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+{"text":"\\section{Introduction}\\label{sec:intro}\nHistorically, graphs as finite metric spaces have been extensively studied~\\cite{Ha}. Even though we approach them differently, we would like to emphasise, amongst others~\\cite{I,P1,P2,S}, the classical result provided by Buneman~\\cite{B}. In short, a metric on a finite set can be realised by the shortest path metric in a positive-weighted tree if and only if it satisfies the four-point condition.\nNot only is it frequently quoted in the context of evolutionary trees~\\cite{SS}, but\nit is also known for its direct connection to the theory of Gromov hyperbolic metric spaces~\\cite{G}. Approximately two decades later after Buneman's theorem, Hendy~\\cite{H} proved the existence of a unique tree representation for every metric satisfying the four-point condition.\n\nGiven this background, a metric space that satisfies the four-point condition is commonly considered tree-like. However, an important caveat should be addressed: \nthe four-point condition is necessary and sufficient to ensure the existence of a \\emph{partially labelled} tree that realises a given metric~\\cite{Ha,H,SS}.\nFor example, a complete graph with a uniform edge length clearly satisfies the four-point condition, but it only becomes tree-like after an extra vertex is added. In this case, the four-point condition does not ensure that  a metric  is  realised by  a \\emph{fully labelled} tree on \\emph{the same} set. It does not characterise the distance within trees, in general, but rather the shortest path metrics induced by graphs of a certain class, called \\emph{block graphs} (\\textit{i.e.}, graphs in which all biconnected components  are complete subgraphs)~\\cite{Ba}.\n\n\nThis may not create an issue in the field of conventional phylogenetics, but considering the recent surge of renewed biological interest in minimum spanning tree (MST)-based tree estimation~\\cite{Q}, determining when a metric space is realised by a positive-weighted tree on the same set is not only a natural undertaking but also a meaningful one.\nThus far, this problem has not been properly recognised, much less addressed. The only two exceptions to this are the recent work provided in~\\cite{A} and in~\\cite{HEF}. It seems to be a non-trivial question not only because it cannot be answered using Buneman's theorem, but also because it is equivalent to determining a method for recognising a special case of the metric travelling salesman problem (TSP). If an input---a metric on a set of cities---is the shortest path metric in a tree on the city set,  \nthe length of the optimal tour must equal twice the length of the MST.\n\n\nIn this paper, we examine the sub-type of tree metrics without relying on the four-point condition. \nOur work is based on three ingredients: \nthe  so-called tie-breaking assumption, which has been popular in algorithmic applications since the work provided by Kruskal in~\\cite{K}; \nwhat we call  the fourth-point condition, which can typically be found in the definition of median metric spaces~\\cite{DD}; and a simple trick for metric-preserving edge removal, which applies to any finite metric space.\nThese concepts, which are part of our original results, are defined and discussed in  Section~\\ref{sec:preliminaries}.\n\nAs expected, if it exists, a fully labelled positive-weighted tree that realises a finite metric space is the unique MST in its associated weighted complete graph (Proposition~\\ref{prop:1}).  Our goal is to prove the following: \nA finite metric space under the tie-breaking rule is realised by the MST if and only if it satisfies the fourth point condition (Theorem~\\ref{thm:1}). This implies that every finite median graph, in which the shortest path lengths between all pairs of vertices are distinct, is necessarily a tree (Corollary~\\ref{cor:2}). This result also yields a stronger condition for understanding when a finite metric space is realised, especially by a spanning path graph (Corollary~\\ref{cor:1}).\nWe define and discuss the notion of a spanning tree-likeness of a finite metric space in Section~\\ref{sec:discussion}.\n\n\\section{Preliminaries}\\label{sec:preliminaries}\nWe apply the metric-related terminology provided in \\cite{DD} throughout this paper.\nLet $(X,d_M)$ be a \\emph{finite metric space}, that is, a finite set, $X$, equipped with metric $d_M$.  \nFor two distinct points $x$ and $x^{\\prime}$ in $X$, the \\emph{closed metric interval} between them is defined to be the set \n\\[\nI(x,x^{\\prime}):=\\{i\\in X: d_M(x,x^{\\prime})=d_M(x,i)+d_M(i,x^{\\prime})\\}.\n\\]\n\nAll graphs considered in this paper will be  simple, undirected,  \n\\emph{fully labelled} (\\textit{i.e.}, each vertex is labelled), and \\emph{positive weighted} (\\textit{i.e.}, each edge has a positive length).\nA graph is denoted $(V,E;w)$ for a set, $V$,  of labelled vertices and a set, $E$,  of edges that are associated with  a positive edge-weighting function, $w: E\\mapsto \\mathbb{R}^{+}$.\nGiven graph $G$, the sets of vertices and edges are denoted $V(G)$ and $E(G)$,  respectively. Moreover, graph $G$ is said to be a graph \\emph{on} $V(G)$.\nVertices may be renamed as needed, assuming no confusion arises, and a vertex labelled `$x$' is referred to as vertex $x$. \nThe distance in graph $G$ is defined to be the shortest path metric and is represented using $d_G$.\n\nAssume $M$ is a finite metric space, $(X,d_M)$. \nLet $K_M$ be the associated weighted complete graph with $M$.\nAn edge of $K_M$ that joins two distinct vertices, $x$ and $x^{\\prime}$, is denoted $e(x,x^{\\prime})$. This paper uses the terms `\\emph{points}' and `\\emph{vertices}' interchangeably because there is a one-to-one correspondence between $X$ and $V(K_M)$ for any finite metric space $M$.\n\n\\subsection{Tie-breaking rule}\\label{subsec:tiebreak}\n\\begin{dfn}\\label{dfn:tiebreak}\nA finite metric space, $(X,d_M)$, is said to satisfy \\emph{the tie-breaking rule} if the values of $d_M$ are distinct for all pairs in $X$.\n\\end{dfn}\n\n\\subsection{The fourth point condition}\\label{subsec:4thpc}\n\\begin{dfn}[Figure~\\ref{fig:satisfy4thpc}]\\label{dfn:4thPC}\nA finite metric space, $(X,d_M)$, is said to satisfy \\emph{the\nfourth-point condition} if, for every (not necessarily distinct) three\npoints $x,y,z\\in X$, there exists a point, $p^{*}\\in X$, such that\n\\[\nd_M(x,p^{*})+d_M(y,p^{*})+d_M(z,p^{*})=\\frac{1}{2} \\{d_M(x,y)+d_M(y,z)+d_M(z,x)\\}.\n\\]\n\\end{dfn}\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.18\\textwidth]{4thpc.eps}\n\\caption{Fourth point $p^*$ for triplet $\\{x,y,z\\}$ \\label{fig:satisfy4thpc}}\n\\end{figure}\n\n\n\\begin{prop}\\label{prop:4th_unique}\nIf a finite metric space, $(X,d_M)$, satisfies the fourth-point condition, \n fourth point $p^*\\in X$ is unique for each triplet in $X$. \n\\end{prop}\n\\begin{proof}\nSuppose that there are two quartets, $\\{x,y,z,p_1^*\\}$ and $\\{x,y,z,p_2^*\\}$ ($p_1^*\\neq p_2^*$), in $X$ such that \n\\[d_M(x,p_1^{*})+d_M(y,p_1^{*})+d_M(z,p_1^{*})= d_M(x,p_2^{*})+d_M(y,p_2^{*})+d_M(z,p_2^{*}).\\]\nBecause $d_M$ is a metric on $X$, we have \n\\begin{align*}\nd_M(x,p_2^{*})&\\leq d_M(x,p_1^{*})+d_M(p_1^{*},p_2^{*});\\\\\nd_M(y,p_2^{*})&\\leq d_M(y,p_1^{*})+d_M(p_1^{*},p_2^{*});\\\\\nd_M(z,p_2^{*})&\\leq d_M(z,p_1^{*})+d_M(p_1^{*},p_2^{*}).\n\\end{align*}\nTherefore, $p_1^{*}=p_2^{*}$, but this is a contradiction. \nHence, if $p^*$ exists for $\\{x, y, z\\}$, it is unique.\n\\end{proof}\n\n\n\\begin{prop}\\label{lem:4thPC_alternative}\nThe following is equivalent to saying that finite metric space $(X,d_M)$ satisfies the fourth-point condition:  \nFor every (not necessarily distinct) three points $x, y, z \\in X$, there\nexists only one point  $p^*\\in I(x,y)\\cap I(y,z)\\cap I(z,x)$.\n\\end{prop}\n\n\\begin{proof}\nBecause $d_M$ is a metric, for all $x,y,z,p\\in X$, we have \n$d_M(x,p)+d_M(y,p)+d_M(z,p)\\geq \\frac{1}{2}\\{d_M(x,y)+d_M(y,z)+d_M(z,x)\\}$. \nThe equality holds if and only if\n$I(x,y)\\cap I(y,z)\\cap I(z,x)=\\{p^*\\}$. \nProposition \\ref{prop:4th_unique} ensures the uniqueness of $p^*$.\n\\end{proof}\n\n\n\\begin{rem}\\label{rem:median}\nFourth point $p^*$ is also known as \\emph{the median} for $\\{x,y,z\\}$  \nbecause it minimises the sum of the distances to the three points, and \na metric space satisfying the fourth-point condition (or a graph inducing this kind of metric space) is said to be \\emph{median}~\\cite{Ba,DD}.\nAlthough a discussion of this topic is provided in~\\cite{Ba,BCE}, it should be noted that median graphs include multiple types of graphs other than trees, such as grid and square graphs.\n\\end{rem}\n\n\n\\begin{lem}\\label{lem:cycle}\nLet $C$ be a cycle graph, $(V,E;w)$, with $\\sum_{e\\in E} w(e) =c$. \nAlso, let $d_C$ be the shortest path metric in $C$.\nGiven three distinct points  $x,y,z\\in V$  such that \n$d_C(x,y)+d_C(y,z)+d_C(z,x)=c$, \nthe fourth point, $p^*$, exists in $V$  \nif and only if $\\max\\{d_C(x,y),d_C(y,z),d_C(z,x)\\}=c\/2$.\n\\end{lem}\n\\begin{proof}\nWithout loss of generality, we can assume $d_C(z,x)=\\max\\{d_C(x,y), d_C(y,z), d_C(z,x)\\}$. Clearly, $y\\in I(x,y)\\cap I(y,z)$. Therefore, $I(x,y)\\cap I(y,z)\\cap I(z,x)=\\emptyset$ if and only if $y\\not\\in I(z,x)$. \nUnder the assumption that the length of $C$ is fixed at $c$, \nthis is equivalent to stating that $d_C(z,x)\\neq c\/2$. \nThus, $I(x,y)\\cap I(y,z)\\cap I(z,x)= \\emptyset$ if and only if $d_C(z,x)\\neq c\/2$. Applying Proposition~\\ref{lem:4thPC_alternative} completes the proof.\n\\end{proof}\n\n\\subsection{Basic geodesic graphs}\\label{subsec:basic}\nIn this subsection, we present a simple trick for metric-preserving edge removal, which can be used to represent an arbitrary finite  metric space  as a graph with the fewest edges. \nLet $M$ be a finite metric space, $(X,d_M)$, \nand assume $K_M$ is the weighted complete graph associated with $M$.\n\n\\begin{dfn}\\label{dfn:realise}\nSuppose $G$ is a connected graph on finite set $X$ with shortest path metric $d_G$.\nGraph $G$ is said to \\emph{realise $M$} if\n$d_G(x,x^{\\prime}) = d_M(x,x^{\\prime})$  \nfor all  $x,x^{\\prime}\\in X$.\n\\end{dfn}\n\n\\begin{dfn}\\label{dfn:circularperm}\nGiven $x,x^{\\prime}\\in X$, \nthe edge, $e(x,x^{\\prime})$, of $K_M$ is said to be \\emph{non-basic} \nif there is a  permutation, $(x_1, \\ x_2, \\ \\cdots, \\ x_k)$, on a non-empty subset of $X\\setminus\\{x,x^{\\prime}\\}$ such that cyclic permutation $(x, \\ x_1, \\ x_2, \\ \\cdots , \\ x_k, \\ x^{\\prime})$  satisfies \n\\[\nd_M(x,x^{\\prime})=d_M(x,x_1)+d_M(x_1,x_2)+...+d_M(x_k,\\,x^{\\prime}). \n\\]\nThe edge is called \\emph{basic} otherwise.\n\\end{dfn}\n\n\\begin{prop}\\label{rem:4thpoint_notexist}\nLet $x,y,z$ be  three different vertices of $K_M$.\nWhen the three edges, $e(x,y)$, $e(y,z)$, and $e(z,x)$, of $K_M$ are basic, the fourth point, $p^*$,   does not exist for $\\{x,y,z\\}$. \nIf a non-basic edge exists, say $e(x,y)$,   \npoints  $x$ and $y$ are the only two candidates for $p^*$.\n\\end{prop}\nThe proof of this proposition is straightforward.\n\n\\begin{dfn}\\label{dfn:basicgeodesicgraph}\nAssume $B_M$ is the set of all basic edges of $K_M$, \nand suppose $\\lambda$ is  a restriction of $d_M$ to $B_M$. \nA subgraph, $G_M:=(X, B_M; \\lambda)$, in $K_M$ is called  \n\\emph{the basic geodesic graph in $K_M$}. \n\\end{dfn}\n\n\n\\begin{lem}\\label{lem:connected}\nThe basic geodesic graph, $G_M$, in $K_M$ is a connected graph on $X$ that realises $M$.\n\\end{lem}\n\n\\begin{proof}\nIt suffices to prove that $G_M$ is connected. \nAssuming that $e(x,x^{\\prime})$ is non-basic, we show that there is a path of basic edges joining $x$ and $x^{\\prime}$ in $K_M$. We also note that they are obviously connected in $G_M$ if $e(x,x^{\\prime})\\in E(K_M)$ is basic. \nLet $C$ be a cycle with the greatest number  of vertices (or edges) of all cycles in $K_M$ that share edge $e(x,x^{\\prime})$ and overall length $2d_M(x,x^{\\prime})$. Let  $V(C)=\\{x,x^{\\prime}\\}\\cup Y$, where   $Y:=\\{x_1,\\cdots,x_k\\}$ is a non-empty subset of $X\\setminus\\{x,x^{\\prime}\\}$, as in  Definition~\\ref{dfn:circularperm}. Furthermore, suppose $d_C$ is the shortest path metric induced by $C$ and $x_i, x_j\\in V(C)$. \nIf a path existed in $K_M$ joining $x_i$ and $x_j$ that was shorter than  $d_C(x_i,x_j)$, then edge $e(x,x^{\\prime})$ would be longer than the path connecting  $x$ and $x^{\\prime}$ through  $x_i$ and $x_j$. Therefore, any path in $K_M$ joining two vertices in $V(C)$  must have a length  greater than or equal to $d_C(x_i,x_j)$.  We use this fact  at the end of the proof.\n\nIn order to obtain a contradiction, we suppose $e(y,y^{\\prime})\\in E(C)\\setminus e(x,x^{\\prime})$  is  non-basic. We define   $C^{\\prime}$  to be a cycle in $K_M$ of overall length $2d_M(y,y^{\\prime})$  with   $e(y,y^{\\prime})\\in E(C^{\\prime})$, which is similar to our previous case except that $|V(C^{\\prime})|$ is unimportant. Let $V(C^{\\prime})=\\{y,y^{\\prime}\\}\\cup Z$, where $Z:=\\{y_1,\\cdots,y_l\\}\\subseteq X\\setminus\\{y, y^{\\prime}\\}$.  By Definition \\ref{dfn:circularperm}, if a cycle contains a non-basic edge, then it must be strictly longer than the other edges in the cycle. This implies that the number of non-basic edges contained in each cycle is zero or one. Thus, $e(y,y^{\\prime})$ is shorter than $e(x,x^{\\prime})$, and $e(y,y^{\\prime})$ is the longest edge in $E(C^{\\prime})$. Therefore, we can conclude that $e(x,x^{\\prime})$ is not in $E(C^{\\prime})$.\nThe assumption on $|V(C)|$ provides $Y\\cap Z\\neq\\emptyset$. Our hypothesis ensures a path in $K_M$ of length $d_C(y,y^{\\prime})$ that connects  $y$ and $y^{\\prime}$ via $y^{\\prime\\prime}\\in Y\\cap Z$. This implies that  $K_M$ contains a path joining  $y$ and $y^{\\prime\\prime}$  of length less than $d_C(y,y^{\\prime})$. If we  assume that  $y^{\\prime}$ lies in the shortest path joining $y$ and $y^{\\prime\\prime}$ in $C$ (note that the roles of $y$ and $y^{\\prime}$ can be exchanged), then we have  $d_C(y,y^{\\prime})<d_C(y,y^{\\prime\\prime})$. It follows that there is a path that joins $y$ and $y^{\\prime\\prime}$ in $K_M$ of length less than $d_C(y,y^{\\prime\\prime})$. This is a contradiction. Hence, $e(y,y^{\\prime})$ is  basic, which completes the proof.\n\\end{proof}\n\n\n\\begin{dfn}\\label{dfn:treepath}\nFinite metric space $M$ is said to be a \\emph{spanning tree metric space} if the basic geodesic graph, $G_M$,  in the weighted complete graph, $K_M$, is a spanning subtree in $K_M$.  In particular, $M$ is said to be a \\emph{spanning path metric space} if $G_M$ is a \\emph{path graph} (\\textit{i.e.}, a tree with two vertices of degree one and remaining vertices of degree two) that spans all the vertices of $K_M$. \n\\end{dfn}\n\n\\begin{prop}\\label{prop:1}\nLet $M:=(X,d_M)$ be a spanning tree metric space and \n$G_M$ be the basic geodesic graph in  $K_M$. Then the following statements hold:\n\\begin{enumerate}\n\\item $G_M$ is the unique minimum spanning tree in $K_M$;\n\\item $G_M$ is the unique fully labelled tree  on $X$ that realises $M$.\n\\end{enumerate}\n\\end{prop}\n\\begin{proof}\n(1) Assume $B:=E(G_M)$ and let $\\overline{B}:=E(K_M)\\setminus B$. Because $|B|=|X|-1$,  $G_M$ is the only spanning tree in $K_M$ such that all edges are basic.\nIn addition, let $e(x,x^{\\prime})\\in \\overline{B}$.  Because $G_M$ is a tree, there is a  unique path  joining $x$ and $x^{\\prime}$, denoted $P$. Each edge of $P$ must be  strictly shorter than $d_M(x,x^{\\prime})$ for the following reasons: the length of $P$ equals $d_M(x,x^{\\prime})$; the number of edges of $P$ exceeds one; and the edge weights are  all positive. Therefore,  replacing an arbitrary edge of $P$  with $e(x,x^{\\prime})$ results in a spanning tree in $K_M$ of greater length. Hence, $G_M$ is shorter than any other spanning trees in $K_M$. \n(2) Suppose that $M$ is realised by fully labelled tree $T$ on $X$.  This implies that each edge of $T$ has a positive weight. \nWe can recover $K_M$ from $T$ by summing the weights along every path in $T$ that has  two or more edges.\nThis process indicates that  $T$ is isomorphic to  the basic geodesic graph  in $K_M$. Hence, given (1), we know $T$ is unique.\n\\end{proof}\n\\begin{rem}\nProposition~\\ref{prop:1} states that a metric space is uniquely realised by the only MST if it is a spanning tree metric space. Note that we do not need Buneman's four-point condition in  the argument ({\\it cf.}~\\cite{A}).\nConcerning the uniqueness of the MST, the tie-breaking rule is a well-known sufficient condition established by Bor\\r{u}vka~\\cite{Bo} (cited in~\\cite{K}) and by Kruskal~\\cite{K}. The next section explores its relation to spanning tree metric spaces.\n\\end{rem}\n\n\\section{Main results}\\label{sec:spanningtree}\n\n\\begin{thm}\\label{thm:1}\nLet $M$ be a finite metric space, $(X,d_M)$, under the tie-breaking rule.  \nThen $M$  is a spanning tree metric space if and only if it satisfies the fourth-point condition. \n\\end{thm}\n\\begin{proof}\n(i)  The fourth-point condition clearly holds for all spanning tree metric spaces.\n(ii) If $d_M$ is not a spanning tree  metric on $X$, then  \nwe will show that there is a triplet in $X$ that violates the fourth-point condition. \nAccording to Lemma \\ref{lem:connected}, our assumption  implies that the basic geodesic graph, $G_M=(X,B;\\lambda)$, in $K_M$ contains at least one cycle. \nSuppose $C:=(X_k,B_k; \\lambda_k)$ is the shortest cycle  in $G_M$,  \nwhere  $X_k\\subseteq X$, $B_k\\subseteq B$, $|X_k|=|B_k|=k$, and $\\lambda_k$ is the restriction of $\\lambda$ to $B_k$. Then Proposition \\ref{rem:4thpoint_notexist} yields $k\\geq 4$. Let $c$ denote the sum of the $\\lambda_k$ over all elements in $B_k$. Also, assume that $d_C$ is the shortest path metric in $C$.\nFor all $i,j\\in X_k$, no path in $G_M$ joining $i$ and $j$ has a shorter length than  $d_C(i,j)$  (otherwise, $C$ would not be the shortest cycle in $G_M$).\nTherefore, $d_C(i,j)=\\min \\{a_{ij},c-a_{ij}\\}$,  \nin which $a_{ij}$  represents the length of the path in $C$ that travels from $i$ to $j$ in a clockwise direction. \n\nConsider a route in which we visit the points in $X_k$. Let $s\\in X_k$ be the starting point from which we travel along the circle in a clockwise direction. \nWe assign a label, `$\\mathrm{L}$' or `$\\mathrm{R}$', to every point $i\\in X_k\\setminus\\{s\\}$: label `$\\mathrm{L}$' is assigned if $a_{si} < c\/2$, and we use label  `$\\mathrm{R}$'  if $a_{si}\\geq c\/2$.\nIf every point in $X_k\\setminus\\{s\\}$ was labelled `$\\mathrm{L}$', the last edge we would traverse returning to $s$ would be non-geodesic or non-basic. \nTherefore, there exists one and only one basic edge between vertices labelled `$\\mathrm{L}$' and `$\\mathrm{R}$'. Suppose that $t$ signifies the last point with label `$\\mathrm{L}$' and $u$ indicates the first point with label `$\\mathrm{R}$' as on the left  in Figure \\ref{fig:thm1}. Note that  $d_M(s,t)+d_M(t,u)+d_M(u,s)=c$. \n\nWe  assume that  $p^*$ exists for $\\{s,t,u\\}$ (otherwise, the assertion of the theorem immediately follows). Lemma~\\ref{lem:cycle} gives us $\\max\\{d_M(s,t),d_M(t,u),d_M(u,s)\\}=c\/2$. Thus,  $d_M(u,s)=c\/2$ (the edge joining $t$ and $u$ is basic, and $d_M(s,t)<c\/2$). Let $v (\\neq u)$ be a point  in $X_k$ with label `$\\mathrm{R}$' that is between $u$ and $s$ as on the right in Figure \\ref{fig:thm1}. We know point $v$ exists because $e(u,s)$ would be non-basic otherwise. According to the tie-breaking rule, we note that $a_{tv} \\neq c-a_{tv}$. We can also set $a_{tv} < c-a_{tv}$ in order to select $\\{s,t,v\\}$. Although we should select $\\{t,u,v\\}$ when  $a_{tv} > c-a_{tv}$, we limit our consideration to the former case. Therefore, we have $d_M(s,t)+d_M(t,v)+d_M(v,s)=c$ again, but each of the three terms does not equal $c\/2$ (recall that  $d_M(s,u)=c\/2$). Hence, Lemma~\\ref{lem:cycle} implies that  $p^*$ does not exist for $\\{s,t,v\\}$, and this completes the proof.\n\\end{proof}\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.4\\textwidth]{points.eps}\n\\caption{Points in the proof of Theorem~\\ref{thm:1} \\label{fig:thm1}}\n\\end{figure}\n\n\\begin{rem}\nGiven a finite metric space on $X$, we can determine in $O(|X|^4)$ time whether it is a spanning tree metric space.\n\\end{rem}\n\n\\begin{cor}\\label{cor:2}\nLet $G$ be a median graph on finite set $X$ and  let $d_G$ be the shortest path metric of $G$. If each pair in $X$ has a different value for $d_G$, then $G$ is a tree.\n\\end{cor}\n\n\\begin{rem}\\label{rem:insufficient}\nAs was mentioned in Remark~\\ref{rem:median}, \nthe fourth-point condition \\textit{per se} is not a sufficient condition, but it is a  necessary condition in order to ensure that a finite metric space \nis induced by the shortest path metric in a tree \n(\\textit{cf.}\\ a  cycle graph on four vertices with a uniform edge length).\n\\end{rem}\n\n\\begin{cor}\\label{cor:1}\nSuppose $M:=(X,d_M)$ is a finite metric space  under the tie-breaking rule.  \nThen $M$  is a spanning path metric space (Definition~\\ref{dfn:treepath}) if and only if it satisfies \\emph{the three-point condition}: for every (not necessarily distinct) three points $x,y,z\\in X$, we have \n\\[\n\\max\\{d_M(x,y),d_M(y,z),d_M(z,x)\\}=\\frac{1}{2}\\{d_M(x,y)+d_M(y,z)+d_M(z,x)\\}. \n\\]\nThe condition can be confirmed in $O(|X|^3)$ time. If $M$ is a spanning path metric space, it is realised by the unique shortest path that joins the farthest two points  in $X$. \n\\end{cor}\n\\begin{proof}\nWe only prove the first statement. The three-point condition obviously holds for all spanning path metric spaces. Therefore, we assume that the three-point condition holds and show that the basic geodesic graph, $G_M$, in $K_M$ is a path graph on $X$. It is clear that $y$ is the fourth point, $p^{*}$, for $\\{x,y,z\\}$ when  the left-hand side equals $d_M(z,x)$. This means that the fourth-point condition automatically holds for any finite metric space that satisfies the three-point condition. Therefore, our assumption implies that $G_M$  is a tree on $X$. The three-point condition also indicates that every vertex in $G_M$ has a degree of one or two. In other words, if vertex $x$ has degree three or more, then any three distinct vertices adjacent to $x$ would violate the three-point condition. Hence, $G_M$ is a path graph on $X$, which completes the proof.\n\\end{proof}\n\n\\section{Discussion}\\label{sec:discussion}\nThe hyperbolicity of finite metric spaces (or graphs) is a concept provided by Gromov~\\cite{G,SS} and measures the deviance of a metric space from Buneman's four-point condition. If a metric space, $M$, satisfies the four-point condition, then the hyperbolicity of $M$ equals $0$, and $M$ is said to be $0$-hyperbolic. As was previously discussed, any complete graph with a uniform edge length is $0$-hyperbolic. Because the four-point condition is a stronger version of the triangular inequality, all metric triangles are also $0$-hyperbolic. Therefore, although the value of hyperbolicity is usually called the `tree-likeness' of $M$, a more precise interpretation refers to the \\emph{partially labelled} tree-likeness of $M$. \nTherefore, as a final remark, we provide the notion of a \\emph{fully labelled} tree-likeness of $M$.\n\nLet us say that finite metric space $M$  is  \\emph{$\\rho$-roundabout}. Here,  $\\rho$ is defined to be \n\\[\n\\max_{x,y,z\\in X}\\min_{i\\in X}\\;\\frac{d_M(x,i)+d_M(y,i)+d_M(z,i)}{d_M(x,y)+d_M(y,z)+d_M(z,x)}-\\frac{1}{2}.\n\\]\nThis measures how far $M$ deviates from the fourth-point condition.   Provided that the tie-breaking rule holds, the value of $\\rho$ can be regarded as the spanning tree-likeness of $M$ or the circuitousness of $d_M$ as illustrated in Figure~\\ref{fig:roundabout}. Note that $\\rho$ is invariant under multiplication of $d_M$ by a constant. As we have already seen, $M$ is $0$-roundabout if and only if there is an exact fit between $M$ and the MST. \n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.4\\textwidth]{roundabout.eps}\n\\caption{Illustrations of spanning tree-likeness ($\\rho=0$ and $\\rho>0$)\\label{fig:roundabout}}\n\\end{figure}\n\n\nThe degree of violation of the three-point condition similarly provides the spanning path-likeness of $M$---the maximum discrepancy between the left and right-hand sides of the triangular inequality. On the other hand, hyperbolicity does not provide any information because all metric triangles are $0$-hyperbolic. \n\n\n\\section*{Acknowledgement}\nThis work was supported in part by JSPS KAKENHI Grant Number 25120012 and 26280009. \nSpecial thanks are extended to Yoshimasa Uematsu for his insights concerning the concept of median.\n\n\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\n\\vskip 0.1cm\n\nSince the establishment of quantum mechanics ($QM$) and the quantization of electromagnetic field, the quantum electrodynamics ($QED$) in particular and the quantum field theory ($QFT$) in general have been developed for over 80 years. A common prominent feature of $QFT$ is the emergence of the divergence in calculations beyond the tree level. To handle\nthese divergences, various regularization and renormalization methods ($RRMs$) have been proposed.\n\nDespite the great success of $QFT$, the present status of $RRMs$ remains ambiguous to some extent. For example, in the\ntheory of Chromodynamics ($QCD$) for describing the strong interactions of colored quarks and gluons, a commonly used\nrenormalization scheme ($RS$) is the modified minimal substraction ($\\bar{MS}$) scheme (see a summary in the Review of\nParticle Physics in 2008, \\cite{1}, p.157) where an arbitrary renormalization mass scale $\\mu$ is introduced (see next\nsection). Physicists believe the fundamental theorem of $RS$ dependence: Physical quantities, such as the cross-section\ncalculated to all orders in perturbation theory, should not depend on the $RS$. However, it follows that a truncated\nseries does exhibit $RS$ dependence. In practice, $QCD$ cross-section are known to different orders, depending on the\nchoice of $RS$ (and $\\mu$) in different sensitive ways. We still don't know what is the \"best\" choice for $\\mu$ within\na given scheme (usually $\\bar{MS}$). There is no definite answer to this question yet.\n\n\\section{What a divergence means?}\n\\label{sec:divergence}\n\n\\vskip 0.1cm\n\nPhysicists often talk about different orders of a divergence, based on its dimension with respect to mass (\\ie, momentum,\nwe use natural unit system with $\\hbar=c=1$). For example, if a Feynman diagram integral ($FDI$) in $QFT$ reads\n\\begin{equation}\\label{1}\nI=\\int\\dfrac{d^4K}{(2\\pi)^4}\\dfrac{K^n}{(K^2-M^2)^2}\n\\end{equation}\nwhere $K$ corresponds to the (4-dimensional)momentum of virtual particle (say, virtual photon in $QED$) and $M$ is a\nmass parameter (maybe in a complex form) characterizing the $QFT$ under consideration. Then\n\\begin{equation}\\label{2}\nn=\\left\\{\\begin{array}{l}\n           0,\\quad \\text{logarithmic divergence} \\\\\n           1,\\quad \\text{linear divergence} \\\\\n           2,\\quad \\text{quadratic divergence}\n         \\end{array}\\right.\n\\end{equation}\nHowever, among these categories, only the first one is really meaningful in mathematics. This is because the definition\nof a number sequence $A_i\\;(i=1,2,\\ldots)$ having a limit being $\\infty$ is as follows: Given arbitrarily a large\nnumber $M$, one can always find such a number $N$ so that $A_i>M$, when $i>N$. Here $A_i$ and $M$, let alone $i$ and $N$,\nare all dimensionless numbers. A number $M\\gg1$ is called a large number, whereas $\\varepsilon\\ll1$ a small number.