diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzmwlb" "b/data_all_eng_slimpj/shuffled/split2/finalzzmwlb" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzmwlb" @@ -0,0 +1,5 @@ +{"text":"\\section{History}\n\\noindent\nThe \\emph{Julia set} $J(f)$ of $f$ is the closure of the set of repelling periodic points. It is also the smallest closed set containing at least three points which is completely invariant under $f^{-1}$. For the example $f(z)=z^2$, the Julia set of $f$ is the unit circle. The complement $F(f)=\\widehat\\C\\setminus J(f)$ of the Julia set, called the \\emph{Fatou set}, is the largest open set such that the iterates of $f$ restricted to it form a normal family. The Julia set and Fatou set are both invariant under $f$ and $f^{-1}$.\n\nThe \\emph{postcritical set} $\\post(f)$ of $f$ is the closure of the forward orbits of the critical points\n\\begin{eqnarray*}\n\\post(f)=\\overline{\\bigcup_{n\\geq 1}\\{f^n(c)\\: c\\in {\\rm crit}(f)\\}}.\n\\end{eqnarray*}\nThe postcritical set plays a crucial role in terms of understanding the expanding and contracting features of a rational map.\nIf the postcritical set $\\post(f)$ is finite, we say that the map $f$ is \\emph{postcritically finite}. In the postcritically finite case,\n\\begin{eqnarray*}\n\\post(f)=\\bigcup_{n\\geq 1}\\{f^n(c)\\: c\\in {\\rm crit}(f)\\}.\n\\end{eqnarray*}\n\nIn 1918, Samuel Latt\\`es described a special class of rational maps which have a simultaneous linearization for all of their periodic points (see \\cite{LatSur}). This class of maps is named after Latt\\`es, even though similar examples had been studied by Ernst Schr\\\"oder much earlier (see \\cite{SchUeber}). A \\emph{Latt\\`es map} $f\\:\\widehat\\C\\ra \\widehat\\C$ is a rational map that is obtained from a finite quotient of a conformal torus endomorphism, i.e., the map $f$ satisfies the following commutative diagram:\n\\begin{equation}\\label{lat}\n \\begin{CD}\n\\T @>\\bar{A}>> \\T\\\\\n@V\\Theta VV @VV\\Theta V\\\\\n\\widehat\\C @>f>> \\widehat\\C\n\\end{CD}\n\\end{equation}\nwhere $\\bar A$ is a map of a torus $\\T$ that is a quotient of an affine map of the complex plane, and $\\Theta$ is a finite-to-one holomorphic map. Latt\\`es maps were the first examples of rational maps whose Julia set is the whole sphere $\\widehat \\C$, and the postcritical set of a Latt\\`es map is finite. More importantly, Latt\\`es maps play a central role as exceptional examples in complex dynamics. We will discuss this further in the following section.\n}\n\n\\section{Introduction}\n\\noindent\nA \\emph{rational map} $f\\:\\widehat\\C \\ra \\widehat\\C$ is a map on the Riemann sphere $\\widehat\\C=\\C \\cup \\{\\infty\\}$ which can be written as a quotient of two relatively prime complex polynomials $p(z)$ and $q(z)$, with $q(z)\\not=0$,\n\\begin{eqnarray}\\label{rationalmap}\nf(z)=\\frac{p(z)}{q(z)}=\n\\frac{a_0z^{m}+\\ldots+a_{m}}{b_0z^l+\\ldots+b_l},\n\\end{eqnarray}\nwhere $a_i,b_j \\in \\C$ for $i=0,\\ldots,m$ and $j=0,\\ldots, l$.\nThe \\emph{postcritical set} $\\post(f)$ of $f$ is defined to be the forward orbits of the critical points\n\\begin{eqnarray*}\n\\post(f)=\\bigcup_{n\\geq 1}\\{f^n(c)\\: c\\in {\\rm crit}(f)\\}.\n\\end{eqnarray*}\nIf the postcritical set $\\post(f)$ is finite, we say that the map $f$ is \\emph{postcritically finite}.\n\nThurston introduced a topological analog of a postcritically finite rational map, now known as a \\emph{Thurston map} (see \\cite{DHThurston}). A \\emph{Thurston map} $f\\:\\S^2\\ra \\S^2$ is a branched covering map with finite postcritical set $\\post(f)$.\nThe notion of an expanding Thurston map was introduced in \\cite{BMExpanding} as a topological analog of a postcritically finite rational map whose Julia set is the whole sphere $\\widehat{\\C}$. Roughly speaking, a Thurston map is called \\emph{expanding} if all the connected components of the preimage under $f^{-n}$ of any open Jordan region disjoint from $\\post(f)$ become uniformly small as $n$ tends to infinity. We refer the reader to Definition~\\ref{expandingmap} for a more precise statement. A related and more general notion of expanding Thurston maps was introduced in \\cite{HPCoarse}. Latt\\`es maps are among the simplest examples of expanding Thurston maps.\n\n\nLet $f$ be an expanding Thurston map, and let $\\mathcal C$ be a Jordan curve containing $\\post(f)$.\nThe Jordan Curve Theorem implies that $\\S^2\\setminus\\mathcal C$ has precisely two connected components, whose closures we call \\emph{$0$-tiles}. We call the closure of each connected component of the preimage of $\\S^2\\setminus\\mathcal C$ under $f^n$ an \\emph{$n$-tile}. In Section 5 of \\cite{BMExpanding}, it is proved that the collection of all $n$-tiles gives a cell decomposition of $\\S^2$.\n\nEvery expanding Thurston map $f\\:\\S^2\\ra \\S^2$ induces a natural class of metrics on $\\S^2$, called \\emph{visual metrics} (see Definition \\ref{visual}), and each visual metric $d$ has an associated \\emph{expansion factor} $\\Lambda > 1$. This visual metric is essentially characterized by the geometric property that the diameter of an $n$-tile is about $\\Lambda^{-n}$, and the distance between two disjoint $n$-tiles is at least about $\\Lambda^{-n}$. The supremum of the expansion factors of all visual metrics is called the \\emph{combinatorial expansion factor} $\\Lambda_0$ (see \\cite[Theorem 1.5]{BMExpanding}). For Latt\\`es maps, the supremum is obtained. In general, the supremum is not obtained.\n\n\\excise{\n--------------------------\nThe points in $\\post(f)$ divide $\\mathcal C$ into several subarcs. Let $D_n=D_n(f,\\mathcal{C})$ be the minimum number of $n$-tiles needed to join two of these subarcs that are non-adjacent (see Definition \\ref{joinoppositesides} and \\eqref{defdn}). Even though $D_n=D_n(f,\\mathcal C)$ depends on the Jordan curve $\\mathcal C$, its growth rate is independent of the $\\mathcal C$. So the limit\n\\begin{equation}\\label{comb}\n \\Lambda_0(f)=\\lim_{n\\ra \\infty}\\big(D_n(f,\\mathcal C)\\big)^{1\/n}\n\\end{equation}\nexists and only depends on the map $f$ itself (see \\cite[Prop.~17.1]{BMExpanding}). We call this limit $\\Lambda_0(f)$ the \\emph{combinatorial expansion factor} of $f$. This quantity $\\Lambda_0(f)$ is invariant under topological conjugacy and multiplicative in the sense that $\\Lambda_0(f)^n$ is the combinatorial expansion factor of $f^n$.\n\nThe combinatorial expansion factor is closely related to the notion of \\emph{visual metrics and their expansion factors}. Every expanding Thurston map $f\\:\\S^2\\ra \\S^2$ induces a natural class of metrics on $\\S^2$, called \\emph{visual metrics} (see Definition \\ref{visual}), and each visual metric $d$ has an associated \\emph{expansion factor} $\\Lambda > 1$. This visual metric is essentially characterized by the geometric property that the diameter of an $n$-tile is about $\\Lambda^{-n}$, and the distance between two disjoint $n$-tiles is at least about $\\Lambda^{-n}$. The supremum of the expansion factors of all visual metrics is equal to the combinatorial expansion factor $\\Lambda_0$ (see \\cite[Theorem 1.5]{BMExpanding}). For Latt\\`es maps, the supremum is obtained. In general, the supremum is not obtained.\n----------------------------\n}\n\\bigskip\n\nA geodesic metric space $(X,d)$ is called a \\emph{Gromov hyperbolic} space if every geodesic triangle in it is ``very thin''. It can also defined in terms of \\emph{Gromov products}.\nFor any points $x,y,p\\in X$, the \\emph{Gromov product} $(x,y)_p$ of $x$ and $y$ with respect to the base point $p$ is defined as\n\\begin{eqnarray} \\label{gproduct}\n (x,y)_p \\:= \\frac12 \\left[d(x,p)+d(y,p)-d(x,y) \\right].\n\\end{eqnarray}\nThe space $X$ is called \\emph{$\\delta$-hyperbolic} (or \\emph{Gromov hyperbolic}) for some $\\delta\\geq 0$ if there exists a base point $p\\in X$ such that for all $x,y,z\\in X$, we have\n\\begin{eqnarray} \\label{trianglein}\n (x,y)_p\\geq \\min\\{(x,z)_p,(z,y)_p\\}-\\delta.\n\\end{eqnarray}\n\nWe construct a graph $\\G=\\G(f,\\mathcal C)$ by letting the tiles in the cell decompositions of $(f, \\mathcal C)$ be vertices of $\\G$.\nThere is an edge between the two vertices $X^n,Y^m\\in V$, denoted\n$X^n\\sim Y^m$ if as underlying tiles\n\\[|n-m|\\leq 1 \\mbox{ and } X^n\\cap Y^m\\not= \\emptyset.\\]\nIt turns out that the graph $\\G$ with the path metric is a Gromov hyperbolic space (see Theorem \\ref{gh}).\n\\begin{thm}\nLet $f\\: \\S^2 \\ra \\S^2$ be an expanding Thurston map\nand let $\\mathcal C \\subset \\S^2$ be a Jordan curve containing $\\post(f)$. Then the graph $\\G(f,\\mathcal C)$ equipped with the path metric $\\eta$ is a Gromov hyperbolic space.\n\\end{thm}\n\n\n\nThere is a natural boundary at infinity of a Gromov hyperbolic space. Roughly speaking, the \\emph{boundary at infinity} is the set of equivalence classes of geodesic rays in the Gromov hyperbolic space. It can also be equipped with a \\emph{Gromov product} by taking infimum of the infimum limit of the Gromov product along all the geodesic rays among the corresponding equivalence classes.\n A \\emph{visual metric} $\\rho$ on the boundary at infinity of a Gromov hyperbolic space is a metric that has a bounded ratio\n\\[ \\rho(\\xi,\\xi')\/ \\Lambda^{-(\\xi,\\xi')_p}\\]\nfor some fixed $\\Lambda>1$ and for all points $\\xi$ and $\\xi'$ on the boundary.\n\n\nIn Proposition \\ref{samevisual}, we show the following:\n\\begin{pro}\nFor an expanding Thurston map $f$ and a Jordan curve $\\mathcal C\\subset \\S^2$ containing $\\post (f)$, the boundary at infinity $\\partial_{\\infty}\\G$ of the graph tile $\\G(f,\\mathcal C)$ can be identified with $\\S^2$.\nUnder this identification, a metric $d$ is a visual metric on $\\S^2$ with respect\nto the expanding Thurston map $f$ if and only if $d$ is a visual metric on $\\partial_{\\infty}\\G$ (in the sense of Gromov hyperbolic spaces).\n\\end{pro}\n\nWe deduce that\nfor any Jordan curves $\\mathcal C$ and $\\mathcal C'$ containing $\\post (f)$, the classes of visual metrics on $\\partial_{\\infty}\\G(f,\\mathcal C)$ and $\\partial_{\\infty}\\G(f,\\mathcal C')$ can also be identified (see Corollary \\ref{corsamevisual}). A similar graph to $\\G(f,\\mathcal C)$ has also been studied by Kevin Pilgrim in \\cite{PilJulia}, from a somewhat different point of view. Our results overlap in some special cases. They consider the map $f$ being $C^1$ and $\\S^2\\setminus \\post{f}$ equipped with a special Riemannian metric, and prove that the Julia set of $f$ can be identified as the Gromov boundary of a certain Gromov hyperbolic one-complex.\n\n\nIn \\cite{BFAsymptotic}, the \\emph{asymptotic upper curvature} of a Gromov hyperbolic space is introduced. It is the analog of sectional curvature on Riemannian manifolds. Fix $\\kappa\\in [-\\infty,0)$. We call a metric space $X$ an \\emph{AC$_u(\\kappa)$-space} if there exists $p\\in X$ and a constant $c\\geq 0$ such that for all $x,x'\\in X$ and all finite sequences $x_0=x,x_1,\\ldots,x_n=x'$ in $X$,\n\\begin{eqnarray}\\label{acspaceeq}\n(x,x')_p\\geq \\min_{i=1,\\ldots,n}(x_{i-1},x_i)_p-\\frac1{\\sqrt{-\\kappa}}\\log n-c.\n\\end{eqnarray}\nHere we use the convention $1\/\\sqrt{\\infty}=0$.\nWe call\n\\begin{eqnarray*}\nK_u(X):=\\inf\\{\\kappa\\ \\in [-\\infty,0): X \\mbox{ is an AC$_u(\\kappa)$-space} \\}\n\\end{eqnarray*}\nthe \\emph{asymptotic upper curvature} of $X$. It is invariant under rough-isometry.\n\nFor any Jordan curves $\\mathcal C$ and $\\mathcal C'$ containing $\\post (f)$, the metric spaces $\\G=\\G(f,\\mathcal C)$ and $\\G'=\\G(f,\\mathcal C')$ are rough-isometric (see Proposition~\\ref{roughgg}). Hence we may define the \\emph{asymptotic upper curvature} $K_u(f)$ of an expanding Thurston map $f$ as\n\\begin{eqnarray} \\label{asyf}\nK_u(f):= K_u(\\G(f,\\mathcal C)),\n\\end{eqnarray}\nwhere $\\mathcal C \\subseteq \\S^2$ is any Jordan curve containing $\\post(f)$. Using the notation above, we have the following theorem (see Theorem~\\ref{main2}).\n\n\\begin{thm} \\label{main3}\nLet $f \\: \\S^2 \\ra \\S^2$ be an expanding Thurston map. The asymptotic upper curvature of $f$ satisfies\n\\[K_u(f)\\geq-\\frac14\\log^2(\\deg f). \\]\nIf in addition, the map $f$ has no periodic critical points, then the tile graph $\\G=\\G(f)$ is an AC$_u(\\kappa)$-space with\n\\[ \\kappa= -\\frac14\\log^2(\\deg f),\\]\nif and only if the map $f$ is topologically conjugate to a Latt\\`es map.\n\\end{thm}\n\nRecall that in \\cite{HamEntropy}, the Hamenst{\\\"a}dt's entropy rigidity theorem establishes a connection between the curvature of a compact manifold $M$ and the topological entropy of the geodesic flow on the tangent space of $M$. Corollary 20.8 in \\cite{BMExpanding} shows that the topological entropy of an expanding Thurston map $f$ is $\\log(\\deg(f))$. Hence\nTheorem \\ref{main3} establishes a connection between the asymptotic upper curvature of an expanding Thurston map $f$ and the topological entropy of $f$, and provides a counterpart to Hamenst{\\\"a}dt's entropy rigidity theorem in the Sullivan dictionary.\n\n\\bigskip\n\\noindent\n\\textbf{Acknowledgements.} This paper is part of the author's PhD thesis under the supervision of Mario Bonk. The author would like to thank Mario Bonk for introducing her to and teaching her about the subject of Thurston maps and its related fields. The author is inspired by his enthusiasm and mathematical wisdom, and is especially grateful for his patience and encouragement.\nThe author would like to thank Dennis Sullivan for valuable conversations and sharing his mathematical insights.\nThe author also would like to thank Michael Zieve and Alan Stapledon for useful comments and feedback.\n\n\n\n\\section{Expanding Thurston maps and Cell Decompositions} \\label{expanding}\n\\noindent\nIn this section we review some definitions and facts on expanding Thurston maps. We refer the reader to Section 3 in \\cite{BMExpanding} for more details. We write $\\N$ for the set of positive integers, and $\\N_0$ for the set of non-negative integers. We denote the identity map on $\\S^2$ by ${\\rm id}_{\\S^2}$.\n\nLet $\\S^2$ be a topological 2-sphere with a fixed orientation. A continuous map $f\\:\\S^2\\ra \\S^2$ is called \\emph{a branched covering map} over $\\S^2$ if $f$ can be locally written as\n\\[z\\mapsto z^d\\]\nunder certain orientation-preserving coordinate changes of the domain and range. More precisely, we require that for any point $p\\in \\S^2$, there exists some integer $d>0$, an open neighborhood $U_p\\subseteq \\S^2$ of $p$, an open neighborhood $V_q\\subseteq \\S^2$ of $q=f(p)$, and orientation-preserving homeomorphism\n\\[\\phi\\: U_p\\ra U\\subseteq \\C\\]\nand\n\\[\\psi \\: V_p\\ra V\\subseteq \\C\\]\nwith $\\phi(p)=0$ and $\\psi(q)=0$ such that\n\\[(\\psi\\circ f \\circ \\phi^{-1} )(z)=z^d\\]\nfor all $z\\in U$. The positive integer $d=\\deg_f(p)$ is called the \\emph{local degree} of $f$ at $p$ and only depends on $f$ and $p$. A point $p\\in \\S^2$ is called a \\emph{critical point} of $f$ if $\\deg_f(p)\\geq 2$, and a point $q$ is called \\emph{critical value} of $f$ if there is a critical point in its preimage $f^{-1}(q)$. If $f$ is a branched covering map of $\\S^2$, $f$ is open and surjective. There are only finitely many critical points of $f$ and $f$ is \\emph{finite-to-one} due to the compactness of $\\S^2$. Hence, $f$ is a covering map away from the critical points in the domain and critical values in the range. The \\emph{degree $\\deg(f)$} of $f$ is the cardinality of the preimage over a non-critical value. In addition, we have\n\\[\\deg(f)=\\sum_{p\\in f^{-1}(q)}\\deg_f(p)\\]for every $q\\in \\S^2$.\n\nFor $n\\in \\N$, we denote the $n$-th iterate of $f$ as\n\\[f^n=\\underbrace{f\\circ f\\circ \\cdots \\circ f}_{\\textstyle{n} \\mbox{ factors}}.\\]\nWe also set $f^0={\\rm id}_{\\S^2}$.\n\nIf $f$ is a branched cover of $\\S^2$, so is $f^n$, and\n\\[\\deg(f^n)=\\deg(f)^n.\\]Let crit$(f)$ be the set of all the critical points of $f$. We define the set of \\emph{postcritical points} of $f$ as\n\\[\\post(f)=\\bigcup_{n\\in \\N}\\{f^n(c)\\: c\\in {\\rm crit}(f)\\}.\\]We call a map $f$ \\emph{postcritically-finite} if the cardinality of $\\post(f)$ is finite. Notice that $f$ is postcritically-finite if and only if there is some $n\\in \\N$ for which $f^n$ is postcritically-finite.\n\nLet $\\mathcal{C}\\subseteq \\S^2$ be a Jordan curve containing $\\post(f)$. We fix a metric $d$ on $\\S^2$ that induces the standard metric topology on $\\S^2$.\nDenote by \\emph{${\\rm mesh}(f,n,{\\mathcal C})$} the supremum of the diameters of all connected components of the set $f^{-n}(\\S^2\\setminus {\\mathcal C})$.\n\n\\begin{de} \\label{expandingmap}\nA branched covering map $f\\:\\S^2\\ra \\S^2$ is called a \\emph{Thurston map} if $\\deg(f)\\geq 2$ and $f$ is postcritically-finite. A Thurston map $f\\:\\S^2\\ra \\S^2$ is called \\emph{expanding} if there exists a Jordan curve $\\mathcal{C}\\subseteq \\S^2$ with $\\mathcal{C} \\supseteq \\post(f)$ and\n\\begin{equation} \\label{mesh}\n\\lim_{n\\ra \\infty}{\\rm mesh}(f,n,{\\mathcal C})=0.\n\\end{equation}\n\\end{de}\n\nThe relation \\eqref{mesh} is a topological property, as it is independent of the choice of the metric, as long as the metric induces the standard topology on $\\S^2$. Lemma 8.1 in \\cite{BMExpanding} shows that if the relation \\eqref{mesh} is satisfied for one Jordan curve ${\\mathcal C}$ containing $\\post(f)$, then it holds for every such curve. One can essentially show that a Thurston map is expanding if and only if all the connected components in the preimage under $f^{-n}$ of any open Jordan region not containing $\\post(f)$ become uniformly small as $n$ goes to infinity.\n\nThe following theorem (Theorem 1.2 in \\cite{BMExpanding}) says that there exists an invariant Jordan curve for some iterates of $f$.\n\\begin{thm} \\label{invariantJordancurve}\nIf $f \\: \\S^2\\ra \\S^2$ is an expanding Thurston map, then for some $n\\in \\N$ there exists a Jordan curve $\\mathcal{C}\\subseteq \\S^2$ containing $\\post(f)$ such that $\\mathcal C$ is invariant under $f^n$, i.e., $f^n({\\mathcal C})\\subseteq {\\mathcal C}$.\n\\end{thm}\n\nRecall that an \\emph{isotopy} $H$ between two homeomorphisms is a homotopy so that at each time $t\\in [0,1]$, the map $H_t$ is a homeomorphism. An \\emph{isotopy $H$ relative to a set $A$} is an isotopy satisfying\n\\[H_t(a)=H_0(a)=H_1(a)\\]\nfor all $a\\in A$ and $t\\in [0,1]$.\n\n\n\\begin{de}\nConsider two Thurston maps $f\\:\\S^2\\ra \\S^2$ and $g\\:\\S^2_1\\ra \\S^2_1$, where $\\S^2$ and $\\S^2_1$ are $2$-spheres. We call the maps $f$ and $g$ \\emph{(Thurston) equivalent} if there exist homeomorphisms $h_0,h_1\\:\\S^2\\ra \\S^2_1$ that are isotopic relative to $\\post(f)$ such that $h_0\\circ f=g\\circ h_1$.\nWe call the maps $f$ and $g$ \\emph{topologically conjugate} if there exists a homeomorphism $h\\:\\S^2\\ra \\S^2_1$ such that $h\\circ f=g\\circ h$.\n\\end{de}\nFor equivalent Thurston maps, we have the following commutative diagram\n\\[\\begin{CD}\n\\S^2 @>h_1>>\\S^2_1 \\\\\n@Vf VV @VVg V\\\\\n\\S^2 @>h_0>> \\S^2_1 .\n\\end{CD} \\]\n\n\n\nWe now consider the cardinality of the postcritical set of $f$. In Remark 5.5 in \\cite{BMExpanding}, it is proved that there are no Thurston maps with $\\#\\post(f)\\leq 1$. Proposition 6.2 in \\cite{BMExpanding} shows that all Thurston maps with $\\#\\post(f)=2$ are Thurston equivalent to a \\emph{power map} on the Riemann sphere,\n\\[z\\mapsto z^k, \\mbox{ for some }k\\in \\Z\\setminus\\{-1,0,1\\}.\\]\nCorollary 6.3 in \\cite{BMExpanding} states that if $f\\:\\S^2\\ra\\S^2$ is an expanding Thurston map, then $\\#\\post(f)\\geq 3$.\n\nLet $f\\:\\S^2\\ra\\S^2$ be a Thurston map, and let ${\\mathcal C}\\subseteq \\S^2$ be a Jordan curve containing $\\post(f)$. By the Sch\\\"onflies theorem, the set $\\S^2\\setminus {\\mathcal C}$ has two connected components, which are both homeomorphic to the open unit disk. Let $T_0$ and $T_0'$ denote the closures of these components. They are cells of dimension $2$, which we call \\emph{$0$-tiles}. The postcritical points of $f$ are called \\emph{$0$-vertices} of $T_0$ and $T_0'$, singletons of which are cells of dimension $0$. We call the closed arcs between vertices \\emph{$0$-edges} of $T_0$ and $T_0'$, which are cells of dimension $1$. These $0$-vertices, $0$-edges and $0$-tiles form a cell decomposition of $\\S^2$, denoted by $\\D^0=\\D^0(f,{\\mathcal C})$. We call the elements in $\\D^0$ $0$-cells. Let $\\D^1=\\D^1(f,{\\mathcal C})$ be the set of connected subsets $c\\subseteq \\S^2$ such that $f(c)$ is a cell in $\\D^0$ and $f|_c$ is a homeomorphism of $c$ onto $f(c)$. Call $c$ a $1$-tile if $f(c)$ is a $0$-tile, call $c$ a $1$-edge if $f(c)$ is a $0$-edge, and call $c$ a $1$-vertex if $f(c)$ is a $1$-vertex. Lemma 5.4 in \\cite{BMExpanding} states that $\\D^1$ is a cell decomposition of $\\S^2$. Continuing in this manner, let $\\D^n=\\D^n(f,{\\mathcal C})$ be the set of all connected subsets of $c\\subseteq \\S^2$ such that $f(c)$ is a cell in $\\D^{n-1}$ and $f|_c$ is a homeomorphism of $c$ onto $f(c)$, and call these connected subsets $n$-tiles, $n$-edges and $n$-vertices correspondingly, for $n\\in\\N_0$. By Lemma 5.4 in \\cite{BMExpanding}, $\\D^n$ is a cell decomposition of $\\S^2$, for each $n\\in \\N_0$, and we call the elements in $\\D^n$ $n$-cells. The following lemma lists some properties of these cell decompositions. For more details, we refer the reader to Proposition 6.1 in \\cite{BMExpanding}.\n\\begin{lem}\\label{tilenumber}\nLet $k,n\\in \\N_0$, let $f\\:\\S^2\\ra \\S^2$ be a Thurston map, let $\\mathcal C\\subset~ \\S^2$ be a Jordan curve with $\\mathcal C\\supseteq \\post(f)$, and let $m=\\#\\post(f)$.\n\\begin{enumerate}\n \\item\n If $\\tau$ is any $(n+k)$-cell, then $f^k(\\tau)$ is an $n$-cell, and $f^k|_{\\tau}$ is a homeomorphism of $\\tau$ onto $f^k(\\tau)$.\n \\item Let $\\sigma$ be an $n$-cell. Then $f^{-k}(\\sigma)$ is equal to the union of all $(n+k)$-cells $\\tau$ with $f^k(\\tau)=\\sigma$.\n \n \\item The number of $n$-vertices is less than or equal to $m\\deg(f)^n$, the number of $n$-edges is $m\\deg(f)^n$, and the number of $n$-tiles is $2\\deg (f)^n$.\n \\item The $n$-edges are precisely the closures of the connected components of\n $f^{-n}(\\mathcal C)\\setminus f^{-n}(\\post(f))$. The $n$-tiles are precisely the closures of the connected components of $\\S^2\\setminus f^{-n}(\\mathcal C)$.\n \\item Every $n$-tile is an $m$-gon, i.e., the number of $n$-edges and $n$-vertices contained in its boundary is equal to $m$.\n\\end{enumerate}\n\\end{lem}\n\n\\excise{\n------------------------\nLet $\\sigma$ be an $n$-cell. Let $W^n(\\sigma)$ be the union of the interiors of all $n$-cells intersecting with $\\sigma$, and call $W^n(\\sigma)$ the \\emph{$n$-flower} of $\\sigma$. In general, $W^n(\\sigma)$ is not necessarily simply connected. The following lemma (from Lemma 7.2 in \\cite{BMExpanding}) says that if $\\sigma$ consists of a single $n$-vertex, then $W^n(\\sigma)$ is simply connected.\n\\begin{lem} \\label{flower}\nLet $f\\: \\S^2\\ra \\S^2$ be a Thurston map, and let $\\mathcal C$ be a Jordan curve containing $\\post(f)$. If $\\sigma$ is an $n$-vertex, then $W^n(\\sigma)$ is simply connected. In addition, the closure of $W^n(\\sigma)$ is the union of all $n$-tiles containing the vertex $\\sigma$.\n\\end{lem}\nOne of the most important properties of $n$-flowers is that they build a connection between $n$-tiles of different Jordan curves due to the following lemma in \\cite[Lemma 7.12]{BMExpanding}.\n\\begin{lem}\\label{flowertile}\nLet $\\mathcal C$ and $\\mathcal C'$ be Jordan curves in $\\S^2$ both containing\n$\\post(f)$. Then there exists a number $M$ such that each $n$-tile for $(f, \\mathcal C)$\nis covered by $M$ $n$-flowers for $(f, \\mathcal C')$.\n\\end{lem}\n\\begin{re}\nThe exact same proof for this lemma shows that for $n'\\geq n$, there exists a number $M$ such that each $n'$-tile $(f, \\mathcal C)$ is covered by $M$ $n$-flowers for $(f, \\mathcal C')$.\n\\end{re}\n-------------------------\n}\n\nWe obtain a sequence of cell decompositions of $\\S^2$ from a Thurston map and a Jordan curve on $\\S^2$. It would be nice if the local degrees of the map $f$ at all the vertices were bounded, and this can be obtained by the assumption of no periodic critical points (see \\cite[Lemma 16.1]{BMExpanding}).\n\\begin{lem} \\label{noperiodic}\nLet $f : \\S^2\\ra \\S^2$ be a branched covering map. Then f has no\nperiodic critical points if and only if there exists $N\\in \\N$ such that\n\\[\\deg_{f^n}(p)\\leq N,\\]\nfor all $p\\in \\S^2$ and all $n\\in \\N$.\n\\end{lem}\nHenceforth we assume that \\emph{all Thurston maps have no periodic critical points}.\n\nLet $f\\:\\S^2\\ra \\S^2$ be an expanding Thurston map and let $\\mathcal{C}$ be a Jordan curve containing $\\post(f)$.\n\\begin{de}\\label{joinoppositesides}\nA set $K\\subseteq \\S^2$ \\emph{joins opposite sides} of $\\mathcal C$ if $\\#$$\\post(f)\\geq 4$\nand $K$ meets two disjoint $0$-edges, or if $\\#$$\\post(f) = 3$ and $K$ meets all\nthree $0$-edges.\n\\end{de}\n\nLet $D_n=D_n(f,\\mathcal C)$ be the minimum number of $n$-tiles needed to join opposite sides of a Jordan curve $\\mathcal{C}$. More precisely,\n\\begin{eqnarray}\\label{defdn}\nD_n =\\min\\{N\\in \\N: \\mbox{ there exist $n$-tiles } X_1, . . . ,X_N \\mbox{ such that } \\\\\n\\bigcup_{j=1}^N X_j\\mbox{ is connected and joins opposite sides of }\\mathcal C\\}. \\nonumber\n\\end{eqnarray}\nOf course, $D_n$ depends on $f$ and $\\mathcal C$.\n\n\n\nLet $f$ be an expanding Thurston map. For any two Jordan curves $\\mathcal{C}$ and $\\mathcal{C'}$ with $\\post(f)\\subset \\mathcal C,\\mathcal C'$, inequality (17.1) in \\cite{BMExpanding} states that there exists a constant $c>0$ such that for all $n>0$,\n\\[ \\frac1{c}D_n(f,\\mathcal C)\\leq D_n(f,\\mathcal C')\\leq c D_n(f,\\mathcal C).\\]\nProposition 17.1 in \\cite{BMExpanding} says that:\n\\begin{pro} \\label{expansionfactor}\nFor an expanding Thurston map $f\\: \\S^2\\ra \\S^2$, and a Jordan curve $\\mathcal C$ containing $\\post(f)$,\nthe limit\n\\[\\Lambda_0=\\Lambda_0(f):=\\lim_{n\\ra\\infty}D_n(f,\\mathcal{C})^{1\/n}\\] exists and is independent of $\\mathcal C$.\n\\end{pro}\n\nWe call $\\Lambda_0(f)$ the \\emph{combinatorial expansion factor} of $f$.\n\nProposition 17.2 in \\cite{BMExpanding} states that:\n\\begin{pro}\nIf $f\\:\\S^2\\ra \\S^2$ and $g\\:\\S^2_1\\ra \\S^2_1$ are expanding Thurston maps that are topologically conjugate, then $\\Lambda_0(f)=\\Lambda_0(g)$.\n\\end{pro}\n\n\n\n\\begin{de}\nLet $f \\: \\S^2 \\ra \\S^2$ be an expanding Thurston map, and let\n${\\mathcal C}\\subseteq \\S^2$ be a Jordan curve containing $\\post(f) $. Let $x, y \\in \\S^2$.\nFor $x \\not= y$ we define\n\\begin{eqnarray*}\nm_{f,\\mathcal C}(x, y) = \\min\\{n\\in \\N_0 :\\mbox{ there exist disjoint $n$-tiles }X \\mbox{ and } Y \\\\\n\\mbox{ for }(f, \\mathcal C) \\mbox{ with } x\\in X \\mbox{ and } y\\in Y \\}.\n\\end{eqnarray*}\nIf $x = y$, we define $m_{f,\\mathcal C}(x, x)= \\infty$.\n\\end{de}\n\nThe minimum in the definition above is always obtained since the\ndiameters of $n$-tiles go to $0$ as $n\\ra \\infty$. We usually drop one or both\nsubscripts in $m_{f,\\mathcal C}(x, y)$ if $f$ or $\\mathcal C$ is clear from the context. If\nwe define for $x,y\\in \\S^2$ and $x \\not= y$,\n\\begin{eqnarray*}\nm'_{f,\\mathcal C}(x, y) = \\max\\{n\\in \\N_0 : \\mbox{ there exist nondisjoint $n$-tiles }X \\mbox{ and } Y \\\\\n\\mbox{ for } (f, \\mathcal C) \\mbox{ with } x\\in X \\mbox{ and } y\\in Y \\},\n\\end{eqnarray*}\nthen $m_{f,\\mathcal C}$ and $m'_{f,\\mathcal C}$ are essentially the same up to a constant (see Lemma 8.6 (v) in \\cite{BMExpanding}).\n\\begin{lem} \\label{twom}\nLet $m_{f,\\mathcal C}$ and $m'_{f,\\mathcal C}$ as defined above. There exists a constant $k>0$, such that for any $x,y\\in \\S^2$ and $x\\not=y$,\n\\[ m_{f,\\mathcal C}(x,y)-k\\leq m'_{f,\\mathcal C}(x,y)\\leq m_{f,\\mathcal C}(x,y)+1.\\]\n\\end{lem}\n\n\\begin{de}\\label{visual}\nLet $f \\: \\S^2\\ra \\S^2$ be an expanding Thurston map and\n$d$ be a metric on $\\S^2$. The metric $d$ is called a \\emph{visual metric} for $f$ if there\nexists a Jordan curve $\\mathcal C\\subseteq \\S^2$ containing $\\post(f) $, constants $\\Lambda > 1$ and $C \\geq 1$ such that\n\\[\\frac1{C}\\Lambda^{-m_{f,\\mathcal C}(x, y)} \\leq d(x, y) \\leq C\\Lambda^{-m_{f,\\mathcal C}(x, y)}\\]\nfor all $x, y \\in \\S^2$.\n\\end{de}\n\nProposition 8.9 in \\cite{BMExpanding} states that for any expanding Thurston map $f\\:\\S^2\\ra \\S^2$, there exists a visual metric for $f$, which induces the standard topology on $\\S^2$. Lemma 8.10 in the same paper gives the following characterization of visual metrics.\n\\begin{lem} \\label{charvisual}\nLet $f \\: \\S^2 \\ra \\S^2$ be an expanding Thurston map. Let $\\mathcal C\\subseteq \\S^2$\nbe a Jordan curve containing $\\post(f)$, and $d$ be a visual metric for $f$ with\nexpansion factor $\\Lambda > 1$. Then there exists a constant $C > 1$ such that\n\\begin{enumerate}\n \\item $d(\\sigma,\\tau)\\geq (1\/C)\\Lambda^{-n}$ whenever $\\sigma$ and $\\tau$ are disjoint $n$-cells,\n \\item $(1\/C)\\Lambda^{-n}\\leq \\diam (\\tau)\\leq C\\Lambda^{-n}$ for $\\tau$ as any $n$-edge or $n$-tile.\n\\end{enumerate}\nConversely, if $d$ is a metric on $\\S^2$ satisfying conditions $(1)$ and $(2)$\nfor some constant $C>1$, then $d$ is a visual metric with expansion\nfactor $\\Lambda > 1$.\n\\end{lem}\n\n\n\\section{Gromov Hyperbolic Spaces}\n\\noindent\nIn this section, we review the definitions of Gromov hyperbolic spaces and the asymptotic upper curvature for Gromov hyperbolic spaces.\n\nLet us first review some basic facts about Gromov hyperbolic spaces. We refer the reader to \\cite{BSElements} as a general source on Gromov hyperbolic spaces. Let $(X,d)$ be a geodesic metric space. For any points $x,y,p\\in X$, the \\emph{Gromov product} $(x,y)_p$ of $x$ and $y$ with respect to base point $p$ is defined as\n\\begin{eqnarray} \\label{gproduct}\n (x,y)_p := \\frac12 \\left[d(x,p)+d(y,p)-d(x,y) \\right].\n\\end{eqnarray}\nThe space $X$ is called \\emph{$\\delta$-hyperbolic} (or Gromov hyperbolic) for some $\\delta\\geq 0$ if there exists a base point $p\\in X$, such that for all $x,y,z\\in X$ we have\n\\begin{eqnarray} \\label{trianglein}\n (x,y)_p\\geq \\min\\{(x,z)_p,(z,y)_p\\}-\\delta.\n\\end{eqnarray}\nIf this inequality holds for some base point $p\\in X$, then it also holds for any other $p'\\in X$ with $\\delta$ being replaced by $2\\delta$.\n\nLet $(X,d)$ be a Gromov hyperbolic metric space with a fixed base point $p\\in X$. A sequence of points $\\{x_i\\}\\subseteq X$ \\emph{converges to infinity} if\n\\begin{eqnarray*}\n\\lim_{i,j\\ra\\infty} (x_i,x_j)_p=\\infty.\n\\end{eqnarray*}\nThis property of a sequence $\\{x_i\\}$ does not depend on the base point $p\\in X$. We say two sequences converging to infinity $\\{x_i\\}$ and $\\{x_i'\\}$ are \\emph{equivalent} if\n\\begin{eqnarray*}\n\\lim_{i\\ra\\infty} (x_i,x_i')_p=\\infty.\n\\end{eqnarray*}\nThe \\emph{boundary at infinity} $\\partial_{\\infty} X$ of $X$ is defined to be the set of equivalence classes of sequences of points converging to infinity. One can also define the \\emph{Gromov product} for points $\\xi,\\xi'\\in \\partial_{\\infty}X$ and $p\\in X$ as\n\\begin{eqnarray*}\n(\\xi,\\xi')_p:=\\inf \\liminf_{i\\ra\\infty}(x_i,x_i')_p\n\\end{eqnarray*}\nwhere the infimum is taken over all sequences $\\{x_i\\}\\in \\xi$ and $\\{x_i'\\}\\in \\xi'$. Here $(\\xi,\\xi')_p=\\infty$ if and only if $\\xi=\\xi'$.\n\nA metric $\\rho$ on the boundary at infinity $\\partial_{\\infty} X$ of a Gromov hyperbolic space $X$ is called \\emph{visual} if there exist $p\\in X$, $\\Lambda>1$ and $k\\geq 1$ such that for all $\\xi,\\xi'\\in \\partial_{\\infty}X$, we have that\n\\begin{eqnarray} \\label{visualg}\n\\frac1{k}\\Lambda^{-(\\xi,\\xi')_p}\\leq \\rho(\\xi,\\xi') \\leq k \\Lambda^{-(\\xi,\\xi')_p}.\n\\end{eqnarray}\nWe call the constant $\\Lambda$ in this inequality the \\emph{expansion factor} of the visual metric $\\rho$.\nRecall that we also defined a visual metric for an expanding Thurston map (see Definition 2.11).\nWhen it is not clear from context, we will refer to the visual metric defined in \\eqref{visualg} as a `visual metric in the Gromov hyperbolic sense'.