diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzicmj" "b/data_all_eng_slimpj/shuffled/split2/finalzzicmj" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzicmj" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\\label{intro}\n\nMotivated by a problem in the theory of extremal hypergraphs,\nRuzsa-Szemer\\'edi \\cite{RuS} proved the following two theorems.\n\n\\begin{theo}[Ruzsa-Szemer\\'edi \\cite{RuS}]\\label{removallemma}\nIf $G$ is an $n$ vertex graph from which one should remove at least\n$\\epsilon n^2$ edges in order to destroy all triangles, then $G$\ncontains at least $f(\\epsilon) n^3$ triangles.\n\\end{theo}\n\n\\begin{theo}[Ruzsa-Szemer\\'edi \\cite{RuS}]\\label{RuzSzeBeh}\nSuppose $S \\subseteq [n]$ is a set of integers containing no 3-term\narithmetic progression. Then there is a graph $G=(V,E)$ with\n$|V|=6n$ and $|E|=3n|S|$, whose edges can be (uniquely) partitioned\ninto $n|S|$ edge disjoint triangles. Furthermore, $G$ contains no\nother triangles.\n\\end{theo}\n\nThese two theorems turned out to be two of the most influential\nresults in extremal combinatorics. First, a simple application of\nthese two theorems gives a short proof of Roth's Theorem \\cite{Roth}\nstating that a subset of $[n]$ of size $\\epsilon n$ contains a\n$3$-term arithmetic progression. The results in \\cite{RuS} were\nfollowed by a long line of investigations leading to the recent\nhypergraph removal lemmas \\cite{Gowers,NRS,RS,Tao}, that also lead\nto new proofs of Szemer\\'edi's Theorem \\cite{Sztheo} and some of its\nextensions.\n\nBesides the above applications to additive number theory and\nextremal hypergraph theory, which were the original motivation for Theorems\n\\ref{removallemma} and \\ref{RuzSzeBeh}, they also turned out to have\nmany additional surprising applications. In particular, these\ntheorems also had applications in extremal combinatorics\n\\cite{Fure,CEMZ}, in the study of probabilistically checkable proofs\nand analysis of linearity tests \\cite{HW}, in communication\ncomplexity \\cite{PS}, as well as in testing monotonicity\n\\cite{FLNRRS} and testing graph properties \\cite{Alon,AS}.\n\nTheorem \\ref{removallemma}, also known as the {\\em triangle removal\nlemma}, was originally proved for simple graphs, that is, graphs\ncontaining no parallel edges. The proof of Theorem\n\\ref{removallemma} applies the regularity lemma \\cite{Sz}, which can\nonly handle graphs with constant edge multiplicity\\footnote{The edge\nmultiplicity of a graph is the maximum number of parallel edges\nbetween any pair of vertices.}. In many applications one thus has to\nbe careful and argue that the graph (or hypergraph) on which one\ntries to apply Theorem \\ref{removallemma} is indeed simple; see\n\\cite{S} for one such example. It is thus natural to ask if the\nremoval lemma also holds for multi-graphs with possibly {\\em\nunbounded} edge multiplicity. Another way of thinking about this\nquestion is whether the removal lemma holds when the edges of a\ngraph have {\\em arbitrary} weights. Note that if we were to identify\ntriangles with their vertex sets, then a simple counter example to\nsuch a removal lemma would be to take three vertices, and connect\neach pair with $n^2$ edges. In multi-graphs, however, we identify a\ntriangle with its set of edges\\footnote{Note that in simple graphs\nthere is no difference between identifying a triangle with its edge\nset or its vertex set.}, so the above example actually has $n^6$\ntriangles. We first show that even with this way of counting the\nnumber of triangles, the removal lemma does not hold in\nmulti-graphs\\footnote{It was actually stated, without proof, in some\npapers (see, e.g., \\cite{Kral2}) that the removal lemma holds only\nin simple graphs, although we are not aware of any proof of this\nfact. }.\n\n\\begin{theo}\\label{theoparaupper}\nThere exists a multi-graph $G$ on $n$ vertices, which contains only\n$n^22^{\\sqrt{8\\log n}}=n^{2+o(1)}$ triangles, and yet one should\nremove $n^2$ edges from $G$ in order to make it triangle-free.\n\\end{theo}\n\nWe note that the edge multiplicity of the multi-graph we use in the\nproof of Theorem \\ref {theoparaupper} is $2^{\\sqrt{8\\log\nn}}=n^{o(1)}$, so we see that the removal lemma fails even when the\nedge multiplicity is sub-linear in the size of the graph.\n\nObserve that if we need to remove $n^2$ edges from a graph in order to\nmake it triangle-free, then it trivially contains at least $ n^2$\ntriangles. While Theorem \\ref{theoparaupper} states that this\ntrivial lower bound cannot be substantially improved, we can still\nask if a minor improvement is possible. The main motivation is that in some\ncases (e.g., the original one in \\cite{RuS}) one actually only needs to know\nthat if a graph is far from being triangle free then it contains asymptotically\nmore\nthan $n^2$ triangles. The following theorem answers this question positively.\n\n\\begin{theo}\\label{theoparalower} If $G$ is an $n$-vertex multi-graph\nfrom which one should remove at least $n^2$ edges in order to\ndestroy all triangles, then $G$ contains $\\omega(n^2)$ triangles.\n\\end{theo}\n\nWe note that because the proof of Theorem \\ref{theoparalower}\napplies Theorem \\ref{removallemma}, the improvement we obtain is\nvery minor and gives a lower bound of roughly $n^2(\\log^*n)^c$ for\nsome $c>0$ on the number of triangles in the graph. We also remind the reader\nthat Theorem \\ref{theoparalower} can also be stated with respect to weighted\n(simple) graphs rather than multigraphs.\n\n\\section{The proofs}\n\nFor completeness we start with the short proof of Theorem \\ref{RuzSzeBeh}.\n\n\\paragraph{Proof of Theorem \\ref{RuzSzeBeh}:} We define a 3-partite\ngraph $G$ on vertex sets $A$, $B$ and $C$, of sizes $n$, $2n$ and\n$3n$ respectively, where we think of the vertices of the sets $A$,\n$B$ and $C$ as representing the sets of integers $[n]$, $[2n]$ and\n$[3n]$. For every $1 \\leq i \\leq n$ and $s \\in S$ we put a triangle\n$T_{i,s}$ in $G$ containing the vertices $i \\in A$, $i+s \\in B$ and\n$i+2s \\in C$. It is easy to see that the above $n|S|$ triangles are\nedge disjoint, because every edge determines $i$ and $s$. To see\nthat $G$ does not contain any more triangles, let us observe that\n$G$ can only contain a triangle with one vertex in each set. If the\nvertices of this triangle are $a \\in A$, $b \\in B$ and $c \\in C$,\nthen we must have $b=a+s_1$ for some $s_1 \\in S$,\n$c=b+s_2=a+s_1+s_2$ for some $s_2 \\in S$, and\n$a=c-2s_3=a+s_1+s_2-2s_3$ for some $s_3$. This means that\n$s_1,s_2,s_3 \\in S$ form an arithmetic progression, but because $S$\nis free of 3-term arithmetic progressions it must be the case that\n$s_1=s_2=s_3$ implying that this triangle is one of the triangles\n$T_{i,s}$ defined above. $\\hspace*{\\fill} \\rule{7pt}{7pt}$\n\n\\bigskip\n\nFor the proof of Theorems \\ref{theoparaupper}\nwe will need to combine Theorem \\ref{RuzSzeBeh}\nwith the following well known result of Behrend \\cite{B} that was recently slightly improved by Elkin \\cite{El}.\n\n\\begin{theo}[Behrend \\cite{B}, Elkin \\cite{El}]\\label{Behrend}\nFor every $n$, there exists $S \\subseteq [n]$ of size\n$n\/2^{\\sqrt{8\\log n}}=n^{1-o(1)}$ containing no 3-term arithmetic\nprogression.\n\\end{theo}\n\n\\paragraph{Proof of Theorem \\ref{theoparaupper}:} Let\n$G'$ be the graph of Theorem \\ref{RuzSzeBeh} when taking $S\n\\subseteq [n]$ to be a $3AP$-free set of size $n\/2^{\\sqrt{8\\log n}}$\nas guaranteed by Theorem \\ref{Behrend}. Let $G$ be the graph\nobtained by replacing every edge of $G'$ with $n\/|S|=2^{\\sqrt{8\\log\nn}}$ parallel edges. Observe that as $G'$ contains $n|S|$ edge\ndisjoint triangles, one must remove at least $n|S|$ edges from it in\norder to make it triangle-free. As $G$ contains $n\/|S|$ parallel\nedges for every edge of $G'$ we infer that one must remove $n^2$\nedges from $G$ in order to make it triangle-free. Finally, as $G'$\ncontains only $n|S|$ triangles, we infer that $G$ contains only\n$n|S|(n\/|S|)^3=n^22^{2\\sqrt{8\\log n}}$ triangles, as needed. $\\hspace*{\\fill} \\rule{7pt}{7pt}$\n\n\n\\paragraph{Proof of Theorem \\ref{theoparalower}:} Given a\nmulti-graph $G$, let $T$ be the simple graph on the same vertex set\nthat contains an edge $(u,v)$ if and only if $G$ has at most\n$g^2(n)$ edges connecting $u$ and $v$ for some function\n$g(n)=\\omega(1)$ to be chosen shortly. Let's first consider the case\nthat one needs to remove at least $\\frac{1}{2g^2(n)}n^2$ edges from\n$T$ in order to make it triangle-free. In this case, by Theorem\n\\ref{removallemma}, we know that $T$ contains at least\n$f(\\frac{1}{2g^2(n)})n^3$ triangles. Let us now choose a function\n$g(n)=\\omega(1)$ such that $f(\\frac{1}{2g^2(n)})n^3=\\omega(n^2)$.\nThis is clearly possible no matter how fast $f(\\epsilon)$ goes to 0\nwith $\\epsilon$. Specifically, given the known bounds on\n$f(\\epsilon)$ in Theorem \\ref{removallemma} (see, e.g., \\cite{KS}),\none can take $g(n)=(\\log^*n)^c$ for some constant $c>0$. Fixing this\nchoice of $g(n)$ guarantees that in this case $T$ contains\n$\\omega(n^2)$ triangles and so $G$ contains at least this many\ntriangles as well.\n\nSo we can assume that we can remove from $T$ a set of edges $E$ of\nsize $\\frac{1}{2g^2(n)}n^2$ and thus make it triangle-free. Let us\nnow remove from $G$ all the edges connecting pairs of vertices that\nare connected by $E$ in $T$. Note that we thus remove from $G$ at\nmost $g^2(n) \\cdot \\frac{1}{2g^2(n)}n^2 \\leq n^2\/2$ edges, hence the\nnew graph we obtain, let's call it $G'$, has the property that we\nshould remove at least $n^2\/2$ edges from it in order to make it\ntriangle-free. Furthermore, each edge in $G'$ has multiplicity at\nleast $g^2(n)$.\n\nLet $T'$ be the simple graph underlining $G'$, that is, the graph on\nthe same vertex set, with an edge $(u,v)$ if and only if $G'$ has an\nedge between $u$ and $v$. Assume first that $T'$ contains at least\n$n^2\/g(n)$ edges that belong to a triangle. In this case $T'$\ncontains at least $n^2\/3g(n)$ triangles, and as the edge multiplicity\nof $G'$ is at least $g^2(n)$ this means that $G'$ contains at least\n$n^2g(n)\/3$ triangles. As $G'$ is a subgraph of $G$ we infer that\n$G$ also contains $n^2g(n)\/3$ triangles.