diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzznfzq" "b/data_all_eng_slimpj/shuffled/split2/finalzznfzq" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzznfzq" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nThis article is concerned with a problem which\narises in the study of the Kardar-Parisi-Zhang (KPZ) equation\\cite{kpz}\nof interface growth\\cite{rev1}\\cite{rev2}. Namely, we consider the\ndeterministic version of the KPZ equation under the influence of a\nlocal, constant force. Although this problem is in principle rather simple,\na detailed investigation has not been carried out to our knowledge. The\nresults, although interesting in their own right, assume greater significance\nwhen related to the weak\/strong-coupling transition in the noisy KPZ equation.\nBefore entering into further details of this problem, we shall\ngive a brief overview of the key properties of the KPZ equation.\nIn, by now, standard notation this\nequation for the evolution of the interface height $h({\\bf x},t)$\n(above a $d$-dimensional substrate) takes the form\n\\begin{equation}\n\\label{e1}\n\\partial_t h = \\nu \\nabla ^2 h + (\\lambda \/2)(\\nabla h)^{2} + \\eta\n\\end{equation}\nwhere $\\eta ({\\bf x},t)$ is a white noise, generally taken to be\ngaussian distributed.\nThe Cole-Hopf transformation $h({\\bf x},t) = (2\\nu\/\\lambda )\\ln (w({\\bf x},t))$\nyields a linear equation for $w$ albeit with multiplicative\nnoise:\n\\begin{equation}\n\\label{e2}\n\\partial_t w = \\nu \\nabla ^2 w + (\\lambda \/ 2\\nu)w\\eta .\n\\end{equation}\nThis equation may be re-expressed by means of the Feynman-Kac formula,\nsuch that $w$ represents the restricted partition function of a directed\npolymer in the presence of quenched random defects. Naturally, the above\nequation may also be interpreted as an imaginary-time Schr\\\"odinger equation\nfor a quantum mechanical (QM) particle in a time-dependent\nrandom potential $(-\\eta )$.\n\nIt is well-known\\cite{kpz} that $d=2$ plays the role of a lower critical\ndimension.\nFor $d \\le 2$, the interface will be asymptotically (i.e. for large times)\nrough for arbitrarily small $\\lambda $. For $d>2$ there exists a\ntransition between asymptotically smooth and rough phases, depending on\nthe value of $\\lambda $. To access the rough (strong-coupling) phase,\n$\\lambda $ must exceed a critical value $\\lambda _c$. Amongst the most\nimportant questions in this field are: i) what are\nthe scaling exponents associated with the strong-coupling phase; ii) does\nthere exist an upper critical dimension $d_u$, above which these exponents\ntake on mean-field values; iii) what are the critical properties\nof the weak\/strong-coupling transition? There are no\nanswers to the first question, bar predictions from extensive numerical\nwork (except for the case $d=1$ where exact analytic results are\navailable \\cite{rev1}).\nAs regards the second question, the community seems split between\nbeliefs in $d_u = 4$ and $d_u = \\infty$. The former belief is based\nprimarily on the predictions of various\nself-consistent `mode-coupling' theories (see for example \\cite{mam}),\nwhilst the latter belief arises\nfrom numerical evidence for $d$-dependent strong-coupling exponents\\cite{rev2}\n(for all accessible $d \\stackrel{<}{\\scriptstyle\\sim } 7$).\nWith regard to the final question, a recent\ntwo-loop renormalization group calculation\\cite{ft} (formulated as an\n$\\epsilon$-expansion, with $\\epsilon = d-2$)\npredicts that at the transition, the roughness exponent\n$\\chi = 0 + O(\\epsilon ^{3})$ (to be interpreted as logarithmic roughness),\nwhilst the correlation length exponent has the\nform ${\\tilde \\nu} = (1\/\\epsilon) + O(\\epsilon^{3})$. For the one-loop\nversions of these results, see Ref. \\cite{nt}.\n\nWe now concentrate on a much simpler problem.\nAs mentioned before, we retain the essential\nnon-linearity of the KPZ equation, but replace the noise $\\eta $ by a\nlocal applied force: $\\eta ({\\bf x},t) = f_{0} \\delta ^{d}({\\bf x})$. In\ndirected polymer language this corresponds to replacing the bulk disorder\nby a single constant energy columnar defect. In terms of the QM analogue,\nwe have replaced the random potential by a delta-function potential with\namplitude $(-f_{0})$.\nThere exists a well-known result\\cite{qm}, which is most\neasily stated in QM language: for $d \\le 2$ the particle will have a\nbound state for arbitrarily small $f_{0}>0$,\nwhereas for $d>2$ there only exists a bound state for $f_{0} > f_{0,c} >0$.\nIn terms of the interface, these results imply that for $d\\le2$ an arbitrarily\nsmall force is sufficient to induce a non-zero velocity to the surface,\nwhereas for $d>2$ the driving force must exceed a critical value in order\nto achieve this. The identification of a lower critical dimension\n(along with the properties of the system for $d\\le2$) may be obtained from a\nperturbative analysis (in terms of $\\lambda$, or equivalently $f_{0}$).\n\nIn order to explore the properties of the\n`binding\/unbinding' transition for $d>2$, a non-perturbative analysis is\nrequired --\nsuch an analysis constitutes the body of\nthe present work. The main results which arise are the following (which we\nshall describe in terms of the interface formulation). Whereas for $d<2$ an\narbitrarily small driving force will induce a velocity $v(f_{0})$ to the\ninterface, for $d>2$ the applied force must exceed a critical value in order\nto achieve this. The dependence of the velocity on the excess force\n$\\delta f_{0} \\equiv f_{0}-f_{0,c}$ is found\nto be $v \\sim (\\delta f_{0})^{\\alpha (d)}$ where $\\alpha (d) = 2\/(d-2)$ for\n$24$. For $d<2$ the applied\nforce will create a non-linear distortion of the surface over a critical\nradius $R_{c} \\sim f_{0}^{-1\/(2-d)}$, but the amplitude of the distortion\nis of order unity. For $d>2$ the distortion scales logarithmically with\ndistance and exists within a critical radius\n$R_{c} \\sim (\\delta f_{0})^{-\\nu(d)}$, where $\\nu(d) = \\alpha(d)\/2$.\nTherefore, for $d>2$ the distortion is a macroscopic rupture\nof the surface with a logarithmic profile (by `rupture' we mean a\ngross distortion of the interface, leaving the continuous\nproperties of the surface in tact). We shall\nleave the interpretation of these results for the final sections of the paper.\n\nThe outline of the paper is as follows. In the next section we shall give a\nprecise definition of the model to be solved, along with some simple\nsteps which lead to an accessible strong-coupling (i.e. non-perturbative)\nanalysis of the model. In section 3, we proceed with this analysis and derive\nthe velocity-force characteristics. The spatial profiles of the surface\ndistortion for various dimensions are evaluated in section 4. Connections\nbetween the properties of this simple model, and the weak\/strong-coupling\ntransition in the noisy KPZ model are pursued in section 5. We end the paper\nwith our conclusions, and a description of possible extensions to the present\nwork.\n\n\\section{Definition and reformulation of the model}\n\nReplacing the stochastic source of the noisy KPZ equation with a local applied\nforce leads us to consider the equation\n\\begin{equation}\n\\label{e3}\n\\partial_t h = \\nu \\nabla ^2 h + (\\lambda \/2)(\\nabla h)^{2} +\nf_{0}\\delta ^{d}({\\bf x}) .\n\\end{equation}\nAs for the noisy problem, we may apply the transformation\n$h({\\bf x},t) = (2\\nu\/\\lambda )\\ln (w({\\bf x},t))$ which yields the linear\nequation\n\\begin{equation}\n\\label{e4}\n\\partial_t w = \\nu \\nabla ^2 w +\n(\\lambda f_{0}\/2\\nu) \\ w \\ \\delta^{d}({\\bf x}).\n\\end{equation}\nSince we are primarily interested in the interface application of this problem,\nwe choose the initial condition $w({\\bf x},0) = 1$, which corresponds to\nan initially flat interface ($h=0$). [The natural initial condition for the\ndirected polymer application is $w({\\bf x},0) = \\delta ^{d}({\\bf x})$. We shall\nnot consider this problem here.]\n\nWe may formally integrate Eq.(\\ref{e4}) using\nthe Green function of the diffusion equation, $g({\\bf x},t)\n= (4\\pi \\nu t)^{-d\/2}\\exp(-x^{2}\/4\\nu t)$, leading to\n\\begin{equation}\n\\label{e5}\nw({\\bf x},t) = 1 + (\\lambda f_{0}\/ 2\\nu) \\int \\limits _{0}^{t}\ndt' \\ g({\\bf x},t-t')w({\\bf 0},t') .\n\\end{equation}\nTherefore, knowledge of the function $w({\\bf 0},t)$ is sufficient to\ndetermine $w$ throughout all space. For convenience we shall define\n$\\phi (t) \\equiv w({\\bf 0},t)$. Setting ${\\bf x}={\\bf 0}$ in the above\nequation leads to a Volterra integral equation for $\\phi \\ $ :\n\\begin{equation}\n\\label{e6}\n\\phi (t) = 1 + f\\int \\limits _{0}^{t} dt' \\ \\sigma (t-t') \\phi (t')\n\\end{equation}\nwhere $\\sigma (t) = t^{-d\/2}$ and we have defined an effective force\n$f \\equiv (\\lambda f_{0}\/ 2\\nu)(4 \\pi \\nu)^{-d\/2}$. [Such a Volterra equation\nmay also be derived for the more complicated situation\nof a time-dependent force $f(t)$. The case where\n$f(t)$ is a stochastic variable is of particular interest in the directed\npolymer picture \\cite{kl}.]\n\nClearly this equation is ill-defined for $d>2$, since the kernel is\nnon-integrable over the range $(0,t)$.\nTo regularize the equation we have many possibilities.\nThe most convenient is to smear the delta-function force over a small region\nof size $l^{d}$ with a gaussian envelope, i.e. we replace\n$f_{0} \\delta ^{d}({\\bf x})$ by $f_{0} (\\pi l^{2})^{-d\/2} \\exp (-x^{2}\/l^{2})$.\nThe length scale $l$ constitutes the smallest physical scale in the problem.\nAssuming that $w({\\bf x},t)$ varies slowly over the core region $|{\\bf x}|< l$\nallows us to rederive the above Volterra equation, where now\n$\\sigma (t) = (t + l^{2})^{-d\/2}$ (after rescaling $l \\rightarrow l\n= l\/(4\\nu )$).\n\nLaplace transforming Eq.(\\ref{e6}) and using the convolution theorem\nyields an explicit solution for this Laplace transform ${\\cal L}[\\phi (t)]$.\nInverting the transform\ngives the result\n\\begin{equation}\n\\label{e7}\n\\phi(t) = {1 \\over 2\\pi i} \\int _{\\gamma } \\ {ds \\over s} \\\n{e^{st\/l^{2}} \\over [1 - l^{2-d} f D(s)] }\n\\end{equation}\nwhere $\\gamma $ is the usual integration\ncontour along the line $s = c+iy$ where $c$ is a real constant\nchosen to lie to the right of any singularities in the complex plane.\nThe function $D(s)$ is given by\n\\begin{equation}\n\\label{e8}\nD(s) = \\int \\limits _{0}^{\\infty} dy \\ {e^{-sy} \\over (1+y)^{d\/2}}\n\\end{equation}\n(which may be expressed in terms of the incomplete Gamma function\\cite{as}).\n\nWe are now in a position to obtain all physical properties of the model\nfrom the analysis of Eq.(\\ref{e7}). We stress once again that this\nanalysis is non-perturbative in $f_{0}$ (or equivalently $\\lambda $). All\nthe results for $d \\le 2$ may be obtained by perturbation theory (although\none must evaluate all orders), whereas none of the results concerning\nthe binding\/unbinding transition may be obtained by such a technique for $d>2$.\n\nIn the next section we shall concentrate on evaluating the asymptotic\n(large time) form of $\\phi(t)$ which will enable us to determine whether\nthe interface has a non-zero velocity. The spatial\nprofile of the interface distortion requires the evaluation of the convolution\nof the diffusion equation Green function with the function $\\phi (t)$ (as\nillustrated in Eq. (\\ref{e5})) - this analysis is described in section 4.\n\n\\section{Velocity-force characteristics}\n\nThe singularities in the integrand of Eq.(\\ref{e7}) consist of a branch cut\nalong the negative real axis, and (depending on the value of $f$) a pole\nin the right-half plane (RHP). The large-time behaviour of $\\phi(t)$ will\nbe dominated by the contribution from the pole (if it exists). One may easily\nconvince oneself that the existence of this pole is required for the\ninterface at the forcing site to have a non-zero velocity, i.e. for\n$H(t) \\sim vt$\n(where $H(t) \\equiv (\\lambda\/2\\nu )h({\\bf 0},t) = \\ln \\ \\phi (t)$.)\n\nBy studying the\nsmall-$s$ behaviour of $D(s)$ one may show the following properties of\nthe integrand in Eq.(\\ref{e7}). For $d < 2$ there always exists a pole\nin the RHP for $f>0$, which for $f \\searrow 0$ is situated at\n$s_{p}=f^{2\/(2-d)}$. For $d>2$ the pole will only exist in the RHP for\n$f > f_{c} = l^{d-2}(d-2)\/2$. For $f \\searrow f_{c}$, the\nposition of the pole is given by $s_{p}=A(d)(f-f_{c})^{2\/(d-2)}$ for\n$24$\none has $s_{p} = [(d-4)\/(d-2)](f-f_{c})$. This abrupt change of\nbehaviour for $s_{p}$ at $d=4$ will have interesting consequences.\n\nEvaluating the contribution to $\\phi (t)$ from the pole leads us to the\nfollowing results for the velocity-force characteristics: i) for $d<2$ one\nhas for $f \\searrow 0$ the relation $v(f) \\sim f^{2\/(2-d)}$; ii) for\n$d>2$, the velocity is zero unless $f$ exceeds $f_{c}$ --\none then has for $f \\searrow f_{c}$ the relation\n$v(f) \\sim (\\delta f)^{\\alpha (d)}$ where $\\delta f \\equiv f-f_{c}$ and\n\\begin{equation}\n\\label{e9}\n\\nonumber\n\\alpha (d) = \\left \\{\n\\begin{array}{ll}\n\\frac{2}{(d-2)}\n\\hspace{20mm} & 2 < d < 4 \\\\\n1 & d > 4 \\ ,\n\\end{array}\n\\right.\n\\end{equation}\nwhich signals the existence of an upper critical dimension $d_{u}=4$.\n\nIn the absence of a pole in the RHP, the asymptotics of $\\phi (t)$ arise\nfrom the contribution from the branch cut. The results for this case (along\nwith the $v(f)$ characteristics at the critical dimensions 2 and 4) are\ndisplayed for completeness in Table I.\n\n\\section{Spatial profiles}\n\nIn this section we wish to evaluate the spatial profile of the\ninterface for $f > f_{c}$. We define the positive quantity\n$\\Delta H({\\bf x},t) \\equiv (\\lambda \/2\\nu)(h({\\bf 0},t)-h({\\bf x},t))$. Making\nuse\nof the Cole-Hopf transformation, along with Eq. (\\ref{e5}), we have (after\nrescaling\n${\\bf x} \\rightarrow {\\bf x} = {\\bf x}\/(4\\nu)$)\n\\begin{equation}\n\\label{e10}\n\\Delta H = -\\ln \\left \\lbrace {1\\over \\phi(t)} + {f\\over \\phi(t)}\\int \\limits\n_{0}^{t}\ndt' \\ g({\\bf x},t'+l^{2}) \\ \\phi(t-t') \\right \\rbrace .\n\\end{equation}\nFor $f>f_{c}$ and $vt \\gg 1$, we have from the previous analysis\n$\\phi (t) \\sim e^{vt}$. In this case, the above expression reduces to\n\\begin{equation}\n\\label{e11}\n\\Delta H = -\\ln [fF({\\bf x})]\n\\end{equation}\nwhere\n\\begin{equation}\n\\label{e12}\nF({\\bf x}) = \\int \\limits _{0}^{\\infty} dt' \\ e^{-vt'} \\ g({\\bf x},t'+l^{2}) .\n\\end{equation}\nThis expression is valid for $|{\\bf x}| \\ll v^{1\/2}t$. For this entire range\nwe see that the height deviation $\\Delta h$ is independent of time. For\nextremely large distances from the forcing site ($|{\\bf x}| \\gg v^{1\/2}t$),\nthe influence of the force is insufficient to induce a velocity,\nand the surface distortion is essentially zero.\n\nWe now concentrate on examining the form of the steady-state profile\ngiven by Eqs.(\\ref {e11}) and (\\ref {e12}). In fact there are {\\it two}\nspatial regimes for this profile, separated by the length scale\n$R_{c} = v^{-1\/2}$ which will emerge as a {\\it critical radius}.\nWe shall denote the regime $R_{c} \\ll |{\\bf x}| \\ll v^{1\/2}t$\nas region A, and the regime $l \\ll |{\\bf x}| \\ll R_{c}$ as region B.\nIt is important to stress that for $vt \\gg 1$ and $v \\ll 1$ (which is\nthe case when $f \\searrow f_{c}$), regions A and B are both large and\nwell-separated. Region A is less interesting and simpler to analyse, so\nwe shall focus on this first.\n\n\\subsection{$\\Delta H$ in region A}\n\nIn this region, $F({\\bf x})$ may be evaluated using a steepest-descents\ntreatment. A straight-forward calculation yields the result\n\\begin{equation}\n\\label{e13}\n\\nonumber\n\\Delta H({\\bf x}) = \\left \\{\n\\begin{array}{ll}\n2|{\\bf x}|\/R_{c} + {\\rm O}\\left ( \\ln (|{\\bf x}|\/R_{c}) \\right ),\n\\hspace{25mm} & d<2 \\\\\n2|{\\bf x}|\/R_{c} + {\\rm O}\\left ( \\ln (|{\\bf x}|\/R_{c}) \\right )\n+ {\\rm O}\\left ( \\ln (R_{c}\/l) \\right ), & d > 2 \\\n\\end{array}\n\\right.