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+{"text":"\\section{Introduction and summary of results}\n\nThe existence of dark matter and dark energy is now firmly established phenomenologically \\cite{Bradac,Spergel} but the theoretical understanding is far from complete.  Einstein equations require ``exotic\" components in the right hand side corresponding  to about $\\%96$ of the total energy density today. Understanding the microscopic nature of these extra components is one of the most challenging and important problems faced by theoretical physics at present.\n\nSupersymmetric and exotic particles in the standard model are the best candidates for dark matter (for a review see \\cite{Jungman}).  Several experiments are now being devised for a direct detection of these particles. Alternative descriptions based on modifications to gravity have also been explored with interesting results. See \\cite{Sanders,Bekenstein,Moffat,Ferreira,Ferreira2,Carroll}, and references quoted therein, for some of these efforts and its consequences. For a recent review of the Einstein aether theory see \\cite{Jacobson}.\n\nThe problem of dark energy is somehow more recent, although the issue of the cosmological constant has been around for a long time. The discovery of an accelerating Universe \\cite{Expansion} resulted in deep changes in cosmology. The simplest explanation for this phenomena is a small positive cosmological constant, but many other possibilities have been explored (see \\cite{Carroll2,Lima} for recent reviews).\n\n~\n\nIn this paper we consider an action for general relativity coupled to a Born-Infeld theory. The Born-Infeld theory has as fundamental variable a symmetric connection  $C^\\rho_{\\ \\mu\\nu}(x)$. $C^{\\mu}_{\\ \\nu\\rho}$ has the same symmetries and transformation properties of the Christoffel symbol but is independent from it. The action is\n\\begin{equation}\\label{I}\n  I[g_{\\mu\\nu},C^{\\mu}_{\\ \\nu\\rho},\\Psi]  = {1 \\over 16 \\pi G} \\int d^4x \\left[ \\sqrt{|g_{\\mu\\nu}|} R  + {2 \\over \\alpha l^2}\\sqrt{\\left|g_{\\mu\\nu}-l^2 K_{(\\mu\\nu)}\\right| } \\right] + \\int d^4x\\, {\\cal L}_m(\\Psi,g_{\\mu\\nu}),\n\\end{equation}\nwhere $|A_{\\mu\\nu}|$, for any $A_{\\mu\\nu}$, denotes the absolute value of the determinant of $A_{\\mu\\nu}$. $K_{\\mu\\nu}$ is the ``Ricci\" curvature associated to $C^{\\mu}_{\\ \\nu\\rho }(x)$,\n\\begin{equation}\\label{Kmn}\nK_{\\mu\\nu} \\equiv K^{\\alpha}_{\\ \\mu\\alpha\\nu } \\ \\ \\ \\   \\ \\  (K^{\\mu}_{\\ \\nu\\, \\alpha\\beta} = C ^{\\mu}_{\\ \\nu\\beta,\\alpha} + C^{\\mu}_{\\ \\sigma\\alpha} C^{\\sigma}_{ \\ \\nu\\beta} - [\\alpha \\leftrightarrow \\beta]).\n\\end{equation}\nBesides Newton's constant, the action (\\ref{I}) has two extra parameters: $l$ is a length and $\\alpha$ is dimensionless.  $\\Psi$ denotes all baryonic fields and ${\\cal L}_{{ m}}$ the baryonic Lagrangian.\n\nThe action (\\ref{I}) is similar in spirit although different in interpretation to the Born-Infeld gravity action proposed by Deser and Gibbons \\cite{Deser-Gibbons},\n\\begin{equation}\\label{IDG}\nI[g_{\\mu\\nu}]=\\int \\sqrt{|g_{\\mu\\nu} - l^2 R_{\\mu\\nu} + X_{\\mu\\nu}(R)|}\n\\end{equation}\nand elaborated in \\cite{DG2}.  As discussed in \\cite{Deser-Gibbons}, the term $X_{\\mu\\nu}(R)$ must be chosen such that the action is free of ghost, and free of Schwarzschild-like singularities. The action (\\ref{IDG}) is an action for pure gravity, and it can be seen as a natural extension to spin two of the scalar $\\sqrt{|g_{\\mu\\nu} + \\partial_\\mu \\phi \\partial_\\nu\\phi| }$ and vector $\\sqrt{|g_{\\mu\\nu} + F_{\\mu\\nu}|}$ Born-Infeld (BI) theories.  For the scalar and vector BI theories the equations of motion are of second order.  For the spin two theory this is not automatic and requires the addition of $X_{\\mu\\nu}$.\n\nThe action (\\ref{I}), on the other hand, gives rise to second order equations because $K_{\\mu\\nu}(C)$ depends on first derivatives of the field $C^{\\mu}_{\\ \\nu\\rho }$. This action, however, is not an action for pure gravity but gravity coupled to $C^{\\mu}_{\\ \\nu\\rho }$. The equations of motion are discussed in the appendix and in Sec. \\ref{Eq\/Sec} below.\n\nIt is known (e.g. \\cite{Fradkin-T}) that general relativity with cosmological constant is dual to Eddington's action \\cite{Eddington} $I[C]\\sim\\int \\sqrt{|K_{\\mu\\nu}|}$. The action (\\ref{I}) can then be interpreted as general relativity interacting with its own dual field theory.\n\nThe action (\\ref{I}) can also be motivated by looking at general relativity without metric \\cite{B}. This interpretation will be discussed in Sec. \\ref{Sec\/g=0}.\n\nOur main goal in this paper is to argue that the field $C^{\\mu}_{\\ \\nu\\rho}$ has good properties to represent dark matter and dark energy. We shall study the equations of motion following from (\\ref{I}) and prove the following properties.\n\\begin{enumerate}\n\\item\nFor a cosmological model, there exist solutions where the expansion factor $a(t)$ behaves as $a(t) \\sim e^{Ht}$ for large $t$, and as $a(t)\\sim t^{2\/3}$ for small $t$.\nThe equation of state for the fluid interpolates between $p=0$ and $p=-\\rho$.    The parameters in the solution can be adjusted such that this field contributes to $\\sim 23\\%$ of the total matter energy density and $\\sim \\%73$ of vacuum energy density, as required by observations \\footnote{Couplings between dark matter and energy have appeared in \\cite{Comelli}, and in \\cite{Bertolami} involving a Chapligyn gas.}.\n\n\\item\nFor a spherically symmetric configurations, the action (\\ref{I}) predicts asymptotically flat rotation curves, as required by galactic dynamics.  The parameters involved in this solution can also be adjusted to deal with realistic situations.\n\\end{enumerate}\n\n\nWe would like to stress the simplicity of this proposal. The ``Born-Infeld\" term is all we need to account for both dark energy and dark matter, at least for the problems described above.  More complicated tests, like lensing, fluctuations, and others will be discussed elsewhere \\cite{BFS,BRR}. See also \\cite{Davi}.\n\n\n\n\n\n\n\n\\section{The Equations of Motion}\n\n\n\\label{Eq\/Sec}\n\n\\subsection{A bi-metric theory}\n\nThe fields varied in the action (\\ref{I}) are the metric $g_{\\mu\\nu}$ and the connection $C^{\\mu}_{\\ \\nu\\rho }$.  Both fields are independent. At the level of the equations of motion, the connection $C^{\\mu}_{\\ \\nu\\rho }$ can be written in terms of a second metric $q_{\\mu\\nu}$. (The full action can also be written as a bi-metric theory \\cite{andy}.) This action then represent a bi-metric theory if gravity. This result follows closely the structure of Eddington's theory \\cite{Eddington}. We shall postpone a detailed derivation for the appendix and include here only the result.\n\nLet $q_{\\mu\\nu}(x)$ be a rank two invertible symmetric tensor satisfying the metricity condition\n\\begin{equation}\nD_\\rho q_{\\mu\\nu}=0\n\\end{equation}\n{\\it with respect to} $C^{\\mu}_{\\ \\nu\\rho }$.  Since $C^{\\mu}_{\\ \\nu\\rho }$ is symmetric  this implies $C^{\\mu}_{\\ \\nu\\rho} = {1 \\over 2} q^{\\mu\\alpha} ( q_{\\alpha\\nu,\\rho} + q_{\\alpha\\rho,\\nu} - q_{\\nu,\\rho,\\alpha} )$, and for every $q_{\\mu\\nu}$ there is a unique $C^{\\mu}_{\\ \\nu\\rho }$.\n\nThe equations of motion derived from the action (\\ref{I}) can be written completely in terms of $g_{\\mu\\nu}$ and $q_{\\mu\\nu}$, and take the very simple form\n\\begin{eqnarray}\nG_{\\mu\\nu} &=& - {1 \\over l^2} \\sqrt{{q}\\over g}\\, g_{\\mu\\alpha}\\,q^{\\alpha\\beta}\\,  g_{\\beta\\nu} + 8\\pi G\\, T^{{\\scriptscriptstyle (m)}}_{\\ \\ \\mu\\nu} \\label{ee} \\\\\nK_{\\mu\\nu} &=& {1 \\over l^2}( g_{\\mu\\nu} + \\alpha\\, q_{\\mu\\nu})  \\label{Ke}\n\\end{eqnarray}\n$T^{{\\scriptscriptstyle (m)}}_{\\ \\mu\\nu}$ is the energy momentum tensor associated to the baryonic Lagrangian ${\\cal L}_{{\\scriptscriptstyle (m)}}$. $q^{\\mu\\nu}$ is the inverse of $q_{\\mu\\nu}$. The derivation of these equations is left for the appendix.\n\n~\n\nEquation (\\ref{ee}) is the Einstein equation.  The first term in the right hand side is the contribution from the Born-Infeld action.  Our main goal will be to prove that this fluid can account for dark matter and dark energy.\n\n\n\\subsection{The de-Sitter solution} \\label{SecdeSitter}\n\nThe de-Sitter spacetime is an exact solution to this theory.  This can be seen as follows. (The de-Sitter spacetime is expected to be relevant after matter becomes negligible so we set here $T^{(m)}_{\\ \\mu\\nu}=0$.)\n\nSuppose there exists solutions of the equations of motion with $R_{\\mu\\nu}=\\Lambda g_{\\mu\\nu}$.  It is direct to see that this implies that both metrics must be proportional,\n\\begin{equation}\nq_{\\mu\\nu}(x) = \\gamma\\,  g_{\\mu\\nu}(x)\n\\end{equation}\nwith $\\gamma$ a constant. The constant $\\gamma$ can be computed as follows.\nReplacing in (\\ref{Ke}) we derive,\n\\begin{equation}\nR_{\\mu\\nu} = {1 \\over l^2}\\left( \\gamma \\alpha + 1 \\right) g_{\\mu\\nu}.\n\\end{equation}\nReplacing in (\\ref{ee}) (with $T^{(m)}_{\\mu\\nu}=0$) we derive\n\\begin{equation}\nR_{\\mu\\nu} = {\\gamma \\over l^2} g_{\\mu\\nu}\n\\end{equation}\nConsistency determines $\\gamma$,\n\\begin{equation}\n\\gamma = {1 \\over 1-\\alpha}.\n\\end{equation}\nThus, the Born-Infeld field can behave as a cosmological constant with the value\n\\begin{equation}\n\\Lambda = {1 \\over 1-\\alpha}\\, {1 \\over l^2}.\n\\end{equation}\nThe value $\\alpha=1$ is a critical point where cosmological solutions ceases to exist.\nCuriously, we shall see that a good fit for the Friedman equation requires $\\alpha$ to be close, but not equal, to one.\n\n\n\n\\section{Friedman cosmological models}\n\nThe evolution equation for the scale factor in flat cosmological models is given by the Friedman equation (neglecting radiation)\n\\begin{equation}\\label{Fr}\n{\\dot a^2 \\over a^2} = {\\Omega_{bm} + \\Omega_{dm} \\over a^3} + \\Omega_{\\Lambda}.\n\\end{equation}\nCurrent values for the (relative) densities of barionic matter $\\Omega_{bm}$, dark matter $\\Omega_{dm}$ and vacuum energy $\\Omega_\\Lambda$ are,\n\\begin{equation}\\label{Omega}\n\\Omega_{bm} \\simeq 0.04, \\ \\ \\ \\ \\ \\ \\  \\Omega_{dm} \\simeq 0.23, \\ \\ \\ \\ \\ \\ \\Omega_\\Lambda \\simeq 0.73.\n\\end{equation}\nAmong the components appearing in the right hand side of (\\ref{Fr}), only the $\\sim 0.04$ fraction of baryonic matter is theoretically well-understood. The other $0.23+0.73=0.96$ fraction remains a great mystery.\n\n\\subsection{Goal of this section}\n\n\n\nThe goal of this section is to demonstrate that the field $C^{\\mu}_{\\ \\nu\\rho }$ behaves like dark matter for small times, and as dark energy for larger times. In other words, its equation of state evolves from $p=0$ into $p=-\\rho$.  Adjusting the parameters $\\alpha$ and $l$, plus initial conditions, the Born-Infeld field can account for both the $\\Omega_{dm}$ and $\\Omega_{\\Lambda}$ contributions in (\\ref{Fr}). Thus, the action (\\ref{I}), is capable to reproduce the correct evolution of the scale factor without adding neither dark matter nor dark energy.\n\nOur approach does not shed any light into the particular values for $\\Omega_\\Lambda,\\Omega_{dm},\\Omega_{bm}$ and other cosmological parameters. We shall only prove that $l$ and $\\alpha$ can be chosen such that the predictions from (\\ref{I}) are consistent with the Friedman equation (\\ref{Fr}).   In particular we have chosen here to set $k=0$ and consider only flat models. There is no particular reason for the choice other than simplicity. A full analysis with a varying $k$ and including other developments will be reported in \\cite{BFS}.\n\n\n\\subsection{The ansatz and equations}\n\\label{Equations}\n\n\nTo solve (\\ref{ee}) and (\\ref{Ke}) we assume that both $g_{\\mu\\nu}$ and $q_{\\mu\\nu}$ are homogeneous, isotropic and with flat spatial sections.  Using the gauge freedom in the time coordinate to fix $g_{tt}=-1$, the ansatz for $g_{\\mu\\nu}$ and $q_{\\mu\\nu}$ is then,\n\\begin{eqnarray}\\label{FRW}\ng_{\\mu\\nu}dx^\\mu dx^\\nu &=& -  dt^2 + a(t)^2 (dx^2 + dy^2 + dz^2), \\\\\nq_{\\mu\\nu}dx^\\mu dx^\\nu &=& - X(t)^2 dt^2 + Y(t)^2 (dx^2 + dy^2 + dz^2)\\label{qFRW}\n\\end{eqnarray}\nwhere $a(t),X(t),Y(t)$ are arbitrary functions of time to be fixed by the equations of motion and initial conditions.\n\nAs usual for flat models, and to match the choice made in (\\ref{Fr}), we set\n\\begin{equation}\na(t) |_{ \\scriptscriptstyle today }=1, \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\  H_0 = \\dot  a(t) |_{ \\scriptscriptstyle today }\n\\end{equation}\nand use $H_0$ to define a natural dimensionless time coordinate $H_0 t$.  The time coordinate in all expressions from now on refer to this choice.\n\nEquations (\\ref{ee},\\ref{Ke}) for the ansatz (\\ref{FRW},\\ref{qFRW}) become,\n\\begin{eqnarray}\n{\\dot a^2 \\over a^2} &=& {1 \\over 3l^2 H_0^2 } {Y^3\\over X}{1 \\over a^3} + {\\rho \\over \\rho_c} \\label{F} \\label{c1} \\\\\n\\left( {Y^3 \\over X} \\right)^. &=& 3 X Y a \\dot a  \\label{c2} \\\\\n{1 \\over X^2} {\\dot Y^2 \\over Y^2}  &=& {1 \\over 3l^2 H_0^2}\\left(  - {1 \\over 2 X^2} + \\alpha + {3 \\over 2} {a^2\\over Y^2}\\right), \\label{c3}\n\\end{eqnarray}\nplus second order equations related to (\\ref{c1}-\\ref{c3}) by Bianchi identities.\nWe have introduced the usual notation $\\rho_c = {3H_0^2 \\over 8\\pi G}$. $\\rho$ is the baryonic matter and we  shall assume\n\\begin{equation}\n{\\rho \\over \\rho_c} = {\\Omega_{bm} \\over a^3 }.\n\\end{equation}\n\nThe interpretation of equations (\\ref{c1}-\\ref{c3}) is straightforward. Equation (\\ref{F}) is the Friedman equation determining the time evolution of the scale factor $a(t)$.  The first term in the right hand side of (\\ref{c1}) is the contribution from the Born-Infeld field $C^{\\mu}_{\\ \\nu\\rho}$. Defining the density and pressure for the Born-Infeld field,\n\\begin{equation}\\label{rhop}\n\\rho_{{\\scriptscriptstyle BI}} = {1 \\over 8\\pi G l^2}{Y^3 \\over X} {1 \\over a^3}, \\ \\ \\ \\ \\ \\ \\ p_{{\\scriptscriptstyle BI}} = - {1 \\over 8\\pi G l^2} {XY \\over a}\n\\end{equation}\nthe right hand side of (\\ref{c1}) is simply ${1 \\over \\rho_{c}}(\\rho_{{\\scriptscriptstyle BI}} + \\rho)$.  Furthermore, in terms of $\\rho_{{\\scriptscriptstyle BI}}$ and $p_{{\\scriptscriptstyle BI}}$, equation (\\ref{c2}) takes the usual conservation form\n\\begin{equation}\n(\\rho_{{\\scriptscriptstyle BI}} a^3)^. = - p_{{\\scriptscriptstyle BI}} (a^3)^. .\n\\end{equation}\n\nEq. (\\ref{c3}) (``the Friedman equation for the metric $q_{\\mu\\nu}$\") provides the equation of state for $\\rho_{{\\scriptscriptstyle BI}}$ and $p_{{\\scriptscriptstyle BI}}$ allowing a full solution to the problem.  Note that using (\\ref{rhop}) the functions $X(t),Y(t)$ can be written in terms of $\\rho_{{\\scriptscriptstyle BI}}(t),p_{{\\scriptscriptstyle BI}}(t)$ and (\\ref{c3}) becomes a (differential) relation between these two functions. This equation of state thus have one free parameter represented as an initial condition.\n\nWe shall now show that $\\rho_{{\\scriptscriptstyle BI}}$ behaves like dark matter for small times, and like dark energy for large times.\n\n\n\\subsection{Asymptotic $a \\rightarrow 0$ and $a \\rightarrow \\infty$ behavior}\n\n\nDue to the complicated and non-linear character of equations (\\ref{c1}-\\ref{c3}) we shall study them by series expansions and numerically.\n\n~\n\nWe first  study the behavior for large values of $a$. In this regime, the baryonic matter density $\\rho \\sim a^{-3}$ does not contribute. (A radiation component would not contribute either.)  Neglecting the term $\\rho\/\\rho_c$, it is direct to see that the functions,\n\\begin{equation}\na(t) = a_0\\, e^{t\/C }, \\ \\ \\ \\ \\    X(t) = {1 \\over \\sqrt{1-\\alpha}} \\ \\ \\ \\ \\  Y(t) = {a_0 \\over \\sqrt{1-\\alpha}} \\, e^{t\/C},\n\\end{equation}\nwith $C = \\sqrt{3(1-\\alpha)}\\,l H_0$ provides an exact solution to (\\ref{c1}-\\ref{c3}). Thus, de-Sitter\\footnote{The existence of this exact solution is not at all surprising because  we already know that the general equations (\\ref{ee}) and (\\ref{Ke}) accepts solutions of the form $R_{\\mu\\nu} = \\Lambda g_{\\mu\\nu}$ when $q_{\\mu\\nu}$ is proportional to $g_{\\mu\\nu}$} space is a solution to (\\ref{c1}-\\ref{c3}) for large times. The constant $C$ measures the value of the associated vacuum density. In order for this solution to approach de-Sitter space with the correct exponent, we must impose\n\\begin{equation}\\label{LO}\n{1 \\over 3(1-\\alpha)l^2 H_0^2} = \\Omega_{\\Lambda}.\n\\end{equation}\n$H_0$ and $\\Omega_\\Lambda$ are determined by observations. This provides a first constraint on the parameters $l$ and $\\alpha$ entering in the action. We shall use (\\ref{LO}) to solve $l$ in terms of $\\alpha$.\n\n~\n\nNow, we study the $a(t) \\simeq 0$ region. In this regime, an exact solution is not available, but one can display a series expansion with the desired properties. The following series\n\\begin{equation}\na(t)= a_0\\, t^{2\/3} (1 + {\\cal O}(t^{4\/3})), \\ \\ \\ \\ \\ X(t) = x_0^3( 1 + {\\cal O}(t)), \\ \\ \\ \\ \\ Y(t) = x_0 (1 + {\\cal O}(t) )\n\\end{equation}\nprovide a solution to (\\ref{c1}-\\ref{c3}). The crucial point here is the exponent $t^{2\/3}$ in $a(t)$ meaning that $C^{\\mu}_{\\ \\nu\\rho}$ does indeed behave like matter for small times. The amount of dark matter is controlled by $a_0$.\n\n\\subsection{Numerical interpolation}\n\nOur final goal is to display a solution for $a(t)$ interpolating between $a(t)\\simeq t^{2\/3}$ for small $a(t)$ and $a(t)\\simeq e^{Ht}$ for large $a(t)$.  Furthermore, we would like this solution to exhibit the right amount of dark matter and dark energy.\nThis will be done by a numerical analysis.\n\n\n\nEquations (\\ref{c1}-\\ref{c3}) are of first order and thus we need to give three conditions $a_1=a(1)$, $X_1=X(1)$ and $Y_1=Y(1)$, plus the values of $\\alpha$ and $l$ to integrate them.  These are 5 parameters. However only two of them are independent. This can be seen as follows.\n\n\nFirst of all, for a flat model, we can choose $a(1)=1$. Second, in (\\ref{LO}), we already encounter one condition on the parameters to achieve the right evolution. Eq. (\\ref{LO}) allows to solve $l$ in terms of $\\alpha$. One extra condition follows by evaluating Eq. (\\ref{c1}) today,\n\\begin{equation}\n1 = {1 \\over 3 l^2 H_0^2} { Y_1^3 \\over X_1}  +  \\Omega_{bm},\n\\end{equation}\nfrom where we can solve $X_1$ in terms of $Y_1$ and $l$.  The remaining parameters are thus $\\alpha$ and $Y_1$.\n\n~\n\nWe have integrated (\\ref{c1}-\\ref{c3}) numerically varying $\\alpha$ and $Y_1$.  The resulting curve is compared with the evolution predicted by (\\ref{Fr},\\ref{Omega}). Our conclusions are the following.\n\n\\begin{enumerate}\n\n\\item\nFirst of all, there exists values of $\\alpha,Y_1$ such that the evolution predicted by (\\ref{Fr}) is almost undistinguishable from that following from (\\ref{c1}-\\ref{c3}), at least for the part of the Universe we can observe $0<t<1$. In Fig. \\ref{fig1}, the continuous line represents the Born-Infeld theory with $\\alpha=0.99$ and $Y_1 = 10.59$. The dots represents the Friedman evolution dictated by (\\ref{Fr}).\n\\begin{figure}[h]\n\\centerline{\\psfig{file=friedman.eps,width=6cm,angle=-90}}\n\\caption{ \\ $\\alpha=0.99, \\ Y_1=10.59$}\n\\label{fig1}\n\\end{figure}\nThe parameters were adjusted for a best fit, and in particular the big-bang occurs at $t=0.00731..$ in both theories.\n\n\\item  If $\\alpha$ is not close to one, there is no value of $Y_1$ to achieve a good fit with the Friedman equation.  The above picture corresponds to $\\alpha=0.99$. The value $\\alpha=0.9$ also gives a good fit, but smaller ones do not. $\\alpha>1$ does not work either.\n\nThe fact that $\\alpha \\sim 1$ to have a good fit is quite peculiar because the actual value $\\alpha=1$ is singular and the de-Sitter solution does not exist (See Sec. \\ref{SecdeSitter}).  In any case, recall that $\\alpha$ enters in the action as a coupling constant and is not subject to variations.  More testings on the theory should narrow the actual value of this parameter.\n\n\n\\item\nOf course no measurements exist for $t>1$, but it is interesting to explore the predictions of Born-Infeld theory to larger times. If one chooses the parameters such that the Big-Bang occurs at the same value of $t$ in both theories, then for large $t$ the expansion factor $a(t)$ grows slightly slower in the Born-Infeld theory.   Further details on this issue will be reported  elsewhere.\n\\end{enumerate}\n\n\n\n\\subsection{The evolution of the equation of state}\n\nAs we mention in Sec. \\ref{Equations}, the field $C^{\\mu}_{\\ \\nu\\rho}$ can be characterized by an energy density $\\rho_{{\\scriptscriptstyle BI}}$ and pressure $p_{{\\scriptscriptstyle BI}}$ whose expressions are given in (\\ref{rhop}).  The corresponding equation of state is,\n\\begin{equation}\n{p_{{\\scriptscriptstyle BI}} \\over \\rho_{{\\scriptscriptstyle BI}}} = - \\left( {a X \\over Y} \\right)^2\n\\end{equation}\nand we observe that the pressure is always negative.  Fig. \\ref{eqst} shows the evolution $0<t<3$ of the quotient $p_{{\\scriptscriptstyle BI}}\/\\rho_{{\\scriptscriptstyle BI}}$.\n\\begin{figure}[h]\n \\centerline{\\psfig{file=EquationState.eps,width=6cm,angle=-90 }}\n\\label{eqst}\n\\caption{ Evolution of the equation of state}\n\\end{figure}\nWe see clearly the interpolation between $p_{{\\scriptscriptstyle BI}}=0$ for small times, and $p_{{\\scriptscriptstyle BI}}=-\\rho_{{\\scriptscriptstyle BI}}$ for larger times.\n\n\n~\n\n\n\\section{Spherical symmetry and galactic rotation curves}\n\nIn this section we explore the action (\\ref{I}) on galactic scales where dark matter also plays an important role. (Dark energy is less relevant at this scale.) Stars orbiting galaxies have rotation curves not matching the observed luminous matter and this implies the existence of `dark matter'. See \\cite{Rubin} for a recent review. If the Born-Infeld theory (\\ref{I}) can be regarded as a good candidate for this exotic form of matter, it must account for these flat rotation curves.\n\nNumerical studies of matter interacting only gravitationally suggests the 2-parameter NFW \\cite{NFW} density profile for the dark matter halo,\n\\begin{equation}\\label{NFWp}\n\\rho_{{\\scriptscriptstyle NFW}}(r) = {a \\over r( 1 + r\/r_0)^2}.\n\\end{equation}\nThe density diverges at the origin but the total mass is finite. The rotation curve associated to this profile has a peak and decreases slowly with $r$.\n\nAnother popular density function is the (pseudo) Isothermic profile,\n\\begin{equation}\\label{ISOp}\n\\rho_{{\\scriptscriptstyle ISO}}(r) = {b \\over 1 + r^2\/r_0^2}\n\\end{equation}\nhaving a finite density at the origin. This halo leads to a rotation curve increasing  monotonically with an asymptotically flat region. See \\cite{deBlok} for an observational comparison of both profiles.\n\n~\n\n\nOur proposal is that the Born-Infeld field appearing in (\\ref{I}) can account for the dark matter present in galaxies. We shall prove that the action (\\ref{I}) gives rise to a rotation curve behaving like the NFW halo for small $r$, and as the (pseudo) Isothermal halo for large $r$.  In particular, the Born-Infeld theory yields an asymptotically flat rotation curve (at least to first order in the coupling constant).  To this end, we study in this section solutions with spherical symmetry to equations (\\ref{ee}-\\ref{Ke}).\n\n\\subsection{The ansatz and equations of motion}\n\n\nOur main interest is the `dark matter' contribution to the gravitational potential induced by $C^{\\mu}_{\\ \\nu\\rho}$.   We shall then neglect here the visible matter, set $T^{{\\scriptscriptstyle (m)}}_{\\ \\ \\mu\\nu}=0$ and study the solutions of\n\\begin{eqnarray}\nG_{\\mu\\nu} &=& - {1 \\over l^2} \\sqrt{{q}\\over g}\\, g_{\\mu\\alpha}\\,q^{\\alpha\\beta}\\,  g_{\\beta\\nu}\\, \\label{oee} \\\\\nK_{\\mu\\nu} &=& {1 \\over l^2}( g_{\\mu\\nu} + \\alpha\\, q_{\\mu\\nu})  \\label{oKe}\n\\end{eqnarray}\nwith spherical symmetry.\n\nFirst, note that since $g_{\\mu\\nu}$ and $q_{\\mu\\nu}$ are invertible, the right hand side of (\\ref{oee}) is different from zero at all points.  This represents a non-compact source.  Since for a galactic problem we expect the curvature to be very small, the right hand side of (\\ref{oee}) must be very small.  This is indeed true because\nthe parameter $l$ is of cosmological scale, and thus ${1 \\over l^2}$ is very small compared to any galactic scale.\n\nThis also means that we can treat the right hand side of (\\ref{oee}) and (\\ref{oKe}) as perturbations.  The parameter ${1 \\over l^2}$ measures the interaction between $g_{\\mu\\nu}$ and $q_{\\mu\\nu}$. If ${1 \\over l^2}\\rightarrow 0$, these two fields are decoupled and satisfy the order zero equations,\n\\begin{equation}\\label{zero}\nG_{\\mu\\nu}=0, \\ \\ \\ \\ \\ \\  K_{\\mu\\nu}=0\n\\end{equation}\n\nThe interactions can be incorporated perturbatively by expansions of the form,\n\\begin{eqnarray}\ng_{\\mu\\nu} &=& g_{\\mu\\nu}^{\\scriptscriptstyle{(0)}} + {1 \\over l^2 } \\, g_{\\mu\\nu}^{\\scriptscriptstyle{(1)}}  + {1 \\over l^4 } \\, g_{\\mu\\nu}^{\\scriptscriptstyle{(2)}} + \\cdots\\\\\nq_{\\mu\\nu} &=& q_{\\mu\\nu}^{\\scriptscriptstyle{(0)}} + {1 \\over l^2 } \\,q_{\\mu\\nu}^{\\scriptscriptstyle{(1)}}  + {1 \\over l^4}\\,  q_{\\mu\\nu}^{\\scriptscriptstyle{(2)}}\\, + \\cdots.\n\\end{eqnarray}\nwhere $g^{{\\scriptscriptstyle (0)}}_{\\mu\\nu}$ and $q^{{\\scriptscriptstyle (0)}}_{\\mu\\nu}$ satisfy the order zero equations (\\ref{zero}). [There exists a different perturbative scheme leading to a different sector of this theory. Instead of treating ${1 \\over l^2}$ as very small, one could start with the exact solution described in Sec. \\ref{SecdeSitter}, and study linearized fluctuations around that background. This yields a different set of solutions that will be studied elsewhere.]\n\n~\n\nWe start our discussion with the order zero fields  $g^{{\\scriptscriptstyle (0)}}_{\\mu\\nu}$ and $q^{{\\scriptscriptstyle (0)}}_{\\mu\\nu}$. The static, spherically symmetric solution to the order zero equation $G_{\\mu\\nu}=0$ is the Schwarzschild metric\n\\begin{equation}\\label{g0}\ng_{\\mu\\nu}^{\\scriptscriptstyle{(0)}}\\, dx^\\mu dx^\\nu  = -c^2\\left( 1 - {2MG \\over c^2 r} \\right)  dt^2 +  \\left( 1 - {2MG \\over c^2 r} \\right)^{-1} dr^2 + r^2d\\Omega^2.\n\\end{equation}\nHowever, $M$ represents the total baryonic mass and since we have set $T^{{\\scriptscriptstyle (m)}}_{\\ \\ \\mu\\nu}=0$ we also set\n\\begin{equation}\nM=0.\n\\end{equation}\nOur order zero $g^{{\\scriptscriptstyle (0)}}_{\\ \\mu\\nu}$ metric is thus flat space. The effects of baryonic matter can easily be incorporated at the end and will be studied in \\cite{BRR}.\n\n\nThe order zero equation for $q_{\\mu\\nu}$ is $K_{\\mu\\nu}=0$. The solution with spherical symmetry is also the Schwarzschild metric in the ``reciprocal\" metric $q_{\\mu\\nu}$,\n\\begin{equation}\\label{q0}\nq_{\\mu\\nu}^{\\scriptscriptstyle{(0)}}\\, dx^\\mu dx^\\nu    = -\\beta^2 c^2\\left( 1 - {w_0 \\over  \\tilde k(r)} \\right) dt^2 + \\left( 1 - {w_0 \\over \\tilde k(r)} \\right)^{-1}  \\tilde k'^2 dr^2  +  \\tilde k^2(r) d\\Omega^2.\n\\end{equation}\nHere, $\\tilde k(r)$ is an arbitrary function of $r$, $w_{0}$ is an arbitrary constant with dimensions of length (the $q_{\\mu\\nu}$ ``Schwarzschild radius\"), and $\\beta$ is a dimensionless constant.  This tensor solves $K_{\\mu\\nu}=0$ everywhere except at $\\tilde k=0$. Some comments on the metric (\\ref{q0}) are in order:\n\n\\begin{enumerate}\n\\item\nWe have chosen the radial coordinate $r$ such that $g^{{\\scriptscriptstyle (0)}}_{\\mu\\nu}$ has $r^2d\\Omega^2$ in the angular part. It is then {\\it not} correct to assume that the metric $q^{{\\scriptscriptstyle (0)}}_{\\mu\\nu}$ can also be written in terms of $r$ and with the same $r^2 d\\Omega^2$ in its angular part. This is role of the function $\\tilde k(r)$ which represents an arbitrary radial re-parametrization.  This function will be determined by the equations of motion.\n\n\\item\nIn the same way, the time scale $t$ is fixed in terms of the metric $g_{\\mu\\nu}$, and does not need to be the same for the metric $q_{\\mu\\nu}$. This the role of the dimensionless constant $\\beta$ entering in (\\ref{q0}). This constant will be important below.\n\n\\item\nIf $w_0\\neq 0$, the metric (\\ref{q0}) solves $K_{\\mu\\nu}=0$ everywhere, except at $k=0$. As we shall see below, the most interesting solution requires $w_0\\neq 0$, and explores $k(r)$ all the way to the horizon. This means that we are forced to interpret the metric (\\ref{q0}) as a black hole in the $q_{\\mu\\nu}$ space.  We shall restrict our discussion to the region,\n\\begin{equation}\n\\tilde k > w_0.\n\\end{equation}\n\\item\nFor $w_0\\neq 0$ it will be convenient to use a dimensionless radial coordinate,\n\\begin{equation}\nk \\equiv {\\tilde k \\over w_0}.\n\\end{equation}\nIn particular the horizon is now located at,\n\\begin{equation}\nk=1, \\ \\ \\ \\ \\ (\\mbox{horizon}).\n\\end{equation}\nFrom now on, all formulas refer to this coordinate.\n\n\n\\end{enumerate}\n\n\n~\n\nHaving chosen the zero order solutions to (\\ref{oee}) and (\\ref{oKe}), we now discuss the corrections induced but the right hand side of these equations. We only discuss here the first order correction to $g_{\\mu\\nu}$, proportional to ${1 \\over l^2}$. Since the right hand side of (\\ref{oee}) is already of order ${1 \\over l^2}$, it is enough to know $q_{\\mu\\nu}$ to order zero. [Note that $q_{\\mu\\nu}^{{\\scriptscriptstyle (0)}}$ contributes to $g_{\\mu\\nu}^{{\\scriptscriptstyle (1)}}$, $q_{\\mu\\nu}^{{\\scriptscriptstyle (1)}}$ contributes to $g_{\\mu\\nu}^{{\\scriptscriptstyle (2)}}$, and so on.]\n\n\nOur problem then reduces to replacing $q_{\\mu\\nu}$ given by (\\ref{q0}) in (\\ref{oee}) and solve for the metric $g_{\\mu\\nu}$ to first order in ${1 \\over l^2}$. The metric $g_{\\mu\\nu}$ must be spherically symmetric.  We then write,\n\\begin{eqnarray}\\label{metric}\nds^2 &=& -c^2\\left( 1 + {1 \\over c^2}\\Phi(r)\\right) dt^2 +\\left(1 - {2m(r) \\over c^2 r} \\right)^{-1} dr^2  + r^2d\\Omega^2,\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\n  \\Phi &=& \\Phi^{{\\scriptscriptstyle (0)}} + {1 \\over l^2}\\, \\Phi^{{\\scriptscriptstyle (1)}} + {1 \\over l^4}\\, \\Phi^{{\\scriptscriptstyle (2)}} + \\cdots \\\\\n   m &=& m^{{\\scriptscriptstyle (0)}} + {1 \\over l^2}\\, m^{{\\scriptscriptstyle (1)}}  + {1 \\over l^4}\\, m^{{\\scriptscriptstyle (2)}} + \\cdots.\n\\end{eqnarray}\nAs we have already discussed, in the approximation with no baryonic matter, the zero order solution is simply flat space and thus\n\\begin{equation}\n \\Phi^{{\\scriptscriptstyle (0)}}=0, \\ \\ \\ \\  m^{{\\scriptscriptstyle (0)}}=0.\n\\end{equation}\n\nTo first order we obtain the equations,\n\\begin{eqnarray}\n    {dm^{{\\scriptscriptstyle (1)}} \\over dr} \\left(1 - {1 \\over k}\\right) - {w_0^3  c_0^2 \\over 2 \\beta}\\, k^2{d k \\over dr} &=& 0, \\label{g11}\\\\\n  \\beta c^2w_0 k^2\\left(1 - {1 \\over k}\\right)  + 2 {dk\\over dr} u^{{\\scriptscriptstyle (1)}} &=& 0, \\label{g22} \\\\\n  {du^{{\\scriptscriptstyle (1)}} \\over dr} + \\beta c^2 w_0\\,r\\, {dk \\over dr}  &=& 0, \\label{g33}\n\\end{eqnarray}\nwhere we have re-defined $\\Phi^{{\\scriptscriptstyle (1)}}(r)$ in terms of a new function $u^{{\\scriptscriptstyle (1)}}(r)$ by\n\\begin{equation}\\label{phi1}\nr{d\\Phi^{{\\scriptscriptstyle (1)}} \\over dr}=   u^{{\\scriptscriptstyle (1)}}(r) + {m^{{\\scriptscriptstyle (1)}}(r) \\over r}.\n\\end{equation}\n[To first order, the equations only depend on $\\Phi'$ and this is why this redefinition does not spoil locality.]\n\n\n\n\n\\subsection{Full parametric solution. Two branches}\n\nEquations (\\ref{g11}-\\ref{g33}) are three non-linear equations for the three unknowns $m^{{\\scriptscriptstyle (1)}}(r),u^{{\\scriptscriptstyle (1)}}(r)$ and $k(r)$.  