text stringlengths 14 5.77M | meta dict | __index_level_0__ int64 0 9.97k ⌀ |
|---|---|---|
class MidsagittalRotation{
private:
Mesh pointCloud;
public:
MidsagittalRotation(const Mesh& pointCloud) :
pointCloud(pointCloud) {
}
void estimate(arma::vec& origin, double& angle) {
angle = estimate_angle();
origin = this->pointCloud.get_center();
return;
}
private:
double estimate_angle() {
// project point cloud into y-x-plane
std::vector<arma::vec> points;
for(arma::vec position: this->pointCloud.get_vertices()) {
position(2) = 0;
points.push_back(position);
}
// use PCA to find most dominant principal direction
arma::vec dominantAxis;
perform_pca(points, dominantAxis);
// projected midsagittal data should vary the most in y-direction
// compute angle in degrees between y-axis and most dominant principal direction
// and compute angle that rotates principal direction back to y-axis
const arma::vec yAxis({0, 1, 0});
const double angle = - acos(arma::norm_dot(dominantAxis, yAxis)) * 180 / M_PI;
return angle;
}
arma::vec compute_mean(const std::vector<arma::vec>& points) const {
arma::vec mean = arma::zeros(3);
for(const arma::vec& point: points) {
mean += point;
}
return mean / points.size();
}
void center_points(std::vector<arma::vec>& points, const arma::vec& mean) const {
for(arma::vec& point: points) {
point -= mean;
}
}
void perform_pca(
std::vector<arma::vec> points,
arma::vec& dominantAxis
) const {
const arma::vec mean = compute_mean(points);
center_points(points, mean);
arma::mat C = construct_covariance_matrix(points);
arma::vec eigval;
arma::mat eigvec;
arma::eig_sym(eigval, eigvec, C, "std");
dominantAxis = eigvec.col(2);
}
arma::mat construct_covariance_matrix(const std::vector<arma::vec>& points) const {
arma::mat result = arma::zeros(3, 3);
for(const arma::vec point: points) {
result += point * point.t();
}
return result;
}
};
#endif
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,247 |
{"url":"https:\/\/math.stackexchange.com\/questions\/1117926\/question-about-series-convergence-sum-n-1-infty-frac1n-and-sum-n-1","text":"# Question about series convergence $\\sum_{n=1}^\\infty \\frac{1}{n}$ and $\\sum_{n=1}^\\infty \\frac{1}{n^2}$\n\nSo I have been playing around with convergent series recently and I still have a hard time understanding why $\\sum_{n=1}^\\infty \\frac{1}{n}$ diverges and $\\sum_{n=1}^\\infty \\frac{1}{n^2}$ converges.\n\nI can write out the first couple of terms of $\\sum_{n=1}^\\infty \\frac{1}{n}$ and compare them to a second series:\n\n$\\color{red}{1}+\\color{green}{\\frac{1}{2}}+\\color{blue}{\\frac{1}{3}+\\frac{1}{4}}+\\color{maroon}{\\frac{1}{5}+\\frac{1}{6}+\\frac{1}{7}+\\frac{1}{8}}+...$\n\n$\\color{red}{1}+\\color{green}{\\frac{1}{2}}+\\color{blue}{\\frac{1}{4}+\\frac{1}{4}}+\\color{maroon}{\\frac{1}{8}+\\frac{1}{8}+\\frac{1}{8}+\\frac{1}{8}}+...$\n\nThe colored terms in the second series always add up to $\\frac{1}{2}$ and the corresponding colored terms in the first series always add up to more than $\\frac{1}{2}$. Since the second series diverges the first one must also diverge. This still makes perfect sense. However if I write out the first couple of therms of $\\sum_{n=1}^\\infty \\frac{1}{n^2}$\n\n$\\frac{1}{1}+\\frac{1}{4}+\\frac{1}{9}+\\frac{1}{16}+\\frac{1}{25}+\\frac{1}{36}+\\frac{1}{49}+\\frac{1}{64}...$\n\nthen how does this converge?\n\nWhy is it not possible to group them similar to the first series? It seems to me that if I have infinitely many numbers I will be able to group $n$ of them to get to some number $k$ that I can add up forever.\n\n\u2022 They decrease too fast in the series in question. Eventually, for no matter what real number you pick, you will never reach that number by adding the last terms. \u2013\u00a0Eoin Jan 24 '15 at 17:52\n\u2022 @SimonS My mistake. I fixed it. \u2013\u00a0qmd Jan 24 '15 at 17:54\n\u2022 Have you tried doing the same with the $\\sum_{n=1}^\\infty \\frac{1}{n^2}$, like what you did with $\\sum_{n=1}^\\infty \\frac{1}{n}$? It's the Cauchy condensation test, you can show $\\sum_{n=1}^\\infty \\frac{1}{n^2}$ converges. \u2013\u00a0sciona Jan 24 '15 at 17:57\n\n$$\\frac{1}{2} + \\frac{1}{2^2} + \\frac{1}{2^3} + \\dots$$\n\nYou want to walk a mile.\n\nFirst you walk half of it. Then take a bit of rest. Then walk half of the remaining, then take a bit of rest. And so on.\n\nWill you ever go beyond the mile?\n\nYou have an infinite number of numbers (of the form $\\frac{1}{2^n}$), but you can never go beyond the mile. Basically, the numbers are decreasing too fast, that no matter how many you take, you still are bounded.\n\nIn this case, you can prove that (using $\\frac{1}{n^2} \\lt \\frac{1}{(n-1)(n+1)}$)\n\n$$1 + \\frac{1}{2^2} + \\frac{1}{3^2} + \\dots + \\frac{1}{n^2} \\lt 2 -\\frac{1}{n}$$\n\nSo it is in fact bounded, and you cannot go beyond $2$, no matter how many terms you take.\n\n(And any bounded monotonic sequence converges).\n\n\u2022 Now I can actually visualize it. Thanks \u2013\u00a0qmd Jan 24 '15 at 18:18\n\u2022 @Rzeta: You are welcome! \u2013\u00a0pedant Jan 24 '15 at 18:21\n\nAn easy way to see that $\\sum \\frac{1}{n^{2}}$ converges is to compare with the larger telescoping series $$1 + \\sum_{n=2}^{\\infty} \\frac{1}{n(n-1)} = 1 + \\sum_{n=2}^{\\infty} \\left[\\frac{1}{n - 1} - \\frac{1}{n}\\right] = 2.$$\n\nThis ad hoc trick may be unsatisfying for a couple of reasons:\n\n\u2022 How does one come up with similar estimates for other series? (Answer: One doesn't, at least not in an algorithmic way.)\n\n\u2022 If someone hands you a general infinite series, how do you tell whether or not it converges? (Answer: That's difficult.)\n\nBut at least this estimate shows $\\displaystyle\\sum_{n=1}^{\\infty} \\frac{1}{n^{2}}$ converges, and the sum is between $1$ and $2$.","date":"2020-05-25 12:01:12","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8240628242492676, \"perplexity\": 166.243908026702}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-24\/segments\/1590347388427.15\/warc\/CC-MAIN-20200525095005-20200525125005-00056.warc.gz\"}"} | null | null |
\section{Introduction}
In his pioneering work \cite{Riesz}, Riesz studied fine properties of the so-called Hardy spaces ${\mathcal{H}}^p(\mathbb{D})$, which are the spaces of holomorphic functions \footnote{In 1915 Hardy observed that if $f$ is holomorphic in $\mathbb{D}$ then $r\mapsto M(r)=\int_0^{2\pi}|f(re^{i\theta})|^p d\theta$ is a nondcreasing function.
}
$f\colon \mathbb{D}\to\mathbb{C}$ such that
\begin{equation}\label{limcondr}
\sup_{0<r<1}\int_0^{2\pi}|f(re^{i\theta})|^p d\theta<+\infty
\end{equation}
for $p>0$. Under condition \eqref{limcondr}, it is known that $f(e^{i\theta})$ exists and
\begin{equation}
\lim_{r\to 1^-}\int_0^{2\pi}|f(re^{i\theta})-f(e^{i\theta})|^p d\theta=0~~\mbox{as well as}~~\lim_{r\to 1^-} f(re^{i\theta})=f(e^{i\theta}),
\end{equation}
for almost every $\theta.$ For $f\in {\mathcal{H}}^p(\mathbb{D})$, one defines
$\|f\|_{{\mathcal{H}}^p(\mathbb{D})}:=\|f\|_{L^p(S^1)}.$
\par
We can independently consider holomorphic functions in $L^2(\mathbb{D})$ which corresponds to the well-known Bergman space ${\mathcal{A}}^2(\mathbb{D})$
\footnote{We recall that ${\mathcal{A}}^2(\mathbb{D}):=\{f\colon\mathbb{D}\to\mathbb{C}:~ f ~\mbox{holomorphic and $\|f\|_{L^2(\mathbb{D})}<+\infty$}\}$}, see e.g \cite{DS}.\par
The connection between Hardy spaces and the Bergman space ${\mathcal{A}}^2(\mathbb{D})$ is given by the embedding ${\mathcal{H}}^1(\mathbb{D}) \hookrightarrow {\mathcal{A}}^2(\mathbb{D})$ together with
the estimate
\begin{equation}\label{embHA}
\|f\|_{L^2(\mathbb{D})}\le C\|f\|_{{\mathcal{H}}^1(\mathbb{D})}:=\|f\|_{L^1(S^1)}.
\end{equation}
In the case $\lim_{r\to 1^-}\|f(re^{i\theta})\|_{{H}^{-1/2}(S^1)}<+\infty$, then, by definition, the following inequality holds as well:
\begin{equation}\label{embHA2}
\|f\|_{L^2(\mathbb{D})}\le C \|f\|_{H^{-1/2}(S^1)}:=\lim_{r\to 1^-}\|f(re^{i\theta})\|_{ {H}^{-1/2}(S^1)}.
\end{equation}
In this note, we prove the following combination of \eqref{embHA} and \eqref{embHA2}:
\begin{thm}\label{BBB}
Let $f\colon \mathbb{D}\to \mathbb{C}$ be an analytic function. Then $f$ belongs
to the Bergman space ${\mathcal{A}}^2(\mathbb{D})$ if and only if $$\|f\|_{ L^1+ {H}^{-1/2}(S^1)}:=\limsup_{r\to 1^-}\|f(re^{i\theta})\|_{ L^1+ {H}^{-1/2}(S^1)}<+\infty.$$
Moreover, it holds
\begin{equation}\label{bergman}
\|f\|_{L^2(\mathbb{D})}\le C\|f\|_{L^1+{H}^{-1/2}(S^1)}.\end{equation}
\end{thm}
This type of inequalities takes its roots in the
pioneering work \cite{bourgain1}, where Bourgain and Brezis proved the following striking result:
\begin{thm}[Lemma 1 in \cite{bourgain1}]\label{ThBB}
Let $u$ be a $2\pi$-periodic function in $\mathbb{R}^n$ such that $\int_{\mathbb{R}^n} u\, dx=0$, and let
$\nabla u=f + g,$ where $f\in \dot{W}^{-1,\frac{n}{n-1}}(\mathbb{R}^n)$\footnote{For $1< p < +\infty$, we will denote by $\dot{W}^{1,p}(\mathbb{R}^2)$ the homogeneous Sobolev space defined as the space of $f\in L^{1}_{loc}(\mathbb{R}^n)$ such that
$\nabla f\in L^{p}(\mathbb{R}^n)$ and by $\dot{W}^{-1,p^{\prime}}(\mathbb{R}^n)$ the corresponding dual space ($p^{\prime}$ is the conjugate of $p$). Every
function $f\in \dot{W}^{-1,p^{\prime}}(\mathbb{R}^n)$ can be represented as $f=\sum_{i=1}^n\partial_{x_i} f^j$ with $f^j\in L^{p'}(\mathbb{R}^n)$.} and $g\in L^1(\mathbb{R}^n)$ are $2\pi$-periodic vector-valued functions. Then
\begin{equation}\label{BBineq}
\|u\|_{L^{\frac{n}{n-1}}}\le c \left(\|f\|_{\dot W^{-1,\frac{n}{n-1}}}+\|g\|_{L^1}\right).\
\end{equation}
\end{thm}
By duality, this implies the following corollary:
\begin{cor}[Theorem 1 in \cite{bourgain1}]\label{corBB}
For every $2\pi$-periodic function $h\in L^n(\mathbb{R}^n)$ with $\int_{\mathbb{R}^n} h=0$, there exists a $2\pi$-periodic $v\in \dot{W}^{1,n}\cap L^{\infty}(\mathbb{R}^n)$
satisfying
$$\div v = h~~~\mbox{in $\mathbb{R}^n$}$$
and
\begin{equation}
\|v\|_{L^{\infty}}+\|v\|_{\dot{W}^{1,n}}\le C(n)\|h\|_{L^n}.
\end{equation}
\end{cor}
One of the main result of this note is a fractional type Bourgain-Brezis inequality on the circle $S^1$. More precisely, we have the following:
\begin{thm}\label{nonlocalBB1D}
Let $u\in{\mathcal{D}}^{\prime}(S^1)$ be such that $(-\Delta)^{\frac{1}{4}}u ,\mathcal{R}(-\Delta)^{\frac{1}{4}}u \in \dot{H}^{-\frac{1}{2}}(S^1) + L^{1}(S^1).$\footnote{We denote by ${\mathcal{R}}$ and ${\mathcal{R}}_j$ the Riesz transform respectively on $S^1$ and with respect to the $x_j$ variable on $T^n$, for $j\in\{1,\ldots, n\}$ and by $\dot{H}^{-\frac{n}{2}}(T^n)$ the space of $f\in \mathcal{D}^\prime(S^1)$ such that $f=(-\Delta)^{n/4}g$, with $g\in L^2(T^n)$. Recall that $L^2_*(S^1):=\{u\in L^2(S^1):~~\Xint-_{S^1}u =0\}$.}
Then \\$u-\Xint-_{S^1}u\in L^2_*(S^1)$ and the following estimate holds true:
\begin{equation}\label{fractineq}
\left\|u-\Xint-_{S^1}u\right\|_{L^2}\le C\left(\|(-\Delta)^{1/4} u\|_{\dot{H}^{-1/2}(S^1)+L^1(S^1)}+\|{\mathcal{R}}(-\Delta)^{1/4} u\|_{\dot{H}^{-1/2}(S^1)+L^1(S^1)}\right),
\end{equation}
for some $C > 0$ independent of $u$.
\end{thm}
We also show the equivalence between Theorems \ref{BBB} and \ref{nonlocalBB1D}, establishing the connection between fractional Bourgain-Brezis inequalities and Bergman spaces.\\
\par
We would like to add some comments about Bourgain-Brezis' inequality.
Bourgain-Brezis' inequality in its general form is of interest in the study of the PDE $\operatorname{div} Y = f$ for $f \in L^{n}_{\ast}(T^{n})$, where for finite $p\ge 1$, $L^{p}_{\ast}(T^n)$ denotes the Banach subspace of $L^p$-functions with vanishing mean over the torus. Precisely, they found that $Y$ can be chosen to be continuous and in $\dot{W}^{1,n}(T^n)$, a result which is non-trivial due to the fact that $\dot{W}^{1,n}(T^n)$ does not continuously embed into $L^{\infty}(T^n)$.
The key ingredient in the proof is a duality argument based on an estimate similar to \eqref{BBineq} and some general results from functional analysis regarding closedness properties of the image space. This motivates the general interest in inequalities of the same type, as improved regularity results in limit cases can be invaluable. Indeed, later, such estimates have been considered by a variety of further authors, see \cite{bourgain2} again by Bourgain and Brezis for results regarding Hodge decompositions, \cite{mazya} due to Maz'ya for an inequality on $H^{1-\frac{n}{2}}(\mathbb{R}^{n})$ leading to a different existence result for the PDE $\operatorname{div} Y = f$ and \cite{mironescu} by Mironescu using a PDE-approach to a $2$D-special case involving explicitly computing fundamental solutions with appropriate boundedness properties.\par
In \cite{dalioriv}, the first two authors provide an alternative proof of \eqref{BBineq} in dimension $2$ without
assuming the periodicity of the function $u$. The proof is related to some compensation phenomena observed first in \cite{De} in the analysis of $2$-dimensional perfect incompressible fluids and then also applied by the second author in \cite{Riv3} in the analysis of {\it isothermic surfaces}.
For an overview of the results in the literature regarding variations of Theorem \ref{ThBB} and Corollary \ref{corBB}, we refer for instance to \cite{vs}.
The inequality \eqref{BBineq} also represents the first key step in the study of the regularity of $ L^2(\mathbb{D},{\mathbb{R}}^n)$ solutions
$u$ to a linear elliptic system of the following form
\begin{eqnarray}
\label{intro-6}
\mbox{div}\,(S\,\nabla u)&=&\sum_{j=1}^n\mbox{div}\,(S_{ij}\,\nabla u^j)=\sum_{j=1}^n\sum_{\alpha=1}^2\frac{\partial}{\partial {x_{\alpha}}}\left(S_{ij}u_{x_{\alpha}}^j\right)=0,
\end{eqnarray}
where $S$ is a $ W^{1,2}(\mathbb{D})$ symmetric $n\times n$ matrix, such that $S^2=id_n$, as seen in \cite{dalioriv}.\\
We would like to mention some results on Riesz potentials showing that
the $1$-dimensional case plays a particular role in the $L^1$-estimates for Riesz potentials. More precisely, one can deduce from the results in \cite{sw} that for all
$0<\alpha<1$, we have:
\begin{equation}\label{steinineq}\|I_{\alpha}u\|_{L^{\frac{1}{1-\alpha}}}\le C(\|{\mathcal{R}}u\|_{L^1}+\|u\|_{L^1}),\end{equation}
for all $u$ in the Hardy space ${\mathcal{H}}^1(\mathbb{R})$.
It follows in particular that:
$$\| u\|_{L^{\frac{1}{1-\alpha}}}\le C(\|{\mathcal{R}}(-\Delta)^{\alpha/2}u\|_{L^1}+\|(-\Delta)^{\alpha/2}u\|_{L^1}).$$
In \cite{ssv}, the authors show that if $N\ge 2$ and $0<\alpha<N$, then there is a constant $C=C(\alpha,N)>0$ such that
\begin{equation}\label{estschr}
\|I_{\alpha}u\|_{L^{\frac{N}{N-\alpha}}}\le C\|{\mathcal{R}}u\|_{L^1} \end{equation}
for all $u\in C^{\infty}_c(\mathbb{R}^N)$, such that ${\mathcal{R}}u\in L^1(\mathbb{R}^N).$
The estimate \eqref{estschr} is however false in $1$-D, as seen in \cite{ssv}.
\par
The inequality \eqref{fractineq} generalizes the inequality \eqref{steinineq} in the case $\alpha=1/2$ and the counter-example
in \cite{ssv} for the estimate \eqref{estschr} in $1$-D shows that the estimate \eqref{fractineq} is in some sense optimal.\\
The paper is organized as follows: In section \ref{prel}, we recall the definitions of the fractional Laplacian on the unit circle and on the torus and of Clifford Algebras. In section \ref{proof}, we provide two distinct proofs of Theorem \ref{nonlocalBB1D} .
In section \ref{equiva}, we establish the equivalence of Theorem \ref{BBB} and Theorem \ref{nonlocalBB1D}.
In section \ref{torus}, we extend the fractional Bourgain-Brezis inequality using Clifford algebras to the $n$-dimensional torus $T^n$.
In section \ref{exist}, we prove
existence results for certain fractional PDE-operators in the same spirit as Corollary \ref{corBB}. Lastly, in section \ref{appendix}, we provide a proof of the inequalities \eqref{embHA} and \eqref{embHA2}.
\par
\section{Preliminaries}\label{prel}
\subsection{Fractional Laplacian on the unit circle and on the torus}
Before we enter the discussion and the proofs of the main results, let us recall a few notions essential in our later arguments. We mainly focus on fractional Laplacians, fractional Sobolev spaces and Clifford algebras.\par
Throughout this note, we shall denote by $T^n$ the torus of dimension $n \in \mathbb{N}$. This means:
\begin{equation}
T^n = \underbrace{S^1 \times \ldots \times S^1}_\text{$n$ times} = \mathbb{R}^n / (2\pi \mathbb{Z})^n
\end{equation}
where $ S^1=\mathbb{R}/2\pi\mathbb{Z}$.
We denote by $\mathcal{D}( T^n):=C^\infty(T^n)$ the Fr\'echet space of smooth functions on $T^n$ and by $\mathcal{D}'(T^n)$ its topological dual. The natural duality paring is denoted by $\langle \cdot, \cdot \rangle$.
For $u\in\mathcal{D}'(T^n)$ and $m\in\mathbb{Z}^{n}$, we define the Fourier coefficients of $u$ as follows:
\begin{equation}
\widehat{u}(m) := \frac{1}{(2 \pi)^n} \int_{T^n} u(x) e^{-i \langle m, x \rangle} dx = \Big{\langle} u, e^{-i \langle m, \cdot \rangle} \Big{\rangle}.
\end{equation}
The Fourier coefficients completely determine $u$ as a distribution on $T^n$ and convergence in the sense of distributions obviously implies convergence of the Fourier coefficients. Notice that, for all $u\in\mathcal{D}'(T^n)$, there exists some $N>0$ such that $\abs{\hat u(m)}\lesssim(1+\abs{m})^N$. Moreover, we recall that $v\in C^\infty(T^n)$ if and only if the Fourier coefficients $\hat v(m)$ have rapid decay, i.e. $\sup_m(1+\abs{m})^N\abs{\hat v(m)}<\infty$ for all $N>0$.
Given $s\in\mathbb{R}$, we define the non-homogeneous and homogeneous Sobolev spaces respectively by
\[ H^s( T^n):=\set{v\in\mathcal{D}'(T^n):\norm{v}_{H^s}^2:=\sum_{k\in\mathbb{Z}^{n}}(1+\abs{k}^2)^s\abs{\hat v(k)}^2<\infty}, \]
and
\[ \dot{H}^s( T^n):=\set{v\in\mathcal{D}'(T^n):\norm{v}_{\dot H^s}^2:=\sum_{k\in\mathbb{Z}^{n}}|k|^{2s}\abs{\hat v(k)}^2<\infty}, \]
where $\mathcal{D}'(T^n)$ is again the space of distributions on $T^n$. Notice that if $s = 0$, we have $L^{2}(T^n) = H^{0}(T^n)$ and $L^{2}_{\ast}(T^n) \simeq \dot H^{0}(T^n)$.
An important family of operators throughout our considerations are the so-called \textit{fractional Laplacians}. Let $s > 0$ be real, then we define for $u: T^n \to \mathbb{C}$ smooth the $s$-Laplacian of $u$ by the following multiplier property:
\begin{equation}
\widehat{(-\Delta)^{s} u}(\xi) = \sum_{m\in\mathbb{Z}^{n}}|m|^{2 s} \widehat{u}(m)e^{i\langle m,\xi\rangle} , \quad \forall \xi \in T^n.
\end{equation}
Clearly, this definition can immediately be extended to the spaces $H^{s}(T^{n})$ or even $\mathcal{D}^\prime (T^n)$ as a multiplier operator on Fourier coefficients, possibly defining merely a distribution on $T^n$.
Finally, we recall the definition of the $j$-Riesz transform on $T^n$ as a multiplier operator:
\begin{equation}
\widehat{{\mathcal{R}}_ju}(\xi) = \sum_{m\in\mathbb{Z}^{n}}i \frac{m_j}{|m|} \widehat{u}(m)e^{i\langle m,\xi\rangle} , \quad \forall \xi \in T^n.
\end{equation}
In particular, in the case $n = 1$, we have:
\begin{equation}
\widehat{{\mathcal{R}}u}(\xi) = \sum_{m\in\mathbb{Z}}i \sign(m) \widehat{u}(m)e^{i m \cdot \xi} , \quad \forall \xi \in S^1.
\end{equation}
\subsection{Clifford Algebras}
The material covered here is due to \cite{gilbert} and \cite{hamilton} and we refer to them for further details on the topics introduced. For the remainder of this subsection, let $\mathbb{K} \in \{ \mathbb{R}, \mathbb{C} \}$ denote a scalar field and $V$ a finite dimensional $\mathbb{K}$-vector space. Let $Q: V \to \mathbb{K}$ be a map, such that:
\begin{itemize}
\item[1.)] For all $\lambda \in \mathbb{K}$ and $v \in V$, we have: $Q(\lambda v ) = \lambda^2 \cdot Q(v)$.
\item[2.)] The map $B(v,w) := \frac{1}{2} \big{(} Q(v+w) - Q(v) - Q(w) \big{)}$ defines a $\mathbb{K}$-bilinear map on $V \times V$.
\end{itemize}
Such a $Q$ will be called a \textit{quadratic form} and the pair $(V,Q)$ a \textit{quadratic space}. Standard examples include real vector spaces equipped with scalar products, but not complex vector spaces with scalar products due to complex anti-linearity in the second argument. Inspired by this example, we say that a basis $e_1, \ldots, e_n$ of a quadratic space $(V,Q)$ is \textit{$B$-orthonormal}, if for all $j \in \{ 1, \ldots, n \}$, we have $| Q(e_j) | = 1$ as well as:
\begin{equation}
B(e_j, e_k) = 0, \quad \forall j \neq k \in \{ 1, \ldots, n \}.
\end{equation}
Given such a quadratic space $(V, Q)$, we call a pair $(\mathcal{A}, \nu)$ a \textit{Clifford algebra} for $(V,Q)$, if the following holds, see \cite[p.8, (2.1)]{gilbert}:
\begin{itemize}
\item[i.)] $\mathcal{A}$ is an associative algebra with unit $1$ and $\nu: V \to \mathcal{A}$ is $\mathbb{K}$-linear and injective.
\item[ii.)] $\mathcal{A}$ is generated as an algebra by $\nu(V)$ and $\mathbb{K} \cdot 1$.
\item[iii.)] For every $v \in V$, we have: $\nu(v)^{2} = - Q(v) \cdot 1$
\end{itemize}
An important immediate corollary of the definition is the following commutation relation:
\begin{equation}
\nu(v) \nu(w) + \nu(w) \nu(v) = - 2 B(v,w) \cdot 1, \quad \forall v, w \in V.
\end{equation}
Thus, pairs of {\em orthogonal} vectors with respect to $B$ anti-commute as elements in $\mathcal{A}$. We usually omit explicitly mentioning $\nu$ and therefore identify $v$ with $\nu(v)$, which is justified due to $\nu$ being injective.\\
For the remainder of the section, let us focus on $(V,Q)$ non-degenerate, i.e. for all $v \in V$, there is a $w \in V$, such that $B(v,w) \neq 0$. In this case, there actually exists a basis $e_1, \ldots, e_n$, where $n = \operatorname{dim}_{\mathbb{K}} V$, orthonormal with respect to $B$ and, consequently, such that:
\begin{equation}
e_j e_k + e_k e_j = \pm 2 \delta_{jk} \cdot 1, \quad \forall j,k \in \{ 1, \ldots, n\},
\end{equation}
(see e.g. Theorem 1.5 in \cite{gilbert}). The signs are determined by the signature of the quadratic form $Q$ and may vary for different choices $j,k$. Provided $\mathbb{K} = \mathbb{C}$, we may assume that all signs are the same, see \cite{gilbert}.\par
It can be shown that every Clifford algebra has $\mathbb{K}$-dimension at most $2^{n}$. If the dimension is equal to $2^n$, the Clifford algebra is called \textit{universal}.\footnote{This definition is justified, as universal Clifford algebras $\mathcal{A}$ have an extension property for linear maps from $V$ to any Clifford algebra respecting the characteristic multiplication relation in $\mathcal{A}$, see \cite{gilbert}.} An important result in \cite[Thm. 2.7]{gilbert} states that there always exists a universal Clifford algebra for any given quadratic space. Moreover, there exist explicit descriptions of all universal Clifford algebras up to isomorphisms in terms of matrices, see \cite{gilbert}.\\
To conclude this brief treatment of Clifford algebras, let us provide an explicit example: Let $V = \mathbb{C}^{n}$, $\mathbb{K} =\mathbb{C}$ and define $Q$ as follows:
\begin{equation}
Q(z_1, \ldots, z_n) := \sum_{j=1}^{n} z_j^2, \quad \forall (z_{1}, \ldots, z_n) \in \mathbb{C}^{n}.
\end{equation}
It is clear that $(V,Q)$ is a non-degenerate quadratic space, as $B$ is the standard scalar product up to a complex conjugation in the second argument. In this case, the standard basis $e_1, \ldots, e_n$ already is $B$-orthonormal. Thus, we have:
\begin{equation}
e_j e_k + e_k e_j = - 2 \delta_{jk} \cdot 1, \quad \forall j,k \in \{ 1, \ldots, n \}.
\end{equation}
The universal Clifford algebra is then spanned by the finite products $e_{\alpha}$ of the basis elements, where $\alpha \subset \{ 1, \ldots, n \}$ is an ordered subset and we define:
$$e_\alpha = \prod_{j \in \alpha} e_{j}$$
In particular, $e_{\emptyset} = 1$ by definition. It can be seen that every complex universal Clifford algebra associated with a non-degenerate quadratic space of dimension $n$ is isomorphic to this one, see \cite{gilbert} and the definition of universal Clifford algebra presented there.\\
Lastly, let us introduce a few definitions from Chapter 1, Section 7 in \cite{gilbert}: We may identify the universal Clifford algebra ${\mathcal{A}}$ as a vector space with $\mathbb{K}^{2^{n}}$, if $\operatorname{dim}_{\mathbb{K}} V = n$. This allows us to generalize the natural scalar product-induced norm on $\mathbb{K}^{2^{n}}$ to the Clifford algebra and we shall denote this norm by $\| \cdot \|$. Moreover, there is a notion of conjugation on Clifford algebras defined by:
\begin{equation}
\label{cliffconjdef}
\overline{e_{j_1} \ldots e_{j_{k}}} := (-1)^{k} Q(e_{j_1}) \ldots Q(e_{j_k}) \cdot e_{j_k} \ldots e_{j_1} = (-1)^{\frac{k(k+1)}{2}} Q(e_{j_1}) \ldots Q(e_{j_k}) \cdot e_{j_1} \ldots e_{j_k},
\end{equation}
and extending linearily. If $\mathbb{K} = \mathbb{C}$, we also conjugate the complex coefficients in the usual manner, i.e. we extend complex anti-linearily. We highlight the following key property of the conjugation:
\begin{equation}
\overline{xy} = \overline{y} \cdot \overline{x}, \quad \forall x,y \in \mathcal{A}.
\end{equation}
This is due to the inversion of factors in \eqref{cliffconjdef}. We emphasise that the definition in \eqref{cliffconjdef} is precisely made with the identity below in mind:
\begin{equation}
\overline{e_{j_1} \ldots e_{j_{k}}} \cdot e_{j_1} \ldots e_{j_{k}} = 1.
\end{equation}
The following property will be useful later as well: Let $x \in \mathcal{A}$ be given and denote by $P_{0}$ the linear projection of an element in the Clifford algebra to the coefficient associated with the neutral element $1$. More precisely, $P_0: \mathcal{A} \to \mathbb{K}$ is the following linear map:
$$P_0 \Big{(} \sum_{\alpha} x_{\alpha} e_{\alpha} \Big{)} = x_{\emptyset}$$
We have by a direct computation:
\begin{align}
P_{0}(\overline{x}x) &= \sum_{\alpha \subset \{1, \ldots, n\}} \overline{x_{\alpha}} x_{\alpha} \notag \\
&= \| x \|^2,
\end{align}
where we wrote explicitly $x = \sum_{\alpha \subset \{1, \ldots, n\}} x_{\alpha} e_{\alpha}$ with $x_{\alpha} \in \mathbb{K}$. It suffices to observe that $e_\alpha \cdot e_\beta$ has non-vanishing contribution in the $e_{\emptyset} = 1$-direction, if and only if $\alpha = \beta$. The formula then follows.
\section{Fractional Bourgain-Brezis inequality on the unit circle ${\mathbf{S}^1}$}\label{proof}
In this section, we provide two distinct proofs of Theorem \ref{nonlocalBB1D}. The first proof is in the spirit of the one presented in \cite{bourgain1}, while the second one is inspired by that in \cite{dalioriv} and is based on some particular compensation phenomena. We assume for simplicity that $u$ is real valued (the proof for complex-valued function is completely analogous, see Remark \ref{complex}).\par
First, we would like to observe that if $u\in C^{\infty} (S^1)$, then by definition:
\begin{eqnarray}\label{nlest}
\displaystyle\left\|u-\Xint-_{S^1}u\right\|_{L^2}&\le &C \|(-\Delta)^{1/4} u\|_{\dot{H}^{-1/2}(S^1)}.
\end{eqnarray}
On the other hand, as we have already observed in the introduction, we also have\footnote{Actually, an even sharper inequality than \eqref{nlestbis} holds true with $L^2(S^1)$ being replaced by the smaller Lorentz space $L^{2,1}(S^1)$.}:
\begin{eqnarray}
\label{nlestbis}
\displaystyle\left\|u-\Xint-_{S^1}u\right\|_{L^2(S^1)}&\le &C\left(\|(-\Delta)^{1/4} u\|_{L^1(S^1)}+\|{\mathcal{R}}(-\Delta)^{1/4} u\|_{L^1(S^1)}\right)\simeq \|(-\Delta)^{1/4} u\|_{{\mathcal H}^1(S^1)}.
\end{eqnarray}
\subsection{A first proof of Theorem \ref{nonlocalBB1D}}
Let us suppose that $u\in C^{\infty} (S^1)$, $\Xint-_{S^1}u=0$. We assume for simplicity that $u$ is real-valued, (see Remark \ref{complex} for the complex-valued case). The proof below follows the main arguments of the original proof by Bourgain and Brezis.
We write:
\begin{equation}\label{estnl1}
\left\{\begin{array}{c}
(-\Delta)^{1/4} u=f^{1}+g^{1}\\[5mm]
{\mathcal{R}} (-\Delta)^{1/4} u=f^{2}+g^{2}
\end{array}\right.
\end{equation}
where $f^{1},f^{2}\in \dot{H}^{-1/2}(S^1)$, $g^{1},g^{2}\in L^1(S^1)$.
We set $u=\sum_{n\in\mathbb{Z}^*} u_n e^{in \theta}$. Since $u$ is real-valued, it holds $\bar u_n=u_{-n}.$
We see:
\begin{eqnarray}\label{estnl2}
\sum_{n\in\mathbb{Z}^*} |u_n|^2&=& \sum_{n\in\mathbb{Z}^*} |n|^{1/2}u_n\frac{u_{-n}}{|n|^{1/2}}= \sum_{n\in\mathbb{Z}^*} \frac{f^{1}_n+g^{1}_n}{|n|^{1/2}}u_{-n},
\end{eqnarray}
\begin{eqnarray}\label{estnl3}
\sum_{n\in\mathbb{Z}^*}\frac{f^{1}_n\, u_{-n}}{|n|^{1/2}}&\le & \left[ \sum_{n\in\mathbb{Z}^*}\frac{|f^{1}_n|^2}{|n|}\right]^{1/2}\left[ \sum_{n\in\mathbb{Z}^*}|u_n|^2\right]^{1/2},
\end{eqnarray}
\begin{eqnarray}\label{estnl4}
\sum_{n\in\mathbb{Z}^*}\frac{g^{1}_n\, u_{-n}}{|n|^{1/2}}&= & \sum_{n>0}\frac{g^{1}_n\, u_{-n}}{|n|^{1/2}}+ \sum_{n<0}\frac{g^{1}_n\, u_{-n}}{|n|^{1/2}}.
\end{eqnarray}
Observe that by definition of the Riesz transform:
\begin{eqnarray}\label{estnl5}
{\mathcal{R}} (-\Delta)^{1/4} u&=&i\left[- \sum_{n<0} |n|^{1/2} u_n e^{in\theta}+\sum_{n>0} |n|^{1/2} u_n e^{in\theta}\right].
\end{eqnarray}
Therefore:
\begin{equation}\label{estnl6}
u_n = \left\{\begin{array}{cc}
\frac{f_n^2+g_n^2}{-i|n|^{1/2}}&~~~\mbox{if $n<0$}\\[5mm]
\frac{f_n^2+g_n^2}{i|n|^{1/2}}&~~~\mbox{if $n>0$}
\end{array}\right.\end{equation}
By combining \eqref{estnl4} and \eqref{estnl6}, we obtain:
\begin{eqnarray}\label{estnl7}
\sum_{n\in\mathbb{Z}^*}\frac{g^{1}_n\, u_{-n}}{|n|^{1/2}}&= &\sum_{n>0}g_n^1\ \frac{f_{-n}^2+g_{-n}^2}{-i|n|}+\sum_{n<0}g_n^1\ \frac{f_{-n}^2+g_{-n}^2}{i|n|}.
\end{eqnarray}
Let us estimate the different parts of the sum \eqref{estnl7} individually:\\
\noindent
{ 1. We first estimate}
\begin{eqnarray}\label{estnl8}
\sum_{n\in\mathbb{Z}^*}{\mbox{sign$(n)$}}\frac{g^{1}_n\ f^2_{-n}}{|n|}&=& \sum_{n\in\mathbb{Z}^*}{\mbox{sign$(n)$}}\frac{| n |^{1/2}u_n-f^1_n}{|n|^{1/2}}\frac{f_{-n}^2}{|n|^{1/2}}\nonumber\\[5mm]
&\le& \left( \sum_{n\in\mathbb{Z}^*}|u_n|^2\right)^{1/2} \left( \sum_{n\in\mathbb{Z}^*}\frac{|f^2_n|^2}{|n|}\right)^{1/2} +\left( \sum_{n\in\mathbb{Z}^*}\frac{|f^1_n|^2}{|n|}\right)^{1/2}\left( \sum_{n\in\mathbb{Z}^*}\frac{|f^2_n|^2}{|n|}\right)^{1/2}\nonumber\\[5mm]
&\le& \|u\|_{L^2}\|f^2\|_{\dot{H}^{-1/2}}+ \|f^1\|_{\dot{H}^{-1/2}}\|f^2\|_{\dot{H}^{-1/2}}.
\end{eqnarray}
{ 2. It remains to estimate}
$$ \sum_{n\in\mathbb{Z}^*} {\mbox{sign$(n)$}}\frac{g^{1}_n\, g^2_{-n}}{i|n|}.$$
For this purpose, we consider the following operator:
\begin{eqnarray*}
{\mathbf{A}}\colon L^1(S^1)\times L^1(S^1)&\to &\mathbb{C} ,
~~~~~~(g^1,g^2)\mapsto \sum_{n\in\mathbb{Z}^*}{\mbox{sign$(n)$}}\frac{g^{1}_n\, g^2_{-n}}{i|n|}.
\end{eqnarray*}
{\bf Claim 1. } The operator $ {\mathbf{A}}$ is continuous, i.e. we have the following estimate:
\begin{equation}\label{estn9} | {\mathbf{A}}(g^1,g^2)|\le C \|g^1\|_{L^1}\|g^2\|_{L^1}.\end{equation}
{\bf Proof of Claim 1.} It is sufficient to prove the claim in the case where $g^1$ and $g^2$ are arbitrary Dirac-delta measures.\footnote{We recall that the linear span of Dirac measures is dense in the space of Radon measures ${\mathcal{M}}(S^1)$ equipped with the weak-* topology.} Therefore, we consider $g^{1}=\sum_{i\in I}\lambda_i\delta_{a_i}$ and $g^{2}=\sum_{j\in J}\mu_j\delta_{b_j}$. We have
$\|g^{1}\|_{{\mathcal{M}}(S^1)}=\sum_{i\in I}|\lambda_i|,$ $\|g^{2}\|_{{\mathcal{M}}(S^1)}=\sum_{j\in J}|\mu_j|.$
By bilinearity, we deduce:
\begin{eqnarray}\label{estnl10}
| {\mathbf{A}}(g^1,g^2)|&=&| {\mathbf{A}}(\sum_{i\in I}\lambda_i\delta_{a_i},\sum_{j\in J}\mu_j\delta_{b_j})|\nonumber\\[5mm]
&\le& \sum_{i\in I, j\in J}|\lambda_i||\mu_j|| {\mathbf{A}}(\delta_{a_i},\delta_{b_j})|\nonumber\\[5mm]
&\le& \sup_{(a,b)\in S^1\times S^1}| {\mathbf{A}}(\delta_a,\delta_{b})| \sum_{i\in I}|\lambda_i| \sum_{j\in J}|\mu_j|\nonumber\\[5mm]
&=& \sup_{(a,b)\in S^1\times S^1}| {\mathbf{A}}(\delta_{a},\delta_{b})|\|g^1\|_{{\mathcal{M}}(S^1)}|\|g^2\|_{{\mathcal{M}}(S^1)}.
\end{eqnarray}
If $ \sup_{(a,b)\in S^1\times S^1}| {\mathbf{A}}(\delta_{a},\delta_{b})|<+\infty$, then the claim holds for linear combinations of Dirac measures.
By a density argument, we get the claim 1 for arbitrary $g^1,g^2\in L^1(S^1).$ Hence, claim 1 is a consequence of the following:
\noindent{\bf Claim 2}. $ \sup_{(a,b)\in S^1\times S^1}| {\mathbf{A}}(\delta_{a},\delta_{b})|<+\infty$.\par
\noindent{\bf Proof of Claim 2.} For $g^1=\delta_a$ and $g^2=\delta_b$ , we have
$g_n^1=e^{ina}$ and $g^2_n=e^{inb}$.
In this case, we observe:
\begin{eqnarray}\label{estnl111}
{\mathbf{A}}(\delta_a,\delta_b)&=&\sum_{n\in Z^*}\mbox{sign$(n)$}\frac{g_n^1 g_{-n}^2}{i|n|}\nonumber\\
&=&\sum_{n\in \mathbb{Z}^*}\mbox{sign$(n)$}\frac{e^{in(a-b)}}{{i|n|}}=2 \sum_{n>0}\frac{\sin(n(a-b))}{n }<+\infty.\footnotemark
\end{eqnarray}
This proves claim 2 and from \eqref{estnl111}, we can deduce claim 1 as well.
\footnotetext{The value of such a series is deduced from the Fourier series of $f(x)=\frac{x}{2\pi}$ for $0<x<2\pi$ and $f(x+2\pi)=f(x)$.}\\
\noindent
By combining \eqref{estnl2}-\eqref{estnl10} we get
\begin{eqnarray}\label{estnl11}
\|u\|^2_{L^2}&\lesssim & \|u\|_{L^2}\left(\|f^{1}\|_{\dot{H}^{-1/2}}+\|f^{2}\|_{\dot{H}^{-1/2}}\right)+ \|f^{1}\|_{\dot{H}^{-1/2}}\|f^{2}\|_{\dot{H}^{-1/2}}+C\|g^1\|_{L^1}\|g^2\|_{L^1} \nonumber\\[5mm]
&\lesssim & \frac{1}{2} \|u\|_{L^2}^2+\frac{1}{2} \left(\|f^{1}\|^2_{\dot{H}^{-1/2}}+\|f^{2}\|^2_{\dot{H}^{-1/2}}\right)+ \|f^{1}\|_{\dot{H}^{-1/2}}\|f^{2}\|_{\dot{H}^{-1/2}}+C\|g^1\|_{L^1}\|g^2\|_{L^1} \nonumber \\[5mm]
&\lesssim&\frac{1}{2} \|u\|_{L^2}^2+\left(\|f^{1}\|^2_{\dot{H}^{-1/2}}+\|f^{2}\|^2_{\dot{H}^{-1/2}}\right)+\frac{1}{2}\left(\|g^1\|^2_{L^1}+\|g^2\|^2_{L^1}\right).
\end{eqnarray}
This estimate permits us to conclude the proof of Theorem \ref{nonlocalBB1D}.
Since $f^1,f^2,g^1,g^2$ were arbitrary, one can deduce \eqref{fractineq}.
In the general case where
$u\in{\mathcal{D}}^{\prime}(S^1)$, one argues by approximation (see section \ref{secproof} for further details).~~\hfill$\Box$
\bigskip
\medskip
\subsection{A second proof of Theorem \ref{nonlocalBB1D}}\label{secproof}
As in the first proof, we will show the following:
Let $u \in \mathcal{D}'(S^{1})$ be such that:
\begin{align}
\label{condition01}
(-\Delta)^{\frac{1}{4}}u &= f_{1} + g_{1} \\
\label{condition02}
\mathcal{R}(-\Delta)^{\frac{1}{4}}u &= f_{2} + g_{2},
\end{align}
where $f_{1}, f_{2} \in \dot{H}^{-\frac{1}{2}}(S^{1})$ and $g_{1}, g_{2} \in L^{1}(S^{1})$. Under these conditions, we prove:
\begin{equation}
u - \int_{S^{1}} u dx \in L^{2}_{\ast}(S^{1})=\left\{u\in L^2(S^1):~~\Xint-_{S^1} u=0\right\},
\end{equation}
together with the following estimate:
\begin{equation}
\label{estimateforuinthm}
\Big{\|} u - \int_{S^{1}} u dx \Big{\|}_{L^{2}} \leq C \big{(} \| f_{1} \|_{\dot{H}^{-\frac{1}{2}}} + \| f_{2}\|_{\dot{H}^{-\frac{1}{2}}} + \| g_{1}\|_{L^{1}} + \| g_{2}\|_{L^{1}} \big{)},
\end{equation}
where $C > 0$ is independent of $f_{1}, f_{2}, g_{1}, g_{2}$ and $u$.
We may assume for simplicity that $u$ is real-valued (see Remark \ref{complex} for the complex-valued case).\par
Firstly, observe that it suffices to consider the case:
\begin{equation}
\int_{S^{1}} udx = 2\pi \cdot \hat{u}(0) = 0,
\end{equation}
by merely changing $u$ by a constant. Similarly, by the conditions in \eqref{condition01} and \eqref{condition02}, we see that $f_{j}, g_{j}$ have vanishing integral over $S^{1}$ and consequently vanishing Fourier coefficient for $n = 0$.\footnote{It would be possible to treat $f_j, g_j$ with non-vanishing integral, i.e. treat the case $(-\Delta)^{{1/4}} u, \mathcal{R} (-\Delta)^{{1/4}} u \in L^{1} + H^{-{1/2}}(S^1)$ by reducing to vanishing Fourier coefficient at $n = 0$: We have by the conditions $\widehat{f_j}(0) =-\widehat{g_j}(0)$. Note that $| \widehat{g_j}(0) | \lesssim \| g_j \|_{L^1}$. Note that $\| f_j \|_{{H}^{-1/2}}^2 \simeq | \widehat{f_{j}}(0) |^2 + \| \tilde{f}_j \|_{\dot{H}^{-1/2}}^2 $, where $\tilde{f}_j$ denotes the corrected $f_j$ with vanishing $0$th Fourier coefficient. Thus, we could reduce to the case of vanishing integral.}
For now, let us assume that $u, f_{j}, g_{j}$ are all smooth on $S^{1}$. The general case can be dealt with using convolution with an appropriate smoothing kernel and approximation arguments as specified at the end of the proof.\\
First, let us define the following operators on $\mathcal{D}'(S^{1})$:
\begin{align}
\label{operatord}
Dv &:= (-\Delta)^{\frac{1}{4}} \big{(} Id + \mathcal{R} \big{)}v \\
\label{operatordbar}
\overline{D}v &:= (-\Delta)^{\frac{1}{4}} \big{(} Id - \mathcal{R} \big{)}v,
\end{align}
for every $v \in \mathcal{D}'(S^{1})$. Consequently, using \eqref{condition01} and \eqref{condition02}, we have:
\begin{align}
\label{rearr1}
Du &= f_{1} + f_{2} + g_{1} + g_{2} = f + g \\
\label{rearr2}
\overline{D}u &= f_{1} - f_{2} + g_{1} - g_{2} = \tilde{f} + \tilde{g}.
\end{align}
Let us calculate the Fourier multipliers associated with $D, \overline{D}$. For every $n \in \mathbb{Z}$, we have:
\begin{align}
\mathcal{F}\big{(} Dv\big{)}(n) &= |n|^{\frac{1}{2}}(1+ i \sign(n))\hat{v}(n) \\
\mathcal{F}\big{(} \overline{D}v\big{)}(n) &= |n|^{\frac{1}{2}}(1- i \sign(n))\hat{v}(n)
\end{align}
{\bf Claim 1:} Given $f\in \dot{H}^{-1/2}(S^1)$, there is a real-valued function $F \in L^{2}_{\ast}(S^{1})$\footnote{The observation that $F$ may be chosen real-valued is due to $\widehat{F}(-n) = \overline{\widehat{F}(n)}$ for all $n$.}, such that $DF = f$.
\par
\noindent
{\bf Proof of the Claim 1} In order to solve $DF=f$, we should have:
\begin{equation}
\label{fourierF}
\hat{F}(n) = \frac{1}{1 + i \sign(n)} \frac{\hat{f}(n)}{\sqrt{|n|}}, \quad \text{if } n \neq 0 \\
\end{equation}
Using the fact that the $L^{2}$-norm of $F$ can be characterized in terms of the $l^{2}$-norm of the Fourier coefficients, we obtain:
\begin{align}
\label{fouriercoeffest}
\| F \|_{L^{2}}^{2} &= \sum_{n \neq 0} \frac{1}{| 1 + i \sign(n) |^{2}} \frac{| \hat{f}(n) |^{2}}{|n|} \notag \\
&\leq \sum_{n \neq 0} \frac{| \hat{f}(n) |^{2}}{|n|} \notag \\
&= \| f \|_{\dot{H}^{-\frac{1}{2}}}^{2},
\end{align}
\noindent
where we used the definition of the $\dot{H}^{-\frac{1}{2}}$-norm. Observe that a converse inequality could be obtained along the same lines.
Next, by defining $\tilde{u} := u - F$, we observe that due to \eqref{rearr1}:
\begin{equation}
\label{equationgbyD}
D\tilde{u} = g.
\end{equation}
Let now $w \in \mathcal{D}'(S^{1})$ real-valued be such that $Dw = \tilde{u}$ and $\hat{w}(0) = 0$. Once more, existence of such a distribution $w$ is easily deduced using Fourier coefficients. We would like to emphasise at this point that due to the assumed smoothness of $u, f_j, g_j$, $w$ is smooth as well, as is $F$.\par
By \eqref{equationgbyD}, we thus notice:
\begin{equation}
\label{eqforw}
D^{2}w = g.
\end{equation}
Going over to Fourier coefficients, we see that for every $n \in \mathbb{Z}^\ast$:
\begin{equation}
\mathcal{F}\big{(} D^{2}w \big{)}(n) = (1 + i \sign(n))^{2}|n| \hat{w}(n) = 2i \sign(n)|n| \hat{w}(n) = 2in \hat{w}(n) = \hat{g}(n),
\end{equation}
or by rearranging:
\begin{equation}
\label{fouriercoeffwing}
\hat{w}(n) = -\frac{i}{2} \frac{\hat{g}(n)}{n}.
\end{equation}
Next, we are going to find a suitable distribution $K$ with coefficients $\hat{K}(n) = -\frac{i}{2n}$, in order to express $w$ as a convolution of $g$ with $K$. To this end, let us consider the function $k: [-\pi, \pi] \rightarrow \mathbb{R}$ defined by:
\begin{equation}
\label{deffunck}
k(x)=
\begin{cases}
x+ \pi, \quad \text{if } x < 0 \\
x - \pi, \quad \text{if } x > 0
\end{cases}
\end{equation}
By slight abuse of notation, let us identify $k$ with its $2\pi$-periodic extension, therefore $k: S^{1} \rightarrow \mathbb{R}$. We calculate the Fourier coefficients of $k$: If $n = 0$, it is obvious due to anti-symmetry that $\hat{k}(0) = 0$. Otherwise, we have $n \neq 0$ and so by using integration by parts:
\begin{align}
\hat{k}(n) &= \frac{1}{2\pi} \int_{-\pi}^{\pi} k(x) e^{-inx}dx \notag \\
&= \frac{1}{2\pi} \Big{(} \int_{-\pi}^{0} (x + \pi) e^{-inx}dx + \int_{0}^{\pi} (x - \pi) e^{-inx}dx \Big{)} \notag \\
&= \frac{1}{2\pi} \int_{0}^{\pi} (\pi - x)e^{inx} - (\pi - x)e^{-inx}dx \notag \\
&= \frac{i}{\pi} \int_{0}^{\pi} (\pi - x) \sin(nx)dx \notag \\
&= \frac{i}{n} - \frac{i}{\pi} \int_{0}^{\pi} \frac{\cos(nx)}{n} dx = \frac{i}{n}.
\end{align}
Consequently, observe that $K = -\frac{1}{2}k$ precisely yields the desired distribution. Let us notice that $K$ is therefore bounded and measurable on $S^{1}$, thanks to the explicit formula for $k$.\\
Using \eqref{fouriercoeffwing} and the convolution formula for Fourier coefficients, namely:
\begin{equation}
\label{convform}
\widehat{f \ast g}(n) = 2\pi \hat{f}(n) \hat{g}(n), \quad \forall n \in \mathbb{Z},
\end{equation}
it is clear that:
\begin{equation}
\label{wasconv}
w =\frac{1}{2\pi} K\ast g.
\end{equation}
From \eqref{wasconv}, we obtain by using Young's inequality on $S^{1}$:
\begin{equation}
\label{wbounded}
\| w \|_{L^{\infty}} \leq \frac{1}{2\pi} \| K \|_{L^{\infty}} \| g \|_{L^{1}} = \frac{1}{4} \| g \|_{L^{1}}.
\end{equation}
To conclude the first part of the proof, let us observe the following\footnote{This will actually be the first and only point in the proof where we use the fact that $u$ is real-valued in a meaningful way. See Remark \ref{complex} for an extension to complex-valued distributions.}:
\begin{align}
\label{simplifynormexpression}
\int_{S^{1}} (u - F)^{2}dx &= \int_{S^{1}} Dw (u - F)dx \notag \\
&\simeq \sum_{n \in \mathbb{Z}} \widehat{Dw}(n) \widehat{u - F}(-n) \notag \\
&= \sum_{n \in \mathbb{Z}} |n|^{\frac{1}{2}}(1+i \sign(n)) \widehat{w}(n) \widehat{u - F}(-n) \notag \\
&= \sum_{n \in \mathbb{Z}} \widehat{w}(n) \cdot |n|^{\frac{1}{2}}(1-i \sign(-n)) \widehat{u - F}(-n) \notag \\
&\simeq \int_{S^{1}} w \overline{D}(u - F)dx \notag \\
&= \int_{S^{1}} w \overline{D}u dx - \int_{S^{1}} w \overline{D}F dx \notag \\
&= \int_{S^{1}} w \tilde{g} dx + \int_{S^{1}} w \tilde{f} dx - \int_{S^{1}} w \overline{D}F dx
\end{align}
where we used the Fourier representation of the distribution $u - F$ to justify the second equation. Observe that this enables us to estimate:
\begin{equation}
\label{est1}
\Big{|} \int_{S^{1}} w \tilde{g} dx \Big{|} \leq \| w \|_{L^{\infty}} \| \tilde{g} \|_{L^{1}} \leq \frac{1}{4} \big{(} \| g_{1} \|_{L^{1}} + \| g_{2} \|_{L^{1}} \big{)}^{2},
\end{equation}
and:
\begin{equation}
\label{est2}
\Big{|} \int_{S^{1}} w \overline{D}F dx \Big{|} \leq \| w \|_{\dot{H}^{\frac{1}{2}}} \| \overline{D}F\|_{\dot{H}^{-\frac{1}{2}}} \leq C \| u - F \|_{L^{2}} \| f \|_{\dot{H}^{-\frac{1}{2}}}.
\end{equation}
The remaining summand may be estimated completely analogous to \eqref{est2}. Notice that we used the explicit definition of the norms of Sobolev spaces with negative exponents and the Fourier multipliers to obtain \eqref{est2}, see \eqref{fouriercoeffest} for the main ideas. Using \eqref{est1} and \eqref{est2} yields:
\begin{align}
\| u - F \|_{L^{2}}^{2} &\leq \frac{1}{4} \big{(} \| g_{1} \|_{L^{1}} + \| g_{2} \|_{L^{1}} \big{)}^{2} + 2C \| u - F \|_{L^{2}} \| f \|_{\dot{H}^{-\frac{1}{2}}} \notag \\
&\leq \frac{1}{4} \big{(} \| g_{1} \|_{L^{1}} + \| g_{2} \|_{L^{1}} \big{)}^{2} + \frac{1}{2} \| u - F \|_{L^{2}}^{2} + \frac{4C^{2}}{2} \| f \|_{\dot{H}^{-\frac{1}{2}}}^{2},
\end{align}
using the arithmetic-geometric mean inequality. Note that the factor $2C$ is due to estimate \eqref{est2} also applying to the integral of $w\tilde{f}$. By absorbing the $L^{2}$-norm of $u - F$, we arrive at:
\begin{align}
\| u - F \|_{L^{2}}^{2} &\leq \frac{1}{2} \big{(} \| g_{1} \|_{L^{1}} + \| g_{2} \|_{L^{1}} \big{)}^{2} + 4C^{2} \| f \|_{\dot{H}^{-\frac{1}{2}}}^{2} \notag \\
&\leq \max \{ \frac{1}{2}, 4C^{2} \} \big{(} \| g_{1} \|_{L^{1}} + \| g_{2} \|_{L^{1}} + \| f_{1} \|_{\dot{H}^{-\frac{1}{2}}} + \| f_{2} \|_{\dot{H}^{-\frac{1}{2}}} \big{)}^{2}.
\end{align}
Consequently, by estimating the $L^{2}$-norm of $F$ by the $\dot{H}^{-\frac{1}{2}}$-norm of $f_{1}, f_{2}$ using \eqref{fouriercoeffest}, we immediately conclude:
\begin{equation}
\label{l2boundu}
\| u \|_{L^{2}} \leq \tilde{C} \big{(} \| g_{1} \|_{L^{1}} + \| g_{2} \|_{L^{1}} + \| f_{1} \|_{\dot{H}^{-\frac{1}{2}}} + \| f_{2} \|_{\dot{H}^{-\frac{1}{2}}} \big{)}.
\end{equation}
The constant $\tilde{C} > 0$ appearing in the estimate is independent of $u, f_{j}, g_{j}$.\\
Now, for a general distribution $u \in \mathcal{D}'(S^{1})$ with $\hat{u}(0) = 0$, let us observe that if we convolute $u$ with a smooth function $\varphi$, the resulting distribution $\varphi \ast u$ will be a smooth function as well (in the sense of regular distributions). By a direct computation, \eqref{condition01} and \eqref{condition02} will continue to hold true if we replace $u, f_{j}, g_{j}$ by their corresponding convolutions with $\varphi$. This is an immediate consequence of the fact that the operators $(-\Delta)^{\frac{1}{4}}, \mathcal{R}$ are Fourier multipliers as well as the linearity of convolutions. Choosing $\varphi$ to be supported on arbitrarily small neighbourhoods of the neutral element in $S^{1}$ (i.e. an approximation of the identity $\varphi_{\varepsilon}$) ensures that the convolutions of $\varphi$ with $f_{j}, g_{j}$ converge in the respective norms as we collapse the support of $\varphi$ (i.e. let the parameter $\varepsilon$ in $\varphi_{\varepsilon}$ tend to $0$) and the approximations of $u$ converge in the distributional sense. As a result, we obtain uniform bounds in the respective spaces. This results in an uniform $L^{2}$-bound for $u$ convoluted with $\varphi_{\varepsilon}$ independent of $\varepsilon$, which can be seen to imply $u \in L^{2}_{\ast}(S^{1})$ by using a weak-$L^2$-convergent subsequence. The estimate follows by the lower semi-continuity of the norm. This concludes our proof.~~\hfill $\Box$\\
\begin{Rem}\label{complex}
Before we enter the discussion of applications and later a generalisation of Theorem \ref{nonlocalBB1D}, let us quickly discuss the assumption that $u$ is real-valued. In fact, this is merely used at a single point in the proof, namely in \eqref{simplifynormexpression}. However, if we proceed similar to the proof of the generalised result in the next section, i.e. we use:
\begin{equation}
\int_{S^1} | u - F|^2 dx = \int_{S^1} (u - F) \cdot \overline{u - F} dx = \int_{S^1} Dw \cdot \overline{u-F} dx \sim \sum_{n \in \mathbb{Z}} \widehat{Dw}(n) \cdot \overline{\widehat{u - F}(n)},
\end{equation}
we can easily avoid the use of properties of real-valued distributions. The remainder of the proof follows then completely analogous, i.e. we can remove the assumption of $u$ being real-valued effortlessly. Indeed, this slight generalisation will be key to our applications to Bergman spaces below.
\end{Rem}
\section{ Fractional Bourgain-Brezis inequality in the Bergman space ${\mathcal{A}}^2(\mathbb{D})$} \label{equiva}
We start with the
{\bf Proof of Theorem \ref{BBB}.}\\
\noindent
Let us consider an analytic function $f\colon \mathbb{D}\to \mathbb{C}$ such that $ \limsup_{r\to 1^-}\|f(re^{i\theta})\|_{L^1+{H}^{-1/2} (S^1)}<+\infty.$ \par
\noindent
\textbf{1.} Now let us write $f(z)=\sum_{n\ge 0} f_n z^n$ and
$u(e^{i\theta})=\sum_{n\ge 1}\frac{ f_n}{\sqrt{n}} e^{in\theta}$. We first observe that $f - f(0) = \sum_{n \ge 1} f_n z^n$ and if $f(e^{i \theta}) = g(e^{i\theta}) + h(e^{i\theta})$ with $g \in L^1(S^1)$ and $h \in H^{-1/2}(S^1)$, then:
$$f(e^{i\theta}) - f(0) = g - \Xint-_{S^1} g + h - \widehat{h}(0).$$
Note that $h - \widehat{h}(0) \in \dot{H}^{-1/2}(S^1)$ with the norm being controlled by $\| h \|_{H^{-1/2}(S^1)}$. We observe that, using the explicit definitions of the norm:
$$\Big{\|} g - \Xint-_{S^1} g \Big{\|}_{L^1(S^1)} + \| h - \widehat{h}(0) \|_{\dot{H}^{-1/2}(S^1)} \lesssim \|g \|_{L^1(S^1)} + \| h \|_{H^{-1/2}(S^1)}$$
Therefore, we may conclude by taking the infimum over all such $g,h$:
\begin{equation}
\label{homogenisationineq}
\| f - f(0) \|_{L^1 + \dot{H}^{-1/2}(S^1)} \leq C \| f \|_{L^1 + H^{-1/2}(S^1)}.
\end{equation}
Assume therefore first that
$$f(e^{i\theta}) - f(0) =\sum_{n\ge 1}{ f_n} e^{in\theta}\in L^1+\dot{H}^{-1/2}(S^1).$$
In this case, we get
$(-\Delta)^{1/4}u = f - f(0) \in L^1+\dot{H}^{-1/2}(S^1) $. Additionally, we observe that since $u$ contains only positive frequencies, we trivially have ${\mathcal{R}}(-\Delta)^{1/4}u\in L^1+\dot{H}^{-1/2}(S^1)$ as well with
$$\|{\mathcal{R}}(-\Delta)^{1/4}u\|_{L^1+\dot{H}^{-1/2}(S^1) }=\|(-\Delta)^{1/4}u\|_{L^1+\dot{H}^{-1/2}(S^1)}.$$
From the inequality \eqref{fractineq}, observing that $\Xint-_{S^1} u =0$, we deduce that
$$\|u\|_{L^2(S^1)}\le C\|(-\Delta)^{1/4}u\|_ {(L^1+\dot{H}^{-1/2})(S^1)} = C\|f - f(0) \|_ {L^1+\dot{H}^{-1/2}(S^1)} \leq C^\prime \| f \|_ {L^1+{H}^{-1/2}(S^1)},$$
where we used \eqref{homogenisationineq}. Hence $\sum_{n> 0}\frac{{ f_n}}{n} e^{in\theta}\in H^{1/2}(S^1)$ and $g(z)=\sum_{n>0} \frac{f_n}{n} z^n\in H^1(\mathbb{D})$. We have
$g'(z)=\sum_{n\ge 0} {f_{n+1}} z^{n}\in L^2(\mathbb{D})$ and
\begin{eqnarray}\label{bergman4}
\|f(z)-f(0)\|_{L^2(\mathbb{D})}&=&\|zg'(z)\|_{L^2(\mathbb{D})}\nonumber\\
&\le& C \|g\|_{H^{1/2}(S^1)}=C\|u\|_{L^2(S^1)}\nonumber\\
&\le& C\|f\|_{L^1+{H}^{-1/2}(S^1)}.\end{eqnarray}
The desired estimate follows by the triangle inequality, if we can show:
$$| f(0) | \leq C \| f \|_{L^1+{H}^{-1/2}(S^1)}$$
To achieve this, let us decompose $f = g + h$ with $g \in L^1(S^1)$ as well as $h \in H^{-1/2}(S^1)$. Then we denote as usual the Fourier coefficients of $g,h$ by $g_n, h_n$ for all $n \in \mathbb{Z}$ and define:
$$G(z) := \sum_{n \geq 0} g_n z^n + \sum_{n < 0} g_n \overline{z}^{|n|}, \quad H(z) := \sum_{n \geq 0} h_n z^n + \sum_{n < 0} h_n \overline{z}^{|n|}.$$
By the summability properties, these define harmonic functions on $\mathbb{D}$ having boundary values $g, h$ respectively. By comparison of the coefficients, we also observe:
$$f(z) = G(z) + H(z),$$
in particular for $z=0$. Moreover, by the mean value property of harmonic functions over the boundary of the disc, we can deduce:
$$| G(0) | \lesssim \| g \|_{L^1(S^1)}.$$
Using the mean value property over the entire disc, we similarily see by H\"older's inequality:
$$| H(0) | \lesssim \| H \|_{L^{1}(\mathbb{D})} \lesssim \| H \|_{L^{2}(\mathbb{D})}.$$
It is easy to verify by a direct computation analogous to the same characterisation of the norm in $\mathcal{A}^2(\mathbb{D})$ that:
$$\| H \|_{L^{2}(\mathbb{D})}^2 \sim \sum_{n \in \mathbb{Z}} \frac{| h_n|^2}{| n | + 1} \leq \| h \|_{H^{-{1/2}}(S^1)}.$$
In conclusion, we have:
$$| f(0)| \leq C \left( \| g \|_{L^1(S^1)} + \| h \|_{H^{-{1/2}}(S^1)} \right).$$
By taking the infimum over $g,h$ such that $f=g+h$ we get
\begin{equation}\label{bergman4bis}
| f(0)| \leq C \| f \|_{L^1 + H^{-{1/2}}(S^1)}\end{equation}
By combining \eqref{bergman4} and \eqref{bergman4bis}, we obtain the desired estimate:
\begin{equation}
\label{bergman4tris}
\|f\|_{L^2(\mathbb{D})} \leq C\|f\|_{L^1+{H}^{-1/2}(S^1)}.
\end{equation}
\noindent
\textbf{2.} In the general case when $ \limsup_{r\to 1^-}\|f(re^{i\theta})\|_{L^1+{H}^{-1/2} (S^1)}<+\infty$, we consider
for every $0<r<1$ the function $f_r(z)=f(rz)\in C^{\infty}(\bar{B}(0,1)).$ We can apply
\eqref{bergman4} to $f_r$ and obtain that
\begin{equation}\label{bergman5}
\|f_r\|_{L^2(\mathbb{D})}\le C\|f_{r}\|_{ L^1+ {H}^{-1/2}(S^1)}.\end{equation}
Since by assumption $\limsup_{r\to 1^-}\|f(re^{i\theta})\|_{ L^1+ {H}^{-1/2}(S^1)}<+\infty,$
we deduce that
\begin{equation}\label{bergman6}
\sup_{0<r<1}\|f_r\|_{L^2(\mathbb{D})}<+\infty.
\end{equation}
The inequality implies that actually $f\in L^2(\mathbb{D})$ as well as\footnote{
Let $f(z)=\sum_{n\ge 0} f_n z^n$
We observe that
\begin{eqnarray}
\|f_r\|^2_{L^2(\mathbb{D})}&=&\int_0^1\int_0^{2\pi}|f(\rho re^{i\theta})|^2 \rho d\theta d\rho\nonumber\\
&=&2\pi\int_0^1\sum_{n=0}^{\infty}|a_n|^2r^{2n}\rho^{2n+1} d\rho=
2\pi \sum_{n=0}^{\infty}\frac{|a_n|^2r^{2n}}{2n+2}.
\end{eqnarray}
and similarily
\begin{equation}
\|f\|^2_{L^2(\mathbb{D})}=2\pi \sum_{n=0}^{\infty}\frac{|a_n|^2}{2n+2}.
\end{equation}
From \eqref{bergman6} and extracting a weakly convergent subsequence which by convergence of the Fourier coefficients must have limit $f$, it follows that
$\|f\|^2_{L^2(\mathbb{D})}<+\infty$ and Abel's Theorem on power series yields that
$$
\lim_{r\to 1^-}\|f_r\|^2_{L^2(\mathbb{D})}=\|f\|^2_{L^2(\mathbb{D})}.$$
}
and
\begin{equation}\label{bergman2}
\|f\|_{L^2(\mathbb{D})}\le C\|f\|_{ L^1+ {H}^{-1/2}(S^1)}.\end{equation}
\noindent
{\bf Conversely}, let $f\colon\mathbb{D}\to\mathbb{C}$ be in ${\mathcal{A}}^2(\mathbb{D}^2)$. We write $f(z)=\sum_{n=0}^{\infty} a_nz^n$. We prove the following:\par
\noindent
{\bf Claim:} $\limsup_{r\to 1^-}\|f(re^{i\theta})\|_{ L^1+{H}^{-1/2}(S^1)}<+\infty$.
\par
\noindent
{\bf Proof of the claim.} We show that $\limsup_{r\to 1^-}\|f(re^{i\theta})\|_{H^{-1/2}(S^1)}<+\infty$. For every
$0<r<1$, we set $f_r(z)=f(rz)\in C^{\infty}(\bar{B}(0,1)).$ Since $f\in L^2(\mathbb{D})$, we have
\begin{equation}\label{frf}
\limsup_{r\to 1^-}\|f_r\|_{L^2(\mathbb{D})}=\|f\|_{L^2(\mathbb{D})}.
\end{equation}
Moreover
\begin{equation}
\|f_r\|^2_{H^{-1/2}(S^1)}=\sum_{n\ge 0} \frac{|f_n|^2}{1+n} r^{2n}
\end{equation}
and
\begin{eqnarray}\label{h12}
\int_{\mathbb{D}}|f_{r}|^2&=&2\pi\int_0^1\sum_{n\ge0} |f_n|^2 r^{2n}s^{2n} s ds\simeq \sum_{n\ge 0} \frac{|f_n|^2}{2n+2} r^{2n}\nonumber\\
&\simeq& \frac{1}{2} \sum_{n\ge 0} \frac{|f_n|^2}{n+1} r^{2n}\simeq \|f_r\|_{H^{-1/2}(S^1)}.
\end{eqnarray}
By combining \eqref{frf} and \eqref{h12}, we get that
\begin{equation}
\limsup_{r\to 1^-} \|f_r\|_{ H^{-1/2}(S^1)}\lesssim \|f\|_{L^2(\mathbb{D})}<+\infty.
\end{equation}
\noindent
We conclude the proof.~~\hfill $\Box$\par
\medskip
Next we show that Theorem \ref{BBB} is actually equivalent to Theorem \ref{nonlocalBB1D}.
\begin{prop}\label{equiv}
Theorem \ref{BBB} implies Theorem \ref{nonlocalBB1D}. Therefore, they are equivalent.
\end{prop}
\noindent
{\bf Proof.} We have already seen in the proof of Theorem \ref{BBB} that Theorem \ref{nonlocalBB1D} implies the fact that a holomorphic function with the property that
$\limsup_{r\to 1^-}\|f(re^{i\theta})\|_{ L^1+ {H}^{-1/2}(S^1)}<+\infty$ is in $L^2(\mathbb{D})$, namely it belongs to the Bergman space ${\mathcal{A}}^2(\mathbb{D}).$
Conversely, let us consider $u\in C^{\infty}(S^1)$ such that $(-\Delta)^{1/4}u,{\mathcal{R}}(-\Delta)^{1/4}u\in L^1+\dot {H}^{-1/2}(S^1)$. We assume that $\int_{0}^{2\pi} u(e^{i\theta}) d\theta=0$. We decompose $u=u^++u^-$, where
$$u^+=\sum_{n>0}u_n e^{in\theta},~~~u^-=\sum_{n<0}u_n e^{in\theta}.$$
Let us first consider $u^+$. By assumption we have $\sum_{n\ge 1}n^{1/2}u_n e^{in\theta} = 1/2 ((-\Delta)^{1/4}u -i {\mathcal{R}}(-\Delta)^{1/4}u)\in L^1+\dot {H}^{-1/2}(S^1).$
Let $f(z)=\sum_{n\ge 1}n^{1/2} u_n z^n$ be the harmonic extension of $v=(-\Delta)^{1/4} u^+$ in $\mathbb{D}.$
From Theorem \ref{BBB}, it follows that $f^+=\sum_{n> 0}n^{1/2} u_n z^n\in L^2(\mathbb{D})$ and
$$\|f^+\|_{L^2(\mathbb{D})}\le C\|f^+\|_{ L^1+ {H}^{-1/2}(S^1)}\le \|(-\Delta)^{1/4} u^+\|_{ L^1+\dot {H}^{-1/2}(S^1)}.$$
Switching to the homogeneous Sobolev space is possible, as we have $\dot{H}^{-1/2}(S^1) \subset H^{-1/2}(S^1)$ continuously embedded. Since $f^+(z)=\sum_{n>0}n^{1/2} u_n z^n\in L^2(\mathbb{D})$, it follows that $\sum_{n>0} \frac{u_n}{n^{1/2} }z^n\in H^1(\mathbb{D})$ and therefore
$\sum_{n>0} \frac{u_n}{n^{1/2} }e^{in\theta}\in \dot{H}^{1/2}(S^1)$.
Hence $u^+\in L^2(S^1)$ with
\begin{eqnarray}
\|u^{+}\|_{L^2(S^1)}&\le& C \|(-\Delta)^{1/4} u^+\|_{ L^1+\dot {H}^{-1/2}(S^1)}\nonumber \\
&\le& C\left( \|(-\Delta)^{1/4}u\|_{ L^1+\dot {H}^{-1/2}(S^1)}+\|{\mathcal{R}}(-\Delta)^{1/4}u\|_{ L^1+\dot {H}^{-1/2}(S^1)} \right).
\end{eqnarray}
The same arguments hold for $u^-.$ We conclude the proof. ~\hfill $\Box$
\section{The Bourgain-Brezis Inequality on the Torus ${\mathbf{T^n, n \geq 2}}$}\label{torus}
Next, we would like to generalise the result from Theorem \ref{nonlocalBB1D} to domains of dimension $n \geq 2$. To achieve this while retaining the general structure of the proof, we first have to determine the right set of conditions and the appropriate domain. Observe that it is clear, due to the proof for $S^1$ heavily relying on Fourier series, that the natural domain for such a generalisation is the torus $T^n$. In investigating generalisations of the proof, we have to focus on two aspects: Clifford algebras and boundedness of the kernel. The first subsection introduces complex Clifford algebras as a useful tool in our proof and shows how to generalize the argument presented in section \ref{secproof}. The results and properties of Clifford algebras are due to \cite{gilbert} and \cite{hamilton} and are briefly discussed in the preliminary section \ref{prel}. The second subsection fills in the gaps in the proof by showing that the kernel used is actually bounded, following an argument presented in \cite[p.405-406]{bourgain1}. In the case $n=1$, we have seen that $k$ has an explicit description as a sawtooth function. In higher dimensions, unfortunately, we are not aware of an explicit formula for the kernel. However, due to some estimates on alternating sums, we can remedy this lack of explicit representation and derive the crucial properties abstractly.
\begin{thm}
\label{nonlocalBBnD}
Let $u \in \mathcal{D}'(T^{n})$ be complex-valued and such that:
$$(-\Delta)^{\frac{n}{4}}u,\mathcal{R}_j(-\Delta)^{\frac{n}{4}}u\in (L^1 + \dot{H}^{-\frac{n}{2}})(T^n), \quad \forall j \in \{ 1, \ldots n \}.$$
Then we have
$u - \Xint-_{T^{n}} u dx \in L^{2}_{\ast}(T^{n})$ with
\begin{equation}
\label{estimateforuinthm}
\Big{\|} u - \Xint-_{T^{n}} u dx \Big{\|}_{L^{2}} \leq C\left(\|(-\Delta)^{n/4} u\|_{L^1 + \dot{H}^{-n/2}(T^n)}+\sum_{j=1}^n\|{\mathcal{R}}_j(-\Delta)^{n/4} u\|_{L^1 + \dot{H}^{-n/2}(T^n)}\right),
\end{equation}
for some $C > 0$ independent of $u$.
\end{thm}
\noindent
{\bf Proof of Theorem \ref{nonlocalBBnD}.}\par
Let us first note that, if we take $T^n$ with $n \geq 2$, there are $n$ different Riesz transforms, one for each basis direction. This suggests that the right conditions should involve some restriction on each of the Riesz transforms. In addition, considering the symbol of $D^2$, we see that we rely some cancellation property stemming from the complex nature of $i$. \footnote{This refers to the property $i^2 = -1$ which was key to reduce the multiplier of $D^2$ to a simpler form.}
Therefore, a natural way to obtain a generalisation would involve Clifford algebras to include sufficiently many anticommuting complex units.
\par
Firstly, it is immediate that the same simplifications as in the case $n=1$ apply here. So we may assume $\hat{u}(0) = 0$. Throughout most of this proof, the coefficient $m=0$ will be implicitly omitted, as it will be vanishing for all functions/distributions considered. Moreover, the reduction to smooth functions applies equally well in this case. Therefore, we may assume without loss of generality that $u, f_j, g_j$ are all smooth.\\
The heart of the argument lies in the correct definition of $D$ and $\overline{D}$ on $T^n$. As mentioned in the introduction of the current section, Clifford algebras and their set of complex units actually provide the desired framework. Let $\mathbb{C}_n$ denote the universal complex Clifford algebra associated with the quadratic space $(\mathbb{C}^{n}, Q)$,
where:
\begin{equation}
Q(z_1, \ldots, z_n) := -\sum_{j=1}^{n} z_j^2, \quad \forall (z_1, \ldots, z_n) \in \mathbb{C}^{n}.
\end{equation}
We emphasise that the particular choice of $Q$ is at odds with usual conventions for complex Clifford algebras, but using our quadratic form, we obtain the appropriate basis commutation relations while remaining isomorphic to the usual convention. One could reduce to the usual defining quadratic form by choosing $i \cdot e_j$ instead of the standard basis $e_j$ throughout our proof. In fact, the main reason why we decided to use our convention is to use the Riesz operators in their usual form.\\
Observe that we then have, for the standard basis denoted by $e_1, \ldots, e_n$:
\begin{equation}
e_j e_k + e_k e_j = 2 \delta_{jk}, \quad \forall j, k \in \{1, \ldots, n\},
\end{equation}
simply by the definition of Clifford algebras and the quadratic form $Q$. We define now for any $v\in C^{\infty}(T^n, \mathbb{C})$:
\begin{align}
\label{diffop1}
Dv &= \Delta^{\frac{n}{4}}( Id + \sum_{j=1}^{n} e_j \mathcal{R}_{j})v \\
\label{diffop2}
\overline{D}v &= \Delta^{\frac{n}{4}}( Id - \sum_{j=1}^{n} e_j \mathcal{R}_{j})v.
\end{align}
We emphasise the similarity with \cite[(5.14)]{gilbert} used in the context of Hardy spaces. The crucial observation for our purposes is the following multiplier property for Fourier series for every $m \in \mathbb{Z}^{n}$:
\begin{align}
\mathcal{F}(Dv)(m) &= | m |^{\frac{n}{2}} \big{(} 1 + \sum_{j=1}^{n} e_j \cdot i \frac{m_j}{| m |} \big{)} \mathcal{F}(v)(m) \\
\mathcal{F}(\overline{D}v)(m) &= | m|^{\frac{n}{2}} \big{(} 1 - \sum_{j=1}^{n} e_j \cdot i \frac{m_{j}}{| m |} \big{)} \mathcal{F}(v)(m),
\end{align}
where $| m|$ denotes the Euclidean norm on $\mathbb{Z}^{n}$. We highlight that at this point, we know that $Du$ and $\overline{D}u$ are functions in $L^1 + \dot{H}^{-\frac{n}{2}}(T^n, \mathbb{C}_{n})$. Completely analogous to the proof of Theorem \ref{nonlocalBB1D}, we may find $F \in L^{2}$ (due to the invertibility of non-zero vectors $v \in \mathbb{R}^{n}$ in $\mathbb{C}_{n}$\footnote{Observe that for real vectors in $\mathbb{R}^n$, we find $m^2 = |m|^2$. For general vectors in $\mathbb{C}^{n}$, this fails, as can be seen in the counterexample: $$(e_1 + i e_2 )^2 = 0$$}). To be precise, observe that if $DF = f$, $f$ and $g$ are defined to satisfy $Du = f + g$ by splitting the terms $f_j, g_j$ in the natural way, then:
\begin{equation}
\forall m \neq 0: \quad | m |^{\frac{n}{2}} \big{(} 1 + \sum_{j=1}^{n} e_j \cdot i \frac{m_j}{| m |} \big{)} \widehat{F}(m) = \widehat{f}(m),
\end{equation}
which may be rewritten as:
\begin{equation}
\widehat{F}(m) = \frac{1}{2|m|^{\frac{n}{2}}} \Big{(} 1 - \sum_{j=1}^{n} e_j \cdot i \frac{m_j}{|m|} \Big{)} \widehat{f}(m),
\end{equation}
by using the multiplication relations and associativity on $\mathbb{C}_{n}$. To conclude that $F \in L^{2}$, it suffices to check summability of the Fourier coefficients:
\begin{align}
\sum_{m \in \mathbb{Z}^{n} \setminus \{ 0 \}} \| \widehat{F}(m) \|^{2} &= \sum_{m \neq 0} \Big{\|} \frac{1}{2|m|^{\frac{n}{2}}} \big{(} 1 - \sum_{j=1}^{n} e_j \cdot i \frac{m_j}{|m|} \big{)} \widehat{f}(m) \Big{\|}^2 \notag \\
&\lesssim \sum_{m \neq 0} \frac{1}{|m|^{n}} \| \widehat{f}(m) \|^2 \notag \\
&\lesssim \| f \|_{\dot{H}^{-\frac{n}{2}}}^2 < +\infty.
\end{align}
We mention here that the characterisations for regularity and integrability carry over without problem, even if we use Clifford algebra-valued functions by verifying componentwise regularity.\\
Consequently, as in the case $n=1$, we may define $\tilde{u} = u - F$ and observe that $D\tilde{u} =: g \in L^1$. Solving $Dw = \tilde{u}$ in the sense of distributions leaves us with $D^{2}w = g$.\\
The key point behind the second proof of Theorem \ref{nonlocalBB1D} lies in the fact, that $D^2$ has an inverse given by the convolution with a bounded function. By a direct computation, we arrive at the following expression for the multiplier associated with $D^2$:
\begin{equation}
\label{intermediate1}
\mathcal{F}(D^2w)(m) = | m |^{n} \big{(} 1 + \sum_{j=1}^{n} e_j \cdot i \frac{m_j}{| m |} \big{)}^2 \mathcal{F}(w)(m) = 2i \cdot | m |^{n} \big{(} \sum_{j=1}^{n} e_j \frac{m_j}{| m |} \big{)} \mathcal{F}(w)(m),
\end{equation}
for every $m \in \mathbb{Z}^{n}$. Observe that we used the fact that the complex unit $i$ of $\mathbb{C}$ commutes with all $e_j$ (as the Clifford algebra is a complex algebra)
and that:
\begin{equation}
(i \cdot e_j)^2 = i^2 \cdot e_j^2 = i^2 = -1.
\end{equation}
Let us identify $m = \sum m_j e_j$, i.e. we consider the vector $m \in \mathbb{Z}^{n} \subset \mathbb{C}^{n}$ as an element in $\mathbb{C}_n$. Therefore, \eqref{intermediate1} becomes:
\begin{equation}
\mathcal{F}(D^2w)(m) = 2i | m |^{n-1} \cdot m \mathcal{F}(w)(m), \quad \forall m \in \mathbb{Z}^{n}.
\end{equation}
As stated before, all vectors $\mathbb{R}^{n} \subset \mathbb{C}_n$
are invertible due to:
\begin{equation}
z^2 = -Q(z), \quad \forall z \in \mathbb{C}^{n}.
\end{equation}
So, for the real vector $m$, we have due to $m \cdot m = - Q(m)$:
\begin{equation}
m^{-1} = \frac{m}{| m |^2}, \quad \forall 0 \neq m \in \mathbb{Z}^{n}.
\end{equation}
This means that $D^2 w = g$ can be restated as:
\begin{equation}
\mathcal{F}(w)(m) = \frac{1}{2i} \cdot \frac{m}{| m |^{n+1}} \mathcal{F}(g)(m),
\end{equation}
for every $0 \neq m \in \mathbb{Z}^{n}$.\\
For now, let us assume that a function $K$ on the torus exists, such that:
\begin{equation}
\widehat{K}(m) = \frac{1}{2i} \cdot \frac{m}{| m |^{n+1}}, \quad \forall m \in \mathbb{Z}^{n} \setminus \{0\}.
\end{equation}
In this case, we may check using Fourier coefficients that (keeping in mind that the order of factors in the convolution matters for products in Clifford algebras):
\begin{equation}
w = \frac{1}{(2\pi)^n} K \ast g
\end{equation}
Thus, we have the following inequality:
\begin{equation}
\| w \|_{L^{\infty}} \lesssim \| K \|_{L^{\infty}} \| g \|_{L^{1}}.
\end{equation}
This is an immediate consequence of the definition, Minkowski's inequality and continuity of the Clifford multiplication in the Clifford algebra norm.\\
Moreover, we may deduce:
\begin{align}
\| u - F \|_{L^2}^2 &= \int_{T^{n}} P_{0}\big{(} \overline{(u-F)} \cdot (u-F) \big{)}dx \notag \\
&= P_{0} \Big{(} \int_{T^{n}} \overline{(u-F)} \cdot (u-F) dx \Big{)} \notag \\
&\leq \Big{\|} \int_{T^{n}} \overline{(u-F)} \cdot (u-F) dx \Big{\|} \notag \\
&= \Big{\|} \int_{T^{n}} \overline{Dw} \cdot (u-F) dx \Big{\|} \notag \\
&= \Big{\|}\int_{T^{n}} \Big{(} \sum_{m} \overline{\widehat{Dw}(m)} e^{-i\langle m, x \rangle} \Big{)} \cdot \Big{(}\sum_{\tilde{m}} \widehat{u-F}(m) e^{i \langle \tilde{m}, x \rangle} \Big{)} dx \Big{\|} \notag \\
&\simeq \Big{\|} \sum_{m} \overline{\widehat{Dw}(m)} \cdot \widehat{u-F}(m) \Big{\|} \notag \\
&= \Big{\|} \sum_{m} \overline{| m |^{\frac{n}{2}} \big{(} 1 + \sum_{j=1}^{n} e_j \cdot i \frac{m_j}{| m |} \big{)}\widehat{w}(m)} \cdot \widehat{u-F}(m) \| \notag \\
&= \Big{\|} \sum_{m} \overline{\widehat{w}(m)} \cdot \overline{| m |^{\frac{n}{2}} \big{(} 1 + \sum_{j=1}^{n} e_j \cdot i \frac{m_j}{| m |} \big{)}}\widehat{u-F}(m) \| \notag \\
&= \Big{\|} \sum_{m} \overline{\widehat{w}(m)} \cdot | m |^{\frac{n}{2}} \big{(} 1 - \sum_{j=1}^{n} e_j \cdot i \frac{m_j}{| m |} \big{)}\widehat{u-F}(m) \| \notag \\
&= \Big{\|} \sum_{m} \overline{\widehat{w}(m)} \cdot \widehat{{\overline{D}}(u-F)}(m) \| \notag \\
&\simeq \Big{\|} \int_{T^{n}} \overline{w} \cdot {\overline{D}}(u-F) dx \Big{\|}.
\end{align}
Observe that in the first inequality, we used that the norm squared of $u - F$ actually appears as the coefficient associated with $1$ in the product $\overline{u-F} \cdot (u-F)$. In addition, the conjugation in the ninth line can easily deduced from our definition in the preliminary section of the paper, see \eqref{cliffconjdef}. The remainder of the argument then follows completely analogous to the $1D$-proof, up to the obvious modifications. Again, simple considerations show that we even have the following inequality:
\begin{equation}
\label{ineqonsum}
\Big{\|} u - \Xint-_{T^{n}} u dx \Big{\|}_{L^{2}} \lesssim \sum_{j=0}^{n} \big{\|} (-\Delta)^{\frac{n}{2}} \mathcal{R}_{j} u \big{\|}_{\dot{H}^{-\frac{n}{2}} + L^{1}},
\end{equation}
where $\mathcal{R}_{0} = Id$.\\
To complete the proof in the same way as for Theorem \ref{nonlocalBB1D}, we still need to find a bounded kernel $K$ satisfying:
\begin{equation}
\widehat{K}(m) = \frac{1}{2i} \cdot \frac{m}{| m |^{n+1}}, \quad \forall 0 \neq m \in \mathbb{Z}^{n}.
\end{equation}
This is the purpose of the next subsection, so we may conclude the proof of Theorem \ref{nonlocalBBnD} at this point.~~\hfill $\Box$
\subsection{Boundedness of the Kernel}
Lastly, let us find an appropriate kernel. We first notice that due to linearity, symmetry and the splitting into different directions, it is enough to find a bounded function $k$, such that:
\begin{equation}
\widehat{k}(m) = \frac{m_1}{| m |^{n+1}}, \quad \forall 0 \neq m \in \mathbb{Z}^{n}.
\end{equation}
Consequently, we want to study the boundedness of the following conditionally convergent series:
\begin{equation}
\label{formulakernel1}
k(x) = \sum_{m \in \mathbb{Z}^{n} \setminus \{ 0\}} \frac{m_1}{| m |^{n+1}} e^{i \langle m, x \rangle}.
\end{equation}
Let us fix some notation. We usually identify $m \in \mathbb{Z}^{n}$ with $m = (m_1, \tilde{m})$, where $\tilde{m} \in \mathbb{Z}^{n-1}$. We will sometimes use the same notation for $x \in \mathbb{R}^{n}$. Moreover, for any $m$, we define $m^\prime = (-m_1, \tilde{m})$. This allows us to immediately see:
\begin{equation}
\widehat{k}(m^\prime) = - \widehat{k}(m), \quad \forall m \in \mathbb{Z}^{n} \setminus \{ 0 \}.
\end{equation}
This observation enables us to rewrite \eqref{formulakernel1} as follows:
\begin{equation}
\label{formulakernel2}
k(x) = 2i \cdot \sum_{m_1 > 0} \sum_{\tilde{m} \in \mathbb{Z}^{n-1}} \frac{m_1}{| m |^{n+1}} \sin(m_1 x_1) e^{i \langle \tilde{m}, \tilde{x} \rangle}.
\end{equation}
The strategy of the proof is based on \cite[p.405-406]{bourgain1}. Thus, the main point is to split the sum into partial sums involving $m_1$ and $| \tilde{m} |$ being comparable to some dyadic $2^{k_1}$ and $2^{\tilde{k}}$ respectively. Then, we distinguish $k_1 \leq \tilde{k}$ and $k_1 \geq \tilde{k}$ to conclude. Thus, we consider the following sum derived from \eqref{formulakernel2}:
\begin{equation}
\label{estimatekernel1}
| k(x) | \leq \sum_{k_1 \geq 0} \sum_{\tilde{k} \geq 0} \Big{|} \sum_{m_1 \sim 2^{k_1}} \sum_{| \tilde{m} | \sim 2^{\tilde{k}}} \frac{m_1}{| m |^{n+1}} \sin(m_1 x_1) e^{i \langle \tilde{m}, \tilde{x} \rangle} \Big{|}.
\end{equation}
Let us mention an uniform estimate for fixed $k_1, \tilde{k}$. To achieve this, we distinguish two cases: $k_1 \geq \tilde{k}$ and $k_1 < \tilde{k}$. We shall need the following estimate that can be found in \cite[(4.22)]{bourgain1}
\begin{equation}
\label{sinest}
\Big{|} \sum_{\ell \in I} \sin(\ell x) \Big{|} \lesssim 4^k |x| \wedge \frac{1}{|x|},
\end{equation}
for every $k \in \mathbb{N}$, $x \in S^1$ and subinterval $I \subset [2^{k - 1}, 2^{k}]$. Here, $\wedge$ denotes the minimum of two functions. Let us provide the argument in a more abstract manner: Consider a finite sum of the form:
\begin{equation}
\label{abstractseries}
\sum_{m_1} \sum_{\tilde{m}} a_{m_1} b_{\tilde{m}} c_{m_1, \tilde{m}}.
\end{equation}
Observe that the summands in \eqref{estimatekernel1} inside the absolute value clearly have this form. Let us denote by $A_{m_1}$ the partial sum of all $a_l$ up to the $m_1$-th element. In the case of \eqref{estimatekernel1}, this would be a sum of $\sin(l x)$ over an interval with $l$ comparable to $2^{k_1}$, hence we may use the bound \eqref{sinest}. Therefore, we may rewrite \eqref{abstractseries} as:
\begin{align}
\label{simplification}
\sum_{m_1} \sum_{\tilde{m}} a_{m_1} b_{\tilde{m}} c_{m_1, \tilde{m}} &= \sum_{m_1} \sum_{\tilde{m}} (A_{m_1} - A_{m_1 - 1} ) b_{\tilde{m}} c_{m_1, \tilde{m}} \notag \\
&= \sum_{m_1} \sum_{\tilde{m}} A_{m_1} b_{\tilde{m}} (c_{m_1, \tilde{m}} - c_{m_1+1, \tilde{m}}),
\end{align}
which, in the case of \eqref{estimatekernel1}, can be estimated using the bound on sums of sinus functions in \eqref{sinest}, the boundedness of the $b_{\tilde{m}}$ which are merely $e^{i\langle \tilde{m}, \tilde{x} \rangle}$ and finally the estimate:
\begin{equation}
\label{differentiationest}
\Big{|} \frac{m_1}{| m |^{n+1}} - \frac{m_1 + 1}{( (m_1 + 1)^2 + | \tilde{m} |^{2})^{\frac{n+1}{2}}} \Big{|} \lesssim \frac{1}{| m |^{n+1}}.