\n\\end{equation}\nThe form of the corrections is of interest. We see that for $d<2$, the\nleading term dominates for $|{\\bf x}| \\gg R_{c}$ as expected. This indicates\nthat the overall height deviation within the critical radius is of\norder unity. For $d>2$ the microscopic length scale enters into the\ncorrections and the condition for the dominance of the leading term\nis now $|{\\bf x}| \\gg R_{c}\\ln(R_{c}\/l)$. We are therefore given the\nhint that for $d>2$ the overall deviation of the height within the critical\nradius may be large (of order $\\ln(R_{c}\/l)$). To see this explicitly\nwe now turn our attention to the form of $\\Delta H$ for\n$|{\\bf x}| \\ll R_{c}$.\n\n\\subsection{$\\Delta H$ in region B}\n\nAlthough the integral determining $F({\\bf x})$ appears quite simple,\nthe analysis in region B is rather complicated. We therefore refer\nthe reader to Appendix A where this analysis is described in some\ndetail. The end result for $\\Delta H$ in region B takes the form\n\\begin{equation}\n\\label{e14}\n\\nonumber\n\\Delta H = \\left \\{\n\\begin{array}{ll}\nc(d)f \\ |{\\bf x}|^{2-d} + {\\rm O}\\left ( (|{\\bf x}|\/R_{c})^{2} \\right ),\n\\hspace{25mm} & d < 2 \\\\\n2f\\ln(|{\\bf x}|\/l) + {\\rm O}\\left ( f \\right ),\n\\hspace{25mm} & d=2 \\\\\n(d-2)\\ln(|{\\bf x}|\/l) + {\\rm O}\\left ( 1 \\right ), & d > 2 \\\n\\end{array}\n\\right.\n\\end{equation}\nwhere $c(d) = 2\\Gamma (d\/2)\/(2-d)$. [It is instructive for $d=2$ to rewrite\nthe prefactor of the height deviation exclusively in terms of $R_{c}$ --\nmaking use of the relation $2f = 1\/\\ln (R_{c}\/l)$].\n\nThere are two main features of these results we wish to stress.\nFirst, there exists for all dimensions a critical radius $R_{c}$\nwithin which the surface profile undergoes a distortion from the\nlinear form shown in Eq.(\\ref{e13}). The critical radius scales\nwith $\\delta f \\equiv f-f_{c}$ as $R_{c} \\sim (\\delta f)^{-\\nu}$\nwhere $\\nu (d) = \\alpha (d)\/2$. In particular we note that\nfor $24$, $\\nu(d) = 1\/2$.\n\nThe second important point is that by calculating the overall\nheight distortion within the critical radius, i.e. $\\Delta H(R_{c})$,\nwe see that\n\\begin{equation}\n\\label{e15}\n\\nonumber\n\\Delta H(R_{c}) \\sim \\left \\{\n\\begin{array}{ll}\n{\\rm O}(1)\n\\hspace{20mm} & 0 \\le d \\le 2 \\\\\n\\ln (R_{c}\/l) & d > 2 \\\n\\end{array}\n\\right.\n\\end{equation}\nThese results confirm the ideas propounded earlier, i.e. for $d \\le 2$\nthe distortion of the surface within the critical radius is negligible,\nwhereas for $d>2$ the distortion is a macroscopic rupture which\nscales logarithmically with distance.\n\n\\section{Applications to KPZ weak\/strong-coupling transition}\n\nAs we mentioned in the Introduction, there are two main\ncharacteristics of the weak\/strong coupling transition for the noisy\nKPZ equation (which arise from a RG study in powers of $\\epsilon =\nd-2$) for $d>2$. These are the vanishing of the roughness exponent,\nwhich implies a logarithmic roughness; and the existence of a\ncorrelation length $\\xi \\sim (g-g_{c})^{-{\\tilde \\nu}}$, where $g$ is the\ndimensionless coupling constant, and ${\\tilde \\nu} = 1\/\\epsilon +\nO(\\epsilon )^{3}$. There is a remarkable correspondence between these\nresults, and those found in the previous section for the simple model\nof a deterministic KPZ interface under a local applied force. To\nreiterate these results, we found that for $d>2$, the applied force\ninduces a logarithmic distortion of the surface within a critical\nradius which scales as $R_{c} \\sim (f-f_{c})^{-\\nu}$ with $\\nu =\n1\/\\epsilon $ for $24$.\n\nThe natural question arises: are the critical properties of the noisy\nKPZ equation directly related to the simple problem discussed above?\nIf so, then we have a much easier way to understand the weak\/strong\ncoupling phase transition, and also we may identify an upper critical\ndimension of 4 (for the unstable fixed point characterising the\ntransition). A quantitative connection between the two sets of\nresults is beyond our reach. However we shall make several points\nwhich certainly make the connection more plausible. Since we are\ninterested in the weak\/strong coupling transition, we henceforth\nrestrict our attention to $d>2$.\n\nThe coupling constant $g$ in the noisy KPZ equation is proportional to\nthe noise amplitude $D$ which we shall take to be our control\nparameter. For $D$ well below the critical value, the non-linearity\nin the KPZ equation is irrelevant. Thus, the surface behaves as an\nEdwards-Wilkinson (EW) interface \\cite{ew}, which is smooth. As we\nincrease the value of D, it becomes more and more likely for small\nregions in the interface to experience strong fluctuations due to rare\nevents in the noise spectrum. If the smooth\/rough transition is\ntriggered by an inherent instability of the interface with respect to strong\nfluctuations, it is plausible to assume that such an instability is\nlocalized due to the local nature of rare events in the noise.\nHaving established this, we may connect the two models by arguing that\nthese strong local fluctuations resemble an effective force over long\ntime scales. There are three essential steps in this argument which\nwe discuss in turn.\n\nFirst, we must argue that the fluctuations appear as a constant force\nover some temporal interval $\\tau $. Clearly any fluctuations in the `downward'\ndirection (i.e. anti-parallel to the direction of positive $h$) will\nhave no effect since they are completely suppressed by the\nnon-linearity. We therefore consider a noise path (or noise `history') which\ntends on\naverage to be positive over the interval $\\tau $ within some region\n$\\Lambda $ of size $l^{d}$. This path may be considered as a force\nof magnitude $f$ so\nlong as it lies between the bounds $f \\pm \\sigma $. The fluctuations\nwithin these bounds are irrelevant. This fact may be established by\nconsidering the delta-function model Eq.(\\ref{e3}) but allowing for\nthe force to be a stochastic variable. By power counting one may show\nthat the fluctuations of the force are irrelevant (about the constant\nforce fixed point) for $d>1$. For the noise path to exceed $f+\\sigma$\nis extremely unlikely as the path is already a rare event even to have\nentered between these bounds. The path is much more likely to drop\nbelow $f-\\sigma$. This is deemed as switching off the `force'. The\nforce will be switched on again when the noise path returns within the\nbounds. Thus the interval $\\tau$ over which the force acts is a random\nvariable.\n\nSecond, we must argue that this random switching off and on of the\neffective force is unimportant. We assume that the force has been able\nto create some disturbance in the interface over the region $\\Lambda$,\nbefore it is switched off. How will the surface evolve? To answer this\nquestion one may simply solve the deterministic KPZ equation in the\nabsence of any source, but with an initial condition corresponding to\na localized disturbance. The result is that for any disturbance\nexceeding a critical height $h_{c} \\sim \\nu \/\\lambda$, the centre of\nthe disturbance decays logarithmically slowly -- it is essentially\nfrozen. This effect is due to the strong influence of the\nnon-linearity driving the disturbance upwards against the smoothing\neffect of the diffusion term. So during the periods of zero force, the\ndisturbance in the region $\\Lambda $ is frozen and on switching on the\nforce, the evolution continues as if there were no interruption.\n\nThe third and final point is to argue that the effective force remains\nlocalized in the region $\\Lambda $. This is in general not the case.\nThe noise has short-range correlations in space and time and rare\nevents are equally likely to occur anywhere in the surface. A simple\nresolution would be that once the rupture has been formed in $\\Lambda $,\nthen all other rare events in the vicinity may be rotationally averaged\nto give an effective force within $\\Lambda $. A more subtle point\nis the following. Once the logarithmic distortion\ncentered in $\\Lambda$ has been established, it is essentially frozen\nduring the periods of zero force. Therefore any rare event occurring\nin $\\Lambda $ will continue the evolution of the distortion. On the\nother hand, if the critical height for freezing is not small, then\nmost rare events will fail to seed such a distortion of the interface,\nsince the disturbances due to rare events which are of a height less\nthan $h_{c}$ will simply diffuse away. We therefore have a picture of\nextremely strong fluctuations being required to seed a distortion of\nheight $> h_{c}$, with less strong fluctuations being required to\nsustain the evolution of the logarithmic rupture once it has been seeded.\n\nIf this rough physical picture is correct then the\ncorrespondence between the local\nforce model considered here, and the noisy KPZ equation at the\nweak\/strong-coupling transition is established. The surface morphology\nresulting from such a picture is that of a dilute system of\nmacroscopic distortions, each with a logarithmic profile. This is in\ncontradistinction to the more conventional view of weak uniform\nfluctuations giving rise to logarithmic roughness. Such a difference\nin morphology should be discernible from numerical simulations of\nmodels believed to lie in the KPZ universality class.\n\n[A physically plausible connection between these two models may\nbe drawn also in the directed polymer\nrepresentation. The strong-coupling phase of the KPZ model\ncorresponds to the low-temperature phase of the directed polymer,\nin which the polymer is stongly localized onto low-energy\npaths, which have characteristic transverse fluctuations. Conversely,\nthe weak-coupling phase of KPZ corresponds to the high-temperature\nphase of the directed polymer in which the polymer is liberated from\nthe low energy path and performs thermal wandering.\nHence the transition between these two phases for the directed polymer\nmay be viewed as the localization transition to the lowest energy path\n(at this temperature). In this case the polymer is essentially subjected\nto a `columnar' defect, although the defect is not of constant\nenergy, and also has transverse fluctuations. Using similar arguments\nto those presented above in the interface language, one can\nmotivate the idea that i) the energy fluctuations on the column are\nirrelevant within some bounds, and ii) the transverse fluctuations\nof the column are thermally averaged such that the effective column\nis straight on some transverse scale of order $l^{d}$. We believe a more\nquantitative connection may be tractable in this directed polymer picture.]\n\n\\section{Conclusions}\n\nWe have presented a non-perturbative analysis of the deterministic KPZ\nequation under the influence of a local driving force f, which for\n$d>2$ must exceed a critical value $f_{c}$ in order to induce a\nnon-zero velocity in the surface. The central result is that for $d>2$\nand for $f \\searrow f_{c}$, the force creates a macroscopic distortion\nof the surface. This distortion has a logarithmic profile and exists\nwithin a critical radius of size $R_{c} \\sim (f-f_{c})^{-\\nu}$ where\n$\\nu (d) = 1\/(d-2)$ for $24$. In the\npenultimate section we offered some physical arguments as to why one\nmight consider this simple model to underlie the weak\/strong-coupling\ntransition in the noisy KPZ model. The main characteristics of such a\nscenario are that i) the logarithmic roughness at the transition is\ndue to a dilute system of logarithmic distortions, as opposed to a\nuniform sea of fluctuations with logarithmic variance, and ii) the\ncritical radius $R_{c}$ is a physical realization of the correlation\nlength found from the RG treatment. A more quantitative connection\nbetween these two models is presently beyond our reach, although one future\npossibility is to estimate the $d$-dependence of the weak\/strong-coupling\nphase boundary based on the exact results obtained in this paper for\nthe local force model, in conjunction with the statistical arguments\npresented above.\n\nIt would be useful to apply the non-perturbative methods presented\nhere to the case of a random local force. Such an analysis would\nallow one to tighten some of the arguments given in section 5.\nThis problem is also of substantial interest in its own right as it may be\nmapped to a system of two directed polymers with random\ncontact interactions\\cite{kl}\\cite{lip}\\cite{more} which has a wide\nrange of applications.\n\n\\vspace{10mm}\n\nThe authors wish to thank M.A.Moore and L-H. Tang for illuminating\ndiscussions. TJN acknowledges financial support under SFB 341.\n\n\\newpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\noindent\n\nSugino gave a presentation [1] in order to continue to construct\nlattice super Yang-Mills theories along the line discussed in the\nprevious papers [2-3]. In his construction of $N$ = 2, 4 theories\nin four dimensions, the problem of degenerate vacua seen in [2] is\nresolved without affecting exact lattice supersymmetry, while in\nthe weak coupling expansion some surplus modes appear both in\nbosonic and fermionic sectors reflecting the exact supersymmetry.\nA slight modi- fication to the models is made such that all the\nsurplus modes are eliminated in two- and three-dimensional models\nobtained by dimensional reduction thereof. $N$ = 4, 8 models in\nthree dimensions need fine-tuning of three and one parameters\nrespectively to obtain the desired continuum theories, while\ntwo-dimensional models with $N$ = 4, 8 do not require any\nfine-tuning. \\newline Some of Sugino's results, say, in page 3 and\nin page 10 for the equations (2.10), (3.5) and (3.6) are based on\nthe derivations in Appendix A (Resolution of Vacuum Degeneracy).\nThe minimum $\\vec{\\Phi} (x)+\\Delta \\vec{\\Phi} (x) = 0$ is realized\nby the unique configuration $U_{\\mu\\nu} (x) = 1$ for any choice of\nthe constant $r$ in the range $r=\\cot(\\phi)$ : $0<\\phi \\le\n\\pi\/(2N)$. \\newline The present author, after following his\nderivations in Appendix A., would like to point out that there are\nother possibilities except the unique solution $U=V=1$ which\nSugino claimed in [1]. \\newline In fact, Sugino missed one other\ncondition : $[u_k + v_{i(k)}]\/2=-\\pi$! \\newline To easily\nillustrate our reasoning, we plot a schematic figure (shown as\nFig. 1) of which the possible regions $\\overline{A'A}$ and\n$\\overline{B'B}$ are related to the equation (A.9) in [1] for\n$|\\cos(\\frac{u_k + v_{i(k)}}{2}-\\phi)| \\ge \\cos \\phi$. This\nequation comes from\n\\begin{equation}\n \\cos(\\frac{u_k + v_{i(k)}}{2}-\\phi) \\cos(\\frac{u_k - v_{i(k)}}{2}) = \\cos\n \\phi, \\hspace*{12mm} k=1,\\cdots, N,\n\\end{equation}\nonce we impose $|\\cos(\\frac{u_k - v_{i(k)}}{2})|\\le 1$ (follow\nSugino's reasoning in [1]), where $0<\\phi \\le \\pi\/(2N)$, and\n\\begin{displaymath}\n e^{i u_1} \\cdots e^{i u_N}=1, \\hspace*{24mm} e^{i v_{i(1)}} \\cdots e^{i\n v_{i(N)}}=1.\n\\end{displaymath}\nThere will be two solutions for N=1 : $u_1 =v_{i(1)}=\\pi$; $u_2\n=v_{i(2)}=-\\pi$ or $u_2 =v_{i(2)}=\\pi$ once $\\phi=\\pi\/2$. Under\nthis case, $e^{i u_1}=e^{i\\pi}=-1$ in the expression of the\nequation (A.4) in [1].\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nIn the prevailing picture of hierarchical early-type-galaxy formation,\nsmall fragments form first and then merge into larger and larger pieces\nuntil the system resembles a large, smooth, anisotropically supported\nelliptical galaxy. The study of globular clusters has an important\nrole to play in testing the predictions of this theory and in\nanswering the question about if and when the bulk of this merging took\nplace. Studies of the Galactic globular cluster system have been\nfundamental to the development of ideas on how the Galactic halo, and\nperhaps the entire galaxy, has been assembled from merging fragments\n(Searle \\& Zinn 1978; Da Costa \\& Armandroff 1995). Studies of\nextragalactic cluster systems have revealed evidence for multiple\npopulations of clusters and allow us to use them to connect the\nvarious phases of galaxy evolution.