A much simpler set of equations can be obtained by changing the independent variable from $r$ to $k$.\n\nWe define the functions $u^{{\\scriptscriptstyle (1)}}(k),m^{{\\scriptscriptstyle (1)}}(k)$ and $r(k)$. Also, for any $f(r)$,\n\\begin{equation}\n{df\\!(r) \\over dr}  = \\left. {df(k) \\over dk}\\right\/ {dr \\over dk}.\n\\end{equation}\nPerforming these substitutions,  equations (\\ref{g1}-\\ref{g3}) become linear for the unknowns $m^{{\\scriptscriptstyle (1)}}(k),u^{{\\scriptscriptstyle (1)}}(k)$ and $r(k)$,\n\\begin{eqnarray}\n    {dm^{{\\scriptscriptstyle (1)}} \\over dk} \\left(1 - {1 \\over k}\\right) - {w_0^3  c_0^2 \\over 2 \\beta}\\, k^2  &=& 0 \\label{g1}\\\\\n  \\beta c^2 w_0 k^2\\left(1 - {1 \\over k}\\right){dr \\over dk}  + 2\\, u^{{\\scriptscriptstyle (1)}} &=& 0 \\label{g2} \\\\\n  {du^{{\\scriptscriptstyle (1)}} \\over dk} + \\beta c^2 w_0\\,r   &=& 0 . \\label{g3}\n\\end{eqnarray}\nNote in particular that $m^{{\\scriptscriptstyle (1)}}$ has decoupled from $u^{{\\scriptscriptstyle (1)}}$ and $r(k)$. The general solution can be found in closed form,\n\\begin{eqnarray}\n  r(k) &=& A_0 \\left( -\\left(k - {1 \\over 2}\\right)\\ln\\left(1-{1 \\over k}\\right) -1\\right) + B_0 \\left(k - {1 \\over 2} \\right) \\label{rk} \\\\\n  u^{{\\scriptscriptstyle (1)}}(k) &=& {1 \\over 2} \\beta c^2 w_0 \\left[ A_0 \\left( k^2 \\left( 1- {1 \\over k} \\right) \\ln\\left( 1 - {1 \\over k}\\right) +  k - {1 \\over 2} \\right) - B_0 ( k^2 - k) \\right] \\label{uk} \\\\\n  m^{\\scriptscriptstyle (1)}(k) &=& {w_0^3c^2\\over 2\\beta}\\left( {1 \\over 3}k^3 + {1 \\over 2}k^2 + k + \\ln(k-1) - h_0 \\right) \\label{mk}\n\\end{eqnarray}\nwhere $A_0,B_0$ and $h_0$ are integration constants.   This solution is real for $k>1$, that is outside the horizon in the reciprocal space $q_{\\mu\\nu}$.\n\n~\n\nTo explore the properties of the different solutions we first note that the function $r(k)$ displayed in (\\ref{rk}) diverges at two different values of $k$,\n\\begin{equation}\nk = \\infty, \\ \\ \\ \\ \\ \\ \\ \\ \\mbox{and} \\ \\ \\  \\ \\ \\ \\ \\ k=1.\n\\end{equation}\n\nSince the function $r(k)$ is a coordinate change and must be globally defined at least in the range $0<r<\\infty$, the derivative $dr\/dk$ must be different from zero everywhere. Note that if $A_0$ and $B_0$ have the same sign, then the function $r(k)$ has a maximum or minimum, which is not allowed.  This leaves two simple cases:\n\n\\begin{itemize}\n\\item {\\bf Linear branch:} $A_0<0,B_0>0$. In this case, $r(k)$ diverges for large $k$, and becomes zero at some finite value $k=k_0$.  For $k<k_0$ the coordinate $r(k)$ is negative and thus this region is not physical. For large $k$ the solution rapidly approaches a linear behavior. The physical range of the coordinate $k$ in this case is\n\\begin{equation}\nk_0 \\leq k < \\infty.\n\\end{equation}\nHowever, it can be easily seen that $m^{{\\scriptscriptstyle (1)}}\/r$ diverges quadratically for large $r$. This behavior is unacceptable.  The same divergency is observed for the potential $\\Phi(r)$. From now on we shall exclude this case.\n\n\\item {\\bf Logarithmic branch:} $A_0>0,B_0<0$. In this case, $r(k)$ diverges at $k=1$ and becomes zero at some finite value $k=k_0$. The physical range of the coordinate $k$ in this case is\n\\begin{equation}\n1 > k \\geq k_0\n\\end{equation}\nThe most salient and peculiar property of this branch is that infinity is mapped to the horizon in the metric $q_{\\mu\\nu}$. There is a strong\/weak relationship between both fields. The details of this branch are studied in the following paragraphs.\n\n\n\n\\end{itemize}\n\nFig. \\ref{bran} shows the behavior of the function $r(k)$ for each branch.\n\n\\begin{figure}[h]\n\\centerline{\\psfig{file=branches.eps,width=10cm,angle=0}}\n\\caption{Two branches}\n\\label{bran}\n\\end{figure}\n\n\n\n\n\n\\subsection{The logarithmic branch and asymptotically flat rotation curves}\n\nThe most important property of this branch is that the rotation curves are asymptotically flat.  Let us recall the relation between the Newtonian potential appearing in (\\ref{metric}) and the rotation speed of a (non-relativistic) object at distance $r$,\n\\begin{equation}\\label{prof}\nv(r) = \\sqrt{ r {d\\Phi(r) \\over dr}}.\n\\end{equation}\n(This follows from the geodesic equation.) On the other hand, the derivative of the potential $\\Phi$, to first order in ${1 \\over l^2}$, is given in terms of $u^{{\\scriptscriptstyle (1)}}$ and $m^{{\\scriptscriptstyle (1)}}$ in (\\ref{phi1}).  The rotation curve can be expressed as a parametric function ,\n\\begin{equation}\nv(k) = {1 \\over l} \\sqrt{ u^{{\\scriptscriptstyle (1)}}(k) + {m^{{\\scriptscriptstyle (1)}}(k) \\over r(k)}}, \\ \\ \\ \\ \\ \\   r=r(k)\n\\end{equation}\nwhere $u^{{\\scriptscriptstyle (1)}}(k),m^{{\\scriptscriptstyle (1)}}(k)$ and $r(k)$ are given in (\\ref{rk}-\\ref{mk}).\n\nFrom these expression is it direct to compute the limit,\n\\begin{eqnarray}\\label{vinf0}\nv_{\\infty}^2 & \\equiv & \\lim_{k\\rightarrow 1} v^2(k) \\nonumber\\\\\n&=&  {w_0 (\\beta^2 A_0^2 - 4 w_0^2) \\over 4A_0 \\beta l^2}\\, c^2\n\\end{eqnarray}\nwhich is indeed finite.\n\nHowever, this is not the whole story. We need to impose boundary conditions at $r=0$ ($k=k_{0}$) to ensure that the solution and in particular the rotation curve (\\ref{prof}) is well-behaved there too. This will imply the following constraints and redefinitions of  the parameters $A_0,B_0$ and $h_0$.\n\n\\begin{enumerate}\n\\item\nWe first express $B_0$ in terms of $k_0$, the point where $r(k_0)=0$. This gives the following expression for $B_0$,\n\\begin{equation}\nB_0 = {A_0 \\over 2k_0-1} \\left((2k_0-1) \\ln\\left(1 - {1 \\over k_0} \\right) + 2\\right).\n\\end{equation}\n\\item\nSecond, ${m^{{\\scriptscriptstyle (1)}}}\\over r$ must be finite at $r=0$.  This implies  that $m^{{\\scriptscriptstyle (1)}}(k)$ must vanish at $k=k_0$ and this fixes $h_0$ to be \\begin{equation}\n h_0 = {1 \\over 3} k_0^3 + {1 \\over 2}k_0^2 + k_0 + \\ln(k_0-1)\n\\end{equation}\n\\item\nFinally, the orbital velocity of an object at $r=0$ must be zero. This implies that $u^{{\\scriptscriptstyle (1)}} + m^{{\\scriptscriptstyle (1)}}\/r$ evaluated at $k=k_0$ must vanish. This is achieved by choosing the constant $A_0$ to be\n\\begin{equation}\n A_0 = {2 w_0k_0^2 (2k_0-1) \\over \\beta}\n\\end{equation}\n\\end{enumerate}\nIn summary, boundary conditions at $r=0$ fix $B_0,A_0$ and $h_0$ in terms of a new parameter $k_0$. The full solution is then characterized by three remaining constants. The length scale $w_0$, and two dimensionless numbers $\\beta$ and $k_0$.\n\n\n\\subsection{A better parametrization and examples}\n\nThe solution we have found is still parameterized by several numbers. The functions $r(k),v(k)$ depend on $l,c,\\beta,w_0,k_0$.   The first two, $l,c$ enter in the action and cannot be varied. In fact $l$ has been already constrained by the cosmological analysis.  The other three remaining parameters can be chosen to match a desired physical situation.  Before plotting examples is it convenient to choose a different basis for these three arbitrary parameters.\n\nFirst, the asymptotic velocity $v_{\\infty}$ computed in (\\ref{vinf0}) in terms of $k_0$ is\n\\begin{equation}\\label{vinf}\nv^2_\\infty  = {4k_0^6-4k_0^5+ k_0^4 - 1 \\over 2(2k_0-1)k_0^2}\\, {w_0^2 \\over l^2}\\, c^2\n\\end{equation}\nThis parameter is of course a natural observable which can be identified easily for most galaxies. We use this equation and express $w_0$ in terms of $v_\\infty$,\n\\begin{equation}\\label{w0}\nw_0 = \\,\\sqrt{{2(2k_0-1)k_0^2 \\over 4k_0^6-4k_0^5+ k_0^4 - 1}}\\, {l\\, v_\\infty \\over c}\n\\end{equation}\nSecond, the dimensionless parameter $\\beta$, which enter in (\\ref{q0}), can be redefined as\n\\begin{equation}\n\\beta = {l \\, \\over r_0} { v_\\infty \\over c}.\n\\end{equation}\nwhere $r_0$ is an arbitrary parameter with dimensions of length.\n\n\nWith these definitions, the functions $r(k),v(k)$ take the convenient form\n\\begin{equation}\nr(k) = r_0 f_1(k,k_0), \\ \\ \\ \\ \\ \\   v(k) = v_\\infty f_2(k,k_0).\n\\end{equation}\nThe arbitrary constant $r_0$ sets the length scale while $v_\\infty$ set the velocity scale. Since both are arbitrary, they can be fixed to any desired values to fit realistic curves.   The constant $k_{0}$ controls the shape of the curve and how fast it grows. Since there are three independent parameters, there will be a degeneracy when fitting these curves with observational data (this will be discussed in \\cite{BRR}).\nThe explicit expressions for $f_1,f_2$ are not very illuminating, and can be derived directly from the solution (\\ref{rk}-\\ref{mk}). Of course $f_2$ satisfies $f_{2}(1,k_0)=1$.\n\n~\n\nFig. (\\ref{figg1}) shows examples of the curve with $v_\\infty=100km\/sec$, $r_0$ fixed, and varying $k_{0}$. The top curve corresponds to $k_0=1.5$. As $k_0$ increases we observe a slower growth of the rotation curve. All curves asymptotically reach the value $v_\\infty=100km\/sec$.  The horizontal axis is expressed in terms of $r\/r_0$, and choosing $r_0$ one can fit any desired length scale.\n\n\n\\begin{figure}[h]\n\\centerline{\\psfig{file=galactic.eps,width=6cm,angle=270}}\n\\caption{Rotation curves for $k_0=50,15,5,1.5$.      }\n\\label{figg1}\n\\end{figure}\n\nIt is interesting to note that for values of $k_0$ smaller than $k_0 \\simeq 1.5$, the curves change shape.  Fig. (\\ref{figg2}) shows the rotation curve for $k_0=1.5,1.03,1.003,1.0005$. The top curve corresponds to $k_{0}=1.5$. As $k_{0}$ becomes smaller, the rotation curves growths more slowly.\n\\begin{figure}[h]\n\\centerline{\\psfig{file=galactic2.eps,width=6cm,angle=270}}\n\\caption{Rotation curves for $k_0=1.5,1.03,1.003,1.0005$.      }\n\\label{figg2}\n\\end{figure}\n\nNote that one does not expect the curves to be asymptotically flat to all orders. The solutions discussed here are only the first order approximation in the coupling ${1 \\over l^2}$. The next orders are necessary to extrapolate the result to large values of $r$, comparable with $l$.  Also, the near horizon region for the metric  $q_{\\mu\\nu}$ is singular in Schwarzschild  coordinates and thus a proper analysis in regular coordinates may also change the behavior near infinity.\n\n\\subsection{Final remarks}\n\nWe end this section with two extra comments regarding the solutions with spherical symmetry.\n\n\n\n\\sub{Orders of magnitude and Solar System:} The solutions we have considered contain a length scale, $w_0$. This parameter was replaced in  (\\ref{w0}) by the final speed $v_\\infty$, which is a better observable.  It is however interesting to estimate the values of $w_0$ for a realistic situation. We set  $l \\sim 10^6 kpc$ (cosmological length), and ${v_{\\infty} \\over c} \\sim {1 \\over 3} 10^{-3}$, for a typical situation with $v_\\infty \\sim 100 km\/sec$. Fig. \\ref{figw0} shows $w_0$ as a function of $k_0$.\n\n\\begin{figure}[h]\n\\centerline{\\psfig{file=w0.eps,width=6cm,angle=270}}\n\\caption{$w_0$(kpc) as a function of $k_0$.      }\n\\label{figw0}\n\\end{figure}\n\nFor $k_{0} > 3 $, $w_0$ is equal to a few kpc.  This is a natural galactic scale. With an optimistic viewpoint one can thus assign to $w_0$ some physical meaning determined by the length of the object observed. In other words, the tensor $q_{\\mu\\nu}$ is a field whose natural length scale of variation is determined by the object.\n\nNow, the natural dimensionless parameter which controls the corrections from flat space is ${w_0 \\over l}$. If we believe that the value of $w_0$ is comparable to the object of study, then for Solar System experiments ${w_0 \\over l}$ is too small, and the effects of $C^{\\mu}_{\\ \\nu\\rho}$ should not contribute.\n\n~\n\n\\sub{Central density:} The central density associated to $C^{\\mu}_{\\ \\nu\\rho }$ diverges linearly, as the NFW profile (\\ref{NFWp}). This can be seen by solving (\\ref{g11}-\\ref{g33}), for small values of $r$, as a series expansion. The series,\n\\begin{eqnarray}\n  k(r) &=& k_0 - {\\beta (k_0-1) \\over w_0k_0 }\\ r + {\\cal O}(r^2) \\\\\n  m^{{\\scriptscriptstyle (1)}}(r) &=& -{w_0^2k_0^2 c^2 \\over 2} \\ r  + {\\cal O}(r^2) \\\\\n  u^{\\scriptscriptstyle (1)}(r) &=& {w_0^2k_0^2c^2 \\over 2} + {\\cal O}(r^2)\n\\end{eqnarray}\nsolve (\\ref{g11}-\\ref{g33}) with the boundary condition $v(r)\\rightarrow 0$ as $r\\rightarrow 0$. With this solution at hand we can compute the behavior of the associated mass density,\n\\begin{eqnarray}\n  4\\pi G\\, \\rho(r) &=& {1 \\over r^2}( r^2 \\Phi' )'  \\\\\n   &\\simeq & { 2(k_0-1)w_0 c^2 \\beta \\over l^2\\, r} + {\\cal O}(1)\n\\end{eqnarray}\nwith a linear divergency, as anticipated.\n\n\\section{Eddington action, the equivalence principle and $g_{\\mu\\nu}=0$}\n\n\\label{Sec\/g=0}\n\n\nOur proposal for dark matter and dark energy is summarized in the action (\\ref{I}). Once the action is written one can ``roll down\" exploring its predictions and consequences by usual methods. This is what we have done so far. However, it is also interesting to ``climb up\" and attempt a derivation, or at least a good motivation to include the Born-Infeld term in the gravitational action.\n\nWe start this section recalling a well-known effect. Consider a system of $N$ spins. If no external field is applied (and the temperature is not too small) the macroscopic average is $\\langle \\vec{S} \\rangle = 0$.  On the contrary, in the presence of an external field, $H_{ext}$, the symmetry is broken, the spins align and produce a non-zero macroscopic average $\\langle \\vec{S} \\rangle_{\\vec{H}_{ext}}\\neq 0$. It then follows that the total magnetic field felt by a charge $q$ is\n\\begin{equation}\n\\vec{H}_T = \\vec{H}_{ext} + \\langle \\vec{S} \\rangle_{\\vec{H}_{ext}}.\n\\end{equation}\nThe orbit of the charge will obey the Lorentz equation with $\\vec{H}_T$ not $\\vec{H}_{ext}$. If we did not know about spins the contribution $\\langle \\vec{S} \\rangle_{\\vec{H}_{ext}}$ would be interpreted as a sort of `dark' magnetic field. If the temperature is below the Curie temperature, the external field could be removed and the spins remain in their `ordered' state with $\\langle \\vec{S} \\rangle_0 \\neq 0$.\n\nLet us now describe an analog of this effect in the theory of gravity.  Topological manifolds are invariant under the full diffeomorphism group. Riemannian manifolds are invariant only under the subgroup of isometries of the metric. The state $g_{\\mu\\nu}=0$ represents the unbroken state of general relativity \\cite{Witten88}, and the  introduction of a metric breaks the symmetry. The natural geometrical analog of the external field $\\vec{H}_{ext}$ is the metric tensor $g_{\\mu\\nu}$. (See \\cite{Horowitz,Giddings,Guendelman} for other discussions on the state $g_{\\mu\\nu}=0$, and \\cite{Witten07} for a recent critical viewpoint.)\n\nWe shall treat the metric as an external field which can be switched on and off\\footnote{In this picture, the big-bang could be understood as a smooth transition from a manifold without metric into a Riemanian manifold.}. Our first goal is to explore fields that can be defined in the absence of a metric. The simplest example is given by a connection $C^{\\mu}_{\\ \\nu\\rho}(x)$.  In fact, Eddington introduced a purely affine theory a long time ago \\cite{Eddington},\n\\begin{equation}\\label{edd0}\nI_0[C] = \\kappa \\int d^4 x\\, \\sqrt{K_{\\mu\\nu}(C)}\n\\end{equation}\nwhere $K_{\\mu\\nu}$ is the curvature associated to the connection $C^{\\mu}_{\\ \\nu\\rho}(x)$ (see Eqn. (\\ref{Kmn})).  This action is invariant under spacetime diffeomorphism and yields second order differential equations for the field $C^{\\mu}_{\\ \\nu\\rho}$.  Eddington action was extensively studied as a purely affine theory of gravity, and also as a possible unification of gravity and electromagnetism \\cite{Eddington,Poplawski}. We take here a different interpretation and let the field $C^{\\mu}_{\\ \\nu\\rho}$ be an independent degree of freedom.\n\nWe now turn on the external field $g_{\\mu\\nu}$ and study the effects of both $g_{\\mu\\nu}$ and $C^{\\mu}_{\\ \\nu\\rho}$ on particles. The first problem is to determine  the action for the coupled system.  We do not want to introduce ghost or higher derivatives. The action  (\\ref{edd0}) is already free of anomalies. So we start by adding the standard Einstein-Hilbert action for $g_{\\mu\\nu}$ and consider\n\\begin{equation}\n\\int d^4 x  \\left( \\sqrt{g}R  + \\kappa \\sqrt{K_{\\mu\\nu}}\\  \\right).\n\\end{equation}\nWith this action, the fundamental fields $g_{\\mu\\nu}$ and $C^{\\mu}_{\\ \\nu\\rho}$ are decoupled. To make the theory more interesting we add interactions. The most attractive  theory (although not unique) having second order field equations is the Einstein-Born-Infeld action introduced in Eq. (\\ref{I}).\n\nAn important point now is to define the geodesic equation for the coupled system. In the presence of a metric $g_{\\mu\\nu}$ there is a natural affine connection $\\Gamma^{\\mu}_{\\ \\nu\\rho}$ represented by the Christoffel symbol,\n\\begin{equation}\\label{chr}\n\\Gamma^\\mu_{\\ \\nu\\rho} = {1 \\over 2}g^{\\mu\\sigma} ( g_{\\sigma\\nu,\\rho} + g_{\\sigma\\rho,\\nu} - g_{\\nu\\rho,\\sigma}).\n\\end{equation}\nThe question is, should geodesics be defined with respect to $C^{\\mu}_{\\ \\nu\\rho}$, $\\Gamma^{\\mu}_{\\ \\nu\\rho}$, both?  In order to comply with the equivalence principle we shall postulate that particles only couple to the metric and not to the connection $C^{\\mu}_{\\ \\nu\\rho }$.  The geodesic equation then take the usual form\n\\begin{equation}\\label{geo}\n\\ddot x^{\\mu} + \\Gamma^{\\mu}_{\\ \\alpha\\beta} \\dot x^\\alpha \\dot x^\\beta=0,\n\\end{equation}\nwhere $\\Gamma^{\\mu}_{\\ \\alpha\\beta}$ is the Christoffel symbol (\\ref{chr}).  Observe that the metric satisfies the equations (\\ref{ee}) and is coupled to the field $C^{\\mu}_{\\ \\nu\\rho }$. In this sense, $C^{\\mu}_{\\ \\nu\\rho }$ does contribute to $g_{\\mu\\nu}$ and indirectly affects the motion of particles. This is how the field $C$ can explain flat rotation curves.\n\nNow, the analogy with spin systems can be pushed a little bit further.  We have seen in the cosmological analysis that for large times the system approaches the de-Sitter solution (see Sec. \\ref{SecdeSitter}), and in particular the metric $q_{\\mu\\nu}$ becomes proportional to $g_{\\mu\\nu}$, $q_{\\mu\\nu}\\rightarrow \\lambda g_{\\mu\\nu}$. One can interpret this fact as analogous to the alignment of spins along the direction of the applied field, $\\langle \\vec{S} \\rangle  \\rightarrow \\lambda \\vec{H}_{ext}$. Of course, to support this interpretation one would need to consider generic initial conditions. This will be analyzed elsewhere.\n\nFinally, recall that when the external  magnetic field is removed, spins can have a spontaneous non-zero average $\\langle \\vec{S} \\rangle$, and this vector generates forces on charged particles.  Is there a gravitational analogue to this effect? The gravitational force is measured by the connection (\\ref{chr}), entering in the geodesic equation. The external field is the metric. Now, as the metric is removed, the Christoffel connection becomes ${0 \\over 0}$, with the same scaling weight in the numerator and denominator. For a large class of paths the limit is a finite function.  Since the only connection available at $g_{\\mu\\nu}=0$ is $C^{\\mu}_{\\ \\nu\\rho}$, it is tempting to conjecture that $\\Gamma^{\\mu}_{\\ \\nu\\rho} \\rightarrow C^{\\mu}_{\\ \\nu\\rho}$, as the metric is removed. In this way, the geodesic equation has a non-trivial limit when the metric vanishes, and particles will feel `forces'.  These are not forces in the usual sense because there is no metric.  (Although note that a geodesic equation, defined by parallel transport, can be introduced without a metric.)  The limit $g_{\\mu\\nu}\\rightarrow 0$ was the key ingredient employed in \\cite{B} for a different approach to understand dark matter as an effect associated to a topological manifold. To make these ideas precise a theory describing the process $g_{\\mu\\nu}\\rightarrow 0$ is necessary. We hope to come back to this interpretation in the future.\n\n\n\n\\section{Conclusions}\n\nDark matter and dark energy have unique properties and their understanding in one of the most crucial problems faced by theoretical physics today.   Dark matter does not interact with normal matter and this property has motivated us to look for fields which have this property somehow ``built in\". We have explored gravity coupled to connection $C^{\\mu}_{\\ \\nu\\alpha}$ field with a Born-Infeld action.\n\nThis theory comply with the main background properties normally attributed to dark matter and dark energy. First, the evolution of the scale factor in cosmological models has the right time dependence interpolating between pressureless matter and a cosmological constant.\n\nAt galactic scales dark energy is less relevant but dark matter still plays an important role. By an approximation valid for distances much smaller to the Hubble radius we have solved the equations of motion for spherical objects and find the expected rotation curves.  These curves satisfy the basic asymptotic flatness observed in galaxies providing new support for this proposal.\n\nWe have left several topics for the future. The stability of this theory and the study of primordial fluctuations are important to determine the CMB anisotropies. This will be reported in \\cite{BFS}.  On galactic scales a systematic fit with observational curves is necessary. This issue is presently under study and will be reported in \\cite{BRR}.\n\n\n\n\n\n\n\\section{Appendix. Derivation of the equations of motion}\n\n\nThe fields which are varied in the action (\\ref{I}) are the metric $g_{\\mu\\nu}$ and the connection $\\Gamma^{\\mu}_{\\ \\nu\\rho}$. The equations of motion for the metric follow by a straightforward variation of the action. The result is\n\\begin{equation}\\label{1}\nG_{\\mu\\nu} = \\sqrt{{|g_{\\mu\\nu}- l^2 K_{(\\mu\\nu)}| \\over |g_{\\mu\\nu}| }} \\ g_{\\mu\\alpha}\\left({1 \\over g-l^2 K }\\right)^{\\alpha\\beta} g_{\\beta\\nu} + 8\\pi G \\, T^{{\\scriptscriptstyle (m)}}_{\\mu\\nu}\n\\end{equation}\nThis equation can be drastically simplified by using the equation of motion for the connection $\\Gamma^{\\mu}_{\\ \\nu\\rho}$. This equation is derived in two steps.  First, since the action only depends on the curvature $K_{\\mu\\nu}$ once can compute the variation using the chain rule,\n\\begin{equation}\n{\\delta I \\over \\delta \\Gamma^{\\mu}_{ \\nu\\rho}} = \\int {\\delta I \\over \\delta K_{(\\alpha\\beta)}} \\, {\\delta K_{(\\alpha\\beta)} \\over \\delta \\Gamma^{\\mu}_{\\nu\\rho}}\n\\end{equation}\nJust like in Eddington \\cite{Eddington} theory one finds  by direct variation that the combination\n\\begin{equation}\\label{K00}\n\\sqrt{q}{q}^{\\mu\\nu}  \\equiv -{1 \\over \\alpha}\\sqrt{|g_{\\mu\\nu} - l^2 K_{\\mu\\nu}|} \\left({1 \\over g-l^2 K }\\right)^{\\mu\\nu}\n\\end{equation}\nsatisfies\n\\begin{equation}\nD_\\rho(\\sqrt{q}{q}^{\\mu\\nu})=0\n\\end{equation}\nwhere $D_\\rho$ is the covariant derivative built with the connection $\\Gamma_{{\\scriptscriptstyle 0}}$.  Since $\\Gamma^{\\mu}_{\\ \\nu\\rho}$ is symmetric, this equation imply\n\\begin{equation}\n\\Gamma^{\\mu}_{ \\nu\\rho} = {1 \\over 2} q^{\\mu\\alpha} ( q_{\\alpha\\nu,\\rho} + q_{\\alpha\\rho,\\nu} - q_{\\nu,\\rho,\\alpha} )\n\\end{equation}\nWe thus write $\\Gamma^{\\mu}_{\\ \\nu\\rho}$ in terms of $q_{\\mu\\nu}$. The equation (\\ref{K00})  now depends only on $q_{\\mu\\nu}$. Taking the determinant at both sides, and inverting one readily derives (\\ref{Ke}).\n\nThe final simplification follows by noticing that the right hand side of (\\ref{1}) contains $\\sqrt{q}q^{\\mu\\nu}$.  Thus, using (\\ref{K00}), Eq. (\\ref{1}) is transformed into (\\ref{ee}).\n\nThe analysis of these equations is greatly simplified by using the bi-metric formalism \\cite{andy}.\n\n~\n\n~\n\n\n\\section{Acknowledgements}\n\nThe author would like to thank S. Carlip, P. Ferreira, A.Gomberoff, M. Henneaux, A. Reisenegger, D. Rodrigues,  N. Rojas, C. Skordis and S. Theisen for useful comments and discussions.  The author was partially supported by Fondecyt Grants (Chile) \\#1060648 and \\#1051084.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nBerry in his pioneer work raised a fundamentally important concept known as\ngeometric phase (GP) in addition to the usual dynamic phase accumulated on\nthe wave function of a quantum system, provided that the Hamiltonian varies\nwith multi-parameters cyclically and adiabatically \\cite{Berry}. At the\npresent time the GP with extensive generalization along many directions has\nwide applications in various branches of physics \\cite{Berry2,Shapere,Bohm}.\n\nRecently, the close relation between GP and quantum phase transition (QPT)\nhas been gradually revealed \\cite{Carollo,Zhu,Hamma} and increasing interest\nhas been drawn to the role of GP in detecting QPT for various many-body\nsystems \\cite{Zhu08,Chen,interests}, which, as a matter of fact, is also a\nnew research field in condensed matter physics \\cite{Sachdev,Sondhi}. QPT\nusually describes an abrupt change in the ground state of a many-body system\ninduced by quantum fluctuations. The phase transition between ordered and\ndisordered phases is accompanied by symmetry breaking, which can also be\ncharacterized by Landau-type order parameters.\n\nOn the other hand, a new type of QPT called topological quantum phase\ntransitions (TQPT) has attracted much attention. The first non-trivial\nexample is the fractional quantum Hall effect \\cite{Tsui,Laughlin}. In the\nlast decade, several exactly soluble spin-models with the TQPT, such as the\ntoric-code model \\cite{Kitaev03}, the Wen-plaquette model \\cit\n{Wen-plaquette,Wen08} and the Kitaev model on a honeycomb lattice \\cit\n{Kitaev}, were found. In contrast to the conventional QPT governed by local\norder parameters \\cite{Sachdev}, the TQPT can be characterized only by the\ntopological order \\cite{WenBook}. As good examples to illustrate the\nunderlying physics, different methods are developed to describe the TQPT in\nthe Kitaev honeycomb model \\cite{T. Xiang07,T. Xiang08,H. D. Chen07,H. D.\nChen08,S. Yang,Gu}. In Ref. \\cite{T. Xiang07}, Feng \\textit{et.al.} obtained\nthe local order parameters of Landau type to characterize the phase\ntransition by introducing Jordan-Wigner and spin-duality transformations\ninto the Majorana representation of the honeycomb model. Gu \\textit{et.al.}\nshowed an exciting result of the ground-state fidelity susceptibility \\cit\n{S. Yang,Gu}, which can be used to identify the TQPT from the gapped $A$\nphase with Abelian anyon excitations to gapless $B$ phase with non-Abelian\nanyon excitations.\n\nQuite recently, Zhu \\cite{Zhu} showed that the ground-state GP in the $XY$\nmodel is non-analytic with a diverged derivative with respect to the field\nstrength at the critical value of magnetic field. Thereupon, the relation\nbetween the GP and the QPT is established. Nevertheless, much attention has\nbeen paid to the QPT, while effort devoted to the relation between the GP\nand the TQPT is very little. The present paper is devoted to exploiting the\nGP of the Kitaev honeycomb model as an essential tool to establish a\nrelation between the GP and the TQPT and reveal the novel quantum\ncriticality. Unlike the GP in the usual lattice-spin model for the QPT,\nwhich is generated by the single-spin rotation of each lattice-site, the\nsimultaneous rotation of linked two spins in one unit-cell seems crucial to\ndescribe the TQPT in the honeycomb model. The non-analyticity of GP at the\ncritical points with a divergent second-order derivative with respect to the\ncoupling parameters shows that the TQPT is the second-order transition and\ncan be well described by the GP.\n\nIn Sec. II, the ground state wave function and energy spectrum of the Kitaev\nhoneycomb model are presented. After introducing a correlated rotation of\ntwo $z$-link spins in each unit-cell, the ground-state GP and its\nderivatives are obtained explicitly in Sec. III. Sec. IV is devoted to\ninvestigating the scaling behavior of the GP. A brief summary and discussion\nare given in Sec. V.\n\n\\section{The Kitaev honeycomb model and spectrum}\n\nThe Kitaev honeycomb model shown in Fig. \\ref{fig1}(a) is firstly introduced\nto illustrate the topologically fault-tolerant quantum-information\nprocessing \\cite{Kitaev03,Kitaev,Nayak}. In this model, each spin located at\nvertices of the lattice interacts with three nearest-neighbor spins through\nthree types of bonds, depending on their directions. By using the Pauli\noperators $\\sigma ^{a}$ $(a=x,y,z)$, the corresponding Hamiltonian is\nwritten a\n\\begin{equation}\nH=-J_{x}\\!\\!\\!\\sum_{x\\mbox{-links}}\\!\\!\\!\\sigma _{j}^{x}\\sigma\n_{k}^{x}-J_{y}\\!\\!\\!\\sum_{y\\mbox{-links}}\\!\\!\\!\\sigma _{j}^{y}\\sigma\n_{k}^{y}-J_{z}\\!\\!\\!\\sum_{z\\mbox{-links}}\\!\\!\\!\\sigma _{j}^{z}\\sigma\n_{k}^{z},  \\label{Kitaev model}\n\\end{equation\nwhere $j$, $k$ denote the two ends of the corresponding bond, and $J_{a}$\nare coupling parameters. After introducing a special notation $K_{jk}=\\sigma\n_{j}^{a}\\sigma _{k}^{a}$, where the indexes $a$ depend on the types of links\nbetween sites $j$ and $k$ (so we also write it into $a_{jk}$ in the\nfollowing text for perspicuousness), Hamiltonian (\\ref{Kitaev model}) can be\nrewritten into a compact form\n\\begin{equation}\nH=-\\frac{1}{2}\\sum_{\\langle j,k\\rangle }J_{a_{jk}}K_{jk}.\n\\label{Kitaev modelK}\n\\end{equation}\n\n\\begin{figure}[h]\n\\centering \\vspace{0cm} \\hspace{0cm} \\scalebox{0.8}\n\\includegraphics{fig1.eps}}\n\\caption{(Color online) (a) Kitaev honeycomb model, in which one spin\ninteracts with three nearest-neighbor spins through three types of bonds,\ndepending on their direction. A unit-cell with $x$, $y$ and $z$ links and\ngraphic representation of Hamiltonian (\\protect\\ref{HM}) with Majorana\noperators are marked by the red dotted line. (b) Phase diagram of the model,\nwhere $A$ phase is gapped and $B$ phase is gapless.}\n\\label{fig1}\n\\end{figure}\n\nIt has been known that the Kitaev honeycomb model can be solved exactly by\nintroducing Majorana fermion operators, which are defined as \\cite{Kitaev,S.\nYang}\n\\begin{equation}\n\\sigma ^{x}=ib^{x}c,\\quad \\sigma ^{y}=ib^{y}c,\\quad \\sigma ^{z}=ib^{z}c.\n\\label{MajoranaFermion}\n\\end{equation\nGenerally, a set of Majorana operators $M=\\left\\{\nb^{x},b^{y},b^{z},c\\right\\} $ can be employed to describe a spin by two\nfermionic modes. They are Hermitian and obey the relations $m^{2}=1$ and \nmm^{\\prime }=-m^{\\prime }m$ for $m,m^{\\prime }\\in M$ and $m\\neq m^{\\prime }\n. Moreover, in the Hilbert space with a spin described by two fermionic\nmodes, the relation $b^{x}b^{y}b^{z}c\\left\\vert \\Psi \\right\\rangle\n=\\left\\vert \\Psi \\right\\rangle $\\ must be satisfied to ensure the obeying of\nthe same algebraic relations as $\\sigma^{x}$, $\\sigma ^{y}$, and $\\sigma\n^{z} $ \\cite{Kitaev}.\n\nDrawing on the operators (\\ref{MajoranaFermion}) to the Kitaev honeycomb\nmodel, Hamiltonian (\\ref{Kitaev modelK}) is given b\n\\begin{equation}\nH=\\frac{i}{2}\\sum_{\\langle j,k\\rangle}\\hat{u}_{jk}J_{a_{jk}}c_{j}c_{k},\n\\label{HM}\n\\end{equation}\nwhere $\\hat{u}_{jk}=ib_{j}^{a_{jk}}b_{k}^{a_{jk}}$. Fig. \\ref{fig1}(a) also\nshows the structure of Hamiltonian (\\ref{HM}), from which it can be seen\nthat $\\hat{u}_{jk}=-\\hat{u}_{kj}$. Since these operators $\\hat{u}_{jk}$\ncommute with the Hamiltonian (\\ref{HM}) and with each other, the Hilbert\nspace splits into two common eigenspaces of $\\hat{u}_{jk}$ with eigenvalues \nu_{jk}=\\pm1$. Thus, Hamiltonian (\\ref{HM}) is reduced to a quadratic\nMajorana fermionic Hamiltonian\n\\begin{equation}\nH=\\frac{i}{2}\\sum_{\\langle j,k\\rangle}u_{jk}J_{a_{jk}}c_{j}c_{k}.  \\label{HU}\n\\end{equation}\n\nWith a Fourier transformation\n\\begin{equation}\nc_{s,\\lambda}=\\frac{1}{\\sqrt{2L^{2}}}\\sum_{\\mathbf{q}}e^{i\\mathbf{q}\\cdo\n\\mathbf{r}_{s}}a_{\\mathbf{q},\\lambda},  \\label{Fourier}\n\\end{equation}\nwhere $s$ denotes a unit cell shown in Fig. \\ref{fig1}(a), $\\lambda $ refers\nto a position inside the cell, $r_{s}$ represents the coordinate of the unit\ncell, and $\\mathbf{q}$ are momenta of the system with finite system-size \n2L^{2}$, and a Bogoliubov transformation\n\\begin{equation}\n\\left\\{\n\\begin{array}{c}\nC_{\\mathbf{q},1}^{\\dag}=\\frac{1}{\\sqrt{2}}a_{-\\mathbf{q},1}-\\frac{1}{\\sqrt{2\n}A_{\\mathbf{q}}^{\\ast}a_{-\\mathbf{q},2}, \\\\\nC_{\\mathbf{q},2}^{\\dag}=\\frac{1}{\\sqrt{2}}A_{\\mathbf{q}}a_{-\\mathbf{q},1}\n\\frac{1}{\\sqrt{2}}a_{-\\mathbf{q},2}\n\\end{array}\n\\right.\n\\end{equation}\nwhere $A_{\\mathbf{q}}=\\sqrt{\\epsilon_{\\mathbf{q}}^{2}+\\Delta_{\\mathbf{q}}^{2\n}\/(\\Delta_{\\mathbf{q}}+i\\epsilon_{\\mathbf{q}})$, Hamiltonian (\\ref{HU}) is\ntransformed int\n\\begin{equation}\nH=\\sum_{\\mathbf{q}}\\sqrt{\\epsilon_{\\mathbf{q}}^{2}+\\Delta_{\\mathbf{q}}^{2}\n\\left( C_{\\mathbf{q},1}^{\\dagger}C_{\\mathbf{q},1}-C_{\\mathbf{q},2}^{\\dagger\n}C_{\\mathbf{q},2}\\right)  \\label{HC}\n\\end{equation}\nwith $\\epsilon_{\\mathbf{q}}=J_{x}\\cos q_{x}+J_{y}\\cos q_{y}+J_{z}$, and \n\\Delta_{\\mathbf{q}}=J_{x}\\sin q_{x}+J_{y}\\sin q_{y}$. In Hamiltonian (\\re\n{HC}), the momenta take the values \\cite{S. Yang\n\\begin{equation}\nq_{x\\left( y\\right) }=\\frac{2n\\pi}{L},n=-\\frac{L-1}{2},\\cdots,\\frac{L-1}{2},\n\\label{Values of q}\n\\end{equation}\nwhen the system size is chosen as $N=2L^{2}$ with $L$ being an odd integer.\nThus, the ground and the first-excited states are obtained by\n\n\\begin{align}\n\\left\\vert \\Psi_{0}\\right\\rangle & \\!\\!=\\prod_{\\mathbf{q}}C_{q,2}^{\\dagger\n}\\left\\vert 0\\right\\rangle =\\prod_{\\mathbf{q}}\\frac{1}{\\sqrt{2}}\\left( A_\n\\mathbf{q}}a_{-\\mathbf{q},1}+a_{-\\mathbf{q},2}\\right) \\left\\vert\n0\\right\\rangle ,  \\label{wavefunction} \\\\\n\\left\\vert \\Psi_{1}\\right\\rangle & \\!\\!=\\prod_{\\mathbf{q}}C_{q,1}^{\\dagger\n}\\left\\vert 0\\right\\rangle =\\prod_{\\mathbf{q}}\\frac{1}{\\sqrt{2}}\\left( a_{\n\\mathbf{q},1}-A_{\\mathbf{q}}^{\\ast}a_{-\\mathbf{q},2}\\right) \\! \\left\\vert\n0\\right\\rangle ,\n\\end{align}\nwith the energy eigenvalue\n\\begin{equation}\nE_{0,1}=\\pm\\sum_{\\mathbf{q}}\\sqrt{\\epsilon_{\\mathbf{q}}^{2}+\\Delta _{\\mathbf\nq}}^{2}},  \\label{EP}\n\\end{equation}\n\nIt has been shown that the Kitaev honeycomb model (\\ref{Kitaev model}) has a\nrich phase diagram including a gapped phase with Abelian anyonic excitations\n(called $A$ phase) and a gapless phase with non-Abelian anyonic excitations \n$B$ phase) \\cite{Kitaev}. In Fig. \\ref{fig1}(b), the two phases $A $ and $B$\nare separated by three transition lines, \\textit{i.e.}, $J_{x}=1\/2$, \nJ_{y}=1\/2$, and $J_{z}=1\/2$, which form a small triangle surrounding the $B$\nphase. Here, we only plot the energy spectrum (\\ref{EP}) as a function of \nJ_{z}$ for $J_{x}=J_{y}$ (the vertical dot-and-dash line in Fig. \\ref{fig1\n(b)) in Fig. \\ref{fig2}. It can be seen from Fig. \\ref{fig2} that the\nenergy-level degeneracy arises or lifts at certain points, which can be\nregarded as the possible critical points of QPT \\cite{Zhu,Sachdev}. In Fig.\n\\ref{fig2}(b, c, d), the degenerate points occur in the $B$ phase, but\ndisappear in the $A$ phase as shown in Fig. \\ref{fig2}(f). Moreover, the\nenergy spectrum may have asymptotic degeneracy at the phase diagram edge\nseen from Fig. \\ref{fig2}(a) when the size of system tends to infinity. The\nnon-analyticity points of ground state in $B$ phase are actual\nlevel-crossing points.\n\n\\begin{figure}[t]\n\\centering \\vspace{0cm} \\hspace{0cm} \\scalebox{0.88}\n\\includegraphics{fig2.eps}}.\n\\caption{Energy spectrum for the parameters $J_{x}=J_{y}$ (a) $J_{z}=0$, the\nspectrum indeed is degenerate in a larger system size; (b), (c), and (d) in \nB$ phase, for $J_{z}<1\/3$, $J_{z}=1\/3$, and $1\/3<J_{z}<1\/2$, respectively;\n(e) at the critical point of $J_{z}=1\/2$ and (f) in $A$ phase. It is clearly\nthat there is level-crossing in (b), (c) and (d).}\n\\label{fig2}\n\\end{figure}\n\n\\section{Geometric phase}\n\nIn general spin-chain systems characterized by Landau-type order parameters,\nthe GPs can be generated by adiabatic and cyclic evolutions with a rotation\nof each spin coupled with its neighbors in the certain parameter spaces \\cit\n{Zhu}. However, in the Kitaev honeycomb model, the two spins \n\\sigma_{s,\\lambda }$ and $\\sigma _{s,\\bar{\\lambda}}$ in the unit cell $s$ is\ncoupled with each other only by a $z$-link, as Fig. \\ref{fig1}(a) shows. As\na result, a simple rotation of a single-spin with the generator $\\sigma\n_{s,\\lambda }^{z}$ can only \"propagate\" along the horizontal spin-chain\nthrough the $x$- and $y$-link couplings but not along the vertical direction\nsince the rotation-generator of a single-spin $\\sigma _{s,\\lambda }^{z}$\ncommutes with the $z$-link spin-operators in the Hamiltonian between the\nhorizontal spin-chains. On other words, if using the above single-spin\nrotation, the Kitaev honeycomb model would be equivalent to a system of\nindependent multi-single-spin-chains. In order to obtain the GP of the\nKitaev honeycomb model, it should be introduced a correlated rotation of two\n$z$-link spins in each unit-cell\n\\begin{equation}\nU\\left( \\phi \\right) =\\exp \\left[ i\\phi R\\right] ,  \\label{Rotation}\n\\end{equation\nwher\n\\begin{equation}\nR=\\sum_{s}\\sum_{\\lambda }\\sigma _{s,\\lambda }^{z}\\sigma _{s,\\bar{\\lambda\n}^{z}  \\label{R expression}\n\\end{equation\nis the correlated rotation-generator, and $\\phi $ is the \"co-rotation\nangle\". Indeed, when the rotation operator is applied on the spin of $\\left(\ns,\\lambda \\right) $-th set, the spin operator $\\sigma _{s,\\bar{\\lambda}}^{z}$\nacts as a c-number because the spin operators on different sets commute each\nother. As a matter of fact, the correlated rotation-generator $R$ can\ngenerate a \"co-rotation\" of the $z$-link spins in each unit cell.\nFurthermore, under the gauge transformation (\\ref{Rotation}), the rotations\nof each spins in the 2D honeycomb Kitaev model become correlative and\nglobal. Hence it is possible to reveal the relation between the GP and the\nnonlocal TQPT by means of the correlated rotation. Then, the ground state\nwave function becomes\n\\begin{equation}\n\\left\\vert \\Psi _{0}^{\\prime }(\\phi )\\right\\rangle =U\\left( \\phi \\right)\n\\left\\vert \\Psi _{0}\\right\\rangle ,\n\\end{equation\nand correspondingly, the GP is given by \\cite{Liang\n\\begin{align}\n\\gamma & =-i{\\int\\nolimits_{0}^{2\\pi }}\\left\\langle \\Psi _{0}\\right\\vert\nU^{^{\\dag }}\\frac{d}{d\\phi }U\\left\\vert \\Psi _{0}\\right\\rangle d\\phi  \\notag\n\\\\\n& =-2\\pi \\left\\langle \\Psi _{0}\\right\\vert R\\left\\vert \\Psi\n_{0}\\right\\rangle .  \\label{BPOrigin}\n\\end{align\nIt can be seen that the GP for the Kitaev honeycomb model is proportional to\nthe expectation value of the correlated rotation $R$. Different from the\nlocal spin-correlation $\\sigma _{s,\\lambda }^{z}\\sigma _{s,\\bar{\\lambda\n}^{z} $, the sum over whole lattice-sites in $R$ leads to a global property,\nwhich is crucial for the topological QPT.\n\nIn terms of the Majorana fermion operators (\\ref{MajoranaFermion}), the\nrotation operator $R$ is given, after using the Fourier transformation (\\re\n{Fourier}), by $R=\\sum_{s,\\lambda}\\left( ib_{s,\\lambda}^{z}c_{s,\\lambda\n}\\right) \\left( ib_{s,\\bar{\\lambda}}^{z}c_{s,\\bar{\\lambda}}\\right)\n=\\sum_{s}\\left( -iu_{s,1;s,2}^{z}c_{s,1}c_{s,2}\\right) +\\sum_{s}\\left(\n-iu_{s,2;s,1}^{z}c_{s,2}c_{s,1}\\right) $. Under the restriction of vortex\nfree subspace, $u_{s,1;s,2}^{z}$ and $u_{s,2;s,1}^{z}$ are the good quantum\nnumbers, that is to say, $u_{s,1;s,2}^{z}=1$, and $u_{s,2;s,1}^{z}=-1$.\nThus, the rotation operator $R$ is finally obtained by\n\n\\begin{equation}\nR=\\frac{-i}{\\sqrt{2L^{2}}}\\sum_{\\mathbf{q}}\\left( a_{-\\mathbf{q},1}a_\n\\mathbf{q},2}-a_{-\\mathbf{q},2}a_{\\mathbf{q},1}\\right) .  \\label{R}\n\\end{equation}\nBy means of Eqs. (\\ref{wavefunction}), (\\ref{BPOrigin}), and (\\ref{R}), the\nground-state GP is given formally by\n\\begin{equation}\n\\gamma=\\frac{-2\\pi}{\\sqrt{2L^{2}}}\\sum_{\\mathbf{q}} \\mathop{\\rm Im} \\left(\n-A_{\\mathbf{q}}e^{2i\\mathbf{q}\\cdot\\mathbf{d}}+A_{\\mathbf{q}}^{\\ast}\\right) ,\n\\label{BP of R}\n\\end{equation}\nwhere $\\mathbf{d}$\\ is a vector from one site to another inside one cell. It\ncan be seen easily from Fig. \\ref{fig1}(a) that the vector is given by \n\\mathbf{d=}\\left( \\mathbf{n}_{1}+\\mathbf{n}_{2}\\right) \/3$. Inserting this\nvector $\\mathbf{d}$ into Eq. (\\ref{BP of R}), we hav\n\\begin{align}\n\\gamma & \\!\\! =\\!\\!\\frac{-2\\pi}{\\sqrt{2L^{2}}}\\sum_{\\mathbf{q}}\\mathop{\\rm\nIm}\\left( -A_{\\mathbf{q}}e^{\\frac{i}{3}\\left( 2q_{x}+2q_{y}\\right) }+A_\n\\mathbf{q}}^{\\ast}\\right)  \\notag \\\\\n& \\!\\! =\\!\\!\\frac{-2\\pi}{\\sqrt{2\\!L^{2}}}\\!\\!\\sum_{\\mathbf{q}}\\!\\frac \n\\epsilon_{\\mathbf{q}}\\!\\!\\left[ 1\\!\\!+\\!\\cos\\!\\frac{1}{3}\\!\\left(\n2q_{x}\\!\\!+\\!2q_{y}\\!\\right) \\right] \\!\\!-\\!\\!\\Delta_{\\mathbf{q}}\\sin \\\n\\frac{1}{3}\\!\\left( 2q_{x}\\!\\!+\\!2q_{y}\\right) }{\\sqrt{\\epsilon _{\\mathbf{q\n}^{2}+\\Delta_{\\mathbf{q}}^{2}}}  \\label{gammafinal}\n\\end{align}\n\n\\begin{figure}[h]\n\\centering \\vspace{0cm} \\hspace{0cm} \\scalebox{0.5}\n\\includegraphics{fig3.eps}}.\n\\caption{(Color online) Scaled GP with system size parameter $L=101$.}\n\\label{fig3}\n\\end{figure}\n\nEq. (\\ref{gammafinal}) is the main result of this paper. The scaled (or\naverage) GP $\\gamma\/L$ as a function of coupling parameters is shown in Fig.\n\\ref{fig3}, which distributes symmetrically with respect to the coupling\nparameters $J_{x}$ and $J_{y}$. For sake of simplicity, the scaled GP \n\\gamma\/L$ is replaced by $\\gamma$ in the following discussions. In order to\nsee the difference between $A$ and $B$ phases, $\\gamma$-variation with \nJ_{z} $ along a selected line $J_{x}=J_{y}$ in the phase diagram of Kitaev\n\\cite{Kitaev} (see Fig. \\ref{fig1}(b), the vertical dot-and-dash line) is\nplotted in Fig. \\ref{fig4}(a). It is shown that $\\gamma$ is smooth in $A$\nphase ($J_{z}>1\/2$) while becomes saltant in $B$ phase ($J_{z}<1\/2$) for the\nsize parameters $L=11$ (dark yellow line)$,$ $33$ (red), and $99$ (blue),\nrespectively. Moreover, all the data fall onto a single curve in $A$ phase,\nwhile the number of saltation increases with the system-size $L$ in $B$\nphase (see insets (1) and (2) of Fig. \\ref{fig4}(a) ). To be specific, the\nnumber of saltation for $L=33$ is three times than that for $L=11$ (see Fig.\n\\ref{fig4}(a)), and the same situation occurs in turn for $L=99$ and $33$.\n\n\\begin{figure*}[t]\n\\centering \\vspace{0cm} \\hspace{0cm} \\scalebox{1.0}\n\\includegraphics{fig4.eps}}\n\\caption{(Color online) (a) $\\protect\\gamma$ curve along the selected\nvariation path $J_{x}=J_{y}$ for $L=11,$ $33,$ and $99$. Both insets reveal\nthe increasing number of saltation in $B$ phase proportional to the system\nsize parameter $L$. (b) $g_{x}$ and (c) $g_{xx}$ as a function of $J_{z}$\nalong the variation path $J_{x}=J_{y}$ for system size parameters $L=101,303$\nand $909$. The two insets are local enlarged-pictures, which show the\nvibration in $B$ phase and the circumstances in the critical point\nrespectively.}\n\\label{fig4}\n\\end{figure*}\n\nIt is meaningful to consider the first-order partial derivative \ng_{_{\\beta}}=\\partial\\gamma\/\\partial J_{\\beta}$ ($\\beta=x,y$) of the GP \n\\gamma$. Since the GP $\\gamma$ in Eq. (\\ref{gammafinal}) is symmetric with\nrespect to $J_{x}$ and $J_{y}$, we only need to investigate $g_{x}$ (or\nequivalently $g_{y}$). The variation of $g_{x}$ with respect to $J_{z}$\nalong the selected path of $J_{x}=J_{y}$ from $B$ phase to $A$ phase is\nshown in Fig. \\ref{fig4}(b) for different system-size parameters $L=101$, \n303$ and $909$. It can be seen from Fig. \\ref{fig4}(b) that $g_{x}$\noscillates in $B$ phase with frequency (or number of peaks), which is\nproportional to $L$ (see inset (1) of Fig. \\ref{fig4}(b)). This rapid\nvariation of the GP $\\gamma$ (in the $B$ phase) has not yet been found, to\nour knowledge. However, a very similar behavior of fidelity susceptibility\nin the Kitaev model has been reported \\cite{S. Yang}. On the other hand, the\nvalue of $g_{x}$ at the critical point $J_{z}=1\/2$ increases with the\nsystem-size and sharply decays in $A$ phase (see inset (2) of Fig. \\ref{fig4\n(b) for detail). It is interesting to remark that the saltation of GP \n\\gamma $ in $B$ phase due to the complex structure of ground state with\ndegeneracy (Fig. \\ref{fig2} (b),(c),(d)) is not random rather has\nregulation, especially it tends to a regular oscillation above the point, \nJ_{z}=1\/3$ , (see inset (1) of Fig. \\ref{fig4}(b)). The oscillation\nfrequency depends linearly on the system size.\n\nTo show the non--analyticity of GP at the critical points explicitly the\nsecond-order derivative $g_{xx}$ of the GP $\\gamma$ with respect to the\ncoupling parameters is calculated. Fig. \\ref{fig4}(c) shows the variation of\n$g_{xx}$ with respect to $J_{z}$ along the variation path of $J_{z}=J_{y}$\nfor different system-size parameters $L=101$, $303$ and $909$. Inset (1)\nreveals the increase of peak-number of $g_{xx}$ in $B$ phase along with the\nsystem-size $L$ similar to $\\gamma$ and $g_{x}$ in behavior. The\nsecond-order derivative $g_{xx}$ is divergent at the critical point \nJ_{z}=1\/2$ as shown in inset (2) indicating that the TQPT of the Kitaev\nhoneycomb model is a second-order transition, While the QPT of the XY spin\nchain \\cite{Zhu,Zhu08} and the Dicke model \\cite{Zhu08,Chen} has been shown\nto be the first-order transition with the divergent first-order derivative\nof the GP. We conclude that the non-analytic GP $\\gamma$ can very well\ndescribe the TQPT in terms of the Landau phase-transition theory.\n\nSimilarly, we can also choose the variation path as $J_{z}=1\/4$ (dashed line\nin Fig. \\ref{fig1}(b)) with two critical points $J_{x}=1\/4$ and $J_{x}=1\/2$.\nQualitatively similar results are shown in Fig. \\ref{fig5} for different\nsize parameters $L=101$ (red line), $303$ (blue) and $707$ (dark),\nrespectively, where $\\gamma$-plot is a smooth curve in $A$ phase for \nJ_{x}<1\/4$ or $J_{x}>1\/2$ and becomes saltant in $B$ phase when \n1\/4<J_{x}<1\/2$ (Fig. \\ref{fig5}(a)). The plots of $g_{x}$ and $g_{xx}$ are\nrespectively shown in Fig. \\ref{fig5}(b) and (c) displaying the clear\nvibration in the $B$ phase. $g_{x}$ has a sharp peak at the critical points\nand $g_{xx}$ is divergent in the thermodynamic limit showing the\ncharacteristic of second-order phase transition. Inset of Fig. \\ref{fig5}(c)\nshows the detail of the vibration in the $B$ phase with the peak-number\nproportional to the system size.\n\n\\begin{figure}[h]\n\\centering \\vspace{0cm} \\hspace{0cm} \\scalebox{0.3}\n\\includegraphics{fig5.eps}}\n\\caption{(Color online) (a) $\\protect\\gamma$ curve, which is smooth in $A$\nphase and saltant in $B$ phase, (b) $g_{x}$ curve and (c) $g_{xx}$ as a\nfunction of $J_{x}$ for $J_{x}$ varying along the path $J_{z}=1\/4$, with\ncritical points at $J_{x}=1\/4$ and $J_{x}=1\/2$.}\n\\label{fig5}\n\\end{figure}\n\n\\section{Finite-size scaling}\n\nIn order to quantify the non-analytic nature of GP through the critical\npoints, and to further understand the quantum criticality, we investigate\nthe scaling behavior of the GP and its first-order and second-order\nderivatives by the finite size scaling analysis \\cite{Barber}. Since the GP\nof a many-body system is an extensive quantity, it usually depends on the\nsystem size in the non-critical region. In the vicinity of critical point (\nJ_{z}^{C}=1\/2$ here), the GP $\\gamma$ follows a power law\\cite{Lin}\n\n\\begin{equation}\n\\gamma \\propto {\\left\\vert J_{z}-J_{z}^{C}\\right\\vert }^{-\\alpha _{\\gamma }}\n\\label{alpha}\n\\end{equation\nwith $\\alpha $ being corresponding exponent. ${\\left\\vert\nJ_{z}-J_{z}^{C}\\right\\vert }$-dependence of $\\gamma $ in the left-hand side\nof critical point ( $J_{z}<J_{z}^{C}$ ) and right-hand side ( \nJ_{z}>J_{z}^{C})$ is respectively plotted in Fig. \\ref{fig6}(a) and (b) for\ndifferent system-size parameters $L=101$, $303$ and $909$. The corresponding\nexponents $\\alpha _{\\gamma }^{-}=-0.99934\\pm 0.00033$ (left-hand side) and \n\\alpha _{\\gamma }^{+}=-0.83538\\pm 0.00008$ (right-hand side) are obtained\nfrom Fig. \\ref{fig6}(a) and (b). Similarly, $g_{x}$ and $g_{xx}$ as a\nfunction of $|J_{z}-J_{z}^{C}|$ are plotted in Fig. \\ref{fig6}(c), (d) and\n(e), (f) for the gapless and gapped phases respectively, from which the\ncritical exponents $\\alpha _{g_{x}}^{-}=-0.17529\\pm 0.02056$, $\\alpha\n_{g_{xx}}^{-}=1.32523\\pm 0.01719$ (left) for B phase and $\\alpha\n_{g_{x}}^{+}=-0.48375\\pm 0.00033$, $\\alpha _{g_{xx}}^{+}=0.60687\\pm 0.00526$\n(right) for A phase are found. The fact of negative exponents $\\alpha\n_{\\gamma }^{\\pm }$, $\\alpha _{g_{x}}^{\\pm }$ and positive $\\alpha\n_{g_{xx}}^{\\pm }$ indicates that $\\gamma $ and $g_{x}$ are finite while \ng_{xx}$ is divergent at the critical point in the thermodynamic limit. Thus\nthe TQPT is a second-order phase transition characterized by the GP $\\gamma \n.\n\n\\begin{figure}[h]\n\\centering \\vspace{0cm} \\hspace{0cm} \\scalebox{1.0}\n\\includegraphics{fig6.eps}}\n\\caption{(Color online) Finite-size scaling analysis of the power-law\ndivergence for (a) GP $\\protect\\gamma$, (c) $g_{x}$, (e) $g_{xx}$ as a\nfunction of $|J_{z}-J_{z}^{C}|$ in the vicinity of critical point with\nsystem sizes $L=301,901$ and $1501$ on the left-hand side ($J_{z}<J_{z}^{C}\n) and (b), (d), (f) on the right-hand side ($J_{z}>J_{z}^{C}$) respectively.}\n\\label{fig6}\n\\end{figure}\n\n\\begin{figure}[h]\n\\centering \\vspace{0cm} \\hspace{0cm} \\scalebox{0.35}\n\\includegraphics{fig7.eps}}\n\\caption{(Color online) $L$-dependence of $J_{z}^{C}-J_{z}^{m}$ in\nlogarithmic coordinate for $L=901,951,...,1901$.}\n\\label{fig7}\n\\end{figure}\n\nThe position of maximum value of $\\gamma $ denoted by $J_{z}^{m}$ may not be\nlocated exactly at the critical point $J_{z}^{C}=1\/2$, but tends to it in\nthe thermodynamic limit $L\\rightarrow \\infty $, which is regarded as the\npseudocritical point \\cite{Barber}. The $J_{z}^{C}-J_{z}^{m}$ versus\ndifferent system-size $L=901,951,...,1901$ in a logarithmic coordinate is\nplotted in Fig. \\ref{fig7}, which is a straight line of slope $-0.37851\\pm\n0.00226$. It means that the $J_{z}^{m}$ tends toward to the critical point \nJ_{z}^{C}$ following the power-law decay\n\\begin{equation}\nJ_{z}^{C}-J_{z}^{m}\\propto L^{-0.37851}.\n\\end{equation}\n\nOn the other hand, the maximum value of $\\gamma $ at $J_{z}=J_{z}^{m}$ for a\nfinite-size system behaves as\n\\begin{equation}\n\\gamma ({J_{z}^{m}})\\propto {L}^{\\mu _{\\gamma }},\n\\end{equation\nwhich is shown in the inset of Fig. \\ref{fig8}(a) with a straight line in\nlogarithmic coordinate. The corresponding size-exponent is given by $\\mu\n_{\\gamma }=0.01148\\pm 0.00054$.\n\n\\begin{figure*}[t]\n\\centering \\vspace{0cm} \\hspace{0cm} \\scalebox{1.0}\n\\includegraphics{fig8.eps}}\n\\caption{(Color online) (a) $F_{\\protect\\gamma}$, (b) $F_{g_x}$, and (c) \nF_{g_{xx}}$ as a function of $L^{\\protect\\nu}\\left( J_{z}-J_{z}^{m}\\right)$\nfor $L=401,601,...,1601$. All the data fall on to a single curve\nrespectively. And the three insets in turn shows the variation of $\\protec\n\\gamma(J_{z}^{m})$, $g_{x}(J_{z}^{m})$, and $g_{xx}(J_{z}^{m})$ with respect\nto the system size-parameters $L=101,151,...,1901$.}\n\\label{fig8}\n\\end{figure*}\n\nSince the GP $\\gamma$ around its maximum position $J_{z}^{m}$ can be written\nas a simple function of $J_{z}^{m}-J_{z}$, it is possible to make all the\nvalue-data defined by a universal scaling function $F_{\\gamma}=\\left( {\\gamm\n}({J_{z}^{m}})-\\gamma\\right) \/\\gamma$ versus $L^{\\nu_{\\gamma}}\\left(\nJ_{z}-J_{z}^{m}\\right) $, namely,\n\\begin{equation}\nF_{\\gamma}=f\\left[ L^{\\nu_{\\gamma}}\\left( J_{z}^{m}-J_{z}\\right) \\right] ,\n\\end{equation}\nwhere $\\nu_{\\gamma}$ is a critical exponent that governs the divergence of\nthe correlation length. The values of $F_{\\gamma}$ for different system-size\nparameters $L$ fall onto a single curve as shown in Fig. \\ref{fig8}(a), from\nwhich we can extract the critical exponent $\\nu_{\\gamma}=-0.015$ numerically.\n\nIn fact, according to the scaling ansatz of a finite system \\cite{Barber,Lin\n, the critical exponent $\\nu $ can be determined by the relation $\\nu =\\mu\n\/\\alpha $. In terms of this relation, the critical exponents in $B$ and $A$\nphases are found as ${\\nu }_{\\gamma }^{-}=-0.01149$ and ${\\nu }_{\\gamma\n}^{+}=-0.01374$, which are consistent with the numerical result $\\nu\n_{\\gamma }$ extracted from Fig. \\ref{fig8}(a). Inset of Fig. \\ref{fig8}(b)\nis a plot of the maximum value of $g_{x}({J_{z}^{m}})$ as a function of $L$\n\\begin{equation}\ng_{x}({J_{z}^{m}})\\propto {L}^{\\mu _{g_{x}}}\n\\end{equation}\nin logarithmic coordinate, from which the size exponent $\\mu\n_{g_{x}}=0.01612\\pm 0.00094$ is found. The universal scaling function\n\\begin{equation}\nF_{g_{x}}=f\\left[ L^{\\nu _{g_{x}}}(J_{z}^{m}-J_{z})\\right]\n\\end{equation}\nis shown in Fig. \\ref{fig8}(b). We have the numerical value $\\nu\n_{g_{x}}=-0.040$ and the results determined by the relation $\\nu =\\mu\n\/\\alpha $ that $\\nu _{g_{x}}^{-}=-0.09196$, $\\nu _{g_{x}}^{+}=-0.03332$. The\ndeviation between $\\nu _{g}$ and $\\nu _{g_{xx}}^{-}$ may be due to the rapid\noscillation in the gapless $B$ phase. Similarly, the critical exponents of \ng_{xx}$ can be obtained from Fig. \\ref{fig8}(c) as $\\mu _{g_{xx}}=0.54570\\pm\n0.00237$, $\\nu _{g_{xx}}^{-}=0.410$ and $\\nu _{g_{xx}}^{+}=0.908$, and the\nresults determined by $\\nu =\\mu \/\\alpha $ are $\\nu _{g_{xx}}^{-}=0.41178$\nand $\\nu _{g_{xx}}^{+}=0.89920$ respectively.\n\n\\section{Summary and discussion}\n\nWe demonstrate that the ground-state GP generated by the correlated rotation\nof two linked-spins in a unit-cell indeed can be used to characterize the\nTQPT for the Kitaev honeycomb model. The non-analytic GP with a divergent\nsecond-order derivative at the critical points shows that the TQPT is a\nsecond-order phase-transition different from the $XY$ spin-chain \\cit\n{Zhu,Zhu08}, in which the first-order derivative of GP is divergent, and the\nLMG model\\cite{Zhu,Zhu08}, in which the GP itself is shown to be divergent.\nMoreover it is found that the GP is zigzagging with oscillating derivatives\nin the gapless $B$ phase, but is a smooth function in the gapped $A$ phase.\nThe scaling behavior of the non-analytic GP in the vicinity of critical\npoint is shown to exhibit the universality with negative exponents of both \n\\gamma$ and $g_{x}$ while a positive exponent of $g_{xx}$ indicating the\ncharacteristic of second-order phase transition.\n\n\\section*{Acknowledgments}\n\nThis work is supported by the NNSF of China under Grant Nos. 11075099 and\n11074154, ZJNSF under Grant No. Y6090001, and the 973 Program under Grant\nNo. 2006CB921603.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\t\\label{section1}\n\t\n\\begin{figure}\t\n\t\\begin{center}\n\t\t\\includegraphics[width=90mm]{Figure1.png}\n\t\t\\caption{Panel (a): Spatial positions of the stars in our sample, with the tidal radius ($r_t=10$\\arcmin) of M~54 over-plotted with a solid line. The open red symbols designate N-rich stars (the diamond symbol refers to a field star, while the open circle highlights the extra-tidal member of M~54). The lime circles designate the M~54 population analyzed in this work, while the black plus symbols designate the stars analyzed by \\citet{Nataf2019}. The empty grey `star' symbols designate the potential Sgr population from \\citet{Hayes2020}. The two concentric circles indicate 5 $r_{t}$ and 7 $r_{t}$ for reference. Panel (b): \\textit{Gaia} EDR3 proper motions of stars that have been associated with the Sgr stream: blue symbols for the  \\citet{Antoja2020} stars and open black `star' symbols for \\citet{Hayes2020} stars. The orbital path of Sgr is shown by the dotted (backward) and solid (forward) purple line in panels (a) and (b), with the thick and thin lines showing the central orbit, and one hundred ensemble of orbits that shows the more probable regions of the space, which are crossed more frequently by the simulated orbit, respectively. Panel (c): Color magnitude diagram from \\textit{Gaia} EDR3 photometry of our sample. The symbols are the same as in panels (a) and (b), except the white circles, which denotes the M~54 members from \\textit{Gaia} EDR3, selected on proper motions and within 3$\\arcmin$ from the cluster center. Panel (d): Radial velocities versus [Fe\/H] ratios determined from APOGEE-2\/\\texttt{ASPCAP} (black symbols) and our [Fe\/H] ratio determinations from \\texttt{BACCHUS} (green and red symbols) in the field around M~54. The [Fe\/H] APOGEE-2\/\\texttt{ASPCAP} determinations have been systematically offset by $\\sim$0.11 dex in order to compare with our [Fe\/H] \\texttt{BACCHUS}  determinations, as suggested in \\citep{Fernandez-Trincado2020c}.}\n\t\t\\label{Figure1}\n\t\\end{center}\n\\end{figure}\t\n\t\nThe Sagittarius (Sgr) dwarf spheroidal (dSph) galaxy is one of the closest massive satellites of the Milky Way (MW) \\citep{Ibata1994}, and has yielded a wealth of observational evidence of ongoing accretion by the MW in the form of persistent stellar debris and tidal streams discovered by \\citet{Mateo1996}, and extensively studied with photometric and spectroscopic observations over a huge range of distances ($\\sim$10--100 kpc) \\citep[see, e.g.,][]{Ibata2001, deBoer2015} using different stellar tracers--including Carbon stars \\citep{Totten1998}, the first all-sky map of the tails using 2MASS M-giants \\citep{Majewski2003}, red clump Stars \\citep{Correnti2010}, RR Lyrae stars \\citep{Newberg2003, Ramos2020}, and CN-strong stars \\citep{Hanke2020}, among other tracers, usually in small patches along the stream (see, e.g., \\citealt{Li2019}). These studies have been followed-up by numerical studies \\citep[see, e.g.,][]{Law2005, Vasiliev2020}, as well as by using precise astrometry from the \\textit{Gaia} second data release \\citep[Gaia DR2;][]{Brown2018}, based on proper motions alone \\citep{Antoja2020}. Its proximity provides a unique laboratory to study accretion in detail, through the tidally stripped streams that outflow from the Sgr system \\citep[][]{Hasselquist2017, Hasselquist2019, Hayes2020}.\n\t\t\nAs a natural result of such an accretion event, there is a claim in the literature that not only field stars but also GCs have been accreted \\citep[see, e.g.,][]{Massari2019}.  Some have been speculated to be lost in the disruption process, and may lie immersed in the Sgr stream.  Candidates include: M~54, Terzan 7, Arp 2, Terzan 8, Pal 12, Whiting 1, NGC 2419, NGC 6534, and NGC 4147 \\citep[e.g.,][]{Law2010a, Bellazzini2020}, but a firm connection is still under debate \\citep[e.g.,][]{Villanova2016, Tang2018, Huang2020, Yuan2020}. In this context, ``chemical tagging\" \\citep[e.g.,][]{Freeman2002}, which is based on the principle that the photospheric chemical compositions of stars reflect the site of their formation, is a promising route for investigation of this question.   \n\nWhile the abundances of light and heavy elements for individual stars in GCs have been widely explored \\citep[e.g.,][]{Pancino2017, Meszaros2020}, little is known about these abundances in disrupted GCs likely associated with the closest dwarf galaxies, such as Sgr \\citep{Karlsson2012}. Although some evidence for chemical anomalies has been detected towards the inner bulge and halo of the MW \\citep[see, e.g.,][]{Fernandez-Trincado2016b, Recio-Blanco2017, Schiavon2017, Fernandez-Trincado2017} and Local Group dwarf galaxies \\citep[see, e.g.,][]{Fernandez-Trincado2020b}, suggesting the presence of GCs in the form of disrupted remnants, alternative ways to produce these stars have been recently discussed \\citep{Bekki2019}.\n\nThis paper is outlined as follows. The high-resolution spectroscopic observations are discussed in Section \\ref{section2}. Section \\ref{section3} describes the sample associated with M~54, including a comparison with data from the literature. Section \\ref{section4} presents our estimated stellar parameters and derived chemical-abundance determinations. Section \\ref{section5} discusses the results, and our concluding remarks are presented in Section \\ref{section6}.\n  \n\\section{Data}\n\\label{section2}\n\n  We make use of the internal dataset (which includes all data taken through March 2020) of the second-generation Apache Point Observatory Galactic Evolution Experiment \\citep[APOGEE-2;][]{Majewski2017}, which includes the first observations from the Ir\\'en\\'ee du Pont 2.5-m Telescope at Las Campanas Observatory \\citep[APO-2S;][]{Bowen1973} in the Southern Hemisphere (Chile), and more observations from the Sloan 2.5-m Telescope at Apache Point Observatory \\citep[APO-2N;]{Gunn2006} in the Northern Hemisphere (New Mexico). The survey operates with two nearly identical spectrographs \\citep{Eisenstein2011, Wilson2012, Wilson2019}, collecting high-resolution ($R\\sim22,000$) spectra in the near-infrared textit{H}-band (1.5145--1.6960 $\\mu$m, vacuum wavelengths). This data set provides stellar parameters, chemical abundances, and radial velocity (RV) information for more than 600,000 sources, which include $\\sim$437,000 targets from the sixteenth data release \\citep[DR16;][]{Ahumada2020} of the fourth generation of the Sloan Digital Sky Survey \\citep[SDSS-IV;][]{Blanton2017}. APOGEE-2 target selection is described in full detail in \\citet{Zasowski2017} (APOGEE-2), Santana et al. (in prep.) (APO-2S), and Beaton et al. (in prep.) (APO-2N). \n  \n  APOGEE-2 spectra were reduced \\citep{Nidever2015} and analyzed using the APOGEE Stellar Parameters and Chemical Abundance Pipeline \\citep[ASPCAP;][]{Garcia2016, Holtzman2015, Holtzman2018, Henrik2018, Henrik2020}. The model grids for APOGEE-2 internal dataset are based on a complete set of \\texttt{MARCS} stellar atmospheres \\citep{Gustafsson2008}, which now extend to effective temperatures as low as 3200 K, and spectral synthesis using the \\texttt{Turbospectrum} code \\citep{Plez2012}. The APOGEE-2 spectra provide access to more than 26 chemical species, which are described in \\citet{Smith2013}, \\citet{Shetrone2015}, \\citet{Hasselquist2016}, \\citet{Cunha2017}, and \\citet{Holtzman2018}.  \n\n\\subsection{M~54 field}\n\\label{section3}\n\nThe APOGEE-2 field toward M~54 was previously examined in \\citet{Meszaros2020} based on public DR16 spectra. In that work, 22 stars were identified as potential members linked to M~54 based in the APOGEE-2 radial velocities \\citep{Nidever2015}, i.e., stars with RV within $3\\sigma _{RV,cluster}$, metallicity within $\\pm$0.5 dex around the cluster average, proper motion from the \\textit{Gaia} Early Data Release 3 \\citep[\\textit{Gaia} EDR3;][]{Brown2020} within 2.5$\\sigma$ around the cluster average proper motion, and located inside the cluster tidal radius, $r_{t}\\lesssim10$ arcmin, \\citep[][2010 edition]{Harris1996} were classified as potential members of M~54. However, only 7 out of 22 stars were spectroscopically examined with the \\texttt{BACCHUS} code in \\citet{Masseron2016}, since only these stars achieved a signal-to-noise (S\/N$>$70) sufficient to provide reliable abundance determinations.  \n\nThe post-APOGEE DR16 dataset provides incremental visits toward M~54, which has allowed to increase the signal-to-noise for 20 out of 22 of the potential cluster members. As a result, nitrogen, titanium, and nickel abundances can be now obtained from the stronger absorption features (as shown for $^{12}$C$^{14}$N lines as shown in Figure \\ref{Figure4}), and other chemical species can also be studied. \n\n\\citet{Nataf2019}, using APOGEE-2 DR14 data \\citep{Abolfathi2018} and abundance determinations from the \\texttt{Payne} pipeline \\citep{Ting2019}, have catalogued eight possible members from M~54. Two of those objects (2M18544275$-$3029012 and 2M18550740$-$3026052) were included in our study. The remaining six stars were rejected from our analysis for the following reasons. Six objects in \\citet{Nataf2019} were found to have low S\/N ($<$70) spectra, resulting in very uncertain CNO abundance ratios for many chemical species, since the molecular lines ($^{16}$OH, $^{12}$C$^{16}$O, and $^{12}$C$^{14}$N) are very weak. Secondly, 6 out of the 8 objects in \\citet{Nataf2019} exhibit [Fe\/H] $> -1.1$, and were recently classified as Sgr stars \\citep[see, e.g.,][]{Hayes2020}, which make them unlikely members of M~54.\n\nIn this study, we make use of the more recent spectra to examine the chemical composition of added stars to the abundance average of M~54. As in \\citet{Meszaros2020}, we also limit our discussion only to stars with S\/N$>70$.\n\n\\subsection{Extra-tidal stars}\n\nWe also report on the serendipitous discovery of two nitrogen-enhanced (N-rich) metal-poor stars beyond the tidal radius of M~54, as shown in pane (a) of Figure \\ref{Figure1}. APOGEE-2 stars in the stream$+$core Sagittarius (Sgr) system \\citep[see, e.g.,][]{Hasselquist2017, Hasselquist2019, Hayes2020} are highlighted as black open `star' symbols in pane (a) of Figure \\ref{Figure1}, while potential star members (blue symbols) of the stream$+$core Sgr system from \\citet[][]{Antoja2020} are also displayed in panel (a) of Figure \\ref{Figure1}. It is important to note that the [Fe\/H] abundance of APOGEE-2 Sgr stars are provided by the \\texttt{ASPCAP} pipeline \\citep[see][]{Hasselquist2017, Hasselquist2019, Hayes2020}. In order to compare with our [Fe\/H] determinations, an offset of $\\sim$0.11 dex was applied to \\texttt{ASPCAP} metallicities in panel (d) of Figure \\ref{Figure1}, as suggested in \\citet{Fernandez-Trincado2020c}.\n\nPanels (a) to (d) of Figure \\ref{Figure1} reveal that one (2M18565969$-$3106454) of the newly discovered N-rich stars meets the minimum criterion to be considered a potential extra-tidal star which has likely escaped the cluster potential, while the second N-rich star (2M18533777$-$3129187) has physical properties that are clearly offset from the M~54 population. In particular, this star is brighter than the typical population of M~54 (see panel (c) in Figure \\ref{Figure1}), and both proper motions and RV differ from the nominal proper motion and RV of the cluster as shown in panels (b) and (d) of Figures \\ref{Figure1}. It is likely that 2M18533777$-$3129187 is a foreground field star (hereafter N-rich field star). \n\n\n\\section{Stellar parameters and chemical-abundance determinations}\n \\label{section4}\n\nThe chemical analysis is very similar to that carried out by \\citet[][]{Fernandez-Trincado2019a, Fernandez-Trincado2019b, Fernandez-Trincado2019c, Fernandez-Trincado2019d, Fernandez-Trincado2020a, Fernandez-Trincado2020b, Fernandez-Trincado2020c, Fernandez-Trincado2020d, Fernandez-Trincado2021a}. The stellar parameters ($T_{\\rm eff}$, $\\log$ \\textit{g}, and first guess on metallicity) for the 20 cluster members with S\/N$>$70 were extracted from \\citet{Meszaros2020}, while we adopt the atmospheric parameters from the uncalibrated post-APOGEE DR16 values for the two stars beyond the cluster tidal radius. The elemental abundances and final errors in [Fe\/H] and [X\/Fe], astrometric and kinematic properties of our sample are listed in Tables \\ref{Table1},  \\ref{Table11}, and \\ref{Table2}, respectively. \n\nA consistent chemical-abundance analysis was then carried out with the \\texttt{BACCHUS} code \\citep{Masseron2016}, from which we obtained the metallicities from Fe I lines, and abundances for twelve other chemical species belonging to the light- (C, N), $\\alpha$- (O, Mg, Si, Ca, and Ti), Fe-peak (Ni), odd-Z (Al, K) and \\textit{s}-process (Ce, Nd) elements.\n\n \\begin{figure}\t\n \t\\begin{center}\n \t\t\\includegraphics[width=88mm]{Figure2.png}\n \t\t\\includegraphics[width=92mm]{Figure3.png}\n \t\t\\caption{{\\bf \\texttt{BACCHUS} elemental abundances}. Panel (a): The observed [X\/H] and [Fe\/H] abundance-density estimation (violin representation) of M~54 stars, and the observed abundance ratios of newly identified N-rich stars. The extra-tidal star from M~54 and a field star is highlighted with a black open circle and diamond, respectively. Each violin indicates with horizontal lines the median and limits of the distribution. The lime and dark violet violin representation refer to the abundance ratios of 20 stars (this work) and 7 stars from \\citet{Meszaros2020}, respectively. Panels (b)--(e): Distributions of light- (C,N), $\\alpha$- (Mg, Si) and odd-Z (Al) elements in different abundance planes. In each panel, the planes [Al\/Fe] -- [Mg\/Fe], [N\/Fe]--[C\/Fe], [Al\/Fe]--[Si\/Fe], [Si\/Fe]--[Mg\/Fe] are shown, respectively, for GCs from \\citet{Meszaros2020}. The black dotted line at [Al\/Fe] $=+0.3$ indicates the separation of FG and SG stars as proposed in \\citet{Meszaros2020}. The distribution of M~54 stars (lime squares) analyzed in this work are overlaid. The black open circle and diamond refer to the extra-tidal and field N-rich star, respectively. The plotted error bars show the typical abundance uncertainties.}\n \t\t\\label{Figure2}\n \t\\end{center}\n \\end{figure}\t\n \n   \\section{Results and discussion}\n \\label{section5}\n \n Panel (a) of Figure \\ref{Figure2} summarizes the chemical enrichment seen in M~54 stars analyzed in this work, and compares to the \\citet{Meszaros2020} determinations. The chemical composition of the two newly identified N-rich stars beyond the cluster tidal radius is also shown in the same figure. Overall, the chemical abundance of M~54 based on the added cluster stars is within the typical errors, and does not affect the science results presented in \\citet{Meszaros2020}, while the two external N-rich stars share chemical patterns similar to the M~54 population.\n \n For M~54, we find a mean metallicity $\\langle$[Fe\/H]$\\rangle = -1.30\\pm0.12$, which agrees well with \\citet{Meszaros2020}\\footnote{Note that here, and for the abundances described below, the number following the average abundance represents the one-sigma dispersion, not the error in the mean.}. The spread in [Fe\/H] increased from 0.04 to 0.12 dex, but it is still smaller than that reported in \\citet{Carretta2010}. Even if the measured scatter is larger than that reported by \\citet{Meszaros2020}, it does not seem to indicate the presence of a significant spread in [Fe\/H], and is similar to that observed in Galactic globular clusters (GCs) at similar metallicity, such as M~10 \\citep[see, e.g.,][]{Meszaros2020}. Nickel (an element that belongs to the Fe-group), exhibits a flat distribution as a function of [Fe\/H], similar to that observed in \\citet{Carretta2010}, and at odds with that observed in Sgr stars. \n \nRegarding the other chemical species, we find excellent agreement with the values provided by \\citet{Meszaros2020}, as can be seen in panel (a) of Figure \\ref{Figure2}, with the main difference that the added stars introduce a larger star-to-star scatter than previously measured. M~54 exhibits a modest enhancement in $\\alpha$-elements, with mean values for [O\/Fe], [Mg\/Fe], [Si\/Fe], [Ca\/Fe], and [Ti\/Fe] which is similar to what is seen in halo GCs: $\\langle$[O\/Fe]$\\rangle = +0.64\\pm0.36$ (14 stars); $\\langle$[Mg\/Fe]$\\rangle = +0.18\\pm0.11$ (18 stars); $\\langle$[Si\/Fe]$\\rangle = +0.26\\pm0.10$ (20 stars); $\\langle$[Ca\/Fe]$\\rangle = +0.25\\pm0.07$ (16 stars); and the new measured $\\langle$[Ti\/Fe]$\\rangle = +0.21\\pm0.21$ (16 stars), indicating a fast enrichment provided by supernovae (SNe) II events. Mean values are in good agreement with \\citet{Meszaros2020}, with the exception of oxygen, which displays the larger star-to-star spread expected in likely second-generation stars. \n\nWe also find that the [O\/Fe], [Mg\/Fe], and [Si\/Fe] ratios are almost flat as a function of the metallicity, while [Ca\/Fe] and [Ti\/Fe] ratios slightly increases as [Fe\/H] increases, similar to the behaviour found by \\citet{Carretta2010}. On the contrary, the $\\alpha$-element trend observed in Sgr stars \\citep[see, e.g.,][]{Carretta2010, McWilliam2013, Hasselquist2017, Hasselquist2019} differ from those seen in the population of M~54. Overall, the $\\alpha$-elements in the cluster are higher than seen in Sgr stars. In conclusion, the measured $\\alpha$-enrichment in this work support the previous hypothesis suggesting that the $\\alpha$-element in M~54 stars formed before the typical $e$-folding time for SN Ia contributing their ejecta to the gas pool \\citep[e.g.,][]{Carretta2010}. \n\nWe also found that some stars in M~54 appear to be quite Mg poor, with strong enrichment in aluminum and nitrogen, providing further evidence for the presence of second-generation stars in M~54, and the signature of very high temperatures achieved during H-burning \\citep[e.g.,][]{Carretta2010, Meszaros2020}. The odd-Z elements (Al and K) in M~54 exhibit an average $\\langle$[Al\/Fe]$\\rangle = +0.14\\pm0.37$ (19 stars) and $\\langle$[K\/Fe]$\\rangle = +0.15\\pm0.18$ (17 stars), with a clear anti-correlation in Al-Mg, as can be seen in in panel (b) of Figure \\ref{Figure2}, with moderate Mg depletions related to the enrichment in Al abundances, as the result of the conversion of Mg into Al during the Mg-Al cycle \\citep[e.g.,][]{Carretta2010, Denissenkov2015, Renzini2015, Pancino2017}. This pattern is evidently not present in the Sgr stars, where, on the contrary, \\texttt{ASPCAP} Mg and Al abundances are positively correlated with each other \\citep[see, e.g.,][]{Hasselquist2017, Hasselquist2019, Hayes2020}. \n\nWe derived average abundances for C and N in M~54, of $\\langle$[C\/Fe]$\\rangle = -0.36\\pm0.25$ (13 stars) and $\\langle$[N\/Fe]$\\rangle = +1.12\\pm0.48$ (17 stars). Most of the stars in M~54 are C deficient ([C\/Fe]$\\lesssim$+0.3) and N enhanced ([N\/Fe]$>+0.5$), but they do not exhibit the typical N-C anti-correlation (see panel (c) of Figure \\ref{Figure2}) seen in other GCs at similar metallicity \\citep[e.g.][]{Meszaros2020}, most probably due the lack of stars with low nitrogen abundances. On the contrary, an apparent continuous distribution of N abundances is present in M~54. This result indicates the prevalence of the multiple-population phenomenon in M~54 as previously suggested in the literature \\citep{Carretta2010, Milone2017, Sills2019, Meszaros2020}.\n\nAdditionally, we do not find any evidence for the presence of the K-Mg anti-correlation in M~54, as have been suggested to be present in a few Galactic GCs at similar metallicity \\citep{Meszaros2020}. Furthermore, a Si-Al correlation is slightly evident in M~54, as shown in panel (d) of Figure \\ref{Figure2}, and has a stubby Mg-Si distribution (see panel (e) of Figure \\ref{Figure2}), which is an indication of $^{28}$Si production from the result of a secondary leakage in the main Mg-Al cycle, which is instead absent in the Sgr stars.\n  \nFor the elements produced by neutron(\\textit{n})-capture processes (Ce II and Nd II), we find, on average, $\\langle$[Ce\/Fe]$\\rangle = +0.18\\pm0.13$ (10 stars) and [Nd\/Fe]$=+0.44$ (1 star). Overall, M~54 exhibits a modest enrichment in \\textit{s}-process elements, with a few stars as enhanced as $+0.4$, similar to that observed in Galactic GC stars at similar metallicity \\citep[see, e.g.,][]{Meszaros2020}, suggesting that it is possible that the \\textit{s}-process enrichment has been produced by a different source than the progenitor of the Mg-Al anti-correlations, possibly by low-mass asymptotic giant branch stars. Lastly, we find that [Ce\/Fe] ratios in M~54 are almost flat as a function of metrallicity. Unfortunately, Nd II is measured in only one star, which has been found to exhibit the modest enhancement, consistent with a moderate enrichment of \\textit{s}-process elements.\n\nFurthermore, we report the serendipitous discovery of two N-enhanced stars identified within $\\sim$7$r_t$ from M~54, as shown in panel (a) of Figure \\ref{Figure1}. Panel (a) of Figure \\ref{Figure2} show the collection of [X\/Fe] and [Fe\/H] abundance ratios for the two newly identified N-rich stars beyond the tidal radius of M~54. Both stars exhibit very similar chemical-abundance patterns as those seen in the population of M~54. A plausible explanation is that both stars were previous members of M~54, from which they have been ejected. However, this possibility seems unlikely for one of these extra-tidal stars (2M18533777$-$3129187), which was ruled out as a possible member of M~54. \n\nAs can be appreciated from inspection of panels (a) to (d) of Figure \\ref{Figure1}, the current position of 2M18533777$-$3129187 does not resemble the kinematic and astrometric properties \\citep[e.g.,][]{Antoja2020, Hayes2020} of Sgr+M~54 stars, nor the orbital path of Sgr\\footnote{The Sgr orbit was computed with the \\texttt{GravPot16} model, \\url{https:\/\/gravpot.utinam.cnrs.fr}, by adopting the same model configurations as described in \\citet{Fernandez-Trincado2020c}. For the Sgr centre, we adopt the heliocentric distance $d_{\\odot} =$ 26.5 kpc and heliocentric radial velocity $RV = 142$ km s$^{-1}$ from \\citet{Vasiliev2020b}, and proper motions from \\citet{Helmi2018}: $\\mu_{\\alpha}\\cos{\\delta} = -2.692$ mas yr$^{-1}$ and $\\mu_{\\delta} = -1.359$ mas yr$^{-1}$, with uncertainties assumed of the order of 10\\% in $d_{\\odot}$, $RV$, and proper motions.} . It is also the most luminous star in our sample, making it a likely foreground star. The possibility that this star was disrupted from M~54 and deposited in the inner Galaxy seems unlikely, as the perigalacticon of M~54 is located well beyond the solar radius \\citep[see, e.g.,][]{Baumgardt2019}. We   conclude that 2M18533777$-$3129187 is a N-enhanced field star born in a different progenitor than M~54, but with a similar chemical-enrichment history to this cluster.\n\n Aside from 2M18533777$-$3129187, there is another N-enhanced field star (2M18565969$-$3106454) located $\\sim{}5\\times{}r_{t}$ from the cluster center, which exhibits a stellar atmosphere strongly enriched in nitrogen ([N\/Fe]$>+1.4$), as extreme as M~54 stars, accompanied by a very low carbon abundance ([C\/Fe]$<-0.7$), and with discernible contributions from the \\textit{s}-process elements (Ce II). Since the [Al\/Fe] ratio is $>+0.5$, which is a 'typical' value for stars in GCs, and unlikely in dwarf galaxy populations, we conclude that 2M18565969$-$3106454 shares the same nucleosynthetic pathways of second-generation stars in M~54. \n \n2M18565969$-$3106454 is a potential extra-tidal star with kinematics and astrometric properties similar to that of M~54 stars, and exhibits unique chemical patterns comparable to that of genuine second-generation GC stars, which makes it very different from Sgr stars. On the other hand, N-rich stars are commonly observed to be more centrally concentrated in GCs \\citep[e.g.][]{Dalessandro2019} and as a consequence they have smaller probabilities to be tidally stripped. Thus, it is likely that the extra-tidal star could well be just a stripped M~54 star as many others in its surroundings. Our finding demonstrate that N-rich stars are a promising route for identifying the unambiguous chemical signatures of stars formed in GC-like environment which may lie immersed in the M~54+Sgr core and\/or Sgr stream, as well as confirm or discard the possible association of GCs to the Sgr stream \\citep{Bellazzini2020}. \n \nFollowing the same methodology as described in \\citet{Fernandez-Trincado2021b}, we compute the predicted number ($N_{N-rich}$) of N-rich field stars observed in APOGEE-2 toward M~54\/Sgr using the smooth halo density relations presented in \\citet{Horta2021}, and by adopting the same Monte Carlo implementation of the Von Neumann Rejection Technique \\citep[see e.g.,][]{Press2002} as in Eq. 7 in \\citep{Fernandez-Trincado2015a}. We find the expected number of observed N-rich halo stars beyond $d_{\\odot}\\gtrsim15$ kpc over the sky area of 1.5 degree radius centred in M~54, and with both astrometric and kinematic properties as M~54 to be $N_{N-rich}< 0.1$ (from 1000 Monte Carlo realisations). This yield a very low probability that the new identified extra-tidal N-rich star associated with M~54 is due to random fluctuations in the field. Furthermore, we also use the Besan\\c{}con galactic model \\citep{Robin2003} and the \\texttt{GravPot16} model \\citep{Fernandez-Trincado2020e} to explore the expectations for a \"default\" Milky Way along the RVs to the Sgr+M\\~54 surrounding field beyond  $d_{\\odot}\\gtrsim$ 15 kpc. The \"all\" sample is dominated by halo kinematics with a negligible contribution from the thin and thick disk beyond $RV \\gtrsim 120$ km s$^{-1}$.  Thus, our Milky Way simulated sample act to guide us in $RV$ space, confirming that the kinematics of the newly identified extra-tidal N-rich star differs from the disk population, with practically low contribution form the expected halo. , \n\n \\section{Concluding remarks}\n \\label{section6}\n\nWe present a spectroscopic analysis for 20 out 22 red giant stars that are members of M~54 from the internal APOGEE DR16 dataset. This study doubles the sample of stars with spectroscopic measurements for this cluster, and the new post-APOGEE DR16 spectra achieve high signal-to-noise (S\/N$>70$), allowing the addition of new chemical species not examined in previous studies \\citep[e.g.,][]{Meszaros2020} in the \\textit{H}-band--APOGEE-2 footprint. \n \n Overall, the chemical species re-examined in M~54 were found to be consistent with previous studies \\citep{Meszaros2020}, although most of them exhibit a large star-to-star scatter. We find that 15 out of the 20 stars investigated show a high [N\/Fe] abundance ratio ([N\/Fe]$\\gtrsim+0.5$), confirming the prevalence of the MPs phenomenon in M~54. Both [Ni\/Fe] and [Ti\/Fe], not previously examined in \\citet{Meszaros2020}, were found to be in good agreement with measurements in the literature. In particular, we confirm the [Ti\/Fe]~ratio slightly increases as [Fe\/H] increases, as has been reported in \\citet{Carretta2010}. We also find a large spread in [Al\/Fe], and the presence of a genuine second-generation star in M~54, which exhibits Mg deficiency ([Mg\/Fe]$<$0) accompanied with large enhancements in nitrogen and aluminum. In general, all chemical species examined in the M~54 members present distinguishable chemical behaviour compared with Sgr stars, suggesting a different chemical-evolution history that resembles other Galactic halo GCs at similar metallicity.\n\nFurthermore, we report on the serendipitous discovery of a potential extra-tidal star toward the surrounding regions of the M~54+Sgr core, which exhibits a strong enrichment in nitrogen comparable to that seen in M~54 stars. As far as we know this is the first study reporting on the unambiguous chemical signatures of stars formed in GC-like environment into a nearby satellite dwarf galaxy around the Milky Way. Finding out how many of such chemical unusual stars likely originated in GCs are present in dwarf galaxy systems, help to understand the link between GCs and their stellar streams \\citep[see e.g.,][]{Bellazzini2020}. \n\n\t\\begin{acknowledgements}  \n\tThe author is grateful for the enlightening feedback from the anonymous referee.\n\t J.G.F-T is supported by FONDECYT No. 3180210. \n\t T.C.B. acknowledges partial support for this work from grant PHY 14-30152: Physics Frontier Center \/ JINA Center for the Evolution of the Elements (JINA-CEE), awarded by the US National Science Foundation. \n\t D.M. is supported by the BASAL Center for Astrophysics and Associated Technologies (CATA) through grant AFB 170002, and by project FONDECYT Regular No. 1170121. \n\t S.V. gratefully acknowledges the support provided by Fondecyt regular No. 1170518. \n    D.G. gratefully acknowledges support from the Chilean Centro de Excelencia en Astrof\\'isica y Tecnolog\\'ias Afines (CATA) BASAL grant AFB-170002. D.G. also acknowledges financial support from the Direcci\\'on de Investigaci\\'on y Desarrollo de la Universidad de La Serena through the Programa de Incentivo a la Investigaci\\'on de Acad\\'emicos (PIA-DIDULS). \n\t A.R.-L. acknowledges financial support provided in Chile by Agencia Nacional de Investigaci\\'on y Desarrollo (ANID) through the FONDECYT project 1170476.\n\t B.B. acknowledge partial financial support from the Brazilian agencies CAPES-Financial code 001, CNPq, and FAPESP. \n\t \\\\\n\t\n\tThis work has made use of data from the European Space Agency (ESA) mission Gaia (\\url{http:\/\/www.cosmos.esa.int\/gaia}), processed by the Gaia Data Processing and Analysis Consortium (DPAC, \\url{http:\/\/www.cosmos.esa.int\/web\/gaia\/dpac\/consortium}). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.\\\\\n\t\n\tFunding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions. SDSS- IV acknowledges support and resources from the Center for High-Performance Computing at the University of Utah. The SDSS web site is www.sdss.org. SDSS-IV is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration including the Brazilian Participation Group, the Carnegie Institution for Science, Carnegie Mellon University, the Chilean Participation Group, the French Participation Group, Harvard-Smithsonian Center for Astrophysics, Instituto de Astrof\\`{i}sica de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and Mathematics of the Universe (IPMU) \/ University of Tokyo, Lawrence Berkeley National Laboratory, Leibniz Institut f\\\"{u}r Astrophysik Potsdam (AIP), Max-Planck-Institut f\\\"{u}r Astronomie (MPIA Heidelberg), Max-Planck-Institut f\\\"{u}r Astrophysik (MPA Garching), Max-Planck-Institut f\\\"{u}r Extraterrestrische Physik (MPE), National Astronomical Observatory of China, New Mexico State University, New York University, University of Notre Dame, Observat\\'{o}rio Nacional \/ MCTI, The Ohio State University, Pennsylvania State University, Shanghai Astronomical Observatory, United Kingdom Participation Group, Universidad Nacional Aut\\'{o}noma de M\\'{e}xico, University of Arizona, University of Colorado Boulder, University of Oxford, University of Portsmouth, University of Utah, University of Virginia, University of Washington, University of Wisconsin, Vanderbilt University, and Yale University.\\\\\n\\end{acknowledgements}\n\t\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe magneto-rotational instability (MRI) is a powerful process to drive\nturbulence and angular momentum transport in protoplanetary disks, \nultimately enabling the accretion of matter onto the central object \\citep{bal91,haw91,bal98}.\nThere is vast literature studying this mechanism in local shearing box\nsimulations with an ideal MHD description\n\\citep{bra95,haw95,haw96,mat95,sto96,san04}. \\\\\nThe effect of non-ideal MHD on MRI, regarding the issue of resistive \nprotoplanetary disks, was mostly studied in\nlocal box simulations \\citep{bla94,jin96,san00,fle00,san02I,san02II,fle03,inu05,tur07,tur08,tur10}. \nThe various studies showed that at a \ncertain level of resistivity the MRI will be suppressed.\n\nUp to now there is no prescription of resistive profile \nin protoplanetary disks which applies for longer timescales.\nIt is known that the dust grains control the ionization level\nin protoplanetary disk. The particle cross section and dust-to-gas\nratio are most important parameters for defining the ionization level of\nthe gas \\citep{tur06,war07}.\nMost studies of non-ideal MRI turbulence use a static dust\ndistribution and neglect dust\ngrowth and evolution \\citep{sim09,tur06,tur07,dzy10}. But exactly in non-turbulent regions, the dust\n particles can grow, reducing quickly the cross section and so driving to\nbetter ionization levels enabling again MRI \\citep{zso10}. \nIn our study we focus on ideal MHD, which applies to sufficiently\nionized disk regions depleted of small dust grains. \nThis also applies to the innermost hot parts of \nprotoplanetary accretion disk and even extended radial regions \nas expected for transitional disks \\citep{chi07}.\nHere, an MRI turbulent ionization front starting at the inner rim of the \ndisk propagates radially outward and the disk get evacuated from\ninside-out.\\\\\n\nRecent results on the MRI obtained in local unstratified box simulations by \\citet{les07},\n\\citet{froII07}, \\citet{sim09} and \\citet{fro10} show that occurrence and saturation level of MRI in zero net flux simulations is controlled by the magnetic Prandtl number \\footnote{The ratio of viscosity vs.\\\nresistivity}. \nIdeal MHD simulations including stratification found at least convergence for zero-net flux local\nsimulations \\citep{dav10,fla10}. Here, the vertical resolution plays the key role for the convergence\n\\citep{shi10}. Despite the simplicity of box simulations and their interesting results obtained over the\nlast years, the study of radial extended structures are very restricted.\nFor instance, the aspect ratio of the boxes is known to influence the saturation level of turbulence \\citep{bod08}.\nBesides local box simulations, global simulations of MRI have also been performed\n\\citep{arm98,haw01I,ste01,haw01,arl01,fro06,lyr08,fro09}.\nThey confirmed the picture of a viscously spreading disk as a proxy for the action of MHD turbulence. \nRecently, the first global non-ideal MHD simulation \\citep{dzy10}, \nwhich included a radial dead-zone \/ active zone interface demonstrated the importance of the inner edge of the\ndead zone as a trap for planetesimals and even small planets.\\\\\n\nSo far only finite difference schemes as implemented in the ZEUS and Pencil\ncodes have been used to perform global simulations. \nHowever, a Godunov code would have several advantages over a finite difference scheme. \nWithout using any artificial viscosity the code solves \nthe MHD Riemann problem and can better handle the supersonic MHD turbulence in corona regions of the disk. Several papers recognized the importance of Godunov-type shock capturing upwind schemes for future astrophysical \nsimulations \\citep{sto05,fro06b,mig07,flo10}.\nIn this project we use the Godunov code PLUTO \\citep{mig09} to perform\nisothermal global MHD simulations of protoplanetary disks. \nIn future work, we\nwill switch on the total energy conservation property of this scheme and\ninclude radiation MHD to follow the temperature evolution in global disk.\\\\\nAn important issue in stratified global simulations is the\nrelatively low resolution per scale height compared to what is possible in local box simulations.\nIn addition, the extent of the azimuthal domain is often restricted to save computational time.\nThe first $2\\pi$ global disk simulations were performed by\n\\citet{arm98}, \\citet{haw00} and\n\\citet{arl01} for a short period of time.\nBut most of the global simulations were performed in restricted azimuthal domains like\n$\\pi\/4$ or $\\pi\/2$ \\citep{fro06,fro09,dzy10}. \nBy restricting the azimuthal domain one also restricts the largest possible mode in the\ndomain. One of the goal of this paper is to investigate whether these\nmodes affects the nonlinear state of the turbulence.\\\\\n\nThe standard viscous $\\alpha-$disk theory \\citep{sha73} introduces an effective turbulent viscosity \n $\\eta = \\rho \\nu = \\alpha P \/ \\Omega$ with the local thermal pressure $P$\nand the orbital frequency $\\Omega$,\narising from undefined magnetic or hydrodynamic processes,\ntransporting angular momentum outward and allowing mass accretion onto the star.\n\\citet{lyn74} calculated the radial mass accretion rate and the\nradial accretion velocity for a 1D viscous disk model as a local function of surface density $\\Sigma$ and the value for $\\alpha$.\nInterestingly 2D viscous disk models \\citep{kle92} showed the appearance of meridional\noutflows. Here, the mass flows radially outward near the\nmidplane compensated by increased radial inflow at upper layers of the disk to allow for net-accretion,\nfor $\\alpha < 0.05$. Much emphasis was given to this radial outflow and its role\nfor the radial transport for grains and chemical species over large distances and relative short time scales\n\\citep{kel04,cie09}.\\\\\nIn addition, we will investigate the onset of a vertical outflow\nas it was described in local box simulations \\citep{suz09,suz10} using a net\nflux vertical field. Such outflows can be launched in the\nmagnetized corona region of the MRI turbulent disk \\citep{mil00,mac00}.\nThey could have an important effect on the dissipation timescales of\naccretion disks and may be related to jet production from accretion\ndisks \\citep{fer06}.\nAn interesting property of MRI in stratified zero-net flux simulations is the emergence of\na \"butterfly\" pattern, an oscillating mean azimuthal magnetic field with an\nperiod of 10 local orbits. It was found in many local MRI simulations,\nrecently again by \\citet{dav10}, \\citet{gre10}, \\citet{fla10} and in global\nsimulations by \\citet{sor10} and \\citet{dzy10}.\nWe indeed identify such a \"butterfly\" pattern in our global runs, which was suggested to be linked to magnetic dynamo action in accretion disks \\citep{sor10,gre10}.\n\nOur paper is organized in the following way.\nIn Section 2 we will present our model setup and the numerical configuration.\nSection 3 contains the results\nSection 4 and 5 will provide a discussion, summary and outlook for our work.\n\\section{Model setup}\nThe setup follows closely the disk model which is presented by\n\\citet{fro06,fro09}.\nWe define the cylindrical radius with $R = r \\sin{(\\theta)}$ with the spherical\nradius $r$ and polar angle $\\theta$.\nThe initial density, pressure and azimuthal velocity are set to be in hydrostatic equilibrium.\n$$\\rho = \\rho_{0} \u00a0R^{-3\/2}\\exp{}\\Bigg(\\frac{\\sin{(\\theta)}-1}{(H\/R)^2}\\Bigg) $$\nwith $\\rho_{0} = 1.0$, $\\rm H\/R = c_0 = 0.07$.\\\\ \nWe choose an isothermal equation\nof state. The pressure is set to $P = c_{s}^2\\rho$ with $\\rm c_{s} =\nc_0\\cdot1\/\\sqrt{R}$.\nThe azimuthal velocity follows  $$V_{\\phi} = \\sqrt{\\frac{1}{r}}\\Bigg(1- \\frac{2.5}{\\sin(\\theta)}c^2_0 \\Bigg).$$\nFor the initial velocities $V_{R}$ and $V_{\\theta}$ we use a white noise\nperturbation amplitude of $V_{R,\\theta}^{Init} = 10^{-4} c_{s}$.\ufffd\nWe start the simulation with a pure toroidal magnetic seed field with constant plasma beta\n$\\beta = 2P \/ B^{2} = 25$.\nThe radial domain extends from 1 to 10 radial code units (CU)\\footnote{We refer to CU instead of a physical length unit because ideal MHD simulations without radiation transport\nare scale free. Thus our simulations could represent a disk from 1 to 10 AU as much as a disk from\n$0.1$ to 1 AU. Only explicit dust physics and radiative transfer will introduce a realistic physical scale.} with radial buffer zones from 1 to 2 CU\nand 9 to 10 CU.\nIn the buffer zones we use a linearly increasing resistivity. This damps \nthe magnetic field fluctuations and suppresses boundary interactions,\nespecially for the closed boundary runs. Our buffer\nzone follows mainly the ones used in global simulations by \\citep{fro06,fro09,dzy10}. \nThe $\\theta$ domain is set to $\\theta = \\pi\/2 \\pm 0.3 $, corresponding to $\\pm\n4.3 \\rm $ scale heights.\nWe calculated in total five disk models. Three models cover the complete $2\\pi$ azimuthal  domain (FC, FO and BO in Table 1) and \ntwo models are constrained to $\\pi\/4$ (PC, PO in Table 1). The simulation FO\nis also used for a test model FOR which is described later.\nThe simulation BO has the best resolution.\nOne subset of models has a closed boundary (FC and PC). Here we\nuse a reflective radial boundary with a sign flip\nfor the tangential magnetic fields and a periodic boundary condition for the $\\theta$ direction.