{W\\nu}}-\\overline{\\ln^2t_{WZ}})\n\\nonumber\\\\\n&&+~{12(t-s)\\over s}\\ln r_{ts}+({4t\\over s}+8)\\ln r_{us}\n-~({2\\over c^2_W})\\ln r_{ts}-{4s\\over u}\\ln r_{tu}\n\\nonumber\\\\\n&&-~{1\\over c^2_W}\n\\Big \\{\\sum_j|Z^N_{1j}s_W+Z^N_{2j}c_W|^2\\overline{\\ln t_{\\chi^0_j\\tilde{e_L}}}\n+2c^2_W\\sum_j|Z^+_{1j}|^2\\overline{\\ln t_{\\chi^+_j\\tilde{\\nu}}} \\Big \\}\n\\nonumber\\\\\n&&+\\overline{\\ln^2r_{ts}}[{u-t\\over uc^2_W}+{6(u-t)\\over u}]\n+{4s\\over u}\\overline{\\ln^2r_{ut}}+{4(t^2+u^2)\\over ut}\\overline{\\ln^2r_{us}}\n\\Bigg\\} ~~, \\label{simMSSM-+-}\n\\eqa\nwhere the indices $(i,j)$ in (\\ref{simMSSM--+}, \\ref{simMSSM-+-}) and (\\ref{simMSSM+00}, \\ref{simMSSM-00}) below, refer to chargino and neutralino contributions, defined as in \\cite{Rosiek}.\n\n\nNote the constant terms at the end of the r.h.s. of the SM results\n(\\ref{simSM--+}, \\ref{simSM-+-}).\nNo such constants appear in the corresponding MSSM amplitudes\n(\\ref{simMSSM--+}, \\ref{simMSSM-+-}).\\\\\n\n In the $m_\\gamma \\neq m_Z$ case, the correction to be added to\n (\\ref{simSM--+}-\\ref{simMSSM-+-}),\nis given by\n\\bqa\n&& \\delta F_{-{1\\over2}\\mp \\pm}= F^{\\rm Born}_{-{1\\over2}\\mp \\pm}\n\\left ({\\alpha\\over16\\pi s^2_W}\\right )\n\\Bigg [ \\Bigg \\{-2s^2_W( \\overline{\\ln^2t_{\\gamma e}}+\\overline{\\ln^2t_{\\gamma W}})\n+16 s^2_W {t\\over s} \\overline{\\ln s_{\\gamma e}}\\nonumber\\\\\n&&+2s^2_W[-2\\overline{\\ln^2s_{\\gamma e}}+8\\overline{\\ln t_{\\gamma e}}]\n -2s^2_W[2\\overline{\\ln^2s_{\\gamma W}}+\\overline{\\ln^2 t_{W\\gamma}}]\\nonumber\\\\\n&&+2s^2_W[-2\\overline{\\ln^2s_{W\\gamma}}-\\overline{\\ln^2 t_{\\gamma e}}\n-\\overline{\\ln^2 t_{\\gamma W}}+4(1-{t\\over s})\\overline{\\ln t_{\\gamma e}}]\\nonumber\\\\\n&&+2s^2_W \\Big [-2{t\\over u}\\overline{\\ln^2s_{W\\gamma}}-{t\\over u}\n(\\overline{\\ln^2 u_{\\gamma e}}\n+\\overline{\\ln^2 u_{\\gamma W}})+4({t\\over u}-{t\\over s})\\overline{\\ln u_{\\gamma e}}\\Big ]\n\\nonumber\\\\\n&&-2s^2_W \\Big [{s-u\\over u}(\\overline{\\ln^2u_{\\gamma e}}+\\overline{\\ln^2 u_{\\gamma W}})\n+{s-t\\over u}\\overline{\\ln^2 t_{W\\gamma}}+4(2+{t\\over u})\\overline{\\ln t_{\\gamma e}} \\Big ]\n\\Bigg\\}\n\\nonumber \\\\\n&& -\\Big \\{ m_\\gamma \\to m_Z \\Big \\} \\Bigg ]~~, \\label{deltaFTT}\n\\eqa\nwhere (\\ref{Born-asym-TT-HC}) is again used.