\n\nSo we can now assume that $T'$ has at most $n^2\/g(n)$ edges that\nbelong to a triangle. Let $E'$ be a set of minimal size whose\nremoval from $G'$ makes it triangle-free. Let $B$ denote the set of\npairs $(u,v)$ for which $E'$ contains at least one edge connecting\n$u$ and $v$, and note that by our assumption on $T'$ we have that\n$|B| \\leq n^2\/g(n)$. For each pair of vertices $(u,v) \\in B$ let\n$m_{u,v}$ be the number of edges connecting $u$ and $v$ that belong\nto $E'$. We claim that for every $(u,v)$ there are at least\n$m_{u,v}$ paths of length exactly 2 connecting $u$ and $v$. Indeed,\nif $G'$ contains less than $m_{u,v}$ such paths, then we can remove\nthe $m_{u,v}$ edges connecting $u$ and $v$ from $E'$ and replace\nthem by one edge from each of the paths of length 2 connecting $u$\nand $v$. The new set has fewer edges and it still makes $G'$\ntriangle-free, which contradicts the minimality of $E'$. We thus\nconclude that for every pair $u,v$ the graph $G'$ has at least\n$m^2_{u,v}$ triangles containing $u$ and $v$. Recall that $G'$ still\nhas the property that one should remove at least $n^2\/2$ edges from\nit in order to make it triangle-free. Therefore we have $\\sum\nm_{u,v} =|E'| \\geq n^2\/2$. Combining the above facts, and using\nCauchy-Schwartz, we conclude that the number of triangles in $G'$\n(and so also in $G$) is at least\n$$\n\\sum_{(u,v) \\in B} m^2_{u,v} \\geq \\frac{1}{|B|}\\left(\\sum_{(u,v) \\in\nB}m_{u,v}\\right)^2 \\geq \\frac{1}{4|B|}n^4 \\geq \\frac14g(n)n^2\\;,\n$$\nthus completing the proof. $\\hspace*{\\fill} \\rule{7pt}{7pt}$\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} Let $T$ be an infinite Toeplitz matrix with nonnegative real entries\n\\begin{equation}\\label{1}\nT=\\left(\\begin{array}{cccc}t_0 &t_{-1} &t_{-2} &\\cdots\\\\\nt_1 &t_0 &t_{-1} &\\cdots\\\\\nt_2 &t_1 &t_0 &\\cdots\\\\\n\\vdots &\\vdots &\\vdots &\\ddots\n\\end{array}\\right).\n\\end{equation}\n\nThe aim of this paper is to find the conditions under which\n\\begin{equation}\\label{2}\n\\boldsymbol{x}=T\\boldsymbol{x}, \\quad \\boldsymbol{x}=\\left(\\begin{array}{c}x_0\\\\ x_1\\\\ \\vdots\\end{array}\\right)\n\\end{equation}\nhas a bounded positive solution. By bounded positive solution we mean \\emph{a positive solution satisfying the property $\\limsup_{k\\to\\infty}x_k<\\infty$.}\n\nThe general matrix equations in the form $\\boldsymbol{x}=A\\boldsymbol{x}$ or $\\boldsymbol{x}=A\\boldsymbol{x}+\\boldsymbol{b}$, where $A$ is a finite or infinite positive matrix, $\\boldsymbol{x}$ is the vector of unknowns and $\\boldsymbol{b}$ is a known vector have been known for a long time. They are widely used for solution of various linear equations by the fixed point method, and the area of their application is wide. They are studied by different mathematical means including functional and numerical analysis (e.g. \\cite{Kelley, Krasnoselskii et al}), while the methods that are typically used for the solution of matrix equations are iterative methods.\nThe detailed discussion of various iteration methods can be found in \\cite{Varga}. A widely known application of the matrix equation $\\boldsymbol{x}=A\\boldsymbol{x}+\\boldsymbol{b}$ in economy is the input-output model or Leontief model. It describes a quantitative economic model for the interdependencies between different sectors of a national economy or different regional economies \\cite{Leontief}.\n\nThe areas of application of matrix equation \\eqref{2}, where the matrix $T$ is specified as an infinite Toeplitz matrix differ from those mentioned above.\nThe possible areas of application of \\eqref{2} include problems from the theory of stochastic processes that are closely related to the earlier studies in \\cite{Takacs}. As well, they can include applied problems from other areas that use a similar type of analytic equations.\n\nThe problems considered in \\cite{Takacs} are based on the convolution type recurrence relations. In particular, they describe the probability problems that appear as an extension of the classic ruin and ballot problems and the problems on fluctuations of sums of random variables. Further applications of the convolution type recurrence relations are known in queueing theory (e.g. \\cite{Takacs1}), dams theory (e.g. \\cite{Abramov}) and many other areas that also considered in \\cite{Takacs}.\n\n\nEquation \\eqref{2} itself with finite or infinite Toeplitz matrix $T$ has not been earlier studied, and the techniques for the study of equation \\eqref{2} come from the theory of the convolution type recurrence relations. Asymptotic analysis of those equations uses analytic techniques of generating functions with further application of Abelian or Tauberian theorems in their asymptotic analysis.\n\nFor the further discussion, let us recall a theorem in \\cite[p. 17]{Takacs} presented here in a slightly reformulated form.\n\n\\begin{thm}\\label{thm0} Let $\\nu_1$, $\\nu_2$,\\ldots, $\\nu_r$,\\ldots be mutually independent, and identically distributed random variables taking nonnegative integer values, and $N_r=\\sum_{j=1}^{r}\\nu_j$. Let $t_j=\\mathsf{P}\\{\\nu_1=j\\}$, $j=0,1,\\ldots$. If $\\mathsf{E}\\nu_1<1$, then\n\\begin{equation}\\label{14}\n\\mathsf{P}\\left\\{\\sup_{1\\leq r< \\infty}(N_r-r)\\mathsf{E}\\nu_1$. Then $\\sup_{1\\leq r<\\infty}(N_{r}-M_r)$ is a proper random variable, and $x_k$ satisfies the recurrence relations\n\\begin{equation}\\label{15}\n\\begin{array}{lllllllll}\nx_0 &= &t_0x_0 &+t_{-1}x_1 &+\\ldots &+t_{-n}x_{n}, & & &\\\\\nx_1 &= &t_1x_0 &+t_0x_1 &+t_{-1}x_2 &+\\ldots &+t_{-n}x_{n+1}, & &\\\\\n\\vdots & &\\vdots &\\vdots &\\vdots &\\vdots &\\vdots &\\ddots &\\\\\n\\end{array}\n\\end{equation}\nwhere $t_j$, $j=-n, -n+1,\\ldots$ satisfy the property $t_{-n}+t_{-n+1}+\\ldots=1$.\nThe above recurrence relations are derived by the same way as those given in \\cite[p. 17, relation (22)]{Takacs} under the proof of the theorem and based on the total probability formula. The justification of \\eqref{15} is given in the appendix.\n\n\nIn the presentation of \\eqref{15} the boundary equations for the initial values of $x_0$, $x_1$,\\ldots, $x_{n-1}$ from which the recurrence relations start are not explicitly presented. Either, we did not provide the connection between the entries of the matrix $T$ and the distributions of $\\nu_1$ and $\\mu_1$ in the explicit form, and in fact it is not a simple problem. The exact information about all these parameters is not important for the purpose of this paper. The paper is only aimed to study the fixed point equations for Toeplitz matrices, and the specific problem formulated above is an example of a possible application. While for this specific problem the values $x_k$, $k=0,1,\\ldots$ satisfy the normalization condition $\\sum_{k=0}^{\\infty}x_k=1$, in our further studies of recurrence relations \\eqref{15}, the positive values $x_0$, $x_1$,\\ldots, $x_{n-1}$ can be chosen with a higher freedom.\n\n\nThe system of equations \\eqref{15} can be represented in the form of equation \\eqref{2}, where the matrix $T$ takes the form\n\\begin{equation}\\label{8}\nT=\\left(\\begin{array}{cccccccc}t_0 &t_{-1} &\\cdots &t_{-n} &0 &0 &0 &\\cdots\\\\\nt_1 &t_0 &t_{-1} &\\cdots &t_{-n} &0 &0 &\\cdots\\\\\nt_2 &t_1 &t_0 &t_{-1} &\\cdots &t_{-n} &0 &\\cdots\\\\\n\\vdots &\\vdots &\\vdots &\\vdots &\\vdots &\\vdots &\\vdots &\\ddots\n\\end{array}\\right), \\quad t_{-n}>0,\n\\end{equation}\nand the study of this equation is central in the paper.\n\n\nThe plan of our study is as follows. Let $n=\\max\\{j: t_{-j}>0\\}$. In Section \\ref{S2}, we formulate the main results. In Section \\ref{S3}, we derive the explicit representations for the generating function for equation \\eqref{2} in the case $n=1$ and then in the case of arbitrary fixed $n$. As well, we discuss the existence of a positive solution of equation \\eqref{2} under the assumptions on the entries of the matrix $T$.\nIn Section \\ref{S4}, we prove the main results of this paper.\n\n\\section{Main results}\\label{S2}\n\nThe theorem below assumes that\n$n=\\max\\{j: t_{-j}>0\\}<\\infty.$\nUnder this assumption there are infinitely many positive solutions of equation \\eqref{2}. However, if the first $n$ positive elements $x_0$, $x_1$,\\ldots, $x_{n-1}$ of the vector $\\boldsymbol{x}$ are given, then the recurrence relations provide a unique solution of equation \\eqref{2}.\nDenote\n\\[\n\\tau_{-n}(z)=\\sum_{k=0}^{\\infty}t_{k-n}z^k, \\quad |z|\\leq1.\n\\]\n\n\\begin{thm}\\label{thm1} Assume that $n=\\max\\{j: t_{-j}>0\\}<\\infty$,\nand\n\\begin{equation}\\label{24}\n\\frac{\\mathrm{d}}{\\mathrm{d}z}\\sqrt[n]{\\tau_{-n}(z)} \\quad \\text{increases.}\n\\end{equation}\n\n\\begin{enumerate}\n\\item [(i)] If $\\sum_{k=0}^{\\infty}t_{k-n}>1$, then for any positive parameters $x_0$, $x_1$,\\ldots $x_{n-1}$ under which the solution of equation \\eqref{2} is positive, the solution depending on these parameters is bounded, and $\\lim_{k\\to\\infty}x_k=0$.\n\\item [(ii)] If $\\sum_{k=0}^{\\infty}t_{k-n}=1$, then for any positive parameters $x_0$, $x_1$,\\ldots $x_{n-1}$ under which the solution of equation \\eqref{2} is positive, the solution depending on these parameters is bounded if and only if\n\\begin{equation}\\label{19}\n\\sum_{k=1}^{\\infty}kt_{k-n}0\\}$ does not exist, we set $n=\\infty$. Then the corresponding results can be derived from the asymptotic relations as $n$ increases to infinity. In this case condition \\eqref{19} and \\eqref{24}\nwill be transformed as follows.\n\nInstead of condition \\eqref{19} we shall require that\n\\[\n\\lim_{n\\to\\infty}\\frac{1}{n}\\sum_{k=1}^{\\infty}kt_{k-n}<1.\n\\]\nInstead of \\eqref{24}, we shall assume that for all large $n$\n\\[\n\\frac{\\mathrm{d}}{\\mathrm{d}z}\\sqrt[n]{\\tau_{-n}(z)}\n\\]\nare increasing functions. Note that a slightly stronger assumption than \\eqref{24} is the requirement that\n\\[\n\\frac{\\mathrm{d}}{\\mathrm{d}z}\\log{\\tau_{-n}(z)}\n\\]\nincreases.\n\\end{rem}\n\n\\section{Case studies of equation \\eqref{2}}\\label{S3}\n\n\\subsection{The case when $T$ is presented by \\eqref{7}}\\label{S3.1}\n\nIn the particular case when $T$ is presented by \\eqref{7} we have the recurrence relations are defined by \\eqref{20}. To make further derivations clearer, we present them in the expanded form:\n\\begin{equation}\\label{9}\n\\begin{array}{lllllll}\nx_0 &= &t_0x_0 &+t_{-1}x_1,& & & \\\\\nx_1 &= &t_1x_0 &+t_0x_1 &+t_{-1}x_2,& &\\\\\n\\vdots & &\\vdots &\\vdots &\\vdots &\\ddots &\n\\end{array}\n\\end{equation}\nUsing generating functions and combining the terms of \\eqref{17} by columns, we obtain\n\\begin{equation}\\label{10}\n\\begin{aligned}\n\\chi_{-1}(z)=\\sum_{k=0}^\\infty x_kz^k&=z^0(t_0x_0+t_1x_0z+t_2x_0z^2+\\ldots)\\\\\n+&z^0(t_{-1}x_1+t_0x_1z+t_1x_1z^2+\\ldots)\\\\\n+&z^1(t_{-1}x_2+t_0x_2z+t_1x_2z^2+\\ldots)\\\\\n+&z^2(t_{-1}x_3+t_0x_3z+t_1x_3z^2+\\ldots)\\\\\n+&\\ldots\\\\\n&=\\frac{1}{z}\\big(\\tau_{-1}(z)\\chi_{-1}(z)-t_{-1}x_0\\big), \\quad |z|\\leq1.\n\\end{aligned}\n\\end{equation}\nFrom \\eqref{10} we obtain\n\\begin{equation}\\label{17}\n\\chi_{-1}(z)=\\frac{t_{-1}x_0}{\\tau_{-1}(z)-z}, \\quad |z|<1.\n\\end{equation}\n\n\nAs in \\eqref{6}, the generating function $\\chi_{-1}(z)$ depends on the choice of $x_0$. The subindex $(-1)$ in the functions $\\chi_{-1}(z)$ and $\\tau_{-1}(z)$ is $\\max\\{j: t_{-j}>0\\}$.\n\n\n\\subsection{The case when $T$ is presented by \\eqref{8}}\\label{S.2}\n\nIn the case when the system of equations is presented by \\eqref{8}, the system of recurrence relations is as follows:\n\\begin{equation}\\label{11}\n\\begin{array}{lllllllll}\nx_0 &= &t_0x_0 &+t_{-1}x_1 &+\\ldots &+t_{-n}x_n, & & &\\\\\nx_1 &= &t_1x_0 &+t_0x_1 &+t_{-1}x_2 &+\\ldots &+t_{-n}x_{n+1}, & &\\\\\n\\vdots & &\\vdots &\\vdots &\\vdots &\\vdots &\\vdots &\\ddots &\\\\\n\\end{array}\n\\end{equation}\n\nSimilarly to that given before, we are to derive the expression for the generating function $\\chi_{-n}(z)=\\sum_{k=0}^{\\infty}x_kz^k$, where subindex $(-n)$ of the function $\\chi_{-n}(z)$ is $\\max\\{j: t_{-j}>0\\}$. The derivation of $\\chi_{-n}(z)$ is provided by the same scheme as that in \\eqref{10}. The expression for $\\chi_{-n}(z)$ is\n\\begin{equation}\\label{18}\n\\chi_{-n}(z)=\\frac{\\sum_{k=0}^{n-1}x_k\\sum_{j=k+1}^{n}t_{-j}z^{n-j+k}}{\\tau_{-n}(z)-z^n}, \\quad |z|<1.