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+{"text":"\\section{Introduction}\n\nIt is not difficult to see that for the 2-element group $\\mathbb{Z}_2 =\n\\langle\\{0,1\\}, + \\rangle$, the term operation $m(x,y,z) = x + y + z$ satisfies\nthe equations\n\\begin{equation}\\label{min-eq}\n  m(y,x,x) \\approx m(x,y,x) \\approx m(x,x,y) \\approx y.\n\\end{equation}\nA slightly more challenging exercise is to show that a~finite Abelian group\nwill have such a~term operation if and only if it is isomorphic to a~Cartesian\npower of~$\\mathbb{Z}_2$.\n\nA ternary operation $m(x,y,z)$ on a~set $A$ is called a~\\emph{minority\noperation on $A$} if it satisfies the identities~(\\ref{min-eq}).  A ternary\nterm $t(x,y,z)$ of an algebra $\\m a$ is a~\\emph{minority term of $\\m a$} if its\ninterpretation as an operation on $A$, $t^{\\m a}(x,y,z)$, is a~minority\noperation on $A$.  Given a~finite algebra $\\m a$, one can decide if it has\na~minority term by constructing all of its ternary term operations and checking\nto see if any of them satisfy the equations~(\\ref{min-eq}).  Since the set of\nternary term operations of $\\m a$ can be as big as $|A|^{|A|^3}$, this\nprocedure will have a~runtime that in the worst case will be exponential in the\nsize of~$\\m a$.\n\nIn this paper we consider the computational complexity of testing for the\nexistence of a~minority term for finite algebras that are  {idempotent}.  An\n$n$-ary operation $f$ on a~set $A$ is \\emph{idempotent} if it satisfies the\nequation $f(x, x, \\dots, x) \\approx x$ and an algebra is \\emph{idempotent} if\nall of its basic operations are. We observe that every minority operation is\nidempotent. While idempotent algebras are rather special, one can always form\none by taking the \\emph{idempotent reduct} of a~given algebra $\\m a$.  This is\nthe algebra with universe $A$ whose basic operations are all of the idempotent\nterm operations of $\\m a$.  It turns out that many important properties of an\nalgebra and the variety that it generates are governed by its idempotent\nreduct~\\cite{kearnes-kiss-book}.\n\nThe condition of an algebra having a~minority term is an example of a~more\ngeneral existential condition on the set of term operations of an algebra\ncalled a~\\emph{strong Maltsev condition}.  Such a~condition consists of\na~finite set of operation symbols along with a~finite set of equations\ninvolving them.  An algebra is said to satisfy the condition if for each\n$k$-ary operation symbol from the condition, there is a~corresponding $k$-ary\nterm operation of the algebra so that under this correspondence, the equations\nof the condition hold.  For a~more careful and complete presentation of this\nnotion and related ones, we refer the reader to~\\cite{Garcia-Taylor}.\n\nGiven a~strong Maltsev condition $\\Sigma$, the problem of determining if\na~finite algebra satisfies $\\Sigma$ is decidable and lies in the complexity\nclass \\compEXPTIME.  As in the minority term case, one can construct all term\noperations of an algebra up to the largest arity of an operation symbol in\n$\\Sigma$ and then check to see if any of them can be used to witness the\nsatisfaction of the equations of $\\Sigma$.  In general, we cannot do any better\nthan this, since for some strong Maltsev conditions, it is known that the\ncorresponding decision problem is {\\compEXPTIME}-complete~\\cite{freese-valeriote}.\n\nThe situation for finite idempotent algebras appears to be better than in the\ngeneral case since there are a~number of strong Maltsev conditions for which\nthere are polynomial-time  procedures to decide if a~finite  idempotent algebra\nsatisfies them~\\cite{freese-valeriote, horowitz-ijac, kazda-valeriote}.  At\npresent there is no known characterization of these strong Maltsev conditions\nand we hope that the results of this paper may help to lead to a~better\nunderstanding of them.  We refer the reader to~\\cite{Bu-Sa} or\nto~\\cite{bergman-book} for background on the basic algebraic notions  and\nresults used in this work.\n\n\n\\section{Formulation of the problem}\n\nIn this section, we formally introduce the considered problem. In all the\nproblems mentioned in the introduction, we assume that the input algebra is\ngiven as a list of tables of its basic operations. In particular, this implies\nthat the input algebra has finitely many operations. We also assume that the\ninput algebra has at least one operation (i.e., the input is non-empty) and we\nforbid nullary operations on the input.\nThe main concern of this paper is the following decision problem.\n\n\\begin{definition}\nDefine \\minority\\ to be the following decision problem:\n\\begin{itemize}\n  \\item INPUT: A~list of tables of basic operations of an idempotent algebra~$\\m A$.\n  \\item QUESTION: Does $\\m a$ have a~minority term?\n\\end{itemize}\n\\end{definition}\n\nThe size of an input is measured by the following formula. For a finite\nalgebra~$\\m a$, let\n\\[\n  \\|\\m a\\| = \\sum_{i = 1}^\\infty k_i|A|^i,\n\\]\nwhere $k_i$ is the number of $i$-ary basic operations of $\\m a$. Since we\nassume that $\\m a$ has only finitely many operations, the sum is finite. Also\nnote that $\\lVert \\m a\\rVert \\geq \\lvert A \\rvert$ since we assumed that $\\m a$\nhas a non-nullary operation.\n\n\n\\section{Minority is a join of two weaker conditions}\n  \\label{join}\n\nOne approach to understanding the minority term condition is to see if maybe\nthere exist two weaker Maltsev conditions $\\Sigma_1$ and $\\Sigma_2$ such that a\nfinite algebra $\\m A$ has a minority term if and only if $\\m a$ satisfies both\n$\\Sigma_1$ and $\\Sigma_2$. In this situation, we would say that the minority\nterm condition is the join of $\\Sigma_1$ and $\\Sigma_2$. Were this the case, we\ncould decide if $\\m a$ has a minority term by deciding $\\Sigma_1$ and\n$\\Sigma_2$.\n\nOn the surface, the minority term condition is already quite concise and\nnatural; it is not clear if having a minority term can be expressed as a join\nof weaker conditions. In this section, we show that it is a join of having a\nMaltsev term with a condition which we call having a minority-majority term\n(not to be confused with the `generalized minority-majority' terms\nfrom~\\cite{GMM-paper}). Maltsev terms are a classical object of study in\nuniversal algebra -- deciding if an\nalgebra has them is in \\compP{} for finite idempotent algebras. The\nminority-majority terms are much less understood.\n\n\\begin{definition}\n  A ternary term $p(x,y,z)$ of an algebra $\\m a$ is a~\\emph{Maltsev term for\n  $\\m a$} if it satisfies the equations\n  \\[\n    p(x,x,y)\\approx p(y,x,x)\\approx y\n  \\]\n  and a~6-ary term $t(x_1, \\dots, x_6)$ is a~\\emph{minority-majority term} of\n  $\\m a$ if it satisfies the equations\n  \\begin{align*}\n    t(y,x,x,z,y,y)&\\approx y\\\\\n    t(x,y,x,y,z,y)&\\approx y\\\\\n    t(x,x,y,y,y,z)&\\approx y.\n  \\end{align*}\n\\end{definition}\n\nWe point out that if an algebra has a~minority term then it also, trivially,\nhas a~Maltsev term, but that the converse does not hold (as witnessed by the\ncyclic group $\\mathbb{Z}_4$).  Our definition of a~minority-majority term is\na~strengthening of the term condition found by Ol\\v{s}\\'{a}k\nin~\\cite{Olsak2017}. Ol\\v{s}\\'{a}k has shown that his terms are a~weakest\nnon-trivial strong Maltsev condition whose terms are all idempotent.\n\nWe observe that by padding variables, any algebra that has a~minority term or\na~majority term (just replace the final occurrence of the variable $y$ in the\nequations~(\\ref{min-eq}) by the variable $x$ to define such a~term) also has\na~minority-majority term.  Since the 2-element lattice has a~majority term but\nno minority term, it follows that having a~minority-majority term is strictly\nweaker than having a~minority term.\n\n\\begin{theorem}\\label{thm:join}\n  An algebra has a~minority term if and only if it has a~Maltsev term and\n  a~minority-majority term.\n\\end{theorem}\n\n\\begin{proof}\nThe discussion preceding this theorem establishes one direction of this\ntheorem.  For the other we need to show that if an algebra $\\m a$ has a~Maltsev\nterm  $p(x,y,z)$,  and a~minority-majority term $t(x_1, \\dots, x_6)$ then $\\m\na$ has a~minority term.  Given such an algebra $\\m a$, define\n\\[\n  m(x,y,z)=t(x,y,z,p(z,x,y),p(x,y,z),p(y,z,x)).\n\\]\nVerifying that $m(x,y,z)$ is a~minority term for $\\m a$  is\nstraightforward; we show one of the three required equalities here as an example:\n\\begin{align*}\n  m(x,x,y)&\\approx t(x,x,y,p(y,x,x),p(x,x,y),p(x,y,x))\\\\\n  &\\approx t(x,x,y,y,y,p(x,y,x))\\approx y.\n\\end{align*}\n\\end{proof}\n\n\\begin{corollary}\n  The problem of deciding if a~finite algebra has a~minority term can be\n  reduced to the problems of deciding if it has a~Maltsev term and if it has\n  a~minority-majority term.\n\\end{corollary}\n\nAs was demonstrated in~\\cite{freese-valeriote, horowitz-ijac}, there is\na~polynomial-time algorithm to decide if a~finite idempotent algebra has\na~Maltsev term. Therefore, should testing for a~minority-majority term for\nfinite idempotent algebras prove to be tractable, then this would lead to\na~fast algorithm for testing for a~minority term, at least for finite\nidempotent algebras.  From the hardness results found\nin~\\cite{freese-valeriote} it follows that in general, the problem of deciding\nif a~finite algebra has a~minority-majority term is \\compEXPTIME-complete; the\ncomplexity of this problem restricted to idempotent algebras is unknown.\n\n\n\\section{Local Maltsev terms}\n  \\label{maltsev}\n\nIn~\\cite{freese-valeriote, horowitz-ijac,  kazda-valeriote,\nValeriote_Willard_2014} polynomial-time algorithms are presented for deciding\nif certain Maltsev conditions hold in the variety generated by a~given finite\nidempotent algebra.  One particular Maltsev condition that is addressed by all\nof these papers is that of having a~Maltsev term.  In all but\n\\cite{freese-valeriote}, the polynomial-time algorithm produced is based on\ntesting for the presence of enough `local' Maltsev terms in the given\nalgebra.\n\n\\begin{definition}\n  Let $\\m a$ be an algebra and $S \\subseteq A^2\\times\\{0,1\\}$.  A term\n  operation $t(x,y,z)$ of $\\m a$ is a~\\emph{local Maltsev term operation for\n  $S$} if:\n  \\begin{itemize}\n    \\item whenever $((a,b), 0) \\in S$, $t(a,b,b) = a$, and\n    \\item whenever $((a,b), 1) \\in S$, $t(a,a,b) = b$.\n  \\end{itemize}\n\\end{definition}\n\nClearly, if $\\m a$ has a~Maltsev term then it has a~local Maltsev term\noperation for every subset $S$ of $A^2 \\times\\{0,1\\}$ and conversely, if $\\m a$\nhas a~local Maltsev term operation for $S = A^2 \\times\\{0,1\\}$ then it has\na~Maltsev term. In~\\cite{horowitz-ijac,  kazda-valeriote,\nValeriote_Willard_2014} it is shown that if a~finite idempotent algebra $\\m a$\nhas local Maltsev term operations for all two element subsets of $A^2\n\\times\\{0,1\\}$ then $\\m a$ will have a~Maltsev term. This fact is then used as\nthe basis for a~polynomial-time test to decide if a~given finite idempotent\nalgebra has a~Maltsev term.\n\nIn this section we extract an additional piece of information from this\napproach to testing for a~Maltsev term, namely that if a~finite idempotent\nalgebra has a~Maltsev term, then we can produce an operation table or a~circuit\nfor a~Maltsev term operation in time polynomial in the size of the algebra.\n We will first prove that there is an\nalgorithm for producing circuits for a Maltsev function; the algorithm for\nproducing the operation table will then be given as a~corollary. However, for\nthe reduction presented in Section~\\ref{sec:np} we need only the algorithm\nfor producing a function table.\n\n\nLet us first briefly describe how to get a~global Maltsev operation from local\nones. Assume we know (circuits of) a~local Maltsev term operation\n$t_{a,b,c,d}(x,y,z)$ for each two element subset\n\\[\n  \\{((a,b), 0), ((c,d), 1)\\}\n\\]\nof $A^2 \\times\\{0,1\\}$. These are required for $\\m A$ to have a~Maltsev term.\nA global Maltsev term can be constructed from them in two stages: First, we construct, for each $a,b\\in\nA$, an operation $t_{a,b}$ such that $t_{a,b}(a,b,b) = a$ and $t_{a,b}(x,x,y) = y$ for all\n$x,y \\in A$. This is done by fixing an enumeration $(a_1, b_1)$, $(a_2,\nb_2)$, \\dots, $(a_{n^2}, b_{n^2})$ of $A^2$, and then defining, for $1 \\le j\n\\le n^2$, the operation $t_{a,b}^j(x,y,z)$ on $A$ inductively as follows:\n\\begin{itemize}\n  \\item $t_{a,b}^1(x,y,z) = t_{a,b,a_1, b_1}(x,y,z)$, and\n  \\item for $1 \\le j < n^2$, $t_{a,b}^{j+1}(x,y,z) =\n    t_{a,b,u,v}(t_{a,b}^j(x,y,z), t_{a,b}^j(y,y,z), z)$, where $u =\n    t_{a,b}^j(a_{j+1}, a_{j+1}, b_{j+1})$ and $v = b_{j+1}$.\n\\end{itemize}\nAn easy inductive argument shows that  $t_{a,b}^j(a,b,b) = a$ and\n$t_{a,b}^j(a_i, a_i, b_i) = b_i$ for all $i \\le j \\le n^2$, and so setting\n$t_{a,b}(x,y,z) = t_{a,b}^{n^2}(x,y,z)$ works.\n\nIn the second stage, we construct a~term $t_j(x,y,z)$ such that $t_j(a,a,b) =\nb$ for all $a$, $b \\in A$ and $t_j(a_i, b_i, b_i) = a_i$ for all $i \\le j$.  We\ndefine this sequence of operations inductively again:\n\\begin{itemize}\n  \\item $t_1(x,y,z) = t_{a_1, b_1}(x,y,z)$, and\n  \\item for $1 \\le j < n^2$, $t_{j+1}(x,y,z) = t_{u,v}(x, t_j(x,y,y),\n    t_j(x,y,z))$, where $u = a_{j+1}$ and $v = t_j(a_{j+1}, b_{j+1},\n    b_{j+1})$.\n\\end{itemize}\nAgain, it can be shown that for $1 \\le j \\le n^2$, the operation $t_j(x,y,z)$\nsatisfies the claimed properties and so $t_{n^2}(x,y,z)$ will be a~Maltsev term\noperation for $\\m a$.\n\n\nFrom the above construction, one can obtain a~term that\n represents a Maltsev term operation of the algebra $\\m A$, starting with terms representing the operations $t_{a,b,c,d}$. But there is\nan efficiency problem with this approach:\nthe term is extended by one layer\nin each step, which results in a term of exponential size. Therefore, the\nbookkeeping of this term would increase the running time of the algorithm\nbeyond polynomial. Nevertheless, this can be circumvented by constructing\na~succint representation of the term operations, namely by considering circuits\ninstead of terms.\n\nInformally, a~circuit over an algebraic language (as a~generalization of\nlogical circuits) is a~collection of gates labeled by operation symbols, where\nthe number of inputs of each gate corresponds to the arity of the operation\nsymbol. The inputs are either connected to outputs of some other gate, or\ndesignated as inputs of the circuit; an output of one of the gates is\ndesignated as an~output of the circuit. Furthermore, these connections allow\nfor straightforward evaluation, i.e., there are no oriented cycles.\n\nFormally, we define an $n$-ary \\emph{circuit} in the language of an algebra $\\m A$ as a~directed acyclic graph with possibly multiple edges that has two kinds of vertices: \\emph{inputs} and \\emph{gates}. There are exactly $n$ inputs, labeled by variables $x_1,\\dots, x_n$, and each of them is a~source, and a~finite number of gates. Each gate is labeled by an~operation symbol of $\\m A$, the in-degree corresponds to the arity of the operation, and the in-edges are ordered. One of the vertices is designated as the \\emph{output} of the circuit. We define the size of the circuit to be the number of its vertices.\n\nThe value of a~circuit given an input tuple $a_1,\\dots,a_n$ is defined by the following\nrecursive computation: The value on an input vertex labeled by $x_i$ is $a_i$,\nthe value on a~gate labeled by $g$ is the value of the operation $g^{\\m A}$\napplied to the values of its in-neighbours in the specified order. Finally, the\noutput value of the circuit is the value of the output vertex. It is easy\nto see that the value of a~circuit on a~given tuple can be computed in linear\ntime (in the size of the circuit) in a~straightforward way. For a~fixed circuit\nthe function that maps the input tuple to the output is a~term function of $\\m\nA$. Indeed, to find such a~term it is enough to evaluate the circuit in the\nfree (term) algebra on the tuple $x_1,\\dots,x_n$. The converse is also true\nsince any term can be represented as a~`tree' circuit (it is an oriented tree\nif we omit all input vertices). Many terms can be expressed by considerably\nsmaller circuits. We give one such example in\nFigure~\\ref{fig:term-and-circuit}.\n\n\\begin{figure}\n  \\[ \\begin{tikzpicture}[ split\/.style = { shape = circle, draw, fill, inner\n      sep = 0, minimum size = 2.5 }, circuit logic US ]\n\n    \\node [and gate, inputs = {nnn}, point right, minimum size = .7cm] (f) at (3,0) {$f$};\n    \\node [and gate, inputs = {nnn}, point right, minimum size = .7cm] (g) at (0,0) {$g$};\n\n    \\node (x) at (-3,.7) {$x$};\n    \\node (y) at (-3,0) {$y$};\n    \\node (z) at (-3,-.7) {$z$};\n\n    \\draw (f.output) -- ++(right:.5) node [right] {$f(g(x,y,y),g(x,y,y),z)$} ;\n\n    \\draw (g.output) -- (f.input 2);\n    \\draw ($(g.output)!.5!(f.input 2)$) |- (f.input 1);\n    \\node [split] at ($(g.output)!.5!(f.input 2)$) {};\n\n    \\draw (x.east) -| ($(x.east)!.5!(g.input 1)$) |- (g.input 1);\n\n    \\draw (y.east) -- (g.input 2);\n    \\draw ($(y.east)!.5!(g.input 2)$) |- (g.input 3);\n    \\node [split] at ($(y.east)!.5!(g.input 2)$) {};\n\n    \\draw (z.east) -| ($(g.output)!.5!(f.input 2) + (-90:.5) $) |- (f.input 3);\n\n  \\end{tikzpicture} \\]\n  \\caption{A~succinct circuit representation of the term $f(g(x,y,y),g(x,y,y),z)$.}\n    \\label{fig:term-and-circuit}\n\\end{figure}\n\nIn the proof of the theorem below, we will also use circuits with multiple outputs. The only difference in the definition is that several vertices are designated as outputs. Any such circuit then computes a~tuple of term functions.\n\n\\begin{theorem}\\label{maltsevcircuit}\nLet $\\m a$ be a~finite idempotent algebra.  There is an algorithm whose\nruntime can be bounded by a~polynomial in the size of $\\m a$ that will either\n(correctly) output that $\\m a$ has no Maltsev term operation, or output\na~circuit for some Maltsev term operation of $\\m a$.\n\\end{theorem}\n\n\\begin{proof}\n Let $n =\n  |A|$.  Recall that $\\m A$ has at least one\n  basic operation of positive arity and hence $\\|\\m A\\|\\geq n$. Let $m\\geq 1$ be the\n  maximal arity of an operation of $\\m A$.\n\n  We construct a~circuit representing a~Maltsev operation in three steps: The\n  first step produces, for each $a$, $b$, $c$, $d$ from $A$, a circuit that computes a local Maltsev term operation $t_{a,b,c,d}$ as defined near the beginning of this section, the second step\n  produces circuits that compute $t_{a,b}$, and the final step produces\n  a~circuit for a~Maltsev operation $t$. We note that the algorithm can fail\n  only in the first step.\n\n  \\begin{figure}\n  \\[\n  \\begin{tikzpicture}[ split\/.style = { shape = circle, draw, fill, inner sep = 0,\n      minimum size = 2.5 }, node distance = 1.7cm, circuit logic US  ]\n    \\node [and gate, inputs = {nnn}, point right] (out1) at (0,1) {$t_{a,b,u,v}$};\n    \\node [and gate, inputs = {nnn}, point right] (out2) at (0,-1) {$t_{a,b,u,v}$};\n\n    \\draw (out1.output) -- ++(right:.5) node [right] {$t^{j+1}_{a,b}(x,y,z)$} ;\n    \\draw (out2.output) -- ++(right:.5) node [right] {$t^{j+1}_{a,b}(y,y,z)$} ;\n\n    \\node at ($(out1.input 1) + (left:2.5)$) (j1) {$t^j_{a,b}(x,y,z)$};\n    \\node at ($(out2.input 1) + (left:2.5)$) (j2) {$t^j_{a,b}(y,y,z)$};\n\n    \\draw (j1.east) -- (out1.input 1);\n\n    \\draw (j2.east) -- (out2.input 1);\n    \\node (split1) [ split ] at ($(j2.east)!.5!(out2.input 1)$) {};\n    \\draw (split1) |- (out1.input 2);\n    \\draw (split1) |- (out2.input 2);\n\n    \\node [split] (split2) at ($(split1) + (-.25,0) - (out1.input 1) + (out1.input 3)$) {};\n    \\draw (split2) |- (out1.input 3);\n    \\draw (split2) -- (out2.input 3);\n\n    \\node [ left of = j1 ] (out1j){};\n    \\draw (j1.west) -- (out1j.east);\n    \\node [ left of = j2 ] (out2j){};\n    \\draw (j2.west) -- (out2j.east);\n\n    \\node (x) at (-8,1.5) {$x$};\n    \\node (y) at (-8,0) {$y$};\n    \\node (z) at (-8,-1.5) {$z$};\n\n    \\node [ right of = x ] (in1j){};\n    \\node [ right of = y ] (in2j){};\n    \\node [ right of = z ] (in3j){};\n\n    \\draw (x.east) -- (in1j);\n    \\draw (y.east) -- (in2j);\n    \\draw (z.east) -- (in3j);\n\n    \\node (split3) [split] at ($(z.east) !.5! (in3j.west)$) {};\n    \\node (mid) [ below of = j2 ] {$z$};\n    \\draw (split3) |- (mid) -| (split2);\n\n    \\node [ draw, dashed, fit = (in1j) (in2j) (in3j) (out1j) (out2j) ] {$C^j_{a,b}$};\n  \\end{tikzpicture}\n  \\]\n    \\caption{Recursive definition of circuit $C^{j+1}_{a,b}$.}\n    \\label{fig:C-j-ab}\n  \\end{figure}\n\n\n  Step 1: Circuits for $t_{a,b,c,d}$. For each $a,b,c,d$, we aim to produce a~circuit that computes a local Maltsev term\n  operation $t_{a,b,c,d}$. To do this, we consider the subuniverse $R$ of $\\m\n  A^2$ generated by $\\{(a,c), (b,c), (b,d)\\}$.\n  According to Proposition~6.1 from~\\cite{freese-valeriote} $R$ can be generated in time $O(||\\m a||^2m)$.\n  It is clear that $\\m A$ has\n  a~local Maltsev term operation $t_{a,b,c,d}$ if and only if $(a,d) \\in R$. Our algorithm produces a circuit for $t_{a,b,c,d}$ by generating elements of $R$ one at a time and keeping track of circuits that witness\n  the membership of these elements.\n\n  More precisely, we employ a subuniverse generating algorithm to produce a sequence\n  $r_1 = (a,c), r_2 = (b,c), r_3 = (b,d), r_4, \\dots$ of elements of $R$ (in time $O(||\\m a||^2m)$) such\n  that each $r_{k+1}$, for $k \\ge 3$, is obtained from $r_1,\\dots,r_{k}$ by a~single\n  application of an operation $f$ of $\\m A^2$.  Our algorithm will also produce  a~sequence of\n  ternary circuits $C_{a,b,c,d}^3 \\subseteq C_{a,b,c,d}^4 \\subseteq \\dots$ such\n  that each $C_{a,b,c,d}^k$ has $k$ outputs, and the values of $C_{a,b,c,d}^k$\n  on $r_1,r_2,r_3$ give $r_1,\\dots,r_k$. We define $C_{a,b,c,d}^3$ to be\n  the~circuit with no gates, and outputs $x_1$, $x_2$, $x_3$. The circuit\n  $C_{a,b,c,d}^{k+1}$ is defined inductively from $C_{a,b,c,d}^k$: Consider an\n  operation $f$ and $r_{i_1},\\dots,r_{i_p}$ with $i_j \\leq k$ such that\n  $r_{k+1} = f(r_{i_1},\\dots,r_{i_p})$; add a~gate labeled $f$ to\n  $C_{a,b,c,d}^k$ connecting its inputs with the outputs of $C_{a,b,c,d}^k$\n  numbered by $i_j$ for $j = 1,\\dots, p$. We designate the output of this gate\n  as the $(k+1)$-st output of $C_{a,b,c,d}^{k+1}$.