\n\n\nGiven two metric spaces $(X,d_X)$ and $(Y,d_Y)$, a map $f\\:X\\ra Y $ is called a \\emph{quasi-isometry} if there are constants $\\lambda\\geq 1$ and $k\\geq 0$ such that for all $x,x'\\in X$\n\\[\\frac1{\\lambda}d_X(x,x')-k\\leq d_Y(f(x),f(x'))\\leq \\lambda d_X(x,x')+k \\] and for all $y\\in Y$,\n\\[\\inf_{x\\in X} d_Y(f(x),y)\\leq k. \\]\nIf $\\lambda=1$, we call the map $f$ a \\emph{rough-isometry}. We say that the spaces $X$ and $Y$ are \\emph{quasi-isometric (rough-isometric)} if there is a quasi-isometry (rough-isometry) between them.\n\n\nIn \\cite{BFAsymptotic}, Bonk and Foertsch introduced the notion of upper curvature bounds for Gromov hyperbolic spaces up to rough-isometry (see \\cite[Definition 1.1 and 1.2]{BFAsymptotic}).\n\\begin{de}\nLet $\\kappa\\in [-\\infty,0)$. We call a metric space $X$ an \\emph{AC$_u(\\kappa)$-space} if there exists $p\\in X$ and a constant $c\\geq 0$ such that for all $x,x'\\in X$ and all finite sequences $x_0=x,x_1,\\ldots,x_n=x'$ in $X$ with $n>0$,\n\\begin{eqnarray}\\label{acspaceeq}\n(x,x')_p\\geq \\min_{i=1,\\ldots,n}(x_{i-1},x_i)_p-\\frac1{\\sqrt{-\\kappa}}\\log n-c.\n\\end{eqnarray}\nHere we use the convention $1\/\\sqrt{\\infty}=0$.\nWe call\n\\begin{eqnarray*}\nK_u(X):=\\inf\\{\\kappa\\: X \\mbox{ is an AC$_u(\\kappa)$-space}\\in [-\\infty,0) \\}\n\\end{eqnarray*}\nthe \\emph{asymptotic upper curvature} of $X$.\n\\end{de}\nRough-isometric Gromov hyperbolic spaces have the same asymptotic upper curvature since under rough-isometries, Gromov products only change by a fixed additive amount, which can be absorbed in the constant $c$ in \\eqref{acspaceeq}.\n\nThe asymptotic upper curvature is related to the expansion factors of visual metrics in Gromov hyperbolic spaces, due to the following theorem \\cite[Theorem 1.5]{BFAsymptotic}.\n\\begin{thm} \\label{acku}\nLet $X$ be a Gromov hyperbolic metric space. If there exists a visual metric on $\\partial_{\\infty}X$ with expansion factor $\\Lambda>1$, then $X$ is an AC$_u(\\kappa)$-space with $\\kappa=-\\log^2\\Lambda$. Conversely, if $X$ is an AC$_u(\\kappa)$-space, then for every $1<\\Lambda1$.\n\nWe define a graph by the cell decompositions of $(f, \\mathcal C)$ as follows.\nLet\n\\[V=V(f,\\mathcal C)\\]\nbe the set of all tiles in the cell decompositions $\\D^n(f,\\mathcal C)$ of $(f, \\mathcal C)$ for $n\\geq -1$, where $\\D^{-1}(f,\\mathcal C)$ contains a single $(-1)$-tile $\\S^2$.\nLet $V$ be the set of vertices of the graph. Define the edge set $E$ as follows:\nthere is an edge between the two vertices $X^n,Y^m\\in V$, which we indicate by the notation\n$X^n\\sim Y^m$ if for the underlying tiles we have\n\\[|n-m|\\leq 1 \\mbox{ and } X^n\\cap Y^m\\not= \\emptyset.\\]\nWe call the graph\n\\[\\G(f,\\mathcal C):=G(V,E)\\]\nthe \\emph{tile graph} of $(f, \\mathcal C)$. We usually drop one or both\nparameters in $\\G(f,\\mathcal C)$ if $f$ or $\\mathcal C$ are clear from the context.\nWe call \\[\\ell\\: V \\ra \\Z\\]\nthe \\emph{level function}, where\nfor an $n$-tile $X^n$, we have $\\ell(X^n)=n$.\n\nIf $X\\cap Y=\\emptyset$, let\n\\begin{align*}\n\\bar m_{f,\\mathcal C}(X,Y) :=\\max\\{m\\in \\N_{-1}\\:&\\mbox{\nthere exist non-disjoint $m$-tiles $X^m$ and} \\\\\n&\\mbox{$Y^m$, such that } X\\cap X^m\\not= \\emptyset,\nY\\cap Y^m\\not= \\emptyset\\};\n\\end{align*}\nif $X\\cap Y\\not=\\emptyset$, let\n\\begin{align*}\n\\bar m_{f,\\mathcal C}(X,Y) :=\\infty.\n\\end{align*}\nHere we assume that the $\\infty$-tile is the empty set.\nFor $X,Y\\in \\G$, define\n\\begin{eqnarray} \\label{mG}\nm(X,Y)=m_{f,\\mathcal C}(X,Y)=\\min\\{\\ell(X),\\ell(Y), \\bar m_{f,\\mathcal C}(X,Y)\\}.\n\\end{eqnarray}\nThe tile graph $\\G$ is path connected since any tile can be connected to the $(-1)$-tile $\\S^2$.\nWe give $\\G$ the path metric $\\eta$. Notice that $\\G$ is a\ngeodesic space under this metric. The distance of $X\\in V$ to the base point $\\S^2$ is\n\\[\\eta(X,\\S^2)= \\ell(X)+1.\\]\nFor $X,Y\\in \\G$, we let\n\\begin{eqnarray} \\label{gp}\n(X,Y)&:=& (X,Y)_{X^{-1}} = (X,Y)_{\\S^2} \\nonumber \\\\\n&=& 1\/2[\\eta(X,\\S^2)+\\eta(Y,\\S^2)-\\eta(X,Y)]\\\\\n&=&1\/2[\\ell(X)+\\ell(Y)-\\eta(X,Y)]+1, \\nonumber\n\\end{eqnarray} be the\n\\emph{Gromov product} of $X$ and $Y$ with respect to $X^{-1}=\\S^2$.\n\nIn the following, we are going to prove that the tile graph $\\G$ equipped with the path metric $\\eta$ is a Gromov\nhyperbolic space.\n\n\\excise{\n\\begin{eg}\nWe define a map $f$ as follows (see the picture below): we glue along the boundary of two unit squares $[0,1]^2$, and get a pillow-like space which is homeomorphic to $\\widehat{\\C}$; we color one of the squares black and the other white; we divide each of the squares into 4 smaller squares of half the side length, and color them with black and white in checkerboard fashion; we map one of the small black pillows to the bigger black pillows by Euclidean similarity, and extend the map to the whole pillow-like space by reflection. In fact, the map $f$ is an expanding Thurston map (see Example 4.13 in \\cite{YinLattes}), and the postcritical set $\\post (f)$ consists of the four common corner points of the two big squares.\n\\begin{center}\n\\mbox{ \\scalebox{0.7}{\\includegraphics{2by2.eps}}}\n\\end{center}\nLet $\\mathcal C$ be the common boundary of the two big squares, then $\\mathcal C$ contains $\\post (f)$.\nOn the $n$-th level, the set of $n$-tiles corresponding to\n\\end{eg}\n}\n\n\n\n\\begin{lem} \\label{mdiam}\nThere exists a constant $C>1$ such that for any tiles $X,Y\\in \\G$,\n\\[\\frac1{C}\\Lambda^{-m(X,Y)}\\leq \\diam (X\\cup Y) \\leq C \\Lambda^{-m(X,Y)}.\\]\n\\end{lem}\nHere and in the following, the diameter function $\\diam(\\cdot)$ is with respect to the visual metric $d$ on $\\S^2$.\n\n\\begin{proof}\nLet $m=m(X,Y)$, and let $X^m, Y^m$ be non-disjoint $m$-tiles such that\n\\[X\\cap X^m\\not=\\emptyset \\mbox{ and } Y\\cap Y^m\\not=\\emptyset.\\]\nWe have that\n\\begin{eqnarray*}\n\\diam (X\\cup Y) &\\leq &\\diam (X)+\\diam (X^m)+\\diam (Y^m)+\\diam (Y)\\\\\n&\\leq& 4 C' \\Lambda^{-m},\n\\end{eqnarray*}\nwhere $C'>1$ is the same as the constant in Lemma \\ref{charvisual}, which only depends on $f$.\nLet $\\bar m= \\bar m_{f,\\mathcal C}(X,Y)$, and let $X^{\\bar m+1}, Y^{\\bar m+1}$ be disjoint $(\\bar m+1)$-tiles such that\n\\[X\\cap X^{\\bar m+1}\\not=\\emptyset,\\quad Y\\cap Y^{\\bar m+1}\\not=\\emptyset.\\] Then\n\\begin{eqnarray*}\n\\diam (X\\cup Y) &\\geq& \\max\\{ \\diam (X), \\diam (Y), d(X^{\\bar m+1}, Y^{\\bar m+1}) \\} \\\\\n&\\geq & \\frac1{C'}\\max\\{\\Lambda^{-\\ell(X)}, \\Lambda^{-\\ell(Y)}, \\Lambda^{-\\bar m}\\} \\\\\n&\\geq &\\frac1{C'}\\Lambda^{-\\min\\{\\ell(X),\\ell(Y),\\bar m\\}}\\\\\n&= &\\frac1{C'}\\Lambda^{-m},\n\\end{eqnarray*}\nwhere $C'>1$ is the same $C$ as in Lemma \\ref{charvisual}, which only depends on $f$. Let $C=4C'$, and the lemmas follows.\n\\end{proof}\n\n\\begin{lem} \\label{pdiam}\nThere exists a constant $k\\geq 1$ such that for any tiles $X,Y\\in \\G$,\n\\[\\diam (X\\cup Y) \\leq k\\Lambda^{-(X,Y)}.\\]\n\\end{lem}\n\n\\begin{proof}\nLet $\\eta=\\eta(X,Y)$. Pick any path $X_0=X, X_1,\\ldots, X_{\\eta}=Y$. Then\n\\begin{eqnarray*}\n\\diam (X\\cup Y)& \\leq & \\sum_{i=0}^{\\eta}\\diam(X_i)\\\\\n&\\leq & C \\sum_{i=0}^{\\eta}\\Lambda^{-\\ell(X_i)} \\\\\n&\\leq & C \\min_{0\\leq l\\leq \\eta}\\left\\{\\sum_{i=0}^{l}\\Lambda^{-\\ell(X)+i}+ \\sum_{i=l+1}^{\\eta}\\Lambda^{-\\ell(Y)+(\\eta-i)}\\right\\}\\\\\n&\\leq & \\frac{C\\Lambda}{\\Lambda-1} \\min_{0\\leq l\\leq \\eta}\\left\\{\\Lambda^{-\\ell(X)+l}+ \\Lambda^{-\\ell(Y)+(\\eta-l)}\\right\\}.\n\\end{eqnarray*}\nNotice that on the right hand-side the minimum is obtained when the two exponents of $\\Lambda$ are the same:\n\\[ -\\ell(X)+l= -\\ell(Y)+(\\eta-l),\\]\nso we let\n\\[l=\\left[\\frac12(\\ell(X)-\\ell(Y)+\\eta)\\right]\\]\nbe the integer part of $\\frac12(\\ell(X)-\\ell(Y)+\\eta)$. Hence, we have\n\\begin{eqnarray*}\n\\diam (X\\cup Y) &\\leq & \\frac{2C\\Lambda}{\\Lambda-1} \\Lambda^{-[1\/2(\\ell(X)+\\ell(Y)-\\eta)]} \\\\\n&\\leq &k\\Lambda^{-(X,Y)},\n\\end{eqnarray*}\nwhere $C>1$ is the same $C$ as in Lemma \\ref{charvisual}, and\n\\[k=\\frac{2C\\Lambda^3}{\\Lambda-1}\\] also only depends on $f$.\n\\end{proof}\n\n\n\\begin{pro} \\label{mp}\nThere exists a constant $C'>0$, such that for any tiles $X,Y\\in \\G$,\n\\begin{eqnarray*}\nm(X,Y)-1\\leq (X,Y)\\leq m(X,Y)+C'.\n\\end{eqnarray*}\n\\end{pro}\n\n\\begin{proof}\nBy Lemma \\ref{mdiam} and Lemma \\ref{pdiam}, we have that\n\\[ \\frac1{C}\\Lambda^{-m(X,Y)}\\leq \\diam (X\\cup Y) \\leq k\\Lambda^{-(X,Y)}\\] for some constants $C,k>1$ which only depend on $f$. Hence, there exists a constant $C'>0$, such that for any tiles $X,Y\\in \\G$,\n\\[(X,Y)\\leq m(X,Y)+C'.\\]\n\nFor the other inequality, let $m=m(X,Y)$, and let $X^m, Y^m$ be non-disjoint $m$-tiles such that\n\\[X\\cap X^m\\not=\\emptyset,\\quad Y\\cap Y^m\\not=\\emptyset.\\]\nSo \\[\\eta(X,X^m)\\leq \\ell(X)-m+1 ,\\]and\n\\[\\eta(Y,Y^m)\\leq \\ell(Y)-m+1 .\\]\nBy the triangle inequality, we have that\n\\begin{eqnarray*}\n\\eta(X,Y)&\\leq& \\eta(X,X^m)+\\eta(X^m,Y)\\\\\n&\\leq & \\eta(X,X^m)+\\eta(Y^m,Y)+1\\\\\n&\\leq & (\\ell(X)-m)+(\\ell(Y)-m)+3.\n\\end{eqnarray*}\nHence, we obtain that\n\\begin{eqnarray*}\n(X,Y)&= &(X,Y)_{X^{-1}} =1\/2[\\ell(X)+\\ell(Y)-\\eta(X,Y)]+1\\\\\n&\\geq & 1\/2[\\ell(X)+\\ell(Y)-(\\ell(X)-m)-(\\ell(Y)-m)-3]+1\\\\\n&\\geq &m(X,Y)-1.\n\\end{eqnarray*}\n\\end{proof}\n\n\\excise{\n----------------------------------\n\\begin{cor}\nThere exists a constant $C'>0$, such that for any tiles $X,Y\\in \\G$, we have\n\\[ (X,Y)\\leq m(X,Y)+C'.\\]\n\\end{cor}\n\\begin{proof}\nBy the Lemma \\ref{mdiam} and Lemma \\ref{pdiam}, we have\n\\[ \\frac1{k}\\Lambda^{-m(X,Y)}\\leq \\diam (X\\cup Y) \\leq k'\\Lambda^{-(X,Y)},\\] for some constant $k,k'>1$ which only depends on $f$. The corollary follows easily.\n\\end{proof}\n\n\\begin{lem}\n\\[ m(X,Y)\\leq (X,Y).\\]\n\\end{lem}\n\n\\begin{proof}\nLet $m=m(X,Y)$, and let $X^m, Y^m$ be non-disjoint $m$-tiles such that\n\\[X\\cap X^m\\not=\\emptyset,\\quad Y\\cap Y^m\\not=\\emptyset.\\]\nBy triangle inequality, we have\n\\begin{eqnarray*}\n\\eta(X,Y)&\\leq& \\eta(X,X^m)+\\eta(X^m,Y)\\\\\n&\\leq & \\eta(X,X^m)+\\eta(Y^m,Y)+1\\\\\n&\\leq & (\\ell(X)-m)+(\\ell(Y)-m)+1.\n\\end{eqnarray*}\nHence,\n\\begin{eqnarray*}\n(X,Y)&= &(X,Y)_{X^{-1}} =1\/2[\\ell(X)+\\ell(Y)-\\eta(X,Y)]+1\\\\\n&\\geq & 1\/2[\\ell(X)+\\ell(Y)-(\\ell(X)-m)-(\\ell(Y)-m)-1]+1\\\\\n&\\geq &m(X,Y).\n\\end{eqnarray*}\n\\end{proof}\n-------------------------------}\n\n\n\\begin{lem} \\label{triangleineq}\nThere exists a number $c\\geq 0$ such that for any tiles $X,Y,Z\\in \\G$,\n\\[m(X,Y)\\geq \\min\\{m(X,Z),m(Y,Z)\\} -c .\\]\n\\end{lem}\n\n\\begin{proof}\nFor any $X,Y,Z\\in \\G$,\n\\begin{eqnarray*}\n\\diam(X\\cup Y)&= &\\max\\{d(x,y),d(x,x'),d(y,y')\\: x,x'\\in X, y,y'\\in Y\\}\\\\\n&\\leq & \\max\\{d(x,z)+d(z,y),d(x,x'),d(y,y')\\: \\\\\n&& \\hspace{1.5cm} x,x'\\in X, y,y'\\in Y, z\\in Z\\}\\\\\n&\\leq & \\max\\{d(x,z),d(x,x')\\: x\\in X,z\\in Z\\}\\\\\n&& +\\max\\{d(z,y),d(y,y')\\: y\\in Y,z\\in Z\\}\\\\\n&\\leq & \\diam(X\\cup Z)+\\diam(Z \\cup Y),\n\\end{eqnarray*}\nand so\n\\begin{eqnarray} \\label{diamtri}\n\\diam(X\\cup Y)&\\leq & 2\\max\\{ \\diam(X\\cup Z),\\diam(Z \\cup Y)\\}.\n\\end{eqnarray}\n\nBy Lemma \\ref{mdiam}, there exists a constant $k>1$, such that for any $X,Y\\in \\G$,\n\\begin{eqnarray*}\n\\frac1{k}\\Lambda^{-m(X,Y)}\\leq \\diam (X\\cup Y) \\leq k \\Lambda^{-m(X,Y)}.\n\\end{eqnarray*}\nHence, by the inequalities above and inequality \\eqref{diamtri}, we have that\n\\begin{eqnarray*}\nm(X,Y)&\\geq & -\\log_{\\Lambda}\\big(k\\diam(X\\cup Y)\\big) \\\\\n&\\geq & -\\log_{\\Lambda}\\Big(2k\\max\\big\\{ \\diam(X\\cup Z),\\diam(Z \\cup Y) \\big\\}\\Big)\\\\\n&\\geq & \\min\\Big\\{ -\\log_{\\Lambda}\\big(2k\\diam(X\\cup Z)\\big),-\\log_{\\Lambda}\\big(2k\\diam(Z \\cup Y)\\big) \\Big\\}\\\\\n&\\geq & \\min\\{m(X,Z),m(Y,Z)\\} -c\n\\end{eqnarray*}\nfor some $c\\geq 0$ that only depends on $f$.\n\\end{proof}\n\n\\begin{thm} \\label{gh}\nLet $f\\: \\S^2 \\ra S^2$ be an expanding Thurston map\nand let $\\mathcal C \\subset \\S^2$ be a Jordan curve containing $\\post(f)$. Then the tile graph $\\G(f,\\mathcal C)$ equipped with the path metric $\\eta$ is a Gromov hyperbolic space.\n\\end{thm}\n\n\\begin{proof}\nFor any tiles $X,Y\\in \\G$, by Proposition \\ref{mp}, the Gromov product $(X,Y)$ defined in equation \\eqref{gp} is equal to $m(X,Y)$ up to a constant which only depends on $f$. So by Lemma \\ref{triangleineq}, there exists a constant $c'>0$, such that for any tiles $X,Y,Z\\in \\G$,\n\\[(X,Y)\\geq \\min\\{(X,Z), (Y,Z)\\} -c' .\\] Therefore, the graph $G(f,\\mathcal C)$ equipped with the path metric $\\eta$ is a Gromov hyperbolic space.\n\\end{proof}\n\n\\begin{re}\nIn the proofs of Proposition \\ref{mp} and Lemma \\ref{triangleineq}, we used visual metrics as a bridge to connect $m(\\cdot,\\cdot)$ and the Gromov product $(\\cdot,\\cdot)$. This idea is contained in \\cite{BPCohomologie}. Theorem \\ref{gh}, Proposition \\ref{mp} and Lemma \\ref{triangleineq} can also be proved combinatorially without using visual metrics.\n\\end{re}\n\n\\begin{pro} \\label{roughgg}\nFor any Jordan curves $\\mathcal C$ and $\\mathcal C'$ containing $\\post (f)$, the tile graphs $\\G=\\G(f,\\mathcal C)$ and $\\G'=\\G(f,\\mathcal C')$ equipped with path metric respectively are rough-isometric.\n\\end{pro}\n\n\\begin{proof}\nBy equation \\eqref{gp}, for any $X,Y\\in \\G(f,\\mathcal C)$, we have\n\\begin{eqnarray} \\label{dp}\n \\eta(X,Y) &=& \\ell(X)+\\ell(Y)+2-2(X,Y).\n\\end{eqnarray}\nWe have similar relations for the path metric $\\eta'$ of $\\G'$. Let $m=m_{f,\\mathcal C}$ and $m'=m_{f,\\mathcal C'}$ as defined in equation \\eqref{mG}.\nWe know that $m(X,Y)$ and $(X,Y)$ are equal up to a constant that only depends on $f$ by Proposition \\ref{mp}. So\nif we can show that there exists a level-preserving bijection $g\\:\\G\\ra \\G'$ and a constant $\\lambda\\geq 0$, such that for any $X,Y\\in \\G$,\n\\begin{eqnarray*}\n m(X,Y)-\\lambda\\leq m'(g(X),g(Y))\\leq m(X,Y)+\\lambda,\n\\end{eqnarray*}\nthen by equation \\eqref{dp}, the map $g$ will be a rough isometry between the path metrics of $\\G$ and $\\G'$.\n\nFix $p\\in {\\rm post}(f)$. We will define\n\\[g\\: \\G(f,\\mathcal C)\\ra \\G(f,\\mathcal C')\\]\nby specifying a bijection between $n$-tiles of $(f,\\mathcal C)$ and $(f,\\mathcal C')$ for all $n\\geq -1$. For $n=-1$, let\n\\[g(\\S^2)=\\S^2.\\]\nFor $n\\geq 0$, and for any $q\\in f^{-n}(p)$, we claim that there exists a bijection $g_{n,q}$ between $n$-tiles of $(f,\\mathcal C)$ containing $q$ and $n$-tiles of $(f,\\mathcal C')$ containing $q$,\n\\begin{equation*}\n g_{n,q}\\: \\Big\\{n\\mbox{-tile } X\\in \\D^n(f,\\mathcal C): q\\in X \\Big\\}\\ra \\Big\\{n\\mbox{-tile } X'\\in \\D^n(f,\\mathcal C'): q\\in X' \\Big\\}.\n\\end{equation*}\nIndeed, the number of tiles containing $q$ is equal to the degree of $f^n$ at $q$, and this justifies the existence of the bijection $g_{n,q}$. Since every $n$-tile contains exactly one point in $f^{-n}(p)$, we get a bijection of all $n$-tiles by $g_{n,q}$ for $q\\in f^{-n}(p)$.\n\nFor any $X, Y\\in \\G$, let $X',Y' \\in \\G'$ be their images under $g$. It follows from the definition of $g$ that\n\\[X\\cap X'\\not=\\emptyset \\mbox{ and } Y\\cap Y'\\not=\\emptyset.\\]\nNow we are going to show that there exists $k\\geq 1$, such that for any $X,Y\\in \\G$,\n\\begin{eqnarray*}\n\\frac1{k}\\diam (X'\\cup Y') \\leq \\diam (X\\cup Y)\\leq k\\diam (X'\\cup Y').\n\\end{eqnarray*}\nLet $m=m(X,Y)$. We have that\n\\begin{eqnarray*}\n\\diam (X\\cup Y)&\\leq &\\diam(X) + \\diam (X'\\cup Y') + \\diam (Y)\\\\\n&\\leq & C^2 \\diam (X')+C^2\\diam(Y') + \\diam(X'\\cup Y')\\\\\n&\\leq & (C^2+1) \\diam (X'\\cup Y'),\n\\end{eqnarray*}\nwhere $C>1$ is the same $C$ as in Lemma 2.12, which only depends on $f$.\nThis implies that\n\\[\\diam (X\\cup Y)\\leq k\\diam (X'\\cup Y'), \\]for some $k>1$ only depending on $f$. Similarly, we get that\n\\[\\diam (X'\\cup Y')\\leq k\\diam (X\\cup Y). \\]\n\nSince $\\diam (X\\cup Y)$ and $\\Lambda^{-m(X,Y)}$ are the same up to a scaling by Lemma \\ref{mdiam}, there exists a constant $\\lambda>0$, such that\n\\[m(X,Y)-\\lambda\\leq m'(g(X),g(Y))\\leq m(X,Y)+\\lambda \\] for all $X,Y\\in \\G(f,\\mathcal C)$.\n\\end{proof}\n\n\\begin{re}\nIn the proof of Proposition \\ref{roughgg}, the bijective rough-isometry $g$ between tile graphs of two different Jordan curves induces a bijection $g_{\\infty}$ on the boundary at infinity of these two tile graphs.\n\\end{re}\n\n\\begin{pro} \\label{samevisual}\nThe boundary at infinity $\\partial_{\\infty}\\G$ of a graph tile $\\G(f,\\mathcal C)$ can be identified with $\\S^2$.\nUnder this identification, a metric $d$ is a visual metric on $\\S^2$ with respect\nto the expanding Thurston map $f$ if and only if $d$ is a visual metric on $\\partial_{\\infty}\\G$ (in the sense of Gromov hyperbolic spaces).\n\\end{pro}\n\nHere the metric $d$ on $\\partial_{\\infty}\\G$ means the pull-pack metric of $d$ under the identification.\n\n\\begin{proof}\nLet $d$ be a visual metric with expansion factor $\\Lambda$ of $\\S^2$ with respect to $f$.\n\nFor any sequence $\\{X_n\\}$ converging to $\\infty$\n\\[\\lim_{i,j\\ra \\infty} (X_i,X_j)=\\infty,\\] we have a filtration\n\\begin{eqnarray*}\n\\bigcup_{i={1}}^{\\infty} X_i\\supset \\bigcup_{i={2}}^{\\infty} X_i\\supset\\ldots \\supset\\bigcup_{i={n}}^{\\infty} X_i \\supset \\bigcup_{i=n+1}^{\\infty} X_i\\supset\\ldots\n\\end{eqnarray*}\nwith\n\\[ \\diam \\left(\\bigcup_{i={n}}^{\\infty} X_i\\right)\\ra 0\\mbox{ as } n\\ra \\infty.\\]\nHence, there exists a limit point $x\\in \\S^2$ such that for any $\\epsilon>0$, there exists $N>0$ such that for all $n>N$,\n\\begin{eqnarray} \\label{inepsilonn}\n\\bigcup_{i={n}}^{\\infty} X_i\\subset N_{\\epsilon}(x),\n\\end{eqnarray}\nwhere $N_{\\epsilon}(x)$ is an $\\epsilon$-neighborhood of $x$ in $\\S^2$, i.e.,\n\\[ X_n\\subset N_{\\epsilon}(x),\\] or\n\\begin{eqnarray*}\nd(x,X_n)< \\epsilon.\n\\end{eqnarray*}\nWe claim that the limit point is unique. Indeed, if there exists $y\\in \\S^2$ also satisfying \\eqref{inepsilonn}, then\n\\[d(x,y)\\leq d(x,\\diam(X_n))+d(y,\\diam(X_n))\\ra 0\\mbox{ as } n\\ra\\infty.\\]\nHence, $x=y$.\nLet $\\{Y_n\\}$ be an sequence converging to infinity equivalent to $\\{X_i\\}$, i.e.,\n\\[\\lim_{i\\ra \\infty} (X_i,Y_i)=\\infty.\\]\nWe claim that the limit point of $\\{Y_n\\}$ is $x$. Indeed, by Lemma \\ref{pdiam}, we have\n\\[d(x,Y_n)\\leq d(x,X_n)+d(Y_n,X_n)\\leq d(x,X_n)+k{\\Lambda}^{-(Y_n,X_n)}\\ra 0 \\]\nas $n$ goes to infinity since $(Y_n,X_n)\\ra \\infty$. Hence, any two equivalent sequences converging to infinity have the same limit point, and we can assign a limit point to an equivalence class of sequences converging to infinity.\n\nWe define\n\\[h\\: \\partial_{\\infty}\\G\\ra \\S^2\\]\nby mapping any equivalence class of sequences converging to infinity to its limit point. For any $x\\in \\S^2$, there exists $X_i$ with $\\ell(X_i)=i$ containing $x$, for any $i\\geq -1$. Then by Lemma \\ref{pdiam}, we have that\n\\begin{eqnarray*}\n(X_i,X_j)&\\geq& -\\log_{\\Lambda}\\diam(X_i\\cup X_j)+\\log k\\\\\n&\\geq&-\\log_{\\Lambda} \\big(\\min\\{ \\diam(X_i),\\diam(X_j)\\} \\big) +\\log k \\ra \\infty\n\\end{eqnarray*}\nas $i,j\\ra\\infty$, where $k\\geq 1$ is a constant as in Lemma \\ref{pdiam}.\nSo $\\{X_i\\}$ is a converging sequence with limit point $x$. Hence, the map $h$ is surjective.\nIn order to prove the injectivity, for any two\nsequences converging to infinity $\\{X_i\\}$ and $\\{Y_i\\}$, we let $x$ and $y$ be their limit points respectively. If $x=y$, then\n\\[\\diam(X_n\\cup Y_n) \\ra 0 \\mbox{ as } n\\ra \\infty.\\]\nSo by Lemma \\ref{mdiam} and \\ref{mp},\n\\[(X_n,Y_n) \\geq m(X_n,Y_n)-1 \\geq -\\log_{\\Lambda}\\big(C\\diam(X_n\\cup Y_n)\\big) -1\\ra \\infty \\]\nas $n$ goes to infinity, which implies that $\\{X_i\\}$ and $\\{Y_i\\}$ are equivalent.\nHence,\n$h$ is injective.\n\nWe only need to show that that there exists a constant $C>0$ such that for any $\\xi,\\xi'\\in \\partial_{\\infty}\\G$, $x=h(\\xi)$ and $y=h(\\xi')$,\n\\[\\frac1{C} \\Lambda^{-(\\xi,\\xi')}\\leq d(x,y)\\leq C\\Lambda^{-(\\xi,\\xi')}. \\]\nPick any $\\{X_n\\}\\in \\xi$ and $\\{Y_n\\}\\in \\xi'$. By Lemma \\ref{mdiam}\n\\begin{eqnarray*}\n\\frac1{C}\\Lambda^{-(X_n,Y_n)}\\leq \\diam(X_n\\cup Y_n) \\leq C \\Lambda^{-(X_n,Y_n)}.\n\\end{eqnarray*}\nTaking the limit superior, we get\n\\begin{eqnarray*}\n\\frac1{C}\\limsup_{n\\ra\\infty} \\Lambda^{-(X_n,Y_n)}\\leq \\lim_{n\\ra\\infty}\\diam(X_n\\cup Y_n)\n\\leq C\\limsup_{n\\ra\\infty} \\Lambda^{-(X_n,Y_n)}.\n\\end{eqnarray*}\nHence, we have\n\\begin{eqnarray} \\label{infdiam}\n\\frac1{C}\\Lambda^{-\\liminf_{n\\ra\\infty} (X_n,Y_n)}&\\leq& \\lim_{n\\ra\\infty}\\diam(X_n\\cup Y_n)\\\\\n &&\\hspace{2cm}\\leq C \\Lambda^{-\\liminf_{n\\ra\\infty} (X_n,Y_n)}. \\nonumber\n\\end{eqnarray}\nSince\n\\[d(x,y)=\\lim_{n\\ra\\infty}\\diam(X_n,Y_n)\\] and\n\\[(\\xi,\\xi')\\leq \\liminf_{n\\ra\\infty} (X_n,Y_n),\\]\nby inequality \\eqref{infdiam}\n\\begin{eqnarray} \\label{dleq}\nd(x,y)&=&\\lim_{n\\ra\\infty}\\diam(X_n\\cup Y_n) \\nonumber\\\\\n &\\leq& {C} \\Lambda^{-\\liminf_{n\\ra\\infty} (X_n,Y_n)}\\\\\n &\\leq& {C} \\Lambda^{- (\\xi,\\xi')}.\\nonumber\n\\end{eqnarray}\nSince \\[(\\xi,\\xi')= \\inf \\liminf_{n\\ra\\infty} (X_n,Y_n)\\] where infimum is taken for all $\\{X_n\\}\\in \\xi$ and $\\{Y_n\\}\\in \\xi'$, by inequality \\eqref{infdiam},\n\\begin{eqnarray*}\n\\frac1{C}\\sup\\Lambda^{-\\liminf_{n\\ra\\infty} (X_n,Y_n)}\\leq \\lim_{n\\ra\\infty}\\diam(X_n\\cup Y_n) =d(x,y)\n\\end{eqnarray*}\nwhere supremum is taken for all $\\{X_n\\}\\in \\xi$ and $\\{Y_n\\}\\in \\xi'$. Hence,\n\\begin{eqnarray} \\label{dgeq}\n\\frac1{C}\\Lambda^{-(\\xi,\\xi')}=\\frac1{C}\\Lambda^{-\\inf\\liminf_{n\\ra\\infty} (X_n,Y_n)}\\leq d(x,y).\n\\end{eqnarray}\nCombining equations \\eqref{dleq} and \\eqref{dgeq}, we get that\n\\begin{eqnarray} \\label{dleqgeq}\n\\frac1{C}\\Lambda^{-(\\xi,\\xi')}\\leq d(x,y)\\leq 4C\\Lambda^{-(\\xi,\\xi')},\n\\end{eqnarray}\nso\n\\begin{eqnarray*}\n\\frac1{C}\\Lambda^{-(\\xi,\\xi')}\\leq d(h(\\xi),h(\\xi'))\\leq C\\Lambda^{-(\\xi,\\xi')}\n\\end{eqnarray*}\nfor all $\\xi,\\xi'\\in \\partial_{\\infty}\\G$.\nTherefore, the pull-back of the metric $d$ on $\\S^2$ under $h$ is a visual metric on $\\partial_{\\infty}\\G$.\n\nSince $d$ is a visual metric with respect to $f$, equation \\eqref{dleqgeq} implies that there exists a constant $c\\geq 0$ such that for all $x,y\\in \\S^2$, and $\\xi=h^{-1}(x)$, $\\xi=h^{-1}(y)$,\n\\begin{eqnarray} \\label{mxi}\n(\\xi,\\xi')-c\\leq m(x,y)\\leq (\\xi,\\xi')+c.\n\\end{eqnarray}\nLet $\\rho$ be a visual metric on $\\partial_{\\infty}\\G$ on the Gromov hyperbolic space, so there exists constant $k\\geq 1$, such that for any $\\xi,\\xi'\\in \\partial_{\\infty}\\G$,\n\\[ \\frac1{k}\\Lambda^{-(\\xi,\\xi')}\\leq \\rho(\\xi,\\xi' ) \\leq k\\Lambda^{-(\\xi,\\xi')}.\\]\nBy equation \\eqref{mxi}, there exists a constant $k'\\geq 1$, such that\n\\[\\frac1{k'}\\Lambda^{-(x,y)}\\leq \\frac1{k}\\Lambda^{-(\\xi,\\xi')}\\leq \\rho(h^{-1}(x),h^{-1}(y) ) \\leq k\\Lambda^{-(\\xi,\\xi')}\\leq k'\\Lambda^{-m(x,y)},\\]\nwhere $x,y\\in \\S^2$, $\\xi=h^{-1}(x)$ and $\\xi=h^{-1}(y)$.\nTherefore, the pull-back of the metric $\\rho$ on $\\partial_{\\infty}\\G$ under $h^{-1}$ is a visual metric on $\\S^2$.\n\\end{proof}\n\nFor any Jordan curves $\\mathcal C$ and $\\mathcal C'$ containing $\\post (f)$, let $\\partial_{\\infty}\\G=\\partial_{\\infty}\\G(f,\\mathcal C)$ and $\\partial_{\\infty}\\G'=\\partial_{\\infty}\\G(f,\\mathcal C')$ be the boundary at infinity of the tile graphs $\\G(f,\\mathcal C)$ and $\\G(f,\\mathcal C')$ respectively. By the proposition above, there exist identifications\n\\[h\\: \\partial_{\\infty}\\G \\ra \\S^2\\]\nand\n\\[h'\\: \\partial_{\\infty}\\G' \\ra \\S^2.\\]\nSo we have the following diagram\n\\[\n\\xymatrix@R=0.5cm{\n \\partial_{\\infty}\\G \\ar[dd]_{g_{\\infty}} \\ar[dr]^{h} \\\\\n & \\S^2\\ar[dl]^{(h')^{-1}} \\\\\n \\partial_{\\infty}\\G' }\n\\]\nThis induced bijection $g_{\\infty}=(h')^{-1}\\circ h$ should be the same as $g_{\\infty}$ as in the remark after Proposition \\ref{roughgg}. In addition, under this identification, visual metrics on $\\partial_{\\infty}\\G$ and $\\partial_{\\infty}\\G'$ are also identified. This is the following corollary.\n\n\\begin{cor} \\label{corsamevisual}\nFor any Jordan curves $\\mathcal C$ and $\\mathcal C'$ containing $\\post (f)$, there exists a natural identification between $\\G=\\partial_{\\infty}\\G(f,\\mathcal C)$ and $\\G'=\\partial_{\\infty}\\G(f,\\mathcal C')$. Under this identification, a metric $\\rho$ is a visual metric on $\\partial_{\\infty}\\G$ if and only if it is a visual metric on $\\partial_{\\infty}\\G'$.\n\\end{cor}\n\n\n\n\n\\section{Asymptotic Upper Curvature}\n\\noindent\nIn this section, we define the asymptotic upper curvature for an expanding Thurston map. After review the definition of Latt\\`es maps, we give a curvature characterization of Latt\\`es maps.\n\nLet $f \\: \\S^2 \\ra S^2$ be an expanding Thurston map.\nWe define the \\emph{asymptotic upper curvature} of $f$ as\n\\begin{eqnarray} \\label{asyf}\nK_u(f)\\:=K_u(\\G(f,\\mathcal C)),\n\\end{eqnarray}\nwhere $\\mathcal C \\subset \\S^2$ is any Jordan curve containing $\\post(f)$ and $\\G=\\G(f,\\mathcal C)$ denotes\nthe Gromov hyperbolic graph constructed from the cell decompositions of $(f, \\mathcal C)$.\nFor any Jordan curves $\\mathcal C$ and $\\mathcal C'$, the Gromov hyperbolic space $\\G(f,\\mathcal C)$ and $\\G(f,\\mathcal C')$ are rough-isomeric by Proposition \\ref{roughgg}, and the asymptotic upper curvature is invariant under rough-isometry, so\n\\[K_u(\\G(f,\\mathcal C))=K_u(\\G(f,\\mathcal C')). \\]\nTherefore, the asymptotic upper curvature $K(f)$ is well-defined in equation \\eqref{asyf}.\n\nA \\emph{Latt\\`es map} $f\\:\\widehat\\C\\ra \\widehat\\C$ is a rational map that is obtained from a finite quotient of a conformal torus endomorphism, i.e., the map $f$ satisfies the following commutative diagram:\n\\begin{equation}\\label{lat}\n \\begin{CD}\n\\T @>\\bar{A}>> \\T\\\\\n@V\\Theta VV @VV\\Theta V\\\\\n\\widehat\\C @>f>> \\widehat\\C\n\\end{CD}\n\\end{equation}\nwhere $\\bar A$ is a map of a torus $\\T$ that is a quotient of an affine map of the complex plane, and $\\Theta$ is a finite-to-one holomorphic map. Latt\\`es maps were the first examples of rational maps whose Julia set is the whole sphere $\\widehat \\C$, and a Latt\\`es map is an expanding Thurston map. In \\cite{YinLattes}, we have the following combinatorial characterization of Latt\\`es maps:\n\\begin{thm}[Yin, 2011] \\label{main0}\nA map $f\\:\\S^2\\ra \\S^2$ is topologically conjugate to a Latt\\`es map if and only if the following conditions hold:\n\\begin{itemize}\n \\item $f$ is an expanding Thurston map;\n \\item $f$ has no periodic critical points;\n \\item there exists $c>0$ such that $D_n\\geq c(\\deg f)^{n\/2}$ for all $n>0$.\n\\end{itemize}\n\\end{thm}\n\nThis leads to an curvature characterization of Latt\\`es maps as follows.\n\n\\begin{thm} \\label{main2}\nLet $f \\: \\S^2 \\ra \\S^2$ be an expanding Thurston map\nThe asymptotic upper curvature of $f$ satisfies\n\\[K_u(f)\\geq-\\frac14\\log^2(\\deg f). \\]\nIf in addition, the map $f$ has no periodic critical points, then the tile graph $\\G=\\G(f)$ is an AC$_u(\\kappa)$-space with\n\\[ \\kappa= -\\frac14\\log^2(\\deg f),\\]\nif and only if the map $f$ is topologically conjugate to a Latt\\`es map.\n\\end{thm}\n\n\\begin{proof}\nThe first part follows directly from the definition of asymptotic upper curvature of $f$ and from Theorem \\ref{acku}.\n\nIf $f$ is topologically conjugate to a Latt\\`es map, then by Corollary \\ref{cormain}, there exists a visual metric on $\\S^2$ with respect to $f$ with expansion factor $\\Lambda=\\deg(f)^{1\/2}$. By Proposition \\ref{samevisual}, there exists a visual metric on $\\partial_{\\infty}\\G$ in the sense of Gromov hyperbolic spaces with expansion factor $\\Lambda=\\deg(f)^{1\/2}$. By Theorem \\ref{acku}, the Gromov hyperbolic space $\\G$ is an AC$_u(\\kappa)$-space with \\[\\kappa=-\\frac14\\log^2(\\deg f).\\]\n\nIf $\\G$ is an AC$_u(\\kappa)$-space with\n\\[ \\kappa= -\\frac14\\log^2(\\deg f),\\]\nthen for all $X,X'\\in \\G$ and all finite sequences $X_0=X,X_1,\\ldots,X_n=X'$ in $X$,\n\\begin{eqnarray}\\label{acspaceeq1}\n(X,X')\\geq \\min_{i=1,2,\\ldots,n}(X_{i-1},X_i)-\\frac{\\log n}{\\log(\\deg f)^{1\/2}}-c.\n\\end{eqnarray}\n\nLet $D_n$ be the minimum number of $n$-tiles needed to join opposite sides of Jordan curve $\\mathcal C$ as defined in \\label{defdn}, for $n>0$. Let $P_n=X_1\\ldots X_{D_n}$ be an $n$-tile chain joining opposite sides of $\\mathcal C$.\nBy the equation \\eqref{acspaceeq1}, we have\n\\begin{eqnarray*}\n(X_1,X_{D_n})\\geq \\min_{i=1,2,\\ldots,D_n}(X_{i-1},X_i)-\\frac{\\log D_n}{\\log(\\deg f)^{1\/2}}-c,\n\\end{eqnarray*}\nso\n\\begin{eqnarray} \\label{logdndegf}\n\\frac{\\log D_n}{\\log(\\deg f)^{1\/2}}\\geq \\min_{i=1,2,\\ldots,D_n}(X_{i-1},X_i)-(X_1,X_{D_n})-c.\n\\end{eqnarray}\n\nBy equation \\eqref{gp}, we have\n\\begin{eqnarray} \\label{XiXi}\n(X_{i-1},X_i) &=&\\frac12[\\ell(X_{i-1})+\\ell(X_{i})-\\eta(X_{i-1},X_i)]+1 \\nonumber\\\\\n&=& \\frac12[2n-\\eta(X_{i-1},X_i)]+1\\\\\n&\\geq & n+1-\\frac14N\n\\end{eqnarray}\nwhere $N>0$ is a constant only depending on $f$ as in Lemma \\ref{noperiodic}, i.e. the constant $N$ is the upper bound of the degree of $f^n$ at any point in $\\S^2$.\nApplying equation \\eqref{XiXi} and Lemma \\ref{pdiam} to equation \\eqref{logdndegf}, we have\n\\begin{eqnarray*}\n\\frac{\\log D_n}{\\log(\\deg f)^{1\/2}}&\\geq &\\min_{i=1,2,\\ldots,D_n}(X_{i-1},X_i)-(X_1,X_{D_n})-c\\\\\n&\\geq& n+1-\\frac14N- \\log_{\\Lambda}\\big(\\diam(X_1\\cup X_{D_n})\\big)+\\log_{\\Lambda}k-c, \\\\\n\\end{eqnarray*}\nwhere $k\\geq 1$ only depends on $f$ and $\\mathcal C$ as in Lemma \\ref{pdiam}, and $N>0$ only depends on $f$.\nLet $D$ be the maximum of diameters of the two $0$-tiles, then\n\\[\\log_{\\Lambda}\\big(\\diam(X_1\\cup X_{D_n})\\big)\\leq D. \\]\nSo\n\\begin{eqnarray*}\n\\frac{\\log D_n}{\\log(\\deg f)^{1\/2}}\n&\\geq& n+1-\\frac14N- \\log_{\\Lambda}(D)+\\log_{\\Lambda}k-c \\\\\n&=& n+C.\n\\end{eqnarray*}\nHere the constant\n\\[C=1-\\frac14N- \\log_{\\Lambda}(D)+\\log_{\\Lambda}k-c \\]\nonly depends on $f$ and $\\mathcal C$.\nHence, we have\n\\begin{eqnarray*}\n{\\log D_n}&\\geq & (n+C) {\\log(\\deg f)^{1\/2}}.\n\\end{eqnarray*}\nTherefore, we have\n\\begin{eqnarray*}\n{D_n}&\\geq & C'(\\deg f)^{n\/2},\n\\end{eqnarray*}\nwhere $C'=(\\deg f)^{C\/2}$ only depends on $f$ and $\\mathcal C$.\nBy Theorem \\ref{main0}, the map $f$ is topologically conjugate to a Latt\\`es map.\n\\end{proof}\n\n\\newpage\n\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThe circum-galactic medium (CGM) is the interface between cold flows from the intergalactic medium onto a galaxy, and hosts hot halo gas and material ejected from galaxies \\citep[for reviews, see][]{Putman12,Tumlinson17}. With various processes in galaxy evolution consuming (e.g. star formation) and removing (e.g. winds) gas, the CGM is shaped by the processes internal to the galaxy. Early progress in the study of the CGM came from connecting absorption lines in quasar (QSO) spectra with galaxies imaged in the foreground, tracing the extent and properties of the CGM gas as a function of the host galaxy's properties \\citep[e.g.][]{Bergeron86,Bowen95,Lanzetta95,Adelberger05,Chen10,Steidel10,Bordoloi11,Prochaska11,Turner14}. Building on these foundations, our understanding of the CGM has been significantly improved through surveys with the Hubble Space Telescope (HST) Cosmic Origins Spectrograph \\citep[COS;][]{Green12}. The first of several surveys of the CGM surrounding low redshift galaxies was the COS-Halos survey \\citep{Tumlinson13} which targetted the CGM around 44 $\\sim$L$^{\\star}$ galaxies, demonstrating that the properties of the CGM differ depending on whether the central galaxy is passive or star-forming \\citep[defined using a specific star formation rate cut of sSFR$\\rm{=10^{-11} yr^{-1}}$;][]{Tumlinson11,Werk13,Borthakur16}. The COS-Halos team found a distinct lack of O\\ion{vi} around passive galaxies, while H\\ion{i}{} was found at the same strength around all galaxies \\citep{Tumlinson11,Thom12}. Additionally, connections have been made between the CGM and properties of the host galaxy, including: increased H\\ion{i}{} content of the CGM with larger interstellar medium (ISM) gas masses \\citep[COS-GASS;][]{Borthakur15}, the presence of extended gas reservoirs around galaxies of all stellar mass \\citep[COS-Dwarfs;][]{Bordoloi14}, and enhanced metal content around starbursting hosts \\citep[COS-Burst;][]{Borthakur13,Heckman17}.