\n\nGCs in galaxies more distant than about a megaparsec cannot be\nresolved into stars, even with HST. However, the fact that they\ncontain simple stellar populations means that their integrated\nproperties can be used to determine metallicities and ages. \nSurveys of large numbers of globular cluster systems reveals that $>$90\\% of all Es\n have bimodal color distributions, and only 10-20\\% in S0s (Kundu \\& Whitmore (2001a, 2001b)).\n Almost all galaxies host a GC population with a peak in their color distribution at (V-I)=0.95.\nAvailable metallicity measurements suggest that these are old,\nmetal-poor clusters like the Galactic halo GCs (Geisler, Lee, \\& Kim\n1996). The second peak is redder than the first and implies that the\nred GCs are more metal-rich. In general, these GCs are also old, but\nthe age-metallicity degeneracy in broad-band colors makes age\ndifferences hard to determine unless the GCs are rather young (see\nWhitmore et al. 1997).\n\nThere are several competing scenarios for how the globular cluster\nsystems of galaxies formed. \\cite{ashan92} predicted that the\nmerger of two disk galaxies would produce an elliptical with a bimodal\nGC color distribution. The blue clusters are Galactic-halo like\nclusters from the progenitor galaxies. These clusters would have\nformed like those in the Galaxy, perhaps from the accretion of dwarf\ngalaxies with large numbers of GCs. Dwarf elliptical galaxies were\nefficient at forming clusters and these clusters resemble those in the\nGalactic halo (Miller et al. 1998). In the Ashman \\& Zepf scenario,\nthe red population is formed during the merger process from the\nmetal-enriched gas in the disks. The formation of new clusters also\nalleviates the problem that ellipticals have specific globular cluster\nfrequencies (number per unit luminosity) about a factor of two higher\nthan spirals do (Schweizer 1987). An alternative scenario is that all\nthe clusters\nwere formed ``in situ,'' in a multi-phase collapse of a single potential\nwell (Forbes et al. 1997) similar to the early monolithic collapse\npicture of the formation of the Galaxy (Eggen, Lynden-Bell, \\& Sandage\n1962). In this case, the metal-poor clusters form first in the halo\nand and metal-rich clusters form later during the final collapse, or\nrecollapse, stages from metal-enriched gas. Both these scenarios fit\ninto the overall picture of hierarchical galaxy formation, but they\ndiffer in when the bulk of the merging takes place.\n\nAn important result of the merger scenario is that\nthe merger of two spiral galaxies can cause the formation of many\nyoung star clusters in the starburst resulting from the collision of the\ntwo gas disks (Schweizer 1987; Ashman \\& Zepf 1992).\nThe creation of star clusters may alleviate the\nproblem that elliptical galaxies have specific globular cluster\nfrequencies (S$_N$, the luminosity-weighted number of GCs) about a\nfactor of two higher than in spirals. The evidence is growing that\nsignificant numbers of star clusters are formed during galaxy mergers.\nHubble Space Telescope observations of NGC1275 (Holtzman et al. 1992;\nCarlson et al. 1998) were some of the first to find very luminous blue\nobjects with the properties of young GCs in a galaxy that may\nhave had a recent merger. Large young cluster populations with ages\nbetween 10 and 500 Myr were soon found in obvious merger remnants like\nNGC7252 (Whitmore et al. 1993; Miller et al. 1997), NGC3921 (Schweizer\net al. 1996), NGC3256 \\citep{zepf99}, NGC4038\/4039 (Whitmore \\&\nSchweizer 1995; Whitmore et al. 1999). Spectra of the brightest young\nclusters in NGC1275 (Zepf et al. 1995) and NGC7252 (Schweizer \\&\nSeitzer 1993,1998) confirmed that the clusters were between 0.5 and 1\nGyr old with roughly solar metallicities (Figure 2). Since these\nclusters are many internal crossing times old, they would seem likely to\nsurvive to become the red GC populations of the faded merger remnants\n(e.g., Whitmore et al.\\ 1997; Goudfrooij et al.\\ 2001a).\n\nThese observations allow us to proceed to the next level of\nunderstanding the evolution of globular cluster systems; we must\nunderstand in more detail how star clusters form and how systems of\nstar clusters evolve. It is thought that star clusters are formed in\nhigh-pressure environments (Harris \\& Pudritz 1994; Elmegreen \\& Efrimov\n1997), be they caused by the collisions of giant molecular clouds\nor by a general pressure increase in the gas surrounding molecular clouds.\nObservations of\nthe Antennae can now provide some key parameters about the state of\nthe ISM during clusters formation and the feedback produced by the\nyoung clusters. Zhang, Fall, \\& Whitmore (2001) compared the\nlocations of different cluster populations with emission from the ISM\nat wavelength from X-rays to radio and found that the youngest\nclusters are associated with molecular cloud complexes and may lie in regions\nof high HI velocity dispersion. However, the small velocity dispersion\nof the clusters among themselves strongly suggest that it is not\nhigh-velocity cloud-cloud collisions that drive cluster formation, but\nthe general pressure increase experienced by gas during the merger\n(Whitmore et al.\\ 2005).\nFeedback, seen in the form of H$\\alpha$ bubbles around young-cluster\ncomplexes, may enhance this process.\n\nAn interesting question is how globular cluster systems evolve.\nMost very young star-cluster systems studied in merging galaxies have\npower-law luminosity functions of the form M$^{-2}$ (e.g. Schweizer et\nal. 1996; Miller et al. 1997; Whitmore et al. 1999; Zhang \\& Fall\n1999). However, the luminosity (or mass) function of old globular\nclusters has a log-normal shape with a peak at $M_V^0 \\sim -7.3$\n($\\sim10^5$ M$_{\\odot}$; Harris 1991). Therefore, either the initial\nmass function of star clusters was different in the past, perhaps due\nto low metallicity, or a substantial fraction of the young clusters\nmust be destroyed for the young mass functions to evolve into what we\nsee in older systems. A great deal of theoretical work has gone into\nglobular cluster destruction processes (e.g. Fall \\& Rees 1977; Gnedin\n\\& Ostriker 1997; Vesperini 1997\/98; Fall \\& Zhang 2001). The main\nprocesses that can destroy globular clusters are stellar evolution,\n2-body relaxation, dynamical friction, and disk shocking, or if the clusters\n formed in gas rich environments, interactions with GMCs can play \n a significant role (Gieles et al.~2006). The models\nof Fall \\& Zhang (2001) predict that there may be radial variations in\nthe peak of the mass function within a galaxy and that the peak will\nshift to higher masses with time. Many of these processes depend on\nthe relative velocities of the clusters and field stars and on the\ncluster orbits.\n\nIn this paper we present some first results of a large spectroscopic survey of\nstar clusters in merging and interacting galaxies. We focus on 23\nbright star-cluster candidates in the main body of \\n3256 observed\nwith GMOS on Gemini South. Other targets in our survey, to be\npresented in future papers, include NGC~4038\/39 (The Antennae),\n\\n6872, Stephan's Quintet, and M82.\n\n\n\\n3256 was classified as an intermediate-stage merger in Toomre's list\nof nearby merging systems \\citep{toomre77}. The merger is more\nadvanced than systems like NGC 4038\/39, in which the two original disks\nare still distinct, but the two nuclei (Moorwood \\& Oliva 1994;\nNorris \\& Forbes 1995; Lira et al.\\ 2002; Neff et al.\\ 2004) have not\nmerged yet either. They have a separation in projection by 5\", or\n$\\sim900$~pc. Hence, \\n3256 is not as relaxed a system as \\n7252 is.\nThe outer parts of the system are characterized by shell-like features\nand two\nextended tidal tails that are typical of merging galaxies. The body\nof the system is criss-crossed by dust lanes that enshroud an on-going\nstarburst: the far-infared luminosity, X-ray luminosity, and star\nformation rate are the highest of all the systems in the Toomre\nsequence. This starburst has created over 1000 star clusters in the\ncentral region \\citep{zepf99} as well as in the tidal tails\n\\citep{knier03, trancho07a}.\n\n The current paper is organized as follows: Section~\\ref{sec:obs}\ndescribes the observations. In \\S~\\ref{sec:results} and\n\\S~\\ref{sec:kinematics} we derive the ages, masses, extinctions,\nmetallicities, and line-of-sight velocities of the 23 clusters.\nFinally, \\S~\\ref{sec:disc} discusses and summarizes the results.\n\n\n\\n3256 is located at $\\alpha_{\\rm J2000}=10^{\\rm h}27^{\\rm m}51\\fs3$,\n$\\delta_{{\\rm J}2000}=-43\\degr54\\arcmin14\\arcsec$ and has a recession\nvelocity relative to the Local Group of $cz_{_{\\rm Helio}} = +2804\\pm 6$\n\\kms\\ , which places it at a distance of\n36.1 Mpc for $H_0 = 70$ \\kms\\ Mpc$^{-1}$.\nAt that distance, adopted throughout the present paper, $1\\arcsec = 175$ pc.\nThe corresponding true distance modulus is $(m-M)_0 = 32.79$.\nBecause of the low galactic latitude of \\n3256, $b = +11\\fdg7$, the\nMilky Way foreground extinction is relatively high, $A_V=0.403$ (Schlegel\net al.~1998), whence the apparent visual distance modulus is $V-M_V = 33.19$.\n\n\n\\section{Observations and Reductions}\n\\label{sec:obs}\n\nImaging and spectroscopic observations of star clusters in \\n3256 were\nmade with GMOS-S in semesters 2003A and 2004A.\nThe data were obtained as part of two Director's Discretionary Time\nprograms, GS-2003A-DD-1 and GS-2004A-DD-3.\nOur images cover the typical GMOS-S field, which measures approximately\n$5\\farcm5\\times 5\\farcm5$.\nThey were obtained through the \\gprime\\ and \\rprime\\ filters.\nFour GMOS masks with slitlets were used for the spectroscopic observations.\nWe used the B600 grating and a slitlet width of $0\\farcs75$, resulting in\nan instrumental resolution of 110 km\/s at 5100 \\AA.\nThe spectroscopic observations were obtained as 8 individual exposures\nwith exposure times of 3600 sec each.\nOur spectroscopy of 70 cluster candidates yielded only 26 objects that\nwere bonafied star clusters in \\n3256.\nOf these, three are located in the western tidal tail and have been\ndescribed in Trancho et al.\\ (2007).\nIn the present paper we focus on the 23 star clusters located in or near\nthe main body of NGC~3256.\nTable~\\ref{table:properties1} lists these star clusters.\nColumn (1) gives the adopted cluster ID, columns (3)--(4) the coordinates,\nand columns (5) and (6) the absolute magnitudes $M_{g'}$ and $M_{r'}$ and\ntheir errors.\nThe magnitudes have been corrected for Galactic extinction, but not for\nany internal extinction.\n\nFigure~\\ref{fig:image} shows an {\\em HST}\/ACS image of the main body of\n\\n3256, with the observed candidate clusters marked by their ID numbers. \n\nThe basic reductions of the data were done using a combination of the\nGemini IRAF package and custom reduction techniques, as described in\nAppendix A.\n\n\n\n\\section{Derivation of Cluster Properties}\n\\label{sec:results}\n\nThe derivation of cluster properties (such as age and metallicity) based on the strengths of \nstellar absorption lines through optical spectroscopy is not straightforward, due to \ndegeneracies between age, metallicity, and extinction.\nMultiple studies have addressed this problem (e.g., Schweizer \\& Seitzer\n1998; Schweizer, Seitzer, \\& Brodie 2004; Puzia et al.\\ 2005), and here we extend previous studies.\n\nAlthough in some of our cluster spectra the strengths of stellar absorption lines cannot be measured due to strong emission lines, the\nfluxes\/equivalents width(EW) of the emission lines of the surrounding \\hii region can be\nmeasured. In these cases a chemical abundance can be estimated for the \\hii\nregion, in which the cluster has recently formed, from line-emission measurements (e.g., Kobulnicky \\& Kewley\n 2004; Kobulnicky \\& Phillips 2003; Vacca \\& Conti 1992). \n This abundance is expected to be the same as that of the young stellar cluster itself, and\nhence complements abundance measurements of the absorption-line\ndominated clusters. \n\nBelow, we outline the method adopted in the present study.\n\n\\subsection{Extinction, Age, and Metallicity}\n\\label{sec:ages}\n\n\\subsubsection{Absorption-Line Clusters}\nWe first select the model spectra by Bruzual \\&\nCharlot (2003, hereafter BC03) and by Gonz\\'alez-Delgado et al.\\ (2005,\nhereafter GD05) . Then we smooth the model spectra to the same resolution\nas the observed spectra. Then we compare the cluster spectra with the models\nfor clusters of solar metallicity, to which we have applied\nvarious amounts of extinction (using de Galactic extinction law for Savage \\& Mathis 1979) , $A_V = 0$--10 in steps of 0.1 mag.\nWe select the best fitting model via minimized $\\chi^2$,\nModel$_{\\rm best}$(age,\\av), and correct the observed spectrum for the\nderived extinction $\\av$ to yield Cluster$_{\\rm obs,ext}$.\n\n\nThis procedure was carried out for the BC03 and GD05 models independently,\nand we note that for individual clusters the results are very similar.\nDue to the finer grid of young ages in the GD05 cluster models we adopted\nthese for our further analysis.\n\nWe then inserted the extinction-corrected spectra into the IDL\nimplementation of the Penalized Pixel Fitting routine (pPxF) of\nCappellari \\& Emsellem(2004)\\footnote{We used the GD05 spectra which have a resolution of 0.3 \\AA ~at 5100\\AA\n, corresponds to $\\sigma \\sim 0.3\/5100*3e5\/2.35 = 7.5$ km\/s and \nour spectra have a $\\sigma \\sim 3.96\/5100*3e5\/2.35 = 99.12$ km\/s.\nThe quadratic difference is sigma necessary to use in ppxf is $\\sigma =\\sqrt(99.12^2-7.5^2)= $ 98.83 km\/s\n}.\nThis routine determines the best linear combination of template spectra\nplus an analytic polynomial to reproduce the observed spectra and returns,\nin addition, the radial velocity of each cluster.\nFor template spectra we used all available models (both in age and\nmetallicity) of GD05.\nWe suppressed the use of any additional polynomial in order to preserve the continuum\nshape of each observed cluster.\nEmission lines and regions of the spectra affected by artifacts were masked\nduring the fits, though care was taken to minimize the number of such areas.\nVia this procedure we obtained a template spectrum for each of our clusters,\nCluster$_{\\rm temp}$, as well as an emission spectrum (the difference between\nCluster$_{\\rm obs,ext}$ and Cluster$_{\\rm temp}$).\n\nIn order to find the age and metallicity of each cluster, we measured\nthe line strengths of the hydrogen Balmer lines as well as of prominent\nmetal lines (See Table 2).\nFor this we used the Lick indices (Faber et al.\\ 1985; Gonz\\'alez 1993;\nTrager et al.\\ 1998) as well as the indices defined by Schweizer \\& Seitzer\n(1998) for young stellar populations.\nTo make the measurements, we used the routine {\\em Indexf} (Cardiel et\nal.\\ 1998), which finds the line strengths and errors by performing\nMonte-Carlo simulations on the spectra, using information derived from\nthe error spectra (i.e., the placement of the continuum bands and noise\nin the data).\n{\\em Indexf} was run on Cluster$_{\\rm temp}$ instead of\nCluster$_{\\rm obs,ext}$.\nThe reason for this is that we found that the measured index strengths of\nthe Cluster$_{\\rm obs,ext}$ depended on the S\/N ratio of the spectrum.\nThis is shown in Fig.~\\ref{fig:test-ppxf-obs}, where we plot the nine\nmeasured indices of an observed cluster (T130 in The Antennae, which is our best S\/N cluster and\n has identical setup and was observed in the same way) degraded to various S\/N ratios. \nThe open symbols represent the measurement of the line strength for each\nindex when {\\em Indexf} is run on Cluster$_{\\rm obs,ext}$, while the filled\nsymbols represent the measurements carried out on Cluster$_{\\rm temp}$.\nThe lines show the average index strength for the five highest S\/N\nmeasurements.