\n\\end{equation}\n\n\\subsection{Existence of a positive solution of equation \\eqref{2} under different assumptions on the matrix $T$}\\label{S3.3}\nThe generating function $\\chi_{-n}(z)$ in \\eqref{18} is expressed via arbitrary chosen $n$ positive parameters $x_0$, $x_1$,\\ldots, $x_{n-1}$ such that a solution of \\eqref{2} is positive.\n\nWe now consider the two cases $\\sum_{k=0}^{\\infty}t_{k-n}\\leq1$ and $\\sum_{k=0}^{\\infty}t_{k-n}>1$.\n\nDemonstrate first that in the case $\\sum_{k=0}^{\\infty}t_{k-n}\\leq1$, positive parameters $x_0$, $x_1$,\\ldots, $x_{n-1}$ guaranteeing a positive solution of equation \\eqref{2} always exist.\n\nIndeed, setting $x_0=x_1=\\ldots=x_{n-1}$, we obtain\n\\begin{equation}\\label{26}\nx_n=\\frac{(1-t_0-t_{-1}-\\ldots-t_{-n+1})x_{n-1}}{t_{-n}}\\geq x_{n-1}.\n\\end{equation}\nThen,\n\\[\n\\begin{aligned}\nx_{n+1}&=\\frac{(1-t_1-t_0-\\ldots-t_{-n+2})x_{n-1}-t_{-n+1}x_n}{t_{-n}}\\\\\n&\\geq\\frac{(1-t_1-t_0-\\ldots-t_{-n+1})x_{n-1}}{t_{-n}}\\\\\n&= x_n.\n\\end{aligned}\n\\]\nThis procedure continues, and by induction we obtain $x_{n-1}\\leq x_{n}\\leq\\ldots$. So, the sequence $x_k$, $k=0,1,\\ldots$ is monotone increasing.\nNote that in the case $n=1$, due to the same derivation as above, if $\\sum_{k=0}^{\\infty}t_{k-1}\\leq1$, then the sequence $x_k$, $k=0,1,\\ldots$ is always monotone increasing.\n\n\nWithin the same case $\\sum_{k=0}^{\\infty}t_{k-n}\\leq1$, assume now that $x_0$, $x_1$,\\ldots, $x_{n-1}$ are chosen in some free way under which the solution of \\eqref{2} is positive.\nLet $x_{m_0}$, $0\\leq m_0\\leq n-1$, be a largest among $x_0$, $x_1$,\\ldots, $x_{n-1}$. Then among the values $x_n$, $x_{n+1}$,\\ldots, $x_{n+m_0}$ there is a value that is not smaller than $x_{m_0}$. Indeed, if we assume that $x_n< x_{m_0}$, $x_{n+1}< x_{m_0}$,\\ldots, $x_{n+m_0-1}< x_{m_0}$, then for $x_{n+m_0}$ we obtain\n\\[\n\\begin{aligned}\nx_{n+m_0}&=\\frac{x_{m_0}(1-t_0)-\\sum_{\\substack{0\\leq j\\leq n+m_0-1\\\\ j\\neq m_0}}x_jt_{m_0-j}}{t_{-n}}\\\\\n&>\\frac{x_{m_0}\\left(1-\\sum_{j=0}^{n+m_0-1}t_{m_0-j}\\right)}{t_{-n}}\\\\\n&> x_{m_0}.\n\\end{aligned}\n\\]\nSimilarly, if $m_1=\\{\\min n: x_n\\geq x_{m_0}\\}$, then among $x_{m_1}$, $x_{m_1+1}$,\\ldots, $x_{m_1+n}$ there exists the value that is not smaller than $x_{m_1}$ that is denoted by $x_{m_2}$. This procedure can be continued, and one finds the limit $x^*=\\lim_{i\\to\\infty}x_{m_i}$, that is $\\limsup_{k\\to\\infty}x_k$. The conditions under which this upper limit is finite follows from Theorem \\ref{thm1}, which is proved in the next section. Note that $m_{i+1}-m_i\\leq n$ for any $i\\geq1$.\n\n In the case when $\\sum_{k=0}^{\\infty}t_{k-n}>1$ a positive solution of equation \\eqref{2} generally does not exist. For instance, if $t_0>1$, then from the first equation of \\eqref{11} we obtain\n\\[\nt_{-1}x_1+\\ldots+t_{-n}x_n<0,\n\\]\nthat means that at least one of $x_1$, $x_2$,\\ldots, $x_n$ must be negative.\n\nBelow we demonstrate a particular case of the matrix $T$ under which a positive solution of \\eqref{2} does exist.\n\nIndeed, assume that $\\sum_{k=1}^{\\infty}t_{k-n}<1$, and as earlier set $x_0=x_1=\\ldots=x_{n-1}$, $x_0>0$. If $\\sum_{k=0}^{\\infty}t_{k-n}>1$, we set $t_{-n}^\\prime=1-\\sum_{k=1}^{\\infty}t_{k-n}$, and denote $t_{-n}=ct_{-n}^\\prime$, $c>1$.\n\nNow consider two equations. The first original equation $\\boldsymbol{x}=T\\boldsymbol{x}$, in which $\\sum_{k=0}^{\\infty}t_{k-n}>1$, and the second one $\\boldsymbol{u}=T^\\prime\\boldsymbol{u}$ in which $\\sum_{k=1}^{\\infty}t_{k-n}+t_{k-n}^\\prime=1$, and $\\boldsymbol{u}=\\left(\\begin{matrix}u_0\\\\ u_1\\\\ \\vdots\\end{matrix}\\right).$ That is, the elements $t_j$, $j=-n+1, j=-n+2,\\ldots$ in both matrices are the same, but the element $t_{-n}$ in the matrix $T$ and the corresponding element $t_{-n}^\\prime$ in the matrix $T^\\prime$ are distinct. For positive initial values set $u_0=u_1=\\ldots=u_{n-1}=x_0=x_1=\\ldots=x_{n-1}$. Then, a positive solution of the equation $\\boldsymbol{u}=T^\\prime\\boldsymbol{u}$ exists and not decreasing, i.e. $u_{n-1}\\leq u_n\\leq\\ldots$. From the equations\n\\begin{eqnarray*}\nu_k&=&t_ku_0+t_{k-1}u_1+\\ldots+t_0u_k+\\ldots+t_{-n+1}u_{k+n-1}+t_{-n}^\\prime u_{k+n}\\\\\nx_k&=&t_kx_0+t_{k-1}x_1+\\ldots+t_0x_k+\\ldots+t_{-n+1}x_{k+n-1}+t_{-n}^\\prime (x_{k+n}\/c),\n\\end{eqnarray*}\nwe obtain a clear dependence between the terms $x_k$, $k=n, n+1\\ldots$ and corresponding terms $u_k$, $k=n, n+1\\ldots$ through the constant $c$. This dependence enables us to conclude all the terms $x_k$ are positive, and hence there exists a positive solution of equation \\eqref{2}.\n\n\n\n\n\n\\section{Proof of Theorem \\ref{thm1}}\\label{S4}\n\\subsection{Lemmas}\nThe proof of the major statements of the theorem are based on the Tauberian theorem of Hardy and Littlewood \\cite{Hardy, HL}, the formulation of which is as follows.\n\n\\begin{lem}\\label{lem1} Let series\n\\[\n\\sum_{j=0}^{\\infty}a_jz^j\n\\]\nconverges for $|z|<1$ and there exists $\\gamma>0$ such that\n\\[\n\\lim_{z\\uparrow1}(1-z)^\\gamma\\sum_{n=0}^{\\infty}a_jz^j=A.\n\\]\nSuppose also that $a_j\\geq0$. Then, as $N\\to\\infty$,\n\\[\n\\sum_{j=0}^Na_j\\asymp\\frac{A}{\\Gamma(1+\\gamma)}N^\\gamma,\n\\]\nwhere $\\Gamma(x)$ is Euler's Gamma-function.\n\\end{lem}\n\nIn addition to these two lemmas we need one more lemma presented below, where we assume that $\\sum_{k=0}^{\\infty}t_{k-n}=1$.\n\n\\begin{lem}\\label{lem3} Let $w$ be a positive value. Consider the equation\n\\begin{equation}\\label{23}\nz^n=w\\tau_{-n}(z),\n\\end{equation}\nand assume that \\eqref{24} is fulfilled.\n\n\\begin{enumerate}\n\\item [$(a_1)$] If $w=1$ and $\\gamma=\\sum_{k=1}^{\\infty}kt_{k-n}\\leq n$, then there are no roots of equation \\eqref{23} in the interval $(0,1)$.\n\\item [$(a_2)$] If $w=1$ and $\\gamma=\\sum_{k=1}^{\\infty}kt_{k-n}> n$, then there is a root of equation \\eqref{23} in the interval $(0,1)$.\n\\item [$(a_3)$] If $w>1$, then there are no roots of equation \\eqref{23} in the interval $(0,1)$.\n\\item [$(a_4)$] If $w<1$, then there is a root of equation \\eqref{23} in the interval $(0,1)$.\n\\end{enumerate}\n\\end{lem}\n\n\n\\begin{proof} From \\eqref{23} we have the equation $z=\\sqrt[n]{w\\tau_{-n}(z)}$. The function $\\sqrt[n]{w\\tau_{-n}(z)}$ is increasing since $\\tau_{-n}(z)$ is increasing, and its derivative, according to assumption of the lemma, is increasing as well. Taking into account that $t_{-n}>0$, in cases $(a_1)$ and $(a_2)$ we easily arrive at the required statements, since the difference $z-\\sqrt[n]{\\tau_{-n}(z)}$ in point $z=0$ is negative and in point $z=1$ is zero. The derivative of this difference in point $z=1$ is equal to $1-(1\/n)\\tau_{-n}^\\prime(1)$. It is nonnegative in case $(a_1)$ and strictly negative in case $(a_2)$. In case $(a_3)$ the required result follows from the fact that under condition \\eqref{24} the difference $z-w\\sqrt[n]{\\tau_{-n}(z)}$ is negative for all $0\\leq z\\leq 1$. In case $(a_4)$ the result trivially follows, since the differences $z-w\\sqrt[n]{\\tau_{-n}(z)}$ in points $z=0$ and $z=1$ are of opposite signs.\n\\end{proof}\n\n\\subsection{Proof of the theorem}\n\nUnder assumption (i) of the theorem, we have\n\\begin{equation}\\label{25}\n\\sum_{k=0}^{\\infty}t_{k-n}=w\\sum_{k=0}^{\\infty}t_{k-n}^\\prime,\n\\end{equation}\nwhere $w>1$ and $\\sum_{k=0}^{\\infty}t_{k-n}^\\prime=1$. Then, according to statement $(a_3)$ of Lemma \\ref{lem3}, the denominator of the fraction on the right-hand side of \\eqref{18} is nonzero for all $z\\in[0,1]$ and hence the series $\\chi_{-n}(z)$ is continuous in $[0,1]$. As $z\\to1$, we have $\\sum_{k=0}^{\\infty}x_k=\\lim_{z \\uparrow1}\\chi_{-n}(z)<\\infty$. Then\n$\n\\lim_{k\\to\\infty}x_k=0,\n$\nand the statement of the theorem under assumption (i) is proved.\n\nUnder assumption (ii), Lemma \\ref{lem1} is applied with $\\gamma=1$. We have\n\\begin{equation}\\label{22}\n\\lim_{z\\uparrow1}(1-z)\\chi_{-n}(z)=\\lim_{z\\uparrow1}(1-z)\\frac{\\sum_{k=0}^{n-1}x_k\\sum_{j=k+1}^{n}t_{-j}z^{n-j+k}}{\\tau_{-n}(z)-z^n}.\n\\end{equation}\n\nIf condition \\eqref{19} of the theorem is satisfied, then the L'Hospital rule yields\n\\[\n\\lim_{z\\uparrow1}(1-z)\\chi_{-n}(z)=\\frac{\\sum_{k=0}^{n-1}x_k\\sum_{j=k+1}^{n}t_{-j}}{n-\\sum_{k=1}^{\\infty}kt_{k-n}}.\n\\]\nThen conditions of Lemma \\ref{lem1} are satisfied, and according to that lemma for large $N$ we have\n\\begin{equation}\\label{27}\n\\sum_{j=0}^Nx_j\\asymp\\frac{\\sum_{k=0}^{n-1}x_k\\sum_{j=k+1}^{n}t_{-j}}{n-\\sum_{k=1}^{\\infty}kt_{k-n}}N.\n\\end{equation}\nNext, it was shown in Section \\ref{S3.3} that there is an increasing sequence of indices $m_0$, $m_1$, \\ldots such that $\\lim_{i\\to\\infty}x_{m_i}=\\limsup_{k\\to\\infty}x_k$, and for any $i$, $m_{i+1}-m_i\\leq n$. This enables us to conclude that $\\sum_{j=0}^Nx_{m_j}=O(N)$, and hence\n\\eqref{27} implies\n$\n\\limsup_{k\\to\\infty}x_k<\\infty.\n$\n\nIn the particular case $n=1$, the sequence $x_k$, $k=0,1,2,\\ldots$ is non-decreasing (see Section \\ref{S3.3}), and hence there is the limit of this sequence as $k\\to\\infty$. This limit is finite, if $\\sum_{k=1}^{\\infty}kt_{k-1}<1$, and according to Abel's theorem\n\\[\n\\lim_{k\\to\\infty}x_k=\\lim_{z\\uparrow1}(1-z)\\chi_{-1}(z)=\\frac{x_0t_{-1}}{1-\\sum_{j=1}^{\\infty}jt_{j-n}}.\n\\]\nRelation \\eqref{21} follows.\n\nIf $\\sum_{k=1}^{\\infty}kt_{k-n}=n$, then the L'Hospital rule yields infinite value in limit. So, under the assumption $\\sum_{k=1}^{\\infty}kt_{k-n}=n$, the sequence $x_0$, $x_1$,\\ldots diverges for any positive initial values of $x_0$, $x_1$,\\ldots, $x_{n-1}$. If $\\sum_{k=1}^{\\infty}kt_{k-n}>n$, then according to statement $(a_2)$ of Lemma \\ref{lem3}, the fraction of the right-hand side of \\eqref{22} has a pole, and hence the sequence $x_0$, $x_1$,\\ldots diverges. Statements (ii) of the theorem are proved.\n\nUnder assumption (iii) of the theorem we have \\eqref{25}, where $w<1$ and $\\sum_{k=0}^{\\infty}t_{k-n}^\\prime=1$.\nThen, according to statement $(a_4)$ of Lemma \\ref{lem3}, the fraction of the right-hand side of \\eqref{22} has a pole. Hence the sequence $x_0$, $x_1$,\\ldots diverges in this case as well. Statement (iii) follows.\nThe theorem is proved.