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+{"text":"\\section{Introduction}\n\\label{sec:1}\nThe understanding of diffusion and transport of passive \ntracers in a given velocity field has both theoretical and practical \nrelevance in many fields of science and engineering, \ne.g. mass and heat transport in geophysical flows \n(for a review see \\cite{davis,davis2}),\ncombustion and chemical engineering \\cite{Moffatt}.\n\n\nOne common interest is the study of the mechanisms\nwhich lead to transport enhancement as a fluid is driven \nfarther from the motionless state. \nThis is related to the fact that the Lagrangian motion of \nindividual tracers can be rather complex even in simple laminar flows\n\\cite{Ottino,lagran}.\n\nThe dispersion of passive scalars in a given velocity field is the result,\nusually highly nontrivial, of two different contributions:\nmolecular diffusion and advection.\nIn particular, one can have rather fast transport, even without\nmolecular diffusion, in presence of {\\it Lagrangian chaos \\\/}, \nwhich is the sensitivity to initial conditions of\nLagrangian trajectories.\nIn addition, also for a 2D stationary velocity field, \nwhere one cannot have Lagrangian chaos \\cite{Licht}, in presence of \na particular geometry of the streamlines the diffusion can \nbe much larger than the one due only to the molecular \ncontribution, as in the case of spatially periodic stationary\nflows \\cite{Rosenbluth,Shraiman}.\n\nTaking into account the molecular diffusion, the motion of \na test particle (the tracer) is described by the following Langevin\nequation:\n\\begin{equation}\n\\frac{d{\\bf x}}{dt}= {\\bf u}({\\bf x},t)+\n\\mbox{\\boldmath $\\eta$}(t),\n\\label{eq:langevin}\n\\end{equation}\nwhere \n${\\bf u}({\\bf x},t)$ \nis the Eulerian \nincompressible velocity field at the point ${\\bf x}$ \nand time $t$, $\\mbox{\\boldmath $\\eta$}(t)$ is a Gaussian white noise \nwith zero mean and\n\\begin{equation}\n<\\eta_{i}(t) \\eta_{j}(t^{'}) >= 2 D_{0} \\delta_{ij} \\delta(t-t^{'})\\,,\n\\label{eq:whitenoise}\n\\end{equation}\nwhere $D_{0}$ is the (bare) molecular diffusivity.\n\nDenoting \n$\\Theta({\\bf x},t)$ \nthe concentration of tracers,\none has:\n\\begin{equation}\n{\\partial}_{t} \\Theta+ \n\\left( {\\bf u} \\cdot \\mbox{\\boldmath $\\nabla$} \\right) \\Theta=\nD_{0} \\,\\Delta \\Theta \\,.\n\\label{eq:fokker}\n\\end{equation}\n\nFor an Eulerian velocity field  \nperiodic in space, or anyway  defined in infinite domains, \nthe long-time, large-distance behavior of the diffusion process\nis described by the effective diffusion tensor $D_{ij}^{E}$\n({\\it eddy-diffusivity tensor}\\\/):\n\\begin{equation}\nD_{ij}^{E}= \\lim_{t\\rightarrow \\infty} \\frac {1}{2t}\n<(x_{i}(t)-<x_{i}>)(x_{j}(t)-<x_{j}>)>\\,,\n\\label{def:eddydiff}\n\\end{equation}\nwhere now ${\\bf x}(t)$ is the position of the the tracer \nat time $t$, $i,j=1,\\cdots,d$ (being $d$ the spatial dimension) , and\nthe average is taken over the initial positions\nor, equivalently, over an ensemble of test particles.\nThe tensor $D^{E}_{ij}$ gives the long-time, large-distance\nequation for $< \\! \\Theta \\!>$ i.e. the concentration field locally averaged\n over a volume of linear distance much larger than the\ntypical length $l_{u}$ of the velocity field, according to\n\\begin{equation}\n{\\partial}_{t} <\\Theta>= \\sum_{i,j=1}^{d} D_{ij}^{E}\\, \n\\frac{{\\partial}^{2}}{\\partial x_{i} \\partial x_{j}} <\\Theta>\\,.\n\\label{eq:eddydiff}\n\\end{equation}\nThe above case, with finite $D_{ij}^{E}$, is\nthe typical situation where the diffusion, for very large \ntimes, is a standard diffusion process. \nHowever there are also cases \nshowing the so-called {\\it anomalous diffusion}\\\/: the spreading \nof the particles does not behave linearly with time but\nhas a power law $t^{2\\nu}$ with $\\nu \\neq 1\/2$.\nTransport anomalies are, in general, indicators of the \npresence of strong \ncorrelation in the dynamics, even at large time and space scales\n\\cite{georges}.\n\nIn the case of infinite spatial domains and periodic \nEulerian fields the powerful multiscale technique \n(also known as homogenization in mathematical literature) \ngives a useful tool for studying standard diffusion\nand, with some precautions, also the anomalous situations \n\\cite{BCVV}.\n\nOn the other hand we have to stress the fact that \ndiffusivity tensor (\\ref{def:eddydiff}) is \nmathematically well defined only in the limit of infinite times, \ntherefore it gives \na sensible result only if the characteristic length\n$l_{u}$ of the velocity field is much smaller than the size \n$L$ of the domain. \n\nThe case when $l_{u}$ and $L$ are not well separated\nis rather common in many geophysical problems, e.g.\nspreading of pollutants in Mediterranean or Baltic sea, \nand also in plasma physics.\nTherefore it is important to introduce some other \ncharacterizations of the diffusion properties\nwhich can be used also in non ideal cases.\nFor instance, \\cite{Zambia} propose to employ exit times \nfor the study of transport in basins with complicated geometry.\n\nIn Section \\ref{sec:2} we introduce a characterization of\nthe diffusion behavior in terms of the typical time\n$\\tau(\\delta)$ at scale $\\delta$;\nthis allows us to define a finite size diffusion \ncoefficient \n$D(\\delta) \\sim \\delta^{2}\/\\tau(\\delta)$.\n>From the shape of $\\tau(\\delta)$ \nas a function of $\\delta$, one can distinguish different\nspreading regimes.\n\nIn Section \\ref{sec:3} we present the results of numerical experiments\nin closed basins and present new results\nrelative to the behavior of the diffusion coefficient \nnear the boundary (a detailed discussion \nis in the appendix).\n\nIn Section \\ref{sec:4} we summarize our results and\npresent conclusions and we discuss \nthe possibility of treatment of experimental data \naccording to the method introduced in Section \\ref{sec:2}.\n\n\\section{Finite size diffusion coefficient}\n\\label{sec:2}\nBefore a general discussion let us start with a simple example.\nConsider the relative diffusion of a cloud of N test particles\nin a smooth, spatially periodic velocity field with characteristic\nlength $l_{u}$.  We assume that the Lagrangian motion is chaotic \ni.e. the maximum Lyapunov exponent $\\lambda$ is positive.\nDenoting with $R^{2}(t)$ the square of the typical radius of\nthe cloud\n\\begin{equation}\nR^{2}(t)= \n\\ll|{\\bf x}_{i}(t)-\\ll{\\bf x}_{i}(t)\\gg|^{2}\\gg\\,,\n\\label{def:disprel}\n\\end{equation}\nwhere\n\\begin{equation}\n\\ll{\\bf x}_i(t)\\gg={1 \\over N} \\sum_{i=1}^N {\\bf x}_i(t)\n\\end{equation}\nwe expect the following regimes to hold\n\\begin{equation}\n\\overline{R^{2}(t)} \\simeq \\left\\{ \n\\begin{array}{ll}\nR^{2}(0)\\exp(L(2)t) & \\;\\;\\;\\;\n{\\mbox {if    $\\overline{R^{2}(t)}^{1\/2} \\ll l_{u}$}}\n \\\\\n2 D t & \\;\\;\\;\\;\n{\\mbox {if    $\\overline{R^{2}(t)}^{1\/2} \\gg l_{u}$}}\n\\end{array}\n\\label{eq:regimiperR}\n\\right.\n\\,,\n\\label{example1} \n\\end{equation}\nwhere $L(2) \\geq 2\\lambda$ is the generalized Lyapunov exponent\n\\cite{BPPV85,PV87}, $D$ is the diffusion coefficient and \nthe overbar denotes the average over initial conditions.\n\nIn this paper we prefer to study the relative diffusion \n(\\ref{def:disprel}) instead of the usual absolute diffusion.\nFor spatially infinite cases, without mean drift\nthere is no difference;\nfor closed basins the relative dispersion is,\nfor many aspects, more interesting than the absolute one\nand, in addition, the latter is dominated by \nthe sweeping induced by large scale flow.\n\nFurthermore we underline \nthat although the dynamics of the ocean circulation is dominated\nby large mesoscale gyres, the smaller scales \nactivities within the gyres\ncontrol important local phenomena as deep water \nformation in North Atlantic and in Mediterranean \nbasin \\cite{marshal}.\nTherefore the study of relative diffusion could be \nrelevant to describe this small-scale motion\nand can give crucial informations on the way \nto parameterize the subgrid scales \nin ocean numerical global model \\cite{garret}. \n\nAnother, at first sight rather artificial, way to describe\nthe above behavior is by introducing the ``doubling \ntime\\\/'' $\\tau(\\delta)$ at scale $\\delta$ as follows:\nwe define a series of thresholds $\\delta^{(n)}= r^{n} \\delta^{(0)}$,\nwhere $\\delta^{(0)}$ is the initial size of the cloud, defined according\nto (\\ref{def:disprel}), and then we measure the time $T(\\delta^{(0)})$ \nit takes for the growth\nfrom $\\delta^{(0)}$ to $\\delta^{(1)}= r \\delta^{(0)}$, and so on\nfor $T(\\delta^{(1)})\\,,\\;T(\\delta^{(2)})\\,,\\ldots$\nup to the largest scale under consideration.\nFor the threshold rate $r$ any value can be chosen but too large ones\nmight not separate different scale contributions, \nthough strictly speaking the term ``doubling time''\nrefers to the threshold rate $r=2$.\n\nPerforming ${\\cal N} \\gg 1$ experiments with\ndifferent initial conditions for the cloud, we define the \ntypical doubling time $\\tau(\\delta)$ at scale \n$\\delta$ as\n\\begin{equation}\n\\tau(\\delta) = < T(\\delta) >_e =\\frac{1}{{\\cal N}}\n \\sum_{i=1}^{{\\cal N}} T_{i}(\\delta)\\,.\n\\label{def:taudelta}\n\\end{equation}\nLet us stress the fact that the average \nin (\\ref{def:taudelta}) is different from the usual\ntime average.\n\nFrom the average doubling time we can define the finite size \nLagrangian Lyapunov exponent as\n\\begin{equation}\n\\lambda(\\delta)=\\frac{\\ln r}{\\tau(\\delta)}\\,,\n\\end{equation}\nwhich is a measure of the average rate of separation of two\nparticles at a distance $\\delta$. Let us remark that $\\lambda(\\delta)$\nis independent of $r$, for $r \\rightarrow 1^{+}$. \nFor very small separations (i.e. $\\delta \\ll l_u$) one recovers the standard \nLagrangian Lyapunov exponent $\\lambda$,\n\\begin{equation}\n\\lambda=\\lim_{\\delta \\rightarrow 0} \\frac{1}{\\tau(\\delta)}\n\\ln r\\,.\n\\label{def:liapfromtau}\n\\end{equation}\nSee \\cite{ABCPV} for a detailed discussion about \nthese points.\nIn this framework the\nfinite size diffusion coefficient $D(\\delta)$ dimensionally turns out to be\n\\begin{equation}\nD(\\delta)=\\delta^{2}\\lambda(\\delta)\\,.\n\\label{def:fsd}\n\\end{equation}\nNote the absence of the factor $2$, as one can expect by\nthe definition (\\ref{def:eddydiff}), in the denominator of\n$D(\\delta)$ in equation (\\ref{def:fsd}); this is due \nto the fact that $\\tau(\\delta)$ is a difference of times.\nFor a standard diffusion process $D(\\delta)$ approaches the diffusion\ncoefficient $D$ (see eq. (\\ref{eq:regimiperR})) in the limit of \nvery large separations ($\\delta \\gg l_u$). This result stems from \nthe scaling of the doubling times $\\tau(\\delta) \\sim \\delta^2$ for \nnormal diffusion. \n\nThus the finite size Lagrangian Lyapunov exponent $\\lambda(\\delta)$, or\nits counterpart $D(\\delta)$, embody the asymptotic behaviors \n\\begin{equation}\n\\lambda(\\delta) \\sim \\left\\{ \n\\begin{array}{ll}\n\\lambda & \\;\\;\\;\\;\n{\\mbox {if    $\\delta \\ll l_{u}$}}\n \\\\\nD\/\\delta^{2} & \\;\\;\\;\\;\n{\\mbox {if    $\\delta \\gg l_{u}$}}\n\\end{array}\n\\right.\n\\,,\n\\label{eq:regimipertau} \n\\end{equation}\nOne could naively conclude, matching the behaviors \nat $\\delta \\sim l_{u}$, that $D \\sim \\lambda l_{u}^{2}$.\nThis is not always true, since one can have a rather large range\nfor the crossover due to the \nfact that nontrivial correlations can be present in \nthe Lagrangian dynamics \\cite{FV89}.\n\nAnother case where the \nbehavior of $\\tau(\\delta)$ as a function of $\\delta$ \nis essentially well understood\nis 3D fully developed turbulence. \nFor sake of simplicity we neglect intermittency \neffects. \nThere are then three different ranges:\n\\begin{enumerate}\n\\item \n$\\delta \\ll \\eta ={\\mbox {Kolmogorov length}}$ :\n$1\/\\tau(\\delta) \\sim \\lambda$;\n\\item\n$\\eta \\ll \\delta \\ll l={\\mbox{ typical size of the \nenergy containing eddies}}$: \nfrom the Richardson law \n$\\overline{R^{2}(t)} \\sim t^{3}$ \none has \n$1\/\\tau(\\delta) \\sim \\delta^{-2\/3}$;\n\\item\n$\\delta \\gg l$ : usual diffusion behavior \n$1\/\\tau(\\delta) \\sim \\delta^{-2}\\,.$\n\\end{enumerate}\n\nOne might wonder that the proposal to introduce\nthe time $\\tau(\\delta)$ is just another way \nto look at\n$\\overline{R^{2}(t)}$ as a function of $t$.\nThis is true only in limiting cases, when\nthe different characteristic lengths are \nwell separated and intermittency is weak.\nIn \\cite{previouswork1,previouswork2,sabot} rather \nclose techniques are used for the computation of\nthe diffusion coefficient in nontrivial cases.\n\nThe method of working at fixed scale $\\delta$,\nallows us to extract the physical information at that spatial\nscale avoiding unpleasant troubles of the method of\nworking at a fixed delay time $t$.\nFor instance, if one has a strong intermittency, and this is a rather\nusual situation, $R^{2}(t)$ as a function of \n$t$ can appear very different in each realization.\nTypically one can have, see figure \\ref{fig1}a,\ndifferent exponential rates of growth for different\nrealizations, producing a rather odd behavior\nof the average $\\overline{R^{2}(t)}$ without\nany physical meaning. For instance in figure \\ref{fig1}b we show\nthe average  $\\overline{R^{2}(t)}$ versus time $t$; at large times we\nrecover the diffusive behavior but at intermediate times \nappears an apparent anomalous regime which is only due to \nthe superposition of exponential and diffusive contributions\nby different samples at the same time.\nOn the other hand exploiting the tool of doubling times one has \nan unambiguous result (see figure \\ref{fig1}c).\n\nOf course the interesting situations are those where\nthe different characteristic lengths ($\\eta\\,,\\;l\\,,\\;L$) \nare not very different and therefore each \nscaling regime for $\\overline{R^2(t)}$ is not well evident.\n\n\\section{Numerical results}\n\\label{sec:3}\nHere we present some numerical experiments \nin simple models with\nLagrangian chaos in the zero molecular diffusion limit.\nBefore showing the results, we describe the numerical \nmethod adopted.\n\nWe choose a passive tracers trajectory having a chaotic behavior, \ni.e. with a positive maximum Lyapunov \nexponent, computed by using standard algorithms \\cite{BeneGalg}.\nThen we place $N-1$ passive tracers around the first one\nin a cloud of initial size \n\\begin{eqnarray}\nR(0)=\\delta(0)=\\delta^{(0)}\\,,\n\\nonumber\n\\end{eqnarray} \nwith $R(0)$ defined by equation (\\ref{def:disprel}).  \nIn order to have average properties we repeat this procedure \nreconstructing the passive cloud around the last\nposition reached by the reference chaotic tracer in the previous \nexpansion.\nThis ensures that the initial expansion of the cloud \nis exponential\nin time, with typical exponential rate equal to the\nLyapunov exponent.\n\nFurther we define a series of thresholds $\\delta^{(n)}=r^{n}\\delta^{(0)}$\n(as described in Section 2) \n$n=1,\\cdots,n_{max}$ and we measure the time $T_{n}$ \nspent in  expanding from $\\delta^{(n)}$ to $\\delta^{(n+1)}\\,$.\nThe value of $n_{max}$ has to be chosen in such a way that\n$\\delta^{(n_{max})}\\sim \\delta_{max}$, where $\\delta_{max}$\ncorresponds to the uniform distribution of the tracers in the basin\n (see forthcoming discussion and the Appendix). Each realization stops\nwhen $\\delta(t)=\\delta^{(n_{max})}$.\n\nTherefore following \\cite{ABCPV} we define a scale dependent\nLagrangian Lyapunov exponent as:\n\\begin{equation}\n\\lambda(\\delta^{(n)}) = \\frac{1}{<{T_{n}}>_e} \\ln r =\n\\frac{1}{\\tau(\\delta^{(n)})} \\, \\ln r.\n\\label{def:lambdadidelta}\n\\end{equation}\nIn equation (\\ref{def:lambdadidelta}) we have implicitly assumed that\nthe evolution of the size $\\delta(t)$ of the cloud is continuous in time.\nThis is not true in the case of discontinuous processes such as maps or\nin the analysis of experimental data taken at \nfixed delay times.\nDenoting $T_{n}$ the time to reach size \n$\\tilde{\\delta} \\geq \\delta^{(n+1)}$ from $\\delta^{(n)}$ ,\nnow $\\tilde{\\delta}$ is a fluctuating quantity,\n equation (\\ref{def:lambdadidelta}) has to be modified as follows \n\\cite{ABCPV}:\n\\begin{equation}\n\\lambda(\\delta^{(n)}) = \\frac{1}{<{T_{n}}>_e}   \n\\left< \\ln \\left( \\frac{\\tilde{\\delta}}{\\delta^{(n)}}\\right) \\right>_e\n\\,.\n\\label{def:lambdadidelta1}\n\\end{equation}\n\nIn our numerical experiments we have the regimes \ndescribed in sect. 2: exponential regime\n, i.e. $\\lambda(\\delta)=\\lambda$, and diffusion-like regime\ni.e. $\\lambda(\\delta)=D\/\\delta^{2}$, at least if the size $L$ \nof the basin is large enough.\n\nFor cloud sizes close to the saturation value $\\delta_{max}$\nwe expect the following behavior to hold for a broad class \nof systems:\n\\begin{equation}\n \\lambda(\\delta)=\\frac{D(\\delta)}{\\delta^{2}} \\propto\n\\frac{(\\delta_{max}-\\delta)}{\\delta} \\,.\n\\label{eq:nearbound}\n\\end{equation}\nThe constant of proportionality\nis given by  the second eigenvalue of the \nPerron-Frobenius operator which is related to the typical time \nof exponential relaxation of tracers' density to the uniform distribution\nActually, the analytical evaluation of this eigenvalue can be \nperformed only for extremely simple dynamical systems\n(for instance random walkers, as shown in the Appendix).\nAs a consequence the range of validity for (\\ref{eq:nearbound})\ncan be assessed only by numerical simulation.\n\n\\subsection{A model for transport in Rayleigh-B\\'enard convection}\nThe advection in two dimensional incompressible flows is described,\nin absence of molecular diffusion, by Hamiltonian equation of motion  \nwhere the Hamilton function is the stream function $\\psi$:\n\\begin{equation}\n\\frac{dx}{dt}=\\frac{\\partial \\psi}{\\partial y}\\,, \\;\\;\\;\n\\frac{dy}{dt}=-\\frac{\\partial \\psi}{\\partial x}\\,.\n\\label{eq:hamilton}\n\\end{equation}\nIf $\\psi$ is time-dependent the system (\\ref{eq:hamilton})\nis non-autonomous and in general non-integrable, then\nchaotic trajectories may exist. \n\nOne example is the model introduced in \\cite{gollub}\nto describe the chaotic advection\nin the time-periodic Rayleigh-B\\'enard convection.\nIt is defined by the stream function:\n\\begin{equation}\n\\psi(x,y,t)=\\frac{A}{k} \n\\sin\\left\\{ k \\left[ x+B \\sin(\\omega t)\\right]\\right\\}\nW(y)\\,,\n\\label{eq:gollubinf}\n\\end{equation}\nwhere $W(y)$ is a function that satisfies rigid \nboundary conditions on the surfaces $y=0$ and $y=a$ \n(we use $W(y)=\\sin(\\pi y\/a)$).\nThe direction $y$ is identified with the vertical direction\nand the two surfaces $y=a$ and $y=0$ are the top and bottom\nsurfaces of the convection cell.\nThe time dependent term $B\\sin(\\omega t)$ represents \nlateral oscillations of the roll pattern  \nwhich mimic the even oscillatory instability \\cite{gollub}.\n\nTrajectories starting near the roll\nseparatrices could have positive Lyapunov exponent and thus\ndisplay chaotic motion and diffusion in the x direction. \nIt is remarkable that in spite of the simplicity of the model,\nthe agreement of the numerical results with experimental ones is quite\ngood \\cite{gollub}.\n\nDefining a passive cloud in the $x$ direction (i.e. a\nsegment) and performing the expansion experiment described \nin the previous section\nwe have that, until $\\delta$ is below a fraction of the\ndimension of the cell, $\\lambda(\\delta)=\\lambda$ (figure \\ref{fig2}a).\nFor larger values of $\\delta$ we have \nthe standard diffusion $\\lambda(\\delta)=D\/\\delta^{2}$ \nwith good quantitative agreement with the value of the \ndiffusion coefficient evaluated by the standard technique, i.e.\nusing $\\overline{R^{2}(t)}$ as a function of time $t$\n(compare figure \\ref{fig2}a with figure \\ref{fig2}b).\n\nTo confine the motion of tracers in a closed domain,\ni.e. $x \\in [-L,L]$, we must slightly modify the streamfunction\n(\\ref{eq:gollubinf}). \nWe have modulated the oscillating term\nin such a way that for $|x|=L$ the amplitude of the oscillation\nis zero, i.e. $B \\rightarrow B \\sin(\\pi x\/L)$ with $L=2\\,\\pi n\/k$\n($n$ is the number of convective cells).\nIn this way\nthe motion is confined in $[-L,L]$.\n\nIn figure \\ref{fig3} we show $\\lambda(\\delta)$ for two values of $L$. \nIf $L$ is large enough one can well see the three regimes, \nthe exponential one, the diffusive one and the saturation given\nby equation (\\ref{eq:nearbound}).\nDecreasing $L$ the range\nof the diffusive regime decreases, and for small values of\n$L$ it disappears. \n\n\\subsection{Modified Standard Map}\nOne of the simplest deterministic dynamical system displaying both\nexponential growth of separation for close trajectories \nand asymptotic diffusive behavior \nis the standard (Chirikov - Taylor) mapping \\cite{Chi79}.\nIt is customarily defined as \n\\begin{equation}\n\\left\\{\n\\begin{array}{ll}\nx_{n+1}=x_n+K \\sin y_n & \\\\\n y_{n+1}=y_n+x_{n+1}& \\mbox{ mod $2 \\pi $}\n\\end{array}\n\\right.\n\\label{eq:standard}\n\\end{equation}\nThis mapping conserves the area in the phase space.\nIt is widely known that for large enough values of the \nnonlinearity strength parameter $K \\gg K_c \\simeq 1$ the motion\nis strongly chaotic in almost all the phase space.\nIn this case the standard map, in the $x$-direction\n mimics the behavior of a one-dimensional random walker,\nstill being deterministic, and so one expects the behavior of\n$\\lambda(\\delta)$ to be quite similar to the one already \nencountered in the model for Rayleigh-B\\'enard convection\nwithout boundaries. \nNumerical iteration of (\\ref{eq:standard}) for a cloud of particles\nclearly shows the two regimes described in \n(\\ref{eq:regimipertau}), similar to that showed for the \nmodel discussed in the previous section.\n\nWe turn now to the more interesting case in which the domain is limited\nby boundaries reflecting back the particle. \nTo achieve the confinement of the trajectory\ninside a bounded region we modify the standard map in the following\nway\n\\begin{equation}\n\\left\\{\n\\begin{array}{ll}\nx_{n+1}=x_n+K f(x_{n+1})\\sin y_n & \\\\ \ny_{n+1}=y_n+x_{n+1}-K f'(x_{n+1}) \\cos y_n & \\mbox{ mod $ 2 \\pi$}.\n\\end{array}\n\\right.\n\\label{eq:modified}\n\\end{equation}\nwhere $f(x)$ is a function which has its only zeros in $\\pm L$.  \nSince the mapping is defined in implicit form, \nthe shape of $f$ must be chosen in such a way to assure\na unique definition for $(x_{n+1},y_{n+1})$ given $(x_n,y_n)$. \nFor any $f$ fulfilling this request the mapping \n(\\ref{eq:modified}) conserves the area.\nA trial choice could be\n\\begin{equation}\nf(x)=\n\\left\\{\n\\begin{array}{ll}\n1 & |x|<\\ell \\\\\n\\begin{displaystyle}\n{L-|x| \\over L-\\ell}\n\\end{displaystyle}\n & \\ell<|x|<L\n\\end{array}\n\\right.\n\\label{eq:f}\n\\end{equation}\nStrictly speaking this is not quite an appropriate choice, since it renders\nthe map discontinuous at $|x| = \\ell$, but \nthis is an irrelevant point and it is easy to bypass this\nobstacle by assuming a suitably smoothed version of (\\ref{eq:f}).\n\nPerforming the doubling times computation (\\ref{def:taudelta})\none recovers both\nthe exponential and diffusive regimes for \n$\\lambda(\\delta)$, and in addition one has the saturation regime\n(\\ref{eq:nearbound}). Figure \\ref{fig4} shows the behavior of\nthe scale dependent diffusion coefficient $D(\\delta)$ (\\ref{def:fsd}).\nApproaching the saturation value $\\delta_{max}$ the diffusion \ncoefficient quickly drops to zero, following the asymptotic law \n(\\ref{eq:nearbound}) derived in the appendix.\nThe qualitative behaviors in figure \\ref{fig4} do not depend on the details \nof the function $f$. \n                   \n\\subsection{Point vortices in a Disk}\nAs another example, we consider the  two-dimensional  \ntime-dependent flow generated by the motion of $N$ point vortices \nin a closed domain \\cite{Aref}.