\nFrom this numerical experiment we conclude that the measurements on\nCluster$_{\\rm temp}$ reproduce those of Cluster$_{\\rm obs,ext}$ for\nhigh S\/N, but remain accurate to S\/N $<$10, while measurements on\nCluster$_{\\rm obs,ext}$ begin to show significant scatter below\nS\/N $\\approx$ 15.\n\nHowever, we note that with this adopted procedure the measured H$\\beta$\nindex is always systematically off.\nThe models never reproduce an absorption\nfeature on the red side of the line.\nAdditionally, similar tests were performed where we inserted the model\nspectrum (i.e. treating the models as observations, degrading\nthem in S\/N and finding the indices). These tests showed a systematic offset\n in the H$\\delta_A$ measurements although the other lines were well reproduced.\nTherefore, we removed both the H$\\beta$ and H$\\delta_A$ indices from\nour list for further analysis.\n\nUsing {\\em Indexf}\\,\\ we also measured the line strengths of the GD05 models,\nfor all model ages and metallicities.\nThe age and metallicity of each observed cluster was then determined\nby comparing its line indices to that of models, weighted by the respective \nerrors, in a least $\\chi^2$ sense.\nIn order to test the robustness of this technique we added random errors\nto the measured indices (using a normal distribution with a dispersion\ncorresponding to the 1-$\\sigma$ measurement error) and re-did the analysis.\nThis was done 5000 times for each cluster.\nThe final age was then determined by creating a histogram of the derived\nages and fitting a gaussian to it (in logarithmic space), where the adopted\nage is the peak age of the gaussian and its error is the standard deviation.\nThe cluster metallicity was found by simply averaging the derived\nmetallicity of the 5000 runs.\nExamples of this process are shown in Fig.~\\ref{fig:ages}.\n\n\nFinally, to check the consistency of our results, we plotted the spectra\nof Cluster$_{\\rm obs, ext}$ and the best fitting model (i.e., the\nmodel closest in age and metallicity).\nIf the fit was not satisfactory, then we began the entire process again,\neliminating the initially derived extinction from the options.\nThis was the case for only a handful of young clusters that were initially\nfit with low extinctions and higher ages.\nTwo examples of cluster spectra are shown in Fig.~\\ref{fig:examples}.\n\nThe black lines are the Cluster$_{\\rm obs, ext}$ spectra, while the red\nand green lines represent the best fitting model (age, metallicity) and\nthe residual (Cluster$_{\\rm obs, ext}$ $-$ template $-$ constant).\nT1002 in the right panel is clearly very young and has, as such, still\nionized gas associated with it.\nIn the observed and residual spectra, we clearly see emission lines from\nH$\\gamma$, H$\\beta$ and [OIII]$\\lambda\\lambda$4959,5007.\n\nAs a further test of the derived cluster ages, we chose a subset of\nindices that are good at distinguishing between the age and metallicity\nof a cluster.\nFor illustrative purposes, Fig.~\\ref{fig:indices} shows the [MgFe] index\n(Thomas et al. 2003) plotted versus the H$\\gamma$ index for both the GD05 and BC03\nmodels. The indices from the models are also shown. From these\n diagrams we check for consistency in the derived ages and\n metallicities of the clusters.\n\nThe derived ages, extinctions, and metallicities are given in\nTable~\\ref{table:properties1}.\n\n\n\\subsubsection{Emission-Line Clusters}\n\nFor the youngest clusters with little or no absorption features in their\nspectra the task is much easier.\nFirst, we assign ages to these clusters of less than 10~Myr, due to the\npresence of large amounts of ionized gas around the cluster.\nAge dating can be refined to some degree by the presence or absence of\nWolf-Rayet features (e.g., Cluster T2005, see Fig~\\ref{fig:wr}); however,\nthat is beyond the scope of the present paper.\nThe extinction of these cluster is calculated from the H$\\gamma$ to H$\\beta$\nemission-line ratio.\n\nWe adopt the chemical analysis method from Kobulnicky \\& Kewley 2004 (hereafter KK04) to determine \nthe metallicity.\nWe measure the EW ratio of the collisionally excited \n[OII]$\\lambda$3727 and [OIII]$\\lambda\\lambda$4959,5007 emission lines relative to the H$\\beta$\nrecombination line (known as R$_{23}$) and [OIII]$\\lambda\\lambda$4959,5007 relative to [OII]$\\lambda$3727 \n (knows as O$_{32}$) , along with the calibrations on KK04 (their Fig. 7 - upper brach) and\nthe solar abundances by Edmunds \\& Pagel (1984).\nInstead of the traditional flux ratio, the KK04 method uses EW ratios that have the\n advantage of being reddening independent.\n\nAs can be seen in Table~\\ref{table:properties1} the metallicites found\nfor absorption-line and emission-line clusters agree well, giving us\nconfidence in the robustness of the diagnostic methods and results. \n\n\\subsection{Masses}\n\nIn order to calculate the mass of each cluster we compared the photometry\n(\\gprime\\ and \\rprime) with the BC03 SSP models, assuming a Chabrier (2003)\nstellar initial mass function and solar metallicity.\nWe then used the age dependent mass-to-light ratio from the models to\nconvert our derived absolute magnitudes (observed magnitudes corrected for\nGalactic extinction, internal extinction, and the assumed distance modulus)\nto mass.\nErrors on the mass were estimated from the derived errors on the age and\nphotometric errors.\nSystematic errors (e.g., errors associated with the distance to \\n3256) are\nnot included.\nTable~\\ref{table:properties2} gives the derived masses of the clusters.\n\n\\subsection{Velocities}\n\nIn the case of absorption-line clusters, we used the IRAF task\n{\\em rvsao.xcsao} for the determination of the redshift from the\nindividual spectra, using three different type (A, O, B) radial-velocity standard stars (HD~100953,\nHD~126248, and HD~133955) observed at the same resolution as the clusters.\nThe three template stars were employed to reduce the\nsystematic errors introduced by the effect of template mismatch when\ncomputing the redshift using the cross-correlation technique.\n\n\nFor the emission-line clusters, velocities were measured from the observed\nemission lines using the IRAF task {\\em rvsao.emsao}.\n\nIn both cases the velocities were corrected to heliocentric (see Table 3).\n\nFigure~\\ref{fig:positions} shows the positions of the observed clusters\nwithin \\n3256 (shown in contours to highlight its main features), marked\nwith the cluster metallicities, extinctions, ages, and velocities,\nrespectively.\n\n\n\n\\section{Cluster Kinematics: Two Populations?}\n\\label{sec:kinematics}\n\n\n\nThere is strong evidence that the molecular gas in the central region\nof \\n3256 lies in a disk that rotates (Sakamoto et al.\\ 2006).\nThe rotation axis of this gas disk lies approximately along the\nnorth-south direction, which is also the apparent minor axis of the main\noptical disk.\nIt is interesting to compare the observed cluster radial velocities to\nthe molecular-gas velocities at each cluster's position.\n\nFigure~\\ref{fig:rot-vel} shows the measured radial velocities of the clusters\nplotted versus the cluster right ascension (RA), corresponding roughly\nto their projected position along the major axis.\nSuperimposed is the rotation curve for the molecular gas measured by\nSakamoto et al.\\ (2006).\nThe figure suggests that the majority of the clusters are still associated\nwith the gas from which they formed (see Table~\\ref{table:properties2}).\nAt least 15 of the 23 observed clusters show clear evidence of corotating\nwith the molecular-gas disk. Hence, we will refer to these clusters as ``disk clusters.''\n\n\n\\subsection{Origin of the Disk}\n\nSakamoto et al.\\ (2006) suggest that the molecular-gas disk may have\nformed during the merger of the parental spiral galaxies. \n By using the ages of the disk clusters we can put a lower limit on the\nlongevity of the disk. These ages range from recently formed (e.g., T761: $<$10~Myr) to 100~Myr\nold (T112). Thus the molecular-gas disk must have existed for at least 100~Myr.\n\nThe \\n3256 merger probably began approximately $\\sim500$ Myr ago (English et al 2003).\nIt is not yet completed, as the two nuclei have yet to merge (Moorwood \\&\nOliva 1994; Norris \\& Forbes 1995; English \\& Freeman 2003).\nTherefore, {\\em if}\\, the observed molecular-gas disk and clusters formed\nduring the merger, the disk must have begun forming early in the merger.\nSakamoto et al.\\ (2006) compare the \\n3256 system to that of \\n7252, a\nrecently formed merger remnant which also hosts a molecular-gas disk.\nHowever, in contrast to the situation in \\n7252 the two nuclei of \\n3256\nhave yet to merge, which may disrupt any current large-scale gaseous\ndisk (Barnes 2002). \nHence, the two merger systems may not presently be comparable.\nAn alternative hypothesis, however, is that the observed gas disk belongs\nto one of the two original spiral galaxies, so that the observed disk\nclusters simply formed in that disk as part of the enhanced star-formation\nactivity caused by the gravitational interaction. \n\nIn either scenario, we would expect older star clusters to be present as\nwell. The fact that they are not detected in the present study is readily\nexplained by selection bias: we selected the brightest clusters, which\ntend to be young, for spectroscopy.\n\n \n\\subsection{Non-Disk Clusters}\n\nIn the western section of the galaxy we find five clusters which have\nvelocities apparently inconsistent with an extrapolation of the\nmolecular-gas disk velocities. The clusters have ages \n between $<7$~Myr (e.g. T96) and $\\sim150$~Myr (e.g. T1078). These clusters may belong\neither to the other spiral disk (which may lie behind the observed disk\nin projection) or to material which has become dissociated from the\noriginal disks due to the interaction\/merger. These clusters are also located spatially near the\n beginning of the western tail (see Fig.~1 in Paper~I).\nA detailed comparison with the HI position-velocity diagram of English et\nal.~(2003) shows that the HI tail begins approximately $45\\arcsec$ to the\nwest of the observed clusters. However, as Fig.~\\ref{fig:english-himap}\nshows, these clusters are coincident spatially and kinematically with\nHI gas that has a very different radial-velocity distribution from that of\nthe molecular gas. The H I radial velocities reach a minimum near the\nCO rotating disk and a maximum at the kinematic center of \\n3256, whereupon\nthey begin dropping again inside the western tidal tail.\nOne possible interpretation is that the gas-velocity anomaly, noted already\nby English et al.\\ (2003), is caused by gas falling back into the central\nparts of \\n3256 from one of the tidal tails (perhaps the eastern one).\n\nTwo other clusters located closer to the observed galactic center stand out in terms of their \nkinematics. These clusters (T779 and T343) have velocities larger than that expected \nif they were part of the rotating molecular-gas disk (although T343 is only incompatible with the disk velocity at the\n 1.5$\\sigma$ level). Both clusters are very young and have only modest extinction (\\av = 0.4--0.8). \n We note that these two clusters are located in a part of the galaxy where there is a rather large \n scatter in the measured velocities of the clusters (e.g. T201, T343, T356, T779) and thus their \n deviation may be part of a larger trend. It is possible that we are seeing a heating or beginning destruction of the disk\n due to the interaction\/merger, and that star-formation is proceeding from an ordered phase, i.e. \n in a disk, to that of a more chaotic phase where dispersion dominates over rotation.\n\n\n\\section{Discussion}\n\\label{sec:disc}\n\n\\subsection{Environment of the Clusters}\nSome of the emission line clusters that appeared isolated in our ground based images\nare clearly resolved into multiple subcomponents in the {\\em HST}\/ACS\nimages.\nThe same phenomenon has been observed in the Antennae galaxies (e.g.,\nWhitmore \\& Schweizer 1995; Bastian et al.~2006), showing that clusters\nare often not formed in isolation but instead tend to form in larger\ngroupings, or cluster complexes.\nThese complexes are thought to be a short-lived phenomenon as they\ndisperse on short timescales, although merging within the central parts\nof the groupings is possible (e.g., Fellhauer \\& Kroupa 2002).\nThe remnants of such cluster-cluster mergers are an attractive means to\nform extremely large clusters, such as W3 and W30 in NGC~7252\n(Kissler-Patig et al.\\ 2006).\n\nHence, it is possible that in a few cases (which are in very crowded\nregions, e.g. T2005 and T116) we may be over-estimating the mass of an apparent ``cluster''\nif, in fact, it is made up of several clusters.\n\n\n\n\\subsection{Star\/Cluster Formation Rates}\n\n\\n3256 contains the most molecular gas among the merging galaxies and merger\nremnants of the Toomre sequence ($1.5\\times 10^{10}$~\\msun: Casoli et al.\\\n1991; Aalto et al.\\ 1991; Mirabel et al.\\ 1990, from Zepf et al.\\ 1999).\nThus, there will be plenty of gas left to fuel a massive starburst when the\nnuclei merge (e.g., Mihos \\& Hernquist 1996).\nAt that time, one may expect the star\/cluster formation rate to increase\nsubstantially (between 3 and 10 times depending on the encounter parameters\nand the time to nuclear coalescence) (Cox et al.~2006).\nThe present star-formation rate in \\n3256 is estimated to be 33 \\msun\/yr\nfrom the far-infrared luminosity (Knierman et al.\\ 2003).\nIt is known that a tight relation between the most massive star cluster\nin a galaxy and the galaxy's star-formation rate exists (e.g., Larsen 2004).\nIn \\n3256 we find about 10 clusters with masses in excess of $10^6 M_{\\odot}$.\nIf the star (cluster) formation rate does increase substantially as the\nnuclei merge, we may expect the \\n3256 system to create clusters\nwith masses significantly above 10$^7$\\,\\msun, such as those found in\n\\n7252 and \\n1316 (e.g., Schweizer \\& Seitzer 1998; Maraston et al.\\ 2004;\nBastian et al.\\ 2006).\n\nArp 220 is comparable to NGC~3256, as it also has an extremely high infrared luminosity, \nis thought to have formed through a merger, and has two distinct nuclei separated by only 300~pc in projection\n(Scoville et al.~1998, Wilson et al. 2006). The nuclei in NGC~3256 are separated (in projection) \nby $\\sim900$~pc indicating that it is possibly slightly younger (in terms of merger stage) than Arp 220.\nThe star-formation rate in Arp~220 ($240~M_{\\odot}$\/yr; Wilson et al.~2006)\nis approximately seven times higher than that in NGC~3256. It is then\npossible that NGC~3256 is currently \npoised to enter the regime of ultra-luminous infrared galaxies (ULIRGs) as its star-formation \nrate increases due to the merging of the two nuclei. Arp 220 also closely follows the relation \nbetween global star-formation rate in the galaxy and the magnitude of the most massive cluster\n (Wilson et al.~2006), arguing further that NGC~3256 is going to form clusters in excess of $10^7 M_{\\odot}$.\n\n\\subsection{Metallicities}\n\nAs Figure~\\ref{fig:hist} shows the young clusters in the NGC~3256 system appear to\n have rather high metallicities, with the average being $\\sim1.5 \\zo$. \n This was also found for the clusters in the tidal tails described in Paper~I.\nWe do not see any major spread in metallicities among our clusters and,\nspecifically, no differences coming from ages or emission \nversus absorption. The fact that the majority of these young clusters formed in a \ndisk and their age spread is small fits in nicely with the notion of a normal starbust process.\n\n\n\\section{Comparison of NGC~3256 with Other Merging Galaxies}\n\n\nMany of the clusters observed to be associated with the molecular gas\nare quite massive ($>10^5 \\msun$), have survived for many internal\ncrossing times ($t_{\\rm cr} \\approx 2$--4 Myr), and are therefore\ngravitationally bound. \nThis justifies calling them young globular clusters. Such young globulars\nhave been found in many merging galaxies, from beginning mergers (e.g.,\n\\n4038\/39: Whitmore \\& Schweizer 1995) to completed mergers (e.g., \\n1316:\nGoudfrooij et al.