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:intro}\nLately, in the invariant mass spectrum of $\\Xi_c^{+} K^{-}$, very narrow excited $\\Omega_c$ states ($\\Omega_c(3000)$, $\\Omega_c(3050)$, $\\Omega_c(3066)$, $\\Omega_c(3090)$, and $\\Omega_c(3119)$) have been observed at LHCb~\\cite{Aaij:2017nav}. Quantum numbers of these newly observed states have not been determined in the experiments yet. Hence, various possibilities about the quantum numbers of these states have been speculated in recent works. In~\\cite{Agaev:2017jyt}, the states $\\Omega_c(3050)$ and $\\Omega_c(3090)$ are assigned as radial excitation of ground state $\\Omega_c(3000)$ and\n$\\Omega^*(3066)$ baryons with the $J^P= \\frac{1}{2}^{+}$ and $\\frac{3}{2}^{+}$, respectively. On the other hand, in \\cite{Karliner:2017kfm,Wang:2017vnc,Padmanath:2017lng,Wang:2017zjw,Aliev:2017led} these new states are assumed as the $P$-wave states with $J^P = \\frac{1}{2}^{-}, \\frac{1}{2}^{-}, \\frac{3}{2}^{-}, \\frac{3}{2}^{-}$ and $\\frac{5}{2}^{-}$ respectively. Moreover the new states are assumed as pentaquarks in~\\cite{Yang:2017rpg}. Similar quantum numbers of these new states are assigned in~\\cite{Wang:2017hej}. Analysis of these states is also studied with lattice QCD, and the results indicated that most probably these states have $J^P = \\frac{1}{2}^{-},~\\frac{3}{2}^{-},~\\frac{5}{2}^{-}$ quantum numbers~\\cite{Padmanath:2017lng}. Another set of quantum number assignments, namely $\\frac{3}{2}^{-}$, $\\frac{3}{2}^{-}$, $\\frac{5}{2}^{-}$, and $\\frac{3}{2}^{+}$ is given in~\\cite{Karliner:2017kfm}. In~\\cite{Wang:2017xam}, it is obtained that the prediction on mass supports assigning $\\Omega_c(3000)$ as $J^P = \\frac{1}{2}^{-}$, $\\Omega_c(3090)$ as $J^P = \\frac{3}{2}^{-}$ or the $2S$ state with $J^P = \\frac{1}{2}^+$, and $\\Omega_c(3119)$ as $J^P = \\frac{3}{2}^+$.\n\nIn this work, we estimate the strong coupling constants of $\\Omega_c^0 \\rightarrow \\Xi_c^{+} K^{-}$ in the framework of light-cone QCD sum rules.\nIn our calculations, two different possibilities on quantum numbers of $\\Omega_c$ baryons are explored:\n\\begin{enumerate}[label=\\alph*)]\n\\item All newly observed $\\Omega_c$ states have negative parities. More precisely, $\\Omega_c(3000)$, $\\Omega_c(3050)$ have quantum numbers $J^P = \\frac{1}{2}^{-}$, $\\Omega_c(3066)$, $\\Omega_c(3090)$ have $\\frac{3}{2}^{-}$, and $\\Omega_c(3119)$ has quantum numbers $\\frac{5}{2}^{-}$.\n\\item Part of newly observed $\\Omega_c$ baryons have negative parity, and another part represents as a radial excitations of ground state baryons, i.e., $\\Omega_c(3000)$ has $J^P = \\frac{1}{2}^{-}$, $\\Omega_c(3050)$ has $J^P = \\frac{1}{2}^+$; $\\Omega_c(3066)$ and $\\Omega_c(3090)$ states have quantum numbers $J^P = \\frac{3}{2}^{-}$ and $\\frac{3}{2}^{+}$, respectively.\n\\end{enumerate}\n\nNote that the strong coupling constants of $\\Omega_c \\rightarrow \\Xi_c^{+} K^{-}$ decays within the same framework are studied in~\\cite{Agaev:2017lip}, and in chiral quark model~\\cite{Ball:2006wn} respectively. However, the analysis performed in \\cite{Agaev:2017lip} is incomplete. First of all, the contribution of negative parity $\\Xi_c$ baryons is neglected entirely. Second, in our opinion the numerical analysis presented in~\\cite{Agaev:2017lip} is inconsistent.\n\nThe article is organized as follows. In section~\\ref{sec:2} the light cone sum rules for the coupling constants of $\\Omega_c \\rightarrow \\Xi_c^{+} K^{-}$ decays are derived. Section~\\ref{sec:numeric} is devoted to the analysis of the sum rules obtained in the previous section. In this section, we also estimate the widths of corresponding decays, and comparison with the experimental data is presented.\n\\section{Light cone sum rules for the strong coupling constants of $\\Omega_c \\rightarrow \\Xi_c^{+} K^-$ transitions} \n\\label{sec:2}\nFor the calculation of the strong coupling constants of $\\Omega_c \\rightarrow \\Xi_c^{+} K^-$ transitions we consider the following two correlation functions in both pictures,\n\\begin{equation}\n \\label{eq:1}\n \\Pi = i \\int d^4x e^{ipx} \\langle K(q) | \\eta_{\\Xi_c}(x) \\bar{\\eta}_{\\Omega_c}(0) | 0 \\rangle,\n\\end{equation}\nand\n\\begin{equation}\n \\label{eq:1a}\n \\Pi^\\mu = i \\int d^4x e^{ipx} \\langle K(q) | \\eta_{\\Xi_c}(x) \\bar{\\eta}_{\\Omega^{*}_{c}}^\\mu(0) | 0 \\rangle,\n\\end{equation}\nwhere $\\eta_{\\Xi_c}~(\\eta_{\\Omega_c})$ is the interpolating current of $\\Xi_c~(\\Omega_c)$ baryon and $\\eta_{\\Omega^{*}_{c}}^\\mu$ is the interpolating current of $ J^P = \\frac{3}{2}$ $\\Omega_c^*$ baryon:\n\\begin{equation}\n \\label{eq:2}\n \\begin{split}\n \\eta_{\\Xi_c} = \\frac{1}{\\sqrt{6}} \\epsilon^{abc} \\bigg\\{ & 2(u^{a^T}Cs^b) \\gamma_5 c^c + 2\\beta (u^{a^T} C \\gamma_5 s^b)c^c + (u^{a^T}C c^b) \\gamma_5 s^c \\\\\n & +\\beta (u^{a^T}C \\gamma_5 c^b) s^c + (c^{a^T}Cs^b) \\gamma_5 u^c + \\beta (c^{a^T} C \\gamma_5 s^b) u^c \\bigg\\}\n \\end{split}\n\\end{equation}\n\n\\begin{equation}\n \\label{eq:3}\n \\eta_{{\\Omega_c}} = \\eta_{\\Xi_c} ( u \\rightarrow s) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\\end{equation}\n\n\\begin{equation}\n \\label{eq:4}\n \\eta_{\\Omega_c^{*}}^\\mu = \\frac{1}{\\sqrt{3}} \\epsilon^{abc} \\bigg\\{ (s^{a^T} C \\gamma^\\mu s^b) c^c + (s^{a^T} C \\gamma^\\mu c^b) s^c + (c^{a^T} C \\gamma^\\mu s^b) s^c \\bigg\\} ~~~~~~~~~\n\\end{equation}\nwhere $a,~b$ and $c$ are color indices, $C$ is the charge conjugation operator and $\\beta$ is arbitrary parameter.\n\nWe calculate $\\Pi$ ($\\Pi_\\mu$) employing the light cone QCD sum rules (LCSR). According to the sum rules method approach, the correlation functions in Eqs.~\\eqref{eq:1} and \\eqref{eq:1a} can be calculated in two different ways:\n\\begin{itemize}\n\\item In terms of hadron parameters,\n \\item In terms of quark-gluons in the deep Euclidean domain. \n \\end{itemize}\n These two representations are then equated by using the dispersion relation, and we get the desired sum rules for corresponding strong coupling constant. The hadronic representations of the correlation functions can be obtained by saturating Eqs.~\\eqref{eq:2} and \\eqref{eq:3} with corresponding baryons.\n\n Here we would like to note that the currents $\\eta_{\\Xi_c}$, $\\eta_{\\Omega_c}$, and $\\eta_{\\Omega_c^{*}}$ interact with both positive and negative parity baryons. Using this fact for the correlation functions from hadronic part we get\n \\begin{equation}\n \\label{eq:5}\n \\Pi^{(\\text{had})} = \\sum_{\\substack{i= +,- \\\\ j= +,-}} \\frac{\\langle 0 | \\eta_{\\Xi_c} | \\Xi_c^j(p) \\rangle \\langle K(q) \\Xi_c^{j}(p)| \\Omega_c^{i}(p+q) \\rangle\\langle \\Omega_c^{i}(p+q) | \\bar{\\eta}_{\\Omega_c} |0 \\rangle}{(p^2 - m_{\\Xi_{c_j}}^2) \\big( (p+q)^2 - m_{\\Omega_{c}(i)}^2\\big)},\n \\end{equation}\n \\begin{equation}\n \\label{eq:6}\n \\Pi^{\\mu^{(\\text{had})}} = \\sum_{\\substack{i= +,- \\\\ j= +,-}} \\frac{\\langle 0 | \\eta_{\\Xi_c} | \\Xi_c^j(p) \\rangle \\langle K(q) \\Xi_c^{j}(p)| \\Omega_{c^*}^{i}(p+q) \\rangle\\langle \\Omega_{c^*}^{i}(p+q) | \\bar{\\eta}_{\\Omega_{c}}^\\mu | 0 \\rangle}{(p^2 - m_{\\Xi_{c_j}}^2) \\big( (p+q)^2 - m_{\\Omega^{*}_{c}(i)}^2\\big)}.\n \\end{equation}\n The matrix elements in Eqs.~\\eqref{eq:5} and \\eqref{eq:6} are determined as\n \\begin{equation}\n \\label{eq:32}\n \\begin{split}\n \\langle 0 | \\eta_{B} | B^{(+)}(p) \\rangle = \\lambda_{+} u(p), \\\\\n \\langle 0 | \\eta_{B} | B^{(-)}(p) \\rangle = \\lambda_{-} u(p),\n \\end{split}\n \\end{equation}\n\n \\begin{equation}\n \\label{eq:33}\n \\begin{split}\n \\langle K(q) B(p) | B(p+q) \\rangle = g \\bar{u}(p) i \\Gamma u(p+q), \n \\end{split}\n \\end{equation}\n \\begin{equation}\n \\label{eq:35}\n \\begin{split}\n \\langle K(q) B(p) | B^{*}(p+q) \\rangle = g^* \\bar{u}(p) \\Gamma^\\prime u_\\mu(p+q) q^\\mu, \n \\end{split}\n \\end{equation}\n where,\n \\begin{equation}\n \\label{eq:34}\n \\Gamma = \n \\begin{cases}\n \\gamma_5 & \\text{for}~ -(+) \\rightarrow -(+) \\\\\n 1 & \\text{for}~ -(+) \\rightarrow +(-) ~~~\\text{transitions},\n \\end{cases}\n \\end{equation}\n\n\n \\begin{equation}\n \\label{eq:36}\n\\Gamma^\\prime =\n\\begin{cases}\n1 & \\text{for}~ -(+) \\rightarrow -(+)\\\\\n\\gamma_5 & \\text{for}~ -(+) \\rightarrow +(-)~~~\\text{transitions},\n\\end{cases}\n \\end{equation}\n and $g$ is strong coupling constant of the corresponding decay, $\\lambda_{B^{(i)}}$ are the residues of the corresponding baryons, $u_\\mu$ is the Rarita-Schwinger spinors. Here the sign $+(-)$ corresponds to positive (negative) parity baryon. In further discussions, we will denote the mass and residues of ground and excited states of $\\Omega_c(\\Omega_c^*)$ baryons as: $m_0, \\lambda_0 (m_0^*, \\lambda_0^*)$, $m_1, \\lambda_1 (m_1^*, \\lambda_1^*)$, $m_2, \\lambda_2 (m_2^*, \\lambda_2^*)$ for scenario a); and for scenario b), the same notation is used as in previous case by just replacing $m_2, \\lambda_2 (m_2^*, \\lambda_2^*)$ to $m_3, \\lambda_3 (m_3^*, \\lambda_3^*)$. Moreover, the mass and residues of $\\Xi_c$ baryons are denoted as $m^\\prime_0$, $\\lambda_0^\\prime$ and $m^\\prime_1, \\lambda_1^\\prime$. Using the matrix elements defined in \\Cref{eq:32,eq:33,eq:35} for the correlation functions given in \\Cref{eq:1,eq:1a} we get (for case a):\n\n \n \n \n \n \n \n \n \n \n \n \n \n\n \\begin{equation}\n \\label{eq:26}\n \\begin{split}\n \\Pi &= i A_1 (\\slashed{p} + m_{0}^{\\prime}) \\gamma_5 (\\slashed{p} + \\slashed{q} + m_{0}) \\\\\n &+ i A_2 (\\slashed{p} + m_{0}^{\\prime})(\\slashed{p} + \\slashed{q} + m_{1})(-\\gamma_5) \\\\\n &+ i A_3 \\gamma_5 (\\slashed{p} + m_{1}^{\\prime}) (\\slashed{p} + \\slashed{q} + m_{0}) \\\\\n &+i A_4 \\gamma_5 (\\slashed{p} + m_{1}^{\\prime}) \\gamma_5 (\\slashed{p} + \\slashed{q} + m_{1})(-\\gamma_5) \\\\\n &+i A_5(\\slashed{p} + m_{0}^{\\prime}) (\\slashed{p} + \\slashed{q} + m_{2})(-\\gamma_5) \\\\\n &+i A_6 \\gamma_5 (\\slashed{p} + m_{1}^{\\prime}) \\gamma_5 (\\slashed{p} + \\slashed{q} + m_{2})(-\\gamma_5)\n \\end{split}\n \\end{equation}\n\n \\begin{equation}\n \\label{eq:27}\n \\begin{split}\n \\Pi^\\mu &= +A_1^{*} (\\slashed{p} + m_{0}^{\\prime}) (\\slashed{p} + \\slashed{q} + m_{0}^*) (-q^\\mu) \\\\\n &+ A_2^{*} (\\slashed{p} + m_{0}^{\\prime}) \\gamma_5 (\\slashed{p} + \\slashed{q} + m_{1}^*)(-\\gamma_5)(-q^\\mu)\\\\\n &+ A_3^{*} \\gamma_5 (\\slashed{p} + m_{1}^{\\prime}) \\gamma_5 (\\slashed{p} + \\slashed{q} + m_{0}^*) (-q^\\mu) \\\\\n &+ A_4^{*} \\gamma_5 (\\slashed{p} + m_{1}^{\\prime}) (\\slashed{p} + \\slashed{q} + m_{1}^*)(-\\gamma_5) (-q^\\mu) \\\\\n &+ A_5^{*} (\\slashed{p} + m_{0}^{\\prime}) \\gamma_5 (\\slashed{p} + \\slashed{q} + m_{2}^*) (-\\gamma_5) (-q^\\mu) \\\\\n &+ A_6^{*} \\gamma_5 (\\slashed{p} + m_{1}^{\\prime}) ( \\slashed{p} + \\slashed{q} + m_{2}^*)(-\\gamma_5) (-q^\\mu) + \\text{other structures},\n \\end{split}\n \\end{equation}\n where\n\n \\begin{equation}\n \\label{eq:13}\n \\begin{split}\n A_1^{(*)} =& \\frac{\\lambda_{0}^{(*)} \\lambda_{0}^{\\prime} g_1^{(*)}}{ ({m_{0}^{\\prime}}^2 - p^2 ) \\big({m_{0}^{(*)}}^2 -(p+q)^2\\big)} \\\\\n A_2^{(*)} =& A_1 \\big( m_0^{(*)} \\rightarrow m_{1}^{(*)}, \\hspace{4mm} g_{1}^{(*)} \\rightarrow g_{2}^{(*)}, \\hspace{4mm} \\lambda_{0}^{(*)} \\rightarrow \\lambda_{1}^{(*)} \\big)\\\\\n A_3^{(*)} =& A_1 \\big( m_0^{\\prime} \\rightarrow m_{1}^{\\prime}, \\hspace{4mm} g_{1}^{(*)} \\rightarrow g_{3}^{(*)}, \\hspace{4mm} \\lambda_{0}^{\\prime} \\rightarrow \\lambda_{1}^{\\prime} \\big)\\\\\n A_4^{(*)} =& A_3 \\big( m_0^{(*)} \\rightarrow m_{1}^{(*)}, \\hspace{4mm} g_{3}^{(*)} \\rightarrow g_{4}^{(*)}, \\hspace{4mm} \\lambda_{0}^{(*)} \\rightarrow \\lambda_{1}^{(*)} \\big)\\\\\n A_5^{(*)} =& A_2 \\big( m_{1}^{(*)} \\rightarrow m_{2}^{(*)}, \\hspace{4mm} g_{2}^{(*)} \\rightarrow g_{5}^{(*)}, \\hspace{4mm} \\lambda_{1}^{(*)} \\rightarrow \\lambda_{2}^{(*)} \\big)\\\\\n A_6^{(*)} =& A_5 \\big( m_0^{\\prime} \\rightarrow m_{1}^{\\prime}, \\hspace{4mm} g_{5}^{(*)} \\rightarrow g_{6}^{(*)}, \\hspace{4mm} \\lambda_{0}^{\\prime} \\rightarrow \\lambda_{1}^{\\prime} \\big)\n \\end{split}\n \\end{equation}\n \n \n \n \n \n \n The result for the scenario b) can be obtained from Eqs.