\nFor a unitary disk the positions of the vortices $(x_i=r_i \\cos \\,\\theta_i\n,y_i=r_i \\sin \\,\\theta_i)$, with circulation\n$\\Gamma_i$, evolve according to the Hamiltonian dynamics \n\\begin{equation}\n\\dot{x_i} = {1 \\over \\Gamma_i} {\\partial H \\over \\partial y_i}, \\: \\: \\:\n\\dot{y_i} = - {1 \\over \\Gamma_i} {\\partial H \\over \\partial x_i}\n\\end{equation}\nwhere the Hamiltonian is\n\\begin{equation}\nH = - {1 \\over 4\\pi} \\sum_{i>j} \\Gamma_i \\Gamma_j \\log \\left[\n{r_i^2+r_j^2-2 r_i r_j \\cos (\\theta_i-\\theta_j) \\over 1 + r_i^2 r_j^2 -\n2 r_i r_j \\cos (\\theta_i-\\theta_j)} \\right] +\n{1 \\over 4 \\pi} \\sum_{i=1}^N \\Gamma_i^2 \\log (1-r_i^2)\n\\label{}\n\\end{equation}\n\nPassive tracers evolve according to (\\ref{eq:hamilton}) with $\\psi$ given\nby\n\\begin{equation}\n\\psi(x,y) = - {1 \\over 4\\pi} \\sum_{i}^{N} \\Gamma_i\n\\log \\left[{r^2+r_i^2-2 r r_i \\cos(\\theta-\\theta_i) \\over\n1 + r^2 r_i^2 - 2 r r_i \\cos(\\theta-\\theta_i)} \\right]\n\\label{}\n\\end{equation}\nwhere $(x=r \\cos \\,\\theta,y=r \\sin \\,\\theta)$ denote the tracer\nposition.\n\nFigure \\ref{fig5} shows the relative diffusion as a function of \ntime in a system with 4 vortices. \nApparently there is an intermediate regime of anomalous diffusion.\nOn the other hand from figure \\ref{fig6} one can see rather clearly \nthat, with the method of working at fixed scale, only\ntwo regimes survive: the exponential one and that one \ndue to the saturation. \nComparing figure \\ref{fig5} and figure \\ref{fig6} one understands that\nthe mechanism described in Section 2 \nhas to be held for responsible of this spurious anomalous diffusion.\nWe stress the fact that these misleading behaviors are \ndue to the superposition of different regimes and that \nthe method of working at fixed scale has the advantage \nto eliminate this trouble.\n\nThe absence of the diffusive range \n$\\lambda(\\delta) \\sim \\delta^{-2}$\nis due to the fact that the characteristic \nlength of the velocity field, which is comparable with \nthe typical distance between two close vortices, is not \nmuch smaller than the size of the basin.\n\n\\section{Conclusions}\n\\label{sec:4}\nIn this paper we investigated the relative dispersion of passive tracers\nin closed basins. Instead of the customary approach based on \nthe average size of the cloud of tracers as a function of time,\nwe introduced a typical inverse time $\\lambda(\\delta)$ which \ncharacterizes the diffusive process at fixed scale $\\delta$.\n\nFor very small values of $\\delta$, $\\lambda(\\delta)$ coincides with the\nmaximum Lagrangian Lyapunov exponent which is positive in\nthe case of chaotic Lagrangian motion.\nFor larger $\\delta$ the shape of $\\lambda(\\delta)$ \ndepends on the detailed mechanism of spreading which is given\nby the structure of the advecting flow, which is in turn conditioned \nby the presence of boundaries. In the case of diffusive regime, one\nexpects the scaling $\\lambda(\\delta) \\simeq \\delta^{-2}$, which leads to a\nnatural generalization of the diffusion coefficient as\n$D(\\delta)=\\lambda(\\delta) \\delta^2$. \n\nThe effectiveness of finite size quantities $\\lambda(\\delta)$ \nor $D(\\delta)$ in characterizing the dispersion properties of\na cloud of particles is demonstrated by several numerical examples.\n\nFurthermore, when $\\delta$ gets close to its saturation value\n(i.e. the characteristic size of the basin), a simple argument gives \nthe shape of $\\lambda(\\delta)$ which is expected to be universal\nwith respect to a wide class of dynamical systems.\n\nIn the limiting case when the characteristic length of\nthe Eulerian velocity $l_u$ and the size of the basin $L$ are\nwell separated, the customary approach and the proposed method\ngive the same information. \nIn presence of strongly intermittent Lagrangian motion, or when\n$l_u\/L$ is not much smaller than one, the traditional method\ncan give misleading results, for instance apparent anomalous\nscaling over a rather wide time interval, as demonstrated by\na simple example.\n\nWe want to stress out that our method is very \npowerful in separating the different scales acting on diffusion \nand consequently it could give improvement about the parameterization \nof small-scale motions of complex flows.\nThe proposed method could be also relevant in the analysis of\ndrifter experimental data or in numerical models for Lagrangian\ntransport, in particular for addressing the question about the\nexistence of low dimensional chaotic flows.\n\n\\section{Acknowledgments}\nWe thank E. Aurell and A. Crisanti for useful suggestions and \nfirst reading of the paper. G.B. and A.C. thank the Istituto di\nCosmogeofisica del CNR, Torino, for hospitality.\nThis work was partially supported by INFN {\\it Iniziativa specifica\nMeccanica Statistica FI11}, by CNR (Progetto speciale coordinato\n{\\it Variabilit\\`a e Predicibilit\\`a del Clima}) and by EC-Mast contract\nMAS3-CT95-0043 (CLIVAMP).\n\n\\section*{Appendix}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \nRecently, the HADES collaboration has measured $\\Lambda(1405)$ production in proton-proton reactions at a beam kinetic energy of 3.5~GeV \\cite{Agakishiev:2012xk}, where the $\\Lambda(1405)$ hyperon has been reconstructed in the charged $\\Sigma^{\\pm}\\pi^{\\mp}$ decay channels. By investigating the $\\Sigma\\pi$ invariant mass distributions, clear peak structures below 1400~MeV\/$c^2$ were observed. These structures were interpreted by a small contribution of $\\Sigma(1385)^0$ and a large contribution of a low mass $\\Lambda(1405)$ signal (see also \\cite{Agakishiev:2012ja}). Besides this, the spectra also showed a considerable contribution by $\\Lambda(1520)$ production and by non$\\--$resonant $\\Sigma\\pi$ production, resulting in a phase space like background below the $\\Lambda(1405)$. The experimental data were finally described by an incoherent sum of Monte Carlo simulations, where the $\\Lambda(1405)$ was simulated to follow a Breit-Wigner type distribution with a Breit-Wigner mass of 1385~MeV\/$c^2$ and a width of 50~MeV\/$c^2$. With help of these simulations the experimental data were corrected for the effects of acceptance and efficiency. These corrected data allow to compare any (more advanced) model to the obtained $\\Sigma\\pi$ invariant mass distributions. \\\\\nThe experimental data, where the maximum of the $\\Sigma\\pi$ missing mass (see Fig.~\\ref{fig:LA1405_HADES_Thomas_Comp}) lies below the nominal value of $1405\\,\\mathrm{MeV\/}c^2$ associated to the $\\Lambda(1405)$, suggest a shift of this resonance towards lower masses.\nIn this paper we want to address different possible explanations for the observed mass shift of the $\\Lambda(1405)$ in the new HADES data and aim to stimulate theorists to further investigate the production of this resonance in $p+p$ reactions.\nThe $\\Lambda(1405)$ spectral function measured in $p+p$ collisions by the HADES collaboration differs from the predictions by theoretical models and also from some of the experimental observations. These differences reside in the complex nature of the resonance.\n\n From the theoretical point of view, the $\\Lambda(1405)$ is normally treated in a unitarized coupled channel framework based on chiral SU(3) dynamics \\cite{Hyodo:2007jq,Borasoy:2006sr,Kaiser:1995eg,Oset:1997it}, where this resonance is generated dynamically as a coherent sum of two different poles. The first pole, $z_1$, is located at higher energies of around 1420~MeV and is mainly associated with a narrow quasi-bound $\\bar{K}N$ state. The second pole, $z_2$, is found at lower energies of about 1390~MeV and this pole strongly couples to a broad $\\Sigma\\pi$ resonance. As the relative contribution of these two states depends on the entrance channel, also the observed properties of the $\\Lambda(1405)$ could differ for different reactions. Therefore, in order to understand the complex formation process of the $\\Lambda(1405)$, it is important that experiments measure this resonance in different collision systems, and that, at the same time, theory provides appropriate models for each of those systems. \n \nFirst we refer to the measurement presented by the ANKE collaboration in \\cite{Zychor:2007gf}, where the $\\Lambda(1405)$ spectral shape is reconstructed in $p+p$ collisions at 2.83~GeV beam kinetic energy out of the decay into $\\Sigma^0$ and $\\pi^0$ pairs.\nThese data are within the systematic errors and the statistical significance of the mass bins around 1400~MeV\/$c^2$ consistent with the HADES results.\\\\\nFrom the theory side, the authors of \\cite{Geng:2007vm} followed a unitarized coupled channel approach based on the chiral Lagrangian in order to \npredict the $\\Lambda(1405)$ line shape in $p+p$ reactions for the $\\Sigma^0\\pi^0$ decay channel. In their Ansatz the $\\Lambda(1405)$ was \ngenerated from pion, kaon and $\\rho$ meson exchange mechanisms, all of them leading to a different coupling to the two $\\Lambda(1405)$ poles. The \ncoherent sum of all contributions results in a $\\Lambda(1405)$ line shape with a maximum in the $\\Sigma\\pi$ mass distribution at around \n$1410\\,\\mathrm{MeV\/}c^2$. With this approach the authors of \\cite{Geng:2007vm} delivered a result compatible with the $\\Lambda(1405)$ signal, measured by the ANKE collaboration in the $\\Sigma^0\\pi^0$ decay channel \\cite{Zychor:2007gf}. \nThis calculation is the only one available for $p+p$ reactions but since the HADES data refer to the charged decay channels\na quantitative comparison results difficult. Nevertheless, the results by \\cite{Geng:2007vm} have been used in this work as a starting point to model the $\\Lambda(1405)$ as the combination of two Breit-Wigner functions.\\\\\nIn general, the HADES data show a larger contribution by the non$\\--$resonant $\\Sigma\\pi$ production in comparison with ANKE. In particular, it has \nbeen shown in \\cite{Agakishiev:2012qx} that the $\\Delta^{++}$ is strongly coupling to the $\\Sigma^- \\pi^+ p K^+$ final state via the reaction $p+p\n\\rightarrow \\Sigma^- + \\Delta^{++} + K^+$ and it cannot be excluded that $N^*\/\\Delta^0$ resonances contribute to the $\\Sigma^+\\pi^-pK^+$ channel\n as well. These contributions might appear in the $I=0$ channels and hence interfere with the resonant amplitude.\n\nAside from the predictions by models that consider the $\\Lambda(1405)$ as the combination of two main poles, one has to consider those models where \na single pole is associated to the formation of the resonance. \nIn \\cite{Hassanvand:2012dn} a phenomenological  ${\\mathrm{\\bar{K}}p}$ interaction was employed to derive the mass distribution of $\n\\Lambda(1405)$.\nThis approach was used to fit the HADES data. For this purpose all simulated contributions, mentioned in\n\\cite{Agakishiev:2012xk}, were subtracted so that only the pure $\\Lambda(1405)$ signal was left. The fit within this phenomenological approach\n results into a good description of the experimental data and allowed to extract the mass and width of the $\\Lambda(1405)$ to\n$1405^{+11} _{-9}\\mathrm{ MeV\/c^2}$ and $62\\pm 10 \\mathrm{MeV\/}c^2$, which is in good agreement with the PDG values \\cite{PDG} and\ndoes not imply any evident shift of the spectral function.\n\nIn the sector of $\\gamma$-induced reactions, the CLAS collaboration has recently published new high quality data on the $\\Lambda(1405)$ \\cite{Moriya:2013eb,Moriya:2013hwg,Schumacher:2013vma}. In these reports, all three $\\Sigma\\pi$ decay channels have been investigated simultaneously for different incident photon energies. The observed $\\Lambda(1405)$ spectral shape partially appears at higher masses, clearly above 1400 MeV\/$c^2$. \n\\begin{figure}[tbp]\n\t\\centering\n\t\t\\includegraphics[width=0.45\\textwidth]{LA1405_HADES_Thomas_Comp}\n\t\\caption{Comparison between the HADES data and the data of \\cite{Thomas:1973uh} and \\cite{Engler:1965zz}, measured in $\\pi^-+p$ reactions. All three spectra show the sum of the $\\Sigma^+\\pi^-$ and $\\Sigma^-\\pi^+$ data samples.}\n\t\\label{fig:LA1405_HADES_Thomas_Comp}\n\\end{figure}\nThe structures measured by CLAS show a dependency on the photon incident energy and also differ among the three $\\Sigma\\pi$ decay channels. \nIndeed, one has to consider that for photon-induced reactions the interference between the $I=0$ and $I=1$  channels is not negligible as for proton- and pion-induced reactions. \nRecent theoretical works \\cite{PhysRevC.87.055201,Roca:2013cca} employ parameters fitted to the CLAS experimental data which \nallow for a small SU(3) breaking. This study shows that more precise calculations including higher order corrections could be needed in this sector \nand also suggests the possible existence of a $I=1$  bound state in the vicinity of the  \n$\\bar{\\mathrm{K}N}$ threshold. Additionally, the CLAS collaboration reported recently on the first observation of the $\\Lambda(1405)$ in electron-\ninduced reactions \\cite{Lu:2013nza} showing very different features respect to the photon-induced results. \n\nIn the sector of pion-induced reactions, Thomas et al. \\cite{Thomas:1973uh} and Engler et al. \\cite{Engler:1965zz} have measured $\\pi^-+p$ \ncollisions at a beam momentum of 1.69~GeV\/$c$ and have reconstructed the $\\Lambda(1405)$ from its decay into $\\Sigma^{\\pm}\\pi^{\\mp}$. The \nresults for the $\\Sigma^+\\pi^-+\\Sigma^-\\pi^+$ invariant mass spectra are shown in the black and open data points in Fig.~\n\\ref{fig:LA1405_HADES_Thomas_Comp}, respectively.\nAccording to \\cite{Thomas:1973uh}, the spectra consist of several contributions, namely 46\\% $\\Lambda(1405)$, 8\\% $\\Sigma(1385)^0$, 3\\% $\\Lambda(1520)$ and 43\\% non$\\--$resonant $\\Sigma\\pi$ production. The broad peak structure around 1400~MeV\/$c^2$ is mainly identified with the $\\Lambda(1405)$ signal. \nThis experimental spectrum, however, is not fully understood from the theoretical side, which expects a large contribution from the first $\\Lambda(1405)$ pole, shifting the expected $\\Lambda(1405)$ distribution to higher mass values \\cite{Hyodo:2003jw}. Therefore, the question arises if, in case of $\\pi^-$-induced reactions, the coupling to the second, broad pole at $\\approx 1390$~MeV is underestimated by theory.\\\\\nIt is interesting to compare the results from $\\pi^-$-induced reactions to the new HADES data. The gray histogram in Fig.~\\ref{fig:LA1405_HADES_Thomas_Comp} shows the summed invariant mass spectrum of $\\Sigma^+\\pi^-+\\Sigma^-\\pi^+$ of \\cite{Agakishiev:2012xk}. As the relative contributions from $\\Lambda(1405)$, $\\Sigma(1385)^0$, $\\Lambda(1520)$ and non$\\--$resonant $\\Sigma\\pi$ channels in these data are quite similar to the ones in the considered $\\pi^-+p$ reactions, a comparison between the data sets is justified. In order to allow such a comparison, the data from \\cite{Thomas:1973uh} and \\cite{Engler:1965zz} have been scaled appropriately. The agreement between the three spectra in the region around 1400~MeV\/$c^2$ is excellent, indicating that all measurements observe a similar low mass $\\Lambda(1405)$ signal. Furthermore, this suggests that in both, $p+p$ and $\\pi^-+p$ reactions, the broad $\\Sigma\\pi$ pole might be dominant in the coupling to the $\\Lambda(1405)$ state. \n\\noindent However, since the measured $\\Sigma\\pi$ final state does not contain the pure signature of the $\\Lambda(1405)$, but also contains a \nconsiderable contribution of non$\\--$resonant background, the interpretation of this result is not straight forward.\n \\section{Influence of interference effects}\n\\begin{figure}[tbp]\n\t\\centering\n\t\t\\includegraphics[width=0.45\\textwidth]{OsetFitted}\n\t\\caption{The $\\Lambda(1405)$ spectral shape calculated in \\cite{Geng:2007vm} (gray histogram). The spectrum is fitted with Eq.~(\\ref{equ:DoubleBW}). The two phase space modified Breit-Wigner functions $C_{\\mathrm{p.s.}}(m)\\left|BW_1(m)\\right|^2q_{\\mathrm{c.m.}}$ and $C_{\\mathrm{p.s.}}(m)\\left|BW_2(m)\\right|^2q_{\\mathrm{c.m.}}$ are shown in the dotted and dashed lines.}\n\t\\label{fig:OsetFitted}\n\\end{figure}  \nInterference between the resonant and the non$\\--$resonant amplitudes can affect the observed mass distribution considerably. In order to evaluate this scenario, we take the result of the chiral Ansatz \\cite{Geng:2007vm} as a starting point to develop a simple model, which we use to parametrize the $\\Lambda(1405)$ amplitude and to interpret the HADES data. The predicted $\\Sigma\\pi$ spectrum of \\cite{Geng:2007vm} is shown in the gray band of Fig.~\\ref{fig:OsetFitted}, with a peak at 1410~MeV and with the typical $\\Lambda(1405)$ shape, having a sharp drop to the $\\bar{K}N$ threshold. The idea is to reconstruct  this spectrum by a coherent sum of two Breit-Wigner functions (BW), where each of these BW amplitudes represents one of the two $\\Lambda(1405)$ poles. This approach is similar to what has been proposed in \\cite{Jido:2003cb} to represent via Breit-Wigner distributions the results of the unitarized coupled channel calculations. It is clear that any variation of the parameters associated to the two poles does not fulfill unitarity anymore and hence our procedure is not equivalent to a full-fledged calculation. Still it is interesting to see how much the parameters have to be modified to fit the experimental data.\n Within this approach the  total $\\Lambda(1405)$ amplitude reads then as follows:\n\\begin{eqnarray}\n\\frac{d \\sigma}{d m}&=&\\left|T_{\\Lambda(1405)}\\right|^2= \\nonumber \\\\\n&=&C_{\\mathrm{p.s.}}(m)\\left|BW_1(m)e^{i\\varphi_1}+BW_2(m) \\right|^2q_{\\mathrm{c.m.}} \\label{equ:DoubleBW} \\\\\n&&\\mbox{with } BW_i=A_i\\frac{1}{\\left(m-m_{0,i}\\right)^2+im_{0,i}\\Gamma_{0,i}} \\nonumber\n\\end{eqnarray}\n$C_{\\mathrm{p.s.}}(m)$ is a dimensionless weight function, normalized to unity in the mass range of 1280-1730~MeV\/$c^2$. This function considers the limited production phase space of the $\\Lambda(1405)$ in $p+p$ reactions. $q_{\\mathrm{c.m.}}$ is the decay momentum of $\\Sigma$ and $\\pi$ in the $\\Lambda(1405)$ rest frame (in units of [MeV\/c]). The Breit-Wigner function is a simple relativistic parametrization with amplitude $A_i$ in units of $\\left[\\sqrt{\\mu b\/\\mbox{c}}\\cdot \\mbox{MeV\/c}^2\\right]$, mass $m_{0,i}$ and width $\\Gamma_{0,i}$ in units of [MeV\/$c^2$]. Thus, the whole expression has dimensions of $\\left[\\frac{\\mu b}{\\mbox{MeV\/c}^2}\\right]$. We also introduce a free phase $e^{i\\varphi_1}$, which determines the interference between the two Breit-Wigner functions. Furthermore, we make use of the recent coupled channel calculations by Ikeda et al. \\cite{Ikeda:2012au}, which are constrained by the new SIDDARTHA data on kaonic hydrogen \\cite{Bazzi:2011zj}. In this work the $\\mathrm{\\bar{K}N}$ pole was found at $z_1=1424^{+7}_{-23}+i26^{+3}_{-14}$~MeV, while the $\\Sigma\\pi$ pole was found at $z_2=1381^{+18}_{-6}+i81^{+19}_{-8}$~MeV. With these values we constrain the Breit-Wigner mass $m_ic^2=Re(z_i)$ and the Breit-Wigner width $\\Gamma_{0,i}c^2=2Im(z_i)$ of our model to vary only within the given ranges.\\\\\nAlthough the parametrization of Eq.~(\\ref{equ:DoubleBW}) is simplified compared to the advanced calculations in \\cite{Geng:2007vm}, it still allows us to reconstruct the spectral shape in Fig.~\\ref{fig:OsetFitted} (gray band).\nBy fitting the Eq.~(\\ref{equ:DoubleBW}) to the theoretical prediction,  we obtain the black distribution with the fit parameters listed in Table~\\ref{tab:tableInter}.\n\\begin{table}[tbp]\n\\caption{\\label{tab:tableInter}%\nTable with fit parameters obtained by fitting Eq.~(\\ref{equ:DoubleBW}) to the theoretical prediction of \\cite{Geng:2007vm} shown in Fig.~\\ref{fig:OsetFitted}.}\n\\begin{ruledtabular}\n\\begin{tabular}{cccccc}\n\\textrm{$m_1$}&\n\\textrm{$\\Gamma_1$}&\n\\textrm{$m_2$}&\n\\textrm{$\\Gamma_2$}&\n\\textrm{$A_1\/A_2$}&\n\\textrm{$\\varphi_1$}\\\\\n\\colrule\n1426 & 28 & 1375 & 147 & 0.23 & 205$^{\\circ}$  \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table} \nThis distribution is consistent with the gray band in Fig.~\\ref{fig:OsetFitted} . Especially the peak structure around 1410~MeV\/$c^2$ and the drop to the $\\bar{K}N$ threshold is reproduced correctly. Additionally included in the figure are the absolute contributions of the two Breit-Wigner functions $C_{p.s.}(m)\\left|BW_1(m)\\right|^2q_{c.m.}$ and $C_{p.s.}(m)\\left|BW_2(m)\\right|^2q_{c.m.}$ (dotted and dashed lines).\\\\ \nIn this way we have fixed the parametrization of the $\\Lambda(1405)$ and can now study the maximal interference effects with the non$\\--$resonant background. \nFor this purpose, we fit the HADES results for the $\\Sigma^+\\pi^-$ and $\\Sigma^-\\pi^+$ invariant mass distributions simultaneously with the following two functions:\n\\begin{widetext}\n\\begin{eqnarray}\n\\left(\\frac{d\\sigma}{dm}\\right)_{\\Sigma^+\\pi^-}=\\left|A_{\\Lambda(1405)}T_{\\Lambda(1405)}+e^{i\\alpha}A_{\\Sigma^+\\pi^-}T_{\\Sigma^+\\pi^-}\\right|^2+\\left|BW_{\\Sigma(1385)^0}\\right|^2 + \\left|BW_{\\Lambda(1520)}\\right|^2 \\label{equ:DistSpPm} \\\\\n\\left(\\frac{d\\sigma}{dm}\\right)_{\\Sigma^-\\pi^+}=\\left|A_{\\Lambda(1405)}T_{\\Lambda(1405)}+e^{i\\beta}A_{\\Sigma^-\\pi^+}T_{\\Sigma^-\\pi^+}\\right|^2+\\left|BW_{\\Sigma(1385)^0}\\right|^2 + \\left|BW_{\\Lambda(1520)}\\right|^2 \\label{equ:DistSmPp}\n\\end{eqnarray} \n\\end{widetext}\nThe contributions from $\\Sigma(1385)^0$ and $\\Lambda(1520)$ are parameterized as Breit-Wigner functions so that they match the extracted shapes and yields reported in \\cite{Agakishiev:2012xk}. We assume here that they do not interfere with the other contributions to the $\\Sigma\\pi$ invariant mass spectra and thus add them incoherently in the Eq.s~(\\ref{equ:DistSpPm}) and (\\ref{equ:DistSmPp}). \\\\\nThe $\\Lambda(1405)$ is parameterized as described above. $A_{\\Lambda(1405)}$ is a free fit parameter which determines the absolute yield of $\\Lambda(1405)$.\\\\      \nThe non$\\--$resonant background shapes for the $\\Sigma^+\\pi^-$ and $\\Sigma^-\\pi^+$ channels ($T_{\\Sigma^+\\pi^-}$ and $T_{\\Sigma^-\\pi^+}$) are described by modified polynomials of 4th order, which read as follows: \n\\begin{eqnarray}\nT_{\\Sigma\\pi}(m)=\\left[C_{p.s.}(m)q_{c.m.}\\sum^{4}_{n=0}a_nm^n\\right]^{\\frac{1}{2}} \\label{equ:NonRes}\n\\end{eqnarray}\nThis parametrization has no physical meaning but it was chosen such to describes the simulated $\\Sigma\\pi$ invariant mass distributions in \\cite{Agakishiev:2012xk}. The parameters $a_n$ are given in units of $\\left[\\left(\\mbox{MeV\/c}^2\\right)^{-n}\\right]$ and their values are listed in Table~\\ref{tab:table1}.\n\\begin{table}[tbp]\n\\caption{\\label{tab:table1}%\nTable with coefficients for the description of the non$\\--$resonant background according to Eq.~(\\ref{equ:NonRes}).}\n\\begin{ruledtabular}\n\\begin{tabular}{cccc}\n\\textrm{$T_{\\Sigma\\pi}$}&\n\\textrm{$a_0$}&\n\\textrm{$a_1$}&\n\\textrm{$a_2$}\\\\\n\\colrule\n$T_{\\Sigma^+\\pi^-}$ & $7.949\\cdot10^{-3}$ & $-4.412\\cdot10^{-7}$ & $-1.558\\cdot10^{-9}$    \\\\\n$T_{\\Sigma^-\\pi^+}$ & $-2.387\\cdot10^{0}$ & $2.512\\cdot10^{-3}$ & $1.315\\cdot10^{-6}$   \\\\\n\\colrule\\\\\n\\textrm{$T_{\\Sigma\\pi}$}&\n\\textrm{$a_3$}&\n\\textrm{$a_4$}\\\\\n\\colrule\n$T_{\\Sigma^+\\pi^-}$ & $2.942\\cdot10^{-12}$ & $-1.121\\cdot10^{-15}$ \\\\\n$T_{\\Sigma^-\\pi^+}$ & $-2.254\\cdot10^{-9}$ & $6.481\\cdot10^{-13}$ \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\nThe modified polynomials are multiplied by constant factors $A_{\\Sigma^+\\pi^-}$ and $A_{\\Sigma^-\\pi^+}$ (see~Eq. (\\ref{equ:DistSpPm}) and (\\ref{equ:DistSmPp})), which have dimensions of $\\left[\\frac{\\sqrt{\\mu b\/\\mbox{c}}}{\\mbox{MeV\/c}^2}\\right]$. These are again free fit parameters, determining the absolute yields of the non$\\--$resonant channels, where a value of $A_{\\Sigma\\pi}=1$ corresponds to the yield extracted in \\cite{Agakishiev:2012xk}. Furthermore, complex phases $e^{i\\alpha}$ and $e^{i\\beta}$ have been included so that the modeled background can interfere with the $\\Lambda(1405)$ amplitude. The values of these phases are determined by the fitting procedure as well. Hence, the simultaneous fit of the two functions (\\ref{equ:DistSpPm}) and (\\ref{equ:DistSmPp}) to the experimentally determined $\\Sigma^+\\pi^-$ and $\\Sigma^-\\pi^+$ invariant mass distributions is characterized by five free parameters.\\\\\nWe consider here a scenario of maximal interference, which means that the whole non$\\--$resonant background interferes with the $\\Lambda(1405)$ amplitude.