\\ 2001a). The formation of these clusters is \nthought to trace the major star-formation events in these galaxies and they must form with approximately \nthe same kinematics as the gas out of which they form. It is therefore interesting to compare the \\n3256 \ncluster population to those of younger (in terms of dynamical stage) and older merging systems.\n\nThe majority of the cluster population of the beginning merger \\n4038\/39\nis still unambiguously confined to the disks of the two progenitor galaxies\n(Whitmore et al.~2005; Bastian et al.~2006; Trancho et al.~2007).\nTherefore, \\n3256 appears to predominately fall into this category (see\n\\S~\\ref{sec:kinematics}).\n\nOlder systems, on the other hand, such as \\n3921 (Schweizer, Seitzer, \\&\nBrodie~2004), \\n7252 (Schweizer \\& Seitzer 1998), and \\n1316 (Goudfrooij\net al.~2001b) are all characterized by the kinematics of their clusters\nbeing dominated by non-circular, halo-type orbits.\nWhen does the transition happen? It will be interesting to determine\nwhether the majority of the star formation happens in the disks of the\nprogenitors and their orbits are subsequently randomized (i.e., turned\ninto pressure supported systems rather than rotational supported systems),\nor whether the star-formation events which precede the destruction of the \ngalactic disks pale in comparison with the star-formation rate during and\nafter the destruction.\n\nNote that shortly after the merger a gaseous disk can reform around the\nnucleus of the merger remnant, which can harbor subsequent star formation\n(e.g., \\n7252: Miller et al.\\ 1997; Wang, Schweizer, \\& Scoville 1992),\nalthough such star formation appears to occur at a much lower intensity\nthan previous star-forming episodes during the merger.\n\n\n\n\\section{Summary and Conclusions}\n\nWe have studied the ages, metallicities, masses, extinctions, and velocities\nof 23 clusters in \\n3256 based on the Lick index system in conjunction\nwith CO and HI maps.\nThe main results are:\n\\begin{itemize}\n\\item The clusters have rather high metallicities, with the average being $\\sim1.5$\\zo\\ (Fig.\\ 9) and are massive, with masses in the range (2--40) $\\times 10^{5}~\\msun$. \nThe ages of the clusters are between a few Myr and $\\sim\\,$150 Myr.\n\n\\item There is strong evidence for a rotating molecular-gas disk in\n\\n3256 (Sakamoto et al.~2006). \nThe majority of the clusters in our sample follow the same rotation curve\nas the gas and hence were presumably formed in the molecular-gas disk.\nHowever, a western subsample of five clusters has velocities that deviate\nsignificantly from the gas rotation curve. These clusters may either\nbelong to the second spiral galaxy of the merger or may have formed in\ntidal-tail gas falling back into the system.\n\n\\item Although the merger began $\\sim\\,$500 Myr ago (English et al.\\ 2003),\nwe found the clusters to be $\\lesssim150$ Myr old. Since there are still\ntwo distinct nuclei marking the presence of two galaxies, we conclude that\nthe gas disk probably belongs to one of the galaxies and is not yet a disk\nof pooled gas\nproduced in the merger itself. Presumably clusters older than the ones\npresent in our sample do exist in \\n3256. However, these older clusters\nwould not have been selected as spectroscopic candidates due their fainter\nmagnitudes (i.e., only the brightest candidates were selected).\n\n\\item By comparing of the NGC~3256 cluster population with other known galactic mergers, we suggest that this system is akin to Arp~220, although slightly dynamically younger. If this is the case, the we may expect the star\/cluster formation rate to increase significantly as the two galactic nuclei merge. This in turn may push \\n3256 into the category of ULIRGs (it is currently a LIRG). Due to the expected large increase the in the star\/cluster formation rate, a few clusters above $10^7$~\\msun are predicted to form before this merger is through.\n\n\\item Some of the clusters which appeared as isolated in our ground-based\nimages are clearly resolved into multiple sub-components in the HST-ACS\nimages. The same effect has been observed in the Antennae galaxies,\nshowing that clusters are often not formed in isolation, but instead tend\nto form in larger groups or cluster complexes. \n\nWith these new results, i.e.~cluster ages, metallicities, extinctions and kinematics, as well as recent CO and HI maps, N-body simulations of this merger would be the best way to fully understand this wealth of data. The models would have important implications for (globular) cluster formation and destruction as well as the star-formation history of the merger (through the age\/metallicity of the clusters) with respect to other mergers like the Antennae, \\n7252 and Arp 220. Finally, the details of the models may present important implications of the formation of ellipticals galaxies through the major mergers of spiral galaxies.\n\n\n\n\n\\end{itemize}\n\n\n\n\\begin{figure}\n \\begin{center}\n \\epsscale{1.0}\n \\caption{ ACS F555W image of \\n3256 (central region) with the\n observed candidate cluster ID numbers. Green labels denote clusters with spectra dominated by emission lines,\n while red labels denote absorption line dominated cluster spectra. The line with an arrow points north, while the line\nwithout one points east.}\n \\label{fig:image}\n \\end{center} \n \\end{figure}\n \n\n\\begin{figure}\n \\epsscale{1.1}\n\t\\plotone{f2a.eps}\n\t\\plotone{f2b.eps}\n \\caption{Top: Histograms of ages derived by simulating the\n effect of errors on the age-fitting routine. The derived age and\n error (in logarithmic units) is given in each panel. Bottom: \n Same as top, except now for metallicity. See text for details of\n the simulations.}\n \\label{fig:ages}\n \\end{figure}\n\n\n\\begin{figure}\n \\epsscale{1.0}\n \\plotone{f3.eps}\n \\caption{Tests showing the effect of the S\/N ratio on the measured\n line indices. We used a high-S\/N cluster spectrum, degraded its S\/N\n ratio, and then measured line indices. Open symbols mark measurements\n made from the observed spectrum (corrected for extinction) directly,\n while solid symbols represent measurements made from a template\n spectrum derived for the cluster using the pPxF technique. Solid\n lines represent averages of the highest S\/N experiments on the\n observed cluster spectrum (Cluster$_{\\rm obs,ext}$). For further\n details, see text.}\n \\label{fig:test-ppxf-obs}\n \\end{figure}\n \n \n\\begin{figure}\n \\epsscale{1.10}\n \\plottwo{f4a.eps}{f4b.eps}\n \\caption{Determination of cluster ages and metallicities.\n (left) H$\\gamma$ vs. [MgFe] from the Gonz\\'alez-Delgado et al.\\\n (2005) SSP models for four different metallicities are shown.\n Data points with error bars mark observed clusters and their\n 1-$\\sigma$ errors. In addition, we show the position of the\n massive cluster W3 in NGC~7252 and the three tidal tail cluster in NGC~3256 from previous studies.\n (right) Same, but for Bruzual-Charlot (2003) models.}\n \\label{fig:indices}\n \\end{figure}\n\n\n \\begin{figure}\n \\epsscale{1.10}\n \\plottwo{f5a.eps}{f5b.eps}\n \\caption{Examples of spectra for two clusters in our sample.\n The observed spectra have been corrected for the estimated\n interstellar extinction. The red lines represent the best\n fitting (see \\S~\\ref{sec:ages} for a discussion of the method)\n model template (age and metallicity). The green lines represent\n the residual (observed cluster $-$ best fitting template $-$ constant).\n The parameters for the best fitting template are given in each panel.}\n \\label{fig:examples}\n \\end{figure}\n\n\n\\begin{figure}\n \\epsscale{0.80}\n \\plotone{f6.eps}\n \\caption{Example spectrum of an emission-line cluster, T2005, which\n also shows strong Wolf-Rayet features.}\n \\label{fig:wr}\n \\end{figure}\n\n\n\\begin{figure}\n \\begin{center}\n \\epsscale{1.0}\n \\plotone{f7.eps}\n \\caption{ The position of the clusters in NGC 3256 for different\n velocities (top left), ages(top right,where the ages are given in logarithmic units in years), extinctions (bottom left),\n and metallicities (bottom right). The contours are shown to highlight the main features\n of the galaxy. The upper and lower (magenta) circles mark the nucleus\n and the second brightest optical source in the galaxy, respectively.\n Triangles (red) mark clusters whose spectra are dominated by absorption\n lines while blue marks emission line clusters (see also\n Table~\\ref{table:properties1}).} \n \\label{fig:positions}\n \\end{center} \n \\end{figure}\n\n\n\\begin{figure}\n \\epsscale{1.10}\n \\plotone{f8.eps}\n \\caption{A position--velocity diagram for the observed clusters in\n \\n3256. The dotted line shows the rotation curve of the molecular gas,\n as measured by Sakamoto et al.~(2006).}\n \\label{fig:rot-vel}\n \\end{figure}\n \n\\begin{figure}\n \\epsscale{1.0}\n \\caption{ Metallicity distribution of the clusters in NGC3256. The figure includes the \n metallicities derived from both the absorption and emission line clusters. Note that all the\n clusters are fairly metal rich, with a mean around 1.5 \\zo}\n \\label{fig:hist}\n \\end{figure}\n \n\\begin{figure}\n \\epsscale{1.0}\n \\caption{HI position--velocity plot by English et al. (2003), with\nthe observed clusters superposed. Open red circles mark the clusters that\nkinematically follow the rotating CO disk, while filled blue circles mark those that\ndo not follow the CO disk, but follow the HI velocities instead.}\n \\label{fig:english-himap}\n \\end{figure}\n \n\n\n\n\n\\begin{deluxetable}{lcccccccc}\n\n\\def\\phs\\phn{\\phs\\phn}\n\\def\\phn\\phn{\\phn\\phn}\n\\tablecolumns{10}\n\\tablewidth{-20pt}\n\\tablecaption{Cluster properties. (The magnitudes have been only corrected for Galactic extinction)}\n\\tablehead{\n\\colhead{ID} & \\colhead{A\/E\\tablenotemark{a}} & \\colhead{$\\Delta$RA\\tablenotemark{b}} & \\colhead{$\\Delta$DEC\\tablenotemark{b}} & \\colhead{$M_{g'}$} & \\colhead{$M_{r'}$} & \\colhead{A$_V$\\tablenotemark{c} }& \\colhead{Z}& \\colhead{Log(age)}\\\\\n\\colhead{ } & \\colhead{ } & \\colhead{(sec) } & \\colhead{(arcsec) } & \\colhead{(mag)} & \\colhead{(mag)} & \\colhead{(mag)} & \\colhead{(\\zo) }& \\colhead{(year)}\n}\n\n\\startdata\nT88 &0 & 49.13 & 22.04 &-12.8$\\pm$0.1 &-12.6$\\pm$0.1 &0.00 &1.5$\\pm$0.5 &7.5$\\pm$0.1 \\\\ \nT96 &1 & 49.92 & 32.90 &-13.1$\\pm$0.1 &-12.4$\\pm$0.1 &1.30 &1.7$\\pm$0.2 &$<$6.8 \\\\ \nT99 &0 & 49.72 & 21.03 &-14.8$\\pm$0.1 &-14.5$\\pm$0.1 &0.00 &1.7$\\pm$0.4 &7.5$\\pm$0.1 \\\\ \nT112 &0 & 50.56 & 28.61 &-14.0$\\pm$0.1 &-13.1$\\pm$0.1 &1.70 &1.9$\\pm$0.1 &7.96$\\pm$0.08 \\\\ \nT116\\tablenotemark{e} &1 & 50.41 & 22.51 &-15.4$\\pm$0.1 &-14.7$\\pm$0.1 &1.09 &1.4$\\pm$0.2 &$<$6.8 \\\\\nT141 &1 & 50.82 & 17.22 &-14.6$\\pm$0.1 &-13.7$\\pm$0.1 &3.31 &1.4$\\pm$0.2 &$<$6.8 \\\\\nT161 &1 & 50.94 & 8.01 &-15.1$\\pm$0.1 &-14.2$\\pm$0.1 &0.58 &1.4$\\pm$0.2 &$<$6.8 \\\\ \nT199 &0 & 51.43 & -4.44 &-14.3$\\pm$0.1 &-14.1$\\pm$0.1 &0.00 &1.6$\\pm$0.3 &6.5$\\pm$0.1 \\\\ \nT201 &0 & 52.28 & 29.03 &-12.5$\\pm$0.1 &-12.4$\\pm$0.1 &0.00 &1.2$\\pm$0.2 &7.0$\\pm$0.1 \\\\ \nT306 &1 & 51.50 & 10.62 &-15.8$\\pm$0.1 &-15.3$\\pm$0.1 &0.00 &1.4$\\pm$0.2 &$<$6.8 \\\\\nT343 &0 & 51.83 & 10.91 &-16.1$\\pm$0.1 &-16.0$\\pm$0.1 &0.40 &1.5$\\pm$0.5 &6.6$\\pm$0.1 \\\\\nT356 &1 & 52.26 &10.38 &-15.2$\\pm$0.1 &-14.9$\\pm$0.1 &0.80 &1.4$\\pm$0.2 &$<$6.8 \\\\\nT374 &1 & 52.77 & 7.60 &-12.8$\\pm$0.1 &-12.5$\\pm$0.1 &0.43 &1.5$\\pm$0.2 &$<$6.8 \\\\ \nT492 &0 & 48.96 & 30.92 &-12.4$\\pm$0.1 &-11.3$\\pm$0.1 &1.70 &1.4$\\pm$0.2 &6.5$\\pm$0.1 \\\\\nT654 &1 & 51.61 & 3.67 &-12.8$\\pm$0.1 &-12.3$\\pm$0.1 &2.70 &1.4$\\pm$0.2 &$<$6.8 \\\\\nT661 &0 & 51.37 & 9.22 &-15.2$\\pm$0.1 &-14.8$\\pm$0.1 &0.20 &1.2$\\pm$0.2 &7.7$\\pm$0.1 \\\\\nT744 &0 & 51.84 & 21.17 &-12.4$\\pm$0.1 &-12.0$\\pm$0.1 &0.50 &1.1$\\pm$0.2 &6.8$\\pm$0.1 \\\\\nT761 &2 & 52.63 & 3.53 &-14.2$\\pm$0.1 &-13.6$\\pm$0.1 &2.20 &1.5$\\pm$0.2 &5.9$-$6.7 \\tablenotemark{d} \\\\\nT779 &1 & 52.53 & 14.00 &-12.9$\\pm$0.1 &-12.6$\\pm$0.1 &0.80 &1.4$\\pm$0.2 &$<$6.8 \\\\\nT799 &1 & 53.05 & 7.78 &-13.0$\\pm$0.1 &-12.9$\\pm$0.1 & 4.20 & 1.6$\\pm$0.2 &$<$6.8 \\\\\nT1002 &0 & 54.74 & 12.46 &-10.6$\\pm$0.1 &-10.2$\\pm$0.1 &0.30 &1.3$\\pm$0.1 &6.9$\\pm$0.1 \\\\\nT1078 &0 & 48.96 & 21.17 &-12.9$\\pm$0.1 &-12.7$\\pm$0.1 &0.50 &1.6$\\pm$0.3 &8.2$\\pm$0.1 \\\\ \nT2005 \\tablenotemark{e} &2 & 53.08 & 13.20 &-15.4$\\pm$0.1 &-14.8$\\pm$0.1 &0.00 &1.4$\\pm$0.2 &5.9$-$6.7 \\tablenotemark{d} \\\\\n\n\n\n\\enddata\n\\tablenotetext{a}{\\,0=absorption, 1=emission, 2=emission with WR features}\n\\tablenotetext{b}{\\,From Base position RA=10:27:00 DEC=$-$43:54:00 (J2000)}\n\\tablenotetext{c}{\\,These extinction have been calculated on spectra already corrected by Galactic extinction ($A_V=0.403$) }\n\\tablenotetext{d}{\\,Ages calculated using the Starburst 99 models}\n\n\\label{table:properties1}\n\n\n\\end{deluxetable}\n\n\n\\begin{deluxetable}{lccccccc}\n\\def\\phs\\phn{\\phs\\phn}\n\\def\\phn\\phn{\\phn\\phn}\n\\tablecolumns{8}\n\\tablewidth{0pt}\n\\tablecaption{Measured indices for the absorption line clusters.}\n\\tablehead{\n\\colhead{ID} &\\colhead{ $H+He$\\tablenotemark{a} } &\\colhead{ $K$\\tablenotemark{a} } &\\colhead{ $H8 $\\tablenotemark{a} } &\\colhead{$H\\gamma_A$\\tablenotemark{b}} &\\colhead{ $Mgb5177$\\tablenotemark{b}} &\\colhead{ $Fe5270$\\tablenotemark{b}} &\\colhead{ $Fe5335$\\tablenotemark{b}} \\\\\n\\colhead{} & \\colhead{ (\\AA) } & \\colhead{ (\\AA) } & \\colhead{ (\\AA) } & \\colhead{( \\AA)}& \\colhead{( \\AA)}& \\colhead{( \\AA)}& \\colhead{( \\AA)} \n}\n\\startdata\nT88 &6.45$\\pm$0.31 &0.34$\\pm$0.25 &5.75$\\pm$0.31 &6.26$\\pm$0.31 &0.22$\\pm$0.17 &0.84$\\pm$0.22 &1.48$\\pm$0.28 \\\\ \nT99 &6.40$\\pm$0.44 &0.35$\\pm$0.26 &5.64$\\pm$0.43 &6.25$\\pm$0.28 &0.25$\\pm$0.17 &0.88$\\pm$0.20 &1.59$\\pm$0.28 \\\\ \nT112 &8.53$\\pm$0.40 &0.79$\\pm$0.23 &7.52$\\pm$0.42 &7.88$\\pm$0.22 &0.41$\\pm$0.10 &1.18$\\pm$0.12 &1.81$\\pm$0.18 \\\\ \nT199 &3.09$\\pm$0.31 &0.03$\\pm$0.17 &2.40$\\pm$0.31 &3.18$\\pm$0.21 &0.16$\\pm$0.12 &0.50$\\pm$0.16 &0.99$\\pm$0.23 \\\\ \nT201 &4.21$\\pm$1.51 &0.11$\\pm$0.85 &3.39$\\pm$1.49 &4.19$\\pm$1.01 &0.45$\\pm$0.11 &0.81$\\pm$0.26 &1.08$\\pm$1.09 \\\\ \nT343 &2.96$\\pm$0.44 &0.40$\\pm$0.24 &2.17$\\pm$0.41 &3.33$\\pm$0.33 &0.43$\\pm$0.18 &0.61$\\pm$0.23 &0.77$\\pm$0.34 \\\\ \nT492 &2.31$\\pm$0.21 &0.57$\\pm$0.21 &2.62$\\pm$0.41 &2.87$\\pm$0.34 &0.35$\\pm$0.08 &0.54$\\pm$0.10 &0.71$\\pm$0.15 \\\\ \nT661 &7.29$\\pm$0.49 &0.63$\\pm$0.29 &6.44$\\pm$0.48 &7.00$\\pm$0.36 &0.39$\\pm$0.21 &0.96$\\pm$0.26 &1.40$\\pm$0.38 \\\\ \nT744 &4.24$\\pm$0.46 &0.52$\\pm$0.24 &3.10$\\pm$0.52 &3.27$\\pm$0.30 &0.41$\\pm$0.18 &0.75$\\pm$0.22 &0.92$\\pm$0.33 \\\\ \nT1002 &4.26$\\pm$1.60 &0.87$\\pm$0.23 &3.55$\\pm$0.73 &3.44$\\pm$0.47 &0.36$\\pm$0.03 &0.65$\\pm$0.14 &0.79$\\pm$0.17 \\\\ \nT1078 &9.16$\\pm$0.33 &0.94$\\pm$0.19 &8.23$\\pm$0.33 &8.55$\\pm$0.22 &0.53$\\pm$0.14 &1.20$\\pm$0.17 &1.66$\\pm$0.26 \\\\ \n\n\n\\enddata\n\\tablenotetext{a}{\\,Index definition by Schweizer \\& Seitzer (1998).}\n\\tablenotetext{b}{\\,Lick index.}\n\\label{table:indices}\n\\end{deluxetable}\n \n\n\n\n\\begin{deluxetable}{lccccc}\n\\tablecaption{Kinematics and Masses of the clusters.