\n\n\n\n\\vspace*{1cm}\n\\subsection{The $e^-e^+\\to W^-_L W^+_L$ HC amplitudes}\n\nIn the $m_\\gamma=m_Z$ case, using the asymptotic Born LL amplitudes\n(\\ref{Born-asym-LL-HC}), \\\\\n\\noindent\nthe high energy supersimple {\\bf sim SM} results are written as\\\\\n\\bqa\nF_{+{1\\over2}00}&=&F^{\\rm Born}_{+{1\\over2}00}\\Bigg\\{\\left ({\\alpha\\over4\\pi} \\right )\n\\Big \\{{1\\over c^2_W}\n\\left [-\\overline{\\ln^2s_{Ze}}+3\\overline{\\ln s_{Ze}}-1 \\right ]\n+{1\\over 4s^2_Wc^2_W}\\left [-\\overline{\\ln^2s_{ZW}}+4\\overline{\\ln s_{ZW}} \\right ]\n\\nonumber\\\\\n&&+{1\\over 2s^2_W}\\left [-{1\\over2}(\\overline{\\ln^2s_{WZ}}+\\overline{\\ln^2s_{WH_{SM}}})\n+2\\overline{\\ln s_{WZ}} +2\\overline{\\ln s_{WH_{SM}}}\\right ]\n\\nonumber\\\\\n&&-{3(m^2_t+m^2_b)\\over 2s^2_Wm^2_W}\\overline{\\ln s_{tb}}\n-{1\\over 4c^2_W} \\Big [4(\\overline{\\ln^2t_{ZW}}-\\overline{\\ln^2u_{ZW}})\n+{2(u-t)\\over u}\\overline{\\ln^2r_{ts}}\n\\nonumber\\\\\n&& -{2(t-u)\\over t}\\overline{\\ln^2r_{us}} \\Big ]\\Big \\}\n+\\Sigma^{\\rm seSM}\\left (+{1\\over2},0,0 \\right )\\Bigg\\}~~,\n\\label{simSM+00} \\\\[0.5cm]\nF_{-{1\\over2}00}&=&F^{\\rm Born}_{-{1\\over2}00}\\Bigg\\{\\left ({\\alpha\\over4\\pi} \\right)\n\\Big \\{{1\\over 4s^2_Wc^2_W}\n\\left[-\\overline{\\ln^2s_{Ze}}+3\\overline{\\ln s_{Ze}}-1 \\right ]\n\\nonumber\\\\\n&&-{(1-2s^2_W)\\over 2s^2_W}[-\\overline{\\ln^2s_{W\\nu}}+3\\overline{\\ln s_{W\\nu}}-1]\n\\nonumber\\\\\n&& +{2c^2_W\\over s^2_W}\\left [{1\\over2}\\overline{\\ln s_{W\\nu}}+{1\\over2}\n+2\\overline{\\ln s_{WW}}\\right ]\n+{1\\over 4s^2_Wc^2_W}\\left [-\\overline{\\ln^2s_{ZW}}+4\\overline{\\ln s_{ZW}} \\right ]\n\\nonumber\\\\\n&&+{(1-2c^2_W)\\over 2s^2_W}\n\\left [-{1\\over2}(\\overline{\\ln^2s_{WZ}}+\\overline{\\ln^2s_{WH_{SM}}})+2\\overline{\\ln s_{WZ}}\n+2\\overline{\\ln s_{WH_{SM}}} \\right ]\n\\nonumber\\\\\n&&+{c^2_W\\over s^2_W}\\left [\\overline{\\ln s_{WZ}}\n+\\overline{\\ln s_{WH_{SM}}}\\right ]-{3(m^2_t+m^2_b)\\over 2s^2_Wm^2_W}\\overline{\\ln s_{tb}}\n\\nonumber\\\\\n&&-{c^2_W\\over 4s^2_W} \\left [4\\overline{\\ln^2t_{W\\nu}}\n+2\\overline{\\ln^2t_{WZ}}+2\\overline{\\ln^2t_{WH_{SM}}}\n-4(1-{t\\over u})\\overline{\\ln^2r_{ts}}\\right ]\n\\nonumber\\\\\n&&-{1\\over 8c^2_Ws^2_W}\\left [4(\\overline{\\ln^2t_{ZW}}-\\overline{\\ln^2u_{ZW}})\n+{2(u-t)\\over u}\\overline{\\ln^2r_{ts}}\n-{2(t-u)\\over t}\\overline{\\ln^2r_{us}} \\right ]\\Big \\}\n\\nonumber\\\\\n&& +\\Sigma^{\\rm seSM}\\left (-{1\\over2},0,0 \\right )\\Bigg\\}~~, \\label{simSM-00}\n\\eqa\n\n\\vspace*{0.5cm}\n\n\\noindent\nwhile the supersimple {\\bf sim MSSM} results are\n\\bqa\nF_{+{1\\over2}00}&=&F^{\\rm Born}_{+{1\\over2}00}\n\\Bigg\\{\\left ({\\alpha\\over4\\pi} \\right ) \\Big \\{{1\\over c^2_W}\n \\left [-\\overline{\\ln^2s_{Ze}}+3\\overline{\\ln s_{Ze}}\n-\\Sigma_i|Z^N_{1i}|^2 \\overline{\\ln s_{\\chi^0_i\\tilde{e_R}}} \\right ]\n\\nonumber\\\\\n&&+{1\\over 4s^2_Wc^2_W}\\left [-\\overline{\\ln^2s_{ZW}}+4\\overline{\\ln s_{ZW}} \\right ]\n+{1\\over 2s^2_W}\\left [-{1\\over2} \\overline{\\ln^2s_{WZ}} +2\\overline{\\ln s_{WZ}} \\right ]\n\\nonumber\\\\\n&&-{1\\over 4s^2_W} \\left [\\cos^2(\\beta-\\alpha)\\overline{\\ln^2 s_{WH^0}}+\\sin^2(\\beta-\\alpha)\n\\overline{\\ln^2 s_{Wh^0}} \\right ]\n\\nonumber\\\\\n&&+{1\\over 2s^2_W} \\left [2\\cos^2(\\beta-\\alpha)\\overline{\\ln s_{WH^0}}\n+2\\sin^2(\\beta-\\alpha)\\overline{\\ln s_{Wh^0}} \\right ]\n\\nonumber\\\\\n&&-{1\\over 2s^2_Wc^2_W}\\Sigma_{ij}\n\\Big [\\Big |{1\\over\\sqrt{2}}Z^-_{2i}(Z^N_{1j}s_W+Z^N_{2j}c_W)-Z^-_{1i}Z^N_{3j}c_W \\Big |^2\n\\nonumber\\\\\n&& +\\Big |{1\\over\\sqrt{2}}Z^+_{2i}(Z^N_{1j}s_W+Z^N_{2j}c_W)\n+Z^+_{1i}Z^N_{4j}c_W \\Big |^2 \\Big ] \\overline{\\ln s_{\\chi^+_i\\chi^0_j}}\n\\nonumber\\\\\n&&-{3(m^2_t+m^2_b)\\over 2s^2_Wm^2_W}\\overline{\\ln s_{tb}}\n-{\\cos^2\\beta\\over2c^2_W} \\left [{s\\over u}\\overline{\\ln^2r_{ts}}-\n{s\\over t}\\overline{\\ln^2r_{us}} \\right ]\n\\nonumber\\\\\n&& -{1\\over 4c^2_W}\\left [4(\\overline{\\ln^2t_{ZW}}-\\overline{\\ln^2u_{ZW}})\n+{2(u-t)\\over u}\\overline{\\ln^2r_{ts}}\n-{2(t-u)\\over t}\\overline{\\ln^2r_{us}} \\right ] \\Big \\}\n\\nonumber\\\\\n&& +\\Sigma^{\\rm seMSSM}\\left (+{1\\over2},0,0 \\right )\\Bigg\\} ~~,\n\\label{simMSSM+00} \\\\[0.5cm]\nF_{-{1\\over2}00}&=&F^{\\rm Born}_{-{1\\over2}00}\\Bigg\\{\\left ({\\alpha\\over4\\pi} \\right )\n\\Big \\{{1\\over 4s^2_Wc^2_W} \\Big [-\\overline{\\ln^2s_{Ze}}+3\\overline{\\ln s_{Ze}}\n-\\overline{\\ln^2s_{ZW}}+4\\overline{\\ln s_{ZW}}\n\\nonumber\\\\\n&& -\\Sigma_i|Z^N_{1i}s_W+Z^N_{2i}c_W|^2\\overline{\\ln s_{\\chi^0_i\\tilde{e_L}}} \\Big ]\n\\nonumber\\\\\n&&-{(1-2s^2_W)\\over 2s^2_W}\\left [-\\overline{\\ln^2s_{W\\nu}}+3\\overline{\\ln s_{W\\nu}}\n- {1\\over2}\\overline{\\ln^2s_{WZ}}+2\\overline{\\ln s_{WZ}}-\\Sigma_i|Z^+_{1i}|^2\n\\overline{\\ln s_{\\chi^+_i\\tilde{\\nu_L}}}\\right ]\n\\nonumber\\\\\n&&+{c^2_W\\over s^2_W} \\left [{\\overline{\\ln s_{W\\nu}}}\n+4\\overline{\\ln s_{WW}}-\\Sigma_i|Z^+_{1i}|^2\\overline{\\ln s_{\\chi^+_i\\tilde{\\nu_L}}} \\right ]\n\\nonumber\\\\\n&& -{(1-2c^2_W)\\over 4s^2_W} \\left [\\cos^2(\\beta-\\alpha)\\overline{\\ln^2 s_{WH^0}}\n+\\sin^2(\\beta-\\alpha) \\overline{\\ln^2 s_{Wh^0}} \\right ]\n\\nonumber\\\\\n&&+{(1-2c^2_W)\\over s^2_W} \\left [ \\cos^2(\\beta-\\alpha)\\overline{\\ln s_{WH^0}}\n+\\sin^2(\\beta-\\alpha)\\overline{\\ln s_{Wh^0}} \\right ]\n\\nonumber\\\\\n&&+{c^2_W\\over