\n \n\n\n  It is straightforward to\n  check that the circuits $C_{a,b,c,d}^k$ satisfy the requirements. We also\n  note that the size of $C_{a,b,c,d}^k$ is exactly $k$. We stop this inductive\n  construction at some step $k$ if $r_k = (a,d)$, in which case we produce the circuit\n  $C_{a,b,c,d}$  from $C_{a,b,c,d}^k$ by indicating a~single output to be the\n  $k$-th output of $C_{a,b,c,d}^k$.  If, on the other hand, we have generated all of $R$ without producing $(a,d)$ at any step then the algorithm halts and outputs that $\\m a$ does not have a Maltsev term operation.  The soundness of our algorithm follows from the fact that $\\m a$ has a~local Maltsev term $t_{a,b,c,d}$ if and only if $(a,d) \\in R$\n  and that $\\m a$ has a Maltsev term if and only if it has local Maltsev terms $t_{a,b,c,d}$ for all $a$, $b$, $c$, $d \\in A$.\n   The algorithm produces circuits of\n  size $\\bigO(n^2)$ and spends most of its\n  time generating new elements of $R$;\n \n \n  generating each $C_{a,b,c,d}$ takes time\n  $O(\\|\\m a\\|^2m)$, making the total time complexity of Step 1 to be $\\bigO(\\|\\m\n  a\\|^2mn^4)$.\n\n\n  Step 2: Circuits for $t_{a,b}$. At this point we assume that the functions $t_{a,b,c,d}$ are part of\n  the signature. It is clear that the full circuit can be obtained by\n  substituting the circuits $C_{a,b,c,d}$ for gates labeled by $t_{a,b,c,d}$,\n  and this can be still done in polynomial time.\n\n  Our task is to obtain a~circuit for $t_{a,b}$. We do this by inductively constructing circuits $C^j_{a,b}$ that compute two\n  values of the terms $t_{a,b}^j$, namely $t_{a,b}^j(x,y,z)$ and\n  $t_{a,b}^j(y,y,z)$. Starting with $j = 0$ and $t^0(x,y,z) = x$, we define\n  $C_{a,b}^0$ to be the circuit with no gates and outputs $x,y$. Further, we\n  define circuit $C_{a,b}^{j+1}$ inductively from $C_{a,b}^j$ by adding two\n  gates labeled by $t_{a,b,u,v}$, where $u = t_{a,b}^j(a_{j+1}, a_{j+1},\n  b_{j+1})$ and $v = b_{j+1}$: the first gate has as inputs the two outputs of\n  $C_{a,b}^j$ and $z$, the second gate has as inputs two copies of the second\n  output of $C_{a,b}^j$ and $z$. See Figure~\\ref{fig:C-j-ab} for a~graphical\n  representation. Again, it is straightforward to check that these circuits\n  have the required properties.  Also note that the size of $C^j_{a,b}$ is\n  bounded by $2j+3$ which is a~polynomial. The final circuit $C_{a,b}$\n  computing $t_{a,b}$ is obtained from $C_{a,b}^{n^2}$ by designating the first\n  output of $C_{a,b}^{n^2}$ to be the only output of $C_{a,b}$. Once we have\n  $t_{a,b,c,d}$ in the signature, this process will run in time $\\bigO(n^2)$.\n\n  Step 3: Circuit for a Maltsev term. Again, we assume that $t_{a,b}$ are basic operations, and construct\n  circuits $C_j$ computing two values $t_j(x,y,y)$ and $t_j(x,y,z)$ of $t_j$\n  inductively. The proof is analogous to Step 2, with the only difference that\n  we use Figure~\\ref{fig:C-j} for the inductive definition. Again the time\n  complexity is $\\bigO(n^2)$.\n\n  \\begin{figure}\n    \\begin{centering}\n    \\begin{tikzpicture}[ split\/.style = { shape = circle, draw, fill, inner sep = 0,\n        minimum size = 2.5 }, node distance = 1.7cm, circuit logic US  ]\n      \\node [and gate, inputs = {nnn}, point right, minimum size = .7cm] (out1) at (0,1) {$t_{u,v}$};\n      \\node [and gate, inputs = {nnn}, point right, minimum size = .7cm] (out2) at (0,-1) {$t_{u,v}$};\n\n      \\draw (out1.output) -- ++(right:.5) node [right] {$t_{j+1}(x,y,y)$} ;\n      \\draw (out2.output) -- ++(right:.5) node [right] {$t_{j+1}(x,y,z)$} ;\n\n      \\node at ($(out1.input 3) + (left:2.5)$) (j1) {$t_j(x,y,y)$};\n      \\node at ($(out2.input 3) + (left:2.5)$) (j2) {$t_j(x,y,z)$};\n      \\node (mid) [ above of = j1 ] {$x$};\n\n      \\draw (j1.east) -- (out1.input 3);\n      \\draw (j2.east) -- (out2.input 3);\n\n      \\node (split1) [ split ] at ($(j1.east)!.5!(out1.input 3)$) {};\n      \\draw (split1) |- (out1.input 2);\n      \\draw (split1) |- (out2.input 2);\n\n      \\node [split] (split2) at ($(split1) + (-.25,0) + (out1.input 1) - (out1.input 3)$) {};\n      \\draw (split2) -- (out1.input 1);\n      \\draw (split2) |- (out2.input 1);\n\n      \\node [ left of = j1 ] (out1j){};\n      \\draw (j1.west) -- (out1j.east);\n      \\node [ left of = j2 ] (out2j){};\n      \\draw (j2.west) -- (out2j.east);\n\n      \\node (x) at (-8,1.5) {$x$};\n      \\node (y) at (-8,0) {$y$};\n      \\node (z) at (-8,-1.5) {$z$};\n\n      \\node [ right of = x ] (in1j){};\n      \\node [ right of = y ] (in2j){};\n      \\node [ right of = z ] (in3j){};\n\n      \\draw (x.east) -- (in1j);\n      \\draw (y.east) -- (in2j);\n      \\draw (z.east) -- (in3j);\n\n      \\node (split3) [split] at ($(x.east) !.5! (in1j.west)$) {};\n      \\draw (split3) |- (mid) -| (split2);\n\n      \\node [ draw, dashed, fit = (in1j) (in2j) (in3j) (out1j) (out2j) ] {$C_j$};\n    \\end{tikzpicture}\n    \\end{centering}\n    \\caption{Recursive definition of circuit $C_{j+1}$.}\n    \\label{fig:C-j}\n  \\end{figure}\n\n  Each step runs in time polynomial in $\\|\\m a\\|$ (the time complexity is\n  dominated by Step 1) and outputs a~polynomial\n  size circuit. This also implies that expanding the gates according to their\n  definitions in Steps 2 and 3 can be done in polynomial time; the final size of\n  the output circuit will be bounded by $\\bigO(n^6)$.\n\\end{proof}\n\n\\begin{corollary}\\label{maltsevterm}\n  Let $\\m a$ be a~finite idempotent algebra.  There is an algorithm whose\n  runtime can be bounded by a~polynomial in the size of $\\m a$ that will\n  produce the table of some Maltsev term operation of $\\m a$, should one exist.\n\\end{corollary}\n\n\\begin{proof}\n  The polynomial-time algorithm is as follows. First, generate a~polynomial size\n  circuit for some Maltsev term operation of $\\m a$. This can be done in polynomial time by\n  the above theorem. Second, evaluate this circuit at all $\\lvert A\\rvert^3$\n  possible inputs. The second step runs in polynomial time since evaluation of\n  a~circuit is linear in the size of the circuit.\n\\end{proof}\n\nWe note that there is also a more straightforward algorithm for producing the\noperation table of a Maltsev term which follows the circuit construction but\ninstead of circuits, it remembers the tables for each of the relevant term\noperations.\n\n\n\\section{Local minority terms}\n\nIn contrast to the situation for Maltsev terms highlighted in the previous\nsection, we will show that having plenty of `local' minority terms does not\nguarantee that a~finite idempotent algebra will have a~minority term.  One\nconsequence of this is that an approach along the lines in~\\cite{horowitz-ijac,\nkazda-valeriote, Valeriote_Willard_2014} to finding an efficient algorithm to\ndecide if a~finite idempotent algebra has a~minority term will not work.\n\nIn this section, we will construct for each odd natural number $n > 2$ a~finite\nidempotent algebra $\\m a_n$ with the following properties: The universe of $\\m\na_n$ has size $4n$ and $\\m a_n$ does not have a~minority term, but for every\nsubset $E$ of $A_n$ of size $n-1$ there is a~term of $\\m a_n$ that acts as\na~minority term on the elements of $E$.\n\nWe start our construction by fixing some odd $n > 2$ and some minority\noperation $m$ on the set $[n] = \\{1, 2, \\dots, n\\}$. To make things concrete\nwe set\n\\[\n  m(x,y,z)=\\begin{cases}\n    x& y=z\\\\\n    y& x=z\\\\\n    z& \\text{else,}\n  \\end{cases}\n\\]\nbut note that any minority operation on $[n]$ will do.\n\nSince there are two nonisomorphic groups of order 4, we have two different\nnatural group operations on $\\{0,1,2,3\\}$: addition modulo~4, which we will\ndenote by `$+$' (its inverse is `$-$'), and bitwise XOR, which we denote by\n`$\\oplus$' (this operation takes bitwise XOR of the binary representations of\ninput numbers, so for example $1\\oplus 3=2$). Throughout this section, we will\nuse arithmetic modulo 4, e.g., $6x = x + x$, for all expressions except those\ninvolving indices.\n\nThe construction relies on similarities and subtle differences of the two group\nstructures, and the derived Maltsev operations, $x-y+z$ and $x\\oplus y\\oplus z$.\nBoth these operations share a congruence $\\equiv_2$ that is given by taking\nthe remainder modulo 2. We note that $x\\equiv_2 y$ if and only if $2x = 2y$.\n\n\\begin{observation}\\label{obs:maltsev-diff}\n  Let $x,y,z\\in \\{0,1,2,3\\}$. Then\n\\[\n  (x\\oplus y\\oplus z) - (x - y + z) \\in \\{0,2\\}\n,\\]\nand moreover the result depends only on the classes of\n  $x$, $y$, and $z$ in the congruence $\\equiv_2$ (i.e., the least significant\nbinary bits of $x$, $y$, and $z$).\n\\end{observation}\n\\begin{proof}\nBoth Maltsev operations agree modulo $\\equiv_2$, hence the difference lies in\nthe $\\equiv_2$-class of 0.\n\nTo see the second part, it is enough to observe that $x\\oplus 2=x+2=x-2$ for all\n$x$. Hence changing, say $x$ to $x'=x\\oplus 2$ simply flips the most\n  significant binary bit\nof both  $x\\oplus y\\oplus z$ and $x - y + z$, keeping the difference the same.\n\\end{proof}\n\n\n\\begin{definition}\nLet $A_n=[n]\\times [4]$. For $i \\in [n]$, we define  $t_i(x,y,z)$ to be the\nfollowing operation on $A_n$:\n\\[\n  t_i((a_1,b_1), (a_2,b_2), (a_3,b_3)) =\n  \\begin{cases}\n    (i,b_1 - b_2 + b_3)\n      \\quad\\text{if $a_1 = a_2 = a_3 = i$, and} \\\\\n    (m(a_1,a_2,a_3),b_1\\oplus b_2 \\oplus b_3),\n      \\quad\\text{otherwise.}\n  \\end{cases}\n\\]\nThe algebra $\\m a_n$ is defined to be the algebra with universe $A_n$ and\nbasic operations $t_1,\\dots,t_n$.\n\\end{definition}\n\nBy construction, the following is true.\n\n\\begin{claim}\\label{local}\n  For every $(n-1)$-element subset $E$ of $A_n$, there is a~term operation of\n  $\\m a_n$ that satisfies the minority term equations when restricted to\n  elements from $E$.\n\\end{claim}\n\n\\begin{proof}\n  Pick $i\\in [n]$ such that no element of $E$ has its first coordinate equal to\n  $i$; the operation $t_i$ is a local minority for this $E$.\n\\end{proof}\n\n\n\\begin{proposition}\\label{prop:An}\n  For $n > 1$ and odd, the algebra $\\m a_n$ does not have a~minority term.\n\\end{proposition}\n\n\\begin{proof}\n  Given some $(i,a)\\in A_n$, we will refer to $a$ as the \\emph{arithmetic part}\n  of $(i,a)$. This is to avoid talking about `second coordinates' in\n  the confusing situation when $(i,a)$ itself is a~part of a~tuple of elements\n  of $A_n$.\n \n \n\n  To prove the proposition, we will define a~certain subuniverse $R$ of $(\\m\n  a_n)^{3n}$ and then show that $R$ is not closed under any minority operation\n  on $A_n$ (applied coordinate-wise). We will write $3n$-tuples of elements of\n  $A_n$ as $3n\\times 2$ matrices where the arithmetic parts of the elements\n  make up the second column.\n\n  Let $R \\subseteq (A_n)^{3n}$ be the set of all $3n$-tuples of the form\n  \\[\n  \\begin{pmatrix}\n    1&x_1\\\\ 2&x_2\\\\ \\vdots\\\\ n&x_n\\\\\n    1&x_{n+1}\\\\ 2&x_{n+2}\\\\ \\vdots\\\\ n&x_{2n}\\\\\n    1&x_{2n+1}\\\\ 2&x_{2n+2}\\\\ \\vdots\\\\ n&x_{3n}\\\\\n  \\end{pmatrix}\n  \\]\n  such that\n  \\begin{align}\n    &x_{kn+1} \\equiv_2x_{kn+2} \\equiv_2 \\dots \\equiv_2 x_{kn+n},\n      &\\text{for $k=0,1,2$, and} \\label{eqn:bits} \\\\\n    &\\sum_{i=1}^{3n} x_i = 2.\\label{eqn:strange-sum}\n  \\end{align}\n  The three equations from (\\ref{eqn:bits}) mean that the least significant bits of the\n  arithmetic parts of the first $n$ entries agree and similarly for the second\n  and the last $n$ entries; equation (\\ref{eqn:strange-sum}) can be viewed\n  as a~combined parity check on all involved bits.\n\n  \\begin{claim}\n    The relation $R$ is a~subuniverse of $(\\m a_n)^{3n}$.\n  \\end{claim}\n  \\begin{proof}\n    By the symmetry of the $t_i$'s and $R$, it is enough to show that $t_1$\n    preserves $R$. Let us take three arbitrary members of $R$:\n    \\[\n    \\begin{pmatrix}\n      1&x_{1,1}\\\\ 2&x_{1,2}\\\\ \\vdots\\\\ n&x_{1,n}\\\\\n      1&x_{1,n+1}\\\\ 2&x_{1,n+2}\\\\ \\vdots\\\\ n&x_{1,2n}\\\\\n      1&x_{1,2n+1}\\\\ 2&x_{1,2n+2}\\\\ \\vdots\\\\ n&x_{1,3n}\\\\\n    \\end{pmatrix},\n    \\begin{pmatrix}\n      1&x_{2,1}\\\\ 2&x_{2,2}\\\\ \\vdots\\\\ n&x_{2,n}\\\\\n      1&x_{2,n+1}\\\\ 2&x_{2,n+2}\\\\ \\vdots\\\\ n&x_{2,2n}\\\\\n      1&x_{2,2n+1}\\\\ 2&x_{2,2n+2}\\\\ \\vdots\\\\ n&x_{2,3n}\\\\\n    \\end{pmatrix},\n    \\begin{pmatrix}\n      1&x_{3,1}\\\\ 2&x_{3,2}\\\\ \\vdots\\\\ n&x_{3,n}\\\\\n      1&x_{3,n+1}\\\\ 2&x_{3,n+2}\\\\ \\vdots\\\\ n&x_{3,2n}\\\\\n      1&x_{3,2n+1}\\\\ 2&x_{3,2n+2}\\\\ \\vdots\\\\ n&x_{3,3n}\\\\\n    \\end{pmatrix}\n    \\]\n    and apply $t_1$ to them to get:\n    \\begin{equation}\n      \\vec r =\n      \\begin{pmatrix}\n        1&x_{1,1}-x_{2,1}+x_{3,1}\\\\\n        2&x_{1,2}\\oplus x_{2,2}\\oplus x_{3,2}\\\\\n         &\\vdots\\\\\n        n&x_{1,n}\\oplus x_{2,n}\\oplus x_{3,n}\\\\\n        1&x_{1,n+1}-x_{2,n+1}+x_{3,n+1}\\\\\n        2&x_{1,n+2}\\oplus x_{2,n+2}\\oplus x_{3,n+2}\\\\\n         &    \\vdots\\\\\n        n&x_{1,2n}\\oplus x_{2,2n}\\oplus x_{3,2n}\\\\\n        1&x_{1,2n+1}-x_{2,2n+1}+x_{3,2n+1}   \\\\\n        2&x_{1,2n+2}\\oplus x_{2,2n+2}\\oplus x_{3,2n+2}\\\\\n         &   \\vdots\\\\\n        n&x_{1,3n}\\oplus x_{2,3n}\\oplus x_{3,3n}\\\\\n      \\end{pmatrix}\n      \\label{eqn:r}\n    \\end{equation}\n    We want to verify that $\\vec r\\in R$. First note that (\\ref{eqn:bits}) is\n    satisfied: This follows from the fact that $x-y+z$ and $x\\oplus y\\oplus z$\n    give the same result modulo 2, and the assumption that the original three\n    tuples satisfied (\\ref{eqn:bits}).\n\n    What remains is to verify the property~(\\ref{eqn:strange-sum}). If in the\n    equality~(\\ref{eqn:r}) above we replace the operations $\\oplus$ by $-$ and\n    $+$, verifying~(\\ref{eqn:strange-sum}) is easy: The sum of the\n    arithmetic parts of such a modified tuple is\n    \\begin{equation}\n      \\sum_{j=1}^{3n} (x_{1,j}-x_{2,j}+x_{3,j})=\n      \\sum_{j=1}^{3n}\n      x_{1,j}-\\sum_{j=1}^{3n}x_{2,j}+\\sum_{j=1}^{3n}x_{3,j}=2-2+2=2.\n      \\label{eqn:2}\n    \\end{equation}\n    This is why we need to examine the difference between the $\\oplus$-based\n    and $+$-based Maltsev operations. For $k\\in \\{0,1,2\\}$ and $i\\in\n    \\{1,\\dots,n\\}$ we let\n    \\[\n      c_{k,i} = (x_{1,kn+i} \\oplus x_{2,kn+i} \\oplus x_{3,kn+i}) -\n      (x_{1,kn+i} - x_{2,kn+i} + x_{3,kn+i})\n    \\]\n   By the second part of Observation~\\ref{obs:maltsev-diff}, $c_{k,i}$ does not\n   depend on $i$ (changing $i$ does not change\n    the $x_{j,kn+i}$'s modulo $\\equiv_2$ by condition~(\\ref{eqn:bits}) in the\n    definition of $R$). Hence we can write just $c_k$ instead of $c_{k,i}$.\n\n    Using $c_0$, $c_1$, and\n    $c_2$ to adjust for the differences between the two Maltsev operations, we can express the\n    sum of the arithmetic parts of the tuple $\\vec{r}$ as\n    \\[\n      \\sum_{j=1}^{3n} (x_{1,j}-x_{2,j}+x_{3,j})+\\sum_{i=2}^{n}\n      c_0+\\sum_{i=2}^{n} c_1+\\sum_{i=2}^{n} c_2\n      = 2+(n-1)(c_0+c_1+c_2)\n    \\]\n    where we used~(\\ref{eqn:2}) to get the right hand side. We chose $n$ odd,\n    hence $n-1$ is even and each $c_k$ is even by\n    Observation~\\ref{obs:maltsev-diff}, so $(n-1)c_k=0$ for any $k=0,1,2$. We see that the sum of the\n    arithmetic parts of $\\vec{r}$ is equal to 2 which concludes the proof\n    of~(\\ref{eqn:strange-sum}) for the tuple~$\\vec r$ and we are done.\n  \\end{proof}\n\n  It is easy to see that\n  \\[\n  \\begin{pmatrix}\n    1&0\\\\ 2&0\\\\ \\vdots\\\\ n&0\\\\\n    1&1\\\\ 2&1\\\\ \\vdots\\\\ n&1\\\\\n    1&1\\\\ 2&1\\\\ \\vdots\\\\ n&1\\\\\n  \\end{pmatrix},\n  \\begin{pmatrix}\n    1&1\\\\ 2&1\\\\ \\vdots\\\\ n&1\\\\\n    1&0\\\\ 2&0\\\\ \\vdots\\\\ n&0\\\\\n    1&1\\\\ 2&1\\\\ \\vdots\\\\ n&1\\\\\n  \\end{pmatrix},\n  \\begin{pmatrix}\n    1&1\\\\ 2&1\\\\ \\vdots\\\\ n&1\\\\\n    1&1\\\\ 2&1\\\\ \\vdots\\\\ n&1\\\\\n    1&0\\\\ 2&0\\\\ \\vdots\\\\ n&0\\\\\n  \\end{pmatrix}\\in R,\n  \\quad\\text{and}\\quad\n  \\begin{pmatrix}\n    1&0\\\\ 2&0\\\\ \\vdots\\\\ n&0\\\\\n    1&0\\\\ 2&0\\\\ \\vdots\\\\ n&0\\\\\n    1&0\\\\ 2&0\\\\ \\vdots\\\\ n&0\\\\\n  \\end{pmatrix}\\notin R.\n  \\]\n  However, the last tuple can be obtained from the first three by applying any\n  minority operation on the set $A_n$ coordinate-wise. From this we conclude\n  that $\\m a_n$ does not have a~minority term.\n\\end{proof}\n\nWe note that the above construction of $\\m a_n$ makes sense for $n$ even as well\nand claim that these algebras also have the same key features, namely, by\nconstruction, they have plenty of `local' minority term operations but they do\nnot have minority terms.  The verification of this last fact for $n$ even is\nsimilar, but slightly more technical than for $n$ odd, and we omit the proof here.\n\nThe algebras $\\m a_n$ can also be used to witness that having a~lot of local\nminority-majority terms does not guarantee the presence of an actual\nminority-majority term. By padding with dummy variables, any local minority\nterm of an algebra $\\m a_n$ is also a~term that locally satisfies the\nminority-majority term equations.  But since each $\\m a_n$ has a~Maltsev term\nbut not a~minority term, then by Theorem~\\ref{thm:join} it follows that $\\m\na_n$ cannot have a~minority-majority term.\n\n\\section{Deciding minority in idempotent algebras is in \\compNP}\\label{sec:np}\n\nThe results from the previous section imply that one cannot base an efficient\ntest for the presence of a~minority term in a~finite idempotent algebra on\nchecking if it has enough local minority terms.   This does not rule out that\nthe problem is in the class \\compP, but to date no other approach to showing\nthis has worked.  As an intermediate result, we show, at least, that this\ndecision problem is in \\compNP{} and so cannot be \\compEXPTIME-complete (unless\n$\\compNP=\\compEXPTIME$).\n\n\n\nWe first show that an instance $\\m a$ of the decision problem \\minority\\  can\nbe expressed as a~particular instance of the subpower membership problem for\n${\\m a}$.\n\n\\begin{definition}\\label{defSMP}\nGiven a~finite algebra $\\m a$, the \\emph{subpower membership problem} for $\\m\na$, denoted by $\\smp(\\m a)$, is the following decision problem:\n\\begin{itemize}\n  \\item INPUT: $\\vec a_1, \\dots, \\vec a_k, \\vec b \\in A^n$\n  \\item QUESTION: Is $\\vec b$ in the subalgebra of $\\m a^n$ generated by\n    $\\{\\vec a_1, \\dots, \\vec a_k\\}$?\n\\end{itemize}\n\\end{definition}\n\nTo build an instance of $\\smp(\\m a)$ expressing that $\\m a$ has a~minority\nterm, let $I =\\{(a,b,c)\\mid \\mbox{$a, b, c \\in A$ and $|\\{a,b,c\\}| \\le 2$}\\}$.\nSo $|I| = 3|A|^2 - 2|A|$. For $(a,b,c) \\in I$, let $\\min(a,b,c)$ be the minority\nelement of this triple.  So\n\\[\n  \\min(a,b,b) = \\min(b,a,b) = \\min(b,b,a) = \\min(a,a,a) = a.\n\\]\nFor $1 \\le i \\le 3$, let $\\vec \\pi_i \\in A^I$ be defined by $\\vec \\pi_i(a_1,\na_2, a_3) = a_i$ and define $\\vec \\mu_A \\in A^I$ by $\\vec \\mu_A(a_1, a_2, a_3)\n= \\min(a_1, a_2, a_3)$, for all $(a_1, a_2, a_3) \\in I$.  Denote the instance\n$\\vec \\pi_1$, $\\vec \\pi_2$, $\\vec \\pi_3$, and $\\vec \\mu_A$ of $\\smp(\\m a)$ by\n$\\min(\\m a)$.\n\n\\begin{proposition}\\label{min-instance}\n  An algebra $\\m a$ has a~minority term if and only if  $\\vec \\mu_A$ is\n  a~member of the subpower of $\\m a^I$ generated by $\\{\\vec \\pi_1, \\vec \\pi_2,\n  \\vec \\pi_3\\}$, i.e., if and only if $\\min(\\m a)$ is a~`yes' instance of\n  $\\smp(\\m a)$ when $\\m a$ is finite.\n\\end{proposition}\n\n\\begin{proof}\n  If $m(x,y,z)$ is a~minority term for $\\m a$, then applying $m$\n  coordinatewise to the generators $\\vec \\pi_1$, $\\vec \\pi_2$, $\\vec \\pi_3$\n  will produce the element $\\vec \\mu_A$.  Conversely, any term that produces\n  $\\vec \\mu_A$ from these generators will be a~minority term for $\\m a$.\n\\end{proof}\n\nExamining the definition of $\\min(\\m a)$, we see that the parameters from\nDefinition~\\ref{defSMP} are $k=3$ and $n=3|A|^2-2|A|$, which is (for algebras\nwith at least one at least unary basic operation) polynomial in $\\|\\m A\\|$.\nFor $\\m a$ idempotent, we can in fact improve $n$ to $3|A|^2-3|A|$,\nsince then we do not need to include in $I$ entries of the form $(a,a,a)$.\n\nIn general, it is known that for some finite algebras the subpower membership problem can be\n\\compEXPTIME-complete~\\cite{Kozik2008} and that for some others, e.g., for any\nalgebra that has only trivial or constant basic operations,  it lies in the\nclass \\compP.  In~\\cite{Mayr2012}, P.\\ Mayr shows that when $\\m a$ has a~Maltsev\nterm, then $\\smp(\\m a)$ is in \\compNP. We claim that a careful reading of Mayr's proof reveals that in fact the following uniform version of the subpower membership problem, where the algebra $\\m a$ is considered as part of the input, is also in \\compNP.\n\n\n\\begin{definition}\nDefine \\smpun\\ to be the following decision problem:\n\\begin{itemize}\n  \\item INPUT:  A~list of tables of basic operations of an algebra~$\\m A$ that includes a~Maltsev operation, and $\\vec a_1, \\dots, \\vec a_k, \\vec b \\in A^n$.\n  \\item QUESTION: Is $\\vec b$ in the subalgebra of $\\m a^n$ generated by\n    $\\{\\vec a_1, \\dots, \\vec a_k\\}$?\n\\end{itemize}\n\\end{definition}\nWe base the main result of this section\non the following.\n\\begin{theorem}[see \\cite{Mayr2012}]\\label{smpun}\n  The decision problem \\smpun\\ is in the class \\compNP.\n\\end{theorem}\n\nWhile this theorem is not explicitly stated in \\cite{Mayr2012}, it can be seen\nthat the runtime of the verifier that Mayr constructs for the problem $\\smp(\\m\na)$, when $\\m a$ has a Maltsev term, has polynomial dependence on the size of\n$\\m a$ in addition to the size of the input to $\\smp(\\m a)$.  We stress that\nMayr's verifier requires that the table for a Maltsev term of $\\m a$ is given as part of\nthe description of $\\m a$.\n\n\n\n\n\n\n\n\n\\begin{theorem}\\label{NP} The decision problem \\minority\\ is in the class \\compNP.\n\\end{theorem}\n\n\\begin{proof}\nTo prove this theorem, we provide a polynomial reduction $f$ of \\minority\\ to \\smpun.  By Theorem~\\ref{smpun}, this will suffice.\nLet $\\m a$ be an instance of \\minority, i.e., a~finite\n  idempotent algebra that has at least one operation.