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+{"text":"\\section{Introduction}\n\nThe results of this article strengthen the connection between invariants of Legendrian knots in standard contact $\\mathbb {R}^3$ and the $2$-variable Kauffman polynomial. Relations between the $2$-variable knot polynomials (HOMFLY-PT and Kauffman) and Legendrian knot theory were first realized in the work of Fuchs and Tabachnikov \\cite{FT} who observed, based on results of Bennequin \\cite{Be}, Franks--Morton--Williams \\cite{FW}, and Rudolph \\cite{Ru}, that these polynomials provide upper bounds on the Thurston--Bennequin number of a Legendrian knot. At that time, it was still unknown whether Legendrian knots in $\\mathbb {R}^3$ were determined up to Legendrian isotopy by their Thurston--Bennequin number, rotation number, and underlying smooth knot type (the so-called ``classical invariants'' of Legendrian knots). This question was soon resolved with the introduction of several non-classical invariants in the late 90's and early 2000's including the Legendrian contact homology algebra which is a differential graded algebra (DGA) coming from $J$-holomorphic curve theory that was constructed by Chekanov in \\cite{Chekanov02} and discovered independently by Eliashberg and Hofer, and combinatorial invariants introduced by Chekanov and Pushkar \\cite{ChP} defined by counting certain decompositions of front diagrams called normal rulings.\\footnote{Around the same time, generating family homology invariants capable of distinguishing Legendrian links with the same classical invariants were introduced by Traynor~\\cite{Tr}.} Interestingly, normal rulings were discovered independently by Fuchs in connection with augmentations of the Le\\-gendrian contact homology DGA. Moreover, Fuchs again pointed toward a connection between Legendrian invariants and topological knot invariants by conjecturing in \\cite{Fuchs} that a Legendrian knot should have a normal ruling if and only if the Kauffman polynomial estimate for the Thurston--Bennequin number is sharp. This conjecture was resolved affirmatively in \\cite{R1} by interpreting Chekanov and Pushkar's combinatorial invariants as polynomials, and showing that the ungraded ruling polynomial, $R^1_K(z)$, of a Legendrian knot $K \\subset \\mathbb {R}^3$ arises as a specialization\n\\begin{gather} \\label{eq:specialization}\nR^1_K(z) = F_K(a,z)|_{a^{-1}=0}\n\\end{gather}\nof the framed version of the Kauffman polynomial $F_K \\in \\mathbb {Z}\\big[a^{\\pm1}, z^{\\pm1}\\big]$; the specialization has the property that it is non-zero if and only if the Kauffman polynomial estimate for $\\operatorname{tb}(K)$ is sharp. An analogous result also established in~\\cite{R1} holds for the $2$-graded ruling polynomial, $R^2_K(z)$, and the HOMFLY-PT polynomial.\n\nInitially, the Legendrian invariance of the ruling polynomials, based on establishing bijections between rulings during bifurcations of the front diagram occurring in a Legendrian isotopy, was somewhat mysterious from the point of view of symplectic topology. Building on the earlier works \\cite{Fuchs, FI,Henry,NgSab, Sab}, Henry and the second author showed in~\\cite{HenryRu} that the ruling polynomials are in fact determined by the Legendrian contact homology DGA, $(\\mathcal{A}, \\partial)$, since their specializations at $z = q^{1\/2} -q^{-1\/2}$ with $q$ a prime power agree with normalized counts of augmentations of $(\\mathcal{A}, \\partial)$ to the finite field $\\mathbb{F}_q$, i.e., DGA representations from $(\\mathcal{A},\\partial)$ to $(\\mathbb{F}_q,0)$. Thus, in the ungraded case (\\ref{eq:specialization}) shows that counts of ungraded augmentations are actually topological (depending only on the underlying framed knot type of $K$), as they arise from a specialization of the Kauffman polynomial. In this article we extend this result by relating counts of higher dimensional (ungraded) representations of $(\\mathcal{A}, \\partial)$ with the $n$-colored Kauffman polynomials.\n\nTo give a statement of our main result, for $n \\geq 1$, let $\\operatorname{Rep}_1\\big(K,\\mathbb{F}_q^n\\big)$ denote the {\\it ungraded total $n$-dimensional representation number} of $K$ as defined in~\\cite{LeRu}; see Definition \\ref{def:total}. Let $F_{n,K}(a,q)$ denote the {\\it $n$-colored Kauffman polynomial} (for framed knots); see Definition~\\ref{def:Kauffman}. In Section~\\ref{sec:ncolored}, we define an {\\it ungraded $n$-colored ruling polynomial} denoted~$R^1_{n,K}(z)$.\n\n\\begin{Theorem} \\label{thm:main} For any Legendrian knot $K$ in $\\mathbb {R}^3$ with its standard contact structure and any $n \\geq 1$, there is a well-defined specialization $F_{n,K}(a,q)|_{a^{-1}=0}$, and we have\n\\[\n\\operatorname{Rep}_1\\big(K,\\mathbb{F}_q^n\\big)=R^1_{n,K}(z)=F_{n,K}(a,q)|_{a^{-1}=0}.\n\\]\n\\end{Theorem}\n\nAs an immediate consequence, we get:\n\n\\begin{Corollary}\nThe ungraded total $n$-dimensional representation number $\\operatorname{Rep}_1\\big(K,\\mathbb{F}_q^n\\big)$ depends only on the underlying framed knot type of $K$.\n\\end{Corollary}\nThe corollary is a significant strengthening of a result from~\\cite{NgR2012} that the existence of an ungraded representation of~$(\\mathcal{A}, \\partial)$ on $\\mathbb{F}_2^n$ depends only on the Thurston--Bennequin number and topological knot type of~$K$. Precisely how much of the Legendrian contact homology DGA is determined by the framed knot type of~$K$ remains an interesting question. See~\\cite{NgR2012} and~\\cite{Ng1} for some open conjectures along this line.\n\nA previous article \\cite{LeRu} establishes analogous results in the case of $2$-graded representations and the HOMFLY-PT polynomial, and in fact establishes the equality $\\operatorname{Rep}_m\\big(K,\\mathbb{F}_q^n\\big)=R^m_{n,K}(z)$ between the $m$-graded total representation numbers and $m$-graded colored ruling polynomials for all $m \\in \\mathbb {Z}_{\\geq 0}$ {\\it except for the ungraded case where $m=1$}.  The $m=1$ case is more involved for a number of reasons. In the following we briefly review the argument from~\\cite{LeRu} and then outline our approach to Theorem~\\ref{thm:main}.\n\nFor $m \\neq 1$, the $n$-colored ruling polynomial is defined as a linear combination of satellite ruling polynomials of the form\n\\begin{gather} \\label{eq:mgraded}\nR^m_{n,K}(q) = \\frac{1}{c_n}\\sum_{\\beta \\in S_n} q^{\\lambda(\\beta)\/2} R^m_{S(K,\\beta)}(z)\\big|_{z = q^{1\/2}-q^{-1\/2}}, \\qquad m \\neq 1,\n\\end{gather}\nwhere \\looseness=1 $S(K,\\beta)$ is the Legendrian satellite of $K$ with a Legendrian positive permutation braid associated to $\\beta \\in S_n$. The same linear combination of HOMFLY-PT polynomials defines the $n$-colored HOMFLY-PT polynomial. In~\\cite{LeRu}, the total $n$-dimensional representation number is recovered from~(\\ref{eq:mgraded}) via a bijection between $m$-graded augmentations of $S(K,\\beta)$ and $n$-dimensional representations of the DGA of $K$ mapping a distinguished invertible generator into $B_\\beta \\subset {\\rm GL}(n, \\mathbb{F})$ where ${\\rm GL}(n,\\mathbb{F}) = \\sqcup_{\\beta \\in S_n}B_\\beta$ is the Bruhat decomposition. Thus, summing over all $\\beta \\in S_n$ corresponds to considering all $n$-dimensional representations of the DGA of~$K$ on~$\\mathbb{F}^n$.\n\nWhen $m=1$, the above bijection becomes modified in an interesting way, as augmentations of $S(K,\\beta)$ now correspond to (ungraded) representations of $(\\mathcal{A}, \\partial)$ on differential vector spaces of the form $\\big(\\mathbb{F}^n,d\\big)$ where $d$ varies over all (ungraded) upper triangular differential on $\\mathbb{F}^n$. (When $m \\neq 1$, $d =0$ is automatic for grading reasons.) The total $n$-dimensional representation number, $\\operatorname{Rep}_1\\big(K,\\mathbb{F}_q^n\\big)$, only counts representations with $d=0$, so the definition of $R^1_{n,K}$ needs to be changed to only take into account normal rulings corresponding to representations with $d=0$. This is done by replacing each $R_{S(K,\\beta)}^1$ in~(\\ref{eq:mgraded}) with the corresponding {\\it reduced ruling polynomial}~$\\widetilde{R}^1_{S(K,\\beta)}$ as introduced in~\\cite{NgR2012} that only counts normal rulings of $S(K,\\beta)$ that never pair two strands of the satellite that correspond to a single strand of $K$. Up to a technical point about the use of different diagrams in~\\cite{LeRu} and~\\cite{HenryRu} that the bulk of Section~\\ref{sec:5} is spent addressing, this leads to the equality $\\operatorname{Rep}_1\\big(K,\\mathbb{F}_q^n\\big)=R^1_{n,K}(z)$.\n\nEstablishing that $R^1_{n,K}(z)=F_{n,K}(a,q)|_{a^{-1}=0}$ requires a much more involved argument than for the case of the colored HOMFLY-PT polynomial and $R^2_{n,K}(z)$ (where the result is immediate from~\\cite{R1} and the definition). The $n$-colored Kauffman polynomial is defined by satelliting $K$ with the symmetrizer in the BMW algebra, $\\mathcal{Y}_n \\in \\operatorname{BMW}_n$. In addition to a sum over permutation braids as in the HOMFLY-PT case, $\\mathcal{Y}_n$ also has terms of a less explicit nature (though, see~\\cite{D}) involving tangle diagrams in $[0,1]\\times \\mathbb {R}$ with turn-backs, i.e., components that have both endpoints on the same boundary component of $[0,1]\\times \\mathbb {R}$. To relate $R^1_{n,K}$ and $F_{n,K}$, we use the combinatorics of normal rulings to find an inductive characterization of $R^1_{n,K}$ in terms of ordinary ruling polynomials rather than reduced ruling polynomials, and then compare this with an inductive characterization of $\\mathcal{Y}_n$ due to Heckenberger and Sch{\\\"u}ler~\\cite{Heck}.\n\nThe remainder of the article is organized as follows. In Section \\ref{sec:ncolored}, we define the ungraded $n$-colored ruling polynomial and establish an inductive characterization of it in Theorem \\ref{thm:main1}. In Section \\ref{sec:Kauffman}, we recall the definition of the colored Kauffman polynomial and prove the second equality of Theorem \\ref{thm:main} (see Theorem~\\ref{thm:main2}). Section \\ref{sec:5} reviews definitions of representation numbers from \\cite{LeRu} and then establishes the first equality of Theorem~\\ref{thm:main} (see Theorem~\\ref{thm:Rep}). In Section \\ref{sec:multicomp}, we close the article with a brief discussion of a modification of Theorem \\ref{thm:main} for the case of multi-component Legendrian links.\n\n\\section[The $n$-colored ungraded ruling polynomial]{The $\\boldsymbol{n}$-colored ungraded ruling polynomial} \\label{sec:ncolored}\n\nIn this section, after a brief review of ruling polynomials and Legendrian satellites, we define the ungraded $n$-colored ruling polynomial $R^1_{n,K}$ as a linear combination of {\\it reduced} ruling polynomials indexed by permutations $\\beta \\in S_n$. Reduced rulings, considered earlier in~\\cite{NgR2012}, form a restricted class of normal rulings of satellite links, so that it is not immediately clear how to describe $R^1_{n,K}$ in terms of ordinary ruling polynomials. For this purpose, we work in a Legendrian version of the $n$-stranded BMW algebra, $\\operatorname{BMW}_n^\\mathfrak{Leg}$, and inductively construct elements $L_n \\in \\operatorname{BMW}_n^\\mathfrak{Leg}$ that can be used to produce $R^1_{n,K}$ via (non-reduced) satellite ruling polynomials.\n\n\\subsection{Legendrian fronts and ruling polynomials}\nIn this article we consider Legendrian links and tangles in a $1$-jet space $J^1M$ where $M$ is one of~$\\mathbb {R}$,~$S^1$, or~$[0,1]$. In all cases, we can view $J^1M = T^*M\\times \\mathbb {R}$ as $M \\times \\mathbb {R}^2$, and using coordinates $(x,y,z)$ with $x \\in M$ and $y,z \\in \\mathbb {R}$ the contact form is ${\\rm d}z - y\\,{\\rm d}x$. Legendrian curves can be viewed via their {\\it front projection} $\\pi_{xz}\\colon J^1M \\rightarrow M \\times \\mathbb {R}$, $(x,y,z) \\mapsto (x,z)$ which is a collection of curves having cusp singularities and transverse double points but no vertical tangencies. The original Legendrian is recovered via $y= \\frac{{\\rm d}z}{{\\rm d}x}$, so in front diagrams (implicitly) the over-strand at a crossing is the strand with lesser slope (as the $y$-axis is oriented away from the viewer). Legendrian links have a {\\it contact framing} which is the framing given by the upward unit normal vector to the contact planes.\n\n\\begin{figure}[t] \\centering\n \\includegraphics{images\/normality.png}\n \\caption{Each closed curve of a normal ruling consists of a pair of companion paths with monotonically increasing $x$-coordinate beginning and ending at a common left and right cusp of~$\\pi_{xz}(L)$. At switches, paths from two different closed curves of~$\\rho$ meet and both turn a corner at a crossing. The normality condition requires that near switches the switching paths and their companion paths match one of the pictured configurations. At crossings that are not switches paths from two different closed curves cross transversally.} \\label{fig:normality}\n\\end{figure}\n\n\\begin{figure}[h!] \\centering\n \\begin{align*}\n \\raisebox{-.5cm}{\\includegraphics[scale=.7]{images\/mruling1.png}}-\\raisebox{-.5cm}{\\includegraphics[scale=.7]{images\/mruling2.png}} &=z\\left( \\;\\raisebox{-.5cm}{\\includegraphics[scale=.7]{images\/mruling3.png}}-\\; \\raisebox{-.5cm}{\\includegraphics[scale=.7]{images\/mruling4.png}}\\right), \\tag{R1} \\\\\n \\raisebox{-.5cm}{\\includegraphics[scale=.7]{images\/mruling5.png}}& =\\raisebox{-.5cm}{\\includegraphics[scale=.7]{images\/mruling6.png}}=0, \\tag{R2} \\\\\n \\raisebox{-.5cm}{\\includegraphics[scale=.7]{images\/unknot.png}} \\;\\sqcup K &=z^{-1}K. \\tag{R3}\n \\end{align*}\\vspace{-0.7cm}\n \\caption{The ungraded ruling polynomial skein relations.}\\label{fig:psuedo-gen}\n\\end{figure}\n\nRecall that for a Legendrian link $K \\subset J^1\\mathbb {R}$ a {\\it normal ruling} $\\rho$ of $K$ is a decomposition of the front diagram of $K$ into a collection of simple closed curves with corners at a left and right cusp and at {\\it switches} (adhering to the normality condition, see Fig.~\\ref{fig:normality}). For each $x=x_0$ where the front projection of $K$ does not have crossings or cusps, a normal ruling $\\rho$ divides the strands of~$K$ at $x=x_0$ into pairs, so that $\\rho$ can be viewed as a sequence of pairings of strands of~$K$. For a~more detailed discussion of normal rulings see for instance \\cite{Fuchs,NgR2012, R1, Sab}. The {\\it ungraded ruling polynomial} of $L$ (also called the $1$-graded ruling polynomial) is defined as\n\\[\nR_K^1(z):=\\sum_{\\rho\\in\\Gamma(K)}z^{j(\\rho)} \\in \\mathbb {Z}\\big[z^{\\pm1}\\big],\n\\] where the sum is over all normal rulings of $K$ and $j(\\rho)=\\#\\text{switches}-\\#\\text{right cusps}$. For Legendrian links in $J^1\\mathbb {R}$, the ungraded ruling polynomial of $K$ satisfies and is uniquely characterized by the skein relations in Fig.~\\ref{fig:psuedo-gen} and the normalization $R^1_{\\includegraphics[scale=.2]{images\/Unknot2}} = z^{-1}$. (See~\\cite{R1}.) The relations in Fig.~\\ref{fig:psuedo-gen} imply two additional relations that we will make use of, cf.\\ \\cite[Section~6]{R2}.\n\\begin{gather} \\label{eq:fish}\n \\text{fishtail relation:} \\ \\raisebox{-.35cm}{\\includegraphics[scale=.5]{images\/lfish.png}} =\\raisebox{-.35cm}{\\includegraphics[scale=.5]{images\/rfish.png}}=0,\n \\vspace{.5cm}\\\\\n \\text{double-crossing relation:} \\ \\raisebox{-.35cm}{\\includegraphics[scale=.5]{images\/dblesigma.png}}= \\raisebox{-.35cm}{\\includegraphics[scale=.5]{images\/id2.png}}+ z\\cdot\\raisebox{-.35cm}{\\includegraphics[scale=.5]{images\/cross.png}}- z\\cdot\\raisebox{-.35cm}{\\includegraphics[scale=.5]{images\/cusps.png}}.\\nonumber\n\\end{gather}\n\\begin{Remark} The double crossing relation can be used to show that the first relation of Fig.~\\ref{fig:psuedo-gen} also hold with right cusps. Moreover, the third relation from Fig.~\\ref{fig:psuedo-gen} is implied by the first two as long as $K \\neq \\varnothing$.\n\\end{Remark}\n\n\\subsection{A Legendrian BMW algebra}\n\nA {\\it Legendrian $n$-tangle} is a properly embedded Legendrian $\\alpha \\subset J^1[0,1]$ (i.e., compact with $\\partial \\alpha \\subset \\partial J^1[0,1]$) whose front projection agrees with the collection of horizontal lines $z=i$, $1\\leq i \\leq n$, near $\\partial J^1[0,1]$.  Legendrian isotopies of $n$-tangles are required to remain fixed in a~neighborhood of $\\partial J^1[0,1]$. At $x=0$ and $x=1$ we enumerate the endpoints and strands of a Legendrian tangle from $1$ to $n$ with {\\it descending} $z$-coordinate. For any permutation $\\beta \\in S_n$, there is a corresponding {\\it positive permutation braid} that is a Legendrian $n$-tangle, also denoted $\\beta \\subset J^1[0,1]$, that connects endpoint $i$ at $x=0$ to endpoint $\\beta(i)$ at $x=1$ for $1 \\leq i \\leq n$. Up to Legendrian isotopy, $\\beta$ is uniquely characterized by requiring that\n\\begin{itemize}\\itemsep=0pt\n\\item[(i)] the front projection does not have cusps and,\n\\item[(ii)] for $i<j$, the front projection has no crossings (resp.\\ exactly one crossing) between the strands with endpoints $i$ and $j$ at $x=0$ if $\\beta(i) < \\beta(j)$ (resp.\\ $\\beta(i)> \\beta(j)$).\n\\end{itemize}\nThe number of crossings in such a front diagram for $\\beta$ is called the {\\it length} of $\\beta$ and will be deno\\-ted~$\\lambda(\\beta)$. For Legendrian $n$-tangles $\\alpha,\\beta \\subset J^1[0,1]$, we define their multiplication $\\alpha\\cdot\\beta \\subset J^1[0,1]$ by stacking $\\beta$ to the {\\it left} of $\\alpha$ (as in composition of permutations). Diagrammatically:\n\\[\n\\labellist\n\\small\n\\pinlabel $\\alpha$ at 45 38\n\\endlabellist\n\\includegraphics[scale=.5]{images\/Tangle1} \\,\\, \\raisebox{.6cm}{$\\cdot$} \\,\\,\n\\labellist\n\\small\n\\pinlabel $\\beta$ at 45 38\n\\endlabellist\n\\includegraphics[scale=.5]{images\/Tangle1}\n\\quad\\raisebox{.6cm}{$=$} \\quad\n\\labellist\n\\small\n\\pinlabel $\\beta$ at 45 38\n\\pinlabel $\\alpha$ at 120 38\n\\endlabellist\n\\includegraphics[scale=.5]{images\/Tangle2}\n\\]\n\n\\begin{Definition} Let $\\mathcal{R}$ be a coefficient ring containing $\\mathbb {Z}[z^{\\pm1}]$ as a subring. Define the {\\it Legendrian BMW algebra}, $\\operatorname{BMW}_n^\\mathfrak{Leg}$, as an $\\mathcal{R}$-module to be the quotient $\\mathcal{R} \\mathfrak{Leg}_n\/\\mathcal{S}$ where $\\mathcal{R} \\mathfrak{Leg}_n$ is the free $\\mathcal{R}$-module generated by Legendrian isotopy classes of Legendrian $n$-tangles in $J^1[0,1]$, and $\\mathcal{S}$ is the $\\mathcal{R}$-submodule generated by the ruling polynomial skein relations from Fig.~\\ref{fig:psuedo-gen}. Multiplication of $n$-tangles induces an $\\mathcal{R}$-bilinear product on $\\operatorname{BMW}_n^\\mathfrak{Leg}$.\n\\end{Definition}\n\nIn the remainder of the article, we fix the coefficient ring $\\mathcal{R}$ to be $\\mathbb {Z}\\big[s^{\\pm1}\\big]$ localized to include denominators of the form $s^n-s^{-n}$ for $n\\geq 1$ where $z = s- s^{-1}$. In Section~\\ref{sec:5}, we will work with the alternate variable $s= q^{1\/2}$.\n\n\\begin{figure} \\centering\n $\\sigma_i:=\\raisebox{-.75cm}{\\includegraphics[scale=.5]{images\/sigmai.png}}\\hspace{.75cm} \\hspace{.75cm} e_i:=\\raisebox{-.75cm}{\\includegraphics[scale=.5]{images\/ei.png}}$\n \\caption{Crossing and hook elements in $\\operatorname{BMW}_n^\\mathfrak{Leg}$.} \\label{fig:sigma}\n\\end{figure}\n\nFig.~\\ref{fig:sigma} indicates crossing and hook elements, $\\sigma_i, e_i \\in \\operatorname{BMW}_n^\\mathfrak{Leg}$, for $1 \\leq i < n$. Note that the fishtail and double-crossing relations imply\n\\begin{gather}\n \\sigma_i e_i = e_i \\sigma_i =0, \\qquad \\mbox{and} \\label{eq:fish2} \\\\\n \\sigma_i^2 = 1 + z \\sigma_i - z e_i, \\qquad \\mbox{for $1 \\leq i <n$.} \\label{eq:double}\n\\end{gather}\n\n\\begin{Remark} \\label{rem:BMW}\\samepage\\quad\n\\begin{enumerate}\\itemsep=0pt\n\\item Occasionally, we will multiply an element $\\alpha \\in \\operatorname{BMW}_i^\\mathfrak{Leg}$ by an element of $\\operatorname{BMW}_j^\\mathfrak{Leg}$ with $i<j$. Unless indicated otherwise, we do so by extending $i$-tangles to $j$-tangles by placing $j-i$ horizontal strands below $\\alpha$.\n\n\\item We do not give a complete set of generators and relations for $\\operatorname{BMW}_n^\\mathfrak{Leg}$, as the main role of $\\operatorname{BMW}_n^\\mathfrak{Leg}$ in this article is to provide a convenient setting for ruling polynomial calculations. We leave finding a presentation for $\\operatorname{BMW}_n^\\mathfrak{Leg}$ as an open problem.\n\\end{enumerate}\n\\end{Remark}\n\n\\subsection{Legendrian satellites and reduced ruling polynomials} \\label{sec:sat} We will be considering rulings of satellites. Given a connected Legendrian knot $K \\subset J^1\\mathbb {R}$ and a Legendrian $n$-tangle $L \\subset J^1[0,1]$, a~front diagrammatic description of the {\\it Legendrian satellite} $S(K,L) \\subset J^1\\mathbb {R}$ is the following: Form the $n$-copy of~$K$, i.e., take $n$ copies of $K$ each shifted up a~small amount in the $z$-direction, then insert a rescaled version of the $n$-tangle $L$ into the $n$-copy at a small rectangular neighborhood $J \\cong [0,1] \\times [-\\epsilon, \\epsilon]$ of part of a strand of $K$ that is oriented from left-to-right. We refer to $J$ as the {\\it $J^1[0,1]$-part} of the satellite. See Fig.~\\ref{fig:Sat}. It can be shown that the Legendrian isotopy type of~$S(K,L)$ depends only on~$K$ and the closure of $L$ in $J^1S^1$, cf.~\\cite{NgTraynor}. (See also Section~\\ref{sec:satellite} for additional discussion.)\n\n\\begin{figure} \\centering\n\t\t\\labellist\n\\small\n\\pinlabel $K$ [t] at 36 30\n\\pinlabel $\\beta$ [t] at 170 44\n\\pinlabel $S(K,\\beta)$ [t] at 352 0\n\\endlabellist\n\n \\includegraphics{images\/Sat}\n\t\t\n\\caption{The Legendrian satellite $S(K,\\beta)$ where $\\beta = \\sigma_1 \\sigma_2 \\subset J^1[0,1]$ is the positive permutation braid associated to the $3$-cycle $(1 \\, 2 \\,3)$. The $J^1[0,1]$-part of the satellite is indicated by the dotted rectangular box.} \\label{fig:Sat}\n\\end{figure}\n\nIn \\cite{NgR2012} a variant of the ruling polynomial was introduced for satellites using reduced normal rulings. Let $L \\subset J^1[0,1]$ be a Legendrian $n$-tangle. A normal ruling $\\rho$ of $S(K,L)$ is said to be {\\it reduced} if, outside of the $J^1[0,1]$-part of $S(K,L)$, parallel strands of $S(K,L)$ corresponding to a~single strand of~$K$ are not paired by $\\rho$. See Fig.~\\ref{fig:reduced}.\n\\begin{figure}[h] \\centering\n \\includegraphics{images\/reducedtrefoil.png}\n \\caption{A reduced normal ruling (left) and a non-reduced normal ruling (right) of $S(K,\\beta)$ where $K$ is a right-handed trefoil and $\\beta=1$.} \\label{fig:reduced}\n\\end{figure}\n We denote the set of reduced normal rulings of $S(K,L)$ by $\\widetilde{\\Gamma}(K,L)$ and define the {\\it reduced ruling polynomial} of $S(K,L)$ as $\\widetilde{R}_{S(K,L)}(z):=\\sum\\limits_{\\rho\\in\\widetilde{\\Gamma}(K,L)}z^{j(\\rho)}$.\n\n\\begin{Remark} \\label{rem:thin}If the reduced condition holds for a normal ruling $\\rho$ of $S(K,L)$ for the parallel strands of $S(K,L)$ corresponding to a single point $k_0 \\in K$ outside of the $J^1[0,1]$-part of $S(K,L)$, then $\\rho$ will be reduced. Indeed, the involution of parallel strands of $S(K,L)$ at $k_0 \\in K$ that arises from restriction the pairing of $\\rho$ (strands of $S(K,L)$ at $k_0$ that are paired with strands in a part of the satellite away from~$k_0$ are fixed points of the involution) is called the ``thin part'' of~$\\rho$ at~$k_0$ in~\\cite{NgR2012}, and Lemmas~3.3 and~3.4 from~\\cite{NgR2012} show that the thin part is independent of~$k_0$.\n\\end{Remark}\n\nWe are now prepared to make the following key definition.\n\n\\begin{Definition} \\label{def:nruling}For any (connected) Legendrian knot $K\\subset J^1\\mathbb {R}$ we define the {\\it ungraded $n$-colored ruling polynomial} as\n\\begin{gather*}\n R_{n,K}^1(s):=\\frac{1}{c_n}\\sum_{\\beta\\in S_n}s^{\\lambda(\\beta)}\\widetilde{R}_{S(K,\\beta)}(z),\n\\end{gather*}\nwhere the sum is over positive permutation braids, $z=s-s^{-1}$, $\\lambda(\\beta)$ is the length of $\\beta$, and $c_n=s^{n(n-1)\/2}\\prod\\limits_{i=1}^n\\frac{s^i-s^{-i}}{s-s^{-1}}$.\n \\end{Definition}\nNote that $R_{n,K}^1$ belongs to coefficient ring $\\mathcal{R}$ defined above.\n\n\\begin{Remark}It is proved in \\cite{NgR2012} that with $L$ fixed the reduced ruling polynomial $\\widetilde{R}_{S(K,L)}(z)$ is a Legendrian isotopy invariant of~$K$. Alternatively, the Legendrian isotopy invariance of $R_{n,K}^1$ also follows from Theorem~\\ref{thm:main1} below and invariance of the ordinary ruling polynomials.\n\\end{Remark}\n\n\\subsection[Inductive characterization of $R_{n,K}^1$]{Inductive characterization of $\\boldsymbol{R_{n,K}^1}$} \\label{sec:Ln}\n\nIt is convenient to extend the concept of satellite ruling polynomials slightly by defining $R^1_{S(K,\\eta)}$ for $\\eta \\in \\operatorname{BMW}_n^{\\mathfrak{Leg}}$ represented as an $\\mathcal{R}$-linear combination of $n$-tangles $\\sum\\limits_{i=1}^k r_i \\alpha_i$ to be\n\\begin{gather} \\label{eq:BMWsatellite}\nR^1_{S(K,\\eta)} = \\sum_{i=1}^kr_i R^1_{S(K,\\alpha_i)}.\n\\end{gather}\n(This is well-defined since the front diagram of each $\\alpha_i$ appears as a~subset of $\\pi_{xz}(S(K,\\alpha_i))$ and~$R^1$ satisfies the skein relations that define $\\operatorname{BMW}_n^\\mathfrak{Leg}$.) The goal in the remainder of this section is to give such a characterization of the ungraded $n$-colored ruling polynomials via appropriate elements $L_n \\in \\operatorname{BMW}_n^\\mathfrak{Leg}$.\n\nLet\n\\[\n\\gamma_n=1+\\sum_{j=1}^{n-1}s^j\\sigma_{n-1}\\sigma_{n-2}\\cdots \\sigma_{n-j} \\in \\operatorname{BMW}^\\mathfrak{Leg}_n,\n\\]\n where $\\sigma_i$ is as in Fig.~\\ref{fig:sigma} and consider the elements of $\\operatorname{BMW}^\\mathfrak{Leg}_n$\n defined inductively for $n\\geq1$ by $L_1 =1$ and\n \\[L_n=L_{n-1}\\beta_n, \\qquad n \\geq 2,\\]\nwhere\n\\[ \\beta_n:=\\left(1-z\\sum_{k=2}^n\\alpha_{k,n}\\right)\\gamma_n.\\]\nHere, $\\alpha_{k,n}\\in \\operatorname{BMW}^\\mathfrak{Leg}_n$ is defined inductively on the {\\it bottom} $k$ strands by $\\alpha_{2,n} = e_{n-1}$ and\n \\begin{gather} \\alpha_{k,n}=\\raisebox{-.25cm}{\\includegraphics{images\/alpha1}}-z\\cdot\\raisebox{-.25cm}{\\includegraphics{images\/alpha2}}-z\\cdot\\raisebox{-.25cm}{\\includegraphics{images\/alpha3}}+z^2\\cdot\\raisebox{-.25cm}{\\includegraphics{images\/alpha4}} \\label{alpha}\\end{gather}\n\t\twith the identity tangle appearing as the top $n-k$ strands of $\\alpha_{k,n}$. For example, using (\\ref{eq:fish2}) we have $L_2=\\mathbf{1}+s\\sigma_1-ze_1$.\n\t\tThe double subscript notation on $\\alpha_{k,n}$ is to emphasize that $\\alpha_{k,n}$\n\t\tinvolves the bottom $k$ strands, $n-k+1, n-k+2, \\ldots, n$, out of $n$ total strands.\n\t\t(By contrast, when multiplying $L_{n-1} \\beta_n$ an extra strand is placed at the bottom of $L_{n-1}$; see Remark~\\ref{rem:BMW}.)\n\nIn Section~\\ref{sec:Kauffman}, we will also make use of a non-inductive characterization of the~$\\alpha_{k,n}$. Given a~front diagram~$D$, let $\\operatorname{cr}(D)$ denote the set of crossings appearing in $D$. We define the {\\it resolution of $D$ with respect to $X\\subset \\operatorname{cr}(D)$} to be the tangle $r_X(D)$ obtained from resolving all crossings of~$X$ as depicted in Fig.~\\ref{fig:res}.\n\n \\begin{figure}[h] \\centering\n \\includegraphics[scale=.4]{images\/xres.png}\\raisebox{.9cm}{$\\;\\longrightarrow$} \\includegraphics[scale=.4]{images\/res.png}\n \\caption{The resolution of a crossing in $X\\subset \\operatorname{cr}(D)$.} \\label{fig:res}\n \\end{figure}\n \\begin{Lemma} For $1<k\\leq n$, \\[\\raisebox{.6cm}{$\\alpha_{k,n}=\\sum\\limits_{X\\subset \\operatorname{cr}(C_{k,n})} (-z)^{\\vert X \\vert} r_X(C_{k,n})$, \\quad where $C_{k,n}=$ }\\includegraphics[scale=.65]{images\/Ck.png}\\]\\label{lemma:alpha}\n \\end{Lemma}\n\t\t\n\\begin{proof}This is a straightforward induction on~$k$.\n\\end{proof}\n\n\\begin{Theorem}\\label{thm:main1}For any Legendrian $K\\subset J^1\\mathbb {R}$, $R_{n,K}^1=\\frac{1}{c_n}R^1_{S(K,L_n)}$.\n\\end{Theorem}\n\nThe proof will be given below after Lemma~\\ref{lemma:THMB}.\n\nGiven $K\\subset J^1\\mathbb {R}$, and $\\eta$ a Legendrian $n$-tangle, recall that the $J^1[0,1]$-part of $S(K,\\eta)$ refers to a rectangular region, $J\\cong [0,1] \\times [-\\epsilon,\\epsilon]$, where the front diagram of $\\eta$ appears. We will call a normal ruling~$\\rho$ of~$S(K,\\eta)$ {\\it $k$-reduced} if {\\it at the right boundary of~$J$} none of the top~$k$ parallel strands of $\\eta$ within $S(K,\\eta)$ are paired with one another. We denote by $\\widetilde{\\Gamma^k}(K,\\eta)$, the set of $k$-reduced normal rulings of~$S(K,\\eta)$. Similarly, given a~location $*$ within $J$ corresponding to an $x$-value in $J^1[0,1]$ without double points of $\\eta$ a~normal ruling $\\rho$ of $S(K,\\eta)$ is said to be {\\it $(k,m)$-paired at $*$} if $\\rho$ is $(m-1)$-reduced (at the right boundary of~$J$) and $\\rho$ pairs strand $k$ of~$\\eta$ with strand~$m$ of $\\eta$ at $*$ (where strands are numbered from top to bottom as they appear in~$\\eta$ at~$*$). The set of $(k,m)$-paired rulings of $S(K,\\eta)$ is denoted by $\\Gamma_{*}^{\\tau(k,m)}(K,\\eta)$. The {\\it $k$-reduced ruling polynomial} and {\\it $(k,m)$-paired ruling polynomial} (at $*$) are defined respectively by \\[ \\widetilde{R}^{(k)}_{(K,\\eta)}(z)=\\sum_{\\rho\\in\\widetilde{\\Gamma^k}(K,\\eta)}z^{j(\\rho)} \\qquad \\text{and} \\qquad R^{\\tau(k,m)}_{(*;K,\\eta)}(z)=\\sum_{\\rho\\in\\Gamma_{*}^{\\tau(k,m)}(K,\\eta)}z^{j(\\rho)}.\\]\nEither polynomial can be extended by linearity to allow $\\eta$ to be an $\\mathcal{R}$-linear combination of Legendrian $n$-tangles.\n\n\\begin{Remark}\\label{rem:reduced}If $\\eta$ is a $n$-stranded positive braid, then any $n$-reduced normal ruling is reduced. This is because if parallel strands of the satellite $S(K,\\eta)$ corresponding to a single strand of $K$ are not paired at the right side of the $J^1[0,1]$-part of the satellite, then such parallel strands cannot be paired anywhere outside of the $J^1[0,1]$-part. See Remark~\\ref{rem:thin}.\n\\end{Remark}\n\n\\begin{Lemma}\\label{lemma:THMB} For any $1<k \\leq m \\leq n$, let $\\beta \\in S_{m-1}$ be a positive permutation braid extended to an Legendrian $n$-tangle by placing the $n-m+1$ stranded identity tangle below $\\beta$, and let $\\nu$ be a linear combination of Legendrian $n$-tangles. Then, we have\n\\[z\\widetilde{R}^{(m-1)}_{(K,\\beta\\alpha_{k,m}\\nu)}(z)=R^{\\tau(m-k+1,m)}_{(*;K,\\beta\\nu)}(z),\\]\nwhere $*$ is the location between $\\beta$ and $\\nu$ $($immediately to the left diagrammatically of~$\\beta)$.\n\\end{Lemma}\n\n\\begin{proof}The proof is by induction on $k$ with $m$ and $n$ fixed. The base case is clear since $\\alpha_{2,m}=e_{m-1}$.\n\nSuppose the statement is true for~$k$. For clarity of this proof, we emphasize the location of our paired rulings within our notation, and abbreviate $\\alpha_{k,m}$ as~$\\alpha_k$. Using~(\\ref{alpha}) and the inductive hypothesis, we compute\n\\begin{gather*}\n z\\widetilde{R}^{(m-1)}_{(K,\\beta\\alpha_{k+1}\\nu)}(z)\\\\\n =z\\widetilde{R}^{(m-1)}_{(K,\\beta\\sigma_{m-k}\\alpha_{k}\\sigma_{m-k}\\nu)}(z) -z^2\\widetilde{R}^{(m-1)}_{(K,\\beta\\sigma_{m-k}\\alpha_{k}\\nu)}(z) -z^2\\widetilde{R}^{(m-1)}_{(K,\\beta\\alpha_{k}\\sigma_{m-k}\\nu)}(z)+z^3\\widetilde{R}^{(m-1)}_{(K,\\beta\\alpha_{k}\\nu)}(z)\\\\\n =R^{\\tau(m-k+1,m)}_{(*;K,\\beta\\sigma_{m-k}*\\sigma_{m-k}\\nu)}(z) -zR^{\\tau(m-k+1,m)}_{(*;K,\\beta\\sigma_{m-k}*\\nu)}(z) -zR^{\\tau(m-k+1,m)}_{(*;K,\\beta*\\sigma_{m-k}\\nu)}(z)+z^2R^{\\tau(m-k+1,m)}_{(*;K,\\beta*\\nu)}(z) \\hypertarget{star}{}\\\\\n \\overset{\\star}{=}R^{\\tau(m-k,m)}_{(*;K,\\beta*\\nu)}(z).\n\\end{gather*}\nWe now establish equality $\\star$. For the diagrams $D$ that follow, when considering $(k,m)$-paired rulings, we indicate the location $*$ with a green vertical segment having endpoints on the paired strands~$k$ and~$m$. When we encode such information in $D$ we suppress the paired notation $R^{\\tau(k,m)}_{(*;K,D)}$ as $R^\\tau_{K,D}$ (and similarly for sets of rulings: $\\Gamma_{*}^{\\tau(k,m)}(K,D)$ becomes $\\Gamma^{\\tau}(K,D)$). For instance,\n\\[\nR^{\\tau(m-k,m)}_{(*;K,\\beta*\\nu)}(z) =\n R^\\tau_{\\left(K,\\; \\raisebox{-.25cm}{\\includegraphics[scale=.6]{images\/paired1.png}}\\right)}(z),\n\\]\nwhere the diagram only pictures strands with numberings in the range $m-k$ to $m$ at $*$. Observe the bijection\n\\begin{gather}\\label{eq:color}\n \\Gamma^\\tau\\left(K, \\raisebox{-.25cm}{\\includegraphics[scale=.6]{images\/paired1.png}}\\right)=\\left\\{\\rho \\in \\Gamma^\\tau\\left(K, \\raisebox{-.25cm}{\\includegraphics[scale=.6]{images\/paired2.png}}\\right) \\,\\bigg|\\, \\mbox{$r_1$ and $r_2$ are not switches of $\\rho$}\\right\\},\n\\end{gather}\n where $r_1$ and $r_2$ denote the left and right crossings of the diagram on the right. In addition, we have the following (separate) bijections:\n\\begin{gather}\\label{eq:bijectionsec2}\n\\begin{array}{@{}lrcl}\n &\\rho&\\mapsto &\\Phi(\\rho),\\\\\n \\Big\\{ \\rho\\in\\Gamma^\\tau\\left(K,\\raisebox{-.25cm}{\\includegraphics[scale=.65]{images\/paired3.png}}\\right) \\,\\big|\\,& \\mbox{ $r_1$ is a switch} \\Big\\} &{\\longleftrightarrow}\n & \\Gamma^\\tau\\left(K,\\raisebox{-.25cm}{\\includegraphics[scale=.65]{images\/paired5.png}}\\right), \\vspace{1mm}\\\\\n \\Big\\{\\rho\\in\\Gamma^\\tau\\left(K,\\raisebox{-.25cm}{\\includegraphics[scale=.65]{images\/paired3.png}}\\right) \\, \\big|\\, & \\mbox{$r_2$ is a switch}\\Big\\} &\n \\longleftrightarrow &\n \\Gamma^\\tau\\left(K,\\raisebox{-.25cm}{\\includegraphics[scale=.65]{images\/paired4.png}}\\right), \\vspace{1mm}\\\\\n \\Big\\{\\rho\\in\\Gamma^\\tau\\left(K,\\raisebox{-.25cm}{\\includegraphics[scale=.65]{images\/paired3.png}}\\right) \\,\\big|\\, & \\mbox{$r_1$ and $r_2$ are switches }\\Big\\} & \\longleftrightarrow & \\Gamma^\\tau\\left(K,\\raisebox{-.25cm}{\\includegraphics[scale=.65]{images\/paired6.png}}\\right) ,\n \\end{array}\n\\end{gather}\nwhere $\\Phi(\\rho)$ is the unique ruling that agrees with $\\rho$ outside of a neighborhood of the resolved crossing(s). (Any ruling of a diagram $D$ with switches at a set of crossings~$X$ gives rise to a~ruling of $r_{X}(D)$.) Each $\\Phi$ is clearly injective. To verify surjectivity, observe that any ruling $\\rho'$ in one of the sets on the right side of~(\\ref{eq:bijectionsec2}) indeed comes from a ruling on the left side by replacing the resolved crossing(s) with switches. Here, it is crucial to note that the normality condition is automatically satisfied. If the normality condition was not satisfied, then $\\rho'$ must pair strand $m-k$ with a strand numbered in the range $m-k+2$ to $m-1$ at the location~$*$ indicated by the vertical segment. Since $\\beta \\in S_{m-1}$, it follows that at least~2 of the top $m-1$ strands are paired at the right side of the $J^1[0,1]$-part of the satellite, contradicting that $\\rho'$ is $(m-1)$-reduced. Finally, the inclusion-exclusion principle and the above bijections~\\eqref{eq:color} and~\\eqref{eq:bijectionsec2} imply\n\\begin{gather*}\n R^\\tau_{\\left(K, \\;\n \\raisebox{-.25cm}{\\includegraphics[scale=.54]{images\/paired1.png}}\\right)}(z)\\\\\n =R^\\tau_{\\left(K,\\; \\raisebox{-.25cm}{\\includegraphics[scale=.54]{images\/paired3.png}}\\right)}(z)\n -zR^\\tau_{\\left(K, \\; \\raisebox{-.25cm}{\\includegraphics[scale=.54]{images\/paired4.png}}\\right)}(z)\n -zR^\\tau_{\\left(K,\\; \\raisebox{-.25cm}{\\includegraphics[scale=.54]{images\/paired5.png}}\\right)}(z)\n +z^2R^\\tau_{\\left(K,\\; \\raisebox{-.25cm}{\\includegraphics[scale=.54]{images\/paired6.png}}\\right)}(z).\n\\end{gather*}\n(The factors of $z$ and $z^2$ appear since the first two bijections in (\\ref{eq:bijectionsec2}) decrease the number of switches by $1$, while the last bijection decreases the number by 2.) This establishes \\hyperlink{star}{$\\star$}, so we are done.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:main1}] With $n\\geq 1$ fixed we prove the following statement by induction on~$m$. The theorem follows from the special case where $m=n$ and $\\mu = 1$ (in view of Remark~\\ref{rem:reduced}).\n\n{\\bf Inductive statement:} For any (linear combination of) Legendrian $n$-tangle(s) $\\mu \\subset J^1[0,1]$\n and any $1 \\leq m \\leq n$,\n\\[\nR_{S(K,L_{m}\\mu)}(z)=\\sum_{\\beta\\in S_{m}}s^{\\lambda(\\beta)}\\widetilde{R}^{(m)}_{(K,\\beta\\mu)}(z)\n\\]\n(with the products $L_m\\mu$ and $\\beta\\mu$ formed as in Remark~\\ref{rem:BMW}).\n\nThe base case of $m=1$ is immediate since any normal ruling is $1$-reduced. For the inductive step, with $m\\geq 2$ and the statement assumed for $m-1$ we compute\n\\begin{align*}\n \\notag R_{S(K,L_m\\mu)}(z)&=R_{S(K,L_{m-1}(1-z\\sum\\limits_{k=2}^m\\alpha_{k,m})\\gamma_m\\mu)}(z)\\\\\n \\notag &=R_{S(K,L_{m-1}\\gamma_m\\mu)}-z\\sum_{k=2}^m R_{S(K,L_{m-1}\\alpha_{k,m}\\gamma_m\\mu)}(z)\\\\\n \\notag &=\\sum_{\\beta\\in S_{m-1}}s^{\\lambda(\\beta)}\\left(\\widetilde{R}^{(m-1)}_{(K,\\beta\\gamma_m\\mu)}(z) -z\\sum_{k=2}^m\\widetilde{R}^{(m-1)}_{(K,\\beta\\alpha_{k,m}\\gamma_m\\mu)}(z)\\right)&&(\\text{inductive hyp})\\\\\n &=\\sum_{\\beta\\in S_{m-1}}s^{\\lambda(\\beta)}\\left(\\widetilde{R}^{(m-1)}_{(K,\\beta\\gamma_m\\mu)}(z) -\\sum_{k=2}^mR^{\\tau(m-k+1,m)}_{(*;K,\\beta*\\gamma_m\\mu)}(z)\\right). &&(\\text{Lemma~\\ref{lemma:THMB}})\n\t\\end{align*}\nNote that since there are no crossings in $\\beta$ involving the $m$-th strand, an $(m-1)$-reduced ruling is $m$-reduced unless at $*$ strand $m$ is paired\n with one of the strands numbered $1, \\ldots, m-1$ in the $J^1[0,1]$-part of the satellite, i.e., a strand numbered $m-k+1$ with $2 \\leq k \\leq m$. Thus, continuing the above computation, we have\n\\begin{equation*}\n \\sum_{\\beta\\in S_{m-1}}s^{\\lambda(\\beta)}\\left(\\widetilde{R}^{(m-1)}_{(K,\\beta\\gamma_m\\mu)}(z)-\\sum_{k=2}^mR^{\\tau(m-k+1,m)}_{(*;K,\\beta*\\gamma_m\\mu)}(z)\\right)=\\sum_{\\beta\\in S_{m-1}}s^{\\lambda(\\beta)}\\widetilde{R}^{(m)}_{(K,\\beta\\gamma_m\\mu)}.\n\\end{equation*}\nNext note that any $\\beta\\in S_m$ can be written uniquely as $\\beta=\\beta'\\sigma_{m-1}\\cdots \\sigma_{m-j}$ with $\\beta'\\in S_{m-1}$ and $0 \\leq j \\leq m-1$. (Here, $m-j = \\beta^{-1}(m)$.) Moreover, such a factorization realizes $\\beta$ as a~positive permutation braid, so that $\\lambda(\\beta)=\\lambda(\\beta')+j$. Using the definition of~$\\gamma_m$, we see that\n\\[\\sum_{\\beta\\in S_{m-1}}s^{\\lambda(\\beta)}\\widetilde{R}^{(m)}_{(K,\\beta\\gamma_m\\mu)}=\\sum_{\\beta\\in S_m}s^{\\lambda(\\beta)}\\widetilde{R}^{(m)}_{(K,\\beta\\mu)}.\\] This completes the inductive step.\n\\end{proof}\n\n\\section{Relation to the Kauffman polynomial} \\label{sec:Kauffman}\n We begin this section with a review of the definition of the $n$-colored Kauffman polynomial~$F_{n,K}$ via satelliting with the symmetrizer $\\mathcal{Y}_n \\in \\operatorname{BMW}_n$ in the BMW algebra. Then, using an inductive characterization of $\\mathcal{Y}_n$ from~\\cite{Heck}, combined with the earlier characterization of $R^1_{n,K}$ we show how to recover $R^1_{n,K}$ as a specialization $F_{n,K}\\big\\vert_{a^{-1}=0}$. This is accomplished by relating the Legendrian~BMW element $L_n \\in \\operatorname{BMW}^\\mathfrak{Leg}_{n}$ from Theorem~\\ref{thm:main1} to~$\\mathcal{Y}_n$ in a certain sub-quotient of~$\\operatorname{BMW}_n$ defined in terms of Legendrian tangles. Along the way we pause to make a conjecture relating ungraded ruling polynomial skein modules with Kauffman skein modules in general.\n\n\\subsection{The Kauffman polynomial and the BMW algebra} Recall that the (framed) Kauffman polynomial (Dubrovnik version) is a regular isotopy invariant that assigns a Laurent polynomial to framed links $L\\subset\\mathbb {R}^3$ characterized by the skein relations (shown with blackboard framing)\n\\begin{gather*}\n \\raisebox{-.5cm}{\\includegraphics[scale=.6]{images\/Kauf1.png}} -\\raisebox{-.5cm}{\\includegraphics[scale=.6]{images\/Kauf2.png}}=z\\left(\\; \\raisebox{-.5cm}{\\includegraphics[scale=.6]{images\/Kauf4.png}}- \\raisebox{-.5cm}{\\includegraphics[scale=.6]{images\/Kauf3.png}}\\;\\right), \\tag{F1} \\\\\n \\raisebox{-.5cm}{\\includegraphics[scale=.6]{images\/Kauf5.png}}=a^{-1}\\raisebox{-.5cm}{\\includegraphics[scale=.6]{images\/Kauf.png}} \\hspace{1cm} \\raisebox{-.5cm}{\\includegraphics[scale=.6]{images\/Kauf6.png}} =a\\raisebox{-.5cm}{\\includegraphics[scale=.6]{images\/Kauf.png}}\\,, \\tag{F2} \\\\\n \\raisebox{-.5cm}{\\includegraphics[scale=.6]{images\/Kauf7.png}} \\;\\sqcup L =\\left(\\frac{a-a^{-1}}{z}+1\\right)L, \\tag{F3}\n\\end{gather*}\nwhere $F_L:=1$ if $L$ is the empty link.\\footnote{This is equivalent to choosing the normalization $F_U=\\frac{a-a^{-1}}{z}+1$, where $U$ is the $0$-framed unknot.}\n\nThe ordinary BMW algebra, $\\operatorname{BMW}_n$ (see \\cite{BW, Mu}) can be defined as the Kauffman skein module for framed tangles in $[0,1]\\times \\mathbb {R}^2 \\cong J^1[0,1]$ with $n$ boundary points on each component. In more detail, let $\\mathfrak{Fr}_n$ denote the set of isotopy classes of {\\it framed $n$-tangles} where we require that near $\\partial J^1[0,1]$ framed $n$-tangles agree with the horizontal lines $z=i$, $y =0$, $1\\leq i \\leq n$ with framing vector $\\frac{\\partial}{\\partial y}$; this requirement is maintained during isotopies. Working over the field of rational functions $\\mathbb{F} = \\mathbb {Z}(a,s)$ with $z=s-s^{-1}$, $\\operatorname{BMW}_n= \\mathbb{F}\\mathfrak{Fr}_n\/\\mathcal{T}$ where $\\mathcal{T}$ is the $\\mathbb{F}$-submodule generated by the Kauffman polynomial skein relations. Multiplication in $\\operatorname{BMW}_n$ is as in the Legendrian case, i.e., the product $\\alpha \\cdot \\beta$ is $\\beta$ stacked to the left of~$\\alpha$.\n\nFor viewing framed $n$-tangles diagrammatically, we continue to use the $xz$-plane for projections, and we require the framing to be globally given by $\\frac{\\partial}{\\partial y}$, that is, perpendicular to the projection plane and away from the viewer. This framing becomes isotopic to the blackboard framing when $n$-tangles are closed to become links in $J^1S^1$ (by identifying the left and right boundaries of $J^1[0,1]$). To view a Legendrian $n$-tangle $L \\subset J^1[0,1]$ as a framed $n$-tangle $\\operatorname{fr}(L) \\in \\mathfrak{Fr}_n$ we form a diagram for $\\operatorname{fr}(L)$ from the front diagram of $L$ by smoothing left cusps and adding a small loop with a negative crossing at right cusps. See Fig.~\\ref{fig:framed}. This has the property that the closure of $\\operatorname{fr}(L)$ in $J^1S^1$ is framed isotopic to the Legendrian closure of~$L$ with its contact framing.\n\n\\begin{figure}\n \\centering\n\t\t\\labellist\n\\small\n\\pinlabel $L$ [t] at 64 0\n\\pinlabel $\\operatorname{fr}(L)$ [t] at 250 0\n\\endlabellist\n\n \\includegraphics{images\/FramedL}\n\t\t\n\t\t\\quad\n\t\t\n \\caption{The framed $n$-tangle $\\operatorname{fr}(L)$ associated to a Legendrian $n$-tangle $L \\subset J^1[0,1]$.}\n \\label{fig:framed}\n\\end{figure}\n\nThe topological satellite operation produces from a framed knot $K \\subset J^1\\mathbb {R}$ and a framed $n$-tangle $L \\subset J^1[0,1]$ a satellite link $S(K,L) \\subset J^1\\mathbb {R}$. A diagram for $S(K,L)$ arises from taking a blackboard framed diagram for $K$ and placing a blackboard framed diagram for the closure of $L$ in an (immersed) annular neighborhood of $K$ in the projection plane. When $K$ and $L$ are Legendrian, the framed knot type of $S(K,L)$ agrees with that of the previously defined Legendrian satellite (with contact framing).\n For $K \\subset J^1\\mathbb {R}$ (a framed link) and $\\eta \\in \\operatorname{BMW}_n$, the Kauffman polynomial of the satellite $F_{S(K,\\eta)}$ is defined by linearity as in (\\ref{eq:BMWsatellite}).\n\n\\subsection[Symmetrizer in $\\operatorname{BMW}_n$ and the $n$-colored Kauffman polynomial]{Symmetrizer in $\\boldsymbol{\\operatorname{BMW}_n}$ and the $\\boldsymbol{n}$-colored Kauffman polynomial}\n\n The $n$-colored Kauffman polynomial is defined by satelliting with the {\\it symmetrizer} $\\mathcal{Y}_n \\in \\operatorname{BMW}_n$. (More general, colored Kauffman polynomials, where the coloring is by a partition $\\lambda$, can be defined using other idempotents in $\\operatorname{BMW}_n$ and are related to quantum invariants of type~$B$,~$C$, and~$D$; see, e.g.,~\\cite{BlBe}.) The symmetrizer $\\mathcal{Y}_n$ is characterized as the unique non-zero element of $\\operatorname{BMW}_n$ that is idempotent, i.e., has $\\mathcal{Y}_n^2 = \\mathcal{Y}_n$, and satisfies\n\\begin{gather}\n \\mathcal{Y}_n\\sigma_i =s\\mathcal{Y}_n, \\qquad \\text{for }1\\leq i <n. \\quad \\quad \\quad (\\mbox{crossing absorbing property})\\label{eats}\n\\end{gather}\nThe following inductive formula for $\\mathcal{Y}_n$ is due to Heckenberger and Sch{\\\"u}ler in \\cite{Heck}. Consider $Y_n \\in \\operatorname{BMW}_n$ defined by\n \\begin{gather}\nY_1 = 1, \\notag\\\\\n\\label{inductiveform}\n Y_n=Y_{n-1}\\gamma_n+Y_{n-1}\\frac{z}{1-s^{2n-3}a} \\left(\\sum_{i=1}^{n-1}as^{2n-2i-1}\\left(\\sum_{j=0}^{i-1}s^{j}D_{i-j,i+1}\\right)\\right),\n\\end{gather}\nwhere\n \\[D_{i,j}:=\\begin{cases} {\\includegraphics[scale=.8]{images\/DIJ}}\\,, &\\raisebox{.7cm}{ $i<j$,}\\\\\n0,&\\;i\\geq j.\n\\end{cases}\n\\] The symmetrizer is then obtained as the normalization\n$\\mathcal{Y}_n=\\frac{Y_n}{c_n}$ (with $c_n$ as in Definition~\\ref{def:nruling}). For example, $\\mathcal{Y}_2=\\frac{1}{c_2}\\big({\\rm id}+s\\sigma_1+\\frac{sza}{1-sa}e_1\\big)$. Note that $Y_n$ is quasi-idempotent (i.e., $Y_n^2 = c_n Y_n$) and also has the crossing absorbing property~(\\ref{eats}).\n\n\\begin{Remark} \\looseness=-1 Our notations $\\sigma_i$, $s$, $a$, $e_i$ translate into the notations from \\cite{Heck} as $g_i$, $q$, $r$, $r^{-1} e_i$. (The~$r^{-1}$ is because our $e_i$ is a Legendrian hook, so that $\\operatorname{fr}(e_i)$ has an extra loop that produces the $r^{-1}$ factor.) Our $D_{i,j}$ is such that $a \\cdot D_{i-j, i+1}$ is the $j$-th term in $d^+_{n,i}$ from~\\cite{Heck}, the factor of $a$ again arising from our use of Legendrian tangles in representing the quasi-idempotent~$Y_n$. Note that \\cite[Proposition~1]{Heck} gives an inductive formula for $\\mathcal{Y}_n$ (notated there as~$S_n$) rather than~$Y_n$, and the denominator $q^n [\\![ n ]\\!]$ that appears there is accounted for by our factor of $\\frac{1}{c_n}$ relating~$Y_n$ and~$\\mathcal{Y}_n$.\n\\end{Remark}\n\n\\begin{Definition} \\label{def:Kauffman}\nGiven a framed knot $K\\subset\\mathbb {R}^3$, we define the {\\it $n$-colored Kauffman polynomial} of~$K$ by\n\\[F_{n,K}(a,s)=F_{S(K,\\mathcal{Y}_n)}(a,z)\\vert_{z=s-s^{-1}},\\]\n where $\\mathcal{Y}_n$ is the symmetrizer in $\\operatorname{BMW}_n$.\n\\end{Definition}\n\n\\subsection{Ruling polynomials via specializations of the Kauffman polynomial}\nFor a Legendrian link in $L \\subset J^1\\mathbb {R}$ the Thurston--Bennequin number satisfies the inequality $\\operatorname{tb}(L) \\leq -\\deg_a \\widehat{F}_L(a,z)$, where $\\widehat{F}_L = a^{-w(L)} F_L$ is the framing independent version of the Kauffman polynomial obtained by normalizing $F_L$ using the writhe of a diagram for~$L$. (See~\\cite{FT, Ng3, T}.) This is equivalent to the inequality\n\\begin{gather} \\label{eq:Estimate}\n\\deg_a F_L(a,z) \\leq 0.\n\\end{gather}\n Thus, when $L$ is Legendrian $F_L \\in \\mathbb {Z}\\big[a^{\\pm 1}, z^{\\pm}\\big]$ does not contain positive powers of~$a$, and a~specialization $F_L(a,z)|_{a^{-1}=0}$ arises from simply setting $a^{-1}=0$.\n\n\\begin{Theorem}[\\cite{R1}]\\label{thm:rulings and the Kauffman poly}\nFor any Legendrian link $L\\subset J^1\\mathbb {R}$, the ungraded ruling polynomial is the specialization \\[R^1_L(z)=F_L(a,z)|_{a^{-1}=0}.\\]\n\\end{Theorem}\nWe will want to perform a similar specialization on elements of $\\operatorname{BMW}_n$ whose coefficients in $\\mathbb{F} = \\mathbb {Z}(a,s)$ may not be Laurent polynomials in $a$.\n To do so, note that the notion of degree in $a$ extends to arbitrary non-zero rational functions via\n $\\deg_a\\colon \\mathbb {Z}(a,s) \\setminus \\{0\\} \\rightarrow \\mathbb {Z}$,\n\\[\n \\deg_a\\left(\\frac{f}{g}\\right) = \\deg_a f - \\deg_a g, \\qquad \\mbox{for $f,g \\in \\mathbb {Z}[a,s]$}.