\nA second subset of models has an outflow boundary condition (FO,PO and BO). Here, we use a relaxation function in the radial buffer zones which reestablishes gently the initial value of density over a time period of one local orbit. \nIn the buffer zones we set: $\\rho^{new} = \\rho - (\\rho-\\rho^{Init})\\cdot \\Delta\nt \/ T_{Orbits}$. \nOur outflow boundary condition projects the radial gradients\nin density, pressure and azimuthal velocity into the radial boundary and the\nvertical gradients in density and pressure at the $\\theta$ boundary. \nWe ensure to have no inflow velocities. For an inward pointing velocity\nwe mirror the values in the ghost cell to ensure no inward mass flux. \nThe $\\theta$ boundary condition for the magnetic field are also set up \nto be zero gradient, which approximates \"force-free\" - outflow conditions. \nWe also ensure the force free character of the tangential components for the radial boundary\nby adjusting the $1\/r$ profile in the magnetic field components in the ghost\ncells.\nThe normal component of the magnetic field in the ghost cells is always \nset to have $\\nabla \\cdot \\vec{B}$ = 0.\nWe set the CFL value to 0.33. Also higher CFL values were successfully tested and\nwill be used for future calculations.\nWe use a uniform grid with an aspect ratio of the individual cells at 5 CU of $1:0.67:1.74$\n$(\\Delta r: r\\Delta\\theta:r\\Delta\\phi)$.\nUsing a uniform grid instead of a logarithmic grid, where\n$\\Delta r\/r$ is constant, has the disadvantage that it will reduce the\naccuracy in the sense that the inner part of the disk is poorly resolved,\ncompared to the outer part of the disk: $H(1AU)\/\\Delta r < H(10AU)\/\\Delta r  $.\\\\ \nHowever for the uniform grid, the relative broad radial inner \nbuffer zone lies in the poorly resolved disk part and is excluded from\nanalysis.\nThe outer parts of the disk are, compared\nto a logarithmic grid with the same resolution, better resolved.  \nLogarithmic grid requires a much smaller buffer\nzone, e.g. a logarithmic grid would place one third of the total number of grid\ncells in the first ninth of the domain, between 1 and 2 AU.\nOf course, using a uniform grid will always restrict the range of the\nradial domain and for more radially extended simulation the need of a logarithmic grid is mandatory.\n\nFor all runs we employ the second order scheme in\nPLUTO with the HLLD Riemann solver \\citep{miy05}, piece-wise linear\nreconstruction and $2^{nd}$ order Runge Kutta time integration. \nWe treat the induction equation with the \"Constrained Transport\" (CT) method in combination with the upwind CT method described in\n\\citet{gar05}.\nThe detailed numerical configuration is presented in \\citet{flo10}.\nOur high resolution run BO was performed on a Blue-gene\/P cluster with 4096\ncores and was calculated\nfor over 1.5 million time steps which corresponds to 1.8 million CPU hours.\n\\begin{table}[th]\n\\begin{center}\n\\begin{tabular}{ccccc}\nModel name & Resolution ($R$ $\\theta$ $\\phi$) \u00a0& $\\phi$-range & Boundary & Orbits at 1 AU (Years)\\\\\n\\hline\n\\hline\nPC & 256 128 64 & $\\pi\/4$ & closed & 1435 \\\\\nPO & 256 128 64 & $\\pi\/4$ & open & 1519 \\\\\nFC & 256 128 512 & $2\\pi$ & closed & 1472 \\\\\nFO & 256 128 512 & \u00a0$2\\pi$ & open & 1526 \\\\\n\\hline\nFOR& 256 128 512 & $2\\pi$ & open & $1000-1448$ \\\\ \n\\hline\nBO& 384 192 768 & $2\\pi$ & open & $1247$ \n\\\\\n\\end{tabular}\n\\caption{MHD runs performed. (P - $\\pi\/4$; F - $2\\pi$; O - open boundary; C -\nclosed boundary; B - best resolved run.)}\n\\end{center}\n\\end{table}\n\\subsection{Code Units vs. Physical Units}\nIsothermal ideal MHD simulations are scale-invariant. \nOne has to define unit-variables to transform from code to cgs units. \nWe can set three independent values to \ndefine our problem. \nThis is gas density for which we choose for instance $\\rm \\rho_u = 10^{-10} g\/cm^3$, the radial\ndistance unity as length $1 CU = 1 AU$\nand the Keplerian velocity $\\rm v_u = \\sqrt{ G\\cdot M_{\\sun} \/ l_u}$ with the gravitational\nconstant $G$ and the solar mass $M_{\\sun}$.\n\nWith those three quantities, we translate the values for our measured surface density and mass accretion\nrate into cgs units. Using this values we derive at 1 AU at the midplane a gas density of\n$\\rm \\rho = 10^{-10} g\/cm^3$ with a Keplerian velocity of $\\rm v_K= 2.98* 10^6 cm\/s$. \nWith this the surface density becomes $524 g\/cm^2$ at 1AU.\nGas velocities and the Alfv\\'enic speed are always presented in units of the sound\nspeed for convenience.\n\n\n\\subsection{Turbulent stresses}\nThe $\\alpha$ parameter relates the turbulent stresses to \nthe local thermal pressure. For the calculation of the $\\alpha$ values\nwe measure the Reynolds and Maxwell\nstresses, which are the $R-\\phi$ components of the respective stress tensors. \nThe Reynolds stress is calculated as $\\rm T_{R}= \\overline{\\rho v'_{\\phi}v'_{R}} $\nand the Maxwell stress as $\\rm T_{M}= \\overline{B'_{\\phi}B'_{R}}\/ 4 \\pi $\nwith the turbulent velocity or magnetic fields, e.g., $\\rm v'_{\\phi} = v_{\\phi} - \\overline{v_{\\phi}}$.\nThe mean component of the velocity and magnetic field are always calculated\nonly along the azimuthal direction because of the radial and vertical gradients in the disk.\nIn our simulations, the amplitude of Maxwell stress is about three times the Reynolds stress.\nFor the total $\\alpha$ value we integrate the mass weighted stresses over the total domain\n$$ \\alpha = \\frac{ \\int \\rho \\Bigg( \\frac{v'_{\\phi}v'_{R}}{c^2_s} - \\frac{B'_{\\phi}B'_{R}}{4 \\pi \\rho c^2_s}\\Bigg)dV} {\\int \\rho dV}.$$\nThe respective turbulence enhanced viscosity can now be represented as $\\nu = \\alpha H c_s$\nwith the height of the disk $H$ and the sound speed $c_s$.\n\\begin{figure}\n\\hspace{-0.6cm}\n\\begin{minipage}{5cm}\n\\psfig{figure=FIG-1A.ps,scale=0.46}\n\\psfig{figure=FIG-1C.ps,scale=0.46}\n\\end{minipage}\n\\hspace{4.0cm}\n\\begin{minipage}{5cm}\n\\psfig{figure=FIG-1B.ps,scale=0.46}\n\\psfig{figure=FIG-1D.ps,scale=0.46}\n\\end{minipage}\n\\label{totalal}\n\\caption{Top left: Total stresses expressed as $\\alpha$-parameters for the $\\pi\/4$ models PC\nand PO and the $2\\pi$ models FC, FO and BO. The parameter is mass weighted\nand integrated for the domain (3-8 AU).\nTop right: Radial $\\alpha$ profile, time averaged between 300 and 1200 inner\norbits.\nFor the best resolution model BO, the profile follows roughly $\\sqrt{r}$ in the region B. \nRegion A and C are affected by boundary conditions and buffer zones.\nBottom left:  Vertical $\\alpha$ profile, averaged over time and space region ($II\/B$). \nBottom right: Evolution of the vertical distribution of azimuthally averaged Maxwell and Reynolds stress\n$T_S$ for a radius of 4 AU with\ntime. Colors are logarithmic values of the corresponding dynamical viscosity including the density profile.}\n\\end{figure}\n\\section{Results}\n\\subsection{Disk evolution}\nWe first describe the typical evolution for an azimuthal MRI (AMRI) \nin global disk simulation with open boundaries. \nThe AMRI simulation starts with a purely toroidal net magnetic field which \nbecomes MRI unstable on timescales of around 10 local orbits.\nAfter approximately 250 inner orbits, the disk reaches its maximum\n$\\alpha$ value of $0.01$.\nAt this time (equivalent to 10 local orbits at\nthe outer boundary of the undamped region) the disk has become fully turbulent.\nDuring this evolution, the initial magnetic flux decreases as it leaves the computational\ndomain in the vertical direction. \nStarting at approximately 250 inner orbits, there is an evolution of oscillating mean fields.\nFig. 1, top left, presents the mass weighted and domain integrated $\\alpha$ value \nover time for all models.\nDuring the time period between 800 and 1200 inner orbits, we get a\nrelatively constant $\\alpha$ value of $5\\cdot10^{-3}$ (model BO). \nWe mark three different time stages of the turbulent\ndisk evolution:\nIn period I (0 to 800 inner orbits), the turbulence is not yet saturated.\nAfter a strong initial rise due to the net azimuthal field the turbulent decays\nto a level where self-sustained turbulence is possible, e.g., the loss of magnetic flux\nin the vertical direction is balanced by the generation of magnetic flux in the turbulent flow.\nThe nature of this generation of magnetic fields can be an indication for dynamo action,\ne.g., an $\\alpha \\Omega$ dynamo\n\\citep{bra05,gre10}, but detailed studies of this effect will be subject to future work.\nIn period II, we have a quasi steady state of at least 400 inner orbits. \nDuring this time period, we do all our analysis. \nIn the period III, a comparison of the models becomes less useful.\nThe models with lower resolution and open boundary (PO and FO) \nshow a decreasing $\\alpha$-stress in time. Thus they are not useful for long time integrations past\n1200 inner orbits. In the closed models, on the other hand, the magnetic flux cannot escape vertically \n(PC and FC) and therefore, turbulence does not decay. \nOn the contrary, turbulence even increases in these runs as the flux in the box \ncannot efficiently escape.\\\\\nA closer view shows that the stress can oscillate locally on shorter time  scales.\nIn Fig. 1, bottom right, we plot the mass weighted stresses $\\alpha\\rho$ at 4 AU over height\nand local orbits. The strength of stresses locally oscillates with a period of around 5 local\norbits. The maxima in the stresses always appear first in the midplane and then propagate\nvertically. These oscillations in the stresses are connected to the $B_\\phi$ \"butterfly\" structures\nwhere the azimuthal mean field oscillates with a frequency of 10 local orbits (Fig. 13). \nEvery change of sign in the mean $B_\\phi$ is now correlated with a minimum in the stresses, \nwhich both occur every 5 local orbits.\nAt the same time this plot shows the importance of the stresses in a region up\nto 3 disk scale heights.\n\nAlthough one can always define a total $\\alpha$-parameter in the disk, the spatial\nvariations are enormous, especially in the vertical direction. \nThe vertical $\\alpha$ profile is plotted in Fig.1, bottom left. For model\nBO, the turbulent stress at the midplane increases from $2.0\\cdot 10^{-3}$\nup to $8\\cdot10^{-2}$ at 4 scale heights.\nThe simulations with moderate resolution show significant lower values around the midplane due\nto the lack of resolution there. For the closed models, the stresses in the\ncorona are artificially increased due to the periodic boundary.\\\\\n\n\\subsection{Radial profile of turbulent stress}\nBeside the vertical profile of turbulent stress, which has been already studied in local\nbox simulations, the radial profile of the turbulent stress can only be addressed in global\nsimulations.\nIn Fig. 1, top right, we present the radial $\\alpha$ profile, averaged\nbetween 300 and 1200 inner orbits.\nIn the inner buffer zones (1 - 2 AU) the $\\alpha$ values are practically zero because of the\nresistive damping. Starting from 2 AU $\\alpha$ rises until it levels off at\naround 3 AU. From 3 to 8 AU we obtain a radial $\\alpha$ profile which can be\napproximated by a $\\sqrt{r}$ dependence. Beyond 9 AU $\\alpha$ is again close to zero because of the damping applied there.\nWe mark three regions in radius (Fig. 1, top right, green lines).\nRegion A, extending from 1 to 3 AU, is affected by the buffer zone.\nRegion B, ranging from 3 to 8 AU shows the $\\sqrt{r}$ slope.\nRegion C, covering 8 to 10 AU, is again affected by the buffer zone.\nIn the following analysis, we will therefore concentrate on region B.\\\\\nIn order to have a radial force-free accretion disk, \nfields have to drop radially as $B \\propto r^{-1}$ (Fig. 12, top left).\nThis was also observed for magnetic fields in galactic disks \\citep{bec01} (Fig. 1).\nIf the most important toroidal field follows $\\propto r^{-1}$, \nthe radial Lorentz force vanishes:\n$$\\rm F_{radial} = - \\frac{1}{r^2\\rho} \\frac{\\partial r^2 B_{\\phi}^2}{\\partial\nr}.$$\nIn the case of $\\rm \\partial \\log{\\rho}\/\\partial \\log{r} = -1.5$\nand $\\partial \\log{c_s}\/\\partial \\log{r} = -0.5$ the $\\alpha$ value,\ndominated by the Maxwell stresses will then \nscale as $\\sqrt{r}$, which is actually matching the value that \nwe measure in our best resolved model (see Fig. 1, top right).\n\n\n\\subsection{Mass loss}\nThe models with open boundaries show a considerable mass loss over the course of our\nsimulations.\nA vertical outflow removes a substantial amount of mass. The\ntotal mass loss over time is presented in Fig. 5. The mass loss is\ndetermined in space region B. The closed models FC and PC loose there mass\ndue to radial mass movement. The open models loose there mass mainly due to\nthe vertical outflow. We will discuss this outflow in section 3.5 .\nTo check the possible impact of this mass loss onto the properties of the\nturbulence, we restarted run FO after 1000 inner orbits with the current\nvelocity and magnetic field configuration but the initial density distribution.\nWe call this model FOR (FO Restarted, Table 1).\nAfter restarting the simulation, the turbulence needs a couple of\ninner orbits to readjust the fully turbulent state.\nWe compared the mean total $\\alpha$ stress of the runs FO and FOR and found a comparable evolution \nof the $\\alpha$ values.  We measure $\\alpha = 1.4\\cdot10^{-3}$ for FOR and $\\alpha = 1.3\\cdot10^{-3}$ for FO in the time period\nfrom 1000 up to 1400 inner orbits. We conclude that the mass loss is not\nyet influencing the development and strength of turbulence. \n\\begin{figure}\n\\hspace{-0.6cm}\n\\begin{minipage}{5cm}\n\\psfig{figure=FIG-2A.ps,scale=0.46}\n\\psfig{figure=FIG-2C.ps,scale=0.46}\n\\end{minipage}\n\\hspace{4.0cm}\n\\begin{minipage}{5cm}\n\\psfig{figure=FIG-2B.ps,scale=0.46}\n\\psfig{figure=FIG-2D.ps,scale=0.46}\n\\end{minipage}\n\\label{surf}\n\\caption{Top left: The surface density profile after 1000 inner orbits for the $\\pi\/4$ model\nPO, the $2\\pi$ model FO and the high-resolution model BO. Dashed lines represent the surface\ndensity profile for the respective viscous disk model and the dotted line the initial profile. \nTop right: 2D contour plot of time and azimuthal averaged radial mass flow for model BO.\nThe red color indicates inward accretion to the star, blue color shows outward motion.\nWe do not observe a meridional flow.\nBottom left: Radial profile of the time-averaged radial mass flow for\nthe high-resolution model BO (solid line) and the viscous models. \nBottom right: 2D contour plot of $(\\Omega - \\Omega_0) \/ \\Omega_0$ over\nradius and time, averaged over azimuth at the midplane. The orbital frequency\nremains sub-Keplerian $(\\Omega_K - \\Omega_0) \/ \\Omega_0 = 0.012$.}\n\n\\end{figure}\n\\subsection{Viscous disk models}\nThe classical $\\alpha$ viscous disk model should reproduce the radial mass flow \nas it occurs in global MHD simulations of MRI turbulent disks.\n\\citet{bal99} have\nargued that the mean flow dynamics in MRI turbulence follows the\n$\\alpha$ prescription. \nTo further test this supposition, we performed a series of 2D HD viscous comparison simulations with the PLUTO code \nfor several of our 3D MHD runs. \nWe use the same resolution and the same initial setup. \nYet, the magnetic field evolution is now replaced by an explicit shear \nviscosity from the time averaged radial $\\alpha$ profile $\\nu (R) = \\alpha(R) H c_s$\nobtained from the MHD simulations (see Fig. 1 top right).\nFig. 2, top left, shows the surface density profile for the models PO,\nFO, model BO and the corresponding viscous\nmodel.\nThe surface density profile of the viscous runs follow nicely the respective MHD model\nprofile (Fig. 2, top left, dashed line) for the region that we use for analysis (3-8 AU).\nAll viscous models show a higher surface\ndensity profile than the open models BO, PO and FO, but of course\nin contrast to the MHD models the viscous models do not show any \nsubstantial vertical mass outflow.\nThe total radial mass flow (e.g. azimuthally and vertically integrated) is plotted as\na time average (0 - 1000 inner orbits) for \nthe high-resolution model B0 (Fig. 2, bottom left, solid line) and the respective viscous\nruns (Fig. 2, bottom left, dashed and dotted line). \nThe radial mass flow of the viscous run matches very well the flow obtained in the MHD model.\nA constant $\\alpha$ value will not reproduce the proper\nevolution of the MRI run. If we adopt for instance a constant $\\alpha$ value of $5\\cdot10^{-3}$, \nwhich would be the global mean value of the MHD run, we get a globally constant accretion rate of\n$\\rm 5.1\\cdot10^{-9} M_\\sun\/yr$.\nAs a sanity check for our viscosity module we compare this value to the \nanalytical estimates by \\citet{lyn74}: \n$$\\dot M(r) = 3 \\pi \\Sigma_g \\nu + 6 \\pi r \u00a0\\frac{\\partial(\\Sigma_g\\nu)}{\\partial r}$$\nand find a value very close to the time dependent viscous run of $\\dot M = 6\\cdot10^{-9}\n\\frac{M_{\\sun}}{yr}$,\nbased on a surface density profile of $\\rm \\Sigma_g = 524\\cdot R^{-0.5} [g\/cm^2]$ and our disk parameter\n$H=0.07*R$.\\\\\nIn Fig. 2, top right, we show the time and azimuthal average of \nthe accretion rate over radius and height. \nThere is a dominant inward accretion at the midplane (Fig. 2, top right, red\ncolor).\nThis result is in contrast to the viscous runs where we see the minimum of\naccretion and even a small outflow at the midplane \\citep{kle92,tak02}.\nAfter \\citet{tak02} (eq. 8) there are several possibilities which could change\nthe vertical profile of the radial velocity, and therefore the mean accretion\nflow. Radial and vertical gradients in the orbital frequency as well as a\nspatially varying $\\alpha$ will affect the vertical profile of the meridional outflow.\nFor the MHD simulations one has to include the vertical gradient as well as the time derivative of the orbital\nfrequency. Fig. 2, bottom right, demonstrate the change of the orbital frequency\nwith a period of around 50 local orbits at 5 AU.\\\\\nRadial mass flow and surface density evolution have shown that we can fit\nour MHD global models with a viscous disk model as long as we use \nan $\\alpha$ profile compatible with our MHD run.\nOf course, the disk spreading that we observe in our MHD run\nis partly due to the existence of our radial buffer zones,\nin which not only the fields decay, but also the $\\alpha$ stresses\nvanish. In a larger radial domain we can expect that also a larger\nregion of the disk will get into a steady state of accretion.\nHowever, one could also argue that in a realistic protoplanetary accretion disk\none will ultimately reach dead zones which behave similar as our\nbuffer zones. In that sense the active part of our global disk is \nembedded between two dead zones.\n\\begin{figure}\n\\hspace{-1.2cm}\n\\begin{minipage}{5cm}\n\\psfig{figure=FIG-3A.ps,scale=0.56}\n\\end{minipage}\n\\hspace{4.2cm}\n\\begin{minipage}{5cm}\n\\psfig{figure=FIG-3B.ps,scale=0.46}\n\\end{minipage}\n\\label{vel_spec3}\n\\caption{Left: Angle between the cylindrical radial and vertical velocity \nwith respect to the midplane axis ($V_R = -1$ and $V_Z = 0$) for the upper hemisphere.\nRight: Vertical mass outflow $\\rho v_z dA_z$ in units of $\\rm M_\\sun\/yr$ at 5 AU. \nThere is a mass outflow present above 3 scale\nheights. The evaporation time, $\\rm \\tau_{ev} = \\Sigma\/(\\rho v_z)$, was determined to 2070 local orbits.}\n\\end{figure}\n\n\\begin{figure}\n\\hspace{-1.2cm}\n\\begin{minipage}{5cm}\n\\psfig{figure=FIG-4A.ps,scale=0.49}\n\\end{minipage}\n\\hspace{4.2cm}\n\\begin{minipage}{5cm}\n\\psfig{figure=FIG-4B.ps,scale=0.46}\n\\end{minipage}\n\\label{vel_spec4}\n\\caption{Left: Logarithmic contour plot of the density, over plotted\nwith the velocity vector in the $R-\\theta$ plane for model BO. Both are averaged over azimuth and time\nand plotted for the upper disk hemisphere. The velocity vectors show a\n outflow pattern above two disk scale heights. The red line marks the optical depth of $\\tau = 1$ for our setup.\nRight: Vertical density profile at 5 AU after 1000 inner orbits for\nmodel BO (solid line), the respective viscous model (dashed line) and the \ninitial profile (dotted line). The optical depth of $\\tau = 1$ is around 2.8 scale\nheights for our model.}\n\\end{figure}\n\n\\begin{figure} \n\\hspace{-0.6cm}\n\\psfig{figure=FIG-5.ps,scale=0.46}\n\\label{totalmass}\n\\caption{Total mass plotted over time for all models. The mass is integrated in the\nspace region B (3-8 AU). The change of the mass for the closed models is due\nto radial movement. The mass loss for the open models is dominated by the vertical outflow.}\n\\end{figure}\n\n\n\\subsection{Vertical outflow}\nIn the previous section, we have seen that the MHD simulations point to the\npresence of an additional process besides turbulent \"viscous\" spreading that removes gas from the disk.\nFig. 4, right, shows the initial vertical density profile at 5 AU (dotted\nline), the respective profile for the MHD model BO (solid line) and the viscous HD run (dashed line) after 1000 inner orbits (90 local orbits).\nFor our model, the vertically integrated optical depth of $\\tau = 1$ is around 2.8 scale heights (Fig. 4, red line).\nFor the calculation of the optical depth, we used Rosseland mean opacities \\citep{dra84} with the temperature at 5 AU.\nThe additional magnetic pressure as well as the vertical mass flow, present in the MHD run, generates higher gas density \nabove 3 scale heights compared to the hydrostatic equilibrium.\nIn Fig. 4, left, we plotted a snapshot of the azimuthally averaged velocity \nfield, taken after 1000 inner orbits. The plot indicates a vertical outflow in the disk starting above \n2 scale heights. We measured the angle \nbetween the cylindrical radial velocity $V_R$ and the vertical velocity\n$V_Z$ for the mean and turbulent components (Fig. 3, left). \nThe angle is measured with respect to the midplane axis\n(pointing to the star, see Fig. 3, left, $V_R = -1$ and $V_Z = 0$).\nThe upper (red solid line) and the lower hemisphere (blue solid line) \npresent similar profiles. \nFrom the midplane up to $1.8$ scale heights, the turbulent velocity field is directed\nupwards but still pointing to the star. \nThe low angle of $10^{o}$ for the mean velocities shows the gas motion\npointing to the star and towards the midplane. \nAt $1.8$ scale heights the turbulent velocity is pointing away from the\nmidplane and the star.\nAlso the mean velocity angle changes quickly in the region between 1.6 and 2 scale heights \nto an outflow angle, e.g., steeper than $90^{o}$. This region coincides with the region where\nthe vertical outflow is launched \\citep{suz10}. \nAbove 2 scale heights the angle between the turbulent and the mean velocity\ncomponents stays above $90^{o}$, leading a vertical outflow with a small radial outward component.\nThe so-called dynamical evaporation time is the time to evacuate\nthe gas completely from the disk assuming no supply of matter. In our\nmodel the value is slightly larger than 2000 local orbits (Fig. 3, right)\nwhich provides a confirmation for the vertical outflow obtained by local box\nsimulations from \\citet{suz10} with a vertical net flux field.\n\n\nIn Fig. 3, right, we plotted the vertical mass flow over height at 5 AU.\nThe outflow starts at 2 scale heights and reaches mass fluxes of $\\rm 10^{-10} M_\\sun\/yr$ at 5 AU (model BO, solid line). The influence of the\npure outflow boundary is observed to cause the small outflow in the HD viscous run (dashed line).\nIn the midplane region, the disk reestablishes the hydrostatic \nequilibrium due to the radial mass loss at the midplane (Fig. 6, bottom\nright, red solid line). This drives to a small mean vertical motions visible\nin the vertical velocity (Fig. 6, bottom right, green dotted line).\n\nThe gas does leave the grid with Mach numbers of only\n0.5, which is significantly lower that the local escape velocity, which would be about Mach 20.\nEven the results indicate a stable vertical outflow, \nwithout including the sonic point and the Alfv\\'enic point in the simulation\nit is not possible to make prediction about the flow, leaving or returning to\nthe disk at larger radii.  \nThus the fate of the vertical outflow to be a disk wind or not will have to be determined in more detail \nin future simulations with a much broader vertically extent. For this study one would\nmost probably need an additional vertical field in the corona, which could support \nadditional propulsion effects like magneto-centrifugal acceleration.\n\\subsection{Velocity analysis}\nPlanet  formation processes in circumstellar accretion disk are strongly dependent on the\nstrength of the turbulence. Turbulence mixes gas and particles, diffuses or concentrates them \nand makes them collide \\citep{ilg04,joh05,joh07,bra08,cuz08,car10,bir10}.\nThe property of MHD turbulence that is important for planet formation are the turbulent velocity\nand density fluctuations of the gas.\nThe density fluctuations are around $10\\%$ and follow the results by\n\\citet{fro06}.\nThe spatial distribution of the fluctuating and mean part of the velocities \nis presented in Fig. 6.\nAll results are obtained for time averages from 800 to 1200 inner orbits\nand are given in units of the sound speed.\nSpatial averaging is performed in azimuth and between 3 and 7 AU for the vertical\nprofiles. The radial profiles are mass\nweighted. \nFig. 6, top left, shows the turbulent RMS velocity over radius. \nThe profile is roughly constant with a total RMS velocity\nof $0.1 c_s$, dominated by the radial turbulent velocity.\nThe vertical dependence of the turbulent velocity (Fig. 6 - right - top) shows a flat profile \naround $\\pm 1$ scale height above and below the\nmidplane for the radial and azimuthal velocity.\nBoth components increase above one scale height by an order of\nmagnitude.\nThe radial component dominates with $0.07 c_{s} $ around the midplane up to $0.3 c_{s} $ at 4 scale\nheights. \nThe azimuthal component follows with $0.05 c_{s} $ up to $0.2 c_{s}$ at 4 scale heights.\nOnly the $\\theta$-component does not show a flat profile around the\nmidplane and increases steadily from $0.02 c_{s} $ to $0.2 c_{s}$ at 4 scale heights,\nwhich is an effect of the density stratification.\nThe small decrease of the $\\theta$ component near the vertical boundary is due to the \noutflow boundary because it does not allow inflow velocities.\\\\\n\nA global picture of the total rms velocity is presented in Fig. 7.\nThe 3D picture is taken after 750 inner orbits and shows again the \ndifferent turbulent structures of the midplane and coronal region.  \nThere are also localized supersonic turbulent motions in the disk\ncorona (Fig. 7, white color). Compared to the turbulent velocity, the mean\nvelocities of the gas are two\norder of magnitude smaller. They show small but steady gas motions in the\ndisk. \nThe vertical dependence for the mean velocity (Fig. 6 right - bottom) \nshows the small inward motion (red solid line) as well as the change of $r$ and $\\theta$-velocity\ncomponents to an outflow configuration around 1.6 scale heights.\n\\begin{figure}\n\\hspace{-0.6cm}\n\\begin{minipage}{5cm}\n\\psfig{figure=FIG-6A.ps,scale=0.46}\n\\psfig{figure=FIG-6C.ps,scale=0.46}\n\\end{minipage}\n\\hspace{4.0cm}\n\\begin{minipage}{5cm}\n\\psfig{figure=FIG-6B.ps,scale=0.46}\n\\psfig{figure=FIG-6D.ps,scale=0.46}\n\\end{minipage}\n\\label{vel_turb}\n\\caption{Top left: RMS fluctuations of the velocity versus radius for model BO,\naveraged over time and azimuth. \nAll components show a roughly flat profile, dominated by the radial turbulent velocity.\nThe radial profiles are mass weighted. The time average is performed during time period II (Fig.1, top\nleft, green line).\nTop right: Turbulent velocity profile versus scale height for model\nBO, averaged over time and azimuth.\nThere is a flat profile visible in the range $\\pm 1.5$\nscale heights above and below the midplane.\nStarting at $1.5$ scale heights the turbulent velocity increases.\nBottom left: Energy power spectra $E_m\\cdot m^2$ for $\\pi\/4$ model PO (red dotted line),\n$2\\pi$ model FO (blue dashed line) and the high-resolution model BO (black solid line). \nBottom right: \u00a0Time average of the mean velocity over scale height.\n}\n\\end{figure}\n\n\n\\subsubsection{Kinetic spectra}\nNot all particles do couple alike to the turbulent gas flow. \nIn fact particles have a size-dependent friction or stopping time \\citep{wei77}.\nThis stopping time is also the time a particle needs to couple to the\nturbulent gas flow. \nBest coupled to turbulence are those particles, which have a coupling\ntime shorter then turbulent correlation time. \nParticle collision velocities are maximized for particles whose stopping time\ncoincides with the turbulent correlation time, e.g., the eddy turn over time.\nThis means that particles of different sizes couple to different length scales of the\nturbulent spectrum. Therefore, a study of planet formation processes needs \nnot only the mean turbulent velocity but also its spectral distribution.\\\\\nIn the global domain, only the $k_{\\phi}$ space of the spectra is\nwithout modifications accessible as only the $\\phi$ direction is periodic in space.\nThe investigation of the complete $k$-space for this model goes beyond the\nscope of this work.\nThe classical Kolmogorov theory predicts the scaling of the energy\nspectra per wavenumber: $E(k) \\propto v_k^2 k^{-1} \\propto \\epsilon^{2\/3} k^{-5\/3}$. \nWe calculate along azimuth $|v(k_\\phi)|^2 = |v_r(k_\\phi)|^2 +\n|v_\\theta(k_\\phi)|^2 + |v_\\phi(k_\\phi)|^2 $ with $v_r(k_\\phi) = \\left\n\\langle \\int_\\phi\nv_r(r,\\theta,\\phi)e^{-ik_{\\phi}\\phi} d\\phi \\right \\rangle$.\nThe average is done in radius (region B, Fig. 1) and height ($\\pm 0.5$ disk scale heights).\nFor our spectra we use the azimuthal wavenumber $\\rm m$ instead of $k$\nto be independent from radius: $k = 2\\pi\/\\lambda = m\/R$.\nIn Fig. 6, bottom left, we plotted the energy power spectra $E_m =\nv(m)^{2}\\cdot m$ with time and\nspace averaged over $\\pm 0.5$ scale heights around the midplane.\nIn our models we do not observe Kolmogorov inertial like range, $E_m\n\\cdot m^2 \\sim m^{1\/3}$.\nThe $2\\pi$ runs F0 and BO have most of the energy placed at $m=5$.\nThe high resolution model BO present a $k^{-1.2}$ dependence, starting from $m=5$ until\n$m=30$.\nThe $\\pi\/4$ run PO piles up the energy at its domain size ($m=8$), reaching higher\nenergy levels compared to the $2\\pi$ models FO and BO.\nThe velocity spectra for each component along the azimuth, \nplotted in Fig. 8, left, indicate that all velocity components have \nsimilar amplitude for the small scales and do not deviate \nby more than a factor of two at the largest scales.\nThe radial velocities peak at m equals 5, but \noverall the entire spectrum above m=20 is essentially flat.\nThe peak at m=5 could be connected to the production of shear waves in the simulations.\nThese shear or density waves are described in \\citet{hei09}.\nOn top of the shorter time scale of MRI turbulence, these long \"time scale\" shear waves\nare visible in the contour plot of the radial velocity in the $r-\\phi$ midplane (Fig. 8 right).\nThe shear wave structures drive the radial velocity up to $0.3 c_s$.\nIn the velocity spectra we see the start of the dissipation regime\nat $m=30-40$ for the high-resolution run BO. For the model BO, \nthis corresponds to 26 or rather 19 grid cells per wavelength, \nwhich is still well resolved by the code \\citep{flo10}.\nShearing waves are also visible in a $r-\\theta$ snapshot \nof the velocity (Fig. 9, left). Here we plot the azimuthal \nvelocity $V_\\phi - V_K$ as contour color, over-plotted with the velocity vectors.\nRed contour lines show Keplerian azimuthal velocities.\nSuper-Keplerian regions are important for dust particle migration.