\end{equation}
We mention the slight imprecision, as in \eqref{simplification}, the extremal partial sums $A_l$ require further attention. However, in the case we are considering, similar techniques can be applied (since we no longer sum over $m_1$) and we omit further details.\\
Therefore, we arrive at the following estimate:
\begin{equation}
\label{baseestimate}
\Big{|} \sum_{m_1 \sim 2^{k_1}} \sum_{| \tilde{m} | \sim 2^{\tilde{k}}} \frac{m_1}{| m |^{n+1}} \sin(m_1 x_1) e^{i \langle \tilde{m}, \tilde{x} \rangle} \Big{|} \lesssim 2^{k_1}\big{(} 2^{k_1} |x_1| \wedge \frac{1}{2^{k_1}|x_1|} \big{)} \Big{\|} \frac{1}{| m |^{n+1}} \Big{\|}_{l^1(m_1 \sim 2^{k_1}, | \tilde{m} | \sim 2^{\tilde{k}})}
\end{equation}
If $k_1 \geq \tilde{k}$, we may simplify \eqref{estimatekernel1} using \eqref{baseestimate} as follows:
\begin{align}
| k(x)| &\lesssim \sum_{k_1 \geq 1} 2^{\tilde{k}(n-1)} 2^{k_1} \frac{1}{2^{k_1 (n+1)}} \cdot 2^{k_1}\big{(} 2^{k_1} |x_1| \wedge \frac{1}{2^{k_1}|x_1|} \big{)} \notag \\
&\leq \sum_{k_1 \geq 0} 2^{k_1} |x_1| \wedge \frac{1}{2^{k_1}|x_1|} \lesssim C < \infty,
\end{align}
which can be easily bounded by the definition of the minimum.\\
\noindent
If $\tilde{k} > k_1$, we find:
\begin{align}
\sum_{k_1 \geq 0} \sum_{\tilde{k} \geq 0} &\Big{|} \sum_{m_1 \sim 2^{k_1}} \sum_{| \tilde{m} | \sim 2^{\tilde{k}}} \frac{m_1}{| m |^{n+1}} \sin(m_1 x_1) e^{i \langle \tilde{m}, \tilde{x} \rangle} \Big{|} \notag \\
&\lesssim \sum_{k_1} \sum_{\tilde{k} > k_1} 4^{k_1}\big{(} 2^{k_1} |x_1| \wedge \frac{1}{2^{k_1}|x_1|} \big{)} \cdot \frac{1}{2^{k_1 (n+1)}} \sum_{| \tilde{m} | \sim 2^{\tilde{k}}} \frac{1}{(1 + \frac{| \tilde{m}|^2}{2^{2 k_1}})^{\frac{n+1}{2}}} \notag \\
&\lesssim \sum_{k_1 \geq 0} \frac{2^{k_1 (n-1)} 4^{k_1}}{2^{k_1 (n+1)}} \big{(} 2^{k_1} |x_1| \wedge \frac{1}{2^{k_1}|x_1|} \big{)} \notag \\
&\leq \sum_{k_1 \geq 0} 2^{k_1} |x_1| \wedge \frac{1}{2^{k_1}|x_1|} \leq C < \infty,
\end{align}
where we estimated the sum over $\tilde{m}, \tilde{k}$ by a dominating integral. This shows that $k(x)$ is actually bounded and possesses the required Fourier coefficients, hence adding the last ingredient missing in our proof of Theorem \ref{nonlocalBBnD}.
\section{Existence Result for a certain Fractional PDE}\label{exist}
Similar to \cite{bourgain1}, the estimates in Theorem \ref{nonlocalBB1D} and \ref{nonlocalBBnD} may be used to derive existence results for a particular differential operator. However, before turning to the PDE itself, let us briefly provide an alternative formulation of our main theorems for a more general class of distributiions:
\begin{thm}
\label{mainres3}
Let $u \in \mathcal{D}^\prime(T^{n}, \mathbb{C}_n)$ be $\mathbb{C}_n$-valued and assume that:
\begin{equation}
Du, \overline{D}u \in \dot{H}^{-\frac{n}{2}} + L^{1}(T^{n}, \mathbb{C}_{n}).
\end{equation}
Here, $D$ and $\overline{D}$ are the operators defined in the proof of Theorem \ref{nonlocalBBnD}. Then $u \in L^{2}(T^n, \mathbb{C}_n)$ and we have the following estimate:
\begin{equation}
\Big{\|} u - \int_{T^{n}} u dx \Big{\|}_{L^2} \lesssim \| Du \|_{\dot{H}^{-\frac{n}{2}} + L^{1}} + \| \overline{D}u \|_{\dot{H}^{-\frac{n}{2}} + L^{1}}.
\end{equation}
\end{thm}
This result is an immediate corollary of the proof of Theorem \ref{nonlocalBBnD}, as we always work with $Du$ and $\overline{D}u$ rather than the $\mathcal{R}_{j} (-\Delta)^{\frac{n}{2}}$. The possibility to generalise to Clifford algebra-valued distributions follows directly, as all arguments involved behave well with respect to the Clifford algebra product. One could also rewrite the estimate by separating the identity operator from the Riesz operators.\\
Let us now turn to the existence result. We would like to consider the following problem:
\begin{equation}
\label{decompositionofg}
g = (-\Delta)^{\frac{n}{4}}f_0+\sum_{j=1}^{n} (-\Delta)^{\frac{n}{4}} \bar{\mathcal{R}}_{j} f_{j},
\end{equation}
where $g \in L^{2}_{\ast}(T^{n})=\left\{u\in L^2(T^n):~~\Xint-_{T^n} u=0\right\}$.\footnote{The conjugate operator $\bar{\mathcal{R}}_j$ appears due to the duality used in the proof. This ensures, that we can apply the result in Theorem \ref{nonlocalBBnD}. It is simpel to see that by suitably exchanging $\mathcal{R}_j$ by $\bar{\mathcal{R}}_j$ throughout the proof of Theorem \ref{nonlocalBBnD}, the same inequality can be obtained for the dual operators and thus yields the same result as in Corollary \ref{cor1} for the usual Riesz operators.} Obviously, the PDE admits solutions $f_{0}, \ldots, f_{n}$ in $\dot{H}^{\frac{n}{2}}(T^n)$. Again, using Sobolev embeddings, it is also clear that there is a-priori no way to deduce that the $f_j$ may be chosen to be bounded or even continuous. We shall remedy this apparent lack of regularity:
\begin{cor}
\label{cor1}
Let $g \in L^{2}_{\ast}(T^{n})$. Then there exist $f_{0}, \ldots, f_{n} \in \dot{H}^{\frac{n}{2}} \cap C^{0}(T^{n})$, such that \eqref{decompositionofg} holds.
\end{cor}
\noindent{\bf Proof of Corollary \ref{cor1}.}
The proof is completely analogous to the one in \cite[Proof of Theorem 1]{bourgain1}: Let us define the following operator:
\begin{equation}
T: \bigoplus_{j=0}^{n} \dot{H}^{\frac{n}{2}} \cap C^{0}(T^{n}) \rightarrow L^{2}_{\ast}(T^{n}), \quad T(u_0, \ldots, u_n) := (-\Delta)^{\frac{n}{4}}u_0+\sum_{j=1}^{n} (-\Delta)^{\frac{n}{4}} \bar{\mathcal{R}}_{j} u_{j}. \end{equation}
It is clear that $T$ is a bounded, linear operator. Moreover, we have that its dual operator is given by:
\begin{equation}
T^{\ast}: L^{2}_{\ast}(T^{n}) \rightarrow \bigoplus_{j=0}^{n} \dot{H}^{-\frac{n}{2}} + \mathcal{M}(T^{n}), \quad T^{\ast}(v) := \big{(} (-\Delta)^{\frac{n}{4}} v, \mathcal{R}_{1}(-\Delta)^{\frac{n}{4}} v, \ldots, \mathcal{R}_{n}(-\Delta)^{\frac{n}{4}} v \big{)}.
\end{equation}
Here, $\mathcal{M}(T^{n})$ denotes the collection of Radon measures on $T^n$. As in \cite[(4.3)]{bourgain1}, it can be easily seen (using convolutions) that:
\begin{equation}
\| \cdot \|_{\dot{H}^{-\frac{n}{2}} + \mathcal{M}} = \| \cdot \|_{\dot{H}^{-\frac{n}{2}} + L^{1}} \quad \text{on } \dot{H}^{-\frac{n}{2}} + L^{1}(T^{n}).
\end{equation}
Therefore, we know by \eqref{ineqonsum} that:
\begin{equation}\label{ineqonsum2}
\| u \|_{L^{2}} \lesssim \| T^{\ast} u \|_{\bigoplus \dot{H}^{-\frac{n}{2}} + \mathcal{M}(T^{n})}.
\end{equation}
This implies that $T$ is surjective (see Theorem 2.20 in \cite{Brez}). The open mapping Theorem yields that
there is $C>0$ such that
$B^{L^2_*}(0,C)\subseteq T(B^{E}(0,1))$, where $E=\bigoplus_{j=0}^{n} \dot{H}^{\frac{n}{2}} \cap C^{0}(T^{n}).$
Therefore, for every $g\in L^2_*(S^1)$, there are $(f_0,\ldots,f_n)\in E$ such that
$(-\Delta)^{1/4} f_0+\sum_{i=1}^n(-\Delta)^{1/4} \bar{\mathcal{R}}_j f_j=g$ and
\begin{equation}
\sum_{j=0}^{n} \| f_{j} \|_{\dot{H}^{\frac{n}{2}} \cap L^{\infty}} \leq C \| g \|_{L^2},
\end{equation}
for some fixed $C > 0$. This concludes the proof.~~\hfill $\Box$\\
Using Corollary \ref{cor1}, we may derive the following simple result:
\begin{cor}
\label{cor2}
Let $f \in \dot{H}^{\frac{n}{2}}(T^{n})$. Then there exist $f_{0}, \ldots, f_{n} \in \dot{H}^{\frac{n}{2}} \cap C^{0}(T^{n})$ as well as a smooth function $\varphi \in C^{\infty}(T^{n})$, such that:
\begin{equation}
f = \varphi + \sum_{j=0}^{n} \mathcal{R}_{j} f_{j}.
\end{equation}
\end{cor}
\noindent{\bf Proof of Corollary \ref{cor2}.}
Take $g = (-\Delta)^{\frac{n}{4}}f \in L^{2}_{\ast}(T^{n})$. By Corollary \ref{cor1}, we see that there exist $f_{0}, \ldots, f_{n} \in \dot{H}^{\frac{n}{2}} \cap C^{0}(T^{n})$, such that \eqref{decompositionofg} is satisfied. Therefore, we know:
\begin{equation}
(-\Delta)^{\frac{n}{4}} \Big{(} f - \sum_{j=0}^{n} \mathcal{R}_{j} f_{j} \Big{)} = 0.
\end{equation}
But this implies that the difference lies in the kernel of $(-\Delta)^{m}$, where $m$ is the smallest integer larger or equal than $\frac{n}{4}$. Thus the difference is smooth, leading to the desired decomposition.~~\hfill $\Box$
\section{Appendix}\label{appendix}
In this section, we provide for the reader's convenience a proof of the two inequalities \eqref{embHA} and \eqref{embHA2},
since the authors have not found a precise reference in the literature. \\
\par
\noindent
{\bf 1.} Assume first that $f(z)=\sum_{n\ge 0} a_nz^n$ is an analytic function such that $\lim_{r\to 1^-}\|f(re^{i\theta})\|_{L^1(S^1)}<+\infty$.
Let $h\in L^{1}(S^1)$ be such that $\lim_{r\to 1^-}\|f(re^{i\theta})-h\|_{L^1(S^1)}.$
We set $g(z)=\sum_{n\ge 0}\frac{ a_n}{n+1}z^{n+1}$. We observe that $g^{\prime}(z)=f(z)$. From our hypothesis, we have
$ \lim_{r\to 1^-}\|g'(re^{i\theta})\|_{L^1(S^1)}<+\infty.$ Observe that this implies that
$\lim_{r\to 1^-} (\|\partial_{\theta}g(re^{i\theta})\|_{L^1(S^1)}+\|\partial_{r}g(re^{i\theta})\|_{L^1(S^1)})<+\infty$.
Define $g_r(z)=g(rz)$ for $0<r<1$. Since $g$ is harmonic in $\mathbb{D}$,
we have
\begin{eqnarray}\label{estL1}
0&=&\int_{\mathbb{D}}(\Delta g_r \bar g_r+g_r\Delta \bar g_r )dx =\int_{\partial\mathbb{D}}(\partial_r g_r\cdot \bar g_r+ \partial_r\bar g_r g_r)d\sigma -2 \int_{\mathbb{D}}|\nabla g_r|^2 dx\nonumber\\
&=& \int_{\partial\mathbb{D}}(\partial_r g\cdot \bar g+ \partial_r\bar g g)d\sigma -\int_{\mathbb{D}}|g^{\prime}_r|^2 dx.
\end{eqnarray}
We first have (observe that $\Xint-_{S^1} g_r=0$)
\begin{equation}\label{estLinfty}
\|g_r\|_{L^{\infty}{(S^1)}}\lesssim \|\partial_{\theta} g_r\|_{L^1(S^1)}
\end{equation}
and
from \eqref{estL1} it follows that
\begin{equation}\label{estL2}
\|f_r\|_{L^2(\mathbb{D})}\simeq\|g^{\prime}_r\|_{L^{2}{(\mathbb{D})}}\lesssim \|g_r\|_{L^{\infty}{(S^1)}}\|\partial_{r} g\|_{L^1(S^1)}\lesssim \| g_r'\|^2_{L^1(S^1)}.
\end{equation}
We let $r\to 1$ in \eqref{estL2} and get
\begin{equation}\label{estL3}
\|f\|_{L^2(\mathbb{D})} \lesssim \|h\|_{L^1(S^1)}.
\end{equation}
\noindent
{\bf 2.} Assume now that $f(z)=\sum_{n\ge 0} a_nz^n$ is an analytic function such that:
$$\lim_{r\to 1^-}\|f(re^{i\theta})\|_{H^{-1/2}(S^1)}<+\infty.$$\par
\noindent
{\bf Claim.} Assume $a_0 = 0$ in the power series above. Then the series $\sum_{n \ge 1} \frac{|a_n|^2}{n}<+\infty$ and
$$\sum_{n\ge 1} \frac{|a_n|^2}{n}=\lim_{r\to 1^-}\|f(re^{i\theta})\|^2_{\dot H^{-1/2}(S^1)}.$$
\noindent
{\bf Proof of the claim.} We set $A=\lim_{r\to 1^-}\|f(re^{i\theta})\|^2_{H^{-1/2}(S^1)} \simeq \lim_{r\to 1^-}\|f(re^{i\theta})\|^2_{\dot H^{-1/2}(S^1)}.$ We observe that:
$$\|f(re^{i\theta})\|^2_{\dot{H}^{-1/2}(S^1)}=\sum_{n >0} \frac{|a_n|^2r^{2n}}{n}.$$
For every $N>1$, we have
\begin{eqnarray}
A\ge \lim_{r\to 1^-} \sum_{n=1}^N \frac{|a_n|^2r^{2n}}{n} &=&\sum_{n=1}^N \lim_{r\to 1^-}\frac{|a_n|^2r^{2n}}{n}\nonumber\\
&=&\sum_{n=1}^N \frac{|a_n|^2}{n}.
\end{eqnarray}
By letting $N\to +\infty$, we get $\sum_{n=1}^\infty \frac{|a_n|^2}{n}<+\infty$ and by Abel's theorem on power series, we deduce that the norms converge
$$
\lim_{r\to 1-}\sum_{n> 0} \frac{|a_n|^2r^{2n}}{n}=\sum_{n=1}^\infty \frac{|a_n|^2}{n}.$$
Therefore, $f(e^{i\theta})\in \dot{ H}^{-1/2}(S^1)$
and $\lim_{r\to 1^-}\|f(re^{i\theta})-f(e^{i\theta})\|_{\dot H^{-1/2}(S^1)}=0$, by observing that the convergence holds weakly and the norms converge, which is an equivalent characterisation for convergence with respect to the norm in Hilbert spaces. This proves the claim.\\
Consider the function $g_r(z)=\sum_{n\ge 0}\frac{ a_n}{n+1}(rz)^{n+1}$. In this case we have $g_r\in \dot H^{1/2}(S^1)$. We have
$r \cdot f_r(z)=g'_r(z)$.
Since $g$ is harmonic in $\mathbb{D}$ we have
\begin{equation}\label{estL4}
\|f_r\|_{L^2(\mathbb{D})}\simeq\|\nabla g_r\|_{L^{2}{(\mathbb{D})}}\lesssim \|g_r\|_{\dot H^{1/2}}\equiv \|f_r\|_{ H^{-1/2}}
\end{equation}
We let $r\to 1$ in \eqref{estL4} and get
\begin{equation}\label{estL5}
\|f\|_{L^2(\mathbb{D})} \lesssim \| f \|_{H^{-1/2}(S^1)}.
\end{equation}
Both inequalities \eqref{embHA} and \eqref{embHA2} have been proved.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,382 |
{"url":"http:\/\/jeremyray.askbot.com\/question\/2896\/bitcoin-mining-components-comparison\/","text":"# Bitcoin Mining Components Comparison\n\nEach and every DragonMint 16T will generate about $one,five hundred per year (calculated with 1 BTC=$eight,857. 02). Mining profitability may possibly fluctuate. You can use this free of charge profitability calculator to figure out your projected earnings.The trick, although, was discovering a location the place you could set all that cheap energy to perform. You necessary an existing creating, due to the fact in individuals times, when bitcoin was buying and selling for just a number of dollars, no one could manage to build some thing new. You essential room for a handful of hundred higher-speed personal computer servers, and also for the large-duty cooling technique to keep them from melting down as they churned out the trillions of calculations needed to mine bitcoin. Previously mentioned all, you necessary a place that could take care of a great deal of electricitya quarter of a megawatt, maybe, or even a half a megawatt, enough to gentle up a pair hundred properties.Soon after you mine your 1st couple of dollars value of bitcoin, you must develop a private bitcoin wallet to securely keep your newly acquired cryptocurrency money. Its not simple to choose the very best bitcoin wallets simply because there are so a lot of of them, but here are out a few favourite ones.","date":"2019-02-20 04:13:21","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.22331778705120087, \"perplexity\": 3114.0595932822407}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-09\/segments\/1550247494424.70\/warc\/CC-MAIN-20190220024254-20190220050254-00027.warc.gz\"}"} | null | null |
By: John Curran
House Bill Would Create Minimum Standards for Federal Websites
A bill introduced in the House on Thursday would create a series of minimum functionality and security requirements for all Federal government agency public-facing websites and digital services. The measure also would place on the shoulders of Federal agency chief information officers the responsibility to ensure funding and implementation of the new requirements.
The bill–The 21st Century Integrated Digital Experience Act–was introduced by Reps. Ro Khanna, D-Calif., and John Ratcliffe, R-Texas. A spokesperson for Khanna said the House bill does not yet have a Senate companion measure.
The congressmen said their bill would dramatically reduce agency costs for providing assistance to citizens, and improve citizens' online interactions with the Federal government. They cited Internal Revenue Service (IRS) data from 2015 that found in-person or live assistance calls to IRS cost taxpayers an average of $40 to $60, versus an average cost of 22 cents for "digital transactions."
A summary of the bill says it would give agencies one year to meet new requirements for existing public-facing websites and digital services including: improving "effective and efficient delivery of digital services"; consolidating and personalizing web content; making data "searchable and discoverable"; ensuring secure connection; ensuring accessibility for people with disabilities; increasing use of web and data analytics "to improve website operation and address user needs"; and ensuring compliance with "GSA TTS U.S. Website standards."
The bill says that new Federal agency websites would have to meet the new standards upon launch, and that agency intranets would have to conform to the new requirements "to the greatest extent practicable."
In addition, the bill would give Federal agencies two years to provide a "digital option" for any in-person government service, and would give agencies one year to ensure that any public-facing, paper-based form, application, or service is made available in an "intuitive and adaptable" digital form.
The measure would give agencies 180 days to submit plans to increase the use of electronic signatures on contracts and related documents.
On the funding and implementation front, Federal agency CIOs would be required to coordinate with executive agency management leaders–including agency secretaries, chief financial officers, and digital service program leads–"to ensure proper funding and management alignment" to support implementation.
The CIOs also would be charged with: monitoring digital service delivery and recommending necessary changes to agency heads; providing advice and assistance to improve digital service delivery and customer experience; using customer experience and satisfaction data; and coordinating and ensuring executive agency compliance with "Public Law 115-115 'Connected Government Act'–Federal Websites Required to be Mobile Friendly."
"Without a responsible agency official, the requirements [of the bill] won't get carried out effectively," the bill summary states.
"Government exists to serve citizens, and this bill ensures government leverages available technology to provide the cohesive, user-friendly online service that people around this country expect and deserve," Khanna said in a statement.
Added Ratcliffe, "Our bill takes advantage of new and emerging technologies that can drastically improve the way our federal agencies provide critical services to folks across the country, including people with disabilities or those who live in rural areas with limited access to traditional, in-person assistance services."
The bill's introduction drew statements of praise from officials at Oracle and Adobe, and several trade groups including the Information Technology and Innovation Foundation, BSA-the Software Alliance, and the Software & Information Industry Association.
John Curran
John Curran is MeriTalk's Managing Editor covering the intersection of government and technology.
John Ratcliffe
Ro Khanna | {
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Душан Милутиновић (Веље Брдо, код Подгорице, 25. новембар 1920 — Београд, 15. октобар 1944) био је учесник Народноослободилачке борбе и народни херој Југославије.
Биографија
Рођен је 25. новембра 1920. у селу Веље Брдо, код Подгорице. Одрастао је у сиромашној сељачкој породици. Његов отац Ристо је имао осморо деце — петоро синова и три ћерке. У тешким материјалним условима завршио је основну школу. Преко дана је морао да помаже у кућним пословима, а увече је учио и писао домаће задатке. Иако је био одличан ђак, због тешког материјалног положаја, није могао да настави школовање.
Његов брат Машан Милутиновић био је члан Комунистичке партије Југославије (КПЈ) од 1936, па је Душан од њега и његових партијских другова, веома млад добио прва сазнања о револуционарном радничком покрету и учествовао у разним акцијама КПЈ, чији је члан постао 1941. године. Након окупације Југославије, 1941. активно је учествовао у припреми оружаног устанка. Када је избио Тринаестојулски устанак, заједно са двојицом браће и сестром, постао је борац Пиперског устаничког одреда. Већ у првим борбама исказао се као храбар и сналажљив борац, па је био узет за курира Привремене Врховне команде Црне Горе.
Када је почетком 1942. формиран Први црногорски омладински батаљон, постављен је за заменика командира чете. У овом батаљону је постао познат, због своје храбрости, али и због доброг односа према младим борцима, који су били у његовој чети. Његова храброст и издржљивост посебно су дошле до изражаја у зиму 1941/42, у даноноћним борбама против четника Павла Ђуришића на Сињајевини, које су трајале више од месеца дана. Маја 1942. донета је одлука да се Омладински батаљон расформира, па је Душан, заједно са педесет младића и девојака, ушао у састав Прве пролетерске ударне бригаде, односно њеног Првог црногорског батаљона. У овој бригади су се још од њеног формирања налазили његова сестра Станка и брат Машан.
У првој борби са Првом пролетерском бригадом, 3. јуна 1942. на Дурмитору, истакао се храброшћу и заробио неколико четника. Потом се истакао у борбама на Коњицу, Бугојну, Дувну, као и у вишедневним борбама на Цинцару, где се истакао као бомбаш. У борби за Ливно, са својим одељењем је уништио бункер код тврђаве и заробио пушкомитраљез. Касније се истакао и у борбама за Кључ, Јајце, Чичево, Балиновац, Миљевину. Истицао се и у борбама током Четврте и Пете непријатељске офанзиве.
Године 1944. постављен је за заменика команданта Првог црногорског батаљона Прве пролетерске ударне бригаде. Био је слушалац Официрске школе Врховног штаба НОВ и ПОЈ у Дрвару. Као слушалац ове школе, затекао се 25. маја 1944. у ослобођеном Дрвару, када је учествовао у борбама на сламању немачког ваздушног десанта и одбрани Врховног штаба, који се налазио у пећини. Након десанта, вратио се у Прву пролетерску бригаду, са којом је у јесен 1944. учествовао у борбама за ослобођење западне Србије и Београда.
Погинуо је у 15. октобра 1944. у борбама за ослобођење Београда, у тешким борбама са немачким снагама на Славији. Након ослобођења Београда, сахрањен је на Славији, заједно са Блажом Попиводом и Драгишом Радовићем, а 1954. њихови посмртни остаци су пренети на Гробље ослободилаца Београда 1944.
Указом председника ФНР Југославије Јосипа Броза Тита 27. новембра 1953. проглашен је за народног хероја.
Референце
Литература
Рођени 1920.
Умрли 1944.
Подгоричани
Комунисти Црне Горе
Југословенски партизани
Борци Прве пролетерске бригаде
Официри НОВЈ
Омладина у Народноослободилачкој борби
Народни хероји - М | {
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Kenneth Richard Purpur (March 1, 1932 – June 5, 2011) was an ice hockey player who played for the American national team. He won a silver medal at the 1956 Winter Olympics. He is the younger brother of Fido Purpur who coached him at the University of North Dakota.
References
1932 births
2011 deaths
Ice hockey players at the 1956 Winter Olympics
Medalists at the 1956 Winter Olympics
Sportspeople from Grand Forks, North Dakota
Olympic silver medalists for the United States in ice hockey
American men's ice hockey right wingers
Ice hockey people from North Dakota
North Dakota Fighting Hawks men's ice hockey players | {
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Miguel Ramírez (Huacho, 5 de abril del 1991) es un futbolista peruano. Juega de portero y su equipo actual es Deportivo Garcilaso que participa en la Liga 1. Tiene .
Trayectoria
Realizó las divisiones menores en Alianza Lima, club al que llegó a los 11 años.
El 2019 consigue el ascenso con Deportivo Llacuabamba, sin embargo a finales del 2020 desciende de categoría.
Clubes
Campeonatos nacionales
Futbolistas del Club Deportivo Alfonso Ugarte
Futbolistas del Club Deportivo Sport Loreto
Futbolistas del Club Defensor San Alejandro
Futbolistas del Club Deportivo Aurora Chancayllo
Futbolistas del Club Unión Tarapoto
Futbolistas del Sport Chavelines Juniors
Futbolistas de la Asociación Deportiva Tarma
Futbolistas del Cultural Santa Rosa
Futbolistas del Club Deportivo Garcilaso
Guardametas de fútbol | {
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Q: transmit touch from uiview to uiwebview Here is the structure of my xib
uiview
--uiview
--uiwebview
I have made changes so that the uiview is transparent. What is the way to send the touch events which are delivered to touchesBegan of uiview to the touchesBegan of uiwebview ?
A: Please go the following link and read the section under "Basics of Touch-Event Handling", i hope after you will get the idea how events trigged down to other/sub views.
http://developer.apple.com/library/ios/documentation/EventHandling/Conceptual/EventHandlingiPhoneOS/MultitouchEvents/MultitouchEvents.html
| {
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\chapter{Anomalous Dimensions}
\label{chpt:App-AnoDim}
Here we present $A$~\cite{Moch:2004pa, Moch:2005tm, Vogt:2004mw,
Vogt:2000ci}, $f$~\cite{Ravindran:2004mb, Moch:2005tm},
and $B$~\cite{Vogt:2004mw, Moch:2005tm} up to three loop level. The
$A$'s are given by
\begin{align}
\label{eq:App-AnoDimA}
A_{gg,1} &= {{C_A}} \Big\{4\Big\} \,,
\nonumber\\
A_{gg,2} &= {{C_A^{2}}} \left\{ \frac{268}{9} - 8 \zeta_2 \right\} + {{C_A n_f}} \left\{ -\frac{40}{9} \right\} \,,
\nonumber\\
A_{gg,3} &= {{C_A^3}} \left\{ \frac{490}{3}
- \frac{1072 \zeta_2 }{9}
+ \frac{88 \zeta_3}{3}
+ \frac{176 \zeta_2^2}{5} \right\}
+ {{C_{A} C_F n_f}} \left\{ - \frac{110}{3} + 32 \zeta_3 \right\}
\nonumber\\ \nonumber
& + {{C_A^{2} n_f}} \left\{ - \frac{836}{27}
+ \frac{160 \zeta_2}{9}
- \frac{112 \zeta_3}{3} \right\}
+ {{C_A n_f^2}} \left\{ - \frac{16}{27} \right\}
\intertext{and}
&A_{q{\bar q},i} = A_{b{\bar b},i} = \frac{C_{F}}{C_{A}} A_{gg,i}\,.
\end{align}
The $f$'s are obtained as
\begin{align}
\label{eq:App-AnoDimf}
f_{gg,1} &= 0 \,,
\nonumber \\
f_{gg,2} &= {{C_A^{2}}} \left\{ -\frac{22}{3} {\zeta_2} - 28 {\zeta_3} + \frac{808}{27} \right\}
+ {{C_A n_f}} \left\{ \frac{4}{3} {\zeta_2} - \frac{112}{27} \right\} \,,
\nonumber \\
f_{gg,3} &= {{{C_A}^3}} \left\{ \frac{352}{5} {\zeta_2}^2 + \frac{176}{3} {\zeta_2} {\zeta_3}
- \frac{12650}{81} {\zeta_2} - \frac{1316}{3} {\zeta_3} + 192 {\zeta_5}
+ \frac{136781}{729}\right\}
\nonumber \\
&
+ {{{C_A^{2}} {n_f}}} \left\{ - \frac{96}{5} {\zeta_2}^2
+ \frac{2828}{81} {\zeta_2}
+ \frac{728}{27} {\zeta_3} - \frac{11842}{729} \right\}
\nonumber \\ \nonumber
&
+ {{C_{A} {C_F} {n_f}}} \left\{ \frac{32}{5} {\zeta_2}^2 + 4 {\zeta_2}
+ \frac{304}{9} {\zeta_3} - \frac{1711}{27} \right\}
+ {{{C_A} {n_f}^2}} \left\{ - \frac{40}{27} {\zeta_2} + \frac{112}{27} {\zeta_3}
- \frac{2080}{729} \right\}
\intertext{and}
&f_{q{\bar q}, i} = f_{b{\bar b}, i} = \frac{C_{F}}{C_{A}} f_{gg, i}\, .
\end{align}
Similarly the $B$'s are given by
\begin{align}
\label{eq:App-AnoDimB}
B_{gg,1} &= {{C_A}} \left\{\frac{11}{3}\right\} - {{n_f}} \left\{\frac{2}{3}\right\}\, ,
\nonumber\\
B_{gg,2} &= {{C_A^2}} \left\{\frac{32}{3} + 12 \zeta_3\right\} - {{n_f C_A}} \left\{\frac{8}{3} \right\} - {{n_f C_F}} \Big\{2 \Big\}\, ,
\nonumber\\
B_{gg,3} &= {{C_A C_F n_f}} \left\{-\frac{241}{18}\right\}
+ {{C_A n_f^2}} \left\{\frac{29}{18}\right\}
- {{C_A^2 n_f}} \left\{\frac{233}{18} + \frac{8}{3} \zeta_2+ \frac{4}{3} \zeta_2^2 + \frac{80}{3} \zeta_3\right\}
\nonumber\\
&+ {{C_A^3}} \left\{\frac{79}{2} - 16 \zeta_2 \zeta_3 + \frac{8}{3} \zeta_2 + \frac{22}{3} \zeta_2^2
+ \frac{536}{3} \zeta_3 - 80 \zeta_5\right\}
+ {{C_F n_f^2}} \left\{\frac{11}{9}\right\}
+ {{C_F^2 n_f}} \Big\{{1}\Big\}\, ,
\nonumber\\
B_{q{\bar q},1} &= {{C_F}} \Big\{{3}\Big\}\,,
\nonumber \\
B_{q{\bar q},2} &= {{C_F^2}} \Bigg\{ \frac{3}{2} - 12 \zeta_2 + 24 \zeta_3 \Bigg\}
+ {{C_A C_F}} \Bigg\{ \frac{17}{34} + \frac{88}{6} \zeta_2 - 12 \zeta_3 \Bigg\}
+ {{n_f C_F}}T_{F} \Bigg\{ - \frac{2}{3} - \frac{16}{3} \zeta_2 \Bigg\}\,,
\nonumber \\
B_{q{\bar q},3} &= {{{C_A}^2 {C_F}}} \Bigg\{ - 2 {\zeta_2}^2 + \frac{4496}{27} {\zeta_2}
- \frac{1552}{9} {\zeta_3} + 40 {\zeta_5} - \frac{1657}{36} \Bigg\}
+ {{{C_A} {C_F}^2}} \Bigg\{ -\frac{988}{15} {\zeta_2}^2
\nonumber\\
&
+ 16 {\zeta_2} {\zeta_3}
- \frac{410}{3} {\zeta_2} +\frac{844}{3} {\zeta_3}
+ 120 {\zeta_5} + \frac{151}{4} \Bigg\}
+ {{{C_A} {C_F} {n_f}}} \Bigg\{ \frac{4}{5} {\zeta_2}^2 - \frac{1336}{27} {\zeta_2}
+ \frac{200}{9} {\zeta_3}
\nonumber\\
&+ 20 \Bigg\}
+ {{{C_F}^3}} \Bigg\{ \frac{288}{5} {\zeta_2}^2 - 32 {\zeta_2} {\zeta_3}
+ 18 {\zeta_2} + 68 {\zeta_3} - 240 {\zeta_5} + \frac{29}{2} \Bigg\}
\nonumber\\
&
+ {{{C_F}^2 {n_f}}} \Bigg\{ \frac{232}{15} {\zeta_2}^2 + \frac{20}{3} {\zeta_2}
-\frac{136}{3} {\zeta_3} - 23 \Bigg\}
+ {{{C_F} {n_f}^2}} \Bigg\{ \frac{80}{27} {\zeta_2} - \frac{16}{9}
{\zeta_3} - \frac{17}{9} \Bigg\} \, ,
\nonumber
\intertext{and}
&B_{q{\bar q},i}=B_{b{\bar b}, i}\, .
\end{align}
\chapter{Results of the Unrenormalised Three Loop Form Factors for the Pseudo-Scalar}
\label{App:pScalar-Results}
In this appendix, we present the unrenormalised quark and gluon form
factors for the pseudo-scalar production up to three loops for the operators $\left[ O_{G} \right]_{B}$
and $\left[ O_{J} \right]_{B}$. Specifically, we present
${\hat{\cal F}}^{G,(n)}_{\beta}$ and ${\hat{\cal F}}^{J,(n)}_{\beta}$
for $\beta=q,g$ up to $n=3$ which are defined in
Sec.~\ref{sec:FF}. One and two loop results completely agree with the
existing literature~\cite{Ravindran:2004mb}. It should be noted
that the form factors at $n=2$ for ${\hat{\cal F}}^{G,(n)}_{q}$ and
${\hat{\cal F}}^{J,(n)}_{g}$ correspond to the contributions arising
from three loop diagrams since these processes start at one loop order.
\begin{align}
\label{eq:FgG1}
{\hat{\cal F}}^{G,(1)}_{g} &= {{C_{A}}} \Bigg\{ - \frac{8}{\epsilon^2} + 4 + \zeta_2 +
\epsilon \Bigg( - 6 - \frac{7}{3} \zeta_3 \Bigg) + \epsilon^2 \Bigg(
7 - \frac{\zeta_2}{2} + \frac{47}{80}
\zeta_2^2 \Bigg) + \epsilon^3 \Bigg( -
\frac{15}{2} + \frac{3}{4} \zeta_2
\nonumber\\
&+ \frac{7}{6} \zeta_3 +
\frac{7}{24} \zeta_2 \zeta_3 - \frac{31}{20} \zeta_5 \Bigg) \Bigg\}\,,
\end{align}
\begin{align}
\label{eq:FgG2}
{\hat{\cal F}}^{G,(2)}_{g} &= {{2 C_{A} n_{f} T_{F}}} \Bigg\{ - \frac{8}{3 \epsilon^3} + \frac{20}{9 \epsilon^2} +
\Bigg( \frac{106}{27} + 2 \zeta_2
\Bigg) \frac{1}{\epsilon} -
\frac{1591}{81} - \frac{5}{3} \zeta_2
- \frac{74}{9} \zeta_3 +
\epsilon \Bigg( \frac{24107}{486}
\nonumber\\
&-
\frac{23}{18} \zeta_2
+ \frac{51}{20} \zeta_2^2 + \frac{383}{27} \zeta_3 \Bigg) +
\epsilon^2 \Bigg( - \frac{146147}{1458}
+ \frac{799}{108} \zeta_2 - \frac{329}{72} \zeta_2^2 -
\frac{1436}{81} \zeta_3 + \frac{25}{6}
\zeta_2 \zeta_3
\nonumber\\
&- \frac{271}{30} \zeta_5 \Bigg) \Bigg\}
+
{{C_{A}^2}} \Bigg\{ \frac{32}{\epsilon^4}
+ \frac{44}{3 \epsilon^3} + \Bigg( -
\frac{422}{9} - 4 \zeta_2 \Bigg) \frac{1}{\epsilon^2} +
\Bigg( \frac{890}{27} - 11 \zeta_2 +
\frac{50}{3} \zeta_3 \Bigg)
\frac{1}{\epsilon}
\nonumber\\
&+ \frac{3835}{81} +
\frac{115}{6} \zeta_2 - \frac{21}{5}
\zeta_2^2 + \frac{11}{9} \zeta_3 +
\epsilon \Bigg( - \frac{213817}{972} -
\frac{103}{18} \zeta_2 +
\frac{77}{120} \zeta_2^2 +
\frac{1103}{54} \zeta_3
\nonumber\\
&-
\frac{23}{6} \zeta_2 \zeta_3 -
\frac{71}{10} \zeta_5 \Bigg) +
\epsilon^2 \Bigg( \frac{6102745}{11664}
- \frac{991}{27} \zeta_2 -
\frac{2183}{240} \zeta_2^2 +
\frac{2313}{280} \zeta_2^3 -
\frac{8836}{81} \zeta_3 -
\frac{55}{12} \zeta_2 \zeta_3
\nonumber\\
&+
\frac{901}{36} \zeta_3^2 + \frac{341}{60} \zeta_5 \Bigg) \Bigg\}
+
{{2 C_{F} n_{f} T_{F}}} \Bigg\{ \frac{12}{\epsilon}
- \frac{125}{3} + 8 \zeta_3 + \epsilon
\Bigg( \frac{3421}{36} - \frac{14}{3}
\zeta_2 - \frac{8}{3} \zeta_2^2
\nonumber\\
&-
\frac{64}{3} \zeta_3 \Bigg)
+ \epsilon^2 \Bigg( - \frac{78029}{432}
+ \frac{293}{18} \zeta_2 +
\frac{64}{9} \zeta_2^2 +
\frac{973}{18} \zeta_3 - \frac{10}{3}
\zeta_2 \zeta_3 + 8 \zeta_5 \Bigg) \Bigg\}\,,
\end{align}
\begin{align}
\label{eq:FgG3}
{\hat{\cal F}}^{G,(3)}_{g} &= {{4 C_{F} n_{f}^2 T_{F}^{2}}} \Bigg\{
\frac{16}{\epsilon^2} + \Bigg( -
\frac{796}{9} + \frac{64}{3} \zeta_3
\Bigg) \frac{1}{\epsilon} +
\frac{8387}{27} - \frac{38}{3} \zeta_2
- \frac{112}{15} \zeta_2^2 -
\frac{848}{9} \zeta_3 \Bigg\}
\nonumber\\
&+ {{2 C_{F}^2 n_{f} T_{F}}} \Bigg\{
\frac{6}{\epsilon}
- \frac{353}{6} + 176 \zeta_3 - 160 \zeta_5 \Bigg\} +
{{2 C_{A}^2 n_{f} T_{F}}} \Bigg\{ \frac{64}{3
\epsilon^5} - \frac{32}{81 \epsilon^4}
+ \Bigg( - \frac{18752}{243}
\nonumber\\
&- \frac{376}{27} \zeta_2 \Bigg) \frac{1}{\epsilon^3} +
\Bigg( \frac{36416}{243}
-
\frac{1700}{81} \zeta_2 +
\frac{2072}{27} \zeta_3 \Bigg)
\frac{1}{\epsilon^2} + \Bigg(
\frac{62642}{2187} + \frac{22088}{243} \zeta_2 - \frac{2453}{90}
\zeta_2^2
\nonumber\\
&-
\frac{3988}{81} \zeta_3 \Bigg)
\frac{1}{\epsilon} -
\frac{14655809}{13122}
- \frac{60548}{729} \zeta_2 +
\frac{917}{60} \zeta_2^2 -
\frac{772}{27} \zeta_3 - \frac{439}{9}
\zeta_2 \zeta_3
+ \frac{3238}{45}
\zeta_5 \Bigg\}
\nonumber\\
&+
{{4 C_{A} n_{f}^2 T_{F}^{2}}} \Bigg\{ - \frac{128}{81
\epsilon^4} + \frac{640}{243
\epsilon^3}
+ \Bigg(
\frac{128}{27} + \frac{80}{27} \zeta_2
\Bigg) \frac{1}{\epsilon^2} +
\Bigg( - \frac{93088}{2187} -
\frac{400}{81} \zeta_2
\nonumber\\
&-
\frac{1328}{81} \zeta_3 \Bigg)
\frac{1}{\epsilon} +
\frac{1066349}{6561} - \frac{56}{27} \zeta_2
+
\frac{797}{135} \zeta_2^2
+\frac{13768}{243} \zeta_3 \Bigg\}
+
{{2 C_{A} C_{F} n_{f} T_{F}}} \Bigg\{ -
\frac{880}{9 \epsilon^3}
\nonumber\\
&+ \Bigg(
\frac{6844}{27} - \frac{640}{9} \zeta_3
\Bigg) \frac{1}{\epsilon^2} + \Bigg( -
\frac{16219}{81}
+ \frac{158}{3} \zeta_2
+ \frac{352}{15} \zeta_2^2 +
\frac{1744}{27} \zeta_3 \Bigg)
\frac{1}{\epsilon} - \frac{753917}{972}
\nonumber\\
&- \frac{593}{6} \zeta_2 - \frac{96}{5} \zeta_2^2 +
\frac{4934}{81} \zeta_3 + 48 \zeta_2
\zeta_3
+ \frac{32}{9} \zeta_5 \Bigg\}
+
{{C_{A}^3}} \Bigg\{ - \frac{256}{3
\epsilon^6} - \frac{352}{3 \epsilon^5}
+ \frac{16144}{81 \epsilon^4}
\nonumber\\
&+
\Bigg( \frac{22864}{243} +
\frac{2068}{27} \zeta_2 - \frac{176}{3}
\zeta_3 \Bigg) \frac{1}{\epsilon^3}
+
\Bigg( - \frac{172844}{243} -
\frac{1630}{81} \zeta_2 +
\frac{494}{45} \zeta_2^2 -
\frac{836}{27} \zeta_3 \Bigg)
\frac{1}{\epsilon^2}
\nonumber\\
&+ \Bigg( \frac{2327399}{2187} -
\frac{71438}{243} \zeta_2
+
\frac{3751}{180} \zeta_2^2 -
\frac{842}{9} \zeta_3
+ \frac{170}{9} \zeta_2 \zeta_3 +
\frac{1756}{15} \zeta_5 \Bigg)
\frac{1}{\epsilon}
+ \frac{16531853}{26244}
\nonumber\\
&+
\frac{918931}{1458} \zeta_2 +
\frac{27251}{1080} \zeta_2^2
- \frac{22523}{270} \zeta_2^3 -
\frac{51580}{243} \zeta_3 +
\frac{77}{18} \zeta_2 \zeta_3
- \frac{1766}{9} \zeta_3^2 +
\frac{20911}{45} \zeta_5 \Bigg\} \,,
\end{align}
\begin{align}
\label{eq:FgJ1}
{\hat{\cal F}}^{J,(1)}_{g} &= {{C_{A}}} \Bigg\{ - \frac{8}{\epsilon^2} +
4 + \zeta_2 +
\epsilon \Bigg( - \frac{15}{2} +
\zeta_2 - \frac{16}{3} \zeta_3 \Bigg)
+ \epsilon^2 \Bigg( \frac{287}{24} - 2
\zeta_2 + \frac{127}{80} \zeta_2^2
\Bigg)
\nonumber\\
&+
\epsilon^3 \Bigg( - \frac{5239}{288} +
\frac{151}{48} \zeta_2 +
\frac{19}{120} \zeta_2^2 +
\frac{\zeta_3}{12} +
\frac{7}{6} \zeta_2 \zeta_3 -
\frac{91}{20} \zeta_5 \Bigg) \Bigg\} +
{{C_{F}}} \Bigg\{ 4 +
\epsilon \Bigg( - \frac{21}{2}
\nonumber\\
&+ 6
\zeta_3 \Bigg) + \epsilon^2 \Bigg(
\frac{155}{8} - \frac{5}{2} \zeta_2 -
\frac{9}{5} \zeta_2^2 - \frac{9}{2}
\zeta_3 \Bigg) + \epsilon^3 \Bigg( -
\frac{1025}{32} + \frac{83}{16}
\zeta_2 +
\frac{27}{20} \zeta_2^2 + \frac{20}{3}
\zeta_3
\nonumber\\
&- \frac{3}{4} \zeta_2 \zeta_3
+ \frac{21}{2} \zeta_5 \Bigg) \Bigg\}\,,
\end{align}
\begin{align}
\label{eq:FgJ2}
{\hat{\cal F}}^{J,(2)}_{g} &= {{2 C_{A} n_{f} T_{F}}} \Bigg\{ - \frac{8}{3
\epsilon^3} + \frac{20}{9 \epsilon^2} +
\Bigg( \frac{106}{27} + 2 \zeta_2
\Bigg) \frac{1}{\epsilon} -
\frac{1753}{81} - \frac{\zeta_2}{3}
- \frac{110}{9} \zeta_3 +
\epsilon \Bigg( \frac{14902}{243}
\nonumber\\
&-
\frac{103}{18} \zeta_2
+ \frac{241}{60} \zeta_2^2 +
\frac{599}{27} \zeta_3 \Bigg) +
\epsilon^2 \Bigg( - \frac{411931}{2916}
+ \frac{2045}{108} \zeta_2 -
\frac{2353}{360} \zeta_2^2 -
\frac{3128}{81} \zeta_3
\nonumber\\
&+ \frac{43}{6}
\zeta_2 \zeta_3
- \frac{167}{10} \zeta_5 \Bigg) \Bigg\} +
{{C_{A} C_{F}}} \Bigg\{ - \frac{32}{\epsilon^2} +
\Bigg( \frac{208}{3} - 48 \zeta_3
\Bigg) \frac{1}{\epsilon} -
\frac{451}{9} + 24 \zeta_2 +
\frac{72}{5} \zeta_2^2
\nonumber\\
&- 8 \zeta_3
+ \epsilon \Bigg( - \frac{16385}{108} - \frac{52}{3} \zeta_2 +
\frac{12}{5} \zeta_2^2 + 32 \zeta_3 +
10 \zeta_2 \zeta_3 - 14 \zeta_5 \Bigg) + \epsilon^2 \Bigg(
\frac{1073477}{1296}
\nonumber\\
&- \frac{815}{9} \zeta_2
+
\frac{19}{20} \zeta_2^2 +
\frac{17}{70} \zeta_2^3 -
\frac{1915}{36} \zeta_3 + 9 \zeta_2
\zeta_3 -
34 \zeta_3^2 - \frac{2279}{6} \zeta_5
\Bigg) \Bigg\} +
{{2 C_{F} n_{f} T_{F}}} \Bigg\{ \frac{26}{3
\epsilon}
\nonumber\\
&- \frac{709}{18} + 16 \zeta_3
+ \epsilon \Bigg( \frac{26149}{216} - \frac{65}{6} \zeta_2 -
\frac{76}{15} \zeta_2^2 - 44 \zeta_3 \Bigg) +
\epsilon^2 \Bigg( - \frac{828061}{2592}
+ \frac{3229}{72} \zeta_2
\nonumber\\
&+ \frac{212}{15} \zeta_2^2 +
\frac{1729}{18} \zeta_3
- 4 \zeta_2 \zeta_3 + \frac{166}{3} \zeta_5 \Bigg) \Bigg\} +
{{C_{A}^2}} \Bigg\{ \frac{32}{\epsilon^4}
+ \frac{44}{3 \epsilon^3} + \Bigg( -
\frac{422}{9} - 4 \zeta_2 \Bigg)
\frac{1}{\epsilon^2}
\nonumber\\
&+
\Bigg( \frac{1214}{27} - 19 \zeta_2
+ \frac{122}{3} \zeta_3 \Bigg)
\frac{1}{\epsilon} + \frac{1513}{81} +
\frac{143}{6} \zeta_2 - \frac{61}{5}
\zeta_2^2 + \frac{209}{9} \zeta_3 +
\epsilon \Bigg( - \frac{202747}{972}
\nonumber\\
&+
\frac{59}{36} \zeta_2 - \frac{349}{24}
\zeta_2^2
- \frac{2393}{108} \zeta_3 -
\frac{53}{6} \zeta_2 \zeta_3 +
\frac{369}{10} \zeta_5 \Bigg) +
\epsilon^2 \Bigg( \frac{7681921}{11664}
- \frac{35255}{432} \zeta_2 +
\frac{1711}{180} \zeta_2^2
\nonumber\\
&-
\frac{7591}{840} \zeta_2^3
-
\frac{5683}{1296} \zeta_3 -
\frac{407}{12} \zeta_2 \zeta_3 +
\frac{775}{36} \zeta_3^2 +
\frac{4013}{30} \zeta_5 \Bigg) \Bigg\}
+
{{C_{F}^2}} \Bigg\{ - 6 + \epsilon \Bigg(
\frac{259}{12} + 41 \zeta_3
\nonumber\\
&- 60
\zeta_5 \Bigg)
+
\epsilon^2 \Bigg( - \frac{7697}{144} +
\frac{\zeta_2}{3} - \frac{184}{15}
\zeta_2^2 + \frac{120}{7} \zeta_2^3 -
163 \zeta_3 + 4 \zeta_2 \zeta_3 + 30
\zeta_3^2 + \frac{470}{3} \zeta_5 \Bigg) \Bigg\}\,,
\end{align}
\begin{align}
\label{eq:FqG1}
{\hat{\cal F}}^{G,(1)}_{q} &= {{2 n_{f} T_{F}}} \Bigg\{ \frac{4}{3 \epsilon} -
\frac{19}{9} + \epsilon \Bigg(
\frac{355}{108} - \frac{\zeta_2}{6}
\Bigg) + \epsilon^2 \Bigg( -
\frac{6523}{1296} + \frac{19}{72}
\zeta_2 + \frac{25}{18} \zeta_3 \Bigg) +
\epsilon^3 \Bigg( \frac{118675}{15552}
\nonumber\\
&- \frac{355}{864} \zeta_2 -
\frac{191}{480} \zeta_2^2 -
\frac{475}{216} \zeta_3 \Bigg) \Bigg\}
+
{{C_{F}}} \Bigg\{ - \frac{8}{\epsilon^2} +
\frac{6}{\epsilon} - \frac{11}{2} +
\zeta_2 +
\epsilon \Bigg( \frac{25}{8} -
\frac{3}{4} \zeta_2 - \frac{7}{3}
\zeta_3 \Bigg)
\nonumber\\
&+
\epsilon^2 \Bigg( - \frac{11}{32} -
\frac{21}{16} \zeta_2 + \frac{47}{80}
\zeta_2^2 + \frac{7}{4} \zeta_3 \Bigg) +
\epsilon^3 \Bigg( - \frac{415}{128} +
\frac{223}{64} \zeta_2 -
\frac{141}{320} \zeta_2^2 -
\frac{155}{48} \zeta_3
\nonumber\\
&+
\frac{7}{24} \zeta_2 \zeta_3 -
\frac{31}{20} \zeta_5 \Bigg) \Bigg\}
+
{{C_{A}}} \Bigg\{ - \frac{22}{3 \epsilon}
+ \frac{269}{18} +
\epsilon \Bigg( - \frac{5045}{216} +
\frac{23}{12} \zeta_2 + 3 \zeta_3
\Bigg) + \epsilon^2 \Bigg(
\frac{90893}{2592}
\nonumber\\
&- \frac{485}{144}
\zeta_2 -
\frac{4}{5} \zeta_2^2 - \frac{275}{36}
\zeta_3 \Bigg) +
\epsilon^3 \Bigg( -
\frac{1620341}{31104} +
\frac{8861}{1728} \zeta_2 +
\frac{751}{320} \zeta_2^2 +
\frac{4961}{432} \zeta_3 +
\frac{\zeta_2 \zeta_3}{8}
\nonumber\\
&+
\frac{15}{2} \zeta_5 \Bigg) \Bigg\}\,,
\end{align}
\begin{align}
\label{eq:FqG2}
{\hat{\cal F}}^{G,(2)}_{q} &= {{4 n_{f}^2 T_{F}^{2}}} \Bigg\{ \frac{16}{9
\epsilon^2} - \frac{152}{27 \epsilon}
+ \frac{124}{9} - \frac{4}{9} \zeta_2 +
\epsilon \Bigg( - \frac{7426}{243} +
\frac{38}{27} \zeta_2 + \frac{136}{27}
\zeta_3 \Bigg) +
\epsilon^2 \Bigg( \frac{47108}{729}
\nonumber\\
&-
\frac{31}{9} \zeta_2
- \frac{43}{30}
\zeta_2^2 - \frac{1292}{81} \zeta_3 \Bigg) \Bigg\}
+
{{C_{A}^2}} \Bigg\{ \frac{484}{9
\epsilon^2} - \frac{6122}{27 \epsilon}
+ \frac{1865}{3} - \frac{319}{9}
\zeta_2 - 66 \zeta_3
\nonumber\\
&+
\epsilon \Bigg( - \frac{702941}{486}
+
\frac{14969}{108} \zeta_2 +
\frac{299}{20} \zeta_2^2 +
\frac{31441}{108} \zeta_3 + 5 \zeta_2
\zeta_3 - 30 \zeta_5 \Bigg) +
\epsilon^2 \Bigg( \frac{18199507}{5832}
\nonumber\\
&- \frac{5861}{16} \zeta_2
-
\frac{63233}{720} \zeta_2^2 -
\frac{691}{140} \zeta_2^3 -
\frac{995915}{1296} \zeta_3 +
\frac{52}{3} \zeta_2 \zeta_3 -
\frac{39}{2} \zeta_3^2 - \frac{1343}{12} \zeta_5 \Bigg) \Bigg\}
\nonumber\\
&+
{{2 C_{F} n_{f} T_{F}}} \Bigg\{ - \frac{40}{3
\epsilon^3}
+ \frac{280}{9 \epsilon^2} +
\Bigg( - \frac{1417}{27} + 2 \zeta_2
\Bigg) \frac{1}{\epsilon} +
\frac{22021}{324} - \frac{14}{3}
\zeta_2 - \frac{82}{9} \zeta_3
\nonumber\\
&+
\epsilon \Bigg( - \frac{238717}{3888} -
\frac{73}{12} \zeta_2
+ \frac{25}{12}
\zeta_2^2 + \frac{394}{27} \zeta_3 \Bigg) +
\epsilon^2 \Bigg( -
\frac{290075}{46656} +
\frac{6181}{144} \zeta_2 -
\frac{499}{180} \zeta_2^2
\nonumber\\
&-
\frac{9751}{324} \zeta_3 + \frac{13}{6}
\zeta_2 \zeta_3
- \frac{29}{6} \zeta_5 \Bigg) \Bigg\}
+
{{C_{F}^2}} \Bigg\{ \frac{32}{\epsilon^4} -
\frac{48}{\epsilon^3} + \Bigg( 62 - 8
\zeta_2 \Bigg) \frac{1}{\epsilon^2} +
\Bigg( - \frac{113}{2}
\nonumber\\
&+ \frac{128}{3}
\zeta_3 \Bigg) \frac{1}{\epsilon} +
\frac{581}{24} + \frac{27}{2} \zeta_2
-
13 \zeta_2^2 - 58 \zeta_3 +
\epsilon \Bigg( \frac{12275}{288} -
\frac{331}{24} \zeta_2 +
\frac{493}{30} \zeta_2^2 +
\frac{587}{6} \zeta_3
\nonumber\\
&-
\frac{56}{3} \zeta_2 \zeta_3 +
\frac{92}{5} \zeta_5 \Bigg)
+
\epsilon^2 \Bigg( - \frac{456779}{3456}
- \frac{2011}{96} \zeta_2 -
\frac{1279}{80} \zeta_2^2 +
\frac{223}{20} \zeta_2^3 -
\frac{13363}{72} \zeta_3
\nonumber\\
&- \frac{5}{2}
\zeta_2 \zeta_3 +
\frac{652}{9} \zeta_3^2
-
\frac{193}{30} \zeta_5 \Bigg) \Bigg\}
+
{{2 C_{A} n_{f} T_{F}}} \Bigg\{ - \frac{176}{9
\epsilon^2} + \frac{1972}{27 \epsilon}
- \frac{1708}{9} + \frac{80}{9} \zeta_2 +
4 \zeta_3
\nonumber\\
&+
\epsilon \Bigg( \frac{104858}{243} -
\frac{853}{27} \zeta_2
- \frac{2}{3}
\zeta_2^2 - \frac{1622}{27} \zeta_3
\Bigg) + \epsilon^2 \Bigg( -
\frac{5369501}{5832} + \frac{1447}{18}
\zeta_2 + \frac{817}{45} \zeta_2^2
\nonumber\\
&+
\frac{31499}{162} \zeta_3 + \frac{7}{3}
\zeta_2 \zeta_3
+ 19 \zeta_5 \Bigg) \Bigg\}
+
{{C_{A} C_{F}}} \Bigg\{ \frac{220}{3 \epsilon^3}+
\Bigg( - \frac{1804}{9} + 4 \zeta_2
\Bigg) \frac{1}{\epsilon^2} + \Bigg(
\frac{20777}{54}
\nonumber\\
&- 19 \zeta_2 - 50
\zeta_3 \Bigg) \frac{1}{\epsilon}
-
\frac{397181}{648 } + \frac{161}{3}
\zeta_2 + \frac{76}{5} \zeta_2^2 +
\frac{1333}{9} \zeta_3 + \epsilon
\Bigg( \frac{6604541}{7776} -
\frac{669}{8} \zeta_2
\nonumber\\
&-
\frac{5519}{120} \zeta_2^2 -
\frac{8398}{27} \zeta_3
+ \frac{89}{6}
\zeta_2 \zeta_3 -
\frac{51}{2} \zeta_5 \Bigg) +
\epsilon^2 \Bigg( -
\frac{93774821}{93312} +
\frac{20035}{288} \zeta_2 +
\frac{33377}{360} \zeta_2^2
\nonumber\\
&+
\frac{1793}{840} \zeta_2^3 +
\frac{390731}{648} \zeta_3
-
\frac{445}{12} \zeta_2 \zeta_3 -
\frac{425}{12} \zeta_3^2 +
\frac{641}{12} \zeta_5 \Bigg) \Bigg\}\,,
\end{align}
\begin{align}
\label{eq:FqJ1}
{\hat{\cal F}}^{J,(1)}_{q} &= {{C_{F}}} \Bigg\{ - \frac{8}{\epsilon^2}
+ \frac{6}{\epsilon} - 2 + \zeta_2 +
\epsilon \Bigg( - 1 - \frac{3}{4}
\zeta_2 - \frac{7}{3} \zeta_3 \Bigg) +
\epsilon^2 \Bigg( \frac{5}{2} +
\frac{\zeta_2}{4} + \frac{47}{80}
\zeta_2^2 + \frac{7}{4} \zeta_3 \Bigg)
\nonumber\\
&+
\epsilon^3 \Bigg( - \frac{13}{4} +
\frac{\zeta_2}{8} - \frac{141}{320}
\zeta_2^2 - \frac{7}{12} \zeta_3 +
\frac{7}{24} \zeta_2 \zeta_3 -
\frac{31}{20} \zeta_5 \Bigg) \Bigg\}\,,
\end{align}
\begin{align}
\label{eq:FqJ2}
{\hat{\cal F}}^{J,(2)}_{q} &={{2 C_{F} n_{f} T_{F}}} \Bigg\{ - \frac{8}{3
\epsilon^3} + \frac{56}{9 \epsilon^2}
+ \Bigg( - \frac{47}{27} - \frac{2}{3}
\zeta_2 \Bigg) \frac{1}{\epsilon} -
\frac{4105}{324} +
\frac{14}{9} \zeta_2 - \frac{26}{9}
\zeta_3 + \epsilon \Bigg(
\frac{142537}{3888}
\nonumber\\
&- \frac{695}{108}
\zeta_2 +
\frac{41}{60} \zeta_2^2 +
\frac{182}{27} \zeta_3 \Bigg) +
\epsilon^2 \Bigg( -
\frac{3256513}{46656} +
\frac{21167}{1296} \zeta_2
- \frac{287}{180} \zeta_2^2 -
\frac{2555}{324} \zeta_3
\nonumber\\
&-
\frac{13}{18} \zeta_2 \zeta_3 -
\frac{121}{30} \zeta_5 \Bigg) \Bigg\}
+
{{C_{F}^2}} \Bigg\{ \frac{32}{\epsilon^4}
- \frac{48}{\epsilon^3} + \Bigg( 34 -
8 \zeta_2 \Bigg) \frac{1}{\epsilon^2}
+ \Bigg( - \frac{5}{2} + \frac{128}{3}
\zeta_3 \Bigg) \frac{1}{\epsilon} -
\frac{361}{8}
\nonumber\\
&+ \frac{9}{2} \zeta_2 -
13 \zeta_2^2 - 58 \zeta_3 +
\epsilon \Bigg( \frac{3275}{32} +
\frac{3}{8} \zeta_2 + \frac{171}{10}
\zeta_2^2 + \frac{503}{6} \zeta_3 -
\frac{56}{3} \zeta_2 \zeta_3 +
\frac{92}{5} \zeta_5 \Bigg)
\nonumber\\
&+
\epsilon^2 \Bigg( - \frac{20257}{128}
- \frac{793}{32} \zeta_2 -
\frac{2097}{80} \zeta_2^2 +
\frac{223}{20} \zeta_2^3 -
\frac{4037}{24} \zeta_3 + \frac{27}{2}
\zeta_2 \zeta_3 +
\frac{652}{9} \zeta_3^2 -
\frac{231}{10} \zeta_5 \Bigg) \Bigg\}
\nonumber\\
&+
{{C_{A} C_{F}}} \Bigg\{ \frac{44}{3
\epsilon^3} +
\Bigg( - \frac{332}{9} + 4 \zeta_2
\Bigg) \frac{1}{\epsilon^2} + \Bigg(
\frac{2545}{54} + \frac{11}{3} \zeta_2
- 26 \zeta_3 \Bigg) \frac{1}{\epsilon}
- \frac{18037}{648} - \frac{47}{9}
\zeta_2
\nonumber\\
&+ \frac{44}{5} \zeta_2^2 +
\frac{467}{9} \zeta_3 + \epsilon \Bigg(
- \frac{221963}{7776} -
\frac{263}{216} \zeta_2 -
\frac{1891}{120} \zeta_2^2 -
\frac{2429}{27} \zeta_3 + \frac{89}{6}
\zeta_2 \zeta_3 - \frac{51}{2} \zeta_5
\Bigg)
\nonumber\\
&+
\epsilon^2 \Bigg(
\frac{11956259}{93312} +
\frac{38987}{2592} \zeta_2 +
\frac{9451}{360} \zeta_2^2 -
\frac{809}{280} \zeta_2^3 +
\frac{92701}{648} \zeta_3 -
\frac{397}{36} \zeta_2 \zeta_3 -
\frac{569}{12} \zeta_3^2
\nonumber\\
&+ \frac{3491}{60} \zeta_5 \Bigg) \Bigg\}\,,
\end{align}
\begin{align}
\label{eq:FqJ3}
{\hat{\cal F}}^{J,(3)}_{q} &= {{4 C_{F} n_{f}^2 T_{F}^{2}}} \Bigg\{ - \frac{128}{81
\epsilon^4} + \frac{1504}{243
\epsilon^3} +
\Bigg( - \frac{16}{9} - \frac{16}{9}
\zeta_2 \Bigg) \frac{1}{\epsilon^2} +
\Bigg( - \frac{73432}{2187} +
\frac{188}{27} \zeta_2
\nonumber\\
&-
\frac{272}{81} \zeta_3 \Bigg)
\frac{1}{\epsilon}
+
\frac{881372}{6561} - 26 \zeta_2 -
\frac{83}{135} \zeta_2^2 + \frac{3196}{243} \zeta_3 \Bigg\}
+
{{C_{F}^3}} \Bigg\{ - \frac{256}{3
\epsilon^6} + \frac{192}{\epsilon^5} +
\Bigg( - 208
\nonumber\\
&+ 32 \zeta_2 \Bigg)
\frac{1}{\epsilon^4}
+
\Bigg( 88 + 24 \zeta_2 - \frac{800}{3}
\zeta_3 \Bigg) \frac{1}{\epsilon^3} +
\Bigg( 254 - 98 \zeta_2 +
\frac{426}{5} \zeta_2^2 + 552 \zeta_3
\Bigg) \frac{1}{\epsilon^2}
\nonumber\\
&+
\Bigg( - \frac{5045}{6} + 83 \zeta_2
- \frac{1461}{10} \zeta_2^2 -
\frac{2630}{3} \zeta_3 +
\frac{428}{3} \zeta_2 \zeta_3 -
\frac{1288}{5} \zeta_5 \Bigg)
\frac{1}{\epsilon} + \frac{38119}{24}
+ \frac{1885}{12} \zeta_2
\nonumber\\
&+
\frac{8659}{40} \zeta_2^2
-
\frac{9095}{252} \zeta_2^3 + 1153
\zeta_3 - 35 \zeta_2 \zeta_3 -
\frac{1826}{3} \zeta_3^2 - \frac{562}{5} \zeta_5 \Bigg\}
+
{{2 C_{F}^2 n_{f} T_{F}}} \Bigg\{ \frac{64}{3
\epsilon^5}
\nonumber\\
&- \frac{592}{9 \epsilon^4}
+ \Bigg( \frac{1480}{27}
+ \frac{8}{3}
\zeta_2 \Bigg) \frac{1}{\epsilon^3} +
\Bigg( \frac{7772}{81} - \frac{266}{9}
\zeta_2 + \frac{584}{9} \zeta_3 \Bigg)
\frac{1}{\epsilon^2} +
\Bigg( - \frac{116735}{243} +
\frac{2633}{27} \zeta_2
\nonumber\\
&-
\frac{337}{18} \zeta_2^2
-
\frac{5114}{27} \zeta_3 \Bigg)
\frac{1}{\epsilon} +
\frac{3396143}{2916} -
\frac{32329}{162} \zeta_2 +
\frac{8149}{216} \zeta_2^2 +
\frac{39773}{81} \zeta_3 -
\frac{343}{9} \zeta_2 \zeta_3
\nonumber\\
&+ \frac{278}{45} \zeta_5 \Bigg\}
+
{{C_{A}^2 C_{F}}} \Bigg\{ - \frac{3872}{81
\epsilon^4} + \Bigg( \frac{52168}{243}
- \frac{704}{27} \zeta_2 \Bigg)
\frac{1}{\epsilon^3} +
\Bigg( - \frac{117596}{243} -
\frac{2212}{81} \zeta_2
\nonumber\\
&-
\frac{352}{45} \zeta_2^2
+
\frac{6688}{27} \zeta_3 \Bigg)
\frac{1}{\epsilon^2} + \Bigg(
\frac{1322900}{2187} +
\frac{39985}{243} \zeta_2 -
\frac{1604}{15} \zeta_2^2 -
\frac{24212}{27} \zeta_3 +
\frac{176}{9} \zeta_2 \zeta_3
\nonumber\\
&+
\frac{272}{3} \zeta_5 \Bigg)
\frac{1}{\epsilon}
+
\frac{1213171}{13122} -
\frac{198133}{729} \zeta_2 +
\frac{146443}{540} \zeta_2^2 - \frac{6152}{189} \zeta_2^3 +
\frac{970249}{486} \zeta_3 -
\frac{926}{9} \zeta_2 \zeta_3
\nonumber\\
&-
\frac{1136}{9} \zeta_3^2
+ \frac{772}{9} \zeta_5 \Bigg\}
+
{{2 C_{A} C_{F} n_{f} T_{F}}} \Bigg\{
\frac{1408}{81 \epsilon^4} + \Bigg( -
\frac{18032}{243} + \frac{128}{27}
\zeta_2 \Bigg) \frac{1}{\epsilon^3} +
\Bigg( \frac{24620}{243}
\nonumber\\
&+
\frac{1264}{81} \zeta_2
-
\frac{1024}{27} \zeta_3 \Bigg)
\frac{1}{\epsilon^2} +
\Bigg( \frac{212078}{2187} -
\frac{16870}{243} \zeta_2 +
\frac{88}{5} \zeta_2^2 +
\frac{12872}{81} \zeta_3 \Bigg)
\frac{1}{\epsilon} -
\frac{5807647}{6561}
\nonumber\\
&+
\frac{299915}{1458} \zeta_2 -
\frac{5492}{135} \zeta_2^2 -
\frac{42941}{81} \zeta_3 +
\frac{422}{9} \zeta_2 \zeta_3 - \frac{28}{3} \zeta_5 \Bigg\}
+
{{C_{A} C_{F}^2}} \Bigg\{ - \frac{352}{3 \epsilon^5} +
\Bigg( \frac{3448}{9}
\nonumber\\
&- 32 \zeta_2
\Bigg) \frac{1}{\epsilon^4} + \Bigg( -
\frac{16948}{27} + \frac{28}{3}
\zeta_2 + 208 \zeta_3 \Bigg)
\frac{1}{\epsilon^3} + \Bigg(
\frac{44542}{81} + \frac{1127}{9}
\zeta_2 - \frac{332}{5} \zeta_2^2
\nonumber\\
&-
840 \zeta_3 \Bigg) \frac{1}{\epsilon^2}
+ \Bigg( \frac{149299}{486} -
\frac{12757}{54} \zeta_2 +
\frac{9839}{36} \zeta_2^2 +
\frac{5467}{3} \zeta_3 - \frac{430}{3}
\zeta_2 \zeta_3 + 284 \zeta_5 \Bigg)
\frac{1}{\epsilon}
\nonumber\\
&-
\frac{15477463}{5832} +
\frac{21455}{324} \zeta_2 -
\frac{1002379}{2160} \zeta_2^2 -
\frac{18619}{1260} \zeta_2^3 -
\frac{51781}{18} \zeta_3 +
\frac{910}{9} \zeta_2 \zeta_3 +
\frac{1616}{3} \zeta_3^2
\nonumber\\
&-
\frac{3394}{45} \zeta_5 \Bigg\}\,.
\end{align}
\chapter{Harmonic Polylogarithms}
\label{chpt:App-HPL}
The logarithms, polylogarithms (Li$_n(x)$) and Nielsen's polylogarithm
(S$_{n,p}(x)$) appear naturally in the analytical expressions of
radiative corrections in pQCD which are defined through
\begin{align}
\label{eq:App-HPL-1}
&\text{ln}(x) = \int_{1}^{x} \frac{dt}{t}\,,
\nonumber\\
&\text{Li}_n(x) \equiv \sum_{k=1}^{\infty} \frac{x^k}{k^n} =
\int_{0}^{x} \frac{dt}{t} \text{Li}_{n-1}(t)\,, \quad\quad
\text{e.g.} \quad \text{Li}_1(x) = - \text{ln}(1-x)\,,
\nonumber\\
&S_{n,p}(x) \equiv \frac{(-1)^{n+p-1}}{(n-1)! p!} \int_{0}^{1}
\frac{dt}{t} [\text{ln} (t)]^{n-1} [\text{ln}(1-x t)]^p\,,
\nonumber\\
&\quad\text{e.g.} \quad S_{n-1,1}(x) = \text{Li}_n(x)\,.
\end{align}
However, for higher order radiative corrections (2-loops and beyond), these functions are not
sufficient to evaluate all the loop integrals appearing in
the Feynman graphs. This is overcome by introducing a new set of
functions which are called \textbf{Harmonic Polylogarithms
(HPLs)}. These are essentially a generalisation of Nielsen's
polylogarithms. In this appendix, we briefly describe the definition
and properties of HPL~\cite{Remiddi:1999ew} and 2dHPL. HPL is
represented by $H(\vec{m}_w;y)$
with a $w$-dimensional vector $\vec{m}_w$ of parameters and its argument $y$. $w$ is called the weight of the HPL. The elements of $\vec{m}_w$ belong to $\{ 1, 0, -1 \}$ through which
the following rational functions are represented
\begin{equation}
f(1;y) \equiv \frac{1}{1-y}, \qquad f(0;y) \equiv \frac{1}{y}, \qquad f(-1;y) \equiv \frac{1}{1+y} \, .
\end{equation}
The weight 1 $(w = 1)$ HPLs are defined by
\begin{equation}
H(1, y) \equiv - \ln (1 - y), \qquad H(0, y) \equiv \ln y, \qquad H(-1, y) \equiv \ln (1 + y) \, .
\end{equation}
For $w > 1$, $H(m, \vec{m}_{w};y)$ is defined by
\begin{equation}\label{1dhpl}
H(m, \vec{m}_w;y) \equiv \int_0^y dx ~ f(m, x) ~ H(\vec{m}_w;x), \qquad \qquad m \in 0, \pm 1 \, .
\end{equation}
The 2dHPLs are defined in the same way as Eq.~(\ref{1dhpl}) with the new elements $\{ 2, 3 \}$ in $\vec{m}_w$ representing a new
class of rational functions
\begin{equation}
f(2;y) \equiv f(1-z;y) \equiv \frac{1}{1-y-z}, \qquad f(3;y) \equiv f(z;y) \equiv \frac{1}{y+z}
\end{equation}
and correspondingly with the weight 1 $(w = 1)$ 2dHPLs
\begin{equation}
H(2, y) \equiv - \ln \Big(1 - \frac{y}{1-z} \Big), \qquad H(3, y) \equiv \ln \Big( \frac{y+z}{z} \Big) \, .
\end{equation}
\subsection{Properties}
\underline{Shuffle algebra} : A product of two HPL with weights $w_1$ and $w_2$ of the same argument $y$ is a combination of HPLs with weight
$(w_1 + w_2)$ and argument $y$, such that all possible permutations of the elements of $\vec{m}_{w_1}$ and $\vec{m}_{w_2}$ are considered preserving the
relative orders of the elements of $\vec{m}_{w_1}$ and $\vec{m}_{w_2}$,
\begin{equation}
H(\vec{m}_{w_1};y) H(\vec{m}_{w_2};y) = \sum_{\text{\tiny $\vec{m}_{w} = \vec{m}_{w_1} \uplus \vec{m}_{w_2}$}} H(\vec{m}_{w};y).
\end{equation}
\underline{Integration-by-parts identities} : The ordering of the elements of $\vec{m}_{w}$ in an HPL with weight $w$ and argument $y$ can be reversed using
integration-by-parts and in the process, some products of two HPLs are generated in the following way
\begin{eqnarray}
H(\vec{m}_{w};y) \equiv H(m_1, m_2, ... , m_w; y ) &=& H(m_1, y) H(m_2, ... , m_w; y )
\nonumber\\
&-& H(m_2, m_1, y) H(m_3, ... , m_w; y )
\nonumber\\
&+& ... + (-1)^{w+1} H ( m_w, ... , m_2, m_1 ; y ) \, .
\end{eqnarray}
\chapter{Inclusive Production Cross Section}
\label{App:CX}
In QCD improved parton model, the inclusive cross-section for the
production of a colorless particle can be computed using
\begin{align}
\label{eq:App-1}
\sigma^{I}(\tau,q^{2}) = \sum\limits_{a,b=q,{\bar q},g} \int\limits_0^1 dx_{1} \int\limits_0^1 dx_{2}
f_{a}(x_{1},\mu_{F}^{2}) f_{b}(x_{2}, \mu_F^{2}) \sigma^{I}_{ab}
\left( z, q^{2}, \mu_{R}^{2}, \mu_F^{2} \right)
\end{align}
where, $f$'s are the partonic distribution functions factorised at the
mass scale $\mu_{F}$. $\sigma^{I}_{ab}$ is the partonic cross section
for the production of colorless particle $I$ from the partons $a$ and
$b$. This is UV renormalised at renormalisation scale $\mu_{R}$ and
mass factorised at $\mu_{F}$. The other quantities are defined as
\begin{align}
\label{eq:App-2}
&q^{2} = m_{I}^{2}\,,
\nonumber\\
&\tau = \frac{q^2}{S}\,,
\nonumber\\
&z = \frac{q^2}{{\hat s}}\,.
\end{align}
In the above expression, $S$ and ${\hat s}$ are square of the hadronic
and partonic center of mass energies, respectively, and they are related
by
\begin{align}
\label{eq:App-3}
{\hat s} = x_{1} x_{2} S\,.
\end{align}
By introducing the identity
\begin{align}
\label{eq:App-4}
\int dz \delta(\tau - x_{1} x_{2} z) = \frac{1}{x_{1} x_{2}} =
\frac{S}{{\hat s}}
\end{align}
in Eq.~(\ref{eq:App-1}), we can rewrite the Eq.~(\ref{eq:App-1}) as
\begin{align}
\label{eq:App-5}
\sigma^{I}(\tau, q^{2}) = \sigma^{I,(0)}(\mu_R^2) \sum\limits_{ab=q,{\bar q},g}
\int\limits_{\tau}^{1} dx \;\Phi_{ab}(x,\mu_F^{2})\;
\Delta^I_{ab}\left(\frac{\tau}{x}, q^{2}, \mu_R^2, \mu_F^2\right)\,.
\end{align}
The partonic flux $\Phi_{ab}$ is defined through
\begin{align}
\label{eq:App-6}
\Phi_{ab}(x, \mu_{F}^2) = \int\limits_x^1 \frac{dy}{y} f_a(y, \mu_F^2)
\;f_b\left(\frac{x}{y}, \mu_F^2 \right)
\end{align}
and the dimensionless quantity $\Delta^{I}_{ab}$ is called the
coefficient function of the partonic level cross section. Upon
normalising the partonic level cross section by the born one, we
obtain $\Delta^I_{ab}$ i.e.
\begin{align}
\label{eq:App-7}
\Delta^I_{ab} \equiv \frac{\sigma^I_{ab}}{\sigma^{I,(0)}}\,.
\end{align}
\chapter{Solving KG Equation}
\label{chpt:App-KGSoln}
The form factor satisfies the KG differential equation (See Sec.~\ref{ss:bBH-FF}):
\begin{equation}
\label{eq:App-KG}
Q^2 \frac{d}{dQ^2} \ln {\cal F}^{I}_{ij} (\hat{a}_s, Q^2, \mu^2, \epsilon)
= \frac{1}{2} \left[ K^{I}_{ij} \left(\hat{a}_s, \frac{\mu_R^2}{\mu^2}, \epsilon
\right) + G^{I}_{ij} \left(\hat{a}_s, \frac{Q^2}{\mu_R^2},
\frac{\mu_R^2}{\mu^2}, \epsilon \right) \right]\,.
\end{equation}
In this appendix we demonstrate the procedure to solve the KG
equation. RG invariance of the ${\cal F}$ with respect to the
renormalisation scale $\mu_{R}$ implies
\begin{align}
\label{eq:App-KG-muRInd}
\mu_R^2 \frac{d}{d\mu_R^2} K^I_{ij}\left(\hat{a}_s, \frac{\mu_R^2}{\mu^2}, \epsilon
\right) = - \mu_R^2 \frac{d}{d\mu_R^2} G^{I}_{ij} \left(\hat{a}_s, \frac{Q^2}{\mu_R^2},
\frac{\mu_R^2}{\mu^2}, \epsilon \right) = - A^I_{ij}\left(a_{s}(\mu_R^{2})\right)
\end{align}
where, $A^I_{ij}$'s are the cusp anomalous dimensions. Unlike the
previous cases, we expand $K^I_{ij}$ in powers of unrenormalised
${\hat a}_{s}$ as
\begin{align}
\label{eq:App-Kexpand}
K^I_{ij}\left(\hat{a}_s, \frac{\mu_R^2}{\mu^2}, \epsilon
\right) &= \sum\limits_{k=1}^{\infty} {\hat a}_s^k
S_{\epsilon}^{k} \left(
\frac{\mu_R^2}{\mu^2} \right)^{k \frac{\epsilon}{2}}
{\hat K}^I_{ij,k}(\epsilon)
\end{align}
whereas we define the components $A^I_{ij, k}$ through
\begin{align}
\label{eq:App-Aexpand}
A^I_{ij} = \sum\limits_{k=1}^{\infty} a_s^k\left( \mu_R^2 \right)
A^I_{ij, k}\,.
\end{align}
Following the methodology discussed in
Appendix~\ref{chpt:App-SolRGEZas}, we can solve for ${\hat K}^I_{ij,k}(\epsilon)$
\begin{align}
\label{eq:App-SolnK}
{\hat K}^I_{ij,1}(\epsilon) &= \frac{1}{\epsilon} \Bigg\{ - 2 A^I_{ij,
1}\Bigg\}\,,
\nonumber\\
{\hat K}^I_{ij,2}(\epsilon) &= \frac{1}{\epsilon^2} \Bigg\{ 2 \beta_0
A^I_{ij,1} \Bigg\} + \frac{1}{\epsilon} \Bigg\{
- A^I_{ij, 2}\Bigg\}\,,
\nonumber\\
{\hat K}^I_{ij,3}(\epsilon) &= \frac{1}{\epsilon^3} \Bigg\{ - \frac{8
}{3} \beta_0^2 A^I_{ij,1} \Bigg\} +
\frac{1}{\epsilon^2} \Bigg\{ \frac{2}{3} \beta_1
A^I_{ij,1} + \frac{8}{3} \beta_0 A^I_{ij,2}
\Bigg\} + \frac{1}{\epsilon} \Bigg\{ - \frac{2
}{3} A^I_{ij, 3} \Bigg\}\,,
\nonumber\\
{\hat K}^I_{ij,4}(\epsilon) &= \frac{1}{\epsilon^4} \Bigg\{ 4 \beta_0^3
A^I_{ij,1} \Bigg\} + \frac{1}{\epsilon^3}
\Bigg\{ -\frac{8}{3} \beta_0 \beta_1 A^I_{ij,1}
- 6 \beta_0^2 A^I_{ij,2} \Bigg\} +
\frac{1}{\epsilon^{2}} \Bigg\{ \frac{1}{3} \beta_2
A^I_{ij,1} + \beta_1 A^I_{ij,2} + 3 \beta_0
A^I_{ij,3} \Bigg\}
\nonumber\\
&+ \frac{1}{\epsilon} \Bigg\{
- \frac{1}{2} A^I_{ij,4} \Bigg\}
\end{align}
Due to dependence of $G^I_{ij}$ on $Q^2$, we need to handle it
differently. Integrating the RGE of $G^I_{ij}$,
(\ref{eq:App-KG-muRInd}), we get
\begin{align}
\label{eq:App-SolveG}
&G^I_{ij} \left( {\hat a}_s, \frac{Q^2}{\mu_R^2},
\frac{\mu_R^2}{\mu^2}, \epsilon \right) - G^I_{ij} \left( {\hat
a}_s, 1, \frac{Q^2}{\mu^2}, \epsilon \right) =
\int\limits_{Q^2}^{\mu_R^2}
\frac{d\mu_R^2}{\mu_R^2}
A^I_{ij}
\nonumber\\
\Rightarrow \quad&G^I_{ij} \left( a_s(\mu_R^2), \frac{Q^2}{\mu_R^2}, \epsilon \right) =
G^I_{ij} \left( a_s(Q^2), 1, \epsilon \right) + \int\limits_{Q^2}^{\mu_R^2}
\frac{d\mu_R^2}{\mu_R^2}
A^I_{ij}
\end{align}
Consider the second part of the above Eq.~(\ref{eq:App-SolveG})
\begin{align}
\label{eq:App-SolveG-1}
\int\limits_{Q^2}^{\mu_R^2} \frac{d\mu_R^2}{\mu_R^2} A^I_{ij}
&= \int\limits_{Q^2}^{\mu_R^2} \frac{d\mu_R^2}{\mu_R^2}
\sum\limits_{k=1}^{\infty} a_s^k A^I_{ij,k}
\nonumber\\
&= \sum\limits_{k=1}^{\infty} \int\limits_{\frac{Q^2}{\mu^2}}^{\frac{\mu_R^2}{\mu^2}}
\frac{dX^2}{X^2} {\hat a}_s^k S_{\epsilon}^k \left(
X^2\right)^{k\frac{\epsilon}{2}} \left( Z_{a_s}^{-1} (X^2) \right)^k A^I_{ij,k}
\end{align}
where we have made the change of integration variable from $\mu_R$ to
$X$ by $\mu_R^2 =
X^2 \mu^2$. By using the $Z_{a_s}^{-1} (X^2)$ from
Eq.~(\ref{eq:App-ZashatInv}) and evaluating the integral we obtain
\begin{align}
\label{eq:App-SolveG-2}
\int\limits_{Q^2}^{\mu_R^2} \frac{d\mu_R^2}{\mu_R^2} A^I_{ij} =
\sum\limits_{k=1}^{\infty} {\hat a}_s^k S_{\epsilon}^k \left(
\frac{\mu_R^2}{\mu^2}\right)^{k\frac{\epsilon}{2}} \left[ \left(
\frac{Q^2}{\mu_R^2} \right)^{k \frac{\epsilon}{2}} -1 \right] {\hat K}^I_{ij,k}(\epsilon)\,.
\end{align}
The first part of $G^I_{ij}$ in Eq.~(\ref{eq:App-SolveG}) can be
expanded in powers of $a_s(Q^2)$ as
\begin{align}
\label{eq:App-SolveG-3}
G^I_{ij} \left( a_s(Q^2), 1, \epsilon \right) =
\sum\limits_{k=1}^{\infty} a_s^k(Q^2) G^I_{ij}(\epsilon)\,.
\end{align}
By putting back the Eq.~(\ref{eq:App-Kexpand}),
(\ref{eq:App-SolveG-1}) and (\ref{eq:App-SolveG-2}) in the original KG
equation~(\ref{eq:App-KG}), we solve for $\ln {\cal F}^{I}_{ij}
(\hat{a}_s, Q^2, \mu^2, \epsilon)$:
\begin{align}
\label{eq:App-lnFSoln}
\ln {\cal F}^{I}_{ij}(\hat{a}_s, Q^2, \mu^2, \epsilon) =
\sum_{k=1}^{\infty} {\hat a}_{s}^{k}S_{\epsilon}^{k} \left(\frac{Q^{2}}{\mu^{2}}\right)^{k
\frac{\epsilon}{2}} \hat {\cal L}_{ij, k}^{I}(\epsilon)
\end{align}
with
\begin{align}
\label{eq:App-lnFitoCalLF}
\hat {\cal L}_{ij,1}^{I}(\epsilon) &= { \frac{1}{\epsilon^2} } \Bigg\{-2 A^{I}_{{ij},1}\Bigg\}
+ { \frac{1}{\epsilon}
}
\Bigg\{G^{I}_{{ij},1}
(\epsilon)\Bigg\}\, ,
\nonumber\\
\hat {\cal L}_{{ij},2}^{I}(\epsilon) &= { \frac{1}{\epsilon^3} } \Bigg\{\beta_0 A^{I}_{{ij},1}\Bigg\}
+ {
\frac{1}{\epsilon^2} }
\Bigg\{- {
\frac{1}{2} } A^{I}_{{ij},2}
- \beta_0 G^{I}_{{ij},1}(\epsilon)\Bigg\}
+ { \frac{1}{\epsilon}
} \Bigg\{ {
\frac{1}{2} } G^{I}_{{ij},2}(\epsilon)\Bigg\}\, ,
\nonumber\\
\hat {\cal L}_{{ij},3}^{I}(\epsilon) &= { \frac{1}{\epsilon^4}
} \Bigg\{- {
\frac{8}{9} } \beta_0^2 A^{I}_{{ij},1}\Bigg\}
+ {
\frac{1}{\epsilon^3} }
\Bigg\{ { \frac{2}{9} } \beta_1 A^{I}_{{ij},1}
+ { \frac{8}{9} }
\beta_0 A^{I}_{{ij},2} + { \frac{4}{3} }
\beta_0^2 G^{I}_{{ij},1}(\epsilon)\Bigg\}
\nonumber\\
&
+ { \frac{1}{\epsilon^2} } \Bigg\{- { \frac{2}{9} } A^{I}_{{ij},3}
- { \frac{1}{3} } \beta_1 G^{I}_{{ij},1}(\epsilon)
- { \frac{4}{3} } \beta_0 G^{I}_{{ij},2}(\epsilon)\Bigg\}
+ { \frac{1}{\epsilon}
} \Bigg\{ { \frac{1}{3} }
G^{I}_{ij,3}(\epsilon)\Bigg\}\,,
\nonumber\\
\hat {\cal L}_{{ij},4}^{I}(\epsilon) &= \frac{1}{\epsilon^5} \Bigg\{ A^I_{ij,1} \beta_0^3 \Bigg\}
+ \frac{1}{\epsilon^4} \Bigg\{ - \frac{3}{2} A^I_{ij,2} \beta_0^2 - \frac{2}{3}
A^I_{ij,1} \beta_0 \beta_1 - 2
\beta_0^3 G^I_{ij,1}(\epsilon))
\Bigg\}
\nonumber\\
&
+ \frac{1}{\epsilon^3} \Bigg\{ \frac{3}{4} A^I_{ij,3} \beta_0 +
\frac{1}{4} A^I_{ij,2} \beta_1 +
\frac{1}{12} A^I_{ij,1} \beta_2 + \frac{4}{3}
\beta_0 \beta_1
G^I_{ij,1}(\epsilon) +
3 \beta_0^2 G^I_{ij,2}(\epsilon)) \Bigg\}
\nonumber\\
& + \frac{1}{\epsilon^2} \Bigg\{ -\frac{1}{8} A^I_{ij,4} - \frac{1}{6} \beta_2
G^I_{ij,1}(\epsilon) - \frac{1}{2}
\beta_1 G^I_{ij,2}(\epsilon)
- \frac{3}{2}\beta_0
G^I_{ij,3}(\epsilon) \Bigg\}
\nonumber\\
& + \frac{1}{\epsilon} \Bigg\{ \frac{1}{4} G^I_{ij,4}(\epsilon) \Bigg\}\,.
\end{align}
This methodology can easily be generalised to all orders in
perturbation theory.
\chapter{Soft-Collinear Distribution for Rapidity}
\label{chpt:App-Rap-Soft-Col-Dist}
In Sec.~\ref{ss:Rap-SCD}, we introduced the soft-collinear
distribution $\Phi^I_{i~{\bar i}}$ in the context of computing SV
correction to the differential rapidity distribution of a colorless
particle at Hadron collider. In this appendix, we intend to elaborate
the methodology
of finding this distribution. For simplicity, we will omit the
partonic indices for our further calculation. To understand
the underlying logics behind finding $\Phi^{I}$, let us consider an
example at one loop level. The generalisation to higher loop is
straightforward. The whole discussion of this appendix is closely
related to the Appendix~\ref{chpt:App-Soft-Col-Dist} where we
discussed the soft-collinear distribution for inclusive production
cross section for a colorless particle.
As discussed in the Sec.~\ref{sec:Rap-ThreResu}, the SV cross section in
$z$-space can be computed in $d=4+\epsilon$
dimensions using
\begin{align}
\label{eq:App-Rap-sigma}
\Delta^{I, \text{SV}}_{Y} (z_1, z_2, q^2, \mu_{R}^{2}, \mu_F^2) =
{\cal C} \exp \Big( \Psi^I_{Y} \left(z_1, z_2, q^2, \mu_R^2, \mu_F^2, \epsilon
\right) \Big) \Big|_{\epsilon = 0}
\end{align}
where, $\Psi^I_{Y} \left(z_1, z_2, q^2, \mu_R^2, \mu_F^2, \epsilon \right)$ is a
finite distribution and ${\cal C}$ is the double Mellin convolution defined
through Eq.~(\ref{eq:Rap-conv}). The $\Psi^I$ is given by, Eq.~(\ref{eq:Rap-psi})
\begin{align}
\label{eq:App-Rap-psi}
\Psi^{I}_{Y,ij} \left(z_1, z_2, q^2, \mu_R^2, \mu_F^2, \epsilon \right)
= &\left( \ln \Big[ Z^I_{ij} (\hat{a}_s, \mu_R^2, \mu^2, \epsilon) \Big]^2
+ \ln \Big| {\cal F}^I_{ij} (\hat{a}_s, Q^2, \mu^2, \epsilon)
\Big|^2 \right) \delta(1-z_1) \delta(1-z_2)
\nonumber\\
& + 2 \Phi^I_{Y,ij} (\hat{a}_s, q^2, \mu^2, z_1, z_2, \epsilon) - {\cal C} \ln
\Gamma^{I}_{ij} (\hat{a}_s, \mu^2, \mu_F^2, z_1,
\epsilon)\delta(1-z_2)
\nonumber\\
&
- {\cal C} \ln
\Gamma^{I}_{ij} (\hat{a}_s, \mu^2, \mu_F^2, z_2,
\epsilon)\delta(1-z_1) \, .
\end{align}
For all the details about the notations, see
Sec.~\ref{sec:Rap-ThreResu}. Considering only the poles at ${\cal
O}(a_{s})$ with $\mu_{R} =
\mu_{F}$ we obtain,
\begin{align}
\label{eq:App-Rap-Psi-Comp-1}
&\ln\left(Z^{I, (1)}\right)^{2} = a_{s}(\mu_{F}^{2}) \frac{4\gamma_{1}^{I}}{\epsilon}\,,
\nonumber\\
&\ln|{\cal F}^{I, (1)}|^{2} = a_{s}(\mu_{F}^{2}) \left(
\frac{q^{2}}{\mu_{F}^{2}} \right)^{\frac{\epsilon}{2}} \left[
-\frac{4A_{1}^{I}}{\epsilon^{2}} + \frac{1}{\epsilon} \left( 2f_{1}^{I}
+ 4B_{1}^{I} - 4\gamma_{1}^{I}\right)\right]\,,
\nonumber\\
&{\cal C} \ln\Gamma^{I, (1)}(z_1) = a_{s}(\mu_{F}^{2}) \left[
\frac{2B^{I}_{1}}{\epsilon} \delta(1-z_1) +
\frac{2A_{1}^{I}}{\epsilon} {\cal D}_{0}\right]\,,
\nonumber\\
&{\cal C} \ln\Gamma^{I, (1)}(z_2) = a_{s}(\mu_{F}^{2}) \left[
\frac{2B^{I}_{1}}{\epsilon} \delta(1-z_2) +
\frac{2A_{1}^{I}}{\epsilon} \overline{\cal D}_{0}\right]
\end{align}
where, the components are defined through the expansion of these
quantities in powers of $a_s(\mu_F^{2})$
\begin{align}
\label{eq:App-Rap-psi-comp-2}
&\Psi^{I}_{Y} = \sum\limits_{k=1}^{\infty} a_s^k \left( \mu_F^2 \right) \Psi^{I,(k)}_{Y}\,,
\nonumber\\
&\ln (Z^I)^2 = \sum\limits_{k=1}^{\infty} a_s^k \left( \mu_F^2 \right)
Z^{I,(k)}\,,
\nonumber\\
&\ln |{\cal F}^{I}|^2 = \sum\limits_{k=1}^{\infty} a_s^k\left( \mu_F^2
\right) \left( \frac{q^2}{\mu_F^2} \right)^{k \frac{\epsilon}{2}}
\ln |{\cal F}^{I,(k)}|^2\,,
\nonumber\\
&\Phi^{I}_{Y} = \sum\limits_{k=1}^{\infty} a_s^k(\mu_F^2) \Phi^I_{Y,k}\,,
\nonumber\\
&\ln \Gamma^I = \sum\limits_{k=1}^{\infty} a_s^k(\mu_F^2) \ln
\Gamma^{I,(k)}
\intertext{and}
&{\cal{D}}_{i} \equiv \left[ \frac{\ln^{i}(1-z_1)}{1-z_1} \right]_{+}\,,
\nonumber\\
&\overline{\cal{D}}_{i} \equiv \left[ \frac{\ln^{i}(1-z_2)}{1-z_2} \right]_{+}\,.
