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+{"text":"\\section{Introduction}\n\nTesting planarity of graphs with additional constraints is a popular theme in\nthe area of graph visualizations.\nOne of most the prominent such planarity variants, c-planarity, raised in 1995 by Feng, Cohen and Eades~\\cite{FCEa95,FCEb95} \nasks for a given planar graph $G$ equipped with a hierarchical structure on its vertex \nset, i.e., clusters, to decide if a planar embedding $G$ with the following property exists:\nthe vertices in each cluster are drawn inside a disc so that the discs\nform a laminar set family corresponding to the given hierarchical structure\nand the embedding has the least possible number of edge-crossings with the boundaries of the discs.\nShortly after, several groups of researchers tried to settle\nthe main open problem formulated by Feng et al. asking to decide its complexity\nstatus, i.e., either provide\na polynomial\/sub-exponential-time  algorithm for c-planarity or show its \\hbox{\\rm \\sffamily NP}-hardness.\nFirst, Biedl~\\cite{B98} gave\n a polynomial-time algorithm for c-planarity with two clusters. A different approach\nfor two clusters was considered by Hong and Nagamochi~\\cite{HN16}\nand quite recently in~\\cite{FKMP15}.\n The result also follows from a work by Gutwenger et al.~\\cite{GJL+02}.\nBeyond two clusters a polynomial time algorithm for c-planarity was obtained only in special cases,\ne.g.,~\\cite{BFPP08,GLS05,GJL+02,JJK+09,JKK+09}, and most recently in~\\cite{BR14+,CBFK14+}. Cortese et al.~\\cite{CDPP05} shows that c-planarity is solvable in polynomial\ntime if the underlying  graph is a cycle and the number of clusters\nis at most three.\n\n In the present work we generalize the result of Cortese et al. to the class of all planar graphs\nwith a given combinatorial embedding. In a recent pre-print~\\cite{F14+} we established \na strengthening for trees, where we do not fix the embedding.\nIn the general case (including already the case of three clusters) of so-called flat clustered graphs a similar result was obtained\nonly in very limited cases. Specifically, either when every face of $G$ is incident\nto at most five vertices~\\cite{BF07,FKMP15}, or when there exist at most two vertices of a cluster incident to a single face~\\cite{CBFK14+}.\nWe remark that the techniques of the previously mentioned papers do not give\na polynomial-time algorithm for the case of three clusters, and also do not seem to be adaptable\nto this setting. Our result and the technique used to achieve it suggest that, for a fairly general class of clustered graphs, c-planarity could be tractable\/solvable in  sub-exponential time at least with a fixed combinatorial embedding. \n\n {\\bf Notation.}\nLet $G=(V,E)$ denote a connected planar graph possibly with multi-edges.\nFor  standard graph theoretical definitions such as path, cycle, walk etc.,\nwe refer reader to~\\cite[Section 1]{D05}. \nA \\emph{drawing} of $G$ is a representation of $G$ in the plane where every vertex\n in $V$ is represented by a unique point and every\nedge $e=uv$ in $E$ is represented by a Jordan arc joining the two points that represent $u$ and $v$. \nWe assume that in a drawing no edge passes through a vertex,\nno two edges touch and every pair of edges cross in finitely many points.\nAn \\emph{embedding} of $G$ is an  edge-crossing free drawing.\nIf it leads to no confusion, we do not distinguish between\na vertex or an edge and its representation in the drawing and we use the words ``vertex'' and ``edge'' in both\n contexts.\nA  \\emph{face} in an embedding is a connected component of the complement of the embedding \nof $G$ (as a topological space) in the plane.\n The \\emph{facial walk} of $f$ is the closed walk in $G$ with a fixed orientation that we obtain by traversing the boundary of $f$ counter-clockwise.\nIn order to simplify the notation we sometimes denote the facial walk of a face $f$ by $f$. \n  A pair of consecutive edges $e$ and $e'$ in a facial walk $f$ creates a \\emph{wedge} incident to $f$ at their common vertex.\n  A vertex or an edge is \\emph{incident} to a face $f$, if it appears on its facial walk.\n\nThe \\emph{rotation} at a vertex is the counter-clockwise cyclic order of the end pieces of its incident edges\nin a drawing of $G$.\nAn embedding of $G$ is up to an isotopy and the choice of an \\emph{outer} (unbounded) face described by the rotations at its vertices. We call such a description of an embedding of $G$ a \\emph{combinatorial embedding}. Remaining faces are \\emph{inner faces}.\nThe \\emph{interior} and \\emph{exterior} of a cycle in an embedded graph is the bounded and unbounded, respectively, connected component\nof its complement in the plane. \nSimilarly, the \\emph{interior} and \\emph{exterior} of an inner face in an embedded graph is the bounded and unbounded, respectively, connected component\nof the complement of its facial walk in the plane, and vice-versa for the outer face.\nWhen talking  about interior\/exterior or area of a cycle  \nin a graph $G$ with a combinatorial embedding and a \\emph{designated} outer face  we mean it with respect to an embedding in the isotopy class that $G$ defines.\nFor $V'\\subseteq V$ we denote by $G[V']$ the sub-graph of $G$ induced by $V'$. \n\n\n\n\n\n\n\n \n\n\nA \\emph{flat clustered graph}, shortly  \\emph{c-graph}, is a pair $(G,T)$, where $G=(V,E)$ is a graph and $T=\\{V_0, \\ldots, V_{c-1}\\}$, $\\biguplus_i V_i=V$, is a partition of the\nvertex set into \\emph{clusters}. See Figure~\\ref{fig:treeEx} for an illustration.\nA  c-graph $(G,T)$ is \\emph{clustered planar} (or briefly \\emph{c-planar}) if $G$ has an\n embedding in the plane such that (i)\nfor every $V_i\\in T$ there is a topological disc $D(V_i)$, where $\\mathrm{interior}(D(V_i))\\cap \\mathrm{interior} (D(V_j))=\\emptyset$, if $i\\not=j$,\n containing all the vertices of $V_i$ in its interior, and (ii)\n every edge of $G$ intersects the boundary of $D(V_i)$ at most once for every $D(V_i)$.\nA c-graph  $(G,T)$ with a given combinatorial embedding of $G$ is \\emph{c-planar} \nif additionally the embedding is combinatorially described as given.\n A \\emph{clustered drawing and embedding} of a flat clustered graph $(G,T)$ is a drawing and embedding, respectively,\n of $G$ satisfying (i) and (ii).\nIn 1995\n Feng, Cohen and Eades~\\cite{FCEa95,FCEb95} introduced the notion of clustered planarity for clustered graphs, shortly c-planarity, (using, a more general, hierarchical clustering)\nas a natural generalization of graph planarity. (Under a different name\nLengauer~\\cite{L89} studied a similar concept in 1989.)\n\n\n\\begin{wrapfigure}{r}{.5\\textwidth}\n  \\centering\n\\centering\n\\subfloat[]{\n\\includegraphics[scale=0.4]{treeEx}\n    \t}\n\\subfloat[]{\n\\includegraphics[scale=0.4]{treeEx+}\n\t\t}\n\\caption{A c-graph that is not c-planar (left); and a c-planar c-graph (right).}\n\\label{fig:treeEx}\n\\end{wrapfigure}\n\nBy slightly abusing the notation for the rest of the paper $G$ denotes  a flat c-graph $(G,T)=(V_0 \\uplus V_1 \\uplus \\ldots \\uplus V_{c-1}, E)$ with $c$ clusters\n$V_0,V_1,\\ldots $ and $V_{c-1}$, and a given combinatorial embedding, and we assume that $G$ is \\emph{cyclic}~\\cite[Section 6]{FKMP15}. Thus, every $e=uv$ of $G$ is such that\n$u\\in V_i$ and $v\\in V_j$ where $j-i \\mod c \\le 1$ and for every\n$i$ there exists an edge in $G$ between $V_i$ and $V_{i+1 \\mod c}$.\nIn the case of three clusters, the first condition is redundant.\nIf the second condition is violated, the problem was essentially solved for three clusters as discussed in\nSection~\\ref{sec:wnwn}.\nWe assume that $G$ is connected, since in the problem that we are studying, the connected components of $G$ can be treated separately. Indeed, \\iflong as we show in Section~\\ref{sec:fan} \\fi without loss of generality  we  assume throughout the paper that in a clustered embedding of $G$ the clusters are unbounded wedges defined by pairs of rays emanating from the origin (see Figure~\\ref{fig:wedges}) that is disjoint from all the edges (see Appendix). We call such a clustered drawing  a \\emph{fan drawing}. \\\\\n\n\n\n\n\n\n\n\nThus, a connected component in a clustered embedding \ncan be drawn so that it is disjoint from a ball $B$ centered at the origin of radius $\\epsilon>0$\nfor any $\\epsilon$. The rest of the graph is then embedded inductively inside $B$.\nThe aim of the present work is to prove the following.\n\n\\begin{theorem}\nThere exists a quadratic-time algorithm in $|V(G)|$ to test if a cyclic c-graph $(G,T)$ is c-planar.\n\\end{theorem}\n\n\n\n\n{\\bf Further research directions.}\nWe think that our technique should be extendable by means of Euler's formula to resolve the c-planarity in more general situations than the one treated in the present paper. In particular, we suspect that\nthe technique should yield a generalization of the characterization of strip planar clustered graphs~\\cite[Section 5]{F14+}. \nThat would allow us to work with graphs without a fixed embedding. We mention that the tractability in a special case \nof our problem known as cyclic level planarity, when the embedding is not fixed, follows from a recent work of Angelini et al.~\\cite{angelini2015beyond}.\n\n{\\bf Organization.} In Section~\\ref{sec:pre} we introduce concepts used in the proof of\nour result. We give an outline of our approach in Section~\\ref{sec:out}.\nA more detailed description and a proof of correctness of our algorithm is in Section~\\ref{sec:alg}.\n\n\n \n\\section{Preliminaries}\n\\label{sec:pre}\n\n\\iflong\n\\subsection{Fan drawings}\n\\label{sec:fan}\nWe show that the clusters can be drawn as regions, each bounded by a pair of rays emanating from the origin.\nSuppose that $G=(V_0\\uplus \\ldots \\uplus V_{c-1},E) $ is given by a clustered embedding\nliving in the $xy$ plane of $\\mathbb{R}^3$.\nWe assume that boundries of discs representing clusters do not touch.\nConsider a stereographic projection from the north pole of a two-dimensional sphere $S$ \nsitting at the origin of $\\mathbb{R}^3$.\nLet $D$ be a stereographical pre-image of the embedding of $G$ on $S$.\nLet $S'$ denote the union of $G$ (as a topological space) with the boundaries of the clusters in $D$.\nLet $R_n$ and $R_s$ be a connected component of the complement of $S'$ in $S$, respectively, containing the north pole and south pole.\nIf necessary, we apply an isotopy  to $D$ (a continuous deformation keeping $D$ to be a clustered embedding all the time)  so that in the resulting embedding $D$ of $G$ on $S$ every boundary of a cluster intersects (in fact touches) the closure of $R_n$ and  the closure of $R_s$. \n\nWe show that a desired isotopy exists. We contract every cluster to a point thereby\ntreating clusters as vertices in an embedding $D'$ of a cycle $C$ of length $c$ having multi-edges. \nFormally, this can be viewed as a quotient $S\/\\sim$, where $x\\sim y$ iff\n$x$ and $y$ belong to the same cluster.\nIn $D'$ there must be a pair of faces $f$ and $f'$ whose facial walk is $C$ since any cycle in the corresponding multi-graph is obtained as a symmetric difference of facial walks. Apply an isotopy to $D'$ such that $f$ contains\nthe north pole in its interior and $f'$ contains the south pole in its interior. Finally, we decontract clusters in the end. The above procedure can be easily turned into an isotopy of $D$. \n\nBy projecting the resulting spherical embedding back to the plan we can also assume that we have a clustered embedding of $G$\nsuch that clusters are represented by small discs of diameter $\\epsilon>0$ each drawn in a close vicinity\nof a different vertex of a regular convex $c$-gon with the center at the origin, and the edges\nbetween clusters $V_i$ and $V_{i+1 \\mod c}$, for every $i$, are closely following the edge\nof the $c$-gon between the corresponding pair of vertices. \nThe desired rays bounding clusters are those from the origin orthogonal to the sides of the $c$-gon. \\\\\n\\fi\n\n\n\n\n\\subsection{Outline of the approach}\n\\label{sec:out}\nBy~\\cite[Theorem 1]{FCEb95} deciding c-planarity of instances $G$ in which all $G[V_i]$'s are connected amounts to \nchecking if an outer face of $G$ can be chosen so that every $V_i$ is embedded in the outer face \nof $G[V\\setminus V_i]$. On the other hand, once we have a clustered embedding of $G$ we can augment $G$ by adding edges drawn inside clusters without creating an edge-crossing so that clusters become connected.\nThese observations suggest that c-planarity of $G$ could be viewed as a connectivity augmentation problem, for example as in~\\cite{CBFK14+,FKMP15},\nin which we want to decide if it is possible to make clusters connected  while maintaining the planarity of $G$.\nOne minor problem with this viewpoint is the fact that if $G$ is c-planar we do not allow\na cluster $V_i$ to induce a cycle such that clusters $V_j$ and $V_{j'}$, $i\\not=j,j'$, are drawn on its opposite \nsides. However, this cannot happen  if $G$ is cyclic.\nFollowing the above line of thought our algorithm tries to augment $G$ by subdividing its faces with\npaths and edges. We proceed in two steps. In the first step, Section~\\ref{sec:norm}, we either \ndetect that $G$ is not c-planar or similarly as in~\\cite{ADDF13} and~\\cite{F14+} by \nturning clusters into independent sets and adding certain paths we normalize the instance. In the second step, Section~\\ref{sec:const}, we decide if the normalized instance can be further augmented by  edges as desired.\n\nIn order to prove the correctness of the second step of the algorithm\nwe use the notion of the \\emph{winding number} $\\mathrm{wn}(W)\\in \\mathbb{Z}$ of a walk $W$ of $G$, as defined\nin Section~\\ref{sec:wnwn}. The parameter $\\mathrm{wn}(W)$ says how many times and in which sense \na walk $W$ of $G$ winds around the origin in a clustered drawing of $G$.\nThus, $G$ is not c-planar if there exists a face $f$ such that for its facial walk $|\\mathrm{wn}(f)|>1$ or\nif there exists at least two inner faces $f$ with $|\\mathrm{wn}(f)|>0$.\nHowever, it can be easily seen that this necessary condition of c-planarity is not sufficient\nexcept when $G$ is a cycle~\\cite{CDPP05}.\nThe  necessary condition allows us to reduce the c-planarity testing problem of a normalized instance to the problem\nof finding a perfect matching in an auxiliary face-vertex incidence graph which is polynomially solvable.\nThe novelty of our work lies in the use of the winding number in the context of connectivity augmentation guided\nby the flow and matching in the auxiliary face-vertex incidence graph \\`a la~\\cite{ADDF13} and~\\cite{F14+}, respectively. \n\nWe remark that the approach of~\\cite{ADDF13} via a variant of upward embeddings\nfor directed graphs in our settings has several problems that seem quite hard to overcome,\nthe main one being the fact that the result of Bertolazzi et al.~\\cite{BBLM94} does not extend, at least not in a natural way, to the drawings on the rolling cylinder, see e.g.,~Auer et al.\\cite{Auer201536} for the definition of these drawings.\nWe are not aware of a polynomial-time algorithm for the corresponding  problem,\nnor a corresponding \\hbox{\\rm \\sffamily NP}-hardness result, and\nfind the corresponding algorithmic question interesting and related to our problem.\n\n\\begin{figure}\n  \\centering\n\\centering\n\\subfloat[]{\n\\label{fig:wedges}\n\\includegraphics[scale=0.5]{3ex}\n    \t}\n    \t\\hspace{10px}\n    \t\\subfloat[]{\n    \t\\label{fig:semiSimple}\n    \t\\includegraphics[scale=0.5]{semiSimple}\n    \t}\n\\caption{(a) A clustered graph $G=(V_0 \\uplus V_1 \\uplus V_2,E)$ with clusters represented by wedges bounded by rays meeting at the origin. The highlighted wedge at $u$ is concave and at $v$ convex. (b) A semi-simple face $f$\nand the outer face $f_o$ with an incident concave wedge.}\n\\end{figure}\n\n\n\\subsection{Winding number} \n\\label{sec:wnwn1}\nWe define the winding number $\\mathrm{wn}(W)$ of a closed oriented walk $W$ in a drawing disjoint from the origin of a graph $G$ (possibly with crossings). In what follows  facial walks are understood with the orientations as in\n  an embedding of $G$ with the given rotations and a face $f_o$ being a designated outer face.\nBy viewing a closed walk $W$ in the drawing  as a continuous  function $w$ from the unit circle $S^1$ to $\\mathbb{R}^2\\setminus {\\bf 0}$,\nthe winding number $\\mathrm{wn}(W)\\in \\mathbb{Z}$ corresponds to the element of the fundamental group of  $S^1$~\\cite[Chapter 1.1]{Hatch02}  represented by $\\frac{w(x)}{||w(x)||_2}$.\nLet $W_1$ and $W_2$ denote a pair of oriented closed walks meeting in a vertex $v$.\nLet $W$ denote the closed oriented walk from $v$ to $v$ obtained by concatenating $W_1$ and $W_2$.\nBy the definition of $\\mathrm{wn}$ we have $\\mathrm{wn}(W)=\\mathrm{wn}(W_1)+\\mathrm{wn}(W_2)$.\nLet $f_1$ and $f_2$, $f_o\\not = f_1, f_2$, denote a pair of faces of $G$ whose walks intersect in a single walk.\nLet $G'$ denote a graph we get from $G$ by deleting edges incident to both $f_1$ and $f_2$.\nLet $f$ denote the new face thereby obtained. Since $f_1$ and $f_2$ intersect in a single walk, the boundary of $f$ is connected.\nIn the drawing of $G'$ inherited from the drawing of $G$ we have $\\mathrm{wn}(f)=\\mathrm{wn}(f_1)+\\mathrm{wn}(f_2)$, since common edges of $f_1$ and $f_2$ are traversed\nin opposite directions by $f_1$ and $f_2$.\nA face  or a vertex is in the interior of a closed walk $W$ in $G$ if it is in the interior of \na cycle induced by the edges of $W$ in an embedding of $G$ with the given rotations and  $f_o$ as the outer face.\nThe previous observation is easily generalized by a simple inductive argument \nas follows \\\\\n$ {\\bf (*)} \\ \\ \\ \\   \\sum_f \\mathrm{wn}(f)=\\mathrm{wn}(W)$ \\\\\n where we sum over all  faces~$f$ of $G$ in the interior of the\nclosed walk $W$ in $G$. In particular, $\\sum_f \\mathrm{wn}(f)=\\mathrm{wn}(f_o)$, where we sum over all \n faces  $f\\not=f_o$ of $G$.\n\n\\subsection{Labeling vertices} \n\\label{sec:wnwn}\nLet $\\gamma:V \\rightarrow \\{0,1,\\ldots c-1\\}$ be a labeling of the vertices $V$ by integers\nsuch that $\\gamma(v) = i$ if $v\\in V_i$.\n Let $W$ denote an oriented closed  walk in a clustered drawing of  $G$. We put $\\mathrm{height}(W)=\n\\sum_{{v'u'}\\in E(W)} g(\\gamma(u')-\\gamma(v'))$,\n where $g(0)=0, \\ g(1)=g(1-c)=1$ and $g(c-1)=g(-1)=-1$.\nWe have the following.\n\n\\begin{lemma}\n\\label{lemma:wn}\nFor a walk $W$ in a fan drawing of $G$ we have $\\mathrm{wn}(W)=\\mathrm{height}(W)\/c$.\n\\end{lemma}\n\n\\begin{proof}\nThe number of times\nthe walk $W$ crosses the ray between $V_i$ and $V_{i+1 \\mod c}$ from right\nto left w.r.t. to the direction of the ray is $\\mathrm{wn}_i^+(W)=\\sum_{{v'u'}} g(\\gamma(u')-\\gamma(v'))$, \nwhere we sum over the edges $v'u'$ in the walk $W$, where\n $v'\\in V_i$ immediately precedes $u'\\in V_{i+1 \\mod c}$ in the walk.\n Similarly, we define \\\\  $\\mathrm{wn}_i^-(W)=\\sum_{{v'u'}} g(\\gamma(u')-\\gamma(v'))$, \nwhere we sum over the edges $v'u'$ in $W$, where\n $v'\\in V_{i+1 \\mod c}$ immediately precedes $u'\\in V_{i}$ in the walk.\nWe have, $\\mathrm{wn}(W) = \\mathrm{wn}_i^+(W) + \\mathrm{wn}_i^-(W)$\nwhich in turn implies $c\\cdot\\mathrm{wn}(W) = \\sum_{i} (\\mathrm{wn}_i^+(W) + \\mathrm{wn}_i^-(W))=\\mathrm{height}(W) $.\n \\ \\vrule width.2cm height.2cm depth0cm\\smallskip\\end{proof}\n\n \nBy the previous lemma $\\mathrm{wn}(W)$ is determined already by the c-graph $G$ and is the same in all  clustered drawings of $G$, and hence, putting $\\mathrm{wn}(W):=\\mathrm{height}(W)\/c$, for a walk $W$ with a fixed orientation, allows us to speak about $\\mathrm{wn}(W)$ without\nreferring to a particular drawing of $G$.\n Thus, $\\mathrm{wn}(W)$ tells us the winding number of $W$ in any clustered \ndrawing.\nBy Jordan-Sch\\\"onflies theorem $G$ the following holds. \n\n\\begin{lemma}\n\\label{lem:2}\n$G$ is not c-planar if there exists a face $f$ such that $|\\mathrm{wn}(f)|>1$ or if there exists more than one inner face $f'$ with $|\\mathrm{wn}(f')|=1$.\n\\end{lemma}\n\\begin{proof}\nIn a crossing free drawing  $|\\mathrm{wn}(f)|\\le 1$ for every face $f$.\nIf $|\\mathrm{wn}(f')|=1$ the origin ${\\bf 0}$ lies in the interior of $f'$ since\notherwise the facial walk is null-homotopic, i.e., homotopic to a constant map, in $\\mathbb{R}^2\\setminus {\\bf 0}$ (contradiction). However, interiors of faces are disjoint.\n \\ \\vrule width.2cm height.2cm depth0cm\\smallskip\\end{proof}\n If $\\mathrm{wn}(f)=0$ for all faces $f$,~\\cite[Lemma 1.2]{F14+} extends easily to this case,\nreducing the problem to the work of Angelini et al.~\\cite{ADDF13}.\nThus, by Lemma~\\ref{lem:2} and for the sake of simplicity of the presentation, throughout the paper we assume that there exists a pair  of faces $f_o,f_o'$, $\\mathrm{wn}(f_o)=\\mathrm{wn}(f_o')\\not=0$  (by\n~$(*)$ there cannot be just one such face) one of which, let's say $f_o$,\nwe designate as an \\emph{outer face}.  The roles of $f_o$ and $f_o'$ are, in fact, interchangeable.\nAlso such a restriction is by no means crucial in our problem, and alternatively, it is always possible\nto choose and subdivide the outer face in the normalized instance (defined later) by a path so that the restriction is satisfied.\n\n\nViewing a facial walk $f$ as a sequence of vertices and edges $w_0e_0w_1e_2\\ldots e_mw_m$, where $e_{i-1}=w_{i-1}w_i$,\nlet $V_f$ be the set $\\{w_0,\\ldots, w_m\\}$ of \\emph{vertex occurrences} along~$f$.\nWe treat $V_f$ also as a multi-set of vertices, and thus, $\\gamma$ is defined on its elements.\nLet $\\gamma_f:V_f \\rightarrow \\mathbb{N}$, for $f\\not=f_o,f_o'$, be a labeling of the elements of $V_f$ by integers defined as follows.\nWe mark all the vertex occurrences  in $V_f$ as unprocessed.\nWe pick an arbitrary vertex occurrence  $v\\in V_f$, set $\\gamma_f(v):=\\gamma(v)$\nand mark $v$ as processed.\nWe repeatedly pick an unprocessed vertex occurrence  $u\\in V_f$ that has its predecessor or successor $v$ along the boundary walk of $f$ in $V_f$  processed.\n We put  $\\gamma_f(u):=\\gamma_f(v)+g(\\gamma(u)-\\gamma(v))$.\n Intuitively, $\\gamma_f$ records \n the distance  in terms of ``winding around origin'' of vertex occurrences \n along the boundary walk of $f$ from a single chosen vertex occurrence.\n Since $\\mathrm{wn}(f)=0$ the function  $\\gamma_f(u)$ is completely determined by\nthe choice of the  first occurrence of a vertex we processed. \nThis choice is irrelevant for our use of $\\gamma_f$ as we see later.\nAlso notice that $\\gamma(v) = \\gamma_f(v) \\mod c$ for all vertices incident to $f$.\n\nA normalized instance allows only the faces of the types defined next.\nAn element $v$ in $V_f$ is a \\emph{local minimum} (\\emph{maximum}) of a face $f$ if in the  facial walk  $f$ the value of $\\gamma(v)$ is not bigger (not smaller)\nwith respect to the  relation $0<1<\\ldots <c-1<0$  than the value of its successor and predecessor.\nA walk $W$ in $G$ is \\emph{(strictly) monotone with respect to $\\gamma$} if the labels of the occurrences of vertices on $W$ form a (strictly) monotone sequence\nwith respect to the  relation $0<1<\\ldots <c-1<0$ when ordered\nin the correspondence with their appearance on  $W$.\nThe  face $f$ is \\emph{simple} if $f$ has at most one local minimum. It follows \nthat a simple face $f$ has also at most one local maximum.\nThe inner face   $f\\not=f_o'$ is \\emph{semi-simple}  if $f$ has exactly two local minima and maxima and these minima and maxima, respectively,  have the same $\\gamma_f$ value.\n\n\n\\section{Algorithm}\n \\label{sec:alg}\n\nA cyclic c-graph $G$ is \\emph{normalized } if \n\n\\noindent\n(i) $G$ is connected; \\\\\n(ii) each cluster $V_i$ induces an independent set; and \\\\\n(iii) each face of $G$ is simple or semi-simple, and $f_o$ and $f_o'$ are both simple. \n\nSuppose that (i)--(iii) are satisfied. \nBy~(ii) we  put directions on all the edges in $G$ as follows.\nLet $\\overrightarrow{G}$ denote the directed c-graph obtained from $G$ by orienting every edge\n$uv$ from the vertex with the smaller label $\\min (\\gamma(u), \\gamma(v))$ to the vertex with the bigger label $\\max (\\gamma(u), \\gamma(v))$ with respect to the relation $0<1<\\ldots < c-1<0$.\nA \\emph{sink} and \\emph{source} of $\\overrightarrow{G}$ is\na vertex with no outgoing and incoming, respectively, edges.\n\nLet $e$ denote an edge of $G$ not contained in a single \ncluster.\nGiven a clustered embedding $\\mathcal{D}$ of $G$ let ${\\bf p_{e}}:={\\bf p_{e}}(\\mathcal{D})$ denote the intersection point of $e$ with a ray separating a pair of clusters.\nLet $e_0,\\ldots,e_{k-1}$ be  the  edges\n incident to a sink or source $u$.\nBy Jordan curve theorem it is not hard to see that (i)--(iii) imply that a clustered embedding $\\mathcal{D}$ of $G$ is ``combinatorially'' determined once we order the set \n  of intersection points\n ${\\bf p_{e_0}}, \\ldots , {\\bf p_{e_{k-1}}}$ along\n rays separating clusters for every sink and sources $u$ in $G$. Moreover, the set of intersection points corresponding to a sink or source $u$ admits in\n an embedding only orders that are cyclic shifts of\n one another, since we have the rotations at vertices \n of $G$ fixed.\n The wedge in $\\mathcal{D}$ formed by a pair of edges $e_i$ and $e_{i+1}$ incident to a face $f$ at its local extreme $u$ is \\emph{concave} (see Figure~\\ref{fig:wedges} \n for an illustration) if \n$u$ is a sink or source of $\\overrightarrow{G}$ and\nthe line segment ${\\bf p_{e_i}}{\\bf p_{e_{i+1}}}$ contains all the other points ${\\bf p_{e_j}}$ or\nin other words the order of intersection points\ncorresponding to $u$ is ${\\bf p_{e_{i+1}},p_{e_{i+2}}, \\ldots, p_{e_{k-1}},p_{e_{0}}, \\ldots , p_{e_i}}$.\nA non-concave wedge is \\emph{convex}.\\iflong\\footnote{The terminology comes from the fact that in an embedding that is ``combinatorially the same'' with straight line edges, wedges become convex and concave in the usual sense.}\\fi\nNote that in $\\mathcal{D}$ every sink or source is\nincident to exactly one concave wedge that in\nturn determines the order of intersection points.\nThus, combinatorially $\\mathcal{D}$ is also determined\nby a prescription of concave wedges at sink and sources.\n\nLet  $S$ be the set of sinks and sources of $\\overrightarrow{G}$. Let $F$ denote the union of the set of \nsemi-simple faces of $G$ with a subset of $\\{f_o,f_o'\\}$ containing faces incident to a sink and a source.\nWe construct a planar bipartite graph $I=(S\\cup F, E(I))$ with parts $S$ and $F$,\n where $s\\in S$ and $f\\in F$\nis joined by an edge if $s$ is incident to $f$. \nGiven that (i)--(iii) are satisfied, the existence of a perfect matching $M$ in $I$ is a necessary condition for $G$ being c-planar. Indeed, as we just said,\nin a clustered embedding, each source or sink has exactly one of its wedges concave.\nOn the other hand, by Jordan curve theorem  it  can be easily checked that in the clustered embedding  \\\\\n{\\bf (A)} every semi-simple face is incident to exactly one concave wedge \\\\\n{\\bf (B)} faces $f_o$ and $f_o'$ are incident to one concave wedge if they are incident to a sink and source, and \\\\ {\\bf (C)} all the other faces are not incident to any concave wedges at the minimum and maximum. \\\\\nThis is fairly easy to see if $G$ is vertex two-connected, see Figure~\\ref{fig:semiSimple} for an illustration.\nThe cycle $C$ \\emph{corresponding to a closed walk} is obtained\nby traversing the walk and introducing a new vertex for each\nvertex occurrence in the walk.\nFor a face $f$ incident to cut-vertices, {\\bf (A)--(C)} follows by considering the cycle corresponding to the facial walk of $f$ (treated as a face) embedded in a close vicinity of the boundary of $f$. \n Thus, a desired matching $M$ is obtained by matching each source or sink with\nthe face incident to its concave wedge.\n\n \nWe show in Section~\\ref{sec:const}  that if $M$ exists $G$  is c-planar by augmenting \n$G$ with edges as described in Section~\\ref{sec:out}.\nTesting the existence, but even counting perfect matchings in a planar bipartite graph can be carried out in a polynomial time~\\cite[Section 8]{L09}.\n\n\nThe running time of our algorithm is $O(|V|^2)$ since finding the perfect matching \ncan be done in $O(|V|^2)$  time, due to $|E(I)|=O(|V|)$, and the pre-processing step including the construction of $I$ and the normalization will be easily seen to have this time complexity. Also computing the winding number for all the faces can be performed in a linear time by Lemma~\\ref{lemma:wn}.\nFirst, we explain and prove the correctness of the algorithm for\n instances satisfying~(i)--(iii). In Section~\\ref{sec:norm},\n we show a polynomial-time reduction of the general case  to instances  satisfying~(i)--(iii). We often use Jordan--Sch\\\"onflies theorem without\n explicitly mentioning~it.\n\n\n\\iflong\\else\n\\fi\n\\subsection{Constructing a clustered embedding}\n \\label{sec:const}\n \n\\begin{wrapfigure}{r}{.5\\textwidth}\n\\centering\n\\includegraphics[scale=0.65]{fuk}\n\\caption{Subdividing a semi-simple face   (left). Subdividing a simple face $f_o'$ (right).}\n\n\\label{fig:figfig}\n\\end{wrapfigure}\n\n \n Given a normalized instance $G$ and a matching $M$ between sources and sinks in $S$, and  faces in $F$ of $G$ we construct a  clustered embedding of $G$ as follows. Recall that we assume that $G$ does not have a face \n $f$ with $|\\mathrm{wn}(f)|>0$ besides $f_o$ and $f_o'$.\n  We start with $\\overrightarrow{G}$ defined\nabove and add edges to it thereby eliminating all the sinks and sources, see Figure~\\ref{fig:figfig}.\nLet $u\\in S$ be a source matched in $M$ with $f$. If $f$ is a semi-simple inner face\nlet $u'$ denote another local minimum  incident to $f$. We add to  $\\overrightarrow{G}$\n  an edge $\\overrightarrow{u'u}$ embedded in the interior of $f$. \n  If $f=f_o$ or $f=f_o'$ we join $u$ by $\\overrightarrow{u'u}$ with the vertex in the same cluster $u'$\nso that we subdivide $f$ into two simple faces $f'$ and $f''$ such that\n$\\mathrm{wn}(f')=0$ and $\\mathrm{wn}(f'')=\\mathrm{wn}(f)$. If $f=f_o$  face $f''$ is the new outer face. By Lemma~\\ref{lemma:wn}, such a vertex $u'$ exists and it is unique.\n\n\n\nWe proceed with $u\\in S$ that are sinks analogously thereby eliminating all the sinks and source in the resulting graph $\\overrightarrow{G'}$, where by $G'$ we denote its underlying undirected graph.\nBy Lemma~\\ref{lemma:wn}, there still exists exactly one inner face $f_o'$ with a non-zero winding number in the resulting graph $G'$.\n\n\n\n\\begin{lemma}\n\\label{lemma:keyFact0}\n$G'$ has exactly one inner face $f_o'$ such that $|\\mathrm{wn}(f_o')|=1$.\n\\end{lemma}\n\nSince $\\gamma(v) = \\gamma_f(v) \\mod c$ for every face $f\\not=f_o,f_o'$ and $v$ incident to $f$, \nevery edge we added joins a pair of vertices in the same cluster.\n\n\n    \\begin{lemma}\n\\label{lemma:keyFact1}\nThe induced sub-graph $G'[V_i]$ of (undirected) $G'$ does not contain a cycle for $i=0,1,\\ldots, c-1$.\n\\end{lemma}\n\\begin{proof}\nFor the sake of contradiction suppose that a cycle $C$ is contained in $G'[V_{j'}]$.\nLet us choose $C$ such that the area of its interior is minimized.\nSince $G[V_{j'}]$ is an independent set all the edges of $C$ are newly added.\nThus, by looking at the rotation of an arbitrary vertex $v'$ of $C$ we see that $v'$ is incident to a vertex \n $v$ from  $V_j$, $j\\not=j'$,  in the interior of $C$. Indeed, no two edges of $C$ subdivide the same face of $G$.\n \n \n\n\n\n \nUsing the fact that $\\overrightarrow{G'}$ does not contain any source or sink, we show that \na vertex $w$ in the interior of $C$ belongs to an oriented cycle $C'$ (by chance also directed in $\\overrightarrow{G'}$), whose interior is contained in the interior of $C$   \nsuch that $\\mathrm{wn}(C')>0$.\nThe cycle $C'$ is obtained by following a directed path in $\\overrightarrow{G'}$ (from which it inherits its orientation)\n passing through  $v$.\nEither both ends of the path meet each other, they both meet $C$, or the path meet itself in the interior. \nIn the first two cases we can take $w:=v$ in the last case it can happen that the directed path gives rise to a cycle $C'$ not containing $v$. However, $C'$ is not induced by a single cluster by the choice of $C$, and thus, $\\mathrm{wn}(C')>0$ by Lemma~\\ref{lemma:wn} and $C'$ contains a vertex $w$ from $V_j$.\n Let $F'$ denote the set of faces\nin the interior of $C$ and not in the interior of $C'$. \nIn all cases it can be seen   by Lemma~\\ref{lemma:wn} that $\\mathrm{wn}(C')>0$.\n\nIndeed, as we proved in  the proof of Lemma~\\ref{lemma:wn} \n$\\mathrm{wn}(C') = \\mathrm{wn}_j^+(C') + \\mathrm{wn}_j^-(C')$.\nSince $C'$ follows a directed path and is not induced by a single cluster we have $\\mathrm{wn}_j^+(W)>0$ and $\\mathrm{wn}_j^-(W)=0$.\nHence, $\\mathrm{wn}(C') = \\mathrm{wn}_j^+(C') + \\mathrm{wn}_j^-(C')>0$. \n\n\n\n By~(*) it follows that $C'$ contains the unique\ninner face with a non-zero winding number in its interior.\nThen Lemma~\\ref{lemma:keyFact0} with~(*) yields the following  contradiction\n$$0=\\mathrm{wn}(C)=\\mathrm{wn}(C')+\\sum_{f\\in F'} \\mathrm{wn}(f)=\\mathrm{wn}(C')\\not=0$$\n \\ \\vrule width.2cm height.2cm depth0cm\\smallskip\\end{proof}\n\nLet $E'\\subseteq \\bigcup_i{V_i \\choose 2}\\setminus E(G')$ such that each edge in $E'$ can be added to \nthe embedding of $G'$ without creating a crossing or increasing the number of inner faces with a non-zero winding number.\nWe do not put any direction on the edges in $E'$.\nSince every inner face $\\not=f_o'$ in $G'$ is simple, and its outer face and the face $f_o'$ are not adjacent\nto a source or sink, all the edges in $E'$ can be \nintroduced simultaneously without creating a crossing. \nIn particular, no edge of $E'$ subdivides $f_o'$ or the outer face.\nLet $E''$ denote a maximal subset of $E'$\nthat does not introduce a cycle in $(G'\\cup E'')[V_i]$ for every $i=0,1,\\ldots, c-1$ (see Figure~\\ref{fig:simpleFace}), where $G' \\cup E'' = (V(G'), E(G') \\cup E'')$.\nBy Lemma~\\ref{lemma:keyFact1}, $E''$ is well-defined.\n\n\n\\begin{figure\n\\centering\n\\includegraphics[scale=0.65]{simple_face}\n\\caption{A simple face $f$ of $G'$ (left). The face $f$ subdivided with edges of $E''$ (right). Labels at vertices are their $\\gamma$ values (or indices of their clusters).}\n\\label{fig:simpleFace}\n\\end{figure}\n\n\n\n\n\n\\begin{lemma}\n\\label{lemma:keyFact2}\n  $(G'\\cup E'')[V_i]$  is a tree for $i=0,1,\\ldots, c-1$.\n  \\end{lemma}\n\\begin{proof}\nSuppose for the sake of contradiction that $(G'\\cup E'')[V_i]$ for some $i$ is not a tree, and thus, it is just a  forest with more than one connected component.\nIt follows that either (1) there exists a cycle in $(G'\\cup E'')[V\\setminus V_i ]$\ncontaining a vertex $v$ of $V_i$ in its interior\nor (2) a pair of vertices of $V_i$ in different\nconnected components of $(G'\\cup E'')[V_i]$  are incident to the same face of $(G'\\cup E'')$.  \nThe claim (1) or (2) implies that there exists a cycle\n$C$ in $(G'\\cup E')[V\\setminus V_i]$ containing a vertex $w$ of $V_i$ in its interior.\nSimilarly as in the proof of Lemma~\\ref{lemma:keyFact1}, by following\na directed path through $v$ we obtain \nan oriented cycle $C'$ (this time not necessarily directed) in $G$,  whose interior is contained in the interior of $C$ with $\\mathrm{wn}(C')>0$ yielding a contradiction.\n\nIndeed, as we proved in  the proof of Lemma~\\ref{lemma:wn} \n$\\mathrm{wn}(C') = \\mathrm{wn}_i^+(C') + \\mathrm{wn}_i^-(C')$.\nSince $C'$ is not induced by a single cluster and follows in the interior of $C$ a directed path, and $C$ does not have any vertex in $V_i$ we have\n $\\mathrm{wn}_i^+(W)>0$ and $\\mathrm{wn}_i^-(W)=0$.\nHence, $\\mathrm{wn}(C') = \\mathrm{wn}_i^+(C') + \\mathrm{wn}_i^-(C')>0$. \n \\ \\vrule width.2cm height.2cm depth0cm\\smallskip\\end{proof}\n\n\n\n\n\n\nBy Lemma~\\ref{lemma:keyFact2}, every $F_i$ is a tree. Taking a close neighborhood of each such $F_i$ as a disc representing\nthe cluster $V_i$ we obtain a desired clustered embedding of $(G'\\cup E'')$. In the obtained embedding we just delete edges not belonging to $G$ and that concludes the proof of the correctness of our algorithm. \\\\\n\n\n\n\n\n\n\\subsection{Normalization}\n \\label{sec:norm}\n\n \n \n In the present section we normalize the instance so that~(i)-(iii) are satisfied.\nWe argued the connectedness in Introduction, and hence, (i) is taken care of.\n\\iflong \n\nA \\emph{contraction} of an  edge $e=uv$ in a topological graph is an operation that turns\n$e$ into a vertex\nby moving $v$ along $e$ towards $u$ while dragging all the other edges incident to $v$ along $e$.\nBy a contraction we can introduce multi-edges or loops at the vertices.\nWe will also  use the following operation which can be thought of as the inverse operation of the edge contraction\nin a topological graph.\nA \\emph{vertex split} in a drawing of a graph $G$ is an operation that replaces a vertex $v$ by two vertices $v'$ and $v''$\ndrawn in a small neighborhood of $v$ joined by a short crossing free edge so that the neighbors of $v$ are partitioned into two parts\naccording to whether they are joined with $v'$ or $v''$ in the resulting drawing, the rotations at $v'$ and $v''$ are inherited from the\nrotation at $v$, and the new edges are drawn in the small  neighborhood of the edges they correspond to in $G$.\n\n\n\nRegarding~(ii), by a series of successive edge contractions we contract each connected component of $G[V_i]$'s to a vertex.\nWe delete any created loop.\nIf a loop at a vertex from $V_i$ contains a vertex from a different cluster $V_j$, $j\\not=i$, in its interior we know that the \ninstance is not c-planar, since for every $j$ all the vertices in $V_j$ must be contained in the outer face of $G[V\\setminus V_j]$ in a positive instance. This all can be easily checked in  polynomial time.\nOtherwise, a contraction preserves c-planarity of $G$, since\ndeleted empty loops can be introduced in a c-planar embedding of the reduced graph,\nand contracted edges recovered via vertex splits.\nFrom now on we assume that clusters of $G$ form independent sets. \nIt remains to satisfy (iii).\n\\else\nTo achieve~(ii) is fairly standard by contracting components induced by clusters to vertices. \\fi\nThus, it remains to satisfy (iii).\n\n\nWe want to sub-divide a non-simple face $f$  into \na pair of faces one of which is\nsemi-simple by a monotone path $P'$ w.r.t. $\\gamma$.\nLet $uPv$ denote an oriented monotone  sub-walk of $f$ w.r.t. $\\gamma$ joining a local minimum $u$ and maximum $v$ of $f$ minimizing  $|\\mathrm{height}(P)|$.\nLet $vQv'$ denote the oriented  monotone walk with $|\\mathrm{height}(P)|=|\\mathrm{height}(Q)|$ immediately following $P$ on the facial walk of $f$, and let $u'Q'u$ be such walk  immediately preceding $P$ on the facial walk of $f$. Note that $Q$ and $Q'$ exists due to the minimality of $P$ and that we have\n$\\mathrm{height}(Q)=\\mathrm{height}(Q') = - \\mathrm{height}(P)$.\nSimilarly as in~\\cite{F14+} we subdivide $f$ into two faces $f'$ and $f''$\nby a strictly monotone path $v'P'u'$  w.r.t. $\\gamma$. Hence,  $\\mathrm{height}(P) = \\mathrm{height}(P')$.\nWe have $\\mathrm{height}(Q)=\\mathrm{height}(Q') = - \\mathrm{height}(P)= -\\mathrm{height}(P')$.\nThus, by Lemma~\\ref{lemma:wn} if $f$ with $\\mathrm{wn}(f)\\not=0$ is semi-simple we obtain a simple face $f'$ with \n$\\mathrm{wn}(f')\\not=0$ and a semi-simple face $f''$ with $\\mathrm{wn}(f'')=0$ as desired.\nIndeed, $\\mathrm{wn}(f'') = \\mathrm{height}(P') +\\mathrm{height}(Q') + \\mathrm{height}(P)+\\mathrm{height}(Q)=0$\nand $c\\cdot\\mathrm{wn}(f) = \\mathrm{height}(v'P''u') + \\mathrm{height}(Q')+\\mathrm{height}(P) + \\mathrm{height}(Q) =\\mathrm{height}(v'P''u') - \\mathrm{height}(P')= c\\cdot\\mathrm{wn}(f')$. It remains to show the following lemma, since both $f'$ and $f''$ \nare incident to less local minima and maxima than $f$ if $f$ is not semi-simple.\nHence, after $O(|V|)$ facial subdivisions we obtain a desired instance, since $|E(I)|=O(|V|)$.\n\n\\begin{lemma}\n\\label{lemma:norm}\nIf the c-graph $G$ is c-planar then by subdividing $f$ of $G$ by $P'$ into\na pair of faces $f'$ and $f''$, where $f''$ is semi-simple we obtain a c-planar c-graph. Moreover, \n$\\mathrm{wn}(f')=\\mathrm{wn}(f)$ and $\\mathrm{wn}(f'')=0$.\n\\end{lemma}\n\\begin{proof}\nThe second statement is proved above.\nHence, we deal just with the first one.\nLet $e_{u}$ and $e_{u}'$ denote the first edge on $P$ and the last edge on $Q'$, respectively. Let $e_{v}$ and $e_v'$\ndenote the last edge on $P$ and the first edge on $Q$, respectively.\nLet $e_{v'}$ and $e_{u'}$ denote the last edge on $Q$ and the first edge on $Q'$.\nLet  ${\\bf p_{u}=p_{e_{u}}}, \\iflong {\\bf p_{u}'=p_{e_{u}'}}, {\\bf p_{u'}=p_{e_{u'}}}, {\\bf p_{v}}={\\bf p_{e_v}},{\\bf p_{v}'}={\\bf p_{e_{v}'}}\\fi$  and ${\\bf p_{v'}=p_{e_{v'}}}$\n denote the intersection of the edges $e_u,\\iflong e_{u}',e_{u'},e_v,e_v'\\fi$ and $e_{v'}$, respectively,  with a ray separating a pair of clusters.\nLet $\\omega_u$ and $\\omega_v$ denote the wedge between $e_u,e_u'$ and $e_v,e_v'$, respectively, in $f$.\n\n\nWe presently show that subdividing $f$ with $P'$  preserves c-planarity, since a clustered\nembedding without $P'$ can be deformed so that $P'$ can be added to a clustered planar embedding without creating a crossing, while keeping the embedding clustered. \nThis is not hard to see if, let's say $\\omega_v$, is convex and the line segment ${\\bf p_up_{v'}}$ is not crossed by an edge. Since $\\omega_v$ is\nconvex, the relative interior of ${\\bf p_up_{v'}}$ is contained in the interior of $f$. Note that $u'Q'PQv'$ is a sub-walk of $f$ since $f$ is not simple. We draw a curve $C$ joining $u'$ with $v'$ following the walk $u'Q'PQv'$ in its small neighborhood in the interior $f$; we cut $C$ at its (two) intersection points with ${\\bf p_up_{v'}}$ and reconnected the severed ends on both sides by a curve following ${\\bf p_up_{v'}}$ in its small neighborhood thereby obtaining a closed curve, and a curve $C'$ joining  $v'$ and $u'$. Finally, $C'$ can be subdivided by vertices thereby \nyielding a desired embedding of $G\\cup P'$.\nOtherwise, if $\\omega_v$ is concave\nor ${\\bf p_up_{v'}}$ is crossed by an edge of $G$ we need to deform the clustered\nembedding of $G$ so that this is not longer the case.\n\n \\begin{figure}[h]\n  \\centering\n\\centering\\textbf{\u2022}\n{\n\\includegraphics[scale=0.6]{stork}\\textbf{\u2022}\n    \t}\n\n\\caption{A pair of deformations of the clustered embedding of $G$ so that $f$ can be subdivided by $P'$. For the sake of clarity clusters are drawn as horiznotal strips rather than wedges.}\n \\label{fig:stork}\n\\end{figure}\n\n\nBy a \\emph{spur} with the \\emph{tip} $u$ we understand a closed curve  obtained  as  a concatenation of a line segment contained in a ray separating clusters\nand a curve contained in the boundary of $f$ passing through exactly one extreme $u$\nof $f$ such that the curve is longest possible. The \\emph{length} is the spur\nis one plus the number of its crossings with rays separating clusters divided by two.\n If $\\omega_u$ is concave, the vertex $u$ is a tip of a spur whose length is the distance of $u$ to a closest other extreme along the face. Note that both $P$ and $Q'$ must be paths in this case.\nThe rough idea in the omitted part of the proof is that shortest spurs have room around them to be deformed while maintaining the embedding\nclustered such that $P'$ can be added.\nSpurs  are deformed  as illustrated in Fig.~\\ref{fig:stork}. (see Appendix for the rest of the proof)\n\\iflong\n\n\n \\begin{figure}[h]\n  \\centering\n\\centering\n{\n\\includegraphics[scale=0.65]{deform1}\n    \t}\n\n\\caption{Deformation in the case when both $u$ and $v$ have concave wedges incident to $f$ that is indicated by grey.\nThe dashed curve represents the path $P'$ subdividing $f$.\nOn the left,  point ${\\bf p_v'}\\in{\\bf p_{v}p_{u'}}$. In the middle, \n   point ${\\bf p_v'}\\not\\in{\\bf p_{v}p_{u'}}$. On the right, the corresponding\n deformation.}\n\\label{fig:deform1}\n\\end{figure}\n\n\n First, we suppose that $\\omega_u$ is concave.\n W.l.o.g. we assume that ${\\bf p_{v}'}\\not\\in{\\bf p_{v}p_{u'}}$. This holds when $\\omega_v$ is convex,\n Figure~\\ref{fig:deform2}. Otherwise,\nwe exchange the roles of $u$ and $v$, see Figure~\\ref{fig:deform1}. Combinatorially, there are two\ncases depending on whether $v$ is concave, but we treat them\nsimultaneously.\nWe isolate a part of the embedding of $G$ inside a spur represented\nby a topological disc $D$. In order to get a desired deformed clustered embedding of $G$ we define a homeomorphism from $D$ that we use to redraw the corresponding part of $G$  thereby disconnecting some edges\nthat are reconnected in the end.