\n\\]\nIn addition, we use the convention that $\\deg_a0:=-\\infty$. Moreover, on the subring $\\mathbb{F}^- := \\{F\\in \\mathbb{F} \\,|  \\allowbreak \\deg_a F\\leq 0 \\}$ we can define a specialization $\\vert_{a^{-1}=0}\\colon \\mathbb{F}^-\\rightarrow \\mathbb{Z}(s)$, by\n\\[\n\\frac{f}{g}\\bigg\\vert_{a^{-1}=0}:=\\begin{cases}0, & \\deg_a(f\/g)<0,\\\\\n\\dfrac{c_a(f)}{c_a(g)},&\\deg_a(f\/g)= 0,\\end{cases}\n\\]\nwhere $c_a(f) \\in \\mathbb {Z}[s]$ denotes the leading coefficient in $a$ of $f$.\n\\begin{Proposition} The specialization $\\vert_{a^{-1}=0}\\colon \\mathbb{F}^-\\rightarrow \\mathbb{Z}(s)$ is a well-defined, unital ring homomorphism.\n \\end{Proposition}\n\\begin{proof}Suppose $\\frac{f_1}{g_1}=\\frac{f_2}{g_2}$ in $\\mathbb {F}^-$. Then $f_1g_2=g_1f_2$ and so $c_a(f_1)c_a(g_2)=c_a(g_1)c_a(f_2)$, since $c_a(fg)=c_a(f)c_a(g)$ is true for polynomials $f$ and $g$. It follows that $\\frac{f_1}{g_1}\\big\\vert_{a^{-1}=0}=\\frac{f_2}{g_2}\\big\\vert_{a^{-1}=0}$ and so the specialization $\\big\\vert_{a^{-1}=0}$ is well-defined. Note that $1\\vert_{a^{-1}=0}=1$ follows from the definition. Now suppose $\\frac{f_1}{g_1},\\frac{f_2}{g_2}\\in\\mathbb {F}^-$. If $\\deg_a({f_1}\/{g_1})=\\deg_a({f_2}\/{g_2})=0$, then $\\deg_a(f_1g_2)=\\deg_a(g_1f_2)=\\deg_a(g_1g_2)$. In which case, \\begin{align*}\n\\left(\\frac{f_1}{g_2}+\\frac{f_2}{g_2}\\right)\\bigg\\vert_{a^{-1}=0} & =\\frac{f_1g_2+g_1f_2}{g_1g_2}\\bigg\\vert_{a^{-1}=0}\\\\\n& =\\frac{c_a(f_1)c_a(g_2)+c_a(g_1)c_a(f_2)}{c_a(g_1)c_a(g_2)} =\\frac{f_1}{g_1}\\bigg\\vert_{a^{-1}=0}+\\frac{f_2}{g_2}\\bigg\\vert_{a^{-1}=0}.\n\\end{align*} If $\\deg_a({f_1}\/{g_1}),  \\deg_a({f_2}\/{g_2})<0$, then\n\\[\n\\deg_a(f_1g_2+g_1f_2)\\leq \\max\\{\\deg_a(f_1g_2),\\deg_a(g_1f_2)\\}<\\deg_a(g_1g_2)\n\\] and so $\\big(\\frac{f_1}{g_1}+\\frac{f_2}{g_2}\\big)\\big\\vert_{a^{-1}=0}= 0 = \\frac{f_1}{g_1}\\big\\vert_{a^{-1}=0}+\\frac{f_2}{g_2}\\big\\vert_{a^{-1}=0}$ holds by definition. Lastly, if $\\deg_a(f_1\/g_1)<\\deg_a(f_2\/g_2)=0$, then $\\deg_a(f_1g_2+g_1f_2)=\\deg_a(g_1f_2)=\\deg(g_1g_2)$ and \\[\\left(\\frac{f_1}{g_2}+\\frac{f_2}{g_2}\\right)\\bigg\\vert_{a^{-1}=0}=\\frac{f_1g_2+g_1f_2}{g_1g_2}\\bigg\\vert_{a^{-1}=0}=\\frac{c_a(g_1)c_a(f_2)}{c_a(g_1)c_a(g_2)}=\\frac{f_1}{g_1}\\bigg\\vert_{a^{-1}=0}+\\frac{f_2}{g_2}\\bigg\\vert_{a^{-1}=0}.\\] It remains to show $\\vert_{a^{-1}=0}$ preserves multiplication. In the case where\n $\\deg_a(f_1\/g_1)<0$, it follows that $\\deg_a(f_1f_2)<\\deg_a(g_1g_2)$ and so\n \\[\\left(\\frac{f_1f_2}{g_1g_2}\\right)\\bigg\\vert_{a^{-1}=0}=0 =\\frac{f_1}{g_1}\\bigg\\vert_{a^{-1}=0}\\cdot\\frac{f_2}{g_2}\\bigg\\vert_{a^{-1}=0}.\\]\n The remaining case where $\\deg_a(f_1\/g_1)=\\deg_a(f_1\/g_1)=0$ immediately follows from the fact that $c_a$ preserves multiplication for then\n\\begin{gather*}\n\\left(\\frac{f_1f_2}{g_1g_2}\\right)\\bigg\\vert_{a^{-1}=0}=\\frac{c_a(f_1f_2)}{c_a(g_1g_2)} =\\frac{c_a(f_1)c_a(f_2)}{c_a(g_1)c_a(g_2)} =\\frac{f_1}{g_1}\\bigg\\vert_{a^{-1}=0}\\cdot\\frac{f_2}{g_2}\\bigg\\vert_{a^{-1}=0}.\\tag*{\\qed}\n\\end{gather*}\\renewcommand{\\qed}{}\n\\end{proof}\n\nIn the following we will make a connection between the Legendrian BMW algebra from Section~\\ref{sec:ncolored} and $\\operatorname{BMW}_n$. Towards this end, define\n\\[\n\\operatorname{BMW}^-_n= \\operatorname{Span}_{\\mathbb{F}^-}\\mathfrak{Leg}_n \\subset \\operatorname{BMW}_n\n\\]\nto be the $\\mathbb{F}^-$-submodule generated by (framed isotopy classes of) Legendrian $n$-tangles. Next, let $\\mathfrak{m} \\subset \\mathbb{F}^-$ denote the maximal ideal $\\mathfrak{m} = \\{F \\in \\mathbb{F}^-\\,|\\, \\deg_a F < 0\\}$, that is the kernel of $\\vert_{a^{-1}=0}$, and consider the quotient $\\mathbb{F}^-$-module\n\\[\n\\operatorname{BMW}^\\infty_n = \\operatorname{BMW}^-_n\/\\mathfrak{m} \\cdot \\operatorname{BMW}^-_n,\n\\]\nwith the projection map notated as\n\\[\n\\operatorname{BMW}^-_n \\rightarrow \\operatorname{BMW}^\\infty_n, \\qquad y \\mapsto y\\vert_{a^{-1}=0}.\n\\]\n\nNote that the estimate~(\\ref{eq:Estimate}) shows that for any fixed Legendrian knot $K \\subset J^1\\mathbb {R}$ the $\\mathbb{F}$-module homomorphism $\\operatorname{BMW}_n \\rightarrow \\mathbb{F}$, $L \\mapsto F_{S(K,L)}(a,s)$ maps $\\operatorname{BMW}_n^-$ to $\\mathbb{F}^-$. Moreover, the specialization $\\operatorname{BMW}^-_n \\rightarrow \\mathbb{Z}(s)$, $L \\mapsto F_{S(K,L)}(a,s)\\big\\vert_{a^{-1}=0}$ induces a well defined map \\mbox{$\\operatorname{BMW}^\\infty_n \\rightarrow \\mathbb{Z}(s)$}.\n\\begin{Proposition} \\label{prop:algebra}\nThere is an $\\mathcal{R}$-algebra homomorphism $\\varphi\\colon \\operatorname{BMW}_n^{\\mathfrak{Leg}} \\rightarrow \\operatorname{BMW}_{n}^\\infty$ induced by the map $\\mathfrak{Leg}_n \\rightarrow \\mathfrak{Fr}_n$. Moreover, we have a commutative diagram\n\\[\n\\xymatrix{ \\operatorname{BMW}^-_n \\ar[rr]^{F_{S(K,\\cdot)}} \\ar[d]_{|_{a^{-1}=0}} & & \\mathbb{F}^- \\ar[rr]^{|_{a^{-1}=0}} & & \\mathbb{Z}(s). \\\\ \\operatorname{BMW}^\\infty_n \\ar[rrrru] & & & & \\\\ \\operatorname{BMW}^\\mathfrak{Leg}_n \\ar[u]^{\\varphi} \\ar[rrrruu]_{R^1_{S(K,\\cdot)}} & & & &}\n\\]\n\\end{Proposition}\n\\begin{proof} To see that $\\varphi$ is well-defined, note that the ruling polynomial relation (R1) holds in $\\operatorname{BMW}_n$ since it is implied by the Kauffman relation (F1). Moreover, using the (F2), we see that when $L \\subset J^1[0,1]$ has a zig-zag, and $L' \\subset J^1[0,1]$ is the Legendrian obtained by removing the zig-zag from $L$, we have $L = a^{-1}L'$ in $\\operatorname{BMW}^-$, and this implies that $[L]=0$ in $\\operatorname{BMW}^\\infty$ as required by (R2). Finally, to verify (R3) when $K \\in \\mathfrak{Leg}_n$ we can compute in $\\operatorname{BMW}^\\infty_n$\n\\begin{align*}\nK \\sqcup \\raisebox{-.3cm}{\\includegraphics[scale=.5]{images\/unknot}} & = a^{-1} K\\sqcup \\raisebox{-.3cm}{\\includegraphics[scale=.4]{images\/Kauf7}} = a^{-1}\\left(\\frac{a-a^{-1}}{z} +1\\right) K \\\\\n& = z^{-1}K + \\big(a^{-1}-a^{-2}\/z\\big) K = z^{-1}K,\n\\end{align*}\nsince $\\big(a^{-1}-a^{-2}\/z\\big) \\in \\mathfrak{m}$.\n\nThe upper triangle of the diagram is commutative by definitions, and the commutativity of the lower triangle follows from Theorem~\\ref{thm:rulings and the Kauffman poly}.\n\\end{proof}\n\n\\begin{Conjecture}When $\\operatorname{BMW}^\\mathfrak{Leg}_n$ is defined over $\\mathbb{Z}(s)$, the map $\\varphi$ is an algebra isomorphism.\n\\end{Conjecture}\n\n\\begin{Remark} A similar conjecture can be made involving the (suitably defined) $1$-graded ruling polynomial skein module and the Kauffman skein module of any contact $3$-manifold, $M$.\n \\end{Remark}\n\nRecall the element $L_n \\in \\operatorname{BMW}^\\mathfrak{Leg}_n$ from Section~\\ref{sec:Ln}.\n\\begin{Proposition} \\label{prop:BMW}\nFor any $n\\geq 1$, we have\n\\begin{enumerate}\\itemsep=0pt\n\\item[$1)$] $\\mathcal{Y}_n \\in \\operatorname{BMW}^-_n$, and\n\\item[$2)$] $\\varphi\\big(\\frac{1}{c_n} L_n\\big) = \\mathcal{Y}_n\\big\\vert_{a^{-1}=0}$ holds in $\\operatorname{BMW}^\\infty_n$.\n\\end{enumerate}\n\\end{Proposition}\nWe prove (1) now; the proof of~(2) is deferred until Section~\\ref{sec:prop}.\n\\begin{proof}[Proof of (1)] Note that all the tangles involved in the inductive characterization of $Y_n$ from (\\ref{inductiveform}) are Legendrian with coefficients in~$\\mathbb{F}^-$, and the normalizing factor $\\frac{1}{c_n}$ also belongs to~$\\mathbb{F}^-$.\n\\end{proof}\n\nThe following is the second equality in the statement of Theorem~\\ref{thm:main} from the introduction.\n\\begin{Theorem} \\label{thm:main2}\nFor any Legendrian knot $K \\subset J^1\\mathbb {R}$, the $n$-colored Kauffman polynomial has $F_{n,K} \\in \\mathbb{F}^-$ and satisfies\n\\[\nF_{n,K}|_{a^{-1}=0} = R^1_{n,K}.\n\\]\n\\end{Theorem}\n\\begin{proof}Since $\\mathcal{Y}_n \\in \\operatorname{BMW}^-_n$ we get that $F_{n,K} = F_{S(K,\\mathcal{Y}_n)} \\in \\mathbb{F}^-$, and since $\\varphi\\big(\\frac{1}{c_n}L_n\\big) = [\\mathcal{Y}_n]$ the commutativity of the diagram in Proposition~\\ref{prop:algebra} together with Proposition~\\ref{thm:main1}\n shows that\n\\begin{gather*}\nF_{n,K}|_{a^{-1}=0} = F_{S(K,\\mathcal{Y}_n)}|_{a^{-1}=0} = R^1_{S(K, \\frac{1}{c_n} L_n)} = R^1_{n,K}.\\tag*{\\qed}\n\\end{gather*}\\renewcommand{\\qed}{}\n\\end{proof}\n\n\\begin{Corollary}For each $n\\geq2$, $\\varphi\\big(\\frac{1}{c_n}L_n\\big)$ is a central idempotent in $\\operatorname{BMW}_n^\\infty$.\n\\end{Corollary}\n\\begin{proof}This follows from Proposition~\\ref{prop:BMW} since $\\mathcal{Y}_n$ has this property already in $\\operatorname{BMW}_n$. (See, e.g.,~\\cite{BlBe, Heck}.)\n\\end{proof}\n\n\\subsection{Establishing (2) of Proposition \\ref{prop:BMW}} \\label{sec:prop}\n\nWe now embark on showing $\\varphi\\big(\\frac{1}{c_n}L_n\\big)=\\mathcal{Y}_n\\vert_{a^{-1}=0}$. Throughout this section we will work in $\\operatorname{BMW}^{\\infty}_n$, but we will simplify notation by writing $L_n$ for $\\varphi(L_n)$. We begin with some preparatory lemmas that provide formulas for $L_n$ that are closer to the inductive formula for $Y_n$ from (\\ref{inductiveform}). Lemma~\\ref{lemma:beta} uses the hypothesis that $L_{n-1}$ has the crossing absorbing property. This assumption is later verified to be true (see Proposition~\\ref{prop:almostmainthm}).\n\n\\begin{Lemma}\\label{lemma:beta}Let $n\\geq2$ and assume that, in $\\operatorname{BMW}^\\infty_n$, $L_{n-1}$ has the crossing absorbing pro\\-per\\-ty~\\eqref{eats}. Then, in $\\operatorname{BMW}^\\infty_n$ we have\n \\begin{equation*}\n {L_{n-1}\\beta_n}=L_{n-1}\\left(1-z\\sum_{k=2}^n s^{2-k}\\left[\\sum_{j=1}^{k-2}-zs^{2+j-k} D_{n-j,n}+ D_{n-k+1,n}\\right]\\right)\\gamma_n.\n\\end{equation*}\n\\end{Lemma}\n\n\\begin{proof}We use Lemma \\ref{lemma:alpha}. For each $2\\leq k\\leq n$, let $\\operatorname{cr}_\\ell(C_{k,n})$ (respectively $\\operatorname{cr}_r(C_{k,n})$) denote the set of crossings appearing in the left (resp.\\ right) half of $C_{k,n}$, and abbreviate $C_k:=C_{k,n}$. Then,\n\\begin{align*}\n {L_{n-1}\\beta_n}&=L_{n-1}\\left(1-z\\sum_{k=2}^n \\sum_{X\\subset \\operatorname{cr}(C_k)}(-z)^{|X|}r_X(C_k)\\right)\\gamma_n\\\\\n &=L_{n-1}\\left(1-z\\sum_{k=2}^n \\sum_{X\\subset \\operatorname{cr}_\\ell(C_k)}\\sum_{Y\\subset \\operatorname{cr}_r(C_k)}(-z)^{|X|+|Y|}r_{X\\bigsqcup Y}(C_k)\\right)\\gamma_n\\\\\n &=L_{n-1}\\left(1-z\\sum_{k=2}^n \\sum_{X\\subset \\operatorname{cr}_\\ell(C_k)}(-z)^{|X|}\\sum_{Y\\subset \\operatorname{cr}_r(C_k)}(-z)^{|Y|}r_{X\\bigsqcup Y}(C_k)\\right)\\gamma_n \\, .\n\\end{align*}\nSince $L_{n-1}$ absorbs crossings $L_{n-1}r_{X \\bigsqcup Y}(C_k)=L_{n-1}s^{k-2-|Y|}r_X(D_{n-k+1,n})$; see Fig.~\\ref{fig:Absorbing}. Furthermore, because there are $\\binom{k-2}{|Y|}$ subsets of $\\operatorname{cr}_r(C_k)$ having $|Y|$ crossings, summing over $|Y|$ one obtains\n\\begin{gather*}\n L_{n-1}\\sum_{Y\\subset \\operatorname{cr}_r(C_k)}(-z)^{|Y|}r_{X\\bigsqcup Y}(C_k) =L_{n-1}\\sum_{j=0}^{k-2}\\binom{k-2}{j}(-z)^j s^{k-2-j}r_X(D_{n-k+1,n})\\\\\n\\qquad{} =L_{n-1}(s-z)^{k-2}r_X(D_{n-k+1,n})\n=L_{n-1}s^{2-k}r_X(D_{n-k+1,n}).\n\\end{gather*}\nTherefore, \\[L_{n-1}\\beta_n=L_{n-1}\\left(1-z\\sum_{k=2}^ns^{2-k}\\sum_{X\\subset \\operatorname{cr}_\\ell(C_k)}(-z)^{|X|}r_X(D_{n-k+1,n})\\right)\\gamma_n.\\]\n\n\\begin{figure}\n \\centering\n\t\t\\labellist\n \\small\n\\pinlabel $n$ [r] at 0 0\n\\pinlabel $n-k+1$ [r] at 0 112\n\\pinlabel $L_{n-1}$ [b] at 168 56\n\\pinlabel $L_{n-1}$ [b] at 448 56\n\\pinlabel $=\\,s^{k-2-|Y|}$ [b] at 236 48\n\\endlabellist\n\n \\quad \\includegraphics[scale=.8]{images\/Absorbing}\n\t\t\n \\caption{An illustration of the identity $L_{n-1}r_{X \\bigsqcup Y}(C_k)=s^{k-2-|Y|}L_{n-1}r_X(D_{n-k+1,n})$, with the resolved crossings from $X \\bigsqcup Y$ indicated in dotted ovals. The number of crossings in the right half of $r_{X\\sqcup Y}(C_{k,n})$ is $k-2-|Y|$.} \\label{fig:Absorbing}\n\\end{figure}\n\nIt remains to establish the innermost sum satisfies\n\\begin{gather} \\label{eq:Xsum}\nL_{n-1}\\sum_{X\\subset \\operatorname{cr}_\\ell(C_k)}(-z)^{|X|}r_X(D_{n-k+1,n}) = L_{n-1}\\left[\\sum_{j=1}^{k-2}-zs^{2+j-k} D_{n-j,n}+ D_{n-k+1,n}\\right].\n\\end{gather}\nWhen we perform the resolution by subsets of $\\operatorname{cr}_\\ell(C_k)$ we will not be able to feed all of the remaining crossings into $L_{n-1}$ (in fact, when $X$ is the empty set there are no crossings that we can push into $L_{n-1}$). We remedy this by partitioning the nonempty subsets of $\\operatorname{cr}_\\ell(C_k)$. Label the crossings in $\\operatorname{cr}_\\ell(C_k)$ by ascending $z$-coordinate as $c_1, \\ldots, c_{k-2}$, and define $\\chi_j=\\{X\\subset \\operatorname{cr}_\\ell(C_k)\\colon \\allowbreak j=\\min\\{i\\,|\\,c_i \\in X\\}\\}$, for $1\\leq j\\leq k-2$. Let $X\\in\\chi_j$ be given. By definition $c_j\\in X$ is the lowest crossing that is resolved in $r_X(D_{n-k+1,n})$. Isotopy allows us to push the $k-2-j-(|X|-1)$ remaining crossings lying above $c_j$ into $L_{n-1}$. Hence, using $i=|X|-1$ and that the requirement that $X \\in \\chi_j$ leaves $i$ choices from the $k-2-j$ crossings above $c_j$ to determine $X$, we have\n \\begin{align*}\n L_{n-1}\\sum_{X\\in\\chi_j}(-z)^{|X|}r_X(D_{n-k+1,n})&=L_{n-1}\\sum_{i=0}^{k-2-j}\\left(\\begin{array}{c} k-2-j \\\\ i \\end{array} \\right)(-z)^{i+1}s^{(k-2-j)-i}D_{n-j,n} \\\\\n\t\t& = -zL_{n-1}(-z+s)^{k-2-j} D_{n-j,n} \\quad \\mbox{(binomial theorem)} \\\\\n\t\t &=-zL_{n-1}s^{2+j-k}D_{n-j,n}.\n \\end{align*}\nSumming over all $j$, and adding the term $D_{n-k+1,n}$ for the case when $X$ is empty, establishes~(\\ref{eq:Xsum}) and completes the proof.\n\\end{proof}\n\\begin{Lemma}\\label{lemma:mult}For any $1\\leq i\\leq n$, in $\\operatorname{BMW}^\\infty_n$ we have\n$D_{i,n}\\gamma_n=\\sum\\limits_{r=0}^{n-i-1}s^rD_{i,n-r}$.\n\\end{Lemma}\n\\begin{proof}\nThis is just applying type II Reidemeister moves to see that\n\\[\nD_{i,n}s^r\\sigma_{n-1}\\sigma_{n-2}\\cdots\\sigma_{n-r} =  \\begin{cases} s^r D_{i,n-r}, &   r < n-i, \\\\ 0, & r \\geq n-i,\\end{cases}\n\\]\n since the fishtail relation (\\ref{eq:fish}) may be applied when $r \\geq n-i$. See Fig.~\\ref{fig:DIJprod}.\n\\end{proof}\n\n\\begin{figure}[h]\n \\centering\n \\raisebox{-1.25cm}{\\includegraphics[scale=.9]{images\/DIJprod1.png}}\\quad=\\quad\\raisebox{-1.25cm}{\\includegraphics{images\/DIJprod2.png}}\n\n \\caption{An Illustration of the Type II Reidemeister moves used in Lemma \\ref{lemma:mult}.}\n\t\t \\label{fig:DIJprod}\n\\end{figure}\nProposition \\ref{prop:BMW} (2) follows from the following.\n\\begin{Proposition} \\label{prop:almostmainthm}\nFor all $n\\geq1$, $L_n$ has the following properties in $\\operatorname{BMW}^\\infty_n$:\n\\begin{itemize}\\itemsep=0pt\n \\item[$(i)$] $L_n=Y_n\\vert_{a^{-1}=0}$,\n \\item[$(ii)$] $L_n$ has the crossing absorbing property~\\eqref{eats}.\n\\end{itemize}\n\\end{Proposition}\n\\begin{proof} The proof is by induction on $n$. In the base case $n=1$, (i) is immediate from definitions and (ii) is vacuous. Assume the result for $n-1$. Note that it suffices to establish (i) since the fact that $Y_n$ has the crossing absorbing property and $\\vert_{a^{-1}=0}$ is a $\\mathbb{Z}(s)$-algebra homomorphism would then allow us to verify (ii) via\n\\[\nL_n\\sigma_i=(Y_n\\vert_{a^{-1}=0})\\sigma_i=(Y_n\\sigma_i)\\vert_{a^{-1}=0}=(sY_n)\\vert_{a^{-1}=0}=sL_n.\n\\]\n Showing $L_n=Y_n\\vert_{a^{-1}=0}$ is the following a computation (with Lemmas \\ref{lemma:beta} and \\ref{lemma:mult} used at the 2nd and 3rd equality):\n\\begin{align*}\n L_n&=L_{n-1}\\beta_n =L_{n-1}\\left(1-z\\sum_{k=2}^n s^{2-k}\\left[\\sum_{j=1}^{k-2}-zs^{2+j-k} D_{n-j,n}+ D_{n-k+1,n}\\right]\\gamma_n\\right)\\\\\n &=L_{n-1}\\left(\\gamma_n-z\\sum_{k=2}^n s^{2-k}\\left[\\sum_{j=1}^{k-2}-zs^{2+j-k}\\sum_{r=0}^{j-1}s^rD_{n-j,n-r} + \\sum_{r=0}^{k-2}s^rD_{n-k+1,n-r}\\right]\\right)\\\\\n &=L_{n-1}\\left(\\gamma_n-z\\sum_{k=2}^n s^{2-k}\\left[\\sum_{r=0}^{k-3}\\sum_{j=r+1}^{k-2}-zs^{2+j-k+r}D_{n-j,n-r} + \\sum_{r=0}^{k-2}s^rD_{n-k+1,n-r}\\right]\\right)\\\\\n &=L_{n-1}\\left(\\gamma_n-z\\left[\\sum_{k=2}^n \\sum_{r=0}^{k-3}\\sum_{j=r+1}^{k-2}-zs^{4+j-2k+r}D_{n-j,n-r} + \\sum_{k=2}^n\\sum_{r=0}^{k-2}s^{2-k+r}D_{n-k+1,n-r}\\right]\\right)\\\\\n\t &=L_{n-1}\\left(\\gamma_n-z\\left[\\sum_{r=0}^{n-3} \\sum_{k=r+3}^{n}\\sum_{j=r+1}^{k-2}-zs^{4+j-2k+r}D_{n-j,n-r} \\right.\\right.\\\\\n& \\left.\\left. \\quad{} + \\sum_{r=0}^{n-2}\\sum_{k=r+2}^{n}s^{2-k+r}D_{n-k+1,n-r}\\right]\\right)\\\\\n &=L_{n-1}\\left(\\gamma_n-zD_{1,2}-z\\sum_{r=0}^{n-3}s^r\\left[ \\sum_{k=r+3}^{n}\\sum_{j=r+1}^{k-2}-zs^{4+j-2k}D_{n-j,n-r}\\right.\\right.\\\\\n& \\left.\\left. \\quad{} +\\sum_{j=r+1}^{n-1}s^{2-j-1}D_{n-j,n-r}\\right]\\right)\\\\\n &=L_{n-1}\\left(\\gamma_n-zD_{1,2}-z\\sum_{r=0}^{n-3}s^r\\left[ \\sum_{j=r+1}^{n-2}s^j D_{n-j,n-r}\\left(\\sum_{k=j+2}^{n}-zs^{4-2k}\\right)\\right.\\right.\\\\\n& \\left.\\left. \\quad{}\n +\\sum_{j=r+1}^{n-1}s^{2-j-1} D_{n-j,n-r}\\right]\\right)\\\\\n &=L_{n-1}\\left(\\gamma_n-z D_{1,2}-z\\sum_{r=0}^{n-3}s^r\\left[ \\sum_{j=r+1}^{n-2}s^j D_{n-j,n-r}\\left(z\\frac{s^{4-2n}-s^{2-2j}}{s^2-1}\\right)\\right.\\right.\\\\\n& \\left.\\left. \\quad{} +\\sum_{j=r+1}^{n-1}s^{2-j-1} D_{n-j,n-r}\\right]\\right)\\\\\n &=L_{n-1}\\left(\\gamma_n-z D_{1,2}-z\\sum_{r=0}^{n-3}s^r\\left[ \\sum_{j=r+1}^{n-2}\\left(s^{j-2n+3}-s^{1-j}\\right) D_{n-j,n-r}\\right.\\right.\\\\\n& \\left.\\left. \\quad{}+\\sum_{j=r+1}^{n-1}s^{2-j-1} D_{n-j,n-r}\\right]\\right)\\\\\n &=L_{n-1}\\left(\\gamma_n-z D_{1,2}-z\\sum_{r=0}^{n-3}s^r\\Bigg[s^{2-n} D_{1,n-r}\n  \\right.\\\\\n& \\left.\\left. \\quad{} +\\sum_{j=r+1}^{n-2}\\left(s^{j-2n+3}-s^{1-j}+s^{2-j-1}\\right) D_{n-j,n-r}\\right]\\right)\\\\\n &=L_{n-1}\\left(\\gamma_n-z D_{1,2}-z\\sum_{r=0}^{n-3}s^r\\left[s^{2-n} D_{1,n-r} +\\sum_{j=r+1}^{n-2}\\left(s^{j-2n+3}\\right) D_{n-j,n-r}\\right]\\right)\\\\\n &=L_{n-1}\\left(\\gamma_n-z D_{1,2}-z\\sum_{r=0}^{n-3}s^r\\sum_{j=r+1}^{n-1}s^{j-2n+3} D_{n-j,n-r}\\right)\\\\\n &=L_{n-1}\\left(\\gamma_n-\\frac{z}{s^{2n-3}}\\left(s^{2n-3} D_{1,2}+\\sum_{r=0}^{n-3}s^r\\sum_{j=r+1}^{n-1}s^{j} D_{n-j,n-r}\\right)\\right)\\\\\n &=L_{n-1}\\left(\\gamma_n-\\frac{z}{s^{2n-3}}\\left(s^{2n-3} D_{1,2}+\\sum_{i=2}^{n-1}s^{n-i-1}\\sum_{j=n-i}^{n-1}s^j D_{n-j,i+1}\\right)\\right)\\\\\n &=L_{n-1}\\left(\\gamma_n-\\frac{z}{s^{2n-3}}\\left(s^{2n-3} D_{1,2}+\\sum_{i=2}^{n-1}s^{n-i-1}\\sum_{j=0}^{i-1}s^{n-i+j} D_{i-j,i+1}\\right)\\right)\\\\\n &=L_{n-1}\\left(\\gamma_n-\\frac{z}{s^{2n-3}}\\left(s^{2n-3} D_{1,2}+\\sum_{i=2}^{n-1}s^{2n-2i-1}\\sum_{j=0}^{i-1}s^{j} D_{i-j,i+1}\\right)\\right)\\\\\n &=L_{n-1}\\left(\\gamma_n-\\frac{z}{s^{2n-3}}\\left(\\sum_{i=1}^{n-1}s^{2n-2i-1}\\sum_{j=0}^{i-1}s^{j} D_{i-j,i+1}\\right)\\right) \\\\\n\t\t& = Y_{n-1}\\big\\vert_{a^{-1}=0} \\cdot \\left(\\gamma_n+\\frac{z}{1-s^{2n-3}a}\\left(\\sum_{i=1}^{n-1}as^{2n-2i-1} \\sum_{j=0}^{i-1}s^{j} D_{i-j,i+1}\\right) \\right)\\bigg\\vert_{a^{-1}=0} = Y_n\\big\\vert_{a^{-1}=0}.\n\\end{align*}\nAt the last equality, we used \\eqref{inductiveform}.\n\\end{proof}\n\n\\section[The $n$-colored ruling polynomial and representation numbers]{The $\\boldsymbol{n}$-colored ruling polynomial and representation numbers} \\label{sec:5}\n\nIn this section, we show that the $1$-graded $n$-colored ruling polynomial agrees with the $1$-graded, total $n$-dimensional representation number defined in~\\cite{LeRu}; see Theorem~\\ref{thm:Rep}. After a brief review of Legendrian contact homology and relevant material from~\\cite{LeRu}, the remainder of the section contains the proof of Theorem~\\ref{thm:Rep}.\n\n\\subsection{Review of the Legendrian contact homology DGA} We assume familiarity with the Legendrian contact homology differential graded algebra, (abbrv. LCH DGA), aka.\\ the Chekanov--Eliashberg algebra, in the setting of Legendrian links in $J^1M$ with $M = \\mathbb {R}$ or $S^1$, and refer the reader to any of \\cite{Chekanov02, EN, ENS, Ng1, NgR2012} for this background material. We continue to use coordinates $(x,y,z) \\in J^1M = T^*M \\times \\mathbb {R} = M \\times \\mathbb {R}^2$, and to view projections to $S^1\\times \\mathbb {R}$ in $[0,1]\\times \\mathbb {R}$ with the left and right boundary identified.\n The Reeb vector field is $\\frac{\\partial}{\\partial z}$, so Reeb chords of $K$ are in bijection with double points of the {\\it Lagrangian projection} aka. the {\\it $xy$-diagram} of $K$ which is the projection to $T^*M$.\n Representation numbers are defined in \\cite{LeRu} using the fully non-commutative version of the LCH DGA associated to a Legendrian knot or link $K$ equipped with a collection of base points, $*_1, \\ldots, *_\\ell$, with the requirement that every component of $K$ has at least one base point. The resulting DGA, notated $(\\mathcal{A}(K), \\partial)$ or $(\\mathcal{A}(K, *_1, \\ldots, *_\\ell), \\partial)$ when the choice of base points should be emphasized, is an associative, non-commutative algebra with identity generated over $\\mathbb {Z}$ by\n\\begin{itemize}\\itemsep=0pt\n\\item[(i)] the Reeb chords of $K$, denoted $b_1, \\ldots, b_r$, and\n\\item[(ii)] invertible generators $t_1^{\\pm1}, \\ldots, t_\\ell^{\\pm1}$ corresponding to the base points $*_1, \\ldots, *_\\ell$.\n\\end{itemize}\nThere are no relations other than $t_it_i^{-1} = t_i^{-1}t_i=1$. The differential $\\partial$ vanishes on the $t_i$; for a Reeb chord, $a$, the differential $\\partial a$ is defined via a signed count of rigid holomorphic disks in~$T^*M$ with boundary on the Lagrangian projection of $K$ and having a single positive boundary puncture at $a$ and an arbitrary number of negative boundary punctures. Each such a disk $u$ contributes a term $\\pm w(u)$ to $\\partial a$ where~$w(u)$ is the product of base point generators and negative punctures as they appear in counter-clockwise order along the boundary of the domain of~$u$, starting from the positive puncture at~$a$. Occurrences of~$t_i$ appear with exponent~$\\pm1$ according to the oriented intersection number of $\\partial u$ with $*_i$. See Fig.~\\ref{fig:PointedDisk}. For associating $\\pm1$ signs to disks, we use the conventions as in \\cite{HenryRu, LeRu}. Most results of this section concern DGA representations defined over a field of characteristic~$2$, and in this case it suffices to work with the version of~$\\mathcal{A}(K)$ defined over $\\mathbb {Z}\/2$ where the~$\\pm1$ signs become irrelevant.\n\n\\begin{figure}\n\\labellist\n\\small\n\\pinlabel $a$ [l] at 186 102\n\\pinlabel $+$ [r] at 166 102\n\\pinlabel $b_1$ [b] at 136 180\n\\pinlabel $b_2$ [b] at 24 182\n\\pinlabel $b_3$ [tl] at 84 20\n\\pinlabel $-$ [t] at 132 152\n\\pinlabel $-$ [t] at 28 162\n\\pinlabel $-$ [b] at 86 34\n\\endlabellist\n\\centerline{\\includegraphics[scale=.6]{images\/PointedDisk}}\n\n\\caption{A holomorphic disk contributing the term $\\partial a = \\pm b_1 b_2 t^{-1} b_3 + \\cdots$\n to the differential of $\\mathcal{A}(K)$.}\n\\label{fig:PointedDisk}\n\\end{figure}\n\n\\subsection{1-graded representation numbers} In~\\cite{LeRu}, Legendrian invariant $m$-graded representation numbers are defined for any non-negative integer $m\\geq 0$ by considering DGA homomorphisms that are only required to preserve grading mod $m$. In the current article, we are concerned only with $1$-graded representations, which we will refer to as {\\it ungraded} representations since the grading condition becomes vacuous when $m=1$. We review definitions from~\\cite{LeRu} in the ungraded setting.\n\nLet $V$ be a vector space over a field, $\\mathbb{F}$, with $\\operatorname{char}(\\mathbb{F}) = 2$, and let $d\\colon V \\rightarrow V$ be an ungraded differential on~$V$, i.e., $d$ is just a linear map satisfying $d^2=0$.\n Then, $d$ induces a differential on the endomorphism algebra\n\\[\n\\delta\\colon \\ \\operatorname{End}(V) \\rightarrow \\operatorname{End}(V), \\qquad \\delta(T) = d \\circ T + T \\circ d\n\\]\nmaking $(\\operatorname{End}(V),\\delta)$ into an ungraded DGA, i.e.,\n $\\delta$ satisfies $\\delta^2=0$ and $\\delta(T_1T_2) = \\delta(T_1)T_2 + T_1\\delta(T_2)$. An {\\it ungraded representation} of a DGA, $(\\mathcal{A}, \\partial)$, on $(V,d)$ is an ungraded DGA homomorphism\n\\[\nf\\colon \\ (\\mathcal{A}, \\partial) \\rightarrow (\\operatorname{End}(V), \\delta),\n\\]\ni.e., a ring homomorphism satisfying $f(1)=1$ and $f \\circ \\partial = \\delta \\circ f$.\nIn the $1$-dimensional case where $V = \\mathbb{F}$ and $d=0$, an ungraded representation on $(\\mathbb{F},0)$ is also called an {\\it ungraded augmentation}.\n\nFor $\\mathcal{A} = \\mathcal{A}(K)$, we use the notation $\\overline{\\operatorname{Rep}}_1(K, (V, d))$ for the set of all ungraded representations of $(\\mathcal{A},\\partial)$ on $(V,d)$ and $\\overline{\\operatorname{Aug}}_1(K, \\mathbb{F})$ for the set of augmentations to $\\mathbb{F}$.\n In the case where $K$ is connected with base points $*_1, \\ldots, *_\\ell$ appearing in order starting at $*_1$ and following the orientation of $K$, given a subset $T \\subset {\\rm GL}(V)$ we will use the notation\n $\\overline{\\operatorname{Rep}}_1(K,(V,d),T)$ to denote the set of those $f \\in \\overline{\\operatorname{Rep}}_1(K, (V, d))$ such that $f(t_1\\cdots t_\\ell) \\in T$. In particular, $\\overline{\\operatorname{Rep}}_1(K, (V, d)) =\\overline{\\operatorname{Rep}}_1(K,(V,d), {\\rm GL}(V))$.\n\n\\begin{Definition} \\label{def:total} Let $\\mathbb{F}_q$ denote the finite field of order $q$ with $q$ a power of $2$. The ($1$-graded) {\\it total $n$-dimensional representation number} of $K$ is defined by\n\\begin{align*}\n\\operatorname{Rep}_1\\big(K, \\mathbb{F}^n_q\\big) & := \\big|\\operatorname{End}\\big(\\mathbb{F}^n_q\\big)\\big|^{-{\\rm rb}(K)\/2}  |{\\rm GL}(n, \\mathbb{F}_q) |^{-\\ell} \\big| \\overline{\\operatorname{Rep}}_1\\big(K, \\big(\\mathbb{F}^n_q, 0\\big)\\big) \\big| \\\\\n & = \\big(q^{n^2}\\big)^{-{\\rm rb}(K)\/2}   \\left(q^{n(n-1)\/2}   \\prod_{m=1}^n\\big(q^m-1\\big)\\right)^{-\\ell} \\big| \\overline{\\operatorname{Rep}}_1\\big(K, \\big(\\mathbb{F}^n_q, 0\\big)\\big) \\big|,\n\\end{align*}\n where ${\\rm rb}(K)$ is the number of Reeb chords of $K$ and $\\ell$ is the number of basepoints.\n\\end{Definition}\n\nThat $\\operatorname{Rep}_1\\big(K, \\mathbb{F}^n_q\\big)$ only depends on the Legendrian isotopy type of $K$ is established in \\cite[Proposition~3.10]{LeRu}. The following theorem shows that the $1$-graded $n$-colored ruling polynomial and the total $n$-dimensional representation numbers of~$K$ are equivalent Legendrian invariants.\n\n\\begin{Theorem} \\label{thm:Rep}\nLet $K \\subset J^1 \\mathbb {R}$ be a $($connected$)$ Legendrian knot and $\\mathbb{F}_q$ a finite field of order $q$ with characteristic~$2$. Then,\n\\[\nR^1_{n,K}(q) = \\operatorname{Rep}_1(K, \\mathbb{F}_q),\n\\]\nwhere $R^1_{n,K}(q) = R^1_{n,K}(s)\\big\\vert_{s=q^{1\/2}}$.\n\\end{Theorem}\n\nThe proof of Theorem \\ref{thm:Rep} rests on the following relation between representation counts and reduced ruling polynomials which we will establish in Sections~\\ref{sec:strat}--\\ref{sec:augsat} using extensions of results from~\\cite{HenryRu} and~\\cite{LeRu}. In \\cite[Section~4]{LeRu}, a ``path subset''\\footnote{The {\\it path subset} $B_\\beta$ is the subset of ${\\rm GL}(n,\\mathbb{F}_q)$ arising from specializing the path matrix $P^{xy}_\\beta$ using arbitrary ring homomorphisms from $\\mathcal{A}(\\beta)$ to $\\mathbb{F}_q$. The {\\it path matrix} $P^{xy}_\\beta$ is a matrix whose entries belong to $\\mathcal{A}(\\beta)$ and record certain left-to-right paths through the $xy$-diagram of $\\beta$ that reflect the possible behavior of boundaries of holomorphic disks bordering $\\beta$ from above. See \\cite[Section~4.1]{LeRu}. Note that~\\cite{LeRu} uses the opposite convention for composing Legendrian $n$-tangles, with $\\alpha \\cdot \\beta$ defined as $\\alpha$ stacked to the {\\it left} of~$\\beta$. Consequently, for a permutation $\\beta \\in S_n$, the permutation braid $\\beta \\subset J^1S^1$ used in the present article corresponds to $\\beta^{-1} \\subset J^1S^1$ in~\\cite{LeRu}. As a~result, the notations $B_\\beta$ and $P^{xy}_\\beta$ used here correspond to $B_{\\beta^{-1}}$ and $P_{\\beta^{-1}}^{xy}$ in~\\cite{LeRu}.} $B_\\beta \\subset {\\rm GL}(n, \\mathbb{F}_q)$ is associated to a reduced\\footnote{A positive permutation braid $\\beta\\subset J^1S^1$ is {\\it reduced} if its front diagram corresponds to a reduced braid word, i.e., one where the product $\\sigma_i\\sigma_{i+1}\\sigma_i$ does not appear for any~$i$.} positive permutation braid, $\\beta \\in S_n$.\n\n\\begin{Lemma} \\label{lem:count}\nLet $\\beta \\in S_n$ be an $n$-stranded reduced positive permutation braid, and suppose that the $($connected$)$ Legendrian knot $K \\subset J^1\\mathbb {R}$ has its front diagram in plat position\\footnote{A front diagram is in {\\it plat position} if all left cusps appear at a common $x$-coordinate at the far left of the diagram and all right cusps appear at a common $x$-coordinate at the far right of the diagram.} and is equipped with $\\ell$ base points where~$\\ell$ is the number of right cusps of~$K$. Then,\n\\begin{gather*}\n\\big|\\overline{\\operatorname{Rep}}_1\\big(K, \\big(\\mathbb{F}_q^n,0\\big), B_\\beta\\big)\\big| = |{\\rm GL}(n, \\mathbb{F}_q)|^{\\ell-1}  q^{n(n-1)\/2}  (q-1)^n   q^{n^2   {\\rm rb}(K)\/2}   q^{\\lambda(\\beta)\/2}\\widetilde{R}_{S(K, \\beta)}(z),\n\\end{gather*}\nwhere ${\\rm rb}(K)$ is the number of Reeb chords of $K$, $\\lambda(\\beta)$ is the length of $\\beta$, i.e., the number of crossings of $\\beta$, and $z=q^{1\/2}-q^{-1\/2}$.\n\\end{Lemma}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:Rep}] Since both sides of the equation are Legendrian isotopy invariant, we may assume $K$ is in plat position.\nIn \\cite[Proposition 4.14]{LeRu}, it is shown that ${\\rm GL}(n, \\mathbb{F}_q) = \\sqcup_{\\beta \\in S_n} B_\\beta$. (Actually, this coincides with the Bruhat decomposition of ${\\rm GL}(n, \\mathbb{F}_q)$.) Thus,\n\\[\n\\overline{\\operatorname{Rep}}_1\\big(K, \\big(\\mathbb{F}_q^n,0\\big), {\\rm GL}(n, \\mathbb{F}_q)\\big) = \\sqcup_{\\beta \\in S_n} \\overline{\\operatorname{Rep}}_1\\big(K, \\big(\\mathbb{F}_q^n,0\\big),B_\\beta\\big),\n\\]\n and using Lemma \\ref{lem:count} and Definition \\ref{def:nruling} we compute\n\\begin{align*}\n\\operatorname{Rep}_1\\big(K, \\mathbb{F}_q^n\\big) & = \\big(q^{n^2}\\big)^{-{\\rm rb}(K)\/2}  |{\\rm GL}(n)|^{-\\ell}  \\sum_{\\beta \\in S_n} \\overline{\\operatorname{Rep}}_1\\big(K, \\big(\\mathbb{F}_q^n,0\\big),B_\\beta\\big) \\\\\n & = |{\\rm GL}(n)|^{-1}   q^{n(n-1)\/2}(q-1)^n \\sum_{\\beta \\in S_n}q^{\\lambda(\\beta)\/2} \\widetilde{R}_{S(K, \\beta)}(z) \\\\\n& = \\left( q^{n(n-1)\/2}  \\prod_{m=1}^n\\big(q^m-1\\big)\\right)^{-1}  q^{n(n-1)\/2}(q-1)^n \\sum_{\\beta \\in S_n} q^{\\lambda(\\beta)\/2} \\widetilde{R}_{S(K, \\beta)}(z) \\\\\n& = \\left(\\prod_{m=1}^n\\left(\\frac{q^m-1}{q-1} \\right) \\right)^{-1} \\sum_{\\beta \\in S_n} q^{\\lambda(\\beta)\/2} \\widetilde{R}_{S(K, \\beta)}(z) \\\\\n& = \\frac{1}{c_n} \\sum_{\\beta \\in S_n} q^{\\lambda(\\beta)\/2} \\widetilde{R}_{S(K, \\beta)}(z) = R^1_{n,K}(q).\n\\end{align*}\n(The notation $c_n$ is as in Definition \\ref{def:nruling} with $s = q^{1\/2}$.)\n\\end{proof}\n\n\\subsection{Strategy of the proof of Lemma \\ref{lem:count}} \\label{sec:strat}\n\nThe relation between counts of representations on $\\big(\\mathbb{F}_q^n,0\\big)$ and reduced ruling polynomials stated in Lemma \\ref{lem:count} is based on refinements of two results:\n\\begin{enumerate}\\itemsep=0pt\n\\item In \\cite[Theorem~6.1]{LeRu}, it is shown that for any $n$-stranded reduced positive permutation braid $\\beta \\in S_n$,\n (using $1$ base point on $K$ and a particular $xy$-diagram of $S(K,\\beta)$) when $\\operatorname{char}(\\mathbb{F}) = 2$ there is a bijection\n\\begin{gather} \\label{eq:LeRu}\n\\bigsqcup_{d}\\overline{\\operatorname{Rep}}_1\\big(K, \\big(\\mathbb{F}^n,d\\big),B_\\beta\\big) \\quad \\longleftrightarrow \\quad \\overline{\\operatorname{Aug}}_1(S(K, \\beta), \\mathbb{F}),\n\\end{gather}\nwhere the union is over all strictly upper triangular differentials on $\\mathbb{F}^n$.\n\n\\item In  \\cite[Theorem 3.2]{HenryRu}, for any Legendrian $K' \\subset J^1\\mathbb {R}$ (using a particular $xy$-diagram of~$K'$) the set of augmentations of~$K'$ is decomposed into pieces\n\\begin{gather} \\label{eq:HenryRu}\n\\operatorname{Aug}_1(K', \\mathbb{F}) = \\bigsqcup_{\\rho} (\\mathbb{F}^*)^{a(\\rho)} \\times \\mathbb{F}^{b(\\rho)},\n\\end{gather}\nwhere the disjoint union is indexed by all normal rulings of $K'$ and the exponents~$a(\\rho)$ and $b(\\rho)$ are specified by the combinatorics of~$\\rho$. Theorem~1.1 of~\\cite{HenryRu} then applies the decomposition to relate the Legendrian invariant augmentation numbers with the ruling polynomial.\n\\end{enumerate}\nThe idea behind the proof of Lemma~\\ref{lem:count} is then to check that the subset of $\\operatorname{Aug}_1(S(K,\\beta), \\mathbb{F})$ corresponding under~(\\ref{eq:LeRu}) to $\\overline{\\operatorname{Rep}}_1\\big(K, \\big(\\mathbb{F}^n, 0\\big), B_\\beta\\big)$, i.e., those representations with $d=0$, is the part of the disjoint union~(\\ref{eq:HenryRu}) with $K' = S(K,\\beta)$ that is indexed by {\\it reduced} normal rulings of~$S(K, \\beta)$. This is roughly what we shall do. However, complications arise as different (but Legendrian isotopic) $xy$-diagrams for the satellite $S(K,\\beta)$ having different (but stable tame isomorphic) DGAs are used in~(\\ref{eq:LeRu}) and~(\\ref{eq:HenryRu}). As a result, we need to also keep track of the way the set $\\operatorname{Aug}_1(S(K, \\beta), \\mathbb{F})$ changes when transitioning between these different diagrams for $S(K,\\beta)$. This contributes to the factor appearing in front of~$\\widetilde{R}_{S(K,\\beta)}$ in the statement of Lemma~\\ref{lem:count}.\n\n\\begin{Remark}\n In the case of $m$-graded representations with $m \\neq 1$, $d=0$ is the only term in the disjoint union (\\ref{eq:LeRu}) for grading reasons.\n\\end{Remark}\n\n\\subsection[Four diagrams for the satellite, $S(K, \\beta)$]{Four diagrams for the satellite, $\\boldsymbol{S(K, \\beta)}$} \\label{sec:satellite}\n\nIn establishing Lemma~\\ref{lem:count} we will make use of four different (but Legendrian isotopic) versions of the satellite $S(K,\\beta)$. The four $xy$-diagrams are denoted $S^1_{xy}(K,\\beta)$, $S^2_{xy}(K, \\beta)$, $S^1_{xz}(K,\\beta)$, $S^2_{xz}(K,\\beta)$ and will be defined momentarily; see Fig.~\\ref{fig:SKbxy}. As a~preliminary, we consider $xz$-diag\\-rams (front projection) and $xy$-diagrams (Lagrangian projection) for the companion \\mbox{$K \\subset J^1\\mathbb {R}$} and pattern $\\beta \\subset J^1S^1$.\n\n\\subsubsection[Diagrams for $K$ and $\\beta$]{Diagrams for $\\boldsymbol{K}$ and $\\boldsymbol{\\beta}$}\n\nFor the companion knot $K \\subset J^1\\mathbb {R}$, apply Ng's resolution procedure (see~\\cite{Ng1}) so that the $xy$-diagram for $K$ is related to the front projection of~$K$ by placing the strand with smaller slope on top at crossings and adding an extra loop at right cusps. Enumerate the Reeb chords of~$K$ by $a_1, \\ldots, a_m$, and $c_1, \\ldots, c_\\ell$ where the $a_i$ correspond to crossings of the front projection of $K$ and the $c_i$ are the extra crossings near right cusps that arise from the resolution procedure. Choose an initial base point~$*$ of~$K$ not located on any of the loops at right cusps and at a point where the front diagram of~$K$ is oriented left-to-right. For convenience, we assume the $c_1, \\ldots, c_\\ell$ are enumerated in the order they appear when following along~$K$ according to its orientation, starting at~$*$.\n\nFor the braid $\\beta \\subset J^1S^1$, we form an $xy$-diagram as indicated in Fig.~\\ref{fig:Dips} having $\\ell$ dips in addition to the original crossings coming from the front projection of $\\beta$. This is done by applying the resolution procedure to the front diagram of $\\beta$ as in \\cite[Section~2.2]{NgTraynor} (this amounts to adding a dip to the right of the crossings of $\\beta$), and then adding $\\ell-1$ extra dips to the $xy$-diagram. In addition, a collection of $n$ basepoints, $*_1, \\ldots, *_n$, are placed immediately to the left of the crossings of $\\beta$, one on each strand. Note that the addition of the dips can be accomplished by Legendrian isotopy as indicated in \\cite[Section 3.1]{Sab}, and we have used a minor variation on the resolution procedure of \\cite[Section 2.2]{NgTraynor} so that the dip arising from the resolution procedure has the same form as the others. Numbering the dips $1,\\ldots, \\ell$ as they appear from left to right in $T^*S^1$ (viewed as $[0,1]\\times \\mathbb {R}$ with boundaries identified) the $k$-th dip consists of two groups of crossings $x_{i,j}^k$ and $y_{i,j}^k$ for $1 \\leq i < j \\leq n$ that appear in the left and right half of the dip respectively. For $1 \\leq k \\leq \\ell$, we collect these crossings as the entries of strictly upper triangular matrices denoted $X_k$ and $Y_k$. The crossings that correspond to $xz$-crossings of $\\beta$ are labeled $p_1, \\ldots, p_\\lambda$, and DGA generators associated to the basepoints $*_1,\\ldots, *_n$ will be labelled $s_1, \\ldots, s_n$.\n\n\\begin{figure}\n\\labellist\n\\small\n\\pinlabel $z$ [l] at 4 26\n\\pinlabel $x$ [l] at 28 2\n\\pinlabel $y$ [l] at 250 26\n\\pinlabel $x$ [l] at 274 2\n\\pinlabel $X_1$ at 392 24\n\\pinlabel $Y_1$ at 452 24\n\\pinlabel $X_2$ at 542 24\n\\pinlabel $Y_2$ at 602 24\n\\pinlabel $\\leadsto$ at 200 96\n\\pinlabel $*_3$ at 264 82\n\\pinlabel $*_2$ at 264 98\n\\pinlabel $*_1$ at 264 112\n\n\\endlabellist\n\\centerline{\\includegraphics[scale=.7]{images\/Dips}}\n\n\\caption{A Lagrangian diagram for the positive braid $\\beta = \\sigma_2 \\sigma_1\\sigma_2$ with $\\ell =2$ dips.}\n\\label{fig:Dips}\n\\end{figure}\n\nIn constructing the diagrams $S^1_{xy}(K,\\beta)$ and $S^2_{xy}(K,\\beta)$ and comparing their DGAs, it is useful to be able to move the locations of dips around via a Legendrian isotopy. This may be accomplished by converting an isotopy $\\phi_t\\colon S^1 \\rightarrow S^1$, $0\\leq t \\leq 1$, to an ambient contact isotopy\n\\begin{gather} \\label{eq:contact}\n\\Phi_t\\colon J^1S^1 \\rightarrow J^1S^1,\\qquad \\Phi_t(x,y,z) = \\left(\\phi_t(x), \\frac{y}{\\phi'_t(x)}, z\\right), \\qquad 0\\leq t \\leq 1.\n\\end{gather}\nIn particular, given any cyclically ordered collection of points $x_1, \\ldots, x_\\ell \\in S^1$, after a Legendrian isotopy the $X_k$ and $Y_k$ crossings can be arranged to appear above an arbitrarily small neighborhood of $x_k$ in $S^1$ for $1\\leq k \\leq \\ell$.\n\n\\subsubsection{Diagrams for Legendrian satellites}\n\nThe Legendrian satellite $S(K,\\beta) \\subset J^1\\mathbb {R}$ is formed by scaling the $y$ and $z$ coordinates of $J^1S^1$ so that $\\beta$ sits in a small neighborhood, $N_0 \\subset J^1S^1$, of the $0$-section and then applying a~contactomorphism $\\Psi\\colon N_0 \\rightarrow N(K)$ of $N_0$ onto a Weinstein tubular neighborhood of~$K$. Two general methods for forming $xy$-diagrams of satellites are as follows:\n\n\\begin{itemize}\\itemsep=0pt\n\\item {\\it The $xy$-method.} Use the orientations of $K$ and $\\mathbb {R}^2$ to identify an (immersed) annular neighborhood, $N_{xy}$, of the $xy$-diagram of $K$ with $S^1 \\times (-\\epsilon,\\epsilon)$. Then, (after scaling the $y$-coordinate appropriately) place an $xy$-diagram for $\\beta$ into $N_{xy}$. As $\\Psi$ can be chosen to preserve the Reeb vector field (see \\cite{Geiges}), this indeed produces an $xy$-diagram for $S(K,\\beta)$.\n\n\\item {\\it The $xz$-method.} First form an $xz$-diagram for $S(K,\\beta)$ as in Section \\ref{sec:sat}: start with the $n$-copy of $K$ (this is $n$-parallel copies of $K$ shifted a small amount in the $z$-direction), then insert the $xz$-projection of $\\beta$\n at the location of the initial base point of $K$.\n Finally, produce an $xy$-diagram for $S(K,\\beta)$ by applying Ng's resolution procedure.\n\\end{itemize}\n\n{\\bf Definition of $\\boldsymbol{S^1_{xy}(K,\\beta)}$:} To form $S^1_{xy}(K,\\beta)$ expand $*$ to a cluster of base points $*_1, \\ldots, *_\\ell$ all appearing in order (according to the orientation of $K$) in a small neighborhood of~$*$. Then, form the satellite with the $xy$-diagram of $\\beta$ described above in such a way that the crossings $p_1, \\ldots, p_\\lambda$ from $\\beta$ and the crossings from $X_1$, $Y_1$ appear in a neighborhood of~$*_1$, and the crossings from $X_k, Y_k$ appear in a neighborhood of $*_k$ for $2 \\leq k \\leq \\ell$.\n\n{\\bf Definition of $\\boldsymbol{S^2_{xy}(K,\\beta)}$:} This $xy$-diagram is formed from $S^1_{xy}(K,\\beta)$ by using a Legendrian isotopy of $\\beta$ (constructed from an isotopy of~$S^1$ as in~(\\ref{eq:contact})) to relocate each collection of~$X_i$ crossings to a neighborhood of a right cusp of~$K$ (so that exactly one $X_i$ appears near each right cusp); relocate each group of $Y_i$ crossings to a neighborhood of a left cusp; and leave the crossings $p_1, \\ldots, p_\\lambda$ of $\\beta$ (and the basepoints $*_1, \\ldots, *_n$) in place at the location of the initial base point $*$ for~$K$. (Recall that~$\\ell$ is both the number of dips and the number of right cusps of~$K$.)\n\n{\\bf Definition of $\\boldsymbol{S^1_{xz}(K,\\beta)}$ and $\\boldsymbol{S^2_{xz}(K,\\beta)}$:} Both $S^1_{xz}(K,\\beta)$ and $S^2_{xz}(K,\\beta)$ are formed using the $xz$-method. The only difference between the two is the placement of base points. For~$S^1_{xz}(K,\\beta)$, base points $*_1, \\ldots, *_n$ appear, one on each parallel strand of the $n$-copy, just before~$\\beta$ (with respect to the orientation of $K$). For $S^2_{xz}(K,\\beta)$, we have base points $*_1, \\ldots, *_c$, one on each component of $S(K,\\beta)$ placed on a loop near some right cusp of the component.\n\nSee Fig.~\\ref{fig:SKbxy}.\n\n\\begin{figure}\n\\labellist\n\\small\n\\pinlabel $S^1_{xy}(K,\\beta)$ [t] at 130 0\n\\pinlabel $S^2_{xy}(K,\\beta)$ [t] at 474 0\n\n\\endlabellist\n\\centerline{\\includegraphics[scale=.6]{images\/SKbxy}}\n\n\\quad\n\n\\quad\n\n\\labellist\n\\small\n\\pinlabel $S^1_{xz}(K,\\beta)$ [t] at 126 0\n\\pinlabel $S^2_{xz}(K,\\beta)$ [t] at 460 0\n\\endlabellist\n\n\\centerline{\\includegraphics[scale=.6]{images\/SKbxz}}\n\n\\vspace{2mm}\n\n\\caption{The Lagrangian ($xy$)-diagrams $S^1_{xy}(K,\\beta)$, $S^2_{xy}(K,\\beta)$, $S^1_{xz}(K,\\beta)$, and $S^2_{xz}(K,\\beta)$ where $K$ is a Legendrian trefoil and $\\beta = \\sigma_1 \\in S_2$.}\n\\label{fig:SKbxy}\n\\end{figure}\n\n\\subsubsection{DGA generators} We set notations for the generators of the DGAs arising from the various $xy$-diagrams of $S(K,\\beta)$ that have been defined. Many generators are indexed with a pair of subscripts $i$,~$j$. Such a~subscript indicates that at the overstrand of the crossing belongs to the $i$-th copy of $K$ and the understrand belongs to the $j$-th copy of $K$. Here,\n outside of an arc $A \\subset K$ where the $p_1, \\ldots, p_\\lambda$ crossings of $\\beta$ appear, $S(K,\\beta)$ consists of $n$ copies of $K\\setminus A$, which we label from $1$ to $n$ according to the descending order of their $y$-coordinates at the boundary of $A$.\n\n{\\bf Generators of $\\boldsymbol{\\mathcal{A}(S^1_{xy}(K,\\beta))}$ and $\\boldsymbol{\\mathcal{A}(S^2_{xy}(K,\\beta))}$:} The generating sets for these DGAs are in bijection. Both contain the DGA of $\\beta$ as a sub-DGA, and this accounts for generators of the form $p_1, \\ldots, p_\\lambda$, $x_{i,j}^k$, $y^k_{i,j}$ for $1\\leq k \\leq \\ell$ and $1\\leq i < j \\leq n$ as well as invertible generators $s_1, \\ldots, s_n$ associated to the base points $*_1, \\ldots, *_n$ on $\\beta$. In addition, for each of the Reeb chords $a_1, \\ldots, a_m$, and $c_1,\\ldots, c_\\ell$ of $K$ there are $n^2$ Reeb chords for $S^1_{xy}(K,\\beta)$ and $S^2_{xy}(K,\\beta)$ that we denote by $a^{k_1}_{i,j}$ and $c^{k_2}_{i,j}$, $1\\leq k_1 \\leq m$, $1\\leq k_2 \\leq \\ell$, $1 \\leq i,j \\leq n$.\n\n{\\bf Generators of $\\boldsymbol{\\mathcal{A}(S^1_{xz}(K,\\beta))}$ and $\\boldsymbol{\\mathcal{A}(S^2_{xz}(K,\\beta))}$:} Both of these diagram have Reeb chords\n\\begin{itemize}\\itemsep=0pt\n\\item $p_1, \\ldots, p_\\lambda$ from the $xz$-crossings of $\\beta$;\n\\item $a^{k_1}_{i,j}$, $1 \\leq k_1 \\leq m$, $1 \\leq i,j \\leq n$, from the $xz$-crossings of~$K$;\n\\item $y^{k_2}_{i,j}$, $1 \\leq k_2 \\leq \\ell$, $1 \\leq i<j \\leq n$, from crossings near left cusps;\n\\item $c^{k_2}_{i,j}$, $1 \\leq k_2 \\leq \\ell$, $1 \\leq j \\leq i \\leq n$, from crossings near right cusps.\n\\end{itemize}\nThe invertible generators coming from base points will be denoted as $s_1, \\ldots, s_n$ for $S^1_{xz}(K,\\beta)$ and as $r_1, \\ldots, r_c$ for $S^2_{xz}(K,\\beta)$.\n\nIn particular, note that for any of the four diagrams for $S(K,\\beta)$ there is a collection of generators of the form $y^{k}_{i,j}$ which we will refer to as {\\it $Y$-generators}.\n\\begin{Definition} Let $K'$ denote one of $S^1_{xy}(K,\\beta)$, $S^2_{xy}(K,\\beta)$, $S^1_{xz}(K,\\beta)$, or $S^2_{xz}(K,\\beta)$. We denote by\n\\[\n\\overline{\\operatorname{Aug}}_1(K', \\mathbb{F})_{Y=0}\n\\]\nthe set of augmentations of $\\mathcal{A}(K')$ to $\\mathbb{F}$ that map all $Y$-generators to~$0$.\n\\end{Definition}\n\nThe subsets $\\overline{\\operatorname{Aug}}_1(K', \\mathbb{F})_{Y=0} \\subset \\overline{\\operatorname{Aug}}_1(K', \\mathbb{F})$ play an important role in the proof Lemma \\ref{lem:count} as they correspond to representations with $d=0$ as well as to reduced normal rulings.\n\n\\begin{Remark}Computations of the differential for similar DGAs of $1$-dimensional Legendrian satellites are presented in detail in several places, and it is not difficult to extend these computations to give complete formulas for differentials in the present setting. See especially~\\cite{LeRu} for satellites formed via the $xy$-method and~\\cite{NgR2012} and~\\cite{NRSSZ} for satellites formed via the $xz$-method. Rather than giving a complete description of differentials here, we state several partial formulas as they become useful in the following proofs.\n\\end{Remark}\n\n\\subsection{Representations and augmentations of the satellite} \\label{sec:augsat}\n\nWe will use the following mild variation of \\cite[Theorem 6.1]{LeRu} to transition between $n$-dimensional representations and augmentations of satellites.\nAs in the construction of $S^1_{xy}(K,\\beta)$, expand the initial base point $*$ of $K$ into a cluster of base points $*_1, \\ldots, *_\\ell$ and consider the DGA $(\\mathcal{A}(K), \\partial)$ with invertible generators $t_1, \\ldots, t_\\ell$.\n\n\\begin{Proposition} \\label{prop:LeRu} Let $\\beta$ be a reduced positive permutation braid, and construct $S^1_{xy}(K, \\beta)$ as above.\nThere is a bijection\n\\[\n\\big\\{ f \\in \\overline{\\operatorname{Rep}}_1\\big(K,\\big(\\mathbb{F}^n,0\\big)\\big) \\,|\\, f(t_1) \\in B_\\beta \\mbox{ and } f(t_i) \\in N_+ \\mbox{ for $i \\geq 2$}\\big\\} \\quad \\leftrightarrow \\quad \\overline{{\\rm Aug}}_1(S^1_{xy}(K, \\beta), \\mathbb{F})_{Y=0},\n\\]\nwhere $B_\\beta$ is the path subset of $\\beta$ and $N_+ \\subset {\\rm GL}(n, \\mathbb{F})$ is the group of upper triangular matrices with~$1$'s on the diagonal.\n\\end{Proposition}\n\n\\begin{proof}The proof is similar to \\cite[Theorem~6.1]{LeRu} which implies the case of only~$1$ base point. We sketch the argument and highlight the modifications to the proof for the case of more than one basepoint. To avoid considering signs, we only treat the case where $\\operatorname{char}(\\mathbb{F}) = 2$ (which is the only case needed for Lemma~\\ref{lem:count}). This allows us to work with the LCH DGA defined over~$\\mathbb {Z}\/2$ rather than over~$\\mathbb {Z}$ (since any representation with $\\operatorname{char}(\\mathbb{F})=2$ factors through the change of coefficients map from the DGA over~$\\mathbb {Z}$ to the DGA over~$\\mathbb {Z}\/2$).