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+{"text":"\n\\section{Introduction}\n\nNested sampling (NS)~\\citep{2004AIPC..735..395S,Skilling:2006gxv} is a popular algorithm for Bayesian inference in cosmology, astrophysics and particle physics. The algorithm handles multimodal and degenerate problems, and returns weighted samples for parameter inference as well as an estimate of the Bayesian evidence for model comparison. It was recently emphasised independently by \\citet[appendix B.5.4]{riley} and \\citet{2020arXiv200508602S}, however, that assumptions in NS are violated by plateaus in the likelihood, that is regions of parameter space that share the same likelihood. We should not be surprised by subtleties caused by ties in the likelihood as NS is based on order statistics and this problem and possible solutions were in fact discussed by \\citet{2004AIPC..735..395S,Skilling:2006gxv}. We believe, however, that it was underappreciated prior to \\citet{riley,2020arXiv200508602S}.\n\nPlateaus most often occur in discrete parameter spaces in which every possible likelihood value is associated with a finite prior mass, or in physics problems when regions of a model's parameter space make predictions that are in such severe disagreement with observations that they are assigned a likelihood of zero. For example, in particle physics, parameter points that fail to predict electroweak symmetry breaking would be vetoed and in cosmology portions of parameter space may be excluded if they result in unphysically large power spectra for the purposes of applying lensing, or if it is impossible to trace a consistent cosmic history (see section XIII.C of \\citealt{Hergt:2020} for more detail). We  generically refer to such points as unphysical and the observation that renders them unphysical as $U$. In fact, within popular implementations of NS, such as  \\code{MultiNest}\\xspace~\\citep{Feroz:2007kg,Feroz:2008xx,Feroz:2013hea} and \\code{PolyChord}\\xspace~\\citep{Handley:2015fda,Handley:2015xxx}, unphysical points can be assigned a likelihood of zero or a prior of zero through the \\texttt{logZero} setting. Any log likelihoods below it are treated as if the prior were in fact zero. Statistically, this makes a difference to the evidences and Bayes factors, as it changes whether we consider $U$ to be prior knowledge or information with which we are updating, i.e.\\ whether we wish to compute\n$\\pg{D}{M, U}$ or $\\pg{D, U}{M}$,\nwhere $D$ represents experimental data and $M$ represents a model. The latter is problematic within NS. We consider both cases interesting: on the one hand, taking a basic observation, e.g., electroweak symmetry breaking, as prior knowledge is reasonable, but, on the other, so is judging models by their ability to predict a basic observation. Although the problem with plateaus can in general lead to faulty posterior distributions as well, when plateaus occur only at zero likelihood, they do not impact posterior inferences about the parameters of the model. There are, furthermore, realistic situations in which plateaus could occur at non-zero likelihood, e.g., if in some regions of parameter space, the likelihood function or the physical observables on which it depends are approximated by a constant.\n\nIn \\citet{riley,2020arXiv200508602S}, the problem caused by plateaus was formally demonstrated. After reviewing the relevant aspects of NS in \\cref{sec:ns}, in \\cref{sec:plateaus}, we instead make an informal explanation of the problem. We continue in \\cref{sec:modified_ns} by proposing a modified NS algorithm that deals with plateaus and reduces to ordinary NS in their absence. We show examples in \\cref{sec:examples} before concluding in \\cref{sec:conclusions}. An implementation of the modified NS algorithm that can be used to correct evidences and posteriors found from \\code{MultiNest}\\xspace and \\code{PolyChord}\\xspace runs is implemented in \\textsf{anesthetic} starting from version \\code{\\href{https:\/\/github.com\/williamjameshandley\/anesthetic\/releases\/tag\/2.0.0-beta.2}{2.0.0-beta.2}}~\\citep{Handley:2019mfs}.\n\n\\section{Nested sampling}\\label{sec:ns}\n\nTo establish our notation and the assumptions in ordinary NS, let us briefly review the NS algorithm. NS works by computing evidence integrals,\n\\begin{equation}\\label{eq:Z}\n\\mathcal{Z} \\equiv \\int_{\\Omega_\\mathbf{\\Theta}}  \\mathcal{L}(\\mathbf{\\Theta}) \\, \\pi(\\mathbf{\\Theta}) \\,\\text{d} \\mathbf{\\Theta},  \n\\end{equation}\nwhere $\\mathcal{L}(\\mathbf{\\Theta})$ is the so-called likelihood function for the relevant experimental data and $\\pi(\\mathbf{\\Theta})$ is the prior density for the model's parameters,\nas Riemann sums of a one-dimensional integral,\n\\begin{equation}\n\\mathcal{Z} = \\int_0^1 \\mathcal{L}(X) \\,\\text{d} X,\n\\end{equation}\nwhere \n\\begin{equation}\\label{eq:X}\nX(\\lambda) = \\int_{\\mathcal{L}(\\mathbf{\\Theta}) > \\lambda}  \\pi(\\mathbf{\\Theta}) \\,\\text{d} \\mathbf{\\Theta},\n\\end{equation}\nis the prior volume contained within the iso-likelihood contour at $\\lambda$ and $\\mathcal{L}(X)$ is the inverse of $X(\\lambda)$, i.e. $\\mathcal{L}(X(\\lambda)) = \\lambda$. This evidently requires that such an inverse exists over the range of integration.\n\nTo tackle the one-dimensional integral, we first sample $n_\\text{live}$ points from the prior --- the live points. Then, at each iteration of NS, we remove the live point with the worst likelihood $\\like^\\star$ and replace it with one drawn form the prior subject to the constraint that $\\mathcal{L} > \\like^\\star$. Thus we remove a sequence of samples of increasing likelihood $\\mathcal{L}_i$. In NS we estimate $X_i \\equiv X(\\mathcal{L}_i)$ by the properties of how that sequence was generated. Indeed, at each iteration the volume contracts by a factor $t$, where the arithmetic and geometric expectations of $t$ are\n\\begin{align}\\label{eq:t}\n    \\expectation{t} &= \\frac{n_\\text{live}}{n_\\text{live} + 1},\\\\\n    \\expectation{\\ln t} &= -\\frac{1}{n_\\text{live}}.\n\\end{align}\nWe can then make a statistical estimate of the prior volume at the $i$-th iteration, $X_i = e^{-i\/n_\\text{live}}$. This enables us to compute\n\\begin{equation}\n\\mathcal{Z} \\approx \\sum \\mathcal{L}_i (X_{i - 1} - X_{i}).\n\\end{equation}\nThe estimates in \\cref{eq:t} assume that the live points are uniformly distributed in $X$. In \\citet{10.1093\/mnras\/staa2345} we proposed a technique for testing the veracity of this assumption within the context of a numerical implementation such as \\textsf{MultiNest} or \\textsf{PolyChord}. In the following section we discuss why plateaus violate that assumption.\n\n\\section{Plateaus}\\label{sec:plateaus}\n\nIn \\citet{Skilling:2006gxv}, plateaus were recognised as a problem, since they offer no guidance about the existence or location of points of greater likelihood and since they cause ambiguity in the ranking of samples by likelihood. The latter problem was addressed by breaking ties by assigning a rankable second property to each live point, that is expected to be unique, such as a cryptographic hash of the parameter coordinates or just a random variate from some pre-specified distribution. This was implemented by extending the likelihood to include that tie-breaking second property, $\\ell$, suppressed by a tiny factor, $\\epsilon$, so that it doesn't impact the evidence estimate,\n\\begin{equation}\\label{eq:extened_like}\n    \\mathcal{L} \\to \\mathcal{L} + \\epsilon \\ell.\n\\end{equation}\nIt was not stated explicitly that plateaus violate an assumption in NS, as formally shown by \\citet{riley,2020arXiv200508602S}, though it was known and it was mentioned in \\citet[section 4.4.6]{murray}. \\citet{murray}, furthermore, noted that hashes of the parameter coordinates in discrete parameter spaces would not be unique and thus would fail to break ties.\n\nIn the presence of plateaus, the outermost contour could contain more than one point with the same likelihood. When we replace one of the points in the plateau by a point with a greater likelihood, the volume cannot contract at all, since the outermost contour still contains other points at the same likelihood. Once we've replaced all the points in the plateau, the volume finally contracts. Crucially, however, it was not possible for any of the replacement points to affect the contraction, as the replacement points could never be the worst point whilst points in the plateau remain in the live points. The latter subtlety changes statistical estimates of the volume. Without plateaus, it is possible that a replacement point is or soon becomes the worst point, slowing the volume contraction. Without that possibility, the volume contracts faster and thus ordinary NS underestimates volume contraction and overestimates the evidence when there are plateaus.\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=0.7\\textwidth]{infographic.pdf}\n    \\caption{Infographic showing the impact of plateaus on assumptions in NS. We show an ordinary Gaussian (orange) and a  modified Gaussian with plateaus in the tails (blue). In the lower right panel, we see that the distribution of $X$ is no longer uniform, breaking assumptions in NS.}\n    \\label{fig:plateau_infographic}\n\\end{figure*}\n\nThe problem is illustrated by \\cref{fig:plateau_infographic}. We show a Gaussian likelihood with mean $\\mu = 0.5$ and standard deviation $\\sigma = 0.25$ (orange) and a modified Gaussian likelihood with a plateau in its tails at $\\ensuremath{\\like_P} = 0.5$ (blue). In each case we consider a flat prior for the parameter $x$ from $0$ to $1$. The upper right panel shows the prior distribution of the likelihood. The plateau manifests as an atom in that distribution at \\ensuremath{\\like_P}. The  lower left panel shows the resulting enclosed prior volume as a function of the likelihood. The plateau causes a jump discontinuity at \\ensuremath{\\like_P}. Finally, the lower right panel shows the distribution of $X(\\mathcal{L})$. The plateau leads to an inaccessible region between about $0.8$ and $1$ and an atom of probability mass at $X(0.5) \\approx 0.8$, and so $X$ is not uniformly distributed.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{compression.pdf}\n    \\caption{The compression from ordinary and modified NS when $q$ of 100 live points lie in a plateau. The correct linear compression is shown for reference (solid blue).}\n    \\label{fig:approximation}\n\\end{figure}\n\nIndeed, in ordinary NS, if $q$ outermost live points were in a plateau, we would compress by \n\\begin{equation}\\label{eq:ns_q_compression}\n    e^{-\\frac{q}{n_\\text{live}}}.\n\\end{equation}\nWe should, however, compress by about\n\\begin{equation}\\label{eq:expected_compression}\n   1 - \\frac{q}{n_\\text{live}},\n\\end{equation}\nwhich is an unbiased estimate of the volume outside the plateau based on binomial statistics (see \\cref{sec:distributions} for further discussion). The compressions are only similar for $q \\lesssim 0.5 n_\\text{live}$, since\n\\begin{equation}\n    e^{-\\frac{q}{n_\\text{live}}} \\approx 1 - \\frac{q}{n_\\text{live}} + \\mathcal{O}\\left(\\frac{q^2}{n_\\text{live}^2}\\right).\n\\end{equation}\nThe breakdown in the NS compression in \\cref{eq:ns_q_compression} is shown in \\cref{fig:approximation}. Note that this problem doesn't impact importance nested sampling \\citep{Feroz:2013hea}, since it does not use the estimated volumes in \\cref{eq:ns_q_compression}.\n\nIn fact, the arguments above show that in the presence of plateaus the inverse of $X(\\lambda)$, denoted $\\mathcal{L}(X)$ in overloaded notation, does not exist for all $ 0 \\le X \\le 1$ (see the lower right panel in \\cref{fig:plateau_infographic}). As shown in \\citet{2020arXiv200508602S}, in this case we should instead consider the generalised inverse\n\\begin{equation}\n    \\bar\\mathcal{L}(X) \\equiv \\left\\{\\sup \\lambda \\in \\image \\mathcal{L} : X(\\lambda) > X \\right\\},\n\\end{equation}\nwith which the evidence may be written as\n\\begin{equation}\\label{eq:generalized}\n    \\mathcal{Z} = \\int_0^1 \\bar\\mathcal{L}(X) dX.\n\\end{equation}\nWe now introduce a modified NS algorithm that correctly computes the evidence even in the presence of plateaus via \\cref{eq:generalized}.  We summarise the treatment of plateaus in existing popular NS software in \\cref{tab:codes}.\n\n\\begin{table*}\n\\begin{tabular}{llc}\n\\hline\nCode & Handles plateaus & Definition of constrained prior\\\\\n\\hline\n\\textsf{nestle-0.2.0}~\\citep{nestle} & No & $\\mathcal{L} \\ge \\like^\\star$\\\\\n\\textsf{dynesty-1.0.0}~\\citep{2020MNRAS.493.3132S} & No & $\\mathcal{L} \\ge \\like^\\star$\\\\\n\\textsf{DIAMONDS-\\href{https:\/\/github.com\/EnricoCorsaro\/DIAMONDS\/commit\/76409b22c9da782436b52e454a8b36bc78fca6f6}{\\#76409b2}}~\\citep{diamonds} & No & $\\mathcal{L} \\ge \\like^\\star$\\\\\n\\textsf{MultiNest-3.1.2}~\\citep{Feroz:2007kg,Feroz:2008xx,Feroz:2013hea}  & No but compatible with \\code{anesthetic} & $\\mathcal{L} > \\like^\\star$\\\\\n\\textsf{PolyChord-1.18.2}~\\citep{Handley:2015fda,Handley:2015xxx} & No but compatible with \\code{anesthetic} & $\\mathcal{L} > \\like^\\star$\\\\\n\\textsf{DNest4-0.2.4}~\\citep{2016arXiv160603757B} & Yes by $\\mathcal{U}(0, 1)$ tie-breaking labels & $\\mathcal{L} > \\like^\\star$\\\\\nSkilling's implementation~\\citep{skilling_code} & Yes by user-specified tie-breaking labels & $\\mathcal{L} > \\like^\\star$\\\\\n\\hline\n\\end{tabular}\n\\caption{\\label{tab:codes} Comparison of treatment of plateaus in popular NS software. The example program in Skilling's implementation specifies $\\mathcal{U}(0, 1)$ tie-breaking labels.}\n\\end{table*}\n\n\\section{Modified NS algorithm}\\label{sec:modified_ns}\n\nIn our modification to NS, we remove \\emph{all} live points at the contour $\\mathcal{L} = \\like^\\star$ \\emph{one by one without replacement}, contracting the volume after each removal. If there is a plateau, there may be more than one such point; if not, our algorithm reduces to ordinary NS. After removing all such points, we finally replenish the live points by adding points sampled from the prior subject to $\\mathcal{L} > \\like^\\star$, as usual. \n\nAfter removing a point, the number of live points drops by one, such that if we were to remove $i=1, \\ldots, q$ such points (i.e., if there were $q$ points in the plateau) we would compress by\n\\begin{equation}\\label{eq:arithmetic}\n\\begin{split}\n\\prod_{i=1}^q \\expectation{t_i} &= \\frac{n_\\text{live}}{n_\\text{live} + 1} \\cdot \\frac{n_\\text{live} - 1}{n_\\text{live}} \\cdots \\frac{n_\\text{live} - (q - 1)}{n_\\text{live} - (q - 2)}\\\\\n                                    &= 1 - \\frac{q}{n_\\text{live} + 1} \n\\end{split}\n\\end{equation}\nif using an arithmetic estimate of the compression, and by\n\\begin{align}\\label{eq:geometric}\n\\sum_{i=1}^q \\expectation{\\ln t_i} &= -\\sum_{i=1}^q \\frac{1}{n_\\text{live} - (i - 1)}\\\\ \n                                   &\\approx \\ln\\left(1 - \\frac{q}{n_\\text{live}}\\right) \n                                  \n                                   \\quad\\text{for $n_\\text{live} \\gg q$}\\label{eq:approx_geometric}\n\\end{align}\nif using a geometric one.\nThus we find in both cases that the compression from removing $q$ points would be about $1 - q \/ n_\\text{live}$ in agreement with \\cref{eq:expected_compression}, the difference being of order $1 \/ n_\\text{live}$. The difference is noteworthy when $q \\simeq n_\\text{live}$, in which case the unbiased estimate of the contraction would be zero, but the above estimates are about $1 \/ n_\\text{live}$. \n\n\n\n\\input{algorithms}\n\nIn \\cref{algo:original_ns} and \\cref{algo:modified_ns} we show the original and our modified NS, respectively. For concreteness we show the geometric estimator of the compression. We highlight the parts of the algorithm that are changed in red. The simple difference is that whereas in the original NS we replace a single live point, we instead replace \\emph{all} the points sharing the minimum likelihood and contract the volume appropriately. If there are no plateaus, the algorithms are identical. \n\nThis modified NS algorithm automatically deals with plateaus whenever they occur and can be applied retrospectively to NS runs. It automatically implements the robust NS algorithm of \\citet{2020arXiv200508602S}, but with the advantage that it avoids the requirement that we explicitly decompose the evidence integral into a sum of contributions from plateaus and non-trivial contributions to be computed by NS. Other advantages are that our modification fits elegantly into ordinary NS, since the modifications are small, and that it is in the spirit of dynamic NS~\\citep{2019S&C....29..891H}, since taking into account the changing number of live points is crucial.\n\nWe could, alternatively, evict all live points in the outermost contour, compress by $1 - q \/ n_\\text{live}$, and finally top up the live points. We don't favour this implementation of our idea, as it breaks the equivalent treatment of plateaus and non-plateaus in \\cref{algo:modified_ns} and our use of ordinary NS compression factors, but it remains a valid possibility. This would use the same estimates of the plateau size as \\citet{2020arXiv200508602S}. Lastly, we could alternatively remove plateaus entirely from the likelihood function by e.g., using the extended likelihood function in \\cref{eq:extened_like}. If the tie-breaking terms, $\\ell$, are random variates, they must be independent and identically distributed \\citep[appendix B.5.4.3]{riley}. This, however, cannot be applied retrospectively to NS runs, and, as discussed in \\citet{riley}, might lead to problems in specific NS implementations. \n\nIndeed, when using a random variate to break ties in a plateau, a new live point could lie anywhere in the plateau as for any point in the plateau we are equally likely to draw a tie-breaking random variate that leads to acceptance. Sampling from the constrained prior thus requires sampling from the whole plateau and contours above the plateau. This becomes unavoidably inefficient for large plateaus, as almost all tie-breaking random variates would lead to rejection. NS implementations that attempt to increase the sampling efficiency thus easily lead to faulty estimates of the evidence.\n\nFor example, in ellipsoidal rejection sampling, as used in \\code{MultiNest}\\xspace, one samples from ellipsoids that are constructed to enclose the current live points and with a volume similar to the current estimated volume, $X_i$, so that the ellipsoids typically shrink during an NS run. In the case of plateaus, this could easily lead to sampling from only a subregion of the whole plateau, as the ellipsoids could shrink as we compress through a plateau. In slice sampling, as used in \\code{PolyChord}\\xspace, we step out from a current live point to find the likelihood contour. To sample from the whole plateau, we should step out until at least the edge of the plateau. This, however, could fail, as the tie-breaking random variate when stepping out that far could by chance lead to rejection and thus we wouldn't step out far enough. These problems could possibly be avoided by elevating the tie-breaking random variate to a model parameter with a $\\mathcal{U}(0, 1)$ prior. For ellipsoidal sampling in the presence of plateaus, this would allow the ellipsoids to contract only in that dimension and without encroaching on the plateau.\n\n\\subsection{Distribution of compression factor}\\label{sec:distributions}\n\nIn our modified NS, we apply ordinary NS compression factors taking into account the dynamic number of live points. This assumes beta distributions for the compression factor. When dealing with plateaus, \\cite{2020arXiv200508602S} consider estimating the compression using the fact that the number of points inside the plateau should follow a binomial distribution parameterized by the size of the plateau. Let us consider more carefully the differences in estimates of the compression factor from these two approaches. In the latter, the number of live points, $q$, that fall in the outermost contour plateau follows a binomial\n\\begin{equation}\n    q \\sim \\BinomDist(n_\\text{live}, 1 - t) \n\\end{equation}\nwhere  $1 - t$ is the size of the plateau and thus $t$ is the compression factor. The probability mass function is thus\n\\begin{equation}\n  \\Pg{q}{t} \\propto t^{n_\\text{live} - q} \\, (1 - t)^q,\n\\end{equation}\nfor $q$ points in the plateau and $n_\\text{live} - q$ points above it. We could make inferences about $t$ by computing a posterior distribution,\n\\begin{equation}\\label{eq:binom}\n \\pg{t}{q} \\propto \\Pg{q}{t} \\p{t} \\propto t^{n_\\text{live} - q} \\, (1 - t)^q \\, \\p{t},\n\\end{equation}\nwhere $\\p{t}$ is our prior for the size of the non-plateau region. For $ \\p{t} = \\text{const.}$ this is in fact the probability density for a $t \\sim \\BetaDist(n_\\text{live} + 1 - q, q + 1)$ distribution.\n\nOn the other hand, taking the approach in modified NS, the number of points that lie in the plateau $q$ is no longer treated as a random variable. Instead, the factor $t$ is assumed to follow a beta distribution when $q$ points are removed~\\citep{2014AIPC.1636..100H},\n\\begin{equation}\nt \\sim \\BetaDist(n_\\text{live} + 1 - q, q).\n\\end{equation}\nThe density for $t$ is\n\\begin{equation}\\label{eq:beta}\n  \\pg{t}{q} \\propto t^{n_\\text{live} - q} \\, (1 - t)^{q - 1}\n\\end{equation}\ncorresponding to $q - 1$ points below $\\like^\\star$, $n_\\text{live} - q$ points above it, and \\emph{one point at $\\like^\\star$.} \n\nThe fact that we must consider $q - 1$ points below $\\like^\\star$ and one point at $\\like^\\star$, rather than $q$ points inside a plateau, results in a factor of $(1 - t)$ difference between \\cref{eq:binom,eq:beta}. A further factor of $\\p{t}$ originates from any prior knowledge about the size of the plateau. If the factors may be neglected inferences based on ordinary NS compression may be reliable. \n\nWith a flat prior for the unknown size of the plateau, the difference is that between a $\\BetaDist(n_\\text{live} + 1 - q, q)$ and a $\\BetaDist(n_\\text{live} + 1 - q, q + 1)$ distribution. As shown in in \\cref{fig:uncertainty}, the first two moments are similar, with only moderate differences of order $1 \/ n_\\text{live}$ even when $q \\simeq 1$ and $q\\simeq n_\\text{live}$.\\footnote{The fact that the distributions as functions of $t$ are approximately identical is enough to ensure that our inferences are approximately identical~\\citep{berger1988likelihood}. This holds despite the fact that in the binomial case the number of live points in the outermost plateau is a random variable and the size of the plateau $1 - t$ is fixed, and in the beta case the compression factor $t$ is a random variable and $q$ is fixed. This needn't be true for frequentist estimates of the factor $t$.}\nThe factor vanishes entirely for a logarithmic prior for the unknown size of the plateau,\n\\begin{equation}\n    \\p{t} \\propto \\frac{1}{1 - t} \\quad\\text{and so}\\quad \\p{f} \\propto \\frac1f.\n\\end{equation}\nThus we see that our modified NS treatment is approximately the same as the binomial treatment, the full details of which would depend on a prior for the size of the plateau. If the differences are important, though, we could instead use the binomial statistics, as discussed at the end of \\cref{sec:modified_ns}, and a more careful treatment of the prior.\n\n\\subsection{Error estimates}\\label{sec:errors}\n\nThe classic NS error estimate\n\\begin{equation}\n    \\Delta \\log\\mathcal{Z} \\approx \\sqrt{\\frac{H}{n_\\text{live}}},\n\\end{equation}\nwhere $H$ is the Shannon entropy between the posterior and prior, assumes the ordinary NS compression and a constant number of live points, and so is not applicable to our modified NS. The \\code{anesthetic}\\ error estimates, however, already account for a dynamic number of live points, and so are applicable to the modified NS algorithm. The \\code{anesthetic}\\ error estimates are found  \\citep[as initially suggested by][]{Skilling:2006gxv} through simulations; sequences of possible compression factors are drawn from beta-distributions and used to make a set of estimates $\\ln\\mathcal{Z}$. One could alternatively compute an analytic estimate using the equivalents of the arithmetic expressions in \\cref{eq:t} for the variance, which can be found in Appendix B of \\citet{Handley:2015xxx}. Future versions of \\texttt{anesthetic} will also support such analytic estimates. \n\nClearly, however, compression estimates for plateaus suffer from increased uncertainties, since we dynamically reduce the number of live points during the plateau period of an NS run. In fact, if the plateau at $\\ensuremath{\\like_P}$ makes up a large fraction $f$ of the current prior volume, the error in the estimated compression could be substantial, as few points lie outside the plateau. Indeed, we show that the fractional error in the compression blows up when $q \\simeq n_\\text{live}$ in \\cref{fig:uncertainty}. In such cases, there may exist efficient schemes for dynamically increasing the number of live points to ensure that sufficient points lie above the plateau. For example, in \\cref{algo:modified_ns} we must ultimately replenish the live points such that there are $n_\\text{live}$ live points at  $\\mathcal{L} > \\ensuremath{\\like_P}$. We could instead dynamically increase the number of live points immediately prior to the plateau by sampling them from $\\mathcal{L} \\ge \\ensuremath{\\like_P}$, stopping at $n_\\text{dynamic}$ once $n_\\text{live}$ of the $n_\\text{dynamic}$ points lie at $\\mathcal{L} > \\ensuremath{\\like_P}$. Once we remove points from the plateau, there would be $n_\\text{live}$ live points remaining, and no need to top up the live points.\n\nIn the worst case scenario, no live points land in contours above the plateau, e.g., the likelihood function shows a tiny peak above a broad plateau. At this point, it would be unclear whether to proceed and if so so how to do so efficiently, since the live points in the plateau provide no clues about the presence or location of the peak. This problem would affect all NS variants known to us.\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{uncertainty.pdf}\n    \\caption{Estimates of compression for $n_\\text{live} = 100$ and $q$ points in the plateau from the binomial (blue) and beta (red). The estimates are shown relative to the ones from the binomial. The dashed lines show one standard deviation uncertainties.} \n    \\label{fig:uncertainty}\n\\end{figure}\n\n\\section{Examples}\\label{sec:examples}\n\nWe now consider a few examples. First, in \\cref{sec:examples_simple} we apply our modified NS to examples considered in \\citet{2020arXiv200508602S}. Second, in \\cref{sec:wedding_cake} we construct a `wedding-cake' function that exhibits a series of plateaus and check that our modified NS correctly computes the evidence.\n\n\\subsection{Examples from \\protect\\citet{2020arXiv200508602S}}\\label{sec:examples_simple}\n\nFor plateaus at the base of the likelihood function at $\\mathcal{L} = 0$, we overestimate the evidence by a factor\n\\begin{equation}\n    \\Delta\\log\\mathcal{Z} = \\log\\left(\\frac{e^{-f}}{1 - f}\\right) = -\\log(1 - f) - f\n\\end{equation}\nwhere $f$ is the fraction of the prior volume occupied by the plateau. For example, in Scenario 2 of \\citet{2020arXiv200508602S}, the plateau occupies $f = 2\/3$ of the prior volume at the base of the likelihood, and we find $\\Delta\\log\\mathcal{Z} \\approx 0.432$, in good agreement with the difference found numerically in \\citet{2020arXiv200508602S}, which was $0.433$. \nIn \\cref{fig:hist}, we check the results of this problem with modified NS in 1000 repeat runs with 500 live points. We find that the histogram of $\\ln\\mathcal{Z}$ estimates form a Gaussian peak (red bars) around the analytic result (dashed blue). The spread was well-described by the average uncertainty estimates from single runs of modified NS (green). The original NS results (orange), on the other hand, lie well away from the analytic result, as expected.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{hist.pdf}\n    \\caption{Repeated modified NS runs on Scenario 2 of \\citet{2020arXiv200508602S}. We show the results from 1000 runs (red histogram) and the average mean and uncertainty estimate from the single modified NS runs as a Gaussian (green). For reference, we show the original NS result (orange) and the analytic result (dashed blue). }\n    \\label{fig:hist}\n\\end{figure}\n\nFor plateaus at the peak of the likelihood function at $\\mathcal{L} = \\max\\mathcal{L}$, the error depends on the stopping conditions and treatment of final live points in NS. In e.g., the \\code{MultiNest}\\xspace implementation of NS, the run converges if all live points share the same likelihood, and the remaining evidence is correctly accounted for. If the run halts but the remaining evidence isn't accounted for, we underestimate the evidence by a term\n\\begin{align}\n\\Delta\\mathcal{Z} &= f \\max\\mathcal{L}\\\\\n\\Delta\\log\\mathcal{Z} &= \\log \\left(\\mathcal{Z} + \\Delta\\mathcal{Z}\\right) - \\log\\mathcal{Z}\n\\end{align}\nwhere $f$ is the fraction of the prior volume occupied by the plateau. For example, in Scenario 3 of \\citet{2020arXiv200508602S}, the plateau occupies $f \\simeq 0.161$ of the prior volume at the peak of the likelihood at $\\max\\log\\mathcal{L} \\approx -2.21$, and we find $\\Delta\\log\\mathcal{Z} \\approx 1.006$, in good agreement with the difference found numerically in \\citet{2020arXiv200508602S}, which was $1.007$.\n\n\\subsection{Wedding cake likelihood}\\label{sec:wedding_cake}\n\nWe now construct a semi-analytical example for which we can numerically confirm our approach. Consider\nan infinite sequence of concentric square plateaus of geometrically decreasing volume $\\alpha^i (1-\\alpha)$ for $i = 0, 1, 2, \\ldots$. The edges of the plateaus lie at \n\\begin{equation}\n    r_i = \\alpha^{i\/D}\/2 \\quad\\text{and}\\quad r = \\left|\\boldsymbol{x} - 1\/2\\right|_\\infty\n\\end{equation}\nfor $i = 0, 1, 2, \\ldots$, where $|\\boldsymbol{x}|_\\infty \\equiv \\max_j(|x_j|)$ denotes the infinity norm. We define the height of each plateau to have a Gaussian profile:\n\\begin{equation}\n    \\log\\mathcal{L} = -\\sum_{i=0}^{\\infty} \\frac{r_i^2}{2\\sigma^2}  \\, \\mathds{1}_{r_{i+1} < r \\le r_{i}}\n\\end{equation}\nwhere $\\mathds{1}$ is an indicator function that, for any given $r$, selects a single term in the sum. The resulting likelihood is therefore a set of hypercuboids with a hypercubic base of side length $r_i$ and height $\\exp(-{r_i^2}\/{2\\sigma^2})$. If the base is two-dimensional, this creates a tiered ``wedding cake'' surface, as can be seen in \\cref{fig:wedding_cake}.\nThe $i$ selected by the indicator function is in fact,\n\\begin{equation}\n    i(r) = \\left\\lfloor D\\log_\\alpha 2 r \\right\\rfloor\n\\end{equation}\nwhere $\\left\\lfloor y \\right\\rfloor$ the floor function (namely the greatest integer less than or equal to $y$), \nenabling us to write\n\\begin{equation}\n     \\log\\mathcal{L} = - \\frac{\\alpha^{2 i(r) \/ D} }{8\\sigma^2}   \n\\end{equation}\nGiven that the volume of the region $[r_i < r < r_{i-1}]$ is $\\alpha^{i}(1-\\alpha)$, the evidence can be expressed as:\n\\begin{equation}\n    \\mathcal{Z} = \\sum_{i=0}^\\infty e^{-\\alpha^{2i\/D}\/8\\sigma^2} \\alpha^i (1-\\alpha) \n\\end{equation}\nwhich as an infinite series converges sufficiently rapidly and stably to be evaluated numerically for any number of dimensions $D$, but if speed is a requirement then a Laplace approximation shows that one only needs to consider the terms in the series around  \n\\begin{equation}\n    i \\sim\\sqrt{\\frac{D}{2}}\\frac{\\log (4 D \\sigma^2) - 1 \\pm \\mathcal{O}(\\text{a few})}{\\log \\alpha}.\n\\end{equation}\n\nFor reference, putting all of these equations together, the likelihood can be computed as:\n\\begin{equation}\n    \\log\\mathcal{L}(\\boldsymbol{x}) = - \\frac{\\alpha^{2  \\left\\lfloor D\\log_\\alpha 2 \\left|\\boldsymbol{x} - 1\/2\\right|_\\infty  \\right\\rfloor \/ D} }{8\\sigma^2}   \n\\end{equation}\nwhere $\\alpha$ is a hyperparameter controlling the depth of the plateaus, and $\\sigma$ controls the width of the overall Gaussian profile.\n\nThis likelihood forms part of the \\texttt{anesthetic} test suite, which confirms that the approach suggested in this paper recovers the true likelihood.\n\nThe wedding cake likelihood can be very useful for testing nested sampling implementations as unlike a traditional Gaussian it can be trivially sampled from using a simple random number generator, has no unexpected edge effects as the boundaries of the prior are also a likelihood contour.\n\n\\begin{figure}\n   \\centering\n   \\includegraphics[width=0.95\\linewidth]{wedding_cake_like_2d.pdf}\n\n   \\includegraphics[width=0.95\\linewidth]{nlive.pdf}\n   \\caption{Top: Example of the wedding cake likelihood function in two dimensions for $\\alpha = 0.7$ and $\\sigma = 0.2$. Bottom: The number of live points used in the calculation of the evidence. The target value is $n_\\text{live}=100$, but in plateaus as points are discarded one-by-one without replacement this causes $n_\\text{live}$ to drop until the plateau is passed. The variability in the number of points discarded can be seen by the varying depth of each local minimum in the number of live points.}\n    \\label{fig:wedding_cake}\n\\end{figure}\n\n\\section{Conclusions}\\label{sec:conclusions}\n\nFollowing from \\citet{riley,2020arXiv200508602S}, which showed formally why NS breaks down if there are plateaus in the likelihood, we first presented the problem of plateaus in an informal but accessible way. We then constructed a modified version of the NS algorithm. The simple modification permits it to remove points in a plateau one by one, without replacement.  Once all points in a plateau are removed, the live points are finally replenished. This leads to correct compression in the presence of plateaus and ordinary NS in their absence.\n\nWe discussed examples from \\citet{2020arXiv200508602S}, shedding light on them by finding the impact of plateaus on ordinary NS in simple analytic formulae. The impact was previously shown only numerically. Lastly, our especially constructed wedding-cake problem showed a case with multiple plateaus. The modified NS algorithm successfully dealt with them.\n\nRuns from popular NS software such as \\code{PolyChord}\\xspace and \\code{MultiNest}\\xspace may be resummed retrospectively via the modified NS algorithm using \\textsf{anesthetic} starting from version \\code{\\href{https:\/\/github.com\/williamjameshandley\/anesthetic\/releases\/tag\/2.0.0-beta.2}{2.0.0-beta.2}}. Our modified NS makes a minimal change to ordinary NS, to the extent that we recommend it becomes the canonical version of NS.\n\n\\section*{Acknowledgements}\n\nAF was supported by an NSFC Research Fund for International Young Scientists grant 11950410509.\nWH was supported by a George Southgate visiting fellowship grant from the University of Adelaide, Gonville \\& Caius College, and STFC IPS grant number ST\/T001054\/1. We thank Brendon Brewer for valuable comments and discussion on the manuscript.\n\n\\section*{Data availability}\nThere is little data associated with this paper, though any data or code will be shared on reasonable request to the corresponding author.\n\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and Motivation}\t\\label{sec:intro}\n\nCircular Dichroism (CD) is defined as differential absorption of left and right circularly polarized photons, and it is widely used in the analysis of materials that are non-symmetric under mirror transformation either through preferred chirality of the material structure or through magnetic phenomena \\cite{Berova}.\n\nTwisted photons, or optical vortices, carry OAM along their direction of propagation, and therefore they can be characterized by their own chirality (or topological charge); see, e.g., \\cite{Padgett2015} for a recent review. Spin-dependence of OAM photon flux enters through a spin-orbit interaction \\cite{BliokhAlonso10}.\nInteractions of the twisted photons with non-chiral nano-structures were reported in Ref. \\cite{Zambrana14, Zambrana16}, showing  significant CD, at 80 to 90 per cent level, for varied topological charges. While the observed large CD can be understood in principle by realizing that left and right circular polarization states are not mirror-symmetric for twisted photons, the observed effect still awaits theoretical explanation, likely in terms of surface plasmon dynamics \\cite{Zambrana16}. Theoretical predictions of CD were previously made for twisted X-rays in metals \\cite{PhysRevLett.98.157401}. \n\nIn the present work we apply previously developed formalism for photo-excitation of atoms by the twisted photons\n \\cite{Afanasev:2013kaa,Afanasev2014} and predict CD effects in the atomic matter. We take advantage is the observation \n \\cite{Afanasev:2013kaa,Scholz2014,Afanasev2016} that the transition amplitudes of atomic photo-excitation with twisted photons can be presented in a simple factorized form in terms of plane-wave photon amplitudes, making our predictions applicable to a variety of atomic targets. We present the results as a function of the distance between a given atom (or ion) to the center of the optical vortex, that we define as {\\it an impact parameter} $b$. Precise measurements of $^{40}$Ca$^+$ ion excitations with the twisted light as a function of the impact parameter with sub-wavelength position resolution were performed recently \nRef.\\cite{Schmiegelow2016} using an ion trap. For the most recent review of twisted-light interactions with atoms, see \\cite{Franke-Arnold2017} and references therein.\n\nSections II and III of this paper review the formalism for twisted photons and for calculating atomic photoexcitation with the twisted photons, respectively.  Sec. IV introduces CD and discusses photon spin effects in the the photoexcitation rates and cross sections, Sec.V describes the evolution of the twisted-photon polarization caused by absorption, Sec. VI provides the analytic expressions in paraxial limit,  and Sec. VII offers some closing comments.\n\n\n\\section{Definition of Twisted-Photon States}\t\t\t\\label{sec:one}\n\n\n\nWe define the twisted-photon states as non-paraxial Bessel beams according to~\\cite{Jentschura:2010ap,Jentschura:2011ih}, that can be viewed as extensions of the nondiffractive Bessel modes described in~\\cite{Durnin:1987,Durnin:1987zz}. More detail is given in~\\cite{Afanasev:2013kaa}.   These states correspond to superposition of TE and TM Bessel modes introduced in Ref. \\cite{Jauregui:2004}; see also Appendix of Ref.\\cite{Afanasev2014} for detailed comparison.\n\nA twisted photon state with symmetry axis passing through the origin, can be given as a superposition of plane waves and in Hilbert space can be written as, \n\\begin{align}\n\\label{eq:twisteddefinition}\n| \\kappa m_\\gamma k_z \\Lambda \\rangle \n&= \\sqrt{\\frac{\\kappa}{2\\pi}} \\  \\int \\frac{d\\phi_k}{2\\pi} (-i)^{m_\\gamma} e^{im_\\gamma\\phi_k}  \\,\n\t|\\vec k, \\Lambda\\rangle\t\t\\,.\n\\end{align}\nThe component states on the right are plane-wave states, all with the same longitudinal momentum $k_z$, the same transverse momentum magnitude $\\kappa = |\\vec k_\\perp|$, and the same plane wave helicity $\\Lambda$ (in the directions $\\vec k$).  The angle  $\\phi_k$  is the azimuthal angle of vector $\\vec k$, and with the phasing shown, $m_\\gamma$ is the total angular momentum in the $z$ direction. We also define a pitch angle   $\\theta_k = \\arctan (\\kappa\/k_z)$, and $\\omega = | \\vec k |$.  The pitch angle $\\theta_k$ was first introduced in the definition of Bessel beams in Ref.\\cite{Durnin:1987,Durnin:1987zz}, and it is related to Berry phase of the photon as \\cite{BliokhAlonso10}: $\\Phi_B=2\\pi(1-\\cos\\theta_k)$. The phase singularity of this beam is located on the beam symmetry axis.\n\nThe electromagnetic potential of the twisted photon in coordinate space is \n\\begin{align}\n\\label{eq:twistedwave}\n\\mathcal A^\\mu_{\\kappa m_\\gamma k_z \\Lambda}(t,\\vec r)\n&= \\sqrt{\\frac{\\kappa}{2\\pi}} \\, e^{-i\\omega t} \\nonumber\\\\\n&\\times\t\\int \\frac{d\\phi_k}{2\\pi} (-i)^{m_\\gamma} e^{im_\\gamma\\phi_k}  \\,\n\t\\epsilon^\\mu_{\\vec k,\\Lambda} e^{i \\vec k {\\cdot} \\vec r}\t.\n\\end{align}\nThe polarization vectors are \n\\be\n\\label{eq:epsilonexpand}\n\\epsilon^\\mu_{\\vec k \\Lambda} \\!\\! = \\!\n\te^{-i\\Lambda\\phi_k} \\! \\cos^2\\frac{\\theta_k}{2} \\eta^\\mu_\\Lambda\n\t+ e^{i\\Lambda\\phi_k} \\! \\sin^2\\frac{\\theta_k}{2} \\eta^\\mu_{-\\Lambda}\n\t+ \\frac{\\Lambda}{\\sqrt{2}} \\sin\\theta_k  \\eta^\\mu_0\n\\ee\nwith $4$-dimensional unit vectors,\n\\be\n\\eta^\\mu_{\\pm 1} = \\frac{1}{\\sqrt{2}}  \\left( 0,\\mp 1,-i,0 \\right)\t\\,,\n\\quad \\eta^\\mu_0 =  \\left( 0,0,0,1 \\right)\t\\,.\n\\ee\nThe electromagnetic potential then has a form\n\\begin{align}\n\\label{eq:twistedwf}\n\\mathcal A^\\mu_{\\kappa m_\\gamma k_z \\Lambda}(x) &= e^{-i(\\omega t - k_z z)}\t\n\\sqrt{\\frac{\\kappa}{2\\pi}}  \\, \\nonumber\\\\ &\t\\Bigg\\{\n\t\\frac{\\Lambda}{\\sqrt{2}} e^{im_\\gamma\\phi_\\rho} \\sin\\theta_k\n\tJ_{m_\\gamma}(\\kappa\\rho) \\, \\eta^\\mu_0\t\t\t\\nonumber\\\\[1ex]\n& \\quad + i^{-\\Lambda} e^{i(m_\\gamma-\\Lambda)\\phi_\\rho}  \\cos^2\\frac{\\theta_k}{2} \n\tJ_{m_\\gamma-\\Lambda}(\\kappa\\rho) \\, \\eta^\\mu_\\Lambda\t\\nonumber\\\\[1ex]\n& \\quad + i^{\\Lambda}  e^{i(m_\\gamma+\\Lambda)\\phi_\\rho}  \\sin^2\\frac{\\theta_k}{2} \n\tJ_{m_\\gamma+\\Lambda}(\\kappa\\rho) \\, \\eta^\\mu_{-\\Lambda}\n\t\\Bigg\\}\t\\,.\n\\end{align}\n\nThe energy flux is given by\n\\begin{align}\n\\label{eq:flux}\nf(\\rho)  &= \\cos(\\theta_k) (|E|^2+|B|^2)\/4= \n \\cos(\\theta_k) \\frac{\\kappa\\omega^2 }{2\\pi} \\nonumber\\\\ &\n \\Bigg\\{ \\cos^4\\frac{\\theta_k}{2} J^2_{m_\\gamma-\\Lambda}(\\kappa\\rho) +\\sin^4\\frac{\\theta_k}{2} J^2_{m_\\gamma+\n\\Lambda}(\\kappa\\rho) \\\\ &\n+\\frac{\\sin^2\\theta_k}{2}J^2_{m_\\gamma}(\\kappa\\rho) \\Bigg\\} . \\nonumber \n\\end{align}\nThe use of the above canonical-momentum expression is essential, since Poynting vector alone does not represent the full energy flux of a twisted photon beam \\cite{BliokhAlonso10,Bliokh2015}.\n\n\n\n\n\n\n\\section{Plane-wave factorization for atomic photoexcitation with twisted photons}\t\t\t\n\n\nHere we briefly review the formalism of atomic photoexcitation by the twisted photons worked out previously \\cite{Afanasev:2013kaa,Afanasev2014,Scholz2014, Afanasev2016}, leading to plane-wave factorization property of the twisted-photon absorption.\n\nConsider the excitation by a twisted photon of an atom. The photon's wave front travels in the $z$-direction and the axis of the twisted photon is displaced from the nucleus of the atomic target by some distance in the $x$-$y$ plane which we will call an impact parameter $\\vec b$, Fig.