\n Let $D_0$ denote the topological disc bounded by the closed curve\nobtained by concatenating the line segment ${\\bf p_{v}}{\\bf p_{u'}}$ with the parts\nof $P$ and $Q'$ connecting endpoints of ${\\bf p_{v}}{\\bf p_{u'}}$  with $u$. \nWe assume that $v'\\not\\in D_0$ which holds automatically when $\\omega_v$ is concave due to ${\\bf p_{v}'}\\not\\in{\\bf p_{v}p_{u'}}$. \n\n \n \n\n \\begin{figure}[h]\n  \\centering\n\\centering\n\\subfloat[]{\n\\label{fig:deform3}\n\\includegraphics[scale=0.65]{deform3}\n    \t}\n    \t\\hspace{10px}\n\\subfloat[]{\n\\label{fig:deform5}\n\\includegraphics[scale=0.65]{deform5}\t\n    \t}\n\\caption{(a) A ``covering map'' $g$ from a disc onto a spur. (b) Identifying parts of the boundary of the discs $D'$ \naccording to $D$.}\n\n\\end{figure}\n \n\n  If $v$ does not have a concave wedge incident to $f$ it could happen that the boundary of $D_0$ crosses itself. As we will see later due to this reason we cannot simply put $D:=D_0$.\nWe use a ``covering map'' $g$, see Figure~\\ref{fig:deform3}.\nIn the light of Jordan-Sch\\\"onflies theorem,\n  let a topological disc $D$ be a pre-image of a continuous  map  $g:D \\rightarrow D_0$ such that the map $g$ is injective when restricted (in the target) to the embedding of $G$;\n$g$ maps the boundary of $D$ to the concatenation of ${\\bf p_{v}}{\\bf p_{u'}}$\nwith the parts of $P$ and $Q'$ connecting endpoints of ${\\bf p_{v}}{\\bf p_{u'}}$  with $u$;\nand the pre-image (of  parts) of the rays separating clusters\n consists of a union of a connected part of the boundary of $D$ \n (contained in the pre-image of ${\\bf p_{v}}{\\bf p_{u'}}$) and\na set of pairwise disjoint diagonals of $D$.\nTreating $G$ as a topological space let $G|_D:=g^{-1}(G)$.  \n\n If $G$ contains cut-vertices $D$ can have pairs of boundary points identified, see Figure~\\ref{fig:deform5}.\nLet $\\ell_0'$ denote  a line segment contained inside the intersection of the interior of $f$ with a ray separating clusters containing ${\\bf p_v'}$, whose end vertex is very close to ${\\bf p_v'}$, and $\\ell_0'\\subset{\\bf p_v'p_v}$ if and only if $\\omega_v$ is convex. Let $D'$ denote a disc bounded by $\\ell_0'$ and a curve \njoining the endpoints of $\\ell_0'$ following $Q$ towards $v'$ and back in its small neighborhood in the interior of $f$.\nLet $\\ell_0$ be the connected component on the boundary of $D$ in $g^{-1}({\\bf p_vp_{u'}})$.\nLet $\\ell_i$ for $0<i<|\\mathrm{height}(P)|$, denote a connected component of a pre-image by $g$ of (a part of) a ray separating clusters. The segments \n$\\ell_i$'s are indexed by the order of appearance of their endpoints along the boundary of $D$.\nA line segment $\\ell_i$ is just a point if it joins identified boundary points in $D$.\nSimilarly, let $\\ell_i'$ for $0<i<|\\mathrm{height}(P)|$, denote a connected component of the \nintersection of $D'$ with rays separating clusters.\nWe contract $\\ell_i'$ to a point iff $\\ell_i$ is a single point\nand we  contract  interior parts of $D'$ in the correspondence with $D$ and $\\ell_i$'s as illustrated in Figure~\\ref{fig:deform5}. Since $v'\\not\\in D_0$, we have $(g(D)=D_0)\\cap D' = \\emptyset$.\n\n \n \n\n \\begin{figure}[h]\n  \\centering\n\\centering\n\\subfloat[]{\n\\includegraphics[scale=0.65]{deform2}\n\\label{fig:deform2}    \t}\n    \t\\hspace{3px}\n\\subfloat[]{\n\\includegraphics[scale=0.65]{deform4}\n\\label{fig:deform4}    \t}\n\n\\caption{(a) Deformation in the case when only $u$ has a concave wedge incident to $f$ that is indicated by grey.\nThe dashed curve represents the desired path subdividing $f$.\nOn the left, the line segment ${\\bf p_{v'}p_{u}}$ crosses  edges of $G$. \nOn the right, the corresponding deformation. (b) The wedge at $u$\nin $f$, that is indicated by grey, is convex.  A sink $s$  in the interior of a spur having the vertex $u$\nas the tip.}\n\\end{figure}\n\n\n\n\n\\begin{figure}\n \\centering\n\\includegraphics[scale=0.7]{surgery}\n\\caption{Re-routing $Q'$ past $u$ along the boundary of $f$.}\n\\label{fig:deform8} \n\\end{figure}\n\nWe map by a homeomorphism $h$ the disc $D$ to $D'$ so that $\\ell_i$\nis mapped to $\\ell_i'$ for all $0\\leq i<|\\mathrm{height}(P)|$\nand so that the endpoint ${\\bf x}$ of $\\ell_0$ for which $g({\\bf x})$ is closer to $\\ell_0'$ is mapped to the point of $\\ell_0'$ closest to (farthest from) \n$g({\\bf x})$ if $\\omega_v$ is concave (convex), see Figure~\\ref{fig:stork} for an illustration.\nWe alter the embedding of $G$ by deleting $g(G|_D)$ and replacing it\nby $h(G|_D)$.\nFinally, we  reconnect the severed end pieces of edges intersecting ${\\bf p_vp_{u'}}$ by curves inside the cluster containing $v$ without\ncreating any edge crossing. \n\n\n\nIf $v'\\in D_0$, we redraw the portion of $G$ contained in the disc $D$ (we override the previous $D$) bounded by the line segment ${\\bf p_{v'}p_u}$\nand the parts of $P$ and $Q$ joining end points of ${\\bf p_{v'}p_u}$  with $v$, see Figure~\\ref{fig:deform6}.\nNote that the boundary of the disc $D$ is non-self intersecting.\nIf $Q'$ does not cross ${\\bf p_{v'}p_u}$ we make ${\\bf p_{v'}p_u}$ crossing free by mapping\nthe part of $G$ in $D$ to a long skinny disc $D'$  (we override the previous $D'$) in the vicinity of $Q'$ and reconnecting the severed edges similarly \nas above.\n\n \\begin{figure}\n  \\centering\n\\centering\n\\subfloat[]{\n\\includegraphics[scale=1]{3rdcaseT}\n\\label{fig:deform6}  }\n    \t\\hspace{1px}\n\\subfloat[]{\n\\includegraphics[scale=1]{3rdcase2T}\n\\label{fig:deform7}}\n\\caption{The face $f$ is indicated by grey. (a) The vertex  $v'\\in D_0$ corresponding to the spur with the tip $u$. (b) Corresponding  deformation involving $Q'$ and the dashed path $P'$ subdividing $f$.}\n\\end{figure}\n\n\n\nIf $Q'$ crosses ${\\bf p_{v'}p_u}$ we cannot apply the previous argument since $D \\cap D' \\not=\\emptyset$,\nand thus, we need a different approach.\nWe temporarily add to $G$ a path $u'Q''w$ starting at $u'$, following closely $Q'$ in the interior of $f$\nand ending in a vertex $w$. Let ${\\bf p_w}$ denote the crossing point of the last edge on $Q''$\nseparating clusters with a ray separating clusters. \nWe cut edges at ${\\bf p_{v'}p_u}$ by removing a small $\\epsilon>0$ neighborhood of their crossing points.\nWe reroute the paths $Q'$  (see Figure~\\ref{fig:deform8}) and $Q''$ without crossing an edge of $G$ from their severed ends outside of $D$ past $u$ and along $Q'$ in the interior of $f$ so that we closely follow the boundary of $f$.\nWe  easily avoid creating crossings since we cut edges at ${\\bf p_{v'}p_u}$.\nLet $D''$  be the disc bounded by rerouted parts of $Q'$ and $Q''$ from their first intersections on the way from $u$ and $w$, respectively, with ${\\bf p_wp_{u}'}$ to their common vertex $u'$; and by $\\ell_0''\\subset {\\bf p_wp_{u}'}$ connecting the ends of those parts.\nLet $\\ell_i''$, $0\\leq i < |\\mathrm{height}(P)|-c$, denote the line segments in the intersection of $D''$ with rays separating clusters listed in the order of appearance along the boundary of $D''$.\nWe deform the embedding of $G$ by a mapping $g$ from $D\\setminus (P \\cup Q)$, see Figure~\\ref{fig:deform7} for an illustration, such that $g$ maps the relative interior of $\\ell_0={\\bf p_{v'}p_u}$ to the relative interior of  ${\\bf p_wp_{u}'}$, the parts of $Q'$ and $Q''$ in $D$ to their rerouted counterparts, and for the open line segments $\\ell_i$, $1\\leq i <|\\mathrm{height}(P)|-c$, in the intersection of $D$ with the rays separating clusters we have $g(\\ell_{i+c})\\subset \\ell_i'' \\subset g(\\ell_i)$.\nThe mapping $g$ is then extended to the whole $D\\setminus (P \\cup Q)$ such that (i) $g$ is a homeomorphism when restricted to the interior\nof the slab between $\\ell_i$ and $\\ell_{i+1}$, for all $0\\le i<|\\mathrm{height}(P)|$, where $\\ell_{|\\mathrm{height}(P)|}$ is the \nfinal piece of the boundary of $D''$, and (ii) no edge crossings are introduced, i.e., $g(D|_{G})$ is injective\nand $g(D|_{G}) \\cap G =\\emptyset$. It is not hard to see that $g$ exists. Finally, we reconnect the severed end pieces of edges inside the cluster\ncontaining $u$ and remove $Q''$.\nThe previous deformation is perhaps easier seen as follows. The clustered embedding of $G|_{D}$ can be made  arbitrarily skinny. Thus, for the purpose of  deformation, we can picture that $G|_{D}$ consists just of the part of $Q'$ and $Q''$ in $D$ and a strictly monotone \npath starting at  $u'$ as in Figure\\ref{fig:deform6}. We map $G|_{D}$ as indicated in Figure~\\ref{fig:deform7}.\n \n   \nSecond, if both $\\omega_u$ and $\\omega_v$ are convex we can subdivide $f$ by $P'$ unless ${\\bf p_{v}p_{u'}}$ is intersected by edge(s) of $G$ (we still assume that  ${\\bf p_v'}\\not\\in{\\bf p_{v}p_{u'}}$).\n   However, if this is the case $D_0$ (defined above) contains a sink or source $s$ in its interior, see Figure~\\ref{fig:deform4}. Note that we can assume that $s$ is also a tip of a shortest spur of $f$, and hence, the previous case applies.\\fi\n \\ \\vrule width.2cm height.2cm depth0cm\\smallskip\\end{proof}\n\n\n\t\n\n\n\n\n\n\n\n\n{\\bf Acknowledgment} I would like to thank Jan Kyn\\v{c}l and D\\\"om\\\"ot\\\"or P\\'alv\\\"olgyi for many comments and suggestions that helped to improve the presentation of the result.\n    \n\t\n\t\n\t\n\n\n\n\n \n\n\n \n \n\n \n   \n\n\n\n\\bibliographystyle{plain}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\n\nAt a high energy hadron or ionized nucleus collider, the dynamics of the collision is overwhelmingly dominated by quantum chromodynamics (QCD).  Because of the Landau pole and confinement of QCD at energies comparable to hadron masses, the momentum exchanged in these hadronic collisions is typically small compared to center-of-mass energies, leading to a production of a large number of observed particles with momenta not far above the QCD scale.  With the high luminosities of the experiments at the Large Hadron Collider (LHC) or the Relativistic Heavy Ion Collider (RHIC), it is not currently technically feasible to record all collision events for later analysis so triggers are employed that flag an event as interesting.  While passing triggers does introduce some bias in the events that are recorded, consideration of all events that pass the various triggers provides an ensemble of events that, as close as possible, probes low-energy QCD in a high-energy collision environment.  Developing an understanding of these so-called minimum bias events has been an active area of research for the past several decades \\cite{Sjostrand:1987su,Durand:1988ax,Butterworth:1996zw,Borozan:2002fk,Ryskin:2009tj}.\n\n\nDespite significant research efforts, a first-principles understanding of minimum bias events is lacking precisely because it exists at the scale for which QCD becomes strongly interacting.  Thus, minimum bias events are typically described by models with many parameters, like that used in modern general-purpose event simulation programs like Pythia \\cite{Sjostrand:2006za} or Herwig \\cite{Bahr:2008dy}.  A hydrodynamics limit is often used to model and interpret collisions of heavy ions which result in extremely high particle multiplicities  \\cite{Kolb:2003dz,Romatschke:2009im,deSouza:2015ena},  providing evidence for production of the quark-gluon plasma (QGP) \\cite{BraunMunzinger:2007zz,Shuryak:2008eq,Nouicer:2015jrf,Pasechnik:2016wkt}.  However, with a model, experimental results can only be interpreted within the context of that model and without a more general framework may obscure other consistent descriptions of data.  What is more, such a description relies on unphysical or unmeasurable parameters (like temperature or chemical potential), and properties of collisions are often described in terms of the unmeasurable impact parameter or overlap of the nuclei. In this paper our philosophy is to propose and explore a framework in which certain collider observables may be captured in a model independent fashion, in terms of purely physical\/measurable quantities.\n\nOne instance where not being tied to a particular model(s) may be particularly useful is in the study of the relatively newly discovered phenomena of collective behavior in small systems such as pp collisions at the LHC and pA collisions at RHIC (see, e.g., \\Ref{doi:10.1146\/annurev-nucl-101916-123209} for a review). For instance, in order to disentangle evidence of the production of small `droplets' of QGP from other physics that leads to particle correlations, it would be desirable to have a framework that can interpolate between different system sizes, and collision energies.\n\nThe basic idea of the model independent approach we explore is to work directly with the measured particles, and  constrain the $S$-matrix of the minimum bias events. This philosophy  aligns with a bootstrap approach to QCD which is actively being pursued for low-multiplicity processes in quantum field theory \\cite{Paulos:2016fap,Paulos:2016but,Paulos:2017fhb,Cordova:2018uop,Mazac:2018mdx,Mazac:2018ycv,Cordova:2019lot,Karateev:2019ymz,Correia:2020xtr, Homrich:2019cbt, Komatsu:2020sag,Caron-Huot:2020adz,Guerrieri:2021tak}. The nature and high-multiplicity of minimum bias events offers some simplifications by appealing to the principle of ergodicity: a given particle is representative of any particle in the event. Further, events with $N$ and $N+1$ particles have approximately equal descriptions, and thus we can expect to be able to expand in the multiplicity parameter $N$.\nIn the absence of a first-principles understanding, we look for a detector-level effective  description of minimum bias for which there exists concrete power counting and symmetries that guide the description and has parameters that can be fit to data that are then ultimately described in QCD. \n\nAt a hadron collider, there are always particles that are unobservable as they go straight down the beampipe. We show that  integrating out these particles from the description of events has the effect of smearing the phase space of the observed particles in the detector in a universal fashion, reproducing the flat-in-(pseudo)rapidity distribution argued by Feynman~\\cite{Feynman:1969ej}. The expansion of the matrix element of the observed particles then provides fluctuations about this flat distribution. \n\nImportantly, the approach is applicable to observables that are binned in $N$. That is, it does not provide a first-principles description of  fluctuations in multiplicity. Thus the kind of statements that can be made are, for example, how {\\it normalized} distributions, binned in $N$, may change as a function of $N$. We also assume a fixed collider energy, $Q$, when taking the large $N$ limit, in the sense that we do not consider a `t Hooft coupling-like limit of taking $Q,N\\to\\infty$ at fixed $Q\/N$; statements about the scaling with $Q$, however, are obtained.\n\nOur aim is to explore concrete, quantitative predictions that must follow from consistency of the above-described approach to minimum bias.  We first define the power counting scheme for minimum bias events and identify its symmetries exclusively in terms of measurable quantities, just like the starting point of any effective field theory (EFT).  Through this procedure, the relevant expansion parameter we identify is novel and unlike familiar EFTs.  We formally take the number of observed particles $N\\gg 1$, which is similar to a hydrodynamics limit.  From the general expression for the cross section for production of $N$ final state particles in minimum bias events, we expand the squared matrix element in symmetric polynomials of the phase space coordinates, ordered in powers of $1\/N$.  Among predictions that we validate in collider data include:\n\\begin{itemize}\n\n\\item In the $N\\to\\infty$ \\ limit the symmetries of minimum bias events and the central limit theorem require that the squared matrix element is exclusively a function of the squared total energy of the observed final state particles.\n\n\\item Inspired by the soft and collinear singularities of perturbative QCD, we show that the distribution of particle transverse momentum is universal and depends on a single parameter in the large-$N$ limit.  Further, the transverse momentum distribution implies that particles in minimum bias events have a fractional dispersion relation $\\omega \\propto k_\\perp^{2\/3}$.\n\n\\item By positivity of the squared matrix element, we demonstrate that all pairwise particle azimuthal correlations necessarily vanish as $N\\to \\infty$ at fixed center-of-mass collision energy. \n\n\\item Long-distance pairwise particle pseudorapidity correlations are a consequence of factorization of the squared matrix element as $N\\to\\infty$.\n\n\\item Non-trivial pairwise particle azimuthal correlations are highly suppressed in minimum bias events in $e^+e^-$ collisions because of the full Lorentz invariance of the observed final state particles.\n\n\\end{itemize}\nThese results and others suggest that development of an effective theory for minimum bias would describe a wide range of phenomena and could be matched to perturbative QCD or hydrodynamics.\n\nThis paper is organized as follows.  In \\Sec{sec:ppmin}, we establish the power counting for minimum bias in identical hadron or heavy ion collisions and the corresponding symmetries that those events enjoy.  We show how to construct the squared matrix element in terms of symmetric polynomials of the phase space coordinates.  In \\Sec{sec:datacomp}, we use this effective form of the minimum bias cross section to interpret collider data.  We show that this large-$N$ expansion can provide quantitative and precise predictions that agree with data in its realm of applicability.  In \\Sec{sec:other}, we describe the power counting and symmetries for different collider environments.  Because minimum bias events in $e^+e^-$ collisions enjoy extensive symmetry, we show that these symmetries strongly suppress pairwise particle azimuthal correlations in the large-$N$ limit.  In \\Sec{sec:concs}, we summarize our findings and discuss future avenues to further develop an effective theory of minimum bias.\n\n\\section{Power Counting and Symmetries of pp\/AA Collision Min-Bias}\\label{sec:ppmin}\n\nWe first establish the power counting and symmetries we employ to describe minimum bias events.  This will correspondingly precisely define what we theoretically mean by ``minimum bias'', which is distinct from the experimental definition.  We will focus on minimum bias events at a hadron collider, in particular, proton-proton (pp) or identical heavy ion (AA) collisions.  We assume that the lab frame is also the center-of-mass frame of the collisions.  The center-of-mass collision energy $Q$ is assumed to be much larger than the masses of the initial, colliding particles.  Throughout this paper, we assume that $Q$ is an arbitrary, but fixed, energy, but will occasionally comment on the dependence of distributions of observables as a function of $Q$.  Additionally, we assume that the experimental detector cannot measure all particles and that there is an unmeasureable beam region.  Concretely, we assume that there is a maximum pseudorapidity $\\eta_{\\max}$ of the detector and particles with pseudorapidity $|\\eta|>\\eta_{\\max}$ are lost down the beampipe.  Finally, we assume that our detector can only measure particle momenta, but has perfect angular and energy resolution.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=14cm]{.\/detector}\n\\caption{Schematic illustration of detector (cylinder), its pseudorapidity range $\\eta_{\\max}$, the beam regions around $\\eta = \\pm\\infty$, and the characteristic transverse momentum of detected particles.\n\\label{fig:detect}\n}\n\\end{center}\n\\end{figure}\n\nWith these assumptions, we can establish power counting that defines minimum bias events and the corresponding symmetries of our collision experiment and of the detected final state particles.  The power counting that we take is as follows.\n\\begin{enumerate}\n\n\\item The beam is a small angular region outside the detection apparatus and we restrict our description of the event to far from the beam region, where detected particle pseudorapidity satisfies $|\\eta|\\sim 1\\ll \\eta_{\\max}$.\n\n\\item  We assume that the mass of the particles is irrelevant and so detected particle transverse momentum $p_\\perp$ is parametrically larger than the QCD scale or pion mass, $p_\\perp \\gg m_\\pi$. \n\n\\item The momentum lost down the beam region is an order-1 fraction of the center-of-mass energy $Q$. \n\n\\item The number of detected particles $N$ for which their pseudorapidity $|\\eta|\\ll\\eta_{\\max}$ is large: $N\\gg1$.\n\n\\item We assume that the mean transverse momentum of the detected particles is representative of all particles' momenta and so the mean and the root mean square momenta are comparable: $\\langle p_\\perp\\rangle\\sim \\sqrt{\\langle p_\\perp^2\\rangle}$.\n\\end{enumerate}\nWe will see below how these power counting assumptions then imply that our expansion parameter is $1\/N \\sim p_\\perp\/Q\\ll 1$.\\footnote{Actually, as we will see in the explicit examples below, there is a (dominant) additional fixed factor of $(Q\/p_{\\perp\\text{cut}})^\\frac{1}{2}$ in this scaling relation, with $p_{\\perp\\text{cut}}$ the experimental cut on transverse momentum.  This factor is fixed over an ensemble of events and we absorb it into `$\\sim$' here and in the following section when evaluating the scaling in $1\/N$ of terms in the expansion.} Additionally, these power counting assumptions imply that the minimum bias events we consider satisfy something like an ergodic hypothesis.  That is, we assume that any individual particle in an event is representative of all particles in an event.  Thus, averages over particles in an event are equivalent to averages of events over an ensemble.  This ergodic hypothesis will have important implications for the structure of the squared matrix element in the $N\\to\\infty$ limit.  An illustration of the physical configuration established by these power counting assumptions is provided in \\Fig{fig:detect}.\n\nHadron collision events that satisfy these power counting requirements then enjoy the following symmetries:\n\\begin{enumerate}\n\n\\item O(2) rotation and reflection symmetry  about the beam,\n\n\\item reflection of the beam $\\eta\\to-\\eta$ because of the identical colliding particles,\n\n\\item $S_N$ permutation symmetry in all $N$ detected particles, and\n\n\\item translation symmetry in pseudorapidity, $\\eta\\to \\eta+\\Delta \\eta$.\n\\end{enumerate}\nMost of these symmetries should be apparent as a consequence of the familiar cylindrical shape of particle detectors.  Because we assume only particle momenta are measured, no distinguishing information of particles (like electric charge) is collected, and hence there is complete permutation symmetry. By our power counting, we assume that the beam region is defined by $\\eta_{\\max}\\gg1$.  Taking the formal limit $\\eta_{\\max}\\to\\infty$, a translation in $\\eta$ by a finite amount $\\Delta \\eta$ can never move particles out of the detected region into the beam region, or vice-versa.  \n\nWe then define minimum bias events in hadron collisions as those that satisfy the power counting assumptions, and therefore inherit the symmetries.  Experimentally, minimum bias events are typically defined as those that pass a minimal trigger threshold for, say, activity in a forward calorimeter.  As the name suggests, passing a trigger does bias the event selection somewhat, but this bias can potentially be reduced by performing other cuts on these events.  For example, to force an event into the regime described by our power counting, one could require that a fixed fraction of particles have a transverse momentum within some range about the mean transverse momentum.  Or, one could require that an event does not have any identified jets with transverse momentum larger than some fraction of the center-of-mass energy.  We will return to jets below, and explicitly demonstrate that the existence of hard jets in the final state violates our power counting.\n\nBecause of the high luminosity of modern hadron colliders, truly zero bias events can collected by exploiting pile-up, or secondary hadron collisions per bunch crossing.  A hard hadron collision that produces jets or high-energy leptons, for example, would pass the triggers, but then secondary hadron collision events in the bunch crossing would have no requirements placed on them, and still be measured.  The challenge with extracting these zero bias events is then pushed to the ability to distinguish the point of collision where different particles were produced.  Any number of our power counting assumptions can be relaxed to provide a more realistic description of these events; however, we will find that even these strongly constraining assumptions will be able to explain and understand a wide breadth of data in various collider environments.\n \n\\subsection{Effective Form of the Cross Section}\n\nWe will use the power counting and symmetries to provide an effective description of the scattering cross section for these minimum bias events exclusively in terms of properties of the measured particles.  To derive this description, we start from the completely general formula for the cross section, considering $2\\to N+N_{B_1}+N_{B_2}$ scattering events, where $N_{B_1}+N_{B_2}$ particles are beam remnants; i.e., $N_{B_1}$ particles have pseudorapidity $\\eta > \\eta_{\\max}$, $N_{B_2}$ particles have $-\\eta > \\eta_{\\max}$, and $N$ particles have $|\\eta|< \\eta_{\\max}$.  The cross section $\\sigma$ for this scattering can be expressed in generality as\n\\begin{align}\n\\sigma &\\sim \\int d\\Pi_{N+N_{B_1}+N_{B_2}}\\, |{\\cal M}(1,\\dotsc,N,N+1,\\dotsc, N+N_{B_1},N+N_{B_1}+1,\\dotsc,N+N_{B_1}+N_{B_2})|^2 \\nonumber\\\\\n&\\sim  \\int \\prod_{i=1}^N\\left[p_{\\perp i}\\, dp_{\\perp i}\\, d\\eta_i\\, \\frac{d\\phi_i}{2\\pi}  \\right]\\, \\prod_{j=1}^{N_{B_1}}\\left[dk_j^+\\, dk_j^-\\, \\frac{d\\phi_j}{2\\pi} \\right]\\, \\prod_{k=1}^{N_{B_2}}\\left[dk_k^+\\, dk_k^-\\, \\frac{d\\phi_k}{2\\pi}  \\right]\\,\n\\nonumber\\\\\n&\n\\hspace{0.5cm}\\times|{\\cal M}(1,\\dotsc,N+N_{B_1}+N_{B_2})|^2\\, \\delta\\left(\nQ - \\sum_{i=1}^N p_{\\perp i}e^{\\eta_i} - \\sum_{j=1}^{N_{B_1}} k_j^-- \\sum_{k=1}^{N_{B_2}} k_k^-\n\\right)\\nonumber\\\\\n&\\hspace{0.5cm}\n\\times \\delta\\left(\nQ - \\sum_{i=1}^N p_{\\perp i}e^{-\\eta_i} - \\sum_{j=1}^{N_{B_1}} k_j^+- \\sum_{k=1}^{N_{B_2}} k_k^+\n\\right)\\,\\delta^{(2)}\\left(\n\\sum_{i=1}^N \\vec p_{\\perp i}+\\sum_{j=1}^{N_{B_1}} \\vec p_{\\perp j}+\\sum_{k=1}^{N_{B_2}} \\vec p_{\\perp k}\n\\right)\\,.\n\\end{align}\nIn writing this expression, we have ignored overall factors of the center-of-mass energy $Q$ and powers of $2\\pi$.  In the first line, we implicitly write the cross section as the integral of the squared matrix element over $N+N_{B_1}+N_{B_2}$-body phase space, and then in the remaining lines, expand out the massless Lorentz-invariant integration measure and momentum-conserving $\\delta$-functions.  $k^+_i$ and $k^-_i$ are the plus and minus components of particle $i$'s lightcone momentum with respect to the beam axis (the $z$ axis).  The momentum-conserving $\\delta$-functions fix the lab frame to be the center-of-mass frame of the collisions.\n\nNow, with the assumption that $\\eta_{\\max}\\gg 1$, we will formally take $\\eta_{\\max} \\to\\infty$, so the beam remnants live at $\\pm \\infty$ pseudorapidity.  With that assumption, their transverse momenta are 0 and one component of their lightcone momenta is also 0, depending on whether the particles are in beam 1 or beam 2.  These assumptions simplify the momentum conserving $\\delta$-functions and the cross section becomes  \n\\begin{align}\n\\sigma &\\sim   \\int \\prod_{i=1}^N\\left[p_{\\perp i}\\, dp_{\\perp i}\\, d\\eta_i\\, \\frac{d\\phi_i}{2\\pi}  \\right]\\, \\prod_{j=1}^{N_{B_1}}\\left[dk_j^+\\, dk_j^-\\, \\frac{d\\phi_j}{2\\pi} \\right]\\, \\prod_{k=1}^{N_{B_2}}\\left[dk_k^+\\, dk_k^-\\, \\frac{d\\phi_k}{2\\pi}  \\right]\\,|{\\cal M}(1,\\dotsc, N+N_{B_1}+N_{B_2})|^2\\nonumber \\\\\n&\n\\hspace{0.5cm}\\times \\delta\\left(\nQ - \\sum_{i=1}^N p_{\\perp i}e^{\\eta_i} - \\sum_{j=1}^{N_{B_1}} k_j^-\n\\right)\\, \\delta\\left(\nQ - \\sum_{i=1}^N p_{\\perp i}e^{-\\eta_i} - \\sum_{k=1}^{N_{B_2}} k_k^+\n\\right)\\,\\delta^{(2)}\\left(\n\\sum_{i=1}^N \\vec p_{\\perp i}\n\\right)\\,.\n\\end{align}\nIt may seem like the assumption that the beam remnants carry no net transverse momentum is too constraining.  However, assuming that the number of detected particles $N$ is large and the correlations between the detected particles are relatively weak, the net transverse momentum of the detected particles is necessarily small.  The detected particles in an individual event can be imagined to randomly walk in the transverse momentum plane and the average net transverse momentum in an ensemble of events will be 0.  As a random walk, the standard deviation of the net transverse momentum in this ensemble of events scales like $1\/\\sqrt{N}$ at fixed collision energy $Q$.  Then, to leading approximation in the large-$N$ limit, we assume that the net transverse momentum of the detected particles is 0.  It would be interesting to determine the consequences of relaxing this assumption, but we leave that to future work.\n\nBecause we do not measure the particles in the beam region, we want to integrate over them.  With the assumption of permutation symmetry of the particles and the fact that we assume that all beam remnants travel in the exact same direction down either pipe, the only quantity that we can be sensitive to in experiment is their total momentum.  That is, the squared matrix element can only depend on a function of the total plus or minus lightcone momentum that goes down the beampipes:\n\\begin{align}\n&|{\\cal M}(1,\\dotsc,N,N+1,\\dotsc, N+N_{B_1},N_{B_2})|^2 \\\\\n&\n\\hspace{1cm}\\sim f\\left(\n \\sum_{j=1}^{N_{B_1}} k_j^-\n,\n\\sum_{k=1}^{N_{B_2}} k_k^+\n\\right)\\, \\frac{|{\\cal M}(1,\\dotsc,N,N+1,\\dotsc, N+N_{B_1},N_{B_2})|^2}{f\\left(\n \\sum_{j=1}^{N_{B_1}} k_j^-\n,\n\\sum_{k=1}^{N_{B_2}} k_k^+\n\\right)}\\nonumber\\\\\n&\\hspace{1cm}\\equiv f\\left(\n \\sum_{j=1}^{N_{B_1}} k_j^-\n,\n\\sum_{k=1}^{N_{B_2}} k_k^+\n\\right)\\, |{\\cal M}(1,2,\\dotsc,N)|^2\\nonumber\\,.\n\\end{align}\nHere, $f$ is some function of the total lightcone momentum that goes in either beam direction.  In the final line, we have expressed squared matrix element with dependence on the measured particles $1,\\dotsc, N$ and dependence on the total lightcone momentum in the beam regions is left implicit.  We will return to the explicit form of this matrix element later. The $\\eta\\to -\\eta$ symmetry of the event requires that the function $f$ is symmetric in its arguments: $f(x,y) = f(y,x)$.  Further, by the $\\eta\\to \\eta+\\Delta \\eta$ translation symmetry, the argument of the function $f$ must be the product of the total plus and minus lightcone momenta, because lightcone momentum components transform homogeneously under boosts along the $z$ axis.  Then, the squared matrix element can be expressed as\n\\begin{align}\n|{\\cal M}(1,\\dotsc,N,N+1,\\dotsc, N+N_{B_1},N_{B_2})|^2 \\sim f\\left(\n \\sum_{j=1}^{N_{B_1}} k_j^-\n\\sum_{k=1}^{N_{B_2}} k_k^+\n\\right)\\, |{\\cal M}(1,2,\\dotsc,N)|^2\\,.\n\\end{align}\n\nBy momentum conservation and choosing the collision frame to be the center-of-mass frame, we also know that the total lightcone momentum lost down the beam regions is related to the lightcone momentum of the detected particles.  In particular, \n\\begin{align}\n& \\sum_{j=1}^{N_{B_1}} k_j^- = Q-\\sum_{i=1}^N p_{\\perp i}e^{\\eta_i}\\,, &\\sum_{k=1}^{N_{B_2}} k_k^+ = Q-\\sum_{i=1}^N p_{\\perp i}e^{-\\eta_i}\\,.\n\\end{align}\nThus, we can equivalently express the function $f$ exclusively in terms of directly measurable quantities.  We define\n\\begin{align}\n&k^-\\equiv \\sum_{i=1}^N p_{\\perp i}e^{\\eta_i}\\,, &k^+\\equiv\\sum_{i=1}^N p_{\\perp i}e^{-\\eta_i}\\,,\n\\end{align}\nso that the argument of the function $f$ is just the product $k^+k^-$, by the same reasoning above.  That is, the matrix element can be expressed as\n\\begin{align}\n|{\\cal M}(1,\\dotsc,N,N+1,\\dotsc, N+N_{B_1},N_{B_2})|^2 \\sim f\\left(\nk^+k^-\n\\right)\\, |{\\cal M}(1,2,\\dotsc,N)|^2\\,,\n\\end{align}\nfor some function $f(k^+k^-)$ and our power counting assumes that the momentum lost down the beam regions (or, the $z$-axis boost of the detected particles), is of a comparable size to the center-of-mass energy, $ Q^2$.  \n\nThen, with the power counting and symmetries enforced on the form of the matrix element, in the cross section we can integrate over the momentum lost down the beams.  The cross section can then be expressed as\n\\begin{align}\\label{eq:xsecmaster}\n\\sigma &\\sim \\int_0^Q dk^+ \\int_0^Q dk^- \\int \\prod_{i=1}^N\\left[p_{\\perp i}\\, dp_{\\perp i}\\, d\\eta_i\\, \\frac{d\\phi_i}{2\\pi}  \\right]\\,\\prod_{j=1}^{N_{B_1}}\\left[dk_j^+\\, dk_j^-\\, \\frac{d\\phi_j}{2\\pi} \\right]\\, \\prod_{k=1}^{N_{B_2}}\\left[dk_k^+\\, dk_k^-\\, \\frac{d\\phi_k}{2\\pi}  \\right]\\,f\\left(\nk^+k^-\n\\right)\\nonumber \\\\\n&\n\\hspace{1cm}\\times  |{\\cal M}(1,2,\\dotsc,N)|^2\\, \\delta\\left(\nQ-k^- -\\sum_{j=1}^{N_{B_1}} k_j^-\n\\right)\\, \\delta\\left(\nQ-k^+ - \\sum_{k=1}^{N_{B_2}} k_k^+\n\\right)\\,\\delta^{(2)}\\left(\n\\sum_{i=1}^N \\vec p_{\\perp i}\n\\right)\n\\nonumber\\\\\n&\\hspace{1cm}\n\\times \\delta\\left(\nk^--\\sum_{i=1}^N p_{\\perp i}e^{\\eta_i}\n\\right)\\,\\delta\\left(\nk^+-\\sum_{i=1}^N p_{\\perp i}e^{-\\eta_i}\n\\right)\\nonumber\n\\\\\n&\\sim  \\int_0^Q dk^+ \\int_0^Q dk^- \\int \\prod_{i=1}^N\\left[p_{\\perp i}\\, dp_{\\perp i}\\, d\\eta_i\\, \\frac{d\\phi_i}{2\\pi}  \\right]\\, f\\left(\nk^+k^-\n\\right)\\, |{\\cal M}(1,2,\\dotsc,N)|^2 \\nonumber\\\\\n&\n\\hspace{1cm}\\times \\delta\\left(\nk^- - \\sum_{i=1}^N p_{\\perp i}e^{\\eta_i}\n\\right)\\, \\delta\\left(\nk^+ - \\sum_{i=1}^N p_{\\perp i}e^{-\\eta_i}\n\\right)\\,\\delta^{(2)}\\left(\n\\sum_{i=1}^N \\vec p_{\\perp i}\n\\right)\\,.\n\\end{align}\nIn doing the integrals over the unobservable beam remnant particles, we have ignored overall factors of the center-of-mass energy $Q$, and additional dependence on the product $k^+k^-$ has been absorbed in the definition of $f(k^+k^-)$. Because $f(k^+k^-)$ is a physical squared matrix element, we make the reasonable assumption that it is finite and analytic on its domain.  It can therefore be expanded in an appropriate basis of orthonormal polynomials, and this sum can be truncated for approximation to fit data.\n\nIn analysis that we will present later, it will be useful to know the volume of this $N$-body phase space smeared by a boost along the beam axis.  For a center-of-mass energy $Q$, the volume of $N$-body phase space is\n\\begin{equation}\\label{eq:psvol}\n\\int d\\Pi_N = (2\\pi)^{4-3N}Q^{2N-4}\\frac{2\\pi^{N-1}}{(N-1)!(N-2)!}\\,.\n\\end{equation}\nSetting the function $f(k^+k^-)$ and the squared matrix element $|{\\cal M}(1,2,\\dotsc,N)|^2$ to unity, the volume of this smeared phase space is therefore\n\\begin{align}\n&\\int_0^Q dk^+ \\int_0^Q dk^- \\int \\prod_{i=1}^N\\left[p_{\\perp i}\\, dp_{\\perp i}\\, d\\eta_i\\, \\frac{d\\phi_i}{2\\pi}  \\right]\\,\\\\\n&\n\\hspace{2cm}\\times \\delta\\left(\nk^- - \\sum_{i=1}^N p_{\\perp i}e^{\\eta_i}\n\\right)\\, \\delta\\left(\nk^+ - \\sum_{i=1}^N p_{\\perp i}e^{-\\eta_i}\n\\right)\\,\\delta^{(2)}\\left(\n\\sum_{i=1}^N \\vec p_{\\perp i}\n\\right)\\nonumber\\\\\n&\\hspace{1cm}=(2\\pi)^{4-3N}\\frac{2\\pi^{N-1}}{(N-1)!(N-2)!}\\int_0^Q dk^+ \\int_0^Q dk^-\\, (k^+k^-)^{N-2}\\nonumber\\\\\n&\\hspace{1cm}=(2\\pi)^{4-3N}Q^{2N-2}\\frac{2\\pi^{N-1}}{(N-1)^2(N-1)!(N-2)!}\\nonumber\\,.\n\\end{align}\nWe also note that the topology of the smeared phase space is that of a $(3N-2)$-ball, found by integrating over two of the dimensions of the $N$-body phase space manifold \\cite{Larkoski:2020thc}.\n\n\\subsection{Expansion of the Squared Matrix Element}\n\nHaving established the form of the cross section for minimum bias events expressed exclusively in terms of observable particle momenta, we now use the power counting and symmetries to expand the squared matrix element $|{\\cal M}(1,2,\\dotsc,N)|^2$.  The natural expansion parameter in which to do this according to our power counting is the number of detected particles $N\\gg 1$.  Namely, we would want to determine the momentum dependence of the squared matrix element systematically at every power of $1\/N$, with undetermined parameters that can be fit to data.  We will do this in a few steps.  First, we will present an expansion of the squared matrix element in inverse powers of the center-of-mass energy $Q$, for which terms can be established order-by-order through use of momentum conservation.  From this expansion, we can identify the scaling with $N$ for each of the terms and reorganize the expansion according to our power counting rules.\n\nWe first note that, with this construction, the leading approximation of the squared matrix element is that it is constant.  As we can fix overall normalization later, the leading approximation of squared matrix element in the $1\/Q$ expansion is\n\\begin{equation}\n|{\\cal M}(1,2,\\dotsc,N)|^2 = 1 + {\\cal O}(Q^{-1})\\,.\n\\end{equation}\nA linearly-independent set of momentum-dependent terms at higher powers in $1\/Q$ can be established using momentum conservation.  The transverse and longitudinal momentum conservation equations of the detected particles are:\n\\begin{align}\\label{eq:momconserv}\n0&=\\left(\n\\sum_{i=1}^N \\vec p_{\\perp i}\n\\right)^2 = \\sum_{i=1}^N p_{\\perp i}^2 + \\sum_{i\\neq j}^N p_{\\perp i}p_{\\perp j}\\cos(\\phi_i-\\phi_j)\\,,\\\\\nk^+k^-&=\\left(\n\\sum_{i=1}^N p_{\\perp i}e^{-\\eta_i}\n\\right)\\left(\n\\sum_{j=1}^N p_{\\perp j}e^{\\eta_j}\n\\right) = \\sum_{i=1}^N p_{\\perp i}^2 + \\sum_{i\\neq j}^N p_{\\perp i}p_{\\perp j}\\cosh(\\eta_i-\\eta_j)\\,.\\nonumber\n\\end{align}\nOn the right, we have expanded the equalities in terms of permutation-symmetric polynomials of the phase space coordinates.  The symmetries of minimum bias events forbid terms at order $1\/Q$ (and actually any odd power of this) and these fundamental momentum conservation equations define a linearly-independent set of terms at order $1\/Q^2$.  We can then write\n\\begin{equation}\n|{\\cal M}(1,2,\\dotsc,N)|^2 = 1+\\frac{c_1^{(2)}}{Q^2}\\sum_{i=1}^N p_{\\perp i}^2+{\\cal O}(Q^{-4})\\,,\n\\end{equation}\nwhere $c_1^{(2)}$ is some dimensionless, numerical coefficient.  We will discuss the potential dependence of $c_1^{(2)}$ on the number of particles $N$ later.\n\nTo construct linear relationships between terms at higher mass dimension, we can multiply the momentum conservation equations by symmetric polynomials of the minimum bias phase space coordinates.  For example, two identities that can be used to establish terms at order $1\/Q^4$ are:\n\\begin{align}\n&\\sum_{i=1}^N p_{\\perp i}^2\\left(\n\\sum_{j=1}^N p_{\\perp j}^2 + \\sum_{j\\neq k}^N p_{\\perp j}p_{\\perp k}\\cos(\\phi_j-\\phi_k)\n\\right) = 0 \\\\\n&\n\\hspace{1cm}=\\sum_{i=1}^N p_{\\perp i}^4 +  \\sum_{j\\neq k}^N p_{\\perp j}^3p_{\\perp k}\\cos(\\phi_j-\\phi_k)+\\sum_{i\\neq j\\neq k}^N p_{\\perp i}^2p_{\\perp j}p_{\\perp k}\\cos(\\phi_j-\\phi_k)\\,,\\nonumber\\\\\n&\\sum_{i=1}^N p_{\\perp i}^2\\left(\nk^+k^--\\sum_{j=1}^N p_{\\perp j}^2 - \\sum_{j\\neq k}^N p_{\\perp j}p_{\\perp k}\\cosh(\\eta_j-\\eta_k)\n\\right) = 0 \\nonumber\\\\\n&\\hspace{1cm}=k^+k^-\\sum_{i=1}^N p_{\\perp i}^2-\\sum_{i=1}^N p_{\\perp i}^4 - \\sum_{j\\neq k}^N p_{\\perp j}^3p_{\\perp k}\\cosh(\\eta_j-\\eta_k)-\\sum_{i\\neq j\\neq k}^N p_{\\perp i}^2p_{\\perp j}p_{\\perp k}\\cosh(\\eta_j-\\eta_k)\\,.\\nonumber\n\\end{align}\nAgain, on the right, we have expanded the identities in terms of independent symmetric polynomials.  Then, through the first few orders, the squared matrix element can be expanded as:\n\\begin{align}\n\\hspace{-0.2cm}|{\\cal M}(1,2,\\dotsc,N)|^2 &= 1+\\frac{c_1^{(2)}}{Q^2}\\sum_{i=1}^N p_{\\perp i}^2+\\frac{c_1^{(4)}}{Q^4}k^+k^-\\sum_{i=1}^N p_{\\perp i}^2+\\frac{c_2^{(4)}}{Q^4}\\sum_{i=1}^N p_{\\perp i}^4+\\frac{c_3^{(4)}}{Q^4}\\sum_{i\\neq j}^N p_{\\perp i}^2p_{\\perp j}^2\\\\\n&\n\\hspace{0.5cm}+\\frac{c_4^{(4)}}{Q^4}\\sum_{i\\neq j}^N p_{\\perp i}^3p_{\\perp j}\\cosh(\\eta_i-\\eta_j)+\\frac{c_5^{(4)}}{Q^4}\\sum_{i\\neq j}^N p_{\\perp i}^2p_{\\perp j}^2\\cosh(2(\\eta_i-\\eta_j))\\nonumber\\\\\n&\n\\hspace{0.5cm}+\\frac{c_6^{(4)}}{Q^4}\\sum_{i\\neq j\\neq k}^N p_{\\perp i}^2p_{\\perp j}p_{\\perp k}\\cosh(\\eta_i-\\eta_j)\\cosh(\\eta_i-\\eta_k)\\nonumber\\\\\n&\n\\hspace{0.5cm}+\\frac{c_7^{(4)}}{Q^4}\\sum_{i\\neq j}^N p_{\\perp i}^2p_{\\perp j}\\cos(\\phi_i-\\phi_j)+\\frac{c_8^{(4)}}{Q^4}\\sum_{i\\neq j}^N p_{\\perp i}^2p_{\\perp j}^2\\cos(2(\\phi_i-\\phi_j))\\nonumber\\\\\n&\n\\hspace{0.5cm}+\\frac{c_9^{(4)}}{Q^4}\\sum_{i\\neq j\\neq k}^N p_{\\perp i}^2p_{\\perp j}p_{\\perp k}\\cos(\\phi_i-\\phi_j)\\cos(\\phi_i-\\phi_k)\\nonumber\\\\\n&\n\\hspace{0.5cm}+\\frac{c_{10}^{(4)}}{Q^4}\\sum_{i\\neq  j}^N p_{\\perp i}^2p_{\\perp j}^2\\cosh(\\eta_i-\\eta_j)\\cos(\\phi_i-\\phi_j)\\nonumber\\\\\n&\n\\hspace{0.5cm}+\\frac{c_{11}^{(4)}}{Q^4}\\sum_{i\\neq j\\neq k}^N p_{\\perp i}^2p_{\\perp j}p_{\\perp k}\\cosh(\\eta_i-\\eta_j)\\cos(\\phi_i-\\phi_k) + {\\cal O}(Q^{-6})\\,.\n\\nonumber\n\\end{align}\nThe $c_i^{(4)}$ are dimensionless numerical coefficients.  Linear relationships from conservation of momentum have already been accounted for and (hyperbolic) trigonometric identities have been used. Note also that any terms that exclusively depend on the product $k^+k^-$ can be absorbed into the function $f(k^+k^-)$ and are not included.\n\nThe numerical coefficients $c_i^{(n)}$ in general depend on the number of detected particles $N$.  For the most part, we will be agnostic as to what their precise scaling with $N$ is, but we will make the following weak, but constraining, assumption.  We assume that, in the $N\\to\\infty$ limit, the dependence of the coefficients $c_i^{(n)}$ on $N$ is such that the entire corresponding term in the squared matrix element remains finite.  For example, consider the term at ${\\cal O}(Q^{-2})$.  With our power counting assumption that $p_\\perp \\sim Q\/N$ and ergodicity, the kinematic dependence of this term scales like\n\\begin{equation}\n\\frac{1}{Q^2}\\sum_{i=1}^N p_{\\perp i}^2 \\sim \\frac{1}{N}\\,,\n\\end{equation}\nbecause there are $N$ terms in the sum and each term scales like $1\/N^2$.  Thus, the coefficient $c_1^{(2)}$ can scale at worst linear in $N$ as $N\\to\\infty$ and the whole term will still be finite in the squared matrix element.\\footnote{In matching this expansion to a short-distance description from QCD, the coefficients $c_i^{(n)}$ can generically also depend on the ratio of collision energy $Q$ to the QCD scale $\\Lambda_\\text{QCD}$ or pion mass $m_\\pi$.  Explicitly doing the matching or resumming large logarithms of this ratio requires an effective field theory for minimum bias, which we do not construct here and leave to future work.}\n\nFurther, the assumption of ergodicity implies that in the limit that $N\\to\\infty$, the entire matrix element becomes a constant, independent of the event's dynamics.  Our ergodic assumption implies that the symmetric sums over terms that appear in the squared matrix element reduce to the corresponding mean values for $N\\to\\infty$.  For example, consider the term at ${\\cal O}(Q^{-2})$ again.  The sum of the squared transverse momentum of all particles in the event returns the mean square transverse momentum $\\langle p_\\perp^2\\rangle$ times $N$, as $N\\to\\infty$.  Because the sum consists of $N$ terms, the variance of the sum is also of order $N$ and so the whole expression takes the form\n\\begin{equation}\n\\lim_{N\\to\\infty}\\sum_{i=1}^N p_{\\perp i}^2 \\to N\\langle p_\\perp^2\\rangle + {\\cal O}\\left(\\sqrt{N}\\langle p_\\perp^2\\rangle\\right)\\,.\n\\end{equation}\nIn the strict $N\\to\\infty$ limit, the variance of the sum vanishes, reducing to a constant value on all of phase space.  Every term in the squared matrix element reduces to its mean as $N\\to\\infty$ by ergodicity, and so the whole squared matrix element itself becomes a constant on the $N$-body phase space as $N\\to\\infty$.  We will exploit this limit of the matrix element in our predictions of the following section.\n\nWhile constructive, this $1\/Q$ expansion isn't exactly the expansion that is natural with our power counting.  Terms at a fixed order in $1\/Q$ do not in general have a homogeneous scaling with $1\/N$.  