\n\n To relate the differential on $D\\colon \\mathcal{A}\\big(S^1_{xy}(K,\\beta)\\big) \\rightarrow \\mathcal{A}\\big(S^1_{xy}(K,\\beta)\\big)$ to the differential $\\partial$ on $\\mathcal{A}(K)$ consider a $\\mathbb {Z}\/2$-algebra homomorphism\n\\[\n\\Phi\\colon \\ \\mathcal{A}(K) \\rightarrow \\operatorname{Mat}\\big(n, \\mathcal{A}\\big(S^1_{xy}(K,\\beta)\\big)\\big)\n\\]\nthat sends Reeb chords $a_k$ or $c_k$ to the corresponding $n\\times n$ matrices of Reeb chords $A_k = \\big(a^k_{i,j}\\big)$ or $C_k = \\big(c^k_{i,j}\\big)$ and satisfies\n\\begin{gather*}\n\\Phi(t_1) = P^{xy}_{\\beta},\n\\qquad \\Phi(t_k) = (I+X_k), \\qquad \\mbox{for $2\\leq k \\leq \\ell$},\n\\end{gather*}\nwhere $P^{xy}_{\\beta}$ and $P^{xz}_{\\beta}$ denote the $xy$- and $xz$-path matrices of $\\beta$ as defined in \\cite[Section~4.1]{LeRu}. Over~$\\mathbb {Z}\/2$, with the differential $D$ on $\\mathcal{A}\\big(S^1_{xy}(K,\\beta)\\big)$ extended entry-by-entry to \\linebreak $\\operatorname{Mat}\\big(n, \\mathcal{A}\\big(S^1_{xy}(K,\\beta)\\big)\\big)$ we have the identities,\n\\begin{gather}\nD Y_k = Y_k^2, \\qquad 1 \\leq k \\leq \\ell, \\qquad \\mbox{and} \\label{eq:Y1} \\\\\nD \\circ \\Phi(x) = \\Phi \\circ \\partial(x) + O(Y), \\qquad \\mbox{for any generator $x \\in\\mathcal{A}(K)$}, \\label{eq:Y2}\n\\end{gather}\nwhere $O(Y)$ denotes a term belonging to the $2$-sided ideal generated by the $Y$-generators. This is essentially as in \\cite[Section~5]{LeRu}; see especially Proposition~5.2 and Corollary~5.3 of~\\cite{LeRu}. For generalizing to the case of more than one base point, note that, for $1 \\leq k \\leq \\ell$, $\\Phi(t_k)$ is the left-to-right $xy$-path matrix for the part of the $xy$-diagram of~$\\beta$ that sits in a~neighborhood of the basepoint~$*_k$ on~$K$. As a result, the entries of $\\Phi(t_k)$ (resp.\\ $\\Phi\\big(t_k^{-1}\\big)$)\n record the possibly negative punctures that boundaries of ``thick disks'' of~$S^1_{xy}(K,\\beta)$ can have when they pass through the location of~$*_k$ on~$K$ in a way that agrees (resp.\\ disagrees) with the orientation of~$K$; see \\cite[Sections~4.1 and~5.2]{LeRu}.\n\nNow, the bijection from the statement of the proposition arises from associating to an augmentation $\\epsilon \\in \\overline{{\\rm Aug}}_1\\big(S^1_{xy}(K, \\beta), \\mathbb{F}\\big)_{Y=0}$ the matrix representation $f\\colon  \\mathcal{A}(K) \\rightarrow \\operatorname{Mat}(n,\\mathbb{F})$ given by $f = \\epsilon \\circ \\Phi$. From here~(\\ref{eq:Y1}) and~(\\ref{eq:Y2}) can then be used to show that under the assumption that $\\epsilon(Y)=0$, the augmentation equation $\\epsilon \\circ D = 0$ is equivalent to the representation equation $f \\circ \\partial = 0$, cf.\\ \\cite[Theorem~6.1]{LeRu}. (Note that the hypothesis that $\\beta$ is a reduced positive permutation braid is used as in~\\cite{LeRu} to see that the equation $\\epsilon(D \\Phi(t_1)) =0 $ is equivalent to having $\\epsilon \\circ D(x) =0$ for all generators of the form $p_i$ or $x^1_{i,j}$.)\n \\end{proof}\n\nWe are now prepared to give the proof of Lemma~\\ref{lem:count}.\n\n\\begin{proof}[Proof of Lemma \\ref{lem:count}]\\quad\n\n {\\bf Step 1.} Establish that\n\\begin{gather*}\n\\big|\\overline{\\operatorname{Rep}}_1\\big(K, \\big(\\mathbb{F}_q^n,0\\big), B_\\beta\\big)\\big| = \\frac{|{\\rm GL}(n)|^{\\ell-1}}{(q^{n(n-1)\/2})^{\\ell-1}} \\cdot \\big| \\overline{{\\rm Aug}}_1\\big(S^{1}_{xy}(K, \\beta), \\mathbb{F}_q\\big)_{Y=0}\\big|.\n\\end{gather*}\n\nKeeping in mind that $\\overline{\\operatorname{Rep}}_1(K, (\\mathbb{F}_q^n,0), B_\\beta)$ is defined in terms of the DGA $\\mathcal{A}(K, *_1, \\ldots, *_\\ell)$ of~$K$ equipped with $\\ell$ base points $*_1, \\ldots, *_\\ell$, let $\\overline{\\operatorname{Rep}}_1\\big(\\mathcal{A}(K, *), \\big(\\mathbb{F}_q^n,0\\big), B_\\beta\\big)$ denote the corresponding set of representations with respect to the DGA of~$K$ equipped with only the one initial base point~$*$, i.e., the set of ungraded representations $f\\colon (\\mathcal{A}(K,*), \\partial) \\rightarrow (\\operatorname{Mat}(n, \\mathbb{F}_q),0)$ ha\\-ving \\mbox{$f(t) \\in B_\\beta$}. The differential of a Reeb chord~$b$ of $K$ in $\\mathcal{A}(K, *_1, \\ldots, *_\\ell)$ is obtained from its differential in~$\\mathcal{A}(K,*)$ by replacing all occurrences of~$t$ by the product $t_1\\cdots t_\\ell$. Consequently,\n\\[\n\\big|\\overline{\\operatorname{Rep}}_1\\big(K, \\big(\\mathbb{F}_q^n,0\\big), B_\\beta\\big)\\big| = |{\\rm GL}(n)|^{\\ell -1} \\cdot \\big|\\overline{\\operatorname{Rep}}_1\\big(\\mathcal{A}(K, *), \\big(\\mathbb{F}_q^n,0\\big), B_\\beta\\big)\\big|,\n\\]\nwhere the factor $|{\\rm GL}(n)|^{\\ell-1}$ arises as the number of ways to factor a given matrix $f(t) \\in B_\\beta$ into a product $f(t_1) \\cdots f(t_\\ell)$ with $f(t_i) \\in {\\rm GL}(n)$ for $1 \\leq i \\leq \\ell$. On the other hand, using Proposition~\\ref{prop:LeRu} gives\n\\[\n\\big| \\overline{{\\rm Aug}}_1(S^{1}_{xy}(K, \\beta), \\mathbb{F}_q)_{Y=0}\\big| = \\big(q^{n(n-1)\/2}\\big)^{\\ell-1} \\big|\\overline{\\operatorname{Rep}}_1\\big(\\mathcal{A}(K, *), \\big(\\mathbb{F}_q^n,0\\big), B_\\beta\\big)\\big|,\n\\]\nwhere in this case the factor $\\big(q^{n(n-1)\/2}\\big)^{\\ell-1}$ arises as the number of ways to factor a given matrix $f(t) \\in B_\\beta$ into a product $f(t_1) \\cdots f(t_\\ell)$ with $f(t_1) \\in B_\\beta$ and $f(t_i) \\in N_+$ for $2 \\leq i \\leq \\ell$. Here, we use the connection with the Bruhat decomposition from \\cite[Section~4.3]{LeRu} so that~$B_\\beta$ has the form $BS_{\\beta} B$ where $S_\\beta$ is a permutation matrix and~$B$ is the group of invertible upper triangular matrices. Therefore, given $f(t) \\in B_\\beta$ and strictly upper-triangular matrices \\mbox{$f(t_2), \\ldots, f(t_{\\ell}) \\in N_+$} there is a unique element $f(t_1) \\in B_\\beta$ so that $f(t) = f(t_1) f(t_2)\\cdots f(t_{\\ell})$.\n\n{\\bf Step 2.} Establish that\n\\begin{gather*}\n\\big| \\overline{{\\rm Aug}}_1\\big(S^{1}_{xy}(K, \\beta), \\mathbb{F}_q\\big)_{Y=0}\\big| = \\big| \\overline{{\\rm Aug}}_1\\big(S^{2}_{xy}(K, \\beta), \\mathbb{F}_q\\big)_{Y=0}\\big|.\n\\end{gather*}\n\nA Legendrian isotopy of $\\beta$ as in (\\ref{eq:contact}) can be used to produce a Legendrian isotopy of $S(K,\\beta)$ that moves the $X_k$ and $Y_k$ crossings around the annular neighborhood $N_{xy}$ of the Lagrangian projections of~$K$. In particular, this procedure leads to a Legendrian isotopy $\\Lambda_t$, $0 \\leq t \\leq 1$, from $S^1_{xy}(K,\\beta)$ to $S^2_{xy}(K,\\beta)$. Note that the crossings $p_1, \\ldots, p_\\lambda$ and the base points $*_1, \\ldots, *_n$ can be assumed to remain in place during the isotopy. Moreover, since the $xy$-diagram of $\\beta$ has no vertical tangencies, by scaling the $y$- and $z$- coordinates by an appropriately small factor, it can be assumed that, for all $0\\leq t \\leq 1$, the $xy$-diagram of $\\Lambda_t$ does not have self-tangencies. As a result, the Reeb chords of $\\Lambda_t$ appear in continuous $1$-parameter families parametrized by $t \\in [0,1]$, and (identifying the corresponding generators of all $\\mathcal{A}(\\Lambda_t)$)\n the differential remains constant except for a finite number of handleslide disk bifurcations.\n These occur when an~$X_k$ or~$Y_k$ crossing of $\\beta$ passes over or under another strand of the satellite as in Move~I of \\cite[Section~6]{ENS}. During the move, there are three Reeb chords, one that is a crossing of $\\beta$ and two more of the form $a^k_{i,j}$ or $c^k_{i,j}$, that all come together at a triple point. Label the three Reeb chords as $x$, $y$, $z$ so that their lengths (i.e., the difference of $z$-coordinates at endpoints) satisfy $h(x) > h(y) >h(z)$, and note that $z$ is the crossing from $\\beta$. The DGAs before and after the triple point move are related by a DGA isomorphism $\\phi\\colon (\\mathcal{A}, \\partial) \\rightarrow (\\mathcal{A}', \\partial')$ that maps~$x$ to an element of the form $x \\pm yz$ or $x\\pm zy$ and fixes all other generators. In particular, $\\phi$ restricts to the identity on the sub-algebra generated by the $Y$-generators.\nComposing all of the DGA isomorphisms from handleslide disks, we see that there is a DGA isomorphism $\\varphi\\colon \\big(\\mathcal{A}\\big(S^1_{xy}(K, \\beta)\\big), \\partial_1\\big) \\rightarrow \\big(\\mathcal{A}\\big(S^2_{xy}(K, \\beta)\\big), \\partial_2\\big)$ that restricts to the identity on all $Y$-generators. As a result, $\\varphi^*\\colon \\overline{{\\rm Aug}}_1\\big(S^{2}_{xy}(K, \\beta), \\mathbb{F}_q\\big) \\rightarrow \\overline{{\\rm Aug}}_1\\big(S^{1}_{xy}(K, \\beta), \\mathbb{F}_q\\big)$, $\\varphi^*\\epsilon = \\epsilon \\circ \\varphi$ induces the required bijection between $\\overline{{\\rm Aug}}_1\\big(S^{1}_{xy}(K, \\beta), \\mathbb{F}_q\\big)_{Y=0}$ and $\\overline{{\\rm Aug}}_1\\big(S^{2}_{xy}(K, \\beta), \\mathbb{F}_q\\big)_{Y=0}$.\n\n{\\bf Step 3.} Establish that\n \\begin{gather} \\label{eq:Step3}\n\\big| \\overline{{\\rm Aug}}_1\\big(S^{2}_{xy}(K, \\beta), \\mathbb{F}_q\\big)_{Y=0}\\big| = \\big(q^{n(n-1)\/2}\\big)^\\ell\\big| \\overline{{\\rm Aug}}_1\\big(S^{1}_{xz}(K, \\beta), \\mathbb{F}_q\\big)_{Y=0}\\big|.\n\\end{gather}\n\nThe generators of $S^1_{xz}(K,\\beta)$ are identified with a subset of the generators of $S^2_{xy}(K,\\beta)$, and this leads to an algebra inclusion $i\\colon \\mathcal{A}\\big(S^1_{xz}(K,\\beta)\\big) \\rightarrow \\mathcal{A}\\big(S^2_{xy}(K,\\beta)\\big)$. In fact, it is not hard to check that~$i$ is a DGA homomorphism; see \\cite[Proposition~4.23]{NRSSZ} for a detailed explanation in the case where $\\beta$ is the identity braid. The difference between the two DGAs is that $\\mathcal{A}\\big(S^2_{xy}(K,\\beta)\\big)$ has additional generators of the form $c^k_{i,j}$ and $x^k_{i,j}$ with $1 \\leq k \\leq \\ell$, $1 \\leq i<j \\leq n$ that $\\mathcal{A}\\big(S^1_{xz}(K,\\beta)\\big)$ does not have. Equation~(\\ref{eq:Step3}) then follows from:\n\n{\\bf Claim:} Given any $\\epsilon \\in \\overline{{\\rm Aug}}_1\\big(S^{1}_{xz}(K, \\beta), \\mathbb{F}_q\\big)_{Y=0}$ and arbitrary values $\\epsilon'\\big(c^k_{i,j}\\big) \\in \\mathbb{F}_q$, there exists a unique augmentation $\\epsilon' \\in \\overline{{\\rm Aug}}_1\\big(S^{2}_{xy}(K, \\beta), \\mathbb{F}_q\\big)_{Y=0}$ extending these values and restricting to $\\epsilon$ on $\\mathcal{A}\\big(S^{1}_{xz}(K, \\beta)\\big)$.\n\nGiven $\\epsilon$ and $\\epsilon'\\big(c^k_{i,j}\\big)$ we need to show that there are unique values $\\epsilon'\\big(x^k_{i,j}\\big)$ for which the equations\n\\[\n\\epsilon' \\circ \\partial\\big(c^k_{i,j}\\big) = 0 \\qquad \\mbox{and} \\qquad \\epsilon' \\circ \\partial\\big(x^k_{i,j}\\big) =0, \\qquad 1 \\leq k \\leq \\ell, \\  1 \\leq i<j\\leq n\n\\]\nhold. Since $\\partial x^k_{i,j}$ belongs to the $2$-sided ideal generated by the $Y$-generators, and $\\epsilon$ vanishes on all $Y$-generators, the equations $\\epsilon' \\circ \\partial\\big(x^k_{i,j}\\big) =0$ are satisfied. Note that (again computing over~$\\mathbb {Z}\/2$)\n\\[\n\\partial C_k = (I+X_k)^{\\pm 1} + W_k,\n\\]\nwhere $W_k$ denotes a matrix with entries in the subalgebra generated by $\\mathcal{A}_{xz}^1(S(K,\\beta))$ and by the~$c^k_{i,j}$ generators. The map $\\mathbb{F}_q^{n(n-1)\/2} \\rightarrow \\mathbb{F}_q^{n(n-1)\/2}$ that sends a collection of values $\\big(\\epsilon'\\big(x^k_{i,j}\\big)\\big)_{i<j}$ to the above diagonal part of $\\epsilon'\\big((I+X_k)^{\\pm 1}\\big)$ is a bijection. Hence, there is a unique way to choose $\\epsilon'\\big(x^k_{i,j}\\big)$ so that the equations $\\epsilon \\circ\\partial c^k_{i,j}=0$, $1 \\leq i<j\\neq n$ hold (since this is equivalent to having $\\epsilon'\\big((I+X_k)^{\\pm1}\\big) = \\epsilon'(W_k)$ hold on the upper triangular entries.)\n\n{\\bf Step 4.} Establish that\n \\begin{gather} \\label{eq:Step4}\n\\big| \\overline{{\\rm Aug}}_1\\big(S^{1}_{xz}(K, \\beta), \\mathbb{F}_q\\big)_{Y=0}\\big| = (q-1)^{n-c}\\big| \\overline{{\\rm Aug}}_1\\big(S^{2}_{xz}(K, \\beta), \\mathbb{F}_q\\big)_{Y=0}\\big|.\n\\end{gather}\n\nFirst, note that when the locations of basepoints are moved around the number of $Y=0$ augmentations does not change. (This is because when a basepoint~$s_i$ moves through an $xy$-crossing~$a$, the DGAs before and afterward are related by an isomorphism $\\phi$ that maps~$a$ to an element of the form $s_i^{\\pm1} a$ or $a s_i^{\\pm1}$ (depending on orientation of $K$ and whether $s_i$ passes the upper or lower endpoint of~$a$) and fixes all other generators; see \\cite[Theorem~2.20]{NgR2012}. In particular, the induced bijection between augmentation sets, $\\phi^*$, preserves the $Y=0$ subsets (since $\\epsilon(a) = 0$ if and only if $\\epsilon\\big(t_i^{\\pm1}a\\big) =0$).\n\nTo understand the effect of changing the number of basepoints, suppose that $K_1$ and $K_2$ are two $xy$-diagrams, identical except that on $K_1$ a collection of base points with corresponding generators $u_1, \\ldots, u_s$ appears near the location of a single base point $u$ of $K_2$. Arguing as in Step 1, shows that there is a DGA map $\\phi\\colon \\mathcal{A}(K_2) \\rightarrow \\mathcal{A}(K_1)$ with $\\phi(u) = u_1\\cdots u_s$ and $\\phi(x) = x$ for all other generators, and moreover $\\phi^*\\colon \\overline{{\\rm Aug}}_1(K_1, \\mathbb{F}_q) \\rightarrow \\overline{{\\rm Aug}}_1(K_2, \\mathbb{F}_q)$ is surjective and $(q-1)^{s-1}$-to-$1$. (Here, $(q-1)^{s-1}$ represents the number of ways to factor an element of~$\\mathbb{F}_q^*$ into a product of~$s$ elements in $\\mathbb{F}_q^*$.) Since $\\phi$ is the identity on Reeb chords, when applied to~$S(K,\\beta)$, $\\phi^*$~restricts to a~surjective, $(q-1)^{s-1}$-to-$1$ map between the $Y=0$ augmentation sets. Starting with $S^1_{xz}(K,\\beta)$ and applying this procedure repeatedly with the collection of base points on each component (keeping in mind that $S^1_{xz}(K,\\beta)$ has $n$ base points while $S^2_{xz}(K,\\beta)$ has $c$) leads to the formula~(\\ref{eq:Step4}).\n\n{\\bf Step 5.} Establish a decomposition of the form\n\\[\n \\overline{{\\rm Aug}}_1\\big(S^{2}_{xz}(K, \\beta), \\mathbb{F}_q\\big)_{Y=0} = \\bigsqcup_\\rho W_\\rho,\n\\]\nwhere the disjoint union is over reduced normal rulings, $\\rho$, and\n\\[\n|W_\\rho| = (q-1)^{j(\\rho)+c} \\cdot q^{\\frac{1}{2}(-j(\\rho) +{\\rm rb}(S(K,\\beta)))}\n\\]\nwith $c$ the number of components of $S(K,\\beta)$ and ${\\rm rb}(S(K,\\beta))$ the number of Reeb chords of~$S_{xz}^2(K,\\beta)$.\n\nA decomposition of the entire augmentation variety $\\overline{{\\rm Aug}}_1\\big(S^{2}_{xz}(K, \\beta), \\mathbb{F}_q\\big)$ (without imposing the $Y=0$ condition) as $\\bigsqcup W_\\rho$ with the disjoint union over all normal rulings (without the reduced condition) is established in Theorem~3.4 of~\\cite{HenryRu}. The statement of that theorem implies that $|W_\\rho| = (q-1)^{j(\\rho)-c}q^{b(\\rho)}$ where $b(\\rho)$ is the number of ``returns''\\footnote{Given a normal ruling $\\rho$ for a Legendrian link $K'$, the $xz$-crossings of $K'$ that are not switches are either {\\it departures} or {\\it returns}. At a departure (resp.\\ return) the normality condition holds to the left (resp.\\ right) of the crossing but not to the right (resp.\\ left) of the crossing. See \\cite[Section~3]{NgSab}.} of $\\rho$ plus the number of right cusps of $\\rho$, and Lemma~5 of~\\cite{NgSab} shows that $b(\\rho) = \\frac{1}{2} ( -j(\\rho) + {\\rm rb}(S(K,\\beta)))$. Thus, it suffices to show that an augmentation $\\epsilon \\in \\overline{{\\rm Aug}}_1\\big(S^{2}_{xz}(K, \\beta), \\mathbb{F}_q\\big)$ belongs to $W_\\rho$ with $\\rho$ {\\it reduced} if and only if the $Y=0$ condition is satisfied.\n\nA summary of the construction of the decomposition $\\overline{{\\rm Aug}}_1\\big(S^{2}_{xz}(K, \\beta), \\mathbb{F}_q\\big)=\\bigsqcup W_\\rho$ is as follows. Let $K' \\subset J^1\\mathbb {R}$ be a Legendrian in plat position whose DGA is computed from resolving the front projection of $K'$ and positioning a single base point on each component of $K'$ in the loop near some chosen right cusp. The DGA of $S^2_{xz}(K,\\beta)$ is of this required type. For such a~$K'$, the article~\\cite{HenryRu} considers objects called Morse complex sequences (MCSs) that consist of a sequence of chain complexes and formal handleslide marks (which are vertical segments on the front diagram of $K'$) subject to several axioms motivated by Morse theory. Section~5 of \\cite{HenryRu} gives a bijection between $\\overline{{\\rm Aug}}_1(K', \\mathbb{F}_q)$ and the set of ``$A$-form''\\footnote{An MCS is in {\\it $A$-form} if its handleslides only appear in specified locations on the front diagram of $K'$ to the left of crossings and right cusps. See \\cite[Section~5]{HenryRu}.} MCSs for $K'$, denoted here $\\operatorname{MCS}^A(K')$, where an augmentation $\\epsilon$ corresponds to an $A$-form MCS with one handleslide mark just to the left of every crossing or right cusp, $x$, with $\\epsilon(x) \\neq 0$ with the handleslide coefficient determined by the value $\\epsilon(x)$. In Section~4.1 of~\\cite{HenryRu}, another class of MCSs called ``$SR$-form'' MCSs are considered; we will denote the set of all $SR$-form MCSs for~$K'$ as~$\\operatorname{MCS}^{SR}(K')$. Each $SR$-form MCS, $\\mathcal{C}$, has an associated normal ruling~$\\rho$ of~$K'$, and all handleslide marks of $\\mathcal{C}$ appear in collections of a~standard form near switches, returns, and right cusps of~$\\rho$. In particular, at every switch of~$\\rho$,~$\\mathcal{C}$ must have a collection of handleslides with non-zero coefficients; returns and right cusps may or may not have handleslides. In Section~6 of~\\cite{HenryRu} a bijection $\\Psi\\colon \\operatorname{MCS}^A(K') \\rightarrow \\operatorname{MCS}^{SR}(K')$ and its inverse $\\Phi=\\Psi^{-1}$ are constructed. By definition, $\\epsilon \\in W_\\rho$ if the corresponding $A$-form MCS, $\\mathcal{C}_\\epsilon$, is such that the $SR$-form MCS $\\Psi(\\mathcal{C}_\\epsilon)$ has associated normal ruling~$\\rho$.\n\nNote that for an $A$-form or $SR$-form MCS every handleslide has some associated $xz$-crossing or right cusp. (For an $SR$-form MCS with normal ruling $\\rho$, associated crossings can only be switches or returns of $\\rho$ and a single crossing may have more than one handleslide associated to it.) Let $\\operatorname{MCS}^A(S(K,\\beta))_{Y=0}$ and $\\operatorname{MCS}^{SR}(S(K,\\beta))_{Y=0}$ denote those $A$-form and $SR$-form MCSs for $S^2_{xz}(K,\\beta)$ that do not have any handleslide marks associated to $xz$-crossings corresponding to the $Y$-generators. (These are the groups of $n(n-1)\/2$ crossings that appear in $S^2_{xz}(K,\\beta)$ near the location of left cusps of~$K$.) Step~5 is then completed by the following.\n\n\\begin{Lemma}The above constructions from {\\rm \\cite{HenryRu}} restrict to bijections\n\\[\n \\overline{{\\rm Aug}}_1\\big(S^{2}_{xz}(K, \\beta), \\mathbb{F}_q\\big)_{Y=0} \\leftrightarrow \\operatorname{MCS}^A(S(K,\\beta))_{Y=0} \\leftrightarrow \\operatorname{MCS}^{SR}(S(K,\\beta))_{Y=0}\n\\]\nand an $SR$-form MCS $\\mathcal{C}$ belongs to $\\operatorname{MCS}^{SR}(S(K,\\beta))_{Y=0}$ if and only if the associated normal ruling $\\rho$ is reduced.\n\\end{Lemma}\n\\begin{proof}The first bijection is clear since an $A$-form MCS has no handleslides at $Y$ crossings if and only if the corresponding augmentation vanishes on $Y$-generators. Turning to the second bijection, given an $A$-form or $SR$-form MCS, $\\mathcal{C}$, let $i(\\mathcal{C})$ denote the first (from left to right) $xz$-crossing of $S^2_{xy}(K,\\beta)$ that has a handleslide associated to it. The defining requirement for both $\\operatorname{MCS}^A(S(K,\\beta))_{Y=0}$ and $\\operatorname{MCS}^{SR}(S(K,\\beta))_{Y=0}$ is equivalent to $i(\\mathcal{C})$ not being one of the $Y$-crossings. (The $Y$-crossings all appear to the left of the other $xz$-crossings of $S^2_{xz}(K,\\beta)$ since $K$ is in plat position.) Examining the definition of $\\Psi$ and $\\Phi$ in Section 6 of \\cite{HenryRu}, it is straightforward to see that $i(\\Psi(\\mathcal{C})) = i(\\mathcal{C})$ and $i(\\Phi(\\mathcal{C})) = i(\\mathcal{C})$, so that $\\Psi$ and $\\Phi$ restrict to provide the bijection $\\operatorname{MCS}^A(S(K,\\beta))_{Y=0} \\leftrightarrow \\operatorname{MCS}^{SR}(S(K,\\beta))_{Y=0}$.\n\nFor the final statement of the lemma, note that a normal ruling of $S^2_{xz}(K,\\beta)$ is reduced if and only if it has no switches at $Y$-crossings. See \\cite[Lemma~3.2]{NgR2012}. Moreover, if a $Y$-crossing is a~return then there must be another $Y$-crossing somewhere to its left that is a switch. (If there are no switches in the $Y$-crossings then, it is easy to see that all $Y$-crossings are departures.) Thus, for $\\mathcal{C}$ in $SR$-form, $i(\\mathcal{C})$ is a $Y$-crossing if and only if the corresponding ruling is not reduced.\n\\end{proof}\n\n{\\bf Step 6.} Completion of the proof.\n\nCombining the identities from Steps 1--5, we have\n\\begin{align*}\n\\big|\\overline{\\operatorname{Rep}}_1\\big(K, \\big(\\mathbb{F}_q^n,0\\big), B_\\beta\\big)\\big| & = |{\\rm GL}(n)|^{\\ell-1}  q^{n(n-1)\/2} (q-1)^{n-c}  \\sum_{\\rho} (q-1)^{j(\\rho)+c}q^{\\frac{1}{2}(-j(\\rho) + {\\rm rb}(S(K,\\beta)))} \\\\\n& = |{\\rm GL}(n)|^{\\ell-1}  q^{n(n-1)\/2} (q-1)^{n}   q^{\\frac{1}{2}{\\rm rb}(S(K,\\beta))}  \\sum_{\\rho} (q^{1\/2}-q^{-1\/2})^{j(\\rho)} \\\\\n& = |{\\rm GL}(n)|^{\\ell-1}  q^{n(n-1)\/2} (q-1)^{n}   q^{n^2 {\\rm rb}(K)\/2} q^{\\lambda(\\beta)\/2} \\widetilde{R}_{S(K,\\beta)}(z),\n\\end{align*}\nwhere the summations are over all {\\it reduced} normal rulings and at the last equality we used that the number of Reeb chords of $S^2_{xz}(K,\\beta)$ is $n^2 \\cdot {\\rm rb}(K)+\\lambda(\\beta)$.\n\\end{proof}\n\n\\section{The multi-component case} \\label{sec:multicomp}\n\nFor simplicity, we have restricted the focus of this article to the case where $K$ is a connected Legendrian knot. We close with a discussion of an appropriate modification of Theorem~\\ref{thm:main} for the case of Legendrian links with multiple components.\n\nWhen $K = \\sqcup_{i =1}^c K_i$ is a Legendrian link with $c$ components, for $\\vec{n} = (n_1, \\ldots, n_c)$ with $n_i\\geq 1$ one can consider the $\\vec{n}$-colored Kauffman polynomial defined by satelliting each $K_i$ with the symmetrizer from the $n_i$-stranded BMW algebra in a multi-linear manner. With the $\\vec{n}$-colored $1$-graded ruling polynomial defined by the analogous modification,\n\\[\nR^1_{\\vec{n},K}(q) = \\left(\\prod_{i=1}^c\\frac{1}{c_{n_i}} \\right)\\sum_{\\vec{\\beta} \\in S_{n_1} \\times \\cdots \\times S_{n_c}} \\left(\\prod_{i=1}^c q^{\\lambda(\\beta_i)\/2} \\right) \\widetilde{R}_{S(K,\\vec{\\beta})}(z)\\vert_{z = q^{1\/2}-q^{-1\/2}},\n\\]\nthe second equality of Theorem \\ref{thm:main}, modified to read $R^1_{\\vec{n},K}(z)=F_{\\vec{n},K}(a,q)|_{a^{-1}=0}$, follows by a~mild variation on the arguments of Sections~\\ref{sec:ncolored} and~\\ref{sec:Kauffman}.\n\nTo obtain an additional equality with a total representation number, one should work with the so-called {\\it composable algebra} version of the LCH DGA, $(\\mathcal{A}_{{\\rm comp}}, \\partial)$, cf.~\\cite{BEE, EENS}. The underlying algebra $\\mathcal{A}_{{\\rm comp}}$ has generators $b_1, \\ldots, b_r$ and $t_1^{\\pm1}, \\ldots, t_{\\ell}^{\\pm1}$ from Reeb chords and basepoints as well as idempotent generators $e_1, \\ldots, e_c$ corresponding to the components of $K$. Moreover, $\\mathcal{A}_{{\\rm comp}}$ has relations\n\\begin{alignat*}{3}\n& e_ie_j = \\delta_{i,j}, \\qquad & & \\displaystyle \\sum_{i=1}^c e_i = 1,& \\\\\n& e_i b_k = \\delta_{i,u(k)} b_k, & & b_ke_i = \\delta_{i, l(k)} b_k, & \\\\\n & e_i t_k = t_k e_i = \\delta_{i, s(k)} t_k, \\qquad & & t_kt_k^{-1} = t_k^{-1}t_k = e_{s(k)}, &\n\\end{alignat*}\nwhere the upper (resp.\\ lower) endpoint of the Reeb chord~$b_k$ is on the component $K_{u(k)}$ (resp.\\ $K_{l(k)}$) and the basepoint $t_k$ sits on the~$K_{s(k)}$ component. Note that an algebra representation $f\\colon \\mathcal{A}_{{\\rm comp}} \\rightarrow \\operatorname{End}(V)$ is equivalent to a~collection of vector spaces $V_1, \\ldots, V_c$ together with linear maps\n\\[\nf(b_k)\\colon \\ V_{l(k)} \\rightarrow V_{u(k)}\n\\]\nassigned to Reeb chords, and invertible linear maps\n\\[\nf(t_k)\\colon \\ V_{s(k)} \\rightarrow V_{s(k)}\n\\]\nassigned to base points. Here, the $V_i$ are determined from $V$ via $V_i = f(e_{i}) V$. (This is as in the correspondence between quiver representations and representations of the corresponding path algebra, see, e.g., \\cite[Section~1.2]{Brion}. Except for the relation $t_kt_k^{-1} = t_k^{-1}t_k = e_{s(k)}$, the composable algebra $\\mathcal{A}_{{\\rm comp}}$ is precisely the path algebra associated to the quiver with vertices indexed by components of $K$ and edges corresponding to Reeb chords and base points.) The DGA differential $\\partial$ on $\\mathcal{A}_{{\\rm comp}}$ is defined to satisfy $\\partial e_i = \\partial t_k= 0$ and by the usual holomorphic disk count on Reeb chords.\n\nFor $\\vec{n}= (n_1, \\ldots, n_c)$, with $n_i \\geq 1$ as above, we denote by $\\overline{\\operatorname{Rep}}_1\\big(K, \\big(\\mathbb{F}^{\\vec{n}}_q, 0\\big)\\big)$\nthe set of DGA representations $f\\colon (\\mathcal{A}_{\\rm comp}, \\partial) \\rightarrow \\big(\\operatorname{End}\\big(\\oplus_{i} \\mathbb{F}_q^{n_i}\\big), 0\\big)$ that,\nwhen viewed as quiver representations, assign the collection of vector spaces $\\mathbb{F}_q^{n_1}, \\ldots, \\mathbb{F}_q^{n_c}$ to the components of~$K$, i.e., $f(e_i)$ is the projection to the $\\mathbb{F}_{q}^{n_i}$ component. Note that to obtain a Legendrian isotopy invariant we need to adjust the normalizing factor from~\\cite{LeRu} used in Definition~\\ref{def:total}. This is done by defining the ($1$-graded) {\\it total $\\vec{n}$-dimensional representation number} of~$K$ to be\n\\begin{align*}\n\\operatorname{Rep}_1\\big(K, \\mathbb{F}^{\\vec{n}}_q\\big) & := \\left(\\prod_{i,j} \\big|\\operatorname{Hom}_{\\mathbb{F}_q}\\big(\\mathbb{F}^{n_j}_q, \\mathbb{F}^{n_i}_q\\big)\\big|^{-{\\rm rb}_{i,j}(K)\/2}\\right)\n\\\\\n & \\quad {}\\times \\left( \\prod_i \\big|{\\rm GL}\\big(n, \\mathbb{F}^{n_i}_q\\big)\\big|^{-\\ell_i} \\right)  \\big| \\overline{\\operatorname{Rep}}_1\\big(K, \\big(\\mathbb{F}^{\\vec{n}}_q, 0\\big)\\big)\\big| \\\\\n & = \\left(\\prod_{i,j} \\big(q^{n_in_j}\\big)^{-{\\rm rb}_{i,j}(K)\/2}\\right) \\\\\n & \\quad {} \\times \\left(\\prod_{i} \\left(q^{n_i(n_i-1)\/2}   \\prod_{m=1}^{n_i}\\big(q^m-1\\big)\\right)^{-\\ell_i} \\right)  \\big| \\overline{\\operatorname{Rep}}_1\\big(K, \\big(\\mathbb{F}^{\\vec{n}}_q, 0\\big)\\big) \\big|,\n\\end{align*}\n where ${\\rm rb}_{i,j}(K)$ is the number of Reeb chords, $b_k$, with $u(k) =i$ and $l(k) = j$ and $\\ell_i$ is the number of basepoints on the $K_i$ component of $K$.\n\nWith the composable algebra used for $K$, a suitable modification of Theorem~6.1 from~\\cite{LeRu} relating augmentations of the satellites $S\\big(K, \\vec{\\beta}\\big)$ with higher dimensional representations of $(\\mathcal{A}_{{\\rm comp}}(K), \\partial)$ continues to hold. [The key point being that when components $K_i$ and $K_j$ are satellited with~$n_i$ and~$n_j$ stranded braids respectively, each Reeb chord, $b_k$, with $u(k) = i$ and $l(k) =j$ corresponds to an $n_i \\times n_j$ matrix of Reeb chords in the satellite $S(K, \\vec{\\beta})$.] From this starting point, the analog of the first equality of Theorem~\\ref{thm:main}, $\\operatorname{Rep}_1\\big(K, \\mathbb{F}^{\\vec{n}}_q\\big)= R^1_{\\vec{n},K}(z)$, can be deduced as in Section~\\ref{sec:5}.\n\n\\subsection*{Acknowledgements}\nThis article is dedicated to Dmitry Fuchs to whom the second author is grateful for his generosity and support through years in grad school and beyond. Thank you Dmitry! DR acknowledges support from Simons Foundation grant~\\#429536.\n\n\\pdfbookmark[1]{References}{ref}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nThe Polonyi problem~\\cite{PLB131-59,PRD49-779} is one of the most serious\nproblems in models based on the $N=1$ supergravity~\\cite{NPB212-413}. In a\nwide class of supergravity models, the Polonyi field $\\phi$, which is\na scalar field related to the supersymmetry (SUSY) breaking,\nhas a mass $m_\\phi$ of the order of the gravitino mass. During\ninflation, $\\phi$ takes an amplitude of the order of the gravitational\nscale $M\\equiv M_{\\rm pl}\/\\sqrt{8\\pi}\\simeq 2.4\\times\n10^{18} {\\rm GeV} $. After the inflation, the condensation of $\\phi$ starts to\noscillate when the expansion rate of the universe, $H$, becomes\ncomparable to $m_\\phi$ and finally decays into particles in the\nobservable sector. Since the interactions of the Polonyi field are\nsuppressed by powers of $M^{-1}$, the decay rate of the Polonyi field,\n$\\Gamma_\\phi$, is very small as\n\\begin{equation}\n    \\Gamma_\\phi \\sim N\\frac{m_\\phi^3}{M_{\\rm pl}^2},\n    \\label{decay_rate}\n\\end{equation}\nwhere $N$ is the number of the decay modes. (In the following\ncalculations, we take $N=100$.)  Therefore, the Polonyi field is\nexpected to decay when the temperature of the universe becomes very\nlow. The reheating temperature $T_R$ due to the decay of the Polonyi\nfield is given by\n\\begin{equation}\n    \\label{rtemp}\n    T_R \\sim 1 {\\rm MeV} \n    \\left(\\frac{m_{\\phi}}{10{\\rm TeV}}\\right)^{3\/2}.\n\\end{equation}\nThis causes serious cosmological difficulties; namely the Polonyi\nfield may destroy the great success of the big-bang nucleosynthesis\n(BBN), and the entropy production due to its decay may dilute the\nbaryon number of the universe too much.\n\nIn the previous works~\\cite{PLB174-176,PLB342-105} it has been pointed\nout that the Polonyi problem can be solved if the gravitino mass\n$m_{3\/2}$ (which is the same order of the Polonyi mass) is larger than\n$O(10) {\\rm TeV} $ in order to hasten the decay of the Polonyi field. Thus,\nit is desirable to raise the gravitino mass while keeping the\neffective SUSY breaking scale in the observable sector at\n$O(100) {\\rm GeV} $. In no-scale type supergravity models, such mass\nhierarchy is realized~\\cite{PRD50-2356} and hence the no-scale type\nsupergravity model with $m_{3\/2} \\mathop{}_{\\textstyle \\sim}^{\\textstyle >}  O(10) {\\rm TeV} $ has been suggested as\nan attractive solution to the Polonyi problem.\n\nIn reference~\\cite{PLB342-105}, however, it has been also pointed out\nthat the mass density of the lightest superparticle (LSP) produced by\nthe decay of the Polonyi field may overclose our universe if LSP is\nstable. As we will see below, the mass density of LSP increases as the\nreheating temperature $T_R$ due to the decay of the Polonyi field\ndecreases.  Therefore, a lowerbound on $T_R$ is derived requiring that\nthe present mass density of LSP should not exceed the critical density\nof the universe $\\rho_c$. In this letter, we obtain the lowerbound on\n$T_R$ in the framework of the minimal SUSY SU(5) model with no-scale\ntype boundary conditions on the SUSY breaking parameters.\n\n\n\n\\section{The Model}\n\\label{sec:model}\n\nBefore starting cosmological arguments, let us first describe our\nbasic assumptions. We consider the minimal SUSY SU(5) model with\nno-scale type boundary conditions. This model has three types of Higgs\nfield; $H({\\bf 5})$ and $\\bar{H}({\\bf 5^*})$ which contain flavor\nHiggses $H_f$ and $\\bar{H}_f$, and $\\Sigma ({\\bf 24})$ whose\ncondensation breaks the SU(5) group into the gauge group of the\nminimal SUSY standard model (MSSM), $\\rm SU(3)_C\\times SU(2)_L\\times\nU(1)_Y$. For the Higgs sector, the superpotential is given by\n\\begin{equation}\n    W = \\frac{1}{3} \\lambda {\\rm tr} \\Sigma^3 \n    + \\frac{1}{2} M_\\Sigma {\\rm tr} \\Sigma^2\n    + \\kappa \\bar{H} \\Sigma H\n    + M_H \\bar{H} H,\n    \\label{W_su5}\n\\end{equation}\nwhere $\\lambda$ and $\\kappa$ are dimensionless constants, while\n$M_\\Sigma$ and $M_H$ are mass parameters which are of the order of the\ngrand unified theory (GUT) scale $M_{\\rm GUT} (\\sim\n10^{16} {\\rm GeV} )$. Furthermore, the model also has the soft SUSY breaking\nterms;\n\\begin{equation}\n    {\\cal L}_{\\rm soft} = \n      - \\frac{1}{3} \\lambda A_\\Sigma {\\rm tr} \\Sigma^3 \n      - \\frac{1}{2} M_\\Sigma B_\\Sigma {\\rm tr} \\Sigma^2\n      - \\kappa A_H \\bar{H} \\Sigma H\n      - M_H B_H \\bar{H} H +h.c.,\n    \\label{L_soft_su5}\n\\end{equation}\nwhere $A_\\Sigma$, $B_\\Sigma$, $A_H$ and $B_H$ are SUSY breaking\nparameters.  Minimising the Higgs potential, we find the following\nstationary point;\n\\begin{equation}\n    \\vev{\\Sigma} = \n    \\frac{1}{\\lambda}  \\left \\{  \n    M_\\Sigma + 2  \\left (  A_\\Sigma - B_\\Sigma  \\right ) \n    + O \\left (  \\frac{A_\\Sigma}{M_\\Sigma}, \\frac{B_\\Sigma}{M_\\Sigma}  \\right )  \n     \\right \\}  \\times {\\rm diag}(2,2,2,-3,-3),\n    \\label{vac_su5}\n\\end{equation}\nwhere the SU(5) is broken down to $\\rm SU(3)_C\\times SU(2)_L\\times\nU(1)_Y$. Regarding this stationary point as the vacuum, we obtain MSSM\nas the effective theory below the GUT scale $M_{\\rm GUT}$. Here, the\nmasslessness of the flavor Higgses $H_f$ and $\\bar{H}_f$ is achieved\nby a fine tuning among several parameters; $M_H - 3 \\kappa\nM_\\Sigma\/\\lambda \\simeq \\mu_H$, where $\\mu_H$ is the SUSY-invariant\nHiggs mass in MSSM.\n\nIn the present model, the parameters in MSSM at the electroweak scale\nis obtained by solving renormalization group equations (RGEs). Our\nmethod is as follows. The boundary conditions on the parameters in the\nminimal SUSY SU(5) model are given at the gravitational scale $M$.\nSince we assume the no-scale type supergravity models, all the SUSY\nbreaking parameters except for the gaugino mass vanish at the\ngravitational scale. From the gravitational scale to the GUT scale,\nthe parameters follow the renormalization group flow derived from RGEs\nin the minimal SUSY SU(5) model. Then we determine the parameters in\nMSSM at the GUT scale through an appropriate matching condition\nbetween the parameters in the SUSY SU(5) model and those in MSSM.\nFinally, we use RGEs in MSSM from the GUT scale to the electroweak\nscale in order to obtain the low energy parameters.\n\nAs for the matching condition, we have a comment. In the stationary\npoint (\\ref{vac_su5}), the mixing soft mass term of the two flavor\nHiggs bosons, $m_{12}^2\\bar{H}_fH_f$, is generated at the tree level,\nwhere $m_{12}^2$ is given by\n\\begin{equation}\n    m_{12}^2 (M_{\\rm GUT}) \\simeq \n       \\left [  \\frac{6\\kappa}{\\lambda} \n      (A_\\Sigma - B_\\Sigma)(A_H - B_\\Sigma) \n      -\\mu_H B_H.\n       \\right ] _{\\mu =M_{\\rm GUT}}\n      \\label{m12^2}\n\\end{equation}\nSince the mixing mass term depends on unknown parameters, $\\lambda$\nand $\\kappa$ in eq.(\\ref{W_su5}), we regard $m_{12}^2$ as a free\nparameter taking account of the uncertainty of $\\lambda$ and $\\kappa$\nin our analysis.  Then, the low energy parameters are essentially\ndetermined by the gauge and Yukawa coupling constants and the\nfollowing three parameters; the supersymmetric Higgs mass $\\mu_H$, the\nmixing mass of the two flavor Higgs bosons $m_{12}^2$, and the unified\ngaugino mass.\\footnote\n{In fact, parameters in MSSM slightly depend on the parameters in the\nSUSY GUT such as $\\lambda$, $\\kappa$ and so on. In our numerical\ncalculation, we ignore the effects of these parameters on the \nrenormalization group flow.}\nHowever, it is more convenient to express these parameters by other\nphysical ones. In fact, one combination of them is constrained so that\nthe flavor Higgs bosons have correct vacuum expectation values;\n$\\langle H_f\\rangle^2+\\langle\\bar{H}_f\\rangle^2\\simeq (174 {\\rm GeV} )^2$. As\nthe other two physical parameters, we use the mass of LSP, $m_{\\rm\nLSP}$, and the vacuum angle\n$\\tan\\beta\\equiv\\langle{H_f}\\rangle\/\\langle{\\bar{H}_f}\\rangle$. Thus,\nonce we fix $m_{\\rm LSP}$ and $\\tan\\beta$, we can determine all the\nparameters in MSSM.\\footnote\n{Yukawa coupling constants are determined so that the fermions have\ncorrect masses. The gauge coupling constants are also fixed so that\ntheir correct values at the electroweak scale are reproduced.}\n\nFollowing the above procedure, we solve the RGEs numerically, and\nobtain the low energy parameters in MSSM. Then, we determine the mass\nspectrum and the mixing angles for all superparticles. One remarkable\nthing is that {\\it LSP almost consists of bino $\\tilde{B}$ which is\nthe superpartner of the gauge field for $U(1)_Y$} if we require that\nLSP is neutral. Therefore, in our model, the LSP mass $m_{\\rm LSP}$ is\nessentially equivalent to the bino mass. This fact simplifies the\nfollowing analysis very much.\n\n\\section{Density of LSP}\n\\label{sec:density}\n\nNow we are in a position to discuss the mass density of LSP produced\nby the decay of the Polonyi field. The decay of the Polonyi field\nproduces a large number of superparticles, which promptly decay into\nLSPs. The number density of LSP produced by the decay, $n_{{\\rm\nLSP},i}$, is of the same order of that of the Polonyi field\n$n_\\phi\\equiv\\rho_\\phi \/m_\\phi$ (with $\\rho_\\phi$ being the energy\ndensity of the Polonyi field). Just after the decay of the Polonyi\nfield, the yield variable for LSP, $Y_{\\rm LSP}$, which is defined by\nthe ratio of the number density of LSP to the entropy density $s$, is\ngiven by\n\\begin{eqnarray}\n    m_{\\rm LSP} Y_{\\rm LSP} & \\simeq & \\frac{\\rho_\\phi}{s}\n    \\simeq \\frac{m_{\\rm LSP}\\rho_{{\\rm LSP},i}}{m_\\phi s}\n    \\sim \\left(\\frac{m_{\\rm LSP}T_R}{m_\\phi}\\right)\n    \\nonumber \\\\\n    & \\sim & 10^{-5} {\\rm GeV}  \\left(\\frac{m_{\\rm LSP}}{100 {\\rm GeV} }\\right)\n    \\left(\\frac{T_R}{1 {\\rm MeV} }\\right)\\left(\\frac{10 {\\rm TeV} }{m_\\phi}\\right),\n    \\label{mY_LSP}\n\\end{eqnarray}\nwhere $\\rho_{{\\rm LSP},i}$ is the mass density of LSP just after the\ndecay of the Polonyi field. If LSP is stable and the pair annihilation\nof LSP is not effective, $Y_{\\rm LSP}$ is conserved until today.\nComparing the ratio given in eq.(\\ref{mY_LSP}) with the ratio of the\ncritical density $\\rho_c$ to the present entropy density $s_{0}$,\n\\begin{equation}\n    \\frac{\\rho_c}{s_{0}} \\simeq 3.6 \\times 10^{-9}h^2~ {\\rm GeV} ,\n    \\label{critical}\n\\end{equation}\nwhere $h$ is the Hubble constant in units of 100km\/sec\/Mpc, we see\nthat LSP overcloses the universe in the wide parameter region for\n$m_{\\rm LSP}, m_{\\phi}$ and $T_R$ which we are concerned with.\n\nIf the pair annihilation of LSP takes place effectively, its abundance\nis reduced to \n\\begin{equation}\n    \\frac{n_{\\rm LSP}}{s} \\simeq \n    \\left. \\frac{H}{s\\langle\\sigma_{\\rm ann}v_{\\rm rel}\\rangle} \n       \\right | _{T=T_R},\n      \\label{abundance_LSP}\n\\end{equation}\nwhere $\\sigma_{\\rm ann}$ is the annihilation cross section, $v_{\\rm\nrel}$ is the relative velocity, and $\\langle\\cdots\\rangle$ represents\nthe average over the phase space distribution of LSP. From\neq.(\\ref{abundance_LSP}), we obtain a lowerbound on the annihilation\ncross section,\n\\begin{equation}\n    \\langle\\sigma_{\\rm ann}v_{\\rm rel}\\rangle  \\mathop{}_{\\textstyle \\sim}^{\\textstyle >} \n    3\\times 10^{-8}h^{-2} {\\rm GeV} ^{-2}\n    \\left( \\frac{m_{\\rm LSP}}{100 {\\rm GeV} }\\right)\n    \\left( \\frac{100 {\\rm MeV} }{T_R}\\right),\n    \\label{sv_limit}\n\\end{equation}\nin order that the mass density of LSP does not overclose the universe. \n\nComparing this bound with the annihilation cross section of LSP, we\nderive a bound on the reheating temperature by the decay of the\nPolonyi field. Since LSP is most dominated by bino, it annihilates\ninto fermion pairs. The annihilation cross section is given\nby~\\cite{PLB230-78}\n\\begin{equation}\n    \\langle\\sigma_{\\rm ann}v_{\\rm rel}\\rangle \n    = a + b\\langle v^2\\rangle,\n    \\label{sigma*v}\n\\end{equation}\nwhere $\\langle v^2\\rangle$ is the average velocity of LSP,\nand\n\\begin{eqnarray}\n    a  & \\simeq &\n    \\frac{32\\pi\\alpha_1^2}{27} \n    \\frac{m_t^2}{(m_{\\tilde{t}_R}^2 + m_{\\rm LSP}^2 - m_t^2)^2}\n     \\left (  1 - \\frac{m_t^2}{m_{\\rm LSP}^2}  \\right ) ^{1\/2} \n    \\theta (m_{\\rm LSP}-m_t),\n    \\label{s-wave} \\\\\n    b &\\simeq& \\frac{8\\pi\\alpha_1^2}{3} \\sum_{m_f\\leq T} Y_f^4  \\left \\{ \n    \\frac{m_{\\rm LSP}^2}{(m_{\\rm LSP}^2+m_{\\tilde{f}}^2)^2}\n    - \\frac{2m_{\\rm LSP}^4}{(m_{\\rm LSP}^2+m_{\\tilde{f}}^2)^3}\n    + \\frac{2m_{\\rm LSP}^6}{(m_{\\rm LSP}^2+m_{\\tilde{f}}^2)^4}  \\right \\} .\n    \\label{p-wave}\n\\end{eqnarray}\nHere, $\\alpha_1^2\\equiv g_1^2\/4\\pi\\simeq 0.01$ represents the fine\nstructure constant for U(1)$_{\\rm Y}$, $m_t$ the top-quark mass, $Y_f$\nthe hypercharge of the fermion $f$, and $m_{\\tilde{f}}$ the mass of\nthe sfermion $\\tilde{f}$.  Notice that $a$ and $b$ terms correspond to\n$s$- and $p$-wave contributions, respectively. Taking\n$m_{\\tilde{f}}\\sim m_{\\rm LSP}\\sim 100 {\\rm GeV} $, the annihilation cross\nsection given in eq.(\\ref{sigma*v}) is at most $3\\times\n10^{-8} {\\rm GeV} ^{-2}$. Using this result in the inequality\n(\\ref{sv_limit}), we can see that the reheating temperature must be\nhigher than about 100MeV even if $\\langle v^2\\rangle\\sim 1$. If the\naverage velocity is smaller than 1, the constraint becomes more\nstringent, as we will see below.\n\n\n\n\\section{Thermalization of LSP}\n\\label{sec:thermalization}\n\n\nIn order to obtain the precise lowerbound on the reheating temperature\n$T_R$, we have to know $\\langle v^2\\rangle$, as well as the mass\nspectrum of the superparticles on which the annihilation cross section\ndepends. First, let us discuss the averaged velocity of LSP, $\\langle\nv^2\\rangle$. Since LSP is mostly the bino, it loses its energy by\nscattering off the background fermions. In the model with the no-scale\ntype boundary conditions, right-handed sleptons become the lightest\namong the sfermions, and hence LSP loses its energy mainly by\nscattering off the background electron (and $\\mu$ and $\\tau$, if the\ntemperature is higher than their masses). If LSP is relativistic, the\ncross section for this process, $\\sigma_{\\rm scatt}$, is estimated as\n\\begin{equation}\n    \\langle \\sigma_{\\rm scatt} v_{\\rm rel} \\rangle \n    \\simeq 128\\pi \\alpha_1^2\n    \\frac{E_{\\rm LSP}^2 T_R^2}{m_{\\tilde{e}_R}^4 m_{\\rm LSP}^2},\n    \\label{sigma_scatt}\n\\end{equation}\nwhere $E_{\\rm LSP}$ is the energy of LSP, and $m_{\\tilde{e}_R}$ the\nmass of the right-handed selectron.\\footnote%\n{This cross section is applied for $E_{\\rm LSP} T_{R} \\ll\nm_{\\tilde{e}_R}^2$.}\nThe energy loss rate for\nthe relativistic LSP, $\\Gamma_{\\rm scatt}^{\\rm R}$, is given by\n\\begin{equation}\n    \\label{loss-rate}\n    \\Gamma_{\\rm scatt}^{\\rm R} \\simeq \n     n_e \\langle \\sigma_{\\rm scatt} v_{\\rm rel} \\rangle \n     \\frac{\\Delta E_{\\rm LSP}}{E_{\\rm LSP}},\n\\end{equation}\nwhere $n_e$ represents the number density of the background electron\nand $\\Delta E_{\\rm LSP}$ is the averaged  energy loss of LSP in \none scattering which is  given by\n\\begin{equation}\n    \\label{energy-loss}\n    \\Delta E_{\\rm LSP} \\simeq  \n    -12E_{\\rm LSP}\\left(\\frac{T_R E_{\\rm LSP}}{m_{\\rm LSP}^2}\\right).\n\\end{equation}\nTaking the ratio of the energy loss  rate $\\Gamma_{\\rm\nscatt}^{\\rm R}$ to the expansion rate $H$ of the universe, we find\n\\begin{equation}\n    \\left. \\frac{\\Gamma_{\\rm scatt}^{\\rm R}}{H}\n       \\right | _{E_{\\rm LSP}\\gg m_{\\rm LSP}}\n      \\simeq \n      2\\times10^3 \\left( \\frac{E_{\\rm LSP}}{10^2 {\\rm GeV} } \\right)^3\n      \\left( \\frac{T_R}{100 {\\rm MeV} } \\right)^4\n      \\left( \\frac{100 {\\rm GeV} }{m_{\\tilde{e}_R}} \\right)^4\n      \\left( \\frac{100 {\\rm GeV} }{m_{\\rm LSP}} \\right)^4.\n\\end{equation}\nThus, if $T_R  \\mathop{}_{\\textstyle \\sim}^{\\textstyle >} $ a few $\\times$ 10MeV, the energetic LSP loses its\nenergy through the scattering off thermal electrons efficiently for\n$m_{\\tilde{e}_R}\\sim m_{\\rm LSP}\\simeq O(100\/GEV)$, and becomes a\nnon-relativistic particle.\n\n\nThe non-relativistic LSP further loses its energy by scattering off\nbackground electrons. The averaged loss of the kinetic energy for the\nnon-relativistic LSP in one scattering process, $\\Delta \\epsilon_{\\rm\nLSP}$, is given\nby\\footnote{%\nNaively, it is expected that $\\Delta \\epsilon_{\\rm LSP}$ is $\\sim\nT_R$. However this order of the energy loss is cancelled out when the\naverage is taken over angles of the incident particles and  the actual\nenergy loss is much smaller than the naive expectation.}\n\\begin{equation}\n    \\Delta \\epsilon_{\\rm LSP} \\simeq\n    -\\frac{20\\epsilon_{\\rm LSP} T_R}{m_{\\rm LSP}}\n    \\left( 1 -\\frac{T_R}{\\epsilon_{\\rm LSP}}\\right),\n    \\label{Delta_E}\n\\end{equation}\nwhere $\\epsilon_{\\rm LSP}\\equiv E_{\\rm LSP}-m_{\\rm LSP}$ is the\nkinetic energy of LSP. As one can see in eq.(\\ref{Delta_E}), the LSP\nwhich has a kinetic energy larger than $\\sim T_R$ tends to lose its\nenergy through the scattering process, while LSP receives energy from\nthe thermal bath if its energy is smaller than $\\sim T_R$. Thus, if\nthe scattering processes take place effectively, the averaged kinetic\nenergy of LSP becomes $\\sim T_R$, {\\it i.e.} LSP goes into the\nkinetic equilibrium.\n\nThe energy loss rate $\\Gamma_{\\rm scatt}^{\\rm NR}$ for the \nnon-relativistic LSP is given by\n\\begin{equation}\n    \\Gamma_{\\rm scatt}^{\\rm NR} \\simeq \n    n_e \\langle\\sigma_{\\rm scatt} v_{\\rm rel}\\rangle \\times\n    \\frac{20T_R}{m_{\\rm LSP}}\n    \\simeq \n    \\frac{5760\\alpha_1^2}{\\pi}\n    \\frac{T_R^6}{m_{\\rm LSP}m_{\\tilde{e}_R}^4}.\n    \\label{gamma_scatt}\n\\end{equation}\nThe LSP goes into the kinetic equilibrium if the scattering rate\n$\\Gamma_{\\rm scatt}^{\\rm NR}$ is larger than the expansion rate of the\nuniverse. Taking $m_{\\tilde{e}_R}\\sim m_{\\rm LSP}$, the ratio of\n$\\Gamma_{\\rm scatt}^{\\rm NR}$ to the expansion rate of the universe,\n$H$, is given by\n\\begin{equation}\n    \\left. \\frac{\\Gamma_{\\rm scatt}^{\\rm NR}}{H}  \\right | _{E_{\\rm LSP}\n      \\sim m_{\\rm LSP}}\n      \\simeq 4 \\times 10^{3} ~\n       \\left (  \\frac{m_{\\rm LSP}}{100 {\\rm GeV} }  \\right ) ^{-5}\n       \\left (  \\frac{T_R}{100 {\\rm MeV} }  \\right ) ^{4}.\n\\end{equation}\nThus, if the reheating temperature is higher than about 10MeV,\nproduced LSPs go into kinetic equilibrium as far as $m_{\\rm LSP} \\sim\nO(100)$GeV.  Furthermore, as we discussed in the previous section,\nthe reheating temperature should be higher than at least 100MeV in\norder to decrease the mass density of LSP sufficiently. Thus, we\nconclude that the produced LSPs go into kinetic equilibrium if we\nrequire that the mass density of the relic LSP should not overclose\nthe universe.\\footnote\n{By the numerical calculation we have checked, in fact, that the\nscattering rate given in eq.(\\ref{gamma_scatt}) is always larger than\nthe expansion rate of the universe when the relic LSP does not\noverclose the universe. (See fig.2.)}\nIn this case, the averaged velocity is given by\n\\begin{equation}\n    \\langle v^2\\rangle \\simeq \\frac{3T_R}{m_{\\rm LSP}}.\n\\end{equation}\n>From this we easily see that the LSP abundance given in\neq.(\\ref{abundance_LSP}) decreases as the reheating temperature gets\nhigher. Thus, we obtain the lowerbound on the reheating temperature.\n\n\n\\section{Results}\n\\label{sec:results}\n\nOnce we know the averaged velocity $\\langle v^2\\rangle$, we can\ncalculate the annihilation cross section of LSP, and get the\nlowerbound on the reheating temperature after the decay of the Polonyi\nfield. In this letter, we first solve RGEs based on the minimal SU(5)\nmodel with the no-scale boundary conditions, and determine the mass\nspectrum of the superparticles. We only investigate the parameter\nspace which is not excluded by the experimental or theoretical\nconstraints. The constraints which we use are as follows:\n\\begin{itemize}\n\\item Higgs bosons $H_f$ and $\\bar{H}_f$ have correct vacuum\nexpectation values.\n\\item Perturbative picture is valid below the gravitational scale.\n\\item LSP is neutral.\n\\item Sfermions (especially, charged sleptons) have masses larger than\nthe experimental lower limits~\\cite{PDG}.\n\\item The branching ratio for $Z$-boson decaying into neutralinos is \nnot too large~\\cite{PLB350-109}.\n\\end{itemize}\nThen, with the obtained mass spectrum of superparticles, we calculate\nthe annihilation cross section and determine the lowerbound on the\nreheating temperature from the following equation;\n\\begin{equation}\n    \\left. \\frac{H}{s\\langle\\sigma_{\\rm ann}v_{\\rm rel}\\rangle} \n       \\right | _{T=T_R} \\leq\n      \\frac{\\rho_c}{s_0} \\simeq 3.6h^2 \\times 10^{-9} {\\rm GeV} .\n\\end{equation}\n\nIn fig.~1, we show the lowerbound on the reheating temperature in the\n$\\tan\\beta$ vs. $m_{\\rm LSP}$ plane. In the figures, large or small\n$\\tan\\beta$'s are not allowed since the Yukawa coupling constant for\nthe top quark or bottom quark blows up below the gravitational scale\nfor such $\\tan\\beta$'s. Furthermore, there also exists a lowerbound on\nthe LSP mass. In the case where $\\tan\\beta \\mathop{}_{\\textstyle \\sim}^{\\textstyle <}  20$, charged sfermions\nbecome lighter than the experimental limit if the LSP mass becomes\nlighter than $\\sim 50 {\\rm GeV} $.  