\end{align}
Collecting the coefficients of $a_s(\mu_F^2)$, we
get
\begin{align}
\label{eq:App-Rap-Psi-1}
\Psi^{I, (1)}|_{Y,{\rm poles}}
&= \left[ \left\{ - \frac{4A_{1}^{I}}{\epsilon^{2}} + \frac{
2f_{1}^{I}}{\epsilon} \right\} \delta(1-z_1) \delta(1-z_2) -
\frac{2A_{1}^{I}}{\epsilon} \left\{ \delta(1-z_1) \overline{\cal
D}_{0} + \delta(1-z_2) {\cal D}_0 \right\} \right]
\nonumber\\
&+ {2\Phi^{I}_{Y,1}}
\end{align}
where, we have suppressed the $\ln (q^2/\mu_F^2)$ terms.
To cancel the remaining divergences appearing in the above
Eq.~(\ref{eq:App-Rap-Psi-1}) for obtaining a finite rapidity distribution, we must
demand that $\Phi^{I}_{Y,1}$ has exactly the same poles with
opposite sign:
\begin{align}
\label{eq:App-Rap-Phi-1}
2 \Phi^{I}_{Y,1}|_{\rm poles} = - \left[ \left\{ - \frac{4A_{1}^{I}}{\epsilon^{2}} + \frac{
2f_{1}^{I}}{\epsilon} \right\} \delta(1-z_1) \delta(1-z_2) -
\frac{2A_{1}^{I}}{\epsilon} \left\{ \delta(1-z_1) \overline{\cal
D}_{0} + \delta(1-z_2) {\cal D}_0 \right\} \right]
\end{align}
In addition, $\Phi^{I}_{Y}$ also should be RG invariant with respect to
$\mu_R$:
\begin{align}
\label{eq:App-Rap-Phi-2}
\mu_R^2 \frac{d}{d\mu_R^2} \Phi^I_{Y} = 0\,.
\end{align}
We make an \textit{ansatz}, the above two demands,
Eq.~(\ref{eq:App-Phi-1}) and~(\ref{eq:App-Phi-2}) can be accomplished
if $\Phi^I_Y$ satisfies the KG-type integro-differential equation which we call
$\overline{KG}_{Y}$:
\begin{align}
\label{eq:App-Rap-KGbarEqn}
q^2 \frac{d}{dq^2} \Phi^I_Y\left(\hat{a}_s, q^2, \mu^2, z_1, z_2,
\epsilon\right) = \frac{1}{2} \left[ \overline K^I_Y
\left(\hat{a}_s, \frac{\mu_R^2}{\mu^2}, z_1, z_2,
\epsilon \right) + \overline G^I_Y \left(\hat{a}_s,
\frac{q^2}{\mu_R^2}, \frac{\mu_R^2}{\mu^2}, z_1, z_2, \epsilon \right) \right]\,.
\end{align}
$\overline{K}^I_Y$ contains all the poles whereas $\overline{G}^I_Y$ consists
of only the finite terms in $\epsilon$. RG
invariance~(\ref{eq:App-Rap-Phi-2}) of $\Phi^I_Y$ dictates
\begin{align}
\label{eq:App-Rap-Phi-3}
\mu_R^2 \frac{d}{d\mu_R^2} \overline{K}^{I}_Y = -\mu_R^2 \frac{d}{d\mu_R^2}
\overline{G}^I_Y \equiv X^{I}_{Y}
\end{align}
where, we introduce a quantity $X^I_{Y}$. Following the methodology of
solving the KG equation discussed in the
Appendix~\ref{chpt:App-KGSoln}, we can write the solution of $\Phi^I_Y$
as
\begin{align}
\label{eq:App-Rap-Phi-4}
\Phi^I_Y(\hat{a}_s, q^2, \mu^2, z_1, z_2, \epsilon) =
\sum_{k=1}^{\infty} {\hat a}_{s}^{k}S_{\epsilon}^{k} \left(\frac{q^{2}}{\mu^{2}}\right)^{k
\frac{\epsilon}{2}} \hat {\Phi}^{I}_{Y,k}(z_1, z_{2}, \epsilon)
\end{align}
with
\begin{align}
\label{eq:App-Rap-Phi-5}
\hat {\Phi}^{I}_{Y,k}(z_1, z_2, \epsilon) = {\hat{\cal L}}^I_k \left( A^I_{i}
\rightarrow X^I_{Y,i}, G^I_i \rightarrow \overline{G}^I_{Y,i}(z,\epsilon) \right)\,.
\end{align}
where we define the components through the expansions
\begin{align}
\label{eq:App-Rap-YGexpans}
&X^I_{Y} = \sum\limits_{k=1}^{\infty} a_s^k(\mu_F^2) X^I_{Y,k}\,,
\nonumber\\
&\overline{G}^I_Y(z_1, z_{2},\epsilon) = \sum\limits_{k=1}^{\infty}
a_s^k(\mu_F^2) \overline{G}^I_{Y,k}(z_1, z_2, \epsilon).
\end{align}
This solution directly follows from the Eq.~(\ref{eq:App-lnFSoln}). Hence we get
\begin{align}
\label{eq:App-Rap-Phi-6}
2 \hat {\Phi}^{I}_{Y,1}(z,\epsilon) &= { \frac{1}{\epsilon^2} } \Bigg\{-4 X^{I}_{Y,1}\Bigg\}
+ { \frac{2}{\epsilon}
}
\Bigg\{\overline{G}^{I}_{Y,1}
(z_1, z_2,\epsilon)\Bigg\}\,.
\end{align}
By expressing the components of $\Phi^I_Y$ in powers of $a_s(\mu_F^2)$,
we obtain
\begin{align}
\label{eq:App-Rap-Phi-7}
\Phi^I(\hat{a}_s, q^2, \mu^2, z_1, z_2, \epsilon) &=
\sum_{k=1}^{\infty} {\hat a}_{s}^{k}S_{\epsilon}^{k} \left(\frac{q^{2}}{\mu^{2}}\right)^{k
\frac{\epsilon}{2}} \hat {\Phi}^{I}_{Y,k}(z_1, z_2, \epsilon)
\nonumber\\
&= \sum_{k=1}^{\infty} a_s^k(\mu_F^2) \left( \frac{q^2}{\mu_F^2} \right)^{k
\frac{\epsilon}{2}} Z_{a_s}^k \hat{\Phi}^I_{Y,k}(z_1, z_2,\epsilon)
\nonumber\\
& \equiv \sum_{k=1}^{\infty} a_s^k(\mu_F^2) \left( \frac{q^2}{\mu_F^2} \right)^{k
\frac{\epsilon}{2}} {\Phi}^I_{Y,k}(z,\epsilon)
\end{align}
and at ${\cal O}(a_s(\mu_F^2)), \hat{\Phi}^I_{Y,1}(z,\epsilon) =
\Phi^I_{Y,1}(z,\epsilon)$ upon suppressing the terms like $\log
(q^{2}/\mu_{F}^{2})$. Hence, by comparing the
Eq.~(\ref{eq:App-Rap-Phi-1}) and (\ref{eq:App-Rap-Phi-6}), we conclude
\begin{align}
\label{eq:App-Rap-Phi-8}
&X^I_{Y,1} = -A^I_1 \delta(1-z_1) \delta(1-z_2)\,,
\nonumber\\
&\overline{G}^I_{Y,1}(z_1, z_2, \epsilon) = -f^I_1 \delta(1-z_1)
\delta(1-z_2) + A^I_1 \Bigg\{ \delta(1-z_1) \overline{\cal
D}_0 + \delta(1-z_2) {\cal D}_{0} \Bigg\}
\nonumber\\
&+ \sum\limits_{k=1}^{\infty}
\epsilon^k \overline{g}^{I,k}_{Y,1}(z)\,.
\end{align}
The coefficients of $\epsilon^k, \overline{g}^{I,k}_{Y,1}(z)$ can only be
determined through explicit computations. These do not contribute to
the infrared poles associated with $\Phi^{I}_{Y}$. This uniquely fixes the
unknown soft-collinear distribution $\Phi^I_{Y}$ at one loop order. This
prescription can easily be generalised to higher orders in $a_s$. In
our calculation of the SV correction to rapidity distribution, instead
of solving in this way,
we follow a bit different methodology which is presented below.
Keeping the demands~(\ref{eq:App-Rap-Phi-1}) and~(\ref{eq:App-Rap-Phi-2}) in
mind, we propose the solution of the $\overline{KG}_{Y}$ equation as (See
Eq.~(\ref{eq:App-Rap-Phi-4})) (which is just the extension of the
Eq.~(\ref{eq:App-Phi-9}) from one variable $z$ to a case of two
variables $z_1$ and $z_2$)
\begin{align}
\label{eq:App-Rap-Phi-9}
\hat{\Phi}^I_{Y,k} (z_1, z_2, \epsilon) &\equiv \Bigg\{ (k \epsilon)^2
\frac{1}{4 (1-z_1) (1-z_2)}
\left[
(1-z_1) (1-z_2) \right]^{k \frac{\epsilon}{2}}\Bigg\}
\hat{\Phi}^I_{Y,k}(\epsilon)
\nonumber\\
&= \Bigg\{ \frac{k\epsilon}{2} \frac{1}{(1-z_1)} \left[ (1-z_1)^2 \right]^{k
\frac{\epsilon}{4}} \Bigg\}
\Bigg\{ \frac{k\epsilon}{2} \frac{1}{(1-z_2)} \left[ (1-z_2)^2 \right]^{k
\frac{\epsilon}{4}} \Bigg\} \hat{\Phi}^I_{Y,k}(\epsilon)
\nonumber\\
&=\Bigg\{ \delta(1-z_1) + \sum\limits_{j=0}^{\infty} \frac{(k
\epsilon/2)^{j+1}}{j!} {\cal D}_{j} \Bigg\}
\Bigg\{ \delta(1-z_2) + \sum\limits_{l=0}^{\infty} \frac{(k
\epsilon/2)^{l+1}}{l!} \overline{\cal D}_{l} \Bigg\}
\hat{\Phi}^I_{Y,k}(\epsilon)\,.
\end{align}
The RG invariance of $\Phi^{I}_Y$, Eq.~(\ref{eq:App-Rap-Phi-2}), implies
\begin{align}
\label{eq:App-Rap-Phi-10}
\mu_R^2 \frac{d}{d\mu_R^2} \overline{K}^{I}_Y = -\mu_R^2 \frac{d}{d\mu_R^2}
\overline{G}^I_Y \equiv X'^{I}_Y
\end{align}
where, we introduce a quantity $X'^I_Y$, analogous to $X^I_{Y}$. Hence, the
solution can be obtained as
\begin{align}
\label{eq:App-Rap-Phi-11}
\hat {\Phi}^{I}_{Y,k}(\epsilon) &= {\hat{\cal L}}^I_k \left( A^I_{i}
\rightarrow X'^I_{Y,i}, G^I_{Y,i} \rightarrow \overline{\cal
G}^I_{Y,i}(\epsilon) \right)\,.
\end{align}
Hence, according to the Eq.~(\ref{eq:App-lnFitoCalLF}), for $k=1$ we get
\begin{align}
\label{eq:App-Rap-Phi-12}
2\Phi^{I}_{Y,1} (z_1, z_2, \epsilon) &= 2{\hat \Phi}^{I}_{Y,1} (z_1, z_2,
\epsilon)
\nonumber\\
&=
\Bigg\{ \delta(1-z_1) + \sum\limits_{j=0}^{\infty} \frac{(k
\epsilon/2)^{j+1}}{j!} {\cal D}_{j} \Bigg\}
\Bigg\{ \delta(1-z_2) + \sum\limits_{l=0}^{\infty} \frac{(k
\epsilon/2)^{l+1}}{l!} \overline{\cal D}_{l} \Bigg\}
\nonumber\\
&~~~~\Bigg\{ \frac{1}{\epsilon^2}\left(
- 4X'^I_{Y,1} \right) +
\frac{2}{\epsilon}{\overline{\cal
G}}^I_{Y,1} \left(\epsilon) \right) \Bigg\}
\end{align}
where, $X'^{I}_{Y}$ and $\overline{\cal G}^I_{Y}$ are expanded similar to
Eq.~(\ref{eq:App-Rap-YGexpans}). Comparison between the two solutions
depicted in Eq.~(\ref{eq:App-Rap-Phi-1}) and (\ref{eq:App-Rap-Phi-12}), we
can write
\begin{align}
\label{eq:App-Phi-13}
&X'^I_{Y,1} = - A^I_1
\nonumber\\
&{\overline{\cal G}}^I_{Y,1} \left(\epsilon\right) = - f^I_1 +
\sum_{k=1}^{\infty}
\epsilon^k \overline{\cal
G}^{I, k}_{Y,1}\,.
\end{align}
Explicit computation is required to determine the coefficients of
$\epsilon^k$, $\overline{\cal G}^{I,k}_{Y,1}$. This solution is used in
Eq.~(\ref{eq:Rap-PhiSoln}) in
the context of SV correction to differential rapidity distribution of
Higgs boson production or leptonic pair in DY production. The method
is generalised to higher orders in $a_s$ to obtain the results of the
soft-collinear distribution. Hence, the all order solution of
$\Phi^I_{Y}$ is
\begin{align}
\label{eq:App-Rap-Soln-PhiY}
&\Phi^I(\hat{a}_s, q^2, \mu^2, z_1, z_2, \epsilon) =
\sum_{k=1}^{\infty} {\hat a}_{s}^{k}S_{\epsilon}^{k} \left(\frac{q^{2}}{\mu^{2}}\right)^{k
\frac{\epsilon}{2}} \hat {\Phi}^{I}_{Y,k}(z_1, z_2, \epsilon)
\intertext{with}
&\hat {\Phi}^{I}_{Y,k}(z_1, z_2, \epsilon) = \Bigg\{ (k \epsilon)^2
\frac{1}{4 (1-z_1) (1-z_2)}
\left[
(1-z_1) (1-z_2) \right]^{k \frac{\epsilon}{2}}\Bigg\}
\hat{\Phi}^I_{Y,k}(\epsilon)\,,
\nonumber\\
&\hat {\Phi}^{I}_{Y,k}(\epsilon) = {\hat{\cal L}}^I_k \left( A^I_{i}
\rightarrow -A^I_{i}, G^I_{Y,i} \rightarrow \overline{\cal
G}^I_{Y,i}(\epsilon) \right)\,.
\end{align}
Up to three loop, $\overline{\cal G}^I_{Y,i}(\epsilon)$ are found to be
\begin{align}
\label{eq:App-Rap-Sol-calGbar}
&\overline{\cal G}^I_{Y,i} (\epsilon) = -f^I_i + \overline{C}^I_{Y,i}
+ \sum\limits_{k=1}^{\infty}
\epsilon^k \overline{\cal
G}^{I,k}_{Y,i}
\intertext{where}
&\overline{C}^I_{Y,1} = 0\,,
\nonumber\\
&\overline{C}^I_{Y,2} = - 2 \beta_0 \overline{\cal G}^{I,1}_{Y,1}\,,
\nonumber\\
&\overline{C}^I_{Y,3} = - 2 \beta_1 \overline{\cal G}^{I,1}_{Y,1} - 2
\beta_0 \left( \overline{\cal G}^{I,1}_{Y,2} + 2 \beta_0 \overline{\cal
G}^{I,2}_{Y,1} \right)\,.
\end{align}
These are employed in the computation of rapidity distributions in
Chapter~\ref{chap:Rap}. In the next subsection, we present the results
of the soft-collinear distribution up to three loops.
\subsection{Results}
\label{app:ss-RapSCD-Res}
We define the renormalised components of the
$\Phi^I_{Y,i~{\bar i},k}$ through
\begin{align}
\label{eq:App-Rap-SCD-Re}
\Phi^I_{Y,i~{\bar i}}(\hat{a}_s, q^2, \mu^2, z_1, z_2, \epsilon) &=
\sum_{k=1}^{\infty} {\hat a}_{s}^{k}S_{\epsilon}^{k} \left(\frac{q^{2}}{\mu^{2}}\right)^{k
\frac{\epsilon}{2}} \hat {\Phi}^{I}_{Y,i~{\bar i},k}(z_1, z_2, \epsilon)
\nonumber\\
&= \sum\limits_{k=1}^{\infty} a_s^k\left( \mu_F^2 \right) \Phi^I_{Y,i~{\bar
i},k} \left( z_1, z_2, \epsilon, q^2, \mu_F^2 \right)
\end{align}
where, we make the choice of the renormalisation scale
$\mu_R=\mu_F$. The $\mu_R$ dependence can be easily restored by using
the evolution equation of strong coupling constant,
Eq.~(\ref{eq:bBH-asf2asr}). Below, we present the $\Phi^I_{Y,i~{\bar
i},k}$ for
$I=H$ and $i~{\bar i}=gg$ up to three
loops and the corresponding components for $I={\rm DY}$ and $i~{\bar
i}=q{\bar q}$ can be obtained using maximally non-Abelian property
fulfilled by this distribution:
\begin{align}
\label{eq:App-Rap-SCD-MaxNonAbe}
\Phi^H_{Y,gg,k} = \frac{C_A}{C_F} \Phi^{\rm DY}_{Y,q~{\bar q},k}\,.
\end{align}
The results are given by
\begin{align}
\label{eq:Rap-PhiY}
\Phi^H_{Y,gg,1} &=
\delta(1-z_1) \delta(1-z_2) \Bigg[ \frac{1}{\epsilon^2} C_A \Bigg\{
8
\Bigg\}
+ \frac{1}{\epsilon} \log\left(\frac{q^2}{\mu_F^2}\right) C_A
\Bigg\{
4
\Bigg\}
+ C_A \Bigg\{
- \zeta_2
\Bigg\}
\nonumber\\
& + \log^2\left(\frac{q^2}{\mu_F^2}\right) C_A \Bigg\{
1
\Bigg\}
\Bigg]
+ {\cal D}_0 \delta(1-z_2) \Bigg[ \frac{1}{\epsilon} C_A \Bigg\{
4
\Bigg\}
+ \log\left(\frac{q^2}{\mu_F^2}\right) C_A \Bigg\{
2
\Bigg\}
\Bigg]
\nonumber\\
& + {\cal D}_0 \overline{\cal D}_0 \Bigg[ C_A \Bigg\{
2
\Bigg\}
\Bigg]
+ {\cal D}_1 \delta(1-z_2) \Bigg[ C_A \Bigg\{
2
\Bigg\}
\Bigg]
+ \overline{\cal D}_0 \delta(1-z_1) \Bigg[ \frac{1}{\epsilon} C_A \Bigg\{
4
\Bigg\}
\nonumber\\
& + \log\left(\frac{q^2}{\mu_F^2}\right) C_A \Bigg\{
2
\Bigg\}
\Bigg]
+ \overline{\cal D}_1 \delta(1-z_1) \Bigg[ C_A \Bigg\{
2
\Bigg\}
\Bigg]\,,
\nonumber\\
\Phi^H_{Y,gg,2} &=
\delta(1-z_1) \delta(1-z_2) \Bigg[ \frac{1}{\epsilon^3} C_A^2 \Bigg\{
44
\Bigg\}
+ \frac{1}{\epsilon^3} n_f C_A \Bigg\{
- 8
\Bigg\}
+ \frac{1}{\epsilon^2} C_A^2 \Bigg\{
\frac{134}{9}
- 4 \zeta_2
\Bigg\}
\nonumber\\
& + \frac{1}{\epsilon^2} n_f C_A \Bigg\{
- \frac{20}{9}
\Bigg\}
+ \frac{1}{\epsilon^2} \log\left(\frac{q^2}{\mu_F^2}\right) C_A^2
\Bigg\{
\frac{44}{3}
\Bigg\}
+ \frac{1}{\epsilon^2} \log\left(\frac{q^2}{\mu_F^2}\right) n_f
C_A \Bigg\{
- \frac{8}{3}
\Bigg\}
\nonumber\\
& + \frac{1}{\epsilon} C_A^2 \Bigg\{
- \frac{404}{27}
+ 14 \zeta_3
+ \frac{11}{3} \zeta_2
\Bigg\}
+ \frac{1}{\epsilon} n_f C_A \Bigg\{
\frac{56}{27}
- \frac{2}{3} \zeta_2
\Bigg\}
\nonumber\\
& + \frac{1}{\epsilon} \log\left(\frac{q^2}{\mu_F^2}\right) C_A^2
\Bigg\{
\frac{134}{9}
- 4 \zeta_2
\Bigg\}
+ \frac{1}{\epsilon} \log\left(\frac{q^2}{\mu_F^2}\right) n_f
C_A \Bigg\{
- \frac{20}{9}
\Bigg\}
\nonumber\\
& + C_A^2 \Bigg\{
\frac{1214}{81}
- \frac{55}{9} \zeta_3
- \frac{67}{6} \zeta_2
- 2 \zeta_2^2
\Bigg\}
+ n_f C_A \Bigg\{
- \frac{164}{81}
+ \frac{10}{9} \zeta_3
+ \frac{5}{3} \zeta_2
\Bigg\}
\nonumber\\
& + \log\left(\frac{q^2}{\mu_F^2}\right) C_A^2 \Bigg\{
- \frac{404}{27}
+ 14 \zeta_3
+ \frac{22}{3} \zeta_2
\Bigg\}
+ \log\left(\frac{q^2}{\mu_F^2}\right) n_f C_A \Bigg\{
\frac{56}{27}
- \frac{4}{3} \zeta_2
\Bigg\}
\nonumber\\
& + \log^2\left(\frac{q^2}{\mu_F^2}\right) C_A^2 \Bigg\{
+ \frac{67}{9}
- 2 \zeta_2
\Bigg\}
+ \log^2\left(\frac{q^2}{\mu_F^2}\right) n_f C_A
\Bigg\{
- \frac{10}{9}
\Bigg\}
\nonumber\\
& + \log^3\left(\frac{q^2}{\mu_F^2}\right) C_A^2 \Bigg\{
- \frac{11}{9}
\Bigg\}
+ \log^3\left(\frac{q^2}{\mu_F^2}\right) n_f C_A
\Bigg\{
\frac{2}{9}
\Bigg\}
\Bigg]
\nonumber\\
& + \delta(1-z_2) {\cal D}_0 \Bigg[ \frac{1}{\epsilon^2} C_A^2 \Bigg\{
\frac{44}{3}
\Bigg\}
+ \frac{1}{\epsilon^2} n_f C_A \Bigg\{
- \frac{8}{3}
\Bigg\}
+ \frac{1}{\epsilon} C_A^2 \Bigg\{
\frac{134}{9}
- 4 \zeta_2
\Bigg\}
\nonumber\\
& + \frac{1}{\epsilon} n_f C_A \Bigg\{
- \frac{20}{9}
\Bigg\}
+ C_A^2 \Bigg\{
- \frac{404}{27}
+ 14 \zeta_3
+ \frac{22}{3} \zeta_2
\Bigg\}
+ n_f C_A \Bigg\{
\frac{56}{27}
- \frac{4}{3} \zeta_2
\Bigg\}
\nonumber\\
& + \log\left(\frac{q^2}{\mu_F^2}\right) C_A^2 \Bigg\{
\frac{134}{9}
- 4 \zeta_2
\Bigg\}
+ \log\left(\frac{q^2}{\mu_F^2}\right) n_f C_A \Bigg\{
- \frac{20}{9}
\Bigg\}
+ \log^2\left(\frac{q^2}{\mu_F^2}\right) C_A^2 \Bigg\{
- \frac{11}{3}
\Bigg\}
\nonumber\\
& + \log^2\left(\frac{q^2}{\mu_F^2}\right) n_f C_A \Bigg\{
\frac{2}{3}
\Bigg\}
\Bigg]
\nonumber\\
& + {\cal D}_0 \overline{\cal D}_0 \Bigg[ C_A^2 \Bigg\{
\frac{134}{9}
- 4 \zeta_2
\Bigg\}
+ n_f C_A \Bigg\{
- \frac{20}{9}
\Bigg\}
+ \log\left(\frac{q^2}{\mu_F^2}\right) C_A^2 \Bigg\{
- \frac{22}{3}
\Bigg\}
\nonumber\\
& + \log\left(\frac{q^2}{\mu_F^2}\right) n_f C_A \Bigg\{
\frac{4}{3}
\Bigg\}
\Bigg]
+ {\cal D}_0 \overline{\cal D}_1 \Bigg[ C_A^2 \Bigg\{
- \frac{22}{3}
\Bigg\}
+ n_f C_A \Bigg\{
\frac{4}{3}
\Bigg\}
\Bigg]
\nonumber\\
& + {\cal D}_1 \delta(1-z_2) \Bigg[ C_A^2 \Bigg\{
\frac{134}{9}
- 4 \zeta_2
\Bigg\}
+ n_f C_A \Bigg\{
- \frac{20}{9}
\Bigg\}
+ \log\left(\frac{q^2}{\mu_F^2}\right) C_A^2 \Bigg\{
- \frac{22}{3}
\Bigg\}
\nonumber\\
& + \log\left(\frac{q^2}{\mu_F^2}\right) n_f C_A \Bigg\{
\frac{4}{3}
\Bigg\}
\Bigg]
+ {\cal D}_1 \overline{\cal D}_0 \Bigg[ C_A^2 \Bigg\{
- \frac{22}{3}
\Bigg\}
+ n_f C_A \Bigg\{
+ \frac{4}{3}
\Bigg\}
\Bigg]
\nonumber\\
& + {\cal D}_2 \delta(1-z_2) \Bigg[ C_A^2 \Bigg\{
- \frac{11}{3}
\Bigg\}
+ n_f C_A \Bigg\{
\frac{2}{3}
\Bigg\}
\Bigg]
\nonumber\\
& + \overline{\cal D}_0 \delta(1-z_1) \Bigg[
\frac{1}{\epsilon^2} C_A^2 \Bigg\{
\frac{44}{3}
\Bigg\}
+ \frac{1}{\epsilon^2} n_f C_A \Bigg\{
- \frac{8}{3}
\Bigg\}
+ \frac{1}{\epsilon} C_A^2 \Bigg\{
\frac{134}{9}
- 4 \zeta_2
\Bigg\}
\nonumber\\
& + \frac{1}{\epsilon} n_f C_A \Bigg\{
- \frac{20}{9}
\Bigg\}
+ C_A^2 \Bigg\{
- \frac{404}{27}
+ 14 \zeta_3
+ \frac{22}{3} \zeta_2
\Bigg\}
+ n_f C_A \Bigg\{
\frac{56}{27}
- \frac{4}{3} \zeta_2
\Bigg\}
\nonumber\\
& + \log\left(\frac{q^2}{\mu_F^2}\right) C_A^2 \Bigg\{
\frac{134}{9}
- 4 \zeta_2
\Bigg\}
+ \log\left(\frac{q^2}{\mu_F^2}\right) n_f C_A \Bigg\{
- \frac{20}{9}
\Bigg\}
+ \log^2\left(\frac{q^2}{\mu_F^2}\right) C_A^2 \Bigg\{
- \frac{11}{3}
\Bigg\}
\nonumber\\
& + \log^2\left(\frac{q^2}{\mu_F^2}\right) n_f C_A \Bigg\{
\frac{2}{3}
\Bigg\}
\Bigg]
+ \overline{\cal D}_1 \delta(1-z_1) \Bigg[ C_A^2 \Bigg\{
\frac{134}{9}
- 4 \zeta_2
\Bigg\}
+ n_f C_A \Bigg\{
- \frac{20}{9}
\Bigg\}
\nonumber\\
& + \log\left(\frac{q^2}{\mu_F^2}\right) C_A^2 \Bigg\{
- \frac{22}{3}
\Bigg\}
+ \log\left(\frac{q^2}{\mu_F^2}\right) n_f C_A \Bigg\{
\frac{4}{3}
\Bigg\}
\Bigg]
\nonumber\\
& + \overline{\cal D}_2 \delta(1-z_1) \Bigg[ C_A^2 \Bigg\{
- \frac{11}{3}
\Bigg\}
+ n_f C_A \Bigg\{
\frac{2}{3}
\Bigg\}
\Bigg]\,,
\nonumber\\
\Phi^H_{Y,gg,3} &=
\delta(1-z_1) \delta(1-z_2) \Bigg[ \frac{1}{\epsilon^4} C_A^3 \Bigg\{
\frac{21296}{81}
\Bigg\}
+ \frac{1}{\epsilon^4} n_f C_A^2 \Bigg\{
- \frac{7744}{81}
\Bigg\}
+ \frac{1}{\epsilon^4} n_f^2 C_A \Bigg\{
\frac{704}{81}
\Bigg\}
\nonumber\\
& + \frac{1}{\epsilon^3} C_A^3 \Bigg\{
\frac{49064}{243}
- \frac{880}{27} \zeta_2
\Bigg\}
+ \frac{1}{\epsilon^3} n_f C_A^2 \Bigg\{
- \frac{15520}{243}
+ \frac{160}{27} \zeta_2
\Bigg\}
\nonumber\\
& + \frac{1}{\epsilon^3} n_f C_F C_A \Bigg\{
- \frac{128}{9}
\Bigg\}
+ \frac{1}{\epsilon^3} n_f^2 C_A \Bigg\{
\frac{800}{243}
\Bigg\}
+ \frac{1}{\epsilon^3} \log\left(\frac{q^2}{\mu_F^2}\right) C_A^3
\Bigg\{
\frac{1936}{27}
\Bigg\}
\nonumber\\
& + \frac{1}{\epsilon^3} \log\left(\frac{q^2}{\mu_F^2}\right) n_f
C_A^2 \Bigg\{
- \frac{704}{27}
\Bigg\}
+ \frac{1}{\epsilon^3} \log\left(\frac{q^2}{\mu_F^2}\right) n_f^2
C_A \Bigg\{
\frac{64}{27}
\Bigg\}
\nonumber\\
& + \frac{1}{\epsilon^2} C_A^3 \Bigg\{
- \frac{8956}{243}
+ \frac{2024}{27} \zeta_3
- \frac{692}{81} \zeta_2
+ \frac{352}{45} \zeta_2^2
\Bigg\}
+ \frac{1}{\epsilon^2} n_f C_A^2 \Bigg\{
\frac{4024}{243}
- \frac{560}{27} \zeta_3
\nonumber\\
& - \frac{208}{81} \zeta_2
\Bigg\}
+ \frac{1}{\epsilon^2} n_f C_F C_A \Bigg\{
- \frac{220}{27}
+ \frac{64}{9} \zeta_3
\Bigg\}
+ \frac{1}{\epsilon^2} n_f^2 C_A \Bigg\{
- \frac{160}{81}
+ \frac{16}{27} \zeta_2
\Bigg\}
\nonumber\\
& + \frac{1}{\epsilon^2} \log\left(\frac{q^2}{\mu_F^2}\right) C_A^3
\Bigg\{
\frac{8344}{81}
- \frac{176}{9} \zeta_2
\Bigg\}
+ \frac{1}{\epsilon^2} \log\left(\frac{q^2}{\mu_F^2}\right) n_f
C_A^2 \Bigg\{
- \frac{2672}{81}
+ \frac{32}{9} \zeta_2
\Bigg\}
\nonumber\\
& + \frac{1}{\epsilon^2} \log\left(\frac{q^2}{\mu_F^2}\right) n_f
C_F C_A \Bigg\{
- \frac{16}{3}
\Bigg\}
+ \frac{1}{\epsilon^2} \log\left(\frac{q^2}{\mu_F^2}\right) n_f^2
C_A \Bigg\{
\frac{160}{81}
\Bigg\}
\nonumber\\
& + \frac{1}{\epsilon} C_A^3 \Bigg\{
- \frac{136781}{2187}
- 64 \zeta_5
+ \frac{1316}{9} \zeta_3
+ \frac{12650}{243} \zeta_2
- \frac{176}{9} \zeta_2 \zeta_3
- \frac{352}{15} \zeta_2^2
\Bigg\}
\nonumber\\
& + \frac{1}{\epsilon} n_f C_A^2 \Bigg\{
\frac{11842}{2187}
- \frac{728}{81} \zeta_3
- \frac{2828}{243} \zeta_2
+ \frac{32}{5} \zeta_2^2
\Bigg\}
+ \frac{1}{\epsilon} n_f C_F C_A \Bigg\{
\frac{1711}{81}
- \frac{304}{27} \zeta_3
\nonumber\\
& - \frac{4}{3} \zeta_2
- \frac{32}{15} \zeta_2^2
\Bigg\}
+ \frac{1}{\epsilon} n_f^2 C_A \Bigg\{
\frac{2080}{2187}
- \frac{112}{81} \zeta_3
+ \frac{40}{81} \zeta_2
\Bigg\}
+ \frac{1}{\epsilon} \log\left(\frac{q^2}{\mu_F^2}\right) C_A^3
\Bigg\{
\frac{490}{9}
\nonumber\\
& + \frac{88}{9} \zeta_3
- \frac{1072}{27} \zeta_2
+ \frac{176}{15} \zeta_2^2
\Bigg\}
+ \frac{1}{\epsilon} \log\left(\frac{q^2}{\mu_F^2}\right) n_f
C_A^2 \Bigg\{
- \frac{836}{81}
- \frac{112}{9} \zeta_3
+ \frac{160}{27} \zeta_2
\Bigg\}
\nonumber\\
& + \frac{1}{\epsilon} \log\left(\frac{q^2}{\mu_F^2}\right) n_f
C_F C_A \Bigg\{
- \frac{110}{9}
+ \frac{32}{3} \zeta_3
\Bigg\}
+ \frac{1}{\epsilon} \log\left(\frac{q^2}{\mu_F^2}\right) n_f^2
C_A \Bigg\{
- \frac{16}{81}
\Bigg\}
\nonumber\\
& + C_A^3 \Bigg\{
\frac{5211949}{26244}
- \frac{484}{9} \zeta_5
- \frac{64886}{243} \zeta_3
+ \frac{536}{9} \zeta_3^2
- \frac{299425}{1458} \zeta_2
+ \frac{286}{3} \zeta_2 \zeta_3
+ \frac{691}{135} \zeta_2^2
\nonumber\\
& + \frac{17392}{945} \zeta_2^3
\Bigg\}
+ n_f C_A^2 \Bigg\{
- \frac{412765}{13122}
- \frac{8}{3} \zeta_5
+ \frac{2920}{81} \zeta_3
+ \frac{38237}{729} \zeta_2
- 4 \zeta_2 \zeta_3
\nonumber\\
& - \frac{1064}{135} \zeta_2^2
\Bigg\}
+ n_f C_F C_A \Bigg\{
- \frac{42727}{972}
+ \frac{112}{9} \zeta_5
+ \frac{1636}{81} \zeta_3
+ \frac{275}{18} \zeta_2
- \frac{40}{3} \zeta_2 \zeta_3
\nonumber\\
& + \frac{152}{45} \zeta_2^2
\Bigg\}
+ n_f^2 C_A \Bigg\{
- \frac{128}{6561}
- \frac{40}{243} \zeta_3
- \frac{68}{27} \zeta_2
+ \frac{124}{135} \zeta_2^2
\Bigg\}
\nonumber\\
& + \log\left(\frac{q^2}{\mu_F^2}\right) C_A^3 \Bigg\{
- \frac{297029}{1458}
- 96 \zeta_5
+ \frac{7132}{27} \zeta_3
+ \frac{13876}{81} \zeta_2
- \frac{88}{3} \zeta_2 \zeta_3
- \frac{308}{15} \zeta_2^2
\Bigg\}
\nonumber\\
& + \log\left(\frac{q^2}{\mu_F^2}\right) n_f C_A^2
\Bigg\{
\frac{31313}{729}
- \frac{268}{9} \zeta_3
- \frac{3880}{81} \zeta_2
+ \frac{104}{15} \zeta_2^2
\Bigg\}
\nonumber\\
& + \log\left(\frac{q^2}{\mu_F^2}\right) n_f C_F C_A
\Bigg\{
\frac{1711}{54}
- \frac{152}{9} \zeta_3
- 4 \zeta_2
- \frac{16}{5} \zeta_2^2
\Bigg\}
+ \log\left(\frac{q^2}{\mu_F^2}\right) n_f^2 C_A
\Bigg\{
- \frac{928}{729}
\nonumber\\
& - \frac{16}{27} \zeta_3
+ \frac{80}{27} \zeta_2
\Bigg\}
+ \log^2\left(\frac{q^2}{\mu_F^2}\right) C_A^3 \Bigg\{
\frac{15503}{162}
- 44 \zeta_3
- \frac{170}{3} \zeta_2
+ \frac{44}{5} \zeta_2^2
\Bigg\}
\nonumber\\
& + \log^2\left(\frac{q^2}{\mu_F^2}\right) n_f C_A^2
\Bigg\{
- \frac{2051}{81}
+ \frac{128}{9} \zeta_2
\Bigg\}
+ \log^2\left(\frac{q^2}{\mu_F^2}\right) n_f C_F C_A
\Bigg\{
- \frac{55}{6}
+ 8 \zeta_3
\Bigg\}
\nonumber\\
& + \log^2\left(\frac{q^2}{\mu_F^2}\right) n_f^2 C_A
\Bigg\{
\frac{100}{81}
- \frac{8}{9} \zeta_2
\Bigg\}
+ \log^3\left(\frac{q^2}{\mu_F^2}\right) C_A^3 \Bigg\{
- \frac{1780}{81}
+ \frac{44}{9} \zeta_2
\Bigg\}
\nonumber\\
& + \log^3\left(\frac{q^2}{\mu_F^2}\right) n_f C_A^2
\Bigg\{
\frac{578}{81}
- \frac{8}{9} \zeta_2
\Bigg\}
+ \log^3\left(\frac{q^2}{\mu_F^2}\right) n_f C_F C_A
\Bigg\{
\frac{2}{3}
\Bigg\}
\nonumber\\
& + \log^3\left(\frac{q^2}{\mu_F^2}\right) n_f^2 C_A
\Bigg\{
- \frac{40}{81}
\Bigg\}
+ \log\left(\frac{q^2}{\mu_F^2}\right)^4 C_A^3 \Bigg\{
\frac{121}{54}
\Bigg\}
+ \log\left(\frac{q^2}{\mu_F^2}\right)^4 n_f C_A^2
\Bigg\{
- \frac{22}{27}
\Bigg\}
\nonumber\\
& + \log\left(\frac{q^2}{\mu_F^2}\right)^4 n_f^2 C_A
\Bigg\{
\frac{2}{27}
\Bigg\}
\Bigg]
\nonumber\\
&
+ {\cal D}_0 \delta(1-z_2) \Bigg[ \frac{1}{\epsilon^3} C_A^3 \Bigg\{
\frac{1936}{27}
\Bigg\}
+ \frac{1}{\epsilon^3} n_f C_A^2 \Bigg\{
- \frac{704}{27}
\Bigg\}
+ \frac{1}{\epsilon^3} n_f^2 C_A \Bigg\{
\frac{64}{27}
\Bigg\}
\nonumber\\
& + \frac{1}{\epsilon^2} C_A^3 \Bigg\{
\frac{8344}{81}
- \frac{176}{9} \zeta_2
\Bigg\}
+ \frac{1}{\epsilon^2} n_f C_A^2 \Bigg\{
- \frac{2672}{81}
+ \frac{32}{9} \zeta_2
\Bigg\}
+ \frac{1}{\epsilon^2} n_f C_F C_A \Bigg\{
- \frac{16}{3}
\Bigg\}
\nonumber\\
& + \frac{1}{\epsilon^2} n_f^2 C_A \Bigg\{
\frac{160}{81}
\Bigg\}
+ \frac{1}{\epsilon} C_A^3 \Bigg\{
\frac{490}{9}
+ \frac{88}{9} \zeta_3
- \frac{1072}{27} \zeta_2
+ \frac{176}{15} \zeta_2^2
\Bigg\}
\nonumber\\
& + \frac{1}{\epsilon} n_f C_A^2 \Bigg\{
- \frac{836}{81}
- \frac{112}{9} \zeta_3
+ \frac{160}{27} \zeta_2
\Bigg\}
+ \frac{1}{\epsilon} n_f C_F C_A \Bigg\{
- \frac{110}{9}
+ \frac{32}{3} \zeta_3
\Bigg\}
\nonumber\\
& + \frac{1}{\epsilon} n_f^2 C_A \Bigg\{
- \frac{16}{81}
\Bigg\}
+ C_A^3 \Bigg\{
- \frac{297029}{1458}
- 96 \zeta_5
+ \frac{7132}{27} \zeta_3
+ \frac{13876}{81} \zeta_2
- \frac{88}{3} \zeta_2 \zeta_3
\nonumber\\
& - \frac{308}{15} \zeta_2^2
\Bigg\}
+ n_f C_A^2 \Bigg\{
\frac{31313}{729}
- \frac{268}{9} \zeta_3
- \frac{3880}{81} \zeta_2
+ \frac{104}{15} \zeta_2^2
\Bigg\}
+ n_f C_F C_A \Bigg\{
\frac{1711}{54}
\nonumber\\
& - \frac{152}{9} \zeta_3
- 4 \zeta_2
- \frac{16}{5} \zeta_2^2
\Bigg\}
+ n_f^2 C_A \Bigg\{
- \frac{928}{729}
- \frac{16}{27} \zeta_3
+ \frac{80}{27} \zeta_2
\Bigg\}
\nonumber\\
& + \log\left(\frac{q^2}{\mu_F^2}\right) C_A^3 \Bigg\{
\frac{15503}{81}
- 88 \zeta_3
- \frac{340}{3} \zeta_2
+ \frac{88}{5} \zeta_2^2
\Bigg\}
+ \log\left(\frac{q^2}{\mu_F^2}\right) n_f C_A^2 \Bigg\{
- \frac{4102}{81}
\nonumber\\
& + \frac{256}{9} \zeta_2
\Bigg\}
+ \log\left(\frac{q^2}{\mu_F^2}\right) n_f C_F C_A \Bigg\{
- \frac{55}{3}
+ 16 \zeta_3
\Bigg\}
+ \log\left(\frac{q^2}{\mu_F^2}\right) n_f^2 C_A \Bigg\{
\frac{200}{81}
- \frac{16}{9} \zeta_2
\Bigg\}
\nonumber\\
& + \log^2\left(\frac{q^2}{\mu_F^2}\right) C_A^3 \Bigg\{
- \frac{1780}{27}
+ \frac{44}{3} \zeta_2
\Bigg\}
+ \log^2\left(\frac{q^2}{\mu_F^2}\right) n_f C_A^2 \Bigg\{
\frac{578}{27}
- \frac{8}{3} \zeta_2
\Bigg\}
\nonumber\\
& + \log^2\left(\frac{q^2}{\mu_F^2}\right) n_f C_F C_A \Bigg\{
2
\Bigg\}
+ \log^2\left(\frac{q^2}{\mu_F^2}\right) n_f^2 C_A \Bigg\{
- \frac{40}{27}
\Bigg\}
+ \log^3\left(\frac{q^2}{\mu_F^2}\right) C_A^3 \Bigg\{
\frac{242}{27}
\Bigg\}
\nonumber\\
& + \log^3\left(\frac{q^2}{\mu_F^2}\right) n_f C_A^2 \Bigg\{
- \frac{88}{27}
\Bigg\}
+ \log^3\left(\frac{q^2}{\mu_F^2}\right) n_f^2 C_A \Bigg\{
\frac{8}{27}
\Bigg\}
\Bigg]
\nonumber\\
& + {\cal D}_0 \overline{\cal D}_0 \Bigg[ C_A^3 \Bigg\{
+ \frac{15503}{81}
- 88 \zeta_3
- \frac{340}{3} \zeta_2
+ \frac{88}{5} \zeta_2^2
\Bigg\}
+ n_f C_A^2 \Bigg\{
- \frac{4102}{81}
+ \frac{256}{9} \zeta_2
\Bigg\}
\nonumber\\
& + n_f C_F C_A \Bigg\{
- \frac{55}{3}
+ 16 \zeta_3
\Bigg\}
+ n_f^2 C_A \Bigg\{
\frac{200}{81}
- \frac{16}{9} \zeta_2
\Bigg\}
+ \log\left(\frac{q^2}{\mu_F^2}\right) C_A^3 \Bigg\{
- \frac{3560}{27}
\nonumber\\
& + \frac{88}{3} \zeta_2
\Bigg\}
+ \log\left(\frac{q^2}{\mu_F^2}\right) n_f C_A^2 \Bigg\{
\frac{1156}{27}
- \frac{16}{3} \zeta_2
\Bigg\}
+ \log\left(\frac{q^2}{\mu_F^2}\right) n_f C_F C_A \Bigg\{
4
\Bigg\}
\nonumber\\
& + \log\left(\frac{q^2}{\mu_F^2}\right) n_f^2 C_A \Bigg\{
- \frac{80}{27}
\Bigg\}
+ \log^2\left(\frac{q^2}{\mu_F^2}\right) C_A^3 \Bigg\{
\frac{242}{9}
\Bigg\}
+ \log^2\left(\frac{q^2}{\mu_F^2}\right) n_f C_A^2 \Bigg\{
- \frac{88}{9}
\Bigg\}
\nonumber\\
& + \log^2\left(\frac{q^2}{\mu_F^2}\right) n_f^2 C_A \Bigg\{
\frac{8}{9}
\Bigg\}
\Bigg]
+ {\cal D}_0 \overline{\cal D}_1 \Bigg[ C_A^3 \Bigg\{
- \frac{3560}{27}
+ \frac{88}{3} \zeta_2
\Bigg\}
+ n_f C_A^2 \Bigg\{
\frac{1156}{27}
- \frac{16}{3} \zeta_2
\Bigg\}
\nonumber\\
& + n_f C_F C_A \Bigg\{
4
\Bigg\}
+ n_f^2 C_A \Bigg\{
- \frac{80}{27}
\Bigg\}
+ \log\left(\frac{q^2}{\mu_F^2}\right) C_A^3 \Bigg\{
\frac{484}{9}
\Bigg\}
+ \log\left(\frac{q^2}{\mu_F^2}\right) n_f C_A^2 \Bigg\{
- \frac{176}{9}
\Bigg\}
\nonumber\\
& + \log\left(\frac{q^2}{\mu_F^2}\right) n_f^2 C_A \Bigg\{
\frac{16}{9}
\Bigg\}
\Bigg]
+ {\cal D}_0 \overline{\cal D}_2 \Bigg[ C_A^3 \Bigg\{
\frac{242}{9}
\Bigg\}
+ n_f C_A^2 \Bigg\{
- \frac{88}{9}
\Bigg\}
+ n_f^2 C_A \Bigg\{
\frac{8}{9}
\Bigg\}
\Bigg]
\nonumber\\
& + {\cal D}_1 \delta(1-z_2) \Bigg[ C_A^3 \Bigg\{
\frac{15503}{81}
- 88 \zeta_3
- \frac{340}{3} \zeta_2
+ \frac{88}{5} \zeta_2^2
\Bigg\}
+ n_f C_A^2 \Bigg\{
- \frac{4102}{81}
+ \frac{256}{9} \zeta_2
\Bigg\}
\nonumber\\
& + n_f C_F C_A \Bigg\{
- \frac{55}{3}
+ 16 \zeta_3
\Bigg\}
+ n_f^2 C_A \Bigg\{
\frac{200}{81}
- \frac{16}{9} \zeta_2
\Bigg\}
+ \log\left(\frac{q^2}{\mu_F^2}\right) C_A^3 \Bigg\{
- \frac{3560}{27}
\nonumber\\
& + \frac{88}{3} \zeta_2
\Bigg\}
+ \log\left(\frac{q^2}{\mu_F^2}\right) n_f C_A^2 \Bigg\{
\frac{1156}{27}
- \frac{16}{3} \zeta_2
\Bigg\}
+ \log\left(\frac{q^2}{\mu_F^2}\right) n_f C_F C_A \Bigg\{
4
\Bigg\}
\nonumber\\
& + \log\left(\frac{q^2}{\mu_F^2}\right) n_f^2 C_A \Bigg\{
- \frac{80}{27}
\Bigg\}
+ \log^2\left(\frac{q^2}{\mu_F^2}\right) C_A^3 \Bigg\{
\frac{242}{9}
\Bigg\}
+ \log^2\left(\frac{q^2}{\mu_F^2}\right) n_f C_A^2 \Bigg\{
- \frac{88}{9}
\Bigg\}
\nonumber\\
& + \log^2\left(\frac{q^2}{\mu_F^2}\right) n_f^2 C_A \Bigg\{
+ \frac{8}{9}
\Bigg\}
\Bigg]
+ {\cal D}_1 \overline{\cal D}_0 \Bigg[ C_A^3 \Bigg\{
- \frac{3560}{27}
+ \frac{88}{3} \zeta_2
\Bigg\}
+ n_f C_A^2 \Bigg\{
\frac{1156}{27}
\nonumber\\
& - \frac{16}{3} \zeta_2
\Bigg\}
+ n_f C_F C_A \Bigg\{
4
\Bigg\}
+ n_f^2 C_A \Bigg\{
- \frac{80}{27}
\Bigg\}
+ \log\left(\frac{q^2}{\mu_F^2}\right) C_A^3 \Bigg\{
\frac{484}{9}
\Bigg\}
\nonumber\\
& + \log\left(\frac{q^2}{\mu_F^2}\right) n_f C_A^2 \Bigg\{
- \frac{176}{9}
\Bigg\}
+ \log\left(\frac{q^2}{\mu_F^2}\right) n_f^2 C_A \Bigg\{
\frac{16}{9}
\Bigg\}
\Bigg]
+ {\cal D}_1 \overline{\cal D}_1 \Bigg[ C_A^3 \Bigg\{
\frac{484}{9}
\Bigg\}
\nonumber\\
& + n_f C_A^2 \Bigg\{
- \frac{176}{9}
\Bigg\}
+ n_f^2 C_A \Bigg\{
\frac{16}{9}
\Bigg\}
\Bigg]
+ {\cal D}_2 \delta(1-z_2) \Bigg[ C_A^3 \Bigg\{
- \frac{1780}{27}
+ \frac{44}{3} \zeta_2
\Bigg\}
\nonumber\\
& + n_f C_A^2 \Bigg\{
\frac{578}{27}
- \frac{8}{3} \zeta_2
\Bigg\}
+ n_f C_F C_A \Bigg\{
2
\Bigg\}
+ n_f^2 C_A \Bigg\{
- \frac{40}{27}
\Bigg\}
\nonumber\\
& + \log\left(\frac{q^2}{\mu_F^2}\right) C_A^3 \Bigg\{
\frac{242}{9}
\Bigg\}
+ \log\left(\frac{q^2}{\mu_F^2}\right) n_f C_A^2 \Bigg\{
- \frac{88}{9}
\Bigg\}
+ \log\left(\frac{q^2}{\mu_F^2}\right) n_f^2 C_A \Bigg\{
\frac{8}{9}
\Bigg\}
\Bigg]
\nonumber\\
& + {\cal D}_2 \overline{\cal D}_0 \Bigg[ C_A^3 \Bigg\{
\frac{242}{9}
\Bigg\}
+ n_f C_A^2 \Bigg\{
- \frac{88}{9}
\Bigg\}
+ n_f^2 C_A \Bigg\{
\frac{8}{9}
\Bigg\}
\Bigg]
\nonumber\\
& + {\cal D}_3 \delta(1-z_2) \Bigg[ C_A^3 \Bigg\{
\frac{242}{27}
\Bigg\}
+ n_f C_A^2 \Bigg\{
- \frac{88}{27}
\Bigg\}
+ n_f^2 C_A \Bigg\{
\frac{8}{27}
\Bigg\}
\Bigg]
\nonumber\\
& + \overline{\cal D}_0 \delta(1-z_1) \Bigg[ \frac{1}{\epsilon^3} C_A^3 \Bigg\{
\frac{1936}{27}
\Bigg\}
+ \frac{1}{\epsilon^3} n_f C_A^2 \Bigg\{
- \frac{704}{27}
\Bigg\}
+ \frac{1}{\epsilon^3} n_f^2 C_A \Bigg\{
\frac{64}{27}
\Bigg\}
\nonumber\\
& + \frac{1}{\epsilon^2} C_A^3 \Bigg\{
\frac{8344}{81}
- \frac{176}{9} \zeta_2
\Bigg\}
+ \frac{1}{\epsilon^2} n_f C_A^2 \Bigg\{
- \frac{2672}{81}
+ \frac{32}{9} \zeta_2
\Bigg\}
+ \frac{1}{\epsilon^2} n_f C_F C_A \Bigg\{
- \frac{16}{3}
\Bigg\}
\nonumber\\
& + \frac{1}{\epsilon^2} n_f^2 C_A \Bigg\{
\frac{160}{81}
\Bigg\}
+ \frac{1}{\epsilon} C_A^3 \Bigg\{
\frac{490}{9}
+ \frac{88}{9} \zeta_3
- \frac{1072}{27} \zeta_2
+ \frac{176}{15} \zeta_2^2
\Bigg\}
\nonumber\\
& + \frac{1}{\epsilon} n_f C_A^2 \Bigg\{
- \frac{836}{81}
- \frac{112}{9} \zeta_3
+ \frac{160}{27} \zeta_2
\Bigg\}
+ \frac{1}{\epsilon} n_f C_F C_A \Bigg\{
- \frac{110}{9}
+ \frac{32}{3} \zeta_3
\Bigg\}
\nonumber\\
& + \frac{1}{\epsilon} n_f^2 C_A \Bigg\{
- \frac{16}{81}
\Bigg\}
+ C_A^3 \Bigg\{
- \frac{297029}{1458}
- 96 \zeta_5
+ \frac{7132}{27} \zeta_3
+ \frac{13876}{81} \zeta_2
- \frac{88}{3} \zeta_2 \zeta_3
\nonumber\\
& - \frac{308}{15} \zeta_2^2
\Bigg\}
+ n_f C_A^2 \Bigg\{
\frac{31313}{729}
- \frac{268}{9} \zeta_3
- \frac{3880}{81} \zeta_2
+ \frac{104}{15} \zeta_2^2
\Bigg\}
+ n_f C_F C_A \Bigg\{
\frac{1711}{54}
\nonumber\\
& - \frac{152}{9} \zeta_3
- 4 \zeta_2
- \frac{16}{5} \zeta_2^2
\Bigg\}
+ n_f^2 C_A \Bigg\{
- \frac{928}{729}
- \frac{16}{27} \zeta_3
+ \frac{80}{27} \zeta_2
\Bigg\}
\nonumber\\
& + \log\left(\frac{q^2}{\mu_F^2}\right) C_A^3 \Bigg\{
\frac{15503}{81}
- 88 \zeta_3
- \frac{340}{3} \zeta_2
+ \frac{88}{5} \zeta_2^2
\Bigg\}
+ \log\left(\frac{q^2}{\mu_F^2}\right) n_f C_A^2 \Bigg\{
- \frac{4102}{81}
\nonumber\\
& + \frac{256}{9} \zeta_2
\Bigg\}
+ \log\left(\frac{q^2}{\mu_F^2}\right) n_f C_F C_A \Bigg\{
- \frac{55}{3}
+ 16 \zeta_3
\Bigg\}
+ \log\left(\frac{q^2}{\mu_F^2}\right) n_f^2 C_A \Bigg\{
\frac{200}{81}
- \frac{16}{9} \zeta_2
\Bigg\}
\nonumber\\
& + \log^2\left(\frac{q^2}{\mu_F^2}\right) C_A^3 \Bigg\{
- \frac{1780}{27}
+ \frac{44}{3} \zeta_2
\Bigg\}
+ \log^2\left(\frac{q^2}{\mu_F^2}\right) n_f C_A^2 \Bigg\{
\frac{578}{27}
- \frac{8}{3} \zeta_2
\Bigg\}
\nonumber\\
& + \log^2\left(\frac{q^2}{\mu_F^2}\right) n_f C_F C_A \Bigg\{
2
\Bigg\}
+ \log^2\left(\frac{q^2}{\mu_F^2}\right) n_f^2 C_A \Bigg\{
- \frac{40}{27}
\Bigg\}
+ \log^3\left(\frac{q^2}{\mu_F^2}\right) C_A^3 \Bigg\{
\frac{242}{27}
\Bigg\}
\nonumber\\
& + \log^3\left(\frac{q^2}{\mu_F^2}\right) n_f C_A^2 \Bigg\{
- \frac{88}{27}
\Bigg\}
+ \log^3\left(\frac{q^2}{\mu_F^2}\right) n_f^2 C_A \Bigg\{
\frac{8}{27}
\Bigg\}
\Bigg]
\nonumber\\
& + \overline{\cal D}_1 \delta(1-z_1) \Bigg[ C_A^3 \Bigg\{
\frac{15503}{81}
- 88 \zeta_3
- \frac{340}{3} \zeta_2
+ \frac{88}{5} \zeta_2^2
\Bigg\}
+ n_f C_A^2 \Bigg\{
- \frac{4102}{81}
+ \frac{256}{9} \zeta_2
\Bigg\}
\nonumber\\
& + n_f C_F C_A \Bigg\{
- \frac{55}{3}
+ 16 \zeta_3
\Bigg\}
+ n_f^2 C_A \Bigg\{
\frac{200}{81}
- \frac{16}{9} \zeta_2
\Bigg\}
+ \log\left(\frac{q^2}{\mu_F^2}\right) C_A^3 \Bigg\{
- \frac{3560}{27}
\nonumber\\
& + \frac{88}{3} \zeta_2
\Bigg\}
+ \log\left(\frac{q^2}{\mu_F^2}\right) n_f C_A^2 \Bigg\{
\frac{1156}{27}
- \frac{16}{3} \zeta_2
\Bigg\}
+ \log\left(\frac{q^2}{\mu_F^2}\right) n_f C_F C_A \Bigg\{
4
\Bigg\}
\nonumber\\
& + \log\left(\frac{q^2}{\mu_F^2}\right) n_f^2 C_A \Bigg\{
- \frac{80}{27}
\Bigg\}
+ \log^2\left(\frac{q^2}{\mu_F^2}\right) C_A^3 \Bigg\{
\frac{242}{9}
\Bigg\}
+ \log^2\left(\frac{q^2}{\mu_F^2}\right) n_f C_A^2 \Bigg\{
- \frac{88}{9}
\Bigg\}
\nonumber\\
& + \log^2\left(\frac{q^2}{\mu_F^2}\right) n_f^2 C_A \Bigg\{
\frac{8}{9}
\Bigg\}
\Bigg]
+ \overline{\cal D}_2 \delta(1-z_1) \Bigg[ C_A^3 \Bigg\{
- \frac{1780}{27}
+ \frac{44}{3} \zeta_2
\Bigg\}
+ n_f C_A^2 \Bigg\{
\frac{578}{27}
\nonumber\\
& - \frac{8}{3} \zeta_2
\Bigg\}
+ n_f C_F C_A \Bigg\{
2
\Bigg\}
+ n_f^2 C_A \Bigg\{
- \frac{40}{27}
\Bigg\}
+ \log\left(\frac{q^2}{\mu_F^2}\right) C_A^3 \Bigg\{
\frac{242}{9}
\Bigg\}
\nonumber\\
& + \log\left(\frac{q^2}{\mu_F^2}\right) n_f C_A^2 \Bigg\{
- \frac{88}{9}
\Bigg\}
+ \log\left(\frac{q^2}{\mu_F^2}\right) n_f^2 C_A \Bigg\{
\frac{8}{9}
\Bigg\}
\Bigg]
+ \overline{\cal D}_3 \delta(1-z_1) \Bigg[ C_A^3 \Bigg\{
\frac{242}{27}
\Bigg\}
\nonumber\\
& + n_f C_A^2 \Bigg\{
- \frac{88}{27}
\Bigg\}
+ n_f^2 C_A \Bigg\{
\frac{8}{27}
\Bigg\}
\Bigg]\,.
\end{align}
\chapter{Soft-Collinear Distribution}
\label{chpt:App-Soft-Col-Dist}
In Sec.~\ref{ss:bBH-SCD}, we introduced the soft-collinear
distribution $\Phi^H_{b{\bar b}}$ in the context of computing SV
cross section of the Higgs boson production in $b{\bar b}$
annihilation. In this appendix, we intend to elaborate the methodology
of finding this distribution. For the sake of generalisation, we
use $I$ instead of $H$ and omit the partonic indices. To understand
the underlying logics behind finding $\Phi^{I}$, let us consider an
example at one loop level. The generalisation to higher loop is straightforward.
As discussed in the Sec.~\ref{sec:bBH-ThreResu}, the SV cross section in
$z$-space can be computed in $d=4+\epsilon$
dimensions using
\begin{align}
\label{eq:App-sigma}
\Delta^{I, \text{SV}} (z, q^2, \mu_{R}^{2}, \mu_F^2) =
{\cal C} \exp \Big( \Psi^I \left(z, q^2, \mu_R^2, \mu_F^2, \epsilon
\right) \Big) \Big|_{\epsilon = 0}
\end{align}
where, $\Psi^I \left(z, q^2, \mu_R^2, \mu_F^2, \epsilon \right)$ is a
finite distribution and ${\cal C}$ is the convolution defined
through Eq.~(\ref{eq:bBH-conv}). The $\Psi^I$ is given by, Eq.~(\ref{eq:bBH-psi})
\begin{align}
\label{eq:App-psi}
\Psi^{I} \left(z, q^2, \mu_R^2, \mu_F^2, \epsilon \right) = &\left(
\ln
\Big[
Z^I
(\hat{a}_s,
\mu_R^2,
\mu^2,
\epsilon)
\Big]^2
+ \ln \Big| {\cal F}^I (\hat{a}_s, Q^2, \mu^2, \epsilon) \Big|^2
\right)
\delta(1-z)
\nonumber\\
& + 2 \Phi^I (\hat{a}_s, q^2, \mu^2, z, \epsilon) - 2 {\cal C} \ln
\Gamma^{I} (\hat{a}_s, \mu^2, \mu_F^2, z, \epsilon) \, .
\end{align}
For all the details about the notations, see
Sec.~\ref{sec:bBH-ThreResu}. Considering only the poles at ${\cal O}(a_{s})$ with $\mu_{R} =
\mu_{F}$ we obtain,
\begin{align}
\label{eq:App-Psi-Comp-1}
&\ln\left(Z^{I, (1)}\right)^{2} = a_{s}(\mu_{F}^{2}) \frac{4\gamma_{1}^{I}}{\epsilon}\,,
\nonumber\\
&\ln|{\cal F}^{I, (1)}|^{2} = a_{s}(\mu_{F}^{2}) \left(
\frac{q^{2}}{\mu_{F}^{2}} \right)^{\frac{\epsilon}{2}} \left[
-\frac{4A_{1}^{I}}{\epsilon^{2}} + \frac{1}{\epsilon} \left( 2f_{1}^{I}
+ 4B_{1}^{I} - 4\gamma_{1}^{I}\right)\right]\,,
\nonumber\\
&2{\cal C} \ln\Gamma^{I, (1)} = 2 a_{s}(\mu_{F}^{2}) \left[
\frac{2B^{I}_{1}}{\epsilon} \delta(1-z) +
\frac{2A_{1}^{I}}{\epsilon} {\cal D}_{0}\right]
\end{align}
where, the components are defined through the expansion of these
quantities in powers of $a_s(\mu_F^{2})$
\begin{align}
\label{eq:App-psi-comp}
&\Psi^{I} = \sum\limits_{k=1}^{\infty} a_s^k \left( \mu_F^2 \right) \Psi^{I,(k)}\,,
\nonumber\\
&\ln (Z^I)^2 = \sum\limits_{k=1}^{\infty} a_s^k \left( \mu_F^2 \right)
Z^{I,(k)}\,,
\nonumber\\
&\ln |{\cal F}^{I}|^2 = \sum\limits_{k=1}^{\infty} a_s^k\left( \mu_F^2
\right) \left( \frac{q^2}{\mu_F^2} \right)^{k \frac{\epsilon}{2}}
\ln |{\cal F}^{I,(k)}|^2\,,
\nonumber\\
&\Phi^{I} = \sum\limits_{k=1}^{\infty} a_s^k(\mu_F^2) \Phi^I_k\,,
\nonumber\\
&\ln \Gamma^I = \sum\limits_{k=1}^{\infty} a_s^k(\mu_F^2) \ln
\Gamma^{I,(k)}
\intertext{and}
&
{\cal{D}}_{i} \equiv \left[ \frac{\ln^{i}(1-z)}{1-z} \right]_{+}\, .
\end{align}
Collecting the coefficients of $a_s(\mu_F^2)$, we
get
\begin{align}
\label{eq:App-Psi-1}
\Psi^{I, (1)}|_{\rm poles}
&= \left[ {\left\{ - \frac{4A_{1}^{I}}{\epsilon^{2}} + \frac{ 2f_{1}^{I}}{\epsilon} \right\} \delta(1-z)} - {\frac{4A_{1}^{I}}{\epsilon} {\cal D}_{0}} \right] + {2\Phi^{I}_{1}}
\end{align}
where, we have not shown the $\ln (q^2/\mu_F^2)$ terms.
To cancel the remaining divergences appearing in the above
Eq.~(\ref{eq:App-Psi-1}) for obtaining a finite cross section, we must
demand that $\Phi^{I}_{1}$ have exactly the same poles with
opposite sign:
\begin{align}
\label{eq:App-Phi-1}
2 \Phi^{I}_{1}|_{\rm poles} = - \left[ {\left\{ - \frac{4A_{1}^{I}}{\epsilon^{2}} + \frac{ 2f_{1}^{I}}{\epsilon} \right\} \delta(1-z)} - {\frac{4A_{1}^{I}}{\epsilon} {\cal D}_{0}} \right]
\end{align}
In addition, $\Phi^{I}$ also should be RG invariant with respect to
$\mu_R$:
\begin{align}
\label{eq:App-Phi-2}
\mu_R^2 \frac{d}{d\mu_R^2} \Phi^I = 0\,.
\end{align}
We make an \textit{ansatz}, the above two demands,
Eq.~(\ref{eq:App-Phi-1}) and~(\ref{eq:App-Phi-2}) can be accomplished
if $\Phi^I$ satisfies the KG-type integro-differential equation which we call
$\overline{KG}$:
\begin{align}
\label{eq:App-KGbarEqn}
q^2 \frac{d}{dq^2} \Phi^I\left(\hat{a}_s, q^2, \mu^2, z,
\epsilon\right) = \frac{1}{2} \left[ \overline K^I
\left(\hat{a}_s, \frac{\mu_R^2}{\mu^2}, z,
\epsilon \right) + \overline G^I \left(\hat{a}_s,
\frac{q^2}{\mu_R^2}, \frac{\mu_R^2}{\mu^2}, z, \epsilon \right) \right]\,.
\end{align}
$\overline{K}^I$ contains all the poles whereas $\overline{G}^I$ consists
of only the finite terms in $\epsilon$. RG
invariance~(\ref{eq:App-Phi-2}) of $\Phi^I$ dictates
\begin{align}
\label{eq:App-Phi-3}
\mu_R^2 \frac{d}{d\mu_R^2} \overline{K}^{I} = -\mu_R^2 \frac{d}{d\mu_R^2}
\overline{G}^I \equiv Y^{I}
\end{align}
where, we introduce a quantity $Y^I$. Following the methodology of
solving the KG equation discussed in the
Appendix~\ref{chpt:App-KGSoln}, we can write the solution of $\Phi^I$
as
\begin{align}
\label{eq:App-Phi-4}
\Phi^I(\hat{a}_s, q^2, \mu^2, z, \epsilon) =
\sum_{k=1}^{\infty} {\hat a}_{s}^{k}S_{\epsilon}^{k} \left(\frac{q^{2}}{\mu^{2}}\right)^{k
\frac{\epsilon}{2}} \hat {\Phi}^{I}_{k}(z,\epsilon)
\end{align}
with
\begin{align}
\label{eq:App-Phi-5}
\hat {\Phi}^{I}_{k}(z,\epsilon) = {\hat{\cal L}}^I_k \left( A^I_{i}
\rightarrow Y^I_{i}, G^I_i \rightarrow \overline{G}^I_i(z,\epsilon) \right)\,.
\end{align}
where we define the components through the expansions
\begin{align}
\label{eq:App-YGexpans}
&Y^I = \sum\limits_{k=1}^{\infty} a_s^k(\mu_F^2) Y^I_k\,,
\nonumber\\
&\overline{G}^I(z,\epsilon) = \sum\limits_{k=1}^{\infty}
a_s^k(\mu_F^2) \overline{G}^I_{k}(z,\epsilon).
\end{align}
This solution directly follows from the Eq.~(\ref{eq:App-lnFSoln}). Hence we get
\begin{align}
\label{eq:App-Phi-6}
2 \hat {\Phi}^{I}_{1}(z,\epsilon) &= { \frac{1}{\epsilon^2} } \Bigg\{-4 Y^{I}_{1}\Bigg\}
+ { \frac{2}{\epsilon}
}
\Bigg\{\overline{G}^{I}_{1}
(z,\epsilon)\Bigg\}\,.
\end{align}
By expressing the components of $\Phi^I$ in powers of $a_s(\mu_F^2)$,
we obtain
\begin{align}
\label{eq:App-Phi-7}
\Phi^I(\hat{a}_s, q^2, \mu^2, z, \epsilon) &=
\sum_{k=1}^{\infty} {\hat a}_{s}^{k}S_{\epsilon}^{k} \left(\frac{q^{2}}{\mu^{2}}\right)^{k
\frac{\epsilon}{2}} \hat {\Phi}^{I}_{k}(z,\epsilon)
\nonumber\\
&= \sum_{k=1}^{\infty} a_s^k(\mu_F^2) \left( \frac{q^2}{\mu_F^2} \right)^{k
\frac{\epsilon}{2}} Z_{a_s}^k \hat{\Phi}^I_k(z,\epsilon)
\nonumber\\
& \equiv \sum_{k=1}^{\infty} a_s^k(\mu_F^2) \left( \frac{q^2}{\mu_F^2} \right)^{k
\frac{\epsilon}{2}} {\Phi}^I_{k}(z,\epsilon)
\end{align}
and at ${\cal O}(a_s(\mu_F^2)), \hat{\Phi}^I_1(z,\epsilon) =
\Phi^I_{1}(z,\epsilon)$ upon suppressing the terms like $\log (q^{2}/\mu_{F}^{2})$. Hence, by comparing the
Eq.~(\ref{eq:App-Phi-1}) and (\ref{eq:App-Phi-6}), we conclude
\begin{align}
\label{eq:App-Phi-8}
&Y^I_1 = -A^I_1 \delta(1-z)\,,
\nonumber\\
&\overline{G}^I_1(z,\epsilon) = -f^I_1 \delta(1-z) + 2 A^I_1 {\cal
D}_0 + \sum\limits_{k=1}^{\infty} \epsilon^k \overline{g}^{I,k}_1(z)\,.
\end{align}
The coefficients of $\epsilon^k, \overline{g}^{I,k}_1(z)$ can only be
determined through explicit computations. These do not contribute to
the infrared poles associated with $\Phi^{I}$. This uniquely fixes the
unknown soft-collinear distribution $\Phi^I$ at one loop order. This
prescription can easily be generalised to higher orders in $a_s$. In
our calculation of the SV cross section, instead of solving in this way,
we follow a bit different methodology which is presented below.
Keeping the demands~(\ref{eq:App-Phi-1}) and~(\ref{eq:App-Phi-2}) in
mind, we propose the solution of the $\overline{KG}$ equation as (See
Eq.~(\ref{eq:App-Phi-4}))
\begin{align}
\label{eq:App-Phi-9}
\hat{\Phi}^I_{k} (z,\epsilon) &\equiv \Bigg\{ k \epsilon \frac{1}{1-z}
\left[
(1-z)^2 \right]^{k \frac{\epsilon}{2}}\Bigg\}
\hat{\Phi}^I_k(\epsilon)
\nonumber\\
&=\Bigg\{ \delta(1-z) + \sum\limits_{j=0}^{\infty} \frac{(k
\epsilon)^{j+1}}{j!} {\cal D}_{j} \Bigg\} \hat{\Phi}^I_k(\epsilon)\,.
\end{align}
The RG invariance of $\Phi^{I}$, Eq.~(\ref{eq:App-Phi-2}), implies
\begin{align}
\label{eq:App-Phi-10}
\mu_R^2 \frac{d}{d\mu_R^2} \overline{K}^{I} = -\mu_R^2 \frac{d}{d\mu_R^2}
\overline{G}^I \equiv Y'^{I}
\end{align}
where, we introduce a quantity $Y'$, analogous to $Y$. Hence, the
solution can be obtained as
\begin{align}
\label{eq:App-Phi-11}
\hat {\Phi}^{I}_{k}(\epsilon) &= {\hat{\cal L}}^I_k \left( A^I_{i}
\rightarrow Y'^I_{i}, G^I_i \rightarrow \overline{\cal G}^I_i(\epsilon) \right)\,.
\end{align}
Hence, according to the Eq.~(\ref{eq:App-lnFitoCalLF}), for $k=1$ we get
\begin{align}
\label{eq:App-Phi-12}
2\Phi^{I}_1 (z, \epsilon) &= \Bigg\{ \frac{1}{\epsilon^2}\left(
-4Y'^I_1\right) +
\frac{2}{\epsilon}{\overline{\cal
G}}^I_1 \left(\epsilon) \right) \Bigg\}
\delta(1-z)
+ \Bigg\{ - \frac{4 Y'^I_1}{\epsilon^2}
+ \frac{2}{\epsilon} {\overline{\cal G}}^I_1(\epsilon) \Bigg\}
\sum_{j=0}^{\infty}
\frac{\epsilon^{j+1}}{j!} {\cal D}_j
\end{align}
where, $Y'^{I}$ and $\overline{\cal G}^I$ are expanded similar to
Eq.~(\ref{eq:App-YGexpans}). Comparison between the two solutions
depicted in Eq.~(\ref{eq:App-Phi-1}) and (\ref{eq:App-Phi-12}), we
can write
\begin{align}
\label{eq:App-Phi-13}
&Y'^I_1 = - A^I_1
\nonumber\\
&{\overline{\cal G}}^I_1 \left(\epsilon\right) = - f^I_1 +
\sum_{k=1}^{\infty}
\epsilon^k \overline{\cal
G}^{I, k}_1\,.
\end{align}
Explicit computation is required to determine the coefficients of
$\epsilon^k$, $\overline{\cal G}^{I,k}_{1}$. This solution is used in
Eq.~(\ref{eq:bBH-PhiSoln}) in
the context of SV cross section of Higgs boson production. The method
is generalised to higher orders in $a_s$ to obtain the results of the
soft-collinear distribution. In the next subsection, we present the
results of the soft-collinear distribution up to three loops.
\subsection{Results}
\label{app:ss-SCD-Res}
We define the renormalised components of the
$\Phi^I_{q{\bar q},k}$ through
\begin{align}
\label{eq:App-SCD-Re}
\Phi^I_{q{\bar q}}(\hat{a}_s, q^2, \mu^2, z \epsilon) &=
\sum_{k=1}^{\infty} {\hat a}_{s}^{k}S_{\epsilon}^{k} \left(\frac{q^{2}}{\mu^{2}}\right)^{k
\frac{\epsilon}{2}} \hat {\Phi}^{I}_{q{\bar q},k}(z, \epsilon)
\nonumber\\
&= \sum\limits_{k=1}^{\infty} a_s^k\left( \mu_F^2 \right) \Phi^I_{q{\bar
q},k} \left( z, \epsilon, q^2, \mu_F^2 \right)
\end{align}
where, we make the choice of the renormalisation scale
$\mu_R=\mu_F$. The $\mu_R$ dependence can be easily restored by using
the evolution equation of strong coupling constant,
Eq.~(\ref{eq:bBH-asf2asr}). Below, we present the $\Phi^I_{i~{\bar
i},k}$ for
$i~{\bar i}=q{\bar q}$ up to three
loops and the corresponding components for $i~{\bar
i}=gg$ can be obtained using maximally non-Abelian property
fulfilled by this distribution:
\begin{align}
\label{eq:App-SCD-MaxNonAbe}
\Phi^I_{gg,k} = \frac{C_A}{C_F} \Phi^I_{q{\bar q},k}\,.
\end{align}
The results are given by
\begin{align}
\label{eq:app-SCD-Res-1}
\Phi^I_{q{\bar q},1} &=
\delta(1-z) \Bigg[\frac{1}{\epsilon^2} C_F \Bigg\{ 8 \Bigg\}
+ \frac{1}{\epsilon} \log\left(\frac{q^2}{\mu_F^2}\right) C_F \Bigg\{ 4 \Bigg\}
+ C_F \Bigg\{ - 3 \zeta_2 \Bigg\}
+ \log^2\left(\frac{q^2}{\mu_F^2}\right) C_F \Bigg\{ 1 \Bigg\}
\Bigg]
\nonumber\\
&
+ {\cal D}_0 \Bigg[ \frac{1}{\epsilon} C_F \Bigg\{ 8 \Bigg\}
+ \log\left(\frac{q^2}{\mu_F^2}\right) C_F \Bigg\{ 4 \Bigg\}
\Bigg]
+ {\cal D}_1 \Bigg[ C_F \Bigg\{ 8 \Bigg\} \Bigg]\,,
\nonumber\\
\Phi^I_{q{\bar q},2} &=
\delta(1-z) \Bigg[ \frac{1}{\epsilon^3} C_F C_A \Bigg\{ 44 \Bigg\}
+ \frac{1}{\epsilon^3} n_f C_F \Bigg\{ - 8 \Bigg\}
+ \frac{1}{\epsilon^2} C_F C_A \Bigg\{ \frac{134}{9} - 4 \zeta_2 \Bigg\}
\nonumber\\
& + \frac{1}{\epsilon^2} n_f C_F \Bigg\{ - \frac{20}{9} \Bigg\}
+ \frac{1}{\epsilon^2} \log\left(\frac{q^2}{\mu_F^2}\right) C_F C_A
\Bigg\{ \frac{44}{3} \Bigg\}
+ \frac{1}{\epsilon^2} \log\left(\frac{q^2}{\mu_F^2}\right) n_f C_F
\Bigg\{ - \frac{8}{3} \Bigg\}
\nonumber\\
& + \frac{1}{\epsilon} C_F C_A \Bigg\{ - \frac{404}{27} + 14 \zeta_3 + \frac{11}{3}
\zeta_2 \Bigg\}
+ \frac{1}{\epsilon} n_f C_F \Bigg\{ \frac{56}{27} - \frac{2}{3} \zeta_2 \Bigg\}
\nonumber\\
& + \frac{1}{\epsilon} \log\left(\frac{q^2}{\mu_F^2}\right) C_F C_A
\Bigg\{ \frac{134}{9} - 4
\zeta_2 \Bigg\}
+ \frac{1}{\epsilon} \log\left(\frac{q^2}{\mu_F^2}\right) n_f C_F
\Bigg\{ - \frac{20}{9} \Bigg\}
+ C_F C_A \Bigg\{ \frac{1214}{81}
\nonumber\\
&- \frac{187}{9} \zeta_3 - \frac{469}{18}
\zeta_2 + 2 \zeta_2^2 \Bigg\}
+ n_f C_F \Bigg\{ - \frac{164}{81} + \frac{34}{9} \zeta_3 + \frac{35}{9}
\zeta_2 \Bigg\}
\nonumber\\
& + \log\left(\frac{q^2}{\mu_F^2}\right) C_F C_A \Bigg\{ - \frac{404}{27} + 14
\zeta_3 + \frac{44}{3} \zeta_2 \Bigg\}
+ \log\left(\frac{q^2}{\mu_F^2}\right) n_f C_F \Bigg\{ \frac{56}{27} - \frac{8}{3} \zeta_2
\Bigg\}
\nonumber\\
& + \log^2\left(\frac{q^2}{\mu_F^2}\right) C_F C_A \Bigg\{ \frac{67}{9} - 2 \zeta_2
\Bigg\}
+ \log^2\left(\frac{q^2}{\mu_F^2}\right) n_f C_F \Bigg\{ - \frac{10}{9} \Bigg\}
+ \log^3\left(\frac{q^2}{\mu_F^2}\right) C_F C_A \Bigg\{ - \frac{11}{9} \Bigg\}
\nonumber\\
& + \log^3\left(\frac{q^2}{\mu_F^2}\right) n_f C_F \Bigg\{ \frac{2}{9} \Bigg\}
\Bigg]
+ {\cal D}_0 \Bigg[ \frac{1}{\epsilon^2} C_F C_A \Bigg\{ \frac{88}{3} \Bigg\}
+ \frac{1}{\epsilon^2} n_f C_F \Bigg\{ - \frac{16}{3} \Bigg\}
+ \frac{1}{\epsilon} C_F C_A \Bigg\{ \frac{268}{9}
\nonumber\\
&- 8 \zeta_2 \Bigg\}
+ \frac{1}{\epsilon} n_f C_F \Bigg\{ - \frac{40}{9} \Bigg\}
+ C_F C_A \Bigg\{ - \frac{808}{27} + 28 \zeta_3 + \frac{88}{3} \zeta_2 \Bigg\}
+ n_f C_F \Bigg\{ \frac{112}{27} - \frac{16}{3} \zeta_2 \Bigg\}
\nonumber\\
& + \log\left(\frac{q^2}{\mu_F^2}\right) C_F C_A \Bigg\{ \frac{268}{9} - 8 \zeta_2 \Bigg\}
+ \log\left(\frac{q^2}{\mu_F^2}\right) n_f C_F \Bigg\{ - \frac{40}{9} \Bigg\}
+ \log^2\left(\frac{q^2}{\mu_F^2}\right) C_F C_A \Bigg\{ - \frac{22}{3} \Bigg\}
\nonumber\\
& + \log^2\left(\frac{q^2}{\mu_F^2}\right) n_f C_F \Bigg\{ \frac{4}{3} \Bigg\}
\Bigg]
+ {\cal D}_1 \Bigg[ C_F C_A \Bigg\{ \frac{536}{9} - 16 \zeta_2 \Bigg\}
+ n_f C_F \Bigg\{ - \frac{80}{9} \Bigg\}
\nonumber\\
& + \log\left(\frac{q^2}{\mu_F^2}\right) C_F C_A \Bigg\{ - \frac{88}{3} \Bigg\}
+ \log\left(\frac{q^2}{\mu_F^2}\right) n_f C_F \Bigg\{ \frac{16}{3} \Bigg\}
\Bigg]
\nonumber\\
& + {\cal D}_2 \Bigg[ C_F C_A \Bigg\{ - \frac{88}{3} \Bigg\}
+ n_f C_F \Bigg\{ \frac{16}{3} \Bigg\}
\Bigg]\,,
\nonumber\\
\Phi^I_{q{\bar q},3} &=
\delta(1-z) \Bigg[ \frac{1}{\epsilon^4} C_F C_A^2 \Bigg\{
\frac{21296}{81}
\Bigg\}
+ \frac{1}{\epsilon^4} n_f C_F C_A \Bigg\{
- \frac{7744}{81}
\Bigg\}
+ \frac{1}{\epsilon^4} n_f^2 C_F \Bigg\{
\frac{704}{81}
\Bigg\}
\nonumber\\
& + \frac{1}{\epsilon^3} C_F C_A^2 \Bigg\{
\frac{49064}{243}
- \frac{880}{27} \zeta_2
\Bigg\}
+ \frac{1}{\epsilon^3} n_f C_F C_A \Bigg\{
- \frac{15520}{243}
+ \frac{160}{27} \zeta_2
\Bigg\}
\nonumber\\
& + \frac{1}{\epsilon^3} n_f C_F^2 \Bigg\{
- \frac{128}{9}
\Bigg\}
+ \frac{1}{\epsilon^3} n_f^2 C_F \Bigg\{
\frac{800}{243}
\Bigg\}
+ \frac{1}{\epsilon^3} \log\left(\frac{q^2}{\mu_F^2}\right) C_F C_A^2 \Bigg\{
\frac{1936}{27}
\Bigg\}
\nonumber\\
& + \frac{1}{\epsilon^3} \log\left(\frac{q^2}{\mu_F^2}\right) n_f C_F C_A \Bigg\{
- \frac{704}{27}
\Bigg\}
+ \frac{1}{\epsilon^3} \log\left(\frac{q^2}{\mu_F^2}\right) n_f^2 C_F \Bigg\{
+ \frac{64}{27}
\Bigg\}
\nonumber\\
& + \frac{1}{\epsilon^2} C_F C_A^2 \Bigg\{
- \frac{8956}{243}
+ \frac{2024}{27} \zeta_3
- \frac{692}{81} \zeta_2
+ \frac{352}{45} \zeta_2^2
\Bigg\}
+ \frac{1}{\epsilon^2} n_f C_F C_A \Bigg\{
\frac{4024}{243}
- \frac{560}{27} \zeta_3
\nonumber\\
& - \frac{208}{81} \zeta_2
\Bigg\}
+ \frac{1}{\epsilon^2} n_f C_F^2 \Bigg\{
- \frac{220}{27}
+ \frac{64}{9} \zeta_3
\Bigg\}
+ \frac{1}{\epsilon^2} n_f^2 C_F \Bigg\{
- \frac{160}{81}
+ \frac{16}{27} \zeta_2
\Bigg\}
\nonumber\\
& + \frac{1}{\epsilon^2} \log\left(\frac{q^2}{\mu_F^2}\right) C_F C_A^2 \Bigg\{
\frac{8344}{81}
- \frac{176}{9} \zeta_2
\Bigg\}
+ \frac{1}{\epsilon^2} \log\left(\frac{q^2}{\mu_F^2}\right) n_f C_F C_A \Bigg\{
- \frac{2672}{81}
+ \frac{32}{9} \zeta_2
\Bigg\}
\nonumber\\
& + \frac{1}{\epsilon^2} \log\left(\frac{q^2}{\mu_F^2}\right) n_f C_F^2 \Bigg\{
- \frac{16}{3}
\Bigg\}
+ \frac{1}{\epsilon^2} \log\left(\frac{q^2}{\mu_F^2}\right) n_f^2 C_F \Bigg\{
\frac{160}{81}
\Bigg\}
+ \frac{1}{\epsilon} C_F C_A^2 \Bigg\{
- \frac{136781}{2187}
\nonumber\\
& - 64 \zeta_5
+ \frac{1316}{9} \zeta_3
+ \frac{12650}{243} \zeta_2
- \frac{176}{9} \zeta_2 \zeta_3
- \frac{352}{15} \zeta_2^2
\Bigg\}
+ \frac{1}{\epsilon} n_f C_F C_A \Bigg\{
\frac{11842}{2187}
- \frac{728}{81} \zeta_3
\nonumber\\
& - \frac{2828}{243} \zeta_2
+ \frac{32}{5} \zeta_2^2
\Bigg\}
+ \frac{1}{\epsilon} n_f C_F^2 \Bigg\{
\frac{1711}{81}
- \frac{304}{27} \zeta_3
- \frac{4}{3} \zeta_2
- \frac{32}{15} \zeta_2^2
\Bigg\}
+ \frac{1}{\epsilon} n_f^2 C_F \Bigg\{
\frac{2080}{2187}
\nonumber\\
& - \frac{112}{81} \zeta_3
+ \frac{40}{81} \zeta_2
\Bigg\}
+ \frac{1}{\epsilon} \log\left(\frac{q^2}{\mu_F^2}\right) C_F C_A^2 \Bigg\{
\frac{490}{9}
+ \frac{88}{9} \zeta_3
- \frac{1072}{27} \zeta_2
+ \frac{176}{15} \zeta_2^2
\Bigg\}
\nonumber\\
& + \frac{1}{\epsilon} \log\left(\frac{q^2}{\mu_F^2}\right) n_f C_F C_A \Bigg\{
- \frac{836}{81}
- \frac{112}{9} \zeta_3
+ \frac{160}{27} \zeta_2
\Bigg\}
+ \frac{1}{\epsilon} \log\left(\frac{q^2}{\mu_F^2}\right) n_f C_F^2 \Bigg\{
- \frac{110}{9}
\nonumber\\
& + \frac{32}{3} \zeta_3
\Bigg\}
+ \frac{1}{\epsilon} \log\left(\frac{q^2}{\mu_F^2}\right) n_f^2 C_F \Bigg\{
- \frac{16}{81}
\Bigg\}
+ C_F C_A^2 \Bigg\{
\frac{5211949}{26244}
- \frac{484}{9} \zeta_5
- \frac{128966}{243} \zeta_3
\nonumber\\
& + \frac{536}{9} \zeta_3^2
- \frac{578479}{1458} \zeta_2
+ 242 \zeta_2 \zeta_3
+ \frac{9457}{135} \zeta_2^2
+ \frac{152}{189} \zeta_2^3
\Bigg\}
+ n_f C_F C_A \Bigg\{
- \frac{412765}{13122}
\nonumber\\
& - \frac{8}{3} \zeta_5
+ \frac{9856}{81} \zeta_3
+ \frac{75155}{729} \zeta_2
- \frac{44}{3} \zeta_2 \zeta_3
- \frac{2528}{135} \zeta_2^2
\Bigg\}
+ n_f C_F^2 \Bigg\{
- \frac{42727}{972}
+ \frac{112}{9} \zeta_5
\nonumber\\
& + \frac{2284}{81} \zeta_3
+ \frac{605}{18} \zeta_2
- \frac{88}{3} \zeta_2 \zeta_3
+ \frac{152}{45} \zeta_2^2
\Bigg\}
+ n_f^2 C_F \Bigg\{
- \frac{128}{6561}
- \frac{1480}{243} \zeta_3
- \frac{404}{81} \zeta_2
\nonumber\\
& + \frac{148}{135} \zeta_2^2
\Bigg\}
+ \log\left(\frac{q^2}{\mu_F^2}\right) C_F C_A^2 \Bigg\{
- \frac{297029}{1458}
- 96 \zeta_5
+ \frac{10036}{27} \zeta_3
+ \frac{24556}{81} \zeta_2
- \frac{88}{3} \zeta_2 \zeta_3
\nonumber\\
& - \frac{748}{15} \zeta_2^2
\Bigg\}
+ \log\left(\frac{q^2}{\mu_F^2}\right) n_f C_F C_A \Bigg\{
\frac{31313}{729}
- \frac{620}{9} \zeta_3
- \frac{7348}{81} \zeta_2
+ \frac{184}{15} \zeta_2^2
\Bigg\}
\nonumber\\
& + \log\left(\frac{q^2}{\mu_F^2}\right) n_f C_F^2 \Bigg\{
\frac{1711}{54}
- \frac{152}{9} \zeta_3
- 8 \zeta_2
- \frac{16}{5} \zeta_2^2
\Bigg\}
+ \log\left(\frac{q^2}{\mu_F^2}\right) n_f^2 C_F \Bigg\{
- \frac{928}{729}
+ \frac{80}{27} \zeta_3
\nonumber\\
& + \frac{160}{27} \zeta_2
\Bigg\}
+ \log^2\left(\frac{q^2}{\mu_F^2}\right) C_F C_A^2 \Bigg\{
\frac{15503}{162}
- 44 \zeta_3
- \frac{752}{9} \zeta_2
+ \frac{44}{5} \zeta_2^2
\Bigg\}
\nonumber\\
& + \log^2\left(\frac{q^2}{\mu_F^2}\right) n_f C_F C_A \Bigg\{
- \frac{2051}{81}
+ 24 \zeta_2
\Bigg\}
+ \log^2\left(\frac{q^2}{\mu_F^2}\right) n_f C_F^2 \Bigg\{
- \frac{55}{6}
+ 8 \zeta_3
\Bigg\}
\nonumber\\
& + \log^2\left(\frac{q^2}{\mu_F^2}\right) n_f^2 C_F \Bigg\{
\frac{100}{81}
- \frac{16}{9} \zeta_2
\Bigg\}
+ \log^3\left(\frac{q^2}{\mu_F^2}\right) C_F C_A^2 \Bigg\{
- \frac{1780}{81}
+ \frac{44}{9} \zeta_2
\Bigg\}
\nonumber\\
& + \log^3\left(\frac{q^2}{\mu_F^2}\right) n_f C_F C_A \Bigg\{
\frac{578}{81}
- \frac{8}{9} \zeta_2
\Bigg\}
+ \log^3\left(\frac{q^2}{\mu_F^2}\right) n_f C_F^2 \Bigg\{
\frac{2}{3}
\Bigg\}
+ \log^3\left(\frac{q^2}{\mu_F^2}\right) n_f^2 C_F \Bigg\{
- \frac{40}{81}
\Bigg\}
\nonumber\\
& + \log\left(\frac{q^2}{\mu_F^2}\right)^4 C_F C_A^2 \Bigg\{
\frac{121}{54}
\Bigg\}
+ \log\left(\frac{q^2}{\mu_F^2}\right)^4 n_f C_F C_A \Bigg\{
- \frac{22}{27}
\Bigg\}
+ \log\left(\frac{q^2}{\mu_F^2}\right)^4 n_f^2 C_F \Bigg\{
\frac{2}{27}
\Bigg\}
\Bigg]
\nonumber\\
& + {\cal D}_0 \Bigg[ \frac{1}{\epsilon^3} C_F C_A^2 \Bigg\{
\frac{3872}{27}
\Bigg\}
+ \frac{1}{\epsilon^3} n_f C_F C_A \Bigg\{
- \frac{1408}{27}
\Bigg\}
+ \frac{1}{\epsilon^3} n_f^2 C_F \Bigg\{
\frac{128}{27}
\Bigg\}
\nonumber\\
& + \frac{1}{\epsilon^2} C_F C_A^2 \Bigg\{
\frac{16688}{81}
- \frac{352}{9} \zeta_2
\Bigg\}
+ \frac{1}{\epsilon^2} n_f C_F C_A \Bigg\{
- \frac{5344}{81}
+ \frac{64}{9} \zeta_2
\Bigg\}
+ \frac{1}{\epsilon^2} n_f C_F^2 \Bigg\{
- \frac{32}{3}
\Bigg\}
\nonumber\\
& + \frac{1}{\epsilon^2} n_f^2 C_F \Bigg\{
\frac{320}{81}
\Bigg\}
+ \frac{1}{\epsilon} C_F C_A^2 \Bigg\{
\frac{980}{9}
+ \frac{176}{9} \zeta_3
- \frac{2144}{27} \zeta_2
+ \frac{352}{15} \zeta_2^2
\Bigg\}
\nonumber\\
& + \frac{1}{\epsilon} n_f C_F C_A \Bigg\{
- \frac{1672}{81}
- \frac{224}{9} \zeta_3
+ \frac{320}{27} \zeta_2
\Bigg\}
+ \frac{1}{\epsilon} n_f C_F^2 \Bigg\{
- \frac{220}{9}
+ \frac{64}{3} \zeta_3
\Bigg\}
\nonumber\\
& + \frac{1}{\epsilon} n_f^2 C_F \Bigg\{
- \frac{32}{81}
\Bigg\}
+ C_F C_A^2 \Bigg\{
- \frac{297029}{729}
- 192 \zeta_5
+ \frac{20072}{27} \zeta_3
+ \frac{49112}{81} \zeta_2
- \frac{176}{3} \zeta_2 \zeta_3
\nonumber\\
& - \frac{1496}{15} \zeta_2^2
\Bigg\}
+ n_f C_F C_A \Bigg\{
\frac{62626}{729}
- \frac{1240}{9} \zeta_3
- \frac{14696}{81} \zeta_2
+ \frac{368}{15} \zeta_2^2
\Bigg\}
+ n_f C_F^2 \Bigg\{
\frac{1711}{27}
\nonumber\\
& - \frac{304}{9} \zeta_3
- 16 \zeta_2
- \frac{32}{5} \zeta_2^2
\Bigg\}
+ n_f^2 C_F \Bigg\{
- \frac{1856}{729}
+ \frac{160}{27} \zeta_3
+ \frac{320}{27} \zeta_2
\Bigg\}
\nonumber\\
& + \log\left(\frac{q^2}{\mu_F^2}\right) C_F C_A^2 \Bigg\{
\frac{31006}{81}
- 176 \zeta_3
- \frac{3008}{9} \zeta_2
+ \frac{176}{5} \zeta_2^2
\Bigg\}
\nonumber\\
& + \log\left(\frac{q^2}{\mu_F^2}\right) n_f C_F C_A \Bigg\{
- \frac{8204}{81}
+ 96 \zeta_2
\Bigg\}
+ \log\left(\frac{q^2}{\mu_F^2}\right) n_f C_F^2 \Bigg\{
- \frac{110}{3}
+ 32 \zeta_3
\Bigg\}
\nonumber\\
& + \log\left(\frac{q^2}{\mu_F^2}\right) n_f^2 C_F \Bigg\{
\frac{400}{81}
- \frac{64}{9} \zeta_2
\Bigg\}
+ \log^2\left(\frac{q^2}{\mu_F^2}\right) C_F C_A^2 \Bigg\{
- \frac{3560}{27}
+ \frac{88}{3} \zeta_2
\Bigg\}
\nonumber\\
& + \log^2\left(\frac{q^2}{\mu_F^2}\right) n_f C_F C_A \Bigg\{
\frac{1156}{27}
- \frac{16}{3} \zeta_2
\Bigg\}
+ \log^2\left(\frac{q^2}{\mu_F^2}\right) n_f C_F^2 \Bigg\{
4
\Bigg\}
\nonumber\\
& + \log^2\left(\frac{q^2}{\mu_F^2}\right) n_f^2 C_F \Bigg\{
- \frac{80}{27}
\Bigg\}
+ \log^3\left(\frac{q^2}{\mu_F^2}\right) C_F C_A^2 \Bigg\{
\frac{484}{27}
\Bigg\}
+ \log^3\left(\frac{q^2}{\mu_F^2}\right) n_f C_F C_A \Bigg\{
- \frac{176}{27}
\Bigg\}
\nonumber\\
& + \log^3\left(\frac{q^2}{\mu_F^2}\right) n_f^2 C_F \Bigg\{
\frac{16}{27}
\Bigg\}
\Bigg]
+ {\cal D}_1 \Bigg[ C_F C_A^2 \Bigg\{
\frac{62012}{81}
- 352 \zeta_3
- \frac{6016}{9} \zeta_2
+ \frac{352}{5} \zeta_2^2
\Bigg\}
\nonumber\\
& + n_f C_F C_A \Bigg\{
- \frac{16408}{81}
+ 192 \zeta_2
\Bigg\}
+ n_f C_F^2 \Bigg\{
- \frac{220}{3}
+ 64 \zeta_3
\Bigg\}
+ n_f^2 C_F \Bigg\{
\frac{800}{81}
- \frac{128}{9} \zeta_2
\Bigg\}
\nonumber\\
& + \log\left(\frac{q^2}{\mu_F^2}\right) C_F C_A^2 \Bigg\{
- \frac{14240}{27}
+ \frac{352}{3} \zeta_2
\Bigg\}
+ \log\left(\frac{q^2}{\mu_F^2}\right) n_f C_F C_A \Bigg\{
\frac{4624}{27}
- \frac{64}{3} \zeta_2
\Bigg\}
\nonumber\\
& + \log\left(\frac{q^2}{\mu_F^2}\right) n_f C_F^2 \Bigg\{
16
\Bigg\}
+ \log\left(\frac{q^2}{\mu_F^2}\right) n_f^2 C_F \Bigg\{
- \frac{320}{27}
\Bigg\}
+ \log^2\left(\frac{q^2}{\mu_F^2}\right) C_F C_A^2 \Bigg\{
\frac{968}{9}
\Bigg\}
\nonumber\\
& + \log^2\left(\frac{q^2}{\mu_F^2}\right) n_f C_F C_A \Bigg\{
- \frac{352}{9}
\Bigg\}
+ \log^2\left(\frac{q^2}{\mu_F^2}\right) n_f^2 C_F \Bigg\{
\frac{32}{9}
\Bigg\}
\Bigg]
+ {\cal D}_2 \Bigg[ C_F C_A^2 \Bigg\{
- \frac{14240}{27}
\nonumber\\
& + \frac{352}{3} \zeta_2
\Bigg\}
+ n_f C_F C_A \Bigg\{
\frac{4624}{27}
- \frac{64}{3} \zeta_2
\Bigg\}
+ n_f C_F^2 \Bigg\{
16
\Bigg\}
+ n_f^2 C_F \Bigg\{
- \frac{320}{27}
\Bigg\}
\nonumber\\
& + \log\left(\frac{q^2}{\mu_F^2}\right) C_F C_A^2 \Bigg\{
\frac{1936}{9}
\Bigg\}
+ \log\left(\frac{q^2}{\mu_F^2}\right) n_f C_F C_A \Bigg\{
- \frac{704}{9}
\Bigg\}
+ \log\left(\frac{q^2}{\mu_F^2}\right) n_f^2 C_F \Bigg\{
\frac{64}{9}
\Bigg\}
\Bigg]
\nonumber\\
& + {\cal D}_3 \Bigg[ C_F C_A^2 \Bigg\{
\frac{3872}{27}
\Bigg\}
+ n_f C_F C_A \Bigg\{
- \frac{1408}{27}
\Bigg\}
+ n_f^2 C_F \Bigg\{
\frac{128}{27}
\Bigg\}
\Bigg]\,.