\n\nOn the other hand, the space-time coordinates $\\bf x$ and $t$, mass $m$ and momentum $p$ (or $k$) are physical quantities and each with certain dimension. If treating them as dimensionless numbers, we will run into trouble inevitably.\n\nExample A: Assume that in $QFT$, a $FDI$ with linear divergence is approximately expressed as $I\\sim 10^3M$ with $M$ being a\nmass parameter in the unit of $mg$. If $M=1\\,mg$, we have $I\\sim 10^3mg$ with $10^3\\gg1$ being a large number. But if changing the unit from $mg$ to $kg$, we will have $I\\sim 10^{-3}kg$ with $10^{-3}\\ll1$ being a small number. A mathematician\nwould ask: \"Could you still treat your $I$ as a divergent quantity?\"Who could answer his question?\n\nExample B: In the $\\bar{MS}$ scheme, the renormalization mass scale $\\mu$ is introduced as follows (see p.137 in \\cite{2})\n\\begin{equation}\\label{3}\n\\dfrac{\\Gamma(2-d\/2)}{(4\\pi)^{d\/2}(m^2)^{2-d\/2}}=\\dfrac{1}{(4\\pi)^2}\\left[\\dfrac{2}{\\varepsilon}-\\gamma\n+\\ln(4\\pi)-\\ln(m^2)\\right]\\to \\dfrac{1}{(4\\pi)^2}\\left[-\\ln\\left(\\dfrac{m^2}{\\mu^2}\\right)\\right]\n\\end{equation}\nwhere $m$ is some mass parameter containing in the model. Eq.(\\ref{3}) is derived from the \"dimensional regularization\"\nmethod. The 4-dimensional (Euclidean) space has been analytically continued into $d$-dimensional one with $\\varepsilon\\sim 4-d\\sim 0$\nand Gamma function $\\Gamma(2-d\/2)=\\Gamma(\\varepsilon\/2)=2\/\\varepsilon-\\gamma$ ($\\gamma=0.4772\\ldots$ is the Euler constant).\nObviously, the left-handed-side ($LHS$) has a dimension of $m^{-\\varepsilon}$, whereas in the right-handed-side ($RHS$),\nthe function $\\ln\\left(\\frac{m^2}{\\mu^2}\\right)$ becomes dimensionless after the $\\mu$ is introduced. However, the mathematician would feel quite uncomfortable because $\\varepsilon\\neq0$. He will focus on the middle of Eq.(\\ref{3}) and\nask: \"Why the divergent number $2\/\\varepsilon$ disappears at the $RHS$ and becomes finite? Where the term $\\ln(m^2)$ comes\nfrom? What is its dimension?\"\n\nWe physicists accept Eq.(\\ref{3}) since it could be derived from a \"mathematical formula\" like (see p.57 in \\cite{3})\n\\begin{equation}\\label{4}\n(m^2)^{d\/2-2}=\\exp\\left[\\left(\\frac{d}{2}-2\\right)\\ln m^2\\right]\\simeq 1+\\left(\\frac{d}{2}-2\\right)\\ln m^2\n\\end{equation}\nThen the mathematician would say: \"No! In the mathematical formula\n\\begin{equation}\\label{5}\nx^y=\\exp[y\\ln x]\n\\end{equation}\nboth $x$ and $y$ must be dimensionless numbers. So in the $LHS$ of Eq.(\\ref{3}), you correctly write down:\n\\begin{equation*}\\label{}\n(4\\pi)^{-d\/2}=(4\\pi)^{2-d\/2-2}=\\dfrac{1}{(4\\pi)^2}\\exp\\left[(2-\\frac{d}{2})\\ln(4\\pi)\\right]\\simeq\n\\dfrac{1}{(4\\pi)^2}\\left[1+(2-\\frac{d}{2})\\ln(4\\pi)\\right]\n\\end{equation*}\nBut Eq.(\\ref{4}) is wrong because $x=m^2$ is a physical quantity with dimension. That's why you got a strange result at the $RHS$ of Eq.(\\ref{3})\".\n\nHence if insisting on mathematical rigor, we should admit that the introduction of $\\mu$ via Eq.(\\ref{3}) is groundless.\nThen a question arises: Why the $\\mu$ seems necessary in $QCD$ ?\n\nTo our understanding, the answer lies in the fact that in high energy $QCD$, the quarks' masses were often neglected. Therefore, in order to express the running coupling constant ($RCC$) of strong interaction, $\\alpha_s$, as a function of\n$Q$, the 3-dimensional momentum transfer in collision, one needs $\\mu$ as shown by the solution of renormalization-group-equation ($RGE$) (see p.532 of \\cite{2} and Eq.(\\ref{53}) below):\n\\begin{equation}\\label{6}\n\\alpha_s(Q)=\\dfrac{\\alpha_s(\\mu)}{1+\\alpha_s(\\mu)\\frac{\\beta_0}{2\\pi}\\ln(Q\/\\mu)}\n\\end{equation}\nIt is interesting to solve Eq.(\\ref{6}) for $\\alpha_s(\\mu)$, yielding\n\\begin{equation}\\label{7}\n\\alpha_s(\\mu)=\\dfrac{\\alpha_s(Q)}{1+\\alpha_s(Q)\\frac{\\beta_0}{2\\pi}\\ln(\\mu\/Q)}\n\\end{equation}\\footnotemark[1]\n\\footnotetext[1]{see Appendix of \\cite{4}, where a typing error exists in the denominator of Eq.(A.5), \"$+$\" should be \"$-$\".}\nWe see that Eqs.(\\ref{6}) and (\\ref{7}) are symmetrical with respect to mutual change of $Q\\leftrightarrow \\mu$. $Q$ and $\\mu$ are\nessentially equivalent. Why we need both of them? The answer is: only $Q\/\\mu$ is capable of expressing a dimensionless\n$\\alpha_s$. However, as shown in Eq.(\\ref{3}), the existence of $\\mu$ is doubtful, even superfluous. Once we take the quarks' masses into account, there will be no need of $\\mu$ at all (see section \\ref{sec:hadronization} below).\n\n\\section{Self-Energy Correction of an Electron, Lamb Shift}\n\\label{sec:selfenergy}\n\n\\vskip 0.1cm\n\nAs is well known (see \\eg, Refs.\\cite{4,5,6}), the $FDI$ of a free electron's self-energy at one loop ($L=1$) level of $QED$\nin covariant form reads\n\\begin{equation}\\label{8}\n-i\\Sigma(p)=(ie)^2\\int\\dfrac{d^4k}{(2\\pi)^4}\\dfrac{g_{\\mu\\nu}}{ik^2}\\gamma^\\mu\\dfrac{i}{\\pslash{p}-\\pslash{k}-m}\\gamma^\\nu\n\\end{equation}\nwhere the Bjorken-Drell metric ($\\pslash{p}=\\gamma^\\mu p_\\mu$) and rationalized Gaussian units are adopted with electron charge $-e\\,(e>0)$ and mass $m=m_e$. In Eq.(\\ref{8}), $p$ and $k$ are momenta of electron and (virtual) photon. After\nintroducing the Feynman parameter $x$ and making a shift in momentum integration: $k\\to K=k-xp$, Eq.(\\ref{8}) is recast into\n\\begin{equation}\\label{9}\n-i\\Sigma(p)=-e^2\\int_0^1dx[-2(1-x)\\pslash{p}+4m]I\n\\end{equation}\nwith\n\\begin{equation}\\label{10}\nI=\\int\\dfrac{d^4K}{(2\\pi)^4}\\dfrac{1}{(K^2-M^2)^2},\\quad M^2=p^2x^2+(m^2-p^2)x\n\\end{equation}\nbeing a logarithmically divergent integral, see Eq.(\\ref{1}).\n\nNote that in Eq.(\\ref{10}) we can change the unit of $M$ (and $K$) at our disposal without any change in the value of $I$,\nwhich is just a \"dimensionless\", \"large\" but \"uncertain\" number. However, in the past, we used to pay too much attention to its feature of being \"large\", trying to curb the divergence by means of some regularization method, which led to complicated\nrenormalization schemes ($RS$).\n\nBy contrast, now we believe the more important, even essential feature of a divergence is hiding in its \"uncertainty\".\nTo stress this cognition, we just use a simple trick to regulate the $I$ in Eq.(\\ref{10}) as follows.\n\nTo render it convergent, we perform a differentiation with respect to the mass-square parameter $M^2$, yielding\n\\begin{equation}\\label{11}\n\\dfrac{\\partial I}{\\partial M^2}=2\\int\\dfrac{d^4K}{(2\\pi)^4}\\dfrac{1}{(K^2-M^2)^3}=\\dfrac{-i}{(4\\pi)^2}\\dfrac{1}{M^2}\n\\end{equation}\nThen we reintegrate Eq.(\\ref{11}) with respect to $M^2$ and arrive at\n\\begin{equation}\\label{}\nI=\\dfrac{-i}{(4\\pi)^2}(\\ln M^2+C_1)=\\dfrac{-i}{(4\\pi)^2}\\ln\\dfrac{M^2}{\\mu_2^2}\n\\end{equation}\nwhere an arbitrary constant $C_1=-\\ln \\mu_2^2$ (with $\\mu_2$ a mass scale to be fixed later) is introduced so that the\nambiguity of dimension in the $\\ln M^2$ term can be eliminated.\n\nFurther integration of Eq.(\\ref{9}) with respect to $x$ leads to ($\\alpha=\\frac{e^2}{4\\pi}$)\n\\begin{equation}\\label{13}\\begin{array}{l}\n\\Sigma(p)=A+B\\pslash{p}, \\\\[5mm]\nA=\\dfrac{\\alpha}{\\pi}m\\left[2-2\\ln\\dfrac{m}{\\mu_2}+\\dfrac{(m^2-p^2)}{p^2}\\ln\\dfrac{(m^2-p^2)}{m^2}\\right], \\\\[5mm]\nB=\\dfrac{\\alpha}{4\\pi}\\left\\{2\\ln\\dfrac{m}{\\mu_2}-3-\\dfrac{(m^2-p^2)}{p^2}\\left[1+\\dfrac{(m^2+p^2)}{p^2}\n\\ln\\dfrac{(m^2-p^2)}{m^2}\\right]\\right\\}\n\\end{array}\n\\end{equation}\nUsing the chain approximation, we can derive the modification on the electron propagator as\n\\begin{equation}\\label{14}\n\\dfrac{i}{\\pslash{p}-m}\\to \\dfrac{i}{\\pslash{p}-m}\\dfrac{1}{1-\\frac{\\Sigma(p)}{\\pslash{p}-m}}=\\dfrac{iZ_2}{\\pslash{p}-m_R}\n\\end{equation}\nwhere\n\\begin{equation}\\label{15}\nZ_2=\\dfrac{1}{1-B}\n\\end{equation}\nis the renormalization factor for electron's wave function and\n\\begin{equation}\\label{16}\nm_R=\\dfrac{m+A}{1-B}\n\\end{equation}\nis the renormalized mass of $m$. The increment of mass reads\n\\begin{equation}\\label{17}\n\\delta m=m_R-m=\\dfrac{A+mB}{1-B}\n\\end{equation}\nIn the past, many physicists viewed $\\delta m$ as some real contribution of \"radiation correction\". While $m_R$ should be\nidentified with the observed mass $m_{obs}$, or physical mass $m_e$, the original $m$ (usually denoted by $m_0$ or $m_B$ in\nthe expression of Lagrangian density) was thought to be a \"bare mass\". Both $\\delta m$ and $m_0$ were divergent quantities.\n(see, \\eg, p.220 in \\cite{2}).\n\nWe don't think so. Let us read carefully the seminal paper by Bethe in 1947\\cite{8}. The theory for the hydrogenlike atom begins with a Hamiltonian in the center-of-mass frame\n\\begin{equation}\\label{18}\nH_0=\\dfrac{{\\bf p}^2}{2m}+\\dfrac{{\\bf p}^2}{2m_N}-\\dfrac{Z\\alpha}{r}=\\dfrac{{\\bf p}^2}{2\\mu}-\\dfrac{Z\\alpha}{r}\n\\end{equation}\nBethe pointed out that the effect of electron's interaction with the vector potential $\\bf A$ of radiation field\n\\begin{equation}\\label{19}\nH_{int}=\\dfrac{e}{mc}{\\bf A}\\cdot{\\bf p}\n\\end{equation}\nshould properly be regarded as already included in the $m_{obs}$, which is denoted by $m$ in Eqs.(\\ref{18}) and (\\ref{19}).\n\nIn our understanding on Bethe's claim, the \"self-interaction\" of electron with radiation field is indivisible from the free\nelectron mass $m$. In other words, in the covariant form of $QED$, certain contributions of $FDIs$ for \"self-energy\" (with\nEq.(\\ref{8}) being merely that at $L=1$ order) at all orders (up to $L\\to \\infty$) are already contained in the value of $m$. To show this cognition, we impose the mass-shell condition $p^2=m^2$ in Eq.(\\ref{17}) together with\n\\begin{equation}\\label{20}\n\\delta m |_{p^2=m^2}=\\dfrac{\\alpha m}{4\\pi}(5-6\\ln\\frac{m}{\\mu_2})=0\n\\end{equation}\nwhich in turn fixes the arbitrary constant $\\mu_2$ to be\n\\begin{equation}\\label{21}\n\\mu_2=me^{-5\/6}\n\\end{equation}\nand thus\n\\begin{equation}\\label{22}\nZ_2|_{p^2=m^2}=\\dfrac{1}{1+\\frac{\\alpha}{3\\pi}}\\simeq 1-\\dfrac{\\alpha}{3\\pi}\n\\end{equation}\nNote that $m_R=m=m_{obs}=m_e$ with no bare mass at all and $Z_2$ is fixed and finite, in sharp contrast to that in\nprevious theories.\n\nOur reader may wonder: \"In this case, does the calculation on $FDI$ for the self-energy become worthless ?\" The answer\nis \"No\" due to two reasons. First, at the $QM$ level, the parameters $m$ and $e$ in Eqs.(\\ref{18}) and (\\ref{19}) can be\nregarded as well-defined. But they are not so at the level of $QED$. As discussed before Eq.(\\ref{20}), the new effect of\nradiative corrections of $FDIs$ for self-energy is inevitably confused with that in the mass, the dividing line between\nthem is blurred. In some sense, the appearance of divergence in the $FDI$ is just a warning: the new effect you want to calculate has become entangled with the mass $m$, rendering both of them uncertain. Hence the aim of so-called mass\nrenormalization is nothing but a reconfirmation of $m$ as we did in Eqs.(\\ref{20})-({22}), where the mass $m$ is renormalized on the mass-shell $p^2=m^2$ with $m=m_e$ being fixed by the experimental value and thus well-defined. This is one important thing we must do and at most we can do on the mass-shell for a free electron.\n\nSecond, the increment of mass, $\\delta m$, ceases to be zero once when the electron is moving off-mass-shell ($p^2\\neq m^2$). Then Eq.(\\ref{17}) will provide some information about the new effect of radiation corrections. For example, for a bound electron in a hydrogenlike atom, in Ref.\\cite{7}, we replace the electron mass $m=m_e$ by reduced mass $\\mu=\\frac{m_em_N}{m_e+m_N}$ (not to be confused with the $\\mu$ in $QCD$) and write (see also \\cite{29}):\n\\begin{equation}\\label{23}\np^2=\\mu^2(1-\\zeta)\n\\end{equation}\nHere a dimensionless parameter $\\zeta\\,(>0)$ is introduced to show (on average) how large the extent of \"off-mass-shell\" is.\nSubstitution of Eq.(\\ref{23}) into Eq.(\\ref{17}) yielding\n\\begin{equation}\\label{24}\n\\delta \\mu\\simeq \\dfrac{\\alpha \\mu}{4\\pi}\\dfrac{(-\\zeta+2\\zeta\\ln\\zeta)}{1+\\frac{\\alpha}{3\\pi}}\n\\end{equation}\nwhere some terms of the order of $\\zeta^2$ or $\\zeta^2\\ln\\zeta$ are neglected since $\\zeta\\ll1$.\n\nAs a perturbative calculation at $L=1$ order, we may ascribe $\\delta\\mu$ to the (minus) binding energy $B$ of electron\nin the Bohr theory\n\\begin{equation}\\label{25}\n\\delta \\mu=\\varepsilon_n=-B=-\\dfrac{Z^2\\alpha^2}{2n^2}\\mu\n\\end{equation}\nCombination of Eqs.(\\ref{24}) and (\\ref{25}) gives the value of $\\zeta=\\zeta^{<S>}$ with the superscript $<S>$ referring to\n\"self-energy (at $L=1$ order)\".\n\nAnother \"nonperturbative\" method to fix the $\\zeta$ in Eq.(\\ref{23}) is to resort to the Virial theorem: For an electron in the Coulomb potential $V=-\\frac{Z\\alpha}{r}$, its kinetic energy $T=\\frac{1}{2\\mu}{\\bf p}^2$ can be evaluated on average as\n\\begin{equation}\\label{26}\n<{\\bf p}^2>=2\\mu<T>=2\\mu[-B-<V>]=2\\mu B\n\\end{equation}\n\\begin{equation}\\label{27}\n<p^2>=<E^2-{\\bf p}^2>=<(\\mu-B)^2-{\\bf p}^2>\\simeq \\mu^2(1-\\dfrac{4B}{\\mu})\n\\end{equation}\nComparing Eq.(\\ref{27}) with Eq.(\\ref{23}), we obtain\n\\begin{equation}\\label{28}\n\\zeta^{<V>}=\\dfrac{4B}{\\mu}=\\dfrac{2Z^2\\alpha^2}{n^2}\n\\end{equation}\nwhere the superscript $<V>$ refers to \"Virial theorem\".\n\nIn Ref.\\cite{7}, for explaining the Lamb shift of energy levels in hydrogenlike atoms, we find the result being expressed\nin terms of $\\zeta$. Throughout the entire calculation, all ultraviolet divergences are handled like that in Eqs.(\\ref{9})-({12}) while the infrared divergence disappears due to the introduction of $\\zeta$. However, the formulas\nare still approximate and either one of $\\zeta^{<S>}$ and $\\zeta^{<V>}$ is not reliable. So in the following table $I$,\nnot only $\\zeta^{<S>}$ and $\\zeta^{<V>}$, but also two kinds of \"average\", $\\zeta^{<S+V>}=\\frac{1}{2}(\\zeta^{<S>}+\\zeta^{<V>})$ and $\\zeta^{<SV>}=\\sqrt{(\\zeta^{<S>}\\zeta^{<V>}}$ are given.\n\n\\vskip 0.5cm\n\\begin{small}\\hspace*{-16mm}\\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\n\\multicolumn{8}{c}{Table I. Off-mass-shell parameter $\\zeta$ and $\\ln\\zeta$}\\\\[5pt]\n  \\hline\n  $\\frac{Z^2}{n^2}$ & $\\zeta^{<S>}\\times 10^4$ &-$\\ln\\zeta^{<S>}$ & $\\zeta^{<V>}\\times 10^6$ & -$\\ln\\zeta^{<V>}$ &\n  $\\zeta^{<S+V>}\\times 10^5$ & -$\\ln\\zeta^{<S+V>}$ & $\\zeta^{<SV>}\\times 10^5$ & $-\\ln\\zeta^{<SV>}$ \\\\\n  \\hline\n  $\\frac{1}{16}$ & $1.546093458$ & $8.77461$ & $\\frac{\\alpha^2}{8}=6.6564192$\n   &11.91992886 & $8.0632$ & 9.425609 & 3.2080284 & 10.34727 \\\\\n   \\hline\n  $\\frac{1}{4}$ & $7.446539697$ & 7.20259 & $\\frac{\\alpha^2}{2}=26.6256771$\n  & 10.5336345 & $38.5639$ & 7.860609 & 14.0808 & 8.86816225 \\\\\n   \\hline\n 1 & $37.73719345$ & 5.57969 & $2\\alpha^2=106.502$ &9.147340142 &\n $194.011$ & 6.2450103 & 63.39626 & 7.36351521 \\\\\n  \\hline\n\\end{tabular}\\end{small}\n\\vspace{0.5cm}\n\\normalsize\n\nThere are $8$ cases discussed in \\cite{7}. The first one is the hydrogen atom's \"classical Lamb shift\" measured as:\n\\begin{equation}\\label{29}\nL_H^{exp}(2S-2P)=E_H(2S_{1\/2})-E_H(2P_{1\/2})=1057.845\\;MHz\n\\end{equation}\nTheoretically, the radiative correction (at $L=1$ order) makes the dominant contribution, yielding:\n\\begin{equation}\\label{30}\\begin{array}{l}\n\\Delta E_H^{Rad<S>}(2S-2P)=1000.657\\;MHz\\\\\n\\Delta E_H^{Rad<S+V>}(2S-2P)=1089.651\\;MHz\\\\\n\\Delta E_H^{Rad<SV>}(2S-2P)=1226.087\\;MHz\\\\\n\\Delta E_H^{Rad<V>}(2S-2P)=1451.791\\;MHz\n\\end{array}\n\\end{equation}\nTaking the small contribution from the nuclear size effect into account, we adopt the $<S+V>$ scheme to obtain\n\\begin{equation}\\label{31}\nL_H^{theor}(2S-2P)=1089.794\\;MHz\n\\end{equation}\nwhich is larger than the experimental value, Eq.(\\ref{29}), by $3\\%$.\n\nThe most interesting case is the $1S-2S$ two-photon transition in hydrogen $H$ or deuterium $D$ because its natural width\nis so tiny ($1.3Hz$) and thus allows precision measurement in recent years\\cite{9}:\n\\begin{equation}\\label{32}\n\\Delta E_{H}^{exp}(1S-2S)=2466061413187.34(84)\\;kHz\n\\end{equation}\nThe isotope shift of $1S-2S$ transition between $H$ and $D$ had been measured first by Schmidt-Kalar \\etal \\cite{10} and\nquoted in \\cite{11} as:\n\\begin{equation}\\label{33}\n\\Delta E_{D-H}^{exp}(2S-1S)=670994337(22)\\;kHz\n\\end{equation}\nTheoretically, the above accurate data cannot be explained by the original Dirac equation with nucleus having mass $m_N\\to\\infty$. We propose a reduced Dirac equation ($RDE$) with electron mass replaced  by reduced mass for  $H$ and $D$ being respectively\n\\begin{equation}\\label{34}\n\\mu_H=\\dfrac{m_em_p}{m_e+m_p},\\quad \\mu_D=\\dfrac{m_em_d}{m_e+m_d}\n\\end{equation}\nThen theoretically, the $RDE$ predicts:\n\\begin{equation}\\label{35}\n\\Delta E_H^{RDE}(2S-1S)=2.466067984\\times 10^{15}\\;Hz\n\\end{equation}\n\\begin{equation}\\label{36}\n\\Delta E_{D-H}^{RDE}(2S-1S)=6.7101527879\\times 10^{11}\\;Hz\n\\end{equation}\nwhich are larger than the experimental values by only $3\\times10^{-6}$ and $3\\times10^{-5}$ respectively. Further\nradiative corrections on Eq.(\\ref{35}) will be sensitive to the choice of schemes in Table $I$, the best one is\n$\\Delta E_H^{Theor<SV>}(2S-1S)$, deviating from the experimental data, Eq.(\\ref{32}), by $-1\\times10^{-7}$ only. On the other hand, besides Eq.(\\ref{36}), the $\\Delta E_{D-H}^{Theor}(2S-1S)$ is influenced considerably by the nuclear size effect and so less sensitive to the scheme choice of the smaller radiative correction, bringing the discrepancy between theory\nand experimental data, Eq.(\\ref{33}), from  $3\\times10^{-5}$ down to $3\\times10^{-6}$ approximately.\n\n\n\\section{Renormalization Group Equation ($RGE$) for $QED$ and Its Solution}\n\\label{sec:renormalization}\n\n\\vskip 0.1cm\n\nIn $QED$, the $FDI$ for photon self-energy (\\ie, vacuum polarization) can also be evaluated \\cite{4}, bringing the\ncharge $e$ into its renormalized one:\n\\begin{equation}\\label{37}\ne^2\\to e^2_R=Z_3e^2\n\\end{equation}\n\\begin{equation}\\label{38}\nZ_3=1+\\dfrac{\\alpha}{3\\pi}\\left(\\ln\\dfrac{m^2}{\\mu^2_3}-\\dfrac{q^2}{5m^2}+\\cdots\\right)\n\\end{equation}\nHere $m$ is the fermion (say, electron) mass, $q$ is the momentum of photon and $\\mu_3$ is an arbitrary constant emerging from the treatment on the divergence like that in Eqs.(\\ref{9})-({12}). Although Eq.(\\ref{37}) looks like that in previous theories, it is really a new one: $e$ is the observed (physical) charge, not a \"bare charge\", and $Z_3$ remains finite.\n\nThe vertex function between two fermions' momenta $p$ and $p'$ with $p'-p=q$ will give another $Z_1$ \\cite{4,7}. Adding\nall the $FDIs$, we find the renormalized charge being:\n\\begin{equation}\\label{39}\ne_R=\\dfrac{Z_2}{Z_1}Z_3^{1\/2}e\n\\end{equation}\nBut the Ward-Takahashi identity ($WTI$) implies that\\cite{6}\n\\begin{equation}\\label{40}\nZ_1=Z_2\n\\end{equation}\nHence Eq.(\\ref{37}) remains valid and\n\\begin{equation}\\label{41}\ne_R(Q)=e\\left\\{1+\\dfrac{\\alpha}{2\\pi}\\left[\\dfrac{1}{3}\\ln\\dfrac{m^2}{\\mu^2_3}+\\dfrac{1}{15}\\dfrac{Q^2}{m^2}+\\cdots\\right]\\right\\}\n\\end{equation}\nwhere $Q^2=-q^2>0$, with $Q$ being the 3-dimensional momentum transfer at fermion collision. The observed charge should be defined at $Q\\to 0$ (Thomson scattering limit):\n\\begin{equation}\\label{42}\ne_{obs}=e_R|_{Q=0}=e\n\\end{equation}\nwhich dictates that\n\\begin{equation}\\label{43}\n\\mu_3=m\n\\end{equation}\nAs usual, the beta function is defined as\n\\begin{equation}\\label{44}\n\\beta(\\alpha,Q)\\equiv Q\\dfrac{\\partial}{\\partial Q}\\alpha_R(Q)\n\\end{equation}\nFrom Eq.(\\ref{41}), it is found in \\cite{4} that\n\\begin{equation}\\label{45}\n\\beta(\\alpha,Q)=\\dfrac{2\\alpha^2}{3\\pi}-\\dfrac{4\\alpha^2m^2}{\\pi Q^2}\\left[1+\\dfrac{2m^2}{\\sqrt{Q^4+4m^2Q^2}}\n\\ln\\dfrac{\\sqrt{Q^4+4m^2Q^2}-Q^2}{\\sqrt{Q^4+4m^2Q^2}+Q^2}\\right]\n\\end{equation}\n\\begin{equation}\\label{46}\n\\beta(\\alpha,Q)=\\dfrac{2\\alpha^2}{15\\pi}\\dfrac{Q^2}{m^2},\\quad (\\dfrac{Q^2}{m^2}\\ll1)\n\\end{equation}\nEvidently, the $e_R(Q)$ will increase with $Q$, becoming a running coupling constant ($RCC$). To calculate it, usually\na renormalization-group-equation ($RGE$) was derived for $QED$ by setting $\\alpha\\to\\alpha_R(Q)$ and $Q\\to\\infty$ in\n$\\beta(\\alpha,Q)$ yielding:\n\\begin{equation}\\label{47}\nQ\\dfrac{d}{dQ}\\alpha_R=\\dfrac{2\\alpha_R^2}{3\\pi}\n\\end{equation}\nwith its solution\n\\begin{equation}\\label{48}\n\\alpha_R(Q)=\\dfrac{\\alpha}{1-\\frac{2\\alpha}{3\\pi}\\ln\\frac{Q}{m}}\n\\end{equation}\nHere, the renormalization was forced to be made at $Q^2=m^2$ so that\n\\begin{equation}\\label{49}\n\\alpha_R|_{Q^2=m^2}=\\alpha\n\\end{equation}\nThis is inconsistent with the physical condition, Eq.(\\ref{42}), a defect due to ignoring the mass $m$, which plays a\ndominant role at low $Q$ region as shown by the Eq.(\\ref{46}). As an improvement, a more practical $RGE$ is constructed\nin \\cite{4} by changing $\\alpha$ into $\\alpha_R(Q)$ in Eq.(\\ref{45}) for the entire $Q$ region and adding up contributions from $9$ elementary charged\nfermions ($e,\\mu,\\tau,u,d,s,c,b,t$). Then $\\alpha_R$ can be numerically calculated as a function of $\\ln(Q\/m_e)$,\nstarting from\n\\begin{equation}\\label{50}\n\\alpha_R(Q=0)=\\alpha=(137.03599)^{-1}\n\\end{equation}\nand passing through another experimental data point \\cite{12}\n\\begin{equation}\\label{51}\n\\alpha_R(Q=M_Z=91.1880\\,GeV)=(128.89)^{-1}\n\\end{equation}\nIn this way, after adopting three heavy quarks' masses as $m_c=1.031\\,GeV,\\; m_b=4.326\\,GeV$ (see section 9.5D in \\cite{13}) and $m_t=175\\,GeV$, three light quarks masses\n\\begin{equation}\\label{52}\nm_u=8\\,MeV,\\;m_d=10\\,MeV,\\;m_s=200\\,MeV\n\\end{equation}\n(or averaged mass for $u,d,s$ being $92\\ MeV$) can be fitted as shown in Fig.1 of Ref.\\cite{4}.\n\n\n\\section{$RGE$ for $QCD$, Threshold Energies of Quarks Hadronization}\n\\label{sec:hadronization}\n\n\\vskip 0.1cm\n\nDifferent from the $RCC$ in $QED$, the $RCC$ in $QCD$, denoted by $\\alpha_s(Q)=\\frac{1}{4\\pi}g_s^2(Q)$, is much larger and\ndecreases with the increase of $Q$. Usually, the $RGE$ for $QCD$ reads (see p.551-552 in \\cite{2})\n\\begin{equation}\\label{53}\nQ\\dfrac{d}{dQ}\\alpha_s(Q)=\\beta(\\alpha_s(Q))=-\\dfrac{\\beta_0}{2\\pi}\\alpha_s^2(Q)\n\\end{equation}\nwhere\n\\begin{equation}\\label{54}\n\\beta_0=11-\\dfrac{2}{3}n_f\n\\end{equation}\nwith $n_f$ being the number of quarks' flavor. Eq.(\\ref{53}) looks similar to Eq.(\\ref{47}), but the negative sign in beta\nfunction implies the property of asymptotic freedom in the strong interaction. Solution of Eq.(\\ref{53}) is give by\nEq.(\\ref{6}), also bears some resemblance to Eq.(\\ref{48}).\n\nLet us make a comparison between Eqs.(\\ref{6}) and (\\ref{48}). Besides the difference in sign (\"$+$\" versus \"$-$\") in the\ndenominator, there is another one: The mass scale $\\mu$ remains arbitrary in $QCD$ whereas $m$ in $QED$ means the mass of an\nobserved charged fermion (usually electron).\n\nAs we already see, even for $QED$, one mass ($m_e$) is far from enough, let alone in $QCD$, where quarks' masses are much\nheavier. How can we ignore them?\n\nUsually, to remove the arbitrary mass scale $\\mu$, a parameter $\\Lambda_{QCD}$ is often defined via equation\n\\begin{equation}\\label{55}\n\\dfrac{\\beta_0}{2\\pi}\\alpha_s(\\mu)\\ln\\left(\\dfrac{\\mu}{\\Lambda_{QCD}}\\right)=1\n\\end{equation}\nsuch that a simpler formula for $\\alpha_s(Q)$ can be found as\n\\begin{equation}\\label{56}\n\\alpha_s(Q)=\\dfrac{2\\pi}{\\beta_0\\ln(Q\/\\Lambda_{QCD})}\n\\end{equation}\nThen the precision experimental data \\cite{1}\n\\begin{equation}\\label{57}\n\\alpha_s(M_Z=91.1876\\,GeV)=0.1176\n\\end{equation}\nserves as a substitute for $\\alpha_s(\\mu)$ in Eq.(\\ref{55}), yielding\n\\begin{equation}\\label{58}\n\\Lambda_{QCD}=M_Z\\exp\\left[\\dfrac{-2\\pi}{\\alpha_s(M_Z)\\beta_0}\\right]\n=\\left\\{\\begin{array}{l}\n          240\\;MeV,\\;(n_f=3) \\\\\n          150\\;MeV,\\;(n_f=4) \\\\\n          85.8\\;MeV,\\;(n_f=5) \\\\\n          44.2\\;MeV,\\;(n_f=6)\n        \\end{array}\\right.\n\\end{equation}\nIt is evident from Eq.(\\ref{56}) that\n\\begin{equation}\\label{59}\n\\alpha_s(\\Lambda_{QCD})\\to\\infty\n\\end{equation}\nwhich implies the \"infrared confinement\" of quarks.\n\nHowever, the value of $\\Lambda_{QCD}$ sensitively depends on the flavor number $n_f$ as shown by Eq.(\\ref{58}) but\nis independent of the concrete flavor of a quark under consideration. Moreover, the divergence of $\\alpha_s$ appears\nat $\\Lambda_{QCD}$. These features seem not so reasonable and are not consistent with the experimental fact that the lighter\na quark's mass is, the lower its \"threshold energy for hadronization\" will be.\n\nThe way out of above difficulties is clear. Just like in Eq.(\\ref{45}) for $QED$, where rather than just the first term,\nall terms for $9$ fermions should be added, now for $RGE$ in $QCD$, all masses of $6$ quarks should be preserved. In this\nway, the $\\alpha_{si}(Q)$ are numerically calculated for $i=u,d,s,c,b$ respectively in Ref.\\cite{4}. Starting from\n$\\alpha_s(M_Z)=0.118$ (the common renormalization point), their running curves (as shown by Figs.2 and 3 in \\cite{4}) follow the trend of experimental data (as shown on p.158 of Ref.\\cite{1}) quite well but separate at the low $Q$ region. Each of\nthem rises to a maximum $\\alpha_{si}^{max}$ at $Q=\\Lambda_i$ and then suddenly drops to zero at $Q=0$.\n\nFor example, for $b$ quark, $\\Lambda_b=7.04\\, GeV,\\;\\alpha_{sb}^{max}=0.161$. If tentatively explain $L_b\\sim \\hbar\/\\Lambda_b$\nas some critical length scale of $b\\bar{b}$ pair, then $L_b\\sim 0.02805\\,fm$. In \\cite{4}, it is further guessed that\n$E_b^{theor}\\sim \\alpha_{sb}^{max}\/L_b\\sim 1.133\\, GeV$ being the order of excitation energy for breaking the binding\n$b\\bar{b}$ pair, \\ie, the hadronization threshold energy of Upsilon $\\Upsilon(b\\bar{b})$ against its dissociation into\ntwo bosons. It is indeed the case found experimentally\\cite{1}:\n\\begin{equation}\\label{60}\nM(\\Upsilon(4S))-M(\\Upsilon)=1.12\\,GeV,\\quad M(\\Upsilon(4S))\\to B^+B \\;\\text{or}\\;B^0\\bar{B^0}\n\\end{equation}\nSimilarly, we can estimate from Ref.\\cite{4} that $E_c^{theor}\\sim0.398\\,GeV$ which also corresponds to experiments that\\cite{1}\n\\begin{equation}\\label{61}\nM(\\psi(3770))-M(\\psi(3097))=673\\,MeV,\\quad \\psi(3770)\\to D^+D^-\\;\\text{or}\\;D^0\\bar{D^0}\n\\end{equation}\nIt seems that $E_s^{theor}\\sim90\\,MeV$ and $E_{u,d}^{theor}\\sim 0.4\\,MeV$ are not so reliable but still reasonable.\n\n\n\\section{$\\lambda \\phi^4$ Model and Higgs Mass in the Standard Model}\n\\label{sec:higgsmass}\n\n\\vskip 0.1cm\n\nThe Lagrangian density of $\\lambda\\phi^4$ ($\\phi(x)$ is a real scalar field) model is defined by\n\\begin{equation}\\label{62}\n{\\cal L}=\\dfrac{1}{2}\\partial_\\mu\\phi\\partial^\\mu\\phi+\\dfrac{1}{2}\\sigma\\phi^2-\\dfrac{\\lambda}{4!}\\phi^4\n\\end{equation}\nThe importance of this model lies in the \"wrong sign\" of mass term ($\\sigma=-m^2>0$) which leads to the spontaneous\nsymmetry breaking ($SSB$) at the tree level ($L=0$). The effective potential ($EP$) reads\n\\begin{equation}\\label{63}\nV_0(\\phi)=-\\dfrac{1}{2}\\sigma\\phi^2+\\dfrac{\\lambda}{4!}\\phi^4\n\\end{equation}\n(The subscript \"$0$\" refers to $L=0$). Obviously, $V_0(\\phi)$ has two extremum, one is a maximum:\n\\begin{equation}\\label{64}\n\\phi_0=0\\quad\\text{(symmetric phase)}\n\\end{equation}\nwhile the other one is a minimum:\n\\begin{equation}\\label{65}\n\\phi_1^2=\\dfrac{6\\sigma}{\\lambda}\\quad\\text{(SSB phase)}\n\\end{equation}\nAt the $QFT$ level, the $EP$ evolves into\n\\begin{equation}\\label{66}\nV=V_0+V_1+\\cdots\n\\end{equation}\nThe theory for $EP$ had been developed by various authors \\cite{14,15,16,17,18}, with $L=1$ contribution to $EP$ being\nevaluated as:\n\\begin{equation}\\label{67}\nV_1(\\phi)=\\dfrac{1}{2}\\int\\dfrac{d^4k_E}{(2\\pi)^4}\\ln(k_E^2-\\sigma+\\frac{1}{2}\\lambda\\phi^2)\n\\end{equation}\nThis highly divergent integral (in 4-dimensional Euclidean momentum space) is treated in Ref.