\n The best results of the fitting procedure (gray lines) are illustrated together with the experimental data in Fig.~\\ref{fig:Interferences} panel a) and b). The black lines show the amplitude of the $\\Lambda(1405)$, the red lines the contribution by the non$\\--$resonant $\\Sigma\\pi$ channels and the gray lines correspond to the coherent sum of all contributions. The fit parameters are listed in Table~\\ref{tab:table2}.  \n\\begin{table}[tbp]\n\\caption{\\label{tab:table2}%\nObtained fit parameters after fitting Eq.s~(\\ref{equ:DistSpPm}) and (\\ref{equ:DistSmPp}) to the experimental data points in Fig.~\\ref{fig:Interferences}.}\n\\begin{ruledtabular}\n\\begin{tabular}{ccccc}\n\\textrm{$A_{\\Lambda(1405)}$}&\n\\textrm{$A_{\\Sigma^+\\pi^-}$}&\n\\textrm{$A_{\\Sigma^-\\pi^+}$}&\n\\textrm{$\\alpha$}&\n\\textrm{$\\beta$}\\\\\n\\colrule\n$1.06$ & $0.93$ & $1.04$ & $67^{\\circ}$ & $109^{\\circ}$  \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\\begin{figure}[tbp]\n\t\\centering\n\t\t\\includegraphics[width=0.46\\textwidth]{Interferences_1}\n\t\\caption{ Missing mass spectrum to proton and $K^+$. The black data points are the measurements of \\cite{Agakishiev:2012xk}, for the $\\Lambda(1405)$ in the $\\Sigma^+\\pi^-$ (a) and $\\Sigma^-\\pi^+$ (b) decay channel. Panel c) shows the summed spectrum of a) and b). The black lines in a) and b) are the results from the simultaneous fit with Eq.~(\\ref{equ:DistSpPm}) and (\\ref{equ:DistSmPp}), the gray line represents the sum of all fitted functions. In c) the fit functions corresponding to the  $\\Sigma^+\\pi^-$ and $\\Sigma^-\\pi^+$ channels respectively are shown in gray and the sum of both functions is shown in the black line.}\n\t\\label{fig:Interferences}\n\\end{figure}\nA reasonable description of the data is achieved, expressed in a normalized $\\chi^2$ value of 1.23. In panel c) of Fig.~\\ref{fig:Interferences} we show the sum of the two invariant mass distributions and compare it to the sum of our model (black line). \\\\\nThe main message of the three pictures is that, assuming a maximal interference between the $\\Lambda(1405)$ and the non$\\--$resonant contributions and the large phases given in Table~\\ref{tab:table2}, one obtains a shift of the maximum to lower masses. This is the case although the mass maximum of the initial $\\Lambda(1405)$ amplitude is located around 1400~MeV\/$c^2$. By integrating $\\left|A_{\\Lambda(1405)}T_{\\Lambda(1405)}\\right|^2$ (black lines in panel a) and b)), the total cross section of the reaction $p+p\\rightarrow\\Lambda(1405)+p+K^+$ is determined to 3.3 $\\mu b$. This is considerably smaller than the value extracted in \\cite{Agakishiev:2012xk}, where an incoherent approach was used with a low mass $\\Lambda(1405)$ signal.\\\\    \nAt this point one has to emphasize some details of the model used in this work. First, the parametrization of the $\\Lambda(1405)$ as a simple sum of two Breit-Wigner amplitudes is not equivalent to the full coupled channel calculation of \\cite{Geng:2007vm}. A second point is the description of the non$\\--$resonant background $T_{\\Sigma\\pi}$ as polynomial functions. This is certainly a simplification. Also the assumption that the non$\\--$resonant channels just have a constant complex phase is oversimplified.\nFurthermore, we assume that the whole $\\Sigma\\pi$ non$\\--$resonant background appears in the s-wave channel with I=0, testing here the maximal possible interference.\nHowever, this maximal interference scenario turns out to be unlikely since no comparable mass shifts have been observed in the spectral shape of the $\\Sigma(1385)^+$ in the $\\mathrm{\\Lambda-p}$ final state \\cite{Agakishiev:2011qw} or for the $\\Lambda(1520)$ in the $\\Sigma\\pi$ decay channel. Moreover, it seems to be rather peculiar that interferences between the $\\Lambda(1405)$ and the non$\\--$resonant background should result in the same mass shift for both, the $\\Sigma^+\\pi^-$ and the $\\Sigma^-\\pi^+$ invariant mass distributions, where in both cases the physical origin of the non$\\--$resonant background is quite different. According to \\cite{Agakishiev:2012xk}, the $\\Sigma^-\\pi^+$ non$\\--$resonant background arises from a strong contribution of a $\\Delta^{++}$, whereas other mechanisms, e.g. $N^*\/\\Delta^0$ production via the reaction $p+p\\rightarrow\\Sigma^+ + N^*\/\\Delta^0+K^+\\rightarrow\\Sigma^++(\\pi^-+p)+K^+$, could contribute to the $\\Sigma^+\\pi^-$ non$\\--$resonant spectrum.\\\\\nOn the other hand, the presented results shall just emphasize that interference effects can play a significant role. Indeed, our model, even though it is not a full-fledged theoretical approach, has shown that interference effects can significantly shift the observed mass peak in the experimental spectra. We cannot prove that these effects are indeed responsible for the observed low mass $\\Lambda(1405)$ signal. We just aim to illustrate the importance of a serious treatment of the non$\\--$resonant background in any theoretical approach.\n\\section{Contributions of the two poles}\nHaving illustrated the maximal possible influence of interference effects between the $\\Lambda(1405)$ resonance and the non$\\--$resonant background, we now consider the second extreme case, where this particular interference term is neglected. The observed low mass peaks in the HADES data are then completely attributed to the \"pure'' $\\Lambda(1405)$ signal. With this assumption one can try to determine the parameters and the relative contribution of the two $\\Lambda(1405)$ poles. \nAs a starting point, we again parameterize the $\\Lambda(1405)$ as a coherent sum of the two Breit-Wigner amplitudes (see Eq.~(\\ref{equ:DoubleBW})). This time, however, $m_{0,1}$, $\\Gamma_{0,1}$, $m_{0,2}$ and $\\Gamma_{0,2}$ as well as $A_1$ and $A_2$ shall be determined directly from the HADES data. As before, the real and imaginary part of both poles are constrained by the results of \\cite{Ikeda:2012au} to $z_1=1424^{+7}_{-23}+i26^{+3}_{-14}$~MeV and $z_2=1381^{+18}_{-6}+i81^{+19}_{-8}$~MeV. Now the two functions~(\\ref{equ:DistSpPm2}) and (\\ref{equ:DistSmPp2}) are used to described the experimental data points of Fig.~\\ref{fig:Interferences} a) and b). \n\\begin{widetext}\n\\begin{eqnarray}\n\\left(\\frac{d\\sigma}{dm}\\right)_{\\Sigma^+\\pi^-}=C_{p.s.}(m)\\left|BW_1(m)e^{i\\varphi_1}+BW_2(m) \\right|^2q_{c.m.}+\\left|A_{\\Sigma^+\\pi^-}T_{\\Sigma^+\\pi^-}\\right|^2+\\left|BW_{\\Sigma(1385)^0}\\right|^2 + \\left|BW_{\\Lambda(1520)}\\right|^2 \\label{equ:DistSpPm2} \\\\\n\\left(\\frac{d\\sigma}{dm}\\right)_{\\Sigma^-\\pi^+}=C_{p.s.}(m)\\left|BW_1(m)e^{i\\varphi_1}+BW_2(m) \\right|^2q_{c.m.}+\\left|A_{\\Sigma^-\\pi^+}T_{\\Sigma^-\\pi^+}\\right|^2+\\left|BW_{\\Sigma(1385)^0}\\right|^2 + \\left|BW_{\\Lambda(1520)}\\right|^2 \\label{equ:DistSmPp2}\n\\end{eqnarray} \n\\end{widetext}\nIn these Eq.s all individual contributions besides the two $\\Lambda(1405)$ amplitudes sum up incoherently. The non$\\--$resonant background is fixed in yield to the HADES results \\cite{Agakishiev:2012xk}, by setting $A_{\\Sigma^+\\pi^-}$ and $A_{\\Sigma^-\\pi^+}$ to 1. In this way only $m_{0,1}$, $\\Gamma_{0,1}$, $m_{0,2}$, $\\Gamma_{0,2}$, $A_1$, $A_2$ and $\\varphi_1$ are the free fit parameters in Eq.~(\\ref{equ:DistSpPm2}) and (\\ref{equ:DistSmPp2}). The results for the best fit ($\\chi^2\/ndf=1.04$) are shown in Fig.~\\ref{fig:Fit_Weise} and the obtained fit parameters are listed in Table~\\ref{tab:table3}.     \n\\begin{figure}[tbp]\n\t\\centering\n\t\t\\includegraphics[width=0.46\\textwidth]{Fit_Weise_1}\n\t\\caption{(Color online) Same as Fig.~\\ref{fig:Interferences} but now the data are fitted with the Eq.s~(\\ref{equ:DistSpPm2}) and (\\ref{equ:DistSmPp2}). See text for details.}\n\t\\label{fig:Fit_Weise}\n\\end{figure}\n\\begin{table}[tbp]\n\\caption{\\label{tab:table3}%\nObtained free fit parameters after fit with Eq.s~(\\ref{equ:DistSpPm2}) and (\\ref{equ:DistSmPp2}).}\n\\begin{ruledtabular}\n\\begin{tabular}{cccccc}\n\\textrm{$m_{0,1}$}&\n\\textrm{$\\Gamma_{0,1}$}&\n\\textrm{$m_{0,2}$}&\n\\textrm{$\\Gamma_{0,2}$}&\n\\textrm{$A_1\/A_2$}&\n\\textrm{$\\varphi_1$}\\\\\n\\colrule\n$1418$ & $58$ & $1375$ & $146$ & $0.395$ & $178^{\\circ}$ \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\nA very good description of the data is achieved with the maximum in the $\\Lambda(1405)$ distribution now appearing at around $1385$~MeV\/$c^2$. Integrating this signal results in a total cross section of $\\sigma=9.0$~$\\mu b$ for the reaction $p+p\\rightarrow\\Lambda(1405)+p+K^+$, which is in good agreement with the result quoted in \\cite{Agakishiev:2012xk}. The composition of the $\\Lambda(1405)$ signal itself is illustrated in Fig.~\\ref{fig:L1405_Weise}. The resulting $\\Lambda(1405)$ amplitude differs strongly from the results reported in \\cite{Geng:2007vm} and shown in Fig. \\ref{fig:OsetFitted}. \n\\begin{figure}[tbp]\n\t\\centering\n\t\t\\includegraphics[width=0.45\\textwidth]{L1405_Weise}\n\t\\caption{$\\Lambda(1405)$ signal (black line) obtained by fitting Eq.~(\\ref{equ:DistSpPm2}) and (\\ref{equ:DistSmPp2}) to the experimental data in Fig.~\\ref{fig:Fit_Weise}. The dotted and dashed lines show the contributions of $C_{p.s.}(m)\\left|BW_1(m)\\right|^2q_{c.m.}$ and $C_{p.s.}(m)\\left|BW_2(m)\\right|^2q_{c.m.}$.}\n\t\\label{fig:L1405_Weise}\n\\end{figure}\nThe $z_2$ pole (dashed line) has the dominant contribution with mass and width values at the edge of the allowed fit range ($m_{0,2}=1375$~MeV, $\\Gamma_{0,2}=146$~MeV). This very broad signal interferes with the $z_1$ pole (dotted line) in a way that the high mass region of the $\\Lambda(1405)$ is strongly suppressed, creating a peak below 1400~MeV\/$c^2$ (solid black line). However, with the large amount of free parameters, with the broad ranges in which the mass and width values are constrained and with the limited number of data points, it is difficult to derive strict conclusions about the relative contributions and the positions of the two poles. It can just be claimed that a significant part of the $\\Lambda(1405)$ amplitude could be associated to the second, broad pole. One should also notice here that the $\\Lambda(1405)$ signal shows a tail for masses above 1440~MeV\/$c^2$. This tail was not considered by the authors of \\cite{Agakishiev:2012xk}, who assumed the $\\Lambda(1405)$ to be only located at lower energies and who therefore used the high mass range of the $\\Sigma\\pi$ spectra to determine the contribution of the $\\Sigma\\pi$ non$\\--$resonant channels. From the results in Fig.~\\ref{fig:Fit_Weise} and \\ref{fig:L1405_Weise} one sees, however, that this assumption might be too simple. Thus, a further possibility is to not fix the non$\\--$resonant background but to treat $A_{\\Sigma^+\\pi^-}$ and $A_{\\Sigma^-\\pi^+}$ as two additional fit parameters.\nApplying this new fit to the data, results in the fit values listed in Table~\\ref{tab:table4}.\n\\begin{table}[tbp]\n\\caption{\\label{tab:table4}%\nObtained free fit parameters after fit with Eq.s~(\\ref{equ:DistSpPm2}) and (\\ref{equ:DistSmPp2}) with $A_{\\Sigma^+\\pi^-}$ and $A_{\\Sigma^-\\pi^+}$ as additional free parameters.}\n\\begin{ruledtabular}\n\\begin{tabular}{cccccccc}\n\\textrm{$m_{0,1}$}&\n\\textrm{$\\Gamma_{0,1}$}&\n\\textrm{$m_{0,2}$}&\n\\textrm{$\\Gamma_{0,2}$}&\n\\textrm{$A_1\/A_2$}&\n\\textrm{$\\varphi_1$} &\n\\textrm{$A_{\\Sigma^+\\pi^-}$} & \n\\textrm{$A_{\\Sigma^-\\pi^+}$} \\\\\n\\colrule\n$1431$ & $58$ & $1375$ & $146$ & $0.204$ & $164^{\\circ}$ & $0.857$ & $0.887$\\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table} \nThe fit quality of $\\chi^2\/ndf=1.03$ is as good as before, but the contribution of the $z_1$ pole is further reduced as seen by comparing the $A_1\/A_2$ ratios in Table~\\ref{tab:table3} and \\ref{tab:table4}. However, as the number of free parameters has further increased, the fit to the data is not robust anymore so that the results of Table~\\ref{tab:table4} are not very reliable.\\\\ \nIn summary, it can be concluded that it is rather difficult to precisely determine the relative contribution between $z_1$ and $z_2$ and simultaneously to determine their exact positions in the complex energy plane just by a fit to the HADES data alone. In fact, an appropriate theory model for proton-proton reactions with a serious treatment of all possible background contributions is required to make further conclusions. Nevertheless, the results obtained in this work clearly show that, in order to describe the new HADES data, a rather large contribution of the broad, low mass $\\Lambda(1405)$ pole is needed, provided that no interference with the non$\\--$resonant background is present.\\\\\nIn this context, it would also be important to have more restrictive constraints on the real and imaginary part of $z_1$ and $z_2$. The values quoted above allow a rather large variation of these parameters. However, the situation is even more unclear. In a recent work by Mai and Meissner \\cite{Mai:2012dt}, who also used the latest SIDDARTHA results, the positions of the two poles were derived to $z_1=1428^{+2}_{-1}+i8^{+2}_{-2}$~MeV and $z_2=1497^{+11}_{-7}+i75^{+9}_{-9}$~MeV. \nThe imaginary part of the first pole is much smaller than the one of \\cite{Ikeda:2012au}, but even more spectacular is the totally different value in the real part of $z_2$, which is shifted by about 100~MeV to higher energies. We can also take these values to fit the HADES data. For this purpose, the non$\\--$resonant background amplitudes $A_{\\Sigma^+\\pi^-}$ and $A_{\\Sigma^-\\pi^+}$ are again fixed to 1 and the mass and width values are allowed to vary within the given ranges. The best fit result is shown in Fig.~\\ref{fig:Fit_Meissner} and the obtained fit parameters are listed in Table~\\ref{tab:tableMeissner}. The corresponding decomposition of the $\\Lambda(1405)$ spectrum is shown in Fig.~\\ref{fig:L1405_Meissner1}. \n\\begin{figure}[tbp]\n\t\\centering\n\t\t\\includegraphics[width=0.46\\textwidth]{Fit_Meissner_1}\n\t\\caption{ Same as Fig.~\\ref{fig:Fit_Weise} but the mass and width constraints for the two $\\Lambda(1405)$ poles are taken from \\cite{Mai:2012dt}. See text for details.}\n\t\\label{fig:Fit_Meissner}\n\\end{figure}\n\\begin{figure}\n\t\\centering\n\t\t\\includegraphics[width=0.45\\textwidth]{L1405_Meissner}\n\t\\caption{Same as Fig.~\\ref{fig:L1405_Weise} but the constraints of \\cite{Mai:2012dt} for the mass and width values of the two $\\Lambda(1405)$ poles were used in the fits. See text for details.}\n\t\\label{fig:L1405_Meissner1}\n\\end{figure}\n\n\\begin{table}[tbp]\n\\caption{\\label{tab:tableMeissner}%\nObtained free fit parameters after fit with Eq.s~(\\ref{equ:DistSpPm2}) and (\\ref{equ:DistSmPp2}) and by using the constraints of \\cite{Mai:2012dt} for the mass and width values of the two $\\Lambda(1405)$ poles.}\n\\begin{ruledtabular}\n\\begin{tabular}{cccccc}\n\\textrm{$m_{0,1}$}&\n\\textrm{$\\Gamma_{0,1}$}&\n\\textrm{$m_{0,2}$}&\n\\textrm{$\\Gamma_{0,2}$}&\n\\textrm{$A_1\/A_2$}&\n\\textrm{$\\varphi_1$}\\\\\n\\colrule\n$1427$ & $20$ & $1490$ & $168$ & $0.17$ & $264^{\\circ}$ \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}      \nThe fit result is very poor, which is also expressed in a normalized $\\chi^2$-value of $\\chi^2\/ndf=3.4$. With the $z_2$ pole having such a large real part, it becomes impossible to create a peak structure at around 1385~MeV\/$c^2$ like it was observed by the HADES collaboration. Within our simplified model, we can thus conclude that the pole positions extracted in \\cite{Mai:2012dt} are not compatible with the new HADES data.\n\n\\section{Summary} \n\nWe have presented different interpretations of the low mass $\\Lambda(1405)$ signal, measured by the HADES collaboration in $p+p$ reactions. It was shown that the obtained signal is very similar to the results obtained in $\\pi^-+p$ reactions. One possible explanation for the observed mass shift is based on interference effects. For that purpose we have developed a simple model, where we assumed the $\\Lambda(1405)$ to consist of two poles, parameterized as Breit-Wigner amplitudes. The contribution and position of these poles were first chosen such to match the  $\\Lambda(1405)$ line shape of \\cite{Geng:2007vm} with the mass peak appearing at $\\approx 1410$~MeV\/$c^2$. With this model we could show that maximum interference between the $\\Lambda(1405)$ amplitude and the non$\\--$resonant $\\Sigma\\pi$ background can indeed create a low mass signal and thus describe the HADES data. However, it was argued that this explanation is probably very unrealistic. In a second approach we have neglected interference effects with the non$\\--$resonant background and have used the HADES data themselves to determine the position and the relative contribution of the two $\\Lambda(1405)$ poles. With a relatively large contribution of the second, broad $\\Lambda(1405)$ pole, $z_2$, we could achieve a very good description of the HADES data. However, the limited statistic in the data did not allow to draw firm conclusions on the precise contribution of the two poles. Also their position in the complex energy plane could not be determined precisely just by a fit to this single data set. Nevertheless, it was shown that, within our simple model, a high energy pole position of $z_2=1497^{+11}_{-7}+i75^{+9}_{-9}$~MeV, like it was obtained in \\cite{Mai:2012dt}, is not compatible with the new HADES data. \\\\            \n\n\nThe authors would like to thank Wolfram Weise, Wolfgang Koenig and Piotr Salabura for very useful discussions. This work is supported by the Munich funding agency, BMBF 06DR9059D,05P12WOGHH, TU M\\\"unchen, Garching\n(Germany) MLL M\\\"unchen: DFG EClust 153, VH-NG-330 BMBF 06MT9156 TP6 GSI.     \n     \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nLet $S$ be a set of rectangles in the plane, with vertical and horizontal sides, whose interiors do not intersect.  We say that two rectangles $A$ and $B$ in $S$ \\textit{\\textbf{see each other}} if there is a vertical or horizontal line segment intersecting the interiors of both $A$ and $B$ and intersecting no other (closed) rectangles in $S$, like the dotted lines in Figure~\\ref{fig:RVG-example}. We refer to such segments as \\textit{\\textbf{lines of sight}}, and under this definition we may consider them to have small positive width. For example, there is no line of sight between rectangles $B$ and $F$ in Figure~\\ref{fig:RVG-example}, since a line of sight needs positive width.  \n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[width=.6\\textwidth]{RVG-example.pdf}\n\\caption{A rectangle visibility graph and a corresponding RV-representation with integer rectangles.}\n\\label{fig:RVG-example} \n\\end{figure} \n\nWe construct a graph $G$ with a vertex for every rectangle in $S$, and an edge between two vertices if and only if their corresponding rectangles see each other.  We say that $S$ is a \\textbf{\\textit{rectangle visibility}} (\\textit{\\textbf{RV-}}) \\textit{\\textbf{representation}} of $G$, and $G$ is a \\textit{\\textbf{rectangle visibility graph}} or \\textit{\\textbf{RVG}}.  We allow rectangles in $S$ to share edges. \n\nA similar notion of rectangle visibility graph was first introduced in 1976 by Garey, Johnson and So \\cite{garey76} as a tool to study the design of printed circuit boards.  Their RVGs have only $1\\times 1$ squares, located at a set of lattice points in a grid.  Hutchinson continued work on this problem in 1993 \\cite{hutchinson93}.  In \\cite{bose1996rectangle}, Bose, Dean, Hutchinson and Shermer described the problem of \\textit{two-layer routing} in ``very large-scale integration'' (VLSI) design as follows: \n\\begin{quote} {\\small\nIn two-layer routing, one embeds processing components and their connections (sometimes called \\textit{wires}) in two layers of silicon (or other VLSI material).  The components are embedded in both layers.  The wires are also embedded in both layers, but one layer holds only horizontal connections, and the other holds only vertical ones.  If a connection must be made between two components that are not cohorizontal or covertical, then new components (called \\textit{vias}) are added to connect horizontal and vertical wires together, resulting in bent wires that alternate between the layers.  However, vias are large compared to wires and their use should be minimized.  In this setting, asking if a graph is a rectangle-visibility graph is the same as asking if a set of components can be embedded so that there is a two-layer routing of their connections that uses no vias.  Our requirement that visibility bands have positive width is motivated by the physical constraint that wires must have some minimum width.  A similar problem arises in printed-circuit board design, as printed-circuit boards naturally have two sides, and connecting wires from one side to the other (the equivalent of making vias) is relatively expensive.}\n\\end{quote}\nIn 1995, Dean and Hutchinson began the study of RVGs in their own right.  They focused on bipartite RVGs, and showed that $K_{p,q}$ is an RVG if and only if $p \\leq 4$, and that every bipartite RVG with $n$ vertices has at most $4n-12$ edges \\cite{Dean95}.  In 1996 Bose, Dean, Hutchinson, and Shermer sought to characterize families of graphs that are RVGs.  They proved that every graph with maximum degree four is an RVG, and every graph that can be decomposed into two caterpillar forests is an RVG, among other results \\cite{bose1996rectangle}.  In 1999 Hutchinson, Shermer, and Vince \\cite{hutchinson1999representations} proved that every RVG with $n$ vertices has at most $6n-20$ edges, and this bound is tight for $n\\geq 8$.  RVGs have since been studied by many other authors \\cite{angelini18,biedl16,  Cahit98, GD2014, Dean97,Dean98,dean08,Dean2010,Streinu03}, including generalizations to 3-dimensional boxes \\cite{bose99,develin03,Fekete99,gethner11}, rectilinear polygons with more than four edges \\cite{digiacomo18, liotta21}, and other variations.\n\nIn 1997, Kant, Liotta, Tamassia, and Tollis considered the minimum area, height, and width required to represent a tree as an RVG, as measured by the smallest bounding box containing all of the rectangles in the RV-representation \\cite{kant97}.  They obtained asymptotic bounds on the area, width, and height of these representations and found a linear-time algorithm to construct them.  In this paper we consider a similar problem, but seek exact bounds on the area, width, height, and perimeter of an RV-representation of any graph with $n$ vertices.  We say that $\\area(G)$, $\\perimeter(G)$, $\\height(G)$, and $\\width(G)$ are the minimum area, perimeter, height, and width, respectively, of the bounding box of any integer rectangle visibility representation of the graph $G$. These are the objects of study in this paper.\n\nIn Section~\\ref{Definitions} we specify the rectangle visibility graphs we consider and provide definitions and notation needed for the paper.  We finish the section with lemmas we will use in later sections.\n\nIn Section~\\ref{SeparatingExamples} we show that these four measures of size of a rectangle visibility graph are all distinct, in the sense that there exist two graphs $G_1$ and $G_2$ with $\\area(G_1)<\\area(G_2)$ but $\\perimeter(G_2)<\\perimeter(G_1)$, and analogously for all other combinations of these parameters. \n\nIn Section~\\ref{SeparatingRepresentations} we show that these measures are not necessarily all attained by the same representation; i.e., there is a graph $G_3$ with two RVG representations $S_1$ and $S_2$ with $\\area(G_3)=\\area(S_1)<\\area(S_2)$ but $\\perimeter(G_3)=\\perimeter(S_2)<\\perimeter(S_1)$, and analogously for all other combinations of these parameters.\n\nIn Section \\ref{SmallParameters} we characterize the graphs that have the smallest height, width, area, and perimeter among all graphs with $n$ vertices.\n\nIn Section~\\ref{LargeParameters} we investigate the graphs with largest height, width, area, and perimeter.  