\end{align}
\chapter{Solving Renormalisation Group Equation}
\label{chpt:App-SolRGEZas}
To demonstrate the methodoogy of solving RGE, let us consider a
general form of an RGE with respect to the renormalisation scale $\mu_{R}$:
\begin{align}
\label{eq:App-GenRGE}
\mu_R^2 \frac{d}{d\mu_R^2} \ln M = N\,
\end{align}
where, $M$ and $N$ are functions of $\mu_{R}$. We need to solve for
$M$ in terms of $N$. Our goal is to solve it
order by order in perturbation theory. We start by expanding the
quantities in powers of $a_{s} \equiv a_{s}(\mu_R^2)$:
\begin{align}
\label{eq:App-ExpandMN}
M &= 1 + \sum\limits_{k=1}^{\infty} a_{s}^{k} M^{(k)}\,,
\nonumber\\
N &= \sum\limits_{k=1}^{\infty} a_{s}^{k} N^{(k)}\,.
\end{align}
The $\mu_{R}$ dependence of $M$ and $N$ on $\mu_R$ enters through
$a_s$. All the coefficients $M^{(k)}$ and $N^{(k)}$ are independent of
$\mu_{R}$. Hence, $\ln M$ can be written as
\begin{align}
\label{eq:App-lnM}
\ln M &= \sum\limits_{k=1}^{\infty} a_s^k M_{k}
\intertext{with}
M_{1} &= M^{(1)}\,,
\nonumber\\
M_{2} &= - \frac{1}{2} (M^{(1)})^2 + M^{(2)}\,,
\nonumber\\
M_{3} &= \frac{1}{3} (M^{(1)})^3 - M^{(1)} M^{(2)} + M^{(3)}\,,
\nonumber\\
M_{4} &= - \frac{1}{4} (M^{(1)})^4 + (M^{(1)})^2 M^{(2)} -
\frac{1}{2} (M^{(2)})^2 - M^{(1)} M^{(3)} + M^{(4)}\,.
\end{align}
Using the above expansions in RGE.~(\ref{eq:App-GenRGE}) and using the
RGE of $a_{s}$
\begin{align}
\label{eq:App-RGEas}
\mu_R^2 \frac{d}{d\mu_R^2} a_s = \frac{\epsilon}{2} a_s -
\sum\limits_{k=0}^{\infty} \beta_k \,a_s^{k+2}
\end{align}
we get $M_{k}$'s by comparing the coefficients of $a_s$ as
\begin{align}
\label{eq:App-SolMdk}
M_1 &= \frac{2}{\epsilon} N^{(1)}\,,
\nonumber\\
M_2 &= \frac{2}{\epsilon^2} \beta_0 N^{(1)} +
\frac{1}{\epsilon} N^{(2)}\,,
\nonumber\\
M_3 &= \frac{8}{3 \epsilon^3} \beta_0^2 N^{(1)} + \frac{1}{\epsilon^2}
\Bigg\{ \frac{4}{3} \beta_1 N^{(1)} + \frac{4}{3} \beta_0 N^{(2)}
\Bigg\} + \frac{2}{3 \epsilon} N^{(3)}\,,
\nonumber\\
M_4 &= \frac{4}{\epsilon^4} \beta_0^3 N^{(1)} + \frac{1}{\epsilon^3}
\Bigg\{ 4 \beta_0 \beta_1 N^{(1)} + 2 \beta_0^2 N^{(2)} \Bigg\} +
\frac{1}{\epsilon^2} \Bigg\{ \beta_2 N^{(1)} + \beta_1 N^{(2)} +
\beta_0 N^{(3)} \Bigg\} + \frac{1}{2 \epsilon} N^{(4)}\,.
\end{align}
By equating the Eq.~(\ref{eq:App-lnM}) and (\ref{eq:App-SolMdk}) we
obtain,
\begin{align}
\label{eq:App-SolMuk}
M^{(1)} &= \frac{2}{\epsilon} N^{(1)}\,,
\nonumber\\
M^{(2)} &= \frac{1}{\epsilon^2} \Bigg\{ 2 \beta_0 N^{(1)} + 2
(N^{(1)})^{2} \Bigg\} + \frac{1}{\epsilon} N^{(2)}\,,
\nonumber\\
M^{(3)} &= \frac{1}{\epsilon^3} \Bigg\{ \frac{8}{3} \beta_0^2 N^{(1)} +
4 \beta_0 (N^{(1)})^2 + \frac{4}{3} (N^{(1)})^3 \Bigg\} +
\frac{1}{\epsilon^{2}} \Bigg\{ \frac{4}{3} \beta_1 N^{(1)} +
\frac{4}{3} \beta_0 N^{(2)} + 2 N^{(1)} N^{(2)} \Bigg\}
\nonumber\\
&+
\frac{2}{3 \epsilon} N^{(3)}\,,
\nonumber\\
M^{(4)} &= \frac{1}{\epsilon^4} \Bigg\{ 4 \beta_0^3 N^{(1)} + \frac{22
}{3}\beta_0^2 (N^{(1)})^2 + 4 \beta_0 (N^{(1)})^3 + \frac{2
}{3} (N^{(1)})^4 \Bigg\} + \frac{1}{\epsilon^3} \Bigg\{ 4
\beta_0 \beta_1 N^{(1)} + \frac{8}{3} \beta_1 (N^{(1)})^2
\nonumber\\
&+
2 \beta_0^2 N^{(2)} + \frac{14}{3} \beta_0 N^{(1)} N^{(2)} +
2 (N^{(1)})^2 N^{(2)} \Bigg\}
+ \frac{1}{\epsilon^2} \Bigg\{ \beta_2 N^{(1)} + \beta_1 N^{(2)} +
\frac{1}{2}(N^{(2)})^2 + \beta_0 N^{(3)}
\nonumber\\
&+ \frac{4}{3} N^{(1)}
N^{(3)} \Bigg\} + \frac{1}{2 \epsilon} N^{(4)}\,.
\end{align}
We have presented the solution up to ${\cal O}(a_s^4)$. However, this procedure
can be easily generalised to all orders in $a_s$.
\begin{itemize}
\item \textbf{Example 1: RGE of $Z_{a_{s}}$}
\begin{align}
\label{eq:App-RGEZ}
\mu_{R}^{2} \frac{d}{d\mu_{R}^{2}}\ln Z_{a_{s}} = \frac{1}{a_{s}}
\sum_{k=0}^{\infty} a_{s}^{k+2}\beta_{k}
\end{align}
Comparing this RGE of $Z_{a_{s}}$ with Eq.~(\ref{eq:App-GenRGE}) we
get
\begin{align}
\label{eq:App-N4Zas}
N^{(k)} &= \beta_{k-1} \qquad k \in [1, \infty)\,.
\end{align}
By putting the values of $N^{(k)}$ in the general
solution~(\ref{eq:App-SolMuk}), we get the corresponding solutions of
$Z_{a_s}$ as
\begin{align}
\label{eq:App-Zas}
Z_{a_{s}} &= 1+ \sum\limits_{k=1}^{\infty} a_s^k Z_{a_s}^{(k)}
\end{align}
where
\begin{align}
\label{eq:App-Zas-comp}
Z_{a_{s}}^{(1)} &=\frac{2}{\epsilon} \beta_0\,,
\nonumber\\
Z_{a_{s}}^{(2)} &= \frac{4}{\epsilon^2 } \beta_0^2 + \frac{1}{\epsilon} \beta_1\,,
\nonumber\\
Z_{a_{s}}^{(3)} &= \frac{8}{ \epsilon^3} \beta_0^3 +\frac{14}{3
\epsilon^2} \beta_0 \beta_1 + \frac{2}{3 \epsilon} \beta_2\,,
\nonumber\\
Z_{a_{s}}^{(4)} &= \frac{16}{\epsilon^{4}}\beta_{0}^{4} +
\frac{46}{3\epsilon^{3}}\beta_{0}^{2}\beta_{1} +
\frac{1}{\epsilon^{2}}\left(\frac{3}{2}\beta_{1}^{2} +
\frac{10}{3}\beta_{0}\beta_{2}\right) +
\frac{1}{2\epsilon}\beta_{3}\,.
\end{align}
The $Z_{a_s}$ can also be expressed in powers of ${\hat a}_{s}$ by
utilising the
\begin{align}
\label{eq:App-ashatANDas}
a_s = {\hat a_s} S_{\epsilon} \left( \frac{\mu_{R}^{2}}{\mu^{2}}
\right)^{\epsilon/2} Z_{a_{s}}^{-1}
\end{align}
iteratively. We get
\begin{align}
\label{eq:App-Zashat}
Z_{a_{s}} &= 1+ \sum\limits_{k=1}^{\infty} {\hat a}_s^k S_{\epsilon}^k \left( \frac{\mu_R^2}{\mu^2}
\right)^{k\frac{\epsilon}{2}} {\hat Z}_{a_s}^{(k)}
\end{align}
where
\begin{align}
\label{eq:App-Zashat-comp}
{\hat Z}_{a_{s}}^{(1)} &=\frac{2}{\epsilon} \beta_0\,,
\nonumber\\
{\hat Z}_{a_{s}}^{(2)} &= \frac{1}{\epsilon} \beta_1\,,
\nonumber\\
{\hat Z}_{a_{s}}^{(3)} &= - \frac{4}{3 \epsilon^2} \beta_0 \beta_1 + \frac{2}{3 \epsilon} \beta_2\,,
\nonumber\\
{\hat Z}_{a_{s}}^{(4)} &= \frac{2}{\epsilon^{3}}\beta_{0}^{2}\beta_{1} +
\frac{1}{\epsilon^{2}}\left(-\frac{1}{2}\beta_{1}^{2} -
2 \beta_{0}\beta_{2}\right) +
\frac{1}{2\epsilon}\beta_{3}\,.
\end{align}
To arrive at the above result, we need to use the $Z_{a_s}^{-1}$ in
powers of ${\hat a}_s$:
\begin{align}
\label{eq:App-ZashatInv}
Z_{a_{s}}^{-1} &= 1+ \sum\limits_{k=1}^{\infty} {\hat a}_s^k S_{\epsilon}^k \left( \frac{\mu_R^2}{\mu^2}
\right)^{k\frac{\epsilon}{2}} {\hat Z}_{a_s}^{-1,(k)}
\end{align}
where
\begin{align}
\label{eq:App-ZashatInv-comp}
{\hat Z}_{a_{s}}^{-1, (1)} &= -\frac{2}{\epsilon} \beta_0\,,
\nonumber\\
{\hat Z}_{a_{s}}^{-1, (2)} &= \frac{4}{\epsilon^2} \beta_0^2 - \frac{1}{\epsilon} \beta_1\,,
\nonumber\\
{\hat Z}_{a_{s}}^{-1, (3)} &= - \frac{8}{\epsilon^3} \beta_0^3 +
\frac{16}{3 \epsilon^2} \beta_0 \beta_1 -
\frac{2}{3 \epsilon} \beta_2\,,
\nonumber\\
{\hat Z}_{a_{s}}^{-1, (4)} &= \frac{16}{\epsilon^4} \beta_0^4 -
\frac{58}{3\epsilon^{3}}\beta_{0}^{2}\beta_{1}
+
\frac{1}{\epsilon^{2}}\left(\frac{3}{2}\beta_{1}^{2} +
\frac{14}{3} \beta_{0}\beta_{2}\right) -
\frac{1}{2\epsilon}\beta_{3}\,.
\end{align}
\item \textbf{Example 2: Solution of the Mass Factorisation Kernel}
The mass factorisation kernel satisfies the RG equation~(\ref{eq:bBH-kernelRGE})
\begin{align}
\label{eq:App-kernelRGE}
\mu_F^2 \frac{d}{d\mu_F^2} \Gamma^I_{ij}(z,\mu_F^2,\epsilon) =
\frac{1}{2} \sum\limits_{k} P^I_{ik} \left(z,\mu_F^2\right) \otimes \Gamma^I_{kj} \left(z,\mu_F^2,\epsilon \right)
\end{align}
where, $P^I\left(z,\mu_{F}^{2}\right)$ are Altarelli-Parisi splitting
functions (matrix valued). Expanding $P^{I}\left(z,\mu_{F}^{2}\right)$ and
$\Gamma^{I}(z,\mu_F^2,\epsilon)$ in powers of the strong coupling constant
we get
\begin{align}
\label{eq:App-APexpand}
&P^{I}(z,\mu_{F}^{2}) = \sum_{k=1}^{\infty} a_{s}^{k}(\mu_{F}^{2})P^{I,(k-1)}(z)\,
\intertext{and}
&\Gamma^I(z,\mu_F^2,\epsilon) = \delta(1-z) + \sum_{k=1}^{\infty}
{\hat a}_{s}^{k} S_{\epsilon}^{k} \left(\frac{\mu_{F}^{2}}{\mu^{2}}\right)^{k
\frac{\epsilon}{2}} \Gamma^{I,(k)}(z,\epsilon)\, .
\end{align}
Following the techniques prescribed above, it can be solved. However,
unlike the previous cases here we have to take care of the fact
that $P^I$ and $\Gamma^I$ are matrix valued quantities i.e. they are
non-commutative. Upon solving we obtain the general solution as
\begin{align}
\label{eq:App-Gamma-GenSoln}
\Gamma^{I,(1)}(z,\epsilon) &= \frac{1}{\epsilon} \Bigg\{
P^{I,(0)}(z)
\Bigg\} \,,
\nonumber\\
\Gamma^{I,(2)}(z,\epsilon) &= \frac{1}{\epsilon^2} \Bigg\{
- \beta_0 P^{I,(0)} (z)
+ \frac{1}{2} P^{I,(0)} (z) \otimes P^{I,(0)} (z)
\Bigg\}
+ \frac{1}{\epsilon} \Bigg\{
\frac{1}{2} P^{I,(1)}(z)
\Bigg\}\,,
\nonumber\\
\Gamma^{I,(3)}(z,\epsilon) &= \frac{1}{\epsilon^3} \Bigg\{
\frac{4}{3} \beta_0^2 P^{I,(0)}
- \beta_0 P^{I,(0)} \otimes P^{I,(0)}
+ \frac{1}{6} P^{I,(0)} \otimes P^{I,(0)} \otimes P^{I,(0)}
\Bigg\}
\nonumber\\
&
+ \frac{1}{\epsilon^2} \Bigg\{
- \frac{1}{3} \beta_1 P^{I,(0)}
+ \frac{1}{6} P^{I,(0)} \otimes P^{I,(1)}
- \frac{4}{3} \beta_0 P^{I,(1)}
+ \frac{1}{3} P^{I,(1)} \otimes P^{I,(0)}
\Bigg\}
\nonumber\\
&
+ \frac{1}{\epsilon} \Bigg\{
\frac{1}{3} P^{I,(2)}
\Bigg\}\,,
\nonumber\\
\Gamma^{I,(4)}(z,\epsilon) &= \frac{1}{\epsilon^4} \Bigg\{
- 2\beta_0^3 P^{I,(0)}
+ \frac{11}{6} \beta_0^2 P^{I,(0)} \otimes P^{I,(0)}
- \frac{1}{2} \beta_0 P^{I,(0)} \otimes P^{I,(0)} \otimes P^{I,(0)}
\nonumber\\
&
+ \frac{1}{24} P^{I,(0)} \otimes P^{I,(0)} \otimes P^{I,(0)} \otimes P^{I,(0)}
\Bigg\}
+ \frac{1}{\epsilon^3} \Bigg\{
\frac{4}{3}\beta_0 \beta_1 P^{I,(0)}
- \frac{1}{3} \beta_1 P^{I,(0)} \otimes P^{I,(0)}
\nonumber\\
&
+ \frac{1}{24} P^{I,(0)} \otimes P^{I,(0)} \otimes P^{I,(1)}
- \frac{7}{12} \beta_0 P^{I,(0)} \otimes P^{I,(1)}
+ \frac{1}{12} P^{I,(0)} \otimes P^{I,(1)} \otimes P^{I,(0)}
\nonumber\\
&
+ 3 \beta_0^2 P^{I,(1)}
- \frac{5}{4} \beta_0 P^{I,(1)} \otimes P^{I,(0)}
+ \frac{1}{8} P^{I,(1)} \otimes P^{I,(0)} \otimes P^{I,(0)}
\Bigg\}
\nonumber\\
&
+ \frac{1}{\epsilon^2} \Bigg\{
- \frac{1}{6}\beta_2 P^{I,(0)}
+ \frac{1}{12} P^{I,(0)} \otimes P^{I,(2)}
- \frac{1}{2} \beta_1 P^{I,(1)}
+ \frac{1}{8} P^{I,(1)} \otimes P^{I,(1)}
\nonumber\\
&
- \frac{3}{2} \beta_0 P^{I,(2)}
+ \frac{1}{4} P^{I,(2)} \otimes P^{I,(0)}
\Bigg\}
+ \frac{1}{\epsilon} \Bigg\{
\frac{1}{4} P^{I,(3)}
\Bigg\}\,.
\end{align}
In the soft-virtual limit, only the diagonal parts of the kernels
contribute. Our findings are consistent with the existing diagonal
solutions which can be found in the article~\cite{Ravindran:2005vv}.
\end{itemize}
\chapter{\label{chap:bBCS}Higgs boson production through $b \bar b$ annihilation at threshold in N$^3$LO QCD}
\textit{\textbf{The materials presented in this chapter are the result of an original research done in collaboration with Narayan Rana and V. Ravindran, and these are based on the published article~\cite{Ahmed:2014cha}}}.
\\
\begingroup
\hypersetup{linkcolor=blue}
\minitoc
\endgroup
\section{Prologue}
\setcounter{equation}{0}
\label{sec:intro}
The discovery of Higgs boson by ATLAS \cite{Aad:2012tfa} and CMS \cite{Chatrchyan:2012ufa}
collaborations of the LHC
at CERN has not only shed the light on the dynamics behind the electroweak symmetry breaking
but also put the SM of particle physics on a firmer ground. In the SM, the
elementary particles such as quarks, leptons and gauge bosons, $Z,W^\pm$ acquire
their masses through spontaneous symmetry breaking (SSB).
The Higgs mechanism provides the framework for
SSB. The SM predicts the existence of a Higgs boson whose mass is a
parameter of the model.
The recent discovery of the SM Higgs boson like particle provides a valuable information on this, namely on
its mass which is about 125.5 GeV.
The searches for the Higgs boson have been going on for several decades
in various experiments.
Earlier experiments such as LEP \cite{Barate:2003sz} and Tevatron \cite{Aaltonen:2010yv} played
an important role in the discovery by the LHC collaborations
through narrowing down its possible mass range.
LEP excluded Higgs boson of mass below 114.4 GeV
and their precision electroweak measurements \cite{ALEPH:2010aa}
hinted the mass less than 152 GeV at $95\%$ confidence level (CL), while
Tevatron excluded Higgs boson of mass in the range $162-166$ GeV at $95\%$ CL.
Higgs bosons are produced dominantly at the LHC
via gluon gluon fusion through top quark loop, while
the sub-dominant ones
are vector boson fusion, associated production of
Higgs boson with vector bosons, with top anti-top pairs and also in bottom anti-bottom
annihilation. The inclusive productions of Higgs boson in gluon gluon \cite{ALEPH:2010aa, Djouadi:1991tka, Dawson:1990zj, Spira:1995rr, Catani:2001ic, Harlander:2001is, Harlander:2002wh, Anastasiou:2002yz, Ravindran:2003um},
vector boson fusion processes \cite{Bolzoni:2010xr} and associated production with vector
bosons \cite{Han:1991ia} are known to NNLO
accuracy in QCD.
Higgs production in bottom anti-bottom annihilation is also known to NNLO accuracy
in the variable flavour scheme (VFS) \cite{Dicus:1988cx, Dicus:1998hs, Maltoni:2003pn, Olness:1987ep, Gunion:1986pe, Harlander:2003ai}, while it is known to NLO in the fixed
flavour scheme (FFS) \cite{Reina:2001sf, Beenakker:2001rj, Dawson:2002tg, Beenakker:2002nc, Raitio:1978pt, Kunszt:1984ri}.
In the MSSM,
the coupling of bottom quarks to Higgs
becomes large in the large $\tan\beta$ region, where
$\tan\beta$ is the ratio of vacuum expectation values of up and down type Higgs fields.
This can enhance contributions from bottom anti-bottom annihilation subprocesses.
While the theoretical predictions of NNLO \cite{ALEPH:2010aa, Djouadi:1991tka, Dawson:1990zj, Spira:1995rr, Catani:2001ic, Harlander:2001is, Harlander:2002wh, Anastasiou:2002yz, Ravindran:2003um} and next to next to
leading log (NNLL) \cite{Catani:2003zt} QCD corrections and of two loop
electroweak effects \cite{Aglietti:2004nj, Actis:2008ug} played an important role in the Higgs discovery,
the theoretical uncertainties resulting from
factorization and renormalization scales are not fully under control.
Hence, the efforts to go beyond NNLO are going on intensively.
Some of the ingredients to obtain N$^3$LO QCD corrections
are already available. For example, quark and gluon form
factors \cite{Moch:2005id, Moch:2005tm, Gehrmann:2005pd, Baikov:2009bg, Gehrmann:2010ue},
the mass factorization kernels \cite{Moch:2004pa}
and the renormalization constant \cite{Chetyrkin:1997un} for
the effective operator describing the coupling of
Higgs boson with the SM fields in the infinite top quark mass
limit up to three loop level in dimensional regularization are known for some time.
In addition, NNLO soft contributions are known \cite{deFlorian:2012za} to all
orders in $\epsilon$ for both DY and Higgs productions using dimensional regularization
with space time dimension being $d=4+\epsilon$.
They were used to obtain the partial N$^3$LO
threshold effects \cite{Moch:2005ky, Laenen:2005uz, Idilbi:2005ni, Ravindran:2005vv, Ravindran:2006cg} to
Drell-Yan production of di-leptons and inclusive productions of Higgs boson through gluon gluon
fusion and in bottom anti-bottom annihilation.
Threshold contribution to the inclusive production
cross section is expanded in terms of $\delta(1-z)$ and ${\cal D}_i(z)$ where
\begin{eqnarray}
{\cal D}_i(z) = \left(\frac{\ln^i (1-z)}{1-z}\right)_+
\end{eqnarray}
with the scaling parameter
$z=m_H^2/\hat s$ for Higgs
and $z=m_{l^+l^-}^2/\hat s$ for DY.
Here $m_H$, $m_{l^+l^-}$ and $\hat s$ are mass of the Higgs boson,
invariant mass of the di-leptons and square of the center of mass energy of the partonic reaction responsible for production mechanism respectively.
The missing $\delta(1-z)$ terms for the complete N$^3$LO threshold contributions
to the Higgs production through gluon gluon fusion are now available
due to the seminal work by Anastasiou et al \cite{Anastasiou:2014vaa}
where the relevant soft contributions were obtained from the real radiations at
N${}^3$LO level.
The resummation of threshold effects \cite{Sterman:1986aj, Catani:1989ne}
to infra-red safe observables
resulting from their factorization properties
as well as Sudakov resummation of soft gluons
provides an elegant framework to obtain threshold enhanced
contributions to inclusive
and semi inclusive observables order by order in perturbation theory.
In \cite{Ahmed:2014cla}, using this framework, we exploited the universal
structure of the soft radiations to obtain the corresponding soft
gluon contributions to DY production, which led to the evaluation of missing
$\delta(1-z)$ part of the N$^3$LO threshold corrections.
In \cite{Li:2014bfa}, relevant one loop double real emissions from light-like Wilson lines
were computed to obtain threshold corrections to Higgs as well as Drell-Yan productions
up to N$^3$LO level providing an independent approach.
In \cite{Catani:2014uta} the universality of soft gluon contributions near threshold and
the results of \cite{Anastasiou:2014vaa} were used to obtain general expression
of the hard-virtual coefficient which contributes
to N$^3$LO threshold as well as
threshold resummation at next-to-next-to-next-to-leading-logarithmic (N$^3$LL)
accuracy for the production cross section of a
colourless heavy particle at hadron colliders.
For the Higgs production through $b \bar{b}$ annihilation,
till date, only partial N${}^3$LO threshold corrections are known \cite{Ravindran:2005vv, Ravindran:2006cg, Kidonakis:2007ww} where again the framework of threshold resummation was used.
In both \cite{Ravindran:2005vv, Ravindran:2006cg} and \cite{Kidonakis:2007ww}, it was not possible
to determine the $\delta(1-z)$ at N$^3$LO
due to the lack of information on
three loop finite part of bottom anti-bottom higgs form factor in QCD and the
soft gluon radiation at N$^3$LO level. In \cite{Kidonakis:2007ww}, subleading corrections
were also obtained through the method of Mellin moments.
The recent results on Higgs form factor with bottom anti-bottom by Gehrmann and Kara
\cite{Gehrmann:2014vha} and on the universal soft distribution obtained
for the Drell-Yan production \cite{Ahmed:2014cla} can now be used
to obtain $\delta(1-z)$ part of the threshold N$^3$LO contribution.
For the soft gluon radiations in the $b \bar{b}$ annihilation,
the results from \cite{Ahmed:2014cla} can be
used as they do not depend on the flavour of the incoming quark states.
We have set bottom quark mass to be zero throughout except in the Yukawa coupling.
We begin by writing down the relevant interacting Lagrangian in Sec.~\ref{sec:bBH-Lag}.
In the Sec.~\ref{sec:bBH-ThreResu}, we present the formalism of computing threshold QCD corrections to the cross-section and in Sec.~\ref{sec:bBH-Res}, we present
our results for threshold N$^3$LO QCD contributions to Higgs production through $b\bar{b}$ annihilation at hadron colliders and their numerical impact .
The numerical impact of threshold enhanced N$^3$LO contributions
is demonstrated for the LHC energy $\sqrt{s} = 14$ TeV by studying
the stability of the perturbation theory under factorization and renormalization scales.
Finally we give a brief summary of our findings in Sec.~\ref{sec:bBH-Summary}.
\section{The Effective Lagrangian}
\label{sec:bBH-Lag}
The interaction of bottom quarks and the scalar Higgs
boson is given by the action
\begin{align}
\label{eq:bBH-Lag}
{\cal L}^{b} = \phi(x) {O}^b(x) \equiv - {\lambda \over \sqrt{2}} \phi(x) \overline \psi_b(x) \psi_b(x)
\end{align}
where, $\psi_b(x)$ and $\phi(x)$ denote the bottom quark and scalar
Higgs field, respectively.
$\lambda$ is the Yukawa coupling given by $\sqrt{2} m_b/\nu$, with
the bottom quark mass $m_b$ and
the vacuum expectation value $\nu\approx 246$ GeV. In MSSM,
for the pseudo-scalar Higgs boson, we need to replace $\lambda \phi(x) \overline \psi_b(x) \psi_b(x)$ by
$\tilde \lambda \tilde \phi(x) \overline \psi_b(x) \gamma_5 \psi_b(x)$ in the above
equation. The MSSM couplings are given by
\[
\tilde{\lambda} = \left\{
\begin{array}{ll}
- \frac{\sqrt{2} m_b \sin\alpha}{\nu \cos\beta} \,,& \qquad \tilde{\phi} = h\,,\\
\phantom{-} \frac{\sqrt{2} m_b \cos\alpha}{\nu \cos\beta} \,,& \qquad \tilde{\phi}=H\,,\\
\phantom{-} \frac{\sqrt{2} m_b \tan\beta}{\nu } \,, & \qquad \tilde{\phi}=A\,
\end{array}
\right.
\]
respectively. The angle $\alpha$ measures the mixing of weak and mass eigenstates
of neutral Higgs bosons. We use VFS scheme throughout, hence except in the Yukawa coupling,
$m_b$ is taken to be zero like other light quarks in the theory.
\section{Theoretical Framework for Threshold Corrections}
\label{sec:bBH-ThreResu}
The inclusive cross-section for the production of a colorless
particle, namely, a Higgs boson through gluon fusion/bottom quark
annihilation or a pair of leptons in the Drell-Yan at the hadron
colliders is computed using
\begin{align}
\label{eq:bBH-1}
\sigma^{I}(\tau, q^{2}) = \sigma^{I,(0)}(\mu_R^2) \sum\limits_{i,j=q,{\bar q},g}
\int\limits_{\tau}^{1} dx \;\Phi_{ij}(x,\mu_F^{2})\;
\Delta^I_{ij}\left(\frac{\tau}{x}, q^{2}, \mu_R^2, \mu_F^2\right)
\end{align}
with the partonic flux
\begin{align}
\label{eq:bBH-2}
\Phi_{ij}(x, \mu_{F}^2) = \int\limits_x^1 \frac{dy}{y} f_i(y, \mu_F^2)
\;f_j\left(\frac{x}{y}, \mu_F^2 \right)\,.
\end{align}
In the above expressions, $f_i (y,\mu_F^2)$ and $f_j \left(\frac{x}{y},\mu_F^2\right)$
are the parton distribution functions (PDFs) of the initial state
partons $i$ and $j$ with momentum fractions $y$ and $x/y$,
respectively. These are renormalized at the factorization scale
$\mu_{F}$. The dimensionless quantity $\Delta^{I}_{ij}\left(\frac{\tau}{x}, q^2,
\mu_{R}^{2}, \mu_F^2\right)$ is called the
coefficient function of the partonic cross section for the
production of a colorless particle from partons $i$ and $j$, computed
after performing the UV renormalization at scale $\mu_{R}$ and mass
factorization at $\mu_{F}$. The quantity $\sigma^{I,(0)}$ is a pre-factor of the born level cross section. The variable $\tau$ is defined as
$q^{2}/s$, where
\begin{equation}
\label{eq:q2}
q^{2} =
\begin{cases}
~m_{H}^{2}& ~\text{for}~ I=H\, ,\\
~m_{l^+l^-}^{2}& ~\text{for}~ I=DY\, .
\end{cases}
\end{equation}
$m_{H}$ is the mass of the Higgs boson and $m_{l^+l^-}$is the invariant
mass of the final state dilepton pair ($l^{+}l^{-}$), which can be
$e^{+}e^{-},\mu^{+}\mu^{-}, \tau^{+}\tau^{-}$, in the DY production.
$\sqrt{s}$ and $\sqrt{\hat{s}}$ stand for the hadronic and partonic
center of mass energy, respectively. Throughout this chapter, we
denote $I=H$ for the
productions of the Higgs boson
through gluon ($gg$) fusion (Fig.~\ref{fig:1-ggH}) and bottom quark ($b{\bar b}$)
annihilation (Fig.~\ref{fig:1-bBH}), whereas we write $I=DY$ for the production of a pair of
leptons in the Drell-Yan (Fig.~\ref{fig:1-qQll}).
\begin{figure}[htb]
\begin{center}
\begin{tikzpicture}[line width=1.5 pt, scale=1]
\draw[gluon] (-2.5,0) -- (0,0);
\draw[gluon] (-2.5,-2) -- (0,-2);
\draw[fermion] (0,0) -- (2,-1);
\draw[fermion] (2,-1) -- (0,-2);
\draw[fermion] (0,-2) -- (0,0);
\draw[scalarnoarrow] (2,-1) -- (4,-1);
\node at (-2.8,0) {$g$};
\node at (-2.8,-2) {$g$};
\node at (4.3,-1) {$H$};
\end{tikzpicture}
\caption{Higgs boson production in gluon fusion}
\label{fig:1-ggH}
\end{center}
\end{figure}
\begin{figure}[htb]
\begin{center}
\begin{tikzpicture}[line width=1.5 pt, scale=1]
\draw[fermion] (-2.5,0) -- (0,-1.3);
\draw[fermion] (0,-1.3) -- (-2.5,-2.6);
\draw[scalarnoarrow] (0,-1.3) -- (2,-1.3);
\node at (-2.8,0) {$b$};
\node at (-2.8,-2.6) {${\bar b}$};
\node at (2.3,-1.3) {$H$};
\end{tikzpicture}
\caption{Higgs boson production through bottom quark annihilation}
\label{fig:1-bBH}
\end{center}
\end{figure}
\begin{figure}[htb]
\begin{center}
\begin{tikzpicture}[line width=1.5 pt, scale=1]
\draw[fermion] (-2.5,0) -- (0,-1.3);
\draw[fermion] (0,-1.3) -- (-2.5,-2.6);
\draw[vector] (0,-1.3) -- (2,-1.3);
\draw[fermion] (2,-1.3) -- (4.5,0);
\draw[fermionbar] (2,-1.3) -- (4.5,-2.6);
\node at (-2.8,0) {$q$};
\node at (-2.8,-2.6) {${\bar q}$};
\node at (1.1,-0.7) {$\gamma^{*}/Z$};
\node at (4.8,0) {$l^{+}$};
\node at (4.8,-2.6) {$l^{-}$};
\end{tikzpicture}
\caption{Drell-Yan pair production}
\label{fig:1-qQll}
\end{center}
\end{figure}
One of the \textit{goals} of this chapter is to study the impact of
the contributions arising from the soft gluons to the cross section of
a colorless particle production at Hadron colliders. The infrared safe
contributions from the soft gluons is obtained by adding the soft part
of the cross section with the UV renormalized virtual
part and performing mass factorisation using appropriate counter
terms. This combination is often called the soft-plus-virtual
cross section whereas the remaining portion is known as hard
part. Hence, we write the partonic cross section by decomposing into
two parts as
\begin{align}
\label{eq:bBH-PartsOfDelta}
&{\Delta}^{I}_{ij} (z, q^{2}, \mu_{R}^{2}, \mu_F^2) = {\Delta}^{I,
\text{SV}}_{ij} (z, q^{2}, \mu_{R}^{2}, \mu_F^2) + {\Delta}^{I,
\text{hard}}_{ij} (z, q^{2}, \mu_{R}^{2}, \mu_F^2) \,
\end{align}
with $z \equiv q^{2}/\hat{s}$. The SV contributions
${\Delta}^{I, \text{SV}}_{ij} (z, q^{2}, \mu_{R}^{2}, \mu_F^2)$
contains only the distributions of kind $\delta(1-z)$ and
${\cal{D}}_{i}$, where the latter one is defined through
\begin{align}
\label{eq:bBH-calD}
{\cal{D}}_{i} \equiv \left[ \frac{\ln^{i}(1-z)}{(1-z)} \right]_{+}\, .
\end{align}
This is also known as the threshold contributions. On the other hand,
the hard part ${\Delta}^{I, \text{hard}}_{ij}$ contains all the
regular terms in $z$. The SV cross section in $z$-space is
computed in $d=4+\epsilon$ dimensions, as formulated in
\cite{Ravindran:2005vv, Ravindran:2006cg}, using
\begin{align}
\label{eq:bBH-sigma}
\Delta^{I, \text{SV}}_{ij} (z, q^2, \mu_{R}^{2}, \mu_F^2) =
{\cal C} \exp \Big( \Psi^I_{ij} \left(z, q^2, \mu_R^2, \mu_F^2, \epsilon
\right) \Big) \Big|_{\epsilon = 0}
\end{align}
where, $\Psi^I_{ij} \left(z, q^2, \mu_R^2, \mu_F^2, \epsilon \right)$ is a
finite distribution and ${\cal C}$ is the convolution defined
as
\begin{equation}
\label{eq:bBH-conv}
{\cal C} e^{f(z)} = \delta(1-z) + \frac{1}{1!} f(z) + \frac{1}{2!}
f(z) \otimes f(z) + \cdots \,.
\end{equation}
Here $\otimes$ represents Mellin convolution and $f(z)$ is a
distribution of the kind $\delta(1-z)$ and ${\cal D}_i$. The
equivalent formalism of the SV approximation in the Mellin (or
$N$-moment) space, where instead of distributions in $z$, the dominant
contributions come from the meromorphic functions of the variable $N$
(see~\cite{Sterman:1986aj, Catani:1989ne}) and the threshold limit of
$z \rightarrow 1$ is translated to $N \rightarrow \infty$.
The $\Psi^I_{ij} \left(z, q^2, \mu_R^2, \mu_F^2, \epsilon \right)$ is
constructed from the form factors ${\cal F}^I_{ij} (\hat{a}_s, Q^2, \mu^2,
\epsilon)$ with $Q^{2}=-q^{2}$, the overall operator UV
renormalization constant $Z^I_{ij}(\hat{a}_s,
\mu_R^2, \mu^2, \epsilon)$,
the soft-collinear distribution $\Phi^I_{ij}(\hat{a}_s, q^2, \mu^2, z, \epsilon)$
arising from the real radiations in the partonic subprocesses and the
mass factorization kernels $\Gamma^{I}_{ij} (\hat{a}_s, \mu^2, \mu_F^2, z,
\epsilon)$. In
terms of the above-mentioned quantities it takes the following form,
as presented in \cite{Ravindran:2006cg, Ahmed:2014cla, Ahmed:2014cha}
\begin{align}
\label{eq:bBH-psi}
\Psi^{I}_{ij} \left(z, q^2, \mu_R^2, \mu_F^2, \epsilon \right) = &\left( \ln \Big[ Z^I_{ij} (\hat{a}_s, \mu_R^2, \mu^2, \epsilon) \Big]^2
+ \ln \Big| {\cal F}^I_{ij} (\hat{a}_s, Q^2, \mu^2, \epsilon) \Big|^2
\right)
\delta(1-z)
\nonumber\\
& + 2 \Phi^I_{ij} (\hat{a}_s, q^2, \mu^2, z, \epsilon) - 2 {\cal C} \ln
\Gamma^{I}_{ij} (\hat{a}_s, \mu^2, \mu_F^2, z, \epsilon) \, .
\end{align}
In this expression, ${\hat{a}_s} \equiv {\hat{g}}_{s}^{2}/16\pi^{2}$
is the unrenormalized strong coupling constant which is related to the
renormalized one $a_{s}(\mu_{R}^{2})\equiv a_{s}$ through the renormalization
constant $Z_{a_{s}}(\mu_{R}^{2}) \equiv Z_{a_{s}}$ as
\begin{align}
\label{eq:bBH-ashatANDas}
&{\hat a_s} S_{\epsilon} = \left( \frac{\mu^{2}}{\mu_R^{2}}
\right)^{\epsilon/2} Z_{a_{s}} a_s\,,
\end{align}
where, $S_{\epsilon} = \exp\left[(\gamma_{E}-\text{ln} 4\pi)\epsilon/2)\right]$
and $\mu$ is the mass scale introduced to keep the ${\hat{a}_s}$
dimensionless in $d$-dimensions. ${\hat g}_{s}$ is the coupling
constant appearing in the bare Lagrangian of QCD. $Z_{a_{s}}$ can be obtained by
solving the underlying renormalisation group equation (RGE)
\begin{align}
\label{eq:bBH-RGEZ}
\mu_{R}^{2} \frac{d}{d\mu_{R}^{2}}\ln Z_{a_{s}} = \frac{1}{a_{s}}
\sum_{k=0}^{\infty} a_{s}^{k+2}\beta_{k}
\end{align}
where, $\beta_{k}$'s are the coefficients of the QCD
$\beta$-function. The solution of the above RGE in terms of the
$\beta_{k}$'s and $\epsilon$ up to ${\cal O}(a_s^4)$comes out to be
\begin{align}
\label{eq:bBH-Zas}
Z_{a_{s}} &= 1+ a_s \left[\frac{2}{\epsilon} \beta_0\right]
+ a_s^2 \left[\frac{4}{\epsilon^2 } \beta_0^2 + \frac{1}{\epsilon} \beta_1 \right]
+ a_s^3 \left[\frac{8}{ \epsilon^3} \beta_0^3 +\frac{14}{3
\epsilon^2} \beta_0 \beta_1 + \frac{2}{3 \epsilon} \beta_2 \right]
\nonumber\\
&+ a_s^4 \left[\frac{16}{\epsilon^{4}}\beta_{0}^{4} +
\frac{46}{3\epsilon^{3}}\beta_{0}^{2}\beta_{1} +
\frac{1}{\epsilon^{2}}\left(\frac{3}{2}\beta_{1}^{2} +
\frac{10}{3}\beta_{0}\beta_{2}\right) +
\frac{1}{2\epsilon}\beta_{3} \right]\,.
\end{align}
Results beyond this order involve $\beta_{4}$ and higher order
$\beta_{k}$'s which are not available yet in the literature. The
$\beta_{k}$ up to $k=3$ are given
by~\cite{Tarasov:1980au}
\begin{align}
\label{eq:bBH-beta}
\beta_0&={11 \over 3 } C_A - {2 \over 3 } n_f \, ,
\nonumber \\[0.5ex]
\beta_1&={34 \over 3 } C_A^2- 2 n_f C_F -{10 \over 3} n_f C_A \, ,
\nonumber \\[0.5ex]
\beta_2&={2857 \over 54} C_A^3
-{1415 \over 54} C_A^2 n_f
+{79 \over 54} C_A n_f^2
+{11 \over 9} C_F n_f^2
-{205 \over 18} C_F C_A n_f
+ C_F^2 n_f\,,
\nonumber\\[0.5ex]
\beta_3 &=
N^2 \;\Bigg( - \frac{40}{3} + 352 \zeta_3 \Bigg)
+ N^4 \;\Bigg( - \frac{10}{27} + \frac{88}{9} \zeta_3 \Bigg)
+ n_f N \;\Bigg( \frac{64}{9} - \frac{208}{3} \zeta_3 \Bigg)
\nonumber\\
&+ n_f N^3 \;\Bigg( \frac{32}{27} - \frac{104}{9} \zeta_3 \Bigg)
+ n_f^2 N^{-2} \;\Bigg( - \frac{44}{3} + 32 \zeta_3 \Bigg)
+ n_f^2 \;\Bigg( \frac{44}{9} - \frac{32}{3} \zeta_3 \Bigg)
\nonumber\\
& + n_f^2 N^2 \;\Bigg( - \frac{22}{27} + \frac{16}{9} \zeta_3 \Bigg)
+ C_F n_f^3 \;\Bigg( \frac{154}{243} \Bigg)
+ C_F^2 n_f^2 \;\Bigg( \frac{338}{27} - \frac{176}{9} \zeta_3 \Bigg)
+ C_F^3 n_f \;\Bigg( 23 \Bigg)
\nonumber\\
& + C_A n_f^3 \;\Bigg( \frac{53}{243} \Bigg)
+ C_A C_F n_f^2 \;\Bigg( \frac{4288}{243} + \frac{112}{9} \zeta_3 \Bigg)
+ C_A C_F^2 n_f \;\Bigg( - \frac{2102}{27} + \frac{176}{9} \zeta_3 \Bigg)
\nonumber\\
& + C_A^2 n_f^2 \;\Bigg( \frac{3965}{162} + \frac{56}{9} \zeta_3 \Bigg)
+ C_A^2 C_F n_f \;\Bigg( \frac{7073}{486} - \frac{328}{9} \zeta_3 \Bigg)
+ C_A^3 n_f \;\Bigg( - \frac{39143}{162} + \frac{68}{3} \zeta_3 \Bigg)
\nonumber\\
& + C_A^4 \;\Bigg( \frac{150653}{486} - \frac{44}{9} \zeta_3 \Bigg)
\end{align}
with the SU(N) quadratic casimirs
\begin{equation}
C_A=N,\quad \quad \quad C_F={N^2-1 \over 2 N}\,.
\end{equation}
$n_f$ is the number of active light quark flavors.
In this chapter, we will confine our discussion on the threshold
corrections to the Higgs boson production through bottom quark
annihilation and more precisely our main goal is to compute the SV
cross section of this process at N$^{3}$LO QCD. In the subsequent
sections, we will demonstrate the methodology to obtain the ingredients, Eq.~(\ref{eq:bBH-psi})
for computing the SV cross section of scalar Higgs boson production at
N$^3$LO QCD.
\subsection{The Form Factor}
\label{ss:bBH-FF}
The quark and gluon form factors represent the QCD loop corrections to
the transition matrix element from an on-shell quark-antiquark pair or
two gluons to a color-neutral operator $O$. For the scalar Higgs boson
production through $b {\bar b}$ annihilation, we consider Yukawa
interaction, encapsulated through the operator ${O}^{b}$ present
in the interacting Lagrangian~\ref{eq:bBH-Lag}. For the process under
consideration, we need to consider bottom quark form factors. The
unrenormalised quark form factors at ${\cal O}({\hat a}_{s}^{n})$
are defined through
\begin{align}
\label{eq:bBH-DefFb}
{\hat{\cal F}}^{H,(n)}_{b{\bar b}} \equiv
\frac{\langle{\hat{\cal
M}}^{H,(0)}_{b{\bar b}}|{\hat{\cal M}}^{H,(n)}_{b{\bar b}}\rangle}{\langle{\hat{\cal
M}}^{H,(0)}_{b{\bar b}}|{\hat{\cal M}}^{H,(0)}_{b{\bar b}}\rangle}\, ,
\end{align}
where, $n=0, 1, 2, 3, \ldots$\,\,. In the above expressions
$|{\hat{\cal M}}^{H,(n)}_{b{\bar b}}\rangle$ is the
${\cal O}({\hat a}_{s}^{n})$ contribution to the unrenormalised matrix
element for the production of the Higgs boson from on-shell $b{\bar
b}$ annihilation. In terms of these quantities, the full matrix element and the full
form factors can be written as a series expansion in ${\hat a}_{s}$ as
\begin{align}
\label{eq:bBH-DefFlambda}
|{\cal M}^{H}_{b{\bar b}}\rangle \equiv \sum_{n=0}^{\infty} {\hat
a}^{n}_{s} S^{n}_{\epsilon}
\left( \frac{Q^{2}}{\mu^{2}} \right)^{n\frac{\epsilon}{2}}
|{\hat{\cal M}}^{H,(n)}_{b{\bar b}} \rangle \, ,
\qquad \qquad
{\cal F}^{H}_{b{\bar b}} \equiv
\sum_{n=0}^{\infty} \left[ {\hat a}_{s}^{n} S_{\epsilon}^{n}
\left( \frac{Q^{2}}{\mu^{2}} \right)^{n\frac{\epsilon}{2}}
{\hat{\cal F}}^{H,(n)}_{b{\bar b}}\right]\, ,
\end{align}
where $Q^{2}=-2\, p_{1}.p_{2}=-q^{2}$ and $p_i$ ($p_{i}^{2}=0$) are
the momenta of the external on-shell bottom quarks. The results of the
form factors up to two loop were present for a long time in~\cite{Harlander:2003ai,
Anastasiou:2011qx} and the three loop one was
computed recently in~\cite{Gehrmann:2014vha}.
The form factor ${\cal F}^{H}_{b{\bar b}}(\hat{a}_{s}, Q^{2}, \mu^{2}, \epsilon)$
satisfies the $KG$-differential equation \cite{Sudakov:1954sw,
Mueller:1979ih, Collins:1980ih, Sen:1981sd, Magnea:1990zb} which is
a direct consequence of the facts that QCD amplitudes exhibit
factorisation property, gauge and renormalisation group (RG)
invariances:
\begin{equation}
\label{eq:bBH-KG}
Q^2 \frac{d}{dQ^2} \ln {\cal F}^{H}_{b{\bar b}} (\hat{a}_s, Q^2, \mu^2, \epsilon)
= \frac{1}{2} \left[ K^{H}_{b{\bar b}} \left(\hat{a}_s, \frac{\mu_R^2}{\mu^2}, \epsilon
\right) + G^{H}_{b{\bar b}} \left(\hat{a}_s, \frac{Q^2}{\mu_R^2},
\frac{\mu_R^2}{\mu^2}, \epsilon \right) \right]\,.
\end{equation}
In the above expression, all the poles in dimensional regularisation
parameter $\epsilon$ are captured in the $Q^{2}$ independent function
$K^{H}_{b{\bar b}}$ and the quantities which are finite as
$\epsilon \rightarrow 0$ are encapsulated in $G^{H}_{b{\bar b}}$. The
solution of the above $KG$ equation can be obtained as~\cite{Ravindran:2005vv} (see also
\cite{Ahmed:2014cla,Ahmed:2014cha})
\begin{align}
\label{eq:bBH-lnFSoln}
\ln {\cal F}^{H}_{b{\bar b}}(\hat{a}_s, Q^2, \mu^2, \epsilon) =
\sum_{k=1}^{\infty} {\hat a}_{s}^{k}S_{\epsilon}^{k} \left(\frac{Q^{2}}{\mu^{2}}\right)^{k
\frac{\epsilon}{2}} \hat {\cal L}_{b{\bar b}, k}^{H}(\epsilon)
\end{align}
with
\begin{align}
\label{eq:bBH-lnFitoCalLF}
\hat {\cal L}_{b{\bar b},1}^{H}(\epsilon) =& { \frac{1}{\epsilon^2} } \Bigg\{-2 A^{H}_{{b{\bar b}},1}\Bigg\}
+ { \frac{1}{\epsilon}
}
\Bigg\{G^{H}_{{b{\bar b}},1}
(\epsilon)\Bigg\}\, ,
\nonumber\\
\hat {\cal L}_{{b{\bar b}},2}^{H}(\epsilon) =& { \frac{1}{\epsilon^3} } \Bigg\{\beta_0 A^{H}_{{b{\bar b}},1}\Bigg\}
+ {
\frac{1}{\epsilon^2} }
\Bigg\{- {
\frac{1}{2} } A^{H}_{{b{\bar b}},2}
- \beta_0 G^{H}_{{b{\bar b}},1}(\epsilon)\Bigg\}
+ { \frac{1}{\epsilon}
} \Bigg\{ {
\frac{1}{2} } G^{H}_{{b{\bar b}},2}(\epsilon)\Bigg\}\, ,
\nonumber\\
\hat {\cal L}_{{b{\bar b}},3}^{H}(\epsilon) =& { \frac{1}{\epsilon^4}
} \Bigg\{- {
\frac{8}{9} } \beta_0^2 A^{H}_{{b{\bar b}},1}\Bigg\}
+ {
\frac{1}{\epsilon^3} }
\Bigg\{ { \frac{2}{9} } \beta_1 A^{H}_{{b{\bar b}},1}
+ { \frac{8}{9} }
\beta_0 A^{H}_{{b{\bar b}},2} + { \frac{4}{3} }
\beta_0^2 G^{H}_{{b{\bar b}},1}(\epsilon)\Bigg\}
\nonumber\\
&
+ { \frac{1}{\epsilon^2} } \Bigg\{- { \frac{2}{9} } A^{H}_{{b{\bar b}},3}
- { \frac{1}{3} } \beta_1 G^{H}_{{b{\bar b}},1}(\epsilon)
- { \frac{4}{3} } \beta_0 G^{H}_{{b{\bar b}},2}(\epsilon)\Bigg\}
+ { \frac{1}{\epsilon}
} \Bigg\{ { \frac{1}{3} }
G^{H}_{b{\bar b},3}(\epsilon)\Bigg\}\, .
\end{align}
In Appendix~\ref{chpt:App-KGSoln}, the derivation of the above
solution is discussed in great details.
$A^{H}_{b{\bar b}}$'s are called the cusp anomalous
dimensions. The constants $G^{H}_{{b{\bar b}},i}$'s are the coefficients of $a_{s}^{i}$ in the following
expansions:
\begin{align}
\label{eq:bBH-GandAExp}
G^{H}_{b{\bar b}}\left(\hat{a}_s, \frac{Q^2}{\mu_R^2}, \frac{\mu_R^2}{\mu^2},
\epsilon \right) &= G^{H}_{b{\bar b}}\left(a_{s}(Q^{2}), 1, \epsilon \right)
+ \int_{\frac{Q^{2}}{\mu_{R}^{2}}}^{1}
\frac{dx}{x} A^{H}_{b{\bar b}}(a_{s}(x\mu_{R}^{2}))
\nonumber\\
&= \sum_{i=1}^{\infty}a_{s}^{i}(Q^{2})
G^{H}_{{b{\bar b}},i}(\epsilon) +
\int_{\frac{Q^{2}}{\mu_{R}^{2}}}^{1}
\frac{dx}{x} A^{H}_{b{\bar b}}(a_{s}(x\mu_{R}^{2}))\,.
\end{align}
However, the solutions of the logarithm of the form factor involves
the unknown functions $G^{H}_{{b{\bar b}},i}$ which are observed to fulfill
\cite{Ravindran:2004mb, Moch:2005tm} the following decomposition in
terms of collinear ($B^{H}_{b{\bar b}}$), soft ($f^{H}_{b{\bar b}}$) and UV
($\gamma^{H}_{b{\bar b}}$) anomalous dimensions:
\begin{align}
\label{eq:bBH-GIi}
G^{H}_{{b{\bar b}},i} (\epsilon) = 2 \left(B^{H}_{{b{\bar b}},i} -
\gamma^{H}_{{b{\bar b}},i}\right) + f^{H}_{{b{\bar b}},i} +
C^{H}_{{b{\bar b}},i} + \sum_{k=1}^{\infty} \epsilon^k g^{H,k}_{{b{\bar b}},i} \, ,
\end{align}
where, the constants $C^{H}_{{b{\bar b}},i}$ are given by
\cite{Ravindran:2006cg}
\begin{align}
\label{eq:bBH-Cg}
C^{H}_{{b{\bar b}},1} &= 0\, ,
\nonumber\\
C^{H}_{{b{\bar b}},2} &= - 2 \beta_{0} g^{H,1}_{{b{\bar b}},1}\, ,
\nonumber\\
C^{H}_{{b{\bar b}},3} &= - 2 \beta_{1} g^{H,1}_{{b{\bar b}},1} - 2
\beta_{0} \left(g^{H,1}_{{b{\bar b}},2} + 2 \beta_{0} g^{H,2}_{{b{\bar b}},1}\right)\, .
\end{align}
In the above expressions, $X^{H}_{{b{\bar b}},i}$ with $X=A,B,f$ and
$\gamma^{H}_{{b{\bar b}}, i}$ are defined through the series expansion in powers
of $a_{s}$:
\begin{align}
\label{eq:bBH-ABfgmExp}
X^{H}_{b{\bar b}} &\equiv \sum_{i=1}^{\infty} a_{s}^{i}
X^{H}_{{b{\bar b}},i}\,,
\qquad \text{and} \qquad
\gamma^{H}_{b{\bar b}} \equiv \sum_{i=1}^{\infty} a_{s}^{i} \gamma^{H}_{{b{\bar b}},i}\,\,.
\end{align}
$f_{i~{\bar i}}^{I}$ are introduced for the first time in the
article~\cite{Ravindran:2004mb} where it is shown to fulfill the
maximally non-Abelian property up to two loop level whose validity is
reconfirmed in~\cite{Moch:2005tm} at three loop level:
\begin{align}
\label{eq:bBH-MaxNAf}
f^{H}_{b{\bar b}} = \frac{C_F}{C_A} f^{H}_{gg}\,.
\end{align}
This identity implies the soft anomalous dimensions for the Higgs
boson production in bottom quark annihilation are related to the same
appearing in the Higgs boson production in gluon fusion through a
simple ratio of the quadratic casimirs of SU(N) gauge group. The same property is also obeyed by the
cusp anomalous dimensions up to three loop level:
\begin{align}
\label{eq:bBH-MaxNAA}
A^{H}_{b{\bar b}} = \frac{C_F}{C_A} A^{H}_{gg}\,.
\end{align}
It is not clear whether this nice property holds true beyond this
order of perturbation theory. Moreover, due
to universality of the quantities denoted by $X$, these are
independent of the operators insertion. These are only dependent on the
initial state partons of any process. Moreover, these are independent of
the quark flavors. Hence, being a process of
quark annihilation, we can make use of the existing results up to
three loop which are employed in case of DY pair productions:
\begin{align}
\label{eq:1}
X^{H}_{b{\bar b}} = X^{DY}_{q{\bar q}} = X^{I}_{q{\bar q}}=X_{q{\bar q}}\,.
\end{align}
Here, $q$ denotes the independence of the quantities on the quark flavors and absence of $I$ represents the independence of the quantities on the nature of colorless particles.
$f^{H}_{b{\bar b}}$ can be found in \cite{Ravindran:2004mb, Moch:2005tm},
$A^{H}_{{b{\bar b}}}$ in~\cite{Moch:2004pa, Moch:2005tm, Vogt:2004mw, Vogt:2000ci}
and $B^{H}_{{b{\bar b}}}$ in \cite{Vogt:2004mw, Moch:2005tm} up to three loop
level. For readers' convenience we list them all up to three loop
level in the
Appendix~\ref{chpt:App-AnoDim}.
Utilising the results of these known quantities and comparing
the above expansions of $G^{H}_{{b{\bar b}},i}(\epsilon)$, Eq.~(\ref{eq:bBH-GIi}), with
the results of the logarithm of the form factors, we extract the
relevant $g_{{b{\bar b}},i}^{H,k}$ and $\gamma^{H}_{{b{\bar b}},i}$'s up to three
loop. For soft-virtual cross section at N$^{3}$LO we need
$g^{H,1}_{b{\bar b},3}$ in addition to the quantities arising from one and two
loop. The form factors for the Higgs boson production in $b{\bar b}$
annihilation up to two loop
can be found in~\cite{Harlander:2003ai,
Ravindran:2005vv, Ravindran:2006cg} and the three loop one is
calculated very recently in the article~\cite{Gehrmann:2014vha}.
These results are employed to extract the
required $g^{H,k}_{{b{\bar b}},i}$'s using Eq.~(\ref{eq:bBH-lnFSoln}),
(\ref{eq:bBH-lnFitoCalLF}) and (\ref{eq:bBH-GIi}).
The relevant one loop terms are found to be
\begin{align}
\label{eq:bBH-gk1}
g_{b{\bar b},1}^{H,1} = C_F \Bigg\{ - 2 + \zeta_2 \Bigg\},\, \quad \quad
g_{b{\bar b},1}^{H,2} = C_F \Bigg\{ 2 - \frac{7}{3} \zeta_3 \Bigg\} , \,\quad \quad
g_{b{\bar b},1}^{H,3} = C_F \Bigg\{ - 2 + \frac{1}{4} \zeta_2 + \frac{47}{80} \zeta_2^2 \Bigg\}\,,
\end{align}
the relevant two loop terms are
\begin{align}
\label{eq:bBH-gk2}
g_{b{\bar b},2}^{H,1} &= C_F n_f \Bigg\{ \frac{616}{81} + \frac{10}{9} \zeta_2 - \frac{8}{3} \zeta_3 \Bigg\}
+ C_F C_A \Bigg\{ - \frac{2122}{81} - \frac{103}{9} \zeta_2 +
\frac{88}{5} {\zeta_2}^2 + \frac{152}{3}
\zeta_3 \Bigg\}
\nonumber \\
& + C_F^2 \Bigg\{ 8 + 32 \zeta_2 - \frac{88}{5} {\zeta_2}^2 - 60 \zeta_3 \Bigg\} \,,
\nonumber \\
g_{b{\bar b},2}^{H,2} &=
C_F n_f \Bigg\{ \frac{7}{12} {\zeta_2}^2 - \frac{55}{27} \zeta_2 +
\frac{130}{27} \zeta_3 - \frac{3100}{243}
\Bigg\}
+ C_A C_F \Bigg\{ - \frac{365}{24} {\zeta_2}^2 + \frac{89}{3} \zeta_2 \zeta_3 + \frac{1079}{54} \zeta_2
\nonumber \\
& - \frac{2923}{27} \zeta_3 - 51 \zeta_5 + \frac{9142}{243} \Bigg\}
+ C_F^2 \Bigg\{ \frac{ 96}{5} {\zeta_2}^2 - 28 \zeta_2 \zeta_3
- 44 \zeta_2 + 116 \zeta_3 + 12 \zeta_5 - 24 \Bigg\}
\nonumber
\end{align}
and the required three loop term is
\begin{align}
g_{b{\bar b},3}^{H,1} &=
C_A^2 C_F \Bigg\{ - \frac{6152}{63} {\zeta_2}^3 + \frac{2738}{9} {\zeta_2}^2
+ \frac{976}{9} \zeta_2 \zeta_3 - \frac{342263}{486} \zeta_2
- \frac{1136}{3} {\zeta_3}^2 + \frac{19582}{9} \zeta_3
\nonumber \\
&
+ \frac{1228}{3} \zeta_5
+ \frac{4095263}{8748} \Bigg\}
+ C_A C_F^2 \Bigg\{ - \frac{15448}{105} {\zeta_2}^3 - \frac{3634}{45} {\zeta_2}^2
- \frac{2584}{3} \zeta_2 \zeta_3 + \frac{13357}{9} \zeta_2
\nonumber \\
&
+ 296 \zeta_3^2
- \frac{11570}{9} \zeta_3 - \frac{1940}{3} \zeta_5 - \frac{613}{3} \Bigg\}
+ C_A C_F n_f \Bigg\{ - \frac{1064}{45} {\zeta_2}^2 + \frac{392}{9} \zeta_2 \zeta_3
+ \frac{44551}{243} \zeta_2
\nonumber \\
&
- \frac{41552}{81} \zeta_3
- 72 \zeta_5 - \frac{6119}{4374} \Bigg\}
+ C_F^2 n_f \Bigg\{ \frac{772}{45} {\zeta_2}^2 - \frac{152}{3} \zeta_2 \zeta_3
- \frac{3173}{18} \zeta_2 + \frac{15956}{27} \zeta_3
\nonumber\\
&-\frac{368}{3} \zeta_5
+ \frac{32899}{324}\Bigg\}
+ C_F n_f^2 \Bigg\{ - \frac{40}{9} {\zeta_2}^2 - \frac{892}{81} \zeta_2
+ \frac{320}{81} \zeta_3 - \frac{27352}{2187} \Bigg\}
\nonumber \\
&
+ C_F^3 \Bigg\{ \frac{21584}{105} {\zeta_2}^3 - \frac{1644}{5} {\zeta_2}^2
+ 624 \zeta_2 \zeta_3
- 275 \zeta_2 + 48 \zeta_3^2
- 2142 \zeta_3 + 1272 \zeta_5 + 603 \Bigg\} \,.
\end{align}
The results up to two loop were present in the literature~\cite{Ravindran:2005vv, Ravindran:2006cg}, however the
three loop result is the new one which is computed in this thesis for
the first time. The other constants $\gamma^{H}_{b{\bar b},i}$,
appearing in the Eq.~(\ref{eq:bBH-GIi}), up to three loop
($i=3$) are obtained as
\begin{align}
\label{eq:bBH-gamma}
\gamma^H_{b{\bar b}, 1}&= 3 C_F \, ,
\nonumber \\
\gamma^H_{b{\bar b}, 2}&= \frac{3}{2} C_F^2
+ \frac{97}{6} C_F C_A
- \frac{5}{3} C_F n_f \, ,
\nonumber \\
\gamma^H_{b{\bar b}, 3}&= \frac{129}{2} C_F^3
- \frac{129}{4} C_F^2 C_A
+ \frac{11413}{108} C_F C_A^2
+\Big(-23+24 \zeta_3\Big) C_F^2 n_f
\nonumber \\
& +\left(-\frac{278}{27} -24 \zeta_3\right) C_F C_A n_f
- \frac{35}{27} C_F n_f^2 \, .
\end{align}
These will be utilised in the next subsection to determine overall
operator renormalisation constant.
\subsection{Operator Renormalisation Constant}
\label{ss:bBH-OOR}
The strong coupling constant renormalisation through $Z_{a_{s}}$ is
not sufficient to make the form factor ${\cal F}^{H}_{b{\bar b}}$ completely UV
finite, one needs to perform additional renormalisation to remove the
residual UV divergences which is reflected through the presence of
non-zero $\gamma^{H}_{b{\bar b}}$ in Eq.~(\ref{eq:bBH-GIi}). This additional
renormalisation is called the overall operator renormalisation which
is performed through the constant $Z^{H}_{b{\bar b}}$. This is determined by
solving the underlying RG equation:
\begin{align}
\label{eq:bBH-ZRGE}
\mu_{R}^{2} \frac{d}{d\mu_{R}^{2}} \ln Z^{H}_{b{\bar b}} \left( {\hat a}_{s},
\mu_{R}^{2}, \mu^{2}, \epsilon \right) = \sum_{i=1}^{\infty}
a_{s}^{i}(\mu_R^2) \gamma^{H}_{b{\bar b},i}\,.
\end{align}
Using the results of $\gamma^{H}_{b{\bar b},i}$ from Eq.~(\ref{eq:bBH-gamma}) and
solving the above RG equation following the methodology described in
the Appendix~\ref{chpt:App-SolRGEZas}, we obtain the following overall renormalisation
constant up to three loop level:
\begin{align}
\label{eq:bBH-OOR-Soln}
Z^H_{b{\bar b}} &= 1+ \sum\limits_{k=1}^{\infty} {\hat a}_s^k S_{\epsilon}^k \left( \frac{\mu_R^2}{\mu^2}
\right)^{k\frac{\epsilon}{2}} {\hat Z}_{b{\bar b}}^{H,(k)}
\end{align}
where,
\begin{align}
\label{eq:bBH-OOR-Soln-1}
{\hat Z}_{b{\bar b}}^{H,(1)} &= \frac{1}{\epsilon} \Bigg\{
6 C_F \Bigg\}\,,
\nonumber\\
{\hat Z}_{b{\bar b}}^{H,(2)} &= \frac{1}{\epsilon^2} \Bigg\{
- 22 C_F C_A
+ 18 C_F^2
+ 4 n_f C_F
\Bigg\}
+ \frac{1}{\epsilon} \Bigg\{
\frac{97}{6} C_F C_A
+ \frac{3}{2} C_F^2
- \frac{5}{3} n_f C_F
\Bigg\}\,,
\nonumber\\
{\hat Z}_{b{\bar b}}^{H,(3)} &=
\frac{1}{\epsilon^3} \Bigg\{
\frac{968}{9} C_F C_A^2
- 132 C_F^2 C_A
+ 36 C_F^3
- \frac{352}{9} n_f C_F C_A
+ 24 n_f C_F^2
+ \frac{32}{9} n_f^2 C_F
\Bigg\}
\nonumber\\
&
+ \frac{1}{\epsilon^2} \Bigg\{
- \frac{4880}{27} C_F C_A^2
+ \frac{247}{3} C_F^2 C_A
+ 9 C_F^3
+ \frac{1396}{27} n_f C_F C_A
- \frac{10}{3} n_f C_F^2
- \frac{80}{27} n_f^2 C_F
\Bigg\}
\nonumber\\
&
+ \frac{1}{\epsilon} \Bigg\{
\frac{11413}{162} C_F C_A^2
- \frac{43}{2} C_F^2 C_A
+ 43 C_F^3
- \frac{556}{81} n_f C_F C_A
- \frac{46}{3} n_f C_F^2
- \frac{70}{81} n_f^2 C_F
\nonumber\\
&
- 16 \zeta_3 n_f C_F C_A
+ 16 \zeta_3 n_f C_F^2
\Bigg\}\,.
\end{align}
\subsection{Mass Factorisation Kernel}
\label{ss:bBH-MFK}
The UV finite form factor contains additional divergences arising from
the soft and collinear regions of the loop momenta. In this section,
we address the issue of collinear divergences and describe a
prescription to remove them. The collinear singularities that arise in
the massless limit of partons are removed by absorbing the divergences
in the bare PDF through
renormalisation of the PDF. This prescription is called the mass
factorisation (MF) and is performed at the factorisation scale
$\mu_F$. In the process of performing this, one needs to introduce mass factorisation kernels
$\Gamma^I_{ij}(\hat{a}_s, \mu^2, \mu_F^2, z, \epsilon)$ which
essentially absorb the collinear singularities. More specifically, MF
removes the collinear singularities arising from the collinear
configuration associated with the initial state partons. The final
state collinear singularities are guaranteed to go away once the phase
space integrals are performed after summing over the contributions from
virtual and real emission diagrams, thanks to
Kinoshita-Lee-Nauenberg (KLN) theorem.
The kernels satisfy
the following RG equation :
\begin{align}
\label{eq:bBH-kernelRGE}
\mu_F^2 \frac{d}{d\mu_F^2} \Gamma^I_{ij}(z,\mu_F^2,\epsilon) =
\frac{1}{2} \sum\limits_{k} P^I_{ik} \left(z,\mu_F^2\right) \otimes \Gamma^I_{kj} \left(z,\mu_F^2,\epsilon \right)
\end{align}
where, $P^I\left(z,\mu_{F}^{2}\right)$ are Altarelli-Parisi splitting
functions (matrix valued). Expanding $P^{I}\left(z,\mu_{F}^{2}\right)$ and
$\Gamma^{I}(z,\mu_F^2,\epsilon)$ in powers of the strong coupling constant
we get
\begin{align}
\label{eq:bBH-APexpand}
&P^{I}_{ij}(z,\mu_{F}^{2}) = \sum_{k=1}^{\infty} a_{s}^{k}(\mu_{F}^{2})P^{I,(k-1)}_{ij}(z)\,
\intertext{and}
&\Gamma^I_{ij}(z,\mu_F^2,\epsilon) = \delta_{ij}\delta(1-z) + \sum_{k=1}^{\infty}
{\hat a}_{s}^{k} S_{\epsilon}^{k} \left(\frac{\mu_{F}^{2}}{\mu^{2}}\right)^{k
\frac{\epsilon}{2}} {\hat \Gamma}^{I,(k)}_{ij}(z,\epsilon)\, .
\end{align}
The RG equation of $\Gamma^{I}(z,\mu_F^2,\epsilon)$,
Eq.~(\ref{eq:bBH-kernelRGE}), can be solved in dimensional regularisation
in powers of ${\hat a}_{s}$. In the $\overline{MS}$ scheme, the
kernel contains only the poles in $\epsilon$. The solutions~\cite{Ravindran:2005vv} up to the
required order $\Gamma^{I,(3)}(z,\epsilon)$ in terms of $P^{I,(k)}(z)$
are presented in the Appendix~(\ref{eq:App-Gamma-GenSoln}).
The relevant ones up
to three loop, $P^{I,(0)}(z), P^{I,(1)}(z) ~\text{and}~ P^{I,(2)}(z)$ are
computed in the articles~\cite{Moch:2004pa, Vogt:2004mw}. For the SV
cross section only the diagonal parts of the splitting functions
$P^{I,(k)}_{ij}(z)$ and kernels $\Gamma^{I,(k)}_{ij}(z,\epsilon)$
contribute since the diagonal elements of $P^{I,(k)}_{ij}(z)$ contain
$\delta(1-z)$ and ${\cal D}_{0}$ whereas the off-diagonal elements are
regular in the limit $z \rightarrow 1$.
The most remarkable fact is that these quantities are universal, independent of
the operators insertion. Hence, for the process under consideration,
we make use of the existing process independent results of kernels and splitting
functions:
\begin{align}
\label{eq:bBH-Gamma-P-ProcessInd}
\Gamma^H_{ij} = \Gamma^I_{ij} = \Gamma_{ij} \qquad \text{and} \qquad P^H_{ij} = P^I_{ij}=P_{ij}\,.
\end{align}
The absence of $I$ represents the independence of these quantities on
$I$. In the next subsection, we discuss the only remaining ingredient,
namely, the
soft-collinear distribution.
\subsection{Soft-Collinear Distribution}
\label{ss:bBH-SCD}
The resulting expression from form factor along with the operator
renormalisation constant and mass factorisation kernel is not
completely finite, it contains some residual divergences which get
cancelled against the contribution arising from soft gluon
emissions. Hence, the finiteness of $\Delta_{b{\bar b}}^{H, \text{SV}}$ in
Eq.~(\ref{eq:bBH-sigma}) in the limit
$\epsilon \rightarrow 0$ demands that the soft-collinear distribution,
$\Phi^H_{b{\bar b}} (\hat{a}_s, q^2, \mu^2, z, \epsilon)$, has pole structure
in $\epsilon$ similar to that of residual divergences. In
articles~\cite{Ravindran:2005vv} and \cite{Ravindran:2006cg}, it was
shown that $\Phi^{H}_{b{\bar b}}$ must obey $KG$ type integro-differential
equation, which we call ${\overline{KG}}$ equation, to remove that
residual divergences:
\begin{align}
\label{eq:bBH-KGbarEqn}
q^2 \frac{d}{dq^2} \Phi^H_{b{\bar b}}\left(\hat{a}_s, q^2, \mu^2, z,
\epsilon\right) = \frac{1}{2} \left[ \overline K^H_{b{\bar b}}
\left(\hat{a}_s, \frac{\mu_R^2}{\mu^2}, z,
\epsilon \right) + \overline G^H_{b{\bar b}} \left(\hat{a}_s,
\frac{q^2}{\mu_R^2}, \frac{\mu_R^2}{\mu^2}, z, \epsilon \right) \right] \, .
\end{align}
${\overline K}^{H}_{b{\bar b}}$ and ${\overline G}^{H}_{b{\bar b}}$ play similar roles as those of $K^{H}_{b{\bar b}}$
and $G^{H}_{b{\bar b}}$ in Eq.~(\ref{eq:bBH-KG}), respectively. Also,
$\Phi^H_{b{\bar b}} (\hat{a}_s, q^2, \mu^2, z, \epsilon)$ being independent of
$\mu_{R}^{2}$ satisfy the RG equation
\begin{align}
\label{eq:bBH-RGEphi}
\mu_{R}^{2}\frac{d}{d\mu_{R}^{2}}\Phi^H_{b{\bar b}} (\hat{a}_s, q^2, \mu^2, z, \epsilon) = 0\, .
\end{align}
This RG invariance and the demand of cancellation of all the residual
divergences arising from ${\cal F}^H_{b{\bar b}}, Z^H_{b{\bar b}}$ and $\Gamma^H_{b{\bar b}}$
against $\Phi^{H}_{b{\bar b}}$ implies the solution of the ${\overline {KG}}$
equation as~\cite{Ravindran:2005vv, Ravindran:2006cg}
\begin{align}
\label{eq:bBH-PhiSoln}
\Phi^H_{b{\bar b}} (\hat{a}_s, q^2, \mu^2, z, \epsilon) &=
\Phi^H_{b{\bar
b}}
(\hat{a}_s,
q^2(1-z)^{2},
\mu^2,
\epsilon)
\nonumber\\
&= \sum_{i=1}^{\infty}
{\hat a}_{s}^{i}
\left(\frac{q^{2}(1-z)^{2}}{\mu^{2}}\right)^{i
\frac{\epsilon}{2}}
S_{\epsilon}^{i}
\left(\frac{i\epsilon}{1-z}\right)
{\hat
\Phi}^{H}_{b{\bar b},i}(\epsilon)
\end{align}
with
\begin{align}
\label{eq:bBH-phiHatIi}
{\hat \Phi}^H_{b{\bar b},i}(\epsilon) = {\hat {\cal L}}_{b{\bar b},i}^{H}(\epsilon)\left(A^H_{b{\bar b},j}
\rightarrow - A^H_{b{\bar b},j}, G^H_{b{\bar b},j}(\epsilon) \rightarrow
{\overline{\cal G}}^H_{b{\bar b},j}(\epsilon)\right)
\end{align}
where, ${\hat {\cal L}}_{b{\bar b},i}^{H}(\epsilon)$ are defined in
Eq.~(\ref{eq:bBH-lnFitoCalLF}). In
Appendix~\ref{chpt:App-Soft-Col-Dist}, the derivation of this
solution is depicted in great details.
The z-independent constants ${\overline{\cal G}}^{H}_{b{\bar b},i}(\epsilon)$ can
be obtained by comparing the poles as well as non-pole terms in $\epsilon$
of ${\hat \Phi}^{H}_{b{\bar b},i}(\epsilon)$ with those arising from form factor,
overall renormalisation constant and splitting functions. We find
\begin{align}
\label{eq:bBH-calGexpans}
\overline {\cal G}^{H}_{b{\bar b},i}(\epsilon)&= - f_{b{\bar b},i}^H + {\overline
C}_{b{\bar b},i}^{H} +
\sum_{k=1}^\infty \epsilon^k
{\overline {\cal
G}}^{H,k}_{b{\bar b},i} \, ,
\end{align}
where,
\begin{align}
\label{eq:bBH-overlineCiI}
&{\overline C}_{b{\bar b},1}^{H} = 0\, ,
\nonumber\\
&{\overline C}_{b{\bar b},2}^{H} = -2\beta_{0}{\overline{\cal G}}_{b{\bar b},1}^{H,1}\, ,
\nonumber\\
&{\overline C}_{b{\bar b},3}^{H} = -2\beta_{1}{\overline{\cal G}}_{b{\bar b},1}^{H,1} -
2\beta_{0}\left({\overline{\cal G}}_{b{\bar b},2}^{H,1} + 2\beta_{0}{\overline{\cal G}}_{b{\bar
b},1}^{H,2} \right)\, .
\end{align}
However, due to the universality of the soft gluon contribution,
$\Phi^{H}_{b{\bar b}}$ must be the same as that of the DY pair
production in quark annihilation since this quantity only depends on
the initial state partons, it does not depend on the final state
colorless particle:
\begin{align}
\label{eq:PhiAgPhiHg}
\Phi^H_{b{\bar b}} &= \Phi^{DY}_{q{\bar q}} = \Phi^I_{q{\bar q}}
\nonumber\\
\text{i.e.}~~{\overline{\cal G}}^{H,k}_{b{\bar b},i} &= {\overline{\cal G}}^{DY,k}_{q{\bar q},i} =
{\overline{\cal G}}^{I,k}_{q{\bar q},i}\,.
\end{align}
In the above expression, $\Phi^{I}_{q{\bar q}}$ and ${\overline{\cal G}}^{I,k}_{q{\bar q},i}$ are written
in order to emphasise the universality of these quantities i.e.
$\Phi^{I}_{q{\bar q}}$ and ${\overline{\cal G}}^{I,k}_{q{\bar q},i}$ can be used for
any quark annihilation process, these are independent of the operators
insertion. In the beginning, it was observed
in~\cite{Ravindran:2006cg,Ravindran:2005vv} that these quantities satisfy the maximally
non-Abelian property up to ${\cal O}(a_s^2)$:
\begin{align}
\label{eq:bBH-Phi-MaxNA}
\Phi^{I}_{q{\bar q}} = \frac{C_F}{C_A} \Phi^I_{gg} \qquad \text{and}
\qquad {\overline{\cal G}}^{I,k}_{q{\bar q}, i} = \frac{C_F}{C_A} {\overline{\cal G}}^{I,k}_{gg,i}\,.
\end{align}
Some of the relevant
constants, namely, ${\overline{\cal G}}_{q{\bar q},1}^{I,1},{\overline{\cal G}}_{q{\bar
q},1}^{I,2},{\overline{\cal G}}_{q{\bar q},2}^{I,1}$ are
computed~\cite{Ravindran:2006cg,Ravindran:2005vv} from the
results of the explicit computations of soft gluon emissions to the DY
productions. However, to calculate the SV cross section at N$^3$LO, we
need to have the results of ${\overline{\cal G}}_{q{\bar q},1}^{I,3},{\overline{\cal G}}_{q{\bar
q},2}^{I,2}$. These are obtained by employing the above
symmetry~(\ref{eq:bBH-Phi-MaxNA}). In~\cite{deFlorian:2012za}, the
soft corrections to the production
cross section of the Higgs boson through gluon fusion to ${\cal
O}(a_s^2)$ was computed to
all orders in dimensional regularisation parameter
$\epsilon$. Utilising this all order result, we
extract ${\overline{\cal G}}_{gg,1}^{H,3}$ and ${\overline{\cal G}}_{gg,2}^{H,2}$. These essentially
lead us to obtain the corresponding quantities for DY production by
means of the maximally non-Abelian symmetry. The
third order constant ${\overline{\cal G}}_{gg,3}^{H,1}$ is extracted from the result of
SV cross section for the production of the Higgs boson at
N$^{3}$LO~\cite{Anastasiou:2014vaa}. We conjecture that the symmetry
relation~(\ref{eq:bBH-Phi-MaxNA}) holds true even at the three loop
level! Therefore, utilising that property we obtain the corresponding
quantity for the DY production, ${\overline{\cal G}}_{q{\bar q},3}^{DY,1}$ which was
presented for the first time in the
article~\cite{Ahmed:2014cla}. Later the result was reconfirmed through threshold
resummation in~\cite{Catani:2014uta} and
explicit computations in~\cite{Li:2014bfa}. This, in turn, establishes
our conjecture of maximally non-Abelian property at N$^3$LO. Being
flavor independent, we can employ
all these constants to the problem under consideration. Below, we list the
relevant ones that contribute up to N$^3$LO level:
\begin{align}
\label{eq:bBH-calGres}
{\overline {\cal G}}^{H,1}_{b{\bar b},1} &= C_F \Bigg\{ - 3 \zeta_2 \Bigg\} \,,
\nonumber\\
{\overline {\cal G}}^{H,2}_{b{\bar b},1} &= C_F \Bigg\{ \frac{7}{3}
\zeta_3 \Bigg\} \,,
\nonumber\\
{\overline {\cal G}}^{H,3}_{b{\bar b},1} &= C_F \Bigg\{ - \frac{3}{16}
{\zeta_2}^2 \Bigg\} \,,
\nonumber\\
{\overline {\cal G}}^{H,1}_{b{\bar b},2} &= C_F n_f \Bigg\{ -
\frac{328}{81} +
\frac{70}{9} \zeta_2 +
\frac{32}{3} \zeta_3
\Bigg\}
+ C_A C_F \Bigg\{ \frac{2428}{81} - \frac{469}{9} \zeta_2
+ 4 {\zeta_2}^2 - \frac{176}{3} \zeta_3 \Bigg\}
\,,
\nonumber\\
{\overline {\cal G}}^{H,2}_{b{\bar b},2} &= C_A C_F \Bigg\{
\frac{11}{40} {\zeta_2}^2
- \frac{203}{3} {\zeta_2}
{\zeta_3}
+ \frac{1414}{27} {\zeta_2} + \frac{2077}{27} {\zeta_3} +
43 {\zeta_5} -
\frac{7288}{243} \Bigg\}
\nonumber\\
& + C_F n_f \Bigg\{ -\frac{1}{20} {\zeta_2}^2 -
\frac{196}{27} {\zeta_2} - \frac{310}{27} {\zeta_3} +
\frac{976}{243} \Bigg\}
\nonumber\\
{\overline {\cal G}}^{H,1}_{b{\bar b},3} &=
C_F {C_A}^2 \Bigg\{\frac{152}{63} \;{\zeta_2}^3 + \frac{1964}{9} \;{\zeta_2}^2
+ \frac{11000}{9} \;{\zeta_2} {\zeta_3} - \frac{765127}{486} \;{\zeta_2}
+\frac{536}{3} \;{\zeta_3}^2 - \frac{59648}{27} \;{\zeta_3}
\nonumber\\
&
- \frac{1430}{3} \;{\zeta_5}
+\frac{7135981}{8748}\Bigg\}
+ C_F {C_A} {n_f} \
\Bigg\{-\frac{532}{9} \;{\zeta_2}^2 - \frac{1208}{9} \;{\zeta_2} {\zeta_3}
+\frac{105059}{243} \;{\zeta_2}
\nonumber\\
&+ \frac{45956}{81} \;{\zeta_3}
+\frac{148}{3} \;{\zeta_5} - \frac{716509}{4374} \Bigg\}
+ {C_F^2} {n_f} \
\Bigg\{\frac{152}{15} \;{\zeta_2}^2
- 88 \;{\zeta_2} {\zeta_3}
+\frac{605}{6} \;{\zeta_2} + \frac{2536}{27} \;{\zeta_3}
\nonumber\\
&+\frac{112}{3} \;{\zeta_5}
- \frac{42727}{324}\Bigg\}
+ C_F {n_f}^2
\Bigg\{ \frac{32}{9} \;{\zeta_2}^2 - \frac{1996}{81} \;{\zeta_2}
-\frac{2720}{81} \;{\zeta_3} + \frac{11584}{2187}\Bigg\} \,.
\end{align}
The above ${\overline{\cal G}}^{H,k}_{b{\bar b},i}$ enable us to get the
$\Phi^{H}_{b{\bar b}}$
up to three loop level. This completes all the ingredients required to
compute the SV cross section up to N$^{3}$LO that are presented in the
next section.
\section{Results of the SV Cross Sections}
\label{sec:bBH-Res}
In this section, we present our findings of the SV cross section at
N$^{3}$LO along with the results of previous orders. Expanding the SV
cross section $\Delta^{H, {\rm SV}}_{b{\bar b}}$, Eq.~(\ref{eq:bBH-sigma}), in
powers of $a_{s}(\mu_F^2)$, we obtain
\begin{align}
\label{eq:bBH-SVRenExp}
\Delta_{b{\bar b}}^{H, {\rm SV}}(z, q^{2}, \mu_{F}^{2}) =
\sum_{i=1}^\infty a_s^i(\mu_F^2) \Delta_{b{\bar b},i}^{H, {\rm SV}} (z, q^{2}, \mu_{F}^{2})
\end{align}
where,
\begin{align}
\nonumber
\Delta_{b{\bar b},i}^{H, {\rm SV}} =
\Delta_{b{\bar b},i}^{H, {\rm SV}}|_\delta
\delta(1-z)
+ \sum_{j=0}^{2i-1}
\Delta_{b{\bar b},i}^{H, {\rm SV}}|_{{\cal D}_j}
{\cal D}_j \, .
\end{align}
Before presenting the final result, we present the general
results of the SV cross section
in terms of the anomalous
dimensions $A^H_{b{\bar b}}$, $B^H_{b{\bar b}}$, $f^H_{b{\bar
b}}$, $\gamma^{H}_{b{\bar b}}$ and other quantities arising from form
factor and soft-collinear distribution below:
\input{bBHSV/ResSVabfg}
Upon substituting the values of all the anomalous dimensions, beta
functions and
$g^{H,k}_{b{\bar b},i}$, $\overline{\cal G}^{H,k}_{b{\bar b},i}$,
we obtain the results of the scalar Higgs boson production cross
section at threshold in
$b{\bar b}$ annihilation up to
N$^{3}$LO for the choices of the scale
$\mu_{R}^{2}=\mu_{F}^{2}$:
\input{bBHSV/ResSV}
The results at NLO $\left(\Delta^{H,{\rm SV}}_{b{\bar b},1}\right)$ and NNLO
$\left(\Delta^{H,{\rm SV}}_{b{\bar b},2}\right)$ match with the existing
ones~\cite{Harlander:2003ai}. At N$^3$LO level, only $\Delta^{H,{\rm
SV}}_{b{\bar b},3}|_{{\cal D}_{j}}$ were
known~\cite{Ravindran:2005vv, Ravindran:2006cg}, remaining terms were
not available due to absence of the required quantities
$g^{H,2}_{b{\bar b},2}$, $g^{H,1}_{b{\bar b},3}$ from form factors and
$\overline{\cal G}^{H,2}_{b{\bar b},2}$, $\overline{\cal
G}^{H,1}_{b{\bar b},3}$ from soft-collinear distributions. The
recent results of $g^{H,2}_{b{\bar b},2}$, $g^{H,1}_{b{\bar b},3}$
from~\cite{Gehrmann:2014vha}, $\overline{\cal G}^{H,2}_{b{\bar b},2}$
from~\cite{deFlorian:2012za} and $\overline{\cal G}^{H,1}_{b{\bar
b},3}$ from~\cite{Ahmed:2014cla} are being employed to compute the
missing $\delta(1-z)$ part i.e. $\Delta^{H,{\rm SV}}_{b{\bar
b},3}|_{\delta}$ which completes the full evaluation of the SV
cross section at N$^3$LO $\left(\Delta^{H,{\rm
SV}}_{b{\bar b},3}\right)$ and is presented for the first time
in~\cite{Ahmed:2014cha} by us. For the sake of
completeness, we mention the leading order contribution which is
\begin{align}
\label{eq:bBH-Leading-Order}
\Delta^{H}_{b{\bar b},0} = \delta(1-z)
\end{align}
and the overall factor in Eq.~(\ref{eq:bBH-1}) comes out to be
\begin{align}
\label{eq:bBH-Leading-Order-1}
\sigma^{H,(0)}_{b{\bar b}} \left( \mu_F^2\right) = \frac{\pi \lambda^2\left(\mu_F^2\right)}{12 m_H^2}\,.
\end{align}
The above results are presented for the choice $\mu_R=\mu_F$. The
dependence of the SV cross section on renormalisation scale $\mu_{R}$
can be easily restored by employing the RG evolution of $a_s$ from
$\mu_F$ to $\mu_R$~\cite{Ahmed:2015sna}:
\begin{align}
\label{eq:bBH-asf2asr}
a_s\left(\mu_R^2 \right) &= a_s \left(\mu_F^2\right) \frac{1}{\omega} + a_s^2
\left(\mu_F^2\right) \Bigg\{ \frac{1}{\omega^2} \left(-\eta_1 \log \omega\right)
\Bigg\}
+ a_s^3 \left(\mu_F^2\right) \Bigg\{ \frac{1}{\omega^2}\left(\eta_1^2-\eta_2\right)
\nonumber\\
& + \frac{1}{\omega^3} \left( -\eta_1^2+\eta_2 - \eta_1^2 \log \omega +
\eta_1^2 \log^2 \omega \right) \Bigg\}
\end{align}
where
\begin{align}
\label{eq:bBH-asfasr-1}
&\omega \equiv 1 - \beta_0 a_s\left( \mu_F^2 \right) \log \left(
\frac{\mu_F^2}{\mu_R^2} \right)\,,
\nonumber\\
&\eta_i \equiv \frac{\beta_i}{\beta_0}\,.