On the other hand, for the large\n$\\tan\\beta$ case, unless the bino mass is sufficiently large, the\nlightest charged slepton becomes LSP. (Remember that the dominant\ncomponent of LSP is bino.)  Thus, the lowerbound on $m_{\\rm LSP}$ is\nobtained. As we can see, the reheating temperature should be larger\nthan about 100MeV, even for the case where $m_{\\rm LSP}\\sim 50 {\\rm GeV} $.\nThe constraint becomes more stringent as $m_{\\rm LSP}$ increases,\nsince the masses of the superparticles which mediate the annihilation\nof LSP becomes larger as the LSP mass increases.\n\nIf we translate the lowerbound on the reheating temperature into that\nof the Polonyi mass $m_\\phi$, we obtain $m_\\phi \\mathop{}_{\\textstyle \\sim}^{\\textstyle >}  100 {\\rm TeV} $ (see\neq.(\\ref{rtemp})). We can also see that the lowerbound is almost\nindependent of $\\tan\\beta$. In fig.~2, We show the lowerbound on $T_R$\nas a function of the LSP mass for $\\tan\\beta =10$, and $\\mu_H >0$.\n\nHere, we should comment on the accidental case where the annihilation\nprocess hits the Higgs pole in the $s$-channel. If the LSP mass is\njust half of the lightest Higgs boson mass, the LSP annihilation cross\nsection is enhanced since LSP has small but nonvanishing fraction of\nhiggsino component. If the parameters are well tuned, such a situation\ncan be realized and the lowerbound of $T_R$ decreases to $O(10) {\\rm MeV} $.\nHowever, we consider that such a scenario are very unnatural since a\nprecise adjustment of the\nparameters is required in order to hit the Higgs pole.\\footnote\n{In the case where the annihilation process hits the pole of heavier\nHiggs bosons, the cross section is not enhanced so much, since the\nwidths of the heavier Higgs bosons are quite large.}\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\n\nIn this letter, we have obtained the lowerbound on the reheating\ntemperature due to the decay of the Polonyi field in a framework of\nthe no-scale type supergravity model. As a result, we have seen that\nthe Polonyi mass should be larger than about 100TeV which may raise a\nnew fine-tuning problem~\\cite{PLB173-303}.\n\nWe have assumed the minimal SUSY GUT model in the present\nanalysis. However, the main conclusion is not changed as far as LSP\nis mostly the bino, because the minimum value of the lowerbound ($T_R\n\\simeq 100$MeV) is obtained when the mass of the selectron takes the\nexperimentally allowed lower limit.\n\nWe have assumed that LSP is stable so far. However, if we introduce\n$R$-parity violation, LSP becomes unstable and the allowed $T_R$ is as\nlow as a few MeV. It is also the case if we assume a very light LSP\nsuch as a neutral higgsino~\\cite{PLB131-59} or an\naxino~\\cite{NPB358-447} whose masses are less than about 100MeV.\n\n\\section*{Acknowledgement}\n\nTwo of the authors (T.M. and T.Y.) would like to thank M.~Yamaguchi for\nuseful discussions in the early stage of this work.\n\n\\newpage\n\\newcommand{\\Journal}[4]{{\\sl #1} {\\bf #2} {(#3)} {#4}}\n\\newcommand{Ap. J.}{Ap. J.}\n\\newcommand{Can. J. Phys.}{Can. J. Phys.}\n\\newcommand{Nuovo Cimento}{Nuovo Cimento}\n\\newcommand{Nucl. Phys.}{Nucl. Phys.}\n\\newcommand{Phys. Lett.}{Phys. Lett.}\n\\newcommand{Phys. Rev.}{Phys. Rev.}\n\\newcommand{Phys. Rep.}{Phys. Rep.}\n\\newcommand{Phys. Rev. Lett.}{Phys. Rev. Lett.}\n\\newcommand{Prog. Theor. Phys.}{Prog. Theor. Phys.}\n\\newcommand{Sov. J. Nucl. Phys.}{Sov. J. Nucl. Phys.}\n\\newcommand{Z. Phys.}{Z. Phys.}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction\\label{ch5:int}}\n\nThis work is aimed at the question of what provides the\ndominant power source of ultraluminous infrared galaxies, that\nis galaxies with luminosities in the wavelength range\n8--1000 \\hbox{\\micron~} in excess of $10^{12}$ \\hbox{L$_\\odot$}~($H_0$=75 km\ns$^{-1}$ Mpc$^{-1}$).  This question is of interest because\nit is possible that the power source of these galaxies is\nqualitatively different from the dominant power source of most\ninfrared galaxies of lower luminosity.  The observational\napproach of this work is the use of moderate resolution\n8--13 \\hbox{\\micron~} spectroscopy.  \n\nIn very broad strokes, the context of the debate over the\nquestion addressed here can be summarized as \nan attempt, on the one hand, to explain ultraluminous infrared\ngalaxies as more powerful versions of starburst galaxies,\nthat is as super starbursts (Joseph \\&\nWright~1985), where massive star\nformation provides the power source, and on the other hand,\nto demonstrate that ultraluminous infrared galaxies are\nthe precursors of quasars (Sanders et al. 1988) and are\npowered by active galactic nuclei (AGNs).\n\nThe question has remained open to serious debate for\nessentially one reason: dust.  Infrared galaxies by their\nnature radiate the bulk of their energy through the \ndegradation of the optical and UV photons emitted by the\nprimary power source into infrared photons emitted by dust\nheated to a temperature of $\\sim$50 K.  This has the\neffect of erasing most of the commonly accepted\nfeatures used to distinguish the characteristics of\nstarbursts and AGNs.\n\nOptical and, to a lesser degree, near-infrared studies are\nparticularly susceptible to this difficulty since spectroscopic signatures of AGNs in ultraluminous\ninfrared galaxies should only betray sources that are\nrelatively unobscured and therefore might not contribute\nsubstantially to the heating of the dust that is the final source\nof most of the observed emission.  However, Veilleux et\nal. (1995) have shown in a large sample of infrared galaxies\nthat the fraction of infrared galaxies with AGN-like optical\nspectra increases as a function of increasing luminosity, providing\nimportant circumstantial evidence that AGN-like activity is\nassociated with ultraluminous infrared galaxies.  The strength\nof this evidence might be considered to be weakened, however, by\nthe finding of Goldader et al. (1995) that the ratio of\nBr$\\gamma$ to infrared luminosity decreases as a function of\nluminosity, suggesting that dust obscuration plays an even\ngreater role in ultraluminous infrared galaxies than in\nluminous infrared galaxies.  This suggests the perverse\npossibility that energetically dominant starbursts in\nultraluminous infrared galaxies are so obscured in the optical\nthat only incidental unobscured AGNs are detected.\n\nRadio observations are unaffected by dust obscuration and can\nprovide high spatial resolution.  However, the primarily\nsynchrotron radiation that is detected is only indirectly linked\nto the power source of infrared galaxies.  In starbursts, it is\nthought that the radio emission arises from electrons\naccelerated in supernovae remnants, while in AGN, relativistic\njets are strong radio emitters. Condon et al. (1991; hereafter\nCHYT) have employed the high spatial resolution provided by the\nVery Large Array (VLA) to show that the radio sizes of a large\nfraction of ultraluminous infrared galaxies are comparable to\nthe minimum size required for a blackbody that reproduces the\n25, 60, and 100 \\hbox{\\micron~} flux densities of these sources.  From this\nthey infer that starbursts rather than AGNs may power these\nsources.  At even higher spatial resolution provided by very\nlong baseline interferometry (VLBI), Lonsdale, Smith \\& Lonsdale\n(1993) find evidence for radio structure on the scale of\nmilliarcseconds, some components of which have brightness\ntemperatures well in excess of what could be expected in\nstarburst-related emission.  They deduce from this that AGNs are\npresent in five of eight of the ultraluminous infrared galaxies\nin their sample\\footnote{Smith, Lonsdale \\& Lonsdale (1998)\nsuggest that radio supernova remnants may be responsible for\nsome of the structure they detect}.\n\nIn this work, the dust emission itself is examined for\nspectral clues that can reveal the nature of the power\nsource in infrared galaxies.  In a sample of 60 galaxies,\nRoche et al. (1991; hereafter RASW) have shown that their\n8--13 \\hbox{\\micron~} spectra can be classified into three types:\nthose with prominent polycyclic aromatic hydrocarbon (PAH)\nfeatures generally show \\ion{H}{2} region-like optical\nspectra, those with flat featureless spectra have Seyfert\n1 or quasar-like optical spectra, and those rare galaxies\nwith silicate absorption features usually have Seyfert 2 optical\nspectra.  Dudley \\& Wynn-Williams (1997) have extended\nthe work on galaxies with very deep silicate features to\nshow that the size of the power source can be constrained\nto be smaller than a few parsecs, implying that such sources\nare powered by deeply embedded AGNs.  Here, the 8--13 \\hbox{\\micron~}\nspectra of a well-defined sample of 25 luminous and\nultraluminous infrared galaxies are examined in this\ncontext. \n\nIt should be noted that all of the sources for which new\nspectral observations are reported here can be found in one or\nmore of four {\\em IRAS}-based catalogs: the Bright Galaxy Survey\n(Soifer et al. 1989), the Extended Bright Galaxy Survey\n(Sanders et al. 1995), the Two Jansky Survey (Strauss et\nal. 1992), or the Extended 12 \\hbox{\\micron~} Survey (Rush, Makan \\&\nSpinogio 1993).\n\nObservations and data reduction procedures are described in\nSection 2.  Section 3 presents the new spectral observations.  The\nclassification of the spectra is discussed in\nSection 4.1.  The justification for the classification of the ultraluminous galaxies that fall within the later subsample\nis discussed in some detail in Section 4.2.  In Section 4.3 a subsample\nof galaxies is defined and an apparent trend with luminosity\nwithin it is examined.  The possibility of an\nanti-correlation between the equivalent width of the 11.3\n\\hbox{\\micron~} PAH feature and the ratio of infrared light to molecular\ngas mass is discussed in Section 4.4, and conclusions are\ndrawn in Section 5.\n\n\n\\section{Observations and Data Reduction\\protect\\label{ch5:odr}}\n\n\n\nThe new observations reported here were obtained over the\ncourse of 5 years, from 1991 through 1995, using the 10 and\n20 \\hbox{\\micron~} Cooled Grating Spectrograph (CGS3) mounted at the\nCassegrain focus of the United Kingdom Infrared Telescope\n(UKIRT) on Mauna Kea in Hawaii.  Observations in 1991 were\nobtained by G. Wynn-Williams and J. Goldader, in 1992 by\nG. Wynn-Williams and C. Dudley, and in 1993--1995 by\nC. Dudley. For parts of each observing run a UKIRT\nscientist (J. Davies, T. Geballe, or G. Wright) was present\nat the summit and participated in the observations.  CGS3 has\nbeen described by Cohen \\& Davies (1995).  All but one of\nthe spectra reported here were obtained using the low-resolution spectroscopy mode with $\\lambda\/\\Delta\\lambda\n\\sim 60$ in a series of beam-switched, chopped observations\nwith a chop frequency of $\\sim$10 Hz and a beam switch\nfrequency of $\\sim$0.5 Hz.  To fully sample each\nspectrum, two interlacing grating positions were observed\nwith changes between the two positions occurring after\n10--16 beam-switched pairs.  Table 1 gives a log of\nobservations for the galaxies.  In Table 1, column 1 is\nthe object name, columns 2 and 3 give the aperture centre\nin RA and DEC, column 4 gives the position reference,\ncolumns 5 and 6 give the object redshift and reference,\ncolumn 7 gives the date of observation, column 8 gives the\ncircular aperture size used in arcseconds (full width at\n10 per cent power), column 9 gives the chop throw in arcseconds,\nand column 10 gives the chop direction.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nObservations were calibrated in wavelength by means of\nhigh order spectral measurements of a Kr lamp in the\nmanner discussed by Hanner, Brooke \\& Tokunaga (1995).\nA master calibration spectrum was first selected and\ncalibrated in wavelength by flattening a 2$\\times$\nover-sampled Kr lamp spectrum (4 grating positions) with\nthe spectrum of a soldering iron replacing the Kr lamp in\nthe beam.  Kr lamp spectra from each night's observations\nwere then cross correlated against the unflattened master\nspectrum to give a wavelength calibration for each night.\nTypical shifts from the nominal CGS3 wavelength\ncalibration were less than a resolution element, and were\nnever more than 2 resolution elements.\n\nCorrections for instrumental response, atmospheric\ntransmission, and flux calibration were performed by dividing\nsource spectra by the spectrum of a standard star with a\nknown 10 \\hbox{\\micron~} flux density and an assumed blackbody\ntemperature based on its spectral type, and then multiplying\nby the Planck curve of that temperature scaled to match the\nstandard star in flux density.  Observations of standard\nstars were performed with the same aperture, chop throw,\nposition angle and frequency as the galaxy observations they\ncalibrate. The standard star observations were made to match\nin airmass so that no additional airmass corrections would be\nrequired.  For the standard stars, the number of beam-switched pairs between grating shifts was typically 3--5. The\nstars have been chosen, where possible, to have a spectral\ntype of K0 or earlier to avoid the effects of SiO absorption,\nH$^-$ opacity, and circumstellar dust emission that can\neffect the spectral shape of late-type stars.  Table 2 gives\nthe HR number, 10.1 \\hbox{\\micron~} flux density, photometric reference,\nand assumed blackbody temperature of each standard star, in\ncolumns 4, 5, 6, and 7, respectively, for each of the\nobservations given in Table 1.  In column 2 of Table 2,\nalternative names are given for galaxies that are found in\nmultiple catalogs, while column 3 gives the date of each\nobservation as in Table 1.  It should be noted that an error of $\\sim$1000 K in the assumed blackbody temperature of the standard\nstars leads to a $\\sim$2 per cent error in flux calibration at\neither end of the passband 8--13 \\hbox{\\micron~} which is negligible for\nthe purposes of this work, while for those stars of late\nspectral type, deviations from the assumed blackbody can be\n$\\sim$10 per cent in the SiO band (Cohen et al. 1995).  Systematic\nuncertainties in overall flux calibration of 15--20 per cent can\narise due to telescope pointing errors with the stronger\neffects for the smaller apertures.  For wavelengths shortward\nof 8.2 \\hbox{$\\mu$m}, strong systematic effects can occur as a\nresult of a variable water vapor column in the Earth's\natmosphere.\n\n\n\n\n\nAll spectra were regridded to a common wavelength scale using a\nGaussian filter of $\\sigma$ = 0.064 \\hbox{\\micron~}, which lead to a slight\ncorrelation between neighboring data points.  Those spectra of the\nsame source obtained on two or more nights were combined,\nweighted by integration time.  Data not falling in the ranges\n7.7--9.1 and 10.0--13.4 \\hbox{\\micron~} have been excluded due to poor\natmospheric transmission.\n\n\nSpectra of Wynn-Williams et al.'s (1991) Arp 299B1 were obtained on 1994 February 6 and 7\nusing CGS3 in high-resolution mode ($\\lambda\/\\Delta\\lambda\n\\sim 250$) using a 3\\farcs26 full width at 10 per cent power\ncircular aperture.  The spectra obtained on each night were\nthe result of two sets of observations each consisting of two\ninterlacing grating positions but with different wavelength\nranges to cover the spectral range 10--13.2 \\hbox{$\\mu$m}.  The chop\nthrow was 20\\arcsec, and the chop and beam switch PA were\n45$^\\circ$.  About 30 per cent of the first night's observations were\naffected by clouds; these data were rejected by inspecting the\nchange in the dispersion of the total signal.  The Kr lamp was\nobserved on each night to provide wavelength calibration.  The\nstrongest lines from each night agreed in wavelength to 3 per cent of\na resolution element so that the Kr spectra of the two nights\nwere combined, and a difference in wavelength of 30 per cent of a\nresolution element was measured between the expected (based on\nthe nominal calibration of CGS3) and the measured wavelengths\nof 5 Kr lines.  An additive correction of 0.02 \\hbox{\\micron~} was\napplied to the observations, this being the interpolation of\nthe 2.117 and 2.191 \\hbox{\\micron~} Kr lines measured at 6th order at the\nexpected position of the galaxy [\\ion{Ne}{2}] line.  Flux\ncalibration was performed using HR 4067 with a 10.1 \\hbox{\\micron~} flux\ndensity of 103 Jy and an assumed temperature of 4000 K on the\nfirst night, and HR 5340 using a 10.1 \\hbox{\\micron~} flux density of 738\nJy and an assumed temperature of 4400 K on the second night.\nNo significant difference was found between the two nights'\nobservations, so they were combined according to their\nstatistical weights. The difference in airmass between the\nsource and standard observations was $\\sim$0.5 but monitoring\nof BS 4518 on the first night at airmasses similar to the\nsource observations showed that the level of spectroscopic\nvariation over the observed wavelength range as a function of\nairmass was negligible compared to the final signal-to-noise\nratio.  The overall photometric calibration is estimated to be\ngood to about 25 per cent while the typical signal-to-noise ratio of\nan independent data point is about 10.\n\n\n\n\n\\section{Results\\label{ch5:r}}\n\n\n\n\n\nThe intensity and equivalent width of the 11.3 \\hbox{\\micron~} PAH\nfeature have been measured directly for all of the sources by\nestimating the continuum in the rest wavelength (see Table 1) ranges\n10.5--10.9 and 11.7--12.1 \\hbox{\\micron~} and subtracting the linear\ninterpolation of these two estimates from the spectra to\nmeasure the residual flux in the range 10.9--11.7 \\hbox{$\\mu$m}.\nUncertainties in the feature intensities represent the\ncombined uncertainties in the residual flux in this range but\nnot the uncertainties in continuum subtraction.  For the\nequivalent width estimates, uncertainties in the continuum\nlevel have been included.  In most cases the continuum\nuncertainties are dominant in the equivalent width estimates,\nleading to indeterminate results in the case of limits, but in\na few cases the uncertainty in the residual is the dominant\ncomponent, and upper limits to these equivalent widths can be\nset.\n\nSince model fitting can lead to substantially cleaner\nestimates of feature fluxes (of order the signal-to-noise\nratio in the peak spectral element being fit)\nestimates based on the model that is applied to these \ndata to test for the presence of a silicate absorption\nfeature are also given.  The model consists of a\npower-law continuum and an additive PAH spectrum\nrepresented by the spectrum of the Orion Bar taken from\nRoche (1989).  The fitting of this model was accomplished\nusing a version of Bevington's (1969) program CURVEFIT\nsupplied with IDL ver. 4.0 using three free parameters,\nthe power-law continuum scaling and index ($f_\\nu =\n\\beta\\times\\nu^{-\\alpha}$), and the scaling of the Orion Bar\nspectrum.  The data were fit in the rest wavelength ranges\n10.0--10.9, 11.0--11.4, and 11.7--12.6 \\hbox{\\micron~} using initial\nguesses based on linear fits to the logarithm of data\nbinned in the ranges 10.0--10.5, 10.5--10.9, and\n11.7--12.6 \\hbox{\\micron~} for the power-law scaling and index, and\nbased on the continuum subtracted estimates of the flux in\nthe 11.3 \\hbox{\\micron~} feature.  The weighting of the data was\nstatistical.  As discussed recently by Goldader et\nal. (1995), the use of a power-law as a fitting function\ntends to throw the bulk of the errors in the fit\nparameters into the power-law scaling, so that the formal\nerrors of the remaining parameters are underestimated.\nFurther, the model assumptions introduce further\nsystematic uncertainties.  In particular, by fitting the\ndata in the range 11.2--12.6 \\hbox{$\\mu$m}, it is assumed that\nthe ratio of 11.3 \\hbox{\\micron~} feature emission with respect to the\nunderlying plateau emission that extends to 12.9 \\hbox{\\micron~} is\nthe same as that found at the position of peak of the 11.3\n\\hbox{\\micron~} emission feature in the Orion Bar.  This ratio,\nhowever, is know to vary spatially within the Orion Bar (Roche\net al. 1989), and also within at least one external galaxy\n(Dudley \\& Wynn-Williams 1999).  The consequences\nof this uncertainty are largest in spectra where the PAH\nemission is most powerful relative to the continuum.  Given\nthat the galaxy redshifts make it impossible to estimate the\ncontinuum beyond 13.1 \\hbox{\\micron~} (rest frame) in $\\sim$50 per cent of the\nsources, it has not been possible to quantify the importance of\nthis effect in these data, regardless of obvious\nsignal-to-noise issues.  Due to these multiple and tangled\nuncertainties, both results of direct measurements and model\nfits with formal errors are presented, but it must be\ncautioned that the model parameter errors are seriously\nunderestimated.  In particular, model 3 $\\sigma$ upper limits\nto the 11.3 \\hbox{\\micron~} intensity are roughly the same as measured 1\n$\\sigma$ upper limits to the 11.3 \\hbox{\\micron~} intensity.  The new\nspectra and model fits are presented in Fig. 1.\n\n\n\n\n\n\n\n\n\nTable 3 presents the model power-law index in column 2,\nthe measured (upper) and model (lower) 11.3 \\hbox{\\micron~} feature\nintensity in column 3, the 11.3 \\hbox{\\micron~} feature equivalent width in\ncolumn 4, the ratio of the flux density of the source in\nthe rest wavelength range 8.0--8.4 \\hbox{\\micron~} to the model flux\ndensity in the same range where the errors represent the\nerrors in the estimates of the source flux densities only.\nMeasurements of fine structure lines are given in columns\n6 and 7.  For the measurements of fine structure lines the\ncontinuum was estimated by eye, and for detections of the\n12.8 \\hbox{\\micron~} [\\ion{Ne}{2}] line it must be cautioned that a\nsignificant amount of the intensity could be due to a\nblended PAH emission feature at 12.7 \\hbox{\\micron~} except in the\ncase of Arp 299B1, where the spectral resolution is\nsufficient to distinguish the two features.  Errors for\ndetections and limits are 1 $\\sigma$ and 3 $\\sigma$,\nrespectively, except in the case of the model 11.3 \\hbox{\\micron~}\nequivalent width, which is reported in all cases to give a\nsense of the model continuum uncertainties.  In the case\nof Arp 299B1, the fine structure [\\ion{S}{4}] and\n[\\ion{Ne}{2}] results are derived from the high-resolution\nspectrum.  The redshift of the [\\ion{Ne}{2}] line (2900\n\\hbox{km s$^{-1}$}) agrees within 1.4 $\\sigma$ (1 $\\sigma$ = 170 \\hbox{km s$^{-1}$}\nbased on Gaussian fitting) with the CO redshift measured\nby Sargent \\& Scoville (1991), while much more precise\nagreement between unpublished IRTF CSHELL \\hbox{Br$\\gamma$}~ data and the CO\nobservations is known, so that some\nconfidence in [\\ion{Ne}{2}] identification is warranted. It\nshould be noted\nthat [\\ion{Ne}{2}] is not detected in the low resolution\n8--13 \\hbox{\\micron~} spectrum of Arp 299B1 but that the 3 $\\sigma$\nupper limit for this feature is 2.1 $\\times 10^{-15} $ W\nm$^{-2}$, which is consistent with the intensity measured\nat high resolution.  Further, the ratio of I(11.3) to\nI([\\ion{Ne}{2}]) (1.7) is not unusual for galaxies for\nwhich the ratio has previously been measured (RASW).\n\n\n\n\n\n\n\n\\addtocounter{table}{1}\n\n\n\n\n\n\n\n\nAs is apparent in Table 3, while a wide range of power-law\nindices are necessary to describe the long-wavelength\ncontinuum, the ratio of 8.2 \\hbox{\\micron~} flux to model 8.2 \\hbox{\\micron~}\nflux is not significantly different from unity for a large\nfraction of the galaxies even in cases where the power-law\nindex is large and a minimum near 9.7 \\hbox{\\micron~} is pronounced.\nAs a result, it is not generally necessary to invoke silicate\nabsorption to explain the observed spectra.  \n\n\n\n\\section{Discussion\\label{ch5:d}}\n\n\\subsection{Spectral classification\\label{ch5:d1}}\n\nRASW have found that a three-type\nclassification scheme is sufficient to classify the 8--13\n\\hbox{\\micron~} spectra of most galaxies.  Their scheme consists of\nflat-featureless, silicate absorption or emission, and PAH-type (their UIR)\nspectra.  Further, there is a general correspondence\nbetween the 8--13 \\hbox{\\micron~} spectrum of a galaxy and its optical\nspectral classification based on optical emission lines.  The\ncorrespondence is that those galaxies with flat 8--13 \\hbox{\\micron~}\nspectra are often broad-line AGNs, those with silicate\nabsorption are frequently narrow-line AGNs, and those with PAH\nfeatures typically have optical spectra with line ratios\nsimilar to Galactic\n\\ion{H}{2} regions.  \n\nParenthetically, the identification of the mid-infrared\nemission feature analysed here is not entirely secure.  Other\nlaboratory analogs, including quenched carbonaceous composites\n(Sakata \\& Wada~1989), hydrogenated amorphous\ncarbon (Duley~1989), and anthracite (Ellis et\nal.~1994), have been proposed in addition to\nthe PAH model (Puget \\& L\\'eger~1989) that is\nadopted here for ease of expression.  However, the precise\nidentification of the chemical source of these features is not\nparticularly important for the present work.  What is\nimportant is that the strength of one feature may be predicted\nto some extent from observations of another.  In particular\nthe strength of the 7.7 \\hbox{\\micron~} feature may be predicted from\nobservations of the 11.3 \\hbox{\\micron~} feature to within about a factor of 2\n(Zavagno, Cox \\& Baluteau 1992).\n\nThe model fitting performed at the long-wavelength end of the\nspectra can be used effectively to test for the presence of\nstrong silicate absorption.  Two cases might occur.  First, a\nPAH spectrum might undergo external extinction.  In this case,\ntwo effects will contribute to the model under predicting the\nobserved flux in the shorter wavelength half of a spectrum.  The\nfirst effect is that the 11.3 \\hbox{\\micron~} feature will undergo more\nsevere extinction than the region near 8 \\hbox{\\micron~} due to the\nspectral shape the silicate component of the interstellar\nextinction curve.  This has the effect of reducing the predicted\nPAH strength.  The second effect is that the slope of the model\ncontinuum will have a larger spectral index than would otherwise\nbe the case if no silicate absorption were present.  The model\nwould therefore under-predict the continuum underlying the 7.7\nand 8.6 \\hbox{\\micron~} feature complex.  Both of these effects will cause\nthe model to under-predict the shorter wavelength observations.\nSecond, a PAH spectrum may undergo no particular extinction,\nhowever, a second silicate-absorbed continuum component may also\ncontribute to the spectrum.  In this case, the model should\npredict the PAH component well.  However, if the silicate\nabsorption is strong, the model spectral index will be increased\ndue to the additional component, and the continuum in the short\nwavelength half will be underestimated so that the model will\nalso under-predict the observed flux.  \n\nThe RASW classification scheme is applied to the present data\nin the following way: the ratio of the observed to model flux\ndensity between 8.0 and 8.4 is used as a guide to classification\nsuch that if this ratio is 3 $\\sigma$ above 2.5, then the\nsource is classified as silicate.  This cut may somewhat\nunderestimate the number of galaxies in which silicate\nabsorption is present compared to the approach taken in\nDudley \\& Wynn-Williams (1999), where the cut is taken at\n2.0. For the purposes of definite classification, under the\nworking hypothesis that all infrared galaxies are PAH-type\nunless convincingly shown to be otherwise, this more\nconservative value is appropriate.  In cases where the 11.3\n\\hbox{\\micron~} feature is not formally detected but only allowed (dot\ndashed curves in Fig. 1), and where a minimum is present\nnear 9.7 \\hbox{$\\mu$m}, a question mark is appended to the PAH\ndesignation; however, whether or not there is a formal\ndetection of the 11.3 \\hbox{\\micron~} feature in sources with obviously\nflat spectra, the classification is given as flat.  A mixed\nclassification of PAH and silicate is given where PAH\nemission can explain $\\sim$30 per cent of the observed emission\nbetween 8.0 and 8.4 \\hbox{$\\mu$m}.  Under this condition, the\nclassification of Arp 220 by RASW as both PAH and silicate\nwould be revised to just silicate.  Arp 220 is discussed\nin detail in Section 4.2.5.\n\n\n\n\n\n\n\n\n\\addtocounter{table}{1}\n\n\nTable 4 gives, for each source in column 1, the spectral\nclassification based on the present data and on optical\nobservations, along with 12--100 \\hbox{\\micron~} photometry, the infrared\nluminosity, and the star formation efficiency.  In some cases\nthe optical spectral classification has been inferred from the\ndata of Veilleux et al. (1995), if some diagnostics were\nmissing. It is clear from Table 4 that the general\ncorrespondence between optical spectroscopy and 8--13 \\hbox{\\micron~}\nspectroscopy is quite similar to that found by RASW.  Neglecting\nLINER-type galaxies, four exceptions where AGNs are identified\noptically but which are found to have PAH-type spectra in this\nwork are Mkn 1034, IRAS 04154+1755, IRAS 05189$-$2524, and Zw\n475.056.\n\nMkn 1034 is a known Seyfert 1 galaxy; however its 8--13 \\hbox{\\micron~}\nspectrum can be fit with a PAH plus continuum model.  This may\nbe consistent with the finding of Sopp \\& Alexander (1992) that\nonly half of the 6-cm flux density measured in a 14\\farcs5 beam\nis also contained within a 1\\farcs5 beam, suggesting that\nspatially extended star formation plays a role in this source.\nA similar explanation may be invoked for the presence of PAH\nemission in the spectra of the Seyfert 2 galaxies IRAS\n04154+1755 and Zw 475.056.  Crawford et al. (1996) find IRAS\n04154+1755 to be resolved at 6 cm using both 4-arcsec and\n0.5-arcsec beams.  In the map of Zw 475.056 presented by CHYT,\nlow-level radio emission extends over 1.5-arcsec.  IRAS\n05189$-$2524 is discussed in detail in the next section; however\nit should be noted here that the presence of exceptions, in this\nsample, to the the correspondence between optical and 8--13 \\hbox{\\micron~}\nspectral classification found by RASW should not be taken as\nevidence that the correspondence is merely coincidental.  The\nsources presented by RASW were sometimes selected on the basis\nof optical spectroscopy to examine the question of what kind of\nspectrum they would present at 8--13 \\hbox{$\\mu$m}.  Hence, many of\ntheir sources were archetypical examples of their optical class.\nIn the present sample, selection is based solely on infrared\nproperties, so that mixed cases might well be expected.\n\nIt is probably not coincidental that these four sources are\namong the sources in Table 3. with the smaller values of\n$\\alpha$ suggesting that the AGN in these sources may contribute\nsomewhat to the 8--13 \\hbox{\\micron~} continuum; strong 12 and 25 \\hbox{\\micron~}\nemission is known to select for AGNs that have unambiguous\noptical spectra (de Grijp et al. 1985; Spinoglio \\& Malkan, see also\nthe AGN spectral energy distributions presented by Sanders et\nal. 1989).\n\n\\subsection{Comments on individual ultraluminous infrared galaxies}\n\n\n\n\nThe observed properties of the ultraluminous infrared galaxies\npresented here show a surprising range.  A closer look at these\nproperties will be discussed in this section.  \n\n\\subsubsection{A phenomenological description of IRAS 05189$-$2524}\n\nOne of the greatest surprises in these data is the case of\nIRAS 05189$-$2524, which might have been expected to show a flat\nor silicate absorbed 8--13 micron spectrum based on its\noptical AGN-like spectrum (Sanders et al. 1988).  What is\nobserved however is a spectrum that is well fit by a PAH plus\npower-law continuum model (see Fig. 1).  Additionally, the\nstrength of the 11.3 \\hbox{\\micron~} emission relative to the 8--1000\n\\hbox{\\micron~} emission is quite similar to known starburst galaxies so that\nit seems possible that star formation might be the primary\npower source of the galaxy.  Here, a tentative decomposition\nof its spectral energy distribution is proposed.  In Fig.\n2, a comparison is made between IRAS 05189$-$2524 and two\nknown AGNs to suggest that while the 1--5 \\hbox{\\micron~} emission in IRAS\n05189$-$2524 may be due to hot dust heated by an AGN, the 8--100\n\\hbox{\\micron~} emission may be better explained as arising from star\nformation traced by the observed PAH emission.  In Fig. 2\nthe power-law extrapolations of the dust continuum emission\n(solid lines) found in the 8--13 \\hbox{\\micron~} spectra are projected to 5\nand 25 \\hbox{$\\mu$m}.  For the two sources where the 8--13 micron\nemission is thought to be due to AGN-heated dust (Mkn 231 and\nIRAS 13349+2438) (Roche et al. 1983; Beichman et al. 1986),\nthese extrapolations are reasonable predictors of the observed\nflux densities at both 5 and 25 \\hbox{$\\mu$m}.  For IRAS 05189$-$2524\nthe extrapolation of the continuum to 25 \\hbox{\\micron~} is in reasonable\nagreement with the observed flux density; however the\nextrapolation to 5 \\hbox{\\micron~} fails to predict the 4--5 \\hbox{\\micron~}\nphotometry of IRAS 05189$-$2524 (see double arrow), suggesting\nthat an unrelated power source is responsible for this\nemission.  Given the apparent similarity in the shape of 1--5\n\\hbox{\\micron~} emission in both IRAS 05189$-$2524 and IRAS 13349+2438, it\nis possible that the spectral energy distribution of IRAS\n05189$-$2524 is the result of a starburst producing the majority\nof the 8--100 \\hbox{\\micron~} emission and an AGN with a spectral energy\ndistribution similar to that of IRAS 13349+2438 responsible\nfor the bulk of the shorter wavelength emission but\ncontributing little to the 8--100 \\hbox{\\micron~} emission.  Such a\ndecomposition could reconcile the apparently contradictory\nobservations of both an AGN-like optical spectrum and the\npresent PAH type 8--13 \\hbox{\\micron~} spectrum.  A decomposition of this\nkind could also be consistent with the suggestion of Goldader\net al. (1995) that the 2 \\hbox{\\micron~} continuum is dominated by AGN\n[heated dust] emission based on the small equivalent width of the observed\n\\hbox{Br$\\gamma$}~ emission.  That hot dust typically plays a lesser\nrole in starburst galaxies at 2 \\hbox{\\micron~} may also be deduced\nfrom the frequent detection of stellar continuum with probably\nlittle veiling of the CO absorption features, while in IRAS\n05189$-$2524 CO absorption is not detected (Goldader et\nal. 1995).  Another piece of evidence that supports the view that\nthe AGN in IRAS 05189$-$2524 may not be energetically dominant\nis the failure to detect a radio core of extreme brightness\ntemperature in VLBI observations (Lonsdale, et al.\n1993).\n\nThe VLA radio map of CHYT shows the radio continuum in this\nsource to be extended on a scale of 160 pc, which is\ncomparable to the minimum blackbody diameter of 280 pc based on\nthe observed 60 and 100 \\hbox{\\micron~} flux densities and assuming optically thick\ndust emission.  It is therefore plausible that the far-infrared\nemission arises from optically thick warm dense clouds embedded\nin the starburst with a low volume filling factor but a column\ncovering factor of order unity. The mid-infrared continuum and\nPAH emission present in this spectrum might arise from the\nsurfaces of these clouds and may be optically thin in the sense\nthat the emission arises from cloud surfaces facing our line of\nsight, and from a layer that is only 1--2 $A_V$ thick on any\ngiven line of sight into the starburst.  \n\nGiven the possibility\nof a decomposition of the 1--100 \\hbox{\\micron~} spectral energy\ndistribution into a dominant starburst component and an AGN\ncomponent important only at shorter wavelengths, the\nplausibility of the infrared emission being spatially associated\nwith the radio continuum, and most importantly, the presence of\n11.3 \\hbox{\\micron~} PAH emission in the 8--13 \\hbox{\\micron~} spectrum taken together with the\nadequate fit of the model given in Fig. 1, the 8--1000 \\hbox{\\micron~}\nemission from IRAS 05189$-$2524 is attributed to a starburst for\nthe purposes of this work.\n\n\\subsubsection{Digression on newly available {\\em ISO} data}\n\nThe model applied to the data presented in Fig. 1 for the\npurpose of spectral classification is not required in cases\nwhere the strength of the 7.7 \\hbox{\\micron~} PAH feature has been\nobserved as is now the case for four of the ultraluminous\ninfrared galaxies in this combined sample as a result of {\\em\nInfrared Space Observatory (ISO)} observations (Genzel et\nal. 1998).  With the additional information it is possible to\ncompare observations with a variety of scenarios to better\nunderstand the physical situation with greater confidence than\npossible when only ground-based data are available.  In Fig. 3,\nthree possibilities are considered: (1) For the top spectrum\nin each panel, a PAH plus power-law continuum model such as\nthat employed in Fig. 1 is fit not just in the spectral region\nsurrounding the 11.3 \\hbox{\\micron~} feature, but to the entire 6--13 \\hbox{\\micron~}\nrange.  (2) For the middle spectrum in each panel, a PAH plus\npower-law continuum model is subjected to extinction by cold\ndust that includes the silicate absorption feature, as adopted\nby Dudley \\& Wynn-Williams (1997), and again fit to the entire\n6--13 \\hbox{\\micron~} range.  (3) For the bottom spectrum in each panel, a\nPAH plus power-law continuum model is additively combined with\na silicate-absorbed continuum source, and again fit over the entire\n6--13 \\hbox{\\micron~} range.  The model fits presented in Fig. 3 have been\ncarried out using CURVEFIT with equal weighting.  In the first\nmodel, the index of the power-law was specified either from\nTable 3 (IRAS 17208$-$0014 and Mkn 273) or as 12.8 and 4.9 for\nArp 220 and Mkn 231, respectively.  It should be noted that\nthree of the sources exhibit a large dynamic range, and in the\nfollowing discussion some attention will be drawn to the poor\nfit at low flux density levels, which is also an expected systematic\neffect of the fitting procedure, since differences between\nsmall numbers are small.  It may also be noted that the\nmismatches are apparent for a large number of neighboring data\npoints.\n\n\n\\begin{figure}\n\\end{figure}\n\n\n\n\n\\subsubsection{IRAS 17208$-$0014}\n\nIn IRAS 17208$-$0014, all models give fairly adequate fits\nto the data, and deciding between them depends mainly on\nother observational evidence.  The strongest evidence that\nsupports a preference for the fit shown for the top spectrum\nis that this source is extended on a scale $\\sim$1.7 kpc in\nthe radio (Condon et al. 1996), although higher spatial\nresolution observations reveal the presence of a sub-kpc\nradio continuum source associated with the OH maser emission\n(Martin et al. 1989).  The failure of Martin et al. (1989)\nto detect more extended emission may be due to the\nrestricted set of base lines they employed.  When this\nevidence is taken along with the optical spectral\nclassification of Veilleux et al. (1995) in which the line\nratios in this source are similar to those in \\ion{H}{2}\nregions, there is no strong reason to suspect that anything\nother than star formation is responsible for the observed\nmid-infrared spectrum.  Nor is there any reason to suspect a\nhigh degree of extinction; IRAS 17208$-$0014 is among the\nsources presented by Genzel et al. (1998) for which the\ncontinuum underlying the 12.8-\\hbox{\\micron~} [\\ion{Ne}{2}] feature in\ntheir SWS data can be comfotably plotted on the scale of the\nISOPHOT-S data in their fig. 2, suggesting a lack of strong\nsilicate absorption (see also Fig. 1).  The fit to the upper\nspectrum might be improved somewhat by increasing the\nstrength of the 7.7 and 6.2 \\hbox{\\micron~} features by a factor of 2,\na procedure which is probably allowed for the 7.7 \\hbox{\\micron~}\nfeature at least since the Orion Bar spectrum used as a\ntemplate here is known to have a somewhat weaker than usual\n7.7 \\hbox{\\micron~} feature relative to the 11.3 \\hbox{\\micron~} feature (Zavagno\net al. 1992).\n\n\\subsubsection{Mkn 273}\n\nThe fit to the top spectrum of Fig. 3 for Mkn 273 gives, as\nmight have been anticipated from Fig. 1, a poor representation\nof the data.  The fitted strength of the 11.3 \\hbox{\\micron~} feature is\nclearly too large, which is roughly the converse of the result\nin Fig. 1.  Thus, it may be concluded that a simple PAH plus\npower-law continuum model does not describe these data.  The fit\nshown for the middle spectrum is better, in that it requires\nless PAH emission over all and the strength of the 11.3 \\hbox{\\micron~}\nfeature is not over-predicted; however the continuum around the\n9.7 \\hbox{\\micron~} silicate minimum is not well reproduced.  This\nsuggests that an unaccounted-for emission component is required\nto produce this emission.  The fit to the lower spectrum\novercomes this difficulty by superimposing an unobscured PAH plus power-law\ncontinuum such as that which fit IRAS 17208$-$0014 (top\nspectrum) upon a continuum source that suffers strong silicate\nabsorption.  Since Mkn 273 shows clear signs of nuclear\nactivity (Armus, Heckman \\& Miley 1989; Veilleux et al. 1995),\na silicate-absorbed continuum source might be produced by an\nedge-on view of a disk model such as those proposed by Pier \\&\nKrolik (1992) or Efstathiou \\& Rowan-Robinson (1995), which are\nnatural elaborations of the AGN unification model proposed by\nAntonucci \\& Miller (1985).  The continuum level evident in the\nSWS observations of 12.8-\\hbox{\\micron~} [\\ion{Ne}{2}] in Genzel et\nal. (1998) is consistent with the new CGS3 data presented here;\nand therefore consistent with silicate absorption playing a\nrole in this source.  If the fit to the bottom spectrum is accepted,\nthen the AGN component could well be responsible for the\ngreater fraction of the 60 and 100 \\hbox{\\micron~} emission.\n\n\\subsubsection{Arp 220}\n\nPrior to the new {\\em ISO} observations the analysis of the\nArp 220 spectrum would have proceeded very much along the\nlines that have just been given for Mkn 273 (Smith et\nal. 1989), with the proviso that Arp 220 has less obvious\nsigns of activity in its nucleus so that disk-like models\nfor the silicate-absorbed continuum emission may be less\nhelpful in advancing our understanding, and deeply embedded\nmodels perhaps more descriptive (Dudley \\& Wynn-Williams\n1997).  Applying the model used in Fig. 1 to the data of\nSmith et al. (1989) severely under-predicts the observed 8.2\n\\hbox{\\micron~} flux density.  However, as can be seen in Fig. 3 the\nnew {\\em ISO} data can apparently be fit by a simple PAH\nplus power-law continuum model due to the apparent\ndiscrepancy between the observations of the 11.3 \\hbox{\\micron~}\nfeature reported by Smith et al. (1989) and the {\\em ISO}\ndata of Genzel et al. (1998).  It should be noted that there\nare no continuum points longward of the 11.3 \\hbox{\\micron~} feature in\nthe new {\\em ISO} data, so that determining its strength\nfrom those data is difficult.  It seems safer at this point\nto accept the ground-based data as a guide in this spectral\nregion and thus hold this model in abeyance, pending further\nobservational evidence.  The fit to the middle spectrum\nis essentially the same as that given in fig. 1 of Smith et\nal. (1989) and suffers from the same difficulty noted by\nSmith et al. (1989), namely that the minimum near 9.7 \\hbox{\\micron~}\nis poorly fit.  The fit to the lower plot is similar to that\nshown in Smith et al.'s (1989) fig. 2 or as adopted by Dudley \\&\nWynn-Williams (1997).  The continuum in the 12.8-\\hbox{\\micron~}\n[\\ion{Ne}{2}] SWS data of Genzel et al. (1998) is consistent\nwith the Smith et al. (1989) data but is not apparently\nconsistent with the ISOCAM-CVF spectrum presented by Elbaz\net al. (1998).\n\nFor Arp 220, therefore, the new {\\em ISO} data presented by\nGenzel et al. (1998) seems to allow the possibility that\nunextinguished PAH emission plus a simple power-law\ncontinuum describes this spectrum as a result of the\napparent strength of the 11.3 \\hbox{\\micron~} feature in the new data.\nIt has been suggested that differing spectral resolution\n(roughly a factor of 2) between the Smith et al. (1989) and\nGenzel et al. (1998) data may account for this difference,\nhowever this could only work and if the 11.3 \\hbox{\\micron~} feature\nwere unresolved in both spectra, and that does not appear to\nbe the case.  Gauging the over all PAH emission from the 6.2\n\\hbox{\\micron~} feature which displays continuum on either side, the\nintensity of which is found to be well correlated with the\nthat of the 7.7 \\hbox{\\micron~} feature in Galactic sources (Cohen\net al. 1989), it is clear the PAH emission relative to the\nover-all 8--1000 \\hbox{\\micron~} emission is exceedingly weak, really\nno stronger than what might be allowed in Mkn 231 (see\nnext), if the apparent 6.2 \\hbox{\\micron~} feature in the {\\em ISO}\ndata for Mkn 231 is real.  In contrast, again using the 6.2\n\\hbox{\\micron~} PAH feature as a guide, the strength of the PAH\nemission relative to the 8--1000 \\hbox{\\micron~} emission in Mkn 273 is\ntwice as strong, and in IRAS 17208$-$0014, five times as\nstrong. The Arp 220 spectrum might possibly be produced by a\nstarburst that reprocesses more that 90 per cent of its\nluminosity not once but many times so that the observed PAH\nemission is produced only in a thin outer shell or in a\nsmall number of narrow chinks. Such a model would be even\nmore extreme than what is argued above as plausible for IRAS\n05189$-$2524 in a starburst context.  However, given the\ndisagreement between the two spectra of this source near\n11.3 \\hbox{\\micron~} and in consideration of the arguments of Dudley \\&\nWynn-Williams (1997), a model that has both unobscured PAH\nemission and a silicate-absorbed continuum due to a deeply\nembedded AGN continues to seem more natural. This\ninterpretation also seems to account simultaneously for both\nthe 158-\\hbox{\\micron~} [CII] (Luhman et al. 1998) and 2-10 keV X-ray\n(Iwasawa 1999) deficits in this source if the ``AGNs''\n(Soifer et al. 1999) are sufficiently obscured.  It should\nbe noted that in this preferred model, only a fraction of\nthe flux density at 7.7 \\hbox{\\micron~} can be attributed to the 7.7\n\\hbox{\\micron~} PAH feature, in accordance with either the 12.8 of 14.3\n\\hbox{\\micron~} continuum seen in the SWS data presented by Genzel et\nal. (1998).  As suggested by Dudley \\& Wynn-Williams (1997),\nspatially resolved spectroscopic observations may help to\nclarify this issue.\n\n\\subsubsection{Mkn 231}\n\nFor Mkn 231 any PAH contribution to the spectrum must be\nminimal compared to a continuum contribution (Roche et\nal. 1983), as can also be seen from the fit to the upper\nspectrum in Fig. 3.  This source had been fit adequately\nwith a model consisting of a continuum source suffering\nsilicate absorption by Roche et al. (1983), and the present\nmodels are {\\it not} an improvement.  It would be of\ninterest to know if disk radiative transfer models such as\nthose proposed by Pier and Krolik (1992) would fair as well\nas the cold absorber model of Roche et al. (1983) in the\ndetailed fit to the silicate absorption feature given their\nlikely success in fitting the overall infrared spectral\nenergy distribution of Mkn 231.  The 11.3 \\hbox{\\micron~} emission\nfeature is not detected in this source and it is not clear\nthat the single point that is spectrally coincident with the\n6.2 \\hbox{\\micron~} feature in the {\\em ISO} data should be considered\nas a real indication of the presence of PAH emission.  Some\nPAH emission is not unexpected in this source given the\npresence of star-forming knots near the nucleus of this\nsource that might account for $\\sim$10 per cent of its\ninfrared luminosity (Surace et al. 1998).  The estimate of\nthe 7.7 \\hbox{\\micron~} PAH strength given by Genzel et al. (1998)\nrelies on a continuum point at 10.9 \\hbox{\\micron~} which lies in the\nsilicate absorption feature.  