We show that, among graphs with $n \\leq 6$ vertices, the empty graph $E_n$ has largest area, and for graphs with 7 or 8 vertices, the complete graphs $K_7$ and $K_8$ have larger area than $E_7$ and $E_8$, respectively. Using this, we show that for all $n \\geq 7$, the empty graph $E_n$ does not have largest area among all RVGs on $n$ vertices.  The graphs with more than 6 vertices that maximize these parameters are still unknown.\n\nIn Section~\\ref{Conclusions}, we conclude with a number of open questions.\n\n\\section{Basic Definitions and Results} \\label{Definitions}\n\nA rectangle with horizontal and vertical sides whose corners are integer lattice points is said to be an \\textit{\\textbf{integer rectangle}}.  We consider only integer rectangles for the remainder of the paper. Each rectangle is specified by the two $x$-coordinates and two $y$-coordinates of its corners.  For a set $S$ of rectangles, the smallest rectangle with horizontal and vertical sides containing $S$ is the \\textit{\\textbf{bounding box}} of $S$.\n\nSuppose $G$ is an RVG with RV-representation $S$ contained in the bounding box $R$, and say $R$ has corners with $x$-coordinates $0$ and $u$ and $y$-coordinates $0$ and $v$, with $u,v\\in \\mathbb{Z}$.  We can view $R$ as a $u \\times v$ grid, with $v$ rows and $u$ columns, and with rectangles in $S$ each contained in a consecutive set of rows and columns.  For example, in the representation shown in Figure~\\ref{fig:RVG-example}, rectangle $A$ is contained in rows 3, 4, 5, and 6, and column 1.\n\nWe use the convention that lower case letters are vertices of the graph $G$, and the corresponding upper case letters are rectangles in the RV-representation $S$ of $G$; e.g., $a$ is a vertex in $G$, and $A$ is its corresponding rectangle in $S$.  For a given rectangle $A$ in $S$, we denote the $x$-coordinates of its vertical sides by $x_1^A$ and $x_2^A$ with $x_1^A < x_2^A$, and the $y$-coordinates of its horizontal sides by $y_1^A$ and $y_2^A$, with $y_1^A < y_2^A$. In other words, as a Cartesian product of intervals, we have $$A=[x_1^A,x_2^A] \\times [y_1^A,y_2^A].$$\n\nWe also introduce notation to refer to the set of rectangles in $S$ that are above (respectively, below, to the left of, or to the right of) a given rectangle $A$. Specifically, let the set of rectangles above (north of) $A$ be denoted by\n$$ {\\mathcal{N}}(A) = \\{ X \\in S :  y_1^X \\geq y_2^A \\mbox{ and } (x_1^X,x_2^X) \\cap (x_1^A,x_2^A) \\not= \\emptyset \\}.$$  Similarly define ${\\mathcal{S}}(A)$, ${\\mathcal{W}}(A)$, and ${\\mathcal{E}}(A)$ (rectangles south, west, and east of $A$, respectively). For example, in Figure \\ref{fig:RVG-example}, $\\mathcal{N}(D)=\\{B,E\\},$ while $\\mathcal{E}(A)=\\{B, E, D\\}$ and $\\mathcal{S}(A)=\\varnothing.$  Note that $A$ might not see every rectangle in ${\\mathcal{N}}(A)$ if there are other rectangles obstructing the view (and similarly for rectangles in the other three sets). \n\nLet $R$ be the smallest bounding box having horizontal and vertical sides and containing all the rectangles in a set of integer rectangles $S$.  For the remainder of the paper, we turn $R$ so that $\\height(R) \\leq \\width(R)$. \nGiven a graph $G$, the \\textit{\\textbf{area}}, \\textit{\\textbf{perimeter}}, \\textit{\\textbf{height}}, and \\textit{\\textbf{width}} of $G$ are the minimums of the corresponding parameters taken over all bounding boxes of RV-representations of $G$ with height less than or equal to width.\n\nWe conclude this section with some preliminary results. \nFirst we explore the extent to which we can focus on the parameters of connected graphs, and in what ways the values for disconnected graphs are determined or bounded by the parameters of their connected components.\n\nFor convenience in stating the next result, we introduce the following notation.\nFor any positive integers $h$ and $w$, let ${\\mathcal{F}}_{h,w}$ denote the (finite) set of graphs that have RV-representations in an $h \\times w$ bounding box.\n\n\n\n\n\n\n\\begin{lemma}\\label{NewDisjointLem}\nIf $G$ is the disjoint union of graphs $H$ and $J$, then:\n\\begin{enumerate}[label=\\rm{(\\roman*).}]\n\\item $\\height(G)=\\height(H)+\\height(J),$\n\\item $\\perimeter(G)=\\perimeter(H)+\\perimeter(J),$ \n\\item $\\width(G)=\\min \\{  \\max \\{x+b, y+a\\} \\, | \\, H \\in {\\mathcal{F}}_{x,y}, J \\in {\\mathcal{F}}_{a,b} \\}$,\n\\item $\\area(G)=\\min \\{ (x+a)(y+b) \\, | \\, H \\in {\\mathcal{F}}_{x,y}, J \\in {\\mathcal{F}}_{a,b} \\}$.\n\\end{enumerate} \n\\end{lemma}\n\n\\begin{proof}\nSuppose $G$ is the disjoint union of graphs $H$ and $J$.  Given any RV-representations $S_1$ and $S_2$ of $H$ and $J$, we construct two RV-representations of $G$. As indicated in Figure~\\ref{Glue}, we identify the upper right corner of $S_1$ with the lower left corner of either $S_2$ or  $S_2^T$, where $S_2^T$ denotes the RV-representation of $J$ formed by transposing $S_2$ across its main (top left to lower right) diagonal.  If $S_1$ is $x \\times y$ and $S_2$ is $a \\times b$, then it follows that\n$ G \\in {\\mathcal{F}}_{x+a,y+b} \\cap {\\mathcal{F}}_{x+b,y+a}.$\nThis implies each of the expressions in (i)-(iv) are upper bounds for the parameters of $G$.\n\nTo prove these expressions are also lower bounds, note that \n any RV-representation $S$ of $G$ must have the rectangles corresponding to vertices of $H$ in separate rows and columns from the rectangles corresponding to vertices of $J$.  If $S$ has height smaller than $\\height(H)+\\height(J)$, then either the rectangles in $S$ corresponding to vertices of $H$ must form an RV-representation of $H$ with height less than $\\height(H)$ or the rectangles in $S$ corresponding to vertices of $J$ must form an RV-representation of $J$ with height less than $\\height(J)$.  Therefore no such representation is possible.  Similar arguments apply to the other parameters.\n\\end{proof}\n\n\\begin{remark}\nThe following example illustrates a subtlety captured by the formula in Lemma~\\ref{NewDisjointLem}(iv).  Consider the graphs $P_4+K_{1,4}$ and $K_{1,4}+K_{1,4}$.  We have $\\area(P_4+K_{1,4})=27$, obtained from a $1 \\times 4$ RV-representation of $P_4$ and a $2 \\times 5$ RV-representation of $K_{1,4}$.  But we also have $\\area(K_{1,4}+K_{1,4})=36$, obtained from two copies of a $3 \\times 3$ RV-representation of $K_{1,4}$ (both representations of $K_{1,4}$ are given in Figure~\\ref{fig:width-examples2a}).  So the minimum area of $H + K_{1,4}$ uses different representations of $K_{1,4}$ depending on $H$.\n\\end{remark}\n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[width=.7\\textwidth]{disjoint-union.pdf}\n\\caption{Two options for a combined representation of a disjoint union of two RVGs}\n\\label{Glue} \n\\end{figure} \n\n\\begin{corollary}\\label{NewDisjointCor}\nIf $G$ is the disjoint union of graphs $H$ and $J$, then:\n\\begin{enumerate}[label=\\rm{(\\roman*).}]\n\\item $\\width(G) \\leq \\width(H) + \\width(J)$,\n\\item $\\area(G) \\leq (\\width(H) + \\width(J))^2$.\n\\end{enumerate} \n\\end{corollary}\n\n\\begin{proof}\nThese both follow immediately from the \nconstruction in the proof of Lemma~\\ref{NewDisjointLem} shown in Figure~\\ref{Glue}. \n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\begin{lemma}\nSuppose $G$ is a graph with RV-representation $S$ with bounding box $R$.  If $\\height(G)=\\height(R)$ and $\\width(G)=\\width(R)$, then $\\area(G)=\\area(R)$ and $\\perimeter(G)=\\perimeter(R)$.  In other words, if $S$ realizes the height and width of $G$, then $S$ also realizes the area and perimeter of $G$.  \n\\label{lemma-same-height-width}\n\\end{lemma}\n\n\\begin{proof}\nAny representation with smaller area or perimeter must have smaller height or smaller width, which is impossible by hypothesis.\n\\end{proof}\n\nLater (in Table~\\ref{table-separating2}), we will see that the hypotheses of Lemma~\\ref{lemma-same-height-width} are necessary.  In particular, $G_4$ has two different representations for minimizing area and perimeter.\n\n\n\\section{Height, Width, Area, and Perimeter induce distinct orderings of RVGs} \\label{SeparatingExamples}\n\nIn this section, we consider the various notions of height, width, perimeter, and area of RVGs. We show that these parameters represent independent measures of RVGs, in the sense that they do not always give identical orderings of the sets of graphs on a given number of vertices. \nExamples to illustrate these results are summarized in Tables~\\ref{table-separating1}  and \\ref{table-separating-new}. Graphs $G_1$, $G_2$, $G_3$, and $G_4$ in these tables are shown in Figures~\\ref{fig:width-examplesBB} and~\\ref{fig:width-examplesCC}.  For each pair of parameters, there exists a pair of graphs with an equal number of vertices that are oppositely ordered by those parameters. We have verified the height, width, area, and perimeter of all connected graphs with 6 or fewer vertices by computer search   \\cite{WebList} and the claims regarding $P_6$, $C_6$, $G_1$, and $G_2$ in Tables~\\ref{table-separating1}  and \\ref{table-separating-new} follow easily, see Figures~\\ref{fig:width-examplesAA} and \\ref{fig:width-examplesBB}.\nThe claims regarding $G_3$ and $G_4$, each with 15 vertices, are proved in Theorems~\\ref{width-theorem} and \\ref{width-theorem2}.  \n\n \n\n\\begin{table}[ht!]\n\\begin{tabular}{c|c|cccc} \n Graph & Vertices & Height & Width & Area & Perimeter \\\\ \\hline \\hline\n  $P_6$  & 6 & {\\bf 1} & 4 & {\\bf 6} & 14 \\\\ \n  $C_6$  & 6 & 2 & {\\bf 3} & 8 & {\\bf 12} \\\\ \\hline\n  $G_1$ & 6 & {\\bf 2} & -- & 10 & -- \\\\\n  $G_2$ & 6 & 3 & -- & {\\bf 9} & -- \\\\ \\hline\n $G_3$ &  15 & -- & 6 & -- & {\\bf 18} \\\\\n $G_4$ & 15 & -- & {\\bf 5} & -- & 20 \\\\   \\hline\n\\end{tabular} \n\\smallskip\n\\caption{Separating examples for height, width, area, and perimeter.}\n\\label{table-separating1}\n\\end{table}\n\n\n\\begin{table}[ht!]\n\\begin{tabular}{c|ccc}\n & Perimeter & Height & Width \\\\ \\hline \\hline\n  Area & $P_6,C_6$ & $G_1,G_2$ & $P_6,C_6$ \\\\ \n  Perimeter & -- & $P_6,C_6$ & $G_3,G_4$ \\\\ \n  Height & -- & -- & $P_6,C_6$ \\\\\n  \\hline\n\\end{tabular} \n\\smallskip\n\\caption{Graph pairs that are oppositely ordered by each of the parameter pairs.}\n\\label{table-separating-new}\n\\end{table}\n \n\n\\begin{figure}[ht!]\n    \\centering \\includegraphics[width=.9\\textwidth]{P6-and-C6.pdf}\n    \\caption{Graphs $P_6$ and $C_6$ show that height and area order graphs differently than  width and perimeter.}\n    \\label{fig:width-examplesAA}\n\\end{figure}\n\n\n\n\\begin{figure}[ht!]\n    \\includegraphics[width=.8\\textwidth]{G1-and-G2-new.pdf}\n    \\caption{Graphs $G_1$ and $G_2$ show that height orders graphs differently than area.}\n    \\label{fig:width-examplesBB}\n\\end{figure}\n\n\n\n\\begin{figure}[ht!]\n    \\centering\n    \\includegraphics[width=.4\\textwidth]{width-example1.pdf} \\hspace{1em}\n    \\includegraphics[width=.4\\textwidth]{width-example2.pdf}\n    \\caption{Graphs $G_3$ and $G_4$ show that width orders graphs differently than perimeter.}\n    \\label{fig:width-examplesCC}\n\\end{figure}\n\n\n\n \n\n\n\n\n\n\\begin{theorem}\nThe graph $G_3$ shown on the left in Figure~\\ref{fig:width-examplesCC} has perimeter 18 and width 6. \\label{G1-proof}\n\\label{width-theorem}\n\\end{theorem}\n\n\\begin{proof}\n\\noindent To find the perimeter of $G_3$, we first consider its area. Let $v$ be the vertex of degree 10 in $G_3$. In any RV-representation of $G_3$, the corresponding rectangle $V$ must have perimeter at least 10, so its area is at least 4. Together with the 14 other vertices, we see $\\area(G_3)\\ge 18.$  Now suppose $G_3$ can be represented in a bounding box of height $h$, width $w$, and perimeter $p=2h+2w$. Such a rectangle has maximum area when $h=w=p\/4$, so $\\area(R) \\leq p^2\/16$. But $\\area(R) \\geq 18$, so $p \\geq \\sqrt{18 \\cdot 16} > 16.$  Since $p$ must be even,  $\\perimeter(G_3)=18$ by Figure \\ref{fig:width-examplesCC}.\n\nNext we consider $\\width(G_3)$.  If $\\width(G_3)<6$, then $G_3$ can be represented in a $5\\times 5$ box $R$.   In a $5 \\times 5$ box with 14 other vertices, $V$ has area at most 11, and hence $V$ must be $1 \\times 4$, $1 \\times 5$, $2 \\times 3$, $2 \\times 4$, $2 \\times 5$, or $3 \\times 3$.  We rule out each possibility below.\n \n \n\\begin{figure}[ht!]\n    \\centering\n    \\includegraphics[width=.9\\textwidth]{G1-cases.pdf}\n    \\caption{Diagrams illustrating several of the cases in the proof of Theorem~\\ref{G1-proof}.}\n    \\label{fig:width-proof}\n\\end{figure} \n \n  $\\bullet \\;$ If  $V$ is $1 \\times 4$, at least one unit of the perimeter of $V$ is on the boundary of $R$, and therefore $V$ cannot represent a degree-10 vertex. \n\n  $\\bullet \\;$ If $V$ is $1 \\times 5$ or $2 \\times 5$, it must touch opposite sides of $R$; thus $V$ must see 5 rectangles on each of its other two sides in order to have degree 10. But then $v$ is a cut vertex, and $G_3$ has none. \n\n  $\\bullet \\;$ If  $V$ is $3\\times 3$, it cannot have an edge on the boundary of $R$ and represent a degree-10 vertex. In this case, $V$ occupies the middle of $R$ as shown on the left in Figure~\\ref{fig:width-proof}. The four vertices not adjacent to $v$ must be represented by $1\\times 1$ squares in the corners of $R$.  Among these four, two (disjoint) pairs have a unique common neighbor in $G_3$.  Locating the rectangles for these common neighbors in their respective  $3 \\times 1$ blocks of $R$, we see there now remain only 6 locations for the other 8 vertices.   \n\n  $\\bullet \\;$ If $V$ is $2\\times 4$, it must (by symmetry) appear as in the middle of Figure~\\ref{fig:width-proof}. To have degree 10, $V$ must see a $1\\times 1$ square at $W$. But $v$ has no neighbor of degree less than $3$.   \n\n  $\\bullet \\;$ If $V$ is $2\\times 3$, it must (by symmetry) appear as on the right of Figure~\\ref{fig:width-proof}.  To have degree 10, $V$ must see distinct rectangles on each unit of its perimeter.  Numbering the locations in $R$ as in the right of Figure~\\ref{fig:width-proof}, if location 3 is empty, location 2 must contain a $1\\times 1$ square, but $v$ has no neighbor of degree less than $3$. If a $2 \\times 1$ rectangle $X$ covers location 3, the rectangles in locations 1 through 6 form a path $P_5$ in the neighborhood of $V$, but $G_3$ has no such subgraph.  Thus, locations 3 and (by symmetry) 7 contain $1 \\times 1$ squares. But $G_3 -v$ has no vertices of degree 2 that are at distance 4.\n \\end{proof}           \n \n\n\\begin{theorem}\nThe graph $G_4$ shown on the right in Figure~\\ref{fig:width-examplesCC} has perimeter 20 and width 5. \\label{G2-proof}\n\\label{width-theorem2}\n\\end{theorem}\n               \n\\begin{proof} \nWe use the labeling in Figure~\\ref{fig:width-examplesCC}.  We begin with perimeter.  Note that since $G_4$ has 15 vertices, every RV-representation of $G_4$ has area at least 15.  Then if $\\perimeter(G_4)<20$, it must have a representation that fits in a $3 \\times 6$ or $4 \\times 5$ bounding box. But a $3 \\times 6$ box has area 18, so the 3 vertices of degree more than four in $G_4$ must be represented with $1 \\times 2$ rectangles. But then all the vertices of $G_4$ must have degree $\\leq 6$, a contradiction.\n\nThus $G_4$ must be representable in a $4 \\times 5$ box. The two vertices of degree 5 in $G_4$ must each be represented with a rectangle of area at least 2. In a $4 \\times 5$ box $R$, $U$ then has area 3 or 4 and cannot lie in a corner. All vertices of $G_4$ have degree at least 3, so no corner of $R$ has a $1 \\times 1$ square. Therefore, each corner of $R$ is either empty or occupied by a rectangle of area at least 2. With 15 vertices, this exceeds the total area of 20 in $R$: each vertex contributes at least one, the four corners require at least one additional unit of area each, and $U$ needs two or three additional units of area not lying in a corner. Now $\\perimeter(G_4)=20$ by Figure~\\ref{fig:width-examplesCC}.\n\nWe next consider width. If $\\width(G_4)<5$, then $G_4$ can be represented in a $4 \\times 4$ box. But then it also has a $4 \\times 5$ representation, which we just proved impossible. By Figure~\\ref{fig:width-examplesCC}, we see that $\\width(G_4) =5$. \n\\end{proof}\n\n\n\\section{Minimizing Height, Width, Area, and Perimeter can require distinct representations of an RVG} \\label{area} \\label{SeparatingRepresentations}\n\nIn this section, we further explore our four parameters and we observe that, even for a single RVG, it is possible that the set of representations minimizing one of them may be disjoint from the set of representations minimizing another.  \nExamples to illustrate these results are summarized in Tables~\\ref{table-separating2} and \\ref{table-separating2b}. For each pair of parameters, there exists a graph that requires distinct representations to separately minimize each parameter in that pair. Specifically, the star $K_{1,4}$ has representations minimizing height that are distinct from those minimizing width, area, and perimeter; the graph $G_5$ shown in Figure~\\ref{fig:width-examples2a} has representations minimizing width that are distinct from those minimizing the other parameters; and the graph $G_6$ shown in Figure~\\ref{fig:width-examples2b} has distinct representations minimizing area and perimeter. Since $K_{1,4}$ has fewer than 7 vertices, the claims regarding it are verified by our computer search \\cite{WebList}. The claims regarding $G_5$ and $G_6$, each with 7 vertices, are proved in Theorems~\\ref{G5-theorem} and \\ref{G6-theorem} below.  \n\n\\begin{table}[ht!]\n\\begin{tabular}{c|c|cccc} \nGraph (representation) & Vertices & Height & Width  & Area & Perimeter\\\\ \\hline \\hline \n  $K_{1,4}$ ($S_1$) & 5 & {\\bf 2} & 5  & 10 & 14\\\\\n  $K_{1,4}$ ($S_2$) & 5 & 3 & {\\bf 3}  & {\\bf 9} & {\\bf 12}\\\\ \\hline\n  $G_5$ ($S_1$) & 7 & {\\bf 2} & 5  & {\\bf 10} & {\\bf 14}\\\\\n  $G_5$ ($S_2$) & 7 & 4 & {\\bf 4}  & 16 & 16\\\\ \\hline\n  $G_6$ ($S_1$) & 7 & {\\bf 2} & 7  & {\\bf 14} & 18\\\\\n  $G_6$ ($S_2$) & 7 & 4 & {\\bf 4}  & 16 & {\\bf 16}\\\\  \\hline\n\\end{tabular} \n\\smallskip\n\\caption{Graphs representations used to minimize height, width, area, and perimeter.}\n\\label{table-separating2}\n\\end{table} \n\n\\begin{table}[ht!]\n\\begin{tabular}{c|ccc}\n & Perimeter & Height & Width \\\\ \\hline \\hline\n  Area & $G_6$ & $K_{1,4}$ & $G_5$ \\\\ \n  Perimeter & -- & $K_{1,4}$ & $G_5$ \\\\ \n  Height & --& --& $G_5$ \\\\\n  \\hline\n\\end{tabular} \n\\smallskip\n\\caption{Graphs requiring different representations to minimize area, perimeter, height, and width.}\n\\label{table-separating2b}\n\\end{table}\n\n\\begin{theorem}\nThe graph $G_5$ shown in Figure~\\ref{fig:width-examples2a} has height 2, width 4, area 10, and perimeter 14.  Furthermore, minimizing width requires a different RV-representation from height, area, or perimeter. \\label{G5-theorem}\n\\end{theorem}\n\n\n\n\\begin{figure}[ht!]\n    \\centering\n   \\includegraphics[width=\\textwidth]{K14-and-G5-new.pdf}\n    \\caption{Graphs $K_{1,4}$ and $G_5$ are shown.  For $K_{1,4}$, one representation ($S_1$) minimizes height and another ($S_2$) minimizes area, perimeter, and width.  For $G_5$, one representation ($S_1$) minimizes height, area, and perimeter, and another  ($S_2$) minimizes width.}\n    \\label{fig:width-examples2a}\n\\end{figure}\n\n\n\\begin{figure}[ht!]\n    \\centering\n   \\includegraphics[width=.6\\textwidth]{G6-new.pdf}\n    \\caption{Graph $G_6$ with distinct representations minimizing area and perimeter.}\n    \\label{fig:width-examples2b}\n\\end{figure}\n \n\\begin{proof}\n The representation $S_1$ in Figure \\ref{fig:width-examples2a} has height 2, width 5, area 10, and perimeter 14. The representation $S_2$ has height 4, width 4, area 16, and perimeter 16. \n \nWe claim $G_5$ has no $3 \\times 4$ representation. Each of the 3-cycles in $G_5$ must have at least 2 rows that occupy at least 2 units of area. This implies that some row in a $3 \\times 4$ box has 2 units of each 3-cycle, which therefore must see each other. Since $G_5$ has no edges joining these 3-cycles, this is impossible.\n\nIt follows that $G_5$ has no $3 \\times 3$ or $2 \\times 4$ representation. Since $G_5$ is not a path, $\\height(G_5)\\geq 2$ and these facts together imply that $G_5$ has height 2, width 4, area 10, and perimeter 14.  \nSince the only representations of $G_5$ with width 4 are $4 \\times 4$, any representation minimizing width does not minimize height, area, or perimeter.\n\\end{proof}\n  \n \n\n\n\\begin{theorem}  \nThe graph $G_6$ shown in Figure~\\ref{fig:width-examples2b} has perimeter 16 and area 14.  Furthermore, minimizing perimeter requires a\ndifferent RV-representation than area. \\label{G6-theorem}\n\\end{theorem}\n\n\n\n\n\\begin{proof}\n The first representation in Figure \\ref{fig:width-examples2b} has perimeter 18 and area 14. The second has perimeter 16 and area 16. \n \n The 4 vertices of degree 1  each require at least 3 units of length on the perimeter with free lines of sight. The degree 2 vertex requires at least 2 units, and the 2 degree 3 vertices each require at least 1 unit. So $\\perimeter(G_6)=16$.  \n \n If $\\area(G_6)<14,$ then $G_6$ can be represented in a rectangle of area 7, 8, ..., or 13. Since $G_6$ is not a path, prime areas are not possible, so $G_6$ must have height $>1$. Since $\\perimeter(G_6)=16$, the only possible bounding box must be $2 \\times 6$. We claim this is impossible. Since $G_6$ has 7 vertices, a box with 6 columns would force some column to contain portions of 2 rectangles. Neither of these rectangles can represent a vertex of degree 1, or else the remaining graph must be a path, so they must have degrees 2 and 3, which implies that $D$ has an empty horizontal line of sight. But now, taken together, the 5 vertices of degree 1 or 2 require 5 empty horizontal lines of sight, while the bounding box only has height 2, with at most 4 such lines.   \n \n The only rectangle with height $>1$ and area 14 is $2 \\times 7$, so the perimeter and area must be achieved with distinct representations.\n\\end{proof}\n\n\n\\section{Graphs with small area, perimeter, height, and width} \\label{SmallParameters} \n\nIn this section, we address the question of which graphs on a given number of vertices minimize each parameter. \n\nFor any real number $x$, we use $\\lceil x \\rceil$ and $\\lfloor x \\rfloor$ to denote the integer ceiling and floor of $x$, respectively. We use $[[ x ]] = \\lfloor x + 1\/2 \\rfloor$ to denote $x$ rounded to the nearest integer. Recall that, for any positive integers $h$ and $w$, we let ${\\mathcal{F}}_{h,w}$ denote the (finite) set of graphs that have RV-representations in an $h \\times w$ bounding box.\n\n\\begin{theorem}\\label{lowerbds}\nLet $G$ be any graph with $n$ vertices and suppose $G$ has an RV-representation. Then the following hold.\n\\begin{enumerate}[label=\\rm{(\\roman*).}]\n    \\item The height of $G$ satisfies\n    $$\\height(G) \\geq 1.$$\n     Equality holds if and only if $G \\cong P_n$, the path on $n$ vertices.\n    \\item The area of $G$ satisfies\n    $$\\area(G) \\geq n.$$  \n     Equality holds if and only if $G \\cong P_h \\openbox P_w$, for some positive integers $h$ and $w$, where $n=h \\cdot w$. \n    \\item The width of $G$ satisfies\n    $$\\width(G) \\geq \\lceil \\sqrt{n} \\rceil.$$ \n    Equality holds if and only if $G \\in {\\mathcal{F}}_{w,w}$ where $w = \\lceil \\sqrt{n} \\rceil$.\n    \\item The perimeter of $G$ satisfies\n    $$\\perimeter(G) \\geq 2 \\cdot [[\\sqrt{n}]] + 2 \\cdot \\lceil \\sqrt{n} \\rceil.$$ \n    Equality holds if and only if $G \\in {\\mathcal{F}}_{h,w}$ for some positive integers $h$ and $w$, where $h+w = [[\\sqrt{n}]] + \\lceil \\sqrt{n} \\rceil$ and $hw \\geq n$.\n\\end{enumerate} \n\\end{theorem}\n\n\\begin{proof}\n To see (i) and (ii), note that every RV-representation of a graph must have height at least 1 and area at least $n$, since each vertex requires at least a $1\\times 1$ rectangle. For $G$ to have height exactly 1, every rectangle in such a representation must be on the same row in the representation, so $G$ is a path.  For $G$ to have area exactly $n$,  every rectangle in such a representation must be $1 \\times 1$, with no empty space in the bounding box $R$. Thus $G$ is the grid $P_h \\openbox P_w$, where $h$ is the height of $R$ and $w$ is the width of $R$.  \n\n \n We show (iii) by way of contradiction.  Suppose $G$ can be represented in an $a \\times b$ bounding box $R$, where $a \\leq b < \\lceil \\sqrt{n} \\rceil$. Then $ b \\leq \\lceil \\sqrt{n} \\rceil -1$, so $b <  \\sqrt{n}$. But now the area of $R$ is $ab <  n$, which is impossible since $G$ has $n$ vertices.  It follows  that $\\width(G) = \\lceil \\sqrt{n} \\rceil$ if and only if $G \\in {\\mathcal{F}}_{w,w}$ where $w = \\lceil \\sqrt{n} \\rceil$. \n \n We also show (iv) by way of contradiction.  Suppose $G$ can be represented in an $a \\times b$ bounding box $R$, where $a + b < [[\\sqrt{n}]] + \\lceil \\sqrt{n} \\rceil$. By (ii), we know $ab \\geq n$. We consider cases for whether $\\sqrt{n}$ rounds up or down. In each case, we find that\n $$ (a+b)^2 < 4n \\leq 4ab,$$\n which implies that $(a-b)^2 <0,$ a contradiction.  \n It follows that $\\perimeter(G) = 2 \\cdot [[\\sqrt{n}]] + 2 \\cdot \\lceil \\sqrt{n} \\rceil$ if and only if $G \\in {\\mathcal{F}}_{h,w}$ for some positive integers $h$ and $w$, where $h+w = [[\\sqrt{n}]] + \\lceil \\sqrt{n} \\rceil$ and $hw \\geq n$. \n\n\\end{proof}\n\n\\begin{remark}\nThe condition for equality in Theorem~\\ref{lowerbds}(iv) restricts the bounding box to be very nearly square. Specifically, we can say the following. \nLet $k$ denote the integer such that $k^2 < n \\leq (k+1)^2$.\n\nIf $k^2 < n \\leq k(k+1)$, then equality holds in Theorem~\\ref{lowerbds}(iv) if and only if $G \\in {\\mathcal{F}}_{k-t,k+1+t}$ for some integer $t$ where $$0 \\leq t \\leq \\sqrt{(k+{\\textstyle\\frac{1}{2}})^2-n}-{\\textstyle\\frac{1}{2}}.$$\n\nIf $k(k+1) < n \\leq (k+1)^2$, then equality holds in Theorem~\\ref{lowerbds}(iv) if and only if $G \\in {\\mathcal{F}}_{k+1-t,k+1+t}$ for some integer $t$ where $$0 \\leq t \\leq \\sqrt{(k+1)^2-n}.$$\n\\end{remark}\n\nFor example, any RVG with $n=70$ vertices must have perimeter at least 34, with equality only when $G$ has a $7 \\times 10$ or $8 \\times 9$ representation.\nSimilarly, any RVG with $n=120$ vertices must have perimeter at least 44, with equality only when $G$ has a $10 \\times 12$ or $11 \\times 11$ representation. \n\n\n\\section{Graphs with large area, perimeter, height, and width} \\label{CompleteArea} \\label{LargeParameters}\n\nIn this section we turn to the question of which graphs on a given number of vertices maximize our four parameters.  \n\nRecall that the \\textit{\\textbf{empty graph}} $E_n$ is the graph with $n$ vertices and no edges. Among small graphs (at most 6 vertices), the empty graphs maximize each of the four parameters. When the number of vertices is larger than 7, however, we will see that the empty graph no longer reigns supreme. Our proof is constructive, as we will provide specific graphs that we will prove are larger than $E_n$ in each parameter. But our results here leave open, perhaps for future work, the more difficult question of which graphs on $n$ vertices actually achieve the maximum values for the four parameters.\n\nWe begin with the small graphs.\n\n\\begin{theorem}\nFor $1 \\leq n \\leq 6$, among all graphs with $n$ vertices, the empty graph $E_n$ has largest height, width, area, and perimeter.\n\\end{theorem}\n\n\\begin{proof}\nBy Lemma~\\ref{NewDisjointLem}, the empty graph $E_n$ has height $n$, width $n$, area $n^2$, and perimeter $4n$.  Figures~\\ref{fig:small representations} and \\ref{fig:small representations6} show RV-representations of all connected graphs with at most 6 vertices.  These figures show that no other connected graphs with $2 \\leq n \\leq 6$ vertices exceed any of these values.  For a disconnected graph $G$, Lemma~\\ref{NewDisjointLem} implies that, as long as $G$ has at least one component with more than one vertex, we can combine the representations of the components as in Figure~\\ref{Glue} to obtain height less than $n$, area less than $n^2$, and perimeter less than $4n$. Because $K_2$ has width 2, the graph $K_2 + E_{n-2}$ has width $n$, but no graph has larger width than $E_n$ for $n \\leq 6$.\n\\end{proof}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=.7\\textwidth]{small-representations.pdf}\n\\caption{RV-representations of all connected graphs with between 1 and 5 vertices.}\n\\label{fig:small representations}\n\\end{figure}\n\n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[width=.9\\textwidth]{small-representations6-labeled-updated.pdf}\n\\caption{RV-representations of all connected graphs with 6 vertices, using the labeling from~\\cite{small-graphs-website}.  These representations have smallest area, by computer search.}\n\\label{fig:small representations6}\n\\end{figure} \n\n\nOur next results focus heavily on the RV-representations of the complete graph $K_n$.  Let $S$ be any set of rectangles representing $K_n$ and let $R = [0,u] \\times [0,v]$ denote the smallest bounding box containing them. \nWe define the set ${\\mathcal{T}}_S$ of {\\bf top rectangles of $S$} as follows:\n$$\n{\\mathcal{T}}_S = \\{ X \\in S :  y_1^X \\geq y_1^Y \\mbox{ for all } Y \\in S\\}.$$\nWe now prove that for $K_n$, ${\\mathcal{T}}_S$ contains a single rectangle when $n\\geq 6$.\n\n\\medskip\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[height=2in]{top-box-proof.pdf}\n    \\caption{An example to illustrate the proof of Lemma~\\ref{lonely}}\n    \\label{fig:lonely rectangle}\n\\end{figure}\n \n \n\\begin{lemma}\\label{lonely}  Suppose $n\\geq 6$ and let $S$ be any rectangle visibility representation of $K_n$.\nThen $|{\\mathcal{T}}_S  | =1.$\n\\end{lemma}\n \n\\begin{proof} By way of contradiction, suppose $|{\\mathcal{T}}_S  | \\geq 2.$ Fix any distinct rectangles $A, B \\in {\\mathcal{T}}_S$, and note that ${\\mathcal{N}}(A)$ and ${\\mathcal{N}}(B)$ are empty, as illustrated by the example in Figure~\\ref{fig:lonely rectangle}. Without loss of generality, assume that $x_1^A \\leq x_1^B$ and $y_2^A \\geq y_2^B$ (i.e., the taller rectangle is on the left). We observe the following:\n\\medskip\n \n$\\bullet \\;$ \n\n${\\mathcal{W}}(A)$ is empty:\nIf $F \\in {\\mathcal{W}}(A)$, then $F$ cannot see $B.$\n\\medskip\n \n$\\bullet \\;$ \n$|{\\mathcal{E}}(B)| \\leq 1$:\nOtherwise, fix $I$, $J \\in {\\mathcal{E}}(B)$ with $x_1^I < x_1^J$, and note that $y_1^I, y_1^J\\le y_1^B$. Then $y_2^I > y_2^B$ since $I$ must see $A$. But now $J$ cannot see $B$, a contradiction.\n\\medskip \n \n$\\bullet \\;$ \n${\\mathcal{S}}(A)={\\mathcal{S}}(B)$:\nIf $C \\in {\\mathcal{S}}(A)$ then $y_2^C \\leq y_1^A = y_1^B$. But $C$ sees $B$, so $C \\in {\\mathcal{S}}(B)$. Thus ${\\mathcal{S}}(A) \\subseteq {\\mathcal{S}}(B)$ and, similarly, ${\\mathcal{S}}(B) \\subseteq {\\mathcal{S}}(A)$. \n\\medskip\n \n  \n$\\bullet \\;$ \n$|{\\mathcal{S}}(A)| >1$:  \nOtherwise $|{\\mathcal{E}}(A)| \\geq 4$, since $n \\geq 6$ and ${\\mathcal{N}}(A)$ and ${\\mathcal{W}}(A)$ are empty.\nSince $|{\\mathcal{E}}(B)|\\leq 1$, this implies $|{\\mathcal{E}}(A) \\cap {\\mathcal{W}}(B)| \\geq 2$. Fix distinct $G$, $H \\in {\\mathcal{E}}(A) \\cap {\\mathcal{W}}(B)$ with $x_1^G  \\leq x_1^H$.  Now $y_2^H > y_2^G$ since $H$ sees $A$. But then $G$ cannot see $B$, a contradiction.\n\\medskip\n \n$\\bullet \\;$ \n$|{\\mathcal{S}}(A)| \\leq 1$:  \nOtherwise, fix distinct $C$,$D \\in {\\mathcal{S}}(A)$ with $y_1^C  \\geq y_1^D$. \n  Since ${\\mathcal{S}}(A)={\\mathcal{S}}(B),$ both $C$, $D$ see $A$ and $B$ from below. Now ${\\mathcal{E}}(A)\\cap {\\mathcal{W}}(B)$ is empty, since if $G \\in {\\mathcal{E}}(A) \\cap {\\mathcal{W}}(B)$, then $D$ cannot see $G$. Since ${\\mathcal{N}}(A)$ and ${\\mathcal{W}}(A)$ are empty and $n \\geq 6$, it follows that\n  $|{\\mathcal{E}}(A)| \\geq 3$ and thus ${\\mathcal{E}}(B) \\geq 2$, a contradiction.\n\\medskip\n\nHaving shown that  $|{\\mathcal{S}}(A)| > 1$ and $|{\\mathcal{S}}(A)| \\leq 1$, we have arrived at a contradiction, and we conclude that $|{\\mathcal{T}}_S  | =1.$\n\\end{proof}\n\n\\begin{figure}\n    \\includegraphics[height=1.9in]{extracting-operation.pdf}\n   \n    \\caption{Applying the extracting operation $S \\uparrow A$.}\n    \\label{fig:my_label}\n\\end{figure}\n\nAnother operation that will be useful is an extraction operation that can move a certain rectangle to the top row of the bounding box. Specifically, for a rectangle visibility representation $S$ of $K_n$ with bounding box $R = [0,u] \\times [0,v]$ and a rectangle $A \\in \\mathcal{T}_S$, we define $S \\uparrow A$ to be the set of rectangles in $R$ given by\t$$ S \\uparrow A = \\{f(X) : X \\in S\\},$$\nwhere $f(A) = [0,u] \\times [v-1,v],$ and where, for every $X \\not= A$, $$ f(X) = [x_1^X,x_2^X] \\times [y_1^X, \\min\\{y_2^X,v-1\\}].$$   As illustrated in Figure~\\ref{fig:my_label}, the function $f$ maps the rectangle $A$ to the top row of $R$ and maps no other rectangle to that row. Furthermore, $f(X) \\subseteq X$ for all $X \\not= A$, so the rectangles of $ S \\uparrow A$ do not overlap.\n\n\\begin{lemma}\\label{extract} For any $n \\geq 1$ let $S$ be a rectangle visibility representation of $K_n$ with bounding box $R = [0,u] \\times [0,v]$. If ${\\mathcal{T}}_S =\\{A\\}$ then $S \\uparrow A$ also represents $K_n$ and has bounding box $R$.\n\\end{lemma}\n \n \\begin{proof} Since $f$ bijectively maps the $n$ rectangles of $S$ to the $n$ rectangles of $S\\uparrow A$, and since these representations share the same bounding box $R$, it remains only to show that $f$ preserves adjacency.  \n \t\nSince ${\\mathcal{N}}(A)$ is empty and the graph is complete, $S$ is partitioned as $$ S = \\{A\\} \\cup {\\mathcal{E}}(A) \\cup {\\mathcal{W}}(A) \\cup {\\mathcal{S}}(A).$$ For any distinct $X$ and $Y$ in $S$, we claim  $f(X)$ and $f(Y)$ see each other: \n\n\\smallskip\n\n\\textbf{Case 1. $X=A$ and $Y \\in {\\mathcal{E}}(A)$.} Since ${\\mathcal{T}}_S=\\{A\\}$, note ${\\mathcal{N}}(Y)$ is empty. So ${\\mathcal{N}}(f(Y))=\\{f(A)\\}$, and $f(Y)$ sees $f(X)$ vertically. \n\n\\smallskip\n\n\\textbf{Case 2. $X=A$ and $Y \\in {\\mathcal{S}}(A)$.} Note\n${\\mathcal{S}}(f(A)) \\supseteq {\\mathcal{S}}(A)$\nand so  $f(Y)$ sees $f(X)$ vertically.\n\n\\smallskip\n\n\\textbf{Case 3. $X\\not=A$ and $Y \\in {\\mathcal{S}}(A)$.} Since ${\\mathcal{T}}_S=\\{A\\}$, any line of sight between $X$ and $Y$ must be contained in the region below the top row of $R$. The only change to this region in $S \\uparrow A$ is the removal of $A$, so $f(X)$ still sees $f(Y)$ along the original line of sight.\n\n\\smallskip\n\n\\textbf{Case 4. $\\{X,Y\\} \\subseteq {\\mathcal{E}}(A)$.} If $X$ sees $Y$ vertically, say with $X$ above $Y$, then $y_1(X) > y_1(A)$ so that $Y$ can see $A$.  Since ${\\mathcal{T}}_S=\\{A\\}$, $X$ must see $Y$ horizontally. If the only line of visibility from $X$ to $Y$ were in the top row of $R$, then $y_2^X =y_2^Y = v$ and one of $X$,$Y$ could not see $A$. Therefore, $X$ must see $Y$ in a lower row, and so $f(X)$ still sees $f(Y)$ horizontally.\n\n\\smallskip\n\n\\textbf{Case 5. $X \\in {\\mathcal{E}}(A)$ and $Y \\in {\\mathcal{W}}(A)$.} Since ${\\mathcal{T}}_S=\\{A\\}$, $X$ must see $Y$ horizontally. If $X$ sees $Y$ in any row below the top row of $R$, then $f(X)$ still sees $f(Y)$ in that same row. If $X$ {\\em only} sees $Y$ in the top row of $R$, then $y_2^A < v$. Since ${\\mathcal{T}}_S=\\{A\\}$, both $X$,$Y$ see $A$ horizontally in the top row of $A$. Since the bottom row of $f(A)$ is above the top row of $A$, now $f(X)$ sees $f(Y)$ horizontally in what was the top row of $A$.\n\n\\smallskip\n\nBy symmetry, these cover all possible cases.\n\\end{proof}\n\n\\begin{figure}[ht!]\n \\centerline{\\includegraphics[height=1.9in]{K7-height.pdf} }\n \\caption{The seven rectangles in the proof of Theorem~\\ref{htk7}.}\n\\label{fig:K7proof}\n \\end{figure}\n \n\\begin{lemma}\\label{4out} Assume $n\\geq 6$ and $K_n$ has a rectangle visibility representation with bounding box $R = [0,u] \\times [0,v]$. Then $K_n$ has a rectangle visibility representation with bounding box $R$ in which the boundary of $R$ is covered by 4 rectangles of height or width 1.\n\\end{lemma}\n\n\\begin{proof}\n\tApply\n\tLemma \\ref{extract} successively in each of the four directions. Each time, a rectangle is brought to the corresponding boundary without changing the bounding box $R$.\n\\end{proof}\n\nRecall that a \\textit{\\textbf{bar visibility graph}} $G$ is a graph representable with a set of disjoint horizontal bars in the plane, with edges between bars that have vertical lines of sight between them.  All bar visibility graphs are planar \\cite{Duchet83, Tamassia86, Wismath85}.\n\n\\begin{lemma}\nSuppose $S$ is an RV-representation for a graph $G$.  If we partition the edges of $G$ into those with vertical lines of sight and horizontal lines of sight in $S$, then the subgraphs $G_V(S)$ and $G_H(S)$ of $G$ with these edges are bar visibility graphs, and hence planar graphs. \\label{bar-planar}\n\\end{lemma}\n\n\\begin{proof}\nReplace each rectangle $A$ in $S$ with a horizontal line segment at the top edge of $A$.  This is a bar visibility representation of $G_V$.  Rotating $S$ by 90 degrees and then replacing each rectangle by its new top edge yields a bar visibility representation of $G_H$.\n\\end{proof}\n\n\\begin{theorem} \\label{htk7} The complete graph $K_7$ has $\\height(K_7)=7$.\n\\end{theorem}\n\\begin{proof}\nSuppose $S$ is an RV-representation of $K_7$ with minimum height. By Lemma \\ref{4out}, we may assume the boundary of the bounding box $R$ is covered by 4 rectangles of height or width 1, as in Figure \\ref{fig:K7proof}.  Label these $A$, $B$, $C$, and $D$ clockwise from the top.\n\nThe remaining 3 rectangles $E$, $F$, and $G$ in the interior induce a 3-clique. If the 3 edges of this clique all correspond to horizontal lines of sight, then, together with rectangles $B$ and $D$, the edges of a 5-clique are represented entirely by horizontal lines of sight.  This is a 5-clique in $G_H$, which is impossible by Lemma~\\ref{bar-planar}.  Similarly, the edges among $E$, $F$, and $G$ cannot all be vertical lines of sight. Rotating and renaming if necessary, assume $E$ sees $F$ and $G$ vertically and $F$ sees $G$ horizontally. Then $F$ and $G$ must be on the same side of $E$ and we may assume $F,G \\in {\\mathcal{S}}(E)$ and $G \\in {\\mathcal{E}}(F)$, as shown in Figure~\\ref{fig:K7proof}. \n\nThe following five horizontal lines of sight must occupy five distinct rows in $R$:\n $BD$, $BE$,  $FG$, $BF$, and $DG$. To see why, notice first that edge $BD$ must have its own row to reach all the way across the representation. The row for $BE$ must be distinct from $FG$, $BF$, and $DG$ since $F,G \\in {\\mathcal{S}}(E)$. Rows for $FG$ and $BF$ are distinct since $B,G \\in {\\mathcal{E}}(F)$. Rows for $FG$ and $DG$ are distinct since $F, D \\in {\\mathcal{W}}(G)$. Rows for $BF$ and $DG$ are distinct since $G \\in {\\mathcal{E}}(F)$.\n \nSince $A$ and $C$ each take their own row by construction, it follows that $R$ has height at least 7. The representation of $K_7$ in Figure \\ref{fig:complete-graph-examples} proves equality.\n\\end{proof}\n\n\\begin{theorem}\\label{wk7} The complete graph $K_7$ has $\\width(K_7)=8$.\n\\end{theorem}\n\\begin{proof}  If $\\width(K_7)=7$, then Lemma~\\ref{lowerbds}(iii) would guarantee an RV-representation $S$ in a $7 \\times 7$ box $R$.\nBy Lemma \\ref{4out} and Theorem \\ref{htk7}, we may assume the boundary of $R$ is covered by 4 rectangles of height or width 1. Label these $A,B,C,D$ clockwise from the top.  As before, assume $F,G \\in {\\mathcal{S}}(E)$ and $G \\in {\\mathcal{E}}(F)$. \n\nThe following six vertical lines of sight must occupy six distinct columns in $R$:\n $AC,AF,EF,EG,AG,CE$. To see why, note that edge $AC$ must have its own column to reach all the way across the representation. Any column meeting $F$ cannot meet $G$, since $G \\in {\\mathcal{E}}(F)$. Columns for $CE$,$EF$,$EG$ are distinct since $C,F,G \\in {\\mathcal{S}}(E)$. Columns for $EG$,$AG$ are distinct since $A,E \\in {\\mathcal{N}}(G)$. Columns for $AF$,$EF$ are distinct since $A,E \\in {\\mathcal{N}}(F)$. Columns for $AF$,$CE$ are distinct since $E \\in {\\mathcal{N}}(F)$. Columns for $AG$,$CE$ are distinct since $E \\in {\\mathcal{N}}(G)$.\n \n Thus $R$ has width at least 8.  By Figure \\ref{fig:complete-graph-examples}, $\\text{width}(K_7)=8$. \n\\end{proof}\n\n\\begin{corollary}\n$\\area(K_7)=56$ and $\\perimeter(K_7)=30$.\n\\end{corollary}\n\n\\begin{proof}\nTheorems~\\ref{htk7} and \\ref{wk7} show that any representation of $K_7$ requires height at least 7 and width at least 8.  Figure~\\ref{fig:complete-graph-examples} shows a representation of $K_7$ with height 7 and width 8 exactly.  Therefore this representation also has smallest area and perimeter by Lemma~\\ref{lemma-same-height-width}.\n\\end{proof}\n\n\\begin{theorem} \\label{htk8}\nThe complete graph $K_8$ has $\\rm{height}(K_8)=10$.\n\\end{theorem} \n\n\\begin{proof} Suppose $S$ is an RV-representation of $K_8$ with minimum height. By Lemma \\ref{4out}, we may assume the boundary of the bounding box $R$ is covered by 4 rectangles of height or width 1, as in Figure \\ref{4out}. Label these $A$, $B$, $C$, and $D$ clockwise from the top.\n\nRectangles $E$, $F$, $G$, and $H$ in the interior must induce a 4-clique. By Lemma~\\ref{bar-planar}, the edges in this clique that correspond to vertical lines of sight must form a triangle-free subgraph (and similarly for the horizontal edges).  Up to isomorphism, there are only two decompositions of $K_4$ into a pair of triangle-free graphs, shown in Figure~\\ref{fig:K8proof}.\n \n \\begin{figure}[ht!]\n \\centerline{\\includegraphics[height=1in]{K8-coloring.pdf}}\n \\caption{The partition of the edges of $K_8$ in Theorem~\\ref{htk8}.}\n\\label{fig:K8proof} \n \\end{figure}\n\nFirst we claim that the graph on the right of Figure~\\ref{fig:K8proof} is impossible. To see why, let the solid edges denote vertical lines of sight. Since $FG$ is horizontal and $EF$ and $EG$ are vertical, $F$ and $G$ are on the same side of $E$. Assume $F,G \\in {\\mathcal{S}}(E)$. Since $EH$ is horizontal and $FH$ and $GH$ are vertical, $F, G \\in {\\mathcal{S}}(H)$. Renaming if necessary, $E \\in {\\mathcal{E}}(H)$ and $F \\in {\\mathcal{E}}(G)$. So $x_2^G \\leq x_1^F$. But $x_1^E < x_2^G$ and $x_1^F < x_2^H$. It follows that $x_1^E < x_2^H$, contradicting that $E \\in {\\mathcal{E}}(H)$. A similar argument elimates the case when the solid edges denote horizontal lines of sight.\n\nNext we claim that the graph on the left requires $R$ to contain at least 10 rows. The graph is symmetric in solid and dotted edges, so assume the solid edges denote vertical lines of sight. Since $FG$ is horizontal and $EF$ and $EG$ are vertical, $F$ and $G$ are on the same side of $E$. Assume $F,G \\in {\\mathcal{S}}(E)$.\nSince $EH$ is horizontal and $EF$ and $FH$ are vertical, $E$ and $H$ are on the same side of $F$. So $H \\in {\\mathcal{N}}(F)$.\nSince $FH$ is vertical and $GH$ and $FG$ are horizontal, $F$ and $H$ are on the same side of $G$. Assume $F,H \\in {\\mathcal{W}}(G)$.\nSince $EG$ is vertical and $GH$ and $EH$ are horizontal, $E$ and $G$ are on the same side of $H$. So $H \\in {\\mathcal{W}}(E)$. \n\nThe following 8 horizontal lines of sight occupy distinct rows in $R$: \n $$BD, DE, EH, BH, GH, DG, FG, BF. $$\nEdge $BD$ must have its own row to reach all the way across the representation. Any row meeting $E$ cannot meet $F$ or $G$, since $F,G \\in {\\mathcal{S}}(E)$. Any row meeting $F$ cannot meet $H$, since $H \\in {\\mathcal{N}}(F)$. \nRows $DE$ and $EH$ are distinct since $D,H \\in {\\mathcal{W}}(E)$.\nRows $EH$ and $BH$ are distinct since $B,E \\in {\\mathcal{E}}(H)$.\nRows $BH$ and $GH$ are distinct since $B,G \\in {\\mathcal{E}}(H)$.\nRows $GH$ and $DG$ are distinct since $D,H \\in {\\mathcal{W}}(G)$.\nRows $DG$ and $FG$ are distinct since $F,D \\in {\\mathcal{W}}(G)$.\nRows $FG$ and $BF$ are distinct since $B,G \\in {\\mathcal{E}}(F)$.\nRows $DE$ and $BH$ are distinct since $H \\in {\\mathcal{W}}(E)$.\nRows $BH$ and $DG$ are distinct since $H \\in {\\mathcal{W}}(G)$.\nRows $DG$ and $BF$ are distinct since $F \\in {\\mathcal{W}}(G)$.\n \nSince $A$ and $C$ each take their own row by construction, $R$ has height $\\geq 10$. The representation of $K_8$ shown in Figure~\\ref{fig:complete-graph-examples} proves equality.\n\\end{proof}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=.3\\textwidth]{complete7-rectangles1.pdf} \\hspace{1cm} \\includegraphics[width=.4\\textwidth]{complete8-rectangles1.pdf}\n\\caption{A representation of $K_7$ in a $7 \\times 8$ bounding box, and a representation of $K_8$ in a $10 \\times 10$ bounding box.}\n\\label{fig:complete-graph-examples}\n\\end{figure}\n\n\\begin{theorem} \\label{wk8} \nThe complete graph $K_8$ has $\\width(K_8)=10$.\n\\end{theorem} \n\n\\begin{proof} Since, by definition, $\\width(K_8) \\geq \\rm{height}(K_8) =10$, the representation of $K_8$ shown in Figure \\ref{fig:complete-graph-examples} proves equality.\n\\end{proof}\n\n\\begin{corollary}\nThe complete graph $K_8$ has $\\area(K_8)=100$ and $\\perimeter(K_8)=40$.\n\\end{corollary}\n\n\\begin{proof}\nTheorems~\\ref{htk8} and \\ref{wk8} show that any representation of $K_8$ requires height at least 10 and width at least 10.  Figure~\\ref{fig:complete-graph-examples} shows a representation of $K_8$ with height 10 and width 10 exactly.  Therefore this representation also has smallest area and perimeter by Lemma~\\ref{lemma-same-height-width}.\n\\end{proof}\n\nNote that $K_8$ is the largest complete RVG \\cite{hutchinson1999representations}, so we can't investigate the size of RV-representations of larger complete graphs.  But using the disjoint unions of complete graphs and Lemma~\\ref{NewDisjointLem}, we can construct graphs on $n$ vertices whose RV-representations are larger than the empty graph for all $n \\geq 8$, as follows.\n\n\\begin{corollary} \\label{disjoint complete graphs}\nFix any positive integer $n$ and write $n=8q+r$ for integers $q$ and $r$ with  $0 \\leq r < 8$.  Define the graph $G_n = q K_8 + E_r$, which has $q$ disjoint copies of $K_8$ and $r$ isolated vertices.  Then $G_n$ has $n$ vertices, $\\height(G_n) = \\width(G_n)=n+2q$, $\\area(G_n)=(n+2q)^2$ and $\\perimeter(G_n)=4n+8q$.\n\\end{corollary}\n\n\\begin{proof}\nBy Lemma~\\ref{NewDisjointLem}, $\\height(G_n)= q \\cdot \\height(K_8)+ \\height(E_r)$.  By Theorem~\\ref{htk8}, $\\height(K_8)=10$, and again by Lemma~\\ref{NewDisjointLem}, $\\height(E_r)=r$.  So $\\height(G_n)=10q+r=n+2q$.\n\nSince the bounding boxes of the representations of $K_8$ and $E_r$ are square, the same argument holds for $\\width(G_n)$.  Since the representations of $G_n$ with minimum height and width are the same, these representations also yield the minimum area and perimeter of $G_n$ by Lemma~\\ref{lemma-same-height-width}.\n\\end{proof}\n\n\\begin{corollary}\nAmong all rectangle visibility graphs with $n \\geq 7$ vertices, the empty graph $E_n$ does not have the largest width, area, or perimeter.\nAmong all rectangle visibility graphs with $n \\geq 8$ vertices, the empty graph $E_n$ does not have the largest height.\n\\end{corollary}\n\n\n\\section{Directions for Further Research} \\label{Conclusions}\n\nWe conclude with a number of open problems and questions that could further this line of research.\n\n\\begin{enumerate}[wide]\n\\item We have established that for $n=7,8$ the complete graph exceeds the empty graph in area, perimeter, height, and width. We have also shown that for $n>8$, the empty graph does not maximize any of these parameters. Accordingly, it is natural to ask, in general, which rectangle visibility graph(s) with $n$ vertices have largest height, perimeter, width, and area?  Note that for $n>8$, $K_n$ is not a rectangle visibility graph \\cite{hutchinson1999representations}.  Furthermore, when $r=7$ in Corollary~\\ref{disjoint complete graphs}, we can replace $E_7$ by $K_7$ to obtain a graph with width $n+2q+1$. Are there any other graphs that beat the graph in Corollary~\\ref{disjoint complete graphs}?\n\n\\item Tables~\\ref{table-separating1} and~\\ref{table-separating-new} show that, for each pair of parameters of area, perimeter, height, and width, there are pairs of graphs that share the same number of vertices, but that are ordered oppositely by that pair of parameters.  