diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzjnlk" "b/data_all_eng_slimpj/shuffled/split2/finalzzjnlk" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzjnlk" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\n\\label{sec:intro}\n\nThe resultant of two polynomials and the discriminant of a polynomial are\nknown since long ago in the mathematical literature\\cite{S1851,G81,BPR03}\nand have already been applied to the analysis of several physical problems.\nSome examples are the determination of singularities in the eigenvalues of\nparameter-dependent matrix eigenvalue problems\\cite{HS91}, level degeneracy\nin a quantum two-spin model\\cite{SM98}, the analysis of the properties of\ntwo-dimensional magnetic traps for laser-cooled atoms\\cite{D02}, the\ndescription of optical polarization singularities\\cite{F04}, the exceptional\npoints (EPs) for the eigenvalues of a modified Lipkin model\\cite{HSG05}, the\nlocation of crossings and avoided crossings between eigenvalues of\nparameter-dependent symmetric matrices\\cite{BR06,B07,BR07a,BR07b} and the\nsolution of two equations with two unknowns that appear in the study of\ngravitational lenses\\cite{TB16}.\n\nThe purpose of this paper is to show that these remarkable mathematical\ntools are not foolproof and, consequently, should be applied with care. In\nsection~\\ref{sec:example} we introduce a simple parameter-dependent\nsymmetric matrix and show that the discriminant of its characteristic\npolynomial vanishes for all values of the model parameter. We reveal the\nreason for the apparent failure of the discriminant and show how to overcome\nthis difficulty in order to obtain the crossings and avoided crossings\nbetween eigenvalues. In section~\\ref{sec:symmetry} we discuss the problem\nfrom the point of view of symmetry and, finally, in section~\\ref\n{sec:conclusions} we summarize the main results and draw conclusions.\n\n\\section{Simple example}\n\n\\label{sec:example}\n\nPresent discussion is based on the simple $6\\times 6$ symmetric matrix\n\\begin{equation}\n\\mathbf{H}(\\lambda )=\\left(\n\\begin{array}{llllll}\n0 & 1 & 0 & 0 & 0 & \\lambda \\\\\n1 & 0 & \\lambda & 0 & 0 & 0 \\\\\n0 & \\lambda & 0 & 1 & 0 & 0 \\\\\n0 & 0 & 1 & 0 & \\lambda & 0 \\\\\n0 & 0 & 0 & \\lambda & 0 & 1 \\\\\n\\lambda & 0 & 0 & 0 & 1 & 0\n\\end{array}\n\\right) , \\label{eq:H}\n\\end{equation}\nthat depends on a real parameter $\\lambda $. This model was chosen\nsome time ago for the application of perturbation theory to the\nresonance energy given by the H\\\"{u}ckel model for benzene in\nexercise 6.6, page 347, of Szabo and\nOstlund\\cite{SO96}. Its real eigenvalues $E_{j}(\\lambda )$, $j=1,2,\\ldots ,6\n, are roots of the characteristic polynomial\n\\begin{equation}\np(E,\\lambda )=\\left| \\mathbf{H}-E\\mathbf{I}\\right| =E^{6}-3E^{4}\\left(\n\\lambda ^{2}+1\\right) +3E^{2}\\left( \\lambda ^{4}+\\lambda ^{2}+1\\right)\n-\\lambda ^{6}-2\\lambda ^{3}-1, \\label{eq:charpoly}\n\\end{equation}\nwhere $\\mathbf{I}$ is the $6\\times 6$ identity matrix.\n\nThe values of $\\lambda $ corresponding to level crossings or avoided\ncrossings can be obtained from the discriminant of the characteristic\npolynomial $Disc_{E}\\left( p(E,\\lambda )\\right) $\\cite{BR06,B07,BR07a,BR07b}\n(see Appendix~\\ref{sec:appendix} for definition, notation and properties of\nthe discriminant). According to equation (\\ref{eq:Disc_x(A)}) of the\nAppendix~\\ref{sec:appendix} this discriminant should enable us to obtain the\nvalues of $\\lambda $ for which two (or more) eigenvalues of $\\mathbf{H}$\ncross. However, in the present case this strategy simply produces the\nuseless result $Disc_{E}\\left( p(E,\\lambda )\\right) =0$ for all $\\lambda $.\nIn order to understand the reason for this failure note that the\ncharacteristic polynomial can be factorized as\n\\begin{equation}\np(E,\\lambda )=\\left( E+\\lambda +1\\right) \\left( E-\\lambda -1\\right) \\left(\nE^{2}-\\lambda ^{2}+\\lambda -1\\right) ^{2}. \\label{eq:p_factorized}\n\\end{equation}\nWe arbitrarily organize the eigenvalues as\n\\begin{equation}\nE_{1}=-E_{6}=-1-\\lambda ,\\;E_{2}=E_{3}=-E_{4}=-E_{5}=-\\sqrt{\\lambda\n^{2}-\\lambda +1}. \\label{eq:E_j}\n\\end{equation}\nThe discriminant of the characteristic polynomial vanishes because \nE_{2}(\\lambda )=E_{3}(\\lambda )$ and $E_{4}(\\lambda )=E_{5}(\\lambda )$ for\nall $\\lambda $.\n\nWe can obtain the desired crossings or avoided crossings from the\ndiscriminant of the polynomial\n\\begin{eqnarray}\nq(E,\\lambda ) &=&\\left( E+\\lambda +1\\right) \\left( E-\\lambda -1\\right)\n\\left( E^{2}-\\lambda ^{2}+\\lambda -1\\right) \\nonumber \\\\\n&=&E^{4}-E^{2}\\left( 2\\lambda ^{2}+\\lambda +2\\right) +\\left( \\lambda\n+1\\right) ^{2}\\left( \\lambda ^{2}-\\lambda +1\\right) , \\label{eq:q(E,lambda)}\n\\end{eqnarray}\nwhere we have removed the two-fold degeneracy that is independent of \n\\lambda $. It follows from\n\\begin{equation}\nDisc_{E}\\left( q(E,\\lambda )\\right) =1296\\lambda ^{4}\\left( \\lambda\n+1\\right) ^{2}\\left( \\lambda ^{2}-\\lambda +1\\right) , \\label{eq:Disc_E(q)}\n\\end{equation}\nthat there are actual level crossings at $\\lambda =-1,0$ and an avoided\ncrossing due to the coalescence of eigenvalues in the complex $\\lambda \n-plane at $\\lambda =\\lambda _{EP}^{\\pm }=\\left( 1\\pm \\sqrt{3}i\\right) \/2$.\nThe multiple level crossing at $\\lambda =0$ comes from the fact that the \n6\\times 6$ matrix exhibits a block diagonal form with three\n$2\\times 2$ sub matrices of the form\n\\begin{equation}\n\\mathbf{H}_{j}=\\left(\n\\begin{array}{ll}\n0 & 1 \\\\\n1 & 0\n\\end{array}\n\\right) ,\\;j=1,2,3, \\label{eq:H_j_block}\n\\end{equation}\neach one with eigenvalues $E=\\pm 1$. They are representations of\nthe three\nethylene molecules in the H\\\"{u}ckel model discussed by Szabo and Ostlun\n\\cite{SO96}.\n\nFigure~\\ref{Fig:Enes} shows the eigenvalues of the matrix (\\ref{eq:H}) for a\nrange of $\\lambda $ values.We appreciate the crossing between $E_{1}$ and \nE_{6}$ at $\\lambda =-1$, the crossing between the degenerate pair $\\left(\nE_{2},E_{3}\\right) $ and $E_{1}$ at $\\lambda =0$ and the crossing between\nthe degenerate pair $\\left( E_{4},E_{5}\\right) $ and $E_{6}$ also at \n\\lambda =0$. In addition to it, this figure also shows an avoided crossing\nbetween the two degenerate pairs $\\left( E_{2},E_{3}\\right) $ and $\\left(\nE_{4},E_{5}\\right) $. It is well known that only levels of different\nsymmetry cross, while those with the same symmetry exhibit avoided crossings\n(see \\cite{F14} and references therein).\n\nIn passing, we mention that the poor convergence of the perturbation series\nfor the resonance energy of benzene at $\\lambda =1$ mentioned by Szabo and\nOstlund\\cite{SO96} is due to the fact that its radius of convergence is\ndetermined by the pair of branch points at $\\lambda _{EP}^{\\pm }$ and,\nconsequently, given by $\\left| \\lambda _{EP}^{\\pm }\\right| =1$. In other\nwords, $\\lambda =1$ is located on the boundary of the disk of convergence.\n\n\\section{Symmetry and degeneracy}\n\n\\label{sec:symmetry}\n\nIt is well known that degeneracy can be predicted beforehand. Typically,\ndegeneracy is caused by the symmetry of the Hamiltonian operator. In the\npresent case, we expect the existence of orthogonal matrices $\\mathbf{U}_{j}\n, $j=1,2,\\ldots ,N$, that leave the matrix $\\mathbf{H}$ invariant; that is\nto say: $\\mathbf{U}_{j}^{t}\\mathbf{HU}_{j}=\\mathbf{H}$, where $t$ stands for\ntranspose and $\\mathbf{U}_{j}^{t}=\\mathbf{U}_{j}^{-1}$. We can rewrite this\ninvariance expression in terms of commutators: $\\left[ \\mathbf{H},\\mathbf{U\n_{j}\\right] =\\mathbf{HU}_{j}-\\mathbf{U}_{j}\\mathbf{H}=\\mathbf{0}$, where \n\\mathbf{0}$ is the $6\\times 6$ zero matrix.\n\nIn order to construct the orthogonal matrices just mentioned we resort to\nthe graphical representation of the matrix (\\ref{eq:H}) shown in Figure~\\ref\n{Fig:H6}. The hexagon in this figure is regular when $\\lambda =1$ and\nirregular otherwise. Note that any rotation of $2\\pi \/3$ about an axis\nperpendicular to the center of the figure leaves it invariant. From the\neffect of this rotation: $\\left[ c_{1},c_{2},c_{3},c_{4},c_{5},c_{6}\\right]\n\\rightarrow \\left[ c_{5},c_{6},c_{1},c_{2},c_{3},c_{4}\\right] $, we obtain\nthe orthogonal matrix\n\\begin{equation}\n\\mathbf{U}_{1}=\\left(\n\\begin{array}{llllll}\n0 & 0 & 0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 1 \\\\\n1 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 1 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 1 & 0 & 0\n\\end{array}\n\\right) . \\label{eq:U_1}\n\\end{equation}\nAnalogously, a rotation of $4\\pi \/3$ about the same axis, $\\left[\nc_{1},c_{2},c_{3},c_{4},c_{5},c_{6}\\right] \\rightarrow \\left[\nc_{3},c_{4},c_{5},c_{6},c_{1},c_{2}\\right] $, leaves the figure invariant\nand is produced by the matrix\n\\begin{equation}\n\\mathbf{U}_{2}=\\left(\n\\begin{array}{llllll}\n0 & 0 & 1 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 1 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 1 & 0 \\\\\n0 & 1 & 0 & 0 & 0 & 1 \\\\\n1 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0 & 0 & 0\n\\end{array}\n\\right) . \\label{eq:U_2}\n\\end{equation}\nThe matrices $\\mathbf{U}_{1}$ and $\\mathbf{U}_{2}=\\mathbf{U}_{1}^{2}$ are\nrepresentations of the point-group operations commonly called $C_{3}$ and \nC_{3}^{2}$, respectively\\cite{T64,C90}.\n\nFigure~\\ref{Fig:H6} also shows the existence of three reflection planes\nperpendicular to the plane of the hexagon across the middle of opposite\nsides (commonly called $\\sigma _{v}$\\cite{T64,C90}). The reflection $\\left[\nc_{1},c_{2},c_{3},c_{4},c_{5},c_{6}\\right] \\rightarrow \\left[\nc_{2},c_{1},c_{6},c_{5},c_{4},c_{3}\\right] $ is produced by the matrix\n\\begin{equation}\n\\mathbf{U}_{3}=\\left(\n\\begin{array}{llllll}\n0 & 1 & 0 & 0 & 0 & 0 \\\\\n1 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 1 \\\\\n0 & 0 & 0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & 1 & 0 & 0 \\\\\n0 & 0 & 1 & 0 & 0 & 0\n\\end{array}\n\\right) , \\label{eq:U_3}\n\\end{equation}\nwhile $\\left[ c_{1},c_{2},c_{3},c_{4},c_{5},c_{6}\\right] \\rightarrow \\left[\nc_{6},c_{5},c_{4},c_{3},c_{2},c_{1}\\right] $ leads to\n\\begin{equation}\n\\mathbf{U}_{4}=\\left(\n\\begin{array}{llllll}\n0 & 0 & 0 & 0 & 0 & 1 \\\\\n0 & 0 & 0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & 1 & 0 & 0 \\\\\n0 & 0 & 1 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0 & 0 & 0 \\\\\n1 & 0 & 0 & 0 & 0 & 0\n\\end{array}\n\\right) , \\label{eq:U_4}\n\\end{equation}\nand $\\left[ c_{1},c_{2},c_{3},c_{4},c_{5},c_{6}\\right] \\rightarrow \\left[\nc_{4},c_{3},c_{2},c_{1},c_{6},c_{5}\\right] $ is given by\n\\begin{equation}\n\\mathbf{U}_{5}=\\left(\n\\begin{array}{llllll}\n0 & 0 & 0 & 1 & 0 & 0 \\\\\n0 & 0 & 1 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0 & 0 & 0 \\\\\n1 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 1 \\\\\n0 & 0 & 0 & 0 & 1 & 0\n\\end{array}\n\\right) . \\label{eq:U_5}\n\\end{equation}\n\nThe set of matrices $G_{6}=\\left\\{ \\mathbf{I},\\mathbf{U}_{1},\\mathbf{U}_{2}\n\\mathbf{U}_{3},\\mathbf{U}_{4},\\mathbf{U}_{5}\\right\\} $ is a group isomorphic\nto $C_{3v}$\\cite{T64,C90}. If instead of the three reflection planes we\nconsider rotation axes on the plane of the figure across the middle of\nopposite the hexagon sides, we appreciate that rotations of $\\pi \/2$ about\nthem also leave the figure invariant (they are commonly called rotation axes\n$C_{2}$\\cite{T64,C90}). In such a case the group $G_{6}$ is isomorphic to \nD_{3}$\\cite{T64,C90}. Both groups exhibit irreducible representations $A_{1}\n, $A_{2}$ and $E$, the latter of dimension $2$ that explains the two-fold\ndegeneracy of the eigenvalues of the matrix (\\ref{eq:H}) for all values of \n\\lambda $. It is clear that we can predict a vanishing discriminant from the\nsymmetry of the problem.\n\nBy means of projection operators\\cite{T64,C90}, constructed\nstraightforwardly from the matrices $\\mathbf{U}_{j}$, we can easily\ndetermine the symmetry of the eigenvectors $\\mathbf{v}_{j}$, $\\mathbf{H\n(\\lambda )\\mathbf{v}_{j}=E_{j}\\mathbf{v}_{j}$, $j=1,2,\\ldots ,6$, of the\nmatrix (\\ref{eq:H}). It is not difficult to verify that $\\mathbf{v}_{1}$ and\n$\\mathbf{v}_{6}$ are bases for the irreducible representations $A_{2}$ and \nA_{1}$, respectively. The pairs of eigenvectors $\\left( \\mathbf{v}_{2}\n\\mathbf{v}_{3}\\right) $ and $\\left( \\mathbf{v}_{4},\\mathbf{v}_{5}\\right) $\nare bases for the two-fold degenerate irreducible representation $E$.\n\nAt $\\lambda =-1$ the eigenvalues $E_{1}$ and $E_{6}$ also become degenerate.\nThis crossing is predicted by the discriminant (\\ref{eq:Disc_E(q)}) as\ndiscussed above and is supposed to lead to an accidental degeneracy because\nit does not appear to be caused by an additional symmetry based on\northogonal matrices like those discussed above.\n\nThe highest symmetry is expected when $\\lambda =1$ because the matrix \n\\mathbf{H}(1)$ is invariant under a rotation of $\\pi \/3$ about an axis\nperpendicular to the center of the \\textit{regular} hexagon in Figure~\\ref\n{Fig:H6}. This axis, commonly called $C_{6}$\\cite{T64,C90} leads to two\nadditional matrix representations of $C_{6}$ and $C_{6}^{3}$ (previously we\nhad $C_{6}^{2}=C_{3}\\rightarrow \\mathbf{U}_{1}$, $C_{6}^{4}=C_{3}^{2\n\\rightarrow \\mathbf{U}_{2}$). In addition to the point-group elements just\ndiscussed we should add reflection planes $\\sigma _{d}$ through opposite\nvertices of the regular hexagon. The resulting group of $12$ elements is\nisomorphic to $C_{6v}$. Alternatively, we may consider three axis $C_{2}$\nthrough opposite vertices in which case the group results to be $D_{6}$\\cite\n{T64,C90}. In both cases the irreducible representations are $A_{1}$, $A_{2}\n, $B_{1}$, $B_{2}$, $E_{1}$ and $E_{2}$ that also predict two-fold\ndegeneracy in agreement with the results in Figure~\\ref{Fig:Enes}.\nWe do not discuss this particular case in detail because it does\nnot add anything relevant to what was said above.\n\n\\section{Conclusions}\n\n\\label{sec:conclusions}\n\nThroughout this paper we have shown that the straightforward application of\nthe discriminant to a quantum-mechanical problem may yield a trivial useless\nresult if there is degeneracy for all values of the model parameter. The\ncause of the failure is the symmetry of the problem which can be determined\nbeforehand by means of suitable tools based on group theory\\cite{T64,C90}.\nMore precisely, one expects to face this difficulty if the symmetry of the\nmodel Hamiltonian does not change with the variation of the model parameter.\nBased on the titles of the papers by Bhattacharya and Raman\\cite{BR06} and\nBhattacharya\\cite{B07} we may say: \\textit{Be careful if you do not look at\nthe spectrum}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\\bigskip\n\nWe consider the following problem:\nWhat are possible sizes of Jordan blocks for a pair of commuting nilpotent matrices? \nOr equivalently, for which pairs of nilpotent orbits of matrices (under similarity) there exists a pair of matrices, \none from each orbit, that commute. The answer to the question could be considered as a generalization of Gerstenhaber--Hesselink theorem\non the partial order of nilpotent orbits \\cite{cmc}.\n\nThe structure of the varieties of commuting pairs of matrices and of commuting pairs of nilpotent matrices is not yet well understood. \nIt was proved by Motzkin and Taussky \\cite{mt} (see also Guralnick \\cite{guralnick}), that the variety of pairs of \ncommuting matrices was irreducible. It was Guralnick \\cite{guralnick} who showed that this is no longer the case for \nthe variety of triples of commuting matrices (see also Guralnick and Sethuraman \\cite{gs}, Holbrook and Omladi\\v c \\cite{ho}, \nOmladi\\v c \\cite{omladic}, Han \\cite{han}, \\v Sivic \\cite{sivic}). Recently, it was proved that the variety of \ncommuting pairs of nilpotent matrices was irreducible (Baranovsky \\cite{baranovsky}, Basili \\cite{basili}). \nOur motivation to study the problem is to contribute to better understanding of the structure of this variety\nand which might also help in understanding the (ir)reducibility of the variety of triples of commuting matrices.\n\nWe are also motivated by the problems posed by Binding and Ko\\v sir \\cite{kosiMST} in the multiparameter spectral theory\nand Gustafson \\cite{gustafson} in the module theory over commutative rings. In both problems, certain \npairs of commuting matrices appear. The matrices from the multiparameter spectral theory generate an algebra that \nis a complete intersection, and the matrices from the theory of modules are both functions of another matrix.\n\n\nHere we initiate the study of the problem.\nFirst we list all possible Jordan forms for nilpotent matrices commuting with a single Jordan block.\nIn the Section \\ref{sec:digraphs} we recall the correspondence between directed graphs and \ngeneric matrices (also known as Gansner--Saks Theorem), which is the main tool to prove our main result, i.e.,\nto compute the maximal index of nilpotency of a nilpotent matrix commuting with a given nilpotent matrix with Jordan canonical form \n$\\underline{\\mu}$.\nIn the last section we discuss some further examples.\n\n\n\n\n\n\n \\bigskip\n\n \n\n\f\\section{The one Jordan block case and consequences}\\label{sec:general}\n\n\\bigskip\n\nLet us denote by ${\\mathcal N}={\\mathcal N}(n,\\mathbb{F})$ the variety of all $n \\times n$ nilpotent matrices over \na field $\\mathbb{F}$ of characteristic 0 and write ${\\mathcal N}_2=\\{(A,B) \\in {\\mathcal N} \\times {\\mathcal N}; AB=BA\\}$. \n\nWe follow the notations used in Basili \\cite{basili} and write \n${\\mathcal N}_B=\\{A \\in {\\mathcal N}; \\; AB=BA\\}$ for some $B \\in {\\mathcal N}$. Suppose $\\mu_1 \\geq \\mu_2 \\geq \\ldots \\geq \\mu_t > 0$ are\nthe orders of Jordan blocks in the Jordan canonical form for $B$. \nWe call the partition $\\underline{\\mu}=(\\mu_1,\\mu_2,\\ldots,\\mu_t)$ \nthe \\DEF{shape} of the matrix $B$ and denote it by $\\sh{B}$. \nWe also write $\\sh{B}=(m_1^{r_1},m_2^{r_2},\\ldots,m_l^{r_l})$, where $m_1 >m_2>\\ldots > m_l$.\n\n\n\\medskip\n\n \nLet $\\partition{n}$ denote the set of all partitions of $n \\in \\mathbb{N}$\nand for a subset ${\\cal S} \\subseteq {\\mathcal N}(n,\\mathbb{F})$ write \n$$\\partition{{\\cal S}}=\\{\\underline{\\mu} \\in \\partition{n}; \\; \\underline{\\mu}=\\sh{A} \\text{ for some } A\\in {\\cal S} \\}\\, .$$\nDenote also \n$$\\partition{{\\mathcal N}_2}=\\{(\\sh{A},\\sh{B}); \\; (A,B) \\in {\\mathcal N}_2 \\} \\subseteq \\partition{n} \\times \\partition{n}\\, .$$\nNote that $(\\underline{\\lambda},\\underline{\\mu}) \\in \\partition{{\\mathcal N}_2}$ if and only if\n$(\\underline{\\mu},\\underline{\\lambda}) \\in \\partition{{\\mathcal N}_2}$, i.e. $\\partition{{\\mathcal N}_2}$ is symmetric.\n\n\\medskip\n\nIt is easy to see that for each $t=1,2,\\ldots,n$ there exists a uniquely defined partition \n$\\rpt{n}{t}:=(\\lambda_1,\\lambda_2,\\ldots,\\lambda_t) \\in \\partition{n}$, such that $\\lambda_1-\\lambda_t \\leq 1$.\nIt can be verified that \n$\\rpt{n}{t}=\\left(\\left\\lceil \\frac{n}{t} \\right\\rceil^r,\\left\\lfloor \\frac{n}{t} \\right\\rfloor^{t-r}\\right)$. \nDenote $\\rp{n}=\\{\\rpt{n}{t}; \\; t=1,2,\\ldots,n\\}$.\nFor $\\underline{\\mu}=(\\mu_1,\\mu_2,\\ldots,\\mu_r)$ we define in the same fashion \n$\\rpt{\\underline{\\mu}}{t}=(\\rpt{\\mu_1}{t},\\rpt{\\mu_2}{t},\\ldots,\\rpt{\\mu_r}{t})$.\n\n\\bigskip\n\nTake a matrix $B \\in {\\mathcal N}(n,\\mathbb{F})$ with $\\dim\\ker B=1$ (i.e. $\\sh{B}=(n)$). Then it is well known that any matrix commuting with\n$B$ is a polynomial in $B$. By computing the lengths of the Jordan chains of $B$, we observe that $\\sh{B^k}=\\rpt{n}{k}$ for \n$k=1,2,\\ldots,n$.\t\nThus we have the following.\n\n\\bigskip\n\n\\begin{proposition}\\label{thm:onejorblock}\nFor a matrix $B$ with $\\dim\\ker B=1$ it follows that $\\partition{{\\mathcal N}_B}=\\rp{n}$.\n\\hfill$\\blacksquare$\n\\end{proposition}\n\n\\bigskip\n\n\nFor a sequence $(a_1,a_2,\\ldots,a_k)$, $a_i \\in \\mathbb{N}$, we write\n$\\ord{a_1,a_2,\\ldots,a_k}=(a_{\\pi(1)},a_{\\pi(2)},\\ldots,a_{\\pi(k)})$, where \n$a_{\\pi(1)}\\geq a_{\\pi(2)}\\geq \\ldots \\geq a_{\\pi(k)}$ and $\\pi$ is a permutation of $\\{1,2,\\ldots,k\\}$.\n\nFor $\\underline{\\lambda}=(\\lambda_1,\\lambda_2,\\ldots,\\lambda_t)\\in \\partition{n}$ we write\n\\begin{align*}\n \\rp{\\underline{\\lambda}\n \n \n =&\\{\\ord{\\rpt{\\lambda_1}{s_1},\\rpt{\\lambda_2}{s_2},\\ldots,\\rpt{\\lambda_t}{s_t}}; \\; s_i=1,2,\\ldots,i \\; \\}\\, . \n\\end{align*}\nSo, for example, $(3,3,2) \\in \\rp{(5,3)}$, but $(4,3,1) \\notin \\rp{(5,3)}$.\n\n\n\n\\bigskip\n\n\n\\begin{proposition} \\label{thm:expjor2}\n\n For all $\\underline{\\mu} \\in \\partition{n}$ it follows that\n $\\{(\\underline{\\mu}, \\underline{\\lambda}); \\underline{\\lambda} \\in \\rp{\\underline{\\mu}}\\} \\subseteq \\partition{{\\mathcal N}_2}$.\n\\end{proposition}\n\n\\medskip\n\n\\begin{proof}\n Take an arbitrary $B \\in {\\mathcal N}$ with $\\sh{B}=\\underline{\\mu}=(\\mu_1,\\mu_2,\\ldots,\\mu_t)$ \n and pick $\\underline{\\lambda} \\in \\rp{\\underline{\\mu}}$.\n We want to find a matrix $A \\in {\\mathcal N}_B$ such that $\\sh{A}=\\underline{\\lambda}$.\n \n By definition, $\\underline{\\lambda}$ is of the form $\\underline{\\lambda}=\\ord{\\rpt{\\mu_1}{k_1},\\rpt{\\mu_2}{k_2},\n \\ldots,\\rpt{\\mu_t}{k_t}}$.\n Let $J_{\\mu_i}$ be a $\\mu_i \\times \\mu_i$ Jordan block \n corresponding to eigenvalue 0. Since\n $\\sh{J_{\\mu_i}^{k_i}}=\\rpt{\\mu_i}{k_i} $, it follows that the matrix\n $A=J_{\\mu_1}^{k_1} \\oplus J_{\\mu_2}^{k_2} \\oplus \\ldots \\oplus J_{\\mu_t}^{k_t}$\n has $\\sh{A}=\\underline{\\lambda}$ and clearly $A \\in {\\mathcal N}_B$. \n\\end{proof}\n\n\\bigskip\n\nFor a matrix $A$ denote by $\\nil{A}$ its index of nilpotency, i.e.\n$\\nil{A}=\\min \\{i; \\; A^i=0\\}$. For ${\\cal S} \\subseteq {\\mathcal N}(n,\\mathbb{F})$ write\n$\\nil{{\\cal S}}=\\max \\{i; \\; i=\\nil{A} \\text{ for some } A \\in {\\cal S}\\}$.\n\n\\medskip\n\nThe following Corollaries can be easily obtained from Propositions \\ref{thm:onejorblock} and \\ref{thm:expjor2}.\n\n\\medskip\n\n\n\\begin{corollary} \\label{thm:oneblock}\n If $\\underline{\\lambda} \\in \\partition{n}$, then $\\underline{\\lambda} \\in \\rp{n}$ if and only if \n $((n),\\underline{\\lambda}) \\in \\partition{{\\mathcal N}_2}$.\n\\hfill$\\blacksquare$ \n\\end{corollary}\n\n\\medskip\n\n\\begin{corollary} \\label{thm:rb1}\n For $B \\in {\\mathcal N}$ it follows that $\\sh{B}\\in \\rp{n}$ if and only if $\\nil{{\\mathcal N}_B}=n$. \\hfill \\qedsymbol\n\\end{corollary}\n\n\\medskip\n\n\\begin{example}\nNote that Proposition \\ref{thm:expjor2} gives us a rather large subset of partitions in ${\\mathcal N}_B$.\n\nTake $\\sh{B}=(4,3,2^2,1)$. First we directly observe that \n$(4,3,2^2,1)$, $(4,3,2,1^3)$, $(4,3,1^5)$, $(4,2^3,1^2)$, $(4,2^2,1^4)$, $(4,2,1^6)$, $(4,1^8)$,\n$(3,2^4,1)$, $(3,2^3,1^3)$, $(3,2^2,1^5)$, $(3,2,1^7)$, $(3,1^9)$, $(2^5,1^2)$, $(2^4,1^4)$,\n$(2^3,1^6)$, $(2^2,1^8)$, $(2,1^{10})$, $(1^{12})$ are all in $\\partition{{\\mathcal N}_B}$.