\n \n \n\n  We first check, using the polynomial-time\n  algorithm from Corollary~\\ref{maltsevterm}, to see if $\\m a$ has a~Maltsev\n  term.  If it does not, then $\\m a$ will not have a~minority term, and in this case we\n  set $f(\\m a)$ to be some fixed `no' instance of \\smpun.  Otherwise, we augment the list of basic operations of $\\m a$ by\n  adding the Maltsev operation on $A$ that the algorithm produced.  Denote\n  the resulting (idempotent) algebra by $\\m a'$ and note that $\\m a'$ can be constructed from $\\m a$ by a polynomial-time algorithm.\n   Also, note that $\\m a'$ is term equivalent to\n  $\\m a$ and so the subpower membership problem is the same for both\n  algebras.\n\n  If we set $f(\\m a)$ to be the instance of \\smpun\\ that consists of  the~list of tables of basic operations of~$\\m A'$ along with\n  $\\min(\\m a)$ then we have, by Proposition~\\ref{min-instance}, that $f(\\m a)$ is a `yes' instance of \\smpun\\ if and only if $\\m a$ has a minority term.  Since the construction of $f(\\m a)$ can be carried out by a procedure whose runtime can be bounded by a polynomial in $\\|\\m a\\|$, we have produced a polynomial reduction of  \\minority\\ to \\smpun, as required.\n \\end{proof}\n\n\n\n\n\n\n\\section{Conclusion}\n\nWhile Theorem~\\ref{NP} establishes that testing for a~minority term for finite\nidempotent algebras is not as hard as it could be, the true complexity of this\ndecision problem is still open.  Our proof of this theorem closely ties the\ncomplexity of {\\minority} to the complexity of the subpower membership problem\nfor finite Maltsev algebras and specifically to the problem \\smpun. Thus any progress on determining the complexity of\n$\\smp(\\m a)$ for finite Maltsev algebras may have a~bearing on the complexity\nof {\\minority}.\nThere has certainly been progress on the algorithmic side of $\\smp$;\na~major recent paper has shown in particular that $\\smp(\\m a)$ is tractable for\n$\\m a$ with cube term operations (of which a Maltsev term operation is\na~special case) as long as $\\m a$ generates a residually small\nvariety~\\cite{BMS18} (the statement from the paper is actually stronger than\nthis, allowing multiple algebras in place of $\\m a$).\n\nIn Section~\\ref{join} we introduced the notion of a~minority-majority term and\nshowed that if testing for such a~term for finite idempotent algebras could be\ndone by a~polynomial-time algorithm, then \\minority\\ would lie in the\ncomplexity class \\compP. This is why we conclude our paper with a~question\nabout deciding minority-majority terms.\n\n\\begin{open-problem*}\n  What is the complexity of deciding if a~finite idempotent algebra has\n  a~minority-majority term?\n\\end{open-problem*}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{ Introduction}\nThe discovery of graphene, one layer of carbon atoms, arose high hopes for its application in future of\nelectronic devices~\\cite{Neto-RMP}. One reason for this attention is high  mobility of charge carriers (electrons or holes)\nwhose density are well controlled  by electrical and chemical doping. \nAlthough the application of pure graphene in electronic industry didn't develop because of  the gap-less dispersion.\nAdding atoms and doping in graphene $e. g.$ fluorinated mono-vacancies~\\cite{Kaloni-EPL-100},\nGe-interrelated~\\cite{Kaloni-EPL-99} have been studied for investigation the electrical \ncharacteristics of graphene under impurity doping, which  gives rise to exploring new interesting electronic and magnetic properties.  \n\n The prediction of existence the neutral triplet collective\nmode in a honeycomb lattice is considered to be the new idea~\\cite{Baskaran-Jafari}.\nThis collective mode was predicted by considering random phase approximation (RPA) method in Hubbard model on \nthe honeycomb lattice $e. g.$ undoped graphene and graphite. By considering this collective mode,\ngood agreement between experimental and theoretical results in TRPES experiment on graphite by Momentum Average (MA) \nmethod was achieved~\\cite{Ebrahimkhas-PRB}. In this approximation, neutral triplet spin-1 mode was considered as coupling\nbetween two spinon bound states in undoped graphene but in scalar form not in $2\\times2$ \nmatrix~\\cite{Jafari-Baskaran},~\\cite{Baskaran-Jafari}. \nIf we study and recheck the calculations of ref.~\\cite{{Jafari-Baskaran}, {Ebrahimkhas-IJMPB}},\nwe will find, first: some parts of the calculations between two sub-lattice\nare eliminated and second: effects of overlap factors  are neglected~\\cite{{Peres-comment},{Jafari-recomment}}.\nThe innovation of ref.~\\cite{Jafari-Baskaran} was considering scalar form for  spin susceptibility, instead of $2\\times 2$ tensor\nfor two sub-lattice material.\nThe essential purpose of this research is analyzing the validity of spin-1 collective mode without one-cone approximation\nin Hubbard model and its novelty  is calculating  the effects of effective long range \ninteraction $(U^{eff}_{l-r})$ in RPA method on  neutral triplet collective mode~\\cite{JPHCM.Jafari}.\nWhy  we considered this form for long range interaction?\nBecause by considering only this interaction type\nwe will obtain  Eq.~\\ref{main.eq} which is an essential equation in finding neutral collective mode in RPA formulas\n(without one-cone approximation), in addition we should consider those types of interaction,\nacting between electrons with opposite spins.   \nThe effects of  Coulomb interaction is known to be in the form $e^{2}\/(\\epsilon r)$ which lead to prediction\nof plasmon dispersion in graphene and has been discussed in details in ref.~\\cite{Wunsch}.\nThe coulomb interaction in the honey comb lattice didn't result spin-1 collective mode.\n\n In the first section of this paper, the effects of short range Hubbard interaction in RPA method on honeycomb\nlattice was calculated exactly without neglecting any interaction terms~\\cite{Peres-comment}, in two\nrepresentations. In the second section, we first introduced new effective long range\ninteraction as a new model in graphene and then we used this model for studying the neutral triplet collective\nmode. finally the finding of these calculations are summarized and discussed on validation of neutral collective mode\nin undoped graphene.\n\n\n\\section{ On-site interaction: sub lattice representation}\n\n  The Hubbard Hamiltonian consist of two terms: tight binding (TB) and short range Hubbard interaction (HI),\n \n \\begin{eqnarray}\n H&=&H_{TB}+H_{HI} \n =-t\\sum_{<i,j>,\\sigma}(a_{i}^{\\dagger \\sigma}b_{j}^{\\sigma}+b_{j}^{\\dagger \\sigma}a_{i}^{\\sigma})\\nonumber\\\\\n &+& U\\sum_{i}(a_{i}^{\\dagger \\uparrow}a_{i}^{\\uparrow}a_{i}^{\\dagger \\downarrow}a_{i}^{\\downarrow}+\n b_{i}^{\\dagger \\uparrow}b_{i}^{\\uparrow}b_{i}^{\\dagger \\downarrow}b_{i}^{\\downarrow}),\\\\\n \\label{TB.eq}\\nonumber\n \\end{eqnarray}\n \n where $t$ is the nearest neighbor hopping. $a^{\\sigma}_{i},b^{\\sigma}_{i},(a^{\\dagger \\sigma}_{i},b^{\\dagger \\sigma}_{i})$\n are annihilation (creation) operators for electrons in  sub lattices $A, B$ respectively, and $\\sigma$ stands for spin of electrons,\n and $U$ is the Hubbard on-site interaction potential. The Hubbard interaction term can be written in the exchange channel,\n \n \\begin{equation}\n H_{HI}= -U\\sum_{i}(a_{i}^{\\dagger \\uparrow}a_{i}^{\\downarrow}a_{i}^{\\dagger \\downarrow}a_{i}^{\\uparrow}+\n b_{i}^{\\dagger \\uparrow}b_{i}^{\\downarrow}b_{i}^{\\dagger \\downarrow}b_{i}^{\\uparrow})\n \\label{exHI.eq}\n \\end{equation}\n \n  \n\n  The  interaction term is represented by Feynman diagrams with two kinds of vertices.\n Each vertex corresponds to   the on-site repulsion interaction on the $A$, $B$ sub-lattices Fig.~\\ref{vertices.fig}.\n In each vertex electrons with opposite spins interact via repulsion potential~\\cite{BF}.\n Fig.(\\ref{vertices.fig}-a) shows creation (annihilation) of electrons with up and down spin in each vertex\n  either in $A$ or $B$ sub-lattice. In Fig.(~\\ref{vertices.fig}-b), by adding vertices of \n each sub-lattice, RPA chain is formed. In this chain the vertex with the same or different sub-lattice put to gather,\n so we  define polarization operators, $\\chi_{AA}, \\chi_{AB}$ which include effects of \n overlap factors.\n \n \n \\begin{figure}[h]\n\\begin{center}\n\\vspace{-3mm}\n\\includegraphics[width=8cm,height=3cm,angle=0]{vertices1.eps}\n\\vspace{-3mm}\n\\caption{(a) Each vertex includes creation(destruction) of an electron with up (down) spin. These vertices show\ninteraction between two electrons with opposite spin in same atoms in sub-lattices by short range repulsive potential  $U$. This type of\ninteraction is shown in Eq.~\\ref{exHI.eq}, in each sub-lattice $A(B)$ electrons with up(down) spin interact to electrons with down(up) spin.\n(b) The RPA chain includes summing of $A$, $B$ vertices. This figure is one of the $RPA$ chains.}\n \\label{vertices.fig} \n\\end{center}\n\\vspace{-3mm}\n   \\end{figure}\n \n Each vertex in the chain can be either of $A$ or $B$ type. By summing the RPA series, one can achieve to the\n collection of equations for susceptibilities in the sub-lattice representation~\\cite{Peres-comment},\n \n \\begin{equation}\n\\begin{cases}\n \\chi_{AA}=\\chi^{0}_{AA}+U\\chi^{0}_{AA}\\chi_{AA}+U\\chi^{0}_{AB}\\chi_{BA}\\\\\n \\chi_{AB}=\\chi^{0}_{AB}+U\\chi^{0}_{AA}\\chi_{AB}+U\\chi^{0}_{AB}\\chi_{BB}\\\\\n \\chi_{BA}=\\chi^{0}_{BA}+U\\chi^{0}_{BA}\\chi_{AA}+U\\chi^{0}_{BB}\\chi_{BA}\\\\\n \\chi_{BB}=\\chi^{0}_{BB}+U\\chi^{0}_{BA}\\chi_{AB}+U\\chi^{0}_{BB}\\chi_{BB}\\\\\n\\end{cases}.\n\\label{coupling.eq}\n\\end{equation}\n\n The coupling equations in Eq.~\\ref{coupling.eq} should be decoupled, so giving two subsystems with \n the common determinant, which should be zero at the plasmon dispersion~\\cite{Peres-comment}:\n \n \\begin{equation}\n \\begin{vmatrix} \n 1-U\\chi^{0}_{AA} &    -U\\chi^{0}_{AB}\\\\ \n  -U\\chi^{0}_{BA}  &   1-U\\chi^{0}_{BB}\n  \\end{vmatrix}=0,\n  \\label{determin1.eq}\n\\end{equation}\n \n or \n \n \\begin{equation}\n 1-U(\\chi^{0}_{AA}+\\chi^{0}_{BB})+U^{2}\\chi^{0}_{AA}\\chi^{0}_{BB}-U^2 \\chi^{0}_{AB}\\chi^{0}_{BA}=0.\n \\label{determin2.eq}\n \\end{equation}\n \n  \n  The bare polarization operators or susceptibilities of vertices can be calculated using non-interacting\n  Green's function for graphene's electrons in the sub-lattice representation. By considering the Green's functions\n  in the momentum representation and close to Dirac points, classification of Green's function\n  by electron valleys $\\vec{K}$ and $-\\vec{K}$, by vicinities to the corresponding Dirac points was achieved. In the valleys\n  $\\vec{K}$ and $-\\vec{K}$ the non-interacting Hamiltonians in the representation of sub lattice $A, B$ are\n\n  \\begin{equation}\n H_{\\pm \\vec{K}}=\\begin{pmatrix} \n   0          &    \\pm p_{x}-ip_{y}\\\\ \n  \\pm p_{x}+ip_{y} &   0\n  \\end{pmatrix},\n  \\label{nonintH.eq}\n\\end{equation}\n\n  here by considering $v_{F}=1$, the corresponding Matsubara Green's functions for the undoped graphene ($\\mu=0$) are\n\n\\begin{equation}\n G^{\\pm \\vec{K}}(\\vec{p},i\\epsilon)=\\frac{1}{(i\\epsilon)^2-p^2}\\begin{pmatrix} \n   i\\epsilon         &    \\pm pe^{\\mp i\\phi}\\\\ \n  \\pm pe^{\\pm i\\phi} &       i\\epsilon\n  \\end{pmatrix},\n  \\label{MG.eq}\n\\end{equation}\n\n where $\\phi$ is the azimuthal angle of the momentum vector $\\vec{p}$.\nBy using Eq.~\\ref{MG.eq}, the polarization operators can be calculated as\n\n\\begin{equation}\n\\chi^{0}_{\\alpha,\\beta}(\\vec{q},i\\omega)=-T\\sum_{v=\\pm \\vec{K}}\\sum_{\\epsilon} \\int \\frac{d\\vec{p}}{(2\\pi)^2}\nG^{v}_{\\alpha, \\beta}(\\vec{p},i\\epsilon)G^{-v}_{\\beta, \\alpha}(\\vec{p'},i\\epsilon+i\\omega),\n\\label{chi-RPA.eq}\n\\end{equation}\n\n where $\\vec{p'}=\\vec{p}+\\vec{q}$. By performing the summation over fermionic Matsubara frequencies\n $\\epsilon=\\pi T(2n+1)$ and electron valleys $v=\\pm\\vec{K}$,\n \n \\begin{eqnarray}\n \\chi^{0}_{AA}(\\vec{q}, i\\omega)=\\chi^{0}_{BB}(\\vec{q}, i\\omega)=-\\int{\\frac{d\\vec{p}}{(2\\pi)^2}\\frac{p+p'}{(i\\omega)^2-(p+p')^2}},\\nonumber\\\\\n  \\chi^{0}_{AB}(\\vec{q}, i\\omega)=\\chi^{0}_{BA}(\\vec{q}, i\\omega)=\n  \\int{\\frac{d\\vec{p}}{(2\\pi)^2}\\frac{cos(\\phi-\\phi')(p+p')}{(i\\omega)^2-(p+p')^2}}.\\nonumber\\\\\n  \\label{chi-sublattices.eq}\n \\end{eqnarray} \n \n   Our results in  Eq.~\\ref{determin1.eq} and Eq.~\\ref{chi-sublattices.eq} are in agreement with results of ref.~\\cite{Peres-comment}\n   , the only difference is that the decomposition of the overlap factor $cos(\\phi-\\phi')$ near the Dirac point was performed in \n   Eq.~\\ref{chi-sublattices.eq}. Now by using Eq.~\\ref{determin1.eq} and Eq.~\\ref{chi-sublattices.eq} we should find zeros of \n   Eq.~\\ref{determin2.eq}, but we didn't obtain  zeros or triplet collective excitation for reasonable range of $U$. In other words,\n   the short-range interaction dose not lead to any neutral collective mode. \n   \n\\section{On-site interaction: eigenvector representation}\n\n In this section the same problem of RPA in the triplet channel of the on-site interaction will be considered~\\cite{note},\n but in the representation of the eigenvectors of the Hamiltonians Eq.~\\ref{nonintH.eq}. For the case\n of electron valleys  $\\vec{K}$ and $-\\vec{K}$, the eigenvectors are\n \n \\begin{equation}\n f^{\\vec{K}}_{s} = \\frac{1}{\\sqrt{2}} \\begin{pmatrix} \n  e^{-i\\phi\/2}\n  \\\\ se^{i\\phi\/2}\n   \\end{pmatrix}, f^{-\\vec{K}}_{s} = \\frac{1}{\\sqrt{2}} \\begin{pmatrix} -se^{i\\phi\/2}\n  \\\\   e^{i\\phi\/2} \\end{pmatrix} ,\n  \\label{eignvector2.eq}\n\\end{equation}\n\n  where $f^{\\pm \\vec{K}}$ corresponds  to electron states  in the conduction band with the energy $v_{F}p$ and in the valance band\n with the energy $-v_{F}p$. The interaction Hamiltonian Eq.~\\ref{exHI.eq} in the momentum representation takes\n this form:\n \n \\begin{eqnarray}\n H_{int}=-\\frac{U}{N}\\sum_{\\vec{p}_{1} \\vec{p}_{2} \\vec{q}} \\left\\{ \\Psi^{\\uparrow \\dagger}_{\\vec{p'}_{1}}\n \\begin{pmatrix} 1  &  0  \\\\ 0  &  0 \\end{pmatrix} \\Psi^{\\downarrow}_{\\vec{p}_{1}} \\Psi^{\\downarrow \\dagger}_{\\vec{p'}_{2}}\n    \\begin{pmatrix} 1  & 0 \\\\ 0  & 0  \\end{pmatrix}  \\Psi^{\\uparrow}_{\\vec{p}_{2}}  \\right. \\nonumber\\\\\n   \\left. + \\Psi^{\\uparrow \\dagger}_{\\vec{p'}_{1}}\n \\begin{pmatrix} 0  &  0  \\\\ 0  &  1 \\end{pmatrix} \\Psi^{\\downarrow}_{\\vec{p}_{1}} \\Psi^{\\downarrow \\dagger}_{\\vec{p'}_{2}}\n    \\begin{pmatrix} 0  & 0 \\\\ 0  & 1  \\end{pmatrix}  \\Psi^{\\uparrow}_{\\vec{p}_{2}}   \\right \\},\n    \\label{sum_psi1.eq}\n \\end{eqnarray}\n   \n    where $\\vec{p'}_{1}=\\vec{p}_{1}+\\vec{q}$, $\\vec{p'}_{2}=\\vec{p}_{2}-\\vec{q}$, and \n $\\Psi^{\\uparrow \\downarrow}_{\\vec{p}}=(a^{\\uparrow \\downarrow}_{\\vec{p}}, b^{\\uparrow \\downarrow}_{\\vec{p}})^T$\n is a two-components column matrix, was made from electron destruction operators. The sum in Eq.~\\ref{sum_psi1.eq} over\n $\\vec{p}_{1}, \\vec{p}_{2}, \\vec{q}$ should be distributed among electron valleys. Retaining only the terms corresponding\n to small $\\vec{q}$ (since we search small-momentum collective excitation), we get two intra-valley and two inter-valley terms\n\n\\begin{eqnarray}\n H_{int}=-\\frac{U}{N}\\sum_{\\vec{p}_{1} \\vec{p}_{2} \\vec{q}} \\sum_{\\vec{v}_{1},\\vec{v}_{2}}\n \\left\\{ \\Psi^{\\uparrow \\dagger}_{\\vec{v'}_{1}}\n \\begin{pmatrix} 1  &  0  \\\\ 0  &  0 \\end{pmatrix} \\Psi^{\\downarrow}_{\\vec{v}_{1}} \\Psi^{\\downarrow \\dagger}_{\\vec{v'}_{2}}\n    \\begin{pmatrix} 1  & 0 \\\\ 0  & 0  \\end{pmatrix}  \\Psi^{\\uparrow}_{\\vec{v}_{2}}  \\right. \\nonumber\\\\\n   \\left. + \\Psi^{\\uparrow \\dagger}_{\\vec{v'}_{1}}\n \\begin{pmatrix} 0  &  0  \\\\ 0  &  1 \\end{pmatrix} \\Psi^{\\downarrow}_{\\vec{v}_{1}} \\Psi^{\\downarrow \\dagger}_{\\vec{v'}_{2}}\n    \\begin{pmatrix} 0  & 0 \\\\ 0  & 1  \\end{pmatrix}  \\Psi^{\\uparrow}_{\\vec{v}_{2}}   \\right \\},\n    \\label{sum_psi2.eq}\n \\end{eqnarray}\n\n   where $\\vec{v}_{\\alpha}=\\vec{V}_{\\alpha}+\\vec{p}_{\\alpha}, \\vec{v'}_{\\alpha}=\\vec{V}_{\\alpha}+\\vec{p'}_{\\alpha}$ and \n   $\\vec{V}_{\\alpha}=\\pm \\vec{K}$. Now by changing the basis of the eigenvectors, the Eq.~\\ref{eignvector2.eq}\n   can be performed according to the relation $\\Psi^{\\uparrow \\downarrow}_{\\vec{V}+\\vec{p}}=\n   \\sum_{s=\\pm}f^{\\vec{V}}_{s}c^{\\uparrow \\downarrow}_{\\vec{V}+\\vec{p},s}$, where $c^{\\uparrow \\downarrow}_{\\vec{V}+\\vec{p},s}$ is the\n   destruction operator for electron from the valley $\\vec{V}=\\pm \\vec{K}$, from the conduction $(s=+1)$ or\n   valance $(s=-1)$ band, with the momentum $\\vec{p}$ (measured from the corresponding Dirac point) and\n   with up\/down spin. The result for such transformation of Eq.~\\ref{sum_psi2.eq} is,\n   \n   \\begin{eqnarray}\n   H_{int}=\\sum_{\\vec{p}_{1}\\vec{p}_{2}\\vec{q}}\\sum_{\\vec{V}_{1}\\vec{V}_{2}=\\pm\\vec{K}}\\sum_{s'_1,s'_2,s_1,s_2=\\pm} \\left \\{\n   \\Gamma^{(1)}_{\\vec{V}_{1}\\vec{V}_{2}}+\\Gamma^{(2)}_{\\vec{V}_{1}\\vec{V}_{2}}  \\right \\} \\nonumber \\\\\n   \\times c^{\\uparrow \\dagger}_{\\vec{V}_{1}+\\vec{p'}_{1},s'_1} c^{\\downarrow }_{\\vec{V}_{1}+\\vec{p}_{1},s_1}\n   c^{\\downarrow  \\dagger}_{\\vec{V}_{2}+\\vec{p'}_{2},s'_2} c^{\\uparrow }_{\\vec{V}_{2}+\\vec{p'}_{2},s_2}\n   \\label{H_int_new.eq}\n   \\end{eqnarray}\n   \n   where the vertices are,\n   \n   \\begin{eqnarray}\n   \\Gamma^{(1)}_{++}&=&\\Gamma^{(2)}_{--}=\\frac{u}{4}e^{\\frac{i}{2}(-\\phi_{1}+\\phi'_{1}-\\phi_{2}+\\phi'_{2})},\\nonumber \\\\\n   \\Gamma^{(2)}_{++}&=&\\Gamma^{(1)}_{--}=\\frac{u}{4}s_{1}s'_{1}s_{2}s'_{2}e^{\\frac{i}{2}(\\phi_{1}-\\phi'_{1}+\\phi_{2}-\\phi'_{2})}, \\nonumber\\\\\n   \\Gamma^{(1)}_{+-}&=&\\Gamma^{(2)}_{-+}=\\frac{u}{4}s_{2}s'_{2}e^{\\frac{i}{2}(-\\phi_{1}+\\phi'_{1}+\\phi_{2}-\\phi'_{2})},\\nonumber \\\\\n   \\Gamma^{(2)}_{+-}&=&\\Gamma^{(1)}_{-+}=\\frac{u}{4}s_{1}s'_{1}e^{\\frac{i}{2}(\\phi_{1}-\\phi'_{1}-\\phi_{2}+\\phi'_{2})}\n   \\label{vertices4.eq}\n   \\end{eqnarray}\n    \n    \n  where $u=U\/N$. The interaction Hamiltonian Eq.~\\ref{H_int_new.eq} gives rise to RPA series of the type\n  depicted in Fig.~\\ref{vertices2.fig}. A vertex in each place of these series can be of type $\\Gamma^{(1,2)}$,\n  while electron valleys in the Green function between vertices can be $\\vec{K}$ or $-\\vec{K}$. \n  \n  \\begin{figure}[h]\n\\begin{center}\n\\vspace{-3mm}\n\\includegraphics[width=6cm,height=2cm,angle=0]{vertices2.eps}\n\\vspace{-3mm}\n\\caption{ Each vertex is including $\\Gamma^{(1)}$ or $\\Gamma^{(2)}$. Each vertex is include interaction between\nelectrons of valance or conduction in each valley. In this chain I  considered many types of $\\Gamma$ functions\nwhich described interaction of electrons in each valley. This chain is one of the $RPA$ chains.}\n   \\label{vertices2.fig}\n\\end{center}\n\\vspace{-3mm}\n\\end{figure}\n  \n  \n  The full vertex can be renormalized by RPA summation, $\\Gamma^{(i,j)}_{\\vec{V}_{1}\\vec{V}_{2}}$, where $i,j=1,2$ are the types of\n  left and right vertices in the series. The Bethe-Sal peter equations for this vertex is,\n  \n  \\begin{eqnarray}\n &&\\Gamma^{(ij)}_{\\vec{V}_{1}\\vec{V}_{2}}(\\vec{p'}_{1},s'_{1},\\vec{p}_{1},s_{1},\\vec{p'}_{2},s'_{2}, \\vec{p}_{2},s_{2};i\\omega)= \\nonumber\\\\\n  &&\\delta_{ij}\\Gamma^{(i)}_{\\vec{V}_{1}\\vec{V}_{2}}(\\vec{p'}_{1},s'_{1},\\vec{p}_{1},s_{1},\\vec{p'}_{2},s'_{2}, \\vec{p}_{2},s_{2}) \\nonumber \\\\\n  &&-T\\sum_{k\\vec{V}}\\sum_{\\epsilon ss'}\\int \\frac{d\\vec{p}}{(2\\pi)^2}\\Gamma^{(i)}_{\\vec{V}_{1}\\vec{V}}(\\vec{p'}_{1},s'_{1},\\vec{p}_{1},s_{1},\\vec{p'},s', \\vec{p},s) \\nonumber\\\\\n  &&\\times G^{\\vec{V}}_{s}(\\vec{p},i\\epsilon)G^{\\vec{V}}_{s'}(\\vec{p'},i\\epsilon+i\\omega)\n  \\Gamma^{(kj)}_{\\vec{V}\\vec{V}_{2}}(\\vec{p},s,\\vec{p'},s',\\vec{p'}_{2},s'_{2}, \\vec{p}_{2},s_{2};i\\omega),\\nonumber\\\\\n  \\label{new-vertex1.eq} \n  \\end{eqnarray}\n  \n  where $\\vec{p'}=\\vec{p}+\\vec{q}$. The Green functions for eigenstates of the Hamiltonians Eq.~\\ref{nonintH.eq} at\n  $\\mu=0$ have a simplest form, \n  \n  \\begin{equation}\n  G^{\\pm \\vec{K}}_{-}(\\vec{p}, i\\epsilon)=\\frac{1}{i\\epsilon-p},~~~~~ G^{\\pm \\vec{K}}_{+}(\\vec{p}, i\\epsilon)=\\frac{1}{i\\epsilon+p}\n  \\label{greenf-ne.eq}\n  \\end{equation}\n  \n  By further processing, I should note that the vertices Eq.~\\ref{vertices4.eq} can be decoupled in to $left$ and $right$ parts,\n  \n  \\begin{equation}\n  \\Gamma^{(i)}_{\\vec{V}_{1}\\vec{V}_{2}}(\\vec{p'}_{1},s'_{1},\\vec{p}_{1},s_{1},\\vec{p'}_{2},s'_{2}, \\vec{p}_{2},s_{2})=\n  \\Gamma^{i.\\vec{V}_{1}}_{\\vec{p'}_{1}s'_{1}\\vec{p}_{1}s_{1}} . u .  \\Gamma^{i.\\vec{V}_{2}}_{\\vec{p'}_{2}s'_{2}\\vec{p}_{2}s_{2}},\n  \\label{l-r-gamma.eq}\n  \\end{equation}\n  \n  where \n  \n  \\begin{equation}\n  \\gamma^{a}_{\\vec{p'}s'\\vec{p}s}=\\frac{1}{2}e^{\\frac{i}{2}(-\\phi+\\phi')}, ~~~~~ \\gamma^{b}_{\\vec{p'}s'\\vec{p}s}=\n  \\frac{1}{2}ss' e^{\\frac{i}{2}(\\phi-\\phi')},\n  \\label{new-gamma.eq}\n  \\end{equation}\n  \n  here the brief notations $1.\\vec{K}=2.(-\\vec{K})=a$ and $2.\\vec{K}=1.(-\\vec{K})=b$ are introduced.  Since all bare vertices are decoupled,\n   the summarized vertices also can  be decoupled into the left and right parts. The RPA-renormalized interaction\n   is written, $\\Gamma^{ij}_{\\vec{V}_{1}\\vec{V}_{2}}=\\Gamma^{i.\\vec{V}_{1}}V_{ij}\\Gamma^{i.\\vec{V}_{2}}$. By substituting\n   it into Eq.~\\ref{new-vertex1.eq}, the interaction equations for this system obtained as,\n   \n   \\begin{equation}\n   V_{ij}=u+u\\sum_{k\\vec{V}}\\Pi^{\\vec{V}}_{ik}V_{kj},\n   \\label{int-term.eq}\n   \\end{equation}\n  \n  the polarization operators were introduced as,\n  \n  \\begin{eqnarray}\n  &&\\Pi^{\\vec{V}}_{ij}(\\vec{q},i\\omega)=\\nonumber\\\\\n  &&-T\\sum_{\\epsilon ss'}\\int\\frac{d\\vec{p}}{(2\\pi)2}\\Gamma^{i.\\vec{V}}_{\\vec{p'}s'\\vec{p}s}G^{\\vec{V}}_{s}(\\vec{p},i\\epsilon)\n  \\Gamma^{i.\\vec{V}}_{\\vec{p}s\\vec{p'}s'}G^{\\vec{V}}_{s'}(\\vec{p'},i\\epsilon+i\\omega).\\nonumber\\\\\n  \\label{pol.eq}\n  \\end{eqnarray}\n  \n  \n    A nontrivial solution for the Eq.