For example, at ${\\cal O}(Q^{-4})$, the term\n\\begin{equation}\n\\frac{1}{Q^4}k^+k^-\\sum_{i=1}^N p_{\\perp i}^2 \\sim \\frac{1}{N}\\,,\n\\end{equation}\nwhile the term\n\\begin{equation}\n\\frac{1}{Q^4}\\sum_{i=1}^N p_{\\perp i}^4 \\sim \\frac{1}{N^3}\\,,\n\\end{equation}\nas $N\\to\\infty$.  The particular form of higher-order terms in $1\/N$ is subtle to determine because, in general, terms that arise in the $1\/Q$ expansion don't have a homogeneous scaling in $1\/N$.  For example, assuming that $k^+k^-$ is homogeneous in powers of $N$ with $k^+k^-\\sim N^0$, then from \\Eq{eq:momconserv} the term with hyperbolic cosine is not homogeneous in $N$:\n\\begin{equation}\n\\sum_{i\\neq j}^N p_{\\perp i}p_{\\perp j}\\cosh(\\eta_i-\\eta_j)\\sim {\\cal O}(N^0) + {\\cal O}(N^{-1})\\,.\n\\end{equation}\nThis follows because of the assumption that $\\eta_i\\sim 1$ and so $\\cosh(\\eta_i-\\eta_j)\\sim 1$, the transverse momentum $p_{\\perp i}\\sim 1\/N$ and that the sum consists of $N^2-N$ terms.  We do not attempt to determine the general form of the $1\/N$ expansion of the squared matrix element here, although a systematic approach could be pursued along the lines of \\Refs{Henning:2015daa,Henning:2015alf,Henning:2017fpj,Henning:2019enq,Henning:2019mcv,Graf:2020yxt,Melia:2020pzd}.  However, what we will do in \\Sec{sec:datacomp}, when using the power counting and symmetries to understand collider data, is identify the $1\/N$ expansion for particular subsets of terms in the squared matrix element that are relevant for the specific measurements of interest that we consider in this work.\n\nOne important point to make at this stage is that we are considering fixed, but large, number of particles $N$.  That is, our expansion can make predictions for observables that survive in the $N\\to\\infty$ limit or for observables that are conditioned on $N$.  We will see a number of examples of such observables in \\Sec{sec:datacomp}.  However, with our present formulation of the cross section for minimum bias, we cannot make predictions for which the relative probability of different numbers of detected particles is important.  Perhaps the simplest such observable is just the multiplicity $N$ of detected particles itself.  We leave an understanding of particle multiplicity within this framework to future work.\n\n\\subsection{Where are Jets?}\n\nJets, while the most ubiquitous phenomena of high-energy QCD, are not described within the framework of our minimum bias expansion of the cross section.  From the perspective of our goals with this expansion, this is a good thing, as it implies that the expansion focuses on the physics of interest, namely the low-energy dynamics of QCD.  Specifically, the presence of high-energy jets in the detected final state particles violates the power counting assumption that the mean particle transverse momentum, $\\langle p_\\perp\\rangle$, is representative of all particles' transverse momentum.  Jets are collimated streams of particles which carry an order-1 fraction of the jet's energy.  Very low energy particles fill in the regions at wide angles from the jets.  The distribution of particle transverse momentum then has two dominant populations corresponding to high energy or very low energy particles.  Of course, for any distribution of $N$ particles with total energy $Q$, the mean transverse momentum $\\langle p_\\perp\\rangle \\sim Q\/N$.  However, when jets are present in the final state, the root mean square transverse momentum $\\sqrt{\\langle p_\\perp^2\\rangle}$ will be dominated by the high energy particles of the jets, and so we find the hierarchy that $\\sqrt{\\langle p_\\perp^2\\rangle}\\gg \\langle p_\\perp\\rangle \\sim Q\/N$.  This then means that jets are not described by any low-order truncation of either the $1\/Q$ or $1\/N$ expansion of the squared matrix element. \n\nOf course, this is not surprising as jets arise because of the approximate scale invariance of QCD at high energies, in addition to the small coupling $\\alpha_s$.  Even approximate scale invariance is not a symmetry of our definition of minimum bias, and so is not exploited in the expansion of the squared matrix element.  Further, the smallness of $\\alpha_s$ for controlling jet dynamics means that jets can be described by a perturbative effective field theory, known as soft-collinear effective theory \\cite{Bauer:2000yr,Bauer:2001ct,Bauer:2001yt,Bauer:2002nz}.  For minimum bias, because the only relevant energy scale is the mean transverse momentum $\\langle p_\\perp\\rangle \\sim Q\/N\\ll Q$, we cannot assume that $\\alpha_s$ is small so as to formulate a perturbative effective field theory.  \n\nNevertheless, even in the presence of collimated jets, the low energy particles that fill the detector away from the jets should exhibit many of the properties we have already discussed.  Namely, as long as the number $N$ of low energy particles not in the jets is large, we still expect their net transverse momentum to be approximately 0, up to the relative $1\/\\sqrt{N}$ standard deviation of a random walk.  However, jets in the final state introduce new, relevant directions that spontaneously break the symmetries of minimum bias as well as introduce correlations that would manifest themselves in the form of the squared matrix element.  As discussed, such correlations would not be captured to low orders in our expansion of the cross section. \n\n\\section{Interpretation of Data}\\label{sec:datacomp}\n\nIn this section, we exploit this expansion of the cross section for minimum bias events to understand and re-interpret collider data from this perspective.  Because the expansion as we have developed it thus far can describe any identical hadron or nucleus scattering, we apply it to understand data from both pp and heavy ion collisions.  Particularly in the case of heavy ion collisions, experimental data are very often interpreted or expressed in terms of strictly unobservable quantities as established in some model of the collision, like the centrality or the number of participating nucleons.  By contrast, our expansion is expressed exclusively in terms of directly observable quantities, like the total number of detected particles or the total observed final state energy. \n\n\n\\subsection{Pseudorapidity Distributions}\n\nThe first observable that we consider is the pseudorapidity distribution of observed final state particles.  This can be calculated directly from the form of the minimum bias cross section we established in \\Eq{eq:xsecmaster}.  From the permutation symmetry of the particles, every particle has the same pseudorapidity distribution $p(\\eta)$, so we can just fix the particle of interest to be particle 1.  Then, we have\n\\begin{align}\np(\\eta)&\\sim \\int_0^Q dk^+ \\int_0^Q dk^- \\int \\prod_{i=1}^N\\left[p_{\\perp i}\\, dp_{\\perp i}\\, d\\eta_i\\, \\frac{d\\phi_i}{2\\pi}  \\right]\\, f\\left(\nk^+k^-\n\\right)\\, |{\\cal M}(1,2,\\dotsc,N)|^2 \\\\\n&\n\\hspace{1cm}\\times \\delta(\\eta-\\eta_1)\\, \\delta\\left(\nk^- - \\sum_{i=1}^N p_{\\perp i}e^{\\eta_i}\n\\right)\\, \\delta\\left(\nk^+ - \\sum_{i=1}^N p_{\\perp i}e^{-\\eta_i}\n\\right)\\,\\delta^{(2)}\\left(\n\\sum_{i=1}^N \\vec p_{\\perp i}\n\\right)\\nonumber\\,.\n\\end{align}\nTo proceed, we will make a number of simplifying assumptions.  First, we will work in the $N\\to\\infty$ limit in which the squared matrix element $|{\\cal M}(1,2,\\dotsc,N)|^2$ reduces to a constant on phase space, as discussed earlier.  Because this constant is just an overall scaling, we can take it to be 1, for simplicity.  With this assumption, we can then determine the pseudorapidity distribution on flat phase space $p_\\text{flat}(\\eta)$, in the large-$N$ limit.  We have\n\\begin{align}\\label{eq:flatmaster}\np_\\text{flat}(\\eta) &\\sim \\lim_{N\\to\\infty}\\int \\prod_{i=1}^N \\left[\np_{\\perp i}\\, dp_{\\perp i}\\, d\\eta_i\\, \\frac{d\\phi_i}{2\\pi}\n\\right]\\, \\delta(\\eta - \\eta_1)\\\\\n&\n\\hspace{2cm}\n\\times \\delta\\left(\nk^--\\sum_i p_{i\\perp}e^{\\eta_i}\n\\right)\\delta\\left(\nk^+-\\sum_i p_{i\\perp}e^{-\\eta_i}\n\\right)\\, \\delta^{(2)}\\left(\n\\sum_i \\vec p_{i\\perp}\n\\right)\\nonumber\\\\\n&\\propto \\lim_{N\\to\\infty}\\int dp_{\\perp 1}\\, p_{\\perp 1}\\, d\\eta_1\\, \\delta(\\eta - \\eta_1)\\left[\n(k^+-p_{\\perp 1}e^{-\\eta_1})(k^--p_{\\perp 1}e^{\\eta_1})-p^2_{\\perp 1}\n\\right]^N\\nonumber\\\\\n&\\propto\\lim_{N\\to\\infty}\\int  dp_\\perp\\,p_\\perp \\left(\n1-\\frac{k^+e^{\\eta}+k^-e^{-\\eta}}{k^+k^-}\\,p_\\perp\n\\right)^N\\nonumber\\\\\n&=\\int_0^\\infty  dp_\\perp\\,p_\\perp \\, e^{-\\frac{k^+e^{\\eta}+k^-e^{-\\eta}}{k^+k^-}\\,Np_\\perp}\\nonumber\\\\\n&=\\frac{(k^+k^-)^2}{N^2}\\left(\nk^+e^{\\eta}+k^-e^{-\\eta}\n\\right)^{-2}\\,.\\nonumber\n\\end{align}\nTo get this expression, we used the large-$N$ limit of the volume of phase space from \\Eq{eq:psvol} and ignore overall factors.  The normalized probability distribution of particle pseudorapidity on flat phase space is then\n\\begin{equation}\\label{eq:flateta}\np_\\text{flat}(\\eta) = 2k^+k^-\\left(\nk^+e^{\\eta}+k^-e^{-\\eta}\n\\right)^{-2}\\,.\n\\end{equation}\nIn the large-$N$ limit, this can also be directly derived from assuming that all particles have a distribution flat in $\\cos\\theta$, the polar angle from the beam.  Note that this large-$N$ limit ignores momentum conservation in transforming to the exponential integrand.\n\nWe can then use this result to determine the observed pseudorapidity distribution $p(\\eta)$ smeared against the function $f(k^+k^-)$.  We have\n\\begin{align}\\label{eq:etadistsmear}\np(\\eta)&=\\frac{1}{Q^2}\\int_0^Q dk^+\\int_0^Q dk^-\\, f\\left(k^+k^-\\right)\n\\, p_\\text{flat}(\\eta) \\\\\n&= \\frac{1}{Q^2}\\int_0^Q dk^+\\int_0^Q dk^-\\, f\\left(k^+k^-\\right)\n\\, 2k^+k^-\\left(\nk^+e^{\\eta}+k^-e^{-\\eta}\n\\right)^{-2}\\nonumber\\\\\n&=\\int_0^1 dx\\, f(x)\\, \\frac{1-x^2}{1+x^2+2x \\cosh(2\\eta)}\n\\,.\\nonumber\n\\end{align}\nIn the final line, we have set $xQ^2=k^+k^-$ and integrated over the pseudorapidity of the system of final state particles in the lab frame.  Written in this way, $f(x)$ is itself a probability distribution whose normalization is inherited from its expression in terms of $k^+$ and $k^-$:\n\\begin{align}\n1&=\\frac{1}{Q^2}\\int_0^Q dk^+ \\int_0^Q dk^-\\, f(k^+k^-) = \\int_0^1 dx\\, \\text{log}\\frac{1}{x}\\, f(x)\\,.\n\\end{align}\nWe also note that because $f(x)\\geq 0$ on $x\\in[0,1]$, the pseudorapidity distribution $p(\\eta)$ monotonically decreases as $|\\eta|$ increases away from $\\eta=0$.  The integral in the final line of \\Eq{eq:etadistsmear} is also dominated by the region for $0\\leq x\\lesssim 1\/\\cosh(2\\eta)$.\n\nTo go further, we need an explicit form for the function $f(x)$.  As we noted earlier, $f(x)$ should be analytic on $x\\in[0,1]$ because it is a physical squared matrix element.  Also, because of the collinear singularity of QCD, we expect a very flat distribution of pseudorapidity over a wide range.  From our expression for the pseudorapidity distribution $p(\\eta)$ in \\Eq{eq:etadistsmear}, we note that if $f(x) = \\delta(x)$, then $p(\\eta)$ is flat, but not normalizable.  So, combining these observations suggests that $f(x)$ should be highly-peaked at $x = 0$ and analytic.  This motivates the following form:\n\\begin{align}\\label{eq:cutffunc}\nf(k^+k^-) = \\frac{n+1}{H_{n+1}}\\left(1-\\frac{k^+k^-}{Q^2}\\right)^n\\simeq \\frac{n}{\\gamma_E+\\text{log}\\, n}\\,e^{-n\\frac{k^+k^-}{Q^2}}\\,,\n\\end{align}\nwhere $n\\gg 1$ ensures strong peaking at $x=0$, $H_n$ is the harmonic number and $\\gamma_E$ is the Euler-Mascheroni constant.\\footnote{We have currently described the function $f(k^+k^-)$ as a function that corresponds to smearing the flat phase space distribution.  However, it can equivalently be interpreted as a squared matrix element $|{\\cal M}|^2$ for the $N$ detected particles using the relationship from \\Eq{eq:momconserv}.  With this interpretation, the squared matrix element is\n\\begin{equation}\n|{\\cal M}|^2 \\sim e^{-n\\frac{k^+k^-}{Q^2}} \\sim \\exp\\left[\n-n\\sum_{i\\neq j}^N \\frac{p_{\\perp i}p_{\\perp j}}{Q^2}\\cosh(\\eta_i-\\eta_j)\n\\right]\\,,\n\\end{equation}\nto leading order in $1\/N$ in the exponent.  Of course, regardless of the interpretation, we find the same predictions for the desired measured quantities.} We discuss the independence of our results on the precise form of this function shortly.\n\nTo determine the value $n$, we must use data.  Pseudorapidity distributions have been measured extensively in the ATLAS \\cite{ATLAS:2011ag,Aad:2012mfa,Aad:2015wga,Aad:2016mok}, CMS \\cite{Khachatryan:2010xs,Khachatryan:2010us,Chatrchyan:2011av,Chatrchyan:2014qka,Khachatryan:2015jna,Chatrchyan:2011pb,Sirunyan:2019cgy} and ALICE \\cite{Aamodt:2009aa,Abbas:2013bpa,Adam:2015kda,Adam:2015pza,Adam:2016ddh,Acharya:2018hhy,Aamodt:2010my,Abelev:2013ala,ALICE:2015qqj,Acharya:2018qsh,Acharya:2018eaq} experiments at the LHC, in both pp and heavy ion collisions.  From those data, we can determine the parameter $n$ by noting the following.  The mean value of $x$ from \\Eq{eq:cutffunc} is roughly $1\/n$, which is the region that dominates the integral.  We call $\\eta_{1\/2}$ where the pseudorapidity distribution is approximately half of its value at $\\eta = 0$ which occurs when $2x \\cosh(2\\eta_{1\/2}) \\sim 1$; or when\n\\begin{equation}\\label{eq:etamaxn}\nn = 2\\cosh(2\\eta_{1\/2})\\,.\n\\end{equation}\nFrom the 8 TeV LHC pp experiment data that follows, for instance, the value of $\\eta_{1\/2}\\sim 6$, and so $n \\sim 1.6\\times 10^5$. The parameter $n$ has implicit $Q$ dependence, at least by the connection to maximum value of $\\eta$.\n\nWe emphasize that in what follows that the fit above to the tail around $|\\eta|\\gtrsim \\eta_{1\/2}$ is not important for the observables that are captured in our approach; that is, any step-like function with a cutoff around $\\eta_{1\/2}$ would suffice.  Kinematic observables binned in $N$ and  at fixed $Q$ in the region of validity of this effective framework are insensitive as $n\\to\\infty$; we will see this explicitly below with the single particle $p_\\perp$ distribution, where we also give a quantitative estimate for the validity condition. In order to correctly capture scalings with $Q$, the value of $n$ is important in that it captures the value of $\\eta_{1\/2}$, which scales with $Q$. Any other step-like function, however, would capture the same scaling and, at least at large enough $Q$, this scaling should be independent of the precise form of the tail.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=9cm]{.\/eta_large_dist}\n\\caption{Comparison of particle density in pseudorapidity between 8 TeV pp collider data from the CMS and TOTEM experiments \\cite{Chatrchyan:2014qka} to our prediction of \\Eq{eq:etadistsmear} (solid) and the flat phase space distribution of \\Eq{eq:flateta} (dashed).  The data only include charged particles with transverse momentum greater than 40 MeV and our prediction uses the form of smearing function $f(x)$ in \\Eq{eq:cutffunc} with $n = 1.6\\times 10^5$. \n\\label{fig:etacomp}\n}\n\\end{center}\n\\end{figure}\n\nThe above established value of $n$ can then be used to compare our predicted pseudorapidity distribution to data.  This is done in \\Fig{fig:etacomp} where we compare our prediction to data from the CMS and TOTEM experiments \\cite{Chatrchyan:2014qka} of the charged particle number density as a function of pseudorapidity, $dN\/d\\eta$, in 8 TeV pp collisions.  Our prediction for the pseudorapidity distribution is just a normalized probability distribution, so we multiply by a factor to fit data.  As mentioned earlier, our present formulation of the expansion of the minimum bias cross section does not enable us to predict the multiplicity distribution, so we can't predict the overall normalization here.  In general, we find good agreement between our prediction and data, with a noticeable lack of a dip in our prediction near $\\eta = 0$.  Hadrons are of course massive particles and there is a distinction between their rapidity and pseudorapidity which is manifest as the dip in the pseudorapidity distribution.  We assumed that the transverse momentum of particles is much larger than their mass, so we ignore the pseudo\/rapidity distinction.  However, the data include particles with transverse momentum above 40 MeV which includes hadrons with transverse momentum comparable to their mass.\n\n\\subsection{Transverse Momentum Distributions}\n\nWe now turn to understanding transverse momentum distributions in pp collisions.  The set-up for this analysis will be the same as that for pseudorapidity.  We take the squared matrix element for the detected particles $|{\\cal M}|^2 = 1$ and take the large-$N$ limit of phase space.  The first steps are therefore very similar to that for pseudorapidity, so we won't repeat them here.  From \\Eq{eq:flatmaster}, the flat phase space distribution of the transverse momentum is\n\\begin{equation}\np_\\text{flat}(p_\\perp) \\propto p_\\perp \\int_{-\\infty}^\\infty d\\eta\\, e^{-\\frac{k^+e^{\\eta}+k^-e^{-\\eta}}{k^+k^-}\\,Np_\\perp}= p_\\perp \\, K_0\\left(\n\\frac{2N p_\\perp}{\\sqrt{k^+k^-}}\n\\right)\\,,\n\\end{equation}\nwhere $K_0(z)$ is a modified Bessel function.  The unit normalized distribution is\n\\begin{equation}\\label{eq:flatpspt}\np_\\text{flat}(p_\\perp) = \\frac{4N^2 p_\\perp}{k^+k^-}\\, K_0\\left(\n\\frac{2Np_\\perp}{\\sqrt{k^+k^-}}\n\\right)\\,.\n\\end{equation}\nWith this result, the distribution smeared with the function $f(k^+k^-)$ is, in general,\n\\begin{align}\np(p_\\perp)&=\\frac{1}{Q^2} \\int_0^Q dk^+\\int_0^Q dk^-\\, f\\left(k^+k^-\\right)\\, p_\\text{flat}(p_\\perp)\\\\\n&=\\frac{1}{Q^2} \\int_0^Q dk^+\\int_0^Q dk^-\\, f\\left(k^+k^-\\right)\\, \\frac{4N^2 p_\\perp}{k^+k^-}\\, K_0\\left(\n\\frac{2Np_\\perp}{\\sqrt{k^+k^-}}\n\\right)\\nonumber\\\\\n&=\n\\frac{4N^2 p_\\perp}{Q^2}\\int_0^1 dx\\, \\text{log}\\frac{1}{x}\\, f(x)\\, \\frac{1}{x}\\, K_0\\left(\n\\frac{2Np_\\perp}{Q\\sqrt{x}}\n\\right)\n\\nonumber\\,.\n\\end{align}\n\nUnlike the pseudorapidity distribution, the transverse momentum distribution depends explicitly on the number of detected particles $N$.  This number fluctuates event-by-event and we do not predict the multiplicity distribution. So, instead, we will re-write this distribution in terms of the mean transverse momentum, which we can calculate and is unique over an ensemble of collision events.  This mean is\n\\begin{align}\n\\langle p_\\perp \\rangle &= \\int_0^\\infty dp_\\perp\\, p_\\perp \\, p(p_\\perp) =\\frac{4N^2}{Q^2}\\int_0^1 dx\\, \\text{log}\\frac{1}{x}\\, f(x)\\, \\frac{1}{x} \\int_0^\\infty dp_\\perp \\,p_\\perp^2\\,  K_0\\left(\n\\frac{2Np_\\perp}{Q\\sqrt{x}}\n\\right)\\\\\n&=\\frac{\\pi}{4}\\frac{Q}{N}\\int_0^1 dx\\, \\text{log}\\frac{1}{x}\\,f(x)\\, \\sqrt{x} \\equiv \\frac{\\pi}{4}\\frac{Q}{N} \\langle \\sqrt{x}\\rangle\n\\nonumber\\,.\n\\end{align}\nSubstituting this expression for $N$, the transverse momentum distribution is then\n\\begin{align}\np(p_\\perp)&=\\frac{\\pi^2 \\langle \\sqrt{x}\\rangle^2}{4\\langle p_\\perp\\rangle^2 }\\,p_\\perp\\int_0^1 dx\\, \\text{log}\\frac{1}{x}\\, f(x)\\, \\frac{1}{x}\\, K_0\\left(\n\\frac{\\pi \\langle \\sqrt{x}\\rangle}{2\\langle p_\\perp\\rangle\\sqrt{x}}\\, p_\\perp\n\\right)\n\\nonumber\\,.\n\\end{align}\nWith $f(x)$ from \\Eq{eq:cutffunc}, the mean transverse momentum is\n\\begin{equation}\\label{eq:meanpt}\n\\langle p_\\perp\\rangle = \\frac{\\pi}{4}\\frac{Q}{N} \\langle \\sqrt{x}\\rangle \\simeq \\frac{\\pi^{3\/2}Q}{8\\sqrt{n}N}\\,,\n\\end{equation}\nin the $n\\to\\infty$ limit. The $p_\\perp$ distribution with this form of $f(x)$ is\n\\begin{align}\\label{eq:perpdistfit}\np(p_\\perp)&=\n\\frac{\\pi^3 p_\\perp}{16 \\langle p_\\perp\\rangle^2}\\frac{1}{\\gamma_E+\\text{log}\\,n }\\int_0^1 dx\\, \\text{log}\\frac{1}{x}\\, e^{-nx}\\, \\frac{1}{x}\\, K_0\\left(\n\\frac{\\pi^{3\/2}p_\\perp}{4\\langle p_\\perp\\rangle\\sqrt{x n}}\n\\right)\\,.\n\\end{align}\nData of the transverse momentum distribution are often displayed as a number density per phase space volume, and so what we will plot is actually\n\\begin{equation}\n\\frac{1}{2\\pi p_\\perp}\\frac{dN}{dp_\\perp} \\propto \\frac{p(p_\\perp)}{p_\\perp}\\,.\n\\end{equation}\nNote that once we determine $f(x)$ from pseudorapidity data, the prediction of the transverse momentum distribution only depends on a single parameter, $\\langle p_\\perp\\rangle$.  Additionally, a cut on the maximum pseudorapidity of particles that contribute to this distribution can be incorporated, as such a cut is always imposed in data.  However, any such cut that is relevant experimentally has an exceedingly small effect on the transverse momentum distribution, so we will not include it in what follows.\n\nThe reason for effective independence on a pseudorapidity cut is as follows.  In the large $n$ limit, the transverse momentum distribution of \\Eq{eq:perpdistfit} is itself independent of $n$.  The Bessel function has an asymptotic form of\n\\begin{equation}\nK_0(z) \\to \\sqrt{\\frac{\\pi}{2z}}\\,e^{-z}\\,,\n\\end{equation}\nfor $z\\gg1$.  With this approximation, the distribution is\n\\begin{align}\np(p_\\perp)&\\sim\n\\sqrt{p_\\perp} \\int_0^1 dx\\, \\text{log}\\frac{1}{x}\\, \\frac{e^{-nx-\\frac{\\pi^{3\/2}p_\\perp}{4\\langle p_\\perp\\rangle\\sqrt{x n}}}}{x^{3\/4}}\\,,\n\\end{align}\nignoring overall constant factors.  Now, with $n\\gg1$, we can saddle-point approximate the exponential.  The value of $x$ for which the exponent factor is minimized is\n\\begin{equation}\nx_{\\min} = \\frac{\\pi}{4n}\\frac{p_\\perp^{2\/3}}{\\langle p_\\perp\\rangle^{2\/3}}\\,.\n\\end{equation}\nJust setting $x$ in the integrand equal to this minimum value, taking $n\\to\\infty$ and ignoring non-exponential factors, we have\n\\begin{align}\\label{eq:asymppt}\np(p_\\perp)&\\sim e^{-\\frac{3\\pi}{4}\\frac{p_\\perp^{2\/3}}{\\langle p_\\perp\\rangle^{2\/3}}}\\,.\n\\end{align}\nAs the value of $n$ in turn determines the maximum value of pseudorapidity according to \\Eq{eq:etamaxn}, as long as $n$ is large enough, any dependence on a pseudorapidity cut is eliminated.\n\n\nThis probability distribution is like the Boltzmann factor for the ``gas'' of $N$ detected particles.  Hence, they have a dispersion relation of $\\omega \\propto k_\\perp^{2\/3}$.  Fractional dispersion relations are very strange, but can arise from integrating out a gapless subsystem of a larger system \\cite{Watanabe:2014zza}.  This is essentially what we did here: the beam remnants were gapless, but we exclusively wanted an effective description of the detected particles.  Integrating out gapless modes introduces non-locality in the effective theory, which is manifested as a non-analytic dispersion relation.  In particular, note that the non-analyticity is not present in the flat phase space distribution of \\Eq{eq:flatpspt}.  The asymptotic form of the Bessel function implies that\n\\begin{equation}\np_\\text{flat}(p_\\perp)\\sim e^{-\\frac{\\pi}{2}\\frac{p_\\perp}{\\langle p_\\perp\\rangle}}\\,,\n\\end{equation}\nignoring overall non-exponential factors.  No gapless modes are integrated out on flat phase space, hence the dispersion relation $\\omega \\propto k_\\perp$ is analytic.  The particular form of the dispersion relation implied by \\Eq{eq:asymppt} may provide {a hint towards the construction of an effective field theory for minimum bias based on spontaneously broken symmetry breaking patterns, but we leave that to future work.}\n\nWe can also verify the consistency of our transverse momentum prediction, within the framework of our assumed power counting.  The second moment of the transverse momentum is\n\\begin{align}\n\\langle p_\\perp^2 \\rangle &= \\int_0^\\infty dp_\\perp\\, p_\\perp^2 \\, p(p_\\perp) =\\frac{4N^2}{Q^2}\\int_0^1 dx\\, \\text{log}\\frac{1}{x}\\, f(x)\\, \\frac{1}{x} \\int_0^\\infty dp_\\perp \\,p_\\perp^3\\,  K_0\\left(\n\\frac{2Np_\\perp}{Q\\sqrt{x}}\n\\right)\\\\\n&=\\frac{Q^2}{N^2}\\int_0^1 dx\\, \\text{log}\\frac{1}{x}\\,f(x)\\,x\n= \\frac{Q^2}{n N^2}\\,,\\nonumber\n\\end{align}\nwhere the final equality is the large $n$ limit of the second moment of the distribution of \\Eq{eq:perpdistfit}.  Then, the root mean square and expectation value of the transverse momentum are related by\n\\begin{equation}\n\\sqrt{\\langle p_\\perp^2 \\rangle} = \\frac{8}{\\pi^{3\/2}}\\langle p_\\perp \\rangle \\simeq 1.44 \\langle p_\\perp \\rangle\\,,\n\\end{equation}\nsatisfying our power counting of $\\sqrt{\\langle p_\\perp^2 \\rangle}\\sim\\langle p_\\perp \\rangle$ and demonstrating consistency of our prediction.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=9cm]{.\/pt_dist}\n\\caption{Transverse momentum distribution of charged particles in minimum bias events in $\\sqrt{s}=7$ TeV pp collisions at CMS from \\Ref{Khachatryan:2010us} compared to our prediction.  The data have an $|\\eta|< 2.4$ cut on the charged particles.  Our prediction from the asymptotic expression of \\Eq{eq:asymppt} (solid) uses $\\langle p_\\perp\\rangle = 0.65$ GeV, and also plotted is the distribution on flat phase space of \\Eq{eq:flatpspt} (dashed).\n\\label{fig:ptpred}}\n\\end{center}\n\\end{figure}\n\nWith this prediction for the transverse momentum distribution, we can compare it to collider data.  \\Fig{fig:ptpred} shows the transverse momentum distribution of charged particles in minimum bias events in $\\sqrt{s}=7$ TeV pp collisions at CMS from \\Ref{Khachatryan:2010us}.  A pseudorapidity cut of $|\\eta|<2.4$ is also imposed, but, as mentioned earlier, that will not matter for our prediction.  This figure also plots the asymptotic form of our prediction from \\Eq{eq:asymppt} where we have set the parameter $\\langle p_\\perp\\rangle = 0.65$ GeV, as well as the transverse momentum distribution on flat $N$-body phase space.  Our prediction is in remarkably good agreement with the data over the plotted range of transverse momentum.  We want to emphasize that once the form of the smearing function $f(x)$ is determined from pseudorapidity data, our prediction for the transverse momentum distribution depends on only one parameter, the mean value of the transverse momentum.\n\nWhile we are unable to directly predict the multiplicity distribution from this framework, we can nevertheless demonstrate consistency between parameters in the expansion of the cross section with measured values of the mean multiplicity.  From \\Eq{eq:meanpt}, the expected value of $N$ with our choice for the form of the function $f(x)$ is\n\\begin{equation}\\label{eq:expmult}\nN  \\simeq \\frac{\\pi^{3\/2}Q}{8\\sqrt{n}\\langle p_\\perp\\rangle}\\,.\n\\end{equation}\nFitting the 7 TeV data from CMS in \\Fig{fig:ptpred} fixes $\\langle p_\\perp\\rangle = 0.65$ GeV, and from the 8 TeV CMS+TOTEM data of the pseudorapidity distribution from \\Fig{fig:etacomp}, we fit $n=1.6\\times 10^5$.  While these data are from different collision energies and so are perhaps not directly comparable and interpretable from one to the other, we only anticipate logarithmic dependence on center-of-mass collision energy, so their distinction should be minimal when applied to our (rather coarse) scaling predictions.  Taking these values along with $Q = 8$ TeV, the expected value of the detected multiplicity from \\Eq{eq:expmult} is then\n\\begin{equation}\nN \\simeq 21\\,.\n\\end{equation}\nWhile this is a very simple and crude prediction, it is nevertheless in the same order-of-magnitude as the number of observed charged particles from \\Fig{fig:etacomp}, for example.  In that figure, the number density of charged particles with transverse momentum greater than 40 MeV is roughly 6.5 per unit pseudorapidity for $|\\eta| \\lesssim 5.5$.  So, there are roughly 72 charged particles in each event.  Our prediction is about a factor of 3 smaller, which is likely accounted for by the transverse momentum cut.  40 MeV is less than the mass of the pion, and so violates an assumption of our power counting.  Increasing this cut would correspondingly decrease the number of detected particles, while not affecting our fit value for $\\langle p_\\perp\\rangle$.\n\nFurther, the expression for multiplicity $N$ from \\Eq{eq:expmult} implies a non-trivial dependence on the center-of-mass collision energy.  First, note that the value of $n$ in the form of the function $f(x)$ is related to the collision energy $Q$ through \\Eq{eq:etamaxn}:\n\\begin{equation}\n\\eta_{\\max}\\simeq \\text{log}\\frac{Q}{p_{\\perp\\text{cut}}} \\simeq \\text{log} \\,n\\,.\n\\end{equation}\nHere, $p_{\\perp\\text{cut}}$ is the experimental lower bound on detected particle transverse momentum.  Then, as long as the dependence of the mean transverse momentum $\\langle p_\\perp\\rangle$ on the collision energy $Q$ is relatively weak, the multiplicity scales with a fractional power of $Q$:\n\\begin{equation}\nN \\sim Q^{1\/2} = s^{1\/4}\\,.\n\\end{equation}\nThis particular fractional power scaling is a prediction of the Fermi-Landau model \\cite{Fermi:1950jd,Landau:1953gs,Belenkij:1955pgn,Wong:2008ex}.  In the context of our analysis here, we note that it is a consequence of our particular choice of the function $f(x)$ in the smeared cross section.  Additionally, the inclusion of a squared matrix element with non-trivial dependence on the detected particles will affect this scaling.  Data prefer a slightly smaller power-law scaling of the multiplicity, e.g., \\Ref{ALICE:2015qqj} in which a power law of $s^{0.11}$ fits the charged particle multiplicity over decades of collision energies.  Nevertheless, this simple result within the context of our large-$N$ expansion suggests that an appropriate squared matrix element could fit the data.  We leave a more detailed analysis of the collision energy dependence of the multiplicity to future work.\n\nData of transverse momentum distributions in minimum bias are often compared to Tsallis distributions \\cite{Tsallis:1987eu,Wilk:2008ue,Biro:2008hz} that assume there are fluctuations in the $N$ particle final state that are quantified by a non-extensive form of entropy.  This is an intriguing interpretation and the success of such models may point to fundamental fractal-like structure of particles produced in minimum bias collisions.  While not inconsistent with this interpretation, our expansion of the minimum bias cross section has a more mundane understanding as a consequence of the symmetries of these collision events.  Further, the Tsallis distribution reduces to a power law at large transverse momentum, while our smeared prediction is exponential, though dependent on a fractional power of transverse momentum.  A detailed study to distinguish the consequences of these two (or other) models of minimum bias dynamics may reveal the microscopic description of these events and lead to an effective field theory in which precision calculations can be performed.\n\n\\subsubsection{Limit of Large-$N$ Expansion}\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=9cm]{.\/highpt_dist}\n\\caption{\nTransverse momentum distribution of charged particles in minimum bias events in $\\sqrt{s}=2.76$ TeV pp collisions at ATLAS from \\Ref{Aad:2015wga} compared to our asymptotic prediction and a Tsallis distribution.  The data have an $|\\eta|< 2$ cut on the charged particles.  In our prediction of \\Eq{eq:asymppt}, we set the parameter $\\langle p_\\perp\\rangle = 0.5$ GeV and the parameters of the Tsallis distribution are $n=6.6$ and $T = 0.12$ GeV.\n\\label{fig:highptpred}}\n\\end{center}\n\\end{figure}\n\nWith data that extends to higher values of transverse momentum, we can see the limit of our prediction, with the assumptions we have made thus far.  In particular, we have assumed that the squared matrix element is just 1, which is its expression at lowest order in the $1\/N$ expansion.  However, at higher transverse momentum, higher order terms in the squared matrix element become more important and may be necessary to describe the distribution.  In Fig.~\\ref{fig:highptpred}, we compare charged particle transverse momentum distribution data from $\\sqrt{s}=2.76$ TeV pp collisions to our asymptotic prediction of \\Eq{eq:asymppt} and a Tsallis distribution.  This Tsallis distribution takes the form\n\\begin{equation}\n\\frac{1}{2\\pi p_\\perp}\\frac{d\\sigma}{dp_\\perp}\\propto \\left(\n1+\\frac{p_\\perp}{nT}\n\\right)^{-n}\\,,\n\\end{equation}\nfor parameters $n$ and $T$. In the plot, we have set the parameters in our prediction and the Tsallis distribution to be $\\langle p_\\perp\\rangle = 0.5$ GeV, $n=6.6$ and $T = 0.12$ GeV.  At low transverse momentum, both our prediction and the Tsallis distribution follow the data extremely well, but around 5 GeV, our prediction exponentially drops, while the Tsallis distribution largely follows the power-law distribution at high transverse momentum.  It would be interesting to study the effect of maintaining momentum conservation in our prediction and non-trivial terms in the matrix element to see if they can reproduce the high-$p_\\perp$ tail, but we leave that to future work.  \n\nHowever, with the approximations that we have made thus far, we can estimate where our description of the transverse momentum distribution should break down.  In the derivation of our results, an important approximation that we made in the large-$N$ limit was that momentum conservation could be ignored, which in turn resulted in an exponential appearing in the calculation of distributions on flat phase space in \\Eq{eq:flatmaster}.  By carefully analyzing this approximation, we can identify where it  breaks down.  First, the large-$N$ flat phase space factor can be written as\n\\begin{align}\n\\left(\n1-\\frac{k^+e^\\eta  + k^-e^{-\\eta}}{k^+k^-}p_\\perp\n\\right)^N &= \\exp\\left[\nN\\,\\text{log}\\left(\n1-\\frac{k^+e^\\eta  + k^-e^{-\\eta}}{k^+k^-}p_\\perp\n\\right)\n\\right]\\\\\n&=\\exp\\left[\nN\\left(\n-\\frac{k^+e^\\eta  + k^-e^{-\\eta}}{k^+k^-}p_\\perp-\\frac{1}{2}\\frac{(k^+e^\\eta  + k^-e^{-\\eta})^2}{(k^+k^-)^2}p^2_\\perp + \\cdots\n\\right)\n\\right]\n\\nonumber\\,.\n\\end{align}\nSo far, this is exact, but higher-order terms in the exponent can be safely ignored only if\n\\begin{equation}\np_\\perp \\ll \\frac{k^+k^-}{k^+e^\\eta  + k^-e^{-\\eta}} \\,.\n\\end{equation}\nRecall that $k^+k^-$ is the total squared energy of the detected particles.  In coordinates that we used earlier, we expressed\n\\begin{align}\n&x = \\frac{k^+k^-}{Q^2}\\,, &\\Delta\\eta =\\frac{1}{2}\\, \\text{log}\\frac{k^-}{k^+}\\,,\n\\end{align}\nand so the limit on transverse momentum is\n\\begin{equation}\np_\\perp \\ll \\frac{k^+k^-}{k^+e^\\eta  + k^-e^{-\\eta}} = \\frac{\\sqrt{x}\\,Q}{2\\cosh(\\eta - \\Delta\\eta)}\\,,\n\\end{equation}\nwhere $x\\in[0,1]$ and \n\\begin{equation}\n\\frac{1}{2}\\,\\text{log}\\, x \\leq \\Delta\\eta \\leq \\frac{1}{2}\\,\\text{log}\\frac{1}{x}\\,.\n\\end{equation}\n\nTo set an upper bound for the transverse momentum, we first maximize over $x$ and $\\Delta\\eta$, for which $x = 1$ and $\\Delta \\eta = 0$:\n\\begin{equation}\np_\\perp \\ll \\frac{Q}{\\cosh \\eta}\\,.\n\\end{equation}\nThat is, the energy of an individual particle must be parametrically less than the collision energy.  We had found that for pp collisions with energy of several TeV, the maximum pseudorapidity was about $\\eta_{\\max}\\sim 6$.  For $Q = 2.76$ TeV from \\Fig{fig:highptpred}, the transverse momentum then must be smaller than\n\\begin{align}\np_\\perp \\ll \\frac{Q}{\\cosh \\eta_{\\max}} \\sim 14\\text{ GeV}\\,.\n\\end{align}\nThis estimate of the limit agrees well with \\Fig{fig:highptpred}, in which our prediction diverges from data well before 14 GeV.  Conversely, for a fixed transverse momentum $p_\\perp$, the maximum pseudorapidity is logarithmically related to the collision energy $Q$, which is well known \\cite{Feynman:1969ej,Wilson:1970zzb}.\n\n\n\\subsection{Azimuthal Correlations}\n\nIn this section, we discuss azimuthal correlations between pairs of particles produced in pp or heavy ion collisions.  We will focus on the ellipticity and the long pseudorapidity-distance correlations or ``ridge'' phenomena later, but we will first determine the form of the probability distribution of the pairwise azimuthal angle difference within the context of our minimum-bias expansion.  Non-trivial azimuthal correlations require a non-trivial squared matrix element.  In general, the form of the terms in the squared matrix element relevant for azimuthal correlations are\n\\begin{align}\n|{\\cal M}|^2&\\supset 1+\\sum_{n=1}^\\infty g_{n}(k^+k^-,N)\\sum_{i\\neq j}^N \\frac{(\\vec p_{\\perp i}\\cdot \\vec p_{\\perp j})^n}{Q^{2n}} \\\\\n&\\supset 1+\\sum_{n=1}^\\infty g_{n}(k^+k^-,N)\\sum_{i\\neq j}^N \\frac{p_{\\perp i}^n p_{\\perp j}^n}{Q^{2n}} \\cos(n(\\phi_i-\\phi_j))\\nonumber\\,,\n\\end{align}\nfor some coefficient functions $g_n(k^+k^-,N)$.  On the second line, we have absorbed multiplicative factors in the expansion of $(\\vec p_{\\perp i}\\cdot \\vec p_{\\perp j})^n$ into a redefinition of $g_n(k^+k^-,N)$.  All other terms in the squared matrix element that are independent of azimuthal differences would just contribute to the constant ``1'' term and would therefore just be an overall normalization. With our power counting assumption that $\\langle p_\\perp\\rangle \\sim \\sqrt{\\langle p_\\perp^2\\rangle}\\sim Q\/N$, we replace $p_\\perp \\sim Q\/N$ to establish the scaling with the number of observed particles $N$.  Our squared matrix element then becomes\n\\begin{align}\n|{\\cal M}|^2&\\supset 1+\\sum_{n=1}^\\infty g_{n}(k^+k^-,N)\\sum_{i\\neq j}^N \\frac{p_{\\perp i}^n p_{\\perp j}^n}{Q^{2n}} \\cos(n(\\phi_i-\\phi_j))\\\\\n&\\sim1+\\sum_{n=1}^\\infty \\frac{g_{n}(k^+k^-,N)}{N^{2n}}\\sum_{i\\neq j}^N  \\cos(n(\\phi_i-\\phi_j))\\nonumber\\,.\n\\end{align}\n\nThe maximal scaling with $N$ of the coefficient functions $g_n(k^+k^-,N)$ can be established by demanding that the squared matrix element is non-negative.  To do this, we note that, in the $N\\to\\infty$ limit, transverse momentum conservation is trivially satisfied and so azimuthal angles are uncorrelated and uniformly distributed on flat phase space:\n\\begin{equation}\n\\int d\\Pi_N \\to \\int_0^{2\\pi}\\prod_{i=1}^N \\frac{d\\phi_i}{2\\pi}\\,.\n\\end{equation}\nThus, on flat phase space, the mean of the sum over the cosine factors of the difference of azimuthal angles is 0:\n\\begin{equation}\n\\int_0^{2\\pi}\\prod_{i=1}^N \\frac{d\\phi_i}{2\\pi}  \\sum_{j\\neq k}^N  \\cos(n(\\phi_j-\\phi_k))=0\\,.\n\\end{equation}\nOn the other hand, the central limit theorem states that the variance $\\sigma^2$ of the sum of cosine factors is\n\\begin{align}\n\\sigma^2 \\equiv \\int_0^{2\\pi}\\prod_{i=1}^N \\frac{d\\phi_i}{2\\pi}  \\left[\\sum_{j\\neq k}^N  \\cos(n(\\phi_j-\\phi_k))\\right]^2 = N^2 \\int_0^{2\\pi}\\prod_{i=1}^N \\frac{d\\phi_i}{2\\pi} \\, \\cos^2(n(\\phi_1-\\phi_2)) = \\frac{N^2}{2}\\,,\n\\end{align}\nas there are $N^2$ terms in the sum, in the $N\\to\\infty$ limit.  To ensure that the squared matrix element is non-negative, the sum over cosine factors at each value of $n$ must not be significantly negative to overwhelm the constant ``1'' term.  Therefore, we must enforce that the bulk of the possible values of the sum over cosines is less than 1:\n\\begin{align}\n1\\gtrsim \\frac{g_{n}(k^+k^-,N)}{N^{2n}}\\sum_{i\\neq j}^N  \\cos(n(\\phi_i-\\phi_j)) \\sim \\frac{g_{n}(k^+k^-,N)}{N^{2n}} \\, \\sigma\\sim \\frac{g_{n}(k^+k^-,N)}{N^{2n-1}}\\,.\n\\end{align}\nThat is, the coefficient functions $g_{n}(k^+k^-,N)$ are required to scale with $N$ no greater than\n\\begin{align}\ng_n(k^+k^-,N) \\lesssim N^{2n-1}\\,.\n\\end{align} \nThis property will have important consequences for the large-$N$ predictions of azimuthal correlations.\\footnote{Few general results are known about conditions for positivity of Fourier transforms.  One sufficient condition for positivity of the continuous Fourier transform of a function $u(x)$ for $x>0$ is that it is decreasing and concave-up for all $x>0$ \\cite{tuck_2006,Giraud:2014sba}.  Our simple result based on scaling of terms in the Fourier series is consistent with this result.}\n\nContinuing, we can integrate over the smeared phase space to establish the probability distribution for the pairwise azimuthal angle difference, $\\Delta\\phi$.  We have\n\\begin{align}\\label{eq:azidist}\np(\\Delta \\phi) &\\sim\\frac{1}{Q^2} \\int_0^Q dk^+ \\int_0^Q dk^- \\, f(k^+k^-)\\, \\int d\\Pi_N |{\\cal M}|^2\\, \\delta(\\Delta\\phi -(\\phi_1-\\phi_2))\\\\\n&\\sim \\frac{1}{Q^2} \\int_0^Q dk^+ \\int_0^Q dk^- \\, f(k^+k^-) \\int_0^{2\\pi} \\prod_{i=1}^N \\frac{d\\phi_i}{2\\pi}\\, \\delta(\\Delta\\phi -(\\phi_1-\\phi_2))\\nonumber\\\\\n&\n\\hspace{1cm}\\times\\left(\n1+\\sum_{n=1}^\\infty \\frac{g_{n}(k^+k^-,N)}{N^{2n}}\\sum_{i\\neq j}^N \\cos(n(\\phi_i-\\phi_j))\n\\right)\\nonumber\\\\\n&=\\frac{1}{2\\pi}+\\frac{1}{\\pi}\\sum_{n=1}^\\infty \\frac{d_n(N)}{N^{2n}}\\, \\cos(n\\, \\Delta\\phi)\\,.\\nonumber\n\\end{align}\nHere, we have used the permutation symmetry of particles to just define $\\Delta \\phi$ as the difference of the azimuthal angles of particles 1 and 2.  The Fourier coefficients $d_n(N)$ are defined to be\n\\begin{equation}\\label{eq:fcoeffsazi}\nd_n(N) =  \\frac{1}{2Q^2} \\int_0^Q dk^+ \\int_0^Q dk^- \\, f(k^+k^-)\\, g_{n}(k^+k^-,N)\\,.\n\\end{equation}\nNote that the maximal $N$ scaling of the coefficients $d_n(N)$ is inherited from the form of $g_n(k^+k^-,N)$ established above by positivity; that is,\n\\begin{equation}\nd_n(N) \\lesssim N^{2n-1}\\,.\n\\end{equation}\nThis scaling means that the Fourier coefficients at each $n$ necessarily vanish in the $N\\to\\infty$ limit:\n\\begin{equation}\\label{eq:fourcoefflimit}\n\\lim_{N\\to\\infty}\\frac{d_n(N)}{N^{2n}} = 0\\,.\n\\end{equation}\nThe distribution on the final line of \\Eq{eq:azidist} is normalized on $\\Delta\\phi \\in[0,2\\pi)$.\\footnote{Recall that we are working at fixed center-of-mass energy $Q$.  It is known that azimuthal correlations have a finite, non-zero value at fixed centrality as both $Q$ and $N$ increase (see, e.g., \\cite{Aamodt:2010pa}).  This is not inconsistent with this analysis because the Fourier coefficients defined in \\Eq{eq:fcoeffsazi} have implicit dependence on the center-of-mass energy $Q$ and other mass scales in the system, like the pion mass $m_\\pi$.}\n\nThe particular scaling of the coefficient $d_1(N)$ with $N$ can also be established by momentum conservation.  Transverse momentum conservation states that\n\\begin{align}\n0=\\left(\n\\sum_{i=1}^N \\vec p_{\\perp i}\n\\right)^2=\\sum_{i=1}^N p_{\\perp i}^2+\\sum_{i\\neq j}^N p_{\\perp i}p_{\\perp j}\\cos(\\phi_i-\\phi_j)\\,.\n\\end{align}\nNow, with $p_{\\perp}\\sim Q\/N$, this relationship implies the scaling\n\\begin{equation}\n\\frac{1}{N^2}\\sum_{i\\neq j}^N \\cos(\\phi_i-\\phi_j) \\sim -\\frac{1}{N}\\,.\n\\end{equation}\nWith our ergodic assumption, the term on the left is just the mean value of $\\cos(\\Delta\\phi)$ in the large-$N$ limit and so\n\\begin{equation}\n\\langle \\cos(\\Delta \\phi)\\rangle \\sim -\\frac{1}{N}\\,.\n\\end{equation}\nThis mean value can also be calculated from the probability distribution $p(\\Delta\\phi)$.  We have\n\\begin{equation}\n\\langle \\cos(\\Delta \\phi)\\rangle = \\int_0^{2\\pi} d\\Delta\\phi\\, p(\\Delta\\phi)\\, \\cos(\\Delta\\phi) = \\frac{d_1(N)}{N^{2}}\\,,\n\\end{equation}\nfrom \\Eq{eq:azidist}.  If this is to scale like $-1\/N$, the coefficient $d_1(N)$ must scale like\n\\begin{equation}\nd_1(N) \\sim -N = -N^{2\\cdot 1-1}\\,,\n\\end{equation}\nexactly as predicted from positivity.\\footnote{Yet another way to prove this scaling with $N$ follows from demanding that the squared matrix element is finite in the $N\\to\\infty$ limit.}  Then, the distribution of the azimuthal angle difference can be expressed as\n\\begin{align}\\label{eq:delphidistfinal}\np(\\Delta\\phi)  =\\frac{1}{2\\pi}-\\frac{d_1}{\\pi}\\frac{1}{N}\\cos(\\Delta\\phi) +\\frac{1}{\\pi}\\sum_{n=2}^\\infty \\frac{d_n(N)}{N^{2n}}\\, \\cos(n\\, \\Delta\\phi)\\,,\n\\end{align}\nwhere now $d_1>0$ is some constant value in the $N\\to\\infty$ limit.  A similar relationship for the $n=1$ coefficient of the Fourier expansion using momentum conservation was established in \\Ref{Luzum:2010fb}, but to the best of our knowledge, the scaling with the number of particles $N$ is novel.\n\n\\subsubsection{Ellipticity}\n\n\n\nEspecially in collisions of heavy ions, azimuthal correlations amongst particles are used as evidence for collective flow phenomena and the production of exotic states of QCD matter.  The first non-trivial azimuthal correlation is referred to as elliptic flow and quantifies particle correlations with respect to the reaction plane of the collision, the plane about which particle production is maximized in the plane and minimized orthogonal to it.  