\n\nNext, we see that $(4,3,2^2,1)$ is included in $\\rp{\\underline{\\mu}}$ if\n$\\underline{\\mu}$ is $(7,5)$, $(7,4,1)$, $(7,3,2)$, $(7,2^2,1)$, $(5,4,3)$, $(5,4,2,1)$, $(4^2,3,1)$,\n$(4,3^2,2)$ or $(4,3,2^2,1)$. Thus, by symmetry, also all these partitions are in $\\partition{{\\mathcal N}_B}$.\n\nCorollary \\ref{thm:oneblock} shows that $\\partition{{\\mathcal N}_B} \\ne \\partition{12}$.\nThe natural question is whether the list above is the entire $\\partition{{\\mathcal N}_B}$. It can be verified \nwith {\\tt Mathematica} that the answer is negative and that there are more partitions in $\\partition{{\\mathcal N}_B}$. \nHowever, it would take a lot of time to compute $\\partition{{\\mathcal N}_B}$ without computer.\nFrom results in Section \\ref{sec:mainthm} it follows that there exists a partition \n$(\\mu_1,\\mu_2,\\ldots,\\mu_k) \\in \\partition{{\\mathcal N}_B}$ such that $\\mu_1=9$.\n\\hfill$\\square$\n\\end{example} \n\n\n\n\\bigskip\n\n\n\\section{Directed graphs and the Gansner-Saks theory} \\label{sec:digraphs}\n\n\\bigskip\n\n\n\n\nA \\DEF{digraph} is a directed graph (i.e. a graph each of whose edges are directed). \nA \\DEF{path of length} $k$ in a digraph $\\Gamma$ is a sequence of vertices $a_1,a_2,\\ldots,a_k$ such that $(a_i,a_{i+1})$ \nis an edge in $\\Gamma$ for $i=1,2,\\ldots,k-1$. We do allow a path to consist of a single vertex.\nFor a path ${\\mathcal P}$ we denote by $|{\\mathcal P}|$ its length. We call $a_1$ (resp. $a_k$) the \\DEF{initial} (resp. \\DEF{final}) vertex\nof ${\\mathcal P}$.\nA path $a_1,\\ldots,a_k$ with $a_1=a_k$ is called a \\DEF{cycle}. A digraph without cycles is called an \\DEF{acyclic digraph}.\n \nLet $\\Gamma$ be a finite acyclic digraph on $n$ vertices, with the vertices labeled from 1 to $n$.\nA \\DEF{$k$--path} in $\\Gamma$ is a subset of the vertices that can be partitioned into $k$ or fewer disjoint paths. Let \n$\\hat{d}_k=\\hat{d}_k(\\Gamma)$ be the largest cardinality of a $k$--path in $\\Gamma$ and define $\\hat{d}_0=0$ and \n$\\Delta_k=\\Delta_k(\\Gamma)=\\hat{d}_k-\\hat{d}_{k-1}$. Since $\\hat{d}_k \\leq \\hat{d}_{k+1}$, all the $\\Delta_k \\, 's$\nare nonnegative, so we have the infinite sequence $\\Delta=\\Delta(\\Gamma)=(\\Delta_1,\\Delta_2,\\ldots)$ of nonnegative integers.\n\n\\bigskip\n\n\nWe denote by $M(i;j)$ the entry in the $i$-th row and the $j$-th column of the matrix $M$. We say that matrices $M$ and $N$ \nof the same size \\DEF{have the same pattern} if $M(i;j)=0$ if and only if $N(i;j)=0$ for $i,j=1,2,\\ldots,n$.\n\n\nLet $\\overline{\\mathbb{F}}$ be a field that contains $\\mathbb{F}$ and has at least $n^2$ algebraically independent\ntranscendentals over rational numbers $\\mathbb{Q}$. \nA matrix $M \\in {\\mathcal N}(n,\\overline{\\mathbb{F}})$ is called \\DEF{generic} if all its nonzero entries are algebraically independent\ntranscendentals over $\\mathbb{Q}$. \n\n\n\\smallskip\n\nFor a nilpotent matrix $A \\in {\\mathcal N}(n,\\mathbb{F})$ and a generic $M \\in {\\mathcal N}(n,\\overline{\\mathbb{F}})$ of the same pattern it follows that \n$\\nil{A} \\leq \\nil{M}$.\n\n\\bigskip\n\nGiven a finite acyclic digraph $\\Gamma$ with $n$ vertices, we assign to it a generic\n$n \\times n$ matrix $M_\\Gamma=[m_{ij}]$ such that $m_{ij}=0$ if $(i,j)$ is not an edge in $\\Gamma$ \nand such that the rest of the entries of $M_\\Gamma$ are nonzero complex numbers which are independent\ntranscendentals. \nSince $\\Gamma$ is acyclic, $M_\\Gamma$ is nilpotent.\nConversely, given a nilpotent generic matrix, it corresponds, reversing the above assignment,\nto an acyclic digraph on $n$ vertices.\n\n\n\\begin{example}\\label{ex:gs}\nConsider a generic nilpotent matrix \n$M=\\left[\\begin{array}{cccccccccccc}\n 0&0&a&0&b&0\\\\\n 0&0&0&c&0&d\\\\\n 0&0&0&0&0&0\\\\\n 0&0&0&0&0&0\\\\\n 0&0&0&e&0&f\\\\\n 0&0&0&0&0&0\n \\end{array}\\right]\\, .$\nIts digraph $\\Gamma_M$ is then equal to the digraph on the Figure \\ref{fig1}. \\pagebreak\n\n \\begin{figure}[htb]\n \\begin{center}\n \\includegraphics[height=2.8cm, width=2.8cm]{fig0.eps}\n \\end{center}\n \\caption{}\\label{fig1}\n \\end{figure}\n\nConversely, every digraph $\\Gamma$ as on the Figure \\ref{fig1}, corresponds to a generic matrix with the\nsame pattern as matrix $M$. \n\\hfill$\\square$\n\\end{example}\n \n \\bigskip\n\nThe following theorem was independently proved by Gansner \\cite{gansner} and Saks \\cite{saks}. \n\n\\bigskip\n \n\\begin{theorem}\\label{thm:gansnersaks}\n Let $M$ be a generic nilpotent matrix. Then \n \\begin{equation}\n \\Delta(\\Gamma_M)=\\sh{M}\\, . \\tag*{$\\blacksquare$}\n \\end{equation}\n\\end{theorem}\n\n\\bigskip\n\n\\begin{corollary}\\label{eq:longpath}\n The length of the longest path in an acyclic digraph $\\Gamma$ is equal to $\\nil{M_\\Gamma}$.\n\\hfill$\\blacksquare$\n\\end{corollary}\n \\bigskip\n\n\\bigskip\n\n\\begin{example}\n Take matrix $M$ as in example \\ref{ex:gs} and digraph $\\Gamma$ as on the Figure \\ref{fig1}. \n We easily see that $\\hat{d}_1=3$ (a path of length 3 is 1,5,4), $\\hat{d}_2=5$ (a path of length 3 is 1,5,6 and a path of length 2 \n is 2,4) and $\\hat{d}_3=6$. By Theorem \\ref{thm:gansnersaks} it follows that $\\sh{M}=(3,2,1)$.\n \\hfill$\\square$\n\\end{example}\n\n\n \n\n\n\\bigskip\n\n\n\f \\section{The $({\\mathcal N}_B,A)$--digraph and its paths} \\label{sec:preliminaries}\n\n\\bigskip\n\nLet us fix a nilpotent matrix $B \\in {\\mathcal N}(n,\\mathbb{F})$ with \n$\\sh{B}=\\underline{\\mu}=(\\mu_1,\\mu_2,\\ldots,\\mu_t)=(m_1^{r_1},m_2^{r_2},\\ldots,m_l^{r_l})$.\nBy convention we have $m_1 > m_2 > \\ldots > m_l > 0$.\n\n\\bigskip\n\nIn this section we introduce some special digraphs, corresponding to elements in ${\\mathcal N}_B$.\n\n\\bigskip\n\nFor a pair of matrices $(A,B) \\in {\\mathcal N}_2$ denote $\\sh{A,B}=(\\sh{A},\\sh{B})$.\nThen there exists $P \\in GL_n(\\mathbb{F})$ such that $B=P J_B P^{-1}$, where $J_B$ denotes the Jordan\ncanonical form of matrix $B$.\nThus $\\sh{PAP^{-1},J_B}=(\\sh{A},\\sh{B})$ and $(PAP^{-1},J_B)\\in {\\mathcal N}_2$.\nTherefore we can assume that $B$ is already in its upper triangular Jordan canonical form.\n\n\\bigskip\n\nWrite $A=[A_{ij}]$ where $A_{ij} \\in {\\mathcal M}_{\\mu_i \\times \\mu_j}$. It is well known \n(see e.~g.~\\cite[p. 297]{GLR}) that if $AB=BA$, then $A_{ij}$ are all upper triangular Toeplitz matrices, i.e.\n for $1 \\leq j \\leq i \\leq t$ we have\n \\begin{equation}\\label{eq:toeplitz}\n A_{ij}=\n \\left[ \\begin{matrix}\n 0 & \\ldots & 0 & a_{ij}^0 & a_{ij}^1 & \\ldots & a_{ij}^{\\mu_i-1}\\\\\n \\vdots & & \\ddots & 0 & a_{ij}^0 & \\ddots & \\vdots \\\\\n \\vdots & & & \\ddots & 0 & \\ddots & a_{ij}^{1} \\\\\n 0 & \\ldots & \\ldots & \\ldots & \\ldots & 0 & a_{ij}^{0}\n \\end{matrix}\n \\right]\n \\;\n \\text{ and }\n \\;\n A_{ji}=\n \\left[ \\begin{matrix}\n a_{ji}^0 & a_{ji}^1 & \\ldots & a_{ji}^{\\mu_i-1}\\\\\n 0 & a_{ji}^0 & \\ddots & \\vdots \\\\\n \\vdots & \\ddots & \\ddots & a_{ji}^{1} \\\\\n \\vdots & & 0 & a_{ji}^{0} \\\\\n \\vdots & & & 0 \\\\\n \\vdots & & & \\vdots \\\\\n 0 & \\ldots & \\ldots & 0\n \\end{matrix}\n \\right]\n \\, .\n \\end{equation}\n\nIf $\\mu_{i}=\\mu_j$ then we omit the rows or columns of zeros in $A_{ij}$ or $A_{ji}$ above.\n\n\\bigskip\n\nWe introduce some further notation following Basili \\cite{basili}. For a matrix $B$ with $\\sh{B}=(\\mu_1,\\mu_2,\\ldots,\\mu_t)$ we\ndenote by $r_B$ and $k_i$, $i=1,2,\\ldots,r_B$, the numbers such that \n$k_1=1$, $\\mu_{k_i}-\\mu_{k_{i+1}} \\geq 2$, $\\mu_{k_i}-\\mu_{k_{i+1}-1} \\leq 1$ for $i=1,2,\\ldots,r_B-1$, \n$\\mu_{k_{r_B}}-\\mu_{t} \\leq 1$. Note that $k_i \\in \\{1,2,\\ldots,t\\}$.\nFor example, if $\\sh{A}=(5^3,3^2,1^3)$ and $\\sh{B}=(5^3,4,3^2,2^4,1^3)$, then $r_A=r_B=3$.\nFurthermore, $\\sh{C}\\in \\rp{n}$ if and only if $r_C=1$.\n\nSet $q_0=0$, $q_l=t$ and for $\\alpha=1,2,\\ldots,l$ let \n$q_\\alpha \\in \\{1,2,\\ldots,t\\}$, be such that $\\mu_i=\\mu_{i+1}$ if $q_{\\alpha-1}+1 \\leq i < q_\\alpha$ and\n$\\mu_{q_\\alpha} \\ne \\mu_{q_\\alpha+1}$. For a block matrix with blocks as in\n\\eqref{eq:toeplitz} we define $\\overline{A}_{\\alpha \\alpha}=[a_{ij}^0]$ where $q_{\\alpha-1}+1 \\leq i,j \\leq q_\\alpha$.\n\n\\medskip\n\n\\begin{lemma}\\label{thm:basiliaa} \\cite[Proposition 2.3]{basili} \nFor an $n \\times n$ matrix $A$ such that $AB=BA$ it follows that $A \\in {\\mathcal N}_B$ if and only if\n$\\overline{A}_{\\alpha\\alpha}$ are nilpotent for $\\alpha=1,2,\\ldots,l$.\n\nMoreover, if $A \\in {\\mathcal N}_B$ there exists a Jordan basis for $B$ such that $\\overline{A}_{\\alpha\\alpha}$\nare all strictly upper triangular.\n\\hfill{$\\blacksquare$}\n\\end{lemma}\n\n\\bigskip\n\nTherefore, when we study what are possible shapes of pairs from the set ${\\mathcal N}_2$ (or in particular what \nare the indices of nilpotency in ${\\mathcal N}_B$), we may consider only matrices $A \\in {\\mathcal N}_B$\nsuch that we have $a_{ij}^0=0$ for all $1 \\leq j \\leq i \\leq t$ whenever $\\mu_i=\\mu_j$.\nFrom now, we assume that the latter relations hold for $A$.\n\n\\bigskip\n\nLet $\\Gamma$ be a digraph corresponding to a generic matrix $M \\in {\\mathcal N}(n,\\overline{\\mathbb{F}})$ such that $M$ has \nthe same pattern as a matrix $A \\in {\\mathcal N}_B$. We call $\\Gamma$ an \\DEF{$({\\mathcal N}_B,A)$-digraph}.\n\n\\bigskip\n\n\\begin{example} \n Consider a nilpotent matrix $B$ with $\\sh{B}=(4,2)$ and a matrix\n $A=\\left[\\begin{array}{cccccccccccc}\n 0&0&a&0&b&0\\\\\n 0&0&0&a&0&b\\\\\n 0&0&0&0&0&0\\\\\n 0&0&0&0&0&0\\\\\n 0&0&0&c&0&d\\\\\n 0&0&0&0&0&0\n \\end{array}\\right] \\in {\\mathcal N}_B \\, .$ \n Then the digraph on the Figure \\ref{fig1} is an $({\\mathcal N}_B,A)$-digraph.\n \\hfill$\\square$\n\\end{example}\n\n\n\\bigskip\n\nLet $\\Gamma$ be an $({\\mathcal N}_B,A)$-digraph that corresponds to $A \\in {\\mathcal N}_B$. \nDenote its vertices by $(x,y)$, where $x=1,2,\\ldots,t$ and $y=1,2,\\ldots,\\mu_x$.\nWrite blocks of $A$ as in \\eqref{eq:toeplitz}. If $a_{ij}^k \\ne 0$ for some $1 \\leq i \\leq j \\leq t$, \nthen $\\Gamma$ contains edges $\\left((i,h),(j,h+k)\\right)$ for $h=1,2,\\ldots,\\mu_j-k$.\nIf $a_{ij}^k \\ne 0$ for some $1 \\leq j \\leq i \\leq t$, \nthen $\\Gamma$ contains edges $\\left((i,h),(j,h+k+\\mu_j-\\mu_i+1)\\right)$ for all $h=1,2,\\ldots,\\mu_i-k$.\n\n\\bigskip\n\nWe say that edges $((i_1,j_1),(i_2,j_2))$ and $((i_3,j_3),(i_4,j_4))$ are \\DEF{parallel} if \n$i_1=i_3$, $i_2=i_4$ and $j_1+j_4=j_2+j_3$. We call two paths ${\\mathcal P}_1$ and ${\\mathcal P}_2$ \\DEF{parallel} if\nthey consist of pairwise parallel edges.\n\n\\bigskip\n\n\n\\begin{example} \\label{ex:43221a}\nConsider the case $\\sh{B}=(4,3,2^2,1) \\in \\partition{12}$. Each $({\\mathcal N}_B,A)$-digraph $\\Gamma$ consists of 12 vertices:\n\n \\begin{figure}[htb]\n \\begin{center}\n \\includegraphics[height=3.6cm, width=6.9cm]{fig43221g}\n \\end{center}\n \\end{figure}\n\nDenote by $M$ the generic nilpotent matrix, which corresponds to $\\Gamma$ and has the same pattern as $A \\in {\\mathcal N}_B$.\nSuppose, for example, that there exists an edge $((4,1),(2,2))$ in $\\Gamma$. By the correspondence \nbetween nilpotent matrices and acyclic digraphs it follows that $M_{10,6} \\ne 0$. Since $A$ and \n$M$ have the same pattern, $a_{4,2}^0 \\ne 0$ and thus\n$M_{11,7} \\ne 0$. Therefore, there exists also an edge $((4,2),(2,3))$ in $\\Gamma$, which is parallel to \nan edge $((4,1),(2,2))$.\n\nSimilarly, we can add some other parallel edges to $\\Gamma$ as described above. For example, a possible $({\\mathcal N}_B,A)$-digraph $\\Gamma$ is\n\n \\begin{figure}[htb]\n \\begin{center}\n \\includegraphics[height=4.5cm, width=8.7cm]{fig43221a}\n \\end{center}\n \\caption{}\\label{fig3} \n \\end{figure}\n\nand the corresponding matrix $A \\in {\\mathcal N}_B$ is equal to\n$$\nA=\\left[\\begin{array}{cccc|ccc|cc|cc|c}\n0 & 0 & a & 0 & b & 0 & c & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & a & 0 & b & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & b & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n\\hline\n0 & 0 & d & 0 & 0 & 0 & 0 & e & f & 0 & 0 & 0 \\\\\n0 & 0 & 0 & d & 0 & 0 & 0 & 0 & e & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n\\hline\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & g & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n\\hline\n0 & 0 & 0 & 0 & 0 & h & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & h & 0 & 0 & 0 & 0 & 0 \\\\\n\\hline\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & i & 0 & 0 & 0 \n\\end{array}\\right]\n$$\nfor some $a,b,c,\\ldots,i \\in \\mathbb{F}$.\n\\hfill$\\square$\n\\end{example}\n\n\\bigskip\n \nSuppose that $r_i,r_{i+1} \\ne 0$ for some $i$.\nThen parts $m_i^{r_i}$, $m_{i+1}^{r_{i+1}}$ in $\\sh{B}$ have to be treated differently if \n$m_i-m_{i+1}=1$ or $m_{i}-m_{i+1} \\geq 2$. We introduce some notation that will help us \nunify the treatments.\n\n\\bigskip\n\nFor $i=1,2,\\ldots,l-1$ such that $r_i \\ne 0$ define \n$$s(i)=\\left\\{\n \\begin{array}{cl}\n r_i, & \\text{if } m_{i}-m_{i+1}\\geq 2 \\\\\n r_i+r_{i+1}, & \\text{if } m_{i}-m_{i+1} \\leq 1\n \\end{array}\n \\right.$$\nand $s(l)=r_l$. \n\n\\medskip\n\nWe set $\\mu_0=\\mu_1+1$. Choose any $k$, $1 \\leq k \\leq t$, such that $\\mu_{k-1}> \\mu_k > 1$ \nand let $w$ and $z$ be such that $\\mu_k-\\mu_w \\leq 1$, $\\mu_k-\\mu_{w+1} \\geq 2$ and \n$\\mu_{k}=\\mu_z > \\mu_{z+1}$.\nIf $\\mu_{k-1}> \\mu_k = 1$, then set $w=z=t$.\n\n\\medskip\n\nWe denote by $V_{B,k}$ the set of vertices\n\\begin{align*}\n \\big\\{(x,y); \\; & x=k,k+1,\\ldots,w, y=1,2,\\ldots,\\mu_{x}\n \\big\\} \\, .\n\\end{align*} \nWe call the path that contains vertices \\label{Bpath}\n$$V_{1,k}=\\{(x,1)\\; x=1,2,\\ldots,k \\}, $$\n$$ V_{B,k}$$ \nand \n$$ V_{3,k}=\\{(x,\\mu_x)\\; x=1,2,\\ldots,k-1 \\} \\cup \\{(z,\\mu_z)\\}$$ \nthe \\DEF{$B_k$-path} (or \\DEF{$B$-path} for short). We call $s(k)$ the \\DEF{width} of $B_k$-path (or of the set $V_{B,k}$).\n\n\n\\pagebreak\n\n\\begin{example}\\label{ex:43221c}\nConsider again the case $\\sh{B}=(4,3,2^2,1) \\in \\partition{12}$ and an $({\\mathcal N}_B,A)$-digraph $\\Gamma'$\n\n\n \\begin{figure}[htb]\n \\begin{center}\n \\includegraphics[height=4cm, width=6cm]{fig43221z}\n \\end{center}\n \\caption{}\\label{fig4} \n \\end{figure}\n \nThen for $k=1,2,3,5$, the set $V_{B,k}$ and $B_k$-path consist of the following vertices\n\n\\begin{center}\n\\begin{tabular}{|c||c|c|c|p{5cm}|p{6cm}|}\n \\hline\n $k$ & $s(k)$ & $w$ & $z$ & $V_{B,k}$ & vertices of $B_k$-path \\\\\n \\hline\n \\hline\n $1$ & $2$ & $2$ & $1$ & $(1,1)$, $(1,2)$, $(1,3)$, $(1,4)$, $(2,1)$, $(2,2)$, $(2,3)$ & $(1,1)$, $(1,2)$, $(1,3)$, $(1,4)$, $(2,1)$, $(2,2)$, $(2,3)$ \\\\ \n \\hline\n $2$ & $3$ & $4$ & $2$ & $(2,1)$, $(2,2)$, $(2,3)$, $(3,1)$, $(3,2)$, $(4,1)$, $(4,2)$ & $(1,1)$, $(2,1)$, $(2,2)$, $(2,3)$, $(3,1)$, $(3,2)$, $(4,1)$, $(4,2)$, $(1,4)$ \\\\ \n \\hline\n $3$ & $3$ & $5$ & $4$ & $(3,1)$, $(3,2)$, $(4,1)$, $(4,2)$, $(5,1)$ & $(1,1)$, $(2,1)$, $(3,1)$, $(3,2)$, $(4,1)$, $(4,2)$, $(5,1)$, $(1,4)$, $(2,3)$ \\\\ \n \\hline\n $5$ & $1$ & $5$ & $5$ & $(5,1)$ & $(1,1)$, $(2,1)$, $(3,1)$, $(4,1)$, $(5,1)$, $(1,4)$, $(2,3)$, $(3,2)$, $(4,2)$\\\\ \n \\hline\n\\end{tabular}\n\\end{center}\n\\hfill$\\square$\n\\end{example}\n\n\\bigskip\n\nThe $B_k$-path consists of three subpaths ${\\mathcal P}_i$, $i=1,2,3$, such that \n${\\mathcal P}_1 \\cap {\\mathcal P}_2=\\{(k,1)\\}$ and ${\\mathcal P}_2 \\cap {\\mathcal P}_3 = \\{(z,\\mu_z)\\}$.\nThe vertices of ${\\mathcal P}_1$ are from $V_{1,k}$ and its edges are \n$$((i,1),(i+1,1))$$ \nfor $i= 1,2, \\ldots, k-1$. The ${\\mathcal P}_2$ consists of vertices $V_{B,k}$ and\nedges $$((i,j),(i+1,j))$$ \nfor \n$i=k,k+1,\\ldots,w-1$, $j=1,2,\\ldots,\\mu_{k}-1$ or \n$i=k,k+1,\\ldots,z-1$, $j=\\mu_{k}$ and \n$$((w,j),(k,j+1))$$ \nfor $j=1,2,\\ldots,\\mu_{k}-1$.\nThe subpath ${\\mathcal P}_3$ on vertices $V_{3,k}$ consists of edges\n$$((i,\\mu_{i}),(i+1,\\mu_{i+1}))$$ \nif $\\mu_i=\\mu_{i+1}$, $1 \\leq i \\leq k-2$, and\n$$((j,\\mu_{j}),(i,\\mu_{i}))$$ \nif $\\mu_{k} \\leq \\mu_{j+1}< \\mu_j < \\mu_i < \\mu_{i-1}$ and \nsuch that if $i \\leq h \\leq j$, then it follows that either $\\mu_{h}=\\mu_i \\text{ or } \\mu_h=\\mu_j$.\n\n\\smallskip\n\nFor a $B_k$-path ${\\mathcal P}$ it follows that $|{\\mathcal P}|=2(k-1)+\\mu_{k}+\\mu_{k+1}+\\ldots+\\mu_{w}$. \n\n\n\\bigskip\n\n\n\\begin{example} \\label{ex:43221b}\nRecall the Examples \\ref{ex:43221a} and \\ref{ex:43221c} where $\\sh{B}=(4,3,2^2,1)$. In the $({\\mathcal N}_B,A)$-digraph $\\Gamma$ from\nFigure \\ref{fig3} there are no $B_k$-paths. However the $({\\mathcal N}_B,A)$-digraph $\\Gamma'$ from Figure \\ref{fig4}\nhas three $B$-paths, namely, $B_1$-path (with width 2 and length 7), $B_2$-path (with width 3 and length 9) and $B_3$-path (also of length 9\nand width 3), as shown below\n\n \\begin{figure}[htb]\n \\begin{center}\n \\includegraphics[height=3.2cm, width=12cm]{fig43221b}\n \\end{center}\n \\end{figure}\n\n\n \\begin{figure}[htb]\n \\begin{center}\n \\includegraphics[height=3.2cm, width=4.7cm]{fig43221c}\n \\end{center}\n \\end{figure}\n\nNote that in this case $B_3$-path and $B_5$-path coincide.\n\\hfill$\\square$\n\\end{example}\n\n\n\\bigskip\n\n\\begin{example}\nLet $\\sh{B}=(4,3^2)$. By Corollary \\ref{thm:rb1} it follows that $\\nil{{\\mathcal N}_B}=10$. \nThe generic $M$, such that its digraph $\\Gamma_M$ is equal to\n\n \\begin{figure}[h]\n \\begin{center}\n \\includegraphics[height=3.5cm, width=3.5cm]{fig433}\n \\end{center}\n \\end{figure}\n\n\\pagebreak\nand $A \\in {\\mathcal N}_B$ such that $a_{12}^0$, $a_{23}^0$, $a_{31}^0$ are its only nonzero entries,\nhave the same pattern. It is easy to verify that $\\nil{M}=\\nil{A}=10$. \nThe shown path is the only $B$-path in $\\Gamma_M$.\n\\hfill$\\square$\n\\end{example}\n\n\\bigskip\n\nRecall that $\\nil{A} \\leq \\nil{M}$ for $A \\in {\\mathcal N}_B$ and a generic matrix $M \\in {\\mathcal N}(n,\\overline{\\mathbb{F}})$ with the same pattern. \nBy Corollary \\ref{eq:longpath} it follows that\n$\\nil{{\\mathcal N}_B}$ is less than or equal to the length of the longest path in a $({\\mathcal N}_B,A)$-digraph.\nThe length of the $B_k$-path is equal to \n$2 \\sum_{i=1}^{k-1} r_i + r_k m_k+r_{k+1}m_{k+1}$ if\n$s(k)=r_k+r_{k+1}$ and $2 \\sum_{i=1}^{k-1} r_i + r_k m_k$ otherwise.\n\n \n\n\\bigskip\n \n\\section{Maximal index of nilpotency in ${\\mathcal N}_B$} \\label{sec:mainthm}\n\n\\bigskip\n\nHere, we prove the following theorem:\n\n\\bigskip\n\n\\begin{theorem} \\label{thm:main}\n Let $B$ be a nilpotent matrix with $\\sh{B}=(\\mu_1,\\mu_2,\\ldots,\\mu_t)$. Then\n $$\\nil{{\\mathcal N}_B}= \\max\\limits_{1 \\leq i < t} \\{2i+\\mu_{i+1}+\\mu_{i+2}+\\ldots+\\mu_{i+r}; \\; \\mu_{i+1} - \\mu_{i+r}\\leq 1, \\mu_i \\ne \\mu_{i+1}\\} \n \\, .$$\n\\end{theorem}\n\n\\bigskip\n\nFirst, we show that the longest path in an $({\\mathcal N}_B,A)$-digraph $\\Gamma$ is actually equal\nto the length of the longest $B$-path in $\\Gamma$.\n\n\n\\bigskip\n\n\\begin{lemma}\\label{thm:path2}\n Suppose that for $x=1,2,\\ldots,t$ vertices $(x,1)$ are contained in a path ${\\mathcal P}$ of an $({\\mathcal N}_B,A)$-digraph $\\Gamma$. \n Then there exists a $B$-path ${\\mathcal P}_B$ in $\\Gamma$ such that $|{\\mathcal P}| \\leq |{\\mathcal P}_B|$.\n\\end{lemma}\n \n\\medskip\n \n\\begin{proof}\n Let us write $\\sh{B}=(\\mu_1,\\mu_2,\\ldots,\\mu_t)=(m_1^{r_1},m_2^{r_2},\\ldots,m_l^{r_l})$, \n where $m_i=\\mu_1-i+1$ for $i=1,2,\\ldots,l$ and $m_l=\\mu_t$. (Note that $r_i$ can be 0 \n for some $i$.) \n \n Let $\\Gamma$ be an $({\\mathcal N}_B,A)$-digraph and $k$ be the maximal index such that \n $s(k)=\\max\\{s(i); \\; 1 \\leq i \\leq l \\}$. \n By $\\bask{k}$ we denote the $B_k$-path, i.e. the widest $B$-path in $\\Gamma$. \n \n Suppose first that $s(k)=r_k+r_{k+1}$, where $r_{k+1} \\ne 0$. Then\n $$|\\bask{k}|=2 \\sum_{i=1}^{k-1} r_i + r_k m_k +r_{k+1}m_{k+1}=\n 2 \\sum_{i=1}^{k-1} r_i + (r_k+r_{k+1}) m_{k+1} + r_k \\, .$$\n\n Examine the edge $((i,y_i),(j,y_j))$ of ${\\mathcal P}$, where $\\mu_i \\ne \\mu_j$. \n If $i>j$ (and thus $\\mu_i < \\mu_j$) then $\\mu_i-y_i > \\mu_j-y_j$.\n If $i \\mu_j$) then $y_i < y_j$. \n \n Let ${\\mathcal P}'$ be the subpath of ${\\mathcal P}$ that contains all the vertices $(x,y)$ from ${\\mathcal P}$, where $y > 1$.\n Let $(x_0,y_0)$ be the initial vertex of ${\\mathcal P}'$. Since $((t,1),(x_0,y_0))$, $1 \\leq x_0 \\leq t$, is an edge of ${\\mathcal P}$,\n it follows that $\\mu_t-1 \\geq \\mu_{x_0}-y_0$.\n \n Since $\\bask{k}$ is the widest $B$-path in $\\Gamma$,\n it follows that\n $|{\\mathcal P}'| \\leq (\\mu_{x_0}-y_0) (r_{k}+r_{k+1}) + \\sum_{i=1}^{l-1} r_i \\leq \\sum_{i=1}^{l-1} r_i + (m_l-1) (r_{k}+r_{k+1})$. \n Therefore\n $$|{\\mathcal P}| \\leq 2 \\sum_{i=1}^{l-1} r_i +r_l +(m_l-1) (r_{k+1}+r_k)\\, .$$\n \n Since $m_i=\\mu_1-i+1$ for $i=1,2,\\ldots,l$, it follows that $m_i - m_l = l-i$ for $i=1,2,\\ldots,l$. \n In particular, $m_l=m_k-l+k$. Thus\n $$|{\\mathcal P}| \\leq 2 \\sum_{i=1}^{l-1} r_i +r_l +(m_k-l+k-1) (r_k+r_{k+1})\\, .$$\n \n \n By definition of $k$ it follows that $r_i+r_{i+1} \\leq r_k+r_{k+1}-1$ for all \n $i=k+1,k+2,\\ldots,l-1$.\n Thus \n $r_{k+1}+2r_{k+2}+2r_{k+3}+\\ldots+2r_{l-1}+r_l=\\sum_{i=k+1}^{l-1} (r_i+r_{i+1}) \\leq (l-k-1)(r_k+r_{k+1}-1)$.\n Therefore\n \\begin{equation*}\n \\begin{split}\n |{\\mathcal P}| &\\leq 2 \\sum_{i=1}^{k-1} r_i + r_k + (r_k+r_{k+1}) + (l-k-1)(r_k+r_{k+1}-1)+\\\\\n & \\qquad \\qquad + (m_k-l+k-1) (r_k+r_{k+1}) =\\\\\n &= 2 \\sum_{i=1}^{k-1} r_i + r_k + (m_{k}-1)(r_k+r_{k+1})+k+1-l \\leq \\\\\n &\\leq 2 \\sum_{i=1}^{k-1} r_i + r_k + m_{k+1}(r_k+r_{k+1})\\; = \\; |\\bask{k}| \\, .\n \\end{split}\n \\end{equation*}\n\n \n If $s(k)=r_k$ and $k < l$, then $r_i+r_{i+1} \\leq r_k-1$ for $ i=k+1,k+2,\\ldots,l-1$.\n Thus \n $r_{k+1}+2r_{k+2}+2r_{k+3}+\\ldots+2r_{l-1}+r_l=\\sum_{i=k+1}^{l-1} (r_i+r_{i+1}) \\leq (l-k-1)(r_k-1)$.\n Again, since $\\bask{k}$ is the widest $B$-path in $\\Gamma$, we see similarly as before,\n \\begin{equation*}\n \\begin{split}\n |{\\mathcal P}| & \\leq 2 \\sum_{i=1}^{l-1} r_i + r_l+ (\\mu_t-1) r_k \\leq \\\\\n & \\leq 2 \\sum_{i=1}^{k-1} r_i +(l-k-1)(r_k-1)+ (\\mu_k-l+k+1) r_k \\leq \\\\\n & \\leq 2 \\sum_{i=1}^{k-1} r_i + \\mu_k r_k \\; = \\; |\\bask{k}| \\, ,\n \\end{split}\n \\end{equation*}\n which proves the Lemma.\n\n\n\n \\end{proof}\n \n \\bigskip \n \n\n\\begin{theorem}\\label{thm:premain}\n For $A \\in {\\mathcal N}_B$ it follows that\n $$\\nil{A} \\leq \\max\\limits_{1 \\leq i < t} \n \\{2i+\\mu_{i+1}+\\mu_{i+2}+\\ldots+\\mu_{i+r}; \\; \\mu_{i+1} - \\mu_{i+r}\\leq 1, \\mu_i \\ne \\mu_{i+1}\\} \\, .$$\n \\end{theorem}\n \n \\medskip\n \n \\begin{proof}\n Write $\\sh{B}=(\\mu_1,\\mu_2,\\ldots,\\mu_t)$.\n We show by induction on $t$ that the longest possible path of $({\\mathcal N}_B,A)$-digraphs is of the\n same length as a $B$-path.\n \n Suppose that the longest path of $({\\mathcal N}_B,A)$-digraphs is included in digraph $\\Gamma$ and\n denote it by ${\\mathcal P}$. \n Note that ${\\mathcal P}$ contains vertices $(1,1)$ and $(1,\\mu_1)$.