}\n\\tablewidth{-10pt}\n\n\\tablehead{\n\\colhead{} & \\colhead{} & \\colhead{$cz$(CO)} & \\colhead{$cz_{\\rm hel}$} & \\colhead{$\\Delta cz$} & \\colhead{Mass} \\\\ \n\\colhead{ID} & \\colhead{D\\tablenotemark{a}} & \\colhead{(km\/s)} & \\colhead{(km\/s)} & \\colhead{(km\/s)} & \\colhead{10$^{5}$\\msun} } \n\n\\startdata\nT88 & 1 & ... & 2821.1$\\pm$17.9\t &... & 8.3$\\pm$2.6\\\\\nT96 & 1 & 2725: & 2845.7$\\pm$17.4\t &+120: & 1.7$\\pm$0.9\\\\\nT99\\tablenotemark{b} & 1 & 2660 & 2786.9$\\pm$23.7\t &+126 &49.5$\\pm$1.8\\\\\nT112\\tablenotemark{b} & 0 & 2735 & 2711.4$\\pm$56.2\t &-23\t &45.0$\\pm$2.4\\\\\nT116\\tablenotemark{b} & 0 & 2715 & 2741.4$\\pm$15.9\t &+26\t &13.8$\\pm$7.3\\\\\nT141 & 0 & 2740 & 2736.4$\\pm$14.2\t &-4 \t & 6.7$\\pm$3.5\\\\\nT161 & 0 & 2740 & 2743.6$\\pm$17.0\t &+3\t &11.0$\\pm$5.8\\\\\nT199 & 1 & 2820: & 2865.5$\\pm$10.2\t &+45: & 3.4$\\pm$0.3\\\\\nT201 & 1 &... & 2840.2$\\pm$26.1\t &... & 2.3$\\pm$0.7\\\\\nT306\\tablenotemark{b} & 0 & 2815 & 2813.2$\\pm$14.1\t &-2\t &19.9$\\pm$0.1\\\\\nT343 & 1 & 2865 & 2993.2$\\pm$45.8\t &+128\t &18.6$\\pm$1.6\\\\ \nT356\\tablenotemark{b} & 0 & 2895 & 2914.2$\\pm$7.6\t &+19\t &12.1$\\pm$6.4\\\\\nT374 & 0 & ... & 2882.4$\\pm$24.1\t &... & 1.2$\\pm$0.6\\\\\nT492 & 1 & ... & 2910.3$\\pm$95.9\t &... & 2.2$\\pm$0.0\\\\\nT654 & 0 & 2795 & 2812.8$\\pm$25.6\t &+22\t & 1.3$\\pm$0.7\\\\\nT661\\tablenotemark{b} & 0 & 2785 & 2802.8$\\pm$25.9\t &+17\t &35.5$\\pm$8.3\\\\\nT744 & 0 & 2850 & 2911.7$\\pm$77.1\t &+61\t &2.1$\\pm$0.5\\\\\nT761 & 0 & 2860 & 2884.2$\\pm$25.6\t &+25\t &4.6$\\pm$2.4\\\\\nT779 & 1 & 2880 & 2959.0$\\pm$10.3\t &+79\t &1.4$\\pm$0.7\\\\\nT799 & 0 & ... & 2883.9$\\pm$41.2\t &... &1.6$\\pm$0.8\\\\\nT1002 & 0 & ... & 2857.5$\\pm$91.8\t &... &0.2$\\pm$0.1\\\\\nT1078 & 1 & ... & 2772.7$\\pm$9.1\t &... &15.9$\\pm$0.6\\\\\nT2005\\tablenotemark{b} & 0 & 2865 & 2871.3$\\pm$5.8\t & +4 &14.0$\\pm$7.4\\\\\n\n\n\\enddata\n\\tablenotetext{a}{\\,0=disk, 1=not disk}\n\\tablenotetext{b}{\\,Complexes}\n\n\\label{table:properties2}\n\\end{deluxetable}\n\n\n\n\n\\clearpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nYellow hypergiants are rare objects which are post-red supergiants rapidly evolving in\nblue-ward loops in the Hertzsprung-Russell diagram (de Jager 1998). \nA few of such objects are found to have prodigous mass loss, leading to the formation of thick\ncircumstellar envelopes. The study of the circumstellar envelope around them is very important\nto provide insight into the mass loss process from these massive stars and the evolution of\nstars themselves.\n\nThe star IRC+10420 is classified as yellow hypergiant (de Jager 1998). Based on\nspectroscopic monitoring, the star initially classified as spectral type\nF8 I$^{+}_{\\rm a}$ in 1973 (Humphreys et al. 1973), has evolved into spectral type A in 1990 (Oudmaijer\net al. 1996, Klochkova et al. 1997). Strong emission lines such as the Balmer series and \nCa II triplets are seen in its optical spectrum (Oudmaijer 1998). Broad wings seen in H$\\alpha$\nand Ca II triplets suggest a large outflow velocity in a massive wind close to the stellar \nsurface. High resolution optical images of IRC+10420 (Humphreys et al. 1997) revealed a complex circumstellar environment\nwith many features such as knots, arcs or loops. Lacking kinematic information, however, the real three\ndimensional structure of the envelope can not be inferred. Recently, through integral-field spectroscopy\nof H$\\alpha$ emission line, Davies et al. (2007) show strong evidence for axi-symmetry in the\ncircumstellar envelope.\n\nThe molecular envelope around IRC+10420 is very massive and is a rich source of molecular lines (Quintana-Lacaci et al. 2007).\nThe mass loss rate estimated from CO observations is at the upper end seen toward evolved stars, \nup to a few times 10$^{-4}$ M$_\\odot$ yr$^{-1}$ (Knapp \\& Morris 1985, Oudmaijer et al. 1996).\nHigh denisty tracers such as ammonia (NH$_3$) and HCN lines have been detected \\citep{menten95}.\nMolecules typical of oxygen-rich envelope such as SO and SO$_2$ (Sahai \\& Wannier 1992,\nOmont et al. 1993) are known to be present. \nIRC+10420 is also a strong OH maser source, exhibiting both\nthe main lines (1665 MHZ and 1667 MHz) maser and the satelite line (1612 MHz) maser. By comparison\nwith typical OH\/IR stars, IRC+10420 has the warmest envelope harboring OH masers (Humphreys et al. 1997).\nHigh angular resolution imaging of the OH masers by \\cite{nedoluha92} shows that the maser emissions are distributed\nin an oblate expanding shell measuring 1\\arcsec.5 in radius. Recently, Castro-Carrizo et al. (2007) presented\nhigh angular resolution ($\\sim$3\\arcsec\\, and 1\\arcsec.4) mapping of the envelope around IRC+10420 in $^{12}$CO J=1--0 and J=2--1 lines.\nTheir observations reveal an expanding but complex molecular envelope with an inner\ncavity around the central star, implying a recent dramatic decrease in the mass loss from IRC+10420. In addition,\nthe shell shows some elongation in the North-East to South-West direction, suggesting a non-spherical symmetry in the\nmass loss process. \nFrom excitation analysis of $^{12}$CO lines, Castro-Carrizo et al. (2007) show the \npresence of two separate shells with mass loss in the range 1.2 10$^{-4}$ to\n3 10$^{-4}$M$_\\odot$ yr$^{-1}$. Interestingly, the gas temperature in shells derived from their model fitting\nis unusually high, 100 K even at the large radial distance of $\\sim$2 10$^{17}$ cm.\n\nThe distance to IRC+10420 is still uncertain. From analysis of photometric and polarimetric data,\nJones et al. (1993) conclude that IRC+10420 is located at a large distance, between 4 to 6 kpc.\nHere we adopt a distance to IRC+10420 of 5 kpc, similar to that adopted in Castro-Carrizo et al. (2007).\n\nIn this paper we present high angular resolution observations of the molecular envelope around IRC+10420.\nWe also perform detailed modelling to derive in a self-consistent manner the dust properties and \nmolecular gas temperature profile in the envelope. \n \n\\begin{table}[h] \\begin{center} \\begin{tabular}{lll}\n\\multicolumn{3}{l}{Table 1: Basic data of IRC+10420 and its molecular envelope} \\\\\n\\hline\n\\hline\nR.A. (J2000) & 19$^{\\rm h}$26$^{\\rm m}$48$\\fs$03 & (1) \\\\ \nDec. (J2000) & 11$\\degr$21$\\arcmin$16$\\mbox{\\rlap{.}$''$}$7 & (1)\\\\ \nDistance & 5 kpc & (2) \\\\\nAngular scale & 1$\\arcsec$ $\\sim$ 5000 AU $\\sim$ 7.5 10$^{16}$ cm & \\\\\nLuminosity L$_*$ & $\\sim$ 6 10$^5$ L$_\\odot$ & (2) \\\\\nSpectral Type & mid-A type & (3,4) \\\\\nEffective Temperature T$_{\\rm eff}$ & 7000 K & (3) \\\\\nStellar radius & 1.66 Au \\\\\nSystemic velocity & V$_{\\rm LSR}$ = 74 kms$^{-1}$ & (3) \\\\\nExpansion velocity & V$_{\\rm exp}$ = 40 kms$^{-1}$ & (3)\\\\\nMass loss rate $\\dot{M}$ & $\\sim$ 6.5 10$^{-4}$ M$_\\odot$ yr$^{-1}$ & (5) \\\\\n\\hline\n\\end{tabular}\n\\tablerefs{ (1) SIMBAD database;\n(2) Jones et al. (1993);\n(3) Oudmaijer et al. (1996);\n(4) Klochkova et al. 1997;\n(5) Knapp \\& Morris (1985)}\n\\end{center}\n\\end{table}\n\n\\section{Observations}\n\nWe have observed IRC+10420 in the 1.3mm band with the SubMillimeter Array (SMA)\nat two different epochs and in two different configurations as summarized in Table 2. The first observation\nwas carried out on 2004 June 21$^{st}$, with 8 antennas in an extended configuration, giving projected\nbaselines from 27 to 226 m. The weather was good during the observation with zenith atmospheric opacity of\n $\\tau_{\\rm 225GHz} \\sim 0.2$ at 225 GHz and the system temperatures ranged between 200 and 600 K. \nThe second observation was carried on 2005 July 2$^{nd}$, with 7 antennas in a compact configuration, giving projected baselines\nin the range from 11 to 69 m. The weather was very good with $\\tau_{\\rm 225GHz} \\sim 0.1$ and the system temperatures ranged\nbetween 100 and 200 K.\n\nIn each observation, the correlator of the SMA was configured in the normal mode with two 2-GHz bandwidth windows\nseparated by 10 GHz. This allowed us to cover simultaneously the $^{12}$CO J=2-1 line in the upper sideband (USB)\nthe $^{13}$CO J=2-1 and SO 6$_{\\rm 5}$-5$_{\\rm 4}$ lines in the lower sideband (LSB). \nThe bandpass of individual antennas was calibrated using the quasar 0423-013 for the observation taken\nin 2004 and Uranus for observation taken in 2005.\nTo improve the S\/N of the data,\nwe smoothed data from the instrumental spectral\nresolution of 0.8125 MHz to 3.25 MHz ({\\em i.e.} $\\sim$ 4 km~s$^{-1}$). For comparison, the \nobserved lines have velocity widths of ($\\sim 80$ km~s$^{-1}$).\nThe nearby quasars 1751+096 and\n1925+211 were observed at regular intervals to correct for gain variations of the antennas due to atmospheric fluctuations. \nThe data reduction and calibration were done\nunder MIR\/IDL\\footnote{http:\/\/cfa-www.harvard.edu\/$\\sim$cqi\/mircook.html}. After the gain and bandpass calibrations, the\nvisibilities data from both tracks were combined together.\nImaging and deconvolution were performed under GILDAS\\footnote{http:\/\/www.iram.fr\/IRAMFR\/GILDAS\/} package.\n\nThe 1.3 mm (225 GHz) continuum emission map was derived by averaging the line-free channels from\nboth LSB and USB, resulting in a total bandwidth of 3.4 GHz. For the line channel maps we subtract\nthe continuum visibilities of the relevant\nsideband from the line emission visibilites. In Table~\\ref{tabobs} we summarize the observations of IRC+10420 carried out by the SMA.\n\nFor the sake of presentation clearity, the center of all the maps were shifted in the visibility domain so that\nthe phase reference matches the position of the continuum peak emission,\n{\\em i.e.} R.A. = 19$^{\\rm h}$26$^{\\rm m}$48$\\fs$09 and Dec. = 11$\\degr$21$\\arcmin$16$\\mbox{\\rlap{.}$''$}$75,\nwhich also corresponds to the center of the SiO shell detected by \\citet{cas01}.\nWe note that the lack of interferometric baselines shorter than 11 m makes our observations blind to\nextended structures of size $\\ge 25\\arcsec$. In order to estimate the amount of flux recovered by the SMA, we convolved\nthe channel maps of $^{12}$CO J=2--1 and $^{13}$CO J=2--1 lines to the spatial resolution of 19\\arcsec.7 and 12\\arcsec,\nrespectively. We show in Figure 2 the spectra taken from the these maps and the single dish observations taken by\nKemper et al. (2003) and Bujarrabal et al. (2001) at the same angular resolutions. We estimate that the SMA \nrecovered more than 60\\% of the flux in $^{12}$CO J=2--1 line and almost all the flux in $^{13}$CO J=2--1 line.\n\\begin{table}[h] \\label{tabobs} \\begin{center} \\begin{tabular}{lllcccc}\n\\multicolumn{7}{l}{Table 2: SMA line observations towards IRC+10420.} \\\\\n\\hline\n\\hline\nLine & Rest Freq & Config. & Clean beam & P.A. & $\\Delta$V & 1$\\sigma$ rms \\\\\n & (GHz) & & (FWHM, arcsec) & (deg) & (km~s$^{-1}$) & (mJy\/beam) \\\\\n\\hline\n\n$^{12}$CO(2-1) & 230.538 & Comp & 3.44 x 3.12 & 46$^\\circ$ & 4.2 & 60 \\\\\n & & Comp+Ext & 1.44 x 1.12 & 46$^\\circ$ & 4.2 & 80 \\\\\n\n$^{13}$CO(2-1) & 220.399 & Comp & 3.52 x 3.21 & 63$^\\circ$ & 4.4 & 50 \\\\\nSO(6$_5$-5$_4$) & 219.949 & Comp & 3.51 x 3.19 & 63$^\\circ$ & 4.4 & 50 \\\\\n\\hline\n\\end{tabular} \\end{center} \\end{table}\n\n\\section{Results}\n\\subsection{1.3mm continuum emission}\n\nAs shown in Figure 1, the 1.3mm continuum emission from IRC+10420 was well detected in our SMA data. \nThe emission is not resolved and has a flux density $\\sim$ 45 mJy. A previous single-dish observation by\n\\citet{walmsley91} also at 1.3mm\ngives a flux of 101 mJy. That suggests that the dust emission is extended, with \nabout 55\\% of the continuum flux resolved out by the interferometer.\nAssuming the central star to have a black body spectrum with a temperature of T${\\rm eff}$ = 7000 K, it\ncontributes only a negligible amount of the flux at 1.3 mm of S$_*$ $\\sim$ 0.1 mJy.\nThus most of the 1.3mm continuum emission\nis produced by the warm dust in the envelope around IRC+10420.\n\n\\subsection{$^{12}$CO J=$2-$1 emission}\nIn Figure. 3 we show the channel maps of the $^{12}$CO J=2$-$1 emission\nobtained with the compact configuration of the SMA. At our angular resolution of $\\sim$3\\arcsec, the envelope\nis well resolved. At both blueshifted and redshifted velocities the emission appears to be circularly\nsymmetric, as is expected for a spherically expanding envelope. More interestingly, at velocities\nbetween 59 -- 89 \\mbox{km\\,s$^{-1}$}, centered around the systemic velocity, the envelope is clearly elongated in\napproximately the North-East to South-West direction. By fitting a two-dimensional Gaussian\nto the intensity distribution of the $^{12}$CO J=2$-$1 emission at systemic velocity\nV$_{\\rm LSR}$ = 74 \\mbox{km\\,s$^{-1}$}, we obtain a position angle for the $^{12}$CO J=2$-$1 envelope of PA=70$^\\circ$. \nThe orientation of the\nCO emission is consistent with the elongation found by Castro-Carrizo et al. (2007). The position-velocity (PV)\ndiagrams of the $^{12}$CO J=2--1 emission along the major (PA $\\sim$70$^\\circ$) and minor (PA $\\sim$160$^\\circ$) axis of the\nenvelope as presented in Figure 7 are typical of an expanding envelope at an expansion velocity of $\\sim$40 \\mbox{km\\,s$^{-1}$}.\n\nWe show in Figure. 4 the channel maps of the $^{12}$CO J=2$-$1 emission, which is obtained by combining the\ndata from both compact and extended configurations of the SMA. The $^{12}$CO J=2$-$1 emission is\nfurther resolved into more complex structures. In the velocity channels between 59 -- 89 \\mbox{km\\,s$^{-1}$}, centered\naround the systemic velocity,\nthe $^{12}$CO J=2--1 emission consists of a central prominent hollow shell structure of $\\sim$1\\arcsec\\, to 2\\arcsec\\,\nin radius and a more clumpy arc or shell located between radii of 3\\arcsec\\, to 6\\arcsec, which is more prominent in the South West quadrant of\nthe envelope. Close inspection of the velocity channels around\nthe systemic velocity V$_{\\rm LSR}$ $\\sim$ 74 \\mbox{km\\,s$^{-1}$} indicates that the central hollow shell-like structure is clumpy and shows\nstronger emission toward the South-Western side.\nThe outer arc or shell of emission can be seen more easily in the azimuthal average of the $^{12}$CO J=2$-$1 brightness\ntemperature as shown in Figure. 13. We can clearly see a central depression within a radius of $\\sim$ 1\" and enhanced emission between\nthe radii of 3\\arcsec\\, to 6\\arcsec. \n\\subsection{$^{13}$CO J=$2-$1 emission}\n\nThe channel maps of $^{13}$CO J=2--1 emission obtained from the compact configuration data at\nan angular resolution of $\\sim$3\\arcsec.5\nis shown in Figure 5. The emission in the $^{13}$CO J=2--1 line appears fainter and more compact then seen in the\nmain isotope $^{12}$CO J=2--1 line. \nLike in $^{12}$CO J=2--1 emission, near the systemic velocity, the envelope traced by\n$^{13}$CO J=2--1 emission is elongated at position angle of PA $\\sim$\n70$^\\circ$. The centroid of the emission appears to be slightly offset from the stellar position. \nAlso like in $^{12}$CO J=2--1 emission, at higher velocities\nin both blue-shifted and red-sfhited parts the $^{13}$CO J=2--1 emission again \nshows a roughly circularly symmetric morphology,\nas would be expected for a spherically expanding envelope. The spatial kinematics of the $^{13}$CO J=2--1 emission\ncan be more clearly seen in the position-velocity diagram (Figure 7) along the major axis (PA $\\sim$70$^\\circ$) of\nthe envelope. In the PV diagram, there is a small velocity gradient in the velocity range between 70 to 100 \\mbox{km\\,s$^{-1}$}. \nCloser inspection of the channel maps suggests that the small velocity gradient corresponds to the slight shift \nof the emission centroid toward the North-East quadrant of the envelope in the abovementioned velocity range.\n\n\\subsection{SO J$_{\\rm K}$=6$_5$--5$_4$ emission}\n\nIn Figure. 6 we show channel maps of the SO 6$_5-$5$_4$ emission obtained from the data of the compact configuration. \nThe emission is well resolved\nin some of the velocity channels. A close inspection reveals that the distribution of SO 6$_5-$5$_4$ emission is different from that\nof $^{12}$CO J=2$-$1. At extreme redshifted velocities the centroid of the emission is shifted to the East, whereas\nat extreme blueshifted velocities the centroid of the emission is shifted to the West. \nIn addition, the SO 6$_5-$5$_4$ emission in velocity channels centered around the systemic velocity is\nclearly elongated to the South-West. This elongation is very similar to that seen in $^{12}$CO J=2$-$1 emission\nat the same angular resolution (see Figure 3). More interestingly, at the opposing redshifted velocities of\n84 to 102 \\mbox{km\\,s$^{-1}$} (see Figure 6) the emission is very compact and shifts to the North-East of the central\nstar. Such positional shift is shown more clearly in the position-velocity diagram of the\nSO J$_K$=6$_5$--5$_4$ emission along the major axis (PA $\\sim$70$^\\circ$)\nof the envelope. There is a clearly velocity gradient in the velocity range \nbetween 70 to 90 \\mbox{km\\,s$^{-1}$}, which is similar but more pronounced\nthat that seen in $^{13}$CO J=2--1 line. \n\nThus, the SO 6$_5-$5$_4$ emission in particular but also the $^{13}$CO J=2$-$1\nemission seem to better trace the asymmetric structure inside the envelope.