Of the sources measured in\nthis way by Genzel et al. (1998) Mkn 231 perhaps provides\nthe clearest example of how this method is prone to\noverestimate the PAH strength when silicate absorption is\npresent.  The 12.8 or 14.3 \\hbox{\\micron~} continuum in their fig. 1\nprovide a rough guide as to when there may be a problem in\nthis respect.  For Mkn 231, the 12.8 \\hbox{\\micron~} continuum of\nGenzel et al. (1998) is high in comparison with the data of\nRoche et al. (1983) but it is possible that it would agree\nwith the extrapolation of the long-wavelength ISOPHOT-S data\nwhich also apparently fall significantly above the Roche et\nal. (1983) data as seen in Fig. 3.  It is not clear if these\ndifferences should be attributed to aperture effects, or\nother systematics, or if they reflect intrinsic source\nvariability over 1.4 decade; Roche et al. (1983) found good\nagreement with the narrowband photometric observations of\nReike (1976) made over 0.4 decade earlier.\n\n\n\\subsection{A 60 \\protect\\hbox{$\\mu$m}~ flux-limited subsample\\label{ch5:d2}}\n\n\n\nAmong galaxies with \\hbox{$L_{\\rm IR}$} $\\geq 1.6\\times10^{11}$ \\hbox{L$_\\odot$}, a nearly\ncomplete (all but three sources have 8--13 \\hbox{\\micron~} spectra)\nflux-limited (at 60 \\hbox{$\\mu$m}) sample of 25 sources may now be\ndefined on the combined Bright Galaxy Survey (Soifer et\nal. 1989) and the Extended Bright Galaxy Survey (Sanders et\nal. 1995) (combined BGS) for $F_\\nu$(60) $\\geq$ 11.25 Jy\n(roughly twice the flux limit of  combined BGS).  The only other\nrestriction on this subsample is that the sources be observable\nusing UKIRT, i.e. $-25^\\circ$ $\\leq \\delta \\leq $ 60$^\\circ$.\nTable 5 gives the ultraluminous and luminous samples with the\nsource name in column 1, the 8--13 spectral classification in\ncolumn 2, the reference for the 8--13 \\hbox{\\micron~} spectrum in column 3,\nthe $\\log$ of the 8--1000 \\hbox{\\micron~} luminosity in column 4, and the\n$\\log$ of the ratio of the 12 \\hbox{\\micron~} flux density to the 60 \\hbox{\\micron~}\nflux density in column 5. The ultraluminous sample has 5 members\nand the luminous comparison sample has 20 members.  The\nultraluminous sample is {\\em unlikely} to span all the possible\ncharacteristics of ultraluminous infrared galaxies and therefore\nis not fully representative.  There are two observations that\nconfirm this: (1) if Arp 220 were scaled to the 60 \\hbox{\\micron~} 11.25 Jy\nflux limit used to define this sample, it would not have been\nobservable using CGS3 or other ground-based spectrographs\navailable when these data were collected, and (2) the ratio\ngiven in column 5 of Table 5 for the two brightest ultraluminous\ngalaxies (Arp 220 and Mkn 231) bracket the same ratio for all\nthe galaxies in the sample with $1.6\\times10^{11} \\leq$ \\hbox{$L_{\\rm IR}$}\n$\\leq 10^{12}$ \\hbox{L$_\\odot$}~ except NGC 1068 and NGC 7469.  An analysis\nover the entire Bright Galaxy Survey shows that the dispersion\nof this ratio is indeed larger for ultraluminous infrared\ngalaxies than for luminous infrared galaxies by a factor of 2\n(Soifer \\& Neugebauer 1991).  Thus, conclusions based on the\npresent flux-limited sample of ultraluminous galaxies should be\nviewed with caution not simply because of the small number of\ngalaxies in this sample (counting statistics on a bimodal parent population), but because the\ncharacteristics of the parent population are not well\nrepresented (under-sampling).\n\n\n\nIt is possible that the last of these two difficulties\ncould be somewhat ameliorated by the following\nconsiderations.  Ultraluminous galaxies that fall outside\nthe range of 12 on 60 \\hbox{\\micron~} flux density ratios defined by\nArp 220 and Mkn 231 may be assigned to the AGN-dominated\nbin in the context of exploring the AGN vs. starburst\nhypotheses.  For ultraluminous galaxies with 12\/60 \\hbox{\\micron~}\ncolors redder than Arp 220, PAH emission cannot make a\ncontribution to the 12 \\hbox{\\micron~} flux density that is\nsufficiently large to explain the 60 \\hbox{\\micron~} flux density. \nIt is therefore likely that such sources will have deep\nsilicate absorption features, and by the arguments\npresented by Dudley \\& Wynn-Williams (1997), the size of\ntheir power source should be too small to be due to a\nstarburst.  For galaxies with 12\/60 \\hbox{\\micron~} colors bluer than\nMkn 231, 60 \\hbox{\\micron~} selected samples will overlap 25 or 12\n\\hbox{\\micron~} selected samples that are known to select for AGNs\n(de Grijp et al. 1985; Spinoglio \\& Malkan 1989).  However,\nreliance on these notions would be ill advised given the\ninternal evidence in the present data showing that PAH\nemission can be important in ultraluminous galaxies with\n12 to 60 \\hbox{\\micron~} flux density ratios similar to the two\nextremes, namely, IRAS 17208$-$0014 and IRAS 05189$-$2524.\n\n\n\n\n\n\nInspection of Table 5 shows that 3 out of 5 of the ultraluminous\ngalaxies are powered predominantly by AGNs although if judgment\nis reserved for Arp 220, this fraction would be 2 out of 5. The\ntrend is thus consistent with about half of ultraluminous\ngalaxies being powered by AGNs.  On the other hand, the fraction\nof the more numerous luminous galaxies in this sample that are\npowered predominantly by AGNs is at most 0.25 and could be as\nsmall as 0.05 or smaller if the relative contributions of the\nAGN and starburst in NGC 1068 are approximately equal (Telesco\net al. 1984).  This trend supports the evidence from optical\nspectroscopy that AGNs are present in greater numbers in\nultraluminous infrared galaxies (Veilleux et al. 1995).  It goes\nfurther than optical or radio studies because it is possible to\ndemonstrate with greater certainty which type of power source\n(starburst or AGN) dominates the infrared luminosity.  Further,\nit is apparent that for luminosities less than 10$^{12}$ \\hbox{L$_\\odot$},\nstar formation is far and away the dominant power source, a\nresult that confirms work at many wavelengths.  This analysis\nlends some support to the proposal of Sanders et al. (1988) that\nthere is an evolutionary connection between ultraluminous\ninfrared galaxies and quasars, while at the same time\ndemonstrating that massive star formation can indeed be the\ndominant power source of some ultraluminous infrared galaxies.\nHowever, it must be conceded that the evidence presented\nconstitutes only a trend. The precise proportion of\nAGN-dominated sources among 60 \\hbox{\\micron~} selected ultraluminous\ninfrared galaxies cannot be said to have been established here,\nand thus no statistically meaningful difference between luminous\nand ultraluminous galaxies has truly been shown.  It is\nof interest that the trend does continue with the next step down\nin 60 \\hbox{\\micron~} flux density.  Adding IRAS 12112+0305 and IRAS\n08572+3915 to the present sample of five ultraluminous galaxies adds\none apparently starburst-dominated (based on the apparently\nnormal ratio of the 6.2 \\hbox{\\micron~} PAH feature emission to overall infrared\nemission in the IRAS 12112+0305 spectrum of Genzel et\nal. 1998\\footnote{A crude attempt to correct for the\ncontribution of the silicate absorption edge to the 7.7 \\hbox{\\micron~} PAH\nfeature by fitting a power law to continuum points at 5.9 and\n14.3 \\hbox{\\micron~} suggests a deficit in the strength of the 7.7 \\hbox{\\micron~} feature for\nIRAS 12112+0305.})\nand one AGN-dominated object (Dudley \\& Wynn-Williams 1997).\n\n\\subsection{Anti-correlation between 11.3 \\hbox{\\micron~} eqivalent width and\n$L_{\\rm IR}$\/$M({\\rm H}_2)$\\label{ch5:d3}}\n\n\nIn Fig. 4  a possible anti-correlation between\nthe equivalent width (EW) of the 11.3 \\hbox{\\micron~} feature and the global ratio of $L_{\\rm IR}$\/$M({\\rm H}_2)$ is presented for a number of galaxies.  The\nEW data are based either on the\nmodel fits given in Table 3 or the previously published\nvalues of RASW reproduced in Table 6. The ratios of\ninfrared luminosity to H$_2$ mass are taken from the\nliterature (Tables 4 and 6).  The solid\nline in Fig. 4 indicates inverse proportionality.\n\n\n\n\n\nThe ratio ${L_{\\rm IR} \\over {\\rm M(H}_2)}$ is often taken to be\na measure of star formation efficiency.  When it is large, then\na large fraction of the available gas is involved in star\nformation; when it is small, conditions may be such that the gas\nexists in a more quiescent state.  However, for a very large\nratio of ${L_{\\rm IR} \\over {\\rm M(H}_2)}$, Sanders et\nal. (1991) argued that even a very high star formation\nefficiency cannot account for the infrared luminosity, on the\ngrounds that star formation would too quickly consume the\navailable gas mass, so that AGN-like activity might be the more\nlikely primary power source of the infrared luminosity.  Both\n${L_{\\rm IR} \\over {\\rm M(H}_2)}$ and EW(11.3) are independent\nof distance in so far as EW(11.3) is independent of aperture\nsize.\n\nA tendency for EW(11.3) to be anti-correlated with the star\nformation efficiency might arise if two conditions were met.\nFirst, if the intensity of the 11.3 \\hbox{\\micron~} feature is a roughly\nconstant fraction of the overall infrared emission (proportional\nto $L_{\\rm IR}$) then this will not contribute to the\nanti-correlation.  Second, if the strength of the continuum at\n11.3 \\hbox{\\micron~} increases with decreasing availability of dust\n(inversely proportional to M(H$_2$)) to reprocess the light\nemitted by massive stars so that a larger fraction of the dust\ngrain population is heated to the warm temperatures required for\nthe dust grains to emit at 11.3 \\hbox{\\micron~} then this could be the\nsource of anti-correlation.  Such an explanation requires\nfurther elaboration since the process by which the 11.3 \\hbox{\\micron~}\ncontinuum is produced is likely, in general, to be a local\nphenomenon rather than a global one.\n\nAn elaboration that may be in keeping with recent {\\em HST}\nobservations (Watson et al. 1996; Surace et al. 1998;\nSatyapal et al. 1997) is that as the star formation\nefficiency increases, the proportion of massive stars formed\nin dense clusters may increase.  If this is the case, then\nthe time-scale for clearing the interstellar medium from\naround these clusters to a distance where dust grains would\nno longer be heated sufficiently to contribute to the 11.3\n\\hbox{\\micron~} continuum may be longer that the time-scale for\nclearing the interstellar medium around the groups of\nmassive stars that might form if the star formation\nefficiency is lower.  Two effects may contribute to this.\nFirst, the distance at which a grain could be heated to a\ntemperature high enough to contribute to the 11.3 \\hbox{\\micron~}\ncontinuum would increase as the square root of the number of\nmassive stars in a cluster so that if the velocity of gas\nand dust being cleared from clusters of massive stars is not\ntoo different from the velocity of gas and dust cleared from\ngroups of massive stars, the dust remains heated for a\nlonger period of time.  Second, dense clusters might be\ntemporarily gravitationally bound and behave ballistically\nwith respect to the dissipative gas. They may sometimes move\ninto a fresh reservoir of gas, heating the associated dust\nto higher temperatures.\n\nIn this view, the additional contribution to the 11.3 \\hbox{\\micron~}\ncontinuum emission at higher star formation efficiency would\nbe provided by possibly equilibrium temperature grains\nheated by a combination of direct and Ly$\\alpha$ heating\nrather than small grains heated by a single photon.  To\nzeroth order, this is a sensible proposition, since one\nexpects (at least within photo-dissociation regions) that\nthe PAH emission and the spike-heated grain emission to have\nroughly fixed relative contributions at 11.3 \\hbox{\\micron~} due to the\nrough similarity of the emission time scales for the two\nemission processes, so that the apparent anti-correlation\nwould not necessarily arise.\n\nA number of observations will be required to examine this\nsituation.  Most important are higher signal-to-noise\nobservations that will allow real measurements of EW(11.3)\nfor a number of the sources included in Fig. 4.  These are\nrequired to decide if the anti-correlation is has any merit.\nAnother avenue of investigation would be to make higher\nspatial resolution observations of nearby starbursts where\nclusters of massive stars are observed to exist to establish\nwhether or not EW(11.3) does decrease in the vicinity of\nsome clusters (see Dudley \\& Wynn-Williams 1999 for a\npreliminary attempt).  Finally, high spatial resolution 2\n\\hbox{\\micron~} adaptive optics imaging of a number of starbursts with\na range of star formation efficiency can help to determine\nif the fraction of massive stars that exist in dense\nclusters varies with the star formation efficency parameter\nas conjectured here.\n\n\\section{Conclusions\\label{ch5:c}}\n\nNew 8--13 \\hbox{\\micron~} spectroscopy of 27 infrared galaxies has been\npresented.  The spectra have been fit with a two-component model\nconsisting of a power-law continuum underlying a PAH emission\nspectrum.  These fits are based on the spectral data between 10 and\n13 \\hbox{\\micron~} and are used to predict the flux\nemitted between 8.0 and 8.4 \\hbox{$\\mu$m}.  It is\nfound that for the majority of the observed galaxies, the\nmodel predicts the observed 8.0--8.4 \\hbox{\\micron~} emission to\nwithin an acceptable factor of 2, suggesting that the\nmajority of the sources are powered by starbursts.\n\nOne source, Mkn 273, has been found to show very clear evidence for\nsilicate absorption.\n\nA flux-limited 60 \\hbox{\\micron~} selected subsample that combines the\npresent spectroscopic results and those of RASW had been\ndefined, consisting of 25 galaxies.  In this subsample the fraction of ultraluminous\ninfrared galaxies powered by AGN is $\\sim$50 per cent in\ncontrast to 5--25\\% among the luminous infrared galaxies.  \nOwing to the small sample size for the ultraluminous infrared\ngalaxies, this difference in AGN fraction can be considered only\na trend.\n\nA possible anti-correlation is identified between the\nequivalent width of the 11.3 \\hbox{\\micron~} PAH feature and the star\nformation efficiency in galaxies where both have been\nmeasured.  It is suggested that variation in the continuum\ndust temperature as a function of star formation efficiency\ncould produce this apparent effect.\n\n\n\n\\noindent{ACKNOWLEDGMENTS}\n\nThe author would like to thank the UKIRT staff: Joel Aikock,\nTim Caroll, Dolores Walthers, and Thor Wold, for assistance at\nthe telescope, and John Davies, Tom Geballe, and Gillian Wright\nfor fruitful discussions and calibration data in advance of\npublication.  UKIRT is operated by the Joint Astronomy Centre\non behalf of the U.~K.~Particle Physics and Astronomy Research\nCouncil.  Brian Rush, Reinhart Genzel, Vassilis Chamanadaris,\nJoe Hora, and D.-C. Kim all kindly provided data in advance of\npublication. A series of exchanges with Dieter Lutz has help to\nclarify certain issues, as has a discussion with Gary\nNeugebauer, Keith Matthews and Eichi Egami.  Eric Becklin has\nprovided thoughtful comments at several stages of this work.\nMany members of the Institute for Astronomy have benefited this\nwork through discussion and sharing of work in progress.  A few\nare Gareth Wynn-Williams (dissertation committee chair), Dave Sanders, Bob Joseph, Klaus\nHoddap, Josh Barnes, Alan Tokunaga, Jeff Goldader, Jason\nSurace, and Masatoshi Imanishi; thanks are due to these and\nothers.  Louise Good proofread the manuscript prior to\nsubmission.  This work has been partially supported by NSF\ngrants ASTR-8919563, NASA grant NAGW-3938, and the Frank Orrall\nfund.  The NASA Extragalactic Database (NED) has been a\nconstant aid.  It is run by the Jet Propulsion Laboratory,\nCalifornia Institute of Technology, under contract with NASA.\nThis work has also made use of NASA's Astrophysics Data System\nAbstract Service.\n\n\\newpage\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro}\n\nUnderstanding the interplay of rare-earth local-moment magnetism\nand $3d$ transition metal itinerant magnetism is of fundamental\nphysical interest\\cite{Campbell72} and may help in the design of\nmore efficient permanent magnets.\nThe tetragonal $R$Fe$_4$Al$_8$ ($R=$rare earth) compounds\n(Fig.\\ \\ref{Fig1}) are well suited for studying this interplay\nbecause of simple symmetry conditions and because the interaction\nbetween the two magnetic sublattices is rather weak: Fe moments\nappear to order without a corresponding $R$ $4f$ order.\nConsequently, numerous studies have been performed on\n$R$Fe$_4$Al$_8$ (R148) over the last 30 years and interest in\nthese compounds has remained\nhigh.\\cite{Buschow78,Felner78,Fujiwara87,Duong01,Duong02,Langridge99,Paixao00,Paixao01,Schobinger98,Schobinger99}\n\n\\begin{figure}[tb]\n\\includegraphics[width=0.98\\linewidth]{pixGd148struc.eps}\n\\caption{(Color online) The crystal structure of $R$Fe$_4$Al$_8$.\nThin lines outline the tetragonal (I$4\/$mmm) unit cell, thick\nlines denote the Fe ``cage'' around the rare earth $R$. }\n\\label{Fig1}\n\\end{figure}\n\n\nNeutron scattering studies performed on R148 with various $R$\nindicate that Fe moments, between which the coupling is strongest,\norder between $100$ and $200\\, {\\rm K}$, generally in cycloidal\nstructures with propagation vectors parallel to [110] and moments\nconfined to the $a$$-$$b$\nplane.\\cite{Paixao00,Paixao01,Schobinger98,Schobinger99} For\nmagnetic rare earths $R$, ordering of the $R$ moments, with much\nweaker coupling and at much lower temperature $T$, has been\nreported. However, the critical temperature associated with this\nordering is very poorly defined. For example in Dy148 at\n$\\sim$$50\\,{\\rm K}$ the magnetic dc susceptibility rises faster\nthan expected for Curie-Weiss behavior, but there are no sharp\nfeatures.\\cite{Paixao00} Furthermore, the rare earth moments were\nfound to follow the Fe-moment\nmodulation.\\cite{Paixao00,Schobinger98} This seems to imply that\nthe Fe$-R$ moment interaction is stronger than the $R$$-$$R$\nmoment interaction.\n\nOf all rare earth elements, Gd has two highly interesting specific\nfeatures associated with its high-spin and zero-angular-momentum\n$4f$ state: 1) high spin results in strong magnetic coupling\nbetween the localized $4f$ and the conduction electrons, implying\nlarge magnetic interactions and the largest de Gennes factor of\nall rare earths; 2) zero angular momentum implies a spherical $4f$\ncharge-cloud and no magneto-crystalline anisotropy (MCA) resulting\nfrom crystal-electric-field effects $-$ the direct interplay of\nmagnetic interactions can be studied without\ncrystal-electric-field effects. In view of this, it is somewhat\nsurprising that GdFe$_4$Al$_8$ (Gd148) has been much less\nstudied\\cite{Fujiwara87,Duong01} than R148 with other rare earths.\nIn particular, apart from M{\\\"{o}}ssbauer\nspectroscopy,\\cite{Buschow78} there have been no microscopic\nstudies of Gd148.\n\nHere, we report on the flux-growth of single crystals of the\nGdFe$_4$Al$_8$ phase, and on the characterization of the crystals\nby magnetization, electrical transport, specific heat, and x-ray\ndiffraction. We also report on first synchrotron x-ray resonant\nmagnetic scattering (XRMS) data. We provide evidence for two\nconsecutive phase transitions at low temperature $T$ in addition\nto the N\\'{e}el transition at $155\\, {\\rm K}$, and show that the\ntwo, low-$T$, transitions are associated with the ordering of the\nGd $4f$ moments, resulting in complex magnetic structures\ninvolving both ferro- and antiferromagnetic components. In\ncontrast to other R148 compounds, the $4f$-moment ordering is\nassociated with a propagation vector that is distinct from that\nassociated with the Fe moment order, which at low $T$ has the same\nmodulation as in the vicinity of the {N{\\'{e}}el} temperature. We\nalso present the magnetic phase diagram for two field directions\nand discuss the implications of changes in the Fe stoichiometry.\n\nThe paper is organized into seven sections. In Sec.\\ \\ref{exp} we\ndescribe the flux growth of the single crystalline samples used in\nthe study, and the experimental procedures. In Sec.\\ \\ref{lowH}\nzero and low field electrical transport, magnetization, and\nspecific-heat data are presented, and we evaluate the influence of\nthe iron stoichiometry on the physical properties. In Sec.\\\n\\ref{highH} we present electrical transport and magnetization data\nmeasured in fields for two field directions, and in Sec.\\\n\\ref{xrms} the results of a first XRMS experiment are described.\nFinally, we discuss the low-temperature magnetic phase diagram and\npossible magnetic structures in Sec.\\ \\ref{disc}, before\nsummarizing our main conclusions in Sec.\\ \\ref{conc}.\n\n\\section{Experimental}\n\\label{exp}\n\nWhereas all previous single crystals of $R$Fe$_4$Al$_8$ material\nwere obtained using the Czochralsky-method, we have grown single\ncrystals of Gd148 (and other R148) with a self-flux\nmethod.\\cite{Sol1,Sol2,Sol3} One of the problems with studies on\nsingle crystals of R148 is a width of formation often observed,\ninvolving some Fe atoms occupying nominal Al positions and vice\nversa; this can lead to striking differences in the magnetic\nstructure and phase diagram as compared to the stoichiometric\nmaterial (see, e.g., Refs.\\ \\onlinecite{Paixao01,Waerenborgh00}).\nA flux-growth procedure may allow better control of the\nstoichiometry of the crystals by varying the starting composition.\nUnfortunately, information about the ternary phase diagram\n$R$-Fe-Al is very limited. We used differential thermal analysis\n(DTA; see Ref.\\ \\onlinecite{Janssen05} for a review of the use of\nDTA in the flux growth of crystals) to establish i) that\nGdFe$_4$Al$_8$ is congruently melting and ii) selected solidus\ntemperatures in the ternary around the GdFe$_4$Al$_8$\nstoichiometry. For the crystals used in this study, we selected\nstarting compositions between Gd148 and the also congruently\nmelting Fe$_2$Al$_5$ (Gd$_3$Fe$_{16}$Al$_{34}$ and\nGd$_2$Fe$_{14}$Al$_{31}$), which should not lead to large\nimbalances in the Fe to Al ratio.\n\nElements in ratios corresponding to the starting composition were\nfirst arc-melted together. The resulting ingot was placed in an\nalumina crucible, which was wrapped in Ta foil (to prevent any\nresidual oxygen in the Ar from reacting with the sample) and\nplaced in a vertical tube furnace in flowing Ar. Crystals were\ngrown by heating to $1475^{\\circ}{\\rm C}$ and then slowly\n($2^{\\circ}{\\rm C} \/ {\\rm h}$) cooling to $1180^{\\circ}{\\rm C}$,\nat which point the furnace was turned off. The flux was removed\nfrom the crystals in a second step by heating to $1200^{\\circ}{\\rm\nC}$, keeping the temperature for $30\\,{\\rm min}$ and then\ndecanting, following procedures described in Refs.\\\n\\onlinecite{Sol1,Sol2,Sol3,Janssen05}.\n\nFor both starting compositions, we obtained crystals of the Gd148\nphase, as identified via powder x-ray diffraction. Crystals\ntypically grow prismatically with (110) facets and the long\ndirection parallel to [001], as determined with Laue scattering.\nWe obtained crystals of dimensions up to about $10\\times 2\\times 2\n{\\rm mm}^3$. We determined, via single crystal\nx-ray-diffraction-structure refinement and electron-microprobe\nanalysis (employing single crystals of GdFe$_2$ as standards), the\ncrystals to be slightly iron deficient, but detected no Al on Fe\nsites or vice versa (measured compositions were between\nGdFe$_{3.88(5)}$Al$_8$ and GdFe$_{3.96(1)}$Al$_8$; deviations of\nthe Al stoichiometry from $8$ were always less than 2 standard\ndeviations and are not listed).\n\nWe note that the x-ray-diffraction-structure refinement cannot\ndistinguish between a Fe deficiency due to vacancies and a\n(larger) Fe deficiency due to Al on Fe sites, both scenarios can\ngive an equal electron density. However, assuming an Al\/Fe mixture\nto retain the full occupancy of the Fe site leads (for one of the\ninvestigated crystals) to a GdFe$_{3.92(1)}$Al$_{8.08(1)}$\ncomposition and, thus, to an excess of Al. Since the\nelectron-microprobe analysis results indicated no significant\nvariations in the $1\/8$ ratio of Gd to Al, we did the structure\nrefinements under the assumption that the electron density on the\nFe sites is caused by vacancies, although some Al\/Fe substitution\ncould never be excluded. Whatever scenario is chosen, none of our\nconclusions drawn in this article are affected by the issue of\nwhether or not some Al atoms are present on the Fe site, and for\ndefiniteness subsequently we will assume that the vacancy scenario\nis the correct one. The composition of our samples can thus be\ndescribed with the empirical formula GdFe$_{4-\\delta}$Al$_8$, with\n$\\delta \\approx 0.04-0.12$. Surface scans in the\nelectron-microprobe analysis gave no indications of compositional\nvariations within the same crystal.\n\nSince the exact iron stoichiometry varies slightly from crystal to\ncrystal (even within the same growth batch) and this was found to\nhave a significant influence on the magnetic properties (see Sec.\\\n\\ref{lowH} below) most of the measurements presented below were\nperformed on the same crystal, sample I, which has a refined\ncomposition GdFe$_{3.96(1)}$Al$_8$ ($a=8.7699(9)\\, $\\AA,\n$c=5.0440(6)\\, $\\AA). A bar cut from the crystal by a wire saw was\nconnected with contacts for 4-point electrical-transport\nmeasurements with the current density $j\\|$[110] (sample Ia).\nOther samples were connected with contacts for electrical\ntransport measurements with either $j\\|$[110] or $j\\|$[001]. On a\nsecond bar (sample Ib, which had a mass of $2.22\\,{\\rm mg}$) cut\nfrom crystal I we measured the longitudinal magnetization and the\nzero-field specific heat. In order to investigate the relation\nbetween compositional variations and variations in the transition\ntemperatures, we carried out additional magnetization measurements\non several other crystals, the composition of which was determined\nby x-ray diffraction and electron microprobe. The additional\ncrystals are labelled II to VII in the order of their appearance\nin the text. The above measurements were performed with commercial\n(Quantum Design) laboratory equipment. For the high-field\nmeasurements, the sample orientation was fixed with a\ntwo-component glue (X60 from {W{\\\"{a}}getechnik} GmbH, Darmstadt,\nGermany).\n\nSample Ib was also used in the XRMS experiment, performed on the\n6ID-B undulator beamline in the MUCAT sector at the Advanced\nPhoton Source, Argonne National Laboratory. The incident energy of\nx-rays was tuned to the Gd $L_{\\rm II}$ edge ($E=7.934\\,{\\rm\nkeV}$), using a liquid nitrogen cooled, double crystal Si (111)\nmonochromator and a bent mirror. Sample Ib was mounted on a copper\nrod on the cold finger of a closed cycle displex refrigerator,\nsuch that a natural (110) facet was exposed to the x-ray beam.\nThermal transfer was enhanced by embedding the sample in copper\npaste. The sample, oriented so that the scattering plane of the\nexperiment was coincident with the $a-b$ plane, was encapsulated\nin a Be dome with He exchange gas to further enhance thermal\ntransfer. The incident radiation was linearly polarized\nperpendicular to the scattering plane ($\\sigma$ polarized). In\nthis geometry, only the component of the magnetic moment that is\nin the scattering plane contributes to the resonant magnetic\nscattering arising from electric dipole transitions ($E1$) from\n$2p$ to $5d$ states. Furthermore, the dipole resonant magnetic\nscattering rotates the linear polarization into the scattering\nplane ($\\pi$ polarization). In contrast, charge scattering does\nnot change the polarization of the incident beam\n($\\sigma\\rightarrow\\sigma$ scattering). Pyrolytic graphite PG\n(006) was used as a polarization analyzer, selecting primarily\n$\\pi$ polarized radiation. For $E=7.934\\,{\\rm keV}$, the\npolarization analyzer used reduces the detected intensity\nresulting from $\\sigma\\rightarrow\\sigma$ charge scattering by\nabout $99.9\\%$, whereas the $\\sigma\\rightarrow\\pi$ resonant\nmagnetic scattering is passed with little loss. Thus, the\npolarization analysis suppresses the charge scattering relative to\nthe magnetic scattering signal.\n\n\n\\label{res}\n\\section{Low field measurements and influence of Fe content}\n\\label{lowH}\n\\begin{figure}[tb]\n\\includegraphics[width=0.98\\linewidth]{xlHTP.eps}\n\\caption{(Color online) Zero-field resistivity $\\rho$ (along\n[110], sample Ia) and inverse dc susceptibility $\\chi ^{-1}$ vs\ntemperature $T$ (sample Ib). } \\label{Fig2}\n\\end{figure}\n\nIn this section, we will first present zero-field transport and\nlow field magnetization measurements on our main crystal (samples\nIa\/Ib). Then, we will briefly discuss sample-to-sample differences\nin the characteristic temperatures, which we think are related to\nFe deficiency. Finally, we will present specificheat measurements\non sample Ib indicating that the Gd $4f$ moments are ordered only\nat low temperatures. The main implications of the measurements\nwill be discussed later in Sec.\\ \\ref{disc}.\n\nIn Fig.\\ \\ref{Fig2}, we show, for sample Ia, the $T$ dependence of\nthe resistivity $\\rho$ ($j\\|$[110]). Since the sample is rather\nsmall, the estimated uncertainty in the absolute value of the\nresistivity is about 15\\%. The antiferromagnetic ordering at $T_N\n\\simeq 155\\, {\\rm K}$ is visible as a sharp kink in $\\rho\n(T)$.\\cite{note_Palasyuk04} The inverse dc magnetic susceptibility\n$\\chi^{-1}$ ($M\/H$) of sample Ib, cut from the same crystal as\nsample Ia, is also presented in Fig.\\ \\ref{Fig2}. The dashed and\ndotted curves shown are determined from the magnetization $M(T)$\nin a field of $\\mu_0 \\,H=0.1\\,{\\rm T}$ applied parallel to [110]\nand parallel to [001]. Below this field $M$ vs $H$ is linear for\n$T \\gtrsim 40\\, {\\rm K}$. Since $\\chi$ is slightly anisotropic, we\nalso calculated the polycrystalline average $\\chi_{\\rm\npoly}=(2\\chi_{110}+\\chi_{001})\/3$, shown as squares. In\n$\\chi^{-1}$ for both directions the antiferromagnetic ordering is\nmanifest by a change in slope, but it is much less sharp than the\ncorresponding signature in the resistivity. A weak signature of\nthe {N{\\'{e}}el} transition may be expected if, as we will discuss\nbelow, only the Fe moments order at $T_N$. In agreement with\nmeasurements on polycrystalline samples,\\cite{Buschow78} $\\chi$\nfollows Curie-Weiss behavior both above and, over a limited $T$\nrange, below $T_N$. We found (on the calculated polycrystalline\naverage) above $T_N$ a Weiss temperature of $-165\\,{\\rm K}$ and an\neffective moment of $11.4 \\, \\mu_B \\, \/ {\\rm f.u.}$. Assuming\ncontributions by the magnetic atoms that are additive in the Curie\nconstant and a contribution by Gd of $7.9 \\, \\mu_B \\, \/ {\\rm Gd}$,\nthe free moment per Fe atom in the paramagnetic state is $4.1 \\,\n\\mu_B \\, \/ {\\rm Fe}$. Below $T_N$ (range $70-115\\,{\\rm K}$) we\nfound a Weiss temperature of $17\\,{\\rm K}$ and an effective moment\nof $7.4 \\, \\mu_B \\, \/ {\\rm f.u.}$. The latter is closer to the Gd\nfree-ion value ($7.9 \\, \\mu_B \\, \/ {\\rm f.u.}$) than the one\nreported by Buschow {\\em et al.}\\cite{Buschow78} ($6.2 \\, \\mu_B \\,\n\/ {\\rm f.u.}$).\n\n\\begin{figure}[t!b]\n\\includegraphics[width=0.98\\linewidth]{xlRcomps.eps}\n\\caption{(Color online) Low-temperature normalized resistance\n(panel a) and derivative of high temperature resistance (panel b)\nvs temperature $T$ of various crystals: Ia (thick line) and II\n(dotted line) for $j\\|$[110], III (open circles) and IV (full\ncircles) for $j\\|$[001].} \\label{Fig3}\n\\end{figure}\n\nThe resistivity curve shows additional features just below $30\\,\n{\\rm K}$, magnified in Fig.\\ \\ref{Fig3}a) (thick line): The broad\npeak in $\\rho (T)$ at $\\sim \\! \\! 28\\, {\\rm K}$ with a drop at\n$\\sim \\! \\! 26.5\\,{\\rm K}$ ($T_2$) indicates a phase transition.\nFurthermore, with decreasing $T$, at $\\sim \\! \\! 21\\, {\\rm K}$\n($T_1$) there is a sudden increase in $\\rho$ by $\\sim 10\\%$. This\nfeature is hysteretic in temperature, suggesting that the feature\nis associated with a first-order transition.\\cite{note_Fujiwara}\nNote that the hysteresis visible at $T_2$ is opposite to what is\nexpected for a first-order transition (i.e.,\nsuperheating\/supercooling). It is most likely a remnant of the\nhysteresis associated with the $T_1$ transition. At $T_1$, the\nform of the hysteresis is consistent with\nsuperheating\/supercooling.\n\nAlso shown in Fig.\\ \\ref{Fig3}a) are low-$T$ zero-field\nresistivity curves for three additional samples, one also with\n$j\\|$[110] and two with $j\\|$[001]. Whereas the feature of a\nsudden increase of $\\rho$ with decreasing $T$ is visible in all\ncurves, the other feature at slightly higher temperatures is only\nidentifiable when $j\\|$[110]. Based on the results from sample Ia\nonly, it may be tempting to associate the increase in $\\rho$ (with\ndecreasing $T$) with the opening of a superzone\ngap.\\cite{Mackintosh62} However, the presence of this increase\nwith the same order of magnitude for both $j\\|$[110] and\n$j\\|$[001] makes a superzone gap scenario less likely.\n\nIt is also clear from Fig.\\ \\ref{Fig3}a) that there are\nsignificant variations of the temperatures $T_1$ corresponding to\nthis feature. For comparison, Fig.\\ \\ref{Fig3}b) shows the\nresistivity derivative ${\\rm d} \\rho \/ {\\rm d} T$ for the same\nsamples at higher $T$, in order to make the {N{\\'{e}}el}\ntransition more visible, which manifests itself differently for\nthe two current density directions. The curves indicate a sharp,\nwell defined $T_N$ for each sample and comparing Figs.\\\n\\ref{Fig3}a) and \\ref{Fig3}b) suggests a correlation between\nhigher $T_1$ and lower $T_N$.\n\n\n\\begin{figure}[tb]\n\\includegraphics[width=0.98\\linewidth]{xlMTmicroprobe.eps}\n\\caption{(Color online) Low field ($H\\|[001]$) Magnetization of\nvarious crystals with Fe stoichiometry (as described in Sec.\\\n\\ref{exp}) determined by electron microprobe or\nx-ray-diffraction-structure refinement.} \\label{Fig4}\n\\end{figure}\n\nFigure \\ref{Fig4} displays low field ($H\\|$[001]) magnetization\nmeasurements performed on three samples with different Fe\nstoichiometry as determined by x-ray diffraction and electron\nmicroprobe: samples Ib (GdFe$_{3.96(1)}$Al$_8$, thick line), V\n(GdFe$_{3.94(6)}$Al$_8$, full squares), and VI\n(GdFe$_{3.88(5)}$Al$_8$, dashed line). The data were taken for\nboth decreasing and increasing $T$ (field cooled), and\nnormalized\\cite{note_normalizedM} to the maximum in magnetization.\nA comparison with Fig.\\ \\ref{Fig3}a) indicates that the rise in\n$M$ with $T$ is associated with $T_1$, the subsequent decrease\nwith $T_2$. A decrease in the Fe content seems to systematically\nshift $T_1$ and $T_2$ to higher temperatures. We note that for\ncrystal I $M(T)$ (Fig.\\ \\ref{Fig4}) indicates a higher $T_2$ than\n$\\rho (T)$ [Fig.\\ \\ref{Fig3}a)]. In principle, this might be due\nto a different Fe content of the two pieces (Ia and Ib) cut from\nthe crystal and used for these measurements. However, we do not\nbelieve that this is the case, because the specific-heat peak\nassociated with $T_2$ (see below; measured on Ib) corresponds well\nto $T_2$ deduced from $\\rho (T)$ on sample Ia.\n\nCombining the above data it seems that decreasing the Fe content\nresults in a decrease of $T_N$ and an increase of $T_1$ and $T_2$.\nIn order to confirm this, a study involving samples that are much\nmore Fe deficient, would be highly desirable.\n\n\\begin{figure}[tb]\n\\includegraphics[width=0.98\\linewidth]{nxlHCalt.eps}\n\\caption{(Color online) %\nSpecific heat $C_p$ measured on sample Ib in zero field\n($\\blacksquare$) and Gd $4f$ moment contribution to the specific\nheat ($\\circ$), obtained by subtracting the specific heat of\nYFe$_4$Al$_8$ (full line, from Ref.\\ \\onlinecite{Hagmusa98}).\nInset: Entropy of the Gd $4f$ moments obtained by integrating the\n$4f$ moment contribution to $C_p \/ T$. The two lines shown were\ncalculated using two extrapolations of the specific heat to $T=0$,\nindicated by dashed lines in the main panel (see text for details\nof the extrapolations). The dotted line in the inset indicates the\nfull Gd $4f$ entropy.} \\label{Fig5}\n\\end{figure}\n\n\nFigure \\ref{Fig5} shows the measured specific heat $C_p$ of sample\nIb (closed squares) after subtraction of the addenda contribution\n(the specific heat of sample platform and grease measured before\nmounting the sample). Also shown is the mass-scaled specific heat\nof YFe$_4$Al$_8$ from Ref.\\ \\onlinecite{Hagmusa98} (full line),\nand the difference between the two specific heats (open circles).\nSince the necessary mass-scaling is small, the remaining specific\nheat is close to the magnetic contribution associated with the Gd\nsublattice. The broad, asymmetric, peak at $\\sim \\! \\! 26.5 \\,\n{\\rm K}$ corresponds to the drop in $\\rho$ at $T_2$ [determined on\nsample Ia cut from the same crystal as sample Ib, see Fig.\\\n\\ref{Fig3}a) dashed curve]. The broadness of the specific-heat\npeak may suggest a degree of inhomogeneity of the Fe distribution\nin the sample, although the electron-microprobe analysis provided\nno indications of inhomogeneities within crystals.\n\n\nNo feature in the specific heat is visible around $T_1$, the small\ndiscontinuity at $20\\,{\\rm K}$ indicated by the data was found to\nbe an artifact due to the change of the heat pulse intensity\napplied by the system. We checked the specific-heat raw data, and\ncan exclude any latent heat restricted to a temperature region\nsmaller than the spacing of measurement points. We conclude that\nthe latent heat of the $T_1$ transition has to be small, i.e., the\nphases above and below $T_1$ have similar entropies. The\nalternative explanation of the specific heat, that $T_1$ is not a\nfirst-order transition, seems unlikely in light of not only the\npresence, but also the shape of thermal hysteresis observed in\n$\\rho (T)$, which is particularly clear when a magnetic field is\napplied in the [110] direction (see Sec.\\ \\ref{highH} below).\n\nThe corresponding magnetic entropy $S$, obtained by integration,\nis shown in the inset of Fig.\\ \\ref{Fig5}. The two curves shown\nare calculated with two different extrapolations of $C_p$ to zero\ntemperature (dashed lines in the main panel). One extrapolation is\nobtained by connecting the $C_p$ data point at the lowest $T$\nlinearly to $C_p=0$ at zero temperature, the other by connecting\nthe $C_p\/T$ data at the lowest $T$ linearly to $C_p\/T=0$ at zero\ntemperature. Given typically observed specific heats of Gd\ncompounds,\\cite{note_DuongGdFe2Ge2} the actual specific heat at\nlow temperature will most likely be between these two\nextrapolations, and thus the two curves shown in the inset of\nFig.\\ \\ref{Fig5} may be considered lower and upper limits of the\nGd $4f$ entropy (neglecting additional experimental\nuncertainties). At $30\\, {\\rm K}$ the magnetic entropy already\nreaches 80\\% (78\\% and 86\\% for the two low-$T$ extrapolations\nmade) of the full entropy of Gd $4f$ moments (${\\rm R}\\,\\ln 8$,\ndashed line). Above $45\\, {\\rm K}$ it hardly varies anymore,\nhaving reached $>90\\%$ and $>97\\%$ of the full $4f$ entropy for\nthe two low-$T$ extrapolations made. This strongly suggests that\nthe Gd $4f$ moments are not ordered above $T_2$, and the\ntransition at $T_2$ corresponds to the ordering of the $4f$\nmoments.\n\nBy careful measurement and analysis of the low-field thermodynamic\nand transport data of solution grown single crystals of Gd148 we\nhave shown that there are sharp features associated with two\nmagnetic phase transitions. These transitions appear to be\nprimarily related to the ordering of the Gd sublattice. In\naddition we have been able to establish a clear correlation\nbetween small variations in the Fe stoichiometry and $T_N$, $T_1$,\nand $T_2$. Having done this, we will now focus on the sample that\nis closest to full stoichiometry (crystal I) and try to more fully\ndelineate and understand the field and temperature dependence and\nthe structure of the magnetic phases in this sample.\n\n\n\\section{High-field measurements}\n\\label{highH}\n\nApplying a magnetic field often helps to clarify the nature of\nmagnetic phases observed in zero (or low) field measurements. We\ntherefore measured the field dependent electrical resistivity and\nthe magnetization on several samples. Shown in this section are\nelectrical transport (subsection \\ref{highHrho}) and magnetization\n(subsection \\ref{highHM}) data measured on samples Ia\/Ib at low\ntemperatures in fields applied in-plane and out-of-plane. The\nfeatures in resistivity corresponding to the two low-temperature\ntransitions are much sharper when an in-plane field is applied and\nobserved features suggest complex domain effects. The\nmagnetization data are used to estimate the in-plane and\nout-of-plane components of the spontaneous ferromagnetic moment.\nThe phase diagrams for $H\\|$[110] and $H\\|$[001], which can be\nconstructed from these transport and magnetization data, will be\npresented and discussed in Sec.\\ \\ref{disc}.\n\n\\subsection{Electrical transport}\n\\label{highHrho}\n\\begin{figure}[tb]\n\\includegraphics[width=0.98\\linewidth]{xldomains.eps}\n\\caption{(Color online) resistivity $\\rho$ vs $T$ in fields of\n$3.5\\,{\\rm T}$ applied in-plane parallel or perpendicular to the\ncurrent density $j\\|[110]$ (sample Ia). Arrows indicate the $T$\ndirection in which the measurements were performed. The open\ncircle indicates the value of the $T_1$ transition in $3.5\\,{\\rm\nT}$, averaged for $T$ increasing and decreasing.} \\label{Fig6}\n\\end{figure}\n\nFigure \\ref{Fig6} shows the resistivity $\\rho$ ($j\\|$[110]) of\nsample Ia when a field of $3.5\\,{\\rm T}$ is applied either $\\|j$\n(longitudinal resistivity) or $\\|[1\\overline{1}0]$ (i.e.\\\ntransverse resistivity, $\\perp$$j$, but in a crystallographically\nequivalent in-plane direction). For both field directions, the\nmagnitude of the jump at $T_1$ is enhanced and the hysteresis in\ntemperature greatly increased, making the first-order nature of\nthe transition more apparent. The average\n($T\\uparrow$,$T\\downarrow$) temperature of the $T_1$ feature,\nhowever, is only weakly influenced by an in-plane field. In\ncontrast to this, the $T_2$ feature is shifted to higher $T$ by\n$H\\|$[110], suggesting a ferromagnetic nature of the phase at\n$T<T_2$. Furthermore, the fields remove the peak in $\\rho$ just\nabove the drop at $T_2$, and cause the feature to exhibit\nhysteresis. This different response to $H\\|$[110] confirms that\nthere are indeed two independent magnetic transitions $T_1$ and\n$T_2$, dividing the ordered state below $T_N$ into three magnetic\nphases, a low-temperature phase (LTP), an intermediate-temperature\nphase (ITP), and a high-temperature phase (HTP).\n\n\n\\begin{figure}[tb]\n\\includegraphics[width=0.98\\linewidth]{xlRTgraph.eps}\n\\caption{(Color online) a) Resistivity $\\rho$ vs $T$ (sample Ia,\n$j\\|$[110], $T$ increasing) in fields applied parallel to\n$j\\|$[110]. From top to bottom 0 (thick line), 0.1, 0.2, 0.4, 0.5,\n0.8, 1 (thick line), 1.5, 2, 2.5, $3.5\\,{\\rm T}$. b) As panel a),\nbut in fields applied parallel to [001] ($j\\|$[110]). From top to\nbottom 0 (thick line), 0.1, 0.2, 0.3, 0.4, 0.5, 0.7, 0.8, 0.9, 1,\n2, $5.5\\,{\\rm T}$.} \\label{Fig7}\n\\end{figure}\n\nWhereas there are similarities in $\\rho (T)$ for the two field\ndirections considered, there are also very pronounced differences,\nparticularly for the $T_1$ feature, below which the field cooled\n$\\rho(T)$ is much higher for $H$$\\perp$$j$ than for $H\\|j$. A\ntransverse magnetoresistance higher than the longitudinal one\nmight arise from a bending of the electron trajectory by the\nLorentz force (see, e.g., Ref.\\ \\onlinecite{Harris95}). However,\nthe zero-field-cooled $\\rho (T)$ below $33\\,{\\rm K}$ (black dotted\nline) is the same (within error bars) for $H\\|j$ and $H$$\\perp j$.\nIndeed, as we will see (Fig.\\ \\ref{Fig8} below), at $5\\,{\\rm K}$\n{\\em after cooling in zero field} the resistivity is only very\nweakly field dependent for an in-plane field (both for transverse\nand longitudinal configurations).\n\n\\begin{figure}[tb]\n\\includegraphics[width=0.98\\linewidth]{nxlRHgraph.eps}\n\\caption{(Color online) Resistivity vs $H\\|j\\|$[110] at selected\n$T$ (sample Ia).} \\label{Fig8}\n\\end{figure}\n\nThis indicates that the difference in the $\\rho (T)$ curves of\nFig.\\ \\ref{Fig6} originates from the magnetism. The\nmagnetoresistance then may arise either from spin-disorder\nscattering or scattering of domain walls\\cite{Singh76} and the\ndifference can be due to a different configuration of domains. In\nprinciple, this configuration (domain population, average size of\ndomains, etc.) may be different for $H\\|[1\\overline{1}0]$\n(transverse magnetoresistance) and $H\\|[110]$ (longitudinal\nmagnetoresistance) due to different demagnetizing effects (the\nsample was cut in a rod such that the [110] direction is a bout\nthree times longer than the $[1\\overline{1}0]$ direction). In any\ncase, the fact that the resistivity is almost field-independent\n(for an in-plane field) at $5\\,{\\rm K}$ indicates that the\nrelevant (for resistivity) domain configuration in the LTP is very\nhard to change by applying a magnetic field, whereas a field\napplied in the ITP prepares a certain LTP domain configuration\nupon cooling below $T_1$. We note that application of a field in\nthe ITP also seems to lead to small, but systematic, changes in\nthe resistivity values in the HTP, persisting up to $\\sim$$T_N$.\n\nFigure \\ref{Fig7}a) shows the evolution of $\\rho (T)$ [for clarity\nonly the data measured upon warming, after the lowest temperature\nhad been reached by cooling with the same field applied\n(field-cooled-warming measurement protocol) are displayed] with\nincreasing $H\\|j$ (again sample Ia, $j\\|$[110]). It seems clear\nthat the $T_2$ transition systematically shifts to higher\ntemperatures with increasing $H$. The $T_1$ transition is far less\nfield dependent (for $H\\|$[110]), and as discussed above, quite\nhysteretic. Furthermore, the resistivity in the ITP becomes lower\nin higher $H$, developing into a well-defined weakly\ntemperature-dependent valley above $\\sim$$2\\,{\\rm T}$. This\ndevelopment is also visible in the $\\rho (H)$ dependence at\nconstant $T$ (Fig.\\ \\ref{Fig8}). There is only a minimal field\ndependence of $\\rho$ at $5\\,{\\rm K}$ in the LTP. In contrast to\nthis, a pronounced drop in low fields at $25\\,{\\rm K}$ in the ITP\nis shifted to higher fields by increasing $T$ and seems to\ntransform into the drop associated with the ITP-HTP transition. In\nthe $\\rho (T)$ curves of Fig.\\ \\ref{Fig7}a), for intermediate\nfield ranges, there are indications for an additional feature in\nthe ITP-HTP transition. The details of this feature have to be\nclarified in future work.\n\nThe evolution of $\\rho (T)$ with increasing $H\\|$[001] for the\nsame sample and current density direction is shown in Fig.\\\n\\ref{Fig7}b). The data shown were again obtained using the\nfield-cooled-warming measurement protocol, but we note that\napplication of $H\\|$[001] (unlike $H\\|$[110]) quickly suppresses\nany hysteresis, and broadens the transitions. Apart from a\nbroadening, the $T_2$ transition does not seem to be strongly\naffected. In contrast to this, $H\\|$[001] systematically shifts\nthe $T_1$ transition to lower temperatures and above\n$\\sim$$1\\,{\\rm T}$ there are no longer any indications for a\ntransition to the LTP. This indicates that the LTP is suppressed\nby $H\\|$[001].\n\n\\subsection{Magnetization}\n\\label{highHM}\n\n\n\\begin{figure}[tb]\n\\includegraphics[width=0.98\\linewidth]{xlMH.eps}\n\\caption{(Color online) Magnetization $M$ as a function of field\n$H$ at selected $T$, $H\\|[110]$ (blue) and $H\\|[001]$ (red),\nsample Ib. Thin arrows indicate the $H$ direction of the\nmeasurements, thick arrows indicate the criterion taken for the\nestimation of the spontaneous ferromagnetic moment.