\n If it does not, we can add an edge from $(1,1)$ to the first vertex of ${\\mathcal P}$ or an edge\n from the last vertex of ${\\mathcal P}$ to $(1,\\mu_1)$ and lenghten it.\n \n \\smallskip\n \n If $t=1$ then the longest path contains all vertices and therefore it is a $B$-path.\n \n Suppose that our claim holds for all partitions with at most $t-1$ parts.\n \n Fix an $x$, $1 \\leq x \\leq t$. If ${\\mathcal P}$ does not contain any of the vertices $(x,y)$ for $y=1,2,\\ldots,\\mu_x$,\n then ${\\mathcal P}$ is a path in a $({\\mathcal N}_{B'},A)$-digraph, where \n $$\\sh{B'}=(\\mu_1,\\mu_2,\\ldots,\\mu_{x-1},\\mu_{x+1},\\mu_{x+2},\\ldots,\\mu_t).$$ By induction, there exists a $B'$-path \n ${\\mathcal P}_{B'}$ such that $|{\\mathcal P}| \\leq |{\\mathcal P}_{B'}|$. If a $B'$-path is not already a $B$-path, \n it can be lengthened to a $B$-path. Therefore there exists a $B$-path ${\\mathcal P}_B$ in $\\Gamma$, such that \n its length is greater than or equal to $|{\\mathcal P}|$. By assumption that ${\\mathcal P}$ is the longest path in $\\Gamma$\n it follows that $|{\\mathcal P}| = |{\\mathcal P}_B|$.\n \n Suppose now that for each $x$ the path ${\\mathcal P}$ contains a vertex $(x,y_x)$ for some $y_x$.\n The basic idea of the proof is to show the following claim: \n \n \\emph{In $\\Gamma$ there exists a path ${\\mathcal P}'$ of the same length as ${\\mathcal P}$ such that its first $t$ vertices are \n $(1,y_1)$, $(2,y_2)$,..., $(t,y_t)$.} \n \n Since ${\\mathcal P}'$ is the longest path in $\\Gamma$ it follows that $y_i=1$ for $i=1,2,\\ldots,t$. \n Otherwise the path that contains vertices $(1,1)$, $(2,1)$,..., $(t,1)$, $(t,2)$,..., $(t,y_t)$ and is after \n $(t,y_t)$ equal to ${\\mathcal P}'$ is longer than ${\\mathcal P}'$.\n \n \\smallskip\n \n Let us prove the claim. Let $y_t > 1$ be the smallest integer such that ${\\mathcal P}$ contains vertex $(t,y_t)$. \n Since ${\\mathcal P}$ is the longest path, $(t,y_t)$ is not the last vertex of ${\\mathcal P}$, and so for $x=1,\\ldots,t$,\n there exists a vertex $(x,y^3_x)$ that ${\\mathcal P}$ visits after $(t,y_t)$, i.e. $y^3_x \\geq y_t$. \n \n Next, suppose that there exists some $k$, where $1 \\leq k < t$ and $y_k^1 < y_k^2 \\leq y_t$, such that ${\\mathcal P}$ \n contains vertices $(k,y_k^1)$ and $(k,y_k^2)$ with the following properties: there does \n not exist a vertex $(k,y_k^0)$ of ${\\mathcal P}$ such that $1\\leq y_k^0 \\leq y_k^1$ and $y_k^2$ is such that \n there does not exist a vertex between $(k,y_k^2)$ and $(t,y_t)$ in ${\\mathcal P}$ with $k$ as its first coordinate. \n\n\n Denote by ${\\mathcal P}_1$ (resp. ${\\mathcal P}_2)$ the subpath of ${\\mathcal P}$ with its initial vertex $(k,y_k^1)$ \n (resp. $(k,y_k^2)$) and its final vertex $(k,y_k^2)$ (resp. $(k,y_k^3)$) and define \n ${\\mathcal P}_0={\\mathcal P} \\, \\backslash \\, \\{{\\mathcal P}_1, {\\mathcal P}_2\\}$.\n In $\\Gamma$ there exist the path ${\\mathcal P}_1'$ parallel to \n ${\\mathcal P}_2$ starting at the $(k,y_k^1)$ and the path ${\\mathcal P}_2'$ parallel to \n ${\\mathcal P}_1$ ending at the $(k,y_k^3)$. Then, the initial vertex of ${\\mathcal P}'_2$ coincides with the final vertex of ${\\mathcal P}'_1$.\n Thus ${\\mathcal P}_0 \\cup {\\mathcal P}_1'\\cup {\\mathcal P}_2'$ is a path in $\\Gamma$ of the same length as ${\\mathcal P}$ and it contains only \n one vertex with its first coordinate $k$ before $(t,y_t)$.\n \n By repeating this swap for $k=1,2,\\ldots,t$, we obtain a path ${\\mathcal P}'$ such that it does not contain vertices\n $(x,y_x^1)$ and $(x,y_x^2)$, with $y_x^1 < y_x^2 \\leq y_t$ for all $x=1,2,\\ldots, t-1$. \n As argued above, this forces ${\\mathcal P}'$ to be a path in the $({\\mathcal N}_B,A)$-digraph $\\Gamma$ that contains vertices \n $(x,1)$ for all $x=1,2,\\ldots,t$. By Lemma \\ref{thm:path2} there exists a $B$-path $\\bask{k}$ such that \n $|{\\mathcal P}|=|{\\mathcal P}'| \\leq \n |\\bask{k}|=2k+\\mu_{k+1}+\\mu_{k+2}+\\ldots+\\mu_{k+r}$ for some $r$, $\\mu_k-\\mu_{k+r}\\leq 1$.\n \\end{proof}\n\n\\bigskip\n\n To prove Theorem \\ref{thm:main}, it only remains to show that the maximum of Theorem \\ref{thm:premain} is attained. \n\n\\bigskip\n \n\\begin{proof} (of {\\bf Theorem \\ref{thm:main}})\n Let $\\sh{B}=(\\mu_1,\\mu_2,\\ldots,\\mu_t)=(m_1^{r_1},m_2^{r_2},\\ldots,m_l^{r_l})$. We define the $B$-path ${\\mathcal P}$, for which the maximum \n $$\\max\\limits_{1 \\leq i < t}\\{2i+\\mu_{i+1}+\\mu_{i+2}+\\ldots+\\mu_{i+r}; \\; \\mu_{i+1} - \\mu_{i+r}\\leq 1; \\; \\mu_{i} \\ne \\mu_{i+1}\\}$$\n is attained.\n \n Let $k$ and $s$ be such that \n $\\max\\limits_{1 \\leq i < t}\\{2i+\\mu_{i+1}+\\mu_{i+2}+\\ldots+\\mu_{i+r}; \\; \\mu_{i+1} - \\mu_{i+r}\\leq 1; \\; \\mu_{i} \\ne \\mu_{i+1}\\}=\n 2k+\\mu_{k+1}+\\mu_{k+2}+\\ldots+\\mu_{k+s}$.\n \n \\smallskip\n \n Denote by ${\\mathcal P}$ the $B_{(k+1)}$-path in $\\Gamma$. \n It follows that $|{\\mathcal P}|=2k+\\mu_{k+1}+\\mu_{k+2}+\\ldots+\\mu_{k+s}$. \n The path ${\\mathcal P}$ is such that it can be completed (by drawing parallel edges) \n to an $({\\mathcal N}_B,A)$-digraph. (For examples of such $({\\mathcal N}_B,A)$-digraphs see Example \\ref{ex:2}.) \n By the proof of Theorem \\ref{thm:premain} it follows that ${\\mathcal P}$ is a longest path in $\\Gamma$.\n \n For a generic matrix $M_\\Gamma$, corresponding to $\\Gamma$, it follows that \n $\\nil{M}=|{\\mathcal P}|$. \n Take $A \\in {\\mathcal N}_B$ to be a positive matrix with the same pattern \n as $M$. \n Since $\\ch{\\mathbb{F}}=0$, it follows that $[A^h]_{ij} =0$ if and only if $[M^h]_{ij}=0$ for all $h$, $i$, $j$.\n Therefore $\\nil{A}=\\nil{M}=2k+\\mu_{k+1}+\\mu_{k+2}+\\ldots+\\mu_{k+s}$ which proves the theorem.\n\\end{proof}\n \n \n\\bigskip\n\n\n\f\\section{Examples} \\label{sec:nilex}\n\n\\bigskip\n\n\\begin{example}\\label{ex:2}\nAs in Examples \\ref{ex:43221a} and \\ref{ex:43221b}, we again consider the case $\\sh{B}=(4,3,2^2,1)$. \nBy Theorem \\ref{thm:main} it \nfollows that $\\nil{{\\mathcal N}_B}=\\max \\{7,9,9\\}=9$. \n\nConsider the following matrices:\n$$\nA_1=\\left[\\begin{array}{cccc|ccc|cc|cc|c}\n0 & 0 & 0 & 0 & a & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & a & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & a & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n\\hline\n0 & d & 0 & 0 & 0 & 0 & 0 & b & 0 & 0 & 0 & 0 \\\\\n0 & 0 & d & 0 & 0 & 0 & 0 & 0 & b & 0 & 0 & 0 \\\\\n0 & 0 & 0 & d & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n\\hline\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c & 0 \\\\\n\\hline\n0 & 0 & 0 & 0 & 0 & e & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & e & 0 & 0 & 0 & 0 & 0 \\\\\n\\hline\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \n\\end{array}\\right],$$\n\n$$ A_2=\\left[\\begin{array}{cccc|ccc|cc|cc|c}\n0 & f & 0 & 0 & a & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & f & 0 & 0 & a & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & f & 0 & 0 & a & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n\\hline\n0 & d & 0 & 0 & 0 & 0 & 0 & b & 0 & 0 & 0 & 0 \\\\\n0 & 0 & d & 0 & 0 & 0 & 0 & 0 & b & 0 & 0 & 0 \\\\\n0 & 0 & 0 & d & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n\\hline\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c & 0 \\\\\n\\hline\n0 & 0 & 0 & 0 & 0 & e & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & e & 0 & 0 & 0 & 0 & 0 \\\\\n\\hline\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \n\\end{array}\\right] $$\nand\n$$\nA_3= \\left[\\begin{array}{cccc|ccc|cc|cc|c}\n0 & 0 & 0 & 0 & a & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & a & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & a & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n\\hline\n0 & d & 0 & 0 & 0 & 0 & 0 & b & 0 & 0 & 0 & 0 \\\\\n0 & 0 & d & 0 & 0 & 0 & 0 & 0 & b & 0 & 0 & 0 \\\\\n0 & 0 & 0 & d & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n\\hline\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c & 0 \\\\\n\\hline\n0 & 0 & 0 & 0 & 0 & e & 0 & 0 & 0 & 0 & 0 & f \\\\\n0 & 0 & 0 & 0 & 0 & 0 & e & 0 & 0 & 0 & 0 & 0 \\\\\n\\hline\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & g & 0 & 0 & 0 \n\\end{array}\\right],\n$$\n\nwhere $a,b,c,d,e,f,g \\in \\mathbb{F}$ are nonzero.\nThe generic matrices $M_1$, $M_2$ and $M_3$ with the same patterns as $A_1$, $A_2$ and $A_3$\nhave the following digraphs\n \n \\begin{figure}[htb]\n \\begin{center}\n \\includegraphics[height=8.2cm, width=12cm]{fig43221y}\n \\end{center}\n \\end{figure}\n\nIt can be easily seen that $\\sh{A_1}=\\sh{M_1}=(9,1^3)$, $\\sh{A_2}=\\sh{M_2}=(9,2,1)$\nand $\\sh{A_3}=\\sh{M_3}=(9,3)$.\n \\hfill$\\square$\n\\end{example}\n\n\\bigskip\n\n\\begin{example}\\label{ex:3}\nNot all $\\underline{\\lambda}=(\\lambda_1,\\lambda_2,\\ldots,\\lambda_k) \\in \\partition{n}$, where $\\lambda_1=\\nil{{\\mathcal N}_B}$,\nare in $\\partition{{\\mathcal N}_B}$. For $\\sh{B}=(6,4)$ it follows that $\\nil{{\\mathcal N}_B}=6$. It can be shown that\n$(6,3,1) \\notin \\partition{{\\mathcal N}_B}$.\n \\hfill$\\square$\n\\end{example}\n\n \\bigskip\n \n \n \\begin{example}\n Not all partitions in $\\partition{{\\mathcal N}_B}$ can be obtained from a generic matrix corresponding to\n an $({\\mathcal N}_B,A)$-digraph. \n Let $\\sh{B}=(5,3)$. It can be verified (for example, using {\\tt Mathematica}) that \n \\begin{align*}\n \\partition{{\\mathcal N}_B}=&\n \\big\\{(5,3), (5,2,1), (5,1^3), (4^2), (4,2^2), (4,2,1^2), (4,1^4), \\\\\n & \\; (3^2,2), (3^2,1^2), (3,2^2,1), (3,2,1^3), (3,1^5), \\\\\n & \\; (2^4),(2^3,1^2), (2^2,1^4), (2,1^6), (1^8)\\big\\}=\\\\\n = & \\, \\rp{5,3} \\cup \\big\\{(4^2), (4,2^2), (4,2,1^2), (4,1^4), (3^2,1^2), (2^4) \\big\\}\\, .\n \\end{align*}\n One can check that there is no $({\\mathcal N}_B,A)$-digraph $\\Gamma$ with $\\Delta(\\Gamma)$ equal to \n $(4^2)$, $(3^2,1^2)$ or $(2^4)$. However, there are matrices with these shapes in ${\\mathcal N}_B$.\n For example, for\n $$\n A_1=\\left[\\begin{array}{ccccc|ccc}\n 0 & 1 & 0 & 0 & 0 &-1 & 0 & 0 \\\\\n 0 & 0 & 1 & 0 & 0 & 0 &-1 & 0 \\\\\n 0 & 0 & 0 & 1 & 0 & 0 & 0 &-1 \\\\\n 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n\\hline\n 0 & 0 & 1 & 0 & 0 & 0 &-1 & 0 \\\\\n 0 & 0 & 0 & 1 & 0 & 0 & 0 &-1 \\\\\n 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \n \\end{array}\\right]$$\n and\n $$A_2=\\left[\\begin{array}{ccccc|ccc}\n 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 \\\\\n 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 \\\\\n 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 \\\\\n 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n \\hline\n 0 & 0 &-1 & 1 & 0 & 0 &-1 & 1 \\\\\n 0 & 0 & 0 &-1 & 1 & 0 & 0 &-1 \\\\\n 0 & 0 & 0 & 0 &-1 & 0 & 0 & 0 \n \\end{array}\\right]\\, .\n $$\n \n we have $\\sh{A_1}=(2^4)$, $\\sh{A_2}=(3^2,1^2)$. \n Let $M_i$, $i=1,2$, be generic matrices of the same shape as $A_i$. It can be proved, using\n Gansner-Saks Theorem, that $\\sh{M_1}=\\sh{M_2}=(5,3)$.\n \\hfill$\\square$\n\\end{example}\n\n\\bigskip\n\\bigskip\n\n\n\n\n \n \\bigskip\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\n\n\n\\subsection{Electron-End-Cap Electromagnetic Calorimeter: EEMC}\nThe electron-end-cap calorimeter will to cover a dynamic energy range of 0.1--18 GeV for electromagnetic showers of the scattered electron based on e+p Pythia simulations at 18x275 GeV$^2$. \nThe choice of technology and detector dimensions are therefore optimized to provide the optimal performance for this expected energy range.\n\nThe EEMC is a high-resolution ECal designed for precision measurements of the energy of scattered electrons and final-state photons in the electron-going region. \nThe requirements for energy resolution in the backward region is driven by inclusive DIS where precise determination of the scattered electron properties is critical to constrain the event kinematics. \nThe EEMC is designed to address the requirements outlined in the EIC Yellow Report.\nIts baseline design is based on an array of approximately 3000 lead tungsten crystals (PbWO$_{4}$) $2\\times 2\\times 20$~cm$^3$ in size, which correspond to approximately 20~$X\/X_0$ longitudinally and a transverse size equal to the PbWO$_{4}$ Moli\\`ere radius. \nThe PbWO$_{4}$ crystal light yield is in the range of 15 to 25 photo-electrons per MeV, providing an excellent energy resolution of $\\sigma_E\/E\\approx2\\%\/\\sqrt{E} \\oplus 1\\%$ \\cite{Inaba:1994jd,Prokoshkin:1995rd} within a very compact design.\\\\\nThe EEEMCAL Consortium is leading the efforts to further develop the EEMC design concept and has summarized their intentions in an Expression Of Interest in 2021.\nThey have begun to organize activities into mechanical design, scintillator, readout, and software\/simulation among the collaborating institutions. \nPre-design activities of the mechanical support structure commenced in 2021 and a document on mechanical design and integration has been prepared \\cite{eeemcal:design}. \nThe concept is based existing detectors that the team has constructed, and in particular the Neutral Particle Spectrometer at Jefferson Lab. \\\\\n\\begin{figure}[!t]\n \\centering\n \\includegraphics[width=\\linewidth]{figs\/design\/EIC_EEEMCAL_Beampipe_9.png}\n \\caption{The Electron-End-Cap Calorimeter conceptual design and integration with the beampipe and surrounding detectors as prepared by the EEEMCAL Consortium \\cite{eeemcal:design}. \n The EEMC consists only of PbWO$_{4}$ crystals and uses the displayed design concept. \n }\n \\label{fig:eemcal_config}\n \\label{fig:eemcal_clearance}\n\\end{figure}\n\\Figure{fig:eemcal_config} shows an overview of the different components of the EEMC as prepared by the EEEMCAL Consortium \\cite{eeemcal:design}. \nIt has four main parts: the detector (PbWO$_{4}$ crystals), the mechanical structure (internal and external), cooling, and electronics (SiPM and cables). \nWith crystal dimensions of $2\\times2\\times20$~cm$^3$, a density of 8.28~g\/cm$^3$, and a mass of 0.6624~kg per crystal the total weight of the EEMC is slightly more than two metric tons. \nThe crystals are aligned and separated using carbon plates of thickness 0.5~mm. \nThe configuration for the first ring of PbWO$_{4}$ crystals depends on the final design of the beam pipe. \nIts minimum diameter will be on the order of 22.5~cm with an additional clearance gap. \nAn additional support and cooling structure with a maximum thickness of 5~mm will be needed to support the crystals directly above the beam pipe.\nFor which detailed calculations will have to be carried out once the final beam pipe design is available.\nThe EEMC is located inside the universal support frame, which also houses the Detection of Internally Reflected Cherenkov light (DIRC) detector \\cite{ecce-paper-det-2022-01}, and covers the pseudorapdity region of $-3.4<\\eta<-1.5$. \nThe main constraints for its acceptance are imposed by the surrounding detector systems and passive materials, as seen in \\Fig{fig:eemcal_config}.\nThe integration of the EEMC into the frame is only possible if the beam pipe is removed, which implies that the flange must be disconnected. \nTo improve the inner diameter of the EEMC and to improve the acceptance up to $-3.7<\\eta<-1.5$, an inner calorimeter is being considered. \nThis option also requires the modification of the overall structure of the EEMC to ensure no significant gaps in scattered electron detection between the electron-end-cap and barrel.\nOverall, the inner diameter of the EEMC will depend on the design of the beam pipe, and in particular the angle between the electron and the hadron tube.\\\\\nCurrently, the EEMC readout is based on silicon photomultipliers (SiPMs) of pixel sizes of 10$\\mu$m or 15$\\mu$m and a photosensitive area of $3\\times3$~mm$^2$. \nThere are two configuration options: 4 SiPM per crystal or 16 SiPM per crystal. \nSince a mechanical structure is required for mounting the PCBs, its width in turn will determine the positioning of the SiPMs.\nAssuming a machined grid with a width of about 5~mm the PCBs can be mounted with small screws. \\\\\nPbWO$_{4}$ crystals are sensitive to temperature changes with a variation of 2\\%\/$^\\circ$C in light output. \nThus, the specification is to keep the crystal temperature stable within $\\pm$0.1 $^\\circ$C.\nTo ensure this stability the additional heat generated by the electronics needs to be removed and the following cooling structures are being considered.\nAs internal cooling structure several machined copper blocks with internal coolant circulation will be used around the beam pipe.\nTo reduce the spatial extend support structures the EEEMC consortium is moreover planning to use cooling plates in between the readout cables which are linked to the support structure surrounding the EEMC with tubes.\nThis system is composed of 12 plates with a 5-8~mm spacing in which water can be circulated. \nThe cooling near the crystals will likely not be enough to meet specification.\nThese challenges could be overcome by outside cooling with standard cooling blocks with airflow in front of the electronics or additional cooling added at the back of the assembly. \nThe main constraint is the space available in the electron end-cap. \\\\\nThe mechanical integration of the EEMC presently envisions that the detector is assembled in its own support structure, mounted on a platform, and then inserted into the universal support frame. \nThe detailed steps and main points of the assembly are described in Ref.~\\cite{eeemcal:design}. \nThe mechanical integration starts when the assembly is complete. \nThe platform is adjusted on rails with an additional support to link the support to the detector. \nThe platform is removed once the EEMC is mounted in the universal support frame. \nClearances of at least 5~mm on all sides between the EEMC and the universal support frame are required to perform maintenance without lifting the detector. \n\\begin{figure*}[ht!]\n \\centering\n \\includegraphics[height=0.22\\textheight]{figs\/design\/tower_bemc_unframed.png}\n \\includegraphics[height=0.22\\textheight]{figs\/design\/becal_eta.png}\n \\includegraphics[height=0.22\\textheight]{figs\/design\/becal_phi.png}\n \\caption{Left: Single BEMC tower as implemented in the GEANT4 simulations. \n Middle: BEMC projective tower layout in $\\eta$. \n The towers are centered at $z = -10$~cm. \n Right: BEMC layout layout as a function of $\\varphi$ for one $\\eta$ slice.\n The towers are tilted in $\\varphi$ by $10^\\circ$ to avoid channeling in the gaps between adjacent towers.\n }\n \\label{fig:becal_eta}\n \\label{fig:becal_phi}\n \\label{fig:becal_tower}\n\\end{figure*}\n\n\\subsection{Barrel}\nBased on Pythia simulations of e+p collisions at 18x275 GeV$^2$, the expected energy range of particles at mid-rapidity is 0.1--50 GeV in high $Q^2$ events. \nThe ECCE detector therefore requires calorimeters that can cover these expected energies for electromagnetic and hadronic shower reconstruction.\n\\paragraph{Barrel Electromagnetic Calorimeter: BEMC} \\mbox{ }\\\\\nThe barrel electromagnetic calorimeter is designed to cover the central region of the detector ($-1.72 < \\eta <1.31$). \nIts total length along the $z$-axis is 584~cm and the detector is fully contained within solenoid magnet, but positioned at a larger radial position than the DIRC detector.\nThe absolute radial position of the calorimeter is $851.1$): & $6.6\\times6.6$ cm\\\\\n \n $\\eta$ projectivity point & $z=-10$ cm \\\\\n $\\varphi$ projectivity tilt & $10^\\circ$ \\\\\n Sampling fraction & 0.97\\\\\n Tower depth & $X\/X_0\\approx16.0$ \\\\\n Moli\\`ere radius & $R_\\mathrm{M}=3.58$ cm\\\\\n \\bottomrule\n \\end{tabular}\n \\caption{Design parameters for the barrel electromagnetic calorimeter BEMC.}\n \\label{tab:becalpar}\n\\end{table}\nThe layout for a single tower around $\\eta = 0$ is shown in \\Fig{fig:becal_tower}~(left).\nAll towers currently have an inner size of $4 \\times 4$~cm$^2$ and the same length of 45.5~cm, which corresponds to approximately 16 $X\/X_0$.\nHowever, their outer face dimension varies from $5^2$ to $6.6^2$~cm$^2$ depending on their position in $|\\eta|$.\nIn addition, the considered SciGlass towers have a Moli\\`ere radius of 3.58~cm, which is approximately double the transverse tower size.\nEach tower is composed of a SciGlass core, surrounded by a 1~mm carbon fiber enclosure. \nThe electronics are currently modeled by Kapton, SiO$_{2}$ and carbon fiber layers in the outer part of the blocks. \nThe SciGlass block length is optimized to contain at least $95\\%$ of the energy of a 10 GeV electron, whilst still fitting into the BABAR 1.5T magnet with at most an inner radius of 80~cm and at least 8~cm space for the electronics and support structure.\nThe electron energy mentioned above corresponds to the average scattered electron energy in the BEMC acceptance. \nConstraining the BEMC to not stretch further into the detector allows for more space for other PID and tracking detectors which are necessary for electron, pion, kaon and proton separation. \nIn particular, towards the electron end cap (negative $\\eta$) it could be studied in the future, whether the tower depth could be increased up to 60~cm for higher $|\\eta|$ to decrease the energy leakage for high energetic electrons, which are more probable in this region.\nHere the projective design allows for such an extension at least for parts of the calorimeter. \n\n\\paragraph{Barrel Hadronic Calorimeter: IHCAL \\& OHCAL}\\mbox{ }\\\\\n\\begin{figure*}[!t]\n \\centering\n \\includegraphics[width=0.55\\linewidth]{figs\/design\/OuterHCalSectorWithTiles.png}\n \\includegraphics[width=0.35\\linewidth]{figs\/design\/OuterHcal-Dimensions-v3-2-crop.pdf}\n \\caption{Left: Scintillator tiles in a layer of the OHCAL. \n Right: Transverse cutaway view of an OHCAL module, showing the tilted tapered absorber plates.\n Light collection and cabling is on the outer radius at the top of the drawing.}\n \\label{fig:inner_tile_pattern}\n \\label{fig:hcal-6}\n\\end{figure*}\nThe Outer Hadronic Calorimeter (OHCAL) will be reused from the sPHENIX HCal~\\cite{sPHENIX:2017lqb}, which instruments the large steel-based barrel flux return of the BABAR magnet.\nThe Inner Hadronic Calorimeter (IHCAL), as currently implemented in ECCE, is very similar in design to the sPHENIX inner HCAL in that it instruments the support for the barrel HCal to provide an additional longitudinal segment of hadronic calorimetry. \nThe IHCAL provides useful data for overall calibration of the combined calorimeter system. \\\\\nIn the following, the construction of the scintillating tiles used in the outer and inner HCals is described, followed by a mechanical description of each calorimeter system.