As a proxy for the elliptic flow, a pairwise azimuthal correlation moment is often measured instead.  Here, we will focus on one such moment, the average over all $N$ particles in an event denoted as $\\langle 2\\rangle$ and defined to be\n\\begin{align}\n\\langle 2\\rangle=\\frac{1}{N^2}\\sum_{j,k=1}^N e^{2i(\\phi_j-\\phi_k)}&=\\frac{1}{N}+\\frac{1}{N^2}\\sum_{j\\neq k}^N\\cos\\left(2(\\phi_j-\\phi_k)\\right)\\,,\n\\end{align}\nwhere $\\phi_j$ is the azimuthal angle of particle $j$.  As above, the ergodic assumption establishes that the sum over cosine factors is just the mean value of cosine of the azimuthal angle difference in the large $N$ limit, and so\n\\begin{equation}\\label{eq:azcorrdef}\n\\langle 2\\rangle=\\frac{1}{N}+\\langle\\cos\\left(2\\Delta \\phi\\right)\\rangle\\,.\n\\end{equation}\nWhen further averaged over an ensemble of events, this is often denoted as $\\langle\\langle 2\\rangle\\rangle$ or $c_2(2)$.  The notation $c_2(2)$ is common in the literature with the parenthetical 2 referencing the pairwise azimuthal correlation and the subscript 2 referencing the 2 in the argument of cosine in \\Eq{eq:azcorrdef}.\n\nAzimuthal correlations have been extensively measured in heavy ion collisions for the past few decades, in experiments at ATLAS \\cite{Aad:2015gqa,Aad:2014eoa,ATLAS:2011ah}, CMS \\cite{Sirunyan:2017fts,Chatrchyan:2012ta,Chatrchyan:2013nka,Sirunyan:2017uyl,Sirunyan:2017gyb,Sirunyan:2017pan,CMS:2013bza,Chatrchyan:2013kba,Chatrchyan:2012vqa,Chatrchyan:2012xq,Chatrchyan:2012wg}, and ALICE \\cite{Acharya:2019vdf,Adam:2016izf,Abelev:2014mda,Adam:2015eta,Aamodt:2010pa} at the LHC and at STAR \\cite{Adamczyk:2012ku,Agakishiev:2011id,Abelev:2008ae,Adler:2002pu,Ackermann:2000tr,Adam:2019woz,Adamczyk:2017ird,Adamczyk:2015obl,Abdelwahab:2014sge,Adams:2004bi,Adams:2004wz} and PHENIX \\cite{Adare:2018zkb,Adare:2010ux} at RHIC.  However, outside of an explicit hydrodynamic model, measurements of azimuthal correlations are typically challenging to interpret.  Azimuthal correlations are often binned in unphysical or unobservable quantities that are fit from models of the ion collisions, like the number of nucleons participating in the collision or the centrality.\\footnote{Experimentally, the centrality is measurable as it is defined as a quantile of the multiplicity distribution for a given collision event data set.  However, the centrality is not defined for an individual event in isolation and ``centrality'' connotes the overlap of the ion nuclei in collision, which is not measurable.}  Few of the listed references plot azimuthal correlations as a function of directly observable quantities, such as the number of charged particles in the event \\cite{Aad:2015gqa,Chatrchyan:2013nka,Sirunyan:2017uyl,Acharya:2019vdf,Abelev:2014mda,Adamczyk:2015obl}.  In what follows we will show how  the features of the observables can in fact be elucidated purely in terms of observable quantities, by appealing to the expansion of the minimum bias cross section derived above.\n\nAs mentioned earlier, our expansion of the minimum bias cross section is defined for a fixed collision energy $Q$ and fixed number $N$ of observed particles.  With our current formulation, we are not able to predict the dependence of the azimuthal correlations either as a function of $Q$ or $N$.  Thus, we will focus on predictions for fixed $Q$ binned in the number of observed particles.  Nevertheless, we will be able to make a number of concrete predictions.  For fixed $N$ and $Q$, the average azimuthal correlation $c_2(2)$ in terms of our expansion of minimum bias established in \\Eq{eq:delphidistfinal} is\n\\begin{align}\nc_2(2) &=\\frac{1}{N}+\\int_0^{2\\pi}d\\Delta\\phi\\, p(\\Delta\\phi)\\, \\cos(2\\Delta\\phi) = \\frac{1}{N} + \\frac{d_2(N)}{N^4}\\,.\n\\end{align}\nThis form immediately identifies two distinct contributions.  First, the explicit $1\/N$ term is completely independent of any of the dynamics of the collision (i.e., the squared matrix element and smearing).  Such a contribution is sometimes called ``non-flow'' in the literature.  The second term, by contrast, is only non-zero if there are non-trivial azimuthal correlations; otherwise, the integral over phase space of the sinusoidal function vanishes.  Such a contribution is sometimes called ``flow''.  Whatever its short-distance interpretation, we can make concrete predictions of the flow contribution within our expansion.\n\nFirst, the azimuthal correlation $c_2(2)$ vanishes in the $N\\to\\infty$ limit.  This follows directly from the results established in \\Eq{eq:fourcoefflimit}.  Additionally, this large-$N$ limit is also the limit of small centrality (the quantile of highest multiplicity in an ensemble of collision events).  Thus, azimuthal correlations should also vanish in the limit of small centrality.  In the nucleus-overlap model of heavy ion collisions, it is also predicted that azimuthal correlations vanish in the low centrality limit because a head-on, perfectly overlapping nucleus collision is completely rotationally symmetric and has no preferred particle production axis.  However, we stress that this prediction in our formulation makes no reference to the unobservable nucleus overlap and relies instead exclusively on our power counting of the relevant observable quantities.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=9cm]{.\/ellipN}\n\\caption{Data of the azimuthal correlation $c_2(2)$ as a function of charged particle multiplicity $N_\\text{ch}$ from the ALICE experiment \\cite{Abelev:2014mda}.  Lead ions are collided with a nucleon-nucleon center of mass of $\\sqrt{s_\\text{NN}} = 2.76$ TeV.  Only charged particles with pseudorapidity $|\\eta|<1$ and transverse momentum $0.2 < p_\\perp <3.0$ GeV contribute.  The open circle data include all charged particles, and the open triangle, diamond, and square data only include those pairs of charged particles with a pseudorapidity difference of at least 0.4, 1.0, and 1.4, respectively.\n\\label{fig:ellipNch}}\n\\end{center}\n\\end{figure}\n\nThis prediction is observed in data from heavy ion collisions.  In \\Fig{fig:ellipNch}, we plot data of the azimuthal correlation $c_2(2)$ from PbPb collisions at the ALICE experiment, as a function of the number of charged particles $N_\\text{ch}$ in the pseudorapidity range $|\\eta| < 1.0$ and transverse momentum in the range $0.2 < p_\\perp < 3.0$ GeV \\cite{Abelev:2014mda}.  Only charged particles are used to compute $c_2(2)$, but as discussed earlier, as long as the number of particles is large, we still expect our expansion to accurately describe these events.  For significantly large $N_\\text{ch}$ the azimuthal correlation decreases toward 0, as our expansion predicts.  Though we do not show a plot here, data of the azimuthal correlation binned in centrality also vanishes as centrality decreases (see, e.g., \\Ref{Adam:2016izf}), corresponding to the highest multiplicity events in an ensemble.\n\nThe plot presented in \\Fig{fig:ellipNch} also illustrates another feature described in our expansion.  The four data sets in that plot correspond to the azimuthal correlation of pairs of particles with a minimal pseudorapidity difference, $|\\Delta\\eta|$.  As the pairwise pseudorapidity difference increases, the azimuthal correlation at small multiplicity $N_\\text{ch}$ significantly decreases.  At small multiplicities, momentum conservation induces significant correlations between pairs of particles that is imprinted on $c_2(2)$, especially when further restricted to long-range pseudorapidity correlations.  However, at large values of $N_\\text{ch}$ the data with different pseudorapidity differences converge onto a universal curve.  This is expected within our expansion.  At large $N_\\text{ch}$, correlations induced by momentum conservation are eliminated.  Further, the pseudorapidity distribution of particles is quite flat over the acceptance region of the detector, and we argued that the transverse momentum distribution of particles effectively independent of a pseudorapidity cut.  So, for sufficiently large number of particles, there is nothing special about the particular pseudorapidity of a particle and its correlation to other particles.  Therefore, in the large $N_\\text{ch}$ limit, neighboring particles and far away particles in pseudorapidity will exhibit the same azimuthal correlations.\n\n\n\n\\subsubsection{Two-Particle Angular Correlations; the ``Ridge''}\n\nA more general analysis of two-particle correlations can be studied through measurement of the cross section differential in both pairwise azimuthal differences $\\Delta\\phi$ as well as pairwise pseudorapidity differences $\\Delta\\eta$.  Such double differential cross sections have been measured at RHIC \\cite{Abelev:2009af,Alver:2009id} and other experiments before the LHC, but became exciting evidence of potential collective phenomena in pp collisions in measurements at CMS \\cite{Khachatryan:2010gv} in 2010.  Since then, CMS \\cite{Chatrchyan:2011eka,Chatrchyan:2012wg,Chatrchyan:2013nka,CMS:2013bza,Khachatryan:2015lva,Khachatryan:2016txc}, ATLAS \\cite{ATLAS:2012ap,Aad:2015gqa,Aaboud:2016yar}, and ALICE \\cite{Aamodt:2011by,Adam:2015gda} have measured pairwise particle correlations in pp and PbPb ion collisions.  In heavy ion collisions, it is well-established that there are long range in $\\Delta \\eta$ correlations, which is interpreted as collective phenomena due to the high-density medium produced in collision.  Extending out to $\\Delta\\eta\\sim 4$, the distribution of azimuthal correlations in $\\Delta\\phi$ is nearly independent of $\\Delta\\eta$ and exhibit an over-density in the distribution around $\\Delta\\phi = 0$, the so-called ``ridge''.  A long distance factorization of the $\\Delta\\eta$ and $\\Delta \\phi$ distributions and the ridge are challenging to observe in pp collisions because of the overwhelming dominance of short-distance correlations, or jet phenomena.  However, with significantly restrictive cuts at high particle multiplicity, the CMS analysis from 2010 and later measurements identified a small, but non-negligible ridge in pp collisions suggesting that a dense QCD medium was produced.\n\nIn this section, we will analyze these angular correlations and the ridge from the perspective of our expansion of high multiplicity, minimum bias events.  First, the quantity that is typically measured and plotted to identify the ridge is the cross section ratio\n\\begin{align}\n\\frac{d^2 N^\\text{pair}}{d\\Delta\\eta \\, d\\Delta\\phi} = B(0,0)\\times \\frac{S(\\Delta\\eta,\\Delta\\phi)}{B(\\Delta\\eta,\\Delta\\phi)}\\,.\n\\end{align}\nHere, $N^\\text{pair}$ is the number of pairs of particles from which their pseudorapidity difference $\\Delta\\eta$ and azimuthal angle difference $\\Delta \\phi$ are measured.  $S(\\Delta\\eta,\\Delta\\phi)$ is the double differential cross section for ``signal'' pairs of particles drawn from the same event.  To eliminate trivial correlations to some extent, this is divided by the ``background'' distribution $B(\\Delta\\eta,\\Delta\\phi)$ which consists of pairs of particles drawn from different events.  Thus, the paired particles that contribute to $B(\\Delta\\eta,\\Delta\\phi)$ are truly uncorrelated.  Finally, $B(0,0)$ is a convenient normalization factor.\n\nStarting with the signal distribution $S(\\Delta\\eta,\\Delta\\phi)$, it is defined to be\n\\begin{align}\nS(\\Delta\\eta,\\Delta\\phi) = \\frac{d^2N^\\text{same}}{d\\Delta\\eta \\, d\\Delta\\phi}\\,,\n\\end{align}\nwhere $N^\\text{same}$ is the number of pairs of particles drawn from the same event.  In the limit that the number of detected particles in the event $N\\to\\infty$, correlations between pairs of particles due to momentum conservation are eliminated.  Further, we expect that the pseudorapidity and azimuthal angle differences are uncorrelated, or only very weakly correlated, because the squared matrix element approaches a constant in this limit.  This suggests that the signal distribution factorizes to leading power in $1\/N$ as\n\\begin{align}\nS(\\Delta\\eta,\\Delta\\phi) = \\frac{d^2N^\\text{same}}{d\\Delta\\eta \\, d\\Delta\\phi} \\sim \\frac{dN^\\text{same}}{d\\Delta\\eta}\\times\\frac{1}{N^\\text{same}}\\frac{dN^\\text{same}}{d\\Delta\\phi} = \\frac{dN^\\text{pair}}{d\\Delta\\eta}\\, p(\\Delta\\phi)\\,.\n\\end{align}\nHere, $p(\\Delta\\phi)$ is the probability distribution for azimuthal angle differences established in \\Eq{eq:delphidistfinal}, and $dN^\\text{pair}\/d\\Delta\\eta$ is the pairwise $\\Delta\\eta$ distribution.\n\nContinuing to the background distribution $B(\\Delta\\eta,\\Delta\\phi)$, we expect no correlation between $\\Delta\\eta$ and $\\Delta\\phi$ for pairs of particles from different events, so this distribution would naturally factorize.  Further, the azimuthal difference should be flat because the experiment is symmetric about the colliding beams and each event spontaneously breaks this rotational symmetry, but independent of all other events.  We also expect that the pseudorapidity difference distribution of the background is identical to that of signal because in the large $N$ limit, correlations between pairs of particles in the same event vanish.  Thus, the pseudorapidity difference distribution for same-event or distinct-event pairs is just determined by drawing from the single particle pseudorapidity distribution twice, to leading power in $1\/N$.  That is, the background distribution factorizes to leading power in $1\/N$ as\n\\begin{align}\nB(\\Delta\\eta,\\Delta\\phi) = \\frac{d^2N^\\text{mix}}{d\\Delta\\eta \\, d\\Delta\\phi} \\sim \\frac{dN^\\text{mix}}{d\\Delta\\eta}\\times\\frac{1}{N^\\text{mix}}\\frac{dN^\\text{mix}}{d\\Delta\\phi} = \\frac{1}{2\\pi}\\frac{dN^\\text{pair}}{d\\Delta\\eta}\\,,\n\\end{align}\nwhere $N^\\text{mix}$ is the number of pairs of particles from distinct or ``mixed'' events.\n\nWith these assumptions, our prediction for the double differential pairwise correlation distribution is independent of $\\Delta\\eta$ and just determined by the azimuthal difference distribution:\n\\begin{align}\n\\frac{d^2 N^\\text{pair}}{d\\Delta\\eta \\, d\\Delta\\phi} \\propto p(\\Delta\\phi)\\,.\n\\end{align}\nThe large-$N$ expansion of the minimum bias cross section immediately implies that there are long range in $\\Delta\\eta$ correlations, through the interpretation that the $\\Delta\\phi$ distribution at disparate values of $\\Delta\\eta$ are, to first order, the same.  While these long-range correlations are observed in data and are a requirement for existence of a ridge, it doesn't by itself predict an over-density of pairs of particles with $\\Delta\\phi = 0$.  That requires the distribution $p(\\Delta\\phi)$ to have a maximum at $\\Delta\\phi = 0$.  One can perform an analysis of the coefficients of the Fourier expansion of $p(\\Delta\\phi)$, and compare to data.  Recall that our large-$N$ expansion of minimum bias predicts the form\n\\begin{equation}\np(\\Delta\\phi)  =\\frac{1}{2\\pi}-\\frac{d_1}{\\pi}\\frac{1}{N}\\cos(\\Delta\\phi) +\\frac{1}{\\pi}\\sum_{n=2}^\\infty \\frac{d_n(N)}{N^{2n}}\\, \\cos(n\\, \\Delta\\phi)\\,,\n\\end{equation}\nwhere $d_1>0$ and the $d_n(N)$ coefficients scale at worst like $N^{2n-1}$ in the large $N$ limit.  Additionally, from ellipticity, we know that $d_2(N)>0$, at least for minimum bias events in PbPb collisions. \n\nFor existence of a ridge, we would additionally need that \n\\begin{align}\n&\\left.\\frac{dp(\\Delta\\phi)}{d\\Delta\\phi}\\right|_{\\Delta\\phi = 0}=0\\,,\n&\\left.\\frac{d^2p(\\Delta\\phi)}{d\\Delta\\phi^2}\\right|_{\\Delta\\phi = 0}<0\\,.\n\\end{align}\nThe vanishing of the first derivative at $\\Delta\\phi=0$ is guaranteed by the symmetries of minimum bias collisions.  The concave down requirement constrains the Fourier coefficients, but without a short-distance theory from which to predict the $d_n(N)$ values, we can't say much more within the context of our minimum bias expansion.  However, the fact that the distribution $p(\\Delta\\phi)$ is physical highly constrains the coefficients $d_n(N)$.  Assuming that a physical distribution is infinitely differentiable and the Fourier series of every derivative is absolutely convergent, the $d_n(N)$ must vanish as $n\\to\\infty$ faster than any power of $n$.  For example, the Fourier series could actually just be finite for which there is a maximum $n$ at which $d_n(N)$ is non-zero, or $d_n(N)$ might vanish exponentially fast, like $d_n(N)\\sim e^{-n}$.  Infinite differentiability implies that there are only a finite number of terms in the Fourier expansion that can have an appreciable effect on the first and second derivatives of $p(\\Delta \\phi)$ at $\\Delta \\phi=0$.  This seems to be borne out in data, e.g., figure 6 of \\Ref{Chatrchyan:2011eka}, in which the $d_2(N)$, $d_3(N)$, and $d_4(N)$ coefficients would be positive, but $d_5(N)$ appears to be vanishingly small.  Of course, with any finite dataset, measuring higher Fourier modes is problematic, but the trend is consistent with expectations from differentiability.\n\n\n\\section{Min-Bias in Other Colliders}\\label{sec:other}\n\nWhile our focus in this paper has been the application of the large-$N$ expansion of minimum bias events to understand identical hadron or identical heavy ion collisions, the power counting and symmetries can be appropriately modified to describe nearly any collision environment.  In this section, we will just briefly describe extension to minimum bias at other colliders.  Minimum bias in electron-positron collisions will be significantly different that we will provide more details about its symmetries and predictions.\n\n\\subsection{pA Collisions and Electron-Ion Collisions}\n\nFor different types of colliders, we must modify our power counting and symmetry assumptions as established in \\Sec{sec:ppmin}.  Here, we will just discuss how the power counting and symmetries are modified for proton-ion (pA) and electron-ion collisions, leaving a detailed analysis of unique predictions from our minimum bias expansion in those environments to future work.  First, for pA collisions, as the initial state still consists of hadronic matter, there are still unmeasurable beam regions in which all we can determine is the total momentum lost down the beam.  Thus, all power counting assumptions for minimum bias in pA collisions remain identical to that of pp\/AA collisions.  The only change is to the symmetries, in which pA collisions lack the beam reflection $\\eta\\to -\\eta$ symmetry of pp\/AA collisions.  As a consequence, the expansion of the corresponding matrix element is not reflection symmetric about the beams and so can have terms that consist of the hyperbolic sine of pseudorapidity differences, as well as hyperbolic cosine.  For example, the squared matrix element for the $N$ detected particles of pA collisions can now have terms of the form\n\\begin{equation}\n|{\\cal M}|^2\\supset \\sum_{i\\neq j}^N p_{\\perp i}p_{\\perp j}\\sinh(\\eta_i-\\eta_j)\\,.\n\\end{equation}\n\nFor electron-ion colliders, the power counting and symmetries are even more different.  Now, because the electron is not hadronic, there is only one unmeasurable beam region, in the direction of the momentum of the initial ion.  This isn't necessarily to say that there is perfect experimental resolution in the direction of the electron's momentum, it is instead that because the electron is not hadronic, its direction of momentum is not special for establishing beam remnants that take away an order-1 energy fraction at very high pseudorapidity.  As there is only one unmeasurable beam region, we only smear over one component of lightcone momentum, $k^+$, say.  This also means that the pseudorapidity translation symmetry $\\eta\\to \\eta+\\Delta\\eta$ is broken because we can effectively measure all particles at arbitrary pseudorapidities in the direction of the electron's momentum.  Additionally, as the colliding particles are not identical, there is no beam reflection symmetry, $\\eta\\to-\\eta$.  Thus in this case, the cross section for minimum bias events takes the form:\n\\begin{align}\n\\sigma &\\sim \\frac{1}{Q}\\int_0^Q dk^+\\,f(k^+) \\int d\\Pi_N\\, |{\\cal M}|^2\\\\\n&\n\\hspace{2cm}\\times\\delta\\left(\nk^+ - \\sum_{i=1}^N p_{\\perp i}e^{-\\eta_i}\n\\right)\\, \\delta\\left(\nQ - \\sum_{i=1}^N p_{\\perp i}e^{\\eta_i}\n\\right)\\, \\delta^{(2)}\\left(\n\\sum_{i=1}^N \\vec p_{\\perp i}\n\\right)\\nonumber\\,.\n\\end{align}\nHere, $f(k^+)$ is some function purely of the total $k^+$ momentum lost down the beampipe in the direction of the initial ion's momentum and $|{\\cal M}|^2$ is the squared matrix element of the $N$ detected particles.  Decreasing the constraining symmetries makes the expansion of the squared matrix element significantly more involved, so we leave predictions for electron-ion collisions to future work.\n\n\\subsection{Electron-Positron Collisions}\n\nThe description of hadronic minimum bias events in electron-positron ($e^+e^-$) collisions is sufficiently constraining and simple that we will explicitly construct the large-$N$ expansion and make some predictions.  First and foremost, because electrons are color-singlet, fundamental particles, there are no unmeasurable beam regions whatsoever, so (in principle) all final state particles can be detected.  Consequently, because the initial state is the hadronic vacuum, we assume full Lorentz invariance of the final state, so natural phase space coordinates are the energies of particles and their location on the celestial sphere.  Further, as with our pp\/AA collision analysis, we assume that only momenta are measured; no information on electric charge, etc., is recorded for particles.\n\nWith these assumptions, the power counting for hadronic minimum bias events in $e^+e^-$ collisions are:\n\\begin{enumerate}\n\n\\item All $N$ particles produced in collision can in principle be detected.  Only their four-momenta are measured; no charge information is determined.\n\n\\item  We assume that the mass of the particles is irrelevant and so detected particle energy $E$ is parametrically larger than the QCD scale or pion mass, $E \\gg m_\\pi$.\n\n\\item The number of detected particles $N$ is large: $N\\gg1$.\n\n\\item We assume that the mean energy of the detected particles is representative of all particles' energies and so the mean and the root mean square energies are comparable: $\\langle E\\rangle\\sim \\sqrt{\\langle E^2\\rangle} \\sim Q\/N$.\n\\end{enumerate}\nThese power counting assumptions then imply that these minimum bias events enjoy the following symmetries:\n\\begin{enumerate}\n\n\\item Complete Lorentz invariance O$(3,1)$ and parity symmetry of the final state,\n\n\\item reflection of the electron-positron beams because no charge information is measured, and\n\n\\item $S_N$ permutation symmetry in all $N$ detected particles.\n\\end{enumerate}\nThe reflection symmetry of the beams is already accounted for with the O(3,1) Lorentz invariance, but we mention it explicitly because it would not be true if charge information were retained.\n\nWith these symmetries, the cross section for minimum bias events can just be expressed in the textbook form of Fermi's Golden Rule with four-momentum conservation in the center-of-mass frame:\n\\begin{align}\n\\sigma &\\sim \\int \\Pi_N\\, |{\\cal M}|^2\\sim \\int \\prod_{i=1}^N\\frac{d^3 p_i}{2E_i}\\, |{\\cal M}|^2\\, \\delta^{(4)}\\left(\nQ-\\sum_{i=1}^N p_i\n\\right)\\\\\n&\\sim \\int \\prod_{i=1}^N \\left[E_i\\, d E_i\\, d\\cos\\theta_i\\, \\frac{d\\phi_i}{2\\pi}\\right]\\, |{\\cal M}|^2\\, \\delta\\left(\nQ-\\sum_{i=1}^N E_i\n\\right)\\, \\delta\\left(\n\\sum_{i=1}^N E_i\\cos\\theta_i\n\\right)\\,\\delta^{(2)}\\left(\n\\sum_{i=1}^N \\vec p_{\\perp i}\n\\right)\\,,\n\\nonumber\n\\end{align}\nwhere $|{\\cal M}|^2$ is the Lorentz-invariant squared matrix element for $N$ final state particles, $E_i$ is the energy of particle $i$, and $\\theta_i$ and $\\phi_i$ are its polar and azimuthal angle coordinates on the celestial sphere.  As it is fully Lorentz invariant, the squared matrix element can be expanded in symmetric polynomials of the Mandelstam invariants $s_{ij} = 2p_i\\cdot p_j$, for two particles' four-momenta $p_i$ and $p_j$.  Up through mass dimension 4 terms, the squared matrix element can be expanded in symmetric polynomials as:\n\\begin{align}\n|{\\cal M}|^2 &=1+\\frac{c_1^{(4)}}{Q^4}\\sum_{i\\neq j\\neq k\\neq l}^N s_{ij}s_{kl}+\\frac{c_2^{(4)}}{Q^4}\\sum_{i\\neq j\\neq k}^Ns_{ij}s_{ik}+\\frac{c_3^{(4)}}{Q^4}\\sum_{i\\neq j}^Ns_{ij}^2+{\\cal O}(Q^{-6})\n\\end{align}\nThe $c_i^{(n)}$ factors are constants that may in general have dependence on the number of particles $N$.  The only constraint we impose on them is that in the $N\\to\\infty$ limit, their scaling with $N$ produces a finite contribution to the squared matrix element and that they are not sufficiently negative to make the squared matrix element negative.  For example, assuming that the energy of particle $i$ scales like $E_i\\sim Q\/N$ and so the second non-trivial term scales with $N$ as\n\\begin{equation}\n\\sum_{i\\neq j\\neq k}^Ns_{ij}s_{ik} \\sim N^3\\frac{Q^4}{N^4}\\sim \\frac{Q^4}{N}\\,,\n\\end{equation}\nbecause there are $N^3$ terms in the sum in the $N\\to\\infty$ limit.  Therefore, for finiteness and positivity, the coefficient $c_2^{(4)}$ is bounded as\n\\begin{equation}\n-N \\lesssim c_2^{(4)}\\lesssim N\\,.\n\\end{equation}\n\nAs discussed in the expansion of the squared matrix element for pp\/AA collisions, in the $N\\to\\infty$ limit, the various terms in the squared matrix element relax to their ensemble mean with 0 variance, with our assumption of ergodicity and the central limit theorem.  So, in the $N\\to\\infty$ limit, our minimum bias power counting implies that the matrix element is just a constant, corresponding to $N$ free final state particles only constrained by momentum conservation.\n\n\\subsubsection{Suppression of a Ridge}\n\nWithin this hadronic minimum bias expansion for $e^+e^-$ collision events, we will concretely demonstrate that this predicts a strong suppression of possible ridge phenomena in azimuthal correlations of pairs of particles.  A ridge in two-particle correlations was searched for in $e^+e^-$ collisions from archived ALEPH data recently \\cite{Badea:2019vey}, but no evidence for such a ridge was found.  Lack of azimuthal correlations over a long distance in pseudorapidity in $e^+e^-$ collisions is perhaps not surprising because the initial state is the QCD vacuum and there are no special directions in which particles are produced at arbitrarily high pseudorapidities.  Nevertheless, it is satisfying to see that the power counting and symmetries of $e^+e^-$ collisions immediately predict that such correlations, if they do exist, are very suppressed in the large-$N$ limit.  \n\nTo demonstrate this suppression of azimuthal correlations, we will focus on terms in the squared matrix element that directly correlate pairs of particles:\n\\begin{align}\n|{\\cal M}|^2 &\\supset 1+\\sum_{n=2}^\\infty c_n\\sum_{i\\neq j}^N \\frac{s_{ij}^n}{Q^{2n}} = 1+\\sum_{n=2}^\\infty \\frac{2^nc_n}{Q^{2n}}\\sum_{i\\neq j}^N p_{\\perp i}^np_{\\perp j}^n\\left(\n\\cosh \\Delta \\eta_{ij} - \\cos\\Delta \\phi_{ij}\n\\right)^n\\,.\n\\end{align}\nHere, the $c_n$ are some constants that possibly depend on $N$ but are constrained by finiteness and on the right, we have expanded the Mandelstam invariant $s_{ij}$ in terms of cylindrical detector coordinates.  Now, we would like to simplify this expression, focusing on the scaling with $N$ and the azimuthal correlations explicitly, so we will make some replacements.  First, the transverse momentum $p_\\perp = E\\sin\\theta$ and on flat phase space, most particles are located around the ``equator'' of the celestial sphere, where $\\theta = \\pi\/2$.  So, as $E\\sim Q\/N$, so too does $p_\\perp \\sim Q\/N$ in $e^+e^-$ collisions.  Further, because most particles lie near the equator, the hyperbolic cosine of the pairwise pseudorapidity difference is close to 1, so we can take the scaling $\\cosh \\Delta \\eta \\sim 1$.  With these scaling assumptions, the squared matrix element takes the form:\n\\begin{align}\n|{\\cal M}|^2 &\\supset  1+\\sum_{n=2}^\\infty \\frac{2^nc_n}{Q^{2n}}\\sum_{i\\neq j}^N p_{\\perp i}^np_{\\perp j}^n\\left(\n\\cosh \\Delta \\eta_{ij} - \\cos\\Delta \\phi_{ij}\n\\right)^n\\\\\n&\\sim1+\\sum_{n=2}^\\infty \\frac{2^nc_n}{N^{2n}}\\sum_{i\\neq j}^N \\left(\n1 - \\cos\\Delta \\phi_{ij}\n\\right)^n\\nonumber\\,.\n\\end{align}\n\nNow, with this squared matrix element, we can determine the probability distribution of the pairwise azimuthal angle difference $\\Delta\\phi$, in the large-$N$ limit.  As in our analysis of pp\/AA collisions, correlations from momentum conservation become trivial in the $N\\to\\infty$ limit, and the probability distribution can be calculated from\n\\begin{align}\np(\\Delta\\phi)&\\propto \\int_0^{2\\pi}\\prod_{k=1}^N\\frac{d\\phi_k}{2\\pi}\\, |{\\cal M}|^2 \\, \\delta(\\Delta \\phi-(\\phi_1-\\phi_2))\\\\\n&= \\int_0^{2\\pi}\\prod_{k=1}^N\\frac{d\\phi_k}{2\\pi}\\,\\left(\n1+\\sum_{n=2}^\\infty \\frac{2^nc_n}{N^{2n}}\\sum_{i\\neq j}^N \\left(\n1 - \\cos\\Delta \\phi_{ij}\n\\right)^n\n\\right)\\, \\delta(\\Delta \\phi-(\\phi_1-\\phi_2))\\nonumber\\\\\n&=\\frac{1}{2\\pi}+\\frac{1}{\\pi}\\sum_{n=2}^\\infty \\frac{2^nc_n}{N^{2n}}\\left[\n\\frac{2^{n-1}\\Gamma\\left(\\frac{1}{2}+n\\right)}{\\sqrt{\\pi}\\Gamma(1+n)} \\,(N+1)(N-2)+\\left(\n1 - \\cos\\Delta \\phi\n\\right)^n\n\\right]\n\\nonumber\n\\end{align}\nWe have used permutation symmetry to define $\\Delta\\phi = \\phi_1-\\phi_2$, and the result in the final line is the exact integral over azimuthal angles of the second line.  In this final form, it is obvious that any azimuthal dependence is suppressed by a relative factor of $N^2$ in the large-$N$ limit.  This explicitly follows from Lorentz invariance of the final state, enforcing that the transverse and longitudinal momentum components to the beam must appear in the expansion of the squared matrix element with the same coefficient.  \n\n\\section{Outlook}\\label{sec:concs}\n\nIn this paper, we elucidated a power counting scheme to describe minimum bias events at colliders. The main tenets of this approach are a principle of particle ergodicity---each particle is representative of any other particle and averages over particles in an event are equivalent to averages of events over an ensemble---and the assumption of large particle multiplicity, $N$, in an event which provides the expansion parameter $1\/N$. Under these conditions, the {\\it variance} of the QCD squared matrix element vanishes as $1\/N$, and as $N\\to \\infty$, this effectively reduces all physical observables to their flat phase space limits, with a smearing factor in p or A collisions to account for beam losses.\n\nWe showed that the above assumptions do indeed lead to a remarkably good description of minimum bias within a realm of validity that we quantified, performing an explicit comparison to minimum bias collider data. Notably, the (smeared) flat phase space values reproduce kinematic distributions well. We also studied observable features that can be described by an expansion in higher order kinematic harmonics about flat phase space, focusing in particular on azimuthal correlations, specifically elliptic flow in heavy ion collisions and the ridge phenomena in pp collisions. We showed that the conditions of positivity and momentum conservation fix the sign and scaling with $N$ of the leading harmonic of two particle azimuthal difference $\\Delta \\phi$ beyond the flat limit, and argued that the different symmetries of $e^+e^-$ collisions alone imply additional suppression $1\/N^2$ of any ridge that could be observed there. \n\nThe results of the above study are promising, in that they provide confidence that the $S$-matrix framework to describe minimum bias can be developed. Benefits of this approach are that it equally applies to small and large collision system sizes at high and low energies, and it does not depend on any underlying model or unphysical parameters. This could be utilized to elucidate the nature of small scale collective phenomena of QCD that were newly discovered at the LHC, and which are the subject of some debate. In particular, it would be interesting to study the emergence of jets in our picture, and see whether this can address the question of jet quenching in small (potentially QGP phase) systems. Another avenue would be exploring the use of this formalism at  low-energy heavy ion beam scans that aim to hit  the QCD critical point.\n\nTheoretically, there are a number of interesting directions to pursue. A systematic study of the harmonic expansion of the phase space manifold could be undertaken; power spectra of minimum bias collisions can be envisaged from this, and could be an efficient way of determining flow vs.~non-flow physics. It would be interesting and potentially very natural to incorporate detector resolution into this language via, e.g., a maximum harmonic sensitivity.\n\nWe formally took a limit $N\\to\\infty$ while working at fixed $Q$, while ignoring the mass of the QCD pions. This approximation agrees with data well, giving confidence that it is a good approximation to low-energy QCD dynamics in the minimum bias regime. However, strictly this limit does not exist, a maximum multiplicity being ensured by the explicit breaking of chiral symmetry and the pion mass, $N_{\\text{max}}=Q\/m_{\\pi}$. The obvious connection between the limit we worked in and broader studies of strongly-coupled conformal field theories at colliders~\\cite{Hofman:2008ar} would be interesting to pursue. It might be possible to address whether  the $Q$ and $N$ dependence of the coefficients the appeared in our expansion be fixed by, e.g., consistency conditions that arise from a non-trivial limit of a distribution as $N\\to\\infty$.\n\nFinally, by working purely with the  $S$-matrix of final particles, the hydrodynamic description of the unobservable QGP was sidestepped. Nevertheless one might wonder how hallmarks of transport coefficients and, e.g., the theoretical lower bound on specific shear viscosity may show up in our coarse-grained approach. \n\n\\acknowledgments\n\nWe thank Bryan Webber for valuable feedback regarding our approach. T.M. is supported by the World Premier International Research Center Initiative (WPI) MEXT, Japan, and by JSPS KAKENHI grants JP19H05810, JP20H01896, and JP20H00153. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{{Introduction}}\n\nIn recent years a significant attention has been devoted to\ninvestigation of low-dimensional asymmetric simple exclusion\nprocesses (ASEPs)\n\\cite{derrida98,zia95,schutz,chowdhury00,derrida93}. They play  a\ncritical role for understanding  fundamental properties of\nnon-equilibrium phenomena in Chemistry, Physics and Biology.\nASEPs have been widely utilized for description of traffic\nphenomena \\cite{chowdhury00}, kinetics of biopolymerization\n\\cite{macdonald68}, protein synthesis\n\\cite{shaw03,chou04,shaw04,dong07}, and biological transport of\nmotor proteins \\cite{lipowsky01,parmeggiani03}. The advantage of\nusing asymmetric exclusion processes for studying mechanisms of\nnon-equilibrium phenomena is due to the fact that some\nhomogeneous versions of ASEPs can be solved exactly via\nmatrix-product approach and related methods\n\\cite{derrida98,derrida93,blythe07}. In addition, understanding\nof processes in ASEPs can be achieved by utilizing a\nphenomenological domain wall approach \\cite{DW}. In order to have\na more realistic description of different non-equilibrium\nphenomena ASEPs with inhomogeneous distribution of rates are\nrequired. However, there is a limited number of  studies dealing\nwith ASEPs with disorder in the transition rates at sites (static impurities)\n\\cite{chou04,shaw04,janowsky92,schutz93,schutz93PRE,janowsky94,kolomeisky98,tripathy97,barma98,kolwanker,ha,stinchcombe,derrida04,juhasz05,barma06,pierobon06,foulaadvand07,foulad07,dongPRE,greulich1,greulich2}\nand with disorder associated to particles's hopping rates (moving impurities) \\cite{speers,mallick,evans,kim,fouladvand99}.\nIn this case exact solutions are not obtained, and extensive\nMonte Carlo computer simulations and approximate theories are\nutilized in order to understand particle dynamics. Disorder has a\nstrong effect on the behavior of ASEPs. Even a single defect bond\nfar away from the boundaries lead to dramatic effects in the\nstationary properties both in closed\n\\cite{janowsky92,janowsky94}  and open boundary conditions\n\\cite{kolomeisky98}.  It was shown recently that the dynamics of\nASEPs is also influenced by several defects that are close to\neach other \\cite{chou04,dong07,greulich1}, although the mechanism\nof this phenomenon is not well understood. This interaction\nbetween  defects is important for understanding several\nbiological transport phenomena \\cite{chou04,dong07}.  Recently\nthe particular case of two defects has been extensively\ninvestigated by Monte Carlo simulations \\cite{dong07}. It has been\nshown that the system current exhibits a notable dependence on the\ndistance between defects with equal hopping rates. Moreover, it\nwas found that the density profile is linear between defects\nwhich marks the existence of wandering shock between defects\n\\cite{dong07}. The case of two defective sites with equal rate\nhas been generalized to include extended objects \\cite{dongPRE}.\nTheoretical efforts to analyze ASEPs with disorder have been\nmostly directed to the cases with a single or few defects\n\\cite{kolomeisky98,chou04,greulich1}. In Ref. \\cite{kolomeisky98}\nASEP with open boundaries  and with a local inhomogeneity in\nthe bulk  has been investigated by arguing that the defect bond\ndivides the system into two coupled homogeneous ASEPs. This\ntheoretical approach can be called a {\\it defect mean-field}\n(DMF) because the mean-field assumptions are made only at the\nposition of  local inhomogeneity.  Although a good agreement with\ncomputer simulations has been found, there were significant\ndeviations in statistical properties of the phase with the\nmaximal current that was attributed to the neglect of\ncorrelations at the defect bond in the proposed theory\n\\cite{kolomeisky98}. A related approach called {\\it interacting\nsubsystem approximation} (ISA) has been proposed in Ref.\n\\cite{greulich1} for ASEPs with a single defect or several\nconsecutive defects (bottleneck). Here it was suggested that due\nto the defect bonds there are 3 segments in the system: two\nhomogeneous ASEPs are coupled by a segment that includes all\nsites that surround defect bonds. Explicit results have been\nused  inside the segments, and  mean-field assumptions have been\nutilized for particle dynamics between the segments. A better\nagreement with Monte Carlo computer simulations has been found,\nand the method was also successfully applied to describe\ninteractions of defects with boundaries. It was argued that ISA\ncan be used for analyzing properties of general ASEPs with\ndisorder \\cite{greulich1,greulich2}. However, ISA has not been\napplied for the systems with 2 defects at finite distances from\neach other, and because of this observation it is difficult to\napply ISA for understanding mechanisms of more complex\ninhomogeneous asymmetric exclusion processes. A slightly\ndifferent method of calculations has been proposed by Chou and\nLakatos \\cite{chou04}, who applied a {\\it finite segment\nmean-field theory} (FSMFT). According to this approach, the\nsegment of finite length $n$ that covers the defect and\nsurrounding sites is considered, and its dynamics is fully\ndescribed by solving explicitly for eigenvectors of the\ncorresponding transition rate matrix. The segment is then coupled\nin the mean-field fashion to the rest of the system. However,\nthis approach becomes numerically quite involved for cluster\nsizes  larger than $\\approx$20, and it also limits its\napplicability. Different studies of asymmetric exclusion\nprocesses with disorder point out to importance of correlations\nin the system. It is reasonable to expect that correlations are\nstronger near the slow defect sites. However, it is not clear how\nfar from the local inhomogeneity and how fast these correlations\ndecay. In addition, it is also unclear  how correlations from two\nclose defects affect each other. The goal of this paper is to\ninvestigate the role of correlations in dynamics of ASEPs with\ndisorder. By analyzing several analytical approaches in\ncombination with extensive Monte Carlo computer simulations it\nwill be shown that a successful description of disordered driven\ndiffusive systems can be achieved by properly accounting for\ncorrelations near the defect bonds.\n\n\n\\section{Model and Theoretical Description}\n\nWe investigate a totally asymmetric simple exclusion processes\nwith disorder. In the one-dimensional lattice\nthe particle at the site $i$ can jump forward with the rate\n$p_{i}$ if the next site $i+1$ is unoccupied, otherwise it stays\nat the same place. The particle can enter the system with the\nrate $\\alpha$ if the site is empty, and it can also exit the\nlattice with the rate $\\beta$. When all $p_{i}=1$ we have a\nhomogeneous ASEP for which dynamic properties are known\nexplicitly from exact solutions \\cite{derrida98,schutz}.\nASEPs with disorder correspond to the situation when there is\ninhomogeneities in the transition rates, and $p_{i}$ are drawn\nfrom arbitrary distributions. Numerous theoretical and\ncomputational studies indicate that in the limit of large times\nthe dynamics in the system can be determined by comparing\nentrance rate, exit rates and the transition rate at the slowest\ndefect bond \\cite{kolomeisky98,greulich2}. This observation has a\nsignificant consequence for properties of ASEPs with disorder,\nyielding a generic  phase diagram with 3 phases.  When the\nentrance is a rate-limiting process the system can be found in\nthe low-density phase, while for slow exiting the high-density\nphase governs the system. If the rate-limiting process is the\ntransition via the slowest defect bond the system is in the\nmaximal-current phase. In this maximal-current phase, a segregation\nof density profile into macroscopic high and low regions occurs at the\nlocation of slowest defect bond. Other defects only perturb the density\nprofile on a local scale. However, when the number slowest defect bonds exceeds\ntwo or more the above picture needs modification. Furthermore, the previous studies\non disordered ASEPs lack investigations on correlation effects induced by defects.\nTo address these questions, we analyze the simplest model with 2 identical defects\nin the bulk of the system far away from the boundaries. It was shown earlier\n\\cite{greulich1} that positioning of the slow defects close to the boundaries leads\nonly to rescaling of the effective entrance and\/or exit rates, and we will not\nconsider this possibility in this paper. Note that in this paper we are using\nterms of defect bonds and defect sites. To clarify, we define the defect site as\nthe site $i$ from which the particle hopes to the site $i+1$ with the rate $q<1$.\nCorrespondingly, the bond connecting sites $i$ and $i+1$ is a defect one.\n\n\n\\subsection{ Defect Mean-Field Theory}\n\nConsider a totally asymmetric exclusion processes with open\nboundaries and with 2 slow defective sites at $i=d_1$ and $i=d_2$ at a distance $d$\nwith $d_2-d_1=d$ (separated by $d-1$ normal sites), as shown in Fig. 1.\nAt the defects the particle jump to the right with the rate $q < 1$, in all\nother sites the hopping rate is equal to one. It can be seen that two defects\ndivide the system into three segments.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=7.5cm]{fig1.eps}\n\\caption{ Fig.1: Schematic of ASEP with two defective sites at $i=d_1$ and $i=d_2$ separated by $d-1$ normal sites\ni.e., $d_2-d_1=d$. The reduced hopping rates at each defect is equal to $q$.\nIn normal sites the hoping rates are one. } \\label{fig:bz2}\n\\end{figure}\n\nThe particle dynamics inside each segment can be calculated exactly,\nhowever, it is assumed that there are no correlations between the segments.\nIf entrance to the lattice is the slowest process then the system\ncan be found in low-density (LD) phase with stationary current\nand bulk densities given by\n\n\\begin{equation}\nJ=\\alpha(1-\\alpha), \\quad \\rho_{bulk}=\\alpha.\n\\end{equation}\n\nSimilarly, when the exit becomes a bottleneck process the system\nis in high-density (HD) phase with\n\n\\begin{equation}\nJ=\\beta(1-\\beta), \\quad \\rho_{bulk}=1-\\beta.