\\ref{fig:Atom}. The transition matrix element is\n\\begin{align}\n\\label{eq:Smatrix}\nS_{fi} &= -i \\int dt  \\langle n_f l_f m_f | H_1 \n\t| n_i l_i m_i; \\kappa m k_z \\Lambda \\rangle\t\t\t\t\\,,\n\\end{align}\nwhere the non-relativistic interaction Hamiltonian is given by\n\\be\n\\label{eq:hamiltonian}\nH_1 = - \\frac{e}{m_e} \\vec A \\cdot \\vec p \\,,\n\\ee\nand we use standard notation $(n,l,m)$ for the principal, orbital and magnetic quantum numbers of initial and final states of an atom.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics{offaxiscoord.eps}\n\\caption{Relative positions of atomic state and photon axis, as projected onto the $x$-$y$ plane, with the origin at the nucleus of the atom.}\n\\label{fig:Atom}\n\\end{center}\n\\end{figure}\n\n\nIt can be shown that  Ref.\\cite{Afanasev:2013kaa,Scholz2014,Afanasev2016} that for atomic excitation from the ground state ($l_i=m_i=0$) the above amplitude from Eq.(\\ref{eq:Smatrix}) is proportional to the plane-wave amplitude $\\mathcal M^{\\rm (pw)}$ times Wigner $d$-functions that only depend on the pitch angle $\\theta_k$ and Bessel functions that define the amplitude dependence on the impact parameter $b$. Here, the Bessel factor arises due to the azimuthal phase dependence of the twisted-photon and the excited atomic state, and Wigner $d$-functions is a result of a tilted quantization axis (by an angle $\\theta_k$) with respect to the direction of beam propagation:\n\\begin{align}\n\\label{eq:factorized}\n&|{\\cal M}_{n_f l_f m_f \\Lambda}(b)|   =  \\nonumber \\\\  &\\left |\\sqrt{\\frac{\\kappa}{2\\pi}}J_{m_f-m_\\gamma}(\\kappa b)d^{l_f}_{m_f\\Lambda}(\\theta_k) \\mathcal M^{\\rm (pw)}_{n_f l_f \\Lambda \\Lambda}(\\theta_k=0)\\right |\n\\end{align}\n\nThe factorized form of the transition amplitude facilitates comparison of twisted-photon vs plane-wave absorption by atoms. It contains the details of atomic structure in a common-factor plane-wave amplitude $\\cal M^{\\rm (pw)}$, while the novel features arising from the phase and spatial structure of the twisted light are contained in Wigner and Bessel functions that enter  independently of the specific details of atomic wave functions.  When deriving the factorization property Eq.(\\ref{eq:factorized}), we used the first-order Born matrix element that assumes the interaction proceeds in the linear regime, justifying representation of S-matrix as a linear superposition of plane-wave matrix elements. To further prove total angular momentum conservation under photo-absorption, we previously assumed that the atom is much smaller than the wavelength of absorbed light \\cite{Afanasev2014}, but this assumption is not needed to derive Eq.(\\ref{eq:factorized}). Therefore as long as the linear interaction regime holds, the above formula is applicable to twisted-photon excitation of arbitrary quantum systems, such as atoms, molecules, ions, atomic nuclei, excitons or quantum dots. For example taking the limit $\\theta_k\\to\\pi\/2$ in Eq.(\\ref{eq:factorized}), we recover a similar factorization property recently derived for the absorption of polariton vortices, $c.f.$ Eq.(2) of  \\cite{Machado2016}.\n\n\n\\section{Spin-Dependence and Circular Dichroism of Twisted-Photon Absorption}\n\nThe twisted-photon flux depends on the sign of $\\Lambda$ defining the handedness of plane-wave photons that form a given Bessel beam, and this dependence was discussed previously by Bliokh and collaborators in the context of spin-orbit interaction of light \\cite{BliokhAlonso10,BliokhNature15}.\n\nThe photo-excitation cross section of is given by \n\\begin{equation}\n\\label{eq:xsec}\n\\sigma^{(m_\\gamma)}_{n_f l_f \\Lambda} = 2 \\pi \\delta(E_f -E_i-\\omega_\\gamma)\n \\frac{ \\sum_{m_f = - l_f}^{m_f=l_f}|\\mathcal M^{(m_\\gamma)}_{n_f l_f m_f \\Lambda}(b)|^2 }{f} \\,.\n\\end{equation}\nwhere summation over final spins and averaging over initial spins is implied and the photon flux $f$ may be either unintegrated $f(b)$ (given by Eq.(\\ref{eq:flux})) or integrated over the transverse beam profile. For the excitation rates $\\Gamma$ one removes the flux $f$ from the above expression, $i.e.$\n\\begin{equation}\n\\label{eq:rate}\n\\Gamma^{(m_\\gamma)}_{n_f l_f \\Lambda}=f\\cdot \\sigma^{(m_\\gamma)}_{n_f l_f  \\Lambda} .\n\\end{equation}\n\nCircular dichroism is defined as a differential absorption probability for left- vs right-circularly polarized light. It can be observed either by measurements of polarization dependence of light transmission through matter or by observing ellipticity acquired by a linearly polarized light beam as a result of the difference of absorption in $\\Lambda=+1$ and $\\Lambda=-1$ states.  \n\nLet us compare the rates and the cross sections of photo-absorption of the twisted photons with opposite signs of circular polarization $\\Lambda$, while keeping (paraxial-limit) OAM unchanged. It leads to a definition\n\\begin{equation}\n\\label{eq:CD}\nCD^{(\\overline m_\\gamma,l_f)}=\\frac{\\sigma^{(\\overline m_\\gamma+1)}_{n_f l_f; \\Lambda=1}-\\sigma^{(\\overline m_\\gamma-1)}_{n_f l_f; \\Lambda=-1}}\n{\\sigma^{(\\overline m_\\gamma +1)}_{n_f l_f; \\Lambda=1}+\\sigma^{(\\overline m_\\gamma-1)}_{n_f l_f; \\Lambda=-1}},\n\\end{equation}\nwhere $\\overline m_\\gamma$ defines twisted photon's topological charge that corresponds to its OAM projection in a paraxial limit.\n\nUsing Equations (\\ref{eq:flux},\\ref{eq:factorized}), we first consider electric dipole $E1$-transitions for which OAM of the atomic electron changes by one unit, and find CD to be identically zero for any twisted light beam of a given $\\overline m_\\gamma$. We also find from the same equations that CD is zero for beams with $\\overline m_\\gamma$=0 and any $l_f$, which is expected from parity considerations. \n\\begin{equation}\nCD^{(\\overline m_\\gamma,l_f=1)}=CD^{(\\overline m_\\gamma=0,l_f)}=0.\n\\end{equation}\n\nThe reason behind this null result for $E1$-transitions is that excitation rates $\\Gamma$ for these transitions are proportional to the photon flux $f(b)$, and even though they separately depend on the sign of $\\Lambda$, this dependence cancels in the cross section, yielding zero CD for isotropic atomic targets.\n\nThe calculation results for CD are shown in Fig.\\ref{fig:CD} for electric quadrupole $\\Delta l=2$  (a) and electric octupole $\\Delta l=3$ transitions (b) for a fixed pitch angle $\\theta_k$=0.1 rad. The calculations show large and positive values of CD near the phase singularity of the optical vortex $b \\to 0$, and CD is seen to die off at the atom's positions of about one photon wavelength and larger. It means that a twisted photon whose spin is {\\it aligned}  with its OAM have a higher relative probability to be absorbed by the atom. Further analyzing the dependence on the pitch angle $\\theta_k$ we find that CD becomes independent of this parameter in a broad range of moderately small angles, and remains the same in the paraxial limit $\\theta_k \\to 0$, see Fig.\\ref{fig:CDang}. If the experiment does not resolve the atom's position $b$, we would have to integrate over all impact parameters, which would result in  zero CD due to the fact that total excitation cross sections would not depend on a particular value of $m_\\gamma$ or $\\Lambda$, see Ref.\\cite{Afanasev:2013kaa}.\n \n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width = 84 mm]{xSecAsL2.pdf}\n\\includegraphics[width = 84 mm]{xSecAsL3.pdf}\n\\caption{Circular dichroism (CD) as a function of impact parameter $b$ (in units of photon wavelength $\\lambda$) for different values of $m_\\gamma$ for excitation of the atomic states with $l_f=2$ (a) and $l_f=3$ (b); the angle $\\theta_k$=0.1 rad. The curve styles denote the average photon's angular momentum projection: $\\overline m_\\gamma=0$ is the blue solid curve, $\\overline m_\\gamma=1$ is orange and dashed, $\\overline m_\\gamma=2$ is green and dotted, $\\overline m_\\gamma=3$ is red and dot-dashed. CD is zero for the beams with no OAM ($\\overline m_\\gamma$=0). }\n\\label{fig:CD}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width = 84 mm]{rateAngL1.pdf}\n\\includegraphics[width = 84 mm]{xSecAngL2.pdf}\n\\caption{CD as a function of a pitch angle $\\theta_k$ for a fixed impact parameter $b\/\\lambda=0.25$, for (a) electric dipole $E1$ transitions ($l_f=1$), which coincides with spin asymmetry of the photon flux Eq.(\\ref{eq:flux}) and (b) electric quadrupole $E2$ transitions ($l_f=2$). Different curves correspond to different topological charges $\\overline m_\\gamma$, the notation is as in Fig.\\ref{fig:CD}.}\n\\label{fig:CDang}\n\\end{center}\n\\end{figure}\n\nIn experiments that directly measure the atomic excitation rates as a function of the impact parameter $b$, the results are presented as rates (or Rabi frequencies as in Ref.\\cite{Schmiegelow2016}) normalized to the total laser-beam power within an aperture that is much wider that the wavelength of light. In this case a relevant observable would be $\\Lambda$-dependence of the {\\it photo-excitation rate}  that was analyzed theoretically in Ref.\\cite{Afanasev:2013kaa} (for the case of leading $E1$ transitions).\n\nWe define photon-spin asymmetry of the photo-excitation rate similarly to CD, \n\\begin{equation}\n\\label{eq:SpinAsym}\nA_\\Lambda^{(\\overline m_\\gamma,l_f)}=\\frac{\\Gamma^{(\\overline m_\\gamma+1)}_{n_f l_f;\\Lambda=1}-\\Gamma^{(\\overline m_\\gamma-1)}_{n_f l_f; \\Lambda=-1}}\n{\\Gamma^{(\\overline m_\\gamma +1)}_{n_f l_f; \\Lambda=1}+\\Gamma^{(\\overline m_\\gamma-1)}_{n_f l_f; \\Lambda=-1}}.\n\\end{equation}\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width = 84 mm]{rateAsL1.pdf}\n\\includegraphics[width = 84 mm]{rateAsL2.pdf}\n\\includegraphics[width = 84 mm]{rateAsL3.pdf}\n\\caption{Spin asymmetry of the photoexcitation rates for electric dipole E1 (a) quadrupole $E2$ (b) and octupole E3 (c) transitions. Different curves correspond to different topological charges $\\overline m_\\gamma$, the notation is as in Fig.\\ref{fig:CD}, and the angle $\\theta_k$=0.1 rad.}\n\\label{fig:rateAs}\n\\end{center}\n\\end{figure}\n\nThe results are presented in Fig.\\ref{fig:rateAs} for the atomic transitions into the states of $l_f=1,2,$ and 3 caused by the photons with  topological charges $\\overline m_\\gamma$=0 to 3.\nWe can see that the spin asymmetry of rates behaves differently from CD: The asymmetry is negative within the distance of about one wavelength near the optical vortex center and, with a few exceptions, reaches a value of -1 at the center. It means that the transitions at the vortex center are mainly caused by the twisted photons whose spin and OAM are {\\it anti-aligned}.  An apparent difference from positive-sign CD is due to the fact that there is a relatively denser flux of the {\\it anti-aligned} twisted photons at the vortex center for the same overall beam power.\n\n If the atom's position is not resolved, we would have to integrate over the position, resulting into zero spin asymmetry $A_\\Lambda$, similarly to the above result for CD. It implies that {\\it observation of spin-asymmetric absorption of twisted light by atomic matter requires localization of the target atoms within about light's wavelength}. It can be achieved, for example, by using nano-sized apertures, well-localized ions in Paul traps, or mesoscopic targets.\n\n\n\n\\section{Evolution of the Twisted-Light Polarization under Propagation in Matter}\n\nAbove predictions of nonzero CD for the twisted light being absorbed by atoms would lead to the evolution of twisted-photon polarization states.\nIndeed, let us represent an arbitrary polarization state as a superposition of $\\Lambda=1$ and $\\Lambda=-1$ spin states, or left- and right- circularly polarized states, for a given topological charge:\n\\begin{equation}\n\\label{eq:stateevol}\n|\\kappa \\overline m_\\gamma k_z>=c_{-} |\\kappa \\overline m_\\gamma k_z \\Lambda=-1>+c_{+} |\\kappa \\overline m_\\gamma k_z \\Lambda=1>\n\\end{equation}\nwhere $c_{\\pm}$ are complex coefficients. Their dependence on the propagation distance $z$ is controlled by the attenuation coefficients $\\mu_{\\pm}$, that in turn are proportional to the photoabsorption cross sections $\\mu_{\\pm}= \\sigma_\\pm \\cdot n$, where $n$ is a number of atoms per unit volume, with the expressions for $\\sigma_\\pm$ coming from Eq.(\\ref{eq:xsec}). Since we are interested in comparison with the plane-wave propagation, we can express the attenuation coefficients in terms of their plane-wave values and the cross section ratios $r_\\pm^{tw}(b)$ introduced in \\cite{Afanasev:2013kaa,Afanasev2016}:\n\\begin{equation}\n\\mu_\\pm=\\mu^{pw}\\frac{\\sigma}{\\sigma^{pw}}=\\mu^{pw}r_{\\pm}^{tw}(b),\n\\end{equation}\nwhere the plane-wave attenuation coefficient $\\mu^{pw}$ is independent of $\\Lambda$ for the isotropic atomic matter, while the factor $r_\\pm^{tw}(b)$ depends on  $\\Lambda$ and on the impact parameter $b$. Then\n\\begin{equation}\nc_{\\pm}(z)=c_{\\pm}(0) e^{-\\mu_\\pm z\/2}=c_{\\pm}(0) e^{-\\mu^{pw}z r_\\pm^{tw}(b)\/2}.\n\\end{equation}\n\nFor the case of electric dipole transitions $l_f$=1, it follows from Eqs.(\\ref{eq:flux},\\ref{eq:factorized}) that $r_\\pm^{tw}(b)=1\/\\cos\\theta_k$, independently of $\\Lambda$ and $b$ \\cite{Afanasev:2013kaa,Scholz2014, Afanasev2016}. Therefore, the coefficients $c_{\\pm}(z)$ have the same $z$-dependence resulting in no evolution of the twisted-photon polarization due to atomic absorption via electric-dipole transitions. However for higher-multipole transitions into the states $l_f>1$ the factors $r_\\pm^{tw}(b)$ depend on the photon spin projection due to CD as defined in Eq.(\\ref{eq:CD}) (since the plane-wave cross section is a constant and it cancels in the ratio). \n\nFor example, let us consider a superposition of $\\Lambda=1$ and $\\Lambda=-1$ spin states for the same topological charge $\\overline m_\\gamma=1$, and assume the coefficients to be real and initially equal: $c_-|_{z=0}=c_+|_{z=0}$, where the field potential with a given $\\Lambda$ is defined by Eq.(\\ref{eq:twistedwf}). It results in a state with almost 100\\% linear polarization of transverse fields in the central region of the vortex (except for the small region near the node of Bessel function $J_{\\overline m_\\gamma}$). However, even for small values of propagation distance $z$, the optical vortex develops 100\\% circular polarization in the vortex center ($C$-point), and this region broadens as the beam passes further through the atomic matter. This prediction is shown in Fig.\\ref{fig:PolEvol}, where we used standard definitions for Stokes parameters $S_{0-3}$ \\cite{FieldGuidePolarization}. Development of $C$-type polarization singularity at the vortex center as a result of beam propagation can be observed in a dedicated experiment.\n\nIt should be noted that here we only considered the effects from photon absorption that result in CD. An additional effect would be forward scattering of the twisted photons, see, e.g. \\cite{Davis13}. Corresponding spin dependence would result in circular birefringence showing in rotation of polarization plane of the linearly-polarized twisted light due to spin dependence of refractive index.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width = 84 mm]{LinEv.pdf}\n\\includegraphics[width = 84 mm]{CircEv.pdf}\n\\caption{Evolution of linear polarization $S_2\/S_0$ (a) and circular polarization $S_3\/S_0$ (b) of an optical vortex due to absorption in the atomic matter as a function of impact parameter $b$ for different values of propagation distance $z$=0 (solid blue), z=0.01 (dashed), z=0.1 (dotted) and z=0.2 (dot-dashed line), where $z$ is in units of plane-wave attenuation length $1\/\\mu^{pw}$. A topological charge is $\\overline m_\\gamma=1$ and the angle is $\\theta_k$=0.1 rad. }\n\\label{fig:PolEvol}\n\\end{center}\n\\end{figure}\n\n\\section{Spin-Dependence and Circular Dichroism in Paraxial Limit}\n\nNumerical calculations of spin-dependent observables for the twisted light with a fixed topological charge $\\overline m_\\gamma$ reveal smooth transition to the paraxial limit $\\theta_k\\to 0$. Actually, as seen in Fig.\\ref{fig:CDang}, there is little dependence on this angle for $\\theta_k \\leq $ 0.25 rad. This observation prompts us to apply small-angle Taylor expansion for the expressions of CD and $A_\\Lambda$ using explicit formulae Eqs.(\\ref{eq:flux},\\ref{eq:factorized}). Noticing that the argument of Bessel functions is \n$\\kappa b=k \\sin \\theta_k b \\approx k b \\theta_k$ and defining $k b \\equiv x$, we consider $\\Lambda$-dependence of the photon flux Eq.(\\ref{eq:flux}), which is the same as the rate of $l_f=1$ dipole excitation.\n\nFor the case of topological charge $\\overline m_\\gamma=1$ we have after Taylor-expanding Bessel functions for small values of their arguments:\n\\begin{align}\n\\label{eq:intTaylor}\n& m_\\gamma=2, \\  \\Lambda=1,  \\ \\ f_{(\\Lambda=1)}\\propto \\frac{x^2}{4} \\theta_k^2,  \\nonumber\\\\\n& m_\\gamma=0, \\  \\Lambda=-1,\\ \\ f_{(\\Lambda=-1)}\\propto \\frac{x^2}{4} \\theta_k^2+\\frac{\\theta_k^2}{2}, \\\\ \\nonumber\n\\end{align}\nwhere the $x$-independent term in the last row comes from $J_{m_\\gamma}$ term in the flux Eq.(\\ref{eq:flux}), the term being indicative of spin-orbit interaction \\cite{BliokhAlonso10}.\n\nThe dependence on $\\theta_k $ cancels in the expression for the asymmetry, yielding\n\\begin{equation}\n\\lim_ {\\theta_k\\to 0}A_\\Lambda^{(\\overline m_\\gamma=1,l_f=1)}=\\frac{-1}{1+x^2},\n\\end{equation}\nwhere the $x-independent$ spin-orbit term of Eq.(\\ref{eq:intTaylor}) is a cause for a nonzero asymmetry. In view the relevant discussion in the literature on the spin-orbit interaction of light \\cite{BliokhAlonso10, BliokhNature15}, we can attribute nonzero spin effects to this interaction.\n\nSimilarly, we can obtain the expression for  $A_\\Lambda$ for any topological charge:\n\\begin{equation}\n\\lim_ {\\theta_k\\to 0}A_\\Lambda^{(\\overline m_\\gamma,l_f=1)}=\\frac{-1}{1+\\frac{2 x^4\/\\overline m_\\gamma^2}{(\\overline m_\\gamma-1)^2+2x^2}},\n\\end{equation}\nwhere in general all three terms of Eq.(\\ref{eq:flux}) appear to be of the same order in $\\theta_k$ for the anti-aligned spin and OAM.\n\nThis expression closely matches rate asymmetries in Fig.\\ref{fig:rateAs}(a) for moderately small pitch angles $\\theta_k\\leq 0.25$. It is remarkable that while the beam waist size strongly depends on the angle $\\theta_k$ (which in turn relates to Berry phase \\cite{BliokhAlonso10}), the spin asymmetry in the paraxial limit depends only on the topological charge $\\overline m_\\gamma$.\n\nWe can use the same approach to determine the spin asymmetries of photoexcitation rates for higher transition multipolarities, that in addition requires small-angle expansion of Wigner $d$-functions in Eq.(\\ref{eq:factorized}). For example, for the quadrupole transitions caused by $\\overline m_\\gamma$=1 beam, we identify the terms that are leading-order in small $\\theta_k$-expansion:\n\\begin{align}\n& m_\\gamma=2, \\  \\Lambda=1,  \\nonumber \\\\\n&{\\cal M}{(m_f=2)}\\propto J_0(\\kappa b)\\cdot d^{(2)}_{21}(\\theta_k)\\approx \\theta_k, \\\\ \\nonumber\n&{\\cal M}{(m_f=1)}\\propto J_1(\\kappa b)\\cdot d^{(2)}_{11}(\\theta_k)\\approx k \\theta_k\/2, \\\\\n& m_\\gamma=0, \\  \\Lambda=-1,  \\nonumber \\\\\n&{\\cal M}{(m_f=0)}\\propto J_0(\\kappa b)\\cdot d^{(2)}_{0-1}(\\theta_k)\\approx \\sqrt{\\frac{3}{2}}\\theta_k, \\\\ \\nonumber\n&{\\cal M}{(m_f=-1)}\\propto J_1(\\kappa b)\\cdot d^{(2)}_{-1-1}(\\theta_k)\\approx k \\theta_k\/2.\n\\end{align}\nSquaring the amplitudes and summing over all magnetic quantum numbers $m_f$ according to Eqs.(\\ref{eq:xsec},\\ref{eq:SpinAsym}), we obtain:\n\\begin{equation}\n\\lim_ {\\theta_k\\to 0}A_\\Lambda^{(\\overline m_\\gamma=1,l_f=2)}=\\frac{-1}{5+x^2},\n\\end{equation} \nthat results in -20\\% asymmetry in the vortex center, in agreement with exact results from Fig.\\ref{fig:rateAs}b. We can trace the factors yielding this value of the asymmetry to the differences between Wigner $d$-functions $d^{(2)}_{21}(\\theta_k)$ and $d^{(2)}_{0-1}(\\theta_k)$ multiplying non-vanishing transition amplitudes in the vortex center: the latter is larger by a factor $\\sqrt{3\/2}$ in a small-angle limit.  Experimental observation of this difference in the excitation amplitudes can be made by analyzing normalized Rabi frequencies in ion traps, in a setup similar to Ref.\\cite{Schmiegelow2016}.\n\nTaking the expressions for CD Eq.(\\ref{eq:CD}), we obtain in the paraxial limit, for example:\n\\begin{equation}\nCD^{(\\overline m_\\gamma=1,l_f=2)}= \\frac{4}{x^4+6 x^2+4}.\n\\end{equation}\n\nOther analytic expressions in the paraxial limit for different values of the topological charge $\\overline m_\\gamma$ and the excited-atom OAM $l_f$ are listed in the Appendix.\n\n\n\\section{Summary and Discussion}\t\t\t\\label{sec:disc}\n\nIn this work we applied previously developed theoretical formalism \\cite{Afanasev:2013kaa,Afanasev2014,Scholz2014} to analyze photoexcitation of an atom by Bessel beams with OAM and with different orientations of photon spin.\nWe found that both the photoexcitation rates and cross sections of twisted-photon absorption show strong dependence on the relative orientation of spin and OAM along the beam propagation direction. They can be observed by fixing the spatial structure of the beam (i.e., OAM) and flipping circular polarization with quarter-wave plates. From the parity considerations, it would be equivalent, up to an overall sign, to fixing the circular polarization and analyzing dependence on the sign of OAM projection on the beam propagation direction. Therefore our calculations predict both circular and OAM dichroism of the twisted light. Due to the factorization property of the twisted-light photoexcitation amplitudes in the first Born approximation Eq.(\\ref{eq:factorized}), the plane-wave matrix elements cancel in the spin asymmetries (due to parity conservation), yielding the results independent of the internal structure of the atomic target. \n\nSince the rates of electric dipole transitions are proportional to the local energy flux, the corresponding CD is zero. However, position-dependent photoexcitation rates show strong dependence on the photon spin, in manifestation of spin-orbit interaction of light, {\\it c.f.} \\cite{BliokhAlonso10, BliokhNature15}. The corresponding rate asymmetry becomes independent of the pitch angle $\\theta_k$ for moderately small angles below about 0.25 rad.  This observation provides a possible method for determination of the topological charge of optical vortex.  For the higher-multipole transitions with excitation of $l_f>1$ atomic states, the spin asymmetries are large near the optical vortex center; the asymmetries die off at distances about one wavelength from the vortex center.  For electric quadrupole transitions caused by the beams of topological charge $\\overline m_\\gamma$=1, the maximum spin asymmetry of the excitation rates is predicted to be -20\\%, that may be observed in the experimental setup similar to Ref.\\cite{Schmiegelow2016} by analyzing Rabi frequencies for individual electronic transitions in an ion trap.\n\nWe believe the present analysis of spin dependence of light absorption in atomic matter will be instrumental for quantum computing applications, optical communications, imaging, and characterization of twisted light in a broad range of wavelengths.\n\n\n\n\\begin{acknowledgments}\n\nCEC thanks the National Science Foundation for support under Grant\nPHY-1516509. Work of AA and MS was supported by Gus Weiss Endowment of The George Washington University. \nUseful discussions with K.~Bliokh, N. Litchinitser, V. Serbo, C. Schmiegelow and F. Schmidt-Kaler are gratefully acknowledged.\n\n\\end{acknowledgments}\n\n\\section{Appendix}\n\n\\subsection{Expressions for Spin Asymmetries in a Paraxial Limit}\n\nHere we present the expressions for $CD$ and $A_\\Lambda$ for small values of the pitch angle $\\theta_k\\to 0$, with $x\\equiv k\\cdot b$.\n\n{\\bf Electric Quadrupole transitions $l_f$=2}\n\n{\\it Rate Asymmetries}\n\\begin{align}\nA_\\Lambda^{(\\overline m_\\gamma=1,l_f=2)}&=-\\frac{1}{x^2+5}, \\nonumber \\\\\nA_\\Lambda^{(\\overline m_\\gamma=2,l_f=2)}&=-\\frac{2 \\left(2 x^2+9\\right)}{x^4+20 x^2+18}, \\\\\nA_\\Lambda^{(\\overline m_\\gamma=3,l_f=2)}&=-\\frac{9 \\left(x^4+18 x^2+8\\right)}{x^6+45 x^4+162 x^2+72}, \\nonumber \\\\\nA_\\Lambda^{(\\overline m_\\gamma=4,l_f=2)}&=-\\frac{8 \\left(2 x^4+81 x^2+144\\right)}{x^6+80 x^4+648 x^2+1152}. \\nonumber\n\\end{align}\n{\\it Circular Dichroism}\n\\begin{align}\nCD^{(\\overline m_\\gamma=1,l_f=2)}&= \\frac{4}{x^4+6 x^2+4},  \\nonumber \\\\\nCD^{(\\overline m_\\gamma=2,l_f=2)}&= \\frac{48 x^2+32}{x^6+24 x^4+84 x^2+32},  \\\\\nCD^{(\\overline m_\\gamma=3,l_f=2)}&= \\frac{36 \\left(5 x^2+16\\right)}{x^6+54 x^4+504 x^2+720}, \\nonumber \\\\\nCD^{(\\overline m_\\gamma=4,l_f=2)}&= \\frac{64 \\left(7 x^2+54\\right)}{x^6+96 x^4+1744 x^2+5760}. \\nonumber \\\\ \\nonumber\n\\end{align}\n\n{\\bf Electric Octupole Transitions $l_f$=3}\n\n{\\it Rate Asymmetries}\n\\begin{align}\nA_\\Lambda^{(\\overline m_\\gamma=1,l_f=3)}&= -\\frac{1}{x^2+11}\\nonumber \\\\\nA_\\Lambda^{(\\overline m_\\gamma=2,l_f=3)}&=-\\frac{4 x^2+42}{x^4+44 x^2+102}  \\\\\nA_\\Lambda^{(\\overline m_\\gamma=3,l_f=3)}&= -\\frac{9 \\left(x^4+42 x^2+80\\right)}{x^6+99 x^4+918 x^2+720}\\nonumber \\\\\nA_\\Lambda^{(\\overline m_\\gamma=4,l_f=3)}&= \\nonumber -\\frac{8 \\left(2 x^6+189 x^4+1440 x^2+540\\right)}{x^8+176 x^6+3672 x^4+11520 x^2+4320}\\\\ \\nonumber\n\\end{align}\n{\\it Circular Dichroism}\n\\begin{align}\nCD^{(\\overline m_\\gamma=1,l_f=3)}&= \\frac{10}{x^4+12 x^2+10} \\nonumber \\\\\nCD^{(\\overline m_\\gamma=2,l_f=3)}&=  \\frac{40 \\left(3 x^4+8 x^2+3\\right)}{x^8+48 x^6+264 x^4+320 x^2+120} \\\\\nCD^{(\\overline m_\\gamma=3,l_f=3)}&= \\frac{90 \\left(5 x^4+64 x^2+108\\right)}{x^8+108 x^6+1746 x^4+7200 x^2+9720} \\nonumber \\\\\nCD^{(\\overline m_\\gamma=4,l_f=3)}&=  \\frac{160 \\left(7 x^4+216 x^2+945\\right)}{x^8+192 x^6+6304 x^4+57600 x^2+159840}\\nonumber \\\\ \\nonumber\n\\end{align}\n\\vfill\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:intro}\n\nIt is well documented that the concordant $\\Lambda$ cold dark matter~($\\Lambda$CDM) theory gives a remarkable description of the cosmological and astrophysical observations on large spatial scales such as the cosmic microwave background radiation~\\citep[e.g.,][]{2011ApJS..192...18K,2018arXiv180706209P}, and large-scale structure of galaxies~\\citep[e.g.,][]{2006Natur.440.1137S,2006PhRvD..74l3507T,2014MNRAS.439.2515O}.\nAt galactic and sub-galactic scales however, this theory has several discrepancies between the simulation predictions and observational facts~\\citep[][for a review]{2017ARA&A..55..343B}.\n\nOne of them is the so-called ``core-cusp'' problem.\nDark-matter-only simulations based on the $\\Lambda$CDM model have predicted a universal dark matter density profile with a strong cusp at the center~\\citep[e.g.,][]{1994Natur.370..629M,1996ApJ...462..563N,1997ApJ...490..493N,1997ApJ...477L...9F,2013ApJ...767..146I}.\nBy contrast, the observations of dwarf spheroidal (dSph) and low surface brightness galaxies seem to favor a cored central dark matter density~~\\citep[e.g.,][]{1995ApJ...447L..25B,2001MNRAS.323..285B,2007ApJ...663..948G,2008AJ....136.2761O,2010AdAst2010E...5D}.\n\nTo solve or ameliorate the issue, many possible solutions have been proposed. \nOne of the solutions is to transform a cusped to cored central density through the baryonic effects such as stellar winds and supernova feedback~\\citep[e.g.,][]{1996MNRAS.283L..72N,2002MNRAS.333..299G,2014ApJ...789L..17M,2016MNRAS.459.2573R} or heating of dark matter due to interaction of gas clumps and dark matter via dynamical friction~\\citep[e.g,][]{2001ApJ...560..636E,2011MNRAS.418.2527I,2015MNRAS.446.1820N,DelPopolo:2015nda}.\nMoreover, for the former mechanism, recent advanced simulations have predicted that the effect of core creation depends upon stellar mass and star formation history~\\citep{2012MNRAS.422.1231G,2014MNRAS.441.2986D,2014MNRAS.437..415D,2015MNRAS.454.2092O,2016MNRAS.456.3542T,2017MNRAS.471.3547F,2018MNRAS.480..800H}.\nNote that core formation ability of baryonic feedback is sensitive to the gas density threshold for a star formation, $n_{\\rm sf}$, assumed in simulations.\nA low threshold ($n_{\\rm sf}=0.1$~cm$^{-3}$) is incapable of creating a core, while a high threshold ($n_{\\rm sf}=10$-$1000$~cm$^{-3}$) is able to lead a core~\\citep[e.g.,][]{2010Natur.463..203G,2019MNRAS.486.4790B}.\nAlthough a high threshold is about four orders of magnitude greater than a low one, a whole range of the threshold is acceptable because current understanding of subgrid physics is not complete yet.\n\nAnother solution is, more radically, to replace CDM with other dark matter models that are well motivated from particle physics, such as~self-interacting dark matter~\\citep[e.g.,][see also \\citealt{2014arXiv1402.5143H,2015PhRvL.115b1301H}]{1992ApJ...398...43C,2000PhRvL..84.3760S,2016PhRvL.116d1302K,2018PhR...730....1T,Nadler:2020ulu}, and ultra-light dark matter~\\citep[e.g.,][]{2000PhRvL..85.1158H,2014MNRAS.437.2652M,2014NatPh..10..496S,2016PhR...643....1M,2016PhRvD..94d3513S,2017MNRAS.471.4559M,2017PhRvD..95d3541H}.\nThese dark matter models can create a cored, low-dense central dark matter density profile on less massive-galaxy scales without relying on any baryonic physics.\n\nMeanwhile, current dynamical studies for dSphs are challenged in the measurement of their central density profiles, because of the existence of $\\rho_{\\rm DM}-\\beta_{\\rm ani}$ degeneracy, where $\\rho_{\\rm DM}$ is a dark matter density and $\\beta_{\\rm ani}$ is a velocity anisotropy of stars as an unknown parameter~\\citep[e.g.,][]{1982MNRAS.200..361B,1990AJ.....99.1548M,2009MNRAS.393L..50E}.\nThis degeneracy originates from the assumption that both stars and dark matter are spherically distributed and from the fact that only line-of-sight velocity components of stars are available from observations~\\citep[e.g.,][]{2007ApJ...657L...1S}.\nTo disentangle this degeneracy, many dynamical modelings have been proposed, as exemplified by using higher order velocity moments~\\citep[e.g.,][]{2002MNRAS.333..697L,2009MNRAS.394L.102L,1990AJ.....99.1548M}, virial theorem~\\citep[e.g.,][]{2014MNRAS.441.1584R}, modeling multiple stellar populations~\\citep[e.g.,][]{2008ApJ...681L..13B,2011ApJ...742...20W}, orbit-based dynamical models~\\citep[e.g.,][]{2013ApJ...763...91J,2013MNRAS.433.3173B}, measuring the internal proper motion data~\\citep{2018NatAs...2..156M,2019arXiv190404037M,2018ApJ...860...56S}, and non-parametric analysis~\\citep[e.g.,][]{2017MNRAS.471.4541R,2019MNRAS.484.1401R}.\nHowever, the inferred dark matter density profiles are not completely unified, and some of these models cannot distinguish a cusp from a core from the currently available kinematic data, due to considerable uncertainties in the derived dark matter density profiles and a prior bias of a dark matter inner slope parameter.\nThus, whether the central dark matter densities in dSphs are cored or cusped is yet unclear.\n\nWe emphasize that many of these studies assume spherical symmetry for both the stellar and dark components, even though we know both from observational facts and theoretical predictions that these components are actually non-spherical~\\citep[e.g.,][]{2012AJ....144....4M,2018ApJ...860...66M,2006MNRAS.367.1781A,2014MNRAS.439.2863V}. \nIn this paper, we relax the spherically symmetric assumption and perform the axisymmetric Jeans analysis for the dSphs.\nSuch non-spherical mass models have several advantages that (i)~giving the specific form of the distribution function is not required; (ii)~this analysis can treat {\\it two-dimensional} distributions of line-of-sight velocity dispersions~\\citep[e.g.,][]{2012ApJ...755..145H}, whereas it is impossible for spherical mass models; and (iii)~$\\rho_{\\rm DM}-\\beta_{\\rm ani}$ degeneracy can be mitigated~\\citep{2008MNRAS.390...71C,BATTAGLIA201352,2015ApJ...810...22H}.\nSeveral studies have developed axisymmetric mass models based on the Schwarzschild method~\\citep{2012ApJ...746...89J} and Jeans anisotropic multiple Gaussian expansion model~\\citep{2016MNRAS.463.1117Z}, but many of these assumed that a dark matter halo is still spherical while a stellar system is non-spherical.\n\nOur group constructed, as presented in \\citet{2015ApJ...810...22H}, totally axisymmetric dynamical mass models based on axisymmetric Jeans equations and applied the models to the dSphs with Milky Way and Andromeda galaxies~\\citep[see also ][]{2012ApJ...755..145H}.\n\\citet{2016MNRAS.461.2914H} applied the axisymmetric mass models to the recent kinematic data for the ultra-faint dSphs as well as classical ones to evaluate the astrophysical factors for dark matter annihilation and decay with considering the uncertainties of non-sphericity.\n\nOur previous models were yet incomplete in the point that an outer dark matter profile is fixed as $\\rho_{\\rm DM}\\propto r^{-3}$ for the sake of simplicity.\nHere, to step further from these previous studies, we adopt a generalized Herquist profile to explore a much wider range of physically plausible dark matter profiles and apply these non-spherical models to the latest observational data of the Galactic classical dSphs~(Draco, Ursa~Minor, Carina, Sextans, Leo~I, Leo~II, Sculptor, and Fornax) having a large number of member stars with well-measured radial velocities.\n\nThe paper is organized as follows. \nIn Section 2, we explain axisymmetric models based on an axisymmetric Jeans analysis and our fitting procedure. \nIn Section 3, we describe the photometric and spectroscopic data for the classical dSphs. \nIn Section 4, we present the results of the fitting analysis. We also show the estimated dark matter density profiles and the values of astrophysical factors. \nIn Section 5, we discuss the results of our estimations. \nFinally, conclusions are presented in Section 6.\n\n\n\\input{table1.tex}\n\\section{Models and Jeans analysis} \\label{sec:jeans}\nIn this section, we briefly introduce our dynamical mass models in this work.\nTo show how precisely we are able to recover actual dark matter density profiles from our fitting analysis, we apply our mass models to mock data sets. The details about mock data and the results of mock analysis are shown in Appendix~\\ref{sec:AppA}.\n\n\\subsection{Axisymmetric Jeans equations}\nAssuming that a galaxy is in a dynamical equilibrium and collisionless under a smooth gravitational potential, the dynamics of stars in such a system is described by its phase-space distribution function governed by the steady-state collisionless Boltzmann equation~\\citep{2008gady.book.....B}.\nHowever, it is virtually impossible to solve this equation from the currently available data of stars in the dSphs whose positions along the line of sight are difficult to resolve and accurate proper motions are yet to be measured.\nIn order to alleviate this issue, one of the classical and useful approaches is to take moments of the equation.\nThe equations taking moments of the steady-state collisionless Boltzmann equation are the so-called Jeans equations.\n\nFor an axisymmetric and steady state system, the Jeans equations are expressed as \n\\begin{eqnarray}\n\\overline{u^2_z} &=&  \\frac{1}{\\nu(R,z)}\\int^{\\infty}_z \\nu\\frac{\\partial \\Phi}{\\partial z}dz,\n\\label{AGEb03}\\\\\n\\overline{u^2_{\\phi}} &=& \\frac{1}{1-\\beta_z} \\Biggl[ \\overline{u^2_z} + \\frac{R}{\\nu}\\frac{\\partial(\\nu\\overline{u^2_z})}{\\partial R} \\Biggr] +\nR \\frac{\\partial \\Phi}{\\partial R},\n\\label{AGEb04}\n\\end{eqnarray}\nwhere $\\nu$ is the three-dimensional stellar density and $\\Phi$ is the gravitational potential, which is significantly dominated by dark matter for the Galactic dSphs.\nThe latter means that stellar motions in a system are governed only by a dark matter potential.\nWe assume that the cross terms of velocity moments such as $\\overline{u_Ru_z}$ vanish and the velocity ellipsoid constituted by $(\\overline{u^2_R},\\overline{u^2_{\\phi}},\\overline{u^2_z})$ is aligned with the cylindrical coordinate.\nWe also assume that the density of tracer stars has the same orientation and symmetry as that of a dark halo.\n$\\beta_z=1-\\overline{u^2_z}\/\\overline{u^2_R}$ is a velocity anisotropy parameter introduced by~\\citet{2008MNRAS.390...71C}.\nIn this work, $\\beta_z$ is assumed to be constant for the sake of simplicity\\footnote{Nevertheless, this assumption is roughly in good agreement with dark matter simulations reported by~\\citet{2014MNRAS.439.2863V} who have shown that simulated subhalos have an almost constant $\\beta_z$ or a weak trend as a function of radius along each axial direction.}.\nIn principle, these second velocity moments are defined as $\\overline{u^2}= \\sigma^2+\\overline{u}^2$, where $\\sigma$ and $\\overline{u}$ are dispersion and streaming motions of stars, respectively.\nThe latter streaming motions are small in the dSphs~\\citep[e.g.,][]{2008ApJ...688L..75W}, and thus these galaxies are largely dispersion-supported stellar systems~\\citep[e.g.,][]{2017MNRAS.465.2420W}.\n\nTo compare with the observed second velocity moments, the intrinsic second velocity moments derived by the Jeans equations are integrated along the line-of-sight second velocity moment followed by the previous works~\\citep{1997MNRAS.287...35R,2006MNRAS.371.1269T,2012ApJ...755..145H}.\nThis moment can be written as \n\\begin{equation}\n\\overline{u^2_{\\rm l.o.s}}(x,y) = \\frac{1}{I(x,y)}\\int^{\\infty}_{-\\infty}\\nu(R,z)\\overline{u^2_{\\ell}}(R,z)d\\ell,\n\\label{los3}\n\\end{equation}\nwhere $I(x,y)$ indicates the surface stellar density profile calculated from $\\nu(R,z)$, and $(x,y)$ are the sky coordinates aligned with the major and minor axes, respectively.\n$\\overline{u^2_{\\ell}}(R,z)$ is driven by\n\\begin{equation}\n\\overline{u^2_{\\ell}} = \\overline{u^2_{\\ast}}\\cos^2\\theta + \\overline{u^2_z}\\sin^2\\theta,\n\\label{los2}\n\\end{equation}\nwhere $\\theta$ is the angle between the line of sight and the galactic plane~($\\theta=90^{\\circ}-i$, which $i$ is an inclination angle explained below).\n$\\overline{u^2_{\\ast}}$ is a velocity second moment derived from the projection $\\overline{u^2_R}$ and $\\overline{u^2_{\\phi}}$ to the plane parallel with the galactic plane along the intrinsic major axis.\nThis moment is described as \n\\begin{equation}\n\\overline{u^2_{\\ast}} = \\overline{u^2_{\\phi}}\\frac{x^2}{R^2} + \\overline{u^2_R}\\Bigl(1-\\frac{x^2}{R^2}\\Bigr).\n\\label{los1}\n\\end{equation}\n\n\\subsection{Stellar density profile}\\label{sec:starprof}\nFor the stellar density profile, we adopt a Plummer profile \\citep{1911MNRAS..71..460P} generalized to an axisymmetric shape:\n\\begin{eqnarray}\n\\nu(R,z)=\\frac{3L}{4\\pi b^3_{\\ast}}\\frac{1}{(1+m^2_{\\ast}\/b^2_{\\ast})^{5\/2}}\n\\label{plummer}\n\\end{eqnarray}\nwhere $m^2_{\\ast}=R^2+z^2\/q^2$, so that $\\nu$ is constant on spheroidal shells with an intrinsic axial ratio $q$, and $L$ and $b_{\\ast}$ are the total luminosity and the half-light radius along the major axis, respectively.\nThis profile can be analytically derived from the surface density profile using Abel transformation: $I(x,y)=(L\/\\pi b^2_{\\ast})(1+m^{\\prime 2}_{\\ast}\/b^2_{\\ast})^{-2}$,\nwhere $m^{\\prime 2}_{\\ast}=x^2+y^2\/q^{\\prime 2}$.\n$q^{\\prime}$ is a projected axial ratio and is related to the intrinsic one $q$ through the inclination angle $i$~$(=90^{\\circ}-\\theta)$: $q^{\\prime 2} = \\cos^2i+q^2\\sin^2i$.\nThis equation can be rewritten as $q=\\sqrt{q^{\\prime 2}-\\cos^2i}\/\\sin i$, and thus the allowed range of the inclination angle is bounded with $0\\leq\\cos^2i<q^{\\prime 2}$.\nIn this work, we assume that the stellar distribution has an {\\it oblate} shape only.\nThis is motivated by the result from \\citet{2015ApJ...810...22H} which concluded that most of stellar distributions of the dSphs are much better fitted by the oblate shape than by the prolate ones.\n\n\n\\subsection{Dark matter density profile}\\label{sec:dmprof}\nIn this work, we adopt a generalized Hernquist profile given by~\\citet{1990ApJ...356..359H} and also~\\citet{1996MNRAS.278..488Z} with considering non-spherical dark matter halos,\n\\begin{eqnarray}\n&& \\rho_{\\rm DM}(R,z) = \\rho_0 \\Bigl(\\frac{r}{b_{\\rm halo}} \\Bigr)^{-\\gamma}\\Bigl[1+\\Bigl(\\frac{r}{b_{\\rm halo}} \\Bigr)^{\\alpha}\\Bigr]^{-\\frac{\\beta-\\gamma}{\\alpha}},\n \\label{DMH} \\\\\n&& r^2=R^2+z^2\/Q^2,\n\\label{DMH2}\n\\end{eqnarray}\nwhere $\\rho_0$ and $b_{\\rm halo}$ are the scale density and radius, respectively, $\\alpha$ is\nthe sharpness parameter of the transition from the inner slope $\\gamma$ to the outer slope $\\beta$, \nand $Q$ is a constant axial ratio of a dark matter halo. \nThis model covers a broad range of physically plausible dark matter profiles from the cusped Navarro-Frenk-White~\\citep[hereafter NFW,][]{1997ApJ...490..493N} profile to the cored Burkert profile~\\citep{1995ApJ...447L..25B}.\n\n\n\\input{table2.tex}\n\n\\subsection{Fitting procedure}\\label{sec:fitting}\nIn order to estimate the dark matter density profiles in the dSphs, we explore the most likely parameter values by fitting theoretical and observed second velocity moments of each dSph.\nIn this work, we suppose that the line-of-sight velocity distribution is Gaussian and centered on the systemic velocity of the galaxy $\\langle u \\rangle$.\nGiven that the total number of member stars for each dSph is $N$, and the observed line-of-sight velocity of the $i$th member star and its velocity error is expressed by $u_i\\pm\\delta_{u,i}$ at the sky plane coordinates~$(x_i,y_i)$, the likelihood function is described as\n\\begin{equation}\n{\\cal L} = \\prod^{N}_{i=1}\\frac{1}{(2\\pi)^{1\/2}[(\\delta_{u,i})^2 + (\\sigma_i)^2]^{1\/2}}\\exp\\Bigl[-\\frac{1}{2}\\frac{(u_i-\\langle u \\rangle)^2}{(\\delta_{u,i})^2 + (\\sigma_i)^2} \\Bigr],\n\\label{LF}\n\\end{equation}\nwhere $\\sigma_i$ is the theoretical line-of-sight velocity dispersion at~$(x_i,y_i)$ which is calculated by model parameters and the Jeans equations.\nThe systemic velocity $\\langle u \\rangle$ of the dSph is a nuisance parameter that we marginalize over as a flat prior.\nFor the model parameters, we introduce flat or log-flat priors over the following ranges:\n\\begin{enumerate}[(i)]\n\\item $0.1\\leq Q\\leq2.0$;\n\\item $0.0\\leq \\log_{10}[b_{\\rm halo}\/{\\rm pc}]\\leq5.0$;\n\\item $-5.0\\leq \\log_{10}[\\rho_0\/(M_{\\odot}~{\\rm pc}^{-3})]\\leq5.0$;\n\\item $-1.0\\leq -\\log_{10}[1-\\beta_{z}]<1.0$;\n\\item $0.5\\leq\\alpha\\leq3$\n\\item $3.0\\leq\\beta\\leq10$\n\\item $0.0\\leq\\gamma\\leq2.5$\n\\item $\\cos^{-1}(q^{\\prime})< i\/{\\rm deg} \\leq 90.0$.\n\\end{enumerate}\n\nIn order to obtain the posterior probability distribution function (PDF) of each parameter by the above likelihood function, we perform a Markov Chain Monte Carlo~(MCMC) techniques, based on Bayesian parameter inference, using the Metropolis-Hastings algorithm~\\citep{1953JChPh..21.1087M,10.1093\/biomet\/57.1.97}.\nTo avoid an influence of initial conditions and to generate independent samples, we take several post-processing steps such as burn-in step, the sampling step and length of the chain.\nWe evaluate the percentiles of these PDFs to estimate credible intervals for each parameter straightforwardly.