~\\eqref{eq:26} and \\eqref{eq:27} by following replacements:\n \\begin{equation}\n \\label{eq:29}\n \\begin{split}\n A_5 (\\slashed{p} + m_{0}^{\\prime}) (\\slashed{p} + \\slashed{q} + m_{2})(-\\gamma_5) & \\rightarrow \\tilde{A_5} (\\slashed{p} + m_{0}^{\\prime}) i\\gamma_5 (\\slashed{p} + \\slashed{q} + m_{3}) \\\\\n A_6 \\gamma_5 (\\slashed{p} + m_{1}^{\\prime}) \\gamma_5 (\\slashed{p} + \\slashed{q} + m_{2}) (-\\gamma_5) & \\rightarrow \\tilde{A_6} \\gamma_5 (\\slashed{p} + m_{1}^{\\prime}) (\\slashed{p} + \\slashed{q} + m_{3}) \\\\\n A_5^* (\\slashed{p} + m_{0}^{\\prime}) \\gamma_5 q^\\alpha (\\slashed{p} + \\slashed{q} + m_{2}^{*})(-\\gamma_5)(-g_{\\mu \\alpha}) & \\rightarrow \\tilde{A_5} (\\slashed{p} + m_{0}^{\\prime}) q^\\alpha (\\slashed{p} + \\slashed{q} + m_{3}^{*}) (-g_{\\mu \\alpha}) \\\\\n A_6^* \\gamma_5 (\\slashed{p} + m_{1}^{\\prime}) q^\\alpha (\\slashed{p} + \\slashed{q} + m_{2}^{*})(-g_{\\mu \\alpha})(-\\gamma_5) &\\rightarrow \\tilde{A_6} \\gamma_5 (\\slashed{p} + m_{1}^{\\prime}) q^\\alpha \\gamma_5 (\\slashed{p} + \\slashed{q} + m_{3}^{*}) (-g_{\\mu \\alpha}).\n \\end{split}\n \\end{equation}\n\n Note that to derive Eq.~\\eqref{eq:27}, we used the following formula for performing summation over spins of Rarita-Schwinger spinors\n \\begin{equation}\n \\label{eq:16}\n \\sum u_\\alpha(p) \\bar{u}_\\beta(p) = -(\\slashed{p} + m) \\bigg( g_{\\alpha\\beta}- \\frac{\\gamma_\\alpha\\gamma_\\beta}{3} + \\frac{2p_\\alpha p_\\beta}{3m^2} + \\frac{p_\\alpha \\gamma_\\beta - p_\\beta \\gamma_\\alpha}{3m} \\bigg)\n \\end{equation}\n and in principle one can obtain the expression for the hadronic part of the correlation function. At this stage two problems arise. One of them is dictated by the fact that the current $\\eta_\\mu$ interacts not only with spin 3\/2 but also 1\/2 states. The matrix element of the current $\\eta_\\mu$ with spin $1\/2$ state is defined as\n \\begin{equation}\n \\label{eq:17}\n \\langle 0 | \\eta_\\mu | 1\/2 \\rangle = A \\bigg( \\gamma_\\mu - \\frac{4}{m}p_\\mu \\bigg) u(p),\n \\end{equation}\n i.e., the terms in the RHS of Eq.\\eqref{eq:16} $\\sim \\gamma_\\mu$ and the right end $(p+q)_\\mu$ contain contributions from $1\/2$ states, which should be removed. The second problem is related to the fact that not all structures appearing in Eqs.~\\ref{eq:27} are independent. In order to cure both these problems we need ordering procedure of Dirac matrices. In present work, we use ordering of Dirac matrices as $\\slashed{p} \\slashed{q} \\gamma_\\mu$. Under this ordering, only the term $\\sim g_{\\mu \\alpha}$ contains contributions solely from spin $3\/2$ states. For this reason, we will retain only $~g_{\\mu \\alpha}$ terms in the RHS of Eq.\\eqref{eq:27}.\n \n \n \n \n \n \n \n \n \n \n \n \n In order to find sum rules for the strong coupling constants of $\\Omega_c \\rightarrow \\Xi_c^+ K^-$ transitions we need to calculate the $\\Pi$ and $\\Pi_\\mu$ from QCD side in the deep Euclidean region, $p^2 \\rightarrow - \\infty$, $(p+q)^2 \\rightarrow -\\infty$. The correlation from QCD side can be calculated by using the operator product expansion.\n\n Now let demonstrate steps of calculation of the correlation function from QCD side. As an example let consider one term of correlation $\\Pi_\\mu$, i.e. consider\n\n \\begin{equation}\n \\label{eq:10}\n \\Pi_\\mu \\sim \\epsilon^{abc} \\epsilon^{a_1 b_1 c_1} \\int d^4 x e^{ipx} \\langle K(q) | \\frac{1}{\\sqrt{6}} 2 (u^{a T} C \\gamma^5 s^{b_1}) c^{c_1}(x) \\big[ \\bar{c}_{(0)}^{c_1} (\\bar{s}^{b_1}(0) \\gamma_\\mu C \\bar{s}^{a_1 T}) | 0 \\rangle \\big]\n \\end{equation}\n\n By using Wick's theorem, this term can be written as;\n \\begin{equation}\n \\label{eq:11}\n \\begin{split}\n \\Pi_\\mu \\sim& \\epsilon^{abc} \\epsilon^{a_1 b_1 c_1} \\int d^4 x e^{ipx} \\langle K(q) | \\bigg\\{ \\bar{s}^{a_1}_{1} (\\gamma_\\mu C)^T S_s^{b b_1 T}(x) C^T u^{a_1} (x) \\gamma_5 S_c^{c c_1} (x) \\\\\n &~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - (\\bar{s}^{b_1}(0) \\gamma_\\mu C S_s^{b a_1 T} C^T u^a ) \\gamma_5 S_c^{c c_1}(x) \\bigg\\} | 0 \\rangle\n \\end{split}\n \\end{equation}\n From this formula, it follows that to obtain the correlation function(s) from QCD side, first of all we need the expressions of light and heavy quark propagators. The expressions of the light quark propagator in the presence of gluonic and electromagnetic background fields are derived in \\cite{Balitsky:1987bk}:\n \\begin{equation}\n \\label{eq:20}\n \\begin{split}\n S(x) = \\frac{i \\slashed{x}}{2 \\pi^2 x^4} -\\frac{m_q}{4 \\pi x^2} - \\frac{i g_s}{16 \\pi^2} \\int & du \\bigg\\{ \\frac{\\bar{u} \\slashed{x} \\sigma_{\\alpha \\beta} + u \\sigma_{\\alpha \\beta} \\slashed{x}}{x^2} [g_s G^{\\alpha \\beta}(ux) + e_q F^{\\alpha \\beta}] \\\\\n &-\\frac{i m_q}{2}[g_s G_{\\mu\\nu} \\sigma^{\\mu \\nu} + e q F_{\\mu \\nu} \\sigma_{\\mu \\nu}] (ln \\frac{-x^2 \\Lambda^2}{4} + 2 \\gamma_E) \\bigg\\}. \n \\end{split}\n \\end{equation}\n\n The heavy quark propagator is given as,\n \\begin{equation}\n \\label{eq:21}\n \\begin{split}\n S_Q = \\int \\frac{d^4 k}{(2\\pi)^4} e^{-ikx} \\frac{i(\\slashed{k}+m_Q)}{k^2-m_Q^2} - ig_s \\int \\frac{d^4k}{(2 \\pi)^4 i} \\int_0^{1}du & \\big[ \\frac{\\slashed{k}+m_Q}{2(m_Q^2 - k^2)^2} G^{\\mu \\nu}(ux) \\sigma_{\\mu \\nu} \\\\\n &+ \\frac{i x_\\mu}{m_Q^2 - k^2}G^{\\mu \\nu}(ux) \\gamma_\\nu \\bigg]\n \\end{split}\n \\end{equation}\n where $\\gamma_E$ is the Euler constant.\n \n For calculation of the correlator function(s) we need another ingredient of light-cone sum rules, namely the matrix elements of non-local operators $\\bar{q}(x) \\Gamma q(y)$ and $\\bar{q}(x) \\Gamma G_{\\mu \\nu} q(y)$ between vacuum and the K-meson , i.e. $\\langle K(q) | \\bar{q}(x) \\Gamma q(y) | 0 \\rangle$ and $\\langle K(q) | \\bar{q}(x) \\Gamma G_{\\mu\\nu}q(y) | 0 \\rangle$. Here $\\Gamma$ is the any Dirac matrix, and $G_{\\mu\\nu}$ is the gluon field strength tensor, respectively. These matrix elements are defined in terms of K-meson distribution amplitudes (DA's). The DA's of K meson up to twist-$4$ are presented in \\cite{Ball:2006wn}.\n \n From Eqs.~\\eqref{eq:26} and \\eqref{eq:27} it follows that the different Lorentz structures can be used for construction of the relevant sum rules. Among of six couplings, we need only $A_2 (A_2^{*})$ and $A_5 (A_5^{*})$ and $A_2 (A_2^{*})$ and $\\tilde{A_5} (\\tilde{A_5^{*}})$ for the cases a) and b) respectively. For determination of these coupling constants, we need to combine sum rules obtained from different Lorentz structures. From Eqs.\\eqref{eq:26} and \\eqref{eq:27} (for case a) it follows that the following Lorentz structures $\\slashed{p} \\slashed{q} \\gamma_5$, $\\slashed{p} \\gamma_5$, $\\slashed{q} \\gamma_5$, $\\gamma_5$, and $\\slashed{p} \\slashed{q} q_\\mu$, $\\slashed{p} q_ \\mu$, $\\slashed{q} q_\\mu$, and $q_\\mu$ appear. We denote the corresponding invariant functions $\\Pi_1, \\Pi_2, \\Pi_3, \\Pi_4$ and $\\Pi_1^*, \\Pi_2^*, \\Pi_3^{*}$ and $\\Pi_4^{*}$, respectively. Explicit expressions of the invariant functions $\\Pi_i$ and $\\Pi_i^*$ are very lengthy, and therefore we do not present them in the present study.\n \nThe sum rules for the corresponding strong coupling constants are obtained by choosing the coefficients aforementioned structures and equating to the corresponding results from hadronic and QCD sides. Performing doubly Borel transformation with respect to variable $p^2$ and $(p+q)^2$ in order to suppress the contributions of higher states and continuum we get the following four equations (for each transition).\n \\begin{equation}\n \\label{eq:24}\n \\begin{split}\n \\Pi_1^{B} &= -A_1^{(B)} - A_2^{(B)} + A_3^{(B)} + A_4^{(B)} -A_5^{(B)} +A_6^{(B)}, \\\\\n \\Pi_2^{B} &= A_1^{(B)} (m_0 - m_0^\\prime) + A_2^{(B)}(-m_1 - m_0^\\prime) + A_3^{(B)} (-m_0 -m_{1}^\\prime) + A_4^{(B)}(m_{1} - m_{1}^\\prime) \\\\\n &~~~~~~+ A_5^{(B)}(-m_{2} - m_0^\\prime) + A_6^{(B)} (m_{2} - m_{1}^\\prime) \\\\\n \\Pi_3^{B} &= A_1^{(B)} (-m_0^\\prime) + A_2^{(B)}( -m_0^\\prime) +A_3^{(B)} (-m_{1}^\\prime) + A_4^{(B)}(-m_{1}^\\prime) + A_5^{(B)}(-m_0^\\prime) + A_6^{(B)}(-m_{1}^\\prime) \\\\\n\n \\Pi_4^{B} &= A_1^{(B)} (m_0 m_0^\\prime - {m_0^\\prime}^2) + A_2^{(B)}( -m_0^\\prime m_{1} - {m_0^\\prime}^2 ) + A_3^{(B)} (m_0 m_{1}^\\prime + {m_{1}^\\prime}^2) \\\\\n &~~~~~~+ A_4^{(B)} (-m_{1}^\\prime m_{1} + {m_{1}^\\prime}^2) + A_5^{(B)}(-m_{0}^\\prime m_2 - {m_{0}^\\prime}^2) + A_6^{(B)}(-m_{2} m_{1}^\\prime + {m_{1}^\\prime}^2)\n \\end{split}\n \\end{equation}\n\n \\begin{equation}\n \\label{eq:25}\n \\begin{split}\n \\Pi_1^{*B} &= - \\big\\{ {A_1^*}^{(B)} + {A_2^*}^{(B)} - {A_3^*}^{(B)} -{A_4^*}^{(B)} + {A_5^*}^{(B)} - {A_6^*}^{(B)}\\big\\} , \\\\\n \\Pi_2^{*B} &= -\\big\\{ {A_1^*}^{(B)} (m_0^* + {m_0}^\\prime) + {A_2^*}^{(B)} ({m_0^\\prime} - m_{1}^*) + {A_3^*}^{(B)} (-m_0^* + {m_{1}^\\prime}) \\\\\n &~~~~~~+ {A_4^*}^{(B)} (m_0^* + {m_{1}^\\prime}) + A_5^{(B)}(-{m_{2}}^* +{m_{0}^\\prime}) + A_6^{(B)}({m_{1}^\\prime} + {m_{2}}^* ) \\big\\}\\\\ \n \\Pi_3^{*B} &= -\\big\\{ {A_1^*}^{(B)} {m_{0}^\\prime} + {A_2^*}^{(B)} {m_{0}^\\prime} + {A_3^*}^{(B)} {m_{1}^\\prime} + {A_4^*}^{(B)} {m_{1}^\\prime} + {A_5^*}^{(B)} {m_{0}^\\prime} + {A_6^*}^{(B)} {m_{1}^\\prime} \\big\\}\\\\ \n \\Pi_4^{*B} &= -\\big\\{ {A_1^*}^{(B)} (m_0^* {m_{0}^\\prime} + {{m_{0}^\\prime}}^2) + {A_2^*}^{(B)} ({{m_{0}^\\prime}}^2 - {m_{1}^*} {m_{0}^\\prime}) + {A_3^*}^{(B)} ({{-m_{1}^\\prime}}^2 + {{m_{1}^\\prime}} {{m_{0}^*}}) \\\\\n &~~~~~~ + {A_4^*}^{(B)} (-{{m_{1}^\\prime}}^2 -{{m_{1}^\\prime}} m_{1}^* )+ {A_5^*}^{(B)}({{m_{0}^\\prime}}^2 - {m_{0}^\\prime} {m_{2}^*}) + {A_6^*}^{(B)}(-{{m_{1}^\\prime}}^2 - {m_{1}^\\prime} {m_{2}^*} ) \\big\\}\n \\end{split}\n \\end{equation}%\n where superscript B means Borel transformed quantities\n\n \\begin{equation}\n \\label{eq:22}\n A^{B(*)}_\\alpha = g_{\\alpha}^{(*)} \\lambda_{i} \\lambda_{j}^\\prime e^{-m_{i}^2\/M_1^2 - m_{j}^2\/M_2^2}.\n \\end{equation}\n\n The masses of the initial and final baryons are close to each other, hence in the next discussions, we set $M_1^2 = M_2^2 = 2M^2$. In order to suppress the contributions of higher states and continuum we need subtraction procedure. It can be performed by using quark-hadron duality, i.e. starting some threshold the spectral density of continuum coincide with spectral density of perturbative contribution. The continuum subtraction can be done using formula\n \\begin{equation}\n \\label{eq:12}\n (M^2)^n e^{- m_c^2\/M^2} \\rightarrow \\frac{1}{\\Gamma(n)} \\int_{m^2}^{s_0} ds e^{-s\/M^2} (s-m_c^2)^{n-1}\n \\end{equation}\n For more details about continuum subtraction in light cone sum rules, we refer readers to work~\\cite{Belyaev:1994zk}.