~\\ref{int-term.eq} occurs when it's determinant is zero,\n    \n  \\begin{eqnarray}\n \\begin{vmatrix} \n 1-u\\Pi^{\\vec{K}}_{11}-u\\Pi^{-\\vec{K}}_{11} &    -u\\Pi^{\\vec{K}}_{12}-u\\Pi^{-\\vec{K}}_{12}\\\\ \n  -u\\Pi^{\\vec{K}}_{21}-u\\Pi^{-\\vec{K}}_{21}  &   1-u\\Pi^{\\vec{K}}_{22}-u\\Pi^{-\\vec{K}}_{22}\n  \\end{vmatrix}=\\nonumber\\\\\n  1-u(\\Pi^{\\vec{K}}_{11}+\\Pi^{-\\vec{K}}_{11}+\\Pi^{\\vec{K}}_{22}+\\Pi^{-\\vec{K}}_{22})\\nonumber\\\\\n  +u^{2}(\\Pi^{\\vec{K}}_{11}+\\Pi^{-\\vec{K}}_{11})(\\Pi^{\\vec{K}}_{22}+\\Pi^{-\\vec{K}}_{22})\\nonumber\\\\\n  -u^{2}(\\Pi^{\\vec{K}}_{12}+\\Pi^{-\\vec{K}}_{12})(\\Pi^{\\vec{K}}_{21}+\\Pi^{-\\vec{K}}_{21})=0,\n  \\label{determin3.eq}\n \\end{eqnarray}\n  \n   By using Eqs.~\\ref{greenf-ne.eq},~\\ref{l-r-gamma.eq},~\\ref{new-gamma.eq} and Eq.~\\ref{pol.eq}, the polarization operators can be\n   easily calculated, by performing summation on $\\epsilon=\\pi T (2n+1)$ and on $s,s'=\\pm1$, we find\n   \n   \\begin{eqnarray}\n   \\Pi^{\\vec{K}}_{11}=\\Pi^{-\\vec{K}}_{11}=\\Pi^{\\vec{K}}_{22}=\\Pi^{-\\vec{K}}_{22}=\\frac{1}{2}\\int\\frac{d\\vec{p}}{(2\\pi)^2}\\frac{p+p'}{(i\\omega)^2-(p+p')^2},\\nonumber\\\\\n   \\Pi^{\\vec{K}}_{12}= \\Pi^{-\\vec{K}}_{21}=-\\frac{1}{2}\\int\\frac{d\\vec{p}}{(2\\pi)^2}\\frac{e^{i(-\\phi+\\phi')}(p+p')}{(i\\omega)^2-(p+p')^2},\\nonumber\\\\\n   \\Pi^{\\vec{K}}_{21}= \\Pi^{-\\vec{K}}_{12}=-\\frac{1}{2}\\int\\frac{d\\vec{p}}{(2\\pi)^2}\\frac{e^{i(\\phi-\\phi')}(p+p')}{(i\\omega)^2-(p+p')^2}.\\nonumber\\\\\n   \\label{pol2.eq}\n   \\end{eqnarray}\n   \n     Comparing Eq.~\\ref{pol2.eq} with~\\ref{chi-sub-lattices.eq} and~\\ref{determin1.eq} with~\\ref{determin3.eq}, we find that \n   $\\chi^{0}_{AA}=\\chi^{0}_{BB}=\\Pi^{\\vec{K}}_{11}+\\Pi^{-\\vec{K}}_{11}=\\Pi^{\\vec{K}}_{22}+\\Pi^{-\\vec{K}}_{22}$ and\n   $\\chi^{0}_{AB}=\\chi^{0}_{BA}=\\Pi^{\\vec{K}}_{12}+\\Pi^{-\\vec{K}}_{12}=\\Pi^{\\vec{K}}_{21}+\\Pi^{-\\vec{K}}_{21}$. Therefore,\n   the  sub-lattice representation give the same result as that in the representation of eigenstates in short range interaction.\n   So in this representation we could not find zeros for Eq.~\\ref{determin3.eq}. Finally we found that\n   the on-site Hubbard interaction can not lead to formation of neutral collective mode. The essential reason is overlap factors\n   which resolve sub-lattices. In the first approach the overlap factors reside in the Green functions, and in the second\n   approach these factors are prescribed to interaction vertices.\n    \n    \\section{Long range interaction: effective Coulomb interaction}\n    \n       The Coulomb interaction between electrons in graphene is usually considered as long-range, so both sub-lattices\n     participate equally in it, since characteristic range of the interaction is much larger than the lattice constant.\n     In this section we consider long range interaction for finding it's effect on spin-1 collective mode, which was considered\n     in~\\cite{JPHCM.Jafari}. If the long range (Coulomb) interaction is included interaction between electrons with same and \n     opposite spins on two sub-lattices, we know,  it results the plasmon collective mode~\\cite{Wunsch}. The only long range interaction\n     which lead to the Eq.~\\ref{main.eq} has the form of Eq.~\\ref{long-H-exch.eq}. Another reason for selecting this form for long range interaction is, the spin-1 collective\n     mode make from interaction between electrons with opposite spins.\n\n     If we assume that the Coulomb interaction is represented by  constant $V$ between electrons with opposite spin (because the spin-1 mode\n     considered between two electrons with opposite spin), we introduced this interaction Hamiltonian,\n     \n     \\begin{eqnarray}\n     H_{Coul}=V\\sum_{ij}\\left \\{a^{\\uparrow \\dagger}_{i}a^{\\uparrow}_{i}a^{\\downarrow \\dagger}_{j}a^{\\downarrow }_{j}+\n     a^{\\uparrow \\dagger}_{i}a^{\\uparrow}_{i}b^{\\downarrow \\dagger}_{j}b^{\\downarrow }_{j}\\right. \\nonumber \\\\\n     \\left. +b^{\\uparrow \\dagger}_{i}b^{\\uparrow}_{i}a^{\\downarrow \\dagger}_{j}a^{\\downarrow }_{j} + \n     b^{\\uparrow \\dagger}_{i}b^{\\uparrow}_{i}b^{\\downarrow \\dagger}_{j}b^{\\downarrow }_{j}  \\right \\}.\n     \\label{long-H.eq}\n     \\end{eqnarray}\n   \n   In the exchange channel, we have\n   \n   \\begin{eqnarray}\n     H_{Coul}=-V\\sum_{ij}\\left \\{a^{\\uparrow \\dagger}_{i}a^{\\downarrow}_{j}a^{\\downarrow \\dagger}_{j}a^{\\uparrow }_{i}+\n     a^{\\uparrow \\dagger}_{i}b^{\\downarrow}_{j}b^{\\downarrow \\dagger}_{j}a^{\\uparrow }_{i}\\right. \\nonumber \\\\\n     \\left. +b^{\\uparrow \\dagger}_{i}a^{\\downarrow}_{j}a^{\\downarrow \\dagger}_{j}b^{\\uparrow }_{i} + \n     b^{\\uparrow \\dagger}_{i}b^{\\downarrow}_{j}b^{\\downarrow \\dagger}_{j}b^{\\uparrow }_{i}  \\right \\}.\n     \\label{long-H-exch.eq}\n     \\end{eqnarray}\n   \n    If we perform the same calculations starting from Eq.~\\ref{H_int_new.eq}, as was done starting from Eq.~\\ref{H_int_new.eq}\n    to derive Eq.~\\ref{determin3.eq}, we will arrive to the different system,\n    \n    \n\n    \\begin{equation}\n   \\begin{cases}\n    \\chi_{AA}=\\chi^{0}_{AA}+V\\chi^{0}_{AA}\\chi_{AA}+V\\chi^{0}_{AB}\\chi_{BA}+V\\chi^{0}_{AA}\\chi_{BA}+V\\chi^{0}_{AB}\\chi_{BA}\\\\\n    \\chi_{AB}=\\chi^{0}_{AB}+V\\chi^{0}_{AA}\\chi_{AB}+V\\chi^{0}_{AB}\\chi_{BB}+V\\chi^{0}_{AA}\\chi_{BB}+V\\chi^{0}_{AB}\\chi_{BB}\\\\\n    \\chi_{BA}=\\chi^{0}_{BA}+V\\chi^{0}_{BA}\\chi_{AA}+V\\chi^{0}_{BB}\\chi_{BA}+V\\chi^{0}_{BA}\\chi_{BA}+V\\chi^{0}_{BB}\\chi_{BA}\\\\\n    \\chi_{BB}=\\chi^{0}_{BB}+V\\chi^{0}_{BA}\\chi_{AB}+V\\chi^{0}_{BB}\\chi_{BB}+V\\chi^{0}_{BA}\\chi_{BB}+V\\chi^{0}_{BB}\\chi_{BB}\\\\\n  \\end{cases}.\n     \\label{coupling2.eq}\n  \\end{equation}\n\n  \n    with the solvable condition \n    \n    \\begin{equation}\n    1-V(\\chi^{0}_{AA}+\\chi^{0}_{AB}+\\chi^{0}_{BA}+\\chi^{0}_{BB})=0,\n    \\label{main.eq}\n    \\end{equation}\n    \n    where the susceptibilities are same as Eq.~\\ref{chi-sub-lattices.eq}. This equation is same as with essential \n    equation of~\\cite{Ebrahimkhas-IJMPB} which predicted neutral collective mode in honey comb lattice. \n    If we consider $ \\chi^{0}_{triplet}=\\chi^{0}_{AA}+\\chi^{0}_{AB}+\\chi^{0}_{BA}+\\chi^{0}_{BB}$, the dispersion of\n    the neutral collective mode can be obtained with one of this equations,\n    \n    \\begin{equation}\n    \\Re{\\chi^{0}_{triplet}(\\vec{q},\\omega)}=1\/V~~~~or~~~~~ \\Im{\\chi^{0}_{triplet}(\\vec{q},\\omega)}=0.\n    \\label{condition.eq}\n    \\end{equation}\n    \n    \n     The zeros of the Eqs.~\\ref{condition.eq} are  collections of $(\\vec{q}, \\omega)$ which\n     are plotted in Fig.~\\ref{e-h-dis.fig} for $V=3.2t$. This contour plot displays dispersion of neutral collective mode.\n     The contour plot of $\\Re \\chi^{0}_{triplet}$ for $V=3.2t$ is plotted under the electron-hole $(e-h)$ continuum \n     region in $\\Gamma \\rightarrow K$ direction. But we know this range of $U$ is too large for electrons in graphene.\n     I checked the solution and zeros of Eq.~\\ref{condition.eq} for $0.0t<V<3.0t$, but I didn't find any zeros. For $V>3.2t$\n     we could find dispersions for neutral collective mode near $\\Gamma$ point in $\\Gamma-K$ direction which are meaningless.\n    \n    \n     So  the long range interaction of type Eq.~\\ref{long-H.eq} can not formed the neutral collective mode\n    in undoped graphene without $one-cone approximation$.\n     \n      \\begin{figure}[h]\n\\begin{center}\n\\vspace{-3mm}\n\\includegraphics[width=6cm,height=5cm,angle=0]{e_h_cont.eps}\n\\vspace{-3mm}\n\\caption{The contour plot of $\\Re{\\chi^{0}_{triplet}(\\vec{q},\\omega)}=1\/V$ is displayed \nfor undoped graphene for V=3.2t. The dispersion of spin collective mode appeared below e-h continuum (cyan region) but not in\nreasonable range of $V$.\nThe $y$ axis is energy ($\\epsilon (eV)$)  $vs$  momentum ($q$). The direction in e-h continuum is  $\\Gamma \\rightarrow K$.}\n\\label{e-h-dis.fig}\n\\end{center}\n\\vspace{-3mm}\n   \\end{figure}\n      \n      \n        The prediction of triplet collective mode in Ref.~\\cite{Ebrahimkhas-IJMPB} was due to\n      the $single-cone$ approximation in short range  on-site interaction that led to scalar form of polarization operator\n      without $U^2$ terms in Eq.~\\ref{determin2.eq}. One-cone approximation \n      which lead to elimination of some parts of Eq.~\\ref{determin2.eq}, is the only reason that predicted of the spin-1 collective mode. \n      \n      \n    \\section{Conclusion}\n\n         In this research the effects of short range Hubbard interaction in two representations and new \n     effective long range interaction Hamiltonian was studied and introduced. The prediction of\n     neutral collective mode in on-site and effective long range interaction is analyzed, and  we found that in the RPA method\n     on Hubbard model without single-cone approximation, mixing the overlap functions of two sub-lattices suppressed\n     this collective mode. These two types of interactions have\n     different Hamiltonians, Eq.~\\ref{H_int_new.eq} and Eq.~\\ref{long-H-exch.eq}, in both I considered interaction between\n     electrons from the electrons  with opposite spins. We can't find and  predict neutral collective mode as the effect of\n     effective long range  and short range interaction because the overlap factors in both case resolve sub-lattices and suppress \n     the neutral collective mode.\n\n  \\section{Acknowledgment}\n  \n    I wish to thank N. Sharifzadeh for helpful discussions.\n    The author appreciates and thanks so much from referee of our paper~\\cite{Ebrahimkhas-IJMPB} who guides to reach the main idea of this\n    paper.\n\n      \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nLet $p$ be a prime, and let $I_1, I_2\\subseteq (0,p)$ be subintervals. This paper is motivated by determining conditions on $I_1,I_2$ under which we can ensure the solubility of the  congruence\n\\[\nxy\\=1\\bmod p,\\quad (x,y)\\in I_1\\times I_2.\n\\]\nFrom a heuristic point of view we would expect this congruence to have a solution whenever $|I_1|,|I_2|\\gg p^{1\/2}$. However,  as highlighted by Heath-Brown \\cite{H-B2000}, the best result to date requires that\n$|I_1|\\cdot|I_2|\\gg p^{3\/2}\\log^2 p$. The  proof requires one to  estimate incomplete Kloosterman sums\n\\[\nS(n,H)=\\sum_{\\substack{m=n+1\\\\mathbf{m}\\not\\= 0\\bmod p}}^{n+H}e\\left(\\frac{\\ell\\overline{m}}{p}\\right),\n\\]\nfor $\\ell\\in (\\mathbb{Z}\/\\ell\\mathbb{Z})^*$, for which the Weil bound yields\n\\begin{equation}\\label{eq:weil}\n|S(n,H)|\\le 2(1+\\log p) p^{1\/2}.\n\\end{equation}\nIt has been conjectured by Hooley \\cite{hooley} that\n$S(n,H)\\ll H^{1\/2}q^\\varepsilon,$ for any $\\varepsilon>0$,\nwhich would enable one to handle\nintervals with $|I_1|,|I_2|\\gg p^{2\/3+\\varepsilon}.$\nHowever such a bound appears to remain a distant prospect.\n\nA different approach to this problem involves  considering a sequence of pairs of intervals $I_1^{(j)}, I_2^{(j)}$, for $1\\le j\\le J$, and to ask whether  there is a value of $j$ for which there is a solution to the congruence\n\\begin{equation}\\label{eqn:intro.1}\nxy\\=1\\bmod p,\\quad (x,y)\\in I_1^{(j)}\\times  I_2^{(j)}.\n\\end{equation}\nThere are some obvious degenerate cases here. For example, if we suppose that $I_1^{(j)}=I_2^{(j)}$ for all $j$, and that these run over all intervals of a given length $H$, then we are merely asking whether there is positive integer $h\\le H$ with the property that the congruence $x(x+h)\\=1\\bmod p$ has a solution $x\\in\\mbb{Z}$. This\nis equivalent to deciding  whether  the set\n$\\{h^2+4:1\\le h\\le H\\}$\ncontains a quadratic residue modulo $p$. When $H=2$, therefore, it is clear that this problem has a solution for all primes $p=\\pm 1\\bmod 8$. We avoid considerations of this sort by assuming that at least one of our sequences of intervals is pairwise disjoint. The following is our main result.\n\n\\begin{theorem}\\label{thm:inverses1}\nLet $H,K>0$ and let $I_1^{(j)}, I_2^{(j)}\\subseteq (0,p)$ be subintervals, for\n$1\\le j\\le J$, such that\n\\[|I_1^{(j)}|=H\\quad\\text{and}\\quad |I_2^{(j)}|=K\\]\nand\n\\[I_1^{(j)}\\cap I_1^{(k)}=\\emptyset\\quad\\text{for all}\\quad j\\not= k.\\]\nThen\nthere exists $j\\in \\{1,\\ldots,J\\}$ for which \\eqref{eqn:intro.1} has a solution\nif\n\\[\nJ\\gg \\frac{p^{3}\\log^4 p}{H^2K^2}.\n\\]\n\\end{theorem}\n\nIf we take $J=1$ in the theorem then we retrieve the above result that \\eqref{eqn:intro.1} is soluble when\n$HK\\gg p^{3\/2}\\log^2 p$. Alternatively, if we allow a larger value of $J$, then we can get closer to what would follow on\nHooley's hypothesised bound for $S(n,H)$.\n\n\n\\begin{corollary}\nWith notation as in Theorem \\ref{thm:inverses1}, suppose that $J\\gg p^{1\/3}$.\nThen\nthere exists $j\\in \\{1,\\ldots,J\\}$ for which \\eqref{eqn:intro.1} has a solution provided that\n$H>p^{2\/3}$ and $K>p^{2\/3}(\\log p)^2$.\n\\end{corollary}\n\n\nOur proof of Theorem \\ref{thm:inverses1}  relies upon a mean value estimate for incomplete Kloosterman sums. These types of estimates have been studied extensively for multiplicative characters, especially in connection with variants of Burgess's bounds (see Heath-Brown \\cite{H-B2012} and the discussion therein).\nThe situation for Kloosterman sums is relatively under-developed (see Friedlander and Iwaniec \\cite{FI}, for example). The result we present here appears to be new, although many of our techniques are borrowed directly from the treatment of the analogous multiplicative problem \\cite[Theorem 2]{H-B2012}. The deepest part of our argument is an appeal to Weil's bound for Kloosterman sums.  We will prove the following result in the next section.\n\n\n\\begin{theorem}\\label{thm:mvt1}\nIf $I_1,\\ldots ,I_J\\subseteq (0,p)$ are disjoint subintervals,  with $H\/2<|I_j|\\leq H$ for each $j$, then for any $\\ell\\in (\\mathbb{Z}\/p\\mathbb{Z})^*$,  we have\n\\[\n\\sum_{j=1}^J\\left|\\sum_{n\\in I_j}e\\left(\\frac{\\ell\\overline{n}}{p}\\right)\\right|^2\\leq 2^{12} p\\log^2H.\n\\]\n\\end{theorem}\n\nTaking $J=1$ shows that, up to a constant factor,  this result includes as a special case the bound \\eqref{eq:weil} for incomplete Kloosterman sums.\n\n\\begin{TA}\nWhile working on this paper TDB was supported by\nEPSRC grant  \\texttt{EP\/E053262\/1} and\nAH was supported by EPSRC grant \\texttt{EP\/J00149X\/1}.\nWe are grateful to  Professor Heath-Brown for several useful discussions.\n\\end{TA}\n\n\\section{Proof of Theorem \\ref{thm:mvt1}}\n\nOur starting point is the following mean value theorem for $S(n,H)$.\n\n\\begin{lemma}\\label{lem:mvt1}\nFor $H\\in\\mbb{N}$ and $\\ell\\in (\\mathbb{Z}\/p\\mathbb{Z})^*$, we have\n$$\n\\sum_{n=1}^p\\left|S(n,H)\\right|^2\\leq \\frac{H^2}{p}+8pH.\n$$\n\\end{lemma}\n\\begin{proof}\nAfter squaring out the inner sum and interchanging the order of summation, the left hand side becomes\n$$\n\\sum_{h_1,h_2=1}^H\\sum_{\\substack{n=1\\\\n\\not= -h_1,-h_2\\bmod p}}^pe\\left(\\frac{\\ell(\\overline{n+h_1}-\\overline{n+h_2})}{p}\\right).\n$$\nUsing orthogonality of characters it is easy to see that the inner sum over $n$ is\n\\begin{align*}\n&=\\sum_{\\substack{x,y\\in (\\mathbb{Z}\/p\\mathbb{Z})^*\\\\ \\overline{x}-\\overline{y}\\=h_1-h_2\\bmod p}}\ne\\left(\\frac{\\ell(x-y)}{p}\\right)\\\\\n&=\n\\frac{1}{p}\n\\sum_{\\substack{x,y\\in (\\mathbb{Z}\/p\\mathbb{Z})^*}}\ne\\left(\\frac{\\ell(x-y)}{p}\\right)\n\\sum_{a=1}^p\ne\\left(\\frac{a(  \\overline{x}-\\overline{y} )+a(h_2-h_1)}{p}\\right).\n\\end{align*}\nHence\n\\begin{align*}\n\\sum_{n=1}^p\\left|S(n,H)\\right|^2\n&=\n\\frac{1}{p}\n\\sum_{a=1}^p\n\\sum_{h_1,h_2=1}^H\ne\\left(\\frac{a(h_2-h_1)}{p}\\right)\n\\sum_{\\substack{x,y\\in (\\mathbb{Z}\/p\\mathbb{Z})^*}}\ne\\left(\\frac{\\ell(x-y)+a(  \\overline{x}-\\overline{y} )}{p}\\right)\\\\\n&=\n\\frac{1}{p}\n\\sum_{a=1}^p\n|K(\\ell,a;p)|^2\n\\sum_{h_1,h_2=1}^H\ne\\left(\\frac{a(h_2-h_1)}{p}\\right),\n\\end{align*}\nwhere $K(\\ell,a;p)$ is the usual complete Kloosterman sum. The contribution from $a=p$ is\n$$\n\\frac{|K(\\ell,0;p)|^2 H^2}{p}=\\frac{H^2}{p},\n$$\nsince $p\\nmid \\ell$. The remaining contribution has modulus\n\\begin{align*}\n\\left|\\frac{1}{p}\n\\sum_{a=1}^{p-1}\n|K(\\ell,a;p)|^2\n\\sum_{h_1,h_2=1}^H\ne\\left(\\frac{a(h_2-h_1)}{p}\\right)\\right|\n&\\leq\n4\n\\sum_{0<|a|\\leq p\/2}\n\\left|\\sum_{h_1,h_2=1}^H\ne\\left(\\frac{a(h_2-h_1)}{p}\\right)\\right|\\\\\n&\\leq\n8\n\\sum_{0<a\\leq p\/2}\n\\min\\left\\{H, \\frac{p}{2a}\\right\\}^2\\\\\n&\\leq 8 pH,\n\\end{align*}\nby the Weil bound for the Kloosterman sum and the familiar estimate for a geometric series.\nCombining these contributions, we therefore arrive at the statement of the lemma.\n\\end{proof}\n\n\nThe rest of the proof of Theorem \\ref{thm:mvt1} is taken from the proof of \\cite[Theorem 2]{H-B2012}, and we include\nit only for completeness. We may assume that $H\\geq 4$ in what follows since the result is trivial otherwise. Write $N_j$ for the smallest integer in $I_j$ and suppose that $N_1<\\cdots <N_J$. By separately considering the odd and then the even numbered intervals we may assume without loss of generality that $N_{j+1}-N_j\\ge H$ for $1\\le j <J$.\n\nThe starting point is the observation that\n\\begin{align}\\label{eqn:pf:thm:mvt1.1}\n\\sum_{j=1}^J\\left|\\sum_{n\\in I_j}e\\left(\\frac{\\ell\\overline{n}}{p}\\right)\\right|^2\\le\\sum_{j=1}^J\\max_{1\\le h\\le H}|S(N_j,h)|^2.\n\\end{align}\nFor any $1\\le h\\le H$ and $N_j-H<n\\le N_j$ we have that\n\\[|S(N_j,h)|=|S(n,N_j-n+h)-S(n,N_j-n)|\\le 2\\max_{1\\le k\\le 2H}|S(n,k)|,\\]\nwhence\n\\begin{align*}\n|S(N_j,h)|\\le\\frac{2}{H}\\sum_{N_j-H<n\\le N_j}\\max_{1\\le k\\le 2H}|S(n,k)|.\n\\end{align*}\nCauchy's inequality yields\n\\[|S(N_j,h)|^2\\le\\frac{4}{H}\\sum_{N_j-H<n\\le N_j}\\max_{1\\le k\\le 2H}|S(n,k)|^2.\\]\nTaking the max over $h$ and then summing over $j$ now gives\n\\begin{equation}\n\\label{eqn:pf:thm:mvt1.2}\n\\begin{split}\n\\sum_{j=1}^J\\max_{1\\le h\\le H}|S(N_j,h)|^2&\\le\\frac{4}{H}\\sum_{j=1}^J\\sum_{N_j-H<n\\le N_j}\\max_{1\\le k\\le 2H}|S(n,k)|^2\\\\\n&\\le\\frac{4}{H}\\sum_{n=1}^p\\max_{1\\le k\\le 2H}|S(n,k)|^2,\n\\end{split}\\end{equation}\nthe last inequality coming from our spacing assumption. We now seek an upper bound for the sum on the right hand side.\n\nLet $t$ be the smallest positive integer with $2H\\le 2^t$, so that in particular $2H\\leq 2^t\\leq 4H$ and\n$t+1\\leq 4\\log H$. For each $1\\le n\\le p$ we choose a positive integer $k=k(n)\\le 2H$, with\n\\[\\max_{1\\le h\\le 2H}|S(n,h)|=|S(n,k)|.\\]\nBy writing\n$k=\\sum_{d\\in\\mc{D}}2^{t-d},$\nwhere $\\mc{D}$ is a collection of integers in $[0,t]$, we have that\n\\[S(n,k)=\\sum_{d\\in\\mc{D}}S(n+v_{n,d}2^{t-d},2^{t-d}),\\]\nwhere\n\\[v_{n,d}=\\sum_{\\substack{e\\in\\mc{D}\\\\ e<d}}2^{d-e}< 2^d.\\]\nThen by Cauchy's inequality we deduce that\n\\begin{align*}\n\\max_{1\\le h\\le 2H}|S(n,h)|^2&\\le |\\mc{D}|\\sum_{d\\in\\mc{D}}|S(n+v_{n,d}2^{t-d},2^{t-d})|^2\\\\\n&\\le (t+1)\\sum_{0\\le d\\le t}\\sum_{0\\le v<2^d}|S(n+v2^{t-d},2^{t-d})|^2.\n\\end{align*}\nNow summing both sides over $n$ and applying Lemma \\ref{lem:mvt1} we have that\n\\begin{align*}\n\\sum_{n=1}^p\\max_{1\\le k\\le 2H}|S(n,k)|^2\n&\\leq\n(t+1) \\sum_{0\\le d\\le t}\\sum_{0\\le v<2^d} \\left(\\frac{2^{2t-2d}}{p}+8p2^{t-d}\\right)\\\\\n&\\leq\n(t+1)^2 \\left(\\frac{2^{2t}}{p}+8p2^{t}\\right)\\\\\n&\\leq 2^{8}\n \\left(\\frac{H^2}{p}+2pH\\right)\\log^2H.\n \\end{align*}\nSince $H^2\/p\\leq 2pH$ we\neasily complete the proof of Theorem \\ref{thm:mvt1} by combining this  with (\\ref{eqn:pf:thm:mvt1.1}) and (\\ref{eqn:pf:thm:mvt1.2}).\n\n\\section{Proof of Theorem \\ref{thm:inverses1}}\n\nNow we proceed to the proof of our main theorem. For each $j$ the number of solutions to\n\\eqref{eqn:intro.1} is equal to\n\\begin{align*}\n\\sum_{x\\in I_1^{(j)},~\\overline{x}\\in I_2^{(j)}}1\n&=\\sum_{x\\in I_1^{(j)}}\\sum_{y\\in I_2^{(j)}}\\frac{1}{p}\\sum_{\\ell=1}^pe\\left(\\frac{\\ell(\\overline{x}-y)}{p}\\right)\\\\\n&=\\sum_{x\\in I_1^{(j)}}\\sum_{y\\in I_2^{(j)}}\\frac{1}{p}+\\sum_{x\\in I_1^{(j)}}\\sum_{y\\in I_2^{(j)}}\\frac{1}{p}\\sum_{\\ell=1}^{p-1}e\\left(\\frac{\\ell(\\overline{x}-y)}{p}\\right)=S_{1,j}+S_{2,j},\n\\end{align*}\nsay.\nThe total contribution from the $S_{1,j}$ terms is\n\\begin{equation}\n\\sum_{j=1}^JS_{1,j}\\gg \\frac{JHK}{p}.\\label{eqn:pf:thm:inverses1.1}\n\\end{equation}\nNext, the standard estimate for a geometric series gives\n\\begin{align*}\nS_{2,j}\n&=\\frac{1}{p}\\sum_{0<|\\ell|\\leq p\/2}\n\\left(\\sum_{y\\in I_2^{(j)}}e\\left(\\frac{-\\ell y}{p}\\right)\\right)\\left(\\sum_{x\\in I_1^{(j)}}e\\left(\\frac{\\ell\\overline{x}}{p}\\right)\\right)\\\\\n&\\ll\n\\sum_{0<|\\ell|\\leq p\/2}\n\\frac{1}{|\\ell|}\\left|\\sum_{x\\in I_1^{(j)}}e\\left(\\frac{\\ell\\overline{x}}{p}\\right)\\right|.\n\\end{align*}\nApplying Cauchy's inequality and Theorem \\ref{thm:mvt1} we deduce that\n\\begin{align*}\n\\sum_{j=1}^J|S_{2,j}|\n&\\ll\n\\sum_{0<|\\ell|\\leq p\/2}\n\\frac{1}{|\\ell|}\\sum_{j=1}^J\\left|\\sum_{x\\in I_1^{(j)}}e\\left(\\frac{\\ell\\overline{x}}{p}\\right)\\right|\\nonumber\\\\\n&\\ll J^{1\/2}\\sum_{0<|\\ell|\\leq p\/2}\n\\frac{1}{|\\ell|}\\left(\\sum_{j=1}^J\\left|\\sum_{x\\in I_1^{(j)}}e\\left(\\frac{\\ell\\overline{x}}{p}\\right)\\right|^2\\right)^{1\/2}\\nonumber\\\\\n&\\ll J^{1\/2}(\\log p)\\left(p\\log^2 H\\right)^{1\/2}.\n\\end{align*}\nUnder the conditions of Theorem \\ref{thm:inverses1}, it now follows that (\\ref{eqn:pf:thm:inverses1.1}) dominates this quantity, from which the conclusion of the theorem follows.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{Introduction}\n\nThree-body hadronic decays of $B$ and $D$ mesons are a rich field for\nsearches on CP violation, for tests of the Standard Model and of\nQuantum Chromodynamics (QCD) in particular~\\cite{Ref1LHCb,Ref2LHCb,1412.4269,Ref4LHCb,Ref5LHCb}. \nFurthermore, they provide an interesting ground to study hadron physics, as strong\ninteraction effects, through the presence of two-body resonances and\ntheir interferences, have an impact on weak-decay observables. In\norder to obtain the latter most reliably, the meson-meson\nfinal state interactions must be addressed using theoretical\nconstraints, such as unitarity, analyticity and chiral symmetry and\nexperimental data from processes other than $B$ and $D$ decays. However, in\nDalitz plot analyses the event distributions are often studied using\nthe isobar model in which the decay amplitudes are parametrized by\ncoherent sums of Breit-Wigner amplitudes with a background\ncontribution, in disagreement with the fundamental principles listed\nabove.  