\n\\end{equation}\n\nNote that in both phases particle densities near the defect bonds\nwill deviate from the bulk values. The more interesting case is\nwhen the dynamics in the system is governed by transitions via\nlocal inhomogeneities. In this phase, which has the maximal current, we expect\nto have density phase segregation similar to the case a single defect ASEPs.\nWe emphasize that all our investigation in this paper is on this maximal-current phase.\nFirst consider a lattice segment after the second defect.\nThe dynamics in this part of the system is controlled by the entrance\nof particle via the defect, then it has a low-density profile with unknown\nbulk density $\\rho^{*}<1\/2$. Similar arguments can be used to analyze the density\nprofile in the segment before the first defect. Here the flux is\nlimited by the exit rate via the local inhomogeneity, leading to\nthe  high-density phase. Since at stationary-state condition\nthe flux through any segment should be the same,\n$J=\\rho^{*}(1-\\rho^{*})$, the bulk density in this segment is\nequal to $1-\\rho^{*}$. The region between two defects can be\nviewed as asymmetric exclusion process on a finite lattice with\n$d$ sites. The effective entrance and exit rates to this segment\ncan be easily evaluated using our mean-field assumptions,\n\n\\begin{equation}\\label{boundary_rates}\n\\alpha_{eff}=\\beta_{eff}=q(1-\\rho^{*}).\n\\end{equation}\n\nThe stationary properties of the lattice segment with $d$ sites\nbetween the defects can be evaluated explicitly by utilizing\nexact results for finite-size ASEPs \\cite{derrida93}.\nSpecifically, the particle flux is given by\n\n\\begin{equation}\\label{J0}\nJ_{0}(\\alpha,\\beta,d)=\\frac{R_{d-1}(1\/\\beta)-R_{d-1}(1\/\\alpha)}{R_{d}(1\/\\beta)-R_{d}(1\/\\alpha)},\n\\end{equation}\n\nwhere the function $R_{d}(x)$ is defined as\n\n\\begin{equation}\\label{R_eq}\nR_{d}(x)=\\sum_{p=2}^{d+1} \\frac{(p-1)(2d-p)!}{d!(d+1-p)!} x^{p}.\n\\end{equation}\n\nTo understand the density profile in the segment we can use a\ndomain-wall picture of asymmetric exclusion processes \\cite{DW}.\nSince the entrance and exit rates are the same [see Eq.\n(\\ref{boundary_rates})], the domain wall that separates\nhigh-density and low-density blocks performs an unbiased random,\nleading to a linear density profile with a positive slope.\nExplicit expressions for particle densities can also be found in\nRef. \\cite{derrida93}. The full description of dynamics in ASEPs\nwith two defects is obtained by solving for the unknown parameter\n$\\rho^{*}$. It can be done by applying the condition of\nstationarity in the particle flux,\n\n\\begin{equation}\\label{relation}\nJ=\\rho^{*}(1-\\rho^{*})=J_{0}(\\alpha_{eff},\\beta_{eff},d).\n\\end{equation}\n\nThis equation can always be solved analytically or numerically\nexactly for any number of sites between local inhomogeneities,\nleading to stationary particle currents and density profiles. it\nis important to note that there is a particle-hole symmetry in\nthe system because defects are far away from the boundaries. To\nillustrate our approach let us calculate dynamic properties of\nASEPs with two defects for several values of the parameter $d$.\nFirst, let us analyze the simplest case of $d=1$ with consecutive\ndefects in the bulk. It can be shown that for this system\n\n\\begin{equation}\nJ_{0}(\\alpha,\\beta,d=1)=\\frac{\\alpha \\beta}{\\alpha + \\beta}.\n\\end{equation}\n\nThen Eq. (\\ref{relation}) can be written as\n\n\\begin{equation}\n\\rho^{*}(1-\\rho^{*})=q(1-\\rho^{*})\/2,\n\\end{equation}\n\nwhich produces simple expressions for the bulk density and the\nparticle current,\n\n\\begin{equation}\n\\rho^{*}=q\/2, \\quad J=q(2-q)\/4.\n\\end{equation}\n\nThe density $l$ at the site between the defects can also be found\nfrom the condition that the flux via this site, $J=q l\n(1-\\rho^{*})$, should be equal to the flux through other\nsegments, and this yields\n\n\\begin{equation}\nl=\\rho^{*}\/q=1\/2.\n\\end{equation}\n\nThis result could also be obtained from the particle-hole\nsymmetry arguments. Note that for $q=1$ we obtain $\\rho^{*}=1\/2$\nand  $J=1\/4$ as expected for homogeneous ASEPs in the\nmaximal-current phase. For $d=2$ there are two lattice sites\nbetween the defects, and stationary properties of this system can\nalso be obtained analytically. From Eq. (\\ref{R_eq}) one can\neasily derive\n\n\\begin{equation}\nR_{1}(x)=x^{2}, \\quad R_{2}(x)=x^{2}+x^{3},\n\\end{equation}\n\nwhich produces the following expression for the current in the\nsegment between the defects\n\n\\begin{equation}\\label{eq_2}\nJ_{0}(\\alpha,\\beta,d=2)=\\frac{\\frac{1}{\\alpha}+\n\\frac{1}{\\beta}}{\\frac{1}{\\alpha}+\n\\frac{1}{\\beta}+\\frac{1}{\\alpha^{2}}+\n\\frac{1}{\\beta^{2}}+\\frac{1}{\\alpha \\beta}}.\n\\end{equation}\n\nUsing the expression for the effective entrance and exit rates\nfor the segment between the inhomogeneities [see Eq.\n(\\ref{boundary_rates})], the condition for the stationary current\nleads to\n\n\\begin{equation}\n\\rho^{*}(1-\\rho^{*})=\\frac{2q(1-\\rho^{*})}{3+2q(1-\\rho^{*})}.\n\\end{equation}\n\nThis quadratic equation can be solved, and taking the physically\nreasonable root we obtain\n\n\\begin{equation}\n\\rho^{*}=\\frac{2q+3-\\sqrt{9+12q-12q^{2}}}{4q};\n\\end{equation}\n\\begin{equation}\nJ=\\frac{8q^{2}-6q-9 + 3\\sqrt{9+12q-12q^{2}}}{8q^{2}}.\n\\end{equation}\n\nIt can be checked  that for $q=1$ these equations reduce to\nexpected relations  $\\rho^{*}=1\/2$ and  $J=1\/4$. We can also\ncalculate the densities $l_{1}$ and $l_{2}$ at the sites between\nthe defects. Because of the particle-hole symmetry one can argue\nthat\n\n\\begin{equation}\nl_{2}=1-l_{1},\n\\end{equation}\n\nand the density at the first site  can be found by analyzing the\ncurrent via the first defect,\n\n\\begin{equation}\nJ=q(1-\\rho^{*})(1-l_{1})=\\rho^{*}(1-\\rho^{*}).\n\\end{equation}\nThen we have\n\\begin{equation}\nl_{1}=1-\\frac{\\rho^{*}}{q}=\\frac{4q^{2}-2q-3+\\sqrt{9+12q-12q^{2}}}{4q^{2}}.\n\\end{equation}\n\nWe have solved equation (6) for the case $d=3$. In this case we\nhave:\n\n\\begin{equation}\nR_{3}(x)=2x^{2}+2x^{3}+x^{4},\n\\end{equation}\n\nAfter some lengthy but straightforward algebra we arrive at the\nfollowing cubic equation for $\\rho^{*}$:\n\n\\begin{equation}\n4q^2(\\rho^{*})^{3} -(8q^2+6q)(\\rho^{*})^{2} +\n(6q^2+6q+4)\\rho^{*}-q(3+2q)=0.\n\\end{equation}\n\nFor brevity we avoid writing the answer explicitly. Analytical\nresults for ASEP with 2 defects can also be obtained in the limit\nof very large distances between the inhomogeneities ($d \\gg 1$).\nIn this case the segment between the defects can be viewed as a\nhomogeneous ASEP in the state of the phase transition between\nhigh-density and low-density phases ($\\alpha_{eff}=\\beta_{eff}$).\nThis corresponds to a linear density profile for the segment\nbetween the defects. Then it leads to the following expression\nfor the current\n\n\\begin{equation}\n\\rho^{*}(1-\\rho^{*})=q(1-\\rho^{*})\\left[1-q(1-\\rho^{*})\\right],\n\\end{equation}\nand finally we obtain\n\\begin{equation}\n\\rho^{*}=\\frac{q}{1+q}, \\quad J=\\frac{q}{(1+q)^{2}}.\n\\end{equation}\n\nThese results are identical to stationary properties of ASEP with\nonly one local inhomogeneity far away from the boundaries\nobtained using DMF approximation \\cite{kolomeisky98}, suggesting\nthat 2 defects at large distances do not affect each other\n\\cite{chou04}. For a general $d$ equation (6) leads to a polynomial equation\nof order $d$ for the unknown $\\rho^*$. For $d>3$ this equation can be solved numerically to\nfind the acceptable answer. In figure (2) we have sketched the behavior of current\n$J$ as a function of $q$ for $d=1,2$ and $3$ and have compared them to the results\nobtained via Monte Carlo simulations. As expected $J$ is an\nincreasing function of both $q$ and $d$. DMF notably\nunderestimates the current in comparison to the MC simulation\nespecially in the intermediate values of $q$.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=7.5cm]{fig3-colour.eps}\n\\caption{ Fig.2: (Colour online) $J$ vs $q$ for\n$d=1,2,3$ obtained by DMF method and MC simulation.} \\label{fig:bz2}\n\\end{figure}\n\n\\subsection{Interacting Subsystem Approximation}\n\nInteracting subsystem approximation (ISA) is another method of\ncalculating stationary properties of ASEPs with a single defect\nor a single bottleneck developed by Greulich and Schadschneider\n\\cite{greulich1}.  It can be easily extended to the case of\nasymmetric exclusion processes with 2 defects separated by $d$\nlattice sites. Similarly to DMF this method divides the lattice\ninto several segments. Particle dynamics inside the segments is\ntreated exactly, while between the segments mean-field\nassumptions are made. ISA differs from DMF in the defining of\nsegments. In DMF the position of defects separates different\nparts, and there is always 3 segments in the system. In ISA the\nsites that are connected by the defect bond are put together in\none segment, as shown in Fig. 3.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=7.5cm]{fig2.eps}\n\\caption{ Fig.2: (Colour online) Fig.3: (Colour online) interacting subsystems connected via mean-field assumption.\nFor d$>$2 the system is divided into five segments. } \\label{fig:bz2}\n\\end{figure}\n\nFor $d=1$ there are also 3 parts in the lattice, and the middle segment has 3 sites.\nFor $d=2$ there are 4 segments and 2 middle segments (with 2 lattice sites\neach) border each other.  For any larger distance between local\ninhomogeneities ISA assumes 5 segments: see Fig. 3. Note that the\nsize of the middle segment is equal to $d-2$.\n\nLet us consider a general case of 5 segments ($d>2$) for\ncomputation of stationary properties of ASEPs with 2 defects. As\nwas argued above, the system can be found in one of three phases:\nLD, HD or HD\/LD (maximal-current). Since the derivation of\nproperties in HD and LD phases is the same as for DMF approach,\nwe concentrate on description of the maximal-current phase. As\nbefore we assume that the bulk density in the segments 1 and 5\nare $1-\\rho^{*}$ and $\\rho^{*}$ correspondingly. Let us define\n$l_{1}$ and $l_{2}$ as the probabilities to find the particles at\nthe corresponding sites of the segment around the first defect.\nSimilarly, $l_{3}$ and $l_{4}$ describe densities in the segment\naround the second defect bond. For the middle segment with $d-2$\nlattice sites we define $x_{i}$ for $i=1,\\cdots,d-2$ as the\nparticle density at $i$-th site of this segment. As for DMF\napproach, the existing particle-hole symmetry simplifies\ncalculations significantly. Specifically, it suggests that\n\n\\begin{equation}\nl_{4}=1-l_{1},\\quad l_{3}=1-l_{2}, \\quad x_{i}=1-x_{d+1-i}.\n\\end{equation}\n\nThe overall particle current in the system can be written as\n\n\\begin{equation}\nJ=\\rho^{*}(1-\\rho^{*}),\n\\end{equation}\n\nwhile due to the mean-field assumptions the current between the\nfirst and the second segments is equal to\n\n\\begin{equation}\nJ_{12}=(1-l_{1})(1-\\rho^{*})=\\alpha_{2}(1-\\rho^{*}),\n\\end{equation}\n\nwhere $\\alpha_{2}$ is the effective rate to enter the second\nsegment.  At large times we expect to find the system in the\nstationary state, i.e., $J=J_{12}$, yielding\n\n\\begin{equation}\n1-\\alpha_{2}= l_{1}=(1-\\rho^{*}).\n\\end{equation}\n\nThe current between segments 2 and 3 can be presented in the\nseveral ways,\n\n\\begin{equation}\nJ_{23}=l_{2}(1-x_{1})=\\beta_{2} l_{2}=\\alpha_{3}(1-x_{1}),\n\\end{equation}\n\nwith $\\beta_{2}$ being the effective exit rate from the segment\n2, while $\\alpha_{3}$ is the effective rate to enter the segment\n3. When the system reaches stationary phase, $J=J_{23}$, and we\nobtain\n\n\\begin{equation}\n\\beta_{2}=1-x_{1}, \\quad\n\\alpha_{3}=\\frac{\\rho^{*}(1-\\rho^{*})}{1-x_{1}}.\n\\end{equation}\n\nBecause of the particle-hole symmetry the effective entrance and\nexit rates from the segment 3 are the same,\n$\\alpha_{3}=\\beta_{3}$. The particle current via the ASEP segment\nwith $N$ sites and with entrance and exit rates $\\alpha$ and\n$\\beta$, respectively, $J(\\alpha,\\beta, N)$, can be calculated\nexplicitly \\cite{derrida93}. Then to obtain stationary properties\nof ASEP  with 2 defects in the maximal-current phase the\nfollowing system of equations should be solved,\n\n\n$$\\rho^{*}(1-\\rho^{*}) = qJ(\\frac{1-\\rho^{*}}{q},\\frac{1-x_{1}}{q},2); $$\n\n\\begin{equation}\n\\rho^{*}(1-\\rho^{*}) =\nJ(\\frac{\\rho^{*}(1-\\rho^{*})}{1-x_{1}},\\frac{\\rho^{*}(1-\\rho^{*})}{1-x_{1}},d-2).\n\\end{equation}\n\n\nwhere $\\rho^{*}$ and $x_{1}$ are 2 unknown variables. The\nexpression on the right side of the first equation describes the\ncurrent inside the segment 2 and 4. Because the hopping rate is\n$q<1$, the effective entrance and exit rates must be rescaled by\nthe same factor. The right side of the second equation gives the\ncurrent inside the segment 3. The application of ISA for $d=1$\nand $d=2$ cases is different. In the case of 2 consecutive defect\nbonds the system is divided  only in 3 segments. The middle\nsegment has 3 sites that surround defect bonds. In the HD\/LD\nphase the effective entrance  rate is $\\alpha_{2}=1-\\rho^{*}$,\nand the stationary properties can be obtained by solving only one\nequation\n\n\\begin{equation}\n\\rho^{*}(1-\\rho^{*})=qJ(\\frac{1-\\rho^{*}}{q},\\frac{1-x_{1}}{q},3).\n\\end{equation}\n\nUsing Eqs. (\\ref{J0}) and (\\ref{R_eq}) for the middle segment\nwith equal entrance and exit rates gives us\n\n\\begin{equation}\n\\rho^{*}(1-\\rho^{*})=\\frac{q(1-\\rho^{*})\\left[2(1-\\rho^{*})+3q\\right]}{2\\left[2(1-\\rho^{*})^{2}+3q(1-\\rho^{*})+2q^{2}\\right]},\n\\end{equation}\n\nwhich can be simplified into the following expression,\n\\begin{equation}\n4(\\rho^{*})^{3}-2(3q+4)(\\rho^{*})^{2}+4(q+1)^{2}\\rho^{*}-q(3q+2)=0.\n\\end{equation}\n\nThis cubic equation can be solved explicitly, yielding\n\\begin{equation}\n\\rho^{*}=\\left[3q+4+(4-3q^{2})\/D+D\\right]\/6,\n\\end{equation}\nwhere\n\\begin{equation}\nD=\\left[-8+9q^{2}-27q^{3}+3\\sqrt{3}\n\\sqrt{16q^{3}-q^{4}-18q^{5}+28q^{6}} \\right]^{1\/3}.\n\\end{equation}\n\nISA also works differently in the case of $d=2$. There are 4\nsegments in the system, and because of the neglect of correlations\nbetween segments 2 and 3  we have\n\n\\begin{equation}\nl_{2}(1-l_{3})=l_{2}^{2}=\\rho^{*}(1-\\rho^{*}).\n\\end{equation}\n\nThen the effective entrance rate to the segment 2 is\n$\\alpha_{2}=1-\\rho^{*}$, and the effective exit rate is equal to\n$\\beta_{2}=l_{2}=\\sqrt{\\rho^{*}(1-\\rho^{*})}$. The unknown\nparameter $\\rho^{*}$ is determined from the equation for the\nstationary current,\n\n\\begin{equation}\n\\rho^{*}(1-\\rho^{*})=qJ(\\frac{\\alpha_{2}}{q},\\frac{\\beta_{2}}{q},2).\n\\end{equation}\n\nSubstituting the values of the effective boundary rates and\nutilizing Eq. (\\ref{eq_2}) we obtain\n\n\\begin{equation}\n\\rho^{*} \\sqrt{1-\\rho^{*}}+(\\rho^{*})^{3\/2}=q \\sqrt{1-\\rho^{*}}.\n\\end{equation}\n\nwhich can be recast in the form of a cubic equation:\n\n\n\\begin{equation}\n 2(\\rho^{*})^{3} - (2q+1)(\\rho^{*})^{2} + (q^2+2q)\\rho^{*}-q^2=0,\n\\end{equation}\n\nThis equation can be solved analytically but for brevity we do not\nwrite the solution. It can also be shown that in the limit of $d\n\\gg 1$ ISA method with 2 defects produces the stationary current\nand bulk densities which are indistinguishable form the situation\nwith only one defect \\cite{greulich1},\n$$\n\\rho^{*}=\\left[3q+2-\\sqrt{9q^{2}-4q+4} \\right]\/4,\n$$\n\\begin{equation}\nJ=\\left[2q-9q^2+3q\\sqrt{9q^{2}-4q+4} \\right]\/8.\n\\end{equation}\n\nLet us now exhibit the dependence of $J$ on q in the ISA method.\nIn figure (4) we have drawn $J$ vs $q$ for $d=1,2$ and have\ncompared the results to those obtained by DMF method and MC\nsimulations.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=7.5cm]{fig4-colour.eps}\n\\caption{ Fig.4: (Color online) Current vs q for\n$d=1,2$ and a large $d$ obtained within ISA method and MC simulation.\nIn simulations we have taken $\\alpha=\\beta=0.6$. } \\label{fig:bz2}\n\\end{figure}\n\nIn general, ISA method gives a better estimation of current\ncompared to DMF at least for small values of $d$ we have\nconsidered.\n\n\n\\section{Correlations near defect}\n\n\\subsection{Monte Carlo Simulations}\n\nIn this section we aim to investigate correlations in the\nvicinity of defects. We restrict ourselves to adjacent two-point\ncorrelations and will present our results for the general two-point and\nmulti-point correlations in a future work. Let us now introduce the\nnormalized connected two-point correlation function $C_i$ between the neighbouring\nsites $i$ and $i+1$. This quantity is defined as follows:\n\n\\begin{eqnarray}\nC_i= \\frac{\\langle \\tau_i\\tau_{i+1} \\rangle - \\langle \\tau_i\n\\rangle \\langle \\tau_{i+1} \\rangle}{ \\sqrt{\\langle \\tau_i^2\n\\rangle- \\langle \\tau_{i} \\rangle ^2}\\sqrt{\\langle \\tau_{i+1}^2\n\\rangle- \\langle \\tau_{i+1} \\rangle ^2}} ~ i=1,\\cdots,L-1.\n\\end{eqnarray}\n\nThe function $C_i$ measures the correlation and it lies between\n$-1$ and $1$. Negative values correspond to anti-correlation\nbetween neighboring sites whereas a positive value signifies\ncorrelation. The values near zero are regarded as uncorrelated.\nFig. (5) depicts the simulated profiles of correlation at $d=10$\nand 100 each for three values of $q$. The system size is $L=500$ and we have\ntaken $\\alpha=\\beta=0.6$ in all our simulation results unless stated otherwise.\nThe system has been updated for $T$ Monte Carlo steps. Each step consists of $L$ moves.\nIn each move, we randomly choose a site and update its status according\nto ASEP rules described above. We discard the first $\\frac{T}{5}$ steps to\nensure reaching steady state, and we have accumulated data\nseparated by 10 MC steps to avoid any possible temporal\ncorrelations. The value of $T$ is taken $10^8$ in our simulations.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=7cm]{fig9-colour.eps}\n\\includegraphics[width=7cm]{fig12-colour.eps}\n\\caption{ Fig.5: (Color online) Profile of normalized correlation function at $d=10$ (top)\nand $d=100$ (bottom) for $q=0.1,0.3,0.5$. } \\label{fig:bz2}\n\\end{figure}\n\n\nTwo defects are symmetrically placed with respect to chain mid\npoint. We observed that correlations are large in sites\nbetween the defects. There is a rather strong anti-correlation in\nthe sites immediately after the first defect and before the\nsecond defect. The correlations are growing up for middle sites\nwhere the maximum value is achieved. It can be seen that\ncorrelations are greater when $d$ is increased. This is unexpected\nand counterintuitive because increasing the distance between the defects\nreduces their interaction. It has been observed via MC\nsimulations that when $d$ is increased the current reaches\nasymptotically to its mean-field value\n$J_{MF}=\\frac{q}{(1+q)^2}$  \\cite{dong07,dongPRE}. Therefore, one\nexpects the correlations to exhibit a reducing behavior with\nrespect to distance $d$ but this is not observed in our\nsimulations. To have a deeper understanding, we have sketched the\nbehavior of correlation profile upon varying the distance $d$ for\n$q=0.1$ and $q=0.3$ in Fig. 6.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=7cm]{fig10-colour.eps}\n\\includegraphics[width=7cm]{fig13-colour.eps}\n\\caption{ Fig.6: (Color online) Profile of\nnormalized correlation functions at various values of $d$ for\n$q=0.1$ (top) and $q=0.3$ (bottom). } \\label{fig:bz2}\n\\end{figure}\n\n\nFor fixed values of $q$, increasing the distance $d$ between the\ndefects gives rise to enhancement of\ncorrelations\/antocorrelations. For instance, the correlation value\nin the middle point rises from roughly 0.5 at small $d \\sim 10$\nto 0.65 for $d \\sim 100$. It can be observed that there is no\nnotable difference in correlation values for $d$ larger than\n$100$. Moreover, the correlations are always greater than\nanti-correlations. By increasing $q$, the correlations\/anti-correlations\nare notably reduced in values. This is expected since in the limit of\nhomogeneous ASEP where $q\\rightarrow 1$ the correlation functions become very small.\nHere we wish to make a pause and have a discussion on correlations in normal ASEPs.\nIn fact the middle segment between two defects can be regarded as an ASEP chain with\nlength $d$ with equal entrance and exit rates. To the best of our knowledge, correlations\nin ASEP with random sequential update, has only been analytically discussed by\nDerrida and Evans who obtained exact analytical expression for a general two-point function\nand made a conjecture to generalize their findings to $n$-point function \\cite{evans93}. Their\nstudy was restricted to the special case $\\alpha=\\beta=1$ and they found that\nlong range correlations persist in the bulk which was attributed as a boundary effect.\nIn order to see if the large value of the connected two-point function survives in\nthe normal ASEP with equal entrance and exit rates, we performed MC simulations. Our\nresults show that when $\\alpha=\\beta$, the profile of $C_i$ reaches a small constant (almost zero) in the bulk.\nThe correlations become large near boundaries. This boundary behaviour depends on whether $\\alpha=\\beta <0.5$\nor $\\alpha=\\beta >0.5$. Figure (7) illustrates this aspect.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=7cm]{fig15-colour.eps}\n\\caption{ Fig.7: (Color online) Profile of normalized correlation functions in a normal ASEP chain with $\\alpha=\\beta$ .\n} \\label{fig:bz2}\n\\end{figure}\n\n\nWe recall that correlations in other types of update such as parallel updating has been discussed in\n\\cite{schutz93,schutz93PRE}. It is worthwhile to examine the behavior of density profile between\ndefects. Dong et al have recently shown via extensive MC simulations that\nthe density profile takes a linear shape between defects \\cite{dong07}. This behavior remains\nunchanged in ASEP with extended objects \\cite{dongPRE}. For the\nsake of completeness, we show some typical density profiles in Fig. 8.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=7cm]{fig11-colour.eps}\n\\caption{ Fig.8: (Color online) Profile of density at various values of $d$ for $q=0.1$.\nThe profile exhibits a linear behaviour with positive slope. This behaviour is associated\nto the existence of wandering shock in the region between left and right defects. } \\label{fig:bz2}\n\\end{figure}\n\n\nThe interesting point is the absence of boundary layer in this\nphase-segregated regime. It would be illustrative\nto look at the dependence of two-point correlation\nfunctions at some particular sites on values of $q$ and $d$.\nThese results are sketched in Fig. 9 where correlations at the\nfirst defect site ($d_1$), its rightmost site ($d_1+1$) and in\nthe middle site of the chain are considered.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=7cm]{fig8-colour.eps}\n\\includegraphics[width=7cm]{fig14-colour.eps}\n\\caption{ Fig.9: (Color online) Dependence of normalized correlation functions at three\nselected sites on $q$ (top) and on $d$ (bottom). } \\label{fig:bz2}\n\\end{figure}\n\n\nNote that all correlations\/anti-correlations approach zero when\n$q$ tends to one. Moreover, increasing defect's separation $d$\nincreases the correlations. The dependence on $d$ is more\ninteresting. While the values of correlation functions reach an\nasymptotic value at large d, the behavior is not monotonous.\nEspecially for $C_{d_1+1}$ correlation increases up to a maximum\nand then begin to decrease smoothly towards its asymptotic value.\nThe value of $d$ where $C_{d_1+1}$ is maximized does not show a\nsignificant dependence on $q$. In can be concluded that varying\n$d$ can dramatically affect the system characteristics as far as\ncorrelations are considered.\n\n\n\\subsection{Analytical theory }\n\nOur simulation findings in the preceding section confirms that in\nbetween the defects the correlations are notably higher than other\nsites. In this section we try to develop a theoretical framework\nto capture this feature. Suppose we have two slow defects located\nin the bulk at sites $k$ and $l$ ($k<l$) respectively both with rate\n$q$. The mean occupation at\nsite $i$ in the steady state is denoted by $n_i=\\langle \\tau_i\n\\rangle$, in which $\\tau_i=0,1$ is the occupation number at site\n$i$. We assume that a simple MF assumption, i.e., $\\langle\n\\tau_i\\tau_{i+1}\\rangle =\\langle \\tau_i \\rangle \\langle\n\\tau_{i+1}\\rangle =n_in_{i+1}$ holds for all sites except\n$i=k,\\cdots,l$ i.e.; defective sites themselves and all the sites between them.\nAt these sites the correlations are strong enough to violate the simple\nmean-field assumption. Let us introduce two-point correlation\nfunctions $m_i$ in the following way,\n\n\\begin{eqnarray}\nm_i=\\langle \\tau_{i}\\tau_{i+1}\\rangle.\n\\end{eqnarray}\n\nThere are $L+l-k+2$ unknowns, namely, $\nn_1,n_2,\\cdots,n_L,m_{k},m_{k+1},\\cdots,m_{l}$ and lastly the\ncurrent $J$. In the stationary state there exists $L+1$ equations among these unknowns.\nLet us label them by $A_0$ to $A_{L}$. These\nequations can be obtained by expressing the current $J$ in terms\nof the mean site densities and two point correlators. The first\nequation, $A_0$, is $J=\\alpha(1-n_1)$. The equations $A_i\n$ for $~i=1,\\cdots,k-1$ and $i=l+1,\\cdots,L-1$ have the following form,\n\n\\begin{eqnarray}\nJ= n_i(1-n_{i+1}).\n\\end{eqnarray}\n\nEquation $A_k$ is:\n\\begin{eqnarray}\nJ=q \\langle \\tau_k(1-\\tau_{k+1}) \\rangle =q(n_k-m_k).\n\\end{eqnarray}\nSimilarly, equation $A_l$ is given below\n\\begin{eqnarray}\nJ=q \\langle \\tau_{l}(1-\\tau_{l+1}) \\rangle =q(n_l-m_l).\n\\end{eqnarray}\n\nEquations $A_i~ (i=k+1,\\cdots,l-1) $ have the following form\n\n\\begin{eqnarray}\nJ= \\langle \\tau_{i}(1-\\tau_{i+1}) \\rangle =(n_i-m_{i+1}).\n\\end{eqnarray}\n\nlastly equation $A_L$ is as follows,\n\n\\begin{eqnarray}\nJ=\\beta \\langle \\tau_L\\rangle=\\beta n_L.\n\\end{eqnarray}\n\nWe do not intend to add more equations. Unfortunately the above\nequations are nonlinear and it would be a formidable task to solve\nthem analytically. Alternatively, we shall utilize a numerical\napproach to solve the system of equations by exploiting their recursive structure.\nThis approach was originally introduced in \\cite{foulaadvand07} in the context of\ndisordered ASEP and was later applied to the problem of two\nintersecting ASEP chains \\cite{foulad07}. According to this\nnumerical scheme, we assign a value to $J$. Having $J$, it is possible\nto iterate equations (in forward direction) and obtain $n_1$ up to $n_k$. Then we\nproceed to find $m_k$ by incorporating the relation $J=q(n_k\n-m_k)$. At this stage it is not possible to proceed further because both\n$n_{k+1}$ and $m_{k+1}$ are unknown and we have only one relation\nbetween them : $J=n_{k+1} -m_{k+1}$. In order to proceed, we\napproximate $n_{k+1}$ in the following way. Consider the rate equation for\nthe 2-point function $\\langle \\tau_{k} \\tau_{k+1} \\rangle$ which is governed by\nthe following master equation:\n\n\\begin{eqnarray}\n\\frac{d\\langle \\tau_{k} \\tau_{k+1} \\rangle }{dt}= \\langle\n\\tau_{k-1}(1-\\tau_k)\\tau_{k+1}\\rangle - \\langle\n\\tau_{k}\\tau_{k+1}(1-\\tau_{k+2})\\rangle.\n\\end{eqnarray}\n\nIn the steady state, the left hand side becomes zero, and\ntherefore two terms on the right hand side will be equal. To\nproceed further we have to approximate 3-point functions. This is\nachieved by utilizing the cluster mean-field assumption \\cite{chowdhury02}.\nAccording to this assumption we replace any three point function\nby the product of 2-point functions as follows,\n\n\\begin{eqnarray}\n\\langle n_in_jn_k\\rangle=\\frac{\\langle n_in_j\\rangle\\langle\nn_jn_k\\rangle}{\\langle n_j\\rangle}.\n\\end{eqnarray}\n\nWe then replace all 2-point functions by the product of 1-point\nfunctions except $m_k=\\langle \\tau_{k}\\tau_{k+1} \\rangle$.\nThen it is possible to express $1-n_{k+2}$ in\nterms of $n_{k-1},m_1$ and $n_{k+1}$. After some algebra a quadratic\nequation for $n_{k+1}$ is obtained:\n\n\\begin{eqnarray} n_{k-1}n_{k+1}^2 - m_kn_{k-1}n_{k+1}\n-m_k J=0.\n\\end{eqnarray}\n\nThe physically reasonable solution for this equation is given by\n\n\\begin{eqnarray}\nn_{k+1}= \\frac{ m_k  + \\sqrt{ m_k^2 + \\frac{4m_kJ}{n_{k-1}} }\n}{2}.\n\\end{eqnarray}\n\nNow it is possible to find $m_{k+1}$ via equation $J=n_{k+1} -m_{k+1}$. Analogous to the above\nprocedure we can find $n_{k+2}$ as follows:\n\n\\begin{eqnarray}\nn_{k+2}= \\frac{ qm_km_{k+1}  + \\sqrt{ (qm_km_{k+1})^2 + 4qJn_kn_{k+1}^2m_{k+1} }\n}{2qn_kn_{k+1}}.\n\\end{eqnarray}\n\n\nAfter having $n_{k+2}$ we simply obtain $m_{k+2}$ via equation $J=n_{k+2} -m_{k+2}$. Now it would be possible\nto proceed iteratively after taking into account some approximation. To this end we recall the equality\n\n\\begin{eqnarray}\n\\frac{d\\langle \\tau_{i} \\tau_{i+1} \\rangle }{dt}= \\langle\n\\tau_{i-1}(1-\\tau_i)\\tau_{i+1}\\rangle - \\langle\n\\tau_{i}\\tau_{i+1}(1-\\tau_{i+2})\\rangle.\n\\end{eqnarray}\n\nPutting the left hand side equal to zero in the steady state, utilizing cluster mean-field in three point functions and finally\nsubstituting $m_{i+1}$ by $m_{i+1}=n_{i+1} -J$ we arrive at the following equation:\n\n\\begin{eqnarray}\nn_{i+1}= \\frac{ m_{i-1}m_{i}  + \\sqrt{ (m_{i-1}m_{i})^2 + 4Jn_{i-1}n_{i}^2m_{i} }\n}{2n_{i-1}n_{i}}.\n\\end{eqnarray}\n\nNote that we have approximated $\\langle \\tau_{i-1} \\tau_{i+1} \\rangle$ by the mean-field relation $\\langle \\tau_{i-1} \\rangle \\langle \\tau_{i+1} \\rangle$.\nWe can iterate equation (53) together with $m_{i+1}=n_{i+1}-J$ from $i=k+2$ to $l-2$ to find the corresponding $n_i$ and $m_i$ up to $i=l-1$.\nThe site $i=l$ needs to be treated separately. Following the same strategy we easily find:\n\n\\begin{eqnarray}\nn_{l}= \\frac{ m_{l-2}m_{l-1}  + \\sqrt{ (m_{l-2}m_{l-1})^2 + 4Jn_{l-2}n_{l-1}^2m_{l-1} }\n}{2n_{l-2}n_{l-1}}.\n\\end{eqnarray}\n\n\nFrom which one can compute $m_l$. In a similar fashion, we can obtain $n_{l+1}$. We only should\nshift up all the subscripts in equation (54) by one. Now it is possible again to proceed iteratively to the end of the chain and\nevaluate $n_L$ which gives us the output current $J^{out}$. If the given value of input $J$ were correct,\nthe output current $J^{out}$, which is $\\beta n_L$, should be the same, up to a given precision, as the input value of $J$.\nBy systematically increasing the input $J$ in an small amount $\\delta J$, we can determine the correct $J$ and\ncorrespondingly the mean densities $n_1,\\cdots,n_L$ together with correlators $m_k,\\cdots,m_l$.\nIn Fig. (10) the dependence $J$ on $q$ obtained by the numeric scheme devised above\nis sketched. For the sake of comparison, we have augmented the\nfigure with the analogous graphs obtained by MF, DMF, ISA and MC\nmethods.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=7cm]{fig5-colour.eps}\n\\caption{ Fig.10: (Color online) Current vs q for $d=2$ obtained by various analytical and\nnumerical methods and MC simulation. $J$ approaches to 0.25 when $q$ tends to one in accordance to normal ASEP.\n} \\label{fig:bz2}\n\\end{figure}\n\n\nThe result of the numerical scheme is almost identical to ISA\nmethod. They both are in very good agreement with Monte Carlo\nsimulations. However, the numerical scheme has an advantage over\nISA method in the sense that it can easily be implemented for any\n$d$, whereas finding the solution of the ISA nonlinear set of\nequations, i.e., Eq. (29) is not an easy task even by employing\nadvanced numerical methods. Finally in figure (11) we have sketched the dependence of $J$ on $d$ obtained\nfrom MC simulation and the numeric scheme.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=7cm]{fig7-colour.eps}\n\\caption{ Fig.11: (Color online) Current vs $d$ for $q=0.1,0.3$ and $0.5$ obtained by the\nnumeric method and MC simulation. } \\label{fig:bz2}\n\\end{figure}\n\nThe results of the numeric method are in rather good agreement\nwith those obtained by MC simulations. $J$ is an increasing\nfunction of $d$ and becomes saturated after some short\n$q$-dependent distance. The results confirms the earlier finding\nin \\cite{dong07}. Note that the length scale on which $J$\nrecovers its single-defect value is of the same order of\nmagnitude of the correlation length in the density profile near\nboundaries. Finally we would like to add that our numerical scheme\nis not capable of reproducing the profile of correlators obtained\nvia MC simulations. Figure (12) depicts the profile of\nunnormalised adjacent two-point correlation function $\\langle\n\\tau_{i+1}\\tau_i \\rangle - \\langle \\tau_{i+1} \\rangle \\langle \\tau_i\n\\rangle$ for two methods of simulation and numerical scheme.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=7cm]{fig16-colour.eps}\n\\caption{ Fig.12: (Color online) Profile of corrrelators  for $d=100$ and $q=0.3$. } \\label{fig:bz2}\n\\end{figure}\n\nWe see that within the numerical framework both the value and the extension of correlators are small in comparison to\nthe simulation results. The reason lies in the implementation of a series of approximation in several places in this\nnumerical algorithm.\n\n\n\\section{Summary and Conclusions}\n\nAn open ASEP chain with two defective sites with reduced hopping rates $q<1$ has been investigated. The system current and mean site densities\nat defective sites and their vicinities have been obtained by two analytical methods namely defect mean-field (DMF) and interacting\nsubsystem approximation (ISA). Both methods combine mean-field approach near defects with known exact solutions. Our results are accomplished\nby extensive Monte Carlo simulations. We focus on the phase-segregated phase in which defects globally affect the system properties and\nthe system is not input\/output rate-limited but rather defect-limited. MC simulations have revealed strong short ranged correlations at the\ndefective sites and at all the site between them. Additionally, the profile of density between defects takes a linear form which marks the\nexistence wandering shock in this intermediate region. In order to take into account these correlations, we have introduced a numerical approach\nwhich utilizes a cluster mean-field assumption. Comparison of the three methods show that ISA and the numeric approach give a current value which\nis in a good agreement with MC simulations. DMF method however, only gives a good results compared to MC and other two methods for small $q$ less\nthan $0.1$ which is due to strong correlations. Furthermore, we have obtained the profile of neighbouring two-point correlation function throughout\nthe chain. It is shown that these two point correlators exhibit a rather strong anti correlation at the first defective site then they grow to the\nmiddle of the defects and after that they start diminishing. In general, two point correlators will tend to a tiny value when $q$ approaches one.\nHowever, the dependence of two point correlators for a fixed $q$ as a function of distance between defects is not monotonous. Despite reaching\nto an asymptotic value for large distances, one observes a peak at short distances for the two point correlators near defects. Our theoretical\nanalysis indicates that correlations are critically important for dynamics of particles in disordered ASEPs. It also shows that it is possible\nto devise an approximate method that can take into account these correlations, providing a satisfactory description of stationary properties.\n\n\\section*{Acknowledgments}\n\nABK acknowledges the support from the Welch Foundation (under Grant\nNo. C-1559), and from the US National Science Foundation (grants\nCHE-0237105 and NIRT ECCS-0708765). MEF expresses his gratitude to\nM. F. miri for their useful help.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Solutions}%\n}\n\n\n\n\\section{Introduction} \nThe increasing demand for wireless services and higher data volume per area has brought to the research forefront of both the scientific community (e.g., the METIS project~\\cite{METIS}) and the standardization bodies  a new technology known as massive multiple-input-multiple-output (MIMO) networks~\\cite{Marzetta2010,Rusek2013}. The keystone of this concept, currently considered for the design of  next generation wireless networks~\\cite{5g,Larsson2014,Osseiran2014}, is based on that the numbers of base station (BS) antennas and users are scaled up by 1-2 orders of magnitude under the same time-frequency resources.  This scientific area is motivated by the resulting gains stemming from large channel matrices, i.e., the arising advantages due to the asymptotics of random matrix theory~\\cite{Couillet2011}. Specifically, by  allowing the number of BS antennas tend to infinity,  small-scale fading and thermal noise average out, as well as the analysis is simplified. \n\n\nNotably, the channel state information (CSI)  is  pivotal to the performance of  such systems that take into account for multi-user MIMO (MU-MIMO)~\\cite{Gesbert2007}. Thus, recognizing the need for a realistic study with imperfect CSI, we focus on an important factor-phenomenon, being present in time-varying channels, that deteriorates the quality of CSI available at the BS in \ntime-division-duplex (TDD) systems. Our consideration includes TDD systems and not frequency-division-duplex (FDD) systems, since the  application of the latter in massive MIMO  systems is meaningful only when  \nthe channel matrices are sparse~\\cite{Bjoernson2015}\\footnote{The hypothesis of sparsity is still an open question requiring further research and channel measurements.}. This phenomenon, known as {\\em  channel aging}, describes the mismatch appearing between the current  and the estimated channel due to the relative movement between the users and the BS antennas in addition to any processing delays. In other words, it models the divergence occurring between the channel estimate used in the precoder\/detector, and the channel over which the data transmission actually  takes place. \n\nAlthough the significance of  channel aging due to user mobility, its effect on the performance of massive MIMO systems has not been addressed adequately as  few studies  on this area exist. In fact, only the user mobility has been considered as the only channel aging cause~\\cite{Truong2013,Papazafeiropoulos2014,Papazafeiropoulos2014WCNC,Papazafeiropoulos2015a,Papazafeiropoulos2015,PapazafeiropoulosPIMRC, Kong2015,Kong2015a}. Basically, channel aging  due to only user mobility was the focal point in~\\cite{Truong2013}, where an application of the deterministic equivalent (DE) analysis was presented by considering linear techniques in the uplink and downlink in terms of maximum ratio combining (MRC) detector and  maximum ratio transmission (MRT) precoder, respectively. Further investigations have been performed in~\\cite{Papazafeiropoulos2015a}, where more sophisticated linear techniques are employed, i.e., minimum-mean-square-error (MMSE) receivers (uplink) and regularized zero-forcing precoders (downlink) are considered, and a comparison between the various linear transceiver techniques is performed. Moreover, in~\\cite{Papazafeiropoulos2015}, the optimal linear receiver in the case of cellular massive MIMO  systems has been derived  by exploiting the correlation between the channel estimates and the interference from other cells, while in~\\cite{PapazafeiropoulosPIMRC}, the uplink analysis of a cellular network with zero-forcing (ZF) receivers that holds for any finite as well as infinite number of BS antennas has been provided.\n\nIn general, hardware undergoes several types of impairments such as high power amplifier nonlinearities and I\/Q imbalance~\\cite{Schenk2008,Studer2010,Qi2012,Mehrpouyan2012,Goransson2008,Bjornson2012Optimal,Qi2010,Bjoernson2013,Zhang2015,Pitarokoilis2015,Bjornson2015,Krishnan2015,Bjornson2014,Demir2000,Bittner2007,Wu2004,Petrovic2007,Krishnan2014,Pitarokoilis2013}\n\\footnote{The arising hardware mismatch between the uplink and the downlink has a negative effect on the  channels reciprocity. For example, a recent insightful work (see \\cite{Zhang2015} and  references therein) has presented the performance analysis in the large number of antennas regime as well as pre-precoding and post-precoding calibration schemes. Herein, focusing on the fundamentals of channel aging, where we want to quantify the individual contributory time-dependent factors (Doppler shift and phase noise), we neglect any hardware mismatch. However, future work can\ninvestigate this mismatch in the channel aging.}.  These hardware impairments can be classified into two categories. In the first category, the variable, describing the hardware impairments, is multiplied with the channel vector, and it might cause channel attenuations and phase shifts. In the case of slow variation of these impairments, they can be characterized as sufficiently static, and thus, can be assimilated by the channel vector by an appropriate scaling of its covariance matrix or due to the property of circular symmetry of the channel distribution.      Note that this factor cannot be incorporated by the channel vector by an appropriate scaling of its covariance matrix or due to the property of circular symmetry of the channel distribution, when it changes faster than the channel. Focusing on this scenario, we include in our analysis the phase noise in the case which results to its accumulation within the channel coherence period~~\\cite{Mehrpouyan2012,Bjornson2015,Pitarokoilis2015,Krishnan2015}. Given that additive transceiver impairments are not time-dependent, we have not included them in our analysis~\\cite{Bjoernson2013,Bjornson2014,Bjornson2015}.\n\nHence, deeper consideration reveals that  user mobility is not the only source of channel aging. Interestingly, {the unavoidable \\em phase noise}, being one of the most severe transceiver impairments~\\cite{Demir2000}, contributes to the further degradation of the system performance due to channel aging because it accumulates over time. Ideally, local oscillators (LOs), used during the conversion of the baseband signal to bandpass and vice-versa, should output a sinusoid with stable amplitude, frequency, and phase. Unfortunately, the presence of phase noise  is inevitable due to the inherent imperfections in the circuitry of  LOs that bring up to the sinusoid a random phase drift.  A lot of research has been conducted regarding phase noise in conventional MIMO systems.  In fact, a big part of the literature has dealt with multi-carrier MIMO systems that employ orthogonal frequency-division multiplexing (OFDM)~\\cite{Bittner2007,Wu2004,Petrovic2007,Krishnan2014}. For example, in~\\cite{Wu2004}, the signal-to-interference-and-noise-ratio (SINR) degradation in OFDM is studied, and a method to mitigate the effect of phase noise is proposed. In~\\cite{Mehrpouyan2012}, a method to jointly estimate the channel coefficients and the phase noise in a point-to-point MIMO system is presented. Not to mention that the degradation appears to be more significant in coherent communications such as massive MIMO  systems.  Specifically, although massive MIMO technology is attractive (cost-efficient) for network deployment, only if the antenna elements consist of inexpensive hardware (more imperfections), most of the research contributions are based on the strong assumption of perfect hardware without phase noise, which is quite idealistic in practice. Reasonably, the introduction of a larger number of antennas will have a more severe contribution to the degradation provoked by phase noise, which could be partially avoided in conventional MIMO  systems, where more expensive hardware with less induced phase is affordable. Hence, the  larger the number of antennas and the corresponding employed LOs, the larger the impact of phase noise. Thus, in the case of massive MIMO  systems, it is a dire necessity to study the impact of channel aging induced by phase noise. Furthermore, motivated by the need of employing low-quality RF elements (e.g., oscillators) for cost-effective construction of large BS antenna arrays, it is  expected that the phase noise will be more severe in a massive MIMO setting.  In particular, the large number of antennas regime has been partially     investigated~\\cite{Pitarokoilis2013,Pitarokoilis2015,Bjornson2015,Krishnan2015}. Specifically, the phase noise has been taken into account for single carrier  uplink  massive MIMO by considering ZF and time-reversal MRC  in~\\cite{Pitarokoilis2013} and~\\cite{Pitarokoilis2015}, respectively. \nRegarding the downlink,  an effort to address the effect of phase noise, being of great importance in massive MIMO systems,  has taken place only in~\\cite{Krishnan2015}. However, the analysis therein does not account for fast time-varying conditions due to the mobility of the users.