\n\n\nAn important stage in the evolution of galaxies is when their central supermassive black holes are actively accreting material. This active galactic nucleus (AGN) phase may be responsible for the removal of gas from star forming reservoirs within galaxies via winds and outflows \\citep{Veilleux05,Tremonti07,Sturm11,Woo17}, and has been associated with the evolution of galaxies off the star-forming main sequence to passive galaxies \\citep{Springel05,Schawinski07,Fabian12,Bluck14,Bluck16}. In addition, radio-mode feedback and radiation from the AGN keeps the CGM hot, buoyant, and consistently ionized \\citep[][]{McNamara07,Bower17,Hani17}, as well as preventing gas from returning to the host galaxy. Such processes have been proposed to be responsible for O\\ion{vi} bimodality seen in the CGM by COS-Halos without an active AGN \\citep{Oppenheimer17}.\n\nObservationally linking the environmental and feedback effects of AGN hosts with their CGM has primarily been done through the use of projected QSO-QSO pairs at higher redshifts. This technique has the added benefit of looking at the role of a stronger and weaker QSO radiation fields located in the respective transverse (background QSO) and line-of-sight (foreground QSO; i.e.~along the outflow) CGM \\citep{Bowen06,Farina13,Johnson15}. Cool gas traced by H\\ion{i}{} and Mg\\ion{ii} is anisotropically distributed about the QSO, with larger column densities of H\\ion{i}{} preferentially found along the transverse direction \\citep{QPQ1,Farina13}; suggesting that radiation from the QSO does not affect the transverse medium \\citep[][]{QPQ2,QPQ7,Farina14}. An excess of cool gas (relative to the intergalactic medium) has been found all the way out to one Mpc, with a stronger enhancement at smaller impact parameters \\citep{QPQ6}. When the QSO-QSO pairs are split by the bolometric luminosity of the QSO host, the Mg\\ion{ii} covering fraction is larger for high-luminosity QSOs (covering fraction of $\\approx 60$\\% for luminosities of ${\\rm L_{Bol}\\geq45.5}$ erg s$^{-1}$) compared to low luminosity QSOs \\citep[$\\approx20$\\%, ${\\rm L_{Bol}\\leq45.5}$ erg s$^{-1}$;][]{Johnson15}. All of these observations of excess cool gas around luminous QSOs is suggestive of either a viewing angle effect with the ionizing radiation exciting cool gas along the line of sight to the QSO, or an environmental effect of haloes hosting massive QSOs such as debris from galaxy interactions fuelling QSO activity \\citep[][]{QPQ6,Farina14,Johnson15}.\n\n\n\n\n\n\n\nMost of the work described above has focused on high luminosity quasars. However, there has been little focus on how the less luminous but more common Seyfert-like AGN shape their surrounding CGM. In the only observational study of the CGM surrounding Seyfert galaxies, \\cite{Kacprzak15} found a low (10\\%) Mg\\ion{ii} $\\lambda$ 2796 \\AA{} covering fraction around 14 AGN (between 100 and 200 kpc) in the transverse direction relative to field and QSO host galaxies, but a reservoir of cool gas still exists along the line of sight to the AGN. They suggest that AGN-driven outflows are destroying the cool gas in the transverse direction (i.e.~along the outflow), suggesting that the difference between their observations of the CGM of AGN-dominated galaxies with previous observations of QSOs \\citep[e.g.][]{QPQ6} is caused by the viewing angle of the AGN.\n\nPredictions from zoom-in simulations of galaxies taken from the EAGLE cosmological simulation \\citep{Schaye15} suggest that radiative feedback from the AGN should ionize the gas out to a distance of two virial radii \\citep{Oppenheimer13,Segers17}. After implementing non-equilibrium ionization into their models, \\cite{Oppenheimer13} have predicted that AGN proximity fossil zones exist around galaxies that host (or have hosted) bright AGN, with the metals remaining in an over ionized state for several megayears \\citep[depending on the luminosity duty cycle and lifetime of the AGN;][]{Segers17,Oppenheimer18}. However, the detailed CGM properties in simulations can be quite sensitive to the size of the CGM clouds, implementation of feedback, and different recipes between codes \\citep{Stinson12,Gutcke17,Nelson17}.\n\n\n\n\n\nIn this paper, we investigate the observational properties of the CGM around galaxies hosting Type II Seyfert AGN (which we will henceforth simply refer to as AGN). We measure the rest-frame equivalent widths (EWs) of a range of ionization species present in the CGM material probed by QSO sightlines near 20 AGN-host galaxies. We provide a systematic comparison to non-AGN galaxies observed in the literature to quantify whether the CGM around AGN host galaxies is different from their counterparts. Throughout the paper, we assume a flat $\\Lambda$CDM Universe with $H_{0}=67.8~{\\rm km~s^{-1}~Mpc^{-1}}$ and $\\Omega_{M}=0.308$ \\citep{Planck15}.\n\n\n\n\n\\section{Data}\n\\subsection{Sample selection and properties}\n\\label{sec:SampProps}\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{tb_BPT_Lbol.pdf}\n\\caption{The BPT diagram of all SDSS galaxies with spectroscopic observations (blue shaded region; only showing SDSS galaxies with $>5\\sigma$ detections of diagnostic emission lines). The solid circles show the COS-AGN galaxies, and are coloured based on their AGN luminosity. The dashed pink and green lines denote the~\\citet[K01]{Kewley01} and~\\citet[K03]{Kauffmann03} cuts typically used to select AGN and composite galaxies. LINERS classified using the \\citet{Kewley06} emission line metrics (see Table \\ref{tab:SightProps}) are denoted with a white dot on top of the datapoint.}\n\\label{fig:BPT}\n\\end{center}\n\\end{figure}\n\n\n\nThe QSO sightlines through the CGM of AGN galaxies were selected by cross-matching coordinates of Sloan Digital Sky Survey \\citep[SDSS;][]{Abazajian09} galaxies hosting AGN with the locations of UV-bright QSOs ($17300$m\\AA{}) between 100 and 200 kpc of the AGN. The Mg\\ion{ii} EW threshold adopted by \\cite{Kacprzak15} is typical of that used to select strong H\\ion{i}{} absorbers at low redshifts \\citep[log(N(H\\ion{i}{})\/cm$^{-2}$)$\\gtrsim18.5$;][]{Rao06}. Translating this Mg\\ion{ii} EW threshold into a corresponding Ly$\\alpha$ EW theshold for these column densities of gas yields a much higher than the threshold used in this work (EW $\\gtrsim 1300$ m\\AA{} --- assuming a minimum broadening parameter of 5 km~s$^{-1}${} --- compared to EW $>124$ m\\AA{} for COS-AGN). Using this larger threshold, only one of the COS-AGN sightlines has an EW $\\gtrsim 1300$ m\\AA{}, giving a consistent result with the observations from \\cite{Kacprzak15}. \n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=\\textwidth]{tb_covering_fractions_R.pdf}\n\\caption{Covering fractions (with 1$\\sigma$ errors) of gas within $\\pm500$ km~s$^{-1}${} of the host galaxy for a variety of species measured in COS-AGN and control-matched samples. Top panel: The global covering fractions measured across all impact parameters from Table \\ref{tab:covfracs} are shown for the COS-AGN galaxies (green circles), and the control-matched passive (red squares) and star-forming (blue diamonds) galaxies. Bottom panels: The covering fractions of an individual species as a function of impact parameter ($\\rho_{\\rm imp}${}), split into two bins by the median $\\rho_{\\rm imp}${} of the COS-AGN sample (164 kpc). The EW thresholds (EW$_{\\rm thrsh}$) used to determine the covering fraction for each species (including the measurements presented in the top panel) are given above the corresponding species panel in the bottom two rows. The horizontal error bars represent the entire range of $\\rho_{\\rm imp}${} probed by each sample within the respective bin. The three points are offset from the centre of each bin by a small amount for clarity. No points are shown for a species that do not have spectral coverage of the corresponding absorption line. }\n\\label{fig:CovFracR}\n\\end{center}\n\\end{figure*}\n\n\n\n\n\n\\subsubsection{Relative EW analysis}\n\nFigure \\ref{fig:EWRho} shows the raw EW values for a variety of ionic species as a function of $\\rho_{\\rm imp}${}. The COS-AGN points are colour-coded by their L$_{\\rm AGN}${}. The control-matched sample is shown as black points, while the grey points are the remaining un-matched literature sightlines. The bold COS-AGN points are CGM sightlines that have spectroscopic companions (see Section \\ref{sec:SampProps}). The top left panel demonstrates that the Ly$\\alpha$ EWs for the COS-AGN follow the general trend of decreasing EW as a function of $\\rho_{\\rm imp}${} seen in previous low redshift studies \\citep[][]{Chen10, Werk14,Borthakur16}. For metal species, the lack of detections in both the COS-AGN and control sample makes a comparative analysis difficult. For the remainder of this section, we focus only on the Ly$\\alpha$ EWs.\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=\\textwidth]{tb_EW_rho_all.pdf}\n\\caption{The rest-frame equivalent widths (within $\\pm500$ km~s$^{-1}${} of the host galaxy) as a function of impact parameter. The coloured circles (EW detections) and triangles (EW upper limits) represent the COS-AGN sample, and are colour coded by the bolometric luminosity of the AGN (L$_{\\rm AGN}${}). The median error bar on the COS-AGN EW measurements is given by the black error bar in the top right region of each panel. The black squares show the EW values of the control-matched galaxies, while the grey squares denote the rest of the literature comparison sample. Data outlined with a thick black line represent COS-AGN systems with nearby galaxies. COS-AGN galaxies flagged as LINERs are indicated by small white dots on top of the respective data points.}\n\\label{fig:EWRho}\n\\end{center}\n\\end{figure*}\n\nIn order to quantify any difference between the COS-AGN and control samples, we calculate $\\Delta$log(EW\/m\\AA), which is defined as ${\\rm \\Delta log(EW\/m\\AA) = log[EW_{AGN}\/median(EW_{Controls})]}$, such that a positive ${\\rm \\Delta log(EW\/m\\AA)}${} would imply that the CGM surrounding the AGN has a larger EW than the median of its control-matched galaxies. The left panel of Figure \\ref{fig:deltaEWRho} shows $\\Delta$log(EW) for Ly$\\alpha$ as a function of $\\rho_{\\rm imp}${} for the COS-AGN galaxies. For reference, the grey band represents the interquartile range of $\\Delta$log(EW) for the entire literature sample matched to itself. The right panel shows the distributions of $\\Delta$log(EW) for the COS-AGN (orange) and literature (grey) galaxies, with medians of the distributions indicated by the arrows. \n\nTo include the non-detections of the controls in the analysis, we calculate the median EW of the controls twice: once including limits as if they were detections, and once setting the non-detected EWs to 0~m\\AA{}. These median EWs span the range of true median EW if the absorption lines were actually detected. For this calculation, we only include non-detections when the upper limits are more sensitive (i.e. smaller) than the largest detected EW as these limits are constraining enough to affect the median value. The corresponding $\\Delta$log(EW) range is shown on Figure \\ref{fig:deltaEWRho} as the thick grey errorbars. The 1$\\sigma$ jackknife errors on $\\Delta$log(EW) are typically smaller than the size of the points.\n\nThe median $\\Delta$log(EW) of the COS-AGN sample is enhanced by $+0.10\\pm0.13$ dex relative to the controls. Repeating this control-matching experiment for the literature sample yields a median $\\Delta$log(EW) of $0.00\\pm0.28$. Note that the errors on these median $\\Delta$log(EW) represent the median absolute deviation (MAD) of the distribution. A KS test rejects the null hypothesis that the distributions of $\\Delta$log(EW) for the COS-AGN and control samples are same at 20\\% confidence. When the LINER galaxies are removed from the COS-AGN sample, the median $\\Delta$log(EW) changes to $+0.10\\pm0.15$, and the KS test yields a rejection of the null hypothesis at 14\\% confidence.\n\n\n\n\n\n\n\n\nTo check if there is any effect from splitting the control sample into star-forming and passive galaxies, we measured the $\\Delta$log(EW) for all star-forming galaxies in the non-AGN literature sample to their control-matched passive counterparts. The median $\\Delta$log(EW) obtained for star-forming galaxies relative to the matched passive controls is $+0.03\\pm0.26$, while a KS-test reveals the null hypothesis of the $\\Delta$log(EW) distributions for the star-forming and passive galaxies are the same is rejected at 38\\% confidence. Therefore there is no significant difference in the offset of the AGN hosts from the controls that would be caused by the star formation rate.\n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=\\textwidth]{tb_dEW_rho_HI.pdf}\n\\caption{The difference in the Ly$\\alpha$ equivalent width of the COS-AGN sightlines relative to their control matched counterparts ($\\Delta$log(EW\/m\\AA{})) as a function of impact parameter ($\\rho_{\\rm imp}${}). The EW measure material within $\\pm500$ km~s$^{-1}${} of the host galaxy. The points are colour-coded by the bolometric luminosity of the AGN (L$_{\\rm AGN}${}). The errorbars denote how the maximal shift on including control matched EW upper limits in the calculation of $\\Delta$log(EW\/m\\AA{}). For reference, the horizontal grey band represents the interquartile range of $\\Delta$EW of the literature sample control matched with itself. The normalized distributions of $\\Delta$EW for both the COS-AGN and control galaxies are shown in the right panel, with the median of each histogram given by an arrow. Data outlined by a thick black line are COS-AGN sightlines flagged as having nearby galaxies. COS-AGN galaxies flagged as LINERs are indicated by small white dots on top of the respective data points.}\n\\label{fig:deltaEWRho}\n\\end{center}\n\\end{figure*}\n\n\n\\subsubsection{Stacked Spectra}\n\n\nThe results in the previous sub-section indicate a possible (but not significant) difference in the Ly$\\alpha$ absorption properties of the COS-AGN sample, and possibly in some of the metal species as well. However, that analysis is limited by the modest S\/N of the data, the small sample size and lack of detections of metal species. Therefore, in this section we consolidate the data by stacking all of the COS-AGN spectra and comparing it to a stack of the control sample. In brief, all spectra are shifted to the rest-frame (using the redshift of the strongest component of the H\\ion{i}{} absorption profile) and rebinned to a linear dispersion of 0.064 \\AA{} pixel$^{-1}$ (similar to the resolution of the COS-AGN spectra). These rebinned spectra are mean combined without any weighting. We note that either using the systemic redshifts of the galaxies or using different weighting schemes does not significantly change the results.\n\nTable \\ref{tab:EWStack} gives the measured EWs of the various absorption lines of interest from the final stacked COS-AGN spectrum. We require that the absorption line be detected at $>3\\sigma$, otherwise the EW is set to a $3\\sigma$ upper limit. The stacked EW errors are calculated using a standard jackknife approach by removing each COS-AGN from the stacked spectrum and recalculating the EW. The numbers in brackets in each column give the EW offsets from removing the `strongest' (${jack,min}$) and `weakest' (${jack,max}$) absorber sightline from the stack in a jackknife fashion (i.e. these are the maximal variations in the EW from the jackknife), as well as the number of sightlines that contributed to the stack (N$_{spec}$). The four columns on the right are the measured EWs when the stacking process is only applied to COS-AGN sightlines split into two bins of log(L$_{\\rm AGN}${}) and $\\rho_{\\rm imp}${} at the median value of the COS-AGN sample (log(L$_{\\rm AGN}${}\/erg s$^{-1}$)$=42.9$ and $\\rho_{\\rm imp}${}$=164$ kpc; respectively).\n\n\n\\begin{table*}\n\\begin{center}\n\\caption{Measured EW of stacked spectra}\n\\label{tab:EWStack}\n\\begin{tabular}{lc|c|cc|cc|}\n\\hline\nIon & Line & \\multicolumn{5}{c}{Stacked EW ($_{jack,min}^{jack,max}$; N$_{spec}$) [m\\AA{}]} \\\\\n & [\\AA] & All sight-lines&\tlog(L$_{\\rm AGN}${}\/erg s$^{-1}$)$\\leq$42.9 & log(L$_{\\rm AGN}${}\/erg s$^{-1}$)$>$42.9& $\\rho_{\\rm imp}${}$\\leq$164 kpc& $\\rho_{\\rm imp}$$>$164 kpc \\\\\n\\hline\nH\\ion{i}{}\t & 1215\t & 739$\\pm$25 ($_{-69}^{+44}$; 14)\t & 937$\\pm$53 ($_{-116}^{+77}$; 7)\t & 539$\\pm$38 ($_{-66}^{+70}$; 7)\t & 799$\\pm$41 ($_{-68}^{+73}$; 7)\t & 680$\\pm$66 ($_{-176}^{+87}$; 7)\t\\\\\nC\\ion{ii}\t & 1036\t & $<$80 (2)\t & $<$80 (2)\t & . . . & . . . & $<$80 (2)\t\\\\\nC\\ion{ii}\t &\t1334& $<$37 (11)\t & $<$53 (6)\t & $<$50 (5)\t & $<$54 (5)\t & $<$51 (6)\t\\\\\nC\\ion{iv}\t & 1548\t & 194$\\pm$13 ($_{-40}^{+20}$; 11)\t & 236$\\pm$38 ($_{-89}^{+61}$; 5)\t & $<$194 (6)\t & 185$\\pm$17 ($_{-38}^{+29}$; 6)\t & 205$\\pm$39 ($_{-96}^{+53}$; 5)\t\\\\\nC\\ion{iv}\t & 1550\t & 131$\\pm$9 ($_{-24}^{+15}$; 13)\t & 132$\\pm$21 ($_{-56}^{+21}$; 6)\t & $<$167 (7)\t & $<$132 (7)\t & 159$\\pm$24 ($_{-52}^{+33}$; 6)\t\\\\\nN\\ion{v}\t & 1238\t & $<$32 (13)\t & $<$42 (6)\t & $<$47 (7)\t & $<$47 (7)\t & $<$43 (6)\t\\\\\nN\\ion{v}\t & 1242\t & $<$38 (10)\t & $<$46 (5)\t & $<$60 (5)\t & $<$51 (6)\t & $<$54 (4)\t\\\\\nO\\ion{i}\t &\t1302& $<$31 (12)\t & $<$45 (5)\t & $<$42 (7)\t & $<$47 (6)\t & $<$39 (6)\t\\\\\nO\\ion{vi}\t &\t1037& $<$379 (1)\t & $<$379 (1)\t & . . . & . . . & $<$379 (1)\t\\\\\nSiII\t & 1190\t & $<$29 (10)\t & $<$36 (5)\t & $<$46 (5)\t & $<$61 (3)\t & $<$32 (7)\t\\\\\nSi\\ion{ii}\t & 1190\t & $<$32 (9)\t & $<$44 (4)\t & $<$46 (5)\t & $<$61 (3)\t & $<$37 (6)\t\\\\\nSi\\ion{ii}\t & 1260\t & 88$\\pm$8 ($_{-28}^{+11}$; 15)\t & 94$\\pm$28 ($_{-75}^{+30}$; 6)\t & 84$\\pm$10 ($_{-17}^{+15}$; 9)\t & $<$101 (6)\t & 103$\\pm$18 ($_{-47}^{+21}$; 9)\t\\\\\nSi\\ion{iii}\t & 1206\t & 152$\\pm$16 ($_{-62}^{+18}$; 12)\t & 211$\\pm$37 ($_{-98}^{+33}$; 8)\t & $<$120 (4)\t & $<$98 (6)\t & 238$\\pm$40 ($_{-112}^{+32}$; 6)\t\\\\\nSi\\ion{iv}\t & 1393\t & $<$44 (11)\t & $<$69 (4)\t & $<$57 (7)\t & $<$59 (6)\t & $<$66 (5)\t\\\\\nFe\\ion{ii}\t & 1144\t & $<$29 (12)\t & $<$34 (7)\t & $<$52 (5)\t & $<$49 (5)\t & $<$35 (7)\t\\\\\nFe\\ion{ii}\t & 1608& $<$102 (8)\t & $<$94 (2)\t & $<$134 (6)\t & $<$129 (5)\t & $<$168 (3)\t\\\\\n\\hline\n\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\n\n\\begin{table*}\n\\begin{center}\n\\caption{Measured EW of stacked control spectra}\n\\label{tab:EWStackControl}\n\\begin{tabular}{lc|c|cc|cc|}\n\\hline\nIon & Line & \\multicolumn{5}{c}{Stacked EW ($_{jack,min}^{jack,max}$; N$_{spec}$) [m\\AA{}]} \\\\\n & [\\AA] & All sight-lines&\t log(sSFR\/yr$^{-1}$)$<-11$ & log(sSFR\/yr$^{-1}$)$\\geq-11$ & $\\rho_{\\rm imp}${}$\\leq$164 kpc& $\\rho_{\\rm imp}$$>$164 kpc \\\\\n\\hline\nH\\ion{i}{}\t & 1215\t & 577$\\pm$12 ($_{-39}^{+24}$; 43)\t & 485$\\pm$30 ($_{-84}^{+52}$; 19)\t & 649$\\pm$21 ($_{-65}^{+40}$; 24)\t & 781$\\pm$20 ($_{-50}^{+50}$; 25)\t & 239$\\pm$15 ($_{-37}^{+30}$; 18)\t\\\\\nC\\ion{ii}\t & 1334\t & 36$\\pm$3 ($_{-11}^{+6}$; 43)\t & $<$29 (19)\t & 46$\\pm$5 ($_{-20}^{+9}$; 24)\t & 62$\\pm$5 ($_{-18}^{+10}$; 25)\t & $<$31 (18)\t\\\\\nSi\\ion{ii}\t & 1190\t & $<$22 (15)\t & $<$39 (5)\t & $<$26 (10)\t & $<$22 (15)\t & . . . \\\\\nSi\\ion{ii}\t & 1260\t & 56$\\pm$4 ($_{-14}^{+7}$; 33)\t & 85$\\pm$10 ($_{-28}^{+17}$; 16)\t & $<$42 (17)\t & 103$\\pm$6 ($_{-20}^{+8}$; 23)\t & $<$55 (10)\t\\\\\nSi\\ion{iii}\t & 1206\t & 98$\\pm$4 ($_{-15}^{+9}$; 39)\t & 77$\\pm$9 ($_{-28}^{+14}$; 21)\t & 126$\\pm$8 ($_{-24}^{+15}$; 18)\t & 156$\\pm$7 ($_{-20}^{+14}$; 25)\t & $<$47 (14)\t\\\\\nSi\\ion{iv}\t & 1393\t & $<$35 (37)\t & $<$44 (17)\t & $<$55 (20)\t & 58$\\pm$5 ($_{-15}^{+14}$; 23)\t & $<$65 (14)\t\\\\\nSi\\ion{iv}\t & 1402\t & $<$27 (37)\t & $<$34 (17)\t & $<$41 (20)\t & $<$33 (21)\t & $<$44 (16)\t\\\\\n\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\n\nA similar procedure was completed for all the spectra in the literature sample that were used in the control sample. Each control sightline in the stacked spectrum was weighted by the number of times the sightline was matched to a unique AGN host (such that the most frequently matched control sightline was given a higher weighting). Although there is very little difference in the measured EW without such a weighting, we elect to use this weighting scheme such that the derived jackknife errors represent the true range in EWs when a given sightline is excluded. The EWs measured from the stacked spectrum of the control galaxies is given in Table \\ref{tab:EWStackControl}. We repeat the stacked EW calculations of splitting the controls by the median $\\rho_{\\rm imp}${} of the COS-AGN sample and log(sSFR\/yr$^{-1}$)$=-11$ (i.e. whether the controls are star-forming or passive; see Table \\ref{tab:EWStackControl}). Note that the COS-GASS results \\citep{Borthakur15,Borthakur16} focus on a smaller subset of species (H\\ion{i}{}, C\\ion{ii}, Si\\ion{ii}, Si\\ion{iii}, and Si\\ion{iv}), thus we are only able to provide a stacked spectrum for these species. We point out that the Si\\ion{iv} EW of the control stacked spectrum is poorly constrained due to an uncertain continuum in some of the literature sample spectra.\n\n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=\\textwidth]{tb_dEW_stack_full.pdf}\n\\caption{$\\delta$log(EW) measured from the stacked spectra for a variety of detected species. Top panel: The $\\delta$log(EW) is shown for the COS-AGN sightlines relative to: all controls (black circles), AGN and controls for small ($\\rho_{\\rm imp}$$<164$ kpc; red circles) and large ($\\rho_{\\rm imp}${}$\\geq164$ kpc; blue circles) impact parameters. Errorbars on all points represent the possible range in $\\delta$log(EW) spanned by the maximal jackknife errors (i.e.~$_{-jack,min}^{+jack,max}$). Upper and lower limits are plotted when an absorption line is detected in either the COS-AGN or control stack (respectively), but not the other. We note that the lower errorbar on the $>164$ kpc $\\delta$log(EW) for Si\\ion{iii} 1206~\\AA{} line represents how shallow the lower limit becomes upon including the jackknife errors. The bottom ten panels show the absorption profiles of the stacked spectra (the COS-AGN spectrum in the middle row; the control spectrum in the bottom row) for each element. The colour coding of which sightlines are included is the same as in the top panel. The absorption lines are offset vertically by 0.5 in relative flux for clarity.}\n\\label{fig:StackdEW}\n\\end{center}\n\\end{figure*}\n\n\nWe repeat a similar differential EW analysis as above, where we calculate $\\delta$log(EW)$={\\rm log(EW_{AGN}) - log(EW_{Control})}$ using the EWs derived from the respective stacked spectra (Tables \\ref{tab:EWStack} and \\ref{tab:EWStackControl}). The top panel of Figure \\ref{fig:StackdEW} provides a comparison of EW between all the AGN sightlines and all controls (black points). The errorbars represent the combination of the maximal jackknife errors, providing the entire range of possible $\\delta$log(EW) from removing a single sightline from each sample. The stacking confirms the results from Figure \\ref{fig:deltaEWRho}, where the COS-AGN sightlines have an enhanced Ly$\\alpha$ relative to the controls, with an enhancement of $\\delta$log(EW)$\\approx0.17$ dex. However, there is a negligible difference in $\\delta$log(EW) for the metal species detected at all impact parameters. Removing the absorption surrounding the LINER galaxies does not change the qualitative picture presented in Figure \\ref{fig:deltaEWRho}; the measured $\\delta$EW from the LINER-free stacks shift $\\leq0.08$ dex (in either direction). \n\nThe additional red and blue points in the top panel of Figure \\ref{fig:StackdEW} show the values of $\\delta$log(EW) calculated only when including only sightlines with $\\rho_{\\rm imp}${} smaller or larger than the median value of the COS-AGN sample (164 kpc; respectively). Splitting the stacking into the `inner' and `outer' CGM around AGN hosts uncovers that the enhancement seen in the Ly$\\alpha$ $\\delta$log(EW) is driven by the the COS-AGN sightlines that probe $\\rho_{\\rm imp}${}$\\geq164$ kpc ($\\delta$log(EW)$=+0.45\\pm0.05$~dex; using standard jackknife EW errors). This enhancement in the outer CGM gas is also seen by the detections of the cool gas tracers Si\\ion{ii} ($\\delta$log(EW)$>0.27$~dex) and Si\\ion{iii} ($\\delta$log(EW)$>0.75$~dex) in COS-AGN, even after removing the strongest metal absorber (towards QSO J0852+0313; see Figure \\ref{fig:J0852}), whilst these species are not detected in the stacked spectrum of the control galaxies. We remind the reader that this enhancement was suggested in our covering fraction analysis of Si\\ion{iii} (Figure \\ref{fig:CovFracR}). For the inner CGM, the stacked spectrum hints that there is a deficit of metal species around AGN galaxies relative to the control sample. The combination of the excess Ly$\\alpha$ EW and tentative Si\\ion{ii} and Si\\ion{iii} EW enhancements at high $\\rho_{\\rm imp}${} is suggestive that these EW enhancement are tracing the cool gas phase of the CGM, rather than just the H\\ion{i}{} gas kinematics. We note that the inner CGM distribution has a slightly higher median M$_{\\star}${} than the outer CGM bin (log(M$_{\\star}$\/M$_{\\odot}$){} of 10.8 dex relative to 10.5 dex), and the inner and outer CGM bins contain approximately the same ratio of passive and star-forming galaxies. We remind the reader that this discrepancy in the median M$_{\\star}${} of each bins is on the order of the size of our control matching tolerance.\n\n\n\n\n\n\\section{Discussion}\n\nIn the previous section, we demonstrated that the CGM around AGN hosts is not much different than the control-matched non-AGN hosts. Statistically significant differences are found in the analysis of the stacked spectrum; the COS-AGN systems have a higher EW of Ly$\\alpha$ (and potentially Si\\ion{ii} and Si\\ion{iii} as well) relative to their non-AGN host counterparts at high impact parameters ($\\rho_{\\rm imp}${} $\\geq 164$ kpc; $\\delta$log(EW\/m\\AA{})$=+0.45\\pm0.05$~dex). The kinematics of the gas traced by the absorption show the gas is likely bound to the halo, whilst no strong kinematic offsets relative to their host are present. We now consider whether these observations are a result of the AGN directly influencing the CGM of the host galaxy, or an effect of the environments (either internal or external to the host galaxy) in which AGN are typically found.\n\n\n\n\\subsection{Are we seeing the effects of AGN feedback?}\n\n\\subsubsection*{Mock COS-AGN simulation}\n\\label{sec:Sims}\nThe radiation field of the AGN may be expected to have a profound effect on the ionization structure of the CGM, by enhancing the ionizing radiation field to which the surrounding gas is subjected, and in the case of metals even long after the AGN has turned off \\citep{Oppenheimer13,Segers17,Oppenheimer18}. To quantify the expected effects of turning on and off the AGN ionizing radiation spectrum on the CGM of a COS-AGN galaxy, we created a mock COS-AGN survey using previously run cosmological zoom-in hydrodynamical simulations with an additional AGN ionizing source following previous work on non-equilibrium ionization effects \\citep{Oppenheimer13,Oppenheimer16}. The purpose of these simulations is to isolate the effect of adding a constant AGN ionizing radiation source versus a control sample without AGN ionizing radiation to study the effect of the changed ionization structure of the CGM. This exploration is not meant to model or consider the effect of AGN feedback mechanically transforming the CGM via superwinds as the inclusion of the AGN radiation is not tied to the accretion onto the central super massive black hole. The AGN radiation is added to these simulations at the position of the central super massive black hole and only alters the ionization states of the CGM.\n\nFor our simulation suite we selected three representative galaxy haloes from \\cite{Oppenheimer16} that were chosen from the EAGLE volume \\citep{Crain15,Schaye15}. These three halos are representative of the properties of the COS-AGN sample at $z=0.075$ [log(M$_{\\star}$\/M$_{\\odot}$)$=10.3$, $10.9$, $11.0$, log(sSFR \/ yr$^{-1}$)$=-10.3$, $-10.7$, $-11.0$, residing in haloes log(M$_{200}$\/M$_{\\odot}$)$=12.1$, $12.8$, $13.3$; respectively]. The haloes were ran at EAGLE HiRes resolution (gas particle mass of $2.3\\times10^{5}~{\\rm M_{\\odot}}$, dark matter particle mass of $1.2\\times10^{6}~{\\rm M_{\\odot}}$, and softening length of 350 proper pc) using the Recal feedback prescription \\citep{Schaye15} and are zooms Gal001, Grp003, and Grp008 listed in Table 1 of \\citet{Oppenheimer16} from the initial conditions ($z=127$).\n\nTo include the effects of the AGN ionizing spectrum on the CGM at different luminosities, we inserted an additional AGN ionizing source \\citep[with an ionizing spectrum from][]{Sazonov04} placed at the centre of the galaxy, instantaneously affecting the radiation field from $z=0.1$ onwards and reaches equilibrium at $z=0.075$. We note that this AGN radiation model has no dynamical effect on the gas accretion or outflows from the AGN, and is not tied to the accretion of material onto the central super-massive black hole. A range of AGN luminosities was used to match the COS-AGN luminosities (log(L$_{\\rm AGN}$\/erg s$^{-1}$){}=42--44 dex, in increments of 0.5 dex), and a control run with no AGN radiation was included for creating the control sample. The time-dependent ionization from the AGN in addition to the \\cite{Haardt01}\\footnote{The \\cite{Haardt01} background is adopted as it better reproduces the statistics of the Ly$\\alpha$ forest in the EAGLE volumes and other simulations \\citep{Rahmati15,Kollmeier14}.} ultra-violet background is followed using the \\cite{Richings14a} ionization network. \n\n\nA mock COS-AGN sample was then generated from this simulation suite to match the properties of the COS-AGN sample. For each observed galaxy in COS-AGN, a simulated galaxy was selected by matching the stellar mass (with a log(M$_{\\star}$\/M$_{\\odot}$){}$=\\pm0.5$~dex tolerance), star formation rate (log(sSFR \/ yr$^{-1}$){}$=\\pm0.5$~dex tolerance), and AGN luminosity (log(L$_{\\rm AGN}$\/erg s$^{-1}$){}$=\\pm0.25$~dex) of the simulated haloes to each COS-AGN galaxy. If multiple matches were identified, a random simulated halo was selected such that a mock sample contains the same number of galaxies as the actual COS-AGN sample (20 galaxies). A mock spectrum was generated for each galaxy using the \\textsc{SpecWizard} package \\citep[][]{Theuns98,Schaye03} by placing a randomly oriented quasar sightline through the CGM at the corresponding impact parameter of the matched COS-AGN galaxy sightline. A control sample was generated in a similar fashion, but all AGN ionizing sources in the simulated galaxies were set to L$_{\\rm AGN}${}$=0$~erg~s$^{-1}$. The above procedure was repeated 200 times (including choosing another random halo if multiple haloes were matched), in order to produce 200 mock survey samples for both the AGN and control samples. These 200 realizations of the COS-AGN survey were needed to reproduce the range in measured EW variations for the metal species between the mock samples, and is used to quantify the expected spread in EWs from different orientations and from halo to halo variations. Further details on how these mock spectra were generated will be presented in a forthcoming paper (Horton et al., in prep.).\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=\\textwidth]{tb_sim_dEW_stack_full.pdf}\n\\caption{The top panel shows the measured $\\delta$EW using a stacked spectrum from the zoom-in simulations of AGN hosts, split into bins of $\\rho_{\\rm imp}${} (all $\\rho_{\\rm imp}${}, $<164$ kpc and $\\geq164$ kpc; black, red and blue points, respectively). The respective open symbols show the $\\delta$EW measured in the observations (i.e.