However, it does not consider triples and quadruples of parameters.  For example, is there a pair of graphs $G_1$ and $G_2$ for which $\\area(G_1)<\\area(G_2)$ but both $\\height(G_2)<\\height(G_1)$ and $\\width(G_2)<\\width(G_1)$?\n\n\\item Tables~\\ref{table-separating2} and~\\ref{table-separating2b} show that, for each pair of parameters of area, perimeter, height, and width, there is a graph with two representations that are ordered oppositely by that pair of parameters.  However, it does not consider triples and quadruples of parameters.  For example, is there a graph $G$ that requires three distinct RV-representations to minimize its area, perimeter, and height?\n\n\\item Say that an RV-representation $S$ is \\textit{compressible} if we can delete a row or column of $S$ and still have a representation of the same graph. For a given number of vertices $n$, which graphs have the largest incompressible representations, in terms of area, perimeter, height, or width?  How large are these values?\n\n\\item We might consider additional measures of size in terms of the rectangles in an RV-representation, rather than the bounding box.  For example, for an RV-representation $S$, say $\\recarea(S)$ is the area of the largest rectangle in $S$.  Then $\\recarea(G)$ is the smallest value of $\\recarea(S)$ for any RV-representation of $G$.  Which graphs $G$ with $n$ vertices have largest $\\recarea(G)$?\n\n\\item We conjecture that if $\\height(G)=2$ then $G$ is outerplanar.  Is this true?  Can we characterize other families of graphs of specific area, perimeter, height, or width?\n\n\\item We can consider other dimensions. For dimension 1, what is the minimum length of an integer bar visibility graph on $n$ vertices?  For 3-dimensional box visibility graphs, there are many parameters measuring the size of an integer box visibility representation.  Which graphs require the largest 3-dimensional representation, as measured by these parameters?  Note that in \\cite{Fekete99} Fekete and Meijer proved that $K_{56}$ is a 3-dimensional box visibility graph.\n \n \\end{enumerate}\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{NGC3516: X-ray and UV Observations}\n\n NGC3516 contains the strongest UV absorption system known in a\nSeyfert 1 galaxy. This system contains at least two distinct\ncomponents: a broad (FWHM$\\sim$2000 km\/s) variable component and a\nnarrow ($\\sim$500 km\/s) non-variable component.  Both the broad and\nthe narrow systems contain high as well as low ionization absorption\nlines. Recent observations have shown that the broad high ionization\nabsorption lines have {\\bf \\it disappeared} since $\\sim$1992 (Koratkar\net al.  1996 and references therein, Kriss et al. 1996)\n\n We analyze a high signal-to-noise (S\/N) ROSAT PSPC archival spectrum\nof NGC3516 obtained in 1992.  The high S\/N allows the strong detection\nof both OVII and OVIII edges independently, in spite of the limited\nspectral resolution of the PSPC. A warm absorber fit to the data shows\nthat the absorber is highly ionized (U$=10^{+2.6}_{-2.1}$), and has a\nlarge column density N$_H \\sim 10^{22}$ cm$^{-2}$.\n\n\\section{The XUV Absorber}\n\n In several AGN, the X-ray and the UV absorbers were found to be one\nand the same (the `XUV' absorbers, Mathur et al. 1994, 1995). Is it\nalso true for NGC3516? The absorption systems in NGC3516 are clearly\ncomplex with multiple components (Kriss et al. ~1996). It is the {\\it\nhigh ionization, broad} absorption system that is most likely to be\nassociated with the X-ray warm absorber.  Investigation of this\nquestion is tricky, however, because the broad absorption lines have\ndisappeared. Here we argue that the XUV absorption picture is {\\it\nconsistent} with the presence of a highly ionized X-ray absorber and the\ncurrent non-detection of CIV and NV broad absorption lines (see Fig. 1).\nThe X-ray absorber {\\it MUST} have a UV signature showing OVI\nabsorption lines (Fig. 1). Since there were no simultaneous\nROSAT \\& far-UV observations in 1992, this cannot be directly determined.\nHowever, we note that in the 1995 HUT observations, OVI doublets are\nunresolved, but consistent with being  broad\n(FWHM=1076$\\pm$146 km\/s, Kriss et al. 1996).\n\\begin{figure}\n\\vspace*{-0.9in}\n\\centerline{\n\\epsscale{.75}\n\\plotone{mathurs.eps}\n}\n\\vspace*{-0.7in}\n\\caption{Ionization fractions f of OVI, OVII, OVIII, CIV and NV as a\nfunction of ionization parameter, U (with CLOUDY, Ferland 1991). The\nvertical lines define the range of U for which the ratio\nf$_{\\mbox{OVII}}$\/f$_{\\mbox{OVIII}}$ lies within the observed ROSAT\nrange. The arrows on the CIV and NV curves indicate the lower limits\nof f$_{\\mbox{CIV}}$${_>\\atop^{\\sim}}$ $3\\times 10^{-4}$ and f$_{\\mbox{NV}}$${_>\\atop^{\\sim}}$\n$3.1\\times 10^{-4}$ based on the published IUE data.  The + mark\ncorresponds to the HUT data in Kriss et al. 1996.}\n\\end{figure}\n\n We argue that the XUV absorber in NGC3516 has evolved with time (Fig.\n1). {\\bf Pre-1992:} It showed broad, high ionization CIV and NV\nabsorption lines and an X-ray ionized absorber (U ${_<\\atop^{\\sim}}$ 7). As it\nevolves, outflowing and expanding, the density falls and the\nionization parameter increases. {\\bf 1992:} CIV and NV absorption lines\ndisappeared; X-ray absorber is still present with OVI lines in the UV\n(U$\\sim$10) (No UV data available to verify). {\\bf 1995:} CIV and NV\nabsorption line remain absent; X-ray absorber is present. OVI lines\nare present, and detected with HUT (U$\\sim$13.5). {\\bf Post-1996:} We\npredict that the OVI absorption lines will disappear as the ionization\nparameter increases further (U${_>\\atop^{\\sim}}$ 20). The OVIII edge will continue\nto strengthen relative to the OVII edge.  Eventually, even the X-ray\nabsorber will also disappear.\n\n\\acknowledgments\nSM gratefully acknowledges the financial support of NASA grant\nNAGW-4490 (LTSA) and BW, TA of NASA contract NAS8-39073 (ASC).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nAn ordinary heavy baryon constitutes a pair of light quarks and a\nheavy quark. Since the charm and bottom quarks are very heavy in\ncomparison with the light quarks, it is plausible to take the limit of\nthe infinitely heavy mass of the heavy quark, i.e. $m_Q\\to \\infty$. In\nthis limit, the physics of heavy baryons become simple. The spin of\nthe heavy quark is conserved, because of its infinitely heavy mass. It\nresults in the conservation of the total spin of light quarks:\n$\\bm{J}_L \\equiv \\bm{J}-\\bm{J}_Q$, where $\\bm{J}_L$, $\\bm{J}_Q$, and\n$\\bm{J}$ denote the spin of the light-quark pair, that of the\nheavy quark, and the total spin of the heavy baryon. This is called \nthe heavy-quark spin symmetry that allows $\\bm{J}_L$ to be a\ngood quantum number. Moreover, the physics is kept intact under the\nplacement of heavy quark flavors. This is called the heavy-quark\nflavor symmetry~\\cite{Isgur:1989vq, Isgur:1991wq,\n  Georgi:1990um, Manohar:2000dt}. Then a heavy quark becomes static,\nso that it can be considered as a static color source. Its importance\nis only found in making the heavy baryon a color singlet, and\nin giving higher-order contributions arising from $1\/m_Q$ \ncorrections. Consequently, the dynamics inside a heavy baryon is\nmainly governed by the light quarks. \n\nThe flavor structure of the heavy baryon is also determined by\nthem. Since there are two light quarks inside the heavy baryon, we\nhave two different flavor  $SU_{\\mathrm{f}}(3)$ irreducible\nrepresentations, i.e. $\\bm{3}\\otimes \\bm{3}=\\overline{\\bm{3}} \\oplus\n\\bm{6}$.  In the language of a quark model, the spatial part of the  \nheavy-baryon ground state is symmetric due to the zero orbital angular\nmomentum, and the color part is totally antisymmetric. Since the\nflavor anti-triplet ($\\overline{\\bm{3}}$) is antisymmrtric, the spin\nstate corresponding to {$\\overline{\\bm{3}}$} should be antisymmetric. Thus, the\nbaryons belonging to the anti-triplet should be $J_L=0$. Similarly, the\nflavor-symmetric sextet ($\\bm{6}$) should be symmetric in spin space,\ni.e. $J_L=1$. This leads to the fact that the baryon antitriplet has\nspin $J=1\/2$, while the baryon sextet carries spin $J=1\/2$ or $J=3\/2$,\nwith the spin of the light-quark pair being coupled with the heavy\nquark spin $J_Q=1\/2$.  So, we can classify 15 different lowest-lying\nheavy baryons as shown in Fig.~\\ref{fig:1} in the case of charmed\nbaryons.   \n\\begin{figure}[htp]\n\\centering\n\\includegraphics[scale=0.35]{fig1.pdf}\\qquad\n\\caption{The anti-triplet ($\\overline{\\bm{3}}$) and sextet ($\\bm{6}$)\n  representations of the lowest-lying heavy baryons. The left panel\n  draws the weight diagram for the anti-triplet with the total spin\n  $\\frac{1}{2}$. The centered panel corresponds to that for the sextet\n  with the total spin $1\/2$ and the right panel depicts that\n  for the sextet with the total spin $3\/2$.} \n\\label{fig:1}\n\\end{figure}\n\nRecently, there has been a series of new experimental data on the\nspectra of heavy baryons~\\cite{Aaltonen:2007ar, Chatrchyan:2012ni,\n  Abazov:2008qm, Kuhr:2011up, Aaij:2012da, Aaij:2013qja, Aaij:2014esa,\n  Aaij:2014lxa, Aaij:2014yka}, which renewed interest\nin the physics of the heavy baryons. The lowest-lying singly heavy baryons\nare now almost classified except for $\\Omega_b ^\\ast$. In the meanwhile, the\nLHCb Collaboration has announced the first finding of two heavy\npentaquarks, $P_c(4380)$ and $P_c(4450)$ ~\\cite{Aaij:2015tga,\n  Aaij:2016phn, Aaij:2016ymb, Aaij:2016iza}. Very recently, the\nfive excited $\\Omega_c$ baryons were reported~\\cite{Aaij:2017nav}, among\nwhich the four of them was confirmed by the Belle\nexperiment~\\cite{Yelton:2017qxg}. Interestingly the two of the excited\n$\\Omega_c$s, i.e. $\\Omega_c(3050)$ and $\\Omega_c(3119)$, have very narrow\nwidths: $\\Gamma_{\\Omega_c(3050)}=(0.8\\pm0.2\\pm0.1)\\,\\mathrm{MeV}$ and\n$\\Gamma_{\\Omega_c(3119)}=(1.1\\pm0.8\\pm0.4)\\,\\mathrm{MeV}$.   \n\nWhile there is a great deal of theoretical approaches for the\ndescription of heavy baryons, we will focus on a pion mean-field\napproach in the present short review. This mean-field \napproach was first proposed by E. Witten in this seminal\npapers~\\cite{Witten:1979kh,Witten:1983}, where he asserted that in the\nlimit of the large number of colors ($N_c$) the nucleon can be\nregarded as a bound state of $N_c$ \\textit{valence} quarks in a pion\nmean field with a hedgehog symmetry~\\cite{Pauli:1942kwa,\n  Skyrme:1961vq}. Since a baryon mass \nis proportional to $N_c$ whereas the quantum fluctuation around the\nsaddle point of the pion field is suppressed by $1\/N_c$, the\nmean-field approach is a rather plausible method for explaining\nproperties of baryons. The presence of $N_c$ \\textit{valence} quarks\nin this large $N_c$ limit, which consist of the lowest-lying baryons,    \nproduce the pion mean fields by which they are influenced\n\\emph{self-consistently}. This picture is very similar to a Hartree \napproximation in many-body theories. Witten also showed how to\nconstruct the mean-field theory for the baryon schematically in\ntwo-dimensional quantum chromodynamics (QCD). Though his idea was\ncriticized sometimes ago by S. Coleman~\\cite{Coleman} because of its\ntechnical difficulties, it is worthwhile to pursue it to see how far\nwe can describe the structure of the baryon in the pion mean-field\napproach.   \n\nThe chiral quark-soliton model ($\\chi$QSM)~\\cite{Diakonov:1987ty,\n  Christov:1995vm, Diakonov:1997sj} has been\nconstructed based on Witten's argument. The $\\chi$QSM starts from the\neffective chiral action (E$\\chi$A) that was derived from the instanton\nvacuum~\\cite{Diakonov:1983hh, Diakonov:1985eg}. The E$\\chi$A respects\nchiral symmetry and its spontaneous breakdown, in which the essential\nphysics of the lowest-lying hadrons consists. One can derive the\nclassical energy of the nucleon by computing the nucleon correlation\nfunction in Euclidean space, taking the Euclidean time to go to\ninfinity. Minimizing the classical energy self-consistently in the\nlarge $N_c$ limit with the $1\/N_c$ meson quantum fluctuations\nsuppressed, we obtain the classical mass and the self-consistent\nprofile function of the chiral soliton. While we ignore the $1\/N_c$\nquantum fluctuations around the saddle point of the soliton field, we\nneed to take into account the zero modes that do not change the\nsoliton energy. Since the soliton with hedgehog symmetry is not\ninvariant under translational, rotational and isotopic\ntransformations, we impose these symmetry properties on the\nsoliton and obtain a completely new solution with the same classical\nenergy. Because of the hedgehog symmetry, an  \n$\\mathrm{SU(2)}$ soliton needs to be embedded into the isospin\nsubgroup of the flavor\n$\\mathrm{SU(3)}_{\\mathrm{f}}$~\\cite{Witten:1983}, which was   \nalready utilized by various chiral soliton\nmodels~\\cite{Guadagnini:1983uv, Mazur:1984yf, Jain:1984gp}. This\ncollective quantization of the chiral soliton leads to the collective\nHamiltonian with effects of flavor $\\mathrm{SU(3)}_{\\mathrm{f}}$\nsymmetry breaking. The $\\chi$QSM has one salient feature: the  \nright hypercharge is constrained to be $Y'=N_c\/3$ imposed by the $N_c$ \nvalence quarks. This right hypercharge selects allowed representations\nof light baryons such as the baryon octet ($\\bm{8}$), the decuplet\n($\\bm{10}$), etc. The $\\chi$QSM was successfully applied to the\nproperties of the lowest-lying light baryons such as the mass\nsplittings~\\cite{Blotz:1992pw, Yang:2010fm}, the form\nfactors~\\cite{Kim:1995mr, Silva:2001st, Ledwig:2010tu},\nthe magnetic moments~\\cite{Kim:1995ha, Wakamatsu:1996xm, Kim:1997ip,\n  Kim:2005gz}, hyperon \nsemileptonic decays~\\cite{Ledwig:2008ku, Yang:2015era}, \nparton distributions~\\cite{Diakonov:1996sr, Wakamatsu:2003wg},\ntransversities of the nucleon~\\cite{Kim:1995bq, Kim:1996vk,\n  Schweitzer:2001sr}, generalized parton\ndistributions~\\cite{Goeke:2001tz}, and so on.     \n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[scale=0.25]{fig2.pdf}\n\\caption{Schematic picture of a heavy baryon. The $N_c-1$ valence\n  quarks are filled in the lowest-lying valence level $K^P=0^+$ with\n  the heavy quark stripped off. $K^P$ denotes the grand spin which we\n  will explain later and  $P$ is the corresponding parity of the\n  level. The presence of the valence quarks will interact with the sea\n  quarks filled in the Dirac sea each other. This interaction will\n  bring about the pion mean field.} \n\\label{fig:2}\n\\end{figure}\nVery recently, Ref.~\\cite{Yang:2016qdz} extended a mean-field \napproach to describe the masses of singly heavy baryons, being  \nmotivated by Ref.~\\cite{Diakonov:2010tf}. A singly heavy baryon\nconstitutes a heavy baryon and $N_c-1$ light valence quarks (see\nFig.~\\ref{fig:2}). In the limit of $m_Q\\to\\infty$, the heavy quark can\nbe considered as a static color source. Thus, the dynamics inside a\nheavy baryon is governed by the $N_c-1$ valence quarks. The presence\nof the $N_c-1$ valence quarks will produce the pion mean fields as in\nthe case of the light baryons. However, there is one very significant\ndifference.   \nthe constraint right hyper charge is taken to be $Y' =(N_c-1)\/3$ and\nallows the lowest-lying representations: the baryon anti-triplet\n($\\overline{\\bm{3}}$), the baryon sextet ($\\bm{6}$), the baryon\nanti-decapentaplet ($\\overline{\\bm{15}}$). The model reproduced\nsuccessfully the mass splitting of the baryon anti-triplet and sextet\nin both the charm and bottom sectors. In addition, the mass of the\n$\\Omega_b^\\ast $ baryon, which has not yet found, was predicted. The model\nwas further extended by including the second-order perturbative\ncorrections of flavor $SU_{\\mathrm{f}}(3)$ symmetry\nbreaking~\\cite{Kim:2018xlc}. The magnetic moments \nbaryons~\\cite{Yang:2018uoj} and electromagnetic form\nfactors~\\cite{Kim:2018nqf} of the singly heavy baryons were also\nstudied within the same framework. The $\\chi$QSM was also used to\ninterpret the five $\\Omega_c$ baryons newly found by the LHCb\nCollaboration~\\cite{Kim:2017jpx, Kim:2017khv}. Within the present\nframework, two of the $\\Omega_c$s with the smaller widths are\nclassified as the members of the baryon $\\overline{\\bm{15}}$, whereas\nall other $\\Omega_c$'s belong to the excited baryon sextet. The widths\nwere quantitatively well reproduced without any free parameter. In the\npresent work, we will review briefly these recent investigations on the singly\nheavy baryons. \n\nWe sketch the present work as follows: In Section\nII, we review the general formalism of the $\\chi$QSM for singly heavy\nbaryons. In Section III, we examine the mass splittings of the heavy\nbaryons, emphasizing the discussion of the effects of\n$\\mathrm{SU(3)}_{\\mathrm{f}}$ breaking. In Section IV, we discuss the\nrecent results of the magnetic moments and electromagnetic form\nfactors of the heavy baryons. In Section V, we briefly introduce a\ntheoretical interpretation of the excited $\\Omega_c$ baryons found by the\nLHCb, based on the present mean-field approach. The final Section is\ndevoted to the conclusions and outlook.\n\n\\section{The chiral quark-soliton model for singly heavy baryons} \nIn the present approach, a heavy baryon is considered as a bound state\nof the $N_c-1$ valence quarks in the pion mean field with a heavy\nquark stripped off from the valence level. Thus, the correlation\nfunction of the heavy baryon can be expressed in terms of the $N_c-1$\nvalence quarks\n\\begin{align}\n\\Pi_{B}(0, T) = \\langle J_B (0, T\/2) J_B^\\dagger\n  (0,-T\/2) \\rangle_0 = \\frac{1}{Z}\\int  \\mathcal{D} U\n  \\mathcal{D}\\psi^\\dagger \n  \\mathcal{D}\\psi J_B(0,T\/2) J_B^\\dagger (0,-T\/2)\n  e^{\\int d^4 x\\,\\psi^\\dagger (i\\rlap{\/}{\\partial} + i\n  MU^{\\gamma_5}+ i \\hat{m})\\psi}  ,   \n\\label{eq:corr1}\n\\end{align}\nwhere $J_B$ denotes the light-quark current with the $N_c-1$ \nlight quarks for a heavy baryon $B$\n\\begin{align}\nJ_B(\\bm{x}, t) = \\frac1{(N_c-1)!}\n  \\varepsilon^{\\beta_1\\cdots\\beta_{N_c-1}} \\Gamma_{J'J_3',TT_3}^{\\{f\\}}\n  \\Psi_{\\beta_1f_1}(\\bm{x}, t) \\cdots \\Psi_{\\beta_{N_c-1}f_{N_c-1}}\n  (\\bm{x}, t).      \n\\end{align}\n$\\beta_i$ stand for color indices and\n$\\Gamma_{J'J_3',TT_3}^{\\{f_1\\cdots f_{N_c-1}\\}}$ represents a matrix \nwith both flavor and spin indices. $J'$ and $T$ are the spin and\nisospin of the heavy baryon, respectively. $J_3'$ and $T_3$ are their\nthird components, respectively. The notation $\\langle \\cdots \\rangle_0$ in\nEq.~(\\ref{eq:corr1}) is the vacuum expectation value, $M$\nthe dynamical quark mass, and the chiral field $U^{\\gamma_5}$ is\ndefined as    \n\\begin{align}\nU^{\\gamma_5} = U\\frac{1+\\gamma_5}{2} + U^\\dagger \\frac{1-\\gamma_5}{2}   \n\\end{align}\nwith \n\\begin{align}\nU = \\exp(i\\pi^a \\lambda^a).  \n\\end{align}\nHere, $\\pi^a$ represents the pseudo-Goldstone boson field and\n$\\hat{m}$ denotes the flavor matrix of the current quarks, written as\n$\\hat{m}=\\mathrm{diag}(m_{\\mathrm{u}},\\,m_{\\mathrm{d}},\\,m_{\\mathrm{s}})$. We \nassume isospin symmetry, i.e. $m_{\\mathrm{u}}=m_{\\mathrm{d}}$. Since\nthe strange current quark mass is small enough, we will treat it\nperturbatively.   \n\nIntegrating over the quark fields, we derive the correlation function\nas \n\\begin{align}\n\\Pi_{B}(0, T) =\n  \\frac{1}{Z}\\Gamma_{J'J_3',TT_3}^{\\{f\\}}\\Gamma_{J'J_3',TT_3}^{\\{g\\}*} \\int\n  \\mathcal{D} U \\prod_{i=1}^{N_c-1}  \\left\\langle 0,T\/2\\left|\n  \\frac1{D(U)} \\right|0,-T\/2\\right\\rangle\n  e^{-S_{\\mathrm{eff}}(U)}, \n\\label{eq:corr2}\n\\end{align}  \nwhere the single-particle Dirac operator $D(U)$ is defined as \n\\begin{align}\nD(U) = i\\gamma_4 \\partial_4 + i\\gamma_k \\partial_k + i MU^{\\gamma_5} +\n  i \\hat{m}   \n\\end{align}\nand $S_{\\mathrm{eff}}$ is the effective chiral action written\nas \n\\begin{align}\nS_{\\mathrm{eff}} = -N_c \\mathrm{Tr}\\log D(U).  \n\\label{eq:effecXac}\n\\end{align}\nEquation~(\\ref{eq:corr2}) can be schematically depicted as\nFig.~\\ref{fig:2}. It consists of two different terms: The first and\nsecond ones are respectively called the \\textit{valence-quark\n  contribution} and \\textit{sea-quark contribution} within the\n$\\chi$QSM.  \n\\begin{figure}[htp]\n\\centering\n\\includegraphics[scale=0.3]{fig3.pdf}\\qquad\n\\caption{Correlation function for a heavy baryon} \n\\label{fig:3}\n\\end{figure}\nWhen the Euclidean time $T$ is taken from $-\\infty$ to $\\infty$, \nthe correlation function picks up the ground-state\nenergy~\\cite{Diakonov:1987ty, Christov:1995vm} \n\\begin{align}\n\\lim_{T\\to\\infty} \\Pi_B(T) \\sim \\exp[-\\left\\{(N_c-1) E_{\\mathrm{val}} +\n  E_{\\mathrm{sea}}\\right\\} T],   \n\\end{align}\nwhere $E_{\\mathrm{val}}$ and $E_{\\mathrm{sea}}$ the valence and\nsea quark energies. Minimizing self-consistently the energies around\nthe saddle point of the chiral field $U$\n\\begin{align}\\label{eq:sol}\n\\left.\\frac{\\delta}{\\delta U}[ (N_c-1) E_{\\mathrm{val}} +\n  E_{\\mathrm{sea}}]\\right|_{U_c} = 0,    \n\\end{align}\nwe get the classical soliton mass \n\\begin{align}\\label{eq:solnc}\nM_{\\mathrm{sol}} = (N_c-1) E_{\\mathrm{val}}(U_c) + E_{\\mathrm{sea}}(U_c).  \n\\end{align}\nNote that a singly heavy baryon has a heavy quark, so its classical\nis expressed as the sum of the classical and heavy-quark masses\n\\begin{align}\nM_{\\mathrm{cl}} = M_{\\mathrm{sol}} + m_Q.  \n\\label{eq:classical_mass}\n\\end{align}\nWe want to mention that $m_Q$ is the \\textit{effective} heavy quark\nmass that is different from that of QCD and will be absorbed\nin the center mass of each representation. \n\nThe rotational excitations of the soliton with $N_c-1$ valence quarks\nwill produce the lowest-lying heavy baryons. To keep the hedgehog\nsymmetry, the SU(2) soliton $U_c(\\bm{r})$ will be embedded into\nSU(3)~\\cite{Witten:1983}  \n\\begin{align}\nU(\\bm{r}) = \\begin{pmatrix}\nU_c (\\bm{r}) & 0 \\\\\n0 & 1\n\\end{pmatrix}.\n\\label{eq:embed}\n\\end{align}\nAs mentioned in Introduction, we consider explicitly the rotational zero\nmodes.  Assuming that the soliton $U(\\bm{r})$ in Eq.(\\ref{eq:embed})\nrotates slowly, we apply the rotation matrix $A(t)$ in\n$\\mathrm{SU}_{\\mathrm{f}}(3)$ space    \n\\begin{align}\nU(\\bm{r},\\,t) = A(t) U(\\bm{r}) A^\\dagger (t).  \n\\end{align}\nThen, we can derive the collective Hamiltonian for heavy\nbaryons\n\\begin{align}\nH =& H_{\\mathrm{sym}} + H^{(1)}_{\\mathrm{sb}} +  H^{(2)}_{\\mathrm{sb}},\n\\end{align}\nwhere $H_{\\mathrm{sym}}$ represents the flavor SU(3) symmetric part, \n$H^{(1)}_{\\mathrm{sb}}$ and $H^{(2)}_{\\mathrm{sb}}$ the\nSU(3) symmetry-breaking parts respectively to the first and second\norders. $H_{\\mathrm{sym}}$ is expressed\nas  \n\\begin{align} \nH_{\\mathrm{sym}}=M_{\\mathrm{cl}}+\\frac{1}{2I_{1}}\\sum_{i=1}^{3}\n\\hat{J}^{2}_{i} +\\frac{1}{2I_{2}}\\sum_{a=4}^{7}\\hat{J}^{2}_{a},  \n\\end{align} \nwhere $I_{1}$ and $I_{2}$ are the moments of inertia of the\nsoliton and the operators  $\\hat{J}_{i}$ denote the SU(3) \ngenerators. We get the eigenvalue of the quadratic Casimir operator\n$\\sum_{i=1}^8 J_i^2$ in the $(p,\\,q)$ representation, given as    \n\\begin{align}\nC_2(p,\\,q) = \\frac13 \\left[p^2 +q^2 + pq + 3(p+q)\\right],   \n\\label{eq:Casimir}\n\\end{align}\nwhich leads to the eigenvalues of $H_{\\mathrm{sym}}$ \n\\begin{align} \nE_{\\mathrm{sym}}(p,q) = M_{\\mathrm{cl}}+ \\frac{1}{2I_{1}} J_L(J_L+1) \n+\\frac{1}{2I_{2}}\\left[C_2(p,\\,q) - J_L(J_L+1)\\right] \n-\\frac{3}{8I_{2}} {Y'}^{2}.\n\\label{eq:RotEn}\n\\end{align} \nThe right hypercharge $Y'$ is constrained by the $N_c-1$ valence\nquarks inside a singly heavy baryon, i.e. $Y'=(N_c-1)\/3$. The\ncorresponding collective wave functions of the singly heavy baryon is\nthen obtained as  \n\\begin{align} \n\\psi_B^{({\\mathcal{R}})}(JJ_3,J_L;A)=\n\\sum_{m_{3}=\\pm1\/2}C^{J J_3}_{J_{Q} m_{3} J_L \nJ_{L3}} \\chi_{m_{3}} \\sqrt{\\mathrm{dim}(p,\\,q)}\n(-1)^{-\\frac{ Y'  }{2}+{J}_{L3}}\n  D^{(\\mathcal{R})\\ast}_{(Y,T,T_3)(Y'  ,J_L,-J_{L3})}(A),  \n\\label{eq:waveftn}\n\\end{align} \nwhere \n\\begin{align}\n\\mathrm{dim}(p,\\,q) = (p+1)(q+1)\\left(1+\\frac{p+q}{2}\\right).  \n\\end{align}\n $J$ and $J_3$ in Eq.~(\\ref{eq:waveftn}) are the spin angular\n momentum and its third component of the heavy baryon, respectively.   \n$J_L$ and $J_{Q}$ represent the soliton spin and\nheavy-quark spin, respectively. ${J_{L3}}$ and ${m_{3}}$ are the\ncorresponding third components, respectively. Since the spin operator\nfor the heavy baryon is given as \n\\begin{align}\n\\label{eq:quantum}\n\\bm{J}=\\bm{J}_{Q}+\\bm{J}_L,\n\\end{align} \nthe relevant Clebsch-Gordan coefficients appear in\nEq.(\\ref{eq:waveftn}). The SU(3) Wigner $D$ function in\nEq.(\\ref{eq:waveftn}) means just the wave-function for the quantized\nsoliton with the $N_c-1$ valence quarks, and $\\chi_{m_3}$\nis the Pauli spinor for the heavy quark. $\\mathcal{R}$ designates a\nSU(3) irreducible representation corresponding to $(p,\\,q)$. \nSince the soliton is coupled to the heavy quark, we finally obtain the\nthree lowest-lying representations illustrated in\nFig.~\\ref{fig:1}. In the limit of $m_Q\\to\\infty$, the two sextet\nrepresentations are degenerate. One needs to introduce a \nhyperfine spin-spin interaction to lift this degeneracy. As will be\ndiscussed soon, this hyperfine interaction will be determined by using\nthe experimental data on the masses of heavy baryons.    \n\nIn the present zero-mode quantization scheme, we find the following\nthe two important selection rule. The allowed SU(3) representations\nmust contain states with $Y'=(N_c-1)\/3$ and the isospin $\\bm{T}$ of\nthe states with $Y'=(N_c-1)$\/3 are coupled with the soliton so that we\nhave a singlet $\\bm{K}=\\bm{T}+\\bm{J}_{L}=\\bm{0}$, where $\\bm{K}$ is called \nthe grand spin. The lowest-lying heavy baryons have the grand spin\n$K=0$, that is, we must have always $J_{L}=T$ with $Y'=(N_c-1)\/3$ for the\nground-state heavy baryons as shown in fig.~\\ref{fig:4}. \n\\begin{figure}[htp]\n\\centering\n\\includegraphics[scale=0.2]{fig4.pdf}\\qquad\n\\caption{The baryon anti-triplet has the  $J_{L}=T=0$ state with $Y'=2\/3$ whereas\n  the baryon sextet contains the  $J_{L}=T=1$ state with $Y'=2\/3$.}\n\\label{fig:4}\n\\end{figure}   \n\nAn observable of the heavy baryon can be expressed in general as a\nthree-point correlation function\n\\begin{align}\n\\langle B,\\,p'| J_\\mu(0) |B,\\,p\\rangle  &= \\frac1{\\mathcal{Z}}\n  \\lim_{T\\to\\infty} \\exp\\left(i p_4\\frac{T}{2} - i p_4'\n  \\frac{T}{2}\\right) \\int d^3x d^3y \\exp(-i \\bm{p}'\\cdot \\bm{y} + i\n  \\bm{p}\\cdot \\bm{x}) \\cr\n& \\hspace{-1cm} \\times \\int \\mathcal{D}U\\int \\mathcal{D} \\psi \\int\n  \\mathcal{D} \\psi^\\dagger J_{B}(\\bm{y},\\,T\/2) \\psi^\\dagger(0)\n  \\gamma_4 \\Gamma \\mathcal{O} \\psi(0) J_B^\\dagger (\\bm{x},\\,-T\/2)\n  \\exp\\left[-\\int d^4 z   \\psi^\\dagger iD(U) \\psi\\right],\n  \\label{eq:3corrftn}\n\\end{align}\nwhere $\\Gamma$ and $\\mathcal{O}$ represent respectively generic Dirac\nspin and flavor matrices. Computing Eq.~(\\ref{eq:3corrftn}), one can  \nstudy heavy baryonic observables such as form factors, magnetic moments,\naxial-vector constants, etc. For the detailed formalism, we refer to\nRefs.~\\cite{Kim:1995mr, Christov:1995vm}. \n\n\\section{Mass splittings  of the singly heavy baryons}\nWe first discuss the mass splittings of the singly heavy baryons. In\norder to obtain the mass splittings, one should include the\nsymmetry-breaking part of the collective\nHamiltonian~\\cite{Blotz:1992pw, Christov:1995vm} \n\\begin{align} \nH^{(1)}_{\\mathrm{sb}} \n&= \\overline{\\alpha} D^{(8)}_{88}+ \\beta \\hat{Y}\n+ \\frac{\\gamma}{\\sqrt{3}}\\sum_{i=1}^{3}D^{(8)}_{8i}\n\\hat{J}_{i},\n\\label{sb}\n\\end{align}\nwhere\n\\begin{align} \n\\overline{\\alpha} = \\left (-\\frac{\\overline{\\Sigma}_{\\pi N}}{3m_0}+\\frac{\n  K_{2}}{I_{2}}Y'  \n\\right )m_{\\mathrm{s}},\n \\;\\;\\;  \\beta=-\\frac{ K_{2}}{I_{2}}m_{\\mathrm{s}}, \n\\;\\;\\;  \\gamma = 2\\left (\n  \\frac{K_{1}}{I_{1}}-\\frac{K_{2}}{I_{2}}   \\right ) m_{\\mathrm{s}}.\n\\label{eq:alphaetc}\n\\end{align}\nThe parameters $\\overline{\\alpha}$, $\\beta$, and\n$\\gamma$ are the essential ones in determining the masses of the\nlowest-lying singly heavy baryons, which are expressed in terms of the\nmoments of inertia $I_{1,\\,2}$ and $K_{1,\\,2}$. However, we do not\nneed to fit them, since they are related to $\\alpha$,\n$\\beta$, and $\\gamma$ in the light-baryon sector. \nThe valence parts are only different from those in the light baryon\nsector by the color factor $N_c-1$. So, we need to replace $N_c$ by $N_c-1$\nin the valence parts of all the relevant dynamical parameters\ndetermined in the light-baryon sector. The valence part of\n$\\overline{\\Sigma}_{\\pi N}$ is just the $\\pi N$ sigma term with\ndifferent $N_c$ factor: $\\overline{\\Sigma}_{\\pi N} = (N_c-1)N_c^{-1}\n\\Sigma_{\\pi    N}$, where $\\Sigma_{\\pi N} = (m_u+m_d)\\langle N|\n\\bar{u} u +  \\bar{d} d|N\\rangle = (m_u+m_d) \\sigma$. On the other\nhand, the sea parts should be kept intact as in the light baryon\nsector. \n\nThe dynamical parameters $\\alpha$, $\\beta$ and $\\gamma$ have been\nfixed by using the experimental data on the baryon octet masses and\na part of the baryon decuplet and anti-decuplet masses with  \nisospin symmetry breaking effects~\\cite{Yang:2010id}. The\nvalues of $\\alpha$, $\\beta$, and $\\gamma$ have been obtained by\nthe $\\chi^2$ fit~\\cite{Yang:2010fm}\n\\begin{align}\n\\alpha  =  -255.03\\pm5.82 \\;{\\rm MeV},\\;\\;\\;\n\\beta  =  -140.04\\pm3.20  \\;{\\rm MeV},\\;\\;\\;\n\\gamma  =  -101.08\\pm2.33 \\;{\\rm MeV},\n\\label{eq:abrNumber}\n\\end{align}\nWhile $\\beta$ and $\\gamma$ are not required to be changed in the\nheavy-baryon sector, $\\alpha$ should be modified by\n\\begin{align}\n  \\label{eq:alpha}\n\\overline{\\alpha} = \\rho \\alpha,\n\\end{align}\nwhere $\\rho=(N_c-1)\/N_c$.\nHowever, there is a caveat when one uses the values of\nEq.~\\eqref{eq:abrNumber}. As mentioned above, only the valence parts\nshould be modified, while the scaling in Eq.~\\eqref{eq:alpha} changes\nthe sea part too.  To compensate this we choose $\\rho \\approx 0.9$. If\none computes the parameters $\\overline{\\alpha}$, $\\beta$, and $\\gamma$\nin a self-consistent way, we do not have this problem~\\cite{Kim:2018xlc}.\n\nConsidering the first-order perturbative corrections of $m_{\\mathrm{s}}$, \none can express the masses of the singly heavy baryons in\nrepresentation $\\mathcal{R}$ as  \n\\begin{align}\nM_{B,\\mathcal{R}}^Q = M_{\\mathcal{R}}^Q + M_{B,\\mathcal{R}}^{(1)}  \n\\label{eq:FirstOrderMass}\n\\end{align}\nwith\n\\begin{align}\nM_{\\mathcal{R}}^Q = m_Q + E_{\\mathrm{sym}}(p,q).  \n\\end{align}\nHere, $M_{\\mathcal{R}}^Q$ is the center mass of a heavy baryon in\nrepresentation $\\mathcal{R}$. $E_{\\mathrm{sym}}(p,q)$ is the\neigenvalue energy of the symmetric part of the collective Hamiltonian\ndefined in Eq.~\\eqref{eq:RotEn}. Note that the lower index $B$\ndesignates a certain baryon in a specific representation\n$\\mathcal{R}$. The upper index $Q$ denotes either the charm sector\n($Q=c$) or the bottom sector ($Q=b$). \nThen the center masses for the anti-triplet and sextet representations\nare obtained as \n\\begin{align}\nM_{\\overline{\\bm{3}}}^Q = M_{\\mathrm{cl}} + \\frac1{2I_2}, \\;\\;\\; \nM_{\\bm{6}}^Q = M_{\\overline{\\bm{3}}}^Q +  \\frac1{I_1},\n\\end{align}\nwhere $M_{\\mathrm{cl}}$ was defined in Eq.~\\eqref{eq:classical_mass}.\nThe second term in Eq.~(\\ref{eq:FirstOrderMass}), which arises from\nthe linear-order $m_{\\mathrm{s}}$ corrections, is proportional to the\nhypercharge of the soliton with the light-quark pair \n\\begin{align}\nM^{(1)}_{B,{\\cal{R}}} = \\langle B, {\\cal{R}} | H_{\\mathrm{sb}}^{(1)} \n| B, {\\cal{R}} \\rangle  = Y\\delta_{{\\cal{R}}},\n\\end{align}\nwhere\n \\begin{align} \n\\delta_{\\overline{\\bm{3}}}=\\frac{3}{8}\\overline{\\alpha}+\\beta, \\;\\;\\;\\;\n\\delta_{\\bm{6}}=\\frac{3}{20}\n  \\overline{\\alpha}+\\beta-\\frac{3}{10}\\gamma.  \n\\label{eq:deltas}\n\\end{align}\nFinally, we arrive at the expressions for the masses of the\nlowest-lying baryon anti-triplet and sextet as follows \n\\begin{align}\nM_{B,\\overline{\\bm{3}}}^Q = M_{\\overline{\\bm{3}}}^Q  +\n                     Y \\delta_{\\overline{\\bm{3}}} ,\\;\\;\\;\nM_{B,\\bm{6}}^Q =M_{\\bm{6}}^Q  + Y  \\delta_{\\bm{6}}, \n  \\label{eq:firstms}\n\\end{align}\nwith the linear-order $m_{\\mathrm{s}}$ corrections taken into account. \n\nSince the baryon sextet with spin 1\/2 and 3\/2 are degenerate, we need\nto remove the degeneracy by introducing the hyperfine spin-spin\ninteraction Hamiltonian~\\cite{Zeldovich}. Typically, the hyperfine\nHamiltonian is written as \n\\begin{align}\nH_{LQ} = \\frac{2}{3}\\frac{\\kappa}{m_{Q}\\,M_{\\mathrm{sol}}}\\bm{J}_L\n\\cdot  \\bm{J}_{Q}  \n= \\frac{2}{3}\\frac{\\varkappa}{m_{Q}}\n\\bm{J}_{L} \\cdot \\bm{J}_{Q}, \n\\label{eq:ssinter}\n\\end{align}\nwhere $\\kappa$ stands for the flavor-independent hyperfine coupling.\n$M_{\\mathrm{sol}}$ has been incorporated into an unknown \ncoefficient $\\varkappa$ that will be fixed by using the experimental\ndata. . The Hamiltonian $H_{LQ}$ does not affect the \n$\\overline{\\bm{3}}$ states with $J_L=0$. On the other hand, \nthe baryon sextet acquire additional contribution from $H_{LQ}$ which\nbring about the splitting between different spin states \n\\begin{align}\nM_{B,{\\bm{6}}_{1\/2}}^{Q} =  \nM_{B,\\bm{6}}^Q\\;-\\;\\frac{2}{3}\\frac{\\varkappa}{m_{Q}}, \n\\;\\;\\;\nM_{B,{\\bm{6}}_{3\/2}}^{Q}  = \nM_{B,\\bm{6}}^Q\\;+\\;\\frac{1}{3} \\frac{\\varkappa}{m_{Q}}, \n\\label{eq:Csextet}\n\\end{align}\nwhich leads to the splitting \n\\begin{align}\nM_{B,{\\bm{6}}_{3\/2}}^{Q}\\;-\\; M_{B,{\\bm{6}}_{1\/2}}^{Q}  =\n  \\frac{\\varkappa}{m_{Q}}  . \n\\label{eq:DCsextet}\n\\end{align}\nThe numerical values of $\\varkappa\/m_Q$ were determined by using the center\nvalues of the masses of the baryon sextet~\\cite{Yang:2016qdz} \n\\begin{align}\n\\frac{\\varkappa}{m_c} = (68.1\\pm 1.1)\\,\\mathrm{MeV},\\;\\;\\;\n\\frac{\\varkappa}{m_b} = (20.3\\pm 1.0)\\,\\mathrm{MeV}.\n  \\label{eq:kappavalue}\n\\end{align}\nNote that $\\varkappa$ is flavor-independent. So, knowing the ratio\n$m_c\/m_b$, one can extract the value of $\\varkappa$ from\nEq.~\\eqref{eq:kappavalue}.\n\nWe now present the numerical results of the masses of the heavy\nbaryons~\\cite{Yang:2016qdz}. Using the values of $\\overline{\\alpha}$,\n$\\beta$, and $\\gamma$, we can immediately determine the values of\n$\\delta_{\\overline{\\bm{3}}}$ and $\\delta_{\\bm{6}}$ defined in\nEq.~\\eqref{eq:deltas} \n\\begin{align}\n\\delta_{\\overline{\\bm{3}}} = (-203.8\\pm 3.5)\\,\\mathrm{MeV},\\;\\;\\;\n\\delta_{\\bm{6}} = (-135.2\\pm 3.3)\\,\\mathrm{MeV}.\n\\end{align}\nIncluding the results of $\\varkappa\/m_c$ and $\\varkappa\/m_b$, we can\nobtain the numerical results of the heavy baryon masses. \nIn Table~\\ref{tab:1} and Table~\\ref{tab:2} the numerical results of\nthe charmed and bottom baryon masses are presented, respectively. They\nare in good agreement with the experimental data taken from \nRef.~\\cite{PDG2017}. The mass of $\\Omega_b^*$ is still experimentally\nunknown. Thus, the prediction of its mass is given as \n\\begin{align}\nM_{\\Omega_b^*} = (6095.0\\pm 4.4)\\,\\mathrm{MeV}.  \n\\end{align}\nThe uncertainties in Tables~\\ref{tab:1} and \\ref{tab:2} are due to\nthose in $\\overline{\\alpha}$, $\\beta$, $\\gamma$, and $\\varkappa\/m_Q$. \n\\begin{table}[htp]\n\\begin{centering}\n\\begin{tabular}{c|ccc}\n\\hline \\hline\n$\\mathbf{\\mathcal{R}}_{J}^{Q}$ \n& $B_{c}$ \n& Mass \n& Experiment\n\\tabularnewline[0.1em]\n\\hline \n\\multirow{2}{*}{$\\mathbf{\\overline{3}}_{1\/2}^{c}$} \n& $\\Lambda_{c}$ \n& $2272.5 \\pm 2.3$\n& $2286.5 \\pm 0.1$\n\\tabularnewline\n& $\\Xi_{c}$ \n& $2476.3 \\pm 1.2$\n& $2469.4 \\pm 0.3$\n\\tabularnewline\n\\hline \n\\multirow{3}{*}{$\\mathbf{6}_{1\/2}^{c}$} \n& $\\Sigma_{c}$ \n& $2445.3 \\pm 2.5$\n& $2453.5 \\pm 0.1$\n\\tabularnewline\n& $\\Xi_{c}^{\\prime}$ \n& $2580.5 \\pm 1.6$\n& $2576.8 \\pm 2.1$\n\\tabularnewline\n& $\\Omega_{c}$ \n& $2715.7 \\pm 4.5$\n& $2695.2 \\pm 1.7$\n\\tabularnewline\n\\hline \n\\multirow{3}{*}{$\\mathbf{6}_{3\/2}^{c}$} \n& $\\Sigma_{c}^{\\ast}$ \n& $2513.4 \\pm 2.3$\n& $2518.1 \\pm 0.8$\n\\tabularnewline\n& $\\Xi_{c}^{\\ast}$ \n& $2648.6 \\pm 1.3$\n& $2645.9 \\pm 0.4$\n\\tabularnewline\n& $\\Omega_{c}^{\\ast}$ \n& $2783.8 \\pm 4.5$\n& $2765.9 \\pm 2.0$\n\\tabularnewline\n\\hline \\hline\n\\end{tabular}\n\\par\\end{centering}\n\\caption{The numerical results of the charmed baryon masses in \n  comparison with the experimental data~\\cite{PDG2017}.}\n\\label{tab:1}\n\\end{table}\n\n\\begin{table}[htp]\n\\centering{}\\vspace{3em}%\n\\begin{tabular}{c|ccc}\n\\hline\\hline \n$\\mathbf{\\mathcal{R}}_{J}^{Q}$ \n& $B_{b}$ \n& Mass\n& Experiment\n\\\\\n\\hline \n\\multirow{2}{*}{$\\mathbf{\\overline{3}}_{1\/2}^{b}$} \n& \\textcolor{black}{$\\Lambda_{b}$} \n& $5599.3 \\pm 2.4 $\n& $5619.5 \\pm 0.2$ \n \\tabularnewline\n& \\textcolor{black}{$\\Xi_{b}$} \n& $5803.1 \\pm 1.2 $\n& $5793.1 \\pm 0.7 $  \n \\tabularnewline\n\\hline \n\\multirow{3}{*}{$\\mathbf{6}_{1\/2}^{b}$} \n& \\textcolor{black}{$\\Sigma_{b}$} \n& $5804.3 \\pm 2.4 $\n& $5813.4 \\pm 1.3$  \n\\tabularnewline\n& \\textcolor{black}{$\\Xi_{b}^{\\prime}$} \n& $5939.5 \\pm 1.5 $\n& $5935.0 \\pm 0.05$ \n\\tabularnewline\n& \\textcolor{black}{$\\Omega_{b}$} \n& $6074.7 \\pm 4.5 $\n& $6048.0 \\pm 1.9$  \n \\tabularnewline\n\\hline \n\\multirow{3}{*}{$\\mathbf{6}_{3\/2}^{b}$} \n& \\textcolor{black}{$\\Sigma_{b}^{\\ast}$} \n& $5824.6 \\pm 2.3 $\n& $5833.6 \\pm 1.3$  \n\\tabularnewline\n&\\textcolor{black}{$\\Xi_{b}^{\\ast}$} \n& $5959.8 \\pm 1.2 $\n& $5955.3 \\pm 0.1$ \n \\tabularnewline\n& \\textcolor{black}{$\\Omega_{b}^{\\ast}$} \n& $6095.0 \\pm 4.4 $\n& $-$\n\\tabularnewline\n\\hline \\hline\n\\end{tabular}\n\\caption{The results of the masses of the bottom baryons in comparison\nwith the experimental data~\\cite{PDG2017}.}\n\\label{tab:2}\n\\end{table}\n\n\\section{Magnetic moments of heavy baryons} \nIn this Section, we briefly summarize a recent work on the magnetic\nmoments of the heavy baryons~\\cite{Yang:2018uoj}.\nStarting from Eq.~\\eqref{eq:3corrftn}, one can derive the general\nexpressions of the collective operator for the magnetic moments   \n\\begin{align}\n  \\label{eq:MagMomOp}\n \\hat{\\mu} = \\hat{\\mu}^{(0)}  + \\hat{\\mu}^{(1)}, \n\\end{align}\nwhere $\\hat{\\mu}^{(0)}$ and $\\hat{\\mu}^{(1)}$ denote the leading\nand rotational $1\/N_c$ contributions, and the linear $m_{\\mathrm{s}}$\ncorrections respectively \n\\begin{align}\n\\hat{\\mu}^{(0)} & =  \n\\;\\;w_{1}D_{\\mathcal{Q}3}^{(8)}\n\\;+\\;w_{2}d_{pq3}D_{\\mathcal{Q}p}^{(8)}\\cdot\\hat{J}_{q}\n\\;+\\;\\frac{w_{3}}{\\sqrt{3}}D_{\\mathcal{Q}8}^{(8)}\\hat{J}_{3},\\cr\n\\hat{\\mu}^{(1)} & =  \n\\;\\;\\frac{w_{4}}{\\sqrt{3}}d_{pq3}D_{\\mathcal{Q}p}^{(8)}D_{8q}^{(8)}\n+w_{5}\\left(D_{\\mathcal{Q}3}^{(8)}D_{88}^{(8)}+D_{\\mathcal{Q}8}^{(8)}D_{83}^{(8)}\\right)\n\\;+\\;w_{6}\\left(D_{\\mathcal{Q}3}^{(8)}D_{88}^{(8)}-D_{\\mathcal{Q}8}^{(8)}D_{83}^{(8)}\\right).\n\\label{eq:magop}\n\\end{align}\n $d_{pq3}$ is the SU(3) symmetric tensor of which the indices run over \n$p=4,\\cdots,\\,7$. $\\hat{J_3}$ and $\\hat{J}_{p}$ denote the third\nand the $p$th components of the spin operator acting on the soliton\nwith the light-quark pair. $D_{\\mathcal{Q}3}^{(8)}$ arises from the\nrotation of the electromagnetic current  \n\\begin{align}\nD_{\\mathcal{Q}3}^{(8)} = \\frac12 \\left( D_{33}^{(8)} + \\frac1{\\sqrt{3}}\n  D_{83}^{(8)}\\right).\n\\end{align}\nThe coefficients $w_i$ in Eq.~\\eqref{eq:magop} are independent of\nbaryons involved, which encode the interaction of light quarks with\nthe electromagnetic current. Each term has a physical meaning: $w_1$\nrepresents the leading-order contribution, a part of the rotational\n$1\/N_c$ corrections, and linear $m_{\\mathrm{s}}$ corrections, whereas\n$w_2$ and $w_3$ describe the rest of the rotational $1\/N_c$\ncorrections. $w_1$ includes the $m_s$-dependent term, which is not\nexplicitly involved in the breaking of flavor SU(3) symmetry. So, we\nneed to treat $w_1$ as if it had contained the SU(3) symmetric part.\n On the other hand, $w_4$, $w_5$, and $w_6$ are the\nSU(3) symmetry breaking terms. There are yet another $m_{\\mathrm{s}}$\ncorrections, which arise from the collective wave functions. Though \n$w_i$ can be determined within a specific chiral solitonic model such\nas the $\\chi$QSM~\\cite{Kim:1995mr, Kim:1995ha}, we will use the values\nof $w_i$, which have been already fixed from the experimental data on\nthe magnetic moments of the baryon octet. \n\nThe baryon wave function given in Eq.~\\eqref{eq:waveftn} is not enough\nto compute the magnetic moments, because the collective wave functions\nshould be revised when the perturbation coming from the strange\ncurrent quark mass is considered. In this case, the baryon is no more\nin a pure state but is mixed with higher representations. \nIn Ref.~\\cite{Kim:2018nqf}, the collective baryon wave functions for\nthe heavy baryons have been already derived. Those for the baryon\nanti-triplet ($J_{L}=0$) and the sextet ($J_{L}=1$) are expressed respectively\nas~\\cite{Kim:2018nqf}    \n\\begin{align}\n&|B_{\\overline{\\bm3}_{0}}\\rangle = |\\overline{\\bm3}_{0},B\\rangle + \np^{B}_{\\overline{15}}|\\overline{\\bm{15}}_{0},B\\rangle, \\cr\n&|B_{\\bm6_{1}}\\rangle = |{\\bm6}_{1},B\\rangle +\n  q^{B}_{\\overline{15}}|{\\overline{\\bm{15}}}_{1},B \n\\rangle + q^{B}_{\\overline{24}}|{\n{\\overline{\\bm{24}}}_{1}},B\\rangle,\n\\label{eq:mixedWF1}\n\\end{align}\nwith the mixing coefficients\n\\begin{eqnarray}\np_{\\overline{15}}^{B}\n\\;\\;=\\;\\;\np_{\\overline{15}}\\left[\\begin{array}{c}\n-\\sqrt{15}\/10\\\\\n-3\\sqrt{5}\/20\n\\end{array}\\right], \n& \nq_{\\overline{15}}^{B}\n\\;\\;=\\;\\;\nq_{\\overline{15}}\\left[\\begin{array}{c}\n\\sqrt{5}\/5\\\\\n\\sqrt{30}\/20\\\\\n0\n\\end{array}\\right], \n& \nq_{\\overline{24}}^{B}\n\\;\\;=\\;\\;\nq_{\\overline{24}}\\left[\\begin{array}{c}\n-\\sqrt{10}\/10\\\\\n-\\sqrt{15}\/10\\\\\n-\\sqrt{15}\/10\n\\end{array}\\right],\n\\label{eq:pqmix}\n\\end{eqnarray}\nrespectively, in the basis $\\left[\\Lambda_{Q},\\;\\Xi_{Q}\\right]$ for\nthe anti-triplet and $\\left[\\Sigma_{Q}\\left(\\Sigma_{Q}^{\\ast}\\right),\\;\n  \\Xi_{Q}^{\\prime}\\left(\\Xi_{Q}^{\\ast}\\right),\\;\\Omega_{Q}\n  \\left(\\Omega_{Q}^{\\ast}\\right)\\right]$ for the sextets. The\nparameters $p_{\\overline{15}}$, $q_{\\overline{15}}$, and\n$q_{\\overline{24}}$ are written by \n\\begin{eqnarray}\np_{\\overline{15}}\n\\;\\;=\\;\\;\n\\frac{3}{4\\sqrt{3}}\\overline{\\alpha}{I}_{2}, \n& \nq_{\\overline{15}}\n\\;\\;=\\;\\;\n{\\displaystyle -\\frac{1}{\\sqrt{2}}\n\\left(\\overline{\\alpha}+\\frac{2}{3}\\gamma\\right)\nI_{2}}, \n& \nq_{\\overline{24}}\\;\\;=\\;\\;\n\\frac{4}{5\\sqrt{10}}\n\\left(\\overline{\\alpha}-\\frac{1}{3}\\gamma\\right)\nI_{2}.\n\\label{eq:pqmix2}\n\\end{eqnarray}\n Combining\nEq.~\\eqref{eq:mixedWF1} with the heavy-quark spinor as in\nEq.~\\eqref{eq:waveftn}, one can construct the collective wave\nfunctions for the heavy baryon states~\\cite{Yang:2018uoj}. \n\nComputing the baryon matrix elements of $\\hat{\\mu}$ in\nEq.~\\eqref{eq:MagMomOp}, we get the magnetic moments of the \nheavy baryons  \n\\begin{equation}\n\\mu_{B}=\\mu_{B}^{(0)}+\\mu_{B}^{(\\mathrm{op})}+\\mu_{B}^{(\\mathrm{wf})}\n\\label{eq:mu_B}\n\\end{equation}\nwhere $\\mu_{B}^{(0)}$ is the part of the magnetic moment in\nthe chiral limit and $\\mu_{B}^{(\\mathrm{op})}$ comes from\n$\\hat{\\mu}^{(1)}$ in Eq.~\\eqref{eq:MagMomOp}, which include $w_4$,\n$w_5$, and $w_6$. $\\mu_{B}^{(\\mathrm{wf})}$ is derived from the\ninterference between the $\\mathcal{O}(m_{\\mathrm{s}})$ and\n$\\mathcal{O}(1)$ parts of the collective wave functions in\nEq.~\\eqref{eq:mixedWF1}.   \n\nSince the soliton with the light-quark pair for the baryon\nanti-triplet has spin $J_L=0$, , the magnetic moments of the baryon\nanti-triplet vanish. In this case $1\/m_Q$ contributions are the\nleading ones. However, we will not include them, since we need to go\nbeyond the mean-field approximation to consider the $1\/m_Q$\ncontributions within the present framework. \n\nSince $w_1$ contains both the leading-order contributions and the \n$1\/N_c$ rotational corrections, we have to decompose them. Following\nthe argument of Ref.~\\cite{Kim:2017khv}, we can separately consider\neach contribution. The coefficients $w_1$, $w_2$, and $w_3$ are\nexpressed in terms of the model dynamical parameters  \n\\begin{align}\n  \\label{eq:w123}\nw_{1}  =  \nM_{0}\\;-\\;\\frac{M_{1}^{\\left(-\\right)}}{I_{1}^{\\left(+\\right)}},\\;\\;\\;\nw_{2}  = -2\\frac{M_{2}^{\\left(-\\right)}}{I_{2}^{\\left(+\\right)}},\\;\\;\\;\nw_{3}  = -2\\frac{M_{1}^{\\left(+\\right)}}{I_{1}^{\\left(+\\right)}},\n\\end{align}\nwhere the explicit forms of $M_0$, $M_1^{(\\pm)}$, $M_2^{(-)}$ are\ngiven in Refs.~\\cite{Kim:1995mr, Praszalowicz:1998j}. $I_1^{(+)}$ and \n$I_2^{(+)}$ are the moments of inertia with the notation of\nRef.~\\cite{Praszalowicz:1998j} taken. In the limit of the small soliton\nsize, the parameters in Eq.~\\eqref{eq:w123} can be simplified as  \n\\begin{align}\nM_{0}\\;\\rightarrow\\;-2N_{c}K,\n\\;\\;\\;\n\\frac{M_{1}^{\\left(-\\right)}}{I_{1}^{\\left(+\\right)}}\n\\;\\rightarrow\\;\\frac{4}{3}K, \\;\\;\\;\n   \\frac{M_{1}^{\\left(+\\right)}}{I_{1}^{\\left(+\\right)}}\n  \\;\\rightarrow\\;-\\frac{2}{3}K,\\;\\;\\;\n\\frac{M_{2}^{\\left(-\\right)}}{I_{2}^{\\left(+\\right)}}\n\\;\\rightarrow\\;-\\frac{4}{3}K.\n\\label{eq:sss}  \n\\end{align} \nThese results yield the expressions of the magnetic moments in the\nnonrelativistic (NR) quark model. For example, the ratio of the proton\nand magnetic moments can be correctly obtained as\n$\\mu_p\/\\mu_n=-3\/2$. In the NR limit, we also derive the relation\n$M_{1}^{\\left(-\\right)}\\;=\\;-2M_{1}^{\\left(+\\right)}$. Furthermore, we\nhave to assume that this relation can be also applied to the \ncase of the realistic soliton size. Then, we can write the \nleading-order contribution $M_0$ in terms of $w_1$ and $w_3$\n\\begin{align}\n  \\label{eq:4}\nM_0= w_1 + w_3.  \n\\end{align}\nSince a heavy baryon constitutes $N_c-1$ valence quarks, the original\n$M_0$ is modified by introducing $(N_c-1)\/N_c$. As mentioned\npreviously, only the valence part of $M_0$ should be changed by this\nscaling factor. Since, however, we have determined the values of\n$w_i$ using the experimental data, we can not fix separately the\nvalence and sea parts. Thus, we introduce an additional scaling factor\n$\\sigma$ to express a new coefficient $\\tilde{w}_1$ \n\\begin{align}\n\\label{eq:w1tilde}\n\\tilde{w}_1 = \\left[\\frac{N_c-1}{N_c} (w_1+w_3) - w_3\\right] \\sigma. \n\\end{align}\n$\\sigma$ compensates also possible deviations from the NR relation\n$M_{1}^{\\left(-\\right)}\\;=\\;-2M_{1}^{\\left(+\\right)}$ assumed to be \nvalid in the realistic soliton case. The value of $\\sigma$ is taken to\nbe $\\sigma\\sim0.85$. \n\nConsidering the scaling parameters, we are able to determine the\nfollowing values for $w_i$\n\\begin{align}\n\\tilde{w_{1}} \n& =  \n-10.08\\pm0.24,\n\\cr\nw_{2} & =  4.15\\pm0.93,\n\\cr\nw_{3} & =  8.54\\pm0.86,\n\\cr\n\\overline{w}_{4} & =  -2.53\\pm0.14,\n\\cr\n\\overline{w}_{5} & =  -3.29\\pm0.57,\n\\cr\n\\overline{w}_{6} & =  -1.34\\pm0.56.\n\\label{eq:numW}\n\\end{align}\n\nBefore we carry on the calculation of the magnetic moments, we examine\nthe general relations between them. First, we find the generalized\nColeman and Glashow relations~\\cite{Coleman:1961jn}, which arise from \nthe isospin invariance\n\\begin{align}\n\\mu(\\Sigma_{c}^{++})\\;-\\;\\mu(\\Sigma_{c}^{+})\n& =  \n\\mu(\\Sigma_{c}^{+})\\;-\\;\\mu(\\Sigma_{c}^{0}),\n\\cr\n\\mu(\\Sigma_{c}^{0})\\;-\\;\\mu(\\Xi_{c}^{\\prime0}) \n& =  \n\\mu(\\Xi_{c}^{\\prime0})\\;-\\;\\mu(\\Omega_{c}^{0}),\n\\cr\n2 [\\mu(\\Sigma_{c}^{+})\\,-\\,\\mu(\\Xi_{c}^{\\prime0})]\n& =  \n\\mu(\\Sigma_{c}^{++})\\,-\\,\\mu(\\Omega_{c}^{0}).\n\\label{eq:coleman}  \n\\end{align}\nSimilar relations were also found in Ref.~\\cite{Banuls:1999mu}.\nHowever, there is one very important difference. While the \nColeman-Glashow relations are known to be valid in the chiral \nlimit, the relations in Eq.