In this work, we suggest to replace these sums by parametrizations in\nterms of unitary two-meson form factors, without losing contact with the description of the \nweak-interaction dynamics that governs the underlying flavor-changing process. These parametrizations \nare constructed, in part, from  results published previously~\\cite{Dedonderetal2011,fkll,LZ2014,ElBennich2006,ElBennichetal09,PLB699_102,Boitoetal09,BE2009,JPD_PRD89}  \nand are motivated by  the forthcoming analyses of high-statistics data sets for many three-body decay channels of $B$ and\n$D$ decays, in particular by the LHC$b$ collaboration~\\cite{LHCbWS}.\n\nThe theoretical amplitude expressions in Refs.~\\cite{Dedonderetal2011,LZ2014,fkll,ElBennich2006,ElBennichetal09,PLB699_102,Boitoetal09,BE2009,JPD_PRD89}\nfrom which we derive the present parametrizations are based on models of QCD factorization. The factorization beyond the \nleading approximation can be expressed as an expansion in the strong coupling, $\\alpha_s$, and inverse powers of the bottom\nquark mass, $m_b$, and has been applied with success to charmless nonleptonic\ntwo-body $B$ decays~(see e.g. Ref.~\\cite{Beneke2003}). Parallel analyses of three-body $B$ decays in the contexts of QCD factorization (QCDF) and perturbative QCD (pQCD) can be found in Refs.~\\cite{1308.5139,1401.5514} and \\cite{1502.05483, 1609.04614}, respectively.  In $D$ decays, this\nfactorization approach is less predictive inasmuch, as it does\nnot allow for a systematic improvement owing to the charm quark mass, $m_c \\simeq m_b\/3$,\nwhich enhances significant corrections to the factorized results.  It\nis, therefore, downgraded from an effective theory that can be\nsystematically improved, in the case of $B$ decays, to a phenomenological procedure, in the case\nof $D$ decays.  Nevertheless, as a purely phenomenological approach, based on\nthe seminal work by Bauer, Stech and Wirbel~\\cite{Bauer1987}, the\nfactorization hypothesis has been applied successfully to $D$ decays,\nprovided one treats Wilson coefficients as phenomenological parameters\nto account for nonfactorizable corrections~\\cite{Abbasetal}.\n\nBesides a recent extension of the QCD factorization framework to nonleptonic $B$ decays into three light \nmesons~\\cite{KMV2015}, no rigorous factorization theorem valid for the entire three-body phase space \nand full three light-meson Dalitz plot exists. On the other hand, three-body decays of $B$ and $D$ mesons \nclearly receive important contributions from intermediate resonances --- such as the\n$\\rho(770)$, $K^*(892)$ and $\\phi(1020)$ --- and can therefore be considered as {\\it quasi two-body decays\\\/}.  \nOne  then assumes that two of the three final-state mesons form a single state originating from a\nquark-antiquark pair, which is interpreted as an intermediate\nquasi-two-body final state in which case the factorization can be\napplied.  Then, the three-body final state is reconstructed with the\nuse of two-body mesonic form factors to account for the important\nhadronic final-state interactions.  For instance, in the $D^0 \\to K^0_S \\pi^- \\pi^+ $ decay, the three-meson final state\n$K^0_S \\pi^+ \\pi^-$ is initially preceded by the quasi-two-body pairs, $[K^0_S \\pi^+]_{L} \\ \\pi^-$, $[K^0_S \\pi^-]_{L} \\ \\pi^+$ and $K^0_S \\ [\\pi^+{\\pi^-}]_{L}$, \nwhere two of the three mesons form a  state in an $L= S$ or $P$ wave. This framework has been successfully applied to several hadronic \nthree-body $B$ and $D$ decays~\\cite{MG02,fkll,ElBennich2006,ElBennichetal09,Dedonderetal2011,LZ2014,PLB699_102,Boitoetal09,BE2009,JPD_PRD89, JPD_inprogress}. \n\nThe factorization of a nonleptonic weak $B$ decay into a quasi-two-body state can be schematically described as follows. The decays are mediated by local dimension-six \nfour-quark operators $O_i (\\mu)$ that form the weak effective nonrenormalizable Hamiltonian. However, depending on flavor content, spin, charge and parity symmetry of the final \nstates only specific operators will contribute to a given decay. The $B$-decay amplitude into two mesons, $M_1$ and $M_2^*$ with four momenta $p_1$ and $p_2$, respectively,  can be written as,\n\\begin{equation}\n   \\langle M_1(p_1) M_2^*(p_2) | \\mathcal{H}_\\mathrm{eff} |  B(p_B) \\rangle = \\frac{G_F}{\\sqrt{2}}\\ V_\\mathrm{CKM} \\sum_i C_i(\\mu) \n   \\langle M_1(p_1) M_2^*(p_2) | O_i(\\mu) | B(p_B) \\rangle \\ ,\n \\label{fullamphamilton}\n\\end{equation}\nwhere $p_B = p_1 + p_2$, $G_F$ is the Fermi constant, $V_\\mathrm{CKM}$ is a product of Cabibbo-Kobayashi-Maskawa~(CKM) matrix elements, $C_i(\\mu)$ are Wilson coefficients \nrenormalized at the scale $\\mu$~\\cite{BurasNPB434_606} and $M_2^*(p_2)$ is the resonant quasi-two body state which decays into two lighter mesons.  The hadronic amplitude \n$\\langle M_1(p_1) M_2^*(p_2) | O_i(\\mu) | B(p_B) \\rangle$  describes long-distance physics. In the factorization approach we henceforth employ, this amplitude is the sum of two matrix-element products,\n\\begin{align}\n  \\langle M_1(p_1) M_2^*(p_2) |  O_i(\\mu) | B(p_B) \\rangle & =   \\Big ( \\langle M_1 (p_1) |  {J}_{1}^{\\nu} | B(p_B) \\rangle  \\langle M_2^*(p_2) | J_{2\\nu} | 0 \\rangle   \\nonumber \\\\\n      +  \\langle M_1 (p_1) |  J_{3}^{\\nu} | 0   &   \\rangle  \\langle M_2^*(p_2) | J_{4\\nu} | B (p_B) \\rangle \\Big ) \n    \\left [ 1 + \\sum_n r_n \\alpha_s^n(\\mu) + \\mathcal{O}  \\left ( \\frac{\\Lambda_\\mathrm{QCD}}{m_b} \\right ) \\right ]  \\ ,\n     \\label{facto1}\n \\end{align}\nwhere  the strong coupling is evaluated at a scale $\\mu$, $r_n$ is a combination of constant strong interaction factors,  and $\\ket{0}$ is the vacuum state.\nThus, at leading order, the decay amplitudes factorize into two matrix elements with either the weak quark currents $J_1$ and $J_2$ or $J_3$ and $J_4$.\nRadiative corrections  can be systematically  taken into account to a given order $\\alpha_s^n (\\mu)$, whereas corrections to the heavy-quark limit are of \nnonperturbative nature and therefore much less  controlled. This is in particular true for the charm quark which is neither a light nor a heavy enough\nquark~\\cite{ElBennich:2010ha,ElBennich:2011py,ElBennich:2012tp,Bashir:2012fs}. This fact  makes the systematic improvements of\nEq.~(\\ref{facto1}), enclosed in square brackets, less reliable for $D$ decays. One should keep this limitation in mind but, for lack of a\nbetter theoretical framework, the phenomenological approach to Eq.~(\\ref{facto1})  remains a good starting point to organize the\ndescription of $D$ decays and can be used to provide a first step beyond the isobar model.\n\nThe weak effective Hamiltonian, $\\mathcal{H}_\\mathrm{eff}$, in Eq.~\\eqref{fullamphamilton} is given by the sum of local operators $O_i(\\mu)$ multiplied by\nWilson coefficients $C_i(\\mu)$ which encode the short-distance effects above the renormalization scale $\\mu$. For a $\\Delta B=1$\ntransition, for example, the\nHamiltonian is given by~\\cite{Ali1998, Beneke:2001ev}\n\\begin{eqnarray}\n\\label{HeffB}\n  \\mathcal{H}^{\\Delta B =1}_\\mathrm{eff}  &=& \\frac{G_F}{\\sqrt{2}}   \\sum_{p=u,c}  V_{pq}^* V_{pb} \\left [  C_1(\\mu) O_1^p(\\mu) + C_2(\\mu) O_2^p(\\mu)    + \n                                                                             \\sum_{i=3}^{10}  \\hspace*{-0mm}  C_i(\\mu) O_i(\\mu) \\right. \\nonumber \\\\\n                                                                  &+&  C_{7\\gamma}(\\mu) O_{7\\gamma}(\\mu) +  C_{8g}(\\mu) O_{8g}(\\mu) \\Big ] +\\  \\mathrm{h.c.} \\ ,\n\\end{eqnarray} \nwhere the quark flavor can be $q=d,s$ and $V_{ij}$ are CKM matrix elements. In the decays, the weak interaction $W$-boson exchange diagram gives rise to two \ncurrent-current operators with different color structure owing to QCD corrections and $SU(3)$ color algebra:\n\\begin{eqnarray}\n  O_1^p(\\mu)  & = & \\bar q_i \\gamma^\\mu (1-\\gamma_5) p_i\\  \\bar p_j \\gamma_\\mu (1-\\gamma_5) b_j \n  \\label{q1}  \\\\\n  O_2^p(\\mu)  & = & \\bar q_i \\gamma^\\mu (1-\\gamma_5) p_j\\  \\bar p_j \\gamma_\\mu (1-\\gamma_5) b_i  \\ .\n  \\label{q2}\n\\end{eqnarray}\nIn Eqs.~\\eqref{q1}  and \\eqref{q2} $i,j$ are color indices and for the corresponding Wilson coefficients one has $C_1(\\mu) \\simeq 1 + \\mathcal{O}(\\alpha_s (\\mu)$) and \n$C_2(\\mu) \\simeq \\mathcal{O}(\\alpha_s (\\mu)$). The operators  $O_i$, $i=3-10$ stem from QCD and electroweak penguin diagrams, while $O_{7\\gamma}$ and $O_{8g}$ are\nelectromagnetic and chromomagnetic dipole operators. The explicit tensor  structure of these operators as well as their Wilson coefficients at next-to-leading logarithms can be \nfound, for example, in Ref.~\\cite{Buchalla:1995vs}. With the use of an appropriate Fierz transformation and the  $SU(N_c)$ identity, \n\\begin{equation}\n( \\bar q_i\\,  p_j )\\, (\\bar p_j \\, b_i) \\  = \\  2\\, ( \\bar q_i \\, T^a_{ik}\\, p_k )\\,  ( \\bar p_j\\, T^a_{jl} \\, b_l )   + \\frac{1}{ N_c}\\, ( \\bar q_i \\, p_i )\\, (\\bar p_j\\,  b_j )   \\ ,\n  \\label{coloridentity}  \n\\end{equation}\nwhere $T^a_{ij}$ are the $SU(N_c)$ generators, the quark bilinears can be rearranged to match the flavor and color structure of the final mesons. In this transformation, the color-octet contribution in Eq.~\\eqref{coloridentity}  \nis commonly neglected.  The two resulting combinations of $C_1(\\mu)$  and $C_2(\\mu)$,\n\\begin{equation}\n   a_1(\\mu) = C_1(\\mu) +\\frac{1}{N_c} C_2 (\\mu)  \\  ,  \\quad  a_2(\\mu) = C_2(\\mu) +\\frac{1}{N_c} C_1 (\\mu)  \\  ,\n\\end{equation}\nlead to ``color allowed'' and ``color suppressed\" amplitudes, respectively,  which are topologically different. Typically, the Wilson coefficients are evaluated at a renormalization scale \nof the heavy quark, i.e. $\\mu \\simeq m_c, m_b$.\n\nOn the right-hand side of Eq.~\\eqref{facto1} the two matrix-element products describe different physical processes.  Namely, the creation of a final  two-meson state from \na $\\bar qq$ pair  is described by the form factors $\\langle M_2^*(p_2) | J_{2\\nu} | 0 \\rangle$, where $M_2^* \\to M_3 M_4$ denotes resonant intermediate states in the different \ntwo-meson coupled channels that lead to the final three-body state. As mentioned, these form factors can be constructed such as to preserve two-body unitarity, reproduce asymptotic QCD and\nare constrained by chiral symmetry at low energies. We discuss them in Appendices~\\ref{AppFFs}.  In Eq.~\\eqref{facto1}, the matrix element $\\langle M_1 (p_1) | J_{3}^\\nu | 0 \\rangle$ defines \nthe weak decay constant of a scalar,  pseudoscalar or vector mesons which is either well known from experiment, for instance $f_\\pi$ and $f_K$, or has been evaluated with \nlattice-regularized QCD and other nonperturbative approaches.  The transition $\\langle M_2^*(p_2) | J_{4\\nu} | B \\rangle $ of a $B$ \nmeson to a strongly interacting two-meson pair  via a resonance is a complicated process and the biggest source of uncertainty in our approach.\nIt could be extracted experimentally  from semi-leptonic processes such as $B^0 \\to K^+ \\pi^- \\mu^+ \\mu^-$~\\cite{B0semilep} or $D^0 \\to K^- \\pi^+ \\mu^+ \\mu^-$~\\cite{D0semilep}. It has also been conjectured within  soft-collinear effective theory that the amplitude  can be factorized in terms of a generalized  $B$-to-two-body form factor and two-hadron light-cone distribution amplitudes~\\cite{MW14}.\n In the derivation of the amplitude expressions presented here, we employ a model approximation which relates this matrix element $\\langle M_2^*(p_2) | J_{4\\nu} | B \\rangle $ to the two-body meson form factor $\\langle M_2^*(p_2) [\\to M_3 M_4]| {J}_{2\\nu} | 0 \\rangle $.  \n\n\n\n\nFinally,  the transition amplitudes $\\langle M_1 (p_1) | J_{1}^\\nu | B \\rangle $ ($=\\langle M_1 (p_1) \\overline {B}| J_{1}^\\nu | 0 \\rangle$) are parametrized by heavy-to-light transition form factors \nwhich are discussed in Appendix~\\ref{transitionff}.  \n\nAs a definite example of the procedure outlined above, let us consider the  $D^+ \\to[ K^- \\pi^+]_{S,P}\\  \\pi^+ $ decay, where the $K^-\\pi^+$\npairs are in $S$- or $P$-wave state. The matrix element given by $\\langle [K^-\\pi^+]_{S,P}\\pi^+ | \\mathcal{H}_\\mathrm{eff} | D^+ \\rangle$ receives \ncontributions from the two amplitudes $a_1(\\mu)$ and  $a_2(\\mu)$ and factorizes as,\n\\begin{align}\n\\langle [ K^- \\pi^+]_{S,P}\\ \\pi^+ | \\mathcal{H}_\\mathrm{eff} | D^+ \\rangle & =  \\frac{G_F}{\\sqrt{2}}\\cos^2\\theta_C \\Big [a_1\\langle [K^-\\pi^+_1]_{S,P}\\vert \\bar s\\gamma^\\nu(1-\\gamma_5)c\\vert D^+\\rangle\n                                                      \\langle\\pi^+_2 \\vert \\bar u\\gamma_\\nu(1-\\gamma_5)d\\vert 0\\rangle    \\nonumber    \\\\\n                                               + \\,  a_2 \\langle [K^-\\pi^+_1]_{S,P} & \\vert \\bar s\\gamma^\\nu(1-\\gamma_5)d \\vert 0\\rangle\n                                                    \\langle \\pi^+_2 \\vert \\bar u\\gamma_\\nu(1-\\gamma_5)c \\vert D^+\\rangle \\Big ] + (\\pi_1^+ \\leftrightarrow \\pi_2^+) \\, , \n\\label{DKpiExample}\n\\end{align}\n$\\theta_C$ being the Cabbibo angle. The\n$K\\pi$ form factors appear explicitly in the matrix element $\\langle\n[K^-\\pi^+_1]_{S,P} \\vert \\bar s\\gamma^\\nu(1-\\gamma_5)d \\vert\n0\\rangle$.  The evaluation of $\\langle [K^-\\pi^+_1]_{S,P}\\vert \\bar\ns\\gamma^\\nu(1-\\gamma_5)c \\vert D^+\\rangle$ is less straightforward.\nHowever, assuming this transition to proceed through the dominant\nintermediate resonances, this matrix element can also be written in\nterms of the $K\\pi$ form factors as shown in Ref.~\\cite{MG02, BE2009}. \n This feature  is of crucial importance to the parametrizations that we propose in this work. It is interesting to note that the calculation of a generalized three-body form factor using light-cone sum rules, in the spirit of Ref.~\\cite{MW14}, also leads to the appearance of  the  two-body meson form factors~\\cite{MW14n2,WZ15,SW15,Lietal16,SWZ17}.\nThe\nother matrix elements of Eq.~(\\ref{DKpiExample}) can be written in\nterms of decay constants or transitions form factors that can be\nextracted from semi-leptonic decays, as outlined above.  Strong phases in the mesonic final state interactions are accounted for by the \nhadronic form factors, which makes this type of description particularly suitable for the interpretation of CP asymmetries that have been observed \nin $B$ decays~\\cite{ElBennichetal09,Dedonderetal2011,LZ2014, PLB699_102, Kleinetal}. Amplitude\nexpressions, such as in Eq.~(\\ref{DKpiExample}), are used throughout\nthis paper as a starting point to build parametrizations based on\nunitary two-body hadronic form factors.  Within this approach,\nexplicit forms of parametrizations for $D^+ \\to K^- \\pi^+ \\pi^+$ and\n$D^0 \\to K^0_S \\pi^- \\pi^+$ amplitudes have already been presented in\nRef.~\\cite{LHCbWS} (see p.~27 therein).\n\nThe paper is structured as follows.  In Section~\\ref{3bodyBdecayAmp}\nwe introduce the parametrizations for three-body hadronic $B$-decay\namplitudes based on the quasi two-body factorization approaches of\nRef.~\\cite{Dedonderetal2011} for $B^\\pm \\to \\pi^+ \\pi^- \\pi^\\pm$, of\nRefs.~\\cite{fkll,ElBennich2006,ElBennichetal09} for $B \\to K \\pi^+\n\\pi^-$ and of Ref.~\\cite{PLB699_102} for $B^\\pm \\to K^+ K^-\nK^\\pm$. Section~\\ref{3bodyDdecayAmp} applies the same procedure to $D$-decay\n amplitudes, viz. $D^+ \\to \\pi^+ \\pi^- \\pi^+\n$~\\cite{Boitoetal09}, $D^+ \\to K^- \\pi^+ \\ \\pi^+ $~\\cite{BE2009}, $D^0\n\\to K^0_S \\pi^+\\ \\pi^- $~\\cite{JPD_PRD89} and $D^0 \\to K^0_S\nK^+\\ K^-$~\\cite{JPD_inprogress}.  The meson-meson and heavy-to-light\nmeson form factors which have been used can be found in the original\npapers.  Nevertheless a short reminder on the derivations of unitary\n$S$- and $P$- waves $\\pi \\pi$-, $\\pi K$- and $K \\bar K$-meson form\nfactors entering these parametrizations is given in\nAppendix~\\ref{AppFFs} together with a short review on heavy-to-light\nmeson form factors. We wrap up with some concluding remarks about the\nmerits of the proposed parametrizations in\nSection~\\ref{Conclusion}. The relations between the free parameters of\nthe different proposed parametrizations and the theoretical decay\namplitudes are presented explicitly in Appendix~\\ref{relationfreeoriginal}.\n\n\n\n\\section[Parametrizations of three-body hadronic $B$-decay amplitudes]{\\boldmath Parametrizations of three-body hadronic $B$-decay amplitudes}\n\\label{3bodyBdecayAmp}\n\n\n\\subsection{Amplitudes for $\\boldsymbol{B^\\pm \\to \\pi^\\pm \\pi^+ \\pi^-}$ } \n\\label{QtbAmpBppp}\n\nThe contributions of pion-pion interactions to CP violating phases in $B^\\pm\\to \\pi^\\pm \\pi^\\mp \\pi^\\pm$ decays have been studied~\\cite{Dedonderetal2011} within the quasi two-body \nfactorization approach discussed in the introduction.\\footnote{During the preparation of this manuscript Ref.~\\cite{Kleinetal} has appeared. Their treatment is very similar to the one we describe here.} \nThe amplitudes were derived as matrix elements of the weak effective Hamiltonian \ngiven by Eq.~\\eqref{HeffB} with $q\\equiv d$. The $\\pi \\pi$ effective mass distributions of the  $B^\\pm \\to \\pi^\\pm \\pi^+ \\pi^-$ data~\\cite{Aubert:2009} are well reproduced for \nan invariant mass, $m_{\\pi^+ \\pi^-}\\lesssim$~1.64~GeV~\\cite{Dedonderetal2011}. To parametrize the amplitudes of $B^\\pm\\to \\pi^\\pm [\\pi^+\\pi^-]_{S,P}$\nwe label the momenta of the decay  as $B^\\pm(p_B)  \\to \\pi^\\pm (p_1) \\pi^+(p_2) \\pi^-(p_3)$, where $p_B$,   the $B^\\pm$ meson momentum, satisfies $p_B=p_1+p_2+p_3$. \nThe amplitudes must be symmetrized by exchanging the $\\pi^+(p_2) \\pi^-(p_3)$ and $\\pi^-(p_1) \\pi^+(p_2)$ pairs in case of a $B^-$ decay or  equivalently the \n$\\pi^+(p_2) \\pi^-(p_3)$ and $\\pi^+(p_1) \\pi^-(p_3)$ pairs in case of a $B^+$ decay. Defining the invariants, $s_{ij}= (p_i + p_j)^2$ (for $i\\neq\nj$), with $s_{12}+ s_{13}+s_{23}= m_B^2+3 m_\\pi^2$, the  interacting pairs of pions  in a relative $S$ or $P$ wave are described by \n$s_{12}$ or $s_{23}$, in the case of a $B^-$ decay, and by $s_{13}$ and $s_{23}$ in the case of  a $B^+$ decay.\n\nThe symmetrized amplitude (see also Eq.~(21) in Ref.~\\cite{Dedonderetal2011}) for the $B^- \\to \\pi^- [\\pi^+\\pi^-]_{S,P}$ decay reads, \n\\begin{eqnarray}\n\\label{Mmoinsinv}\n\\hspace*{-3mm}\n\\mathcal{A}^-_\\mathrm{sym}(s_{12}, s_{23})  = \\frac{1}{\\sqrt{2}}\\left [\\mathcal{A}^-_S(s_{12})+\n  \\mathcal{A}^-_S(s_{23}) + (s_{13}-s_{23}) \\mathcal{A}^-_P(s_{12})  + (s_{13}-s_{12})\\mathcal{A}^-_P(s_{23})  \n\\right ],\n\\end{eqnarray}\nand an analogous amplitude holds for the $B^+ \\to \\pi^+ [\\pi^-\\pi^+]_{S,P}$ decay . The amplitudes $\\mathcal{A}^-_{S,P}(s_{ij})$, $ij= 12$ or $23$, \ngiven by Eqs.~(22) and  (23) of Ref.~\\cite{Dedonderetal2011}, can be parametrized in terms of four complex parameters,  $a_{1,2}^{S,P}$ as,\\footnote{In a fit to a Dalitz plot there is always a global phase that cannot be observed. Therefore, the phase of one of the complex parameters can be set to zero. This is also valid for the other channels discussed in the remainder of this paper.}\n\\begin{eqnarray}\n\\label{MSsij}\n\\mathcal{A}^-_S(s_{ij})&=& \\Big  [ a_1^S\\left(M_{B}^2-s_{ij}\\right) + a_2^S F_0^{B\\pi}(s_{ij}) \\Big ] F_{0n}^{\\pi \\pi}(s_{ij}),\\\\\n\\label{MPsij}\n\\mathcal{A}^-_P(s_{ij})&=&\\Big [a_1^P + a_2^P F_1^{B\\pi}(s_{ij})  \\Big ] F_1^{\\pi \\pi}(s_{ij}),\n\\end{eqnarray}\nwhere $M_B$ is the charged $B$-meson mass. As done in the BABAR\ncollaboration analysis~\\cite{Aubert:2009} and in\nRef.~\\cite{Dedonderetal2011}, a contribution from the $f_2(1270)$ resonance\ncan be accounted for by a Breit-Wigner line shape in a $D$-wave\namplitude of the $\\pi^+ \\pi^-$ pair.  The $B \\to \\pi$ scalar and\nvector transition form factors $F_{0,1}^{B\\pi}(s)$ in\nEqs.~(\\ref{MSsij}) and (\\ref{MPsij}) are discussed in\nAppendix~\\ref{transitionff}.  The $\\pi\\pi$ $S$-wave amplitude\n$\\mathcal{A}^-_S(s_{ij})$ includes via the non-strange scalar form\nfactor $F_{0n}^{\\pi \\pi}(s_{ij})$ the contributions of the scalar\n$f_0(500)$, $f_0(980)$ and $f_0(1400)$ resonances. In a Dalitz-plot\nanalysis, one can use, for example, the  pion scalar form factor\nderived in Refs.~\\cite{Dedonderetal2011} and \\cite{Moussallam_2000}. More\ndetails are given in Appendix~\\ref{Fpp}.\n\nThe $P$-wave amplitude $\\mathcal{A}^-_P(s_{ij})$, proportional to the\npion vector form factor $F_1^{\\pi \\pi}(s_{ij})$, contains the\n$\\rho(770)^0$, $\\rho(1450)$ and $\\rho(1700)$ contributions. In\nRef.~\\cite{Dedonderetal2011}, the $(\\pi\\pi)_P$ form factor was extracted\nfrom the Belle Collaboration analysis of $\\tau^-\\to\\pi^-\\pi^0\\nu_\\tau$\ndecay data~\\cite{Fujikawa_PRD78_072006}. Alternatively, one can employ\nthe unitary parametrization of Ref.~\\cite{Hanhart} which fits the\n$(\\pi\\pi)_P$-wave phase shifts and inelasticities, the $e^+ e^- \\to \\pi^+\n\\pi^-$ data and the $\\tau^-\\to\\pi^-\\pi^0\\nu_\\tau$ decay data, as done\nin the $D^0 \\to K^0_S \\pi^+ \\pi^-$ Dalitz plot fit of\nRef.~\\cite{JPD_PRD89}; see Appendix~\\ref{Fpp}.\n\nSetting the phase of $a_1^P$ in $\\mathcal{A}^-_P(s_{ij})$ to zero\nyields a total of 7 real parameters to be fitted. The fully\nsymmetrized CP-conjugate $B^+ \\to \\pi^+ \\pi^- \\pi^+ $ decay\namplitude is given by expressions similar to\nEqs.~(\\ref{Mmoinsinv})--(\\ref{MPsij}) with again 7 free real\nparameters.  The reproduction of the Dalitz-plot data over the full\nphase space, in particular for the high invariant mass regions, might\nrequire some adjustment of the $\\pi \\pi$ form factors. The addition of\nfurther phenomenological amplitudes that represent contributions of\nhigher $\\pi \\pi$-interacting waves and possible three-body\nrescattering terms may be necessary.\n\n\n\n\\subsection{Amplitudes for $\\boldsymbol{ B \\to K \\pi^+ \\pi^-} $}\n\\label{QtbAmpBKpp}\n\nThe amplitude is based on the weak effective Hamiltonian in Eq.~(\\ref{HeffB}) with $q\\equiv s$. The momenta are labelled as $B(p_B)\\to K(p_1) \\pi^+(p_2)\\pi^ -(p_3)$, \nwith $s_{12}=(p_1+p_2)^ 2$, $s_{13}=(p_1+p_3)^ 2$,  $s_{23}=(p_2+p_3)^ 2$ and  $s_{12}+ s_{13}+s_{23}= m_B^2+m_K^2+2 m_\\pi^2$ .\n\n\\subsubsection{Parametrization of  the $B \\to K [\\pi^\\pm \\pi^\\mp]_{S,P}$ amplitude}\n\\label{pipiSPwavepKAmp}\n\nThe isoscalar $S$-wave $\\pi^+\\pi^-$ final state interactions in $B\\to\nK\\pi^+\\pi^-$ decays were studied in Ref.