~identical points presented in the top panel of Figure \\ref{fig:StackdEW}). Errorbars were calculated using the jackknife approach identical to that used for the observations. The bottom panels show the simulated stacked velocity profiles for each species for the AGN (middle row) and non-AGN control run (bottom row). The dashed lines represent the continuum levels of the absorption profiles. The Ly$\\alpha$ are staggered by 0.5 in relative flux for clarity. The relative flux scale has been enlarged by a factor of four for the metal line profiles (indicated by a $\\times4$ in the bottom right of each panel), with the dashed continuum lines separated by a relative flux of 0.125. For the outer CGM, the simulations predict little change in the EW of H\\ion{i}{} and Si\\ion{iii} due to the ionizing radiation from the AGN, which is in stark contrast to the COS-AGN observations similarly presented in Figure \\ref{fig:StackdEW}.}\n\\label{fig:Sims}\n\\end{center}\n\\end{figure*}\n\n\\subsubsection*{Stacked mock COS-AGN spectra}\nWe assessed the relative impact of the AGN ionizing radiation on the measured simulated EWs by repeating the stacking procedure above, but with the simulated data. We computed the relative EW ratio $\\delta$log(EW) as with the observations. Note that this relative EW analysis removes any systematic differences between the observed and simulated EWs caused by assumptions in the physics models (e.g. feedback) or ability to resolve small clouds in the simulations \\citep{Schaye07,Stinson12,Crighton15,Gutcke17,Nelson17}. A quantitative comparison between the simulated and observed EWs will be presented in Horton et al.~(in prep.), although we note that the simulations produce a systematically weaker H\\ion{i}{} EW from the observations, while metal lines show better agreement with the observed data \\citep[see][]{Oppenheimer16,Oppenheimer17}. \n\nThe simulated stacked sightline spectra were created from the mocks by centering the velocity profile on the redshift of the simulated galaxy, rebinning the mock spectra to 15 km~s$^{-1}${}, and mean stacking these rebinned spectra. Three simulated stacked spectra were created for the same three bins of impact parameters that were previously used for the observed data: all $\\rho_{\\rm imp}${}, $\\rho_{\\rm imp}${}$<164$ kpc, and $\\rho_{\\rm imp}${}$\\geq164$ kpc. Jackknife errors on the measured EWs were computed by removing all 200 mock spectra associated with the matched COS-AGN sightline which contributed the most and least to the derived EW. This approach for calculating the simulated jackknife error is identical to that used for the observations. Using the same analysis for our COS-AGN sightlines, Figure \\ref{fig:Sims} presents the $\\delta$log(EW) analysis for these simulated stacked spectra (solid points in the top panel) to quantify the effect of including AGN radiation on the CGM relative to the control simulations. The inclusion of an AGN ionizing spectrum results in a negligible change in the EW as a function of impact parameter for Ly$\\alpha$. At high impact parameters ($\\rho_{\\rm imp}${}$\\geq 164$~kpc), the simulated results are in contrast to the observations (open points on Figure \\ref{fig:Sims}) where the COS-AGN sample shows a significant enhancement in the Ly$\\alpha$ EW relative to the control sample, as well as a potential enhancement for Si\\ion{ii} and Si\\ion{iii}. We do note that for the inner impact parameter bin, the results from the simulation are consistent with the lack of metal species detected of our COS-AGN stacked spectrum ($\\rho_{\\rm imp}${}$<164$ kpc; Figure \\ref{fig:StackdEW}). \n\n\nCompounded by the fact that the H\\ion{i}{} ionizing radiation from a \\cite{Sazonov04} AGN at the observed $\\rho_{\\rm imp}${} of the COS-AGN sample is weaker ($\\lesssim25\\%$ ) than the ionizing radiation from the UV background \\citep{Haardt01} at most of the impact parameters probed by COS-AGN (see the bottom right panel of Figure \\ref{fig:SampDists}), it is unlikely that photoionizing radiation from the AGN is responsible for the observed difference in EWs between the COS-AGN and the control samples. We note that at the impact parameters of COS-AGN, the light travel time of radiation from the AGN ($\\sim5\\times10^{5}$ yr) is comparable to the AGN's lifetime \\citep[$\\lesssim10^{6}$ yr;][]{Schirber04,Goncalves08,Furlanetto11,Keel12}, implying that photoionization events caused by an AGN likely require previous AGN cycle(s) to have ionized the CGM gas, and have remained in the predicted long-lived fossil zone around the AGN \\citep{Oppenheimer13,Segers17} provided an AGN has been active within the last several Myrs. Unfortunately, our simulations predict that low ionization species cannot distinguish the presence of fossil zones in COS-AGN due to the low intensity of the photo-ionizing radiation.\n\n\nDespite being unable to probe the effects of the lower ionization species, ions such as C\\ion{iv}, N\\ion{v}, and O\\ion{vi} are better indicators of these proximity zone fossil. We highlight that in our simulated COS-AGN haloes, C\\ion{iv} and N\\ion{v} are still sensitive to the ionizing radiation of the harder AGN ionizing spectrum relative to the \\cite{Haardt01} UV background \\citep[see figure 3 in][]{Segers17}, as demonstrated by the enhanced EWs for these ionization species out to 300 kpc. However, to observe such an excess with N\\ion{v} 1238~\\AA{} would require spectra with S\/N of $\\sim30$ to detect an absorption line of the predicted strength displayed in Figure \\ref{fig:Sims}. The excess EW of C\\ion{iv} 1548~\\AA{} would be an excellent test of the effects of the AGN ionizing field as we have already detected absorption in our stacked COS-AGN spectra (Table \\ref{tab:EWStack}). Such a test would required observing the C\\ion{iv} 1548~\\AA{} covering fraction in the CGM of non-AGN galaxies from the control-matched sample.\n\n\\subsubsection*{AGN-driven winds and outflows}\nAn alternative form of feedback is AGN-driven winds or outflows \\citep[][]{Concas17, Fiore17,Woo17}. As the typical lifetime of an AGN ($\\lesssim10^6$ yr) is much smaller than the expected travel time of winds out to the impact parameters probed by COS-AGN \\citep[$\\sim10^8$ yr, assuming a constant, maximum velocity of $1000$ km~s$^{-1}${}; e.g.][]{Tremonti07,Veilleux13,Ishibashi15}, any signatures of winds or outflowing material would likely trace material expelled from a previous cycles of AGN activity or star-formation in the host galaxy \\citep[e.g.][]{Nedelchev17,Kauffmann17,Woo17}. The typically observed signatures of outflowing winds from a galaxy manifest as kinematic offsets of ionized emission or absorption lines from the host galaxy \\citep[e.g.][]{Bordoloi14b,Rubin14,Woo16,Heckman17,Perna17}, but as demonstrated in Figure \\ref{fig:Kinematics}, the Ly$\\alpha$ gas that we are probing in the COS-AGN sightlines does not have any strong bulk motion away from the host galaxy relative to the control matched sample. If such winds or outflows were driven by previous AGN or star-forming activity, the lack of kinematic offsets from the host galaxy suggests that these winds have dissipated over time, and should have deposited metals into the CGM \\citep[e.g.][]{Muzahid15,Turner15}. The low metal covering fraction at low impact parameter ($\\leq164$ kpc; Figure \\ref{fig:CovFracR}) suggests an absence of outflowing material polluting the CGM with metals, rejecting the notion that any recent (within 160~Myr) AGN-driven winds have enhanced the CGM. We note that the Ly$\\alpha$ EW at these column densities is more sensitive to the kinematics of the gas than to the amount of gas. Although we have rejected the possibility that AGN-driven winds are responsible for our results, it is possible that the gas is more turbulent in the CGM of AGN hosts compared to their control matches. To estimate the effects of turbulence, we would require observations of unsaturated low ionization metal absorption lines to verify if the EW enhancements are from enhanced column densities or kinematic broadening.\n\nGiven that AGN-driven winds can affect the absorption line profile by up to $\\approx \\pm 1000$ km~s$^{-1}${}, our adopted search window of $\\pm500$ km~s$^{-1}${} (Section \\ref{sec:DataEW}) could potentially miss gas present in a wind. We searched for the Ly$\\alpha$ profiles within $\\pm1000$ km~s$^{-1}${} and found two systems (J1214+0825 and J2133$-$0712) with minimal absorption outside the original window not associated with other absorption systems or the Galaxy. These two additional absorption components do not show any associated metal line absorption. As these missed components are small and narrow, the calculated flux-weighted velocity centroids presented in Figure \\ref{fig:Kinematics} would still be contained within the already identified absorption component. However, we remind the reader that our adopted search window of $\\pm500$ km~s$^{-1}${} is adopted to be consistent with methods used in the literature and control samples.\n\nRather than AGN feedback, it is possible that the effects we are seeing are from a different process co-eval or prior to the onset of AGN accretion. Several works have pointed out that AGN activity coincide with a recent starburst; with the AGN having significant accretion events at least $\\sim200$ Myr after the starburst has occurred \\citep{Wild07,Davies07,Wild10,Yesuf14} giving the neutral material time to propagate out to the impact parameters probed by COS-AGN \\citep{Heckman17}. With a sample of QSO sightlines probing the CGM around 17 low-redshift starburst and post-starburst galaxies, \\cite{Heckman17} have observed a similar signature of enhanced EWs of Ly$\\alpha$, Si\\ion{iii}, and C\\ion{iv} (the latter of which is not measured in our control sample) relative to a control-matched sample (matched in stellar mass and impact parameter). In the range of impact parameters and stellar masses probed by COS-AGN, the strength of our enhanced EW signature is consistent with the values probed by \\cite{Heckman17}. However, the results of \\cite{Heckman17} show strong offsets in the kinematics of the gas from the host galaxy \\citep[$\\approx 100$ km~s$^{-1}${}; see figure 5 from][]{Heckman17}, whereas the COS-AGN sightlines do not (bottom panel of Figure \\ref{fig:Kinematics}). Assuming the AGN activity was triggered by the starburst, a minimum delay time of 200 Myr could allow for any starburst-driven winds to dissipate and kinematic offsets to no longer be present at the impact parameters of the COS-AGN sample. Although this starburst picture provides a possible explanation of our observations, we caution that starbursts are not the only astrophysical event linked to AGN accretion activity. For example, mergers that trigger the AGN \\citep{Ellison11,Ellison13,Satyapal14,Silverman14,Goulding17} could potentially affect the surrounding CGM gas. Past and future work focussing on the CGM of galaxy mergers can further test this result \\citep[][Bordoloi et al. in prep.]{Johnson14,Hani17}.\n\n\n\n\\subsection{Are we seeing the effects from environment or other galaxy properties?}\nIf the AGN (or host galaxy) is not responsible for the observed differences in the CGM, an alternative is that the circumgalactic environment in which an AGN host is found is different. Results from \\emph{Quasars probing Quasars} \\citep{QPQ6}, \\cite{Farina14}, and \\cite{Johnson15} have all suggested that the excess of cool gas seen in the CGM out to 1 Mpc around $z\\approx1$ quasars is a result of residing in group environments. Although the excess of cool gas around quasars goes in the same direction as the 0.1 dex enhancement we find in the outer CGM of $z\\approx0.1$ COS-AGN galaxies (though we \\emph{do not} see an excess in the Ly$\\alpha$ EW in the inner CGM of AGN hosts, as seen for QSOs), the dark matter haloes of the AGN in our sample are typically an order of magnitude smaller than group dark matter haloes that host quasars. As stated in Section \\ref{sec:ControlMatching}, the $\\delta_{5}$ parameter provides an estimate of the environment. Given that differences in the distributions of $\\delta_{5}$ between AGN and star-forming galaxies in the SDSS vary on the order of a percent for a given stellar mass, it is likely that the enhanced EW of cool gas is not due to the contribution from a difference in the galaxy environment.\n\n\n\n\nGiven that we find an excess in the H\\ion{i}{} content of the outer CGM around AGN hosts, the results from \\cite{Borthakur13} which find a connection between the ISM and CGM gas properties would imply that the ISM would also host a large reservoir of H\\ion{i}{} gas. Such an enhancement in the ISM gas mass (relative to non-AGN galaxies) has been previously seen in AGN hosts \\citep[e.g.][]{Vito14}, where an excess of $\\sim0.2$ dex in gas mass is seen for AGN hosts similar to those probed in our sample. If such large gas reservoirs do exist in the ISM of AGN hosts, the AGN could be fuelled by the excess cool gas in the ISM, which in turn is fed by the cool CGM gas surrounding the AGN host. We note that other works such as \\cite{Fabello11} have found that the ISM of optically-selected AGN hosts contain the same H\\ion{i}{} gas mass as their star-forming counterparts. However, $\\sim$50\\% of the \\cite{Fabello11} AGN sample are so-called `composite' AGN (galaxies whose emission lines contain contributions from both star formation and AGN activity), which are not representative of the AGN hosts selected in our COS-AGN sample. \n\n\nA further test of this accretion picture would be to look for any orientation effects. If we are probing the gas reservoirs that are fuelling the AGN, we are likely to find the accreting gas along the major axis of the galaxy \\citep[e.g.][]{Kacprzak12,Nielsen15,Ho17}. Unfortunately, we do not have the ability to measure robust inclinations for many of our COS-AGN galaxies from SDSS imaging (see Figure \\ref{fig:postage}), and have too small of a sample size to produce a significant statistic. In addition, the \\cite{Ho17} sample are at much closer impact parameters ($\\lesssim 50$ kpc) whereas \\cite{Borthakur15} showed that at higher impact parameters (such as those probed by COS-AGN) there is no evidence of orientation effects for galaxies in COS-GASS. However, we do note that for the 4--5 sightlines that are probing along the edge of the disc relative to the 2--3 that are perpendicular to the disc, there is no significant difference in the median $\\Delta$log(EW). A larger sample would be required to test this explicitly.\n\n\n\n\\section{Summary}\nUsing a sample of 19 quasar sightlines through the circumgalactic medium (CGM) of 20 Type II Seyfert AGN and LINERs, we have demonstrated that there are mild differences in the rest-frame equivalent widths (EWs) of cool CGM gas around AGN hosts relative to their non-AGN counterparts. After matching in stellar mass and impact parameter, we find:\n\\begin{itemize}\n\n\\item[1.] The covering fraction of Ly$\\alpha$ gas for the AGN is 94$^{+6}_{-23}$\\%, which is comparable to the star-forming control galaxies (100$^{+0}_{-21}$\\%) and consistent with passive galaxies (75$^{+25}_{-21}$\\%). The covering fractions of metal species (C\\ion{ii}, Si\\ion{ii}, Si\\ion{iii}, C\\ion{iv}, Si\\ion{iv}, and N\\ion{v}) are consistent with the control-matched galaxies (Figure \\ref{fig:CovFracR}).\n\n\\item[2.] An insignificant increase in the Ly$\\alpha$ EW for AGN relative to control-matched galaxies on a sightline by sightline basis. The measured median EW offset between these two population is $+0.10\\pm0.13$ dex (Figure \\ref{fig:deltaEWRho}). \n\n\\item[3.] After stacking the spectra, the observed EW enhancement of low ionization species for AGN is seen only at high impact parameters ($\\rho_{\\rm imp}${}$\\geq164$ kpc; the median impact parameter of COS-AGN) for both Ly$\\alpha$ ($\\delta$log(EW)$=+0.45\\pm0.05$) and cool metal line tracers (Si\\ion{ii} 1260~\\AA{} [$\\delta$log(EW)$>0.27$~dex] and Si\\ion{iii} 1206~\\AA{} [$\\delta$log(EW)$>0.75$~dex]; Figure \\ref{fig:StackdEW}). These results are inconsistent with the expected effects from AGN feedback seen in our zoom-in simulations at high impact parameters (Figure \\ref{fig:Sims}). At lower impact parameters ($\\rho_{\\rm imp}${}$<164$ kpc), our results are consistent with the simulations.\n\n\\item[4.] The Ly$\\alpha$ line kinematics for the COS-AGN sightlines does not differ significantly from what is observed around the control-matched sample, suggesting there is no strong bulk motion in the CGM due to the presence of an AGN (Figure \\ref{fig:Kinematics}). As all but one system show gas within the escape velocity of the host halo, the probed material is likely bound to the AGN host.\n\n\\end{itemize}\n\nThese results suggest that the circumgalactic environments that host AGN show little difference than their non-AGN hosts on a sightline-by-sightline basis, likely attributed to our small sample size. We only detect a significant difference in the amount of cool gas in our stacked spectrum at high impact parameters ($\\rho_{\\rm imp}${}$\\geq164$~kpc), which we use to interpret our results. Given the lack of signatures (in both EW and kinematic diagnostics) of recent AGN feedback on the CGM from winds and ionizing radiation, we speculate that possible causes for these stacked spectrum results could be from the accretion of cool gas to feed the AGN, or a remnant effect from previous evolutionary activity of the host galaxy (such as starbursts) prior to the AGN accretion phase.\n\n\n\\section*{Acknowledgments}\nWe thank the anonymous referee for their comments to improve the clarity of this manuscript. We are grateful for Sanchayeeta Borthakur providing the reduced spectra and all Si species equivalent widths from the COS-GASS survey. This manuscript benefited greatly from discussions with H.W. Chen, T. Heckman, C. Martin, J.X. Prochaska, and J. Werk. BDO's contribution was made possible by the HST observing grant HST-GO-13774. \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} \\label{section-introduction}\n\\noindent Special functions play a prominent role in mathematics and physics. They are solutions of important equations and come in various forms, in terms of series or integrals. We are interested in the way special functions connect to representation theory. That such a connection exists is well known. Special functions appear in the study of spherical functions (see e.g. \\cite{Helgason1984}, Chapter IV), of matrix coefficients of representations (see e.g. \\cite{Vilenkin1993-1}, Section 4.1) and of symmetry breaking operators (see \\cite{Kobayashi-Kubo-Pevzner2016} and \\cite{Kobayashi-Pevzner2016}). In this work, we focus on the $K$-types of degenerate principle series of $\\spg(n,\\C)$, looking for $K$-finite vectors that have explicit formulas in terms of special functions. Here, $K$ refers to $\\spg(n)$.\n\\vskip 8pt\n\\noindent The reason why we choose to work with degenerate principle series of $\\spg(n,\\C)$ is twofold.\n\\vskip 8pt\n\\noindent First, the geometric setting we choose is motivated by the following sequence of groups of isometries of vector spaces over different number fields:\n$$\\xymatrix{\n\\spg(n) \\ar@{~>}[d] & \\subset & \\sug(2n) \\ar@{~>}[d] & \\subset & \\sog(4n) \\ar@{~>}[d] \\\\\n\\quat^n & \\simeq & \\C^{2n} & \\simeq & \\R^{4n}.}$$\nThis sequence enables us to use a refined version of the classical theory of spherical harmonics. Moreover, the non-commutativity of the skew field $\\quat$ of quaternions adds a rich ingredient to the underlying representation theory and harmonic analysis: see for example \\cite{Howe-Tan1993}, \\cite{Kobayashi1992}, \\cite{Pasquale1999} and the very recent paper \\cite{Schlichtkrull-Trapa-Vogan2018}. \n\\vskip 8pt\n\\noindent Second, the degenerate principal series representations of the complex symplectic Lie group $\\spg(n,\\C)$ are \"small\" in the sense of the Kirillov-Gelfand dimension(see e.g. \\cite{Vogan2017}) amongst infinite-dimensional unitary representations of $\\spg(n,\\C)$. According to the guiding principle \"small representation of a group $=$ large symmetries in a representation space\" suggested by T. Kobayashi in \\cite{Kobayashi2013}, explicit models of such representations are a natural source of information on special functions that arise in this framework as specific vectors. This philosophy has been applied to the analysis of minimal representations of $\\org(p,q)$ (see \\cite{Kobayashi-Mano2011} and \\cite{Kobayashi-Orsted2003}) and small principal series representations of the real symplectic group $\\spg(n,\\R)$ (see \\cite{Kobayashi-Orsted-Pevzner2011}).\n\\vskip 8pt\n\\noindent Fix an integer $n \\geq 2$, set $m=n-1$ and write $G=\\spg(n,\\C)$ and $K=\\spg(n)$. The representations we work with are defined by parabolic induction with respect to a maximal parabolic subgroup $Q$ of $G$ whose Langlands decomposition is $Q=MAN$(see Section \\ref{subsection-non_comp._pict._and_heisenberg_group} for explicit description) with:\n$$M \\simeq \\ung(1) \\times \\spg(m,\\C), \\; A \\simeq \\R^{\\times}_{+} \\; {\\rm and} \\; N \\simeq {\\rm H}_{\\C}^{2m+1}.$$\nHere, $\\mathrm{H}_{\\C}^{2m+1}$ refers to the $(2m+1)$-dimensional complex Heisenberg group. We denote the $\\ung(1)$ component of an element $m$ of $M$ by $e^{i \\theta(m)}$ and the positive real scalar that corresponds to an element $a$ of $A$ by $\\alpha(a)$. Now consider for $(\\lambda,\\delta) \\in \\R \\times \\Z$ the character $\\chi_{i\\lambda,\\delta}$ defined on $Q$ by:\n$$\\chi_{i\\lambda,\\delta}(man)= \\big( e^{i \\theta(m)}\\big)^{\\delta} \\, \\big( \\alpha(a)\\big)^{i\\lambda}.$$\nThe corresponding induced representation $\\pild={\\rm Ind}_Q^G \\, \\chi_{i\\lambda,\\delta}$ can be realised on the completion $V_{i\\lambda,\\delta}$ of the complex vector space\n{\\small $$V_{i\\lambda,\\delta}^0 = \\left \\lbrace f \\in C^0 \\big( \\C^{2n} \\setminus \\{0\\} \\big) \\; \\Big\/ \\; \\forall c \\in \\C \\setminus \\{0\\}: \\; f(c \\, \\cdot)=\\left( \\frac{c}{\\vert c \\vert} \\right)^{-\\delta} \\vert c \\vert^{-i\\lambda-2n} f(\\cdot) \\right \\rbrace$$}with respect to the $L^2$-norm on $S^{4n-1}$.\n\\vskip 8pt\n\\noindent Representations $\\pild$ form a degenerate principal series of $G$. It is proved in \\cite{Gross1971} that $\\pild$ is irreducible if $(\\lambda,\\delta) \\neq (0,0)$ and in \\cite{Clare2012} that $\\pi_{0,0}$ decomposes into a sum of two irreducible subrepresentations. The isotypic decomposition of $\\pild \\big|_K$ is multiplicity free and given by (see \\cite{Clare2012} and \\cite{Howe-Tan1993}):\n\\begin{equation} \\label{equation-general_isot._dec.}\n\\pild \\big|_K \\simeq \\sideset{}{^\\oplus}\\sum_{ \\substack{l-l' \\geq |\\delta| \\\\\nl-l' \\equiv \\delta [2]}} \\, \\pi^{l,l'}.\n\\end{equation}\nIn this sum, $l$ and $l'$ are integers such that $l \\geq l' \\geq 0$ and $\\pi^{l,l'}$ is the irreducible representation of $K$ whose highest weight is $(l,l',0,\\cdots,0)$; we denote the irreducible invariant subspace of $V_{i\\lambda,\\delta}$ (with respect to the left action of $K$) that corresponds to $\\pi^{l,l'}$ by $V^{l,l'}$, calling it a \\textit{component} of $V_{i\\lambda,\\delta}$.\n\\vskip 8pt\n\\noindent Our aim is to describe specific elements of components $V^{l,l'}$ in terms of special functions. This will require suitable changes of the carrying space $V_{i\\lambda,\\delta}$ (together with the action of $G$) so as to have a clearer view of $\\pild$, depending on the kind of special functions we have in mind; each point of view is called a \\textit{picture} of $\\pild$ (above definition is the \\textit{induced picture}).\n\\vskip 8pt\n\\noindent This paper is organised as follows:\n\\begin{itemize}\n\\item Section 2: in the compact picture, we study the $K$-type structure, by which we mean the detailed description of the components of $V_{i\\lambda,\\delta}$ and connections that exist between them (Propositions \\ref{proposition-left_action_decomposition}, \\ref{proposition-hwv_for_right_action} and \\ref{proposition-isotypic_decomposition_for_sp(1)}).\n\\item Section 3: we consider the case $l=l'$ and use, again in the compact picture, invariance properties with respect to $\\spg(1)$ and $1 \\times \\spg(n-1)$ to exhibit in components $V^{l,l}$ elements which can be seen as solutions of hypergeometric differential equations (Theorem \\ref{theorem-hypergeometric_equation}).\n\\item Section 4: we define the non-standard picture of $\\pild$ (which was introduced in \\cite{Kobayashi-Orsted-Pevzner2011} and followed in \\cite{Clare2012}) by applying a certain partial Fourier transform $\\mathcal{F}$ to the non-compact picture. For a wide class of components, namely components $V^{l,0}$ (said otherwise, those components such that $l'=0$), this leads to elements that can be expressed in terms of modified Bessel functions (Theorem \\ref{theorem-final_formula}).\n\\end{itemize}\nLet us put together our most important results:\n\\begin{nnthm}[Main results] \\label{thm-quat._sph._harm._thm} \\\n\n\\noindent Let $n \\in \\N$ be such that $n \\geq 2$ and set $m=n-1$.\\\\\n\\noindent Consider the group $G=\\spg(n,\\C)$, its maximal compact subgroup $\\spg(n)$ and the parabolic subgroup $Q=MAN \\simeq \\ung(1) \\times \\spg(m,\\C) \\times \\R^{\\times}_{+} \\times {\\rm H}_{\\C}^{2m+1}$.\\\\\n\\noindent Consider a pair $(\\lambda,\\delta) \\in \\R \\times \\Z$, together with the character $\\chi_{i\\lambda,\\delta}$ defined by $\\chi_{i\\lambda,\\delta}(man)= \\big( e^{i \\theta(m)}\\big)^{\\delta} \\, \\big( \\alpha(a)\\big)^{i\\lambda}$\nand the degenerate principal series representations $\\pi_{i\\lambda,\\delta} = {\\rm Ind}_Q^G \\, \\chi_{i\\lambda,\\delta}$ of $G$. \n\\begin{enumerate}\n\\item For $l \\in \\N$, consider the $K$-type $\\pi^{l,l}$. The corresponding subspace $V^{l,l}$ of $V_{i\\lambda,\\delta}$ contains an element which can be seen as a function of a single variable $\\tau \\in [0,1]$ and the restriction $\\varphi$ of this function to $]0,1[$ satisfies the following hypergeometric equation:\n$$\\quad \\; \\tau(1-\\tau) \\varphi''(\\tau) \\, + \\, 2(1-n\\tau) \\, \\varphi'(\\tau) \\, + \\, l(l+2n-1) \\, \\varphi(\\tau) \\, = \\, 0.$$\n\\item For $(l,\\alpha,\\beta) \\in \\N^3$ such that $l=\\alpha+\\beta$ and $\\delta=\\beta-\\alpha$, consider the $K$-type $\\pi^{l,0}$ of $\\pild$. Then, in the non-standard picture, highest weight vectors of $\\pi^{l,0}$ are proportionnal to a function\n$$\\psi \\, : \\, \\C \\times \\C^m \\times \\C^m \\, \\longrightarrow \\C$$\nwhose expression for $s \\neq 0$ and $v \\neq 0$ is\n$$\\quad \\psi(s,u,v) \\, = \\, R(s,u,v) \\, K_{\\frac{i\\lambda+\\delta}{2}} \\left( \\pi \\sqrt{ 1 + \\Vert u \\Vert^2 } \\sqrt{ \\vert s \\vert^2+4\\Vert v \\Vert^2 } \\right)$$\nwhere we set\n$$\\; R(s,u,v) \\, = \\, \\frac{(-i \\overline{s})^{\\alpha} \\, \\pi^{i\\lambda+\\beta+n}}{2^{\\frac{i\\lambda+l}{2}+1} \\, \\Gamma \\left( \\frac{i\\lambda+l}{2}+n \\right)} \\, \\left( \\frac{ \\sqrt{ |s|^2+4\\Vert v \\Vert^2}}{\\pi \\sqrt{1+\\Vert u \\Vert^2}} \\right)^{\\frac{i\\lambda+\\delta}{2}}$$\nand where $K_{\\frac{i\\lambda+\\delta}{2}}$ denotes a modified Bessel function of the third kind (see appendix for definition).\n\\end{enumerate}\n\\end{nnthm}\n\\noindent Before we enter the details, let us motivate the use of partial Fourier transforms. For one thing, they have proved useful in the study of Knapp-Stein operators (see \\cite{Clare2012}, \\cite{Kobayashi-Orsted-Pevzner2011}, \\cite{Pevzner-Unterberger2007} and \\cite{Unterberger2003}). For another, partial Fourier transforms with respect to appropriate Lagrangian subspaces modify the nature of constraints imposed on specific vectors in representation spaces and have been used to find explicit formulas for $K$-finite vectors in \\cite{Kobayashi-Mano2011}, \\cite{Kobayashi-Orsted2003}, and \\cite{Kobayashi-Orsted-Pevzner2011}. These works establish formulas that involve Bessel functions. This has lead us to apply similar techniques to degenerate principal series of $\\spg(n,\\C)$.\n\\section{$K$-type structure} \\label{section-k_type_structure}\n\\noindent We investiqate the $K$-type structure of the degenerate principal series $\\pild={\\rm Ind}_Q^G \\, \\chi_{i\\lambda,\\delta}$ of $\\spg(n,\\C)$. \n\\subsection{Compact picture and general facts} \\label{subsection-compact_picture_and_general_facts} \n\\\n\\vskip 8pt\n\\noindent Let us fix $(\\lambda,\\delta) \\in \\R \\times \\Z$. In the compact picture (see \\cite{Knapp1986}, Chapter VII), the carrying space of $\\pild$ is a subspace of $L^2(G)$. The natural action of $G$ on $\\C^{2n}$ enables one to identify it with the following Hilbert space:\n$$L^2_{\\delta}(S^{4n-1})=\\left \\lbrace f \\in L^2(S^{4n-1}) \\; \\big\/ \\; \\forall \\theta \\in \\R: \\; f(e^{i\\theta} \\, \\cdot)=e^{-i\\delta \\theta} f(\\cdot) \\right \\rbrace.$$\n\\vskip 8pt\n\\noindent The compact picture is the ideal setting to look for irreducible invariant subspaces with respect to the left action of $K$, because one can benefit from decompositions given by standard harmonic analysis (see e.g. Chapter 9 of \\cite{Faraut2008}, Chapters IV and V of \\cite{Knapp2002} and Chapters 9 and 11 of \\cite{Vilenkin1993-2}).\n\\vskip 8pt\n\\noindent Denote by $\\harm^k$ the complex vector space of polynomial functions $f$ on $\\R^{4n}$ that are harmonic and homogeneous of degree $k$. Let us write $\\sph^k=\\harm^k \\big|_{S^{4n-1}}$ (elements of $\\sph^k$ are called spherical harmonics). Then:\n\\begin{itemize}\n\\item each $\\sph^k$ is invariant under the left action of $\\sog(4n)$ (this is also true for each $\\harm^k$);\n\\item the representations defined by the left action of $\\sog(4n)$ on the various spaces $\\sph^k$ are irreducible and pairwise inequivalent;\n\\item $\\displaystyle L^2\\left( S^{4n-1} \\right) = \\widehat{\\bigoplus_{k \\in \\N}} \\; \\sph^k$.\n\\end{itemize}\nConsider the identification $(x,y) \\in \\R^{2n}\\times \\R^{2n} \\longleftrightarrow z=x+iy \\in \\C^{2n}$. Then:\n\\begin{itemize}\n\\item functions of the variables $x$ and $y$ (in particular elements of the spaces $\\harm^k$) can be regarded as functions of the variables $z$ and $\\bar{z}$; \n\\item matrices $A+iB$ of $\\glg(2n,\\C)$ (resp. $\\sug(2n)$), where $A$ and $B$ denote $2n \\times 2n$ real matrices, can be regarded as matrices $\\left( \\begin{array}{cc}\n A & -B \\\\\n B & A \\\\\n \\end{array}\n\\right)$ of $\\glg(4n,\\R)$ (resp. $\\sog(4n)$);\n\\item accordingly, the left action $L$ of $\\sug(2n)$ on functions of $z$ and $\\bar{z}$ is defined by $L(u)f(z,\\bar{z}) = f(u^{-1}z, \\, \\overline{u^{-1} z}) = f(u^{-1}z,\\, {^t u}\\bar{z})$.\n\\end{itemize}\n\\vskip 8pt\n\\noindent The Laplace operator can be written $\\displaystyle \\Delta =4\\sum_{i=1}^{2n} \\frac{\\partial^2}{\\partial z_i \\partial \\bar{z_i}}$. For $\\alpha,\\beta \\in \\N$, consider the space $\\harm^{\\alpha,\\beta}$\nof polynomial functions $f$ of the variables $z$ and $\\bar{z}$ such that $f$ is homogeneous of degree $\\alpha$ in $z$ and degree $\\beta$ in $\\bar{z}$ and such that $\\Delta f=0$. Let us write $\\sph^{\\alpha,\\beta}=\\harm^{\\alpha,\\beta} \\big|_{S^{4n-1}}$. Then:\n\\begin{itemize}\n\\item each $\\sph^{\\alpha,\\beta}$ is invariant under the left action of $\\sug(2n)$ (this is also true for each $\\harm^{\\alpha,\\beta}$);\n\\item the representations defined by the left action of $\\sug(2n)$ on the various spaces $\\sph^{\\alpha,\\beta}$ are irreducible and pairwise inequivalent;\n\\item $\\displaystyle \\sph^k = \\bigoplus_{\n\\substack{\n(\\alpha,\\beta) \\in \\N^2 \\\\\n\\alpha+\\beta=k}} \\sph^{\\alpha,\\beta}$.\n\\end{itemize}\nThis leads to the Hilbert sum $\\displaystyle L^2_{\\delta}(S^{4n-1})=\\widehat{\\bigoplus_{\n\\substack{(\\alpha,\\beta) \\in \\N^2 \\\\\n\\delta=\\beta-\\alpha}}} \\sph^{\\alpha,\\beta}$. We see that, in order to describe the isotypic decomposition of $\\pild \\big|_K$, we need to understand how each $\\sph^{\\alpha,\\beta}$ breaks into irreducible invariant subspaces under the left action of $K$.\n\\vskip 8pt\n\\noindent From now on, we consider $3$-tuples $(k,\\alpha,\\beta) \\in \\N^3$ such that $\\alpha+\\beta=k$ and denote by $L$ the left action of $K$, be it on $\\harm^k$, $\\sph^k$, $\\harm^{\\alpha,\\beta}$ or $\\sph^{\\alpha,\\beta}$.