\nThe small velocity gradient seen in both lines points to the presence of an ejecta in the envelope of\nIRC+10420. From the abovementioned elongation of the envelope and the orientation of the positional velocity shift, \nwe estimate that the\nejecta is oriented at position angle of PA=70$^\\circ$.\nWe note that SO emission is known to be enhanced in AGB and post-AGB envelopes where collimated high velocity\noutflows are present. \nExamples can be found in the rotten egg nebula OH 231.8+4.2 (S\\'{a}nchez Contreras et al. 2000)\nand VY CMa (Muller et al. 2007). In both cases SO emission is found to be strongly enhanced in the bipolar lobes. \nThe SO enhancement in bipolar outflows could be related to the\nsynthesis of SO in the shocked molecular gas in these outflows. Alternatively, the warm environment around\nsupergiants like VY CMa and IRC+10420 could facilitate the synthesis of SO through chemical pathways such as \nS + OH $\\longrightarrow$ SO + H or HS + O $\\longrightarrow$ SO + H (Willacy \\& Millar 1997).\n\\section{Structure of the envelope}\nIn this section we attempt to build a model for the circumstellar\nenvelope of IRC+10420 using our newly obtained SMA data and also the \nlarge body of observational data on IRC+10420, ranging from optical to milimeter wavelengths.\nOur goal is to better understand the physical conditions of the molecular gas inside the\nenvelope and also to retrace the mass loss history of the central star.\nThe structure of the envelope around IRC+10420 has been previously modelled in the work of Castro-Carrizo et al. (2007),\nby prescribing a radial distribution for the temperature in order to derive \nthe gas density and consequently the mass loss rate. \nIn our model we take into account the properties of\nthe dust inside envelope and the balance of heating and cooling processes. From the discussion in the previous\nsection, the envelope can be approximated as spherically symmetric.\nThe physical conditions such as kinetic temperature, density in the envelope can then be inferred from\ndetailed calculations of the energy balance within the envelope and matching the predictions of the radiative\ntransfer model to the observed strength of the CO rotational lines. This procedure represents an improvement in\ncomparison to previous models, and allows a better understanding of the physical conditions \nin the envelope of IRC+10420.\n\n\\subsection{A simple model for dust continuum emission}\n\nTo determine the heating process in the detected molecular shells requires the knowledge of the\nmomentum transfer coefficient (or the flux average extinction coefficient $$) between the dust \nparticles and the gas molecules.\nTo estimate this quantity, we follow the treatment of Oudmaijer et al. (1996) who modelled in detail\nthe spectral energy distribution (SED) of IRC+10420. We emphasize here that by following\nthe treatment of Oudmaijer et al. (1996), the structure of the molecular envelope as seen in our SMA data and\npreviously in the data of Castro-Carrizo et al. (2007) is not considered explicitly.\n \nWe use the photometric data of IRC+10420 collected\nby Jones et al. (1993) and Oudmaijer et al. (1996), i.e. the 1992 photometric dataset. \nWe also assume that the dust consists of \nsilicate particles with a radius of $a$ = 0.05 $\\mu$m. The dust\nopacity as a function of wavelength is taken from the work of Volk \\& Kwok (1988). Following Oudmaijer et al. (1996),\nwe scale the opacity law to match the absolute value of the opacity at 60$\\mu$m of\n$\\kappa_{60 \\mu m}$ = 150 cm$^2$ g$^{-1}$. We also a use a specific density for dust particles of\n$\\rho$ = 2 g cm$^{-3}$ and assume a dust to gas ratio \n$\\Psi$ = 5 10$^{-3}$ as used by Oudmaijer et al. (1996).\n\nTo fit the SED of IRC+10420, especially in the mid-IR region between 5 to 20 $\\mu$m, Oudmaijer et al. (1996)\nsuggest that two dust shells are needed: a hot inner shell with a low mass loss rate and an outer cooler shell \nwith a higher mass loss rate. We note that Bl\\\"{o}cker et al. (1999) \nalso reached same conclusion in their attempt to fit the SED of IRC+10420.\nIn our model we also use two dust shells: a hot inner shell with a low mass loss rate of \\.{M} = 8 10$^{-5}$ M$_\\odot$\/yr \nlocated between 2.6 10$^{15}$ cm (70 stellar radii) to 1.8 10$^{16}$ cm (486 stellar radii) and\nan outer cooler shell with a higher mass loss rate of \\.{M} = 1.2 10$^{-3}$ M$_\\odot$\/yr extending to a radial\ndistance of 10$^{18}$ cm where the dust particle number density drops to very low values.\nWe also infer a stellar luminosity of 6 10$^5$ L$_\\odot$. The size of the dust shells, the mass loss \nrates and the stellar luminosity are very similar to that derived by Oudmaijer et al. (1996) scaled to the adopted distance of 5 kpc. \nThe results of our radiative transfer model are presented in Figure 8 and 9. The fluxes from the model are corrected\nfor an interstellar extinction of A$_{\\rm V}$=5 as suggested by Oudmaijer et al. (1996). As can be seen in \nFigure 8, the results of our model are in close agreement with that shown in Oudmaijer et al. (1996) and match \nreasonably the observed SED of IRC+10420. As shown in Figure. 9, the dust temperature\ndecreases monotonically from $\\sim$1000 K at the inner radius to $\\sim$50 K at the outer radius\nof the dust envelope, even though there is a jump in mass loss rate between the inner hot dust shell and the\nouter cooler dust shell at a radial distance of 1.8 10$^{16}$ cm. \n\nFrom the flux $F_\\lambda$ of the radiation field at each point in the envelope we can define the flux average \nextinction coefficient $$, which characterizes the transfer of angular momentum from radiation photons to dust\nparticles during the absorption and reemission processes of the continuum radiation. Because in our case \nwe do not consider the scattering of radiation, the extinction coefficient $Q_\\lambda$ is \nsimply the opacity $\\kappa_\\lambda$, where $\\lambda$ is the wavelength of the radiation.\n\\begin{equation}\n = \\frac{\\int_{0}^{\\infty} F_\\lambda Q_\\lambda d\\lambda}{\\int_{0}^{\\infty}F_\\lambda d\\lambda} = \n\\frac{\\int_{0}^{\\infty} F_\\lambda \\kappa_\\lambda d\\lambda}{\\int_{0}^{\\infty}F_\\lambda d\\lambda}\n\\end{equation}\nThe flux average extinction coefficient $$ \nchanges significantly in the inner part of the dust envelope but \nreaches a constant value in the outer part. Because the molecular shells are located in the outer \npart of the dust envelope, at radii larger than a few times 10$^{16}$ cm, \nfor simplicity, we will adopt a constant value $$=0.025 (see Figure 9).\n\n\\subsection{Excitation of CO molecule}\nThe temperature of the molecular gas at any point in the envelope is determined by the balance between cooling\ndue to adiabatic expansion and molecular emissions, and the heating, which is mainly due to collisions of molecules\nwith dust grains (Goldreich \\& Scoville 1976). As the dust grains stream through the \ngas under radiation pressure, the grains transfer angular momentum to\nthe gas by collisions. The heating term is therefore\ndirectly related to the drift velocity:\n\\begin{equation}\nH = \\frac{1}{2}\\rho \\pi a^2 n_d v_{\\rm drift}^3\n\\end{equation} \nwhere $n_d$ is the number density of dust particles. The drift velocity is determined by\nthe relation:\n\\begin{equation}\nv_{\\rm drift} = \\left(\\frac{L_{*}V_{\\rm exp}}{\\dot{M}c}\\right)^{1\/2}\n\\end{equation}\nwhere $L_*$ is the stellar luminosity, $V_{exp}$ is the expansion velocity, $\\dot{M}$ is the mass loss rate,\nand $c$ is the speed of light, $$ is the flux average extinction coefficient of dust particles estimated\nin previous section. The same dust to gas ratio of $\\Psi$ = 5 10$^{-3}$ is used to calculate the heating rate.\n\nGenerally, in oxygen-rich envelopes the main coolants are H$_2$O and CO molecules. \nThe contribution of H$_2$O molecules to the cooling process in IRC+10420 is quite uncertain.\nWe note that far-IR emission lines of H$_2$O are not detected in the ISO data (Molster et al. 2002). \nIn addition, OH masers are seen in a shell of radius 1 to 1.5 arcsec (Nedoluha \\& Bowers 1992). \nBecause OH radical is the photodissociation product\nof H$_2$O, it is likely that H$_2$O molecules can only exist at even smaller radii, i.e. inside the central\ncavity disccused previously. Therefore, we do not expect the H$_2$O molecules to be\npresent in significant quantity in the part of the molecular envelope traced in CO emission\nAs a result, we do not include the cooling due to the H$_2$O molecule in our model\nand calculate the cooling solely due to CO molecule and its isotopomer $^{13}$CO. \n\nWe use the thermal balance and radiative transfer model published by Dinh-V-Trung \\& Nguyen-Q-Rieu (2000) to\nderive the temperature profile of the molecular gas and to\ncalculate the excitation of CO molecules and predict the strength of CO rotational transitions. We take\ninto account all rotational levels of CO molecules up to J=20. The collisional cross sections with H$_2$\nare taken from Flower (2001), assuming an ortho to para ratio of 3 for molecular hydrogen. The local \nlinewidth is determined by the turbulence velocity and the local thermal linewidth. Because the expansion\nvelocity of the gas in the envelope of IRC+10420 is large and the observed CO line profiles have smooth\nslopes at both blueshifted and redshifted edges (see Figure 11), we need to assume a constant\nturbulence velocity of 3 \\mbox{km\\,s$^{-1}$}. We note that this is higher than normally used (about 1 \\mbox{km\\,s$^{-1}$}) for the envelope around AGB stars where\nthe expansion velocity is lower (Justtanont et al. 1994). In our model, the spherical envelope is covered with a \ngrid of 90 radial points. For each rotational \ntransition, the radiative transfer equation is integrated accurately through the envelope\nto determine the average radiation field at each radial mesh point. We use the approximate lambda operator\nmethod together with Ng-acceleration to update the populations of CO molecules of different energy levels. \nOnce convergence is achieved, we convolve the predicted radial distribution of the line intensity with \na Gaussian of specified FWHM to produce simulated data for comparison with observations.\n\nAs discussed in the previous section, there is a clear evidence for the presence of two molecular shells from\nour SMA data. The size and location of each molecular shell are determined from fits to the interferometric data.\nIn our model we adjust both the mass loss rates and the relative abundance of CO molecules in the two molecular shells \nto match both the single dish and interferometric data. \nWe find that a relatively low mass loss rate similar to that\nderived by Castro-Carrizo et al. (2007) would result in a too high gas temperature in the envelope as the heating term\nbecomes more dominant in comparison to the cooling term.\nWe also find that a high\nabundance of CO of a few times 10$^{-4}$ as usually used for oxygen rich envelopes \n(Kemper et al. 2003, Castro-Carrizo et al. 2007) \nwould result in CO lines that are too strong. Instead, a relative abundance of 10$^{-4}$ for CO molecules with\nrespect to H$_2$ for the CO molecules provides reasonable fit to all the available data.\nThe radial profile of the gas temperature is shown in \nFigure 10. The sudden jump in gas temperature at the\ninner radius of the second shell ($\\sim$1.85 10$^{17}$ cm) is due to the change in the mass loss rate. We can see that\nthe gas temperature in the envelope of IRC+10420, although lower than assumed in Castro-Carrizo et al. (2007), is\nhigher than that typically found in circumstellar envelopes. Even at the outermost radius of the envelope, the\ngas temperature is still around 50 K. For comparison, in the envelope around OH\/IR stars, \nthe gas temperature at similar radial distances from the central star is predicted to be much lower, \n10 K or even less (Justtanont et al. 1994,\nGroenewegen 1994). The main reason for the elevated gas temperature in the envelope of IRC+10420 is the\nstrong heating due to the enormous luminosity of the central yellow hypergiant. The cooling due to molecular\nemissions is also affected by the lower abundance of CO, that we find neccessary to match the data.\nA summary of the parameters used in our model is presented in Table 3.\n\nAs shown in Figures 11--14, the results of our model are in reasonable agreement with both the strengths of CO lines measured by\nsingle dish telescopes and the radial distribution of brightness temperature of J=1--0 and J=2--1 lines\nobserved with interferometers. We note that except for the extra emission in the blue part of the\nline profile between 65 and 75 \\mbox{km\\,s$^{-1}$}, the shape of the J=2--1 and J=3--2 is in close agreement with observation.\nHigher transitions J=4--3 and 6--5 show some evidence of narrower linewidth than predicted by the model.\nOne possible explanation is that the expansion velocity in the inner region traced by these high lying transitions has\nslightly lower expansion velocity than in the outer part of the envelope.\n\nBy comparing the model predictions with the strengths of $^{13}$CO J=1--0 and J=2--1 lines observed with\nthe IRAM 30m telescope (Bujarrabal et al. 2001), we derive a relative abundance for $^{13}$CO\/H$_2$ of\n1.5 10$^{-5}$, or an isotopic ratio $^{12}$C\/$^{13}$C $\\sim$ 6. This ratio is very similar to\nthe values $^{12}$C\/$^{13}$C = 6 and 12, respectively, found in the red supergiants $\\alpha$ Ori and $\\alpha$ Sco\n(Harris \\& Lambert 1984, Hinkle et al. 1976). Such a low isotopic ratio suggests\nthat IRC+10420 has experienced significant mixing of H burning products to its surface prior to\nthe ejection of the material in the envelope.\n\nThe difference between the mass-loss rates and gas temperature profile derived from our modelling and that\nobtained by Castro-Carrizo et al. (2007) can be understood as the consequence of the higher gas temperature \nand higher CO abundance adopted in their work. For the same strength of the CO rotational lines, the higher CO abundance \ncan be almost exactly compensated by a corresponding decrease in the mass loss rate.\n\nOur results indicate the presence of two separate shells (shell I and shell II, see Table 3) with slightly \ndifferent mass loss rates. With an expansion velocity of 38 \\mbox{km\\,s$^{-1}$}, the time interval \nbetween the two shells is $\\sim$ 200 yrs. The cavity inside shell I\nalso implies a dramatic decrease in mass loss from IRC+10420 over the last $\\sim$300 yrs. Thus IRC+10420\nloses mass in intense bursts, separated by relatively quiet periods of a few hundreds years in duration.\n\\begin{table}[h] \\begin{center} \\begin{tabular}{lll}\n\\multicolumn{3}{l}{Table 3: Parameters of the model for IRC+10420 envelope.} \\\\\n\\hline\n\\hline\nParameters & Shell I & Shell II \\\\\n\\hline\nInner radius & 3.5 10$^{16}$ cm & 1.85 10$^{17}$ cm \\\\\nOuter radius & 1.5 10$^{17}$ cm & 5 10$^{17}$ cm \\\\\nMass loss rate \\.{M} & 9 10$^{-4}$ M$_\\odot$ yr$^{-1}$ & 7 10$^{-4}$ M$_\\odot$ yr$^{-1}$ \\\\\nExpansion velocity V$_{\\rm exp}$ & 38 kms$^{-1}$ & 38 kms$^{-1}$ \\\\\nDust to gas ratio $\\Psi$ & 5 10$^{-3}$ & 5 10$^{-3}$ \\\\\nMomentum transfer coefficient $$ & 0.025 & 0.025 \\\\\n$^{12}$CO\/H$_2$ & 10$^{-4}$ & 10$^{-4}$ \\\\\n$^{13}$CO\/H$_2$ & 1.5 10$^{-5}$ & 1.5 10$^{-5}$ \\\\\nTurbulent velocity & 3 kms$^{-1}$ & 3 kms$^{-1}$ \\\\\n\\hline\n\\end{tabular} \\end{center} \\end{table}\n\n\\subsection{Caveats}\nIn our model we consider separately the dust and gas component in the envelope of IRC+10420. The dust continuum\nemission model is used as a simple way to estimate the $$ parameter of dust particles, which is needed for the thermal balance \ncalculation in the molecular shells. It turned out that the mass loss rates of the molecular shells between\n9 10$^{-4}$ M$_\\odot$\/yr for shell I and 7 10$^{-4}$ M$_\\odot$\/yr for shell II are lower but quite close to the \nmass loss rate of 1.2 10$^{-3}$ M$_\\odot$\/yr of the cool outer dust shell inferred from the modelling \nthe SED of the envelope. Our approach, although less ad hoc than simply assuming a value of the $$ parameter for dust particles,\nis not fully self-consistent in the treatment of the dust and gas component.