\n\n\\subsection{Results from applying our models to mock data}\nTo scrutinize whether our models can reproduce dark matter density profiles, we apply them to mock data.\nHere, we focus on testing how precisely our models are able to reproduce inner density slopes of dark halos.\nIn this subsection, we briefly describe the procedure for mock analysis and summarize the result from this analysis. The details are shown in Appendix~\\ref{sec:AppA}.\n\nWe use public mock data sets provided by~\\citet{2016MNRAS.463.1117Z}, which generated kinematic samples with axisymmetric stellar systems embedded in spherical dark matter halos and generated two kinds of dark matter halos: one with a cusped halo and one with a cored halo.\n\nWhen we apply our models to these mock data, we first estimate a projected axial ratio and a half-light radius, employing a maximum likelihood analysis~\\citep{2008ApJ...675..201M} with the Plummer stellar density profile.\nThen, we perform a MCMC analysis for mock kinematic samples to estimate the dark matter density profiles. \n\nIn this mock analysis, we carry out the fitting in the three cases: (A) a cusped model with 1000 samples, (B) a cored model with 1000 samples, and (C) a cored one with 4000 samples.\nThe dark matter density profiles estimated from the analysis are nearly reproduced within $1\\sigma$ uncertainties for each case.\nThe estimated values of dark matter inner slope, $\\gamma$, are $\\gamma=1.2^{+0.4}_{-0.5}$, $0.4^{+0.8}_{-0.3}$, and $0.4^{+0.3}_{-0.2}$ for the cusped mock~(A), the cored ones~(B) and (C), respectively~(see Figure~\\ref{fig:Mock}).\nIt is found that the MCMC fitting to the cored mock data results in a somewhat biased dark matter density profile with $\\gamma \\simeq 0.4$, although the model is nearly consistent with a cored halo within $1\\sigma$ confidence.\nSuch a bias is also appeared in several previous works~\\citep[e.g.,][]{2016MNRAS.463.1117Z,2018MNRAS.481..860R}, but the reason for this bias is yet unclear.\n\nConsequently, we should bear in mind the fact that there exits this small amount of bias for a dark matter inner slope in our fitting analysis.\n\n\\section{Data}\\label{sec:data}\nIn this section, we present the basic properties of photometric and spectroscopic data of the eight classical dSphs: Draco, Ursa~Minor, Carina, Sextans, Leo~I, Leo~II, Sculptor, and Fornax.\nThe classical dSphs have a larger number of line-of-sight velocities for the resolved stars ($\\gtrsim200$~stars) than ultra faint dwarf galaxies.\nThese galaxies also have large velocity dispersions~($\\gtrsim10$~km~s$^{-1}$), so that an influence of unresolved binary stars on the velocity dispersion measurements of each galaxy can be negligible~\\citep{2010ApJ...721.1142M,2013ApJ...779..116M,2017ApJ...836..202S,2018AJ....156..257S}.\n\nTable~\\ref{table1} lists the observational properties of the eight dSphs:~the number of member stars with velocity measurements available from the kinematic analysis, the central sky coordinates, distances from the Sun, projected half-light radii, projected stellar axial ratios, systemic velocities, and their references.\nFollowing previous works, we fix the values of distance, half-light radius, and axial ratio of dSphs in this paper.\n\nFor the stellar kinematic data of their member stars, we use the published data as follows.\nFor Carina, Draco, Ursa~Minor, Leo~I, and Leo~II, we use the stellar-kinematic data taken from \\citet{2016ApJ...830..126F}, \\citet{2015MNRAS.448.2717W}, \\citet{2018AJ....156..257S}, \\citet{2008ApJ...675..201M}, \\citet{2017ApJ...836..202S}, respectively.\nFor Sextans, Sculptor, and Fornax, we use the data published by \\citet{2009AJ....137.3100W,2009AJ....137.3109W}.\nThe membership selection criteria for each galaxy follow the methods described by each of the observational papers given above.\n\n\n\\section{Results} \\label{sec:results}\nIn this section, we present the results from the MCMC fitting analysis described above and several trends among the resultant parameters.\nMoreover, as a by-product of the fitting results, we estimate the astrophysical factors for dark matter annihilation and decay.\n\n\\subsection{Best-fitting models} \\label{sec:best}\nTable~\\ref{table2} shows the best-fitting parameters for each dSph.\nThe error values indicate the 68~per~cent credible intervals computed from posterior PDFs of the parameters. \nWe also show the dark matter density at 150~pc, $\\rho_{\\rm DM}(150\\ {\\rm pc})$, to compare with the other works. \n\\citet{2019MNRAS.484.1401R} presented the dark matter density at a common radius of 150~pc from the center of each galaxy, $\\rho_{DM}(150\\ {\\rm pc})$, which is insensitive to the choice of a $\\gamma$'s prior in spherical mass models.\nFurthermore, using this density, \\citet{2019MNRAS.490..231K} pointed out the anti-correlation between $\\rho_{DM}(150\\ {\\rm pc})$ and their orbital pericenter distances, $r_{\\rm peri}$, of the classical dSphs.\nThis implies a survivor bias: while galaxies with low dark matter densities were completely destroyed by strong tidal effects, those with high dark matter densities survive in the present day.\nFollowing these works, we also calculate the dark matter density at 150~pc along the major axes of the sample dSphs, considering non-sphericity of a dark matter halo, and the calculated $\\rho_{DM}(150\\ {\\rm pc})$ are tabulated in the last column of Table~\\ref{table2}.\nWe discuss it in the next section.\n\nFigure~\\ref{drafnx} displays the posterior PDFs implemented by the MCMC fitting for Draco~(left panel) and Fornax~(right panel) dSphs as the representative galaxies in our sample.\nThe results of other galaxies are similar to these and thus we show the other posterior PDFs~(Figure~\\ref{carumi}, \\ref{leo12}, and \\ref{sclsex}) in Appendix~\\ref{sec:AppB}. \nThe contours in these figures show 68, 95, and 99.7 per~cent credible interval levels.\nThe vertical lines in each histogram also show the median and 68 per~cent credible interval levels.\nFrom these posterior PDF maps, the parameters $Q$, $\\alpha$, $\\beta$, and $i$ are widely distributed in these parameter ranges and thus it is difficult to get limits on them.\nOn the other hand, the other parameters $b_{\\rm halo}$, $\\rho_0$, $\\beta_z$, and $\\gamma$ are better constrained than the above parameters, even though there are obvious degeneracies between $b_{\\rm halo}$-$\\rho_0$ and $Q$-$\\beta_z$, which have already been discussed in several previous papers~\\citep[e.g.,][]{2008MNRAS.390...71C,BATTAGLIA201352,2015ApJ...810...22H}.\nIn particular, the velocity anisotropy parameters $\\beta_z$ of all galaxies (see also Table~\\ref{table2}) tend to be a somewhat radially-biased velocity ellipsoid.\nMoreover, it is worth noting that owing to non-spherical models, the inner slope parameter of a dark matter density profile $\\gamma$ can be confined without being distracted by any parameter degeneracies.\nFrom these PDFs and Table~\\ref{table2}, we find that the posteriors of $\\gamma$ show a wide spread from cusped $(\\gamma>1.0)$ to shallower cusped $(\\gamma<0.5)$ inner dark matter density slopes, even though there is a large uncertainty.\nWe discuss this further in Section~\\ref{sec:bestdmprof}.\n\nFigure~\\ref{los} shows the comparison between the observed and the estimated line-of-sight velocity dispersion profiles obtained by the resultant posterior PDFs, to present our fitting analysis successfully reproduced to the binned data\\footnote{The method for calculating these binned profiles along the projected major, middle, and minor axes for the dSphs is the same way as \\citet{2019MNRAS.tmp.2554H}, and thus the details are described in the Section~3.1.2 in that paper.}.\nIn this figure, the colored solid lines and shaded regions denote the median and confidence levels (dark: 68~per~cent, light: 95~per~cent) of our {\\it unbinned} MCMC analysis.\nThe black points with error bars denote {\\it binned} velocity dispersions calculated by the observed data.\nThese errors correspond to the 68~per~cent confidence intervals.\nAs shown in this figure, our mass models and unbinned analysis can provide good fits to the binned data for all dSphs.\n\n\\begin{figure*}\n\t\\begin{minipage}{0.49\\hsize}\n\t\t\\begin{center}\n\t\t\t\\includegraphics[width=\\columnwidth]{PDF_dra.pdf}\n\t\t\\end{center}\n\t\\end{minipage}\n\t\\begin{minipage}{0.49\\hsize}\n\t\t\\begin{center}\n\t\t\t\\includegraphics[width=\\columnwidth]{PDF_fnx.pdf}\n\t\t\\end{center}\n\t\\end{minipage}\n    \\caption{Posterior distributions for the fitting parameters for Draco~(left) and Fornax~(right).\n    The dashed lines in each histogram represent the median and 68~per~cent confidence values. The contours in each panel are the 68, 95, and 99.7~per~cent regions.}\n    \\label{drafnx}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[scale=0.33]{Draco_LOSdisp.pdf}\n\\includegraphics[scale=0.33]{UMi_LOSdisp.pdf}\n\\includegraphics[scale=0.33]{Carina_LOSdisp.pdf}\n\\includegraphics[scale=0.33]{Sextans_LOSdisp.pdf}\n\\includegraphics[scale=0.33]{LeoI_LOSdisp.pdf}\n\\includegraphics[scale=0.33]{LeoII_LOSdisp.pdf}\n\\includegraphics[scale=0.33]{Sculptor_LOSdisp.pdf}\n\\includegraphics[scale=0.33]{Fornax_LOSdisp.pdf}\n\\end{center}\n\\caption{Line-of-sight velocity dispersion along major, middle and minor axes for each dSph. The black squares with error bars in each panel denote the observed ones. The solid lines are the median velocity dispersion of the models and the dark and light shaded regions encompass the 68~per cent and 95~per cent confidence levels from the results of the unbinned MCMC analysis. The vertical dashed lines in each panel correspond to their half-light radii.}\n\\label{los}\n\\end{figure*}\n\n\\subsection{Revisiting the core-cusp problem} \\label{sec:bestdmprof}\n\\subsubsection{Dark matter density profiles}\nUsing the results of the MCMC fitting analysis for the kinematic data of the dSphs, we estimate the dark matter density profiles by marginalizing all free parameters.\nFigure~\\ref{dmpro} shows the inferred dark matter density profiles of all sample dSphs.\nThe solid lines show the medium, and dark and light contours mark the 68~per~cent and 95~per~cent intervals.\nThe vertical black lines mark the projected half light radii of each dSph.\n\nFirstly, it is noteworthy that in our non-spherical models, Draco favors a cusped inner slope for its dark matter density profile, which is consistent with an NFW cusp predicted by $\\Lambda$CDM theory.\nEven if we consider 95~per~cent confidence intervals of the dark matter profile, its inner slope still remains cuspy.\nTherefore, Draco highly likely has a cusped dark matter halo.\n\nSecondly, Ursa~Minor, Leo~I and Leo~II also prefer cusped dark matter halos, although the uncertainties for the inner slope, $\\gamma$, are larger than for Draco.\nOn the other hand, the remaining sample of dSphs (Carina, Sextans, Sculptor, and Fornax) favors smaller $\\gamma$ and thus has less dense than the other dSphs which have cuspy dark matter halos.\nIn particular, Sextans, Sculptor, and Fornax permit $\\gamma=0$, i.e. a cored dark matter density within their 95~per~cent confidence intervals.\n\nNotably, the dark matter density profile in Draco is better constrained than Fornax, though the data volume of Fornax is greater than Draco.\nThis is because while the observed kinematic sample in Draco covers the stars up to its outskirts, that in Fornax is limited only to its inner region.\nActually, \\citet{2015ApJ...810...22H} suggested that the lack of kinematic sample volume in the outer region of a galaxy makes the constraints on the dark matter profile very uncertain.\n\nTherefore, from our dynamical analysis, we propose that there is {\\it no} core-cusp problem in the Galactic classical dSphs.\nMoreover, a {\\it diversity} of the inner density slope, $\\gamma$, is found for these dSphs.\nNote that this result is in agreement with \\citet{2019MNRAS.484.1401R}, which investigated the inner dark matter densities in the Galactic dSphs as well as in low surface brightness galaxies based on non-parametric spherical Jeans analysis.\n\n\\subsubsection{Why do some galaxies prefer cusped dark matter halos?} \\label{sec:demonstration}\nAs shown in the previous section, we present that some dSphs prefer cusped dark matter density profiles.\nThen the question is why these galaxies are regarded to have cusped dark matter halos.\nWe schematically illustrate this reason in Figure~\\ref{los_demo4} in the Appendix~\\ref{sec:AppC}.\nThis figure shows the normalized line-of-sight velocity dispersion profiles along the major~(top panels) and the minor axes~(bottom panels) for the oblate stellar system $(q=0.7)$.\nThe left-hand panels show the dispersion profiles with changing the value of velocity anisotropy parameter, $\\beta_z$, under spherical dark matter halo, $Q=1$, whilst the right-hand ones depict those with changing $Q$ under $\\beta_z=0$.\n\nAs already discussed in \\citet{2008MNRAS.390...71C} and \\citet{2015ApJ...810...22H}, the variation of $Q$ and $\\beta_z$ gives a similar effect on line-of-sight dispersion profiles.\nFor instance, as is shown in the top-left panel of Figure~\\ref{los_demo4}, the effect of $\\beta_z>0$ (i.e., red lines) increases inner line-of-sight velocity dispersions and decreases outer ones, simultaneously, compared with those in the fiducial $(Q=1,\\beta_z=0)$ case which corresponds to the black lines.\nIn the top-right panel, the effect of $Q<1$ (the red lines) is resemblant in the features of line-of-sight velocity dispersion profiles computed by $\\beta_z>0$ (the red lines in the top-left panel), even though there is a difference between these effects at the outer parts (the reason of this difference is already discussed in \\citealt{2015ApJ...810...22H}). \n\nHowever, comparing the dispersion profiles in the cases for cusped (the solid lines) and for cored (the dotted lines) dark matter density profile, we can see a difference in the shape of those profiles at inner parts.\nIn the case of a cusped dark matter halo, the velocity dispersion profiles along both major and minor axes rapidly increase towards the central region, while there is no such trend in the case of a cored one.\nLooking at the observed line-of-sight velocity dispersion profiles in Figure~\\ref{los}, Draco, as an example, seems to have the trend characterized by a cusped dark matter halo, whereas Fornax has the almost flat profiles.\nTherefore, we suggest that Draco highly likely has a cusped dark matter halo.\nIt is also found that the feature of a central velocity dispersion profile can be important in determining an inner slope of a dark matter density profile.\n\n\\subsubsection{The robustness of our results}\nIn order to demonstrate the robustness of our results, especially regarding the inner slope of a dark matter density profile, $\\gamma$,\nwe show the case when a wide range of prior for $\\gamma$ is adopted, compared to our fiducial parameter range of $\\gamma$ ($0\\leq\\gamma\\leq2.5$).\nNamely, we show here the case of a flat prior over range $-2.5\\leq\\gamma^{\\prime}\\leq2.5$, and we impose $\\gamma=0$ if $\\gamma^{\\prime}$ has a negative value and $\\gamma=\\gamma^{\\prime}$ otherwise.\nThis is because the fiducial parameter range of $\\gamma$ ($0\\leq\\gamma\\leq2.5$) might lead to a bias toward cuspy density profiles.\nUsing this new prior, we re-run the same MCMC fitting procedure described in Section~\\ref{sec:fitting}.\nFigure~\\ref{dmpro_bias} shows the comparison of the inferred dark matter density profiles for all sample dSphs for the fiducial~(solid) and wider~(dashed) prior ranges.\nThe thick and thin lines in each panel denote the median and the 68~per~cent confidence intervals.\nIt is found from this figure that the galaxies having a cusped dark matter halo like Draco and Ursa Minor are not so much affected by new prior, whilst the effect of new prior makes Fornax and Sextans less dense core.\nTherefore, we bear in mind that Fornax and Sextans are possible to have a cored dark matter density.\nOn the other hand, we can confirm that our results for Draco and Ursa~Minor have cusped dark matter halos.\n\n\\begin{figure*}[t!]\n\t\\includegraphics[scale=0.4]{DMpro_all.pdf}\n    \\caption{Dark matter density profiles along major axes of the galaxies derived from our Jeans analysis.\n    The solid line in each panel denotes the median value, and the dark and light shaded regions denote the 68 and 95~per~cent confidence intervals.\n    The vertical dashed line in each panel corresponds to the half-light radius of each galaxy.\n    In the panel for Draco, we mark on two power law density profiles, $\\rho_{\\rm DM}\\propto r^{-1}$ (cusp) and $\\rho_{\\rm DM}={\\rm const.}$ (core) under the shaded regions.}\n    \\label{dmpro}\n\\end{figure*}\n\\begin{figure*}[t!]\n\t\\includegraphics[scale=0.4]{DMpro_all_bias.pdf}\n    \\caption{Dark matter density profiles of all dSphs, with taking into account a wider parameter range of $\\gamma$~(described in Section~\\ref{sec:bestdmprof}).\n    The solid lines in each panel denote the median values~(thick) and the 68~per~cent confidence intervals~(thin) calculated by our default parameter range ($0\\leq\\gamma\\leq2.5$), while the dashed ones are calculated by a new parameter range ($-2.5\\leq\\gamma^{\\prime}\\leq2.5$, but if $\\gamma^{\\prime}<0\\rightarrow\\gamma=0$).\n    The vertical dashed lines in each panel correspond to their half-light radii.}\n    \\label{dmpro_bias}\n\\end{figure*}\n\n\n\\subsection{Astrophysical factors} \\label{sec:bestjd}\nThe Galactic dSphs are promising targets for indirect searches for particle dark matter through $\\gamma$-rays or X-rays stemmed from annihilating and decaying dark matters~\\citep[e.g.,][]{1978ApJ...223.1015G,2012AnP...524..479B}, because they contain a good deal of dark matter with low astrophysical backgrounds and are located at relative proximity.\nThe signal flux of the dark matter annihilation or decay depends only on two important factors.\nOne is the particle physics factor which is based on the microscopic physics of particle dark matter, while another is the astrophysical factor derived by line-of-sight integrals over the dark matter distribution within the system.\nThe latter largely depends on the estimate of the signal flux.\nTherefore, an accurate estimation of the astrophysical factor in the dSphs is of crucial importance so that we can set robust constraints on the particle nature of dark matter candidates.\n\nPrevious works have estimated the astrophysical factors for these galaxies considering various uncertainties: the spatial dependence of stellar velocity anisotropy~\\citep{2016JCAP...07..025U}, non-sphericity of a dark matter distribution~\\citep{2015MNRAS.446.3002B,2016MNRAS.461.2914H,2017PhRvD..95l3012K}, halo truncation radius~\\citep{2015ApJ...801...74G}, prior bias of Bayesian analysis~\\citep{2009JCAP...06..014M}, and foreground contamination of stars~\\citep{2016MNRAS.462..223B,2017MNRAS.468.2884I,2018MNRAS.479...64I,2020arXiv200204866H}.\n\nHere we calculate the astrophysics factors of the dSphs focusing only on non-sphericity based on the generalized Hernquist density profile of their dark matter halos.\nIn fact, (sub-) subhalos and substructures can boost the annihilation signals~\\citep[subhalo boost,][]{2017MNRAS.466.4974M,2018PhRvD..97l3002H,2020MNRAS.492.3662I}. However, this boost contributes little to the signals on the dSph's mass scales, and thus we do not include this boost to estimate $J$-factor values.\nTo compare with previous works, we show only the factors integrated within a fixed solid angle $0.5^{\\circ}$.\n\nThe astrophysical factors are written as \n\\begin{eqnarray}\nJ &=& \\int_{\\Delta\\Omega}\\int_{\\rm los}d\\ell d\\Omega\\rho^2_{\\rm DM}(\\ell,\\Omega) \\ \\hspace{5mm}  [{\\rm annihilation}], \\\\\nD &=& \\int_{\\Delta\\Omega}\\int_{\\rm los}d\\ell d\\Omega\\rho_{\\rm DM}(\\ell,\\Omega) \\ \\hspace{5mm}  [{\\rm decay}],\n\\end{eqnarray}\nwhich are so-called $J$- and $D$-factors, defined as the integrated dark matter density squared for annihilation and the dark matter density for decay, respectively, over a distance $\\ell$ along a line-of-sight and a solid angle $\\Delta\\Omega$.\nUsing these equations, we estimate the median and its uncertainties of the astrophysical factors from the posterior PDFs of the dark matter halo parameters.\n\nTable~\\ref{table3} shows the $J$ and $D$ values integrated within $\\Delta\\Omega=0.5^{\\circ}$ of our results.\nFigure~\\ref{JDcomp} displays a comparison of the $J$ (top) and $D$ (bottom) values of our results with those of previous works.\nIn this figure, the red colored points with error bars are the median values in this work with 68~per~cent confidence intervals.\nThe blue ones denote these values reported by \\citet{2015ApJ...801...74G}, which assumed a spherical dark matter halo with a generalized Hernquist density profile and performed Jeans analysis.\nThe green ones are evaluated by \\citet{2016MNRAS.461.2914H}, which assumed an axisymmetric dark matter halo.\nThe differences between them in this figure are caused primarily by the assumption of shapes of dark matter halos (spherical or non-spherical) as already discussed by \\citet{2016MNRAS.461.2914H} and dark matter density profiles.\nThe latter means that \\citet{2016MNRAS.461.2914H} imposed that the outer slope of dark matter profiles is $\\rho\\propto r^{-3}$ and the sharpness parameter $\\alpha$ in Equation~\\ref{DMH} is fixed at $2$ for simplicity, while the dark matter profiles in this work and \\citet{2015ApJ...801...74G} take into account these parameter as free parameters.\n\n\\input{table3}\n\nFrom Figure~\\ref{JDcomp}, we conclude that because of having a cuspy dense dark matter halo and of the close distance to the Sun, Draco is the most promising detectable target for an indirect search of dark matter annihilation and decay among all sample dSphs.\n\n\\begin{figure}\n\t\\begin{center}\n\t\\includegraphics[width=\\columnwidth]{J_comp.pdf}\n\t\\includegraphics[width=\\columnwidth]{D_comp.pdf}\n\t\\end{center}\n    \\caption{Comparison of $J_{0.5}$~(top) and $D_{0.5}$~(bottom) calculated from previous and this works.\n    The blue and green symbols are estimated by~\\citet{2015ApJ...801...74G} and \\citet{2016MNRAS.461.2914H}.\n    The red symbols denote the results of this work.}\n    \\label{JDcomp}\n\\end{figure}\n\n\n\\section{Discussion} \\label{sec:discussion}\n\\subsection{Comparison dark matter profiles with previous works}\nIn this section, we compare our estimated dark matter density profiles with other works based on different methods or assumptions.\n\n\\citet{2019MNRAS.484.1401R} considered non-parametric dynamical mass models based on a spherical Jeans equation, {\\sc GravSphere}~\\citep{2017MNRAS.471.4541R,2018MNRAS.481..860R} to measure the dark matter density profiles of dwarf spheroidal\/irregular galaxies, and then they found the relation between the central densities of dark matter halos and the stellar vestiges of galaxy evolution such as a star formation history, stellar mass, and stellar-to-halo mass ratio.\nRegarding the inner slopes of dark matter density profiles in the dSphs, they showed that Draco favors a cusped dark matter halo which is consistent with an NFW profile, while Fornax has a shallower inner density profile $\\gamma\\sim0.3$.\nThis trend is similar to that in this work.\nThey mentioned, however, the caveat that the estimation of an inner slope of a dark matter profile using their method is largely affected by a choice of priors.\nTherefore, they utilized a dark matter density within 150~pc, $\\rho_{\\rm DM}(150\\ {\\rm pc})$, to discuss a diversity of the central dark matter densities in the dwarf galaxies, instead of their inner slopes. \nWe also discuss $\\rho_{\\rm DM}(150\\ {\\rm pc})$ calculated by our models and then find that this physical quantity is useful to understand the dynamical evolution of dark matter halos in the Universe.\nWe discuss them further in the following subsection.\n\nOwing to recent spectroscopic observations for the dSphs, some of them have multiple stellar populations, in which the metal-rich stars are centrally concentrated and have colder kinematics, while the metal-poor ones are more extended and have hotter kinematics~\\citep[e.g.,][]{2006A&A...459..423B,2008ApJ...681L..13B}.\nUsing the coexistence of such multiple populations, \\citet{2011ApJ...742...20W} statistically separated multiple stellar components by applying their constructed likelihood function for spatial, metallicity, and velocity distributions of the stars, and then inferred the slopes of dark matter densities of Sculptor and Fornax.\nThey concluded that both galaxies have cored dark matter halos and a cuspy profile can be ruled out with high statistical significance.\nHowever, this method imposes that both stellar and dark matter distributions are spherical symmetric.\nThis sphericity can accompany a systematic bias, and an inner slope inferred by this method depends largely on viewing angles~\\citep{2013MNRAS.431.2796K,2013MNRAS.433L..54L,2018MNRAS.474.1398G}.\n\n\\citet{2012ApJ...754L..39A} and \\citet{2013MNRAS.429L..89A} applied the projected virial theorem to these multiple stellar components for Sculptor and Fornax, respectively and concluded that these dSphs do not have cusped dark matter profiles.\nOn the other hand, using these multiple populations, several other works concluded that Sculptor has a cusped dark matter halo based on a phase space distribution function method~\\citep{2017ApJ...838..123S}, whilst it is difficult to distinguish between cusp and core based on a Schwarzschild method~\\citep{2013MNRAS.433.3173B} and Multi-Gaussian expansion model~\\citep{2016MNRAS.463.1117Z}.\nAlthough the dark matter inner slopes in Fornax and Sculptor are still under debated, those inferred by our mass models prefer to be less cuspy than an NFW profile.\n\nAxisymmetric dynamical models based on Schwarzschild technique have been developed and applied to the kinematic data of the dSphs~\\citep{2012ApJ...746...89J,2013ApJ...775L..30J,2013ApJ...763...91J}.\n\\citet{2013ApJ...763...91J} applied these models to the data of Draco and found that its dark matter inner slope is consistent with an NFW profile.\nThis agrees well with our mass models for Draco.\n\\citet{2013ApJ...775L..30J} performed the same analysis with respect to the other classical dSphs (Carina, Fornax, Sculptor and Sextans) and concluded that these galaxies have an unified cusped profile, but there are considerable large uncertainties.\n\n\n\\begin{figure*}[t]\n\t\\begin{center}\n\t\\includegraphics[scale=0.9]{DSPH_gamma_MsMh.pdf}\n\t\\end{center}\n    \\caption{The impact of baryonic feedback on the inner profiles of dark matter halos. \n    The inner dark matter density slope at $1.5$\\%$R_{\\rm vir}$ is shown as a function of the ratio of stellar-to-halo masses.\n    The filled black circles with error bars are the results from this work. \n    The shaded gray band shows the expected range of dark matter profile slopes for NFW as derived from dark matter only simulations~\\citep{2016MNRAS.456.3542T}.\n    The blue and orange points are expected from NIHAO~\\citep{2016MNRAS.456.3542T} and FIRE-2~\\citep{2017MNRAS.471.3547F,2018MNRAS.480..800H} hydrodynamical plus dark mater simulations, respectively.\n    The blue and orange shaded bands are the expected range from NIHAO~\\citep{2016MNRAS.456.3542T} and FIRE-2~\\citep{2020arXiv200410817L} predictions, respectively (to guide the eye).}\n    \\label{GammaMsMh}\n\\end{figure*}\n\n\\subsection{The origin of a diversity of inner dark matter slopes} \\label{sec:diversity}\nIn Figure~\\ref{dmpro}, we show that the classical dSphs have a wide range of central dark matter density profiles.\nIn this section, we discuss what the origin of this diversity is.\nTo this end, we investigate the relation between the central dark matter density profiles and stellar properties of the dSphs.\n\n\\subsubsection{Inner dark matter density slope versus stellar-to-halo mass ratio}\nRecent dark matter plus hydrodynamical simulations have shown that an inner slope of a dark matter density profile depends largely on the ratio of stellar mass to total halo mass.\nFigure~\\ref{GammaMsMh} shows the logarithmic slope of the dark matter density profile at $1.5$\\% of the virial radius, $R_{\\rm vir}$, as a function of the ratio of stellar-to-halo masses, $M_{\\ast}\/M_{\\rm halo}$, predicted from NIHAO~\\citep[][magenta]{2016MNRAS.456.3542T} and FIRE-2~\\citep[][cyan]{2017MNRAS.471.3547F,2018MNRAS.480..800H} simulations.\nNote that baryon feedback for bright dwarf galaxies ($\\log_{10}(M_{\\ast}\/M_{\\rm halo})\\sim-3$~to~$-2$) has a systematic impact on inner slopes, while for the fainter galaxies with $\\log_{10}(M_{\\ast}\/M_{\\rm halo})\\lesssim-3.5$, the impact of baryonic feedback is negligible. \nTherefore, these simulations predict that the efficiency of baryonic feedback for a dark matter halo can provoke the diversity of dark matter inner slopes.\n\nTo test this prediction, we derive the relation between the dark matter inner slopes and $M_{\\ast}\/M_{\\rm halo}$ for the current sample of dSphs, which is shown in Figure~\\ref{GammaMsMh}.\nIn order to calculate the ratio of stellar-to-halo masses, we employ the self-consistent abundance matching model computed by \\citet{2013MNRAS.428.3121M} and adopt the stellar masses of the dSphs taken from \\citet{2012AJ....144....4M}.\nThe filled black circles with error bars in Figure~\\ref{GammaMsMh} show the results of the classical dSphs inferred by our analysis.\nAlthough there are still large uncertainties in both the inner slopes and the stellar-to-halo mass ratios, \nthe systematic trend in the plots is generally in agreement with the predictions from recent numerical simulations, which are presented in blue~(NIHAO: \\citealt{2016MNRAS.456.3542T}) and orange (FIRE-2: \\citealt{2020arXiv200410817L}) shaded region in the figure.\nTo make an attempt to characterize the trend quantitatively, we employ a least squares fitting method to\ndetermine the slope of $\\gamma$ as a function of $M_{\\ast}\/M_{\\rm halo}$, and we find $\\gamma\\propto\\log_{10}(M_{\\ast}\/M_{\\rm halo})^{0.27\\pm0.15}$.\nThus, we confirm that $\\gamma$ is slightly proportional to $M_{\\ast}\/M_{\\rm halo}$ on dwarf-galaxy scales.\n\n\nHowever, comparing between these shaded bands in detail, there is a systematic difference especially on classical dwarf galaxy scales stemmed from the different prescriptions of hydrodynamics regime.\nThus, the predicted relation between dark matter inner slope and stellar-to-halo mass ratio still has large uncertainties.\nRegarding this relation, \\citet{2019arXiv191100544K} have argued that a self-interacting dark matter (SIDM) model combined with the impact of a baryon potential on the halo profile can also reproduce the diversity of the inner dark matter density profiles for low surface brightness galaxies.\nHowever, the corresponding $M_{\\ast}\/M_{\\rm halo}$ in these galaxies are greater than $-3$, and it is thus unclear for the diversity in the current fainter dwarf galaxy scales.  \n\nWe also investigate the relation between the inner density slopes and their stellar masses and the orbital properties of the dSphs~(apocenter radius, orbital eccentricity, angular momentum, the time elapsed since the last apocenter and pericenter) but we find no clear relations.\n\n\n\\begin{figure}[t!]\n\t\\begin{center}\n\t\\includegraphics[scale=0.3]{cumSFH_2.pdf}\n\t\\includegraphics[scale=0.3]{DSPH_gamma_tau7.pdf}\n\t\\end{center}\n    \\caption{{\\it Top panel}: Cumulative star formation history of dwarf satellites in the MW taken from \\citet{2009ApJ...703..692L} for Sextans and \\citet{2014ApJ...789..147W} for the other classical dSphs.\n    {\\it Bottom panel}: Inner slope parameter of dark matter density profile $\\gamma$ as a function of the lookback time of achieving 70~per~cent of current stellar masses, $\\tau_{0.7}$.}\n    \\label{Gamma_esn}\n\\end{figure}\n\n\\subsubsection{Inner dark matter density slope versus SFH}\nThe relation in Figure~\\ref{GammaMsMh} implies that an inner dark matter slope depends on stellar feedback associated with star formation activity.\nIndeed, some high-resolution dark matter and hydrodynamical simulations have shown an inner slope of a dark matter density profile depends on star formation history~(SFH)~\\citep[e.g.,][]{2014ApJ...789L..17M, 2015MNRAS.454.2092O}.\nIn particular, \\citet{2015MNRAS.454.2092O} predicted that the dwarfs with rapid SFHs tend to have cuspy dark matter density profiles, while ones with consecutive SFHs have cored ones at the present day.\nTherefore, we investigate whether this dependence indeed exists by comparing it with the observed SFH of dSphs.\n\nTo this end, we adopt the SFHs derived by \\citet{2009ApJ...703..692L} for Sextans and \\citet{2014ApJ...789..147W} for the other classical dSphs.\nThe top panel in Figure~\\ref{Gamma_esn} displays the cumulative SFHs of the classical dSphs taken from their works.\nAs is shown in the panel, the SFHs of the dSphs can be classified into two groups:\nthe dwarfs~(the dashed lines in the panel) that formed the majority of their stellar component early on (before $z\\simeq2$), and the other ones~(the solid ones) that formed only a small fraction of their stars at early times and continued forming stars over almost a Hubble time~\\citep{2015ApJ...811L..18G,2018MNRAS.479.1514B}.\nTo quantify these properties of the dwarfs, we estimate the lookback time at achieving 70~per~cent of the current stellar mass of these dSphs, $\\tau_{0.7}$ (as indicated as a black horizontal dotted line in the left panel in Figure~\\ref{Gamma_esn}).\n$\\tau_{0.7}$ can characterize the duration and efficiency of star formation in dSphs.\nThe bottom panel in Figure~\\ref{Gamma_esn} shows the comparison between $\\tau_{0.7}$ and dark matter inner slope, $\\gamma$, from our analysis.\nAccording to the prediction from \\citet{2015MNRAS.454.2092O}, we expect that the galaxies with higher $\\tau_{0.7}$ may have cuspy dark matter density profiles.\nFrom this figure, however, we find no clear relation between them within uncertainties of $\\gamma$.\nTherefore, the diversity of the dark matter inner slopes cannot be explained straightforwardly by SFH within the current observation and model uncertainties.\nOne of the possible reasons why there is no relation could be that the cusp-core transition requires the resonance between dark matter particles and a gas density oscillation induced by periodic SN feedbacks.\n\\citet{2014ApJ...793...46O} suggested that to transform cusp into core, at least 50 oscillations with $\\mathcal{O}(100)$~Myr periods are needed.\nUnfortunately, current photometric and spectroscopic observations are difficult to resolve such a oscillatory star formation activity.\n\n\n\\subsubsection{Dark matter density at 150~pc}\n\\citet{2019MNRAS.484.1401R} proposed to use the dark matter density at a common radius of 150~pc from the center of each galaxy, $\\rho_{DM}(150\\ {\\rm pc})$, which is insensitive to the choice of a $\\gamma$'s prior in spherical mass models.\nUsing this density, \\citet{2019MNRAS.490..231K} pointed out the anti-correlation between $\\rho_{DM}(150\\ {\\rm pc})$ and their orbital pericenter distances, $r_{\\rm peri}$, of the classical dSphs.\nThis implies a survivor bias which means that galaxies with low dark matter densities were completely destroyed by strong tidal effects.\nFollowing these works, we also calculate the dark matter density at 150~pc along the major axis of the sample dSphs, considering the non-sphericity of a dark matter halo, and the calculated $\\rho_{DM}(150\\ {\\rm pc})$ are tabulated in the last column of Table~\\ref{table2}. \n\nFirst, we compare their $\\rho_{DM}(150\\ {\\rm pc})$ to stellar masses and stellar-to-halo mass ratios.\n\\citet{2019MNRAS.484.1401R} presented the anti-correlation between them, but we do not find clear relations of $\\rho_{DM}(150\\ {\\rm pc})$-$M_{\\ast}$ and $\\rho_{DM}(150\\ {\\rm pc})$-$M_{\\ast}\/M_{\\rm halo}$.\nThis is caused by the fact that \\citet{2019MNRAS.484.1401R} discussed these correlations by including not only the dSphs but dwarf irregular galaxies which have HI gas rotation curves.\nThese gas-rich galaxies have higher $M_{\\ast}$ and $M_{\\ast}\/M_{\\rm halo}$ and much lower $\\rho_{DM}(150\\ {\\rm pc})$ than those of the dSphs, thereby the galaxies make the correlations conspicuous.\n\n\\begin{figure}[t!]\n\t\\begin{center}\n\t\\includegraphics[width=\\columnwidth]{rho150_rperi08_Vpeak.pdf}\n\t\\end{center}\n    \\caption{Dark matter densities at 150~pc, $\\rho_{DM}(150\\ {\\rm pc})$, versus pericenter radii, $r_{\\rm peri}$, of the dSphs.\n    The colored filled circles with error bars are the classical dSphs from our Jeans analysis.\n    The filled small squares are the individual subhalos predicted from dark matter simulations~(Ishiyama et al. in prep.).\n    The gray scale indicates the maximum circular velocities of subhalos over their formation histories~(the redshift when they were first accreted on to a host).\n    The big black filled squares with error bars are the stacked $\\rho_{DM}(150\\ {\\rm pc})$ and $r_{\\rm peri}$ in each radial bin.\n    The error bars correspond to the 16th and 84th percentiles of the subhalos in each bin.}\n    \\label{rho150_rperi}\n\\end{figure}\n\n\\begin{figure*}[t!]\n\t\\includegraphics[scale=0.43]{Vcric_all.pdf}\n    \\caption{The circular velocity profiles for all sample dSphs.\n    The colored solid line and shaded band in each panel show median and the 68~percent confidence intervals calculated by our non-spherical mass models.\n    The dashed lines depict the results from spherical mass models assuming NFW cusped dark matter density profiles taken from~\\citet{2019MNRAS.490..231K}.\n    The black diamonds correspond to the mass estimator of~\\citet{2010MNRAS.406.1220W}.}\n    \\label{vcirc_all}\n\\end{figure*}\n\nSecond, we investigate the anti-correlation between $\\rho_{DM}(150\\ {\\rm pc})$ and $r_{\\rm peri}$.\nFor the pericenter radius, we adopt the values presented by \\citet{2018A&A...619A.103F}, which estimated using the recent Gaia data~\\citep{2018A&A...616A..12G} and assuming a Milky Way potential model with mass of $0.8\\times10^{12}M_{\\odot}$.\nFigure~\\ref{rho150_rperi} shows the relation between $\\rho_{DM}(150\\ {\\rm pc})$ and $r_{\\rm peri}$.\nThe colored filled circles with error bars are the inferred $\\rho_{DM}(150\\ {\\rm pc})$ of the sample dSphs.\nFrom this plot, we find the anti-correlation between them similar to \\citet{2019MNRAS.490..231K}, even though there are still large uncertainties.\n\nWe also compare with dark matter subhalos predicted from dark matter only simulations.\nIn this work, we utilize a high resolution $N$-body simulation, named Phi-4096,  performed by~\\citet{2020arXiv200714720I}.\nThe detail of the simulation is as below.\nUsing a massively parallel TreePM code\nGreeM~\\footnote{http:\/\/hpc.imit.chiba-u.jp\/~ishiymtm\/greem\/}\n~\\citep{2009PASJ...61.1319I,2012arXiv1211.4406I},\nwe simulated the motion of $4096^3$ dark matter particles\nin a comoving box with the side length of 16$ \\, h^{-1} \\rm Mpc$,\nwhich corresponds to $5.13 \\times 10^{3} \\, h^{-1} M_{\\odot}$ particle mass.\nThe gravitational softening length is 60 comoving $ \\, h^{-1} \\rm pc$.\nThe initial condition was constructed using the MUSIC code \\citep{2011MNRAS.415.2101H}.\nThe cosmological parameters of the simulation\nare $\\Omega_0=0.31$, $\\lambda_0=0.69$, $h=0.68$, $n_s=0.96$, and\n$\\sigma_8=0.83$, which are consistent with the measurement of\ncosmic microwave background by the Planck satellite~\\citep{2018arXiv180706209P}.\nTo identify halos and subhalos and construct merger trees, we used\nROCKSTAR phase space halo\/subhalo finder ~\\citep{2013ApJ...762..109B}\nand consistent trees code ~\\citep{2013ApJ...763...18B}.  We picked up\nMilky Way-sized host halos with the mass of \n$3.4 \\times 10^{11} < M_{\\rm vir} < 2.0 \\times 10^{12} \\, h^{-1} M_{\\odot}$ at z=0, where $M_{\\rm vir}$ is the\nhalo virial mass.\nThe total number of host halos is 27.\n\nTo compute $\\rho_{DM}(150\\ {\\rm pc})$ of the simulated dark subhalos, we use the scale density and radius of each subhalo, supposing spherical NFW dark matter halos.\nIn Figure~\\ref{rho150_rperi}, the small filled squares denote the predicted subhalos associated with these Milky Way-sized dark matter host halos, while the big black squares with error bars are the results from stacked analysis of the subhalos in each $r_{\\rm peri}$ bin.\nIt is found from this plot that dark matter simulations indicate somewhat anti-correlation, and \nthis correlation is similar to the observed one.\nMoreover, we also find that the maximum circular velocities of subhalos over their formation histories, $V_{\\rm peak}$, of subhalos depends slightly on $\\rho_{DM}(150\\ {\\rm pc})$ and $r_{\\rm peri}$.\nIn other words, the subhalos with higher $\\rho_{DM}(150\\ {\\rm pc})$ and smaller $r_{\\rm peri}$ (the left-top area in Figure~\\ref{rho150_rperi}) tend to have large $V_{\\rm peak}$.\nSince subhalos with large $V_{\\rm peak}$ were formed at earlier, most of them have dense central densities, and such subhalos can still survive even suffering from strong tidal effects.\nIn addition, we find that $\\rho_{DM}(150\\ {\\rm pc})$ of subhalos depends on their host dark matter halo masses, which means that the subhalos associated with larger hosts have higher $\\rho_{DM}(150\\ {\\rm pc})$ than those with smaller ones.\nTherefore, this anti-correlation can be also dependence on a host halo mass.\nEven though this anti-correlation seems to support a survivor bias suggested by \\citet{2019MNRAS.490..231K}, we note that adding a stellar disk preferentially reduces the subhalo densities with smaller pericenter distances~\\citep{2019MNRAS.490.2117R}, thereby we cannot make a final conclusion about the anti-correlation without considering the impact of the disk.\n\n\n\n\\subsection{Circular velocity profile}\n\n$\\Lambda$CDM theory has another serious problem that central densities of dark matter halos associated in the bright dSphs in Milky Way are significantly lower than those of the most massive subhalos in MW-sized halos in the $\\Lambda$CDM simulations.\nThis problem is so-called the ``too-big-to-fail (TBTF)'' problem~\\citep{2011MNRAS.415L..40B}.\nIn order to compare with the central densities in the observed and simulated dark matter halos, they adopted the maximum circular velocity, $V_{\\rm max}$, for most massive ten subhalos.\nOn the other hand, for the observed ones, they used the circular velocities at the half-light radii of the dSphs, because this physical value is well-constrained by kinematic data~\\citep{2010MNRAS.406.1220W}.\nInstead of relying on such a single value of a circular velocity at a specific radius, we calculate a circular velocity profile directly from the posterior PDFs of the dark matter halo parameters.