\n\\subsection{Left action of $\\spg(n)$} \\label{subsection-left_action}\n\\\n\\vskip 8pt\n\\noindent Recall that the Lie algebra $\\spa(n,\\C)$ of $G$ is\n$$\\mathfrak{g}=\\left\\lbrace X=\\left(\n\\begin{array}{cc}\nA & C \\\\\nB & -{}^t\\!{A} \\\\\n\\end{array}\n\\right) \\in \\m(2n,\\C)\\ \/ \\ B\\ {\\rm and}\\ C\\ {\\rm are\\ symmetric} \\right\\rbrace$$ \nand that the Lie algebra $\\spa(n)$ of $K$ is\n$$\\mathfrak{k}=\\left\\lbrace X=\\left(\n\\begin{array}{cc}\nA & -\\overline{B} \\\\\nB & \\overline{A} \\\\\n\\end{array}\n\\right) \\in \\m(2n,\\C)\\ \/ \\ A\\ {\\rm is\\ skew \\; and}\\ B\\ {\\rm is\\ symmetric} \\right\\rbrace.$$ \n\\noindent Consider the complexification $\\mathfrak{g}=\\mathfrak{k}\\oplus i\\mathfrak{k}$ of $\\mathfrak{k}$. Let $\\mathfrak{h}$ be the usual Cartan subalgebra of $\\mathfrak{g}$ consisting of diagonal elements of $\\mathfrak{g}$. If $r$ belongs to $\\{1,...,n\\}$, denote by $L_r$ the linear form that assigns to an element of $\\mathfrak{h}$ its $r^{\\rm th}$ diagonal term.\n\\vskip 8pt\n\\noindent The set $\\Delta$ of roots of $\\mathfrak{g}$ (with respect to $\\mathfrak{h}$) consists of the following linear forms:\n\\begin{itemize}\n\\item $L_r-L_s$ and $-L_r+L_s$ ($1 \\leq r 0$ ($I_m$ denotes the $m$-dimensional identity matrix);\n\\item $N$ consists of matrices {\\small $n=\\left(\n\\begin{array}{cccc}\n1 & {}^t\\!{u} & 2s & {}^t\\!{v} \\\\\n0 & I_m & v & 0 \\\\\n0 & 0 & 1 & 0 \\\\\n0 & 0 & -u & I_m \\\\\n\\end{array}\n\\right)$} such that $s\\in \\C$ and $u$ and $v$ both belong to $\\C^m$.\n\\end{itemize}\nThen $Q=MAN$ is indeed a parabolic subgroup of $G$.\n\\vskip 8pt\n\\noindent We regard the $(2m+1)$-dimensional complex Heisenberg group $\\mathrm{H}_{\\C}^{2m+1}$ as $\\C \\times \\C^m \\times \\C^m$ equipped with the following product: \n$$(s,u,v)(s',u',v')=\\left( s+s'+\\frac{\\langle v,u' \\rangle - \\langle u,v' \\rangle}{2},u+u',v+v' \\right).$$\nHere, $\\langle \\cdot,\\cdot \\rangle$ is defined for elements $(x,y) \\in \\C^m \\times \\C^m$ by: \n$$\\langle x,y \\rangle = \\sum_{i=1}^m x_i y_i.$$\nWe point out that, later on, this sum will be denoted by $x \\cdot y$ instead of $\\langle x,y \\rangle$ when we consider $(x,y) \\in \\R^m \\times \\R^m$.\n\\vskip 8pt\n\\noindent We use the following sequence of identifications (with $n$ as above):\n$$\\begin{array}{ccccc}\n\\mathrm{H}_{\\C}^{2m+1} & \\simeq & \\overline{N} & \\simeq & {\\rm H}\\\\\n(s,u,v) & \\longmapsto & {}^t\\!{n} & \\longmapsto & (1,u,2s,v). \\\\\n\\end{array}$$\nwhere ${\\rm H}=\\{1\\} \\times \\C^m \\times \\C \\times \\C^m$ is regarded as a complex hyperplane of $\\C^{2n}$. The first identification map is a group isomorphism (one can also prove that $\\mathrm{H}_{\\C}^{2m+1} \\simeq N$) and the second identication results from the natural action of $\\overline{N}$ on $\\C^{2n}$ (applied to the element $(1,0,\\cdots,0)$ of $\\C^{2n}$).\n\\vskip 8pt\n\\noindent One can now identify $L^2(\\overline{N})$ with $L^2(\\mathrm{H}_{\\C}^{2m+1})$ and, given $f \\in V_{i\\lambda,\\delta}^0$ (in the induced picture), regard the restriction $f|_{\\rm H}$ as an element of $L^2(\\mathrm{H}_{\\C}^{2m+1})$.\n\\vskip 8pt\n\\noindent From now on, we write $\\C^{2m+1}$ instead of $\\mathrm{H}_{\\C}^{2m+1}$.\n\n\\subsection{Definition of the non-standard picture} \\label{subsection-definition_of_the_non_stand._pict.}\n\\\n\\vskip 8pt\n\\noindent We first define two partial Fourier transforms\n\\begin{align*}\n\\mathcal{F}_{\\tau} & : L^2(\\C^{2m+1}) \\longrightarrow L^2(\\C^{2m+1})\\\\\n\\mathcal{F}_{\\xi} &: L^2(\\C^{2m+1}) \\longrightarrow L^2(\\C^{2m+1})\n\\end{align*}\nby setting for functions $g \\in L^2(\\C^{2m+1})$ that fulfill integrability conditions: \n$$\\mathcal{F}_{\\tau}(g)(s,u,v)=\\int_{\\C} g(\\tau,u,v) \\, e^{-2i\\pi {\\rm Re}(s\\tau) } \\, d\\tau$$\n$$\\mathcal{F}_{\\xi}(g)(s,u,v)=\\int_{\\C^m} g(s,u,\\xi)\\,e^{-2i\\pi {\\rm Re} \\langle v,\\xi \\rangle}\\,d\\xi.$$\n\\noindent We then define the partial Fourier transform on which is based the non-standard picture:\n$$\\mathcal{F}= \\mathcal{F}_{\\tau} \\circ \\mathcal{F}_{\\xi}.$$\n\\begin{nndef}\n\\noindent The \\textit{non-standard picture} of $\\pi_{i\\lambda,\\delta}$ has $L^2(\\C^{2m+1})$ as carrying space. The action of $G$ is then the conjugate under $\\mathcal{F}$ of the action of $G$ in the non-compact picture; in other words, $\\mathcal{F}$ intertwines the action of $G$ in the non-compact picture and the action of $G$ in the non-standard picture.\n\\end{nndef}\n\\subsection{Connection with modified Bessel functions} \\label{subsection-connection_with_modified_bessel_functions}\n\\subsubsection{Selected components} \\label{subsubsection-selected_components}\n\\\n\\vskip 8pt\n\\noindent \nBecause we intend to use the right action of $\\spg(1)$, we consider an entire space $\\harm^k$ of harmonic polynomials. We restrict our attention to those components of Proposition \\ref{proposition-left_action_decomposition} that are labelled by $\\gamma=0$ and are subspaces of $\\harm^k$, namely the components $V_0^{\\alpha,\\beta}$ with $\\alpha + \\beta=k$; the corresponding highest weight vectors are the polynomials \n$$P_0^{\\alpha,\\beta}(z,w,\\bar{z},\\bar{w})=w_1^{\\alpha}\\bar{z_1}^{\\beta}.$$\n\\noindent We denote by $g_{\\alpha,\\beta}$ the restriction of $P_0^{\\alpha,\\beta}$ to the unit sphere $S^{4n-1}$. Let us call $g$ the function in the induced picture that corresponds to $g_{\\alpha,\\beta}$. It extends $g_{\\alpha,\\beta}$, meaning that $g_{|_{S^{4n-1}}}=g_{\\alpha,\\beta}$. By definition of the induced picture, $g$ must satisfy for all non-zero complex numbers $c$:\n$$g(c\\,\\cdot)=\\left( \\frac{c}{|c|}\\right)^{-\\delta} \\, \\vert c \\vert^{-i\\lambda-2n} \\, g.$$\n\\noindent We define\n$$a(s,u)=\\sqrt{1+4\\vert s \\vert ^2 + \\Vert u \\Vert^2}$$ \n$$r(s,u,v)= \\sqrt{ a^2(s,u) + \\Vert v \\Vert^2}.$$\n\\noindent By restricting $g$ to the complex hyperplane ${\\rm H} \\simeq \\C^{2m+1}$, we get a function $G_{\\alpha,\\beta}$ defined on $\\C^{2m+1}$ by:\n\\begin{align*}\n& G_{\\alpha,\\beta}(s,u,v) = g(1,u,2s,v)\\\\\n& = \\left( \\frac{1}{r(s,u,v)} \\right)^{i\\lambda+2n} g_{\\alpha,\\beta} \\left( \\frac{1}{r(s,u,v)},\\frac{u}{r(s,u,v)},\\frac{2s}{r(s,u,v)},\\frac{v}{r(s,u,v)} \\right).\n\\end{align*}\n\\noindent Finally, due to the total homogeneity degree $k$ of $P_0^{\\alpha,\\beta}$:\n$$G_{\\alpha,\\beta}(s,u,v)=\\frac{(2s)^{\\alpha}}{ \\left( a^2(s,u) + \\Vert v \\Vert^2 \\right)^{\\frac{i\\lambda+k}{2}+n}} \\, .$$\nWe call $G_{\\alpha,\\beta}$ the \\textit{non-compact form} of $P_0^{\\alpha,\\beta}$. The aim of the rest of Section \\ref{section-non_standard_picture_and_mod._bessel_functions} is to determine the \\textit{non-standard form} of $P_0^{\\alpha,\\beta}$, that is, $\\mathcal{F}(G_{\\alpha,\\beta})$. We apply $\\mathcal{F}_{\\xi}$ in Section \\ref{subsubsection-first_transform} and $\\mathcal{F}_{\\tau}$ in Section \\ref{subsubsection-second_transform}. Calculations involve Bessel functions and various formulas that we have gathered in the appendix. \n\\subsubsection{First transform} \\label{subsubsection-first_transform}\n\\\n\\vskip 8pt\n\\noindent \nBy definition (integrability condition is fulfilled):\n\\begin{equation} \\label{equation-step_1-definition}\n\\displaystyle \\mathcal{F}_{\\xi}(G_{\\alpha,\\beta})(s,u,v)=\\int_{\\C^m} \\frac{(2s)^{\\alpha}}{ \\left( a^2(s,u) + \\Vert \\xi \\Vert^2 \\right)^{\\frac{i\\lambda+k}{2}+n}} \\,e^{-2i\\pi {\\rm Re} \\langle v,\\xi \\rangle}\\,d\\xi.\n\\end{equation}\n\\noindent In real coordinates, writing $\\xi=x+iy$ and $v=a+ib$ (elements $x,y,a,b$ each belong to $\\R^m$) and identifying $\\xi$ and $v$ with the elements $(x,y)$ and $(a,b)$ of $\\R^m \\times \\R^m$, formula (\\ref{equation-step_1-definition}) reads:\n\\begin{multline} \\label{equation-step_2-real_coordinates}\n\\displaystyle \\mathcal{F}_{\\xi}(G_{\\alpha,\\beta})(s,u,v)=\\\\\n\\int_{\\R^m \\times \\R^m} \\frac{(2s)^{\\alpha}}{ \\left( a^2(s,u) + \\Vert x \\Vert^2 + \\Vert y \\Vert^2 \\right)^{\\frac{i\\lambda+k}{2}+n}} \\,e^{-2i\\pi (a \\cdot x-b \\cdot y)}\\,dxdy.\n\\end{multline}\n\\noindent Switch to polar coordinates by writing $(x,y)=rM$ and $(a,-b)=r'M'$, with $M$ and $M'$ in $S^{2m-1} \\subset \\R^{2m}$, $r =\\sqrt{\\Vert x \\Vert^2+\\Vert y \\Vert^2} =\\Vert \\xi \\Vert$ and $r'=\\sqrt{a^2+(-b)^2}=\\Vert v \\Vert$. Then the integral in Formula (\\ref{equation-step_2-real_coordinates}) becomes\n\\begin{equation} \\label{equation-step_3-polar_coordinates}\n\\int_0^{\\infty} \\left( \\int_{S^{2m-1}} \\frac{(2s)^{\\alpha}}{ \\left( a^2(s,u) + r^2 \\right)^{\\frac{i\\lambda+k}{2}+n}} e^{-2i\\pi rr'M \\cdot M'} d\\sigma(M) \\right) r^{2m-1} dr\n\\end{equation}\nwhere $M \\cdot M'$ now denotes the Euclidean scalar product of $\\R^{2m}$ applied to the points $M$ and $M'$ of the sphere $S^{2m-1}$ seen as vectors of $\\R^{2m}$. Integral (\\ref{equation-step_3-polar_coordinates}) can be written:\n\\begin{equation} \\label{equation-step_4-polar_coordinates-2nd_version}\n\\displaystyle \n\\int_{0}^{\\infty} \\frac{(2s)^{\\alpha}}{ \\left( a^2(s,u) + r^2 \\right)^{\\frac{i\\lambda+k}{2}+n}} \\left( \\int_{S^{2m-1}} e^{-2i\\pi rr'M \\cdot M'} d\\sigma(M) \\right) r^{2m-1} dr.\n\\end{equation}\n\\noindent Proposition \\ref{proposition-bochner} then changes (\\ref{equation-step_4-polar_coordinates-2nd_version}) into:\n\\begin{equation} \\label{equation-step_5-bessel_expression_1}\n\\int_{0}^{\\infty} \\frac{(2s)^{\\alpha}}{ \\left( a^2(s,u) + r^2 \\right)^{\\frac{i\\lambda+k}{2}+n}} 2 \\pi (rr')^{1-m} J_{m-1}(2 \\pi rr') r^{2m-1} dr.\n\\end{equation}\n\\noindent Because $r'=\\Vert v \\Vert$, (\\ref{equation-step_5-bessel_expression_1}) becomes:\n\\begin{equation} \\label{equation-step_6-bessel_expression_2}\n\\displaystyle \n2^{\\alpha+1} \\pi s^{\\alpha} \\Vert v \\Vert^{1-m} \\int_{0}^{\\infty} \\frac{r^m}{ \\left( a^2(s,u) + r^2 \\right)^{\\frac{i\\lambda+k}{2}+n}} J_{m-1}(2 \\pi \\Vert v \\Vert r) dr.\n\\end{equation}\n\\noindent We now want to apply Proposition \\ref{proposition-integral_formulas-erdelyi}. But it uses another notation system than ours. To understand how to switch from one system of notation to the other, let us define new variables $x,y,\\mu$: \n$$x=r\\ ;\\ y=2 \\pi \\Vert v \\Vert\\ ;\\ \\mu=\\frac{i\\lambda+k}{2}+n-1\\ ;\\ \\nu=m-1.$$\n\\noindent Then (\\ref{equation-step_6-bessel_expression_2}) becomes:\n$$2^{\\alpha+1} \\pi s^{\\alpha} \\Vert v \\Vert^{1-m} y^{-\\frac{1}{2}} \\int_{0}^{\\infty} \\frac{x^{\\nu+\\frac{1}{2}}}{ \\left( a^2(s,u) + r^2 \\right)^{\\mu+1}} J_{m-1}(xy) \\sqrt{xy} dx.$$\n\\noindent Proposition \\ref{proposition-integral_formulas-erdelyi} (first formula) now gives (as long as $\\Vert v \\Vert > 0$)\n$$2^{\\alpha+1} \\pi s^{\\alpha} \\Vert v \\Vert^{1-m} y^{-\\frac{1}{2}} \\frac{a^{\\nu-\\mu} y^{\\mu+\\frac{1}{2}} K_{\\nu-\\mu}(ay)}{2^{\\mu} \\Gamma(\\mu+1)}$$\nwhich, back to our own notation choices, is equal to\n$$\\frac{2^{\\alpha+1} s^{\\alpha} \\pi^{\\frac{i\\lambda+k}{2}+n}}{\\Gamma\\left( \\frac{i\\lambda+k}{2}+n \\right)} \\left( \\frac{\\Vert v \\Vert}{a(s,u)} \\right)^{\\frac{i\\lambda+k}{2}+1} K_{-\\left( \\frac{i\\lambda+k}{2}+1 \\right)}(2\\pi a(s,u) \\Vert v \\Vert).$$\n\\noindent To make formulas lighter, from now on we will write:\n{\\small $$\\Lambda=\\frac{i\\lambda+k}{2}.$$}\nBecause $K_{-(\\Lambda+1)}=K_{\\Lambda+1}$, we have proved:\n\\begin{prop} \\label{proposition-first_transform-k_bessel_expression}\nGiven any $\\lambda \\in \\R$ and any $(k,\\alpha,\\beta) \\in \\N^3$ such that $\\alpha+\\beta=k$, consider the non-compact form $G_{\\alpha,\\beta}$ of the highest weight vector $P_0^{\\alpha,\\beta}$. Then for all $(s,u,v)$ in $\\C \\times \\C^m \\times \\C^m$ such that $v \\neq 0$:\n$$\\mathcal{F}_{\\xi}(G_{\\alpha,\\beta})(s,u,v)=2^{\\alpha+1} s^{\\alpha} \\frac{\\pi^{\\Lambda+n}}{\\Gamma\\left( \\Lambda + n \\right)} \\left( \\frac{\\Vert v \\Vert}{a(s,u)} \\right)^{\\Lambda+1} K_{ \\Lambda+1 }(2\\pi a(s,u) \\Vert v \\Vert).$$\n\\end{prop}\n\\subsubsection{Second transform} \\label{subsubsection-second_transform}\n\\\n\\vskip 8pt\n\\noindent Consider any $(s,u,v) \\in \\C \\times \\C^m \\times \\C^m$ such that $s \\neq 0$ and $v \\neq 0$. We want to compute:\n\\begin{equation} \\label{equation-second_transform_integral}\n\\mathcal{F}_{\\tau} \\Big( \\mathcal{F}_{\\xi}(G_{\\alpha,\\beta}) \\Big) \\, (s,u,v) = \\int_{\\C} \\ \\mathcal{F}_{\\xi}(G_{\\alpha,\\beta})(\\tau,u,v) \\,e^{-2i\\pi {\\rm Re}(s\\tau)} \\ d\\tau.\n\\end{equation}\n\\noindent Using propositions \\ref{proposition-first_transform-k_bessel_expression} and \\ref{proposition-bessel_asymptotics-erdelyi}, one shows that $\\tau \\longmapsto \\mathcal{F}_{\\xi}(G_{\\alpha,\\beta})(\\tau,u,v)$ is indeed integrable. Let us use letters $a,b,x,y$ again, this time taking them to refer to real numbers: $s=a+ib$ and $\\tau=x+iy$. Then ${\\rm Re}(s\\tau)=ax-by$ and (\\ref{equation-second_transform_integral}) becomes\n$$\\int_{\\R^2} \\frac{2^{\\alpha+1} \\tau^{\\alpha} \\pi^{\\Lambda+n}}{\\Gamma\\left( \\Lambda + n \\right)} \\left( \\frac{\\Vert v \\Vert}{a(\\tau,u)} \\right)^{\\Lambda+1} K_{ \\Lambda+1 }(2\\pi a(\\tau,u) \\Vert v \\Vert) \\,e^{-2i\\pi (ax-by)}\\,dxdy$$\nwhich can be re-organised as\n\\begin{equation} \\label{equation-ax_by_integral}\n\\frac{2^{\\alpha+1} \\pi^{\\Lambda+n} \\Vert v \\Vert^{\\Lambda+1}}{\\Gamma\\left( \\Lambda + n \\right)} \\int_{\\R^2} \\frac{\\tau^{\\alpha} K_{ \\Lambda+1 }(2\\pi a(\\tau,u) \\Vert v \\Vert) }{\\Big( a(\\tau,u) \\Big)^{\\Lambda+1}} \\,e^{-2i\\pi (ax-by)}\\,dxdy.\n\\end{equation}\n\\noindent Let us again use polar coordinates (outside the origin):\n\\begin{itemize}\n\\item $(x,y)= r v_{\\theta}$ with $r>0$, $\\theta \\in \\R$ and $v_{\\theta}=(\\cos \\theta, \\sin \\theta)$; thus, $\\tau =r e^{i\\theta}$.\n\\item $(a,-b)=r' v_{\\theta'}$ with $r'>0$, $\\theta' \\in \\R$ and $v_{\\theta'}=(\\cos \\theta', \\sin \\theta')$; thus, $\\overline{s} =r' e^{i\\theta'}$.\n\\end{itemize}\n\\noindent Let us write $a(r,u)$ instead of $a(\\tau,u)$:\n$$a(r,u)=\\sqrt{1+4r^2+\\Vert u \\Vert^2}.$$\n\\noindent Integral (\\ref{equation-ax_by_integral}) can now be written:\n\\begin{multline} \\label{equation-cos_sin_integral}\n\\frac{2^{\\alpha+1} \\pi^{\\Lambda+n} \\Vert v \\Vert^{\\Lambda+1}}{\\Gamma\\left( \\Lambda + n \\right)} \\int_0^{\\infty} \\frac{r^{\\alpha} K_{ \\Lambda+1 }(2\\pi a(r,u) \\Vert v \\Vert) }{\\Big( a(r,u) \\Big)^{\\Lambda+1}} \\\\\n\\left( \\int_0^{2\\pi} e^{i\\alpha\\theta}\\,e^{-2i\\pi rr' (\\cos \\theta \\cos \\theta'+\\sin \\theta \\sin \\theta')} d\\theta \\right)\\,rdr.\n\\end{multline}\n\\noindent Following Proposition \\ref{proposition-bessel_integral_formula}, the inner integral\n$$\\int_0^{2\\pi} e^{i\\alpha \\theta} \\, e^{-2i\\pi r r' (\\cos \\theta \\, \\cos \\theta' \\, + \\, \\sin \\theta \\, \\sin \\theta')} \\, d\\theta$$\nis equal to:\n\\begin{equation} \\label{equation-inner_integral_computation}\n2 \\pi e^{i\\alpha\\left( \\theta'-\\frac{\\pi}{2} \\right) } \\, J_{\\alpha}(2\\pi r r').\n\\end{equation}\n\\noindent Because $r'=\\vert s \\vert$ and $\\theta'={\\rm Arg}(\\overline{s})$, (\\ref{equation-inner_integral_computation}) is equal to:\n$$2 \\pi e^{i\\alpha \\left( {\\rm Arg}(\\overline{s}) - \\frac{\\pi}{2} \\right) } \\, J_{\\alpha}(2\\pi r \\vert s \\vert ).$$\n\\noindent This turns (\\ref{equation-cos_sin_integral}) into:\n\\begin{multline} \\label{equation-non_standard_version_of_hwv_gamma=0}\n\\frac{2^{\\alpha+2} \\pi^{\\Lambda+n+1} \\Vert v \\Vert^{\\Lambda+1} e^{i\\alpha \\left( {\\rm Arg}(\\overline{s}) - \\frac{\\pi}{2} \\right) } }{\\Gamma\\left( \\Lambda + n \\right)} \\\\\n\\int_0^{\\infty} \\frac{r^{\\alpha + 1} \\, K_{ \\Lambda+1 }(2\\pi a(r,u) \\Vert v \\Vert) }{\\Big( a(r,u) \\Big)^{\\Lambda+1}} \\, J_{\\alpha}(2\\pi r |s|) \\, dr.\n\\end{multline}\n\\noindent We can now apply the second formula of proposition \\ref{proposition-integral_formulas-erdelyi}. To help follow notation choices made in this proposition, we set:\n\\begin{itemize}\n\\item $x=2r$, $dx=2dr$ and $\\beta=\\sqrt{1+\\Vert u \\Vert^2 } > 0$;\n\\item $a=2\\pi\\Vert v \\Vert > 0$ (careful: this variable $a$ is not what we have denoted $a(r,u)$);\n\\item $y=\\pi\\vert s\\vert > 0$, $\\nu=\\alpha$ and $\\mu=\\Lambda+1$.\n\\end{itemize}\n\\noindent Plugging these expressions in (\\ref{equation-non_standard_version_of_hwv_gamma=0}) and using Proposition \\ref{proposition-integral_formulas-erdelyi}, we finally obtain:\n\\begin{thm} \\label{theorem-final_formula}\nGiven any $\\lambda \\in \\R$ and any $(k,\\alpha,\\beta) \\in \\N^3$ such that $\\alpha+\\beta=k$, consider the non-compact form $G_{\\alpha,\\beta}$ of the highest weight vector $P_0^{\\alpha,\\beta}$. Then for all $(s,u,v)$ in $\\C \\times \\C^m \\times \\C^m$ such that $s \\neq 0$ and $v \\neq 0$:\n\\begin{multline*}\n\\mathcal{F}(G_{\\alpha,\\beta})(s,u,v)\\, = \\\\\n\\int_{\\C \\times \\C^m} \\, \\frac{(2\\tau)^{\\alpha}}{ \\left(1+4\\vert \\tau \\vert ^2 + \\Vert u \\Vert^2 + \\Vert \\xi \\Vert^2 \\right)^{\\frac{i\\lambda+k}{2}+n}} \\, e^{-2i\\pi {\\rm Re} \\left( s \\tau + \\langle v,\\xi \\rangle \\right) } \\,d\\tau d\\xi \\, = \\\\\nR(s,u,v) \\, K_{\\frac{i\\lambda+\\delta}{2}} \\left( \\pi \\sqrt{ 1 + \\Vert u \\Vert^2 } \\sqrt{ \\vert s \\vert^2+4\\Vert v \\Vert^2 } \\right)\n\\end{multline*}\nwhere we set\n$$R(s,u,v) \\, = \\, \\frac{(-i \\, \\overline{s})^{\\alpha} \\, \\pi^{i\\lambda+\\beta+n}}{2^{\\frac{i\\lambda+k}{2}+1} \\, \\Gamma \\left( \\frac{i\\lambda+k}{2}+n \\right)} \\left( \\frac{ \\sqrt{ |s|^2+4\\Vert v \\Vert^2}}{\\pi \\sqrt{1+\\Vert u \\Vert^2}} \\right)^{\\frac{i\\lambda+\\delta}{2}}.$$\n\\end{thm}\n\\noindent This theorem establishes the second part of our \"Main results\" theorem (stated in the introduction).\n\\begin{nnsrem} \\\n\\begin{itemize}\n\\item The partial Fourier transform $\\mathcal{F}$ changes, up to a constant $\\frac{-1}{i\\pi}$, multiplication by the $s$ coordinate into differentiation with respect to $s$. This implies $\\mathcal{F}(G_{\\alpha+1,\\beta-1})=\\frac{2}{-i\\pi}\\frac{\\partial}{\\partial s} \\Big( \\mathcal{F}(G_{\\alpha,\\beta}) \\Big)$, which is in fact proved in \\cite{Mendousse2017} for $\\beta \\geq 2$ (due to integrability issues).\n\\item One inevitably notices in the formula of Theorem \\ref{theorem-final_formula} a structure in two variables. Indeed, the particular value $\\alpha=0$ and the square root terms lead us to study the functions\n$$\\begin{array} {cccc}\n\\psi_{\\nu} : & ]0,+\\infty[ \\times ]0,+\\infty[ & \\longrightarrow & \\C \\\\\n & (x,y) & \\longmapsto & \\left( \\frac{x}{y} \\right)^{\\nu} K_{\\nu} (xy) \\\\\n\\end{array}$$\nwhere the parameter $\\nu$ is a complex number. Fix $\\nu \\in \\C$, $x_0 > 0$ and define the function\n$$\\begin{array} {cccc}\n\\varphi_{x_0,\\nu} : & ]0,+\\infty[ & \\longrightarrow & \\C \\\\\n & y & \\longmapsto & \\psi_{\\nu}(x_0,y). \\\\\n\\end{array}$$\n\\noindent One can show:\n$$ \\varphi_{x_0,\\nu}''(y) \\, + \\, \\frac{(1+2\\nu)}{y} \\, \\varphi_{x_0,\\nu}'(y) \\, - \\, x_0^2 \\, \\varphi_{x_0,\\nu}(y) \\, = \\, 0.$$\n\\noindent This equation belongs to the family of \\textit{Emden-Fowler equations} (or \\textit{Lane-Emden equations}) and its solutions can be written as the following combinations of Bessel functions of the first and second kind:\n$$u(t) \\, = \\, C_1 \\, t^{-\\nu} \\, J_{\\nu} \\left( -itx_0 \\right) \\, + \\, C_2 \\, t^{-\\nu} \\, Y_{\\nu} \\left( -itx_0 \\right).$$\nSimilar conclusions hold if one fixes $y_0$ instead of $x_0$.\n\\end{itemize}\n\\end{nnsrem}\n\\section{Appendix: Bessel functions} \\label{section-appendix}\n\\noindent In these definitions, following for instance \\cite{Lebedev1972} (sections 5.3 and 5.7), we take $\\nu \\in \\C$ and $z \\in \\C \\, \\setminus \\{0\\}$ such that $-\\pi < {\\rm Arg}(z) < \\pi$:\n\\begin{enumerate}\n\\item The \\textit{Bessel function of the first kind} is the function $J_{\\nu}$ defined by:\n$$J_{\\nu}(z)=\\sum_{k=0}^{\\infty} \\frac{(-1)^k}{\\Gamma(k+1) \\Gamma(k+\\nu+1)} \\left( \\frac{z}{2} \\right)^{\\nu+2k}.$$\n\\item The \\textit{Bessel function of the second kind} is the function $Y_{\\nu}$ defined by:\n$$Y_{\\nu}(z)=\\frac{J_{\\nu}(z) \\cos(\\nu \\pi) - J_{-\\nu}(z)}{\\sin (\\nu \\pi)}$$\nwhen $\\nu \\notin \\Z $ and, when $\\nu \\in \\Z$, by\n$$Y_{\\nu}(z)=\\lim_{\\substack{\n\\epsilon \\rightarrow \\nu \\\\\n0< |\\epsilon-\\nu| < 1}\n} Y_{\\epsilon}(z).$$\n\\item The \\textit{modified Bessel function of the first kind} is the function $I_{\\nu}$ defined by: \n$$I_{\\nu}(z)=\\sum_{k=0}^{\\infty} \\frac{1}{\\Gamma(k+1) \\Gamma(k+\\nu+1)} \\left( \\frac{z}{2} \\right)^{\\nu+2k}.$$\n\\item The \\textit{modified Bessel function of the third kind} is the function $K_{\\nu}$ defined by \n$$K_{\\nu}(z)=\\frac{\\pi}{2} \\frac{I_{-\\nu}(z) - I_{\\nu}(z)}{\\sin (\\nu \\pi)}$$\nwhen $\\nu \\notin \\Z $ and, when $\\nu \\in \\Z$, by\n$$K_{\\nu}(z)=\\lim_{\\substack{\n\\epsilon \\rightarrow \\nu \\\\\n0 < |\\epsilon-\\nu| < 1}\n} K_{\\epsilon}(z).$$\n\\end{enumerate}\n\\noindent As a consequence of Formula (2) of Section 7.3.1 in Chapter VII of \\cite{Erdelyi-higher_etc.-2}:\n\\begin{prop}[An integral representation of Bessel functions] \\label{proposition-bessel_integral_formula} Given $\\nu \\in \\N$, $\\rho >0$ and $a > 0$:\n$$J_{\\nu}(\\rho) \\, = \\, \\frac{1}{2\\pi e^{i\\nu\\left( a-\\frac{\\pi}{2} \\right)}} \\, \\int_0^{2\\pi} e^{i\\nu \\theta} \\, e^{-i\\rho(\\cos a \\, \\cos \\theta \\, + \\, \\sin a \\, \\sin \\theta)} \\, d\\theta.$$\n\\end{prop}\n\\noindent Formulas in the next proposition are stated in \\cite{Erdelyi-tables_etc.-2} (Chapter VIII: Formula (20) of Section 8.5 and Formula (35) of Section 8.14):\n\\begin{prop}[Two integral formulas involving Bessel functions] \\label{proposition-integral_formulas-erdelyi}\\\n\\begin{itemize}\n\\item For any real number $y > 0$ and any complex numbers $a,\\nu,\\mu$ such that ${\\rm Re}(a) > 0$ and $-1 < {\\rm Re}(\\nu) < 2 {\\rm Re}(\\mu) + \\frac{3}{2}$, one has:\n$$\\int_0^{\\infty} x^{\\nu + \\frac{1}{2}} \\left( x^2 + a^2 \\right)^{-\\mu-1} J_{\\nu}(xy) \\sqrt{xy} \\ dx =$$\n$$ \\frac{a^{\\nu-\\mu} y^{\\mu+\\frac{1}{2}} K_{\\nu-\\mu}(ay)}{2^{\\mu} \\Gamma(\\mu+1)}.$$\n\\item For any real number $y > 0$ and any complex numbers $a,\\beta,\\nu,\\mu$ such that ${\\rm Re}(a) > 0$, ${\\rm Re}(\\beta) > 0$ and ${\\rm Re}(\\nu) > -1$, one has:\n$$\\int_0^{\\infty} x^{\\nu + \\frac{1}{2}} \\left( x^2 + \\beta^2 \\right)^{-\\frac{\\mu}{2}} K_{\\mu} \\left( a(x^2+\\beta^2)^{\\frac{1}{2}} \\right) J_{\\nu}(xy) \\sqrt{xy}\\ dx =$$\n$$a^{-\\mu} \\beta^{\\nu+1-\\mu} y^{\\nu + \\frac{1}{2}} (a^2+y^2)^{\\frac{\\mu}{2}-\\frac{\\nu}{2}-\\frac{1}{2}} K_{\\mu-\\nu-1} \\left( \\beta (a^2+y^2)^{\\frac{1}{2}} \\right).$$\n\\end{itemize}\n\\end{prop}\n\\noindent One can find the next formula in \\cite{Erdelyi-higher_etc.-2} (Section 7.4.1, Formula (4)):\n\\begin{prop}[Asymptotic expansion for modified Bessel functions] \\label{proposition-bessel_asymptotics-erdelyi} \nFor any fixed $P \\in \\N \\setminus \\{0\\}$ and $\\nu \\in \\C$:\n$$K_{\\nu}(z)=\\left( \\frac{\\pi}{2z} \\right)^{\\frac{1}{2}} e^{-z} \\left( \\left[ \\sum_{p=0}^{P-1} \\frac{ \\Gamma \\left( \\frac{1}{2} + \\nu + p\\right) }{p! \\ \\Gamma \\left( \\frac{1}{2} + \\nu - p\\right) } \\ (2z)^{-p} \\right] + {\\rm O} \\left( |z|^{-P} \\right) \\right).$$\n\\end{prop}\n\\noindent The following proposition can be derived from Section 9.6 in \\cite{Faraut2008}:\n\\begin{prop}[Bochner formula] \\label{proposition-bochner} \nConsider any integer $p \\geq 2$. For $\\xi'\\in S^{p-1}$ and $s > 0$:\n$$\\int_{S^{p-1}}e^{-2i\\pi s \\xi \\cdot \\xi'} d\\sigma(\\xi)\n=2 \\pi s^{1-\\frac{p}{2}} J_{\\frac{p}{2}-1}(2\\pi s)$$\nwhere $d\\sigma$ denotes the Euclidean measure of $S^{p-1}$ and $\\xi \\cdot \\xi'$ denotes the Euclidean scalar product of $\\R^p$ applied to the elements of the sphere $\\xi$ and $\\xi'$ seen as elements of $\\R^p$.\n\\end{prop}\n\n\\footnotesize{ \n\\noindent \\textbf{Contact information.}\\\\\n\\noindent Universit{\\'e} de Reims Champagne-Ardenne\\\\\n\\noindent Laboratoire de Math{\\'e}matiques FRE 2011 CNRS\\\\\n\\noindent UFR Sciences Exactes et Naturelles, Moulin de la Housse - BP 1039\\\\\n\\noindent 51687 REIMS Cedex 2, FRANCE\\\\\n\\noindent \\texttt{gregory.mendousse@univ-reims.fr}\n}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nBlazars, active galactic nuclei (AGN) with strong nonthermal\nemission from an aligned\nrelativistic jet \\citep[][]{br78,up95},\nare the most luminous persistent objects in the universe.\nThese sources emit photons\nacross the whole electromagnetic spectrum from the radio to gamma-ray bands.\nTheir spectral energy distributions (SEDs) are well characterized with a\ndouble-hump structure where the low-energy hump, peaking in the IR\/optical\/UV\/X-ray\nband, is thought to be produced by synchrotron emission of the jet electrons.\nTheir high-energy peak in the gamma-ray band is produced by\nsynchrotron self-Compton (SSC) and external Compton (EC) scattering, or possibly\nby hadronic processes \\citep[e.g.,][]{mb92,bms97,gtf+10}.\n\nBlazars are heuristically classified into flat spectrum radio quasars (FSRQs) and\nBL Lacertae objects (BL Lacs). The former show broad optical emission lines associated\nwith clouds surrounding or in the accretion disk. The latter lack such lines and have a jet\ncontinuum strong enough to obscure spectral features of the host galaxy\n\\citep[][]{mbi+96,lpp+04}. \\citet{pg95} further divided BL Lacs\nbased on the synchrotron peak frequency ($\\nu^{\\rm sy}_{\\rm pk}$) into low synchrotron peak\n(LSP, $\\nu^{\\rm sy}_{\\rm pk}<10^{14}\\rm Hz$), intermediate peak\n(ISP, $10^{14}\\rm Hz<\\nu^{\\rm sy}_{\\rm pk}<10^{15}\\rm Hz$), and high peak\n(HSP, $10^{15}\\rm Hz<\\nu^{\\rm sy}_{\\rm pk}$) subclasses. FSRQs are\nalmost all classified as LSP \\citep[][]{fermiblazar10}.\n\n \\citet{fmc+98} found that 5\\,GHz luminosity,\nthe synchrotron peak luminosity ($L^{\\rm sy}_{\\rm pk}$), and\nthe gamma-ray dominance (ratio of the peak\ngamma-ray to peak synchrotron $\\nu F_{\\nu}$ luminosity)\nare correlated with $\\nu^{\\rm sy}_{\\rm pk}$.\nThey characterize this as\na ``blazar sequence'' trend from low-peaked powerful sources (i.e., FSRQs) to high-peaked\nless powerful sources (HSPs). A plausible physical explanation for this\nsequence is provided by \\citet{gcf+98}; more luminous sources tend to have stronger disk\naccretion, and the external photons from the broad line region (BLR) or the disk in these\nsources provide additional seeds for Compton upscattering which cools the jet\nelectrons, lowering $\\nu^{\\rm sy}_{\\rm pk}$, while increasing the Compton luminosity.\nIndeed, as the typical accretion state evolves over cosmic time, this picture\nmay provide an explanation of evolution in the FSRQ\/BL Lac blazar populations\n\\citep[][]{bd02,cd02}. Quantitatively, this may explain the apparent ``negative evolution''\n(increase at low redshift) observed for HSP BL Lacs \\citep[][]{rsp+00,beb+03,arg+14}.\n\n\\newcommand{\\tablenotemark{a}}{\\tablenotemark{a}}\n\\newcommand{\\tablenotemark{b}}{\\tablenotemark{b}}\n\\newcommand{\\tablenotemark{c}}{\\tablenotemark{c}}\n\\begin{table*}[t]\n\\vspace{-0.0in}\n\\begin{center}\n\\caption{Summary of observations used in this work\n\\label{ta:ta1}}\n\\vspace{-0.05in}\n\\scriptsize{\n\\begin{tabular}{cccccccc} \\hline\\hline\nSource & R.A. & Decl. & Redshift & Observatory & Start date & Obs. ID & Exposure \\\\\n & & & & & (MJD) & & (ks) \\\\ \\hline\n\\multirow{4}{*}{J0022} & \\multirow{4}{*}{0$^{\\rm h}$22$^{\\rm m}$09.25$^{\\rm s}$} & \\multirow{4}{*}{$-$18$^\\circ$53$'$34.9$''$} & \\multirow{4}{*}{0.774} & GROND & 57031.1 & $\\cdots$ & 0.25\/0.24\\tablenotemark{a} \\\\\n & & & & {\\em Swift} & 57031.7 & 00080777001 & 1.9\\tablenotemark{b} \\\\\n & & & & {\\em XMM} & 57026.8 & 0740820501 & 15\/9\\tablenotemark{c} \\\\\n & & & & {\\em NuSTAR} & 57026.7 & 60001141002--4 & 110 \\\\ \\hline\n\\multirow{4}{*}{J0630} & \\multirow{4}{*}{6$^{\\rm h}$30$^{\\rm m}$59.515$^{\\rm s}$} &\\multirow{4}{*}{$-$24$^\\circ$06$'$46.09$''$} &\\multirow{4}{*}{$>$1.239} & GROND & 56949.2 & $\\cdots$ & 0.25\/0.24\\tablenotemark{a} \\\\\n & & & & {\\em Swift} & 56948.5 & 00080776001 & 0.27\\tablenotemark{b} \\\\\n & & & & {\\em XMM} & 56948.2 & 0740820401 & 8\/4\\tablenotemark{c} \\\\\n & & & & {\\em NuSTAR} & 56947.7 & 60001140002 & 67 \\\\ \\hline\n\\multirow{4}{*}{J0811} & \\multirow{4}{*}{8$^{\\rm h}$11$^{\\rm m}$03.214$^{\\rm s}$} & \\multirow{4}{*}{$-$75$^\\circ$30$'$27.85$''$} & \\multirow{4}{*}{0.689} & GROND & 56903.3 & $\\cdots$ & 0.25\/0.24\\tablenotemark{a} \\\\\n & & & & {\\em SWIFT} & 56908.2 & 00091903001 & 0.39\\tablenotemark{b} \\\\\n & & & & {\\em XMM} & 56901.2 & 0740820601 & 9\/6\\tablenotemark{c} \\\\\n & & & & {\\em NuSTAR} & 56901.2 & 60001142002 & 113 \\\\ \\hline\n\\end{tabular}}\n\\end{center}\n\\hspace{-2.0 mm}\n$^{\\rm a}${ For {\\it g$'$r$'$i$'$z$'$\/JHK} bands.}\\\\\n$^{\\rm b}${ For the UW1 band. Exposures in the other UVOT bands may differ from this value.}\\\\\n$^{\\rm c}${ For MOS1,2\/PN.}\\\\\n\\vspace{-1.0 mm}\n\\end{table*}\n\n On the other hand, \\citet{gpp+12} used Monte Carlo simulations to argue that the\n$L^{\\rm sy}_{\\rm pk}$ and $\\nu^{\\rm sy}_{\\rm pk}$ anti-correlation may be primarily a\nselection effect. \\citet{pgr12} discuss four sources with high $\\nu^{\\rm sy}_{\\rm pk}$ and\nhigh peak (synchrotron + SSC) power as examples well off of the blazar sequence.\nSuch sources might be FSRQs with unusually strong jet emission along the Earth line-of-sight\nmasking the underlying host components. Thus simultaneous observations and careful SED\nmodeling of such (generally higher-redshift) BL Lac sources is interesting as it can\nhelp us understand the underlying emission zone physics and whether it is truly different\nfrom the bulk of the blazar population. Characterization via less redshift-dependent parameters\n\\citep[e.g. gamma-ray dominance or Compton dominance;\nsee][for example]{fmc+98,f13} may also help clarify\ntheir place in the population. Also, comparing robust SED model fits with\ngamma-ray spectra of high-$z$ blazars can reveal the effect of absorption\nby the extragalactic background light (EBL), which provides important constraints\non evolution of cosmic star formation \\citep[e.g.,][]{fermiEBL, HESSEBL}.\nBL Lacs are believed to have higher Compton dominance and less sensitivity\nto local soft photon fields and so are particularly useful for such study.\n\n Appropriate high-redshift HSP BL Lac objects are rare because they\nare faint especially in the gamma-ray band, and HSPs appear to exhibit negative\nevolution \\citep[][]{arg+14}. We select three\n{\\it Fermi}-detected \\citep[][]{fermi2fgl,fermi2lac} sources,\n3FGL~J0022.1$-$1855 (J0022, $z=0.774$),\n3FGL~J0630.9$-$2406 (J0630, $z>1.239$),\nand 3FGL~J0811.2$-$7529 (J0811, $z=0.689$), whose optical spectra\nare unusual, showing no emission lines but a set of strong low excitation (Mg~I,\nFe~II, Al~II etc) absorption lines on a blue, power-law\ncontinuum. These indicate that the AGN is viewed through the disk of an intervening\nabsorber. In \\citet{src+13}, this was taken to be the host galaxy; indeed for\nJ0630 the photometry of \\citet{rsg+12} supports this as the host redshift. With\nestimated redshifts of 0.774, $>$1.239, and 0.689 \\citep[][]{rsg+12,src+13} for\nJ0022, J0630 and J0811, respectively,\nthese are thus luminous high-peak sources suitable for studying the extreme of the BL Lac\npopulation. At these redshifts, we may also see the effects of\nextragalactic background light absorption at\nthe high end of the {\\it Fermi} band. To probe this absorption, and the high end\nof the jet particle population most sensitive to Compton cooling, we require particularly\ngood characterization of the peak and high-energy cutoff (near-IR to hard X-ray)\nof the synchrotron component. Under classic SSC modeling,\nthis allows us to characterize the high-energy Compton component, as well, thus\nproviding inferences about the Compton cooling at the source and EBL absorption of the\nGeV photons as they propagate to Earth.\n\nIn this paper, we present broadband SEDs of the three high-redshift BL Lacs which are\nsimultaneous across the critical $\\nu >\\nu^{\\rm sy}_{\\rm pk}$ range (Section~\\ref{sec:sec2}).