\end{align}
The above result of the evolution of the $a_s$ is a resummed one and
the fixed order result can be easily obtained by performing the series
expansion of this equation~(\ref{eq:bBH-asf2asr}).
\section{Numerical Impact of SV Cross Sections}
\label{sec:bBH-Numerics}
The numerical impact of our results can be studied using the exact LO, NLO, NNLO
$\Delta^{H}_{b{\bar b},i},~ i=0,1,2$ and the threshold N$^3$LO result
$\Delta^{H,{\rm SV}}_{b{\bar b},3}$.
We have used $\sqrt{s} = 14$ TeV for the LHC, the $Z$ boson mass $M_Z=91.1876$ GeV
and Higgs boson mass $m_H$ = 125.5 GeV throughout.
The strong coupling constant $\alpha_s (\mu_R^2)$ ($a_s=\alpha_s/4\pi$) is evolved
using the 4-loop RG equations with
$\alpha_s^{\text{N$^3$LO}} (m_Z ) = 0.117$ and for parton density sets we use
MSTW 2008NNLO \cite{Martin:2009iq}.
The Yukawa coupling is evolved using 4 loop RG with
$\lambda(m_b)=\sqrt{2} m_b(m_b)/\nu$ and $m_b(m_b)=4.3$ GeV.
The renormalization scale dependence is studied by varying $\mu_{R}$
between $0.1 ~ m_H$ and $10 ~ m_H$
keeping $\mu_{F}=m_{H}/4$ fixed. For the factorization scale, we have fixed $\mu_R=m_H$ and
varied $\mu_F$ between $0.1 ~ m_H$ and $10 ~ m_H$. We find that the perturbation theory
behaves better if we include more and more higher order terms (see
Fig.\ref{fig:murnmuf}).
\begin{figure}[htb]
\centering
\begin{minipage}[c]{0.48\textwidth}
\includegraphics[width=1.0\textwidth]{bBHSV/bbHmur.pdf}
\end{minipage}
\begin{minipage}[c]{0.48\textwidth}
\includegraphics[width=1.0\textwidth]{bBHSV/bbHmuf.pdf}
\end{minipage}
\caption{\label{fig:murnmuf}
Total cross section for Higgs production in $b\bar{b}$ annihilation at
various orders in $a_s$ as a function of $\mu_R/m_H$ (left panel) and
of $\mu_F/m_H$ (right panel)
at the LHC with $\sqrt{s}=14$ TeV.
}
\end{figure}
\section{Summary}
\label{sec:bBH-Summary}
To summarize, we have systematically developed a framework to compute
threshold contributions in
QCD to the production of Higgs boson in bottom anti-bottom annihilation subprocesses at the
hadron colliders. This formalism is applicable for any colorless particle. Factorization of UV, soft and collinear singularities and
exponentiation of their sum allow us to obtain threshold corrections
order by order in perturbation theory.
Using the recently obtained N${}^3$LO soft distribution function for Drell-Yan production
and the three loop Higgs form factor with bottom anti-bottom quarks, we have obtained threshold
N${}^3$LO corrections to Higgs production through bottom anti-bottom annihilation.
We have also studied the stability of our result under renormalization and factorization scales.
\chapter{\label{chap:ConclOutlook}Conclusions and Outlooks}
\begingroup
\hypersetup{linkcolor=blue}
\minitoc
\endgroup
No doubt, the whole particle physics community is standing on the
verge of a crucial era, where the main tasks can be largely
categorized into two parts: testing the SM with unprecedented accuracy
and searching for the physics beyond SM. In achieving these golden tasks,
precise theoretical predictions play a very crucial role. The field of
precision studies at theoretical level is mostly controlled by the
higher order corrections to the scattering amplitudes, that are the
basic building blocks of constructing any observable in QFT. Among all the higher
order corrections, the QCD ones contribute substantially to any
typical observable. This thesis deals with this higher order QCD
radiative corrections to the observables associated with the
Drell-Yan, scalar and pseudo-scalar Higgs boson.
The Higgs boson is among the best candidates at hadron collider, and
hence it is of utmost importance to make the theoretical prediction as
precise as possible to the associated observables. In the first part
of the thesis, Chapter~\ref{chap:bBCS}, we have computed the N$^3$LO QCD
radiative corrections, arising
from the soft gluons, to the
inclusive production cross section of the Higgs boson produced through
bottom quark annihilation~\cite{Ahmed:2014cha}. Of course, this is not the dominant
production channel of the scalar Higgs boson
in the SM,
nonetheless its contribution must also be taken into account in this spectacular
precision studies. In order to achieve this, we have systematically
employed an elegant
prescription~\cite{Ravindran:2005vv,Ravindran:2006cg}. The
factorisation of QCD
amplitudes, gauge invariance, renormalisation group invariance and the
Sudakov resummation of soft gluons are at the heart of this
formalism. The recently available three loop $Hb{\bar b}$ QCD form
factors~\cite{Gehrmann:2014vha} and the soft gluon contributions
calculated~\cite{Ahmed:2014cla}
from the threshold QCD corrections to the Higgs boson at
N$^3$LO~\cite{Anastasiou:2014vaa}, enable us to compute the full
N$^3$LO soft-virtual QCD corrections
to the production cross section of the Higgs boson produced through
bottom quark annihilation. One of the most beautiful parts of this
calculation is that even without evaluating all the hundreds or
thousands of Feynman diagrams contributing to the real emissions, we
have obtained the required contribution arising from the soft gluons! The
universal nature of the soft gluons are the underlying reasons behind
this remarkable feature. We have also demonstrated the numerical
impact of this result at the LHC. This is the most accurate result for
this production channel
existing in the literature till date and it is expected to play an
important role in coming days.
In the second part of the thesis, Chapter~\ref{chap:Rap}, we have dealt with an another very
important observable, namely, the rapidity distributions of the Higgs
boson produced through gluon fusion and the leptonic pair in
Drell-Yan. The importance of these two processes are quite
self-evident! We have computed the threshold enhanced N$^3$LO QCD
corrections~\cite{Ahmed:2014uya}
to these observables employing the formalism developed in the
article~\cite{Ravindran:2006bu}.
The skeleton of this elegant prescription which has been employed is also based on the
properties, like, the factorisation of QCD
amplitudes, gauge invariance, renormalisation group invariance and the
Sudakov resummation of soft gluons. With the help of recently computed inclusive production
cross section of the Higgs boson~\cite{Anastasiou:2014vaa} and
Drell-Yan~\cite{Ahmed:2014cla} at
threshold N$^3$LO QCD, we have computed the contributions arising from
the soft gluons to the processes under consideration. These were the
only missing ingredients to achieve our goal. Our newly calculated
part of this distribution is found to be the most dominant one
compared to the other contributions. We have demonstrated numerically
the impact of this result for the Higgs boson at the LHC. Indeed,
inclusion of this N$^3$LO contributions does reduce the dependence on
the unphysical renormalisation and factorisation scales.
It is worth mentioning that, this beautiful formalism not only
helps us to compute the rapidity distribution at threshold, but also
enhance our understanding about the underlying structures of the QCD
amplitudes.
In the third part of the thesis, Chapter~\ref{chap:Multiloop}, we have
discussed the relatively modern techniques of the multiloop
computations which have been employed to get some of the results
calculated in this thesis. The backbone of this methodology is the
integration-by-parts~\cite{Tkachov:1981wb,
Chetyrkin:1981qh} and Lorentz invariant~\cite{Gehrmann:1999as}
identities. The successful implementation of these in computer codes
revolutionizes the area of multiloop computations.
The last part, Chapter~\ref{chap:pScalar}, is dealt with a particle,
pseudo-scalar,
which is not included in particle spectrum of the SM, but is believed
to be present in the nature. Intensive search for this particle has
been going on for past several years, although nothing conclusive
evidence has been found. However, to make conclusive remark about the
existence of this particle, we need to revamp the understanding about
this particle and improve the precision of the theoretical
predictions. This work arises exactly at this context. In
these articles~\cite{Ahmed:2015qpa, Ahmed:2015pSSV}, we have computed
one of the important ingredients to
calculate the inclusive production cross section or the differential
distributions for the pseudo-scalar at N$^3$LO QCD which is presently
the level of accuracy for the scalar Higgs boson, achieved very
recently~\cite{Anastasiou:2015ema}. In particular, we have derived
the three loop massless QCD corrections to the quark and gluon form
factors of the pseudo-scalar. Unlike the scalar Higgs boson, this
problem involves the $\gamma_5$ which makes the life interesting as
well as challenging. We have handled them under the `t Hooft-Veltman
prescription for the $\gamma_5$ and Levi-Civita tensor in dimensional
regularisation. Employing this prescription, however, brings some
additional complication, namely, it violates the chiral Ward
identity. In order to rectify this, we need to perform an additional
and non-trivial finite renormalisation. By exploiting the universal
behaviour of the infrared pole structure at three loops in QCD, we
were able to independently determine the renormalisation constants and
operator mixing, in agreement with the earlier results that were
obtained in a completely different
approach~\cite{Larin:1993tq,Zoller:2013ixa}. We must emphasize the
approach which we have employed here for the first time is exactly
opposite to the usual one: the infrared pole
structures of the form factors have been taken to be universal that
dictates us to obtain the UV operator renormalisation constants upon
imposing the demand of UV finiteness. With our new results, the
threshold approximation to the N$^3$LO inclusive production cross
section for the pseudo-scalar through gluon fusion are
obtained~\cite{Ahmed:2015pSSV} by us. This is also extended to the N$^3$LL resummed
accuracy in~\cite{Ahmed:2015pSSV}. We have also
computed the hard matching coefficients in the context of
soft-collinear effective theory which are later employed to obtain the
N$^3$LL' resummed cross section~\cite{Ahmed:2016otz}. We have also found some
interesting facts about the form factors in the context of Leading
Transcendentality principle~\cite{hep-th/0611204, hep-th/0404092,
Kotikov:2001sc}: the LT terms of the diagonal
form factors with replacement $C_A = C_F = N$ and
$T_{f} n_{f}=N/2$ are not only identical to
each other but also coincide with the LT terms of the QCD form
factors~\cite{Gehrmann:2010ue} with the same replacement as well as
with the LT terms of the scalar form factors in ${\cal N}=4$ SYM
\cite{Gehrmann:2011xn}, up to a normalization factor of
$2^{l}$. This observation holds true for the finite terms in
$\epsilon$, and could equally be validated for higher-order terms up
to transcendentality 8 (which is the highest order for which all
three-loop master integrals are available~\cite{Lee:2010ik}). In
addition to checking the diagonal form factors, we also examined the
off-diagonal ones,
where we find that the LT terms these two form factors are identical
to each other after the replacement of colour factors. However, the LT
terms of these do not coincide with those of the diagonal ones.
The state-of-the-art techniques, which mostly use our in-house codes,
have been employed extensively to
carry out all the computations presented in this thesis. The prescription of
computing the threshold correction is applicable for any colorless
final state particle. We are in the process of extending this
formalism to the case of threshold resummation of differential
rapidity distributions. The methodology of calculating the
pseudo-scalar form factors can be generalized to the cases involving any number
of operators which can mix among each others under UV renormalisation.
In conclusion, it has been a while the Higgs-like particle has been discovered at the LHC and finally, we are
very close to having enough statistics for precision measurements of the Higgs quantum numbers and coupling
constants to fermions and gauge bosons. This, along with the precise results from theoreticians like us, hopefully,
would help to explore the underlying nature of the electroweak symmetry breaking and possibly open the door
of new physics.
\chapter{\label{chap:Multiloop}A Diagrammatic Approach To Compute
Multiloop Amplitude}
\begingroup
\hypersetup{linkcolor=blue}
\minitoc
\endgroup
\section{Prologue}
\label{sec:Multi-pro}
The scattering amplitudes play the most crucial role in any quantum
field theory. These are the gateway to unveil the elegant structures
associated with the quantum world. At the phenomenological level,
they are the main ingredients in predicting the observables at high energy
colliders for the processes within and beyond the SM. Hence, the efficient evaluation
of the scattering amplitudes is of prime importance at theoretical
as well as experimental level. However, in perturbative QFT, the theoretical predictions based
on the leading order calculation happens to be unreliable. One must go
beyond the leading order to make the predictions more accurate and
reliable. While considering the effects arising from the higher
orders, the contributions coming from the QCD radiations dominate
substantially, in particular, at high energy colliders like Tevatron
or LHC. In this thesis, we are concentrating only on the corrections arising
from the QCD sector. In the process of computing these higher order
QCD corrections, one has to carry out three different types of
contributions, namely, virtual, real and real-virtual processes. Upon clubbing
together all the three contributions appropriately, finite result for
any observable is obtained. As very much expected, the complexity involved in
the calculations grows very
rapidly as we go towards higher and higher orders in perturbation
theory, where more and more different pieces interfere with each other
that eventually contribute to the final physical observables.
In this Chapter, we will confine our discussion only to the higher
order QCD virtual or loop corrections. There exists at least two
different formalisms to compute these.
\begin{enumerate}
\item\label{item:1} \textit{Diagrammatic approach:} one directly evaluates all
the relevant Feynman diagrams appearing at the perturbative order under
consideration.
\item\label{item:2} \textit{Unitary-based approach:} the unitary
properties of the scattering amplitudes are employed extensively to
avoid the direct evaluation of all the Feynman diagrams.
\end{enumerate}
Despite the spectacular beauty of the unitary based approach, its
applicability to the computation of the amplitudes remains confined mostly to one
loop or only few multiloop problems. Its generalisation to any
multiloop computation is still unavailable in the literature. In these
more complicated scenarios, the first methodology of directly evaluating
the Feynman diagrams is more effective and is therefore employed more often.
\section{Feynman Diagrams and Simplifications}
\label{sec:Multi-dia}
For any generic scattering process in QFT, we can expand any
observable in powers of all the coupling
constants present in the underlying Lagrangian. Feynman diagrams are the diagrammatic representations of
this expansion. In this thesis, we confine our discussion into QCD. Let us
consider a scattering process involving $E$ external particles with
momenta $p_1, p_2, \cdots, p_E$. Without loss of generality, we
consider the cross-section which can be expanded in powers of strong
coupling constant:
\begin{align}
\label{eq:Multi-CSexpand}
\sigma_E \left(p_1, p_2, \cdots, p_E\right) = \sum\limits_{l=0}^{\infty}
a_s^l\sigma_E^{(l)}\left(p_1, p_2, \cdots, p_E\right)\,.
\end{align}
For the sake of simplicity, we suppress all the dependence on quantum
numbers of the external particles. The index $l$ denotes the order of
perturbative expansion. The cross section for $l=0$ is called the leading
order (LO), $l=1$ is next-to-leading order (NLO) and so on.
The cross section at at each perturbative
order, $\sigma^{(l)}_E$, is related to the scattering matrix elements through
\begin{align}
\label{eq:Multi-Sigma-MatrixEle}
\sigma_E^{(0)} &= K \int | \, | {\cal M}_E^{(0)} \rangle \, |^2 \, d
\Phi_E\,,
\nonumber\\
\sigma_E^{(1)} &= K \int 2 \, \text{Re} \, ( \, \langle {\cal
M}_E^{(0)} | {\cal M}_E^{(1)} \rangle \, ) \, d
\Phi_E
+ K \int | \, | {\cal M}_{E + 1}^{(0)} \rangle \, |^2 \, d \Phi_{E +
1}\,,
\nonumber\\
\sigma_E^{(2)} &= K \int 2 \, \text{Re} \, ( \, \langle {\cal
M}_E^{(0)} | {\cal M}_E^{(2)} \rangle \, ) d \Phi_E
+ K \int 2 \, \text{Re} \, ( \, \langle {\cal M}_{E + 1}^{(0)} | {\cal
M}_{E + 1}^{(1)} \rangle \, ) \, d \Phi_{E + 1}
\nonumber\\
&+ K \int | \, | {\cal M}_{E + 2}^{(0)} \rangle \, |^2 \, d \Phi_{E +
2}
\nonumber\\
\text{and}&~ \text{so on.}
\end{align}
In the above set of equations, $| {\cal M}_E^{(l)} \rangle$ is the scattering
amplitude at $l^{\rm th}$ order in perturbation theory involving $E$
number of external particles. The quantity $d\Phi_E$ is the phase
space element. "Re" denotes the real part of the amplitude and $K$ is
an overall constant containing various factors. The amplitudes with $E$
number of external particles and $l \ge 1$ represent the contributions
arising from the virtual Feynman diagrams, whereas amplitudes with
more than $E$ number of external particles come from the real emission
diagrams. In this chapter, we address the issue of evaluating the
virtual diagrams. The scattering matrix element can also be expanded perturbatively in
powers of $a_s$ as
\begin{align}
\label{eq:Multi-MatrixEle-Expand}
| {\cal M}_E \rangle = \sum\limits_{l=0}^{\infty}
a_s^l | {\cal M}_E^{(l)} \rangle\,.
\end{align}
Each term in the right hand side can be represented through a set of
Feynman diagrams of same perturbative order. In this chapter, we will
explain the prescription to evaluate the contribution to the matrix
element arising from the virtual diagrams.
The evaluation of the scattering matrix element at any particular order begins with the
generation of associated Feynman diagrams. We make use of a package,
named, QGRAF~\cite{Nogueira:1991ex} to accomplish this job. QGRAF does
not provide the graphical representation of the
Feynman diagrams, rather it generates those symbolically. We use our
in-house codes written in FORM~\cite{Vermaseren:2000nd} to convert the raw
output into a format for further computation. Employing
the Feynman rules derived from the underlying Lagrangian,
which are the languages establishing the connection between the diagrams
and the corresponding formal mathematical expressions, we obtain the
amplitude. The raw amplitude contains series of
Dirac gamma matrices, QCD color factors, Dirac and Lorentz
indices. We simplify those using our in-house codes. We perform the
color simplification in SU(N) gauge theory and follow dimensional
regularisation where the space-time dimension is considered to be
$d=4+\epsilon$. The amplitude, beyond
leading order, consists of a set of tensorial Feynman integrals.
Instead of handling the tensorial integrals, we
multiply the amplitude with appropriate projectors to convert those to
scalar integrals. Hence, the problem essentially boils down to solving
those scalar integrals. Often, at any typical order in perturbation
theory, this involves hundreds or thousands of different scalar loop integrals. Of
course, start evaluating all of these integrals is not a practical way
of dealing with the problem. Remarkably, it has been found that the
appeared integrals are not independent of each other, they can be
related through some set of identities! This drastically reduces the
independent integrals which ultimately need to be computed. In the next section, we will
elaborate this procedure.
\section{Reduction to Master Integrals}
\label{sec:Multi-Reduction}
The dimensionally regularised Feynman loop integrals do satisfy a
large number of relations, which allow one to express most of those
integrals in terms of a much smaller subset of independent integrals
(where ''independent'' is to be understood in the sense of the
identities introduced below), which are now commonly referred to as
the Master Integrals (MIs). For a detailed review on this,
see~\cite{Gehrmann:1999as, Argeri:2007up}. These identities are known
as \textbf{integration-by-parts} and \textbf{Lorentz invariance} identities.
\subsection{Integration-by-Parts Identities (IBP)}
\label{ss:Multi-IBP}
The integration-by-parts identities~\cite{Tkachov:1981wb,
Chetyrkin:1981qh} are the most important class of identities which
establish the relations among the dimensionally regularised scalar
Feynman integrals. These can be seen as a generalisation of Gauss'
divergence theorem in $d$-dimensions. They are based on the fact that,
given a Feynman integral which is a function of space-time dimensions $d$, there
always exists a value of $d$ in the complex plane where the integral is
well defined and consequently convergent. The necessary condition for the
convergence of an integral is the integrand be zero at the
boundaries. This condition can be rephrased as, the integral of the
total derivative with respect to any loop momenta vanishes, that is
\begin{align}
\label{eq:Multi-IBP-1}
\int \prod\limits_{j=1}^l \frac{d^dk_j}{(2\pi)^{d}}
\frac{\partial}{\partial k_{i}^{\mu}} \left( v_{s}^{\mu}
\frac{1}{{\cal D}_1^{b_1} \cdots
{\cal D}_\beta^{b_\beta}} \right) = 0\,.
\end{align}
In the expression, $k_j$ are the loop momenta, $v_s^{\mu}$ can be loop
or external momenta, $v_s^{\mu}=\{ k_1^{\mu}, \cdots, k_l^{\mu};
p_1^{\mu}, \cdots, p_E^{\mu}\}$. ${\cal D}_i$ are the propagators that
depend on the masses, loop and external momenta. To begin with, a
diagram contains a set of propagators as well as scalar products
involving the loop and external momenta. However, we can express all the
scalar products involving loop momenta in terms of propagators. This
is possible since any Lorentz scalar can be written either in terms of
scalar products or propagators. Both of the representations are
equivalent. For our convenience, we choose to work in the propagator
representation. Performing the differentiation on
the left hand side of the above Eq.~(\ref{eq:Multi-IBP-1}), one obtains
set of IBP identities.
Let us demonstrate the role of IBP identities through an one-loop
example.
\begin{itemize}
\item \textbf{Example:} We consider an one loop box diagram, depicted
through Fig.~\ref{fig:Multi-IBP-Dia} where, all
the external legs are taken to be massless, for simplicity, and the
momentum $q=p_1+p_2+p_3$.
\begin{figure}
\begin{center}
\begin{tikzpicture}[line width=0.6 pt, scale=0.7]
\draw (-4,0) -- (4,0);
\draw (-4,-3) -- (4, -3);
\draw (-2,0) -- (-2,-3);
\draw (2,0) -- (2,-3);
\draw[->] (-3.5,0.3) -- (-3,0.3);
\node at (-3.3,0.8) {$q$};
\draw[->] (-0.2,0.3) -- (0.3,0.3);
\node at (0,0.8) {$k_{1}$};
\draw[->] (3,0.3) -- (3.5,0.3);
\node at (3.2,0.8) {$p_{1}$};
\draw[->] (3,-3.3) -- (3.5,-3.3);
\node at (3.2,-3.8) {$p_{2}$};
\draw[<-] (-0.2,-3.3) -- (0.3,-3.3);
\node at (0,-3.8) {$k_{1}-p_1-p_2$};
\draw[<-] (-3.5,-3.3) -- (-3,-3.3);
\node at (-3.3,-3.8) {$p_{3}$};
\draw[->] (2.3,-1.3) -- (2.3,-1.8);
\node at (3.5,-1.5) {$k_{1}-p_{1}$};
\draw[<-] (-2.3,-1.3) -- (-2.3,-1.8);
\node at (-4.7,-1.5) {$k_{1}-p_{1}-p_2-p_3$};
\end{tikzpicture}
\caption{One loop box}
\label{fig:Multi-IBP-Dia}
\end{center}
\end{figure}
The corresponding dimensionally regularised Feynman integral can be
cast into the following form
\begin{align}
\label{eq:Multi-IBP-Ex-1}
&\int \frac{d^d k}{(2 \pi)^d} \frac{1}{{\cal D}_1^{b_1} \, {\cal D}_2^{b_2} \,
{\cal D}_3^{b_3} \, {\cal D}_4^{b_4}} \equiv I\left[b_1, b_2, b_3, b_4\right]
\intertext{with}
&{\cal D}_1 \equiv k_1\,,
\nonumber\\
&{\cal D}_2 \equiv (k_1 - p_1)\,,
\nonumber\\
&{\cal D}_3 \equiv (k_1 - p_1 - p_2)\,,
\nonumber\\
&{\cal D}_4 \equiv (k_1 - p_1 - p_2 - p_3)\,.
\end{align}
We can obtain 4-set of IBP identities for each choice of the set
$\{b_1, b_2, b_3, b_4\}$. For a choice of $v_s^{\mu}=p_1^{\mu}$ in
Eq.~(\ref{eq:Multi-IBP-1}), we obtain the corresponding IBP identities
as
\begin{align}
\label{eq:Multi-IBP-Ex-2}
0 = \int \frac{d^d k}{(2 \pi)^d} &\bigg[b_1 \left( - 1 +
\frac{{\cal D}_2}{{\cal D}_1}\right) + b_2 \left(1 -
\frac{{\cal D}_1}{{\cal D}_2}\right) - b_3
\left(\frac{{\cal D}_1}{{\cal D}_3} -
\frac{{\cal D}_2}{{\cal D}_3} -
\frac{s}{{\cal D}_3}\right)
\nonumber\\
& - b_4 \left(\frac{{\cal D}_1}{{\cal D}_4} - \frac{{\cal D}_2}{{\cal D}_4} - \frac{s}{{\cal D}_4}
- \frac{u}{{\cal D}_4}\right) \bigg] \frac{1}{{\cal D}_1^{b_1} \,
{\cal D}_2^{b_2} \, {\cal D}_3^{b_3} \, {\cal D}_4^{b_4}}\,.
\end{align}
It can be symbolically expressed as
\begin{align}
\label{eq:Multi-IBP-Ex-3}
0 &= b_1 ( - 1 + 1^+ 2^-) + b_2 (1 - 2^+ 1^-) - b_3 (3^+ 1^- - 3^+ 2^- - s
\, 3^+)
\nonumber\\
&- b_4 (4^+ 1^- - 4^+ 2^- - s \, 4^+ - u \, 4^+)
\end{align}
where, we have made use of the convention as $1^+ 2^- I[b_1, b_2, b_3, b_4] =
I[b_1 + 1, b_2 - 1, b_3, b_4]$ and the associated Mandelstam variables
are defined as $s \equiv (p_1+p_2)^2=2p_1.p_2$, $t \equiv
(p_2+p_3)^2=2p_2.p_3$, $u \equiv (p_1+p_3)^2=2p_1.p_3$. From the
Eq.~(\ref{eq:Multi-IBP-Ex-3}), it is clear
that the IBP identities provide recursion relations among the
integrals of a topology and/or its sub-topologies. Similarly, we
can get the IBP identities corresponding to other external as well as
internal momenta. Upon employing all of these identities, it can be
shown that there exists only three MIs, which are
$I[1,0,1,0]$, $I[1,0,0,1]$ and $I[1,1,1,1]$. Hence, at the end we need
to evaluate only three independent integrals corresponding to the
problem under consideration. For higher loop and more number of
external legs, the IBP identities often become too clumsy to handle
manually. Hence, these identities are generated systematically with
the help of some computer algorithms in some packages, such as
AIR~\cite{Anastasiou:2004vj}, FIRE~\cite{Smirnov:2008iw},
REDUZE~\cite{Studerus:2009ye,vonManteuffel:2012np},
LiteRed~\cite{Lee:2012cn,Lee:2013mka}.
\end{itemize}
\subsection{Lorentz Invariant Identities (LI)}
\label{ss:Multi-LI}
The Lorentz invariance of the scalar Feynman integrals can be used in
order to obtain more set of identities among the integrals, which are
known as Lorentz invariant identities~\cite{Gehrmann:1999as}:
\begin{align}
\label{eq:Multi-LI-1}
p_{j}^{[\mu} p_{k}^{\nu]} \sum_i p_{i,[\mu} \frac{\partial}{\partial p_i^{\nu]}} I(p_i) = 0\,.
\end{align}
It has been recently found~\cite{Lee:2008tj} that the LI identities
are not independent from IBP ones, since these can be reproduced
generating and solving larger systems of IBPs. However, use of LI
identities along with the IBP helps to speed up the solution. Hence,
in almost all of the computer codes for performing automated reduction
to MIs, LI identities are therefore extensively used.
Employing the IBP and LI identities, we obtain a set of MIs which
ultimately need to be evaluated. Upon evaluation of the MIs, we can
obtain the final unrenormalised result of the virtual
corrections. Often these contain UV as well as soft and collinear
divergences. The UV divergences are removed through UV
renormalisation. The UV renormalised result of the virtual corrections exhibit
a universal infrared pole structures which serve a crucial check on
the correctness of the computation. In the next chapter, we employ
this methodology to compute the three loop quark and gluon form
factors in QCD
for the production of a pseudo-scalar.
\section{Summary}
\label{ss:Multi-Summary}
We have discussed the techniques largely used for the computations of the multiloop amplitudes which is mostly based on the IBP and LI identities. These are employed in the computer codes to automatise the reduction process. Among some packages, we utilise LiteRed~\cite{Lee:2013mka, Lee:2012cn} for our computations. In these articles~\cite{Ahmed:2014gla, Ahmed:2014pka, Ahmed:2015qia, Ahmed:2015qpa}, we have applied this methodology successfully to compute the 2- and 3-loop QCD corrections. In the next chapter, we will present the computation of 3-loop QCD form factors for the pseudo-scalar production where we have essentially made use of the methodology discussed in this chapter.
\chapter{\label{chap:pScalar}Pseudo-Scalar Form Factors at Three Loops
in QCD}
\textit{\textbf{The materials presented in this chapter are the result of an original research done in collaboration with Thomas Gehrmann, M. C. Kumar, Prakash Mathews, Narayan Rana and V. Ravindran, and these are based on the published articles~\cite{Ahmed:2015qpa, Ahmed:2015pSSV}}}.
\\
\begingroup
\hypersetup{linkcolor=blue}
\minitoc
\endgroup
\section{Prologue}
\setcounter{equation}{0}
\label{sec:intro}
Form factors are the matrix elements of local composite operators
between physical states. In the calculation of scattering cross
sections, they provide the purely virtual corrections. For example,
in the context of hard scattering processes such as
Drell-Yan~\cite{Altarelli:1979ub,Hamberg:1990np} and the Higgs boson
production in gluon
fusion~\cite{Dawson:1990zj,Djouadi:1991tka,Graudenz:1992pv,Spira:1995rr,
Djouadi:1995gt, Spira:1997dg,Catani:2001ic, Harlander:2002wh,
Anastasiou:2002yz, Ravindran:2003um, Ravindran:2004mb,
Harlander:2005rq,Anastasiou:2015ema}, the form factors corresponding
to the vector current operator $\overline \psi \gamma_\mu \psi$ and
the gluonic operator $G^{a}_{\mu \nu} G^{a,\mu\nu}$ contribute,
respectively. Here $\psi$ is the fermionic field operator and
$G^{a}_{\mu \nu}$ is the field tensor of the non-Abelian gauge field
$A_\mu^a$ corresponding to the gauge group SU(N). In QCD the form factors can be computed order by order
in the strong coupling constant using perturbation theory. Beyond
leading order, the UV renormalisation of the form
factors involves the renormalisation of the composite operator itself,
besides the standard procedure for coupling constant and external
fields.
The resulting UV finite form factors still contain divergences of
infrared origin, namely, soft and collinear divergences due to
the presence of massless gluons and quarks/ antiquarks in the theory.
The inclusive hard scattering cross sections require, in addition to
the form factor, the real-emission partonic subprocesses as well as
suitable mass factorisation kernels for incoming partons. The soft
divergences in the form factor resulting from the gluons cancel
against those present in the real emission processes and the mass
factorisation kernels remove the remaining collinear divergences
rendering the hadronic inclusive cross section IR finite. While these
IR divergences cancel among various parts in the perturbative
computations, they can give rise to logarithms involving physical
scales and kinematic scaling variables of the processes under study.
In kinematical regions where these logarithms become large, they may
affect the convergence and reliability of the perturbation series
expansion in powers of the coupling constant. The solution for this
problem goes back to the pioneering work by
Sudakov~\cite{Sudakov:1954sw} on the asymptotic behaviour of the form
factor in Quantum Electrodynamics: all leading logarithms can be
summed up to all orders in perturbation theory. Later on, this
resummation was extended to non-leading
logarithms~\cite{Collins:1980ih} and systematised for non-Abelian
gauge theories~\cite{Sen:1981sd}. Ever since, form factors have been
central to understand the underlying structure of amplitudes in gauge
theories.
The infrared origin of universal logarithmic corrections to form
factors~\cite{Magnea:1990zb} and scattering amplitudes results in a
close interplay between resummation and infrared pole structure.
Working in dimensional regularisation in $d=4+\epsilon$ dimensions,
these poles appear as inverse powers in the Laurent expansion in
$\epsilon$. In a seminal paper, Catani~\cite{Catani:1998bh} proposed a
universal formula for the IR pole structure of massless two-loop QCD
amplitudes of arbitrary multiplicity (valid through to double pole
terms). This formula was later on justified systematically from
infrared factorization~\cite{Sterman:2002qn}, thereby also revealing
the structure of the single poles in terms of the anomalous dimensions
for the soft radiation. In \cite{Ravindran:2004mb}, it was shown that
the single pole term in quark and gluon form factors up to two loop
level can be shown to decompose into UV ($\gamma_{I}, I=q,g$) and
universal collinear ($B_I$), color singlet soft ($f_I$) anomalous
dimensions, later on observed to hold even at three loop level
in~\cite{Moch:2005tm}. An all loop conjecture for the pole structure
of the on-shell multi-loop multi-leg amplitudes in SU(N) gauge theory
with $n_f$ light flavors in terms of cusp ($A_I$), collinear ($B_{I}$)
and soft anomalous dimensions ($\Gamma_{IJ},f_I$ - colour non-singlet
as well as singlet) was proposed by Becher and
Neubert~\cite{Becher:2009cu} and Gardi and Magnea~\cite{Gardi:2009qi},
generalising the earlier results~\cite{Catani:1998bh,Sterman:2002qn}.
The validity of this conjecture beyond three loops depends on the
presence/absence of non-trivial colour correlations and crossing
ratios involving kinematical invariants~\cite{Almelid:2015jia} and
there exists no all-order proof at present. The three-loop expressions
for cusp, collinear and colour singlet soft anomalous dimensions were
extracted~\cite{Moch:2005ba,Laenen:2005uz} from the three loop flavour
singlet~\cite{Vogt:2004mw} and non-singlet~\cite{Moch:2004pa}
splitting functions, thereby also predicting~\cite{Moch:2005tm} the
full pole structure of the three-loop form factors.
The three-loop quark and gluon form factors through to finite terms
were computed
in~\cite{Baikov:2009bg,Gehrmann:2010ue,Gehrmann:2011xn,Gehrmann:2014vha}
and subsequently extended to higher powers in the $\epsilon$
expansion~\cite{Gehrmann:2010tu}. These results were enabled by
modern techniques for multi-loop calculations in quantum field theory,
in particular integral reduction methods. These are based on
IBP~\cite{Tkachov:1981wb,Chetyrkin:1981qh} and
LI~\cite{Gehrmann:1999as} identities which reduce
the set of thousands of multi-loop integrals to the one with few
integrals, so called MIs in dimensional
regularisation. To solve these large systems of IBP and LI identities,
the Laporta algorithm~\cite{Laporta:2001dd}, which is based on
lexicographic ordering of the integrals, is the main tool of
choice. It has been implemented in several computer algebra
codes~\cite{Anastasiou:2004vj, Smirnov:2008iw, Studerus:2009ye,
vonManteuffel:2012np, Lee:2012cn,Lee:2013mka}. The MIs relevant to
the form factors are single-scale three-loop vertex functions, for
which analytical expressions were derived in
Refs.~\cite{Gehrmann:2005pd, Gehrmann:2006wg, Heinrich:2007at,
Heinrich:2009be, Lee:2010cga, Gehrmann:2010ue}.
Recently, some of us have applied these state-of-the-art methods to
accomplish the task of computing spin-2 quark and gluon form factors
up to three loops~\cite{Ahmed:2015qia} level in SU(N) gauge theory
with $n_f$ light flavours. These form factors are ingredients to the
precise description of production cross sections for graviton
production, that are predicted in extensions of the SM.
In addition, the spin-2 form factors relate to operators with higher
tensorial structure and thus provide the opportunity to test the
versatility and robustness of calculational techniques for the vertex
functions at three loop level. The results~\cite{Ahmed:2015qia} also
confirm the universality of the UV and IR structure of the gauge
theory amplitudes in dimensional regularisation.
In the present work, we derive the three-loop corrections to the quark
and gluon form factors for pseudo-scalar operators. These operators
appear frequently in effective field theory descriptions of extensions
of the SM. Most notably, a pseudo-scalar state coupling to
massive fermions is an inherent prediction of any two-Higgs doublet
model~\cite{Fayet:1974pd,Fayet:1976et, Fayet:1977yc,
Dimopoulos:1981zb, Sakai:1981gr,Inoue:1982pi, Inoue:1983pp,
Inoue:1982ej}. In the limit of infinite fermion mass, this gives
rise to the operator insertions considered here. The recent discovery
of a Standard-Model-like Higgs boson at the LHC~\cite{Aad:2012tfa,
Chatrchyan:2012xdj} has not only revived the interest in such Higgs
bosons but also prompted the study of the properties of the discovered
boson to identify either with lightest scalar or pseudo-scalar Higgs
bosons of extended models. Such a study requires precise predictions
for their production cross sections. In the context of a CP-even
scalar Higgs boson, results for the inclusive production cross section
in the gluon fusion are available up to N$^{3}$LO
QCD~\cite{Anastasiou:2002yz,Harlander:2002wh,Ravindran:2003um,
Anastasiou:2015ema}, based on an effective scalar coupling that
results from integration of massive quark loops that mediate the
coupling of the Higgs boson to
gluons~\cite{Ellis:1975ap,Shifman:1979eb,Kniehl:1995tn}. On the other
hand for the CP-odd pseudo-scalar, only NNLO QCD
results~\cite{Kauffman:1993nv,Djouadi:1993ji,Harlander:2002vv,Anastasiou:2002wq,
Ravindran:2003um} in the effective theory~\cite{Chetyrkin:1998mw}
are known. The exact quark mass dependence for scalar and
pseudo-scalar production is known to NLO
QCD~\cite{Spira:1993bb,Spira:1995rr}, and is usually included through
a re-weighting of the effective theory results. Soft gluon resummation
of the gluon fusion cross section has been performed to N$^3$LL for
the scalar
case~\cite{Catani:2003zt,Moch:2005ky,Ravindran:2005vv,Ravindran:2006cg,
Idilbi:2005ni,Ahrens:2008nc,deFlorian:2009hc,Bonvini:2014joa,Catani:2014uta}
and to NNLL for the pseudo-scalar case~\cite{deFlorian:2007sr}. A
generic threshold resummation formula valid to N$^3$LL accuracy for
colour-neutral final states was derived in~\cite{Catani:2014uta},
requiring only the virtual three-loop amplitudes as process-dependent
input. The numerical impact of soft gluon resummation in scalar and
pseudo-scalar Higgs boson production and its combination with mass
corrections is reviewed comprehensively in~\cite{Schmidt:2015cea}.
The three-loop corrections to the pseudo-scalar form factors computed
in this thesis are an important ingredient to the N$^3$LO and N$^3$LL
gluon fusion cross sections~\cite{Ahmed:2015pSSV} for pseudo-scalar
Higgs boson production, thereby enabling predictions at the same level
of precision that is attained in the scalar case.
The framework of the calculation is outlined in
Section~\ref{sec:frame}, where we describe the effective
theory~\cite{Chetyrkin:1998mw}. Due to the pseudo-scalar coupling,
one is left with two effective operators with same quantum number and
mass dimensions, which mix under renormalisation. Since these
operators contain the Levi-Civita tensor as well as $\gamma_5$, the
computation of the matrix elements requires additional care in
$4+\epsilon$ dimensions where neither Levi-Civita tensor nor
$\gamma_5$ can be defined unambiguously. We use the prescription by
't Hooft and Veltman~\cite{'tHooft:1972fi,Larin:1993tq} to define
$\gamma_5$. We describe the calculation in Section~\ref{sec:FF},
putting particular emphasis on the UV renormalisation. Exploiting the
universal IR pole structure of the form factors, we determine the UV
renormalisation constants and mixing of the effective operators up to
three loop level. We also show that the finite renormalisation
constant, known up to three loops~\cite{Larin:1993tq}, required to
preserve one loop nature of the chiral anomaly, is consistent with
anomalous dimensions of the overall renormalisation constants. As a
first application of our form factors, we compute the hard matching
functions for N$^3$LL resummation in soft-collinear effective theory
(SCET) in Section~\ref{sec:scet}. Section~\ref{sec:conc} summarises
our results and contains an outlook on future applications to
precision phenomenology of pseudo-scalar Higgs production.
\section{Framework of the Calculation}
\label{sec:frame}
\subsection{The Effective Lagrangian}
\label{sec:ThreResu}
A pseudo-scalar Higgs boson couples to gluons only indirectly through
a virtual heavy quark loop. This loop can be integrated out in the
limit of infinite quark mass. The resulting effective
Lagrangian~\cite{Chetyrkin:1998mw} encapsulates the interaction
between a pseudo-scalar $\Phi^A$ and QCD particles and reads:
\begin{align} {\cal L}^{A}_{\rm eff} = \Phi^{A}(x) \Big[ -\frac{1}{8}
{C}_{G} O_{G}(x) - \frac{1}{2} {C}_{J} O_{J}(x)\Big]
\end{align}
where the operators are defined as
\begin{equation}
O_{G}(x) = G^{\mu\nu}_a \tilde{G}_{a,\mu
\nu} \equiv \epsilon_{\mu \nu \rho \sigma} G^{\mu\nu}_a G^{\rho
\sigma}_a\, ,\qquad
O_{J}(x) = \partial_{\mu} \left( \bar{\psi}
\gamma^{\mu}\gamma_5 \psi \right) \,.
\label{eq:operators}
\end{equation}
The Wilson coefficients $C_G$ and $C_J$ are obtained by integrating
out the heavy quark loop, and $C_G$ does not receive any QCD
corrections beyond one loop due to the Adler-Bardeen
theorem~\cite{Adler:1969gk}, while $C_J$ starts only at second order
in the strong coupling constant. Expanded in
$a_s \equiv {g}_{s}^{2}/(16\pi^{2}) = \alpha_s/(4\pi)$,
they read
\begin{align}
\label{eq:const}
& C_{G} = -a_{s} 2^{\frac{5}{4}} G_{F}^{\frac{1}{2}} {\rm \cot} \beta
\nonumber\\
& C_{J} = - \left[ a_{s} C_{F} \left( \frac{3}{2} - 3\ln
\frac{\mu_{R}^{2}}{m_{t}^{2}} \right) + a_s^2 C_J^{(2)} + \cdots \right] C_{G}\, .
\end{align}
In the above expressions, $G^{\mu\nu}_{a}$ and $\psi$ represent
gluonic field strength tensor and light quark fields, respectively and
$G_{F}$ is the Fermi constant and ${\rm \cot}\beta$ is the mixing angle in a generic
Two-Higgs-Doublet model.
$a_{s} \equiv a_{s} \left( \mu_{R}^{2} \right)$ is the strong coupling
constant renormalised at the scale $\mu_{R}$ which is related to the
unrenormalised one, ${\hat a}_{s} \equiv {\hat g}_{s}^{2}/(16\pi^{2})$
through
\begin{align}
\label{eq:asAasc}
{\hat a}_{s} S_{\epsilon} = \left( \frac{\mu^{2}}{\mu_{R}^{2}} \right)^{\epsilon/2}
Z_{a_{s}} a_{s}
\end{align}
with
$S_{\epsilon} = {\rm exp} \left[ (\gamma_{E} - \ln 4\pi)\epsilon/2
\right]$
and $\mu$ is the scale introduced to keep the strong coupling constant
dimensionless in $d=4+\epsilon$ space-time dimensions. The
renormalisation constant $Z_{a_{s}}$~\cite{Tarasov:1980au} is given by
\begin{align}
\label{eq:Zas}
Z_{a_{s}}&= 1+ a_s\left[\frac{2}{\epsilon} \beta_0\right]
+ a_s^2 \left[\frac{4}{\epsilon^2 } \beta_0^2
+ \frac{1}{\epsilon} \beta_1 \right]
+ a_s^3 \left[\frac{8}{ \epsilon^3} \beta_0^3
+\frac{14}{3 \epsilon^2} \beta_0 \beta_1 + \frac{2}{3
\epsilon} \beta_2 \right]
\end{align}
up to ${\cal O}(a_{s}^{3})$. $\beta_{i}$ are the coefficients of the
QCD $\beta$ functions which are given by~\cite{Tarasov:1980au}
and presented in Eq.~(\ref{eq:bBH-beta}).
\subsection{Treatment of $\gamma_5$ in Dimensional Regularization}
\label{sec:gamma5}
Higher order calculations of chiral quantities in dimensional regularization face the problem of
defining a generalization of
the inherently four-dimensional objects $\gamma_5$ and $\varepsilon^{\mu\nu\rho\sigma}$
to values of $d\neq 4$.
In this thesis, we have followed the most practical and
self-consistent definition of $\gamma_{5}$ for multiloop calculations
in dimensional regularization which was introduced by 't~Hooft and
Veltman through \cite{'tHooft:1972fi}
\begin{align}
\gamma_5 = i \frac{1}{4!} \varepsilon_{\nu_1 \nu_2 \nu_3 \nu_4}
\gamma^{\nu_1} \gamma^{\nu_2} \gamma^{\nu_3} \gamma^{\nu_4} \,.
\end{align}
Here, $\varepsilon^{\mu\nu\rho\sigma}$ is the Levi-Civita tensor which is contracted as
\begin{align}
\label{eqn:LeviContract}
\varepsilon_{\mu_1\nu_1\lambda_1\sigma_1}\,\varepsilon^{\mu_2\nu_2\lambda_2\sigma_2}=
\large{\left |
\begin{array}{cccc}
\delta_{\mu_1}^{\mu_2} &\delta_{\mu_1}^{\nu_2}&\delta_{\mu_1}^{\lambda_2} & \delta_{\mu_1}^{\sigma_2}\\
\delta_{\nu_1}^{\mu_2}&\delta_{\nu_1}^{\nu_2}&\delta_{\nu_1}^{\lambda_2}&\delta_{\nu_1}^{\sigma_2}\\
\delta_{\lambda_1}^{\mu_2}&\delta_{\lambda_1}^{\nu_2}&\delta_{\lambda_1}^{\lambda_2}&\delta_{\lambda_1}^{\sigma_2}\\
\delta_{\sigma_1}^{\mu_2}&\delta_{\sigma_1}^{\nu_2}&\delta_{\sigma_1}^{\lambda_2}&\delta_{\sigma_1}^{\sigma_2}
\end{array}
\right |}
\end{align}
and all the Lorentz indices are considered to be $d$-dimensional~\cite{Larin:1993tq}.
In this scheme, a finite renormalisation of the
axial vector current is required in order to fulfill chiral Ward identities and the Adler-Bardeen theorem. We discuss this
in detail in Section~\ref{ss:UV} below.
\section{Pseudo-scalar Quark and Gluon Form Factors}
\label{sec:FF}
The quark and gluon form factors describe the QCD loop corrections to
the transition matrix element from a color-neutral operator $O$ to an
on-shell quark-antiquark pair or to two gluons. For the pseudo-scalar
interaction, we need to consider the two operators $O_{G}$ and
$O_{J}$, defined in Eq.~(\ref{eq:operators}), thus yielding in total
four form factors.
We define the unrenormalised gluon form factors at
${\cal O}({\hat a}_{s}^{n})$ as
\begin{align}
\label{eq:DefFg}
{\hat{\cal F}}^{G,(n)}_{g} \equiv \frac{\langle{\hat{\cal
M}}^{G,(0)}_{g}|{\hat{\cal M}}^{G,(n)}_{g}\rangle}{\langle{\hat{\cal
M}}^{G,(0)}_{g}|{\hat{\cal M}}^{G,(0)}_{g}\rangle}\, ,
\qquad \qquad
{\hat{\cal F}}^{J,(n)}_{g} \equiv \frac{\langle{\hat{\cal
M}}^{G,(0)}_{g}|{\hat{\cal M}}^{J,(n+1)}_{g}\rangle}{\langle{\hat{\cal
M}}^{G,(0)}_{g}|{\hat{\cal M}}^{J,(1)}_{g}\rangle}
\end{align}
and similarly the unrenormalised quark form factors through
\begin{align}
\label{eq:DefFq}
{\hat{\cal F}}^{G,(n)}_{q} \equiv \frac{\langle{\hat{\cal
M}}^{J,(0)}_{q}|{\hat{\cal M}}^{G,(n+1)}_{q}\rangle}{\langle{\hat{\cal
M}}^{J,(0)}_{q}|{\hat{\cal M}}^{G,(1)}_{q}\rangle}\, ,
\qquad \qquad
{\hat{\cal F}}^{J,(n)}_{q} \equiv \frac{\langle{\hat{\cal
M}}^{J,(0)}_{q}|{\hat{\cal M}}^{J,(n)}_{q}\rangle}{\langle{\hat{\cal
M}}^{J,(0)}_{q}|{\hat{\cal M}}^{J,(0)}_{q}\rangle}
\end{align}
where, $n=0, 1, 2, 3, \ldots$\,. In the above expressions
$|{\hat{\cal M}}^{\lambda,(n)}_{\beta}\rangle$ is the
${\cal O}({\hat a}_{s}^{n})$ contribution to the unrenormalised matrix
element for the transition from the bare operator $[O_{\lambda}]_B$
$(\lambda = G,J)$ to a quark-antiquark pair ($\beta=q$) or to two
gluons ($\beta=g$). The expansion of these quantities in powers of
${\hat a}_{s}$ is performed through
\begin{tabular}{p{5cm}p{8.5cm}}
\begin{align*}
|{\cal M}^{\lambda}_{\beta}\rangle \equiv \sum_{n=0}^{\infty} {\hat
a}^{n}_{s} S^{n}_{\epsilon}
\left( \frac{Q^{2}}{\mu^{2}} \right)^{n\frac{\epsilon}{2}}
|{\hat{\cal M}}^{\lambda,(n)}_{\beta} \rangle
\end{align*}
&
\begin{equation}
\label{eq:Mexp}
\hspace{-0.5cm}\text{and}\qquad
{\cal F}^{\lambda}_{\beta} \equiv
\sum_{n=0}^{\infty} \left[ {\hat a}_{s}^{n}
\left( \frac{Q^{2}}{\mu^{2}} \right)^{n\frac{\epsilon}{2}}
S_{\epsilon}^{n} {\hat{\cal F}}^{\lambda,(n)}_{\beta}\right]\,.
\end{equation}
\end{tabular}
\\
where, $Q^{2}=-2\, p_{1}.p_{2}$ and $p_{i}'s$ $(p_{i}^{2}=0)$ are the
momenta of the external quarks and gluons. Note that
$|{\hat{\cal M}}^{G,(n)}_{q}\rangle$ and
$|{\hat{\cal M}}^{J,(n)}_{g}\rangle$ start from $n=1$ i.e. from one
loop level.
\subsection{Calculation of the Unrenormalised Form Factors}
\label{ss:CalcFF}
The calculation of the unrenormalised pseudo-scalar form factors up to
three loops follows closely the steps used in the derivation of the
three-loop scalar and vector form factors
\cite{Gehrmann:2010ue,Gehrmann:2014vha}. The Feynman diagrams for all
transition matrix elements (Eq.~(\ref{eq:DefFg}),
Eq.~(\ref{eq:DefFq})) are generated using
QGRAF~\cite{Nogueira:1991ex}. The numbers of diagrams contributing to
three loop amplitudes are 1586 for
$|{\hat{\cal M}}^{G,(3)}_{g}\rangle$, 447 for
$|{\hat{\cal M}}^{J,(3)}_{g}\rangle$, 400 for
$|{\hat{\cal M}}^{G,(3)}_{q}\rangle$ and 244 for
$|{\hat{\cal M}}^{J,(3)}_{q}\rangle$ where all the external particles
are considered to be on-shell. The raw output of QGRAF is converted to
a format suitable for further manipulation. A set of in-house routines
written in the symbolic manipulating program FORM
\cite{Vermaseren:2000nd} is utilized to perform the simplification of
the matrix elements involving Lorentz and color indices. Contributions
arising from ghost loops are taken into account as well since we use
Feynman gauge for internal gluons. For the external on-shell gluons,
we ensure the summing over only transverse polarization states by
employing an axial polarization sum:
\begin{equation}
\label{eq:PolSum}
\sum_{s} {\varepsilon^{\mu}}^{\, *}(p_{i},s)
\varepsilon^{\nu}(p_{i},s) = - \eta^{\mu\nu} + \frac{p_{i}^{\mu}
q_{i}^{\nu} + q_{i}^{\mu} p_{i}^{\nu}}{p_{i}.q_{i}} \quad ,
\end{equation}
where $p_{i}$ is the $i^{\rm th}$-gluon momentum, $q_{i}$ is the
corresponding reference momentum which is an arbitrary light like
4-vector and $s$ stands for spin (polarization) of gluons. We choose
$q_{1}=p_{2}$ and $q_{2}=p_{1}$ for our calculation. Finally, traces
over the Dirac matrices are carried out in $d$ dimensions.
The expressions involve thousands of three-loop scalar
integrals. However, they are expressible in terms of a much smaller
set of scalar integrals, called master integrals (MIs), by use of
IBP~\cite{Tkachov:1981wb, Chetyrkin:1981qh} and LI~\cite{Gehrmann:1999as} identities. These
identities follow from the Poincare invariance of the integrands, they
result in a large linear system of equations for the integrals
relevant to given external kinematics at a fixed loop-order. The LI
identities are not linearly independent from the IBP
identities~\cite{Lee:2008tj}, their inclusion does however help to
accelerate the solution of the system of equations. By employing
lexicographic ordering of these integrals (Laporta
algorithm,~\cite{Laporta:2001dd}), a reduction to MIs is accomplished.
Several implementations of the Laporta algorithm exist in the
literature: AIR~\cite{Anastasiou:2004vj}, FIRE~\cite{Smirnov:2008iw},
Reduze2~\cite{vonManteuffel:2012np, Studerus:2009ye} and
LiteRed~\cite{Lee:2013mka, Lee:2012cn}. In the context of the present
calculation, we used LiteRed~\cite{Lee:2013mka, Lee:2012cn} to perform
the reductions of all the integrals to MIs.
Each three-loop Feynman integral is expressed in terms of a list of
propagators involving loop momenta that can be attributed to one of
the following three sets (auxiliary
topologies,~\cite{Gehrmann:2010ue})
\begin{align}
\label{eq:Basis}
{\rm A}_1 &: \{ {\cal D}_1, {\cal D}_2, {\cal D}_3, {\cal D}_{12}, {\cal D}_{13},
{\cal D}_{23}, {\cal D}_{1;1}, {\cal D}_{1;12}, {\cal D}_{2;1}, {\cal D}_{2;12}, {\cal D}_{3;1},
{\cal D}_{3;12} \}
\nonumber\\
{\rm A}_2 &: \{ {\cal D}_1, {\cal D}_2, {\cal D}_3, {\cal D}_{12}, {\cal D}_{13}, {\cal D}_{23},
{\cal D}_{13;2}, {\cal D}_{1;12}, {\cal D}_{2;1}, {\cal D}_{12;2},
{\cal D}_{3;1}, {\cal D}_{3;12} \}
\nonumber\\
{\rm A}_3 &: \{ {\cal D}_1, {\cal D}_2, {\cal D}_3, {\cal D}_{12}, {\cal D}_{13}, {\cal D}_{123},
{\cal D}_{1;1}, {\cal D}_{1;12}, {\cal D}_{2;1}, {\cal D}_{2;12}, {\cal D}_{3;1},
{\cal D}_{3;12} \}\, .
\end{align}
In the above sets
\begin{align*}
{\cal D}_{i} = k_{i}^2, {\cal D}_{ij} = (k_i-k_j)^2, {\cal D}_{ijl} = (k_i-k_j-k_l)^2,
\end{align*}
\vspace{-0.8cm}
\begin{align*}
{\cal D}_{i;j} = (k_i-p_j)^2, {\cal D}_{i;jl} = (k_i-p_j-p_l)^2, {\cal D}_{ij;l} = (k_i-k_j-p_l)^2
\end{align*}
To accomplish this, we have used the package
Reduze2~\cite{vonManteuffel:2012np, Studerus:2009ye}. In each set in
Eq.~(\ref{eq:Basis}), ${\cal D}'$s are linearly independent and form a
complete basis in a sense that any Lorentz-invariant scalar product
involving loop momenta and external momenta can be expressed uniquely
in terms of ${\cal D}'$s from that set.
As a result, we can express the unrenormalised form factors in terms
of 22 topologically different master integrals (MIs) which can be
broadly classified into three different types: genuine three-loop
integrals with vertex functions ($A_{t,i}$), three-loop propagator
integrals ($B_{t,i}$) and integrals which are product of one- and
two-loop integrals ($C_{t,i}$).
Defining a generic three loop
master integral through
\begin{align}
A_{i, m_{1}^i m_{2}^i \cdots m_{12}^i} = \int \frac{d^d k_1}{(2 \pi)^d}
\int \frac{d^d k_2}{(2 \pi)^d}
\int \frac{d^d k_3}{(2 \pi)^d}
\frac{1}{\prod_{j} D_j^{m_j^i} } , \quad \quad \quad i=1,2,3
\end{align}
where $D_j$ is the $j^{\rm th}$ element of the basis set $A_i$. We
identify the resulting master integrals appeared in our computation to
those given in \cite{Gehrmann:2010ue} and they are listed in the
figures below.
\begin{figure}[h!]
\centering \hspace{-0.5cm}
\begin{subfigure}[b]{0.43\textwidth}
\includegraphics[width=\textwidth]{pScalar/Diag/b61.jpg}
\caption*{$B_{6,1} \equiv A_{1,111000010101}$}
\end{subfigure}
\quad\quad
\begin{subfigure}[b]{0.23\textwidth}
\includegraphics[width=\textwidth]{pScalar/Diag/b62.jpg}
\caption*{$B_{6,2} \equiv A_{1,011110000101}$}
\end{subfigure}
\quad\quad
\begin{subfigure}[b]{0.23\textwidth}
\includegraphics[width=\textwidth]{pScalar/Diag/b81.jpg}
\caption*{$B_{8,1} \equiv A_{3,011111010101}$}
\end{subfigure}
\end{figure}
\begin{figure}[h!]
\centering \hspace{-1cm}
\begin{subfigure}[b]{0.23\textwidth}
\includegraphics[width=\textwidth]{pScalar/Diag/b41.jpg}
\caption*{$B_{4,1} \equiv A_{1,001101010000} $}
\end{subfigure}
\quad\quad\quad
\begin{subfigure}[b]{0.35\textwidth}
\includegraphics[width=\textwidth]{pScalar/Diag/b51.jpg}
\caption*{$B_{5,1} \equiv A_{1,011010010100}$}
\end{subfigure}
\quad\quad
\begin{subfigure}[b]{0.23\textwidth}
\includegraphics[width=\textwidth]{pScalar/Diag/b52.jpg}
\caption*{$B_{5,2} \equiv A_{1,001011010100}$}
\end{subfigure}
\end{figure}
\begin{figure}[h!]
\centering \hspace{-1cm}
\begin{subfigure}[b]{0.31\textwidth}
\includegraphics[width=\textwidth]{pScalar/Diag/c61.jpg}
\caption*{$C_{6,1} \equiv A_{1,011100100101}$}
\end{subfigure}
\quad\quad
\begin{subfigure}[b]{0.31\textwidth}
\includegraphics[width=\textwidth]{pScalar/Diag/c81.jpg}
\caption*{$C_{8,1} \equiv A_{2,111100011101}$}
\end{subfigure}
\quad\quad\quad
\begin{subfigure}[b]{0.22\textwidth}
\includegraphics[width=\textwidth]{pScalar/Diag/a51.jpg}
\caption*{$A_{5,1} \equiv A_{1,001101100001}$}
\end{subfigure}
\end{figure}
\begin{figure}[h!]
\centering
\begin{subfigure}[b]{0.22\textwidth}
\includegraphics[width=\textwidth]{pScalar/Diag/a52.jpg}
\caption*{$A_{5,2} \equiv A_{1,001011011000}$}
\end{subfigure}
\qquad\qquad
\begin{subfigure}[b]{0.22\textwidth}
\includegraphics[width=\textwidth]{pScalar/Diag/a61.jpg}
\caption*{$A_{6,1} \equiv A_{1,010101100110}$}
\end{subfigure}
\hspace{0.3cm} \quad~~~
\begin{subfigure}[b]{0.23\textwidth}
\includegraphics[width=\textwidth]{pScalar/Diag/a62.jpg}
\caption*{$A_{6,2} \equiv A_{1,001111011000}$}
\end{subfigure}
~~
\end{figure}
\begin{figure}[h!]
\centering \hspace{-1.8cm}
\begin{subfigure}[b]{0.22\textwidth}
\includegraphics[width=\textwidth]{pScalar/Diag/a63.jpg}
\caption*{$A_{6,3} \equiv A_{1,001110100101}$}
\end{subfigure}
\qquad\qquad
\begin{subfigure}[b]{0.22\textwidth}
\includegraphics[width=\textwidth]{pScalar/Diag/a71.jpg}
\caption*{$A_{7,1} \equiv A_{2,011110011100}$}
\end{subfigure}
\quad\quad\quad
\begin{subfigure}[b]{0.22\textwidth}
\includegraphics[width=\textwidth]{pScalar/Diag/a72.jpg}
\caption*{$A_{7,2} \equiv A_{2,011011001101}$}
\end{subfigure}
\hspace{-1.3cm}
\end{figure}
\begin{figure}[h!]
\centering \hspace{-1.8cm}
\begin{subfigure}[b]{0.22\textwidth}
\includegraphics[width=\textwidth]{pScalar/Diag/a73.jpg}
\caption*{$A_{7,3} \equiv A_{1,011011110100}$}
\end{subfigure}
\qquad\qquad
\begin{subfigure}[b]{0.22\textwidth}
\includegraphics[width=\textwidth]{pScalar/Diag/a74.jpg}
\caption*{$A_{7,4} \equiv A_{2,011110001101}$}
\end{subfigure}
\end{figure}
\begin{figure}[h!]
\centering \hspace{-1.8cm}
\begin{subfigure}[b]{0.22\textwidth}
\includegraphics[width=\textwidth]{pScalar/Diag/a75.jpg}
\caption*{$A_{7,5} \equiv A_{2,011011010101}$}
\end{subfigure}
\qquad\qquad
\begin{subfigure}[b]{0.22\textwidth}
\includegraphics[width=\textwidth]{pScalar/Diag/a81.jpg}
\caption*{$A_{8,1} \equiv A_{2,001111011101}$}
\end{subfigure}
\end{figure}
\begin{figure}[h!]
\centering \hspace{-1.8cm}
\begin{subfigure}[b]{0.22\textwidth}
\includegraphics[width=\textwidth]{pScalar/Diag/a91.jpg}
\caption*{$A_{9,1} \equiv A_{1,011111110110}$}
\end{subfigure}
\qquad\qquad
\begin{subfigure}[b]{0.22\textwidth}
\includegraphics[width=\textwidth]{pScalar/Diag/a92.jpg}
\caption*{$A_{9,2} \equiv A_{2,011111011101}$}
\end{subfigure}
\quad\quad\quad
\begin{subfigure}[b]{0.22\textwidth}
\includegraphics[width=\textwidth]{pScalar/Diag/a94.jpg}
\caption*{$A_{9,4} \equiv A_{2,111011111100}$}
\end{subfigure}
\end{figure}
These integrals were computed
analytically as Laurent series in $\epsilon$
in~\cite{Gehrmann:2005pd,Gehrmann:2006wg,Heinrich:2007at,Heinrich:2009be,Lee:2010cga}
and are collected in the appendix of~\cite{Gehrmann:2010ue}.
Inserting those, we obtain the final expressions for the
unrenormalised (bare) form factors that are listed in
Appendix~\ref{App:pScalar-Results}.
\subsection{UV Renormalisation}
\label{ss:UV}
To obtain ultraviolet-finite expressions for the form factors, a
renormalisation of the coupling constant and of the operators is
required. The UV renormalisation of the operators
$\left[ O_{G} \right]_{B}$ and $\left[ O_{J} \right]_{B}$ involves
some non-trivial prescriptions. These are in part related to the
formalism used for the $\gamma_{5}$ matrix, section~\ref{sec:gamma5}
above.
This formalism fails to preserve the anti-commutativity of
$\gamma_{5}$ with $\gamma^{\mu}$ in $d$ dimensions. In addition, the
standard properties of the axial current and Ward identities, which
are valid in a basic regularization scheme like the one of
Pauli-Villars, are violated as well. As a consequence, one fails to
restore the correct renormalised axial current, which is defined as
\cite{Larin:1993tq, Akyeampong:1973xi}
\begin{align}
\label{eq:J5}
J^{\mu}_{5} \equiv \bar{\psi}\gamma^{\mu}\gamma_{5}\psi = i
\frac{1}{3!} \varepsilon^{\mu\nu_{1}\nu_{2}\nu_{3}} \bar{\psi}
\gamma_{\nu_{1}} \gamma_{\nu_{2}}\gamma_{\nu_{3}} \psi
\end{align}
in dimensional regularization. To rectify this, one needs to introduce
a finite renormalisation constant $Z^{s}_{5}$
\cite{Adler:1969gk,Kodaira:1979pa} in addition to the standard overall
ultraviolet renormalisation constant $Z^{s}_{\overline{MS}}$ within
the $\overline{MS}$-scheme:
\begin{align}
\label{eq:J5Ren}
\left[ J^{\mu}_{5} \right]_{R} = Z^{s}_{5} Z^{s}_{\overline{MS}} \left[ J^{\mu}_{5} \right]_{B}\,.
\end{align}
By evaluating the appropriate Feynman diagrams explicitly,
$Z^{s}_{\overline{MS}}$ can be computed, however the finite
renormalisation constant is not fixed through this calculation. To
determine $Z^{s}_{5}$ one has to demand the conservation of the one
loop character \cite{Adler:1969er} of the operator relation of the
axial anomaly in dimensional regularization:
\begin{align}
\label{eq:Anomaly}
\left[ \partial_{\mu}J^{\mu}_{5} \right]_{R} &= a_{s} \frac{n_{f}}{2} \left[ G\tilde{G} \right]_{R}
\nonumber\\
\text{i.e.}~~~ \left[ O_{J} \right]_{R} &= a_{s} \frac{n_{f}}{2} \left[ O_{G} \right]_{R}\,.
\end{align}
The bare operator $\left[ O_{J} \right]_{B}$ is renormalised
multiplicatively exactly in the same way as the axial current
$J^{\mu}_{5}$ through
\begin{align}
\label{eq:OJRen}
\left[ O_{J} \right]_{R} = Z^{s}_{5} Z^{s}_{\overline{MS}} \left[ O_{J}\right]_{B}\,,
\end{align}
whereas the other one $\left[ O_{G} \right]_{B}$ mixes under the
renormalisation through
\begin{align}
\left[ O_{G} \right]_{R} = Z_{GG} \left[ O_{G}\right]_B +
Z_{GJ} \left[ O_{J} \right]_B
\end{align}
with the corresponding renormalisation constants $Z_{GG}$ and
$Z_{GJ}$. The above two equations can be combined to express them
through the matrix equation
\begin{align}
\label{eq:OpMat}
\left[ O_{i} \right]_{R} &= Z_{ij} \left[ O_{j}\right]_{B}
\end{align}
with
\begin{align}
\label{eq:ZMat}
i,j &= \{G, J\}\,,
\nonumber\\
O \equiv
\begin{bmatrix}
O_{G}\\
O_{J}
\end{bmatrix}
\qquad\quad &\text{and} \qquad\quad
Z \equiv
\begin{bmatrix}
Z_{GG} & Z_{GJ}\\
Z_{JG} & Z_{JJ}
\end{bmatrix}\,.
\end{align}
In the above expressions
\begin{align}
\label{eq:ZJGZJJ}
Z_{JG} &= 0 \qquad \text{to all orders in perturbation theory}\,,
\nonumber\\
Z_{JJ} &\equiv Z^{s}_{5} Z^{s}_{\overline{MS}}\,.
\end{align}
We determine the above-mentioned renormalisation constants
$Z^{s}_{\overline{MS}},Z_{GG},Z_{GJ}$ up to
${\cal{O}}\left( a_{s}^{3} \right)$ from our calculation of the bare
on-shell pseudo-scalar form factors described in the previous
subsection. This procedure provides a completely independent approach
to their original computation, which was done in the operator product
expansion~\cite{Zoller:2013ixa}.
Our approach to compute those $Z_{ij}$ is based on the infrared
evolution equation for the form factor, and will be detailed in
Section~\ref{ss:IR} below. Moreover, we can fix $Z^{s}_{5}$ up to
${\cal O}(a_{s}^{2})$ by demanding the operator relation of the axial
anomaly (Eq.~(\ref{eq:Anomaly})). Using these overall operator
renormalisation constants along with strong coupling constant
renormalisation through $Z_{a_{s}}$, Eq.~(\ref{eq:Zas}), we obtain the
UV finite on-shell quark and gluon form factors.
To define the UV renormalised form factors, we introduce a quantity
${\cal{S}}^{\lambda}_{\beta}$, constructed out of bare matrix
elements, through
\begin{align}
\label{eq:CalSG}
{\cal{S}}^{G}_{g} &\equiv Z_{GG} \langle {\hat{\cal
M}}^{G,(0)}_{g}|{{\cal M}}^{G}_{g}\rangle + Z_{GJ} \langle {\hat{\cal
M}}^{G,(0)}_{g}|{{\cal M}}^{J}_{g}\rangle
\nonumber\\
\intertext{and}
{\cal{S}}^{G}_{q} &\equiv Z_{GG} \langle {\hat{\cal
M}}^{J,(0)}_{q}|{{\cal M}}^{G}_{q}\rangle + Z_{GJ} \langle {\hat{\cal
M}}^{J,(0)}_{q}|{{\cal M}}^{J}_{q}\rangle \,.
\end{align}
Expanding the quantities appearing on the right hand side of the above
equation in powers of $a_{s}$ :
\begin{align}
\label{eq:MZExpRenas}
|{\cal M}^{\lambda}_{\beta}\rangle &= \sum_{n=0}^{\infty} {a}^{n}_{s}
|{\cal M}^{\lambda,(n)}_{\beta}\rangle\,,
\nonumber\\
Z_{I} &= \sum_{n=0}^{\infty} a^{n}_{s} Z^{(n)}_{I} \qquad \text{with}
\qquad I=GG, GJ\,\,\,,
\end{align}
we can write \\
\begin{equation}
{\cal S}^{G}_{g} = \sum_{n=0}^{\infty} a^{n}_{s} {\cal S}^{G,(n)}_{g}\qquad
\text{and}\qquad {\cal S}^{G}_{q} = \sum_{n=1}^{\infty} a^{n}_{s} {\cal S}^{G,(n)}_{q}\,.
\label{eq:CalSG}
\end{equation}
\\
Then the UV renormalised form factors corresponding to $O_{G}$ are
defined as
\begin{align}
\label{eq:RenFFG}
\left[ {\cal F}^{G}_{g} \right]_{R} \equiv \frac{{\cal S}^{G}_{g}}{{\cal
S}^{G,(0)}_{g}}
&=
Z_{GG} {\cal
F}^{G}_{g} + Z_{GJ} {\cal F}^{J}_{g}
\frac{\langle {\cal
M}^{G,(0)}_{g}|{\cal M}^{J,(1)}_{g}\rangle}{\langle {\cal
M}^{G,(0)}_{g}|{\cal M}^{G,(0)}_{g}\rangle}
\nonumber\\
&\equiv 1 + \sum^{\infty}_{n=1} a^{n}_{s} \left[ {\cal
F}^{G,(n)}_{g} \right]_{R} \,,
\nonumber\\ \nonumber\\
\left[ {\cal F}^{G}_{q}
\right]_{R} \equiv \frac{{\cal S}^{G}_{q}}{a_{s} {\cal
S}^{G,(1)}_{q}}
&= \frac{Z_{GG} {\cal F}^{G}_{q} \langle {\cal
M}^{J,(0)}_{q}|{\cal M}^{G,(1)}_{q}\rangle + Z_{GJ} {\cal
F}^{J}_{q} \langle {\cal
M}^{J,(0)}_{q}|{\cal M}^{J,(0)}_{q}\rangle}{a_{s} \left[ \langle {\cal
M}^{J,(0)}_{q}|{\cal M}^{G,(1)}_{q}\rangle + Z^{(1)}_{GJ} \langle {\cal
M}^{J,(0)}_{q}|{\cal M}^{J,(0)}_{q}\rangle \right]}
\nonumber\\
&
\equiv 1 + \sum^{\infty}_{n=1} a^{n}_{s}
\left[ {\cal
F}^{G,(n)}_{q} \right]_{R} \,
\end{align}
where
\begin{align}
\label{eq:SGg0SGq1}
{\cal
S}^{G,(0)}_{g} &= \langle {\cal
M}^{G,(0)}_{g}|{\cal M}^{G,(0)}_{g}\rangle \,,
\nonumber\\
{\cal
S}^{G,(1)}_{q} &=
\langle {\cal M}^{J,(0)}_{q}|{\cal M}^{G,(1)}_{q}\rangle
+ Z^{(1)}_{GJ}\langle {\cal M}^{J,(0)}_{q}|{\cal M}^{J,(0)}_{q}\rangle\,.
\end{align}
Similarly, for defining the UV finite form factors for the other
operator $O_{J}$ we introduce
\begin{align}
\label{eq:CalSJ}
{\cal S}^{J}_{g} &\equiv Z^{s}_{5} Z^{s}_{\overline{MS}} \langle
{\hat{\cal M}}^{G,(0)}_{g}| {\cal M}^{J}_{g} \rangle\,
\nonumber\\
\intertext{and}
{\cal S}^{J}_{q} &\equiv Z^{s}_{5} Z^{s}_{\overline{MS}} \langle
{\hat{\cal M}}^{J,(0)}_{q}| {\cal M}^{J}_{q} \rangle\,.
\end{align}
Expanding $Z^{s}_{\overline{MS}}$ and
$|{\cal{M}}^{\lambda}_{\beta}\rangle$ in powers of $a_{s}$, following
Eq.~(\ref{eq:MZExpRenas}), we get
\begin{align}
\label{eq:CalSJExpand}
{\cal S}^{J}_{g} = \sum_{n=1}^{\infty} a^{n}_{s} {\cal S}^{J,(n)}_{g}
\qquad\quad \text{and} \qquad\quad
{\cal S}^{J}_{q} = \sum_{n=0}^{\infty} a^{n}_{s} {\cal S}^{J,(n)}_{q}\,.
\end{align}
\\
With these we define the UV renormalised form factors corresponding to
$O_{J}$ through
\begin{align}
\label{eq:RenFFJ}
\left[ {\cal F}^{J}_{g} \right]_{R} &\equiv \frac{{\cal S}^{J}_{g}}{a_{s}{\cal
S}^{J,(1)}_{g}} = Z^{s}_{5}
Z^{s}_{\overline{MS}} {\cal
F}^{J}_{g} \equiv 1 +
\sum^{\infty}_{n=1} a^{n}_{s} \left[ {\cal
F}^{J,(n)}_{g} \right]_{R} \,,
\nonumber\\
\left[ {\cal F}^{J}_{q}
\right]_{R} &\equiv \frac{{\cal S}^{J}_{q}}{{\cal
S}^{J,(0)}_{q}} = Z^{s}_{5}
Z^{s}_{\overline{MS}} {\cal
F}^{J}_{q} = 1 + \sum^{\infty}_{n=1} a^{n}_{s}
\left[ {\cal
F}^{J,(n)}_{q} \right]_{R} \,
\end{align}
where
\begin{align}
\label{eq:SGg0SGq1}
{\cal
S}^{J,(1)}_{g} &= \langle {\cal
M}^{G,(0)}_{g}|{\cal M}^{J,(1)}_{g}\rangle \,,
\nonumber\\
{\cal
S}^{J,(0)}_{q} &=
\langle {\cal M}^{J,(0)}_{q}|{\cal M}^{J,(0)}_{q}\rangle\,.
\end{align}
The finite renormalisation constant $Z^{s}_{5}$ is multiplied in
Eq.~(\ref{eq:CalSJ}) to restore the axial anomaly equation in
dimensional regularisation. We determine all required renormalisation
constants from consistency conditions on the universal structure of
the infrared poles of the renormalised form factors in the next
section, and use these constants to derive the UV-finite form factors
in Section~\ref{ss:Ren}.
\subsection{Infrared Singularities and Universal Pole Structure}
\label{ss:IR}
The renormalised form factors are ultraviolet-finite, but still
contain divergences of infrared origin. In the calculation of
physical quantities (which fulfill certain infrared-safety
criteria~\cite{Sterman:1977wj}), these infrared singularities are
cancelled by contributions from real radiation processes that yield
the same observable final state, and by mass factorization
contributions associated with initial-state partons. The pole
structures of these infrared divergences arising in QCD form factors
exhibit some universal behaviour. The very first successful proposal
along this direction was presented by Catani~\cite{Catani:1998bh} (see
also \cite{Sterman:2002qn}) for one and two-loop QCD amplitudes using
the universal subtraction operators. The factorization of the single
pole in quark and gluon form factors in terms of soft and collinear
anomalous dimensions was first revealed in \cite{Ravindran:2004mb} up
to two loop level whose validity at three loop was later established
in the article \cite{Moch:2005tm}. The proposal by Catani was
generalized beyond two loops by Becher and
Neubert~\cite{Becher:2009cu} and by Gardi and
Magnea~\cite{Gardi:2009qi}. Below, we outline this behaviour in the
context of pseudo-scalar form factors up to three loop level,
following closely the notation used in~\cite{Ravindran:2005vv}.
The unrenormalised form factors
${\cal F}^{\lambda}_{\beta}(\hat{a}_{s}, Q^{2}, \mu^{2}, \epsilon)$
satisfy the so-called $KG$-differential equation \cite{Sudakov:1954sw,
Mueller:1979ih, Collins:1980ih, Sen:1981sd} which is dictated by the
factorization property, gauge and renormalisation group (RG)
invariances:
\begin{equation}
\label{eq:KG}
Q^2 \frac{d}{dQ^2} \ln {\cal F}^{\lambda}_{\beta} (\hat{a}_s, Q^2, \mu^2, \epsilon)
= \frac{1}{2} \left[ K^{\lambda}_{\beta} (\hat{a}_s, \frac{\mu_R^2}{\mu^2}, \epsilon
) + G^{\lambda}_{\beta} (\hat{a}_s, \frac{Q^2}{\mu_R^2}, \frac{\mu_R^2}{\mu^2}, \epsilon ) \right]
\end{equation}
where all poles in the dimensional regulator $\epsilon$ are contained in
the $Q^{2}$ independent function $K^{\lambda}_{\beta}$ and the finite
terms in $\epsilon \rightarrow 0$ are encapsulated in
$G^{\lambda}_{\beta}$. RG invariance of the form factor implies
\begin{equation}
\label{eq:KIA}
\mu_R^2 \frac{d}{d\mu_R^2} K^{\lambda}_{\beta}(\hat{a}_s, \frac{\mu_R^2}{\mu^2},
\epsilon ) = - \mu_R^2 \frac{d}{d\mu_R^2} G^{\lambda}_{\beta}(\hat{a}_s,
\frac{Q^2}{\mu_R^2}, \frac{\mu_R^2}{\mu^2}, \epsilon )
= - A^{\lambda}_{\beta} (a_s (\mu_R^2)) = - \sum_{i=1}^{\infty} a_s^i (\mu_R^2) A^{\lambda}_{\beta,i}
\end{equation}
where, $A^{\lambda}_{\beta,i}$ on the right hand side are the $i$-loop
cusp anomalous dimensions. It is straightforward to solve for
$K^{\lambda}_{\beta}$ in Eq.~(\ref{eq:KIA}) in powers of bare strong
coupling constant $\hat{a}_{s}$ by performing the following expansion
\begin{align}
K^{\lambda}_{\beta}\left({\hat a}_{s}, \frac{\mu_{R}^{2}}{\mu^{2}}, \epsilon\right) =
\sum_{i=1}^{\infty} {\hat a}_{s}^{i}
\left(\frac{\mu_{R}^{2}}{\mu^{2}}\right)^{i\frac{\epsilon}{2}} S_{\epsilon}^{i} K^{\lambda}_{\beta,i}(\epsilon)\, .
\end{align}
The solutions $K^{\lambda}_{\beta,i}(\epsilon)$ consist of simple poles in
$\epsilon$ with the coefficients consisting of $A_{\beta, i}^{\lambda}$ and
$\beta_{i}$. These can be found in \cite{Ravindran:2005vv,
Ravindran:2006cg}. On the other hand, the RGE of
$G^{\lambda}_{\beta,i}(\hat{a}_s, \frac{Q^2}{\mu_R^2},
\frac{\mu_R^2}{\mu^2}, \epsilon )$
can be solved. The solution contains two parts, one is dependent on
$\mu_{R}^{2}$ whereas the other part depends only the boundary point
$\mu^{2}_{R}=Q^{2}$. The $\mu_{R}^{2}$ dependent part can eventually
be expressed in terms of $A^{\lambda}_{\beta}$:
\begin{align}
\label{eq:GSoln}
G^{\lambda}_{\beta}(\hat{a}_s, \frac{Q^2}{\mu_R^2},
\frac{\mu_R^2}{\mu^2}, \epsilon ) = G^{\lambda}_{\beta}({a}_s(Q^{2}),
1, \epsilon ) + \int_{\frac{Q^{2}}{\mu_{R}^{2}}}^{1} \frac{dx}{x}
A^{\lambda}_{\beta}(a_{s}\left(x\mu_{R}^{2})\right)\,.
\end{align}
The boundary term can be expanded in powers of $a_{s}$ as
\begin{align}
G^{\lambda}_{\beta}(a_s(Q^2), 1, \epsilon) = \sum_{i=1}^{\infty} a_s^i(Q^2) G^{\lambda}_{\beta,i}(\epsilon)\, .
\end{align}
The solutions of $K^{\lambda}_{\beta}$ and $G^{\lambda}_{\beta}$
enable us to solve the $KG$ equation (Eq.~(\ref{eq:KG})) and thereby
facilitate to obtain the
$\ln {\cal F}^{\lambda}_{\beta}(\hat{a}_s, Q^2, \mu^2, \epsilon)$ in terms
of $A^{\lambda}_{\beta, i}, G^{\lambda}_{\beta, i}$ and $\beta_{i}$
which is given by~\cite{Ravindran:2005vv}
\begin{align}
\label{eq:lnFSoln}
\ln {\cal F}^{\lambda}_{\beta}(\hat{a}_s, Q^2, \mu^2, \epsilon) =
\sum_{i=1}^{\infty} {\hat a}_{s}^{i} \left(\frac{Q^{2}}{\mu^{2}}\right)^{i
\frac{\epsilon}{2}} S_{\epsilon}^{i} \hat {\cal L}_{\beta,i}^{\lambda}(\epsilon)
\end{align}
with
\begin{align}
\label{eq:lnFitoCalLF}
\hat {\cal L}_{\beta,1}^{\lambda}(\epsilon) =& { \frac{1}{\epsilon^2} } \Bigg\{-2 A^{\lambda}_{\beta,1}\Bigg\}
+ { \frac{1}{\epsilon}
}
\Bigg\{G^{\lambda}_{\beta,1}
(\epsilon)\Bigg\}\, ,
\nonumber\\
\hat {\cal L}_{\beta,2}^{\lambda}(\epsilon) =& { \frac{1}{\epsilon^3} } \Bigg\{\beta_0 A^{\lambda}_{\beta,1}\Bigg\}
+ {
\frac{1}{\epsilon^2} }
\Bigg\{- {
\frac{1}{2} } A^{\lambda}_{\beta,2}
- \beta_0 G^{\lambda}_{\beta,1}(\epsilon)\Bigg\}
+ { \frac{1}{\epsilon}
} \Bigg\{ {
\frac{1}{2} } G^{\lambda}_{\beta,2}(\epsilon)\Bigg\}\, ,
\nonumber\\
\hat {\cal L}_{\beta,3}^{\lambda}(\epsilon) =& { \frac{1}{\epsilon^4}
} \Bigg\{- {
\frac{8}{9} } \beta_0^2 A^{\lambda}_{\beta,1}\Bigg\}
+ {
\frac{1}{\epsilon^3} }
\Bigg\{ { \frac{2}{9} } \beta_1 A^{\lambda}_{\beta,1}
+ { \frac{8}{9} }
\beta_0 A^{\lambda}_{\beta,2} + { \frac{4}{3} }
\beta_0^2 G^{\lambda}_{\beta,1}(\epsilon)\Bigg\}
\nonumber\\
&
+ { \frac{1}{\epsilon^2} } \Bigg\{- { \frac{2}{9} } A^{\lambda}_{\beta,3}
- { \frac{1}{3} } \beta_1 G^{\lambda}_{\beta,1}(\epsilon)
- { \frac{4}{3} } \beta_0 G^{\lambda}_{\beta,2}(\epsilon)\Bigg\}
+ { \frac{1}{\epsilon}
} \Bigg\{ { \frac{1}{3} } G^{\lambda}_{\beta,3}(\epsilon)\Bigg\}\, .
\end{align}
All these form factors are observed to satisfy \cite{Ravindran:2004mb,
Moch:2005tm} the following decomposition in terms of collinear
($B^{\lambda}_{\beta}$), soft ($f^{\lambda}_{\beta}$) and UV
($\gamma^{\lambda}_{\beta}$) anomalous dimensions:
\begin{align}
\label{eq:GIi}
G^{\lambda}_{\beta,i} (\epsilon) = 2 \left(B^{\lambda}_{\beta,i} -
\gamma^{\lambda}_{\beta,i}\right) + f^{\lambda}_{\beta,i} +
C^{\lambda}_{\beta,i} + \sum_{k=1}^{\infty} \epsilon^k g^{\lambda,k}_{\beta,i} \, ,
\end{align}
where the constants $C^{\lambda}_{\beta,i}$ are given by
\cite{Ravindran:2006cg}
\begin{align}
\label{eq:Cg}
C^{\lambda}_{\beta,1} &= 0\, ,
\nonumber\\
C^{\lambda}_{\beta,2} &= - 2 \beta_{0} g^{\lambda,1}_{\beta,1}\, ,
\nonumber\\
C^{\lambda}_{\beta,3} &= - 2 \beta_{1} g^{\lambda,1}_{\beta,1} - 2
\beta_{0} \left(g^{\lambda,1}_{\beta,2} + 2 \beta_{0} g^{\lambda,2}_{\beta,1}\right)\, .
\end{align}
In the above expressions, $X^{\lambda}_{\beta,i}$ with $X=A,B,f$ and
$\gamma^{\lambda}_{\beta, i}$ are defined through
\begin{align}
\label{eq:ABfgmExp}
X^{\lambda}_{\beta} &\equiv \sum_{i=1}^{\infty} a_{s}^{i}
X^{\lambda}_{\beta,i}\,,
\qquad \text{and} \qquad
\gamma^{\lambda}_{\beta} \equiv \sum_{i=1}^{\infty} a_{s}^{i} \gamma^{\lambda}_{\beta,i}\,\,.
\end{align}
Within this framework, we will now determine this universal structure
of IR singularities of the pseudo-scalar form factors. This
prescription will be used subsequently to determine the overall
operator renormalisation constants.
We begin with the discussion of form factors corresponding to
$O_{J}$. The results of the form factors ${\cal F}^{J}_{\beta}$ for
$\beta=q,g$, which have been computed up to three loop level in this
article are being used to extract the unknown factors,
$\gamma^{J}_{\beta,i}$ and $g^{J,k}_{\beta,i}$, by employing the $KG$
equation. Since the ${\cal F}^{J}_{\beta}$ satisfy $KG$ equation, we
can obtain the solutions Eq.~(\ref{eq:lnFSoln}) along with
Eq.~(\ref{eq:lnFitoCalLF}) and Eq.~(\ref{eq:GIi}) to examine our
results against the well known decomposition of the form factors in
terms of the quantities $X^{J}_{\beta}$. These are universal, and
appear also in the vector and scalar quark and gluon form
factors~\cite{Moch:2005tm, Ravindran:2004mb}. They are
known~\cite{Vogt:2004mw, Catani:1990rp, Vogt:2000ci, Ravindran:2004mb,
Ahmed:2014cha} up to three loop level in the literature. Using
these in the above decomposition, we obtain
$\gamma^{J}_{\beta,i}$. The other process dependent constants, namely,
$g^{J,k}_{\beta,i}$ can be obtained by comparing the coefficients of
$\epsilon^{k}$ in Eq.~(\ref{eq:lnFitoCalLF}) at every order in
${\hat a}_{s}$. We can get the quantities $\gamma^{J}_{g,i}$ and
$g^{J,k}_{g,i}$ up to two loop level, since this process starts at one
loop. From gluon form factors we get
\begin{align}
\label{eq:gmJqQ}
\gamma^{J}_{g,1} &= 0\,,
\nonumber\\
\gamma^{J}_{g,2} &= {{C_{A} C_{F}}} \Bigg\{- \frac{44}{3} \Bigg\}+
{{C_{F} n_{f}}} \Bigg\{- \frac{10}{3}
\Bigg\}\,. \hspace{6cm}
\end{align}
Similarly, from the quark form factors we obtain
\begin{align}
\label{eq:gmJqQ}
\gamma^{J}_{q,1} &= 0\,,
\nonumber\\
\gamma^{J}_{q,2} &= {{C_{A} C_{F}}} \Bigg\{- \frac{44}{3} \Bigg\}+
{{C_{F} n_{f}}} \Bigg\{- \frac{10}{3} \Bigg\}\,,
\nonumber\\
\gamma^{J}_{q,3} &= {{C^{2}_{A} C_{F}}} \Bigg\{ -
\frac{3578}{27}\Bigg\} +
{{C^{2}_{F}n_{f}}} \Bigg\{\frac{22}{3}\Bigg\}
- {{C_{F} n^{2}_{f}}}
\Bigg\{\frac{26}{27}\Bigg\} + {{C_{A}
C^{2}_{F}}} \Bigg\{\frac{308}{3}\Bigg\}
\nonumber\\
&+
{{C_{A} C_{F} n_{f}}} \Bigg\{-\frac{149}{27}\Bigg\}\,.
\end{align}
Note that $\gamma^{J}_{q,i} = \gamma^{J}_{g,i}$ which is expected
since these are the UV anomalous dimensions associated with the same
operator $[O_{J}]_{B}$. The $\gamma^{J}_{\beta,i}$ are further used to
obtain the overall operator renormalisation constant
$Z^{s}_{\overline{MS}}$ through the RGE:
\begin{align}
\label{eq:RGEZMS}
\mu_{R}^{2}\frac{d}{d\mu_{R}^{2}}
\ln Z^{\lambda}(a_{s},\mu_{R}^{2},\epsilon) = \sum_{i=1}^{\infty}
a_{s}^{i} \gamma^{\lambda}_{i}.
\end{align}
\\
The general solution of the RGE is obtained as
\begin{align}
\label{eq:GenSolZI}
Z^{\lambda} &= 1 + a_s \Bigg[ \frac{1}{\epsilon} 2 {\gamma^{\lambda}_{1}} \Bigg] +
a_s^2 \Bigg[ \frac{1}{\epsilon^2} \Bigg\{ 2 \beta_0
{\gamma^{\lambda}_{1}} + 2 ({\gamma^{\lambda}_{1}})^2 \Bigg\} +
\frac{1}{\epsilon} {\gamma^{\lambda}_{2}} \Bigg]
+ a_s^3 \Bigg[
\frac{1}{\epsilon^{3}} \Bigg\{ 8 \beta_0^2 {\gamma^{\lambda}_{1}} +
4 \beta_0 ({\gamma^{\lambda}_{1}})^2
\nonumber\\
&+
\frac{4 ({\gamma^{\lambda}_{1}})^3}{3} \Bigg\} + \frac{1}{\epsilon^{2}} \Bigg\{ \frac{4
\beta_1 {\gamma^{\lambda}_{1}}}{3} + \frac{4 \beta_0
{\gamma^{\lambda}_{2}}}{3} + 2 {\gamma^{\lambda}_{1}}
{\gamma^{\lambda}_{2}} \Bigg\} +
\frac{1}{\epsilon} \Bigg\{ \frac{2 {\gamma^{\lambda}_{3}}}{3} \Bigg\}
\Bigg]\,.
\end{align}
By substituting the results of $\gamma^{J}_{\beta,i}$ in the above
solution we get $Z^{s}_{\overline{MS}}$ up to ${\cal O}(a_{s}^{3})$:
\begin{align}
\label{eq:ZMS}
Z^{s}_{\overline{MS}} &= 1 + a^{2}_{s} \Bigg[C_{A} C_{F} \Bigg\{-
\frac{44}{3 \epsilon} \Bigg\} + C_{F} n_{f} \Bigg\{ -
\frac{10}{3 \epsilon} \Bigg\}
\Bigg]
+ a^{3}_{s} \Bigg[ C_{A}^2 C_{F} \Bigg\{ -
\frac{1936}{27 \epsilon^2} - \frac{7156}{81 \epsilon}
\Bigg\}
\nonumber\\
&+ C_{F}^2 n_{f} \Bigg\{ \frac{44}{9 \epsilon} \Bigg\} +
C_{F} n_{f}^2 \Bigg\{ \frac{80}{27 \epsilon^2} - \frac{52}{81 \epsilon} \Bigg\} +
C_{A} C_{F}^2 \Bigg\{ \frac{616}{9 \epsilon}\Bigg\} +
C_{A} C_{F} n_{f} \Bigg\{ - \frac{88}{27 \epsilon^2} -
\frac{298}{81 \epsilon} \Bigg\}
\Bigg],
\end{align}
which agrees completely with the known result in \cite{Larin:1993tq}.
In order to restore the axial anomaly equation in dimensional
regularization (see Section~\ref{ss:UV} above), we must multiply the
$Z^{s}_{\overline{MS}} \left[ O_{J} \right]_{B}$ by a finite
renormalisation constant $Z^{s}_{5}$, which reads \cite{Larin:1993tq}
\begin{align}
\label{eq:Z5s}
Z^{s}_{5} = 1 + a_{s} \{-4 C_{F}\} + a^{2}_{s} \left\{ 22 C^{2}_{F} -
\frac{107}{9} C_{A} C_{F} + \frac{31}{18} C_{F} n_{f} \right\}\,.
\end{align}
Following the computation of the operator mixing constants below, we
will be able to verify explicitly that this expression yields the
correct expression for the axial anomaly.