} \\label{Fig9}\n\\end{figure}\n\n\nIn order to investigate the magnetic nature of these phases, we\nmeasured the magnetization $M$ as a function of $T$ and $H$\n($\\|$[110] and $\\|$[001]). In Fig.\\ \\ref{Fig9}, we display for\nsample Ib $M(H)$ loops at selected $T$. For $H\\|$[110], at\n$40\\,{\\rm K}$, $M(H)$ shows almost linear behavior, but at\n$2\\,{\\rm K}$ $M(H)$ has a behavior typical for soft ferromagnets:\nsaturation in a field which is of the order of the estimated\ndemagnetizing field, but no hysteresis or remanence. However, the\nsaturation value of $4.5 \\mu_B\/{\\rm f.u.}$ at $2\\, {\\rm K}$ is\nmuch lower than the full Gd moment of $7 \\mu_B\/{\\rm f.u.}$. At\n$26\\,{\\rm K}$, pronounced hysteresis is visible in low $H$. This\nis ubiquitous for $M(H)$ in the ITP and correlates with the\nhysteresis in the drop in $\\rho$ visible in Fig.\\ \\ref{Fig8}a).\nThe hysteresis may suggest a first-order metamagnetic transition.\nIn higher fields $M(H)$ is nearly constant, resembling the\nbehavior at $2\\,{\\rm K}$. In low fields $M(H)$ indicates a\nspontaneous ferromagnetic moment, unlike at $2\\,{\\rm K}$, at\n$26\\,{\\rm K}$ we did observe remanence. The $T$ dependence of the\nspontaneous ferromagnetic moment, as estimated from Fig.\\\n\\ref{Fig9} taking into account demagnetizing effects (we took the\nmagnetization at the first kink in low $H$, indicated by thick\narrows in Fig.\\ \\ref{Fig9}), is shown in Fig.\\ \\ref{Fig9N} (black\nsquares). The LTP-ITP transition in $M(H)$, $H\\|$[110] is\naccompanied by a drop of the spontaneous moment to $\\sim \\! \\! 1\\,\n\\mu_B$, and by the appearance of hysteresis.\n\nFor $H\\|$[001] at $26\\,{\\rm K}$ (ITP), the form of $M(H)$ (Fig.\\\n\\ref{Fig9} red curves) is consistent with a spontaneous\nferromagnetic moment of about $2\\, \\mu_B$. For the LTP consider\n$M(H)$ at $2\\,{\\rm K}$. Below $0.5\\, {\\rm T}$, $M(H)$ is linear,\nabove it rises more strongly before slowly starting to saturate.\nThe field $H_1$ where $M(H)$ starts to increase faster than linear\nis lower at higher $T$, extrapolating to zero at $T_1$. For\nsamples with a higher $T_1$, $H_1$ at low temperatures is\nproportionally higher (not shown). Whereas the feature at $H_1$ as\ndefined above is rather sharp, the slow saturation of $M$ above\n$H_1$ may be due to a small inhomogeneity of the Fe content in the\nsample, as discussed in Sec.\\ \\ref{lowH}.\n\n\\begin{figure}[tb]\n\\includegraphics[width=0.98\\linewidth]{xlMsat.eps}\n\\caption{(Color online) Spontaneous ferromagnetic moment $M_s$,\nestimated from the $M(H)$ curves in Fig.\\ \\ref{Fig9} vs\ntemperature $T$, $H\\|[110]$ (black squares) and $H\\|[001]$ (red\ncircles), sample Ib. The square root of the sum of the squares for\nboth field directions is shown as open triangles.} \\label{Fig9N}\n\\end{figure}\n\n\nFor $H\\|$[001], the saturation value of $M \\lesssim 5\\, \\mu_B\/{\\rm\nf.u.}$ is higher than for $H\\|$[110], but still far below the full\nGd $4f$ moment. Note that the difference to $7\\, \\mu_B\/{\\rm f.u.}$\nis much too large to be explained by a negative contribution of\nthe Gd conduction electrons. The low saturated moment implies that\nthe magnetization does not correspond to the magnetization of a\nsimple ferromagnet. For $H\\|$[001], the temperature dependence of\nthe estimated spontaneous ferromagnetic moment is shown in Fig.\\\n\\ref{Fig9N} as red circles. It is similar to $M(T)$ in low $H$\n(c.f.\\ Fig.\\ \\ref{Fig4}).\n\nThe linear $M(H)$ behavior (below $0.5 \\, {\\rm T}$ at $2\\,{\\rm\nK}$) may be due in-plane ferromagnetic moments resisting a\nreorientation by an out-of-plane field. The LTP-ITP transition\ncould correspond to a reorientation of ferromagnetically aligned\n$4f$ moments (as suggested in Ref.\\ \\onlinecite{Fujiwara87}).\nFigure \\ref{Fig9N} indicates that in the LTP the ferromagnetically\naligned moments are confined to the $a$$-$$b$ plane, whereas in\nthe ITP the moments are mainly out-of-plane, pointing into a\ndirection with an angle of about $25^{\\circ}$ to [001]. A crude\nestimate of the magnitude of the spontaneous moment is given by\n$(M_{s,110}^2 + M_{s,001}^2)^{1\/2}$, shown in Fig.\\ \\ref{Fig9N} as\nopen triangles. Apart from one point near the LTP-ITP transition\n(where the estimation of $M_s$ is less clear than in other\nregions) the temperature dependence of $(M_{s,110}^2 +\nM_{s,001}^2)^{1\/2}$ is consistent with the LTP-ITP transition not\naffecting the magnitude of the ferromagnetic component, but only\nit's direction. Finally, we note that, like the low-field\nmagnetization (Fig.\\ \\ref{Fig4} full line), the estimated\nspontaneous moment extrapolates to zero around $30-35\\,{\\rm K}$,\ni.e., slightly higher than $T_2$ as indicated by resistivity and\nthe specific heat maximum. This roughly correlates with the {\\em\nonset} of the $4f$ contribution to the specific heat (Fig.\\\n\\ref{Fig5}), which is not sharp at all. We therefore think it is\nnot an indication of an additional transition, rather it may be\ndue to an inhomogeneous Fe distribution or due to significant\nprecursor-effects (e.g.\\ fluctuations).\n\nThe electrical-transport and magnetization data presented in this\nsection will allow us to construct $H$$-$$T$ phase diagrams for\n$H\\|$[110] and $H\\|$[001]. We will do this in Sec.\\ \\ref{disc}.\nFurthermore, we have established that the Gd $4f$ moment order\nsetting in at $T_2$ has a large ferromagnetic component, and the\nfirst-order (see Fig.\\ \\ref{Fig6}) transition at $T_1$ involves a\nreorientation of the ferromagnetic component into the $a$$-$$b$\nplane for $T<T_1$. In order to elucidate the underlying magnetic\nstructure we will now focus on scattering data.\n\n\\section{X-ray magnetic resonant scattering}\n\\label{xrms}\n\nIn order to prove the long-range nature of the magnetic order, and\nto learn more about the magnetic structure than the information\nthat is obtainable from thermodynamic and transport measurements,\nscattering techniques can be applied, typically neutron\nscattering. For Gd148, neutron scattering is less feasible because\nof the large neutron absorption cross-section of the natural\nisotope of Gd. An alternative technique, which relies on intense\nsynchrotron radiation and exploits a large increase in the\nscattered intensity due to resonant processes when the energy of\nthe radiation is close to an absorption edge of a magnetically\nordered element, is x-ray resonant magnetic scattering\n(XRMS).\\cite{Gibbs88,Hannon88,XRMSreview} To gain further insight\ninto the magnetism of Gd148, we therefore performed a first XRMS\nexperiment on the (together with piece Ia from the same crystal)\nbest characterized sample Ib.\n\nBased on the neutron scattering and XRMS results on R148 with\nother rare\nearths,\\cite{Langridge99,Paixao00,Paixao01,Schobinger98,Schobinger99}\nas well as on the thermodynamic and transport results presented in\nearlier sections, we expected at temperatures between $T_2$ and\n$T_N$ a modulation of Fe moments with propagation along the [110]\ndirection to be present, with a feasible resonant enhancement of\nthe intensity scattered at the corresponding satellite reflections\nat the Gd $L_{\\rm II}$ edge. Figure \\ref{Fig11}a) shows reciprocal\nspace scans along [$h$,$h$,$0$] with $\\sigma \\rightarrow \\pi$\npolarization analysis and the energy tuned to the Gd $L_{\\rm II}$\nedge ($7.934\\,{\\rm keV}$) at $37$ and $160\\,{\\rm K}$. At $37\\,{\\rm\nK}$, the charge reflection at (4,4,0) is accompanied by small\nsatellites at ($4\\pm \\tau$,$4\\pm \\tau$,$0$) with $\\tau=0.086$. The\nsatellite reflections are not present at $160\\,{\\rm K}$,\nindicating that they are due to the magnetic ordering at the\n{N{\\'{e}}el} temperature (c.f.\\ Fig.\\ \\ref{Fig2}). We also found\nsatellite reflections at ($3\\pm \\tau$,$3\\pm \\tau$,$0$), ($2\\pm\n\\tau$,$2\\pm \\tau$,$0$), and ($1\\pm \\tau$,$1\\pm \\tau$,$0$), with\n$0.06\\leq \\tau \\leq 0.135$, depending on $T$. The full width at\nhalf maximum (FWHM) of the satellite reflections is approximately\ntwice the FWHM of the charge reflections. The corresponding\nsmaller correlation length of the magnetic order may be an\nindication of the presence of small domains. We roughly estimate\nthe average size of the domains (in [110] direction) to be\n$\\sim$$0.17\\,\\mu{\\rm m}$ or about $140$ unit cells. Rearrangements\nof these domains might be the cause of the small changes in\nresistivity values in the HTP (see the discussion of Fig.\\\n\\ref{Fig6} in Sec.\\ \\ref{highHrho}).\n\n\\begin{figure}[t!b]\n\\includegraphics[width=0.98\\linewidth]{xlnewXRMS.eps}\n\\caption{(Color online) a) Reciprocal space scans along\n[$h$,$h$,0] at $37$ and $160\\,{\\rm K}$ (sample Ib, Gd $L_{\\rm II}$\nedge). The data in the range $h=3.96$ to $4.04$ have been divided\nby a factor of $10000$.\nb) Energy scans of the intensity of the ($4-\\tau$,$4-\\tau$,$0$)\nsatellite reflections at $9$, $37$, and $130\\,{\\rm K}$. Note that\nintensities in panel a) are not comparable to intensities in panel\nb).} \\label{Fig11}\n\\end{figure}\n\n\nThe intensity of the ($4\\pm \\tau$,$4\\pm \\tau$,$0$) satellite\nreflections is about four orders of magnitude lower than the\nintensity of the (4,4,0) charge reflection, despite the use of the\npolarization analysis, which suppresses the charge reflection\nintensity by three orders of magnitude. Due to the small intensity\nof the satellite reflections and their vicinity to the charge\nreflections, there is considerable intensity due to the charge\ntail at the position of the satellite reflections. A slight\nasymmetry in the charge tail leads to a larger charge contribution\nto the scattered intensity at the ($4+\\tau$,$4+\\tau$,$0$)\nreflection, and we therefore further analyzed only the\n($4-\\tau$,$4-\\tau$,$0$) satellite reflection.\n\n\n\nFigure \\ref{Fig11}b) shows, at selected $T$, energy scans over the\nGd $L_{\\rm II}$ edge for the satellite reflections at ($4-\\tau ,\n4-\\tau , 0$). The intensity of the resonance is largest at\n$37\\,{\\rm K}$. The lower intensity at $130\\,{\\rm K}$ is due to the\nvicinity of the {N{\\'{e}}el} transition, the lower intensity at\n$9\\,{\\rm K}$ will be discussed below in Sec.\\ \\ref{disc}. The\nposition and the (amplitude-normalized) peak shape of the\nresonance is $T$ independent within our resolution and consistent\nwith dipolar ($E1$)\ntransitions.\\cite{Langridge99,Gibbs88,Hannon88,XRMSreview} We note\nthat a resonant enhancement at the Gd $L_{\\rm II}$ edge was\nobserved only at the ($4\\pm\\tau , 4\\pm\\tau , 0$) satellite\nreflections, there was no enhancement at the (4,4,0) charge\nreflection, consistent with resonant magnetic scattering.\nReciprocal space scans and the resonant enhancement at the Gd\n$L_{\\rm II}$ edge are consistent with a magnetic modulation with\npropagation vector ($\\tau$,$\\tau$,0) associated with Gd.\n\nThe specific-heat measurement discussed in Sec.\\ \\ref{lowH}\nimplies that in the HTP the Gd $4f$ moments are not ordered. Since\nXRMS is element-sensitive, this might seem in contradiction to our\nobservation of a resonant magnetic signal at the Gd $L_{\\rm II}$\nedge at $37$ and $130\\, {\\rm K}$ (Fig.\\ \\ref{Fig11}). However,\nXRMS ($E1$) of the Gd $L$ edges is sensitive to the polarization\nof the Gd $5d$ bands, not (directly) to the $4f$\nmoments.\\cite{Langridge99,Gibbs88,Hannon88,XRMSreview} A Gd $5d$\nband polarization can also be induced by the Fe $3d$ moments. For\nthis situation, a very weak intensity of the corresponding\nmagnetic satellites is expected.\\cite{Everitt95} Compared to other\nGd compounds,\\cite{Granado04,Good05,Tan05,Kim05branch} the\nintensity of the satellite reflections we observed is indeed very\nweak $-$ more than three orders of magnitude lower. This low\nintensity might be due to the ordered moments pointing into a\ndirection close to [001], since in our scattering geometry we are\nnot sensitive to moments in [001] direction. However, changing the\norientation of the sample such that [001] is in the scattering\nplane\\cite{Langridge99,Tan05} did not significantly change the\nintensity, suggesting that its low value is not related to the\ndirection of magnetic moments. The behavior is similar to the\nsituation in Dy148 at high temperatures.\\cite{Langridge99}\n\nAt this point it may be useful to briefly comment on an XRMS\nsignal coming directly from the Fe moments. This would of course\nbe very important to confirm the above considerations.\nUnfortunately, the energies of the $L$ absorption edges of $3d$\ntransition metals do not allow diffraction experiments for regular\ncrystalline materials.\\cite{Tonnerre95} Resonances at the $K$ edge\nmay be used instead, but the x-ray resonant process at transition\nmetal $K$ edges is neither well understood nor efficient. Reported\nresonant enhancements at transition metal $K$ edges of magnetic\nscattering intensities were typically very small and often broad\n(see, e.g., Refs.\\\n\\onlinecite{Hill97,Stunault99,Hill00,Neubeck01,Paolasini02}),\ncompared to the substantial enhancements often observed at rare\nearth $L$ edges (see for example Refs.\\\n\\onlinecite{Granado04,Good05,Tan05,Kim05branch}).\n\nAn additional problem in our case is the large charge scattering\nbackground due to the vicinity of the magnetic satellite\nreflections to the charge reflections. This is more problematic\nfor Fe $K$ edge measurements, because at the Fe $K$ edge, the\nangle of the analyzer crystal for the polarization analysis is\nfurther away from the ideal $90^{\\circ}$ than in the case of the\nGd $L_{\\rm II}$ edge, resulting in a significantly higher charge\nscattering background at the Fe $K$ edge. Performing energy scans\nin the range of the Fe $K$ edge ($7.11\\,{\\rm keV}$) we did not\nobserve a clear resonant enhancement. In view of the mentioned\ndifficulties, this does not, however, imply that the Fe $3d$\nmoments are not ordered. Rather, we conclude that the direct\nobservation of an XRMS signal from the Fe moments will need a\nseparate experiment with specialized equipment and enough time to\ngather high statistics.\n\nAlthough we did not observe a clear resonant enhancement at the Fe\n$K$ edge, based on the above discussion of the intensity of the\nXRMS at the Gd $L_{\\rm II}$ edge and the comparison with other\nR148 compounds, we conclude that the ($\\tau ,\\tau , 0$)\npropagation vector describes the order of the Fe moments. A Fe\nmoment modulation along [110] is common for R148 compounds, but in\nR148 with other rare earths $\\tau$ is typically larger ($0.127 <\n\\tau < 0.243$) and varies less with\n$T$.\\cite{Paixao00,Paixao01,Schobinger98,Schobinger99}\n\nAt $9\\,{\\rm K}$, the satellite reflections are present as well\n[Fig.\\ \\ref{Fig11}b)], with the propagation vector in the same\ndirection. This demonstrates that the Fe moments maintain their\nmodulation from the HTP, where the Gd $4f$ moments are not\nordered, in the LTP, where the Gd $4f$ moments {\\em are} ordered.\nThis is similar to the situation in other R148 compounds. However,\nthe low intensity of the ($4-\\tau , 4-\\tau , 0$) satellite\nreflections at low $T$ indicate that for Gd148, unlike the\nsituation in other R148\ncompounds,\\cite{Langridge99,Paixao00,Schobinger98} the rare-earth\n$4f$ moments do {\\em not} follow the Fe moment modulation. In\norder to find possible modulations involving the Gd $4f$ moments,\nwe searched at $9\\,{\\rm K}$ also for magnetic satellite\nreflections in other high-symmetry directions. However, full scans\nalong [110], [100], and [001] (for scattering geometries mentioned\nabove) did not turn up any reflections with intensity high enough\nto correspond to a modulation of a significant part of the $4f$\nmoments.\n\nFurther XRMS experiments, including tracking the observed magnetic\nsatellite reflections in detail through temperature and applied\nfield, and comparing the resonances at absorption edges other than\nGd $L_{\\rm II}$ are planned.\n\n\\section{Discussion and conclusions}\n\\label{disc}\n\nIn the following we will discuss our results, starting with the\nmagnetic ordering at the {N{\\'{e}}el} temperature and the nature\nof the resulting phase. We will then discuss the $H-T$ phase\ndiagrams (for two field directions) at low temperature and the\nmagnetic order in these phases.\n\nOur resistivity and magnetization data indicate that\nGdFe$_4$Al$_8$ orders magnetically at $T_N \\approx 155\\,{\\rm K}$\n(see Sec.\\ \\ref{lowH}). Even small ($1-3\\%$) iron deficiency\nsignificantly lowers $T_N$, indicating that magnetic interactions\ninvolving the Fe $3d$ moments drive the transition. In the x-ray\nresonant magnetic scattering (XRMS) experiment (Sec.\\ \\ref{xrms}),\nthe order below $T_N$ was identified as an incommensurate\nantiferromagnetic order with propagation in [110] direction.\nAlthough corresponding satellite reflections were measured on the\nGd $L_{\\rm II}$ edge, the $T$ dependence of entropy associated\nwith the Gd $4f$ moments, estimated from specific-heat data (see\nFig.\\ \\ref{Fig5} in Sec.\\ \\ref{lowH}), indicates that at high\ntemperature, the Gd $4f$ moments do not participate in this\nantiferromagnetic order, which is thus driven by Fe-Fe moment\ninteractions. This can explain the weak intensity of the measured\nsatellite reflections. Antiferromagnetic order propagating in\n[110] direction, at high $T$ confined to Fe $3d$ moments, seems to\nbe rather typical for $R$Fe$_4$Al$_8$ compounds.\n\nIn addition to the {N{\\'{e}}el} transition, we found two more\nphase transitions at low temperature ($T_1$ and $T_2$), dividing\nthe ordered state into three phases: LTP (low-temperature phase),\nITP (intermediate-temperature phase), and HTP (high-temperature\nphase). Due to the proximity of the two low-$T$ transitions it\nproved to be very useful to apply magnetic fields in two\ndirections and thereby more clearly separate these phases.\n\n\\begin{figure}[tb]\n\\includegraphics[width=0.98\\linewidth]{xlnewphd.eps}\n\\caption{(Color online) Low-temperature phase diagrams of\nGdFe$_{3.96(1)}$Al$_8$ (sample Ia\/b) for $H\\|$[110] (a) and\n$H\\|$[001] (b). Triangles in a) are from resistivity measurements\non sample Ia, the other symbols from magnetization measurements on\nsample Ib (see text for details). The three magnetic phases LTP\n(low-temperature phase), ITP (intermediate-temperature phase), and\nHTP (high-temperature phase), as well as the zero-field transition\ntemperatures $T_1$ and $T_2$ are indicated in both panels. A\npossible fourth magnetic phase existing in fields applied parallel\nto [110] (HFP) is indicated in panel a). Underlaid is a contour\nplot of the magnetization $M$ (sample Ib).} \\label{Fig10}\n\\end{figure}\n\nLow-temperature phase diagrams established from resistivity and\nand magnetization measurements (see Sec.\\ \\ref{highH}) on crystal\nI (samples Ia and Ib) are presented in Fig.\\ \\ref{Fig10} [for\n$H\\|$[110] in panel a) and for $H\\|$[001] in panel b)]. The phase\ndiagrams are underlaid with a contour-plot of the magnetization\n[data determined from $M(H)$ measurements as displayed in Fig.\\\n\\ref{Fig9}, for $H$ increasing].\n\nFor $H\\|$[110] [Fig.\\ \\ref{Fig10}a)] the phase boundaries\n($\\bigtriangleup$,$\\bigtriangledown$) were determined from $\\rho\n(T)$ measurements [steepest slope, c.f.\\ Fig.\\ \\ref{Fig7}a); shown\nis the average between points determined on curves measured with\n$T$ increasing and decreasing]. In $M(T,H)$ the ITP-HTP transition\ncorresponds to, upon decreasing $T$, a steep rise in $M$ and\nsaturation. For definiteness, we used the extrema in ${\\rm\nd}M\/{\\rm d}H$ [${\\rm d}(MT)\/{\\rm d}T$], plotted in Fig.\\\n\\ref{Fig10}a) as $\\circ$ ($\\Box$). In case of hysteresis we again\ntook the average between $T$ or $H$ increasing and decreasing.\nAbove $0.5\\,{\\rm T}$, this tracks the ITP-HTP transition deduced\nfrom $\\rho (T)$ well, but in lower $H$ the smallness of the\nmagnetization component in [110] direction leads the transition\nline established by these criteria two lower $T$, in fact\nextrapolating to $T_1$ for $H=0$. The $M(H)$ curve measured at\n$26\\,{\\rm K}$ (Fig.\\ \\ref{Fig9}) and the $\\rho (H)$ curve measured\nat $25\\,{\\rm K}$ (Fig.\\ \\ref{Fig8}) {\\em do} suggest an additional\nmetamagnetic transition in the temperature region of the ITP. This\ntransition is what is determined from the criteria above below\n$0.5\\,{\\rm T}$, starting from $T_1$ in zero field and apparently\nmerging with the boundary to the HTP at about $28\\,{\\rm K}$ in\n$0.5\\,{\\rm T}$.\n\nThis seems to separate the ITP as deduced from resistivity\nmeasurements into two phases, we call the tentative high field\nphase HFP. In the ({\\em longitudinal}, $H\\|$[110]) magnetization\nthe HFP is indistinguishable from the LTP, and thus it could be\nthat the HFP is better described as a ``modified LTP'' phase\nrather than as a ``modified ITP'' one. However, the electrical\ntransport data (see, e.g., Fig.\\ \\ref{Fig6}) clearly indicate a\nfirst-order transition separating LTP and HFP. This transition\nexhibits a wide hysteresis in temperature when $H\\|$[110] is\nsubstantial (see Fig.\\ \\ref{Fig6}) and likely involves complex\ntransformation of domains present in the LTP and in the HFP (see\nthe brief discussion in Sec.\\ \\ref{highHrho}; note that the HFP\nwas not yet introduced there) warranting further investigation.\nThe connection of the LTP-HFP transition with pronounced domain\neffects might indicate that domains are a significant force\ndriving this transition, but presently this is tentative at best.\nThe large hysteresis associated with the LTP-HFP transition\ncomplicates the determination of the equilibrium transition line.\nEstimating the equilibrium transition by taking the average of the\ntransitions for $T$ increasing and decreasing [for $3.5\\,{\\rm T}$\nthis is indicated by an open circle in Fig.\\ \\ref{Fig6}, in the\nphase diagram, the transition is indicated by open down triangles]\nsuggests that $H\\|$[110] favors neither the LTP nor the HFP\nstrongly. Similarly, the transition between the ITP and the HTP is\nhard to pin down from $M(T)$ measurements with $H\\|$[110] because\nof the smallness of the component along [110] of the\nmagnetization. We observed no second ``extremum in $[{\\rm d}(MT) \/\n{\\rm d}T]$'', using this criterion only leads to the transition\nbetween the HFP and the ITP, rather than to the one between the\nITP and the HTP. For a field applied along [110], resistivity vs\ntemperature measurements clearly indicate two transitions, whereas\n(longitudinal) magnetization vs temperature measurements indicate\nonly one. Only when examining the spontaneous magnetization along\n[110] (Fig.\\ \\ref{Fig9N}) is the presence of two separate\ntransitions clearly indicated.\n\nFor $H\\|$[001] [Fig.\\ \\ref{Fig10}b)], three distinct regions in\nthe contour-plot of the magnetization are readily identified with\nLTP (low $M$), ITP (high $M$) and HTP (low $M$). The transition\nlines between these phases are qualitatively consistent with what\nmay be deduced from the resistivity measurements presented in\nFig.\\ \\ref{Fig7}b). In contrast to the phase diagram for\n$H\\|$[110], there are no indications for the presence of\nadditional phases for $H\\|$[001]. Since $H\\|$[001] significantly\nbroadens the features in $\\rho (T)$, we rely here on magnetization\nmeasurements (see Figs.\\ \\ref{Fig4} and \\ref{Fig9}) only to draw\nthe transition lines in Fig.\\ \\ref{Fig10}b). To fix the ITP-HTP\ntransition, we used the same procedure as for $H\\|$[110]\n(squares). For $H\\|$[001] too, the $H$ dependence of the\ntransition and a saturation consistent (considering demagnetizing\neffects) with a spontaneous moment indicate a ferromagnetic\ncomponent in the ITP (see also Fig.\\ \\ref{Fig9N}). The LTP-ITP\ntransition is determined from the $M(H)$ loops (c.f.\\ Fig.\\\n\\ref{Fig9}), taking the field value where $M$ starts to increase\nfaster than linear as the criterion for the transition field. This\ntransition is displayed by yellow circles in Fig.\\ \\ref{Fig10}b).\nFor $H\\rightarrow 0$ the transition line extrapolates to $T_1$,\nreaffirming it's identity as the LTP-ITP transition. It can be\nseen that $H\\|$[001] quickly suppresses the LTP.\n\nBased on the specific-heat data, we can conclude that the HTP to\nITP transition corresponds to the ordering of the Gd $4f$ moments.\nSince the magnetization data indicate the presence of a\nspontaneous ferromagnetic moment in the ITP, this Gd $4f$ ordering\nis either ferromagnetic or has a ferromagnetic component. The\nincrease of the transition temperature $T_2$ upon application of a\nmagnetic field (Fig.\\ \\ref{Fig10}) is consistent with this. Thus,\nGd moments do not simply follow the Fe moment order from higher\ntemperatures [this is in contrast to, e.g., DyFe$_4$Al$_8$ (Ref.\\\n\\onlinecite{Paixao00})].\n\nThe transition from the LTP to the ITP seems to be of first-order\nas indicated by the observation of thermal hysteresis in the\nresistivity (see Fig.\\ \\ref{Fig3}) as well as the magnetization\n(see Fig.\\ \\ref{Fig9}). Surprisingly though, no feature could be\nidentified in the specific heat corresponding to this transition\nand any latent heat would have to be very small. The LTP-ITP\ntransition is associated with a lock-in of the observed\nspontaneous ferromagnetic component (Fig.\\ \\ref{Fig9N}) into the\n$a$$-$$b$ plane in the LTP, but there are no clear indications\nthat its magnitude is changed by the transition. The appearance of\nan additional antiferromagnetic modulation is possible, at least\nit is not clear why a pure lock-in transition of a ferromagnetic\ncomponent should result in a sizeable increase of the resistivity\n(both for $j\\|$[110] and $j\\|$[001]).\n\nThe presence of the HFP in $H\\|$[110] complicates the situation.\nIn fact, there are clear indications  (meta-stability) for the\nfirst-order nature of the transitions at all boundaries of the\nHFP. In contrast to this, there are no indications for\nmeta-stability in the ITP-HTP transition. Meta-stability seems to\nbe present in the LTP-ITP transition, but it is much weaker than\nin any transition from or to the HFP. We point out that the\nfirst-order transition lines LTP-HFP and HFP-HTP both seem to\nextrapolate to $T_1$ in zero field, and therefore the observed\nsmaller meta-stability of the LTP-ITP transition might well be\nconnected with the HFP phase. This is a strong incentive for\nfurther investigations of the HFP.\n\nWe now attempt to draw a picture accounting for the full magnetic\nmoment of both Gd and Fe. The full ordered moment of Gd is\n$7\\,{\\mu_B}\/{\\rm Gd}$. Since the valence of Gd is fixed and there\nare no crystal-electric-field effects, this is the same for\ndifferent compounds, with only small corrections, up to $\\sim\n0.5\\,\\mu_B$, due to contributions by itinerant electrons (such as\nthe Gd $5d$ electrons). The full moment of Fe can vary from\ncompound to compound.\n\nWe know from the specific heat and XRMS measurements that in the\nHTP Gd moments (the Gd $4f$ moments at least) are not ordered, and\nFe moments are antiferromagnetically ordered with propagation\nalong [110]. Magnetization measurements further show that there is\nno significant ferromagnetic component present [see the $40\\,{\\rm\nK}$ $M(H)$ loops in Fig.\\ \\ref{Fig9}]. We recall that, according\nto the x-ray-diffraction-structure refinement and specific-heat\nmeasurements, the Gd lattice is fully occupied and that at low\ntemperatures the Gd $4f$ moments are well ordered. The appearance\nof a spontaneous ferromagnetic moment below the temperature $T_2$\nwhere the Gd moments start to order (compare Figs.\\ \\ref{Fig5} and\n\\ref{Fig9N}) indicates that the ordering of the Gd moments is\nferromagnetic, but the saturation moments for both field\ndirections are far below the $7\\,{\\mu_B}\/{\\rm f.u.}$ (see Fig.\\\n\\ref{Fig9}) expected for a full ferromagnetic alignment of Gd\nmoments. What could be the reason for this?\n\nOne possibility is that the Gd $4f$ moments are not fully\nferromagnetically ordered, but rather also have an\nantiferromagnetic component. However, since we did not find any\nsatellite reflections with high enough intensity to account for\nthe necessary fraction of the full Gd $4f$ moment in any of the\nhigh symmetry directions (see Sec.\\ \\ref{xrms}, note that the\nintensity of the ($4-\\tau,4-\\tau,0$) reflection is much too low)\nthis scenario is not very likely.\n\nFrom high-field measurements on finely ground free powder\nparticles of Gd148 (likely, at least roughly, corresponding to\n$H\\|$[001], ITP), Duong {\\em et al.}\\cite{Duong01} suggested for\n$H\\rightarrow 0$ a ferrimagnetic structure with Gd moments\nparallel, Fe moments antiparallel to the field, with an\nantiferromagnetic Fe moment order re-established only in high\nfields. On the one hand, such a simple ferrimagnetic arrangement\nof Gd and Fe moments is inconsistent with our observation in the\nXRMS experiment of the presence of the ($4-\\tau,4-\\tau,0$)\nsatellite reflection characteristic for the antiferromagnetic Fe\nmodulation even at $9\\,{\\rm K}$ [Fig.\\ \\ref{Fig11}b)]. On the\nother hand, a related idea would be consistent with both the XRMS\nand the magnetization measurements: upon the ferromagnetic\nordering of the Gd $4f$ moments, the Fe moments {\\em could} pick\nup a ferromagnetic component, leading to a canted\nantiferromagnetic structure. Such a canted magnetic structure with\na large ferromagnetic component may also explain why the intensity\nof the ($4-\\tau,4-\\tau,0$) satellite reflection is about three\ntimes smaller at $9\\,{\\rm K}$ (in the LTP) than at $37\\,{\\rm K}$\n(in the HTP). Furthermore, it would also allow Fe to influence the\nlow-$T$ transitions, as indicated by the influence of Fe\ndeficiencies on all characteristic temperatures (see Sec.\\\n\\ref{lowH}). The results of a recent dichroism experiment (on a\ncrystal ground to powder), which will be discussed in detail\nelsewhere, are consistent with such a scenario as well.\n\nThis picture covers, at least for the LTP and the HTP, all the\ndata presented in the paper, but without more detailed scattering\ndata it remains speculative. Particularly concerning the ITP and\nHFP, the absence of scattering data makes detailed conclusions\nelusive.\n\nSince the direction of the spontaneous ferromagnetic moment is\ndifferent for the ITP and the LTP, magnetocrystalline anisotropy\n(MCA) is likely one of the forces driving this transition. In rare\nearth compounds, MCA typically arises from crystal-electric-field\neffects. As we mentioned in Sec.\\ \\ref{intro}, this source of\nanisotropy is absent for Gd. Significant MCA that was nevertheless\nobserved for both elemental Gd and Gd compounds has been ascribed\nto a combination of dipole-dipole interactions and (mainly)\nanisotropy in the $5d$ bands.\\cite{ColarietiTosti03,KimPhD} The\ncorresponding magnetic-anisotropy energies are much lower than the\nmagnetic-anisotropy energies resulting from crystal-electric-field\neffects in other rare earths. A weak magnetic-anisotropy energy\nmight also, at least partly, explain why no latent heat could be\nobserved at $T_1$. In Gd148, anisotropy in the Fe $3d$ bands might\nbe an additional source of MCA.\\cite{note_Lu148} In any case, the\ndifferent spontaneous moment direction in the ITP likely results\nin a different structure of the Fe moments. The apparent shift of\n$T_1$ upon changing the Fe stoichiometry may be taken as an\nindication of the involvement of Fe moments in the LTP-ITP\ntransition. Before going any further in the discussion of the\ndriving forces (there is likely a delicate balance between\nmultiple energy scales) of the transitions between LTP, ITP, and\nHFP, the magnetic structures of these phases need to be solved.\nWith this aim in mind, additional scattering experiments are\nplanned.\n\nRegardless of the details of the magnetic structure at low\ntemperature, it is important to point out that i) on the one hand,\nthe Fe moments have at least a component antiferromagnetically\nmodulated in [110] direction ii) The Gd $4f$ moments, on the other\nhand can {\\em not} have a component ordered with the same\npropagation as the Fe $3d$ moments, because then the intensity of\nthe reflections measured at $9\\,{\\rm K}$ would have to be orders\nof magnitude higher (see Sec.\\ \\ref{xrms}). This is strikingly\ndifferent from the behavior for example in Dy148, where the rare\nearth moments order at lower temperature, but then follow the Fe\nmoment modulation, and it implies a {\\em co-existence of two\ndistinct orders} associated with Gd $4f$ and Fe $3d$ moments.\nWhereas compounds in which one magnetic sublattice orders at\ntemperatures much lower than the other magnetic sublattice are not\nvery rare (indeed this was well known to be the case for R148\ncompounds in general), we are aware of only one example,\nPrBa$_2$Cu$_3$O$_{6.92}$,\\cite{Hill00} in which the magnetic\nsublattice ordering at lower $T$ has no component with the same\nmodulation that the other sublattice has at all $T$.\n\nThe at low temperatures persisting antiferromagnetic modulation of\nthe Fe moments coexisting with an order of the Gd moments with a\nlarge ferromagnetic component and without a component having the\nFe moment modulation is the central point of our paper. It may\nindicate a modified relation between the strength of the Gd-Gd and\nGd-Fe moment interactions in Gd148 as compared to other R148\ncompounds. As we mentioned, apart from the stronger couplings\nbetween $4f$ and conduction electrons in general, the most\npeculiar property of Gd is the absence of crystal-electric-field\neffects.\n\nIt seems remarkable that in the absence of the MCA due to crystal\nelectric field effects, the magnetic behavior in Gd148 is {\\em\nmore} complex than in other R148. Likely, strong\ncrystal-electric-field effects play a central role in the\nmagnetism in R148 (except for Gd148). E.g., in Dy148,\ncrystal-electric-field effects press the $4f$ moments into the\ncrystallographic plane, making an order following the Fe-moment\nmodulation more probable, since the Fe moments are already\nin-plane. Our results on Gd148, in contrast, indicate a complex\ninterplay of local $4f$ and itinerant $3d$ moments when\ncrystal-electric-field effects are absent.\n\n\\section{Summary}\n\\label{conc}\n\nWe presented an extensive set of data including magnetization,\nelectrical transport, specific heat, and x-ray resonant magnetic\nscattering, measured on flux-grown single crystals of\nGdFe$_4$Al$_8$. We found that two transitions at $T_1 \\sim 21\\,\n{\\rm K}$ and $T_2 \\sim 27\\, {\\rm K}$ at low temperature divide the\nordered state below $T_N \\sim 155\\, {\\rm K}$, where Fe moments\norder antiferromagnetically with modulation along [110], into\nthree phases. The corresponding phase diagrams for fields applied\nin [110] and [001] direction were presented.\n\nGd $4f$ moments order, mainly ferromagnetically, below $T_2$.\nAbove $T_1$ the ferromagnetic component of the moments points into\na direction close to [001], whereas below $T_1$ the ferromagnetic\ncomponent is locked into the $a$$-$$b$ plane. The Gd $4f$ moment\norder is distinct from the Fe-moment order At low temperature $T$:\nthe Fe moments still have at least a component modulated along\n[110], but the Gd $4f$ moments apparently do not have such a\ncomponent. We proposed a canted antiferromagnetic structure of the\nFe moments at low $T$, which can cover all the low-$T$ data\npresented in the paper, but to some extent still is speculative.\nAbove  The transition at $T_1$ likely involves a delicate balance\nof multiple energy scales associated with both Gd and Fe.\n\nThe complex magnetism in GdFe$_4$Al$_8$ as compared to\n$R$Fe$_4$Al$_8$ compounds with other rare earths $R$ likely is\nrelated to a modified ratio of coupling strengths and to the\nabsence of crystal-electric-field effects. In order to further\nelucidate the structure of the magnetic phases of GdFe$_4$Al$_8$\nadditional scattering experiments are planned.\n\n\\begin{acknowledgments}\n\\label{ack} We thank S.~L. {Bud'ko}, P.~Ryan, and R.~W. McCallum\nfor useful discussions, P.~Ryan also for technical assistance\nduring the synchrotron experiment. Ames Laboratory is operated for\nthe U.S. Department of Energy by Iowa State University under\nContract No.\\ W-7405-Eng-82. This work was supported by the\nDirector for Energy Research, Office of Basic Energy Sciences.\nSynchrotron work was performed at the MuCAT sector at the Advanced\nPhoton Source supported by the U.S. DOE, BES, and OS under\nContract No.\\ W-31-109-Eng-38.\n\\end{acknowledgments}\n\n\\newcommand{\\noopsort}[1]{} \\newcommand{\\printfirst}[2]{#1}\n  \\newcommand{\\singleletter}[1]{#1} \\newcommand{\\switchargs}[2]{#2#1}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{intro}\nIn perturbative QCD the two well-known parton evolution equations DGLAP \\cite{63,64,65} and BFKL \\cite{66,67} serve as the basic tools for prediction of parton distribution functions (PDFs) on their respective kinematic domains. The DGLAP approach is valid for the large interaction scale $Q^2$ since it sums higher order $\\alpha_s$ contributions enhanced by the leading logarithmic powers of $\\ln Q^2$. On the other hand, BFKL evolution deals with the small-$x$ and semi-hard $Q^2$ kinematic region by summing the leading logarithmic contributions of $(\\alpha_s \\ln 1\/x)^n$. However, both the linear evolutions turn out to be inadequate to comply with the unitary bound or the conventional Froissart bound \\cite{68}. To restore the unitarity, a higher order pQCD correction is required to shadow the rapid parton growth which eventually leads to the saturation phenomena at very small-$x$. In this respect, several nonlinear equations have been proposed in recent years to entertain this shadowing correction in DGLAP and BFKL equation.\n\\par In pQCD interactions, the origin of the shadowing correction is primarily considered as the gluon recombination ($g+g\\rightarrow  g$) which is simply the inverse process of gluon splitting ($g\\rightarrow g+g$). The modification of gluon recombination to the original DGLAP equation was first performed in GLR-MQ equation \\cite{16,75,86,87} proposed by Gribov-Levin-Ryskin and Mueller-Qui. It is considered that the interferant cut diagrams of the recombination amplitude are the origin  for the negative shadowing effects in the gluon recombination processes \\cite{59,69}. The GLR-MQ equation was originally derived using AGK-cutting rules \\cite{71} to compute the contributions from these interference processes. However, GLR-MQ does not possess any antishadowing correction term since in this equation momentum conservation is violated in the gluon recombination processes. The shadowing and antishadowing effects are expected to happen in a complementary fashion which in fact ensure the momentum conservation during the process of gluon recombination \\cite{70}. In consequence of gluon recombination, a part of gluon momentum lost due to shadowing is believed to be compensated in terms of new gluons with comparatively larger $x$. This causes an enhanced density of larger momentum gluons responsible for the antishadowing effect. In this respect, to take care of momentum conservation and antishadowing effects in gluon evolution, a modified DGLAP equation was proposed \\cite{69} to replace the GLR-MQ equation. In the derivation of the modified DGLAP equation, the author has used TOPT-cutting rules based on time ordered perturbation theory (TOPT) to calculate the contribution of the gluon recombination, instead of using AGK-cutting rules unlike in the derivation of GLR-MQ equation. This equation naturally comes with two separate nonlinear terms, quadratic in gluon distribution function which are identified as the positive antishadowing and negative shadowing components in the gluon evolution. This suggests that negative shadowing effects are suppressed by antishadowing effects in the gluon evolution process.\n\\par   However, above mentioned modified DGLAP approach cannot predict the evolution of unintegrated gluon distribution because DGLAP is based on collinear factorization scheme while the later deals with $k_T$-factorization. In this respect, a new unitarized BFKL equation was proposed by Ruan, Shen, Yang and Zhu \\cite{1} incorporating both shadowing and antishadowing correction. This modified BFKL (MD-BFKL) equation allows one to study antishadowing effects in unintegrated gluon distribution platform.\n\\par However, there are several other BFKL based nonlinear evolution equations that consider corrections from gluon fusion. One of the most widely studied models is Balitsky-Kovchegov (BK) equation \\cite{10,11} which is originally derived in terms of scattering amplitude. It can be explained by the dipole model in which the nonlinear term comes from the dipole splitting while the screening effect origins from the double scattering of the dipole on the target \\cite{85}. The solution of the BK equation suggests an intense shadowing leading to the so-called saturation of the scattering amplitude showing a complete flat spectrum. However, like GLR-MQ equation the BK equation is also irrelevant to the gluon antishadowing. On the other hand, besides inclusion of antishadowing correction term, the nonlinear terms in MD-BFKL equation hold a simple form which is quadratic in unintegrated gluon distribution as well as running coupling constant with some constant coefficients. These unique features of MD-BFKL equation motivate our current studies on small-$x$ physics.\n\\par A numerical solution of MD-BFKL equation suggests a sizable impact of antishadowing effect on gluon evolution particularly in the pre-asymptotic regime \\cite{1}. In the literature \\cite{73} the phenomenology of the MD-BFKL equation is extended to study the implicit nuclear shadowing and antishadowing effects. In our current work, we try to construct a modified evolution equation by implanting the so-called kinematic constraint or consistency constraint on the MD-BFKL equation. The kinematic constraint (see Fig.~\\ref{2x} ) is implemented in different forms:\n\\begin{align}\n\t\\label{1.1}\n\tq_T^2&<\\frac{k_T^2}{z} \\text{     } \\text{ LDC \\cite{6,74}},\\\\\n\t\\label{1.2}\n\tq_T^2&<\\frac{(1-z)k_T^2}{z}\\text{  \\cite{3}},\\\\\n\t\\label{1.3}\n\tk_T^{'2}&<\\frac{k_T^2}{z}\\text{      }\\text{  BFKL \\cite{3}}.\n\\end{align}\nThis constraint arises as a consequence of BFKL multi-Regge kinematics which suggests the exchanged gluon virtuality is dominated by transverse components while the longitudinal components of the gluon momentum are required to be small i.e. $k^{'2}\\approx k_T^{'2}$. The kinematic constraint gives an implicit cutoff on $k_T^{'2}$ as depicted by \\eqref{1.3}. The inequality \\eqref{1.3} can be considered as a special case of \\eqref{1.1} recalling the fact that for a given value of $k_T^2$, a high $q_T^2$ implies an equally high $k_T^{'2}$ \\cite{3}. Although there are other constraints coming from energy-momentum conservation \\cite{7}, this bound is considerably tighter than the later \\cite{3}. Besides the introduction of this cutoff on the upper limit of integration found to preserve the scale invariance of the BFKL equation.\n\\par Our primary goal of this work is to investigate the nonlinear effects on gluon evolution in terms of MD-BFKL equation supplemented by the kinematic constraint. Our studies are particularly focused on the near saturation region where shadowing effects are dominant over antishadowing effects. The content of the paper is organized as follows. In Sect. \\ref{analytic} we present our kinematic constraint improved MD-BFKL equation starting from the construction of the equation and a brief discussion on some of its main features (Sect. \\ref{analytic1}). We suggest an analytical solution to this equation and sketch a particular solution in terms of $x$ and $k_T^2$ evolution followed by three-dimensional realization of gluon evolution in $x\\text{-}k_T^2$ phase space (Sect. \\ref{analytic2}). We have also extracted collinear gluon distribution $xg(x,Q^2)$ from unintegrated gluon distribution $f(x,k_T^2)$ and compared our prediction with that of global parameterization groups NNPDF 3.1sx and CT 14. In Sect. \\ref{critical} we suggest a differential geometric approach towards finding the equation of the critical boundary between high and low gluon density regime. In this section, a phenomenological insight of the geometrical scaling is discussed as well.\n\\par In Sect. \\ref{hera} we explore the phenomenological implication of our solution for unintegrated gluon distribution towards prediction of DIS structure function $F_2$ and longitudinal structure function $F_L$ at HERA. We show the comparisons of reduced cross sections (Sect. \\ref{hera1}) as well as virtual photon-proton cross sections in the transition region (Sect. \\ref{hera2}) with HERA H1 and ZEUS data. Finally, in Sect. \\ref{con} we summarize and outline our conclusion as well as possible future prospects.\n\n\n\n\n\\section{Nonlinear effects in kinematic constraint improved MD-BFKL equation}\\label{analytic}\n\n\\subsection{Construction of kinematic constraint improved MD-BFKL}\\label{analytic1}\nThe modified BFKL (MD-BFKL) equation reads \\cite{1},\n\\begin{equation}\n\t\\label{1}\n\t\\begin{split}\n\t\t&-x\\frac{\\partial f \\left(x, k_T^2\\right)}{\\partial x}=\\\\&\\frac{\\alpha_s N_c k_T^2}{\\pi}\\int_{k_{T_{\\min }}^{'2}}^{\\infty}\\frac{dk_T^{'2}}{k_T^{'2}}\\left[\\frac{f (x, k_T^{'2})-f \\left(x, k_T^2\\right)}{|k_T^{'2}-k_T^2|} + \\frac{f \\left(x, k_T^2\\right)}{\\sqrt{k_T^4+4k_T^{'4}}}\\right]\\\\\n\t\t&-\\frac{36\\alpha_s^2}{\\pi k_T^2 R^2}\\frac{N_c^2}{N_c^2-1}f^2(x,k_T^2)+\\frac{18\\alpha_s^2}{\\pi k_T^2 R^2}\\frac{N_c^2}{N_c^2-1}f^2(\\frac{x}{2},k_T^2),\n\t\\end{split}\n\\end{equation}\nwhere $f \\left(x, k_T^2\\right)$ denotes the gluon distribution unintegrated over the transverse momentum of gluon $k_T$ and $k_{T_{\\min }}^{'2}$ is the infrared cutoff of the evolution. The first part of \\eqref{1} linear in $f(x,k_T^{'2})$ (or $f(x,k_T^2)$) is the BFKL kernel at leading logarithmic of $1\/x$ (LLx) accuracy. A diagrammatic representation for probing gluon content of the proton at high photon virtuality $Q^2$ is sketched in Fig.~\\ref{2x} (a). The BFKL kernel corresponds to the sum of gluon ladder diagram Fig.~\\ref{2x}(b) generated by squaring the amplitude of Fig.~\\ref{2x}(a). The 1st term in the BFKL kernel, involving $f(x,k_T^{'2})$   corresponds to diagrams with real gluon emission while the second term takes care of diagrams with virtual corrections. Note that the apparent singularity that is observed at $k_T^{'2} =k_T^2$ cancels between real and virtual contributions. \n\n\\par The quadratic terms in \\eqref{1} viz.  \n$\\frac{\\partial f_{\\text{shad}} }{\\partial \\ln{1\/x}}=-\\frac{36\\alpha_s^2}{\\pi k_T^2 R^2}\\frac{N_c^2}{N_c^2-1}f^2$\nand\n$\\frac{\\partial f_{\\text{antishad}} }{\\partial \\ln{1\/x}}$ $=\\frac{18\\alpha_s^2}{\\pi k_T^2 R^2}\\frac{N_c^2}{N_c^2-1}f^2$ depict shadowing and antishadowing correction to the original BFKL equation respectively. The factor $\\pi R^2$ represents the transverse area populated by gluons. The radius parameter $R$ arises in the QCD cut diagram coupling 4 gluons to 2 gluons (see Fig.~\\ref{3x}), the value of which depends on how exactly the gluon ladders couple to hadron. If the gluon ladder couples to two different constituents of hadron, $R$ is characterized by hadronic radius i.e. $R\\approx R_H=5 \\text{ GeV}^{-1}$ whereas if we consider the possibility of ladder coupling to the same parton then the appropriate choice of $R$ is the radius of valence quark i.e. $R=2\\text{ GeV}^{-1}$. In the latter case $(R=2\\text{ GeV}^{-1})$ we have \"hot spots\" \\cite{75,72}.\n\\begin{figure*}[tbp]\n\t\\centering\n\t\\includegraphics[width=.21\\textwidth,clip]{GluonLadderII.eps}\\hspace{5mm}\n\t\\includegraphics[width=.23\\textwidth,clip]{Structure_box.eps}\\hspace{5mm}\n\t\\includegraphics[width=.15\\textwidth,clip]{f011.eps}\\hspace{5mm}\n\t\\includegraphics[width=.15\\textwidth,clip]{f026.eps}\n\t\n\n\t\\caption{\\label{2x} From left: (a) chain of sequential gluon emission which forms the basis of BFKL equation. On squaring the amplitude of (a) the ladder diagram (b) is generated which when summed gives the BFKL kernel. (c) and (d) are two simple examples of the inhomogeneous driving term of \\eqref{4} which correspond to the shaded region of gluon-virtual photon coupling in (b). }\n\\end{figure*}\n\n\\par The interpretation of the real emission term in the BFKL kernel is that we have a splitting $k^{'}\\rightarrow k + q$ inside hadron resulting an infinite chain of reggeized gluons labeled as $k_1,k_2,k_3,\\text{...