\\\\\n\\begin{table}[!t]\n \\centering\n \\begin{tabular}{lr}\n \\toprule\n Parameter & Value \\\\\n \\midrule\n Inner radius (envelope) & 1820 mm\\\\\n Outer radius (envelope) & 2700 mm\\\\\n Length (envelope) & 6316 mm\\\\\n Material & 1020 steel \\\\\n \\# towers in azimuth ($\\Delta \\varphi$) & 64 \\\\\n \\# tiles per tower & 5 \\\\\n \\# towers in pseudorapidity ($\\Delta \\eta$) & 24 \\\\\n \\# electronic channels (towers) & $64 \\times 24 = 1536$ \\\\\n \\# optical devices (SiPMs) & 5 $\\times$ 1536 = 7680 \\\\\n \\# modules (azimuthal slices) & 32 \\\\\n \\# towers per module & $2 \\times 24 = 48$ \\\\\n Total \\# absorber plates & $5 \\times 64 = 320$ \\\\\n Tilt angle (relative to radius) & 12$^{\\circ}$ \\\\\n Absorber plate thickness at inner radius & 10.2 mm\\\\\n Absorber plate thickness at outer radius & 14.7 mm\\\\\n Gap thickness & 8.5 mm\\\\\n Scintillator thickness & 7 mm\\\\\n Module weight & 12247 kg\\\\\n Sampling fraction & 0.035\\\\\n Calorimeter depth & 4.0$\\lambda\/\\lambda_0$ \\\\\n Moli\\`ere radius $R_M$ for $\\pi^\\pm$ & 14.4 cm\\\\\n \\bottomrule\n \\end{tabular}\n \\caption{Design parameters for the Outer Hadronic Calorimeter (OHCAL).}\n \\label{tab:ohcalpar}\n\\end{table}\n\\begin{table}[!t]\n \\centering\n \\begin{tabular}{lr}\n \\toprule\n Parameter & Value \\\\\n \\midrule\n Inner radius (envelope) & 1350 mm\\\\\n Outer radius (envelope) & 1385 mm\\\\\n Material & 310 stainless steel \\\\\n \\# towers in azimuth ($\\Delta \\varphi$) & 64 \\\\\n \\# towers per module & $2 \\times (12+15) = 56$ \\\\\n \\# tiles per tower & 4 \\\\\n \\# towers in pseudorapidity ($\\eta > 0$) & 24 \\\\\n \\# towers in pseudorapidity ($\\eta < 0$) & 30 \\\\\n \\# electronic channels (towers) & $64 \\times 27 = 1728$ \\\\\n \\# optical devices (SiPMs) & 4 $\\times$ 1728 = 6912 \\\\\n Tilt angle (relative to radius) & 32 $^{\\circ}$ \\\\\n Absorber plate thickness & 13 mm\\\\\n Gap thickness & 8.5 mm\\\\\n Scintillator thickness & 7 mm\\\\\n \\# modules (azimuthal slices) & 32 \\\\\n Sampling fraction & 0.059\\\\\n Calorimeter depth & 0.17$\\lambda\/\\lambda_0$ \\\\\n \\bottomrule\n \\end{tabular}\n \\caption{Design parameters for the Inner Hadronic Calorimeter (instrumented frame) for ECCE.}\n \\label{tab:ihcalpar}\n\\end{table}\nThe basic calorimeter concept for the IHCAL\/OHCAL is a sampling calorimeter with absorber plates tilted in the radial direction.\nThis design provides more uniform sampling in azimuth and provides information on the longitudinal shower development. \nThe current design uses tapered plates for the OHCAL and non-tapered plates for the IHCAL.\nBased on detailed studies, this design choice lowers the IHCAL machining cost without decreasing its performance.\nExtruded tiles of plastic scintillator with an embedded wavelength shifting fiber are interspersed between the absorber plates and read out at the outer radius with SiPMs.\nThe tilt angle is chosen so that a radial track from the center of the interaction region traverses at least four scintillator tiles. \nEach tile is read out by a single SiPM, and the analog signal from each tile in a tower (five for the OHCAL, four for the IHCAL) are ganged to a single preamplifier channel to form a calorimeter tower. \nTiles are divided in slices of $\\eta$ so that the overall segmentation is $\\Delta \\eta \\times \\Delta \\varphi \\approx 0.1 \\times 0.1$.\\\\\nThe scintillating tiles are similar to the design of the scintillators for the T2K experiment by the INR group (Troitzk, Russia) who designed and built 875~mm long scintillation tiles with a serpentine wavelength shifting fiber readout~\\cite{Izmaylov:2009jq}.\nSimilar extruded scintillator tiles were also developed by the MINOS experiment. \nThe properties of the HCal scintillating tiles and of the WLS fibers are detailed in Ref.~\\cite{sPHENIX:2017lqb}.\nThe Kuraray single clad fiber is chosen due to its flexibility and longevity, which are critical in the geometry with multiple fiber bends.\\\\\nThe OHCAL is north-south symmetric and requires 24 tiles along the $\\eta$ direction, whereas the IHCAL is asymmetric and has 12 towers in the forward direction and 15 towers in the backward direction. \nThe OHCAL design therefore requires 12 different shapes of tiles for each longitudinal segment.\n\\Figure{fig:inner_tile_pattern} shows the tile and embedded fiber pattern for the OHCAL.\\\\\nThe major components of the OHCAL are tapered steel absorber plates and 7680 scintillating tiles which are read out with SiPMs along the outer radius of the detector.\nThe detector consists of 32 modules, which are wedge-shaped sectors containing 2 towers in $\\varphi$ and 24 towers in $\\eta$ equipped with SiPM sensors, preamplifiers, and cables carrying the differential output of the preamplifiers to the digitizer system on the floor and upper platform of the detector.\nEach module comprises nine full-thickness absorber plates and two half-thickness absorber plates, so that as the modules are stacked, adjoining half-thickness absorber plates have the same thickness as the full-thickness absorber plates.\nThe tilt angle is chosen to be 12 degrees relative to the radius, corresponding to the geometry required for a ray from the vertex to cross four scintillator tiles.\nTable~\\ref{tab:ohcalpar} summarizes the major design parameters of the OHCAL, which are illustrated in \\Figure{fig:hcal-6}.\nSince the OHCAL will serve as the flux return of the solenoid, the absorber plates are single, long plates running along the field direction.\nThe IHCAL occupies a radial envelope bounded by a 50~mm clearance inside the solenoid cryostat and the outer radius of the BEMC.\nThe inner radius provides support for the BEMC and the HCal, while the end of the structure carries load to the OHCAL.\\\\\nTable~\\ref{tab:ihcalpar} shows the basic mechanical parameters of the IHCAL reference design.\nThe detector is designed to be built in 32 modules, which are wedge-shaped sectors comprising 8 gaps with 7 full-thickness plates and 2 half-thickness plates (so that as the modules are stacked, adjoining half-thickness plates have the same thickness as the full-thickness plates).\nThe modules contain 2 towers in $\\varphi$ and 27 towers in $\\eta$ equipped with SiPM sensors, preamplifiers, and cables carrying the differential output of the preamplifiers to the digitizer system on the floor and upper platform of the detector.\nThe instrumentation consists of 6912 scintillating tiles and optical devices, 1728 preamplifiers, and cabling.\n\n\\subsection{Hadron-End-Cap}\nWe envision the forward calorimeter system as an integrated ECal and HCal, where the installation units, where appropriate, are constructed in a common casing. \nThese so-called modules consist of an electromagnetic calorimeter segment in the front which is part of the forward EMCal (FEMC) followed by a HCal segment which is part of the longitudinally separated HCal (LFHCAL). \nIn between these segments a read-out section is foreseen for the ECal. \nThe modules of up to 4 different sizes will be installed in half shells surrounding the beam pipe, which are movable on steel trolleys to give access to the inner detectors in the barrel in the hadron going direction. \nEach of these trolley should carry about 150 metric tons of weight.\nThis integrated ECal and HCal design reduces the dead material in the detector acceptance and allows for an easier installation in the experimental hall.\nThis implies that the construction of the modules has to happen in the same location to reduce shipping and assembly costs.\nIn the following, details on the FEMC will be discussed, followed by the design considerations and plans for the longitudinally separated HCal. \\\\\n\\begin{table}[t!]\n \\centering\n \\small\n \\begin{tabular}{lll}\n \\toprule\n parameter &\\textbf{FEMC} &\\textbf{LFHCAL}\\\\\n \\midrule\n inner radius (envelope) & 17 cm & 17 cm\\\\\n outer radius (envelope) & 170 cm & 270 cm\\\\\n $\\eta$ acceptance & $1.3 < \\eta < 3.5$ & $1.2 < \\eta < 3.5$\\\\\n tower information \\\\\n \\hspace{1em} x, y ($R <$\/$> 0.8$ m) & 1 cm\/ 1.65 cm & 5 cm\\\\\n \\hspace{1em} z (active depth) & 37.5 cm & 140 cm\\\\\n \\hspace{1em} z read-out & 5 cm & 20 cm\\\\\n \\hspace{1em} $\\#$ scintillor plates & 66 (0.4 cm each) & 70 (0.4 cm each)\\\\\n \\hspace{1em} $\\#$ aborber sheets & 66 (0.16 cm Pb) & 60 (1.6 cm steel)\\\\\n & & 10 (1.6 cm tungsten)\\\\\n \\hspace{1em} weight & $\\sim 6.4$ kg & $\\sim 30.6$ kg\\\\\n \\hspace{1em} radiation lengths & 18.5 $X\/X_0$ & - \\\\\n \\hspace{1em} interaction lengths & 1.0 $\\lambda\/\\lambda_0$ & 6.9 $\\lambda\/\\lambda_0$\\\\\n Moli\\`ere radius $R_M$ & 5.2 cm (e$^\\pm$ shower) & 21.1 cm ($\\pi^\\pm$ shower)\\\\\n Sampling fraction $f$ & 0.220 & 0.040\\\\\n $\\#$ towers (inner\/outer) & 19,200\/ 34,416 & 9040\\\\\n $\\#$ read-out channels & 53,616 & 7 x 9,040 = 63,280\\\\\n \\bottomrule\n \\end{tabular}\n \\caption{Overview of the calorimeter design properties for the FEMC and the LFHCAL.}\n \\label{tab:fwdcaloproperties}\n \\centering\n \\small\n \\begin{tabular}{lll}\n \\toprule \n \\textbf{Assembly Module Type} & \\textbf{\\# modules} \\\\\n \\midrule\n 8 LFHCAL tower modules (8M) & 1091 (total)\\\\\n \\hspace{1em} no FEMC towers in front & 538\\\\\n \\hspace{1em} 200 FEMC towers (inner) & 87\\\\\n \\hspace{1em} 72 FEMC towers (outer) & 466\\\\\n 4 LFHCAL tower modules (4M) & 76 (total)\\\\\n \\hspace{1em} no FEMC towers in front & 36\\\\\n \\hspace{1em} 100 FEMC towers (inner) & 16\\\\\n \\hspace{1em} 36 FEMC towers (outer) & 24\\\\\n 2 LFHCAL tower modules (2M) & 2 (total)\\\\\n \\hspace{1em} 50 FEMC towers (inner) & 2\\\\\n 1 LFHCAL tower modules (1M) & 4 (total)\\\\\n \\hspace{1em} 25 FEMC towers (inner) & 4\\\\\n \\bottomrule\n \\end{tabular}\n \\caption{Number of assembly modules for the full combined FEMC and LFHCAL detector.}\n \\label{tab:fwdcalomodules}\n\\end{table}\nBoth detector systems need to be able to handle the expected energies of incoming particles up to 150 GeV, based on simulated Pythia events for e+p collisions at 18x275 GeV$^2$. \nDue to the asymmetric collision system, these calorimeters are therefore focused strongly on high energetic particle shower containment while still providing good energy resolution down to lower energies.\n\\begin{figure*}[t]\n \\centering\n \\includegraphics[width=\\linewidth]{figs\/design\/LFHCAL_FEMC_compilation_propsal_modified.png} \n \\caption{Design pictures of the forward calorimeter assembly (left), 8-tower module design (top right) and single scintillator plates of the LFHCAL (bottom middle) and FEMC (bottom right) for an 8M tower module with embedded wavelength shifting fibers.}\n \\label{fig:FCALdesign}\n\\end{figure*}\n\n\\paragraph{Hadron End-Cap Electromagnetic Calorimeter: FEMC}\\mbox{ }\\\\\n\\label{sec:FEMC}\nThe forward ECal (FEMC) is a Pb-Scintillator shashlik calorimeter. \nIt is placed at a distance of $z=3.07$ m from the interaction point in the hadron-going direction after the tracking and particle identification detectors. \nThe detector is made up of two half disks with a radius of about 1.7m. \nThe calorimeter is based on traditional Pb-Scint-Shashlik calorimeter designs like they have previously been used in ALICE, STAR and PHENIX.\nHowever, it employs more modern techniques for the readout and the scintillation tile separation. \\\\\nIts towers have an active depth of 37.5~cm with additional space for the readout of about 5~cm.\nEach tower consists of 66 layers of alternating 0.16~cm Pb sheets and 0.4~cm scintillator material, as listed in \\Table{tab:fwdcaloproperties}.\nDue to the high occupancy of the detector at large pseudorapities and the collimation of the particles in this area in physical space, the tower size varies depending on the radial position with respect to the beam axis.\nTowers which are close to the beam pipe ($R < 0.8$~m) have an approximate tower size of $1 \\times 1 \\times 37.5$~cm$^3$. \nFor the outer radii this granularity is not necessary and thus the size is increased to $1.65 \\times 1.65 \\times 37.5$~cm$^3$.\nThese numbers are intentionally well below the Moli\\`ere radius of $R_m=5.18$~cm, thus showers will spread transversely over multiple towers.\nIn order to collect the light produced in the scintillator tiles, each scintillator and Pb-plate is pierced by four 0.2~mm diameter wavelength shifting fibers. \nThese fibers are used to collect the light generated in the scintillators across all 66 layers. \nAll four fibers are read out together by a single SiPM. \\\\\nMultiple towers are contained in modules of either $20 \\times 10$~cm$^2$ (8M), $10 \\times 10$~cm$^2$ (4M), $5 \\times 10$~cm$^2$ (2M) or $5 \\times 5$~cm$^2$ (1M) in size. \nThese module sizes match the 8-, 4-, 2- and 1-tower modules of the LFHCAL with which they share a 1.5~mm thin steel enclosure. \nDepending on the radial position, the FEMC packs 72 or 200 read-out towers in an 8M module. \nDue to the integration of the FEMC towers in the LFHCAL modules, the combined ECal and HCal modules are about 2.05 m long. \nA detailed drawing of the 8M inner scintillator tile design for the FEMC can be found in \\Figure{fig:FCALdesign} (bottom right). \nThe full 8M tile is made out of one piece.\nIn order to separate the light produced in different segments of the 8M-tile, the tile surface is subdivided into $1\\times1$~cm$^2$ readout segments by CNC cutting or edging into the scintillator using a laser.\nThese 0.37~mm deep gaps (about 92\\% of the tile thickness) are then refilled with a mixture of epoxy and Titanium-oxide (TiO$_{2}$) in order to reduce the light cross talk among different towers. \nThe 4 fibers per tower are combined in a small light-collecting prism, which is directly attached to the SiPM with an effective photosensitive area of 9-16~mm$^2$ (ie. Hamamatsu S14160-3050HS). \nThese SiPMs are most sensitive around wavelengths of 450~nm, thus the wave length shifting fibers have to be chosen accordingly to peak in a similar region.\\\\\nThe first signal processing happens after the ECal part of the module within the 5~cm space currently assigned for the FEMC read-out, realized using CMS HGCROC chips mounted on custom PCBs \\cite{Thienpont:2020wau}, which can simultaneously process 72 channels. \nThe signals are then transmitted via fiber optic cables to the end of the module for further processing.\\\\\nA first full mechanical design for the joint LFHCAL and FEMC inner 8M module can be seen in \\Figure{fig:FCALdesign}. \nAdditionally, a first full illustration of a half shell is shown. \nThe higher granular 8M and 4M FEMC-LFHCAL modules are indicated in green and red respectively, while the yellow and dark blue towers show the lower granularity 8M and 4M FEMC-LFHCAL modules. \nThe lighter blue and orange modules reflect the modules only containing LFHCAL towers.\\\\\nThe majority of the FEMC is build of 8M modules, supplemented by 4M, 2M and 1M modules as outlined in \\Table{tab:fwdcalomodules} to come closer to the beam pipe and allow for a vertical separation of the two half shells. \nThe entire detector consists of approximately 53600 readout channels and provides a measurement of the energy of photons and electrons created in the collision going in the hadron-going (forward) direction. \n\n\\paragraph{Hadron-End-Cap Hadronic Calorimeter: LFHCAL}\\mbox{ }\\\\\n\\label{sec:LFHCAL}\nThe longitudinally separated forward HCal (LFHCAL) is a Steel-Tungsten-Scintillator calorimeter. \nThe initial idea is based on the PSD calorimeter employed in the forward direction for the NA61\/SHINE experiment \\cite{Guber:109059}, but it has been extensively modified to meet the desired physics performance laid out in the Yellow Report. \nThis longitudinally separated HCal is positioned after the tracking and PID detectors at $z=3.28$m from the center of the detector and is made up of two half disks with a radius of about 2.6m. \\\\\nThe LFHCAL towers have an active depth of $\\Delta z =1.4$ m with an additional space for the readout of about 20-30~cm depending on their radial position, as summarized in \\Table{tab:fwdcaloproperties}. \nEach tower consists of 70 layers with alternating 1.6~cm absorber and 0.4~cm scintillator material and has transverse dimensions of $5 \\times 5$~cm$^2$.\nFor the first 60 layers the absorber material is steel, while the last 10 layers serve as tail catcher and are thus made of tungsten to maximize the interaction length within the available space. \\\\\nIn each scintillator, a loop of wavelength shifting fiber is embedded, as can be seen in \\Figure{fig:FCALdesign} (bottom center). \nTen consecutive fibers in a tower are read out together by a single SiPM, leading to 7 samples at different depth per tower. \nThe towers are constructed in units of 8-, 4-, 2- and 1-tower modules to ease the construction and to reduce the dead space between the towers. \nSimilar to the FEMC, the scintillator tiles in the larger modules are made out of one piece and then separated by gaps refilled with epoxy and Titanium oxide to reduce light cross-talk among the different readout towers. \nThe wavelength shifting fibers running on the sides of the towers are grouped early on according to their readout unit and separated by thin plastic pieces over the full length.\nThe corresponding fiber bundles are indicated in \\Figure{fig:FCALdesign} by different colors. \nThey terminate in one common light collector, which is directly attached to a SiPM with an effective photosensitive area of 9-16~mm$^2$ (ie.\\ Hamamatsu S14160-3050HS).\nThese 7 SiPMs per tower are then read out by a common readout design between the FEMC and LFHCAL based on the CMS HGCROC chips.\nAlternatively, a common readout board which could be used for nearly all ECCE calorimeters is being pursued.\nThe entire detector consists of 63280 readout channels grouped in 9040 read-out towers and provides a measurement of the energy of hadronic particles created in the collision in the hadron-going (forward) direction.\\\\\nThe majority of the calorimeter is built of 8-tower modules ($\\sim$1091) which are stacked in the support frame using a lego-like system for alignment and internal stability.\nThe remaining module sizes are necessary to fill the gaps at the edges and around the beam pipe to allow for maximum coverage. \nThe absorber plates in the modules are held to their frame by 4 screws each. To leave space for the read-out fibers, the steel and scintillator plates are not entirely square but equipped with 1.25~mm notches, creating the fiber channels on the sides, as can be seen in \\Figure{fig:FCALdesign} (bottom center) for a scintillator plate. \nIn order to protect the fragile fibers, the notched fiber channels are covered by 0.5~mm thin steel plates after module installation and testing\nFor internal alignment we rely on the usage of 1-2~cm steel pins in the LFHCAL part which are directly anchored to the steel or tungsten absorber plates. \nConsequently, the modules are self-supporting within the outer support frame. \nThe support frame for the half disks is arranged on rails which allows the HCal and ECal to slide out to the sides and gives access to the inner detectors.\nIn addition, the steel in the LFHCAL serves as flux return for the central 1.5T BABAR magnet.\nAs a consequence, a significant force is exerted on the calorimeter, which needs to be compensated for by the frame and internal support structure. \n\n\\section{Calorimeter Performance}\n\\label{sec:performance}\nThe ECCE electromagnetic and hadronic calorimeters are designed to meet the criteria outlined in the Yellow Report.\nIn the following, the expected performance of the different systems is presented based on standalone and full detector GEANT4 simulations.\n\\begin{figure}[t]\n \\includegraphics[width=\\linewidth]{figs\/analysis\/clusterizationEIC_paper.pdf}\n \\caption{Clusterization algorithms visualized on an example energy deposit from two particles ($E_1^\\mathrm{true}=6$ GeV and $E_2^\\mathrm{true}=4$ GeV) in calorimeter towers of arbitrary size. \n Presented are the AA (Aggregate-All) and the MA (Modified-Aggregation) clusterizers.\n The found clusters are outlined in color and their reconstructed energy is indicated in the figure. \n The same seed and aggregation energy thresholds are assumed for both algorithms in this example. }\n \\label{fig:clusterizersEIC}\n\\end{figure}\n \n\\subsection{Clusterization}\n The energy deposit from an electromagnetic or hadronic shower is generally spread over multiple towers.\n The magnitude of this effect depends on the tower size relative to the Moli\\`ere radius ($R_M$) of the material used and is more prominent for hadronic showers.\n The Moli\\`ere radius is defined as the radius in which 90\\% of the shower energy is contained, where electron-induced showers are used for the ECals and charged pion-induced showers are used for the HCals.\n Since $R_M$ is in all cases larger than the individual tower sizes in the different calorimeters, it becomes apparent that the full shower can only be reconstructed when the information of multiple towers is combined.\n Different reconstruction algorithms can be employed in order to group towers containing energy deposits into so-called clusters, which are the main objects used in physics analyses.\n The performance of these algorithms mostly depends on the calorimeter occupancy for a given event. \n While showers from single electromagnetic particles are mostly trivial to reconstruct, a significant challenge is posed by overlapping particle showers, for example in a jet or from high energetic neutral meson decays. \n In the latter case, the decay photon showers, e.g. from $\\pi^0\\rightarrow\\gamma\\gamma$, can not be separated within the calorimeter granularity above a certain particle energy due to the decay kinematics.\n Thus, extensive studies were performed to increase the separation power between single and multi-particle showers and to absorb as much of the deposited energy as possible during the so-called clusterization procedure.\n This procedure always starts with the highest energetic tower in the calorimeter, which is required to contain an energy deposit above a seed energy threshold ($E_\\mathrm{seed}$).\n \\begin{figure}[t!]\n \\centering\n \n \\includegraphics[width=0.48\\textwidth]{figs\/analysis\/NTowAndMeanClus_Paper.pdf}\n \\caption{Mean number of clusters per generated particle (a) and average number of towers aggregated within a cluster (b) as a function of generated particle energy using the MA clusterizer for the different ECCE calorimeters. }\n \\label{fig:ntowandncluspartFEMC}\n \\end{figure}\n \\begin{figure*}[t!]\n \\centering\n \n \\includegraphics[width=\\textwidth]{figs\/analysis\/ClusRecEffEta_E+HCals_PaperPlotAlt.pdf}\\\\\n \\caption{Cluster reconstruction efficiencies in the EEMC, BEMC and FEMC for electrons (a--c) and in the OHCAL and LFHCAL for charged pions (b and c) reconstructed with the MA-Clusterizer. \n The efficiencies are calculated according to Equation~\\ref{eq:caloeffi}.}\n \\label{fig:clusterreceffiEBF}\n \\label{fig:clusterreceffiHCLF}\n \\end{figure*}\n Additional neighboring towers are added to the cluster if their energy exceeds a certain aggregation threshold ($E_\\mathrm{agg}$).\n The thresholds ($E_\\mathrm{seed}$ and $E_\\mathrm{agg}$) for the different ECals and HCal have been optimized to reduce false seeding from minimum ionizing particles and to suppress noise during aggregation.