\\cite{19} like that for\nEqs.(\\ref{10})-({12}). First, three times of differentiation with respect to $M^2=-\\sigma+\\frac{1}{2}\\lambda\\phi^2$\nare needed before it becomes just convergent.\n\\begin{equation}\\label{68}\n\\dfrac{\\partial^3V_1}{\\partial(M^2)^3}=\\int\\dfrac{d^4k_E}{(2\\pi)^4}\\dfrac{1}{(k_E^2+M^2)^3}=\\dfrac{1}{2(4\\pi)^2M^2}\n\\end{equation}\nSecond, three times integration with respect to $M^2$ are performed, yielding\n\\begin{equation}\\label{69}\nV_1(\\phi)=\\dfrac{1}{2(4\\pi)^2}\\left\\{\\dfrac{M^4}{2}(\\ln M^2-\\dfrac{1}{2})-\\dfrac{1}{2}M^4+\\dfrac{1}{2}C_1M^4+C_2M^2+C_3\\right\\}\n\\end{equation}\nAs expected, three arbitrary constants $C_1,C_2,C_3$ appear. The renormalization amounts to fix them at our disposal.\n\nThird, like that in Eq.(\\ref{3}), for eliminating the ambiguity of dimension in the first term involving $\\ln M^2$,\nthe only possible choice of $C_1=-\\ln\\mu^2$ is fixed. Then the choice of $C_2=\\mu^2=2\\sigma$ and $C_3=-\\sigma^2+(4\\pi)^2\\frac{3\\sigma^2}{\\lambda}$ leads to $V=V_0+V_1$ with its derivatives being given at the Table II\n\\cite{19}.\n\n\\renewcommand\\arraystretch{1.5}\n\\begin{center}\n\\hspace*{-16mm}\\begin{tabular}{|c|c|c|}\n\\multicolumn{2}{c}{Table II. Effective potential of $\\lambda\\phi^4$ model with $SSB$}\\tabularnewline\n  \\hline\n    & $SSB$ phase & symmetric phase \\tabularnewline\n  \\hline\n  $\\phi$ & $\\phi_1=\\sqrt{\\dfrac{6\\sigma}{\\lambda}}$  & $\\phi_0=0$ \\tabularnewline\n  \\hline\n  $V$ & $0$  & $-\\dfrac{\\sigma^2}{2(4\\pi)^2}\\left[\\dfrac{15}{4}+\\dfrac{1}{2}\\ln2-i\\dfrac{\\pi}{2}\\right]+\\dfrac{3}{2}\\dfrac{\\sigma^2}{\\lambda}$ \\tabularnewline\n   \\hline\n  $\\dfrac{dV}{d\\phi}$ & $0$ & $0$ \\tabularnewline\n   \\hline\n  $\\dfrac{d^2V}{d\\phi^2}$ & $2\\sigma$  & $-\\sigma\\left[1-\\dfrac{\\lambda}{2(4\\pi)^2}(3+\\ln2-i\\pi)\\right]$\\tabularnewline\n   \\hline\n  $\\dfrac{d^3V}{d\\phi^3}$ & $\\lambda\\sqrt{\\dfrac{6\\sigma}{\\lambda}}\\left[1+\\dfrac{3\\lambda}{2(4\\pi)^2}\\right]$  & $0$ \\tabularnewline\n   \\hline\n $\\dfrac{d^4V}{d\\phi^4}$ & $\\lambda\\left[1+\\dfrac{9\\lambda}{2(4\\pi)^2}\\right]$  & $\\lambda\\left[1-\\dfrac{3\\lambda}{2(4\\pi)^2}(\\ln2-i\\pi)\\right]$ \\tabularnewline\n  \\hline\n\\end{tabular}\n\\end{center}\n\nNote that, with the above assignment of $C_i\\,(i=1,2,3)$, both the position of the $SSB$ phase, $\\phi_1$, and the mass\n$m_\\sigma$ excited above it take the same expression as that at the tree level\n\\begin{equation}\\label{70}\nm_\\sigma^2=\\dfrac{d^2V}{d\\phi^2}|_{\\phi=\\phi_1}=2\\sigma\n\\end{equation}\nHowever, the renormalization coupling constant\n\\begin{equation}\\label{71}\n\\lambda_R=\\dfrac{d^4V}{d\\phi^4}|_{\\phi=\\phi_1}=\\lambda\\left[1+\\dfrac{9\\lambda}{2(4\\pi)^2}\\right]\n\\end{equation}\ndoes receive some quantum correction on its classical value $\\lambda$. Hence it is suddenly realized that the invariant meaning of $\\lambda$ in the Lagrangian, Eq.(\\ref{62}), is by no means a \"coupling constant\", but the ratio of two mass scales \\cite{21}.\n\\begin{equation}\\label{72}\n\\lambda=3\\dfrac{m_\\sigma^2}{\\phi_1^2}\n\\end{equation}\nTwo parameters, $\\sigma$ and $\\lambda$, together with Eqs.(\\ref{70}) and (\\ref{72}), should all be preserved through out\nhigh loop ($L$) evaluations in perturbation theory until $L\\to\\infty$, \\ie, in any nonperturbative treatment.\n\nOn the other hand, the above assignment of $C_i$ renders the appearance of imaginary part in $V$ and its derivatives at\nthe symmetric phase ($\\phi_0=0$). It means the instability of symmetric phase at the presence of stable $SSB$ phase.\n\nIt is interesting to see that an alternative choice of\n\\begin{equation}\\label{73}\nC_1=-\\ln(-\\sigma),\\;C_2=-\\sigma,\\;C_3=-\\dfrac{1}{4}\\sigma^2\n\\end{equation}\nwould leads to the survival of $\\phi_0=0$ as a semistable state with\n\\begin{equation}\\label{74}\nV(0)=\\dfrac{dV}{d\\phi}|_{\\phi=0}=\\dfrac{d^3V}{d\\phi^3}|_{\\phi=0}=0\n\\end{equation}\n\\begin{equation}\\label{75}\n\\dfrac{d^2V}{d\\phi^2}|_{\\phi=0}=-\\sigma,\\;\\dfrac{d^4V}{d\\phi^4}|_{\\phi=0}=\\lambda\n\\end{equation}\nwhereas no real $SSB$ solution exists. Hence we see that two different choices of $C_i$ lead two separable sectors in\nthe effective potential \\cite{19}.\n\nIn 1989, we had estimated the upper and lower bounds of Higgs mass $M_H$ in the standard model of particle physics by\nusing a nonperturbative approach in $QFT$ --- the Gaussian effective potential ($GEP$) method, yielding\\cite{20}\n\\begin{equation}\\label{76}\n76\\,GeV<M_H<170\\,GeV\n\\end{equation}\nLike many authors, we were bothered a lot by divergences. After a deeper understanding on $\\lambda\\phi^4$ model\\cite{19},\nthis problem was restudied in 1998 by a combination of $GEP$ with our $RRM$, yielding\n\\begin{equation}\\label{77}\nM_H=138\\,GeV\n\\end{equation}\nwhich is based on the following input of experimental data:\n\\begin{equation}\\label{78}\n\\begin{array}{l}\n  M_W=80.359\\,GeV,\\;M_Z=91.1884\\,GeV, \\\\\n  \\alpha^{-1}=\\dfrac{4\\pi}{g^2\\sin^2\\theta_W}=128.89,\\;\\sin^2\\theta_W=0.2317\n\\end{array}\n\\end{equation}\nwhere $\\theta_W$ is the weak mixing (Weinberg) angle. As now the search for Higgs particle becomes so urgent experimentally\nwhile the theoretical estimation about its mass still remains uncertain\\cite{1}, we think our method \\cite{21} with its\nprediction, Eq.(\\ref{77}), deserves to be reconsidered.\n\n\n\\section{Summary and Discussion}\n\\label{sec:discussion}\n\n\\vskip 0.1cm\n\nOur $RRM$ was first proposed by J. F. Yang in 1994\\cite{22}, then elaborated in a series of papers since 1998 (\\cite{19,21,\n23,24,4,7} \\etc). What we have been thinking about is: Why the divergence emerges inevitably in $QFT$? What is the essential meaning of a regularization and renormalization procedure?\n\nFor instance, in a pioneering work to explain the Lamb shift, Welton (\\cite{25}, see also section 9.6B in Ref.\\cite{13})\nencountered a divergent integral $I=\\int\\frac{d\\omega}{\\omega}$ with $\\omega$ being the (angular) frequency of virtual photon (vacuum fluctuation). He simply set the lower and upper bounds by $\\omega_{min}\\sim mZ\\alpha=\\frac{Z}{a}$ ($a$ is Bohr\nradius) and $\\omega_{max}\\sim m$ respectively, arriving at $I\\simeq \\ln(\\frac{1}{Z\\alpha})=4.92\\,(Z=1)$ which leads to\nan estimation of Lamb shift $L_H^{theor}(2S_{1\/2}-2P_{1\/2})\\simeq 668\\,MHz$. However, if instead of Bohr radius, the lower cutoff is provided by the electron's binding energy, one would get $I\\simeq \\ln(Z\\alpha)^{-2}$ and\n$L_H^{theor}(2S_{1\/2}-2P_{1\/2})\\simeq 1336\\,MHz$ (see Eq.(\\ref{30}) in the Ref.\\cite{26}). We see the integral $I$ being\na dimensionless number, not very large ($I\\leq 10$), but uncertain indeed. The root cause of uncertainty lies in the fact\nthat a reconfirmation process of electron mass like Eq.(\\ref{20}) was missing.\n\nFor further clarity, a study on Lamb shift in the form of noncovariant $QED$ was performed in \\cite{24} (see also Appendix 9A in \\cite{13}). Beginning with Eqs.(\\ref{18}) and (\\ref{19}) (with $m\\to\\mu=\\frac{mm_N}{m+m_N}$), the perturbative\ncalculation of electron's self-energy at second order in noncovariant form (corresponding to the one-loop ($L=1$) order in\ncovariant form) leads to an energy increase of an electron with momentum $\\bf p$, $\\Delta E_p$, which contains divergence\nand can be handled just like that in Eqs.(\\ref{10}) and (\\ref{67}), yielding\n\\begin{equation}\\label{79}\n\\begin{array}{l}\n  \\Delta E_p=b_1p^2+b_2p^4+\\cdots\\\\[5mm]\n  b_1=\\dfrac{\\alpha}{\\pi\\mu}(\\dfrac{4}{3}\\ln2+\\dfrac{4}{3}\\ln\\mu-\\dfrac{4}{3}C_1)\\\\[5mm]\n  b_2=\\dfrac{\\alpha}{\\pi\\mu^3}(-\\dfrac{2}{15})\n\\end{array}\n\\end{equation}\nThe only choice of arbitrary constant $C_1$ is to make $b_1=0$ such that the reduced mass $\\mu$ in Eq.(\\ref{18}) can be\nreconfirmed. However, Eq.(\\ref{19}) must be supplemented by the interaction between electron's spin and the radiation field\n\\begin{equation}\\label{80}\nH'_{int}=\\dfrac{ge\\hbar}{4\\mu c}{\\boldsymbol \\sigma}\\cdot\\nabla\\times{\\bf A}\n\\end{equation}\nwhere $g=2\\times 1.0011596522$ is gyromagnetic ratio of electron. Similar treatment leads to\n\\begin{equation}\\label{81}\n\\begin{array}{l}\n  \\Delta E'_p=b'_0+b'_1p^2+b'_2p^4+\\cdots\\\\[4mm]\n  b'_0=\\dfrac{g^2}{4}\\dfrac{\\alpha\\mu}{\\pi}[4(\\ln2+\\ln\\mu)-4C_2-\\dfrac{2C_3}{\\mu}-\\dfrac{C_4}{\\mu^2}]\\\\[4mm]\n  b'_1=\\dfrac{g^2}{4}\\dfrac{\\alpha}{\\pi\\mu}(\\dfrac{4}{3}\\ln2+2+\\dfrac{4}{3}\\ln\\mu-\\dfrac{4}{3}C_2)\\\\[4mm]\n  b'_2=\\dfrac{g^2}{4}\\dfrac{\\alpha}{\\pi\\mu^3}(-\\dfrac{1}{15})\n\\end{array}\n\\end{equation}\nBecause $\\mu$ has already been reconfirmed (by $b_1=0$), the only choice of arbitrary constant $C_2$ is to cancel the\nambiguous term with $\\ln\\mu,\\;C_2=\\ln\\mu$, leaving a nonzero $b'_1p^2$ and combining with $\\frac{1}{2\\mu}{\\bf p}^2$. Hence\n$\\mu$ really acquires a modification as\n\\begin{equation}\\label{82}\n\\mu\\to \\mu_{obs}=\\dfrac{\\mu}{1+\\beta},\\qquad \\beta=\\dfrac{g^2\\alpha}{2\\pi}(\\dfrac{4}{3}\\ln2+2)\n\\end{equation}\nwhere $\\mu_{obs}=\\frac{m_em_N}{m_e+m_N}$. Then constants $C_3$ and $C_4$ must be chosen such that $b'_0=0$.\n\nNotice that, however, the spin induced interaction, Eq.(\\ref{80}), endows electron with relativistic feature, creating\na term ($-\\frac{1}{8\\mu^3}p^4$) in its kinetic energy. Yet the modification on $\\mu$ shown in Eq.(\\ref{82}) does\ninduce a corresponding change $-\\frac{1}{8}(\\frac{1}{\\mu_{obs}^3}-\\frac{1}{\\mu^3})p^4$, which should be regarded as an invisible \"background\" and subtracted from the $p^4$ term induced by the radiative corrections. Hence the \"renormalized\"\n$b_2$ should be\n\\begin{equation}\\label{83}\nb_2^R=b_2+b'_2+\\frac{1}{8\\mu^3}(3\\beta+3\\beta^2+\\beta^3)\\simeq \\dfrac{\\alpha}{\\pi\\mu_{obs}^3}(1.99808)\n\\end{equation}\nwhere only the lowest approximation is kept ($\\mu\\simeq\\mu_{obs}$). Hence the radiative correction on the energy\nlevel of a stationary state $|Z,n,l\\rangle$ in hydrogenlike atom simply reads\n\\begin{equation}\\label{84}\n\\Delta E^{rad}(Z.n,l)=\\langle Z,n,l|b_2^Rp^4|Z,n,l\\rangle=[\\dfrac{8n}{3l+1}-3]\\dfrac{b_2^RZ^4\\alpha^4}{n^4}\\mu_{obs}^4\n\\end{equation}\nThis contribution, together with that from the vacuum polarization (borrowed from covariant theory) and nuclear size effect, gives a theoretical value for the Lamb shift \\cite{24}\n\\begin{equation}\\label{85}\nL_H^{theor}(2S_{1\/2}-2P_{1\/2})= 1056.52\\,MHz\n\\end{equation}\nwhich is smaller than the experimental value, Eq.(\\ref{29}), by $0.13\\%$ only.\n\nDespite the approximation involved, the above method clearly shows that our regularization is by noo means a trick to curb the divergence. Rather, it is a natural way to transform a divergence into some arbitrary constants, revealing that the essential meaning of divergence is just the \"uncertainty\" in the theory. Thus so-called renormalization turn out to be nothing but a process of reconfirmation to fix these constants via experiments. We must reconfirm a mass before it could\nbe modified via radiative corrections. Either \"skipping over the first step\" or \"combining two steps into one\" is not allowed.\n\nIn deeper understanding, our $RRM$ is based on a \"principle of relativity\" in epistemology\\cite{27}: Every thing is\nmoving and becomes recognizable only in relationship with other things. What we can understand is either no mass scale\nor two mass scales, but never one mass scale. This scenario is clearly displayed in the Gross-Neveu model\\cite{28}, also\nin the $\\lambda\\phi^4$ model with $SSB$ as shown by Eqs.(\\ref{65}), (\\ref{70}) and (\\ref{72}). The vacuum expectation\nvalue of field, $\\phi_1$, just provides a \"mass unit\" for the mass $m_\\sigma$ excited on the vacuum with $SSB$.\n\nSimilarly, in perturbative $QFT$, we will be able to calculate various radiative corrections on a particle only when its mass $m$ can be reconfirmed again and again throughout any high loop ($L$) order of theory until $L\\to\\infty$. Just\nlike one has to reconfirm his plane ticket before his departure from the airport, he  must use the same name throughout\nhis entire jouney\\cite{4}.\n\n\\section*{Acknowledgements}\n\n\\vskip 0.1cm\n\nWe thank S. Q. Chen, Y. S. Duan, R. T. Fu, S. S. Feng, T. Huang, P. T. Leung, W. F. Lu,\nX. T. Song, F. Wang, H. B. Wang, K. Wu, Y. L. Wu, J. Yan, G. H. Yang,\nJ. F. Yang and Z. X. Zhang for close collaborations and\/or helpful discussions.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{intro} \n\\setcounter{equation}{0}\n\nThe uninitiated reader of the table of integrals by \nI. S. Gradshteyn and I. M. Ryzhik \\cite{gr} will surely be \npuzzled by choice of integrands. In this note we provide an elementary proof \nof the evaluation {$\\mathbf{4.353.3}$} \n\\begin{equation}\n\\int_{0}^{1} ( ax + n+1)x^{n} e^{ax} \\ln x \\, dx = \ne^{a} \\sum_{k=0}^{n} (-1)^{k-1} \\frac{n!}{(n-k)! a^{k+1}} +\n(-1)^{n} \\frac{n!}{a^{n+1}}.\n\\label{iden1}\n\\end{equation}\n\\noindent\nWe also consider the integrals\n\\begin{equation}\nq_{n} := \\int_{0}^{1} x^{n} e^{-x} \\ln x \\, dx \n\\end{equation}\n\\noindent\nand the companion family \n\\begin{equation}\np_{n} := \\int_{0}^{1} x^{n} e^{-x} \\, dx.\n\\end{equation}\n\\noindent\nThe integral $q_{n}$ corresponds to the case $a=-1$ in (\\ref{iden1}). \nSection \\ref{new} provides closed-form\nexpressions for $p_{n}$ and $q_{n}$. Section \\ref{sec-param} considers the \ngeneralization\n\\begin{equation}\nP_{n}(a) = \\int_{0}^{1} x^{n} e^{-ax} \\, dx \\text{ and }\nQ_{n}(a) = \\int_{0}^{1} x^{n} e^{-ax} \\ln x \\, dx. \n\\end{equation}\n\\noindent\nThe main result of this section is the closed-form expressions\n\\begin{equation}\nP_{n}(a) := \\int_{0}^{1}x^{n}e^{-ax} \\, dx = \n\\frac{n!}{a^{n+1}} \\left( 1 - e^{-a} \\sum_{k=0}^{n} \\frac{a^{k}}{k!} \\right),\n\\end{equation}\n\\noindent\nand\n\\begin{equation}\nQ_{n}(a) := \\int_{0}^{1}x^{n}e^{-ax} \\ln x \\, dx = \n\\frac{n!}{a^{n+1}} \\left[  \\sum_{k=1}^{n} \\frac{1}{k} \n\\left( 1 - e^{-a} \\sum_{j=0}^{k-1} \\frac{a^{j}}{j!} \\right) + a Q_{0}(a) \n\\right], \n\\nonumber \n\\end{equation}\nwhere \n\\begin{equation}\nQ_{0}(a) = \\int_{0}^{1} e^{-ax} \\ln x \\, dx = -\\frac{1}{a} \n\\left( \\gamma + \\ln a + \\Gamma(0,a) \\right),\n\\end{equation}\n\\noindent\nand $\\Gamma(0,a)$ is the incomplete gamma function defined by\n\\begin{equation}\n\\Gamma(a,x) := \\int_{x}^{\\infty} t^{a-1} e^{-t} \\, dt.\n\\end{equation}\n\n\\section{The evaluation of {$\\mathbf{4.353.3}$}} \\label{simple} \n\\setcounter{equation}{0}\n\nThe identity \n\\begin{equation}\n\\frac{d}{dx} \\left( x^{n+1}e^{ax} \\right) = (ax+n+1)x^{n}e^{ax} \n\\end{equation}\n\\noindent\nand integration by parts yield \n\\begin{equation}\n\\int_{0}^{1} ( ax + n+1)x^{n} e^{ax} \\ln x \\, dx = \n- \\int_{0}^{1}x^{n} e^{ax} \\, dx. \\label{form2}\n\\end{equation}\n\\noindent\nThis last integral appears as $\\mathbf{3.351.1}$ in \\cite{gr}. We have obtained\na closed-form expression for it in \\cite{moll-gr7}. A new proof is \npresented in Section \\ref{sec-param}. \n\nA closed form expression for the right hand side of  (\\ref{form2}) \nis obtained from \n\\begin{equation}\n\\int_{0}^{1} x^{n} e^{ax} \\, dx = \\left( \\frac{d}{da} \\right)^{n} \n\\frac{e^{a}-1}{a}. \n\\label{form3}\n\\end{equation}\n\\noindent\nThe symbolic evaluation of (\\ref{form3}) for small values of \n$n \\in \\mathbb{N}$ suggests the existence of a polynomial $P_{n}(a)$\nsuch that \n\\begin{equation}\n\\int_{0}^{1}x^{n}e^{ax} \\, dx = \n\\frac{(-1)^{n+1} \\, n!}{a^{n+1}} + \\frac{P_{n}(a)}{a^{n+1}} e^{a}. \n\\label{pn-def}\n\\end{equation}\n\\noindent \nThe next lemma confirms the existence of this polynomial. \n\n\\begin{lemma}\nThe function $P_{n}(a)$ defined by \n\\begin{equation}\nP_{n}(a) =a^{n+1}e^{-a} \\left( \n\\left( \\frac{d}{da} \\right)^{n} \\frac{e^{a}-1}{a} - \n\\frac{(-1)^{n+1} n!}{a^{n+1}} \\right) \n\\end{equation}\n\\noindent\nis a polynomial of degree $n$. \n\\end{lemma}\n\\begin{proof}\nLet $D = \\frac{d}{da}$. Then $D^{n+1} = D(D^{n})$ produces the recurrence\n\\begin{equation}\nP_{n+1}(a) = a P_{n}'(a) + (a-n-1)P_{n}(a).\n\\label{recu1}\n\\end{equation}\n\\noindent\nThe initial condition $P_{0}(a) = 1$ and (\\ref{recu1}) show that \n$P_{n}$ is a polynomial of degree $n$. \n\\end{proof}\n\n\\begin{theorem}\nThe polynomial \n\\begin{equation}\nQ_{n}(a) := (-1)^{n} P_{n}(-a)\n\\end{equation}\n\\noindent\nhas positive integer coefficients, written as \n\\begin{equation}\nQ_{n}(a) = \\sum_{k=0}^{n} b_{n,k}a^{k}.\n\\end{equation}\n\\noindent\nThese coefficients satisfy\n\\begin{eqnarray}\nb_{n+1,0} & = & (n+1)b_{n,0} \\label{recu3} \\\\\nb_{n+1,k} & = & (n+1-k)b_{n,k} + b_{n,k-1}, \\quad 1 \\leq k \\leq n  \\nonumber \\\\\nb_{n+1,n+1} & = & b_{n,n}. \\nonumber\n\\end{eqnarray}\n\\noindent\nMoreover, the polynomial $Q_{n}(a)$ is given by\n\\begin{equation}\nQ_{n}(a) = n! \\, \\sum_{k=0}^{n} \\frac{a^{k}}{k!}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nThe recurrence (\\ref{recu1}) yields\n\\begin{equation}\nQ_{n+1}(a) = -a Q_{n}'(a) + (a+n+1)Q_{n}(a).\n\\label{recu5}\n\\end{equation}\n\\noindent\nThe recursion for the coefficients $b_{n,k}$ follows directly from here.\nMorover, it is clear that $b_{n,n}=1$ and $b_{n,0} = n!$. A little \nexperimentation suggets that $b_{n,k} = n!\/k!$, and this can be established \nfrom (\\ref{recu3}). \n\\end{proof}\n\n\nThis proposition amounts to the evaluation of ${\\mathbf{3.351.1}}$ in \n\\cite{gr}:\n\\begin{equation}\n\\int_{0}^{u} x^{n} e^{ax} \\, dx = \n\\frac{(-1)^{n+1} \\, n!}{a^{n+1}} + \n\\frac{e^{au}}{a^{n+1}} \\sum_{k=0}^{n} \\frac{n!}{k!} (-1)^{n-k} u^{k}a^{k}. \n\\end{equation}\n\\noindent\nThe reader will find a proof of this formula in \\cite{moll-gr7}.\n\n\\section{A new family of integrals} \\label{new} \n\\setcounter{equation}{0}\n\nIn this section we consider the family of integrals \n\\begin{equation}\nq_{n}  := \\int_{0}^{1} x^{n} e^{-x} \\ln x \\, dx, \\label{qn-def}\n\\end{equation}\n\\noindent\nand its companion \n\\begin{equation}\np_{n}  := \\int_{0}^{1} x^{n} e^{-x} \\, dx. \\label{p-def}\n\\end{equation}\n\n\\begin{lemma}\nThe integrals $p_{n}, \\, q_{n}$ satisfy the recursion\n\\begin{eqnarray}\np_{n+1} & = & (n+1)p_{n} - e^{-1} \\label{recu8} \\\\\nq_{n+1} & = & (n+1)q_{n} + p_{n} \\label{recu8a}\n\\end{eqnarray}\n\\end{lemma}\n\\begin{proof}\nIntegrate by parts. \n\\end{proof}\n\nThe initial conditions are \n\\begin{equation}\np_{0} = 1 - e^{-1} \\text{ and } q_{0} = \\int_{0}^{1} e^{-x} \\ln x \\, dx  = \\gamma - \\text{Ei}(-1). \\label{initq}\n\\end{equation}\n\\noindent\nHere $\\gamma$ is Euler's constant defined by\n\\begin{equation}\n\\gamma := \\lim\\limits_{n \\to \\infty} \\sum_{k=1}^{n} \\frac{1}{k} - \\ln n \n\\end{equation}\n\\noindent\nwith integral representation \n\\begin{equation}\n\\gamma = \\int_{0}^{\\infty} e^{-x} \\ln x \\, dx \n\\end{equation}\n\\noindent\ngiven as ${\\mathbf{4.331.1}}$. The \nreader will find in \\cite{irrbook}  a proof of \nthis identity. The second term in (\\ref{initq}) is converted into\n\\begin{equation}\n\\int_{1}^{\\infty} e^{-x} \\ln x \\, dx = \\int_{1}^{\\infty} \\frac{e^{-x}}{x} \\, dx\n\\end{equation}\n\\noindent\nand this last form is identified as $\\text{Ei}(-1)$, where $\\text{Ei}$ is the \nexponential integral defined by\n\\begin{equation}\n\\text{Ei}(z) = - \\int_{-z}^{\\infty} \\frac{e^{-x}}{x} \\, dx. \n\\end{equation}\n\\noindent\nIn the current context, the value of $\\text{Ei}(-1)$ will be simply one of \nthe terms in the initial condition $q_{0}$.\n\nWe determine first an explicit expression for $p_{n}$. The recursion \n(\\ref{recu8}) shows the existence of integers $a_{n}, \\, b_{n}$ such that\n\\begin{equation}\np_{n} = a_{n} + b_{n}e^{-1},\n\\end{equation}\n\\noindent\nwith $a_{0} =1, \\, b_{0} = -1$. From (\\ref{recu8}) we obtain\n\\begin{equation}\na_{n+1} + b_{n+1}e^{-1} = (n+1)a_{n} + \\left[ (n+1)b_{n} - 1 \\right] e^{-1}.\n\\end{equation}\n\\noindent\nThe irrationality of $e$ produce the system \n\\begin{eqnarray}\na_{n+1} & = & (n+1)a_{n}, \\text{ with } a_{0}=1, \\label{system1} \\\\\nb_{n+1} & = & (n+1)b_{n}-1, \\text{ with } b_{0}=-1. \\label{system2}\n\\end{eqnarray}\n\\noindent\nThe expression $a_{n} = n!$ follows directly from (\\ref{system1}). To solve \n(\\ref{system2}), define $B_{n} := b_{n}\/n!$ and observe that\n\\begin{equation}\nB_{n+1} = B_{n} - \\frac{1}{(n+1)!}, \n\\end{equation}\n\\noindent\nthat telescopes to \n\\begin{equation}\nb_{n} = -n! \\sum_{k=0}^{n} \\frac{1}{k!}. \n\\end{equation}\n\n\\noindent\nWe have shown:\n\n\\begin{proposition}\nThe integral $p_{n}$ in (\\ref{p-def}) is given by\n\\begin{equation}\np_{n} = \\int_{0}^{1} x^{n} e^{-x} \\, dx = \n\\frac{n!}{e}  \\left( e - \\sum_{k=0}^{n} \\frac{1}{k!} \\right).\n\\end{equation}\n\\end{proposition}\n\nWe now determine a similar closed-form for $q_{n}$.  The recursion \n(\\ref{recu8a}) shows the existence of integers $c_{n}, \\, d_{n}, \\, f_{n}$ \nsuch that\n\\begin{equation}\nq_{n} = c_{n} + d_{n}e^{-1} + f_{n} q_{0}. \n\\end{equation}\n\\noindent\nIn order to produce a system similar to (\\ref{system1},\\ref{system2}) we will \nassume that the constants $1, \\, e^{-1}$ and $q_{0} = -(\\gamma + \n\\text{Ei}(-1))$ are linearly independent over $\\mathbb{Q}$.  Under this\nassumption (\\ref{recu8a}) produces\n\\begin{eqnarray}\nc_{n+1} & = & (n+1)c_{n} + n!, \\label{system3} \\\\\nd_{n+1} & = & (n+1)c_{n}  - n! \\sum_{k=0}^{n} \\frac{1}{k!},  \\label{system4} \\\\\nf_{n+1} & = & (n+1)f_{n}, \\label{system5} \n\\end{eqnarray}\n\\noindent\nwith the initial conditions $c_{0} = 0, \\, d_{0} = 0$ and $f_{0} = 1$. \n\nThe expression $f_{n} = n!$ follows directly from (\\ref{system5}).  To solve \n(\\ref{system3}) and (\\ref{system4}) we employ the following result \nestablished in \\cite{moll-gr5}. \n\n\\begin{lemma}\n\\label{lemrec}\nLet $a_{n}, \\, b_{n}$ and $r_{n}$ be sequences with $a_{n}, \\, \nb_{n} \\neq 0$. Assume that\n$z_{n}$ satisfies \n\\begin{equation}\na_{n}z_{n} = b_{n} z_{n-1} + r_{n}, \\, n \\geq  1\n\\label{recu0}\n\\end{equation}\n\\noindent\nwith initial condition $z_{0}$. Then \n\\begin{equation}\n\\label{valuex}\nz_{n} = \\frac{b_{1}b_{2} \\cdots b_{n}}{a_{1}a_{2} \\cdots a_{n}} \n\\left( z_{0} + \\sum_{k=1}^{n} \\frac{a_{1}a_{2} \\cdots a_{k-1}}\n{b_{1} b_{2} \\cdots b_{k}} r_{k} \\right). \n\\end{equation}\n\\end{lemma}\n\nWe conclude that\n\\begin{equation}\nc_{n} = n! \\sum_{k=1}^{n} \\frac{1}{k},\n\\end{equation}\n\\noindent\nand \n\\begin{equation}\nd_{n} = -n! \\sum_{k=1}^{n} \\frac{1}{k} \\sum_{j=0}^{k-1} \\frac{1}{j!}. \n\\end{equation}\n\\noindent\nThe expression for $c_{n}$ shows that they coincide with the  Stirling numbers\nof the first kind: $c_{n} = | s(n+1,2)|$. \\\\\n\nWe have established \n\n\\begin{proposition}\nThe integral $q_{n}$ in (\\ref{qn-def}) is given by\n\\begin{equation}\nq_{n} = \\int_{0}^{1} x^{n} e^{-x} \\ln x \\, dx = \nn! \\left[ \\frac{1}{e} \\sum_{k=1}^{n} \\frac{1}{k} \n\\left( e - \\sum_{j=0}^{k-1} \\frac{1}{j!}\n\\right) + q_{0} \\right].\n\\end{equation}\n\\end{proposition}\n\n\\begin{example}\nThe expressions for $p_{n}$ and $q_{n}$ provide the evaluation of \n$\\mathbf{4.351.1}$ in \\cite{gr}\n\\begin{equation}\n\\int_{0}^{1} (1-x)e^{-x} \\ln x \\, dx = \\frac{1-e}{e},\n\\end{equation}\n\\noindent\nby identifying the integral as $q_{0}-q_{1}$. The recurrence (\\ref{recu8a}) \nshows that \n\\begin{equation}\nq_{0}-q_{1} = -p_{0} = e^{-1}-1,\n\\end{equation}\n\\noindent\nas claimed. \n\\end{example}\n\n\\begin{example}\nThe evaluation of\n$\\mathbf{4.362.1}$ in \\cite{gr} \n\\begin{equation}\n\\int_{0}^{1} xe^{x} \\ln(1-x) \\, dx = \n\\int_{0}^{1} (1-t)e^{1-t} \\ln t \\, dt\n\\end{equation}\n\\noindent\nis achieved by observing that this integral is $e( q_{0} - q_{1}) \n= 1-e$. \n\\end{example}\n \n\\section{A parametric family} \\label{sec-param} \n\\setcounter{equation}{0}\n\nIn this section we consider the evaluation of \n\\begin{eqnarray}\nP_{n}(a) & := & \\int_{0}^{1} x^{n} e^{-ax} \\, dx \\label{generalp} \\\\\nQ_{n}(a) & := & \\int_{0}^{1} x^{n} e^{-ax} \\ln x \\, dx. \\label{generalq}\n\\end{eqnarray}\n\\noindent\nThe integrals $q_{n}$ considered in Section \\ref{new} corresponds to the \nspecial case: $q_{n} = Q_{n}(1)$. \n\nWe now establish a recursion for $Q_{n}$ by differentiating \n(\\ref{generalq}).\n\n\\begin{lemma}\nThe integral $Q_{n}(a)$ satisfies the relation\n\\begin{equation}\nQ_{n+1}(a) = -\\frac{d}{da} Q_{n}(a).\n\\end{equation}\n\\end{lemma}\n\nTo obtain a closed-form expression for $Q_{n}(a)$ we need to determine the \ninitial condition \n\\begin{equation}\nQ_{0}(a) = \\int_{0}^{1} e^{-ax} \\ln x \\, dx. \n\\end{equation}\n\\noindent\nThis is expressed in terms of the {\\em incomplete gamma function}\ndefined in $\\mathbf{8.350.1}$ by\n\\begin{equation}\n\\Gamma(a,x) := \\int_{x}^{\\infty} t^{a-1} e^{-t} \\, dt.\n\\end{equation}\n\\noindent\nObserve that $\\Gamma(a,0) = \\Gamma(a)$, the usual gamma function. \n\n\\begin{lemma}\nThe initial condition $Q_{0}(a)$ is given by\n\\begin{equation}\nQ_{0}(a) = \\int_{0}^{1} e^{-ax} \\ln x \\, dx = -\\frac{1}{a} \n\\left( \\gamma + \\ln a + \\Gamma(0,a) \\right).\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nThe change of variables $t = ax$ yields\n\\begin{equation}\nQ_{0}(a) = \\frac{1}{a} \\int_{0}^{a} e^{-t} \\ln t \\, dt - \\frac{\\ln a}{a} \n\\left( 1 - e^{-a} \\right). \n\\end{equation}\n\\noindent\nThen \n\\begin{equation}\n\\int_{0}^{a} e^{-t} \\ln t \\, dt = \n\\int_{0}^{\\infty} e^{-t} \\ln t \\, dt  - \n\\int_{a}^{\\infty} e^{-t} \\ln t \\, dt. \n\\end{equation}\n\\noindent\nThe first integral is \n\\begin{equation}\n\\int_{0}^{\\infty} e^{-t} \\ln t \\, dt   = - \\gamma,\n\\end{equation}\n\\noindent\nthat simply reflects the fact that $\\gamma  = - \\Gamma'(1)$. \nIntegrating by parts yields\n\\begin{equation}\n\\int_{a}^{\\infty}  e^{-t} \\ln t \\, dt = e^{-a} \\ln a + \\Gamma(0,a). \n\\end{equation}\n\\noindent\nThe formula for $Q_{0}(a)$ is established.\n\\end{proof}\n\nWe now determine a closed-form expression for $P_{n}(a)$ and $Q_{n}(a)$ \nfollowing the procedure employed in Section \\ref{new}. \n\n\\begin{lemma}\nThe integrals $P_{n}$ and $Q_{n}(a)$ satisfy the recursion\n\\begin{eqnarray}\nP_{n+1}(a) = \\frac{1}{a} \\left( (n+1)P_{n}(a) - e^{-a} \\right) \\label{recP} \\\\\nQ_{n+1}(a) = \\frac{1}{a} \\left( (n+1)Q_{n}(a) + P_{n}(a) \\right). \\label{recQ}\n\\end{eqnarray}\n\\noindent\nThe initial conditions are given by \n\\begin{equation}\nP_{0}(a) = \\frac{1}{a}( 1 - e^{-a}), \\text{ and } \nQ_{0}(a) = - \\frac{1}{a}( \\gamma + \\Gamma(0,a) + \\ln a ). \n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nIntegrate by parts.\n\\end{proof}\n\nWe conclude that we can write \n\\begin{equation}\nP_{n}(a) = A_{n}(a) - B_{n}(a) e^{-a},\n\\end{equation}\n\\noindent\nand \n\\begin{equation}\nQ_{n}(a) = C_{n}(a) - D_{n}(a)e^{-a} - E_{n}(a) \n( \\gamma + \\Gamma(0,a) + \\ln a ). \n\\end{equation}\n\n\\begin{lemma}\nThe recursions (\\ref{recP}) and (\\ref{recQ}) imply that\n\\begin{eqnarray}\nA_{n+1}(a) & = & \\frac{1}{a} (n+1) A_{n}(a), \\label{recA1} \\\\\nB_{n+1}(a) & = & \\frac{1}{a} \\left[ (n+1) B_{n}(a) + 1 \\right], \\nonumber  \\\\\nC_{n+1}(a) & = & \\frac{1}{a} \\left[ (n+1) C_{n}(a) + A_{n}(a)  \\right], \\nonumber  \\\\\nD_{n+1}(a) & = & \\frac{1}{a} \\left[ (n+1) D_{n}(a) + B_{n}(a)  \\right], \\nonumber  \\\\\nE_{n+1}(a) & = & \\frac{1}{a} (n+1) E_{n}(a) \\nonumber \n\\end{eqnarray}\n\\noindent\nwith initial conditions \n\\begin{equation}\nA_{0}(a) = B_{0}(a) = E_{0}(a) = \\frac{1}{a} \\text{ and } \nC_{0}(a) = D_{0}(a) = 0. \n\\end{equation}\n\\end{lemma}\n\nThese recursion can now be solved as in Section \\ref{new} to produce a \nclosed-form expression for the integrals $P_{n}(a)$ and $Q_{n}(a)$. We \nemploy the notation \n\\begin{equation}\nH_{n} = \\sum_{k=1}^{n} \\frac{1}{k}\n\\end{equation}\n\\noindent\nfor the harmonic numbers and \n\\begin{equation}\n\\text{Exp}_{n}(x) = \\sum_{k=0}^{n} \\frac{x^{k}}{k!