Now, we move towards the discussion of $O_{G}$ form factors. Similar
to previous case, we consider the form factors
$Z_{GG}^{-1} [ {\cal F}^{G}_{\beta} ]_{R}$, defined through
Eq.~(\ref{eq:RenFFG}), to extract the unknown constants,
$\gamma^{G}_{\beta,i}$ and $g^{G,k}_{\beta,i}$, by utilizing the $KG$
differential equation. Since, $[{\cal F}^G_{\beta}]_{R}$ is UV finite,
the product of $Z_{GG}^{-1}$ with $[{\cal F}^G_{\beta}]_{R}$ can
effectively be treated as unrenormalised form factor and hence we can
demand that $Z_{GG}^{-1} [{\cal F}^{G}_{\beta}]_{R}$ satisfy $KG$
equation. Further we make use of the solutions Eq.~(\ref{eq:lnFSoln})
in conjunction with Eq.~(\ref{eq:lnFitoCalLF}) and Eq.~(\ref{eq:GIi})
to compare our results against the universal decomposition of the form
factors in terms of the constants $X^{G}_{\beta}$. Upon substituting
the existing results of the quantities
$A^{G}_{\beta,i}, B^{G}_{\beta,i}$ and $f^{G}_{\beta,i}$ up to three
loops, which are obtained in case of quark and gluon form factors, we
determine the anomalous dimensions $\gamma^{G}_{\beta,i}$ and the
constants $g^{G,k}_{\beta,i}$. However, it is only possible to get the
factors $\gamma^{G}_{q,i}$ and $g^{G,k}_{q,i}$ up to two loops because
of the absence of a tree level amplitude in the quark initiated
process for the operator $O_{G}$. Since $[{\cal F}^G_{\beta}]_{R}$ are
UV finite, the anomalous dimensions $\gamma^{G}_{\beta,i}$ must be
equal to the anomalous dimension corresponding to the renormalisation
constant $Z_{GG}$. This fact is being used to determine the overall
renormalisation constants $Z_{GG}$ and $Z_{GJ}$ up to three loop level
where these quantities are parameterized in terms of the newly
introduced anomalous dimensions $\gamma_{ij}$ through the matrix
equation
\begin{align}
\label{eq:ZijDefn}
\mu_{R}^{2}\frac{d}{d\mu_{R}^{2}}Z_{ij} \equiv \gamma_{ik} Z_{kj}\,
\qquad \text{with} \qquad i,j,k={G,J}
\end{align}
This can be equivalently written as
\begin{align}
\label{eq:ZijDefn-1}
\gamma_{ij} = \left( \mu_{R}^{2}\frac{d}{d\mu_{R}^{2}}Z_{ik} \right) \left( Z^{-1} \right)_{kj}\,.
\end{align}
The general solution (See Example 2 in Appendix~\ref{chpt:App-SolRGEZas}) of the RGE up to $a_{s}^{3}$ is obtained as
\begin{align}
\label{eq:ZCoupSoln}
Z_{ij} &= \delta_{ij}
+ {a}_{s} \Bigg[ \frac{2}{\epsilon}
\gamma_{ij,1} \Bigg]
+ {a}_{s}^{2} \Bigg[
\frac{1}{\epsilon^{2}} \Bigg\{ 2
\beta_{0} \gamma_{ij,1} + 2 \gamma_{ik,1} \gamma_{kj,1} \Bigg\} + \frac{1}{\epsilon} \Bigg\{ \gamma_{ij,2}\Bigg\}
\Bigg]
+ {a}_{s}^{3} \Bigg[ \frac{1}{\epsilon^{3}} \Bigg\{
\frac{8}{3} \beta_{0}^{2} \gamma_{ij,1}
\nonumber\\
&+ 4 \beta_{0} \gamma_{ik,1}
\gamma_{kj,1} + \frac{4}{3} \gamma_{ik,1} \gamma_{kl,1}
\gamma_{lj,1} \Bigg\} + \frac{1}{\epsilon^{2}} \Bigg\{ \frac{4}{3} \beta_{1} \gamma_{ij,1} +
\frac{4}{3} \beta_{0} \gamma_{ij,2}
+ \frac{2}{3}
\gamma_{ik,1} \gamma_{kj,2}
\nonumber\\
&+ \frac{4}{3}
\gamma_{ik,2} \gamma_{kj,1} \Bigg\} + \frac{1}{\epsilon}
\Bigg\{ \frac{2}{3} \gamma_{ij,3} \Bigg\} \Bigg]
\end{align}
where, $\gamma_{ij}$ is expanded in powers of $a_{s}$ as
\begin{align}
\label{eq:gammaijExp}
\gamma_{ij} = \sum_{n=1}^{\infty} a_{s}^{n} \gamma_{ij,n}\,.
\end{align}
Demanding the vanishing of $\gamma^{G}_{\beta,i}$, we get
\begin{align}
\label{eq:gammaGG}
\gamma_{GG} &= a_{s} \Bigg[ \frac{11}{3} C_{A} - \frac{2}{3} n_{f}\Bigg] +
a_{s}^{2} \Bigg[ \frac{34}{3} C_{A}^{2} - \frac{10}{3} C_{A} n_{f} -
2 C_{F} n_{f} \Bigg]
+ a_{s}^{3} \Bigg[ \frac{2857}{54} C_{A}^3 - \frac{1415}{54} C_{A}^2
n_{f}
\nonumber\\
&- \frac{205}{18} C_{A} C_{F} n_{f} + C_{F}^2 n_{f} +
\frac{79}{54} C_{A} n_{f}^2 + \frac{11}{9} C_{F} n_{f}^2\Bigg]\,,
\nonumber\\
\gamma_{GJ} &= a_{s} \Bigg[ - 12 C_{F} \Bigg] + a_{s}^{2} \Bigg[
- \frac{284}{3} C_{A} C_{F} + 36 C_{F}^2 +
\frac{8}{3} C_{F} n_{f} \Bigg]
+ a_{s}^{3} \Bigg[ - \frac{1607}{3} C_{A}^2 C_{F}
\nonumber\\
&+ 461 C_{A} C_{F}^2 - 126 C_{F}^3 - \frac{164}{3} C_{A} C_{F} n_{f} +
214 C_{F}^2 n_{f} + \frac{52}{3} C_{F} n_{f}^2
+ 288 C_{A} C_{F}
n_{f} \zeta_3
\nonumber\\
&- 288 C_{F}^2 n_{f} \zeta_3
\Bigg]\,.
\end{align}
In addition to the demand of vanishing $\gamma^{G}_{\beta,i}$, it is
required to use the results of $\gamma_{JJ}$ and $\gamma_{JG}$, which
are implied by the definition, Eq.~(\ref{eq:ZijDefn}), up to
${\cal O}(a_{s}^{2})$ to determine the above-mentioned $\gamma_{GG}$
and $\gamma_{GJ}$ up to the given order. This is a consequence of the
fact that the operators mix under UV renormalisation. Following
Eq.~(\ref{eq:ZijDefn}) along with Eq.~(\ref{eq:ZJGZJJ}),
Eq.~(\ref{eq:ZMS}) and Eq.~(\ref{eq:Z5s}), we obtain
\begin{align}
\label{eq:gammaJJJG}
\gamma_{JJ} &= a_{s} \Bigg[ - \epsilon 2 C_{F} \Bigg] + a_{s}^{2}
\Bigg[ \epsilon \Bigg\{ - \frac{107}{9} C_A C_F + 14
C_F^2 + \frac{31}{18} C_F n_f
\Bigg\} - 6
C_F n_f \Bigg]
\intertext{and}
\gamma_{JG} &=0\,.
\end{align}
As it happens, we note that $\gamma_{JJ}$'s are $\epsilon$-dependent
and in fact, this plays a crucial role in determining the other
quantities. Our results are in accordance with the existing ones,
$\gamma_{GG}$ and $\gamma_{GJ}$, which are available up to
${\cal O}(a_{s}^{2})$ \cite{Larin:1993tq} and ${\cal O}(a_{s}^{3})$
\cite{Zoller:2013ixa}, respectively. In addition to the existing ones,
here we compute the new result of $\gamma_{GG}$ at
${\cal O}(a_{s}^{3})$. It was observed through explicit computation in
the article \cite{Larin:1993tq} that
\begin{align}
\label{eq:gammaGGbt}
\gamma_{GG} = - \frac{\beta}{a_{s}}
\end{align}
holds true up to two loop level but there was no statement on the
validity of this relation beyond that order. In \cite{Zoller:2013ixa},
it was demonstrated in the operator product expansion that the
relation holds even at three loop. Here, through explicit calculation,
we arrive at the same conclusion that the relation is still valid at
three loop level which can be seen if we look at the $\gamma_{GG, 3}$
in Eq.~(\ref{eq:gammaGG}) which is equal to the $\beta_{2}$.
Before ending the discussion of $\gamma_{ij}$, we examine our results
against the axial anomaly relation. The renormalisation group
invariance of the anomaly equation (Eq.~(\ref{eq:Anomaly})), see
\cite{Larin:1993tq}, gives
\begin{align}
\label{eq:AnomalyAlt}
\gamma_{JJ} = \frac{\beta}{a_{s}} + \gamma_{G{G}} + a_{s}
\frac{n_{f}}{2} \gamma_{GJ}\,.
\end{align}
Through our calculation up to three loop level we find that our
results are in complete agreement with the above anomaly equation
through
\begin{align}
\label{eq:AnomalyHold}
\gamma_{GG} &= - \frac{\beta}{a_{s}}
\qquad \text{and} \qquad
\gamma_{GJ} = \left( a_{s} \frac{n_{f}}{2} \right)^{-1} \gamma_{JJ}
\end{align}
in the limit of $\epsilon \rightarrow 0$. This serves as one of the
most crucial checks on our computation.
Additionally, if we conjecture the above relations to hold beyond
three loops (which could be doubted in light of recent
findings~\cite{Almelid:2015jia}), then we can even predict the
$\epsilon$-independent part of the $\gamma_{JJ}$ at
${\cal O}(a_{s}^{3})$:
\begin{align}
\label{eq:gammaJJ4}
\gamma_{JJ}|_{\epsilon \rightarrow 0} &= a_{s}^{2} \Bigg[ - 6 C_{F} n_{f} \Bigg] + a_{s}^{3} \Bigg[
- \frac{142}{3} C_{A} C_{F} n_{f} + 18 C_{F}^2 n_{f} +
\frac{4}{3} C_{F} n_{f}^{2} \Bigg]\,.
\end{align}
The results of $\gamma_{ij}$ uniquely specify $Z_{ij}$, through
Eq.~(\ref{eq:ZCoupSoln}). We summarize the resulting expressions of
$Z_{ij}$ below:
\begin{align}
\label{eq:ZGGtZGJ}
Z_{GG} &= 1 + a_s \Bigg[ \frac{22}{3\epsilon}
C_{A} -
\frac{4}{3\epsilon} n_{f} \Bigg]
+
a_s^2 \Bigg[ \frac{1}{\epsilon^2}
\Bigg\{ \frac{484}{9} C_{A}^2 - \frac{176}{9} C_{A}
n_{f} + \frac{16}{9} n_{f}^2 \Bigg\}
+ \frac{1}{\epsilon} \Bigg\{ \frac{34}{3} C_{A}^2
\nonumber\\
&-
\frac{10}{3} C_{A} n_{f} - 2 C_{F} n_{f} \Bigg\} \Bigg]
+
a_s^3 \Bigg[ \frac{1}{\epsilon^3}
\Bigg\{ \frac{10648}{27} C_{A}^3 - \frac{1936}{9}
C_{A}^2 n_{f} + \frac{352}{9} C_{A} n_{f}^2 -
\frac{64}{27} n_{f}^3 \Bigg\}
\nonumber\\
&+ \frac{1}{\epsilon^2}
\Bigg\{ \frac{5236}{27} C_{A}^3 - \frac{2492}{27}
C_{A}^2 n_{f} - \frac{308}{9} C_{A} C_{F} n_{f} +
\frac{280}{27} C_{A} n_{f}^2 + \frac{56}{9} C_{F}
n_{f}^2 \Bigg\}
\nonumber\\
&
+ \frac{1}{\epsilon} \Bigg\{ \frac{2857}{81} C_{A}^3 -
\frac{1415}{81} C_{A}^2 n_{f} - \frac{205}{27} C_{A} C_{F} n_{f} +
\frac{2}{3} C_{F}^2 n_{f} + \frac{79}{81} C_{A}
n_{f}^2 + \frac{22}{27} C_{F} n_{f}^2 \Bigg\}
\Bigg]
\nonumber \intertext{and}
Z_{GJ} &= a_s \Bigg[ - \frac{24}{\epsilon} C_{F} \Bigg]
+
a_s^2 \Bigg[ \frac{1}{\epsilon^2}
\Bigg\{ - 176 C_{A} C_{F} + 32 C_{F} n_{f} \Bigg\}
+ \frac{1}{\epsilon} \Bigg\{ - \frac{284}{3} C_{A} C_{F} +
84 C_{F}^2
\nonumber\\
&+ \frac{8}{3} C_{F} n_{f} \Bigg\} \Bigg]
+ a_{s}^{3} \Bigg[
\frac{1}{\epsilon^3} \Bigg\{ - \frac{3872}{3} C_{A}^2 C_{F} +
\frac{1408}{3} C_{A} C_{F} n_{f} - \frac{128}{3} C_{F}
n_{f}^2 \Bigg\}
\nonumber\\
&+
\frac{1}{\epsilon^2} \Bigg\{ - \frac{9512}{9} C_{A}^2 C_{F} +
\frac{2200}{3} C_{A} C_{F}^2 + \frac{2272}{9} C_{A} C_{F}
n_{f} -
\frac{64}{3} C_{F}^2 n_{f} - \frac{32}{9} C_{F} n_{f}^2 \Bigg\}
\nonumber\\
&+
\frac{1}{\epsilon} \Bigg\{ - \frac{3214}{9} C_{A}^2 C_{F} +
\frac{5894}{9} C_{A} C_{F}^2 - 356 C_{F}^3 - \frac{328}{9}
C_{A} C_{F} n_{f} +
\frac{1096}{9} C_{F}^2 n_{f} + \frac{104}{9} C_{F} n_{f}^2
\nonumber\\
&+ 192
C_{A} C_{F} n_{f} \zeta_3 -
192 C_{F}^2 n_{f} \zeta_3 \Bigg\}
\Bigg]\,.
\end{align}
$Z_{GG}$ and $Z_{GJ}$ are in agreement with the results already
available in the literature up to ${\cal{O}}(a^{2}_{s})$
\cite{Larin:1993tq} and ${\cal{O}}(a_{s}^{3})$ \cite{Zoller:2013ixa},
where a completely different approach and methodology was used.
\subsection{Results of UV Renormalised Form Factors}
\label{ss:Ren}
Using the renormalisation constants obtained in the previous section,
we get all the UV renormalised form factors
$\left[ {\cal F}^{\lambda}_{\beta} \right]_{R}$, defined in
Eq.~(\ref{eq:RenFFG}) and Eq.~(\ref{eq:RenFFJ}), up to three loops. In
this section we present the results for the choice of the scales
$\mu_R^2=\mu_F^2=q^2$.
\begin{align}
\label{eq:FFRen1Ggg}
\left[ {\cal F}^{G,(1)}_{g} \right]_{R} &= {{2 n_{f} T_{F}}} \Bigg\{ - \frac{4}{3
\epsilon} \Bigg\} + {{C_{A}}} \Bigg\{ -
\frac{8}{\epsilon^2} +
\frac{22}{3 \epsilon} + 4 + \zeta_2 + \epsilon \Bigg( - 6 -
\frac{7}{3} \zeta_3 \Bigg) +
\epsilon^2 \Bigg( 7 -
\frac{\zeta_2}{2}
\nonumber\\
&+ \frac{47}{80} \zeta_2^2
\Bigg) +
\epsilon^3 \Bigg( - \frac{15}{2}
+ \frac{3}{4} \zeta_2 +
\frac{7}{6} \zeta_3 +
\frac{7}{24} \zeta_2 \zeta_3 -
\frac{31}{20} \zeta_5 \Bigg) \Bigg\}\,,
\\ \left[ {\cal F}^{G,(2)}_{g} \right]_{R} &= {{4 n_{f}^2 T_{F}^{2}}} \Bigg\{ \frac{16}{9
\epsilon^2} \Bigg\}
+
{{C_{A}^2}} \Bigg\{
\frac{32}{\epsilon^4} -
\frac{308}{3 \epsilon^3} +
\Bigg( \frac{62}{9} - 4 \zeta_2
\Bigg) \frac{1}{\epsilon^2} +
\Bigg( \frac{2780}{27} +
\frac{11}{3} \zeta_2 +
\frac{50}{3} \zeta_3 \Bigg)
\frac{1}{\epsilon}
\nonumber\\
&-
\frac{3293}{81} +
\frac{115}{6} \zeta_2 -
\frac{21}{5} \zeta_2^2 - 33 \zeta_3 +
\epsilon \Bigg( -
\frac{114025}{972} -
\frac{235}{18} \zeta_2 +
\frac{1111}{120} \zeta_2^2 +
\frac{1103}{54} \zeta_3
\nonumber\\
&-
\frac{23}{6} \zeta_2 \zeta_3 -
\frac{71}{10} \zeta_5 \Bigg) +
\epsilon^2 \Bigg(
\frac{4819705}{11664} -
\frac{694}{27} \zeta_2 -
\frac{2183}{240} \zeta_2^2 +
\frac{2313}{280} \zeta_2^3 -
\frac{7450}{81} \zeta_3
\nonumber\\
&-
\frac{11}{36} \zeta_2 \zeta_3 +
\frac{901}{36} \zeta_3^2 -
\frac{341}{20} \zeta_5 \Bigg)
\Bigg\}
+
{{2 C_{A} n_{f} T_{F}}} \Bigg\{ \frac{56}{3
\epsilon^3} - \frac{52}{3
\epsilon^2} + \Bigg( -
\frac{272}{27} - \frac{2}{3}
\zeta_2 \Bigg) \frac{1}{\epsilon}
\nonumber\\
& - \frac{295}{81}
-
\frac{5}{3} \zeta_2 - 2 \zeta_3 +
\epsilon \Bigg( \frac{15035}{486}
+ \frac{\zeta_2}{18} +
\frac{59}{60} \zeta_2^2 +
\frac{383}{27} \zeta_3 \Bigg) +
\epsilon^2 \Bigg( -
\frac{116987}{1458} +
\frac{583}{108} \zeta_2
\nonumber\\
&-
\frac{329}{72} \zeta_2^2 -
\frac{1688}{81} \zeta_3 +
\frac{61}{18} \zeta_2 \zeta_3 -
\frac{49}{10} \zeta_5 \Bigg)
\Bigg\}
+
{{2 C_{F} n_{f} T_{F}}} \Bigg\{ - \frac{2}{\epsilon} -
\frac{71}{3} + 8
\zeta_3 +
\epsilon \Bigg( \frac{2665}{36}
\nonumber\\
&-
\frac{19}{6} \zeta_2 -
\frac{8}{3} \zeta_2^2 -
\frac{64}{3} \zeta_3 \Bigg) +
\epsilon^2 \Bigg( -
\frac{68309}{432} +
\frac{505}{36} \zeta_2 +
\frac{64}{9} \zeta_2^2 +
\frac{455}{9} \zeta_3 -
\frac{10}{3} \zeta_2 \zeta_3
\nonumber\\
&+ 8 \zeta_5 \Bigg) \Bigg\}\,,
\\ \left[ {\cal F}^{G,(3)}_{g} \right]_{R} &= {{8 n_{f}^3 T_{F}^{3}}} \Bigg\{ -
\frac{64}{27 \epsilon^3} \Bigg\}
+
{{4 C_{F} n_{f}^2 T_{F}^{2}}} \Bigg\{ \frac{56}{9
\epsilon^2} + \Bigg(
\frac{874}{27} - \frac{32}{3}
\zeta_3 \Bigg) \frac{1}{\epsilon}
-
\frac{418}{27} + 2 \zeta_2 +
\frac{16}{5} \zeta_2^2
\nonumber\\
&- \frac{80}{9} \zeta_3 \Bigg\}
+
{{2 C_{F}^2 n_{f} T_{F}}} \Bigg\{
\frac{2}{3
\epsilon} + \frac{457}{6} + 104 \zeta_3 - 160
\zeta_5 \Bigg\}
+
{{2 C_{A}^2 n_{f} T_{F}}} \Bigg\{ -
\frac{320}{3 \epsilon^5}
\nonumber\\
&+
\frac{28480}{81 \epsilon^4}
+
\Bigg( - \frac{608}{243} +
\frac{56}{27} \zeta_2 \Bigg)
\frac{1}{\epsilon^3} +
\Bigg( - \frac{54088}{243} +
\frac{676}{81} \zeta_2 +
\frac{272}{27} \zeta_3 \Bigg)
\frac{1}{\epsilon^2}
\nonumber\\
&+
\Bigg( - \frac{623293}{2187}
-
\frac{7072}{243} \zeta_2 -
\frac{941}{90} \zeta_2^2 -
\frac{7948}{81} \zeta_3 \Bigg)
\frac{1}{\epsilon} +
\frac{6345979}{13122} -
\frac{42971}{729} \zeta_2 +
\frac{687}{20} \zeta_2^2
\nonumber\\
&+
\frac{652}{3} \zeta_3
-
\frac{301}{9} \zeta_2 \zeta_3 +
\frac{4516}{45} \zeta_5 \Bigg\}
+
{{4 C_{A} n_{f}^2 T_{F}^{2}}} \Bigg\{ -
\frac{2720}{81 \epsilon^4} +
\frac{7984}{243 \epsilon^3} +
\Bigg( \frac{560}{27}
\nonumber\\
&+
\frac{8}{27} \zeta_2 \Bigg)
\frac{1}{\epsilon^2}
+ \Bigg(
\frac{10889}{2187} +
\frac{140}{81} \zeta_2 +
\frac{328}{81} \zeta_3 \Bigg)
\frac{1}{\epsilon} +
\frac{9515}{6561} +
\frac{10}{27} \zeta_2 -
\frac{157}{135} \zeta_2^2 -
\frac{20}{243} \zeta_3 \Bigg\}
\nonumber\\
&+
{{2 C_{A} C_{F} n_{f} T_{F}}} \Bigg\{
\frac{272}{9 \epsilon^3} +
\Bigg( \frac{4408}{27} -
\frac{640}{9} \zeta_3 \Bigg)
\frac{1}{\epsilon^2} +
\Bigg( - \frac{65110}{81} +
\frac{74}{3} \zeta_2 +
\frac{352}{15} \zeta_2^2
\nonumber\\
&+
\frac{6496}{27} \zeta_3 \Bigg)
\frac{1}{\epsilon} +
\frac{1053625}{972} -
\frac{311}{2} \zeta_2 -
\frac{1168}{15} \zeta_2^2 -
\frac{24874}{81} \zeta_3 + 48 \zeta_2 \zeta_3 +
\frac{32}{9} \zeta_5 \Bigg\}
\nonumber\\
&+
{{C_{A}^3}} \Bigg\{ -
\frac{256}{3 \epsilon^6} +
\frac{1760}{3 \epsilon^5} -
\frac{62264}{81 \epsilon^4} +
\Bigg( - \frac{176036}{243} -
\frac{308}{27} \zeta_2 -
\frac{176}{3} \zeta_3 \Bigg)
\frac{1}{\epsilon^3} +
\Bigg( \frac{207316}{243}
\nonumber\\
&-
\frac{8164}{81} \zeta_2 +
\frac{494}{45} \zeta_2^2 +
\frac{9064}{27} \zeta_3 \Bigg)
\frac{1}{\epsilon^2} +
\Bigg( \frac{2763800}{2187} +
\frac{36535}{243} \zeta_2 -
\frac{12881}{180} \zeta_2^2 -
\frac{3988}{9} \zeta_3
\nonumber\\
&+
\frac{170}{9} \zeta_2 \zeta_3 +
\frac{1756}{15} \zeta_5 \Bigg)
\frac{1}{\epsilon} -
\frac{84406405}{26244} +
\frac{617773}{1458} \zeta_2 +
\frac{144863}{1080} \zeta_2^2 -
\frac{22523}{270} \zeta_2^3
\nonumber\\
&+
\frac{44765}{243} \zeta_3 -
\frac{1441}{18} \zeta_2 \zeta_3 -
\frac{1766}{9} \zeta_3^2 +
\frac{13882}{45} \zeta_5 \Bigg\}\,,
\\ \left[ {\cal F}^{G,(1)}_{q} \right]_{R} &= {{C_{F}}} \Bigg\{ -
\frac{8}{\epsilon^2} +
\frac{6}{\epsilon} -
\frac{33}{4} + \zeta_2 +
\epsilon \Bigg( \frac{29}{16}
+ \frac{25}{48}
\zeta_2 - \frac{7}{3}
\zeta_3 \Bigg) +
\epsilon^2 \Bigg( \frac{299}{192}
- \frac{1327}{576} \zeta_2
\nonumber\\
&+
\frac{1387}{2880} \zeta_2^2 +
\frac{143}{48} \zeta_3 \Bigg) + \epsilon^3 \Bigg( -
\frac{13763}{2304} +
\frac{32095}{6912} \zeta_2 -
\frac{1559}{3456} \zeta_2^2 + \frac{61}{6912} \zeta_2^3 -
\frac{1625}{576} \zeta_3
\nonumber\\
&+
\frac{377}{864} \zeta_2 \zeta_3 - \frac{31}{20} \zeta_5 \Bigg)
\Bigg\}
+
{2 {n_{f} T_{F}}} \Bigg\{ -
\frac{445}{162} + \epsilon \Bigg( \frac{8231}{1944} -
\frac{239}{1944} \zeta_2 -
\frac{2}{3} \zeta_3 \Bigg)
\nonumber\\
&+
\epsilon^2 \Bigg( -
\frac{50533}{7776} + \frac{1835}{7776} \zeta_2
+ \frac{22903}{116640} \zeta_2^2 +
\frac{9125}{5832} \zeta_3 +
\frac{1}{18} \zeta_2 \zeta_3 \Bigg) +
\epsilon^3 \Bigg(
\frac{2754151}{279936}
\nonumber\\
&- \frac{35083}{93312}
\zeta_2 - \frac{316343}{699840} \zeta_2^2 -
\frac{22903}{1399680} \zeta_2^3
- \frac{61121}{23328} \zeta_3 +
\frac{2053}{34992} \zeta_2
\zeta_3 - \frac{1}{216} \zeta_2^2
\zeta_3
\nonumber\\
&- \frac{7}{54} \zeta_3^2 -
\frac{7}{6} \zeta_5 \Bigg) \Bigg\}
+
{{C_{A}}} \Bigg\{ \frac{7115}{324}
- \frac{2}{3} \zeta_2 - 2 \zeta_3 +
\epsilon \Bigg( -
\frac{114241}{3888} + \frac{7321}{3888} \zeta_2
+ \frac{53}{90} \zeta_2^2
\nonumber\\
&+ \frac{13}{3} \zeta_3 +
\frac{1}{6} \zeta_2 \zeta_3 \Bigg)
+ \epsilon^2 \Bigg(
\frac{692435}{15552} -
\frac{55117}{15552} \zeta_2 -
\frac{326369}{233280} \zeta_2^2
- \frac{53}{1080} \zeta_2^3 -
\frac{90235}{11664} \zeta_3
\nonumber\\
&-
\frac{41}{108} \zeta_2 \zeta_3 -
\frac{1}{72} \zeta_2^2 \zeta_3 -
\frac{7}{18} \zeta_3^2 - 5 \zeta_5
\Bigg) +
\epsilon^3 \Bigg( - \frac{37171073}{559872} +
\frac{1013165}{186624} \zeta_2
\nonumber\\
&+
\frac{3399073}{1399680} \zeta_2^2
+ \frac{34037663}{19595520}
\zeta_2^3 +
\frac{53}{12960} \zeta_2^4 +
\frac{585439}{46656} \zeta_3 -
\frac{56159}{69984} \zeta_2 \zeta_3
+
\frac{3223}{12960} \zeta_2^2
\zeta_3
\nonumber\\
&+ \frac{1}{864} \zeta_2^3 \zeta_3 + \frac{8}{9}
\zeta_3^2 +
\frac{7}{108} \zeta_2 \zeta_3^2 +
8 \zeta_5 + \frac{5}{12} \zeta_2 \zeta_5 \Bigg) \Bigg\}\,,
\\ \left[ {\cal F}^{G,(2)}_{q} \right]_{R} &= {{4 n_{f}^2 T_{F}^{2}}}
\Bigg\{
\frac{9505}{1458} +
\epsilon \Bigg( -
\frac{146177}{5832} +
\frac{12419}{17496}
\zeta_2 + \frac{38}{9}
\zeta_3 \Bigg) \Bigg\}
+
{{2 C_{F} n_{f} T_{F}}} \Bigg\{
\frac{8}{\epsilon^3} +
\frac{1636}{81 \epsilon^2}
\nonumber\\
&+
\Bigg( - \frac{12821}{243} -
\frac{247}{243} \zeta_2 +
\frac{16}{3} \zeta_3 \Bigg)
\frac{1}{\epsilon} +
\frac{20765}{324} + \frac{35}{486} \zeta_2 +
\frac{85}{2916} \zeta_2^2 +
\frac{6265}{729} \zeta_3 -
\frac{4}{9} \zeta_2 \zeta_3
\nonumber\\
&+
\epsilon \Bigg(-
\frac{1457425}{34992} -
\frac{11146}{729} \zeta_2 -
\frac{232457}{174960} \zeta_2^2 -
\frac{85}{34992} \zeta_2^3
+ \frac{9907}{1458} \zeta_3 -
\frac{7723}{4374} \zeta_2 \zeta_3
\nonumber\\
&+
\frac{1}{27} \zeta_2^2
\zeta_3 +
\frac{28}{27} \zeta_3^2 - \frac{20}{9} \zeta_5
\Bigg) \Bigg\}
+
{{C_{A}^2}} \Bigg\{
\frac{2796445}{5832} -
\frac{587}{18} \zeta_2 +
\frac{53}{30} \zeta_2^2
-
\frac{185}{2} \zeta_3 -
\frac{10}{3} \zeta_2 \zeta_3
\nonumber\\
&+
20 \zeta_5 +
\epsilon \Bigg( -
\frac{34321157}{23328} +
\frac{10420379}{69984} \zeta_2 +
\frac{589}{20} \zeta_2^2 +
\frac{7921}{2520} \zeta_2^3
+
\frac{8411}{24} \zeta_3 -
\frac{329}{72} \zeta_2 \zeta_3
\nonumber\\
&+
\frac{5}{18} \zeta_2^2 \zeta_3 +
13 \zeta_3^2 - \frac{757}{18} \zeta_5 -
\frac{5}{3} \zeta_2 \zeta_5
\Bigg) \Bigg\}
+
{{2 C_{A} n_{f} T_{F}}} \Bigg\{ -
\frac{178361}{1458} + \frac{44}{9} \zeta_2 -
\frac{76}{45} \zeta_2^2
\nonumber\\
&-
\frac{44}{9} \zeta_3
+
\epsilon \Bigg(
\frac{2357551}{5832} -
\frac{478171}{17496} \zeta_2 -
\frac{137}{135} \zeta_2^2 +
\frac{19}{135} \zeta_2^3 -
\frac{1621}{27} \zeta_3 -
\frac{40}{27} \zeta_2
\zeta_3
\nonumber\\
&+
\frac{22}{3} \zeta_5 \Bigg)
\Bigg\}
+
{{C_{A} C_{F}}} \Bigg\{ -
\frac{44}{\epsilon^3}
+
\Bigg( - \frac{13654}{81} +
\frac{28}{3} \zeta_2 + 16
\zeta_3 \Bigg)
\frac{1}{\epsilon^2} +
\Bigg( \frac{186925}{486}
\nonumber\\
&-
\frac{3919}{486} \zeta_2 -
\frac{212}{45} \zeta_2^2
-
\frac{218}{3} \zeta_3
-
\frac{4}{3} \zeta_2 \zeta_3
\Bigg) \frac{1}{\epsilon} -
\frac{61613}{81} +
\frac{59399}{972} \zeta_2 +
\frac{749513}{29160}
\zeta_2^2
\nonumber\\
&+
\frac{53}{135} \zeta_2^3
+ \frac{213517}{1458} \zeta_3 +
\frac{91}{27} \zeta_2 \zeta_3
+
\frac{1}{9} \zeta_2^2
\zeta_3 + \frac{28}{9} \zeta_3^2 +
\epsilon \Bigg(
\frac{35327209}{34992} -
\frac{2158003}{23328}
\zeta_2
\nonumber\\
&-
\frac{3532645}{69984}
\zeta_2^2 -
\frac{11307767}{2449440}
\zeta_2^3 -
\frac{53}{1620} \zeta_2^4
- \frac{1030169}{2916} \zeta_3 +
\frac{191915}{8748}
\zeta_2 \zeta_3 -
\frac{817}{405} \zeta_2^2
\zeta_3
\nonumber\\
&- \frac{1}{108} \zeta_2^3
\zeta_3 -
\frac{121}{9} \zeta_3^2 -
\frac{14}{27} \zeta_2
\zeta_3^2 - \frac{43}{6} \zeta_5
\Bigg) \Bigg\}
+
{{C_{F}^2}} \Bigg\{
\frac{32}{\epsilon^4} -
\frac{48}{\epsilon^3} +
\Bigg( 84 - 8 \zeta_2 \Bigg)
\frac{1}{\epsilon^2}
\nonumber\\
&+ \Bigg(
- \frac{125}{2} - \frac{61}{6}
\zeta_2 + \frac{128}{3} \zeta_3
\Bigg) \frac{1}{\epsilon} +
\frac{6881}{216} +
\frac{193}{12} \zeta_2
-
\frac{281}{24} \zeta_2^2 -
\frac{1037}{18} \zeta_3
\nonumber\\
&+
\epsilon \Bigg(
\frac{166499}{2592} -
\frac{3761}{648} \zeta_2
+ \frac{3451}{480}
\zeta_2^2 - \frac{31}{288}
\zeta_2^3 +
\frac{10607}{108} \zeta_3
- \frac{1081}{108}
\zeta_2 \zeta_3
+ \frac{328}{45} \zeta_5
\Bigg)
\Bigg\}\,,
\\ \left[ {\cal F}^{J,(1)}_{g} \right]_{R} &= {{2 n_{f} T_{F}}} \Bigg\{ - \frac{4}{3
\epsilon} \Bigg\}
+
{{C_{A}}} \Bigg\{ -
\frac{8}{\epsilon^2} +
\frac{22}{3 \epsilon} + 4 + \zeta_2 +
\epsilon \Bigg( - \frac{15}{2} +
\zeta_2 - \frac{16}{3} \zeta_3
\Bigg)
\nonumber\\
&+
\epsilon^2 \Bigg( \frac{287}{24}
- 2 \zeta_2 + \frac{127}{80}
\zeta_2^2 \Bigg) +
\epsilon^3 \Bigg( -
\frac{5239}{288} +
\frac{151}{48} \zeta_2 +
\frac{19}{120} \zeta_2^2 +
\frac{\zeta_3}{12} +
\frac{7}{6} \zeta_2
\zeta_3
\nonumber\\
&-
\frac{91}{20} \zeta_5 \Bigg)
\Bigg\}
+
{{C_{F}}} \Bigg\{
\epsilon \Bigg( - \frac{21}{2} +
6 \zeta_3 \Bigg) + \epsilon^2 \Bigg(
\frac{155}{8} - \frac{5}{2}
\zeta_2 - \frac{9}{5} \zeta_2^2
- \frac{9}{2} \zeta_3
\Bigg)
\nonumber\\
&+ \epsilon^3
\Bigg( - \frac{1025}{32} +
\frac{83}{16} \zeta_2
+
\frac{27}{20} \zeta_2^2 +
\frac{20}{3} \zeta_3 -
\frac{3}{4} \zeta_2 \zeta_3 + \frac{21}{2} \zeta_5 \Bigg) \Bigg\}\,,
\\ \left[ {\cal F}^{J,(2)}_{g} \right]_{R} &= {{4 n_{f}^2 T_{F}^{2}}} \Bigg\{ \frac{16}{9 \epsilon^2} \Bigg\}
+
{{C_{A} C_{F}}} \Bigg\{\Bigg( 84 - 48
\zeta_3 \Bigg) \frac{1}{\epsilon}
- 232 +
20 \zeta_2 + \frac{72}{5}
\zeta_2^2 + 80 \zeta_3
\nonumber\\
&+
\epsilon \Bigg( \frac{17545}{108}
- 58 \zeta_2 - 24 \zeta_2^2 -
\frac{38}{3} \zeta_3 + 10
\zeta_2 \zeta_3 -
14 \zeta_5 \Bigg) +
\epsilon^2 \Bigg(
\frac{402635}{1296} -
\frac{233}{36} \zeta_2
\nonumber\\
&+
\frac{72}{5} \zeta_2^2 +
\frac{17}{70} \zeta_2^3
+
\frac{535}{12} \zeta_3 - 2
\zeta_2 \zeta_3 - 34 \zeta_3^2 -
\frac{1355}{6} \zeta_5 \Bigg) \Bigg\}
+ {{2 C_{A} n_{f} T_{F}}}
\Bigg\{
\frac{56}{3 \epsilon^3} -
\frac{52}{3 \epsilon^2}
\nonumber\\
&+
\Bigg( - \frac{272}{27} -
\frac{2}{3} \zeta_2 \Bigg)
\frac{1}{\epsilon}
- \frac{133}{81} - 3 \zeta_2 +
2 \zeta_3 +
\epsilon \Bigg( \frac{7153}{243}
- \frac{7}{18} \zeta_2 -
\frac{13}{60} \zeta_2^2 +
\frac{599}{27} \zeta_3
\Bigg)
\nonumber\\
&+
\epsilon^2 \Bigg( -
\frac{135239}{1458}
+
\frac{1139}{108} \zeta_2 -
\frac{167}{24} \zeta_2^2 -
\frac{3146}{81} \zeta_3 +
\frac{73}{18} \zeta_2 \zeta_3 -
\frac{137}{30} \zeta_5 \Bigg)
\Bigg\}
\nonumber\\
&+
{{2 C_{F} n_{f} T_{F}}} \Bigg\{
- \frac{2}{\epsilon} -
\frac{29}{3}
+ \epsilon
\Bigg( \frac{14989}{216} -
\frac{25}{6} \zeta_2 -
\frac{4}{15} \zeta_2^2 -
32 \zeta_3 \Bigg) + \epsilon^2
\Bigg( -
\frac{606661}{2592}
\nonumber\\
&+
\frac{2233}{72} \zeta_2 +
\frac{158}{15} \zeta_2^2
+
\frac{1409}{18} \zeta_3 - 2
\zeta_2 \zeta_3 + \frac{82}{3}
\zeta_5 \Bigg) \Bigg\}
+
{{C_{A}^2}} \Bigg\{
+ \frac{32}{\epsilon^4} -
\frac{308}{3 \epsilon^3}
\nonumber\\
&+
\Bigg( \frac{62}{9} - 4 \zeta_2
\Bigg) \frac{1}{\epsilon^2} +
\Bigg( \frac{3104}{27}
-
\frac{13}{3} \zeta_2 +
\frac{122}{3} \zeta_3 \Bigg)
\frac{1}{\epsilon} - \frac{7397}{81} +
\frac{77}{2} \zeta_2 -
\frac{61}{5} \zeta_2^2 - 55
\zeta_3
\nonumber\\
&+
\epsilon \Bigg( -
\frac{32269}{972} -
\frac{997}{36} \zeta_2
+
\frac{1049}{120} \zeta_2^2 -
\frac{2393}{108} \zeta_3 -
\frac{53}{6} \zeta_2 \zeta_3 +
\frac{369}{10} \zeta_5
\Bigg)
\nonumber\\
&+
\epsilon^2 \Bigg(
\frac{4569955}{11664} -
\frac{15323}{432} \zeta_2 +
\frac{2129}{180} \zeta_2^2
-
\frac{7591}{840} \zeta_2^3 -
\frac{4099}{1296} \zeta_3 -
\frac{605}{36} \zeta_2 \zeta_3 +
\frac{775}{36} \zeta_3^2
\nonumber\\
&+
\frac{2011}{30} \zeta_5 \Bigg)
\Bigg\}
+
{{C_{F}^2}} \Bigg\{ \epsilon \Bigg(
\frac{763}{12} + 17 \zeta_3
- 60
\zeta_5 \Bigg) +
\epsilon^2 \Bigg( -
\frac{18857}{144} + \frac{31}{3}
\zeta_2 - \frac{76}{15}
\zeta_2^2
\nonumber\\
&+ \frac{120}{7}
\zeta_2^3 -
145 \zeta_3 + 4 \zeta_2 \zeta_3 +
30 \zeta_3^2
+ \frac{470}{3} \zeta_5 \Bigg) \Bigg\}\,,
\\ \left[ {\cal F}^{J,(1)}_{q} \right]_{R} &= {{C_{F}}} \Bigg\{ -
\frac{8}{\epsilon^2} +
\frac{6}{\epsilon} - 6 + \zeta_2 +
\epsilon \Bigg( - 1 -
\frac{3}{4} \zeta_2 \frac{7}{3}
\zeta_3 \Bigg) +
\epsilon^2 \Bigg( \frac{5}{2} +
\frac{\zeta_2}{4} +
\frac{47}{80} \zeta_2^2 +
\frac{7}{4} \zeta_3 \Bigg)
\nonumber\\
&+
\epsilon^3 \Bigg( - \frac{13}{4}
+ \frac{\zeta_2}{8} -
\frac{141}{320} \zeta_2^2 -
\frac{7}{12} \zeta_3 +
\frac{7}{24} \zeta_2 \zeta_3 -
\frac{31}{20} \zeta_5 \Bigg) \Bigg\}\,,
\\ \left[ {\cal F}^{J,(2)}_{q} \right]_{R} &= {{2 C_{F} n_{f} T_{F}}} \Bigg\{
\frac{8}{\epsilon^3} -
\frac{16}{9 \epsilon^2} + \Bigg(
- \frac{65}{27} - 2 \zeta_2
\Bigg) \frac{1}{\epsilon} -
\frac{3115}{324} +
\frac{23}{9} \zeta_2 +
\frac{2}{9} \zeta_3 + \epsilon
\Bigg( \frac{129577}{3888}
\nonumber\\
&-
\frac{731}{108} \zeta_2 -
\frac{\zeta_2^2}{10} +
\frac{119}{27} \zeta_3 \Bigg) +
\epsilon^2 \Bigg( -
\frac{3054337}{46656} +
\frac{20951}{1296} \zeta_2 -
\frac{145}{144} \zeta_2^2 -
\frac{2303}{324} \zeta_3
\nonumber\\
&-
\frac{10}{9} \zeta_2 \zeta_3 -
\frac{59}{30} \zeta_5 \Bigg)
\Bigg\}
+
{{C_{F}^2}} \Bigg\{
\frac{32}{\epsilon^4} -
\frac{48}{\epsilon^3} + \Bigg( 66
- 8 \zeta_2 \Bigg)
\frac{1}{\epsilon^2} +
\Bigg( - \frac{53}{2} +
\frac{128}{3} \zeta_3 \Bigg)
\frac{1}{\epsilon}
\nonumber\\
&- \frac{121}{8} +
\frac{\zeta_2}{2} -
13 \zeta_2^2 - 58 \zeta_3 +
\epsilon \Bigg( \frac{3403}{32}
+ \frac{27}{8} \zeta_2 +
\frac{171}{10} \zeta_2^2 +
\frac{559}{6} \zeta_3 -
\frac{56}{3} \zeta_2 \zeta_3
\nonumber\\
&+
\frac{92}{5} \zeta_5 \Bigg) +
\epsilon^2 \Bigg( -
\frac{21537}{128} -
\frac{825}{32} \zeta_2 -
\frac{457}{16} \zeta_2^2 +
\frac{223}{20} \zeta_2^3 -
\frac{4205}{24} \zeta_3 +
\frac{27}{2} \zeta_2
\zeta_3
\nonumber\\
&+
\frac{652}{9} \zeta_3^2
-
\frac{231}{10} \zeta_5 \Bigg)
\Bigg\}
+
{{C_{A} C_{F}}} \Bigg\{ -
\frac{44}{\epsilon^3} +
\Bigg( \frac{64}{9} + 4 \zeta_2
\Bigg) \frac{1}{\epsilon^2} +
\Bigg( \frac{961}{54} + 11
\zeta_2
\nonumber\\
&- 26 \zeta_3 \Bigg)
\frac{1}{\epsilon}
-
\frac{30493}{648} -
\frac{193}{18} \zeta_2 +
\frac{44}{5} \zeta_2^2 +
\frac{313}{9} \zeta_3 + \epsilon
\Bigg( - \frac{79403}{7776} +
\frac{133}{216} \zeta_2 -
\frac{229}{20} \zeta_2^2
\nonumber\\
&-
\frac{4165}{54} \zeta_3
+
\frac{89}{6} \zeta_2 \zeta_3 -
\frac{51}{2} \zeta_5 \Bigg) +
\epsilon^2 \Bigg(
\frac{9732323}{93312} +
\frac{41363}{2592} \zeta_2 +
\frac{33151}{1440} \zeta_2^2 -
\frac{809}{280} \zeta_2^3
\nonumber\\
&+
\frac{89929}{648} \zeta_3
-
\frac{80}{9} \zeta_2 \zeta_3 -
\frac{569}{12} \zeta_3^2 +
\frac{2809}{60} \zeta_5 \Bigg) \Bigg\}\,,
\\ \left[ {\cal F}^{J,(3)}_{q} \right]_{R} &= Z^{s,(3)}_{5}
+ {{4 C_{F}
n_{f}^2 T_{F}^{2}}} \Bigg\{ -
\frac{704}{81 \epsilon^4} +
\frac{64}{243 \epsilon^3} +
\Bigg( \frac{184}{81} +
\frac{16}{9} \zeta_2 \Bigg)
\frac{1}{\epsilon^2} +
\Bigg( - \frac{4834}{2187} +
\frac{40}{27} \zeta_2
\nonumber\\
&+
\frac{16}{81} \zeta_3 \Bigg)
\frac{1}{\epsilon} +
\frac{538231}{13122} -
\frac{680}{81} \zeta_2 -
\frac{188}{135} \zeta_2^2-
\frac{416}{243} \zeta_3 \Bigg\}
+
{{C_{F}^3}} \Bigg\{ -
\frac{256}{3 \epsilon^6} +
\frac{192}{\epsilon^5}
\nonumber\\
&+
\Bigg( - 336 + 32 \zeta_2 \Bigg)
\frac{1}{\epsilon^4} +
\Bigg( 280 + 24 \zeta_2 -
\frac{800}{3} \zeta_3 \Bigg)
\frac{1}{\epsilon^3} + \Bigg(
- 58 - 66 \zeta_2 +
\frac{426}{5} \zeta_2^2
\nonumber\\
&+ 552
\zeta_3 \Bigg)
\frac{1}{\epsilon^2} +
\Bigg( - \frac{4193}{6} + 83
\zeta_2 - \frac{1461}{10}
\zeta_2^2 - \frac{3142}{3} \zeta_3 +
\frac{428}{3} \zeta_2 \zeta_3 -
\frac{1288}{5} \zeta_5 \Bigg)
\frac{1}{\epsilon}
\nonumber\\
&+ \frac{41395}{24} +
\frac{1933}{12} \zeta_2 +
\frac{10739}{40} \zeta_2^2 -
\frac{9095}{252} \zeta_2^3 + 1385
\zeta_3 - 35 \zeta_2 \zeta_3 -
\frac{1826}{3} \zeta_3^2 -
\frac{562}{5} \zeta_5 \Bigg\}
\nonumber\\
&+
{{2 C_{F}^2 n_{f} T_{F}}} \Bigg\{ -
\frac{64}{\epsilon^5} +
\frac{560}{9 \epsilon^4} +
\Bigg( - \frac{680}{27} + 24
\zeta_2 \Bigg)
\frac{1}{\epsilon^3} +
\Bigg( \frac{5180}{81} -
\frac{266}{9} \zeta_2 -
\frac{440}{9} \zeta_3 \Bigg)
\frac{1}{\epsilon^2}
\nonumber\\
&+
\Bigg( - \frac{78863}{243} +
\frac{2381}{27} \zeta_2 +
\frac{287}{18} \zeta_2^2 -
\frac{938}{27} \zeta_3 \Bigg)
\frac{1}{\epsilon} +
\frac{1369027}{1458} -
\frac{16610}{81} \zeta_2 -
\frac{8503}{1080} \zeta_2^2
\nonumber\\
&+
\frac{22601}{81} \zeta_3 +
\frac{35}{3} \zeta_2 \zeta_3 -
\frac{386}{9} \zeta_5 \Bigg\}
+
{{C_{A}^2 C_{F}}} \Bigg\{ -
\frac{21296}{81 \epsilon^4} +
\Bigg( - \frac{22928}{243} +
\frac{880}{27} \zeta_2 \Bigg)
\frac{1}{\epsilon^3}
\nonumber\\
&+ \Bigg(
\frac{23338}{243} +
\frac{6500}{81} \zeta_2 -
\frac{352}{45} \zeta_2^2 -
\frac{3608}{27} \zeta_3 \Bigg)
\frac{1}{\epsilon^2} +
\Bigg( \frac{139345}{4374} +
\frac{14326}{243} \zeta_2 +
\frac{332}{15} \zeta_2^2
\nonumber\\
& -
\frac{7052}{27} \zeta_3 +
\frac{176}{9} \zeta_2 \zeta_3 +
\frac{272}{3} \zeta_5 \Bigg)
\frac{1}{\epsilon} -
\frac{10659797}{52488} -
\frac{207547}{729} \zeta_2 +
\frac{19349}{270} \zeta_2^2 -
\frac{6152}{189} \zeta_2^3
\nonumber\\
&+
\frac{361879}{486} \zeta_3 +
\frac{344}{3} \zeta_2 \zeta_3 -
\frac{1136}{9} \zeta_3^2 -
\frac{2594}{9} \zeta_5 \Bigg\}
+
{{2 C_{A} C_{F} n_{f} T_{F}}} \Bigg\{ +
\frac{7744}{81 \epsilon^4} +
\Bigg( \frac{6016}{243}
\nonumber\\
&-
\frac{160}{27} \zeta_2 \Bigg)
\frac{1}{\epsilon^3} + \Bigg( -
\frac{8272}{243} -
\frac{1904}{81} \zeta_2 +
\frac{848}{27} \zeta_3 \Bigg)
\frac{1}{\epsilon^2} +
\Bigg( \frac{17318}{2187} -
\frac{5188}{243} \zeta_2 -
\frac{88}{15} \zeta_2^2
\nonumber\\
&+
\frac{1928}{81} \zeta_3 \Bigg)
\frac{1}{\epsilon} -
\frac{4158659}{13122} +
\frac{81778}{729} \zeta_2 -
\frac{17}{135} \zeta_2^2 -
\frac{5881}{27} \zeta_3 +
\frac{22}{3} \zeta_2 \zeta_3 +
\frac{176}{3} \zeta_5 \Bigg\}
\nonumber\\
&+
{{C_{A} C_{F}^2}} \Bigg\{ \frac{352}{\epsilon^5} +
\Bigg( - \frac{2888}{9} - 32
\zeta_2 \Bigg)
\frac{1}{\epsilon^4} + \Bigg(
\frac{4436}{27} - 108 \zeta_2 +
208 \zeta_3 \Bigg)
\frac{1}{\epsilon^3}
\nonumber\\
&+
\Bigg( \frac{39844}{81} +
\frac{983}{9} \zeta_2 -
\frac{332}{5} \zeta_2^2 -
\frac{1928}{9} \zeta_3 \Bigg)
\frac{1}{\epsilon^2} + \Bigg( -
\frac{97048}{243} -
\frac{12361}{54} \zeta_2 +
\frac{2975}{36} \zeta_2^2
\nonumber\\
&+
\frac{3227}{3} \zeta_3 -
\frac{430}{3} \zeta_2 \zeta_3 +
284 \zeta_5 \Bigg) \frac{1}{\epsilon} -
\frac{709847}{729} +
\frac{36845}{324} \zeta_2 -
\frac{536683}{2160} \zeta_2^2 -
\frac{18619}{1260} \zeta_2^3
\nonumber\\
&-
\frac{31537}{18} \zeta_3 -
\frac{518}{3} \zeta_2 \zeta_3 +
\frac{1616}{3} \zeta_3^2 +
\frac{1750}{9} \zeta_5 \Bigg\}\,.
\end{align}
\subsection{Universal Behaviour of Leading Transcendentality
Contribution}
\label{sec:SUYM}
In \cite{Gehrmann:2011xn}, the form factor of a scalar composite
operator belonging to the stress-energy tensor super-multiplet of
conserved currents of ${\cal N}=4$ super Yang-Mills (SYM) with gauge
group SU(N) was studied to three-loop level. Since the theory is UV
finite in $d=4$ space-time dimensions, it is an ideal framework to
study the IR structures of amplitudes in perturbation theory. In this
theory, one observes that scattering amplitudes can be expressed as a
linear combinations of polylogarithmic functions of uniform degree
$2 l$, where $l$ is the order of the loop, with constant coefficients.
In other words, the scattering amplitudes in ${\cal N}=4$ SYM exhibit
uniform transcendentality, in contrast to QCD loop amplitudes, which
receive contributions from all degrees of transcendentality up to
$2l$.
The three-loop QCD quark and gluon form factors~\cite{Gehrmann:2010ue}
display an interesting relation to the SYM form factor. Upon
replacement~\cite{hep-th/0611204} of the color factors $C_A = C_F = N$
and $T_{f} n_{f}=N/2$, the
leading transcendental (LT) parts of the quark and gluon form factors
in QCD not only coincide with each other but also become identical, up
to a normalization factor of $2^{l}$, to the form factors of scalar
composite operator computed in ${\cal N}=4$ SYM
\cite{Gehrmann:2011xn}.
This correspondence between the QCD form factors and that of the
${\cal N}=4$ SYM can be motivated by the leading transcendentality
principle~\cite{hep-th/0611204, hep-th/0404092, Kotikov:2001sc} which
relates anomalous dimensions of the twist two operators in
${\cal N} =4$ SYM to the LT terms of such operators computed in QCD.
Examining the diagonal pseudo-scalar form factors ${\cal F}^G_g$ and
${\cal F}^J_q$, we find a similar behaviour: the LT terms of these
form factors with replacement $C_A = C_F = N$ and
$T_{f} n_{f}=N/2$ are not only identical to
each other but also coincide with the LT terms of the QCD form
factors~\cite{Gehrmann:2010ue} with the same replacement as well as
with the LT terms of the scalar form factors in ${\cal N}=4$ SYM
\cite{Gehrmann:2011xn}, up to a normalization factor of
$2^{l}$. This observation holds true for the finite terms in
$\epsilon$, and could equally be validated for higher-order terms up
to transcendentality 8 (which is the highest order for which all
three-loop master integrals are available~\cite{Lee:2010ik}). In
addition to checking the diagonal form factors, we also examined the
off-diagonal ones namely, ${\cal F}^{G}_{q}$, ${\cal F}^{J}_{g}$,
where we find that the LT terms these two form factors are identical
to each other after the replacement of colour factors. However, the LT
terms of these do not coincide with those of the diagonal ones.
\section{Gluon Form Factors for the Pseudo-scalar Higgs Boson Production}
\label{sec:pScalar-FullFF}
The complete form factor for the production of a pseudo-scalar Higgs boson through gluon
fusion, ${\hat {\cal F}}^{A,(n)}_{g}$, can be written in terms of the
individual gluon form factors, Eq.~(\ref{eq:Mexp}), as follows:
\begin{align}
\label{eq:FFDefMatEle}
{{\cal F}}^{A}_{g} = {{\cal F}}^{G}_{g} + \Bigg(
\frac{Z_{GJ}}{Z_{GG}} + \frac{4 C_{J}}{C_{G}} \frac{Z_{JJ}}{Z_{GG}} \Bigg)
{{\cal F}}^{J}_{g} \frac{\langle{\hat{\cal
M}}^{G,(0)}_{g}|{\hat{\cal M}}^{J,(1)}_{g}\rangle}{\langle{\hat{\cal
M}}^{G,(0)}_{g}|{\hat{\cal M}}^{G,(0)}_{g}\rangle}\,.
\end{align}
In the above expression, the quantities $Z_{ij} (i,j= G, J)$ are the
overall operator renormalization constants which are required to
introduce in the context of UV renormalization. These are discussed in Sec.~\ref{ss:UV} in great
detail. The ingredients of the form factor ${{\cal F}}^{A}_{g}$, namely,
${{\cal F}}^{G}_{g}$ and ${{\cal F}}^{J}_{g}$ have been calculated up to three
loop level by us~~\cite{Ahmed:2015qpa} and are presented in the Appendix~\ref{App:pScalar-Results}. Using those results we obtain the three
loop form factor for the pseudo-scalar Higgs boson production through gluon
fusion.
In this section, we present the unrenormalized form factors
${\hat {\cal F}}^{A,(n)}_{g}$ up to three loop where the components are
defined through the expansion
\begin{align}
\label{eq:FF3}
{{\cal F}}^{A}_{g} \equiv
\sum_{n=0}^{\infty} \left[ {\hat a}_{s}^{n}
\left( \frac{Q^{2}}{\mu^{2}} \right)^{n\frac{\epsilon}{2}}
S_{\epsilon}^{n} {\hat{{\cal F}}}^{A,(n)}_{g}\right] \, .
\end{align}
We present the unrenormalized results for the choice of the scale
$\mu_{R}^{2}=\mu_{F}^{2}=q^{2}$\, as follows:
\begin{align}
\label{eq:FF}
{\hat {\cal F}}^{A,(1)}_{g} &= {{C_{A}}} \Bigg\{ - \frac{8}{\epsilon^2}
+ 4 +
\zeta_2
+
\epsilon \Bigg( - 6 - \frac{7}{3} \zeta_3
\Bigg)
+ \epsilon^2 \Bigg( 7 -
\frac{\zeta_2}{2} + \frac{47}{80} \zeta_2^2
\Bigg)
+
\epsilon^3 \Bigg( - \frac{15}{2}
\nonumber\\
&+
\frac{3}{4} \zeta_2 + \frac{7}{6} \zeta_3 +
\frac{7}{24} \zeta_2 \zeta_3 -
\frac{31}{20} \zeta_5 \Bigg)
+ \epsilon^4 \Bigg( \frac{31}{4} -
\frac{7}{8} \zeta_2 - \frac{47}{160}
\zeta_2^2 +
\frac{949}{4480} \zeta_2^3 - \frac{7}{4}
\zeta_3 - \frac{49}{144} \zeta_3^2 \Bigg)
\nonumber\\
&+
\epsilon^5 \Bigg( - \frac{63}{8} +
\frac{15}{16} \zeta_2 + \frac{141}{320}
\zeta_2^2 + \frac{49}{24} \zeta_3 -
\frac{7}{48} \zeta_2 \zeta_3 +
\frac{329}{1920} \zeta_2^2 \zeta_3 +
\frac{31}{40} \zeta_5 +
\frac{31}{160} \zeta_2 \zeta_5
\nonumber\\
&-
\frac{127}{112} \zeta_7 \Bigg)
+ \epsilon^6
\Bigg( \frac{127}{16} - \frac{31}{32}
\zeta_2 - \frac{329}{640} \zeta_2^2 -
\frac{949}{8960} \zeta_2^3 +
\frac{55779}{716800} \zeta_2^4 -
\frac{35}{16} \zeta_3 +
\frac{7}{32} \zeta_2 \zeta_3
\nonumber\\
&+ \frac{49}{288}
\zeta_3^2 + \frac{49}{1152} \zeta_2 \zeta_3^2 -
\frac{93}{80} \zeta_5 - \frac{217}{480}
\zeta_3 \zeta_5 \Bigg)
+
\epsilon^7 \Bigg( - \frac{255}{32} +
\frac{63}{64} \zeta_2 + \frac{141}{256}
\zeta_2^2 + \frac{2847}{17920} \zeta_2^3
\nonumber\\
&+
\frac{217}{96} \zeta_3 - \frac{49}{192}
\zeta_2 \zeta_3 - \frac{329}{3840} \zeta_2^2
\zeta_3 +
\frac{949}{15360} \zeta_2^3 \zeta_3 -
\frac{49}{192} \zeta_3^2 - \frac{343}{10368}
\zeta_3^3 +
\frac{217}{160} \zeta_5
\nonumber\\
&- \frac{31}{320}
\zeta_2 \zeta_5 + \frac{1457}{12800}
\zeta_2^2 \zeta_5 +
\frac{127}{224} \zeta_7 + \frac{127}{896}
\zeta_2 \zeta_7 - \frac{511}{576} \zeta_{9} \Bigg) \Bigg\}\,,
\nonumber\\
{\hat {\cal F}}^{A,(2)}_{g} &= {{C_{F} n_{f}}} \Bigg\{ - \frac{80}{3} +
6 \ln
\left(\frac{q^2}{m_t^2}\right) + 8
\zeta_3
+ \epsilon \Bigg(
\frac{2827}{36} - 9 \ln
\left(\frac{q^2}{m_t^2}\right) -
\frac{19}{6} \zeta_2 -
\frac{8}{3} \zeta_2^2
\nonumber\\
&- \frac{64}{3} \zeta_3
\Bigg)
+
\epsilon^2 \Bigg( - \frac{70577}{432} +
\frac{21}{2} \ln
\left(\frac{q^2}{m_t^2}\right) +
\frac{1037}{72} \zeta_2 - \frac{3}{4}
\ln \left(\frac{q^2}{m_t^2} \right)
\zeta_2 + \frac{64}{9} \zeta_2^2 +
\frac{455}{9} \zeta_3
\nonumber\\
&-
\frac{10}{3} \zeta_2 \zeta_3 + 8 \zeta_5
\Bigg)
+
\epsilon^3 \Bigg( \frac{1523629}{5184} -
\frac{45}{4} \ln
\left(\frac{q^2}{m_t^2}\right) -
\frac{14975}{432} \zeta_2 +
\frac{9}{8} \ln
\left(\frac{q^2}{m_t^2}\right) \zeta_2
\nonumber\\
&-
\frac{70997}{4320} \zeta_2^2 + \frac{22}{35}
\zeta_2^3 -
\frac{3292}{27} \zeta_3 + \frac{7}{4}
\ln \left(\frac{q^2}{m_t^2}\right)
\zeta_3 + \frac{80}{9} \zeta_2 \zeta_3 +
15 \zeta_3^2 - \frac{64}{3} \zeta_5 \Bigg)
\nonumber\\
&+ \epsilon^4 \Bigg( -
\frac{30487661}{62208} + \frac{93}{8} \ln
\left( \frac{q^2}{m_t^2}\right) +
\frac{43217}{648} \zeta_2 - \frac{21}{16}
\ln \left(\frac{q^2}{m_t^2}\right)
\zeta_2 + \frac{1991659}{51840} \zeta_2^2
\nonumber\\
&-
\frac{141}{320} \ln \left(\frac{q^2}{m_t^2}\right) \zeta_2^2 -
\frac{176}{105} \zeta_2^3 +
\frac{694231}{2592} \zeta_3 -
\frac{21}{8} \ln \left(\frac{q^2}{m_t^2}\right) \zeta_3 -
\frac{9757}{432} \zeta_2 \zeta_3 -
\frac{1681}{180} \zeta_2^2 \zeta_3
\nonumber\\
&- 40
\zeta_3^2 + \frac{8851}{180} \zeta_5 -
2 \zeta_2 \zeta_5 - \frac{127}{8} \zeta_7
\Bigg) \Bigg\}
+
{{C_{A} n_{f}}} \Bigg\{ -
\frac{8}{3 \epsilon^3} + \frac{20}{9
\epsilon^2} +
\Bigg( \frac{106}{27} + 2 \zeta_2 \Bigg)
\frac{1}{\epsilon}
\nonumber\\
&- \frac{1591}{81} -
\frac{5}{3} \zeta_2 - \frac{74}{9}
\zeta_3
+
\epsilon \Bigg( \frac{24107}{486} -
\frac{23}{18} \zeta_2 + \frac{51}{20}
\zeta_2^2 + \frac{383}{27} \zeta_3 \Bigg)
+
\epsilon^2 \Bigg( - \frac{146147}{1458}
\nonumber\\
&+
\frac{799}{108} \zeta_2 - \frac{329}{72} \zeta_2^2 -
\frac{1436}{81} \zeta_3 + \frac{25}{6}
\zeta_2 \zeta_3 - \frac{271}{30} \zeta_5
\Bigg)
+
\epsilon^3 \Bigg( \frac{6333061}{34992} -
\frac{11531}{648} \zeta_2
\nonumber\\
&+
+ \frac{1499}{240}
\zeta_2^2
\frac{253}{1680} \zeta_2^3 +
\frac{19415}{972} \zeta_3 - \frac{235}{36}
\zeta_2 \zeta_3 -
\frac{1153}{108} \zeta_3^2 + \frac{535}{36}
\zeta_5 \Bigg)
+
\epsilon^4 \Bigg( - \frac{128493871}{419904}
\nonumber\\
&+ \frac{133237}{3888} \zeta_2 -
\frac{21533}{2592} \zeta_2^2 +
\frac{649}{1440} \zeta_2^3 -
\frac{156127}{5832} \zeta_3 + \frac{215}{27}
\zeta_2 \zeta_3 +
\frac{517}{80} \zeta_2^2 \zeta_3 +
\frac{14675}{648} \zeta_3^2
\nonumber\\
&-
\frac{2204}{135} \zeta_5 +
\frac{171}{40} \zeta_2 \zeta_5 +
\frac{229}{336} \zeta_7 \Bigg) \Bigg\}
+
{{C_{A}^2}} \Bigg\{
\frac{32}{\epsilon^4} + \frac{44}{3
\epsilon^3} + \Bigg( - \frac{422}{9} - 4
\zeta_2 \Bigg) \frac{1}{\epsilon^2}
+
\Bigg( \frac{890}{27}
\nonumber\\
&- 11 \zeta_2 +
\frac{50}{3} \zeta_3 \Bigg)
\frac{1}{\epsilon} + \frac{3835}{81} +
\frac{115}{6} \zeta_2 - \frac{21}{5}
\zeta_2^2 + \frac{11}{9} \zeta_3
+
\epsilon \Bigg( - \frac{213817}{972} -
\frac{103}{18} \zeta_2 + \frac{77}{120}
\zeta_2^2
\nonumber\\
&+ \frac{1103}{54} \zeta_3 -
\frac{23}{6} \zeta_2 \zeta_3 - \frac{71}{10}
\zeta_5 \Bigg)
+
\epsilon^2 \Bigg( \frac{6102745}{11664} -
\frac{991}{27} \zeta_2 - \frac{2183}{240} \zeta_2^2 +
\frac{2313}{280} \zeta_2^3 - \frac{8836}{81}
\zeta_3
\nonumber\\
&- \frac{55}{12} \zeta_2 \zeta_3 +
\frac{901}{36} \zeta_3^2 + \frac{341}{60}
\zeta_5 \Bigg)
+
\epsilon^3 \Bigg( - \frac{142142401}{139968}
+ \frac{75881}{648} \zeta_2 +
\frac{79819}{2160} \zeta_2^2 -
\frac{2057}{480} \zeta_2^3
\nonumber\\
&+
\frac{606035}{1944} \zeta_3 - \frac{251}{72}
\zeta_2 \zeta_3 -
\frac{1291}{80} \zeta_2^2 \zeta_3 -
\frac{5137}{216} \zeta_3^2 +
\frac{14459}{360} \zeta_5 +
\frac{313}{40} \zeta_2 \zeta_5 -
\frac{3169}{28} \zeta_7 \Bigg)
\nonumber\\
&+
\epsilon^4 \Bigg( \frac{2999987401}{1679616}
- \frac{1943429}{7776} \zeta_2 -
\frac{15707}{160} \zeta_2^2 -
\frac{35177}{20160} \zeta_2^3 +
\frac{50419}{1600} \zeta_2^4 -
\frac{16593479}{23328} \zeta_3
\nonumber\\
&+
\frac{1169}{27} \zeta_2 \zeta_3 +
\frac{22781}{1440} \zeta_2^2 \zeta_3 +
\frac{93731}{1296} \zeta_3^2 -
\frac{1547}{144} \zeta_2 \zeta_3^2 -
\frac{8137}{54} \zeta_5 -
\frac{1001}{80} \zeta_2 \zeta_5 +
\frac{845}{24} \zeta_3 \zeta_5
\nonumber\\
&-
\frac{33}{2} \zeta_{5,3} +
\frac{56155}{672} \zeta_7 \Bigg) \Bigg\}\,,
\nonumber\\
{\hat {\cal F}}^{A,(3)}_{g} &= {{n_{f} C^{(2)}_{J}}} \Bigg\{ - 2+3 \epsilon \Bigg\}
+
{{C_{F} n_{f}^2}} \Bigg\{ \Bigg( -
\frac{640}{9} + 16 \ln
\left(\frac{q^2}{m_t^2}\right) + \frac{64}{3}
\zeta_3 \Bigg) \frac{1}{\epsilon} +
\frac{7901}{27}
\nonumber\\
&- 24 \ln
\left(\frac{q^2}{m_t^2}\right)
- \frac{32}{3}
\zeta_2 - \frac{112}{15} \zeta_2^2 -
\frac{848}{9} \zeta_3 \Bigg\}
+
{{C_{F}^2 n_{f}}} \Bigg\{ \frac{457}{6} + 104
\zeta_3 - 160 \zeta_5 \Bigg\}
\nonumber\\
&+
{{C_{A}^2 n_{f}}} \Bigg\{
\frac{64}{3 \epsilon^5} - \frac{32}{81
\epsilon^4}
+
\Bigg( - \frac{18752}{243} - \frac{376}{27}
\zeta_2 \Bigg) \frac{1}{\epsilon^3} +
\Bigg( \frac{36416}{243} - \frac{1700}{81}
\zeta_2 + \frac{2072}{27} \zeta_3 \Bigg)
\frac{1}{\epsilon^2}
\nonumber\\
&+ \Bigg(
\frac{62642}{2187} + \frac{22088}{243}
\zeta_2
- \frac{2453}{90} \zeta_2^2 -
\frac{3988}{81} \zeta_3 \Bigg)
\frac{1}{\epsilon} -
\frac{14655809}{13122} -
\frac{60548}{729} \zeta_2 +
\frac{917}{60} \zeta_2^2
\nonumber\\
&- \frac{772}{27} \zeta_3
- \frac{439}{9} \zeta_2 \zeta_3
+ \frac{3238}{45}
\zeta_5 \Bigg\}
+
{{C_{A} n_{f}^2}} \Bigg\{ - \frac{128}{81
\epsilon^4} + \frac{640}{243 \epsilon^3} + \Bigg(
\frac{128}{27} + \frac{80}{27} \zeta_2
\Bigg) \frac{1}{\epsilon^2}
\nonumber\\
&+
\Bigg( - \frac{93088}{2187}
- \frac{400}{81}
\zeta_2
- \frac{1328}{81} \zeta_3 \Bigg)
\frac{1}{\epsilon} +
\frac{1066349}{6561} -
\frac{56}{27} \zeta_2 +
\frac{797}{135} \zeta_2^2 +
\frac{13768}{243} \zeta_3 \Bigg\}
\nonumber\\
&+
{{C_{A} C_{F}
n_{f}}} \Bigg\{ -
\frac{16}{9 \epsilon^3}
+
\Bigg( \frac{5980}{27} - 48 \ln \left(\frac{q^2}{m_t^2}\right) -
\frac{640}{9} \zeta_3 \Bigg)
\frac{1}{\epsilon^2} + \Bigg( -
\frac{20377}{81}
\nonumber\\
&- 16 \ln \left(\frac{q^2}{m_t^2}\right) + \frac{86}{3}
\zeta_2 +
\frac{352}{15} \zeta_2^2
+ \frac{1744}{27}
\zeta_3 \Bigg) \frac{1}{\epsilon} + 72 \ln \left(\frac{q^2}{m_t^2}\right)
- \frac{587705}{972} -
\frac{551}{6} \zeta_2
\nonumber\\
&+ 12 \ln \left(\frac{q^2}{m_t^2}\right) \zeta_2 -
\frac{96}{5} \zeta_2^2 + \frac{12386}{81}
\zeta_3
+
48 \zeta_2 \zeta_3 +
\frac{32}{9} \zeta_5 \Bigg\}
+
{{C_{A}^3}} \Bigg\{ -
\frac{256}{3 \epsilon^6} - \frac{352}{3
\epsilon^5}
\nonumber\\
&+ \frac{16144}{81 \epsilon^4} +
\Bigg( \frac{22864}{243} + \frac{2068}{27}
\zeta_2 - \frac{176}{3} \zeta_3 \Bigg)
\frac{1}{\epsilon^3}
+
\Bigg( - \frac{172844}{243} - \frac{1630}{81}
\zeta_2 + \frac{494}{45} \zeta_2^2
\nonumber\\
&-
\frac{836}{27} \zeta_3 \Bigg)
\frac{1}{\epsilon^2} + \Bigg(
\frac{2327399}{2187} - \frac{71438}{243}
\zeta_2 +
\frac{3751}{180} \zeta_2^2
- \frac{842}{9}
\zeta_3 + \frac{170}{9} \zeta_2 \zeta_3 +
\frac{1756}{15} \zeta_5 \Bigg)
\frac{1}{\epsilon}
\nonumber\\
&+ \frac{16531853}{26244} +
\frac{918931}{1458} \zeta_2 +
\frac{27251}{1080} \zeta_2^2 -
\frac{22523}{270} \zeta_2^3
- \frac{51580}{243}
\zeta_3 + \frac{77}{18} \zeta_2 \zeta_3 -
\frac{1766}{9} \zeta_3^2
\nonumber\\
&+
\frac{20911}{45} \zeta_5 \Bigg\}\,.
\end{align}
The results up to two loop level is consistent with the existing
ones~\cite{Ravindran:2004mb} and the three loop result is the new
one. These are later used to compute the SV cross-section for the production
of a pseudo-scalar particle through gluon fusion at N$^{3}$LO QCD~\cite{Ahmed:2015pSSV}.
This is an essential ingredient to compute all the other associated observables.
\section{Hard Matching Coefficients in SCET}
\label{sec:scet}
Soft-collinear effective theory (SCET, \cite{hep-ph/0005275,
hep-ph/0011336, hep-ph/0107001, hep-ph/0109045, hep-ph/0206152,
hep-ph/0211358, hep-ph/0202088}) is a systematic expansion of the
full QCD theory in terms of particle modes with different infrared
scaling behaviour. It provides a framework to perform threshold
resummation. In the effective theory, the infrared poles of the full
high energy QCD theory manifest themselves as ultraviolet
poles~\cite{Korchemsky:1985xj, Korchemsky:1987wg, Korchemsky:1988pn},
which then can be resummed by employing the renormalisation group
evolution from larger scales to the smaller ones. To ensure matching
of SCET and full QCD, one computes the matrix elements in both
theories and adjusts the Wilson coefficients of SCET accordingly. For
the on-shell matching of these two theories, the matching coefficients
relevant to pseudo-scalar production in gluon fusion can be obtained
directly from the gluon form factors.
The UV renormalised form factors in QCD contain IR
divergences. Since the IR poles in QCD turn into UV ones in SCET, we
can remove the IR divergences with the help of a renormalisation
constant $Z^{A,h}_{g}$, which essentially absorbs all residual IR poles
and produces finite results. The result is the matching coefficient
$C^{A, {\rm eff}}_{g}$, which is defined through the following
factorisation relation:
\begin{align}
\label{eq:MatchCof}
C^{A, {\rm eff}}_{g}\left(Q^{2},\mu_h^{2}\right) &\equiv \lim_{\epsilon
\rightarrow 0} (Z_{g
}^{A,h})^{-1}(\epsilon, Q^{2},\mu_h^{2})
\left[ {\cal F}^{A}_{g} \right]_{R}(\epsilon,Q^{2})
\end{align}
where, the UV renormalised form factor
$\left[ {\cal F}^{A}_{g} \right]_{R}$, is defined as
\begin{align}
\label{eq:UVRenFF}
\left[ {\cal F}^{A}_{g} \right]_{R} = \left[ {\cal F}^{G}_{g}
\right]_{R} + \frac{4 C_J}{C_G} \left[ {\cal F}^{J}_{g}
\right]_{R} { \left( a_{s} \frac{S^{J,(1)}_g}{S^{G,(0)}_{g}}
\right)} \,.
\end{align}
The parameter $\mu_h$ is the newly introduced mass scale at which the
above factorisation is carried out. For the UV renormalised form
factors $[{\cal F}^{A}_{g}]_{R}$ in Eq.~(\ref{eq:MatchCof}), we fixed
the other scales as $\mu_{R}^{2}=\mu_{F}^{2}=\mu_h^2$. Upon
expanding the $Z^{A,h}_{g}$ and $C^{A,{\rm eff}}_{g}$ in powers
of $a_{s}$ as
\begin{align}
\label{eq:MatchCofExp}
Z^{A,h}_{g}(\epsilon, Q^{2},\mu_h^{2}) &= 1 + \sum_{i=1}^{\infty}
a_{s}^{i}(\mu_h^{2})
Z^{A, h}_{g,i}(\epsilon,
Q^{2}, \mu_h^{2})\,,
\nonumber\\
C^{A, {\rm eff}}_{g}\left(Q^{2},\mu_h^{2}\right) &= 1+\sum_{i=1}^{\infty}
a_{s}^{i}(\mu_h^{2})
C^{A, {\rm eff}}_{g,i}\left(Q^{2},\mu_h^{2}\right)
\end{align}
and utilising the above Eq.~(\ref{eq:MatchCof}), we compute the
$Z^{A,h}_{g,i}$ as well as $C^{A, {\rm eff}}_{g,i}$ up to three
loops ($i=3$). Demanding the cancellation of the residual IR poles of
$\left[ {\cal F}^{A}_{g} \right]_{R}$ against the poles of
$(Z^{A,h}_{g,i})^{-1}$, we compute $Z^{A,h}_{g,i}$ which
comes out to be
\begin{align}
\label{eq:ZIR}
Z^{A,h}_{g,1} &= {{C_{A}}} \Bigg\{ - \frac{8}{\epsilon^2} +
\Bigg( - 4 L +
\frac{22}{3} \Bigg) \frac{1}{\epsilon}
\Bigg\} - {{n_{f}}} \Bigg\{ \frac{4}{3 \epsilon} \Bigg\}\,,
\nonumber\\
Z^{A,h}_{g,2} &= C_{F} n_{f} \Bigg\{ - \frac{2}{\epsilon} \Bigg\}
+ n_{f}^2 \Bigg\{ \frac{16}{9
\epsilon^2} \Bigg\}
+
C_{A} n_{f} \Bigg\{ \frac{56}{3 \epsilon^3} +
\Bigg( - \frac{52}{3} + 8 L \Bigg) \frac{1}{\epsilon^2} +
\Bigg( - \frac{128}{27} + \frac{20}{9} L
\nonumber\\
&+
\frac{2}{3} \zeta_2 \Bigg)
\frac{1}{\epsilon} \Bigg\}
+
C_{A}^2 \Bigg\{ \frac{32}{\epsilon^4} + \Bigg(
- \frac{308}{3} + 32 L \Bigg) \frac{1}{\epsilon^3} +
\Bigg( \frac{350}{9} - 44 L + 8 L^2 + 4
\zeta_2 \Bigg) \frac{1}{\epsilon^2} +
\Bigg( \frac{692}{27}
\nonumber\\
&- \frac{134}{9} L -
\frac{11}{3} \zeta_2 + 4 L \zeta_2 - 2
\zeta_3 \Bigg) \frac{1}{\epsilon} \Bigg\}\,,
\nonumber\\
Z^{A,h}_{g,3} &= C_{F}^2 n_{f} \Bigg\{ \frac{2}{3 \epsilon}
\Bigg\}
+ C_{F} n_{f}^2
\Bigg\{ \frac{56}{9 \epsilon^2} +
\frac{22}{27 \epsilon} \Bigg\}
-
n_{f}^3 \Bigg\{ \frac{64}{27 \epsilon^3}
\Bigg\}
+ C_{A}^2
n_{f} \Bigg\{ - \frac{320}{3 \epsilon^5} +
\Bigg( \frac{28480}{81}
\nonumber\\
&- 96 L \Bigg) \frac{1}{\epsilon^4} +
\Bigg( - \frac{18752}{243} + \frac{3152}{27}
L - \frac{64}{3} L^2 - \frac{448}{27}
\zeta_2 \Bigg) \frac{1}{\epsilon^3} +
\Bigg( - \frac{32656}{243} + \frac{7136}{81}
L
\nonumber\\
&- \frac{80}{9} L^2 + \frac{1000}{81} \zeta_2 -
\frac{104}{9} L \zeta_2 + \frac{344}{27}
\zeta_3 \Bigg) \frac{1}{\epsilon^2} +
\Bigg( - \frac{30715}{2187} + \frac{836}{81}
L + \frac{2396}{243} \zeta_2 -
\frac{160}{27} L \zeta_2
\nonumber\\
&-
\frac{328}{45} \zeta_2^2 - \frac{712}{81}
\zeta_3 + \frac{112}{9} L \zeta_3 \Bigg)
\frac{1}{\epsilon} \Bigg\}
+
C_{A} n_{f}^2 \Bigg\{ - \frac{2720}{81
\epsilon^4} + \Bigg( \frac{7984}{243} -
\frac{352}{27} L \Bigg) \frac{1}{\epsilon^3} +
\Bigg( \frac{368}{27}
\nonumber\\
&- \frac{400}{81} L -
\frac{40}{27} \zeta_2 \Bigg)
\frac{1}{\epsilon^2} +
\Bigg( \frac{269}{2187} + \frac{16}{81} L -
\frac{40}{81} \zeta_2 + \frac{112}{81}
\zeta_3 \Bigg) \frac{1}{\epsilon} \Bigg\}
+ C_{A}
C_{F} n_{f} \Bigg\{ \frac{272}{9 \epsilon^3}
\nonumber\\
&+
\Bigg( - \frac{704}{27} + \frac{40}{3} L -
\frac{64}{9} \zeta_3 \Bigg)
\frac{1}{\epsilon^2} +
\Bigg( - \frac{2434}{81} + \frac{110}{9} L +
\frac{4}{3} \zeta_2 + \frac{32}{15}
\zeta_2^2 +
\frac{304}{27} \zeta_3
\nonumber\\
&- \frac{32}{3} L
\zeta_3 \Bigg) \frac{1}{\epsilon} \Bigg\}
+
C_{A}^3 \Bigg\{ - \frac{256}{3 \epsilon^6} +
\Bigg( \frac{1760}{3} - 128 L \Bigg)
\frac{1}{\epsilon^5} +
\Bigg( - \frac{72632}{81} + 528 L - 64 L^2
\nonumber\\
&- 32 \zeta_2 \Bigg) \frac{1}{\epsilon^4} +
\Bigg( - \frac{29588}{243} - \frac{5824}{27}
L + \frac{352}{3} L^2 - \frac{32}{3} L^3 +
\frac{2464}{27} \zeta_2 - 48 L \zeta_2 + 16
\zeta_3 \Bigg) \frac{1}{\epsilon^3}
\nonumber\\
&+
\Bigg( \frac{80764}{243} - \frac{25492}{81}
L + \frac{536}{9} L^2 - \frac{1486}{81}
\zeta_2 +
\frac{572}{9} L \zeta_2 - 16 L^2 \zeta_2 -
\frac{352}{45} \zeta_2^2 - \frac{836}{27}
\zeta_3
\nonumber\\
&+
8 L \zeta_3 \Bigg) \frac{1}{\epsilon^2} +
\Bigg( \frac{194372}{2187} - \frac{490}{9}
L - \frac{12218}{243} \zeta_2 +
\frac{1072}{27} L \zeta_2 + \frac{1276}{45}
\zeta_2^2 - \frac{176}{15} L \zeta_2^2 -
\frac{244}{9} \zeta_3
\nonumber\\
&- \frac{88}{9} L
\zeta_3 + \frac{80}{9} \zeta_2 \zeta_3 +
\frac{32}{3} \zeta_5 \Bigg) \frac{1}{\epsilon} \Bigg\}\,.
\end{align}
After cancellation of the IR poles, we are left with the following
finite matching coefficients:
\begin{align}
\label{eq:SCETCof}
C^{A, {\rm eff}}_{g,1} &= {{C_{A}}} \Bigg\{ - L^2 + 4 + \zeta_2 \Bigg\}\,,
\nonumber\\
C^{A, {\rm eff}}_{g,2} &= C_{A}^2 \Bigg\{ \frac{1}{2} L^4 +
\frac{11}{9} L^3 + L^2
\Bigg( - \frac{103}{9} + \zeta_2 \Bigg)
+ L \Bigg( - \frac{10}{27} - \frac{22}{3}
\zeta_2 - 2 \zeta_3 \Bigg) + \frac{4807}{81} +
\frac{91}{6} \zeta_2
\nonumber\\
&+ \frac{1}{2}\zeta_2^2 -
\frac{143}{9} \zeta_3 \Bigg\}
+ C_{A} n_{f}
\Bigg\{ - \frac{2}{9} L^3 + \frac{10}{9}
L^2 + L \Bigg(
\frac{34}{27} + \frac{4}{3} \zeta_2 \Bigg)
- \frac{943}{81} -
\frac{5}{3} \zeta_2 - \frac{46}{9} \zeta_3 \Bigg\}
\nonumber\\
& +
C_{F} n_{f} \Bigg\{ - \frac{80}{3} + 6
\ln \left(\frac{\mu_h^2}{m_t^2}\right) + 8 \zeta_3 \Bigg\}\,,
\nonumber\\
C^{A, {\rm eff}}_{g,3} &= n_{f} C^{(2)}_{J} \Bigg\{ - {2}
\Bigg\}
+ C_{F} n_{f}^2 \Bigg\{ L \Bigg( -
\frac{320}{9} + 8 \ln \left(\frac{\mu_h^2}{m_t^2}\right) + \frac{32}{3}
\zeta_3 \Bigg) +
\frac{749}{9} - \frac{20}{9} \zeta_2 -
\frac{16}{45} \zeta_2^2
\nonumber\\
&-
\frac{112}{3} \zeta_3 \Bigg\}
+
C_{F}^2 n_{f} \Bigg\{ \frac{457}{6} + 104
\zeta_3 - 160 \zeta_5 \Bigg\}
+
C_{A}^2 n_{f} \Bigg\{
\frac{2}{9} L^5 - \frac{8}{27} L^4 +
L^3 \Bigg( - \frac{752}{81}
\nonumber\\
&- \frac{2}{3}
\zeta_2 \Bigg) +
L^2 \Bigg( \frac{512}{27} - \frac{103}{9}
\zeta_2 + \frac{118}{9} \zeta_3 \Bigg) +
L \Bigg( \frac{129283}{729} +
\frac{4198}{81} \zeta_2 - \frac{48}{5}
\zeta_2^2 + \frac{28}{9} \zeta_3 \Bigg)
\nonumber\\
&-
\frac{7946273}{13122} - \frac{19292}{729} \zeta_2
+ \frac{73}{45} \zeta_2^2 -
\frac{2764}{81} \zeta_3 - \frac{82}{9}
\zeta_2 \zeta_3 +
\frac{428}{9} \zeta_5 \Bigg\}
+
C_{A}^3 \Bigg\{ - \frac{1}{6} L^6 -
\frac{11}{9} L^5
\nonumber\\
&+ L^4
\Bigg( \frac{389}{54} - \frac{3}{2}
\zeta_2 \Bigg) +
L^3 \Bigg( \frac{2206}{81} + \frac{11}{3}
\zeta_2 + 2 \zeta_3 \Bigg) +
L^2 \Bigg( - \frac{20833}{162} +
\frac{757}{18} \zeta_2 - \frac{73}{10}
\zeta_2^2
\nonumber\\
&+ \frac{143}{9} \zeta_3 \Bigg) +
\frac{2222}{9} \zeta_5 + L \Bigg( -
\frac{500011}{1458} - \frac{16066}{81}
\zeta_2 + \frac{176}{5} \zeta_2^2 +
\frac{1832}{27} \zeta_3 + \frac{34}{3}
\zeta_2 \zeta_3
\nonumber\\
&+ 16 \zeta_5 \Bigg) + \frac{41091539}{26244} +
\frac{316939}{1458} \zeta_2 -
\frac{1399}{270} \zeta_2^2 -
\frac{24389}{1890} \zeta_2^3 -
\frac{176584}{243} \zeta_3 - \frac{605}{9}
\zeta_2 \zeta_3
\nonumber\\
&- \frac{104}{9} \zeta_3^2 \Bigg\}
+
C_{A} n_{f}^2 \Bigg\{
- \frac{2}{27} L^4 + \frac{40}{81} L^3 +
L^2 \Bigg( \frac{80}{81} + \frac{8}{9}
\zeta_2 \Bigg) +
L \Bigg( - \frac{12248}{729} -
\frac{80}{27} \zeta_2
\nonumber\\
&- \frac{128}{27}
\zeta_3 \Bigg) + \frac{280145}{6561} +
\frac{4}{9} \zeta_2 + \frac{4}{27} \zeta_2^2 + \frac{4576}{243} \zeta_3
\Bigg\}
+
C_{A} C_{F} n_{f} \Bigg\{ -
\frac{2}{3} L^3 + L^2 \Bigg( \frac{215}{6}
\nonumber\\
&- 6 \ln \left(\frac{\mu_h^2}{m_t^2}\right)
- 16 \zeta_3 \Bigg) +
L \Bigg( \frac{9173}{54} - 44 \ln
\left(\frac{\mu_h^2}{m_t^2}\right) + 4 \zeta_2 +
\frac{16}{5} \zeta_2^2 - \frac{376}{9}
\zeta_3 \Bigg)
\nonumber\\
&+ 24 \ln
\left(\frac{\mu_h^2}{m_t^2}\right) -
\frac{726935}{972} -
\frac{415}{18} \zeta_2 +
6 \ln \left(\frac{\mu_h^2}{m_t^2}\right)
\zeta_2 - \frac{64}{45} \zeta_2^2 +
\frac{20180}{81} \zeta_3 + \frac{64}{3}
\zeta_2 \zeta_3
\nonumber\\
&+ \frac{608}{9} \zeta_5 \Bigg\}\,.
\end{align}
In the above expressions,
$L=\ln \left( Q^{2}/\mu_h^{2} \right)=\ln \left( -q^{2}/\mu_h^{2}
\right)$.
These matching coefficients allow to perform the matching of the
SCET-based resummation onto the full QCD calculation up to three-loop
order.
Before ending the discussion of this section, we demonstrate the
universal factorisation property fulfilled by the anomalous dimension
of the $Z^{A,h}_{g}$ which is defined through the RG equation
\begin{align}
\label{eq:RGEZIR}
\mu_{h}^{2}\frac{d}{d\mu_{h}^{2}} \ln Z^{A,h}_{g}(\epsilon, Q^{2},\mu_{h}^{2}) \equiv
\gamma^{A,h}_{g}(Q^{2},\mu_{h}^{2}) = \sum_{i=1}^{\infty} a_{s}^{i}(\mu_{h}^{2})
\gamma^{A,h}_{g,i}(Q^{2},\mu_{h}^{2})\,.
\end{align}
The renormalisation group invariance of the UV renormalised
$[F^{A}_{g}]_{R}(\epsilon, Q^{2})$ with respect to the scale $\mu_{h}$
implies
\begin{align}
\label{eq:RGEZIRC}
\mu_{h}^{2}\frac{d}{d\mu_{h}^{2}} \ln Z^{A,h}_{g} +
\mu_{h}^{2}\frac{d}{d\mu_{h}^{2}} \ln C^{A,{\rm eff}}_{g} = 0\,.
\end{align}
By explicitly evaluating the $\gamma^{A,h}_{g,i}$ using the
results of $Z^{A,h}_{g}$ (Eq.~(\ref{eq:ZIR})) up to three loops
($i=3$), we find that these satisfy the following decomposition in
terms of the universal factors $A_{g,i}, B_{g,i}$ and $f_{g.i}$:
\begin{align}
\label{eq:gammaIRdecom}
\gamma^{A,h}_{g,i} = - \frac{1}{2} A_{g,i} L + \left(B_{g,i} +
\frac{1}{2} f_{g,i}\right)\,.
\end{align}
This in turn implies the evolution equation of the matching
coefficients as
\begin{align}
\label{eq:RGECeff}
\mu_{h}^{2}\frac{d}{d\mu_{h}^{2}} \ln C^{A,{\rm eff}}_{g,i} = \frac{1}{2} A_{g,i} L - \left(B_{g,i} +
\frac{1}{2} f_{g,i}\right)
\end{align}
which is in complete agreement with the existing
results~\cite{Becher:2006nr} upon identifying
\begin{align}
\label{eq:gammaV}
\gamma^{V} = B_{g.i} + \frac{1}{2} f_{g,i}\,.
\end{align}
\section{Summary}
\label{sec:conc}
In this part of the thesis, we derived the three-loop massless QCD corrections to
the quark and gluon form factors of pseudo-scalar operators. Working
in dimensional regularisation, we used the 't~Hooft-Veltman
prescription for $\gamma_5$ and the Levi-Civita tensor, which requires
non-trivial finite renormalisation to maintain the symmetries of the
theory. By exploiting the universal behaviour of the infrared pole
structure at three loops in QCD, we were able to independently
determine the renormalisation constants and operator mixing, in
agreement with earlier results that were obtained in a completely
different approach~\cite{Larin:1993tq,Zoller:2013ixa}.
The three-loop corrections to the pseudo-scalar form factors are an
important ingredient to precision Higgs phenomenology. They will
ultimately allow to bring the gluon fusion cross section for
pseudo-scalar Higgs production to the same level of accuracy that has
been accomplished most recently for scalar Higgs production with fixed
order N$^3$LO~\cite{Anastasiou:2015ema} and soft-gluon resummation at
N$^3$LL~\cite{Ahrens:2008nc,Bonvini:2014joa,Catani:2014uta,Schmidt:2015cea}.
With our new results, the soft-gluon resummation for pseudo-scalar
Higgs production~\cite{deFlorian:2007sr,Schmidt:2015cea} can be
extended imminently to N$^3$LL
{accuracy~\cite{Ahmed:2015pSSV}}, given the established
formalisms at this order~\cite{Ahrens:2008nc,Catani:2014uta}.
With the derivation of the three-loop pseudo-scalar form factors
presented here, all ingredients to this calculation are now available.
Another imminent application is
the threshold approximation to the N$^3$LO cross {section~\cite{Ahmed:2015pSSV}}. By
exploiting the universal infrared structure~\cite{Catani:2014uta}, one
can use the result of an explicit computation of the threshold
contribution to the N$^3$LO cross section for scalar Higgs
production~\cite{Anastasiou:2014vaa} to derive threshold results for
other processes essentially through the ratios of the respective form
factors (which is no longer possible beyond
threshold~\cite{Anastasiou:2014lda,Anastasiou:2015ema}, where the
corrections become process-specific), as was done for the Drell-Yan
process~\cite{Ahmed:2014cla} and for Higgs production from bottom
quark annihilation~\cite{Ahmed:2014cha}.
\chapter{\label{chap:Rap}Rapidity Distributions of Drell-Yan and Higgs Boson
at Threshold in N$^{3}$LO QCD}
\textit{\textbf{The materials presented in this chapter are the result of an original research done in collaboration with Manoj K. Mandal, Narayan Rana and V. Ravindran, and these are based on the published article~\cite{Ahmed:2014uya}}}.