s^2_W} \\left [\\overline{\\ln s_{WZ}}\n+\\cos^2(\\beta-\\alpha) \\overline{\\ln s_{WH^0}}\n+\\sin^2(\\beta-\\alpha) \\overline{\\ln s_{Wh^0}}\\right ]\n\\nonumber\\\\\n&&-{3(m^2_t+m^2_b)\\over 2s^2_Wm^2_W}\\overline{\\ln s_{tb}}-{1\\over 2s^2_Wc^2_W}\\Sigma_{ij}\n\\Big [\\Big |{1\\over\\sqrt{2}}Z^-_{2i}(Z^N_{1j}s_W+Z^N_{2j}c_W)-Z^-_{1i}Z^N_{3j}c_W \\Big |^2\n\\nonumber\\\\\n&&+\\Big |{1\\over\\sqrt{2}}Z^+_{2i}(Z^N_{1j}s_W+Z^N_{2j}c_W)\n+Z^+_{1i}Z^N_{4j}c_W \\Big |^2 \\Big ] \\overline{\\ln s_{\\chi^+_i\\chi^0_j}}\n\\nonumber\\\\\n&&- {c^2_W\\over 4s^2_W} \\left [4\\overline{\\ln^2t_{W\\nu}}\n+2\\overline{\\ln^2t_{WZ}}\n-4\\left (1-{t\\over u} \\right )\\overline{\\ln^2r_{ts}} \\right ]\n\\nonumber\\\\\n&&- {c^2_W\\over 2s^2_W}\n\\left [\\cos^2(\\beta-\\alpha)\\overline{\\ln^2 t_{WH^0}}+\\sin^2(\\beta-\\alpha)\n\\overline{\\ln^2 t_{Wh^0}} \\right ]\n\\nonumber\\\\\n&&-{1\\over 8c^2_Ws^2_W}\\left [4(\\overline{\\ln^2t_{ZW}}-\\overline{\\ln^2u_{ZW}})\n+{2(u-t)\\over u}\\overline{\\ln^2r_{ts}}\n-{2(t-u)\\over t}\\overline{ln^2r_{us}} \\right ]\n\\nonumber\\\\\n&&-{\\sin^2\\beta\\over2c^2_Ws^2_W} \\left [{s\\over u}\\overline{\\ln^2r_{ts}}-\n{s\\over t}\\overline{\\ln^2r_{us}} \\right ]\n-{c^2_W \\sin^2\\beta \\over s^2_W} {s\\over u}\\overline{\\ln^2r_{ts}} \\Big \\}\n\\nonumber\\\\\n&& +\\Sigma^{\\rm seMSSM}\\left (-{1\\over2},0,0 \\right )\\Bigg\\}~~. \\label{simMSSM-00}\n\\eqa\\\\\n\n In the $m_\\gamma \\neq m_Z$ case,\n the correction to be added to (\\ref{simSM+00}-\\ref{simMSSM-00}) is given by\n \\bqa\n \\delta F_{\\pm {1\\over2} 00}&= &F^{\\rm Born}_{\\pm{1\\over2}00}\\left ({\\alpha\\over4\\pi} \\right )\n\\Bigg [ \\Big \\{ -\\overline{\\ln^2s_{\\gamma e}}+3\\overline{\\ln s_{\\gamma e}}\n -\\overline{\\ln^2s_{\\gamma W}}\n \\nonumber\\\\\n&& +4\\overline{\\ln s_{\\gamma W}}\n-2 \\overline{\\ln^2t_{\\gamma W}} +2 \\overline{\\ln^2u_{\\gamma W}} \\Big \\}\n- \\Big \\{ m_\\gamma \\to m_Z \\Big \\} \\Bigg ]~~ , \\label{deltaF00}\n\\eqa\nwhere (\\ref{Born-asym-LL-HC}) is again used.\n\n\n\n\nThe $\\Sigma^{\\rm se}$-contributions in either (\\ref{simSM+00}-\\ref{simSM-00})\nor (\\ref{simMSSM+00}-\\ref{simMSSM-00}), respectively appearing in SM and MSSM,\n come from the photon and Z self-energy contributions together with\n their renormalization counter terms. Their\n explicit expressions are\n\\bqa\n\\Sigma^{\\rm se}\\left (-{1\\over2},0,0 \\right )&= &\n{-4s^2_Wc^2_W\\over s}\\left \\{\\hat{\\Sigma}_{\\gamma\\gamma}(s)\n+{1-2s^2_W\\over s_Wc_W}\\hat{\\Sigma}_{Z\\gamma}(s)+{(1-2s^2_W)^2\\over4s^2_Wc^2_W}\n\\hat{\\Sigma}_{ZZ}(s) \\right \\} \\nonumber \\\\\n& + & C_P ~~, \\label{Cse-00} \\\\[0.5cm]\n\\Sigma^{\\rm se}\\left (+{1\\over2},0,0 \\right )&=&\n{-2c^2_W\\over s} \\left \\{\\hat{\\Sigma}_{\\gamma\\gamma}(s)\n+{1-4s^2_W\\over 2s_Wc_W}\\hat{\\Sigma}_{Z\\gamma}(s)-{(1-2s^2_W)\\over2c^2_W}\n\\hat{\\Sigma}_{ZZ}(s)\\right \\}~~, \\label{Cse+00}\n\\eqa\nwhere the renormalized gauge self energies $\\hat{\\Sigma}$ can be found in\n\\cite{ttbar}, together with their supersimple approximations. The last term in\n(\\ref{Cse-00}), given by\n\\bq\nC_P=-{\\alpha c^2_W\\over\\pi s^2_W}\\overline{\\ln s_{WW}}~~, \\label{pinch-part}\n\\eq\ncomes from the pinch part that had been previously\nremoved from the left and right triangular contributions, and is here restored\n\\cite{pinch,pinch1}.\n\nNote that no such $\\Sigma^{\\rm se}$-contributions exist for the transverse amplitudes\n in (\\ref{simSM--+}-\\ref{simMSSM-+-}).\n\nAs it should, the high energy ln and ln-squared parts of all expressions\n(\\ref{simSM--+}- \\ref{simMSSM-00}), agree with the usual Sudakov\nrules and the renormalization group results\n\\bqa\n&& A^{RG}=-{ln\\over 4\\pi^2} (g^4\\beta{dA^{Born}\\over dg^2}\n+g^{-4}\\beta'{dA^{Born}\\over dg^{'2}}) ~~, \\nonumber \\\\\n&& \\beta^{SM}={43\\over24}-{N_f\\over3}~~,~~\\beta^{SUSY}=-{13\\over24}-{N_f\\over6}\n~~,~~N_f=3 ~~, \\nonumber \\\\\n&& \\beta^{'SM}=-{1\\over24}-{5N_f\\over9}~~,~~\\beta^{'SUSY}=-~{5\\over24}-{5N_f\\over18}\n~~, \\label{renorm-group}\n\\eqa\ndiscussed in \\cite{MSSMrules1,MSSMrules2,MSSMrules3,MSSMrules4}.\n\n\n\n\\vspace*{1cm}\n\n\\renewcommand{\\thesection}{B}\n\\renewcommand{\\theequation}{B.\\arabic{equation}}\n\\setcounter{equation}{0}\n\n\n\\section{ Appendix: AGC and $Z'$ amplitudes}\n\n\n\\subsection{The AGC amplitudes}\n\n\nAs an Anomalous Gauge Coupling (AGC) model induced by\ns-channel $\\gamma $ and $Z$ exchanges with\n 5 anomalous couplings $\\delta_Z$, $x_{\\gamma,Z}$, $y_{\\gamma,Z}$, we consider\n the one presented in \\cite{heliWW} and Table V of \\cite{Andreev}.\nIn terms of these couplings and the SM ones in (\\ref{e-couplings}),\nthe induced AGC contributions to the TT, TL, LT and LL amplitudes,\nto lowest order, are\\footnote{Compare with\n(\\ref{FBorn-TT}, \\ref{FBorn-TL}, \\ref{FBorn-LT},\\ref{FBorn-LL}).