\n \n As we have already noted in case a) we need to determine two coupling constants $g_2$($g_2^{*}$) and $g_5$($g_5^{*}$) for each class of transitions. From Eqs \\eqref{eq:24} and \\eqref{eq:25} it follows that we have six unknown coupling constants but have only four equations. Two extra equations can be obtained by performing derivative over ($1\/M^2$) of the any two equations. In result, we have six equations and six unknowns and the relevant coupling constants $g_2(g_2^*)$ and $g_5(g_5^{*})$ can be determined by solving this system of equations. \n\n The results for scenario b) can be obtained from the results for scenario a) with the help of aforementioned replacements.\n \n From Eqs \\eqref{eq:24} and \\eqref{eq:25}, it follows that to estimate strong coupling constants $g_2(g_2^*)$ and $g_5(g_5^*)$ responsible for the decay of $\\Omega_c \\rightarrow \\Xi_c K$ and $\\Omega_c^{*} \\rightarrow \\Xi_c K$, we need the residues of $\\Omega_c$ and $\\Xi_c$ baryons. For calculation of these residues for $\\Omega_c$, we consider the following two point correlation functions\n\n \\begin{equation}\n \\label{eq:37}\n \\begin{split}\n \\Pi (p) &= \\int d^4 x e^{ipx} \\langle 0 | T \\{\\eta_{\\Omega_c}(x) \\bar{\\eta}_{\\Omega_c}(0) \\} | 0 \\rangle, \\\\ \n \\Pi^{\\mu \\nu} (p) &= \\int d^4 x e^{ipx} \\langle 0 | T \\{\\eta^{\\mu}_{ \\Omega_c}(x) \\bar{\\eta}^\\nu_{\\Omega_c}(0) \\} | 0 \\rangle . \n \\end{split}\n \\end{equation}\n\n The interpolating currents $\\eta_{\\Omega_c}$ and $\\eta_{\\mu \\Omega_c}$ couples not only to ground states, but also to negative (positive) parity excited states, therefore their contributions should be taken into account. In result, for physical parts of the correlation functions we get,\n \\begin{equation}\n \\label{eq:38}\n \\begin{split}\n \\Pi(p) & = \\frac{\\langle 0 | \\eta | \\Omega_{c}(p,s) \\rangle \\langle \\Omega_{c}(p,s) | \\bar{\\eta}(0) | 0 \\rangle}{-p^2 + m_{0}^2}\n + \\frac{\\langle 0 | \\eta | \\Omega_{1c}(p,s) \\rangle \\langle \\Omega_{1c}(p,s) | \\bar{\\eta}(0) | 0 \\rangle}{-p^2 + m_{1}^2}, \\\\\n & + \\frac{\\langle 0 | \\eta | \\Omega_{2(3)c}(p,s) \\rangle \\langle \\Omega_{2(3)c}(p,s) | \\bar{\\eta}(0) | 0 \\rangle}{-p^2 + m_{2}^2} + ...~,\n \\end{split}\n \\end{equation}\n\n \\begin{equation}\n \\label{eq:39}\n \\begin{split}\n \\Pi^{\\mu \\nu } &= \\frac{\\langle 0 | \\eta_\\mu | \\Omega_c^{*}(p,s) \\rangle \\langle \\Omega_c^{*}(p,s) | \\bar{\\eta}_\\nu(0) | 0 \\rangle}{m_{0}^{*^2} - p^2}\n + \\frac{\\langle 0 | \\eta_\\mu | \\Omega_{1c}^{*}(p,s) \\rangle \\langle \\Omega_{1c}^{*}(p,s) | \\bar{\\eta}_\\nu(0) | 0 \\rangle}{m_{1}^{*^2} - p^2} \\\\\n &+ \\frac{\\langle 0 | \\eta_\\mu | \\Omega_{2(3)c}^{*}(p,s) \\rangle \\langle \\Omega_{2(3)c}^{*}(p,s) | \\bar{\\eta}_\\nu(0) | 0 \\rangle}{m_{2}^{*2} - p^2} + ...\n \\end{split}\n \\end{equation}\nwhere the dots denote contributions of higher states and continuum.\n The matrix elements in these expressions are defined as\n \\begin{equation}\n \\label{eq:40}\n \\begin{split}\n \\langle 0 | \\eta | \\Omega_c(p,s) \\rangle & = \\lambda_{0} u(p), \\\\\n \\langle 0 | \\eta | \\Omega_{1(2)c}(p,s) \\rangle & = \\lambda_{1(2)} \\gamma_5 u(p) \\\\\n \\langle 0 | \\eta | \\Omega_c^{*}(p,s) \\rangle & = \\lambda_{0}^{*} u_\\mu(p), \\\\\n \\langle 0 | \\eta | \\Omega_{1(2)c}(p,s) \\rangle & = \\lambda_{1(2)}^{*} \\gamma_5 u_\\mu(p) . \n \\end{split}\n \\end{equation}\n\n \n \n \n \n \n \n \n \n\n As we already noted, only the structure $g_{\\mu \\nu}$ describes the contribution coming from $3\/2$ baryons. Therefore we retain only this structure.\n \nFor the physical parts of the correlation function, we get\n \\begin{equation}\n \\label{eq:42}\n \\begin{split}\n \\Pi^{phy} &= \\frac{(\\slashed{p} + m_0) \\lambda_0^2 }{m_0^2 - p^2} + \\frac{(\\slashed{p} - m_{1}) \\lambda_{1}^2 }{m_{1}^2 - p^2} + \n \\frac{(\\slashed{p} \\mp m_{2(3)} )\\lambda_{2(3)}^{2} }{m_{2(3)}^{2} - p^2} \\\\\n \\Pi^{phy}_{\\mu \\nu} &= \\frac{(\\slashed{p} + m_0^*) g_{\\mu \\nu} \\lambda_0^{*^2} }{m_0^{*^2} - p^2} + \\frac{(\\slashed{p} - m^*_{1}) g_{\\mu \\nu} \\lambda_{1}^{*^2} }{m_{1}^{*^2} - p^2} +\n \\frac{(\\slashed{p} \\mp m_{2(3)}^{*}) \\lambda_{2(3)}^{2} }{m_{2(3)}^{*^2} - p^2}. \n \\end{split}\n \\end{equation}\n\n Here in the last term, upper(lower) sign corresponds to case a) (case b).\n\n Denoting the coefficients of the Lorentz structures $\\slashed{p}$ and $I$ operators $\\Pi_1$, $\\Pi_2$ and $\\slashed{p} g_{\\mu \\nu}$, $g_{\\mu \\nu}$ as $\\Pi_1^*$, $\\Pi_2^*$ respectively and performing Borel transformations with respect to $- p^2$, for spin $1\/2$ case we find,\n\n \\begin{equation}\n \\label{eq:43}\n \\begin{split}\n \\Pi_1^{B} &= \\lambda_{0}^2 e^{-m_{0}^2\/M^2} + \\lambda_{1}^2 e^{-m_{1}^2\/M^2} + \\lambda_{2(3)}^{2} e^{-m_{2}^{2}\/M^2}, \\\\\n \\Pi_2^{B} &= \\lambda_{0}^2 m_{0}e^{-m_{0}^2\/M^2} - \\lambda_{1}^2 m_{1} e^{-m_{1}^2\/M^2} \\mp \\lambda_{2(3)}^{2} m_{2(3)} e^{-m_{2(3)}^{ 2}\/M^2}. \n \\end{split}\n \\end{equation}\n\n The expressions for spin $3\/2$ case formally can be obtained from these expressions by replacing $\\lambda \\rightarrow \\lambda^{*}$, $m \\rightarrow m^{*}$ and $\\Pi \\rightarrow \\Pi^*$.\n The invariant functions $\\Pi_i$, $\\Pi_i^*$ from QCD side can be calculated straightforwardly by using the operator product expansion. Their expressions are presented in~\\cite{Aliev:2009jt} (see also \\cite{Aliev:2017led}).\n\n Similar to the determination of the strong coupling constant, for obtaining the sum rule for residues we need the continuum subtraction. It can be performed in following way. In terms of the spectral density $\\rho(s)$ the Borel transformed $\\Pi^B$ can be written as\n \\begin{equation}\n \\label{eq:14}\n \\begin{split}\n \\Pi_i^B = \\int_{m_c^2}^{\\infty} \\rho_i(s) e^{-s \/ M^2} ds \n \\end{split}\n \\end{equation}\n The continuum subtraction can be done by using the quark-hadron duality and for this aim it is enough to replace\n \\begin{equation}\n \\label{eq:15}\n \\int_{m_c^2}^{\\infty} \\rho_i(s) e^{-s \/ M^2} ds \\rightarrow \\int_{m_c^2}^{s_0} \\rho_i(s) e^{-s \/ M^2} ds. \n \\end{equation}\n \n It follows from the sum rules we have only two equations, but six (three masses and three residues) unknowns. In order to simplify the calculations, we take the masses of $\\Omega_c$ as input parameters. Hence, in this situation, we need only one extra equation, which can be obtained by performing derivatives over ($\\frac{-1}{M^2}$) on both sides of the equation. Note that the residues of $\\Xi_c$ baryons are calculated in a similar way.\n \\section{Numerical Analysis}\n\\label{sec:numeric}\nIn this section we present our numerical results of the sum rules for the strong coupling constants responsible for $\\Omega_c(3000) \\rightarrow \\Xi_c^{+} K^-$ and $\\Omega_c(3066) \\rightarrow \\Xi_c^{+} K^-$ decays derived in previous section. The Kaon distribution amplitudes are the key non-perturbative inputs of sum rules whose expressions are presented in \\cite{Ball:2006wn}. The values of other input parameters are:\n\\begin{equation}\n \\label{eq:7}\n \\begin{split}\n f_K & = 0.16~\\rm{GeV}, \\\\\n m_0^2 & = (0.8 \\pm 0.2) ~\\rm{GeV^2}, \\\\\n \\langle \\bar{q} q \\rangle & = -(0.240 \\pm 0.001)^3~\\rm{GeV^3}, \\\\\n \\langle \\bar{s}s \\rangle & = 0.8~ \\langle \\bar{q} q \\rangle . \n \\end{split}\n\\end{equation}\n\nThe sum rules for $g_{-+}$ and $g_{-+}^{*}$contain the continuum threshold $s_0$, Borel variable $M^2$ and parameter $\\beta$ in interpolating current for spin $1\/2$ particles. In order to extract reliable values of these constants from QCD sum rules, we must find the working regions of $s_0,~M^2$ and $\\beta$ in such a way that the result is insensitive to the variation of these parameters. The working region of $M^2$ is determined from conditions that the operator product expansion (OPE) series be convergent and higher states and continuum contributions should be suppressed. More accurately, the lower bound of $M^2$ is obtained by demanding the convergence of OPE and dominance of the perturbative contributions over the non-perturbative one. The upper bound of $M^2$ is determined from the condition that the pole contribution should be larger than the continuum and higher states contributions. We obtained that both conditions are satisfied when $M^2$ lies in the range\n\\begin{equation}\n \\label{eq:8}\n 2.5~\\rm{GeV^2} \\leq M^2 \\leq 5~\\rm{GeV^2}.\n\\end{equation}\nThe continuum threshold $s_0$ is not arbitrary and related with the energy of the first excited state i.e. $s_0 = (m_{\\text{ground}} + \\delta)^2$. Analysis of various sum rules shows that $\\delta$ varies between $0.3$ and $0.8$ \\rm{GeV}, and in this analysis $\\delta=0.4~\\rm{GeV}$ is chosen.\nAs an example, in Figs.~\\ref{fig:1} and \\ref{fig:2} we present the dependence of the residues of $\\Omega_c(3000)$ and $\\Omega_c(3050)$ on $\\cos{\\theta}$ for the scenario a) at $s = 11~\\rm{GeV^2}$ and several fixed values of $M^2$ , respectively. From these figures, we obtain that when $\\cos{\\theta}$ lies between $-1$ and $-0.5$ the residues exhibits good stability with respect to the variation of $\\cos{\\theta}$ and the results are practically insensitive to the variation of $M^2$. And we deduce the following results for the residues\n\\begin{equation}\n \\label{eq:44}\n \\begin{split}\n \\lambda_{1} &= (0.08 \\pm 0.03)~~\\rm{GeV^3}, \\\\\n \\lambda_{2} &= (0.11 \\pm 0.04)~~\\rm{GeV^3}.\n \\end{split}\n\\end{equation}\n\nPerforming similar analysis for $\\Omega_c$ baryons in scenario b) we get (Figs.~\\ref{fig:3} and \\ref{fig:4})\n\n\\begin{equation}\n \\label{eq:45}\n \\begin{split}\n \\lambda_{1} &= (0.030 \\pm 0.001)~~\\rm{GeV^3}, \\\\\n \\lambda_{3} &= (0.04 \\pm 0.01)~~\\rm{GeV^3}. \n \\end{split}\n\\end{equation}\nThe detailed numerical calculations lead to the following results for spin $3\/2$ $\\Omega_c$ baryon residues:\n\n\\begin{equation}\n \\label{eq:46}\n \\begin{split}\n \\lambda_{1}^* &= (0.18 \\pm 0.02)~~\\rm{GeV^3}, \\\\\n \\lambda_{2}^* &= (0.17 \\pm 0.02)~~\\rm{GeV^3}, \\\\\n \\end{split}\n\\end{equation}\n\n\\begin{equation}\n \\label{eq:47}\n \\begin{split}\n \\lambda_{1}^* &= (0.024 \\pm 0.002)~~\\rm{GeV^3}, \\\\\n \\lambda_{3}^* &= (0.05 \\pm 0.01)~~\\rm{GeV^3}. \n \\end{split}\n\\end{equation}\n\nFrom these results we observe that the residues of $\\Omega_c$ baryons in scenario a) is larger than that one for the scenario b). This leads to the larger strong coupling constants for scenario b) because it is inversely proportional to the residue.\n\nHaving obtained the values of the residues, our next problem is the determination of the corresponding coupling constants using the values of $M^2$ and $s_0$ in their respective working regions which are determined from mass sum rules . In Figs.