\nJ0630 has been previously discussed as a high-$\\nu^{\\rm sy}_{\\rm pk}$,\nhigh-power source \\citep{pgr12}; our improved data allow more refined modeling,\nwhich is discussed in Section~\\ref{sec:sec3}, including EBL constraints. The implications\nof our inferred model parameters are discussed in Section~\\ref{sec:sec4}.\nWe use $H_0=70\\rm \\ km\\ s^{-1}\\ Mpc^{-1}$, $\\Omega_m=0.3$, $\\Omega_{\\Lambda}=0.7$ \\citep[e.g.,][]{ksd+11},\nand redshift values given in Table~\\ref{ta:ta1} ($z=1.239$ for J0630) throughout.\n\n\\section{Observations and Data Reduction}\n\\label{sec:sec2}\n\nBL Lac objects can be variable on all timescales from minutes to years \\citep[][]{HESSvariable},\nso coordinated broad-band coverage is important for characterizing the instantaneous SED.\nWe therefore carried out nearly contemporaneous observations of the sources using the\nGamma-Ray burst Optical\/Near-Infrared Detector (GROND)\ninstrument at the 2.2-m MPG telescope at the ESO La Silla Observatory \\citep{gbc+08}\nas well as the {\\it Swift} \\citep{gcg+04}, {\\it XMM-Newton} \\citep{jla+01}\nand {\\it NuSTAR} \\citep{hcc+13} satellites, covering the upper range of the synchrotron component.\nOur sources showed relatively modest variability in the {\\it Fermi} \\citep[][]{fermimission} band\nand so we average over 6 years of Large Area Telescope (LAT) data to best characterize the\nmean Compton component of these relatively faint (but luminous, for BL Lacs) sources.\nArchival radio, optical, and near-IR observations are provided for comparison\nalthough we do not use them in the SED fitting.\n\n\\subsection{Contemporaneous observations: GROND, Swift, XMM-Newton, and NuSTAR}\n\\label{sec:sec2_1}\n\nThe GROND data were reduced and analyzed with the standard tools and methods\ndescribed in \\citet{kkg+08}. The photometric data were obtained using\nFWHM-matched PSF ($g^\\prime r^\\prime i^\\prime z^\\prime$) or aperture photometry ({\\it JHK}).\nThe $g^\\prime$, $r^\\prime$, $i^\\prime$, and $z^\\prime$ photometric\ncalibration was obtained via standard star fields observed on the\nsame nights as the target integrations. The {\\it J}, {\\it H}, and {\\it Ks} photometry was\ncalibrated against selected in-field 2MASS stars \\citep[][]{scs+06}.\n\nFor {\\it Swift} UVOT data, we performed aperture photometry for the six\n{\\it Swift} filters \\citep[][]{pbp+08} using the {\\tt uvotsource} tool\nin HEASOFT 6.16\\footnote{http:\/\/heasarc.nasa.gov\/lheasoft\/}. We measured photometric magnitude of the sources using\na $R=5^{\\prime\\prime}$ aperture. Backgrounds were estimated using a $R=20^{\\prime\\prime}$ circle\nnear the source.\n\nX-ray SEDs of the sources were measured with {\\it XMM-Newton} and {\\it NuSTAR}.\nThe sources were detected with very high significance ($>20\\sigma$)\nwith {\\it XMM-Newton} but with relatively low significance ($\\ifmmode\\stackrel{>}{_{\\sim}}\\else$\\stackrel{>}{_{\\sim}}$\\fi 6\\sigma$)\nwith {\\it NuSTAR}.\nFor the {\\em XMM-Newton} data, we processed the observation data files\nwith {\\ttfamily epproc} and {\\ttfamily emproc} of Science Analysis System (SAS)\nversion 14.0.0\\footnote{http:\/\/xmm.esac.esa.int\/sas\/} and then applied\nstandard filters. The {\\it NuSTAR} data were processed with the standard pipeline\ntool {\\tt nupipeline} of {\\tt nustardas}\n1.4.1 integrated in the HEASOFT 6.16.\nWe used {\\it NuSTAR} CALDB version 20140414 and applied the standard\nfilters.\\footnote{See http:\/\/heasarc.gsfc.nasa.gov\/docs\/nustar\/analysis\/nustar\\\\\\_swguide.pdf for more details}\nWe then extracted source events using circular regions with $R=20^{\\prime\\prime}$ and $R=30^{\\prime\\prime}$\nfor the {\\it XMM-Newton} and the {\\it NuSTAR} data, respectively. Backgrounds were\nextracted from nearby source-free regions.\n\n\\subsection{Gamma-ray observations}\n\\label{sec:sec2_2}\nFor the gamma-ray data, we used the {\\it Fermi} observations taken between\n2008 August 4 and 2015 January 31. The Pass 8 data \\citep[][]{fermiP8}, based on a complete\nand improved revision of entire LAT event-level analysis, were downloaded\nfrom Fermi Science Support Center\\footnote{http:\/\/fermi.gsfc.nasa.gov\/ssc\/},\nand we analyzed the data using the {\\it Fermi}\nScience tool 10-00-04 along with the instrument response functions (IRFs) P8R2\\_SOURCE\\_V6.\nWe extracted source class events in the 100\\,MeV--500\\,GeV band in a\n$R=5^\\circ$ region of interest (ROIs)\nand $<80^\\circ$ zenith angle and $<52^\\circ$ rocking angle cuts.\nThese events were analyzed using the background models ({\\tt gll\\_iem\\_v06} and\n{\\tt iso\\_P8R2\\_SOURCE\\_V6\\_v06}) and all 3FGL sources within $15^\\circ$.\nWe first modeled fluxes on a one-month cadence to check for strong source\nvariability using the standard {\\it Fermi} likelihood analysis with {\\tt gtlike}\n(see Figure~\\ref{fig:fig1} and Section~\\ref{sec:sec3_1}).\nNo strong flares were seen and so we combined all the LAT\ndata, modeling the mission-averaged spectrum.\nIn Figure~\\ref{fig:fig1}, we mark the epochs of the\ncontemporaneous campaign and historical spectra. For J0630 we also have access\nto optical monitoring from the KAIT program \\citep[][]{crf+14}, shown on the top panel.\nVariability is clearly seen in the optical band.\n\n\\begin{figure}\n\\includegraphics[width=3.65 in]{fig1.eps}\n\\vspace{-3mm}\n\\figcaption{Optical ({\\it R} band) and gamma-ray (100\\,MeV--500\\,GeV) light curves.\nThe top panel shows KAIT (right scale)\nand {\\it Fermi} (left scale) fluxes for J0630. Our contemporaneous observation\nepoch and the optical spectrum epochs are marked. The lower panel shows the LAT light curves and\nmultiwavelength epochs for J0022 and J0811. The modest LAT variability justifies\nthe use of mission-averaged spectra.\n\\label{fig:fig1}\n}\n\\vspace{0mm}\n\\end{figure}\n\n\\subsection{Archival observations}\n\\label{sec:sec2_3}\nFor comparison, we also collected archival data in the radio to UV band. We assembled data\nfrom various catalogs (e.g., {\\it WISE} and 2MASS for IR data) or reanalyze the\narchival data (e.g., VLT\/Keck spectra and {\\it Swift} UVOT). For the catalog data, we convert the\nmagnitude to flux appropriately.\nThe VLT\/Keck data reduction and calibration were described in \\citet{src+13}.\nThe archival UVOT data are processed as described above (Section~\\ref{sec:sec2_1}).\nThe measurements are corrected for Galactic extinction in constructing\nthe SED (Section~\\ref{sec:sec3_2}).\nArchival measurements are used only in flux variability studies.\n\n\\subsection{Discovery of a serendipitous source}\n\\label{sec:sec2_4}\n\n\\begin{figure*}\n\\centering\n\\begin{tabular}{ccc}\n\\hspace{-3.0 mm}\n\\includegraphics[width=2.1 in]{fig2a.eps} &\n\\hspace{1.0 mm}\n\\includegraphics[width=2.05 in]{fig2b.eps} &\n\\hspace{-3.0 mm}\n\\includegraphics[width=2.7 in]{fig2c.eps} \\\\\n\\end{tabular}\n\\figcaption{{\\it Left}: {\\it NuSTAR} image\nof the field containing J0811. The color scale is arbitrarily adjusted\nfor better visibility. {\\it Fermi}\/LAT 3FGL ellipse (95\\%, white) and the best-fit circle (95\\%, magenta)\nare shown, and a $R=30''$ circle is drawn around the serendipitous source (denoted as J0810.0$-$7527).\n{\\it Middle}: Location of the sources we are studying in the {\\it WISE}\n[3.4]$-$[4.6]$-$[12]$\\mu m$ color-color diagram \\citep[Figure taken from][]{dma+12}.\nThe four sources, including J0810, are located in the middle of the BZB\n(naming convention for BL Lac in the ROMA-BZCAT catalog) distribution.\nSee \\citet{dma+12} for more detail.\n{\\it Right}: Observed SED of the serendipitous source. Note that we used\n$N_{\\rm H}=6.9\\times 10^{20}\\rm \\ cm^{-2}$, the optical extinction inferred value,\nfor constructing the SED. Notice that this new source is quite\nhard, emitting more strongly in the {\\it NuSTAR} band than in the {\\it XMM-Newton} band.\n\\label{fig:fig2}\n}\n\\vspace{0mm}\n\\end{figure*}\n\nWe discovered a serendipitous X-ray source (J0810) in the field of J0811 (Figure~\\ref{fig:fig2}).\nThe X-ray ({\\it XMM-Newton}) position of the source is\nR.A. = 08\\textsuperscript{h}10\\textsuperscript{m}03\\textsuperscript{s}\nand decl. = $-$75$^\\circ$27$'$21$''$ (J2000, $\\delta_{\\rm R.A.,\\ decl.} =2''$ statistical only),\nonly 6$'$ from J0811 (Figure~\\ref{fig:fig2} left).\nWe find that the spectrum cannot be described with a simple absorbed power law\n($\\chi^2$\/dof=185\/118, $p=7\\times10^{-5}$).\nA broken power-law model\\footnote{http:\/\/heasarc.gsfc.nasa.gov\/docs\/xanadu\/xspec\/manual\/XS\\\\modelBknpower.html}\nexplains the data ($\\chi^2$\/dof=116\/116, $p=0.47$) and\nthe best-fit parameters are $N_{\\rm H}=1.4\\pm0.3\\times10^{21}\\rm \\ cm^{-2}$,\nlow-energy photon index $\\Gamma_{\\rm 1}=3.4\\pm0.3$, high-energy photon index\n$\\Gamma_{\\rm 2}=1.74\\pm0.07$, break energy $E_{\\rm break}=1.46\\pm0.08$\\,keV\nand 3--10\\,keV flux $F_{\\rm 3-10 keV}=2.7\\pm0.2\\times 10^{-13}\\rm \\ erg \\ s^{-1}\\ cm^{-2}$.\n\nTogether with archival radio, optical, and {\\it Swift} UV data, we\nconstruct the SED of the source (Figure~\\ref{fig:fig2} right). If we use the best-fit X-ray\n$N_{\\rm H}$, the extrapolated spectrum matches poorly to the optical. Instead\nwe de-absorb using the value from the optical\/UV extinction $N_{\\rm H}=6.9\\times10^{20}\\rm \\ cm^{-2}$.\nX-ray fits with absorption fixed at this value are statistically acceptable\n(null hypothesis probability $p=0.3$).\nThe SED of this source suggests a blazar with $\\nu^{\\rm sy}_{\\rm pk}$ in the optical range,\nand a rise to a Compton component in the hard X-ray band. Its location in the\n{\\it WISE} color-color diagrams \\citep[Figure~\\ref{fig:fig2} middle; see also][]{dma+12}\nsuggests that the source should be a BL Lac.\nIf the Compton component peaks at $>100$\\,MeV, this source may contribute to the J0811 SED,\nsince the source is within the aperture we used for J0811. If we free the position of J0811 in the\n{\\it Fermi} analysis, we find a maximum likelihood coincident with J0811\n(magenta circle in Figure~\\ref{fig:fig2} left). Also, a second source\nat the J0810 position does not significantly increase the model test statistic (TS).\n\nWe then increased the zenith angle cut to $<100^\\circ$ to have more events\nand used a small spatial bin size (0.05$^\\circ$) to see if J0810 is detected\nin the {\\it Fermi} band. We performed binned likelihood analysis with\nthe new data. In this case, a gamma-ray counterpart\nof J0810 is detected significantly (TS=56); the model without J0810 is only 0.03\\% as\nprobable as the one with J0810.\nIn the 0.1--500\\,GeV band, J0810 has $\\sim$20\\% of the flux (with 40\\% flux uncertainty)\nof J0811 with a similar power-law index ($\\Gamma_{\\gamma}=1.8\\pm0.1$).\nThese spectral parameters for J0810 may not be very accurate because of mixing from\nthe brighter source, J0811.\nSince J0811 is brighter than J0810 in the gamma-ray band,\nwe attribute all of the LAT flux to J0811 in SED modeling\nand discuss implication of J0810 contamination on the model (see Section~\\ref{sec:sec3_3}).\n\n\\begin{figure*}\n\\centering\n\\vspace{-75.0 mm}\n\\hspace{-12.0 mm}\n\\includegraphics[width=5.7 in,angle=90]{fig3.eps} \\\\\n\\vspace{-5.0 mm}\n\\figcaption{Observed broadband SED and best-fit models for\n({\\it a}) J0022, ({\\it b}) J0630, and ({\\it c}) J0811.\nData points with an error bar are taken from the contemporaneous\nobservations (Sections~\\ref{sec:sec2_1} and \\ref{sec:sec2_2})\nand diamonds are from the archival observations (Section~\\ref{sec:sec2_3}).\nThe dashed lines are the best-fit SSC SED models\nof \\citet{bms97} with (black) and without (red) EBL absorption \\citep[][]{frd10}.\nNote that the archival data are not taken contemporaneously even if they\nare plotted in the same color and symbol. The insets plot the VLT\/Keck spectra of\n\\citet{src+13}, showing the lack of emission lines and the low excitation\nabsorption complexes placing lower limits on the redshift. These observations\nappear to have been in a brighter, harder optical state.\n\\vspace{2.0 mm}\n\\label{fig:fig3}\n}\n\\vspace{0mm}\n\\end{figure*}\n\n\\section{Data Analysis and Results}\n\\label{sec:sec3}\n\n\\subsection{Variability}\n\\label{sec:sec3_1}\n\n We have examined the collected data for variability, since short\ntimescales can give useful constraints on the characteristic size of the emission zone\nin the various wavebands. We first examined our contemporaneous data sets for\nshort-timescale variations. For the {\\it XMM-Newton} and {\\it NuSTAR} data, spanning\n$\\sim$10--100\\,ks, we constructed exposure-weighted light curves using various time bin sizes\n($\\sim$100--20,000~s), ensuring $> 20$ counts in each time bin, and calculated $\\chi^2$\nfor a constant flux. The probability for constancy was always high ($\\ifmmode\\stackrel{>}{_{\\sim}}\\else$\\stackrel{>}{_{\\sim}}$\\fi$10\\%),\nimplying no significant short-term variability for any of the three sources at this epoch.\nSimilarly, the optical\/UV data from the contemporaneous epoch did not show sub-day variability.\n\n However, on longer time scales, the optical synchrotron peak flux does show\nsubstantial variability, as can be seen by comparing the contemporaneous and archival points\nin Figures~\\ref{fig:fig1} and \\ref{fig:fig3}. J0022, for example varies by $\\sim 6\\times$.\nAs noted, the VLT\/Keck spectra also appear to represent brighter epochs, although slit losses limit\nthe precision of the flux calibration. In general, the brighter epochs appear to have harder\nnear-IR to UV spectra, suggesting increased electron energy (or increased bulk Doppler factor)\nin flaring events. A much better characterization of J0630's optical variability is available from\nthe KAIT {\\it Fermi} AGN monitoring data \\citep[][]{crf+14}.\\footnote{http:\/\/brando.astro.berkeley.edu\/kait\/agn\/}\nThe dominant modulation is slow on $\\sim$year timescales; this is of modest amplitude compared\nto other BL Lacs ($\\sim$50\\%). KAIT resolves times as short as the $\\sim$3d cadence and\nwe do see statistically significant ($\\ifmmode\\stackrel{>}{_{\\sim}}\\else$\\stackrel{>}{_{\\sim}}$\\fi$6$\\sigma$) changes between consecutive observations.\nThis suggests that at least some of the jet flux arises in compact $r<10^{16}$cm structures.\n\n We can use the LAT band to probe variability in the Compton peak emission. Since these\nsources are not very bright, we were able to only probe $\\sim$month timescales. To\nthis end we generated lightcurves by fitting source fluxes to 100\\,MeV--500\\,GeV\nphotons from a 5 degree ROI about each source using the {\\tt gtlike} tool for each time bin.\nFor this we fixed the background model\nnormalization and the background source spectral parameters at the mission-averaged values (see below),\nallowing only the source flux to vary with the spectral index held fixed at the values given\nin Table~\\ref{ta:ta2} .\nFigure~\\ref{fig:fig1} shows the corresponding light curves. The variability\nis not strong ($\\chi^2$\/dof values for a constant light curve of \n5\/8, 92\/72, and 28\/33 for J0022, J0630 and J0811, respectively).\nWe confirm the results of the 3FGL catalog \\citep[][]{fermi3fgl};\nour sources are not flagged as variable in the 3FGL catalog at a 99\\% confidence.\nFinally, examination of light curves assembled by the\nAgenzia Spaziale Italiana science data center\\footnote{http:\/\/www.asdc.asi.it\/fermi3fgl\/}\nalso shows no significant variability in any source. We conclude that the three sources have\nbeen relatively quiescent for BL Lacs -- this gives us confidence that the mission-averaged LAT\nspectrum may be usefully compared with our contemporaneous campaign fluxes for SED fitting.\n\n\\subsection{Constructing broadband SEDs}\n\\label{sec:sec3_2}\n\n\\newcommand{\\tablenotemark{a}}{\\tablenotemark{a}}\n\\newcommand{\\tablenotemark{b}}{\\tablenotemark{b}}\n\\newcommand{\\tablenotemark{c}}{\\tablenotemark{c}}\n\\begin{table}[t]\n\\vspace{-0.0in}\n\\begin{center}\n\\caption{Galactic foreground reddening values and X-ray\/gamma-ray fit results\n\\label{ta:ta2}}\n\\vspace{-0.05in}\n\\scriptsize{\n\\begin{tabular}{cccccccc} \\hline\\hline\nSource\t\t\t&\t\t\t\t& J0022 & J0630 & J0811 \\\\ \\hline\n$E(B-V)$\t\t& (mag)\t\t\t\t& 0.024 & 0.056 & 0.125 \\\\\n$N_{\\rm H,\\ Dust}$\\tablenotemark{a}\t& ($10^{20}\\rm \\ cm^{-2}$)\t& 1.3 & 3.1 & 7 \\\\\n$N_{\\rm H}$\t\t& ($10^{20}\\rm \\ cm^{-2}$)\t& 4(1) & 13(1) & 7(1) \\\\\n$\\Gamma_{\\rm X}$\t& $\\cdots$\t\t\t& 2.55(6) & 2.98(7) & 2.45(7) \\\\\n$F_{\\rm X}$\\tablenotemark{b}\t& $\\cdots$\t\t\t& 0.93(8) & 1.6(1) & 1.4(1) \\\\\n$\\Gamma_{\\rm \\gamma}$\t& $\\cdots$\t\t\t& 1.86(6) & 1.83(3) & 1.93(4) \\\\\n$F_{\\rm \\gamma}$\\tablenotemark{c}\t& $\\cdots$\t\t\t& 6.3(9) & 25(2) & 23(2) \\\\ \\hline\n\\end{tabular}}\n\\end{center}\n\\vspace{-0.5 mm}\n$^{\\rm a}$ Dust-extinction equivalent $N_H$, converted with\n$N_{\\rm H}=1.8\\times 10^{21} A(V)\\rm \\ cm^{-2}\\ mag^{-1}$ and $R_V=3.1$ \\citep{ps95}.\\\\\n$^{\\rm b}$ 3--10\\,keV flux in units of $10^{-13}\\rm \\ erg\\ s^{-1}\\ cm^{-2}$.\\\\\n$^{\\rm c}$ 0.1--500\\,GeV flux in units of $10^{-9}\\rm \\ photons \\ s^{-1}\\ cm^{-2}$.\n\\end{table}\n\nNext we assembled broadband SEDs for the sources using the data described in Section~\\ref{sec:sec2}.\nThe optical\/UV magnitudes were corrected for the dust map extinction in these directions\n(Table~\\ref{ta:ta2}) obtained from the NASA\/IPAC extragalactic database, using\nthe \\citet{sf11} calibration. We show the SEDs in Figure~\\ref{fig:fig3}.\nNote that Lyman-$\\alpha$ forest absorption was visible in J0630 at frequencies above\n$\\sim10^{15}$\\,Hz in the UVOT data, as expected from its large redshift;\nwe do not use the high-frequency UVOT data $\\ifmmode\\stackrel{>}{_{\\sim}}\\else$\\stackrel{>}{_{\\sim}}$\\fi 10^{15}$\\,Hz in the J0630 SED modeling.\n\nThe X-ray response files are produced with the standard tools in SAS and in {\\tt nustardas}\nfor the {\\it XMM-Newton} and {\\it NuSTAR} spectra, respectively.\nWe fit the spectra in the 0.3--79\\,keV band with an absorbed power-law model\nin {\\tt XSPEC} 12.8.2 and found that the model describes the data well, having\n$\\chi^2$\/dof$\\ifmmode\\stackrel{<}{_{\\sim}}\\else$\\stackrel{<}{_{\\sim}}$\\fi$1 for all three sources. The fact that all X-ray spectra are\nwell modeled by a single absorbed power law is important to the modeling below.\nThe absorption corrections for the X-ray data were obtained from the $N_{\\rm H}$\nin the power-law fits. The fit results are presented in Table~\\ref{ta:ta2}.\n\n While the X-ray fit and extinction map values for the absorption agree well for J0811,\nJ0022 and especially J0630 show stronger X-ray absorption. Given the modest dust map resolution,\nand the $\\sim$50\\% conversion uncertainties \\citep[e.g.,][]{g75,w11,fgos15}, the discrepancy\nfor J0022 may be reconciled. However the large value for J0630 seems difficult to accommodate\nand we have no clear explanation. The Galactic H{\\scriptsize I} column\ndensity\\footnote{https:\/\/heasarc.gsfc.nasa.gov\/cgi-bin\/Tools\/w3nh\/w3nh.pl}\ntoward J0630 is 7--12$\\times 10^{20}\\rm cm^{-2}$, consistent with the X-ray inferred value.\nIf we assume the X-ray value for de-extinction of the optical, we find an unnatural UV flux rise\n(similarly, using the optical value makes an unnatural cutoff in the low energy X-ray spectrum).\nThus we can only accommodate the X-ray fit value if the optical\/UV flux has an extra\nblue, narrow-band component. This seems unnatural. Alternatively the dust map extinction might\nbe correct and the X-ray component may be spatially separated from the optical emission,\nexperiencing extra local (host) absorption.\nMeasuring the J0630 VLT absorption line strengths indicated that the intervening\/host\ngalaxy supplies negligible extinction $E(B-V)<0.01$ to the optical component, which is\nconsistent with the low effective $E(B-V)$.\nAcknowledging this inconsistency, we use the two values in Table~\\ref{ta:ta2}\nwhen constructing the SED.\n\nFor the {\\it Fermi} SED, we performed binned likelihood analysis\nusing the same configuration as described in Section~\\ref{sec:sec2_2} with the 6.5-yr data.\nIn doing so, we fit spectra for all bright sources (detected with $\\gapp5\\sigma$) in the ROI\nand the background amplitudes. Spectral parameters for faint sources or those outside the ROI\nare held fixed at the 3FGL values.\nThe results are shown in Table~\\ref{ta:ta2}.\nThe highest-energy bands in which a significant detection (TS$>15$)\nwas made are 29--75\\,GeV, 75--194\\,GeV, and 75--194\\,GeV for J0022,\nJ0630 and J0811, respectively (see Figure~\\ref{fig:fig3}).\nWe then derive the SEDs using the best-fit power-law model, and show the inferred spectrum in\nFigure~\\ref{fig:fig3}, where the TS is greater than 15\nfor each data point. We performed the analysis using different ROI sizes,\nfinding consistent results.\nIn Figure~\\ref{fig:fig3} we show the results obtained for the 5$^\\circ$ extraction\nas it gives the highest TS value.\n\nWe show the broadband SEDs in Figure~\\ref{fig:fig3}.\nA non-contemporaneous broadband SED for J0630 with sparser X-ray and gamma-ray data\nhas been previously reported \\citep[][]{gtf+12, pgr12}; the results are broadly similar to\nour measurements.\n\n\\subsection{SED modeling}\n\\label{sec:sec3_3}\n\n\\newcommand{\\tablenotemark{a}}{\\tablenotemark{a}}\n\\newcommand{\\tablenotemark{b}}{\\tablenotemark{b}}\n\\begin{table*}[]\n\\vspace{-0.0in}\n\\begin{center}\n\\caption{Best-fit parameters for the SSC model of B97 with single power-law injection\n\\label{ta:ta3}}\n\\vspace{-0.05in}\n\\scriptsize{\n\\begin{tabular}{ccccc} \\hline\\hline\nParameter & Symbol & 3FGL~J0022.1$-$1855 & 3FGL~J0630.9$-$2406 & 3FGL~J0811.2$-$7529 \\\\ \\hline\nRedshift & $z$ & 0.774 & $>1.239$ & 0.689 \\\\\nDoppler factor & $\\delta_{\\rm D}$ & 19 & 71 & 33 \\\\\nBulk Lorentz factor & $\\Gamma$ & $>9.6$ & $>35.3$ & $>16.5$ \\\\\nViewing angle (deg.) & $\\theta_v$ & $<3.0$ & $<0.81$ & $<1.74$ \\\\\nMagnetic field (mG) & $B$ & 60 & 1016 & 7 \\\\\nComoving radius of blob (cm) & $R'_b$ & $1.12\\times10^{14}$ & $1.78\\times10^{13}$ & $1.52\\times10^{14}$ \\\\ \nEffective radius of the blob ($cm$) & $R'_{\\rm E}=(3 R'^{2}_b t_{\\rm evol} c\/4)^{1\/3}$ & $1.4\\times10^{15}$ & $1.9\\times10^{14}$ & $1.7\\times10^{15}$ \\\\ \\hline\nInitial electron spectral index & $p_{\\rm 1}$ & 3.14 & 4.26 & 3.19 \\\\ \nInitial minimum electron Lorentz factor & $\\gamma'_{\\rm min}$ & $2.88\\times 10^{4}$ & $1.41\\times 10^4$ & $1.18\\times 10^{4}$ \\\\\nInitial maximum electron Lorentz factor & $\\gamma'_{\\rm max}$ & $1.5\\times10^{6}$ & $2.7\\times10^7$ & $3\\times10^{7}$ \\\\\nInjected particle luminosity (erg s$^{-1}$)\\tablenotemark{a} & $L_{\\rm inj}$ & $9\\times10^{42}$ & $7\\times10^{41}$ & $8\\times10^{42}$ \\\\ \n$\\chi^2$\/dof & $\\cdots$ & 151.1\/122 & 186\/140 & 128.5\/94 \\\\ \\hline\nSynchrotron peak frequency (Hz)\\tablenotemark{b} & $\\nu^{\\rm sy}_{\\rm pk}$ & $5.6\\times10^{14}$ & $1.5\\times10^{15}$ & $5.8\\times10^{14}$ \\\\\nSynchrotron peak luminosity($\\rm erg\\ s^{-1}$)\\tablenotemark{b} & $L^{\\rm sy}_{\\rm pk}$ & $4.6\\times10^{45}$ & $6.7\\times10^{46}$ & $5.1\\times10^{45}$ \\\\\nCompton dominance & CD & 1.2 & 1.4 & 2.1 \\\\ \\hline \\hline\n\\end{tabular}}\n\\end{center}\n\\hspace{-2.0 mm}\n$^{\\rm a}${Energy injected into the jet in the jet rest frame \\citep[see][]{bc02}.}\\\\\n$^{\\rm b}${Quantities in the observer frame.\nThe luminosity quoted is that inferred assuming isotropic emission.}\\\\\n\\end{table*}\n\n We use the one-zone synchro-Compton model of \\citet[][hereafter B97]{bms97} to model\nthe SEDs of the sources. The code evolves a spherical blob of electron\/positron\nplasma with a power-law injected energy distribution,\nfollowing the e$^+$\/e$^-$ population over $10^7$\\,s ($t_{\\rm evol}$)\nassuming that the particle energy loss is dominated by radiative cooling\nas the blob zone flows along a jet axis.\nAs blobs are continuously injected, the emission zone forms\na cylindrical shape (i.e., jet)\nelongated along the jet axis ($l=ct_{\\rm evol}=3\\times10^{17}$\\,cm)\nand the time-integrated spectrum determines the jet emission.\nThe effect of pair-absorption is calculated and included in the model.\nThe full model has 16 parameters including those for disk and BLR emission;\nto simplify we start with standard BL Lac assumption\nthat self-Compton emission dominates so that the seed photons from BLR\nand disk are negligible. The seven remaining parameters we adjust are\nthe low-energy and high-energy cutoffs ($\\gamma'_{\\rm min,max}$)\nand spectral index of the power-law electron distribution ($p_{\\rm 1}$),\nthe magnetic field strength ($B$), the bulk Lorentz factor of the jet ($\\Gamma$)\n(this is done for a fixed viewing angle $\\theta_{\\rm v}$,\nhence equivalent to adjusting the Doppler factor $\\delta_{\\rm D}$)\nand the blob rest frame size ($R'_b$) and electron density ($n_{e}$),\nwhich serve to normalize the total flux.\nThis model has also been used for modeling SED of other blazars \\citep[e.g.,][]{hba+01,r06}.\n\n We use the following steps to find best-fit SED parameters:\n(1) adjust the parameters to visually match the SED for initial values,\n(2) vary each individual parameter over a range\n(a factor of $\\sim$2 initially and decreased with iterations) with\nten grid points while holding the other parameters fixed,\n(3) find the parameter value that provides the minimum $\\chi^2$,\n(4) update the parameter found in step (3) with the best-fit value,\n(5) repeat (2)--(4) until the fit does not improve any more.\nBecause the X-ray spectra are so well described by a simple power law,\nwe initially identify their spectra with synchrotron emission of a cooled electron population,\nstrongly constraining the fit parameter set.\nWe do not include the highest energies ($\\ifmmode\\stackrel{>}{_{\\sim}}\\else$\\stackrel{>}{_{\\sim}}$\\fi 40$\\,GeV) LAT points in the initial\nfits, as we will use them later for EBL constraints as done by \\citet{dfp+13}. We update\nonly one parameter each iteration although we vary all seven parameters.\nWe present the best-fit parameters in Table~\\ref{ta:ta3}.\nWe also measured $\\nu^{\\rm sy}_{\\rm pk}$, $L^{\\rm sy}_{\\rm pk}$ and CD\nusing the best-fit SED model, and present them in Table~\\ref{ta:ta3}.\n\n In the model $\\Gamma$ and $\\theta_{\\rm v}$\nappear only in combination through the Doppler factor $\\delta_{\\rm D} = [\\Gamma(1-\\beta\\mu)]^{-1}$,\nwhere $\\beta=\\sqrt{1-1\/\\Gamma^2}$ and $\\mu=\\mathrm{cos}(\\theta_{\\rm v})$. Hence, the model determines\nonly $\\delta_{\\rm D}$ unless one has external constraints on one of $\\Gamma$ or $\\theta_{\\rm v}$.\nTherefore, for a given $\\delta_{\\rm D}$, only lower and upper limit for $\\Gamma$ and $\\theta_{\\rm v}$ can\nbe inferred, also given in Table~\\ref{ta:ta3}.\n\n While the procedure above converges well to a local minimum, there is always a risk\nthat quite distinct solutions could provide better fits. The high dimensionality of the fit space,\nplus the incomplete SED coverage makes it difficult to locate such minima. To aide our exploration of\nparameter space, we used the initial scans to define the covariance between the various\nquantities. We find that simple power-law co-dependencies capture most of the covariance\ntrend around the fit minimum. We fit an amplitude and slope for each parameter pair. Thus,\nby varying one\ncontrol parameter, say $B$, and then setting the others to the covariance-predicted values, we can\ntake larger steps without wandering too far from the $\\chi^2$ minimum surface. For each\nsuch trial solution, we then compute small test grids to rapidly converge to the local\nminimum (with the control parameter held fixed). In this way we explored the minima\nconnected to the `best fit' solution tabulated above. This gave us larger ranges for\n`acceptable' (i.e. null hypothesis probability $p>0.01$) solutions. For example for J0630\nacceptable solutions were found for $0.3$\\,G$40$\\,GeV here.\nNot unexpectedly, EBL absorption provides no significant improvement to the fits\nof the lower redshift sources J0022 and J0811. However, we see clear\nimprovements ($\\Delta\\chi^2\\sim10$ corresponding to $\\sim5\\sigma$)\nfor J0630. Only the high UV model provides no improvement.\nThe $\\chi^2$ decrease is similar for the more conventional models.\n\n\\begin{table}\n\\vspace{-0.0in}\n\\begin{center}\n\\caption{Best-fit $\\chi^2$ values for the EBL models tested in this work\n\\label{ta:ta4}}\n\\vspace{-0.05in}\n\\scriptsize{\n\\begin{tabular}{cccccccc} \\hline\\hline\nModel & J0022 & J0630 & J0811 & reference \\\\ \\hline\nNo EBL & 151.1 & 197.4 & 128.9 & $\\cdots$ \\\\\nDom{\\'{\\i}}nguez & 151.1 & 186.2 & 129.6 & [1] \\\\\nFranceshini & 151.1 & 186.2 & 129.6 & [2] \\\\\nGilmore Fiducial & 151.0 & 189.2 & 130.0 & [3] \\\\\nGilmore Fixed & 151.1 & 186.5 & 129.6 & [3] \\\\\nHelgason & 151.1 & 186.3 & 129.5 & [4] \\\\\nKneiske04 best fit & 151.1 & 191.4 & 130.6 & [5] \\\\\nKneiske \\& Dole & 151.1 & 187.4 & 129.8 & [6] \\\\\nKneiske high UV & 150.3 & 205.1 & 132.8 & [5] \\\\\nStecker high opac. & 151.0 & 194.0 & 131.6 & [7] \\\\\nStecker low opac. & 151.0 & 187.4 & 130.2 & [7] \\\\\nFinke `C' & 151.1 & 187.0 & 129.7 & [8] \\\\ \\hline\n\\end{tabular}}\n\\end{center}\n\\footnotesize{\nReferences: [1] \\citet{dpr+11}\n[2] \\citet{frv08}\n[3] \\citet{gsp+12}\n[4] \\citet{hk12}\n[5] \\citet{kbm+04}\n[6] \\citet{kd10}\n[7] \\citet{sms12}\n[8] \\citet{frd10}\n}\\\\\n\\end{table}\n\n Since the redshift measurement for J0630 is only a lower limit,\nwe attempted to fit $z$ in the EBL model fits.\nAllowing one more free parameter (holding the other parameters fixed)\nimproves the fit in general but the improvement\nis small except for the case of the disfavored models.\nFor all models the best-fit $z$ is less than the spectroscopic lower limit, although\nthis is within errors for the best-fit models. Accordingly, we hold $z$ fixed at 1.239.\n\n Although the LAT observations continue, unless there is a strong flare,\nwe are unlikely to greatly improve the J0630 EBL constraints without going to higher\nenergy. This will be challenging with present and future generation air Cerenkov telescopes;\nwe predict an absorbed 200\\,GeV energy flux\nof $\\nu F_{\\nu} \\sim4\\times10^{-14}\\rm \\ erg\\ cm^{-2}\\ s^{-1}$ which is\nan order of magnitude lower than the 5-$\\sigma$ sensitivity of the Cherenkov\nTelescope Array\\footnote{https:\/\/portal.cta-observatory.org\/Pages\/Home.aspx}.\nFurther LAT study of other high-redshift BL Lacs\ncan certainly probe the EBL evolution at $z>1.5$.\n\n\\subsection{Alternative Fits}\n\\label{sec:sec3_5}\n\n\\begin{figure}\n\\centering\n\\hspace{-5.0 mm}\n\\includegraphics[width=3.5 in]{fig4.eps} \\\\\n\\figcaption{An SED model fit with (black dotted line) and without (red dotted line)\nthe EBL absorption model (Finke `C' in Table~\\ref{ta:ta4}) for the J0630 data\nwith a hard injection spectrum. The parameters\nfor this model are:\n$\\delta_{\\rm D}=73$,\n$\\theta_{\\rm v}=0.74^\\circ$, $B=10$\\,mG,\n$R'_{\\rm b}=2\\times10^{14}\\rm cm$, $p_{\\rm 1}=2.35$,\n$\\gamma_{\\rm min}=5\\times 10^{3}$,\nand $\\gamma_{\\rm max}=2\\times10^5$.\n\\label{fig:fig4}\n\\hspace{-10mm}\n}\n\\end{figure}\n\nThe best-fit parameters for our BL Lacs are unusual with steep $p_{\\rm 1}>3$ injection spectra.\nJ0630 is the most extreme, with $p_{\\rm 1}\\approx 4.3$ and a strong $\\sim 1$\\,G magnetic field.\nThe excellent power-law fits to the {\\it XMM-NuSTAR} X-ray data drive these values.\nWe have attempted to fit J0630 with more conventional $24\\times10^{4}$, $\\gamma_{\\rm max}'=5\\times10^{6}$, $p_{\\rm 1}=4.1$\nand a small electron density $\\sim 10^{-1}\\rm cm^{-3}$ in order not to\noverproduce the optical and the Compton emission.\n\n We are focused on the LAT band fit, so it is interesting to see that\nthis model has a very similar cutoff to that of Figure~\\ref{fig:fig3}b, requiring a similar EBL absorption.\nThe $\\chi^2$ values (18 data points ignoring the X-ray data) are 62 and 86 with and without the\nEBL absorption, respectively.