\n\nIn this paper, we provide a novel realistic CSI model that allows studying channel aging  thoroughly by means of the simultaneous impact of    user mobility and phase noise at the LOs. Taking into account that   next generation wireless systems need to be cost-efficient, implying that real systems may include low-quality RF elements with more imperfections, makes the significance of the proposed model in describing the ineluctable time-varying nature of realistic scenarios remarkable.  Under these conditions, we analyze the downlink performance of a massive MIMO system with imperfect CSI using linear precoders (MRT and RZF). To this end, we invoke random matrix theory (RMT) tools, and we derive the DEs (asymptotically tight approximations as system's dimensions increase). Actually, these approximations agree with simulation results as has been  already shown in the literature~\\cite{Couillet2011,Truong2013,Papazafeiropoulos2015a}. As a result, these deterministic approximations make any lengthy simulations\nunnecessary. Furthermore, as far as phase noise is concerned, we address two system arrangements: a) all BS antennas are connected to a common LO (CLO), and b) each BS antenna includes its own separate LO (SLOs setup). The contributions of this paper can be summarized as follows:\n\\begin{itemize}\n \\item Given a single cell MU-MIMO realistic system model with path loss and shadowing, we present a joint channel-phase noise estimate describing the impact of channel aging by incorporating the time-varying effects of Doppler shift and phase noise at the BS and user elements (UEs).\n \\item Using tools of large RMT, we obtain the DEs of the downlink SINRs, when the BS applies MRT and RZF precoders under imperfect CSI due to uplink training and time variation of the channel (generalized channel aging). Moreover, we account for special cases of the channel covariance, corresponding to various interesting effects  described by the large-scale component, that simplify a lot the resulting expressions.\n \\item We demonstrate that the square root power-law, generally holding in imperfect CSI scenarios, stands for the case of generalized channel aging as well. Hence, the transmit power per user can be cut down by scaling the number of BS antennas to the inverse of their square root.\n \\item We corroborate our analysis concerning deterministic approximations by performing simulations coinciding with the analytical expressions.  \n \\item We elaborate on insightful conclusions, drawn from our analysis. For example, Doppler shift has a far more severe impact than phase noise even for low vehicle velocities. \nIn addition, massive MIMO  systems outperform conventional MIMO even in generalized channel aging conditions with RZF precoder behaving better than MRC for various Doppler shifts and phase noise severities.  \n \\end{itemize}\n\nThe remainder of this paper is organized as follows. Section II introduces the system  and signal models accounting for both the effects of user mobility and phase noise, which allow a more complete characterization of channel aging. Next, Section III presents the downlink transmission, in terms of the received signal, the applied precoders, and the achievable user rate.  In Section IV, we proceed with the main results that include the derivation of the DEs of the downlink SINRs, as well as the investigation of certain simpler cases such as the scenario where the large-scale fading component is absent. Section V reports a set of numerical results that confirms the validity of our analytical results, and sheds light on the behavior of channel aging for various mobility environments, phase noise settings, and a number of antennas in a massive MIMO system. We summarize our results and observations in Section VI. In Appendix A, some useful results from the literature are presented, while the following appendices provide the proofs for the main analytical results.\n\n\\textit{Notation:} Vectors and matrices are represented by boldface lower and upper case symbols. $(\\cdot)^{\\scriptscriptstyle\\mathsf{T}}$, $(\\cdot)^\\H$, and $\\mathop{\\mathrm{tr}}\\nolimits{\\cdot}$ denote the transpose, Hermitian  transpose, and trace operators, respectively. The expectation and variance operators, as well as the spectral norm of a matrix are denoted by $\\mathbb{E}\\left[\\cdot\\right]$ and $\\mathrm{var}\\left[\\cdot\\right]$, as well as $\\| \\cdot \\|$,\nrespectively, while $\\mathrm{Re}\\{\\cdot\\}$  is the real part of a complex number. The  $\\mathrm{diag}\\{\\cdot\\}$ operator generates a diagonal matrix from a given vector, and the symbol $\\triangleq$ declares definition. The notations $\\mathcal{C}^{M}$ and $\\mathcal{C}^{M\\times N}$ refer to complex $M$-dimensional vectors and  $M\\times N$ matrices, respectively. Finally, ${\\mathbf{b}} \\sim {\\cal C}{\\cal N}{({\\mathbf{0}},\\mathbf{\\Sigma})}$  or ${\\mathbf{b}} \\sim {\\cal N}{({\\mathbf{0}},\\mathbf{\\Sigma})}$ denote a circularly symmetric complex Gaussian  or a real Gaussian vector ${\\mathbf{b}}$ with zero-mean and covariance matrix $\\mathbf{\\Sigma}$.\n\n\n\n\n\n\n\n\\section{System Model}\nThis work considers a single-cell system, where a BS communicates with a number of single-antenna non-cooperative UEs belonging to the set $\\mathcal{K}$ with cardinality $K=|\\mathcal{K}|$ being the number of UEs. In particular,  each UE is assigned an index $k$ in $\\mathcal{K}$, while the BS is deployed with an array of $M$ antennas\\footnote{The antennas can be co-located at the BS or can be distributed as fully-coordinated, multiple, smaller BSs~\\cite{Bjornson2015}.}. Since each UE has a single antenna, the terms user and user element, are used interchangeably in the following analysis. Our interest focuses on large-scale topologies, where both number of antennas $M$ and users $K$ grow infinitely large while keeping a finite ratio $\\beta$, i.e., $M, K\\to \\infty$ with $K\/M=\\beta$ fixed such that $\\mathrm{lim\\,sup}_{M} K\/M < \\infty$\\footnote{ In the following, in order to simplify notation, $M\\to \\infty$ will be used instead of $M, K\\to \\infty$, unless  stated otherwise.}.\n\nDuring the transmission  of the $n$th symbol, the channel gain vector between the BS and user $k$, denoted by ${\\mathbf{h}}_{k,n}  \\in {\\mathbb{C}}^{M}$ exhibits flat-fading, and it is assumed constant for the symbol period, while it may vary slowly from symbol to symbol. Basically, given the time-frequency resources of the system, the symbol duration is assumed smaller or equal to the coherence time of all the users. Moreover, in order to account for certain inevitable effects such as path loss and lognormal shadowing, which are dependent on the distance of the users from the BS,  we express the channel between user $k$ and the BS as\n\\begin{align}\n {\\mathbf{h}}_{k} = {\\mathbf{R}}^{1\/2}_{k}{\\mathbf{w}}_{k},\n\\end{align}\nwhere ${\\mathbf{R}}_{k} =\\mathbb{E}\\left[ {\\mathbf{h}}_{k,n} {\\mathbf{h}}_{k,n}^{\\H}\\right]\\in {\\mathbb{C}}^{M \\times M}$ is a deterministic Hermitian-symmetric positive-definite matrix representing the aforementioned effects\\footnote{The independence of ${\\mathbf{R}}_{k}$ from the time index, $n$, originates from the assumption that the  shadowing vary with time in a longer slower pace  than the coherence time~\\cite{Truong2013,Papazafeiropoulos2014,Papazafeiropoulos2014WCNC}. Interestingly, this model is quite versatile, since  ${\\mathbf{R}}_{k}$ can also describe the antenna correlation due to either insufficient antenna spacing or a lack of scattering.However, the following results focus only on diagonal ${\\mathbf{R}}_{k}$ matrices describing only large-scale fading, and ${\\mathbf{w}}_{k,n} \\in {\\mathbb{C}}^{M}$ is an uncorrelated fast-fading Gaussian channel vector drawn as  a realization from a zero-mean circularly symmetric complex Gaussian distribution, i.e., ${\\mathbf{w}}_{k,n} \\sim {\\cal C}{\\cal N}({\\mathbf{0}},{\\mathbf{I}}_{M})$.}. \n\n\\subsection{Phase Noise Model}\nIn practice, both  the transmitter and the receiver are impaired by phase noise induced during the up-conversion of the baseband signal to passband and vice-versa\\footnote{The conversion takes place by multiplying the signal with the LO's output.}. As a result, the received signal is distorted during the reception processing, while at the transmitter side, a mismatch appears between the signal that  is intended to be transmitted and the generated signal.\n\nIn our analysis regarding the $n$th time slot, we take into account for non-synchronous operation at the BS, i.e., the BS antennas have independent phase noise processes $\\phi_{m,n}, m=1,\\ldots,M$ with $\\phi_{m,n}$ being the phase noise process at the $m$th antenna. Note that the  phase noise processes are considered as mutually independent, since each antenna has its own oscillator, i.e., separate LOs (SLOs) at each antenna~\\cite{Pitarokoilis2015,Krishnan2015}.  In case that all the antennas have a common LO (CLO), the phase noise processes $\\phi_{m,n}$ are identical for all $m = 1, \\ldots, M$ (synchronous operation). Similarly, we denote $\\varphi_{k,n}$, $k=1,\\ldots, K$ the phase noise process at the single-antenna user $k$. \n\n\nPhase noise during the $n$th symbol can be described by a discrete-time  independent Wiener process, i.e., the phase noises at the  LOs of the $m$th antenna of the BS and $k$th user are modeled as\n\\begin{align}\n \\phi_{m,n}&=\\phi_{m,n-1}+\\delta^{\\phi_{m}}_{n}\\label{phaseNoiseBS}\\\\\n \\varphi_{k,n}&=\\varphi_{k,n-1}+\\delta^{\\varphi_{k}}_{n},\\label{phaseNoiseuser}\n\\end{align}\nwhere $\\delta^{\\phi_{m}}_{n}\\sim {\\cal N}(0,\\sigma_{\\phi_{m}}^{2}) $ and $\\delta^{\\varphi_{k}}_{n}\\sim {\\cal N}(0,\\sigma_{\\varphi_{k}}^{2})$~\\cite{Demir2000,Pitarokoilis2015,Krishnan2015}. Note that $\\sigma_{i}^{2}=4\\pi^{2}f_{\\mathrm{c}} c_{i}T_{\\mathrm{s}}$, $i=\\phi_{m}, \\varphi_{k}$ describes the phase noise increment variance with $T_{\\mathrm{s}}$, $c_{{i}}$, and $f_{\\mathrm{c}}$ being the  symbol interval, a constant dependent on the oscillator, and the carrier frequency, respectively. \n\nThe phase noise, being the distortion in the phase due to the random phase drift  in the signal coming from the LOs of the BS and user $k$,  is expressed as a multiplicative factor to the channel vector as \n\\begin{align}\n{\\mathbf{g}}_{k,n}={\\boldsymbol{\\Theta}}_{k,n}{\\mathbf{h}}_{k,n},\n\\end{align}\nwhere ${\\boldsymbol{\\Theta}}_{k,n}\\!\\triangleq\\!\\mathrm{diag}\\!\\left\\{ e^{j \\theta_{k,n}^{(1)}}, \\ldots, e^{j \\theta_{k,n}^{(M)}} \\!\\right\\}=e^{j \\varphi_{k,n}}{\\boldsymbol{\\Phi}}_{n}\\in \\mathbb{C}^{M\\times M}$ with ${\\boldsymbol{\\Phi}}_{n}\\!\\triangleq\\!\\mathrm{diag}\\!\\left\\{\\! e^{j \\phi_{1,n}}, \\ldots, e^{j \\phi_{M,n}} \\!\\right\\}$ being the phase noise sample matrix  at time $n$ because of the imperfections in the LOs of the BS,  while $e^{j \\varphi_{k,n}}$ corresponds to the phase noise contribution by the LO of user $k$ \\cite{Pitarokoilis2015,Krishnan2015}.\n\\subsection{Channel Estimation}\nIn realistic conditions,  where no perfect CSI is available, the BS needs to estimate the uplink channel~\\cite{Papazafeiropoulos2015a,Ngo2013,Hoydis2013}. We take advantage of the TDD operation scheme, where  an uplink training phase during each transmission block (coherence period) exists by means of transmit pilot symbols. Since in TDD the number of required pilots scales with the number of terminals $K$, but not the number of BS antennas $M$, not only the analysis of systems with large number of BS antennas is  possible because of the finite duration of the coherence time\\footnote{TDD is the only viable solution in fast time-varying channel conditions with massive MIMO systems, where the limitations for the coherence time are more stringent, while FDD can be considered only when there is some kind of channel sparsity~\\cite{Bjoernson2015}.}, but also the knowledge of the downlink channel is considered known due to the property of reciprocity.   \n\n\nBy assuming that the channel estimation takes place at time $0$, we now derive the joint channel-phase noise linear minimum mean-square\nerror detector  (LMMSE) estimate of the effective channel ${\\mathbf{g}}_{k,0}={\\boldsymbol{\\Theta}}_{k,0}{\\mathbf{h}}_{k,0}$ in the presence of phase noise, small-scale fading, and channel impairments such as path loss. During this phase, we neglect the channel aging effect by assuming that both the channel and phase noise remain constant~\\cite{Truong2014}. This is a valid assumption since the duration of the training phase is small, and the consequent time-variation of the channel is unnoticeable.  However, in the data transmission phase, taking place for ($T_{\\mathrm{c}} -\\tau$ ) symbols, the channel is supposed to vary from symbol to symbol, where $\\tau$ and $T_{\\mathrm{c}}$ are  the duration of the  training sequence during the training phase and   the channel coherence time, respectively.\n\\begin{proposition}\\label{LMMSE} \nThe LMMSE estimator of ${\\mathbf{g}}_{k,0}$, obtained during the training phase, is\n \\begin{align}\\label{estimatedChannel}\n \\hat{{\\mathbf{g}}}_{k,0}=\\left(  \\mat{\\mathrm{I}}_{M}+\\frac{\\sigma_{\\mathrm{b}}^{2}}{p_{{\\mathrm{p}}}}{\\mathbf{R}}_{k}^{-1}\\right)^{-1}\\tilde{{\\mathbf{y}}}_{k,0}^{{\\mathrm{p}}},\n\\end{align}\nwhere $\\sigma_{\\mathrm{b}}^{2}$ is the variance of the post-processed noise at\nbase station, $\\tilde{{\\mathbf{y}}}_{,k,0}^{{\\mathrm{p}}}$ is a noisy observation of the effective channel from user $k$ to the BS, and $p_{\\mathrm{p}}=\\tau p_{\\mathrm{u}}$  with $p_{\\mathrm{u}}$   being the power per user in the uplink data transmission phase.\n\\end{proposition}\n\\proof: The proof of Proposition~\\ref{LMMSE} is given in Appendix~\\ref{proposition1}.\\endproof\n\n\\subsection{Channel Aging}\nThe relative movement between the users and the BS inflicts a phenomenon, known as channel aging in the literature~\\cite{Truong2013,Papazafeiropoulos2014,Papazafeiropoulos2014WCNC}. Specifically, the  Doppler shift arising from this movement causes the channel to vary between when it is learned by estimation, and when the estimate is applied for detection\/precoding. Especially, channel aging is more severe in massive MIMO systems, where  narrow transmit beams appear due to the high angular resolution. Thus, the precoders that are used for achieving this spatial focus, bring a critical performance loss, which worsens as the users move with increasing velocity~\\cite{Ying2014}. An autoregressive model of order $1$ enables  the  joint channel-phase noise process ${\\mathbf{g}}_{k,n}$ between the BS and the $k$th user at time $n$ (current time) to be approximated as\\footnote{Contrary to other works~\\cite{Vu2007,Truong2013,Papazafeiropoulos2014,Papazafeiropoulos2014WCNC}, ${\\mathbf{A}}_{n}$, being the factor of ${\\mathbf{g}}_{k,0}$,   is described by a matrix  instead of a scalar to account for the setting of SLOs. Moreover, in this work, ${\\mathbf{A}}_{n}$ is time-dependent by the  delay  $n$ in terms of the number of symbols transmitted, in order to include the accumulative channel aging effect~\\cite{Kong2015a}.  Therefore, the longer the delay, the smaller the ${\\mathbf{A}}_{n}$, which in turn results in less accurate CSI. Hence, the channel estimated at time $0$, is used at time $1$, $2$, and so on till $n$. It is intuitive that the CSI will be much less reliable at time $n$, which results to a more realistic model. We assume that\nit can be embodied in the channel vector ${\\mathbf{g}}_{k,0}$ by employing the circular symmetry of the channel distribution, or mainly, by the inclusion of an appropriate scaling of the covariance matrix.}\n\\begin{align}\n{\\mathbf{g}}_{k,n}  = {\\mathbf{A}}_{n} {\\mathbf{g}}_{k,0} + {\\mathbf{e}}_{k,n},\\label{eq:GaussMarkoModel}\n\\end{align}\nwhere, initially, ${\\mathbf{A}}_{n}=\\mathrm{J}_{0}(2 \\pi f_{\\mathrm{D}}T_{\\mathrm{s}}n) \\mat{\\mathrm{I}}_{M}$ is supposed to model 2-D isotropic scattering~\\cite{WCJr1974}\\footnote{Normally, the BS antennas are deployed in a fixed space instead of considering  distantly distributed small BSs constituting one massive MIMO BS. In such case, ${\\mathbf{A}}_{n}$ is a scalar, since it can be  reasonably assumed that all the BS antennas have the same relative movement comparing to the user.}, ${\\mathbf{g}}_{k,0}$ is the channel vector received during the training phase, and ${\\mathbf{e}}_{k,n} \\in {\\mathbb{C}}^{M}$ is the uncorrelated channel error vector due to the channel variation modelled as a stationary Gaussian random process with independent and identically distributed (i.i.d.)~entries and distribution ${\\cal C}{\\cal N}({\\mathbf{0}},{\\mathbf{R}}_{k}-{\\mathbf{A}}_{n}{\\mathbf{R}}_{k}{\\mathbf{A}}_{n})$. \n\nThis model can be enriched by incorporating other interesting effects. By introducing the effect of channel estimation to the autoregressive model of~\\eqref{eq:GaussMarkoModel}, the BS  takes into account for both estimated and delayed CSI, in order to   design the detector ${\\mathbf{W}}_{n}\\in \\mathbb{C}^{M\\times M}$ or the precoder ${\\mathbf{F}}_{n}\\in \\mathbb{C}^{M\\times M}$ at time instance $n$ for the uplink and downlink, respectively. Hence, the effective channel at time $n$ is expressed by\n\\begin{align}\n{\\mathbf{g}}_{k,n} \n= & {\\mathbf{A}}_{n} {\\mathbf{g}}_{k,0} + {\\mathbf{e}}_{k,n}\\\\\n= & {\\mathbf{A}}_{n}\\hat{{\\mathbf{g}}}_{k,0} + \\tilde{{\\mathbf{e}}}_{k,n},\\label{eq:GaussMarkov2}\\end{align} \nwhere $\\hat{{\\mathbf{g}}}_{k,0}\\sim {\\cal C}{\\cal N}\\left({\\mathbf{0}},{\\mathbf{D}}_{k}\\right)$ with ${\\mathbf{D}}_{k}=\\left(  \\mat{\\mathrm{I}}_{M}+\\frac{\\sigma_{b}^{2}}{p_{{\\mathrm{p}}}}{\\mathbf{R}}_{k}^{-1}\\right)^{-1}{\\mathbf{R}}_{k}$ and  $\\tilde{{\\mathbf{e}}}_{k,n}\\triangleq {\\mathbf{A}}_{n} \\tilde{{\\mathbf{g}}}_{k,n} +  {\\mathbf{e}}_{k,n}\\sim {\\cal C}{\\cal N}({\\mathbf{0}}, {\\mathbf{R}}_{k} - {\\mathbf{A}}_{n}{\\mathbf{D}}_{k}{\\mathbf{A}}_{n})$  are mutually independent, while ${\\mathbf{A}}_{n}$, is assumed to be known to the BS, now incorporates both the effects of 2-D isotropic scattering, user mobility,  and phase noise, as shown below by Theorem~\\ref{LMMSE}. In other words, the combined error $\\tilde{{\\mathbf{e}}}_{k,n}$ depends on both the imperfect and the general delayed CSI effects, allowing the elicitation of interesting outcomes during the ensuing analysis in Section~\\ref{Deterministic}.  \n  \nWe turn our attention to the derivation of ${\\mathbf{A}}_{n}$ by accounting for user mobility, phase noise, and the general channel effects modelled by  ${\\mathbf{R}}_{k}$.\n\\begin{Theorem}\\label{LMMSE}\n The  effective joint channel-phase noise  vector at time $n$  is expressed as\n \\begin{align}\n  {{\\mathbf{g}}}_{k,n}^{\\mathrm{Joint}}={\\mathbf{A}}_{n} {{\\mathbf{g}}}_{k,0}\n \\end{align}\nwith \n\\begin{align}\n{\\mathbf{A}}_{n}=\\mathrm{J}_{0}(2 \\pi f_{\\mathrm{D}}T_{\\mathrm{s}}n)e^{-\\frac{\\sigma_{\\varphi_{k}}^{2}}{2}n}\\Delta{\\boldsymbol{\\Phi}}_{n}, \\label{agingParameter}\n\\end{align}\nwhere $\\Delta{\\boldsymbol{\\Phi}}_{n}=\\mathrm{diag}\\left\\{e^{-\\frac{\\sigma_{\\phi_{1}}^{2}}{2}n} ,\\ldots,e^{-\\frac{\\sigma_{\\phi_{M}}^{2}}{2}n}\\right\\}$.\n \\end{Theorem}\n\\proof\nThe total mean square error (MSE), which is actually  given by means of the error covariance matrix, is $\\mathrm{MSE}=\\mathop{\\mathrm{tr}}\\nolimits {{\\mathbf{E}}}_{k,n}$, where ${{\\mathbf{E}}}_{k,n}$ is the error covariance written as\n\\begin{align}\n&\\!\\!{{\\mathbf{E}}}_{k,n}=\\mathbb{E}\\left[ \\left( {\\mathbf{g}}_{k,n}-{\\mathbf{A}}_{n} {{\\mathbf{g}}}_{k,0} \\right) \\left( {\\mathbf{g}}_{k,n}-{\\mathbf{A}}_{n} {{\\mathbf{g}}}_{k,0} \\right)^{\\H}\\right]\\nonumber\\\\\n&\\!\\!=\\mathbb{E}| {\\mathbf{g}}_{k,n}|^{2}+{\\mathbf{A}}_{n} \\mathbb{E}|{{\\mathbf{g}}}_{k,0}|^{2}{\\mathbf{A}}_{n}^{\\H}-2\\mathbb{E}\\left[ \\mathrm{Re}\\left\\{{\\mathbf{A}}_{n}  {{\\mathbf{g}}}_{k,0}{\\mathbf{g}}_{k,n}^{\\H} \\right\\} \\right]\\nonumber\\\\\n&\\!\\!={\\mathbf{R}}_{k}+{\\mathbf{A}}_{n}{\\mathbf{R}}_{k}{\\mathbf{A}}_{n}^{\\H}-2\\mathbb{E}\\left[ \\mathrm{Re}\\left\\{{\\mathbf{A}}_{n}  {{\\mathbf{g}}}_{k,0}{\\mathbf{g}}_{k,n}^{\\H} \\right\\} \\right]\\nonumber\\\\\n&\\!\\!\\!\\!={\\mathbf{R}}_{k}\\!+\\!{\\mathbf{A}}_{n}{\\mathbf{R}}_{k}{\\mathbf{A}}_{n}^{\\H}\\!-\\!2 {\\mathrm{J}_{0}}(2 \\pi f_{\\mathrm{D}}T_{\\mathrm{s}}n)e^{-\\frac{\\sigma_{\\varphi_{k}}^{2}}{2}n}\\!{\\mathbf{A}}_{n} {\\mathbf{R}}_{k}\\Delta{\\boldsymbol{\\Phi}}_{n}.\\label{error_Cov_mat}\n\\end{align}\nNote that the expectation is taken over all channel and phase noise  realizations. Also,  we have used that ${\\mathbf{g}}_{k,n}={\\boldsymbol{\\Theta}}_{k,n}{\\mathbf{h}}_{k,n}$ with ${\\boldsymbol{\\Theta}}_{k,n}$ and ${\\mathbf{h}}_{k,n}$ being uncorrelated. Also, we have denoted $\\Delta{\\boldsymbol{\\Phi}}_{n}=\\mathrm{diag}\\left\\{e^{-\\frac{\\sigma_{\\phi_{1}}^{2}}{2}n} ,\\ldots,e^{-\\frac{\\sigma_{\\phi_{M}}^{2}}{2}n}\\right\\}$, and  \n\\begin{align}\n\\mathbb{E} [{{\\mathbf{h}}}_{k,0}{{\\mathbf{h}}}_{k,n}^{\\H}] = \\mathrm{J}_{0}(2 \\pi f_{D}T_{s}n){\\mathbf{R}}_{k}.\\label{eq:autoCorrelation2}\n\\end{align}\nThe multiplcative parameter matrix ${\\mathbf{A}}_{n}$ is determined by minimizing the MSE in~\\eqref{error_Cov_mat}. Thus, if we differentiate~\\eqref{error_Cov_mat} with respect to ${\\mathbf{A}}_{n}$, and equate the resulting expression to zero,  we obtain the Hermitian  matrix ${\\mathbf{A}}_{n}$ as in~\\eqref{agingParameter}.\n\\endproof\n\\begin{remark}\n During the data transmission phase, the available CSI at time $n$ is obtained in terms of the estimated CSI at time 0. The parameter matrix ${\\mathbf{A}}_{n}$  has a physical meaning, since it describes the effects of 2-D isotropic scattering, user mobility, and phase noise impairments at  both the UE and the BS. \n\\end{remark}\n\\begin{remark}\nNotably, phase noise induces an extra loss due to the additional deviation between the actual channel at time n, and the channel estimate from time 0 because of the symbol-by-symbol random phase drift. In other words, the phase noise contributes to the channel aging phenomenon that  imposes a further challenge to  investigate the realistic potentials of massive MIMO systems.\n\\end{remark}\n\n\n\\begin{corollary}\n In the case where  all the BS antennas are connected to the same oscillator or  all the oscillators are identical, ${\\mathbf{A}}_{n}$ degenerates to a scalar $\\alpha_{n}= \\mathrm{J}_{0}(2 \\pi f_{\\mathrm{D}}T_{\\mathrm{s}}n)e^{-\\frac{\\sigma_{\\varphi_{k}}^{2}+\\sigma_{\\phi_{k}}^{2}}{2}n}$ or a scaled identity matrix $\\alpha_{n}\\mat{\\mathrm{I}}_{M}$, respectively. \n\\end{corollary}\n\n\n\n\\section{Downlink Transmission}\nThe BS transmits data simultaneously to all users in its cell by employing space-division multiple access (SDMA). Applying reciprocity, the downlink channel is written as the Hermitian  transpose of the uplink channel. The coherence time,  the received signal $y_{k,n}\\in{\\mathbb{C}}$ by the $k$th UE during the data transmission phase ($n=\\tau+1,\\ldots, T_{\\mathrm{c}}$) is\n\\begin{align}\n\\!\\!y_{k,n}=\\sqrt{p_{\\mathrm{d}}}{\\mathbf{h}}^\\H_{k,n}{\\boldsymbol{\\Theta}}_{k,n}{\\mathbf{s}}_{n}+z_{k,n}\\label{eq:DLreceivedSignal}\n\\end{align}\nwhere  ${\\mathbf{s}}_{n}=\\sqrt{\\lambda}{\\mathbf{F}}_{n}{\\mathbf{x}}_{n}$ denotes the signal vector transmitted by the BS with $\\lambda$, ${\\mathbf{F}}_{n} \\in {\\mathbb{C}}^{M \\times K}$ and  ${\\mathbf{x}}_{n} = \\big[x_{1,n},~x_{2,n},\\cdots,~x_{K,n}\\big]^{\\scriptscriptstyle\\mathsf{T}} \\in {\\mathbb{C}}^{K}\\sim {\\cal C}{\\cal N}({\\mathbf{0}},{\\mathbf{I}}_{K})$ being a normalization parameter, the linear precoding matrix,  and the data symbol vector to its $K$ served users, respectively. Moreover, we have assumed that the BS transmits data to all the users with the same power $p_{\\mathrm{d}}$, and $z_{k,n} \\sim {\\cal C}{\\cal N}(0,\\sigma_{k}^{2})$ is complex Gaussian noise at user $k$.  The need for constraining the transmit power per user to $p_{\\mathrm{d}}$, i.e., $\\mathbb{E}\\left[\\frac{p_{\\mathrm{d}}}{K}{\\mathbf{s}}_{n}^{\\H}{\\mathbf{s}}_{n}\\right]=p_{\\mathrm{d}}$, provides the normalization parameter as\n\\begin{align}\n\\lambda=\\frac{1}{\\mathbb{E} \\left[\\frac{1}{K} \\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{F}}_{n}{\\mathbf{F}}^\\H_{n}\\right]}. \\label{eq:lamda} \n\\end{align}\n\nHenceforth, by taking into account that ${\\mathbf{g}}_{k,n}={\\boldsymbol{\\Theta}}_{k,n}{\\mathbf{h}}_{k,n}$, we proceed with the following presentation of the received signal, obtained by~\\eqref{eq:DLreceivedSignal}.  Thus, we have\n\\begin{align}\n&\\!\\!y_{k,n}\\!\\!=\\!\\!{\\sqrt{\\!\\lambda p_{\\mathrm{d}}}}{\\mathbf{g}}^\\H_{k,n}\\!{\\boldsymbol{\\Theta}}_{k,n}^{2} {{\\mathbf{f}}}_{k,n}x_{k,n}\n\\!\\!+\\!\\!\\sum_{i \\neq k}\\!\\! {\\sqrt{\\!\\lambda p_{\\mathrm{d}}}}{\\mathbf{g}}^\\H_{k,n}\\!{\\boldsymbol{\\Theta}}_{k,n}^{2} { {\\mathbf{f}}}_{i,n}x_{i,n}\\!+\\!z_{k,n}\\label{eq:DLreceivedSignal1}\\\\\n\\!\\!&=\\!\\!\\underbrace{{ \\sqrt{\\!\\lambda p_{\\mathrm{d}}}}\\mathbb{E}\\!\\!\\left[{\\mathbf{g}}^\\H_{k,n} \\!{\\boldsymbol{\\Theta}}_{k,n}^{2} {{\\mathbf{f}}}_{k,n}\\right]\\!x_{k,n}}_{\\mathrm{desired~signal}}\\!+\\!\\underbrace{z_{k,n}}_{\\mathrm{noise}}\n\\!\\!+\\!\\!\\sum_{i \\neq k} \\!{\\sqrt{\\!\\lambda p_{\\mathrm{d}}}}{\\mathbf{g}}^\\H_{k,n}\\! {\\boldsymbol{\\Theta}}_{k,n}^{2} {{\\mathbf{f}}}_{i,n}x_{i,n}\\nonumber\\\\\n\\!\\!&+\\!{ \\sqrt{\\!\\lambda p_{\\mathrm{d}}}}\\left({\\mathbf{g}}^\\H_{k,n}{\\boldsymbol{\\Theta}}_{k,n}^{2} {{\\mathbf{f}}}_{k,n}x_{k,n}\\!-\\!\\mathbb{E}\\!\\!\\left[{\\mathbf{g}}^\\H_{k,n} {\\boldsymbol{\\Theta}}_{k,n}^{2} {{\\mathbf{f}}}_{k,n}\\right]\\!x_{k,n}\\right)\\!.\\label{eq:DLreceivedSignalgeneral}\n\\end{align}\n\nIn~\\eqref{eq:DLreceivedSignal1}, we have  applied a similar technique to~\\cite{Medard2000} because UEs do not have access to instantaneous CSI during the downlink phase, but we can  assume that, in particular, UE $k$  is aware of only $\\mathbb{E}[{\\mathbf{g}}^\\H_{k,n} {\\mathbf{f}}_{jm} ]$.\n\nGiven that the input symbols are Gaussian, we provide the achievable user rate based on the analysis in~\\cite[Theorem $1$]{Hassibi2003}. Specifically, the worst case uncorrelated additive noise is also zero-mean with variance equal to the variance of interference plus noise. In addition, since ${\\mathbf{g}}_{k,n}$ is circularly symmetric, we let ${\\mathbf{g}}_{k,n}={\\boldsymbol{\\Theta}}_{k,0}^{2}{\\mathbf{g}}_{k,n}$. Thus, by denoting $\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}=\\Delta{{\\boldsymbol{\\Theta}}}_{k,n}$,  where $\\Delta{{\\boldsymbol{\\Theta}}}_{k,n}=\\left( {\\boldsymbol{\\Theta}}_{k,n}{{\\boldsymbol{\\Theta}}_{k,0}}^{\\!\\!\\!\\H} \\right)^{2}=\\mathrm{diag}\\left\\{ e^{2 j \\left(\\theta_{k,n}^{(1)}-\\theta_{k,0}^{(1)}  \\right)}, \\ldots, e^{2 j \\left(\\theta_{k,n}^{(1)}-\\theta_{k,0}^{(1)}  \\right)} \\right\\}$, we may consider a SISO model with the desired signal power and the interference plus noise power at UE $k$ given by \n \\begin{align}\nS_{k,n}\n=  \\lambda\\Big|\\mathbb{E}\\left[{\\mathbf{g}}^\\H_{k,n}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}{{\\mathbf{f}}}_{k,n}\\right]  \\Big|^{2},\\label{eq:DLgenSignalPower}\n\\end{align}\nand \n\\begin{align}\nI_{k,n}\n&\\!=\\!   \\lambda{\\mathrm{var}}\\!\\left[{\\mathbf{g}}^\\H_{k,n}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n} {\\mathbf{f}}_{k,n}\\right] \\! +  \\! \\frac{\\sigma_{k}^{2}}{p_{\\mathrm{d}}} \\!+\\!    \\sum_{i\\ne k}   \\!  \\lambda\\mathbb{E}\\bigg[\\Big|{\\mathbf{g}}^\\H_{k,n}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n} {\\mathbf{f}}_{i,n}\\Big|^{2}\\bigg] .   \\label{eq:DLgenIntfPower}\n\\end{align}\n\nSince the available CSI at the BS at time $n$ is ${\\mathbf{A}}_{n}\\hat{{\\mathbf{g}}}_{k,0}$,   as can be seen from~\\eqref{eq:GaussMarkov2}, the expression corresponding to the MRT precoder is\n\\begin{align}\n{\\mathbf{F}}_{n}&=  {\\mathbf{A}}_{n}{\\hat{{\\mathbf{G}}}}_{0}. \\label{eq:precoderMRT}\n\\end{align}\n\nIn a similar way, \nthe BS  designs its RZF precoder as\\! \\cite{Hoydis2013}\n\\begin{align}\n{\\mathbf{F}}_{n}\n&= \\left({\\mathbf{A}}_{n}{\\hat{{\\mathbf{G}}}}_{0}{\\hat{{\\mathbf{G}}}}_{0}^\\H {\\mathbf{A}}_{n} \\!+\\! {\\mathbf{Z}} + M  a \\mat{\\mathrm{I}}_M\\right)^{-1}{\\mathbf{A}}_{n}{\\hat{{\\mathbf{G}}}}_{0}\\nonumber\\\\\n&= {{\\mathbf{\\Sigma}}} {\\mathbf{A}}_{n}{\\hat{{\\mathbf{G}}}}_{0}, \\label{eq:precoderRZF}\n\\end{align}\nwhere we define ${{\\mathbf{\\Sigma}}}\\triangleq\\left({{\\mathbf{A}}_{n}{\\hat{{\\mathbf{G}}}}_{0}{\\hat{{\\mathbf{G}}}}_{0}^{\\H}{\\mathbf{A}}_{n}} + {\\mathbf{Z}} +   a M \\mat{\\mathrm{I}}_M\\right)^{-1}$ with ${\\mathbf{Z}} \\in {\\mathbb{C}}^{M \\times M}$ being an arbitrary Hermitian\nnonnegative definite matrix and $a$ being a regularization scaled by $M$, in order to converge to a constant, as $M$, $K\\to \\infty$. Note that both ${\\mathbf{Z}}$ and $a$ could be optimized, but this is outside the\nscope of this paper and left to future work. One possible common choise for these parameters is provided in~\\cite{Hoydis2013}.\n\n\nFollowing a similar approach to~\\cite{Bjornson2015,Pitarokoilis2015}, we compute the achievable rate for each time instance of the data transmission phase. Thus, the downlink ergodic achievable rate of user $k$, is given by\n\\begin{align}\nR_{k}& = \\frac{1}{T_{c}}\\sum_{n=1}^{T_{c}-\\tau}R_{k,n}\\nonumber\\\\\n &=\\frac{1}{T_{c}}\\sum_{n=1}^{T_{c}-\\tau}\\mathrm{log}_{2}\\left(1+ \\gamma_{k,n} \\right),\n\\label{eq:DLrate}\n\\end{align}\nwhere $\\gamma_{k,n}=\\frac{S_{k,n} }{I_{k,n}}$ is the instantaneous downlink SINR at time $n$.  In other words, the mutual information between the received signal and the transmitted symbols is lower bounded by the achievable rate per user provided in~\\eqref{eq:DLrate}. \n\n\n\\section{Deterministic Equivalent Analysis}\\label{Deterministic}\nIn this section, we provide the derivations of the downlink SINRs after applying  MRT and RZF precoders in the presence of imperfect  and delayed CSI due to user mobility and phase noise. Use of RMT in terms of DEs enables the extraction of asymptotic expressions as $K,M \\rightarrow \\infty$, while keeping their ratio ${K}\/{M}=\\beta$ finite. Given that the theory of  DEs provides tight approximations  even for moderate system dimensions, the significance of our  results is of great importance. Confirmation of this tightness is provided in Section~\\ref{results} by  simulations. Interestingly, the results mirror the dependence of the harmful effects described by channel aging.\n\nThe DE of the SINR $\\gamma_{k,n}$ is such that $\\gamma_{k,n}-\\bar{\\gamma}_{k,n}\\xrightarrow[M \\rightarrow \\infty]{\\mbox{a.s.}}0$\\footnote{Note that $\\xrightarrow[ M \\rightarrow \\infty]{\\mbox{a.s.}}$ denotes almost sure convergence, and  $a_n\\asymp b_n$ expresses the equivalence relation $a_n - b_n  \\xrightarrow[ M \\rightarrow \\infty]{\\mbox{a.s.}}  0$ with $a_n$  and $b_n$  being two infinite sequences.}, while the deterministic rate of user $k$ is obtained by the dominated convergence~\\cite{Billingsley2008} and the continuous mapping theorem~\\cite{Vaart2000} by means of~\\eqref{eq:DLrate} \n\\begin{align}\nR_{k}-\\frac{1}{T_{\\mathrm{c}}}\\sum_{n=1}^{T_{\\mathrm{c}}-\\tau}\\log_{2}(1 + \\bar{\\gamma}_{k,n}) \\xrightarrow[ N \\rightarrow \\infty]{\\mbox{a.s.}}0.\\label{DeterministicSumrate}\n\\end{align}\n\n\n\n\n\nThe DE downlink achievable  user rates corresponding to MRT and RZF can be obtained by means of the following theorems. Hereafter, for the sake of exposition and without loss of generality, we assume that in the case of  SLOs, the phase noises obey to identical statistics. However, in Appendix~\\ref{theorem2}, we scrutinize the general setting, where  phase noises across different LOs  are independent, but not identically distributed. First, we consider MRT precoding. \n\\subsection{MRT}\nThis section presents the general DE regarding the SINR with MRT precoding. Then, this asymptotic result particularizes to a number of cases of interest. In additions, it is investigated the dependence of the transmit power per user with the number of BS antennas (power scaling law), in order to support a specific desired rate.\n\\begin{Theorem}\\label{theorem:DLagedCSIMRT}\nThe downlink DE of the SINR of user $k$ at time $n$ with MRT precoding , accounting for imperfect CSI and  delayed CSI due to phase noise and user mobility, is given by \n\\begin{align}\n\\bar{\\gamma}_{k,n} = \\frac{  e^{-2\\left( \\sigma_{\\varphi_{k}}^{2}+\\sigma_{\\phi}^{2} \\right)n} {\\delta}_{k}^{2}}{{\\frac{1}{M}{\\delta}_{k}^{'}}+  \\frac{\\sigma_{k}^{2}}{p_{\\mathrm{d}}  \\bar{\\lambda}M}\n + \\sum_{i\\neq k}\\frac{1}{M}\\delta_{i}^{''}},\\label{eq:DLdelayedCSIetaMRT1}\n \\end{align}\n with \n  \\begin{align}\n  \\bar{\\lambda}&=\\left( \\frac1K\\sum_{i=1}^{K}\\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits {\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k} \\right)^{-1},\\nonumber\n  \\end{align}\n${\\delta}_{k}=\\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k}$, ${\\delta}_{k}^{'}=\\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{A}}_{n}^{2} {\\mathbf{D}}_{k} \\left( {\\mathbf{R}}_{k} - {\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k} \\right)$, and $\\delta_{i}^{''}=\\frac{1}{M} \\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{i} {\\mathbf{R}}_{k}$.\n\\end{Theorem}\n\\proof: The proof of Theorem~\\ref{theorem:DLagedCSIMRT} is given in Appendix~\\ref{theorem2}.\\endproof\n\\begin{corollary}\\label{NoCorrelation}\nIn  case that ${\\mathbf{R}}_{k}=\\mat{\\mathrm{I}}_{M}$ (no large-scale component), and ${\\mathbf{A}}_{n}$ degenerates to a scalar $\\alpha_{n}$ or a $\\alpha_{n}\\mat{\\mathrm{I}}_{M}$, i.e., the BS employs a CLO or a SLOs setting with identical LOs, respectively, we obtain\n \\begin{align}\n  \\bar{\\gamma}_{k,n} =\\frac{\\alpha_{n}^{2} d  e^{-2\\left( \\sigma_{\\varphi_{k}}^{2}+\\sigma_{\\phi}^{2} \\right)n} }{  \\frac{1-\\alpha_{n}^{2} d}{M}+  \\frac{\\sigma_{k}^{2}}{p_{\\mathrm{d}} M}\n +{  \\frac{ K-1 }{M} }},\\label{eq:DLdelayedCSIetaMRT3}\n \\end{align}\n where $d=\\left( \\frac{\\tau p_{ \\mathrm{u}}}{\\tau p_{\\mathrm{u}}+\\sigma_{b}^{2}}\\right)^{-1}$ and $\\alpha_{n}= \\mathrm{J}_{0}(2 \\pi f_{\\mathrm{D}}T_{\\mathrm{s}}n)e^{-\\frac{\\sigma_{\\varphi_{k}}^{2}+\\sigma_{\\phi}^{2}}{2}n}$.\n \\end{corollary}\n\\proof Using ${\\mathbf{R}}_{k}=\\mat{\\mathrm{I}}_{M}$ leads to ${\\mathbf{D}}_{k}=d\\,\\mat{\\mathrm{I}}_{M}=\\left( \\frac{\\tau p_{ \\mathrm{u}}}{\\tau p_{\\mathrm{u}}+\\sigma_{b}^{2}}\\right)^{-1}\\mat{\\mathrm{I}}_{M}$. Moreover, if ${\\mathbf{A}}_{n}$ is replaced by the scalar $\\alpha_{n}$ or the scaled identity matrix $\\alpha_{n}\\mat{\\mathrm{I}}_M$,   Theorem~\\ref{theorem:DLagedCSIMRT} allows to simplify enough the deterministic SINR. Hence, by substituting  $\\bar{\\lambda}=\\left(\\alpha_{n}^{2} d\\right)^{-1}$, ${\\delta}=\\alpha_{n}^{2} d$, ${\\delta}^{'}=\\alpha_{n}^{2} d \\left( 1 - \\alpha_{n}^{2} d \\right)$, and $\\delta^{''}=\\alpha_{n}^{2} d$, we obtain $\\bar{\\gamma}_{k,n}$ as in~\\eqref{eq:DLdelayedCSIetaMRT3}.\n\\endproof\n\nNevertheless, it is interesting to investigate the effect of the combined impairments of phase noise and user mobility  on the asymptotic power scaling law. For the sake of exposition, we focus on MRT precoding, however similar results can be obtained in the case of RZF precoding as well.\n\\begin{proposition}\\label{powerLaw}\n For the identical SLOs or CLO setting, the downlink achievable SINR of user $k$ at time $n$ with MRT precoding  subject to imperfect CSI, delayed CSI due to phase noise  and user mobility, and collocated  BS antennas  becomes\n \\begin{align}\n  \\gamma_{k,n}=\\frac{\\tau E_{\\mathrm{d}}E_{\\mathrm{u}}}{\\sigma^{4}M^{2 q-1}}\\mathrm{J}_{0}^{2}(2 \\pi f_{\\mathrm{D}}T_{\\mathrm{s}}n)e^{-3\\left( {\\sigma_{\\varphi_{k}}^{2}+\\sigma_{\\phi}^{2}} \\right)n}[{\\mathbf{R}}_{k}^{2}]_{mm},\\label{scalingPower}\n \\end{align}\nwhere the transmit uplink and downlink powers are scaled proportionally to $1\/M^{q}$, i.e.,  $p_{\\mathrm{u}} = E_{\\mathrm{u}}\/M^{q}$ and $p_{\\mathrm{d}}= E_{\\mathrm{d}}\/M^{q}$\nfor fixed $E_{\\mathrm{u}}$ and $E_{\\mathrm{d}}$, and $q>0$.\n\\end{proposition}\n\\proof\nLet us first substitute the MRT precoder, given by~\\eqref{eq:precoderMRT},  in~\\eqref{eq:DLreceivedSignalgeneral}. Then, we divide both the desired and the interference parts by $\\lambda$ and $1\/M^{2}$. The desired signal power is written as\n\\begin{align}\nS_{k,n}^{\\mathrm{MRT}}\n&=  \\frac{1}{M^{2}}\\Big|{\\mathbf{g}}^\\H_{k,n} \\widetilde{{\\boldsymbol{\\Theta}}}_{k,n} {\\mathbf{A}}_{n} {\\hat{{\\mathbf{g}}}}_{k,0}\\Big|^{2}\\nonumber\\\\\n&=  \\frac{1}{M^{2}}\\Big|\\hat{{\\mathbf{g}}}^\\H_{k,0}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}  {\\mathbf{A}}_{n}^{2} {\\hat{{\\mathbf{g}}}}_{k,0}\\Big|^{2}\\label{desiredMid}\\\\\n&=\\frac{1}{M^{2}}\\mathrm{J}_{0}^4{2}(2 \\pi f_{\\mathrm{D}}T_{\\mathrm{s}}n)e^{-4 \\left( {\\sigma_{\\varphi_{k}}^{2}+\\sigma_{\\phi}^{2}} \\right)n}[{\\mathbf{D}}_{k}]_{mm}^{2},\\label{desired}\n\\end{align}\nwhere $[{\\mathbf{D}}_{k}]_{mm}$ is the $m$th diagonal element of the matrix  ${\\mathbf{D}}_{k}$ expressing the variance of the $m$th element. Given that both $\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}$ and $  {\\mathbf{A}}_{n}$ are diagonal matrices,   we have taken into account in~\\eqref{desired} that they can commute. Moreover, in the last step of~\\eqref{desired}, we have used the law of large numbers for large $M$ in the case of collocated  BS antennas, as well as that ${\\mathbf{A}}_{n}$ is a scaled identity matrix  or a scalar in the case of identical SLOs or CLO, respectively. In other words, $ {\\hat{{\\mathbf{g}}}}_{k,0}$ has i.i.d. elements with variance $[{\\mathbf{D}}_{k}]_{mm}$. As far as the interference is concerned, the first and third terms of~\\eqref{eq:DLgenIntfPower} vanish to zero as $M \\to \\infty$ by means of the same law. Thus, we have\n\\begin{align}\nI_{k,n}^{\\mathrm{MRT}}&=\\frac{\\sigma^{2}_{k}}{M^{2}p_{\\mathrm{d}} \\lambda}\\nonumber\\\\\n&=\\frac{\\sigma^{2}_{k}\\mathrm{J}_{0}^{2}(2 \\pi f_{\\mathrm{D}}T_{\\mathrm{s}}n)e^{-\\left( {\\sigma_{\\varphi_{k}}^{2}+\\sigma_{\\phi}^{2}} \\right)n}[{\\mathbf{D}}_{k}]_{mm}}{M p_{\\mathrm{d}} }, \n\\end{align}\nwhere  we have applied the law of large numbers to obtain $ \\lambda\\!=\\!\\left(\\frac{1}{M}\\mathrm{J}_{0}^{2}(2 \\pi f_{\\mathrm{D}}T_{\\mathrm{s}}n)e^{-\\left( {\\sigma_{\\varphi_{k}}^{2}\\!+\\sigma_{\\phi}^{2}} \\right)n}[{\\mathbf{D}}_{k}]_{mm}\\! \\right)^{-1}\\!$ from~\\eqref{eq:lamda}, and we have substituted  ${\\mathbf{D}}_{k}=\\frac{\\tau E_{\\mathrm{u}}}{M^{q}\\sigma^{2}_{k}}{\\mathbf{R}}_{k}^{2}$, since  $p_{\\mathrm{p}}$ depends on $p_{\\mathrm{u}}$, the result in~\\eqref{scalingPower} is obtained. \\endproof\n\nProposition~\\ref{powerLaw} reveals that the selection of $q$ affects heavily the achievable rate per user. In fact, proper selection allows maintaining the same performance, even by further scaling down  the transmit power of each user. Specifically, if $q$ is set less than $1\/2$, $\\gamma_{k,n}$ is unbounded, i.e., it tends to infinity.  On the contrary, the SINR of user $k$ diminishes to zero, if $q>1\/2$. This clearly indicates that the transmit powers of each user during the training and downlink phases have been reduced over the required value. Most importantly, in the special case that $q=1\/2$, the SINR approaches a non-zero limit given by the following corollary.\n\\begin{corollary}\nIn the presence of phase noise and user mobility, when the transmit uplink and downlink powers are scaled down by $p_{\\mathrm{u}} = E_{\\mathrm{u}}\/\\sqrt{M}$ and $p_{\\mathrm{d}} = E_{\\mathrm{d}}\/\\sqrt{M}$ for fixed $E_{\\mathrm{u}}$ and $E_{\\mathrm{d}}$, the downlink SINR per user with MRT can be finite  as\n\\begin{align}\n   \\gamma_{k,n}=\\frac{\\tau^{2} E_{\\mathrm{d}} E_{\\mathrm{u}}}{\\sigma_{k}^{4}}\\mathrm{J}_{0}^{2}(2 \\pi f_{\\mathrm{D}}T_{\\mathrm{s}}n)e^{-3\\left( {\\sigma_{\\varphi_{k}}^{2}+\\sigma_{\\phi}^{2}} \\right)n}[{\\mathbf{R}}_{k}^{2}]_{mm}.\n\\end{align}\n\\end{corollary}\n\nEvidently, phase noise and user mobility reduce the SINR, but the power scaling law is not affected. \n\\subsection{RZF}\nAs far as RZF is concerned, the analysis is more complex, as shown below. First we present the DE of the SINR in general channel aging conditions by means of the following theorem.\n\\begin{Theorem}\\label{theorem:DLagedCSIRZF}\nThe downlink DE of the SINR user $k$ at time $n$ with RZF precoding, accounting for imperfect CSI and delayed CSI due to phase noise and user mobility, is given by \n\\begin{align}\n\\bar{\\gamma}_{k,n} = \\frac{  e^{-2\\left( \\sigma_{\\varphi_{k}}^{2}+\\sigma_{\\phi}^{2} \\right)n}  {\\delta}_{k}^{2}}{{\\frac{1}{M}{\\delta}_{k}^{'}}+  \\frac{\\sigma_{k}^{2}\\left( 1+\\delta_{k} \\right)^{2}}{p_{\\mathrm{d}}  \\bar{\\lambda}}\n + \\sum_{i\\neq k}{ \\frac{{Q}_{ik}\\left( 1+\\delta_{k} \\right)^{2}}{M\\left(1+{\\delta_{i}}\\right)^{2}} }},\\label{eq:DLdelayedCSIetaRZF}\n \\end{align}\n with \n  \\begin{align}\n  \\bar{\\lambda}&=\\frac{K}{ \\left(\\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits{\\bf T} - \\frac{1}{M} \\mathop{\\mathrm{tr}}\\nolimits \\left(\\frac{{\\bf Z}}{M} +a \\mat{\\mathrm{I}}_M\\right) {{\\mathbf{C}}}\\right)}\\nonumber\\\\\n{Q}_{ik}&\\asymp \\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{i}{\\mathbf{C}}^{'''}\n+\\frac{\\left|{\\delta_{k}}\\right|^{2}\\delta_{k}^{''}}{\\left( 1+\\delta_{k} \\right)^{2}}-2\\mathrm{Re}\\left\\{  \\frac{\\delta_{k}\\delta_{k}^{''} }{\\left( 1+\\delta_{k} \\right)}\\right\\},\\nonumber\n  \\end{align}\n  ${\\delta}_{k}=\\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k}{\\mathbf{T}},\n {\\delta}_{k}^{'}=\\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k}{{\\mathbf{C}}}^{'}, \\mathrm{and}~\n \\delta_{k}^{''}=\\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k}{\\mathbf{C}}^{''}$,\n where\n\\begin{itemize}\n\\renewcommand{\\labelitemi}{$\\ast$}\n\\item ${\\mathbf{T}}={\\mathbf{T}}(a)$ and ${\\hbox{\\boldmath$\\delta$}}=[{\\delta}_{1},\\cdots,{\\delta}_{K}]^{\\scriptscriptstyle\\mathsf{T}}={\\hbox{\\boldmath$\\delta$}}(a)={\\vect{e}}(a)$ are given by Theorem~\\ref{th:detequ} for ${\\mathbf{L}}={\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k}$, ${\\mathbf{S}}={\\mathbf{Z}}\/M$, ${\\mathbf{R}}_k={\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k}\\, \\forall k \\in \\mathcal{K}$,\n\\item  ${\\mathbf{C}}={\\mathbf{T}}^{'}(a)$ is given by Theorem~\\ref{th:detequder} for ${\\mathbf{L}}=\\mat{\\mathrm{I}}_M$, ${\\mathbf{S}}={\\mathbf{Z}}\/M$, ${\\mathbf{K}}=\\mat{\\mathrm{I}}_M$, ${\\mathbf{R}}_k={\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k}\\, \\forall  k \\in \\mathcal{K}$,\n\\item  ${\\mathbf{C}}^{'}={\\mathbf{T}}^{''}(a)$ is given by Theorem~\\ref{th:detequder} for ${\\mathbf{L}}={\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k}$, ${\\mathbf{S}}={\\mathbf{Z}}\/M$, ${\\mathbf{K}}= {\\mathbf{R}}_{k} - {\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k}$, ${\\mathbf{R}}_k={\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k}\\, \\forall  k \\in \\mathcal{K}$,\n\\item  ${\\mathbf{C}}^{''}={\\mathbf{T}}^{'''}(a )$ is given by Theorem~\\ref{th:detequder} for ${\\mathbf{L}}={\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k}$, ${\\mathbf{S}}={\\mathbf{Z}}_j\/M$, ${\\mathbf{K}}={\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{i}$, ${\\mathbf{R}}_k={\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k}\\, \\forall  k \\in \\mathcal{K}.$\n\\item  ${\\mathbf{C}}^{'''}={\\mathbf{T}}^{''''}(a)$ is given by Theorem~\\ref{th:detequder} for ${\\mathbf{L}}={\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{i}$, ${\\mathbf{S}}={\\mathbf{Z}}_j\/M$, ${\\mathbf{K}}={\\mathbf{R}}_{k}$, ${\\mathbf{R}}_k={\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k}\\, \\forall  k \\in \\mathcal{K}.$\n\\end{itemize} \n\\end{Theorem}\n\\proof: The proof of Theorem~\\ref{theorem:DLagedCSIRZF} is given in Appendix~\\ref{theorem3}.