\n Their values are tower size and calorimeter type dependent, with approximate values of $E_\\mathrm{seed}^\\mathrm{ECal} = 100$ MeV and $E_\\mathrm{agg}^\\mathrm{ECal}=$ 5--10 MeV or $E_\\mathrm{seed}^\\mathrm{HCal} =$ 100--500 MeV and $E_\\mathrm{agg}^\\mathrm{HCal}=$ 5--100 MeV for the ECals or HCals, respectively.\n Two main algorithms have been explored for the cluster reconstruction: the so-called aggregate-all (AA) and modified-aggregation (MA) clusterizers. \n \n The AA clusterizer associates all towers sharing a common side with already aggregated towers in the cluster and only stops the aggregation when no further tower above $E_\\mathrm{agg}$ can be found.\n At this point, the already aggregated towers are removed from the sample and a new seeding starting from the next highest energetic tower is performed.\n Since this approach can aggregate energy deposits from multiple particles depending on the occupancy, a subsequent splitting of the cluster should be performed based on the number of maxima found in the energy distribution.\n This cluster splitting procedure is necessary when AA clusters are meant to be used for single particle analyses.\n \n \n \n The MA clusterizer works similar to the AA clusterizer, however the algorithm stops when a neighboring tower with larger energy than the already aggregated tower is found.\n It also aggregates towers that share a common corner, thus a $3 \\times 3$ tower window around each already aggregated tower is inspected.\n This algorithm is preferred for the reconstruction of hadronic showers in high granularity calorimeters, since the energy deposits can fragment over a large amount of towers.\n \\Figure{fig:clusterizersEIC} shows these algorithms applied to an example energy deposit in a calorimeter, where different clusters are reconstructed based on the various aggregation conditions.\n The MA clusterizer is also the only clusterizer employed in the LFHCAL cluster reconstruction due to the additional z-segmentation of the calorimeter.\n For this, the MA clusterizer also allows the inclusion of neighboring towers in z-direction sharing an edge or corner with already aggregated towers.\\\\\n As can be seen in \\Fig{fig:ntowandncluspartFEMC}, the various ECCE calorimeters show visible differences in the average number of towers they aggregate per MA-based cluster.\n In addition, the top panel of the same figure shows the mean number of clusters per generated particle, which is approximately one at low energies for the MA clusterizer, but reaches up to two clusters per particle at higher energies for the LFHCAL and FEMC.\\\\ \n Overall it was found that the MA clusterizer performs slightly better than the AA clusterizer for all calorimeters and especially in events with a higher occupancy in the different detectors.\n The MA clusterizer is therefore chosen as default for the following detector performance studies.\\\\\n An important property of any clusterization algorithm and calorimeter is the efficiency with which a cluster can be reconstructed for any given particle.\n \\Fig{fig:clusterreceffiEBF} shows the reconstruction efficiencies for electrons and charged pions for the different ECals and HCal as a function of generated particle pseudorapidity calculated according to Equation~\\ref{eq:caloeffi}.\n \\begin{equation}\n \\epsilon \\cdot a= \\frac{N_{\\mathrm{clus,Ntow}>1} \\text{ in calorimeter}}{N_\\text{MC gen. particles} \\text{ in acceptance}}\n \\label{eq:caloeffi}\n \\end{equation}\n In the calculation, only clusters formed according to the seeding and aggregation thresholds are used and additionally required to be made of more than a single tower.\n Furthermore, only one cluster per generated particle is considered for the calculation of $\\epsilon \\cdot a$ to avoid counting multiple clusters of a single particle (e.g. due to an induced pre-shower).\n The latter requirement is necessary to reject secondary low energy clusters or showers that are not contained in the calorimeter (e.g.\\ on the outer and\/or inner edges).\n The efficiencies show that at low energies, the seed and aggregation thresholds decrease the reconstruction efficiency.\n Moreover, an edge effect at low and high pseudorapidity (e.g. strong shower leakage) can be observed even at higher energies also reflecting a small remaining effect from acceptance losses in the calorimeters mostly due to deflection of the original particles in the inner detectors. \n Additionally, the non-uniform $\\varphi$-coverage of the LFHCAL and FEMC is clearly visible, yielding a lower average efficiency for $\\eta > 3$. \n \\begin{figure*}[!t]\n \\centering\n \n \\includegraphics[width=\\textwidth]{figs\/analysis\/EReso_PaperPlot_ECals.pdf}\\\\ \n \\includegraphics[width=0.66\\textwidth]{figs\/analysis\/EReso_PaperPlot_HCals.pdf} \n \\caption{Energy resolution for different particles generated in single particle simulations at fixed energies as measured by the electromagnetic calorimeters EEMC, BEMC, and FEMC (a--c) and the hadronic calorimeters OHCAL and LFHCAL (d and e). }\n \\label{fig:eresodistributioneemcbecalfemc}\n \\end{figure*}\n\n\\subsection{Energy resolution}\n The energy resolution for the ECals and HCals is evaluated based on single particle simulations for photons, electrons, pions and protons generated for $0.2 < E < 30 (50)$ GeV. \n For these studies the reconstructed energy deposits in the towers are combined into clusters using the MA clusterizer with the aforementioned seed and aggregation energy settings for each calorimeter. \n The energy scale of the calorimeters is calibrated such that in simulations without material in front of the calorimeter the reconstructed electron energy over the generated energy is approximately unity. \n Thus, this calibration corrects the ECals and HCal to approximately the same energy scale.\n No $\\eta$ dependent corrections for the energy response are introduced so far.\n \\begin{figure}[t!]\n \\centering\n \\includegraphics[width=0.4\\textwidth]{figs\/analysis\/BECAL_electron__ExampleBinHighest_1.pdf}\n \\vspace{-0.2cm}\n \\caption{Comparison of the energy resolution for electrons generated in single particle simulations at $E= 1$ GeV (top) and $E= 8$ GeV (bottom) as measured by the BEMC (left) and FEMC (right) without additional material in front of the calorimeter and in the full detector setup.}\n \\label{fig:materialresopeak}\n \\end{figure}\n \\begin{figure}[t!]\n \\centering\n \\includegraphics[width=0.45\\textwidth]{figs\/analysis\/Resolution_electron_Fitted_wFWHM.pdf}\\\\\n \\includegraphics[width=0.45\\textwidth]{figs\/analysis\/Resolution_pion_Fitted_wTB}\n \\caption{Energy resolution for electrons (charged pions) generated in single particle simulations with energies between 0.2 and 20 (50) GeV as measured by the different ECals (top) and the different HCals (bottom) in the central acceptance of the corresponding detectors.\n The shaded bands show the requirements as extracted from the yellow report for the different calorimeters. \n The data points and fits indicated as $\\sigma_g\/E$ are based on the Gaussian width of the resolution peaks, while $\\sigma_F\/E$ is based on the FWHM.\n The energy resolution based on a test beam for the OHCAL is shown for comparison \\cite{sPHENIX:2017lqb}.}\n \\label{fig:eresocalos}\n \\end{figure}\n \\Figure{fig:eresodistributioneemcbecalfemc} shows the energy response $E^\\mathrm{rec}\/E^\\mathrm{MC}$ for the various particle species and in each calorimeter.\n By construction, the electron and photon response in the ECals peaks around unity with a strong excess that is accompanied by a visible tail towards lower values. \n This tail is a result of multiple effects.\n First, the clusterization in the calorimeter is not perfect (see clusterization chapter) and thus not all energy of an incoming particle is reconstructed.\n In addition, for these studies only the highest energetic cluster in each event is selected, which combined with the clusterizer performance leads to a smearing to lower $E^\\mathrm{rec}\/E$ values.\n Further smearing comes from bremsstrahlung losses of the electrons in the magnetic field as well as from material interaction of photons that could lead to photon conversions, as seen in \\Fig{fig:materialresopeak}.\n The figure shows a comparison of the energy response for the BEMC with and without the remaining ECCE detector material in front, highlighting an increasing tail at lower $E^\\mathrm{rec}\/E$ due to the additional material.\n In the following studies, contributions from photon conversions are not rejected and thus are still contained in the photon sample.\n The left side tails of the resolution peaks can also arise through particles hitting the support material in between the towers.\n The reconstructed energy loss from hitting and subsequently channeling in the passive support structures is a major factor to be considered for the calorimeter design.\n Initial studies have shown that already a 2 mm carbon fiber support structure between the EEMC towers is enough to significantly deteriorate the energy resolution.\n As such, the supports were optimized to the current design of 0.5 mm carbon sheets, which greatly improves the energy resolution.\n Further improvements are possible with carbon support grids holding multiple crystals that are further separated by a thin foil.\n Similar support material considerations are to be made for the BEMC, where the current design employs 2 mm carbon fiber sheets.\n Charged hadrons deposit in the majority of cases only a minimum ionizing signal in the ECals, which is visible as a strong peak at low $E^\\mathrm{rec}\/E$ values.\n However, there is also a non-negligible amount of charged hadrons that deposit 40\\% or more of their energy in the ECals, which can negatively impact the HCal energy resolution.\n For the HCals, the charged pions and protons peak around unity, whereas remaining shower leakage from electron showers out of the ECals is mostly negligible.\n \\Figure{fig:eresodistributioneemcbecalfemc} also highlights a shifted peak for protons compared to pions in the HCals which can be explained by a loss of visible energy for baryons.\n In future studies, this effect could be counteracted for the LFHCAL by shower depth analyses and subsequent application of a correction factor for the loss of visible energy.\\\\\n In order to determine the energy resolutions of the different calorimeters, the $E^\\mathrm{rec}\/E$ distributions are fitted with crystalball functions in order to determine the peak width.\n This width can either be taken from the Gaussian component or from the full width at half maximum (FWHM).\n The slightly larger values of the latter are a reflection of the asymmetric $E^\\mathrm{rec}\/E$ distribution as described above.\n Based on the fit values, \\Fig{fig:eresocalos} shows the energy resolution for electrons in their generated energy range in the ECals and for charged pions in the HCals.\\\\\n All ECal resolutions, based either on the Gaussian sigma ($\\sigma_\\mathrm{g}$) or the FWHM ($\\sigma_\\mathrm{F}$), are well within the limits imposed by the YR and even exceed the requirements in the case of the BEMC by a significant amount.\n Thus, despite the smeared $E^\\mathrm{rec}\/E$ peaks from the full ECCE detector simulation, the resolution is still within the imposed limits.\n In addition, a minimal pseudorapidity dependence for all calorimeters is observed, but none of the $\\eta$-regions fail to deliver the required YR performance.\\\\ \n For the HCals, I\/OHCAL and LFHCAL, a similar behavior is observed, where the resolutions are found to be better than the YR requirements with $\\sigma\/E=(31-34\\%)\/\\sqrt{E}\\oplus(17-18\\%)$ and $\\sigma\/E=(33-44\\%)\/\\sqrt{E}\\oplus(1.4\\%)$ by about 1--20\\% and constant 8\\%, respectively.\n This also holds true for both tested particle species ($\\pi^\\pm$ and protons) and in each $\\eta$ region individually.\n In addition, the HCal resolution is compared to the sPHENIX test beam data and shows a better resolution in the presented simulations \\cite{sPHENIX:2017lqb}, which can be explained by an imperfect simulation setup of the details of the calorimeter response.\n \n\\subsection{Position resolution}\n\\label{sec:PositionResolution}\n A significant fraction of physics observables either directly or indirectly require a good position resolution of the reconstructed clusters in the calorimeters. \n For example, the jet reconstruction clusters objects which are reconstructed in a given radial cone and thus position inaccuracies especially in difficult pseudorapdity regions can deteriorate the physics performance. \n Moreover, charged particle association or neutral cluster determination via track matching (see next section) depends on the cluster position resolution as much as on the tracking resolution.\n \\begin{figure}[t!]\n \\centering\n \\includegraphics[width=\\linewidth]{figs\/analysis\/PositionResolution_Paper.pdf} \n \\caption{ Position resolution in $\\eta$ (top) and $\\varphi$ (bottom) for electrons or charged pions generated in single particle simulations with energies between 0.2 and 40 GeV as measured by the different calorimeters in the central acceptance of the corresponding destectors without a magnetic field. }\n \\label{fig:posreso}\n \\end{figure}\n\n \\begin{figure*}[t!]\n \\centering\n \\includegraphics[width=\\textwidth]{figs\/analysis\/TMEffEta_E+HCal_PaperPlotAlt.pdf}\n \\caption{Track matching efficiencies for electrons reconstructed with the MA-Clusterizer in the EEMC (a), the BEMC (b) and the FEMC (c) and for charged pions reconstructed in the OHCAL (b) and the LFHCAL (c). The efficiencies are relative to the number of reconstructed tracks according to \\Equation{eq:tmeffitrk}. }\n \\label{fig:tmeffiEBF}\n \\end{figure*}\n To determine the pure position resolution of the clusterization algorithm and intrinsic calorimeter granularity single particle simulations without a magnetic field have been used.\n This setup allows to separate between the intrinsic position resolution in the respective calorimeters and effects arrising from a larger inclination angle at the calorimeter surface as well as inaccuracies in the particle propagation through the material due to the 1.4T magnetic field.\n For the track-to-cluster matching under realisitic conditions within a magnetic field the $\\eta$ and $\\varphi$ coordinates for charged particles are calculated by propating the tracks through the detector material to approximately half the depth of each calorimeter.\n The median cluster depth is however $\\eta$ dependent for non projective calorimeters.\n Consequently, the mean shift in the $\\eta$-position has to be corrected for the forward and backward calorimeters based on the zero-field data.\n \\Figure{fig:posreso} presents the width of the difference of the generated particle $\\eta$($\\varphi$) and the reconstructed cluster position in $\\eta$($\\varphi$) in the different calorimeters.\n For all electro-magnetic calorimeters an excellent resolution of about $0.01-0.015$ in pseudorapidity is observed which only degrades slightly towards lower energies.\n The $\\varphi$-resolution for highly energetic particles is similarly good with $\\Delta \\varphi = 0.02$ (corresponding to $1.15$ degrees). \n It is mainly determined by the size of the single towers in $\\Delta \\varphi$ of the respective calorimeter and the width of the electro magnetic shower.\n Due to the larger tower sizes and wider spread of hadronic showers without a very well defined core the $\\eta$ and $\\varphi$ resolutions of the hadronic calorimeters are slightly worse in both dimensions. \n The resolutions for the LFHCAL could be further improved in the future by taking into account the correct depth of the shower as well, which so far has not been considered in the position calculation.\n\\subsection{Track-Cluster matching}\n The position resolution described in the previous chapter is a necessary ingredient for performance studies of the cluster-to-track matching.\n This matching is needed for particle identification studies, like electron selection via charged pion rejection or cluster neutralization for photon analyses.\n Moreover, the track matching procedure is a crucial ingredient for particle flow-based jet measurements.\\\\\n The track matching efficiency can be calculated as the number of track-matched clusters relative to the number of reconstructed tracks in the full calorimeter acceptance via\n \\begin{equation}\n \\epsilon_\\mathrm{TM} = N_\\mathrm{clus}^{\\text{matched}} \/ N_\\mathrm{tracks}^{\\text{in acc.}},\n \\label{eq:tmeffitrk}\n \\end{equation}\n or relative to the number of reconstructed clusters via\n \\begin{equation}\n \\kappa_\\mathrm{TM} = N_\\mathrm{clus}^{\\text{matched}} \/ N_\\mathrm{clus}.\n \\end{equation}\n \\Figure{fig:tmeffiEBF} shows the track matching efficiencies ($\\epsilon_\\mathrm{TM}$) for the different calorimeters for single particle simulations of either electrons or charged pions in the full ECCE GEANT4 detector setup.\n For a majority of the ECal acceptance, an excellent efficiency of $\\epsilon_\\mathrm{TM}>95\\%$ is observed.\n Expected deviations towards lower particle momenta are observed, where the track to cluster association breaks down due to a cut-off in the particle cluster reconstruction imposed by the minimum seeding and aggregation thresholds.\n An additional pseudorapdity dependence for the track matching efficiencies is expected due to the previously observed cluster position resolution, which deteriorates for certain $\\eta$ regions as wellas the reducted cluster finding efficiency due to remaining acceptance effects.\n Moreover, the particles might be stronger deflected in the $\\eta$-regions with higher amounts of material due to support structures.\\\\\n Further insights into the track matching efficiency are given by \\Fig{fig:tmeffitrkclus}, where the track matching efficiencies $\\kappa_\\mathrm{TM}$ are shown for all calorimeters in their nominal acceptance.\n The comparison of $\\epsilon_\\mathrm{TM}$ and $\\kappa_\\mathrm{TM}$ highlights that the track matching efficiency depends equally on the cluster finding efficiency and the track finding efficiency.\n This can clearly be seen in the electron matching efficiency for the ECals, where $\\epsilon_\\mathrm{TM}$ is nearly unity when calculated according to \\Equation{eq:tmeffitrk}, meaning that if a track is found, it is nearly always matched to a cluster.\n On the other hand, $\\kappa_\\mathrm{TM}$ shows a reduced efficiency, meaning that for a large portion of clusters no track is found for matching, especially in the forward region.\n For the HCals, the performance is generally worse as particles can pre-shower in the ECals, resulting in clusters with distorted positions on the HCals, thus not for all tracks a matching cluster is found.\n\n \\begin{figure}[!ht]\n \\centering\n \n \\includegraphics[width=0.49\\textwidth]{figs\/analysis\/ClusterTMEfficiency_KappaOnly_Paper.pdf}\n \\caption{Track matching efficiencies calculated relative to the number of reconstructed clusters ($\\kappa_\\mathrm{TM}$). Single particle simulations of electrons in the detector acceptance of the ECals and of charged pions in the acceptance of the HCals are used to obtain the efficiencies.}\n \\label{fig:tmeffitrkclus}\n \\end{figure}\n\\subsection{Particle Identification}\n The information provided by the ECals and HCal can help distinguish between particle species and thus provide highly efficient particle identification, which is crucial for a variety of physics analyses.\n This section therefore focuses first on the PID capabilities of the ECals and subsequently the additional benefits from the HCals.\n \n The electromagnetic calorimeters (EEMC, BEMC and FEMC) are most commonly used to identify electromagnetic showers coming from a single particle.\n They can differentiate between photons and their background from merged $\\pi^0$ decay photons.\n If tracking information is used in addition, the calorimeters can be used to provide a strong separation power between electrons and charged hadrons like $\\pi^\\pm$, kaons or protons.\n In the following, the different PID approaches are briefly explained and the expected performance is shown based on full detector GEANT4 simulations.\n \\begin{figure}[!t] \n \\centering\n \\includegraphics[width=0.4\\textwidth]{figs\/analysis\/Pi0_InvMassBin_AllCalos.pdf}\n \\caption{Invariant mass $M_{\\gamma\\gamma}$ distribution for generated $\\pi^0$ mesons in the energy range from 2.5 to 3.0 GeV for EEMC, BEMC, and FEMC including a composite Gaussian fit function that includes a left-sided exponential tail component.}\n \\label{fig:invmasspi0sep}\n \\end{figure}\n \\begin{figure}[!t] \n \\centering\n \\includegraphics[width=0.4\\textwidth]{figs\/analysis\/Pi0_MassAndWidth.pdf}\n \\caption{Invariant mass $M_{\\gamma\\gamma}$ peak width (top) and position (bottom) obtained from a composite Gaussian fit including a left-sided exponential tail.}\n \\label{fig:invmassfittedvalues}\n \\end{figure}\n\n \\begin{figure*}[!t] \n \\centering\n \\includegraphics[width=\\textwidth]{figs\/analysis\/EoP_ExampleBin_PaperPlot.pdf}\n \\caption{$E\/p$ distribution for electrons (blue) and charged pions (red) in the EEMC (right) and the BEMC (left). The $E\/p$ distribution is shown for two different approaches where $E\/p$ is either calculated using the generated (true) particle momentum or the reconstructed tracking based (rec.) momentum. For both distributions, the full ECCE detector has been simulated using its GEANT4 implementation.}\n \\label{fig:eopdistribution}\n \\end{figure*}\n\n \\begin{figure}[!ht] \n \\centering\n \\includegraphics[width=0.4\\textwidth]{figs\/analysis\/Pi0_MergingFraction.pdf}\n \\caption{Fraction of neutral pions for which the showers from their decay photons are merged into a single cluster and can not be reconstructed using an invariant-mass-based approach for the different ECals.\n }\n \\label{fig:pi0merging}\n \\end{figure}\n\n\\subsubsection{Single photon and neutral pion separation}\n A significant background for photon analyses originates from $\\pi^0$ meson decay photons, which end up in the same reconstructed cluster due to their close proximity.\n In general, when the decay photons can still be reconstructed separately, their calculated invariant mass ($M_{\\gamma\\gamma}=\\sqrt{2E_{\\gamma_1}E_{\\gamma_2}(1-\\cos{\\theta_{12}})}$) can be used to veto decay photon clusters if the mass falls in a certain window around the nominal $\\pi^0$ mass.\n Example invariant mass distributions for the ECals are shown in \\Fig{fig:invmasspi0sep} for a selected energy range of the two-photon meson candidates including a composite Gaussian fit with a left-sided exponential tail.\n The BEMC invariant mass distribution is significantly wider than that of the EEMC or FEMC, as can also be seen in \\Fig{fig:invmassfittedvalues}, where the peak width (obtained from the width of the Gaussian fit component) is shown as a function of $\\pi^0$ energy.