}k_n$ (Fig.~\\ref{2x}(a)). In high energy limit, the longitudinal components of the gluon momentum are strongly ordered while there is no ordering on the transverse components of the gluon momentum i.e.\n\t\\begin{align}\n\t\t\\label{2.2a}\n\t\t&x_1\\ll x_2\\ll x_3\\text{.....}\\ll x_{n-1}\\ll x_n,\\\\\n\t\t\\label{2.2b}\n\t\t&k_{1 T}\\sim k_{2 T}\\sim k_{3 T}\\sim \\text{...}.\\sim k_{n-1 T}\\sim k_{\\text{nT}}.\n\t\\end{align}\nIn addition, the small-$x$ regime where the BFKL is valid, the gluon virtuality along the chain must be dominated by the transverse components of the gluon momentum, \n\\begin{align}\n\t\\label{2.3}\n\tk^2=2 k^+ k^--k_T^2\\approx k_T^2.\n\\end{align}\\\\\nThe above kinematics corresponding to \\eqref{2.2a},\\eqref{2.2b} and \\eqref{2.3} is referred as multi-Regge kinematics. The inequality \\eqref{2.2a} implies $z=\\frac{x}{\\frac{x}{z}}=\\frac{x_{n-1}}{x_n}\\ll1$ and since transverse momenta are of the same order: $k_T\\simeq k_T{}^{'}$, depict a cutoff \n\\begin{equation}\n\t\\label{2.4}\n\tk_T{}^{'}{}^2< \\frac{k_T{}^2}{z},\n\\end{equation}\nwhich is so-called  kinematic constraint or consistency constraint for real gluon emission \\cite{3,4,5,6}. The BFKL kernel in \\eqref{1} is at leading logarithmic $1\/x$ (LLx) accuracy. However, higher order corrections to the BFKL equation are already been evaluated up to NLLx accuracy, which turns out to be quite large \\cite{2}. Implementation of the constraint \\eqref{2.4} on the evolution of BFKL based equations makes the theory more realistic in the sense that this ensures the participation of only the LLx part of higher order correction in the evolution.  This, in fact, portrays the importance NLLx correction in BFKL kernel.\n\n\\par Recalling that BFKL equation can be written as an integral equation of  $f(x,k_T^2)$ \\cite{15,16}, the kinematic  constraint \\eqref{2.4} can be imposed onto the real emission part of the BFKL kernel as follows\n\\begin{equation}\n\t\\label{2.5}\n\t\\begin{split}\n\t\t\\int_{x}^{1}\\frac{dz}{z}\\int _{k_{T_{\\min }}^{'2}}\\frac{dk_T^{'2}}{k_T^{'2}}\\bigg[&\\frac{\\Theta \\left(\\frac{k_T^2}{z}-k_T^{'2}\\right)f\\left(\\frac{x}{z},k_T^{'2}\\right)-f\\left(\\frac{x}{z},k_T^2\\right)}{\\left| k_T^{'}{}^2-k_T^2\\right| }\\\\\n\t\t&+\\frac{f\\left(\\frac{x}{z},k_T^2\\right)}{\\sqrt{k_T^4+4k_T^{'4}}}\\bigg],\n\t\\end{split}\n\\end{equation}\nwhere the heaviside function $\\Theta \\left(\\frac{k^2}{z}-k^{'}{}^2\\right)$ in \\eqref{2.5} serves the purpose of the kinematic cutoff. The upper limit of the integration over $k_T^{'2}$ is implicit in $\\Theta$. The BFKL kernel gains one more degree of freedom after implementation of kinematic constraint for real emission. Now expressing  the MD-BFKL equation \\eqref{1} in terms of the KC improved BFKL kernel \\eqref{2.5} yields\n\\begin{equation}\n\t\\label{4}\n\t\\begin{split}\n\t\t&f\\left(x,k_T^2\\right)=f^{(0)}\\left(x,k_T^2\\right)+\\\\&\\frac{\\alpha _s k_T^2 N_c }{\\pi }{\\int _x^1\\frac{dz}{z}\\int _{k_{T_{\\min }}^{'2}}}\\frac{dk_T^{'2}}{k_T^{'2}}\\bigg[\\frac{\\Theta \\left(\\frac{k_T^2}{z}-k_T^{'2}\\right)f\\left(\\frac{x}{z},k_T^{'2}\\right)-f\\left(\\frac{x}{z},k_T^2\\right)}{\\left| k_T^{'2}-k_T^2\\right| }\\\\\n\t\t&+\\frac{f\\left(\\frac{x}{z},k_T^2\\right)}{\\sqrt{k_T^4+4k_T^{'4}}}\\bigg]-\\frac{36 \\alpha _s^2}{\\pi  k_T^2 R^2}\\frac{N_c^2}{N_c^2-1}\\int _x^1\\frac{dz}{z}f^2\\left(x,k_T^2\\right)\\\\\n\t\t&+\\frac{18 \\alpha _s^2}{\\pi  k_T^2 R^2}\\frac{N_c^2}{N_c^2-1}\\int _x^1\\frac{dz}{z}f^2\\left(\\frac{x}{2},k_T^2\\right).\n\t\\end{split}\n\\end{equation}\nThe above equation \\eqref{4} is the integral form of our KC improved MD-BFKL equation. The inhomogeneous driving term $f^{(0)}(x,k_T^2)$ depicts gluon-proton coupling corresponding to the shaded region in Fig.~\\ref{2x}(b). The Fig.~\\ref{2x}(c, d) represents two simple possible contributions to $f^{0}(x,k_T^2)$ which indicates radiation of gluons from valence quark.\n\n\\par To sketch an integro-differential form of KC improved MD-BFKL equation out of \\eqref{4} we differentiate \\eqref{4} w.r.t. $\\ln(1\/x)$, then using properties of $\\Theta$ and Dirac-$\\delta$ function, to be specific  $\\Theta '(t)=\\delta  (t)$ and $f( t)\\delta  (t-a)=f(a)\\delta  (t-a)$ and doing some algebra one can show\n\n\\begin{equation}\n\t\\label{5}\n\t\\begin{split}\n\t\t\\frac{\\partial}{\\partial \\ln\\frac{1}{x}}\\int_{x}^{1}\\frac{\\text{d}z}{z}\\Theta \\left(\\frac{k_T^2}{z}-k_T^{'2}\\right)&f(x,k_T^{'2})\\longrightarrow          \\Theta(k_T^2-k_T^{'2})f(x,k_T^{'2})\\\\\n\t\t&+\\Theta(k_T^{'2}-k_T^{2})f\\left(\\frac{k_T^{'2}}{k_T^{2}}x,k_T^{'2}\\right).\n\t\\end{split}\n\\end{equation}\nAbove prescription allows one to express \\eqref{4} in the following integro-differential form:\n\\begin{align}\n\t\\label{2.8}\n\t\\begin{split}\n\t\t&-x\\frac{\\partial f\\left(x,k_T^2\\right)}{\\partial x}=\\frac{\\alpha _s k_T^2 N_c }{\\pi }{\\int _{k_{T_{\\min }}^{'2}}}\\frac{dk_T^{'2}}{k_T^{'2}}\\\\&\\times\\bigg[\\frac{\\Theta \\left(k_T^2-k_T^{'2}\\right)f\\left(x,k_T^{'2}\\right)+\\Theta (k_T^{'2}-k_T^{2})f\\left(\\frac{k_T^{'2}}{k_T^{2}}x,k_T^2\\right)}{\\left| k_T^{'2}-k_T^2\\right| }\\\\&-\\frac{f(x,k_T^2)}{|k_T^{'2}-k_T^2|}+\\frac{f\\left(x,k_T^2\\right)}{\\sqrt{k_T^4+4k_T^{'4}}}\\bigg]-\\frac{36 \\alpha _s^2}{\\pi  k_T^2 R^2}\\frac{N_c^2}{N_c^2-1}f^2\\left(x,k_T^2\\right)\\\\&+\\frac{18 \\alpha _s^2}{\\pi  k_T^2 R^2}\\frac{N_c^2}{N_c^2-1}f^2\\left(\\frac{x}{2},k_T^2\\right).\n\t\\end{split}\n\\end{align}\nIn derivation of \\eqref{2.8} we have neglected the term $x\\frac{\\partial f^{(0)}}{\\partial x}$. This is justified in the sense that $x\\frac{\\partial f^{(0)}}{\\partial x}$ is much less singular than $x\\frac{\\partial f}{\\partial x}$ at small-$x$. Moreover if we consider $f^{(0)}$ is particularly contributed by the diagrams shown in Fig.~\\ref{2x} (c, d) then $f^{(0)}$ would be independent of $x$ (or, $\\frac{\\partial f^{(0)}}{\\partial x}=0$ ) at small-$x$ limit \\cite{6,15}.\n\\begin{figure}[tbp]\n\t\\centering\n\t\\includegraphics[width=.27\\textwidth,clip]{Shadow.eps}\t\n\n\t\\caption{\\label{3x} QCD cut diagram representing nonlinear terms in \\eqref{1} with $4\\rightarrow 2$ gluon recombination kernel (dashed box). The dashed box includes all possible pQCD diagrams which couple 4 gluons to 2 gluons.}\n\\end{figure}\n\n\\par To simplify the distribution function $f(\\frac{k_T^{'2}}{k_T^{'2}}x,k_T^{2})$ corresponding to real emission term in \\eqref{2.8} we have incorporated Regge like behavior of gluon distribution. Before the advent of pQCD, Regge theory is been considered as a very successful theory regarding the phenomenological analysis of total hadronic and  photoproduction cross sections. These non-perturbative processes can economically described by two Regge contributions namely pomerons with intercept slightly above unity (${\\alpha_{P}(0)\\approx 1.08}$) and Reggons with intercept $\\alpha_{R}(0)\\approx 0.5$. The Regge behavior corresponding to sea quark and anti-quark distributions is given by $q_{sea}\\sim x^{-\\alpha_{P}}$ whereas that of valence quark distribution is given by $q_{v}\\sim x^{-\\alpha_{R}}$. On the other hand, in pQCD also, particularly the small-$x$ region is considered to have higher possibility towards exploring Regge limit of pQCD. Note that the BFKL dynamics itself is based on the concept of pomeranchuk theorem or pomeron: the Regge-pole carrying the quantum-numbers of the vacuum. However, the BFKL pomeron (also called hard pomeron) should be contrasted with non perturbative description of pomeron (soft pomeron) in the sense that they pose relatively different magnitude of intercept. For BFKL-pomeron the intercept is\n\\begin{equation*}\n\t\\alpha_{P}^{BFKL}(0) = 1 + \\lambda_{\\text{BFKL}},\n\\end{equation*}\nwhere  $\\lambda_{\\text{BFKL}}= \\frac{3 \\alpha_s}{\\pi} 28  \\zeta(3)$, $\\zeta$ being Reiman zeta function \\cite{9}. The typical value of $\\lambda_{\\text{BFKL}}$ for $\\alpha_s$=0.2 is $\\sim 0.5$ implies $\\alpha_{P}^{BFKL}(0)\\approx1.5$ which is potentially large in magnitude compared to soft pomeron $(\\alpha_{P}(0)=1.08)$.\n\\par In pQCD the small-$x$ behavior of the parton distribution is supposed to be controlled by intercept of appropriate Regge trajectory. Regge model provides parametrizations of DIS distribution functions, $f_i(x,Q^2)=A_i(Q^2)x^{-\\lambda_i}$ (i=$\\sum$ (singlet structure function)and g (gluon distribution)), where$\\lambda_i$ is the pomeron intercept minus one $(\\alpha_{P}(0)-1)$.  At small-$x$, the leading order calculations in \\(\\text{ln}(1\/x)\\) with fixed \\(\\alpha _s\\) predicts a steep power law behavior of \\(\\textit{f}\\left(x,k^2\\right)\\sim x^{-\\lambda_\\text{BFKL}}\\) \\cite{7,8}. This motivates us to consider a simple form of Regge factorization as follows,\n\\begin{equation}\n\t\\label{6}\n\t\\begin{split}\n\t\tf\\left(\\frac{k_T^{'2}}{k_T^2}x,k_T^2\\right)&\\simeq x^{-\\lambda_{\\text{BFKL}}}\\left(\\frac{k_T^{2}}{k_T^{'2}}\\right)^{\\lambda_{\\text{BFKL}}} H(k_T^2)\\\\\n\t\t&= \\left(\\frac{k_T^2}{k_T^{'2}}\\right)^{\\lambda }f\\left(x,k_T^2\\right),\n\t\\end{split}\n\\end{equation}\nwhere we have dropped the subscript on $\\lambda$ and in the rest of the text we will follow this notation. Similar way we can express \n\\begin{equation}\n\t\\label{2.10}\n\t\\begin{split}\n\t\tf\\left(\\frac{x}{2},k^2\\right)&\\simeq 2^{\\lambda }f\\left(x,k^2\\right).\n\t\\end{split}\n\\end{equation}\nThis type of Regge like form considered in \\eqref{6} and \\eqref{2.10} is supported by various literatures \\cite{76,77,78,79}. But note that Regge factorization can't be taken as good ansatz for the entire kinematic region of $x$ and $k_T^2$ \\cite{80}. This type of Regge behavior is considered to be valid only in the vicinity of the saturation scale, $Q_s^2$ where scattering amplitude depends only on a single dimensionless variable, $Q^2\/Q_s^2$. The Regge theory is applicable if the quantity invariant mass, $W$ $(=\\sqrt{Q^2(1-x)\/x})$ is much greater than all other variables. Therefore, we expect it to be valid if $x$ is enough small, for any value of $Q^2$.\n\n\n\\par Now recalling $\\Theta(t)=1-\\Theta(-t)$ and substituting \\eqref{6} and \\eqref{2.10} in \\eqref{2.8} we get\n\\begin{equation}\n\t\\label{7}\n\t\\begin{split}\n\t\t&-x\\frac{\\partial f\\left(x,k_T^2\\right)}{\\partial x}=\\\\&\\frac{\\alpha _s N_c k_T^2 }{\\pi }\\int _{k_{T_{\\min }}^{'2}}\\frac{dk'^2}{k_T^{'}{}^2}\\bigg[\\Theta \\left(k_T^2-k_T^{'2}\\right)\\frac{1-\\left(\\frac{k_T^2}{k_T^{'2}}\\right)^{\\lambda }}{\\left| k_T^{'2}-k_T^2\\right| }f\\left(x,k_T^{'2}\\right)\\\\\n\t\t&+\\left(\\frac{k_T^2}{k_T^{'2}}\\right)^{\\lambda }\\frac{f\\left(x,k_T^2\\right)}{\\left| k_T^{'2}-k^2\\right|}-\\frac{f\\left(x,k_T^2\\right)}{\\left| k_T^{'2}-k^2\\right|} +\\frac{f\\left(x,k_T^2\\right)}{\\sqrt{k_T^4+4k_T^{'4}}}\\bigg]\\\\\n\t\t&-\\frac{36 \\alpha _s^2}{\\pi  k_T^2 R^2}\\frac{N_c^2}{N_c^2-1}(1-2^{2 \\lambda -1})f^2\\left(x,k_T^2\\right).\n\t\\end{split}\n\\end{equation}\nThe above equation \\eqref{7} is our kinematic constraint improved MD-BFKL equation in LLx accuracy.\n\n\\par Before going into in depth phenomenological analysis and consistency of the equation towards experimental result, we want to highlight two important features of the evolution equation \\eqref{7}. As we follow, in the region far below saturation limit, for the canonical choice of $\\lambda\\sim0.5$, the quadratic term in \\eqref{7} tends to vanish since $(1-2^{2\\lambda-1}) \\rightarrow0$. This indicates that below saturation region both the contribution from shadowing correction $\\frac{\\partial f_{\\text{shad}}}{\\partial \\ln{\\frac{1}{x}}}$ and antishadowing correction $\\frac{\\partial f_{\\text{antishad}}}{\\partial \\ln{\\frac{1}{x}}}$ coexists but seem to balance each other for which nonlinear effect becomes negligible. Therefore, in the below saturation regime, we can revert  \\eqref{7} back to the original BFKL equation choosing $\\lambda = 0.5$.\n\nOn the other hand, in the vicinity of saturation limit, our interpretation of gluon distribution for the nonlinear term has to be reviewed. In this region, the gluon distribution becomes flat which makes our Regge factorization for the nonlinear term in  \\eqref{2.10} invalid. Rather essentially, we should take the approximation\n\\begin{equation*}\n\tf\\left(\\frac{x}{2},k_T^2\\right) \\simeq f(x,k_T^2),\n\\end{equation*} \nwhich is supposed to justify our understanding of the saturation region. Taking this approximation in \\eqref{2.8} one can arrive at\n\\begin{equation}\n\t\\label{8}\n\t\\begin{split}\n\t\t&-x\\frac{\\partial f\\left(x,k_T^2\\right)}{\\partial x}\\\\\n\t\t&=\\frac{N_c \\alpha _sk_T^2 }{\\pi }\\int _{k_{T_{\\min }}^{'2}}\\frac{dk'^2}{k_T^{'}{}^2}\\bigg[\\Theta \\left(k_T^2-k_T^{'2}\\right)\\frac{1-\\left(\\frac{k_T^2}{k_T^{'2}}\\right)^{\\lambda }}{\\left| k_T^{'2}-k_T^2\\right| }f\\left(x,k_T^{'2}\\right)\\\\\n\t\t&+\\left(\\frac{k_T^2}{k_T^{'2}}\\right)^{\\lambda }\\frac{f\\left(x,k_T^{'2}\\right)}{\\left| k_T^{'2}-k^2\\right|}-\\frac{f\\left(x,k_T^2\\right)}{\\left| k_T^{'2}-k^2\\right|} +\\frac{f\\left(x,k_T^2\\right)}{\\sqrt{k_T^4+4k_T^{'4}}}\\bigg]\\\\\n\t\t&-\\frac{18 \\alpha _s^2}{\\pi  k_T^2 R^2}\\frac{N_c^2}{N_c^2-1}f^2\\left(x,k_T^2\\right).\n\t\\end{split}\n\\end{equation}\nEquation \\eqref{8} serves the evolution of gluon near the saturation limit. Here the $\\lambda$ dependence of the nonlinear term has been completely wiped up, unlike \\eqref{7}. In this regime, the shadowing contribution becomes twice as that of antishadowing effect forecasting a net shadowing effect in the evolution. \n\n\n\n\n\n\n\n\\subsection{Analytical solution of KC improved MD-BFKL}\\label{analytic2}\n\nIn this section, we present an analytical solution of our KC improved MD-BFKL equation, particularly in the saturation region where shadowing effect is the dominant one. Since our calculations are limited to fixed strong coupling $\\alpha_s$, so do fix $\\lambda$, therefore,  the solution for both \\eqref{7} and \\eqref{8} will exhibit the same form  only differ by the coefficients of their respective quadratic terms.\n\n\\par Recalling that there is no ordering for transverse momenta $k_T^{'}\\backsimeq k_T$ in BFKL multi-Regge kinematics, this allows us to write the gluon distribution in Taylor series,\n\\begin{equation}\n\t\\label{9}\n\tf\\left(x,k_T^{'2}\\right)=f\\left(x,k_T^2\\right)+\\frac{\\partial f\\left(x,k_T^2\\right)}{\\partial k_T^2}\\left(k_T^{'2}-k_T^2\\right) +  \\EuScript{O}\\left(k_T^{'2}-k_T^2\\right),\n\\end{equation}\nwhere  $\\EuScript{O}\\left(k_T^{'2}-k_T^2\\right)$ denotes the higher order terms. Above series is a convergent series in $\\left(k_T^{'2}-k_T^2\\right)$ as no ordering of the transverse momenta in BFKL kinematics $k_T^{'}- k_T\\backsimeq 0$ implies the higher order terms  $\\EuScript{O}\\left(k_T^{'2}-k_T^2\\right)\\rightarrow 0$, thereby ensures the higher order terms to become insignificant and can be neglected. Thus this assumption would hold good as long as no ordering condition of the transverse momenta in BFKL kinematics is concerned.  This type of series expansion of distribution function is well supported in the literature \\cite{12}.\n\\par Now neglecting these higher order terms  we can express \\eqref{8} as\n\\begin{equation}\n\t\\label{10}\n\t\\begin{split}\n\t\t-x\\frac{\\partial f\\left(x,k_T^2\\right)}{\\partial x}=&\\xi(k_T^2) \\frac{\\partial f\\left(x,k_T^2\\right)}{\\partial k_T^2}\n\t\t+\\zeta(k_T^2) f\\left(x,k_T^2\\right)\\\\&-\\Delta(k_T^2) f^2\\left(x,k_T^2\\right),\n\t\\end{split}\n\\end{equation}\nwhere\n\\begin{equation}\n\t\\label{11}\n\t\\begin{split}\n\t\t\\xi(k_T^2)=\\frac{\\alpha _s N_c }{\\pi }k_T^2\\bigg[&\\int _{k_{T_{\\min }}^{'2}}^{k_T^2}\\frac{dk_T^{'2}}{k_T^{'2}}\\frac{k_T^{'2}-k_T^2}{\\left| k_T^{'2}-k_T^2\\right| }\\\\&+\\int _{k_T^{2}}^\\infty\\frac{dk_T^{'2}}{k_T^{'2}}\\left(\\frac{k_T^2}{k_T^{'2}}\\right)^{\\lambda }\\frac{k_T^{'2}-k_T^2}{\\left| k_T^{'2}-k_T^2\\right| }\\bigg],\\\\\n\t\\end{split}\n\\end{equation}\n\n\\begin{equation}\n\t\\label{12}\n\t\\begin{split}\n\t\t\\zeta(k_T^2)&=\\frac{\\alpha _s N_c }{\\pi }k_T^2\\bigg[-\\int _{k_T^2}^{\\infty }\\frac{dk_T^{'2}}{k_T^{'2}}\\frac{1}{\\left| k_T^{'2}-k_T^2\\right| }\\\\&+\\int _{k_T^2}^\\infty\\frac{dk_T^{'2}}{k_T^{'2}}\\left(\\frac{k_T^2}{k_T^{'2}}\\right)^{\\lambda }\\frac{1}{\\left| k_T^{'2}-k_T^2\\right| }+\\int _{k_{T_{\\min }}^{'2}}^{\\infty }\\frac{dk_T^{'2}}{k_T^{'2}}\\frac{1}{\\sqrt{k_T^4+4k_T^{'4}}}\\bigg],\n\t\\end{split}\n\\end{equation}\nand $\\Delta(k_T^2)=\\frac{18 \\alpha _s^2}{\\pi  k_T^2 R^2}\\frac{N_c^2}{N_c^2-1}$.\n\\par Note that the integrals in \\eqref{12}, $I_1=\\int _{k_T^2}^{\\infty }\\frac{dk_T^{'2}}{k_T^{'2}}\\frac{1}{\\left| k_T^{'2}-k_T^2\\right| }$ and $I_2=\\int _{k_T^2}\\frac{dk_T^{'2}}{k_T^{'2}}\\left(\\frac{k_T^2}{k_T^{'2}}\\right)^{\\lambda }\\frac{1}{\\left| k_T^{'2}-k_T^2\\right| }$ are improper since both blow-up at the lower limit $k_T^2$.\nTo get rid of the singularity in $I_1$ we have performed some angular integral prescription. To be specific we have used the standard trigonometric integral\n\n\\begin{equation}\n\t\\label{13}\n\t\\int_{0}^{2\\pi}\\frac{\\text{d}\\theta}{p+q\\cos\\theta}=\\frac{2\\pi}{\\sqrt{p^2+q^2}},\n\\end{equation} \nvalid for $p+q>0$. Replacing the variable $k_T^{'} \\rightarrow k_T^{'} +k_T$ one can write\n\\begin{equation}\n\t\\label{14}\n\t\\int\\frac{\\text{d}^2k_T^{'}}{(k_T^{'}-k_T)^2}\\frac{1}{(k_T^{'}-k_T)^2+k_T^{'2}}=\\int\\frac{\\text{d}^2k_T^{'}}{k_T^{'2}}\\frac{1}{k_T^{'2}+(k_T^{'}+k_T)^2}.\n\\end{equation}\nNow using \\eqref{13} in the r.h.s. of \\eqref{14} we get\n\\begin{equation}\n\t\\label{15}\n\t\\begin{split}\n\t\t\\int\\frac{\\text{d}k_T^{'}  k_T^{'} \\text{d}\\theta}{k_T^{'2}(2k_T^{'2}+k_T^2+2k_T^{'}k_T\\cos\\theta)}&=2\\pi\\int\\frac{\\text{d}k_T^{'} k_T^{'}}{k_T^{'2} \\sqrt{4k_T^{'4}+k_T^4}}\\\\&=\\pi\\int\\frac{\\text{d}k_T^{'2}}{k_T^{'2} \\sqrt{4k_T^{'4}+k_T^4}}.\n\t\\end{split}\n\\end{equation}\nSimilarly using \\eqref{13} in the l.h.s. of \\eqref{14} we get\n\n\\begin{equation}\n\t\\label{16}\n\t\\begin{split}\n\t\t\\int\\frac{\\text{d}^2k_T^{'}}{(k_T^{'}-k_T)^2}\\frac{1}{(k_T^{'}-k_T)^2+k_T^{'2}} &=\\pi \\int \\frac{dk_T^{'2}}{k_T^{'2}}\\frac{1}{\\left| k_T^{'2}-k_T^2\\right| }\\\\& -\\pi\\int\\frac{\\text{d}k_T^{'2}}{k_T^{'2} \\sqrt{4k_T^{'4}+k_T^4}}.\n\t\\end{split}\n\\end{equation}\nEquating \\eqref{15} and \\eqref{16} we obtain\n\n\\begin{equation}\n\t\\label{17}\n\t\\int\\frac{dk_T^{'2}}{k_T^{'2}}\\frac{1}{\\left| k_T^{'2}-k_T^2\\right| }= 2 \\int\\frac{\\text{d}k_T^{'2} }{k_T^{'2} \\sqrt{4k_T^{'4}+k_T^4}},\n\\end{equation}\nwhich turns out to be well-behaved for both integration limit for $I_1$. On the other hand, in the limit $k_{T_{\\text{min}}}^{'2} \\leq k_T^{'2}\\leq k_T^2$ i.e. for  not too large $k_T^{'2}$, the contribution from the longitudinal component to the gluon virtuality $k^{'2}$ becomes negligible which in turn preserves the no ordering condition of transverse momentum of BFKL kinematics i.e. $k_T^{'2}\\approx k_T^2$ very strictly. Therefore, in this limit $k_{T_{\\text{min}}}^{'2} \\leq k_T^{'2}\\leq k_T^2$, it is justified to implant a factor $\\left(k_T^{2}\/k_T^{'2}\\right)^\\lambda$ inside the integrals which will in fact make our calculation simpler without altering the underlying physics. Moreover, this will fix the infrared divergence problem of the second improper integral $I_2$ by allowing us to evaluate the integral down to $k_{T_\\text{min}}^{'2}$.\n\\par Now considering these approximations in \\eqref{11} and \\eqref{12} and putting $I_1$ in \\eqref{12} we obtain\n\\begin{align}\n\t\\label{18}\n\t\\begin{split}\n\t\t\\xi(k_T^2)=&\\frac{\\alpha _s N_c k_T^2}{\\pi }\\bigg[\\int _{k_{T_{\\min }}^{'2}}^{k_T^2}\\frac{dk_T^{'2}}{k_T^{'2}}\\left(\\frac{k_T^2}{k_T^{'2}}\\right)^\\lambda\\frac{k_T^{'2}-k_T^2}{\\left| k_T^{'2}-k_T^2\\right| }\\\\&+\\int _{k_T^{2}}^{\\infty}\\frac{dk_T^{'2}}{k_T^{'2}}\\left(\\frac{k_T^2}{k_T^{'2}}\\right)^{\\lambda }\\frac{k_T^{'2}-k_T^2}{\\left| k_T^{'2}-k_T^2\\right| }\\bigg]\\\\=&\\frac{\\alpha_s N_c k_T^2}{\\pi}\\frac{1}{\\lambda}\\left(2-k_T^{2\\lambda}\\right)\\approx-\\frac{\\alpha_s N_c}{\\pi\\lambda}(k_T^2)^{\\lambda+1},\n\t\\end{split}\n\\end{align}\n\n\\begin{align}\n\t\\label{19}\n\t\\begin{split}\n\t\t\\zeta(k_T^2)=&\\frac{\\alpha _s N_c }{\\pi }k_T^2\\bigg[\\int _{k_{T_\\text{min}}^{'2}}^{\\infty }\\left(\\frac{k_T^2}{k_T^{'2}}\\right)^\\lambda\\frac{dk_T^{'2}}{k_T^{'2}}\\frac{1}{\\left| k_T^{'2}-k_T^2\\right| }\n\t\t\\\\&-\\int _{k_{T_{\\min }}^{'2}}^{\\infty }\\frac{dk_T^{'2}}{k_T^{'2}}\\frac{1}{\\sqrt{k_T^4+4k_T^{'4}}}\\bigg]\\\\\n\t\t=&\\frac{\\alpha _s N_c }{\\pi }\\bigg[\\frac{k_T^{2\\lambda}}{\\lambda}- 2^{2-\\frac{\\lambda }{2}}  \\lambda^{-1}k_T^{2\\lambda}\\bigg(1-\\frac{\\sqrt{k_T^4+4}}{k_T^2}\\bigg)^{\\lambda \/2} \\, _2F_1\\\\&-\\lambda\\ln \\left(\\frac{k_T^2}{2}+\\sqrt{1+\\frac{k_T^4}{4}}\\right)\\bigg]\\approx \\frac{\\alpha _s N_c }{\\pi }\\left(\\epsilon+\\frac{(k_T^2)^\\lambda }{\\lambda}\\right),\n\t\\end{split}\n\\end{align}\nwhere $\\, _2F_1$=$\\, _2\\text{F}_1\\left(-\\frac{\\lambda }{2},\\frac{\\lambda }{2};1-\\frac{\\lambda }{2};\\frac{k_T^2+\\sqrt{k_T^4+4}}{2 k_T^2}\\right)$ is a standard hypergeometric function. In the above calculations we have taken infrared cutoff $k_{T_\\text{min}}^{'2}= 1 \\text{ GeV}^2$ since for unified BFKL-DGLAP framework  this provided a very consistent result towards HERA DIS  data for proton structure function $F_2$ \\cite{14}.  In \\eqref{19} $(k_T^2)^\\lambda\/\\lambda$ is the only dominant term since other terms are significantly small.  The contribution from the logarithmic term in \\eqref{19} is negligible in comparison to the net contribution for all $k_T^2$. However, phenomenological studies shows the term involving hypergeometric function in \\eqref{19} becomes irrelevant towards change in $k_T^2$ i.e. it possess a constant value $(\\approx -4.79)$ (see Fig.~\\ref{reg}(a)). This constant contribution is insignificant to the net contribution for $\\zeta(k_T^2)$ if $k_T^2$ is enough high. But for small $k_T^2$ this contribution can not be neglected since at this range, the $k_T^{2\\lambda}\/\\lambda$ contribution  itself is small. In consequence, this constant contribution can be treated as a small perturbation $\\epsilon$ to the dominant term $k_T^{2\\lambda}\/\\lambda$ . We have performed phenomenological determination of the constant perturbation parameter $\\epsilon$ using standard non-linear regression method (see Fig.~\\ref{reg}(b)).\n\n\\begin{figure*}[tbp]\n\t\\centering\n\t\\includegraphics[width=.38\\textwidth,clip]{epsilon.eps}\\hspace{1cm}\n\t\\includegraphics[width=.38\\textwidth,clip]{N11.eps}\n\t\\caption{\\label{reg} (a) $k_T^2$ dependence of the small perturbation $\\tilde{\\epsilon}=|\\epsilon|$ corresponding to the term with the hypergeometric function in \\eqref{9} (left). (b) A phenomenological calculation of $\\epsilon$ using standard non-linear regression (right). Comparison between $\\tilde{\\zeta}=\\frac{\\pi}{\\alpha_sN_c}\\zeta$ (green dotted line) and $\\epsilon+2\\lambda^{-1}k_T^{2\\lambda}$ (red dotted line) is shown.}\n\\end{figure*}\n\n\nNow we are set to solve our original equation \\eqref{10} which is indeed a 1st order  semilinear (nonlinear) PDE. Our analytical approach of solving the same involves two steps: first we will express the nonlinear PDE in terms of a linear PDE then we will solve the linear PDE via. change of coordinates.\n\\par Substitution of $f(x, k_T^2)$  by its inverse function $\\omega(x,k_T^2)$ i.e. $f=\\omega^{-1}$ in \\eqref{10} yields\n\\begin{equation}\n\t\\label{20}\n\t-x\\frac{\\partial \\omega }{\\partial x}=\\xi \\frac{\\partial \\omega }{\\partial k_T^2}-\\zeta  \\omega +\\Delta,\n\\end{equation}\nwhich is in fact a linear PDE in $\\frac{\\partial \\omega}{\\partial x}$, $\\frac{\\partial \\omega}{\\partial k_T^2}$\nand $\\omega$. Now we construct a new set of  co-ordinate  $\\sigma\\equiv\\sigma(x, k_T^2)$ and $\\eta\\equiv\\eta(x,k_T^2)$ such that it transforms \\eqref{20} into an ODE. To be specific we define this transformation in  such a way that it is one to one for all $(x,k_T^2)$ in some set of points D in $x$-$k_T^2$ plane. This will allow us to solve \\eqref{20} for $x$ and $k_T^2$ as functions of $\\sigma$ and $\\eta$. The only requirement is that we should ensure  the Jacobian of the transformation does not vanish  i.e.\n$\\text{J}=\\left|\n\\begin{array}{cc}\n\\sigma_x & \\eta_x  \\\\\n\\sigma_{k_T^2} & \\eta_{k_T^2}  \n\\end{array} \\right| \\neq 0$ in D.\\\\\n\\par Next we want to recast \\eqref{20} in $(\\sigma,\\eta)$ plane computing the derivatives via chain rule:\n\\begin{equation}\n\t\\label{21}\n\t\\begin{split}\n\t\t&\\omega_x=\\omega_\\sigma\\sigma_x+\\omega_\\eta\\eta_x, \\\\\n\t\t&\\omega_{k_T^2}=\\omega_\\sigma\\sigma_{k_T^2}+\\omega_\\eta\\eta_{k_T^2}.\n\t\\end{split}\n\\end{equation}\nSubstitution of \\eqref{21} into \\eqref{20} yields\n\\begin{equation}\n\t\\label{22}\n\t-(x \\sigma_x+\\xi\\sigma_{k_T^2})\\omega_\\sigma-(x\\eta_x+\\xi\\eta_{k_T^2})\\omega_\\eta+\\zeta\\omega-\\Delta=0.\n\\end{equation}\nSince we want above equation to be expressed as an ODE, we require either,\n\\begin{equation*}\n\tx \\sigma_x+\\xi\\sigma_{k_T^2}=0 \\text{    or  } x\\eta_x+\\xi\\eta_{k_T^2}=0.\n\\end{equation*}\nIf we consider $x \\sigma_x+\\xi\\sigma_{k_T^2}=0$, the solution $\\sigma$ is constant along the curves that satisfy\n\\begin{equation}\n\t\\label{23}\n\t\\frac{\\text{d}k_T^2}{\\text{d}x}=\\frac{\\xi}{x} \\implies \\ln (x C_1)=\\int\\frac{\\text{d} k_T^2}{\\xi}\n\t\\implies C_1=\\frac{e^{-\\frac{k_T^{-2\\lambda}}{l \\lambda}}}{x},\n\\end{equation}\nwhere $l=-\\frac{\\alpha_s N_c}{\\pi \\lambda}$ and $C_1$ is the constant of integration. Now we can choose the new coordinates as $\\sigma =\\frac{e^{-\\frac{k_T^{-2\\lambda}}{l \\lambda}}}{x}$ and $\\eta=x$. Here $\\text{J}=\\left|\n\\begin{array}{cc}\n\\sigma_x & \\eta_x  \\\\\n\\sigma_{k_T^2} & \\eta_{k_T^2}  \n\\end{array} \\right| \\neq 0$ as required.\n\\par We rewrite \\eqref{21} as\n\\begin{align}\n\t\\label{24}\n\t\\begin{split}\n\t\t&\\omega_x=\\omega_\\sigma\\sigma_x+\\omega_\\eta\\eta_x=-\\frac{e^{-\\frac{k_T^{-2\\lambda}}{l \\lambda}}}{x^2}\\omega_\\sigma+\\omega_\\eta \\text{,}\\\\\n\t\t&\\omega_{k_T^2}=\\omega_\\sigma\\sigma_{k_T^2}+\\omega_\\eta\\eta_{k_T^2}=\\frac{(k_T^2)^{-1-\\lambda}e^{-\\frac{k_T^{-2\\lambda}}{l \\lambda}}}{lx}.\n\t\\end{split}\n\\end{align}\nNow putting \\eqref{24} in \\eqref{22} we get\n\\begin{align}\n\t\\label{25}\n\t-\\eta \\omega_\\eta+\\zeta\\omega-\\Delta=0.\n\\end{align}\nEquation \\eqref{25} is an ODE and it can be solved using standard ODE solving techniques. Now solving \\eqref{25} and then transforming it to the original coordinates $(\\sigma,\\eta)\\rightarrow (x,k_T^2)$ we get the general solution of the KC improved MD-BFKL equation, \n\\begin{widetext}\n\t\\begin{align}\n\t\\begin{split}\n\t\\label{26}\n\t&f(x,k_T^2)=\\omega^{-1}(x,k_T^2)=\\frac{k_T^{-2\\frac{n}{l}} (-1)^{\\frac{n}{\\lambda  l}} l^{-\\frac{n}{\\lambda  l}} \\lambda ^{-\\frac{n}{\\lambda  l}}}{\\text{G}\\left(\\frac{e^{-\\frac{k_T^{-2\\lambda}}{l \\lambda}}}{x}\\right) x^m+\\frac{18\\alpha_s ^2 (-1)^{1\/\\lambda } \\lambda ^{1\/\\lambda } N_c^2 l^{1\/\\lambda } e^{-\\frac{m k_T^{-2\\lambda }}{\\lambda  l}} m^{-\\frac{\\lambda  l+l+n}{\\lambda  l}} \\Gamma \\left(\\frac{l+n}{l \\lambda }+1,-\\frac{k_T^{-2\\lambda } m}{l \\lambda }\\right)}{\\pi  R^2 \\left(N_c^2-1\\right)}},\n\t\\end{split}\n\t\\end{align}\n\\end{widetext}\nwhere  $\\Gamma \\left(\\frac{l+n}{l \\lambda }+1,-\\frac{k_T^{-2\\lambda } m}{l \\lambda }\\right)$ is a standard Gamma function and $\\text{G}\\left(\\frac{e^{-\\frac{k^{-2\\lambda}}{l \\lambda}}}{x}\\right)$ is an arbitrary continuously differentiable function. The parameters $m=\\frac{\\epsilon\\alpha_sN_c }{\\pi}$ and $n=\\frac{\\alpha_sN_c}{\\pi\\lambda}$ are coming from \\eqref{19}. In the following section, we try to get particular solutions for the PDE applying some initial boundary condition and present an analysis of the phenomenological aspects of \\eqref{26}.\n\n\n\n\\subsubsection{$x$ and $k_T^2$ evolution}\n\nIn this section, we study the small-$x$ dependence of gluon distribution by picking an appropriate input distribution at some high $x$, as well as the $k_T^2$ dependence of gluon distribution setting input distribution at some low $k_T^2$.\n\\par Let us rewrite \\eqref{26} rearranging a bit \n\\begin{widetext}\n\\begin{align}\n\t\\label{27}\n\t\\begin{split}\n\t\t\\text{G}\\left(\\frac{e^{-\\frac{k_T^{-2\\lambda}}{l \\lambda}}}{x}\\right)=\\frac{1}{N_c^2-1}\\Bigg[&x^{-m}N_C^2 \\bigg(\\frac{18\\alpha_s ^2 (-1)^{\\frac{1}{\\lambda }+1} \\lambda ^{1\/\\lambda }  l^{1\/\\lambda } e^{-\\frac{m k_T^{-2\\lambda }}{\\lambda  l}} m^{-\\frac{\\lambda  l+l+n}{\\lambda  l}} \\Gamma \\left(\\frac{l+n}{l \\lambda }+1,-\\frac{k_T^{-2\\lambda } m}{l \\lambda }\\right)}{\\pi  R^2}\\\\\n\t\t&+\\frac{1}{f(x,k_T^2)} k_T^{-2\\frac{n}{l}} (-1)^{\\frac{n}{\\lambda  l}} l^{-\\frac{n}{\\lambda  l}} \\lambda ^{-\\frac{n}{\\lambda  l}}\\bigg)+\\frac{1}{f(x,k_T^2)}x^{-m} k_T^{-2\\frac{n}{l}} (-1)^{\\frac{n}{\\lambda  l}+1} l^{-\\frac{n}{\\lambda  l}} \\lambda ^{-\\frac{n}{\\lambda  l}}\\Bigg],\n\t\\end{split}\n\\end{align}\n\\end{widetext}First we will try to evaluate the functional form of the arbitrary differentiable function G applying some initial boundary condition on it. For $k_T^2$ evolution we set the initial distribution at $(x,k_0^2)$ where $x$ is fixed throughout the evolution. At $(x,k_0^2)$  let the argument of G be denoted by\n$t=\\frac{e^{-\\frac{k_0^{-2\\lambda}}{l \\lambda}}}{x}$.\n\nNow writing \\eqref{27} for $(x,k_0^2)$ and putting $x=\\frac{e^{-\\frac{k_0^{-2\\lambda}}{l \\lambda}}}{t}$ we get\n\\begin{widetext}\n\\begin{equation}\n\t\\label{28}\n\t\\begin{split}\n\t\t\\text{G}\\left(t\\right)=\\frac{1}{N_c^2-1}\\Bigg[&\\frac{e^{\\frac{mk_0^{-2\\lambda}}{l \\lambda}}}{t^{-m}}N_c^2 \\bigg(\\frac{18\\alpha_s ^2 (-1)^{\\frac{1}{\\lambda }+1} \\lambda ^{1\/\\lambda }  l^{1\/\\lambda } e^{-\\frac{m k_0^{-2\\lambda }}{\\lambda  l}} m^{-\\frac{\\lambda  l+l+n}{\\lambda  l}} \\Gamma \\left(\\frac{l+n}{l \\lambda }+1,-\\frac{k_0^{-2\\lambda } m}{l \\lambda }\\right)}{\\pi  R^2}\\\\\n\t\t&+\\frac{1}{F_0} k_0^{-2\\frac{n}{l}} (-1)^{\\frac{n}{\\lambda  l}} l^{-\\frac{n}{\\lambda  l}} \\lambda ^{-\\frac{n}{\\lambda  l}}\\bigg)+\\frac{1}{F_0}\\frac{e^{\\frac{mk_0^{-2\\lambda}}{l \\lambda}}}{t^{-m}} k_0^{-2\\frac{n}{l}} (-1)^{\\frac{n}{\\lambda  l}+1} l^{-\\frac{n}{\\lambda  l}} \\lambda ^{-\\frac{n}{\\lambda  l}}\\Bigg],\n\t\\end{split}\n\\end{equation}\n\\end{widetext}\nwhere $F_0$ is the unintegrated gluon distribution at $(x,k_0^2)$.\nEquation \\eqref{28} is the functional form of G. Replacing $t$ with any other argument will give us the value of G at that particular argument. We put $t=\\frac{e^{-\\frac{k_T^{-2\\lambda}}{l \\lambda}}}{x}$ in \\eqref{28} which gives us the l.h.s. of \\eqref{27} and then we solve  \\eqref{27} for $f(x,k_T^2)$. The solution for $k_T^2$ evolution turns out to be\n\\begin{widetext}\n\\begin{equation}\n\t\\label{29}\n\t\\begin{split}\n\t\tf(x,k_T^2)=\\dfrac{(-1)^{\\frac{n}{\\lambda  l}} \\big(N_c^2-1\\big) k_T^{-2\\frac{n}{l}} k_0^{2n\/l}   \\big( e^{\\frac{k_T^{-2\\lambda }-k_0^{-2\\lambda }}{\\lambda  l}}\\big)^m\\text{A}[x,k_T^2]}{(-1)^{\\frac{1}{\\lambda }+1}N_c^2 \\text{B}[ k_0^2 ] +(-1)^{\\frac{n}{\\lambda  l}}N_c^2 F_0 \\text{A}[x,k_T^2] +(-1)^{1\/\\lambda } N_c^2 \\text{B}[k_T^2]+ (-1)^{\\frac{n}{\\lambda  l}+1}F_0 \\text{A}[x,k_T^2]},\n\t\\end{split}\n\\end{equation}\n\\end{widetext}\nwhere\n\\begin{align*}\n\t\\begin{split}\n\t\t&\\text{A}[x,k_T^2]=\\pi  R^2 x^m m^{\\frac{\\lambda  l+l+n}{\\lambda  l}} e^{\\frac{m \\big(k_T^{-2\\lambda }+k_0^{-2\\lambda }\\big)}{\\lambda  l}}; \\\\\n\t\t&\\text{B}[i]=18 \\alpha_s ^2 x^m l^{\\frac{l+n}{\\lambda  l}} \\lambda ^{\\frac{l+n}{\\lambda  l}} k_0^{2n\/l} \\Gamma\\left. \\left(\\frac{l+n}{l \\lambda }+1,-\\frac{m (i)^{-\\lambda }}{l \\lambda }\\right)\\right\\vert_{i=k_T^2 , k_0^2}.\n\t\\end{split}\n\\end{align*}\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[trim=25 5 21.5 0,width=.46\\textwidth,clip]{x-4.eps}\\hspace{5mm}\n\t\\includegraphics[trim=25 5 21.5 0,width=.46\\textwidth,clip]{x-5.eps}\n\t\n\n\t\\caption{\\label{k1}$k_T^2$ evolution of unintegrated gluon distribution $f(x,k_T^2)$. Our results of KC improved MD-BFKL are shown for conventional $R=5 \\text{ GeV}^{-1}$ (gluons are distributed throughout the nucleus) and $R=2 \\text{ GeV}^{-1}$(at \"hot-spots\" within proton disk). Prediction from modified BK equation is plotted for comparison.}\n\\end{figure*}\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[trim=25 5 21.5 0,width=.46\\textwidth,clip]{x-3.eps}\\hspace{5mm}\n\t\\includegraphics[trim=25 5 21.5 0,width=.46\\textwidth,clip]{x-6.eps}\n\t\n\n\t\\caption{\\label{k2}$Q^2$ evolution of collinear gluon distribution $xg(x,Q^2)$. Our results of KC improved MD-BFKL are shown for conventional $R=5 \\text{ GeV}^{-1}$ (gluons are distributed throughout the nucleus) and $R=2 \\text{ GeV}^{-1}$(at \"hot-spots\" within proton disk). Theoretical prediction is compared with global datasets NNPDF 3.1sx and CT 14.}\n\\end{figure*}\nSimilarly setting the initial distribution at $(x_0,k_T^2)$, we obtain the solution for x-evolution as follows\n\\begin{widetext}\n\\begin{equation}\n\t\\label{30}\n\tf(x,k_T^2)=\\dfrac{(-1)^{\\frac{3}{\\lambda }+\\frac{2 n}{\\lambda  l}}\\left(N_c^2-1\\right) (\\frac{x_0}{x})^m k_T^{-2\\frac{n}{l}}  \\left(-\\frac{k_T^{-2\\lambda }}{\\lambda  l}-\\ln \\frac{x}{x_0}\\right)^{-\\frac{n}{\\lambda l} }\\text{C}[x,k_T^2]}\n\t{\\Bigg[\\splitdfrac{(-1)^{\\frac{3}{\\lambda }+\\frac{3 n}{\\lambda  l}}N_c^2 F_0^{'} \\text{C}[x,k_T^2] +\\frac{ (-1)^{\\frac{2}{\\lambda }+\\frac{n}{\\lambda  l}} }{k_T^2} \\Gamma_1 \\text{D}[x,k_T^2]}{+\n\t\t\t(-1)^{\\frac{2}{\\lambda }+\\frac{n}{\\lambda  l}+1}  k_T^{2\\left(\\frac{n}{l}-\\lambda  \\left(\\frac{1}{\\lambda }+\\frac{n}{\\lambda  l}\\right)\\right)} \\Gamma_2 \\text{D}[x,k_T^2]+(-1)^{\\frac{3}{\\lambda }+\\frac{3 n}{\\lambda  l}+1}\\pi  R^2 F_0^{'} x^m \\text{C}[x,k_T^2]}\\Bigg]},\n\\end{equation}\n\\end{widetext}\nwhere, $F_0^{'}$ is the initial gluon distribution at $(x_0,k_T^2)$,\n\\begin{equation*}\n\t\\begin{split}\n\t\t&\\Gamma_1=\\Gamma\\left(\\frac{l+n}{l \\lambda }+1,-\\frac{k_T^{-2\\lambda } m}{l \\lambda }\\right),\\text{     }\n\t\t\\\\&\\Gamma_2=\\Gamma\\left(\\frac{l+n}{l \\lambda }+1,m \\left(-\\frac{k_T^{-2\\lambda }}{l \\lambda }-\\ln \\frac{x}{x_0}\\right)\\right),\\\\\n\t\t&\\text{C}[x,k_T^2]=\\pi  R^2 x^ml^{-\\frac{1}{\\lambda }-\\frac{n}{\\lambda  l}} \\lambda ^{-\\frac{1}{\\lambda }-\\frac{n}{\\lambda  l}} k_T^{2\\left(\\frac{n}{l}-\\lambda  \\left(\\frac{1}{\\lambda }+\\frac{n}{\\lambda  l}\\right)\\right)}\\\\\n\t\t&\\times m^{\\frac{2}{\\lambda }+\\frac{2 n}{\\lambda  l}+1} \\left(-\\frac{k_T^{-2\\lambda }}{\\lambda  l}-\\ln \\frac{x}{x_0}\\right)^{\\frac{1}{\\lambda }+\\frac{n}{\\lambda  l}},\\\\\n\t\t&\\text{D}[x,k_T^2]=18 \\alpha_s ^2 N_c^2 x_0^m  l^{-\\frac{n}{\\lambda  l}} \\lambda ^{-\\frac{n}{\\lambda  l}} e^{-\\frac{m k_T^{-2\\lambda }}{\\lambda  l}}\\\\\n\t\t&\\times m^{\\frac{1}{\\lambda }+\\frac{n}{\\lambda  l}} \\left(-\\frac{k_T^{-2\\lambda }}{\\lambda  l}-\\ln \\frac{x}{x_0}\\right)^{1\/\\lambda }.\n\t\\end{split}\n\\end{equation*}\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[trim=25 5 21.5 0,width=.46\\textwidth,clip]{k_5.eps}\\hspace{5mm}\n\t\\includegraphics[trim=25 5 21.5 0,width=.46\\textwidth,clip]{k_50.eps}\n\t\n\n\t\\caption{\\label{x1}$x$ evolution of unintegrated gluon distribution $f(x,k_T^2)$. Our results of KC improved MD-BFKL are shown for conventional $R=5 \\text{ GeV}^{-1}$ (gluons are distributed throughout the nucleus) and $R=2 \\text{ GeV}^{-1}$(at \"hot-spots\" within proton disk). Prediction from modified BK equation is plotted for comparison.}\n\\end{figure*}\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[trim=25 5 21.5 0,width=.46\\textwidth,clip]{k_35.eps}\\hspace{5mm}\n\t\\includegraphics[trim=25 5 21.5 0,width=.46\\textwidth,clip]{k_100.eps}\n\t\n\n\t\\caption{\\label{x2}$x$ evolution of collinear gluon distribution $xg(x,k_T^2)$. Our results of KC improved MD-BFKL are shown for conventional $R=5 \\text{ GeV}^{-1}$ (gluons are distributed throughout the nucleus) and $R=2 \\text{ GeV}^{-1}$(at \"hot-spots\" within proton disk). Theoretical prediction is compared with global datasets NNPDF 3.1sx and CT 14.}\n\\end{figure*}\n\nWe have plotted the solution for both $x$ evolution \\eqref{30} and $k_T^2$ evolution \\eqref{29} in Fig.~\\ref{k1} and Fig.~\\ref{x1}. Our prediction of unintegrated gluon distribution $f(x,k_T^2)$ is contrasted with that of modified BK equation \\cite{5},\n\n\\begin{equation}\n\\label{bk}\n\\begin{split}\n&-x\\frac{\\partial f \\left(x, k_T^2\\right)}{\\partial x}=\\\\&\\frac{\\alpha_s N_c k_T^2}{\\pi}\\int_{k_{T_{\\min }}^{'2}}^{\\infty}\\frac{dk_T^{'2}}{k_T^{'2}}\\left[\\frac{f (x, k_T^{'2})-f \\left(x, k_T^2\\right)}{|k_T^{'2}-k_T^2|} + \\frac{f \\left(x, k_T^2\\right)}{\\sqrt{k_T^4+4k_T^{'4}}}\\right]\\\\\n&-\\alpha_{s}\\left(1-k_T^2\\frac{\\text{d}}{\\text{d}k_T^2}\\right)^2\\frac{k_T^2}{R^2}\\left[\\int_{k_T^2}^{\\infty}\\frac{\\text{d}k_T^{'2}}{k_T^{'2}}\\ln \\left(\\frac{k_T^{'2}}{k_T^2}\\right)f(x,k_T^2)\\right]\n\\end{split}\n\\end{equation}\nwhich is BFKL equation supplemented by the negative nonlinear term, derived in approximation of infinite and uniform target. In \\cite{5} the perturbative parton saturation is studied to a vast extent, including modification of \\eqref{bk} in terms of kinematic constraint, DGLAP, $\\text{P}_{gg}$ splitting function and running coupling constant, then solving the same numerically. We have also extracted collinear gluon distribution from unintegrated gluon distribution using the standard relation,\n\\begin{equation}\n\\label{f2g}\nxg(x,Q^2)=\\int_{0}^{Q^2}\\frac{\\text{d}k_T^2}{k_T^2}f(x,k_T^2)\n\\end{equation}\nsketched in Fig.~\\ref{k2} and Fig.~\\ref{x2}. Our predicted collinear gluon density is compared with that of LHAPDF global parameterization groups NNPDF 3.1sx \\cite{36} and CT 14 \\cite{53}. Both of the LHAPDF datasets  include HERA as well as recent LHC data with high precision PDF sensitive measurements.\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[trim=25 0 21.5 0,width=.47\\textwidth,clip]{compar_x.eps}\n\t\\hspace{1cm}\n\t\\includegraphics[trim=25 0 21 0,width=.46\\textwidth,clip]{compar_k.eps}\n\t\n\n\t\\caption{\\label{3a} (a) Comparison between linear BFKL, MD-BFKL without KC and MD-BFKL with KC for $k_T^2$ evolution at $x=10^{-4}$ and $10^{-6}$ (b) Comparison between linear BFKL, MD-BFKL without KC and MD-BFKL with KC for $x$ evolution at $k_T^2=5\\text{ GeV}^2$ and $50\\text{ GeV}^2$. Results are shown for $R= 5 \\text{ GeV}^{-1}$. }\n\\end{figure*} \n\\par Our prediction of both unintegrated and collinear gluon distribution is obtained for  two different  form of shadowing: conventional $R= 5 \\text{ GeV}^{-1}$ (order of proton radius) where gluons are spread throughout the nucleus and extreme $R= 2 \\text{ GeV}^{-1}$ where gluons are expected to concentrated in hotspots within the proton disk,  recalling that  $\\pi R^2$ is the transverse area within which gluons are concentrated inside proton. From Fig.~\\ref{k1}-\\ref{x2} it is clear that shadowing correction are more prominent when gluons are concentrated in hotspots within proton. \n\n\n\\par The $k_T^2$ (and $Q^2$) evolution in Fig.~\\ref{k1}-\\ref{k2} is studied for the kinematic range $1 \\text{ GeV}^2\\leq k_T^2(\\text{or }Q^2)\\leq 10^3 \\text{ GeV}^2$ corresponding to four different values of $x$ as indicated in the figure. Our evolution for both $f(x,k_T^2)$ and $xg(x,Q^2)$ shows a similar growth as modified BK as well as datasets respectively for all $x$. It is also observed that the growth of $f(x,k_T^2)$ (or $xg(x,Q^2)$) is almost linear for the entire kinematic range of $k_T^2$ (or $Q^2$). This is expected since the net shadowing term in \\eqref{8} has $1\/k_T^2$ dependence, which suppresses the contribution from the shadowing term at large $k_T^2$.\n\n\n\\par The $x$ evolution of $f(x,k_T^2)$ and $xg(x,Q^2)$ is shown in Fig.~\\ref{x1}-\\ref{x2} for  two $k_T^2$ values viz. $5 \\text{ GeV}^2$, $50 \\text{ GeV}^2$ and two $Q^2$ values viz. $35 \\text{ GeV}^2$, $100 \\text{ GeV}^2$. The input is taken at higher $x$ value $x=10^{-2}$ and then evolved down to smaller $x$ value upto $x=10^{-6}$ thereby setting the kinematic range of evolution $10^{-6}\\leq x\\leq10^{-2}$. We observed that the singular $x^{-\\lambda}$ growth of the gluon is eventually suppressed by the net shadowing effect. KC improved MD-BFKL seem to poses a more intense shadowing then modified BK. However, it is hard to establish the existence of shadowing for $x\\geq 10^{-3}$. The obvious distinction between the two form of shadowing $R=  \\text{5 GeV}^{-1}$ (conventional) and $R= 2\\text{ GeV}^{-1}$ (\"hotspot\") is also observed towards small-$x$ $(x\\leq 10^{-3})$.  Interestingly towards very  small-$x$ $(x\\leq 10^{-5})$, at small $k_T^2$ (or $Q^2$) values (viz. $k_T^2$=$5\\text{ GeV}^2$, $Q^2$=$35 \\text{ GeV}^2$) the gluon distribution becomes almost irrelevant of change in $x$. This could be a strong hint for the possible saturation phenomena in the small-$x$ high density regime.\n\n\\par In Fig.~\\ref{3a}(a) we have shown a comparison between the linear BFKL equation, MD-BFKL without kinematic constraint and MD-BFKL with kinematic constraint for $x$ evolution. The three solutions are compared for two different values of $k_T^2$ i.e. $ 5 \\text{ GeV}^2$ and $50 \\text{ GeV}^2$. Similarly in Fig.~\\ref{3a}(b) we have shown similar comparison but for $k_T^2$ evolution for two different values of $x$ i.e. $x=10^{-4}$ and $x=10^{-6}$. In Fig.~\\ref{3a}  the singular $x^{-\\lambda}$  behavior of $f(x,k_T^2)$ is distinct for unshadowed linear BFKL equation. On the other hand, the deviation of the two different forms of the MD-BFKL equation from linear BFKL equation reflects the underlying shadowing correction. It is also observed that shadowing effect is more intense in MD-BFKL with KC than MD-BFKL without KC.\n\n\n\\subsubsection{Complete solution of KC improved MD-BFKL}\nIn this section we implant a functional form of the input distribution (more likely a dynamic one) on the general solution of our KC improved MD-BFKL equation \\eqref{26} and try to obtain a parametric form of the solution. The underlying motivation towards doing so is that this allows us to evolve our solution for both $x$ and $k_T^2$ simultaneously in $x\\text{-}k_T^2$ phase space which help us to portray a three-dimensional realization of the gluon evolution. \n\\par Recall the well-known solution of the linear BFKL equation \\cite{9}\n\\begin{equation}\n\t\\label{32}\n\tf(x,k_T^2)=\\beta \\frac{x^{-\\lambda}\\sqrt{k_T^2}}{\\sqrt{\\ln\\frac{1}{x}}}\\exp \\left(-\\frac{\\ln^2(k_T^2\/k_s^2)}{2\\Omega\\ln (1\/x)}\\right),\n\\end{equation}\nwhere $\\lambda= \\frac{3 \\alpha_s}{\\pi} 28  \\zeta(3)$, $\\zeta$ being Reimann zeta function and $\\Omega=32.1 \\alpha_s$. The nonperturbative parameter $k_s^2= 1 \\text{ GeV}^2$ and the normalization constant $\\beta\\sim 0.01$ \\cite{1}. Since far below saturation region both linear and nonlinear equation should give the same solution, therefore we can take \\eqref{32} as the input distribution for our general solution of KC improved MD-BFKL equation.\n\nFirst we try to find the functional form of the arbitrary differentiable function $\\text{G}\\left(\\frac{e^{-\\frac{k_T^{-2\\lambda}}{l \\lambda}}}{x}\\right)$ present in the general solution \\eqref{26} applying the initial distribution \\eqref{32}.\nLet us rewrite \\eqref{27} which is the rearranged form of the general solution \\eqref{26}\n\n\n\\begin{widetext}\n\\begin{equation}\n\t\\label{33}\n\t\\begin{split}\n\t\t\\text{G}\\left(\\frac{e^{-\\frac{k_T^{-2\\lambda}}{l \\lambda}}}{x}\\right)=&\\frac{1}{N_c^2-1}\\Bigg[x^{-m}N_c^2 \\bigg(\\frac{18\\alpha_s ^2 (-1)^{\\frac{1}{\\lambda }+1} \\lambda ^{1\/\\lambda }  l^{1\/\\lambda } e^{-\\frac{m k_T^{-2\\lambda }}{\\lambda  l}} m^{-\\frac{\\lambda  l+l+n}{\\lambda  l}} \\Gamma \\left(\\frac{l+n}{l \\lambda }+1,-\\frac{k_T^{-2\\lambda } m}{l \\lambda }\\right)}{\\pi  R^2}\\\\\n\t\t&+\\frac{1}{f(x,k_T^2)} k_T^{-2\\frac{n}{l}} (-1)^{\\frac{n}{\\lambda  l}} l^{-\\frac{n}{\\lambda  l}} \\lambda ^{-\\frac{n}{\\lambda  l}}\\bigg)+\\frac{1}{f(x,k_T^2)}x^{-m} k_T^{-2\\frac{n}{l}} (-1)^{\\frac{n}{\\lambda  l}+1} l^{-\\frac{n}{\\lambda  l}} \\lambda ^{-\\frac{n}{\\lambda  l}}\\Bigg].\n\t\\end{split}\n\\end{equation}\n\\end{widetext}\nSetting initial parameters at $(x_0,k_T^2)$ we get initial distribution \\eqref{32} as\n\\begin{equation}\n\t\\label{34}\n\tf(x_0,k_T^2)=\\beta \\frac{x_0^{-\\lambda}\\sqrt{k_T^2}}{\\sqrt{\\ln\\frac{1}{x_0}}}\\exp \\left(-\\frac{\\ln^2(k_T^2\/k_s^2)}{2\\Omega\\ln (1\/x_0)}\\right).\n\\end{equation}\n\nWe denote the argument of G at $(x_0,k_T^2)$ as  $\\tau=\\frac{e^{-\\frac{k_T^{-2\\lambda}}{l \\lambda}}}{x_0}$ which implies $k_T^2=( -l\\lambda  \\ln (\\tau  x_0))^{-1\/\\lambda }$. Now writing \\eqref{33} for $(x_0,k_T^2)$  we obtain\n\\begin{widetext}\n\\begin{equation}\n\t\\label{35}\n\t\\begin{split}\n\t\t\\text{G}(\\tau)=&x^{-m} k_T^{-\\frac{n}{l}} (-l\\lambda)^{\\frac{n}{\\lambda  l}}  \\Bigg(\\frac{x^{\\lambda } \\sqrt{\\ln \\frac{1}{x}} (\\lambda  (-l) \\ln (\\tau  x_0))^{1\/\\lambda } \\exp \\left(\\frac{\\ln ^2\\left((\\lambda  (-l) \\ln (\\tau  x_0))^{-1\/\\lambda }\\right)}{2 \\Omega  \\ln \\frac{1}{x}}\\right)}{\\beta }\\\\\n\t\t&-\\frac{18\\alpha_s ^2  N_c^2 \\left(k_T^{-\\lambda }\\right)^{1\/\\lambda } \\left(e^{-\\frac{k_T^{-\\lambda }}{\\lambda  l}}\\right)^m \\left(-\\frac{m k_T^{-\\lambda }}{\\lambda  l}\\right)^{-\\frac{l+n}{\\lambda  l}} \\Gamma \\left(\\frac{l+n}{l \\lambda }+1,-\\frac{k_T^{-\\lambda } m}{l \\lambda }\\right)}{\\pi  m R^2 \\left(N_c^2-1\\right)}\\Bigg),\n\t\\end{split}\n\\end{equation}\n\\end{widetext}\nwhere, $k_T^2=( -l\\lambda  \\ln (\\tau  x_0))^{-1\/\\lambda }$. Note that \\eqref{35} is the functional form of G. We substitute $\\tau=\\frac{e^{-\\frac{k_T^{-2\\lambda}}{l \\lambda}}}{x}$ in \\eqref{35} which gives us the l.h.s. of \\eqref{33} and then solve \\eqref{33} for $f(x,k_T^2)$,\n\\begin{widetext}\n\\begin{equation}\n\t\\label{36}\n\t\\begin{split}\n\t\tf(x,k_T^2)=\\frac{\\beta  m q^{n\/l} \\left(-\\frac{m k_T^{-2\\lambda }}{\\lambda  l}\\right)^{\\frac{l+n}{\\lambda  l}} \\left(-\\frac{m q^{-\\lambda }}{\\lambda  l}\\right)^{\\frac{l+n}{\\lambda  l}}}\n\t\t{\\left(-\\frac{m q^{-\\lambda }}{\\lambda  l}\\right)^{\\frac{l+n}{\\lambda  l}} \\left(\\tilde{\\Delta}  q^{n\/l} \\tilde{A}[ k_T^2 ] +\\chi\\right)-\\tilde{\\Delta} \\beta   k_T^{2n\/l} \\tilde{A}(q) \\left(-\\frac{m k_T^{-2\\lambda }}{\\lambda  l}\\right)^{\\frac{l+n}{\\lambda  l}}},\n\t\\end{split}\n\\end{equation}\n\n\n\nwhere,\n\\begin{equation*}\n\t\\begin{split}\n\t\t&\\chi=m x^{\\lambda } \\sqrt{\\ln \\frac{1}{x}} k_T^{2n\/l} e^{\\frac{\\ln ^2(q)}{2 \\Omega  \\ln \\frac{1}{x}}} \\left(-\\frac{m k_T^{-2\\lambda }}{\\lambda  l}\\right)^{\\frac{l+n}{\\lambda  l}} q^{-1},\\text{    } \\text{   }\\tilde{A}[i]= i^{-1}\\beta  \\left(e^{-\\frac{q^{-\\lambda }}{\\lambda  l}}\\right)^m  \\Gamma \\left[i\\right], \\text{    }\\text{    } \\text{   }\n\t\tq=\\left(-l\\lambda \\ln \\frac{x_0 e^{-\\frac{k_T^{-2\\lambda }}{\\lambda  l}}}{x}\\right)^{-1\/\\lambda },\\\\\n\t\t&\\Gamma \\left[i\\right]=\\Gamma\\left(1+\\frac{l+n}{l\\lambda},\\frac{m (i)^{-\\lambda}}{l\\lambda}\\right), \\text{  }\\tilde{\\Delta}=\\frac{18 \\alpha _s^2}{\\pi  R^2}\\frac{N_c^2}{N_c^2-1}.\n\t\\end{split}\n\\end{equation*}\n\\end{widetext}\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=.40\\textwidth,clip]{MD-BFKL3D1.eps}\n\t\\caption{\\label{5x} Three dimensional representation of (a) TMDlib HERA data fit: PB-NLO-HERAI+II+2018 (left)   (b) our solution for KC improved MD-BFKL in $x$-$k_T^2$ phase space (right).}\n\\end{figure}\n\n\n\nFor simplicity let us denote r.h.s. of \\eqref{36} by $\\gamma$ i.e.\n\n\\begin{equation}\n\t\\label{37}\n\tf(x,k_T^2) = \\gamma(x,k_T^2).\n\\end{equation}\nAt $x=x_0$ and $k_T^2=k_0^2$\n\\begin{equation}\n\t\\label{38}\n\tf(x_0,k_0^2) = \\gamma(x_0,k_0^2).