\nIt would be desirable in the future to treat both the dust and gas component together when high angular resolution\ndata on the dust continuum emission and higher J lines of CO become available.\n\nAnother caveat of our model is the assumption that the heating of molecular gas is mainly due to collision between dust\nparticles and gas molecules. The high luminosity of the central star of IRC+10420 suggests that \nradiation pressure on dust particles is a possible mechanism for driving the\nwind as assumed in our mode. However, given the high expansion\nvelocity and the presence of many small scale features in high angular resolution optical images of the envelope\n(Humphreys et al. 1997), which has been interpreted as jets or condensations, other mechanisms\nsuch as shocks produced by interaction between higher velocity ejecta and the envelope should be considered.\nThe heating due shocks within the envelope might contribute to the thermal balance of the molecular gas in IRC+10420.\n\n\\section{Comparison with optical imaging data}\nThe circumstellar envelope around IRC+10420 is known to be very complex, with a number of\npeculiar structures (Humphreys et al. 1997). In the outer part of the envelope at radial distance\nof $\\sim$3\\arcsec\\, to 6\\arcsec, several arc-like features can be seen. These features have been interpreted as\nrepresenting different mass loss episodes of IRC+10420. In addition, condensations or blobs are seen\ncloser (about 1\\arcsec\\, to 2\\arcsec\\, in radius) to the central star. That might represent a more recent mass loss\nepisode. These features correspond spatially to the two molecular shells (shell I and II) identified in our \nobservations. In addition, Humphreys et al. (1997) also identified several broad fan-shaped features to the\nSouth-West side of the central star, between radius of 0\".5 to 2\". From their higher surface brightness in comparison\nwith other parts of the envelope,\nHumphrey et al. (1997) suggest that these features are ejecta moving obliquely toward the observer.\nAs discussed in previous section, the channel maps of $^{13}$CO J=2--1 and SO J$_{\\rm K}$=6$_5$--5$_4$ (Figures 5 and 6)\nreveal enhanced emissions in blueshifted velocity channels together with a positional shift of the emission centroid \nalso to the South-West quadrant of the envelope. Therefore, the spatial kinematics obtained from our observations and \nthe properties of these fan-shaped features are consistent with the presence of an ejecta in the South-West\nside of the envelope around IRC+10420. \n\n\\section{Summary}\n\nWe have used the sub-millimeter array to image and study the structure of the molecular envelope\naround IRC+10420. Our observations reveal a large expanding envelope with a clumpy and very complex\nstructure. The envelope shows clear asymmetry in $^{12}$CO J=2--1 emission in the South-West direction\nat position angle PA$\\sim$70$^\\circ$. The elongation of the envelope is found even more pronounced in\nthe emission of $^{13}$CO J=2--1 and SO J$_{\\rm K}$=6$_5$--5$_4$. A positional shift with velocity is\nseen in the above emission lines, suggesting the presence of a weak bipolar outflow in the envelope of\nIRC+10420. \n\nIn the higher resolution data of $^{12}$CO J=2--1, we find that \nthe envelope has two components: (1) a inner shell (shell I) located between radius\nof about 1\"-2\"; (2) an outer shell (shell II) between radius 3\" to 6\". These shells represernt two previous \nmass loss episodes from IRC+10420. \n\nWe constrain the physical conditions inside the envelope\nby modelling the dust properties, the heating and cooling of molecular gas. From comparison with observations\nwe derive the size and the location of each molecular shell. We estimate\na mass loss rate of $\\sim$ 9 10$^{-4}$ M$_\\odot$ yr$^{-1}$ for shell I and 7 10$^{-4}$ M$_\\odot$ yr$^{-1}$ for\nshell II. The gas temperature is found to be an usually high in IRC+10420 in comparison with other oxygen rich envelopes,\nmainly due to the large heating induced by the large luminosity of the central star.\n\nWe also derive a low isotopic ratio $^{12}$C\/$^{13}$C = 6 for IRC+10420, which suggests a strong mixing of processed\nmaterial from stellar interior to the surface of the star.\n\n\\acknowledgements\n\nWe are grateful to the Sub-millimeter array staff for their help in carrying out the observations. \nThe SMA is a collaborative project between the Smithsonian Astrophysical Observatory and \nAcademia Sinica Institute of Astronomy and Astrophysics of Taiwan. \nWe thank F. Kemper\nand D. Teyssier for providing single dish CO spectra of IRC+10420. This research has made use of \nNASA's Astrophysics Data System Bibliographic Services\nand the SIMBAD database, operated at CDS, Strasbourg, France.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nPerturbative study of many quantum field models has lead to many impressive results: beta functions for QED and for the $\\Phi^4$ theory are known up to the fifth order whereas that of QCD up to the fourth order \\cite{Vladimirov}-\\cite{Baikov}. These studies however are based on the expansion in the small coupling constant such that the behavior at large couplings is completely missed. Although the possibility of a perturbative expansion in the strong coupling constant has been explored over the years \\cite{Kovesi}-\\cite{Bender3} no definite method emerged.\n\nOf significant importance is the $\\Phi^4$ theory at strong coupling whose triviality \\cite{Wilson}-\\cite{Jora} has been the subject of intensive debate. In essence the triviality of the $\\Phi^4$ theory means that the renormalized coupling constant $\\lambda_R$ vanishes in the limit of large cut-off and the model behaves like a non-interacting field theory. Many previous theoretical studies suggested that the $\\Phi^4$ theory is trivial claiming it in $d\\neq 4$ \\cite{Frohlich}, \\cite{Aizenmann}, \\cite{Jaffe}, in computer simulations \\cite{Callaway} or for $O(N)$ symmetric model \\cite{Bardeen}. In \\cite{Frasca}, \\cite{Jora} it was proved, based on non perturbative methods, that the $\\Phi^4$ theory is trivial for a large bare coupling constant.\n\nIn the present work we will give a proof of triviality of the $\\Phi^4$ theory valid for any value of the the bare coupling constant. Thus we will show that the all order propagator of the theory is that of the free Lagrangian:\n\\begin{eqnarray}\n\\langle\\Phi(x)\\Phi(y)\\rangle=\\int \\frac{d^4p}{(2\\pi)^4}e^{-ip(x-y)}\\frac{1}{p^2-m_0^2}.\n\\label{trtr657}\n\\end{eqnarray}\nThis definition is more restrictive than that suggested in \\cite{Frasca} but it is particular case of it.\n\n\n\n\n\n Consider a simple $\\Phi^4$ theory without spontaneous symmetry breaking with the Lagrangian:\n \\begin{eqnarray}\n {\\cal L}=\\frac{1}{2}\\partial^{\\mu}\\Phi\\partial_{\\mu}\\Phi-\\frac{1}{2}m_0^2\\Phi^2-\\frac{\\lambda}{4}\\Phi^4.\n \\label{lagr4567}\n \\end{eqnarray}\n The all order two point function has the well known expression:\n \\begin{eqnarray}\n \\langle [\\Phi(x)\\Phi(y)]\\rangle=\n \\int \\frac{d^4p}{(2\\pi)^4}e^{-ip(x-y)}\\frac{i}{p^2-m_0^2-M(p^2)},\n \\label{rez435}\n \\end{eqnarray}\n where $M(p^2)$ is the all order correction to the scalar mass.\nHere it is assumed that a cut-off procedure is used and no attempt at renormalization is made. Let us write the two point function explicitly in the Fourier space:\n\\begin{eqnarray}\n\\langle [\\Phi(x)\\Phi(y)]\\rangle=\n\\int \\frac{d^4p}{(2\\pi)^4}\\frac{d^4q}{(2\\pi)^4}e^{-ipx}e^{-ipy}\\langle\\Phi(p)\\Phi(q)\\rangle.\n\\label{four5467}\n\\end{eqnarray}\nNote that the quantity $\\langle \\Phi(p)\\Phi(q) \\rangle$ is given in the path integral formalism by:\n\\begin{eqnarray}\n\\langle\\Phi(p)\\Phi(q)\\rangle=\n\\frac{\\int \\prod_k d\\Phi(k)\\Phi(p)\\Phi(q)\\exp[i \\int d^4x {\\cal L}]}{\\int \\prod_k d\\Phi(k)\\exp[i \\int d^4x {\\cal L}]}.\n\\label{ev5534567}\n\\end{eqnarray}\n\n Assume that instead the quantity in Eq. (\\ref{four5467}) we consider in the Fourier space the function:\n \\begin{eqnarray}\n I_{(x-y)}=\\int \\frac{d^4p}{(2\\pi)^4}e^{-ip(x-y)}\\langle\\Phi(p)\\Phi(-p)\\rangle.\n \\label{func546789}\n \\end{eqnarray}\nWe need to find the significance of this function in the coordinate space:\n\\begin{eqnarray}\n&&\\int \\frac{d^4p}{(2\\pi)^4}e^{-ip(x-y)}\\langle\\Phi(p)\\Phi(-p)\\rangle=\n\\nonumber\\\\\n&&\\int \\frac{d^4p}{(2\\pi)^4}\\int d^4 z_1 d^4 z_2\\langle \\Phi(z_1)\\Phi(z_2) \\rangle e^{ipz_1}e^{-ipz_2}e^{-ip(x-y)}=\n\\nonumber\\\\\n&&\\int d^4 z_1 d^4 z_2\\delta(z_1-z_2-x+y)\\langle \\Phi(z_1)\\Phi(z_2) \\rangle=\\int d^4 z_2\\langle \\Phi(z_2+x-y)\\Phi(z_2)\\rangle.\n\\label{rez43567}\n\\end{eqnarray}\nKnowing that,\n\\begin{eqnarray}\n\\langle [\\Phi(z_2+x-y)\\Phi(z_2)]\\rangle=\\int \\frac{d^4p}{(2\\pi)^4}e^{-ip(x-y)}\\frac{i}{p^2-m_0^2-M^2(p^2)}= \\langle [\\Phi(x)\\Phi(y)]\\rangle\n\\label{equal7689}\n\\end{eqnarray}\nwe obtain the following relation between $I_{(x-y)}$ and the two point function:\n\\begin{eqnarray}\n I_{(x-y)}=\\int d^4 z_2\\langle [\\Phi(x)\\Phi(y)]\\rangle=\\int d^4 z_2 \\int \\frac{d^4p}{(2\\pi)^4}e^{-ip(x-y)}\\frac{i}{p^2-m_0^2-M(p^2)}.\n \\label{rez615789}\n\\end{eqnarray}\nNote that $\\int d^4 z_2$ is independent of the rest of the expression and if one considers the functional integration over a lattice with the volume V one can simply write $\\int d^4 z_2=V$.\n\nNow we shall calculate the function $I_{(x-y)}$ in the path integral formalism.\n\nFirst we write the action for the Lagrangian in Eq. (\\ref{lagr4567}) in the Fourier space.\n\\begin{eqnarray}\n\\int d^4 x {\\cal L}=\\frac{1}{V}\\sum_p \\frac{1}{2}(p^2-m_0^2)\\Phi(p)\\Phi(-p)-\\frac{\\lambda}{4}\\frac{1}{V^3}\\sum_{k,n,m}\\Phi(p_n)\\Phi(p_m)\\Phi(p_k)\\Phi(-p_k-p_m-p_n).\n\\label{rez3245}\n\\end{eqnarray}\nThen we compute in the path integral formalism:\n\\begin{eqnarray}\n\\langle\\Phi(p)\\Phi(-p)\\rangle=\n\\frac{\\int \\prod_k d\\Phi(k)\\Phi(p)\\Phi(-p)\\exp[i \\int d^4x {\\cal L}]}{\\int \\prod_k d\\Phi(k)\\exp[i \\int d^4x {\\cal L}]}=\\frac{V}{i}\\frac{\\delta Z}{\\delta p^2}\\frac{1}{Z},\n\\label{ev34567}\n\\end{eqnarray}\nwhere Z is the zero current partiton function:\n\\begin{eqnarray}\nZ=\\int \\prod_k d \\Phi(k) \\exp[i \\int d^4x {\\cal L}].\n\\label{part4567}\n\\end{eqnarray}\nHere we used the fact that the following relations hold:\n\\begin{eqnarray}\n\\int \\prod_k d\\Phi(k)\\Phi(p)\\Phi(-p)\\exp[i \\int d^4x {\\cal L}]=\\int \\prod_k {\\rm Re}\\Phi(k) {\\rm Im}\\Phi(k)\\exp[\\frac{i}{V}\\sum_{p^0>0}(p^2-m_0^2)[({\\rm Re}\\Phi(p))^2+({\\rm Im}\\Phi(p))^2]+...],\n\\label{wri777}\n\\end{eqnarray}\nand,\n\\begin{eqnarray}\n\\langle\\Phi(p)\\Phi(-p)\\rangle=\\langle [({\\rm Re}\\Phi(p))^2+({\\rm Im}\\Phi(p))^2]\\rangle.\n\\label{rez32455}\n\\end{eqnarray}\n\n\\section{Partition function}\n\nBefore going further we need to establish some facts about the zero current partition function Z. For that we write explicitly:\n\\begin{eqnarray}\nZ=\\int \\prod_k d \\Phi(k)\\exp[\\frac{i}{2V}\\sum_p(p^2-m_0^2)\\Phi(p)\\Phi(-p)][1-i\\frac{\\lambda}{4}\\frac{1}{V^3}\\sum_{k,n,m}\\Phi(p_n)\\Phi(p_m)\\Phi(p_k)\\Phi(-p_k-p_m-p_n)+...],\n\\label{part111}\n\\end{eqnarray}\nwhere an infinite expansion in $\\lambda$ (the interaction term) is considered. First we note that any term in the expansion gives contributions only if it contains pairs of the type $\\Phi(k)\\Phi(-k)$ and any such pair can be written as:\n\\begin{eqnarray}\n\\Phi(k)\\Phi(-k)=\\frac{V}{i}\\frac{\\delta \\exp[i\\frac{1}{2V}\\sum_p(p^2-m_0^2)\\Phi(p)\\Phi(-p)]}{\\delta k^2}.\n\\label{rez32456}\n\\end{eqnarray}\nThen one can write:\n\\begin{eqnarray}\n&&Z=\\int \\prod_k d \\Phi(k)\\exp[\\frac{i}{2V}\\sum_p(p^2-m_0^2)\\Phi(p)\\Phi(-p)][1-i\\frac{3\\lambda}{4}\\frac{1}{V^2}\\sum_{k,n}\\Phi(p_n)\\Phi(-p_n)\\Phi(p_k)\\Phi(-p_k)+...]=\n\\nonumber\\\\\n&&=\\int \\prod_k d \\Phi(k)[1-i\\frac{3\\lambda}{4}\\frac{1}{V^2}\\frac{V^2}{i^2}\\sum_{k,n}\\frac{\\delta}{\\delta p_n^2}\\frac{\\delta}{\\delta p_k^2}+...]\\exp[\\frac{i}{2V}\\sum_p(p^2-m_0^2)\\Phi(p)\\Phi(-p)]\n\\nonumber\\\\\n&&=[1-i\\frac{3\\lambda}{4}\\frac{1}{V^2}\\frac{V^2}{i^2}\\sum_{k,n}\\frac{\\delta}{\\delta p_n^2}\\frac{\\delta}{\\delta p_k^2}+...]\\frac{1}{\\det[\\frac{i}{V}(p_m^2-m_0^2)]}.\n\\label{calc657890}\n\\end{eqnarray}\nSince the determinant is diagonal one obtains:\n\\begin{eqnarray}\n&&\\frac{V}{i}\\frac{\\delta }{\\delta p^2}\\frac{1}{p^2-m_0^2}=iV\\frac{1}{p^2-m_0^2}\\frac{1}{p^2-m_0^2}\n\\nonumber\\\\\n&&\\frac{V}{i}\\frac{\\delta }{\\delta p^2}\\frac{1}{\\det[\\frac{i}{V}(p_m^2-m_0^2)]}=iV\\frac{1}{p^2-m_0^2}\\frac{1}{\\det[\\frac{i}{V}(p_m^2-m_0^2)]}\n\\label{calc65789}\n\\end{eqnarray}\nAlthough the procedure is more intricate this type of results are valid for any terms in the expansion of the interaction Lagrangian. Noting from the last line in Eq. (\\ref{calc657890}) that these terms are summed (or integrated) over the momenta one can conclude that besides the determinant that appears in the expression of the partition function there is no other contribution that depends on individual momenta as in all other contributions the momenta are summed over. Thus one can determine that the all orders partition function has the expression:\n\\begin{eqnarray}\nZ=\\frac{1}{\\det_{p^0>0}[\\frac{i}{V}(p_m^2-m_0^2)]}\\times {\\rm const},\n\\label{final54678}\n\\end{eqnarray}\nwhere the factor $const$ depends on the regularization procedure but it is independent on the individual momenta.\n\n\n\\section{Conclusion}\n\nFrom Eqs. (\\ref{ev34567}) and (\\ref{final54678}) we then determine:\n\\begin{eqnarray}\n\\langle\\Phi(p)\\Phi(-p)\\rangle=\\frac{V}{i}\\frac{\\delta Z}{\\delta p^2}\\frac{1}{Z}=\\frac{iV}{p^2-m_0^2}.\n\\label{ev345678}\n\\end{eqnarray}\nNote that this is an all orders result.\n\nFurthermore Eqs. (\\ref{func546789}), (\\ref{rez615789}) and (\\ref{ev345678}) lead to:\n \\begin{eqnarray}\n &&I_{(x-y)}=\\int \\frac{d^4p}{(2\\pi)^4}e^{-ip(x-y)}\\langle\\Phi(p)\\Phi(-p)\\rangle=\n \\nonumber\\\\\n &&=\\int \\frac{d^4p}{(2\\pi)^4}e^{-ip(x-y)}\\frac{iV}{p^2-m_0^2}=\\int d^4 z_2 \\int \\frac{d^4p}{(2\\pi)^4}e^{-ip(x-y)}\\frac{i}{p^2-m_0^2-M(p^2)}\n \\label{func11546789}\n \\end{eqnarray}\nConsidering the assumption that we work on a lattice with the volume V ($\\int d^4 z_2=V$) Eq. (\\ref{func11546789}) yields the all orders relation:\n\\begin{eqnarray}\n\\frac{i}{p^2-m_0^2}=\\frac{i}{p^2-m_0^2-M(p^2)},\n\\label{rez32456}\n\\end{eqnarray}\nwhich shows that the all order $\\Phi^4$ theory is trivial in the sense that the nonperturbative complete two point function receives no corrections from the $\\lambda$ term or the all order $M(p^2)=0$. Note that this result does not contradict the all order result for the mass corrections computed in \\cite{Jora} since $m^2=m_0^2$ where $m$ is the physical mass is also a solution of the recurrence relations found there.\n\nThus we completed the proof that the $\\Phi^4$ theory is trivial in the sense that it behaves like a free non interacting theory. No assumption about the value of $\\lambda$ is made so this result is valid for both small couplings and large couplings regimes.\nThe method in this work needs adjustments in order to be applicable for a scalar with spontaneous symmetry breaking or for other cases in quantum field theories.\n\n\n\n\n\n\n\\section*{Acknowledgments} \\vskip -.5cm\n\n\nThe work of R. J. was supported by a grant of the Ministry of National Education, CNCS-UEFISCDI, project number PN-II-ID-PCE-2012-4-0078.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}