}\n\\end{equation}\n\\noindent\nfor the partial sums of the exponential function. \n\n\\begin{theorem}\n\\label{mainPQ}\nLet $a \\in \\mathbb{R}$ and $n \\in \\mathbb{N}$. Then \n\\begin{equation}\nP_{n}(a) := \\int_{0}^{1}x^{n}e^{-ax} \\, dx = \n\\frac{n!}{a^{n+1}} \\left[ 1 - e^{-a} \\text{Exp}_{n}(a) \\right],\n\\label{formulaP}\n\\end{equation}\n\\noindent\nand\n\\begin{equation}\nQ_{n}(a) := \\int_{0}^{1}x^{n}e^{-ax} \\ln x \\, dx = \n\\frac{n!}{a^{n+1}} \\left[  H_{n} - G(a) - e^{-a} \\sum_{k=1}^{n} \n\\frac{1}{k} \\text{Exp}_{k-1}(a) \\right],\n\\label{formulaQ}\n\\nonumber \n\\end{equation}\n\\noindent\nwhere  $G(a) = -aQ_{0}(a) = \\gamma + \\Gamma(0,a) + \\ln a$. \n\\end{theorem}\n\nThese expressions provide the evaluations of two integrals in \\cite{gr}. \n\n\\begin{example}\nFormula ${\\mathbf{4.351.2}}$ states that\n\\begin{equation}\n\\int_{0}^{1} e^{-ax} (-ax^{2} + 2x) \\ln x \\, dx = \n\\frac{1}{a^{2}} \\left[ -1 + (1+a)e^{-a} \\right]. \n\\end{equation}\n\\noindent\nIn order to verify this, observe that the stated integral is \n\\begin{equation}\n-a \\int_{0}^{1} x^{2} e^{-ax}  \\ln x \\, dx + \n2 \\int_{0}^{1} x e^{-ax}  \\ln x \\, dx  = -a Q_{2}(a) + 2 Q_{1}(a). \n\\end{equation}\n\\noindent\nThe expressions in Theorem \\ref{mainPQ} now complete the evaluation.  \n\\end{example}\n\n\\begin{example}\nFormula ${\\mathbf{4.353.3}}$ in \\cite{gr} gives the value of \n\\begin{equation}\nI_{n}(a) := \\int_{0}^{1} (-ax+ n+1) x^{n} e^{-ax} \\ln x \\, dx. \n\\end{equation}\n\\noindent\nObserve that \n\\begin{equation}\nI_{n}(a) = -aQ_{n+1}(a) + (n+1)Q_{n}(a),\n\\end{equation}\n\\noindent\nand using the recursion (\\ref{recQ}) we conclude that $I_{n}(a) = \n-P_{n}(a)$. The expression in Theorem \\ref{mainPQ} is precisely what \nappears in \\cite{gr}. \n\\end{example}\n\nWe conclude with the evaluation of a series shown to us by \nTewodros Amdeberhan. Expand the exponential \nterm in (\\ref{formulaQ}) and integrate term by term to obtain\n\\begin{equation}\n\\sum_{k=0}^{\\infty} \\frac{(-a)^{k}}{k! \\, (n+1+k)^{2}} = \n\\frac{n!}{a^{n+1}} \\left( -\\psi(n+1) + \\ln a + \\Gamma(0,a) +\ne^{-a} \\sum_{k=0}^{n} \\frac{1}{k} \\text{Exp}_{k-1}(a) \\right). \n\\label{series1}\n\\end{equation}\n\\noindent\nHere \n\\begin{equation}\n\\psi(x) = \\frac{\\Gamma'(x)}{\\Gamma(x)}\n\\end{equation}\n\\noindent\nis the {\\em digamma} function defined in $\\mathbf{8.360.1}$ of \\cite{gr}. \nthe identity \n\\begin{equation}\n\\psi(n+1) = H_{n} - \\gamma,\n\\end{equation}\n\\noindent\nthat is a direct consequence of the functional equation $\\Gamma(x+1) = \nx \\Gamma(x)$ and $\\Gamma'(1)= - \\gamma$, was used to transform \n(\\ref{series1}). \n\n\\medskip\n\nThe identity (\\ref{series1}) can be used to provide multiple expressions \nfor the incomplete gamma function, such as \n\\begin{equation}\n\\int_{a}^{\\infty} \\frac{e^{-x}}{x} \\, dx = \n\\sum_{k=0}^{\\infty} \\frac{(-1)^{k} a^{n+1+k}}{n! \\, k! \\, (n+1+k)^{2}} + \n\\psi(n+1) - \\ln a - e^{-a} \\sum_{k=1}^{n} \\frac{\\text{Exp}_{k-1}(a)}{k}, \n\\end{equation}\n\\noindent\nand the special case for $n=0$:\n\\begin{equation}\n\\int_{a}^{\\infty} \\frac{e^{-x}}{x} \\, dx = \n-\\gamma - \\ln a + \\sum_{k=0}^{\\infty} \\frac{(-1)^{k} a^{k+1}}\n{  (k+1)! \\, (k+1)}.   \n\\end{equation}\n\n\\medskip\n\nThese issues will be explored in a future publication. \n\n\\vskip 1.0in\n\n\\bigskip\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\\section{Conclusion}\nIn this paper we proposed an algorithm to efficiently compute the full-CP set of SHIM. Through numerical experiments, we demonstrated that SHIM  is statistically superior than the vanilla lasso and comparable to other complex models (NN, RF). The confidence interval along with a point estimation is better than a point estimation alone. The computation of a full-CP set for other complex models (NN, RF) remains an open question. SHIM is interpretable and it generates the decision function as a weighted combination of \\emph{decision sets} (if-then rules). The homotopy-mining based exact full-CP of SHIM is better than the split-CP of SHIM. The efficient pruning strategy makes the computation of exact full-CP set tractable for SHIM. \n\n\\section{Submission of papers to NeurIPS 2022}\n\n\n\n\n\\section{Introduction}\nThe uncertainty in data-driven analysis is a major concern, particularly in risk-sensitive automated decision-making problems (for example, in medical diagnosis and criminal justice). Several strategies exist to quantify the uncertainty of a point estimators. For example, the Bayesian approach can provide a strong confidence bound, but requires the assumption on prior distribution. \nThe PAC analysis is another approach that provides bounds on the probability of error. The conformal prediction (CP) is one such uncertainty quantification method that is very generic, and it is applicable to almost any point estimators \\cite{vovk2005algorithmic,shafer2008tutorial}. The CP method has recently gained significant attention as it can provide valid finite sample statistical coverage guarantee at any nominal level as long as data are independently and identically distributed (i.i.d.). The coverage guarantee provided is valid even when the model is misspecified. In this paper we are interested in the \\emph{full-CP} where the full data is used to compute the CP set. An alternative to this approach is the \\emph{split-CP} in which the data is split into two parts: one part is used for fitting and the remaining part is used to compute the CP set.\nThe essential idea of the full-CP framework can be stated as follows: Given a training set \\(\\mc{D}_n = \\{(x_1, y_1), \\ldots, (x_n, y_n)\\}\\) and a test instance $x_{n+1}$ which are both i.i.d., the goal of CP is to construct a $100(1-\\alpha)\\%$ confidence set that contains the unobserved $y_{n+1}$. Here, $\\alpha \\in [0,1]$ represents the confidence level. In other words, we are interested in all possible completions of the augmented dataset in the form of \\(\\mc{D}_n \\cup (x_{n+1}, \\tau)\\), such that $y_{n+1}=\\tau$ is \\textit{typical} for the known data  \\(\\{\\mc{D}_n, x_{n+1}\\}\\). This typicality is measured by a typicalness function $\\pi(\\cdot)$, which is also called the \\emph{$p$-value} in analogy to the classical hypothesis testing. In other words, the CP set for $x_{n+1}$ is defined as the set of all $\\tau$ for which the null hypothesis $H_0: y_{n+1}=\\tau$ is not rejected against the alternative hypothesis  $H_1: y_{n+1}\\neq\\tau$.\n\n\\paragraph{Related work:} Since its inception, there have been several extensions and application of the CP framework in diverse directions. Examples include the choice of conformity score and statistical efficiency \\cite{lei2013distribution,lei2014distribution}, high-dimensional regression \\cite{lei2018distribution}, classification \\cite{lei2014classification,sadinle2019least}, active learning \\cite{ho2008query}, time series \\cite{xu2021conformal}, few-shot learning \\cite{fisch2021few}, text and speech completion \\cite{dey2021conformal}, image classification \\cite{angelopoulos2020uncertainty}, outlier detection \\cite{laxhammar2015inductive,bates2021testing}. \nRecently, some approximate computation of the conformal set was proposed for regression in \\cite{ndiaye2019computing,ndiaye2021root} and in classification, significant advancement has been made in \\cite{cherubin2021exact} to compute exact full-CP.\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=\\linewidth]{figures\/fig1.pdf}\n  \\caption{(A) An intuitive illustration of a SHIM solution space, which is a hyper rectangle (i.e., generalization of a rectangle for higher dimensions that essentially represents a cartesian product of intervals). The decision sets, the predicted response $(\\hat{y})$ and the associated confidence interval for a randomly chosen example of data from the ProPublica two-year recidivism criminal justice (COMPAS) dataset are shown. (B) A tree of patterns has been constructed by exploiting the hierarchical structure of high-order interaction features (Note: not all patterns appear due to pruning).\n}\n\\label{fig1}\n \\end{figure}\nDespite its attractive properties, the application of exact, full-CP in many practical problems  remains an open problem owing to its high computational cost. By definition, the full-CP framework requires the refitting of model by augmenting the data for every possible candidate $\\tau \\in \\mb{R}$ of the unobserved $y_{n+1}$. In a regression setting, one needs to refit the model an infinite number of times for all possible candidates on the real line $(\\forall \\tau \\in \\mb{R})$, and check the conformity by computing the $p$-values $\\pi(\\tau)$. \n\nHence, an efficient computation of the full-CP set is possible only for a handful of simple models (e.g., ordinary LS regression, ridge regression) in which the solution is explicitly represented as a function of $\\tau$.\nThis enables a closed-form derivation of full-CP set and avoids an exhaustive search over the real line \\cite{nouretdinov2001ridge}. \nRecently, \\citet{lei2019fast} proposed a homotopy method to efficiently compute the full-CP set of the LASSO. The homotopy method exploits the piece-wise linearity of the LASSO solutions as a function of $\\tau$, and avoids the exhaustive search by keeping track of the transition points along the solution path and the use of a linear interpolation between transition points.\nUnfortunately, such a structure does not exist for most of the models, and the application of full-CP is still an open question for complex models such as a random forest (RF), neural network (NN) etc. For those complex models we need to rely on split-CP.\n\nIn CP, although the coverage property is satisfied even when the model is misspecified, it is better to use sufficiently complex models for complex data because this enables us to obtain a more compact confidence set (i.e., shorter confidence intervals). Furthermore, it is important to note that the confidence set obtained by a full-CP is more compact than that obtained by a split-CP because only a part of instances in the available dataset is used for constructing the confidence set in split-CP. This means that it is valuable to construct a full-CP algorithm for sufficiently complex models.\nIn this paper, we considered the \\emph{sparse high-order interaction model (SHIM)} which is complex but interpretable, and proposed a \\emph{homotopy-mining} method to efficiently compute the \\emph{full-CP} set. We call the resulting machine as \\emph{SHIM confidence machine}. SHIM represents the model as a weighted sum of conjunction rules that are highly interpretable as well as accurate decision sets \\cite{das2019interpretable,lakkaraju2016interpretable}. SHIM can capture the combinatorial interactions of multiple factors, which can prove to be beneficial for deciphering complex data. A conjunction rule of a SHIM looks like \\(I(-1.5 \\leq x_1 \\leq 2.3)\\wedge I(x_3 \\geq 20.0)\\wedge I(x_5 \\leq 7.5) \\), where $I(\\cdot)$ refers to the indicator function. An intuitive illustration of a SHIM is shown in Fig.~\\ref{fig1}A. \n\n\\textbf{Contribution:}\nOur main contribution in this paper is to develop an efficient algorithm to conduct \\emph{exact full-CP} for SHIM which can capture complex structures in the data by considering high-order interaction features.\nThe exact full-CP for complex black box models such as RF and NN are intractable and an efficient computational method does not exist. For such black box models only split-CP is possible.\nThe proposed method enables us to obtain a more compact confidence set (much better than linear and comparable to non linear models) by the full-CP for sufficiently flexible and complex SHIM.\nBecause SHIM uses higher-order interaction features, the full-CP is computationally challenging, but we overcome this difficulty by introducing a method called \\emph{homotopy mining} which exploits the best of both homotopy and (pattern) mining methods. The computation of exact full-CP for SHIM by homotopy mining can be interpreted as an extension of LASSO's exact full-CP in \\cite{lei2019fast}.\nThe use of black box or explainable machine learning models in high-stakes decision-making such as in healthcare, criminal justice, and other domains is highly criticized in the literature (Rudin, C. 2019). We believe that the exact full-CP of a SHIM has significant importance in practice where accuracy, statistical reliability, and interpretability are important. If a practitioner chooses SHIM as a regressor, how to compute the corresponding exact full-CP set?. The proposed method provides a computationally efficient solution to this. \\looseness=-1\n\n\\paragraph{Notation:}\nIn our notation, $y\\in \\mathbb{R}^n$ represents the response vector of $n$ instances, $y(\\tau) \\in \\mathbb{R}^{n+1}$ is the augmented response vector constructed by augmenting the possible response value ($\\tau \\in \\mb{R}$) of the $(n+1)^{th}$ instance with $y$. Later we defined $y(\\tau)$ as a function of the variable $\\tau$ as the vector $y(\\tau)$ changes for different possible response value ($\\tau$) of the $(n+1)^{th}$ instance. The $y_i\\in \\mathbb{R}, \\forall i \\in [n+1]$, represents the scalar response of the $i^{th}$ instance, $X \\in \\mb{R}^{(n+1) \\times p}$ is the design matrix of all $(n+1)$ instances, each having $p$ features, $x_\\ell \\in \\mb{R}^{n+1}$ is the column vector corresponding to some $\\ell \\in [p]$, where $[p] = \\{1, \\ldots, p\\}$, $X_{\\mc{A}_\\tau} \\in \\mb{R}^{(n+1) \\times |\\mc{A}_\\tau|} \\subseteq X$ is a smaller design matrix in which the columns are restricted to the elements of some subset $\\mc{A}_\\tau \\subseteq [p]$.  \\looseness=-1\n\n\\section{Problem Statement}\nConsider a regression problem with a response vector $y \\in \\mb{R}^n$ and \n$m$ original covariate vectors $z_1, \\ldots, z_m$, where \\(z_{\\ell} \\in \\mb{R}^n\\) and $\\ell \\in [m] = \\{1, ..., m\\}$. A high-order interaction model up to the $d^{\\rm th}$ order is then written as follows:\n\\begin{equation}\\label{eq:shim_model}\n\\begin{split}\n\\hspace{-2.5mm} y = \n\\sum_{\\ell_1 \\in [m]} \\theta_{\\ell_1} z_{\\ell_1} \n+ \\sum_{ \\substack{ (\\ell_1, \\ell_2)  \\in [m] \\times [m]  \\\\ \\ell_1 \\neq \\ell_2}} \\hspace{-2.5mm} \\theta_{\\ell_1, \\ell_2} z_{\\ell_1} z_{\\ell_2}\n+  \\cdots\n+ \\sum_{\\substack{(\\ell_1, ..., \\ell_d)  \\in [m]^d \\\\ \\ell_1 \\neq ... \\neq \\ell_d}} \\hspace{-2.5mm} \\theta_{\\ell_1, \\ldots, \\ell_d} z_{\\ell_1} \\cdots  z_{\\ell_d},\n\\end{split}\n\\end{equation}\nwhere $z_{\\ell_1} \\cdots  z_{\\ell_d}$ is the element-wise product and scalar $\\theta$ represents the coefficient.\nIn this study, we consider each element of the original covariate vector $z_{\\ell} \\in \\{0, 1\\}^n$ for $\\ell \\in [m]$. However, our model is equally applicable to covariate vectors defined in the domain $[0, 1]^n$.\nTo simplify the notation, it is convenient to write the high-order interaction model in (\\ref{eq:shim_model}) using the following matrix of concatenated vectors of all high-order interactions:\n\\[\nX=[\\underbrace{z_1, \\ldots, z_m}_{1\\text{\\textsuperscript{st} order}}, \n\\cdots, \\underbrace{z_1 \\ldots z_d, \\ldots, z_{m-d+1} \\ldots z_m}_{d\\text{\\textsuperscript{th} order}}] \\in \\mb{R}^{n \\times p},\n\\]\nwhere \\(p := \\sum_{\\kappa=1}^d{m \\choose \\kappa}\\).\nSimilarly, the coefficient vector associated with all possible high-order interaction terms can be written as follows:\n\\[\n\\beta:= [ \\underbrace{ \\theta_1, \\ldots, \\theta_m}_{1\\text{\\textsuperscript{st} order}},\n\\cdots, \\underbrace{\\theta_{1, \\ldots ,d},  \\ldots, \\theta_{m-d+1, \\ldots, m}}_{d\\text{\\textsuperscript{th} order}}\n]^\\top \\in \\mb{R}^p.\n\\]\nThe high-order interaction model (\\ref{eq:shim_model}) is then simply written as a linear model\n$\ny = X \\beta.\n$\nUnfortunately, $p$ can be prohibitively large unless both $m$ and $d$ are fairly small.\nIn SHIM, we consider a sparse estimation of a high-order interaction model.\nAn example of SHIM is as follows:\n\\begin{equation}\\label{eq:shim_eg}\n y = \\theta_3z_{3} +  \\theta_5z_{5} +  \\theta_{2,6}z_{2}z_{6} +  \\theta_{1,2,5,9}z_{1}z_{2}z_{5}z_{9}.\n\\end{equation}\n\nBefore delving into our proposed method, we briefly overview the conformal prediction framework.\n\\subsection{Conformal prediction}\nA mere point estimation is insufficient for risk-sensitive automated decision-making problems \\cite{rudin2019stop,angelino2018learning,das2019interpretable}, such as in medical diagnosis and criminal justice. In such high-stake decision-making problems, if the estimators are equipped with the associated confidence information, the decision maker will be more informed and sufficiently confident to make a prudent decision when the stakes are high.   \n\\paragraph{Full-CP:}\nGiven a labelled dataset ${\\mc{D}}_n$ and a new observation $x_{n+1}$, the goal of full-CP framework is to construct a set of likely values $\\mc{C}(x_{n+1})$ of unobserved $y_{n+1}$ with a valid statistical coverage guarantee \\cite{vovk2005algorithmic,shafer2008tutorial}, {\\em i.e.~},\n\\begin{equation}\\label{eqn:CP_guarantee}\n    \\mathbb{P}(y_{n+1} \\in \\mc{C}(x_{n+1})) \\geq 1 - \\alpha \\enspace,\n\\end{equation}\nwhere \\(\\alpha \\in [0, 1]\\) determines the level of confidence. \nIf we define a prediction function  $\\mu(\\cdot)$ that maps the input $X$ to the output $y$, then the essential idea of constructing a full-CP set is to fit a model $\\mu_{\\tau}(\\cdot)$ with the augmented data $\\mc{D}_n \\cup (x_{n+1},\\tau)$ for every possible candidate $\\tau \\in \\mb{R}$ and compare the prediction error of each instances.\nMore precisely, let \\( y(\\tau)  = [y_1, \\ldots, y_n, \\tau]^\\top \\in \\mb{R}^{n+1} \\) be a vector augmented with $\\tau$. We define a score function that measures how well the model can predict each output variables; with the constraint that it should not depend on the order of the data instances \\cite{lei2019fast}. One such conformity score function generally used in linear regression setting is the (coordinate-wise) absolute residual  i.e.\n $$S(\\tau) = | y(\\tau) - \\mu_{\\scaleto{\\tau\\mathstrut}{6pt}}(X)| \\enspace,$$\n where we stack the input vectors in the design matrix $X = [x_1, \\ldots, x_{n+1}]^\\top$ in $\\mb{R}^{(n+1) \\times p}$.\nThe conformity function can now be defined as\n$$ \\pi (\\tau) = 1 - \\frac{1}{n+1}\\sum_{i=1}^{n+1} \\mathbbm{1}_{S_i(\\tau) \\leq S_{n+1}(\\tau)},$$\nwhich evaluates how the prediction of the candidate $\\tau$ using the fitted model $\\mu_{\\tau}(x_{n+1})$ is ranked compared to the prediction of the previously observed data $y_{i}$ using $\\mu_{\\tau}(x_{i})$. The conformal set is defined as:\n\\begin{equation}\\label{eqn:CP2}\n    \\mc{C}(x_{n+1}) = \\{\\tau:  \\pi (\\tau) \\geq \\alpha\\}\\enspace,\n\\end{equation}\nwhich is merely the collection of candidates whose conformity is large enough. \\cite{vovk2005algorithmic} showed that the defined conformal set $\\mc{C}(x_{n+1})$ satisfies the coverage guarantee \\ref{eqn:CP_guarantee} as long as the data are i.i.d.\n\\paragraph{Split-CP:} In split-CP \\cite{papadopoulos2002inductive}, the data set \\(\\mc{D}_n\\) is split into two parts, defined as the training set \\(\\mc{D}_{\\text{tr}}=\\{(x_1, y_1), \\ldots, (x_m, y_m)\\}\\) and the calibration set \\(\\mc{D}_{\\text{cal}}=\\{(x_{m+1}, y_{m+1}), \\ldots, (x_n, y_n)\\}\\), with \\(m < n\\). Then the model ($\\mu^{\\text{tr}}(\\cdot)$) is fit with the training set $\\mc{D}_{\\text{tr}}$ only once, and the $p$-values, denoted as $\\pi_{\\text{split}}(\\cdot)$, are determined using the calibration set $\\mc{D}_{\\text{cal}}$ as follows.\n\\begin{equation}\\label{eqn:pi_z_split}\n    \\pi_{\\text{split}} (\\tau) = 1 - \\frac{1}{n-m}\\sum_{i=m+1}^{n} \\mathbbm{1}_{S^{\\text{cal}}_i(\\tau) \\leq S_{n+1}(\\tau)},\n\\end{equation}\nwhere \\(S^{\\text{cal}}_i(\\tau) = |y_i - \\mu^{\\text{tr}}(x_i)|, \\forall i \\in [m+1, n]\\).\nTherefore, the split-CP can be defined as follows:\n\\begin{align}\\label{eqn:CP_split}\n    \\mc{C}_{\\text{split}}(x_{n+1}) &= \\{\\tau:  \\pi_{\\text{split}} (\\tau) \\geq \\alpha\\}\n    =\\{\\tau:  S^{\\text{cal}}_{n+1}(\\tau) \\leq Q^{\\text{cal}}_{1-\\alpha}\\},\n\\end{align}\nwhere \\(Q^{\\text{cal}}_{1-\\alpha}\\) is the \\((1-\\alpha)\\) quantile of the calibration scores \\(S^{\\text{cal}}_i, \\forall i \\in [m+1, n]\\). When the conformity score is defined as the absolute residual, then the split-CP set can be conveniently written as \\(\\mc{C}_{\\text{split}}(x_{n+1}) = [\\mu^{\\text{tr}}(x_{n+1}) \\pm Q^{\\text{cal}}_{1-\\alpha}] \\).\n\nAlthough split-CP enjoys the computational efficiency of single model fitting, it suffers from the poor statistical efficiency owing to the smaller sample size both in the model fitting and calibration phases. \n\n\\paragraph{Cross-conformal Predictors} introduced in \\cite{Vovk15} follow a similar trade-off as bias reduction versus variance reduction while trying to exploit the entire dataset for calibration and a significant proportions for training the model. The dataset is partitioned into $K$ folds and one performs a split conformal set by sequentially defining the $k$th fold as the calibration set and the remaining as the training set for $k \\in \\{1, \\ldots, K\\}$. The main difficulty lies in aggregating the different $p$-values while maintaining the validity of the method see \\cite{Carlsson_Eklund_Norinder14, Linusson_Norinder_Bostrom_Johansson_Lofstrom17}. In general, it can be shown that the confidence level is inflated by a factor of $2$ {\\em i.e.~} the (not improvable) theoretical coverage level is $1 - 2\\alpha$ instead of $1 - \\alpha$, see \\cite{Barber_Candes_Ramdas_Tibshirani21}. Nevertheless, the practical performance is fairly acceptable both computationally and statistically. \n\nOn the other hand, to compute the exact full-CP set for a regression problem where \\(y \\in \\mb{R}\\), one needs to refit the regression model $\\mu_\\tau(\\cdot)$ and evaluate $\\pi(\\tau)$ for an infinite number of candidates $\\tau$. Hence, efficient computation of the exact full-CP set is possible only for a handful of predictors (e.g., ridge regression) where it is possible to have a closed form expression of $\\mu_{\\tau}(\\cdot)$. \nHowever, such structure does not exist for most of the machine learning models and computing the exact full-CP set remains an open question for many models. Recently, \\citet{lei2019fast} proposed a homotopy method to efficiently compute the exact full-CP set of the LASSO which was not trivial. \\citet{lei2019fast} considered the response $y$ as a function of a scalar and derived a homotopy method to construct the exact LASSO solution path as function of that scalar. After constructing the exact path, \\citet{lei2019fast} used the solutions at the transition points of the path to efficiently compute the exact full-CP set of the LASSO. In this paper we extend that framework for the SHIM. \n\n\n\n\n\n\\section{Proposed Method}\nWe propose a \\emph{homotopy-mining} method to compute the exact full-CP set of a SHIM. \nThe homotopy method refers to an optimization framework for solving a sequence of parameterized optimization problems.  \nThe basic idea of our method is to consider the following optimization problem: \n\\begin{equation}\\label{obj:primal_lasso}\n    \\beta(\\tau) \\in \\mathop{\\mathrm{arg\\,min}}_{\\beta \\in \\mb{R}^p} \\frac{1}{2}\\norm{y(\\tau) - X\\beta}_2^2  + \\lambda \\norm{\\beta}_1 \\enspace,\n\\end{equation}\nAt optima we can write \\(X^\\top \\big(X\\beta(\\tau) - y(\\tau)\\big) + \\lambda s(\\tau) = 0,\\)\n\\begin{equation*}\\label{eq:opt_condn_shim_homotopy}\n    \\text{ where} \\hspace{0.2cm}  s_{\\ell}(\\tau) \\in \\begin{cases}\n        \\{ -1, +1 \\} \\hspace{0.2cm} \\hspace{0.2cm}    \\text{if} \\hspace{0.2cm} \\beta_\\ell(\\tau) \\neq 0 \\enspace,\\\\\n        [-1, +1]  \\hspace{0.2cm} \\hspace{0.2cm} \\hspace{0.2cm} \\text{if} \\hspace{0.2cm}   \\beta_\\ell(\\tau) = 0 \\enspace,\n\\end{cases}\n\\end{equation*}\nand \\(\\ell \\in [p]\\). Let us define the active set\n\\(\\label{eq:active_set}\n\\mc{A}_\\tau = \\left\\{\\ell \\in [p]:\\; \\big\\lvert x_{\\ell}^{\\top} w(\\tau) \\big\\rvert = \\lambda \\right\\}\n\\), where \\(w(\\tau)  = y(\\tau) - X\\beta(\\tau)\\) is the residual~\\citep{efron2004least}.\nIn \\citet{lei2019fast}, the author showed that for a fixed $\\lambda$, the exact solution path of LASSO with a variable response $y(\\tau)$, characterized by $\\tau$ can be shown to be piece-wise linear as stated in Proposition \\ref{prop:homotopy}.\n\\begin{proposition}\\label{prop:homotopy}\nIf $\\tau_1$ and $\\tau_2$ are any two points, where \\(\\tau_2 = \\tau_1 + \\Delta, \\Delta>0\\) such that the active set $\\mc{A}_\\tau$ and the sign of the coefficients $\\beta_\\ell(\\tau), \\forall \\ell \\in [p]$ remain the same \\(\\forall \\tau \\in [\\tau_1, \\tau_2)\\) and \\(X^{\\top}_{\\mc{A}_{\\tau}}X_{\\mc{A}_{\\tau}}\\) is invertible \\(\\forall \\tau \\in [\\tau_1, \\tau_2)\\), then one can show that\n\\begin{align*}\n&\\beta_{\\mc{A}_{\\tau}} (\\tau_2) - \\beta_{\\mc{A}_{\\tau}} (\\tau_1) = \\nu_{\\mc{A}_{\\tau}} (\\tau) \\times \\Delta, \n&s_{\\mc{A}^c_{\\tau}} (\\tau_2) - s_{\\mc{A}^c_{\\tau}} (\\tau_1) &= \\gamma_{\\mc{A}^c_{\\tau}} (\\tau) \\times  \\Delta \/ \\lambda,\n\\end{align*}\n\\begin{align}\n\\label{direction_vec}\n   \\text{where} \\hspace{0.5cm} &\\nu_{\\mc{A}_{\\tau}}(\\tau) = (X^{\\top}_{\\mc{A}_{\\tau}}X_{\\mc{A}_{\\tau}})^{-1}x_{n+1,\\mc{A}_{\\tau}},\n  &\\gamma_{\\mc{A}^c_{\\tau}} (\\tau) = x_{n+1, \\mc{A}_{\\tau}^c} - (X^{\\top}_{\\mc{A}_{\\tau}^c}X_{\\mc{A}_{\\tau}})\\nu_{\\mc{A}_{\\tau}}(\\tau) \n \n\\end{align}\nare the direction vectors. Here, \\(\\nu_{\\mc{A}_{\\tau}}(\\tau)\\)  and \\(\\gamma_{\\mc{A}^c_{\\tau}} (\\tau)\\) remain constant \\(\\forall \\tau \\in [\\tau_1, \\tau_2)\\), and hence, \\(\\beta(\\tau)\\) is linear \\(\\forall \\tau \\in [\\tau_1, \\tau_2)\\). Now, for any $t, t+1 \\in \\{1, \\ldots, T\\}$, where $T$ is a finite number, if $\\tau_{t+1} > \\tau_t$ is the next zero crossing point and there is only one feature $\\ell \\in [p]$ that enters or leaves the active set $\\mc{A}_{\\tau_t}$ at $\\tau=\\tau_{t+1}$, then \\citet{lei2019fast} showed that \\(\\tau \\mapsto \\beta(\\tau)\\) is well defined and piece-wise linear in $\\tau$.\n\n\\end{proposition}\n  We call the mapping \\(\\tau \\mapsto \\beta(\\tau)\\) the \\emph{$\\tau$-path} and the number of linear pieces of this $\\tau$-path is finite (see complexity analysis in the appendix). This $\\tau$-path can be computed exactly using homotopy method,\nwhere the sequences of $\\tau$ represent the breakpoints of the homotopy method  \\citep{efron2004least,mairal2012complexity}, the indices of which are represented by $t \\in \\{1, \\cdots, T\\}$. \nThe basic idea of homotopy method is stated below. At $\\tau_{t+1}$ which is the next zero-crossing point of the $\\tau$-path, either of the following two events occurs \\citep{efron2004least,rosset2007piecewise} \n\\begin{itemize}\n \\item A zero variable becomes non-zero, that is, \\( \\exists \\ell \\in \\mc{A}^c_{\\tau_t} \\text{ s.t. } \\big\\lvert x_{\\ell}^{\\top} w(\\tau_{t+1}) \\big\\rvert = \\lambda \\hspace{0.2cm} \\text{or,}\\)\n\\item A non-zero variable becomes zero, that is,\n\\(\\exists \\ell \\in \\mc{A}_{\\tau_t} \\text{ s.t. } \\beta_\\ell(\\tau_t) \\neq 0 \\text{, but } \\beta_\\ell(\\tau_{t+1}) = 0 \\enspace.