\\endproof\n\\begin{corollary}\\label{equal_correlation}\n Let ${\\mathbf{R}}_{k}={\\mathbf{R}}$, i.e., the large-scale effects (path loss and shadowing) affect the same all users, then ${\\mathbf{D}}_{k}={\\mathbf{D}}$. In such case, the deterministic SINR of Theorem~\\ref{theorem:DLagedCSIRZF}  $\\bar{\\gamma}_{k,n}$ can be simplified to\n \\begin{align}\n  \\bar{\\gamma}_{k,n} = \\frac{  e^{-2\\left( \\sigma_{\\varphi_{k}}^{2}+\\sigma_{\\phi}^{2} \\right)n}    {{\\delta}}^{2}}{  {\\frac{1}{M}{\\delta}^{'}}+  \\frac{\\sigma_{k}^{2}\\left( 1+\\delta \\right)^{2}}{p_{\\mathrm{d}} \\bar{\\lambda}}\n + \\left( K-1 \\right){  \\frac{{Q}}{M} }},\\label{eq:DLdelayedCSIetaRZF2}\n \\end{align}\nwith  ${\\delta}=\\frac{1}{M}{\\mathbf{A}}_{n}^{2}{\\mathbf{D}}{\\mathbf{T}}\\triangleq e,\n {\\delta} ^{'}=\\frac{1}{M}{\\mathbf{A}}_{n}^{2}{\\mathbf{D}}{{\\mathbf{C}}}^{'},  \\delta ^{''}=\\frac{1}{M}{\\mathbf{A}}_{n}^{2}{\\mathbf{D}}{\\mathbf{C}}^{''}$,  $ {\\mathbf{T}} = ({{\\mathbf{A}}_{n}^{2}{\\mathbf{D}} }\/{\\beta\\left( 1+\\delta \\right)}$  $  +{\\mathbf{Z}}\/M + a\\mat{\\mathrm{I}}_M)^{-1}$, $\\bar{\\lambda}={K}\/{ \\left(\\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits{\\bf T} - \\frac{1}{M} \\mathop{\\mathrm{tr}}\\nolimits \\left(\\frac{{\\bf Z}}{M} +a \\mat{\\mathrm{I}}_M\\right) {{\\mathbf{C}}}\\right)}$, and\n \\begin{align}\n  {Q}\\asymp \\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{R}} {\\mathbf{C}}^{''}+\\frac{\\left|{\\delta }\\right|^{2}\\delta ^{''}}{\\left( 1+\\delta  \\right)^{2}}-2\\mathrm{Re}\\left\\{  \\frac{\\delta \\delta ^{''} }{\\left( 1+\\delta  \\right)}\\right\\},\\nonumber\n \\end{align}\nwhile the general expression of ${\\mathbf{T}}^{'}$ is given by \n \\begin{align}\n{\\mathbf{T}}^{'}={\\mathbf{T}}{\\mathbf{K}}{\\mathbf{T}}+e^{'}_{{\\mathbf{K}}}\/{\\beta\\left(1+\\delta\\right)^2}{\\mathbf{T}}{{\\mathbf{A}}_{n}^{2}{\\mathbf{D}}  }{\\mathbf{T}},  \\nonumber\n \\end{align}\nwhere  $e^{'}_{{\\mathbf{K}}}=\\beta \\left( 1+\\delta \\right)^{2}e_{111}^{{\\mathbf{K}}}\/\\left( \\beta-e_{201}^{{\\mathbf{K}}} \\right)$  with  $e_{ijm}^{{\\mathbf{K}}}=1\/\\left( 1+\\delta \\right)^{j+m}\\frac1M \\mathop{\\mathrm{tr}}\\nolimits\\left( {\\mathbf{A}}_{n}^{2}{\\mathbf{D}} \\right)^{i}{\\mathbf{T}}{\\mathbf{K}}^{j}{\\mathbf{T}}^{m}$.\n \\end{corollary}\n \\begin{corollary}\\label{NoCorrelation1}\nIf no large-scale component $({\\mathbf{R}}_{k}=\\mat{\\mathrm{I}}_{M}$) are assumed, and the phase noise  from all the oscillators is considered identical or the BS has only a CLO, i.e., ${\\mathbf{A}}_{n}$ becomes $\\alpha_{n}\\mat{\\mathrm{I}}_M$ or a scalar $\\alpha_{n}$, we obtain\n \\begin{align}\n  \\bar{\\gamma}_{k,n} = \\frac{    e^{-2\\left( \\sigma_{\\varphi_{k}}^{2}+\\sigma_{\\phi}^{2} \\right)n}  {{\\delta}}^{2}}{  {\\frac{1}{M}{\\delta}^{'}}+  \\frac{\\sigma_{k}^{2}\\left( 1+\\delta \\right)^{2}}{p_{\\mathrm{d}} \\bar{\\lambda}}\n + \\left( K-1 \\right){  \\frac{{Q}}{M} }},\\label{eq:DLdelayedCSIetaRZF3}\n \\end{align}\n where $d=\\left( \\frac{\\tau p_{ \\mathrm{u}}}{\\tau p_{\\mathrm{u}}+\\sigma_{b}^{2}}\\right)^{-1}$ and $\\alpha_{n}= \\mathrm{J}_{0}(2 \\pi f_{\\mathrm{D}}T_{\\mathrm{s}}n)e^{-\\frac{\\sigma_{\\varphi_{k}}^{2}+\\sigma_{\\phi}^{2}}{2}n}$.\n \\end{corollary}\n\\proof When ${\\mathbf{R}}_{k}=\\mat{\\mathrm{I}}_{M}$, ${\\mathbf{D}}_{k}$  becomes a scaled identity matrix, i.e., ${\\mathbf{D}}_{k}=d \\mat{\\mathrm{I}}_{M}=\\left( \\frac{p_{{\\mathrm{p}}}\\tau}{p_{{\\mathrm{p}}}\\tau+\\sigma_{b}^{2}}\\right)\\mat{\\mathrm{I}}_{M}$. In addition, replacing ${\\mathbf{A}}_{n}$ by the scalar $\\alpha_{n}=\\mathrm{J}_{0}(2 \\pi f_{\\mathrm{D}}T_{\\mathrm{s}}n)e^{-\\frac{\\sigma_{\\varphi_{k}}^{2}+\\sigma_{\\phi_{k}}^{2}}{2}n}$, we obtain $\\delta=\\alpha_{n}^{2} d  t$, \n ${\\delta} ^{'}=\\alpha_{n}^{2}dt^{'}_1$,  $\\delta ^{''}=\\alpha_{n}^{2} d t^{'}_{2}$,   $\\bar{\\lambda}=\\frac{K}{ \\left(t -t^{'}_{3}a- \\frac{ t^{'}_{3}z}{M}  \\right)}$, \n \\begin{align}\n  {Q}&\\asymp\\frac{\\beta d^4 \\delta^2 \\left(\\delta^2+\\delta+1\\right)}{\\beta (\\delta+1)^2-d^4 \\delta^2}.\\nonumber\n  \\end{align}\n By considering the unique positive root of the quadratic equation  $\\delta=\\alpha_{n}^{2}d t$, we obtain~\\eqref{delta}.\n \\begin{figure*}\n \\begin{align}\n \\delta =\\frac{\\sqrt{4 \\alpha_{n}^2 \\beta^2 d (a +z)+\\left(\\alpha_{n}^2 d (1-\\beta )+\\beta (a +z)\\right)^2}+\\alpha_{n}^2 (\\beta-1) d+\\alpha_{n} \\beta-\\beta (a +z)}{2 \\beta (a +z)}.\\label{delta}\n \\end{align}\n \\line(1,0){470}\n\\end{figure*}\nNote that  the general expression of $t^{'}_{i}$ for $i=1,2,3$ is given by \n \\begin{align}\nt^{'}_{i}=\\frac{\\beta d^2 \\delta^{2}(\\delta+1)^2 k_i}{a^4 \\left(\\beta (\\delta+1)^2-d^4 \\delta^2\\right)}, \\nonumber\n\\end{align}\nsince  $e^{'}_{k_{i}}=e^{'}_{{\\mathbf{K}}}=\\beta d \\delta^2 k_i\/(\\alpha_{n}^2 (\\beta - (d^2 \\delta^2)\/(1 + \\delta)^2)) $. Here, $k_{1}=(1  - \\alpha_{n}^{2}d)$, $k_{2}=\\alpha_{n}^{2} d$, and $k_{3}=1$. \n\\endproof\n\n\n\n\n\\section{Numerical Results}\\label{results}\nThe purpose of this section is to present some representative numerical examples enabling the study of the time variation of the channel due to the effects of phase noise and  user mobility on the performance of massive MIMO systems with MRT and RZF precoders. The metrics under study are the achievable sum-rates and the required transmit power achieving a certain user rate. Note that the achievable sum-rates are described by means of the SINRs, given by Theorems~\\ref{theorem:DLagedCSIMRT} and~\\ref{theorem:DLagedCSIRZF}. In addition, the correctness of the proposed results is validated by Monte-Carlo simulations. Notably, despite that the analytical results are obtained by  assuming that $M,K \\to \\infty$, they coincide with the simulations even for finite values of $M$ and $K$. Actually, this is a known observation in the literature concerning the DEs~\\cite{Couillet2011,Hoydis2013,Truong2013,Papazafeiropoulos2014,Papazafeiropoulos2014WCNC}.\n\\subsection{Simulation Setup}\nGiven that the interest of this work is based  on the investigation of the impact of practical channel impairments, the setting of our scenario includes long-term evolution (LTE) system specifications~\\cite{Marzetta2010}. Specifically,  the simulation setup considers a single cell with a radius of $R = 1000$ meters, and  a guard range of $r_{0} = 100$ meters specifying the distance between the nearest user and the BS. The BS, comprised of  $M$ antennas, broadcasts to $K$ users that are  uniformly distributed within the cell. According to the system model, the channel vector between the BS and the UE $k$ in the $n$th time slot describes the large-scale fading and path loss or spatial correlation by means of ${\\mathbf{R}}_{k}$. Herein,  ${\\mathbf{R}}_{k}$ describes  large-scale fading  modelled as ${\\mathbf{R}}_{k}=l_{k}\\mat{\\mathrm{I}}_{M}$ with $l_{k}=q_{k}\/\\left( r_{k}\/r_{0} \\right)^{\\upsilon}$. Especially, $q_k$ is a log-normal random variable with standard deviation $\\sigma$ ($\\sigma = 8$ dB) expressing the shadow-fading effect, $r_{k}$ denotes the distance between UE $k$ and the BS, and $\\upsilon$ ($\\upsilon = 3.8$) is the path loss exponent. The uplink and downlink powers are $p_{\\mathrm{u}}=p_{\\mathrm{d}}=46$dBm, while the thermal noise density is $-174$ dBm\/Hz. The length of the training duration is $\\tau=K$ symbols, and the phase noises  at the  BS and user LOs are simulated as discrete Wiener processes given by~\\eqref{phaseNoiseBS} and~\\eqref{phaseNoiseuser}, with increment variances in the interval $\\operatorname{0\\deg-2\\deg}$~\\cite{Colavolpe2005}. The coherence time is $T_{\\mathrm{c}}=1\/4 f_{\\mathrm{D}}=1~\\mathrm{ms}$, where $f_{\\mathrm{D}}=250~\\mathrm{Hz}$ is the Doppler spread corresponding to a relative velocity of  $135$ km\/h between the BS and the users, if  the  center frequency is assumed to be $f_\\mathrm{c}= 2~\\mathrm{GHz}$. Moreover, given that the bandwidth for LTE-A is $\\mathrm{W}=20\\mathrm{MHz}$, the symbol time is $T_s=1\/(2\\mathrm{W})=0.025~\\mathrm{\\mu s}$. In order to account for fast varying channels, where high mobility occurs, the coherence block is assumed to include $T=196$ channel uses corresponding to the coherence bandwidth ${B}_{\\mathrm{c}}=196~\\mathrm{KHz}$.\n\nOne useful metric, providing the means for study the considered system, is\n\\begin{align} \\label{eq num 1}\n \\mathcal{S} \\triangleq\\sum_{k=1}^K\n \\bar{R}_{k},\n\\end{align}\nwhere $\\mathcal{S}$ is the sum-rate and $\\bar{R}_{k}$ is the DE rate of UE $k$ given by~\\eqref{DeterministicSumrate}, and \nTheorems~\\eqref{theorem:DLagedCSIMRT} and~\\eqref{theorem:DLagedCSIRZF} for the cases of MRT and RZF, respectively.\n\nIn the following figures, the red and black line patterns correspond to the rates with RZF and MRT precoding for various phase noise nominal values, respectively. Thus, the ``solid'', ``dash'', and ``dot'' lines designate the analytical results with no channel aging, $\\phi_{m,n}=0\\deg,\n \\varphi_{k,n}=2\\deg$, and $\\phi_{m,n}= \\varphi_{k,n}=2\\deg$, respectively.   The bullets represent the simulation results. Finally,  the green ``solid'' and ``dot'' lines, where applicable, depict a scenario with channel aging but not phase noise in the cases of MRT and RZF, respectively.\n\\subsection{Rate Comparison between MRT and RZF}\nThe rate comparison between MRT and RZF considers the impact of both Doppler shift and phase noise. Clearly, the RZF outperforms the MRT for the scenarios considered. Especially, Fig.~\\ref{fig:1} depicts the achievable rates by varying the number of BS antennas for different values of phase noise, when the users are assumed to be static ($v=0~\\mathrm{Km\/h}$), or equivalently, in the case of no Doppler shift ($f_{\\mathrm{D}}=0~\\mathrm{Hz}$). By increasing the hardware imperfection in terms of phase noise, the sum-rate decreases. As far as the RZF is concerned, when the phase noise at the user side has a variance of $\\sigma_{\\varphi_{k}}^{2}=2\\deg$ the rate loss is $75\\%$ with the BS having $M=30$ antennas, but when $M$ reaches $300$, the loss is smaller, i.e.,  $52\\%$.  \nMoreover, when the phase noise increases, the sum-rate saturates faster. Similar conclusions can be made in the case of MRT precoding. The perfect match between Monte Carlo simulations and analytical results validates the analytical results.\n\n\\begin{figure}[t]\n    \\centering\n    \\centerline{\\includegraphics[width=0.5\\textwidth]{figures\/VaryingRateVsAntennas.pdf}}\n    \\caption{Simulated and DE downlink sum-rates with MRT and RZF precoders in a static environment versus the number of BS antennas for  various values of phase noise. Red and black lines correspond to the theoretical sum-rates with RZF and MRT precoding, respectively, while the  black bullets refer to the simulation results.}\n    \\label{fig:1}\n\\end{figure}\n\nOn the other hand, Fig.~\\ref{fig:2} provides the variation of $\\mathcal{S}$ versus the normalized Doppler shift $f_{\\mathrm{D}}T_{\\mathrm{s}}$ when $M=60$ for different values of phase noise starting from perfect LOs (no phase noise) to high phase noise ($\\sigma_{\\phi_{k}}^{2}=\\sigma_{\\varphi_{k}}^{2}=2\\deg$). It is revealed that the effect of the Doppler shift on the achievable rate is more detrimental than that of phase noise. In fact, for low velocities in the order of $30~\\mathrm{km\/h}$ equivalent to $f_{\\mathrm{D}}T_{\\mathrm{s}}\\approx 0.2$, the degradation due to phase noise starts to become  insignificant, but then the achievable rate can become so low that it is inadequate for practical applications, i.e., it is not worthy to investigate the impact of phase noise in high mobility environments. Hence, the higher the mobility, the less important role the phase noise plays. In the same figure, the straight lines illustrate the sum-rate with imperfect CSI, but with no user mobility and with perfect LOs at both BS and user ends.  Furthermore, it is evident that as phase noise increases, the performance worsens. However, the loss due to phase noise is prominent in static environments. \n\\begin{figure}[t]\n    \\centering\n    \\centerline{\\includegraphics[width=0.5\\textwidth]{figures\/VaryingRateVsDoppler.pdf}}\n    \\caption{Simulated and DE downlink sum-rates with MRT and RZF precoders when $M=60$ as a function of  the normalized Doppler shift for  various values of phase noise. Red and black lines correspond to the theoretical sum-rates with RZF and MRT precoding, respectively, while the black bullets refer to the  simulation results. The green ``solid'' and ``dot'' lines mirror a scenario with channel aging but not phase noise in the cases of MRT and RZF, respectively. Lines parallel to x-axis represent scenarios with no channel aging.}\n    \\label{fig:2}\n\\end{figure}\n\\subsection{Required Transmit Power Comparison between MRT and RZF}\nRegarding the required transmit power achieving a specific rate per user equal to $1$ bit\/s\/Hz, it is quite insightful to investigate how it changes by varying the number of BS antennas, the amount of Doppler shift, and  the severity of phase noise.\n\nHence, Fig.~\\ref{fig:3} illustrates the variation of the transmit power $p_{\\mathrm{d}}$ versus the number of BS antennas in a static environment ($f_{\\mathrm{D}}T_{\\mathrm{s}}=0$). Specifically,  $p_{\\mathrm{d}}$ decreases considerably when we increase $M$. Especially, a closer observation shows a reduction in the transmit power by approximately 1.5dB  after doubling the number of BS antennas, which agrees with previously known results e.g.,~\\cite{Ngo2013}. Notably, the more severe is the phase noise at the user, the more the required transmit power should be. \n\n\\begin{figure}[t]\n    \\centering\n    \\centerline{\\includegraphics[width=0.5\\textwidth]{figures\/VaryingPowerVsAntennas.pdf}}\n    \\caption{Required transmit power  to achieve $1$ bit\/s\/Hz per user  with MRT and RZF precoders in a static environment versus the number of BS antennas and various values of phase noise. Red and black lines correspond to the theoretical sum-rates with RZF and MRT precoding, respectively, while the black bullets refer to the simulation results.}\n    \\label{fig:3}\n\\end{figure}\n\nFig.~\\ref{fig:4} depicts $p_{\\mathrm{d}}$ versus the varying normalized Doppler shift $f_{\\mathrm{D}}T_{\\mathrm{s}}$ and phase noise,  when $M=60$. In particular, the straight lines represent the required transmit power with no channel aging. As a result, these do not depend on $f_{\\mathrm{D}}T_{\\mathrm{s}}$ or phase noise. However, as the Doppler shift increases,  $p_{\\mathrm{d}}$ becomes higher, and very soon (low velocities), it saturates to a constant. In other words, after a specific value of  $f_{\\mathrm{D}}T_{\\mathrm{s}}$, any increase of  $p_{\\mathrm{d}}$ cannot achieve any benefit. Furthermore, the higher the phase noise, the faster the saturation ensues.\n\n\\begin{figure}[t]\n    \\centering\n    \\centerline{\\includegraphics[width=0.5\\textwidth]{figures\/VaryingPowerVsDoppler.pdf}}\n    \\caption{Required transmit power  to achieve $1$ bit\/s\/Hz per user with MRT and RZF precoders  when $M=60$ as a function of the normalized Doppler shift for  various values of phase noise.  Red and black lines correspond to the theoretical sum-rates with RZF and MRT precoding, respectively. The green ``solid'' and ``dot'' lines mirror a scenario with channel aging but not phase noise in the cases of MRT and RZF, respectively. Lines parallel to x-axis represent scenarios with no channel aging.}\n    \\label{fig:4}\n\\end{figure}\n\\subsection{Extension to Multi-Cell Large MIMO Systems}\nHerein, we focus on the impact of channel aging in the practical case of a hexagonal cellular system. Especially, we consider the downlink of a cellular MIMO system with $L$ cells\noperating under the same frequency band, i.e., the system is impaired by pilot contamination. Actully, each cell can be assumed as an instance of the single-cell setting provided in Section II. The received signal at the $k$th user, located at the $i$th cell, is given by\n\\begin{align}\n\\!\\!y_{ik,n}=\\sqrt{p_{\\mathrm{d}}}\\sum_{l=1}^{L}{\\mathbf{h}}^\\H_{lik,n}{\\boldsymbol{\\Theta}}_{lk,n}{\\mathbf{s}}_{l,n}+z_{ik,n}\\label{eq:DLreceivedSignal}\n\\end{align}\nwhere  ${\\mathbf{s}}_{l,n}$ denotes the unit-norm signal vector transmitted by the BS. Moreover, we have assumed that the BS transmits data to all the users with the same power $p_{\\mathrm{d}}$, and $z_{ik,n} \\sim {\\cal C}{\\cal N}(0,\\sigma_{k}^{2})$ is \ncomplex Gaussian noise at user $k$, found in the $i$th cell.\n\nThe LMMSE estimator of ${\\mathbf{g}}_{ik,0}$, obtained during the training phase, is\n \\begin{align}\\label{estimatedChannel}\n \\hat{{\\mathbf{g}}}_{ik,0}=\\left(  \\mat{\\mathrm{I}}_{M}+\\frac{\\sigma_{\\mathrm{b}}^{2}}{p_{{\\mathrm{p}}}}\\sum_{l}{\\mathbf{R}}_{lik}^{-1}\\right)^{-1}\\tilde{{\\mathbf{y}}}_{iik,0}^{{\\mathrm{p}}},\n\\end{align}\nwhere $\\sigma_{\\mathrm{b}}^{2}$ is the variance of the post-processed noise at\nbase station, $\\tilde{{\\mathbf{y}}}_{ik,0}^{{\\mathrm{p}}}$ is a noisy observation of the effective channel from user $k$ to the $i$th BS, and $p_{\\mathrm{p}}=\\tau p_{\\mathrm{u}}$  with $p_{\\mathrm{u}}$   being the power per user in the uplink data transmission phase.\n\nGiven that the focal point of this work is to conduct an investigation of a generalized channel aging model, and  shed deeper light on channel aging as well as reveal new properties, we present only simulation results and not the corresponding deterministic equivalent analysis, which is straightforward and will distract the reader from the main objective of this part. In particular, we consider the same design parameters with the setup of the single cell, but now we employ $L=7$ cells. Moreover, the average sum-rate is going to be investigated for the central cell, while it has to be mentioned that in the multi-cell setting the large-scale fading depends on the distance of not only the distance of $k$th user from its associated BS, but also from the neighbor BSs. In such scenario, the equivalent SISO model for UE $k$ in the central cell has \ndesired signal power and interference plus noise power given by \n \\begin{align}\nS_{ik,n}\n=  \\lambda\\Big|\\mathbb{E}\\left[{\\mathbf{g}}^\\H_{iik,n}\\widetilde{{\\boldsymbol{\\Theta}}}_{ik,n}{{\\mathbf{f}}}_{ik,n}\\right]  \\Big|^{2}\\label{eq:DLgenSignalPower5}\n\\end{align}\nand \n\\begin{align}\nI_{ik,n}\n&=   \\lambda{\\mathrm{var}}\\left[{\\mathbf{g}}^\\H_{iik,n}\\widetilde{{\\boldsymbol{\\Theta}}}_{ik,n} {\\mathbf{f}}_{ik,n}\\right]  +   \\frac{\\sigma_{k}^{2}}{p_{\\mathrm{d}}} \\nonumber\\\\\n&+    \\sum_{(l,m \\ne i,k)}     \\lambda\\mathbb{E}\\bigg[\\Big|{\\mathbf{g}}^\\H_{lim,n}\\widetilde{{\\boldsymbol{\\Theta}}}_{lm,n} {\\mathbf{f}}_{lm,n}\\Big|^{2}\\bigg],   \\label{eq:DLgenIntfPower5}\n\\end{align}\nwhere $\\widetilde{{\\boldsymbol{\\Theta}}}_{ik,n}=\\Delta{{\\boldsymbol{\\Theta}}}_{ik,n}$ and ${{\\mathbf{f}}}_{ik,n}$ represent the accumulated phase noise and the precoder applied by the $i$th BS.\n\\begin{figure}[t]\n    \\centering\n    \\centerline{\\includegraphics[width=0.5\\textwidth]{figures\/VaryingRateVsDopplerCells.pdf}}\n    \\caption{Simulated downlink sum-rates with MRT and RZF precoders in a cellular setting with $L=7$ when $M=60$ as a function of  the normalized Doppler shift for  various values of phase noise. Red and black lines correspond to the simulated sum-rates with RZF and MRT precoding, respectively. The green ``solid'' and ``dot'' lines mirror a scenario with channel aging but not phase noise in the cases of MRT and RZF, respectively. Lines parallel to x-axis represent scenarios with no channel aging.}\n    \\label{fig:2}\n\\end{figure}\n\nObviously, the difference between the single-cell and multi-cell analyses is limited to the term describing the inter-cell interference, or in other words, the pilot contamination. Fig.~5 illustrates the sum-rate of the $i$th (central) cell versus the normalized Doppler shift. As can be seen, the impact of channel aging on the inter-cell interference is similar to the intra-cell interference, and results only in an extra degradation of the  system performance. In addition, the system is more sensitive to the normalized Doppler shift. In other words, the more severe the channel aging, and basically the user mobility, the higher the degradation of the achievable  sum-rate because it affects the inter-cell interference term.\n\n\\section{Conclusions}\nIn this work, we  modeled channel aging by incorporating both effects of Doppler shift coming from user mobility and phase noise due to circuitry imperfections of LOs, since both the effects contribute to the time variation of the effective channel multiplying\/affecting the transmitted data. Given this novel integrated framework, we provided a joint channel-phase noise estimate. Next, the new CSI model was exploited to construct the MRT  and the RZF precoders, and derived the DEs of the corresponding downlink SINRs. As attested by Monte Carlo simulations, these DEs are tight approximations for the rate performance of the studied system. In addition, the numerical results showed that the degradation due to Doppler shift dominates against phase noise. As a result, the detrimental effect of phase noise is meaningful only in low mobility conditions. \nNotably, the use of massive MIMO systems should be preferred even in general channel aging conditions,  where, in addition, RZF behaves better than MRT as expected. Finally, we showed that in the case of MRT precoding, the required transmit power per user to achieve a certain rate can be scaled down at most by the inverse of the square root of the number of antennas, while a similar result should hold for RZF precoding as well.\n\n\\begin{appendices}\n\\section{Useful Lemmas}\n\\begin{lemma}[Matrix inversion lemma (I) {\\cite[Eq.~2.2]{Bai1}}]\\label{lemma:inversion}\n\\\\Let ${\\mathbf{B}}\\in\\mathbb{C}^{M\\times M}$ be Hermitian  invertible. Then, for any vector ${\\mathbf{x}}\\in\\mathbb{C}^{M}$, and any scalar $\\tau\\in\\mathbb{C}^{M}$ such that ${\\mathbf{B}}+\\tau{\\mathbf{x}}\\bx^\\H$ is invertible,\n\\begin{align}\n{\\mathbf{x}}^\\H({\\mathbf{B}}+\\tau{\\mathbf{x}}\\bx^\\H)^{-1}=\\frac{{\\mathbf{x}}^\\H{\\mathbf{B}}^{-1}}{1+\\tau{\\mathbf{x}}^\\H{\\mathbf{B}}^{-1}{\\mathbf{x}}}.\\nonumber\n\\end{align}\n \\end{lemma}\n\n\n\n\\begin{lemma}[Matrix inversion lemma (II) {\\cite[Lemma~2]{Hoydis2013}}]\\label{lemma:inversion2}\n\\\\Let ${\\mathbf{B}}\\in\\mathbb{C}^{M\\times M}$ be Hermitian  invertible. Then, for any vector ${\\mathbf{x}}\\in\\mathbb{C}^{M}$, and any scalar $\\tau\\in\\mathbb{C}^{M}$ such that ${\\mathbf{B}}+\\tau{\\mathbf{x}}\\bx^\\H$ is invertible,\n\\begin{align}\n({\\mathbf{B}}+\\tau{\\mathbf{x}}\\bx^\\H)^{-1}={\\mathbf{B}}-\\frac{{\\mathbf{B}}^{-1}\\tau{\\mathbf{x}}\\bx^\\H{\\mathbf{B}}^{-1}}{1+\\tau{\\mathbf{x}}^\\H{\\mathbf{B}}^{-1}{\\mathbf{x}}}.\\nonumber\n\\end{align}\n \\end{lemma}\n\n\n\n\n\\begin{lemma}[Rank-1 perturbation lemma {\\cite[Lemma~2.1]{Bai2}}]\n\\\\Let $z\\in<0$, ${\\mathbf{B}}\\in\\mathbb{C}^{M\\times M}$, ${\\mathbf{B}}\\in\\mathbb{C}^{M\\times M}$ with ${\\mathbf{B}}$ Hermitian  nonnegative-definite, and ${\\mathbf{x}}\\in\\mathbb{C}^{M}$. Then,\n\\begin{align}\n|\\mathop{\\mathrm{tr}}\\nolimits\\left(({\\mathbf{B}}-z\\mat{\\mathrm{I}}_M)^{-1} -({\\mathbf{B}}+{\\mathbf{x}}\\bx^\\H-z\\mat{\\mathrm{I}}_M)^{-1}{\\mathbf{B}}\\right)|\\leq\\frac{\\|{\\mathbf{B}}\\|}{|z|}.\\nonumber\n\\end{align}\n \\end{lemma}\n\n\n\n\n\\begin{lemma}[{\\cite[Thm. 3.7]{Hoydis2013}},{\\cite[Lem. 1]{Truong2013}}]\\label{lemma:asymptoticLimits}\nLet ${\\mathbf{B}} \\in {\\mathbb{C}}^{M \\times M}$ with uniformly bounded spectral norm (with respect to $M$). Consider ${\\mathbf{x}}$ and ${\\mathbf{y}}$, where ${\\mathbf{x}}, {\\mathbf{y}} \\in {\\mathbb{C}}^{M}$, ${\\mathbf{x}} \\sim {\\cal C}{\\cal N}({\\mathbf{0}}, {\\boldsymbol{\\Phi}}_{x})$ and ${\\mathbf{y}}  \\sim {\\cal C}{\\cal N}({\\mathbf{0}}, {\\boldsymbol{\\Phi}}_{y})$, are mutually independent and independent of ${\\mathbf{B}}$. Then, we have\n\\begin{align}\n\\frac{1}{M}{\\mathbf{x}}^{\\H}{\\mathbf{B}}{\\mathbf{x}} - \\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits {\\mathbf{B}}{\\boldsymbol{\\Phi}}_{x} & \\xrightarrow[ M \\rightarrow \\infty]{\\mbox{a.s.}} 0 \\label{eq:oneVector}\\\\\n\\frac{1}{M}{\\mathbf{x}}^{\\H}{\\mathbf{B}}{\\mathbf{y}} & \\xrightarrow[ M \\rightarrow \\infty]{\\mbox{a.s.}} 0 \\label{eq:twoVector}\\\\\n\\mathbb{E}\\!\\!\\left[\\left|\\left(\\frac{1}{M}{\\mathbf{x}}^{\\H}{\\mathbf{B}}{\\mathbf{x}}\\right)^{2}\\!\\! - \\!\\left(\\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits {\\mathbf{B}} {\\boldsymbol{\\Phi}}_{x} \\right)^{2} \\right|\\right] \\!\\!&\\xrightarrow[ M \\rightarrow \\infty]{\\mbox{a.s.}}  0\\label{eq:squared}\\\\\n\\frac{1}{M^{2}} |{\\mathbf{x}}^{\\H}{\\mathbf{B}}{\\mathbf{y}}|^{2} - \\frac{1}{M^{2}} \\mathop{\\mathrm{tr}}\\nolimits {\\mathbf{B}} {\\boldsymbol{\\Phi}}_{x} {\\mathbf{B}}^{\\H} {\\boldsymbol{\\Phi}}_{y}  & \\xrightarrow[ M\\rightarrow \\infty]{\\mbox{a.s.}} 0. \\label{eq:twoVectorGeneral}\n\\end{align}\n\\end{lemma}\n\n\\begin{lemma}[{\\cite[p. 207]{Tao2012}}]\\label{lemma:asymptoticproduct}\nLet ${\\mathbf{A}}\\mathrm{,}~{\\mathbf{B}}\\in \\mathcal{C}^{M\\times M}$ be freely independent random\nmatrices  with uniformly bounded spectral norm for all $M$. Further, let all the moments of the entries of ${\\mathbf{A}}\\mathrm{,}~{\\mathbf{B}}$ be finite. Then,\n\\begin{align}\n\\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits {\\mathbf{A}}{\\mathbf{B}}-\\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits {\\mathbf{A}} \\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits {\\mathbf{B}} \\xrightarrow[ M\\rightarrow \\infty]{\\mbox{a.s.}} 0.\n\\end{align}\n\\end{lemma}\n\n\n\\begin{Theorem}[{\\cite[Theorem 1]{Wagner2012}}]\\label{th:detequ}\nLet ${\\mathbf{L}}\\in\\mathbb{C}^{M\\times M}$ and ${\\mathbf{S}}\\in\\mathbb{C}^{M\\times M}$ be Hermitian  nonnegative definite matrices, and let ${\\mathbf{H}}\\in\\mathbb{C}^{M\\times K}$ be a random matrix with columns ${\\mathbf{v}}_k\\sim {\\cal C} {\\cal N}\\left(0,\\frac{1}{M}{\\mathbf{R}}_k\\right)$. Assume that ${\\mathbf{L}}$ and the matrices ${\\mathbf{R}}_k$, $k=1,\\dots,K$, have uniformly bounded spectral norms (with respect to $M$). Then, for any $\\rho>0$,\n\\begin{align*}\n\\frac1M\\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{L}}\\left({\\mathbf{H}}\\bH^\\H +{\\mathbf{S}} +\\rho\\mat{\\mathrm{I}}_M\\right)^{-1} - \\frac1M\\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{L}}{\\mathbf{T}}(\\rho) \\xrightarrow[ M_{{\\mathrm{t}}} \\rightarrow \\infty]{\\mbox{a.s.}} 0,\n\\end{align*}\nwhere ${\\mathbf{T}}(\\rho)\\in\\mathbb{C}^{M\\times M}$ is defined as\n$$ {\\mathbf{T}}(\\rho) = \\left(\\frac1M\\sum_{k=1}^K\\frac{{\\mathbf{R}}_k}{1+e_k(\\rho)}  +{\\mathbf{S}} + \\rho\\mat{\\mathrm{I}}_M\\right)^{-1},$$\nand the elements of $\\boldsymbol{e}(\\rho)=\\left[e_1(\\rho)\\cdots e_K(\\rho)\\right]^{\\scriptscriptstyle\\mathsf{T}}$ are defined as $ e_k(\\rho)  = \\lim_{t \\to \\infty}e_k^{(t)}(\\rho),$ where for $t=1,2,\\ldots$\n\\begin{align}\n\\!\\!\\!e_k^{(t)}(\\rho)\\!=\\!\\frac1M\\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{R}}_k\\!\\left(\\!\\!\\frac1M\\!\\!\\sum_{j=1}^{K}\\!\\!\\frac{{\\mathbf{R}}_j}{1+e_j^{(t-1)}(\\rho)}+{\\mathbf{S}}+\\rho\\mat{\\mathrm{I}}_M\\!\\!\\right)\\!\\!^{-1}\n\\end{align}\nwith initial values $e_k^{(0)}(\\rho)=\\frac{1}{\\rho}$ for all $k$.\n\\end{Theorem}\n\\vspace{5pt}\n\n\\begin{Theorem}[{\\cite[Theorem 2]{Hoydis2013}}]\\label{th:detequder}\nLet ${\\boldsymbol{\\Theta}}\\in\\mathbb{C}^{M\\times M}$ be a Hermitian  nonnegative definite matrix with uniformly bounded spectral norm (with respect to $M$). Under the same conditions as in Theorem~\\ref{th:detequ}, we have  \n\\begin{align*}\n\\frac1M\\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{L}}\\left({\\mathbf{H}}\\bH^\\H + {\\mathbf{S}}+\\rho\\mat{\\mathrm{I}}_M\\right)^{-1}&{\\mathbf{K}}\\left({\\mathbf{H}}\\bH^\\H  +{\\mathbf{S}}+\\rho\\mat{\\mathrm{I}}_M\\right)^{-1}\\nonumber\\\\\n&- \\frac1M\\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{L}}{\\mathbf{T}}^{'}(\\rho) \\xrightarrow[]{\\text{a.s.}} 0,\n\\end{align*}\nwhere ${\\mathbf{T}}^{'}(\\rho)\\in\\mathbb{C}^{M\\times M}$ is defined as\n$$ {\\mathbf{T}}^{'}(\\rho) = {\\mathbf{T}}(\\rho){\\mathbf{K}} {\\mathbf{T}}(\\rho) + {\\mathbf{T}}(\\rho)\\frac1M\\sum_{k=1}^K\\frac{{\\mathbf{R}}_k e^{'}_k(\\rho) }{\\left(1+e_k(\\rho)\\right)^2}{\\mathbf{T}}(\\rho)$$\nwith ${\\mathbf{T}}(\\rho)$ and $\\boldsymbol{e}_k(\\rho)$ as defined in Theorem~\\ref{th:detequ} and $\\boldsymbol{e}^{'}(\\rho) = \\left[e^{'}_1(\\rho)\\cdots e^{'}_K(\\rho)\\right]^{\\scriptscriptstyle\\mathsf{T}}$ given by\n\\begin{align}\n \\boldsymbol{e}^{'}(\\rho) &= \\left(\\mat{\\mathrm{I}}_K - {\\mathbf{J}}(\\rho)\\right)^{-1}{\\mathbf{v}}(\\rho).\n\\end{align}\nThe elements of ${\\mathbf{J}}(\\rho)\\in\\mathbb{C}^{K\\times K}$ and ${\\mathbf{v}}(\\rho)\\in\\mathbb{C}^{K}$ are defined as\n\\begin{align}\n[{\\mathbf{J}}(\\rho)]_{kl} = \\frac{\\frac1M\\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{R}}_k{\\mathbf{T}}(\\rho){\\mathbf{R}}_l{\\bf T}(\\rho)}{M\\left(1+e_k(\\rho)\\right)^2}\\nonumber\n\\end{align}\nand\n\\begin{align}\n[{\\mathbf{v}}(\\rho)]_k = \\frac1M\\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{R}}_k{\\bf T}(\\rho){\\mathbf{K}}{\\bf T}(\\rho).\\nonumber\n\\end{align}\n\n\\end{Theorem}\n\n\n\n\n\\section{Proof of Proposition~\\ref{LMMSE}}\\label{proposition1} \nDuring the training phase,  users transmit mutually orthogonal training sequences consisting of $\\tau$ symbols, and it is assumed that the channel remains  constant during this phase.  In particular,  the pilot sequences can be represented by  ${\\boldsymbol{\\Psi}} = [{\\boldsymbol{\\psi}}_{1}; \\cdots; {\\boldsymbol{\\psi}}_{K}] \\in {\\mathbb{C}}^{K \\times \\tau}$ with ${\\boldsymbol{\\Psi}}$ normalized, i.e., ${\\boldsymbol{\\Psi}}\\bPsi^\\H = {\\mathbf{I}}_K$. Given that the channel estimation takes place at time $0$, the received signal at the BS is written as\n\\begin{align}\\label{eq:Ypt}\n{\\mathbf{Y}}_{{\\mathrm{p}},0}= & \\sqrt{p_{{\\mathrm{p}}}}{\\boldsymbol{\\Theta}}_{k,0} {\\mathbf{H}}_{0}{\\boldsymbol{\\Psi}} + {\\mathbf{Z}}_{{\\mathrm{p}},0},\n\\end{align}\nwhere $p_{{\\mathrm{p}}}$ is the common average transmit power for all users, ${\\boldsymbol{\\Theta}}_{k,0}=\\mathrm{diag}\\left\\{ e^{j \\theta_{k,0}^{(1)}}, \\ldots, e^{j \\theta_{k,0}^{(M)}} \\right\\}$ is the phase noise because of the BS and user $k$ LOs at time $0$, and ${\\mathbf{Z}}_{{\\mathrm{p}},0} \\in {\\mathbb{C}}^{M \\times \\tau}$ is the spatially white additive Gaussian noise matrix at the  BS   during the training phase. Note that $ \\theta_{k,0}^{(m)}=\\phi_{m,0}+\\varphi_{k,0},~m=1,\\ldots, M$. By correlating the received signal with the training sequence $\\frac{1}{\\sqrt{p_{{\\mathrm{p}}}}}{\\boldsymbol{\\psi}}^\\H_k$ of user $k$, and by substituting ${\\mathbf{g}}_{k,0}={\\boldsymbol{\\Theta}}_{k,0} {{\\mathbf{h}}_{k,0}}$ the BS obtains\n\\begin{align}\n\\tilde{{\\mathbf{y}}}_{k,0}^{{\\mathrm{p}}}\n= {\\mathbf{g}}_{k,0}  + {\\frac{1}{\\sqrt{p_{{\\mathrm{p}}}}}\\tilde{{\\mathbf{z}}}_{{\\mathrm{p}},0}},\\label{eq:Ypt3}\n\\end{align}\nwhere $\\tilde{{\\mathbf{z}}}_{{\\mathrm{p}},0}\\triangleq {\\mathbf{Z}}_{{\\mathrm{p}},0}{\\boldsymbol{\\psi}}^\\H_{k}\\sim {\\cal C}{\\cal N}({\\mathbf{0}},\\sigma_{\\mathrm{b}}^{2}{\\mathbf{I}}_{M})$.  After applying the MMSE estimation method~\\cite{Kay}, the  effective channel estimated at the BS  can be written as~\\eqref{estimatedChannel}.\nEmploying the orthogonality principle, the  channel decomposes as\n\\begin{align}\n{\\mathbf{g}}_{k,0} = \\hat{{\\mathbf{g}}}_{k,0} + \\tilde{{\\mathbf{g}}}_{k,0},\\label{eq:MMSEorthogonality}\n\\end{align}\nwhere $\\hat{{\\mathbf{g}}}_{k,0}$ is distributed as ${\\cal C}{\\cal N}\\left({\\mathbf{0}},{\\mathbf{D}}_{k}\\right)$ with ${\\mathbf{D}}_{k}=\\left(  \\mat{\\mathrm{I}}_{M}+\\frac{\\sigma_{b}^{2}}{p_{{\\mathrm{p}}}}{\\mathbf{R}}_{k}^{-1}\\right)^{-1}{\\mathbf{R}}_{k}$, and $\\tilde{{\\mathbf{g}}}_{k,0} \\sim {\\cal C}{\\cal N}({\\mathbf{0}}, {\\mathbf{R}}_{k} - {\\mathbf{D}}_{k})$ is the channel estimation error vector. Note that $\\hat{{\\mathbf{g}}}_{k}$ and $\\tilde{{\\mathbf{g}}}_{k}$ are statistically independent because they are uncorrelated and jointly Gaussian.\n \\section{Proof of Theorem~\\ref{theorem:DLagedCSIMRT}}\\label{theorem2}\n We consider the desired signal power \n\\begin{align}\nS_{k,n}\n&= \\frac{\\lambda}{M}  \\Big|\\mathbb{E}\\left[{{\\mathbf{g}}}^\\H_{k,n}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}   {\\mathbf{A}}_{n} \\hat{{\\mathbf{g}}}_{k,0} \\right]\\!\\!\\Big|^{2}.\\label{normalizedMRT}\n\\end{align}\nFirst, we  derive the DE of the normalization parameter $\\lambda$. Hence, the normalization parameter can be written by means of~\\eqref{eq:lamda} and~\\eqref{eq:precoderMRT} as\n\\begin{align}\n\\lambda&=\n\\frac{K}{\\mathbb{E}\\Big[\\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits {\\mathbf{A}}_{n}{\\hat{{\\mathbf{G}}}}_{0}{\\hat{{\\mathbf{G}}}}_{0}^{\\H} {\\mathbf{A}}_{n}   \\Big]} \\nonumber\\\\\n&= \\frac{K}{\\mathbb{E}\\Big[\\sum_{i=1}^{K}\\frac{1}{M}{\\mathbf{A}}_{n}^{2}  \\hat{{\\mathbf{g}}}_{k,0} \\hat{{\\mathbf{g}}}_{k,0}^{\\H}  \\Big]}\\nonumber\\\\\n&\\asymp \\left( \\frac1K\\sum_{i=1}^{K}\\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits {\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k} \\right)^{-1},\\label{desired1MRT}\n\\end{align}\nwhere we have applied Lemma~\\ref{lemma:asymptoticLimits}.\nAs far as the other term in~\\eqref{normalizedMRT} is concerned, we obtain\n\\begin{align}\n \\frac{1}{M}\\mathbb{E}\\!\\left[{{\\mathbf{g}}}^\\H_{k,n}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}  {\\mathbf{A}}_{n} \\hat{{\\mathbf{g}}}_{k,0} \\right]\n &\\!=\\! \\frac{1}{M}\\mathbb{E}\\!\\left[(\\hat{{\\mathbf{g}}}^\\H_{k,0}{\\mathbf{A}}_{n}  \\!+ \\!\\tilde{{\\mathbf{e}}}^\\H_{k,n} ) \\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}{\\mathbf{A}}_{n} \\hat{{\\mathbf{g}}}_{k,0} \\right]\\nonumber\\\\\n &=\\frac{1}{M}\\mathbb{E}\\Big[{\\hat{{\\mathbf{g}}}^\\H_{k,0}{\\mathbf{A}}_{n}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n} {\\mathbf{A}}_{n}\\hat{{\\mathbf{g}}}_{k,0} }\\Big],\\nonumber\\\\\n &\\asymp\\frac{1}{M}\\mathbb{E}\\left[\\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{A}}_{n}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}{\\mathbf{A}}_{n}{\\mathbf{D}}_{k}\\label{desired4MRT}\\right]\\\\\n &\\asymp \\frac{e^{-\\left( \\sigma_{\\varphi_{k}}^{2}+\\sigma_{\\phi}^{2} \\right)n}}{M}\\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k},\\label{desired2MRT}\n\\end{align}\nwhere in~\\eqref{desired4MRT} and~\\eqref{desired2MRT}, we have applied Lemmas~\\ref{lemma:asymptoticLimits} and~\\ref{lemma:asymptoticproduct}, respectively. Note that ${\\mathbf{A}}_{n}$ and $\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}$ commute because both are diagonal matrices. In particular, regarding the CLO setting, it holds that $\\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits \\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}=e^{ 2 j \\left( \\varphi_{k, n}-\\varphi_{k, 0} +\\phi_{k, n}-\\phi_{k, 0}\\right)}$, while in the case of the SLOs setup, we obtain $\\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits \\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}=e^{2 j \\left( \\varphi_{k, n}-\\varphi_{k, 0} \\right)-\\sigma_{\\phi}^{2}n}$ or  $\\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits \\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}=e^{2 j \\left( \\varphi_{k, n}-\\varphi_{k, 0} \\right)}\\frac{1}{M}\\prod_{i=1}^{M}e^{-\\sigma_{\\phi_{i}}^{2}n}$, when the LOs obey to identical or non-identical statistics, respectively. Herein, we focus on the former scenario regarding the BS antennas. Especially, in such case their increment variance is $\\sigma_{\\phi}^{2}$. Application of the expectation operator to  $\\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits \\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}$ gives~$e^{-\\left( \\sigma_{\\varphi_{k}}^{2}+\\sigma_{\\phi}^{2} \\right)n}$ for both the  CLO and SLOs settings. Finally, after simple algebraic maipulations, we lead to ~\\eqref{desired2MRT}. By denoting $\\bar{S}_{k,n} = \\lim_{M \\rightarrow \\infty} S_{k,n}$ the DE signal power, and by using~\\eqref{desired1MRT} and~\\eqref{desired2MRT}, we have\n\\begin{align}\n\\bar{S}_{k,n}\\asymp  \\bar{\\lambda}e^{-2\\left( \\sigma_{\\varphi_{k}}^{2}+\\sigma_{\\phi}^{2} \\right)n}  {\\delta}_{k}^{2},\\label{eq:theorem4.5MRT}\n\\end{align}\nwhere ${\\delta}_{k}=\\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k}$.\nNow, we proceed with the derivation of each term in~\\eqref{eq:DLgenIntfPower}. The first term on the right hand side can be written as \n\\begin{align}\n\\frac{1}{M} {\\mathrm{var}}&\\left[{\\mathbf{g}}^\\H_{k,n} \\widetilde{{\\boldsymbol{\\Theta}}}_{k,n} {\\mathbf{A}}_{n} \\hatvg_{k,0} \\right]\\nonumber\\\\\n&-\\frac{1}{M^{2}} \\mathbb{E}\\bigg[\\Big|{\\tilde{{\\mathbf{e}}}_{k,n}^{\\H}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n} {\\mathbf{A}}_{n}\\hatvg\n_{k,0} }\\Big|^2\\bigg]\\xrightarrow[ M \\rightarrow \\infty]{\\mbox{a.s.}} 0,\n\\end{align}\nwhere the property of the variance operator $\\mathrm{var}\\left[ x \\right]=\\mathbb{E}[x^{2}]- \\mathbb{E}^{2}[x] $ together with~\\eqref{eq:GaussMarkov2} have been used. Lemmas~\\ref{lemma:asymptoticLimits} and~\\ref{lemma:asymptoticproduct} enable us to derive this DE as\n\\begin{align}\n \\frac{\\lambda}{M} {\\mathrm{var}}\\left[{\\mathbf{g}}^\\H_{k,n} \\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}  {\\mathbf{A}}_{n} \\hatvg_{k,0} \\right] \n\\asymp&  \\frac{\\bar{\\lambda}}{M}{\\delta}_{k}^{'}, \\label{eq:theorem4.7MRT}\n\\end{align}\nwhere ${\\delta}_{k}^{'}=\\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k} \\left( {\\mathbf{R}}_{k} - {\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k} \\right)$.\nSimilarly, use of Lemmas~\\ref{lemma:asymptoticLimits} and~\\ref{lemma:asymptoticproduct} in the last term of~\\eqref{eq:DLgenIntfPower} completes the proof. Thus, we have\n\\begin{align}\n  \\frac{\\lambda}{M^{2}}\\mathbb{E}\\Big[\\left|{\\mathbf{g}}^\\H_{k,n} \\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}{\\mathbf{A}}_{n} \\hatvg_{i,0} \\right|^{2}\\Big] \n&\\asymp \\frac{\\bar{\\lambda}}{M^{2}} \\mathop{\\mathrm{tr}}\\nolimits\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}{\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{i}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n} ^{\\H}{\\mathbf{R}}_{k},\\nonumber\\\\\n&=\\frac{\\bar{\\lambda}}{M}\\delta_{i}^{''},\\label{lasttermMRT}\n\\end{align}\nsince ${\\mathbf{g}}_{k,n} $ and $\\hatvg_{i,0} $ are mutually independent. Note here that $\\delta_{i}^{''}=\\frac{1}{M} \\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{i} {\\mathbf{R}}_{k}$. \n\n \\section{Proof of Theorem~\\ref{theorem:DLagedCSIRZF}}\\label{theorem3}\nFirst, we obtain  the DE of the normalization parameter $\\lambda$. Taking into account for~\\eqref{eq:lamda} and~\\eqref{eq:precoderRZF}, and making a simple algebraic manipulation, we lead to\n\\begin{align}\n\\lambda&=\n\\frac{K}{\\mathbb{E}\\Big[\\mathop{\\mathrm{tr}}\\nolimits{ {\\mathbf{\\Sigma}}} {\\mathbf{A}}_{n}{\\hat{{\\mathbf{G}}}}_{0}{\\hat{{\\mathbf{G}}}}_{0}^{\\H} {\\mathbf{A}}_{n} {{\\mathbf{\\Sigma}}} \\Big]} \\label{eq:theorem4.6}\\\\\n&= \\frac{K}{\\mathbb{E}\\Big[\\mathop{\\mathrm{tr}}\\nolimits {{\\mathbf{\\Sigma}}}  - \\mathop{\\mathrm{tr}}\\nolimits \\left( {\\bf Z}+ a M\\mat{\\mathrm{I}}_M\\right)  {{\\mathbf{\\Sigma}}}^{2} \\Big]},\n\\end{align}\nwhile its DE is \n\\begin{align}\n \\bar{\\lambda}=\\frac{K}{ \\left(\\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits{\\bf T} - \\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits \\left(\\frac{{\\bf Z}}{M} + a \\mat{\\mathrm{I}}_M\\right) {{\\mathbf{C}}}\\right)},\\label{desired1}\n\\end{align}\nwhere we have applied Theorems 1 and 2 for ${\\mathbf{K}}=\\mat{\\mathrm{I}}_M$.\nRegarding the other term of the desired signal power, we have\n\\begin{align}\n  &\\mathbb{E}\\!\\left[{{\\mathbf{g}}}^\\H_{k,n} \\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}{{\\mathbf{\\Sigma}}} {\\mathbf{A}}_{n} \\hat{{\\mathbf{g}}}_{k,0} \\right]\n\\!=\\! \\mathbb{E}\\!\\left[(\\hat{{\\mathbf{g}}}^\\H_{k,0}{\\mathbf{A}}_{n}\\!  + \\!\\tilde{{\\mathbf{e}}}^\\H_{k,n} )\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}{{\\mathbf{\\Sigma}}} {\\mathbf{A}}_{n}\\hat{{\\mathbf{g}}}_{k,0} \\right]\\label{desiredsig1}\\nonumber\\\\\n &=\\mathbb{E}\\Big[\\frac{\\hat{{\\mathbf{g}}}^\\H_{k,0} {\\mathbf{A}}_{n} \\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}{{\\mathbf{\\Sigma}}}_k{\\mathbf{A}}_{n} \\hat{{\\mathbf{g}}}_{k,0} }{1+{\\hatvg}^\\H_{k,0}{\\mathbf{A}}_{n}  \\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}{{\\mathbf{\\Sigma}}}_k \\widetilde{{\\boldsymbol{\\Theta}}}_{k,n} ^{\\H}{\\mathbf{A}}_{n}{\\hatvg}_{k,0} }\\Big],\n\\end{align}\nwhere Lemmas~\\ref{lemma:inversion} and~\\ref{lemma:asymptoticLimits} have been applied in \\eqref{desiredsig1}, while $ {{\\mathbf{\\Sigma}}}_k $ is defined as\n\\begin{align}\n {{\\mathbf{\\Sigma}}}_k\\! &=\\!\\left(\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n} ^{\\H}{\\mathbf{A}}_{n}   \\hat{{\\mathbf{G}}}_{0} \\hat{{\\mathbf{G}}}^\\H_{0} {\\mathbf{A}}_{n}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n} \\!-\\!\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n} ^{\\H}{\\mathbf{A}}_{n}\\hat{{\\mathbf{g}}}_{k,n}\\hat{{\\mathbf{g}}}^\\H_{k,n}{\\mathbf{A}}_{n}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}\n \\right.\\nonumber\\\\\n & \\left.+ {\\mathbf{Z}} + M  a\\mat{\\mathrm{I}}_M\\right)^{-1}.\\nonumber\n\\end{align}\nExploiting Lemmas~\\ref{lemma:asymptoticLimits},~\\ref{lemma:asymptoticproduct}, and Theorem~\\ref{th:detequ} gives\n\\begin{align}\n   \\mathbb{E}\\!\\left[{{\\mathbf{g}}}^\\H_{k,n}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}{{\\mathbf{\\Sigma}}} {\\mathbf{A}}_{n} \\hat{{\\mathbf{g}}}_{k,0} \\right]\\asymp \\mathbb{E}\\!\\left[\\frac{\\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}{\\mathbf{T}}}{1+\\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k}{\\mathbf{T}}}\\right].\\label{desired2}\n\\end{align}\n\nHence, use of~\\eqref{desired1} and~\\eqref{desired2} provides the DE signal power ${\\bar{S}_{k,n}}$ as\n\\begin{align}\n\\bar{S}_{k,n}\\asymp  \\bar{\\lambda}  \\bigg(\\frac{e^{-\\left( \\sigma_{\\varphi_{k}}^{2}+\\sigma_{\\phi}^{2} \\right)n}{\\delta}_{k}}{1+{\\delta}_{k}}\\bigg)^{2},\\label{eq:theorem4.5}\n\\end{align}\nwhere ${\\delta}_{k}=\\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k}{\\mathbf{T}}$. Note that the manipulation regarding $\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}$ follows a similar analysis to~\\eqref{desired2MRT}.\nWe continue with the derivation of each term of~\\eqref{eq:DLgenIntfPower}. Specifically, after applying  Lemmas~1 and~\\ref{lemma:asymptoticLimits}, as well as~\\eqref{eq:GaussMarkov2} to the first term in~\\eqref{eq:DLgenIntfPower}, we have\n\\begin{align}\n &{\\mathrm{var}}\\left[{\\mathbf{g}}^\\H_{k,n}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}  {{\\mathbf{\\Sigma}}}{\\mathbf{A}}_{n} \\hatvg_{k,0} \\right]\\nonumber\\\\\n & - \\mathbb{E}\\bigg[\\Big|\\frac{\\tilde{{\\mathbf{e}}}_{k,n}^{\\H}   \\widetilde{{\\boldsymbol{\\Theta}}}_{k,n} {{\\mathbf{\\Sigma}}}_k {\\mathbf{A}}_{n}\\hatvg\n_{k,0} }{1+\\hatvg^\\H_{k,0}  {\\mathbf{A}}_{n} \\widetilde{{\\boldsymbol{\\Theta}}}_{k,n} {{\\mathbf{\\Sigma}}}_k \\widetilde{{\\boldsymbol{\\Theta}}}_{k,n} ^{\\H}{\\mathbf{A}}_{n} \\hatvg_{k,0} }\\Big|^2\\bigg]\\xrightarrow[ M \\rightarrow \\infty]{\\mbox{a.s.}} 0,\n\\end{align}\nwhich yields\n\\begin{align}\n \\lambda {\\mathrm{var}}\\left[{\\mathbf{g}}^\\H_{k,n} \\widetilde{{\\boldsymbol{\\Theta}}}_{k,n} {{\\mathbf{\\Sigma}}} {\\mathbf{A}}_{n} \\hatvg_{k,0} \\right] \n\\asymp \\bar{\\lambda} \\frac{\\frac{1}{M}{\\delta}_{k}^{'}}{\\left(1+{\\delta}_{k}\\right)^2}, \\label{eq:theorem4.7}\n\\end{align}\nwhere we have used Theorems~\\ref{th:detequ} and~\\ref{th:detequder} as well as Lemmas~\\ref{lemma:asymptoticLimits} and~\\ref{lemma:asymptoticproduct}. Note that ${\\delta}_{k}^{'}=\\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k}{{\\bf T}}^{'}$ and ${\\mathbf{K}}= {\\mathbf{R}}_{k} - {\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k}$.