\nThey reverse the radial migration of particles, leading to their efficient\nconcentration and triggering parasitic instabilities in the dust layer, like\nthe streaming instability leading potentially to gravoturbulent planetesimal formation\n\\citep{kla08,joh07}.\nIn our simulation these super-Keplerian regions are not completely\naxisymmetric, but have\na large extension in the azimuthal direction of several scale heights.\nThe variation of the orbital frequency over time and space, presented in\nFig. 9, left, and Fig. 2, bottom right, are connected to zonal flows. They\nare observed and discussed in several local and global studies\n\\citep{joh09,dzy10}.\n\\subsection{Magnetic field analysis}\nThe azimuthal MRI generates a turbulent zero-net field configuration in the\ndisk.\nDespite the loss of mass and magnetic flux through the vertical\nboundary there is no sign of decay for the highest resolution case BO \n(Fig. 1, top left, bottom right).\nWe find well established turbulence. Fig. 9, right, presents a snapshot of the magnetic fields \nafter 750 inner orbits. \nThe $r-\\theta$ components are shown as vectors   \nwith the azimuthal magnetic field as background color. \n\\subsubsection{Magnetic energy spectrum}\nTo understand the magnetic turbulence at the midplane, we\ninvestigated the spectral distribution of the magnetic energy.\nThe magnetic energy power spectrum (Fig. 12 - bottom left) is plotted along\nthe azimuthal direction with the same time and space average as for the \nkinetic energy power spectra.\nWe plot the magnetic energy power spectra times the wave-number \n$m\\cdot B_m^2\/2P_{Init-5AU}$ to show where most of the \nmagnetic energy is located.\nFor all runs, most of the magnetic energy is\ndeposited in small scale magnetic turbulence. This was found in several\nrecent MRI simulations, latest in local box simulations by \\citet{dav10} and \\citep{fro10}.\nThe peak of the magnetic energy lies just above the dissipation regime.\nFor the $2\\pi$ model FO, the peak is located between \n$m=10$ and $m=20$, whereas for the high-resolution run BO this regime\nis shifted to $m=20$ and $m=30$. \nThe spectra follows closely the $m^{1.0}$ slope until the dissipation regime is reached.\nThe $\\pi\/4$ run does not resolve the scales where we observe this $m$ dependence. \nIn the restricted model PO most of the magnetic energy is again located \nat the scale of the domain size.\\\\\n\\subsubsection{Convergence}\nThe convergence of MRI is an important aspect in ongoing MRI research in\nlocal and global simulations. In local boxes, there was found convergence for\nthe large scale turbulence between 32 and 64 grid cells per scale height\n\\citep{dav10}. Due to the large domain in global simulations, it was up to now not feasible to reach\nsuch resolutions per scale height. Here the first resolution level is needed \nto reach a self sustaining turbulence, at least for\nsimulations with a zero-net flux toroidal field \\citep{fro06}. Comparing the\nresults from stratified local box simulations we can already give\npredictions for global simulations with such high resolutions per scale\nheight. \nIn comparison with the local box simulations by \\citet{dav10} we get a very\nsimilar profile of the magnetic energy with increasing resolution.\nWith higher resolution (FO to BO, Fig. 12, bottom left) the large scale magnetic energy decreases \nwhile the small scale energy increases. A doubled resolution as model BO should also\nshow convergence for the large scales. Doing this, we expect\nonly a weak decrease for the large scale modes, as presented in\n\\citet{dav10}, Fig. 3.\n\n\n\\subsubsection{Plasma beta}\nThe overall strength of the magnetic fields is best analyzed by this plasma beta\nvalue $\\beta = 2P\/B^2$.\nFig. 10 presents a 3D picture of the logarithmic plasma beta for the \n$r-\\theta$ components, taken at 750 inner orbits. \nThe two-phase structure of the disk is again visible.\nThe well established turbulence at the midplane has a broad distribution \nof high plasma beta values (Fig. 11 , top left).\nIn contrast, there are regions in the corona of the disk with plasma beta below unity (Fig. 10, black regions).\nThe azimuthal and time averaged plasma beta at the midplane lies around 400 (Fig. 11, bottom right).\nIn Fig. 11, top left, we plot the correlation of plasma beta over height in a scatter plot of all grid cells.\nWe find the distribution of beta values to be very narrow in the disk corona (1-10) but on the\nother hand to be much broader (10 - $10^4$) around the midplane, but strongly peaked around $\\beta$ = 500. \nThe value of plasma beta in the disk corona depends on several issues. A\nzero-net flux MRI turbulence with toroidal field produces lower magnetic\nfields in the corona. This was already shown in a very similar simulation by \\citet{fro06} (Fig. 8, solid line, model S2). \nIn contrast, a vertical initial field produces a stronger turbulence level\nwith plasma beta values below unity in the corona.\nThe boundary condition also affects the values in the corona.\nA closed boundary condition, e.g. periodic in the vertical direction \nwill accumulate large amount of magnetic flux in the corona and drive to a \nplasma beta value smaller then one (observed in model FC and PC). \nThe small increase of plasma beta above 3 disk scale\nheights is connected to the vertical outflow and the increase of gas\npressure and density in this area (Fig. 4, right). This effect has to be investigated in future work\nwith a much broader vertical extent.\nVery high plasma beta values in the midplane (Fig. 11, top left) indicate reconnection.\nTwo magnetic fields with different sign and comparable strength coming\ntoo close to each other, e.g., in the same grid cell, do reconnect. \nSuch reconnections are visible in single grid cells with nearly no magnetic field.\nFor our BO model, the reconnection zones reach plasma beta values up to $10^{11}$. \nThe heating due to reconnection in those regions is not covered in our\nisothermal model, but shall be a subject for future studies.\n\n\\begin{figure}\n\\begin{minipage}{5cm}\n\\psfig{figure=FIG-7.ps,scale=0.22}\n\\end{minipage}\n\\label{vrm}\n\\caption{3D picture of turbulent RMS velocity at 750 inner orbits for model BO.\nThe white regions in the corona present super sonic turbulence.}\n\\end{figure}\n\\begin{figure}\n\\hspace{-0.6cm}\n\\begin{minipage}{5cm}\n\\psfig{figure=FIG-8A.ps,scale=0.46}\n\\end{minipage}\n\\hspace{4.0cm}\n\\begin{minipage}{5cm}\n\\psfig{figure=FIG-8B.ps,scale=0.46}\n\\end{minipage}\n\\label{vel_spec2}\n\\caption{Left: Velocity spectra in units of the sound speed for all three components\nat the midplane. Space and time averaged is again between 3 and 8 AU\nand between 800 and 1200 inner orbits. The radial velocity peaks at $m=3-5$\nfor both $2\\pi$ models.\nRight: Contour plot of the radial velocity at the midplane ($R-\\phi$ plane).\nLarge shear wave structures become visible. This snapshot is taken after 750\ninner orbits.}\n\\end{figure}\n\\begin{figure}\n\\hspace{-0.6cm}\n\\begin{minipage}{5cm}\n\\psfig{figure=FIG-9A.ps,scale=0.46}\n\\end{minipage}\n\\hspace{4.0cm}\n\\begin{minipage}{5cm}\n\\psfig{figure=FIG-9B.ps,scale=0.46}\n\\end{minipage}\n\\label{pow_spec}\n\\caption{Left: Contour plot of $V_\\phi - V_K$ for an azimuthal slice. The red\ncontour line encloses regions with Super-Keplerian velocity.\nOverplotted are the $r-\\theta$ velocity fields.\nRight: \u00a0Contour plot of $B_\\phi$ for an azimuthal slice.\nOverplotted are the $r-\\theta$ magnetic fields fields.\nBoth snapshots are taken after 750 inner orbits.\n}\n\\end{figure}\n\\begin{figure}\n\\begin{minipage}{5cm}\n\\psfig{figure=FIG-10.ps,scale=0.22}\n\\end{minipage}\n\\label{pbeta}\n\\caption{3D picture of plasma beta after 750 inner orbits for model BO.\nThe black regions in the corona present plasma beta values below unity.}\n\\end{figure}\n\\begin{figure}\n\\hspace{-0.6cm}\n\\begin{minipage}{5cm}\n\\psfig{figure=FIG-11A.ps,scale=0.46}\n\\psfig{figure=FIG-11C.ps,scale=0.46}\n\\end{minipage}\n\\hspace{4.0cm}\n\\begin{minipage}{5cm}\n\\psfig{figure=FIG-11B.ps,scale=0.46}\n\\psfig{figure=FIG-11D.ps,scale=0.46}\n\\end{minipage}\n\\label{mag_turb2}\n\\caption{Top left: Distribution of plasma beta, $N(\\beta)\/N_{Total}$, over height at 750 inner orbits for model BO.\nThe color represents the relative number of grid cells, containing specific\nplasma beta values. At the midplane,\nthere is a wide distribution of plasma beta values between 10 and 10000.\nIn the coronal region the distribution becomes more narrow with values between 1 and 10.\nTop right: Contour plot of azimuthal and time averaged plasma beta of BO\nwith radial (bottom left) and vertical profile (bottom right).}\n\\end{figure}\n\\subsubsection{Spatial distribution}\nAs we already mentioned in section 3.2, the radial profile of the turbulent magnetic field\nhas a direct effect onto the radial profile of the Maxwell stress in the $\\alpha$ parameter.\nThe dominant turbulent azimuthal magnetic field goes as $1\/r$, as shown in a\nazimuthal and time average in Fig. 12, top left.\nThe saturated turbulent field is 4 times lower than the initial azimuthal field.\nAll values are normalized to the initial gas pressure at 5 AU at the midplane\nand the radial profiles are again mass weighted.\nThe vertical profile shows a constant distribution around $\\pm 2$ scale heights \nfrom the midplane until it decreases with height (Fig. 12, top right).\nIn contrast, the radial and $\\theta$ component show a local\nminimum at the midplane with a peak of turbulent magnetic field slightly\nabove 2 scale heights. \\\\\nThe turbulent magnetic fields are around 2 orders of magnitude larger than the\nmean fields. \nThe vertical profiles of mean magnetic fields over height are presented in Fig. 12, bottom right.\nThe radial magnetic field is anti-symmetric to the midplane and correlated\nwith the dominating azimuthal component.\nThe distribution of mean magnetic fields are connected to the \"butterfly\"\noscillations.\n\n\\subsubsection{Butterfly structure}\nThe butterfly pattern is a general property of MRI turbulence and  \nwas found in many local and global simulations, latest by\n\\citet{gre10}, \\citet{fla10}, \\citet{sor10} and \\citet{dzy10} .\nThe \"butterfly\" pattern becomes visible for the mean $B_\\phi$ evolution, \nplotted over disk height and time.\nIn Fig. 13, bottom, we plotted the $B_\\phi$ component of the magnetic field\naveraged over a small radius (4 - 5 AU) and over azimuth for model FO, left,\nand PO, right. \nWe see a clear \"butterfly\" pattern in both models. \nThis pattern is also visible in the total accretion stress with\ndoubled period (Fig. 1, bottom right). In comparison, the $\\pi\/4$ run shows \nno systematic and more violent picture of the butterfly. The amplitudes are stronger and \nit has mixed symmetry (Fig. 13, bottom right). Also the total magnetic flux\nevolution shows these properties for model PO (Fig. 13, top right). \nThe FO run presents a similar amplitude and period as the BO run. \nThe effect of the narrow azimuthal domain on the mean fields will be investigated in a follow-up work.\nThe reason of this butterfly structure and its role for the MRI is still under discussion. \nRecent studies show the connection to the MHD dynamo \\citep{gre10}\nand magnetic buoyancy \\citep{shi10}.\n\n\\begin{figure}\n\\hspace{-0.6cm}\n\\begin{minipage}{5cm}\n\\psfig{figure=FIG-12A.ps,scale=0.46}\n\\psfig{figure=FIG-12C.ps,scale=0.46}\n\\end{minipage}\n\\hspace{4.0cm}\n\\begin{minipage}{5cm}\n\\psfig{figure=FIG-12B.ps,scale=0.46}\n\\psfig{figure=FIG-12D,scale=0.46}\n\\end{minipage}\n\\label{mag_turb}\n\\caption{Top left: Time averaged turbulent magnetic field over radius for\nmodel BO. The turbulent field adjusts to the force-free $r^{-1}$ profile.\nTop right: Time-averaged turbulent magnetic field over scale height for\nmodel BO. The dominating turbulent azimuthal field represents the same flat profile\n$\\pm 1.5$ scale heights around the midplane as the velocity (Fig. 6, top\nright). The turbulent radial and $\\theta$ components represent a different profile with\nmaximum at 2.3 scale heights.\nBottom left: \u00a0Magnetic energy power spectra $B_m^2\\cdot m$ for $\\pi\/4$ model PO (red dotted line), $2\\pi$ model FO\n(blue dashed line) and the high-resolution model BO (black solid line).\nThe profile follows the $m^{1.0}$ slope until the dissipation range.\nBottom right: \u00a0Time-averaged mean magnetic field over height for \u00a0\u00a0\u00a0\u00a0\nmodel BO. The radial and azimuthal field show again anti-correlation.\nThe anti-symmetry for the upper and lower hemisphere could be correlated\nwith a $\\alpha$-$\\Omega$ MHD dynamo.\nAll radial profiles are mass weighted. The time averaged is performed in time\nperiod II (Fig.1, top left, green line).}\n\\end{figure}\n\\begin{figure} \n\\hspace{-0.6cm}\n\\begin{minipage}{5cm}\n\\psfig{figure=FIG-13A.ps,scale=0.55}\n\\psfig{figure=FIG-13C.ps,scale=0.46}\n\\end{minipage}\n\\hspace{4.0cm}\n\\begin{minipage}{5cm}\n\\psfig{figure=FIG-13B.ps,scale=0.46}\n\\psfig{figure=FIG-13D.ps,scale=0.46}\n\\end{minipage}\n\\label{mag_spec2}\n\\caption{\nTop left: Total magnetic flux evolution integrated over the entire computational \ndomain (without the buffer zones) normalized over initial flux $B_\\phi$\nin the $2\\pi$ run FO and $\\pi\/4$ run PO (top right).\nBottom: Contour plot of $B_{\\phi}$ over disk height and time. The value is averaged over\nazimuth and radius (4 AU to 5 AU). Local \norbits are calculated at 4.5 AU. Bottom left: model FO. Bottom right: model\nPO. The butterfly pattern becomes visible. The $\\pi\/4$ model shows irregular and stronger amplitudes.}\n\\end{figure}\n\n\n\n\n\n\\section{Discussion}\nFor a number of aspects, our Godunov method confirms results previously\nobtained with a finite difference method as presented by \\citet{fro06}.\n\\begin{itemize}\n\\item A minimum amount of grid cells per scale height, which is about 25 grid\ncells, is needed to sustain turbulence, which was to be expected as both methods \npresent similar numerical behavior \\citep{flo10}. \nOtherwise the turbulent magnetic energy slowly decays in the nonlinear MRI\nevolution \\citep{fro06}. \nOur highest resolution model BO was able to sustain a constant level of\nturbulent stress for more than 400 inner orbits.\n\n\\item The toroidal magnetic net flux is quickly lost via an \n      open vertical boundary. Then, there is a oscillating zero-net\nflux field present in the disk.\n\n\\item The disks show a two layer structure of turbulence.\n\\end{itemize}\n\n\\subsection*{$\\alpha$-stress evolution}\nWe obtained a steady state $\\alpha$ value of about $5\\cdot10^{-3}$, which\nis comparable to the results obtained in \\citet{fro06}.\nThe time averaged radial profile of $\\alpha$ follows $\\sqrt{r}$.\nThis profile can be explained by the choice of our\nradial pressure (density and temperature) profile, in combination\nwith the resulting magnetic field profile which is\nforce free, $|B'_\\phi| \\propto r^{-1}$.\nFor this magnetic field profile, the net radial magnetic force\nvanishes. \nAny quasi steady state of disk turbulence must display this profile,\notherwise large scale radial readjustments in the density profile would occur.\nBoth $B'_\\phi$ and $B'_r$ determine the Maxwell stress $B'_\\phi B'_r $ to be $\\sim 1\/r^2$.\nFor the chosen $ \\partial \\ln{P} \/ \\partial \\ln {R} = -2.5$ this results in  $\\alpha \\sim B'_\\phi\nB'_r\/P \\sim \\sqrt{r}$.\\\\\nOf course, this profile is only valid for well-ionized accretion disk\nregions.\nThe radial $\\alpha$ profile in protoplanetary disks remains an open\nquestion. \nThe ionization rate, and possibly the MRI\nactivity, will be a function of radius and height \\citep{sem04,dzy10}. Furthermore, the pressure scale height \nwill vary with radius, and this also changes the MRI evolution.\nBoth effects will lead to different saturation levels for MRI turbulence\nat different radii and thus also to different $\\alpha$ values. \n\\subsection*{Magnetic energy convergence}\n\n\\citet{dav10}, \\citet{shi10} and \\citet{fla10} show in local box simulations \nthat the large scale magnetic energy converges for a resolution\nbetween 32 and 64 grid cells per pressure scale height (\\citet{dav10}, Fig. 3). \nOtherwise, the large scale magnetic energy decreases with increasing\nresolution.\nAt the current state we could only handle 25 cells per scale height in our global\nsimulations. We expect also a large scale \nconvergence of magnetic energy with 1.5(2.5) higher resolution.\nFuture calculations shall complete this point, but will be five times more computationally\nexpensive.\\\\\nWe can already conclude that our global magnetic energy spectra as well as the effect of increasing \nresolution (Fig, 12, bottom left, FO to BO) look very similar\nto the results presented in local box simulations. \nThe magnetic energy power spectra reveals that most of\nthe magnetic energy is deposited in the small turbulent scales.\nFor the model with a restricted azimuthal domain of $\\pi\/4$,\nthe largest energy scale is always the domain size.\n\\subsection*{Turbulent velocity}\nRecently observed turbulent velocities in TW Hya and HD 163296\n\\citep{hug10} fit nicely to our computed Mach numbers of 0.1 and 0.4 in the midplane and corona of the\ndisks.\\\\\nThe kinetic energy spectra as well as the velocity spectra along\nazimuth ($k_\\phi$ space) show a peak between $\\rm m=5$ and $6$ due to radial shear\nwaves.\nLatest results in local box MRI simulations presented\na $k^{-3\/2}$ slope for the kinetic energy power spectra \\citep{fro10}.\nSimilar as in \\citet{fro10}, our power-law fit \napplies only for a small range in k space.\nWe do not find a Kolmogorov type slope of $k^{-5\/3}$ (Fig. 6).\nFurther studies are needed, including the $k_r$ and $k_{\\theta}$.\\\\\nA Kolmogorov scaling was predicted for magnetic ISM turbulence by\n\\citet{gol95} and recently confirmed in numerical simulations by \\citet{ber10}.\nHowever, it only applies for the inertial range of incompressible\nisotropic turbulence. The driving of the turbulence via MRI,\nthe anisotropy of the turbulent eddies, the geometry and rotation of the disk \nand the compressibility of the gas\nmake it difficult to argue for a Kolmogorov scaling. \nWe expected a spectrum to be more or less deviating from this simple law.\n\n\\subsection*{Two-phase disk structure}\nWe observe that the accretion disks establish a two-phase structure:\\\\\n- The midplane region between $\\pm$ two scale heights shows a pretty\nconstant turbulent RMS velocity of about $10\\%$ of the local sound speed\n independent of radius or height.\nPart of the RMS velocity occurs due to global shear waves which have radial\npeak velocities of up to $30\\%$ of the local sound speed. \nThe amplitude of the azimuthal fluctuations in the magnetic\nfield is also independent of height ($\\pm$ two scale heights around the midplane) but develops a \n$1\/r$ profile in radius.\nThe midplane region shows a broad distribution of plasma beta values, \n$\\beta = \\frac{2P}{B^2}$, with a mean value of about 500 and a full width at\nhalf maximum of two order of magnitude.\\\\\n- In the coronal region, more than two scale heights above the midplane,\n the mean turbulent velocity reaches a Mach number of 0.5 with supersonic peaks up to 1.5. \nThe mean magnetic fields decrease in this region with height.\nThe disk corona shows a narrower distribution of the plasma beta values with most values between 1\nand 10. Here the magnetic fields are buoyant, gas and fields are\nexpelled from the disc.\nRelative high plasma beta ($\\beta > 1$) in the corona have been\nreported in \\citet{fro06} for global models of AMRI with open boundary. The\nmagnetic flux escapes through the vertical boundary with a remaining zero-net flux\nin the computational domain. This leads to the weakly magnetized corona (below equipartition).\\\\\n\n\n\\subsection*{Vertical outflow}\nOur models show a MRI driven vertical outflow.\nAbove 2 scale heights, the gas flow is directed vertically and radially \noutward, Fig. 3. The outflow velocity of the gas (measured at the vertical\nboundary) is still subsonic.\nThe disk evaporation time was determined at 5 AU to 2000 local orbits.\nThe launching region is located between 1.6 and 2 scale heights. \nThis results matches values obtained in local box simulations \\citep{suz09,suz10} with a\nvertical net-flux field.\\\\\nHowever, we are aware that a detailed study of the vertical outflows\nrequires \nmuch broader vertical extended\nsimulations to confirm that the gas is evacuated from the disk and not\nreturning.\nThese simulations should then include the sonic point or even the Alfv\\'enic point to give\nfurther insight into disk-wind and disk-jet interacting regions.\n\n\\subsection*{Meridional flows}\nOur present work shows that the meridional\noutflow at the midplane is only present in HD simulations, e.g., in\nviscous simulations with an $\\alpha$ value assumed to be constant in time and space.\nFor our MHD models, we find time variations of the orbital frequency of\naround 50 local orbits, which are not present in the viscous disk models\nand which prevent a steady radial outflow.\nA similar result, the absence of a meridional flow in global MHD simulations \nwas recently found by \\citet{fro11}.\\\\\nHowever, we confirm the more general picture of a viscous disk and \nshow that viscous disk models with a radial viscosity\nprofile can reproduce successfully the radial mass flow rate in global MRI turbulent stratified disks.\nClearly, the vertical mass flow cannot be described with such an HD\nmodel.\\\\\n\n\\subsection*{Mean field evolution}\nThe azimuthal MRI is self-sustaining in our zero net flux simulations with\nopen boundaries.\nThe fact that the total flux oscillates around zero could be due\nto the generation of a mean poloidal magnetic field by a turbulent toroidal\nfield. \nWe observe also an antisymmetric distribution of the mean magnetic fields in the upper and\nlower hemisphere which could be an indication for the action of an MHD dynamo in our global\nsimulations.\\\\\nThe existence of an $\\alpha$-$\\Omega$ MHD dynamo and its role for accretion\ndisks was investigated by \\citet{bra95}, \\citet{zie00}, \\citet{arl01},\n\\citet{bra07}, \\citet{les08} and \\citet{bla10}.\nThe temporal oscillations of the mean azimuthal field, plotted \nover height and time (Fig. 13), generates a butterfly pattern. \nLatest results connect the butterfly pattern with \na dynamo mechanisms \\citep{gre10}. \nWe present also a butterfly pattern with a period of 10 local orbits,\nindependent of the azimuthal extent.\nAdditionally, the butterfly structure is reflected in the temporal\nspatial fluctuations of the mean turbulent stresses with double period. \nA change of sign of the mean azimuthal field occurs every five local orbits, at the same time the \n$\\alpha$-stresses show a minimum (Fig. 1, bottom right).\\\\\nThe magnetic energy as well as the mean field evolution have shown\nthat the $\\pi\/4$ model does not capture the correct properties \nof the larger scale simulations. \n\\citet{haw00} also studied full $2\\pi$ and restricted $\\pi\/2$ models of\naccretion tori. However, a detailed study of the impact of different \nazimuthal domain extents is still needed and will be covered in future work.\n\n\\section{Summary}\nWe have performed full $2\\pi$ 3D stratified global MHD simulations of \naccretion disks with the Godunov code PLUTO.\nOur chosen disk parameter represent well-ionized proto-planetary disk\nregions. We obtain a quasi steady state zero-net flux MRI turbulence \nafter around 250 inner orbits.\n\n\\begin{itemize}\n\n\n\\item The second order Godunov scheme PLUTO including the HLLD Riemann\nsolver presents a similar nonlinear MRI evolution as finite difference\nschemes. There is also a need of about 25 grid cells per pressure\nscale height to reach a self-sustaining MRI turbulence in global zero net flux\nazimuthal MRI simulations.\n\n\\item We observe a total $\\alpha$ parameter of about $5\\cdot10^{-3}$,\nwhich remains constant for at least 400 inner orbits and scales with $\\sqrt{r}$ for our used pressure and\ndensity profile.\n\n\\item The turbulent magnetic fields show a $1\/r$ profile in radius, mainly\nvisible in the dominating toroidal magnetic field. This configuration is force free\nin the sense that there exist no large scale net force on the gas. This\nprofile determines the slope of the $\\alpha$ parameter.\n\n\\item The magnetic energy spectra is similar as in local box simulations.\nMost of the magnetic energy is placed in the smallest resolved turbulent\nscale. \n\n\\item The kinetic energy spectra as well as the velocity spectra peak for an\nazimuthal wavenumber between $\\rm m=3$ and $5$ due to shear waves, driving the\nradial velocity up to a Mach number of 0.3. We do not find a Kolmogorov type\nscaling in the $k_\\phi$ space.\n\n\\item The model with an azimuthal extent of only $\\pi\/4$ has most of the energy at the domain\nsize and does not show the same mean field evolution. \n\n\\item We observe a butterfly pattern with then local orbits independent of\nthe azimuthal extent. The butterfly period becomes also visible in the\nMaxwell stress with double period. The mean magnetic fields are\nantisymmetric for the two hemispheres.\n\n\\item At the midplane ($\\pm 2 $ disk scale heights), \nour turbulent RMS velocity presents a constant Mach number of 0.1\nindependent on radius. At the corona ($> 2$ disk scale heights), the\nturbulent velocity increases up to a Mach number of 0.5 at 4 scale heights.\n\n\\item The turbulent magnetic fields at the midplane present a broad plasma\nbeta distribution with a mean of about 500 $\\pm$ one order of magnitude.\nIn the corona the plasma beta is between unity and ten.\n\n\\item The turbulent and the mean velocities are pointing vertically and\nradially outward in the disk corona ($> 2$ disk scale heights). We observe a\nsteady vertical outflow for the open boundary models, dominating the radial\naccretion flow.\n\n\\item We do not see a meridional flow pointing radially outward at the\nmidplane in our MHD models. However, we reproduce our total radial mass flow\nin 2D viscous disk simulations with a radial dependent $\\alpha$-viscosity.\n\n\n\n\n \n\n\n\n\\end{itemize}\n\n\\section{Outlook}\n\nThis paper presents a huge data set of about 10 TBytes.\nThis means we will continue to analyze the data for different goals. \nOne study will deal with a closer investigation of dynamo properties in\nour global disk models. Another one will analyze the turbulent spectra in a\nbetter way to derive correlation times for the turbulence.\nWe will also fill the parameter space with $\\pi$ and $\\pi\/2$ models to\nidentify whether a subsection will be sufficient. Higher resolution is\nenvisioned to reach resolution per scale height comparable to recent stratified local\nbox simulations. Finally, our global MHD model will be the work horse for our\nfuture investigations of planet formation processes in circumstellar disks,\nlike collisions of boulders, planetesimal formation and planet migration.\\\\\nIn future runs, we also plan to use non-ideal MHD to include more realistic\nmagnetic Prandtl numbers and magnetic Reynolds numbers to understand the occurrence and\nsaturation level of the turbulence. Improving the thermodynamics is also\na must in future work, dealing with the proper ionization of the disk, like\ncapturing $p\\Delta V$ terms or magnetic dissipation as heat input.\n\n\\acknowledgments\nWe thank Andrea Mignone for very useful discussions about the numerical setup.\nWe thank Sebastien Fromang for the helpful comments on the global models and \non the manuscript.\nWe thank Alexei Kritsuk for the discussion about turbulent spectra.\nWe also thank Willy Kley for the comments on the viscous model.\nWe thank Frederic A. Rasio and an anonymous referee for the fast and very professional processing of\nthis work.\nH. Klahr, N. Dzyurkevich and M. Flock have been supported in part by the\nDeutsche Forschungsgemeinschaft DFG through grant DFG Forschergruppe 759\n\"The Formation of Planets. Neal Turner was supported by a\nNASA Solar Systems Origins grant through the Jet Propulsion\nLaboratory, California Institute of Technology, and by an Alexander\nvon Humboldt Foundation Fellowship for Experienced Researchers.\nThe Critical First Growth Phase\". Parallel\ncomputations have been performed on the PIA cluster of the MaxPlanck\nInstitute for Astronomy Heidelberg as well as the GENIUS Blue Gene\/P cluster\nboth located at the computing center of the MaxPlanck Society in Garching.\n\\bibliographystyle{apj}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{s_intro}\n\n\n\nGlobular clusters (GCs) host multiple populations of stars characterized by different spectroscopic and photometric signatures. Anti-correlations between the abundances of several elements such as C-N, Na-O, and sometimes Mg-Al have been reported at the surface of various types of stars, from the main sequence (MS) to the red giant branch (RGB) and the asymptotic giant branch (AGB) \\citep[e.g.,][]{Cohen1978,Peterson1980,sneden92,Kraft1994,Thevenin2001,gratton07,Gratton2019,Meszaros2015,carretta15,Carretta2019,Johnson2016,Pancino2017,Wang2017,Masseron2019}. These differences in chemical composition are thought to explain the various sequences observed in color-magnitude diagrams (CMDs) built with specific filters sensitive to atomic and molecular lines of the elements listed above \\citep[e.g.,][]{bedin04,piotto07,Bowman2017,nardiello18,marino19}. The discovery of these peculiarities (variations in chemical composition and discrete sequences in CMDs) turned GCs from textbook examples of clusters born in a single star forming event into complex structures the origin of which remains elusive.\n\n\nThe standard explanation\\footnote{Alternative possibilities exist but all rely at least partly on nucleosynthesis products of AGB or massive stars, see \\citet{marcolini09,elmegreen17}.} to these puzzling properties is the pollution of the original proto-cluster gas by a population of rapidly evolving stars more massive than the present-day GC members. Indeed, the abundance patterns observed are typical outcomes of nucleosynthesis at about 75 MK. More precisely, hydrogen burning through the CNO cycle, together with the Ne-Na and the Mg-Al chains, fully explains the anti-correlations \\citep{1988ATsir1525...11K,Denisenkov1990,prantzos07,Prantzos2017}. Potential polluters experiencing such nucleosynthesis phases are $\\sim$5-7.5~M$_{\\sun}$\\ AGB stars \\citep[e.g.,][]{ventura01,Ventura2009}, massive stars either rotating fast \\citep{decressin07,DecressinCM2007}, in binary systems \\citep{demink09,Izzard2013}, or in a red supergiant phase \\citep{szecsi19}, and super-massive stars \\citep{denis14}. Whatever the nature of the polluter, it should be objects present at early times of the GC history that have spread their nucleosynthesis products in the surrounding gas and out of which the stellar populations we see today were formed. Depending on the degree of mixing of the nucleosynthesis products with pristine gas, a range of chemical composition is predicted for the newly formed stars. The polluters have long disappeared because of their relatively high mass, and only low-mass stars either from the initial population formed out of original proto-GC gas or from the second population formed out of polluted material are still present. We usually refer to these two populations of stars as 1P (or 1G) and 2P (or 2G), standing for first and second population (generation), respectively.\n\nOne of the key differences in the predictions of the scenarios that invoke the different polluters listed above is the amount of helium that is inevitably ejected together with other nucleosynthesis products. Indeed, as hot hydrogen-burning is needed to explain the observed abundance patterns, helium is naturally produced and released too. On one hand, the fast-rotating massive star (FRMS) models of \\citet{decressin07}, developed  specifically to reproduce the observations of the GC NGC~6752, predict a wide range of Y for the ejecta, up to Y=0.8, and the test case\\footnote{The study focuses on one binary system made of a 20~M$_{\\sun}$\\ primary and a 15~M$_{\\sun}$\\ secondary star orbiting each other on a 12-day orbit, and with individual rotation periods synchronized on the orbital period.