}\n\\bqa\nF^{\\rm AGC}_{\\lambda\\mu\\mu}(\\theta)&=& {(2\\lambda)se^2\\over8}(1+\\mu\\mu')\\beta_W\\sin\\theta\n\\Bigg \\{ {\\delta_Z(a_{eL}\\delta_{\\lambda,-}+a_{eR}\\delta_{\\lambda,+})\\over\ns^2_W(s-m^2_Z)}\n\\nonumber\\\\\n&&-\\left [{y_{\\gamma}\\over s}-{y_{Z}(a_{eL}\\delta_{\\lambda,-}+a_{eR}\\delta_{\\lambda,+})\n\\over 2s^2_W(s-m^2_Z)} \\right ]{s\\over m^2_W} \\Bigg \\} ~~, \\label{FAGC-TT} \\\\[0.5cm]\nF^{\\rm AGC}_{\\lambda \\mu 0}(\\theta)&=&\n- {(2\\lambda)s\\beta_W\\sqrt{s}e^2\\over4\\sqrt{2}m_W}(2\\lambda+\\mu\\cos\\theta)\n\\Bigg \\{ {\\delta_Z(a_{eL}\\delta_{\\lambda,-}+a_{eR}\\delta_{\\lambda,+})\\over\ns^2_W(s-m^2_Z)}\n\\nonumber\\\\\n&&-\\left [{(x_{\\gamma}+y_{\\gamma})\\over s}-{(x_{Z}+y_{Z})\n(a_{eL}\\delta_{\\lambda,-}+a_{eR}\\delta_{\\lambda,+})\\over\n2s^2_W(s-m^2_Z)}\\right ] \\Bigg \\} ~~, \\label{FAGC-TL}\\\\[0.5cm]\nF^{\\rm AGC}_{\\lambda 0\\mu'}(\\theta)&=&\n- {(2\\lambda)s\\beta_W\\sqrt{s}e^2\\over4\\sqrt{2}m_W}(2\\lambda-\\mu'\\cos\\theta)\n\\Bigg \\{ {\\delta_Z(a_{eL}\\delta_{\\lambda,-}+a_{eR}\\delta_{\\lambda,+})\\over\ns^2_W(s-m^2_Z)}\n\\nonumber\\\\\n&&-\\left [{(x_{\\gamma}+y_{\\gamma})\\over s}-{(x_{Z}+y_{Z})\n(a_{eL}\\delta_{\\lambda,-}+a_{eR}\\delta_{\\lambda,+})\\over\n2s^2_W(s-m^2_Z)} \\right ] \\Bigg \\} ~~, \\label{FAGC-LT}\\\\[0.5cm]\nF^{\\rm AHC}_{\\lambda 00}(\\theta)&=& {(2\\lambda)s^2e^2\\over4m^2_W}\\beta_W\\sin\\theta\n\\Bigg \\{ {\\delta_Z(a_{eL}\\delta_{\\lambda,-}+a_{eR}\\delta_{\\lambda,+})\\over\ns^2_W(s-m^2_Z)}\\left (1+{s\\over2m^2_W} \\right )\n\\nonumber\\\\\n&&-\\left [{x_{\\gamma}\\over s}-x_{Z}{a_{eL}\\delta_{\\lambda,-}+a_{eR}\\delta_{\\lambda,+}\\over\n2s^2_W(s-m^2_Z)}\\right ]{s\\over m^2_W} \\Bigg \\} ~~. \\label{FAGC-LL}\n\\eqa\nNote that $\\delta_Z$ contributes to all amplitudes, except the two\nTT HC ones (because of the vanishing of the overall coefficient $(1+\\mu\\mu')$ in\n (\\ref{FAGC-TT}) in such a case);\n $x_{\\gamma,Z}$ contribute to all TL, LT and LL amplitudes; while\n $y_{\\gamma,Z}$ contribute only to the HV TT, TL and LT amplitudes.\n\nIn the figures, and under the name AGC1, we present illustrations\nfor the purely arbitrary choice\n\\bq\n{\\rm AGC1}~~~~\\Rightarrow ~~~ \\delta_Z=x_{\\gamma}=x_Z=0.003 ~~~,~~~ y_\\gamma=y_Z=0~~~.\n\\label{AGC1-choice}\n\\eq\nFor AGC1, the HV TT anomalous amplitudes behave like constants at high energy;\nthe HC LL ones explode like $s\/ m^2_W$; while the LT ones increase like\n$\\sqrt{s}\/ m^2_W$.\\\\\n\nIn the figures we also present results for an alternative AGC2 model in which the\n$s\/ m^2_W$ behavior of the HC LL anomalous amplitudes is canceled by a $t$-channel\ncontribution; much like it is done in the Born SM case. So we construct an\n ad-hoc model with an anomalous contribution in the\nt-channel which would lead to a similar cancelation.