\nEvidently inverse Compton emission from the X-ray component, if any, is in the highly absorbed\nTeV band. We can speculate that the soft X-ray component rises in\na different zone of the jet \\citep[e.g.,][]{m14},\narguably with large $B$ and a steep, highly cooled spectrum. Whether this connects to the apparently\ndifferent absorption for this component is unclear.\n\n\\begin{table*}[t]\n\\vspace{-0.0in}\n\\begin{center}\n\\caption{Best-fit parameters of the FDB08 model\n\\label{ta:ta6}}\n\\vspace{-0.05in}\n\\scriptsize{\n\\begin{tabular}{ccccc} \\hline\\hline\nParameter & Symbol & J0022.1$-$1855 & 3FGL~J0630.9$-$2406 & 3FGL~J0811.2$-$7529 \\\\ \\hline\nRedshift & $z$ & 0.774 & $>$1.239 & 0.689 \\\\\nDoppler factor & $\\delta_{\\rm D}$ & 29 & 110 & 49 \\\\\nMagnetic field (mG) & $B$ & 37 & 4.7 & 7.9 \\\\\nVariability timescale (s) & $t_v$ & $10^5$ & $10^5$ & $10^5$ \\\\\nComoving radius of blob (cm) & $R'_b$ & $4.9\\times10^{16}$ & $1.5\\times10^{17}$ & $8.7\\times10^{16}$ \\\\ \\hline\nLower-energy electron spectral index & $p_{\\rm 1}$ & 2.5 & 2.4 & 2.6 \\\\\nHigh-energy electron spectral index & $p_{\\rm 2}$ & 4.0 & 4.5 & 4.0 \\\\\nMinimum electron Lorentz factor & $\\gamma'_{\\rm min}$ & $6\\times10^{3}$ & $10^3$ & $3\\times10^{3}$ \\\\\nBreak electron Lorentz factor & $\\gamma'_{brk}$ & $3.9\\times10^{4}$ & $6.9\\times10^4$ & $4.9\\times10^{4}$ \\\\\nMaximum electron Lorentz factor & $\\gamma'_{\\rm max}$ & $3.0\\times10^{6}$ & $3.0\\times10^6$ & $6\\times10^{6}$ \\\\ \\hline\n\\end{tabular}}\n\\end{center}\n\\hspace{-2.0 mm}\n\\vspace{2.0 mm}\n\\end{table*}\n\n\\begin{figure*}\n\\centering\n\\vspace{-80.0 mm}\n\\hspace{-12.0 mm}\n\\includegraphics[width=5.7 in,angle=90]{fig5.eps} \\\\\n\\vspace{-4.0 mm}\n\\figcaption{SED models with the disk component for J0630.\n({\\it a}): A model with the disk component added to the baseline synchrotron+SSC model\nin Figure~\\ref{fig:fig3}b.\n({\\it b}): Similar to (a), but baseline model is that in Figure~\\ref{fig:fig4}.\n({\\it c}): Same as (b) with larger $B$ and lower $\\gamma_{\\rm max}$.\nThe model parameters are further adjusted from the baseline ones to match the SED.\nThe EBL model we used for the plot is the ``Finke C'' model in Table~\\ref{ta:ta4}.\nSee text for more details.\n\\label{fig:fig5}\n}\n\\end{figure*}\n\nIf we allow an additional X-ray emitting component, we might also consider a more\ncomplex injection model \\citep[][hereafter FDB08]{fdb08}.\nWe try an electron distribution that is a broken power law or a log parabola.\nTo compare parameters, we fit to this model by first choosing\na variability timescale and then adjusting the other parameters ($\\delta_{\\rm D}$, $B$, \nand the electron distribution) until a good fit was obtained.\nWe assumed $t_v=10^{5}$\\,s which is consistent with the timescale for the optical flux variability\nin J0630 ($t_v\\lapp3$\\,days). The broken power-law model is always more satisfactory than\nthe log-parabola version and we show the best-fit parameters for our three BL Lacs in Table~\\ref{ta:ta6}.\nIt is interesting to compare to our cooling model fits. In particular, the power law breaks strongly\nto large $p_{\\rm 2}$ values. This is imposed by fiat here, but the drive to such large break\nis difficult to accommodate in self-consistent cooling and can require large\nmagnetic field strengths (Table~\\ref{ta:ta3}).\nWe conclude that if conventional $p_{\\rm 1}\\sim2-3$ electron injection spectra\nare adopted, we will always require an additional steep component not easily achieved by\nradiative cooling.\n\n We have noted that the $>$GeV LAT spectrum is not affected by this extra electron component\n(and thus our EBL conclusions for J0630 are robust). However this is in the context of SSC models.\n\\citet{gtf+12} and \\citet{pgr12} noted that HSP BL Lacs\ncan also have low level disk\/BLR emission, overwhelmed by\n(and invisible behind) the jet synchrotron component along the Earth line-of-sight, yet\nproviding substantial seed photons for Compton up-scatter. These may have significant impact on\nthe high-energy hump of the SED \\citep[blue FSRQ model;][]{gtf+12, pgr12}.\nThus, we explore B97 model for J0630\nwith a disk component (orders of magnitude fainter than the baseline synchrotron emission)\nwhich can produce additional Compton emission at $\\sim 10^{24}-10^{26}$\\,Hz (Figure~\\ref{fig:fig5}).\nWe assume a small BL covering fraction given the strong limits\non broad line equivalent widths \\citep[][]{src+13}.\n\n In Figure~\\ref{fig:fig5}a, we add disk EC emission to the model of\nFigure~\\ref{fig:fig3} with a soft ($p_{\\rm 1}=4.26$) injection spectrum. The strong constraint\nof the X-ray data preclude any large change in the SSC component. We find that the additional\nEC emission contributes primarily at high LAT energies. The net effect is to under-produce\nthe low energy gamma-rays leading to an excessively hard LAT spectrum, while not significantly\nchanging the high-energy spectral shape. Thus the EC is not statistically demanded by this model,\nbut even if EC is added, significant EBL absorption should be present;\nimprovement of the fit when the EBL models in Table~\\ref{ta:ta4} are included is typically\n$\\Delta \\chi^2\\sim20$.\n\nAddition of the disk\/EC component to the model in Figure~\\ref{fig:fig4}\n(hard injection spectrum) provides more flexibility since we do not need to match the\nX-ray spectrum, having assumed above that the X-ray emission in this model\nis from a different region than the peak jet emission.\nIn this case, the shape of the SSC component can be adjusted to match the low-energy\ngamma-ray data and the EC emission accounts for the higher energy data (Figure~\\ref{fig:fig5}b);\nthis model reproduces\nthe optical\/UV and gamma-ray data better than the baseline model (Figure~\\ref{fig:fig4})\ndoes. Nevertheless, the effect of EBL absorption is clearly visible in Figure~\\ref{fig:fig5}b,\nand including the EBL models improves the fit by $\\Delta\\chi^2\\sim40$.\n\nIt may be imagined that the sharp drop above $10^{25}$\\,Hz in the unabsorbed model\n(dashed magenta line in Figure~\\ref{fig:fig5}b) may be able to reproduce\nthe sharp drop in the SED without a visible effect of the EBL absorption\nif the peak frequency of the EC component can be lowered.\nThis can be done by lowering $\\gamma_{\\rm max}'$, but merely adjusting\n$\\gamma_{\\rm max}'$ will damage the goodness of fit in the optical-UV band.\nHowever, by adjusting $B$, $\\gamma'_{\\rm max}$, and $\\Gamma$ ($\\delta_{\\rm D}$),\nlowering only $\\nu^{\\rm IC}_{\\rm pk}$ without affecting\n$\\nu^{\\rm sy}_{\\rm pk}$ is possible since the latter is $\\propto \\Gamma B\\gamma_{\\rm max}'^2$\nwhile the former is $\\propto \\Gamma^2 \\gamma_{\\rm max}'^2$.\nWe first adjust $B$ (decrease) and $\\gamma_{\\rm max}'$ (increase), and find that\n$\\nu^{\\rm sy}_{\\rm pk}$ is also lowered in this case owing to stronger cooling caused\nby the stronger magnetic field strength. So we lower $\\Gamma$, and adjusted $B$ and $\\gamma_{\\rm max}'$.\nIn this way, we were able to match the steep fall in the SED at $\\ifmmode\\stackrel{>}{_{\\sim}}\\else$\\stackrel{>}{_{\\sim}}$\\fi 10^{25}$\\,GHz\nwithout invoking EBL absorption (Figure~\\ref{fig:fig5}c). For this model, we use\n$B=15$\\,mG, $\\gamma_{\\rm max}'=8\\times 10^{4}$ and $\\delta_{\\rm D}=27$\n(corresponding to $\\Gamma>14$ and $\\theta_v<2.1^\\circ$).\nIn this case, as we intended, the fit\nis better when the EBL absorption is not considered; the EBL effect makes\nthe model underpredict the data, and including the EBL models increases\n$\\chi^2$ by $\\sim3$ typically.\nNote that for models in Figures~\\ref{fig:fig5}b and c,\nwe assumed that there is a sharp high-energy cutoff in the synchrotron emission.\nHowever, if such a sharp cutoff does not exist,\nthe high-frequency SSC\/EC component should be enhanced, perhaps similar to that\nin figure~\\ref{fig:fig5}a, requiring the EBL absorption.\n\nNote that we can also add BLR-reflected disk photons to this\nmodel \\citep[see][for example]{r06}. The EC emission of the reflected photons\nonly appears at higher frequencies than the direct disk component and thus\nsuffers from severe EBL absorption. Therefore, we do not consider this component here.\n\n\\section{Discussion and Conclusions}\n\\label{sec:sec4}\nWe constructed broadband SEDs for three high-redshift BL Lac objects, J0022,\nJ0630, and J0811, using nearly contemporaneous observations in the\noptical to X-ray band. Studying the LAT data, we conclude that the variability\non day to year timescales is fairly low for these three systems. This allows us to use the\n6-year (mission averaged) LAT spectrum in forming our SED.\nWe fit the SEDs with a synchrotron\/Compton model to infer physical properties of the sources.\n\nInterestingly, Figure~\\ref{fig:fig3} shows that there is a\ntrend for high-flux optical states to be spectrally harder. Similar trends have\nbeen seen in other blazars \\citep[e.g.,][]{zlz+12}.\nOur contemporaneous data (and SED modeling) are for the low, relatively quiescent state. We\nlack the broad-band high state coverage to study the physical properties imposing this variation\nvia separate SED fits.\nStill, if the variation (increase in $L^{\\rm sy}_{\\rm pk}$ and $\\nu^{\\rm sy}_{\\rm pk}$)\nwere produced by an increase in the external photon field, one expects $\\nu^{\\rm sy}_{\\rm pk}$\nto decrease as the jet particles should cool more efficiently. This is not observed and so\nwe infer that the variation is likely produced in the injection particle spectrum or in the\njet blob flow (e.g., increase in $\\delta_{\\rm D}$) and $B$ field. This suggests correlated optical\nGeV variability, which may be too weak for the LAT to detect.\n\n The basic B97 modeling constrains the emission parameters well under\nthe assumptions of pure SSC emission and radiative cooling of the injected\nelectrons (Figure~\\ref{fig:fig3}). The SED fits assuming only the assigned statistical\nerrors is adequate (probabilities $pr = 10^{-2}$--$10^{-3}$)\nHowever there are almost certainly additional\nsystematic errors including extinction uncertainty and inter-instrument calibrations.\nFor example, increasing the measurement uncertainties by 5\\% (all the SED data points)\nmakes the fit acceptable, with $pr\\sim$10\\%.\n\n The SED parameters are, however, somewhat unusual, giving\nparticularly soft injection spectra, with $p_{\\rm 1}$ well above that expected for relativistic\nshock acceleration, $p_{\\rm 1}\\sim 2-2.5$. For J0022 and J0811,\nhigher $p_{\\rm 1}$ are required because of\nthe flat SED ($\\alpha=0$ in $\\nu F_{\\rm \\nu}\\propto \\nu^{\\alpha}$)\nin the optical band, which requires $p_{\\rm 1}\\sim3$. If we identify this with the cooled\nspectrum, allowing harder injection, then we cannot accommodate the steeper X-ray spectrum\nsince radiative cooling produces only a $\\Delta \\alpha=0.5$ break\n(if the electrons were in the Klein-Nishina regime the break would be even weaker). Similarly,\nmatching the J0630 optical spectrum ($\\alpha \\sim 0.2$) and X-ray spectrum ($\\alpha \\sim -1$)\nis not possible if we let the electrons cool with the break between the optical and\nthe X-ray bands (Figure~\\ref{fig:fig4}). Thus we are forced to very steep injection spectra\nif the X-rays are produced by the same population as the optical emission.\nThis conclusion is supported by fitting with more complex heuristic electron spectra (FDB08 model).\nWith such models we can avoid the very high magnetic field strength\nrequired for J0630 to implement the rapid X-ray cooling and use lower\n10\\,mG fields.\n\n The minimum electron energies for the sources are rather high. While these values\nare not unusual when compared to those in other works \\citep[e.g.,][]{tgg+10},\nit is not clear what environments\/conditions are required in the acceleration site\nto achieve such high minimum electron energies and\nfurther investigations are needed to tell whether or not\nsuch values are realistic. Note that we do not\nuse the equipartition magnetic-field strength in our modeling,\nand the particle energy is much larger than the\nmagnetic energy in our models. In particular, the inferred magnetic field strength for J0811\nis very low compared to those for previously studied BL~Lacs\n\\citep[see][for example]{fdb08, tgg+10, zlz+12},\nalthough there are several objects in the literature with lower inferred $B$\n(and lower magnetic-to-particle-energy ratio). As we already noted (Section~\\ref{sec:sec3_3}),\nit may be possible to find another solution with lower $\\gamma_{\\rm min}$ and higher $B$.\nCovering the SED more completely will help to infer the parameters\nmore precisely. Nevertheless, the SED at the high-energy end is primarily\ndetermined by the X-ray spectrum in our model, and thus our conclusion\non the EBL would not change.\n\n By excluding the X-rays from the SED fit we can indeed accommodate lower injection $p_{\\rm 1}$,\nbut the cost is that the X-ray must be an independent, steep spectrum component. Heuristic\nmodeling with inferred stationary e$^+$e$^-$ spectra confirm that a very steep population is\nneeded to model the X-ray component. Thus a simple, single-zone SSC model with typical\nparticle acceleration spectra is inadequate. The additional ingredient may be a separate, steep\ncooled jet population for the X-ray emission. There is some indication for separate X-ray\/optical\ncomponents seen in the different absorption columns inferred from the two bands for J0630.\nHowever other effects (e.g. adiabatic expansion cooling) may also be relevant.\n\n We find that the $\\ifmmode\\stackrel{>}{_{\\sim}}\\else$\\stackrel{>}{_{\\sim}}$\\fi$100\\,GeV LAT points for our highest redshift source J0630 are generally\nsignificantly over-predicted by our SED models and take this to be strong evidence of the\neffect of EBL absorption. Standard EBL models do a good job of producing the observed spectral\ncutoff, but high UV models are not satisfactory\n\\citep[see also][]{fermiEBL, HESSEBL}.\nThis conclusion is fairly robust, and\nEBL absorption is still required if we allow the observed X-ray emission to be a separate jet\ncomponent. Introduction of EC components from faint (unobserved) disk emission\naffects the shape of the LAT spectrum. In general the harder EC spectrum does not match the LAT data\nand it is difficult to arrange components to mimic the high-energy cutoff; EBL\nabsorption is still preferred unless the synchrotron cutoff is extraordinarily\nsharp. We can approximate this with\nan abrupt cut-off in the electron energy distribution (Figure~\\ref{fig:fig5}c), but\nsuch a sharp feature is unlikely to be realized in physical\nacceleration models.\nNote that the effects of EBL absorption are not clearly visible\nin the low redshift sources as expected in EBL models; optical depth at 50\\,GeV for $z=0.7$\nis only 0.08 estimated with the Dom{\\'{\\i}}nguez model in Table~\\ref{ta:ta4}.\n\n\\begin{figure*}\n\\centering\n\\begin{tabular}{cc}\n\\hspace{-0.0 mm}\n\\includegraphics[width=3.2 in]{fig6a.eps} &\n\\hspace{4 mm}\n\\includegraphics[width=3.1 in]{fig6b.eps} \\\\\n\\end{tabular}\n\\figcaption{{\\it Left}: Synchrotron peak luminosity vs. synchrotron peak frequency.\n{\\it Right}: Compton dominance vs. synchrotron peak frequency.\nWe use black filled circles for FSRQs, red empty circles for BL Lacs, and green\nsquares for sources which are not clearly classified.\nThe three BL Lacs we study are shown as purple diamonds. Note that the red circle\nfor J0630 shows the position of the source reported in a previous study \\citep[][]{f13}.\n\\label{fig:fig6}\n}\n\\vspace{0mm}\n\\end{figure*}\n\nWe conclude with a few comments about the place of our sources in the BL Lac population.\nOur objects are luminous with high $\\nu_{\\rm pk}^{\\rm sy}$ so it is natural to consider\ntheir relation to the `blazar sequence'. In Figure~\\ref{fig:fig6}, we plot $L^{\\rm sy}_{\\rm pk}$\nand CD \\citep[][]{f13} vs. $\\nu_{\\rm pk}^{\\rm sy}$ (in the source rest frame) for\nblazars from the 3LAC sample, including our three sources.\nThe general trend is commonly attributed to the effect of an increased external photon field\n(e.g., from the BLR or disk) for blazars with lower $\\nu^{\\rm sy}_{\\rm pk}$ and magnetic field strength\n\\citep[e.g.,][]{gcf+98, f13}.\nOur three sources are HSPs\/ISPs,\nbut are relatively close to the ISP border. They show higher $L_{\\rm pk}^{\\rm sy}$\nand higher CD than the general population, but only J0630 is a true outlier, in the\n$L_{\\rm pk}^{\\rm sy}$ plot. In fact with the quiescent state SED assembled here, it is\nsomewhat less extreme than in previous studies. Still, as one of the four high-redshift BL Lacs\ncalled out by \\citet[][]{pgr12} it does present some challenges to the simple blazar sequence.\nA more complete study of the high-redshift LAT BL Lacs is needed to see if such sources are\na robust population and thus conflict with the blazar sequence correlation. If so, sources\nsuch as J0630 may be FSRQs viewed very close to the jet axis ($\\theta_{\\rm v}<0.81$\\,deg;\nTable~\\ref{ta:ta3}) so that the disk\/BLR emission is overwhelmed by the beamed jet emission.\nA detailed study along the lines of the blue FSRQ model \\citep[][]{gtf+12}\nusing our high-quality contemporaneous SEDs\nwould be quite interesting.\n\n Since $L_{\\rm pk}^{\\rm sy}$ is redshift-dependent, it is more subject to\nselection effects in a survey study. Thus it is argued \\citep[e.g.,][]{f13} that CD is a more\nrobust classifier of the blazar status, being redshift independent (although still\nsensitive to viewing angle effects, if EC components contribute). In\nFigure~\\ref{fig:fig6} right \\citep[see][for more details]{f13}, we see that our three sources lie\nnear the upper edge of the HSP population. These are highly Compton-dominated sources but\nnot really distinct from the rest of the HSP population. Since our three sources, and the other\nhigh-peak\/high-power BL Lacs, still follow a general correlation in this plot, it suggests that\nthe blazar sequence scenario may still be robust to inclusion of high-power, high-redshift BL Lacs.\n\nNonetheless, the Doppler factors ($\\delta_{\\rm D}$) of these three sources are fairly large.\nFollowing the cosmic evolution, \\citet{arg+14} inferred the distribution of\nthe Lorentz factor ($\\Gamma$) and the viewing angle ($\\theta_{\\rm v}$) for the LAT blazar\npopulation. We note that the distribution for $\\theta_{\\rm v}$\nderived by \\citet{arg+14} (their Figure~9) is broad and the values we inferred with the\nmodels (Tables~\\ref{ta:ta3}) are not exceptional. However, the\nbest-fit Lorentz factors are very high considering the power-law distribution\nwith the slope $k=-2.03\\pm0.70$ for BL Lacs \\citep[][]{arg+14}. In order for\nthe chance probability of having $\\Gamma>35.3$ (for J0630) to be greater than 1\\%,\n$k$ should be greater than $-2.49$. So perhaps our sources do represent a high velocity,\ntightly beamed wing of the BL Lac population and their unusual properties are due to beaming\neffects.\n\n Whether or not BL Lacs at $z>1$ contradict our present picture of the source evolution,\nour SED measurements, particularly that for J0630, show that these sources can be a powerful\nprobe of the EBL and its evolution. We anticipate more striking EBL constraints, pushing to\nthe peak of cosmic star formation via further study of high-redshift {\\it Fermi}-detected BL Lacs.\n\n\\bigskip\n\nThis work was supported under NASA Contract No. NNG08FD60C,\nand made use of data from the {\\it NuSTAR} mission,\na project led by the California Institute of Technology, managed by the Jet Propulsion Laboratory,\nand funded by the National Aeronautics and Space Administration. We thank the {\\it NuSTAR} Operations,\nSoftware and Calibration teams for support with the execution and analysis of these observations.\nThis research has made use of the {\\it NuSTAR} Data Analysis Software (NuSTARDAS) jointly developed by\nthe ASI Science Data Center (ASDC, Italy) and the California Institute of Technology (USA).\n\nThe \\textit{Fermi} LAT Collaboration acknowledges generous ongoing support\nfrom a number of agencies and institutes that have supported both the\ndevelopment and the operation of the LAT as well as scientific data analysis.\nThese include the National Aeronautics and Space Administration and the\nDepartment of Energy in the United States, the Commissariat \\`a l'Energie Atomique\nand the Centre National de la Recherche Scientifique \/ Institut National de Physique\nNucl\\'eaire et de Physique des Particules in France, the Agenzia Spaziale Italiana\nand the Istituto Nazionale di Fisica Nucleare in Italy, the Ministry of Education,\nCulture, Sports, Science and Technology (MEXT), High Energy Accelerator Research\nOrganization (KEK) and Japan Aerospace Exploration Agency (JAXA) in Japan, and\nthe K.~A.~Wallenberg Foundation, the Swedish Research Council and the\nSwedish National Space Board in Sweden.\n\nAdditional support for science analysis during the operations phase is gratefully\nacknowledged from the Istituto Nazionale di Astrofisica in Italy and\nthe Centre National d'\\'Etudes Spatiales in France.\n\nH.A. acknowledges supports provided by the NASA sponsored {\\it Fermi}\nContract NAS5-00147 and by\nKavli Institute for Particle Astrophysics and Cosmology (KIPAC).\nPart of the funding for GROND (both hardware as well as personnel)\nwas generously granted from the Leibniz-Prize to Prof. G. Hasinger\n(DFG grant HA 1850\/28-1).\n\n\\bibliographystyle{apj}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{intro}\nEffective field theories (EFTs) have become a standard tool in nuclear few-body\nphysics to construct the interactions between the considered degrees of\nfreedom~\\cite{Epelbaum:2008ga,Hammer:2012id}. For example, chiral effective\ntheory is a low-energy expansion of the nucleon-nucleon ($NN$) interaction that\nemploys only nucleons and pions as degrees of freedom and that uses the pion\nmass $m_\\pi$ (or a small momentum) over a large scale $\\Lambda$ that can be\nassociated with the lightest degree of freedom not included in the EFT (e.g.~the\n$\\rho$-meson). This framework is then used to derive the nuclear Hamiltonian in\na systematic low-energy expansion. The resulting potential has been used\nextensively in few-nucleon studies and ab initio nuclear structure calculations.\nIt was pointed out that the most singular piece of the one-pion exchange (OPE)\nin the deuteron channel is an inverse cube\npotential~\\cite{Sprung:1994,PavonValderrama:2005gu}. The renormalization of\nthis leading order (LO) potential has been studied repeatedly in the two- and\nthree-nucleon\nsector~\\cite{PavonValderrama:2004nb,Nogga:2005hy,Birse:2005um,Long:2011xw,Song:2016ale}.\nHere, we study the renormalization of the finite range inverse cube\npotential (FRIC) in the much simpler three-boson system thereby\nremoving the complications due to\nthe spin-dependent tensor force. In particular, we examine whether the\nthree-body system with pairwise inverse cube interactions requires a three-body\ncounterterm for renormalization, and whether residual cutoff corrections can be\nused as a reliable tool to build a power counting scheme as suggested in\nRef.~\\cite{Griesshammer:2015osb}. We note that there is also interest in atomic\nphysics regarding the inverse cube interaction. However, most attention is\nfocused on the low-energy properties in the \\textit{infinite} range\nlimit~\\cite{Mueller:2013,Gao:1999}.\n\nSince the residual cutoff dependence to some extent can be influenced\nby the chosen regularization scheme, we carry out this analysis\nfor various schemes that are currently used by the\ncommunity. Specifically, we consider a \\textit{local}\nregularization scheme~\\cite{Gezerlis:2013ipa} that cuts off the\npotential in coordinate space at a small distance $R$, a non-local\nregularization scheme~\\cite{Epelbaum:2008ga} that cuts off the high momenta\nin the momentum space form of the two-body interaction $V(p,p')$\nseparately, and a semi-local regularization scheme~\\cite{Epelbaum2015}\nthat applies these strategies separately to the long-range inverse\ncube part of the interaction and the short-distance regulator.\n\nThese different regularization schemes have different advantages for\ndifferent methods that are used to diagonalize the nuclear\nHamiltonian. For example, local interactions are commonly used in\nquantum Monte Carlo calculations, though progress has been made\nincluding nonlocal interactions\n(e.g.~\\cite{Roggero:2014kea,Lynn:2012}). However, while these have\nbeen used extensively in the literature, a detailed comparison of\nthese approaches is missing.\n\nWe find that the regularization schemes analyzed can be used to obtain\nregulator-independent results at large cutoffs. We find however that the regulator\ndependence of the short-distance counterterm is different for the\nregulation schemes we apply. In agreement with findings in the\nthree-nucleon sector\\cite{Nogga:2005hy, Song:2016ale}, we find that\nthree-body observables are completely renormalized without the\ninclusion of an additional three-body counterterm. However, an\nanalysis of the cutoff dependence of three-body observables shows also\nthat observables converge more slowly than expected from previous\nstudies of the three-nucleon sector~\\cite{Song:2016ale}.\n\nIn Sec.~\\ref{sec:theory}, we discuss the regularization schemes as well as the\nrenormalization and calculation of observables. In\nSec.~\\ref{sec:results}, we present the results obtained for the two- and\nthree-boson system as well as quantitative analyses of the remaining cutoff\ncorrections. We conclude with a summary and an outlook.\n\n\\section{Theory}\n\\label{sec:theory}\n\nIn the following subsections, we describe the interaction that is used\nin this work, how it is regulated, and how it is renormalized. We\ncomment also briefly on technical details such as the normalization of\nstates and the calculation of observables through the Schr\\\"odinger,\nLippmann-Schwinger, and Faddeev equations.\n\nThe non-regulated and singular potential $V_S$ that we consider is a FRIC\npotential of the form\n\\begin{equation}\n\t\\label{eq:fric_pot}\n V_{\\rm{S}}(r) = -C_3 \\frac{e^{-m_\\pi r}}{r^3}~.\n\\end{equation}\nWe choose $m_\\pi = 138$ MeV and $C_3 = 0.8$ fm$^2$ such that a\ndeuteron-like state ($B_2=2.2$ MeV) exists when we regulate the\npotential at $\\sim 1~\\textrm{fm}$. This potential has to be regulated\nat short distances and observables will depend strongly on the\nregularization scale as the interaction is too singular~\\cite{Frank:1971xx}.\nBelow we display how a (\\textit{smeared out}) short-distance counterterm can be\nintroduced to address this problem.\n\nWe perform our calculations in momentum space, and we Fourier transform the\ninteraction $V$ and carry out a partial-wave projection\n\\begin{equation}\n \\label{eq:ft_pwp}\n \\tilde{V}_{l}(p,k) \\equiv FT\\left[V(r)\\right] =\n \\frac{2}{\\pi}\\int_0^\\infty drr^2 j_l(pr) V(r) j_{l}(kr)~,\n\\end{equation}\nwhere $j_l(z)$ are the spherical Bessel functions of order $l$.\n\n\n\\subsection{Regulator Formulations}\\label{sec:regs}\n\n\\subsubsection{Local Regulation}\\label{sec:local_reg}\n\nFor a local, singular potential, $V_S(r)$, we have\nimplemented three different forms of regulation: local, semi-local, and\nnonlocal. The locally regulated potential has the form\n\\begin{equation}\n \\label{eq:local}\n V(r) = \\rho(r;R)V_S(r) + g(R)\\chi(r;R)~,\n\\end{equation}\nwhere $\\rho(r;R)$ is an arbitrary function that minimally fulfills two requirements.\nFirst, it must overcome $V_S(r)$ in the $r\\rightarrow 0$ limit such that the\nproduct $\\rho(r;R)V_S(r)$ is finite.\nSecond, in the limit of $r\\rightarrow\\infty$, $\\rho(r;R)$ must go to one.\nFor the locally regulated case we use\n\\begin{equation}\n \\label{eq:local_reg}\n \\rho(r;R) = {\\left(1-e^{-{(r\/R)}^2}\\right)}^4~,\n\\end{equation}\nwhere $R$ is the range at which the characteristic behavior of $V_S(r)$ is cut\noff. The counterterm\n\\begin{equation}\n \\label{eq:local_cterm}\n g(R)\\chi(r;R)~,\n\\end{equation}\nhas two components. The first, $g(R)$ is an $R$-dependent coupling\nstrength. We tune this parameter to match some low-energy, two-body\nobservable such as the two-body binding energy. The second,\n$\\chi(r;R)$, is a contact-like interaction or a \\textit{smeared} $\\delta$\nfunction such that\n\\begin{equation}\n\t\\lim_{R\\rightarrow 0}\\chi(r;R) \\sim \\delta(r)~.\n\\end{equation}\nFor the locally regulated case we use\n\\begin{equation}\n \\label{eq:locally_xterm}\n \\chi(r;R) = e^{-{(r\/R)}^3}~.\n\\end{equation}\nWe discuss below that the RG flow of the locally-regulated\ncounterterm strength, $g(R)$, contains multiple\nbranches~\\cite{Beane:2000wh}. To ensure consistency between our\nresults and others', we also implement a semi-local regulation\nscheme.\n\n\\subsubsection{Semi-Local Regulation}\\label{sec:semi_local_reg}\n\nThe difference between local regulation and semi-local regulation\nlies in the definition of the counterterm. In Eq.~\\eqref{eq:local}\nwe defined the counterterm in coordinate space. This counterterm,\nthat regulates the relative distance in the two-body system and\nthereby the momentum exchange, has multiple solutions (provided the\nshort-distance cutoff is small enough) for which the two-body binding\nenergy $B_2$ is reproduced.\n\nIf we instead define the counterterm in momentum space as\n\\begin{equation}\n \\label{eq:momspace_cterm}\n g(R)\\tilde{\\chi}(p;R)\\tilde{\\chi}(k;R)~,\n\\end{equation}\nsuch that, by itself, only permits one state, we obtain a unique RG flow.\nThe full potential in momentum space is then\n\\begin{equation}\n \\label{eq:momspace_int_sl}\n \\tilde{V}(p,k) = FT\\left[\\rho(r;R)V_S(r)\\right] +\n g(R)\\tilde{\\chi}(p;R)\\tilde{\\chi}(k;R)~,\n\\end{equation}\nwhere $FT$ represents the Fourier transform and partial-wave projection\nshown in Eq.~\\eqref{eq:ft_pwp}.\n\nFor the semi-locally regulated case, similar to~\\cite{Epelbaum2015}, we use\n\\begin{equation}\n \\label{eq:semilocal_reg}\n \\rho(r;R) = {\\left[1-e^{-{(r\/R)}^2}\\right]}^4~,\n\\end{equation}\nand\n\\begin{equation}\n \\label{eq:mom_reg}\n \\tilde{\\chi}(p;R) = e^{-{(pR\/2)}^2} = e^{-{(p\/\\Lambda)}^2}~,\n\\end{equation}\nwhere $\\Lambda\\equiv 2\/R$. For a brief discussion on the different $\\rho(r;R)$\nfunctions used for the locally and semi-locally regulated cases,\nsee Appendix~\\ref{sec:rho_choice}.\n\n\\subsubsection{Nonlocal Regulation}\\label{sec:nonlocal_reg}\n\nFor the fully nonlocal interaction, we take the semi-local\ninteraction Eq.~\\eqref{eq:momspace_int_sl}, including the forms of $\\rho(r;R)$ and\n$\\tilde{\\chi}(p;R)$, and modify the first term as follows\n\\begin{equation}\n \\label{eq:momspace_int_nl}\n \\tilde{V}(p,k) = \\tilde{\\chi}(p;R) FT\\left[\\rho(r;R_<)V_S(r)\\right]\n\\tilde{\\chi}(k;R) + g(R)\\tilde{\\chi}(p;R)\\tilde{\\chi}(k;R)~.\n\\end{equation}\nThe momentum-space regulators multiplying the first term suppress the diagonal\nmatrix elements where the incoming and outgoing momenta are large but similar,\nremoving some sensitivity to the choice of $\\rho(r;R)$ that we discuss\nin~\\ref{sec:rho_choice}. The short-distance cutoff used before we take the\nFourier transform, $R_<$, is chosen to be much less than $R$. This allows us to\nensure that the resulting cutoff dependence in the observables is attributable\nto the regulator function, $\\tilde{\\chi}(p;R)$, rather than the Fourier\ntransform.\n\n\n\\subsection{Two-Body Bound States}\\label{sec:two_body_bound_states}\nWe calculate two-body binding energies by solving the Schr\\\"odinger equation\n\\begin{equation}\n \\label{eq:shroedinger}\n (\\hat{H}_0 + \\hat{V})\\ket{\\psi} = E\\ket{\\psi}~,\n\\end{equation}\nin coordinate and momentum space. In coordinate space, we tune the counterterm\nsuch that for a desired value $E$, the radial equation\n\\begin{equation}\n \\label{eq:shroedinger_radial}\n -\\frac{1}{m}\\frac{d^2u}{dr^2} + V(r) u(r) = E\\,u(r)~,\n\\end{equation}\nis solved where $u(r)\\equiv rR_0(r)$. We have dismissed the centrifugal term as\nonly s-waves are considered. In momentum space, we\nrearrange Eq.~\\eqref{eq:shroedinger} such that we have\n\\begin{equation}\n \\label{eq:determinant}\n \n \\hat{G}_0(E)\\hat{V}\\ket{\\psi} = \\ket{\\psi}~,\n\\end{equation}\nwhere $G_0(z)\\equiv 1\/(z-\\hat{H}_0)$. After discretization with the basis states\n$|p_i\\rangle$, Eq.