~\\cite{fkll} in the quasi\ntwo-body factorization approach with an extension in\nRef.~\\cite{ElBennich2006} to include the $\\pi^+\\pi^-$ isovector $P$\nwave. These studies reproduce very well the Belle and BABAR data in an\neffective $\\pi \\pi$ mass range up to about 1.2~GeV. Following Eq.~(1) of Ref.~\\cite{fkll} the $B \\to K [\\pi^+ \\pi^-]_{S}$\ndecay amplitude can be parametrized in terms of three complex\nparameters, $b_i^S, i=1,2,3$, for the different charges $B= B^\\pm, K =\nK^\\pm$ and $B= B^0 (\\bar B^0), K = K^0(\\bar K^0)$ or $K^0_S$,\n\\begin{equation}\n\\label{paracs}\n  \\mathcal{A}_S(s_{23})  \\equiv  \\langle K \\ [\\pi^+\\pi^-]_S \\vert \\mathcal{H}_{\\rm eff} \\vert B \\rangle = \n   b_1^S \\left (M_B^2-s_{23} \\right ) F_{0n}^{\\pi\\pi} (s_{23})  + \\left ( b_2^S F_0^{B K}(s_{23})+b_3^S \\right ) F_{0s}^{\\pi\\pi}  (s_{23}).\n\\end{equation}\nFor the scalar-isoscalar strange form factor $ F_{0s}^{\\pi\\pi} (s)$ in Eq.~(\\ref{paracs}), one can employ its numerical expression\ngiven in Ref.~\\cite{Dedonderetal2011} or that in Ref.~\\cite{Moussallam_2000}~(see Appendix~\\ref{SFpp}). Parametrizations for the $B \\to K$ \nscalar transition form factors $F_{0}^{BK}(s)$ in Eq.~\\eqref{paracs} are reviewed in Appendix~\\ref{transitionff}.\n\nThe amplitude for $B \\to K [\\pi^+\\pi^-]_{P}$ decays can be written in\nterms of the complex parameter $b_1^P$ as,\n\\begin{equation}\n\\label{paracp}\n \\mathcal{A}_P(s_{12}, s_{13}, s_{23})  \\equiv \\langle K \\ [\\pi^+\\pi^-]_P \\vert \\mathcal{ H}_{\\rm eff} \\vert B  \\rangle=  b_1^P \\left (s_{13} - s_{12}\\right ) F_1^{\\pi \\pi}(s_{23}).\n\\end{equation}\nIn Ref.~\\cite{ElBennich2006}, the pion vector form factor, $\nF_{1}^{\\pi\\pi} (s)$, is approximated by a Breit-Wigner form. However,\nwe recommend the use of the unitary vector form factor derived in\nRef.~\\cite{Hanhart} described in Appendix~\\ref{VFpp}. We stress that\nthe $b_i^S$ in Eq.~(\\ref{paracs}) and $b_1^P$ in Eq.~(\\ref{paracp})\nrepresent different parameters for each charge state.\n\nAs in the $B^\\pm \\to \\pi^+ \\pi^- \\pi^\\pm$ case (see\nSection~\\ref{QtbAmpBppp}), addition of the $[\\pi^+ \\pi^-]$ $D$-wave\ncontribution, parametrized in terms of the $f_2(1270)$ resonance, is\nrequired; higher invariant-mass phenomenological amplitudes may also\nbe necessary.\n\n\n\\subsubsection{Parametrization of  the $B \\to [K \\pi^\\pm]_{S,P} \\pi^\\mp$ amplitude}\n\\label{SwavepiKAmp}\n \nA parametrization of the $B \\to [K \\pi^\\pm]_{S} \\pi^\\mp$ channel was\nintroduced in Ref.~\\cite{ElBennichetal09} (see Eq.~(68) therein),\nwhere in the center-of-mass of the $K\\pi$ pair the $S$-wave amplitude\nin case of the $B^- \\to [K^-\\pi^+]_S\\pi^-$ decay can be represented by,\n\\begin{equation}\n\\label{MSparam}\n   \\mathcal{A}_S (s_{12}) \\equiv \\langle \\pi^- \\ [ K^- \\pi^+]_S \\vert \\mathcal{H}_\\mathrm{eff}\\vert B^- \\rangle = \n    \\big ( c_1^S + c_2^S s_{12} \\big )\\, \\frac {F_0^{ B \\pi }  (s_{12} ) F_0^{K \\pi} (s_{12} )}{s_{12}} \\ ,\n\\end{equation}\nwhich follows from Eq.~(10) of Ref.~\\cite{ElBennichetal09}. In\nEq.~(\\ref{MSparam}), $s_{12}$ is the invariant mass squared of the interacting\n$K^-\\pi^+$ pair, whereas for $B^+$ and $\\bar B^0$ decays the kinematic\nvariable is $s_{13}$.  The complex parameters, $c_1^S$ and $c_2^S$,\ncan be determined through the Dalitz-plot analysis for each given\ncharge state.  We note that the isolated $K_0^*(1430)$ resonance\ncontribution can be obtained by replacing, once the parameters $c_1^S$\nand $c_2^S$ are determined, $F_0^{K\\pi} (s )$ by its pole part\n$F_0^\\mathrm{pole} (s)$ given in Eqs.~(45)$-$(47) of\nRef.~\\cite{ElBennichetal09}.\n\nFollowing the momentum conventions of the $S$-wave above, the  $K \\pi$ $P$-wave amplitude of the  $B^- \\to [K^- \\pi^+]_P\\pi^-$ decays can be parametrized as,\n\\begin{eqnarray}\n\\label{MPparam}\n\\mathcal{A}_P (s_{12}, s_{23}) & \\equiv & \\langle \\pi^- \\ [ K^- \\pi^+ ]_P  \\vert \\mathcal{H}_\\mathrm{eff}\\vert B^- \\rangle \n          \\nonumber \\\\ \n        & = &  c_1^P  \\left( s_{13} - s_{23} - (M_B^2-m_\\pi^2)\\ \\dfrac{m_K^2-m_\\pi^2}{s_{12}}  \\right  )  F_1^{B \\pi } ( s_{12} ) F_1^{K\\pi} (s_{12}) \\ .\n\\end{eqnarray}\nThe parametrizations for the transition form factors, $F_{0(1)}^{ B\\pi\n}(s)$, are discussed in Appendix~\\ref{transitionff} and those for the\n$K\\pi$ scalar and vector form factors $F_{0(1)}^{K \\pi } (s)$ in\nAppendix~\\ref{Kp}.\n     \nThe then available Belle and BABAR data were well reproduced in the\n$\\mathcal{A}_{S,P}$ amplitude analysis of Ref.~\\cite{ElBennichetal09}\nin a $m_{K\\pi}$ range from threshold up to 1.8~GeV. Within the\nfactorization approximation there is no contribution from $[K\\pi]$\npartial waves, $l\\geq 2$, thus one expects the $[K\\pi]$ $D$-wave\ncontribution to be small. However, in order to analyze the Dalitz-plot\ndata over the full energy ranges, additional phenomenological\namplitudes are required.\n\n\n\n\\subsection{Amplitudes for  $\\boldsymbol{ B^\\pm \\to K^\\pm K^+ K^- }$}\n\\label{QtbAmpBKKK}\n\nThe weak effective Hamiltonian that describes this decay channel is\ngiven by Eq.~(\\ref{HeffB}) with $q\\equiv s$.  Explicit factorized\nexpressions of $B^- \\to K^- K^+ K^-$ amplitudes can be found in\nAppendix~A of Ref.~\\cite{Cheng0704.1049}.  The effective invariant\n$K^+ K^-$ mass distributions of the decays $B^\\pm \\to K^+ K^-\nK^\\pm$~\\cite{Belle,BABAR} up to $1.8$~GeV were shown to be well\nreproduced in the factorization approach of Ref.~\\cite{PLB699_102},\nwhere the $B^\\pm (p_B) \\to K^\\pm(p_1) K^+(p_2) K^-(p_3)$ amplitudes\nwere derived for interacting $K^+ K^-$ pairs in a relative $S$ or $P$\nstate.  The symmetrized term for a $B^-$ decay is obtained by exchange\nof the $K^+(p_2) K^-(p_3)$ pair with the $K^-(p_1) K^+(p_2)$ one (or\nexchanging the $K^+(p_2) K^-(p_3)$ and $K^+(p_1) K^-(p_3)$ pairs in\ncase of a $B^+$ meson) and is added to the amplitude.  The totally\nsymmetrized amplitude using the Lorentz invariants $s_{ij}= (p_i +\np_j)^2$ for $i\\neq j$ is given by,\n\\begin{eqnarray}\n\\label{Amoinsinv}\n\\hspace*{-5mm}\n      \\mathcal{A}^-(s_{12}, s_{13}, s_{23})& = & \\frac{1}{\\sqrt{2}}\\big [ \\mathcal{A}^-_S(s_{12})+  \\mathcal{A}^-_S(s_{23}) + \\mathcal{A}^-_P(s_{12}) (s_{13}-s_{23})\n   + \\mathcal{A}^-_P(s_{23}) (s_{13}-s_{12}) \\big ] ,\n\\end{eqnarray}\nwith $s_{12}+ s_{13}+s_{23}= m_B^2+3 m_K^2$.\n\nFollowing Eqs.~(2) and (3) of Ref.~\\cite{PLB699_102}, the amplitudes $\\mathcal{A}^-_{S,P}(s_{ij})$, $ij= 12$ or $23$, can be also written in terms of six complex \nparameters  $d_{1,2}^S$ and  $d_{1,2,3,4}^P$\n\\begin{eqnarray}\n\\label{ASsij} \n\\mathcal{A}^-_S (s_{ij})&=&  d_1^S \\left(M_{B}^2-s_{ij}\\right) F^{KK}_{0n} (s_{ij})  +  d_2^S F_0^{BK}(s_{ij})  F_{0s}^{KK} (s_{ij}) \\ , \\\\\n\\label{APsij}\n \\mathcal{A}^-_P (s_{ij})&=& d_1^P F^{KK}_{1u}(s_{ij}) \n                         + F^{BK}_1(s_{ij}) \\left [d_2^P F^{KK}_{1u} (s_{ij}) + d_3^P  F^{KK}_{1d} (s_{ij})+ d_4^P F^{KK}_{1s} (s_{ij}) \\right ].\n\\end{eqnarray}\nwhere the $B \\to K$ scalar and vector transition form factors $F_{0,1}^{BK}(s)$ in Eqs.~(\\ref{ASsij}) and (\\ref{APsij}) are discussed  in Appendix~\\ref{transitionff} and\nthe scalar and vector form factors $F^{KK}_{0n}, F^{KK}_{0s}, F^{KK}_{1u}$, $F^{KK}_{1d}$ and $F^{KK}_{1s}$ are introduced in Appendix~\\ref{KK}. \n\nBecause of its small branching fraction to $K \\bar K$ (4.6\\%) the $f_2(1270)$ contribution was not introduced in Ref.~\\cite{PLB699_102}.\nHowever, in their study of the Dalitz-plot dependence of CP asymmetry, the authors of Ref.~\\cite{LZ2014} have included it and an amplitude analysis of the full Dalitz\nplot should also add it together with a phenomenological term representing\nthe high invariant-mass contributions.\n\n\n\n\n\n\n\n\\section{Parametrizations of three-body hadronic $\\boldsymbol{D}$-decay amplitudes}\n\\label{3bodyDdecayAmp}\n\n\n\n\\subsection{Amplitudes for  $\\boldsymbol{ D^+ \\to \\pi^+ \\pi^- \\pi^+ }$}\n\\label{QtbAmpDppp}\n\nThe decay $D^+ \\to \\pi^- \\pi^+ \\pi^+$ is a Cabibbo suppressed mode governed by the quark-level transition $c\\to du\\bar d$. The leading contribution to\nthe amplitude arises from the current-current operators and is proportional to $V_{cd}V_{ud}^*$, which is $\\mathcal{O}(\\lambda)$ in\nWolfenstein parametrization, with $\\lambda = 0.2257$.  At next-to-leading order (NLO) in QCD, penguin operators contribute to the \ndecay amplitude. However, CKM unitarity implies that those come with a coefficient $V_{cb}V_{ub}^*$, which is $\\mathcal{O}(\\lambda^5)$. \nIt is therefore safe to neglect the penguin contributions. The dominant contributions to the effective Hamiltonian therefore are, \n\\begin{equation}\n\\label{effHDpppp}\n{\\cal H}_{\\rm eff}=\\frac{G_F}{\\sqrt{2}} V_{cd}V_{ud}^\\ast\\, \\big  [C_1(\\mu)O_1+C_2(\\mu)O_2 \\big ]\\ + \\ \\mathrm{h.c.}  \\ ,\n\\end{equation}\nwhere the two four-quark operators read,\n\\begin{eqnarray}\n\\label{effO12D3pi}\nO_1 &=&  \\big [\\bar d_i\\gamma^\\nu (1-\\gamma_5) c_i][\\bar u_j\\gamma_\\nu (1-\\gamma_5) d_j  \\big ] \\ ,  \\\\\nO_2 &=&  \\big [\\bar u_i\\gamma^\\nu (1-\\gamma_5) c_j][\\bar d_j\\gamma_\\nu (1-\\gamma_5) d_i  \\big ]  \\ .\n\\end{eqnarray}\nIn the description of $D$ decays, however, because of the limitations discussed in the Introduction,  the Wilson coefficients\ndepart from their calculated values due to  nonfactorizable\ncorrections.  In the spirit of Ref.~\\cite{Abbasetal}, it is safe to assume they\ncan be complex numbers and that the corrections will depend on whether\nthe  $\\pi\\pi$ pair  is in $S$- or $P$-wave state (the same applies to the $K\\pi$ pairs in the next section). Our parametrizations below encompass this assumption.\n\n\nThe parametrization we propose here is chiefly based on the work of Ref.~\\cite{Boitoetal09}.  The crucial dynamical ingredient\nto describe the two-body hadronic final state interactions are the scalar and vector $\\pi\\pi$ form factors. The description of the decay proceeds in full\nanalogy to that outlined in the introduction for $D^+ \\to K^- \\pi^+ \\pi^+$.  Within this framework, the main difference between these two decays is that \nthe relevant two-pion matrix element, namely, $\\sand{\\pi^-\\pi^+}{\\bar d\\gamma^\\nu(1-\\gamma_5)d}{0}$ is proportional to the $\\pi\\pi$ vector form factor \nonly --- no scalar contribution appears because the pseudo-scalars involved have the same mass. This implies that the $\\pi^+\\pi^-$ $S$-wave in the \ndecay $D^+ \\to \\pi^- \\pi^+ \\pi^+$ receives no contribution from the diagram proportional to $a_2(\\mu)$. The decay amplitude for $S$- and $P$-wave $\\pi^+\\pi^-$ pairs can be written as\n\\begin{equation}\n\\sand{\\pi^+[\\pi^-\\pi^+]_{S,P}}{\\mathcal{H}_{\\rm eff}}{D^+} = \\mathcal{A}_{S}^{+}+ \\mathcal{A}_P^{+}    ,\\label{eq:DpppSandP}\n\\end{equation}\nwhere $\\mathcal{A}_S^+$, $\\mathcal{A}_P^+$  are respectively the $S$- and  $P$-wave $\\pi^+\\pi^-$ amplitudes. \nThe $S$-wave  is dominated by the intermediate scalar-isoscalar resonances $f_0(500)$ and $f_0(980)$ while the $P$-wave  is largely dominated by the $\\rho(770)^0$. \n\n\nWe label the four-momenta as $D^+(p_D) \\to \\pi^+(p_1)\\pi^-(p_2)\\pi^+(p_3)$,\nand define the following invariant masses squared $s_{12} = (p_1+p_2)^2$, $s_{23} = (p_2+p_3)^2$, and  $s_{13} = (p_1+p_3)^2$, with $s_{12}+ s_{13}+s_{23}= m_D^2+3 m_\\pi^2$.\nResonances occur in the $\\pi^+\\pi^-$ states described in terms of $s_{12}$ and  $s_{23}$ invariants. With these definitions the amplitudes $\\mathcal{A}_{S,P}^+$ of Eq.~(\\ref{eq:DpppSandP}) \ncan be parametrized with three complex  parameters, $e_1^{S}$ and $e_{1,2}^P$, as:\n\\begin{align}\n\\label{AD3piS}\n\\mathcal{A}_{S}^{+} (s_{12}, s_{23})&=  e_1^S (m^2_D - s_{12}) F^{\\pi\\pi}_{0n}(s_{12}) + (s_{12}\\leftrightarrow s_{23}), \\\\\n\\mathcal{A}_{P}^{+} (s_{12}, s_{13}, s_{23})&=   \\left[  e_1^{P} +  e_{2}^{P}  F_1^{D\\pi}(s_{12})   \\right]  (s_{23} - s_{13})F_1^{\\pi\\pi}(s_{12})  + (s_{12}\\leftrightarrow s_{23}) \\ .\n\\label{AD3piP}\n\\end{align}\nWe are implicitly assuming that non-factorizable corrections depend on\nthe spin of the $\\pi^+\\pi^-$ pair and can be absorbed in the parameters\n$e_i^L$.  In this parametrization, the two-body $\\pi^+\\pi^-$\ninteractions are fully taken into account by the scalar and vector\n$\\pi\\pi$ form factors, $F_{0n}^{\\pi\\pi}$ and $F_1^{\\pi\\pi}(s)$,\nrespectively, which are detailed in Appendix~\\ref{Fpp}. The vector $D\\to \\pi$ transition form factor, $F_1^{D\\pi}(s)$, in Eq.~(\\ref{AD3piP}) is discussed in Appendix~\\ref{transitionff}. We observe that the $D$-wave resonance contribution, arising from the $f_2(1270)$, is\nsizeable (with fit fractions of about $20\\%$~\\cite{PDG,E791Dp3pi}) and could be included in data analyses through usual isobar model\nexpressions. Finally, in one of the models employed by the CLEO collaboration~\\cite{CLEODp3pi}, some evidence for a contribution from isospin-2\n$\\pi^+\\pi^+$ interactions is presented, which may have to be included in a realistic analysis.\n\n\n\n\\subsection{Amplitudes for $\\boldsymbol{  D^+ \\to K^- \\pi^+  \\pi^+ }$} \n\\label{QtbAmpDKpp}\n\n\nThe $D^+ \\to K^- \\pi^+ \\pi^+$ decay is  Cabibbo allowed, governed by the quark-level transition  $c\\to su \\bar d$. Since four different quark flavors intervene, the effective Hamiltonian\nfor this processes does not include penguin-type operators. At NLO in QCD, there are only two operators  to be considered,\n\\begin{equation}\n\\label{effHDpKpp}\n{\\cal H}_{\\rm eff}=\\frac{G_F}{\\sqrt{2}}V_{cs}V_{ud}^\\ast \\big [ C_1(\\mu)O_1+C_2(\\mu)O_2 \\big ] \\ +\\ \\mathrm{h.c.} \\ ,\n\\end{equation}\nwhere the relevant four-quark operators are\n\\begin{eqnarray}\n\\label{effO12DKpp}\nO_1 &=& \\big [\\bar c_i\\gamma^\\nu (1-\\gamma_5) s_i][\\bar d_j\\gamma_\\nu (1-\\gamma_5) u_j \\big ] ,\\\\\nO_2 &=& \\big [\\bar c_i\\gamma^\\nu (1-\\gamma_5) s_j][\\bar d_j\\gamma_\\nu (1-\\gamma_5) u_i \\big ] .\n\\end{eqnarray}\n\nIn Ref.~\\cite{BE2009} the $K\\pi$ $S$- and $P$-wave amplitudes in this\ndecay were written in terms of the scalar and vector $K\\pi$ form\nfactors, $F_{0,1}^{K\\pi}(s)$.  We use these results as the basis for\nour suggested parametrization.  We label the momenta as $D^+(p_D) \\to\n\\pi^+(p_1) \\pi^+(p_2)K^-(p_3)$ and define the following invariant\nmasses squared of the final state, $s_{13}=(p_{1}+p_{3})^2$,\n$s_{23}=(p_{2}+p_{3})^2$ and $ s_{12}=(p_{1}+p_{2})^2$, with $s_{12}+\ns_{13}+s_{23}= m_D^2+m_K^2+2 m_\\pi^2$.  Thus, the $S$- and $P$-wave\namplitudes for $D^+ \\to [K^- \\pi^+]_{S,P}\\ \\pi^+$ are,\n\\begin{equation}\n\\sand{[K^-\\pi^+]_{S,P}\\pi^+}{{\\cal H}_{\\rm eff}}{D^+} =  \\mathcal{A}_S^+  + \\mathcal{A}_P^+    .\n\\end{equation}\nContribution from  $D$-wave resonances  are known to be rather small in this decay~\\cite{PDG}. The $S$- and $P$-wave \namplitudes can be parametrized with complex parameters, $f_{1,2}^{S}$ and $f_{1,2}^{P}$, as follows:\n\\begin{align}\n\\label{ASPD+1} \n\\mathcal{A}^+_S(s_{13}, s_{23}) & =  \\left [ f_1^{S}\\, (m_D^2 -s_{13}) + f_2^S \\, \\frac{F_0^{D\\pi}(s_{13})}{s_{13}}   \\right] F_0^{K\\pi}(s_{13}) +(s_{13} \\leftrightarrow s_{23}),  \\\\\n\\mathcal{A}^+_P(s_{12}, s_{13}, s_{23})&  =  \\left[ f_1^P\\,  \\Omega(s_{13},s_{23}) + f_2^P \\left( \\frac{s_{23} - s_{12}}{\\Delta^2}    - \\frac{1}{s_{13} } \\right ) \n  F_1^{D\\pi} (s_{13})  \\right]F_1^{K\\pi}(s_{13})+(s_{13} \\leftrightarrow s_{23}),\n  \\label{ASPD+2} \n\\end{align}\nwith $\\Omega(s_{13},s_{23}) = s_{23}-s_{12}-{\\Delta}^2\/s_{13}$ and ${\\Delta}^2=({m_{K}}^2 -m_\\pi^2)(m^2_{D^+}-m^2_\\pi)$. We assume that nonfactorizable corrections\nare absorbed in the complex  parameters, $f_{1,2}^{S}$ and $f_{1,2}^{P}$, to be fitted to the data.  \n\nThe parametrization we introduce in Eqs.~\\eqref{ASPD+1} and\n\\eqref{ASPD+2} makes the emergence of the scalar and vector $K\\pi$\nform factors, $F_{0,1}^{K\\pi}(s)$, explicit. They are discussed in\nmore detail in Appendix~\\ref{Kp}.  The scalar and vector $D\\pi$\ntransition form factors also appear in Eqs.~(\\ref{ASPD+1}) and\n(\\ref{ASPD+2}). Their variation with energy for the physical values of\n$s$ is not significant, but they affect the shape of the amplitudes\nclose to the edges of the Dalitz plot. Possible parametrizations are\ndiscussed in Appendix~\\ref{transitionff}.\n\nA version of the above description put forward here has been employed\nsuccessfully in Ref.~\\cite{BE2009}. Additional contributions,\ne.g. with higher angular momentum or the isospin-2 $\\pi^+\\pi^+$\ninteractions\\footnote{Most experimental analyses agree this\n  contribution is negligible~\\cite{E791DKpipi,FOCUSDKpipi} with the\n  exception of the CLEO collaboration analysis where a fit fraction of\n  $\\sim 20\\%$ is attributed to the $\\pi^+\\pi^+$\n  interactions~\\cite{CLEODKpipi}.}, are small in this process.  In a\nrealistic high-statistics Dalitz plot analysis, however, they may be\nrequired and would have to be included in the signal function through\nusual isobar model expressions, for instance.\n\n\nA final ingredient that is not present in our\n  parametrizations are the genuine three-body hadronic final state\n  interactions, which are most often neglected in experimental\n  analyses. Their treatment is somewhat involved and only recently\n  this problem started to be dealt with. An approach based on Feynman\n  diagrams from effective lagrangians was introduced in\n  Refs.~\\cite{Magetal2011,MR15} precisely in the case of $D^+\\to\n  K^-\\pi^+ \\pi^+$. Alternatively, a description based on a dispersive treatment\n introduced in \\cite{NKS12} for $(\\omega\/\\phi)\\to\n  \\pi\\pi\\pi$ decays has been applied to $D\\to K \\pi\\pi$ decays in\n  Refs.~\\cite{NK15,NK17}. Finally, a coupled channel description\n  including three-body scattering was performed in\n  Ref.~\\cite{Nakamura}. These treatments do not allow for a simple\n  parametrization of the type we advocate here with the goal of\n  replacing isobar model expressions. These three-body effects, if\n  important, should show as deviations from our description and\n  represent a refinement to the amplitudes discussed here that should be addressed in the future. \n\n\n\n\\subsection{Amplitudes  for $\\boldsymbol{  D^0 \\to K^0_S \\pi^+\\ \\pi^- }$}\n\\label{QtbAmpD0K0pp}\n\nThe decay $D^0 \\to K^0_S \\pi^- \\pi^+ $ was treated within the\nframework of quasi-two body factorization in Ref.~\\cite{JPD_PRD89}.  A\ngood reproduction of the Belle Dalitz plot density distributions~\\cite{A.Poluektov_PRD81_112002_Belle} was\nobtained and so were the distributions produced by the BABAR\nmodel.\\footnote{The model is built from a fit to the BABAR Dalitz plot\n  data; see Ref.~\\cite{JPD_PRD89}.}  The parametrizations that follow\nin the next subsections are based on the quasi-two-body amplitudes\nderived in this study. The Hamiltonian that describes this decay\nchannel is similar to that of Eq.~(\\ref{effHDpKpp}), but besides a\nCabibbo favored term proportional to $V^*_{cs}V_{ud}$ there is also a\ndoubly Cabibbo suppressed contribution proportional to\n$V^*_{cd}V_{us}$. The momenta are labelled as $D^ 0(p_D)\\to K^0_S(p_1)\n\\pi^-(p_2) \\pi^+(p_3)$ where the kinematic configuration is defined by\n$s_{12}=(p_1+p_2)^2$, $s_{13} =(p_1+p_3)^2$ and $s_{23}=(p_2+p_3)^2$,\nwith $s_{12}+ s_{13}+s_{23}= m_{D^0}^2+m_{K^0}^2+2 m_\\pi^2$. We start\nwith the parametrization of the amplitude for the interacting $K^\n0_S\\pi^ -$ in an $S$- or $P$-wave state.\n\n\n\\subsubsection{Parametrization of the $D^0 \\to [K^0_S \\pi^-]_{S,P}\\ \\pi^+ $ amplitudes}\n\\label{paramD0Kp-p+}\n\nThe following parametrizations are derived from Eqs.~(66) and (68) of Ref.~\\cite{JPD_PRD89}.\nIn terms of three complex parameters $g_{1,2}^ S$ and $g_1^ P$, the parametrized amplitudes read\n\\begin{eqnarray}\n\\label{ASD0}\n\\mathcal{A}^{0}_{S,-}(s_{12})&=& \\left( g_{1}^ S+g_{2}^S s_{12} \\right) F_0^{K\\pi}(s_{12}),  \\\\\n\\mathcal{A}^{0}_{P,-}(s_{12}, s_{13}, s_{23})&=& g_{1}^ P\\left( s_{23} - s_{13} +\\frac{\\Delta_0^2}{s_{12}}\\right)  F_1^{K\\pi}(s_{12}),\n\\label{APD0}\n\\end{eqnarray}\nwith ${\\Delta_0}^2=({m_{K^0}}^2 -m_\\pi^2)(m^2_{D^0}-m^2_\\pi)$.  The\n$\\pi K$ $S$-wave $\\mathcal{A}^{0}_{S,-}$ amplitude includes the\ncontribution of the scalar $K^*_0(800)^-$ and $K^*_0(1430)^-$\nresonances and the $P$-wave $\\mathcal{A}^{0}_{P,-}$ that of the vector\n$K^*(892)^-$.  Despite its small fit fraction, the $\\pi K$ $D$-wave\n$D^0 \\to [K^0_S \\pi^-]_{D}\\ \\pi^+ $ amplitude plays an important role\nthrough interference.  The contribution of the tensor $K^{*-}_2(1430)$\nresonance can be parametrized by a relativistic Breit-Wigner formula, with a\nmagnitude and phase that should be obtained through a fit to the data,\nas done successfully in Ref.~\\cite{JPD_PRD89}. This component should\nbe added to the $S$ and $P$ wave amplitudes parametrized above.\n\n\n\n\n\\subsubsection{Parametrization of the $D^0 \\to [K^0_S \\pi^+]_{S,P}\\ \\pi^- $ amplitudes}\n\\label{paramD0Kp+p-}\n\nLikewise, the decay amplitudes for $D^0 \\to [K^0_S \\pi^+]_{S,P}\\ \\pi^- $ are given in Ref.~\\cite{JPD_PRD89}  [see Eqs.~(84) and (85)]\nand can be parametrized as, \n\\begin{eqnarray}\n\\label{A8SPD0}\n\\mathcal{A}^{0}_{S,+}(s_{13})&=& \\left[ g_3^ S(m_\\pi^2-s_{13})+g_4^ S\\ \\frac{\\Delta_0^2}{s_{13}} F_0^{D\\pi}(s_{13}) \\right] F_0^{K\\pi}(s_{13}) \\ , \\\\\n\\label{A9SPD0}\n\\mathcal{A}^{0}_{P,+}(s_{12}, s_{13}, s_{23})&=&\\left[ g_2^ P+ g_3^ P F_1^{D\\pi}(s_{13})\\right] \\left( s_{23}-s_{12} +\\frac{\\Delta_0^2}{s_{13}}\\right)   F_1^{K\\pi}(s_{13}) \\ .\n\\end{eqnarray}\n\n\nThe  $\\pi K$ $S$-wave $\\mathcal{A}^{0}_{S,+}$ amplitude includes the contribution of the  scalar $K^*_0(800)^+$ and $K^*_0(1430)^+$ resonances, \nthe $P$-wave $\\mathcal{A}^{0}_{P,+}$ that of  the vector $K^*(892)^+$. The contribution from\nthe $D$-wave, that stems mainly from the $K^*_2(1430)^+$, could be parametrized by the usual Breit-Wigner expressions. \n \n\n\n\\subsubsection{Parametrization of the $D^0 \\to K^0_S [\\pi^+ \\pi^-]_{S,P}$ amplitudes}\n\\label{paramD0K0p+p-}\n\nThe weak $D^0 \\to K^0_S [\\pi^+ \\pi^-]_{S,P}$  decay amplitudes, following\nRef.