\n\nIn axisymmetric models, the circular velocity along a major axis can be calculated by\n\\begin{equation}\n    V^2_{\\rm circ}(R) = R\\left| -\\frac{\\partial\\Phi}{\\partial R}\\right|,\n\\end{equation}\nwhere $\\Phi$ is a gravitational potential originated from the dark matter density profile~ \\citep{2008gady.book.....B}. \nThe colored solid lines and shaded regions in Figure~\\ref{vcirc_all} show the inferred circular velocity profiles for the classical dSphs from our models.\nFor comparison with our results, we also plot those profiles estimated by \\citet{2019MNRAS.490..231K} and the circular velocities at their half-light radii of the dSphs, $V_{\\rm circ}(r_{\\rm half})$,~\\citep{2010MNRAS.406.1220W}.\nInterestingly, the both circular velocity profiles computed by axisymmetric and spherical models are consistent in the value of $V_{\\rm circ}(r_{\\rm half})$, but the shapes of these profiles, especially quantified with the values of $V_{\\rm max}$, look quite different in different mass models.\nThis implies that $V_{\\rm circ}(r_{\\rm half})$ would not be an adequate tracer for comparison with the central densities in dark matter halos. \nHowever, there are huge uncertainties in our estimated circular velocities, especially their outskirts due to the lack of data sample.\nThus, a sufficient number of stellar kinematic sample out to their outer parts of the dSphs should be needed.\n\n\\section{Conclusion} \\label{sec:conclusion}\nIn this paper, we revisit the core-cusp problem in the Galactic dSphs based on non-spherical Jeans analysis.\nAn advantage in these non-spherical models is that $\\rho_{\\rm DM}-\\beta_{\\rm ani}$ degeneracy occurred under the assumption of spherical symmetry can be mitigated.\n\nApplying our non-spherical mass models to the latest kinematic data of the eight classical dSphs, we estimate their dark matter density profiles by marginalizing posterior distributions of dark matter halo parameters.\nWe find that most of these dSphs favor cusped or mildly cusped dark matter profiles in their centers rather than cored one.\nIn particular, Draco robustly has a cusped dark matter halo even considering a wide prior range.\nTherefore, we conclude that there is no core-cusp problem in the classical dSphs.\n\nWe also find the diversity in the central dark matter density profiles.\nInterestingly, this diversity can be explained if we consider the impact of baryonic feedback on the central dark matter densities, which depends largely on the ratio of stellar-to-halo mass as predicted by recent $N$-body and hydrodynamical simulations.\nTherefore, $\\Lambda$CDM framework combined with baryon physics can explain the observed dark matter densities in the classical dSphs.\n\nWe also investigate the relation between the central dark matter density profiles and their star formation histories, because several high-resolution dark matter and hydrodynamical simulations predicted the correlation between these.\nHowever, we find no clear relation between an inner slope parameter of dark matter density profile and SFH characterized by $\\tau_{0.7}$.\n\nWe confirm that a dark matter density at a radius of 150~pc is anti-correlated with the pericenter distance of a dSph suggested by \\citet{2019MNRAS.490..231K}.  \nFurthermore, this anti-correlations also found in the simulated dark subhalos.\nIn addition, we also find that the maximum circular velocities of subhalos over their formation histories, $V_{\\rm peak}$ of subhalos depends slightly on $\\rho_{DM}(150\\ {\\rm pc})$ and $r_{\\rm peri}$.\nThis implies that the subhalos having dense central densities can survive from strong tidal effects due to being closer to the center of a host halo.\n\nUsing our non-spherical mass models, we calculate the circular velocity profiles of all sample dSphs and compare with those estimated by spherical mass models.\nAs a result, the shapes of circular velocity profiles, especially quantified with the maximum circular velocity, $V_{\\rm max}$, are quite different between spherical and axisymmetric mass models. \nHowever, there are huge uncertainties in our estimated circular velocities, especially their outskirts due to the lack of data sample.\n\nTo ensure our conclusions, it is necessary to determine the dark matter density profiles for much fainter dSphs, namely ultra-faint dSphs, which are believed to have held original dark matter density profiles.\nIt is also important to more precisely estimate the dark matter profiles of the classical dSphs.\nThe next-generation wide-field spectroscopic surveys with the Subaru Prime Focus Spectrograph~\\citep{2014PASJ...66R...1T} will enable us to obtain statistically significant samples of stellar kinematics and chemical abundances for the Galactic dSphs over the wide areas out to their outskirts, thereby allowing us to estimate robustly their dark matter density profiles.\n\n\n\\section*{Acknowledgements}\nWe would like to give special thanks to Manoj Kaplinghat, Ethan Nadler, Hai-Bo Yu, Chervin Laporte, Masahiro Ibe, Shigeki Matsumoto, Evan Kirby for useful discussions.\nThis work was supported in part by the MEXT Grant-in-Aid for Scientific Research on Innovative Areas (No.~18H04359, 18J00277 and 20H01895 for K.H., No.~17H01101, 18H04334 and 18H05437 for~M.C., No.~17H01101, 17H04828 and 18H04337 for~T.I.).\nNumerical computations were partially carried out on Aterui II supercomputer at\nCenter for Computational Astrophysics, CfCA, of National Astronomical\nObservatory of Japan.  T.I. has been supported by MEXT as\n``Priority Issue on Post-K computer'' (Elucidation of the Fundamental\nLaws and Evolution of the Universe), JICFuS, \nand Mext as ``Program for Promoting Researches on the Supercomputer\nFugaku'' (Toward a unified view of the universe: from large scale\nstructures to planets, proposal numbers hp200124). \n\n\n\n\n\n\n\\section{Introduction} \\label{sec:intro}\n\nIt is well documented that the concordant $\\Lambda$ cold dark matter~($\\Lambda$CDM) theory gives a remarkable description of the cosmological and astrophysical observations on large spatial scales such as the cosmic microwave background radiation~\\citep[e.g.,][]{2011ApJS..192...18K,2018arXiv180706209P}, and large-scale structure of galaxies~\\citep[e.g.,][]{2006Natur.440.1137S,2006PhRvD..74l3507T,2014MNRAS.439.2515O}.\nAt galactic and sub-galactic scales however, this theory has several discrepancies between the simulation predictions and observational facts~\\citep[][for a review]{2017ARA&A..55..343B}.\n\nOne of them is the so-called ``core-cusp'' problem.\nDark-matter-only simulations based on the $\\Lambda$CDM model have predicted a universal dark matter density profile with a strong cusp at the center~\\citep[e.g.,][]{1994Natur.370..629M,1996ApJ...462..563N,1997ApJ...490..493N,1997ApJ...477L...9F,2013ApJ...767..146I}.\nBy contrast, the observations of dwarf spheroidal (dSph) and low surface brightness galaxies seem to favor a cored central dark matter density~~\\citep[e.g.,][]{1995ApJ...447L..25B,2001MNRAS.323..285B,2007ApJ...663..948G,2008AJ....136.2761O,2010AdAst2010E...5D}.\n\nTo solve or ameliorate the issue, many possible solutions have been proposed. \nOne of the solutions is to transform a cusped to cored central density through the baryonic effects such as stellar winds and supernova feedback~\\citep[e.g.,][]{1996MNRAS.283L..72N,2002MNRAS.333..299G,2014ApJ...789L..17M,2016MNRAS.459.2573R} or heating of dark matter due to interaction of gas clumps and dark matter via dynamical friction~\\citep[e.g,][]{2001ApJ...560..636E,2011MNRAS.418.2527I,2015MNRAS.446.1820N,DelPopolo:2015nda}.\nMoreover, for the former mechanism, recent advanced simulations have predicted that the effect of core creation depends upon stellar mass and star formation history~\\citep{2012MNRAS.422.1231G,2014MNRAS.441.2986D,2014MNRAS.437..415D,2015MNRAS.454.2092O,2016MNRAS.456.3542T,2017MNRAS.471.3547F,2018MNRAS.480..800H}.\nNote that core formation ability of baryonic feedback is sensitive to the gas density threshold for a star formation, $n_{\\rm sf}$, assumed in simulations.\nA low threshold ($n_{\\rm sf}=0.1$~cm$^{-3}$) is incapable of creating a core, while a high threshold ($n_{\\rm sf}=10$-$1000$~cm$^{-3}$) is able to lead a core~\\citep[e.g.,][]{2010Natur.463..203G,2019MNRAS.486.4790B}.\nAlthough a high threshold is about four orders of magnitude greater than a low one, a whole range of the threshold is acceptable because current understanding of subgrid physics is not complete yet.\n\nAnother solution is, more radically, to replace CDM with other dark matter models that are well motivated from particle physics, such as~self-interacting dark matter~\\citep[e.g.,][see also \\citealt{2014arXiv1402.5143H,2015PhRvL.115b1301H}]{1992ApJ...398...43C,2000PhRvL..84.3760S,2016PhRvL.116d1302K,2018PhR...730....1T,Nadler:2020ulu}, and ultra-light dark matter~\\citep[e.g.,][]{2000PhRvL..85.1158H,2014MNRAS.437.2652M,2014NatPh..10..496S,2016PhR...643....1M,2016PhRvD..94d3513S,2017MNRAS.471.4559M,2017PhRvD..95d3541H}.\nThese dark matter models can create a cored, low-dense central dark matter density profile on less massive-galaxy scales without relying on any baryonic physics.\n\nMeanwhile, current dynamical studies for dSphs are challenged in the measurement of their central density profiles, because of the existence of $\\rho_{\\rm DM}-\\beta_{\\rm ani}$ degeneracy, where $\\rho_{\\rm DM}$ is a dark matter density and $\\beta_{\\rm ani}$ is a velocity anisotropy of stars as an unknown parameter~\\citep[e.g.,][]{1982MNRAS.200..361B,1990AJ.....99.1548M,2009MNRAS.393L..50E}.\nThis degeneracy originates from the assumption that both stars and dark matter are spherically distributed and from the fact that only line-of-sight velocity components of stars are available from observations~\\citep[e.g.,][]{2007ApJ...657L...1S}.\nTo disentangle this degeneracy, many dynamical modelings have been proposed, as exemplified by using higher order velocity moments~\\citep[e.g.,][]{2002MNRAS.333..697L,2009MNRAS.394L.102L,1990AJ.....99.1548M}, virial theorem~\\citep[e.g.,][]{2014MNRAS.441.1584R}, modeling multiple stellar populations~\\citep[e.g.,][]{2008ApJ...681L..13B,2011ApJ...742...20W}, orbit-based dynamical models~\\citep[e.g.,][]{2013ApJ...763...91J,2013MNRAS.433.3173B}, measuring the internal proper motion data~\\citep{2018NatAs...2..156M,2019arXiv190404037M,2018ApJ...860...56S}, and non-parametric analysis~\\citep[e.g.,][]{2017MNRAS.471.4541R,2019MNRAS.484.1401R}.\nHowever, the inferred dark matter density profiles are not completely unified, and some of these models cannot distinguish a cusp from a core from the currently available kinematic data, due to considerable uncertainties in the derived dark matter density profiles and a prior bias of a dark matter inner slope parameter.\nThus, whether the central dark matter densities in dSphs are cored or cusped is yet unclear.\n\nWe emphasize that many of these studies assume spherical symmetry for both the stellar and dark components, even though we know both from observational facts and theoretical predictions that these components are actually non-spherical~\\citep[e.g.,][]{2012AJ....144....4M,2018ApJ...860...66M,2006MNRAS.367.1781A,2014MNRAS.439.2863V}. \nIn this paper, we relax the spherically symmetric assumption and perform the axisymmetric Jeans analysis for the dSphs.\nSuch non-spherical mass models have several advantages that (i)~giving the specific form of the distribution function is not required; (ii)~this analysis can treat {\\it two-dimensional} distributions of line-of-sight velocity dispersions~\\citep[e.g.,][]{2012ApJ...755..145H}, whereas it is impossible for spherical mass models; and (iii)~$\\rho_{\\rm DM}-\\beta_{\\rm ani}$ degeneracy can be mitigated~\\citep{2008MNRAS.390...71C,BATTAGLIA201352,2015ApJ...810...22H}.\nSeveral studies have developed axisymmetric mass models based on the Schwarzschild method~\\citep{2012ApJ...746...89J} and Jeans anisotropic multiple Gaussian expansion model~\\citep{2016MNRAS.463.1117Z}, but many of these assumed that a dark matter halo is still spherical while a stellar system is non-spherical.\n\nOur group constructed, as presented in \\citet{2015ApJ...810...22H}, totally axisymmetric dynamical mass models based on axisymmetric Jeans equations and applied the models to the dSphs with Milky Way and Andromeda galaxies~\\citep[see also ][]{2012ApJ...755..145H}.\n\\citet{2016MNRAS.461.2914H} applied the axisymmetric mass models to the recent kinematic data for the ultra-faint dSphs as well as classical ones to evaluate the astrophysical factors for dark matter annihilation and decay with considering the uncertainties of non-sphericity.\n\nOur previous models were yet incomplete in the point that an outer dark matter profile is fixed as $\\rho_{\\rm DM}\\propto r^{-3}$ for the sake of simplicity.\nHere, to step further from these previous studies, we adopt a generalized Herquist profile to explore a much wider range of physically plausible dark matter profiles and apply these non-spherical models to the latest observational data of the Galactic classical dSphs~(Draco, Ursa~Minor, Carina, Sextans, Leo~I, Leo~II, Sculptor, and Fornax) having a large number of member stars with well-measured radial velocities.\n\nThe paper is organized as follows. \nIn Section 2, we explain axisymmetric models based on an axisymmetric Jeans analysis and our fitting procedure. \nIn Section 3, we describe the photometric and spectroscopic data for the classical dSphs. \nIn Section 4, we present the results of the fitting analysis. We also show the estimated dark matter density profiles and the values of astrophysical factors. \nIn Section 5, we discuss the results of our estimations. \nFinally, conclusions are presented in Section 6.\n\n\n\\input{table1.tex}\n\\section{Models and Jeans analysis} \\label{sec:jeans}\nIn this section, we briefly introduce our dynamical mass models in this work.\nTo show how precisely we are able to recover actual dark matter density profiles from our fitting analysis, we apply our mass models to mock data sets. The details about mock data and the results of mock analysis are shown in Appendix~\\ref{sec:AppA}.\n\n\\subsection{Axisymmetric Jeans equations}\nAssuming that a galaxy is in a dynamical equilibrium and collisionless under a smooth gravitational potential, the dynamics of stars in such a system is described by its phase-space distribution function governed by the steady-state collisionless Boltzmann equation~\\citep{2008gady.book.....B}.\nHowever, it is virtually impossible to solve this equation from the currently available data of stars in the dSphs whose positions along the line of sight are difficult to resolve and accurate proper motions are yet to be measured.\nIn order to alleviate this issue, one of the classical and useful approaches is to take moments of the equation.\nThe equations taking moments of the steady-state collisionless Boltzmann equation are the so-called Jeans equations.\n\nFor an axisymmetric and steady state system, the Jeans equations are expressed as \n\\begin{eqnarray}\n\\overline{u^2_z} &=&  \\frac{1}{\\nu(R,z)}\\int^{\\infty}_z \\nu\\frac{\\partial \\Phi}{\\partial z}dz,\n\\label{AGEb03}\\\\\n\\overline{u^2_{\\phi}} &=& \\frac{1}{1-\\beta_z} \\Biggl[ \\overline{u^2_z} + \\frac{R}{\\nu}\\frac{\\partial(\\nu\\overline{u^2_z})}{\\partial R} \\Biggr] +\nR \\frac{\\partial \\Phi}{\\partial R},\n\\label{AGEb04}\n\\end{eqnarray}\nwhere $\\nu$ is the three-dimensional stellar density and $\\Phi$ is the gravitational potential, which is significantly dominated by dark matter for the Galactic dSphs.\nThe latter means that stellar motions in a system are governed only by a dark matter potential.\nWe assume that the cross terms of velocity moments such as $\\overline{u_Ru_z}$ vanish and the velocity ellipsoid constituted by $(\\overline{u^2_R},\\overline{u^2_{\\phi}},\\overline{u^2_z})$ is aligned with the cylindrical coordinate.\nWe also assume that the density of tracer stars has the same orientation and symmetry as that of a dark halo.\n$\\beta_z=1-\\overline{u^2_z}\/\\overline{u^2_R}$ is a velocity anisotropy parameter introduced by~\\citet{2008MNRAS.390...71C}.\nIn this work, $\\beta_z$ is assumed to be constant for the sake of simplicity\\footnote{Nevertheless, this assumption is roughly in good agreement with dark matter simulations reported by~\\citet{2014MNRAS.439.2863V} who have shown that simulated subhalos have an almost constant $\\beta_z$ or a weak trend as a function of radius along each axial direction.}.\nIn principle, these second velocity moments are defined as $\\overline{u^2}= \\sigma^2+\\overline{u}^2$, where $\\sigma$ and $\\overline{u}$ are dispersion and streaming motions of stars, respectively.\nThe latter streaming motions are small in the dSphs~\\citep[e.g.,][]{2008ApJ...688L..75W}, and thus these galaxies are largely dispersion-supported stellar systems~\\citep[e.g.,][]{2017MNRAS.465.2420W}.\n\nTo compare with the observed second velocity moments, the intrinsic second velocity moments derived by the Jeans equations are integrated along the line-of-sight second velocity moment followed by the previous works~\\citep{1997MNRAS.287...35R,2006MNRAS.371.1269T,2012ApJ...755..145H}.\nThis moment can be written as \n\\begin{equation}\n\\overline{u^2_{\\rm l.o.s}}(x,y) = \\frac{1}{I(x,y)}\\int^{\\infty}_{-\\infty}\\nu(R,z)\\overline{u^2_{\\ell}}(R,z)d\\ell,\n\\label{los3}\n\\end{equation}\nwhere $I(x,y)$ indicates the surface stellar density profile calculated from $\\nu(R,z)$, and $(x,y)$ are the sky coordinates aligned with the major and minor axes, respectively.\n$\\overline{u^2_{\\ell}}(R,z)$ is driven by\n\\begin{equation}\n\\overline{u^2_{\\ell}} = \\overline{u^2_{\\ast}}\\cos^2\\theta + \\overline{u^2_z}\\sin^2\\theta,\n\\label{los2}\n\\end{equation}\nwhere $\\theta$ is the angle between the line of sight and the galactic plane~($\\theta=90^{\\circ}-i$, which $i$ is an inclination angle explained below).\n$\\overline{u^2_{\\ast}}$ is a velocity second moment derived from the projection $\\overline{u^2_R}$ and $\\overline{u^2_{\\phi}}$ to the plane parallel with the galactic plane along the intrinsic major axis.\nThis moment is described as \n\\begin{equation}\n\\overline{u^2_{\\ast}} = \\overline{u^2_{\\phi}}\\frac{x^2}{R^2} + \\overline{u^2_R}\\Bigl(1-\\frac{x^2}{R^2}\\Bigr).\n\\label{los1}\n\\end{equation}\n\n\\subsection{Stellar density profile}\\label{sec:starprof}\nFor the stellar density profile, we adopt a Plummer profile \\citep{1911MNRAS..71..460P} generalized to an axisymmetric shape:\n\\begin{eqnarray}\n\\nu(R,z)=\\frac{3L}{4\\pi b^3_{\\ast}}\\frac{1}{(1+m^2_{\\ast}\/b^2_{\\ast})^{5\/2}}\n\\label{plummer}\n\\end{eqnarray}\nwhere $m^2_{\\ast}=R^2+z^2\/q^2$, so that $\\nu$ is constant on spheroidal shells with an intrinsic axial ratio $q$, and $L$ and $b_{\\ast}$ are the total luminosity and the half-light radius along the major axis, respectively.\nThis profile can be analytically derived from the surface density profile using Abel transformation: $I(x,y)=(L\/\\pi b^2_{\\ast})(1+m^{\\prime 2}_{\\ast}\/b^2_{\\ast})^{-2}$,\nwhere $m^{\\prime 2}_{\\ast}=x^2+y^2\/q^{\\prime 2}$.\n$q^{\\prime}$ is a projected axial ratio and is related to the intrinsic one $q$ through the inclination angle $i$~$(=90^{\\circ}-\\theta)$: $q^{\\prime 2} = \\cos^2i+q^2\\sin^2i$.\nThis equation can be rewritten as $q=\\sqrt{q^{\\prime 2}-\\cos^2i}\/\\sin i$, and thus the allowed range of the inclination angle is bounded with $0\\leq\\cos^2i<q^{\\prime 2}$.\nIn this work, we assume that the stellar distribution has an {\\it oblate} shape only.\nThis is motivated by the result from \\citet{2015ApJ...810...22H} which concluded that most of stellar distributions of the dSphs are much better fitted by the oblate shape than by the prolate ones.\n\n\n\\subsection{Dark matter density profile}\\label{sec:dmprof}\nIn this work, we adopt a generalized Hernquist profile given by~\\citet{1990ApJ...356..359H} and also~\\citet{1996MNRAS.278..488Z} with considering non-spherical dark matter halos,\n\\begin{eqnarray}\n&& \\rho_{\\rm DM}(R,z) = \\rho_0 \\Bigl(\\frac{r}{b_{\\rm halo}} \\Bigr)^{-\\gamma}\\Bigl[1+\\Bigl(\\frac{r}{b_{\\rm halo}} \\Bigr)^{\\alpha}\\Bigr]^{-\\frac{\\beta-\\gamma}{\\alpha}},\n \\label{DMH} \\\\\n&& r^2=R^2+z^2\/Q^2,\n\\label{DMH2}\n\\end{eqnarray}\nwhere $\\rho_0$ and $b_{\\rm halo}$ are the scale density and radius, respectively, $\\alpha$ is\nthe sharpness parameter of the transition from the inner slope $\\gamma$ to the outer slope $\\beta$, \nand $Q$ is a constant axial ratio of a dark matter halo. \nThis model covers a broad range of physically plausible dark matter profiles from the cusped Navarro-Frenk-White~\\citep[hereafter NFW,][]{1997ApJ...490..493N} profile to the cored Burkert profile~\\citep{1995ApJ...447L..25B}.\n\n\n\\input{table2.tex}\n\n\\subsection{Fitting procedure}\\label{sec:fitting}\nIn order to estimate the dark matter density profiles in the dSphs, we explore the most likely parameter values by fitting theoretical and observed second velocity moments of each dSph.\nIn this work, we suppose that the line-of-sight velocity distribution is Gaussian and centered on the systemic velocity of the galaxy $\\langle u \\rangle$.\nGiven that the total number of member stars for each dSph is $N$, and the observed line-of-sight velocity of the $i$th member star and its velocity error is expressed by $u_i\\pm\\delta_{u,i}$ at the sky plane coordinates~$(x_i,y_i)$, the likelihood function is described as\n\\begin{equation}\n{\\cal L} = \\prod^{N}_{i=1}\\frac{1}{(2\\pi)^{1\/2}[(\\delta_{u,i})^2 + (\\sigma_i)^2]^{1\/2}}\\exp\\Bigl[-\\frac{1}{2}\\frac{(u_i-\\langle u \\rangle)^2}{(\\delta_{u,i})^2 + (\\sigma_i)^2} \\Bigr],\n\\label{LF}\n\\end{equation}\nwhere $\\sigma_i$ is the theoretical line-of-sight velocity dispersion at~$(x_i,y_i)$ which is calculated by model parameters and the Jeans equations.\nThe systemic velocity $\\langle u \\rangle$ of the dSph is a nuisance parameter that we marginalize over as a flat prior.\nFor the model parameters, we introduce flat or log-flat priors over the following ranges:\n\\begin{enumerate}[(i)]\n\\item $0.1\\leq Q\\leq2.0$;\n\\item $0.0\\leq \\log_{10}[b_{\\rm halo}\/{\\rm pc}]\\leq5.0$;\n\\item $-5.0\\leq \\log_{10}[\\rho_0\/(M_{\\odot}~{\\rm pc}^{-3})]\\leq5.0$;\n\\item $-1.0\\leq -\\log_{10}[1-\\beta_{z}]<1.0$;\n\\item $0.5\\leq\\alpha\\leq3$\n\\item $3.0\\leq\\beta\\leq10$\n\\item $0.0\\leq\\gamma\\leq2.5$\n\\item $\\cos^{-1}(q^{\\prime})< i\/{\\rm deg} \\leq 90.0$.\n\\end{enumerate}\n\nIn order to obtain the posterior probability distribution function (PDF) of each parameter by the above likelihood function, we perform a Markov Chain Monte Carlo~(MCMC) techniques, based on Bayesian parameter inference, using the Metropolis-Hastings algorithm~\\citep{1953JChPh..21.1087M,10.1093\/biomet\/57.1.97}.\nTo avoid an influence of initial conditions and to generate independent samples, we take several post-processing steps such as burn-in step, the sampling step and length of the chain.\nWe evaluate the percentiles of these PDFs to estimate credible intervals for each parameter straightforwardly.\n\n\\subsection{Results from applying our models to mock data}\nTo scrutinize whether our models can reproduce dark matter density profiles, we apply them to mock data.\nHere, we focus on testing how precisely our models are able to reproduce inner density slopes of dark halos.\nIn this subsection, we briefly describe the procedure for mock analysis and summarize the result from this analysis. The details are shown in Appendix~\\ref{sec:AppA}.\n\nWe use public mock data sets provided by~\\citet{2016MNRAS.463.1117Z}, which generated kinematic samples with axisymmetric stellar systems embedded in spherical dark matter halos and generated two kinds of dark matter halos: one with a cusped halo and one with a cored halo.\n\nWhen we apply our models to these mock data, we first estimate a projected axial ratio and a half-light radius, employing a maximum likelihood analysis~\\citep{2008ApJ...675..201M} with the Plummer stellar density profile.\nThen, we perform a MCMC analysis for mock kinematic samples to estimate the dark matter density profiles. \n\nIn this mock analysis, we carry out the fitting in the three cases: (A) a cusped model with 1000 samples, (B) a cored model with 1000 samples, and (C) a cored one with 4000 samples.\nThe dark matter density profiles estimated from the analysis are nearly reproduced within $1\\sigma$ uncertainties for each case.\nThe estimated values of dark matter inner slope, $\\gamma$, are $\\gamma=1.2^{+0.4}_{-0.5}$, $0.4^{+0.8}_{-0.3}$, and $0.4^{+0.3}_{-0.2}$ for the cusped mock~(A), the cored ones~(B) and (C), respectively~(see Figure~\\ref{fig:Mock}).\nIt is found that the MCMC fitting to the cored mock data results in a somewhat biased dark matter density profile with $\\gamma \\simeq 0.4$, although the model is nearly consistent with a cored halo within $1\\sigma$ confidence.\nSuch a bias is also appeared in several previous works~\\citep[e.g.,][]{2016MNRAS.463.1117Z,2018MNRAS.481..860R}, but the reason for this bias is yet unclear.\n\nConsequently, we should bear in mind the fact that there exits this small amount of bias for a dark matter inner slope in our fitting analysis.\n\n\\section{Data}\\label{sec:data}\nIn this section, we present the basic properties of photometric and spectroscopic data of the eight classical dSphs: Draco, Ursa~Minor, Carina, Sextans, Leo~I, Leo~II, Sculptor, and Fornax.\nThe classical dSphs have a larger number of line-of-sight velocities for the resolved stars ($\\gtrsim200$~stars) than ultra faint dwarf galaxies.\nThese galaxies also have large velocity dispersions~($\\gtrsim10$~km~s$^{-1}$), so that an influence of unresolved binary stars on the velocity dispersion measurements of each galaxy can be negligible~\\citep{2010ApJ...721.1142M,2013ApJ...779..116M,2017ApJ...836..202S,2018AJ....156..257S}.\n\nTable~\\ref{table1} lists the observational properties of the eight dSphs:~the number of member stars with velocity measurements available from the kinematic analysis, the central sky coordinates, distances from the Sun, projected half-light radii, projected stellar axial ratios, systemic velocities, and their references.\nFollowing previous works, we fix the values of distance, half-light radius, and axial ratio of dSphs in this paper.\n\nFor the stellar kinematic data of their member stars, we use the published data as follows.\nFor Carina, Draco, Ursa~Minor, Leo~I, and Leo~II, we use the stellar-kinematic data taken from \\citet{2016ApJ...830..126F}, \\citet{2015MNRAS.448.2717W}, \\citet{2018AJ....156..257S}, \\citet{2008ApJ...675..201M}, \\citet{2017ApJ...836..202S}, respectively.\nFor Sextans, Sculptor, and Fornax, we use the data published by \\citet{2009AJ....137.3100W,2009AJ....137.3109W}.\nThe membership selection criteria for each galaxy follow the methods described by each of the observational papers given above.\n\n\n\\section{Results} \\label{sec:results}\nIn this section, we present the results from the MCMC fitting analysis described above and several trends among the resultant parameters.\nMoreover, as a by-product of the fitting results, we estimate the astrophysical factors for dark matter annihilation and decay.\n\n\\subsection{Best-fitting models} \\label{sec:best}\nTable~\\ref{table2} shows the best-fitting parameters for each dSph.\nThe error values indicate the 68~per~cent credible intervals computed from posterior PDFs of the parameters. \nWe also show the dark matter density at 150~pc, $\\rho_{\\rm DM}(150\\ {\\rm pc})$, to compare with the other works. \n\\citet{2019MNRAS.484.1401R} presented the dark matter density at a common radius of 150~pc from the center of each galaxy, $\\rho_{DM}(150\\ {\\rm pc})$, which is insensitive to the choice of a $\\gamma$'s prior in spherical mass models.\nFurthermore, using this density, \\citet{2019MNRAS.490..231K} pointed out the anti-correlation between $\\rho_{DM}(150\\ {\\rm pc})$ and their orbital pericenter distances, $r_{\\rm peri}$, of the classical dSphs.\nThis implies a survivor bias: while galaxies with low dark matter densities were completely destroyed by strong tidal effects, those with high dark matter densities survive in the present day.\nFollowing these works, we also calculate the dark matter density at 150~pc along the major axes of the sample dSphs, considering non-sphericity of a dark matter halo, and the calculated $\\rho_{DM}(150\\ {\\rm pc})$ are tabulated in the last column of Table~\\ref{table2}.\nWe discuss it in the next section.\n\nFigure~\\ref{drafnx} displays the posterior PDFs implemented by the MCMC fitting for Draco~(left panel) and Fornax~(right panel) dSphs as the representative galaxies in our sample.\nThe results of other galaxies are similar to these and thus we show the other posterior PDFs~(Figure~\\ref{carumi}, \\ref{leo12}, and \\ref{sclsex}) in Appendix~\\ref{sec:AppB}. \nThe contours in these figures show 68, 95, and 99.7 per~cent credible interval levels.\nThe vertical lines in each histogram also show the median and 68 per~cent credible interval levels.\nFrom these posterior PDF maps, the parameters $Q$, $\\alpha$, $\\beta$, and $i$ are widely distributed in these parameter ranges and thus it is difficult to get limits on them.\nOn the other hand, the other parameters $b_{\\rm halo}$, $\\rho_0$, $\\beta_z$, and $\\gamma$ are better constrained than the above parameters, even though there are obvious degeneracies between $b_{\\rm halo}$-$\\rho_0$ and $Q$-$\\beta_z$, which have already been discussed in several previous papers~\\citep[e.g.,][]{2008MNRAS.390...71C,BATTAGLIA201352,2015ApJ...810...22H}.\nIn particular, the velocity anisotropy parameters $\\beta_z$ of all galaxies (see also Table~\\ref{table2}) tend to be a somewhat radially-biased velocity ellipsoid.\nMoreover, it is worth noting that owing to non-spherical models, the inner slope parameter of a dark matter density profile $\\gamma$ can be confined without being distracted by any parameter degeneracies.\nFrom these PDFs and Table~\\ref{table2}, we find that the posteriors of $\\gamma$ show a wide spread from cusped $(\\gamma>1.0)$ to shallower cusped $(\\gamma<0.5)$ inner dark matter density slopes, even though there is a large uncertainty.\nWe discuss this further in Section~\\ref{sec:bestdmprof}.\n\nFigure~\\ref{los} shows the comparison between the observed and the estimated line-of-sight velocity dispersion profiles obtained by the resultant posterior PDFs, to present our fitting analysis successfully reproduced to the binned data\\footnote{The method for calculating these binned profiles along the projected major, middle, and minor axes for the dSphs is the same way as \\citet{2019MNRAS.tmp.2554H}, and thus the details are described in the Section~3.1.2 in that paper.}.\nIn this figure, the colored solid lines and shaded regions denote the median and confidence levels (dark: 68~per~cent, light: 95~per~cent) of our {\\it unbinned} MCMC analysis.\nThe black points with error bars denote {\\it binned} velocity dispersions calculated by the observed data.\nThese errors correspond to the 68~per~cent confidence intervals.\nAs shown in this figure, our mass models and unbinned analysis can provide good fits to the binned data for all dSphs.\n\n\\begin{figure*}\n\t\\begin{minipage}{0.49\\hsize}\n\t\t\\begin{center}\n\t\t\t\\includegraphics[width=\\columnwidth]{PDF_dra.pdf}\n\t\t\\end{center}\n\t\\end{minipage}\n\t\\begin{minipage}{0.49\\hsize}\n\t\t\\begin{center}\n\t\t\t\\includegraphics[width=\\columnwidth]{PDF_fnx.pdf}\n\t\t\\end{center}\n\t\\end{minipage}\n    \\caption{Posterior distributions for the fitting parameters for Draco~(left) and Fornax~(right).\n    The dashed lines in each histogram represent the median and 68~per~cent confidence values. The contours in each panel are the 68, 95, and 99.7~per~cent regions.}\n    \\label{drafnx}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[scale=0.33]{Draco_LOSdisp.pdf}\n\\includegraphics[scale=0.33]{UMi_LOSdisp.pdf}\n\\includegraphics[scale=0.33]{Carina_LOSdisp.pdf}\n\\includegraphics[scale=0.33]{Sextans_LOSdisp.pdf}\n\\includegraphics[scale=0.33]{LeoI_LOSdisp.pdf}\n\\includegraphics[scale=0.33]{LeoII_LOSdisp.pdf}\n\\includegraphics[scale=0.33]{Sculptor_LOSdisp.pdf}\n\\includegraphics[scale=0.33]{Fornax_LOSdisp.pdf}\n\\end{center}\n\\caption{Line-of-sight velocity dispersion along major, middle and minor axes for each dSph. The black squares with error bars in each panel denote the observed ones. The solid lines are the median velocity dispersion of the models and the dark and light shaded regions encompass the 68~per cent and 95~per cent confidence levels from the results of the unbinned MCMC analysis. The vertical dashed lines in each panel correspond to their half-light radii.}\n\\label{los}\n\\end{figure*}\n\n\\subsection{Revisiting the core-cusp problem} \\label{sec:bestdmprof}\n\\subsubsection{Dark matter density profiles}\nUsing the results of the MCMC fitting analysis for the kinematic data of the dSphs, we estimate the dark matter density profiles by marginalizing all free parameters.\nFigure~\\ref{dmpro} shows the inferred dark matter density profiles of all sample dSphs.\nThe solid lines show the medium, and dark and light contours mark the 68~per~cent and 95~per~cent intervals.\nThe vertical black lines mark the projected half light radii of each dSph.\n\nFirstly, it is noteworthy that in our non-spherical models, Draco favors a cusped inner slope for its dark matter density profile, which is consistent with an NFW cusp predicted by $\\Lambda$CDM theory.\nEven if we consider 95~per~cent confidence intervals of the dark matter profile, its inner slope still remains cuspy.\nTherefore, Draco highly likely has a cusped dark matter halo.\n\nSecondly, Ursa~Minor, Leo~I and Leo~II also prefer cusped dark matter halos, although the uncertainties for the inner slope, $\\gamma$, are larger than for Draco.\nOn the other hand, the remaining sample of dSphs (Carina, Sextans, Sculptor, and Fornax) favors smaller $\\gamma$ and thus has less dense than the other dSphs which have cuspy dark matter halos.\nIn particular, Sextans, Sculptor, and Fornax permit $\\gamma=0$, i.e. a cored dark matter density within their 95~per~cent confidence intervals.\n\nNotably, the dark matter density profile in Draco is better constrained than Fornax, though the data volume of Fornax is greater than Draco.\nThis is because while the observed kinematic sample in Draco covers the stars up to its outskirts, that in Fornax is limited only to its inner region.\nActually, \\citet{2015ApJ...810...22H} suggested that the lack of kinematic sample volume in the outer region of a galaxy makes the constraints on the dark matter profile very uncertain.\n\nTherefore, from our dynamical analysis, we propose that there is {\\it no} core-cusp problem in the Galactic classical dSphs.\nMoreover, a {\\it diversity} of the inner density slope, $\\gamma$, is found for these dSphs.\nNote that this result is in agreement with \\citet{2019MNRAS.484.1401R}, which investigated the inner dark matter densities in the Galactic dSphs as well as in low surface brightness galaxies based on non-parametric spherical Jeans analysis.\n\n\\subsubsection{Why do some galaxies prefer cusped dark matter halos?} \\label{sec:demonstration}\nAs shown in the previous section, we present that some dSphs prefer cusped dark matter density profiles.\nThen the question is why these galaxies are regarded to have cusped dark matter halos.\nWe schematically illustrate this reason in Figure~\\ref{los_demo4} in the Appendix~\\ref{sec:AppC}.\nThis figure shows the normalized line-of-sight velocity dispersion profiles along the major~(top panels) and the minor axes~(bottom panels) for the oblate stellar system $(q=0.7)$.\nThe left-hand panels show the dispersion profiles with changing the value of velocity anisotropy parameter, $\\beta_z$, under spherical dark matter halo, $Q=1$, whilst the right-hand ones depict those with changing $Q$ under $\\beta_z=0$.\n\nAs already discussed in \\citet{2008MNRAS.390...71C} and \\citet{2015ApJ...810...22H}, the variation of $Q$ and $\\beta_z$ gives a similar effect on line-of-sight dispersion profiles.\nFor instance, as is shown in the top-left panel of Figure~\\ref{los_demo4}, the effect of $\\beta_z>0$ (i.e., red lines) increases inner line-of-sight velocity dispersions and decreases outer ones, simultaneously, compared with those in the fiducial $(Q=1,\\beta_z=0)$ case which corresponds to the black lines.\nIn the top-right panel, the effect of $Q<1$ (the red lines) is resemblant in the features of line-of-sight velocity dispersion profiles computed by $\\beta_z>0$ (the red lines in the top-left panel), even though there is a difference between these effects at the outer parts (the reason of this difference is already discussed in \\citealt{2015ApJ...810...22H}). \n\nHowever, comparing the dispersion profiles in the cases for cusped (the solid lines) and for cored (the dotted lines) dark matter density profile, we can see a difference in the shape of those profiles at inner parts.\nIn the case of a cusped dark matter halo, the velocity dispersion profiles along both major and minor axes rapidly increase towards the central region, while there is no such trend in the case of a cored one.\nLooking at the observed line-of-sight velocity dispersion profiles in Figure~\\ref{los}, Draco, as an example, seems to have the trend characterized by a cusped dark matter halo, whereas Fornax has the almost flat profiles.\nTherefore, we suggest that Draco highly likely has a cusped dark matter halo.\nIt is also found that the feature of a central velocity dispersion profile can be important in determining an inner slope of a dark matter density profile.\n\n\\subsubsection{The robustness of our results}\nIn order to demonstrate the robustness of our results, especially regarding the inner slope of a dark matter density profile, $\\gamma$,\nwe show the case when a wide range of prior for $\\gamma$ is adopted, compared to our fiducial parameter range of $\\gamma$ ($0\\leq\\gamma\\leq2.5$).\nNamely, we show here the case of a flat prior over range $-2.5\\leq\\gamma^{\\prime}\\leq2.5$, and we impose $\\gamma=0$ if $\\gamma^{\\prime}$ has a negative value and $\\gamma=\\gamma^{\\prime}$ otherwise.\nThis is because the fiducial parameter range of $\\gamma$ ($0\\leq\\gamma\\leq2.5$) might lead to a bias toward cuspy density profiles.\nUsing this new prior, we re-run the same MCMC fitting procedure described in Section~\\ref{sec:fitting}.\nFigure~\\ref{dmpro_bias} shows the comparison of the inferred dark matter density profiles for all sample dSphs for the fiducial~(solid) and wider~(dashed) prior ranges.\nThe thick and thin lines in each panel denote the median and the 68~per~cent confidence intervals.\nIt is found from this figure that the galaxies having a cusped dark matter halo like Draco and Ursa Minor are not so much affected by new prior, whilst the effect of new prior makes Fornax and Sextans less dense core.\nTherefore, we bear in mind that Fornax and Sextans are possible to have a cored dark matter density.\nOn the other hand, we can confirm that our results for Draco and Ursa~Minor have cusped dark matter halos.\n\n\\begin{figure*}[t!]\n\t\\includegraphics[scale=0.4]{DMpro_all.pdf}\n    \\caption{Dark matter density profiles along major axes of the galaxies derived from our Jeans analysis.\n    The solid line in each panel denotes the median value, and the dark and light shaded regions denote the 68 and 95~per~cent confidence intervals.\n    The vertical dashed line in each panel corresponds to the half-light radius of each galaxy.\n    In the panel for Draco, we mark on two power law density profiles, $\\rho_{\\rm DM}\\propto r^{-1}$ (cusp) and $\\rho_{\\rm DM}={\\rm const.}$ (core) under the shaded regions.}\n    \\label{dmpro}\n\\end{figure*}\n\\begin{figure*}[t!]\n\t\\includegraphics[scale=0.4]{DMpro_all_bias.pdf}\n    \\caption{Dark matter density profiles of all dSphs, with taking into account a wider parameter range of $\\gamma$~(described in Section~\\ref{sec:bestdmprof}).\n    The solid lines in each panel denote the median values~(thick) and the 68~per~cent confidence intervals~(thin) calculated by our default parameter range ($0\\leq\\gamma\\leq2.5$), while the dashed ones are calculated by a new parameter range ($-2.5\\leq\\gamma^{\\prime}\\leq2.5$, but if $\\gamma^{\\prime}<0\\rightarrow\\gamma=0$).\n    The vertical dashed lines in each panel correspond to their half-light radii.}\n    \\label{dmpro_bias}\n\\end{figure*}\n\n\n\\subsection{Astrophysical factors} \\label{sec:bestjd}\nThe Galactic dSphs are promising targets for indirect searches for particle dark matter through $\\gamma$-rays or X-rays stemmed from annihilating and decaying dark matters~\\citep[e.g.,][]{1978ApJ...223.1015G,2012AnP...524..479B}, because they contain a good deal of dark matter with low astrophysical backgrounds and are located at relative proximity.\nThe signal flux of the dark matter annihilation or decay depends only on two important factors.\nOne is the particle physics factor which is based on the microscopic physics of particle dark matter, while another is the astrophysical factor derived by line-of-sight integrals over the dark matter distribution within the system.\nThe latter largely depends on the estimate of the signal flux.\nTherefore, an accurate estimation of the astrophysical factor in the dSphs is of crucial importance so that we can set robust constraints on the particle nature of dark matter candidates.\n\nPrevious works have estimated the astrophysical factors for these galaxies considering various uncertainties: the spatial dependence of stellar velocity anisotropy~\\citep{2016JCAP...07..025U}, non-sphericity of a dark matter distribution~\\citep{2015MNRAS.446.3002B,2016MNRAS.461.2914H,2017PhRvD..95l3012K}, halo truncation radius~\\citep{2015ApJ...801...74G}, prior bias of Bayesian analysis~\\citep{2009JCAP...06..014M}, and foreground contamination of stars~\\citep{2016MNRAS.462..223B,2017MNRAS.468.2884I,2018MNRAS.479...64I,2020arXiv200204866H}.\n\nHere we calculate the astrophysics factors of the dSphs focusing only on non-sphericity based on the generalized Hernquist density profile of their dark matter halos.\nIn fact, (sub-) subhalos and substructures can boost the annihilation signals~\\citep[subhalo boost,][]{2017MNRAS.466.4974M,2018PhRvD..97l3002H,2020MNRAS.492.3662I}. However, this boost contributes little to the signals on the dSph's mass scales, and thus we do not include this boost to estimate $J$-factor values.\nTo compare with previous works, we show only the factors integrated within a fixed solid angle $0.5^{\\circ}$.\n\nThe astrophysical factors are written as \n\\begin{eqnarray}\nJ &=& \\int_{\\Delta\\Omega}\\int_{\\rm los}d\\ell d\\Omega\\rho^2_{\\rm DM}(\\ell,\\Omega) \\ \\hspace{5mm}  [{\\rm annihilation}], \\\\\nD &=& \\int_{\\Delta\\Omega}\\int_{\\rm los}d\\ell d\\Omega\\rho_{\\rm DM}(\\ell,\\Omega) \\ \\hspace{5mm}  [{\\rm decay}],\n\\end{eqnarray}\nwhich are so-called $J$- and $D$-factors, defined as the integrated dark matter density squared for annihilation and the dark matter density for decay, respectively, over a distance $\\ell$ along a line-of-sight and a solid angle $\\Delta\\Omega$.\nUsing these equations, we estimate the median and its uncertainties of the astrophysical factors from the posterior PDFs of the dark matter halo parameters.