~\\eqref{eq:determinant} becomes an eigenvalue problem that is\neasily solved by finding the energies that fulfill\n\\begin{equation}\n \\label{eq:determinant1}\n \\det\\left[\\hat{1}-K_{ij}(E)\\right] = 0~,\n\\end{equation}\nwhere $K_{ij}(E) = \\braket{p_i|\\hat{G}_0(E)\\hat{V}|p_j}$ and we tune the counterterm such\nthat the requirement Eq.~\\eqref{eq:determinant1} is satisfied.\n\n\\subsection{Lippmann-Schwinger Equation}\\label{sec:lse}\nTo obtain two-body phase shifts, we solve numerically the\nLippmann-Schwinger Equation for the two-body $t$-matrix\n\\begin{equation}\n\\label{eq:lse}\n\\hat{t} = \\hat{V} + \\hat{V}\\hat{G}_0\\hat{t}~.\n\\end{equation}\nIn the partial-wave projected momentum basis, considering bosons interacting in\n$s$-waves only, we have\n\\begin{align}\n\\nonumber\n \\braket{p\\,|\\hat{t}|p^\\prime }\n &=\\braket{p\\,|\\hat{V}|p^\\prime } + \\braket{p\\,|\\hat{V}\\,\\hat{G}_0(E+i\\epsilon)\\,t|p^\\prime}~,\\\\\nt(p, p^\\prime; E) & = \\tilde{V}(p,p^\\prime) + \\int_0^\\infty\n dq\\,q^2\\,\\frac{\\tilde{V}(p,q) \\,t(q, p^\\prime; E)} {E+i\\epsilon-q^2\/m}\n\\end{align}\nwhere $m$ is the nucleon mass and $\\epsilon \\to +0$. From the on-shell matrix element $t(p,p;E=p^2\/m)$\nwe extract the phase shift via\n\\begin{equation}\n t(p,p;E=p^2\/m) = -\\frac{2}{m\\pi} \\frac{1}{p\\cot{\\delta}-ip}~.\n\\end{equation}\nThe scattering length is defined by the effective range expansion\n\\begin{equation}\n \\label{eq:ere}\n p\\cot\\delta \\approx = -\\frac{1}{a} + \\frac{r_s}{2}p^2~,\n\\end{equation}\nwhich allows us to calculate it exactly from the on-shell $t$-matrix amplitude\nat $p=0$.\n\\begin{equation}\n a = \\frac{m\\pi}{2} t(0,0;0)~.\n\\end{equation}\n\n\n\\subsection{Three-Body Bound States}\\label{sec:3b_bound_states}\nTo calculate three-body binding energies, we start with the equation for a\nsingle Faddeev component of a system containing three identical particles\n\\begin{equation}\n\t\\label{eq:Faddeev_bound_state}\n \\ket{\\psi} = \\hat{G}_0(E)\\hat{t}\\hat{P}\\ket{\\psi}~,\n\\end{equation}\nwhere\n\\begin{equation}\n\t\\hat{P} = \\hat{P}_{12}\\hat{P}_{23} + \\hat{P}_{13}\\hat{P}_{23}~,\n\\end{equation}\nis the permutation operator with $\\hat{P}_{ij}$ interchanging particles\n$i$ and $j$~\\cite{Gloeckle:99109}.\nAfter projecting onto the partial-wave, momentum basis for three identical\nbosons described by two Jacobi momenta $p$ (the relative momentum between\nparticles 1 and 2) and $q$ (the relative momentum between particle 3 and the\ncenter of mass of the 1--2 subsystem), we discretize the equation and solve for\nthe bound state energy $E$ using the same techniques as in the two-body case,\nas long as $E$ remains below the deepest state in the two-body\nspectrum. However, this limitation is in conflict with our goal of\nstudying the cutoff dependence of two- and three-body observables. As\nwe go to higher momentum-space cutoffs (smaller $R$ values), spurious\nbound states enter the two-body spectrum. Three-body states quickly\nbecome resonances in this regime, bounded above and below by two-body\nbound states. There are two ways that we deal with this.\n\nThe first method follows~\\cite{Nogga:2005hy} and is repeated here. It involves\n\\textit{removing} the spurious two-body state from the spectrum by transforming\nthe potential\n\\begin{equation}\n \\label{eq:state_removal}\n \\hat{V} \\rightarrow \\hat{V} + \\ket{\\phi}\\lambda\\bra{\\phi}~,\n\\end{equation}\nwhich takes the eigenvalue of the state $\\phi$ and modifies it by an amount\n$\\lambda$.\nUsing this transformed potential in the Lippmann-Schwinger equation and taking\nthe limit\nof $\\lambda\\rightarrow\\infty$ (removing the state from the spectrum), we have\n\\begin{equation}\n \\lim_{\\lambda\\rightarrow\\infty} \\hat{t}(\\lambda) =\n \\hat{t} - \\ket{\\eta}\\frac{1}{\\braket{\\phi|\\hat{G}_0|\\eta}}\\bra{\\eta}~,\n\\end{equation}\nas our modified $t$-matrix where\n\\begin{equation}\n\t\\label{eq:eta}\n \\ket{\\eta} = \\ket{\\phi} + \\hat{t}\\hat{G}_0\\ket{\\phi}~.\n\\end{equation}\nThis only requires that we have the wave function $\\braket{p|\\phi}$ to\ncalculate the modified $t$-matrix where that state no longer\ncontributes a pole. \nIn practical calculations using a large, finite $\\lambda$ value in\n(\\ref{eq:state_removal}) is sufficient.\nIf there are several spurious two-body states, the procedure is repeated\nfor each of them.\n\nThe second method we employ to study the cutoff dependence of\nthree-body resonances is to look for the resonances in the three-body\nphase shifts.\n\n\\subsection{Three-Body Phase Shifts}\\label{sec:3b_phase_shifts}\nIn the cutoff regime where spurious two-body bound states exist, we\ncan scatter a third particle off the spurious deep two-body state\nand scan the phase shifts in the energy range between the two-body\nstates for a resonance. To do this, we calculate the three-body\n$T$-matrix using~\\cite{Gloeckle:1995jg}\n\\begin{equation}\n\\label{eq:T}\n \\hat{T} = \\hat{t}\\hat{P} + \\hat{t}\\hat{G}_0\\hat{P}\\hat{T}~,\n\\end{equation}\nwhich relates to the elastic scattering operator $\\hat{U}$ by\n\\begin{equation}\n \\label{eq:U}\n \\hat{U} = \\hat{P}\\hat{G}_0^{-1} + \\hat{P}\\hat{T}~.\n\\end{equation}\nIn the partial-wave-projected, momentum basis, considering bosons interacting\nonly via $s$-waves, we have\n\\begin{equation}\n\t\\begin{split}\n \\braket{pq|\\hat{T}|\\phi} & = \\braket{pq|\\hat{t}\\hat{P}|\\phi} + \\\\\n & \\int_0^\\infty dq^\\prime {(q^\\prime)}^2 \\int_{-1}^1 dx\\,\n \\frac{t(p,\\pi_1,E-3q^2\/4m)\\,G(q,q^\\prime, x)}\n {E+i\\epsilon-q^2\/m-{(q^\\prime)}^2\/m-qq^\\prime x\/m}\n \\braket{\\pi_2q^\\prime|\\hat{T}|\\phi}~,\n\t\\end{split}\n\\end{equation}\nwhere the incoming state $\\ket{\\phi}=\\ket{\\varphi k}$ contains the\nwave function $\\varphi(p)$ of the two-body bound state and the relative\nmomentum $k$ between the third particle and the center of mass of the\ntwo-body subsystem,\n$G(q,q^\\prime,x)$ is a geometrical factor introduced by the\npermutation operator, $\\pi_1 = \\sqrt{q^2\/4+{(q^\\prime)}^2+qq^\\prime x}$, and\n$\\pi_2 = \\sqrt{q^2+{(q^\\prime)}^2\/4+qq^\\prime x}$.\n\nThe elastic scattering amplitude $M$ is related to the $U$ operator by\n\\begin{equation}\n M = -\\frac{2m\\pi}{3}\\braket{\\phi|\\hat{U}|\\phi}~,\n\\end{equation}\nand the phase shift by\n \n \n \n\\begin{equation}\n \\label{eq:three-body-pw-amp}\n M = \\frac{1}{k\\cot\\delta-ik}~.\n\\end{equation}\nIn the three-body sector, we have a similar effective range expansion\n\\begin{equation}\n \\label{eq:ere-atom-dimer}\n k\\cot\\delta \\approx -\\frac{1}{a_{AD}} + \\frac{r_{s,AD}}{2}k^2~,\n\\end{equation}\nwhich defines the atom-dimer scattering length $a_{AD}$ and atom-dimer effective\nrange $r_{s,AD}$.\nWe also study the inelasticity parameter given in terms of the $S$-matrix by\n\\begin{equation}\n \\eta = e^{-2\\delta_i}~,\n \\label{eq:inelasticity}\n\\end{equation}\nwhere the phase shift is complex and the usual decomposition\n\\begin{equation}\n \\delta = \\delta_r + i\\delta_i~,\n\\end{equation}\nis taken.\n\n\\subsection{Quantitative Uncertainty Analysis}\nTo analyze the uncertainties induced by short-distance physics of our\nregularization procedure, we study in this section the regulator\ndependence of observables. Similar to the analysis done by Song\n{\\textit{et al.}}~\\cite{Song:2016ale}, our uncertainty analysis is based on a\nsimple power series expansion of observables quantities $\\mathcal{O}$\nof the form\n\\begin{equation}\n \\label{eq:power_series_uncertainties}\n \\mathcal{O}(\\Lambda) \\approx \\mathcal{O}_\\infty \\left[1+\\sum_i^\\infty\n c_i{\\left(\\frac{q}{\\Lambda}\\right)}^i\\right]~,\n\\end{equation}\nwhere $q$ is associated with the low-momentum scale relevant to the calculation;\nhowever, $i$ is \\textit{not} assumed to be an integer. For the purposes of this\nproject, we truncate the summation over $i$ after the first term $i=n$, leaving\n\\begin{equation}\n \\label{eq:lo_correction}\n \\mathcal{O}(\\Lambda) \\approx \\mathcal{O}_\\infty \\left[1 + c_n\n {\\left(\\frac{q}{\\Lambda}\\right)}^n\\right]~,\n\\end{equation}\nWe seek to establish the value of $n$. In Ref.~\\cite{Song:2016ale},\n$n$ was found by fitting the first few terms in the above expansion\nwith integer $n$ to the cutoff dependence of observables. Here, we\nstudy the cutoff dependence at very large cutoffs, focus on the\ndominant term in the expansion, and fit $n$ itself to data and \nallow for non-integer values.\n\nTo extract the power of the leading cutoff correction, we examine both the\n$\\Lambda$ and the $q$ dependence. The first approach we take to investigate the\n$\\Lambda$ dependence is to calculate observable $\\mathcal{O}$ over a range of\n$\\Lambda$ values, and fit the results to Eq.~\\eqref{eq:lo_correction} for a\nrange of $n$ values. For each $n$, we evaluate a penalty function that we define\nas\n\\begin{equation}\n \\label{eq:penalty_function}\n p_n = \\sum_i {\\left(\\frac{\\mathcal{O}_{calc}(\\Lambda_i) -\n \\mathcal{O}_{fit}(\\Lambda_i)}{\\mathcal{O}_{calc}(\\Lambda_i)}\\right)}^2~,\n\\end{equation}\nwhere $\\mathcal{O}_{calc}(\\Lambda)$ is the observable calculated for a specific\nvalue of $\\Lambda$ and $\\mathcal{O}_{fit}(\\Lambda)$ is the value of the\nobservable as it is ``reproduced'' by Eq.~\\eqref{eq:lo_correction} and the fit\nparameters $\\mathcal{O}_\\infty$ and $c_n$. Once we have $p_n$ for a range of $n$\nvalues, we search for a minimum $p_n$ where $n$ is optimal.\n\nGriesshammer has shown~\\cite{Griesshammer:2015osb} that the $q$ dependence of\nobservables provides a necessary though insufficient window into the order of\ncutoff-dependent corrections. To isolate the $q$ dependence, we have to restrict\nthe observables we study to those whose $q$ dependence is well understood. Doing\nso allows us to calculate the observable at two different cutoffs and study the\nrelative difference\n\\begin{equation}\n \\label{eq:gh_diff}\n 1 - \\frac{\\mathcal{O}(\\Lambda_1)}{\\mathcal{O}(\\Lambda_2)}\n \\approx\n q^n c_n \\left[\\frac{1}{\\Lambda_2^n} - \\frac{1}{\\Lambda_1^n}\\right]~.\n\\end{equation}\nTaking the logarithm, we get\n\\begin{equation}\n \\label{eq:gh_diff_log}\n \\ln\\left[1 - \\frac{\\mathcal{O}(\\Lambda_1)}{\\mathcal{O}(\\Lambda_2)}\\right] =\n n\\ln q + b~,\n\\end{equation}\nwhere $n$ and $b$ are the slope and intercept that we fit, respectively.\n\n\n\\section{Results}\\label{sec:results}\n\n\\subsection{Renormalization Group Flow}\\label{sec:rgflow}\nThe first thing we compare between the regulation schemes is the RG\nflow. We choose to fix the shallowest two-body state at\n$B_2 = 2.2$ MeV. Figure~\\ref{fig:rg_flows} shows the stark difference\nbetween the RG flow found using a local counterterm and the RG flows\nfound with nonlocal counterterms. The main difference is the issue of\nuniqueness. For the locally regulated potential, as pointed out\nby~\\cite{Beane:2000wh}, $g(R)$ has multiple solutions that give a two-body bound\nstate at the desired binding energy. There is one branch where there exists one\nstate in the two-body system. Each branch below that branch contains\nsuccessively one additional state. The RG flow shown for the locally regulated\ninteraction connects four of those branches, ``hopping'' downward when it is\neasier to add an additional state than to continue to maintain the shallowness\nof the fixed state. Only two of the ``hops'' are visible in the plot due to the\nscale and the relative difference between the magnitudes of $g$ between the\ndifferent branches. Note also the difference in the units of the upper and\nlowers plots if Fig.~\\ref{fig:rg_flows}. There is a factor of $R^3$ that comes\nfrom the Fourier transform and partial-wave projection of $\\chi(r;R)$.\n\nThe other two functions shown in the lower plot of Fig.~\\ref{fig:rg_flows}\nare qualitatively very\nsimilar. They correspond to the semi-local and nonlocal regulation schemes.\nWhile the same $\\rho(r;R)$ is used in both, the prescription is somewhat\ndifferent as one can see from Eq.~\\eqref{eq:momspace_int_sl}\nand Eq.~\\eqref{eq:momspace_int_nl}. The\nsemi-local regulation scheme brings in spurious bound states faster than the\nnonlocal regulation scheme, but as mentioned before, nonlocal regulation cuts\noff the potential at large incoming and outgoing momenta, suppressing\nhigh-momentum contributions. Still, they are very similar interactions, thus\nthey provide very similar RG flows.\n\n\\begin{figure}\n\t\\includegraphics[width=\\linewidth]{rg_flows.pdf}\n \\caption{RG flows of the counterterm coupling $g$. The yellow circles in the upper\n plot represent $g(R)$ values calculated with a local regulator and local\n counterterm. The red, solid line in the upper plot are the $g(R)$ values\n used to calculate the phase shifts in\n Fig.~\\ref{fig:two_body_phase_shifts}. The blue, dashed line in the lower\n plot corresponds to the semi-locally regulated interaction. The orange,\n dashed line corresponds to the nonlocally regulated interaction.\n }\\label{fig:rg_flows}\n\\end{figure}\n\n\\subsection{Two-Body Scattering}\\label{sec:two-body-scattering}\nAs the different regulation schemes are tuned to reproduce the same\nshallow state at $B_2=2.2$ MeV, we expect that differences in\nlow-energy scattering observables are highly suppressed when large\ncutoffs are employed. We calculate the phase shifts using all three\nregulation schemes and show the results in\nFig.~\\ref{fig:two_body_phase_shifts}. The left plot contains the phase\nshifts of an non-renormalized, nonlocally regulated potential with\n$g(R) = 0$, demonstrating the strong cutoff dependence of low-energy\nobservables and the need for a counterterm. The most important feature\nof the right plot is the agreement between the different regulation\nschemes. \nIt is also worth mentioning the ``turning point'' $\\Lambda$ value at which phase\nshifts clearly begin to flatten out.\nAt low energies, the point is near 2 GeV.\nAs the scattering energy increases that point increases as well.\nImportantly, this behavior agrees with studies of the OPE\npotential~\\cite{Song:2016ale,Nogga:2005hy} where similar convergence behavior is\nfound across a range of partial-wave channels.\nOur $C_3$ value is chosen to mimic the OPE in the bosonic sector such that we\ncan expect similar renormalization behavior.\nObserving this similarity is consistent with the known result that the\none-pion-exchange potential goes like an inverse cube potential at short\ndistances (high cutoffs)~\\cite{Sprung:1994,PavonValderrama:2005gu}.\n\n\\begin{figure}\n\t\\includegraphics[width=\\linewidth]{two_body_phase_shifts.pdf}\n \\caption{[Left] Cutoff dependence of the s-wave phase shifts at $E = 1$ (red,\n dashed), $10$ (green, dotted), and $100$ MeV (blue, dot-dashed) calculated\n via a nonlocally regulated potential without a counterterm. [Right] Cutoff\n dependence of the s-wave phase shifts at (from top to bottom) $E = 1$, $10$,\n and $100$ MeV in the center-of-mass frame. The solid, red lines are the phase\n shifts calculated from a locally regulated potential. The green, dashed\n lines are the phase shifts at the same energies calculated with a\n semi-locally regulated interaction. The blue, dot-dashed lines are the phase\n shifts using a nonlocally regulated interaction. All three schemes include a\n contact-like counterterm.\n }\\label{fig:two_body_phase_shifts}\n\\end{figure}\nIt is clear from Fig.~\\ref{fig:two_body_phase_shifts} that a two-body\ncontact interaction is sufficient to renormalize the two-body phase\nshifts. The corresponding result for the two-body scattering length is\nshown in Fig.~\\ref{fig:two_body_scattering_length}.\n\n\\begin{figure}\n \\includegraphics[width=0.9\\linewidth]{two_body_scattering_length.pdf}\n \\caption{The scattering length is shown as a function of the high-momentum\n (short-distance) cutoff. The blue circles are the numerical results.\n }\\label{fig:two_body_scattering_length}\n\\end{figure}\n\nOne of the advertised, key advantages of EFT is quantifiable\nuncertainty which in turn requires a power counting that orders\ncontributions in the Hamiltonian according to their\nimportance. These uncertainties have usually two sources: (i) the\ntruncation of the low-energy expansion and (ii) uncertainties that are\nintroduced when low-energy counterterms are fitted to data. Here we\nfocus on the first source of uncertainties and some information on\nthis truncation error is contained in the convergence behavior of\nobservables as the short-distance cutoff is increased. To study this\nproblem, we first chose a range of cutoffs over which to fit the\nscattering length to Eq.~\\eqref{eq:lo_correction}. However, as the window of\ncutoffs over which the fit was carried out was narrowed to include only the highest\nvalues of $\\Lambda$, the resulting $n$ was found to be unstable.\nAs a result, we plotted $\\Lambda(da\/d\\Lambda)$, shown in\nFig.~\\ref{fig:a_analysis}.\nThe solid, red line in the left-hand plot of Fig.~\\ref{fig:a_analysis} is the\nexpected $\\Lambda(da\/d\\Lambda)$ dependence based on a fit to\nEq.~\\eqref{eq:lo_correction} with $n=1.5$.\nClearly, there is behavior in the cutoff dependence of the scattering length\nthat is not captured by the simple form assumed in\nEq.~\\eqref{eq:lo_correction}.\n\nEmpirically, we model the residual cutoff dependence by\n\\begin{equation}\n \\label{eq:rgi_correction}\n \\Lambda\\frac{da}{d\\Lambda} \\approx \\frac{1}{\\Lambda^n}\\left[A +\n B\\cos \\left({h\\Lambda^{1\/3}+f}\\right)\\right]~,\n\\end{equation}\nwhere $A, B, h$ and $f$ are treated as fit parameters.\nWe choose a range of $n$ values over which we carry out the fit and evaluate the\nquality of the fit with Eq.~\\eqref{eq:penalty_function} at each value.\nThe right-hand plot of Fig.~\\ref{fig:a_analysis}\nshows $\\Lambda(da\/d\\Lambda)$ in blue circles with $n_{\\min}=1.7$.\nThe red, dashed line in the left-hand plot of Fig.~\\ref{fig:a_analysis}\nrepresents Eq.~\\eqref{eq:rgi_correction} with the fit parameters found when\nusing $n_{min}$.\nThe agreement between the data and the empirical formula is excellent.\n\n\\begin{figure}\n \\includegraphics[width=\\linewidth]{a_analysis.pdf}\n \\caption{[Left] RG analysis of the two-body scattering length as a function of\n the cutoff. Blue circles represent the data. The red, dashed line represents\n a fit to Eq.~\\eqref{eq:lo_correction} with $n=1.5$. [Right] The blue circles\n represent the same data as the left-hand plot. The red, dashed line\n represents a fit to Eq.~\\eqref{eq:rgi_correction} with $n_{\\min}=1.7$.\n }\\label{fig:a_analysis}\n\\end{figure}\n\nWe expect that all low-energy, two-body observables come with similar cutoff\ndependence.\nIn keeping with our study of the cutoff dependence of the scattering length, we\napplied the same analysis to the phase shifts and cross sections.\nIn Fig.~\\ref{fig:delta_sigma_analysis} we plot the results.\nIn both cases, the calculation was performed at a relative, center-of-mass\nmomentum of 106~MeV.\nThe analyses produced minima of the penalty functions\n(Eq.~\\eqref{eq:penalty_function}) near $n_{\\min}=1.7$.\nSimilar analyses performed at different energies produced similar results.\nThe only trend worth mentioning is the slight decrease of $n_{\\min}$ to\napproximately 1.5 as the scattering energy increases.\nOverall, the agreement between the data and Eq.~\\eqref{eq:rgi_correction} found\nfor the scattering length is found for the phase shift and cross section as\nwell.\nThe $n_{\\min}$ values are collected in Table~\\ref{tab:nmins}.\n\\begin{figure}\n \\includegraphics[width=\\linewidth]{delta_sigma_analysis.pdf}\n \\caption{[Left] RG analysis of the phase shift at a center-of-mass momentum of\n 106 MeV as a function of the cube root of the cutoff. Blue circles\n represent the data. The red, dashed line represents a fit to\n Eq.~\\eqref{eq:rgi_correction} with $n_{\\min}\\approx 1.7$. [Right] The same analysis\n of the cross section at a center-of-mass momentum of 106 MeV as a function\n of the cube root of the cutoff. The legend is the same as in the left-hand\n plot.}\\label{fig:delta_sigma_analysis}\n\\end{figure}\n\nInterestingly, the $h$ values vary by less than a few percent around 1.5\nMeV$^{-1\/3}$ between the observables.\nThis fairly constant oscillation frequency matches up with the frequency of new\nbound states in the RG flow.\nAs shown below, this correspondence carries over to the three-body sector as\nwell.\n\nThe order of corrections is independent of the method used to obtain it.\nIn that spirit, we apply in addition to our modified power series expansion the\nmethod proposed by Griesshammer~\\cite{Griesshammer:2015osb}.\nFig.~\\ref{fig:delta_gh_analysis} shows the comparison of the phase shifts at\n$\\Lambda=3408$ and $6704$ MeV.\nBy Eq.~\\eqref{eq:gh_diff_log}, we expect the behavior to be linear.\nIn fact, we are able to extract a reliable slope of $n=1.5$ by fitting the data\nto Eq.~\\ref{eq:gh_diff_log}.\nUnfortunately, we found that other observables such as the cross section and\n$k\\cot\\delta$ provide unreliable results.\nSpecifically, zeros and unpredictable crossings precluded the extraction of\nlinear behavior.\nSelecting the phase shifts as the quantities of interest follows naturally from\nthese unfortunate conditions as discussed by\nGriesshammer~\\cite{Griesshammer:2015osb}.\n\\begin{figure}[ht]\n \\includegraphics[width=\\linewidth]{delta_gh_analysis.pdf}\n \\caption{Residual cutoff corrections to the two-body phase shifts as a\n function of the relative momentum.\n The blue circles represent the numerical calculation.\n The red line represents a fit to Eq.~\\eqref{eq:gh_diff_log}, resulting in\n $n=1.5$.\n The pink, shaded region represents the range of $k$ over which the fit was\n performed.\n The vertical, green line is the binding momentum $\\gamma$.\n }\\label{fig:delta_gh_analysis}\n\\end{figure}\n\n\n\\subsection{Three-Body Scattering}\n\n\nThe first observable in the three-body sector that we study is the atom-dimer\nscattering length.\nFigure~\\ref{fig:a_AD_cutoff_dependence} shows the convergence of $a_{\\textrm{AD}}$\nwith respect to the momentum cutoff $\\Lambda$, clearly demonstrating that a\ntwo-body contact term is sufficient to renormalize three-body observables.\n\\begin{figure}[ht]\n\t\\includegraphics[width=0.9\\linewidth]{a_AD_cutoff_dependence.pdf}\n \\caption{The cutoff dependence of the atom-dimer scattering length.\n }\\label{fig:a_AD_cutoff_dependence}\n\\end{figure}\n\nAgain, we apply the analysis based on Eq.~\\eqref{eq:rgi_correction} to the\natom-dimer scattering length.\nThe results are shown in Fig.~\\ref{fig:a_AD_analysis}.\nAs in the two-body sector, Eq.~\\eqref{eq:rgi_correction} is able to accurately\ndescribe the oscillatory convergence behavior occuring on top of the typically\nexpected $\\Lambda$-dependence.\nThe fit was performed over a range of cutoffs --- from\n$\\Lambda_\\textrm{lower}=3.1$~GeV to $\\Lambda_\\textrm{upper}=8.1$~GeV.\nFor the atom-dimer scattering length, the best fit to\nEq.~\\eqref{eq:rgi_correction} occurs at $n_{\\min}=1.3$.\nBecause this analysis involves the derivative of the observable with respect to\n$\\Lambda$ and three-body observables are particularly difficult to obtain to\narbitrary accuracy, we are often forced to constrain our fit window.\nThe atom-dimer scattering length, as well as the other three-body observables\npresented below, are selected because they provide stable results over a\nsignificant range of cutoffs.\n\\begin{figure}[ht]\n \\includegraphics[width=0.9\\linewidth]{a_AD_analysis.pdf}\n \\caption{$\\Lambda (da_{AD}\/d\\Lambda)$ as a function of the momentum-space\n cutoff. The blue circles are the calculation. The solid, red line is the\n fit to Eq.~\\eqref{eq:rgi_correction} with $n_{\\min}=1.3$.\n }\\label{fig:a_AD_analysis}\n\\end{figure}\n\nIn addition to the atom-dimer scattering length, we also conduct analyses of\nthree-body phase shifts and inelasticities at center-of-mass, kinetic energies\nof 10, 50, and 100~MeV.\nThe results are shown in Fig.~\\ref{fig:3b_scatter_analysis}.\nThe $n_{\\min}$ values, ranging from 1.1 to 1.3, used to plot the solid, red\nlines corresponding to Eq.~\\ref{eq:rgi_correction} are tabulated in\nTable~\\ref{tab:nmins}.\nThe bounds of the cutoff range are included as well to assure the reader that\nthe behavior represents a significant and relevant portion of the cutoff\ndependence.\n\\begin{figure}[ht]\n \\includegraphics[width=\\linewidth]{3b_scatter_analysis.pdf}\n \\caption{[Upper Left] Eq.~\\ref{eq:rgi_correction}-based analysis of the 2+1\n phase shift at $E=10$~MeV. The blue circles are the calculation. The solid,\n red line is the fit to Eq.~\\ref{eq:rgi_correction}. [Upper Right] Same\n analysis and legend applied to the 2+1 phase shift at $E=50$~MeV. [Lower\n Left] Inelasticity at $E=50$~MeV. [Lower Right] Inelasticity at\n $E=100$~MeV.\n }\\label{fig:3b_scatter_analysis}\n\\end{figure}\n\n\\subsection{Three-Body Bound States}\\label{sec:three-body-bound}\n\nOne of the main goals of these efforts has been to examine the\nsufficiency of a two-body counterterm to renormalize three-body\nobservables.\nIn Fig.~\\ref{fig:three_body_states} we plot the cutoff dependence of the\nthree-body binding energies associated with two three-body states that appear in\nthe system defined by the nonlocally regulated interaction\nEq.~\\eqref{eq:momspace_int_nl}.\nThe results shown come from the solution of Eq.~\\eqref{eq:Faddeev_bound_state},\nthough equivalent results were found by calculating the three-body phase shifts\ndefined by Eq.~\\eqref{eq:three-body-pw-amp} and scanning for resonances.\nThe ground state and excited state binding energies at $\\Lambda=10$~GeV are\n-18.086~MeV and -2.2379~MeV, respectively.\nThe primary feature of Fig.~\\ref{fig:three_body_states} is the convergence of\nthe binding energies in the infinite $\\Lambda$ limit.\nAt $\\approx 2$~GeV, the binding energies (or rather, the resonant energies)\nbegin to flatten out, just as in the two-body phase shifts.\n\\begin{figure}[ht]\n\t\\includegraphics[width=\\linewidth]{e3_cutoff_dependence.pdf}\n \\caption{[Left] Three-body ground state\/resonance energy as a function of the\n short-distance cutoff. [Right] Three-body excited state\/resonance energy as\n a function of the short-distance cutoff.\n }\\label{fig:three_body_states}\n\\end{figure}\n\nUnfortunately, small inaccuracies in the three-body binding energies left only\nsmall windows of cutoffs over which a fit to Eq.~\\eqref{eq:rgi_correction} could\nbe performed when all four fit parameters were treated as such.\nUsing the values of $h$ and $f$ from the fit of the atom-dimer scattering length to\nEq.~\\eqref{eq:rgi_correction}, we fit only $A$ and $B$ for the ground state\nbinding energy and show the results in Fig.~\\ref{fig:3b_gs_cutoff_dep}.\nAn $n_{\\min}$ value of 1.4 is found to minimize the penalty function, and \nthe form of Eq.~\\eqref{eq:rgi_correction} is further validated.\n\\begin{figure}[ht]\n\t\\includegraphics[width=\\linewidth]{3b_gs_cutoff_dep.pdf}\n \\caption{RG analysis of the three-body, ground-state binding energy. The blue\n circles are the calculation. The red line represents a fit to\n Eq.~\\eqref{eq:rgi_correction} with $n_{\\min}=1.4$ and the values of $h$ and\n $f$ taken from the same fit of the atom-dimer scattering length.\n }\\label{fig:3b_gs_cutoff_dep}\n\\end{figure}\n\nThroughout all of the three-body observables, we see a consistency among the\n$h$ values.\nNotably, it is enforced manually for the three-body ground state.\nThey range from 1.4 to 1.5 MeV$^{-1\/3}$ which is also consistent with the $h$\nvalues found by fitting the two-body observables.\nThis consistency between the two- and three-body sectors can be seen in\nTable~\\ref{tab:nmins} which establishes the pervasive nature of these oscillations.\n\n\\setlength\\tabcolsep{12pt}\n\\begin{center}\n \\begin{table}\n \\begin{tabular} { c c c c c c }\n \\hline\\hline\n Observable & $n_{\\min}$ & $\\Lambda_{\\textrm{lower}}$ (GeV) &\n $\\Lambda_{\\textrm{upper}}$ (GeV) & h (MeV$^{-1\/3}$) \\\\\n \\hline\n $a(\\Lambda)$ & 1.7 & 3.6 & 10.0 & 1.5 \\\\\n $\\delta(\\Lambda;E=12\\textrm{MeV})$ & 1.7 & 2.6 & 10.0 & 1.5 \\\\\n $\\sigma(\\Lambda;E=12\\textrm{MeV})$ & 1.7 & 2.4 & 10.0 & 1.5 \\\\\n $\\delta(k)$ & 1.5 & 3.4 & 6.7 & --- \\\\\n $a_{AD}(\\Lambda)$ & 1.3 & 3.1 & 8.1 & 1.5 \\\\\n $\\delta_{2+1}(\\Lambda;E=10\\textrm{MeV})$ & 1.3 & 3.7 & 7.7 & 1.4 \\\\\n \n $\\delta_{2+1}(\\Lambda;E=50\\textrm{MeV})$ & 1.2 & 3.7 & 7.0 & 1.4 \\\\\n $\\eta_{2+1}(\\Lambda;E=50\\textrm{MeV})$ & 1.3 & 3.7 & 7.0 & 1.5 \\\\\n $\\eta_{2+1}(\\Lambda;E=100\\textrm{MeV})$ & 1.1 & 3.7 & 7.1 & 1.4 \\\\\n $E_3^{(0)}$ & 1.4 & 3.5 & 7.8 & 1.5* \\\\\n \\hline\n \\end{tabular}\n \\caption{$n_{\\min}$ values for various two- and three-body observables\n alongside the bounds of cutoffs over which the fit to\n Eq.~\\eqref{eq:rgi_correction} was performed as well as the frequency that\n optimizes the fit. * The $h$ value for $E_3^{(0)}$ was taken from the fit of\n $a_{AD}$.}\\label{tab:nmins}\n\\end{table}\n\\end{center}\n\n\\section{Summary}\n\\label{sec:summary}\n\nIn this manuscript, we have set out to understand the renormalization\nproperties of the FRIC potential in the two- and\nthree-body sector.\nIn particular, we have studied the regulator dependence of observables such as\ntwo-body phase shifts, three-body binding energies, the atom-dimer scattering\nlength, phase shifts, and inelasticity parameter.\nMotivated by a recent development in the nuclear theory community, we did these\ncalculations using different, frequently used regulator functions.\n\nOur results in the two-body sector confirm that the two-body sector is\nproperly renormalized. One input parameter is required (at leading\norder) to renormalize one low-energy counterterm and thereby the\ntwo-body sector. In the three-body sector, we have demonstrated that a\nthree-body force is not needed at leading order to renormalize\nthree-body observables for the inverse cube interaction.\n\nIn both the two- and three-body sectors, we have observed significant\noscillatory behavior in the cutoff dependence of observables. These\noscillations are not captured by a simple power series expansion.\n\nInstead, we have empirically found that a generalized oscillatory dependence of\nthe form presented in Eq.~\\eqref{eq:rgi_correction} allows accurate fits of the\ndata to be made and a much clearer picture of the power of the cutoff dependence\nto be revealed.\n\nOur analysis strongly indicates that $n$ is smaller in the three-body sector\nthan in the two-body sector.\nThis would suggest that a three-body force is needed at next-to-leading order.\n\nOur analysis also indicates that $n$ is consistent with approximately 1.5\nfor two-body observables and approximately 1 for three-body\nobservables. It is an interesting question whether this has any\nsignificance for the counting of two- and three-body counterterms in\nan EFT for the inverse cube potential. For example, the singular\n$1\/r^2$ has been considered previously as the starting point for an\nEFT expansion in Ref.~\\cite{Long:2007vp}, however the inverse cube and\nall other singular coordinate space potentials need their own\nindependent analysis.\n\nHaving tested several different local, semi-local, and nonlocal regulators and\nhaving found no significant differences above $\\approx$2~GeV, we conclude that\nthese oscillations are most likely attributable to the singular nature of the\ninverse cube potential in coordinate space.\n\nIn the future, we plan to carry out an analysis of higher order corrections in\nthe three-boson and three-nucleon sector.\nHowever, we plan to also extend our work to the infinite range inverse cube\npotential that is of relevance to the atomic dipole interaction.\nThis will let us combine the results obtained by M\\\"uller~\\cite{Mueller:2013}\nwith three-body observables and study the dependence of three-body observables\non the boundary condition employed in the two-body sector.\nA more detailed analysis of the short-distance behaviour of the three-nucleon\nwave function might also provide novel insights into the power counting of\nelectroweak currents~\\cite{Valderrama:2014vra}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}