~\\ref{fig:5},~\\ref{fig:6},~\\ref{fig:7}, and \\ref{fig:8}, we studied the dependence of the strong coupling constants for $\\Omega_c^{*} \\rightarrow \\Xi_c K^0$ transitions for the scenarios a) and b) on $\\cos{\\theta}$, respectively. We obtained that when $M^2$ varies in its working region the strong coupling constant demonstrates weak dependence on $M^2$, and the results for the spin-$3\/2$ states also practically do not change with the variation of $s_0$. Our results on the coupling constants are:\n\nFor scenario a:\n\\begin{equation}\n \\label{eq:46}\n \\begin{split}\n g_{2} & = 19 \\pm 2 \\hspace{2cm} g_{2}^* = 40 \\pm 10 \\\\ \n g_{5} & = 20 \\pm 2 \\hspace{2cm} g_{5}^* = 42 \\pm 10 \n \\end{split}\n\\end{equation}\n\nFor scenario b:\n\\begin{equation}\n \\label{eq:47}\n \\begin{split}\n g_{2} & = 2.2 \\pm 0.2 \\hspace{2cm} g_{2}^* = 2.0 \\pm 0.5 \\\\ \n \\tilde{g_5} & = 6 \\pm 1 \\hspace{2.65cm} \\tilde{g_5}^* = 8 \\pm 1 \n \\end{split}\n\\end{equation}\n\nThe decay widths of these transitions can be calculated straightforwardly and we we get;\n\\begin{equation}\n \\label{eq:48}\n \\begin{split}\n \\Gamma &= \\frac{g_i^2}{16 \\pi m_i^3} \\big[ (m_i + m_{0}^ \\prime)^2 - m_K^2 \\big] \\lambda^{1\/2}(m_{i}^2, m_{0}^{\\prime ^2}, m_K^2) \\\\\n \\Gamma &= \\frac{g_i^{*2}}{192 \\pi m_i^{*^5}} \\big[ (m_i^* + m_{0}^\\prime)^2 - m_K^2 \\big] \\lambda^{3\/2}(m_{i}^{*^2}, m_{0}^{ \\prime ^2}, m_K^2) \n \\end{split}\n \\end{equation}\n where $m_i (m_i^*)$ and $m_0^\\prime$ are the mass of initial spin $1\/2$ (spin $3\/2$) $\\Omega_c$ baryon and $\\Xi_c$ baryons respectively and $\\lambda(x,y,z) = x^2 + y^2 + z^2 - 2 x y - 2 x z - 2 y z$.\n Having the relevant strong coupling constants, the decay width values for scenario a) and b) are shown in Table~\\ref{tab:1}.\n\\begin{table}\n \\begin{tabular}{ccc}\n \\toprule\n \\multirow{2}{*}{} & Scenario-a & Scenario-b \\\\\n & (GeV) & (GeV) \\\\\n \\midrule\n $ \\Gamma (\\Omega_c(3000) \\rightarrow \\Xi_c K)$ & $8.1 \\pm 1.8 $ & $0.10 \\pm 0.02$ \\\\\n $ \\Gamma (\\Omega_c(3050) \\rightarrow \\Xi_c K)$ & $14.1 \\pm 3$ & $(3.8 \\pm 1.2) \\times 10^{-3}$ \\\\\n $ \\Gamma (\\Omega_c(3066) \\rightarrow \\Xi_c K)$ & $(6.6 \\pm 3) \\times 10^{-3}$ & $(1.6 \\pm 0.6) \\times 10^{-5}$\\\\\n $\\Gamma (\\Omega_c(3090) \\rightarrow \\Xi_c K)$ & $(1.3 \\pm 0.5) \\times 10^{-2}$ & $0.10 \\pm 0.04$ \\\\\n \\bottomrule\n \\end{tabular}\n \\caption{Decay widths for the two-scenarios considered are shown.}\n \\label{tab:1}\n\\end{table}\n\n Our results on the decay widths are also drastically different than the one presented in \\cite{Agaev:2017lip}. In our opinion, the source of these discrepancies are due to the following facts:\n\\begin{itemize}\n\\item In \\cite{Agaev:2017lip}, the contributions coming from $\\Xi_c^-$ baryons are all neglected.\n\\item The second reason is due to the procedure presented in \\cite{Agaev:2017lip}, namely by choosing the relevant threshold $s_0$, isolating the contributions of the corresponding $\\Omega_c$ baryons is incorrect. From analysis of various sum rules, it follows that $s_0 = (m_{\\text{ground}} + \\delta)^2$, where $0.3~\\rm{GeV} \\leq \\delta \\leq 0.8~\\rm{GeV}$. Since the mass difference between $\\Omega_c(3000)$ and $\\Omega_c(3090)$ is around $0.1~\\rm{GeV}$, isolating the contribution of each baryon is impossible while their contributions should be taken into account simultaneously. For these reasons our results on decay widths are different than those one predicted in \\cite{Agaev:2017lip}. From experimental data on the width of $\\Omega_c$ are~\\cite{Olive:2016xmw}:\n\\begin{equation}\n \\label{eq:9}\n \\begin{split}\n \\Gamma (\\Omega_c(3000) \\rightarrow \\Xi_c^+ K^-) &= (4.5 \\pm 0.6 \\pm 0.3)~\\rm{MeV} \\\\\n \\Gamma (\\Omega_c(3050) \\rightarrow \\Xi_c^+ K^-) &= (0.8 \\pm 0.2 \\pm 0.1)~\\rm{MeV} \\\\\n \\Gamma (\\Omega_c(3066) \\rightarrow \\Xi_c^+ K^-) &= (3.5 \\pm 0.4 \\pm 0.2)~\\rm{MeV} \\\\\n \\Gamma (\\Omega_c(3090) \\rightarrow \\Xi_c^+ K^-) &= (8.7 \\pm 1.0 \\pm 0.8)~\\rm{MeV} \\\\\n \\Gamma (\\Omega_c(3119) \\rightarrow \\Xi_c^+ K^-) &= (1.1 \\pm 0.8 \\pm 0.4)~\\rm{MeV} \n \\end{split}\n\\end{equation}\nWe find out that, our predictions strongly differ from the experimental results.\n\nBy comparing our predictions with the experimental data, we conclude that both scenarios are ruled out.\n\\end{itemize}\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\n\nIn conclusion, we calculated the strong coupling constants of negative parity $\\Omega_c$ baryon with spins $1\/2$ and $3\/2$ with $\\Xi_c$ and $K$ meson in the framework of light cone QCD sum rules. Using the obtained results on coupling constants we estimate the corresponding decay widths. We find that our predictions on the decay widths under considered scenarios are considerably different from experimental data as well as theoretical predictions and considered both scenarios are ruled out. Therefore further theoretical studies for determination of the quantum numbers of $\\Omega_c$ states as well as for correctly reproducing the decay widths of $\\Omega_c$ baryons are needed.\n\n\\section{Acknowledgments}\nThe authors acknowledge METU-BAP Grant no. 01-05-2017004.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\nTheory of general relativity (GR) has been successful in describing a wide range of phenomena, from solar system to cosmological scales. In addition to being consistent with various experiments, the mathematical elegance of the theory is very appealing. Diffeomorphism invariance, at the core of GR, gives a straightforward constructive way of building the theory. In fact, GR is the simplest diffeomorphism invariant theory for metric.\n\nFrom observational point of view, there is no reason to abandon this theory. GR is compatible with a wide variety of experimental constraints\\footnote{Although there have been various attempts to solve the problems of dark matter and dark energy with GR modifications, simple solutions to these problem in the context of GR exist. In other words, there is no apparent observational contradiction with GR which necessitates GR modifications.}. On the other hand, many attempts have shown so far that modifying GR is a tricky task, and one often faces physically unacceptable results, e.g. the appearance of Boulware-Deser ghost in massive gravity \\cite{Boulware:1973my} and ghost degrees of freedom in quadratic gravity \\cite{Stelle:1976gc}. \n\nHowever, studying non-GR theories of gravity is still valuable, and the main reason stems from quantizing gravity. GR, while being a very successful classical theory, has failed to cope with quantum mechanics.\nTherefore, one approach to quantum gravity has been to abandon diffeomorphism invariance (at high enough energies), as e.g., done in the celebrated Horava-Liftshitz gravity \\cite{Horava:2009uw}.\n\nIn different examples of theories with broken Lorentz invariance, superluminal degrees of freedom appear (see \\cite{Blas:2010hb,Jacobson:2000xp}). The existence of superluminal excitations (SLE) points out that a different causal structure exists in these theories compared to GR, even when the back-reaction of these excitations on the geometry is negligible. This property is especially of significance in the black hole (BH) solutions. While potentially SLE can escape the traditional killing horizon of a BH and make the classical theory unpredictable, it has been shown in many examples \\cite{Sotiriou:2014gna,Ding:2015kba,Barausse:2012qh,Bhattacharyya:2015gwa,Barausse:2013nwa,Babichev:2006vx,Barausse:2011pu,Blas:2011ni} that a notion of horizon (called universal horizon) still exists in these theories. Moreover, universal horizon (UH) thermally radiate and satisfies the first law of horizon thermodynamics\\footnote{so far only for spherically symmetric solutions} \\cite{Mohd:2013zca,Berglund:2012bu,Berglund:2012fk,Bhattacharyya:2014kta}. Studying the notion of universal horizon and its temperature and entropy is important since it guides us to better understanding the structure of UV theory.\n\nIn this paper, we study the universal horizon formation in dynamical or stationary spacetimes with an inner killing horizon, in the limit of infinite sound speed for excitations (i.e. {\\it incompressible} limit). In order to do so, we make use of the fact that surfaces of global time (defined by the background field), in the incompressible limit, coincide with constant mean curvature (CMC) surfaces of the spacetime. Furthermore, the backreaction of the incompressible field on the spacetime geometry is negligible as long as the event horizon is much smaller than the cosmological horizon \\cite{Saravani:2013}. In the next section, we show how the universal horizon forms in a dynamical setting, in the collapse of a charged shell, and we derive a formula for the radius of the universal horizon in terms of the charge. In Section \\ref{geoUH}, we propose a geometric definition for universal horizon. This allows us to study the universal horizon for spinning black holes. In \\ref{KerrUH} we show that there are three axisymmetric surfaces which satisfy the conditions of a universal horizon. As we show, this means that two families (with infinite numbers) of axi-symmetric universal horizons in Schwarzschild case exist.\nSection \\ref{conclude} concludes the paper. \n\n\n \n\n\n\\section{Universal Horizon in Dynamic Reissner--Nordstrom Geometry}\n\\label{RN}\nWe start this section by finding CMC slicing of dynamic Reissner--Nordstrom (RN) geometry. As we mentioned earlier, CMC surfaces of this spacetime are the constant global time surfaces of the background incompressible field, and they define the new causal structure imposed by this field (see the analysis in \\cite{Saravani:2013}). Once we derive the CMC slicing, we focus on the (universal) horizon formation in this geometry. \n\n\n\\subsection{CMC Surfaces in a Dynamic Reissner--Nordstrom Geometry}\n\\label{CMC-RN}\n\nIn order to examine the formation of the universal horizon in a dynamic Reissner--Nordstrom geometry, one must first describe surfaces of constant mean curvature for a collapsing charged massive spherical shell. An examination of CMC surfaces has been similarly looked at in the restricted case of maximal surfaces ($K=0$) \\cite{Maeda:1980}. The dynamics of the collapse itself is well known and described by Israel \\cite{Israel:1967}. Describing the metric in the standard way has the geometry inside the shell as flat and the RN outside. We write this geometry as:\n\n\\begin{align*}\n &\\mathrm{d} s^2 =f_-(r) \\mathrm{d} t_-^2 -f_-(r)^{-1} \\mathrm{d} r^2 -r^2\\mathrm{d} \\Omega^2 &(rR)\n\\end{align*}\nwhere $f_-(r)=1$ and $f_+(r)=1-\\frac{2M}{r}+\\frac{Q^2}{r^2}$ in $G=c=1$ units. The parameters are the gravitational mass $M$ and shell's charge $Q$. For simplicity we will often use relative charge $\\mathit{q}=Q\/M$. While the spherical coordinates are shared between the inner and outer regions, the time coordinates $t_-$ and $t_+$ correspond to the Minkowski and RN time respectively.\n\nLet the family of spacelike CMC surfaces be denoted by $\\Sigma_K(t_g)$ where $t_g$ is a global time coordinate that is constant for each surface. The timelike normal vector to this surface is labelled $v^\\mu$. The CMC condition implies $\\nabla_\\mu v^\\mu=K$, resulting in:\n\\begin{equation}\\label{CMC_eq}\n\\frac{\\partial}{\\partial t_\\pm}v^{t_\\pm}+\\frac{1}{r^2}\\frac{\\partial}{\\partial r}r^2v^r =K\n\\end{equation}\n\nIf we denote $B \\equiv -r^2v^r$ and use the normalization condition $v_\\mu v^\\mu =1$ then:\n\\begin{equation}\nr^2v^{t_\\pm}=\\pm f_\\pm(r)^{-1}\\sqrt{B^2+f_\\pm r^4}.