\n\nTable~\\ref{table3} shows the $J$ and $D$ values integrated within $\\Delta\\Omega=0.5^{\\circ}$ of our results.\nFigure~\\ref{JDcomp} displays a comparison of the $J$ (top) and $D$ (bottom) values of our results with those of previous works.\nIn this figure, the red colored points with error bars are the median values in this work with 68~per~cent confidence intervals.\nThe blue ones denote these values reported by \\citet{2015ApJ...801...74G}, which assumed a spherical dark matter halo with a generalized Hernquist density profile and performed Jeans analysis.\nThe green ones are evaluated by \\citet{2016MNRAS.461.2914H}, which assumed an axisymmetric dark matter halo.\nThe differences between them in this figure are caused primarily by the assumption of shapes of dark matter halos (spherical or non-spherical) as already discussed by \\citet{2016MNRAS.461.2914H} and dark matter density profiles.\nThe latter means that \\citet{2016MNRAS.461.2914H} imposed that the outer slope of dark matter profiles is $\\rho\\propto r^{-3}$ and the sharpness parameter $\\alpha$ in Equation~\\ref{DMH} is fixed at $2$ for simplicity, while the dark matter profiles in this work and \\citet{2015ApJ...801...74G} take into account these parameter as free parameters.\n\n\\input{table3}\n\nFrom Figure~\\ref{JDcomp}, we conclude that because of having a cuspy dense dark matter halo and of the close distance to the Sun, Draco is the most promising detectable target for an indirect search of dark matter annihilation and decay among all sample dSphs.\n\n\\begin{figure}\n\t\\begin{center}\n\t\\includegraphics[width=\\columnwidth]{J_comp.pdf}\n\t\\includegraphics[width=\\columnwidth]{D_comp.pdf}\n\t\\end{center}\n    \\caption{Comparison of $J_{0.5}$~(top) and $D_{0.5}$~(bottom) calculated from previous and this works.\n    The blue and green symbols are estimated by~\\citet{2015ApJ...801...74G} and \\citet{2016MNRAS.461.2914H}.\n    The red symbols denote the results of this work.}\n    \\label{JDcomp}\n\\end{figure}\n\n\n\\section{Discussion} \\label{sec:discussion}\n\\subsection{Comparison dark matter profiles with previous works}\nIn this section, we compare our estimated dark matter density profiles with other works based on different methods or assumptions.\n\n\\citet{2019MNRAS.484.1401R} considered non-parametric dynamical mass models based on a spherical Jeans equation, {\\sc GravSphere}~\\citep{2017MNRAS.471.4541R,2018MNRAS.481..860R} to measure the dark matter density profiles of dwarf spheroidal\/irregular galaxies, and then they found the relation between the central densities of dark matter halos and the stellar vestiges of galaxy evolution such as a star formation history, stellar mass, and stellar-to-halo mass ratio.\nRegarding the inner slopes of dark matter density profiles in the dSphs, they showed that Draco favors a cusped dark matter halo which is consistent with an NFW profile, while Fornax has a shallower inner density profile $\\gamma\\sim0.3$.\nThis trend is similar to that in this work.\nThey mentioned, however, the caveat that the estimation of an inner slope of a dark matter profile using their method is largely affected by a choice of priors.\nTherefore, they utilized a dark matter density within 150~pc, $\\rho_{\\rm DM}(150\\ {\\rm pc})$, to discuss a diversity of the central dark matter densities in the dwarf galaxies, instead of their inner slopes. \nWe also discuss $\\rho_{\\rm DM}(150\\ {\\rm pc})$ calculated by our models and then find that this physical quantity is useful to understand the dynamical evolution of dark matter halos in the Universe.\nWe discuss them further in the following subsection.\n\nOwing to recent spectroscopic observations for the dSphs, some of them have multiple stellar populations, in which the metal-rich stars are centrally concentrated and have colder kinematics, while the metal-poor ones are more extended and have hotter kinematics~\\citep[e.g.,][]{2006A&A...459..423B,2008ApJ...681L..13B}.\nUsing the coexistence of such multiple populations, \\citet{2011ApJ...742...20W} statistically separated multiple stellar components by applying their constructed likelihood function for spatial, metallicity, and velocity distributions of the stars, and then inferred the slopes of dark matter densities of Sculptor and Fornax.\nThey concluded that both galaxies have cored dark matter halos and a cuspy profile can be ruled out with high statistical significance.\nHowever, this method imposes that both stellar and dark matter distributions are spherical symmetric.\nThis sphericity can accompany a systematic bias, and an inner slope inferred by this method depends largely on viewing angles~\\citep{2013MNRAS.431.2796K,2013MNRAS.433L..54L,2018MNRAS.474.1398G}.\n\n\\citet{2012ApJ...754L..39A} and \\citet{2013MNRAS.429L..89A} applied the projected virial theorem to these multiple stellar components for Sculptor and Fornax, respectively and concluded that these dSphs do not have cusped dark matter profiles.\nOn the other hand, using these multiple populations, several other works concluded that Sculptor has a cusped dark matter halo based on a phase space distribution function method~\\citep{2017ApJ...838..123S}, whilst it is difficult to distinguish between cusp and core based on a Schwarzschild method~\\citep{2013MNRAS.433.3173B} and Multi-Gaussian expansion model~\\citep{2016MNRAS.463.1117Z}.\nAlthough the dark matter inner slopes in Fornax and Sculptor are still under debated, those inferred by our mass models prefer to be less cuspy than an NFW profile.\n\nAxisymmetric dynamical models based on Schwarzschild technique have been developed and applied to the kinematic data of the dSphs~\\citep{2012ApJ...746...89J,2013ApJ...775L..30J,2013ApJ...763...91J}.\n\\citet{2013ApJ...763...91J} applied these models to the data of Draco and found that its dark matter inner slope is consistent with an NFW profile.\nThis agrees well with our mass models for Draco.\n\\citet{2013ApJ...775L..30J} performed the same analysis with respect to the other classical dSphs (Carina, Fornax, Sculptor and Sextans) and concluded that these galaxies have an unified cusped profile, but there are considerable large uncertainties.\n\n\n\\begin{figure*}[t]\n\t\\begin{center}\n\t\\includegraphics[scale=0.9]{DSPH_gamma_MsMh.pdf}\n\t\\end{center}\n    \\caption{The impact of baryonic feedback on the inner profiles of dark matter halos. \n    The inner dark matter density slope at $1.5$\\%$R_{\\rm vir}$ is shown as a function of the ratio of stellar-to-halo masses.\n    The filled black circles with error bars are the results from this work. \n    The shaded gray band shows the expected range of dark matter profile slopes for NFW as derived from dark matter only simulations~\\citep{2016MNRAS.456.3542T}.\n    The blue and orange points are expected from NIHAO~\\citep{2016MNRAS.456.3542T} and FIRE-2~\\citep{2017MNRAS.471.3547F,2018MNRAS.480..800H} hydrodynamical plus dark mater simulations, respectively.\n    The blue and orange shaded bands are the expected range from NIHAO~\\citep{2016MNRAS.456.3542T} and FIRE-2~\\citep{2020arXiv200410817L} predictions, respectively (to guide the eye).}\n    \\label{GammaMsMh}\n\\end{figure*}\n\n\\subsection{The origin of a diversity of inner dark matter slopes} \\label{sec:diversity}\nIn Figure~\\ref{dmpro}, we show that the classical dSphs have a wide range of central dark matter density profiles.\nIn this section, we discuss what the origin of this diversity is.\nTo this end, we investigate the relation between the central dark matter density profiles and stellar properties of the dSphs.\n\n\\subsubsection{Inner dark matter density slope versus stellar-to-halo mass ratio}\nRecent dark matter plus hydrodynamical simulations have shown that an inner slope of a dark matter density profile depends largely on the ratio of stellar mass to total halo mass.\nFigure~\\ref{GammaMsMh} shows the logarithmic slope of the dark matter density profile at $1.5$\\% of the virial radius, $R_{\\rm vir}$, as a function of the ratio of stellar-to-halo masses, $M_{\\ast}\/M_{\\rm halo}$, predicted from NIHAO~\\citep[][magenta]{2016MNRAS.456.3542T} and FIRE-2~\\citep[][cyan]{2017MNRAS.471.3547F,2018MNRAS.480..800H} simulations.\nNote that baryon feedback for bright dwarf galaxies ($\\log_{10}(M_{\\ast}\/M_{\\rm halo})\\sim-3$~to~$-2$) has a systematic impact on inner slopes, while for the fainter galaxies with $\\log_{10}(M_{\\ast}\/M_{\\rm halo})\\lesssim-3.5$, the impact of baryonic feedback is negligible. \nTherefore, these simulations predict that the efficiency of baryonic feedback for a dark matter halo can provoke the diversity of dark matter inner slopes.\n\nTo test this prediction, we derive the relation between the dark matter inner slopes and $M_{\\ast}\/M_{\\rm halo}$ for the current sample of dSphs, which is shown in Figure~\\ref{GammaMsMh}.\nIn order to calculate the ratio of stellar-to-halo masses, we employ the self-consistent abundance matching model computed by \\citet{2013MNRAS.428.3121M} and adopt the stellar masses of the dSphs taken from \\citet{2012AJ....144....4M}.\nThe filled black circles with error bars in Figure~\\ref{GammaMsMh} show the results of the classical dSphs inferred by our analysis.\nAlthough there are still large uncertainties in both the inner slopes and the stellar-to-halo mass ratios, \nthe systematic trend in the plots is generally in agreement with the predictions from recent numerical simulations, which are presented in blue~(NIHAO: \\citealt{2016MNRAS.456.3542T}) and orange (FIRE-2: \\citealt{2020arXiv200410817L}) shaded region in the figure.\nTo make an attempt to characterize the trend quantitatively, we employ a least squares fitting method to\ndetermine the slope of $\\gamma$ as a function of $M_{\\ast}\/M_{\\rm halo}$, and we find $\\gamma\\propto\\log_{10}(M_{\\ast}\/M_{\\rm halo})^{0.27\\pm0.15}$.\nThus, we confirm that $\\gamma$ is slightly proportional to $M_{\\ast}\/M_{\\rm halo}$ on dwarf-galaxy scales.\n\n\nHowever, comparing between these shaded bands in detail, there is a systematic difference especially on classical dwarf galaxy scales stemmed from the different prescriptions of hydrodynamics regime.\nThus, the predicted relation between dark matter inner slope and stellar-to-halo mass ratio still has large uncertainties.\nRegarding this relation, \\citet{2019arXiv191100544K} have argued that a self-interacting dark matter (SIDM) model combined with the impact of a baryon potential on the halo profile can also reproduce the diversity of the inner dark matter density profiles for low surface brightness galaxies.\nHowever, the corresponding $M_{\\ast}\/M_{\\rm halo}$ in these galaxies are greater than $-3$, and it is thus unclear for the diversity in the current fainter dwarf galaxy scales.  \n\nWe also investigate the relation between the inner density slopes and their stellar masses and the orbital properties of the dSphs~(apocenter radius, orbital eccentricity, angular momentum, the time elapsed since the last apocenter and pericenter) but we find no clear relations.\n\n\n\\begin{figure}[t!]\n\t\\begin{center}\n\t\\includegraphics[scale=0.3]{cumSFH_2.pdf}\n\t\\includegraphics[scale=0.3]{DSPH_gamma_tau7.pdf}\n\t\\end{center}\n    \\caption{{\\it Top panel}: Cumulative star formation history of dwarf satellites in the MW taken from \\citet{2009ApJ...703..692L} for Sextans and \\citet{2014ApJ...789..147W} for the other classical dSphs.\n    {\\it Bottom panel}: Inner slope parameter of dark matter density profile $\\gamma$ as a function of the lookback time of achieving 70~per~cent of current stellar masses, $\\tau_{0.7}$.}\n    \\label{Gamma_esn}\n\\end{figure}\n\n\\subsubsection{Inner dark matter density slope versus SFH}\nThe relation in Figure~\\ref{GammaMsMh} implies that an inner dark matter slope depends on stellar feedback associated with star formation activity.\nIndeed, some high-resolution dark matter and hydrodynamical simulations have shown an inner slope of a dark matter density profile depends on star formation history~(SFH)~\\citep[e.g.,][]{2014ApJ...789L..17M, 2015MNRAS.454.2092O}.\nIn particular, \\citet{2015MNRAS.454.2092O} predicted that the dwarfs with rapid SFHs tend to have cuspy dark matter density profiles, while ones with consecutive SFHs have cored ones at the present day.\nTherefore, we investigate whether this dependence indeed exists by comparing it with the observed SFH of dSphs.\n\nTo this end, we adopt the SFHs derived by \\citet{2009ApJ...703..692L} for Sextans and \\citet{2014ApJ...789..147W} for the other classical dSphs.\nThe top panel in Figure~\\ref{Gamma_esn} displays the cumulative SFHs of the classical dSphs taken from their works.\nAs is shown in the panel, the SFHs of the dSphs can be classified into two groups:\nthe dwarfs~(the dashed lines in the panel) that formed the majority of their stellar component early on (before $z\\simeq2$), and the other ones~(the solid ones) that formed only a small fraction of their stars at early times and continued forming stars over almost a Hubble time~\\citep{2015ApJ...811L..18G,2018MNRAS.479.1514B}.\nTo quantify these properties of the dwarfs, we estimate the lookback time at achieving 70~per~cent of the current stellar mass of these dSphs, $\\tau_{0.7}$ (as indicated as a black horizontal dotted line in the left panel in Figure~\\ref{Gamma_esn}).\n$\\tau_{0.7}$ can characterize the duration and efficiency of star formation in dSphs.\nThe bottom panel in Figure~\\ref{Gamma_esn} shows the comparison between $\\tau_{0.7}$ and dark matter inner slope, $\\gamma$, from our analysis.\nAccording to the prediction from \\citet{2015MNRAS.454.2092O}, we expect that the galaxies with higher $\\tau_{0.7}$ may have cuspy dark matter density profiles.\nFrom this figure, however, we find no clear relation between them within uncertainties of $\\gamma$.\nTherefore, the diversity of the dark matter inner slopes cannot be explained straightforwardly by SFH within the current observation and model uncertainties.\nOne of the possible reasons why there is no relation could be that the cusp-core transition requires the resonance between dark matter particles and a gas density oscillation induced by periodic SN feedbacks.\n\\citet{2014ApJ...793...46O} suggested that to transform cusp into core, at least 50 oscillations with $\\mathcal{O}(100)$~Myr periods are needed.\nUnfortunately, current photometric and spectroscopic observations are difficult to resolve such a oscillatory star formation activity.\n\n\n\\subsubsection{Dark matter density at 150~pc}\n\\citet{2019MNRAS.484.1401R} proposed to use the dark matter density at a common radius of 150~pc from the center of each galaxy, $\\rho_{DM}(150\\ {\\rm pc})$, which is insensitive to the choice of a $\\gamma$'s prior in spherical mass models.\nUsing this density, \\citet{2019MNRAS.490..231K} pointed out the anti-correlation between $\\rho_{DM}(150\\ {\\rm pc})$ and their orbital pericenter distances, $r_{\\rm peri}$, of the classical dSphs.\nThis implies a survivor bias which means that galaxies with low dark matter densities were completely destroyed by strong tidal effects.\nFollowing these works, we also calculate the dark matter density at 150~pc along the major axis of the sample dSphs, considering the non-sphericity of a dark matter halo, and the calculated $\\rho_{DM}(150\\ {\\rm pc})$ are tabulated in the last column of Table~\\ref{table2}. \n\nFirst, we compare their $\\rho_{DM}(150\\ {\\rm pc})$ to stellar masses and stellar-to-halo mass ratios.\n\\citet{2019MNRAS.484.1401R} presented the anti-correlation between them, but we do not find clear relations of $\\rho_{DM}(150\\ {\\rm pc})$-$M_{\\ast}$ and $\\rho_{DM}(150\\ {\\rm pc})$-$M_{\\ast}\/M_{\\rm halo}$.\nThis is caused by the fact that \\citet{2019MNRAS.484.1401R} discussed these correlations by including not only the dSphs but dwarf irregular galaxies which have HI gas rotation curves.\nThese gas-rich galaxies have higher $M_{\\ast}$ and $M_{\\ast}\/M_{\\rm halo}$ and much lower $\\rho_{DM}(150\\ {\\rm pc})$ than those of the dSphs, thereby the galaxies make the correlations conspicuous.\n\n\\begin{figure}[t!]\n\t\\begin{center}\n\t\\includegraphics[width=\\columnwidth]{rho150_rperi08_Vpeak.pdf}\n\t\\end{center}\n    \\caption{Dark matter densities at 150~pc, $\\rho_{DM}(150\\ {\\rm pc})$, versus pericenter radii, $r_{\\rm peri}$, of the dSphs.\n    The colored filled circles with error bars are the classical dSphs from our Jeans analysis.\n    The filled small squares are the individual subhalos predicted from dark matter simulations~(Ishiyama et al. in prep.).\n    The gray scale indicates the maximum circular velocities of subhalos over their formation histories~(the redshift when they were first accreted on to a host).\n    The big black filled squares with error bars are the stacked $\\rho_{DM}(150\\ {\\rm pc})$ and $r_{\\rm peri}$ in each radial bin.\n    The error bars correspond to the 16th and 84th percentiles of the subhalos in each bin.}\n    \\label{rho150_rperi}\n\\end{figure}\n\n\\begin{figure*}[t!]\n\t\\includegraphics[scale=0.43]{Vcric_all.pdf}\n    \\caption{The circular velocity profiles for all sample dSphs.\n    The colored solid line and shaded band in each panel show median and the 68~percent confidence intervals calculated by our non-spherical mass models.\n    The dashed lines depict the results from spherical mass models assuming NFW cusped dark matter density profiles taken from~\\citet{2019MNRAS.490..231K}.\n    The black diamonds correspond to the mass estimator of~\\citet{2010MNRAS.406.1220W}.}\n    \\label{vcirc_all}\n\\end{figure*}\n\nSecond, we investigate the anti-correlation between $\\rho_{DM}(150\\ {\\rm pc})$ and $r_{\\rm peri}$.\nFor the pericenter radius, we adopt the values presented by \\citet{2018A&A...619A.103F}, which estimated using the recent Gaia data~\\citep{2018A&A...616A..12G} and assuming a Milky Way potential model with mass of $0.8\\times10^{12}M_{\\odot}$.\nFigure~\\ref{rho150_rperi} shows the relation between $\\rho_{DM}(150\\ {\\rm pc})$ and $r_{\\rm peri}$.\nThe colored filled circles with error bars are the inferred $\\rho_{DM}(150\\ {\\rm pc})$ of the sample dSphs.\nFrom this plot, we find the anti-correlation between them similar to \\citet{2019MNRAS.490..231K}, even though there are still large uncertainties.\n\nWe also compare with dark matter subhalos predicted from dark matter only simulations.\nIn this work, we utilize a high resolution $N$-body simulation, named Phi-4096,  performed by~\\citet{2020arXiv200714720I}.\nThe detail of the simulation is as below.\nUsing a massively parallel TreePM code\nGreeM~\\footnote{http:\/\/hpc.imit.chiba-u.jp\/~ishiymtm\/greem\/}\n~\\citep{2009PASJ...61.1319I,2012arXiv1211.4406I},\nwe simulated the motion of $4096^3$ dark matter particles\nin a comoving box with the side length of 16$ \\, h^{-1} \\rm Mpc$,\nwhich corresponds to $5.13 \\times 10^{3} \\, h^{-1} M_{\\odot}$ particle mass.\nThe gravitational softening length is 60 comoving $ \\, h^{-1} \\rm pc$.\nThe initial condition was constructed using the MUSIC code \\citep{2011MNRAS.415.2101H}.\nThe cosmological parameters of the simulation\nare $\\Omega_0=0.31$, $\\lambda_0=0.69$, $h=0.68$, $n_s=0.96$, and\n$\\sigma_8=0.83$, which are consistent with the measurement of\ncosmic microwave background by the Planck satellite~\\citep{2018arXiv180706209P}.\nTo identify halos and subhalos and construct merger trees, we used\nROCKSTAR phase space halo\/subhalo finder ~\\citep{2013ApJ...762..109B}\nand consistent trees code ~\\citep{2013ApJ...763...18B}.  We picked up\nMilky Way-sized host halos with the mass of \n$3.4 \\times 10^{11} < M_{\\rm vir} < 2.0 \\times 10^{12} \\, h^{-1} M_{\\odot}$ at z=0, where $M_{\\rm vir}$ is the\nhalo virial mass.\nThe total number of host halos is 27.\n\nTo compute $\\rho_{DM}(150\\ {\\rm pc})$ of the simulated dark subhalos, we use the scale density and radius of each subhalo, supposing spherical NFW dark matter halos.\nIn Figure~\\ref{rho150_rperi}, the small filled squares denote the predicted subhalos associated with these Milky Way-sized dark matter host halos, while the big black squares with error bars are the results from stacked analysis of the subhalos in each $r_{\\rm peri}$ bin.\nIt is found from this plot that dark matter simulations indicate somewhat anti-correlation, and \nthis correlation is similar to the observed one.\nMoreover, we also find that the maximum circular velocities of subhalos over their formation histories, $V_{\\rm peak}$, of subhalos depends slightly on $\\rho_{DM}(150\\ {\\rm pc})$ and $r_{\\rm peri}$.\nIn other words, the subhalos with higher $\\rho_{DM}(150\\ {\\rm pc})$ and smaller $r_{\\rm peri}$ (the left-top area in Figure~\\ref{rho150_rperi}) tend to have large $V_{\\rm peak}$.\nSince subhalos with large $V_{\\rm peak}$ were formed at earlier, most of them have dense central densities, and such subhalos can still survive even suffering from strong tidal effects.\nIn addition, we find that $\\rho_{DM}(150\\ {\\rm pc})$ of subhalos depends on their host dark matter halo masses, which means that the subhalos associated with larger hosts have higher $\\rho_{DM}(150\\ {\\rm pc})$ than those with smaller ones.\nTherefore, this anti-correlation can be also dependence on a host halo mass.\nEven though this anti-correlation seems to support a survivor bias suggested by \\citet{2019MNRAS.490..231K}, we note that adding a stellar disk preferentially reduces the subhalo densities with smaller pericenter distances~\\citep{2019MNRAS.490.2117R}, thereby we cannot make a final conclusion about the anti-correlation without considering the impact of the disk.\n\n\n\n\\subsection{Circular velocity profile}\n\n$\\Lambda$CDM theory has another serious problem that central densities of dark matter halos associated in the bright dSphs in Milky Way are significantly lower than those of the most massive subhalos in MW-sized halos in the $\\Lambda$CDM simulations.\nThis problem is so-called the ``too-big-to-fail (TBTF)'' problem~\\citep{2011MNRAS.415L..40B}.\nIn order to compare with the central densities in the observed and simulated dark matter halos, they adopted the maximum circular velocity, $V_{\\rm max}$, for most massive ten subhalos.\nOn the other hand, for the observed ones, they used the circular velocities at the half-light radii of the dSphs, because this physical value is well-constrained by kinematic data~\\citep{2010MNRAS.406.1220W}.\nInstead of relying on such a single value of a circular velocity at a specific radius, we calculate a circular velocity profile directly from the posterior PDFs of the dark matter halo parameters.\n\nIn axisymmetric models, the circular velocity along a major axis can be calculated by\n\\begin{equation}\n    V^2_{\\rm circ}(R) = R\\left| -\\frac{\\partial\\Phi}{\\partial R}\\right|,\n\\end{equation}\nwhere $\\Phi$ is a gravitational potential originated from the dark matter density profile~ \\citep{2008gady.book.....B}. \nThe colored solid lines and shaded regions in Figure~\\ref{vcirc_all} show the inferred circular velocity profiles for the classical dSphs from our models.\nFor comparison with our results, we also plot those profiles estimated by \\citet{2019MNRAS.490..231K} and the circular velocities at their half-light radii of the dSphs, $V_{\\rm circ}(r_{\\rm half})$,~\\citep{2010MNRAS.406.1220W}.\nInterestingly, the both circular velocity profiles computed by axisymmetric and spherical models are consistent in the value of $V_{\\rm circ}(r_{\\rm half})$, but the shapes of these profiles, especially quantified with the values of $V_{\\rm max}$, look quite different in different mass models.\nThis implies that $V_{\\rm circ}(r_{\\rm half})$ would not be an adequate tracer for comparison with the central densities in dark matter halos. \nHowever, there are huge uncertainties in our estimated circular velocities, especially their outskirts due to the lack of data sample.\nThus, a sufficient number of stellar kinematic sample out to their outer parts of the dSphs should be needed.\n\n\\section{Conclusion} \\label{sec:conclusion}\nIn this paper, we revisit the core-cusp problem in the Galactic dSphs based on non-spherical Jeans analysis.\nAn advantage in these non-spherical models is that $\\rho_{\\rm DM}-\\beta_{\\rm ani}$ degeneracy occurred under the assumption of spherical symmetry can be mitigated.\n\nApplying our non-spherical mass models to the latest kinematic data of the eight classical dSphs, we estimate their dark matter density profiles by marginalizing posterior distributions of dark matter halo parameters.\nWe find that most of these dSphs favor cusped or mildly cusped dark matter profiles in their centers rather than cored one.\nIn particular, Draco robustly has a cusped dark matter halo even considering a wide prior range.\nTherefore, we conclude that there is no core-cusp problem in the classical dSphs.\n\nWe also find the diversity in the central dark matter density profiles.\nInterestingly, this diversity can be explained if we consider the impact of baryonic feedback on the central dark matter densities, which depends largely on the ratio of stellar-to-halo mass as predicted by recent $N$-body and hydrodynamical simulations.\nTherefore, $\\Lambda$CDM framework combined with baryon physics can explain the observed dark matter densities in the classical dSphs.\n\nWe also investigate the relation between the central dark matter density profiles and their star formation histories, because several high-resolution dark matter and hydrodynamical simulations predicted the correlation between these.\nHowever, we find no clear relation between an inner slope parameter of dark matter density profile and SFH characterized by $\\tau_{0.7}$.\n\nWe confirm that a dark matter density at a radius of 150~pc is anti-correlated with the pericenter distance of a dSph suggested by \\citet{2019MNRAS.490..231K}.  \nFurthermore, this anti-correlations also found in the simulated dark subhalos.\nIn addition, we also find that the maximum circular velocities of subhalos over their formation histories, $V_{\\rm peak}$ of subhalos depends slightly on $\\rho_{DM}(150\\ {\\rm pc})$ and $r_{\\rm peri}$.\nThis implies that the subhalos having dense central densities can survive from strong tidal effects due to being closer to the center of a host halo.\n\nUsing our non-spherical mass models, we calculate the circular velocity profiles of all sample dSphs and compare with those estimated by spherical mass models.\nAs a result, the shapes of circular velocity profiles, especially quantified with the maximum circular velocity, $V_{\\rm max}$, are quite different between spherical and axisymmetric mass models. \nHowever, there are huge uncertainties in our estimated circular velocities, especially their outskirts due to the lack of data sample.\n\nTo ensure our conclusions, it is necessary to determine the dark matter density profiles for much fainter dSphs, namely ultra-faint dSphs, which are believed to have held original dark matter density profiles.\nIt is also important to more precisely estimate the dark matter profiles of the classical dSphs.\nThe next-generation wide-field spectroscopic surveys with the Subaru Prime Focus Spectrograph~\\citep{2014PASJ...66R...1T} will enable us to obtain statistically significant samples of stellar kinematics and chemical abundances for the Galactic dSphs over the wide areas out to their outskirts, thereby allowing us to estimate robustly their dark matter density profiles.\n\n\n\\section*{Acknowledgements}\nWe would like to give special thanks to Manoj Kaplinghat, Ethan Nadler, Hai-Bo Yu, Chervin Laporte, Masahiro Ibe, Shigeki Matsumoto, Evan Kirby for useful discussions.\nThis work was supported in part by the MEXT Grant-in-Aid for Scientific Research on Innovative Areas (No.~18H04359, 18J00277 and 20H01895 for K.H., No.~17H01101, 18H04334 and 18H05437 for~M.C., No.~17H01101, 17H04828 and 18H04337 for~T.I.).\nNumerical computations were partially carried out on Aterui II supercomputer at\nCenter for Computational Astrophysics, CfCA, of National Astronomical\nObservatory of Japan.  T.I. has been supported by MEXT as\n``Priority Issue on Post-K computer'' (Elucidation of the Fundamental\nLaws and Evolution of the Universe), JICFuS, \nand Mext as ``Program for Promoting Researches on the Supercomputer\nFugaku'' (Toward a unified view of the universe: from large scale\nstructures to planets, proposal numbers hp200124). \n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\\label{introduction}\n\nThe charmonium spectroscopy has been studied extensively during the\nlast few years. The states below the open charm threshold are all\nobserved now while many states above the open charm threshold are\nstill missing. On the other hand, a large number of charmonium-like\nstates (or so called XYZ states) have been observed by experimental\ncollaborations such as Belle, Barbar, CDF, D0, LHCb, BESIII, CLEOc.\nThese XYZ states decay into the conventional charmonium, but some of\nthem do not fit into the quark model charmonium spectrum easily.\nEspecially the charged charmonium-like signals are the good\ncandidates of the exotic states, such as $Z(4430)$ observed in the\n$\\psi' \\pi^{\\pm}$ modes, $Z_1(4050)$, $Z_2(4250)$ in the $\\chi_{c1}\n\\pi^{\\pm}$ modes in the B meson decays\n\\cite{R.Mizuk:2008,Choi:2008,K.Chilikin:2013}, $Z_c(4025)^{\\pm}$ in\nthe $\\pi^{\\pm}$ recoil mass spectrum and $Z_c(4020)^{\\pm}$ in the\n$\\pi^{\\pm}h_c$ mass spectrum \\cite{Ablikim:2013}. Recently, the BES\nCollaboration announced a charged structure $Z_c(3900)$ in the\n$\\pi^{\\pm}J\/\\psi$ invariant mass spectrum of the process $e^+ e^-\n\\rightarrow \\pi^+ \\pi^- J\/\\psi$ at a center-of-mass (CM) energy of\n$\\sqrt{S}=4.260\\pm0.001$GeV\\cite{Ablikim:2013}. How to explain the\nunderlying structure of these charmonium-like states becomes an\nimportant issue.\n\nMany theoretical schemes were proposed to explain these XYZ states,\nincluding the molecular states \\cite{F.E.Close:2004,\nM.B.Voloshin:2004, C.Y.Wong:2004, E.S.Swanson:2004,\nN.A.Tornqvist:2004, Y.R.Liu:2010}, hybrid charmonium\n\\cite{B.A.Li:2005}, tetraquark states \\cite{H.Hogaasen:2006,\nD.Ebert:2006, N.Barnea:2006, Y.Cui:2007, R.D.Matheus:2007,\nT.W.Chiu:2007}, dynamically generated resonances\n\\cite{D.Gamermann:2007}. Among the above schemes, the molecular\npicture provides a plausible explanation since some XYZ states are\nvery close to the thresholds of a pair of charmed meson.\n\nSince the first observation of $X(3872)$ by the Belle Collaboration\n\\cite{Choi:2003} in the exclusive decay process $B^{\\pm}\\!\n\\rightarrow\\! K^{\\pm}\\pi^{+}\\pi^{-} J\/\\psi$, its interpretation as a\nmolecular candidate of the $D \\bar{D}^{*}$ system has been\ninvestigated by many theoretical groups \\cite{C.Y.Wong:2004,\nE.S.Swanson:2004}\\cite{M.T.AlFiky:2006, S.Fleming:2007,\nE.Braaten:2007, C.Hanhart:2007, M.B.Voloshin:2007, P.Colangelo:2007,\nM.Suzuki:2005, S.L.Zhu:2008}. Due to the same intriguing\nnear-threshold nature, the recently observed two charged\nbottomonium-like states $Z_{b}(10610)$ and $Z_{b}(10650)$ by the\nBelle observation \\cite{I.Adachi:2011} were also interpreted as good\ncandidates of the $B \\bar{B}^{*}$ and $B^* \\bar{B}^{*}$ molecular\nstates \\cite{Y.R.Liu:2008, X.Liu:2009, A.E.Bondar:2011,\nD.Y.Chen:2011, Z.F.Sun:2011}. The newly observed $Z_c(3900)$ by\nBESIII collaborations \\cite{Ablikim:2013}, CLEOc \\cite{T.Xiao:2013}\nand Belle with ISR \\cite{Z.Q.Liu:2013} is also close to the\nthreshold of $D \\bar{D}^*$. In many references, it was interpreted\nas the isovector partner of the well established isoscalar state\n$X(3872)$ with the same quantum number $J^{P}=\n1^{+}$.\\cite{Q.Wang:2013, Z.G.Wang:2013, F.Aceti:2013}.\n\nWhen investigating the possibility of $X(3872)$ as the $D\n\\bar{D}^{*}$ molecular state with $J^{PC}= 1^{++}$, the\none-pion-exchange (OPE) model and one-boson-exchange (OBE) model\nwere used to calculate the binding energy of the $D \\bar{D}^{*}$\nsystem in the Ref.\\cite{N.Li:2012}. In Ref.\\cite{Z.F.Sun:2011}, the\nOBE model was applied to investigate the possibility of\n$Z_{b}(10610)$ and $Z_{b}(10650)$ as the molecular states of the $B\n\\bar{B}^{*}$ and $B^* \\bar{B}^{*}$ system.\n\nWith the exchange of the light pseudoscalar, vector and scalar\nmesons, the OBE model provides an effective framework to describe\nthe interaction between two hadrons at different range. In the\nprevious work, the heavy quark symmetry is always invoked to\nsimplify the calculation in the derivation of the interaction\npotential between two heavy mesons such as $D \\bar{D}^{*}$ or $B\n\\bar{B}^{*}$. Moreover, the three momentum of the external particles\nis sometimes ignored. Hence the resulting potential between the\nheavy mesons depends on the exchanged momentum only. All the recoils\ncorrections were omitted.\n\nThe possible $D \\bar{D}^{*}$ or $B \\bar{B}^{*}$ molecular system is\nvery close to the two heavy meson threshold. The binding energy is\nsometime quite small. Especially in the case of X(3872), its binding\nenergy may be less than 1 MeV if it turns out to be a $D\n\\bar{D}^{*}$ molecule. Compared to the tiny binding energy, the\nhigher order recoil corrections may turn out to be non-negligible.\n\nIn the present work, we keep the momentum of the initial and final\nstates explicitly and derive the effective potential using the\nrelativistic Lagrangian. We will keep the recoil corrections up to\nthe order $\\frac{1}{M^2}$, where $M$ is the mass of the component in\nthe system. Especially the spin-orbit force first appears at ${O\n(\\frac{1}{M})}$. With the effective potentials with the explicit\nrecoil corrections ${O (\\frac{1}{M^2})}$, we carefully investigate\nthe $D \\bar{D}^{*}$ system with $I=0$, $J^{PC}= 1^{++}$ to measure\nthe $\\frac{1}{M^2}$ correction for $X(3872)$, $D \\bar{D}^{*}$ system\nwith $I=1$, $J^{PC}= 1^{+}$ for $Z_c(3900)$, and $B \\bar{B}^{*}$\nwith $I=1$, $J^{PC}= 1^{+}$ for $Z_{b}(10610)$. Numerically, these\nrecoil corrections are quite important in the loosely bound heavy\nmeson systems. Especially, the recoil correction is comparable to\nthe binding energy in the case of X(3872).\n\nThis paper is organized as follows. We first introduce the formalism\nof the derivation of the effective potential in Section\n\\ref{potential}. We present our numerical results in Section\n\\ref{Numerical}. The last section is the summary and discussion. We\ncollect some lengthy formulae in the appendix.\n\n\n\\section{The effective potential}\\label{potential}\n\\begin{center}\n\\textbf{A. Wave function, Effective Lagrangian and Coupling\nconstants}\n\\end{center}\n\nFirst, we construct the flavor wave functions of the isovector and\nisoscalar molecular states composed of the $B\\bar{B}^{*}$ and\n$D\\bar{D}^{*}$ as in Refs. \\cite{Y.R.Liu:2008,X.Liu:2009}. The\nflavor wave function of the $B\\bar{B}^{*}$ system reads\n\\begin{equation}\n\\left\\{\n  \\begin{array}{ll}\n    |1,1\\rangle = \\frac{1}{\\sqrt{2}}(|B^{*+}\\bar{B}^0\\rangle +\nc|B^{+}\\bar{B}^{*0}\\rangle),  \\\\\n    |1,-1\\rangle = \\frac{1}{\\sqrt{2}}(|B^{*-}B^0\\rangle +\nc|B^{-}B^{*0}\\rangle),  \\\\\n    |1,0\\rangle = \\frac{1}{2}[(|B^{*+}B^-\\rangle\n-|B^{*0}\\bar{B}^0\\rangle) +\nc(|B^{+}B^{*-}\\rangle-|B^{0}\\bar{B}^{*0}\\rangle)],\n  \\end{array}\n\\right.\n\\end{equation}\n\n\\begin{equation}\n|0,0\\rangle = \\frac{1}{2}[(|B^{*+}B^-\\rangle\n+|B^{*0}\\bar{B}^0\\rangle) +\nc(|B^{+}B^{*-}\\rangle+|B^{0}\\bar{B}^{*0}\\rangle)]\n\\end{equation}\nwhere $c=\\pm$ corresponds to C-parity $C=\\mp$ respectively. For the\n$D\\bar{D}^{*}$ system\n\\begin{equation}\n\\left\\{\n  \\begin{array}{ll}\n    |1,1\\rangle = \\frac{1}{\\sqrt{2}}(|\\bar{D}^{*0}D^{+}\\rangle +\nc|\\bar{D}^{0}D^{*+}\\rangle),  \\\\\n    |1,-1\\rangle = \\frac{1}{\\sqrt{2}}(|D^{*0}D^-\\rangle +\nc|D^{0}D^{*-}\\rangle),  \\\\\n    |1,0\\rangle = \\frac{1}{2}[(|\\bar{D}^{*0}D^0\\rangle\n-|D^{*-}D^+\\rangle) +\nc(|\\bar{D}^{0}D^{*0}\\rangle-|D^{-}D^{*+}\\rangle)],\n  \\end{array}\n\\right.\n\\end{equation}\n\n\\begin{equation}\n|0,0\\rangle = \\frac{1}{2}[(|\\bar{D}^{*0}D^0\\rangle\n+|D^{*-}D^{+}\\rangle) +\nc(|\\bar{D}^{0}D^{*0}\\rangle+|D^{-}D^{*+}\\rangle)]\n\\end{equation}\n\nSince the C-parity of $Z_b(10610)^0$ is odd, we will take the\ncoefficient $c=+$ for the $B\\bar{B}^{*}$ system. While the C parity\nof $X(3872)$ was even, the $I=0$ $D\\bar{D}^{*}$ system will take the\ncoefficient $c=-$. Moreover, we will consider both two C-parity\noption for the $I=1$ $D\\bar{D}^{*}$ system.\n\nThe meson exchange Feynman diagrams for both the $B\\bar{B}^{*}$ and\n$D\\bar{D}^{*}$ systems at the tree level are shown in Fig.\n\\ref{t-channel} and Fig. \\ref{u-channel}.\n\n\\begin{figure}[ht]\n  \\begin{center}\n    \\rotatebox{0}{\\includegraphics*[width=0.30\\textwidth]{t-channel.eps}}\n    \\caption{ The direct-channel Feynman diagrams for both the $D\\bar{D}^{*}$ and\n$B\\bar{B}^{*}$ systems at the tree level. The thick line represents\nthe vector state $D^*$, $B^*$, $\\bar{D}^{*}$ or $\\bar{B}^{*}$ while\nthe thin line stands for $D$, $B$, $\\bar{D}$ and $\\bar{B}$, .}\n    \\label{t-channel}\n  \\end{center}\n\\end{figure}\n\n\\begin{figure}[ht]\n  \\begin{center}\n    \\rotatebox{0}{\\includegraphics*[width=0.35\\textwidth]{u-channel.eps}}\n    \\caption{ The cross-channel Feynman diagrams for both the $D\\bar{D}^{*}$ and\n$B\\bar{B}^{*}$ systems at the tree level. Notations are the same as\nin Fig. \\ref{t-channel}}.\n    \\label{u-channel}\n  \\end{center}\n\\end{figure}\n\n\nBased on the chiral symmetry, the Lagrangian for the pseudoscalar,\nscalar and vector meson interaction with the heavy flavor mesons\nreads\n\\begin{eqnarray}\n\\mathcal{L}_{P}&=&-i\\frac{2g}{f_\\pi}\\bar{M}\nP^{*\\mu}_b\\partial_{\\mu}\\phi_{ba}P^{\\dag}_{a}+i\\frac{2g}{f_\\pi}\\bar{M} P_b\\partial_{\\mu}\\phi_{ba}P^{*\\mu\\dag}_{a} \\nonumber\\\\\n&-& \\frac{g}{f_\\pi}\nP^{*\\mu}_b\\partial^{\\alpha}\\phi_{ba}\\partial^{\\beta}P^{*\\nu\\dag}_{a}\\epsilon_{\\mu\\nu\\alpha\\beta}\n+ \\frac{g}{f_\\pi}\n\\partial^{\\beta}P^{*\\mu}_b\\partial^{\\alpha}\\phi_{ba}P^{*\\nu\\dag}_{a}\\epsilon_{\\mu\\nu\\alpha\\beta},\\label{pseudo-exchange}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\widetilde{\\mathcal{L}_{P}}&=&-i\\frac{2g}{f_\\pi}\\bar{M}\\widetilde{P^{\\dag}_{a}}\n\\partial_{\\mu}\\phi_{ab}\\widetilde{P^{*\\mu}_b}-i\\frac{2g}{f_\\pi}\\bar{M}\\widetilde{P^{*\\mu\\dag}_{a}}\\partial_{\\mu}\\phi_{ab}\\widetilde{P_b} \\nonumber\\\\\n&+& \\frac{g}{f_\\pi}\n\\partial^{\\beta}\\widetilde{P^{*\\mu\\dag}_{a}}\\partial^{\\alpha}\\phi_{ab}\\widetilde{P^{*\\nu}_b}\\epsilon_{\\mu\\nu\\alpha\\beta}\n-\\frac{g}{f_\\pi}\n\\widetilde{P^{*\\mu\\dag}_{a}}\\partial^{\\alpha}\\phi_{ab}\\partial^{\\beta}\\widetilde{P^{*\\nu}_b}\\epsilon_{\\mu\\nu\\alpha\\beta},\\label{anti-pseudo-exchange}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\mathcal{L}_{V}&=&i\\frac{\\beta g_v}{\\sqrt{2}} P_b\nV^{\\mu}_{ba}\\partial_{\\mu}P^{\\dag}_{a}-i\\frac{\\beta g_v}{\\sqrt{2}}\n\\partial_{\\mu}P_b V^{\\mu}_{ba}P^{\\dag}_{a} \\nonumber\\\\\n&-&i\\sqrt{2}\\lambda g_v\\epsilon_{\\mu\\alpha\\beta\\nu}\\partial^{\\mu}P_b\n\\partial^{\\alpha}V^{\\beta}_{ba}P^{*\\nu\\dag}_{a}\\nonumber\\\\\n&-&i\\sqrt{2}\\lambda g_v \\epsilon_{\\mu\\alpha\\beta\\nu}P^{*\\mu}_b\n\\partial^{\\alpha}V^{\\beta}_{ba}\\partial^{\\nu}P^{\\dag}_{a}\\nonumber\\\\\n&-&i\\frac{\\beta g_v}{\\sqrt{2}} P^{*\\nu}_b\nV^{\\mu}_{ba}\\partial_{\\mu}P^{*\\dag}_{\\nu a}+ i\\frac{\\beta\ng_v}{\\sqrt{2}} \\partial_{\\mu}P^{*\\nu}_b V^{\\mu}_{ba}P^{*\\dag}_{\\nu a}\\nonumber\\\\\n&-&i2\\sqrt{2}\\lambda g_v \\bar{M^*}P^{*\\mu}_b\n(\\partial_{\\mu}V_{\\nu}-\\partial_{\\nu}V_{\\mu})_{ba}P^{*\\nu\\dag}_a,\\label{vector-exchange}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\widetilde{\\mathcal{L}_{V}}&=&-i\\frac{\\beta g_v}{\\sqrt{2}}\n\\partial_{\\mu}\\widetilde{P^{\\dag}_{a}} V^{\\mu}_{ab}\\widetilde{P_b}+\ni\\frac{\\beta g_v}{\\sqrt{2}} \\widetilde{P^{\\dag}_{a}} V^{\\mu}_{ab}\\partial_{\\mu}\\widetilde{P_b} \\nonumber\\\\\n&+&i\\sqrt{2}\\lambda g_v\\epsilon_{\\mu\\alpha\\beta\\nu}\n\\widetilde{P^{*\\mu\\dag}_{a}}\\partial^{\\alpha}V^{\\beta}_{ab}\\partial^{\\nu}\\widetilde{P_b}\\nonumber\\\\\n&+&i\\sqrt{2}\\lambda g_v \\epsilon_{\\mu\\alpha\\beta\\nu}\\partial^{\\mu}\n\\widetilde{P^{\\dag}_{a}}\\partial^{\\alpha}V^{\\beta}_{ab}\\widetilde{P^{*\\nu}_b}\\nonumber\\\\\n&+&i\\frac{\\beta g_v}{\\sqrt{2}}\\partial_{\\mu}\n\\widetilde{P^{*\\dag}_{\\nu a}} V^{\\mu}_{ba} \\widetilde{P^{*\\nu}_b}-\ni\\frac{\\beta g_v}{\\sqrt{2}}\\widetilde{P^{*\\dag}_{\\nu a}}\n V^{\\mu}_{ab}\\partial_{\\mu} \\widetilde{P^{*\\nu}_b}\\nonumber\\\\\n&-&i2\\sqrt{2}\\lambda g_v \\bar{M^*}\\widetilde{P^{*\\mu\\dag}_a}\n(\\partial_{\\mu}V_{\\nu}-\\partial_{\\nu}V_{\\mu})_{ab}\\widetilde{P^{*\\nu}_b},\\label{anti-vector-exchange}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\mathcal{L}_{S}=-2g_s \\bar{M}P_b\\sigma P^{\\dag}_{b}+ 2g_s\n\\bar{M^*}P^{*\\mu}_b\\sigma P^{*\\dag}_{\\mu b}\\label{scalar-exchange}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\widetilde{\\mathcal{L}_{S}}=-2g_s\n\\bar{M}\\widetilde{P^{\\dag}_a}\\sigma \\widetilde{P_{a}}+ 2g_s\n\\bar{M^*}\\widetilde{P^{*\\dag}_{\\mu a}}\\sigma\n\\widetilde{P^{*\\mu}_a}\\label{anti-scalar-exchange}\n\\end{eqnarray}\nwhere the heavy flavor meson fields $P$ and $P^*$ represent $P=(D^0,\nD^+)$ or $(B^-, \\bar{B}^0)$ and $P^*=(D^{*0}, D^{*+})$ or $(B^{*-},\n\\bar{B}^{*0})$. Its corresponding heavy anti-meson fields\n$\\widetilde{P}$ and $\\widetilde{P}^*$ represent\n$\\widetilde{P}=(\\bar{D}^0,D^-)$ or $(B^+, B^0)$ and\n$\\widetilde{P}^*=(\\bar{D}^{*0},D^{*-})$ or $(B^{*+}, B^{*0})$.\n$\\phi$, $V$ represent the the exchanged pseudoscalar and vector\nmeson matrices, $\\sigma$ is the only scalar meson interacting with\nthe heavy flavor meson.\n\n\\begin{eqnarray}\n\\phi=\\left(\n         \\begin{array}{cc}\n           \\frac{\\pi^0}{\\sqrt{2}}+\\frac{\\eta}{\\sqrt{6}} & \\pi^+ \\\\\n           \\pi^- & -\\frac{\\pi^0}{\\sqrt{2}}+\\frac{\\eta}{\\sqrt{6}} \\\\\n         \\end{array}\n       \\right)\n\\end{eqnarray}\n\n\\begin{eqnarray}\nV=\\left(\n         \\begin{array}{cc}\n           \\frac{\\rho^0}{\\sqrt{2}}+\\frac{\\omega}{\\sqrt{2}} & \\rho^+ \\\\\n           \\rho^- & -\\frac{\\rho^0}{\\sqrt{2}}+\\frac{\\omega}{\\sqrt{2}} \\\\\n         \\end{array}\n       \\right)\n\\end{eqnarray}\n\n\nAccording the OBE model, five mesons ( $\\pi$, $\\sigma$, $\\rho$,\n$\\omega$ and $\\eta$) contribute to the effective potential. In the\n$D\\bar{D}^{*}$ and $B\\bar{B}^{*}$ systems we considered, the\npotentials are the same for the three isovector states in Eqs.\n(1)$\\sim$(4) with the exact isospin symmetry. Expanding the\nLagrangian densities in Eqs. (5)$\\sim$(10) leads to each meson's\ncontribution for the two coupled channels. These channel-dependent\ncoefficients are listed in Table \\ref{tab:channel-coeff}. The pionic\ncoupling constant $g\\!