~\\cite{JPD_PRD89}, can be parametrized, using the \nsame momentum definition as before, as: \n\\begin{eqnarray}\n\\label{A2SPD0}\n\\mathcal{A}^{0}_{S,0}(s_{23})&=& \\left( g_5^S+g_6^S s_{23}\\right) F_{0n}^{\\pi\\pi}(s_{23})\\ , \\\\\n\\label{A4SPD0}\n\\mathcal{A}^{0}_{P,0}(s_{12}, s_{13}, s_{23})&=& \\left( s_{12} - s_{13} \\right)   \\left [g_{4}^ P F_1^{\\pi\\pi}(s_{23})+g_5^P F_1^{\\omega}(s_{23}) \\right ] \\ .\n\\end{eqnarray}\nThe $\\pi\\pi$ $S$-wave $\\mathcal{A}^{0}_{S,0}$ amplitude includes the\ncontributions of the scalar $ f_0(500)$ (or $\\sigma), f_0(980)$ and\n$f_0(1400)$ resonances. The $[\\pi^+\\pi^-]_P$ pair can originate from\nthe $\\omega$ resonance through isospin violation. This introduces\na term proportional to the vector form factor $F_1^\\omega(s_{23})$ in\nEq.~(\\ref{A4SPD0}).\\footnote{In Eq.~(71) of Ref.~\\cite{JPD_PRD89} this term was explicitly written as $F_1^\\omega(s_{23})= m_\\omega^2\/(m_\\omega^2-s_{23}-i m_\\omega \\Gamma_\\omega)$.}\nThe effects of the vector $\\rho(770)^0$ and\n$\\omega(782)$ resonances are included in the $P$-wave amplitudes,\n$\\mathcal{A}^{0}_{P,0}$, which also contain the contribution of the\n$\\rho(1450)^0$ and $\\rho(1700)^0$; see details in\nAppendix~\\ref{Fpp}. The $D$-wave is dominated by the $f_2(1270)$\ntensor meson and must be included in a realistic amplitude. In\nRef.~\\cite{JPD_PRD89} it was parametrized by the usual relativistic\nBreit-Wigner line shape.\n\n\nThe full decay  amplitude is thus given by,\n\\begin{equation}\n \\mathcal{A}^0 = \\mathcal{A}^{0}_{S,-} + \\mathcal{A}^{0}_{P,-} + \\mathcal{A}^{0}_{S,+} + \\mathcal{A}^{}_{P,+} + \\mathcal{A}^{0}_{S,0} + \\mathcal{A}^{0}_{P,0} \n  +  \\cdots  \\ ,\n\\end{equation}\nwhere the ellipsis denotes $D$- and higher-wave contributions and possible high invariant-mass contribution. \n\n\n\n\n\\subsection{Amplitudes  for  $\\boldsymbol{  D^0 \\to K^0_S K^+ K^- }$}\n\\label{QtbAmpD0K0KK}\n\nThe  $D^0 \\to K^0_S K^+ K^-$ decay channel was measured by the  BABAR Collaboration with high statistics~\\cite{SanchezPRL105_081803}. Using the quasi two-body \nfactorization approach~\\cite{BLoiseau_Rio2015}, we parametrize~\\cite{JPD_inprogress} this decay channel with the definitions of the invariants  $s_{12}$, $s_{13}$ and $s_{23}$ \nsimilar to  those introduced for  $D^0 \\to K^0_S \\pi^+ \\pi^- $ in the previous section, replacing charged pions by charged kaons, the effective Hamiltonian being identical (here the charged kaon mass is denoted $m_K$). \nThe momenta are thus labelled as $D^ 0(p_D)\\to K^0_S(p_1) K^-(p_2) K^+(p_3)$ with  $s_{12}=(p_1+p_2)^2$, $s_{13} =(p_1+p_3)^2$,  $s_{23}=(p_2+p_3)^2$, \nand $s_{12}+ s_{13}+s_{23}= m_{D^0}^2+m_{K^0}^2+2 m_K^2$. The involved three interacting kaon pairs, $[K^+ K^-]_L$, $[K^0_S K^-]_L$ and $[K^0_S K^+]_L$ can be \nin a scalar or vector state with $L=S$ or $P$, respectively. The  isospin of the $[K^+ K^-]_L$ pairs can be either 0 or 1, but that of the $[K^0_S K^\\mp]_L$ pairs is 1.\n\n\n\\subsubsection{Parametrization of the $D^0 \\to K^0_S [K^+ K^-]_S $ amplitudes}\n\\label{KKS}\n\nThe decay amplitude in which the isoscalar $[K^+ K^-]_S$ pair is associated with the $f_0(980)$ and $f_0(1370)$ resonances, and the isovector one is related to \nthe $a_0(980)^0$ and $a_0(1450)^0$ resonances, can be parametrized as: \n\\begin{equation}\n\\label{M1kT}\n{\\mathcal{A}}^0_{S,0}(s_{23}) =  \\left({h}_1^S + {h}_2^S s_{23} \\right) F^{ K\\bar K}_{0n} (s_{23} ) + h_3^S \\left(m_{K^0}^2-s_{23} \\right) F^{ K\\bar K}_{0s} (s_{23} )  +\n\\left({h}_4^S+{h}_5^S  s_{23} \\right) G_0^{K \\bar K}(s_{23})  \\ .\n\\end{equation}\n\n\nThe  amplitude with the isovector $ [K^0 K^-]_S$  pairs in an $S$-wave state, which include the $a_0(980)^-$ and $a_0(1450)^-$ resonances, can be parametrized as,\n\\begin{equation}\n\\label{M3k}\n\\mathcal{A}_{S,-}^0(s_{12})=\\left(h_6^ S+h_7^ S s_{12}\\right) G_0^{K \\bar K}(s_{12}),\n\\end{equation}\nand the corresponding amplitude associated with the $a_0(980)^+$ and $a_0(1450)^+$ resonances as,\n\\begin{equation}\n\\label{M4k}\n\\mathcal{A}_{S,+}^0(s_{13})=\\left[h_8^S \\dfrac{F_0^{D K}(s_{13})}{s_{13}}+h_9^ S(m_{K}^2-s_{13})\\right]  G_0^{K\\bar K}(s_{13}) \\ .\n\\end{equation}\nThe scalar-isocalar form factors, $F^ {K\\bar K}_{0n(s)}(s)$, in Eq.~(\\ref{M1kT}), and the scalar-isovector ones, $G^{K\\bar K}_0(s)$, in Eqs.~(\\ref{M3k}) and~(\\ref{M4k}) are detailed in Appendix~\\ref{KK} \nwhile the scalar $D$ to $K$  transition form factor $F_0^{D K}(s)$ is defined in Appendix~\\ref{transitionff}.\n\n\n\n\\subsubsection{Parametrization of the $D^0 \\to K^0_S [K^+ K^-]_P $ amplitudes}\n\\label{KKP}\n\nWe parametrize this decay amplitude, where the isoscalar and isovector $ [K^+ K^-]_P$  pairs contain contributions from the $\\omega(782)$, $\\omega(1420)$, $\\phi(1020)$,  $\\rho(770)^0$,  $\\rho(1450)^0$, and $\\rho(1700)^0$\nresonances, by the expression~\\cite{JPD_inprogress}:\n\\begin{equation}\n\\label{M5k}\n\\mathcal{A}_{P,0}^0(s_{12}, s_{13}, s_{23})=\\left(s_{12} -s_{13}\\right) \\left(h_1^ P F_{1u}^{K^+  K^-}(s_{23}) +h_{2}^ P F_{1s}^{K^+ K^-}(s_{23}\n\\right)  \\ .\n\\end{equation}\nLikewise, one can express the amplitude in which the isovector  $ [K^0_S K^-]_P$  pair is associated with the three $\\rho^-$ resonances by,\n\\begin{equation}\n\\label{M8k}\n\\mathcal{A}_{P,-}^0(s_{12}, s_{13}, s_{23})= h_3^ P \\left [s_{23} - s_{13} + \\left(m_{D^0}^2- m_{K}^2\\right)\\ \\frac{m_{K^0}^2-m_{K}^2}{s_{12}} \\right ]    F_{1}^{K^-  K^0}(s_{12})  \\ ,\n\\end{equation}\nwhile the parametrization of the amplitude associated with the $\\rho^+$ resonances  reads,\n\\begin{equation}\n\\label{M9k}\n\\mathcal{A}_{P,+}^0 (s_{12}, s_{13}, s_{23})=  \\left[ h_4^ P+h_5^P F_1^{D K}(s_{13}) \\right] \\left[s_{23} - s_{12} + \\left(m_{D^0}^2- m_{K}^2\\right)\\ \\frac{m_{K^0}^2-m_{K}^2}{s_{13}} \\right ]  F_{1}^{K^+\\bar K^0}(s_{13})\\ .\n\\end{equation}\nIn Eqs.~(\\ref{M5k}) to (\\ref{M9k}), the vector-isocalar, $F^ {K\\bar K}_{1u(s)}(s)$, the vector-isovector $F_1^{K^-  K^0}(s)$ and  $F_1^{K^+ \\bar K^0}(s)$ form factors\nare defined in Appendix~\\ref{KK}. The parametrization of the \nvector $D$ to $K$ transition form factor $F_1^{D K}(s)$ is discussed in Appendix~\\ref{transitionff}.\n\n\nThe full decay amplitude is the coherent sum of all the sub-amplitudes  discussed above:\n\\begin{equation}\n \\mathcal{A}^0 = \\mathcal{A}^{0}_{S,-} + \\mathcal{A}^{0}_{P,-} + \\mathcal{A}^{0}_{S,+} + \\mathcal{A}^{0}_{P,+} + \\mathcal{A}^{0}_{S,0} + \\mathcal{A}^{0}_{P,0}  +  \\cdots  \\ ,\n\\end{equation}\nwhere the ellipsis denotes the omission of higher waves that could be included using Breit-Wigner line shapes.\n\n\n\n\n\\section{Concluding remarks}\n\\label{Conclusion}\n\n\n We have introduced alternatives to the isobar-model Dalitz-plot\n parametrizations of weak $D$ and $B$ decays into exclusive final\n states composed of three light mesons, namely the various charge\n states $\\pi\\pi\\pi$, $K\\pi\\pi$ and $KK\\bar K$. Such isobar\n parametrizations have been frequently employed in fits although they\n do not respect unitarity, which leads, amongst other effects, to a\n sum of branching fractions that can exceed the total decay width by\n large amounts.  As a consequence, any strong CP phases that may be\n extracted from these fits must be taken with caution.\n\nOur alternative parametrizations, while not fully three-body unitary,\nare based on a sound theoretical application of QCD factorization to a\nhadronic quasi two-body decay. We thus assume that the final three-meson\nstate is preceded by intermediate resonant states which is justified\nby ample phenomenological and experimental evidence. Analyticity, unitarity, chiral\nsymmetry as well as the correct asymptotic behavior of the two-meson\nscattering amplitude in $S$ and $P$ waves are implemented via analytical and \nunitary $S$- and $P$-wave $\\pi\\pi$, $\\pi K$ and $K\\bar K$ form factors which\nenter the hadronic final states of our amplitude parametrizations. These\namplitudes can be readily used adjusting the parameters in a least-square\nfit to the Dalitz plot --- for a given decay channel --- and employing\ntabulated form factors as functions of momentum squared or\nenergy. The different quasi-two-body $B$- and $D$-decay channels \nfor which we provide explicit amplitude expressions are summarized in \nTables~\\ref{ParamBdecay} and~\\ref{ParamDdecay}, respectively.\nFor each channel the relevant equation for the parametrization  \nis cited and the dominant contributing resonances are listed.\nLet us add a practical remark: in any application of the\nparametrized amplitudes to experimental analyses, one can set to\nzero one phase of the $S$ or $P$ wave amplitude since the Dalitz plot\ndensity is not sensitive to its value.\n\nWith this  ``tool kit\" we strongly hope to contribute to more\nsophisticated experimental extractions of three-body decay observables, in\nparticular CP-violating phases.\n\n\n\n\\begin{table}[t!]\n\\caption{For  each $B$-decay channel  in the first column, the second column refers to the equation of the  proposed amplitude parametrization and the third column lists the dominant contributing resonances.}\n\\label{ParamBdecay}\n\\begin{center}\n\\begin{tabular}{lcr}\n\\hline\n\\hline\nQuasi-two-body channel\\hspace{0.3cm} & see Eq.:& Dominant resonances \\\\\n\\hline\n$B^- \\to \\pi^- [\\pi^+\\pi^-]_{S}$ & (\\ref{MSsij}) & $f_0(500)$, $f_0(980)$, $f_0(1400)$  \\\\\n$B^- \\to \\pi^- [\\pi^+\\pi^-]_{P}$ & (\\ref{MPsij}) & $\\rho(770)^0$, $\\rho(1450)^0$ , $\\rho(1700)^0$ \\\\\n$B \\to K [\\pi^+ \\pi^-]_{S}$ &(\\ref{paracs}) & $f_0(500)$, $f_0(980)$, $f_0(1400)$  \\\\\n$B \\to K [\\pi^+ \\pi^-]_{P}$ & (\\ref{paracp}) &  $\\rho(770)^0$, $\\rho(1450)^0$ , $\\rho(1700)^0$ \\\\\n$B^{-(0)} \\to [K^{-(0)}\\pi^+]_S\\pi^-$ & (\\ref{MSparam}) & $K^*_0(800)^{0(+)}$, $K^*_0(1430)^{0(+)}$\\\\\n$B^{-(0)} \\to [K^{-(0)} \\pi^+]_P\\pi^-$ & (\\ref{MPparam}) & $K^*(892)^{0(+)}$, $K^*(1410)^{0(+)}$ \\\\\n$B^- \\to K^- [K^+ K^-]_S$ & (\\ref{ASsij}) & $f_0(980)$, $f_0(1400)$\\\\\n$B^- \\to K^- [K^+ K^-]_P$ & (\\ref{APsij})& $\\rho(770)^0$, $\\rho(1450)^0$, $\\rho(1700)^0$,   \\\\\n    &  &\n    $\\omega(782)$, $\\omega(1420)$, $\\omega(1650)$, \\\\\n   &  &  $\\phi(1020)$, $\\phi(1680)$\\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\n\\begin{table}[h!]\n\\caption{As in Table~\\ref{ParamBdecay} but for hadronic quasi-two-body $D$ decays.}\n\\label{ParamDdecay}\n\\begin{center}\n\\begin{tabular}{lcr}\n\\hline\n\\hline\nQuasi-two-body channel \\hspace{0.5cm} & see Eq.: & Dominant resonances \\\\\n\\hline\n$D^+ \\to [\\pi^+ \\pi^-]_S \\pi^+ $ & (\\ref{AD3piS})& $f_0(500)$, $f_0(980)$, $f_0(1400)$\\\\\n$D^+ \\to [\\pi^+ \\pi^-]_P \\pi^+ $ & (\\ref{AD3piP}) & $\\rho(770)^0$, $\\rho(1450)^0$ \\\\\n $D^+ \\to [K^- \\pi^+]_S \\ \\pi^+ $ &(\\ref{ASPD+1})&  $K^*_0(800)^{0}$, $K^*_0(1430)^{0}$\\\\\n  $D^+ \\to [K^- \\pi^+]_P \\ \\pi^+ $ &(\\ref{ASPD+2})& $K^*(892)^0$, $K^*(1410)^0$\\\\\n  $D^0 \\to [K^0_S \\pi^-]_{S}\\ \\pi^+ $&(\\ref{ASD0}) & $K^*_0(800)^{-}$, $K^*_0(1430)^{-}$\\\\\n   $D^0 \\to [K^0_S \\pi^-]_{P}\\ \\pi^+ $ &(\\ref{APD0})& $K^*(892)^-$, $K^*(1410)^-$\\\\\n   $D^0 \\to [K^0_S \\pi^+]_{S}\\ \\pi^- $ &(\\ref{A8SPD0})& $K^*_0(800)^{+}$, $K^*_0(1430)^{+}$\\\\\n    $D^0 \\to [K^0_S \\pi^+]_{P}\\ \\pi^- $ & (\\ref{A9SPD0})&$K^*(892)^+$, $K^*(1410)^+$ \\\\\n    $D^0 \\to K^0_S [\\pi^+ \\pi^-]_{S}$ &(\\ref{A2SPD0}) &  $f_0(500)$, $f_0(980)$, $f_0(1400)$\\\\\n  $D^0 \\to K^0_S [\\pi^+ \\pi^-]_{P}$  & (\\ref{A4SPD0})& $\\rho(770)^0$, $\\omega(782)$\\\\\n  $D^0 \\to K^0_S [K^+ K^-]_{S}$  &(\\ref{M1kT})&$f_0(980)$, $f_0(1400)$, $a_0(980)^0$,  $a_0(1450)^0$\\\\\n  $ D^0 \\to K^+[K^0 K^-]_S$ &(\\ref{M3k})& $a_0(980)^-$,  $a_0(1450)^-$\\\\\n  $ D^0 \\to K^-[K^0 K^+]_S$ &(\\ref{M4k})& $a_0(980)^+$,  $a_0(1450)^+$\\\\\n    $D^0 \\to K^0_S [K^+ K^-]_{P}$ &(\\ref{M5k})&  $\\omega(782)$, $\\omega(1420)$, $\\phi(1020)$, $\\rho(770)^0$, $\\rho(1450)^0$  \\\\\n     $ D^0 \\to K^+[K^0 K^-]_P$& (\\ref{M8k})& $\\rho(770)^-$, $\\rho(1450)^-$  \\\\\n     $ D^0 \\to K^-[K^0 K^+]_P$ &(\\ref{M9k})& $\\rho(770)^+$, $\\rho(1450)^+$\\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n \n \n\n\n \n \n \n\n\\section*{Acknowledgements}\nIt is a pleasure to thank the Centro Brasileiro de Pesquisas F\\'isicas\n(CBPF) and the organizers of the ``2015 LHC$b$ workshop on multi-body\ndecays of $D$ and $B$ mesons'' where this work was initiated. BL and\nDB appreciated the hospitality of Universidade Cruzeiro do Sul where\npart of this work was carried out.  Discussions with Alberto dos Reis, Gast\\~ao Krein and Patr\\'icia Magalh\\~aes are gratefully acknowledged.  This work was\npartially supported by the S\\~ao Paulo Research Foundation (FAPESP), grant\nnos. 2015\/20689-9 and 2016\/03154-7; by the Brazilian National Council for\nScientific and Technological Development (CNPq), grant\nnos.~305431\/2015-3 and 458371\/2014-9; by the IN2P3-COPIN collaboration agreement (project No\n08-127); by \nthe Spanish ``Ministerio de Econom\\'ia y Competitividad\" under grants \nCICYT-FEDER-FPA 2014-55613-P and SEV-2012-0234, \nas well as by  Secretaria d'Universitats i Recerca del Departament d'Economia i Coneixement \nde la Generalitat de Catalunya under grant 2014 SGR 1450.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nSurfaces of ferroelectrics can strongly effect their properties. In general\nthis is a much stronger effect than in paraelectric materials, because the\ntermination of a ferroelectric gives rise to a depolarization field, which\nis huge in an ideal, ionic ferroelectric. The depolarization field, $E_{d%\n\\text{,}}$ is opposite in sign to the bulk polarization, and gives a\ncontribution to the energy of $-E_d\\cdot P$ large enough to completely\ndestabilize the bulk ferroelectric state \\cite{1,2}. The ferroelectric\ndistortion in thin films varies with thickness so that the structures become\ncubic at thin edges \\cite{3}. Huge particle size effects are observed \\cite\n{4,5}, orders of magnitude greater than observed for non-ferroelectrics.\nSurfaces are important also during crystal growth, and their properties are\nkey to catalysis and use as a epitaxial substrate. Here we study thin slabs\nof BaTiO$_3$ to better understand the electronic structure of the surface,\nsurface energetics and relaxation, and intrinsic screening of the\ndepolarization field.\n\nExperimental studies of BaTiO$_3$ surfaces are few and many more studies\nhave focused on SrTiO$_3$ which lacks the complications of ferroelectricity.\nPhotoemission studies of sputtered and annealed BaTiO$_3$ show evidence of\nsignificant surface defects \\cite{6}, and a number of papers have addressed\nthe defect properties of the surface \\cite{7,8,9}. Studies of SrTiO$_{3\\text{\n}}$surfaces show evidence for minor relaxations by motions of atoms\nperpendicular to the surfaces \\cite{10,11} and show no evidence for states\nin the gap. \\cite{12} Some recent studies of BaTiO$_3$ show evidence for gap\nstates \\cite{8}, and others do not \\cite{13}, probably indicating\nsensitivity to the surface defect state. Photoemission spectroscopy on BaTiO$%\n_3$ shows excellent agreement with bulk LAPW computations of the LDA band\nstructure \\cite{13}.\n\nFirst-principles electronic structure approaches have been very successful\nfor bulk ferroelectrics \\cite{14,15,16,17,18,19,20,21} but the application\nof first-principles band structure methods to surface properties is daunting\ndue to the giant computational burden. Here the first set of accurate\nelectronic structure calculations for periodic BaTiO$_3$ slabs are\npresented. These results can be considered benchmark results for faster,\nmore approximate methods, and also give some insights into the electronic\nstructure of ferroelectric BaTiO$_3$ surfaces.\n\n\\section{Method}\n\nWe studied six to seven atomic layer (001) and (111) periodic slabs of BaTiO$%\n_{3\\text{ }}$containing 15-18 atoms in the periodic unit, with seven to six\nlayers of vacuum, using the Linearized Augmented Plane Wave method with\nextra local orbitals (LAPW+LO). \\cite{22} Both slabs were studied in\nthe ideal configuration, and the more stable (001) slab was studied\nextensively with tetragonal ferroelectric distortions and with surface\nrelaxations. In each case the in-plane lattice constant was 7.57 bohr\n(=4.006 \\AA ). An ABO$_3$ perovskite slab can have two types of\nterminations, AO (Type I) and BO$_2$ (Type II). Three configurations for the\n(001) slab were studied, asymmetrically terminated (BaO and TiO$_2$\nsurfaces), and symmetrically terminated with BaO and TiO$_2$ surfaces,\nrespectively. The slabs in each case were repeated with periodic boundary\nconditions. The (001) slab had a repeat length of six times the cubic\nlattice constant, or 45.42 bohr (=24.035 \\AA ); whereas for the (111) slab $%\na $=10.70 bohr $c$=26.2 bohr. The ideal slabs are illustrated in\nfigure\\thinspace \\ref{fig_structures}. The (001) slab has P4mm tetragonal\nsymmetry with eight space group operations, and the (111) slab is\nrhombohedral with R3m symmetry. In order to obtain accurate energy\ndifferences and surface energies, bulk calculations were also performed in\nthe same symmetry with a tripled supercell with the same k-points and\nconvergence parameters.\n\nThe ferroelectric distortion in the (001) slab used the experimental\ntetragonal average displacements. The Ba's were displaced by 0.06 \\AA , the\nTi's by -0.1122 \\AA\\ along {\\it c }relative to the O(II)'s, and the O(I)'s\nby 0.0288 \\AA\\ . For the asymmetrically terminated slab, displacements were\nperformed in both directions, with Ti displaced towards the Ba-O surface (+)\nand towards the Ti-O surface (-). The tetragonal phase in bulk BaTiO$_3$\ndoes not consist primarily of displacements towards the perovskite cube\nfaces, as we are studying here, but rather displacements towards the cube\ndiagonals, and the tetragonal structure is a dynamical average with hopping\namong four (111) directions. Such disordered configurations are well beyond\nthe possibilities of present computations for slabs, and must await future\nfaster and probably more approximate methods. Nevertheless, the general\neffects of a surface on ferroelectric properties can be illuminated by the\npresent study.\n\nThe surface layers were relaxed using the LAPW forces \\cite{23} and a\nquasi-Newton method for the ideal (001) Ba-Ti terminated slab, the\nferroelectric (+) Ba-Ti terminated slab, and the ferroelectric Ba-Ba\nterminated slab.\n\nThe convergence parameter RK$_{\\max }$ was set to 7.0; Table II shows the\nmuffin tin radii. A 4$\\times $4$\\times $2 special k-point mesh was used\nwhich gives 3 k-points for the (001) slab and 10 k-points for the (111)\nslab. The ferroelectric Ba-Ti slab was also converged with a 6$\\times $6$%\n\\times $2 mesh (6 k-points) and the total energy for the 15 atom unit cell\nonly differed by 0.2 mRyd. The matrix order (number of basis functions) for\nthe slabs was about 2900 for the first set of muffinb tin radii (see Table II) and 3800\nfor the second set. These calculations are extremely computationally\nintensive. Not only was each iteration in the self-consistent cycle time\nconsuming but an unusually large number of iterations were required to reach\nself-consistency due to charge fluctuations across the slab set up by the\nelectric fields generated across the slab.\n\n\\section{Results and Discussion}\n\nTable I shows the unrelaxed and relaxed (001) slab structures, and total\nenergies for all studied slabs are shown in Table II. Since the (111) slab\nis found to be much more unstable than the (001) slab, most of the\ncomputations and the discussion below concentrate on the (001) slab. Whereas\nthere is a ferroelectric instability, or double well, found for bulk BaTiO$%\n_3 $, there is no ferroelectric instability for the slab. Rather the energy\nincreases with ferroelectric distortion due to the depolarization field that\nforms at the surface. This is the same result that was obtained using an\nionic model for finite clusters. \\cite{2} Furthermore, for the\nasymmetrically terminated Ba-Ti slab, the energy is not symmetric with\nrespect to ``+'' and ``-'' ferroelectric distortions, with a much lower\nenergy being obtained when the Ti atoms are moved so that the partially\ncoordinated surface Ti atoms move in towards the slab (``+'' distortion in\nTable II).\n\n\\subsection{Relaxation}\n\nThe surface relaxations can be described as a combination of relaxation of\nthe surface layer in towards the slab, and a dimpling of the layers. Here we\ndefine the former by the motion of the midpoint between the z coordinate for\nthe surface cation and anion relative to that of the ideal interlayer\nspacing, and the latter by half of the difference in z-coordinate of the\nsurface ions. In agreement with work on other oxides, the sign of the\nrumpling is such that the surface cations move in towards the slab and the\nanions move out (Table III). The sign of the effect is independent of the\nslab termination or whether the slab is ferroelectric or ideal. The\nmagnitude of the rumpling is quite similar to that obtained experimentally\nfor SrTiO$_3$. \\cite{10} However, the experiment did not resolve net\nrelaxation of the layer spacing, whereas we find the that the surface layer\nspacing is contracted relative to that of the bulk spacing. This may be\npartly due to the fact that we did not relax all of the atomic positions in\nthe bulk of the slab, but rather fixed the interlayer spacing at the\nexperimentally observed zero pressure value, and LDA gives a smaller lattice\nconstant than experiment for the bulk. The relaxations are energetically\nsignificant as is shown in Table II: 0.032 Ryd for the ideal (001) Ba-Ti\nterminated slab, 0.016 Ryd for the ferroelectric Ba-Ti terminated slab, and\n0.026 Ryd for the ferroelectric Ba-Ba terminated slab. For the Ba-Ti slab\nthe relaxation energy per surface is about 16\\% of the cleavage energy.