\\
\begingroup
\hypersetup{linkcolor=blue}
\minitoc
\endgroup
\def{\cal D}{{\cal D}}
\def\overline{\cal G}{\overline {\cal G}}
\def\gamma{\gamma}
\def\epsilon{\epsilon}
\def\overline{z}_1{\overline{z}_1}
\def\overline{z}_2{\overline{z}_2}
\def\overline{C}{\overline{C}}
\section{Prologue}
\label{sec:Rap-Intro}
The Drell-Yan production \cite{Drell:1970wh} of a pair of leptons at the LHC is one of the cleanest processes that can
be studied not only to test the SM to an
unprecedented accuracy but also to probe physics beyond
the SM (BSM) scenarios in a very clear environment. Rapidity
distributions of $Z$ boson \cite{Affolder:2000rx} and charge
asymmetries of leptons in $W$ boson decays \cite{Abe:1998rv} constrain various
parton densities and, in addition, possible excess events can provide
hints to BSM physics, namely R-parity violating supersymmetric models,
models with $Z'$ or with contact interactions and large
extra-dimension models.
One of the production mechanisms responsible for
discovering the Higgs boson of the SM at the LHC \cite{Aad:2012tfa, Chatrchyan:2012ufa} is
the gluon-gluon fusion through top quark
loop. Being a dominant one, it will continue to play a major role in studying the properties of the Higgs boson and its coupling to other SM particles.
Distributions of transverse momentum and rapidity of the Higgs boson are going to be
very useful tools to achieve this task. Like the inclusive
rates \cite{Kubar-Andre:1978uy, Altarelli:1978id, Humpert:1980uv, Matsuura:1987wt, Matsuura:1988sm, Hamberg:1990np, Dawson:1990zj, Djouadi:1991tk, Spira:1995rr, Harlander:2001is, Catani:2001ic, Catani:2003zt, Harlander:2002wh, Anastasiou:2002yz, Ravindran:2003um}, the rapidity distribution
of dileptons in DY production
and of the Higgs boson in gluon-gluon fusion are also known to NNLO level in perturbative QCD due to seminal works
by Anastasiou \textit{et al.}~\cite{Anastasiou:2003yy}.
The quark and gluon form factors \cite{Moch:2005id, Moch:2005tm, Baikov:2009bg, Gehrmann:2010ue},
the mass factorization kernels \cite{Moch:2004pa}, and the renormalization
constant \cite{Inami:1982xt, Chetyrkin:1997iv, Chetyrkin:1997un} for the effective operator describing the coupling of the
Higgs boson with the SM fields in the infinite top quark mass limit up to three loop level in dimensional regularization
with space-time dimensions $n = 4 + \epsilon$ were found to be useful to obtain
the N$^3$LO threshold effects \cite{Moch:2005ky, Laenen:2005uz, Idilbi:2005ni, Ravindran:2005vv, Ravindran:2006cg} to the
inclusive Higgs boson and DY productions at the LHC, excluding $\delta(1-z)$ terms,
where the scaling parameter is $z=m_{l^+l^-}^2/\hat s$ for the DY process and
$z=m_H^2/\hat s$ for the Higgs boson.
Here, $m_{l^+l^-}$, $m_H$ and $\hat s$ are the
invariant mass of the dileptons, the mass of the Higgs boson,
and square of the center of mass energy of the partonic reaction responsible for the production mechanism,
respectively.
Recently, Anastasiou \textit{et al.}~\cite{Anastasiou:2014vaa} made an important contribution in computing
the total rate for the Higgs boson production at N$^3$LO resulting from the threshold
region including the $\delta(1-z)$ term. Their result, along with
three loop quark form factors and mass factorization kernels, was used to
compute the DY cross section at N$^3$LO at threshold in \cite{Ahmed:2014cla}.
In this thesis, we will apply the formalism developed in \cite{Ravindran:2006bu} to obtain
rapidity distributions of the dilepton pair and of the Higgs boson at N$^3$LO in the threshold
region using the available information that led to the computation
of the N$^3$LO threshold corrections to the inclusive Higgs boson~\cite{Anastasiou:2014vaa} and DY productions~\cite{Ahmed:2014cla}.
We begin by writing down the relevant interacting Lagrangian in Sec.~\ref{sec:Rap-Lag}.
In the Sec.~\ref{sec:Rap-ThreResu}, we present the formalism of computing threshold QCD corrections to the differential rapidity distribution and in Sec.~\ref{sec:Rap-Res}, we present
our results for the threshold N$^3$LO QCD corrections to the rapidity distributions of the dilepton pairs in DY and Higgs boson. The numerical impact in case of Higgs boson is discussed in brief in Sec.~\ref{sec:Rap-Numerics}.
The numerical impact of threshold enhanced N$^3$LO contributions
is demonstrated for the LHC energy $\sqrt{s} = 14$ TeV by studying
the stability of the perturbation theory under factorization and renormalization scales.
Finally we give a brief summary of our findings in Sec.~\ref{sec:Rap-Summary}.
\section{The Lagrangian}
\label{sec:Rap-Lag}
In the SM, the scalar Higgs boson couples to gluons
only indirectly through a virtual heavy quark loop. This loop can be
integrated out in the limit of infinite quark mass. The resulting
effective Lagrangian encapsulates the interaction between a
scalar $\phi$ and QCD particles and reads:
\begin{align}
\label{eq:Rap-Lag-H}
&{\cal L}^{H}_{\rm eff} = G_H \phi(x) O^{H}(x)
\intertext{with}
&O^H(x) \equiv - \frac{1}{4} G^a_{\mu\nu}(x) G^{a, \mu\nu}(x)\,,
\nonumber\\
&G_H \equiv - \frac{2^{5/4}}{3} a_s(\mu_R^2) G_F^{\frac{1}{2}} C_H
\left( a_s(\mu_R^2), \frac{\mu_R^2}{m_t^2} \right)\,.
\end{align}
$C_H(\mu_R^2)$ is the Wilson coefficient, given as a perturbative
expansion in the $\overline{MS}$ renormalised strong coupling constant
$a_s \equiv a_s(\mu_R^2)$, evaluated at the renormalisation scale
$\mu_R$. This is given by~\cite{Chetyrkin:1997un, Schroder:2005hy,
Chetyrkin:2005ia}
\begin{align}
\label{eq:Rap-Wilson}
C_H \left( a_s(\mu_R^2), \frac{\mu_R^2}{m_t^2} \right) &= 1 + a_s
\Bigg\{ 11
\Bigg\}
+ a_s^2 \Bigg\{ \frac{2777}{18} + 19 L_t + n_f \left(-\frac{67}{6} +
\frac{16}{3}
L_t\right)
\Bigg\}
\nonumber\\
&+
a_s^3 \Bigg\{ -\frac{2892659}{648} +
\frac{897943}{144} \zeta_3 + \frac{3466}{9} L_t + 209 L_t^2
\nonumber\\
&+ n_f \left( \frac{40291}{324} - \frac{110779}{216} \zeta_3 +
\frac{1760}{27} L_t + 46 L_t^2 \right)
\nonumber\\
&+ n_f^2 \left(-\frac{6865}{486} + \frac{77}{27} L_t - \frac{32}{9} L_t^2 \right)
\Bigg\}
\end{align}
up to ${\cal O}(a_s^3)$ with $L_t= \log \left( \mu_R^2/m_t^2 \right)$ and
$n_f$ is the number of active light quark flavors. For the DY process, we work in the framework of exact
SM with $n_f=5$ number of active light quark flavors.
\section{Theoretical Framework for Threshold Corrections to Rapidity}
\label{sec:Rap-ThreResu}
The differential rapidity distribution for the production of a
colorless particle,
namely, a Higgs boson through gluon fusion/bottom quark annihilation
or a pair of leptons in the DY at the hadron colliders can
be computed using
\begin{align}
\label{eq:Rap-RapDefn}
\frac{d}{dY} \sigma_Y^{I} \left( \tau, q^2, Y \right) =
\sigma^{I,(0)}_Y \left( \tau, q^2, \mu_R^2 \right) W^{I} \left(
\tau, q^2, Y, \mu_R^2 \right)\,.
\end{align}
In the above expression, $Y$ stands for the rapidity which is defined
as
\begin{align}
\label{eq:Rap-RapDefn}
Y \equiv \frac{1}{2} \log \left( \frac{P_2.q}{P_1.q} \right)
\end{align}
where, $P_i$ and $q$ are the momentum of the incoming hadrons and the
colorless particle, respectively. The variable $\tau$ equals $q^2/s$
with
\begin{equation}
\label{eq:Rap-q2}
q^{2} =
\begin{cases}
~m_{H}^{2}& ~\text{for}~ I=H\, ,\\
~m_{l^+l^-}^{2}& ~\text{for}~ I={\rm DY}\, .
\end{cases}
\end{equation}
$m_{H}$ is the mass of the Higgs boson and $m_{l^+l^-}$is the invariant
mass of the final state dilepton pair ($l^{+}l^{-}$), which can be
$e^{+}e^{-},\mu^{+}\mu^{-}, \tau^{+}\tau^{-}$, in the DY production.
$\sqrt{s}$ and $\sqrt{\hat{s}}$ stand for the hadronic and partonic
center of mass energy, respectively. Throughout this chapter, we
denote $I=H$ for the
productions of the Higgs boson
through gluon ($gg$) fusion (Fig.~\ref{fig:Rap-ggH}) and bottom quark ($b{\bar b}$)
annihilation (Fig.~\ref{fig:Rap-bBH}), whereas we write $I=$DY for the
production of a pair of
leptons in the DY (Fig.~\ref{fig:Rap-qQll}).
\begin{figure}[htb]
\begin{center}
\begin{tikzpicture}[line width=1.5 pt, scale=1]
\draw[gluon] (-2.5,0) -- (0,0);
\draw[gluon] (-2.5,-2) -- (0,-2);
\draw[fermion] (0,0) -- (2,-1);
\draw[fermion] (2,-1) -- (0,-2);
\draw[fermion] (0,-2) -- (0,0);
\draw[scalarnoarrow] (2,-1) -- (4,-1);
\node at (-2.8,0) {$g$};
\node at (-2.8,-2) {$g$};
\node at (4.3,-1) {$H$};
\end{tikzpicture}
\caption{Higgs boson production in gluon fusion}
\label{fig:Rap-ggH}
\end{center}
\end{figure}
\begin{figure}[htb]
\begin{center}
\begin{tikzpicture}[line width=1.5 pt, scale=1]
\draw[fermion] (-2.5,0) -- (0,-1.3);
\draw[fermion] (0,-1.3) -- (-2.5,-2.6);
\draw[scalarnoarrow] (0,-1.3) -- (2,-1.3);
\node at (-2.8,0) {$b$};
\node at (-2.8,-2.6) {${\bar b}$};
\node at (2.3,-1.3) {$H$};
\end{tikzpicture}
\caption{Higgs boson production through bottom quark annihilation}
\label{fig:Rap-bBH}
\end{center}
\end{figure}
\begin{figure}[htb]
\begin{center}
\begin{tikzpicture}[line width=1.5 pt, scale=1]
\draw[fermion] (-2.5,0) -- (0,-1.3);
\draw[fermion] (0,-1.3) -- (-2.5,-2.6);
\draw[vector] (0,-1.3) -- (2,-1.3);
\draw[fermion] (2,-1.3) -- (4.5,0);
\draw[fermionbar] (2,-1.3) -- (4.5,-2.6);
\node at (-2.8,0) {$q$};
\node at (-2.8,-2.6) {${\bar q}$};
\node at (1.1,-0.7) {$\gamma^{*}/Z$};
\node at (4.8,0) {$l^{+}$};
\node at (4.8,-2.6) {$l^{-}$};
\end{tikzpicture}
\caption{Drell-Yan pair production}
\label{fig:Rap-qQll}
\end{center}
\end{figure}
In Eq.~(\ref{eq:Rap-RapDefn}), $\sigma^I_Y$ is defined through
\begin{equation}
\label{eq:Rap-sigmaIY}
\sigma^I_Y \left( \tau, q^2, Y \right) =
\begin{cases}
~&\sigma^I \left( \tau,q^2, Y \right) ~~\quad\text{for}~ I=H\, ,\\
~&\frac{d}{dq^2}\sigma^{I} \left( \tau,q^2, Y \right) ~\text{for}~
I={\rm DY}\, .
\end{cases}
\end{equation}
where, $\sigma^I\left( \tau,q^2\right)$ is the inclusive production
cross section. $\sigma^{I, (0)}_Y$ is an overall prefactor extracted
from the leading order contribution. The other quantity $W^I$ is given
by
\begin{align}
\label{eq:Rap-WI}
W^I \left( \tau, q^2, Y, \mu_R^2 \right) &= \frac{\left( Z^I(\mu_R^2)
\right)^2
}{\sigma^{I,(0)}_{Y}}
\sum\limits_{i,j=q,{\bar
q},g} \int\limits_0^1 dx_1
\int\limits_0^1 dx_2
{\hat {\cal
H}}_{ij}^{I}\left(x_{1},x_2 \right)
\int\limits_0^1 dz
\delta(\tau-z x_1 x_2)
\nonumber\\
&~~~~\times\int
dPS_{1+X} |{\hat {\cal M}}_{ij
}^{I}|^2 \delta
\left( Y-\frac{1}{2} \log
\left( \frac{P_2.q}{P_1.q} \right) \right)\,,
\nonumber\\
&\equiv \sum\limits_{i,j=q,{\bar q},g} \int\limits_0^1 dx_1
\int\limits_0^1 dx_2
{\hat {\cal H}}_{ij}^{I}\left(x_{1},x_2 \right)
\frac{1}{x_1 x_2}{\hat \Delta}^I_{Y,ij} \left( \tau, Y, {\hat a}_s,
\mu^2, q^2, \mu_R^2, \epsilon \right)
\end{align}
where, we have introduced the dimensionless differential partonic
cross section ${\hat \Delta}^I_{Y,ij}$. $Z^I$ is the
overall operator UV renormalisation constant, $x_k ~(k=1,2)$
are the momentum fractions of the initial state partons i.e. $p_k=x_k
P_k$ and ${\hat {\cal H}}^I_{ij}$ stands for
\begin{align}
{\hat {\cal H}}_{ij}^{I}\left(x_{1},x_2\right) \equiv
\begin{cases}
~& {\hat f}_i \left( x_1 \right) {\hat f}_j \left( x_2 \right) ~{\rm for}~
I={\rm DY}\,,
\nonumber\\
~& {\hat f}_i \left( x_1 \right) {\hat f}_j \left( x_2 \right) ~{\rm for}~
I=H ~{\rm through}~ b{\bar b} ~{\rm annihilation}\,,
\nonumber\\
~& x_1 {\hat f}_i \left( x_1 \right) x_2 {\hat f}_j \left( x_2 \right) ~{\rm for}~
I=H ~{\rm in}~ gg ~{\rm fusion}\,.
\end{cases}
\end{align}
${\hat f}_i(x_k)$ is the unrenormalised PDF of the initial state partons $i$
with momentum fractions $x_k$.
$X$ is the remnants other than the colorless particle $I$, $dPS_{1+X}$
is the phase space element for the $I+X$ system and ${\hat {\cal
M}}_{ij}^{I}$ represents the partonic level scattering matrix
element for the process $ij \rightarrow I$. The renormalised PDF, $f_i
\left( x_1, \mu_F^2 \right)$, renormalised at the factorisation scale
$\mu_F$, is related to the unrenormalised ones through
Altarelli-Parisi (AP) kernel:
\begin{align}
\label{eq:Rap-PDF-Renorm}
f_i \left( x_k, \mu_F^2 \right) = \sum\limits_{j=q,{\bar q},g}
\int\limits_{x_k}^1 \frac{dz}{z} \Gamma_{ij} \left( {\hat a}_s,
\mu^2, \mu_F^2, z, \epsilon \right) {\hat f}_j \left( \frac{x_k}{z} \right)
\end{align}
where the scale $\mu$ is introduced to keep the unrenormalised strong coupling
constant ${\hat a}_s$ dimensionless in space-time dimensions
$d=4+\epsilon$. ${\hat{a}_s} \equiv {\hat{g}}_{s}^{2}/16\pi^{2}$
is the unrenormalized strong coupling constant which is related to the
renormalized one $a_{s}(\mu_{R}^{2})\equiv a_{s}$ through the renormalization
constant $Z_{a_{s}}(\mu_{R}^{2}) \equiv Z_{a_{s}}$,
Eq.~(\ref{eq:bBH-ashatANDas}). The form of e
$Z_{a_s}$ is presented in Eq.~(\ref{eq:bBH-Zas}).
Expanding the AP kernel in powers of ${\hat a}_s$, we
get
\begin{align}
\label{eq:Rap-APKernel-Expand}
\Gamma^I_{ij}({\hat a}_s, \mu^2, \mu_F^2, z, \epsilon) = \delta_{ij}\delta(1-z) +
\sum_{k=1}^{\infty}
{\hat a}_{s}^{k} S_{\epsilon}^{k} \left(\frac{\mu_{F}^{2}}{\mu^{2}}\right)^{k
\frac{\epsilon}{2}} {\hat \Gamma}^{I,(k)}_{ij}(z,\epsilon)\,.
\end{align}
${\hat \Gamma}^{I,(k)}_{ij}(z,\epsilon)$ in terms of the Altarelli-Parisi
splitting functions $P^{I,(k)}_{ij}\left(z,\mu_F^2\right)$ are presented in
the Appendix~(\ref{eq:App-Gamma-GenSoln}). Employing the
Eq.~(\ref{eq:Rap-PDF-Renorm}), we can write the renormalised ${\cal
H}^I_{ij} \left( x_1, x_2, \mu_F^2 \right)$ as
\begin{align}
\label{eq:Rap-RenormH}
{\cal H}^I_{ij} \left( x_1, x_2, \mu_F^2 \right) &= \sum\limits_{k,l}
\int\limits_{x_1}^1
\frac{dy_1}{y_1}
\int\limits_{x_2}^1
\frac{dy_2}{y_2}
\Gamma^I_{ik}({\hat
a}_s, \mu^2,
\mu_F^2, y_1,
\epsilon)
{\hat{\cal
H}}^I_{kl} \left(
\frac{x_1}{y_1},
\frac{x_2}{y_2}\right)
\Gamma^I_{jl}({\hat a}_s,
\mu^2, \mu_F^2, y_2,
\epsilon)\,.
\end{align}
In addition to renormalising the PDF, the AP kernels absorb the
initial state collinear singularities present in the ${\hat
\Delta}^I_{Y,ij}$ through
\begin{align}
\label{eq:Rap-RenormD}
\Delta^I_{Y,ij} \left( \tau, Y, q^2, \mu_R^2, \mu_F^2 \right)
&= \int \frac{dy_1}{y_1} \int \frac{dy_2}{y_2} \left( \Gamma^{I}
\left( {\hat a}_s, \mu^2, \mu_F^2, y_1, \epsilon \right) \right)^{-1}_{ik}
{\hat \Delta}^I_{Y,kl} \left( \tau, Y, {\hat a}_s, \mu^2, q^2,
\mu_R^2, \epsilon \right)
\nonumber\\
& \times
\left( \Gamma^{I} \left( {\hat a}_s, \mu^2, \mu_F^2, y_2, \epsilon \right)
\right)^{-1}_{jl} \,.
\end{align}
The $\Delta^I_{Y,ij}$ is free of UV, soft and collinear
singularities. With these we can express $W^I$ in terms of the
renormalised quantities. Before writing down the renormalised version
of the Eq.~(\ref{eq:Rap-WI}), we introduce two symmetric variables
$x_1^0$ and $x_2^0$ instead of $Y$ and $\tau$ through
\begin{align}
\label{eq:Rap-Def-x10-x20}
Y \equiv \frac{1}{2} \log \left( \frac{x_1^0}{x_2^0} \right)\,, \qquad
\tau \equiv x_1^0 x_2^0\,.
\end{align}
In terms of these new variables, the contributions arising from
partonic subprocesses can be shown to depend on the ratios $z_j =
x_j^0/x_j$ which take the role of scaling variables at the partonic
level. In terms of these newly introduced variables, we get the
renormalised $W^I$ as
\begin{align}
\label{eq:Rap-Ren-WI}
W^I \left( x_1^0, x_2^0, q^2, \mu_R^2 \right)
&= \sum\limits_{i,j=q,{\bar q}, g} \int\limits_{x_1^0}^1 \frac{dz_1}{z_1}
\int\limits_{x_2^0}^1 \frac{dz_2}{z_2} {\cal H}^I_{ij} \left(
\frac{x_1^0}{z_1} \frac{x_2^0}{z_2}, \mu_F^2 \right)
\Delta^I_{Y,ij} \left( z_1, z_2, q^2, \mu_R^2, \mu_F^2 \right)\,.
\end{align}
The \textit{goal} of this chapter is to study the impact of the
contributions arising from the soft gluons to the differential
rapidity distributions of a colorless particle production at Hadron
colliders. The infrared safe contributions from the soft gluons is
obtained by adding the soft part of the distribution with the
UV renormalized virtual part and performing mass
factorisation using appropriate counter terms. This combination is
often called the soft-plus-virtual (SV) rapidity distribution whereas the
remaining portion is known as hard part. Hence, we write the rapidity
distribution by decomposing into two parts as
\begin{align}
\label{eq:Rap-PartsOfDelta}
&{\Delta}^{I}_{Y,ij} (z_1, z_2, q^{2}, \mu_{R}^{2}, \mu_F^2) = {\Delta}^{I,
\text{SV}}_{Y,ij} (z_1, z_2, q^{2}, \mu_{R}^{2}, \mu_F^2) + {\Delta}^{I,
\text{hard}}_{Y,ij} (z_1, z_2, q^{2}, \mu_{R}^{2}, \mu_F^2) \,.
\end{align}
The SV contributions
${\Delta}^{I, \text{SV}}_{Y, ij} (z_1, z_2, q^{2}, \mu_{R}^{2}, \mu_F^2)$
contains only the distributions of kind $\delta(1-z_1)$, $\delta(1-z_2)$ and
${\cal{D}}_{i}$, $\overline{\cal D}_{i}$ where the latter ones are
defined through
\begin{align}
\label{eq:Rap-calD}
{\cal{D}}_{i} \equiv \left[ \frac{\ln^{i}(1-z_1)}{(1-z_1)}
\right]_{+}\,, \qquad \overline{\cal{D}}_{i} \equiv \left[
\frac{\ln^{i}(1-z_2)}{(1-z_2)} \right]_{+} \quad {\rm with} \quad
i=0,1,2,\ldots\,.
\end{align}
This is also known as the threshold contributions. On the other hand,
the hard part ${\Delta}^{I, \text{hard}}_{Y, ij}$ contains all the
regular terms in $z_1$ and $z_2$. The SV rapidity distribution in $z$-space is
computed in $d$-dimensions, as formulated in
\cite{Ravindran:2006bu}, using
\begin{align}
\label{eq:Rap-Delta-Psi}
\Delta^{I, \text{SV}}_{Y,ij} (z_1, z_2, q^2, \mu_{R}^{2}, \mu_F^2) =
{\cal C} \exp \Big( \Psi^I_{Y,ij} \left(z_1, z_2, q^2, \mu_R^2, \mu_F^2, \epsilon
\right) \Big) \Big|_{\epsilon = 0}
\end{align}
where, $\Psi^I_{Y, ij} \left(z_1, z_2, q^2, \mu_R^2, \mu_F^2, \epsilon
\right)$ is a
finite distribution and ${\cal C}$ is the convolution defined
as
\begin{equation}
\label{eq:bBH-conv}
{\cal C} e^{f(z_1, z_2)} = \delta(1-z_1) \delta(1-z_2) + \frac{1}{1!} f(z_1,z_2) + \frac{1}{2!}
f(z_1,z_2) \otimes f(z_1,z_2) + \cdots \,.
\end{equation}
Here, $\otimes$ represents the double Mellin convolution with respect
to the pair of variables $z_1$, $z_2$ and $f(z_1,z_2)$ is a
distribution of the kind $\delta(1-z_j)$, ${\cal D}_i$ and
$\overline{\cal D}_{i}$. The $\Psi^I_{Y, ij} \left(z_1, z_2, q^2, \mu_R^2,
\mu_F^2, \epsilon \right)$ is
constructed from the form factors ${\cal F}^I_{ij} (\hat{a}_s, Q^2, \mu^2,
\epsilon)$ with $Q^{2}=-q^{2}$, the overall operator UV
renormalization constant $Z^I_{ij}(\hat{a}_s,
\mu_R^2, \mu^2, \epsilon)$,
the soft-collinear distribution $\Phi^I_{Y,ij}(\hat{a}_s,$ $q^2,$ $\mu^2,
z_1, z_2, \epsilon)$
arising from the real radiations in the partonic subprocesses and the
mass factorization kernels $\Gamma^{I}_{ij} (\hat{a}_s, \mu^2, \mu_F^2, z_{j},
\epsilon)$. In
terms of the above-mentioned quantities it takes the following form,
as presented in \cite{Ravindran:2006bu, Ahmed:2014uya, Ahmed:2014era}
\begin{align}
\label{eq:Rap-psi}
\Psi^{I}_{Y,ij} \left(z_1, z_2, q^2, \mu_R^2, \mu_F^2, \epsilon \right)
= &\left( \ln \Big[ Z^I_{ij} (\hat{a}_s, \mu_R^2, \mu^2, \epsilon) \Big]^2
+ \ln \Big| {\cal F}^I_{ij} (\hat{a}_s, Q^2, \mu^2, \epsilon)
\Big|^2 \right) \delta(1-z_1) \delta(1-z_2)
\nonumber\\
& + 2 \Phi^I_{Y,ij} (\hat{a}_s, q^2, \mu^2, z_1, z_2, \epsilon) - {\cal C} \ln
\Gamma^{I}_{ij} (\hat{a}_s, \mu^2, \mu_F^2, z_1,
\epsilon)\delta(1-z_2)
\nonumber\\
&
- {\cal C} \ln
\Gamma^{I}_{ij} (\hat{a}_s, \mu^2, \mu_F^2, z_2,
\epsilon)\delta(1-z_1) \, .
\end{align}
In this chapter, we will confine our discussion on the threshold
corrections to the Higgs boson production through gluon fusion and DY
pair productions. More precisely, our main \textit{goal} is to compute the SV corrections to the
rapidity distributions of these two processes at N$^{3}$LO QCD. In the
subsequent
sections, we will demonstrate the methodology to get the ingredients,
Eq.~(\ref{eq:Rap-psi})
to compute the SV rapidity distributions at N$^3$LO QCD.
\subsection{The Form Factor}
\label{ss:Rap-FF}
The quark and gluon form factors represent the QCD loop corrections to
the transition matrix element from an on-shell quark-antiquark pair or
two gluons to a color-neutral particle. For the processes under
consideration, we require gluon form factors in case of
scalar Higgs boson production in $gg$ fusion and quark form factors
for DY pair productions from $q{\bar q}$ annihilation (happens through
intermediate off-shell photon, $\gamma^{*}$ or $Z$-boson). The
unrenormalised quark form factors at ${\cal O}({\hat a}_{s}^{n})$
are defined through
\begin{align}
\label{eq:Rap-DefFb}
&{\hat{\cal F}}^{I,(n)}_{i\,{\bar i}} \equiv
\frac{\langle{\hat{\cal
M}}^{I,(0)}_{i\,{\bar i}}|{\hat{\cal M}}^{I,(n)}_{i\,{\bar
i}}\rangle}{\langle{\hat{\cal M}}^{I,(0)}_{i\,{\bar i}}|{\hat{\cal
M}}^{I,(0)}_{i\,{\bar i}}\rangle}, \qquad n=0,1,2,3, \cdots
\intertext{with}
&i~{\bar i} =
\begin{cases}
gg \qquad {\rm for ~H,}
\nonumber\\
q{\bar q} \qquad {\rm for ~DY}\,.
\end{cases}
\end{align}
In the above expressions
$|{\hat{\cal M}}^{I,(n)}_{i~{\bar i}}\rangle$ is the
${\cal O}({\hat a}_{s}^{n})$ contribution to the unrenormalised matrix
element for the production of the particle $I$ from on-shell $i~{\bar
i}$ annihilation. In terms of these quantities, the full matrix
element and the full
form factors can be written as a series expansion in ${\hat a}_{s}$ as
\begin{align}
\label{eq:Rap-DefFlambda}
|{\cal M}^{I}_{i~{\bar i}}\rangle \equiv \sum_{n=0}^{\infty} {\hat
a}^{n}_{s} S^{n}_{\epsilon}
\left( \frac{Q^{2}}{\mu^{2}} \right)^{n\frac{\epsilon}{2}}
|{\hat{\cal M}}^{I,(n)}_{i~{\bar i}} \rangle \, ,
\qquad \qquad
{\cal F}^{I}_{i~{\bar i}} \equiv
\sum_{n=0}^{\infty} \left[ {\hat a}_{s}^{n} S_{\epsilon}^{n}
\left( \frac{Q^{2}}{\mu^{2}} \right)^{n\frac{\epsilon}{2}}
{\hat{\cal F}}^{I,(n)}_{i~{\bar i}}\right]\, ,
\end{align}
where $Q^{2}=-2\, p_{1}.p_{2}=-q^{2}$ and $p_i$ ($p_{i}^{2}=0$) are
the momenta of the external on-shell quarks or gluons. Gluon form
factors ${\cal F}^{H}_{gg}$ up to three loops in QCD were computed
in~\cite{Harlander:2000mg, Moch:2005tm, Gehrmann:2005pd,
Baikov:2009bg, Lee:2010cga, Gehrmann:2010ue}. The quark form factors
${\cal F}^{\rm DY}_{q{\bar q}}$
up to three loops in QCD are available from~\cite{Kramer:1986sg,
Matsuura:1987wt, Matsuura:1988sm, Moch:2005id, Gehrmann:2005pd,
Moch:2005tm, Baikov:2009bg, Lee:2010cga, Gehrmann:2010ue}.
The form factor ${\cal F}^{I}_{i~{\bar i}}(\hat{a}_{s}, Q^{2}, \mu^{2}, \epsilon)$
satisfies the $KG$-differential equation \cite{Sudakov:1954sw,
Mueller:1979ih, Collins:1980ih, Sen:1981sd, Magnea:1990zb} which is
a direct consequence of the facts that QCD amplitudes exhibit
factorisation property, gauge and renormalisation group (RG)
invariances:
\begin{equation}
\label{eq:Rap-KG}
Q^2 \frac{d}{dQ^2} \ln {\cal F}^{I}_{i~{\bar i}} (\hat{a}_s, Q^2,
\mu^2, \epsilon)
= \frac{1}{2} \left[ K^{I}_{i~{\bar i}} \left(\hat{a}_s,
\frac{\mu_R^2}{\mu^2}, \epsilon
\right) + G^{I}_{i~{\bar i}} \left(\hat{a}_s, \frac{Q^2}{\mu_R^2},
\frac{\mu_R^2}{\mu^2}, \epsilon \right) \right]\,.
\end{equation}
In the above expression, all the poles in dimensional regularisation
parameter $\epsilon$ are captured in the $Q^{2}$ independent function
$K^{I}_{i~{\bar i}}$ and the quantities which are finite as
$\epsilon \rightarrow 0$ are encapsulated in $G^{I}_{i~{\bar i}}$. The
solution of the above $KG$ equation can be obtained
as~\cite{Ravindran:2005vv} (see also
\cite{Ahmed:2014cla,Ahmed:2014cha})
\begin{align}
\label{eq:Rap-lnFSoln}
\ln {\cal F}^{I}_{i~{\bar i}}(\hat{a}_s, Q^2, \mu^2, \epsilon) =
\sum_{k=1}^{\infty} {\hat a}_{s}^{k}S_{\epsilon}^{k}
\left(\frac{Q^{2}}{\mu^{2}}\right)^{k
\frac{\epsilon}{2}} \hat {\cal L}_{i~{\bar i}, k}^{I}(\epsilon)
\end{align}
with
\begin{align}
\label{eq:Rap-lnFitoCalLF}
\hat {\cal L}_{i~{\bar i},1}^{I}(\epsilon) =& { \frac{1}{\epsilon^2} }
\Bigg\{-2 A^{I}_{{i~{\bar
i}},1}\Bigg\}
+ { \frac{1}{\epsilon}
}
\Bigg\{G^{I}_{{i~{\bar i}},1}
(\epsilon)\Bigg\}\, ,
\nonumber\\
\hat {\cal L}_{{i~{\bar i}},2}^{I}(\epsilon) =& { \frac{1}{\epsilon^3} }
\Bigg\{\beta_0
A^{I}_{{i~{\bar
i}},1}\Bigg\}
+ {
\frac{1}{\epsilon^2} }
\Bigg\{- {
\frac{1}{2} } A^{I}_{{i~{\bar i}},2}
- \beta_0 G^{I}_{{i~{\bar i}},1}(\epsilon)\Bigg\}
+ { \frac{1}{\epsilon}
} \Bigg\{ {
\frac{1}{2} } G^{I}_{{i~{\bar i}},2}(\epsilon)\Bigg\}\, ,
\nonumber\\
\hat {\cal L}_{{i~{\bar i}},3}^{I}(\epsilon) =& { \frac{1}{\epsilon^4}
} \Bigg\{- {
\frac{8}{9} } \beta_0^2 A^{I}_{{i~{\bar i}},1}\Bigg\}
+ {
\frac{1}{\epsilon^3} }
\Bigg\{ { \frac{2}{9} } \beta_1 A^{I}_{{i~{\bar i}},1}
+ { \frac{8}{9} }
\beta_0 A^{I}_{{i~{\bar i}},2} + { \frac{4}{3} }
\beta_0^2 G^{I}_{{i~{\bar i}},1}(\epsilon)\Bigg\}
\nonumber\\
&
+ { \frac{1}{\epsilon^2} } \Bigg\{- {
\frac{2}{9} } A^{I}_{{i{\bar i}},3}
- { \frac{1}{3} } \beta_1 G^{I}_{{i~{\bar i}},1}(\epsilon)
- { \frac{4}{3} } \beta_0
G^{I}_{{i~{\bar i}},2}(\epsilon)\Bigg\}
+ { \frac{1}{\epsilon}
} \Bigg\{ { \frac{1}{3} }
G^{I}_{i~{\bar i},3}(\epsilon)\Bigg\}\, .
\end{align}
In Appendix~\ref{chpt:App-KGSoln}, the derivation of the above
solution is discussed in great details.
$A^{I}_{i~{\bar i}}$'s are called the cusp anomalous
dimensions. The constants $G^{I}_{{i~{\bar i}},i}$'s are the
coefficients of $a_{s}^{i}$ in the following
expansions:
\begin{align}
\label{eq:Rap-GandAExp}
G^{I}_{i~{\bar i}}\left(\hat{a}_s, \frac{Q^2}{\mu_R^2}, \frac{\mu_R^2}{\mu^2},
\epsilon \right) &= G^{I}_{i~{\bar i}}\left(a_{s}(Q^{2}), 1, \epsilon \right)
+ \int_{\frac{Q^{2}}{\mu_{R}^{2}}}^{1}
\frac{dx}{x} A^{I}_{i~{\bar i}}(a_{s}(x\mu_{R}^{2}))
\nonumber\\
&= \sum_{k=1}^{\infty}a_{s}^{k}(Q^{2})
G^{I}_{{i~{\bar i}},k}(\epsilon) +
\int_{\frac{Q^{2}}{\mu_{R}^{2}}}^{1}
\frac{dx}{x} A^{I}_{i~{\bar i}}(a_{s}(x\mu_{R}^{2}))\,.
\end{align}
However, the solutions of the logarithm of the form factor involves
the unknown functions $G^{I}_{{i~{\bar i}},k}$ which are observed to fulfill
\cite{Ravindran:2004mb, Moch:2005tm} the following decomposition in
terms of collinear ($B^{I}_{i~{\bar i}}$), soft ($f^{I}_{i~{\bar i}}$) and UV
($\gamma^{I}_{i~{\bar i}}$) anomalous dimensions:
\begin{align}
\label{eq:Rap-GIi}
G^{I}_{{i~{\bar i}},k} (\epsilon) = 2 \left(B^{I}_{{i~{\bar i}},k} -
\gamma^{I}_{{i~{\bar i}},k}\right) + f^{I}_{{i~{\bar i}},k} +
C^{I}_{{i~{\bar i}},k} + \sum_{l=1}^{\infty} \epsilon^l g^{I,l}_{{i~{\bar i}},k} \, ,
\end{align}
where, the constants $C^{I}_{{i~{\bar i}},k}$ are given by
\cite{Ravindran:2006cg}
\begin{align}
\label{eq:Rap-Cg}
C^{I}_{{i~{\bar i}},1} &= 0\, ,
\nonumber\\
C^{I}_{{i~{\bar i}},2} &= - 2 \beta_{0} g^{I,1}_{{i~{\bar i}},1}\, ,
\nonumber\\
C^{I}_{{i~{\bar i}},3} &= - 2 \beta_{1} g^{I,1}_{{i~{\bar i}},1} - 2
\beta_{0} \left(g^{I,1}_{{i~{\bar i}},2} + 2
\beta_{0} g^{I,2}_{{i~{\bar i}},1}\right)\,
.
\end{align}
In the above expressions, $X^{I}_{{i~{\bar i}},k}$ with $X=A,B,f$ and
$\gamma^{I}_{{i~{\bar i}}, k}$ are defined through the series expansion in powers
of $a_{s}$:
\begin{align}
\label{eq:Rap-ABfgmExp}
X^{I}_{i~{\bar i}} &\equiv \sum_{k=1}^{\infty} a_{s}^{k}
X^{I}_{{i~{\bar i}},k}\,,
\qquad \text{and} \qquad
\gamma^{I}_{i~{\bar i}} \equiv \sum_{k=1}^{\infty}
a_{s}^{k} \gamma^{I}_{{i~{\bar i}},k}\,\,.
\end{align}
$f_{i~{\bar i}}^{I}$ are introduced for the first time in the
article~\cite{Ravindran:2004mb} where it is shown to fulfill the
maximally non-Abelian property up to two loop level whose validity is
reconfirmed in~\cite{Moch:2005tm} at three loop:
\begin{align}
\label{eq:Rap-MaxNAf}
f^{I}_{q{\bar q}} = \frac{C_F}{C_A} f^{I}_{gg}\,.
\end{align}
This identity implies the soft anomalous dimensions for the production
of a colorless particle in quark annihilation are related to the same
appearing in the gluon fusion through a
simple ratio of quadratic Casimirs of SU(N) gauge group. The same property is also obeyed by the
cusp anomalous dimensions up to three loop level:
\begin{align}
\label{eq:Rap-MaxNAA}
A^{I}_{q{\bar q}} = \frac{C_F}{C_A} A^{I}_{gg}\,.
\end{align}
It is not clear whether this nice property holds true beyond this
order of perturbation theory. Moreover, due
to universality of the quantities denoted by $X$, these are
independent of the operators insertion. These are only dependent on the
initial state partons of any process:
\begin{align}
\label{eq:Rap-IndOfX}
X^{I}_{i~{\bar i}} = X_{i~{\bar i}}\,.
\end{align}
Moreover, these are independent of
the quark flavors.
Here, absence of $I$ represents the independence of the quantities on
the nature of colorless particles.
$f^{I}_{i~{\bar i}}$ can be found in \cite{Ravindran:2004mb, Moch:2005tm},
$A^{H}_{i~{\bar i}}$ in~\cite{Moch:2004pa, Moch:2005tm, Vogt:2004mw, Vogt:2000ci}
and $B^{H}_{i~{\bar i}}$ in \cite{Vogt:2004mw, Moch:2005tm} up to three loop
level. For readers' convenience we list them all up to three loop
level in the
Appendix~\ref{chpt:App-AnoDim}.
Utilising the results of these known quantities and comparing
the above expansions of $G^{I}_{{i~{\bar i}},k}(\epsilon)$ with
the results of the logarithm of the form factors, we extract the
relevant $g_{{i~{\bar i}},k}^{I,l}$ and $\gamma^{I}_{{i~{\bar i}},k}$'s up to three
loop level using Eq.~(\ref{eq:Rap-lnFSoln}),
(\ref{eq:Rap-lnFitoCalLF}) and (\ref{eq:Rap-GIi}).
The relevant one loop terms for $I=H$ and $i~{\bar i}=gg$ are found to be
\begin{align}
\label{eq:Rap-gk1}
g_{gg,1}^{H,1} &= {{C_A}} \zeta_{2}\, ,
\quad\quad
g_{gg,1}^{H,2} = {{C_A}} \Bigg\{1-\frac{7}{3} {\zeta_{3}}\Bigg\}\, ,
\quad\quad
g_{gg,1}^{H,3} = {{C_A}} \Bigg\{\frac{47}{80}
{\zeta_2}^2-\frac{3}{2}\Bigg\}\, ,
\end{align}
the relevant two loop terms are
\begin{align}
\label{eq:Rap-gk2}
g_{gg,2}^{H,1} &= {{C_A}^2} \Bigg\{\frac{67}{3} {\zeta_2}-\frac{44}{3}
{\zeta_3}+\frac{4511}{81}\Bigg\} + {{{C_A} {n_f}}}
\Bigg\{-\frac{10 }{3}{\zeta_2}-\frac{40}{3}
{\zeta_3}-\frac{1724}{81}\Bigg\}
\nonumber\\
&+ {{{C_F} {n_f}}} \Bigg\{16 {\zeta_3}-\frac{67}{3}\Bigg\}\, ,
\nonumber\\
g_{gg,2}^{H,2} &= {{{C_A}^2}} \Bigg\{\frac{671}{120} {\zeta_2}^2+\frac{5}{3} {\zeta_2}
{\zeta_3}-\frac{142}{9} {\zeta_2}+\frac{1139}{27} {\zeta_3}-39
{\zeta_5}-\frac{141677}{972}\Bigg\} + {{{C_A}
{n_f}}} \Bigg\{\frac{259}{60}{\zeta_2}^2
\nonumber\\
&+\frac{16}{9} {\zeta_2}+\frac{604}{27}
{\zeta_3}+\frac{24103}{486}\Bigg\} + {{{C_F} {n_f}}}
\Bigg\{-\frac{16}{3} {\zeta_2}^2-\frac{7}{3}
{\zeta_2}-\frac{92}{3} {\zeta_3}+\frac{2027}{36}\Bigg\}\, ,
\end{align}
and the required three loop term is
\begin{align}
\label{eq:Rap-gk3}
g_{gg,3}^{H,1} &= {{{C_A}^2 {n_f}}}\Bigg\{ -\frac{128}{45} {\zeta_2}^2-\frac{88}{9}
{\zeta_2} {\zeta_3}-\frac{14225}{243} {\zeta_2}-\frac{11372}{81}
{\zeta_3}+\frac{272}{3}
{\zeta_5}-\frac{5035009}{2187}\Bigg\}
\nonumber\\
&+ {{{C_F} {n_f}^2}} \Bigg\{-\frac{368}{45} {\zeta_2}^2-\frac{88}{9}
{\zeta_2}-\frac{1376}{9} {\zeta_3}+\frac{6508}{27}\Bigg\} +
{{{C_A} {C_F} {n_f}}} \Bigg\{\frac{1568}{45}
{\zeta_2}^2+40{\zeta_2} {\zeta_3}
\nonumber\\
&+\frac{503}{18} {\zeta_2}+\frac{20384}{27}
{\zeta_3}+\frac{608}{3}
{\zeta_5}-\frac{473705}{324}\Bigg\} + {{{C_A} {n_f}^2}}
\Bigg\{\frac{232}{45} {\zeta_2}^2+\frac{100}{27}
{\zeta_2}+\frac{6992}{81}
{\zeta_3}
\nonumber\\
&+\frac{912301}{4374}\Bigg\} + {{{C_A^3}}}
\Bigg\{-\frac{12352}{315} {\zeta_2}^3-\frac{5744}{45}
{\zeta_2}^2-\frac{1496}{9} {\zeta_2} {\zeta_3}+\frac{221521}{486}
{\zeta_2}-\frac{104}{3} {\zeta_3}^2
\nonumber\\
&-\frac{57830}{27}
{\zeta_3}+\frac{3080}{3}
{\zeta_5} + \frac{39497339}{8748}\Bigg\} + {{{C_F}^2 {n_f}}}
\Bigg\{296 {\zeta_3}-480
{\zeta_5}+\frac{304}{3}\Bigg\}\,.
\end{align}
Similarly for $I={\rm DY}$ and $i~{\bar i}=q{\bar q}$, we have for one loop
\begin{align}
\label{eq:Rap-gk1-DY}
g_{q{\bar q},1}^{{\rm DY},1} &= {{{C_F}}} \left\{{\zeta_2}-8\right\}\, ,
\quad\quad
g_{q{\bar q},1}^{{\rm DY},2} = {{{C_F}}} \Bigg\{-\frac{3}{4} {\zeta_2}-\frac{7}{3}
{\zeta_3}+8\Bigg\}\, ,
\nonumber\\
g_{q{\bar q},1}^{{\rm DY},3} &= {{{C_F}}} \Bigg\{\frac{47}{80}
{\zeta_2}^2+{\zeta_2}+\frac{7}{4}
{\zeta_3}-8\Bigg\}\, ,
\end{align}
for two loop we require
\begin{align}
\label{eq:Rap-gk2-DY}
g_{q{\bar q},2}^{{\rm DY},1} &= {{{C_F}^2}} \Bigg\{-\frac{88}{5}
{\zeta_2}^2+58 {\zeta_2}-60
{\zeta_3}-\frac{1}{4}\Bigg\} + {{{C_A} {C_F}}}
\Bigg\{\frac{88}{5}
{\zeta_2}^2-\frac{575}{18}
{\zeta_2}+\frac{260}{3}
{\zeta_3}-\frac{70165}{324}\Bigg\}
\nonumber\\
&+ {{{C_F} {n_f}}} \Bigg\{\frac{37}{9} {\zeta_2}-\frac{8}{3}
{\zeta_3}+\frac{5813}{162}\Bigg\}\, ,
\nonumber\\
g_{q{\bar q},2}^{{\rm DY},2} &= {{{C_F}^2}} \Bigg\{\frac{108}{5} {\zeta_2}^2-28 {\zeta_2}
{\zeta_3}-\frac{437}{4} {\zeta_2}+184 {\zeta_3}+12
{\zeta_5}-\frac{109}{16}\Bigg\} + {{{C_A} {C_F}}} \Bigg\{-\frac{653}{24} {\zeta_2}^2
\nonumber\\
&+\frac{89}{3}
{\zeta_2} {\zeta_3}+\frac{7297}{108} {\zeta_2}-\frac{12479}{54}
{\zeta_3}-51 {\zeta_5}+\frac{1547797}{3888}\Bigg\}
\nonumber\\
&+ {{{C_F}
{n_f}}} \Bigg\{\frac{7}{12} {\zeta_2}^2-\frac{425}{54}
{\zeta_2}+\frac{301}{27}
{\zeta_3}
-\frac{129389}{1944}\Bigg\}
\end{align}
and the only required three loop term is
\begin{align}
\label{eq:Rap-gk3-DY}
g_{q{\bar q},3}^{{\rm DY},1} &= {{{C_F}^3}} \Bigg\{\frac{21584}{105} {\zeta_2}^3-534
{\zeta_2}^2+840 {\zeta_2} {\zeta_3}-206 {\zeta_2}+48
{\zeta_3}^2-2130 {\zeta_3}+1992
{\zeta_5}-\frac{1527}{4}\Bigg\}
\nonumber\\
&+ {{{C_A} {C_F}^2}} \Bigg\{-\frac{15448}{105}
{\zeta_2}^3+\frac{2432}{45} {\zeta_2}^2-\frac{3448}{3} {\zeta_2}
{\zeta_3}+\frac{55499}{18} {\zeta_2}+296
{\zeta_3}^2-\frac{23402}{9} {\zeta_3}
\nonumber\\
&-\frac{3020}{3}
{\zeta_5}+\frac{230}{3}\Bigg\} + {{{C_F}^2 {n_f}}}
\Bigg\{-\frac{704}{45} {\zeta_2}^2-\frac{152}{3}
{\zeta_2} {\zeta_3}-\frac{7541}{18} {\zeta_2}+\frac{19700}{27}
{\zeta_3}-\frac{368}{3}
{\zeta_5}
\nonumber\\
&+\frac{73271}{162}\Bigg\} + {{{C_A}^2 {C_F}}} \Bigg\{-\frac{6152}{63}
{\zeta_2}^3+\frac{37271}{90} {\zeta_2}^2+\frac{1786}{9} {\zeta_2}
{\zeta_3}-\frac{1083305}{486} {\zeta_2}-\frac{1136}{3}
{\zeta_3}^2
\nonumber\\
&+\frac{85883}{18} {\zeta_3}+\frac{688}{3}
{\zeta_5}-\frac{48902713}{8748}\Bigg\} + {{{C_F} {n_f}^2}}
\Bigg\{-\frac{40}{9} {\zeta_2}^2-\frac{3466}{81}
{\zeta_2}+\frac{536}{81}
{\zeta_3}
\nonumber\\
&-\frac{258445}{2187}\Bigg\}
+ {{{C_A} {C_F} {n_f}}} \Bigg\{-\frac{1298}{45}
{\zeta_2}^2+\frac{392}{9} {\zeta_2} {\zeta_3}+\frac{155008}{243}
{\zeta_2}-\frac{68660}{81} {\zeta_3}-72
{\zeta_5}
\nonumber\\
&+\frac{3702974}{2187}\Bigg\}
+ {{{C_F} {n_{f,v}} \left(\frac{N^2-4}{N}\right) }}
\Bigg\{-\frac{6}{5} {\zeta_2}^2+30
{\zeta_2}+14 {\zeta_3}-80 {\zeta_5}+12\Bigg\}\,.
\end{align}
$n_{f,v}$ is proportional to the charge weighted sum of the quark
flavors~\cite{Gehrmann:2010ue}. The other constants $\gamma^{I}_{i~{\bar i},k}$,
appearing in the Eq.~(\ref{eq:Rap-GIi}), up to three loop
($k=3$) are obtained as
\begin{align}
\label{eq:Rap-gamma}
&\gamma_{gg,1}^{H} = \beta_{0}\, ,
\quad\quad
\gamma_{gg,2}^{H} = 2 \beta_{1}\, ,
\quad\quad
\gamma_{gg,3}^{H} = 3 \beta_{2}\,
\nonumber\\
{\rm and} \quad\quad
&\gamma_{q{\bar q}}^{\rm DY} = 0\,.
\end{align}
$\beta_i$ are the coefficient of QCD-$\beta$ function, presented in
Eq.~(\ref{eq:bBH-beta}). These will be utilised in the next subsection
to determine the overall operator renormalisation constants.
\subsection{Operator Renormalisation Constant}
\label{ss:Rap-OOR}
The strong coupling constant renormalisation through $Z_{a_{s}}$ may
not be sufficient to make the form factor ${\cal F}^{I}_{i~{\bar i}}$ completely UV
finite, one needs to perform additional renormalisation to remove the
residual UV divergences which is reflected through the presence of
non-zero $\gamma^{I}_{i~{\bar i}}$. Due to non-zero $\gamma^H_{gg}$ in
Eq.~(\ref{eq:Rap-gamma}), overall UV renormalisation is required for
the Higgs boson production in gluon fusion. However, for DY this is
not required. This additional
renormalisation is called the overall operator renormalisation which
is performed through the constant $Z^{I}_{i~{\bar i}}$. This is determined by
solving the underlying RG equation:
\begin{align}
\label{eq:Rap-ZRGE}
\mu_{R}^{2} \frac{d}{d\mu_{R}^{2}} \ln Z^{I}_{i~{\bar i}} \left( {\hat a}_{s},
\mu_{R}^{2}, \mu^{2}, \epsilon \right) = \sum_{k=1}^{\infty}
a_{s}^{k}(\mu_R^2) \gamma^{I}_{i~{\bar i},k}\,.
\end{align}
Using the results of $\gamma^{I}_{i~{\bar i},k}$ from Eq.~(\ref{eq:Rap-gamma}) and
solving the above RG equation following the methodology described in
the Appendix~\ref{chpt:App-SolRGEZas}, we obtain the following overall renormalisation
constant up to three loop level:
\begin{align}
\label{eq:Rap-OOR-Soln}
Z^I_{i~{\bar i}} &= 1+ \sum\limits_{k=1}^{\infty} {\hat a}_s^k
S_{\epsilon}^k \left( \frac{\mu_R^2}{\mu^2}
\right)^{k\frac{\epsilon}{2}} {\hat Z}_{i~{\bar i}}^{I,(k)}
\end{align}
where,
\begin{align}
\label{eq:bBH-OOR-Soln-1}
{\hat Z}_{gg}^{H,(1)} &= \frac{1}{\epsilon} \Bigg\{ 2 \beta_0
\Bigg\}\,,
\nonumber\\
{\hat Z}_{gg}^{H,(2)} &= \frac{1}{\epsilon} \Bigg\{ 2 \beta_1
\Bigg\}\,,
\nonumber\\
{\hat Z}_{gg}^{H,(3)} &= \frac{1}{\epsilon^2} \Bigg\{ - 2 \beta_0
\beta_1 \Bigg\} + \frac{1}{\epsilon} \Bigg\{ 2
\beta_2\Bigg\}\,
\nonumber\\
{\rm and} \qquad {\hat Z}_{q{\bar q}}^{\rm DY} &= 1\,.
\end{align}
\subsection{Mass Factorisation Kernel}
\label{ss:Rap-MFK}
The UV finite form factor contains additional divergences arising from
the soft and collinear regions of the loop momenta. In this section,
we address the issue of collinear divergences and describe a
prescription to remove them. The collinear singularities that arise in
the massless limit of partons are removed by absorbing the divergences
in the bare PDF through
renormalisation of the PDF. This prescription is called the mass
factorisation (MF) and is performed at the factorisation scale
$\mu_F$. In the process of performing this, one needs to introduce
mass factorisation kernels
$\Gamma^I_{ij}(\hat{a}_s, \mu^2, \mu_F^2, z_j, \epsilon)$ which
essentially absorb the collinear singularities. More specifically, MF
removes the collinear singularities arising from the collinear
configuration associated with the initial state partons. The final
state collinear singularities are guaranteed to go away once the phase
space integrals are performed after summing over the contributions from
virtual and real emission diagrams, thanks to
Kinoshita-Lee-Nauenberg theorem.
The kernels satisfy the following RG equation :
\begin{align}
\label{eq:Rap-kernelRGE}
\mu_F^2 \frac{d}{d\mu_F^2} \Gamma^I_{ij}(z_j,\mu_F^2,\epsilon) =
\frac{1}{2} \sum\limits_{k} P^I_{ik} \left(z_j,\mu_F^2\right) \otimes
\Gamma^I_{kj} \left(z_j,\mu_F^2,\epsilon \right)
\end{align}
where, $P^I\left(z_j,\mu_{F}^{2}\right)$ are Altarelli-Parisi splitting
functions (matrix valued). Expanding $P^{I}\left(z_j,\mu_{F}^{2}\right)$ and
$\Gamma^{I}(z_j,\mu_F^2,\epsilon)$ in powers of the strong coupling constant
we get
\begin{align}
\label{eq:Rap-APexpand}
&P^{I}_{ij}(z_j,\mu_{F}^{2}) = \sum_{k=1}^{\infty}
a_{s}^{k}(\mu_{F}^{2})P^{I,(k-1)}_{ij}(z)\,
\intertext{and}
&\Gamma^I_{ij}(z,\mu_F^2,\epsilon) = \delta_{ij} \delta(1-z) + \sum_{k=1}^{\infty}
{\hat a}_{s}^{k} S_{\epsilon}^{k} \left(\frac{\mu_{F}^{2}}{\mu^{2}}\right)^{k
\frac{\epsilon}{2}} {\hat \Gamma}^{I,(k)}_{ij}(z,\epsilon)\, .
\end{align}
The RG equation of $\Gamma^{I}(z,\mu_F^2,\epsilon)$,
Eq.~(\ref{eq:Rap-kernelRGE}), can be solved in dimensional regularisation
in powers of ${\hat a}_{s}$. In the $\overline{MS}$ scheme, the
kernel contains only the poles in $\epsilon$. The solutions~\cite{Ravindran:2005vv} up to the
required order $\Gamma^{I,(3)}(z,\epsilon)$ in terms of $P^{I,(k)}(z)$
are presented in the Appendix~(\ref{eq:App-Gamma-GenSoln}).
The relevant ones up
to three loop, $P^{I,(0)}(z), P^{I,(1)}(z) ~\text{and}~ P^{I,(2)}(z)$ are
computed in the articles~\cite{Moch:2004pa, Vogt:2004mw}. For the SV
cross section only the diagonal parts of the splitting functions
$P^{I,(k)}_{ij}(z)$ and kernels $\Gamma^{I,(k)}_{ij}(z,\epsilon)$
contribute since the diagonal elements of $P^{I,(k)}_{ij}(z)$ contain
$\delta(1-z)$ and ${\cal D}_{0}$ whereas the off-diagonal elements are
regular in the limit $z \rightarrow 1$.
The most remarkable fact is that these quantities are universal, independent of
the operators insertion. Hence, for the processes under consideration,
we make use of the existing process independent results of kernels and splitting
functions:
\begin{align}
\label{eq:Rap-Gamma-P-ProcessInd}
\Gamma^H_{ij} = \Gamma^{\rm DY}_{ij} = \Gamma^I_{ij} = \Gamma_{ij}
\qquad \text{and} \qquad P^H_{ij} = P^{\rm DY}_{ij} = P^I_{ij} = P_{ij}\,.
\end{align}
The absence of $I$ represents the independence of these quantities on
$I$. In the next subsection, we discuss the only remaining ingredient,
namely, the soft-collinear distribution.
\subsection{Soft-Collinear Distribution for Rapidity}
\label{ss:Rap-SCD}
The resulting expression from form factor along with the operator
renormalisation constant and mass factorisation kernel is not
completely finite, it contains some residual divergences which get
cancelled against the contribution arising from soft gluon
emissions. Hence, the finiteness of $\Delta_{Y,i~{\bar i}}^{I, \text{SV}}$ in
Eq.~(\ref{eq:Rap-Delta-Psi}) in the limit
$\epsilon \rightarrow 0$ demands that the soft-collinear distribution,
$\Phi^I_{Y,i~{\bar i}} (\hat{a}_s, q^2, \mu^2, z_1, z_2, \epsilon)$, has pole structure
in $\epsilon$ similar to that of residual divergences. In
article~\cite{Ravindran:2006bu}, it was
shown that $\Phi^{I}_{Y,i~{\bar i}}$ must obey $KG$ type integro-differential
equation, which we call ${\overline{KG}_Y}$ equation, to remove that
residual divergences:
\begin{align}
\label{eq:Rap-KGbarEqn}
q^2 \frac{d}{dq^2} \Phi^I_{Y,i~{\bar i}}\left(\hat{a}_s, q^2, \mu^2,
z_1, z_2,
\epsilon\right) = \frac{1}{2} \left[ \overline K^I_{Y,i~{\bar i}}
\left(\hat{a}_s, \frac{\mu_R^2}{\mu^2}, z_1, z_2,
\epsilon \right) + \overline G^I_{Y,i~{\bar i}} \left(\hat{a}_s,
\frac{q^2}{\mu_R^2}, \frac{\mu_R^2}{\mu^2}, z_1, z_2, \epsilon \right) \right] \, .
\end{align}
${\overline K}^I_{Y,i~{\bar i}}$ and ${\overline G}^I_{Y,i~{\bar i}}$
play similar roles as those of $K^I_{i~{\bar i}}$
and $G^I_{i~{\bar i}}$ in Eq.~(\ref{eq:Rap-KG}), respectively. Also,
$\Phi^I_{Y,i~{\bar i}} (\hat{a}_s, q^2, \mu^2, z, \epsilon)$ being independent of
$\mu_{R}^{2}$ satisfy the RG equation
\begin{align}
\label{eq:Rap-RGEphi}
\mu_{R}^{2}\frac{d}{d\mu_{R}^{2}}\Phi^I_{Y,i~{\bar i}} (\hat{a}_s,
q^2, \mu^2, z_1, z_2, \epsilon) = 0\, .
\end{align}
This RG invariance and the demand of cancellation of all the residual
divergences arising from ${\cal F}^I_{i~{\bar i}}, Z^I_{i~{\bar i}}$
and $\Gamma^I_{i~{\bar i}}$
against $\Phi^{I}_{Y,i~{\bar i}}$ implies the solution of the ${\overline {KG}}_{Y}$
equation as~\cite{Ravindran:2006bu}
\begin{align}
\label{eq:Rap-PhiSoln}
&\Phi^I_{Y,i~{\bar i}}(\hat{a}_s, q^2, \mu^2, z_1, z_2, \epsilon) =
\sum_{k=1}^{\infty} {\hat a}_{s}^{k}S_{\epsilon}^{k} \left(\frac{q^{2}}{\mu^{2}}\right)^{k
\frac{\epsilon}{2}} \hat {\Phi}^{I}_{Y,i~{\bar i},k}(z_1, z_2, \epsilon)
\intertext{with}
&\hat {\Phi}^{I}_{Y,i~{\bar i},k}(z_1, z_2, \epsilon) = \Bigg\{ (k \epsilon)^2
\frac{1}{4 (1-z_1) (1-z_2)}
\left[
(1-z_1) (1-z_2) \right]^{k \frac{\epsilon}{2}}\Bigg\}
\hat{\Phi}^I_{Y,i~{\bar i},k}(\epsilon)\,,
\nonumber\\
&\hat {\Phi}^{I}_{Y,i~{\bar i},k}(\epsilon) = {\hat{\cal
L}}^I_{i{\bar i},k} \left( A^I_{l}
\rightarrow -A^I_{l}, G^I_{l} \rightarrow \overline{\cal
G}^I_{Y,i~{\bar i},l}(\epsilon) \right)\,.
\end{align}
where, ${\hat {\cal L}}_{i~{\bar i},k}^{I}(\epsilon)$ are defined in
Eq.~(\ref{eq:Rap-lnFitoCalLF}). In
Appendix~\ref{chpt:App-Rap-Soft-Col-Dist}, the derivation of this
solution is depicted in great details.
The $z_j$-independent constants ${\overline{\cal G}}^{I}_{Y,i~{\bar i},l}(\epsilon)$ can
be obtained by comparing the poles as well as non-pole terms in $\epsilon$
of ${\hat \Phi}^{I}_{Y,i~{\bar i},k}(\epsilon)$ with those arising from form factor,
overall renormalisation constant and splitting functions. We find
\begin{align}
\label{eq:Rap-calGexpans}
& \overline {\cal G}^{I}_{Y,i~{\bar i},k}(\epsilon)= -f^I_{i~{\bar i},k} +
\overline{C}^I_{Y,i~{\bar
i}, k}
+ \sum\limits_{l=1}^{\infty}
\epsilon^l \overline{\cal
G}^{I,l}_{Y,i~{\bar i},k}
\intertext{where}
&\overline{C}^I_{Y,i~{\bar i},1} = 0\,,
\nonumber\\
&\overline{C}^I_{Y,i~{\bar i},2} = - 2 \beta_0 \overline{\cal G}^{I,1}_{Y,i~{\bar i},1}\,,
\nonumber\\
&\overline{C}^I_{Y,i~{\bar i},3} = - 2 \beta_1 \overline{\cal G}^{I,1}_{Y,i~{\bar i},1} - 2
\beta_0 \left( \overline{\cal G}^{I,1}_{Y,i~{\bar i},2} + 2 \beta_0 \overline{\cal
G}^{I,2}_{Y,i~{\bar i},1} \right)\,.
\end{align}
In-depth understanding about the pole structures including the
single pole~\cite{Ravindran:2004mb} of the form factors, overall operator
renormalisation constants and mass factorisation kernels helps us to
predict all the poles of the soft-collinear
distribution.
However, to determine the finite part of the SV corrections to the
rapidity distribution, we
need the coefficients of $\epsilon^{k}~(k \ge 1)$, $\overline{\cal
G}^{I,k}_{Y,i~{\bar i},l}$. Now, we address the question of
determining those constants. This is achieved with the help of an
identity which has been found:
\begin{align}
\label{eq:Rap-Reln-CS-Rap-1}
\int\limits_0^1 dx_1^0 \int\limits_0^1 dx_2^0 \left( x_1^0 x_2^0
\right)^{{\cal N}-1} \frac{d\sigma^I_{ij}}{dY} = \int\limits_0^1
d\tau \,\tau^{{\cal N}-1} \sigma^I_{ij}\,.
\end{align}
In the large ${\cal N}$ limit i.e. ${\cal N}
\rightarrow \infty$ the above identity relates~\cite{Ravindran:2006bu} the
$\hat{\Phi}^I_{Y,i~{\bar i},k}(\epsilon)$ with the corresponding
$\hat{\Phi}^I_{i~{\bar i},k}(\epsilon)$ appearing in the computation of
SV cross section, Eq.~(\ref{eq:bBH-PhiSoln}):
\begin{align}
\label{eq:Rap-Reln-CS-Rap-2}
\hat{\Phi}^I_{Y,i~{\bar i},k}(\epsilon) =
\frac{\Gamma(1+k\epsilon)}{\Gamma^2(1+k \frac{\epsilon}{2})}
\hat{\Phi}^I_{i~{\bar i},k}(\epsilon)\,.
\end{align}
Hence, the computation of soft-collinear distribution for the
inclusive production cross section is sufficient to determine the
corresponding one for the rapidity distribution. All the properties
satisfied by $\hat{\Phi}^I_{i~{\bar i},k}(\epsilon)$ are obeyed by
$\hat{\Phi}^I_{Y,i~{\bar i},k}(\epsilon)$ too, see Sec.~\ref{ss:bBH-SCD}
for all the details. Utilising the
relation~(\ref{eq:Rap-Reln-CS-Rap-2}), the relevant constants $\overline {\cal G}_{Y,i~{\bar i},k}^{{I},l}$
to determine
$\hat{\Phi}^I_{Y,i~{\bar i},k}(\epsilon)$ up to N$^3$LO level are found to be
\begin{align}
\label{eq:Rap-calGres}
\overline {\cal G}_{Y,q{\bar q},1}^{{\rm DY},1} &= C_{F} \Bigg\{ -
\zeta_{2} \Bigg\} \,,
\nonumber\\
\overline {\cal G}_{Y,q{\bar q},1}^{{\rm DY},2} &= C_{F} \Bigg\{
\frac{1}{3}
\zeta_{3} \Bigg\}
\,,
\nonumber\\
\overline {\cal G}_{Y,q{\bar q},1}^{{\rm DY},3} &= C_{F} \Bigg\{
\frac{1}{80}
\zeta_{2}^{2}
\Bigg\} \,,
\nonumber\\
\overline {\cal G}_{Y,q{\bar q},2}^{{\rm DY},1} &= {C_A} {C_F} \Bigg\{
-4
{\zeta_2}^2-\frac{67}{3}
{\zeta_2} -\frac{44
}{3} {\zeta_3} +
\frac{2428}{81}
\Bigg\}
+ C_{F} n_{f} \Bigg\{ \frac{8}{3}
\zeta_{3} +
\frac{10}{3}
\zeta_{2} -
\frac{328}{81}
\Bigg\} \,,
\nonumber\\
\overline {\cal G}_{Y,q{\bar q},2}^{{\rm DY},2} &= {C_A} {C_F} \Bigg\{
-\frac{319 }{120}
{\zeta_2}^2 -
\frac{71 \
}{3} {\zeta_2} {\zeta_3} + \frac{202
}{9} {\zeta_2} +
\frac{469 \
}{27} {\zeta_3} + 43 {\zeta_5}-\frac{7288}{243} \Bigg\}
\nonumber\\
& + {C_F} {n_f} \Bigg\{ \frac{29 }{60}
{\zeta_2}^2 - \frac{28 \
}{9} {\zeta_2} - \frac{70 }{27}
{\zeta_3} + \frac{976}{243} \Bigg\} \,,
\nonumber\\%
\overline {\cal G}_{Y,q{\bar q},3}^{{\rm DY},1} &= {C_A}^2 {C_F}
\Bigg\{ \frac{17392
}{315} {\zeta_2}^3 +
\frac{1538 \
}{45} {\zeta_2}^2 + \frac{4136 }{9}
{\zeta_2} {\zeta_3}
- \frac{379417 \
}{486} {\zeta_2} + \frac{536 }{3} {\zeta_3}^2
- 936 {\zeta_3}
\nonumber\\
& - \frac{1430 \
}{3} {\zeta_5} + \frac{7135981}{8748} \Bigg\}
+ {C_A} {C_F} {n_f} \Bigg\{ -\frac{1372 \
}{45} {\zeta_2}^2 -\frac{392}{9}
{\zeta_2} {\zeta_3} + \frac{51053 \
}{243} {\zeta_2}
\nonumber\\
& + \frac{12356}{81} {\zeta_3} + \frac{148 \
}{3} {\zeta_5} - \frac{716509}{4374} \Bigg\}
+ {C_F} {n_f}^2 \Bigg\{ \frac{152}{45}
{\zeta_2}^2 - \frac{316 \
}{27} {\zeta_2} - \frac{320 }{81}
{\zeta_3} + \frac{11584}{2187} \Bigg\}
\nonumber\\
& + {C_F}^2 {n_f} \Bigg\{ \frac{152}{15}
{\zeta_2}^2 - 40 {\zeta_2} \
{\zeta_3}+\frac{275 }{6} {\zeta_2} + \frac{1672 \
}{27} {\zeta_3} + \frac{112 }{3}
{\zeta_5} - \frac{42727}{324} \Bigg\} \,
.
\end{align}
The corresponding constants for the Higgs boson production in gluon
fusion can be obtained by employing the identity
\begin{align}
\label{eq:Rap-calG-MaxNA}
{\overline{\cal G}}^{H,k}_{Y, gg, i} = \frac{C_A}{C_F} {\overline{\cal G}}^{{\rm DY},k}_{Y, q{\bar q},i}\,.
\end{align}
The results up to ${\cal O}(a_s^2)$ were present in the
literature~\cite{Ravindran:2006bu} and the term at ${\cal O}(a_s^3)$
is computed for the first time by us in the article~\cite{Ahmed:2014uya}.
Using these, the $\Phi^I_{Y,i~{\bar i}}$ can be obtained which are
presented up to three loops in the Appendix~\ref{app:ss-RapSCD-Res}.
This completes all the ingredients required to compute the SV
correction to the rapidity distributions up to N$^3$LO that are
provided in the next section.
\section{Results of the SV Rapidity Distributions}
\label{sec:Rap-Res}
In this section, we present our findings of the SV rapidity
distributions at N$^3$LO along with the results of the previous
orders. Expanding the SV rapidity distribution ,
Eq.~(\ref{eq:Rap-Delta-Psi}), in powers of $a_s(\mu_F^2)$, we obtain
\begin{align}
\label{eq:Rap-SV-Expand}
&\Delta^{I,{\rm SV}}_{Y,i~{\bar i}} \left( z_1, z_2, q^2, \mu_F^2
\right) = \sum\limits_{k=1}^{\infty} a_s^k(\mu_F^2) \Delta^{I,{\rm
SV}}_{Y,i~{\bar i},k} \left( z_1, z_2, q^2, \mu_F^2 \right)
\end{align}
where,
\begin{align}
\label{eq:Rap-SV-Expand-1}
\Delta^{I,{\rm SV}}_{Y,i~{\bar i},k} &=
\Delta^{I,{\rm SV}}_{Y,i~{\bar i},k}|_{\delta\delta} \delta(1-z_1)
\delta(1-z_2)
+ \sum\limits_{j=0}^{\infty} \Delta^{I,{\rm SV}}_{Y,i~{\bar
i},k}|_{\delta{\cal D}_j}
\delta(1-z_2) {\cal D}_j
\nonumber\\
&+ \sum\limits_{j=0}^{\infty} \Delta^{I,{\rm SV}}_{Y,i~{\bar
i},k}|_{\delta \overline{\cal D}_j}
\delta(1-z_2) \overline{\cal
D}_j
+ \sum\limits_{j \circledS l} \Delta^{I,{\rm SV}}_{Y,i~{\bar
i},k}|_{{\cal D}_j \overline{\cal D}_l}
{\cal D}_j \overline{\cal D}_l\,.
\end{align}
The symbol $j \circledS l$ implies $j, l \geq 0$ and $ j + l \leq (2 k - 2)$.
Terms proportional to ${\cal D}_j$ and/or $\overline {\cal D}_j$ in
Eq.~(\ref{eq:Rap-SV-Expand-1})
were obtained in \cite{Ravindran:2006bu} and the first term is possible to calculate
as the results for the threshold N$^3$LO QCD corrections to the
production cross section are now available for
DY \cite{Ahmed:2014cla} and the Higgs boson \cite{Anastasiou:2014vaa}
productions. By setting $\mu_R^2=\mu_F^2$ we present the results. For
$I=H$ and $i~{\bar i}=gg$, we obtain~\cite{Ahmed:2014uya}
\input{Rap/RapResHiggs}
and for $I={\rm DY}$ and $i~{\bar i}=q{\bar q}$, we get~\cite{Ahmed:2014uya}
\input{Rap/RapResDY}
For sake of completeness, we mention the leading order contribution
which is
\begin{align}
\label{eq:Rap-DeltaSV0}
\Delta^{I}_{Y,i~{\bar i},0} = \delta(1-z_1) \delta(1-z_2)\,.
\end{align}
The above results are presented for the choice $\mu_R=\mu_F$. The
dependence of the SV rapidity distributions on renormalisation scale $\mu_{R}$
can be easily restored by employing the RG evolution of $a_s$ from
$\mu_F$ to $\mu_R$~\cite{Ahmed:2015sna} using Eq.~(\ref{eq:bBH-asf2asr}).
In the next Sec.~\ref{sec:Rap-Numerics}, we discuss the numerical
impact of the N$^3$LO SV correction to the Higgs rapidity distribution
at the LHC.
\section{Numerical Impact of SV Rapidity Distributions}
\label{sec:Rap-Numerics}
In this section, we confine ourselves to the numerical impact of the
SV rapidity distributions of the Higgs boson production through gluon
fusion. We present the relative contributions in percentage
of the pure N$^3$LO terms in Eq.~(\ref{eq:Rap-SV-Expand-1})
with respect to $\Delta^{H,{\rm SV}}_{Y,gg,3}$, for rapidity $Y$ = 0
in Table~\ref{table:Rap-perc-1} and \ref{table:Rap-perc-2}.
\begin{table}[h!]
\centering
\begin{tabular}{ c c c c c c c c c c c }
\hline\hline
&
$~\delta \delta~$ &
$~\delta \overline{{\cal D}}_0~$ & $~\delta \overline{{\cal D}}_1~$ & $~\delta \overline{{\cal D}}_2~$ &
$~\delta \overline{{\cal D}}_3~$ & $~\delta \overline{{\cal D}}_4~$ & $~\delta \overline{{\cal D}}_5~$ &
$~{\cal D}_0 \overline{{\cal D}}_0~$ & $~{\cal D}_0 \overline{{\cal D}}_1~$
\\
\hline
\% &
73.3 & 16.0 & 9.1 & 31.4 & 1.0 & -9.9 & -23.1 &
-13.7 & -10.7
\\
\hline\hline
\end{tabular}
\vspace*{5mm}
\caption{Relative contributions of pure N$^3$LO terms.}
\label{table:Rap-perc-1}
\end{table}
\begin{table}[h!]
\centering
\begin{tabular}{c c c c c c c c }
\hline\hline
& $~{\cal D}_0 \overline{{\cal D}}_2~$ &
$~{\cal D}_0 \overline{{\cal D}}_3~$ & $~{\cal D}_0 \overline{{\cal D}}_4~$ &
$~{\cal D}_1 \overline{{\cal D}}_1~$ & $~{\cal D}_1 \overline{{\cal D}}_2~$ & $~{\cal D}_1 \overline{{\cal D}}_3~$ &
$~{\cal D}_2 \overline{{\cal D}}_2~$ \\
\hline
\% & -0.3 &
3.1 & 7.3 & -0.2 & 3.8 & 8.6 & 4.2 \\
\hline\hline
\end{tabular}
\vspace*{5mm}
\caption{Relative contributions of pure N$^3$LO terms.}
\label{table:Rap-perc-2}
\end{table}
The notation ${\cal D}_i \overline{{\cal D}}_j$ corresponds to the sum of the
contributions coming from
${\cal D}_i \overline{{\cal D}}_j$ and ${\cal D}_j \overline{{\cal D}}_i$.
We have used $\sqrt{s} = 14$ TeV for the LHC, $G_F = 4541.68$ pb, the $Z$ boson
mass $m_Z$ = 91.1876 GeV,
top quark mass $m_t$ = 173.4 GeV
and the Higgs boson mass $m_H$ = 125.5 GeV throughout.
For the Higgs boson production, we use the effective theory where top
quark is integrated out in the large $m_t$ limit.
The strong coupling constant $\alpha_s (\mu_R^2)$ is evolved
using the 4-loop RG equations with
$\alpha_s^{\text{N$^3$LO}} (m_Z ) = 0.117$ and for parton density sets we use
MSTW 2008NNLO \cite{Martin:2009iq}, as N$^3$LO evolution kernels are not yet available.
In \cite{Forte:2013mda}, Forte \textit{et al.} pointed out that the
Higgs boson cross sections will remain unaffected with this
shortcoming. However, for the DY process, it is not clear whether the
same will be true.
We find that the contribution from the $\delta(1-z_1) \delta (1-z_2)$
part is the largest.
Impact of the threshold NNLO and N$^3$LO contributions to the Higgs boson rapidity distribution
at the LHC is presented in Fig.~\ref{fig:Rap-rapidity_distr}.
\begin{figure}[htb]
\centering
\includegraphics[width=1\textwidth]{Rap/Rap-rapidity_distr.pdf}
\caption{
\label{fig:Rap-rapidity_distr}
Rapidity distribution of Higgs boson}
\vspace{-0.5cm}
\end{figure}
The dependence on the renormalization and factorization scales can by studied by
varying them in the range $m_H/2 <\mu_R,\mu_F<2 m_H$. We find
that the inclusion of the
threshold correction at N$^3$LO further reduces their dependence.
For the inclusive Higgs boson production, we find that
about 50\% of exact NNLO contribution comes from
threshold NLO and NNLO terms. It increases to 80\% if we use exact
NLO and threshold NNLO terms.
Hence, it is expected that the rapidity distribution of the Higgs boson
will receive a significant contribution from the threshold region compared to
inclusive rate due to the soft emission over the entire range of $Y$.
Our numerical study with threshold enhanced NNLO rapidity distribution
confirms our expectation.
Comparing our threshold NNLO results
against exact NNLO distribution using the FEHiP \cite{Anastasiou:2005qj} code ,
we find that about $90\%$ of
exact NNLO distribution comes from the threshold region as can be seen
from Table~\ref{table:Rap-nnlo-1} and \ref{table:Rap-nnlo-2}, in accordance with~\cite{Becher:2007ty},
where it was shown that for low $\tau~(m_H^2/s \approx 10^{-5})$
values the threshold terms are dominant, thanks to the inherent
property of the matrix element, which receives the largest radiative
corrections from the phase-space points corresponding to Born
kinematics.
\begin{table}[h!]
\centering
\begin{tabular}{ l c c c c c }
\hline\hline
$Y$ & ~~0.0~~ & ~~0.4~~ & ~~0.8~~ & ~~1.2~~ & ~~1.6~~ \\
\hline
NNLO & 11.21 & 10.96 & 10.70 & 9.13 & 7.80 \\
NNLO$_{\rm SV}$ & 9.81 & 9.61 & 8.99 & 8.00 & 6.71 \\
NNLO$_{\rm SV}$(A) & 10.67 & 10.46 & 9.84 & 8.82 & 7.48 \\
N$^3$LO$_{\rm SV}$ & 11.62 & 11.36 & 11.07 & 9.44 & 8.04 \\
N$^3$LO$_{\rm SV}$(A) & 11.88 & 11.62 & 11.33 & 9.70 & 8.30 \\
$K$3 & 2.31 & 2.29 & 2.36 & 2.21 & 2.17 \\
\hline\hline
\end{tabular}
\vspace*{5mm}
\caption{Contributions of exact NNLO, NNLO$_{\rm SV}$, N$^3$LO$_{\rm SV}$, and $K3$.}
\label{table:Rap-nnlo-1}
\end{table}
\begin{table}[h!]
\centering
\begin{tabular}{ l c c c c c }
\hline\hline
$Y$ & ~~2.0~~ & ~~2.4~~ & ~~2.8~~ & ~~3.2~~ & ~~3.6~~ \\
\hline
NNLO & 6.10 & 4.23 & 2.66 & 1.40 & 0.54 \\
NNLO$_{\rm SV}$ & 5.21 & 3.66 & 2.25 & 1.14 & 0.42 \\
NNLO$_{\rm SV}$(A) & 5.90 & 4.24 & 2.69 & 1.42 & 0.56 \\
N$^3$LO$_{\rm SV}$ & 6.27 & 4.33 & 2.70 & 1.40 & 0.53 \\
N$^3$LO$_{\rm SV}$(A) & 6.51 & 4.54 & 2.88 & 1.53 & 0.60 \\
$K$3 & 2.07 & 1.89 & 1.70 & 1.63 & 1.51 \\
\hline\hline
\end{tabular}
\vspace*{5mm}
\caption{Contributions of exact NNLO, NNLO$_{\rm SV}$, N$^3$LO$_{\rm SV}$, and $K3$.}
\label{table:Rap-nnlo-2}
\end{table}
Here we have used the exact results up to the NLO level.
Because of an inherent ambiguity in the definition of
the partonic cross section at threshold one can multiply a factor $z
g(z)$, where $z=\tau/x_1 x_2$ and $\lim_{z \rightarrow 1} g(z) = 1$,
with the partonic flux and divide the same in the partonic cross
section for an inclusive rate. In~\cite{Catani:2003zt,Kramer:1996iq}
this was exploited to take into account the subleading collinear logs
also, thereby making the threshold approximation a better
one. Recently, Anastasiou \textit{et al.} used this
in~\cite{Anastasiou:2014vaa} to modify the partonic flux keeping the
partonic cross section unaltered to improve the threshold effects.
Following \cite{Anastasiou:2014vaa,Herzog:2014wja}, we
introduce
$G (z_1,z_2)$ such that $\lim_{z_1,z_2 \rightarrow 1} G = 1$ in Eq.~(\ref{eq:Rap-Ren-WI}):
\begin{align}
\label{eq:Rap-Ren-WI}
W^I \left( x_1^0, x_2^0, q^2, \mu_R^2 \right)
&= \sum\limits_{i,j=q,{\bar q}, g} \int\limits_{x_1^0}^1 \frac{dz_1}{z_1}
\int\limits_{x_2^0}^1 \frac{dz_2}{z_2} {\cal H}^I_{ij} \left(
\frac{x_1^0}{z_1} \frac{x_2^0}{z_2}, \mu_F^2 \right) G(z_1, z_2)
\nonumber\\
& \times \lim\limits_{z_1,z_2 \to 1} \left[ \frac{\Delta^I_{Y,ij} \left(
z_1, z_2, q^2, \mu_R^2, \mu_F^2 \right)}{G(z_1,z_2)} \right] \,.
\end{align}
We also find that with the choice $G(z_1,z_2)=z_1^2 z_2^2$, the threshold NNLO results are
remarkably close to the exact ones for the entire range of $Y$ [see
Table~\ref{table:Rap-nnlo-1} and \ref{table:Rap-nnlo-2}, denoted by $(A)$].
This clearly demonstrates the dominance
of threshold contributions to rapidity distribution of the Higgs boson production at the NNLO level.
Assuming that the trend will not change drastically beyond NNLO, we present numerical values for
N$^3$LO distributions for $G(z_1,z_2)=1, z_1^2 z_2^2$, respectively, as N$^3$LO$_{\rm SV}$ and N$^3$LO$_{\rm SV}(A)$
in Table~\ref{table:Rap-nnlo-1} and \ref{table:Rap-nnlo-1}.
The threshold N$^3$LO terms give $6 \% (Y = 0)$ to $12 \% (Y = 3.6)$ additional
correction over the NNLO contribution to the inclusive DY production.
Finally, in Table~\ref{table:Rap-nnlo-1} and \ref{table:Rap-nnlo-2}, we have presented $K3 =$
N$^3$LO$_{\rm SV}$/LO as a function of $Y$ in order to demonstrate
the sensitivity of higher order effects to the rapidity $Y$.
\section{Summary}
\label{sec:Rap-Summary}
To summarize, we present the full threshold enhanced N$^3$LO QCD corrections to
rapidity distributions of the dilepton pair in the DY process and of
the Higgs boson in gluon fusion at the LHC. These are the most
accurate results for these observables available in the literature.
We show that the infrared structure of QCD amplitudes,
in particular, their factorization properties,
along with Sudakov resummation of soft gluons and renormalization group invariance provide an elegant
framework to compute these threshold corrections systematically for rapidity distributions
order by order in QCD perturbation theory.
The recent N$^3$LO results for inclusive DY and Higgs boson production cross sections
at the threshold provide crucial ingredients to obtain $\delta(1-z_1) \delta(1-z_2)$
contribution of their rapidity distributions for the first time.
We find that this contribution numerically dominates over the rest of
the terms in $\Delta^{H,{\rm SV}}_{Y,gg,3}$ at
the LHC. Inclusion of N$^3$LO contributions reduces the scale dependence further.
We also demonstrate the dominance of the threshold contribution to
rapidity distributions by comparing it against the exact NNLO for two
different choices of $G(z_1,z_2)$. Finally, we find that threshold
N$^3$LO rapidity distribution with $G(z_1,z_2)=1,z_1^2 z_2^2$ shows a
moderate effect over NNLO distribution.
\section{Outlook-Beyond N$^3$LO}
\label{sec:Rap-Outlook}
The results presented above is the most accurate one existing in the
literature. However, in coming future, we may need to go beyond this
threshold N$^3$LO in hope of making more precise theoretical predictions. The
immediate step would be to compute the complete N$^3$LO QCD
corrections to the differential rapidity distributions. No doubt, this
is an extremely challenging goal! Presently, though we are incapable of computing
this result, we can obtain the general form of the threshold N$^4$LO
QCD corrections to the rapidity distributions! However, due to
unavailability of the quantities, namely, form factors, anomalous
dimensions, soft-collinear distributions at 4-loop level, we are
unable to estimate the contributions arising from this. Nevertheless, the general
form of this contribution is available to the authors which can be utilised to make the predictions once
the missing ingredients become available in future.
\chapter{\label{chap:Intro}Introduction}
\begingroup
\hypersetup{linkcolor=blue}
\minitoc
\endgroup
The Standard Model (SM) of Particle Physics is one of most remarkably successful
fundamental theories which encapsulates the governing principles of
elementary constituents of matter and their interactions. Its
development throughout the latter half of the 20th century resulting
from an unprecedented collaborative effort of the brightest minds
around the world is undoubtedly one of the greatest achievements in
human history. Over the duration of many decades around 1970s, the
theoretical predictions of the SM were verified one after
another with a spectacular accuracy and it got the ultimate credence
through the announcement,
made on a fine morning of 4th July 2012 at CERN in Geneva:
\begin{quote}
\textit{``If we combine $ZZ$ and $\gamma\gamma$, this is what we get: they line
up extremely well in a region of 125 GeV with the combine significance
of 5 standard deviation!''}
\end{quote}
The SM relies on the mathematical framework of quantum field theory
(QFT), in which a Lagrangian controls the dynamics
and kinematics of the theory. Each kind of particle is described in
terms of a dynamical field that pervades space-time. The construction
of the SM proceeds through the modern methodology of constructing a QFT,
it happens through postulating a set of underlying symmetries of
the system and writing down the most general renormalisable Lagrangian
from its field content.
The underlying symmetries of the QFT can be largely categorized into
global and local ones. The global Poincar\'e symmetry is postulated for
all the relativistic QFT. It consists of the familiar
translational symmetry, rotational symmetry and the inertial reference
frame invariance central to the special theory of relativity. Being
global, its operations must be simultaneously applied to all points of space-time
On the
other hand, the local gauge symmetry is an internal symmetry that
plays the most crucial role in determining the predictions of the
underlying QFT. These are the symmetries that act independently at each point in space-time.
The SM relies on the local
SU(3)$\times$SU(2)$_{\rm L}\times$U(1)$_{\rm Y}$ gauge symmetry. Each
gauge symmetry manifestly
gives rise to a fundamental interaction: the
electromagnetic interactions are characterized by an U(1), the
weak interactions by an SU(2) and the strong interactions by an SU(3)
symmetry.
In its current formulation of the SM, it includes three different families
of elementary particles. The first ones are called fermions arising
from the quantisation of the fermionic fields. These constitute the
matter content of the theory. The quanta of the bosonic fields, which
form the second family, are the force carriers i.e. the mediators of
the strong, weak, and electromagnetic fundamental interactions. In
addition to the these, there is a third boson, the Higgs boson
resulting from the quantum excitation of the Higgs field. This is the only
known scalar particle that was postulated long ago and
observed very recently at the Large Hadron Collider
(LHC)~\cite{Aad:2012tfa, Chatrchyan:2012ufa}. The presence of this
field, now believed to be confirmed, explains the
mechanisms of acquiring mass of some of the fundamental
particles when, based on the underlying gauge symmetries controlling
their interactions, they should be massless. This mechanism, which is
believed to be one of the most revolutionary ideas of the last
century, is known as Brout-Englert-Higgs-Kibble (BEHK) mechanism.
Two of the four known fundamental forces, electromagnetism and weak forces
which appear very different at low energies,
are actually unified to so called electro-weak force in high
energy. The structure of this unified picture is accomplished under
the gauge group SU(2)$_{\rm L} \times$ U(1)$_{\rm Y}$. The
corresponding gauge bosons are the three $W$ bosons of weak isospin from SU(2) and the
$B$ boson of weak hypercharge from U(1), all of which are
massless. Upon spontaneous symmetry breaking from SU(2)$_{\rm L}
\times$ U(1)$_{\rm Y}$ to U(1)$_{\rm EM}$, caused by the BEHK
mechanism, the three mediators of the electro-weak force, the
$W^{\pm},Z$ bosons acquire mass, leaving the mediator of the
electromagnetic force, the photon, as massless. Finally, the theory of
strong interactions, Quantum Chromo-Dynamics (QCD) is governed by the
unbroken SU(3) gauge group, whose force carriers, the gluons remain
massless.
Although the SM is believed to be theoretically
self-consistent with a spectacular accuracy and has demonstrated huge and continued successes
in providing experimental predictions, it indeed does leave some phenomena
unexplained and it falls short of being a complete theory of
fundamental interactions. It fails to incorporate the full theory of
gravitation as described by general relativity, or account for the
accelerating expansion of the universe (as possibly described by dark
energy). The model does not contain any viable dark matter particle
that possesses all of the required properties deduced from
observational cosmology. It also does not incorporate neutrino
oscillations (and their non-zero masses).
Currently, the high energy physics community is standing on the verge of a
crucial era where the new physics may show up as tiny deviations from
the prediction of the SM! To exploit this possibility it is absolutely
necessary to make the theoretical predictions, along with the
revolutionary experimental progress, to a very high accuracy within the SM and beyond.
The relevance of this thesis arises exactly in this context.
The most crucial quantity in the process of accomplishing the job of
making any prediction based on QFT is undoubtedly the scattering
amplitude. This is the fundamental building block of any observable in
QFT. In the upcoming section, we will elaborate on the idea of
scattering amplitude which will be followed by a brief description of
QCD. We will close the chapter of introduction by introducing the
concept of computing the observables under certain approximation,
known as soft-virtual approximation.
\section{Scattering Amplitudes}
\label{sec:Intro-ScattAmp}
The fundamental quantity of any QFT which encodes all the underlying
symetries of the theory is called the action. This is constructed out
of Lagrangian density, which is a functional of the fields present in
the theory, and integrating over all space time points:
\begin{align}
\label{eq:Intro-action}
S &= \int d^4x \,{\cal L}\left[\phi_i(x)\right]\,.
\end{align}
By construction the QFT is a probabilistic theory and all the
observables calculated based on this theory always carry a
probabilistic interpretation. For example, an important observable is
the total cross section which measures the total probability of any
event to happen in colliders. The computation of the cross section, and in fact, almost
all the observables in QFT requires the evaluation of scattering matrix ($S$-matrix)
elements which describe the evolution of the system from asymptotic initial to final states
due to presence of the interaction. The $S$-matrix elements are defined as
\begin{align}
\langle f|S| i \rangle = \delta_{fi} + i (2\pi)^{4} \delta^{(4)}\left(p_{f}-p_{i}\right) {\cal M}_{i \rightarrow f}
\end{align}
where, the $\delta_{fi}$ represents the unscattered forward scattering states, while the other part
${\cal M}_{i \rightarrow f}$ encapsulates the ``actual'' interaction (For simplicity, we
will call ${\cal M}_{i \rightarrow f}$ as scattering matrix element.). So, the calculation of all those
observables essentially boils down to the computation of the second quantity. However, the exact computation
of this quantity is
incredibly difficult in any general
field theory. The only viable methodology is provided under the
framework of perturbation theory where the matrix elements as well as
the observables are expanded in powers of coupling constants, $c$, present
in the theory:
\begin{align}
\label{eq:Intro-expand-coupling}
{\cal M}_{i \rightarrow f} = \sum\limits_{n=0}^{\infty} c^n {\cal M}_{i \rightarrow f}^{(n)}\,.
\end{align}
If the coupling constant is small enough, the evaluation of only the first term of
the perturbative expansion often turns out to be a very good
approximation that provides a
reliable prediction to any observable. However, in QFT, it is a
well-known fact that the coupling constants are truly not `constants', their
strength depends on the energy scale at which the interaction takes
place. This evolution of the coupling constant may make it
comparatively large at some energy scale. In case of Quantum
Electro-Dynamics (QED), quantum field theory of electromagnetism, the
magnitude of the coupling constant, $c=\alpha_{\rm EM}$, increases with
the increase of momentum transfer:
\begin{align}
\label{eq:Intro-QED-flow}
\alpha_{\rm EM}(Q^2 \approx 0) \approx \frac{1}{137}\,, \qquad {\rm
and} \qquad \alpha_{\rm EM}(Q^2 \approx m_W^2) \approx \frac{1}{128}
\end{align}
where, $m_W \approx 80$ GeV is the invariant mass of the $W$
boson. The smallness of $\alpha_{\rm EM}$ at all typical energy scales
which can be probed in all collider experiments guarantees very fast
convergence of the perturbation series to what we expect to be real
non-perturbative result. However, this picture no longer holds true in
case of QCD where the coupling constant, $c=\alpha_s$, may become quite large at
certain energy scales:
\begin{align}
\label{eq:Intro-QCD-flow}
\alpha_{s} (m_p^2) \approx 0.55, \qquad {\rm and} \qquad \alpha_{s} (m_Z^2) \approx 0.1
\end{align}
where, $m_p \approx 938 MeV$ and $m_Z \approx 90 GeV$ are the masses
of the proton and $Z$ boson. Clearly the magnitude $0.55$ is far from
being small! Hence, computation of only the leading term in
perturbative series often turns out to be a very crude approximation
which fails to deliver a reliable prediction. We must take into
account the contributions arising beyond leading term.