=\\!0.59$ is extracted from the width of\n$D^{*+}$\\cite{S.Ahmed:2001}. $f_{\\pi}=132 MeV$ is the pion decay\nconstant. According the vector meson dominance mechanism, the\nparameters $g_v$ and $\\beta$ can be determined as $g_v=5.8$ and\n$\\beta=0.9$. At the same time, by matching the form factor obtained\nfrom the light cone sum rule and that calculated from the lattice\nQCD, we can get $\\lambda=0.56 GeV^{-1}$\\cite{C.Isola:2003,\nM.Bando:1988}. The coupling constant related to the scalar meson\nexchange is $g_s=g_{\\pi}\/2\\sqrt{6}$ with $g_{\\pi}=3.73$\n\\cite{X.Liu:2009, A.F.Falk:1992}. All these parameters are listed in\nTable \\ref{tab:coupling-constant}.\n\n\\begin{table}[htbp]\n\\caption{The coupling constants and masses of the heavy mesons and\nthe exchanged light mesons used in our calculation. The masses of\nthe mesons are taken from the PDG \\cite{PDG}}\n\\label{tab:coupling-constant}\n\\begin{center}\n\\begin{tabular}{c | c  | c }\n\\hline \\hline  & {mass(MeV)} & {coupling constants} \\\\\n\\cline{2-3} \\hline\n\\multirow{2}{*}{pseudoscalar} & $m_{\\pi}=134.98$  &  $g=0.59$  \\\\\n\n                                      &  $m_{\\eta}=547.85$ &  $f_{\\pi}=132 MeV$  \\\\\n\\hline\n\\multirow{3}{*}{vector} & $m_{\\rho}=775.49$       &  $g_v=5.8$  \\\\\n\n                                & $m_{\\omega}=782.65$     &  $\\beta=0.9$  \\\\\n\n                                &                         &  $\\lambda=0.56 GeV^{-1}$  \\\\\n\\hline\n\\multirow{2}{*}{scalar} & $m_{\\sigma}=600$    &  $g_s=g_{\\pi}\/2\\sqrt{6}$  \\\\\n\n                                &   &  $g_{\\pi}=3.73$  \\\\\n\\hline\\hline\n\\multirow{4}{*}{heavy flavor } & $m_{D}=1864.9$    &    \\\\\n\n                                &  $m_{D^*}=2010.0$     &   \\\\\n                                &  $m_{B}=5279.0$     &   \\\\\n                                &  $m_{B^*}=5325$     &   \\\\\n\\hline\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\begin{table}[htbp]\n\\caption{coefficients } \\label{tab:channel-coeff}\n\\begin{center}\n\\begin{tabular}{c| c  | c c c | c c c c }\n\\hline \\hline & \\multirow{2}{*}{isospin}   & \\multicolumn{3}{c|}{direct-channel }& \\multicolumn{4}{c}{cross-channel} \\\\\n\\cline{3-9}\n& & $~~~\\rho~~~$ & $~~~\\omega~~~$ & $~~~\\sigma~~~$ & $~~~\\rho~~~$ & $~~~\\omega~~~$ & $~~~\\pi~~~$ & $~~~\\eta~~~$\\\\\n\\hline\n\\multirow{2}{*}{$D\\bar{D}^{*}$} & $I=1$ &   ~-1\/2~ &  ~1\/2~ & ~1~ & ~$-c\/2$~  & ~$c\/2$~ & $-c\/2$  &  $c\/6$  \\\\\n\\cline{2-9}\n                                & $I=0$ &   ~3\/2~  & ~1\/2~  & ~1~ & ~$3c\/2$~  & ~$c\/2$~ & $3c\/2$  &  $c\/6$  \\\\\n\\hline\n\\multirow{2}{*}{$B\\bar{B}^{*}$} & $I=1$ &   ~-1\/2~ &  ~1\/2~ & ~1~ & ~$-c\/2$~  & ~$c\/2$~ & $-c\/2$  &  $c\/6$  \\\\\n\\cline{2-9}\n                                & $I=0$ &   ~3\/2~  & ~1\/2~  & ~1~  & ~$3c\/2$~ &  ~$c\/2$~ & $3c\/2$  &  $c\/6$  \\\\\n\\hline\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nIn order to include all the momentum-related terms in our\ncalculation, we need introduce the polarization vectors of the\nvector mesons. The polarization vector at its rest frame is\n\\begin{eqnarray}\\label{pol}\n\\epsilon_{\\lambda}=(0,\\vec{\\epsilon_{\\lambda}})\n\\end{eqnarray}\nWe need to make a lorentz boost to Eq. \\ref{pol} to derive the\npolarization vector in the laboratory frame\n\\begin{eqnarray}\n\\epsilon^{lab}_{\\lambda}=(\\frac{\\vec{p}\\cdot\\vec{\\epsilon_{\\lambda}}}{m},\n\\vec{\\epsilon_{\\lambda}}+\\frac{\\vec{p}\\cdot\n(\\vec{p}\\cdot\\vec{\\epsilon_{\\lambda}})}{m(P_0 + m)})\n\\end{eqnarray}\nwhere $p=(p_0,\\mathbf{p})$ is the particle's 4-momentum in the\nlaboratory frame and $m$ is the mass of the particle.\n\n\\begin{center}\n\\textbf{B. Effective potential }\n\\end{center}\n\nTogether with the wave function and Feynman diagram, we can derive\nthe relativistic scattering amplitude at the tree level\n\\begin{equation}\n\\langle f | S | i \\rangle = \\delta_{fi} + i \\langle f | T | i\n\\rangle = \\delta_{fi} + (2\\pi)^4\\delta^4(p_f-p_i) i M_{fi},\n\\end{equation}\nwhere the T-matrix is the interaction part of the S-matrix and M is\ndefined as the invariant matrix element. After applying Bonn\napproximation on the Lippmann-Schwinger equation, the S-matrix reads\n\\begin{equation}\n\\langle f | S | i \\rangle = \\delta_{fi} - 2\\pi \\delta(E_f-E_i) i\nV_{fi}\n\\end{equation}\nwith $V_{fi}$ being the effective potential. Considering the\ndifferent normalization conventions used for the scattering\namplitude $M_{fi}$, $T$-matrix $T_{fi}$ and $V_{fi}$, we have\n\\begin{equation}\nV_{fi}=-\\frac{M_{fi}}{\\sqrt{ \\mathop\\prod\\limits_{f}2{p_f}^0\n\\mathop\\prod\\limits_{i} 2{p_i}^0}}\\approx -\\frac{M_{fi}}\n{\\sqrt{\\mathop\\prod\\limits_{f} 2{m_f}^0 \\mathop\\prod\\limits_{i}\n2{m_i}^0}}\n\\end{equation}\nwhere $p_{f(i)}$ denotes the four momentum of the final (initial)\nstate.\n\nDuring our calculation, $P_1(E_1,\\vec{p})$ and $P_2(E_2,-\\vec{p})$\ndenote the four momenta of the initial particles in the center mass\nsystem, while $P_3(E_3,\\vec{p'})$ and $P_4(E_4,-\\vec{p'})$ denote\nthe four momenta of the final particles, respectively.\n\\begin{equation}\nq=P_3-P_1=(E_3-E_1,\\vec{p'}-\\vec{p})=(E_2-E_4,\\vec{q})\n\\end{equation}\nis the transferred four momentum or the four momentum of the meson\npropagator. For convenience, we always use\n\\begin{equation}\n\\vec{q}=\\vec{p'}-\\vec{p}\n\\end{equation}\nand\n\\begin{equation}\n\\vec{k}=\\frac{1}{2}(\\vec{p'}+\\vec{p})\n\\end{equation}\ninstead of $\\vec{p'}$ and $\\vec{p}$ in the practical calculation.\n\nIn the OBE model, each vertex in the Feynman diagram needs a form\nfactor to suppress the high momentum contribution. We take the\nconventional form for the form factor as in the Bonn potential\nmodel.\n\\begin{equation}\nF(q)=\\frac{\\Lambda^2-m_{\\alpha}^2}{\\Lambda^2-q^2}=\\frac{\\Lambda^2-m_{\\alpha}^2}{{\\tilde{\\Lambda}}^2+\\vec{q}^2}\n\\end{equation}\n$m_\\alpha$ is the mass of the exchanged meson and\n\\begin{equation}\n\\tilde{\\Lambda}^2=\\Lambda^2-(m^*-m)^2\n\\end{equation}\nwhere $m$ and $m^*$ is the mass of the heavy flavor meson $D$ and\n$D^*$ or $B$ and $B^*$. So far, the effective potential is in the\nmomentum space. In order to solve the time independent\nSchr\\\"{o}dinger equation in the coordinate space, we need to make\nthe Fourier transformation to $V(\\vec{q},\\vec{k})$. The details of\nthe Fourier transformations are presented in the Appendix.\n\nAll the meson exchanged potentials for $B\\bar{B}^{*}$ and\n$D\\bar{D}^{*}$ are the same, except the $\\pi$ exchange potential.\nThe $\\pi$ mass is larger than the mass difference of $B$ and\n$\\bar{B}^*$ but smaller than that of $D$ and $\\bar{D}^*$.\n\nThe expressions of the direct-channel effective potential through\nexchanging the $\\sigma$, $\\rho$ mesons are\n\\begin{eqnarray}\nV_{\\sigma}&=&-C_{\\sigma}g^2_s(\\vec{\\epsilon_b}\\cdot\n\\vec{\\epsilon_a}^{\\dag})F_{1t\\sigma}\\nonumber\\\\\n&~&-C_{\\sigma}g^2_s \\frac{1}{2m^{*2}}(F_{3t1\\sigma}+F_{3t2\\sigma})\\nonumber\\\\\n&~&+C_{\\sigma}g^2_s \\frac{1}{2m^{*2}}\\frac{(\\vec{\\epsilon_b}\\times\n\\vec{\\epsilon_a}^{\\dag})\\cdot \\vec{L}}{i}F_{5t\\sigma}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nV_{\\rho}&=&-C_{\\rho}\\beta^2 g^2_v \\frac{\\vec{\\epsilon_b}\\cdot\n\\vec{\\epsilon_a}^{\\dag}}{2}F_{1t\\rho}\\nonumber\\\\\n&~&+C_{\\rho}(\\frac{\\lambda \\beta g^2_v}{m^*}-\\frac{\\beta^2 g^2_v}{4m^{*2}} )(F_{3t1\\rho}+F_{3t2\\rho})\\nonumber\\\\\n&~&-C_{\\rho}\\beta^2 g^2_v \\frac{\\vec{\\epsilon_b}\\cdot\n\\vec{\\epsilon_a}^{\\dag}}{2m m^*}[F_{4t1\\rho}+\\{-\\frac{1}{2}\\nabla^2, F_{4t2\\rho}\\}]\\nonumber\\\\\n&~&+C_{\\rho}( \\frac{\\beta^2 g^2_v }{4m^{*2}}-\\lambda \\beta\ng^2_v\\frac{m^* + m}{m m^*})\\frac{(\\vec{\\epsilon_b}\\times\n\\vec{\\epsilon_a}^{\\dag})\\cdot \\vec{L}}{i}F_{5t\\rho}\n\\end{eqnarray}\nThe $\\omega$ and $\\rho$ meson exchange potentials have the same form\nexcept that the meson mass and channel-dependent coefficients are\ndifferent.\n\nThe expression of the cross-channel effective potential through\nexchanging the $\\pi$ meson in the $B\\bar{B}^{*}$ system is\n\\begin{eqnarray}\nV_{\\pi}&=&C C_{\\pi} \\frac{g^2_{\\pi}}{f^2_{\\pi}}\\frac{(m^* + m)^2}{4 m^{*2}}(F_{3u1\\pi}+F_{3u2\\pi})\\nonumber\\\\\n&~&+C C_{\\pi} \\frac{g^2_{\\pi}}{f^2_{\\pi}} \\frac{m^{*2}-m^2}{2m^{*2}}\n\\frac{(\\vec{\\epsilon_b}\\times \\vec{\\epsilon_a}^{\\dag})\\cdot\n\\vec{L}}{i}F_{5u\\pi}\\nonumber\\\\\n&~&-C C_{\\pi}\\frac{g^2_{\\pi}}{f^2_{\\pi}}\\frac{(m^* -\nm)^2}{m^{*2}}(F_{6u1}+F_{6u2\\pi}\\nabla + F_{6u3\\pi}\\nabla^2)\n\\end{eqnarray}\n\nThe expression of the cross-channel effective potential through\nexchanging the $\\pi$ meson in the $D\\bar{D}^{*}$ system is\n\\begin{eqnarray}\nV'_{\\pi}&=&C C_{\\pi} \\frac{g^2_{\\pi}}{f^2_{\\pi}}\\frac{(m^* + m)^2}{4 m^{*2}}(F'_{3u1\\pi}+F'_{3u2\\pi})\\nonumber\\\\\n&~&+C C_{\\pi} \\frac{g^2_{\\pi}}{f^2_{\\pi}} \\frac{m^{*2}-m^2}{2m^{*2}}\n\\frac{(\\vec{\\epsilon_b}\\times \\vec{\\epsilon_a}^{\\dag})\\cdot\n\\vec{L}}{i}F'_{5u\\pi}\\nonumber\\\\\n&~&-C C_{\\pi}\\frac{g^2_{\\pi}}{f^2_{\\pi}}\\frac{(m^* -\nm)^2}{m^{*2}}(F'_{6u1}+F'_{6u2\\pi}\\nabla + F'_{6u3\\pi}\\nabla^2)\n\\end{eqnarray}\n\nThe expression of the cross-channel effective potential through\nexchanging the $\\rho$ meson is\n\\begin{eqnarray}\nV_{\\rho}&=&-C C_{\\rho}\\lambda^2 g^2_v \\frac{(m^* + m)^2}{2m m^*}\n(\\vec{\\epsilon_b}\\cdot \\vec{\\epsilon_a}^{\\dag})F_{2u\\rho}\\nonumber\\\\\n&~&+C C_{\\rho}\\lambda^2 g^2_v \\frac{(2m^*-m)(m^* + m)^2}{2m^{*3}}(F_{3u1\\rho}+F_{3u2\\rho})\\nonumber\\\\\n&~&+C C_{\\rho}\\lambda^2 g^2_v \\frac{2(m^*-m)^2}{m m^*}\n\\vec{\\epsilon_b}\\cdot\\vec{\\epsilon_a}^{\\dag} [F_{4u1\\rho}+\\{-\\frac{1}{2}\\nabla^2, F_{4u2\\rho}\\}]\\nonumber\\\\\n&~&-C C_{\\rho}\\lambda^2 g^2_v \\frac{m(m^{*2} - m^2)}{m^{*3}}\n\\frac{(\\vec{\\epsilon_b}\\times \\vec{\\epsilon_a}^{\\dag})\\cdot\n\\vec{L}}{i}F_{5u\\rho}\\nonumber\\\\\n&~&+C C_{\\rho}\\lambda^2 g^2_v \\frac{2(2m^*+m)(m^* -\nm)^2}{m^{*3}}(F_{6u1\\rho}+ F_{6u2\\rho}\\nabla\\nonumber\\\\\n&~&+F_{6u3\\rho}\\nabla^2)\n\\end{eqnarray}\nSimilarly, the $\\eta$ and $\\pi$ meson exchange potential has the\nsame form in the $B {\\bar B}^{*}$ system. The potential from the\n$\\omega$ and $\\rho$ meson exchange is also similar except the meson\nmass and channel-dependent coefficients. The explicit forms of\n$\\mathcal{F}_{\\mu t \\alpha}$,$\\mathcal{F}_{\\mu u\n \\alpha}$,$\\mathcal{F}_{\\mu t \\nu \\alpha}$, $\\mathcal{F}_{\\mu u\n\\nu \\alpha}$, $\\mathcal{F'}_{\\mu u \\nu \\alpha}$ are shown in the\nAppendix.\n\nIn our calculation, we explicitly consider the external momentum of\nthe initial and final states. Due to the recoil corrections, several\nnew terms appear which were omitted in the heavy quark symmetry\nlimit. These momentum dependent terms are related to the momentum\n$\\vec{k}=\\frac{1}{2}(\\vec{p'}+\\vec{p})$:\n\\begin{eqnarray}\n\\frac{\\vec{k}^2}{\\vec{q}^2+m_{\\alpha}^2}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\\label{so}\n{\\frac{i\\vec{S}\\cdot\\vec{k}\\times\\vec{q}}{\\vec{q}^2+m_{\\alpha}^2}}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\\label{t2}\n\\frac{(\\vec{\\epsilon_b}\\cdot\\vec{k})(\\vec{\\epsilon_a}^{\\dag}\n\\cdot\\vec{k})}{\\vec{p}^2+m_{\\alpha}^2}\n\\end{eqnarray}\nwhere $\\vec{S}=-i(\\vec{\\epsilon_b}\\times\\vec{\\epsilon_a}^{\\dag})$.\nThe term in Eq. (\\ref{so}) is the well-known spin orbit force. The\nterm in Eq. (\\ref{t2}) depends on the spin and results in the\nmomentum-related operator $\\nabla$, $\\nabla^2$. The Fourier\ntransformation of the above new interaction terms are also shown in\nthe Appendix. In short, all the terms in the effective potentials in\nthe form of $F_{4t1\\rho}$, $F'_{5u\\pi}$, $(F_{6u1\\rho}+\nF_{6u2\\rho}\\nabla +F_{6u3\\rho}\\nabla^2)$ etc with the sub-indices\n$4,5,6$ arise from the recoil corrections and vanish when the heavy\nmeson mass $m, m^*$ goes to infinity. Especially, the spin orbit\nforce appears at $O(1\/M)$!\n\n\\begin{center}\n\\textbf{C.  Schr\\\"{o}dinger equation }\n\\end{center}\nWith the effective potential $V(\\vec{r})$ in Eqs. (23) $\\sim$ (27),\nwe are able to study the binding property of the system by solving\nthe Schr\\\"{o}dinger Equation\n\\begin{equation}\n(-\\frac{\\hbar^2}{2\\mu}\\nabla^{2}+V(\\vec{r})-E)\\Psi(\\vec{r})=0,\n\\label{eq:schrod}\n\\end{equation}\nwhere $\\Psi(\\vec{r})$ is the total wave function of the system. The\ntotal spin of the system $S=1$ and the orbital angular momenta $L=0$\nand $L=2$. Thus the wave function $\\Psi(\\vec{r})$ should have the\nfollowing form\n\\begin{equation}\n\\Psi(\\vec{r})=\\psi_S(\\vec{r})+\\psi_D(\\vec{r}),\n\\end{equation}\nwhere $\\psi_S(\\vec{r})$ and $\\psi_D(\\vec{r})$ are the $S$-wave and\n$D$-wave functions, respectively. In the matrix method, we use\nLaguerre polynomials as a set of orthogonal basis\n\\begin{equation}\n\\chi_{nl}(r)=\\sqrt{\\frac{(2\\lambda)^(2l+3) n!}{\\Gamma(2l+3+n)}}r^l\ne^{-\\lambda r}L^{2l+2}_n (2\\lambda r), n=1,2,3...\n\\end{equation}\nwith a normalization condition of\n\\begin{equation}\n\\int^\\infty _0 \\chi_{im}(r) \\chi_{in}(r) r^2\ndr=\\delta_{ij}\\delta_{mn}.\n\\end{equation}\nWe expand the total wave function as\n\\begin{equation}\n\\Psi(\\vec{r})=\\sum^{n-1}_{i=0}a_i \\chi_{i 0}(r)\\phi_S +\n\\sum^{n-1}_{p=0}b_p \\chi_{p 2}(r)\\phi_D, \\label{eq:tottrail}\n\\end{equation}\nwhere $\\phi_S$ and $\\phi_D$ are the angular part of the spin and\norbital wave function for the $S$- and $D$-states, respectively.\n$a_i$ and $b_i$ are the corresponding expansion coefficients.\n\nIn the practical calculation, we detach the terms related to the\nkinetic-energy-operator $\\nabla^{2}$ and $\\nabla$ from $V(\\vec{r})$\nand re-write Eq. (\\ref{eq:schrod}) as\n\\begin{eqnarray}\n&~&(-\\frac{\\hbar^2}{2\\mu}\\nabla^{2}-\\frac{\\hbar^2}{2\\mu}[\\nabla^2\n\\alpha(r)+\\alpha(r)\\nabla^2]+ \\alpha_1(r)\\nabla\n+\\alpha_2(r)\\nabla^2\\nonumber\\\\\n&~&+\\widetilde{V}(\\vec{r})-E~)\\Psi(\\vec{r})=0\n\\end{eqnarray}\nwith\n\\begin{equation}\n\\nabla^2=\\frac{1}{r}\\frac{d^2}{dr^2}r-\\frac{\\overrightarrow{L}^2}{r^2},\n\\end{equation}\nin which $\\alpha(r)$,$\\alpha_1(r)$ and $\\alpha_2(r$) are\n\\begin{eqnarray}\n\\alpha(r)&=&(-2\\mu)(-C_{\\rho}\\beta^2 g^2_v\n\\frac{\\vec{\\epsilon_b}\\cdot \\vec{\\epsilon_a}^{\\dag}}{2m m^*}\n\\mathcal{F}_{4t2 \\rho}\n-C_{\\omega}\\beta^2 g^2_v \\frac{\\vec{\\epsilon_b}\\cdot\\vec{\\epsilon_a}^{\\dag}}{2m m^*}\n\\mathcal{F}_{4t2\\omega}\\nonumber\\\\\n&~&+C C_{\\rho}\\lambda^2 g^2_v \\frac{2(m^*-m)^2}{m m^*}\n\\vec{\\epsilon_b}\\cdot\\vec{\\epsilon_a}^{\\dag}F_{4u2\\rho}\\nonumber\\\\\n&~&+C C_{\\omega}\\lambda^2 g^2_v \\frac{2(m^*-m)^2}{m m^*}\n\\vec{\\epsilon_b}\\cdot\\vec{\\epsilon_a}^{\\dag}F_{4u2\\omega})\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\alpha_1(r)&=&C C_{\\pi}\\frac{g^2_{\\pi}}{f^2_{\\pi}}\\frac{(m^* -\nm)^2}{m^{*2}}F'_{6u2\\pi}\\nonumber\\\\\n&~&+C C_{\\eta}\\frac{g^2_{\\eta}}{f^2_{\\eta}}\\frac{(m^* -\nm)^2}{m^{*2}}F_{6u2\\eta}\\nonumber\\\\\n&~&+C C_{\\rho}\\lambda^2 g^2_v \\frac{2(2m^*+m)(m^* - m)^2}{m^{*3}}\nF_{6u2\\rho}\\nonumber\\\\\n&~&+C C_{\\omega}\\lambda^2 g^2_v \\frac{2(2m^*+m)(m^* - m)^2}{m^{*3}}\nF_{6u2\\omega}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\alpha_2(r)&=& C_{\\pi}\\frac{g^2_{\\pi}}{f^2_{\\pi}}\\frac{(m^* -\nm)^2}{m^{*2}}F'_{6u3\\pi}\\nonumber\\\\\n&~&+C C_{\\eta}\\frac{g^2_{\\eta}}{f^2_{\\eta}}\\frac{(m^* -\nm)^2}{m^{*2}}F_{6u3\\eta}\\nonumber\\\\\n&~&+C C_{\\rho}\\lambda^2 g^2_v \\frac{2(2m^*+m)(m^* - m)^2}{m^{*3}}\nF_{6u3\\rho}\\nonumber\\\\\n&~&+C C_{\\omega}\\lambda^2 g^2_v \\frac{2(2m^*+m)(m^* - m)^2}{m^{*3}}\nF_{6u3\\omega}\n\\end{eqnarray}\n\nThen, with the wave function in Eq. (\\ref{eq:tottrail}), the\nHamiltonian matrix can be expressed as\n\\begin{equation}\n\\begin{pmatrix}\nH^{SS} & H^{SD}\\\\\nH^{DS} & H^{DD}\n\\end{pmatrix}\n\\end{equation}\nwith\n\\begin{eqnarray}\nH^{SS}&=&\\langle \\phi_S |\\int^\\infty _0 \\sum^{n-1}_{i,j} a_i \\chi_{i\n0}(r) \\{-\\frac{\\hbar^2}{2\\mu}[1+\\alpha(r)]\\nabla^2 a_j \\chi_{j 0}(r)\\nonumber\\\\\n&~&-\\frac{\\hbar^2}{2\\mu}\\nabla^2 [\\alpha(r)a_j \\chi_{j 0}(r)]\n+\\alpha_1(r)\\nabla a_j \\chi_{j 0}(r)\\nonumber\\\\\n&~&+\\alpha_2(r)\\nabla^2 a_j \\chi_{j 0}(r) + V_{SS} (r) a_j \\chi_{j\n0}(r)\\} r^2 dr | \\phi_S \\rangle,\n\\end{eqnarray}\n\\begin{eqnarray}\nH^{SD}=\\langle \\phi_S |\\int^\\infty _0 \\sum^{n-1}_{i,p} a_i \\chi_{i\n0}(r) V_{SD} (r) b_p \\chi_{p 2}(r) r^2 dr | \\phi_D \\rangle,\n\\end{eqnarray}\n\\begin{eqnarray}\nH^{DS}=\\langle \\phi_D |\\int^\\infty _0 \\sum^{n-1}_{p,i} b_p \\chi_{p\n2}(r) V_{DS} (r)a_i \\chi_{i 0}(r) r^2 dr | \\phi_S \\rangle,\n\\end{eqnarray}\nand\n\\begin{eqnarray}\nH^{DD}&=&\\langle \\phi_D |\\int^\\infty _0 \\sum^{n-1}_{p,q} b_p \\chi_{p\n2}(r) \\{-\\frac{\\hbar^2}{2\\mu}[1+\\alpha(r)]\\nabla^2 b_q \\chi_{q 2}(r)\\nonumber\\\\\n&~&-\\frac{\\hbar^2}{2\\mu}\\nabla^2 [\\alpha(r)b_q \\chi_{q 2}(r)]+\\alpha_1(r)\\nabla b_q \\chi_{q 2}(r)\\nonumber\\\\\n&~&+\\alpha_2(r)\\nabla^2 b_q \\chi_{q 2}(r) + V_{DD} (r) b_q \\chi_{q\n2}(r)\\} r^2 dr  | \\phi_D \\rangle.\n\\end{eqnarray}\nThe total Hamiltonian contains three angular momentum related\noperators $\\hat{\\vec{\\epsilon_b}}\\cdot\n\\hat{\\vec{\\epsilon_a}}^{\\dag}$, $\\hat{S}_{12}$,\n$(\\hat{\\vec{\\epsilon_b}}\\times \\hat{\\vec{\\epsilon_a}}^{\\dag})\\cdot\n\\hat{\\vec{L}}$, which corresponds to the spin-spin interaction, the\nspin orbit force and tensor force respectively. They act on the S\nand D-wave coupled wave functions and split the total effective\npotential $\\widetilde{V}(\\vec{r})$ into the subpotentials\n$V_{SS}(r)$, $V_{SD}(r)$, $V_{DS}(r)$ and $V_{DD}(r)$. The matrix\nform reads\n\\begin{eqnarray}\n\\langle \\phi_S+\\phi_D |\\hat{\\vec{\\epsilon_b}}\\cdot\n\\hat{\\vec{\\epsilon_a}}^{\\dag}\\widetilde{V}(\\vec{r})|\\phi_S+\\phi_D\\rangle=\\left(\n                                                \\begin{array}{cc}\n                                                  V_{SS}(r) & 0 \\\\\n                                                  0 & V_{DD}(r) \\\\\n                                                \\end{array}\n                                              \\right)\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\langle \\phi_S+\\phi_D |\\hat{S}_{12}\\widetilde{V}(\\vec{r})\n|\\phi_S+\\phi_D\\rangle=\\left(\\!\n         \\begin{array}{cc}\n           \\!\\!0 & \\!\\!-\\sqrt{2}V_{SD}(r) \\\\\n           \\!\\!-\\sqrt{2}V_{DS}(r) & \\!\\!1 \\\\\n         \\end{array}\n      \\! \\right)\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\langle \\phi_S+\\phi_D |(\\hat{\\vec{\\epsilon_b}}\\times\n\\hat{\\vec{\\epsilon_a}}^{\\dag})\\cdot\n\\hat{\\vec{L}}\\widetilde{V}(\\vec{r}) |\\phi_S+\\phi_D\\rangle =\\left(\n                                                \\begin{array}{cc}\n                                                  0 & 0 \\\\\n                                                  0 & 3i V_{DD}(r)\\\\\n                                                \\end{array}\n                                              \\right)\n\\end{eqnarray}\nwhere the tensor force operator $\\hat{S}_{12}$ mixes the S-wave and\nD-wave contribution and is defined as\n\\begin{eqnarray}\n\\hat{S}_{12}=3(\\vec{r}\\cdot \\hat{\\vec{\\epsilon_b}})(\\vec{r}\\cdot\n\\hat{\\vec{\\epsilon_a}}^{\\dag})- \\hat{\\vec{\\epsilon_b}}\\cdot\n\\hat{\\vec{\\epsilon_a}}^{\\dag}\n\\end{eqnarray}\n\n\\section{Numerical Results}\\label{Numerical}\n\nWe diagonalize the Hamiltonian matrix to obtain the eigenvalue and\neigenvector. If there exists a negative eigenvalue, there exists a\nbound state. The corresponding eigenvector is the wave function. We\nuse the variation principle to solve the equation. We change the\nvariable parameter to get the lowest eigenvalue. We also change the\nnumber of the basis functions to reach a stable result.\n\n\\subsection{X(3872)}\n\nThe mass of the $\\pi$ meson is smaller than the mass difference of\n$D$ and $\\bar{D}^*$, which causes the Fourier transformation of the\n$\\pi$-meson-exchange potential to be a complex function. The\ndifferent treatment of this complex potential would lead to quite\ndifferent results for the system. In our approach, we drop the the\nimaginary part of the potential.\n\nIn order to distinguish each meson's effect, we plot each meson's\nS-wave contribution to the potential in the first figure in\nFig.\\ref{D-potential-meson}. The $\\pi$ meson provides the most\nattractive force while the $\\sigma$ meson's attraction is relatively\nsmall.\n\nThe main contribution to the binding energy comes form the S-wave\nattractive force. We also plot the effective potential in the first\ndiagram in Fig. \\ref{D-potential}. $V_s$ and $V_d$ are the effective\npotential of the S-wave and D-wave interaction after adding the\nmomentum-related terms. $V'_s$ and $V'_d$ are the effective\npotential of the S-wave and D-wave interaction without the\nmomentum-related terms. We can see a clear difference between $V_s$\nand $V'_s$, which cause an obvious correction to the binding energy\nwhen we consider the momentum-related terms.\n\nWe first used the computation programme to reproduce the deuteron\nsystem successfully. Then we move on to investigate the possibility\nof $X(3872)$ as the $D \\bar{D}^{*}$ molecular state with quantum\nnumber $I=0$, $J^{PC}=1^{++}$. For comparison, we first do not\nconsider the momentum-related terms. Then we add the\nmomentum-related terms and repeat the numerical analysis to\ninvestigate its correction to the system.\n\nConsidering that the binding energy of $X(3872)$ is tiny, the\ninclusion of the momentum-related terms may lead to significant\ncorrections to this very loosely bound system.\n\nWe collect the numerical results of the binding energy with the\nvariation of the cutoff parameter $\\Lambda$ Table \\ref{tab:X3872+}.\n$E$ and $E'$ is the eigen-energy of Hamiltonian with and without the\nmomentum-related terms respectively. Besides the total energy, we\nalso list the separate contribution to the energy from the S-wave,\nD-wave and spin-orbit force components respectively in the fourth,\nfifth  and sixth column. The last column is the mass of X(3872) as a\nmolecular system.\n\nThere exists a bound state solution when the cutoff parameter\nchanges from $1.1 \\sim 1.3 $ GeV. The binding energy with the recoil\ncorrection is around $0.054 \\sim 7.131$ MeV and the binding energy\nwithout the recoil correction is around $0.276 \\sim 9.686$ MeV. When\nthe binding energy is $7.131$ MeV with $\\Lambda =1.3$ GeV, the\nrecoil correction is $2.555$ MeV and the contribution of the\nspin-orbit force is $0.573$ MeV. When the binding energy is $2.361$\nMeV, the recoil correction is $1.075$ MeV and the contribution of\nthe spin-orbit force is $0.213$ MeV. When the binding energy\ndecrease to $0.054$ with $\\Lambda =1.1$ GeV, the recoil correction\nreach $0.222$ MeV, which is even bigger than the binding energy\nitself. Now the contribution of the spin-orbit force is $0.038$ MeV\nand almost as big as the D-wave contribution. Clearly the recoil\ncorrection decrease the binding energy and renders X(3872) to be an\nextremely loosely bound molecular states partly.\n\n\n\\begin{table}[htbp]\n\\caption{The bound state solutions of the $D \\bar{D}^{*}$ system\nwith $I^G=0^+$, $J^{PC}=1^{++}$ (in unit of MeV) with the cutoff\n$\\Lambda$. $E$ and $E'$ is the eigen-energy of the system with and\nwithout the momentum-related terms respectively. We also list the\nseparate contribution to the energy from the S-wave, D-wave and\nspin-orbit force components respectively in the fourth, fifth and\nsixth column. The last column is the mass of X(3872) as a molecular\nsystem.} \\label{tab:X3872+}\n\\begin{center}\n\\begin{tabular}{c c c c c c | c}\n\\hline \\hline \\multirow{2}{*}{$\\Lambda$(GeV)} & \\multirow{2}{*}{~}  &\\multicolumn{4}{c|}{Eigenvalue} & {Mass}  \\\\\n\\cline{2-6}\n                                 &   &total   &S             &D        & LS   &(MeV) \\\\\n\\hline\\hline\n\\multirow{2}{*}{1.10} & $E$    &  ~~-0.054~~  & ~~-4.364~~   & ~~0.052~~  & ~~0.038~~     & ~~3874.846~~\\\\\n                      & $E'$   &  ~~-0.276~~  & ~~-4.458~~   & ~~0.017~~  & ~~-~~     & ~~3874.624~~\\\\\n\\hline\\hline\n\\multirow{2}{*}{1.15} & $E$    &  ~~-0.884~~  & ~~-10.02~~   & ~~0.128~~   & ~~0.104~~   & ~~3874.016~~\\\\\n                      & $E'$   &  ~~-1.449~~  & ~~-10.48~~   & ~~0.031~~   & ~~-~~   & ~~3873.451~~\\\\\n\\hline\\hline\n\\multirow{2}{*}{1.20} & $E$    &  ~~-2.361~~  & ~~-17.23~~   & ~~0.245~~     & ~~0.213~~   & ~~3872.539~~\\\\\n                      & $E'$   &  ~~-3.436~~  & ~~-18.14~~   & ~~0.046~~    & ~~-~~     & ~~3871.464~~\\\\\n\\hline\\hline\n\\multirow{2}{*}{1.25} & $E$    &  ~~-4.469~~  & ~~-24.80~~    & ~~0.401~~    &  ~~0.367~~    & ~~3870.431~~\\\\\n                       & $E'$  &  ~~-6.203~~  & ~~-26.29~~    & ~~0.059~~     & ~~-~~     & ~~3868.697~~ \\\\\n\\hline\\hline\n\\multirow{2}{*}{1.30} & $E$    &  ~~-7.131~~  & ~~-32.45~~   & ~~0.609~~     & ~~0.573~~      & ~~3867.769~~\\\\\n                       & $E'$  &  ~~-9.686~~  & ~~-34.63~~   & ~~0.076~~     &  ~~-~~   & ~~3865.214~~\\\\\n\\hline\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\subsection{The $D \\bar{D}^{*}$ system with $I^G=0^-$,\n$J^{PC}=1^{+-}$}\n\n\nWe also calculate the $D \\bar{D}^{*}$ system with $I=0$,\n$J^{PC}=1^{+-}$. The results with the variation of the cutoff from\n$1.4 \\sim 1.6 GeV$ are shown in the Table.\\ref{tab:X3872-}. There\nmight also exist a bound state with odd C parity. Its binding energy\nis slightly smaller than that of $X(3872)$ with the same cutoff.\nWhen the binding energy is $2.386$ MeV with $\\Lambda =1.4$ GeV, the\ntotal recoil correction reaches $-0.447$ MeV while the contribution\nof the spin-orbit force is $+0.9$ MeV, which is also almost as big\nas the D-wave contribution. Clearly the recoil correction is\nfavorable to the formation of the molecular state in this channel.\n\nThe corresponding effective potential and the exchanged meson's\ncontribution are also shown in the second figure in\nFig.\\ref{D-potential} and Fig.\\ref{D-potential-meson}.\n\n\\begin{table}[htbp]\n\\caption{The bound state solution of the $D \\bar{D}^{*}$ system with\n$I^G=0^-$, $J^{PC}=1^{+-}$ (in unit of MeV) with $\\Lambda$. $E$ and\n$E'$ is the eigen-energy of the system with and without the\nmomentum-related terms respectively. We also list the separate\ncontribution to the energy from the S-wave, D-wave and spin-orbit\nforce components respectively in the fourth, fifth and sixth column.\nThe last column is the mass of the $D \\bar{D}^{*}$ system with\n$I^G=0^-$, $J^{PC}=1^{+-}$ as a molecular state.} \\label{tab:X3872-}\n\\begin{center}\n\\begin{tabular}{c c c c c c | c }\n\\hline \\hline \\multirow{2}{*}{~$\\Lambda$(GeV)~} & \\multirow{2}{*}{~}  &\\multicolumn{4}{c|}{Eigenvalue} & {Mass}   \\\\\n\\cline{2-6}\n                                &   &total  &   S           &D              & LS          &(MeV) \\\\\n\\hline\n\\multirow{2}{*}{1.40} & $E$    &  ~~-2.386~~  & ~~-10.55~~ & ~~-1.587~~   & ~~0.900~~  & ~~3872.514~~ \\\\\n                      & $E'$   &  ~~-1.939~~  & ~~-12.30~~ & ~~-2.371~~   & ~~-~~  & ~~3872.961~~\\\\\n\\hline\\hline\n\\multirow{2}{*}{1.45} & $E$    &  ~~-7.098~~  & ~~-20.90~~ &~~-2.863~~    & ~~2.019~~   & ~~3867.802~~\\\\\n                      & $E'$   &  ~~-6.298~~  & ~~-24.65~~ &~~-4.655~~    & ~~-~~ & ~~3868.602~~\\\\\n\\hline\\hline\n\\multirow{2}{*}{1.50} & $E$    &  ~~-14.62~~  & ~~-33.88~~ &~~-4.236~~    & ~~3.635~~  & ~~3860.28~~\\\\\n                      & $E'$   &  ~~-13.43~~  & ~~-40.41~~ & ~~-7.513~~   & ~~-~~ & ~~3861.47~~\\\\\n\\hline\\hline\n\\multirow{2}{*}{1.55} & $E$    &  ~~-25.20~~  & ~~-49.63~~ &~~-5.657~~   & ~~5.822~~  & ~~3849.70~~\\\\\n                      & $E'$  &  ~~-23.58~~   & ~~-59.87~~ &~~-10.97~~   & ~~-~~  & ~~3851.32~~\\\\\n\\hline\\hline\n\\multirow{2}{*}{1.60} & $E$    &  ~~-39.10~~  & ~~-68.32~~ &~~-7.074~~   & ~~8.65~~  & ~~3835.80~~\\\\\n                       & $E'$  &  ~~-36.95~~  & ~~-83.32~~ & ~~-15.06~~  & ~~-~~ & ~~3837.95~~\\\\\n\\hline\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\subsection{$Z_c(3900)$}\n\nThe newly observed $Z_c(3900)$ was explained as the isovector\npartner of $X(3872)$ with $J^{PC}=1^{+-}$ by some theoretical groups\n\\cite{Q.Wang:2013, Z.G.Wang:2013, F.Aceti:2013}.\n\nWe carefully perform the investigation of the $D \\bar{D}^{*}$ system\nwith $I^=1^+$, $J^{PC}=1^{+-}$. We consider the S-wave and D-wave\nmixing, the spin orbit force at $O(1\/M)$ and all the other possible\nrecoil corrections up to $O(1\/M^2)$. The corresponding effective\npotential and the exchanged meson's contribution are also shown in\nthe fourth diagram in Fig. \\ref{D-potential} and Fig.\n\\ref{D-potential-meson}. Unfortunately, we are unable to obtain a\nbound state solution with the pionic coupling $g=0.59 $ which was\nextracted from the $D^*$ decay width. It seems there probably does\nnot exist a loosely bound isovector molecular state composed of the\n$D \\bar{D}^{*}$ mesons.\n\nOn the other hand, the $\\pi$ meson exchange plays a dominant role.\nConsidering the uncertainty of $g$, we try to increase this coupling\nconstant to check the dependence of the results on $g$. We find when\nthe coupling constant $g$ increases by a factor of $1.6$ , a bound\nstate appears. The results are listed Table \\ref{tab:Zc3900-}.\n\nThe binding energy of the $J^{PC}=1^{+-}$ molecule with\/without the\nrecoil correction is around $0.037 \\sim 15.82$ MeV and $0.322 \\sim\n18.51$ MeV respectively. When the binding energy is $0.037$ MeV, the\nrecoil correction is $0.285$ MeV and the contribution of the\nspin-orbit force is $0.058$ MeV. Clearly the recoil corrections are\nof the same order as the binding energy and unfavorable to the\nformation of the molecular state.\n\n\\begin{table}[htbp]\n\\caption{The $I^G=1^+$, $J^{PC}=1^{+-}$ $D \\bar{D}^{*}$ system with\nwith the enhanced coupling constant $g$ and $\\Lambda = 2.0$ GeV. The\nother notations are the same as in Table \\ref{tab:X3872+}.}\n\\label{tab:Zc3900-}\n\\begin{center}\n\\begin{tabular}{c c c c c c | c}\n\\hline \\hline \\multirow{2}{*}{~$g\\cdot n$~} & \\multirow{2}{*}{~}  &\\multicolumn{4}{c|}{Eigenvalue} & {Mass}   \\\\\n\\cline{2-6}\n                                 &   &total  &   S          &D           & LS       &(MeV) \\\\\n\\hline\\hline\n\\multirow{2}{*}{$g\\cdot1.6$}       & $E$    &  ~~-0.037~~  & ~~-7.244~~ & ~~0.169~~ & ~~0.058~~ & ~~3874.863~~\\\\\n                                   & $E'$   &  ~~-0.322~~  & ~~-7.421~~ &~~0.107~~ & ~~-~~ & ~~3874.578~~\\\\\n\\hline\\hline\n\\multirow{2}{*}{$g\\cdot1.7$}       & $E$    &  ~~-4.293~~  & ~~-42.32~~ &~~0.957~~ & ~~0.359~~ & ~~3870.607~~\\\\\n                                   & $E'$   &  ~~-5.634~~  & ~~-43.21~~ &~~0.579~~ & ~~-~~ & ~~3869.266~~\\\\\n\\hline\\hline\n\\multirow{2}{*}{$g\\cdot1.8$}       & $E$    &  ~~-15.82~~  & ~~-93.30~~ &~~2.007~~ & ~~0.822~~ & ~~3859.08~~\\\\\n                                   & $E'$   &  ~~-18.51~~  & ~~-95.04~~ & ~~1.146~~ & ~~-~~ & ~~3856.39~~\\\\\n\\hline\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\\subsection{The $D \\bar{D}^{*}$ system with $I^G=1^-$,\n$J^{PC}=1^{++}$}\n\nWe also perform the investigation of the $D \\bar{D}^{*}$ system with\n$I^G=1^-$, $J^{PC}=1^{++}$. The corresponding effective potential\nand the exchanged meson's contribution are also shown in the third\ndiagram in Fig. \\ref{D-potential} and Fig. \\ref{D-potential-meson}.\nThere does not exist a bound state solution with the pionic coupling\n$g=0.59 $. If we increase $g$ by a factor $2.4$, there appears a\nbound state. The numerical results are listed in Table\n\\ref{tab:Zc3900+}.\n\nThe binding energy of the possible $J^{PC}=1^{++}$ state with the\nrecoil correction is around $1.777 \\sim 14.49$ MeV while it is\naround $0.524 \\sim 8.67$ MeV without the recoil correction. When the\nbinding energy is $1.777$ MeV, the total recoil correction is\n$1.253$ MeV and the contribution of the spin-orbit force alone is\n$-1.903$ MeV. The recoil corrections are comparable with the binding\nenergy and very favorable to the formation of the possible loosely\nbound molecule.\n\n\n\\begin{table}[htbp]\n\\caption{The $I^G=1^-$, $J^{PC}=1^{++}$ $D \\bar{D}^{*}$ system with\nthe enhanced coupling constant $g$ and $\\Lambda =2$ GeV. The other\nnotations are the same as in Table \\ref{tab:X3872+}. }\n\\label{tab:Zc3900+}\n\\begin{center}\n\\begin{tabular}{c c c c c c | c}\n\\hline \\hline \\multirow{2}{*}{~$g\\cdot n$~} & \\multirow{2}{*}{~}  &\\multicolumn{4}{c|}{Eigenvalue} & {Mass}   \\\\\n\\cline{2-6}\n                                   &   &total       &   S            &D              & LS          &(MeV) \\\\\n\\hline\\hline\n\\multirow{2}{*}{$g\\cdot2.4$}       & $E$    &  ~~-1.777~~  & ~~6.093~~ & ~~-7.074~~ & ~~-1.903~~ & ~~3873.123~~\\\\\n                                   & $E'$   &  ~~-0.524~~  & ~~6.019~~ & ~~-5.202~~ & ~~-~~      & ~~3874.376~~\\\\\n\\hline\\hline\n\\multirow{2}{*}{$g\\cdot2.5$}       & $E$    &  ~~-6.518~~  & ~~12.23~~ & ~~-15.67~~ & ~~-4.313~~ & ~~3868.382~~\\\\\n                                   & $E'$   &  ~~-3.311~~  & ~~12.11~~ & ~~-11.42~~ & ~~-~~      & ~~3871.589~~\\\\\n\\hline\\hline\n\\multirow{2}{*}{$g\\cdot2.6$}       & $E$    &  ~~-14.49~~  & ~~18.99~~ & ~~-26.35~~ & ~~-7.42~~  & ~~3860.41~~\\\\\n                                   & $E'$   &  ~~-8.67~~   & ~~18.86~~ & ~~-19.02~~ & ~~-~~      & ~~3866.23~~\\\\\n\\hline\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\n\\begin{figure}[ht]\n  \\begin{center}\n  \\rotatebox{0}{\\includegraphics*[width=0.38\\textwidth]{DI01++.eps}}\n    \\rotatebox{0}{\\includegraphics*[width=0.38\\textwidth]{DI01+-.eps}}\n      \\rotatebox{0}{\\includegraphics*[width=0.38\\textwidth]{DI11++.eps}}\n    \\rotatebox{0}{\\includegraphics*[width=0.38\\textwidth]{DI11+-.eps}}\n    \\caption{The effective potential of the $D \\bar{D}^{*}$ system.\n    Labels A,B,C,D correspond to the four cases $I=0$, $J^{PC}=1^{++}$; $I=0$, $J^{PC}=1^{+-}$;\n    $I=1$, $J^{PC}=1^{++}$; $I=1$, $J^{PC}=1^{+-}$ respectively\n    from top to bottom. $V_s$ and $V_d$ are the effective potential of the S-wave and D-wave\n    interaction with the momentum-related terms while $V'_s$ and $V'_d$ are\n    the S-wave and D-wave effective potential without the momentum-related terms. }\n    \\label{D-potential}\n  \\end{center}\n\\end{figure}\n\n\\begin{figure}[ht]\n  \\begin{center}\n  \\rotatebox{0}{\\includegraphics*[width=0.38\\textwidth]{DI01++meson.eps}}\n    \\rotatebox{0}{\\includegraphics*[width=0.38\\textwidth]{DI01+-meson.eps}}\n      \\rotatebox{0}{\\includegraphics*[width=0.38\\textwidth]{DI11++meson.eps}}\n    \\rotatebox{0}{\\includegraphics*[width=0.38\\textwidth]{DI11+-meson.eps}}\n    \\caption{The effective potential from the different meson exchange\n    in the $D \\bar{D}^{*}$ system. Labels A,B,C,D are the same as in\n    Fig. \\ref{D-potential}.}\n    \\label{D-potential-meson}\n  \\end{center}\n\\end{figure}\n\n\\subsection{The $B \\bar{B}^{*}$ system}\n\nThe effective potential and meson contributions are shown in Fig.\nref{B-potential}and Fig. \\ref{B-potential-meson}. From Fig.\n\\ref{B-potential-meson}, one can see that the $\\pi$ and $\\rho$ and\n$\\omega$ mesons potentials are comparable.\n\nLet's focus on the the momentum-related correction. From Fig.\n\\ref{B-potential}, we can see that the two curves of $V_s$ and\n$V'_s$ almost overlap. The dominant momentum-related correction\ncomes from the D-wave interaction. In all cases, the\nmomentum-related correction is much smaller than that in the $D\n\\bar{D}^{*}$ system, which is expected because the $B$ meson is much\nheavier than the $D$ meson.\n\nFor the $B \\bar{B}^{*}$ system, there exist bound states with the\nabove three kinds of quantum number when varying the cutoff in an\nappropriate range. We collect the numerical results in Tables\n\\ref{tab:Zb10610-}, \\ref{tab:Zb10610-}, \\ref{tab:BBstar+},\n\\ref{tab:BBstar-}.\n\nThe $I^G=1^+$, $J^{PC}=1^{+-}$ bound state corresponds to the\ncandidate of $Z_b(10610)$. The binding energy with the recoil\ncorrection is around $0.251 \\sim 18.5$ MeV and the binding energy\nwithout recoil correction is about $0.348 \\sim 19.58$ MeV with the\ncutoff from $2.1 \\sim 2.9$ GeV. When the binding energy is $0.251$\nMeV, the recoil correction is $0.097$ MeV.\n\nFor the $I^G=1^+$, $J^{PC}=1^{+-}$ bound state, its binding energy\nwith the recoil correction is around $0.02 \\sim 0.446$ MeV and about\n$0.065 \\sim 0.56$ MeV without the recoil correction with the cutoff\nvaries from $4.9 \\sim 5.1$ GeV. However, this cutoff may be too\nlarger for a loosely bound system. Its binding energy is much\nsmaller than that of $Z_b(10610)$. When the binding energy is $0.02$\nMeV, the recoil correction is $0.045$ MeV and the contribution of\nspin-orbit force is $0.04$ MeV.\n\nThere exist two $I=0$ bound states which might be the isocalar\npartners of $Z_b(10610)$. For the $I^G=0^+$, $J^{PC}=1^{++}$\nmolecule, the binding energy with the recoil correction is about\n$0.28 \\sim 36.87$ MeV when the cutoff varies from $0.7 \\sim 1.1$\nGeV. When the binding energy is $0.28$ MeV, the recoil correction is\n$0.047$ MeV. For the $I^G=0^-$, $J^{PC}=1^{+-}$ molecule, the\nbinding energy with the recoil correction varies from $0.29 \\sim\n21.09$ MeV with the cutoff around $1.0 \\sim 1.2$ GeV.\n\n\\begin{table}[htbp]\n\\caption{The $B \\bar{B}^{*}$ system with $I^G=1^+$, $J^{PC}=1^{+-}$\n(in unit of MeV). The other notations are the same as in Table\n\\ref{tab:X3872+}.} \\label{tab:Zb10610-}\n\\begin{center}\n\\begin{tabular}{c c c c c c | c}\n\\hline \\hline \\multirow{2}{*}{~$\\Lambda (GeV)$~} & \\multirow{2}{*}{~}  &\\multicolumn{4}{c|}{Eigenvalue} & {Mass}   \\\\\n\\cline{2-6}\n                                 &   &total  &   S  &D   & LS &($MeV$) \\\\\n\\hline\\hline\n\\multirow{2}{*}{2.1}  & $E$    &  ~~-0.251~~  & ~~-6.320~~ & ~~0.079~~ & ~~0.0008~~& ~~10603.749~~\\\\\n                      & $E'$   &  ~~-0.348~~  & ~~-6.337~~ &~~0.075~~ & ~~-~~& ~~10603.652~~\\\\\n\\hline\\hline\n\\multirow{2}{*}{2.3}  & $E$    &  ~~-1.766~~  & ~~-18.76~~ &~~0.227~~ & ~~0.011~~& ~~10602.234~~\\\\\n                      & $E'$   &  ~~-2.026~~  & ~~-19.11~~ &~~0.214~~ & ~~-~~& ~~10601.974~~\\\\\n\\hline\\hline\n\\multirow{2}{*}{2.5}  & $E$    &  ~~-4.988~~  & ~~-36.17~~ &~~0.430~~ & ~~0.022~~& ~~10599.012~~\\\\\n                      & $E'$   & ~~-5.461~~   & ~~-36.21~~ &~~0.404~~ & ~~-~~& ~~10598.539~~\\\\\n\\hline\\hline\n\\multirow{2}{*}{2.7}  & $E$    &  ~~-10.39~~  & ~~-59.52~~ &~~0.706~~ & ~~0.038~~& ~~10593.61~~\\\\\n                      & $E'$   &  ~~-11.14~~  & ~~-59.56~~ &~~0.663~~ & ~~-~~& ~~10592.86~~\\\\\n\\hline\\hline\n\\multirow{2}{*}{2.9}  & $E$    &  ~~-18.50~~  & ~~-89.71~~ &~~1.075~~ & ~~0.058~~& ~~10585.50~~\\\\\n                      & $E'$   &  ~~-19.58~~  & ~~-89.74~~ &~~1.009~~ & ~~-~~& ~~10584.42~~\\\\\n\\hline\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\begin{table}[htbp]\n\\caption{The $B \\bar{B}^{*}$ system with $I^G=1^-$, $J^{PC}=1^{++}$\n(in unit of MeV). The other notations are the same as in Table\n\\ref{tab:X3872+}.} \\label{tab:Zb10610+}\n\\begin{center}\n\\begin{tabular}{c c c c c c | c}\n\\hline \\hline \\multirow{2}{*}{~$\\Lambda (GeV)$~} & \\multirow{2}{*}{~}  &\\multicolumn{4}{c|}{Eigenvalue} & {Mass}   \\\\\n\\cline{2-6}\n                     &   &total&   S  &D   & LS      &($MeV$) \\\\\n\\hline\\hline\n\\multirow{2}{*}{4.9}  & $E$    &  ~~-0.02~~   & ~~0.772~~ & ~~-0.932~~ & ~~-0.040~~ & ~~10603.98~~\\\\\n                      & $E'$   &  ~~-0.065~~  & ~~0.764~~ &~~-0.895~~ & ~~-~~       & ~~10603.935~~\\\\\n\\hline\\hline\n\\multirow{2}{*}{4.95} & $E$    &  ~~-0.089~~  & ~~1.049~~ &~~-1.252~~ & ~~-0.054~~ & ~~10603.911~~\\\\\n                      & $E'$   &  ~~-0.148~~  & ~~1.039~~ &~~-1.202~~ & ~~-~~      & ~~10603.852~~\\\\\n\\hline\\hline\n\\multirow{2}{*}{5.0}  & $E$    &  ~~-0.18~~  & ~~1.397~~ &~~-1.665~~ & ~~-0.072~~ & ~~10603.820~~\\\\\n                      & $E'$   &  ~~-0.256~~  & ~~1.384~~ & ~~-1.598~~ & ~~-~~    & ~~10603.744~~\\\\\n\\hline\\hline\n\\multirow{2}{*}{5.05} & $E$    &  ~~-0.298~~  & ~~1.809~~ &~~-2.155~~ & ~~-0.094~~& ~~10603.702~~\\\\\n                       & $E'$  &  ~~-0.392~~   & ~~1.792~~ & ~~-2.068~~ & ~~-~~& ~~10603.608~~\\\\\n\\hline\\hline\n\\multirow{2}{*}{5.1}  & $E$    &  ~~-0.446~~  & ~~2.273~~ &~~-2.710~~ & ~~-0.118~~& ~~10603.554~~\\\\\n                       & $E'$  &  ~~-0.56~~  & ~~2.254~~ & ~~-2.60~~ & ~~-~~& ~~10603.440~~\\\\\n\\hline\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\\begin{table}[htbp]\n\\caption{The $B \\bar{B}^{*}$ system with $I^G=0^+$, $J^{PC}=1^{++}$\n(in unit of MeV). The other notations are the same as in Table\n\\ref{tab:X3872+}.} \\label{tab:BBstar+}\n\\begin{center}\n\\begin{tabular}{c c c c c c | c}\n\\hline \\hline \\multirow{2}{*}{~$\\Lambda (GeV)$~} & \\multirow{2}{*}{~}  &\\multicolumn{4}{c|}{Eigenvalue} & {Mass}   \\\\\n\\cline{2-6}\n                                 &   &total&   S  &D  & LS   &($MeV$) \\\\\n\\hline\n\\multirow{2}{*}{0.7}  & $E$     &  ~~-0.280~~  & ~~-3.174~~ &~~0.039~~ & ~~0.005~~ & ~~10603.720~~\\\\\n                      & $E'$    &  ~~-0.327~~  & ~~-3.178~~ &~~0.034~~ & ~~-~~ & ~~10603.673~~\\\\\n\\hline\\hline\n\\multirow{2}{*}{0.8}  & $E$    &  ~~-0.930~~   & ~~-6.631~~ &~~0.108~~ & ~~0.008~~ & ~~10603.070~~\\\\\n                      & $E'$   &  ~~-1.027~~   & ~~-6.615~~ &~~0.100~~ & ~~-~~ & ~~10602.973~~\\\\\n\\hline\\hline\n\\multirow{2}{*}{0.9}  & $E$    &  ~~-6.631~~  & ~~-22.31~~ &~~0.188~~ & ~~0.050~~ & ~~10597.369~~\\\\\n                      & $E'$   &  ~~-7.705~~  & ~~-22.45~~ &~~0.140~~ & ~~-~~ & ~~10596.295~~\\\\\n\\hline\\hline\n\\multirow{2}{*}{1.0}  & $E$    &  ~~-19.08~~  & ~~-44.46~~ &~~0.034~~ & ~~0.206~~ & ~~10584.920~~\\\\\n                       & $E'$  &  ~~-20.42~~   & ~~-45.08~~ &~~0.663~~ & ~~-~~ & ~~10583.58~~\\\\\n\\hline\\hline\n\\multirow{2}{*}{1.1}   & $E$    &  ~~-36.87~~  & ~~-67.91~~ &~~0.403~~ & ~~0.590~~ & ~~10567.13~~\\\\\n                       & $E'$   & ~~-39.87~~  & ~~-69.45~~  &~~-0.158~~ & ~~-~~ & ~~10643.87~~\\\\\n\\hline\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\\begin{table}[htbp]\n\\caption{The $B \\bar{B}^{*}$ system with $I^G=0^-$, $J^{PC}=1^{+-}$\n(in unit of MeV). The other notations are the same as in Table\n\\ref{tab:X3872+}.} \\label{tab:BBstar-}\n\\begin{center}\n\\begin{tabular}{c c c c c c| c }\n\\hline \\hline \\multirow{2}{*}{~$\\Lambda (GeV)$~} & \\multirow{2}{*}{~}  &\\multicolumn{4}{c|}{Eigenvalue}  & {Mass}  \\\\\n\\cline{2-6}\n                                &   &total&   S  &D  &  LS &($MeV$) \\\\\n\\hline\\hline\n\\multirow{2}{*}{1.0}  & $E$     &  ~~-0.290~~  & ~~-0.042~~ & ~~-0.568~~ & ~~0.023~~& ~~10603.710~~\\\\\n                      & $E'$    &  ~~-0.290~~  & ~~-0.049~~ &~~-0.584~~  & ~~-~~& ~~10603.710~~\\\\\n\\hline\\hline\n\\multirow{2}{*}{1.05}  & $E$    &  ~~-1.838~~   & ~~-0.502~~ &~~-1.832~~ & ~~0.132~~& ~~10602.162~~\\\\\n                      & $E'$    &  ~~-1.841~~   & ~~-0.548~~ &~~-1.936~~ & ~~-~~& ~~10602.159~~\\\\\n\\hline\\hline\n\\multirow{2}{*}{1.1}  & $E$    &  ~~-5.388~~  & ~~-1.992~~ &~~-3.794~~ & ~~0.409~~& ~~10598.612~~\\\\\n                      & $E'$   &  ~~-5.439~~  & ~~-2.145~~ &~~-4.135~~ & ~~-~~& ~~10598.561~~\\\\\n\\hline\\hline\n\\multirow{2}{*}{1.15}  & $E$    &  ~~-11.60~~  & ~~-5.079~~ &~~-6.447~~ & ~~0.939~~& ~~10592.40~~\\\\\n                       & $E'$   &  ~~-11.81~~  & ~~-5.451~~ &~~-7.256~~ & ~~-~~& ~~10592.19~~\\\\\n\\hline\\hline\n\\multirow{2}{*}{1.2}   & $E$    &  ~~-21.02~~  & ~~-10.24~~ &~~9.779~~  & ~~1.798~~& ~~10582.98~~\\\\\n                       & $E'$   &  ~~-21.59~~  & ~~-10.99~~ &~~-11.37~~ & ~~-~~& ~~10582.41~~\\\\\n\\hline\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\\begin{figure}[ht]\n  \\begin{center}\n  \\rotatebox{0}{\\includegraphics*[width=0.38\\textwidth]{BI11+-.eps}}\n    \\rotatebox{0}{\\includegraphics*[width=0.38\\textwidth]{BI11++.eps}}\n  \\rotatebox{0}{\\includegraphics*[width=0.38\\textwidth]{BI01++.eps}}\n    \\rotatebox{0}{\\includegraphics*[width=0.38\\textwidth]{BI01+-.eps}}\n    \\caption{ The effective potential of the $B \\bar{B}^{*}$ system.