\n\n\\subsection{Surface energies}\n\nOnly the symmetrically terminated ideal slabs are expected to give accurate\nestimates of the surface energy, because the asymmetrically terminated and\nferroelectric slabs have potential gradients (net polarization or\nmacroscopic electric field $E_{\\text{mac}}$) which adds a term to the energy\nproportional to $E_{\\text{mac}}^2$. It is necessary to consider both the BaO\nand TiO$_2$ terminated slabs together, since neither alone has bulk\nstoichiometry, and thus its energy cannot be compared with bulk. Thus if we\nadd the energies of the BaO terminated slab and the TiO$_2$ terminated slab,\nwe get -128935.8994 Ryd for 7 BaTiO$_3$ units and 4 surfaces (2 BaO and 2 TiO%\n$_2$). We subtract 7 times the bulk BaTiO$_3$ energy per cell (-128936.1703\nRyd) and get 0.2709 Ryd for 4 surfaces, or an average of 0.0677 Ryd\/surface,\nwhich gives 0.0574 eV\/\\AA $^2$ (=2712 erg\/cm$^2$ = 0.92 J\/m$^2$). For\ncomparison, the unrelaxed surface energy for Cr$_2$O$_{3\\text{ }}$is\nestimated to be 2000 erg\/cm$^2$ for the relaxed surface \\cite{24}. The\nsurface energy of SiO$_2$ glass is much lower, about 300 erg\/cm$^2$. We have\nnot relaxed the symmetric TiO$_2$ terminated slab, but if its relaxation\nenergy is similar to that of the three slabs we did relax, relaxation would\nlower the surface energy by about 12\\%. The very high surface energy for\nBaTiO$_{3\\text{ }}$explains why it does not cleave easily, but rather\nfractures. Real BaTiO$_{3\\text{ }}$surfaces are probably highly defective\nand measured surface energies may be therefore lower.\n\n\\subsection{Electronic structure}\n\nFig.\\thinspace \\ref{fig_band1} shows the calculated band structures the band\nstructure of the unrelaxed asymmetrically terminated Ba-Ti slab, the slab\nwith surface layers relaxed, and for bulk BaTiO$_3$ folded into the tripled\nBrillouin zone of the slab. The relaxation makes small changes in the band\nstructure, but they are difficult to see on the scale shown. The top valence\nband and bottom conduction band of the slab are interface states. The nature\nof these states is clearer from Fig.\\thinspace \\ref{fig_band2} where the\nband structures for the unrelaxed symmetrically terminated BaO and TiO$_2$\nslabs are shown. The occupied interface state is due to the TiO$_2$\nsurfaces, and the interface state for the conduction band is primarily due\nto the BaO surface.\n\nThe density of states was computed using the tetrahedron method with Fourier\ninterpolation \\cite{25} from a k-point mesh of 8$\\times $8$\\times $2 to a 16$%\n\\times $16$\\times $4 mesh in the full Brillouin zone. Fig.\\thinspace \\ref\n{fig_BaO_dos} shows the density of states for the ferroelectric BaO\nsymmetrically terminated slab. There are Ba p -O p hybrid occupied surface\nstates, and an unoccupied O p band also comes down about 0.5 eV lower than\nthe bulk band. The Ba p states look quite different on the surface than in\nthe bulk, and in fact appear to be significantly less hybridized than the\nbulk Ba, indicated than Ba is even more of a perfect 2+ ion on the surface\nthan in the bulk. The main valence band states on the surface O are\nsignificantly less bound than the bulk O, with more weight at the top of the\nvalence band. Fig.\\thinspace \\ref{fig_titi_dos} shows that the occupied\nsurface states on TiO$_2$ surfaces have primarily O p character, with some\nhybridization, interestingly, with the neighboring Ba in the next layer.\nFig.\\thinspace \\ref{fig_ferrodos} shows the density of states for the\nunrelaxed and relaxed ferroelectric BaO symmetrically terminated slab. The\nferroelectric displacement sets up a potential gradient across the slab (a\nnet macroscopic field) that shifts the bands as a function of position in\nthe slab. In fact, the experimental ferroelectric displacement generates a\npotential gradient large enough to make the slab metallic even for the thin\nslabs studied here. The Ti d-states at the bottom of the slab become\npartially occupied and the O p and Ba p states become partially empty at the\nother end of the slab. Most interestingly, the bands do not move rigidly. It\nappears that the O 2p states really do not want to be partially occupied,\nand these states ``pile up'' against the Fermi level. The fact that this is\nenergetically preferred rather than allowing the oxygen to become partially\ncharged and highly nonspherical indicates how strongly the O atoms are\nstabilized in the closed shell configuration, even on the surface of a slab.\nWhat is so surprising about this is that O$^{2-}$ is not even stable in the\nfree state; it is only stabilized due to the Madelung (electrostatic) field\nfrom the rest of the crystal. This is not an theoretical artifact; free O$%\n^{2-}$ is unstable and decays to O$^{-}$ $+$ e$^{-}$. It appears that this\nstabilization is strong enough even on the BaO perovskite surface. The\ncharge transfer across the slab screens partially screens the field, so that\nthe gap is just closed, as is discussed further below.\n\nIn band theory as applied here, there is a constant Fermi level and states\nare occupied up to it. In an ideal insulating slab or crystal in a field,\nhowever, the situation would be different and the Fermi level would vary\nwith macroscopic position and charge would not flow since states would not\ncommunicate over macroscopic distances except by tunneling which is very\nslow. This is similar to a hydrogen atom in an electric field; the ground\nstate for an infinitesimal field is for the electron to be stripped off, but\nthis does not happen even over long times due to a long lifetime for the\nmetastable state. In a real slab or crystal, however, there are extrinsic or\nthermally induced defects so the conductivity is finite. Charge flow thus\nwould occur and the results would be similar to what we obtain here in the\nstatic limit.\n\n\\subsection{Analysis of potential and charge densities}\n\nFigure \\ref{fig_den}a shows the charge density of the asymmetrically\nterminated ideal slab with overlapping spherical ions subtracted. On the TiO$%\n_2$ surface the O p charge density, instead of dangling, bonds back to the\nsurface Ti's. In other words, the surface charge density is self-healing.\nThe bonds between the O and Ti are clearly evident. Most interesting is the\ncollapse of the surface Ti bond. Instead of dangling the charge moves back\nonto the Ti and the Ti-O surface bonds. This self-healing leaves the surface\nhighly reactive but stable (i.e. it doesn't reconstruct), and is probably\nresponsible for the utility of BaTiO$_3$ as a substrate for epitaxial growth\nand for surface catalysis. The Ba-O surface shows much less difference from\nspherical ions, and is found to be highly ionic. These results suggest that\nthe Ti-O surface of BaTiO$_3$ is highly reactive due to the possibility of\ncovalent bonding on the surface, whereas any reactivity Ba-O surface is due\nentirely to ionic bonding. The non-bonding O p surface state may also\nenhance surface reactivity.\n\nFigure \\ref{fig_den}b shows the difference between the LAPW (001) slab\ndensity and the self-consistent LAPW charge density for bulk periodic BaTiO$%\n_3$. Large changes are seen on the surface Ti, and smaller changes on the\nsurface Ba and O. Most interestingly, the differences the interior Ti and O,\nwhich are only one unit cell away from the surface, are almost identical to\nbulk Ti and O. This shows that electronic perturbations due the surface are\nscreened very rapidly in the interior of a crystal. The screening is\naccomplished primarily due to polarization and charge redistributions around\nthe surface Ti, and to a lesser extent, O ions.\n\nIn order to understand the polarization and depolarization fields of the\nslab, we now examine the total potentials averaged in the x-y plane as a\nfunction of z-coordinate in the slabs. Figure \\ref{fig_BaO_potential} shows\nthe potential for the symmetric BaO terminated ideal and ferroelectric\nslabs. There is no net field in the vacuum region in the ideal case, and the\nfield is a constant 0.015 Ryd\/bohr in the vacuum in the ferroelectric case.\nThe results for the asymmetrically terminated slabs are much more\ncomplicated, since a field is present in the vacuum even in the ideal case\n(Fig.\\thinspace \\ref{fig_asym_field}). The existence of the field in this\ncase can be thought of as due to different work functions for the BaO and TiO%\n$_2$ surfaces, but in any case it indicates that the slab has a net dipole\nmoment in spite of the fact that if one considers the nominal charges (2+,\n4+ and 2- for Ba, Ti and O respectively), the TiO$_2$ and BaO planes would\nbe charge balanced and the ideal slab would have no net polarization. One\npossible explanation of the result is that a surface charge develops due to\nthe surface states. This possibility was tested by comparing the charge\ndensities of the asymmetrically terminated slab with the symmetrically\nterminated BaO and TiO$_{2\\text{ }}$slabs, but no evidence for a surface\ncharge could be found. The implication is that even in the bulk the BaO and\nTiO$_2$ planes do not carry the same charge.\n\nThe above analysis allows the field to be easily extracted from the vacuum\nregion, but does not easily allow extraction of the macroscopic field in the\nslab since the potential is dominated by local contributions from the atoms.\nWe can extract macroscopic field effects in the slabs by examining deep core\nstates. In figure \\ref{fig_BaO_core} the O 1s core levels are shown as a\nfunction of position in the symmetrically terminated BaO slab. As expected\nthere is no net field in the ideal slab, but there is a nearly constant\nmacroscopic field as determined from the O 1s levels in the ferroelectric\nslab. The picture in the asymmetrically terminated slab is again complicated\n(Fig.\\thinspace \\ref{fig_asym_core}). The net field as seen in the center of\nthe asymmetrically terminated ideal slab is very small, in spite of the\nlarge field in the vacuum. The field for the ferroelectric relative to the\nideal asymmetric slab is nearly constant, and is about 0.01 Ryd\/bohr.\n\nWe can understand better the observed macroscopic fields by considering the\nideal case of a periodic slab of thickness $L_1$ separated by vacuum regions\nof thickness $L_2$ with constant fields $E_1$ and $E_2$ in the two regions.\nThe electric displacements are given by $D_1=E_1+4\\pi P_1$ in the slab and $%\nD_2=E_2$ in the vacuum, where $P_1$ is the total polarization in the slab,\nincluding the spontaneous polarization $P_s$ and the induced polarization $%\nP_i=(\\varepsilon _1-1)E_1\/4\\pi $. Now $D_1=D_2$ and $E_1L_1+E_2L_2=0$ since\nthe potential must be continuous. For the BaO terminated slab we get $%\nE_1=0.005$ H\/bohr and $E_2=-0.0073$ H\/bohr which gives $L_1\/L_2=1.45$, which\ncompares well with the simple consideration of 7 slab layers and 5 vacuum\nlayers, or $L_1\/L_2=1.4$, indicating that our method of estimating $E_1$ and \n$E_2$ from the vacuum potential and the O 1s core levels is consistent.\nUsing the experimental dielectric constant $\\varepsilon =5.24$ we get $P_s=$%\n15 $\\mu $C\/cm$^2$, which is significantly lower than the experimental value\nof 26 $\\mu $C\/cm$^2$. The screening caused by the metallic band structure is\nmost likely the major cause of this difference. This has important\nimplications, and shows that with finite conductivity, a ferroelectric will\nself-screen the depolarization field, because of the effective closing of\nthe gap from one part of a ferroelectric domain to the other. Many of the\nimplications remain to be worked out.\n\nFinally, we consider the magnitude of the depolarization energy, $W_D=\\frac 1%\n2\\int D\\cdot E$ d$V$ . For the ferroelectric symmetrically terminated BaO\nslab we find $W_D=0.055$ Ryd ($=$ 0.75 eV), a large energy which easily\novercomes the energy lowering for the ferroelectric distortion in bulk BaTiO$%\n_3$ of 0.015 eV per unit cell, or 0.053 eV for the equivalent number of\natoms as the slab contains in a periodic unit by a factor of about 14. Note\nthat both numbers scale up with size, so even macroscopic crystals must\nbreak up into domains, or the depolarization field must be pacified by\nformation of a surface space charge.\n\n\\section{Conclusions}\n\nWe have studied ferroelectric and ideal slabs of BaTiO$_3$ using the\nfull-potential all-electron LAPW method. Surface relaxations are\nsignificant, and similar to what is observed for SrTiO$_3$. There are\nsignificant surface states, especially on the TiO$_2$ surfaces. The dominant\nsurface state has O p character, and seems to be related to the collapse of\nthe dangling bonds back onto the surface Ti ions. The charge density in the\ncenter of the slab is very close to the charge density in bulk BaTiO$_3$. We\nfind the asymmetrically terminated slabs have significant potential\ngradients (electric fields) in the vacuum due to the different net charges\non BaO and TiO$_2$ planes. Most interestingly, we find that the\nferroelectric slabs are metallic due to potential shifts that are greater\nthan the band gap. The bands, however, do not shift in a rigid way; rather\nthe O 2p states pile up at the Fermi level showing that the O atoms prefer\nto remain closed-shelled, even on the surface. The metallic nature of the\nferroelectric slabs screens the macroscopic polarization, so we find a\npolarization smaller than the experimental bulk value. The depolarization\nfield is large and inhibits ferroelectricity. In order to recover a\nferroelectric state, this field must be screened or the crystal must break\nup into domains.\n\n\\acknowledgments{Thanks to V. Heine, I. Mazin, R. Resta, and D. Vanderbilt\nfor helpful discussions and suggestions on how to interpret the slab\nresults. D. Vanderbilt suggested the study of symmetrically terminated slabs\nwhich was very useful. This research is supported by the Office of Naval\nResearch. Computations were performed on the Cray J90 at the Geophysical\nLaboratory.. }\n\n\\begin{table}[tbp]\n\\caption{Structures of (001) BaTiO$_3$ slabs (z coordinates shown only, in\nlattice coordinates, $c = $24.035 \\AA). Only relaxed coordinates are shown\nfor partially relaxed structures.}\n\\begin{tabular}{lllllllllll}\n&  & ideal & relaxed & ferro - & ferro + & relaxed & ideal & ferro & relaxed\n& ideal \\\\ \natom x,y & termination & Ba-Ti & Ba-Ti & Ba-Ti & Ba-Ti & Ba-Ti & Ba-Ba & \nBa-Ba & Ba-Ba & Ti-Ti \\\\ \nBa 0,0 &  & 0 & 0.00715 & 0.0025 & -0.0025 & 0.00596 & 0 & -0.0025 & 0.00762\n& 0 \\\\ \nBa 0,0 &  & 0.16667 &  & 0.16917 & 0.16417 &  & 0.16667 & 0.16417 &  & \n0.16667 \\\\ \nBa 0,0 &  & 0.33333 &  & 0.33583 & 0.33083 &  & 0.33333 & 0.33083 &  & \n0.33333 \\\\ \nBa 0,0 &  &  &  &  &  &  & 0.5 & 0.50128 & 0.49110 &  \\\\ \nTi 0.5,0.5 &  &  &  &  &  &  &  &  &  & -0.08333 \\\\ \nTi 0.5,0.5 &  & 0.08333 &  & 0.088 & 0.07867 &  & 0.08333 & 0.07867 &  & \n0.08333 \\\\ \nTi 0.5,0.5 &  & 0.25 &  & 0.25467 & 0.24533 &  & 0.25 & 0.24533 &  & 0.25 \\\\ \nTi 0.5,0.5 &  & 0.41667 & 0.40868 & 0.42133 & 0.412 & 0.40910 & 0.41667 & \n0.412 &  & 0.41667 \\\\ \nO 0,0 &  &  &  &  &  &  & 0 & 0.00128 &  & -0.08333 \\\\ \nO 0.5,0.5 &  & 0 & 0.00544 & -0.00128 & 0.00128 & 0.00241 & 0.08333 & 0.08333\n& 0.00276 & 0 \\\\ \nO 0,0 &  & 0.08333 &  & 0.08333 & 0.08333 &  & 0.1667 & 0.16795 &  & 0.08333\n\\\\ \nO 0.5,0.5 &  & 0.16667 &  & 0.1653 & 0.16795 &  & 0.25 & 0.25 &  & 0.16667\n\\\\ \nO 0,0 &  & 0.25 &  & 0.25 & 0.25 &  & 0.33333 & 0.33462 &  & 0.25 \\\\ \nO 0.5,0.5 &  & 0.33333 &  & 0.332 & 0.33462 &  & 0.41667 & 0.41667 &  & \n0.33333 \\\\ \nO 0,0 &  & 0.41667 & 0.41219 & 0.41667 & 0.41667 & 0.41312 & 0.5 & 0.50128 & \n0.49199 & 0.41667\n\\end{tabular}\n\\end{table}\n\\begin{table}[tbp]\n\\caption{Summary of total energies for BaTiO$_3$ slabs and bulk.}\n\\begin{tabular}{lllll}\n& Termination & E Ryd\/cell &  &  \\\\ \nsymmetric bulk (3 (001) layers) \\tablenotemark[1] &  & $-55258.3586$ &  & \n\\\\ \nsymmetric bulk (2 (111) layers)\\tablenotemark[1] &  & $-55258.3578$ &  &  \\\\ \nideal (001) slab \\tablenotemark[1] & Ba-Ti & $-55258.1189$ &  &  \\\\ \nideal (111) slab\\tablenotemark[1] & Ba-Ti & $-55257.4260$ &  &  \\\\ \nferroelectric slab ($-$) \\tablenotemark[1] & Ba-Ti & $-55258.0482$ &  &  \\\\ \nferroelectric slab ($+$) \\tablenotemark[1] & Ba-Ti & $-55258.1119$ &  &  \\\\ \nsymmetric bulk (3 (001) layers)\\tablenotemark[2] &  & $-55258.4348$ &  &  \\\\ \nideal (001) slab \\tablenotemark[2] & Ba-Ti & $-55258.2119$ &  &  \\\\ \n(001) slab relaxed \\tablenotemark[2] & Ba-Ti & $-55258.2434$ &  &  \\\\ \nferroelectric slab ($+$) \\tablenotemark[2] & Ba-Ti & $-55258.2063$ &  &  \\\\ \nferroelectric slab ($+$) relaxed \\tablenotemark[2] & Ba-Ti & $-55258.2222$ & \n&  \\\\ \nideal (001) slab \\tablenotemark[2] & Ba-Ba & $-71674.0910$ &  &  \\\\ \nferroelectric (001) slab \\tablenotemark[2] & Ba-Ba & $-71674.0495$ &  &  \\\\ \nferroelectric (001) slab relaxed \\tablenotemark[2] & Ba-Ba & $-71674.0758$ & \n&  \\\\ \nideal (001) slab \\tablenotemark[2] & Ti-Ti & $-57261.8084$ &  & \n\\end{tabular}\n\\tablenotetext[1]{Muffin tin radii: Ba=2.3 bohr, Ti=1.75 bohr,O=1.75} %\n\\tablenotetext[2]{Muffin tin radii: Ba=2.3 bohr, Ti=1.59938 bohr,O=1.59938}\n\\end{table}\n\n\\begin{table}[tbp]\n\\caption{Summary of relaxations of surface layers in \\AA.}\n\\begin{tabular}{llllll}\n&  & Ba-Ti ideal & Ba-Ti ferro + & Ba-Ba ferro & SrTiO$_3$\\tablenotemark[2]\n\\\\ \n& relaxation of Ba-O layer & 0.15 & 0.12 & 0.14 & -0.08 \\\\ \n& change in dimpling of Ba-O layer & 0.02 & 0.088 \\tablenotemark[1] & 0.10 %\n\\tablenotemark[1] & 0.08 \\\\ \n& relaxation of Ti-O layer & 0.15 & 0.078 & 0.23 & -0.05 \\\\ \n& change in dimpling of Ti-O layer & 0.08 & 0.0078 & 0.011 & 0.05\n\\end{tabular}\n\\tablenotetext[1]{Sign of dimpling changes relative to ferroelectric\ndistortion.} \\tablenotetext[2]{Ref.~\\cite{10}}\n\\end{table}\n\n\n\\begin{figure}[tbp]\n\\caption{Structures of slabs studied. (a) (111) slab (b) (001) slab. Blue is\noxygen, green is Ba, and red is Ti. Ti-O bonds are shown in black.}\n\\label{fig_structures}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\caption{Band structures for (a) undistorted bulk BaTiO$_3$ folded into the\nBrillouin zone for the three layer slab, and (b) for the assymetrically\nterminated slab with BaO and TiO$_{2\\text{ }}$surfaces. The bands for the\nunrelaxed ideal slab are shown as solid lines, and for the slab with relaxed\nsurfaces are dashed.}\n\\label{fig_band1}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\caption{Band structures for symmetric (a) BaO and (b) TiO$_2$ terminated\nideal slabs. The highest valence band in (b) is a surface state with O 2p\ncharacter, and the lowest conduction band in (a) is also a surface state\nwith primarily O 2p character.}\n\\label{fig_band2}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\caption{Partial density of states for the symmetric ideal BaO terminated\nslab. Note the different scales for Ba p and Ti d. The Fermi level is shown\nas the vertical dashed line. The amount of Ti hybridization with O 2p is\nabout constant through the slab, but the Ba p hybridization is lower on the\nsurface than in the bulk. The O 2p bands are correspondingly wider in the\nbulk than on the surface. The surface is more ionic than the bulk.}\n\\label{fig_BaO_dos}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\caption{Partial density of states for the symmetric ideal TiO$_2$\nterminated slab. There is a significant surface state with O 2p character as\nwas shown in fig.\\thinspace \\ref{fig_band2}. The surface seems slightly more\nionic than the interior of the slab, and the Ti d states are slightly less\nbound.}\n\\label{fig_titi_dos}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\caption{Partial density of states for the ferroelectric symmetrically BaO\nterminated slab. The Ti displacements were towards the bottom of the slab\n(corresponding to the bottom panels). The solid line is for the ideal\nferrolectric slab with the experimental tetragonal displacement pattern, and\nthe dashed line is for the slab with relaxed surface atoms. The slab is\nmetallic, with conduction bands crossing the Fermi level at the bottom of\nthe slab and O 2p states and Ba p states crossing the Fermi level at the top\nof the slab. Note that the bands are displaced due to the ramp potential\nacross the slab, but the band shifts are not rigid. Rather the O 2p density\npiles up at the Fermi level at the top of the slab, indicating that the O\nion wants to remain closed shell. Only a small amount of charge transfer\noccurs in order to screen the macroscopic field to the point that the gap is\njust closed.}\n\\label{fig_ferrodos}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\caption{Charge density for BaTiO$_3$ slab. (a) Deformation charge density\nfor BaTiO$_3$. The image shows the difference in charge density between the\nself-consistent LAPW charge density and overlapping PIB ions. Green\nrepresents no change in density (i.e. identical to the ionic crystal).\nPartially charged ions, Ba$^{1.63+}$, Ti$^{3.26+}$, O$^{1.63-}$ were used to\ngenerate the ionic charge density. Note the ``healing'' of the surface Ti\ndangling bond. It collapses back onto the Ti. (b) Difference in charge\ndensity between the ideal slab and bulk BaTiO$_3$. Even though the slab is\nonly three layers thick, the central layer has almost an identical charge\ndensity to bulk BaTiO$_3$, although there are still some small differences\non the Ba.}\n\\label{fig_den}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\caption{Total potential averaged in the x-y plane as a function of z for\nthe symmetric ideal and ferroelectric BaO terminated slabs. The average\nfield is zero is zero in the vacuum between slabs in the ideal case, and is\n0.39 eV\/\\AA\\ in the ferroelectric case. The experimental tetragonal\ndisplacement pattern is used.}\n\\label{fig_BaO_potential}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\caption{Electric field shown only in the vacuum region for the\nasymmetrically terminated slabs. Abscissa is for the whole periodic slab\nand vacuum. The field is not shown in the slab interior since it\noscillates widely on this scale.}\n\\label{fig_asym_field}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\caption{Core levels for symmetric BaO terminated slabs. (a) O 1s core\nenergies as a function of layer (z coordinate) in slabs for ideal and\nferroelectric slab. (b) Macroscopic electric field derived from changes in\ncore levels.}\n\\label{fig_BaO_core}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\caption{Core levels for asymmetrically terminated slabs. (a) O 1s core\nenergies as a function of layer (z coordinate) in slabs for ideal and\nferroelectric slabs, relaxed and unrelaxed. (b) Macroscopic electric field\nderived from changes in core levels.}\n\\label{fig_asym_core}\n\\end{figure}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}