\n The broadening of the peak width with increasing energy and the cutoff of the BEMC data at $E\\approx12$ GeV is further elaborated in the following.\n Above a certain energy, the decay kinematics of the $\\pi^0$ together with the granularity and resolution of the calorimeter no longer allow to reconstruct separate decay photons due to a merging of their showers.\n Thus, the separation power between meson decay photons and single photons decreases and the reconstructed meson mass starts to deviate from the nominal meson mass as seen in \\Fig{fig:invmassfittedvalues}.\\\\\n The energy dependence of this cluster merging effect is shown in \\Fig{fig:pi0merging} for the three ECals.\n The close proximity of the BEMC to the interaction point together with its $4 \\times 4$~cm tower size results in a large fraction of merged decay photon clusters already at 5 GeV around $\\eta = 0$.\n For $|\\eta| > 0.9$ within the BEMC the merging starts to set in at around $10$~GeV due to the larger distance of the calorimeter surface from the interaction point and thus a larger average distance of the two decay photons on the calorimeter surface.\n In contrast, the higher granularity and larger distance from the IP of the EEMC and FEMC, respectively, results in a much later onset of the cluster merging. \n For the FEMC this effect becomes only significant above 25 GeV, while the EEMC experiences this effect already above 15 GeV.\n\n\\subsubsection{Electron PID via charged pion rejection}\nSeveral observables of EIC physics require a clean electron sample \\cite{AbdulKhalek:2021gbh}.\nOne of the largest backgrounds for electrons stems from charged pions ($\\pi^\\pm$), which can be distinguished on a statistical basis from electrons with a high efficiency using ECal information.\nThe so-called pion rejection factor is a handle on how strong this $e^\\pm$--$\\pi^\\pm$ separation is for a given calorimeter.\nIt can be calculated by simulating the response for single electron and separate single pion events.\nThe quantity $E\/p$, meaning the reconstructed cluster energy relative to the incident particle momentum exhibits only slightly overlapping distributions for both particles.\nThis is shown in \\Fig{fig:eopdistribution} where electrons (blue) show a strong enhancement around $E\/p\\approx1$, while charged pions (red) are smeared towards lower $E\/p$ values for all three ECCE ECals.\nIn realistic events, e.g. based on the Pythia event generator, one expects significantly more hadrons relative to electrons and thus the hadronic tail overlap is expected to be stronger than shown in \\Fig{fig:eopdistribution}.\nThis effect is further enhanced by the presence of jets which result in shower overlaps in the calorimeters.\nThe track momentum in the following is determined using the full ECCE tracking capabilities \\cite{ecce-paper-det-2022-03}.\n\nDue to the small overlap of the $E\/p$ distribution for different particle species, a minimum $E\/p$ cut can be employed to reject the majority of charged pions in the sample while retaining a high efficiency electron sample.\nIn previous studies, a minimum cut of $\\Delta=1.6\\,\\sigma_E\/E$ has been determined to result in a high electron efficiency of $\\varepsilon_e\\approx 95\\%$.\nHowever, the asymmetric electron resolution distributions of the calorimeters within the full ECCE integration, as shown in \\Fig{fig:eresodistributioneemcbecalfemc} lead to a significantly reduced electron efficiency when applying a 1.6$\\sigma$-based cut.\nEspecially for the BEMC where a strong tail in the energy resolution distribution is visible, the cut results in an electron efficiency of $\\varepsilon_e\\approx70\\%$, while for the other ECals values of about 90--95\\% are observed.\nThus, an additional $E\/p$ cut value has been determined that allows for $\\varepsilon_e= 95\\%$, which is indicated as $\\varepsilon_{95\\%}$ in the following.\nThis cut value corresponds to approximately $2\\sigma$ for the EEMC, $6\\sigma$ for the BEMC, and $3\\sigma$ for the FEMC, highlighting the difference in the energy resolution peak asymmetry for the various ECCE ECals.\n\\begin{figure}[!t] \n \\centering\n \\includegraphics[width=0.4\\textwidth]{figs\/analysis\/EoP_pionrejection_AllCalo_95comp.pdf}\n \\caption{Pion rejection factor for the different ECals with $E\/p>1-1.6\\,\\sigma_e\/E$ or based on a $\\varepsilon_e\\approx 95\\%$ cut.\n }\n \\label{fig:pionrejection}\n\\end{figure}\nApplying these cuts on the single pion event simulations results in the rejection factors shown in \\Fig{fig:pionrejection}.\nValues up to $6\\times10^4$ are reached for the EEMC using the $1.6\\,\\sigma$-based cut, while for the other calorimeters $\\pi^\\pm$ rejection factors ranging from 20 to more than $10^3$ are reached.\nFor the EEMC the pion rejection capabilities are so striking that an accurate pion rejection factor is hard to determine with the currently available single particle production statistics and the reported values should be interpreted as lower limits.\nA significant reduction of about an order of magnitude in the $\\pi^\\pm$ rejection is observed for the $\\varepsilon_e=95\\%$ based cut for the FEMC and BEMC, which therefore stands in no reasonable relation to the efficiency loss observed for the other $E\/p$ cut values.\nThis loss mainly arrises from the significant tails observed for these two calorimeters in their current configuration.\n\n\\begin{figure}[!t] \n \\centering\n \\includegraphics[width=0.4\\textwidth]{figs\/analysis\/M02_ExampleBin_EEMC.pdf} \n \\caption{The shower shape distribution (\\ensuremath{\\sigma_\\mathrm{long}^{2}}) for electrons (blue) and charged pions (red) in the EEMC using the full ECCE detector simulations at a fixed generated energy of $E=1.3$ GeV. \n The gray vertical line indicates the \\ensuremath{\\sigma_\\mathrm{long}^{2}}\\ cut value below which 90\\% of the electrons would be kept.}\n \\label{fig:SSdistribution}\n\\end{figure}\n\n\\subsubsection{Hadron PID}\nBesides using an $E\/p$ cut to differentiate between electrons and hadrons the shape of the shower and thus the cluster can be used.\nThe distribution of energy within a cluster is referred to as ``shower shape'', which is described using a parametrization of the shower surface ellipse axes~\\cite{Alessandro:2006yt, Abeysekara:2010ze}. \nThe shower surface is defined by the intersection of the cone containing the shower with the front plane of the calorimeter. \nThe energy distribution along the $\\eta$ and $\\varphi$ directions is represented by a covariance matrix with terms ${\\sigma_{\\varphi\\varphi}}$, ${\\sigma_{\\eta\\eta}}$ and ${\\sigma_{\\varphi\\eta}}$, which are calculated using logarithmic energy weights $w_i$.\nThe tower dependent weights are expressed as:\n\\begin{equation}\n\\label{eq:w_i}\nw_i = {\\rm Maximum}(0,w_0+\\ln(E_{i}\/E_\\mathrm{cluster}))\n\\end{equation}\nand\n\\begin{equation}\nw_\\mathrm{tot} = \\sum_i w_i,\\\\\n\\end{equation}\nwhere $w_0$~=~4.5 for the EEMC \\cite{Awes:1992yp}, which excludes towers with energy smaller than 1.1\\% of the cluster energy.\nFor the BEMC and FEMC $w_0$~=~4.0 and $w_0$~=~3.5 are used, respectively, in order to compensate for the different Moliere radii and tower size.\nThe covariance matrix terms can then be calculated as follows\n\\begin{equation}\n\\sigma^{2}_{\\alpha\\beta} = \\sum_i \\frac{w_i\\alpha_i\\beta_i}{w_\\mathrm{tot}}-\\sum_i \\frac{w_i\\alpha_i}{w_\\mathrm{tot}}\\sum_i \\frac{w_i\\beta_i}{w_\\mathrm{tot}}\\,,\n\\label{eq:ss_centroid}\n\\end{equation}\nwhere $\\alpha_{i}$ and $\\beta_{i}$ are the tower indices in the $\\eta$ or $\\varphi$ direction. \nSimilarly, also the average cluster position in the $\\eta$ and $\\varphi$ direction in the calorimeter plane is determined using the tower positions weighted logarithmically by their deposited energy \\cite{Awes:1992yp}.\\\\ \nThe shower shape parameters \\ensuremath{\\sigma_\\mathrm{long}^{2}}\\ (long axis) and \\ensuremath{\\sigma_\\mathrm{short}^{2}}\\ (short axis) are defined as the eigenvalues of the covariance matrix, and are calculated as\n\\begin{eqnarray}\n\\ensuremath{\\sigma_\\mathrm{long}^{2}} = 0.5(\\sigma^{2}_{\\varphi\\varphi}+\\sigma^{2}_{\\eta\\eta})+\\sqrt{0.25(\\sigma^{2}_{\\varphi\\varphi}-\\sigma^{2}_{\\eta\\eta})^2+\\sigma^{2}_{\\eta\\varphi}}, \\label{eq:ss1}\\\\\n\\ensuremath{\\sigma_\\mathrm{short}^{2}} = 0.5(\\sigma^{2}_{\\varphi\\varphi}+\\sigma^{2}_{\\eta\\eta})-\\sqrt{0.25(\\sigma^{2}_{\\varphi\\varphi}-\\sigma^{2}_{\\eta\\eta})^2+\\sigma^{2}_{\\eta\\varphi}} \\label{eq:ss2},\n\\end{eqnarray}\nPrevious experiments have determined that the short axis \\ensuremath{\\sigma_\\mathrm{short}^{2}}\\ carries significantly less discriminative power compared to \\ensuremath{\\sigma_\\mathrm{long}^{2}}\\ and thus only the long axis is considered in the following.\n\n\\begin{figure}[!htb] \n \\centering\n \\includegraphics[width=0.4\\textwidth]{figs\/analysis\/RejectPi_AllCalo_true.pdf}\n \\includegraphics[width=0.4\\textwidth]{figs\/analysis\/EoP_pionrejection_AllCalo_withPID.pdf}\n \\caption{Left: Fraction ($R_\\pi$) of cluster originating from charged pions, which can be rejected due to the chosen shower shape cut as a function of the incident pion energy. \n Right: Pion rejection factor for the different ECals with $E\/p>1-1.6\\,\\sigma_e\/E$ (w$\/$o PID) or $E\/p>1-1.6\\,\\sigma_e\/E$ and the additional \\ensuremath{\\sigma_\\mathrm{long}^{2}}\\ selection (w$\/$ PID) applied as a function of the true track momentum.}\n \\label{fig:RejectionPID}\n\\end{figure}\nUsing these parameters symmetric electromagnetic showers with a small spread originating either from photons or electrons can be distinguish from non-symmetric showers caused by hadronic interactions.\nThe shower shape of charged particles can also be elongated by the angle of incidence. calorimeters.\nFurthermore, the merging of showers from electromagnetic processes, i.e. ${\\rm e^{+}e^{-}}$ pairs from conversions within a close distance to the calorimeter or photons from neutral meson decays with high transverse momenta, also lead to asymmetric shower shapes.\\\\\nAn example distribution of the shower shape parameter \\ensuremath{\\sigma_\\mathrm{long}^{2}}\\ for electrons (blue) and pions (red) as seen by the EEMC can be found in \\Fig{fig:SSdistribution}.\nAs can be seen, the energy deposits from an electrons at the same incident energy are significantly more collimated than those of charged pions.\nConsequently, electron clusters have predominantly lower shower shape values. \nAs these distributions strongly change as a function of the incident energy a \\ensuremath{\\sigma_\\mathrm{long}^{2}}\\ cut value function is calculated that preserves 90\\% of the electrons. \nUsing these cut values based on the shower shape alone up to 90\\% of the pions can be rejected in the EEMC as shown in \\Fig{fig:RejectionPID}~(top).\nBy simultaneously using the aforementioned $E\/p$ and \\ensuremath{\\sigma_\\mathrm{long}^{2}}\\ cuts, the pion rejection quoted in \\Fig{fig:pionrejection} is improved by at least a factor two in most momentum bins as seen in \\Fig{fig:RejectionPID}~(bottom).\n\n\\section{Introduction}\n\\label{sec:introduction}\n\nWe report the design and performance of the calorimeter systems for the ECCE detector~\\cite{ecce-paper-det-2022-01}.\nHomogeneous and sampling calorimeter technologies are employed in the different pseudorapidity regions (backwards, central, and forward) aiming to achieve the overall performance requirements outlined in the EIC Yellow Report~(YR)~\\cite{AbdulKhalek:2021gbh} cost effectively and with consideration of technical and schedule risks.\nThe main physics program of the EIC imposes strong detector performance requirements on the calorimeter systems.\nWhile single inclusive DIS, jets and heavy quark reconstruction require an excellent energy resolution for the electromagnetic and hadronic calorimeters, further requirements for $\\pi$ \/ $e$ separation at the $3\\sigma$ level are imposed, for example, by spin asymmetry measurements, TMD evolution, and $XYZ$ spectroscopy.\nIn order to probe the requested kinematic regions for such processes, a large acceptance in pseudorapidity for the calorimeters is required with special focus on continuous coverage from the backward region to the forward region.\nThe key performances of the ECCE calorimeter systems are reported and put in context to their impact on physics analyses.\nThis includes the reconstruction performance, expected energy and position resolution, as well as particle identification via matching to charged particle tracks obtained from the ECCE tracking systems \\cite{ecce-paper-det-2022-03}. \n\n\\section{Acknowledgements}\n\\label{acknowledgements}\n\nWe acknowledge support from the Office of Nuclear Physics in the Office of Science in the Department of Energy, the National Science Foundation, and the Los Alamos National Laboratory Laboratory Directed Research and Development (LDRD) 20200022DR.\n\nWe thank (list of individuals who are not coauthors) for their useful discussions and comments.\n\n\\section{Summary}\n\\label{summary}\n\nIn summary, the ECCE calorimeter systems have been designed to support the full scope of the EIC physics program as presented in the EIC white paper~\\cite{Accardi:2012qut} and in the 2018 report by the National Academies of Science (NAS)~\\cite{NAP25171}. These systems can be built within the budget envelope set out by the EIC project while simultaneously managing cost and schedule risks.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{intro}\n\nCollisions of asymmetric nuclear systems, like proton-nucleus or deuteron-nucleus \ncollisions, attract significant experimental and theoretical attention at present. \nAt the Relativistic Heavy Ion Collider (RHIC), Run 8 included deuteron-gold\ncollisions at 200 $A$GeV with a luminosity increase of about an order of magnitude\ncompared to the deuteron-gold run of 2003. One of the physics goals was to provide \na high-statistics ``cold'' nuclear matter data set to establish a definitive baseline \nfor ``hot'' nuclear matter, created in gold-gold collisions. The benchmark role of the\n``the deuteron-gold control experiment'' for e.g. energy-loss studies has often been \nemphasized~\\cite{Gyulassy:2004jj,Hemmick:2004jc}.\nProton-nucleus and deuteron-nucleus reactions were \nalso used to study the Cronin-effect~\\cite{Cron75,Antr79}. \n\nLight-on-heavy nucleus-nucleus collisions offer unique information about the \nunderlying dynamics, not available in symmetric proton-proton or \nnucleus-nucleus collision systems. The light-on-heavy asymmetry\nmanifests itself in an asymmetric (pesudo-)rapidity distribution of \ncharged particles with respect to\nzero rapidity (or pseudorapidity) as measured by BRAHMS~\\cite{Arsene:2004cn} and \nPHOBOS~\\cite{Back:2004mr}. The asymmetry of the yields can be quantified by \nintroducing the ratio of pseudorapidity densities at a given negative \npseudorapidity relative to that at\nthe positive pseudorapidity of the same magnitude. This backward\/forward ratio\n(forward being the original direction of motion of the light partner) is referred to\nas pseudorapidity asymmetry. The STAR Collaboration published pseudorapidity \nasymmetries in 200 $A$GeV $dAu$ collisions for several identified hadron species \nand total charged hadrons in the pseudorapidity intervals $|\\eta| \\le 0.5$ and \n$0.5 \\le |\\eta| \\le 1.0$~\\cite{Abelev:2006pp}. Asymmetries with the\nbackward\/forward ratio above unity for transverse momenta up to $\\approx 5$ GeV\/c\nare observed for charged pion, proton+anti-proton, and total charged hadron production \nin both rapidity regions. Because of the importance of asymmetric collisions, we \nanticipate that proton-lead or deuteron-lead data will be \ncollected at the Large Hadron Collider (LHC). \n\nThe theoretical relevance of asymmetric collision systems (in particular colliding\nthe lightest nuclei like protons or deuterons on a heavy partner) in \ntesting parameterizations for nuclear shadowing and models for initial multiple \nscattering was recognized prior to the availability of 200 $A$GeV RHIC deuteron-gold \ndata~\\cite{Vitev:2003xu,Wang:2003vy,Barnafoldi:2004kh}. One can take advantage\nof the fact that\npositive (forward) rapidities correspond to large parton momentum fractions from\nthe light partner and small momentum fractions from the heavy nucleus\n(and vice versa for negative rapidities), and that there are more collisions \nsuffered traversing the heavy partner. Wang made predictions for pseudorapidity \nasymmetries on this basis~\\cite{Wang:2003vy}. A subsequent calculation focused on the \nlowest transverse momenta where pQCD may be applicable~\\cite{Barnafoldi:2005rb}.\nMore recently, two versions of a pQCD-based model with nuclear modifications were \nused to address deuteron-gold collisions and results were compared to available\ndata~\\cite{Adeluyi:2008qk,Barnafoldi:2008rb}. The related subject of\nforward-backward rapidity correlations in asymmetric systems is treated in\n~\\cite{Brogueira:2007ub} using a derivative of string percolation models, and \nin \\cite{Armesto:2006bv} under the framework of color glass condensate. \n\nIn the present study we investigate the roles of nuclear shadowing and multiple \nscattering in the generation of rapidity asymmetry \nin the backward\/forward yields\nat intermediate and high transverse-momenta. Here we use the HIJING\nshadowing parameterization~\\cite{Li:2001xa} and the recently released \nEskola--Paukkunen--Salgado (EPS08) nuclear parton distribution functions\n(nPDFs)~\\cite{Eskola:2008ca}. While the former has been applied widely, \nthe latter was not available at the time of the\nearlier studies mentioned above. We calculate the pseudorapidity asymmetry in three \nrepresentative asymmetric light-on-heavy systems: $pBe$ at $30.7$ GeV (Fermilab), \n$dAu$ at $200$ $A$GeV (RHIC), and $dPb$ at $8.8$ $A$TeV (LHC). \nWe concentrate on neutral pion production and compare results to experimental data \nfrom the E706 experiment~\\cite{Apanasevich:2002wt},\nPHENIX~\\cite{PHENIXdAu}, and STAR~\\cite{Abelev:2006pp}.\n\nThe paper is organized as follows: in Sec.~\\ref{col} we review the basic \nformalism of the pQCD-improved parton model with intrinsic transverse momentum and\nmultiple scattering as applied to proton-nucleus ($pA$) and deuteron-nucleus ($dA$) \ncollisions. This section also includes the definition of the pseudorapidity asymmetry \nand a discussion of the roles of nuclear shadowing and multiple scattering in asymmetric \ncollisions. We present the results of our calculation in Sec.~\\ref{res} and conclude \nin Sec.~\\ref{concl}.\n\n\\section{Calculational framework}\n\\label{col}\n\n\\subsection{Description of the model}\n\\label{partmd}\nThe invariant cross section for the production of \nfinal hadron $h$ from the collision of nucleus $A$ and nucleus $B$ \n($A+B \\!\\to\\! h+X$), can be written, in collinearly factorized pQCD, as\n\\begin{eqnarray}\\label{eq:pdA}\nE_h\\f{\\dd ^3 \\sigma_{AB}^{h}}{\\dd^3 p} =\n\\sum_{\\!\\!abcd}\\!\\int\\!\\! \\dd^2b\\ \\dd^2r\\ t_A(b) t_B(|\\vec{b}-\\vec{r}|)\\dd{x_a} \\dd{x_b}\n\\nonumber \\\\ \n\n\\dd{z_c} \\,\nf_{\\!a\/A}(x_a,Q^2)\\ f_{\\!b\/B}(x_b,Q^2)\\\n\\nonumber \\\\ \n\\f{\\dd\\sigma(ab\\!\\to\\!cd)}{\\dd\\hat{t}}\\,\n\\frac{D_{h\/c}({z_c},\\!Q_f^2)}{\\pi z_c^2}\n\\hat{s} \\, \\delta(\\hat{s}\\!+\\!\\hat{t}\\!+\\!\\hat{u}) \\,\\, ,\n\\end{eqnarray}\nwhere $x_a$ and $x_b$ are parton momentum fractions in $A$ and $B$, respectively, \nand $z_c$ is the fraction of the parton momentum carried by the final-state \nhadron~$h$.\nThe factorization and fragmentation scales are $Q$ and $Q_f$, respectively. \nAs usual, $\\hat{s}$, $\\hat{t}$, and $\\hat{u}$ refer to the partonic Mandelstam variables, \nthe massless parton approximation is used, and\n\\begin{equation} \\label{eq:tf}\nt_{A}(\\!{\\vec s}) = \\int\\!\\! \\dd z \\rho_{A}(\\!{\\vec s},z)\n\\end{equation}\nis the Glauber thickness function of nucleus $A$, with the nuclear density distribution,\n$\\rho_{A}({\\vec s},z)$ subject to the normalization condition\n\\begin{equation}\n\\int\\!\\! \\dd^2s \\ \\dd z \\rho_{A}({\\vec s},z) = A \\,\\, .\n\\end{equation}\nThe quantity $\\dd\\sigma(ab\\!\\to\\!cd)\/\\dd\\hat{t}$ in eq.~(\\ref{eq:pdA}) represents the \nperturbatively calculable partonic cross section, and $D_{h\/c}(z_c,\\!Q_f^2)$ stands \nfor the fragmentation function of parton $c$ to produce hadron $h$, evaluated at momentum \nfraction $z_c$ and fragmentation scale $Q_f$. \n\nThe collinear parton distribution functions (PDFs) can be generalized to include a transverse\nmomentum degree of freedom, ${\\vec k}_T$, as required by the uncertainty principle. This\ncan be formally implemented in terms of unintegrated PDFs~\\cite{Collins:2007ph,Czech:2005vy}. \nTo avoid some of the complications associated with using unintegrated\nPDFs, it is expedient to parameterize phenomenologically the ${\\vec\n k}_T$ dependence of the parton distributions. In this light \na phenomenological model for proton-proton ($pp$), proton-nucleus ($pA$), \nand nucleus-nucleus ($AB$) collisions incorporating parton transverse \nmomentum in the collinear pQCD formalism was developed in\nRef.~\\cite{Zhang:2001ce}. In order to make the present study\nreasonably self-contained, we include details relevant to the\npresent treatment.\n\nIn this model, the invariant cross section can be written as\n\\begin{eqnarray}\\label{eq:pdA_kt}\nE_h\\f{\\dd ^3 \\sigma_{AB}^{h}}{\\dd^3 p} =\n\\sum_{\\!\\!abcd}\\!\\int\\!\\! \\dd^2b\\ \\dd^2r\\ t_A(b) t_B(|\\vec{b}-\\vec{r}|)\\dd{x_a} \\dd{x_b}\n\\nonumber \\\\ \n \\dd{\\vec k}_{Ta} \\, \\dd{\\vec k}_{Tb} \\, \\dd{z_c} \\,\nf_{\\!a\/A}(x_a,\\!{\\vec k}_{Ta},Q^2)\\ f_{\\!b\/B}(x_b,\\!{\\vec k}_{Tb},Q^2)\\\n\\nonumber \\\\ \n\\f{\\dd\\sigma(ab\\!\\to\\!cd)}{\\dd\\hat{t}}\\,\n\\frac{D_{h\/c}({z_c},\\!Q_f^2)}{\\pi z_c^2}\n\\hat{s} \\, \\delta(\\hat{s}\\!+\\!\\hat{t}\\!+\\!\\hat{u}) \\,\\, ,\n\\end{eqnarray}\nwhere the $k_T$-broadened parton distribution in the nucleon is\nwritten, in a simple product approximation, as\n\\begin{equation}\nf_{a\/N}(x,{\\vec k}_{T},Q^2) \\longrightarrow \\ g({\\vec k}_T) \\cdot f_{a\/N}(x,Q^2) \\,\\, ,\n\\end{equation} \nwith $f_{a\/N}(x,Q^2)$ denoting the standard collinear PDF in the nucleon.\nThe transverse momentum distribution is taken to be\na Gaussian,\n\\begin{equation}\\label{eq:kT}\ng({\\vec k}_T) = \\frac{\\exp(-k_T^2\/\\langle k_T^2 \\rangle_{pp})} {\\pi \\langle k_T^2 \\rangle_{pp}} \\,\\, ,\n\\end{equation}\nwhere $\\langle k_T^2 \\rangle_{pp}$ is the two-dimensional width of the transverse-momentum\ndistribution in the proton.\nBased on the then-available pion and unidentified hadron production data, an estimate \nfor the model width of the transverse-momentum distribution of partons in the proton\n($\\langle k_T^2 \\rangle_{pp}$) \nwas presented in Ref.~\\cite{Zhang:2001ce}.