~\\eqref{eq:coleman} are justified \neven when the effects of SU(3) flavor symmetry breaking are\nconsidered. We also find the relation according to\nthe $U$-spin symmetry\n\\begin{align}\n\\mu(\\Sigma_{c}^{0})\\;=\\;\n\\mu(\\Xi_{c}^{\\prime0})\\;=\\;\n\\mu(\\Omega_{c}^{0})\\;=\\;\n-2\\mu(\\Sigma_{c}^{+})\\;=\\;\n-2\\mu(\\Xi_{c}^{\\prime+})\\;=\\;\n-\\frac{1}{2}\\mu (\\Omega_{c}^{0}),\n\\label{eq:Usym}\n\\end{align}\nwhich are only valid in the SU(3) symmetric case. \nWe derive also the sum rule given as    \n\\begin{align}\n\\sum_{B_c\\in\\mathrm{sextet}}\\mu(B_c)\\;=\\;0\n\\label{eq:sum}\n\\end{align}\nin the SU(3) symmetric case. \n\n\\begin{table}[htp]\n\\caption{Numerical results of the magnetic moments for the charmed\n  baryon sextet with $J=1\/2$ in units of the nuclear magneton $\\mu_N$.} \n\\renewcommand{\\arraystretch}{1.3\n\\begin{tabular}{ccc}\n\\hline \\hline\n$\\mu\\left[6_{1}^{1\/2},\\;B_{c}\\right]$ \n& $\\mu^{(0)}$ \n& $\\mu^{(\\text{total})}$ \n\\tabularnewline \\hline\n$\\Sigma_{c}^{\\text{++}}$ \n& $2.00\\pm0.09$\n& $2.15\\pm0.1$ \n\\tabularnewline\n$\\Sigma_{c}^{\\text{+}}$ \n& $0.50\\pm0.02$ \n& $0.46\\pm0.03$ \n\\tabularnewline\n$\\Sigma_{c}^{0}$ \n& -$1.00\\pm0.05$ \n& -$1.24\\pm0.05$ \n\\tabularnewline\n\\hline \n$\\Xi_{c}^{\\prime+}$ \n& $0.50\\pm0.02$ \n& $0.60\\pm0.02$ \n\\tabularnewline\n$\\Xi_{c}^{\\prime0}$ \n& -$1.00\\pm0.05$ \n& -$1.05\\pm0.04$ \n\\tabularnewline\n\\hline \n$\\Omega_{c}^{0}$ \n& -$1.00\\pm0.05$ \n& -$0.85\\pm0.05$ \n\\tabularnewline\n\\hline \\hline\n\\end{tabular}\n\\label{tab:3}\n\\end{table}\n\\begin{table}[htp]\n\\renewcommand{\\arraystretch}{1.3\n\\caption{Numerical results of magnetic moments for charmed baryon\n  sextet with $J=3\/2$ in units of the nuclear magneton $\\mu_N$.} \n\\begin{tabular}{ccc}\n\\hline \\hline \n$\\mu\\left[6_{1}^{3\/2},\\;B_{c}\\right]$ \n& $\\mu^{(0)}$ \n& $\\mu^{(\\text{total})}$ \n\\tabularnewline\n\\hline \n$\\Sigma_{c}^{\\ast\\text{++}}$ \n& $3.00\\pm0.14$ \n& $3.22\\pm0.15$ \n\\tabularnewline\n$\\Sigma_{c}^{\\ast\\text{+}}$ \n& $0.75\\pm0.04$ \n& $0.68\\pm0.04$ \n\\tabularnewline\n$\\Sigma_{c}^{\\ast0}$ \n& $-1.50\\pm0.07$ \n& $-1.86\\pm0.07$ \n\\tabularnewline\n\\hline \n$\\Xi_{c}^{\\ast+}$ \n& $0.75\\pm0.04$ \n& $0.90\\pm0.04$ \n\\tabularnewline\n$\\Xi_{c}^{\\ast0}$ \n& $-1.50\\pm0.07$ \n& $-1.57\\pm0.06$ \n\\tabularnewline\n\\hline \n$\\Omega_{c}^{\\ast0}$ \n& -$1.50\\pm0.07$ \n& -$1.28\\pm0.08$ \n\\tabularnewline\n\\hline \\hline\n\\end{tabular}\n\\label{tab:4}\n\\end{table}\nIn Tables~\\ref{tab:3} and \\ref{tab:4}, we list the numerical results\nof the charmed baryon sextet with spin 1\/2 and 3\/2, respectively. We\nobtain exactly the same results for the bottom baryons because of the\nheavy-quark symmetry in the $m_Q\\to\\infty$ limit. In\nRef.~\\cite{Yang:2018uoj}, a detailed discussion can be found, the\npresent results being compared with those from many other models. \n\\section{Excited $\\Omega_c$ baryons}\nThe present mean-field approach was applied to the classification of\nthe excited $\\Omega_c^0$'s that were recently reported by the LHCb\nCollaboration~\\cite{Aaij:2017nav}. The masses and decay widths of the\n$\\Omega_c^0$'s, which were reported by the LHCb Collaboration, are\nlisted in Table~\\ref{tab:5}. The Belle Collaboration has confirmed the\nfour of them~\\cite{Yelton:2017qxg} (see Table~\\ref{tab:6}). The Belle\ndata unambiguously confirmed the existence of the $\\Omega_c(3066)$ and\n$\\Omega_c(3090)$, and $\\Omega_c(3000)$ and $\\Omega_c(3050)$ are also\nconfirmed with reasonable significance. On the other hand the narrow\nresonance $\\Omega_c(3119)$ was not seen in the Belle experiment but\nthe nonobservation of $\\Omega_c(3119)$ is not in disagreement because\nit is due to the small yield. \n\\begin{table}[htp]\n\\renewcommand{\\arraystretch}{1.3\n\\caption{Experimental data on the five $\\Omega_c^0$ baryons reported by the\n  LHCb Collaboration~\\cite{Aaij:2017nav}.} \n\\begin{tabular}{ccc}\n\\hline \\hline \nResonance\n& Mass (MeV)\n& Decay width  (MeV)\n\\tabularnewline\n\\hline \n $\\Omega_c(3000)^0$\n& $3000.4\\pm 0.2\\pm0.1_{-0.5}^{+0.3}$ \n& $4.5\\pm 0.6\\pm 0.3$\n\\tabularnewline\n $\\Omega_c(3050)^0$\n& $3050.2\\pm0.1\\pm0.1_{-0.5}^{+0.3}$\n& $0.8\\pm 0.2\\pm 0.1$\n\\tabularnewline\n $\\Omega_c(3066)^0$\n& $3065.6\\pm 0.1\\pm 0.3_{-0.5}^{+0.3}$ \n& $3.5\\pm0.4\\pm0.2$\n\\tabularnewline\n $\\Omega_c(3090)^0$\n& $3090.2\\pm 0.3\\pm 0.5_{-0.5}^{+0.3}$\n& $8.7\\pm 1.0 \\pm 0.8$\n\\tabularnewline\n $\\Omega_c(3119)^0$\n& $3119.1\\pm 0.3 \\pm 0.9_{-0.5}^{+0.3}$\n& $1.1 \\pm 0.8 \\pm 0.4$\n\\tabularnewline\n$\\Omega_{c}(3188)$ \n& $3188\\pm 5 \\pm 13$\n& $60\\pm 15 \\pm 11$\n\\tabularnewline\n\\hline \\hline\n\\end{tabular}\n\\label{tab:5}\n\\end{table}\n\n\\begin{table}[htp]\n\\renewcommand{\\arraystretch}{1.3\n\\caption{Experimental data on the four $\\Omega_c^0$ baryons reported by the\n  Belle Collaboration~\\cite{Yelton:2017qxg}.}\n\\begin{tabular}{cc}\n\\hline \\hline \nResonance\n& Mass (MeV)\n\\tabularnewline\n\\hline \n $\\Omega_c(3000)^0$\n& $3000.7\\pm 1.0\\pm 0.2$\n\\tabularnewline\n $\\Omega_c(3050)^0$\n& $3050.2\\pm0.4\\pm0.2$\n\\tabularnewline\n $\\Omega_c(3066)^0$\n& $3064.9\\pm 0.6\\pm 0.2$\n\\tabularnewline\n $\\Omega_c(3090)^0$\n& $3089.3\\pm 1.2\\pm 0.2$\n\\tabularnewline\n $\\Omega_c(3119)^0$\n& --\n\\tabularnewline\n$\\Omega_{c}(3188)$ \n& $3199\\pm 9 \\pm 4$\n\\tabularnewline\n\\hline \\hline\n\\end{tabular}\n\\label{tab:6}\n\\end{table} \n\nWhen one examines the excited heavy baryons in the present work, we\nneed to consider states with the grand spin $K=1$. Since we have the\nquantization rule $\\bm{K}=\\bm{J}_L+\\bm{T}$, the possible values of the\nspin are determined by \n\\begin{align}\nJ_L = |K-T|,\\cdots, K+T.  \n\\end{align}\nThus, In the case of $T=0$ which corresponds to the anti-triplet with\n$Y'=2\/3$, we must have $J_L=1$ because of $K=1$. Combining it with the\nheavy-quark spin 1\/2, we have \\textit{two} excited baryon\nanti-triplet. Similarly, $T=1$ corresponds to the sextet. In this\ncase $J_L$ can have the values of 0, 1, and 2. Being coupled with the\nheavy-quark spin $1\/2$, we get \\textit{five} excited baryon sextets:\n$(1\/2)$, $(1\/2,\\,3\/2)$, and $(3\/2,\\,5\/2)$, corresponding to $J_L=0$, and\n$J_L=1$, and $J_L=2$. In each sextet representation, we have a\nisosinglet $\\Omega_c^0$. Thus, is is natural to think that the newly\nfound five $\\Omega_c^0$'s are those in the excited baryon sextets.   \nNote that the representations for each value of $J$ are degenerate in\nthe limit of $m_Q\\to\\infty$. So, we need to introduce an additional\nhyperfine spin-spin interaction as done for the ground-state baryon\nsextet\n\\begin{align}\nH_{LQ}  = \\frac23 \\frac{\\varkappa'}{m_Q} \\bm{J}_L \\cdot \\bm{J}_Q,\n\\end{align}\nwhich is very similar to Eq.~\\eqref{eq:ssinter}. $\\varkappa'$ can be\nfixed by using the experimental data on the masses of the excited\nbaryon anti-triplet.\n\nFollowing Refs.~\\cite{Diakonov:2012zz, Diakonov:2013qta}, we revise the\neigenvalues of the symmetric Hamiltonian for the excited baryons\n($K\\neq 0$) as \nfollows \n\\begin{align}\nM_{\\mathcal{R}}^{(K)\\prime} &=  M_{\\mathrm{cl}}^{(K)\\prime} + \\frac1{2I_2} \\left[\n  C_2(\\mathcal{R}) - T (T+1) - \\frac34 Y^{\\prime 2} \\right]  \\cr\n& + \\frac1{2I_1} \\left[(1-a_K) T(T+1) + a_K J_L(J_L+1) - a_K(1-a_K) K(K+1) \n  \\right], \n\\label{eq:symmass}\n\\end{align}\nwhere $C_2(\\mathcal{R})$ is the eigenvalue of the SU(3) Casimir\noperator, which was already defined in Eq.~\\eqref{eq:Casimir}.  The\nparameter $a_K$ is related to one-quark excitation. \nThe collective wave functions for the soliton are derived as  \n\\begin{align}\n\\Phi_{B,J_L, J_{L3},(T,K)}^{\\mathcal{R}} = \\sqrt{\\frac{2J_L+1}{2K+1}} \\sum_{T_3\n  J_{L3}' K_3'} C_{TT_3J_LJ_{L3}'}^{KK_3}  (-1)^{(T+T_3)}\n  \\Psi_{(\\mathcal{R^*};-Y'TT_3)}^{(\\mathcal{R};B)} D_{J_{L3}'J_{L3}}^{(J_L)*}(S)\n  \\chi_{K_3'}, \n\\end{align}\nwhere index $(\\mathcal{R};YTT_3)$ denotes the SU(3) quantum numbers of\na corresponding baryon in representation $\\mathcal{R}$, and\n$(\\mathcal{R}^*;-Y'TT_3)$ is attached to a fixed value of $Y'$ and\nis formally given in a conjugate representation to $\\mathcal{R}$. The\nfunction $D^{(J_L)}$ represents the SU(2) Wigner $D$ function and\n$\\chi_{K_3}$ is the spinor corresponding to $K$ and $K_3$. The \nwave function for the excited baryons can be constructed by coupling\n$\\Phi_{B,J_L, J_{L3},(T,K)}^{\\mathcal{R}}$ with the heavy-quark\nspinor.  \n\nThe SU(3) symmetry-breaking Hamiltonian in Eq.~\\eqref{sb} also needs to\nbe extended to describe the mass splittings of the excited heavy\nbaryons \n\\begin{align}\nH_{\\mathrm{sb}}^{(K)} = \\overline{\\alpha} D_{88}^{(8)} + \\beta \\hat{Y}\n  + \\frac{\\gamma}{\\sqrt{3}} \\sum_{i=1}^3 D_{8i}^{(8)} \\hat{T}_i +\n  \\frac{\\delta}{\\sqrt{3}} \\sum_{i=1}^3 D_{8i}^{(8)} \\hat{K}_i. \n\\label{eq:excitedsu3br}\n\\end{align}\nThe additional parameter $\\delta$ can be determined by using the mass\nspectrum of excited baryons. \n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[scale=0.2]{fig5.pdf}\\qquad\n\\caption{Schematic picture of the first excited heavy baryons. A\n  possible excitation of a quark from the Dirac sea to the valence\n  level might have $K^P=1^-$.} \n\\label{fig:5}\n\\end{figure}\nAs shown in Fig.~\\ref{fig:5}, the transition from a $K^P=1^-$\nDirac-sea level to an unoccupied $K^P=0^+$ state may correspond to the\nfirst excited heavy baryons~\\cite{Diakonov:2010tf}. Note that such a transition\nis only allowed in the heavy-baryon sector, not in the light-baryon\nsector. As discussed already, there are two baryon anti-triplets and\nfive baryon sextets. From Eq.~\\eqref{eq:symmass}, we can derive the\nfollowing expressions\n\\begin{align}\nM_{\\overline{\\bm{3}}}^{\\prime} &= M_{\\mathrm{cl}}^{\\prime} +\n  \\frac1{2I_2} + \\frac1{I_1} (a_1^2),\\cr\nM_{\\bm{6}}^{\\prime} &= M_{\\overline{\\bm{3}}}^{\\prime} +\n                           \\frac{1-a_1}{I_1} + \\frac{a_1}{I_1}\\times\n                          \\left\\{\n                           \\begin{array}{ll} \n-1 & \\mbox{ for $J_{L}=0$} \\\\\n0 & \\mbox{ for $J_{L}=1$} \\\\\n2 & \\mbox{ for $J_{L}=2$}\n                           \\end{array} \\right. .\n  \\label{eq:excitedM3-6}\n\\end{align}\nConsidering the SU(3) symmetry breaking from\nEq.~\\eqref{eq:excitedsu3br}, we find the splitting parameters for the\n$\\overline{\\bm{3}}$ and $\\bm{6}$ \n\\begin{align}\n\\delta_{\\overline{\\bm{3}}}' &= \\frac38 \\overline{\\alpha} + \\beta =\n  \\delta_{\\overline{\\bm{3}}} = -180\\,\\mathrm{MeV},\\cr\n \\delta_{\\bm{6}J_{L}}' &= \\delta_{\\bm{6}} -\\frac{3}{20}\\delta \\times\n                     \\left\\{\n \\begin{array}{ll} \n-1 & \\mbox{ for $J_{L}=0$} \\\\\n0 & \\mbox{ for $J_{L}=1$} \\\\\n2 & \\mbox{ for $J_{L}=2$}\n                           \\end{array} \\right.,\n\\label{eq:excitedeltas}\n\\end{align}\nwhere we see that $\\delta_{\\overline{\\bm{3}}}'$ is just the same as\n$\\delta_{\\overline{\\bm{3}}}$ given in\nEq.~\\eqref{eq:deltas}. $\\delta_{\\bm{6}}$ is given as $-120$\nMeV. Though we do not know the numerical value of the new parameter\n$\\delta$, we still can analyze the mass splittings of the newly found\n$\\Omega_c$'s, using the splittings between the states with different\nvalues of $J_{L}$. \n\nWe now turn to the hyperfine splittings. The two anti-triplets of spin\n1\/2 and 3\/2 and the two sextets of spin 1\/2 and 3\/2 are split by \n\\begin{align}\n\\Delta_{\\overline{\\bm{3}}}^{\\mathrm{hf}} =\n  \\Delta_{\\bm{6}J_{L}=1}^{\\mathrm{hf}} = \\frac{\\varkappa'}{m_c},    \n\\end{align}\nwhereas another two sextets of spin 3\/2 and 5\/2 are split by\n\\begin{align}\n  \\Delta_{\\bm{6}J_{L}=2}^{\\mathrm{hf}} = \\frac53 \\frac{\\varkappa'}{m_c}.    \n\\label{eq:hf6}\n\\end{align}\nOne sextet of spin 1\/2 from the $J_{L}=0$ case has no hyperfine\nsplitting. The results are depicted in Fig.~\\ref{fig:6}.\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[scale=0.2]{fig6.pdf}\\qquad\n\\caption{Mass splitting of the five excited sextets.}\n\\label{fig:6}\n\\end{figure}\nNote that the $\\Delta_1$ represent the splittings\nbetween the $J_{L}=0$ state and the degenerate $J_{L}=1$ state, whereas\n$\\Delta_2$ denote those between degenerate $J_{L}=1$ and $J_{L}=2$ states \n\\begin{align}\n\\Delta_1 = \\frac{a_1}{I_1} + \\frac{3}{20} \\delta,\\;\\;\\; \\Delta_2 =\n  2\\Delta_1.  \n\\label{eq:Jsplit}\n\\end{align}\nWe will soon see that the relation $\\Delta_1=2\\Delta_2$ will play an\ncritical role \nin identifying the excited $\\Omega_c$'s within the $\\chi$QSM. \n\nIf one identifies $\\Lambda_c(2592)$ and $\\Xi_c(2790)$ as the members\nof the excited baryon anti-triplet of spin $(1\/2)^-$ with negative\nparity, and $\\Lambda_c(2592)$ and $\\Xi_c(2790)$ as those of the\nexcited baryon anti-triplet of spin $(3\/2)^-$, then we find\n$\\delta_{\\overline{\\bm{3}}}=-198$ and $-190$ MeV, which are more or\nless in agreement with the value given in\nEq.~\\eqref{eq:excitedeltas}. The $\\varkappa'\/m_c$ can be also\ndetermined as \n\\begin{align}\n\\frac{\\varkappa'}{m_c} = \\frac13 (M_{\\Lambda_c(2628)} + 2\n  M_{\\Xi_c(2818)}) - \\frac13 (M_{\\Lambda_c(2592)} + 2 M_{\\Xi_c(2790)})\n  = 30\\,\\mathrm{MeV}, \n\\label{eq:exhfvalue}\n\\end{align}\nand $M_{\\overline{\\bm{3}}}$ is also fixed by \n\\begin{align}\nM_{\\overline{\\bm{3}}} = \\frac29 (M_{\\Lambda_c(2628)} + 2\n  M_{\\Xi_c(2818)}) + \\frac19 (M_{\\Lambda_c(2592)} + 2 M_{\\Xi_c(2790)})\n  = 2744\\,\\mathrm{MeV}.  \n\\end{align}\n\nWe now assert that as a minimal scenario the newly found $\\Omega_c$ baryons\nby the LHCb Collaboration belong to the five excited sextets. Then\n$\\Omega_c(3000)$ can be identified as the state with $(J_{L}=0,\\,1\/2^-)$,\nwhich corresponds to the lightest state in Fig.~\\ref{fig:6}. All other\nfour states can be consequently identified as depicted in\nFig.~\\ref{fig:6}. Including the hyperfine interactions, we get the\nresults as summarized in Table~\\ref{tab:7}.\n\\renewcommand{\\arraystretch}{1.3}\n\\begin{table}[thp]\n\\caption{Scenario 1: All five LHCb\n  $\\Omega_{c}$ states are assigned to the excited baryon sextets.\n\\label{tab:7}%\n\\begin{center}%\n\\begin{tabular}\n[c]{ccccc}\\hline\\hline\n$J_{L}$ & $S^{P}$ & $M$~[MeV] & $\\varkappa^{\\prime}\/m_{c}$~[MeV] &\n                                                            $\\Delta_{J_{L}}$~[MeV]\\\\ \\hline\n0 & $\\frac{1}{2}^{-}$ & 3000 & not applicable & not applicable\\\\ \n\\multirow{2}{*}{1} & $\\frac{1}{2}^{-}$ & 3050 & \\multirow{2}{*}{16} &\n\\multirow{2}{*}{61}\\\\\n~ & $\\frac{3}{2}^{-}$ & 3066 &  & \\\\\n\\multirow{2}{*}{2} & $\\frac{3}{2}^{-}$ & 3090 & \\multirow{2}{*}{17} &\n\\multirow{2}{*}{47}\\\\\n& $\\frac{5}{2}^{-}$ & 3119 &  & \\\\\\hline \\hline\n\\end{tabular}\n\\end{center}\n\\par\n\\end{table}\n\\renewcommand{\\arraystretch}{1}\nWe find at least three different contradictions arising from the\nassignment of these $\\Omega_c$ states as the members of the\nexcited sextets within the $\\chi$QSM.  Firstly, this assignment\nrequires that the hyperfine splitting should be almost as twice as \nsmaller than in the $\\overline{\\bm{3}}$ case. Secondly,  the robust\nrelation $\\Delta_2=2\\Delta_1$ given in Eq.~\\eqref{eq:Jsplit}  is badly\nbroken. Finally, there are two orthogonal sum rules\n$\\sigma_1=\\sigma_2=0$ derived from the $\\chi$QSM  \n\\begin{align}\n\\sigma_1&=6\\; \\Omega_c(J_{L}=0,1\/2^-)- \\Omega_c(J_{L}=1,1\/2^-)-8 \\;\n          \\Omega_c(J_{L}=1,3\/2^-)+3\\; \\Omega_c(J_{L}=2,5\/2^-) , \\label{sr}\\\\ \n\\sigma_2&=-4\\; \\Omega_c(J_{L}=0,1\/2^-)+9\\; \\Omega_c(J_{L}=1,1\/2^-)-3 \\;\n          \\Omega_c(J_{L}=1,3\/2^-)-5 \\; \\Omega_c(J_{L}=2,3\/2^-)  \n+3\\; \\Omega_c(J_{L}=2,5\/2^-),  \\notag\n\\end{align}\nwhich are also badly broken. Thus, we come to the conclusion that the\nthe five $\\Omega_c$ baryons is unlikely to belong to the excited\nsextets. A similar conclusion was drawn by\nRef.~\\cite{Karliner:2017kfm} in a different theoretical framework.  \nMoreover, the computed decay widths of the excited $\\Omega_c$'s do not\nmatch with the experimental data. Therefore, the first scenario is\nunrealistic in the present mean-field approach. \n\nSince the first scenario is not suitable for identifying the five\nexcited $\\Omega_c$ baryons, we have to come up with another\nscenario. Observing that two of them have rather narrower decay widths\nthan other three $\\Omega_c$'s, we assert that these narrow\n$\\Omega_c(3050)$ and $\\Omega_c(3119)$ belong to the possible exotic\nanti-decapentaplet ($\\overline{\\bm{15}}$) which is yet another\nlowest-lying allowed representation, whereas three of them belong to\nthe excited sextet. We find in this scenario that two other members of\nthe excited baryon sextet with $J_{L}=2$ have masses above the $\\Xi D$\nthreshold at 3185 MeV. Since they have rather broad widths, they are\nnot clearly seen in the LHCb data and may fall into the bump\nstructures appearing in the LHCb data.  \n\n\\renewcommand{\\arraystretch}{1.3}\n\\begin{table}[thp]\n\\caption{Scenario 2. Only three LHCb states are\nassigned to {the} sextets. }%\n\\label{tab:8}%\n\\begin{center}%\n\\begin{tabular}\n[c]{ccccc}\\hline \\hline\n$J_{L}$& $S^{P}$ & $M$~[MeV] & $\\varkappa^{\\prime}\/m_{c}$~[MeV] & $\\Delta_{J_{L}}%\n$~[MeV]\\\\\\hline\n0 & $\\frac{1}{2}^{-}$ & 3000 & not applicable & not applicable\\\\\n\\multirow{2}{*}{1} & $\\frac{1}{2}^{-}$ & 3066 & \\multirow{2}{*}{24} &\n\\multirow{2}{*}{82}\\\\\n~ & $\\frac{3}{2}^{-}$ & 3090 &  & \\\\\n\\multirow{2}{*}{2} & $\\frac{3}{2}^{-}$ & \\emph{3222} & input & input\\\\\n& $\\frac{5}{2}^{-}$ & \\emph{3262} & 24 & 164\\\\\\hline \\hline\n\\end{tabular}\n\\end{center}\n\\par\n\\end{table}\n\\renewcommand{\\arraystretch}{1}\nThe results of the second scenario are summarized in Table~\\ref{tab:8}\nexcept for the $\\Omega_c(3050)$ and $\\Omega_c(3119)$ which will be\ndiscussed separately.  The italic numbers correspond to the bump\nstructures from which $\\Omega_c(3222)$ used as input. Scenario 2\nprovides a much more plausible prediction than scenario 1\ndoes. Interestingly, the value of $\\varkappa'\/m_c\\approx 24$ MeV is\ncloser to that determined from the excited baryon anti-triplets, given\nin Eq.~\\eqref{eq:exhfvalue}. Moreover, the relation\n$\\Delta_1=2\\Delta_2$ is nicely satisfied in this scenario.  \n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[scale=0.3]{fig7.pdf}\\qquad\n\\caption{Representation of the anti-decapentaplet\n  ($\\overline{\\bm{15}}$). As in the case of the baryon sextet, there\n  are two baryon anti-decapentaplets with spin 1\/2 and 3\/2. The\n  $\\Omega_c$s belong to the isotriplet in the $\\overline{\\bm{15}}$plet.}\n\\label{fig:7}\n\\end{figure}\n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[scale=0.3]{fig8.pdf}\\qquad\n\\caption{The allowed representations for the lowest-lying heavy\n  baryons. }\n\\label{fig:8}\n\\end{figure}\nThe anti-decapentaplet ($\\overline{\\bm{15}}$) was first suggested by\nDiakonov~\\cite{Diakonov:2010tf}. Figure~\\ref{fig:7} illustrates the\nrepresentation of the $\\overline{\\bm{15}}$.  Since the\n$\\overline{\\bm{15}}$ belongs to the allowed representations for the\nground-state heavy baryons, it satisfies the quantization rule\n$\\bm{J}_{L}+\\bm{T}=\\bm{0}$, so $T=J_{L}=1$ (see Fig.~\\ref{fig:8}).  When the\nlight-quark pair with $J_{L}=1$ is coupled to the heavy-quark spin, there\nare two possible $\\overline{\\bm{15}}$ representations that are\ndegenerate in the limit of $m_Q\\to \\infty$. It means that one needs to\nconsider the hyperfine interaction defined in Eq.~\\eqref{eq:ssinter}.\nAs given in Eq.~\\eqref{eq:kappavalue}, the value of $\\varkappa\/m_c$ is\naround 68 MeV. Surprisingly, the mass difference between the\n$\\Omega_c(3050)$ and the $\\Omega_c(3119)$ is  \n\\begin{align}\nM_{\\Omega_c(3\/2^+)} (3119) -  M_{\\Omega_c(1\/2^+)} (3050) =\n  \\frac{\\varkappa}{m_c} \\approx   69\\,\\mathrm{MeV} \n\\end{align}\nwhich is almost the same as what was determined from the lowest-lying\nsextet baryons. The decay widths of the excited $\\Omega_c$ baryons\npredicted within the present framework further support the\nplausibility of scenario 2~\\cite{Kim:2017khv}.  The decay widths for\nthe $\\Omega_c(3050)$ and $\\Omega_c(3119)$ are predicted to be \n\\begin{align}\n\\Gamma_{\\Omega_c(3050)(\\overline{\\bm{15}},1\/2^+)} =\n  0.48\\,\\mathrm{MeV}, \\;\\;\\; \n\\Gamma_{\\Omega_c(3119)(\\overline{\\bm{15}},3\/2^+)} = 1.12\\,\\mathrm{MeV},  \n\\end{align}\nwhich are in good agreement with the LHCb data\n$\\Gamma_{\\Omega_c(3050)}=(0.8\\pm0.2\\pm 0.1)$ MeV and\n$\\Gamma_{\\Omega_c(3119)}=(1.1\\pm0.8\\pm 0.4)$ MeV. For detailed\ndiscussion related to the decay widths of $\\Omega_c$, we refer to\nRef.~\\cite{Kim:2017khv}.   \n\nIn addition to scenarios 1 and 2, we also tried to examine\nseveral other scenarios but find that they all turned out to be\ninconsistent with the experimental data. Finally, we want to emphasize\nthat the $\\Omega_c(3050)$ and $\\Omega_c(3119)$ assigned to the members\nof the $\\overline{\\bm{15}}$ are isotriplets. It implies that if they\nindeed belong to the $\\overline{\\bm{15}}$, charged $\\Omega_c^{\\pm}$\nshould exist. Knowing that the excited $\\Omega_c^0$'s have been\nmeasured in the $\\Xi_c^+K_c^-$ channel, we propose that the $\\Xi_c^+\nK^0$ and $\\Xi_c^0 K^-$ channels need to be scanned in the range of the\ninvariant mass between 3000 MeV and 3200 MeV to find an isovector\n$\\Omega_c$'s. If they do not exist, this will falsify the present\npredictions. \n\\section{Conclusion and outlook}\nIn the present short review, we briefly summarized a series of recent\nworks on the properties of the singly heavy baryons within a pion\nmean-field approach, also known as the chiral quark-soliton model.\nIn the limit of the infinitely heavy quark mass ($m_Q\\to \\infty$), \nthe heavy quark inside a heavy baryon can be treated as a mere static\ncolor source. Then a heavy baryon is portrayed as a state of $N_c-1$\nvalence quarks bound by the pion mean field with a heavy stripped off\nfrom the valence level.  This mean-field approach has a certain\nvirtue since both the light and heavy baryons can be dealt with on an\nequal footing. It means that we can bring all dynamical parameters\nwhich have been already determined in the light-baryon sector to\ndescribe the heavy baryons. Indeed we can simply replace the\n$N_c$-counting prefactor by $N_c-1$ for the valence contributions to\nthe heavy baryons. Accordingly, we were able to explain the masses of\nthe lowest-lying heavy baryons and the magnetic moments of them\nwithout introducing additional parameters except for the hyperfine\nspin-spin interactions. We have employed the same framework to\nidentify the newly found excited $\\Omega_c$ baryons reported by the\nLHCb Collaboration. Assigning the three of them to the excited baryon\nsextets and the two of them with narrower decay widths to the possible\nexotic baryon anti-decapentaplet, we were able to classify the\n$\\Omega_c$'s successfully.  Since the $\\Omega_c$ baryons in the\nanti-decapentaplet are the isovector baryons, we anticipate that\ncharged $\\Omega_c$'s might be found in other channels such as the $\\Xi_c^+\nK^0$ and $\\Xi_c^0 K^-$. \n\nThe present model can be further applied to future investigations on\nvarious properties and form factors of heavy baryons. As already shown\nin Ref.~\\cite{Kim:2018nqf}, the electric form factor of the charged\nheavy baryon indicates that a heavy baryon is an electrically compact\nobject. Transition form factors of heavy baryons will further reveal\ntheir internal structure. Understanding excited heavy baryons is\nanother crucial issue that should be investigated. Related studies are\nunder way. \n\n\\begin{acknowledgments}\nI am very grateful to M. V. Polyakov, M. Prasza{\\l}owicz, and\nGh.-S. Yang for fruitful collaborations and discussions over decades.  \nI am thankful to J.-Y. Kim for the discussion related to the\nelectromagnetic form factors of heavy baryons. \nI want to express the gratitude to the editors of the Journal of the\nKorean Physical Society (JKPS) for giving me an opportunity to join\nthe very special 50th anniversary celebration of the JKPS.   \nThe present work was supported by the Inha University Grant in 2017.\n\\end{acknowledgments}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}