\nA simple phenomenological solution is obtained by keeping only $x_{\\gamma}$ and $x_Z$\n(called now $x'_{\\gamma}$ and $x'_Z$) in (\\ref{FAGC-TT}-\\ref{FAGC-LL}),\nand adding t-channel contributions induced by\n left- and right-handed $We\\nu$ couplings obtained from the\n initial SM one $g_L=e\/( \\sqrt{2} s_W)$, through\n\\bqa\ng^2_L & \\Rightarrow & g^2_L \\left (1+2s^2_W \\left\n[x'_{\\gamma}-{2s^2_W-1\\over2s_Wc_W}x'_Z \\right]\n\\right ) ~~~,\n\\nonumber \\\\\ng^2_R & \\Rightarrow & g^2_L (2s^2_W)\\left [x'_{\\gamma}-{s_W\\over c_W}x'_Z \\right ]\n~~. \\label{AGC2-gLgR}\n\\eqa\n This does not necessarily represent\ntrue anomalous $We\\nu$ couplings; it just represents the new\ncontribution necessary at high energy.\nFor example it may come from additional neutral fermion exchanges or\nfrom any sort of effective interaction. In the illustrations under the AGC2 name, we use\n\\bq\n{\\rm AGC2}~~~~\\Rightarrow ~~~ x'_{\\gamma}=x'_Z=0.03 ~~~;\\label{AGC2-choice}\n\\eq\nthese values are larger than those in (\\ref{AGC1-choice}),\nbecause of the\nglobal suppression effect following from the high energy\ncancelation between t- and s-channel terms.\n\nIf one does not want to introduce an anomalous right-handed contribution\none can just keep a non vanishing $x'_{\\gamma}$ only,\nand add the anomalous left-handed term\n\\bq\ng^2_L ~ \\Rightarrow ~ g^2_L (1+2s^2_Wx'_{\\gamma}) ~~, \\label{AGC2-gL}\n\\eq\n\nIn any case, investigating the origin of such anomalous terms\nis beyond the scope of the present work.\\\\\n\n\n\\subsection{ The $Z'$ New Physics model}\n\nThe general form of helicity amplitudes with a $Z'$ is written in Table VI of \\cite{Andreev}.\nThe $Z'$ contributions are very similar to the SM Z ones, with specific $Z'$ mass,\nwidth and couplings.\n\nIn general, with arbitrary $Z'$ couplings, there is an explosion of the LL, LT and\nTL amplitudes at high energies.\nBut, it is again easy to get high energy cancelation in an ad-hoc manner\nby just replacing the usual $Z$ contribution\ninvolving products of couplings like $g_{Zee}g_{ZWW}$, by $Z+Z'$ exchanges using respectively\n$g_{Zee}g_{ZWW}\\cos^2\\Phi$ for Z and\n$g_{Zee}g_{ZWW}\\sin^2\\Phi$ for $Z'$ (with a small value of $\\Phi$).\nThis way, the s-channel high energy contribution will be\nsimilar to the SM $Z$ one, and will cancel with the SM t-channel contribution.\nOnly around the $Z'$ peak, will the $Z'$ contribution be observable.\n\nFor the illustrations presented in the figures under the name $Z'$,\nwe use $\\sin\\Phi=0.05$ and $m_{Z'}=3$ TeV.\n\n\n\n\\newpage\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}