\nA recent summary of the energy dependence of the model parameter $\\langle k_T^2 \\rangle$, \nrelated to the average transverse momentum of the created pair by\n\\begin{equation}\\label{eq:pT_kT}\n\\langle k_T^2 \\rangle_{pp} = \\frac{\\langle p_T \\rangle^{2}_{pair}}{\\pi} \\,\\, ,\n\\end{equation}\ncan be found in Ref.~\\cite{Barnafoldi:2007uw}. The value of $\\langle k_T^2 \\rangle_{pp}$\nincreases logarithmically with $\\sqrt{s}$, and the data are well described by the function\n\\begin{equation}\\label{eq:pT_rs}\n\\langle p_T \\rangle_{pair} = (1.74 \\pm 0.12) \\centerdot \\log_{10}(\\sqrt s) +\n(1.23 \\pm 0.2) \\,\\, .\n\\end{equation}\nUsing eq.~(\\ref{eq:pT_rs}) one can estimate \n$\\langle k_T^2 \\rangle$ at required cms energies.\nIt should be noted that eq.~(\\ref{eq:pT_rs}) is a fit to the data at\nlower energies. Since no data exists at LHC energies, we\nextrapolate $\\langle k_T^2 \\rangle$. We observe that: \n(a) Since the dependence is logarithmic, the energy step \nwe make from RHIC to LHC is comparable to the one from SPS to RHIC.\nWe thus do not a priori expect a radical change in the trend.\n(b) The $k_T$ effects have been shown to be appreciable at low $p_T$, and since \nour predictions at LHC cover a wide $p_T$ range up to hundreds of GeV\/c in\n$p_T$, potential uncertainties arising from the extrapolation of Eq. 8 will \naffect a small fraction of this range at the low $p_T$ end.\n(c) The present study, at LHC energies, is exploratory. Results of pp\ncollisions at the LHC will be helpful in determining the magnitude of\nthe presently uncertain $\\langle k_T^2 \\rangle_{pp}$.\n\nIt is easy to appreciate the physical necessity of the presence of a \ntransverse-momentum degree of freedom in proton-proton and, therefore, nuclear \ncollisions. However, the handling of the effect of the nuclear environment \non transverse momenta is one of the specific features of the given description. \nMost shadowing parameterizations include at least some of the effects\nof multiple scattering in the nuclear medium, while the HIJING parameterization\n(as we discuss further in Sec.~\\ref{shad_multiscatt}) needs to be augmented\nwith modeling nuclear multiscattering. For this purpose, in $pA$ collisions\nwe use a broadening of the width of the transverse momentum distribution \n(\\ref{eq:kT}) according to \n\\begin{equation}\\label{eq:broad}\n\\langle k_T^2\\rangle_{pA} = \\langle k_T^2\\rangle_{pp} + C\\ h_{pA}(b) \\,\\, ,\n\\end{equation}\nwhere $\\langle k_T^2\\rangle_{pp} $ is the width already present in proton-proton \ncollisions, $h_{pA}(b)$ is the number of effective nucleon-nucleon ($NN$) collisions as a \nfunction of nucleon impact parameter $b$, and $C$ is the average increase in width \nper $NN$ collision. In nucleus-nucleus ($AB$) collisions, the $p_T$ distributions of\nboth nuclei are subject to the broadening represented by eq.~(\\ref{eq:broad}). \n\nThe function $h_{pA}(b)$ can be written in terms of the number of collisions suffered \nby the incoming proton in the target nucleus, $\\nu_{A}(b) =\n\\sigma_{NN}t_{A}(b)$, where $\\sigma_{NN}$ is the inelastic nucleon-nucleon \ncross sections. It was found in \nRef.~\\cite{Zhang:2001ce} that only a limited number of collisions is effective in broadening\nthe transverse momentum distribution. This model scenario was referred to as\n``saturation'', and an optimal description was found with\nthe effective number of nucleon-nucleon collisions maximized at $4$, \nand the average width increase per $NN$ collision, $C$, set to $0.35$ GeV$^2\/c^2$.\nWe do not change the values of these parameters in the present study. The resulting\ntransverse momentum broadening may appear too large relative to what is observed \nin Drell-Yan data at FNAL. However, we focus on meson production in the present\napplication, where the PHENIX experiment extracts intrinsic transverse momenta\nin $pp$ collisions at RHIC energies that are similar in magnitude to the ones\ndiscussed here\\cite{Adler:2006sc}.\nFurther details about this aspect of the model\ncan be found in Ref.~\\cite{Zhang:2001ce}.\n\nThe intrinsic transverse momentum $k_T$ is treated \nphenomenologically in Ref.~\\cite{Zhang:2001ce} and in the \npresent study. While next-to-leading-order (NLO) calculations provide\na more accurate description of the parton-level cross section, they continue\nto rely on the factorization theorem and represent the non-perturbative\ninformation in terms of PDFs and fragmentation functions. Since these\nfunctions are fitted to the same data as in LO, the expected change is a shift\nin responsibilty between the perturbative and non-perturbative sectors in\ndescribing the data used to define the non-perturbative ingredients. In \naplications to a larger set of data this will not eliminate the need for the \nphenomenological use of a transverse momentum distribution, in particular\nconsidering the fact that even NLO may be far from a full perturbative\nexpansion. In addition, above we argue for inclusion of a transverse momentum \ndegree of freedom on fundamental physical grounds as basic as the uncertatinty\nprinciple. The simple Gaussian representation of this physics provides a \nphenomenologically useful additional parameter (the width). This holds true\nboth at the LO and NLO levels. Going to NLO may change the range of transverse\nmomenta where intrinsic $k_T$ is important (see e.g. \nRefs.~\\cite{MRTS:1995,CTEQ2:1995,Vogelsang:2007}).\nNo attempt is made in the present work to discuss the $x$, \nflavor (quarks and gluons), and rapidity dependence of $k_T$. We have \nlimited ourselves here to a simple effective description. This seems adequate for now, \nsince $k_T$ effects are appreciable only at relatively low $p_T$, and a \nmajor focus of this work is asymmetry at high $p_T$ (except at very forward \nrapidities where $p_T$-s are low due to phase space constraints). The intrinsic \ntransverse momentum, $\\langle k_T^2\\rangle_{pp}$, enhances hadron production \nyields at low $p_T$ in both negative and positive pseudorapidity regions, thus tending \nto cancel out in the asymmetry ratio. Multiscattering, on the other hand, does impact \nappreciably on the asymmetry even at low $p_T$.\nOverall, calculations within this framework \nhave proven their value in the interpretation of hadron-production \ndata~\\cite{Zhang:2001ce,Levai:2003at}. (For a comparison of the leading-order \n$k_T$-factorized approach and next-to-leading order collinear approach see \nRef.~\\cite{Szczurek:2007bt}.)\n\nThe collinear nPDFs $f_{a\/A}(x,Q^2)$ are \nexpressible as convolutions of nucleonic parton distribution functions (PDFs)\n$f_{a\/N}(x,Q^2)$ and a shadowing function ${\\cal S}_{a\/A}(x,Q^2)$ which encodes the \nnuclear modifications of parton distributions. We use the MRST2001\nPDFs~\\cite{Martin:2001es} for the nucleon \nparton distributions, and for the shadowing function we employ both \nthe EPS08 shadowing routine~\\cite{Eskola:2008ca} and HIJING~\\cite{Li:2001xa}.\n(Other nPDFs, like FGS~\\cite{Frankfurt:2003zd}, HKN~\\cite{Shad_HKN}, \nand the earlier EKS~\\cite{Eskola:1998df}, are used elsewhere\nto calculate pseudorapidity asymmetries~\\cite{Adeluyi:2008qk}.) \nFor the final hadron fragmentation we utilize the fragmentation functions\nin the AKK set~\\cite{Albino:2005me}. The factorization scale is tied to the \nparton transverse momentum via $Q = (2\/3) p_T\/z_c$, while the\nfragmentation scale varies with the transverse momentum of the outgoing hadron,\naccording to $Q_f = (2\/3) p_T$, following the best-fit results obtained in\nRef.~\\cite{Levai:2006yd}. This scale fixing is not modified from earlier applications,\nand is used consistently throughout the present calculations.\nTo protect against divergences in the partonic cross sections, a cutoff \nregulator mass is necessary. We have tested the sensitivity of our\ncalculations at different cms energies by varying the regulator mass \nbetween $0.5-2.0$ GeV. For $p_T>1.5$ GeV\/c our results show little\nsensitivity to variation of regulator mass. We present results for \n$p_T>2.25$ GeV\/c here, and thus regulator mass effects are expected \nto be minimal.\nWe obtain the density distribution of the deuteron from the Hulthen \nwave function~\\cite{Hulthen1957} (as in Ref.~\\cite{Kharzeev:2002ei}), while a \nWoods-Saxon density distribution is used for gold and lead with parameters from\nRef.~\\cite{DeJager:1974dg}.\n\n\\subsection{Forward and backward nuclear modifications}\n\\label{rapasym}\n\nThe nuclear modification factor is designed to compare, as a ratio, spectra \nof particles produced in nuclear collisions to a hypothetical scenario \nin which the nuclear collision is assumed to be a superposition of the \nappropriate number of nucleon-nucleon collisions. The ratio can be \ndefined as a function of $p_T$\nfor any produced hadron species $h$ at any pseudorapidity $\\eta$:\n\\begin{equation}\nR^h_{AB}(p_T, \\eta) = \\frac{1}{\\langle N_{bin}\\rangle} \\cdot\n\\frac{E_h \\dd^3\\sigma_{AB}^{h}\/\\dd^3 p |_{\\eta}}\n{E_h \\dd^3\\sigma_{pp}^{h}\/\\dd^3 p |_{\\eta}} \\,\\, ,\n\\label{rdau}\n\\end{equation}\nwhere $\\langle N_{bin} \\rangle$ is the average number of binary collisions in \nthe various impact-parameter bins.\nNuclear effects manifest themselves in $R^h_{AB}(p_T, \\eta)$ values greater or \nsmaller than unity, representing enhancement or suppression, respectively, \nrelative to the $NN$ reference.\n\nIn asymmetric collisions, hadron production at forward rapidities\nmay be different from what is obtained at backward \nrapidities. It is thus of interest to study ratios of particle\nyields between a given pseudorapidity value and its negative in these\ncollisions. The pseudorapidity asymmetry $Y_{Asym}(p_T)$ is defined \nfor a hadron species $h$ as \n\\begin{equation}\nY^h_{Asym}(p_T) = \\left. E_h\\f{\\dd ^3 \\sigma_{AB}^{h}}{\\dd^3 p} \\right|_{-\\eta} \n \\left\/ \n\\left. E_h\\f{\\dd ^3 \\sigma_{AB}^{h}}{\\dd^3 p} \\right|_{\\eta} \\right. \\,\\, .\n\\label{yasym}\n\\end{equation}\n\nLet us consider the (double) ratio of the \nforward and backward nuclear modification factors in $dAu$\ncollisions for species $h$:\n\\begin{eqnarray}\nR^h_{\\eta}(p_T) = \\frac{R^h_{dAu}(p_T,-\\eta)}{R^h_{dAu}(p_T,\\eta)}= \\nonumber \\\\ \n\\frac{E_h \\dd^3\\sigma_{dAu}^{h}\/\\dd^3 p |_{-\\eta}}\n{E_h \\dd^3\\sigma_{pp}^{h}\/\\dd^3 p |_{-\\eta}} \\left\/ \n\\frac{E_h \\dd^3\\sigma_{dAu}^{h}\/\\dd^3 p |_{\\eta}}\n{E_h \\dd^3\\sigma_{pp}^{h}\/\\dd^3 p |_{\\eta}} \\right. \\,\\, . \n\\label{y-r:eq}\n\\end{eqnarray}\nAs discussed in Ref.~\\cite{Barnafoldi:2008rb}, since the $pp$ \nrapidity distribution is symmetric around $y=0$,\nif the same backward and forward (pseudo)rapidity ranges are taken\nin both directions (i.e. $ |\\eta_{min}| \\leq |\\eta| \\leq |\\eta_{max}| $), \nthen the $pp$ yields cancel in eq.~(\\ref{y-r:eq}) and one obtains\nthat the ratio defined in (\\ref{y-r:eq}) is identical to \nthe pseudorapidity asymmetry (\\ref{yasym}): \n\\begin{equation}\nY^h_{Asym}(p_T)= R^h_{\\eta}(p_T)=\\frac{R^h_{dAu}(p_T,-\\eta)}{R^h_{dAu}(p_T,\\eta)} \\,\\, .\n\\label{y-r2:eq}\n\\end{equation}\n\nEq. (\\ref{y-r2:eq}) is useful from the experimantal point of view to handle the \nsystematic errors of data and extract the proper pseudorapidity asymmetry. It also \nprovides a connection between measured rapidity asymmetry and the nuclear \nmodification factors. \n\n\n\n\\subsection{Nuclear shadowing and multiple scattering}\n\\label{shad_multiscatt}\n\nWe expect that particle production in $pA$ ($dA$) collisions will have \ndifferent yields in the forward and backward directions. This is because the\nrespective partons have different momentum fractions (shadowing differences)\nand because the forward-going parton has to traverse a large amount of matter.\nWe found in our earlier studies that the HIJING shadowing parameterization\nis particularly useful for the study of multiple scattering, as it requires\nan explicit treatment of $k_T$ broadening (see eq.~(\\ref{eq:broad})) for a \nsuccessful description of data~\\cite{Zhang:2001ce}. In this Section we therefore \nstudy the interplay of shadowing and multiple scattering in more detail \nusing the HIJING parameterization. \n\n\\begin{figure}[!htb]\n\\begin{center} \n\\includegraphics[width=8.5cm, height=8.5cm, angle=0]{lin_etadist_hij_pt_3.eps}\n\\end{center}\n\\caption[...]{(Color Online) Illustration of calculated yields at $p_T=3.0$ GeV\/c in\nminimum bias neutral pion production from $dAu$ collisions at 200 $A$GeV. \nThe solid line represents HIJING shadowing with intrinsic $k_T$ in the\nproton, the dashed line is HIJING with intrinsic $k_T$ and multiple \nscattering, while the dot-dashed curve is obtained by turning off\nany transverse momentum.}\n\\label{fig:multiscatt}\n\\end{figure}\n\nIn Fig.~\\ref{fig:multiscatt} we display the \nneutral pion yield as a function of pseudo-rapidity in $dAu$ collisions \nat a fixed $p_T$, namely at $ 3.0$ GeV\/c. This illustrates how the \nnuclear effects we examine modify the asymmetry of the yields. \nThe figure shows the calculated distribution using the HIJING shadowing\nparameterization without any intrinsic $k_T$ (dot-dashed curve), \nusing HIJING shadowing and including the intrinsic $k_T$ in the nucleon (solid), \nand HIJING shadowing with intrinsic $k_T$ and multiple scattering (dashed).\nThe parameters of eq.~(\\ref{eq:broad}) are unchanged from our previous \nstudies at midrapidity~\\cite{Levai:2003at,Levai:2006yd}, and \nwe have chosen a transverse momentum value comparable to $\\langle k_T \\rangle_{pp}$,\nwhere the various effects are clearly displayed. \n\nWe checked that when shadowing and all nuclear effects are turned off, \nthe distribution is symmetric around midrapidity, as expected. \nWith shadowing only, we find an asymmetry in the distribution: the yield \nat a fixed negative pseudorapidty, say $\\eta=-2$ ($Au$ side or ``backward'') \nis higher than at the corresponding positive pseudorapidity ($\\eta=2$, $d$ side, \nor ``forward''). Thus the pseudorapidity asymmetry, $Y_{Asym}$, is greater \nthan unity in this case. The inclusion of the intrinsic transverse momentum \nin the proton significantly increases the yields on both the $Au$ side and the\n$d$ side at this transverse momentum, as it makes larger $p_T$-s accessible.\nWe tested that adding intrinsic transverse momentum in the proton in the \nabsence of shadowing does not destroy the forward\/backward symmetry. When\nintrinsic $k_T$ in the proton is added to the calculation using shadowing,\nbut without the multiple scattering contribution, the sense of the asymmetry \ngiven by shadowing (backward\/forward $\\geq$ 1) is preserved. However, when \nmultiple scattering is included, the yield shows a stronger increase in \nthe forward direction. This is understandable in the present picture, since \nforward-going products originate in the partons of the deuteron, and have\nto traverse a large amount of nuclear matter resulting in strong multiple \nscattering, while the backward products suffer no or little multiple \nscattering. This has the effect of reversing the asymmetry: the yield on the \n$d$-side is now greater than that on the $Au$-side. The calculated pseudorapidity \nasymmetry, $Y_{Asym}$, turns out to be less than unity in this case.\nAn alternative way to summarize the situation is to say that shadowing suppresses\nthe yield more on the $d$ side (forward) relative to a symmetric collision,\nwhile the multiple scattering contribution is understandably large in the \nforward direction.\n \nWe have carried out similar studies at higher $p_T$ values. When the intrinsic\ntransverse momenta are small relative to the $p_T$ of the final hadron, $k_T$\neffects become naturally smaller. Shadowing effects also become smaller as\nthe antishadowing region of the HIJING parameterization is approached. Thus,\nat $p_T \\gtrsim 15$ GeV\/c the influence of multiple scattering and intrinsic\n$k_T$ become negligible. These phenomena are most important at intermediate $p_T$ \nvalues, 2 GeV\/c $\\lesssim p_T \\lesssim$ 8 GeV\/c at RHIC. It is interesting to note\nthat a similar transverse-momentum region is sensitive to nuclear effects \nat lower (e.g. CERN SPS) energies, due to the $\\sim \\log(\\sqrt{s})$ scaling of the \nCronin peak~\\cite{Barnafoldi:2007uw,e706,Zielinski}. \n\n\\section{Results}\n\\label{res}\n\nTo judge the success of the model in reproducing spectra, \nhere we first present calculated spectra for neutral pion production \nat midrapidity and non-zero rapidities. This will help select the model \nchoices providing the best agreement with the experimental information. \nThe selected model variants are then used to calculate pseudorapidity \nasymmetries.\n \n\\subsection{Midrapidity spectra}\n\\label{centrap}\n\nFigure~\\ref{fig:dAuspect1} displays midrapidity RHIC $\\pi^0$ spectra from \n$dAu$ collisions at $|\\eta| < 0.35$, and for reference, the spectra \nfrom $pp$ collisions at the same energy ($\\sqrt{s}= 200 A$GeV). The results are \ncompared to data from the PHENIX collaboration~\\cite{PHENIXdAu}. In the top left\npanel we show calculated $dAu$ spectra using EPS08 nPDFs with and without intrinsic $k_T$ \nin the proton, but without any multiple scattering contribution. The top right depicts \nthe $dAu$ spectra using HIJING plus $k_T$ with and without multiple scattering.\nThe bottom panels contain data\/theory (i.e. data\/model) ratios. \nAs can be seen clearly in the bottom left $dAu$ data\/theory panel, including \nintrinsic $k_T$ in the proton increases the calculated yield mostly at low $p_T$.\nWe consider the EPS08 description with intrinsic $k_T$ in the proton\nsatisfactory in the 4 GeV\/c \n$\\lesssim p_T \\lesssim$ 10 GeV\/c interval. Comparing the two bottom \ndata\/theory panels it is easy to see that the HIJING parameterization gives\na very similar accuracy for $dAu$ when multiple scattering is included. This is \nalso very close to the data\/theory for $pp$ collisions given by HIJING. \n\n\\begin{figure}[!h]\n\\begin{center} \n\\includegraphics[width=8.5cm, height=8.5cm, angle=270]{dAus1.ps}\n\\end{center}\n\\caption[...]{(Color Online) Top: spectra for $dAu$ $\\pi^0$ production at \n$|\\eta| < 0.35$. The left panel shows EPS08 with (dashed) and without (solid) \nintrinsic $k_T$. The right panel is HIJING plus $k_T$ with (solid) and without \n(dashed) multiple scattering. Filled triangles denote the PHENIX data~\\cite{PHENIXdAu}. \nThe $pp$ spectra from the same experiment and calculated with (dashed) and without \n(solid) intrinsic $k_T$ are also included. Bottom: corresponding data\/theory\nratios for (left) $dAu$ with and without $k_T$ and (right) $pp$ with $k_T$\nand HIJING plus $k_T$ with multiple scattering.}\n\\label{fig:dAuspect1}\n\\end{figure}\n\n\\begin{figure}[!h]\n\\begin{center} \n\\includegraphics[width=8.5cm, height=8.5cm, angle=270]{pBes1.ps}\n\\end{center}\n\\caption[...]{(Color Online) Spectra for $p+Be \\rightarrow \\pi{^0}+X$ at $|\\eta| < 0.2$.\nThe left panel depicts EPS08 calculations with (dashed) and without (solid) intrinsic $k_T$. \nThe right panel is HIJING plus $k_T$ with (solid) and without (dashed)\nmultiple scattering. Filled triangles denote the E706\ndata~\\cite{Apanasevich:2002wt}. Data and calculations for proton-proton (pp)\ncollisions are at $530$ GeV\/c.}\n\\label{fig:pBespect1}\n\\end{figure}\n\nFig.~\\ref{fig:pBespect1} shows the spectra for neutral pion \nproduction from $pBe$ collisions at $|\\eta| < 0.2$. The top left \npanel shows the spectra using EPS08 nPDFs with and without intrinsic\n$k_T$. The top right depicts the spectra using HIJING plus $k_T$ \nwith and without multiple scattering. The agreement with the E706 \ndata from Fermilab is quite good as can be seen from the lower panels\nwhich display the data per theory ratio. The HIJING \nparameterization with and without multiple scattering gives almost \nidentical data\/theory ratios, i.e. multiple scattering has only\na small effect on spectra from $pBe$ \ncollisions. This is reasonable in view of the fact that in \nthe light $Be$ nucleus there are few \nscattering centers. The effect of intrinsic \n$k_T$ is to increase the calculated yield, leading to data\/theory ratios\nless than unity for all relevant $p_T$. \n\nWe conclude from Fig.s~\\ref{fig:dAuspect1} and \\ref{fig:pBespect1} that\nit is necessary to include the intrinsic transverse \nmomentum of partons in the proton in our model to obtain a satisfactory\ndescription of available data. The EPS08 and HIJING parameterizations differ\nto the extent that while HIJING calls for the inclusion of the broadening of\nthe intrinsic transverse momentum distribution via multiple scattering, \nEPS08 appears to incorporate this physics in their nPDFs. We thus \nconcentrate on three model variants (EPS08 with proton intrinsic $k_T$ and \nHIJING with intrinsic $k_T$ and with\/without multiscattering) in the remainder\nof this study.\n\n\\subsection{Spectra at non-zero rapidities}\n\\label{noncentrap}\n\nWe now consider spectra at non-zero pseudorapidities. \nWe display results from EPS08 with proton \nintrinsic $k_T$ and HIJING with intrinsic $k_T$ and multiscattering.\nFig.~\\ref{fig:pBespect2} shows the spectra and corresponding data\/theory ratio \nfor $p+Be \\rightarrow \\pi{^0}+X$ at $-0.7 < \\eta < -0.2$ (``backward'', left panel) \nand $0.2 < \\eta < 0.7$ (``forward'', right panel) using EPS08 nPDFs with $k_T$ \nand HIJING plus $k_T$ with multiscattering. The two sets (EPS08 and HIJING) give\nvery similar data\/theory ratios for both pseudorapidity intervals. There is\nreasonable agreement with the E706 data~\\cite{Apanasevich:2002wt} as is apparent \nfrom the quality of the data\/theory ratios.\n\\begin{figure}[!h]\n\\begin{center} \n\\includegraphics[width=8.5cm, height=8.5cm, angle=270]{pBes2.ps}\n\\end{center}\n\\caption[...]{(Color Online) Spectra for $p+Be \\rightarrow \\pi{^0}+X$ at \n$-0.7 < \\eta < -0.2$ (left panel) and $0.2 < \\eta < 0.7$ (right panel). \nThe solid lines represent the EPS08 nPDFs with intrinsic $k_T$, while \nthe dashed line is obtained from HIJING \nplus $k_T$ with the inclusion of multiscattering. Stars denote \nthe E706 data~\\cite{Apanasevich:2002wt}.}\n\\label{fig:pBespect2}\n\\end{figure}\n\nThe $dAu$ spectra are displayed in Fig.~\\ref{fig:dAuspect2} for both \nEPS08 and HIJING at $-1.0 < \\eta < -0.5$ (left panel) and $0.5 < \\eta < 1.0$\n(right panel). Note that the experimental data from the STAR \ncollaboration~\\cite{Abelev:2006pp} are not separated into negative and\npositive pseudorapidities, but rather averaged over both \nintervals. Therefore, the data points in Fig.~\\ref{fig:dAuspect2} only\nserve to guide the eye; the relevance of the Figure is to highlight the \ndifference between the EPS08 and HIJING results visible in the bottom panels.\nIt can be seen that the EPS08 results do not differ\nmuch in the forward and backward directions, while HIJING gives significantly larger\ndata\/theory ratios forward. Thus, EPS08 and HIJING \ndiffer appreciably at forward pseudorapidities, $0.5 < \\eta < 1.0$.