\n\\end{equation}\nNow dividing \\eqref{37} by \\eqref{38} we get\n\n\\begin{equation}\n\t\\label{39}\n\tf(x,k_T^2) = f(x_0,k_0^2)\\frac{\\gamma(x,k_T^2)}{\\gamma(x_0,k_0^2)}.\n\\end{equation}\nFrom \\eqref{32} we have the input distribution\n\\begin{equation}\n\t\\label{40}\n\tf(x_0,k_0^2)=\\beta \\frac{x_0^{-\\lambda}\\sqrt{k_0^2}}{\\sqrt{\\ln\\frac{1}{x_0}}}\\exp \\left(-\\frac{\\ln^2(k_0^2\/k_s^2)}{2\\Omega\\ln (1\/x_0)}\\right).\n\\end{equation}\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[width=.29\\textwidth,clip]{lambda04.eps}\\hspace{1cm}\n\t\\includegraphics[width=.29\\textwidth,clip]{lambda06.eps}\n\t\\caption{\\label{6y} Density plots showing $\\lambda$ sensitivity of unintegrated gluon distribution $f(x,k_T^2)$, sketched for two canonical choice of $\\lambda$ viz.  $\\lambda=0.4$ (left) and  $\\lambda=0.6$ (right). }\n\\end{figure*}\nNow substituting $f(x,k_0^2)$ from \\eqref{40} into \\eqref{39} we obtain\n\\begin{equation}\n\t\\label{41}\n\t\\begin{split}\n\t\t&f(x,k_T^2)=\\\\&\\frac{(\\ln \\frac{1}{x_0})^{-\\frac{1}{2}}q^{n\/l} k_0^{-2\\frac{n}{l}+4\\frac{l+n}{ l}-2}\\beta  x_0^{-\\lambda }  e^{-\\frac{\\ln ^2(k_0^2)}{2 \\Omega  \\ln \\frac{1}{x_0}}} \\left(k_T^2q\\right)^{-\\frac{l+n}{l}} \\tilde{\\phi}}\n\t\t{\\left(-\\frac{m q^{-\\lambda }}{\\lambda  l}\\right)^{\\frac{l+n}{\\lambda  l}} \\left(\\tilde{\\Delta}  q^{n\/l} \\tilde{A}[k_T^2] +\\chi\\right)-\\tilde{\\Delta} k_T^{2n\/l} \\tilde{A}[q] \\left(-\\frac{m k_T^{-2\\lambda }}{\\lambda  l}\\right)^{\\frac{l+n}{\\lambda  l}}},\n\t\\end{split}\n\\end{equation}\nwhere\n\\begin{equation*}\n\t\\begin{split}\n\t\t\\delta=&m x_0^{\\lambda } \\sqrt{\\ln \\frac{1}{x_0}} k_0^{2n\/l-2} e^{\\frac{\\ln ^2(k_0^2)}{2 \\Omega  \\ln \\frac{1}{x_0}}}   \\left(-\\frac{m k_0^{-2\\lambda }}{\\lambda  l}\\right)^{\\frac{l+n}{\\lambda  l}}, \\\\\n\t\t\\tilde{\\phi}=&\\left(-\\frac{m k_0^{-2\\lambda }}{\\lambda  l}\\right)^{\\frac{l+n}{\\lambda  l}} \\left(\\tilde{\\Delta}k_0^{2n\/l} \\tilde{A}[k_0^2] +\\delta[x,k_T^2]\\right)\\\\\n\t\t&-\\tilde{\\Delta} k_0^{2n\/l}\\tilde{A}[k_0^2]\\left(-\\frac{m k_0^{-2\\lambda }}{\\lambda  l}\\right)^{\\frac{l+n}{\\lambda  l}}.\t\n\t\\end{split}\n\\end{equation*}\n\n\\begin{figure*}[tbp]\n\t\\centering\n\t\\includegraphics[width=.29\\textwidth,clip]{R5.eps}\\hspace{1cm}\n\t\\includegraphics[width=.29\\textwidth,clip]{R2.eps}\n\t\\caption{\\label{6xx} Density plots showing $R$ sensitivity of unintegrated gluon distribution $f(x,k_T^2)$, sketched for two form of shadowing: conventional $R=5 \\text{ GeV}^{-1}$ (left) and extreme $R=2 \\text{ GeV}^{-1}$ (right).} \n\\end{figure*}\n\nEquation \\eqref{41} serves as the parametric form of the solution for KC improved MD-BFKL equation. The input distribution is inclusive in the solution \\eqref{41} itself, therefore, we do not have to depend on the experimental data fits for input distribution unlike we did in the previous section of $x$ and $k_T^2$ evolution. This parametric form of the solution actually helps us to explore the three-dimensional insight of the gluon evolution in $x$-$k_T^2$ phase space. However, using \\eqref{41} one may also study $x$ and $k_T^2$ evolution separately by setting $x$ fixed for $k_T^2$ evolution or $k_T^2$ fixed for $x$ evolution.\n\n\n\\par In Fig.~\\ref{5x} we have shown our solution of KC improved MD-BFKL equation \\eqref{41} in three dimension. The kinematic region for our study is set to be $10^{-6}\\leq x \\leq10^{-2}$ and $1 \\text{ GeV}^2\\leq k_T^2 \\leq10^3\\text{ GeV}^2$. In the 3D surface, the suppression in the rise of the gluon distribution towards small $x$ due to shadowing correction is visible. However, a linear rise of the surface in the $k_T^2$ direction can be seen, attributed to the $1\/k_T^2$ factor in the nonlinear term, which offset the effect of shadowing at large $x$.    \n\n\\par In Fig.~\\ref{6y} we have shown density plots of $f(x,k_T^2)$ in $x\\text{-}k_T^2$ domain to examine the sensitivity of $f(x,k_T^2)$ towards the parameter $\\lambda$. Density plot allows us to visualize and distinguish the kinematic regions with high\/low $f(x,k_T^2)$ in $x\\text{-}k_T^2$ plane which is more informative then any ordinary 3D plot. In Fig.~\\ref{6y} the plots are sketched for two canonical choices of $\\lambda$ viz. $\\lambda=0.4$ and $0.6$ corresponding to two $\\alpha_s$ values 0.15 and 0.23.\nOur solution seems to be very sensitive towards a small change in $\\lambda$. An apparent 50\\% change in $\\lambda$ (0.4 to 0.6) suggests approximately around one order of magnitude rise in gluon distribution $f(x,k_T^2)$ for an approximate limit of $x$ and $k_T^2$: $10^{-6}\\leq x\\leq 10^{-5}$ and $50 \\text{ GeV}^2 \\leq k_T^2\\leq 100 \\text{ GeV}^2$. It is also observed that the range of high $k_T^2$ and very small-$x$ is the high gluon distribution $f(x,k_T^2)$ region where gluons are mostly populated. In Fig.~\\ref{6xx} we have shown $R$ sensitivity of our solution for two choices of the shadowing parameter $R$ viz. $R= 5 \\text{ GeV}^{-1}$ and $2 \\text{ GeV}^{-1}$. A satisfactory shadowing effect is observed  from the comparison of the two plots. The extreme form of shadowing $(R=2 \\text{ GeV}^{-1})$ is found to suppress atleast $10\\text{-}20 \\%$ magnitude of gluon density than the conventional shadowing $(R=5 \\text{ GeV}^{-1})$ in the high $k_T^2$ and small-$x$ region.\n\n\\section{Equation of Critical line and prediction of saturation scale: a differential geometric approach}\\label{critical}\n\\begin{figure*}[tbp]\n\t\\centering\n\t\\includegraphics[width=.4\\textwidth,clip]{Saturation_scale.eps}\\hspace{1.5cm}\n\t\\includegraphics[width=.4\\textwidth,clip]{critical.eps}\n\t\\caption{\\label{6x} (a) Contour plot obtained by solving \\eqref{35x} (left). Solid lines show contours of constant gradients along the curve. (b)Diagram showing critical boundary line separating high and low gluon density region (right). Blue dashed line is obtained from \\eqref{38x} while red dashed line is corresponding to GBW model \\cite{21}.  }\n\\end{figure*}\nThe so-called saturation physics allows one to study the high parton density region in the small coupling regime. The transition from the linear region to saturation region is characterized by the saturation scale. The saturation momentum scale $Q_s$ is the threshold transverse momentum for which non-linearity becomes visible. The boundary in $(x, Q^2)$ or $(x,k_T^2)$ plane along which saturation sets in is characterized by the critical line. An important feature of our analytical solution to the  KC improved MD-BFKL equation is the finding of the equation of the critical line which is supposed to mark the boundary between dilute and dense partonic system in $x\\text{-}k_T^2$ phase space. \n\\par Although the gluon saturation can be achieved only when $Q_s\\sim Q$ (or $k_T^2$), the observables already become sensitive to $Q_s$ during the approach to saturation regime. This property is known as geometrical scaling in DIS inclusive event which means that instead of depending on $k_T^2$ and $x$ separately, the gluon distribution depends on a single dimensionless variable $\\frac{k_T^2}{Q_s^2}$ i.e.\n\\begin{equation}\n\t\\label{31s}\n\t\\phi(x,k_T^2)\\equiv\\phi\\left(\\frac{k_T^2}{Q_s^2}\\right).\n\\end{equation}\nIn recent years, this geometrical scaling property of DIS observables is studied very extensively for various frameworks \\cite{42,56, 57, 58,84}. In \\cite{56} an analysis of the saturation scale has been performed in the platform of resummed NLLx BFKL where the saturation scale was calculated via the relation\n\\begin{equation}\n\t\\label{32s}\n\t-\\frac{d\\omega(\\gamma_c)}{d\\gamma_c}=\\frac{\\omega_s(\\gamma_c)}{1-\\gamma_c},\n\\end{equation}\nwhich has been repeatedly derived in several literature \\cite{59,60,61}. In \\cite{42} the saturation scale $Q_s$ was obtained from the numerical solution of a nonlinear equation by finding the maximum of the momentum distribution of the gluon. Another approach for determination of $Q_s$ can be found in \\cite{5} where a parameter $\\beta$ is defined as the relative difference between the solutions to the linear and nonlinear equation,\n\\begin{equation}\n\t\\label{33s}\n\t\\beta=\\frac{|f^\\text{lin}(x,Q_s^2(x,\\beta))-f^\\text{lin}(x,Q_s^2(x,\\beta))|}{f^\\text{lin}(x,Q_s^2(x,\\beta))}.\n\\end{equation} \nThe crucial parameter $\\beta$ actually depicts the percentage deviation that the non linear solution shows from the linear one and it lies in the order of $0.1 \\text{-} 0.5$ (or $10\\text{-}50 \\%$ deviation).\n\n\n\\par Our approach towards studying geometrical scaling and critical line is primarily based on the basic understanding from differential geometry, in particular gradient of a function which is considered as the direction of steepest ascent of that function. Each component of the gradient gives us the rate of change of the function with respect to some standard basis i.e. it gives us an idea about how fast our function grows or decays or saturates with respect to the change of the variables. One important advantage of choosing gradient is that it is a two dimensional object since it does not possess any component along $f(x,k_T^2)$ axis in $\\mathbb{R}^3$. This ensures that it does not have any direct dependence on the magnitude of gluon density $f(x,k_T^2)$, rather it depends on the rate at which $f(x,k_T^2)$ changes with respect to $x$ and $k_T^2$ change. This property of gradient actually helps us in distinguishing out the saturation region and linear region although the distribution function $f(x,k_T^2)$ has large variation in order of magnitude for different regions in $x\\text{-}k_T^2$ phase space.\n\\par Recall that towards small-$x$, gluon evolution is suppressed due to shadowing effect as sketch in Fig.~\\ref{5x}  which motivates us to evaluate the gradient of $f(x,k_T^2)$ particularly along $x$ basis. For simplicity we consider an unit vector $\\vec{\\nu}$ along $\\tilde{\\eta} \\text{ }( = 1\/x)$ basis, we can project the gradient $\\nabla f(\\tilde{\\eta},k_T^2)$ along $\\vec{\\nu}$ via dot product $\\nabla f(\\tilde{\\eta},k_T^2)$.$\\vec{\\nu}$. This scalar quantity can also be interpreted as the directional derivative along the direction $\\vec{\\nu}$,\n\\begin{equation}\n\t\\label{34x}\n\tg\\equiv \\nabla_{\\vec{\\nu}} f(\\tilde{\\eta},k_T^2) = \\nabla f(\\tilde{\\eta},k_T^2).\\vec{\\nu}.\n\\end{equation}\nTaking the Euclidean norm yields\n\\begin{equation}\n\t\\label{35x}\n\tg = \\pm|\\nabla_{\\vec{\\nu}} f(\\tilde{\\eta},k_T^2)|,\n\\end{equation}\nwhere the negative (-) sign is for the descending function (or negative slope).\n\\par We obtained a family of contours (or level curves)  $\\tilde{\\eta}(k_T^2)$ in $\\tilde{\\eta}\\text{-}k_T^2$ plane solving \\eqref{35x} as shown in Fig.~\\ref{6x}(a). Each contour depicts a constant gradient $g$ along the curves. The set of contours can also be identified as some set of possible saturation scales. As sketch in Fig.~\\ref{6x}(a) the $\\tilde{\\eta}\\text{-}k_T^2$ plane is divided into two regions: low gluon density region where the spacing between two consecutive contours is very small and high gluon density region where the spacing becomes very large compared to previous. This distinction in contour spacing in the two regions comes from the fact that the gradient changes very fast until the saturation is reached and then after reaching the saturation boundary gradient changes very slowly or almost freezes for further increase in $\\tilde{\\eta}(k_T^2)$ as can be seen in Fig.~\\ref{6x}(a). In high gluon density region, the contour curves tend to $\\tilde{\\eta}(k_T^2)\\rightarrow \\infty $ towards high $k_T^2$. For low gluon density region, shadowing effects are negligible and the contours become almost parallel straight lines. \n\n\n\\par Let us try to find the equation of critical line which divides the two regions in $\\tilde{\\eta}\\text{-}k_T^2$ space. The level curves of the function $g=\\pm| \\nabla_{\\vec{\\nu}} f(\\tilde{\\eta},k_T^2)| $ are two dimensional curves that can be obtained by setting $g=k$ where $k$ is a constant $(k\\in \\mathbb{R})$. Therefore, the equations of the level curves are given by \n\\begin{equation}\n\t\\label{36x}\n\t| \\nabla_{\\vec{\\nu}} f(\\tilde{\\eta},k_T^2)|=\\pm k.\n\\end{equation}\nNow for a known initial saturation momentum scale $Q_{s0}(1\/x_0)$ we can predict equation of  the critical line using  \\eqref{35x} and \\eqref{36x}. The equation of the critical line is the equation for the level curve of the function $g=\\pm| \\nabla_{\\vec{\\nu}} f(\\tilde{\\eta},k_T^2)| $ that passes through the point $(Q_{s0}(\\tilde{\\eta}_0),\\tilde{\\eta}_0)$. First we find the value of $k$ by plugging the point $(Q_{s0}(\\tilde{\\eta}_0),\\tilde{\\eta}_0)$ into \\eqref{36x}\n\\begin{equation}\n\t\\label{37x}\n\tk_{s0}=\\pm| \\nabla_{\\vec{\\nu}} f(Q_{s0}^2(\\tilde{\\eta}_0))|.\n\\end{equation}\nNow the level curve passing through $(Q_{s0}(\\tilde{\\eta}_0),\\tilde{\\eta}_0)$ is obtained by setting\n\\begin{equation}\n\t\\label{38x}\n\t| \\nabla_{\\vec{\\nu}} f(\\tilde{\\eta},Q_s^2)|=| \\nabla_{\\vec{\\nu}} f(Q_{s0}^2(\\tilde{\\eta}_0))|,\n\\end{equation}\nwhich is the equation of the critical boundary. The knowledge of an appropriate initial saturation scale $Q_{s0}(1\/x_0)$ allows one to separate out the linear and saturation region using \\eqref{38x}. In Fig.~\\ref{6x}(b) we have sketched a possible critical line obtained from \\eqref{38x} for the choice of initial saturation scale $Q_{s0}^2(\\eta_0)\\backsimeq 2.8 \\text{ GeV}^{2}$ at $\\eta_0=10^6$ (or $x_0=10^{-6}$) which is taken from the calculation from the original saturation model by Golec-Biernat and Wusthoff \\cite{21, 5}. A rough agreement between our prediction and that of GBW model is observed in Fig.~\\ref{6x}(b). However, $Q_s$ given by \\eqref{38x} is found to have weaker $x$ dependence than the one from GBW model $Q_s^{'2}$. The saturation scale has direct dependence on partons per unit transverse area. Smaller $x$ suggests larger parton density giving rise to a larger saturation momentum scale, $Q_s^2$. In other words the saturation scale $Q_s$ depends on $x$ in such a way that with decreasing $x$ one has to probe  smaller distances or higher $Q^2$ in order to resolve the dense partonic structure of the proton which is clear from our analysis.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{KC improved MD-BFKL prediction of HERA DIS data}\\label{hera}\n\\subsection{DIS structure functions and reduced cross section}\\label{hera1}\nIn this section we present a quantitative prediction of proton structure functions $F_2(x,Q^2)$ and longitudinal structure function $F_L(x,Q^2)$ as an outcome of our solution to the KC improved MD-BFKL equation. At small-$x$, the sea quark distribution is driven by gluons via. $g\\rightarrow q\\bar{q}$ process. This component from sea quark distribution can be calculated in perturbative QCD. The relevant diagram for this QCD process is shown in Fig.~\\ref {d1} and the contribution to the transverse and longitudinal components of the structure functions can be written using the $k_T$-factorization theorem \\cite{25,26}\n\n\\begin{equation}\n\t\\label{42}\n\tF_\\text{i}(x,Q^2)=\\int_x^1\\frac{dx^{'}}{x^{'}}\\int\\frac{dk_T^2}{k_T^4}f\\left(\\frac{x}{x^{'}},k_T^2\\right)F_\\text{i}^{(0)}(x^{'},k_T^2,Q^2),\n\\end{equation}\nwhere i= T, L and $\\frac{x}{x^{'}}$ is the fractional momentum carried by gluon which splits into $q\\bar{q}$ pair. $F_\\text{i}^{(0)}$ includes both the quark box and cross box contribution which comes from virtual gluon-virtual photon subprocess leading to $q\\bar{q}$ production (Fig.~\\ref{d1} ). The gluon distribution $f(\\frac{x}{x^{'}},k_T^2)$ in \\eqref{42} represents the sum of the gluon ladder diagrams in the lower part of the box as shown in Fig.~\\ref {d1} is given by BFKL equation \\cite{2}. Here we will study the effect of gluon shadowing using the  solution $f(x,k_T^2)$ of KC improved MD-BFKL in \\eqref{42}.\n\n\n\\par The explicit expression for quark box contribution $F_\\text{i}^{0}$ can be obtained by writing four momentum in terms of the convenient light like momenta $p$ and  $q^{'}\\equiv q +xp$, where $x=\\frac{Q^2}{2p.q}$ and $Q^2=-q^2$ like as usual (see Fig.~\\ref{d1} ). Now we can decompose quark and gluon momentum in terms of sudakov variables\n\\begin{equation}\n\t\\begin{split}\n\t\t&\\kappa=\\alpha p-\\beta q^{'}+\\kappa_T,\\\\\n\t\t&k= a p+bq^{'}+k_T.\n\t\\end{split}\n\\end{equation}\nThe integration should be performed over the box diagram subject to quark mass-shell constraints \\cite{8}\n\n\\begin{align}\n\t\\begin{split}\n\t\t&(\\alpha -x)2p.q(1-\\beta)-\\kappa_T^2=m_q^2,\\\\\n\t\t&(a-\\alpha)2p.q\\beta-(\\kappa_T-k_T)^2=m_q^2,\n\t\\end{split}\n\\end{align}\nwhich leads to the box contribution \\cite{29}\n\\begin{align}\n\t\\label{43}\n\t\\begin{split}\n\t\t&\\tilde{F}_T^{(0)}(k_T^2,Q^2)=2 \\sum_q e_q^2\\frac{Q^2}{4\\pi^2}\\int_{0}^{1}d\\beta\\int d^2\\kappa_T\\alpha_s(\\kappa_T)\\\\&\\times\\big\\{[\\beta^2+(1-\\beta)^2]\\bigg[ \\frac{\\kappa_T^2}{L_1^2}-\\frac{\\kappa_T.(\\kappa_T-k_T)}{L_1L_2}\\bigg]+\\frac{m_q}{L_1^2}-\\frac{m_q^2}{L_1L_2} \\big\\},\n\t\t\\end{split}\n\t\\end{align}\n\t\\begin{align}\n\t\\label{44}\n\t\\begin{split}\n\t\t\\tilde{F}_L^{(0)}(k_T^2,Q^2)=&2 \\sum_q e_q^2\\frac{Q^4}{\\pi^2}\\int_{0}^{1}d\\beta\\int d^2\\kappa_T\\alpha_s(\\kappa_T)\\\\&\\times\\beta^2(1-\\beta)^2\\bigg( \\frac{1}{L_1^2}-\\frac{1}{L_1L_2}\\bigg),\n\t\t\\end{split}\n\\end{align}\nwhere the denominators $L_\\text{i}$ are\n\\begin{equation*}\n\t\\begin{split}\n\t\t&L_1=\\kappa_T^2+\\beta(1-\\beta)Q^2+m_q^2,\\\\\n\t\t&L_2=(\\kappa_T-k_T)^2+\\beta(1-\\beta)Q^2+m_q^2.\n\t\\end{split}\n\\end{equation*}\n\\begin{figure*}[tbp]\n\t\\centering\n\t\\includegraphics[width=.23\\textwidth,clip]{Structure_box1.eps}\n\t\\hspace{2cm}\n\t\\includegraphics[width=.23\\textwidth,clip]{Structure_Cross1.eps}\n\t\n\n\t\\caption{\\label{d1} Diagrammatic representation of the factorization formula \\eqref{42} where gluon couples to virtual photon through the (a) quark box (left) and (b) crossed box (right) diagrams. }\n\\end{figure*}\n\n\nNote that $\\tilde{F}_\\text{i}^{(0)}(k_T^2,Q^2)\\equiv \\int_x^1\\frac{dx^{'}}{x^{'}}F_\\text{i}^{(0)}(x^{'},k_T^2,Q^2)$ i.e. the $x^{'}$ integration of \\eqref{42} is implicit in the $d^2k_T^{'}$ and $d\\beta$ integration in \\eqref{43} and \\eqref{44}. Now plugging \\eqref{43} and \\eqref{44} in the $k_T$-factorization formula \\eqref{42} we get\n\\begin{align}\n\t\\label{45}\n\t\\begin{split}\n\t\tF_T(x,Q^2)=&2 \\sum_q e_q^2\\frac{Q^2}{4\\pi^2}\\int_{k_0^2}^\\infty\\frac{dk_T^2}{k_T^4} \\int_{0}^{1}d\\beta\\int d^2\\kappa_T\\alpha_s(\\kappa_T)\\\\&\\times\\big\\{[\\beta^2+(1-\\beta)^2]\\bigg[ \\frac{\\kappa_T^2}{L_1^2}-\\frac{\\kappa_T.(\\kappa_T-k_T)}{L_1L_2}\\bigg]\\\\\n\t\t&+\\frac{m_q}{L_1^2}-\\frac{m_q^2}{L_1L_2} \\big\\}f(\\frac{x}{x^{'}},k_T^2),\n\t\t\\end{split}\n\t\\end{align}\n\\begin{align}\n\\begin{split}\t\n\t\\label{46}\n\t\tF_L(k_T^2,Q^2)=&2 \\sum_q e_q^2\\frac{Q^4}{\\pi^2}\\int_{k_0^2}^\\infty\\frac{dk_T^2}{k_T^4}\\int_{0}^{1}d\\beta\\int d^2\\kappa_T\\alpha_s(\\kappa_T)\\\\&\\times\\beta^2(1-\\beta)^2\\bigg( \\frac{1}{L_1^2}-\\frac{1}{L_1L_2}\\bigg)f(\\frac{x}{x^{'}},k_T^2).\n\t\\end{split}\n\\end{align}\n\\begin{figure*}[tbp]\n\t\\label{}\n\t\\centering\n\t\\includegraphics[trim=21 0 21.5 0,width=.45\\textwidth,clip]{F2_5.eps}\\hspace{5mm}\n\t\\includegraphics[trim=21 0 21.5 0,width=.45\\textwidth,clip]{F2_15.eps}\\\\\n\t\\includegraphics[trim=21 0 21.5 0,width=.45\\textwidth,clip]{F2_35.eps}\\hspace{5mm}\n\t\\includegraphics[trim=21 0 21.5 0,width=.45\\textwidth,clip]{F2_45.eps}\n\t\n\t\n\t\n\n\t\\caption{\\label{7x} Prediction for proton structure function F$_2$ obtained for two choices of shadowing: conventional ($R= 5 \\text{ GeV}^{-1}$) and \"hotspot\" ($R= 2 \\text{ GeV}^{-1}$). Data are taken from HERA (H1 \\cite{32} and ZEUS \\cite{33}) as well as fixed target experiment (NMC \\cite{31} and BCDMS \\cite{34}). Data-sets from global parametrization groups NNPDF3.1sx \\cite{36} and PDF4LHC 15 \\cite{35} is also included. The background contribution is given by \\eqref{49} with $F_2^{BG}(x_0=0.1)\\approx$ 0.316, 0.384, 0.391 and 0.406 corresponding to four $Q^2$ values viz. $5 \\text{ GeV}^2, 15 \\text{ GeV}^2, 35 \\text{ GeV}^2$ and $45 \\text{ GeV}^2$. Separate plots of $F_2^{\\text{npert.}}$ vs. $F_2^{\\text{npert.+pert.}}$ is also sketched.\n\t}\n\\end{figure*}\nEquations \\eqref{45} and \\eqref{46} serve as the basic tool for calculating structure functions at small-$x$ provided that the gluon distribution $f(x,k_T^2)$ is known. An analytical approach towards the calculation of $F_2$ and $F_L$ can be found in literature \\cite{27,28} for fixed coupling case using linear BFKL equation. In the literature \\cite{9}, structure functions are calculated taking gluon distribution $f(x,k_T^2)$ from the numerical solution of unitarized BFKL equation for running coupling consideration. It has been seen that the analytical approach \\cite{28} considerably overestimates the actual (numerical) solution \\cite{9}. This is because of the fact that analytical approach neglects terms down only by powers of $\\ln (1\/z)^{-1}$ as well as it does not accommodate running of strong coupling. However, our approach is a semi-analytical one, in the sense that we take gluon distribution from our analytical solution while further integrations are performed numerically. This indeed allows us to check the feasibility of our analytical solution in describing DIS data. \n\\par In \\eqref{42}  $m_q$ denotes the quark mass and it is taken to be $m_q=1.28 \\text{ GeV}$ for charm quark while  massless ($m_q=0$) for light quarks (u, d and s). Our phenomenology is limited for light quark therefore putting $m=0$ in \\eqref{43} and replacing the quark transverse momentum by $\\kappa_T=\\kappa_T^{'}+(1-\\lambda)k_T$ we obtain \\cite{28}\n\\begin{equation}\n\t\\label{47}\n\t\\begin{split}\n\t\t&\\tilde{F}_T^{(0)}(k_T^2,Q^2)=\\\\& \\sum_q e_q^2\\frac{\\alpha_s}{\\pi}Q^2k_T^2\\int_{0}^{1}d\\beta\\int_{0}^{1}d\\lambda\\int_0^{\\infty} d\\kappa_T^{'2}[\\beta^2+(1-\\beta)^2]\\\\&\\times\\lambda\\frac{(2\\lambda-1)\\kappa_T^{'2}+(1-\\lambda)[\\lambda(1-\\lambda)k_T^2+\\beta(1-\\beta)Q^2]}{[\\kappa_T^{'2}+\\lambda(1-\\lambda)k_T^2+\\beta(1-\\beta)Q^2]^3}.\n\t\\end{split}\n\\end{equation}\nAfter integrating over $\\kappa_T^{'2}$ one can arrive at\n\\begin{equation}\n\t\\label{48}\n\t\\begin{split}\n\t\t\\tilde{F}_T^{(0)}(k_T^2,Q^2)=& \\sum_q e_q^2\\frac{\\alpha_s}{\\pi}Q^2k_T^2\\int_{0}^{1}d\\beta\\int_{0}^{1}d\\lambda\\\\&\\times\\frac{[\\lambda^2+(1-\\lambda)^2][\\beta^2+(1-\\beta)^2]}{\\lambda(1-\\lambda)k_T^2+\\beta(1-\\beta)Q^2}.\n\t\\end{split}\n\\end{equation}\nEquation \\eqref{47} and \\eqref{48}   are written in terms of Feynman integral which actually eliminates the azimuthal dependence and reduces the two fold integral $d^2\\kappa_T$ of \\eqref{43} to single integral $\\pi d\\kappa_T^{'2}$ . From \\eqref{48} it is clear that $\\tilde{F}_T^{(0)}(k_T^2,Q^2)$ or $F(x^{'},k_T^2,Q^2)$ possess the dimension of $k_T^2$. Therefore, $F(x^{'},$$k_T^2,$ $Q^2)$ or more conveniently $F(x^{'},k_T^2,Q^2)\/k_T^2$ may be considered as the structure function of an off mass shell gluon of approximate virtuality $k_T^2$. In \\cite{27} differential structure functions have been studied for fixed coupling and it is found that the ratio between longitudinal $F_L$ and transverse structure function $F_T$ is 2:9 for fixed coupling approximation. We have considered this ratio directly in our calculation of longitudinal structure function. Finally, we have taken an assumption $f(x\/x^{'}.k_T^2)\\rightarrow f(x,k_T^2)$ i.e. we ignore the $x^{'}$ dependence of $f(x\/x^{'}.k_T^2)$ which is reasonable in LLx accuracy since \n\\begin{equation*}\n\t(\\ln\\frac{x}{x^{'}})^n=(\\ln x)^n[1+O(1\/\\ln x)].\n\\end{equation*}\nThe advantage of taking this assumption is that we do not have to impose the possible constraint \\cite{8} coming from $x\/x^{'}<1$ on the region of integration. In principle, the factorization formula \\eqref{42} require to be run down to $k_T^2=0$. The integral itself is infrared finite as both the functions $f(\\frac{x}{x^{'}},k_T^2)$ and $F_i^{(0)}(x^{'},k_T^2,Q^2)$ vanish at $k_T^2=0$. However, BFKL dynamics is based on perturbative QCD which is not expected to hold the nonperturbative small $k_T^2$ physics. On the other hand, for small $k_T^2$ the gluon distribution vanishes linearly with the decrease in $k_T^2$ on account of gauge invariance \\cite{9} making the contribution small. Therefore, we have neglected this small contribution from small $k_T^2$ region in our calculations of unintegrated gluon distribution $f(x,k_T^2)$.\n\n\\begin{figure*}[tbp]\n\t\\label{}\n\t\\centering\n\t\\includegraphics[trim=21 0 21.5 0,width=.45\\textwidth,clip]{FL_5.eps}\\hspace{5mm}\n\t\\includegraphics[trim=21 0 21.5 0,width=.45\\textwidth,clip]{FL_15.eps}\\\\\n\t\\includegraphics[trim=21 0 21.5 0,width=.45\\textwidth,clip]{FL_35.eps}\\hspace{5mm}\n\t\\includegraphics[trim=21 0 21.5 0,width=.45\\textwidth,clip]{FL_45.eps}\n\t\n\t\n\t\n\n\t\\caption{\\label{77x} Prediction for proton structure function $F_L$ obtained for two choices of shadowing: conventional ($R= 5 \\text{ GeV}^{-1}$) and \"hotspot\" ($R= 2 \\text{ GeV}^{-1}$). Data are taken from HERA H1 \\cite{49} and ZEUS \\cite{33}. Data-sets from global parametrization group PDF4LHC 15 \\cite{35} is also included. The background contribution is given by \\eqref{49} with $F_L^{BG}(x_0=0.1)\\approx$ 0.057, 0.063, 0.071 and 0.073 corresponding to four $Q^2$ values viz. 5 $\\text{ GeV}^2$, 15 $\\text{ GeV}^2$, 35 $\\text{ GeV}^2$ and 45 $\\text{ GeV}^2$. Separate plots of $F_2^{\\text{npert.}}$ vs. $F_2^{\\text{npert.+pert.}}$ is also sketched. }\n\\end{figure*}\n\\begin{figure}[tbp]\n\t\\centering\n\t\\includegraphics[width=.48\\textwidth,clip]{RQ1.eps}\n\t\\caption{\\label{Rx} pQCD prediction from KC improved MD-BFKL for the ratio $R(Q^2)$ in the kinematic range $2 \\text{ GeV}^2\\leq Q^2\\leq 100 \\text{ GeV}^2$. The data \\cite{49} are from H1  ($\\sqrt{s}= 252\\text{ GeV}$) and ZEUS ($\\sqrt{s}= 225\\text{ GeV}$) experiment. The theoretical calculation are for $\\sqrt{s}= 225\\text{ GeV}$ analogous to c.m.s. of ZEUS .\n\t}\t\n\\end{figure}\n\n\\par Before proceeding to the realistic estimation of the structure functions, we add on the \"background\" some non-BFKL contribution, $F_\\text{i}^{\\text{BG}}$ to $F_\\text{i}$  since the above prediction of structure function is not enough for describing DIS data \\cite{9}. This is because \\eqref{45} and \\eqref{46} represent only the LLx gluon contribution and they are not the only contributions to the DIS structure functions. Although gluonic contribution is dominant at small-$x$, towards higher $x$ their effect becomes weak and we cannot neglect other non-BFKL contribution. For instance, we assume that $F_\\text{i}^{\\text{BG}}$ evolves like $x^{-0.08}$ motivated from soft pomeron intercept $\\alpha_{P}(0)=1.08$ \\cite{30}. To be precise we use \n\\begin{equation}\n\t\\label{49}\n\tF_\\text{i}^{\\text{BG}}(x,Q^2)=F_\\text{i}(x_0,Q^2)\\left(\\frac{x}{x_0}\\right)^{-0.08}.\n\\end{equation}\nAn intelligent way of calculating this non-BFKL contribution is to choose $x_0$ at some high $x$ and then take $F_\\text{i}^{\\text{BG}}(x_0,Q^2)$ from data \\cite{31} which is also listed in the figure caption. The recent HERA DIS data taken for comparison with our result can be found in \\cite{32,49} from H1 collaboration and in \\cite{33} from ZEUS collaboration. In the text \\cite{32} from H1 collaboration inclusive neutral current $e^{\\pm}p$ scattering cross section data collected during the years (2003-2007) is presented. The beam energies $E_p$ of corresponding H1 experiment run are $920$, $575$ and $460 \\text{ GeV}^2$. Corresponding $F_L$ data of H1 is taken from \\cite{49} where measurement are performed at c.m.s. energies $\\sqrt{s}=$ 225 and 252 $\\text{GeV}$. On the other hand, in \\cite{33} from ZEUS collaboration reduced cross sections for $ep$ scattering for different c.m.s. energies viz. $318$, $251$ and $225\\text{ GeV}$ is presented. The fixed target data from NMC \\cite{31} and BCDMS \\cite{34} collaboration which exist for $x>10^{-2}$ is also shown in the Figure \\ref{7x}. Finally, we have included data from global parameterization groups viz. NNPDF3.1sx \\cite{36} and PDF4LHC15 \\cite{35} for comparison. \n\n\n\\begin{figure*}[tbp]\n\t\\centering\n\t\\includegraphics[width=.32\\textwidth,clip]{cros2.eps}\n\t\\includegraphics[width=.32\\textwidth,clip]{cros5.eps}\n\t\\includegraphics[width=.32\\textwidth,clip]{cros85.eps}\\\\\n\t\\includegraphics[width=.32\\textwidth,clip]{cros12.eps}\t\n\t\\includegraphics[width=.32\\textwidth,clip]{cros25.eps}\n\t\\includegraphics[width=.32\\textwidth,clip]{cros45.eps}\n\n\t\\caption{\\label{8x} Theoretical prediction from KC improved MD-BFKL for reduced cross section $\\sigma_r (x,Q^2)$. The data are from H1 \\cite{40,83} ($\\sqrt{s}\\approx$ 318, 300 and 252 $\\text{GeV}^2$) and ZEUS \\cite{33} ($\\sqrt{s}\\approx$ 318 $\\text{GeV}^2$). Theoretical calculations are for $\\sqrt{s}= 318 \\text{ GeV}^2$.}\n\\end{figure*}\n\\par The $x$ dependence of the structure functions $F_L$ and $F_2$ is shown in Fig.~\\ref{7x} and Fig.~\\ref{77x} for the four $Q^2$ values $5 \\text{ GeV}^2$, $15 \\text{ GeV}^2$, $35 \\text{ GeV}^2$ and $45 \\text{ GeV}^2$. The nonperturbative contribution, $F_{2\/L}^{\\text{npert.}}$ (or $F_{2\/L}^{\\text{BG}}$) given by \\eqref{49} contrasted with total (nonpert. +pert.) contribution,  $F_{2\/L}^{\\text{npert.+pert.}}$. QCD prediction shows a satisfactory agreement with DIS data for both structure functions $F_2$ and $F_L$. It is clear that perturbative contribution is insignificant towards large $x$  ($\\geq 10^{-2}$), while for small $x$ around $x\\leq 10^{-3}$ this contribution becomes visibly important. Interestingly both data and theory for $F_2$ structure function seem to preserve the $x^{-\\lambda}$ singular behavior rather than showing taming due to net shadowing effect in the asymptotic limit $x\\rightarrow0$. It is difficult to observe the existence of any sizable shadowing effect for $x>10^{-3}$. However, for $x<10^{-3}$, recalling that the shadowing term is proportional to $1\/R^2$ it is seen that for $R=2\\text{ GeV}^{-1}$ (gluons in hotspots) the rise of structure function becomes slower than that of $R=5\\text{ GeV}^{-1}$ ($R=R_H$, hadron radius) as expected. But even the extreme form of the gluon shadowing ($R=2\\text{ GeV}^{-1}$) suppresses $F_2$ only by about $10\\%$ or less at $10^{-3}$. It just depicts that shadowing has the negligible impact on structure functions even towards smaller $x$. This is because of the fact that in this low $x$ regime gluons are expected to drive the sea quark distribution via $g\\rightarrow q\\bar{q}$. Therefore, a similar sea quark contribution to the structure function $F_2$ in addition to the gluon contribution can be expected in low $x$ regime. On the other hand from Fig.~\\ref{77x} it is clear that for $x>10^{-3}$, the size of the longitudinal structure function is negligibly small while for very small-$x$ regime ($x<10^{-3}$) , $F_L$  grows eventually. This is in accord with our expectation since measurement of $F_L$ directly probes the gluonic content of the proton which is dominant in small-$x$ regime. \n\n\n\\par The proton structure functions $F_2$ and $F_L$ are of complementary in nature. These are related to the $\\gamma^{*}p$ cross sections of longitudinally and transversely polarized photons $\\sigma_L$ and $\\sigma_T$ as $F_L\\propto\\sigma_L$ and $F_2\\propto (\\sigma_L+\\sigma_T)$.  Since $\\sigma_L$ and $\\sigma_T$ are positive, that imposes a restriction on $F_L$ and $F_2$ i.e. $0\\leq F_L\\leq F_2$. To circumvent the need of a proper relationship between the structure functions and  polarized cross sections to study $\\gamma^*p$ cross section one can define a ratio between the structure functions or the equivalent cross section ratio\n\\begin{equation}\n\t\\label{R}\n\tR=\\frac{\\sigma_L}{\\sigma_T}=\\frac{F_L}{F_2-F_L}.\n\\end{equation}\nThe advantage behind this formulation is that this is independent of any normalization factors.\n\n\\par In Fig \\ref{Rx} a phenomenological comparison between data and theory is shown to illustrate $Q^2$ dependence of the ratio $R(x,Q^2)$. The HERA data taken for comparison can be found in \\cite{49} where $ep$ cross section data measured for two center of mass energies $\\sqrt{s}=225$ and $252\\text{ GeV}$ is enlisted. From Fig.~\\ref{Rx} it  is clear that the HERA data is in general agreement with the QCD expectation. Available H1 data for very low $Q^2$ $(Q^2\\leq 5\\text{ GeV})$ is also sketched in Fig.~\\ref{Rx}. We have excluded the very low $Q^2$ region in our calculation of $R$ since we are bounded to stick in the perturbative region. However, a more realistic study in this transition region is performed in Sect.~\\ref{hera2}. \n\n\\par Now we try to predict reduced cross section $\\sigma_r$ for $ep$ scattering process from the knowledge of structure functions that we have discussed above. The inclusive deep-inelastic differential cross section for $ep$ scattering can be represented in terms of three structure functions $F_2$, $F_L$ and $xF_3$. The structure functions have direct dependence with DIS kinematic variables, $x$ and $Q^2$ only, while the cross section has additional dependence with inelasticity $y=Q^2\/s x$. The inclusive cross section for neutral current $ep$ scattering is given by\n\\begin{equation}\n\t\\label{50}\n\t\\begin{split}\n\t\t\\frac{d^2 \\sigma^{e^{\\pm}p}}{dx dQ^2}=\\frac{2\\pi\\alpha^2}{x Q^4}Y_{+}\\left(F_2-\\frac{y^2}{Y_+}F_L\\mp\\frac{Y_{-}}{Y_{+}}xF_3\\right),\n\t\\end{split}\n\\end{equation}\nwhere $Y_{\\pm}=1\\pm(1-y)^2$. The cross section for exchanged virtual boson $(\\text{Z} \\text{ or } \\gamma^{*})$ is related to $F_2$ and $F_3$ in which contribution from both longitudinal and transverse boson polarization state exists. On the other hand, only the longitudinal polarized virtual boson exchange processes contribute to $F_L$ which has a significant impact on higher order QCD though it vanishes at lowest order. At small momentum transfer $Q^2$ $(\\text{i.e. } Q^2\\ll M_Z^2\\approx800\\text{ GeV}^2)$, interaction of massless photon is dominant over the exchange of heavy Z boson. Thus, the contribution from Z boson exchange and the influence of the term $xF_3$ is negligible at low and moderate $Q^2$. Therefore, in this range of $Q^2$, in one photon exchange approximation, the differential cross section formula \\eqref{50} can be written as \n\n\\begin{equation}\n\t\\label{51}\n\t\\sigma_r\\equiv\\frac{d^2 \\sigma^{e^{\\pm}p}}{dx dQ^2}\\frac{x Q^4}{2\\pi\\alpha^2}\\frac{1}{Y_{+}}=\\left(F_2-\\frac{y^2}{Y_+}F_L\\right),\n\\end{equation}\nwhere $\\sigma_r$ is the reduced cross section. Note that \\eqref{51} is symmetric under charge exchange i.e. identical for both $e^{+}p$ and $e^{-}p$ processes unlike \\eqref{50}. Additionally, it is independent  of incoming electron  helicity state. Thus, at low and moderate $Q^2$ $(\\leq m_Z^2)$ which is our region of study, the knowledge of $F_2$ and $F_L$ is enough to predict the reduced cross section. We can also express reduced cross section in terms of the ratio $R(x,Q^2)$ defined in \\eqref{R} replacing $F_L$ by $F_L=\\frac{R}{1+R}F_2$ which yields\n\\begin{equation}\n\t\\label{51x}\n\t\\sigma_r=F_2(x,Q^2)\\left[1-\\frac{y^2}{Y_+}\\frac{R}{1+R}\\right].\n\\end{equation}\n\n\\par The $x$ dependence of $e^{\\pm}p$ reduced cross section $\\sigma_r$ calculated for center of mass energy $\\sqrt{s}=318\\text{ GeV}$ is shown in Fig.~\\ref{8x}. Our theoretical expectation is compared with HERA H1 measurement \\cite{40,83} and ZEUS  ($\\sqrt{s}=318\\text{ GeV}$) \\cite{33}. The available H1 data for low $Q^2$ ($\\leq12\\text{ GeV}^2$) measured at $\\sqrt{s}=$ 318 $\\text{GeV}$ (SVX) and 300 $\\text{GeV}$ (NVX-BST) is taken from \\cite{40}, while the same for high $Q^2$ ($>12\\text{ GeV}^2$) is taken from \\cite{83}.\nThe cross section measurement of SVX is found to be slightly higher than that of NVX-BST as expected because of the increase in center of mass energy. Both theory and data agree well for our phenomenology range $Q^2\\leq 100\\text{ GeV}^2$. The distinction in $\\sigma_r$ due to the two forms of shadowing $R=5\\text{ GeV}^{-1}$ and $R=2\\text{ GeV}^{-1}$ is more prominent for smaller $Q^2$ values. For each $Q^2$, starting at some high $x$ the reduced cross section first increases as $x\\rightarrow0$ and then an abrupt fall in cross section can be observed in both theory and data at very small-$x$ region ($x<10^{-4}$) . For all $Q^2$, this region of $x$ corresponds to the highest inelasticity $y\\approx0.65$ ($y=Q^2\/sx$) and thus characteristic turn over of cross section at $y\\approx0.65$ can be attributed to the influence of $F_L$.  In simple words, towards very small-$x$ (or high $y$) the monotonic rise of $F_2$ is suppressed by the contribution from longitudinal structure function $F_L$ thereby causing an overall fall in the cross section. For low inelasticity $y<0.65$, the contribution from the longitudinal structure function is small on the other hand, structure function $F_2$ exhibits a steady increase as $x\\rightarrow 0$. Therefore, in the region where $x$ is not so small, the growth of the cross section is found to be power like as expected. \n\n\n\\subsection{Virtual photon-proton cross section prediction in transition region}\\label{hera2}\n\\par Traditionally the photon-proton interaction is classified into two separate processes depending upon the photon virtuality $Q^2$ viz. photoproduction (at low $Q^2$) and deep inelastic scattering (at high $Q^2$). DIS is considered  as a basic tool for exploring pQCD where the point like virtual photon directly probes the partonic contents of the proton. On the other hand, photoproduction is completely nonperturbative phenomena defined in the limit of vanishing $Q^2$ where real (or quasi-real) photons interact with the proton more likely a hadron-hadron collision. The experiment at HERA collider provides a unique opportunity to study both the processes photoproduction and DIS on their respective kinematic domains. The $Q^2$ dependence of the proton structure functions are well described by pQCD over a wide range of $x$ and $Q^2$ \\cite{37,38} in accordance with HERA data. However, for $Q^2\\lesssim 2\\text{ GeV}^2$ (photo production region) the pQCD breakdowns since the higher order contributions to the perturbative expansion becomes very large. In this region, data can be only described by non-perturbative phenomenological models \\cite{39}. Our present study is especially focused on the transition region $(2\\text{ GeV}^2\\leq Q^2\\leq10\\text{ GeV}^2)$ from photoproduction to deep inelastic scattering. To measure the photon-proton cross section in the transition region two dedicated runs were performed in the years 1999 (Nominal vertex \"NVX'99\") and 2000 (Shifted Vertex \"SVX'00\") by H1 experiment at HERA. The published data can be found in \\cite{40}.\n\n\n\n\\par Recall the neutral current $ep$ double differential cross section formula \\eqref{51} defined in the region $Q^2\\ll M_Z^2$. In this region massive boson $(M_Z)$ exchange is neglected and only one photon exchange is considered, thereby the role of incoming electron reduces to be a source of virtual photon interacting proton. Thus, we can recast the formula \\eqref{51} in terms of photon-proton reaction. In fact the structure function $F_2$ and $F_L$ in \\eqref{51} related to the longitudinally and transversely polarized photon-proton scattering cross sections $\\sigma_L$ and $\\sigma_T$ by the relations\n\\begin{align}\n\t\\label{52}\n\t&F_L = \\frac{Q^2}{4\\pi^2\\alpha}(1-x)\\sigma_L,\\\\\n\t\\label{53}\n\t&F_2 = \\frac{Q^2}{4\\pi\\alpha}(1-x)(\\sigma_L+\\sigma_T),\n\\end{align}\nwhich hold good at low $x$. Considering \\eqref{52} and \\eqref{53} the reduce cross section in \\eqref{51} can be written as\n\\begin{equation}\n\t\\label{54}\n\t\\sigma_r = \\frac{Q^2(1-x)}{4\\pi^2\\alpha}\\sigma_{\\gamma^*p}^{\\text{eff}},\n\\end{equation}\nwhere \\begin{align}\n\t\\label{55}\n\t\\sigma_{\\gamma^*p}^{\\text{eff}}=\\sigma_T+(1-\\frac{y^2}{Y_+})\\sigma_L\n\\end{align}\nis the effective virtual photon-proton cross section. Note that the expression for  $\\sigma_{\\gamma^*p}^{\\text{eff}}$ is similar to the total cross section $\\sigma_{\\gamma^*p}^{\\text{tot}}$ which is linear combination of $\\sigma_L$ and $\\sigma_T$ i.e.  \n$\\sigma_{\\gamma^*p}^{\\text{tot}}=\\sigma_L+\\sigma_T$. In fact the total cross section $\\sigma_{\\gamma^{*}p}^{\\text{tot}}$ and $\\sigma_{\\gamma^*p}^{\\text{eff}}$ can be regarded as the same quantity at low inelasticity $y$ i.e. $\\sigma_{\\gamma^*p}^{\\text{eff}} \\xrightarrow{y\\rightarrow0}\\sigma_{\\gamma^*p}^{\\text{tot}}$ which differ only in the region of high y.\n\n\\begin{figure}[tbp]\n\t\\centering\n\t\\includegraphics[width=.4\\textwidth,clip]{W_cross.eps}\n\t\\caption{\\label{7xx} QCD prediction from KC improved MD-BFKL for virtual photon-proton cross section $\\sigma_{\\gamma^{*}p}^\\text{eff}$ as a function of $Q^2$ at different values of $W$. The cross section for different $W$ values are multiplied by factor multiple of 2 indicated in the figure. The included data are from  H1 \\cite{40} ($\\sqrt{s}= 318\\text{ GeV}$) and ZEUS  \\cite{81} ($\\sqrt{s}= 300\\text{ GeV}$). }\t\n\\end{figure}\n\\par Figure \\ref{7xx} shows the measurement of the virtual photon-proton cross section $\\sigma_{\\gamma^{*}p}^\\text{eff}$ as a function of $Q^2$ corresponding to different values of $W$.  The total cross section $\\sigma_{\\gamma^*p}^{\\text{tot}}$ is often expressed as a function of $Q^2$ and invariant mass $W$. The standard relation between $W$, $x$ and $Q^2$ is $W=\\sqrt{Q^2(1-x)\/x}$. Since $Q^2\\approx syx$, for small-$x$ we can have an approximate relationship between $W$ and $y$ i.e. $W^2\\simeq sy$ which we have used in our calculations. HERA measurement for $\\sigma^{\\text{eff}}_{\\gamma^{*}p}$ from H1 \\cite{40} ($\\sqrt{s}= 318\\text{ GeV}$) and ZEUS  \\cite{81} ($\\sqrt{s}= 300\\text{ GeV}$) are included for comparison with our theoretical prediction. We have chosen  $\\sqrt{s}= 318\\text{ GeV}$ for our calculation analogous to H1 measurement. The precision of the data is such that their errors are hardly visible.  Both the HERA data and theoretically measured cross sections for different values of $W$ are multiplied by the factors multiple of 2 as indicated in the figure. For $Q^2\\gtrsim3 \\text{ GeV}^2$, an excellent agreement between the theory and HERA data can be observed for the wide range of $W$, while for $Q^2<3 \\text{ GeV}^2$, the slop of the QCD prediction seems unsatisfactory. This indicates the inadequacy of perturbative QCD at very low $Q^2$ ($<3 \\text{ GeV}^2$) and roughly provides a lower bound of $Q^2$ to our theory.\n\n\n\n\n\n\n\n\n\n\\section{Conclusion}\\label{con}\n\n\\par In conclusion, we have presented a phenomenological study on the behavior of unintegrated gluon distribution at small-$x$ and moderate $k_T^2$ region particularly $10^{-6}\\leq x\\leq 10^{-2}$ and $2 \\text{ GeV}^2\\leq k_T^2\\leq 1000 \\text{ GeV}^2 $ which is also the accessible kinematic range to experiments performed at HERA $ep$ collider. In the beginning, we have improved the MD-BFKL equation supplementing so-called kinematic constraint on it. Then we have solved this unitarized BFKL equation analytically in order to study $x$ and $k_T^2$ dependence of unintegrated gluon distribution function. Our prediction of gluon distribution is contrasted with that of modified BK equation as well as global datasets NNPDF 3.1sx and CT 14. The $x$ evolution of gluon distribution shows the singular $x^{-\\lambda}$ type behavior of gluon evolution tamed by shadowing correction. We found that for intense shadowing $(R= 2 \\text{ GeV}^{-1})$, towards smaller x, certainly from $x=10^{-4}$, the gluon distribution emerges from rapid BFKL growth which indicates the dominance of gluon shadowing. While in case of conventional shadowing $(R= 5 \\text{ GeV}^{-1}) $ an appreciable modification of BFKL behavior can only be seen from $x= 10^{-5}$. The $k_T^2$ dependence of unintegrated gluon distribution is also studied. Although obvious suppression due to net shadowing correction is seen in our studies, however no significant saturation phenomenon is observed. The reason is the nonlinear contribution term is suppressed by the factor $1\/k_T^2$ at large values of $k_T^2$.\n\n\\par We have obtained a more general solution of KC improved MD-BFKL equation implementing a pre-defined input distribution on it. This has allowed us to visualize the gluon evolution in three dimensions $\\mathbb{R}^3$ into $x\\text{-}k_T^2$ phase space. We have also shown the sensitiveness of unintegrated gluon distribution towards $R$ and $\\lambda$ using density plots in $x\\text{-}k_T^2$ plane.\n\n\\par An important achievement of obtaining an analytical solution in this work is its implication on qualitative studies on geometrical scaling which is presented in Sect. \\ref{critical}. Starting from a basic concept of differential geometry and knowledge of level curves we have managed to obtain an equation of the critical boundary which is supposed to separate low and high gluon density regions.\n\\par In Sect. \\ref{hera} we have studied the small-$x$ dependence of the structure function $F_2$ and $F_L$ obtained via $k_T$-factorization formula $F_i=f\\otimes F_i^{(0)}$. Here the unintegrated gluon distribution $f(x,k_T^2)$ is taken from our analytical solution of KC improved MD-BFKL equation setting boundary condition at $x_0=0.01$. Surprisingly the quantitative size of the shadowing correction to $F_2$ and $F_L$ is found to be very small and the structure functions seem to hold the singular $x^{-\\lambda}$ behavior. Even at intense shadowing condition $(R= 2 \\text{ GeV}^{-1})$ the $F_2$ structure function is found to be suppressed only by $10\\%$. In addition we have measured $e^{\\pm}p$ reduced cross section as well as equivalent $\\gamma^*p$ longitudinal to transverse cross section ratio $R(x,Q^2)$ from the knowledge of $F_2$ and $F_L$. Our results are compared with recent high precision HERA measurements. The comparison reveals a good agreement between our theory and DIS data.\n\\par In Sect. \\ref{hera2} we have examined the virtual photon-proton effective cross section, particularly in the transition region from photoproduction to deep inelastic scattering. The quantity $\\sigma_{\\gamma^{*}p}^\\text{eff}$ serves the role of the total cross section if small inelasticity $y$ is concerned. We have compared our predicted $\\sigma_{\\gamma^{*}p}^\\text{eff}$ with the HERA data of two dedicated runs \"NVX'99\" and \"SVX'00\" by H1 as well as ZEUS for the transition region. Our theoretical prediction shows well-consistency with HERA data particularly upto $Q^2\\sim3 \\text{ GeV}^2$ in the transition region.\n\n\n\\par In summary, there are several attractive features of our present study. First, we have able to predict a wide range of physical quantities, starting right from our solution for unintegrated gluon density. Secondly, all analysis is performed in terms of relatively small numbers of parameters. Moreover, the idea developed in this work for studying geometrical scaling can be implemented in any other framework. Finally, a very significant feature of this analysis is that we have considered two extreme possibilities of shadowing which can be distinguished by experimental data. In the end, we conclude that, as per feasibility towards HERA DIS data is concerned, the KC improved MD-BFKL equation could be a reliable framework for exploring high energy physics over a wide range of $x$ and $k_T^2$ which is also relevant for LHC probe and future collider phenomenology. \n\n\n\\section*{Acknowledgments}\nWe thank Prof.~Johannes Bluemlein, DESY (Hamburg, Germany) and Prof.~Dieter Schildknecht, Bielefeld university (Germany)  for their valuable comments. Two of us ( P.P. and M.L.) acknowledge Department of Science and Technology (DST), India (grant DST\/INSPIRE Fellowship \/2017 \/IF160770) and Council of Scientific and Industrial Research (CSIR), New Delhi (grant 09\/796(0064)2016-EMR-I) respectively for the financial assistantship. \n\n\n\n\n\\bibliographystyle{spphys}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}