} of binary evolution presented by \\citet{demink09} also predicts a wide range of Y, up to 0.63.\nOn the other hand, models of \\citet{doherty14} indicate that nucleosynthesis in massive AGB stars should produce material with helium mass fraction of about 0.35--0.40, as a result of second dredge-up.  On the other hand, supermassive stars\ncan potentially release material rich in hot hydrogen-burning products but only mildly enriched in helium \\citep{denis14}, with a helium mass fraction close to that of the proto-cluster gas in the case where they form through the so-called runaway collision scenario \\citep{Gieles2018}.\n\nMeasuring the helium surface abundance from spectroscopy is not possible in most GC stars. Only hot horizontal branch (HB) stars display weak \\ion{He}{i} lines, and these are difficult to model and interpret. However, the helium surface abundance in these stars, with effective temperatures higher than $\\sim$11 000~K, is affected by atomic diffusion  \\citep[e.g.,][]{Michaud08,Quievy09}. Thus, the measured helium no longer represents the original chemical composition of the stars. The vast majority of GC stars are too cool to display any helium line in their spectra. Nevertheless, helium enrichment is detected in a few hot HB stars with $\\Delta$Y ---the difference between the highest Y of 2P stars and Y of 1P stars--- generally not larger than $\\sim$0.1 \\citep{villanova12,marino14}, although \\citet{pasquini11} and \\citet{dupree13} report a Y difference of up to 0.17 between two HB stars, of NGC~2808 and $\\omega$~Cen, respectively.\n\nA change in helium mass fraction affects the internal structure as well as the color and the brightness of stars because the opacity is modified. As a consequence, stars with the same mass, age, and metallicity but different Y have different effective temperatures ($T_{\\rm{eff}}$; the higher Y, the higher $T_{\\rm{eff}}$; e.g., \\citealt{IbenFaulkner1968,chantereau15,CassisiSalaris2020}). Their spectral energy distributions (SEDs) therefore peak at different wavelengths: stars with high Y have more flux at shorter wavelengths, and thus bluer colors. Consequently they are located to the left of stars with smaller Y in CMDs (e.g., \\citealt{sbordone11} and Sect.~\\ref{s_sed}). Variations in the helium content naturally lead to a widening of classical branches (MS, RGB, HB, AGB) in CMDs \\citep[e.g.,][]{Rood1973,Norris1981,DAntonaCaloi2004,DAntona2005ApJ,chantereau16}. In addition, the associated variations in (among others) C, N, and O abundances impact specific colors built with photometric filters encompassing lines sensitive to these elements. This further increases the separation between stars in CMDs, leading to the observed multiple and discrete sequences \\citep[e.g.,][]{Lardo2012,marino19}. \n\nComparison of observed and theoretical colors in filters mostly sensitive to $T_{\\rm{eff}}$, and thus Y, has been used to   indirectly constrain the amount of helium present at the surface of GC stars \\citep[e.g.,][]{piotto07,milone15}. \\citet{king12} report a maximum helium mass fraction of about 0.39$\\pm$0.02 in $\\omega$~Cen based on the analysis of MS stars. Using the Hubble Space Telescope (HST) ultra-violet (UV) survey of GCs \\citep[HUGS][]{piotto15,nardiello18}, \\citet{milone18} determined Y in 57 GCs. $\\Delta$Y ranges from nearly 0 to 0.124, corresponding to a maximum Y of about 0.38. \\citet{milone18} also show that $\\Delta$Y correlates with the cluster mass, with higher Y being determined in more massive clusters.\n\nAnother indirect constraint on the helium content of GCs comes from the morphology of their HB. Its different shape in clusters with similar general properties, such as M3 and M13 \\citep[e.g.,][]{ferraro97}, is difficult to explain. A spread in Y of HB stars is a viable possibility \\citep{rood73} and quantitative determinations give $\\Delta$Y between 0.02 and 0.15 on the HB, depending on the cluster \\citep{Caloi2005,lee05,dicri10,valcarce16,tailo16,denis17,vandenberg18,chantereau19}. \n\nThe presence of hot HB stars in GCs of early-type galaxies is also thought to be responsible for the existence of a so-called UV upturn \\citep{gr90}. This feature refers to the increase of flux below $\\sim$2500 \\AA\\ in galaxies that no longer form stars and that are made of low-mass MS and post-MS stars. These objects have SEDs that rapidly drop short of about 3500~\\AA. Hot HB stars, such as those seen in Galactic GCs, may explain the UV fluxes. \\citet{ali18} argued that if such hot HB stars are present because of a high helium content (as suggested by \\citealt{Meynet2008} for elliptical galaxies), there should be a redshift above which they would no longer be observed, having not yet sufficiently evolved. The position of this transition redshift can be used to constrain Y in hot HB stars. \\citet{ali18} showed that a value of $\\sim$0.45 would be compatible with the observed disappearance of the UV upturn with redshift.\n\nThus, modern estimates of Y in GCs indicate that the very helium-rich stars predicted in particular by the FRMS scenario are not detected. However, a key specificity of the FRMS model highlighted by \\citet{chantereau15} is that such He-rich objects evolve faster and differently compared to more classical stars. In particular, \\citet{chantereau16} showed that in a GC that would have been formed under the FRMS scenario, the distribution of stars quickly falls as Y increases (at ages typical of GCs, i.e., 9 to 13.5 Gyr). In practice, there should be little to no He-rich stars on the RGB and AGB of the cluster NGC~6752, for which the models of Chantereau et al. were tailored. This also naturally explains the lack of Na-rich AGB stars in some GCs \\citep{Campbell2013,Wang2016,Wang2017}, although observations reveal that the presence of 2P AGB stars can be affected by more than one  factor \\citep{Wang2017}.\n\nThe question therefore naturally arises as to whether the absence of very He-rich stars (i.e., Y$>$0.4) is a robust observational fact, or these stars simply escaped detection due to their exceptionally small number. In view of the discriminating nature of the helium content for various scenarios, it is important to investigate this potential issue, which we aim to do in the present study. In practice, we build on the work of \\citet{chantereau16} to compute synthetic clusters with the predicted distribution of multiple populations for NGC~6752. We compute synthetic spectra consistently with isochrones, produce synthetic photometry and CMDs, and perform determinations of the maximum Y values of these synthetic clusters in order to see if we recover or miss the highly enriched populations predicted by the model of Chantereau et al.\n\nOur paper is organized as follows: Sect.~\\ref{s_specphotom} presents the computation of synthetic spectra and photometry. We then discuss the behavior of our models with special emphasis on the effects of surface abundances on colors. We subsequently compare our predicted photometry to observations in various CMDs before moving to the determination of the maximum helium content in NGC~6752 and synthetic clusters in Sect.~\\ref{s_maxHe}. We discuss our results in Sect.~\\ref{s_disc} and give our conclusions in Sect.~\\ref{s_conc}.\n\n\\section{Spectral synthesis and synthetic photometry}\n\\label{s_specphotom}\n\nIn this section we first present our computations of synthetic spectra and the associated photometry (Sect.~\\ref{s_meth}). We then describe the effects of stellar parameters and surface abundances on both the SEDs and synthetic colors (Sect.~\\ref{s_sed}). We build synthetic CMDs and compare them with observations (Sect.~\\ref{s_compobs}).\n\n\n\\subsection{Method}\n\\label{s_meth}\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=9cm]{hrd_theo.eps}\n\\caption{Hertzsprung-Russell diagram with the isochrones of \\citet{chantereau15}. Different symbols and colors stand for different initial composition, especially different helium mass fraction. Symbols correspond to the points for which atmosphere models have been calculated.}\n\\label{hr_theo}\n\\end{figure}\n\nWe proceeded as in \\citet{martins20} to compute synthetic photometry. We relied on the evolutionary tracks and isochrones computed by \\citet{chantereau15} with the code STAREVOL.\nThese models are based on the FRMS scenario \\citep{decressin07} for the formation of multiple stellar populations in GCs. Each isochrone is characterized by a set of abundances that directly comes from the FRMS predictions. We refer to \\citet{chantereau15} for details. Each set of abundances is tailored to reproducing 1P and various 2P stars in NGC~6752, in which the abundances vary according to different degrees of pollution by FRMS. Figure~\\ref{hr_theo} shows the Hertzsprung-Russell diagram (HRD) with the isochrones we consider in the present work. Each isochrone is labeled according to its helium mass fraction (Y), but we stress that abundances of other elements are also varied (in particular, nitrogen is increased and carbon and oxygen are depleted when Y increases; see Appendix~\\ref{ap_chem}). We stress that the isochrones have been recomputed for a metallicity of $[Fe\/H]=-1.53$, which is slightly higher than the value of -1.75 used in \\citet{chantereau15}. This change was made to better match the metallicity of NGC~6752. \n\nAlong each isochrone we computed atmosphere models and synthetic spectra at ten points (see Fig.~\\ref{hr_theo}) using the codes ATLAS12 \\citep{kur14} and SYNTHE \\citep{kur05}. Our computations include the so-called predicted lines, that is, lines for which at least one of the energy levels comes from quantum mechanics calculation and not from laboratory measurements. These lines thus have approximate wavelengths, but their opacities have been shown to be of prime importance to reproducing observed SEDs \\citep{coelho14}. We adopted a microturbulent velocity of 1~km~s$^{-1}$\\ in all our calculations. \n\nTo convert the HRD into CMDs we computed synthetic photometry in the Vegamag system: \n\n\\begin{equation}\n  mX = -2.5 log (F_X\/F^{Vega}_X) = -2.5 log (F_X) + ZP^{Vega}_X\n\\label{eq_mX}\n,\\end{equation}\n\n\\noindent where $mX$ is the magnitude in the X filter and ZP the zero point. The average flux $F_X$ over the passband X was calculated according to \\citet{bohlin14}:\n\n\\begin{equation}\nF_X = \\frac{\\int \\lambda F_{\\lambda} R_X d\\lambda}{\\int \\lambda R_X d\\lambda}\n,\\end{equation}\n\n\\noindent where $R_X$ is the transmission curve of filter X\\footnote{Transmission curves were retrieved from the Spanish VO \\url{http:\/\/svo2.cab.inta-csic.es\/svo\/theory\/fps3\/}.}.\nThe zero point $ZP^{Vega}_X$ in Eq.~\\ref{eq_mX} was calculated using the Vega STScI reference spectrum\\footnote{File alf\\_lyr\\_stis\\_010.fits from \\url{ftp:\/\/ftp.stsci.edu\/cdbs\/current_calspec\/}.} and the appropriate transmission curve.\n\n\n\nFinally, for comparison with observations, we assumed a distance modulus of 13.18 for NGC~6752 based on the \\emph{Gaia} DR2 determination \\citep{helmi18}. This value is consistent with those found by \\citet{harris96} (13.19), \\citet{renzini96} (13.05) , and \\citet{gratton03} (13.24).\nPrior to synthetic photometry calculations, we reddened our SEDs using a color excess $E(B-V) = 0.060$ and A$_V$=3.2, adopting the extinction law of \\citet{ccm89}.\nThis choice best reproduces the turn-off (TO) and sub-giant region in the  m814W (m606W-m814W) CMD (see for instance Fig.~\\ref{cmd}). This is slightly larger than the value of 0.046$\\pm$0.005 reported by \\citet{gratton05} but close to the value of \\citet{shleg98}:  0.056.\n\n\n\n\\subsection{Spectral energy distributions}\n\\label{s_sed}\n\nIn this section we describe the effects of chemical composition on the SED. The goal is to identify spectral regions that are affected by certain species. We refer to \\citet{sbordone11} or \\citet{milone20}, among other recent studies, for similar descriptions.\n\nFigure~\\ref{comp_sed_L0p85} shows a selection of models at the bottom of the RGB, with luminosities equal to $10^{0.85}$~L$_{\\odot}$ (see Fig.~\\ref{hr_theo}). These spectra have different $T_{\\rm{eff}}$, surface gravities and chemical compositions (see Table~\\ref{tab_chem}). Because of the higher $T_{\\rm{eff}}$\\ in higher Y models, the SED peak is shifted towards shorter wavelengths. In the optical region, the slope of the SED becomes steeper (faster decline with wavelength) which translates into bluer colors. Variations in individual abundances of light elements (CNO) also changes the strength of absorption lines, mostly below 4500~\\AA. \n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=9cm]{comp_sed_L0p85.eps}\n\\caption{Spectral energy distributions of some of the models with \\ifmmode \\log \\frac{L}{L_{\\sun}} \\else $\\log \\frac{L}{L_{\\sun}}$\\fi~=~0.85 but different chemical compositions (and thus different $T_{\\rm{eff}}$\\ and $\\log g$). Different colors correspond to different models, identified by their helium mass fraction.}\n\\label{comp_sed_L0p85}\n\\end{figure}\n\nTo better separate the effects of light elements and helium on the SED, we show in Fig.~\\ref{comp_CNO} examples of spectra in which the abundances of C, N, and O have been varied by a factor of three, all other parameters being kept constant. A reduction of the carbon abundance translates into a weaker CH absorption between 4200 and 4400 \\AA. Conversely, an increase in the nitrogen content leads to a stronger NH absorption at 3300-3500 \\AA. The CN band around 3900 \\AA\\ is less affected. The OH absorption between 2800 and 3300 \\AA\\ reacts to a change in the oxygen content. Consequently, photometry in the HST filters F275W and F336W is affected by C, N, and O abundances \\citep{sbordone11,milone18,CassisiSalaris2020}. These filters have been used to build the so-called super color C$_{410}$=(m275W-m336W)-(m336W-m410M) \\citep{milone13}. This photometric diagnostic has been shown to be a powerful tool for separating multiple populations (see also following section). Filters F395N, F467M, F606W, F814W and to some extent F410M are relatively insensitive to variations in C, N, and O abundances. An important result of Fig.~\\ref{comp_CNO} is that C, N, and O abundances do not affect the global shape of the SED, but only specific wavelength regions that include molecular bands \\citep{milone13,Dotter2015}. \n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=9cm]{comp_sed_effectCNO.eps}\n\\caption{Spectral energy distribution of two models with similar parameters except the C, N, and O abundances: the red line shows a model with C\/H reduced by a factor three (top panel), N\/H increased by a factor three (middle), and O\/H reduced by a factor three (bottom) compared to the initial model in black. The other parameters are: $T_{\\rm{eff}}$\\ = 5375 K, $\\log g$\\ = 3.37, Y=0.248. The gray solid lines indicate the transmission curves of the HST WFC3\/UVIS F275W, F336W, F395N, F410M, F467M, and ACS\/WFC F606W and F814W.}\n\\label{comp_CNO}\n\\end{figure}\n\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=9cm]{comp_sed_L0p85_effectcompo.eps}\n\\caption{Spectral energy distribution of three models with \\ifmmode \\log \\frac{L}{L_{\\sun}} \\else $\\log \\frac{L}{L_{\\sun}}$\\fi=0.85: one with Y=0.248, $T_{\\rm{eff}}$\\ = 5375 and $\\log g$\\ = 3.37 (black line); one with Y=0.400 and the same $T_{\\rm{eff}}$ \\ and $\\log g$\\ (green line); and the model with Y=0.400 and the associated $T_{\\rm{eff}}$\\ and $\\log g$\\ (5549 K and 3.31 respectively, red line).}\n\\label{comp_sed_L0p85_compo}\n\\end{figure}\n\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=9cm]{comp_Tstru_L0p85_effectcompo.eps}\n\\caption{Temperature as a function of Rosseland optical depth in the three models leading to the SEDs shown in Fig. \\ \\ref{comp_sed_L0p85}.}\n\\label{comp_Tstru_L0p85}\n\\end{figure}\n\n\n\nTo further quantify the effect of chemical composition (and in particular of Y) on the SED, in Fig.\\ \\ref{comp_sed_L0p85_compo} we compare  the SEDs of three models with the same luminosity but different chemical composition. We focus on the models with L=$10^{0.85}$~L$_{\\odot}$. We have chosen the models of the Y=0.248 and Y=0.400 sequence, to which we add a third model with the effective temperature and gravity of the Y=0.248 sequence, but the chemical composition of the Y=0.400 sequence. Hence, we are able to disentangle the effect of chemical composition on the effective temperature and logg on one side, and on the SED on the other side. Comparing the two models with $T_{\\rm{eff}}$\\ = 5375 K, we find the same trends as the ones drawn from Fig.\\ \\ref{comp_CNO}. It is mainly the CH, CN, NH and OH bands that are affected. The general shape of the SED is otherwise relatively similar in the two models. Hence, models with different helium content but similar $T_{\\rm{eff}}$\\ and surface gravity have very similar SEDs, except in regions containing lines involving C, N, and O. If we now compare the two models with the same composition but different effective temperature and surface gravity, we see a major modification of the SED. The hottest model has more (less) flux blueward (redward) of 6000 \\AA. This is mainly governed by the change in $T_{\\rm{eff}}$\\ which itself is due to the different helium content. Figure\\ \\ref{comp_Tstru_L0p85} shows the temperature structure in the three models. Changing only the chemical composition leads to a higher temperature in the inner atmosphere, but a similar structure in the outer parts. This effect is entirely dominated by the difference in the helium content: computing a model with $T_{\\rm{eff}}$\\ = 5375 K, $\\log g$\\ = 3.37, and the helium content of the Y=0.400 sequence but the abundances of all elements heavier than He from the Y=0.248 sequence leads to a temperature structure which is almost indistinguishable from that of the green model in Fig.\\ \\ref{comp_Tstru_L0p85}.  \nIf we now change both the chemical composition and the effective temperature\/surface gravity (red model), we obtain a global increase in the temperature at all depths in the atmosphere, which translates into a change of the SED. \n\nFrom these comparisons we conclude that the global shape of the SED is mainly sensitive to the effective temperature which itself depends critically on the helium content and its effect on opacities. The abundances of heavier elements affect certain portions of the SED but not its global shape. Variations in Y thus affect colors sensitive to the global shape of the SED. The filters F395N, F410M, F606W, and F814W should be relatively free of contamination by absorption lines of light elements, and can therefore be used to study the helium content \\citep{milone13,milone18,Dotter2015,CassisiSalaris2020}.\n\n\\subsection{Comparison with observed CMDs}\n\\label{s_compobs}\n\n\n\nIn this section we compare our synthetic photometry to the observed CMDs of NGC~6752. We use the data of \\citet{milone13} and of the HUGS survey \\citep{piotto15,nardiello18}. The goal is to see if synthetic colors reproduce observations and to identify potential failures.\n\nThe bottom-right panel of Fig.~\\ref{cmd} shows the m814W versus (m606W-m814W) CMD. Taking the Y=0.248 isochrone as a reference, we see that on average our synthetic photometry is able to reproduce the shape of the observed sub-giant branch and the bottom of the RGB. More specifically, the synthetic isochrone is located near the red part of the envelopes that define these branches, which are broadened because of intrinsic dispersion and the presence of multiple populations. The synthetic isochrone reproduces the TO relatively well.\nOn the upper RGB, the synthetic isochrone appears slightly too red compared to the observations, which could be due to the fact that the stellar evolution models were computed with an atmosphere treated in the gray, plane-parallel, and Eddington approximations. \n\n\n\n\\begin{figure*}[]\n  \\centering\n\\includegraphics[width=0.4\\textwidth]{cmd_275W_814W.eps}\n\\includegraphics[width=0.4\\textwidth]{cmd_336W_814W.eps}\n\n\\includegraphics[width=0.4\\textwidth]{cmd_410M_814W.eps}\n\\includegraphics[width=0.4\\textwidth]{cmd_C410_814W.eps}\n\n\\includegraphics[width=0.4\\textwidth]{cmd_467M_814W.eps}\n\\includegraphics[width=0.4\\textwidth]{cmd_606W_814W.eps} \n\\caption{Color-magnitude diagrams: In all panels, the ordinate axis is the magnitude in the ACS F814W filter. The abscissa axis is the difference between magnitude in the X filter and the magnitude in the ACS F814W filter, where X is the WFC3 F275W filter (top left panel), WFC3 F336W filter (top right panel), WFC3 F410M filter (middle left panel), WFC3 F467M filter (bottom left panel), and ACS 606W (bottom right panel). In the middle right panel, the abscissa axis is the color difference (m275W-m336W)-(m336W-m410M). In all panels, different symbols and colors correspond to models with different chemical composition, tagged by their helium content (Y). Gray points correspond to the photometric data of \\citet{milone13}, except in the bottom right panel where they are from the HUGS survey \\citep{piotto15,nardiello18}.}\n\\label{cmd}\n\\end{figure*}\n\nFigure~\\ref{cmd} shows several CMDs involving filters F275W, F336W, F410M, F467M and F814W. These filters are sensitive to the chemical composition as described in Sect.~\\ref{s_sed}. \nFocusing on the Y=0.248 isochrone we find, in general, relatively good qualitative agreement with observations. We note that in the m814W versus (m410M-m814W) diagram, the synthetic isochrone is located slightly too much to the red on the RGB (at least when compared to the m814W-(m275W-m814W) and m814W-(m336W-m814W) CMDs). In the (m336W-m814W) color the Y=0.248 isochrone is located roughly in the middle of the observed branch, for reasons that are explained in the following paragraph. The observed RGB is redder than the synthetic isochrone in the m814W versus C$_{410}$ diagram. This is likely the result of the small offsets seen in other CMDs that are amplified by the super color. All these (relatively) small offsets between observations and synthetic isochrones are due to different photometric calibrations between observations and synthetic colors and to limitations in the stellar evolution and spectral modeling. This likely has a small impact on the determination of the helium content of multiple populations, because only relative color differences are used and not absolute ones. However, it is important to keep in mind that full consistency is not achieved between observations and synthetic photometry.\n\n\nIncreasing Y globally leads to a shift of all isochrones to the left because of the increased $T_{\\rm{eff}}$. An exception is the m814W-(m336W-m814W) CMD where the isochrones move first to the right (Y=0.260 to Y=0.300) and then move back to the blue from Y=0.300  to Y=0.600. This is caused by the strong sensitivity of the F336W filter to the nitrogen abundance (see Sect.~\\ref{s_sed}). The N abundance increases rapidly when Y increases, which causes a deepening of the NH absorption band. Consequently, the (m336W-m814W) color is redder. At the same time, $T_{\\rm{eff}}$\\ increases because of the higher helium content. But at first, the effect of nitrogen is stronger, leading to a redder color. When Y reaches $\\sim$0.300, the nitrogen increase is not sufficient to counter-balance the effect of the higher $T_{\\rm{eff}}$\\ and the colors become bluer again. This effect, specific to the F336W filter, is not seen in the TO region of the CMD because at the corresponding $T_{\\rm{eff}}$\\ there is no molecular NH band in the  spectra of the stars investigated here. \n\n\n\n\\vspace{0.5cm}\n\nHaving presented and described our synthetic photometry of NGC~6752, we now turn to the main question tackled by this study: the maximum helium content.\n\n\n\\section{Maximum helium content}\n\\label{s_maxHe}\n\nIn this section, we investigate whether the current estimates of the maximum helium content of stars of the second population are true values or lower limits. More precisely, we study the possibility that highly enriched stars could be missed when studying multiple populations with HST photometry.\n\n\n\\subsection{Helium in NGC~6752}\n\\label{smax_ngc6752}\n\n\nIn a first step, we re-determine the maximum helium mass fraction difference in NGC~6752. We use a  similar method to that of \\citet{milone18} which we describe below. Briefly, we select the extreme populations, that is, the least and most chemically enriched ones, from the chromosome map \\citep{milone17}. We then estimate the helium mass fraction difference between these two populations in various CMDs using the theoretical isochrones and synthetic photometry presented in Sects.~\\ref{s_meth} and \\ref{s_sed}.\n\nTo build the chromosome map we first create the m814W versus (m275W-m814W) and m814W versus C$_{410}$ diagrams shown in the top panels of Fig.~\\ref{fidchmap6752}. We subsequently define the red and blue so-called fiducial lines that bracket the distribution of stars along the RGB. These lines are defined manually by selecting points along the red and blue envelopes and by applying a spline function over the selected points. The width of the RGB is set to the difference between the two fiducial lines at a magnitude of 14.9, corresponding to 2.0 magnitudes above the TO, in agreement with the definition of \\citet{milone17}. For each star on the RGB in the  m814W versus (m275W-m814W) and m814W versus C$_{410}$ diagrams, the quantities $\\Delta$(m275W-m814W) and $\\Delta (C_{410})$ are calculated according to equations 1 and 2 of \\citet{milone17}. We select stars with m814W magnitudes between 15.8 and 12.8 (dashed lines in Fig.~\\ref{fidchmap6752}) to cover the bottom of the RGB but avoid its part above the bump in the luminosity function where internal mixing can affect surface chemical composition \\citep[e.g.,][]{Briley1990,Shetrone2003,CZ2007,Lind2009,Henkel2017}. The resulting chromosome map is shown in the bottom panel of Fig.~\\ref{fidchmap6752}.\n\n\n\\begin{figure}[]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{fid_chmap_ngc6752.eps}\n\\caption{\\textit{Top left panel}: m814W versus (m275W-m814W) CMD of NGC~6752. \\textit{Top right panel}: m814W versus C$_{410}$ diagram. In both panels, the red and blue solid lines are the fiducial lines along the RGB. The horizontal dashed lines indicate the magnitude bin considered to build the chromosome map shown in the bottom panel. The horizontal dotted line marks the m814W value at which the width of the RGB is measured (see text). \\textit{Bottom panel}: Chromosome map ($\\Delta (C_{410})$ versus $\\Delta$ (m275W-m814W)) on the RGB. The dotted lines define boxes in which stars are considered either 1Pe (bottom right) or 2Pe (top left).}\n\\label{fidchmap6752}\n\\end{figure}\n\n\nFollowing the concept of \\citet{milone17}, we manually select the extreme 1Pe and 2Pe stars from the chromosome map by defining boxes respectively near the right-most and left-most parts of the stars' distribution, as illustrated in Fig.~\\ref{fidchmap6752}. We then study the color (and thus Y) differences between these two extreme populations in three CMDs:  m814W versus (m395N-m814W),  m814W versus (m410M-m814W), and  m814W versus (m467M-m814W). According to Sect.~\\ref{s_compobs} and \\citet{milone18} the colors (m395N-m814W), (m410M-m814W), and (m467M-m814W) depend almost exclusively on the helium content at the metallicity of NGC~6752. In addition, the color difference is the largest for the colors that involve filters located at sufficiently blue wavelengths to show sensitivity to effective temperature ---and thus Y--- variations (see Fig.~\\ref{comp_sed_L0p85} and middle panel of Fig.~5 of \\citealt{milone18}).\n\nFigure~\\ref{395m814_1G2G_6752} shows the m814W versus (m395N-m814W) diagram for the 1Pe and 2Pe stars (see Appendix \\ref{ap_dY} for the other CMDs). Building on \\citet{milone18} we define new fiducial lines along the two populations. We divide each distribution into m814W bins of 0.2 mag  in width. In each bin, we calculate the median m814W and (m395N-m814W). We subsequently perform a boxcar averaging, replacing each median point by the average of its three closest neighbours. This gives the filled circles in Fig.~\\ref{395m814_1G2G_6752}. Finally, we run a spline function over these new points to obtain the fiducial lines for each of the two extreme populations. The color difference between these lines is estimated at six m814W magnitudes: 15.2, 14.9, 14.6, 14.3, 14.0, and 13.7. We do not consider brighter stars for reasons that will be explained further below. Section~\\ref{ap_dY} of the Appendix shows the CMDs using the F410M and F467M filters. \n\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{delta395m814_ngc6752.eps}\n\\caption{m814W versus (m395N-m814W) CMD for the 1Pe (black) and 2Pe (red) populations selected in the chromosome map of Fig.~\\ref{fidchmap6752}. The filled circles are the points used to define the fiducial lines, which themselves are shown by the solid lines. See text for details. The horizontal dotted lines indicate the m814W magnitudes at which the color difference between the 2Pe and 1Pe fiducial lines is determined.}\n\\label{395m814_1G2G_6752}\n\\end{figure}\n\n\nOnce obtained, this set of six color differences is compared to theoretical values in Fig.~\\ref{Ydcol395_6752}. The latter are calculated from our synthetic photometry, including distance and extinction corrections appropriate for NGC~6752 as described in Sect.~\\ref{s_meth}. For each m814W magnitude, we determine the Y difference between the isochrones that match the (m395N-m814W) color difference determined from observations. We do this for the six selected m814W magnitudes and finally average the six determinations to yield the final Y difference. The standard deviation is taken as the uncertainty in this measurement. We perform this process for the three selected colors. The results are gathered in Table~\\ref{tab_he_ngc6752}. They are broadly consistent with the determination of \\citet{milone18} who quote a maximum Y difference of 0.042$\\pm$0.004. We note that our uncertainties are much larger. We tested the effect of varying the size of the boxes to select the 1Pe and 2Pe stars in the chromosome map. Increasing the size of 0.2 magnitudes does not affect the results. However, reducing the size (i.e., selecting fewer points, but at even more extreme positions) translates into an increase in $\\Delta$Y by between 0.015 and 0.020. However, at the same time, the uncertainties also increase by the same amount because of\nthe reduced number of stars, \n\n\n\n\\begin{figure}[]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{Y_dcol395_ngc6752.eps}\n\\caption{Determination of the helium mass fraction difference between 1Pe and 2Pe stars for the six magnitudes (see labels in panels) defined in Fig.~\\ref{395m814_1G2G_6752}. For each panel the filled circles are the theoretical values calculated from our isochrones and synthetic photometry. The black solid line is a linear regression to these values. The vertical gray line highlights the measured color difference between 1Pe and 2Pe fiducial lines. The horizontal dashed line indicates the corresponding Y difference read from the black solid line.}\n\\label{Ydcol395_6752}\n\\end{figure}\n\n\n\n\\begin{table}[ht]\n\\begin{center}\n  \\caption{Difference in the helium mass fraction Y between the 1Pe and 2Pe populations, for different colors and for the RGB and MS stars.