\nThis is not unexpected, because multiple scattering influences\nthe $d$-side (forward) more than the $Au$-side (backward).\nTo be able to draw stronger conclusions, separated forward and backward\ndata will be necessary, hopefully forthcoming from the high-statistics Run 8.\n\n\\begin{figure}[!h]\n\\begin{center} \n\\includegraphics[width=8.5cm, height=8.5cm, angle=270]{dAus2.ps}\n\\end{center}\n\\caption[...]{(Color Online) Spectra for $d+Au \\rightarrow \\pi{^0}+X$ at \n$-1.0 < \\eta < -0.5$ (left panel) and $0.5 < \\eta < 1.0$ (right panel). \nThe solid line represents the\nEPS08 with intrinsic $k_T$, while the dashed line is obtained from HIJING \nplus $k_T$ with the inclusion of multiscattering. Stars denote STAR data\naveraged over $0.5 < |\\eta| < 1.0$}\n\\label{fig:dAuspect2}\n\\end{figure}\n\n\\subsection{Pseudorapidity asymmetry}\n\\label{etasmy}\n\n\\subsubsection{Asymmetry in $pBe$ collisions at $30.7$ GeV}\n\\label{asympBe}\n\nThe pseudorapidity asymmetry for the rapidity interval\n$0.2 < |\\eta| < 0.7$ is shown in the upper panel of Fig.~\\ref{fig:asyfnal} \ncompared with the E706 data~\\cite{Apanasevich:2002wt}.\nThe solid line represents EPS08 with proton intrinsic $k_T$, the dot-dashed line \ndisplays HIJING with intrinsic $k_T$, and the dashed is HIJING plus \nintrinsic $k_T$ and multiple scattering. Both EPS08 with $k_T$ and HIJING \nwithout multiple scattering give very small asymmetries and the data are\nalso consistent with $Y_{Asym}=1$ for low $p_T$. The HIJING parameterization\nwith multiple scattering yields somewhat lower values at all transverse momenta.\nThis is in line with the effect of multiple scattering moderately increasing \nthe yield on the $p$-side relative to that of the $Be$-side. In view of the rather \nlarge error bars, all three sets are in reasonable agreement with the data. \n\\begin{figure}[!h]\n\\begin{center} \n\\includegraphics[width=8.5cm, height=8.5cm, angle=270]{asympBe.ps}\n\\end{center}\n\\caption[...]{(Color Online) Pseudorapidity asymmetry,\n$Y_{Asym}$ for $p+Be \\rightarrow \\pi{^0}+X$ at $0.2 < |\\eta| < 0.7$ (top)\nand $1.0 < |\\eta| < 1.5$ (bottom). The solid line represents the\nEPS08 nPDFs, while the dashed line is obtained from HIJING with \nthe inclusion of multiscattering. The dot-dashed line \ncorresponds to HIJING without multiscattering, and filled triangles denote \nthe E706 data~\\cite{Apanasevich:2002wt}.}\n\\label{fig:asyfnal}\n\\end{figure}\nThe lower panel is our prediction for the interval $1.0 < |\\eta| < 1.5$. \nHere, the calculated effects are larger, but have a similar structure to\nwhat is seen at lower $\\eta$. This trend is similar to what will be seen at \nother energies. \n\n\n\\subsubsection{Asymmetry in $dAu$ collisions at $200$ $A$GeV}\n\\label{asymdAu}\n\nFigure~\\ref{fig:asyrhic} shows the pseudorapidity asymmetry for $\\pi^0$ \nproduction from $dAu$ collisions at RHIC, for different pseudorapidity intervals. \nThe two uppermost panels are our results for the asymmetry at $|\\eta| < 0.5$ \nand $0.5 < |\\eta| < 1.0$ compared with the STAR data~\\cite{Abelev:2006pp}. For\n$p_T > 4.0$ GeV\/c, the agreement with data is quite good for all three sets.\nAt lower $p_T$, multiple scattering increases the calculated yield mostly in the \nforward direction as discussed in Sec.~\\ref{shad_multiscatt}, leading to \nasymmetries below unity. At very high $p_T$ we observe a divergence in \nthe model predictions.\n\\begin{figure}[!h]\n\\begin{center} \n\\includegraphics[width=8.5cm, height=8.5cm, angle=270]{asymdAu1.ps}\n\\end{center}\n\\caption[...]{(Color Online) Pseudorapidity asymmetry,\n$Y_{Asym}$ for $d+Au \\rightarrow \\pi{^0}+X$ at different \npseudorapidity intervals. The solid line represents the\nEPS08 nPDFs, while the dashed line is obtained using HIJING shadowing with \nthe inclusion of multiscattering. The dot-dashed line \ncorresponds to HIJING without multiscattering, and filled triangles denote \nthe STAR data~\\cite{Abelev:2006pp}.}\n\\label{fig:asyrhic}\n\\end{figure}\nThe lower four panels are our predictions for the asymmetry as pseudorapidity \nincreases. The first three of these correspond to the BRAHMS\npseudorapidity intervals~\\cite{Arsene:2004ux}. The general trend is that the\nasymmetry becomes larger as $\\eta$ increases. This mainly arises\nfrom the strong shadowing in the larger nucleus at lower $x$ values. Also, \nsince increasing $\\eta$ leads to decreasing accessible $p_T$ \ndue to phase space constraints, the effects of multiple scattering become \nmore pronounced. In fact, for the largest $\\eta$ considered, \n$3.7 < |\\eta| < 4.3$, there is a marked difference between HIJING with \nand without multiple scattering for all $p_T$ considered. \n\nIt is pertinent at this point to make some remarks about\npseudorapidity asymmetry in the BRAHMS pseudorapidity intervals.\nThe BRAHMS Collaboration~\\cite{Arsene:2004ux} has observed a progressive \nsuppression of both minimum bias nuclear modification $R_{dAu}$ and \ncentral-to-peripheral ratios, $R_{CP}$, with increasing\n$\\eta$. The present study is limited to minimum bias pseudorapidity\nasymmetry, and both shadowing parameterizations (EPS08 and HIJING) adequately \ndescribe the existing experimental data at very forward rapidities. \nThe EPS08 parameterization incorporates RHIC data at large rapidities, i.e. at\nlow $x$, and thus reproduces the data. Therefore, at least in the \nminimum bias case, shadowing seems sufficient for a good description of \nthe suppression observed at low $x$. \nThe situation is different for the geometry-dependent $R_{CP}$, \nwhere ($b$-independent) shadowing plus conjectured \nimpact parameter dependencies~\\cite{Adeluyi:2008qk,Vogt:2004hf} \nare clearly inadequate in describing the observed suppression.\n\n\\subsubsection{Asymmetry in $dPb$ collisions at $8.8$ $A$TeV}\n\\label{asymdPb}\n\nLet us now turn to our predictions for the pseudorapidity asymmetry in a potential future \n$dPb$ collision at LHC energy of $8.8$ $A$TeV. \nThe calculated results are displayed in \nFig.~\\ref{fig:asylhc}, where the upper panel is for the interval $|\\eta| < 0.9$ \nand the lower panel is for $2.4 < |\\eta| < 4.0$. These \nintervals correspond to acceptance in the central detector and in the muon \narm, respectively, of the ALICE experiment~\\cite{Alessandro:2006yt}. All three \nsets predict minimal asymmetry of the order of a few percent for the interval \n$|\\eta| < 0.9$. \n\\begin{figure}[!h]\n\\begin{center} \n\\includegraphics[width=8.5cm, height=8.5cm, angle=270]{asymdPb1.ps}\n\\end{center}\n\\caption[...]{(Color Online) Predicted pseudorapidity asymmetry,\n$Y_{Asym}$ for $d+Pb \\rightarrow \\pi{^0}+X$ at $\\sqrt{s} = 8.8$ $A$TeV\nfor $|\\eta| < 0.9$ and $2.4 < |\\eta| < 4.0$. The solid line represents the\nEPS08 nPDFs, while the dashed line is obtained from HIJING with \nthe inclusion of multiscattering. The dot-dashed line \ncorresponds to HIJING without multiscattering.}\n\\label{fig:asylhc}\n\\end{figure}\nAs we move to higher $\\eta$, the predicted asymmetry becomes more significant. \nAs can be seen in the lower panel of Fig.~\\ref{fig:asylhc}, both EPS08 and \nHIJING predict substantial asymmetry up to $\\sim 10$ GeV\/c, and both variants\nof HIJING asymmetries remaining significant up to $\\sim 100$ GeV\/c, in \ncontrast to EPS08. \n\nAt the present level, neither model variant gives agreement with all aspects of \nthe data: in an earlier calculation we have found that shadowing \nparameterizations which do not need to be augmented by a multiple scattering \nprescription~\\cite{Eskola:1998df,Shad_HKN,Frankfurt:2003zd} have difficulty \ndescribing central-to-peripheral ratios at forward rapidity~\\cite{Adeluyi:2008qk}. \nWe have checked that this also holds for the EPS08 nPDFs. On the other hand,\nthe HIJING parameterization with multiscattering yields pseudorapidity \nasymmetries below unity at low transverse momenta. This deficiency may be cured \nby allowing an $\\eta$-dependent multiscattering~\\cite{gergely:2008}. \n\\section{Conclusion}\n\\label{concl}\n\nHere we give a concise summary of the results of our\ncalculations. We have demonstrated the usefulness of asymmetric \n(light-on-heavy) nuclear collisions at relativistic energies. As illustrated\nin Sec.~\\ref{shad_multiscatt}, the physical differences between forward- and \nbackward-going produced particles arise from the different ranges of $x$ \nsampled (different shadowing) and different amount of multiple scattering.\nThis leads to observable pseudorapidity asymmetries\nat some collision energies and transverse momenta.\nWe have considered the effects of nuclear shadowing \nand multiple scattering on pseudorapidity asymmetry for three asymmetric \nsystems: $pBe$, $dAu$, and $dPb$ in a wide energy range.\n\nTo calibrate and fine-tune our model we first examined spectra of\nproduced neutral pions. We found that there are two avenues in the model for \nthe reasonable description of these data: (i) HIJING shadowing, intrinsic\n$k_T$, plus multiscattering, or (ii) some other nPDFs (we use EPS08 here) \nand intrinsic $k_T$ (but no additional multiscattering). We then calculated \npseudorapidity asymmetries with these prescriptions.\nOverall, the calculated asymmetries are in reasonable agreement with available \nexperimental data. Intrinsic transverse momentum in the nucleon is\nseen to be important at low $p_T$. Multiple scattering \nincreases the yield in the ``forward'' or positive pseudorapidity region,\nthus leading to a tendency for asymmetries less than unity at low $p_T$\nin the scheme explicitely relying on multiple scattering,\nat variance with the data. An LHC measurement at a high pseudorapidity \nand high $p_T$ (where $k_T$ effects no longer make a difference) may be \nable to distinguish between strong shadowing (as in the HIJING prescription) \nand an nPDF with a relatively weaker gluon suppression (like e.g. EPS08).\n\nA major constraint in assessing pseudorapidity asymmetries is the limited \navailability of data for direct comparison with theoretical calculations. \nMore data in asymmetric light-on-heavy collisions separated with respect\nto positive and negative pseudorapidities are needed\nto judge calculated pseudorapidity asymmetries. At RHIC, it is expected\nthat the high-statistic $dAu$ Run 8 is going to provide such a large data set.\n\n\n\\section{Acknowledgments}\n\\label{ack}\n\nThis work was supported in part by Hungarian OTKA PD73596,\nT047050, NK62044, and IN71374, by the U.S. Department of Energy under\ngrant U.S. DOE DE-FG02-86ER40251, and jointly by the U.S. and Hungary under\nMTA-NSF-OTKA OISE-0435701.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{Introduction}\nAn important issue in general relativity is the nature of singularities. While it is widely believed that the strong cosmic censorship conjecture holds, which can be loosely stated as the expectation that timelike singularities do not form by evolution from generic spatially compact or asymptotically flat initial data sets \\cite{Penrose1979}, the issue is wide open. From this perspective it is of interest to consider timelike singularities and therefore the ways in which cosmic censorship could be violated.\n\nThere are numerous exact solutions with timelike singularities (e.g. \\cite{Griffiths2009}). Such solutions are typically obtained in searches of solutions with symmetries. This leads naturally to the question, whether there exist solutions with timelike singularities and without symmetries. We prove in this work that this is indeed the case: We construct an analog to the class of the non-chaotic solutions without symmetries and with controlled asymptotics of \\cite{Chrusciel2015,Klinger2015}, by changing the time parameter $\\tau$ from a timelike to a spacelike coordinate. As the behavior of the Hamiltonian differs from the spacelike case only by sign changes which do not affect the analysis in the analytic case, we obtain a family of solutions with the same free functions and asymptotics (in terms of the now spacelike $\u03c4$ coordinate) but with a timelike instead of spacelike singularity.\n\nThe construction of the solutions is based on the cosmological billiard formalism using the Iwasawa decomposition of the metric. This method was introduced by Damour, Henneaux, and Nicolai in \\cite{Damour2003} to give a heuristic argument for the chaotic picture of spacelike singularities provided by the BKL conjecture, and later by Damour and DeBuyl in \\cite{Damour2008} to provide a precise statement of the conjecture.\n\nWe also show that the change of the time parameter $\\tau$ from a timelike to a spacelike coordinate, i.e. considering timelike instead of spacelike singularities, switches the signs of some of the terms (walls) in the Hamiltonian considered. These changes violate the property of the spacelike case that the coefficients of the dominant wall terms are positive, thus rendering the arguments of Damour et al. in \\cite{Damour2003} inapplicable. The affected terms become attractive rather than repulsive, allowing subdominant walls lying behind the dominant ones to become relevant. This does not affect the class of solutions we construct here, as these are non-generic and use an ansatz that suppresses the wall terms asymptotically.\n\n\\section{Derivation of Hamiltonian for spacelike ``time''-variable}\nWe follow the derivation of the Hamiltonian formalism by Wald \\cite[Appendix E.2]{Wald1984}. The spacetime metric is denoted by $\\bar{g}_{\u03b1\u03b2}$ while the induced lorentzian metric on the timelike hypersurfaces of constant $\u03c4$ is denoted by $g_{ij}$. We choose a zero shift gauge, i.e. the metric takes the form\n\\begin{equation}\n\\mathrm{d}s^2=N^2\\mathrm{d}\u03c4^2+g_{ij}\\mathrm{d}x^i\\mathrm{d}x^j\\,.\n\\end{equation}\n\nAs the hypersurfaces of constant $\u03c4$ are timelike, their normal vector is spacelike. This means that the Gauss equation takes the form\n\\begin{equation}\nR\\indices{_{abc}^d}=g_a{}^f g_b{}^g g_c{}^k g^d{}_j\\bar{R}\\indices{_{fgk}^j}+K_{ac}K\\indices{_b^d}-K_{bc}K\\indices{_a^d}\n\\end{equation}\nwhere $R\\indices{_{abc}^d}$ and $\\bar{R}\\indices{_{abc}^d}$ are the Riemann curvature tensors of the induced and full metric respectively and $K_{ab}$ is the second fundamental form of the hypersurface. Compared to the case of spacelike hypersurfaces the signs of the $KK$ terms are interchanged.\n\nUsing $g_{\u03b1\u03b2}=\\bar{g}_{\u03b1\u03b2}-n_\u03b1 n_\u03b2$ with $n^\u03b1$ the unit normal vector of the hypersurface ($n_\u03b1n^\u03b1=1$) gives\n\\begin{equation}\n\\bar{R}_{\u03b1\u03b2\u03b3\u03b4}g^{\u03b1\u03b3}g^{\u03b2\u03b4}=-2\\bar{G}_{\u03b1\u03b2}n^\u03b1n^\u03b2\\,.\n\\end{equation}\n\nThis leads to a change of sign in the constraint equation:\n\\begin{equation}\\label{const}\n0=\\bar{G}_{\u03bc\u03bd}n^\u03bc n^\u03bd = -\\frac{1}{2}R-(K\\indices{^\\mu_\\mu})^2+K_{\u03bc\u03bd}K^{\u03bc\u03bd}\\,.\n\\end{equation}\n\nContracting the Einstein tensor twice with the normal vector $n^a$ gives an expression for the scalar curvature:\n\\begin{equation}\\label{scurv}\n\\bar{R}=-2n^\u03b1n^\u03b2(\\bar{G}_{\u03b1\u03b2}-\\bar{R}_{\u03b1\u03b2})\\,.\n\\end{equation}\n\nThe definition of the Riemann tensor gives for the last term\n\\begin{equation}\\label{riemnn}\n\\begin{split}\n\\bar{R}_{\u03b1\u03b2}n^\u03b1n^\u03b2=&\\bar{R}_{\u03b1\u03b3\u03b2}{}^\u03b3n^\u03b1n^\u03b2=-n^\u03b1(\\nabla_\u03b1\\nabla_\u03b3-\\nabla_\u03b3\\nabla_\u03b1)n^\u03b3\\\\\n=&(\\nabla_\u03b1 n^\u03b1)(\\nabla_\u03b3 n^\u03b3)-(\\nabla_\u03b3 n^\u03b1)(\\nabla_\u03b1 n^\u03b3)\\\\\n&-\\nabla_\u03b1(n^\u03b1\\nabla_\u03b3 n^\u03b3)+\\nabla_\u03b3(n^\u03b1\\nabla_\u03b1n^\u03b3)\\\\\n=&(K^\u03b1_\u03b1)^2-K_{\u03b1\u03b3}K^{\u03b1\u03b3}-\\nabla_\u03b1(n^\u03b1\\nabla_\u03b3 n^\u03b3)+\\nabla_\u03b3(n^\u03b1\\nabla_\u03b1n^\u03b3)\n\\end{split}\n\\end{equation}\nwhere the last two terms are divergences, which will be discarded in the Lagrangian.\n\nUsing \\eqref{const}, \\eqref{scurv}, \\eqref{riemnn} and $\\sqrt{-\\bar{g}}=N\\sqrt{-g}$ to express the Einstein-Hilbert action gives\n\\begin{equation}\n\\mathcal{L}=\\sqrt{-\\bar{g}}\\bar{R}=-\\sqrt{-g}N\\left(R-K_{ab}K^{ab}+(K^a{}_a)^2\\right)\\,.\n\\end{equation}\nThe momenta canonically conjugate to the components $g_{ij}$ are given by\n\\begin{equation}\n\u03c0^{ij}=\\frac{\u2202\\mathcal{L}}{\u2202\\dot{g}_{ij}}=\\sqrt{-g}(K^{ij}-K^k{}_k g^{ij})=N^{-1}\\sqrt{-g}\\frac{1}{2}(\\dot{g}_{kl}g^{ki}g^{lj}-\\dot{g}_{kl} g^{kl}g^{ij})\\,,\n\\end{equation}\nunchanged from the standard case.\n\nThe Hamitonian, expressed in terms of the canonical coordinates $g_{ab}$ and momenta $\u03c0^{ab}$ is finally\n\\begin{equation}\\label{hamiltonian}\n\\mathcal{H}=\u03c0^{ab}\\dot{g}_{ab}-\\mathcal{L}=(-g)^{-1\/2}N\\left(\u03c0^{ab}\u03c0_{ab}-\\frac{1}{2}(\u03c0^a{}_a)^2\\right)+RN\\sqrt{-g}\n\\end{equation}\ni.e. the standard one with the sign of the curvature term changed.\n\n\\section{Iwasawa variable Hamiltonian}\nHere we will describe the changes to the derivation of the Iwasawa variable Hamiltonian, as given in Appendix A of \\cite{Klinger2015}.\n\nSince the level sets of $\\tau$ are timelike, we need to decide which frame vector is the timelike one. As the Iwasawa ansatz breaks the symmetry between the frame vectors, different choices will lead to different dynamical systems. We will use an index $J\\in\\{1,2,3\\}$ to distinguish between those cases: $x^J$ will denote the timelike coordinate.\n\nThe Lorentzian metric $g_{ij}$ on the $\u03c4=\\text{const}$ hypersurfaces is split in Iwasawa variables as\n\\begin{equation}\ng_{ij}=\\sum_a m^J_a e^{-2\u03b2^a}\\N{^a_i}\\N{^a_j}\n\\end{equation}\nwhere $m^J_a=1-2\u03b4_{Ja}$, i.e. $-1$ for $a=J$ and $1$ otherwise. \n\nWe set the lapse function $N$ equal to $\\sqrt{-g}$ where $g$ is the determinant of the metric $g_{ij}$. The (timelike) singularity will be approached as $\u03c4\\to \\infty$.\n\nThe conjugate momenta $\u03c0_a$to the $\u03b2^a$ and $P\\indices{^i_a}$ to the $\\N{^a_i}$ are given by\n\\begin{equation}\n\u03c0_a=\\frac{\u2202\\mathcal L}{\u2202\\dot{\u03b2}^a}=\\frac{\u2202\\mathcal L}{\u2202\\dot{g}_{ij}}\\frac{\u2202\\dot{g}_{ij}}{\u2202\\dot{\u03b2^a}}=-2\u03c0^{ij}m^J_ae^{-2\u03b2^a}\\N{^a_i}\\N{^a_j}\n\\end{equation}\nand\n\\begin{equation}\nP\\indices{^i_a}=2m^J_a \u03c0^{ij}\\N{^a_j}e^{-2\u03b2^a}\\,,\n\\end{equation}\ni.e. the same as in the spacelike case except for the additional factor $m^J_a$.\n\nThe non-curvature terms of the Hamiltonian \\eqref{hamiltonian}, with $N=\\sqrt{-g}$ inserted, are\n\\begin{equation}\\label{2015X30.2}\n\u03c0^{ab}\u03c0_{ab}-\\frac{1}{2}(\u03c0^a{}_a)^2\\,.\n\\end{equation}\n\nThe first term can be split into\n\\begin{equation}\\label{2015X30.1}\n\\frac{1}{4}\\sum_a \u03c0_a^2+\\frac{1}{2}\\sum_{a0\\,.\n\\end{cases}\n\\end{equation}\nIf, however, some of the prefactors are negative the corresponding terms are potential wells instead of walls. In the timelike case in 3+1 dimensions this affects at least one of the dominant symmetry walls.\n\nFigure \\ref{fig:hyp} shows the potentials in the hyperbolic space of the $\u03b3^a$, projected onto the Poincar\\'{e} disk.\n\n\\section{Consequences for solutions constructed in \\cite{Chrusciel2015,Klinger2015}}\nThe class of solutions constructed in \\cite{Chrusciel2015,Klinger2015} for the case of timelike $\u03c4$ also exists for spacelike $\u03c4$. The sign changes in the Hamiltonian have no effect on the arguments concerning the evolution equations in the context of the analytic Fuchs theorem, as the decay of the exponential terms is unchanged.\n\nAn additional factor $m^J_b$ appears in the term $\\tilde{\u03c0}^b{}_a$, which enters in the Iwasawa variable momentum constraint:\n\\begin{equation}\\label{pi_iwa}\n\t\\tilde{\u03c0}\\indices{^b_a}=\\frac{1}{2}\\begin{cases}\n\t-\u03c0_b&\\text{for }a=b\\,,\\\\\n\t\\N{^b_i}P\\indices{^i_a}&\\text{for }b>a\\,,\\\\\n\tm^J_be^{-2(\u03b2^a-\u03b2^b)}\\N{^a_i}P\\indices{^i_b}&\\text{for }a>b\\,.\n\t\\end{cases}\n\\end{equation}\nAs the factor $m^J_b$ is only present in the asymptotically decaying case $a>b$, which is discarded in the asymptotic constraints, this leaves the conditions on the free functions unchanged.\n\nSimilarly there are sign changes in the derivation of the evolution equations for the constraints, given in Appendix D of \\cite{Klinger2015}, which cancel out in the final equations.\n\nAs in the case of a spacelike singularity, the presence of a cosmological constant does not affect the result (see Appendix F of \\cite{Klinger2015}).\n\nThis leads to the following theorem, in close analogy with the results of \\cite{Chrusciel2015,Klinger2015}:\n\n\\begin{theorem}\nFor any choice of $J\\in\\{1,2,3\\}$ and analytic functions $\u03b2_\\circ^2$, $\u03b2_\\circ^3$ and $P\\indices{_\\circ^2_1}$\ndepending on coordinates $x^i\\,,i\\in\\{1,2,3\\}$,\nand for any two analytic functions, $p_\\circ^2$ and $p_\\circ^3$ depending on $x^i$, which satisfy the inequalities\n %\n\\bel{13VI15.1}\n00$, the curvature diverges as $\u03c4\\to \\infty$. Along curves $\u03b3(\u03c4)=(\u03c4,\u03b3^i(\u03c4))$, $\u03c4\\in [\u03c4_0,\\infty)$, fulfilling $|\u03b3'^i(\u03c4)|=O(e^{(p_\\circ^i(\u03b3^j(\u03c4))-\u03b5)\u03c4})$ for some $\u03b5>0$ and for $i=1,2,3$, the curvature diverges in finite proper time \/ length. \n\\end{theorem}\n\n\\section{Conclusion}\nWe have constructed a large class of vacuum spacetimes containing a timelike singularity. The solutions asymptotically approach a timelike Kasner metric at each point $(x^i)$, which can be interpreted as the field of an infinitely extended thin rod, with positive mass for $J\\neq1$ and negative mass for $J=1$ \\cite{Israel1977}. As the Kasner exponents now depend upon the coordinates $x^i$ the solutions might represent the field of more complicated, non-symmetric and non-static, one-dimensional sources.\n\nWe have also noted that the cosmological billiards arguments of Damour, Henneaux, and Nicolai \\cite{Damour2003} are not directly applicable to this case, because of the transformation of asymptotically infinite potential walls into infinite wells. \nOne should keep in mind the results of Parnovsky~\\cite{Parnovsky1980,Parnovsky1999}, who applied the original procedure used by BKL to the timelike case, and concluded that the heuristic construction of chaotic singularities remains applicable. It would be of interest to resolve this apparent contradiction.\n\nIn \\cite{Shaghoulian2016} the authors argue, using a model Bianchi IX spacetime, that the change of sign of some of the wall terms is an artifact of the Iwasawa decomposition and that the affected walls vanish in a different gauge. It is not clear to us whether their arguments apply to the general inhomogeneous case.\n\n\\bigskip\n\n\\noindent{\\sc Acknowledgements:} Supported in part by a uni:docs grant of the University of Vienna.\nWe are grateful to W.~Piechocki for drawing our attention to~\\cite{Parnovsky1980,Parnovsky1999} and to P. Chru\\'{s}ciel for helpful comments and discussions.\n\\printbibliography\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}