\\)\n\\end{itemize}\nOverall, the next change in the active set (or change in direction vector, \\(\\nu_{\\mc{A}_{\\tau_t}}(\\tau_t)\\)  or \\(\\gamma_{\\mc{A}^c_{\\tau_t}} (\\tau_t)\\)) occurs at $\\tau_{t + 1} = \\tau_t + \\Delta_{\\ell^\\ast}$, such that \n\\begin{equation}\\label{eq:step_length}\n\\Delta_{\\ell^\\ast} = \\min(\\Delta_{\\ell^\\ast}^1, \\Delta_{\\ell^\\ast}^2),\n\\end{equation}\n\\begin{align*}\n\\text{where  }\n&\\Delta_{\\ell^\\ast}^1 = \\min_{\\ell \\in \\mc{A}_{\\tau_t}} \\left( - \\frac{\\beta_\\ell(\\tau_t)}{\\nu_\\ell(\\tau_t)} \\right)_{++},\n&\\Delta_{\\ell^\\ast}^2 = \\min_{\\ell \\in \\mathcal{A}^c_{\\tau_t}}\\left( \\lambda \\frac{ \\text{sign}(\\gamma_\\ell (\\tau_t)) - x_{\\ell}^{\\top}w(\\tau_{t})}{\\gamma_\\ell(\\tau_t)} \\right)_{++},\n\\end{align*}\nwhere, we use the convention that for any $a \\in \\mb{R}, (a)_{++} = a$ if $a > 0$ and $\\infty$ otherwise. However, naively (minimization over all possible interaction terms) determining the step size of inclusion $(\\Delta_{\\ell^\\ast}^2)$ will be intractable for the SHIM type problem. Because in SHIM the search space grows exponentially due to the combinatorial effect of high-order interaction terms. \nTherefore, both the fitting and constructing the full-CP set of a SHIM are non-trivial because unless both $m$ and $d$ are very small, a high-order interaction model will have a significantly large number of parameters to be considered.\nSeveral algorithms for fitting a sparse high-order interaction model have been proposed in the literature \\cite{ERP_Tsuda,saigo2009gboost,nakagawa2016safe}.\nA common approach adopted in these existing works is to exploit the hierarchical structure of high-order interaction features.\nIn other words, a tree structure as in Fig.~\\ref{fig1}B is considered and a branch-and-bound strategy is employed in order to avoid handling all the exponentially increasing number of high-order interaction features. Hence, we need efficient computational methods to make the computation practically feasible. \nIn the following section, we present an efficient tree pruning strategy that considers the tree structure of the interaction terms (or patterns). Here, each node of the tree represents an interaction term. The basic idea of tree pruning is that we construct a tree of interaction terms in a ``progressive manner\". Here, we keep track of the current minimum step size (up to the construction of $\\ell^{th}$ pattern) i.e. \\(\\Delta_{\\ell^\\ast}^2 = \\underset{j \\in \\{1, 2, \\ldots, \\ell\\}}{\\min}\\{\\Delta_j^2\\}, \\) as we construct the tree progressively, and prune a large part of the tree if some bound condition fails. Here, $\\Delta_j^2$ is the bracketed quantity in the expression of $\\Delta_{\\ell^\\ast}^2$ in (\\ref{eq:step_length}) for any non-active feature $j \\in \\mc{A}^c_{\\tau_t}$.\n\n\\subsection{Tree pruning ($\\tau$-path)}\n\\paragraph{Definition of Tree}\nA tree is constructed in such a way that for any pair of nodes (\\(\\ell, \\ell^\\prime)\\), where $\\ell$ is the ancestor of $\\ell^\\prime$, i.e., \\(\\ell \\subset \\ell^\\prime\\), the following conditions are satisfied\n\\begin{align*}\n&x_{i \\ell^\\prime} = 1 \\implies x_{i\\ell } = 1,\n     &x_{i\\ell } = 0 \\implies x_{i \\ell^\\prime} = 0, \\forall i \\in [n].\n\\end{align*}\nThe basic idea of our tree pruning condition is stated below. The equicorrelation condition for any active feature \\(k  \\in \\mathcal{A}_{\\tau}\\) at a fixed $\\lambda$ can be written as\n$\\big\\lvert x_{k}^{\\top} w(\\tau) \\big\\rvert = \\lambda$ (see definition of $\\mc{A}_{\\tau}$ below (\\ref{obj:primal_lasso})).\n\nTherefore, at $\\tau = \\tau_{t + 1}$ any non-active feature \\(\\ell \\in \\mathcal{A}^c_{\\tau_{t} } \\) becomes active  if\n\\begin{equation}\\label{eqn:inclusion_condition2}\n    \\big\\lvert x_{\\ell}^{\\top} w(\\tau_{t+1})\\lvert =\n    \\big\\lvert x_{k}^{\\top} w(\\tau_{t+1}) \\big\\rvert .\n\\end{equation}\nNow, using the triangular inequality one can show that (\\ref{eqn:inclusion_condition2}) will not have any solution if\n\\begin{align}\\label{cond:pruning_tau1}\n\\lvert \\rho_{\\ell}(\\tau_t) \\lvert + \\Delta_{\\ell^{\\ast}}^2 (\\lvert \\eta_{\\ell}(\\tau_t) \\lvert + x_{n+1,\\ell}) < \\lvert \\rho_k (\\tau_t) \\lvert   - \\Delta_{\\ell^{\\ast}}^2(\\lvert\\eta_k(\\tau_t) \\lvert +  x_{n+1,k}),\n\\end{align}\nwhere, \\(\\rho_{\\ell} = x_{\\ell}^{\\top}w(\\tau_t) \\) and \\(\\eta_{\\ell} =x_{\\ell}^{\\top} v(\\tau_t), \\forall \\ell \\in \\mathcal{A}^c_{\\tau_t} \\); \\(\\rho_k = x_{k}^{\\top}w(\\tau_t) \\) and \\(\\eta_k = x_k^{\\top} v(\\tau_t), \\forall k \\in \\mathcal{A}_{\\tau_t}, v(\\tau_t) =  X_{\\mc{A}_{\\tau_t}}\\nu_{\\mc{A}_{\\tau_t}}(\\tau_t)\\).\nTherefore, (\\ref{cond:pruning_tau1}) can be used to derive the pruning condition of the $\\tau$-path which is formally stated in Lemma \\ref{lemma:lemma_1}.\n\nSimilar idea has been used in the context of graph mining  \\cite{ERP_Tsuda} and selective inference of SHIM \\cite{das2021fast}. Note that \\citet{ERP_Tsuda} provided a pruning condition for the exact regularization of graph data ($\\lambda$-path) and \\citet{das2021fast} provided a pruning condition in the context of selective inference to characterize the conditional distribution of the test statistics. However, in our case we adapted the similar idea to compute the exact full-CP set of SHIM. \n\\begin{lemma}\\label{lemma:lemma_1}\nIf $\\Delta_{\\ell^\\ast}^2$ is the current minimum step size, that is, \\(\\Delta_{\\ell^\\ast}^2 = \\underset{j \\in \\{1, 2, \\ldots, \\ell\\}}{\\min}\\{\\Delta_j^2\\}, \\)\\newline then\n$\\forall \\ell^\\prime \\supset \\ell, \\Delta_{\\ell^\\prime}^2 > \\Delta_{\\ell^\\ast}^2$ if \n\\begin{align} \\label{eq:lemma_1_eq}\nb_{\\ell, w(\\tau_t)} + \\Delta_{\\ell^\\ast}^2 (b_{\\ell, v(\\tau_t)} + x_{n+1,\\ell} )\n\t< |\\rho_k(\\tau_t)|  - \\Delta_{\\ell^\\ast}^2 ( |\\eta_k(\\tau_t)| + x_{n+1,k}).\n\\end{align}\n\\end{lemma}\nwhere, \\(b_{\\ell, w(\\tau_t)} :=\\max \\big\\{ \\sum \\limits_{w_{i}(\\tau_t) > 0} |w_{i}(\\tau_t)| x_{i \\ell}, \\sum  \\limits_{w_{i}(\\tau_t) < 0} |w_{i}(\\tau_t)| x_{i \\ell} \\big\\}, \\\\ b_{\\ell, v(\\tau_t)}:= \\max \\big \\{ \\sum \\limits_{v_{i}(\\tau_t) > 0} |v_{i}(\\tau_t)| x_{i \\ell}, \\sum  \\limits_{v_{i}(\\tau_t) < 0} |v_{i}(\\tau_t)| x_{i \\ell} \\big \\} \\).\nThe Lemma \\ref{lemma:lemma_1} essentially states that if the condition in (\\ref{eq:lemma_1_eq}) is satisfied, then one can safely ignore the subtree with $\\ell$ as the root node, thereby dramatically improving the computational efficiency. We extended our method to elastic net and call it El-SHIM. The details of El-SHIM, proof of Lemma \\ref{lemma:lemma_1} and the algorithm to compute the exact full-CP of SHIM are provided in the appendix.\n\n\n\\section{Results and Discussions}\nWe evaluated our proposed method using both synthetic and real-world data. For all experiments, we considered a coverage guarantee of $90\\%$, that is, \\(\\alpha = 0.1\\). We compared the statistical efficiency of our proposed method (SHIM) with those of other simple (LASSO) and complex models (NN, RF).  For the complex models, we reported the split-CP because it is the only available method for computing the CP for complex models. We also demonstrated the statistical efficiency of full-CP in comparison to that of split-CP both for LASSO and SHIM.  \n\\paragraph{Synthetic data}\nWe generated random i.i.d. samples $(z_i, y) \\in \\{0,1\\}^m \\times \\mb{R} $ in such a way that $100m(1-\\zeta)\\%$ features of $z_i \\in \\mb{R}^m$ contain a value of 1 on average.\nHere, $\\zeta \\in [0, 1]$ is the sparsity controlling parameter that controls the sparsity of the design matrix, whereas the sparsity in the model coefficients are controlled by the regularizer $\\lambda$.\nThe pruning effectiveness (Table~\\ref{table:comp_eff_time_taken}) depends on the sparsity of the design matrix as it exploits the tree's anti-monotonicity property. High dimensional real-world data is generally very sparse and the choice of $\\zeta$ in our experiments is just for the demonstration purpose.\nThe response $y_i \\in \\mb{R}$ is randomly generated from a normal distribution $N(\\mu(x_i), \\sigma^2)$. For demonstration purposes, we considered a true model of up to fifth-order interactions, which is defined as $\\mu (x_i) = 2.0z_1 +2.0 z_1z_2 + 2.0z_1z_2z_3 + 2.0z_1z_3z_4z_5 + 2.0z_1z_2z_3z_4z_5$, and set $\\sigma=1$. The choice of this model is merely for the demonstration purposes, and the proposed method is equally applicable to any chosen model. We used the same true model $\\mu(x_i)$ in both low and high dimensional settings (Table~\\ref{table:stat_low} \\& ~\\ref{table:stat_high} and Fig.~\\ref{fig:cpl_synthetic}).\n\\begin{table}[t]\n\\centering\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{ |c|c|c|c|c|c|c|c|c| } \n\\hline\n &mlp & rf &lasso\\_s & lasso\\_f &shim\\_2s &shim\\_2f &shim\\_3s &shim\\_3f\\\\\n \\hline\n \n length&\\begin{tabular}{c} 1.93\\\\(0.21) \\end{tabular}  & \\begin{tabular}{c} 1.84\\\\ (0.17) \\end{tabular}   &\\begin{tabular}{c} 2.32\\\\ (0.23)  \\end{tabular}   &\\begin{tabular}{c} 2.18\\\\  (0.10)\\end{tabular}   &\\begin{tabular}{c} 2.00\\\\  (0.23) \\end{tabular}  &\\begin{tabular}{c} 1.82\\\\ (0.17) \\end{tabular}   &\\begin{tabular}{c} 1.96 \\\\ (0.24) \\end{tabular} &\\begin{tabular}{c} 1.76 \\\\ (0.25)  \\end{tabular}  \\\\\n\\hline\n  %\n  cov&\\begin{tabular}{c} 0.91 \\\\(0.05) \\end{tabular} &\\begin{tabular}{c} 0.90 \\\\(0.04) \\end{tabular} &\\begin{tabular}{c} 0.91 \\\\(0.03) \\end{tabular}  &\\begin{tabular}{c} 0.90 \\\\(0.05) \\end{tabular} &\\begin{tabular}{c}0.89 \\\\(0.03) \\end{tabular} &\\begin{tabular}{c}0.89 \\\\(0.05) \\end{tabular}  &\\begin{tabular}{c}0.89 \\\\(0.04) \\end{tabular} &\\begin{tabular}{c}0.88 \\\\(0.05) \\end{tabular} \\\\\n\\hline\n r2&\\begin{tabular}{c}0.64 \\\\(0.10)\\end{tabular} &\\begin{tabular}{c} 0.67 \\\\(0.09)\\end{tabular} &\\begin{tabular}{c}0.52 \\\\(0.13)\\end{tabular}  &\\begin{tabular}{c}0.54 \\\\(0.11)\\end{tabular} &\\begin{tabular}{c}0.62 \\\\(0.09)\\end{tabular} &\\begin{tabular}{c}0.67 \\\\(0.10)\\end{tabular}  &\\begin{tabular}{c}0.63 \\\\(0.09)\\end{tabular} &\\begin{tabular}{c}0.67 \\\\(0.09) \\end{tabular}\\\\\n %\n \\hline\n   \\end{tabular}}\n\\caption{Comparison of the statistical power of the proposed method (shim) with other simple (lasso) and complex models (mlp, rf) using low dimensional synthetic data ($m=10, n=150$). The bracketed values represent the standard deviations.}\n\\label{table:stat_low}\n\\end{table}\n\\begin{table}[t]\n\\centering\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{ |c|c|c|c|c|c|c|c|c| } \n\\hline\n &mlp & rf &lasso\\_s\n      & lasso\\_f & shim\\_2s &shim\\_2f  &shim\\_3s &shim\\_3f\\\\\n \\hline\nlength&\\begin{tabular}{c}3.39 \\\\(0.55)\\end{tabular} &\\begin{tabular}{c} 1.97 \\\\(0.43)\\end{tabular} &\\begin{tabular}{c} 2.43 \\\\(0.37)\\end{tabular}  &\\begin{tabular}{c} 2.26 \\\\(0.19)\\end{tabular} &\\begin{tabular}{c} 2.21 \\\\(0.39)\\end{tabular} &\\begin{tabular}{c}1.92 \\\\(0.26)\\end{tabular}  &\\begin{tabular}{c}2.25 \\\\(0.41)\\end{tabular} &\\begin{tabular}{c}1.89 \\\\(0.30)\\end{tabular} \\\\\n  \\hline\n  cov&\\begin{tabular}{c}0.89 \\\\(0.06)\\end{tabular} &\\begin{tabular}{c} 0.91 \\\\(0.06)\\end{tabular} &\\begin{tabular}{c} 0.91 \\\\(0.05)\\end{tabular}  &\\begin{tabular}{c} 0.91 \\\\(0.05)\\end{tabular} &\\begin{tabular}{c}0.91 \\\\(0.04)\\end{tabular} &\\begin{tabular}{c}0.91 \\\\(0.04)\\end{tabular}  &\\begin{tabular}{c}0.91 \\\\(0.04)\\end{tabular} &\\begin{tabular}{c}0.90 \\\\(0.04)\\end{tabular} \\\\\n  \\hline\n r2&\\begin{tabular}{c}-0.01 \\\\(0.13)\\end{tabular} &\\begin{tabular}{c} 0.68 \\\\(0.08)\\end{tabular} &\\begin{tabular}{c}0.47 \\\\(0.07)\\end{tabular}  &\\begin{tabular}{c}0.53 \\\\(0.07)\\end{tabular} &\\begin{tabular}{c}0.55 \\\\(0.09)\\end{tabular} &\\begin{tabular}{c}0.65 \\\\(0.09)\\end{tabular}  &\\begin{tabular}{c}0.55 \\\\(0.07)\\end{tabular} &\\begin{tabular}{c}0.66 \\\\(0.08)\\end{tabular} \\\\\n \\hline\n   \\end{tabular}}\n\\caption{Comparison of the statistical power of the proposed method (shim) with other simple (lasso) and complex models (mlp, rf) using high dimensional synthetic data ($m=100, n=150$). The bracketed values represent the standard deviations.}\n\\label{table:stat_high}\n\\end{table}\n\\begin{table}[t]\n\\centering\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{ |c|c|c|c|c|c|c| } \n\\hline\n &mlp & rf &lasso\\_s\n      &lasso\\_f & shim\\_2s &shim\\_2f \\\\\n \\hline\n  length&7.48 (0.18) & 7.38 (0.17) & 7.42 (0.16)  & 7.41 (0.09) & 7.39 (0.17) &7.36 (0.13) \\\\\n  \\hline\n  cov&0.90 (0.01)& 0.90 (0.01) & 0.90 (0.01)  & 0.90 (0.01) &0.90 (0.01) &0.90 (0.01) \\\\\n  \\hline\n r2&0.38 (0.01) & 0.39 (0.01) &0.39 (0.01)  &0.39 (0.01) &0.39 (0.01) &0.39 (0.01) \\\\\n \\hline\n   \\end{tabular}}\n\\caption{Comparison of the statistical power of the proposed method (shim) with other simple (lasso) and complex models (mlp, rf) using compas data. shim\\_2s and shim\\_2f respectively represent split-CP and full-CP for a $2^{nd}$ order SHIM. The bracketed values represent the standard deviations.}\n\\label{table:stat_compas}\n\\end{table}\n\\begin{figure}[t]\n   \\centering\n   \\includegraphics[width=\\linewidth]{figures\/fig2.pdf}\n   \\caption{Comparison of the confidence interval lengths (CI length) of the proposed method (shim) with other simple (lasso) and complex models (mlp, rf) using synthetic data for different sample sizes. shim\\_3s and shim\\_3f respectively represent split-CP and full-CP for a $3^{rd}$ order SHIM.}\n   \\label{fig:cpl_synthetic}\n \\end{figure}\n\\paragraph{Real data}\nWe applied our method to a real-world criminal justice data set called the ProPublica two-year recidivism dataset (COMPAS data). See appendix for details of this data.\n\\paragraph{Comparison of statistical powers}\nWe compared the statistical efficiency using both synthetic and real-world data. For the synthetic data, we considered both low- and high-dimensional settings. In the low-dimensional setting, we considered a dataset with $n=150$ instances and $m=10$ original covariates, whereas for the high-dimensional setting, we considered $n=150$ and $m=100$. We kept the sparsity of the design matrix fixed at $\\zeta =0.4$ in both the settings. We varied the order of interaction $d=2, 3, \\cdots$, however, almost in all the experiments we found that the model get saturated after $3^{rd}$ order interactions (i.e. the performance of SHIM did not change much for $d\\geq 3$). Hence, we reported the results of up to $3^{rd}$ order interactions. \nNote that the max order of interaction $d$ is not needed to be specified beforehand. Our pruning condition takes care of the case even when $d$ is not specified that is when the whole search space is considered (Table~\\ref{table:comp_eff_time_taken}).\nFor both the settings (low and high), we generated a dataset of $n=150$ training instances, each accompanied with $n=50$ test instances. We generated 5 such independent random datasets and repeated the experiments 3 times. Hence, in total we reported the average results of 15 independent datasets i.e. \\(3 \\times 5 \\times 50 = 750 \\) test instances. The details of the hyper parameter selection are given in the appendix. We used a multi-layer perceptron (MLP) as a neural network architecture in our experiments.\n\nFrom Table \\ref{table:stat_low} and Table \\ref{table:stat_high} it can be observed that both in low- and high-dimensional settings, all the methods produced perfect (or nearly perfect) coverage i.e cov=$0.90$. The statistical efficiency of individual methods are compared using the length of confidence interval. A statistically efficient model is expected to produce a shorter confidence interval length. Comparing the length of confidence intervals, one can observe that the statistical efficiency of SHIM increases as we increase the order of interactions. A $3^{rd}$ order SHIM produced the shortest average confidence interval lengths both in low- and high-dimensional settings. \nTo compare the fitting power of individual methods we reported the $R$-squared (r2) scores. It can be observed that the r2 scores of a $3^{rd}$ order SHIM are comparable to the best performing complex model (RF) both in low- and high-dimensional settings. We also demonstrated the comparison of the confidence interval lengths (CI length) and $R$-squared (r2) scores of the proposed method (SHIM) with other simple (LASSO) and complex models (MLP, RF) for three different sample sizes, $n \\in \\{100, 150, 200\\}, m=10$ (Figure \\ref{fig:cpl_synthetic}). It can be observed that the full-CP of $3^{rd}$ order SHIM (shim3\\_f ) produced the best results (shortest avg. length, highest avg. r2 score) in all the cases. It can also be observed that the performance of data splitting (in mlp, rf, lasso\\_s, shim3\\_s) is worse compared to that of full-CP. The split-CP tends to produce a longer confidence interval and smaller r2 score, and suffer from high variance due to the smaller data size as well as the additional randomness considered in data splitting. For the real data we randomly split the data into a training set of $n=5000$ and a test set of $n=2214$. We selected the hyper parameters by 5-fold cross-validation using the training data and reported the statistical power using the test data. The range of hyper parameters from which the optimal one is selected is the same as used for the synthetic data experiment. We repeated this experimental setup 100 times and reported the average results. From Table \\ref{table:stat_compas}, one can observe that all the method produced a perfect coverage (cov=0.90) and nearly same r2 scores. A $2^{nd}$-order SHIM produced the shortest average confidence interval length. Here, the performance of SHIM did not improve for ($d> 2$). Hence, we reported the results of a $2^{nd}$-order SHIM. Additional results are provided in the appendix.\n\n\\subsection{Comparison of computational efficiencies.}\\label{sec_comp_eff}\n\\begin{table*}[t]\n\\centering\n\\resizebox{\\textwidth}{!}{\n\\begin{tabular}{ |c|c|c|c|c|c|c|c|c|c|c|c|c|c| } \n\\hline\n \\multirow{4}{*}{$d$} & \\multirow{4}{*}{\\shortstack{Search space \\\\ (\\# nodes)}} &  \\multicolumn{6}{|c|}{$\\lambda=1$} & \\multicolumn{6}{|c|}{$\\lambda=10$}\\\\\n \\cline{3-14}\n& & \\multicolumn{3}{|c|}{With pruning} &\\multicolumn{3}{|c|}{Without pruning} &\\multicolumn{3}{|c|}{With pruning} &\\multicolumn{3}{|c|}{Without pruning}\\\\\n\\cline{3-14}\n& &$\\zeta=0.4$ &$\\zeta=0.7$ &$\\zeta=0.9$ &$\\zeta=0.4$ &$\\zeta=0.7$ &$\\zeta=0.9$ &$\\zeta=0.4$ &$\\zeta=0.7$ &$\\zeta=0.9$ &$\\zeta=0.4$ &$\\zeta=0.7$ &$\\zeta=0.9$\\\\\n \\hline\n \\hline\n  2 & 465 &0.09  &0.10   &0.03  &0.10  &0.08  &0.07   &0.08  &0.08 &0.03  &0.07  &0.07 &0.12 \\\\\n  \\hline\n  3 & 4525 &0.84  &0.67   &0.03  &0.82  &0.79 &0.64 &0.56  &0.36 &0.03  &0.63  &0.65 &0.85  \\\\\n  \\hline\n  4 & 31930 &4.49   &1.51   &0.03  &5.91  &5.32 &4.73 &2.14  &0.90 &0.03  &4.12  &4.42 &3.87  \\\\\n  \\hline\n  5 & 174436 &12.34    &2.74   &0.03  &30.19  &28.76    &23.31  &5.11  &1.38 &0.03  &26.74  &22.70 &25.42  \\\\\n  \\hline\n  10 & 53009101 &112.17    &3.83   &0.03  &$>$ 1 day  &$>$ 1 day &$>$ 1 day &49.88  &2.13 &0.03  &$>$ 1 day  &6891.44 &6861.16  \\\\\n  \\hline\n 15 & 614429671 &126.04   &3.39   &0.03  &$>$ 1 day  &$>$ 1 day &$>$ 1 day &56.22   &2.00 &0.03  &$>$ 1 day  &$>$ 1 day & $>$ 1 day  \\\\\n \\hline\n 20 & 1050777736  &126.86    &3.51   &0.03   &$>$ 1 day  &$>$ 1 day &$>$ 1 day &55.33  &2.13 &0.03   &$>$ 1 day  &$>$ 1 day & $>$ 1 day \\\\\n  \\hline\n 25 &1073709892  &130.28   &3.62   &0.03   &$>$ 1 day  &$>$ 1 day &$>$ 1 day &55.21   &2.02 &0.03   &$>$ 1 day  &$>$ 1 day &$>$ 1 day  \\\\\n \\hline\n\\end{tabular}}\n\\caption{Computation time (in sec) with and without pruning using two different $\\lambda$ values ($\\lambda=1, 10$) for three different sparsity levels ($\\zeta=0.4, 0.7, 0.9$). All computation times were measured on an Intel(R) Xeon(R) Gold 6130 CPU @ 2.10GHz.}\n\\label{table:comp_eff_time_taken}\n\\end{table*}\nTo demonstrate the computational efficiency of the proposed pruning strategy for the $\\tau$-path, we generated a synthetic dataset of $n=100$ and $m=30$ for three different sparsity levels of the design matrix ($\\zeta=0.4, 0.7, 0.9$)  using the same $5^{th}$ order model as used to demonstrate the statistical power. \nWe compared both the fraction of nodes traversed (see appendix) and the time taken (Table \\ref{table:comp_eff_time_taken}) against a different maximum interaction order $d$  for three different sparsity levels ($\\zeta=0.4, 0.7, 0.9$) using two different $\\lambda$ values ($\\lambda=1, 10$). \nIt can be observed that the pruning is more effective at the deeper nodes of the tree and saturates after a certain depth of the tree.\nThis is evident as the sparsity of the data increases at the deeper nodes, and the pruning exploits the anti-monotonicity of high-order interaction terms constructed as tree of patterns.\n\nIn the case of the homotopy method without pruning, we stopped the execution of the program if the $\\tau$-path was not finished in one day.\nFrom Table ~\\ref{table:comp_eff_time_taken}, it can be observed that without the tree pruning, the construction of the $\\tau$-path is not practical as we progress to the deeper nodes of the tree because of the generation of an exponential number of high-order interaction terms.\nThe maximum time taken by the $\\tau$-path with pruning was approximately 130 s, for a ``max-pat\" size of 25, i.e. for 1073709892 number of nodes at $\\lambda=1, \\zeta=0.4$.\nIn our numerical experiments, the number of kink in the $\\tau$-path appears to be polynomial in the number of features (see appendix). \nThe worst-case complexity of $\\tau$-path is exponential (see appendix). However, fortunately, it has been well-recognized that this worst-case rarely happens in practice \\cite{li2018well,le2021parametric} and, this is also evident from our experimental results (see appendix). \n\n\\section{Appendix A}\n\\section{Algorithms and Proofs}\n\\subsection{Proof of Lemma \\ref{lemma:lemma_1}}\nIf $\\Delta^2_{\\ell^\\ast}$ is the current minimum step size, that is, \\(\\Delta^2_{\\ell^\\ast} = \\underset{j \\in \\{1, 2, \\ldots, \\ell\\}}{\\min}\\{\\Delta_j^2\\}, \\) then, we can rewrite (\\ref{eqn:inclusion_condition2}) as follows.\n\\begin{align}\\label{eqn:inclusion_condition3}\n    \\big\\lvert \\rho_{\\ell}(\\tau_t) -   &\\Delta^2_{\\ell^\\ast} (\\eta_{\\ell}(\\tau_t) -  x_{n+1,\\ell})   \\lvert =\n    \\big\\lvert \\rho_{k}(\\tau_t) -  \\Delta^2_{\\ell^\\ast} (\\eta_{k}(\\tau_t) - x_{n+1,k} )\\big\\lvert \\end{align}\nwhere, \\(\\rho_{\\ell} = x_{\\ell}^{\\top}w(\\tau_t) \\) and \\(\\eta_{\\ell} =x_{\\ell}^{\\top} v(\\tau_t), \\forall \\ell \\in \\mathcal{A}^c_{\\tau_t} \\); \\(\\rho_k = x_{k}^{\\top}w(\\tau_t) \\) and \\(\\eta_k = x_k^{\\top} v(\\tau_t), \\forall k \\in \\mathcal{A}_{\\tau_t} \\). Here, we used the fact that $\\beta_{\\mc{A}_{\\tau_t}}(\\tau_{t+1})=\\beta_{\\mc{A}_{\\tau_t}}(\\tau_t) + \\Delta^2_{\\ell^\\ast}\\nu_{\\mc{A}_{\\tau_t}}(\\tau_t)$.\nThe right hand side (r.h.s.) of (\\ref{eqn:inclusion_condition3}) has a lower bound i.e.\n\\begin{align*}\n    \\lvert \\rho_{k}(\\tau_t) -   &\\Delta_{\\ell^\\ast} (\\eta_{k}(\\tau_t) - x_{n+1,k})   \\lvert \\geq \n    \\lvert \\rho_{k}(\\tau_t) \\rvert  -   \\Delta_{\\ell^\\ast} (\\lvert \\eta_{k}(\\tau_t) \\rvert  +   \\lvert x_{n+1,k} \\rvert), \\end{align*}\nand the left hand side (l.h.s.) of (\\ref{eqn:inclusion_condition3}) has an upper bound i.e.\n\\begin{align*}\n    \\lvert \\rho_{\\ell}(\\tau_t) -  &\\Delta_{\\ell^\\ast} (\\eta_{\\ell}(\\tau_t) - x_{n+1,\\ell} )  \\lvert  \\leq \n    \\lvert \\rho_{\\ell}(\\tau_t) \\rvert  + \\Delta_{\\ell^\\ast} (\\lvert \\eta_{\\ell}(\\tau_t) \\rvert  +  \\lvert x_{n+1,\\ell} \\rvert). \\end{align*}\nThe above two bounds are derived by considering the fact that for any $a \\in \\mb{R}, b \\in \\mb{R},  c \\in \\mb{R}$ and $d \\in \\mb{R}>0$ we can write the following:\n\\begin{equation*}\n|a-d(b-c)| \\leq |a| + d (|b|+|c|) \\hspace{0.2cm} \\text{ and } \\hspace{0.2cm} |a-d(b-c)| \\geq |a| - d (|b|+ |c|)\n\\end{equation*}\nTherefore, for equation (\\ref{eqn:inclusion_condition3}) to have a solution the following condition needs to be satisfied.\n\\begin{align}\\label{eqn:inclusion_condition4}\n    \\lvert \\rho_{\\ell}(\\tau_t) \\rvert + \\Delta^2_{\\ell^\\ast} (\\lvert \\eta_{\\ell}(\\tau_t) \\rvert +  \\lvert x_{n+1,\\ell}   \\lvert )\\geq \\lvert \\rho_{k}(\\tau_t) \\rvert  -   \\Delta^2_{\\ell^\\ast} (\\lvert \\eta_{k}(\\tau_t) \\rvert  +  \\lvert x_{n+1,k} \\rvert)\n\\end{align}\nThe equation (\\ref{eqn:inclusion_condition3}) will not have any solution if (\\ref{eqn:inclusion_condition4}) is not satisfied, and that can be used to derive the pruning condition.\nTherefore, the pruning condition of the SHIM $\\tau$-path can be written as follows:\n\\begin{align}\\label{cond:pruning_tau}\n\\lvert \\rho_{\\ell}(\\tau_t) \\lvert + \\Delta^2_{\\ell^\\ast} (\\lvert \\eta_{\\ell}(\\tau_t) \\lvert + x_{n+1,\\ell}) < \\lvert \\rho_k (\\tau_t) \\lvert   - \\Delta^2_{\\ell^\\ast} ( \\lvert\\eta_k(\\tau_t) \\lvert + x_{n+1,k}),\n\\end{align}\nBefore proving the Lemma \\ref{lemma:lemma_1}, we introduce the following two propositions:\n\\begin{proposition}\\label{prop:1} We can write\n\\begin{align*}\n\t|\\rho_\\ell (\\tau_t)| & \\leq b_{\\ell, w(\\tau_t)}, \\\\ \n\t|\\eta_\\ell (\\tau_t)| & \\leq b_{\\ell, v(\\tau_t)},\n\\end{align*}\nwhere \n\\begin{align*}\n \t b_{\\ell, w(\\tau_t)} &= \\max \\big\\{ \\sum_{w_i(\\tau_t) < 0} \\lvert w_i(\\tau_t) \\lvert x_{i\\ell}, \\sum_{w_i(\\tau_t) > 0} \\lvert w_i(\\tau_t) \\lvert x_{i\\ell} \\big\\}  \\\\ \n\t b_{\\ell, v(\\tau_t)} &= \\max \\big\\{ \\sum_{v_i(\\tau_t) < 0} \\lvert v_i(\\tau_t) \\lvert x_{i\\ell}, \\sum_{v_i(\\tau_t) > 0} \\lvert v_i(\\tau_t) \\lvert x_{i\\ell} \\big\\}.\n\\end{align*}\n\n\\end{proposition}\n\n\\paragraph{Proof of Proposition \\ref{prop:1}:}\nWe have \n\\begin{align*}\n\t|\\rho_\\ell (\\tau_t)| &= |x_\\ell^\\top w(\\tau_t)| \\\\ \n\t&=  \\left | \\sum \n\t\\limits_{i=1}^n w_{i}(\\tau_t) x_{i \\ell} \\right |  \\\\ \n\t&=  \\left | \\sum \n\t\\limits_{w_{i}(\\tau_t) > 0} |w_{i}(\\tau_t)| x_{i \\ell} - \\sum  \\limits_{w_{i}(\\tau_t) < 0} |w_{i}(\\tau_t)| x_{i \\ell}\\right | \\\\ \n\t&\\leq  \\max \\left \\{ \\sum \n\t\\limits_{w_{i}(\\tau_t) > 0} |w_{i}(\\tau_t)| x_{i \\ell}, \\sum  \\limits_{w_{i}(\\tau_t) < 0} |w_{i}(\\tau_t)| x_{i \\ell} \\right \\} =: b_{\\ell, w(\\tau_t)}.\n\\end{align*}\nSimilarly, \n\\begin{align*}\n\t|\\eta_\\ell (\\tau_t)| &= |x_\\ell^\\top v(\\tau_t)| \\\\ \n\t&=  \\left | \\sum \n\t\\limits_{i=1}^n v_{i}(\\tau_t) x_{i \\ell} \\right |  \\\\ \n\t&=  \\left | \\sum \n\t\\limits_{v_{i}(\\tau_t) > 0} |v_{i}(\\tau_t)| x_{i \\ell} - \\sum  \\limits_{v_{i}(\\tau_t) < 0} |v_{i}(\\tau_t)| x_{i \\ell}\\right | \\\\ \n\t&\\leq  \\max \\left \\{ \\sum \n\t\\limits_{v_{i}(\\tau_t) > 0} |v_{i}(\\tau_t)| x_{i \\ell}, \\sum  \\limits_{v_{i}(\\tau_t) < 0} |v_{i}(\\tau_t)| x_{i \\ell} \\right \\} =: b_{\\ell, v(\\tau_t)}.\n\\end{align*}\nTherefore, using Proposition \\ref{prop:1} we can further write equation (\\ref{eq:lemma_3_eq}) which implies equation (\\ref{cond:pruning_tau}). \n\\begin{align} \\label{eq:lemma_3_eq}\n\t\\nonumber b_{\\ell, w(\\tau_t)} + &\\Delta^2_{\\ell^\\ast} (b_{\\ell, v(\\tau_t)} + x_{n+1,\\ell})  \\\\\n\t&< |\\rho_k(\\tau_t)|  - \\Delta^2_{\\ell^\\ast} (|\\eta_k(\\tau_t)| + x_{n+1,k}).