\n    Notations are the same as in Fig. \\ref{D-potential}.\n    }\n    \\label{B-potential}\n  \\end{center}\n\\end{figure}\n\n\\begin{figure}[ht]\n  \\begin{center}\n  \\rotatebox{0}{\\includegraphics*[width=0.38\\textwidth]{BI11+-meson.eps}}\n    \\rotatebox{0}{\\includegraphics*[width=0.38\\textwidth]{BI11++meson.eps}}\n  \\rotatebox{0}{\\includegraphics*[width=0.38\\textwidth]{BI01++meson.eps}}\n    \\rotatebox{0}{\\includegraphics*[width=0.38\\textwidth]{BI01+-meson.eps}}\n    \\caption{The effective potential from the different meson exchange\n   in the $B \\bar{B}^{*}$ system. Labels A,B,C,D are the same as in\n    Fig. \\ref{D-potential}.}\n    \\label{B-potential-meson}\n  \\end{center}\n\\end{figure}\n\n\n\\section{Summary and Discussion}\\label{summary}\n\nIn the framework of the one boson exchange model, we have calculated\nthe effective potentials between two heavy mesons from the t- and\nu-channel $\\pi$, $\\eta$, $\\rho$, $\\omega$ and $\\sigma$ meson\nexchange. We keep the recoil corrections to the $B \\bar{B}^{*}$ and\n$D \\bar{D}^{*}$ system up to $O(\\frac{1}{M^2})$. We also keep terms\nrelated to $\\vec{k}=\\frac{1}{2}(\\vec{p'}+\\vec{p})$, which is the sum\nof the initial and final momentum of the system. Especially, the\nspin orbit force appears at $O(\\frac{1}{M})$, which turns out to be\nimportant for the very loosely bound molecular states.\n\nWe have carefully investigated the $B \\bar{B}^{*}$ and $D\n\\bar{D}^{*}$ systems with four kinds of quantum number: $I=0$,\n$J^{PC}=1^{++}$; $I=0$, $J^{PC}=1^{+-}$; $I=1$, $J^{PC}=1^{++}$;\n$I=1$, $J^{PC}=1^{+-}$.\n\nAfter solving the Schr$\\ddot{o}$dinger equation by the variation\nmethod, we notice that there exist two isoscalar $D \\bar{D}^{*}$\nmolecular states with $J^{PC}=1^{++}$ and $J^{PC}=1^{+-}$ with or\nwithout the momentum-related corrections. The first C-parity even\n$1^{++}$ state corresponds to $X(3872)$. In contrast, there exist\nloosely bound states in three channels for the $B \\bar{B}^{*}$\nsystem with the cutoff in a reasonable range.\n\nOur numerical results show that the momentum-related corrections are\nunfavorable to the formation of the loosely bound molecular states\nin the $I=0$, $J^{PC}=1^{++}$ and $I=1$, $J^{PC}=1^{+-}$ channels in\nthe $D \\bar{D}^{*}$ system. Especially the recoil corrections are\nquite large. For example, the recoil correction may be larger than\nthe binding energy of X(3872), which may partly force X(3872) to\nbecome a very shallow bound state. As expected, the recoil\ncorrection in the $D \\bar{D}^{*}$ system is much larger than that in\nthe $B \\bar{B}^{*}$ system.\n\nHowever, we are unable to find a bound state for the $D \\bar{D}^{*}$\nsystem with $I=1$, $J^{PC}=1^{++}$ and $I=1$, $J^{PC}=1^{+-}$ with\nthe pionic coupling $g=0.59 $ which was extracted from the $D^*$\ndecay width and plays a dominant role in the effective potential,\nalthough we have systematically included the S-D wave mixing effect,\nthe spin orbit force and all the other recoil corrections up to\n$O(\\frac{1}{M^2})$. A loosely bound state appears if we increase $g$\nmanually by a factor of $1.6\\sim 1.8$ after the inclusion of the\nrecoil corrections.\n\nIt seems that it's not so easy to accommodate the newly observed\ncharged resonance $Z_c(3900)$ as the candidate of the isovector\nmolecular state of $D \\bar{D}^{*}$. The present investigation shows\nthat the recoil corrections may diminish the binding energy by one\nto several MeV and are unfavorable to the formation of loosely bound\nmolecular states in this channel. Experimentally the mass\n$Z_c(3900)$ seems above the $D \\bar{D}^{*}$ threshold. Our analysis\nshows that there does exist attraction in this channel. One may\nwonder whether $Z_c(3900)$ is a candidate of the molecular-type\nresonance instead of a $D \\bar{D}^{*}$ bound state.\n\nOn the other hand, we should also inspect the framework of the one\nboson exchange model. One obvious uncertainty arises from the cutoff\nparameter, which is commonly used to suppress the high momentum\ncontribution. Moreover, in the derivation of the effective\npotential, we make Fourier transformation to the effective potential\nin the momentum space to derive the potential in the coordinate\nspace. In case of the $D \\bar{D}^{*}$ system, the mass splitting\nbetween $D $ and $ \\bar{D}^{*}$ is larger than the pion mass. Hence\nthe integral contains an imaginary part. The commonly used approach\nis to take the principal value of this integral and omit the\nimaginary part in order to ensure the effective potential and\nHamiltonian to be real. The resulting potential is oscillating. The\nreliability of such a formalism deserves further investigation.\n\nIn short, the XYZ states provide a unique platform to study the\ncomplicated low energy strong dynamics. The charmonium (or Upsilon)\nspectrum above the open charm (or bottom) threshold and those\ncharmonium-like XYZ states as non-conventional candidates are\nparticularly interesting. In order to interpret their underlying\nstructures, we need also investigate their decay pattern and\nproduction mechanisms.\n\n\\section{acknowledgement}\nWe thank Li-Ping Sun, Zhi-Feng Sun and Li Ning for useful\ndiscussions. This project is supported by the National Natural\nScience Foundation of China under Grant No. 11261130311.\n\n\n\n\\section{Appendix}\n\nWe collect the expressions of the functions used in the previous\nsections in the appendix.\n\\begin{equation}\nY(\\tilde{m}_{\\alpha}r)=\\frac{\\exp(\\tilde{m}_{\\alpha}r)}{\\tilde{m}_{\\alpha}r}\n\\end{equation}\n\\begin{equation}\nZ(\\tilde{m}_{\\alpha}r)=(1+\\frac{3}{\\tilde{m}_{\\alpha}r}+\\frac{3}{(\\tilde{m}_{\\alpha}r)^2})Y(\\tilde{m}_{\\alpha}r)\n\\end{equation}\n\\begin{equation}\nZ_1(\\tilde{m}_{\\alpha}r)=(\\frac{1}{\\tilde{m}_{\\alpha}r}+\\frac{1}{(\\tilde{m}_{\\alpha}r)^2})Y(\\tilde{m}_{\\alpha}r)\n\\end{equation}\n\\begin{equation}\nZ'(\\tilde{m}_{\\alpha}r)=\\frac{\\sin(\\tilde{m}_{\\alpha}r)}{\\tilde{m}_{\\alpha}r}-\\frac{3}{\\tilde{m}_{\\alpha}r}\\frac{\\sin(\\tilde{m}_{\\alpha}r)}{\\tilde{m}_{\\alpha}r}+\\frac{1}{(\\tilde{m}_{\\alpha}r)^2}\\frac{\\cos(\\tilde{m}_{\\alpha}r)}{\\tilde{m}_{\\alpha}r}.\n\\end{equation}\n\\begin{equation}\nZ'_1(\\tilde{m}_{\\alpha}r)=\\frac{1}{\\tilde{m}_{\\alpha}r}\\frac{\\sin(\\tilde{m}_{\\alpha}r)}{\\tilde{m}_{\\alpha}r}+\\frac{1}{(\\tilde{m}_{\\alpha}r)^2}\\frac{\\cos(\\tilde{m}_{\\alpha}r)}{\\tilde{m}_{\\alpha}r}\n\\end{equation}\nwhere for the $D \\bar{D}^*$ system\n\\begin{equation}\n\\tilde{m}^2_{\\pi}=(m_D^*-m_D)^2-m^2_{\\pi},\n\\end{equation}\n\\begin{equation}\n\\tilde{m}^2_{\\sigma,\\rho,\\omega,\\eta}=m^2_{\\sigma,\\rho,\\omega,\\eta}-(m_D^*-m_D)^2.\n\\end{equation}\nwhille for the $B \\bar{B}^*$ system\n\\begin{equation}\n\\tilde{m}^2_{\\pi,\\sigma,\\rho,\\omega,\\eta}=m^2_{\\pi,\\sigma,\\rho,\\omega,\\eta}-(m_B^*-m_B)^2.\n\\end{equation}\n\\begin{eqnarray}\n\\mathcal{F}_{1t\\alpha}&=&\\mathcal{F}\\{(\\frac{\\Lambda^2-m_{\\alpha}^2}{{\\Lambda}^2+\\vec{q}^2})\n\\frac{1}{\\vec{q}^2+m_{\\alpha}^2}\\}\\nonumber\\\\\n&=&m_{\\alpha}Y(m_{\\alpha}r)-\\Lambda Y(\\Lambda r)-\n(\\Lambda^2-m_{\\alpha}^2)\\frac{e^{-\\Lambda r}}{2\\Lambda}\n\\end{eqnarray}\n\\begin{eqnarray}\n\\mathcal{F}_{1u\\alpha}&=&\\mathcal{F}\\{(\\frac{\\Lambda^2-m_{\\alpha}^2}{\\tilde{\\Lambda}^2+\\vec{q}^2})\n\\frac{1}{\\vec{q}^2+\\tilde{m}_{\\alpha}^2}\\}\\nonumber\\\\\n&=&\\tilde{m}_{\\alpha}Y(\\tilde{m}_{\\alpha}r)-\\tilde{\\Lambda}\nY(\\tilde{\\Lambda}\nr)-(\\Lambda^2-m_{\\alpha}^2)\\frac{e^{-\\tilde{\\Lambda}\nr}}{2\\tilde{\\Lambda}}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\mathcal{F}_{2t\\alpha}&=&\\mathcal{F}\\{(\\frac{\\Lambda^2-m_{\\alpha}^2}{{\\Lambda}^2+\\vec{q}^2})\n\\frac{\\vec{q}^2}{\\vec{q}^2+m_{\\alpha}^2}\\}\\nonumber\\\\\n&=&m_{\\alpha}^2[\\Lambda Y(\\Lambda r)-m_{\\alpha}Y(m_{\\alpha}r)]\\nonumber\\\\\n&+&(\\Lambda^2-m_{\\alpha}^2)\\Lambda\\frac{e^{-\\Lambda r}}{2}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\mathcal{F}_{2u\\alpha}&=&\\mathcal{F}\\{(\\frac{\\Lambda^2-m_{\\alpha}^2}{{\\tilde{\\Lambda}}^2+\\vec{q}^2})\n\\frac{\\vec{q}^2}{\\vec{q}^2+\\tilde{m}_{\\alpha}^2}\\}\\nonumber\\\\\n&=&\\tilde{m}_{\\alpha}^2[\\tilde{\\Lambda} Y(\\tilde{\\Lambda} r)-\\tilde{m}_{\\alpha}Y(\\tilde{m}_{\\alpha}r)]\\nonumber\\\\\n&+&(\\Lambda^2-m_{\\alpha}^2)\\tilde{\\Lambda}\\frac{e^{-\\tilde{\\Lambda}\nr}}{2}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\mathcal{F}_{3t\\alpha} &=&\n\\mathcal{F}\\{(\\frac{\\Lambda^2-m_{\\alpha}^2}{{\\Lambda}^2+\\vec{q}^2})\n\\frac{(\\vec{\\sigma_1}\\cdot\\vec{q})(\\vec{\\sigma_2}\\cdot\\vec{q})}{\\vec{p}^2+m_{\\alpha}^2}\\}\\nonumber\\\\\n&=&\\frac{1}{3}\\vec{\\sigma_1}\\cdot\\vec{\\sigma_2}[~m_{\\alpha}^2\\Lambda Y(\\Lambda r)-m_{\\alpha}^3 Y(m_{\\alpha}r)\\nonumber\\\\\n&+&(\\Lambda^2-m_{\\alpha}^2)\\Lambda\\frac{e^{-\\Lambda r}}{2}~]\\nonumber\\\\\n&+&\\frac{1}{3}S_{12}[-m_{\\alpha}^3 Z(m_{\\alpha}r)+ \\Lambda^3 Z(\\Lambda r) \\nonumber\\\\\n&+&(\\Lambda^2-m_{\\alpha}^2)(1+\\Lambda r)\n\\frac{\\Lambda}{2}Y(\\Lambda r)]\\nonumber\\\\\n&=&(\\vec{\\sigma_1}\\cdot\\vec{\\sigma_2})\\mathcal{F}_{3t1} +\nS_{12}\\mathcal{F}_{3t2}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\mathcal{F}_{3u\\alpha} &=&\n\\mathcal{F}\\{(\\frac{\\Lambda^2-m_{\\alpha}^2}{{\\tilde{\\Lambda}}^2+\\vec{q}^2})\n\\frac{(\\vec{\\sigma_1}\\cdot\\vec{q})(\\vec{\\sigma_2}\\cdot\\vec{q})}{\\vec{q}^2+\\tilde{m}_{\\alpha}^2}\\}\\nonumber\\\\\n&=&\\frac{1}{3}\\vec{\\sigma_1}\\cdot\\vec{\\sigma_2}[\\tilde{m}_{\\alpha}^2\\tilde{\\Lambda} Y(\\tilde{\\Lambda} r)-\\tilde{m}_{\\alpha}^3Y(\\tilde{m}_{\\alpha}r)\\nonumber\\\\\n&+&(\\Lambda^2-m_{\\alpha}^2)\\tilde{\\Lambda}\\frac{e^{-\\tilde{\\Lambda} r}}{2}]\\nonumber\\\\\n&+&\\frac{1}{3}S_{12}[-\\tilde{m_{\\alpha}}^3 Z(\\tilde{m_{\\alpha}}r)+ \\tilde{\\Lambda}^3 Z(\\tilde{\\Lambda} r) \\nonumber\\\\\n&+&(\\Lambda^2-m_{\\alpha}^2)(1+\\tilde{\\Lambda} r)\\frac{\\tilde{\\Lambda}}{2}Y(\\tilde{\\Lambda} r)~]\\nonumber\\\\\n&=&(\\vec{\\sigma_1}\\cdot\\vec{\\sigma_2})\\mathcal{F}_{3u1\\alpha} +\nS_{12}\\mathcal{F}_{3u2\\alpha}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\mathcal{F'}_{3u\\alpha}&=&\n\\mathcal{F}\\{(\\frac{\\Lambda^2-m_{\\alpha}^2}{{\\tilde{\\Lambda}}^2+\\vec{q}^2})\n\\frac{(\\vec{\\sigma_1}\\cdot\\vec{q})(\\vec{\\sigma_2}\\cdot\\vec{q})}{\\vec{p}^2-\\tilde{m_{\\alpha}}^2}\\}\\nonumber\\\\\n&=&\\frac{1}{3}\\vec{\\sigma_1}\\cdot\\vec{\\sigma_2}[~-\\tilde{m}_{\\alpha}^2\\tilde{\\Lambda} Y(\\tilde{\\Lambda} r)-\\tilde{m}_{\\alpha}^3\\frac{\\cos(\\tilde{m}_{\\alpha}r)}{\\tilde{m}_{\\alpha}r}\\nonumber\\\\\n&+&(\\Lambda^2-m_{\\alpha}^2)\\tilde{\\Lambda}\\frac{e^{-\\tilde{\\Lambda} r}}{2}~]\\nonumber\\\\\n&+&\\frac{1}{3}S_{12}[\\tilde{m}_{\\alpha}^3 Z'(\\tilde{m}_{\\alpha}r)+ \\tilde{\\Lambda}^3 Z(\\tilde{\\Lambda} r)\\nonumber\\\\\n&+&(\\Lambda^2-m_{\\alpha}^2)(1+\\tilde{\\Lambda} r)\\frac{\\tilde{\\Lambda}}{2}Y(\\tilde{\\Lambda} r)~]\\nonumber\\\\\n&=&(\\vec{\\sigma_1}\\cdot\\vec{\\sigma_2})\\mathcal{F'}_{3u1\\alpha} +\nS_{12}\\mathcal{F'}_{3u2\\alpha}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\mathcal{F}_{4t\\alpha}\n&=&\\mathcal{F}\\{{(\\frac{\\Lambda^2-m_{\\alpha}^2}{{\\Lambda}^2+\\vec{q}^2})\n\\frac{\\vec{k}^2}{\\vec{q}^2+m_{\\alpha}^2}}\\}\\nonumber\\\\\n&=&\\frac{m_{\\alpha}^3}{4}Y(m_{\\alpha}r)-\\frac{\\Lambda^3}{4}Y(\\Lambda r)\\nonumber\\\\\n&-&\\frac{\\Lambda^2-m_{\\alpha}^2}{4}(\\frac{\\Lambda r}{2}-1)\\frac{e^{-\\Lambda r}}{r}\\nonumber\\\\\n&-&\\frac{1}{2}\\{\\nabla^2,m_{\\alpha} Y(m_{\\alpha}r)-\\Lambda Y(\\Lambda r)-\\frac{\\Lambda^2-m_{\\alpha}^2}{2}\\frac{e^{-\\Lambda r}}{\\Lambda}\\}\\nonumber\\\\\n&=&\\mathcal{F}_{4t1\\alpha}+\\{-\\frac{1}{2}\\nabla^2,\\mathcal{F}_{4t2\\alpha}\\}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\mathcal{F}_{4u\\alpha}\n&=&\\mathcal{F}\\{{(\\frac{\\Lambda^2-\\tilde{m}_{\\alpha}^2}{{\\tilde{\\Lambda}}^2+\\vec{q}^2})\n\\frac{\\vec{k}^2}{\\vec{q}^2+\\tilde{m}_{\\alpha}^2}}\\}\\nonumber\\\\\n&=&\\frac{\\tilde{m}_{\\alpha}^3}{4}Y(\\tilde{m}_{\\alpha}r)-\\frac{\\tilde{\\Lambda}^3}{4}Y(\\tilde{\\Lambda} r)\\nonumber\\\\\n&-&\\frac{\\Lambda^2-m_{\\alpha}^2}{4}(\\frac{\\tilde{\\Lambda} r}{2}-1)\\frac{e^{-\\tilde{\\Lambda} r}}{r}\\nonumber\\\\\n&-&\\frac{1}{2}\\{\\nabla^2,\\tilde{m}_{\\alpha}Y(\\tilde{m}_{\\alpha}r)-\\tilde{\\Lambda}\nY(\\tilde{\\Lambda} r)-\\frac{\\Lambda^2-m_{\\alpha}^2}{2}\n\\frac{e^{-\\tilde{\\Lambda} r}}{\\tilde{\\Lambda}}\\}\\nonumber\\\\\n&=&\\mathcal{F}_{4u1\\alpha}+\\{-\\frac{1}{2}\\nabla^2,\\mathcal{F}_{4u2\\alpha}\\}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\mathcal{F}_{5t\\alpha}\n&=&\\mathcal{F}\\{{i(\\frac{\\Lambda^2-m_{\\alpha}^2}{{\\Lambda}^2+\\vec{q}^2})\n\\frac{\\vec{S}\\cdot(\\vec{q}\\times\\vec{k})}{\\vec{q}^2+m_{\\alpha}^2}}\\}\\nonumber\\\\\n&=&\\vec{S}\\cdot\\vec{L}[-m_{\\alpha}^3 Z_{1}(m_{\\alpha}r)+\\Lambda^3 Z_{1}(\\Lambda r)\\nonumber\\\\\n&+&(\\Lambda^2-m_{\\alpha}^2)\\frac{e^{-\\Lambda r}}{2r}]\\nonumber\\\\\n&=&\\vec{S}\\cdot\\vec{L}\\mathcal{F}_{5t0\\alpha}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\mathcal{F}_{5u\\alpha}\n&=&\\mathcal{F}\\{{i(\\frac{\\Lambda^2-m_{\\alpha}^2}{{\\tilde{\\Lambda}}^2+\\vec{q}^2})\n\\frac{\\vec{S}\\cdot(\\vec{q}\\times\\vec{k})}{\\vec{q}^2+\\tilde{m}_{\\alpha}^2}}\\}\\nonumber\\\\\n&=&\\vec{S}\\cdot\\vec{L}[-\\tilde{m}_{\\alpha}^3 Z_{1}(\\tilde{m}_{\\alpha}r)+\\tilde{\\Lambda}^3 Z_{1}(\\tilde{\\Lambda} r)\\nonumber\\\\\n&+&(\\Lambda^2-m_{\\alpha}^2)\\frac{e^{-\\tilde{\\Lambda} r}}{2r}]\\nonumber\\\\\n&=&\\vec{S}\\cdot\\vec{L}\\mathcal{F}_{5u0\\alpha}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\mathcal{F'}_{5u\\alpha}\n&=&\\mathcal{F}\\{{i(\\frac{\\Lambda^2-m_{\\alpha}^2}{{\\tilde{\\Lambda}}^2+\\vec{q}^2})\n\\frac{\\vec{S}\\cdot(\\vec{q}\\times\\vec{k})}{\\vec{q}^2+\\tilde{m_{\\alpha}}^2}}\\}\\nonumber\\\\\n&=&\\vec{S}\\cdot\\vec{L}[-\\tilde{m}_{\\alpha}^3 Z'_{1}(\\tilde{m}_{\\alpha}r)+\\tilde{\\Lambda}^3 Z_{1}(\\tilde{\\Lambda} r)\\nonumber\\\\\n&+&(\\Lambda^2-m_{\\alpha}^2)\\frac{e^{-\\tilde{\\Lambda} r}}{2r}]\\nonumber\\\\\n&=&\\vec{S}\\cdot\\vec{L}\\mathcal{F'}_{5u0\\alpha}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\mathcal{F}_{6u\\alpha}&=&\n\\mathcal{F}\\{(\\frac{\\Lambda^2-m_{\\alpha}^2}{{\\tilde{\\Lambda}}^2+\\vec{q}^2})\n\\frac{(\\vec{\\sigma_1}\\cdot\\vec{k})(\\vec{\\sigma_2}\\cdot\\vec{k})}{\\vec{p}^2+\\tilde{m}_{\\alpha}^2}\\}\\nonumber\\\\\n&=&-\\frac{\\vec{\\sigma_1}\\cdot\\vec{\\sigma_2}}{4}[~\\tilde{m}_{\\alpha}^3Y(\\tilde{m}_{\\alpha}r)-(\\tilde{\\Lambda})^3Y(\\tilde{\\Lambda}r)\\nonumber\\\\\n&-&(\\Lambda^2-m_{\\alpha}^2)\\tilde{\\Lambda}\\frac{e^{-\\tilde{\\Lambda} r}}{2}~]\\nonumber\\\\\n&+&\\frac{1}{3}(S_{12}+\\vec{\\sigma_1}\\cdot\\vec{\\sigma_2})[~(1+\\frac{3}{\\tilde{m}_{\\alpha}r})\\tilde{m}_{\\alpha}^2 Y(\\tilde{\\Lambda} r)\\nonumber\\\\\n&-&(1+\\frac{3}{\\tilde{\\Lambda}r})(\\tilde{\\Lambda})^2 Y(\\tilde{\\Lambda} r)\\nonumber\\\\\n&-&(\\Lambda^2-m_{\\alpha}^2)(\\tilde{\\Lambda}+\\frac{2}{r})\\frac{e^{-\\tilde{\\Lambda} r}}{2\\tilde{\\Lambda}}~]\\nabla\\nonumber\\\\\n&-&\\frac{1}{3}(S_{12}+\\vec{\\sigma_1}\\cdot\\vec{\\sigma_2})[\\tilde{m}_{\\alpha} Y(\\tilde{m}_{\\alpha}r)- \\tilde{\\Lambda}Y(\\tilde{\\Lambda} r)\\nonumber\\\\\n&-&(\\Lambda^2-m_{\\alpha}^2)\\frac{e^{-\\tilde{\\Lambda} r}}{2\\tilde{\\Lambda}}~]\\nabla^2\\nonumber\\\\\n&=&-\\frac{\\vec{\\sigma_1}\\cdot\\vec{\\sigma_2}}{4}\\mathcal{F}_{6u1\\alpha}\n+\\frac{1}{3}(S_{12}+\\vec{\\sigma_1}\\cdot\\vec{\\sigma_2})\\mathcal{F}_{6u2\\alpha}\\nonumber\\\\\n&-&\\frac{1}{3}(S_{12}+\\vec{\\sigma_1}\\cdot\\vec{\\sigma_2})\\mathcal{F}_{6u3\\alpha}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\mathcal{F'}_{6u\\alpha}&=&\n\\mathcal{F}\\{(\\frac{\\Lambda^2-m_{\\alpha}^2}{{\\tilde{\\Lambda}}^2+\\vec{q}^2})\n\\frac{(\\vec{\\sigma_1}\\cdot\\vec{k})(\\vec{\\sigma_2}\\cdot\\vec{k})}{\\vec{p}^2-\\tilde{m}_{\\alpha}^2}\\}\\nonumber\\\\\n&=&-\\frac{\\vec{\\sigma_1}\\cdot\\vec{\\sigma_2}}{4}[~\\tilde{m}_{\\alpha}^3 \\frac{\\cos(M_{\\alpha}r)}{\\tilde{m}_{\\alpha}r}-(\\tilde{\\Lambda})^3Y(\\tilde{\\Lambda}r)\\nonumber\\\\\n&-&(\\Lambda^2-m_{\\alpha}^2)\\tilde{\\Lambda}\\frac{e^{-\\tilde{\\Lambda} r}}{2}~]\\nonumber\\\\\n&+&\\frac{1}{3}(S_{12}+\\vec{\\sigma_1}\\cdot\\vec{\\sigma_2})[~(\\frac{\\sin(\\tilde{m}_{\\alpha}r)}{\\tilde{m}_{\\alpha}r}+\\frac{3}{\\tilde{m}_{\\alpha}r}\\frac{\\cos(\\tilde{m}_{\\alpha}r)}\n{\\tilde{m}_{\\alpha}r})\\tilde{m}_{\\alpha}^2\\nonumber\\\\\n&-&(1+\\frac{3}{\\tilde{\\Lambda}r})(\\tilde{\\Lambda})^2\nY(\\tilde{\\Lambda} r)\n-(\\Lambda^2-m_{\\alpha}^2)(\\tilde{\\Lambda}+\\frac{2}{r})\\frac{e^{-\\tilde{\\Lambda} r}}{2\\tilde{\\Lambda}}~]\\nabla\\nonumber\\\\\n&-&\\frac{1}{3}(S_{12}+\\vec{\\sigma_1}\\cdot\\vec{\\sigma_2})[\\tilde{m}_{\\alpha} \\frac{\\cos(\\tilde{m}_{\\alpha}r)}{\\tilde{m}_{\\alpha}r}- \\tilde{\\Lambda}Y(\\tilde{\\Lambda} r)\\nonumber\\\\\n&-&(\\Lambda^2-m_{\\alpha}^2)\\frac{e^{-\\tilde{\\Lambda} r}}{2\\tilde{\\Lambda}}~]\\nabla^2\\nonumber\\\\\n&=&-\\frac{\\vec{\\sigma_1}\\cdot\\vec{\\sigma_2}}{4}\\mathcal{F'}_{6u1\\alpha}\n+\\frac{1}{3}(S_{12}+\\vec{\\sigma_1}\\cdot\\vec{\\sigma_2})\\mathcal{F'}_{6u2\\alpha}\\nonumber\\\\\n&-&\\frac{1}{3}(S_{12}+\\vec{\\sigma_1}\\cdot\\vec{\\sigma_2})\\mathcal{F'}_{6u3\\alpha}\n\\end{eqnarray}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nCommon approaches to studying ``brain rhythms\" can be broadly divided in two categories. The first is based\non correlating the wave's instantaneous phases, amplitudes and frequencies with parameters of cognitive, \nbehavioral or neuronal processes. For example, instantaneous phases of $\\theta$-wave were found to modulate\nneuronal spikings \\cite{Skaggs,Benchenane}, while $\\gamma$-waves' amplitudes and frequencies link the synaptic\n\\cite{Nikoli,ClgMsr,ColginGm} and circuit \\cite{LisBuz,Lismn1,Lismn3} current flow to learning dynamics. \nThe second category of analyses is based on quantifying the brain waves' time-averaged characteristics, e.g.,\nestablishing dependencies between the mean $\\theta$-frequency and the animal's speed \\cite{Richard,BuzTheta2}\nor acceleration \\cite{Kropff} or linking rising mean $\\theta$- and $\\gamma$-power to heightened attention \nstates \\cite{Rangel,Kropff} and so forth.\n\nHowever, little work has been done to examine the waves' overall shape, e.g., the temporal arrangement of \npeaks and troughs, or sequences of ripples or spindles generated over finite periods. Yet, the physiological\nrelevance of the brain wave morphology is well recognized: rigidly periodic or excessively irregular rhythms\nthat contravene a certain ``natural\" level of statistical variability are suggestive of circuit pathologies\n\\cite{Wilkinson,Donoghue,Eissa,Nicola,Hawkins,Blanco,Cole} or may indicate external driving \\cite{Mostafa,Will}.\nFor example, the nearly periodic sequence of peaks shown on Fig.~\\ref{fig:pat}A is common for $\\theta$-waves,\nbut certainly too orderly for the $\\gamma$-waves. Conversely, the intermittent pattern on Fig.~\\ref{fig:pat}B\nis unlikely to appear among $\\theta$-waves, but may represent irregular $\\gamma$-activity or typical sharp waves.\nOn the other hand, the series of clumping peaks shown on Fig.~\\ref{fig:pat}C seem usual for in $\\gamma$ waves,\nbut the cluttered pattern on Fig.~\\ref{fig:pat}D may be a manifestation of a particular process. In contrast,\nthe waveforms shown on Fig.~\\ref{fig:pat}E,F appear to represent a mundane level of arhythmicity and temporal \ndisorder expected in most $\\theta$ and $\\gamma$ waves.\n\n\\begin{figure}[h]\n\n\t\\centering\n\t\\includegraphics[scale=0.84]{Patterns}\n\t\\caption{{\\footnotesize\n\t\t\t\\textbf{Waveform morphologies}. \\textbf{A}. A wave exhibiting a nearly periodic sequence of peak is\n\t\t\tcommonly found among $\\theta$-oscillations, but rarely in higher-frequency waves. Conversely, the\n\t\t\tintermittent patterns shown on panel \\textbf{B} may be exhibited by fast $\\gamma$ or sharp waves,\n\t\t\tbut is atypical for lower-frequencies. The temporal clustering shown on panel \\textbf{C} is not \n\t\t\tunusual for the $\\gamma$-wave, but the uneven pattern on panel \\textbf{D} would be an irregularity.\n\t\t\tIn contrast, the patterns shown on panels \\textbf{E} and \\textbf{F} are all in all ordinary and\n\t\t\tseem to fluctuate with a fixed oscillatory rate. Shown are the peaks exceeding one standard deviation\n\t\t\tfrom the mean.\n\t}}\n\t\\label{fig:pat}\n\n\\end{figure}\n\nIt remains unclear however, how to identify these patterns impartially, how to quantify the intuitive notions \nof ``regularity,\" ``typicality,\" ``orderliness,\" etc. Furthermore, since all brain waves exhibit a certain \nlevel of erraticness, it is unclear how justified are the experiential, visceral classifications of the \nwaveforms. For example, might the ``improbable\" patterns illustrated above be attributed to mere fluctuations\nof otherwise regular waves, or should they be viewed as a structural peculiarity? \n\nIn the following, we address these questions in the context of two independent mathematical frameworks, using\ntwo cognate pattern quantifications that allow understanding the brain rhythms' functional structure at\nintermediate timescales and their role in behavior and cognition.\n\n\\section{Approach}\n\\label{sec:app}\n\\textit{1. Kolmogorov stochasticity, $\\lambda$}, describes deviation of an ordered sequence, $X$, from the \noverall trend, and a remarkable observation made in \\cite{Kolmogorov} is that this score is universally \ndistributed. As it turns out, deviations $\\lambda(X)$ that are too high or too low are rare: sequences with\n$\\lambda(X)\\leq0.4$ or $\\lambda(X)\\geq 1.8$ appear with probability less than $0.3\\%$ (Fig.~\\ref{fig:stochs}A,B, \n\\cite{Stephens,Arnold1,Arnold2,Arnold3,Arnold4,Arnold5}).\nIn other words, typical patterns are consistent with the underlying mean behavior and produce a limited range\nof $\\lambda$-values, with mean $\\lambda^{\\ast}\\approx 0.87$. Thus, the $\\lambda$-score can serve as a universal\nmeasure of \\textit{stochasticity}\\footnote{Throughout the text, terminological definitions and highlights are \n\tgiven in \\textit{italics}.} and be used for identifying statistical biases (or lack of thereof) in various \npatterns \\cite{KolMen,Stark,Gurzadyan1,Gurzadyan2,Brandouy,Ford}.\n\n\\textit{2. Arnold stochasticity, $\\beta$}, is alternative measure that quantifies whether the elements of a\npattern ``repel''  or ``attract\" each other. Repelling elements seek to maximize separations and hence produce\norderly, more equispaced arrangements, while attracting elements tend to cluster together. As shown\nin \\cite{ArnoldB1,ArnoldB2,ArnoldB3,ArnoldB4}, for ordered patterns $\\beta(X)\\approx1$, for clustering ones\n$\\beta(X)$ can be high, while sequences with independent elements yield $\\beta$-values close to the impartial \nmean, $\\beta^{\\ast}\\approx2$ (Fig.~\\ref{fig:stochs}C, Suppl. Sec.~\\ref{sec:met}). Thus, the $\\beta$-score can\nbe used to characterize \\textit{orderliness} of brain rhythms \\cite{ArnoldB1,ArnoldB2,ArnoldB3,ArnoldB4}, \ncomplementing the $\\lambda$-score. \n\n\\begin{figure}[h]\n\n\t\\centering\n\t\\includegraphics[scale=0.84]{StochasticityParams}\n\t\\caption{{\\footnotesize\n\t\t\t\\textbf{Stochasticity parameters}. \\textbf{A}. The elements of an ordered sequence $X=\\{x_1,x_2,\n\t\t\t\\ldots,x_n\\}$ following a linear trend $\\bar{N}(x)=mx+b$ (dashed line). The sequence's deviations\n\t\t\tfrom the mean, $\\lambda(X)$, exhibit statistical universality and can hence impartially \n\t\t\tcharacterize the stochasticity of the individual data sequences $X$.\n\t\t\t\\textbf{B}. The probability distribution of $\\lambda$-scores is unimodal, with mean $\\lambda^{\\ast}\n\t\t\t\\approx 0.87$ (red dot).\tAbout $99.7\\%$ of sequences produce $\\lambda$-scores in\tthe interval\n\t\t\t$0.4\\leq\\lambda(X)\\leq 1.8$ (pink stripe); these sequences are typical and consistent with the \n\t\t\tunderlying mean behavior. In contrast, sequences with smaller or larger $\\lambda$-scores are \n\t\t\tstatistically uncommon. \n\t\t\t\\textbf{C}. A sequence $X$ arranged on a circle of length $L$ produces a set of $n$ arcs. The \n\t\t\tnormalized quadratic sum of the arc lengths is small for orderly sequences, $\\beta\\approx1$ (left),\n\t\t\tas high as $\\beta\\approx n$ for the ``clustering\" sequences (right), and intermediate, $\\beta\\approx\n\t\t\t2$ (middle), for generic sequences. \n\t}}\n\t\\label{fig:stochs}\n\n\\end{figure}\n\n\\textit{3. Time-dependence}. The recurrent nature of brain rhythms suggests dynamic generalization of $\\lambda$ \nand $\\beta$. Given a time window, $L$, containing a sequence of events, $X_t$, such as $\\theta$-peaks or Sharp \nWave Ripples (SWR), evaluate the parameters $\\lambda(X_t)$ and $\\beta(X_t)$, then shift the window over a time\nstep $\\Delta t$, evaluate the next $\\lambda(X_{t+\\Delta t})$ and $\\beta(X_{t+\\Delta t})$, and so on. The \nconsecutive segments, obtained by small window shifts, $X_t, X_{t+\\Delta t}, X_{t+2\\Delta t},\\ldots$, differ \nonly slightly from one another. The resulting time-dependencies $\\lambda(t)$ and $\\beta(t)$ will define the\ndynamics of stochasticity over the signal's entire span.\n\nFor a visualization, one can imagine the elements of a given sample sequence, $X_{t+k\\Delta t}$, as ``beads\" \nscattered over a necklace of length $L$. As the sliding window shifts forward in time, the beads shift back and\nmay disappear at the back of the window, and new beads may appear toward the front, while a majority of the beads\nretain their relative positions. The corresponding $\\lambda$- and $\\beta$-values will then produce semi-continuous\ntime-dependencies $\\lambda(t)$ and $\\beta(t)$ that quantify the ``necklace dynamics\"---gradual pattern changes.\nThe parameter $\\beta$ then describes the orderliness of the beads' distribution over the necklace, while\n$\\lambda$ measures how typical the beads' arrangement is overall. \n\n\\begin{figure}[h]\n\n\t\\centering\n\t\\includegraphics[scale=0.8]{RandStoch}\n\t\\caption{{\\footnotesize\n\t\t\t\\textbf{Pattern dynamics for three kinds of random sequences} in which the intervals between\n\t\t\tconsecutive points are distributed 1) exponentially with the rate $\\nu=2$; 2) uniformly\n\t\t\twith constant density $\\rho=1$; or 3) with Poisson rate $\\mu=5$. Sample intervals are selected\n\t\t\tproportionally to the distribution scales ($L_u=25\\rho$, $L_e=25\\nu$, and $L_p=25\\mu$, so that each\n\t\t\tsample sequence contains about $n=25$ elements) and are shifted by a single data point at a time. \n\t\t\t\\textbf{A}. The Kolmogorov parameter of the exponential sequence (red trace, $\\lambda_e$), uniform\n\t\t\tsequence (blue trace, $\\lambda_u$) and Poisson sequence (orange trace, $\\lambda_p$) remain mostly\n\t\t\twithin the ``pink zone\" of stochastic typicality (same pink stripe as on Fig.~\\ref{fig:stochs}B). \n\t\t\t$\\lambda_u$ is the most volatile and often escapes the expected range, whereas $\\lambda_p$ is more\n\t\t\tcompliant, lingering below the expected mean $\\lambda_p\\lesssim\\lambda^{\\ast}\\approx 0.87$ (black\n\t\t\tdashed line).\t\t\n\t\t\t\\textbf{B}. The corresponding Arnold stochasticity parameters show similar behavior: $\\beta_u=1.93\n\t\t\t\\pm 0.2$ fluctuates around the expected mean $\\beta^{\\ast}(25) = 1.92$ (black dotted line). The \n\t\t\texponential sequence has smaller $\\beta$-variations and a slightly higher mean, $\\beta_e=2\\pm 0.04$.\n\t\t\tThe Poisson sequence is the least stochastic (nearly-periodic), with $\\beta_p=1.22\\pm 0.004$, due \n\t\t\tto statistical suppression of small and large gaps.\n\t\t\t\\textbf{C}. The mean stochasticity scores computed for about $10^4$ random patterns of each type.}\n\t}\n\t\\label{fig:rndstoch}\n\n\\end{figure}\n\nAs an illustration, consider a data series $X$ with random spacings between the adjacent values---intervals\ndrawn from exponential, uniform, and Poisson distributions, with sample subsequences containing about $25$\nconsecutive elements. As shown on Fig.~\\ref{fig:rndstoch}A,C, the $\\lambda(t)$-dependence of the exponential\nsequences remains, for the most part, constrained within the ``typicality band\" (pink stripe on \nFigs.~\\ref{fig:stochs}B and Fig.~\\ref{fig:rndstoch}A), while the uniformly distributed patterns are more \nvariable and Poisson patterns follow the mean most closely. The $\\beta$-scores of exponentially and uniformly\ndistributed patterns are overall mundane, while the Poisson patterns exhibit periodic-like orderliness. \n\nThe mean $\\lambda$- and $\\beta$-scores in the uniform and the exponential sequences are close to universal\nmeans, $\\lambda^{\\ast}$ and $\\beta^{\\ast}$, which shows that, on average, they are statistically unbiased.\nIn contrast, the Poisson-distributed patterns are atypically orderly, due to statistically suppressed small\nand large gaps between neighboring elements (Fig.~\\ref{fig:rndstoch}B).\n\nThe fluctuations of stochasticity scores---the rises and drops of $\\lambda(t)$ and $\\beta(t)$ dependencies on\nFig.~\\ref{fig:rndstoch}---are chancy, since random sequences vary sporadically between instantiations. In \ncontrast, brain wave patterns may carry physiological information, and the dynamics of their stochasticity\nmay serve as an independent characterization of circuit activity at a mesoscopic timescale in different \nbehavioral and cognitive states, as discussed below.\n\n\\section{Results}\n\\label{sec:res}\n\n\\subsection{Stochasticity in time}\nWe analyzed Local Field Potentials (LFP) recorded in the hippocampal CA1 area of wild type male \nmice\\footnote{The data used in this work was outlined in \\cite{ChengJi}.} and studied their $\\theta$-wave,\n$\\gamma$-wave, and SWR patterns \\cite{ColginR}. The recurring nature of brain rhythms suggests that their key\nfeatures distribute uniformly over sufficiently long periods. Therefore, the expected mean used for evaluating\nthe Kolmogorov $\\lambda$-parameter is linear,\n\\begin{equation}\n\\bar{N}(t) = m t + b,\n\\label{lin}\n\\end{equation}\nwith the coefficients $m$ and $b$ obtained via linear regression. The lengths of the sample sequences \nwere then selected to highlight the specific wave's structure and functions, as described below.\n\n\\textbf{1. $\\theta$-waves} ($4-12$ Hz, \\cite{Burgess,BuzTheta1}) are known to correlate with the animal's motion\nstate, which suggests that the sample sequences should be selected at a behavioral scale \\cite{BuzTheta2,Richard,\nKropff}. In the analyzed experiments, the mice shuttled between two food wells on a U-shaped track, spending\nabout $22$ secs per lap (average for $5$ mice, for both inbound and outbound runs) and consumed food reward \nover $17$ secs (Fig.~\\ref{fig:thstoch}A). On the other hand, the intervals between successive $\\theta$-peaks \ndistribute around the characteristic $\\theta$-period, $\\overline{T}_{\\theta}\\approx 1\/m_{\\theta}\\approx 110$ \nmsecs, which defines the timescale of oscillatory dynamics (Fig.~\\ref{fig:thstoch}A). To accommodate both \ntimescales, we used periods required to complete $1\/6^\\textrm{th}$ of the run between the food wells, $L_{\\theta}\n\\approx 3.6$ secs, containing about $20-30$ peaks---large enough to produce stable $\\lambda$- and $\\beta$- scores\n\\cite{Bol1,Vrbik1,Vrbik2}, but short enough to capture the ongoing dynamics of $\\theta$-patterns.\n\n\\begin{figure}[h]\n\n\t\\centering\n\t\\includegraphics[scale=0.8]{ThetaTimeStoch}\n\t\\caption{{\\footnotesize\n\t\t\t\\textbf{$\\theta$-wave's stochasticity}. \n\t\t\t\\textbf{A}. A histogram of intervals between subsequent $\\theta$-peaks concentrates around the\n\t\t\tcharacteristic $\\theta$-period, $\\overline{T}_{\\theta}\\approx 110$ msec: gaps shorter than \n\t\t\t$\\overline{T}_{\\theta}\/2$ or wider than $2\\overline{T}_{\\theta}$ are rare. $\\theta$-amplitude, \n\t\t\t$\\tilde{\\theta}$, oscillates with $T_{\\tilde{\\theta}}\\approx 180$ msec period.\n\t\t\t\\textbf{B}. The animal's lapses (trajectory shown by gray line) between food wells, $F_1$ and \n\t\t\t$F_2$, take on average $22$ secs.\n\t\t\t\\textbf{C}. Due to quasiperiodicity of the $\\theta$-wave and of its envelope, $\\tilde{\\theta}$, \n\t\t\tthe average scores $\\langle\\lambda_{\\theta}\\rangle$, $\\langle\\tilde{\\lambda}_{\\theta}\\rangle$, \n\t\t\t$\\langle\\beta_{\\theta}\\rangle$ and $\\langle\\tilde{\\beta}_{\\theta}\\rangle$ are significantly \n\t\t\tlower than the impartial means $\\lambda^{\\ast}$ and $\\beta^{\\ast}$, with small deviations (data \n\t\t\tfor $5$ mice). \n\t\t\t\\textbf{D}. The dynamics of $\\lambda_{\\theta}(t)$ (red trace, upper panel) correlates with the \n\t\t\tspeed profile (gray line) when the mouse moves methodically. The $\\lambda_{\\theta}(t)$-stochasticity\n\t\t\tremains mostly within the ``typical\" range (pink stripe in the background), falling below it as the\n\t\t\tmouse slows down. For rapid moves there is a clear similarity between the $\\lambda_{\\theta}$-score\n\t\t\tand the speed, e.g., their peaks and troughs roughly match. When the mouse meanders (vertical gray\n\t\t\tstripes), the coupling between speed and $\\lambda_{\\theta}$-stochasticity is lost. \n\t\t\tThe Arnold score $\\beta_{\\theta}(t)$ (blue trace, lower panel) remains close to $\\beta_{\\min}=1$,\n\t\t\taffirming $\\theta$-wave's quasiperiodicity. Note the antiphasic relationship between the \n\t\t\t$\\beta_{\\theta}$-stochasticity and the acceleration $a(t)$ (the latter graph is shifted upwards to\n\t\t\tmatch the mean level $\\langle\\beta_{\\theta}\\rangle$): $\\theta$-periodicity loosens as the animal\n\t\t\tslows down ($\\beta_{\\theta}$-splashes correlate with animal's deceleration) and sharpens as he \n\t\t\tspeeds up. \\textbf{E}. Locally averaged $\\hat{\\lambda}_{\\theta}$-score grows with speed, whereas \n\t\t\t$\\hat{\\beta}_{\\theta}$ tends to drop down with acceleration.\n\t}}\n\t\\label{fig:thstoch}\n\n\\end{figure}\n\nThe resulting mean Kolmogorov score $\\langle\\lambda_{\\theta}\\rangle=0.54\\pm0.12$ is low, indicating\nthat, on average, $\\theta$-cycles closely follow the prescribed trend (\\ref{lin}). The mean Arnold score \n$\\langle \\beta_{\\theta} \\rangle = 1.1\\pm 0.03\\approx\\beta^{\\min}$ also points at the near-periodic behavior \nof the $\\theta$-wave (Fig.~\\ref{fig:thstoch}C). Nonetheless, $\\theta$-patterns exhibit complex dynamics that\ndiffer between quiescence and active states and couple to the animal's speed and acceleration.\n\n\\textit{Fast moves}. As mentioned above, the experimental design enforces recurrent behavior, in  which speed\ngoes up and down repeatedly as the animal moves between the food wells (Fig.~\\ref{fig:thstoch}B). When the mouse\nmoves methodically (lap time less than $25$ sec), $\\lambda_{\\theta}$ rises and falls along with the speed with\nsurprising persistence (Fig.~\\ref{fig:thstoch}D). Yet, the $\\theta$-patterns appearing in this process are \nstochastically generic---the entire sequence of $\\lambda_{\\theta}$-values remains mostly within the ``domain\nof stochastic typicality'' (pink stripe on Figs.~\\ref{fig:thstoch}C, D and \\ref{fig:stochs}B), below the \nuniversal mean $\\lambda^{\\ast}$. However, the patterns become overtly structured as the animal slows \ndown, when the Kolmogorov scores drop below $\\lambda_{\\theta}\\approx 0.1$, exhibiting uncommon compliance with\nthe mean behavior. Such values of $\\lambda_{\\theta}$ can occur by chance with vanishingly small probability \n$\\Phi(0.1)<10^{-17}$ (see formula (\\ref{sml}) in Sec.~\\ref{sec:met}), which, together with small\n$\\beta_{\\theta}$-score, $\\beta_{\\theta}\\approx 1$, imply that limited motor driving reduces $\\theta$-wave \nto a simple nearly-harmonic oscillation with a base frequency $\\nu\\approx 8$ Hz.\n\nIncreasing speed randomizes $\\theta$-patterns: the faster the mouse moves, the higher the \n$\\lambda_{\\theta}$-score. Furthermore, the shape of $\\lambda_{\\theta}(t)$ dependence exhibits an uncanny \nresemblance to the speed profile\n$s(t)$ (Fig.~\\ref{fig:thstoch}D). To quantify this effect, we used the Dynamic Time Warping (DTW) technique \nthat uses a series of local stretches to match two functions---in this case, $\\lambda_{\\theta}(t)$ and \n$s(t)$---so that the net stretch can be interpreted as separation, or distance\\footnote{DTW separation typically\n\tsatisfies the triangle inequality, $D(a,b)+D(b,c)\\geq D(a,c)$, which permits interpreting it geometrically,\n\tas a distance between signals \\cite{Neamtu}.} between functions in ``feature space\" \\cite{Berndt,Salvador}.\nIn our case, the DTW-distance between the speed $s(t)$ and Kolmogorov score $\\lambda_{\\theta}(t)$ during active\nmoves is small, $D(\\lambda_{\\theta},s)=19.6\\%$, indicating that the deviations of the $\\theta$-patterns\nfrom the mean reflect the animals' mobility (SFig.~1).\n\nNote that DTW-affinity between $\\lambda_{\\theta}(t)$ and speed does not necessarily imply a direct functional\ndependence between these quantities. Indeed, plotting points with coordinates $(s,\\lambda_{\\theta})$ yields \nscattered clouds, suggesting a broad trend, rather than a strict relationship (Fig.~\\ref{fig:thstoch}E).\nHowever, if the $\\lambda$-scores and the speeds are \\textit{locally averaged}, i.e., if each individual $s$- \nand $\\lambda$-value is replaced by the mean of itself and its adjacents, then the pairs of such \\textit{local\n\tmeans}, $(\\hat{s}_i,\\hat{\\lambda}_i)$, reveal a core dependence: increasing speed of the animal entails\nhigher variability of the $\\theta$-patterns. \n\nIn the meantime, the Arnold stochasticity score, $\\beta_{\\theta}(t)$, is closely correlated with the mouse's\nacceleration, $a(t)$. As shown on Fig.~\\ref{fig:thstoch}D, the $\\beta_{\\theta}$-score rises as the mouse \ndecelerates ($\\theta$-wave clumps) and falls when he accelerates ($\\theta$-wave becomes more orderly),\nproducing a curious antiphasic $\\beta_{\\theta}\\textrm{-}a$ relationship, which is also captured by the local\naverages $(\\hat{a}_i,\\hat{\\beta}_i)$ (Fig.~\\ref{fig:thstoch}E). The distance between $\\beta_{\\theta}(t)$\nand $-a(t)$ (the minus sign accounts for the antiphase) is $D(\\beta_{\\theta},-a)=33.6\\%$, which means that\nspeed influences the $\\theta$-wave's statistical typicality more than acceleration impacts its orderliness. \n\n\\textit{Slow moves}. When the mouse meanders and slows down (lapse time over $25$ sec), $\\theta$-patterns \nchange: the $\\lambda_{\\theta}$-score increases in magnitude and uncouples from speed, (DTW distance is twice\nthat of the fast moves case, $D(\\lambda_{\\theta},s)=38.8\\%$), suggesting that, without active motor driving,\n$\\theta$-rhythmicity is less controlled by the mean oscillatory rate, i.e., is more randomized. The Arnold's \nparameter $\\beta_{\\theta}$ also slightly increases, $D(\\beta_{\\theta},-a)=36.1\\%$, indicating concomitant \n$\\theta$-disorder.\n\nOverall, the combined $\\lambda_{\\theta}\\textrm{-}\\beta_{\\theta}$ dynamics suggests that during active behavior, \nthe shape of the $\\theta$-wave is strongly controlled by the mouse's moves. Highly ordered, nearly periodic\n$\\theta$-peaks appear when the animal starts running---the $\\theta$-frequency range then narrows to the mean,\nexpected value. The increasing speed stirs up the $\\theta$-patterns; the disorder grows and reaches its maximum\nwhen the animal moves fastest and begins to slow down. During periods of inactivity, the coupling between \n$\\theta$-patterns and speed is weakened and then reinforced as the mouse stiffens his resolve.\n\n\nWe emphasize however, that these dependencies should not be viewed as na\\\"ive manifestations of known couplings\nbetween instantaneous or time-averaged frequency with the animals' speed or acceleration \\cite{Richard,Kropff}. \nIndeed, the $\\theta$-frequency alters at the same rate as the $\\theta$-amplitude---many times over the span of\neach $\\theta$-pattern, providing an instantaneous characterization of the wave \\cite{Vakman,Rice}. In contrast,\nthe stochasticity parameters describe the wave form as a single entity (Fig.~\\ref{fig:thstoch}A,C).\nOn the other hand, time-averaging levels out fluctuations and highlights mean trends, whereas Kolmogorov and \nArnold parameters are sensitive to individual elements in the data sequences. \\textit{Thus, $\\lambda$ and \n$\\beta$ scores describe  of wave shapes without defeaturing, putting each pattern, as a whole, into a \nstatistical perspective}. It hence becomes possible to approach questions addressed in the Introduction: \nidentify typical and atypical wave patterns, quantify levels of their  orderliness, detect deviations from \nnatural behavior and so forth. \n \n\\textbf{2. $\\gamma$-waves} ($30-80$ Hz, \\cite{ColginGm}) exhibit a wider variety of patterns than $\\theta$-waves.\nThe interpeak intervals between consecutive $\\gamma$-peaks, $T_{\\gamma}$, are nearly-exponentially distributed,\nwhich implies that both smaller and wider $\\gamma$-intervals are statistically more common \n(Fig.~\\ref{fig:gstoch}A).\n\n\\textit{Fast moves}. For consistency, the sample sequences, $X_{\\gamma}$, were drawn from the same time windows,\n$L_{\\gamma}=L_{\\theta}\\approx 6$ secs, which contained, on average, about $300$ elements that yield a mean\nKolmogorov score $\\langle\\lambda_{\\gamma}\\rangle=1.84\\pm 1.03$---more than twice higher than the impartial mean\n$\\lambda^{\\ast}$ and three times above the $\\langle\\lambda_{\\theta}\\rangle$ score. Such values can randomly \noccur with probability $1-\\Phi(1.84)\\lesssim 2\\cdot 10^{-3}$, which suggests that generic $\\gamma$-patterns are\nstatistically atypical and may hence reflect organized network dynamics, rather than random extracellular field\nfluctuations (Fig.~\\ref{fig:gstoch}B). The average Arnold parameter also grows compared to the $\\theta$-case, \nbut remains lower than the impartial mean, $\\langle\\beta_{\\gamma}\\rangle=1.61\\pm0.53<\\beta^{\\ast}$, implying \nthat, although $\\gamma$-waves are more disordered than the $\\theta$-waves, they remain overall oscillatory.\n\n\\begin{figure}\n\n\t\\centering\n\t\\includegraphics[scale=0.8]{GammaTimeStoch}\n\t\\caption{{\\footnotesize\n\t\t\t\\textbf{$\\gamma$-wave stochasticity}. \n\t\t\t\\textbf{A}. A histogram of $\\gamma$-interpeak intervals exhibits an exponential-like distribution\n\t\t\twith mean characteristic $\\gamma$-period, $\\overline{T}_{\\gamma}=18.6\\pm 1.9$ msec, about six times\n\t\t\tsmaller than $\\overline{T}_{\\theta}$. \n\t\t\t\\textbf{B}. The average scores $\\langle\\beta_{\\gamma}\\rangle$ and $\\langle\\lambda_{\\gamma}\\rangle$\n\t\t\tare higher than for the $\\theta$-wave, indicating that $\\gamma$-patterns are more diverse than\n\t\t\t$\\theta$-patterns.