}\n\\label{tab_he_ngc6752}\n\\begin{tabular}{lc}\n\\hline  \ncolor & $\\Delta$Y \\\\     \n\\hline\nRGB & \\\\\n(m395N-m814W)  & 0.039$\\pm$0.013 \\\\       \n(m410M-m814W)  & 0.052$\\pm$0.011 \\\\        \n(m467M-m814W)  & 0.068$\\pm$0.025 \\\\\n\\hline\nMS & \\\\\n(m395N-m814W)  & 0.042$\\pm$0.004 \\\\       \n(m410M-m814W)  & 0.047$\\pm$0.004 \\\\        \n(m467M-m814W)  & 0.049$\\pm$0.049 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nIn anticipation of the following section,  in Fig.~\\ref{fidchmap6752MS} and Table~\\ref{tab_he_ngc6752} we present the analysis of extreme populations on the MS of NGC~6752. The details of the analysis are presented in Sect.~\\ref{clu_W16}. The results indicate that the maximum helium mass fraction reported by \\citet{milone18} based on the RGB is also recovered for MS stars. These results are further discussed in Sect.\\ref{s_disc}.\n\n\\begin{figure}[]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{fidchmap6752MS.eps}\n\\caption{Same as Fig.~\\ref{fidchmap6752} but for the MS of NGC~6752.}\n\\label{fidchmap6752MS}\n\\end{figure}\n\n\n\n\\subsection{Synthetic clusters}\n\\label{smax_synthclu}\n\nIn a second step, we build synthetic clusters to investigate whether or not we underestimate the maximum helium difference between extreme populations. \n\nWe first start by building synthetic CMDs. In practice, we draw artificial points along the isochrones in the m814W versus (m275W-m814W), m814W versus (m336W-m814W), m814W versus (m410M-m814W), m814W versus (m395N-m814W), and m814W versus (m467M-m814W) diagrams (see Fig.~\\ref{cmd}). We assume a magnitude distribution in the F814W filter similar to that of NGC~6752 (see Fig.~\\ref{distrib814}). We build clusters made of 9000 stars, ensuring a consistent number of stars on the RGB compared to NGC~6752.\n\n\\begin{figure}[]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{distrib814.eps}\n\\caption{Distribution of stars according to their m814W magnitude in NGC~6752 (black) and in a typical synthetic cluster (red). The latter distribution results from random sampling using a Gaussian distribution.}\n\\label{distrib814}\n\\end{figure}\n\n\nFor each m814W magnitude drawn along the isochrone (in each diagram) we add a correction to the color (x-axis in each diagram) to introduce a dispersion on the theoretical isochrone. We estimate the correction in the following way. We retrieve photometry of NGC~6752 from the HUGS survey\\footnote{\\url{https:\/\/archive.stsci.edu\/prepds\/hugs\/}} and build the m814W versus (m275W-m814W), m814W versus (m336W-m814W), and m814W versus (m435W-m814W) diagrams. We select stars at m814W magnitudes of 14.9 and 16.5 (in a bin of size 0.2 magnitude). For these subgroups of stars and for each color, (m275W-m814W), (m336W-m814W), and (m435W-m814W), we take the standard deviation with respect to the median point as representative of the dispersion on the RGB and MS. We subsequently use these values to introduce a dispersion in the synthetic colors. For that we randomly draw a color correction by means of a Gaussian distribution centered on zero and characterized by the standard deviation determined above.\nIn the absence of photometric error for the m395N, m410M, and m467M magnitudes, we assume the dispersion in the colors (m395N-m814W), (m410M-m814W), and (m467M-m814W) is the same as in (m435W-m814W). We perform this process for all isochrones, that is, for all chemical compositions. We finally end up with a series of synthetic CMDs that are used to build synthetic clusters.\n\n\n\\subsubsection{Two chemical compositions}\n\\label{clu_2pop}\n\nWe build synthetic clusters by mixing populations of different chemical composition. We start with a cluster made up of one-third stars with Y=0.248, and two-thirds stars with Y=0.330. For the m814W versus (m275W-m814W) CMD, we therefore select 3000 stars from Y=0.248 isochrone in the synthetic m814W versus (m275W-m814W) just created, and 6000 stars from the Y=0.330 isochrone. We repeat the process for all five diagrams.\nWe then apply the same method as described in Sect.~\\ref{smax_ngc6752} to determine the maximum helium mass fraction difference in the synthetic cluster. We know by construction that it is equal to 0.082. We find that $\\Delta$Y=0.081$\\pm$0.002, 0.079$\\pm$0.002, and 0.070$\\pm$0.015 using the (m395N-m814W), (m410M-m814W), and (m467M-m814W) colors, respectively. We thus recover the input value ($\\Delta$Y=0.082) with a good level of confidence.\n\n\n\\begin{figure}[]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{fidchmap_unif0.eps}\n\\caption{Same as Fig.~\\ref{fidchmap6752} but for a synthetic cluster with one-third stars with  Y=0.248 and two-thirds stars with Y=0.260 to 0.600 with a flat distribution.}\n\\label{fidchmapunif0}\n\\end{figure}\n\n\n\\begin{figure}[]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{delta395m814_unif0.eps}\n\\caption{Same as Fig.~\\ref{395m814_1G2G_6752} but for a synthetic cluster with one-third stars with Y=0.248  and two-thirds stars with Y=0.260 to 0.600 with a flat distribution.}\n\\label{395m814_1G2G_unif0}\n\\end{figure}\n\n\n\\subsubsection{Uniform distribution among second population}\n\\label{clu_unif}\n\nWe then run another test by building a synthetic cluster still made up of one-third 1P stars (3000 data points) drawn from the Y=0.248 isochrone, but with the remaining two-thirds of 2P stars (i.e., 6000 data points) spread uniformly between isochrones with Y varying from 0.260 to 0.600. For that, we create the isochrones characterized by Y=0.430, 0.470, 0.500, 0.530, and 0.570 by linearly interpolating between the Y=0.400 and Y=0.600 isochrones. We create the five CMDs with these additional isochrones using the same method as described immediately above. We select 500 stars for each isochrone between 0.260 and 0.600, which is 5.5\\% per isochrone, to obtain the final synthetic cluster.\nWe then proceed as previously to determine $\\Delta$Y. Figure~\\ref{fidchmapunif0} shows the definition of the fiducial lines and the chromosome map, together with the selected 1Pe and 2Pe stars. Figure~\\ref{395m814_1G2G_unif0} shows the 1Pe and 2Pe stars in the m814W versus (m395N-m814W) diagram. Here, we face a problem: the number of 2Pe stars is too small to apply the automatic process for defining the fiducial line of the 2Pe population. We therefore select the fiducial line by hand\\footnote{We estimate the uncertainty on $\\Delta$Y resulting from the use of this manual process by performing ten repetitions on the same set of 1Pe and 2Pe populations. We find the uncertainty is of the order 0.004.}, as done in the first step of the process that leads to the chromosome map. We then determine the Y difference and we obtain $\\Delta$Y=0.362$\\pm$0.005. Using the other colors, (m410M-m814W) and (m467M-m814W), we obtain $\\Delta$Y=0.364$\\pm$0.008 and $\\Delta$Y=0.367$\\pm$0.007, respectively. The true Y difference, being 0.352, shows that we are able to measure it with good accuracy. \n\n\n\\begin{figure}[]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{distriW16.eps}\n\\caption{Black solid histogram: Relative fraction of stars as a function of their helium content on the lower RGB in the models of \\citet{chantereau16} and for an age of 13~Gyr. Red dashed histogram: Distribution we adopt to build synthetic clusters with populations as close as possible to those of \\citet{chantereau16}.}\n\\label{distriW16}\n\\end{figure}\n\n\n\\subsubsection{Distribution of Chantereau et al.}\n\\label{clu_W16}\n\n\nWe finally move to clusters made of populations similar to those presented by \\citet{chantereau16} within the framework of the FRMS scenario that fits the O-Na anti-correlation in NGC~6752, and which is the most extreme case in terms of He enrichment. We use the distribution of stars according to their helium content shown in Fig.~6 of \\citet{chantereau16}  for an age of 13 Gyr (at that age, their model predicts that 10$\\%$ of the stars should have Y$>$0.4, from an initial fraction of 21$\\%$). We adopt the distribution of the lower RGB as representative of the entire population of the cluster. Examination of Fig.~6 of \\citet{chantereau16} reveals that this is a fair approximation for the MS up to the upper RGB. As we do not have synthetic photometry for the full distribution of Y presented by Chantereau et al., we group bins of Y. This is illustrated in Fig.~\\ref{distriW16}. For instance,  in the bin with Y=0.30 we gather the stars of Chantereau et al. with Y=0.29, 0.30, and 0.31. In addition, we linearly extrapolated our isochrones beyond Y=0.6 (up to Y=0.72) using our Y=0.4 and Y=0.6 synthetic isochrones. For the populations with Y$>$0.6 in Fig.~\\ref{distriW16} we group stars in bins of  0.2 in width (e.g., the Y=0.64 bin gathers stars with Y=0.64 and 0.65 from the distribution of Chantereau et al.). The most enriched population has Y=0.72 in our clusters, corresponding to $\\Delta$Y=0.472. We built ten clusters.\n\nWe then perform the determination of the maximum helium mass fraction as in the previous example where a uniform distribution of stars with Y$>$0.248 was used. Figures~\\ref{fidchmapW16} and \\ref{395m814_W16} show the synthetic CMDs, chromosome map, and extreme populations in the m814W versus (m395M-m814W) CMD in a representative example. We gather the results in Table~\\ref{tab_he_synth} and Fig.~\\ref{deltaY}. For some realizations of our synthetic clusters, and for some colors, we derive enrichments that are marginally compatible with the input value $\\Delta$Y=0.472: we obtain $\\Delta$Y=0.437$\\pm$0.038 for cluster 8 using the color (m467M-m814W). However, the main result is that, on average, we determine a maximum $\\Delta$Y of about 0.43, which is smaller than the input value of 0.472. This is qualitatively understood by looking at Figs.~\\ref{distriW16} and \\ref{fidchmapW16}. Stars with Y$>$0.6 have an almost flat distribution which translates into the extended population in upper left part of the chromosome map. When picking the stars identified as the 2Pe population, and to ensure a sufficient number of stars in that population, we include the stars with the highest Y, but also some stars with slightly smaller Y. We therefore tend to create a population with an average Y that is smaller than 0.72. \n\n\n\\begin{figure}[]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{fidchmap_W16_000.eps}\n\\caption{Same as Fig.~\\ref{fidchmap6752} but for a synthetic cluster with the populations of \\citet{chantereau16}.}\n\\label{fidchmapW16}\n\\end{figure}\n\n\n\\begin{figure}[]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{delta395m814_W16_000.eps}\n\\caption{Same as Fig.~\\ref{395m814_1G2G_6752} but for a synthetic cluster with the populations of \\citet{chantereau16}.}\n\\label{395m814_W16}\n\\end{figure}\n\n\n\\begin{table}[t]\n\\begin{center}\n  \\caption{Helium difference recovered in ten synthetic clusters with a distribution of the population as in Fig.~\\ref{distriW16}. $\\Delta$Y$_{X}$ stands for the helium difference estimated from color ($X$-m814W) where $X$ is the magnitude in one of the filters F395N, F410M, or F467M.}\n\\label{tab_he_synth}\n\\begin{tabular}{cccc}\n\\hline  \ncluster id & $\\Delta$Y$_{395N}$ & $\\Delta$Y$_{410M}$ & $\\Delta$Y$_{467M}$ \\\\     \n\\hline\n0  &    0.444$\\pm$0.022 &   0.448$\\pm$0.014  &   0.468$\\pm$0.019 \\\\\n1  &    0.409$\\pm$0.007 &   0.411$\\pm$0.006  &   0.416$\\pm$0.007 \\\\\n2  &    0.411$\\pm$0.031 &   0.416$\\pm$0.021  &   0.434$\\pm$0.038 \\\\\n3  &    0.425$\\pm$0.027 &   0.434$\\pm$0.027  &   0.427$\\pm$0.051 \\\\\n4  &    0.420$\\pm$0.013 &   0.429$\\pm$0.008  &   0.436$\\pm$0.010 \\\\\n5  &    0.437$\\pm$0.025 &   0.438$\\pm$0.023  &   0.438$\\pm$0.011 \\\\\n6  &    0.417$\\pm$0.013 &   0.421$\\pm$0.009  &   0.430$\\pm$0.014 \\\\\n7  &    0.472$\\pm$0.010 &   0.472$\\pm$0.017  &   0.488$\\pm$0.011 \\\\\n8  &    0.457$\\pm$0.029 &   0.460$\\pm$0.020  &   0.492$\\pm$0.023 \\\\\n9  &    0.449$\\pm$0.021 &   0.460$\\pm$0.023  &   0.457$\\pm$0.019 \\\\\n\\hline\naverage & 0.430$\\pm$0.004  &  0.426$\\pm$0.004  &  0.441$\\pm$0.004 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\begin{figure}[]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{deltaY.eps}\n\\caption{Maximum helium difference $\\Delta$Y for ten synthetic clusters. Different symbols refer to different colors used for the determination. Blue (red) symbols stand for stars on the MS (RGB). The horizontal black solid line shows the true helium difference used to build the clusters.}\n\\label{deltaY}\n\\end{figure}\n\n\n\nTo further investigate this behavior, we show in Fig.~\\ref{chmapzoom2Ge} and Table~\\ref{tab_he_2Ge} the effect of the selection of 2Pe stars on the determination of the maximum helium difference. Choosing a larger population (population 2Pe(c)) translates into a decrease in $\\Delta$Y by $\\sim$0.03, as expected because of the inclusion of stars with less extreme Y in the 2Pe population. We also note that the values of $\\Delta$Y obtained with that selection of 2Pe stars are consistent with those obtained for other synthetic clusters reported in Table \\ref{tab_he_synth} and for which the 2Pe selection was more strict. Inversely, reducing the 2Pe population (2Pe(a) in Fig.~\\ref{chmapzoom2Ge}), but still keeping a number of stars sufficient to define the 2Pe fiducial line, leads to a larger $\\Delta$Y, which is consistent with the input value. \n\n\nFrom these experiments, we conclude that for distributions of multiple populations with small numbers of stars with large chemical enrichments, such as that of \\citet{chantereau16}, the derived maximum helium enrichment critically depends on the selection of the 2Pe population. We also find that we slightly underestimate the maximum Y of the cluster.\n\n\n\n\n\n\\begin{figure}[]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{plot_chmap_zoom_2Ge_000.eps}\n\\caption{Zoom into the upper left part of the chromosome map shown in the bottom panel of Fig.~\\ref{fidchmapW16}. The gray dotted lines mark the selection of 2Pe stars as in Fig.~\\ref{fidchmapW16}. The blue and red broken lines indicate alternative selections of 2Pe stars.}\n\\label{chmapzoom2Ge}\n\\end{figure}\n\n\\begin{table}[]\n\\begin{center}\n  \\caption{Effect of the selection of 2Pe stars (see Fig.~\\ref{chmapzoom2Ge}) on the determination of the maximum helium difference in synthetic cluster number 0. The first column gives the label of the 2Pe selection. The following columns give the helium difference estimated using the same three colors as in Table~\\ref{tab_he_synth}.}\n\\label{tab_he_2Ge}\n\\begin{tabular}{cccc}\n\\hline  \n2Pe selection & $\\Delta$Y$_{395N}$ & $\\Delta$Y$_{410M}$ & $\\Delta$Y$_{467M}$ \\\\     \n\\hline\n2Pe(a)  &       0.444$\\pm$0.022 &   0.448$\\pm$0.014  &   0.468$\\pm$0.019 \\\\\n2Pe(b)  &       0.458$\\pm$0.018 &   0.471$\\pm$0.011  &   0.489$\\pm$0.006 \\\\\n2Pe(c)  &       0.410$\\pm$0.040 &   0.410$\\pm$0.034  &   0.418$\\pm$0.032 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\\vspace{0.2cm}\n\n\\citet{chantereau16}  showed that the RGB contains less very enriched (Y$>$0.6) stars than the MS (see their Fig.~5).\nWe therefore expect that the determination of the maximum helium content of a GC using MS stars suffers less from the difficulties described above. To test this, we determined $\\Delta$Y as reported for RGB stars in Figs.~\\ref{fidchmapW16} and \\ref{395m814_W16} and Table~\\ref{tab_he_synth} but focusing on MS stars. Figure~\\ref{fidchmapW16_MS} illustrates our selection of stars: those with m814 magnitudes between 16.9 and 18.1. We estimate the width of the MS at m814=17.5. The 1Pe and 2Pe stars are extracted from the chromosome map as shown in the lower panel of Fig.~\\ref{fidchmapW16_MS}. The results are gathered in Table~\\ref{tab_he_synth_MS}. Compared to Table~\\ref{tab_he_synth} we see that the true helium mass fraction difference (0.472) is indeed better recovered when using MS stars. A graphical representation of this result is given in Fig.~\\ref{deltaY} where we see that depending on the cluster simulation, $\\Delta$Y based on RGB stars may be underestimated, while for MS stars, the input value is always recovered. We thus conclude that for GCs that would have stellar populations similar to those of \\citet{chantereau16}, the study of the maximum helium content should be preferentially performed on MS stars, provided sufficient data quality. \n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{fidchmap_W16_000_MS.eps}\n\\caption{Same as Fig.~\\ref{fidchmapW16} but for MS stars.}\n\\label{fidchmapW16_MS}\n\\end{figure}\n\n\n\n\\begin{table}[]\n\\begin{center}\n  \\caption{Same as Table~\\ref{tab_he_synth} but for MS stars.}\n\\label{tab_he_synth_MS}\n\\begin{tabular}{cccc}\n\\hline  \ncluster id & $\\Delta$Y$_{395N}$ & $\\Delta$Y$_{410M}$ & $\\Delta$Y$_{467M}$ \\\\     \n\\hline\n0  &    0.464$\\pm$0.004 &   0.468$\\pm$0.005  &    0.478$\\pm$0.010 \\\\\n1  &    0.472$\\pm$0.007 &   0.478$\\pm$0.004  &    0.472$\\pm$0.009 \\\\\n2  &    0.468$\\pm$0.010 &   0.478$\\pm$0.007  &    0.482$\\pm$0.009 \\\\\n3  &    0.466$\\pm$0.009 &   0.474$\\pm$0.007  &    0.487$\\pm$0.006 \\\\\n4  &    0.473$\\pm$0.007 &   0.472$\\pm$0.010  &    0.473$\\pm$0.009 \\\\\n5  &    0.459$\\pm$0.005 &   0.474$\\pm$0.005  &    0.482$\\pm$0.003 \\\\\n6  &    0.468$\\pm$0.010 &   0.470$\\pm$0.011  &    0.475$\\pm$0.008 \\\\\n7  &    0.464$\\pm$0.008 &   0.464$\\pm$0.007  &    0.476$\\pm$0.013 \\\\\n8  &    0.466$\\pm$0.016 &   0.470$\\pm$0.015  &    0.481$\\pm$0.005 \\\\\n9  &    0.467$\\pm$0.006 &   0.478$\\pm$0.011  &    0.491$\\pm$0.006 \\\\\n\\hline\naverage & 0.465$\\pm$0.002  &  0.473$\\pm$0.002  &  0.482$\\pm$0.002 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\n\\subsubsection{Steep distribution}\n\\label{clu_steep}\n\n\nIn this section we try to answer the question of whether or not we can miss very helium-rich stars if multiple populations follow a steeper distribution than that presented by \\citet{chantereau16}. To this end, we define an artificial distribution as shown in Fig.~\\ref{distribsteep}. We start from the distribution of \\citet{chantereau16} between Y=0.248 and Y=0.29. We linearly interpolate their distribution in this range of Y, and then extrapolate the resulting distribution up to Y=0.400 (blue line in Fig.~\\ref{distribsteep}). We thus obtain a much steeper distribution of populations as a function of Y  than that of Chantereau et al. for Y$>$0.29. According to \\citet{milone18} and Sect.~\\ref{smax_ngc6752}, the maximum helium mass fraction observed in NGC~6752 is about 0.29. Our artificial distribution can therefore be seen as a test case where only a few stars with Y$>$0.29 are present in synthetic clusters. We then try to see if they are recovered or missed when determining $\\Delta$Y.\n\n\\begin{figure}[]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{distribsteep.eps}\n\\caption{Black solid histogram: Relative fraction of stars as a function of their helium content on the lower RGB in the models of \\citet{chantereau16} and for an age of 13~Gyr. Blue solid line: Linear interpolation of Chantereau's distribution between Y=0.248 and 0.290, and extrapolation up to Y=0.400. Red dashed histogram: Adopted distribution based on the previous interpolation and re-scaling so that the sum of relative fractions is equal to 1.}\n\\label{distribsteep}\n\\end{figure}\n\n\nWe built ten synthetic clusters using the above distribution. We use only Y values for which isochrones are available, that is, Y=0.248, 0.260, 0.270, 0.300, 0.330, 0.370, and 0.400 (red bins in Fig.~\\ref{distribsteep}). We perform the $\\Delta$Y determination as in the previous sections. The results are gathered in Fig.~\\ref{deltaYsteep}. Using RGB stars, we are not able to recover the input value ($\\Delta$Y=0.152), but we can still detect stars with Y$>$0.29 (i.e., $\\Delta$Y=0.042). If present, and under the assumption that their distribution follows that described above, stars more helium-rich than Y=0.29 should therefore be detectable. On the MS, we confirm the trend seen in Sect.~\\ref{clu_W16} that higher values of $\\Delta$Y are recovered. However, with the present distribution, the maximum helium mass fraction of the cluster is also missed on the MS due to the very small number of highly enriched stars. This is different from Sect.~\\ref{clu_W16} where on the MS the initial value of $\\Delta$Y could be retrieved. \n\n\n\n\\begin{figure}[]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{deltaYsteep.eps}\n\\caption{Same as Fig.~\\ref{deltaY} but for the distribution of the population described in Sect.~\\ref{clu_steep}. The gray area marks the observed value of $\\Delta$Y in NGC~6752.}\n\\label{deltaYsteep}\n\\end{figure}\n\n\n\\section{Discussion}\n\\label{s_disc}\n\nOur method for determining the maximum helium mass fraction in GCs is slightly less automated than that presented by \\citet{milone18}. In particular, the selection of the extreme 1Pe and 2Pe populations is made visually in our case, while it relies on a more sophisticated treatment in \\citet{milone18}. We tested the effect of the 2Pe selection in Sect.~\\ref{clu_W16} and Fig.~\\ref{chmapzoom2Ge}, and argue that the choice of 2Pe stars can affect the value of the maximum Y. However, qualitatively, stars much more He-rich than the observed limit of 0.042 are easily detected regardless of the 2Pe selection. In addition, the automated method of \\citet{milone18} works well when 2Pe stars are clearly identified by an overdensity in the chromosome map, but there are clusters for which no such overdensity is visible. For instance, the chromosome map of NGC~6752 itself, presented in the bottom panel of Fig.~\\ref{fidchmap6752}, shows a rather uniform distribution along the 2G sequence (i.e., from the bottom right corner up to the upper left corner). Hence, we argue that the choice of 2Pe stars in our study is not drastically different from \\citet{milone18}.\n\n\nHe-rich populations may be contaminated by blue stragglers and their descendants which would populate similar regions of CMDs. \\citet{marino19} showed that candidate evolved blue stragglers should be located preferentially along the 1P sequence, and thus would not contaminate 2Pe stars. However, one may wonder whether blue stragglers resulting from the merger of two 2P stars could produce stars along the 2P sequence too. This needs to be investigated further.  \nBinaries can also contaminate the 2P sequence. As shown by \\citet{martins20}, the presence of a companion among RGB stars will tend to displace a star up and left in the chromosome map. The magnitude of the displacement depends on the relative brightness and effective temperature of the two stars. Hence, stars along the 2G sequence, with medium Y but with a companion, may be moved to the position of single stars with higher Y. Consequently, the extremity of the 2G sequence may contain stars with less extreme Y. This could lead to overestimation of the maximum Y. The binary fraction among GCs is low, usually below 10\\% \\citep{sollima07,milone12,jb15}, and their effect on the maximum Y should therefore be limited; unless the number of very He-rich stars is also small, as in the distribution of \\citet{chantereau16}. In any case, if binaries are present, the maximum Y determined from photometry is probably overestimated.\n\n\nWith these limitations in mind, and given the results of the present study, it is unlikely that  NGC~6752 contains stars with Y$\\gtrsim$0.3. If multiple populations were formed as predicted by the FRMS scenario developed by \\citet{decressin07} and \\citet{chantereau15,chantereau16}, a wide distribution of Y, from 0.248 to 0.72, should be present among stars either on the MS or the RGB, with ~39$\\%$ of the stars with Y$>$0.3 and ~10$\\%$ with Y$>$0.4 at 13~Gyr. Our study reveals that while we may not retrieve the most helium-rich objects on the RGB, we should still be able to detect stars with Y as high as 0.65 (see Fig.~\\ref{deltaY}). This is clearly in contrast with results based on HST photometry indicating Y no higher than 0.3. We have shown that even in the case of a Y distribution much steeper than that predicted by \\citet{chantereau16} we would be able to find stars with helium enrichment beyond the observed value.\n\nA prediction of our study dedicated to NGC~6752 is that the maximum helium content of GC stars, if it follows a distribution where helium-rich stars are less numerous than helium-poor ones, is best determined on the MS rather than on the RGB. This is due to the faster evolution of He-rich stars and consequently the larger number of stars on the MS than on the RGB \\citep{dm73,dantona10,chantereau16}. \nHowever, the maximum Y we obtain on the MS using HST data of NGC~6752 does not indicate a significant difference compared to the value found on the RGB.\n\nAll in all, the present results strongly suggest that stars in NGC~6752 do not follow the distribution predicted by the FRMS model presented in \\citet{chantereau16}, and consequently, multiple populations were not likely formed out of material polluted by this type of object.\nThis conclusion applies only to NGC~6752, and additional studies of clusters with different ages, metallicities, and masses are required to see whether generalizations can be made. A wider study of this kind is necessary to investigate whether or not the current observational limit of Y$\\sim$0.4 is robust and to investigate whether or not it suffers from observational limitations. As recalled in Sect.~\\ref{s_intro}, this limit is a key prediction of some models, but also a building block of some others. In particular, the formation of multiple populations caused by pollution of material ejected from a super-massive star (SMS) originally {assumes} that the SMS stops its evolution when its core helium content reaches 0.4, and is further dislocated by instabilities and\/or strong stellar winds \\citep{denis14}. This assumption was dictated  by the observational fact that Y appears to be no higher than 0.4 in the most extreme GC populations. On the other hand, hot-hydrogen burning products with low helium abundances in agreement with the photometric constraints can be ejected by SMSs in the case where they are continuously rejuvenated by runaway stellar collisions \\citep{Gieles2018}.\n\nPollution by material produced in massive AGB stars predicts that the helium distribution among multiple populations reaches a maximum of $\\sim$0.38 \\citep[e.g.,][]{dercole10}. This is consistent with the current observational limit, and would make these objects good polluter candidates if other problems were not linked to their use, as summarized in \\citet{Renzini2015} and \\citet{bastianlardo18} for instance. Other important factors are the time constraints on the dilution by pristine material, as also highlighted by \\citet[][see also \\citealt{dercole16}]{dercole11}, and the mass budget issue \\citep[e.g.,][]{PC2006,Krause2016}. The transformation of the Na-O correlation in AGB ejecta into a Na-O anti-correlation, as seen in all GCs, also requires specific conditions for mixing of pristine material with nucleosynthesis products \\citep[e.g.,][]{Ventura2008,Karakas2010}. In addition, the C+N+O sum of AGB ejecta is not conserved \\citep{Decressin2009}, contrary to what is deduced from spectroscopic analysis of stars in NGC~6752 \\citep{yong15}. \n\nThe role of massive binaries in the multiple population phenomenon remains to be investigated. \\citet{demink09}  showed that these objects could potentially produce the required abundance patterns. But their study is limited to a 20+15 M$_{\\sun}$\\ system on a 12-day orbit and in which both components have reached synchronization. The average helium mass fraction of the ejecta of this binary system is 0.3. Material ejected at later phases of the system's evolution has Y as large as $\\sim$0.63 (see Fig.~1 of de Mink et al.). The average helium content is therefore consistent with the current maximum Y observed in GCs. However binary evolution depends not only on the properties and evolution of the  components, but also on the parameters of the  system (separation, eccentricity, mass ratio, mass transfer efficiency); see for example \\citet{menon20}. Additionally, the maximum central temperature of the considered mass domain does not reach the high values required to build the Mg-Al anti-correlation \\citep[e.g.,][]{prantzos07,Prantzos2017}. A wider study involving population synthesis is therefore required to assess the impact of massive binaries on the origin of multiple populations in GCs.  \n\n\n\n\n\n\\section{Conclusion}\n\\label{s_conc}\n\nIn this study, we investigated the determination of the maximum helium mass fraction in stars of the GC NGC~6752. Our goal was to decipher whether we really detect the most He-rich stars with present-day\nphotometric methods, or miss them.\nWe relied on the work of \\citet{chantereau16} who produced isochrones with various chemical compositions corresponding to different degrees of pollution by FRMS which is an extreme case in terms of He enrichment. We computed synthetic spectra along these isochrones using the atmosphere code ATLAS12 and the spectral synthesis code SYNTHE. The resulting spectra were used to compute synthetic photometry in the following HST filters: WFC3 F275W, WFC3 F336W, WFC3 F410M, WFC3 F467M, ACS F606W, and ACS F814W. We compared the synthetic colors with data of NGC~6752 obtained by \\citet{milone13}. The different CMDs are usually reasonably well reproduced, although offsets exist between synthetic and observed sequences (MS, TO, RGB).\n\nWe re-determined the maximum helium mass fraction of stars in NGC~6752 using a method very similar to that of \\citet{milone18}. Our results are consistent with those of Milone at al.\nWe built synthetic clusters with various populations characterized by their He content (they also have different composition in light elements). We validated our method on simple population distributions, ensuring that we are able to recover the input maximum Y. We subsequently created synthetic clusters following the distribution presented by \\citet{chantereau16}, that is, with stars with Y of between 0.248 and 0.72. We show that on the RGB, we slightly underestimate the maximum Y, but by a relatively small amount ($\\sim$0.05). On the MS, we retrieve the input value. In any case, even on the RGB the maximum Y we determine in synthetic clusters is higher than the observed value of 0.042$\\pm$0.004 \\citep{milone18}. We tested that even if populations followed a steeper Y distribution than that of \\citet{chantereau16}, stars with Y higher than 0.042 are recovered (in that case the maximum Y is underestimated both on the RGB and MS). Finally, we determined the maximum Y on the MS of NGC~6752 using HST data and find it to be consistent with the value obtained on the RGB.\n\nThese results indicate that multiple populations in NGC~6752 have a Y distribution that is very likely not that assumed by \\citet{chantereau16} in the framework of the FRMS scenario. Our results also show that in the specific case studied here, the maximum helium mass fraction determined from observations is probably the true value (i.e., there are no stars more He-rich in that cluster). \nWe stress that although we have focused on the specific case of pollution by FRMS, our results apply to any scenario that would predict a strong He enrichment among 2P stars \\citep{Salarisetal2006,Pietrinferni2009,Cassisi2013,Dotter2015}.\nOur results need to be extended to other clusters spanning a range of parameters (age, metallicity, mass) in order to confirm that FRMSs are not responsible for the multiple populations in GCs and to better constrain the scenario that produce them. \n\n\n\\section*{Acknowledgments}\n\nWe thank an anonymous referee for a positive report. We thank A. Milone for sharing HST photometry of NGC~6752 and for interesting discussions. This work was supported by the Swiss National Science Foundation (Project 200020-192039 PI C.C.). This work was supported by the Agence Nationale de la Recherche grant POPSYCLE number ANR-19-CE31-0022.\n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}