\n\\end{align}\n\\begin{proposition}\\label{prop:2}\nBy using the definition of tree, we have \n\\begin{align*}\n\tb_{\\ell, w(\\tau_t)} & \\geq b_{\\ell^\\prime, w(\\tau_t)}, \\\\ \n\tb_{\\ell, v(\\tau_t)} & \\geq b_{\\ell^\\prime, v(\\tau_t)}.\n\\end{align*}\nsince $x_{i\\ell} \\geq x_{i\\ell^\\prime}, \\hspace{0.2cm} \\forall \\ell^\\prime \\supset \\ell, \\forall i \\in [n+1]$.\n\\end{proposition}\n\\paragraph{Proof of Proposition \\ref{prop:2}:}\nFrom the definition of tree we have $x_{i\\ell}\\geq x_{i\\ell^\\prime}, \\forall \\ell^{\\prime} \\supset \\ell, \\forall i \\in [n+1]$. Hence, we can write\n\\begin{align*}\n\tb_{\\ell, w(\\tau_t)} &= \\max \\big\\{ \\sum_{w_i(\\tau_t) < 0} \\lvert w_i(\\tau_t) \\lvert x_{i\\ell}, \\sum_{w_i(\\tau_t) > 0} \\lvert w_i(\\tau_t) \\lvert x_{i\\ell} \\big\\} \\\\ \n\t&\\geq \\max \\big\\{ \\sum_{w_i(\\tau_t) < 0} \\lvert w_i(\\tau_t) \\lvert x_{i\\ell^\\prime}, \\sum_{w_i(\\tau_t) > 0} \\lvert w_i(\\tau_t) \\lvert x_{i\\ell^\\prime} \\big\\}\\\\\n\t&=: b_{\\ell^\\prime, w(\\tau_t)}.\n\\end{align*} \nSimilarly, we also have\n\\begin{align*}\n\tb_{\\ell, v(\\tau_t)} &= \\max \\big\\{ \\sum_{v_i(\\tau_t) < 0} \\lvert v_i(\\tau_t) \\lvert x_{i\\ell}, \\sum_{v_i(\\tau_t) > 0} \\lvert v_i(\\tau_t) \\lvert x_{i\\ell} \\big\\} \\\\ \n\t&\\geq \\max \\big\\{ \\sum_{v_i(\\tau_t) < 0} \\lvert v_i(\\tau_t) \\lvert x_{i\\ell^\\prime}, \\sum_{v_i(\\tau_t) > 0} \\lvert v_i(\\tau_t) \\lvert x_{i\\ell^\\prime} \\big\\}\\\\\n\t&=: b_{\\ell^\\prime, v(\\tau_t)}.\n\\end{align*}\nWe now prove Lemma \\ref{lemma:lemma_1} by contradiction, that is we assume that $\\exists \\ell$ such that (\\ref{eq:lemma_3_eq}) holds, and there exists one \\( \\ell^{\\prime} \\supset \\ell:  \\Delta^2_{\\ell^{\\prime}} < \\Delta^2_{\\ell^\\ast} \\); then show that this is a contradiction.\n\\begin{align*}\n   \n    \\therefore \\quad &|\\rho_k(\\tau_t)| - \\Delta^2_{\\ell^{\\prime}} (|\\eta_k(\\tau_t)| + x_{n+1,k})\\\\\n    &> |\\rho_k(\\tau_t)| - \\Delta^2_{\\ell^\\ast} (|\\eta_k(\\tau_t)| + x_{n+1,k}), \\hspace{0.2cm} \\because \\Delta^2_{\\ell^{\\prime}} < \\Delta^2_{\\ell^\\ast}\\\\\n    &> b_{\\ell, w(\\tau_t)} + \\Delta^2_{\\ell^\\ast} (b_{\\ell, v(\\tau_t)} + x_{n+1,\\ell}), \\hspace{0.2cm} \\text{using } (\\ref{eq:lemma_3_eq})\\\\\n    &> b_{\\ell^\\prime, w(\\tau_t)} + \\Delta^2_{\\ell^\\ast} (b_{\\ell^\\prime, v(\\tau_t)} + x_{n+1,\\ell}),  \\hspace{0.2cm} \\text{(Proposition \\ref{prop:2}),}\\\\\n    &> b_{\\ell^\\prime, w(\\tau_t)} + \\Delta^2_{\\ell^\\prime} (b_{\\ell^\\prime, v(\\tau_t)} + x_{n+1,\\ell}), \\hspace{0.2cm} \\because \\Delta^2_{\\ell^{\\prime}} < \\Delta^2_{\\ell^\\ast}.\n   \n\\end{align*}\nTherefore, we got\n\\begin{align*}\n  |\\rho_k(\\tau_t)| - &\\Delta^2_{\\ell^{\\prime}} (|\\eta_k(\\tau_t)|  + x_{n+1,k})\\\\\n  &> b_{\\ell^\\prime, w(\\tau_t)} + \\Delta^2_{\\ell^\\prime} (b_{\\ell^\\prime, v(\\tau_t)} + x_{n+1,\\ell}) \\\\\n  &\\implies \\ell^{\\prime} \\text{ is infeasible,} \\hspace{0.5cm} \\text{ using (\\ref{eqn:inclusion_condition4}), (\\ref{cond:pruning_tau}), (\\ref{eq:lemma_3_eq})}\\\\\n  &\\implies \\Delta^2_{\\ell^{\\prime}} \\nless \\Delta^2_{\\ell^\\ast}.\n\\end{align*}\nThis completes the proof of Lemma \\ref{lemma:lemma_1}.\nHence, if the pruning condition in Lemma \\ref{lemma:lemma_1} holds, then we do not need to search the sub-tree with $\\ell$ as the root node, and hence increasing the efficiency of the search procedure.\n\\subsection{Complexity analysis of the $\\tau$-path:}\\label{complexity_analysis}\nGiven that \\(X^{\\top}_{\\mc{A}_{\\tau}}X_{\\mc{A}_{\\tau}}\\) is full rank for some $\\mc{A}_{\\tau}\\subseteq [p]$ and for a fixed regularization parameter $\\lambda$, the $\\tau$-path \\( \\tau \\mapsto \\beta(\\tau),  \\forall \\tau \\in [-\\infty, +\\infty] \\) is well defined (see Proposition~\\ref{prop:homotopy}), then the worst-case complexity of the $\\tau$-path is $3^p$. Indeed, the sign $s(\\tau) \\in \\{-1, 0, +1\\}^p$ of the coefficients $\\beta(\\tau)$ does not change between any two consecutive kinks (zero-crossing points) of the piece-wise linear path \\(\\tau \\mapsto \\beta(\\tau)\\). Therefore, for all $p$ patterns the number of linear pieces in $\\tau$-path is bounded by $3^p$ (for details see \\citet{mairal2012complexity}). However, this worst-case complexity is too pessimistic and fortunately, it has been well-recognized that this worst-case rarely happens in practice \\cite{le2021parametric} and, this is also evident from our experimental results (Table \\ref{table:comp_eff_kinks}). \n\n\\subsection{Algorithms}\nThe algorithms to compute the $\\tau$-path and the full-CP set of SHIM are given in Algorithm \\ref{algo:tau_path} and Algorithm \\ref{algo:full-CP} respectively. The Algorithm \\ref{algo:tau_path} returns the transition points ($\\mb{T}$), the LASSO solutions ($\\mb{B}$) and the active sets ($\\mb{A}$) at those transition points  which are subsequently provided to Algorithm \\ref{algo:full-CP}. In Algorithm \\ref{algo:full-CP}, for each linear piece of the $\\tau$-path, i.e. \\(\\forall \\tau \\in (\\tau_t, \\tau_{t+1})\\) for any two consecutive kinks $\\tau_t$ and $\\tau_{t+1}$, we need to first identify the points at which $|w_{n+1}(\\tau)| = |w_i(\\tau)|, \\forall i\\in [n]$ (Line 4). Let us denote those points as \\(\\{\\tau_t = u_1, u_2, u_3, \\ldots, u_r = \\tau_{t+1} \\}\\) (for simplicity we slightly abused the notation here). Then we need to check that at which of those points the condition stated in Equation (\\ref{eqn:CP2}) is satisfied (Line 5, 6, 7) to determine the full-CP set of SHIM.\n\\begin{algorithm}[h!]\n\\begin{footnotesize}\n\\caption{Compute $\\tau$-path}\n\\label{algo:tau_path}\n\\begin{algorithmic}[1]\n\\STATE \\textbf{Input:} \\(Z \\in \\mb{R}^{(n+1) \\times m}, y \\in \\mb{R}^n, [y_{min}, y_{max}], \\lambda \\)\n\\STATE Initialization: \n\\STATE\\hspace{\\algorithmicindent} \\( t=1, \\tau_1 = y_{min}\\), \\(\\mc{A}_{\\tau_1} = \\{\\ell: \\beta_\\ell(\\tau_1) \\neq 0\\}\\), \\( \\forall \\ell  \\in [p]\\), \n\\STATE\\hspace{\\algorithmicindent}\\(\\mb{T}= \\{\\tau_1\\},  \\mb{B}= \\{\\beta_{\\mc{A}_{\\tau_1}}(\\tau_1)\\}, \\mb{A}=\\{\\mc{A}_{\\tau_1}\\}\\)\n\\WHILE{\\((\\tau_t < y_{max})\\)}\n\\STATE Compute \\(\\Delta_{\\ell^\\ast} \\) using (\\ref{eq:step_length}) and Lemma \\ref{lemma:lemma_1}\n\\STATE If \\(\\Delta_{\\ell^\\ast} = \\Delta_{\\ell^\\ast}^1\\), \\(\\mc{A}_{\\tau_{t+1}} \\leftarrow  \\mc{A}_{\\tau_t} \\setminus \\ell \\) \\hspace{0.2cm} (remove $\\ell$ from \\(\\mc{A}_{\\tau_t})\\) \n\\STATE If $\\Delta_{\\ell^\\ast} = \\Delta_{\\ell^\\ast}^2$, \\(\\mc{A}_{\\tau_{t+1}} \\leftarrow  \\mc{A}_{\\tau_t} \\cup \\{\\ell\\} \\) \\hspace{0.2cm}  (add  \\(\\ell\\) into \\(\\mathcal{A}_{\\tau_t})\\)\n \\STATE Update:\n %\n \\STATE\\hspace{\\algorithmicindent} \\( \\tau_{t+1} \\leftarrow \\tau_{t} + \\Delta_{\\ell^\\ast} \\)\n\\STATE\\hspace{\\algorithmicindent} \\(\\beta_{\\mc{A}_{\\tau_t}}(\\tau_{t+1}) \\leftarrow \\beta_{\\mc{A}_{\\tau_t}}(\\tau_t) + \\Delta_{\\ell^\\ast} \\nu_{\\mc{A}_{\\tau_t}}(\\tau_t)\\)\n\\STATE\\hspace{\\algorithmicindent} \\(\\mb{T}=\\mb{T} \\cup \\{\\tau_{t+1}\\}\\)\n\\STATE\\hspace{\\algorithmicindent} \\( \\mb{B}=\\mb{B} \\cup \\{\\beta_{\\mc{A}_{\\tau_t}}(\\tau_{t+1})\\} \\)\n\\STATE\\hspace{\\algorithmicindent} \\(\\mb{A}=\\mb{A} \\cup \\{\\mc{A}_{\\tau_{t+1}}\\}\\)\n\\STATE\\hspace{\\algorithmicindent} $\\nu_{\\mc{A}_{\\tau_t}}(\\tau_t)$ and $\\gamma_{\\mc{A}_{\\tau_t}}(\\tau_t)$ using (\\ref{direction_vec})\n\\STATE\\hspace{\\algorithmicindent} t = t +1\n\\ENDWHILE\n\\STATE \\textbf{Output:} $\\mb{T}, \\mb{B}, \\mb{A}$\n\\end{algorithmic}\n\\end{footnotesize}\n\\end{algorithm}\n\\begin{algorithm}[h!]\n\\begin{footnotesize}\n\\caption{Compute full-CP}\n\\label{algo:full-CP}\n\\begin{algorithmic}[1]\n\\STATE \\textbf{Input:} \\(Z \\in \\mb{R}^{(n+1) \\times m}, y \\in \\mb{R}^n, \\alpha \\in [0, 1], \\mb{T}, \\mb{B}, \\mb{A}  \\)\n\\STATE Initialization: \\( t=0, t_{max} = |\\mb{T}|, \\mc{C} = \\{\\emptyset\\}\\)\n\\WHILE{\\((t < t_{max})\\)}\n\\STATE $\\{u_2, \\ldots, u_{r-1} \\}=\\{u \\in (\\tau_t, \\tau_{t+1}):|w_i(u)| = |w_{n+1}(u)|,  \\forall i \\in [n]\\}$\n\\STATE \\(\\mc{T}_t = \\{\\tau_t\\} \\cup \\{u_2, u_3, \\ldots, u_{r-1} \\} \\cup \\{\\tau_{t+1}\\} \\)\n\\STATE \\(\\mc{H}_t = \\{ h \\in \\{1, \\ldots, | \\mc{T}_t | \\} \\, :\\,  \\pi(\\mc{T}_t (h)) \\geq \\alpha\\}\\) using $\\mb{B}$ and  $\\mb{A}$\n\\STATE \\(\\mc{C} = \\mc{C} \\cup_{h \\in \\mc{H}_t} [ \\mc{T}_t(h), \\mc{T}_t(h + 1)) \\)\n\\STATE t = t +1\n\\ENDWHILE\n\\STATE \\textbf{Output:} $\\mc{C}$\n\\end{algorithmic}\n\\end{footnotesize}\n\\end{algorithm}\n\n\\section{Extension for Elastic Net (El-SHIM)}\nA common problem of the LASSO is that if the data has correlated features, then the LASSO picks only one of them and ignores the rest, which leads to instability. To solve this problem \\citet{zou2005regularization} proposed the Elastic Net (ElNet). This feature correlation problem is very much evident in the SHIM-type problem, and hence we extended our framework for the Elastic Net. We solve the following optimization problem to extend our framework for the Elastic Net:\n\\begin{equation}\\label{obj:primal_elnet}\n    \\beta(\\tau) \\in \\mathop{\\mathrm{arg\\,min}}_{\\beta \\in \\mathbb{R}^p} \\frac{1}{2}\\norm{y(\\tau) - X\\beta}_2^2 + \\frac{1}{2}\\alpha \\norm{\\beta}_2^2  + \\lambda \\norm{\\beta}_1 \\enspace .\n\\end{equation}\nThe elastic net optimization problem can actually be formulated as a LASSO optimization problem using augmented data.\nIf we consider an augmented data defined as \\(\\Tilde{X} = \\begin{pmatrix} X\\\\ \\sqrt{\\alpha} I_p \\end{pmatrix}  \\in \\mb{R}^{(n+1+p) \\times p}\\) and \\(\\Tilde{y} (\\tau) = \\begin{pmatrix}y (\\tau) \\\\ \\mathbf {0} \\end{pmatrix} \\in \\mb{R}^{n+1+p}\\), where $I_p \\in \\mb{R}^{p \\times p}$ is an identity matrix and $\\mathbf {0} \\in \\mb{R}^p$ is a zero vector, then solving the elastic net optimization problem (\\ref{obj:primal_elnet}) for a fixed $\\lambda$, is equivalent to solving the following problem.\n\\begin{equation}\n    \\beta(\\tau) \\in \\mathop{\\mathrm{arg\\,min}}_{\\beta \\in \\mathbb{R}^p} \\frac{1}{2}\\norm{\\Tilde{y} (\\tau) - \\Tilde{X}\\beta}_2^2 + \\lambda \\norm{\\beta}_1.\n\\end{equation}\n\\paragraph{Step size ($\\tau$-path)}\nIf we consider two real values $\\tau_t$ and $\\tau_{t+1}$ ( $\\tau_{t+1}>\\tau_t$) at which the active set does not change, and their signs also remain the same, then we can write \n\\begin{align*}\n  \\beta_{\\mathcal{A}_{\\tau_{t}}}(\\tau_{t+1}) - \\beta_{\\mathcal{A}_{\\tau_t}}(\\tau_t) = \\nu_{\\mathcal{A}_{\\tau_t}}(\\tau_t)(\\tau_{t+1} - \\tau_t),\\\\ \n   \\lambda s_{\\mathcal{A}^c_{\\tau_{t}}}(\\tau_{t+1}) - \\lambda s_{\\mathcal{A}^c_{\\tau_t}}(\\tau_t) = \\gamma_{\\mathcal{A}^c_{\\tau_t}}(\\tau_t)(\\tau_{t+1} - \\tau_t),\n\\end{align*}\nwhere, \\( \\nu_{\\mc{A}_{\\tau_t}}(\\tau_t) = (X^{\\top}_{\\mc{A}_{\\tau_t}}X_{\\mc{A}_{\\tau_t}} + \\alpha I_{|\\mathcal{A}_{\\tau_t}|} )^{-1}x_{n+1;\\mc{A}_{\\tau_t}} \\)  and \\( \\gamma_{\\mc{A}^c_{\\tau_t}} (\\tau) = x_{n+1, \\mc{A}_{\\tau_t}^c} - (X^{\\top}_{\\mc{A}_{\\tau_t}^c}X_{\\mc{A}_{\\tau_t}})\\nu_{\\mc{A}_{\\tau_t}}(\\tau_t) \\). \nNote that here the only change compared to the vanilla SHIM is the addition of an \\(\\alpha I_{|\\mathcal{A}_{\\tau_t}|}\\) term to the expression of \\(\\nu_{\\mathcal{A}_{\\tau_t}}\\). Now, one can also derive a similar expression of the step size of inclusion and deletion as done for the vanilla SHIM $\\tau$-path (\\ref{eq:step_length}), considering the updated expression of \\(\\nu_{\\mathcal{A}_{\\tau_t}}(\\tau_t)\\).\n\\paragraph{Tree pruning ($\\tau$-path)}\nThe pruning condition (\\ref{cond:pruning_tau}) with the augmented data (\\(\\Tilde{X}, \\Tilde{y}\\)) can be written as follows:\n\\begin{align}\\label{cond:pruning_elnet1}\n\\lvert \\Tilde{\\rho}_{\\ell}(\\tau_t) \\lvert + \\Delta^2_{\\ell^{\\ast}} (\\lvert \\Tilde{\\eta}_{\\ell}(\\tau_t) \\lvert + x_{n+1,\\ell}) \n< \\lvert \\Tilde{\\rho}_k (\\tau_t) \\lvert   - \\Delta^2_{\\ell^{\\ast}} (\\lvert\\Tilde{\\eta}_k(\\tau_t) \\lvert + x_{n+1,k}),\n\\end{align}\nwhere, \\(\\Tilde{\\rho}_{\\ell} (\\tau_t)= \\Tilde{x}_{\\ell}^{\\top}\\Tilde{w}(\\tau_t) \\) and \\(\\Tilde{\\eta}_{\\ell} (\\tau_t)=\\Tilde{x}_{\\ell}^{\\top} \\Tilde{v}(\\tau_t), \\forall \\ell \\in \\mathcal{A}^c_{\\tau_t} \\); \\(\\Tilde{\\rho}_k (\\tau_t)= \\Tilde{x}_{k}^{\\top}\\Tilde{w}(\\tau_t) \\) and \\(\\Tilde{\\eta}_k (\\tau_t) = \\Tilde{x}_k^{\\top} \\Tilde{v}(\\tau_t), \\forall k \\in \\mathcal{A}_{\\tau_t} \\); and \\(\\Tilde{w}(\\tau_t) = \\Tilde{y} (\\tau_t) - \\Tilde{X}_{\\mc{A}_{\\tau_t}}\\beta_{\\mc{A}_{\\tau_t}}(\\tau_t)  \\in \\mathbb{R}^{n+1+p}\\) and  \\(\\Tilde{v}(\\tau_t) = \\Tilde{X}\\nu(\\tau_t)  \\in \\mathbb{R}^{n+1+p}\\). \nThe above pruning condition (\\ref{cond:pruning_elnet1}) can be further simplified and redefined as stated in Lemma \\ref{lemma:lemma_2}.\n\\begin{lemma}\\label{lemma:lemma_2}\nIf $\\Delta^2_{\\ell^\\ast}$ is the current minimum step size, that is, \\(\\Delta^2_{\\ell^\\ast} = \\underset{j \\in \\{1, 2, \\ldots, \\ell\\}}{\\min}\\{\\Delta^2_j\\}, \\) then\n$\\forall \\ell^\\prime \\supset \\ell, \\Delta^2_{\\ell^\\prime} > \\Delta^2_{\\ell^\\ast}$ if \n\\begin{align} \\label{eq:lemma_2_eq}\n\t b_{\\ell, w(\\tau_t)} + &\\Delta^2_{\\ell^\\ast} (b_{\\ell, v(\\tau_t)} + x_{n+1,\\ell})\n\t< |\\Bar{\\rho}_k(\\tau_t)|  - \\Delta^2_{\\ell^\\ast} ( |\\Bar{\\eta}_k(\\tau_t)| + x_{n+1,k}).\n\\end{align}\n\\end{lemma}\nwhere,  \\(\\Bar{\\rho}_k(\\tau_t) = \\sum_{i=1}^n w_i(\\tau_t) x_{ik} - \\alpha \\beta_k\\), \\(\\Bar{\\eta}_k(\\tau_t) = \\sum_{i=1}^n v_i(\\tau_t) x_{ik} + \\alpha \\nu_k\\).\n\n\nThe proof of Lemma \\ref{lemma:lemma_2} is given below.\n\\begin{proof}\n\\textnormal{Lets, consider} \\(\\Tilde{w}(\\tau_t) = \\Tilde{y} (\\tau_t) - \\Tilde{X}_{\\mc{A}_{\\tau_t}}\\beta_{\\mc{A}_{\\tau_t}}(\\tau_t)  \\in \\mathbb{R}^{n+1+p}\\) \\textnormal{and} \\(w(\\tau_t) = y(\\tau_t) - \\bar{X}_{\\mc{A}_{\\tau_t}}\\beta_{\\mc{A}_{\\tau_t}}(\\tau_t)  \\in \\mathbb{R}^{n+1}\\), \\textnormal{where} \\(p = |\\mathcal{A}_{\\tau_t}| + |\\mathcal{A}^c_{\\tau_t}| \\), \\textnormal{then we can write}\n\\begin{equation}\\label{eqn:aug_w}\n    \\Tilde{w}_i(\\tau_t) = \\begin{cases}\n     w_i(\\tau_t) \\hspace{1.1cm} \\textnormal{if} \\quad  i \\leq n+1, \\\\\n    -\\sqrt{\\alpha}\\beta_{\\ell}(\\tau_t) \\hspace{0.3cm} \\textnormal{if} \\quad n+1 < i \\leq n + 1+ |\\mathcal{A}_{\\tau_t}|,\n    \\hspace{0.5cm} \\forall \\ell \\in \\mathcal{A}_{\\tau_t}, \\\\\n    0 \\hspace{1.8cm}  \\textnormal{if} \\quad n+1 + |\\mathcal{A}_{\\tau_t}| < i \\leq n + 1 + p.\n    \\end{cases}\n\\end{equation}\n\\textnormal{similarly considering \\(\\Tilde{v}(\\tau_t) = \\Tilde{X}\\nu(\\tau_t)  \\in \\mathbb{R}^{n+1+p}\\) and \\(v(\\tau_t) = \\bar{X}\\nu(\\tau_t)  \\in \\mathbb{R}^{n+1}\\), we can write}\n\\begin{equation}\\label{eqn:aug_v}\n    \\Tilde{v}_i(\\tau_t) = \\begin{cases}\n     v_i(\\tau_t) \\quad \\quad \\hspace{0.08cm} \\textnormal{if} \\quad i \\leq n +1,\\\\\n    \\sqrt{\\alpha}\\nu_{\\ell}(\\tau_t)  \\hspace{0.3cm} \\textnormal{if} \\quad n+1 < i \\leq n +1 + |\\mathcal{A}_{\\tau_t}|,\n    \\hspace{0.5cm} \\forall \\ell \\in \\mathcal{A}_{\\tau_t},\\\\\n    0 \\quad \\quad \\quad \\hspace{0.42cm} \\textnormal{if} \\quad n+1 + |\\mathcal{A}_{\\tau_t}| < i \\leq n +1 + p.\n    \\end{cases}\n\\end{equation}\n\\textnormal{and, considering \\(\\Tilde{X} \\in \\mathbb{R}^{(n+1+p)\\times p}\\) and \\(X \\in \\mathbb{R}^{(n+1) \\times p}\\) we can write}\n\\begin{equation}\\label{eqn:aug_X}\n    \\Tilde{x}_{i\\ell} = \\begin{cases}\n    x_{i\\ell} \\quad \\quad \\hspace{0.1cm} \\textnormal{if} \\quad i \\leq n+1, \\\\\n    \\sqrt{\\alpha} \\quad \\quad \\hspace{0.03cm} \\textnormal{if} \\quad i > n+1 \\enspace \\textnormal{and} \\enspace (i-n-1)=\\ell,\\\\\n    0 \\hspace{1.05cm} \\textnormal{otherwise}.\n    \\end{cases}\n\\end{equation}\n\\textnormal{Therefore, \\(\\forall \\ell \\in \\mathbb{R}^p\\)  we can write}\n\\begin{align*}\n    \\Tilde{x}_{\\ell}^{\\top} \\Tilde{w}(\\tau_t) &= \\sum_{i=1}^{n+1+p} \\Tilde{w}_i(\\tau_t) \\Tilde{x}_{i\\ell}, \\\\\n    &=\\sum_{i=1}^{n+1} \\Tilde{w}_i(\\tau_t) \\Tilde{x}_{i\\ell} + \\sum_{\\substack{i=n+2}}^{n+1+|\\mathcal{A}_{\\tau_t}|} \\Tilde{w}_i(\\tau_t) \\Tilde{x}_{i\\ell} + \\sum_{i=n+2+|\\mathcal{A}_{\\tau_t}|}^{n+p} \\Tilde{w}_i(\\tau_t) \\Tilde{x}_{i\\ell}.\n\\end{align*}\n\\textnormal{Now, using (\\ref{eqn:aug_w}) and (\\ref{eqn:aug_X}) the second and the third quantity in the above expression can be written as follows:}\n\\begin{equation*}\n\\sum_{\\substack{i=n+2}}^{n+1+|\\mathcal{A}_{\\lambda_t}|} \\Tilde{w}_i (\\tau_t) \\Tilde{x}_{i\\ell} =\\begin{cases}\n    (-\\sqrt{\\alpha}\\beta_{\\ell}(\\tau_t))(\\sqrt{\\alpha}), \\hspace{0.5cm} \\textnormal{if} \\enspace (i-n-1) = \\ell,\\\\\n    0 \\hspace{0.5cm} \\textnormal{otherwise}.\n    \\end{cases}\n\\end{equation*}\n\\textnormal{and,}\n\\begin{equation*}\n   \\sum_{i=n+2+|\\mathcal{A}_{\\tau_t}|}^{n+p} \\Tilde{w}_i(\\tau_t) \\Tilde{x}_{i\\ell} = 0. \n\\end{equation*}\n\\textnormal{Therefore,}\n\\begin{align*}\n    \\Tilde{x}_{\\ell}^{\\top} \\Tilde{w}(\\tau_t) &= \\sum_{i=1}^{n+1} w_i(\\tau_t) x_{i\\ell} \\\\\n    &= \\rho_{\\ell}(\\tau_t), \\enspace \\forall \\ell \\in \\mathcal{A}^c_{\\tau_t} \\enspace \\textnormal{and,} \n\\end{align*}\n\\begin{align*}\n   \\Tilde{x}_k^{\\top} \\Tilde{w}(\\tau_t) &= \\sum_{i=1}^{n+1} w_i(\\tau_t) x_{ik} - \\alpha \\beta_k (\\tau_t)\\\\\n   &= \\rho_k(\\tau_t) - \\alpha \\beta_k (\\tau_t), \\enspace \\forall k \\in \\mathcal{A}_{\\tau_t},\\\\\n   &=: \\bar{\\rho}_k(\\tau_t) .\n\\end{align*}\n\\textnormal{Similarly, using (\\ref{eqn:aug_v}) and (\\ref{eqn:aug_X}) we can write}\n\\begin{align*}\n    \\Tilde{x}_{\\ell}^{\\top} \\Tilde{v}(\\tau_t) &= \\sum_{i=1}^{n+1} v_i (\\tau_t) x_{i\\ell} \\\\\n    &= \\eta_{\\ell}(\\tau_t), \\enspace \\forall \\ell \\in \\mathcal{A}_{\\tau_t}^c, \\hspace{0.5cm} \\textnormal{and,}\n\\end{align*}\n\\begin{align*}\n  \\Tilde{x}_k^{\\top} \\Tilde{v}(\\tau_t) &= \\sum_{i=1}^{n+1} v_i(\\tau_t) x_{ik} + \\alpha \\nu_k (\\tau_t) \\\\\n  &=  \\eta_k(\\tau_t) + \\alpha \\nu_k (\\tau_t) , \\enspace \\forall k \\in \\mathcal{A}_{\\tau_t}\\\\\n  &= \\bar{\\eta}_k(\\tau_t).\n\\end{align*}\n\\end{proof}\nTherefore, we can write the pruning condition (\\ref{cond:pruning_elnet1}) as follows.\n\\begin{align}\\label{cond:pruning_elnet3}\n\\lvert \\rho_{\\ell}(\\tau_t) \\lvert + &\\Delta^2_{\\ell^{\\ast}} (\\lvert \\eta_{\\ell}(\\tau_t) \\lvert + x_{n+1,\\ell}) < \\lvert \\bar{\\rho}_k (\\tau_t)  \\lvert   - \\Delta^2_{\\ell^{\\ast}} (\\lvert \\bar{\\eta}_k(\\tau_t) \\lvert + x_{n+1,k}),\n\\end{align}\n\nNow, using Proposition\\ref{prop:1}, we can further simplify the above pruning condition as follows.\n\\begin{align}\\label{pruning_elnet3}\n b_{\\ell,w(\\tau_t)} + &\\Delta^2_{\\ell^{\\ast}}  (b_{\\ell,v(\\tau_t)}  + x_{n+1,\\ell}) <  |\\Bar{\\rho}_k(\\tau_t)| - \\Delta^2_{\\ell^{\\ast}} (|\\Bar{\\eta}_k(\\tau_t)| + x_{n+1,k}),\n\\end{align}\nNow similar to the Lemma \\ref{lemma:lemma_1} one can formally prove the Lemma \\ref{lemma:lemma_2} using (\\ref{pruning_elnet3}) and Proposition \\ref{prop:2}.\n\\section{Experimental Setup and Additional Results}\n\\subsection{Hyper parameter selection (synthetic data experiments)}\\label{hyperparameter_selection}\nThe hyper parameter selection for all the methods (e.g. $\\lambda$ in LASSO, SHIM; hidden layer's sizes in MLP etc.) are done based on $5-$fold cross validation using a separate set of 15 independent samples. For MLP, the activations are chosen from \\{identity, relu, logistic, tanh\\} and the hidden layer's sizes are chosen from all possible combinations of \\{50, 100, 150\\} nodes, considering both 2-hidden layers and 3-hidden layers architectures. The most frequent activation and architecture chosen based on separate set of 15 independent samples are considered to report the average results. Similarly for RF, the number of estimators (n\\_estimators) and the min samples in the leaf of a tree (min\\_samples\\_leaf) are chosen from \\{50, 100, 200\\} and \\{0.1, 0.05, 0.01\\} respectively. The median $\\lambda$ value based on the separate set of 15 independent samples is considered to report the results of LASSO and SHIM. For the $\\lambda$ selection we considered a grid of 10 points in the range of $[\\lambda_{max}\/10, 0)$. For, MLP and RF, we used the standard \\emph{scikit-learn} implementation.\n\\subsection{Real data details}\nThis is a dataset of criminal records containing the criminal history, jail and prison time, and demographics of a subject used to predict the potential recidivism. This dataset contains risk scores of 7210 defendants (n = 7210). The dataset consists of seven categorical and integer-valued features: sex (male, female), age, juvenile-felonies, juvenile-misdemeanors, juvenile-crimes, priors, currentcharge-degree (misdemeanor, felony) which are subsequently converted to binary features. Hence, this resulted in 14 binary features which are - sex:male, age:18-20, age:21-22, age:23-25, age:26-45, age:$>$45 juvenile-felonies:$>$0, juvenile-misdemeanors:$>$0, juvenile-crimes:$>$0 priors:2-3, priors:=0, priors:=1, priors:$>$3, and current-charge-degree:misdemeanor. These 14 binary features are used to predict the two years recidivism risk scores.\n\\subsection{Additional results}\\label{additional_results}\nTo further compare the performance of the proposed method we considered highly sparse data ($\\zeta=0.6$) in two different experimental settings. (1) \\textbf{model-1}: We generated a model with strong signals that is we considered a true model of up to third-order interactions, which is defined as $\\mu (x_i) = 5.0z_1 +5.0 z_1z_2 + 5.0z_1z_2z_3$ and (2) \\textbf{model-2}: We generated a model with weak signals that is we considered a true model of up to third-order interactions, which is defined as $\\mu (x_i) = 1.0z_1 +1.0 z_1z_2 + 1.0z_1z_2z_3$. In both the settings we set $\\sigma=1$. The choice of these models is merely for demonstration purposes, and the proposed method is equally applicable to any chosen model. \nFor both the settings (strong and weak), we generated a dataset of $n\\in \\{50, 100, 150\\}$ training instances, each accompanied with $n=10$ test instances. Here we considered a covariate size of $m=5$. We generated 3 such independent random datasets. Hence, in total we reported the average results of  \\(3 \\times 10 = 30 \\) test instances. The hyper parameter were selected based on another independent set of $3$ samples using the same strategy as explained in the hyper parameter selection section in the appendix. \nFor split-CP, we repeated the experiments 30 times to highlight the effect of randomness in the confidence set generation. For SHIM, we did not mention any ``max-pat\" size of $d$, i.e. the entire search space is considered for exploration (tree generation) and the proposed tree pruning condition takes care of it to improve the efficiency of the search. The best shim model is automatically chosen by the algorithm corresponding to the best hyper parameter $\\lambda$ (explained in the hyper parameter selection section in the appendix).\n\n\nThe results are shown in Fig.\\ref{fig:syn_low_signal_sparse} where the left figure corresponds to the model-1 (strong signal) and the right figure corresponds to the model-2 (weak signal). In both the settings, the full-CP methods (lasso\\_f, shim\\_f) produced more compact confidence sets irrespective of sample sizes.\nIt can be observed that in case of highly sparse data, the shim model produces better results, more specifically when the sample size is small and the signal is weak.\n\\begin{figure}[t]\n   \\centering\n   \\includegraphics[width=\\linewidth]{figures\/fig3.pdf}\n   \\caption{Comparison of the confidence interval lengths (CI length) of the proposed method (shim) with other simple (lasso) and complex models (mlp, rf) using synthetic data for different sample sizes. Left figure shows the results using model-1 (strong signal) and the right figure shows the results using model-2 (weak signal).}\n   \\label{fig:syn_low_signal_sparse}\n \\end{figure}\n\\begin{table*}[t]\n\\centering\n\\resizebox{\\textwidth}{!}{\n\\begin{tabular}{ |c|c|c|c|c|c| } \n\\hline\n $y$ &$\\hat{y}$ & confidence interval &length  &rule &coef \\\\\n \\hline\n 1 &1.34 &[-2.30, 5.00] &7.30  &  \\begin{tabular}{@{}c@{}}[[prior crimes=0], [misdemeanor], [prior crimes=0  \\\\ and misdemeanor], [age$>=$46 and misdemeanor]]\\end{tabular} &[ 1.75, -0.20,  -0.12,   -0.08]\\\\\n \\hline \n 9 &6.15 &[2.51, 9.78] &7.27  &  \\begin{tabular}{@{}c@{}} [[prior crimes $>=4$], [$26<=$age$<=45$]]\\end{tabular} &[4.61, 1.54]\\\\\n \\hline\n 4 &2.94 &[-0.70, 6.58] &7.28  &  \\begin{tabular}{@{}c@{}} [[prior crimes=1], [$26<=$age$<=45$], [$26<=$age$<=45$ \\\\\\relax and misdemeanor], [misdemeanor], [male], [male and $26<=$age$<=45$]]\\end{tabular} &[ 2.01,  1.54, -0.30, -0.20,  -0.09, -0.03]\\\\\n \\hline\n 8 &1.25 &[-2.38, 4.90]&7.28  &  \\begin{tabular}{@{}c@{}} [[prior crimes=0], [misdemeanor], [prior crimes=0 and misdemeanor], [male], \\\\\\relax [age$>=46$ and misdemeanor], [male and prior crimes=0]]\\end{tabular} &[ 1.75, -0.20, -0.12, -0.09, -0.08, -0.001]\\\\\n \\hline\n  2 &6.04 &[2.41, 9.67] &7.26  &  \\begin{tabular}{@{}c@{}} [[prior crimes $>=$ 4], [26$<=$age$<=$ 45], \\\\\\relax [male], [male and 26$<=$ age $<=$45 ]] \\end{tabular} &[ 4.61,  1.54, -0.09, -0.03]\\\\\n \\hline\n  \\end{tabular}}\n\\caption{The SHIM model for 5 randomly chosen test instances of the compas data. The true score ($y$), predicted score ($\\hat{y}$), confidence interval (CI), length of confidence interval (length), rule and the coefficient (coef) of the individual component of a rule are shown.}\n\\label{table:SHIM_compas}\n\\end{table*}\n\nTable \\ref{table:SHIM_compas} shows true score ($y$), predicted score ($\\hat{y}$), confidence interval (CI), length of confidence interval (length), rule and the coefficient (coef) of the individual component of a rule of five randomly chosen test instances of the compas data. This demonstrates that the generated SHIM is easy to interpret and, one can easily judge the reliability of model based on the provided confidence interval along with the point estimation.\n\\begin{figure}[h!]\n   \\centering\n      \\includegraphics[width=\\linewidth]{figures\/node_counts.pdf}\n   \\caption{Variation of node counts (``fraction of node counts\") for different \\emph{max\\_pat} ($d$) sizes. Here, the fraction of nodes for a specific $d$, represents the number of nodes traversed divided by the total number of possible combinations of interaction terms using the ``max-pat\" size of $d$. The results are shown for two $\\lambda$ values ($\\lambda = 1, 10$) and for three different sparsity levels ($0.4, 0.7, 0.9$). The pruning is more effective as the data is more sparse. }\n   \\label{fig:node_counts}\n \\end{figure}\n\\begin{table}[t]\n\\centering\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{ |c|c|c|c|c|c|c|c| } \n\\hline\n \\multirow{2}{*}{$d$} & \\multirow{2}{*}{\\shortstack{Search space \\\\ (\\# nodes)}} &  \\multicolumn{3}{|c|}{$\\lambda=1$} & \\multicolumn{3}{|c|}{$\\lambda=10$}\\\\\n \\cline{3-8}\n & &$\\zeta=0.4$ &$\\zeta=0.7$ &$\\zeta=0.9$  &$\\zeta=0.4$ &$\\zeta=0.7$ &$\\zeta=0.9$\\\\\n \\cline{3-8}\n \\hline\n \\hline\n  2 & 465 &200  &134   &7  &9  &7  &2    \\\\\n  \\hline\n  3 & 4525 &317  &129   &7  &9  &7  &2    \\\\\n  \\hline\n  4 & 31930 &314   &129   &7  &9  &7  &2    \\\\\n  \\hline\n  5 & 174436 &312    &129   &7  &9  &7  &2    \\\\\n  \\hline\n  10 & 53009101 &312    &129   &7  &9  &7  &2   \\\\\n  \\hline\n 15 & 614429671 &312   &129   &7  &9  &7  &2    \\\\\n \\hline\n 20 & 1050777736  &312    &129   &7   &9  &7  &2   \\\\\n  \\hline\n 25 &1073709892  &312   &129   &7   &9  &7  &2   \\\\\n \\hline\n\\end{tabular}\n}\n\\caption{Number of homotopy transition points (\\# kinks) along the $\\tau$-path of SHIM using two different $\\lambda$ values ($\\lambda=1, 10$) for three different sparsity levels ($\\zeta=0.4, 0.7, 0.9$) for different value of ``max\\_pat\" ($d$) sizes.