\nNext, we focus on the last term of~\\eqref{eq:DLgenIntfPower}, where we make use of Theorems~\\ref{th:detequ} and~\\ref{th:detequder} as well as Lemmas~1 and~\\ref{lemma:asymptoticLimits} as before. In particular, if $i \\neq k$, we have\n\\begin{align}\n& \\mathbb{E}\\Big[\\left|{\\mathbf{g}}^\\H_{k,n}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}{{\\mathbf{\\Sigma}}}{\\mathbf{A}}_{n} \\hatvg_{i,0} \\right|^{2}\\Big] \n= \\mathbb{E}\\bigg[\\left|\\frac{{\\mathbf{g}}^\\H_{k,n}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n} {\\mathbf{\\Sigma}}_i {\\mathbf{A}}_{n} \\hat{{\\mathbf{g}}}_{i,0}  }{1+\\hatvg^\\H_{i,0} {\\mathbf{A}}_{n} {\\mathbf{\\Sigma}}_i {\\mathbf{A}}_{n} \\hatvg_{i,0} }\\right|^{2}\\bigg]\\nonumber\\\\\n&= \\mathbb{E}\\bigg[\\frac{ {\\mathbf{g}}^\\H_{k,n}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}{\\mathbf{\\Sigma}}_i {\\mathbf{A}}_{n} \\hatvg_{i,0} \\hatvg_{i,0}^{\\H}{\\mathbf{A}}_{n} {\\mathbf{\\Sigma}}_i  \\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}^{\\H}{\\mathbf{g}}_{k,n}}{\\left( 1+\\hatvg^\\H_{i,0} {\\mathbf{A}}_{n} {\\mathbf{\\Sigma}}_i{\\mathbf{A}}_{n} \\hatvg_{i,0} \\right)^{2} }\\bigg]\\nonumber\\\\\n&\\asymp \\mathbb{E}\\bigg[\\frac{ {\\mathbf{g}}^\\H_{k,n}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}{\\mathbf{\\Sigma}}_i {\\mathbf{A}}_{n}{\\mathbf{D}}_{i} {\\mathbf{A}}_{n} {\\mathbf{\\Sigma}}_i\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}^{\\H}{\\mathbf{g}}_{k,n}}{\\left( 1+\\delta_{i}\\right)^{2} }\\bigg],\\label{lastterm}\n\\end{align}\nwhere we have taken into consideration that ${\\mathbf{g}}_{k,n} $ and $\\hatvg_{i,0} $ are mutually independent. Unfortunately, upon inspecting~\\eqref{lastterm}, we observe that ${\\mathbf{\\Sigma}}_i$ is not independent of $\\hat{{\\mathbf{g}}}_{k,0}$. For this reason, we use  Lemma~\\ref{lemma:inversion2}, which gives\n\\begin{align}\n{\\mathbf{\\Sigma}}_i={{\\mathbf{\\Sigma}}}_{ik}-\\frac{{{\\mathbf{\\Sigma}}}_{ik}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}^{\\H}{\\mathbf{A}}_{n}\\hatvg_{k,0}\\hatvg_{k,0}^{\\H}{\\mathbf{A}}_{n}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}{{\\mathbf{\\Sigma}}}_{ik}}{1+\\hatvg^\\H_{k,0} {\\mathbf{A}}_{n}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n} {{\\mathbf{\\Sigma}}}_{ik}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}^{\\H}{\\mathbf{A}}_{n} \\hatvg_{k,0} },\\label{eq:theorem2.I.51}\n\\end{align}which introduces a new matrix ${{\\mathbf{\\Sigma}}}_{ik}$ to~\\eqref{lastterm} defined as\n\\begin{align}\n{{\\mathbf{\\Sigma}}}_{ik}\\!&=\\!\\left(\\!\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}^{\\H}{\\mathbf{A}}_{n}\\hat{{\\mathbf{G}}}_{0}\\hat{{\\mathbf{G}}}_{0}^\\H{\\mathbf{A}}_{n}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}\\! -\\!\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}^{\\H}{\\mathbf{A}}_{n}\\hat{{\\mathbf{g}}}_{i,0}\\hat{{\\mathbf{g}}}^\\H_{i,0}{\\mathbf{A}}_{n}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n} \\right.\\nonumber\\\\\n&\\left.\\!-\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}^{\\H}{\\mathbf{A}}_{n}\\hat{{\\mathbf{g}}}_{k,0}\\hat{{\\mathbf{g}}}^\\H_{k,0}{\\mathbf{A}}_{n}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}\\!+\\! {\\mathbf{Z}}\\! + \\!M a\\mat{\\mathrm{I}}_M\\right)^{\\!-\\!1}.\n\\end{align}\nBy substituting~\\eqref{eq:theorem2.I.51} into~\\eqref{lastterm}, we obtain\n\\begin{align}\n  \\mathbb{E}\\Big[\\left|{\\mathbf{g}}^\\H_{k,n}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}{{\\mathbf{\\Sigma}}}{\\mathbf{A}}_{n} \\hatvg_{i,0} \\right|^{2}\\Big] \n&=\\frac{{Q}_{ik}}{M\\left(1+{\\delta_{i}}\\right)^{2}},\\label{eq:theorem2.I.6}\n\\end{align}           \nwhere ${Q}_{ik}$ is given in~\\eqref{eq:theorem2.I.mu1} with $\\widetilde{{\\mathbf{\\Sigma}}}_{ik}=\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}{\\mathbf{\\Sigma}}_{ik}$.\n\\begin{figure*}\n\\begin{align}\n{Q}_{ik}&= {\\mathbf{g}}^\\H_{k,n}\n\\widetilde{{\\mathbf{\\Sigma}}}_{ik} {\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{i}\\widetilde{ {\\mathbf{\\Sigma}}}_{ik}^{\\H}  {\\mathbf{g}}_{k,n}\\!+\\!\\frac{\\left|  {\\mathbf{g}}^\\H_{k,n}\\widetilde{{{\\mathbf{\\Sigma}}}}_{ik}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}^{\\H}{\\mathbf{A}}_{n} {\\mathbf{g}}_{k,0}\\right|^{2}\\hat{{\\mathbf{g}}}^\\H_{k,0} {\\mathbf{A}}_{n} \\widetilde{{{\\mathbf{\\Sigma}}}}_{ik} {\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{i}\\widetilde{{{\\mathbf{\\Sigma}}}}_{ik}^{\\H}{\\mathbf{A}}_{n}\\hat{{\\mathbf{g}}}_{k,0}}{\\left( 1+\\hatvg^\\H_{k,0} {\\mathbf{A}}_{n}\\widetilde{{{\\mathbf{\\Sigma}}}}_{ik} \\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}^{\\H}{\\mathbf{A}}_{n}\\hatvg_{k,0} \\right)^{2}}\\nonumber\\\\\n&-2\\mathrm{Re}\\left\\{  \\frac{\\hatvg^\\H_{k,0}{\\mathbf{A}}_{n}\\widetilde{{{\\mathbf{\\Sigma}}}}_{ik}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}^{\\H} {\\mathbf{g}}_{k,n}{\\mathbf{g}}_{k,n}^{\\H}\\widetilde{{{\\mathbf{\\Sigma}}}}_{ik}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}^{\\H}{\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{i}\\widetilde{{{\\mathbf{\\Sigma}}}}_{ik}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}^{\\H}{\\mathbf{A}}_{n}\\hatvg_{k,0}}{1+\\hatvg^\\H_{k,0} {\\mathbf{A}}_{n}\\widetilde{{{\\mathbf{\\Sigma}}}}_{ik} \\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}^{\\H}{\\mathbf{A}}_{n}\\hatvg_{k,0}}\\right\\}.\n \\label{eq:theorem2.I.mu1}\n\\end{align}\n\\line(1,0){470}\n\\end{figure*}\nThe DE of each term in~\\eqref{eq:theorem2.I.mu1} is obtained as\n\\begin{align}\n \\!\\!\\!\\!{\\mathbf{g}}^\\H_{k,n}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}{\\mathbf{\\Sigma}}_{ik} {\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{i} {\\mathbf{\\Sigma}}_{ik}  \\widetilde{{\\boldsymbol{\\Theta}}}_{k,n} ^{\\H}{\\mathbf{g}}_{k,n}&\\asymp  \\frac{1}{M^{2}}\\mathop{\\mathrm{tr}}\\nolimits {\\mathbf{R}}_{k}{\\mathbf{C}}^{''}\n \\end{align}\n\\begin{align}\n \\hat{{\\mathbf{g}}}^\\H_{k,0}{\\mathbf{A}}_{n}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}{{\\mathbf{\\Sigma}}}_{ik} {\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{i}{{\\mathbf{\\Sigma}}}_{ik}\\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}^{\\H}{\\mathbf{A}}_{n}\\hat{{\\mathbf{g}}}_{k,0}\n &\\asymp  \\frac{1}{M^{2}}\\mathop{\\mathrm{tr}}\\nolimits {\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k}{\\mathbf{C}}^{''}\\nonumber\\\\\n &=\\frac{ \\delta_{k}^{''}}{M}\n\\end{align}\n\\begin{align}\n{\\mathbf{g}}_{k,n}^{\\H} \\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}{{\\mathbf{\\Sigma}}}_{ik} \\widetilde{{\\boldsymbol{\\Theta}}}_{k,n} ^{\\H}&{\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{i} \\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}{{\\mathbf{\\Sigma}}}_{ik} \\widetilde{{\\boldsymbol{\\Theta}}}_{k,n}^{\\H}{\\mathbf{A}}_{n}\\hatvg_{k,0}\\nonumber\\\\\n&\\asymp \\frac{1}{M^{2}}\\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k}{\\mathbf{C}}^{''}\\nonumber\\\\\n&=\\frac{ \\delta_{k}^{''}}{M}\n\\end{align}\n\\begin{align}\n\\!\\!{Q}_{ik}\\!\\asymp\\! \\frac{1}{M^{2}}\\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{R}}_{k}{\\mathbf{C}}^{''}\\!\\!+\\!\\frac{\\left|{\\delta_{k}}\\right|^{2}\\delta_{k}^{''}}{M\\left( 1\\!+\\!\\delta_{k} \\right)^{2}}\\!-\\!2\\mathrm{Re}\\left\\{ \\! \\frac{\\delta_{k}\\delta_{k}^{''} }{M\\left( 1\\!+\\!\\delta_{k} \\right)}\\!\\right\\}\\!,\n \\label{eq:theorem2.I.mu}\n\\end{align}\nwhere ${\\mathbf{K}}={\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{i}$ with  $\\delta_{k}^{''}=\\frac{1}{M}\\mathop{\\mathrm{tr}}\\nolimits{\\mathbf{A}}_{n}^{2}{\\mathbf{D}}_{k}{\\mathbf{C}}^{''}$.\nThis concludes the proof for the derivation of   $\\bar{\\gamma}_{k,n}$.\n\n\n\\end{appendices}\n\\section*{Acknowledgement}\nThe author would like to express his gratitude to Dr. Rajet Krishnan for his help and support in making this work possible.\n\n\n\\bibliographystyle{IEEEtran}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec1}\n\n\nQuantum entanglement is  a vital resource for quantum information\nprocessing \\cite{r1}. However, isolating a quantum system completely\nfrom its environment is plainly an impossible task and each quantum\nsystem will inevitably interact with its environment. Therefore, it\nis important to investigate the behavior of an entangled quantum\nsystem under the influence of its environment. Recently, Yu and\nEberly \\cite{r7,r8}  investigated the dynamics of two-qubit\nentangled states undergoing various modes of decoherence. They found\nthat it takes an infinite time to complete decoherence locally, the\nglobal entanglement may be lost in  a finite time, and the decay of\na single-qubit coherence can be slower than the decay of a two-qubit\nentanglement. The abrupt disappearance of entanglement in a finite\ntime was named ''entanglement sudden death''(ESD). A geometric\ninterpretation of the phenomenon is given in Ref.\\cite{Geometric}.\nIn addition, experimental evidences of ESD have been reported for\noptical setups \\cite{Optical1,Optical2}  and atomic ensembles\n\\cite{Atomic}.  Clearly, ESD can seriously affect the applications\nof entangled states in  a practical quantum information processing.\nRecently,  dynamics of entanglement has received increasing\nattention \\cite{nomar,zhaopra1,zhaopra2}.\n\n\n\nESD in  finite-dimensional systems is not limited only to two-qubit\nsystems. It may be occurs in a composite quantum system with a\nlarger dimension  and a multiqubit system  as well\n\\cite{Large1,Large2,r9,r10,mul1,mul2,mul3}.  The dissipative\ndynamics for a specific one-parameter class of states in a\nqubit-qutrit ($2\\otimes3$) system interacting with dephasing and\ndepolarizing channels was studied by Ann \\emph{et al.} \\cite{r9} and\nKhan \\cite{r10}, respectively. Ann \\emph{et al.} \\cite{r9}\nconjectured that ESD exists in all bipartite quantum systems. Khan\n\\cite{r10} showed that no ESD happens in any density matrix of a\nqubit-qutrit system when only the qubit is coupled to its local\ndepolarizing channel but the re-birth of entanglement occurs in\nparticular initial states. However, for general qubit-qutrit states\nand other common noise channels, the dissipative dynamics of a\nhybrid qubit-qutrit is not presented.\n\n\n\n\n\nIn this paper, we devote to investigate the behavior of entanglement\nfor a two-parameter class of states in a qubit-qutrit system under\nthe influence of both two independent (multi-local) and only one\n(local) various noise channels, such as dephasing, phase-flip,\nbit-(trit-) flip, bit-(trit-) phase-flip, and depolarizing channels.\nUsing negativity for quantifying entanglement, some analytical or\nnumerical results are presented. We find that ESD is a general\nphenomenon in a  qubit-qutrit system undergoing all these noise\nchannels, not only the case with dephasing and depolarizing channels\nobserved by others \\cite{r9}.  It is interesting to show that ESD\nalways takes place in any density matrix when each subsystem couples\nto its depolarizing channel or only the qutrit couples to its\ntrit-flip or trit-phase-flip channels. ESD can only be avoided in\nsome initial states undergoing particular noise channels. For\nexample, no ESD occurs when the system under the influence of\nmulti-local (local) dephasing, multi-local (local) phase-flip, local\nbit-flip, and local bit-phase-flip channels if it is initially in\nthe state shown in Eq.(\\ref{eq.1}) with the parameter $b=0$.  Our\nresults show that the noise channels affect the entanglement and the\ncoherence of a hybrid qubit-qutrit system in very different ways.\nFor local or multi-local dephasing, phase-flip, and depolarizing\nnoise channels, a time scale of disentanglement is usually shorter\nthan the decay of the off-diagonal dynamics, and coherence\ndisappears in an infinite-time limit. For multi-local and local\nbit-flip and bit-phase-flip channels, disentanglement  occurs in an\ninfinite time, but coherence does not disappear even though\n$t\\mapsto\\propto$.\n\n\n\n\nThis paper is organized as follows. In Sec.\\ref{sec2}, we motivate\nthe choice of a two-parameter class of states in a qubit-qutrit\nsystem, and the physical model  are introduced. In Sec.\\ref{sec3},\nentanglement dynamics of a two-parameter class of states in a\nqubit-qutrit system under the influence of  local and multi-local\ndephasing, phase-flip, bit-(trit-) flip, bit-(trit-) phase-flip, and\ndepolarizing noise channels are discussed, respectively. Discussion\nand summary are shown in Sec.\\ref{sec4}.\n\n\n\n\n\\section{ Initial states and noise model }\\label{sec2}\n\n\nA two-parameter  class of states with real parameters in a hybrid\nqubit-qutrit ($2 \\otimes 3$) quantum system \\cite{r15} can be\ndescribed as\n\\begin{eqnarray}\n\\rho_{bc}(0)&=&\na\\left(|02\\rangle\\langle02|+|12\\rangle\\langle12|\\right)+\nb(|\\phi^{+}\\rangle\\langle\\phi^{+}|+ \\nonumber\n\\\\&&|\\phi^{-}\\rangle\\langle\\phi^{-}|+|\\psi^{+}\\rangle\\langle\\psi^{+}|)+c|\\psi^{-}\\rangle\\langle\\psi^{-}|, \\label{eq.1}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n|\\phi^{\\pm}\\rangle &=& \\frac{1}{\\sqrt{2}}(|00\\rangle\\pm|11\\rangle),\\nonumber\\\\\n|\\psi^{\\pm}\\rangle &=& \\frac{1}{\\sqrt{2}}(|01\\rangle\\pm|10\\rangle),\n\\label{eq.2}\n\\end{eqnarray}\nand  $a$, $b$, and $c$ are three real parameters, and they satisfy\nthe relation $2a+3b+c=1$. $\\vert 0\\rangle$ and $\\vert 1\\rangle$ are\nthe two eigenstates of a two-level quantum system (qubit) or the\neigenstates of a three-level quantum system (qutrit) with the other\neigenstate $\\vert 2\\rangle$. The two-parameter class of states\n$\\rho_{bc}(0)$ can be obtained from an arbitrary state of a $2\n\\otimes 3$ quantum system by means of local quantum operations and\nclassical communication \\cite{r15}.\n\n\n\n\n\nFor an arbitrary mixed state $\\rho^{AB}$ in a $2\\otimes2$ or a\n$2\\otimes3$ system, its entanglement can well be characterized and\nquantified by its\nnegativity,\\textsuperscript{\\cite{negativity1,negativity2}}\n\\begin{eqnarray}                                               \\label{eq.3}\nN(\\rho^{AB})=\\parallel\\rho^{T_{B}}\\parallel_{1}-1,\n\\end{eqnarray}\nwhich corresponds to the absolute value of the sum of negative\neigenvalues of $\\rho^{T_{B}}$(the partial transpose $\\rho^{T_{B}}$\nassociated with an arbitrary product orthonormal basis $f_i \\otimes\nf_j$ is defined by the matrix elements:\n$\\rho_{m\\mu,n\\nu}^{T_B}\\equiv \\langle f_m \\otimes\nf_\\mu|\\rho^{T_B}|f_n \\otimes f_\\nu\\rangle=\\rho_{m\\nu,n\\mu}$), i.e.,\n\\begin{eqnarray}                                        \\label{eq.4}\nN(\\rho^{AB})=2\\max\\{0,-\\lambda_{S}\\},\n\\end{eqnarray}\nwhere $\\lambda_{S}$ represents the sum of all negative eigenvalues\nof $\\rho^{T_{B}}$. $N(\\rho^{AB})=0$ for an unentangled state.\nTherefore, from Eq.(\\ref{eq.1}) one can obtain the range of\nparameters as $3b<c\\leq1-3b$, i.e., $b\\in[0,1\/6)$  for the initial\nentangled states.\n\n\n\n\n\n\n\n\nIn our physical model of noise for a qubit-qutrit system (composed\nof a two-level subsystem $A$ and a three-level subsystem $B$), the\ntwo subsystems interact with their  environments independently. The\nevolved states of the initial density matrix of such a system when\nit is influenced by multi-local environments can be given compactly\nby\n\\begin{eqnarray}                                        \\label{eq.5}\n\\rho_{bc}^{AB}(t)=\\sum_{i=1}^{2}\\sum_{j=1}^{3}F_{j}^{B}E_{i}^{A}\\rho^{AB}_{bc}(0)E_{i}^{A\\dagger}F_{j}^{B\n\\dagger}.\n\\end{eqnarray}\nHere, the operators $E_{i}^{A}$ and $F_{j}^{B}$ are the Kraus\noperators which are used to describe the noise channels acting on\nthe qubit $A$ and the qutrit $B$, respectively. They satisfy the\ncompleteness relations $E_{i}^{A\\dagger}E_{i}^{A}=I$ and $F_{j}^{B\n\\dagger}F_{j}^{B}=I$ for all $t$.\n\n\n\n\\bigskip\n\\section{Dynamics of entanglement under decoherence} \\label{sec3\n\nIt is important to consider the possible degradation of any\ninitially prepared entanglement due to decoherence. In this section\nwe investigate what happens to the entanglement in a qubit-qutrit\nsystem under common noise channels for qubit (qutrit): dephasing,\nphase-flip, bit-(trit-) flip, bit-(trit-) phase-flip, and\ndepolarizing channels. The two specific environment noise situations\nwill be considered: (i) local and (ii) multi-local. In the case (i),\nonly one part of a qubit-qutrit system ($S$) interacts with its\nenvironment. In the case (ii), both the two parts of $S$ interact\nwith their local environments, independently.\n\n\n\n\\subsection{Dephasing channels} \\label{sec3.1\n\nThe set of Kraus operators for a single qubit $A$ and a single\nqutrit $B$ that reproduce the effect of a dephasing channel are\ngiven  by\n\\begin{eqnarray}                                                   \\label{eq.6}\nE_{1}^{A}&=&\\left(\\begin{array}{cc}\n1&0\\\\\n0&\\sqrt{1-\\gamma_{A}}\\\\\n\\end{array}\n\\right)\\otimes I_{3}, \\nonumber \\\\\nE_{2}^{A}&=&\\left(\\begin{array}{cc}\n0&0\\\\\n0&\\sqrt{\\gamma_{A}}\\\\\n\\end{array}\n\\right)\\otimes I_{3},\\\\                       \\label{eq.7}\nF_{1}^{B}&=&I_{2}\\otimes\\left(\\begin{array}{ccc}\n1&0&0\\\\\n0&\\sqrt{1-\\gamma_{B}}&0\\\\\n0&0&\\sqrt{1-\\gamma_{B}}\\\\\n\\end{array}\n\\right), \\nonumber \\\\\nF_{2}^{B}&=&I_{2}\\otimes\\left(\\begin{array}{ccc}\n0&0&0\\\\\n0&\\sqrt{\\gamma_{B}}&0\\\\\n0&0&0\\\\\n\\end{array}\n\\right), \\nonumber \\\\\nF_{3}^{B}&=&I_{2}\\otimes\\left(\\begin{array}{ccc}\n0&0&0\\\\\n0&0&0\\\\\n0&0&\\sqrt{\\gamma_{B}}\\\\\n\\end{array}\n\\right).\n\\end{eqnarray}\nThe time-dependent parameters are defined as\n$\\gamma_{A}=1-e^{-t\\Gamma_{A}}$ and $\\gamma_{B}=1-e^{-t\\Gamma_{B}}$.\nHere $\\gamma_{A}$, $\\gamma_{B}\\in[0,1]$. $\\Gamma_{A}$ ($\\Gamma_{B}$)\ndenotes the decay rate of the subsystem $A$ ($B$).\n\n\n\nAccording to Eq.(\\ref{eq.5}), the time-dependent evolved density\noperator $\\rho^{AB}(t)$ of a hybrid qubit-qutrit system, which is\ninitially in the entangled state $\\rho^{AB}(0)$, is given by\n\\begin{widetext}\n\\begin{center}\n\\begin{eqnarray}                                                   \\label{eq.8}\n\\rho^{AB}(t)=\\left(\\begin{array}{cccccc}\nb&0&0&0&0&0\\\\\n0&\\frac{b+c}{2}&0&\\frac{(b-c)\\sqrt{(1-\\gamma_{A})(1-\\gamma_{B})}}{2}&0&0\\\\\n0&0&\\frac{1-c-3b}{2}&0&0&0\\\\\n0&\\frac{(b-c)\\sqrt{(1-\\gamma_{A})(1-\\gamma_{B})}}{2}&0&\\frac{b+c}{2}&0&0\\\\\n0&0&0&0&b&0\\\\\n0&0&0&0&0&\\frac{1-c-3b}{2}\\\\\n\\end{array}\n\\right).\n\\end{eqnarray}\n\\end{center}\n\\end{widetext}\nIn order to characterize the dynamics of evolution for  the density\nmatrix and consider the entanglement of this system quantified by\nnegativity, we should calculate the eigenvalues of the partial\ntranspose of the time-evolved density matrix $\\rho^{AB}(t)$ and\ndetermine the potential negative eigenvalues.\n\n\n\n\\textsl{(1) Multi-local dephasing channel}. The eigenvalue which can\npotentially be negative is\n\\begin{eqnarray}                                                   \\label{eq.9}\n\\lambda=b-\\frac{c-b}{2}\\sqrt{(1-\\gamma_{A})(1-\\gamma_{B})}.\n\\end{eqnarray}\nThe entanglement of the qubit-qutrit system under a multi-local\ndephasing channel  is\n\\begin{eqnarray}                                                   \\label{eq.10}\nN^{mul-loc}(\\rho^{AB})=2\\max\\{0,\n\\frac{c-b}{2}\\sqrt{(1-\\gamma_{A})(1-\\gamma_{B})}-b\\}.\\nonumber\\\\\n\\end{eqnarray}\nIt is easy to obtain that all the states which are initially\nentangled ($3b<c\\leq1-3b$) become separable when\n$(1-\\gamma_{A})(1-\\gamma_{B})\\leq\\left(\\frac{2b}{c-b}\\right)^2$ and\n$b\\neq0$.\n\n\n\n\\textsl{(2) Qubit dephasing channel only}. If the qutrit field is\nturned off (i.e., $\\gamma_{B}=0$), that is, only a dephasing noise\nacts on qubit $A$ alone, the  entanglement of the qubit-qutrit\nsystem is\n\\begin{eqnarray}                                                     \\label{eq.11}\nN^{bit}(\\rho^{AB})=2\\max\\{0, \\frac{c-b}{2}\\sqrt{1-\\gamma_{A}}-b\\}.\n\\end{eqnarray}\nAll the states which are initially entangled ($3b<c\\leq1-3b$) become\nseparable as soon as $\\gamma_{A}\\geq\n1-\\left(\\frac{2b}{c-b}\\right)^2$ and $b\\neq0$.\n\n\n\n\\textsl{(3) Qutrit dephasing channel only}. By the same argument as\nthat made in the qubit dephasing noise, for a dephasing noise acting\non qutrit $B$ alone, the  entanglement is\n\\begin{eqnarray}                                                   \\label{eq.12}\nN^{trit}(\\rho^{AB})=2\\max\\{0, \\frac{c-b}{2}\\sqrt{1-\\gamma_{B}}-b\\}.\n\\end{eqnarray}\nA single-qutrit dephasing channel will induce ESD in the\nqubit-qutrit system when $\\gamma_{B}\\geq\n1-\\left(\\frac{2b}{c-b}\\right)^2$ and $b\\neq 0$.\n\n\nTogether with these pieces of results, one can see that ESD takes\nplace under local and multi-local dephasing channels in a general\nqubit-qutrit system if and only if the system is initially in the\nstate shown in Eq.(\\ref{eq.1}) with the parameter $b\\neq0$.\nMoreover, the smaller $c\/b$, the longer the death time range.\nHowever, if $b=0$,\nno ESD occurs.\\\\\n\n\n\n\n\n\\subsection{ Phase-flip channels}\\label{sec3.2}\n\nThe Kraus operators describing the phase-flip channel for a single\nqubit $A$ are given by\n\\begin{eqnarray}                                                   \\label{eq.13}\nE_{1}^{A}&=&\\sqrt{1-\\frac{\\gamma_{A}}{2}}\\left(\\begin{array}{cc}\n1&0\\\\\n0&1\\\\\n\\end{array}\n\\right)\\otimes I_{3}, \\nonumber \\\\\nE_{2}^{A}&=&\\sqrt{\\frac{\\gamma_{A}}{2}}\\left(\\begin{array}{cc}\n1&0\\\\\n0&-1\\\\\n\\end{array}\n\\right)\\otimes I_{3},\n\\end{eqnarray}\nand those for a single qutrit $B$ can be written as\n\\begin{eqnarray}                                                   \\label{eq.14}\nF_{1}^{B}&=&I_{2}\\otimes\\sqrt{1-\\frac{2\\gamma_{B}}{3}}\\left(\\begin{array}{ccc}\n1&0&0\\\\\n0&1&0\\\\\n0&0&1\\\\\n\\end{array}\n\\right), \\nonumber \\\\\nF_{2}^{B}&=&I_{2}\\otimes\\sqrt{\\frac{\\gamma_{B}}{3}}\\left(\\begin{array}{ccc}\n1&0&0\\\\\n0&e^{-i2\\pi\/3}&0\\\\\n0&0&e^{i2\\pi\/3}\\\\\n\\end{array}\n\\right),  \\nonumber \\\\\nF_{3}^{B}&=&I_{2}\\otimes\\sqrt{\\frac{\\gamma_{B}}{3}}\\left(\\begin{array}{ccc}\n1&0&0\\\\\n0&e^{i2\\pi\/3}&0\\\\\n0&0&e^{-i2\\pi\/3}\\\\\n\\end{array}\n\\right),\n\\end{eqnarray}\nwhere $\\gamma_{A}=1-e^{-t\\Gamma_{A}}$,\n$\\gamma_{B}=1-e^{-t\\Gamma_{B}}$, and\n$\\gamma_{A},\\gamma_{B}\\in[0,1]$. $\\Gamma_{A}$ ($\\Gamma_{B}$)\nrepresents the decay rate of the subsystem $A$ ($B$).\n\n\n\nWe can obtain the time-evolved density-matrix dynamics\n$\\rho^{AB}(t)$ of the qubit-qutrit system under a phase-flip\nchannel, according to Eq.(\\ref{eq.5}). That is,\n\\begin{widetext}\n\\begin{center}\n\\begin{eqnarray}                                                   \\label{eq.15}\n\\rho^{AB}(t)=\\left(\\begin{array}{cccccc}\nb&0&0&0&0&0\\\\\n0&\\frac{b+c}{2}&0&\\frac{(b-c)(1-\\gamma_{A})(1-\\gamma_{B})}{2}&0&0\\\\\n0&0&\\frac{1-c-3b}{2}&0&0&0\\\\\n0&\\frac{(b-c)(1-\\gamma_{A})(1-\\gamma_{B})}{2}&0&\\frac{b+c}{2}&0&0\\\\\n0&0&0&0&b&0\\\\\n0&0&0&0&0&\\frac{1-c-3b}{2}\\\\\n\\end{array}\n\\right).\n\\end{eqnarray}\n\\end{center}\n\\end{widetext}\nWith the same argument as that made in dephasing channels, some\nresults can be presented as follows.\n\n\n\n\n\\textsl{(1) Multi-local phase-flip channel}. The entanglement of the\nqubit-qutrit system under a multi-local phase-flip channel is\ncalculated as\n\\begin{widetext}\n\\begin{center}\n\\begin{eqnarray}                                                   \\label{eq.16}\n &&N^{mul-loc}(\\rho^{AB})= 2\\max\\left\\{0,\n\\frac{(c-3b)+(c-b)((1-\\gamma_{A})(1-\\gamma_{B})-1)}{2}\\right\\}.\\nonumber\n\\end{eqnarray}\n\\end{center}\n\\end{widetext}\nThat is,  all the states which are initially entangled\n($3b<c\\leq1-3b$) become separable for all values of\n$(1-\\gamma_{A})(1-\\gamma_{B})\\leq \\frac{2b}{c-b}$ and $b\\neq0$.\n\n\n\n\\textsl{(2) Qubit phase-flip channel only}. The entanglement of the\nsystem is\n\\begin{eqnarray}                                                   \\label{eq.17}\nN^{bit}(\\rho^{AB})=2\\max\\{0,\\frac{c-3b-\\gamma_{A}(c-b)}{2}\\}.\n\\end{eqnarray}\nAll the states which are initially entangled ($3b<c\\leq1-3b$) become\nseparable as soon as $\\gamma_{A}\\geq \\frac{c-3b}{c-b}$  and $b \\neq\n0$.\n\n\n\n\\textsl{(3) Qutrit phase-flip channel only}. The negativity for all\nthe states which are initially entangled ($3b<c\\leq1-3b$) is given\nas\n\\begin{eqnarray}                                                   \\label{eq.18}\nN^{trit}(\\rho^{AB})=2\\max\\{0, \\frac{c-3b-\\gamma_{B}(c-b)}{2}\\}.\n\\end{eqnarray}\nAll the states which are initially entangled become separable when\n$\\gamma_{B}\\geq \\frac{c-3b}{c-b}$ and $b\\neq 0$.\n\n\n\nSimilar to the case with a dephasing noise, ESD  also takes place\nwhen the system undergos a phase-flip noise environment but no ESD\noccurs when the parameter $b=0$. Difference from the dephasing\nchannel, the decay time range  under the multi-local phase-flip\nchannel is shorter than that in the local channel if $3b<c<5b$, and\nlonger for $5b<c\\leq 1-3b$.\n\n\n\n\n\\bigskip\n\n\\subsection{ Bit-(Trit-) flip channels} \\label{sec3.3\n\n\n\nThe Kraus operators describing the bit-flip channel for a single\nqubit $A$ are given by\n\\begin{eqnarray}                                                   \\label{eq.19}\nE_{1}^{A}&=&\\sqrt{1-\\frac{\\gamma_{A}}{2}}\\left(\\begin{array}{cc}\n1&0\\\\\n0&1\\\\\n\\end{array}\n\\right)\\otimes I_{3}, \\nonumber \\\\\nE_{2}^{A}&=&\\sqrt{\\frac{\\gamma_{A}}{2}}\\left(\\begin{array}{cc}\n0&1\\\\\n1&0\\\\\n\\end{array}\n\\right)\\otimes I_{3},\n\\end{eqnarray}\nand those for a single qutrit $B$ can be written as\n\\begin{eqnarray}                                                   \\label{eq.20}\nF_{1}^{B}&=&I_{2}\\otimes\\sqrt{1-\\frac{2\\gamma_{B}}{3}}\\left(\\begin{array}{ccc}\n1&0&0\\\\\n0&1&0\\\\\n0&0&1\\\\\n\\end{array}\n\\right), \\nonumber \\\\\nF_{2}^{B}&=&I_{2}\\otimes\\sqrt{\\frac{\\gamma_{B}}{3}}\\left(\\begin{array}{ccc}\n0&0&1\\\\\n1&0&0\\\\\n0&1&0\\\\\n\\end{array}\n\\right), \\nonumber \\\\\nF_{3}^{B}&=&I_{2}\\otimes\\sqrt{\\frac{\\gamma_{B}}{3}}\\left(\\begin{array}{ccc}\n0&1&0\\\\\n0&0&1\\\\\n1&0&0\\\\\n\\end{array}\n\\right),\n\\end{eqnarray}\nwhere $\\gamma_{A}=1-e^{-t\\Gamma_{A}}$,\n$\\gamma_{B}=1-e^{-t\\Gamma_{B}}$, and\n$\\gamma_{A},\\gamma_{B}\\in[0,1]$.\n\n\n\n\nThe matrix elements of $\\rho$ after the interaction time $t$ under a\nmulti-local bit-(trit-) flip channel are given by\n\n\\begin{widetext}\n\\begin{center}\n\\begin{eqnarray}                                                   \\label{eq.21}\n\\rho_{11}(t)&=\\rho_{55}(t)=&\\frac{1}{12}(12b+3(b-c)\\gamma_{A}(\\gamma_{B}-1)\\nonumber\\\\&&+2(1-6b)\\gamma_{B}),\\nonumber\\\\\n\\rho_{22}(t)&=\\rho_{44}(t)=&\\frac{1}{12}(6(b+c)-3(b-c)\\gamma_{A}(\\gamma_{B}-1)\\nonumber\\\\&&+(2-6b-6c)\\gamma_{B}),\\nonumber\\\\\n\\rho_{33}(t)&=\\rho_{66}(t)=&\\frac{1}{6}\\left(3(1-3b-c)+(9b+3c-2)\\gamma_{B}\\right),\\nonumber\\\\\n\\rho_{15}(t)&=\\rho_{51}(t)=&\\frac{1}{12}(b-c)\\gamma_{A}(3-2\\gamma_{B}),\\nonumber\\\\\n\\rho_{35}(t)&=\\rho_{53}(t)=&\\rho_{16}(t)=\\rho_{61}(t)=\\frac{1}{12}(b-c)(2-\\gamma_{A})\\gamma_{B},\\nonumber\\\\\n\\rho_{26}(t)&=\\rho_{62}(t)=&\\rho_{34}(t)=\\rho_{43}(t)=\\frac{1}{12}(b-c)\\gamma_{A}\\gamma_{B},\\nonumber\\\\\n\\rho_{24}(t)&=\\rho_{42}(t)=&\\frac{1}{12}(b-c)(2-\\gamma_{A})(3-2\\gamma_{B}),\n\\end{eqnarray}\n\\end{center}\n\\end{widetext}\nand all the remaining matrix elements are zero. Here\n$\\rho=(\\rho_{ij})$  and $i,j=1,\\ldots,6$.\n\n\n\n\n\\textsl{(1) Qubit bit-flip channel only}. We obtain the same results\nas the case with a phase-flip channel.\n\n\n\n\\textsl{(2) Qutrit trit-flip channel only}. The negativity for all\nthe states which are initially entangled ($3b<c\\leq1-3b$) can be\ngiven by\n\\begin{widetext}\n\\begin{center}\n\\begin{eqnarray}                                                        \\label{eq.22}\nN^{trit}(\\rho^{AB})=2\\max\\left\\{0,\\frac{3b-9c-(1-8b+2c)\\gamma_{B}}{6}\\right\\}.\n\\end{eqnarray}\n\\end{center}\n\\end{widetext}\nThe states  become separable once $\\gamma_{B}\\geq\n\\frac{3c-9b}{1-8b+2c}$.\n\n\n\n\\textsl{(3) Multi-local bit-(trit-) flip channel}. For simplicity,\nwe choose the local asympotic bit-(trit-) flip rates\n$\\Gamma_{A}=\\Gamma_{B}=\\Gamma$ to discuss the effect of this noise\non entanglement. Unfortunately, it is difficult to calculate the\nanalytical eigenvalues of the partial transpose of the time-evolved\ndensity matrix. Therefore, we take the numerical calculation for\nvarious initial states as examples to investigate the dynamics of\nentanglement of the qubit-qutrit system under a multi-local\nbit-(trit-) flip channel.\n\n\n\n\\begin{figure}[!h\n\\begin{center}\n\\includegraphics[width=8 cm,angle=0]{fig1.eps}\n\\caption{Dynamics of entanglement for the system undergoing the\nmulti-local bit- (trit-) flip noise with $a=0$. $N$ represents the\nnegativity of a hybrid qubit-qutrit system. $\\gamma$ is the\ntime-dependent parameter and\n$\\gamma_A=\\gamma_B=\\gamma=1-e^{-t\\Gamma}$. The solid, dashed,\nshort-dashed, dashed-dotted, and dotted lines correspond to $b=0$,\n$b=1\/30$, $b=2\/30$, $b=3\/30$, and $b=4\/30$, respectively.}\n\\label{Fig1}\n\\end{center}\n\\end{figure}\n\n\n\n\n\\begin{figure}[!h\n\\begin{center}\n\\includegraphics[width=8 cm,angle=0]{fig2.eps}\n\\caption{Dynamics of entanglement for the system undergoing the\nmulti-local bit-(trit-) flip noise with parameter $b=0$. The solid,\ndashed,  dashed-dotted, and dotted lines correspond to $c=1$,\n$c=3\/4$, $c=1\/2$, and $c=1\/4$, respectively.}\\label{Fig2}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\\begin{figure}[!h\n\\begin{center}\n\\includegraphics[width=8 cm,angle=0]{fig3.eps}\n\\caption{Dynamics of entanglement for the system undergoing the\nmulti-local bit-(trit-) flip noise with $b=1\/20$. The solid, dashed,\ndashed-dotted, and dotted lines correspond to $c=16\/20$, $c=12\/20$,\n$c=8\/20$, and $c=4\/20$, respectively.}\\label{Fig3}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\nAs the first example, we consider the initial entangled states shown\nin Eq.(\\ref{eq.1}) with the parameter $a=0$ which are equivalent to\nWerner states \\cite{r17}  in a $2 \\otimes 2$ systems but with\ndifferent noise channels. Their dynamics of entanglement is shown in\nFig.\\ref{Fig1}. We note that the time-evolved density matrix for the\ninitial state with the parameters $a=b=0$ is just the case for the\nmaximally entangled Bell state $\\vert \\psi^-\\rangle$ \\cite{r18},\nand the system does not suffer from ESD under a multi-local\nbit-(trit-) flip channel.\n\n\n\nAs a second example, we consider the initial entangled states with\nthe parameter $b=0$. Their dynamics of entanglement is shown in\nFig.\\ref{Fig2}.\n\n\n\n\nFinally, we consider the initial states with $a, b, c >0$. The\ndynamics of entanglement is plotted in Fig.\\ref{Fig3} for a\nparticular value of the parameter $b=0.05$.\n\n\n\n\nFrom the above analysis and numerical results, we conjecture that\nESD in a hybrid qubit-qutrit system under local and multi-local bit-\n(trit-) flip channels is a general phenomenon. No ESD takes place\nunder a qubit-flip channel alone if and only if the qubit-qutrit\nsystem is initially in the states shown in Eq.(\\ref{eq.1}) with the\nparameter $b=0$. However, ESD always occurs when the system\nundergoes a qutrit-flip channel alone.\n\n\n\n\\bigskip\n\n\n\\subsection{Bit-(Trit-) phase-flip channels}\\label{sec3.4}\n\n\n\nThe Kraus operators describing the bit-phase flip channel for a\nsingle qubit $A$ are given by\n\\begin{eqnarray}                                                   \\label{eq.23}\nE_{1}^{A}&=&\\sqrt{1-\\frac{\\gamma_{A}}{2}}\\left(\\begin{array}{cc}\n1&0\\\\\n0&1\\\\\n\\end{array}\n\\right)\\otimes I_{3}, \\nonumber \\\\\nE_{2}^{A}&=&\\sqrt{\\frac{\\gamma_{A}}{2}}\\left(\\begin{array}{cc}\n0&-i\\\\\ni&0\\\\\n\\end{array}\n\\right)\\otimes I_{3},\n\\end{eqnarray}\nand those for a single qutrit $B$ can be written as\n\\begin{eqnarray}                                                   \\label{eq.24}\nF_{1}^{B}&=&I_{2}\\otimes\\sqrt{1-\\frac{2\\gamma_{B}}{3}}\\left(\\begin{array}{ccc}\n1&0&0\\\\\n0&1&0\\\\\n0&0&1\\\\\n\\end{array}\n\\right), \\nonumber \\\\\nF_{2}^{B}&=&I_{2}\\otimes\\sqrt{\\frac{\\gamma_{B}}{6}}\\left(\\begin{array}{ccc}\n0&0&e^{i2\\pi\/3}\\\\\n1&0&0\\\\\n0&e^{-i2\\pi\/3}&0\\\\\n\\end{array}\n\\right), \\nonumber \\\\\nF_{3}^{B}&=&I_{2}\\otimes\\sqrt{\\frac{\\gamma_{B}}{6}}\\left(\\begin{array}{ccc}\n0&0&e^{-i2\\pi\/3}\\\\\n1&0&0\\\\\n0&e^{i2\\pi\/3}&0\\\\\n\\end{array}\n\\right),\\nonumber\\\\\nF_{4}^{B}&=&I_{2}\\otimes\\sqrt{\\frac{\\gamma_{B}}{6}}\\left(\\begin{array}{ccc}\n0&e^{-i2\\pi\/3}&0\\\\\n0&0&e^{i2\\pi\/3}\\\\\n1&0&0\\\\\n\\end{array}\n\\right), \\nonumber \\\\\nF_{5}^{B}&=&I_{2}\\otimes\\sqrt{\\frac{\\gamma_{B}}{6}}\\left(\\begin{array}{ccc}\n0&e^{i2\\pi\/3}&0\\\\\n0&0&e^{-i2\\pi\/3}\\\\\n1&0&0\\\\\n\\end{array}\n\\right),\n\\end{eqnarray}\nwhere $\\gamma_{A}=1-e^{-t\\Gamma_{A}}$,\n$\\gamma_{B}=1-e^{-t\\Gamma_{B}}$, and\n$\\gamma_{A},\\gamma_{B}\\in[0,1]$.\n\n\n\n\nThe  elements of the density matrix $\\rho$ after the interaction\ntime $t$ under multi-local bit-(trit-) phase-flip channels are given\nby\n\\begin{widetext}\n\\begin{center}\n\\begin{eqnarray}                                                   \\label{eq.25}\n\\rho_{11}(t)&=\\rho_{55}(t)=&b+\\frac{1}{4}(b-c)\\gamma_{A}(\\gamma_{B}-1)+(\\frac{1}{6}-b)\\gamma_{B},\\nonumber\\\\\n\\rho_{22}(t)&=\\rho_{44}(t)=&\\frac{1}{12}(6(b+c)-3(b-c)\\gamma_{A}(\\gamma_{B}-1)+(2-6b-6c)\\gamma_{B}),\\nonumber\\\\\n\\rho_{33}(t)&=\\rho_{66}(t)=&\\frac{1}{6}(3(1-3b-c)+(9b+3c-2)\\gamma_{B}),\\nonumber\\\\\n\\rho_{15}(t)&=\\rho_{51}(t)=&-\\frac{1}{12}(b-c)\\gamma_{A}(3-2\\gamma_{B}),\\nonumber\\\\\n\\rho_{16}(t)&=\\rho_{61}(t)=&\\rho_{35}(t)=\\rho_{53}(t)=\\frac{1}{24}(b-c)(\\gamma_{A}-2)\\gamma_{B},\\nonumber\\\\\n\\rho_{24}(t)&=\\rho_{42}(t)=&\\frac{1}{12}(b-c)(2-\\gamma_{A})(3-2\\gamma_{B}),\\nonumber\\\\\n\\rho_{26}(t)&=\\rho_{62}(t)=&\\rho_{34}(t)=\\rho_{43}(t)=\\frac{1}{12}(b-c)\\gamma_{A}\\gamma_{B},\n\\end{eqnarray}\n\\end{center}\n\\end{widetext}\nand all the remaining matrix elements are zero.\n\n\n\n\\textsl{(1) Local bit-(trit-) phase-flip channel only}. We obtain\nthe same results as the case with a bit-(trit-) flip channel.\n\n\n\n\n\n\n\n\\begin{figure}[!h\n\\begin{center}\n\\includegraphics[width=8 cm,angle=0]{fig4.eps}\n\\caption{Dynamics of entanglement for the system undergoing the\nmulti-local bit-(trit-) phase-flip noise with  the parameter $a=0$.\nThe solid, dashed, short-dashed, dashed-dotted, and dotted lines\ncorrespond to $b=0$, $b=1\/30$, $b=2\/30$, $b=3\/30$, and $b=4\/30$,\nrespectively.}\\label{Fig4}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\\begin{figure}[!h\n\\begin{center}\n\\includegraphics[width=8 cm,angle=0]{fig5.eps}\n\\caption{Dynamics of entanglement for the system undergoing the\nmulti-local bit-(trit-) phase-flip noise with the parameter $b=0$.\nThe solid, dashed, dashed-dotted, and dotted lines correspond to\n$c=1$, $c=3\/4$, $c=1\/2$, and $c=1\/4$, respectively.}\\label{Fig5}\n\\end{center}\n\\end{figure}\n\n\n\n\n\\begin{figure}[!h\n\\begin{center}\n\\includegraphics[width=8 cm,angle=0]{fig6.eps}\n\\caption{Dynamics of entanglement for the system undergoing the\nmulti-local bit-(tri-t) phase-flip noise with the parameter\n$b=1\/20$. The solid, dashed, dashed-dotted, and dotted lines\ncorrespond to $c=16\/20$, $c=12\/20$, $c=8\/20$, and $c=4\/20$,\nrespectively. }\\label{Fig6}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\\textsl{(2) Multi-local bit-(trit-) phase-flip channel}. For\nsimplicity, the time dependent parameters are also defined as\n$\\gamma_{A}=\\gamma_{B}=\\gamma$. It is  difficult to obtain the\nanalytical results. The similar work is made as that in the case\nwith a bit-(trit-) flip channel, and the dynamics of entanglement of\nthe initial states with the parameter $a=0$ is displayed in\nFig.\\ref{Fig4}. Different from a multi-local bit-(trit-) flip\nchannel, one can see that there exists ESD for the maximally\nentangled Bell state (solid line). The dynamics of entanglement of\nthe initial states shown in Eq.(\\ref{eq.1}) with the parameters\n$b=0$ and  $a, b, c\n> 0$ are displayed in Fig.\\ref{Fig5} and Fig.\\ref{Fig6}, respectively.\n\n\n\n\n\n\n\\subsection{Depolarizing channels}\\label{sec3.5\n\n\n\n\nA depolarizing channel represents the process in which the density\nmatrix is dynamically replaced by the maximally mixed state $I\/d$.\nHere $I$ is the identity matrix of a single qudit. The set of Kraus\noperators that reproduces the effect of the depolarizing channel for\na single qubit $A$  are given by\n\\begin{eqnarray}                                                   \\label{eq.26}\nE_{1}^{A} &=& \\sqrt{1-\\frac{3\\gamma_{A}}{4}}I_{6},\\nonumber\\\\\nE_{2}^{A} &=& \\sqrt{\\frac{\\gamma_{A}}{4}}\\sigma_{1}\\otimes I_{3},\\nonumber\\\\\nE_{3}^{A} &=& \\sqrt{\\frac{\\gamma_{A}}{4}}\\sigma_{2}\\otimes\nI_{3},\\nonumber\\\\\nE_{4}^{A}  &=& \\sqrt{\\frac{\\gamma_{A}}{4}}\\sigma_{3}\\otimes I_{3},\n\\end{eqnarray}\nwhere $\\sigma_i$ ($i=1,2,3$) are the three Pauli matrices. The Kraus\noperators describing a single-qutrit depolarizing noise are given\nby\\textsuperscript{\\cite{19}}\n\\begin{eqnarray}                                                   \\label{eq.27}\nF_{1}^{B} &=& I_{2}\\otimes\\sqrt{1-\\frac{8\\gamma_{B}}{9}}I_{3},\\nonumber \\\\\nF_{2}^{B} &=& I_{2}\\otimes \\frac{\\sqrt{\\gamma_{B}}}{3}Y,\\nonumber \\\\\nF_{3}^{B} &=& I_{2}\\otimes\n\\frac{\\sqrt{\\gamma_{B}}}{3}Z,\\nonumber \\\\\nF_{4}^{B} &=& I_{2}\\otimes \\frac{\\sqrt{\\gamma_{B}}}{3}Y^{2},\\nonumber \\\\\nF_{5}^{B} &=& I_{2}\\otimes \\frac{\\sqrt{\\gamma_{B}}}{3}YZ,\\nonumber \\\\\nF_{6}^{B} &=& I_{2}\\otimes \\frac{\\sqrt{\\gamma_{B}}}{3}Y^{2}Z,\\nonumber \\\\\nF_{7}^{B} &=& I_{2}\\otimes\\frac{\\sqrt{\\gamma_{B}}}{3}YZ^{2},\\nonumber \\\\\nF_{8}^{B} &=& I_{2}\\otimes \\frac{\\sqrt{\\gamma_{B}}}{3}Y^{2}Z^{2},\\nonumber \\\\\nF_{9}^{B} &=& I_{2}\\otimes\\frac{\\sqrt{\\gamma_{B}}}{3}Z^{2},\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}                                                   \\label{eq.28}\nY &=& \\left(\\begin{array}{ccc}\n0&1&0\\\\\n0&0&1\\\\\n1&0&0\\\\\n\\end{array}\n\\right), \\nonumber\\\\ Z &=& \\left(\\begin{array}{ccc}\n1&0&0\\\\\n0&e^{i2\\pi\/3}&0\\\\\n0&0&e^{-i2\\pi\/3}\\\\\n\\end{array}\n\\right),\n\\end{eqnarray}\nand $\\gamma_{A}=1-e^{-t\\Gamma_{A}}$,\n$\\gamma_{B}=1-e^{-t\\Gamma_{B}}$, $\\gamma_{A}, \\gamma_{B}\\in[0,1]$.\n\n\n\n\n\n\nThe matrix elements of $\\rho$ after the interaction time $t$ under a\nmulti-local depolarizing channel are given by\n\\begin{widetext}\n\\begin{center}\n\\begin{eqnarray}                                                   \\label{eq.29}\n\\rho_{11}(t)&=\\rho_{55}(t)=&\\frac{1}{12}(12b+3(b-c)(\\gamma_B-1)\\gamma_{A} + 2(1-6b)\\gamma_{B}),\\nonumber\\\\\n\\rho_{22}(t)&=\\rho_{44}(t)=&\\frac{1}{12}(6(b+c)-3(b-c)(\\gamma_B-1)\\gamma_A+ (2-6b-6c)\\gamma_B),\\nonumber\\\\\n\\rho_{33}(t)&=\\rho_{66}(t)=&\\frac{1}{6}(3(1-3b-c)+(9b+3c-2)\\gamma_B),\\nonumber\\\\\n\\rho_{24}(t)&=\\rho_{42}(t)=&\\frac{1}{2}(b-c)(1-\\gamma_{A})(-\\gamma_{B}),\n\\end{eqnarray}\n\\end{center}\n\\end{widetext}\nand all the remaining matrix elements are zero.\n\n\n\n\n\n\n\n\\textsl{(1) Multi-local depolarizing channel}. The negativity for\nthe composite system with initial entangled states ($3b<c\\leq1-3b$)\nis given by\n\\begin{eqnarray}                                                   \\label{eq.30}\nN^{mul-loc}(\\rho^{AB})=2\\max\\{0,-\\lambda\\},\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\lambda=\\frac{9(b-c)\\gamma_{A}(\\gamma_{B}-1)+2\\gamma_{B}(1-9b+3c)+18b-6c}{12}.\\nonumber\\\\\n\\label{eq.31}\n\\end{eqnarray}\nIt is easy to obtain the result that all the states, which are\ninitially entangled ones, become separable if and only if\n$9(b-c)\\gamma_{A}(\\gamma_{B}-1)+2\\gamma_{B}(1-9b+3c)\\geq 6(c-3b)$.\n\n\n\n\n\\textsl{(2) Qubit depolarizing channel only}. The negativity for all\nthe states which are initially entangled ($3b<c\\leq1-3b$) is given\nby\n\\begin{eqnarray}                                                   \\label{eq.32}\nN^{bit}(\\rho^{AB})=2\\max\\left\\{0,\\frac{2c-6b-3\\gamma_{A}(c-b)}{4}\\right\\}.\n\\end{eqnarray}\nThe states become separable for all values of $\\gamma_{A}\\geq\n\\frac{2c-6b}{3(c-b)}$.\n\n\n\n\n\\textsl{(3) Qutrit depolarizing channel only}. The negativity for\nall the states which are initially entangled ($3b<c\\leq1-3b$) can be\nwritten as\n\\begin{eqnarray}                                                   \\label{eq.33}\nN^{trit}(\\rho^{AB})=2\\max\\{0, \\frac{3c-9b-(1-9b+3c)\\gamma_{B}}{6}\\}.\n\\end{eqnarray}\nThe states become separable for all values of  $\\gamma_{B}\\geq\n\\frac{3c-9b}{1-9b+3c}$.\n\n\n\n\\begin{widetext}\n\\begin{center}\n\\begin{table}[!h]\n\\tabcolsep 0pt \\caption{ESD in $2\\otimes3$ systems under certain\nnoises. }\n\\begin{center}\n\\def\\temptablewidth{0.7\\textwidth}\n{\\rule{\\temptablewidth}{1pt}}\n\\begin{tabular*}{\\temptablewidth}{@{\\extracolsep{\\fill}}cccc}\n                              & multi-local           & qubit noise only           & qutrit noise only  \\\\\\hline\n      dephasing               & ESD with $b\\neq 0$    & ESD with $b\\neq 0$         & ESD with $b\\neq 0$   \\\\\n       phase-flip             & ESD with $b\\neq 0$    & ESD with $b\\neq 0$         & ESD with $b\\neq 0$    \\\\\n       bit-(trit-) flip       & exist ESD             & ESD with $b\\neq0$          & ESD  \\\\\n      bit-(trit-) phase-flip  & exist ESD             &ESD with $b\\neq0$           & ESD  \\\\\n      depolarizing            & exist ESD             &ESD                         & ESD\n       \\end{tabular*}\n       {\\rule{\\temptablewidth}{1pt}}\n       \\end{center}\\label{tab1}\n       \\end{table}\n       \\end{center}\n\\end{widetext}\n\n\n\n\n\\section{Discussion and summary}\\label{sec4\n\n\n\n\nPutting all the pieces of our results together, one can see that ESD\nis a general phenomenon in a qubit-qutrit system undergoing various\nindependent noise channels and we show the outcomes explicitly in\nTable.\\ref{tab1}. ``ESD with $b\\neq 0$'' denotes  ESD takes places\nin a qubit-qutrit system if and only if $b\\neq 0$, that is, if b=0,\nno ESD occurs.``exist ESD'' denotes ESD may occurs in a qubit-qutrit\nsystem, but not necessary. ``ESD'' denotes ESD always occurs.\n\n\nDecoherence which is characterized by the decay of the off-diagonal\nelements of the density matrix describing the system\n\\cite{Decoherence},  results from the unwanted interactions of a\nquantum system with its environment. According to\nEqs.(\\ref{eq.8},\\ref{eq.15},\\ref{eq.21},\\ref{eq.25},\\ref{eq.29}),\none can see that the coherence of a qubit-qutrit system can be\ndestroyed completely when the system undergos the multi-local or\nlocal\n dephasing, phase-flip and depolarizing channels, while\nthe disappearance of coherence does not occur even though\n$\\gamma_{A}, \\gamma_{B}\\mapsto 1$ for multi-local or local the\nbit-(trit-)flip and bit-(trit-)phase-flip\n channels.\n\n\n\n\nUsing the negativity criterion for quantifying entanglement, it is\nshown that the density matrix of a hybrid qubit-qutrit system always\nsuffers from ESD and decoherence under both local and multi-local\ndepolarizing channels, which is different from the result that no\nESD in any density matrix of a qubit-qutrit system occurs when only\nthe qubit is coupled to its depolarizing channel presented in Ref.\n\\cite{r10}.\n\n\n\n\n\n\n\n\n\n\nThe  entanglement dynamics of a specific one-parameter class of\nstates interacting with a local and multi-lcoal dephasing noise in\nhybrid qubit-qutrit systems has been studied by Ann \\emph{et al.}\n\\cite{r9}  in 2008. We have generalized their study to a more\ngeneral case, that is, a two-parameter class of states under the\ninfluence of local and multi-local dephasing, phase-flip,\nbit-(trit-) flip, bit-(trit-) phase-flip, and depolarizing channels.\nWith analytical and numerical analysis, we have shown that ESD is a\ngeneral phenomenon in a qubit-qutrit system under various noise\nchannels,\n not only the case with dephasing and\ndepolarizing ones \\cite{r9}.  It can only be avoided in some initial\nstates undergoing particular noise channels.\n\n\n\n\n\n\nIn summary, our results show that the  environment, which causes\ndephasing, phase-flip, bit-(trit-) flip, bit-(trit-) phase-flip, and\ndepolarizing, affects the entanglement and the coherence of  a\nhybrid qubit-qutrit system in a two-parameter class of entangled\nstates in very different ways. ESD is a general phenomenon in a\nqubit-qutrit system undergoing various independent noise channels\nand it can only be avoided in some initial states undergoing\nparticular noise channels. Moreover, we can divide those noise\nchannels into two groups. For multi-local and local dephasing,\nphase-flip, and depolarizing noise channels, a time scale of\ndisentanglement is usually shorter than the decay of the\noff-diagonal dynamics, and coherence disappears in an infinite-time\nlimit $t\\mapsto\\propto$ in which $\\gamma_{A}, \\gamma_{B}\\mapsto 1$.\nFor multi-local or local bit-(trit-) flip and bit-(trit-) phase-flip\nchannels, disentanglement may occur in the infinite-time limit, but\nthe disappearance of coherence does not occur even though\n$\\gamma_{A}, \\gamma_{B}\\mapsto 1$.\n\n\n\n\n\n\\section*{ACKNOWLEDGEMENTS}\n\n\nThis work is supported by the National Natural Science Foundation of\nChina under Grant Nos. 10974020 and 11174039,  NCET-11-0031, and the\nFundamental Research Funds for the Central Universities.\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}