\n\\end{equation}\nFor now we use the '$+$' case so that for $v_t>0$ for $r\\gg M$. Additional explanation and the cases where '$-$' is relevant will be seen in Section \\ref{foliation_struc}. Combining this result with \\eqref{CMC_eq} we get\n\n\\begin{equation}\n \\label{eq:pde}\n \\frac{B}{f_\\pm(r)\\sqrt{h(r,B)}}\\frac{\\partial}{\\partial t_\\pm}B-\\frac{\\partial}{\\partial r}B=Kr^2\n\\end{equation}\n\nwith $h(r,B)=B^2+f_\\pm r^4$. The characteristic equations of (\\ref{eq:pde}) are simply:\n\n\\begin{equation}\n \\label{eq:chareq}\n \\frac{\\mathrm{d} t_\\pm}{\\mathrm{d} s}= \\frac{B}{f_\\pm(r)\\sqrt{h(r,B)}},~\\: \\frac{\\mathrm{d} r}{\\mathrm{d} s}=-1, \\: ~\\text{and}~ \\: \\frac{\\mathrm{d} B}{\\mathrm{d} s}= K r^2,\n\\end{equation}\nfor some parameter $s$. Using the second equation of (\\ref{eq:chareq}) to integrate the first and third equations results in:\n\n\\begin{equation}\n \\label{eq:coordeq}\n t_\\pm=(t_\\pm)_0-\\int_{r_0}^r \\frac{B\\mathrm{d} r}{f_\\pm(r)\\sqrt{h(r,B)}},~~ \\text{and}~~ B= \\frac{K}{3} (r^3-r_0^3)+B_0,\n\\end{equation}\n where $(t_\\pm)_0$, $r_0$, and $B_0$ are integration constants. In order to fix these constants, we examine the internal and external cases separately. \n\n\\paragraph{1) Inside the shell:}\n\nIf $r_0=0$ then $B(r=0)=B_0$. If $B_0 \\neq 0$ this would lead to a contradiction, as $v^r=\\frac{-B}{r^2}$ should be finite in the flat geometry. Therefore with $r_0=0$, the equation reduces to:\n\n\n\\begin{equation}\n \\label{eq:coordeqinside}\n t_-=(t_-)_0+\\int_{0}^r \\frac{Kr^3\\mathrm{d} r}{3\\sqrt{(\\frac{Kr^3}{3})^2+r^4}} \\: \\text{and}\\: B= \\frac{K}{3} r^3\n\\end{equation}\n\n\\paragraph{2) Ouside the shell:}\n\nLet $r_0=R((t_+)_0)$, we can determine $B_0$ by looking at the boundary between the flat and RN spaces. Projecting the vector $v^\\mu$ along the shell should give us continuous observable values. The shell timelike path comes from $S=R(t_\\pm)-r=0$ which creates the unit normal vector and tangent vector labelled as $n^\\mu$ and $u^\\mu$ respectively. If we choose the sign of the normalization factors such that $u^r<0$ , the vectors take the form of:\n\\begin{eqnarray}\n n_-^\\mu&&=\\frac{g^{\\mu\\nu}}{N_-}(\\nabla_-)_\\nu S=\\frac{1}{N_-}(\\frac{\\mathrm{d} R}{\\mathrm{d} t_-},1,0,0),\\\\ \nu_-^\\mu &&=\\frac{1}{N_-}(1,\\frac{\\mathrm{d} R}{\\mathrm{d} t_-},0,0),\\\\\nN_-^2&&=1-(\\frac{\\mathrm{d} R}{\\mathrm{d} t_-})^2,\n\\end{eqnarray}\ninside the shell, while outside takes the from of:\n\\begin{eqnarray}\nn_+^\\mu&&=\\frac{g^{\\mu\\nu}}{N_+}(\\nabla_+)_\\nu S=\\frac{1}{N_+}(f_+^{-1}\\frac{\\mathrm{d} R}{\\mathrm{d} t_+},f_+,0,0),\\\\\nu_+^\\mu&&=\\frac{1}{N_+}(1,\\frac{\\mathrm{d} R}{\\mathrm{d} t_+},0,0),\\\\\nN_+^2&&=\\frac{f_+^2-(\\frac{\\mathrm{d} R}{\\mathrm{d} t_+})^2}{f_+}.\n\\end{eqnarray}\nWe wish to find functions $C(R)$ and $D(R)$, such that:\n\\begin{equation}\nv_-^\\mu=C n_-^\\mu+D u_-^\\mu.\n\\end{equation}\n From inside the shell $v^\\mu(R)=(1,0,0,0)$ which means $C=\\frac{-1}{N_-}\\frac{\\mathrm{d} R}{\\mathrm{d} t_-}$ and $D=\\frac{1}{N_-}$. Requiring projections ($C$ and $D$) to be the same from outside, we get: \n\\begin{equation}\nB_0=-R^2(C (n_+)^r+D (u_+)^r)=\\frac{R^2}{N_+N_-}\\left(\\frac{\\mathrm{d} R}{\\mathrm{d} t_+}-f_+ \\frac{\\mathrm{d} R}{\\mathrm{d} t_-}\\right).\n\\end{equation}\nSo, if we specify the dynamics of the shell $\\frac{dR}{dt_{\\pm}}$, all the parameters are fixed.\n\nThe description of the radial velocity comes from Israel and De La Cruz \\cite{Israel:1967}:\n\\begin{eqnarray}\n\\left( \\frac{\\mathrm{d} R}{\\mathrm{d} t_-}\\right)^2&&=\n1-\\frac{R^2}{(\\epsilon R-b)^2}\\label{drdt+},\\\\\n\\left( \\frac{\\mathrm{d} R}{\\mathrm{d} t_+}\\right)^2&&\nf_+^2-\\frac{f_+^3 R^2}{(\\epsilon R-b-\\frac{m}{\\epsilon})^2},\\label{drdt-}\n\\end{eqnarray}\nwhere $\\epsilon=\\frac{M}{\\mathcal{M}}$ and $b=\\frac{M(\\epsilon^2\\mathit{q}^2-1)}{2\\epsilon}$ with $\\mathcal{M}$ denoting the total rest mass. \nWe can use (\\ref{drdt+}) and (\\ref{drdt-}) to reduce $N_+=\\frac{Rf_+}{\\epsilon R-b-\\frac{M}{\\epsilon}}$ and $N_-=\\frac{R}{\\epsilon R-b}$. Note that $N_+$ changes signs to enforce $u^r<0$, becoming negative only when $\\frac{\\mathrm{d} R}{\\mathrm{d} t_\\pm}$ flips signs. These choices simplifies $B_0$ to:\n\\begin{equation}\nB_0=\\frac{M}{\\epsilon}\\sqrt{(\\epsilon R-b)^2-R^2}.\n\\end{equation}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.5\\textwidth]{hplot}\n \\caption{$h(r,B)$ with sub-critical, post-critical and critical $B$.}\n \\label{fig:hsols}\n\\end{figure}\n\n \n\\subsection{Horizon Formation}\n\nFollowing the analysis of \\cite{Saravani:2013}, we examine the properties of $t_+$. The behaviour of $t_+$ heavily depends on $h(r,B)$. While $B$ is large, which corresponds to large $R$, $h(r,B)$ is never vanishing. However when a critical value $B_c$ is reached, $ h(r,B_c)$ has double root at a particular value of $r$ labelled $r_h$ (see Figure \\ref{fig:hsols}). Something interesting will occur when $r_h$ is larger than the radius of the shell for which $B_c$ occurs named $R_{lc}$ or radius of {\\it last contact} ($B(R_{lc})=B_c$). A signal sent out from the shell at $R_{lc}$ will proceed out to $r_h$ but takes infinitely long time to ever reach this radius. In fact, signals sent just outside $R_{lc}$ will form an envelope around $r_h$ staying at this radius longer and longer, as $R_{lc}$ is approached, before escaping to infinity (see Figure \\ref{fig:plot0log}). The values of $B_c$ and $r_h$ can be found by finding the solutions to $h(r,B)=\\frac{\\partial h(r,B)}{\\partial \\mathrm{r}}=0$. \nWe examine this equation in two different cases. \n\n\\subsubsection*{Case 1: $K=0$}\n \nEquations for the double root reduce to: \n\n\\begin{eqnarray}\n&&r_h^4-2Mr_h^3+Q^2r_h^2+B_c^2=0\\label{eq1_rh}\\\\\n&&2r_h^3-3Mr_h^2+ Q^2r_h=0\\label{eq2_rh}\n\\end{eqnarray}\n\nto which the solutions with non-negative real $B_c$ are the trivial $r_h=B_c=0$ and\n\\begin{eqnarray}\nr_h&&=\\frac{3M}{4}+\\frac{M}{4}\\sqrt{9-8\\mathit{q}^2} \\label{r_h_ns}\\\\\nB_c&&=r_h\\sqrt{-r_h^2+2mr_h-Q^2}=r_h^2\\sqrt{-f_+(r_h)}.\n\\end{eqnarray}\n\nIt is of interest to not that the UH is always between the inner and outer killing horizons of the metric (see Figure \\ref{fig:uni-hor}).\n\n\\begin{figure}[b]\n \\centering\n \\includegraphics[width=0.5\\textwidth]{unihor}\n \\caption{The outer, inner, and universal horizons for $K=0$ and varying $Q$ }\n \\label{fig:uni-hor}\n\\end{figure}\n\n\\subsubsection*{Case 2: $K\\not=0$ }\n\n \nEquations for the double root are written as:\n\\begin{eqnarray}\n&&\\frac{K^2}{9}r_h^6+r_h^4+(\\frac{2 K B_0}{3}-2M)r_h^3+Q^2r_h^2+B_0^2=0,\\notag\\\\\n&&\\frac{ K^2}{3}r_h^5+ 2r_h^3+(k B_0-3M)r_h^2+ Q^2r_h=0.\\notag\n\\end{eqnarray}\n\n\\begin{figure}[b]\n \\centering\n \\includegraphics[width=0.5\\textwidth]{plot0logprime}\n \\caption{The universal horizon formation for $Q=0$ in Schwarzschild coordinates. The blue lines represent CMC surfaces, the lowest brown line is the shell's surface, the red line is the universal horizon (UH) and the dotted black is the radius that UH asymptotes to. Here, and in all the subsequent diagrams, the red region lies behind the UH. }\n \\label{fig:plot0log}\n\\end{figure}\n\nThe non-trivial solution for $B_c$ is:\n\\begin{equation}\nB_c=r_h\\sqrt{-r_h^2+2Mr_h-Q^2}-\\frac{Kr_h^3}{3}=r_h^2\\sqrt{-f_+(r_h)}-\\frac{Kr_h^3}{3},\n\\end{equation}\nhowever the solution for $r_h$ can be at best expressed perturbatively in $K$. To linear order the solution is:\n\n\\begin{equation}\nr_h=r_h^0-K\\frac{(r_h^0)^3\\sqrt{-f_+(r_h^0)}}{M\\sqrt{9-8q^2}}+\\mathcal{O}(K^2),\n\\end{equation}\nwhere $r_h^0=\\frac{3M}{4}+\\frac{M}{4}\\sqrt{9-8\\mathit{q}^2}$. Assuming that the expansion of the background field is negligible (for example fixed by cosmology, as $K=3 \\times$Hubble constant), in the region of interest $01$. \n \n\n\\begin{figure}\n\\begin{subfigure}[t]{0.45\\textwidth}\n\\centering\n\\includegraphics[width=\\hsize]{plot0krprime}\n\\caption{The universal horizon formation for $Q=0$ in Kruskal-Szekeres coordinates. The coloured lines\/region have the same meaning as Figure \\ref{fig:plot0log}, additionally the thick black line represents the null horizons and the dashed line the singularity. }\n\\label{fig:plot0kr}\n\\end{subfigure}\n\\begin{subfigure}[t]{0.5\\textwidth}\n\\centering\n \\includegraphics[width=\\hsize]{UHschwprime}\n \\caption{The Penrose diagram for a $Q=0$ collapsing shell depicting the UH horizon. Coloured lines\/region have the same meaning as Figure \\ref{fig:plot0log} with the inclusion of sub-UH CMC surfaces in blue.\n}\n\\label{fig:plot0pn}\n\\end{subfigure}\n\\caption{Surfaces for constant global time and formation of the universal horizon in Kruskal-Szekeres coordinates and Penrose diagram for $Q=0$.}\n\\end{figure}\n\n \n\n\n\nWe take the larger of the solutions to the quadratic, as the first instance of $B_c$ will create the behaviour desired. \n\n\n\\subsection{Inside the Universal Horizon}\n\nThe foliation can be extended for $BR_{lc}$ and the final well defined CMC surface that creates the universal horizon in Schwarzchild and Kruskal-Szekeres coordinates. Once a conformal compactification has been performed the casual structure is clear in Figure \\ref{fig:plot0pn} with the additional sub-UH CMC surfaces (which end in singularity, rather than the space-like infinity $i^0$).\n\n\n\\subsubsection{Case 2: $Q\\neq0$ \\& $b\\leq0$}\nFor $\\epsilon$ small enough such that the numerator of $b$ remains negative, the shell is unable to rebound before collapsing to a singularity. In Schwarzchild and the charged generalization of Kruskal-Szekeres remain almost identical in their analogous charts. The causal structure in Figure \\ref{fig:plotqpn} reveals the distinction from case 1. The collapse ends in the coordinates, colloquially called the {\\it parallel universe}.\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.5\\textwidth]{UHRN1prime}\n \\caption{The Penrose diagram for $Q=0.99 M$ and $\\epsilon=1$ collapsing shell depicting the UH. Coloured lines\/region have the same meaning as Figure \\ref{fig:plot0log}} with the inclusion of sub-UH CMC surfaces in blue.\n \\label{fig:plotqpn}\n\\end{figure}\n\n\n\\subsubsection{Case 3: $Q\\neq 0$ \\& $0b\/\\epsilon+1\/\\epsilon^2$. Coloured lines\/region have the same meaning as Figure \n\\ref{fig:plot0log} with the inclusion of sub-UH CMC surfaces in blue.\n } \n \\label{fig:UHRN2}\n\\end{figure}\n\n\n\n\\subsubsection{Case 4: $Q\\neq0$ \\& $b\\geq b\/\\epsilon+1\/\\epsilon^2$}\nSubsequently for $b>\\frac{M}{ep}$ shell rebounds at the radius of $\\frac{b}{\\epsilon-1}$ in the original coordinate charts or exactly where the original and parallel coordinates meet . In particular this means that $t(R)$ has a stationary point at or inside $r_-$. The Schwarzchild and Kruskal-Szekeres coordinates are again nearly indistinguishable from case 1 except when the placement of $R_{lc}$ requiring that the UH being in a second charts however this does not reveal any new structure. Figure \\ref{fig:UHRN4} and \\ref{fig:UHRN2} represents paths within this case, $b=b\/\\epsilon+1\/\\epsilon^2$ and $b>b\/\\epsilon+1\/\\epsilon^2$ respectively. In this final case $R_{lc}