}\n\\label{table:comp_eff_kinks}\n\\end{table}\n\nFig.~\\ref{fig:node_counts} shows the variation of node counts (``fraction of node counts\") for different \\emph{max\\_pat} ($d$) size during the construction of $\\tau$-path. One can observe that our pruning condition is more effective at the deeper nodes of the tree as it exploits the tree's anti-monotonicity property. Table ~\\ref{table:comp_eff_kinks} shows the number of homotopy transition points (\\# kinks) along the $\\tau$-path. Although the worst-case complexity of the $\\tau$-path is exponential (see complexity analysis for details), it appears to be polynomial in practice. In fact, from our numerical results, one can observe that it saturates after a certain depth of the tree. For the experimental setup details of Fig.~\\ref{fig:node_counts} and Table ~\\ref{table:comp_eff_kinks} please see section~\\ref{sec_comp_eff} of the main paper.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe calculation of high order terms in low-energy effective field theories\n (EFTs) is a difficult task. Nowadays, most interesting observables have\nbeen calculated at the second order of the expansion, and the difficulty of \nthese calculations shows little hope\nfor any further expansion. The main problem which restricts the potential of EFTs is their non-renormalizability. The non-renormalizability does\nnot bring any problem, in principle, for the calculation by means of counting schemes for EFTs, first introduced in \\cite{Weinberg:1978kz}.\nHowever, the rapidly increasing number of low-energy coupling constants (LECs), makes very high order applications practically of little use in\ngeneral.\n\nNevertheless, there are contributions of the higher order terms which are free from higher order LECs. In particular this is true for the\nleading logarithmical (LLog) contributions. LLogs are not in general dominant for a generic observable. However, for some observables the LLog\ncontribution is dominant. Examples of such observables are the generalized parton distributions at small-$x$\n\\cite{Kivel:2007jj,Moiseeva:2013qoa} and certain $\\pi\\pi$ scattering lengths \\cite{Colangelo:1995np}. In addition, the LLog terms are of great\ntheoretical interest because they allow us to judge the behaviour of a whole series of corrections in EFTs. We therefore\nconsider the calculation of LLog terms in EFTs as an interesting and useful task.\n\nIn renormalizable field theories the LLog terms can be calculated to all orders using the renormalization group (RG) and (simple) one-loop\ncalculations of the beta functions. In EFTs, as e.g. Chiral Perturbation Theory (ChPT), they can also be calculated using one-loop calculations\nas was suggested already in \\cite{Weinberg:1978kz}, and proven in \\cite{Buchler:2003vw}. In contrast to renormalizable theories, in EFTs the\nLLog terms cannot be obtained simultaneously for all orders, and every order of the perturbative expansion requires an additional\ncalculation. However, the evaluation of LLog terms is considerably simpler than a full calculation. As an example, the full two-loop leading\nlogarithms in bosonic ChPT were known long before the full results \\cite{Bijnens:1998yu}.\n\nWithin bosonic EFTs the LLogs have been studied extensively. It has been shown that for EFTs with massless particles the LLog behaviour is\ndescribed by a closed set of equations with known kernels, which were elaborated in\n\\cite{Kivel:2008mf,Kivel:2009az,Koschinski:2010mr,Polyakov:2010pt}. Although, the analytical solution of these equations is not known, one can\ngenerate numerically the first few hundreds coefficients rather fast, and use the approximate numerical solution in applications. An example is\nthe exploration of the ``chiral inflation'' of the pion radius within ChPT \\cite{Perevalova:2011qi}. Taking into account the mass of the fields\nallows for non-zero tadpole diagrams, which leads to a rapidly increasing number of equations with the chiral order since one has to consider\none-loop diagrams with an ever increasing number of external legs. Therefore, one needs to incorporate new processes at every new\norder. As a result, the difficulty of the calculation grows extremely fast with the chiral order. By automatizing the procedure for a large\nnumber of processes the LLogs are known up to seven loops for some quantities\n\\cite{Bijnens:2009zi,Bijnens:2010xg,Bijnens:2012hf,Bijnens:2013yca}.\n\nThe main goal of this paper is to generalize the methods used for bosonic EFTs\nwith masses to the nucleon case. As mentioned earlier, it\nis not only interesting from the theoretical side, but also necessary for the\nevaluation of nucleon parton distributions at $x\\sim{m_\\pi}\/{M_N}$\n\\cite{Moiseeva:2013qoa,Moiseeva:2012zi}. In the paper we present the extension\nof the RG method of \\cite{Buchler:2003vw} to nucleon-pion ChPT. With its help,\nwe calculate the LLog coefficients for the chiral expansion of the nucleon mass\nin the heavy-baryon formulation of ChPT. The main results are presented in\nSects.~\\ref{sec:resultmass} and \\ref{sec:conjectures}.\nAn earlier application of LLogs in the nucleon sector was the calculation of\nthe two-loop LLog contribution to the axial nucleon coupling constant $g_A$\n\\cite{Bernard:2006te}.\n\nThe paper is organized as follows: In Sect.~\\ref{sec:RGO} we introduce the concept of renormalization group order (RGO). This is needed since in\nthe nucleon sector chiral counting and loop counting are not identical. Sect.~\\ref{sec:meson} shows how the RGO concept\nworks in the meson sector and quotes some known results. Sect.~\\ref{sec:HB_Lagrangian} introduces the heavy baryon ChPT Lagrangian in its two\nmost common variants and the different meson parametrizations we have used as a check on our result. Sect.~\\ref{sec:general_comm} shows how the\nRGO can be used to prove the calculation of the leading logarithms using only one-loop diagrams also in the nucleon sector. This is then used to\ncalculate the LLogs for the nucleon mass in Sect.~\\ref{sec:nucl_mass}. Some technicalities are discussed in Sects.~\\ref{sec:propagator} and\n\\ref{sec:pole}. We then calculate the LLogs for the nucleon mass as well as the odd-power next-to-leading logarithms (NLLogs) in\nSect.~\\ref{sec:resultmass} up to four respectively five loops. The observed regularity in the leading logarithm allows to also calculate the\nfive loop result with a mild assumption. The LLogs, then essentially known to five loops, show a remarkable regularity when rewritten in the\nphysical pion mass. We conjecture that this regularity holds to all orders and in that case using the known results for the pion LLogs we have a\nresult for the nucleon mass LLogs up to 7 loops. This is described in Sect.~\\ref{sec:conjectures}. A short numerical discussion of our results\nis given in Sect.~\\ref{sec:numerics}. We summarize our conclusions in Sect.~\\ref{sec:conclusions}. The LLogs for a general heavy baryon one-loop\nintegral are discussed in App.~\\ref{app:loopintegrals}.\n\n\n\\section{Renormalization group and order}\n\\label{sec:RGO}\n\n\\subsection{Renormalization group operator}\n\nIn this section we present a short, hopefully self-contained, introduction\nto the renormalization group approach in EFTs. Our main goal is to\npresent the method of obtaining the dependence of observables on\nthe renormalization or subtraction scale $(\\mu)$. The material is presented in\na form transparent for the application at higher orders. More extensive\ndiscussions can be found in \\cite{Buchler:2003vw,Bijnens:2009zi,AVthesis}.\nIn particular, what we call LLogs is the contribution with the highest power\nof $\\log\\mu$ at a given order of the expansion.\n\nTo start with, we remind the reader that the Lagrangian of an EFT is the most\ngeneral local Lagrangian satisfying given symmetry properties with a given set of\ndegrees of freedom or fields. Such a Lagrangian contains an infinite number\nof terms. In the absence of additional restrictions, every independent operator\nis multiplied by an unknown coupling constant, usually called low-energy\nconstant (LEC).\n\nIt is convenient to multiply every operator by the counting parameter $\\hbar$\nto the power which reflects the minimal order of the perturbative\nexpansion the operator contributes to. In this way, the constant $\\hbar$\nresembles the coupling constant in a renormalizable field theory as a way to\nkeep track of (loop) orders in the expansion.\nTherefore, an EFT Lagrangian takes the form\n\\begin{eqnarray}\n\\label{1:L=sum L^(n)}\n\\mathcal{L}^{\\text{EFT}}_{\\text{bare}}\n=\\sum_{n=0}^\\infty \\hbar^n\\mathcal{L}^{(n)}_{\\text{bare}}.\n\\end{eqnarray}\nThe Lagrangian $\\mathcal{L}^{(n)}$ we call the Lagrangian of $n$'th\n$\\hbar$-order\\footnote{The Lagrangian which contains the propagator of\nfields, must be included in the zeroth $\\hbar$-order.} and its LECs are\nconsequently called LECs of $n$'th $\\hbar$-order. Let us, following\n\\cite{Bijnens:2009zi}, denote LECs of $n$'th $\\hbar$-order as $c_i^{(n)}$,\nwhere the index $i$ enumerates independent operators. In this way, the\n$n$'th $\\hbar$-order Lagrangian reads\n\\begin{eqnarray}\n\\mathcal{L}^{(n)}_{\\text{bare}}=\\sum_{i} c_{(\\text{bare})i}^{(n)}\\mathcal{O}^{(n)}_i.\n\\end{eqnarray}\nFor some low-energy EFTs, like mesonic ChPT, the $\\hbar$-ordering of operators\nis in one-to-one correspondence with the chiral ordering. However, the\ndefinition (\\ref{1:L=sum L^(n)}) is more general. It can be applied to any EFT,\nand, even, to renormalizable theories, some examples can be\nfound in \\cite{Buchler:2003vw,AVthesis}. We should mention that there is no\nunique definition of the $\\hbar$-ordering for a theory. The only\nconstraint is that the $\\hbar$-order of an operator should increase with\nincreasing perturbative order. The choice made is for EFTs often referred to\nas the choice of power counting.\n\nThe bare Lagrangian is now split into a part with\nrenormalized couplings $c_i^{(n)}$, which depend on $\\mu$, and the counterterms.\nThe renormalization scale independence of the Lagrangian leads to the set of RG\nequations for the LECs $c_i^{(n)}$. These equations are of the form\n\\begin{eqnarray}\n\\label{1:dc=beta}\n\\mu^2\\frac{d}{d\\mu^2}c_i^{(n)}(\\mu^2)=\\beta^{(n)}_{i}(\\{c_j^{(m)}(\\mu^2)\\}),\n\\end{eqnarray}\nwhere the beta-function is a polynomial in the LECs and we have indicated\nexplicitly the $\\mu$-dependence. An important point, used later, is\nthat the right-hand side of (\\ref{1:dc=beta}) contains only combinations of\nLECs with total $\\hbar$-order strictly less then $n$.\n\nThe general formal solution of the system of equations (\\ref{1:dc=beta}) is\n\\begin{eqnarray}\n\\label{defR}\nc_i^{(n)}(\\mu^2)=\\hat R\\(\\frac{\\mu}{\\mu_0}\\)c_{i}^{(n)}(\\mu^2_0)\n=\\exp\\(\\log\\(\\frac{\\mu^2}{\\mu^2_0}\\)\\hat H\\)c_{i}^{(n)}(\\mu^2_0).\n\\end{eqnarray}\nThis defines also $\\hat R$. The operator $\\hat H$ is defined as\n\\begin{eqnarray}\n\\label{defH}\n\\hat H= \\int d\\rho^2\\sum_{n,i}\\beta^{(n)}_i(\\{c^{(m)}_j(\\rho^2)\\})\n\\frac{\\delta}{\\delta c_i^{(n)}(\\rho^2)}.\n\\end{eqnarray}\nThe derivative in (\\ref{defH}) is defined by\n\\begin{eqnarray}\n\\frac{\\delta}{\\delta c_i^{(n)}(\\rho^2)} c_j^{(m)}(\\mu^2)\n= \\delta_{ij}\\delta^{mn}\\delta\\left(\\rho^2-\\mu^2\\right)\n\\end{eqnarray}\nsuch that\n\\begin{eqnarray}\n\\hat H c_i^{(n)}(\\mu^2) = \\beta_i^{(n)}(\\{c_j^{(m)}(\\mu^2)\\})\\,.\n\\end{eqnarray}\nWith the help of $\\hat H$ or $\\hat R$, one can obtain the coefficients of the\nLLog for any observable, without actual calculation of loop\ndiagrams, if the beta-functions are already known. We will demonstrate this\nexplicitly in the next sections.\n\n\\subsection{Renormalization group order}\n\\label{sec:defRGO}\n\nThe crucial property of the operator $\\hat H$ is that the repetitive action of $\\hat H$ nullifies any given LEC (or products of LECs). This is\nthe direct consequence of two features of $\\hbar$-counting. The first one is that the lowest order couplings, with $\\hbar$-order equal to zero,\nhave zero beta-function, and therefore $\\hat H c_i^{(0)}=0$. The second one is that the $\\beta$-function of LEC $c_i^{(n)}$, as defined in\n(\\ref{1:dc=beta}), contains only products of couplings with total $\\hbar$-order lower then $n$. Thus, every application of the operator\n$\\hat H$ onto a product of LECs lowers the total $\\hbar$-order of that product, until it becomes zero.\n\nFor future convenience, we introduce the concept of renormalization group\norder (RGO). A product $P_c$ of LECs has RGO $g$ if\n\\begin{eqnarray}\n\\hat H^g P_c \\ne 0\\quad\\text{and}\\quad\\hat H^{g+1}P_c=0\\,.\n\\end{eqnarray}\nFor a generic\\footnote{We will use this term below to indicate that there are\nexceptions where the beta-functions are zero ``accidentally.'' An\nexample of this is the constant $L_7^r$ in three-flavour bosonic ChPT. This\ndoes not invalidate our later use of the RGO.} quantity with a tree\nlevel contribution of $\\hbar$-order $n$, the RGO is the same as the maximum\nloop order that can appear when calculating that quantity to $\\hbar^n$.\n\nIn the bosonic EFTs treated in the earlier works,\ne.g.\\cite{Kivel:2008mf,Bijnens:2009zi,Bijnens:2010xg,Bijnens:2012hf}, there is\na one-to-one correspondence between $\\hbar$-order and RGO, namely $g=n$.\nTherefore, the notion of RGO is unnecessary and was not used in these works.\nHowever, such a relation does not hold in general, i.e. LECs of different\n$\\hbar$-order can have the same RGO. For example, in the nucleon ChPT,\nthe $\\hbar$-counting as is related to RGO as $g=[n\/2]$, where $[x]$ indicates\nthe integer part of $x$\n(see detailed discussion in Sec. \\ref{sec:general_comm}). In\nsuch a case the use of the RGO concept is convenient.\n\nIt is natural to split the beta-function into terms with the same RGO. For\nthe beta-function of a coupling constant $c_i^{(n)}$ with RGO $g$ we can write\n\\begin{eqnarray}\n\\beta^{(n)}_{i}= \\sum_{p=0}^{g-1}\\beta^{(n,p)}_i(c).\n\\end{eqnarray}\nHere $g=n$ or $[n\/2]$ for the two cases mentioned above.\nOne can show that the part of the beta function with the highest RGO,\n$\\beta^{(n,g-1)}_i$, contains only contributions from one-loop. The next part,\n$\\beta^{(n,g-2)}_{i}$, contains contributions from one and two-loop diagrams\nand so on. Thus, the expression for the operator $\\hat H$ can be ordered by\nRGO as\n\\begin{eqnarray}\n\\hat H=\\sum_{p=1}^\\infty \\hat H_p\\,.\n\\end{eqnarray}\n$\\hat H_p$ contains the beta-functions $\\beta^{(n,g-p)}$ of the coupling\nconstants $c_i^{(n)}$ of RGO $g$. As a consequence,\nacting with $\\hat H_p$ on an expression reduces its RGO by $p$.\n\n\\section{LLog in mesonic ChPT}\n\\label{sec:meson}\n\nIn mesonic ChPT the choice of $\\hbar$-counting versus chiral counting relates\nboth as $\\hbar^n\\sim \\mathcal{O}(p^{2n+2})$. The\nlowest order Lagrangian is of the second chiral order and\nreads\\footnote{We write here only the terms relevant for the mass and\nneglect external fields.}\n\\begin{eqnarray}\n\\mathcal{L}^{(0)}_{\\pi}=\\frac{F^2}{4}\\tr\\[u_\\mu u^\\mu+\\chi_+\\],\n\\end{eqnarray}\nwhere we use the standard notation\n\\begin{eqnarray}\n\\label{2:u_mu_def}\nu_\\mu=i\\(u^\\dagger \\partial_\\mu u-u\\partial_\\mu u^\\dagger\\),~~~~\\chi_+=u^\\dagger \\chi u^\\dagger+u\\chi^\\dagger u=m^2\\(u^2+u^{\\dagger2}\\).\n\\end{eqnarray}\nHere and though the text, $F$ is the bare pion decay constant and $m$ is the\nbare pion mass, $m^2 = 2B\\hat m$ in the notation of\n\\cite{Gasser:1983yg}. $u$ contains the meson fields, \na few examples of possible parametrizations are given in\n(\\ref{u_param1}-\\ref{u_param3}). The next order Lagrangian $\\mathcal{L}^{(1)}$\nis of fourth chiral order. The absence of odd chiral order\nLagrangians is guaranteed by Lorentz invariance.\n\nIn mesonic ChPT, the generic RGO of a LEC $c^{(n)}$ is equal to $n$. The one-to-one correspondence between generic RGO and the $\\hbar$-order is\nthe result of the absence of odd-chiral-order Lagrangians. Any product of LECs is also in one-to-one correspondence with its generic RGO, which\nis equal to the sum of the LECs' $\\hbar$-orders. That, in turn, results in the simple ordering of beta-functions: the beta-function\n$\\beta^{(n,n-l)}$ contains only $l$-loop beta functions.\n\nAs an example of using the operator $\\hat H$ of (\\ref{defH}) to obtain the\nLLog, we look at the physical pion mass. In order to obtain the\nphysical pion mass one should solve the equation\n$m_\\pi^2-m^2+\\Sigma_\\pi(m_\\pi^2,m^2)=0$, where $\\Sigma_\\pi(p^2,m^2)$ is a series\nof perturbative corrections to the pion propagator. The expression for\n$\\Sigma_\\pi$ has the general form\n\\begin{eqnarray}\n\\label{2:Sigma_pi} \\Sigma_\\pi(p^2=m_\\pi^2,m^2)\n  = m^2\\sum_{n=1}^\\infty \\(\\frac{m^2}{(4\\pi F)^2}\\)^{n}\n   \\Sigma^{(n)}_\\pi\\(\\frac{\\mu^2}{m^2},c(\\mu^2)\\),\n\\end{eqnarray}\nwhere $\\Sigma^{(n)}_\\pi$ is a dimensionless expression of maximum\n$\\hbar$-order $n$.\nThe first argument of $\\Sigma_\\pi^{(n)}$ appears only as\nthe argument of logarithms. We have suppressed the arguments $p^2\/m^2$.\nNote, that $p^2\/m^2$ can also enter the arguments of logarithms,\nmoreover there can be logarithms of more complicated expressions of it.\nSuch logarithms are not RG logarithms, and can not in general be obtained by\nany procedure based on RG.\n\nThe expression for $\\Sigma_\\pi$ is renormalization scale\nindependent\\footnote{We use here a scheme where all one-particle irreducible\ndiagrams are made finite, otherwise one should apply the argument to a well\ndefined Green function of external currents.}.\nMoreover, it is renormalization scale invariant at every chiral order\nindependently:\n\\begin{eqnarray}\n\\left[\\mu^2\\frac{\\partial}{\\partial\\mu^2}\n+\\sum_{i,n}\\beta^{(n)}_i\\frac{\\partial}{\\partial c_i^{(n)}(\\mu^2)}\\right]\n\\Sigma^{(n)}_\\pi\\(\\frac{\\mu^2}{m^2},c(\\mu^2)\\)\n=\n\\left[\\mu^2\\frac{\\partial}{\\partial\\mu^2}\n+\\hat H\\right]\n\\Sigma^{(n)}_\\pi\\(\\frac{\\mu^2}{m^2},c(\\mu^2)\\)\n= 0\\,.\n\\end{eqnarray}\nTherefore, we again have as solution, similar to (\\ref{defR}),\n\\begin{eqnarray}\n\\label{2:Sigma=R Sigma}\n\\Sigma^{(n)}_\\pi\\(\\frac{\\mu^2}{m^2},c(\\mu^2)\\)=\\hat\nR\\(\\frac{\\mu^2}{\\mu_0^2}\\)\\Sigma^{(n)}_\\pi\\(\\frac{\\mu_0^2}{m^2},c(\\mu_0^2)\\).\n\\end{eqnarray}\nChoosing $\\mu_0^2=m^2$, one neglects all the RG logarithms in\n$\\Sigma^{(n)}_\\pi$ on the right-hand-side.\nThus, all RG logarithms are collected in\nthe action of the operator $\\hat R$.\n\nThe expression for $\\Sigma^{(n)}_\\pi$ has the following form\n\\begin{eqnarray}\n\\Sigma^{(n)}_\\pi=\\sum_{i}\\{c^{(n)}_{i}\\}V^{(n)}_i+\\text{terms with lower RGO},\n\\end{eqnarray}\nwhere $\\{c^{(n)}_{i}\\}V^{(n)}_i$ form the tree contribution to $\\Sigma^{(n)}$.\nThe symbol $\\{c^{(n)}_{i}\\}V^{(n)}_i$ denotes here the products of $c_j^{(m)}$\nwith the highest possible RGO for the product. The $V^{(n)}_i$ depend on\n $p^2\/m^2$. The highest power of the RG logarithm\n$\\log\\mu^2$ in the expression (\\ref{2:Sigma=R Sigma}) accompanies the highest\npower of $\\hat H$, in $\\hat R = \\exp(\\log(\\mu^2\/\\mu_0)\\hat H)$, which gives a\nnon-zero result acting on $\\Sigma^{(n)}$. Since every action of $\\hat H$\nreduces the RGO of expression, the coefficient of the\nLLog is\n\\begin{eqnarray}\n\\label{2:LL_mass_pi} \\Sigma^{(n)}_\\pi\\(\\frac{\\mu^2}{m^2},c(\\mu^2)\\)= \\frac{1}{n!}\\log^{n}\\(\\frac{\\mu^2}{m^2}\\)\\hat H_1^{n}\n\\sum_{i}c^{(n)}_iV^{(n)}_i+\\mathcal{O}\\(\\text{NLLog}\\),\n\\end{eqnarray}\nwhere NLLog is the the acronym for the next-to-leading logarithms, and\nhence $\\mathcal{O}(\\text{NLLog})$ denotes the part of expression without\nLLog.\n\nTherefore, constructing the higher chiral order Lagrangians,\nand calculating the one-loop beta-functions of their LECs, one can obtain\nthe LL coefficients without actual calculation of multi-loop diagrams.\nMoreover, the result is independent on the details of the higher order\nLagrangians, as long as they are sufficiently general for the process at\nhand \\cite{Bijnens:2009zi}.\nPractically it is convenient to use a non-minimal Lagrangian generated\n``on-the-fly'' by the counterterms to one-loop diagrams only.\nThis was the approach used\nin \\cite{Bijnens:2009zi,Bijnens:2010xg,Bijnens:2012hf,Bijnens:2013yca} for\nthe mesonic theory for several processes.\n\nThe solution of the pole equation gives us the expression for the physical\npion mass to LLog accuracy. It is known up to sixth power of\nlogarithms \\cite{Bijnens:2012hf} and reads\n\\begin{eqnarray}\n\\label{2:mPhys}\nm^2_\\text{phys}=m^2\\(\n  1-\\frac{1}{2}L+\\frac{17}{8}L^2-\\frac{103}{24}L^3+\\frac{24367}{1152}L^4\n  -\\frac{8821}{144}L^5+\\frac{1922964667}{6220800}L^6+\\cdots\\),\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\label{2:L_def}\nL=\\frac{m^2}{(4\\pi F)^2}\\log\\(\\frac{\\mu^2}{m^2}\\).\n\\end{eqnarray}\n\n\n\\section{Heavy baryon Lagrangian}\n\\label{sec:HB_Lagrangian}\n\nThe nucleon-meson ChPT in the naive form has the problem of the large nucleon\nmass $M$. There are several ways of dealing with the presence\nof this large scale, each with advantages and disadvantages. In this article,\nwe use the heavy baryon approach to meson-nucleon ChPT since in this approach\nall scales that explicitly appear are soft and there are no divergences\nnor $\\mu$-dependence associated directly with the scale $M$.\n\nFor the LLog calculation, we have to determine the Lagrangians of zero RGO.\nFor the pion-nucleon system, these are Lagrangians of the first and\nthe second chiral orders. The first chiral order Lagrangian, neglecting terms\nwith external fields, reads\n\\begin{eqnarray}\\label{3:L0}\n\\mathcal{L}^{(0)}_{N\\pi}=\\bar N\\(i v^\\mu D_\\mu+g_A S^\\mu u_\\mu\\)N,\n\\end{eqnarray}\nwhere $S^\\mu$ is a spin vector. We use the standard notation for the field\n combinations (see also the definitions in (\\ref{2:u_mu_def})):\n\\begin{eqnarray}\nD_\\mu&=&\\partial_\\mu+\\Gamma_\\mu, \\qquad u^2=U, \\qquad\n\\Gamma_\\mu= \\frac{1}{2}\\(u^\\dagger \\partial_\\mu u+u \\partial_\\mu u^\\dagger\\).\n\\end{eqnarray}\n\n\nThe second order Lagrangian is sensitive to redefinitions of the nucleon\nfield. The most standard form of the second chiral order heavy\nbaryon Lagrangian reads \\cite{Bernard:1992qa,Bernard:1995dp}\n\\begin{eqnarray}\n\\label{3:BKKM}\n\\mathcal{L}_{\\pi N}^{(1)}&=&\\bar N_v\\Big[\n \\frac{(v\\cdot D)^2-D\\cdot D-ig_A \\{S\\cdot D,v\\cdot u\\}}{2M}\n +c_1\\tr\\(\\chi_+\\)+\\(c_2-\\frac{g_A^2}{8M}\\)(v\\cdot u)^2\n \\nn\\\\&&\n \\hspace*{1cm}\n  +c_3 u\\cdot u+\\(c_4+\\frac{1}{4M}\\)i\\epsilon^{\\mu\\nu\\rho\\sigma}u_\\mu u_\\nu v_\\rho\n          S_\\sigma \\Big]N_v.\n\\end{eqnarray}\nA different but equivalent version of the second order chiral Lagrangian is\ngiven in \\cite{Ecker:1995rk}, and it reads\n\\begin{eqnarray}\\label{3:EM}\n\\mathcal{L}^{(1)}_{N\\pi}&=&\n\\frac{1}{M}\\bar N\\Big[\n -\\frac{1}{2}\\(D_\\mu D^\\mu+i g_A\\{S_\\mu D^\\mu,v_\\nu u^\\nu\\}\\)\n +A_1 \\Tr\\(u_\\mu u^\\mu\\)+A_2 \\Tr\\((v_\\mu u^\\mu)^2\\)\n\\nn\\\\&&\n\\hspace*{1cm}\n  +A_3 \\Tr\\(\\chi_+\\)+A_5i\\epsilon^{\\mu\\nu\\rho\\sigma}v_\\mu S_\\nu u_\\rho u_\\sigma\n  \\Big]N.\n\\end{eqnarray}\nThe relation between the LECs $A_i$ and $c_i$ is the following\n\\begin{eqnarray}\nA_1=\\frac{Mc_3}{2}+\\frac{g_A^2}{16},~~A_2=\\frac{Mc_2}{2}-\\frac{g^2_A}{8},~~A_3=Mc_1,~~A_5=Mc_4+\\frac{1-g_A^2}{4}.\n\\end{eqnarray}\n\nAlthough the S-matrix elements are independent of the parametrization of\nthe nucleon field, the contributions of individual diagrams, and\nexpressions for the beta-functions are dependent on the field parametrization.\nTherefore, the comparison of results for calculations performed\nin different parameterizations is a very strong check of a calculation. The\ncalculations presented in the next sections have been done in both\nparametrizations of the nucleon field. Additionally for further cross-checks,\nwe used different parameterizations of the pion field $u$. We have\nused\n\\begin{eqnarray}\\label{u_param1}\nu=\\exp\\(i\\frac{\\pi^a\\tau^a}{2F}\\),\n\\end{eqnarray}\n\\begin{eqnarray}\\label{u_param2}\nu=\\sqrt{1-\\frac{\\vec\\pi^2}{4F^2}}+i\\frac{\\pi^a\\tau^a}{2F},\n\\end{eqnarray}\nand\n\\begin{eqnarray}\\label{u_param3}\nu=\\sqrt{\\frac{Y}{\\sqrt{2}}}+i\\frac{\\pi^a\\tau^a}{F}\\sqrt{\\frac{1}{2Y}}\n\\qquad\\text{with}\\qquad Y=1+\\sqrt{1-\\frac{\\vec\\pi^2}{F^2}}.\n\\end{eqnarray}\n\n\n\\section{LLog in nucleon-meson ChPT: general comments}\n\\label{sec:general_comm}\n\nThe consideration of the LLog behavior of nucleon-meson systems is similar to\nmeson systems, but with some additional features. The main additional feature\nof meson-nucleon systems is the presence of operators with odd number of\nderivatives. Therefore, the relation between $\\hbar$-order and chiral\norder is $\\hbar^n\\sim \\mathcal{O}(p^{n+1})$ for the single-nucleon sector of\nthe ChPT Lagrangian, and $\\hbar^n\\sim \\mathcal{O}(p^{n+2})$ for the meson\nsector of EFT Lagrangian. At the same time, every loop increases the chiral\norder by at least two. Therefore, the RGO is not in\none-to-one correspondence with $\\hbar$-order. A LEC of $n$'th $\\hbar$-order\n$c^{(n)}$ has generically an RGO $\\[\\frac{n}{2}\\]$. This has\nimportant consequences in the RG and LLog structure of the theory.\n\nThe first consequence is the contribution of diagrams with different\nRGO to the same chiral order. Indeed, a loop diagram with several\nvertices of even chiral order (i.e. odd $\\hbar$ order) has an RGO less\nthen a diagram with the same chiral order but with fewer\neven-chiral-order vertices. An example of diagrams with the same chiral\norder but different RGO is shown in Fig.~\\ref{fig:1}.\n\\begin{figure}[tb!]\n\\centering\n\\includegraphics[width=0.5\\textwidth]{1loopamb.eps}\n\\caption{An example of diagrams which contribute to the renormalization of the nucleon mass at the same chiral order (here, the fifth chiral\norder), but which have different RGO. The left diagram has zero RGO, while the right diagram has RGO equal to one. Therefore, the left diagram\ndoes not contribute to the LLogs. The number in the box indicates the chiral order of the vertex. Thick arrowed lines indicate nucleon\npropagators, thin lines indicate pion propagators.}\n\\label{fig:1}\n\\end{figure}\nUsing the relation between chiral order and RGO, one can see that every\ntwo even-chiral-order vertices reduce the RGO of a diagram by one, from\nthe possible maximum. For example: for any diagram with two even-chiral-order\nvertices with certain RGOs, there exists diagrams of the same chiral order\nand of the same topology, but with these two vertices replaced by\nodd-chiral-order vertices, one with a chiral order one lower and one\nchiral-order one higher. The lower chiral order has the same generic RGO as\nthe even vertex but the higher chiral order vertex has generic RGO one higher.\nThe latter diagram has thus a higher RGO. We conclude that at a given chiral\norder the highest RGO contribution is given by the diagrams with\n\\textit{zero or one} vertex of even chiral order.\n\n\nThe second consequence is the ambiguity of the definition of a leading\nlogarithm. The natural definition of LLogs, the logarithm of a maximum\npower of $\\log\\mu^2$ at a given chiral order, does not always coincides with\nthe RG definition for the same observable. In Sect.~\\ref{sec:nucl_mass} we\nwill show that in the chiral expansion of the physical nucleon mass, the LLog\nterms of odd chiral order are actually of NLLog origin when seen\nfrom an RGO perspective.