\n\t\t\t\\textbf{C}. Locally averaged $\\hat{\\lambda}_{\\gamma}$-score grows with speed, while \n\t\t\t$\\hat{\\beta}_{\\gamma}$ switches from higher to lower level with increasing acceleration. \n\t\t\t\\textbf{D}. The dynamics of the $\\lambda_{\\gamma}$-score (top panel) correlates with changes in the\n\t\t\tspeed when the animal moves actively. Note that $\\lambda_{\\gamma}$ often exceeds the upper bound of\n\t\t\tthe ``pink stripe,'' i.e., $\\gamma$-waves often produce statistically uncommon patterns, especially \n\t\t\tduring rapid moves. The $\\beta_{\\gamma}$-score (bottom panel) correlates with the animal's acceleration,\n\t\t\twhich is lost when lap times increase (gray stripes).\n\t}}\n\t\\label{fig:gstoch}\n\n\\end{figure}\n\nOn average (for all five mice), the Kolmogorov score, $\\lambda_{\\gamma}(t)$, escapes the domain of ``stochastic\ntypicality\" approximately half of time through transitions that closely follow speed dynamics. Together with the\nobservation that $\\lambda_\\gamma$ takes a larger range of values than $\\lambda_\\theta$ ($\\gamma$-patterns deviate\nmore from the average as speed increases), this suggests that there is a greater diversity of $\\gamma$-responses\nto movements (Fig.~\\ref{fig:gstoch}C). In particular, the dependence between locally averaged $\\hat{\\lambda}\n_{\\gamma}(t)$ and $\\hat{s}(t)$ is less tight: the average DTW distance, $D(\\lambda_{\\gamma},s)\\approx 23.4\\%$, is\nslightly higher than the distance in the $\\lambda_{\\theta}(t)$ dynamics, which illustrates that \n$\\gamma$-patterns are less sensitive to speed than $\\theta$-patterns.\n\n\nFrom a structural, $\\beta$-perspective, $\\gamma$-wave becomes closer to periodic when an actively moving animal\nslows down: during these periods, $\\beta_{\\gamma}$-score reduces close to its minimal value, when the \ncorresponding Kolmogorov score also drops to $\\lambda_{\\gamma}\\approx0.2$. Since the latter is unlikely to occur\nby chance (cumulative probability of that is $\\Phi(0.2)\\lesssim10^{-12}$), these changes may represent structured\nnetwork dynamics. The highest deviations of $\\gamma$-patterns from the mean ($\\lambda_{\\gamma}\\gtrsim 3$) are \naccompanied by high $\\beta_{\\gamma}$-scores, which happen as the mice slow down from maximal speed and implies\nthat circuit activity is least structured during these periods (Fig.~\\ref{fig:gstoch}D).\nThe relation between $\\gamma$-orderliness, $\\beta_{\\gamma}(t)$, and acceleration is also similar to the \ncorresponding dependence in the $\\theta$-case: acceleration induces stricter $\\gamma$-rhythmicity and\ndeceleration randomizes $\\gamma$-patterns, with about the same overall DTW distance, $D(\\beta_{\\gamma},-a)\\approx\n34.4\\%$. \n\n\\textit{During slower movements}, the $\\gamma$-dynamics change qualitatively: the magnitudes of both \n$\\lambda_{\\gamma}(t)$ and $\\beta_{\\gamma}(t)$ grow higher, indicating that decoupling from motor activity\nenforces statistically atypical $\\gamma$-rhythmicity in the hippocampal network, as in the $\\theta$-waves. \nIn particular, the uncommonly high $\\beta_{\\gamma}$ scores point at frequent $\\gamma$-bursting during quiescence. \n\nOnce again, we emphasize that these results do not represent known correlations between instantaneous or \ntime-averaged $\\gamma$-characteristics and motion parameters \\cite{Ahmed,Montgomery}. Rather, the outlined \n$\\lambda_{\\gamma}(t)$ and $\\beta_{\\gamma}(t)$ dependencies capture pattern-level dynamics of $\\gamma$-waves\nthat reflect circuit activity at an intermediate timescale. As an illustration, note that the amplitude of\n$\\gamma$-waves, $\\tilde{\\gamma}$, along with the instantaneous $\\gamma$-frequency, $\\omega_{\\gamma}$, have low\nstochasticity scores, comparable to the ones produced by the Poisson process, $\\langle\\beta_{\\tilde{\\gamma}}\n\\rangle= 1.15\\pm 0.08$ and $\\langle\\lambda_{\\tilde{\\gamma}}\\rangle=0.52\\pm 0.26$ (Fig.~\\ref{fig:rndstoch}).\nThus, although instantaneous characteristics exhibit restrained, quasiperiodic behavior, they allow a rich \nmorphological variety of the underlying $\\gamma$-oscillations.\n\n\\begin{figure}\n\n\t\\centering\n\t\\includegraphics[scale=0.8]{SWRTimeStoch}\n\t\\caption{{\\footnotesize\n\t\t\t\\textbf{Sharp Wave-Ripples' stochasticity}. \n\t\t\t\\textbf{A}. A histogram of intervals between SWR events is nearly exponential. \n\t\t\t\\textbf{B}. The averages $\\langle\\lambda_{\\swr}\\rangle$ and $\\langle\\beta_{\\swr}\\rangle$ are high, \n\t\t\tindicating both frequent deviation of SWR events from the mean and higher temporal clustering than\n\t\t\tfor the $\\theta$ and $\\gamma$-patterns. \n\t\t\t\\textbf{C}. The animal's speed (gray line, top panel) correlates with the Kolmogorov parameter \n\t\t\t$\\lambda_{\\swr}$ during fast exploratory lapses. During inactivity (vertical gray stripes) the \n\t\t\t$\\lambda_{\\swr}$-stochasticity uncouples from speed, exhibiting high spikes that mark strong\n\t\t\t``fibrillation'' of SWR patterns. The antiphasic relationship between the animal's acceleration\n\t\t\t$a(t)$ (gray line, bottom panel) and Arnold score $\\beta_{\\swr}(t)$ shows that SWRs tend to cluster\n\t\t\twhen as the animal decelerates, while acceleration enforces periodicity. During slower moves (gray\n\t\t\tstripes), the relationship between speed, acceleration, and stochasticity is washed out and stochastically\n\t\t\timprobable patterns dominate.\n\t\t\t\\textbf{D}. Locally averaged $\\hat{\\lambda}_{\\swr}$ grows with speed and $\\hat{\\beta}_{\\swr}$ drops\n\t\t\twith acceleration.\n\t}}\n\t\\label{fig:swr}\n\n\\end{figure} \n\n\\textbf{3. Sharp Wave Ripples} (SWRs), the high amplitude splashes (over $2-3$ standard deviations from the \nmean) of high frequency waves  ($150-250$ Hz, \\cite{ColginR}), exhibit the richest pattern dynamics.\n\n\\textit{During fast moves}, SWR events appear at approximately the same exponential rate as the \n$\\tilde{\\gamma}$-peaks, $\\overline{T}_{\\swr}\\approx \\overline{T}_{\\tilde{\\gamma}}$ msec, but exhibit higher\n$\\lambda$-scores, $\\langle \\lambda_{\\swr}\\rangle=2.40\\pm1.57$, over the same sampling periods $L_{\\swr}\\approx\n6$ sec (Fig.~\\ref{fig:swr}A,B). The low probability of these patterns ($1-\\Phi(2.5)\\lesssim 10^{-6}$) and the\nrelatively high mean $\\beta$-score, $\\langle\\beta_{\\swr}\\rangle=1.71\\pm0.64$, indicate that SWRs tend to \nexhibit intermittent clustering that may reflect brisk, time-localized circuit activity, such as rapid replays\nand preplays of the hippocampal place cells \\cite{Girardeau2,Singer,Roux,Sadowski2,Denovellis,BarneSpars,Wu}.\n\nInterestingly, SWR-patterns also correlate with the animal's speed profile about as much as $\\gamma$-patterns,\n$D(\\lambda_{\\swr},s)\\approx 23.8\\%$ \\cite{Denovellis}. The $\\beta_{\\swr}(t)$-dependence displays the familiar\nantiphasic relationship with the animal's acceleration---SWR events tend to cluster more when the animal slows\ndown (Fig.~\\ref{fig:swr}C,D). However, orderliness of SWRs is driven by acceleration stronger than orderliness\nof $\\gamma$-patterns: the range of $\\beta_{\\swr}$-scores is twice as wide as the range of $\\beta_{\\gamma}$-scores\n(broader pattern variety), with a similar DTW distance $D(\\beta_{\\swr},-a)\\approx39.7\\%$. \n\n\\textit{During quiescent periods}, both $\\lambda_{\\swr}$ and $\\beta_{\\swr}$ grow and exhibit extremely high\nspikes, indicating that endogenous network dynamics produce stochastically improbable, highly clustered SWR\nsequences. Physiologically, these statistically uncommon SWR patterns may indicate sleep or still wakefulness\nreplay activity, known to play an important role in memory consolidation \\cite{Kudrimoti,ONeil}.\n\nOverall, the temporal clumping comes forth as a characteristic feature of the SWR events, suggesting that SWRs\nare manifestations of fast, targeted network dynamics that brusquely ``ripple\" the extracellular field, unlike\nthe rhythmic $\\theta$ and $\\gamma$-undulations \\cite{ColginR}.\n\n\\subsection{Stochasticity in space}\n\\label{sec:space}\n\nDistributing the $\\lambda$ and $\\beta$ scores along the animal's trajectory yields \\textit{spatial maps of\nstochasticity} for each brain rhythm and reveals a curious spatial organization of LFP patterns with similar\nmorphology. As shown on Fig.~\\ref{fig:space}, higher $\\lambda$-values for all waves are attracted to segments\nwhere the mouse is actively running with maximal speed, furthest away from the food wells. Patterns that are \nclose to the expected average (low-$\\lambda$) concentrate in the vicinity of food wells where the animal moves\nslowly. The latter domains also tend to host high $\\beta$-scores that appear as the animal approaches the food\nwells, as well as the lowest $\\beta$s, which appear as the animal accelerates away \\cite{BarneSpars}. In other\nwords, the LFP waves become more ``trendy'' and, at the same time, more structured (either more periodic or more\nclustered) over the behaviorally important places (e.g., food wells) that require higher cognitive activity. \nOn the other hand, the outer parts of the track, where the brain waves are less controlled by the mean and \nremain moderately disordered, are marked by uncommon patterns.\n\n\\begin{figure}[h]\n\n\t\\centering\n\t\\includegraphics[scale=0.79]{BehaviorBL}\n\t\\caption{{\\footnotesize\n\t\t\t\\textbf{Spatial stochasticity maps} were obtained by plotting $\\lambda$ and $\\beta$ parameters along\n\t\t\tthe\ttrajectory.\n\t\t\t\\textbf{A}. The $\\lambda$-maps show that $\\theta$-wave, $\\gamma$-wave and SWRs generally follow the\n\t\t\tmean trend near the food wells (with scattered wisps of high stochasticity) and deviate from the mean\n\t\t\tmostly over the areas most distant from the food wells. The smaller maps in \n\t\t\tthe gray boxes represent slow lapses: the overall layout of high-$\\lambda$ and low-$\\lambda$ fields\n\t\t\tis same as during the fast moves, which suggests spatiality of $\\lambda$-stochasticity.\n\t\t\t\\textbf{B}. The behavior of $\\beta_{\\theta}$ is opposite: the ``uneventful,'' distant run segments \n\t\t\tattract nearly-periodic behavior, while the food wells attract time-clumping wave patterns.\n\t}}\n\t\\label{fig:space}\n\n\\end{figure}\n\nIntriguingly, the same map structure is reproduced during slow lapses, when the motor control of the patterns\nweakens, suggesting that speed and acceleration are not the only determinants of the LFP patterns. As shown \non Fig.~\\ref{fig:space}, even when the mouse dawdles, the waves tend to deviate from the mean around the outer \ncorners and follow the mean in the vicinity of the food wells. Similarly, the patterns start clumping as the\nmouse approaches the food wells, and distribute more evenly as he moves away.\n\nThese results suggest that spatial context may, by itself, influence hippocampal brain rhythm structure, which\nis reminiscent of the place-specific activity exhibited by spatially tuned neurons, e.g., place cells \n\\cite{MosRev} or parietal neurons \\cite{Nitz1}. For example, the ``bursting'' (high-$\\beta$) fields and \n``domains of evenness\" (small $\\lambda$) surround food wells; the quasiperiodicity fields (small $\\beta$s) \nas well as ``wobbling-waves'' (large $\\lambda$s) stretch over the outer segments (Fig.~\\ref{fig:space}). \nPhysiologically, this ``spatiality of stochasticity\" may reflect a coupling between the hippocampal\nplace-specific spiking activity and extracellular field oscillations. \n\n\\subsection{$\\lambda$-$\\beta$ relationships}\n\\label{sec:lb}\n\nThe definitions of the stochasticity scores $\\beta$ and $\\lambda$ do not imply an \\textit{a priori} \n$\\beta\\textrm{-}\\lambda$ relationship. Indeed, plotting sets of points with coordinates $(\\beta,\\lambda)$ \nfor all sample sequences, $X_{\\theta}$, $X_{\\gamma}$ and $X_{\\swr}$, yields scattered clouds, rather than\ncurve-like graphs (Fig.~\\ref{fig:bl}A). However, computational studies of number-theoretic sequences carried\nin \\cite{ArnoldB1,ArnoldB2,ArnoldB3,ArnoldB4} suggest that such behavior may be caused by the occasional large\ncontributions from atypical sequences and that local smoothing may yield much tighter couplings between \nthe stochasticity parameters.\n\n\\begin{figure}[h]\n\n\t\\centering\n\t\\includegraphics[scale=0.8]{BetaLambda}\n\t\\caption{{\\footnotesize\n\t\t\t\\textbf{Dependencies between stochasticity parameters}. \n\t\t\t\\textbf{A}. Points with coordinates $(\\beta_i,\\lambda_i)$ computed for each individual sample \n\t\t\tsequence produce clouds that imply no strict $\\beta(\\lambda)$ dependence. However, locally \n\t\t\taveraged stochasticity parameters $(\\hat{\\lambda},\\hat{\\beta})$ exhibit much tighter \n\t\t\trelationships (black dots). The growing $\\hat{\\beta}(\\hat{\\lambda})$ trends indicate that ordered,\n\t\t\tsemiperiodic sequences (low $\\hat{\\beta}$) tend to accompany samples that comply with the expected\n\t\t\tbehavior (low $\\hat{\\lambda}$). Conversely, patterns tend to fibrillate more as they deviate farther\n\t\t\tfrom the mean and abandon the pink stripe of ``stochastic typicality.''\n\t\t\t\\textbf{B}. The $(\\beta,\\lambda)$ pairs for sequences drawn from the three random distributions \n\t\t\t(Fig.~\\ref{fig:rndstoch}) produce similar clouds (uniform, blue dots; exponential, red dots; \n\t\t\tPoisson, orange dots). \n\t\t\tLocally averaged scores indicate tight $(\\hat{\\beta}\\textrm{-}\\hat{\\lambda})$ relations  for all\n\t\t\tthese series trend similarly to the brain waves' local averages.\n\t\t\t\\textbf{C}. The $\\hat{\\beta}(\\hat{\\lambda})$ trends evaluated for individual mice may exhibit\n\t\t\tnon-generic features. In this case, the oscillatory rate of semi-periodic $\\theta$-, $\\gamma$- and\n\t\t\tSWR sequences (low $\\hat{\\beta}$s) deviates significantly from the predicted mean (large \n\t\t\t$\\hat{\\lambda}$s). As the disorder increases, the oscillatory rate gets closer to the predicted mean,\n\t\t\treaching the minimal $(\\hat{\\beta}_{\\theta}\\textrm{-}\\hat{\\lambda}_{\\theta})$ and \n\t\t\t$(\\hat{\\beta}_{\\gamma}\\textrm{-}\\hat{\\lambda}_{\\gamma})$ combinations (cyan dot) before retaking a\n\t\t\tjoint growth trend. In case of the SWRs, the minimum is spread into a plateau where changes in\n\t\t\t$\\hat{\\beta}_{\\swr}$ do not affect $\\hat{\\lambda}_{\\swr}$.\n\t}}\n\t\\label{fig:bl}\n\n\\end{figure}\n\nTo study whether similar phenomena take place in the LFP sequences, we evaluated the the local averages\n$(\\hat{\\beta}_i,\\hat{\\lambda}_i)$, which revealed $\\hat{\\beta}\\textrm{-}\\hat{\\lambda}$ relationships\nillustrated on Fig.~\\ref{fig:bl}. For the full dataset that includes all mice, we observe that orderly \nsequences (low $\\hat{\\beta}$-scores) tend to follow the prescribed mean trend more closely (low $\\hat{\n\t\\lambda}$s). In contrast, patterns that deviate from the expected mean tend to produce disordered\nand clumping sequences, notably for $\\gamma$-waves and SWRs.\n\nCuriously, similar $\\hat{\\beta}\\textrm{-}\\hat{\\lambda}$ behavior are exhibited by random sequences (uniformly,\nexponentially and Poisson-distributed, Fig.~\\ref{fig:bl}B), which suggests that the tendency of $\\lambda$ to\nrise with growing $\\beta$, observed in large volumes of heterogeneous data (different mice) may not be of a \nspecifically physiological nature, but may reflect mathematical connections between the \nstochasticity indices \\cite{Arnold3,Arnold4,Arnold5}.\n\nIn view of these results, it is surprising that individual mice can exhibit personalized dependencies between\npredictability, $\\lambda$, and orderliness, $\\beta$, of their LFP patterns. In the case illustrated on \nFig.~\\ref{fig:bl}, nearly periodic (small $\\hat{\\beta}_{\\theta}$) $\\theta$-patterns tend to deviate from the \npredicted behavior quite significantly. As the disorder increases, the patterns become more compliant with the \nunderlying mean, until a trough of $\\hat{\\beta}_{\\theta}\\textrm{-}\\hat{\\lambda}_{\\theta}$ dependence is reached.\nThen the tendency is reversed: growing $\\theta$-disorder is accompanied by further deviation from the mean. Note\nthat the entire stochasticity dynamics remain within the ``typicality zone,\" $0.4\\leq\\hat{\\lambda}_{\\theta}\\leq\n1.8$. Analogous behavior is exhibited by the $\\gamma$-wave but at a larger scale: orderly $\\gamma$-waves, \n$\\hat{\\beta}_{\\gamma}\\approx1.4$, are concomitant with $\\lambda$-scores as high as $\\lambda_{\\gamma}\\approx6.5$ \n(highly improbable patterns, $1-\\Phi(6.5)\\lesssim 10^{-50}$). As $\\hat{\\beta}_{\\gamma}$ grows, $\\hat{\\lambda}_{\\gamma}$\ndecreases, approaching stochastic commonality at the lowest point. Then, as $\\hat{\\beta}_{\\gamma}$ grows further,\nthe disordered sequences increasingly deviate from the expected mean. The $\\hat{\\beta}_{\\swr}\\textrm{-}\\hat{\\lambda}_{\\swr}$\ndependence is less tight but exhibits a similar trend: at first, small $\\hat{\\beta}_{\\swr}$'s pair with higher\n$\\hat{\\lambda}_{\\swr}$'s, then level out over an intermediate range of $\\beta$'s (the minimum is flattened out\nin contrast with $\\theta$- and $\\gamma$-waves), and then grows again with the increasing SWR-disorder.\n\nImportantly, we found no relationship between the $\\lambda_{\\theta}$, $\\lambda_{\\gamma}$, and $\\lambda_{swr}$,\nnor between $\\beta_{\\theta}$, $\\beta_{\\gamma}$, and $\\beta_{swr}$ (SFig.~2), which implies that the $\\lambda_\n{\\ast}$ and the $\\beta_{\\ast}$-scores associated with different waves are largely independent, providing their\nown, autonomous characterizations of the wave shapes.\n\n\\section{Discussion}\n\nThe recorded LFP signals are superpositions of locally induced extracellular fields and inputs transmitted from\nanatomically remote networks. The undulatory appearance of the LFP is often interpreted as a sign of structural\nand functional regularity\\footnote{A succinct expression of this view is provided in \\cite{Fransen}:\n\t\\textit{``rhythmicity is the extent to which future phases can be predicted from the present one.''}}, \nbut the dynamics of these oscillations is actually highly complex. Understanding the balance between \ndeterministic and stochastic components in LFPs, as well as questions about their continuity and discreteness\npose significant conceptual challenges, as it happened previously in other disciplines\\footnote{In his 1955 \ndiscussion of the foundations of Quantum Mechanics, John von Neumann attributes a great significance to the\nfact that \\textit{``...the general opinion in theoretical physics had accepted the idea that ...continuity\n\t\t...is merely simulated by an averaging process in a world which in truth discontinuous by its very \n\t\tnature. This simulation is such that man generally perceives the sum of many billions of elementary \n\t\tprocesses simultaneously, so that the leveling law of large numbers completely obscures the real \n\t\tnature of the individual processes.''} \\cite{Neumann}}.\n\nStructurally, LFP rhythms may be described through discrete sequences of wave features (heights of peaks, \nspecific phases, interpeak intervals, etc.), or viewed as transient series---bursts---of events, as in the\ncase of SWRs \\cite{Ede}. It is well recognized that such sequences are hard to decipher and to forecast, e.g.,\na recent discussion of a possible role of bursts in brain waves' genesis posits: \\textit{``An important feature\n\tthat \nsets the burst scenario apart is the lack of continuous phase-progression between successive time points---and\ntherefore the ability to predict the future phase of the signal---at least beyond the borders of individual bursts\"}\n\\cite{Ede}. In other words, the nonlinearity of LFP dynamics, as well as its transience and sporadic external\ndriving, result in effective stochasticity of LFP patterns---an observation that opens a new round of inquiries \n\\cite{Ede,Jones}. \nFor example, how exactly should one interpret the ``unpredictability'' of a temporal sequence? Does it mean that\nits pattern cannot be resolved by a particular algorithm, or that it is unpredictable in principle, ``genuinely\nrandom,\" such as a gambling sequence? What is the difference between the two? How is the apparent randomness of\nLFP rhythms manifested physiologically? Are the actual network computations based on ``overcoming the apparent \nrandomness\" and somehow deriving the upcoming phases or amplitudes from the preceding ones, or may there be \nalternative ways of extracting information? Does the result depend on the ``degree of randomness\" and if so, \nthen how to distinguish between a ``more random'' and a ``less random'' patterns? These questions are not \ntechnical, pertaining to a specific mechanism, nor specifically neurophysiological; rather, these are \nfundamental problems that transcend the field of neuroscience. Historically, similar questions have motivated\nmathematical definitions of randomness that are still debated to this day \\cite{Mises,Uspenskii,Volchan}. \n\nOne approach to addressing these issues was suggested by Kolmogorov in 1933 (also the year when brain waves\nwere discovered \\cite{Berger1}), based on the statistical universality of stochastic deviations from the \nexpected behavior \\cite{Kolmogorov,Stephens}. From Kolmogorov's perspective, randomness is contextual: if a\nsequence $X$ deviates from an expected mean behavior within bounds established by the distribution (\\ref{phi}),\nthen $X$ is \\textit{effectively} random. In other words, an individual sequence may be viewed as random if it\ncould be randomly drawn from a large pool of similarly trending sequences, with sufficiently high probability.\nThis view permits an important conceptual relativism: even if a sequence is produced by a specific mechanism\nor algorithm, it can still be viewed as random as long as its $\\lambda$-score is ``typical\" according to the\nstatistics (\\ref{phi}). For example, it can be argued that geometric sequences are typically more random than\narithmetic ones, although both are defined by explicit formulae \\cite{Arnold1,Arnold2,Arnold3,Arnold4,Arnold5}.\nBy analogy, individual sample sequences of $\\theta$, $\\gamma$, or SWR events may be generated by specific \nsynchronization mechanisms at a precise timescale, and yet they may be empirically classified and quantified\nas stochastic.\n\nA practical advantage of Kolmogorov's approach is that mean trends, such as (\\ref{lin}), can often be reliably\nestablished, interpreted, and then used for putting the stochasticity of the underlying sample patterns into a\nstatistical perspective. Correspondingly, assessments based on $\\lambda$-scores were previously applied in a \nvariety of disciplines from genetics \\cite{KolMen,Stark,Gurzadyan1} to astronomy \\cite{Gurzadyan2}, and from \neconomics \\cite{Brandouy} to number theory \\cite{Arnold2,Arnold3,Arnold1,Arnold4,Arnold5,ArnoldB1,ArnoldB2,ArnoldB3,\n\tArnoldB4,Christoph,Ford}. Some work has also been done in brain wave analyses, e.g., for testing normality \nof electroencephalograms' long-term statistics \\cite{Weiss1,Weiss2,Weiss3,McEwen}. Arnold's $\\beta$-score \nprovides an independent assessment of orderliness (whether elements of an arrangement tend to attract, repel\nor be independent of each other) and it has not been, to our knowledge, previously used in applications.\n\nShifting window analyses ground the $\\lambda$- and $\\beta$-values in the context of preceding and upcoming\nobservations. Since neighboring patterns change only marginally, the time-dependent $\\lambda(t)$ and \n$\\beta(t)$ describe quasi-continuous pattern dynamics at a \\textit{mesoscale}---over the span of several \nundulations---which complements the microscale (instantaneous) and macroscale (time-averaged) assessments. \nWhile the individual, ``stroboscopically selected\" patterns can be viewed as stochastic, the continuous \n$\\lambda(t)$ and $\\beta(t)$ dependencies describe ongoing pattern dynamics. \n\nImportantly, Kolmogorov's and Arnold's scores are impartial and independent from physiological specifics\nor contexts, thus providing self-contained semantics for describing the LFP data and a novel venue for\nanalyzing the underlying neuronal mechanisms. It becomes possible to distinguish ``statistically mundane\" LFP\npatterns from exceptional ones and to capture the transitions between them, as well as to link pattern dynamics\nto changes in the underlying network's dynamics (Figs.~\\ref{fig:thstoch}-\\ref{fig:space}). For example, since\n$\\theta$-bursts are physiologically linked to long-term synaptic potentiation \\cite{Greenstein,Hinder}, \n$\\theta$-patterns with high $\\beta$-scores may serve as markers of plasticity processes taking place in the\nhippocampal network at specific times and places \\cite{Larson,Sheridan,Nguyen}. \nFurthermore, high-$\\beta_{\\theta}$ regions near food wells indicate that reward proximity may trigger\nhippocampal plasticity and, since hippocampal neurons' spiking is coupled to $\\theta$-cycles \\cite{Skaggs}, \nhave a particular effect on memory (Fig.~\\ref{fig:space}). On the other hand, low-$\\beta_{\\theta}$ indicates\nlimit cycles in the network's phase space that uphold simple\nperiodicity. $\\gamma$-bursts (high $\\beta_{\\gamma}$) mark heightened attention and learning periods \n\\cite{ClgMsr,Lundqvist}. In our observations, they appear during the mouse's approach to the reward locations\nand disappear as it ventures away from them (Fig.~\\ref{fig:space}). Clustering SWR events reflect dense replay\nactivity \\cite{Roux,Singer}, indicative of periods of memory encoding, retrieval, and network \nrestructurings \\cite{Replays,Sadowski2}. \n\nOverall, the proposed approach allows studying brain rhythms from a new perspective that complements existing\nmethodology, which may lead to a deeper understanding of the synchronized neuronal dynamics and its physiological\nfunctions at temporal mesoscale.\n\n\\vspace{7pt}\n\n\\textbf{Acknowledgments}. We are grateful to Dr. A. Babichev for fruitful discussions. \nC.H. and Y.D. are supported by NIH grant R01NS110806 and NSF grant 1901338.\nC.J. and D.J. are supported by NIH grants R01MH112523 and R01NS097764.\n\n\n\\newpage\n\\section{Mathematical Supplement}\n\\label{sec:met}\n\n\\textbf{Computational algorithms}. \n\n\\textit{1. Kolmogorov score}. Let $X={x_1\\leq x_2\\leq\\ldots\\leq x_n}$, be an ordered sequence and let\n$N(X,L)$ be the number of elements smaller than $L$,\n\\begin{equation*}\n\tN(X,L) = \\{\\textrm{number of} \\,\\, 0\\leq x_k < L\\}.\n\t\\label{N}\n\\end{equation*}\nLet $\\bar{N}(X,L)$ be the expected number of elements that interval (Fig.~\\ref{fig:stair}). The closer $X$\nfollows the prescribed behavior, the smaller the normalized deviation\\footnote{The supremum, rather\tthan \nmaximum, is required in formula (\\ref{lam}) due to discontinuity of the counting function $N(X,L)$ at the\nstepping points.}\n\\begin{equation}\n\t\\lambda(X)= \\sup_L|N(X,L) - \\bar{N}(X,L)|\/\\sqrt{n}.\n\t\\label{lam}\n\\end{equation}\n\n\\begin{wrapfigure}{c}{0.4\\textwidth}\n\t\\centering\n\t\\includegraphics[scale=0.8]{Stair}\n\t\\caption{{\\footnotesize\n\t\t\t\\textbf{Counting function} $N(X)$ (red staircase) makes unit steps at each point of a sequence\n\t\t\t$X=\\{x_1,x_2,\\ldots,x_n\\}$ (tick marks on the $x$-axis). The normalized maximal deviations \n\t\t\t$\\lambda(X)$ from the expected mean $\\bar{N}(x)$ (straight line) exhibit statistical universality\n\t\t\tand can hence be used for characterizing stochasticity\tof the individual data sequences $X$.\n\t}}\n\t\\label{fig:stair}\n\t\\end{wrapfigure}\n\nA remarkable observation made in \\cite{Kolmogorov} is that the cumulative probability of having $\\lambda(X)$ \nsmaller than a given $\\lambda$ converges to the function\n\\begin{equation}\n\t\\Phi(\\lambda) = \\sum_{k=-\\infty}^{\\infty}(-1)^k e^{-2k^2 \\lambda^2},\n\t\\label{phi}\n\\end{equation}\nthat starts at $\\Phi(0)=0$ and grows to $\\Phi(\\infty)=1$. The derivative of the cumulative density (\\ref{phi})\ndefines the probability distribution for $\\lambda$, $P(\\lambda)=\\partial_{\\lambda}\\Phi(\\lambda)$\n(Fig.~\\ref{fig:stochs}B). Even though the range of $P(\\lambda)$ includes arbitrarily small or large $\\lambda$s,\nthe shape of the distribution implies that excessively high or low $\\lambda$-values are rare, e.g., sequences \nwith $\\lambda(X)\\leq 0.4$ or $\\lambda(X)\\geq 1.8$ appear with probability less than $0.3\\%$, $\\Phi(0.4)\\approx\n0.003$ and $\\Phi(1.8)\\approx0.997$. Since these statistics are universal, i.e., apply to any sequence $X$, the\n$\\lambda$-score can serve as a universal measure of ``stochastic typicality\" of a pattern\n \\cite{Stephens,Arnold1,Arnold2,Arnold3,Arnold4,Arnold5}.\n\n\\textit{2. Corrections to Kolmogorov score} up to the order $n^{-3\/2}$,\n\\begin{equation}\n\\lambda(X)\\to\\lambda(X)\\left(1+\\frac{1}{4n}\\right)+\\frac{1}{6n}-\\frac{1}{4n^{3\/2}},\n\\label{lamn}\n\\end{equation}\nallows for an increase in the accuracy of the finite-sample estimates to over $0.01\\%$ for sequences containing\nas little as $10$-$20$ elements \\cite{Bol1,Vrbik1,Vrbik2}. In this study, all $\\lambda$-evaluations are based\non the expression (\\ref{lamn})  and use data sequences that contain more than $25$ elements. \n\n\\textit{3. Mean Kolmogorov stochasticity score}. The mean $\\lambda$ can then be computed as\n\\begin{equation*}\n\t\\lambda^{\\ast}=\\int_0^{\\infty} \\lambda P(\\lambda)d\\lambda=\\int_0^{\\infty} \\Phi(\\lambda)d\\lambda,\n\\end{equation*}\nwhere we used integration by parts and the fact that the distribution $P(\\lambda)$ starts at $0$, $P(0)=0$, \nand approaches $0$ at infinity, $P(\\infty)=0$ (Fig.~\\ref{fig:stochs}B). \nIntegrating the Gaussian terms in expansion (\\ref{phi}) yields Mercator series\n\\begin{equation*}\n\t\\lambda^{\\ast}=\\sqrt{\\frac{\\pi}{2}}\\sum_{k=1}^{\\infty}(-1)^{k+1}\\frac{1}{k}=\\sqrt{\\frac{\\pi}{2}}\\ln2\n\t\\approx 0.8687.\n\\end{equation*}\n\n\\textit{4. $\\Phi(\\lambda)$ estimates}. For small $\\lambda$s, the Kolmogorov's $\\Phi$-function (\\ref{phi}) can\nbe approximated by \n\\begin{equation}\n\t\\Phi(\\lambda)\\approx\\frac{\\sqrt{2\\pi}}{\\lambda}e^{-\\pi^2\/8\\lambda^2},\n\t\\label{sml}\n\\end{equation} \nand for large $\\lambda$s, it is approximated by the two lowest-order terms in (\\ref{phi}), $\\Phi(\\lambda)\n\\approx 1-2 e^{-2\\lambda^2}$ \\cite{Kolmogorov,Stephens}. These formulae allow quick evaluations of the \n$\\lambda$-scores' cumulative probabilities outside of the ``stochastic typicality band,\" $\\lambda<0.4$ or\n$\\lambda>1.8$.\n\n\\textit{5. Arnold score}. Let us arrange the points of the sequence $X$ on a circle of length $L$ and consider\nthe arcs between pairs of consecutive elements, $x_i$ and $x_{i+1}$ (Fig.~\\ref{fig:stochs}C). If the lengths\nof these arcs are $l_1,l_2,\\ldots,l_{n}$, then the sum \n\\begin{equation}\n\tB =l_1^2+l_2^2+\\ldots+l_n^2\n\t\\label{B}\n\\end{equation} \ngrows monotonically from its smallest value $B_{\\min}=n(L\/n)^2=L^2\/n$, produced when the points $x_k$ lay\nequidistantly from each other, to its largest value, $B_{\\max}=L^2$, attained when all elements share the same\nlocation, with the mean $B^{\\ast}= B_{\\min}2n\/(n+1)\\approx 2B_{\\min}$ \\cite{ArnoldB1,ArnoldB2,ArnoldB3,ArnoldB4}.\n\nIntuitively, orderly arrangements appear if the elements ``repel''  each other, ``clumping'' is a sign of \nattraction, while independent elements are placed randomly. Hence the ratio $\\beta = B\/B_{\\min}$ can be used\nto capture the orderliness of patterns:\n\\begin{equation}\n\t\\begin{cases}\n\t\t\\,\\,\\beta(X)\\approx1, \\,\\,\\, & \\mbox{indicates atypically ordered, nearly equidistant sequences;} \\\\ \n\t\t\\,\\,\\beta(X)\\approx\\beta^{\\ast}\\approx2 \\,\\,\\, & \\mbox{marks statistically typical, commonly scattered sequences;}\\\\\n\t\t\\,\\,\\beta(X)\\gg\\beta^{\\ast} \\,\\,\\, & \\mbox{corresponds to clustering sequences.}\n\t\\end{cases}\n\t\\label{betas}\n\\end{equation}\n\n\\textit{6. The length $L$ of the circle} accommodating a random sample sequence $X$ in Arnold's method was \nselected so that the distance between the end points, $x_0$ and $x_n$, became equal to the mean arc length\nbetween the remaining pairs of neighboring points, $$l_n=|x_n-x_0|_{\\mod L}=\\frac{1}{n-1}\\sum_{i=1}^{n-1}l_i.$$\n\n\n\\begin{figure}[h]\n\n\t\\centering\n\t\\includegraphics[scale=0.75]{SupplFig4}\n\t\\captionsetup{width=.9\\linewidth}\n\t\\caption{{\\footnotesize\n\t\t\t\\textbf{Averaging over a simplex}. \\textbf{A}. If two coordinates $l_1$ and $l_2$ of a two-element\n\t\t\tsequence could independently vary between $0$ and $L$, then the pair $(l_1,l_2)$ would cover a \n\t\t\t$2D$ square. However, if the elements $(x_1,x_2)$ remain on a circle (orange dots below) then the \n\t\t\tequation (\\ref{L}) restricts $(l_1,l_2)$-values to the cube's diagonal (orange cross on the top panel),\n\t\t\ti.e., to a $1$-dimensional simplex.\n\t\t\t\\textbf{B}. A configuration of three points on a circle corresponds to a point on the diagonal section\n\t\t\tof a $L$-cube.\n\t\t\t\\textbf{C}. Tetrahedron---a section of a $4D$ cube---is the highest dimensional ($3D$) depictable \n\t\t\tsimplex $\\sigma^{(3)}$, which is used to schematically represent $n$-dimensional simplexes, $\\sigma^{(n)}$. \n\t\t\tAveraging over $l_i^2$ in (\\ref{Lmean}) involves integrating over it the $(n-1)$-dimensional layers of\n\t\t\t$\\sigma^{(n)}$.\n\t}}\n\t\\label{fig:cub}\n\n\\end{figure}\n\n\t\\textit{7. Mean Arnold stochasticity score}. A short derivation of $\\beta^{\\ast}$ is provided below for \n\tcompleteness, following the exposition given in \\cite{ArnoldB4}.\n\n\t\\begin{itemize}[leftmargin=0.34cm]\n\t\t\\item The $n$ arcs lengths $l_1,l_2,\\ldots,l_n$ produced by $n$ points, $X=\\{x_1,x_2,\\ldots,x_n\\}$, can\n\t\tbe viewed as the ``coordinates\" of $X$ in a $n$-dimensional sequence space.\tIf these coordinates could\n\t\tvary independently on a\tcircle of length $L$, then the sequences would be in one-to-one correspondence\n\t\twith the points of a $n$-dimensional hypercube with the side $L$. However, since the sum of $l_i$s must\n\t\tremain fixed,\n\t\t\\begin{eqnarray}\n\t\t\tl_1+l_2+...+l_n=L,\n\t\t\t\\label{L}\n\t\t\\end{eqnarray}\n\t\tthe admissible $l$-values occupy a hyperplane that cuts between the vertices $(0,0,\\ldots,0)$ and $(L,\n\t\tL,\\ldots,L)$. For example, the two-element sequences described by the coordinates $l_1$ and $l_2=L-l_1$\n\t\tcorrespond to the points on the diagonal of a $L$-square (Fig.~\\ref{fig:cub}A) and three-element sequences\n\t\tcorrespond to the points of a ``diagonal\" equilateral triangle in the $L$-cube (Fig.~\\ref{fig:cub}B). \n\t\tThe four-element sequences are represented by the points of a regular tetrahedron (Fig.~\\ref{fig:cub}C)\n\t\tand so forth. Thus, a generic $n$-sequence is represented by a point in a polytope spanned by $n$ \n\t\tvertices in $(n-1)$-dimensional Euclidean space---a $(n-1)$-\\textit{simplex}, $\\sigma^{(n-1)}$ \n\t\t\\cite{Alexandrov}.\n\t\t\n\t\t\\item The \\textit{defining property} of a simplex is that any sub-collection of its vertices spans a\n\t\tsub-simplex: a tetrahedron, $\\sigma^{(3)}$, is spanned by four vertices, any three of which span a triangle \n\t\t$\\sigma^{(2)}$---a ``face\" of $\\sigma^{(3)}$; any two vertices span an edge, $\\sigma^{(1)}$, between them,\n\t\tetc. \\cite{Alexandrov}. Correspondingly, a generic section of the $\\sigma^{(n-1)}$-simplex by a hyperplane\n\t\tis also a $\\sigma^{(n-2)}$-simplex (Fig.~\\ref{fig:cub}C).\n\t\t\n\t\t\\item Averaging the sum (\\ref{B}) requires evaluating the mean of each $l_i^2$,\n\t\t\\begin{equation}\n\t\t\t\\langle l_i^2\\rangle=\\frac{1}{V_{n-1}}\\int_{\\sigma^{(n-1)}} l_i^2 dV,\n\t\t\t\\label{Lmean}\n\t\t\\end{equation}\n\t\tfor $i=1,2,\\ldots,n$. Here $V_{n-1}$ refers to the volume of $\\sigma^{(n-1)}$ and ``$dV$\" refers to the\n\t\tvolume of a thin layer positioned at a distance $l_i$ away from the $i^{\\textrm{th}}$ face (Fig.~\\ref{fig:cub}C).\n\t\tBy the defining property of simplexes mentioned above, the base of this layer is a $(n-2)$-simplex \n\t\tspecified by the equation\n\t\t\\begin{equation*}\n\t\t\t\\sum_{j\\neq i}^n l_j=L-l_i,\n\t\t\\end{equation*}\n\t\twhich implies that the sides of this base have length $L-l_i$ (just as the sides of $\\sigma^{(n-1)}$\n\t\tdefined\tby (\\ref{L}) have length $L$). The volume of the thin layer is $dV=C(L-l_{i})^{n-2}dl_i$, so\n\t\tthat\n\t\t\\begin{equation*}\n\t\t\\langle l_i^2\\rangle=\\frac{C}{V_{n-1}}\\int_{0}^{L} l_i^2(L-l_{i})^{n-2}dl_i=\\frac{CL^{n+1}}{V_{n-1}}\n\t\t\t\\int_0^1 u^2(1-u)^{n-2}du,\n\t\t\n\t\t\\end{equation*}\n\t\twhere $u=l_i\/L$. Using the variable $v=1-u$ the latter integral yields:\n\t\t\\begin{equation*}\n\t\t\t\\langle l_i^2\\rangle=\\frac{CL^{n+1}}{V_{n-1}}\\int_0^1 (1-v)^2v^{n-2}dv=C\\left(\\frac{1}{n-1}-\n\t\t\t\\frac{2}{n}+\\frac{1}{n+1}\\right).\n\t\t\n\t\t\\end{equation*}\n\t\tThe volume of the  $\\sigma^{(n-1)}$-simplex is\n\t\t\\begin{equation*}\n\t\t\tV_{n-1}=\\int_0^LdV=\\frac{CL^{n-1}}{n-1};\n\t\t\t\\nonumber\n\t\t\n\t\t\\end{equation*}\n\t\thence the sum (\\ref{B}) divided by $B_{\\min}=L^2\/n$ yields\n\t\t\\begin{equation}\n\t\t\t\\beta_n=n^2\\left(1-2\\frac{n-1}{n}+\\frac{n-1}{n+1}\\right)=2\\frac{n}{n+1}\\approx \\beta^{\\ast}=2.\n\t\t\t\\label{dbb}\n\t\t\\end{equation}\n\t\\end{itemize}\n\t\\textit{8. Probability distributions of $\\beta$-values} form a family parameterized by the number of\n\telements in the sequence. As shown on Fig.~\\ref{fig:pbeta}, these distributions have a well-defined peak\n\tat $\\beta_{n}\\approx 2\\frac{n}{n+1}$ (see below) and rapidly decay as $\\beta$ approaches $1$ or for \n\t$\\beta>3.5$, which illustrates that typical $\\beta$-values, for all $n$, remain near the impartial\n\tmean $\\beta^{\\ast}\\approx 2$.\n\n\\begin{wrapfigure}{c}{0.55\\textwidth}\n\t\\centering\n\t\\includegraphics[scale=0.458]{Pbeta}\n\t\\caption{{\\footnotesize\n\t\t\t\\textbf{$\\beta$-distributions}. \\textbf{A}. Histograms of $\\beta$-values obtained for $10^6$\n\t\t\tsequences containing $n=17$, $25$, $50$, $125$ and $250$ elements peak in a vicinity of the \n\t\t\timpartial mean $\\beta^{\\ast}$ and rapidly decay for $\\beta\\lesssim 1.5$ and $\\beta\\gtrsim 3.5$.\n\t\t\t\\textbf{B}. The distribution of $\\beta'=\\beta-1$ for sequences containing about $n=25$ points\n\t\t\tis close to the universal Kolmogorov distribution $P(\\lambda)$ (red line, Fig.~\\ref{fig:stochs}A). \n\t}}\n\t\\label{fig:pbeta}\n\\end{wrapfigure}\n\n\t\\textit{9. The sliding window algorithm} can be implemented in two ways:\n\t\\begin{itemize}[nosep,leftmargin=0.34cm]\n\t\t\\item using a \\textit{fixed window} that may capture different numbers of events at each step, i.e.,\n\t\t$L_t=L$, but $n_t$ and $n_{t+1}$ may differ;\n\t\t\\item using a \\textit{fixed number of events} per window, i.e., $n_t=n$, but $L_t$ and $L_{t+1}$ may differ.\n\t\\end{itemize}\n\t\n\tIn both cases, the mean window width $\\bar{L}$ is proportional to the mean separation between nearest events, \n\t\\begin{equation}\n\t\t\\bar{l}\\equiv\\langle l_i\\rangle=\\langle x_{i+1}-x_i\\rangle,\n\t\t\\label{meangap}\n\t\\end{equation} \n\tand the mean number $\\bar{n}$ of the data points in the sample sequence, $\\bar{L}=\\bar{l}\\bar{n}$.\n\tThe resulting estimates for $\\lambda(t)$ and $\\beta(t)$ are nearly identical and, for qualitative assessments,\n\t can be used interchangeably.\n\t \n\t \\textit{10. Local averaging}. To build the dependencies between local averages, we ordered the values \n\t assumed by the independent variable, e.g., the speeds, from smallest to largest, $$\\{s_1,s_2,\\ldots\\}\\to\n\t \\{s'_1\\leq s'_2\\leq\\ldots\\},$$ subdivided the resulting sequence into consecutive groups containing $100$\n\t elements and averaged each set, \n\t  \\begin{equation*}\n\t \t\\hat{s}_i=\\frac{1}{100}\\sum_{k=-50}^{50}s'_{i+k}.\n\t \\end{equation*}\n\tSince each $s_{i_k}$ is associated with a particular moment of time $t_{i_k}$, we computed the averages of \n\tthe corresponding dependent variable, e.g., $\\lambda_{i_k}$,\n\t \\begin{equation*}\n\t \t\\hat{\\lambda}_i=\\frac{1}{100}\\sum_{k=1}^{100}\\lambda_{i_k}.\n\t \\end{equation*}\nSimilarly, ordering the $\\beta$-scores and evaluating their local means produces the $\\hat{\\beta}_i$-values,\nalong with the means of their $\\lambda_{i_k}$-counterparts that occur at the corresponding moments $t_{i_k}$,\nyielding $\\hat{\\beta}\\textrm{-}\\hat{\\lambda}$ dependence.\n\n\n\\newpage\n\\section{Supplementary Figures}\n\n\\renewcommand{\\figurename}{Suppl. Fig.}\n\\setcounter{figure}{0}\n\\begin{figure}[h]\n\n\t\\centering\n\t\\includegraphics[scale=0.95]{DTWdyn}\n\t\\caption{{\\footnotesize\n\t\t\\textbf{Matching waves using DTW}. \\textbf{A}. Top panels show the original shapes of the speed ($s(t)$,\n\t\tgray trace) and the $theta$-wave's Kolmogorov stochasticity score ($\\lambda_{\\theta}(t)$, red trace).\n\t\tBottom panel shows same functions, matched up by a sequences of local DTW-stretches. Clearly, the speed\n\t\tand the\t$\\lambda_{\\theta}$-stochasticity have the same qualitative shape during active behavior, while\n\t\tduring inactive moves (domains marked by light gray stripe) the connection is lost. \n\t\t\\textbf{B}. Same analyses carried for the mouse's acceleration ($a(t)$, gray trace) and Arnold \n\t\tstochasticity parameter ($\\beta_{theta}$, blue trace). The net amount of stretch required to match speed\n\t\tand $\\lambda_{\\theta}$ in this case (including the inactivity periods) is 26$\\%$, while the net stretch \n\t\tmatching $\\beta_{\\theta}$ and the acceleration is $\\sim 18\\%$.\n\t}}\n\t\\label{fig:bela}\n\n\\end{figure}\n\n\n\n\n\\renewcommand{\\figurename}{Suppl. Fig.}\n\\begin{figure}[h]\n\n\t\\centering\n\t\\includegraphics[scale=0.8]{LamLam}\n\t\\caption{{\\footnotesize\n\t\t\t\\textbf{Coupling between stochasticity parameters of different waves}. \n\t\t\t\\textbf{A}. The geometric layout of points with coordinates $(\\lambda_{\\theta},\\lambda_{\\gamma})$, \n\t\t\t$(\\lambda_{\\theta},\\lambda_{\\swr})$ and $(\\lambda_{\\gamma},\\lambda_{\\swr})$ shows that a given\n\t\t\t$\\lambda$-value produced by one wave may pair with any $\\lambda$-value that another wave is capable\n\t\t\tof producing, i.e., wave patterns deviate from their respective means largely independently from \n\t\t\teach other. Correspondingly, the locally averaged scores $\\hat{\\lambda}$ lay approximately \n\t\t\thorizontally, at the level of the corresponding means $\\langle\\lambda_{\\theta}\\rangle$, $\\langle\n\t\t\t\\lambda_{\\gamma}\\rangle$ and $\\langle\\lambda_{\\swr}\\rangle$ (see Fig.~\\ref{fig:thstoch}C, \n\t\t\tFig.~\\ref{fig:gstoch}B and Fig.~\\ref{fig:swr}B). \n\t\t\t\\textbf{B}. The $\\beta$-scores reveal similar lack of coupling between waves. Changes in \n\t\t\tlocally averaged $\\hat{\\beta}$-values of one wave not entrain consistent \n\t\t\t$\\hat{\\beta}$-changes of another wave. Thus, the (dis)orderliness of one wave does not\n\t\t\tenforce the (dis)orderliness of the other and the stochasticity dynamics discussed above provide \n\t\t\tindependent characterizations of of the LFP waves.\n\t}}\n\t\\label{fig:bb}\n\n\\end{figure}\n\n\n\\newpage\n\\clearpage\n\n\\section{References}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}