\n\n\\section{Nucleon mass at LLog accuracy}\n\\label{sec:nucl_mass}\n\n\\subsection{Propagator at LLog accuracy}\n\\label{sec:propagator}\n\nThe physical mass of the nucleon is given by  the position of the pole in the\nDyson propagator. In the heavy baryon approach the inverse Dyson\npropagator reads (we remind the reader that superscripts $n$ refer to the\n$\\hbar$-order of quantities, and that the meson and meson-nucleon\nsectors of the action have different chiral counting)\n\\begin{eqnarray}\n\\label{5:inverse_prop}\nS^{-1}=(rv)+\\sum_{n=1}^\\infty \\Sigma^{(n)}((r\\cdot v),r^2),\n\\end{eqnarray}\nwhere $r^\\mu=p^\\mu-Mv^\\mu$, $p$ is the momentum of the nucleon, and $v_\\mu$\nis the reference four velocity of the heavy nucleon. There are some\nintricacies of defining the propagator and renormalization of the nucleon\nwave function in heavy baryon theory, see e.g.\n\\cite{Ecker:1995rk,Kambor:1998pi}. However for the determination of the\nnucleon mass the straightforward usage of (\\ref{5:inverse_prop}) is sufficient.\n\nThe expressions for $\\Sigma^{(n)}$ are the result of the calculation of\none-particle irreducible diagrams with a nucleon line at the $(n+1)$'th\nchiral order. The maximum power of the logarithm $\\log\\mu^2$ which can appear\nin $\\Sigma^{(n)}$ is $\\[\\frac{n}{2}\\]$.\n\nThe derivation of the LLog coefficient is the same as the derivation of\nexpression (\\ref{2:LL_mass_pi}). We collect all the RG-logarithms by the\naction of the normalization point rescaling operator $\\hat R$, again\nsuppressing the other arguments of $\\Sigma^{(n)}$,\n\\begin{eqnarray}\n\\label{5:Sigma=R Sigma(nucl)}\n\\Sigma^{(n)}\\(\\frac{\\mu^2}{m^2},c(\\mu^2)\\)=\\hat\nR\\(\\frac{\\mu^2}{\\mu_0^2}\\)\\Sigma^{(n)}\\(\\frac{\\mu_0^2}{m^2},c(\\mu_0^2)\\).\n\\end{eqnarray}\nThe highest RGO in $\\Sigma^{(n)}$ has the tree diagram with only a $c_i^{(n)}$\nvertex, therefore, the LLog coefficient is given by\n\\begin{eqnarray}\\label{5:LL_mass_nucl}\n\\text{LLog coef.}=\n\\(\\[\\frac{n}{2}\\]!\\)^{-1}\\hat H_1^{\\[n\/2\\]} \\sum_{i}c^{(n)}_iV^{(n)}_i,\n\\end{eqnarray}\nwhere $V^{(n)}_i$ are expression for the $c^{(n)}_i$-vertices.\n\nIt is also interesting to look at the NLLog contribution. The NLLog coefficient\ncomes from the $([n\/2]-1)$ term of the exponent series in\n(\\ref{5:Sigma=R Sigma(nucl)}). There are several different parts\nof $\\Sigma^{(n)}$ which survive after the action by $\\hat H^{[n\/2]-1}$.\nThese are the terms with RGO $\\[\\frac{n}{2}\\]$ and $\\[\\frac{n-2}{2}\\]$. While\nthe first are given by tree diagrams, the second are given by\none-loop diagrams:\n\\begin{eqnarray}\n\\Sigma^{(n)}\\(\\frac{\\mu^2}{m^2},c(\\mu^2)\\)=\\sum_i c^{(n)}_i(\\mu^2) V^{(n)}_i\n+\\sum_{\\text{1-loop diag.}}W\\(\\frac{\\mu^2}{m^2},c^{(k)}(\\mu^2)\\)+\\cdots,\n\\end{eqnarray}\nwhere the $V_i$ are the expressions for the tree level diagrams and $W$\nindicates the  expressions for the one-loop diagrams at a fixed\nrenormalization scale. The dots represents the contributions with lower RGO.\nWe note that $W$ contains only one-loop diagrams with RGO equal to\n$\\[\\frac{n-2}{2}\\]$, but not all possible one-loop diagrams.\nThe NLLog coefficient is given by\n\\begin{eqnarray}\n\\label{5:NLL} \\text{NLLog coef.}&=&\n   \\frac{1}{([n\/2]-1)!}\\hat H_1^{[n\/2]-1}\\sum_i c^{(n)}_i V^{(n)}_i\n\\nn\\\\&&\n +\\frac{1}{([n\/2]-1)!}\\[\\sum_{k=0}^{[n\/2]-1}\n     \\hat H_1^{k}\\hat H_2 \\hat H_1^{[n\/2]-1-k}\\]\\sum_i c^{(n)}_i V^{(n)}_i\n\\nn\\\\&&\n  +\\frac{1}{([n\/2]-1)!}\\hat H_1^{[n\/2]-1} \\sum_{\\text{1-loop diag.}}\n      W\\(\\frac{\\mu_0^2}{m^2},c^{(k)}\\),\n\\end{eqnarray}\nwhere $\\hat H_2$ contains two-loop beta-functions in addition to\none-loop beta-functions.\n\nThe expressions given by the different terms on the right-hand-side\nin (\\ref{5:NLL}) have significantly different properties. The first term\nof (\\ref{5:NLL}) gives the contribution to NLLogs with LECs from the\nnext-to-leading chiral Lagrangian. These terms are LLog terms of\n$([n\/2]-1)$-loop diagrams with insertion of higher order vertices.\nThe second line of (\\ref{5:NLL}) gives the ``true'' NLLog contribution with\nLECs of the lowest order Lagrangian only. These are the NLLog terms\nof $[n\/2]$-loop diagrams. The third line represents the non-analytical\ncontribution of $[n\/2]$-loop diagrams to the NLLog coefficient. We should\nmention that the part of NLLog coefficient given by the second line is\nrenormalization-scheme-dependent, while the parts given by the first and the\nthird lines are scheme-independent.\n\nIf the quantity has no tree-order contribution, the only non-zero part\nof (\\ref{5:NLL}) is the last line. In this case the NLLog can be\ncalculated from one-loop diagrams only. An example of such behavior\nare the non-analytic in quark mass terms. These terms result only from\nthe loops and therefore, their contribution to NLLog can\nbe calculated with one-loop diagrams only.\nThe methods of \\cite{Bijnens:2009zi} can also be used to prove this. The\nabsence of the tree level contribution allows the NLLog to be determined from\nthe set of equations relating the different loop-order contributions.\n\n\n\\subsection{Pole equation at LLog accuracy}\n\\label{sec:pole}\n\nIn this section we discuss the properties of the solution of the pole equation\nat LLog accuracy. From the previous discussion it follows that it is\nalso valid for the NLLog multiplied by a nonanalytic power of the quark mass.\n\nThe position of the pole in the propagator (\\ref{5:inverse_prop}) is a\nLorentz invariant quantity, when evaluated to all orders in the expansion.\nTherefore, one can choose\\footnote{If one chooses $v^\\mu=(1,0,0,0)$\nthis corresponds to $\\vec r=\\vec p=0$.}\n$r^\\mu$ or $p^\\mu$ such that $r^2=(r\\cdot v)^2$.\nThen, $\\omega = (r\\cdot v)$ gives the difference between\nthe physical mass and the bare mass, $(r\\cdot v)=\\delta M=M_{\\text{phys}}-M$.\nIn this regime we can expand the expression for $\\Sigma^{(n)}$ in\npowers of $\\delta M$\n\\begin{eqnarray}\n\\label{5:sigma_def}\n\\Sigma^{(n-1)}(\\delta M)=\\sum_{k=0}^n \\sigma^{k,n-k}\\delta M^k m^{n-k},\n\\end{eqnarray}\nwhere the coefficients $\\sigma^{(i,j)}$ contain logarithms. The coefficients\n$\\sigma^{k,n-k}$ have mass-dimension $(1-n)$.\n\nThe solution of the pole equation $S^{-1}=0$, (\\ref{5:inverse_prop}),\ncan be found perturbatively in $m$\n\\begin{eqnarray}\n\\label{5:delta_expansion}\n\\delta M=\\sum_{n=2}^\\infty a_n m^n\\,\n\\end{eqnarray}\nwhere again the $a_i$ contain logarithms. Inserting the expansions (\\ref{5:delta_expansion}) and (\\ref{5:sigma_def}) into the Dyson propagator\n(\\ref{5:inverse_prop}), and considering the pole equation for every power of $m$ independently, we obtain a system of equations for the\ncoefficients $a_n$,\n\\begin{eqnarray}\n\\label{5:eqns}\na_{n}+\\sum_{\\{i\\},j\\leqslant n-2}a_{i_1}a_{i_2}...a_{i_j}\\sigma^{j,n-\\sum i}=0,\n\\end{eqnarray}\nwhere summation runs over all possible sets of indices including empty set and permutations.\n\nLet us consider the system of equations (\\ref{5:eqns}) in the LLog regime.\nWe recall that the power of the LLog is $\\[\\frac{n}{2}\\]$ for $\\Sigma^{(n)}$.\nHowever, the coefficients $a_n$ have different logarithm counting.\nThe reason is the presence of terms non-analytical in quark masses. The\nLagrangian of ChPT is necessarily analytical in the quark masses, i.e. it\ncontains only even powers of $m$. The terms non-analytic in quark masses\nappear only through loop-integrals, and, therefore, they can not appear in\nthe expression (\\ref{5:LL_mass_nucl}) or in the first two lines of the\nexpression (\\ref{5:NLL}). In this way, the number of logarithms in front of\nthe pion mass in the odd power is suppressed by one (at least). Summarizing,\nwe obtain the following LLog counting for the coefficients $a$ and $\\sigma$\n\\begin{eqnarray}\\label{5:LL_counting}\n a_n \\sim \\log^{\\[(n-2)\/2\\]}(\\mu),~~~~\\sigma^{s,t}\\sim\n \\left\\{\\begin{array}{cc} \\log^{[(s+t-1)\/2]}(\\mu) & t\\in\\text{even}\\\\\n                 \\log^{[(s+t-3)\/2]}(\\mu) & t\\in\\text{odd}\n        \\end{array}\n \\right. .\n\\end{eqnarray}\n\nUsing the counting (\\ref{5:LL_counting}), we neglect the NLLog terms in the\nequations (\\ref{5:eqns}) and obtain the system of equations in the\nLLog regime:\n\\begin{eqnarray}\\label{5:a_even}\n&&a_n+\\sigma^{0,n}+\\sum_{k=2,4,..}^{n-2}a_k \\sigma^{1,n-k}=0,\n\\qquad n\\in \\text{even},\n\\\\\n\\label{5:a_odd}\n&& a_n+\\sigma^{0,n}+a_{n-1}\\sigma^{1,1}+\\sum_{k=3,5,..}^{n-2}a_k \\sigma^{1,n-k}=0,\n\\quad n\\in\\text{odd}.\n\\end{eqnarray}\nThis is a system of linear equations. The important result is that the\neven-$n$ coefficients allow LLog evaluation only, because they\ninvolve only the LLog coefficient of analytical in quark mass terms. At the\nsame time, the odd coefficients involve the terms non-analytical in\nquark masses. These coefficients are really NLLog.. However, they can\nbe obtained from a one-loop calculation as well, because they follow from\nthe third line of (\\ref{5:NLL}).\n\nOne can see that the system (\\ref{5:a_even}-\\ref{5:a_odd}) involves only the\ncoefficients $\\sigma^{0,n}$ and $\\sigma^{1,n}$, which are the\ncoefficients of the zeroth and the first powers of $(r\\cdot v)$ in the\npropagator diagrams. It is a reflection of the fact that according to\n(\\ref{5:LL_counting}) the quantity $(r\\cdot v)^2=\\delta M^2$ is of NLLog\norder. Therefore, the powers of $\\omega$ can be eliminated from the\nequation $S^{-1}=0$. The solution can be presented in the simple form\n\\begin{eqnarray}\\label{5:mass_sol}\n\\delta M= \\frac{-\\Sigma(0)}{1+\\Sigma'(0)}+\\mathcal{O}(\\text{NLLog}),\n\\end{eqnarray}\nwhere $\\Sigma(r\\cdot v)=\\sum_{n}\\Sigma^{(n)}(r\\cdot v)$ and\n$\\Sigma'$ is its derivative with respect to $(r\\cdot v)$.\n\n\\subsection{Expression for the physical mass}\n\\label{sec:resultmass}\n\nWe have performed the calculation of the nucleon mass up to the fourth power\nof RG logarithms. We present the results in the form:\n\\begin{eqnarray}\n\\label{mainresult}\nM_{\\text{phys}}&=&M+k_2 \\frac{m^2}{M}+k_3 \\frac{\\pi m^3}{(4\\pi F)^2}\n  +k_4 \\frac{m^4}{(4\\pi F)^2 M}\\log\\(\\frac{\\mu^2}{m^2}\\)\n  +k_5 \\frac{\\pi m^5}{(4\\pi F)^4}\\log\\(\\frac{\\mu^2}{m^2}\\)+\\cdots\n\\nn\\\\\n&=&\nM+\\frac{m^2}{M}\\sum_{n=1}^\\infty k_{2n} L^{n-1}\n+\\pi m\\frac{m^2}{(4\\pi F)^2}\\sum_{n=1}^\\infty k_{2n+1} L^{n-1},\n\\end{eqnarray}\nwhere $L$ is defined in (\\ref{2:L_def}). The coefficients up to $k_{11}$ are\npresented in Tab.~\\ref{table_mass}. This corresponds to the\nfour-loop calculation of LLog and five-loop calculation for the terms\nnon-analytical in quark masses.\n\nThe presented results have been obtained via the different parametrizations of the Lagrangians (see sec.\\ref{sec:HB_Lagrangian}), which gives a\nvery strong check of calculation. Additionally, the coefficients up to $k_6$ agree with known results. The one-loop coefficients $k_{3,4}$ are\nwell known, see e.g. \\cite{Bernard:1995dp}. The two-loop coefficient $k_{5}$ was first derived in \\cite{McGovern:1998tm}. The two-loop\ncoefficients $k_6$ and $k_5$ are known from the full two-loop calculation for the nucleon mass performed in the EOMS scheme\n\\cite{Schindler:2007dr}.\n\n\\begin{table}[tbh!]\n\\begin{center}\n\\begin{tabular}{|c||l|}\n\\hline\n\\rule{0ex}{2.5ex} $k_2$ & $-4c_1 M$\n\\\\[1mm]\\hline\n\\rule{0ex}{2.5ex}$k_3$ & $-\\frac{3}{2}g_A^2$\n\\\\[1mm]\\hline\n\\rule{0ex}{2.5ex}$k_4$ & $\\frac{3}{4}\\(g_A^2+(c_2+4c_3-4c_1)M\\)-3c_1M $\n\\\\[1mm]\\hline\n\\rule{0ex}{2.7ex}$k_5$ & $\\frac{3g_A^2}{8}\\(3-16 g_A^2\\)$\n\\\\[1mm]\\hline\n\\rule{0ex}{2.5ex}$k_6$ & $-\\frac{3}{4}\\(g_A^2+(c_2+4c_3-4c_1)M\\)+\\frac{3}{2}c_1M $\n\\\\[1mm]\\hline\n\\rule{0ex}{2.9ex}$k_7$ & $g_A^2\\(-18 g_A^4+\\frac{35 g_A^2}{4}-\\frac{443}{64}\\)$\n\\\\[2mm]\\hline\n\\rule{0ex}{2.5ex}$k_8$ & $\\frac{27}{8}\\(g_A^2+(c_2+4c_3-4 c_1)M\\)-\\frac{9}{2}c_1M $\n\\\\[1mm]\\hline\n\\rule{0ex}{2.9ex}$k_9$ & $\\frac{g_A^2}{3}\\(-116 g_A^6+\\frac{2537 g_A^4}{20}-\\frac{3569 g_A^2}{24}+\\frac{55609}{1280}\\)$\n\\\\[2mm] \\hline\n\\rule{0ex}{2.5ex}$k_{10}$ & $-\\frac{257}{32}\\(g_A^2+(c_2+4c_3-4c_1)M\\)+\\frac{257}{32}c_1M $\n\\\\[1mm]\\hline\n\\rule{0ex}{2.9ex}$k_{11}$ & $\\frac{g_A^2}{2}\\(-95 g_A^8+\\frac{5187407 g_A^6}{20160}-\\frac{449039\n   g_A^4}{945}+\\frac{16733923 g_A^2}{60480}-\\frac{298785521}{1935360}\\)$\n\\\\[2mm]\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{The coefficients $k_i$ defined in (\\ref{mainresult}) of the LLog expansion of the nucleon mass.} \\label{table_mass}\n\\end{table}\n\nThe generation of the higher order Lagrangians and the evaluation of one-loop\nbeta-functions has been done automatically using the computer algebra system\nFORM \\cite{FORM}. The algorithm we used is similar to that used and described\nin \\cite{Bijnens:2009zi,Bijnens:2010xg}. The main integral needed for\nevaluation of beta-functions is presented in the appendix. Although the\ncalculation involves only one-loop diagrams, it is very demanding in\nmachine time and memory. The most demanding factor is the length of the \nexpression for the high order effective vertices and\nthe number of diagrams to compute.\nThese quantities grow rapidly with chiral order. For example, in order\nto calculate the $k_{10}$ coefficient one needs to evaluate nearly $10^4$\none-loop diagrams.\n\nThe calculation of the even coefficients $k$ can be significantly simplified by using the conjectures discussed below in\nSect.~\\ref{sec:conjectures}. So, by neglecting higher powers of $g_A$ during the evaluation of the diagrams, we could also evaluate the\nfive-loop coefficient $k_{12}$. Adding the further conjecture about the relation with the LLog in the pion mass, we can obtain the six and\nseven-loop coefficients $k_{14}$ and $k_{16}$. However, these coefficients are the result of conjectures and, therefore, are presented in\nTab.~\\ref{table_mass_conjecture}, separately from the results of the full calculation.\n\n\\subsection{Properties of the result and conjectures}\n\\label{sec:conjectures}\n\nThe straightforward calculation, limited by the available computer power,\ngives us\nthe coefficients $k_1,\\ldots,k_{11}$ presented in Tab.~\\ref{table_mass}.\nConsidering the presented coefficients a number of regularities show up\nimmediately. Some of the regularities we can explain easily, while some of\nthem we cannot.\n\nThe first observation is that only even powers of $g_A$ show up.\nThis can be easily understood. The meson Lagrangian and the nucleon Lagrangian\nare invariant under the transformation $u\\leftrightarrow u^\\dagger$\nand $g_A\\leftrightarrow -g_A$. As a consequence only terms even in $g_A$ can\nappear in the nucleon mass. This explains the pattern occurring in the\nodd coefficients $k_{2n+1}$.\n\nThe second observation is that the coefficients $k_{2n}$ contains a very\nparticular combination of LECs. The pattern appearing in $k_{2n}$ is not\nwell understood yet. Let us consider it in detail.\n\nAs shown in Sec.~\\ref{sec:general_comm}, one can have at most one\ninsertion of the order $p^2$ Lagrangian. The expression for $k_{2n}$ are\nthus at most linear in $c_1,c_2,c_3$ and the other terms\nin $\\mathcal{L}^{(1)}_{\\pi N}$. The coupling constant $c_4$ or $A_5$ cannot\nenter the nucleon mass at LLog since it produces\nan $\\epsilon_{\\mu\\nu\\alpha\\beta}$. However, we found no simple argument why\n$g_A$ only appears up to order $g_A^2$.\n\nFor the powers of $g_A$,\nthere are two sources for factors of $g_A$ in the loop diagrams, namely\nfrom the vertices $\\mathcal{L}^{(0)}$ (\\ref{3:L0}) and from the vertices\n$\\mathcal{L}^{(1)}$ (\\ref{3:BKKM}-\\ref{3:EM}). While the number of vertices\nfrom $\\mathcal{L}^{(1)}$ is restricted to one, the number of\nvertices from $\\mathcal{L}^{(0)}$ is naturally unrestricted. Moreover, the\nexpression for $\\Sigma$ contains all allowed powers of $g_A$. These\npowers cancels within the solution (\\ref{5:mass_sol}). We have checked that if\none introduces new LECs for the terms proportional to\n$g_A$ in $\\mathcal{L}^{(1)}$ (say coefficients $B_{1,2}$ in front of the first\ntwo terms in (\\ref{3:EM})) the coefficients $k_{2n}$ would\ncontain higher powers of $g_A$. These induced higher powers are proportional\nto $(B_1-B_2)$ and disappear when $B_1=B_2$. Since these operators\nappear in $\\mathcal{L}^{(1)}$ as the compensation of the non-relativistic\nnucleon reference frame, we conclude that absence of higher powers\nof $g_A$ in coefficients $k_{2n}$ is a consequence of Lorentz invariance.\n\nSupposing that the cancelation of the higher powers of $g_A$ takes place at\nall orders, one can neglect these powers during the computation of\ndiagrams. This procedure significantly reduces the demands for computer time\nand allows us to calculate the coefficient $k_{12}$, which is\npresented in Tab.~\\ref{table_mass_conjecture}.\n\n\nConsidering the equation (\\ref{5:a_even}) one can see that the coefficients\n$k_{2n}$ consist from the terms proportional to exactly the first\npower of $\\sigma^{0,k}$ where $k$ is even. We remind that $\\sigma^{0,\\text{even}}$ are the result of diagrams with even chiral order and hence\nproportional to a single vertex from $\\mathcal{L}^{(1)}$ (which is also checked by explicit calculation with coefficients $B_{1,2}$). Therefore,\nthe term $g_A^2$ which appears in the coefficients $k_{2n}$ resulted solely from $\\mathcal{L}^{(1)}$. In its own turn, it implies that there is\nno contribution from the diagrams with vertices proportional to $g_A$ only from $\\mathcal{L}^{(0)}$. All such vertices have an odd number of\npions. Absence of such vertices implies that \\textit{diagrams with more than\n two odd-number-of-pion vertices do not contribute to the LLog\ncoefficient of nucleon mass}. Undoubtedly such a structure is a consequence\nof the additional subtractions of infrared (heavy\nmass) singularities into renormalization counterterms within heavy baryon\ntheory, but we have not been able to prove this.\n\nConsidering the first six coefficients $k_{2n}$ one can observe that they have\nthe pattern\n\\begin{eqnarray}\n\\label{5:pattern}\nk_{2n}=b_n\\(\\frac{-3c_1M}{n-1}+\\frac{3}{4}\\(g_A^2+(c_2+4c_3-4c_1)M\\)\\),\n\\end{eqnarray}\nwhere $b_n$ are some rational numbers. The coefficients $b_n$ can be obtained\nfrom the calculation of the physical pion mass as we demonstrate\nbelow.\n\nThe nucleon mass LLog coefficient in terms of the physical pion mass $m_{\\text{phys}}$ has the form\n\\begin{eqnarray}\n\\label{mainresult2} M_{\\text{phys}}&=&\nM+\\frac{m_\\text{phys}^2}{M} \\sum_{n=1}^\\infty r_{2n} L_\\pi^{n-1}\n +\\pi m_\\text{phys}\\frac{m_\\text{phys}^2}{(4\\pi F)^2}\n     \\sum_{n=1}^\\infty r_{2n+1} L_\\pi^{n-1},\n\\end{eqnarray}\nwhere\n$$\nL_\\pi=\\frac{m^2_\\text{phys}}{(4\\pi F)^2}\\log\\(\\frac{\\mu^2}{m_\\text{phys}^2}\\).\n$$\nThe coefficients $r_n$ of this expansion are presented in\nTab.~\\ref{table_mass_phys}.\n\\begin{table}[tb!]\n\\begin{center}\n\\begin{tabular}{|c||l|}\n\\hline\n\\rule{0ex}{2.5ex}$r_2$ & $-4c_1 M$\n\\\\[1mm]\\hline\n\\rule{0ex}{2.5ex}$r_3$ & $-\\frac{3}{2}g_A^2$\n\\\\[1mm]\\hline\n\\rule{0ex}{2.5ex}$r_4$ & $\\frac{3}{4}\\(g_A^2+(c_2+4c_3-4c_1)M\\)-5c_1M $\n\\\\[1mm]\\hline\n\\rule{0ex}{2.5ex}$r_5$ & $-6 g_A^4 $\n\\\\[1mm]\\hline\n\\rule{0ex}{2.5ex}$r_6$ & $5c_1M $\n\\\\[1mm]\\hline\n\\rule{0ex}{2.7ex}$r_7$ & $\\frac{g_A^2}{4}\\(-8+5g_A^2-72 g_A^4 \\)$\n\\\\[1mm]\\hline\n\\rule{0ex}{2.5ex}$r_8$ & $\\frac{25}{3}c_1M $\n\\\\[1mm]\\hline\n\\rule{0ex}{2.9ex}$r_9$ & $\\frac{g_A^2}{3}\\(-116 g_A^6+\\frac{647 g_A^4}{20}-\\frac{457 g_A^2}{12}+\\frac{17}{40}\\)$\n\\\\[2mm]\\hline\n\\rule{0ex}{2.5ex}$r_{10}$ & $\\frac{725}{36}c_1M $\n\\\\[1mm]\\hline\n\\rule{0ex}{2.9ex}$r_{11}$ & $\\frac{g_A^2}{2}\\(-95 g_A^8+\\frac{1679567 g_A^6}{20160}-\\frac{451799 g_A^4}{3780}+\\frac{320557\n   g_A^2}{15120}-\\frac{896467}{60480}\\)$\n\\\\[2mm]\\hline\n\\end{tabular}\n\\end{center}\n\\caption{The coefficients $r$ of LLog expansion of the nucleon mass using the physical pion mass as defined in (\\ref{mainresult2}).}\n\\label{table_mass_phys}\n\\end{table}\n\nOne can see that the non-analytical in quark mass terms $r_\\text{odd}$ do not\nsimplify in this form of expansion, while the expressions for the\ncoefficient $r_\\text{even}$ are significantly simplified. Moreover the\ncombination of the LECs proportional to $b_n$ in (\\ref{5:pattern})\ncompletely disappears from the higher order terms. We conclude that the\ncoefficients $b_n$ are the coefficients of the LLog expansion of $m^4$\nin the terms of physical pion mass. Thus, assuming that the\n pattern~(\\ref{5:pattern}) holds for all orders we conjecture the LLog part of\nthe expression for the nucleon bare mass via the physical\nmasses\\footnote{This expression should be understood as not rewriting the term\n $k_2m^2\/M$ in the physical pion mass and the integral\nover $\\mu^2$ should be done after applying (\\ref{2:mPhys})\nto $m^4_\\text{phys}(\\mu')$.} \\textit{at all orders} to be\n\\begin{eqnarray}\\label{6:M(m)}\nM=\nM_\\text{phys} +\\frac{3}{4}m_\\text{phys}^4 \\frac{\\log\\(\\frac{\\mu^2}{m_\\text{phys}^2}\\)}{(4\\pi F)^2} \\(\\frac{g_A^2}{M_\\text{phys}}-4c_1+c_2+4c_3\\)\n\\qquad\\qquad\n\\nn\\\\\n -\\frac{3c_1}{{(4\\pi F)^2}}\\int\\limits_{m^2_\\text{phys}}^{\\mu^2}m_\\text{phys}^4(\\mu')~\\frac{d\\mu'^2}{\\mu'^2}.\n\\end{eqnarray}\nThe expression for the physical pion mass is known up to 6-loop order, Eq.~(\\ref{2:mPhys}), therefore, we can guess two more LLog coefficients for\nthe physical nucleon mass. These are presented in the table \\ref{table_mass_conjecture} and indicated by the double-star marks.\n\n\\begin{table}[tb!]\n\\begin{center}\n\\begin{tabular}{|c||l|}\n\\hline\n\\rule{0ex}{2.5ex}$k_{12}$(*) & $\\frac{115}{3}\\(g_A^2+(c_2+4c_3-4c_1)M\\)-\\frac{92}{3}c_1M $\n\\\\[1mm]\\hline\n\\rule{0ex}{2.5ex}$k_{14}$(**) & $-\\frac{186515}{1536}\\(g_A^2+(c_2+4c_3-4c_1)M\\)+\\frac{186515}{2304}c_1M $\n\\\\[1mm]\\hline\n\\rule{0ex}{2.5ex}$k_{16}$(**) & $\\frac{153149887}{259200}\\(g_A^2+(c_2+4c_3-4c_1)M\\)-\\frac{153149887}{453600}c_1M $\n\\\\[1mm]\\hline\n\\rule{0ex}{2.5ex}$r_{12}$(*) & $\\frac{175}{4}c_1M $\n\\\\[1mm]\\hline\n\\rule{0ex}{2.5ex}$r_{14}$(**) & $\\frac{4153903}{24300}c_1M $\n\\\\[1mm]\\hline\n\\end{tabular}\n\\end{center}\n\\caption{The coefficients $k_i$ and $r_i$ defined in (\\ref{mainresult}) and\n(\\ref{mainresult2}), that are obtained by using the conjectures described in Sect.~\\ref{sec:conjectures}. (*): The coefficients $k_{12}$ and $r_{12}$ have\nbeen calculated within the simplified scheme by neglecting higher powers in $g_A$. (**):\nThe coefficients $k_{14,16}$ and $r_{14}$ are the result suggested by\nthe expression (\\ref{6:M(m)}). $r_{16}$ would require the knowledge of the $L^7$ term in the expression for the pion mass.}\n\\label{table_mass_conjecture}\n\\end{table}\n\n\\subsection{Numerical results}\n\\label{sec:numerics}\n\nAs mentioned in the introduction, the LLog are not necessarily dominant.\nThey do however give an indication of the size of corrections to be expected.\nWe use here one set of inputs to show an example. The input we use uses\nthe $c_i$ as determined in \\cite{Bernard:1995dp} and reasonable values for the\nother quantities. The actual values we use are:\n\\begin{eqnarray}\nM = 938~\\text{MeV}, & c_1 = -0.87~\\text{GeV}^{-1}, & c_2 = 3.34~\\text{GeV}^{-1}, \\quad \\mu=0.77~\\text{GeV},\n\\nn\\\\\nF = 92.4~\\text{MeV}, & c_3 = -5.25~\\text{GeV}^{-1},& g_A= 1.25\\,.\n\\end{eqnarray}\nWe plot in Fig.~\\ref{figloop} the total correction $M_\\text{phys}-M$\nof (\\ref{mainresult}) by loop order. We have included the results up to the\n$k_{12}$ term since we do not have the odd powers higher than five loops.\nAs can be seen there is a reasonable convergence for the range given.\n\\begin{figure}[tb!]\n\\centering\n\\includegraphics[width=0.5\\textwidth]{masslooporder.eps}\n\\caption{The contribution of the terms in mass correction of (\\ref{mainresult}) with the terms included up to a given loop-order.}\n\\label{figloop}\n\\end{figure}\n\nTo see the convergence better, we have plotted in Fig.\\ref{figparts} the\nabsolute value of the individual terms containing $k_i$ of (\\ref{mainresult})\nfor $m=138$~MeV.  Note the excellent convergence.\n\\begin{figure}[tb!]\n\\centering\n\\includegraphics[width=0.5\\textwidth]{individualterms.eps}\n\\caption{The absolute value of the contribution of the individual terms ($\\sim k_n$) in (\\ref{mainresult}) at $m=138$~MeV. Open symbols are the odd orders. Filled symbols are the even orders.}\n\\label{figparts}\n\\end{figure}\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nIn this paper we have presented the application of the renormalization group\nmethod for nucleon-pion chiral perturbation theory. The theoretical\nbasis of the method was developed in \\cite{Buchler:2003vw}. The method has\nbeen applied before only for bosonic theories, see\n\\cite{Kivel:2008mf,Polyakov:2010pt,Bijnens:2009zi,Bijnens:2010xg,Bijnens:2012hf,Bijnens:2013yca}. In particular, we have calculated the physical\nmass of the nucleon within the heavy baryon formulation in the LLog\napproximation (analytical and non-analytical in quark mass terms) up to\nfive-loop order. The results of the calculation are presented in\n(\\ref{mainresult}, \\ref{mainresult2}) and tables\n\\ref{table_mass}, \\ref{table_mass_phys}.\n\nThe theories with fermions (or more precisely, the theories involving\nLagrangians of odd chiral order) have a more involved\nstructure of RG equations. In contrast to the bosonic theories, where all\none-loop beta functions contribute to LLog coefficients, in theories\nwith fermions some one-loop beta functions do not contribute to the LLog\ncoefficients. For resolving the RG hierarchy, we have\nintroduced the concept of renormalization group order (RGO), see\nSec.~\\ref{sec:defRGO}. Using the RGO allows us to extract the diagrams which\ncontribute to the LLog approximation (or any other order of RG logarithms).\n\nThe calculation of the necessary one-loop beta-functions has been performed\nsymbolically with the help of the FORM computer symbolic computation system.\nThe results of our calculation agree with known one- and two-loop results,\nsee e.g. \\cite{Schindler:2007dr}. The calculations were performed in several\ndifferent parametrizations of the nucleon and pion fields with the same result,\nproviding a very strong check for the computational algorithm. The analytical\npart of the computation, namely the expression for a basis one-loop-integral\nin the heavy baryon theory, is presented in the appendix.\n\nThe obtained LLog coefficients show a number of regularities. Some of which\ncan be easily understood, while the rest is more involved. The most\nintriguing regularity is the absence of higher powers of axial coupling\nconstant $g_A$ in the LLog coefficients $k_{2n}$ (see (\\ref{mainresult})\nand table \\ref{table_mass}). Moreover, the pattern of the LLog coefficients\nallows us to guess the \\textit{all-order} expression for the LLog\ncontribution to the nucleon mass in terms of \\textit{physical} pion mass \n(\\ref{6:M(m)}). Although the latter is only a conjecture, we consider it\nas an exact result, most likely a consequence of the subtraction of\nheavy-mass-singularities within heavy baryon theory and Lorentz invariance.\n\nWe also showed some numerical results in Sect.~\\ref{sec:numerics}.\n\n\\section*{Acknowledgements}\n\nA.V. thanks J.~Relefors for stimulating discussions and technical help.\nThis work is supported in part by the European Community-Research\nInfrastructure Integrating Activity Study of Strongly Interacting Matter\"\n(HadronPhysics3, Grant Agreement No. 28 3286) and the Swedish Research\nCouncil grants 621-2011-5080 and 621-2013-4287.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}