In perturbation theory, the most acceptable and well known
prescription to compute the terms in a perturbative series is
provided by Feynman diagrams. Every term of a series is represented
through a set of Feynman diagrams and each diagram corresponds to a
mathematical expression. Hence, evaluation of a term in any perturbative
series boils down to the computation of all the corresponding Feynman
diagrams. Given an action of a QFT, one first requires to derive a set
of rules, called Feynman rules, which essentially establish the correspondence
between the Feynman diagrams and mathematical expressions. With the
rules in hand, we just need to draw all the possible Feynman diagrams
contributing to the order of our interest and eventually evaluate those using the
rules. Needless to say, as the perturbative order increases, the
number of Feynman diagrams to be drawn grows so rapidly that after certain order
it becomes almost prohibitively large to draw.
In this thesis, we will concentrate only on the aspects of
perturbative QCD. We will start our discussion of QCD by introducing
the basic aspects of this QFT which will be followed by the writing
down the quantum action and corresponding Feynman rules. Then we will
discuss how to compute amplitudes beyond leading order in QCD and
eventually get reliable numerical predictions at hadron colliders for
any process.
\section{Quantum Chromo-Dynamics}
\label{sec:QCD}
Quantum Chromo-dynamics, familiarly called QCD, is the modern theory
of strong interactions, a fundamental force describing the
interactions between quarks and gluons which make up hadrons such as
the protons, neutrons and pions. QCD is a type of QFT called non-Abelian
gauge theory that has underlying SU(3) gauge symmetry. It appears as an
expanded version of QED. Whereas in QED there is just
one kind of charge, namely electric charge, QCD has three different kinds of
charge, labeled by ``colour''. Avoiding chauvinism, those are
chosen as red, green, and blue. But, of course, the colour
charges of QCD have nothing to do with optical colours.
Rather, they have properties analogous to electric charges in QED.
In particular, the colour charges are conserved in all physical
processes. There are also photon-like massless gauge bosons,
called gluons, that act as the mediators of the strong interactions between
spin-1/2 quarks. Unlike the photons, which mediate the electromagnetic
interaction but lacks an electric charge, the gluons themselves carry
color charges. Gluons, as a consequence, participate in the strong
interactions in addition to mediating it, making QCD substantially
harder to analyse than QED.
In sharp contrast to other gauge theories, QCD enjoys two salient
features: confinement and asymptotic freedom. The force among
quarks/gluons fields does not diminish as they are separated from each
others. With the increase in mutual distance between them, the
mediating gluon fields gather enough energy to create a
pair of quarks/gluons which forbids them to be found as free
particles; they are thus forever bound into hadrons such as the
protons, neutrons, pions or kaons. Although literature lacks the
satisfactory theoretical explanation, confinement is believed to be
true as it explains the consistent failure of finding free quarks or
gluons. The other interesting property, the asymptotic
freedom~\cite{Gross:1973id, Gross:1973ju, Politzer:1973fx, Gross:1974cs,
Politzer:1974fr}, causes
bonds between quarks/gluons become asymptotically weaker as energy
increases or distance decreases which allows us to perform the
calculation in QCD using the technique of perturbation theory. The Nobel
prize was awarded for this remarkable discovery of last century.
In perturbative QCD, the basic building blocks of performing any
calculation are the Feynman rules, which will be discussed in next
subsection.
\subsection{The QCD Lagrangian and Feynman Rules}
\label{ss:QCD-Feynman-Rules}
The first step in performing perturbative calculations in a
QFT is to work out the Feynman rules. The SU(N) gauge invariant classical
Lagrangian density encapsulating the interaction between fermions and non-Abelian gauge
fields is
\begin{align}
\label{eq:Intro-Lag-QCD-1}
{\cal L}_{classical} = - \frac{1}{4} F^a_{\mu\nu} F^{a,\mu\nu} +
\sum\limits_{f=1}^{n_f} {\overline \psi}_{\alpha,i}^{(f)} \left( {i\mkern1mu}
{\slashed D}_{\alpha\beta,ij} -
m_f \delta_{\alpha\beta} \delta_{ij} \right) \psi_{\beta,j}^{(f)}\,.
\end{align}
In the above expression,
\begin{align}
\label{eq:Intro-Lag-QCD-2}
&F^a_{\mu\nu} = \partial_{\mu}A_{\nu}^a - \partial_{\nu}A_{\mu}^a +
g_s f^{abc} A_{\mu}^b A_{\nu}^c\,,
\nonumber\\
&{\slashed D}_{\alpha\beta,ij} \equiv \gamma^{\mu}_{\alpha\beta} D_{\mu,ij} = \gamma^{\mu}
\left( \delta_{ij} \partial_{\mu} - {i\mkern1mu} g_s T^a_{ij} A^a_{\mu} \right)
\end{align}
where, $A^a_{\mu}$ and $\psi_{\alpha,i}^{(f)}$ are the guage and
fermionic quark fields, respectively. The indices represent the following things:
\begin{align}
\label{eq:Intro-indices}
&a, b, \cdots: \quad \text{color indices in the adjoint
representation} \Rightarrow [1, N^2-1]\,,
\nonumber\\
&i,j, \cdots: \quad \text{~color indices in the fundamental
representation} \Rightarrow [1, N]\,,
\nonumber\\
&\alpha,\beta, \cdots: \quad \text{Dirac spinor indices} \Rightarrow
[1, d]\,,
\nonumber\\
&\mu, \nu, \cdots: \quad \text{\,Lorentz indices} \Rightarrow [1,d]\,.
\end{align}
Numbers within the `[]' signifies the range of the corresponding
indices. $d$ is the space-time dimensions. $f$ is the quark flavour index which runs from 1 to $n_f$. $m_f$ and
$g_s$ are the mass of the quark corresponding to $\psi^{(f)}$ and
strong coupling constant,
respectively. $f^{abc}$ are the structure constants of SU(N)
group. These are related to the Gellmann matrices $T^a$, generators of the
fundamental representations of SU(N), through
\begin{align}
\label{eq:Intro-Gellman-fabc}
\left[T^a,T^b\right]=i f^{abc} T^{c}\,.
\end{align}
The $T^a$ are traceless, Hermitian matrices and these are
normalised with
\begin{align}
\label{eq:Intro-Gellman-Norm}
Tr\left(T^{a}T^{b}\right) = T_{F} \delta^{ab}
\end{align}
where, $T_F=\frac{1}{2}$.
They satisfy the following completeness relation
\begin{align}
\label{eq:Intro-Gellman-Complete}
\sum\limits_{a} T^a_{ij} T^a_{kl} = \frac{1}{2} \left( \delta_{il} \delta_{kj} -
\frac{1}{N} \delta_{ij} \delta_{kl} \right)\,.
\end{align}
In addition to the above three parent identities expressed through the
Eq.~(\ref{eq:Intro-Gellman-fabc}, \ref{eq:Intro-Gellman-Norm},
\ref{eq:Intro-Gellman-Complete}), we can have some auxiliary ones
which are often useful in simplifying colour algebra:
\begin{align}
\label{eq:Intro-aux-identities}
&\sum\limits_a (T^a T^a)_{ij} = C_F \delta_{ij}\,,
\nonumber\\
&f^{acd}f^{bcd} = C_A \delta^{ab}\,.
\end{align}
The $C_{A}=N$ and $C_F=\frac{N^2-1}{2N}$ are the quadratic Casimirs of
the SU(N) group in the adjoint and fundamental representations,
respectively. For QCD, the SU(N) group index, $N=3$ and the flavor number
$n_f=6$.
The quantisation of the non-Abelian gauge theory or the Yang-Mills (YM) theory
faces an immediate problem, namely, the propagator of gauge fields
cannot be obtained unambiguously. This is directly related to the
presence of gauge degrees of freedom inherent into the ${\cal
L}_{classical}$. We need to perform the gauge fixing in order to get
rid of this problem. The gauge fixing in a covariant way, when done
through the path integral formalism, generates new particles called
Faddeev-Popov (FP) ghosts having spin-0 but obeying fermionic
statistics. The absolute necessity of introducing
the ghosts in the process of quantising the YM theory is a horrible
consequence of the Lagrangian formulation of QFT. There is no
observable consequence of these particles, we just need them in order
to describe an interacting theory of a massless spin-1 particle using
a local manifestly Lorentz invariant Lagrangian. These particles never
appear as physical external states
but must be included in internal lines to cancel the unphysical
degrees of freedom of the gauge fields. Some alternative formulations of
non-Abelian gauge theory (such as the lattice) also do not
require ghosts. Perturbative gauge theories in certain non-covariant
gauges, such as light-cone or axial gauges, are also ghost free. However, to
maintain manifest Lorentz invariance in a perturbative gauge theory,
it seems ghosts are unavoidable and in this thesis we will be remained
within the regime of covariant gauge and consequently will include
ghost fields consistently into our computations.
Upon applying this technique to quantise the YM theory, we end up with
getting the following full quantum Lagrangian density:
\begin{align}
\label{eq:Intro-Lag-Quant}
{\cal L}_{YM} &= {\cal L}_{classical} + {\cal L}_{gauge-fix} + {\cal L}_{ghost}
\end{align}
where, the second and third terms on the right hand side correspond to
the gauge fixing and FP contributions, respectively. These are
obtained as
\begin{align}
\label{eq:Intro-Lag-GF-FP}
&{\cal L}_{gauge-fix} = - \frac{1}{2 \xi}
\left( \partial^{\mu}A^a_{\mu} \right)^2\,,
\nonumber\\
&{\cal L}_{ghost} = \left( \partial^{\mu} \chi^{a*} \right) D_{\mu,ab} \,\chi^b
\end{align}
with
\begin{align}
\label{eq:Intro-Lag-Dmuab}
D_{\mu,ab} \equiv \delta_{ab} \partial_{\mu} - g_s f_{abc} A^c_{\mu}\,.
\end{align}
The gauge parameter $\xi$ is arbitrary and it is introduced in order
to specify the gauge in a covariant way. This prescription of fixing
gauge in a covariant way is known as $R_{\xi}$ gauge. A typical choice
which is often used is $\xi=1$, known as Feynman gauge. We will be
working in this Feynman gauge throughout this thesis, unless otherwise
mentioned specifically. However, we emphasize that the physical
results are independent of the choice of the gauges. The field
$\chi^a$ and $\chi^{a*}$ are ghost and anti-ghost fields,
respectively.
All the Feynman rules can be read off from the quantized Lagrangian
${\cal L}_{YM}$ in Eq.~\ref{eq:Intro-Lag-Quant}. We will denote the
quarks through straight lines, gluons through curly and ghosts through
dotted lines. We provide the rules in $R_{\xi}$ gauge.
\begin{itemize}
\item The propagators for quarks, gluons and ghosts are obtained as
respectively:
\begin{figure}[H]
\qquad
\begin{minipage}{2in}
\begin{tikzpicture}[line width=0.6 pt, scale=0.7]
\draw[fermion] (-2,0) -- (2,0);
\node at (-2.2,0.8) {$j,\beta$};
\node at (2.2,0.8) {$i,\alpha$};
\draw[->] (-1.5,-0.5) -- (-2,-0.5);
\node at (-1.7,-1) {$p_2$};
\draw[->] (1.5,-0.5) -- (2,-0.5);
\node at (1.7,-1) {$p_1$};
\end{tikzpicture}
\end{minipage}
\hspace{-0.5cm}
\begin{minipage}{3in}
\begin{align*}
{i\mkern1mu} \left( 2\pi \right)^4 \delta^{(4)}\left(p_1+p_2\right) \delta_{ij} \left(
\frac{1}{{\slashed p_1}-m_f+{i\mkern1mu} \varepsilon} \right)_{\alpha\beta}
\end{align*}
\end{minipage}
\vspace{1cm}
\qquad
\begin{minipage}{2in}
\begin{tikzpicture}[line width=0.6 pt, scale=0.7]
\draw[gluon] (-2,0) -- (2,0);
\node at (-2.2,0.8) {$b,\nu$};
\node at (2.2,0.8) {$a,\mu$};
\draw[->] (-1.5,-0.5) -- (-2,-0.5);
\node at (-1.7,-1) {$p_2$};
\draw[->] (1.5,-0.5) -- (2,-0.5);
\node at (1.7,-1) {$p_1$};
\end{tikzpicture}
\end{minipage}
\begin{minipage}{3in}
\begin{align*}
{i\mkern1mu} \left( 2\pi \right)^4 \delta^{(4)}\left(p_1+p_2\right) \delta_{ab}
\frac{1}{p_1^2} \left[ -g_{\mu\nu} + (1-\xi) \frac{p_{1\mu} p_{1\nu}}{p_1^{2}} \right]
\end{align*}
\end{minipage}
\vspace{1cm}
\qquad
\begin{minipage}{2in}
\begin{tikzpicture}[line width=0.6 pt, scale=0.7]
\draw[scalar] (-2,0) -- (2,0);
\node at (-2.2,0.8) {$b$};
\node at (2.2,0.8) {$a$};
\draw[->] (-1.5,-0.5) -- (-2,-0.5);
\node at (-1.7,-1) {$p_2$};
\draw[->] (1.5,-0.5) -- (2,-0.5);
\node at (1.7,-1) {$p_1$};
\end{tikzpicture}
\end{minipage}
\hspace{-1.8cm}
\begin{minipage}{3in}
\begin{align*}
{i\mkern1mu} \left( 2\pi \right)^4 \delta^{(4)}\left(p_1+p_2\right) \delta_{ab} \frac{1}{p_1^2}
\end{align*}
\end{minipage}
\label{fig:Intro-QCD-Prop}
\end{figure}
\item The interacting vertices are given by:
\begin{figure}[H]
\qquad
\begin{minipage}{2in}
\begin{tikzpicture}[line width=0.6 pt, scale=0.7]
\draw[fermion] (-2,-1) -- (0,1);
\draw[fermion] (0,1) -- (2,-1);
\draw[gluon] (0,1) -- (0,3);
\draw[->] (0.5,3) -- (0.5,2.5);
\draw[->] (-2.6,-0.9) -- (-2.2,-0.5);
\draw[->] (2.6,-0.9) -- (2.2,-0.5);
\node at (2.9,-0.6) {$p_1$};
\node at (-2.9,-0.6) {$p_2$};
\node at (1,2.8) {$p_3$};
\node at (2,-1.5) {$i,\alpha$};
\node at (-2,-1.5) {$j,\beta$};
\node at (0,3.5) {$a,\mu$};
\end{tikzpicture}
\end{minipage}
\hspace{-0.5cm}
\begin{minipage}{3in}
\begin{align*}
{i\mkern1mu} g_s \left( 2\pi \right)^4 \delta^{(4)}\left(p_1+p_2+p_3\right) T^{a}_{ij}
\,\left(\gamma^{\mu}\right)_{\alpha\beta}
\end{align*}
\end{minipage}
\end{figure}
\begin{figure}[H]
\qquad
\begin{minipage}{2in}
\begin{tikzpicture}[line width=0.6 pt, scale=0.7]
\draw[gluon] (-2,-1) -- (0,1);
\draw[gluon] (0,1) -- (2,-1);
\draw[gluon] (0,1) -- (0,3);
\draw[->] (0.5,3) -- (0.5,2.5);
\draw[->] (-2.6,-0.9) -- (-2.2,-0.5);
\draw[->] (2.6,-0.9) -- (2.2,-0.5);
\node at (2.9,-0.6) {$p_1$};
\node at (-2.9,-0.6) {$p_2$};
\node at (1,2.8) {$p_3$};
\node at (2,-1.5) {$a,\mu$};
\node at (-2,-1.5) {$b,\nu$};
\node at (0,3.5) {$c,\rho$};
\end{tikzpicture}
\end{minipage}
\quad
\begin{minipage}{3in}
\begin{align*}
&\frac{g_s}{3!} \left( 2\pi \right)^4 \delta^{(4)}\left(p_1+p_2+p_3\right) f^{abc}
\\& \times \left[ g^{\mu\nu}(p_{1}-p_2)^{\rho} + g^{\nu\rho}(p_{2}-p_3)^{\mu} +
g^{\rho\mu}(p_{3}-p_1)^{\nu} \right]
\end{align*}
\end{minipage}
\end{figure}
\begin{figure}[H]
\qquad
\begin{minipage}{2in}
\begin{tikzpicture}[line width=0.6 pt, scale=0.7]
\draw[scalar] (-2,-1) -- (0,1);
\draw[scalar] (0,1) -- (2,-1);
\draw[gluon] (0,1) -- (0,3);
\draw[->] (0.5,3) -- (0.5,2.5);
\draw[->] (-2.6,-0.9) -- (-2.2,-0.5);
\draw[->] (2.6,-0.9) -- (2.2,-0.5);
\node at (2.9,-0.6) {$p_1$};
\node at (-2.9,-0.6) {$p_2$};
\node at (1,2.8) {$p_3$};
\node at (2,-1.5) {$b$};
\node at (-2,-1.5) {$c$};
\node at (0,3.5) {$a,\mu$};
\end{tikzpicture}
\end{minipage}
\hspace{-0.5cm}
\begin{minipage}{3in}
\begin{align*}
- g_s\left( 2\pi \right)^4 \delta^{(4)}\left(p_1+p_2+p_3\right) f^{abc} p_1^{\mu}
\end{align*}
\end{minipage}
\end{figure}
\begin{figure}[H]
\qquad
\begin{minipage}{2in}
\begin{tikzpicture}[line width=0.6 pt, scale=0.7]
\draw[gluon] (-2,-1) -- (0,1);
\draw[gluon] (0,1) -- (2,-1);
\draw[gluon] (-2,3) -- (0,1);
\draw[gluon] (0,1) -- (2,3);
\draw[->] (-2.6,-0.9) -- (-2.2,-0.5);
\draw[->] (2.6,-0.9) -- (2.2,-0.5);
\draw[->] (-2.6,3) -- (-2.2,2.6);
\draw[->] (2.6,3) -- (2.2,2.6);
\node at (2.9,-0.6) {$p_1$};
\node at (-2.9,-0.6) {$p_2$};
\node at (-2.9,2.6) {$p_3$};
\node at (2.9,2.6) {$p_4$};
\node at (2,-1.5) {$a,\mu$};
\node at (-2,-1.5) {$b,\nu$};
\node at (-2,3.5) {$c,\rho$};
\node at (2,3.5) {$d,\sigma$};
\end{tikzpicture}
\end{minipage}
\quad
\begin{minipage}{3in}
\begin{align*}
&-\frac{g_s^2}{4!} \left( 2\pi \right)^4
\delta^{(4)}\left(p_1+p_2+p_3+p_4\right)
\nonumber\\
&\Bigg\{ \left( f^{ac,bd} -f^{ad,cb} \right) g^{\mu\nu} g^{\rho\sigma}
+ \left( f^{ab,cd} -f^{ad,bc} \right) g^{\mu\rho} g^{\nu\sigma}
\nonumber\\
&+ \left( f^{ac,db} -f^{ab,cd} \right) g^{\mu\sigma} g^{\nu\rho} \Bigg\}
\intertext{with}
&f^{ab,cd} \equiv f^{abx} f^{cdx}
\end{align*}
\end{minipage}
\end{figure}
\end{itemize}
In addition to these rules, we have keep in mind the following points:
\begin{itemize}
\item For any Feynman diagram, the symmetry factor needs to be
multiplied appropriately. The symmetry factor is defined as the
number of ways one can obtain the topological configuration of the Feynman diagram
under consideration.
\item For each loop momenta, the integration over the loop momenta,
$k$, needs to be performed with the integration measure
$d^dk/(2\pi)^d$ in d-dimensions (in dimensional regularisation).
\item For each quark/ghost loop, one has to multiply a factor of (-1).
\end{itemize}
\section{Perturbative Calculations in QCD}
\label{sec:Intro-pQCD}
The asymptotic freedom of the QCD allows us to perform the calculations in
high energy regime using the techniques of perturbative QCD (pQCD). In
pQCD, we make the theoretical predictions through the computations of
the scattering matrix (S-matrix) elements. The S-matrix elements are
directly related to the scattering amplitude which is formally
expanded, within the framework of perturbation theory, in powers of
coupling constants. This expansion is represented through the set of Feynman
diagrams and the Feynman rules encapsulate the connection between
these these two. Hence, the theoretical predictions boil down to
evaluate the set of Feynman diagrams. Using the Feynman rules
presented in the previous Sec.~\ref{ss:QCD-Feynman-Rules}, we can
evaluate all the Feynman diagrams.
Achieving precise theoretical predictions demand to go beyond leading
order which consists of evaluating the virtual/loop as well as real emission
diagrams. However, the contribution arising from the individual one is
not finite. The resulting expressions from the evaluation of loop
diagrams contain the ultraviolet (UV), soft and collinear
divergences. For simplicity, together we call the soft and collinear as
infrared (IR) divergence.
The UV divergences arise from the region of large
momentum or very high energy (approaching infinity) of the Feynman
integrals, or, equivalently, because of the physical phenomena at very
short distances. We get rid of this through UV renormalisation. Before
performing the UV renormalisation, we need to regulate the Feynman
integrals which is essentially required to identify the true nature of
divergences. There are several ways to regulate the integrals. The
most consistent and beautiful way is the framework of dimensional
regularisation~\cite{'tHooft:1972fi,Cicuta:1972jf,Bollini:1972ui}. Within
this, we need to perform the integrals in general
$d$-dimensions which is taken as $4+\epsilon$ in this thesis. Upon
performing the integrals, all the UV singularities arise as poles in
$\epsilon$. The UV renormalisation, which is performed through redefining
all the quantities present in the Lagrangian, absorbs these poles and
gives rise a UV finite result. The UV renormalisation is done at
certain energy scale, known as renormalisation scale, $\mu_{R}$. On
the other hand, the soft divergences
arise from the low momentum
limit (approaching zero) of the loop integrals and the collinear ones
arise when any loop momentum becomes collinear to any of
the external massless particles. The collinear divergence is a property of
theories with massless particles. Hence, even after performing the UV
renormalisation, the resulting expressions obtained through the
evaluation the loop integrals are not finite, they contain poles
arising from soft and collinear regions of the loop integrals.
To remove the residual IR divergences, we need to add the
contributions arising from the real emission diagrams. The latter
contains soft as well as collinear divergences which have the same
form as that of loop integrals. Once we add the virtual and real
emission diagrams and evaluate the phase space integrals, the
resulting expressions are guaranteed to be freed from UV, soft and
final state collinear singularities, thanks to the
Kinoshita-Lee-Nauenberg (KLN) theorem. An analogous result for quantum electrodynamics alone is known as Bloch?Nordsieck cancellation. However, the collinear
singularities arising from the collinear configurations involving
initial state particles remain. Those are removed at the hadronic level through the
techniques, known as mass factorisation, where the residual
singularities are absorbed into the bare parton distribution
functions (PDF). So, the observables at the hadronic level are finite
which are compared with the experimental outcomes at the hadron
colliders. Just like UV renormalisation, mass factorisation is
done at some energy scale, called factorisation scale, $\mu_F$. The
$\mu_R$ as well as $\mu_F$ are unphysical scales. The dependence of
the fixed order results on these scale is an artifact of
the truncation of the perturbative series to a finite order. If we can
capture the results to all order, then the dependence goes away.
The core part of this thesis deals with the higher order QCD
corrections employing the methodology of perturbation theory to some
of the very important processes within the SM and
beyond. More specifically, the thesis contains
\begin{itemize}
\item the soft-virtual QCD corrections to the inclusive cross section of the Higgs
boson production through bottom quark annihilation at
next-to-next-to-next-to-leading order
(N$^3$LO)~\cite{Ahmed:2014cha},
\item the soft-virtual QCD corrections at N$^{3}$LO to the differential rapidity
distributions of the productions of the Higgs boson in gluon fusion
and of the leptonic pair in Drell-Yan~\cite{Ahmed:2014uya},
\item the three loop QCD corrections to the pseudo-scalar form
factors~\cite{Ahmed:2015qpa,Ahmed:2015pSSV}.
\end{itemize}
In the subsequent subsections, we will discuss the above things in
brief.
\subsection{Soft-Virtual Corrections To Cross Section at N$^3$LO QCD}
\label{ss:Intro-SV}
The Higgs bosons are produced dominantly at the LHC
via gluon fusion through top quark loop, while
one of the sub-dominant ones take place through bottom quark
annihilation. In the SM, the interaction between the Higgs boson and
bottom quarks is controlled through the Yukawa coupling which is
reasonably small at typical energy scales. However, in
the minimal super symmetric SM (MSSM), this
channel can contribute substantially due to enhanced coupling between the
Higgs boson and bottom quarks in the large $\tan\beta$ region, where
$\tan\beta$ is the ratio of vacuum expectation values of the up
and down type Higgs fields. In the present run of LHC, the measurements
of the various coupling constants including this one are underway which can
shed light on the properties of the newly discovered Higgs boson~\cite{Aad:2012tfa,
Chatrchyan:2012xdj}.
Most importantly, for the precision studies we
must take into account all the contributions, does not matter how tiny
those are, arising from sub-dominant
channels along with the dominant ones to reduce the dependence on the
unphysical scales and make a reliable prediction.
The computations of the higher order QCD corrections beyond leading
order often become quite challenging because of the large number of Feynman
diagrams and, presence of the complicated loop and phase space
integrals. Under this circumstance, when we fail to compute the
complete result at certain order, it is quite natural to try an
alternative approach to capture
the dominant contributions from the missing higher order corrections.
It has been observed for many processes that the dominant contributions to an observable
often comes from the soft gluon emission diagrams. The contributions
arising from the
associated soft gluon emission along with the virtual Feynman
diagrams are known as
the soft-virtual (SV) corrections. \textit{The goal of the works published in the
article~\cite{Ahmed:2014cha} is to
discuss the SV
QCD corrections to the production cross section of the Higgs boson,
produced through bottom quark annihilation.}
The next-to-next-to-leading order (NNLO) QCD corrections to this channel are already present
in the variable flavour scheme (VFS) \cite{Dicus:1988cx, Dicus:1998hs,
Maltoni:2003pn, Olness:1987ep, Gunion:1986pe, Harlander:2003ai},
while it is known to NLO in the fixed
flavour scheme (FFS) \cite{Reina:2001sf, Beenakker:2001rj,
Dawson:2002tg, Beenakker:2002nc, Raitio:1978pt, Kunszt:1984ri}.
In addition, the partial result for the N$^3$LO
corrections~\cite{Ravindran:2005vv,Ravindran:2006cg,Kidonakis:2007ww} under the SV
approximation were also computed long
back. In both~\cite{Ravindran:2005vv,Ravindran:2006cg} and~\cite{Kidonakis:2007ww},
it was not possible to determine the complete contribution at
N$^{3}$LO due to the lack of information on three loop finite part of
bottom anti-bottom Higgs form factor in QCD and the soft gluon
radiation at N$^{3}$LO level. \textit{In this
work~\cite{Ahmed:2014cha}, we have computed the missing part and
completed the full SV corrections to the cross section at N$^3$LO}.
The infrared safe contributions from the soft gluons are obtained by
adding the soft part of the cross section with the UV
renormalized virtual part and performing mass factorisation using
appropriate counter terms. The main ingredients are the
form factors, overall operator UV renormalization constant,
soft-collinear distribution arising from the real radiations in the partonic
subprocesses and mass factorization kernels. The computations of
SV cross section at N$^3$LO QCD require all of these above quantities
up to 3-loop order. The relevant form factor becomes available very
recently in~\cite{Gehrmann:2014vha}. The soft-collinear distribution at
N$^3$LO was computed by
us around the same time in~\cite{Ahmed:2014cla}. This was calculated
from the recent result of
N$^3$LO SV cross section of the Higgs boson productions in gluon
fusion~\cite{Anastasiou:2014vaa} by employing a symmetry (maximally
non-Abelian property). Prior to this, this symmetry was verified
explicitly up to NNLO order. However, neither there was any clear
reason to believe that the symmetry would fail
nor there was any transparent indication of holding it beyond this
order. Nevertheless, we conjecture~\cite{Ahmed:2014cla} that the relation would hold true
even at N$^3$LO order! This is inspired by the universal properties of
the soft gluons which are the
underlying reasons behind the existence of this remarkable
symmetry. Later, this conjecture is verified by explicit computations
performed by two different groups on Drell-Yan
process~\cite{Catani:2014uta, Li:2014bfa}. This symmetry
plays the most important role in
achieving our goal. With these, along with the existing results of the
remaining required ingredients, we obtain the complete analytical
expressions of N$^3$LO SV
cross section of the Higgs boson production through bottom quark
annihilation~\cite{Ahmed:2014cha} employing the methodology prescribed
in~\cite{Ravindran:2005vv,Ravindran:2006cg}. It reduces the scale dependence and
provides a more precise result. We demonstrate the impact of this result numerically at
the LHC briefly. \textit{This is the most accurate
result for this channel which exists in the literature till date and
it is expected to play an important role in coming days at the LHC.}
\subsection{Soft-Virtual QCD Corrections to Rapidity at N$^3$LO}
\label{ss:Intro-SV-Rap}
The productions of the Higgs boson in gluon fusion and leptonic pair
in Drell-Yan (DY) are among the most important processes at the LHC which are studied not
only to test the SM to an unprecedented accuracy but also to explore
the physics beyond Standard Model (BSM). During the present
run at the LHC, in addition to the inclusive production cross section,
the differential rapidity distribution is among the most important
observables,
which is expected to be measured in upcoming days. This immediately
calls for very precise theoretical predictions.
In the same spirit of the SV corrections to
the inclusive production cross section, the dominant contributions to
the differential rapidity distributions often arise from the soft
gluon emission diagrams. Hence, in the absence of complete fixed
order result, the rapidity distribution under SV approximation is the
best available alternative in order to capture the dominant contributions from
the missing higher orders and stabilise the dependence on unphysical
scales. For the Higgs boson production through gluon fusion, we work in the effective
theory where the top quark is integrated out.
\textit{This work published in the article~\cite{Ahmed:2014uya} is devoted to demonstrate the SV corrections
to this observable at N$^3$LO for the Higgs boson, produced through
gluon fusion, and leptonic pair in DY production.}
For the processes under considerations, the NNLO QCD
corrections are present~\cite{Anastasiou:2003yy, Anastasiou:2004xq,
Anastasiou:2005qj}, computed long back, and in addition, the partial
N$^3$LO SV
results~\cite{Ravindran:2006bu} are also available. However, due to reasonably large scale
uncertainties and crying demand of uplifting the accuracy of theoretical
predictions, we must push the boundaries of existing
results. \textit{In this work~\cite{Ahmed:2014uya},
we have computed the missing part and completed the SV corrections
to the rapidity distributions at N$^3$LO QCD}.
The prescription~\cite{Ravindran:2006bu} which has been employed to calculate the SV QCD
corrections is similar to that of the inclusive cross section, more
specifically, it is a generalisation of the other one. The infrared
safe contributions under SV approximation
can be computed by adding the soft part of the rapidity distribution
with the UV renormalised virtual part and performing the mass
factorisation using appropriate counter terms. Similar to the
inclusive case, the main ingredients to perform this computation are
the form factors, overall UV operator renormalisation constant,
soft-collinear distribution for rapidity and mass factorisation
kernels. These quantities are required up to N$^3$LO to calculate the
rapidity at this order. The three loop quark and gluon form
factors~\cite{Moch:2005tm, Baikov:2009bg, Gehrmann:2010ue, Gehrmann:2010tu} were
calculated long back. The operator renormalisation constants are also
present. For DY, this constant is not required or equivalently equals
to unity. The mass factorisation kernels are also available in the
literature to the required order. The only missing part was the
soft-collinear distribution for rapidity at N$^3$LO. This was
not possible to compute until very recently. Because of the universal
behaviour of the soft gluons, the soft-collinear distributions for
rapidity and inclusive cross section can be related to all orders in
perturbation theory~\cite{Ravindran:2006bu}. Employing this beautiful
relation, we obtain this
quantity at N$^3$LO
from the results of soft-collinear distribution of the inclusive cross
section~\cite{Ahmed:2014cla}. Using this, along with the existing results of the other
relevant quantities, we compute the complete analytical expressions of
N$^3$LO SV correction to the rapidity distributions for the Higgs
boson in gluon fusion and leptonic pair in DY~\cite{Ahmed:2014uya}. We demonstrate the
numerical impact of this correction for the case of Higgs boson at the
LHC. This indeed reduces the scale dependence significantly and
provides a more reliable theoretical predictions. \textit{These are the
most accurate results for the rapidity distributions of the Higgs
boson and DY pair which exist in the literature and undoubtedly,
expected to play very important role in the upcoming run at the LHC.}
\subsection{Pseudo-Scalar Form Factors at Three Loops in QCD}
\label{ss:Intro-PS}
One of the most popular extensions of the SM, namely, the MSSM and two
Higgs doublet model have richer Higgs sector containing more than one
Higgs boson and there have been intense search strategies to observe
them at the LHC. In particular, the production of CP-odd Higgs
boson/pseudo-scalar at the LHC has been studied in detail, taking into
account higher order QCD radiative corrections, due to similarities
with its CP-even counter part. Very recently, the N$^{3}$LO QCD corrections
to the inclusive production cross section of the CP-even Higgs boson
becomes available~\cite{Anastasiou:2015ema}. So, it is very natural to
extend the theoretical accuracy
for the CP-odd Higgs boson
to the same order of N$^{3}$LO. This requires the 3-loop quark and gluon form factors for
the pseudo-scalar which are the only missing ingredients to achieve this goal.
Multiloop and multileg computations play a crucial role to achieve the golden
task of making precise theoretical predictions. However, the
complexity of these computations grows very rapidly
with the increase of number of loops and/or external
particles. Nevertheless, it has become a reality due to several
remarkable developments in due course of time. \textit{These
articles~\cite{Ahmed:2015qpa, Ahmed:2015pSSV} are
devoted to demonstrate the computations of the 3-loop quark and gluon
form factors for the pseudo-scalar operators in QCD}.
The coupling of a pseudo-scalar Higgs boson to gluons is mediated
through a heavy quark loop. In the limit of large quark mass, it is
described by an effective Lagrangian~\cite{Chetyrkin:1998mw} that only
admits light degrees of freedom.
In this effective theory, we compute the 3-loop massless QCD
corrections to the form factor that describes the coupling of a
pseudo-scalar Higgs boson to gluons.
The evaluation of this 3-loop
form factors is truly a non-trivial task not only because of the
involvement of a large number of Feynman diagrams but also due to
the presence of the axial vector coupling. We work in dimensional
regularisation and use the 't Hooft-Veltman
prescription~\cite{'tHooft:1972fi} for the axial vector current,
The state-of-the-art techniques including
integration-by-parts~\cite{Tkachov:1981wb,Chetyrkin:1981qh} and
Lorentz invariant~\cite{Gehrmann:1999as} identities
have been employed to accomplish this task. The UV
renormalisation is quite involved since the two operators, present in
the Lagrangian, mix under UV renormalization due to the axial anomaly and
additionally, a finite renormalisation constant needs to be introduced
in order to fulfill the chiral Ward identities.
Using the universal infrared factorization properties, we
independently derive~\cite{Ahmed:2015qpa} the three-loop operator
mixing and finite operator renormalisation from the renormalisation
group equation for the form factors, thereby confirming recent
results~\cite{Larin:1993tq, Zoller:2013ixa}, which were computed
following a completely different methodology, in the operator product
expansion.
This form factor~\cite{Ahmed:2015qpa,Ahmed:2015pSSV} is an important
ingredient to the precise
prediction of the pseudo-scalar Higgs boson production cross section
at hadron colliders. We derive the hard matching coefficient in
soft-collinear effective theory (SCET). We
also study the form factors in the context of leading
transcendentality principle and we find that the
diagonal form factors become identical to those of ${\cal N}=4$ upon
imposing some identification
on the quadratic Casimirs. Later, these form factors are used to
calculate the SV corrections~\cite{Ahmed:2015pSSV} to the pseudo-scalar production cross
section at N$^3$LO and next-to-next-to-next-to-leading logarithm (N$^3$LL) QCD.
\chapter*{SYNOPSIS}
\addcontentsline{toc}{chapter}{Synopsis}
The Standard Model (SM) of particle physics is one of the most
remarkably successful fundamental theories of all time which got its
finishing touch on the eve of July 2012 through the discovery of the
long-awaited particle, ``the Higgs boson'', at the biggest underground
particle research amphitheater, the Large Hadron Collider (LHC). It
would take a while to make the conclusive remarks about the true
identity of the newly-discovered particle. However,
after the discovery of this SM-like-Higgs boson, the high
energy physics community is standing on the verge of a very crucial era
where the new physics may show up as tiny deviations from the
predictions of the SM. To exploit this possibility, it is a crying need to make the
theoretical predictions, along with the revolutionary experimental
progress, to a spectacularly high accuracy within the SM and beyond
(BSM).
The most successful and celebrated methodology to perform the theoretical
calculations within the SM and BSM are based on the perturbation
theory, due to our inability to solve the theory exactly. Under the
prescriptions of perturbation
theory, all the observables are expanded in powers of the coupling
constants present in the underlying Lagrangian. The result obtained
from the first term of
perturbative series is called the leading order (LO), the next one
is called next-to-leading order (NLO) and so on. In most of the cases,
the LO results fail miserably to deliver a reliable theoretical prediction of the
associated observables, one must go beyond the wall of
LO result to achieve a higher accuracy.
Due to the presence of three fundamental forces within the SM, any
observable can be expanded in powers of the coupling constants
associated with the corresponding forces, namely, electromagnetic
($\alpha_{\rm EM}$), weak ($\alpha_{\rm EW}$) and strong ($\alpha_s$)
ones and consequently, perturbative calculations can be
performed with respect to each of these constants. However, at typical
energy scales, at which the hadron colliders undergo operations, the
contributions arising from the $\alpha_s$ expansion dominate over the
others due to comparatively large values of $\alpha_s$. Hence, to
catch the dominant contributions to any observables,
we must concentrate on
the $\alpha_s$ expansion and evaluate the terms beyond LO. These are
called Quantum Chromo-dynamics (QCD) radiative or perturbative QCD (pQCD) corrections.
In addition, the pQCD predictions depend on two
unphysical scales, the renormalisation ($\mu_R$) and factorisation ($\mu_F$) scales, which
are required to introduced in the process of renormalising the
theory. The $\mu_R$ arises from the ultraviolet (UV) renormalisation,
whereas the mass factorisation (removes collinear
singularities) introduces the $\mu_F$. Any fixed order results do
depend on these unphysical scales which happens due to the truncation of
the perturbative expansion at any finite order. As we include the
contributions from higher and higher orders, the dependence of any
physical observable on these unphysical scales gradually goes down.
Hence, to make a reliable theoretical prediction, it is absolutely
necessary to take into account the contributions arising from the
higher order QCD corrections to any observable at the hadron
colliders.
\textit{This thesis arises exactly in this context. The central
part of this thesis deals with the QCD radiative corrections to some
important observables associated with the Drell-Yan, scalar and pseudo
scalar Higgs boson production at three loop or N$^3$LO order}. In the
subsequent discussions, we will concentrate only on these three
processes.
\section{Soft-Virtual QCD Corrections to Cross Section at N$^3$LO}
\label{sec:Synop-SV-CS}
The Higgs bosons are produced dominantly at the LHC
via gluon fusion through top quark loop, while
one of the sub-dominant ones take place through bottom quark
annihilation. In the SM, the interaction between the Higgs boson and
bottom quarks is controlled through the Yukawa coupling which is
reasonably small at typical energy scales. However, in
the minimal super symmetric SM (MSSM), this
channel can contribute substantially due to enhanced coupling between the
Higgs boson and bottom quarks in the large $\tan\beta$ region, where
$\tan\beta$ is the ratio of vacuum expectation values of the up
and down type Higgs fields. In the present run of LHC, the measurements
of the various coupling constants including this one are underway which can
shed light on the properties of the newly discovered Higgs boson.
Most importantly, for the precision studies we
must take into account all the contributions, does not matter how tiny
those are, arising from sub-dominant
channels along with the dominant ones to reduce the dependence on the
unphysical scales and make a reliable prediction.
The computations of the higher order QCD corrections beyond leading
order often becomes quite challenging because of the large number of Feynman
diagrams and, presence of the complicated loop and phase space
integrals. Under this circumstance, when we fail to compute the
complete result at certain order, it is quite natural to try an
alternative approach to capture
the dominant contributions from the missing higher order corrections.
It has been observed for many processes that the dominant contributions to an observable
often comes from the soft gluon emission diagrams. The contributions
arising from the
associated soft gluon emission along with the virtual Feynman
diagrams are known as
the soft-virtual (SV) corrections. \textit{The goal of this section is to
discuss the SV
QCD corrections to the production cross section of the Higgs boson,
produced through bottom quark annihilation.}
The NNLO QCD corrections to this channel are already present in the
literature. In addition, the partial result for the N$^3$LO
corrections under the SV approximation were also computed long
back. \textit{In this work, we have computed the missing part and
completed the full SV corrections to the cross section at N$^3$LO}.
The infrared safe contributions from the soft gluons are obtained by
adding the soft part of the cross section with the UV
renormalized virtual part and performing mass factorisation using
appropriate counter terms. The main ingredients are the
form factors, overall operator UV renormalization constant,
soft-collinear distribution arising from the real radiations in the partonic
subprocesses and mass factorization kernels. The computations of
SV cross section at N$^3$LO QCD require all of these above quantities
up to 3-loop order. The relevant form factor becomes available very
recently. The soft-collinear distribution at N$^3$LO was computed by
us around the same time. This was calculated from the recent result of
N$^3$LO SV cross section of the Higgs boson productions in gluon
fusion by employing a symmetry (maximally
non-Abelian property). Prior to this, this symmetry was verified
explicitly up to NNLO order. However, neither there was any clear
reason to believe that the symmetry would fail
nor there was any transparent indication of holding it beyond this
order. Nevertheless, we postulate that the relation would hold true
even at N$^3$LO order! This is inspired by the universal properties of
the soft gluons which are the
underlying reasons behind the existence of this remarkable
symmetry. Later, this conjecture is verified by explicit computations
performed by two different groups on Drell-Yan process. This symmetry plays the most important role in
achieving our goal. With these, along with the existing results of the
remaining required ingredients, we obtain the complete analytical
expressions of N$^3$LO SV
cross section of the Higgs boson production through bottom quark
annihilation. It reduces the scale dependence and
provides a more precise result. We demonstrate the impact of this result numerically at
the Large Hadron Collider (LHC) briefly. \textit{This is the most accurate
result for this channel which exists in the literature till date and
it is expected to play an important role in coming days at the LHC.}
\section{Soft-Virtual QCD Corrections to Rapidity at N$^3$LO}
The productions of the Higgs boson in gluon fusion and leptonic pair
in DY are among the most important processes at the LHC which are studied not
only to test the SM to an unprecedented accuracy but also to explore
the new physics under BSM. During the present run at the LHC, in addition to the inclusive production cross section, the differential rapidity distribution is among the most important observables,
which is expected to be measured in upcoming days. This immediately
calls for very precise theoretical predictions.
In the same spirit of the SV corrections to
the inclusive production cross section, the dominant contributions to
the differential rapidity distributions often arise from the soft
gluon emission diagrams. Hence, in the absence of complete fixed
order result, the rapidity distribution under SV approximation is the
best available alternative in order to capture the dominant contributions from
the missing higher orders and stabilise the dependence on unphysical
scales. For the Higgs boson production through gluon fusion, we work in the effective
theory where the top quark is integrated out.
\textit{This section is devoted to demonstrate the SV corrections
to this observable at N$^3$LO for the Higgs boson, produced through
gluon fusion, and leptonic pair in Drell-Yan (DY) production.}
For the processes under considerations, the NNLO QCD
corrections are present, computed long back, and in addition, the partial N$^3$LO SV
results are also available. However, due to reasonably large scale
uncertainties and crying demand of uplifting the accuracy of theoretical
predictions, we must push the boundaries of existing results. \textit{In this work,
we have computed the missing part and completed the SV corrections
to the rapidity distributions at N$^3$LO QCD}.
The prescription which has been employed to calculate the SV QCD
corrections is similar to that of the inclusive cross section, more
specifically, it is a generalisation of the other one. The infrared
safe contributions under SV approximation
can be computed by adding the soft part of the rapidity distribution
with the UV renormalised virtual part and performing the mass
factorisation using appropriate counter terms. Similar to the
inclusive case, the main ingredients to perform this computation are
the form factors, overall UV operator renormalisation constant,
soft-collinear distribution for rapidity and mass factorisation
kernels. These quantities are required up to N$^3$LO to calculate the
rapidity at this order. The three loop quark and gluon form factors were
calculated long back. The operator renormalisation constants are also
present. For DY, this constant is not required or equivalently equals
to unity. The mass factorisation kernels are also available in the
literature to the required order. The only missing part was the
soft-collinear distribution for rapidity at N$^3$LO. This was
not possible to compute until very recently. Because of the universal
behaviour of the soft gluons, the soft-collinear distributions for
rapidity and inclusive cross section can be related to all orders in
perturbation theory. Employing this beautiful relation, we obtain this
quantity at N$^3$LO
from the results of soft-collinear distribution of the inclusive cross
section. Using this, along with the existing results of the other
relevant quantities, we compute the complete analytical expressions of
N$^3$LO SV correction to the rapidity distributions for the Higgs
boson in gluon fusion and leptonic pair in DY. We demonstrate the
numerical impact of this correction for the case of Higgs boson at the
LHC. This indeed reduces the scale dependence significantly and
provides a more reliable theoretical predictions. \textit{These are the
most accurate results for the rapidity distributions of the Higgs
boson and DY pair which exist in the literature and undoubtedly,
expected to play very important role in the upcoming run at the LHC.}
\section{Pseudo-Scalar Form Factors at Three Loops in QCD}
One of the most popular extensions of the SM, namely, the MSSM and two
Higgs doublet model have richer Higgs sector containing more than one
Higgs boson and there have been intense search strategies to observe
them at the LHC. In particular, the production of CP-odd Higgs
boson/pseudo-scalar at the LHC has been studied in detail, taking into
account higher order QCD radiative corrections, due to similarities
with its CP-even counter part. Very recently, the N$^{3}$LO QCD corrections
to the inclusive production cross section of the CP-even Higgs boson is
computed. So, it is very natural to extend the theoretical accuracy for the CP-odd Higgs boson
to the same order of N$^{3}$LO. This requires the 3-loop quark and gluon form factors for
the pseudo-scalar which are the only missing ingredients to achieve this goal.
Multiloop and multileg computations play a crucial role to achieve the golden
task of making precise theoretical predictions. However, the
complexity of these computations grows very rapidly
with the increase of number of loops and/or external
particles. Nevertheless, it has become a reality due to several
remarkable developments in due course of time. \textit{This section is
devoted to demonstrate the computations of the 3-loop quark and gluon
form factors for the pseudo-scalar operators in QCD}.
The coupling of a pseudo-scalar Higgs boson to gluons is mediated through a heavy quark loop. In the limit of large quark mass, it is described by an effective Lagrangian that only admits light degrees of freedom.
In this effective theory, we compute the 3-loop massless QCD corrections to the form factor that describes the coupling of a pseudo-scalar Higgs boson to gluons.
The evaluation of this 3-loop
form factors is truly a non-trivial task not only because of the
involvement of a large number of Feynman diagrams but also due to
the presence of the axial vector coupling. We work in dimensional regularisation and use the 't Hooft-Veltman prescription for the axial vector current,
The state-of-the-art techniques including
integration-by-parts (IBP) and Lorentz invariant (LI) identities
have been employed to accomplish this task. The UV
renormalisation is quite involved since the two operators, present in
the Lagrangian, mix under UV renormalization due to the axial anomaly and
additionally, a finite renormalisation constant needs to be introduced
in order to fulfill the chiral Ward identities.
Using the universal infrared (IR) factorization properties, we independently derive the three-loop operator mixing and finite operator renormalisation from the renormalisation group equation for the form factors, thereby confirming recent results, which were computed following a completely different methodology, in the operator product expansion.
This form factor is an important ingredient to the precise
prediction of the pseudo-scalar Higgs boson production cross section
at hadron colliders. We derive the hard matching coefficient in soft-collinear effective theory (SCET). We
also study the form factors in the context of leading transcendentality principle and we find that the
diagonal form factors become identical to those of ${\cal N}=4$ upon imposing some identification
on the quadratic Casimirs. Later, these form factors are used to
calculate the SV corrections to the pseudo-scalar production cross
section at N$^3$LO and N$^3$LL QCD.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 367 |
Cheritra is een geslacht van vlinders van de familie Lycaenidae, uit de onderfamilie Theclinae.
Soorten
C. aenea Semper, 1890
C. aenigma Cowan, 1967
C. freja (Fabricius, 1793)
C. jafra (Godart, 1823)
C. orpheus (Felder, 1862)
C. pseudojafra Moore, 1881
C. regia Evans, 1925
C. teunga Grose-Smith | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,642 |
<html>
<head>
<link rel="icon" href="img/locked.png">
<!-- Main CSS stylesheet -->
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<link rel="stylesheet" href="https://maxcdn.bootstrapcdn.com/bootstrap/3.3.7/css/bootstrap.min.css" integrity="sha384-BVYiiSIFeK1dGmJRAkycuHAHRg32OmUcww7on3RYdg4Va+PmSTsz/K68vbdEjh4u" crossorigin="anonymous">
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<script src="https://maxcdn.bootstrapcdn.com/bootstrap/3.3.7/js/bootstrap.min.js" integrity="sha384-Tc5IQib027qvyjSMfHjOMaLkfuWVxZxUPnCJA7l2mCWNIpG9mGCD8wGNIcPD7Txa" crossorigin="anonymous"></script>
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<title>Results</title>
</head>
<body>
<div class="container">
<div class="row" id="header">
<?php
$url = $_POST['url'];
#Creating the search bar with the entered url as placeholder
echo "<h2 class='col-sm-3'>Showing results for</h2><form method='post' action='results.php'><div class='form-group'><input name='url' id='url' class='form-control col-sm-3' type='text' placeholder=$url/><input name='submit' type='submit' class='btn btn-primary col-sm-1' id='submit' value='Submit'/>";
?>
</div>
</form>
</div>
</div>
<!-- creating table and div that allows text to be shortened using an ellipsis -->
<div class="ellipsis">
<div class="container" id="tableContain">
<table class='table'>
<th>Link Name</th><th>URL</th>
<?php
;# Use the Curl extension to query the site and get back a page of results
$ch = curl_init();
$timeout = 5;
curl_setopt($ch, CURLOPT_URL, $url);
curl_setopt($ch, CURLOPT_RETURNTRANSFER, 1);
curl_setopt($ch, CURLOPT_CERTINFO, true);
curl_setopt($ch, CURLOPT_CONNECTTIMEOUT, $timeout);
$html = curl_exec($ch);
curl_close($ch);
# Create a DOM parser object
$dom = new DOMDocument();
# Parse the HTML from the url.
# The @ before the method call suppresses any warnings that
# The @ before the method call suppresses any warnings that
# loadHTML might throw because of invalid HTML in the page.
@$dom->loadHTML($html);
# Iterate over all the <a> tags
foreach($dom->getElementsByTagName('a') as $link) {
# Show the <a href>
$href = $link->getAttribute('href');
# Grabbing the link name
$linkValue = $link->nodeValue;
#if the href attribute is blank, then ignore
if (strlen(trim($href)) == 0) {
}
#if href attr. starts with /, then ignore (relative filepath)
elseif (substr($href, 0, 1) === '/') {
}
#if href attr. starts with #, ignore. opening up a module etc.
elseif (substr($href, 0, 1) === '#') {
}
#if doesn't meet any of those and doesn't contain https:// then create a row with the link name as well as the href attribute
elseif (strpos($href, 'https://') === false) {
echo "<tr>";
echo "<td class='linkValue'>$linkValue</td>";
echo "<td class='href'>$href</td>";
echo"</tr>";
}
}
# Looking over the img tags
foreach($dom->getElementsByTagName('img') as $node) {
$src = $node->getAttribute('src');
$nodeValue = $node->nodeValue;
if (strlen(trim($src)) == 0) {
}
elseif (substr($src, 0, 1) === '/') {
}
#Display the image and the src attribute if doesn't contain https://
elseif (strpos($src, 'https://') === false) {
echo "<tr>";
echo "<td class='nodeValue'><img src='$src' class='srcValue'/></td>";
echo "<td class='src'>$src</td>";
echo"</tr>";
}
}
# Looking at script tags
foreach($dom->getElementsByTagName('script') as $script) {
$scriptsrc = $script->getAttribute('src');
#Doesn't work from what I am able to see.
$scriptValue = $script->nodeValue;
if (strlen(trim($scriptsrc)) == 0) {
}
elseif (substr($scriptsrc, 0, 1) === '/') {
}
elseif (strpos($scriptsrc, 'https://') === false) {
echo "<tr>";
echo "<td class='scriptValue'>$scriptValue</td>";
echo "<td class='scriptsrc'>$scriptsrc</td>";
echo"</tr>";
}
}
?>
</table>
</div>
</div>
</body
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| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,291 |
Green investment trust with all-female board plans £150m IPO
Show caption Atrato is chaired by Juliet Davenport, the founder of Good Energy. Photograph: Emily Whitfield-Wicks/PA
Atrato, which aims to raise £150m, is run by three women including Good Energy founder Juliet Davenport
An investment trust focused on renewable energy due to float next month is understood to be the first company with an all-female board to list on the London Stock Exchange.
Atrato Onsite Energy, which said it wants to raise £150m by listing in London next month, is run by three women who make up its board, surpassing voluntary targets for getting more women into senior roles across most of FTSE companies.
It marks another victory for diversity advocates pushing for greater representation for women across the top tier of British business. While there is limited data covering the gender makeup of boards throughout the entirety of the London Stock Exchange's history, which dates back to 1608, it is understood that there are currently no companies with all-female boards listed in London.
The government-backed Hampton-Alexander review revealed in February that it had achieved its target of 33% of board positions at FTSE 100 and FTSE 250 firms being held by women by the end of 2020.
Twenty-one percent of FTSE 100 boards and 32% of FTSE 250 boards had yet to reach the target, while progress across executive leadership teams – covering executives directors and those who report to them – also lagged behind the 33% target at 29.4%.
The Atrato chair, Juliet Davenport, the head of its audit committee, Marlene Wood, and the non-executive director Faye Goss will oversee Atrato's operations. It will primarily invest in solar photovoltaic panels installed on the roofs of warehouses, factories and other industrial buildings.
Atrato said its investment in solar panels and related infrastructure would help companies reduce their carbon footprint and energy bills while also offering investors a stable income stream based on green investments. The firm said investors could expect annual dividends of 5p a share, as part of a larger return on investments of 8%-10%.
"The UK's binding net zero emissions target in 2050 and the resulting future demand for green energy means that additional generation from low carbon sources such as rooftop solar is growing," Davenport said. "The company will play a leading role in providing new green power capacity, delivering businesses a dedicated clean energy supply at a low fixed cost."
The trust is expected to launch its initial public offering by the end of November under the ticker name ROOF.
{{#ticker}}
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Judge overturns Trump decision to end tariff exemption for imported solar panels - Reuters
There is no question that hyd...
Hydro Versus Batteries: Tasmania Pushes Its Undersea Cable Plan | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,501 |
namespace Votter.Models
{
using System;
using System.ComponentModel.DataAnnotations;
using System.ComponentModel.DataAnnotations.Schema;
using System.Linq;
/// <summary>
/// Picture uploaded by given user for given category
/// One user can upload one picture for one category
/// </summary>
public class Picture
{
[Key]
public int PictureId { get; set; }
[ForeignKey("ApplicationUser")]
public string ApplicationUserId { get; set; }
public virtual ApplicationUser ApplicationUser { get; set; }
public int CategoryId { get; set; }
public virtual Category Category { get; set; }
public string Link { get; set; }
public int ScoreId { get; set; }
public virtual Score Score { get; set; }
}
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 251 |
Q: PyQt5 How to apply different QStyles to different widgets? I am trying to make multiple widgets with different styles from the built in styles provided by QStyleFactory, but when I run my code they all look the same. How can I fix this?
from PyQt5 import QtWidgets, QtCore, QtGui
import sys
class Demo(QtWidgets.QMainWindow):
def __init__(self, parent=None):
super().__init__(parent)
self.container = QtWidgets.QWidget()
self.setCentralWidget(self.container)
self.layout = QtWidgets.QVBoxLayout()
self.container.setLayout(self.layout)
self.btn = QtWidgets.QPushButton("button")
self.lw = QtWidgets.QListWidget()
self.lw.addItems(["one", "two", "three"])
self.layout.addWidget(self.btn)
self.layout.addWidget(self.lw)
self.resize(400, 150)
self.show()
if __name__ == '__main__':
app = QtWidgets.QApplication(sys.argv)
widgets = []
for style_name in QtWidgets.QStyleFactory.keys():
demo = Demo()
demo.setWindowTitle(style_name)
style = QtWidgets.QStyleFactory.create(style_name)
demo.setStyle(style)
widgets.append(demo)
sys.exit(app.exec_())
A: An important aspect of setting a QStyle on widgets (instead of setting it on the whole application) is reported in the QWidget.setStyle() documentation:
Setting a widget's style has no effect on existing or future child widgets.
So, what's happening is that you're setting the style on the QMainWindow only, while the children will always use the QApplication style.
What you could try to do is to manually set the style for the existing children:
for style_name in QtWidgets.QStyleFactory.keys():
demo = Demo()
demo.setWindowTitle(style_name)
style = QtWidgets.QStyleFactory.create(style_name)
demo.setStyle(style)
for child in demo.findChildren(QtWidgets.QWidget):
child.setStyle(style)
widgets.append(demo)
In any case, the above approach has a drawback: any new children created after setting the style will still inherit the QApplication style. The only way to avoid this is to watch for childEvent() by (recursively) installing an event filter on the parent, and set the styles accordingly; note that you need to watch for StyleChange events too.
class ChildEventWatcher(QtCore.QObject):
def __init__(self, parentWidget):
super().__init__()
self.parentWidget = parentWidget
self.parentWidget.installEventFilter(self)
def eventFilter(self, source, event):
if event.type() == QtCore.QEvent.ChildAdded and isinstance(event.child(), QtWidgets.QWidget):
event.child().installEventFilter(self)
event.child().setStyle(self.parentWidget.style())
for child in event.child().findChildren(QtWidgets.QWidget):
child.installEventFilter(self)
child.setStyle(self.parentWidget.style())
elif event.type() == QtCore.QEvent.StyleChange and source == self.parentWidget:
for child in self.parentWidget.findChildren(QtWidgets.QWidget):
child.setStyle(self.parentWidget.style())
return super().eventFilter(source, event)
class Demo(QtWidgets.QMainWindow):
def __init__(self, parent=None):
super().__init__(parent)
# this *must* be created before adding *any* child
self.childEventWatcher = ChildEventWatcher(self)
# ...
Also remember another important aspect the documentation warns about:
Warning: This function is particularly useful for demonstration purposes, where you want to show Qt's styling capabilities. Real applications should avoid it and use one consistent GUI style instead.
While the above code will do what you're expecting, installing an event filter on all child QWidgets is not a good thing to do, especially if you only need to do the style change (which is something that should normally be done just once, possibly at the start of the program). Considering the warning about using different styles, I highly suggest you to do this exactly as suggested: for demonstration purposes only.
A: You can set it on the application.
from PyQt5 import QtWidgets, QtCore, QtGui
import sys
class Demo(QtWidgets.QMainWindow):
def __init__(self, parent=None):
super().__init__(parent)
self.container = QtWidgets.QWidget()
self.setCentralWidget(self.container)
self.layout = QtWidgets.QVBoxLayout()
self.container.setLayout(self.layout)
self.btn = QtWidgets.QPushButton("button")
self.lw = QtWidgets.QListWidget()
self.lw.addItems(["one", "two", "three"])
self.layout.addWidget(self.btn)
self.layout.addWidget(self.lw)
self.resize(400, 150)
self.show()
if __name__ == '__main__':
app = QtWidgets.QApplication(sys.argv)
win = Demo()
app.setStyle(QtWidgets.QStyleFactory.create("Windows")) # Set style theme on app
sys.exit(app.exec_())
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,898 |
\section{Introduction}
For a complex number $z$ from the upper half-plane let
\begin{equation}
m(z)=\frac{1}{2\pi i}\int_{C}\frac{e^{sz}}{\zeta(s)}ds,
\end{equation}
where $\zeta(s)$ denotes the classical Riemann zeta function, and the path of integration consists of the half-line $s=-\frac{1}{2}+it, \infty>t\geq0,$ the line segment $\left[-\frac{1}{2}, \frac{3}{2}\right]$ and the half-line $s=\frac{3}{2}+it, 0\leq t<\infty.$ This function was considered in \cite{bartz1} and \cite{kaczorowski1} where the following theorems were proved.
\begin{thrm}[Bartz \cite{bartz1}]\label{bartz}
The function $m(z)$ can be analytically continued to a meromorphic function on the whole complex plane and satisfies the following functional equation
\begin{equation}
m(z)+\overline{m(\overline{z})}=-2\sum_{n=1}^\infty\frac{\mu(n)}{n}\cos\left(\frac{2\pi}{n}e^{-z}\right).
\end{equation}
The only singularities of $m(z)$ are simple poles at the points $z=\log{n}$, where $n$ is a square-free natural number. The corresponding residues are
$$\text{Res}_{z=\log n} m(z)=-\frac{\mu(n)}{2\pi i}.$$
\end{thrm}
J. Kaczorowski in \cite{kaczorowski1} simplified the proof of this result and gave an explicit formula for $m(z)$ in the strip $|\Im{z}|<\pi.$
\begin{thrm}[Kaczorowski \cite{kaczorowski1}]\label{kaczorowski}
For $|\Im{z}|<\pi, z\neq\log{n}, \mu(n)\neq0$ we have
\begin{equation}
\begin{split}
m(z)&=-\sum_{n=1}^\infty\frac{\mu(n)}{n}e\left(-\frac{1}{ne^z}\right)-\frac{e^z}{2\pi i}m_0(z)\\
&\quad-\frac{1}{2i}(m_1(z)+\overline{m_1}(z))+\frac{1}{2i}(F_m(z)+\overline{F_m}(z)),
\end{split}
\end{equation}
where
\begin{align*}
m_0(z)&=\sum_{n=1}^\infty\frac{\mu(n)}{n}\frac{1}{z-\log{n}}\\
\intertext{is meromorphic on $\mathbb{C}$ and}\\
m_1(z)&=\frac{1}{2\pi i}\int\limits_C\left(\tan{\frac{\pi s}{2}}-i\right)\frac{e^{sz}}{\zeta(z)}ds,\\
F_m(z)&=\frac{1}{2\pi i}\int\limits_{1}^{1+i\infty}\left(\tan{\frac{\pi s}{2}}-i\right)\frac{e^{sz}}{\zeta(z)}ds
\end{align*}
are holomorphic in the half-plane $\Im{z}>-\pi.$
\end{thrm}
In this paper we prove analogous results for the M\"obius function of an elliptic curve over $\mathbb{Q}$ defined by the Weierstrass equation
$$E/\mathbb{Q}: y^2=x^3+ax+b, \quad a, b\in\mathbb{Q}.$$
Let $L(s, E)$ denote the $L$-function of $E$ (see for instance \cite{iwaniec1}, p.365-366). For $\sigma=\Re{s}>3/2$ we have
\begin{equation}\label{Lfunkcja}
L(s,E)=\prod_{p|N}\left(1-a_pp^{-s}\right)^{-1}\prod_{p\nmid N}\left(1-a_pp^{-s}+p^{1-2s}\right)^{-1},
\end{equation}
where $N$ is the conductor of $E.$ It is well-known that coefficients $a_p$ are real and for $p\not| N$ one has
$$a_p=p+1-\#E(\mathbb{F}_p),$$
\noindent
where $\#E(\mathbb{F}_p)$ denotes the number of points on $E$ modulo $p$ including the point at infinity, and $a_p\in\{-1, 0, 1 \},$ when $p|N$ (for details see \cite{iwaniec1}, p.365).
The M\"obius function of $E$ is defined as the sequence of the Dirichlet coefficients of the inverse of the shifted $L(s, E)$:
$$\frac{1}{L\left(s+\frac{1}{2}, E\right)}=\sum_{n=1}^\infty\frac{\mu_E(n)}{n^s}, \quad \sigma>1.$$
Using (\ref{Lfunkcja}) and the well-known Hasse inequality (see \cite{iwaniec1}, p.366, (14.32)) we easily show that $\mu_E$ is a multiplicative function satisfying Ramanujan's condition ($\mu_E(n)\ll n^\epsilon$ for every $\epsilon>0$), and moreover
$$\mu_E(p^k)=
\begin{cases}
-\frac{a_p}{\sqrt{p}}, &k=1 \\
1, &k=2 \text{ and } p\not|\Delta\\
0, &k\geq3 \text{ or } k=2 \text{ and } p|\Delta
\end{cases}$$
for every prime $p$ and positive integer $k$.
Furthermore, C. Breuil , B.Conrad, F.Diamond and R.Taylor, using the method pioneered by A. Wiles, proved in \cite{breuil1} that every $L-$function of an elliptic curve analytically continues to an entire function and satisfies the following functional equation
\begin{equation}\label{rownaniefunk}
\left(\frac{\sqrt{N}}{2\pi}\right)^s\Gamma(s)L(s, E)=\eta\left(\frac{\sqrt{N}}{2\pi}\right)^{2-s}\Gamma(2-s)L(2-s, E),
\end{equation}
where $\eta=\pm1$ is called the root number.
\noindent
In analogy to $m(z)$ we define $m(z, E)$ as follows:
$$m(z, E)=\frac{1}{2\pi i}\int_C\frac{1}{L\left(s+\frac{1}{2}, E\right)}e^{sz}ds,$$
where the path of integration consists of the half-line $s=-\frac{1}{4}+it, \infty>t\geq0$, the simple and smooth curve $l$ (which is parametrized by $\tau:[0,1]\rightarrow\mathbb{C}$ such that $\tau(0)=-\frac{1}{4}, \tau(1)=\frac{3}{2}, \Im{\tau(t)}>0$ for $t\in(0,1)$ and $F(s)$ has no zeros on $l$ and between $l$ and the real axis) and the half-line $s=\frac{3}{2}+it, 0\leq t<\infty.$
\noindent
Using (\ref{rownaniefunk}) and the Stirling's formula (see \cite{iwaniec1}, p.151, (5.112)) it is easy to see that $m(z, E)$ is holomorphic on the upper half-plane.
Our main goal in this paper is to prove the following results which are extensions of Theorems \ref{bartz} and \ref{kaczorowski}.
\begin{thrm}\label{maintheorem1}
The function $m(z, E)$ can be continued analytically to a meromorphic function on the whole complex plane and satisfies the following functional equation
\begin{equation}\label{mrownaniefunk} m(z, E)+\overline{m}(z, E)=-\frac{2\pi}{\eta\sqrt{N}}\sum_{n=1}^{\infty}\frac{\mu_E(n)}{n}J_1\left(\frac{4\pi}{\sqrt{Nn}}e^{-\frac{z}{2}}\right)-R(z),
\end{equation}
\noindent
where $R(z)=\sum \text{Res}_{s=\beta}\frac{e^{sz}}{L\left(s+\frac{1}{2}, E\right)}$ (summation is over real zeros of $L\left(s+\frac{1}{2}, E\right)$ in $(0,1)$, if there are any) and $J_1(z)$ denotes the Bessel function of the first kind
$$J_1(z)=\sum_{k=1}^\infty\frac{(-1)^k(z/2)^{2k+1}}{k!\Gamma(k+2)}.$$
\noindent
The only singularities of $m(z, E)$ are simple poles at the points $z=\log{n}, \mu_E(n)\neq0$ with the corresponding residues
$$\text{Res}_{z=\log{n}}m(z, E)=-\frac{\mu_E(n)}{2\pi i}.$$
\end{thrm}
Let $Y_1(z)$ be the Bessel function of the second kind and let
\begin{equation}\label{hankel}
H_1^{(2)}(z)=J_1(z)-iY_1(z)
\end{equation}
denote the classical Hankel function (see \cite{bateman2}, p.4). Moreover, let
\begin{align*}
R^{*}(z)&=\text{Res}_{s=\frac{1}{2}}\left(\tan{\pi s}\frac{e^{sz}}{L\left(s+\frac{1}{2}, E\right)}\right)+\sum\text{Res}\left(\tan{\pi s}\frac{e^{sz}}{L\left(s+\frac{1}{2}, E\right)}\right),\\
\intertext{summation is over real zeros of $L\left(s+\frac{1}{2}, E\right)$ in $(0,1)\setminus\left\{\frac{1}{2}\right\}$ (if there are any),}\\
m_0(z, E)&=\sum_{n=1}^\infty\frac{\mu_E(n)}{n^{\frac{3}{2}}}\frac{1}{z-\log n},\\
m_1(z, E)&=\frac{1}{2\pi i}\int_\textbf{\itshape{C}}\left(\tan{\pi s}-i\right)\frac{e^{sz}}{L\left(s+\frac{1}{2}, E\right)}ds,\\
H(z, E)&=\frac{1}{2\pi i}\int_\frac{3}{2}^{\frac{3}{2}+i\infty}\left(\tan{\pi s}-i\right)\frac{e^{sz}}{L\left(s+\frac{1}{2}, E\right)}ds.\\
\end{align*}
It is plain to see that $R(z)$ and $R^{*}(z)$ are entire functions, $m_0(z, E)$ is meromorphic on the whole plane, whereas $m_1(z, E)$ and $H(z, E)$ are holomorphic for $\Im{z}>-2\pi$. With this notation we have the following result.
\begin{thrm}\label{maintheorem2}
For $z=x+iy, |y|<2\pi, x\in\mathbb{R}, z\neq\log n, \mu_E(n)\neq0$ we have
\begin{equation}\label{formula}
\begin{split}
m(z, E)&=\frac{-\pi}{\eta\sqrt{N}}\sum_{n=1}^\infty\frac{\mu_E(n)}{n}\left(H_1^{(2)}\left(\frac{4\pi}{\sqrt{Nn}}e^{-\frac{z}{2}}\right)-\frac{2}{\pi}i\left(\frac{4\pi}{\sqrt{Nn}}e^{-\frac{z}{2}}\right)^{-1}\right)\\
&-\frac{1}{2}(R(z)-iR^{*}(z))+\frac{1}{2i}\left(H(z, E)+\overline{H}(z, E)\right)-\frac{e^{\frac{3}{2}z}}{2\pi i}m_0(z, E)\\
&-\frac{1}{2i}\left(m_1(z, E)+\overline{m_1}(z, E)\right).
\end{split}
\end{equation}
\end{thrm}
\section{An auxiliary lemma}
We need the following technical lemma.
\begin{lem}\label{oszacowanie}
Let $z=x+iy, y>0, s=Re^{i\theta}, R\sin{\theta}\geq1,\frac{\pi}{2}\leq\theta\leq\pi.$ Then for $R\geq R(x, y)$ we have
$$\left|\frac{e^{sz}}{L\left(s+\frac{1}{2}, E\right)}\right|\leq e^{-y\frac{R}{2}}.$$
\begin{proof}
Using (\ref{rownaniefunk}), the Stirling's formula and estimate \newline $\log{L\left(\sigma+it, E\right)}\ll\log(|t|+2),\quad |\sigma|\geq\frac{3}{2},\quad|t|\geq1$ (see \cite{perelli2}, p. 304) we obtain
\begin{equation}\label{rownosc}
\log\left|\frac{e^{sz}}{L\left(s+\frac{1}{2}, E\right)}\right|=\Re \log\frac{e^{sz}}{L\left(s+\frac{1}{2}, E\right)}=2R\log{R}\cos{\theta}+Rf(\theta, x, y)+O(\log{R}),
\end{equation}
where $f(\theta, x, y)=\left(x+2\log\frac{\sqrt{N}}{2\pi}-2\right)\cos\theta-(y+2\theta-\pi)\sin\theta.$
\noindent
For $\frac{\pi}{2}\leq\theta\leq\frac{\pi}{2}+\frac{1}{\sqrt{\log R}}$ we have
$$f(\theta, x, y)=-(y+2\theta-\pi)+O\left(\frac{1}{\sqrt{\log R}}\right)$$
\noindent
and hence
$$\log\left|\frac{e^{sz}}{L\left(s+\frac{1}{2}, E\right)}\right|\leq-\frac{yR}{2}.$$
For $\frac{\pi}{2}+\frac{1}{\sqrt{\log R}}\leq\theta\leq\pi$ we have
$$|\cos{\theta}|\gg\frac{1}{\sqrt{\log R}}$$
and consequently
$$\log\left|\frac{e^{sz}}{L\left(s+\frac{1}{2}, E\right)}\right|=-2|\cos{\theta}|R\log{R}+O(R)\leq-yR\leq-\frac{yR}{2}$$
for sufficently large $R$, and the lemma easily follows.
\end{proof}
\end{lem}
\section{Proof of Theorem \ref{maintheorem1}}
We shall first prove that $m(z, E)$ has meromorphic continuation to the whole complex plane.
\noindent
Let us write
\begin{equation}
\begin{split}\label{trzycalki}
2\pi i m(z, E)&=\int\limits_{-\frac{1}{4}+i\infty}^{-\frac{1}{4}}\frac{e^{sz}}{L\left(s+\frac{1}{2}, E\right)}ds+\int\limits_{l}\frac{e^{sz}}{L\left(s+\frac{1}{2}, E\right)}ds+\int\limits_{\frac{3}{2}}^{\frac{3}{2}+i\infty}\frac{e^{sz}}{L\left(s+\frac{1}{2}, E\right)}ds\\
&=n_1(z)+n_2(z)+n_3(z),\\
\end{split}
\end{equation}
say.
\noindent
Notice that $n_2(z)$ is an entire function.
\noindent
We compute $n_3(z)$ explicitly. Term by term integration gives
\begin{equation}
n_3(z)=-e^{\frac{3}{2}z}\sum_{n=1}^\infty\frac{\mu_E(n)}{n^\frac{3}{2}(z-\log{n})}.
\end{equation}
This shows that $n_3(z)$ is meromorphic on the whole complex plane and has simple poles at the points $z=\log{n}, \mu_E(n)\neq0$ with residues
\begin{equation}\label{residua}
\text{Res}_{z=\log{n}}n_3(z)=-\mu_E(n).
\end{equation}
\noindent
Let us now consider $n_1(z)$. Let $C_1$ consist of the half-line $s=\sigma+i, -\infty<\sigma\leq-\frac{1}{4}$ and the line segment $[-\frac{1}{4}+i, -\frac{1}{4}]$. Using lemma \ref{oszacowanie} we can write
$$n_1(z)=\int\limits_{C_1}\frac{e^{sz}}{L\left(s+\frac{1}{2}, E\right)}ds.$$
\noindent
Putting $s=\sigma+i,\quad\sigma\leq0$ in (\ref{rownosc}) we obtain
$$\left|\frac{e^{(\sigma+i)z}}{L\left(\frac{1}{2}+\sigma+i, E\right)}\right|\ll e^{-c_0|\sigma|\log(|\sigma|+2)},$$
hence $n_1(z)$ is an entire function.
\noindent
Then for $z\in\mathbb{C},$ $z\neq\log{n},$ $\mu_E(n)\neq0$ we have
\begin{equation}\label{malefunk}
m(z, E)+\overline{m}(z, E)=-\frac{1}{2\pi i}\int\limits_{\overline{C_1}\cup(-C_1)}\frac{e^{sz}}{L\left(s+\frac{1}{2}, E\right)}ds-R(z)
\end{equation}
and minus before contour denotes opposite direction.
\noindent
Using the equality (\ref{rownaniefunk}) we get
\begin{equation}
\frac{1}{2\pi i}\int\limits_{\overline{C_1}\cup(-C_1)}\frac{e^{sz}}{L\left(s+\frac{1}{2}, E\right)}ds=\frac{\pi}{\eta\sqrt{N}}\sum_{n=1}^\infty\frac{\mu_E(n)}{n}\cdot\frac{1}{2\pi i}\int\limits_{\overline{C_1}\cup(-C_1)}\frac{\Gamma\left(s+\frac{1}{2}\right)}{\Gamma\left(\frac{3}{2}-s\right)}\left(\frac{Nne^z}{(2\pi)^2}\right)^s ds.
\end{equation}
\noindent
The last integrand has simple poles at $s=-\frac{1}{2}, -\frac{3}{2}, -\frac{5}{2}, \ldots$. Computing residues we obtain
$$\frac{1}{2\pi i}\int\limits_{\overline{C_1}\cup(-C_1)}\frac{\Gamma\left(s+\frac{1}{2}\right)}{\Gamma\left(\frac{3}{2}-s\right)}\left(\frac{Nne^z}{(2\pi)^2}\right)^s ds=J_1\left(\frac{4\pi}{\sqrt{Nn}}e^{-\frac{z}{2}}\right).$$
\section{Proof of Theorem \ref{maintheorem2}}
\noindent
Let us now consider the function
$$m^{*}(z, E)=\frac{1}{2\pi i}\int\limits_\textbf{\itshape{C}}\tan(\pi s)\frac{e^{sz}}{L\left(s+\frac{1}{2}, E\right)}ds.$$
Using lemma \ref{oszacowanie} we can write
\begin{equation}
m^{*}(z, E)=\frac{1}{2\pi i}\left(\int\limits_{C_1\cup l} +\int\limits_{\frac{3}{2}}^{\frac{3}{2}+i\infty}\right)\tan{\pi s}\frac{e^{sz}}{L\left(s+\frac{1}{2}, E\right)}ds=m_a^*(z, E)+m_b^*(z, E).
\end{equation}
\noindent
Using again estimation
$$\left|\frac{e^{(\sigma+i)z}}{L\left(\frac{1}{2}+\sigma+i, E\right)}\right|\ll e^{-c_0|\sigma|\log(|\sigma|+2)},\quad\sigma\leq0$$
and $\tan(\pi(\sigma+i))\ll1$ it is easy to see that $m_a^*(z, E)$ is an entire function.
\noindent
Moreover
$$m_b^*(z, E)=H(z, E)-\frac{e^{\frac{3}{2}z}}{2\pi}m_0(z, E).$$
\noindent
This gives the meromorphic continuation of $m^*(z, E)$ to the half-plane $\Im{z}>-2\pi$ and $m^*(z, E)$ has poles at the points $\log{n}, n=1,2,3,\ldots, \mu_E(n)\neq0,$ with residues
$$\text{Res}_{s=\log{n}}m^*(z, E)=-\frac{\mu_E(n)}{2\pi}.$$
\noindent
Now we consider the function $\overline{m^*}(z, E)$ Changing $s$ to $\overline{s}$ we get
$$\overline{m^{*}}(z, E)=\frac{1}{2\pi i}\int_{-\overline{C}}\tan{\pi s}\frac{e^{sz}}{L\left(s+\frac{1}{2}, E\right)}ds, \quad \Im z<2\pi.$$
\noindent
Further we have
$$ \overline{m^{*}}(z, E)=\frac{1}{2\pi i}\int_{-\left(\overline{C_1}\cup\overline{l}\right)}\tan{\pi s}\frac{e^{sz}}{L\left(s+\frac{1}{2}, E\right)}ds+\overline{H}(z, E)-\frac{e^{\frac{3}{2}z}}{2\pi}m_0(z, E).$$
\noindent
Then for $|\Im(z)|<2\pi$ we have
\begin{equation}
m^{*}(z, E)+\overline{m^{*}}(z, E)=-J(z, E)-\frac{e^{\frac{3}{2}z}}{\pi}m_0(z, E)+H(z, E)+\overline{H}(z, E)-R^{*}(z),
\end{equation}
where
\begin{equation}
J(z, E)=\frac{1}{2\pi i}\int\limits_{\overline{C_1}\cup(-C_1)}\tan{\pi s}\frac{e^{sz}}{L\left(s+\frac{1}{2}, E\right)}ds
\end{equation}
\noindent
Using functional equation (\ref{rownaniefunk}) we get
\begin{equation*}
\begin{split}
&J(z, E)=\frac{1}{2\pi i}\int\limits_{\overline{C_1}\cup(-C_1)}\tan(\pi s)\frac{e^{sz}\Gamma\left(s+\frac{1}{2}\right)}{\eta\Gamma\left(\frac{3}{2}-s\right)L\left(\frac{3}{2}-s\right)}\left(\frac{\sqrt{N}}{2\pi}\right)^{2s-1}ds\\
&=\frac{-2\pi}{\eta\sqrt{N}}\sum_{n=1}^\infty\frac{\mu_E(n)}{n}\left(\frac{1}{2\pi i}\int\limits_{\overline{C_1}\cup(-C_1)}\frac{\Gamma\left(s+\frac{1}{2}\right)\Gamma\left(s-\frac{1}{2}\right)}{\Gamma(s)\Gamma(1-s)}\left(\frac{e^z Nn}{4\pi^2}\right)^sds\right).
\end{split}
\end{equation*}
\noindent
The last integral we can compute using inverse Mellin transform (see \cite{paris1}, p.407)
$$\frac{1}{2\pi i}\int\limits_{\overline{C_1}\cup(-C_1)}\frac{\Gamma\left(s+\frac{1}{2}\right)\Gamma\left(s-\frac{1}{2}\right)}{\Gamma(s)\Gamma(1-s)}\left(\frac{e^z Nn}{4\pi^2}\right)^s ds=-Y_1\left(\frac{4\pi}{\sqrt{Nn}}e^{-\frac{z}{2}}\right)-\frac{2}{\pi}\left(\frac{4\pi}{\sqrt{Nn}}e^{-\frac{z}{2}}\right)^{-1}.$$
\noindent
Therefore
$$J(z, E)=\frac{2\pi}{\eta\sqrt{N}}\sum_{n=1}^\infty\frac{\mu_E(n)}{n}\left(-Y_1\left(\frac{4\pi}{\sqrt{Nn}}e^{-\frac{z}{2}}\right)-\frac{2}{\pi}\left(\frac{4\pi}{\sqrt{Nn}}e^{-\frac{z}{2}}\right)^{-1}\right).$$
\noindent
For $x\in\mathbb{R}, x\neq\log{n}$ we have
\begin{equation}
\begin{split}
\Re(m^*(x, E))=\frac{\pi}{\eta\sqrt{N}}\sum_{n=1}^\infty\frac{\mu_E(n)}{n}\left(-Y_1\left(\frac{4\pi}{\sqrt{Nn}}e^{-\frac{x}{2}}\right)-\frac{2}{\pi}\left(\frac{4\pi}{\sqrt{Nn}}e^{-\frac{x}{2}}\right)^{-1}\right)\\
-\frac{e^{\frac{3}{2}x}}{2\pi}m_0(x, E)+\frac{1}{2}\left(H(x, E)+\overline{H}(x, E)\right)-\frac{1}{2}R^{*}(x).
\end{split}
\end{equation}
Obviously
$$m^*(z, E)=im(z, E)+m_1(z, E),$$
therefore we get
\begin{equation}\label{urojone}
\begin{split}
\Im(m(x, E))=&-\frac{\pi}{\eta\sqrt{N}}\sum_{n=1}^\infty\frac{\mu_E(n)}{n}\left(-Y_1\left(\frac{4\pi}{\sqrt{Nn}}e^{-\frac{x}{2}}\right)-\frac{1}{\pi}\frac{e^{\frac{x}{2}}\sqrt{Nn}}{2\pi}\right)\\
&+\frac{e^{(\frac{3}{2}x}}{2\pi}m_0(x, E)-\frac{1}{2}\left(H(x, E)+\overline{H}(x, E)\right)\\
&+\frac{1}{2}\left(m_1(x, E)+\overline{m_1}(x, E)\right)+\frac{1}{2}R^{*}(x).
\end{split}
\end{equation}
On the other hand
\begin{equation}\label{rzeczywiste}
\Re(m(x, E))=-\frac{\pi}{\eta\sqrt{N}}\sum_{n=1}^\infty\frac{\mu_E(n)}{n}J_1\left(\frac{4\pi}{\sqrt{Nn}}e^{-\frac{x}{2}}\right)-\frac{1}{2}R^(x).
\end{equation}
The equations (\ref{urojone}) and (\ref{rzeczywiste}) imply the formula for $z\in\mathbb{R},\quad z\neq\log{n},\newline \mu_E(n)\neq0,$ and by the analytic continuation, formula (\ref{formula}) is valid in the strip $|\Im{z}|<2\pi.$
\proof[Acknowledgements]
This paper is a part of my PhD thesis. I thank my thesis advisor Prof. Jerzy Kaczorowski for
suggesting the problem and helpful discussions.
\vspace{3cm}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,480 |
// Copyright 2010 Google Inc.
//
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
// http://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.
package com.google.enterprise.connector.otex;
import com.google.common.annotations.VisibleForTesting;
import com.google.enterprise.connector.otex.CacheMap.CacheStatistics;
import com.google.enterprise.connector.otex.client.Client;
import com.google.enterprise.connector.otex.client.ClientValue;
import com.google.enterprise.connector.spi.RepositoryException;
import java.util.ArrayList;
import java.util.HashSet;
import java.util.List;
import java.util.Set;
import java.util.StringTokenizer;
import java.util.logging.Level;
import java.util.logging.Logger;
/** A class that knows about the node hierarchy in DTree. */
class Genealogist {
/** The logger for this class. */
private static final Logger LOGGER =
Logger.getLogger(Genealogist.class.getName());
/**
* Duplicates the entries to include their negations: (a) becomes (a,-a).
*
* @return a comma-separated string of the entries and their negations
* @see #getAncestorSet
*/
public static String getAncestorNodes(String startNodes) {
// Projects, Discussions, Channels, and TaskLists have a rather
// strange behavior. Their contents have a VolumeID that is the
// same as the container's ObjectID, and an AncestorID that is the
// negation the container's ObjectID. To catch that, I am going
// to create a superset list that adds the negation of everything
// in the specified list. We believe this is safe, as negative
// values of standard containers (folders, compound docs, etc)
// simply should not exist, so we shouldn't get any false
// positives.
StringBuilder buffer = new StringBuilder();
StringTokenizer tok = new StringTokenizer(startNodes, ":;,. \t\n()[]\"\'");
while (tok.hasMoreTokens()) {
String objId = tok.nextToken();
if (buffer.length() > 0)
buffer.append(',');
buffer.append(objId);
try {
int intId = Integer.parseInt(objId);
buffer.append(',');
buffer.append(-intId);
} catch (NumberFormatException e) {}
}
return buffer.toString();
}
/**
* Duplicates the entries to include their negations: (a) becomes (a,-a).
*
* @return a set of the entries and their negations
* @see #getAncestorNodes
*/
private static Set<Integer> getAncestorSet(String nodes) {
HashSet<Integer> set = new HashSet<Integer>();
if (nodes == null || nodes.length() == 0)
return set;
StringTokenizer tok = new StringTokenizer(nodes, ":;,. \t\n()[]\"\'");
while (tok.hasMoreTokens()) {
String objId = tok.nextToken();
try {
int intId = Integer.parseInt(objId);
set.add(intId);
set.add(-intId);
} catch (NumberFormatException e) {
}
}
return set;
}
/**
* Gets a new instance of the given Genealogist class.
*
* @param className the {@code Genealogist} class to instantiate
* @param matching the candidates matching the non-hierarchical filters
* @param startNodes the includedLocationNodes property value
* @param excludedNodes the excludedLocationNodes property value
* @param minCacheSize the initial size of the node cache
* @param maxCacheSize the maximum size of the node cache
* @return a new instance of the configured Genealogist class
* @throws RepositoryException if the class cannot be instantiated
* or initialized
*/
public static Genealogist getGenealogist(String className, Client client,
String startNodes, String excludedNodes, int minCacheSize,
int maxCacheSize) throws RepositoryException {
Genealogist genealogist;
try {
genealogist = Class.forName(className)
.asSubclass(Genealogist.class)
.getConstructor(Client.class, String.class, String.class, int.class,
int.class)
.newInstance(client, startNodes, excludedNodes, minCacheSize,
maxCacheSize);
} catch (Exception e) {
throw new LivelinkException(e, LOGGER);
}
return genealogist;
}
/** The logging level for orphan nodes in the Livelink database. */
protected static final Level LOG_ORPHANS_LEVEL = Level.WARNING;
/** The Livelink client to use to execute SQL queries. */
protected final Client client;
/** The SQL queries resource bundle wrapper. */
protected final SqlQueries sqlQueries;
/** A set based on the includedLocationNodes property value. */
private final Set<Integer> includedSet;
/** A set based on the excludedLocationNodes property value. */
private final Set<Integer> excludedSet;
/** A cache of items known to be included. */
@VisibleForTesting
final Cache<Integer> includedCache;
/** A cache of items known to be excluded. */
@VisibleForTesting
final Cache<Integer> excludedCache;
/** For logging statistics, the number of nodes processed by this instance.*/
protected int nodeCount = 0;
/** For logging statistics, the number of queries run by this instance.*/
protected int queryCount = 0;
public Genealogist(Client client, String startNodes, String excludedNodes,
int minCacheSize, int maxCacheSize) {
this.client = client;
// TODO(jlacey): We're saying this is SQL Server, because we aren't
// wired to Connector.isSqlServer. All the genealogist queries are
// DB agnostic at the moment, so this is (just barely) OK.
this.sqlQueries = new SqlQueries(true);
this.includedSet = getAncestorSet(startNodes);
this.excludedSet = getAncestorSet(excludedNodes);
// If the included set is empty, then everything that hits the
// top-level (ParentID = -1) without being excluded should be
// included. If the included set is not empty, then everything
// that hits the top-level without being included should excluded.
(includedSet.isEmpty() ? includedSet : excludedSet).add(-1);
if (LOGGER.isLoggable(Level.FINEST)) {
LOGGER.finest("DESCENDANTS: includedSet = " + includedSet);
LOGGER.finest("DESCENDANTS: excludedSet = " + excludedSet);
LOGGER.finest("DESCENDANTS: minCacheSize = " + minCacheSize);
LOGGER.finest("DESCENDANTS: maxCacheSize = " + maxCacheSize);
}
this.excludedCache = new Cache<Integer>(minCacheSize, maxCacheSize);
this.includedCache = new Cache<Integer>(minCacheSize, maxCacheSize);
}
/**
* Finds the included nodes from among the matching candidates. This
* is the core algorithm behind {@link getMatchingDescendants}. This
* implementation looks up each parent for each node individually.
*
* @param matching the matching nodes to check for inclusion
* @param descendants a buffer to write a comma-separated list of
* included node IDs to
*/
protected void matchDescendants(ClientValue matching,
StringBuilder descendants) throws RepositoryException {
for (int i = 0; i < matching.size(); i++) {
// We do not cache the matches, which are probably mostly documents.
ArrayList<Integer> cachePossibles = new ArrayList<Integer>();
final int matchingId = matching.toInteger(i, "DataID");
Integer parentId = matchingId;
// TODO: Check for an interrupted traversal in this loop?
while (!matchParent(matchingId, parentId, cachePossibles, descendants)) {
parentId = getParent(matchingId, parentId);
if (parentId == null) {
break;
}
cachePossibles.add(parentId);
}
}
}
/**
* Matches the given parent against the included and excluded nodes.
*
* @param matchingId the original node or nodes
* @param parentId the current ancestor node to match
* @param cachePossibles the ancestors between the two
* @param descendants a buffer to write the {@code matchingId} and
* a trailing comma to if the node should be included
* @return {@code true} if a match was found, whether it is included
* or excluded, or {@code false} if nothing is known about
* {@code parentId}
*/
protected final boolean matchParent(Object matchingId, int parentId,
List<Integer> cachePossibles, StringBuilder descendants) {
// We add the cachePossibles to the appropriate cache when we
// determine an answer for matchingId. In the case of a cache
// hit, parentID is obviously already in the cache, but the
// other possibles are not.
if (excludedCache.contains(parentId)) {
if (LOGGER.isLoggable(Level.FINEST)) {
LOGGER.finest("DESCENDANTS: Excluding " + matchingId
+ ", found " + parentId + " in cache after " + cachePossibles);
}
excludedCache.addAll(cachePossibles);
return true;
} else if (excludedSet.contains(parentId)) {
if (LOGGER.isLoggable(Level.FINEST)) {
LOGGER.finest("DESCENDANTS: Excluding " + matchingId
+ ", found " + parentId + " in set after " + cachePossibles);
}
excludedCache.addAll(cachePossibles);
return true;
} else if (includedCache.contains(parentId)) {
if (LOGGER.isLoggable(Level.FINEST)) {
LOGGER.finest("DESCENDANTS: Including " + matchingId
+ ", found " + parentId + " in cache after " + cachePossibles);
}
includedCache.addAll(cachePossibles);
descendants.append(matchingId).append(',');
return true;
} else if (includedSet.contains(parentId)) {
if (LOGGER.isLoggable(Level.FINEST)) {
LOGGER.finest("DESCENDANTS: Including " + matchingId +
", found " + parentId + " in set after " + cachePossibles);
}
includedCache.addAll(cachePossibles);
descendants.append(matchingId).append(',');
return true;
} else {
return false;
}
}
/**
* Gets the ParentID of the given node.
*
* @param matchingId the original descendent node or nodes
* @param objectID the object ID of the node
* @return the parent ID of the node
*/
protected Integer getParent(Object matchingId, int objectId)
throws RepositoryException {
// Get the parent of the current node, and also check for the
// parent of the related node (with a negated object ID) in the
// case of volumes such as projects. The query can return 0, 1, or
// 2 results, since we are asking for the parents of two nodes,
// one or both of which might not exist.
queryCount++;
ClientValue parent = sqlQueries.execute(client, null,
"Genealogist.getParent", objectId, -objectId);
switch (parent.size()) {
case 0:
// The nodes do not exist.
logOrphans(matchingId, objectId);
return null;
case 1:
// The usual case.
return parent.toInteger(0, "ParentID");
case 2:
// Both nodes exist, so one is a volume with a parent of -1.
// Return the other one.
if (parent.toInteger(0, "ParentID") != -1) {
return parent.toInteger(0, "ParentID");
} else {
return parent.toInteger(1, "ParentID");
}
default:
throw new AssertionError(String.valueOf(parent.size()));
}
}
/**
* Logs a warning for orphans discovered in the hierarchy.
*
* @param orphans the discovered descendants of the missing node
* @param parentId the missing node
*/
protected final void logOrphans(Object orphans, int parentId) {
// We've already detected an orphan at this point, which is very
// rare, so don't bother checking the logging level.
LOGGER.log(LOG_ORPHANS_LEVEL, "DESCENDANTS: Excluding " + orphans
+ ", ancestor " + parentId + " does not exist.");
}
/**
* Finds the included nodes from among the matching candidates.
*
* @param matching the matching nodes to check for inclusion
* @return a comma-separated list of included node IDs to
*/
public final String getMatchingDescendants(ClientValue matching)
throws RepositoryException {
StringBuilder descendants = new StringBuilder();
matchDescendants(matching, descendants);
if (LOGGER.isLoggable(Level.FINEST)) {
nodeCount += matching.size();
LOGGER.finest("DESCENDANTS: Query statistics: " + nodeCount + " nodes, "
+ queryCount + " queries");
LOGGER.finest("DESCENDANTS: Excluded cache statistics: "
+ excludedCache.statistics());
LOGGER.finest("DESCENDANTS: Included cache statistics: "
+ includedCache.statistics());
}
if (descendants.length() > 0) {
descendants.deleteCharAt(descendants.length() - 1);
if (LOGGER.isLoggable(Level.FINER))
LOGGER.finer("DESCENDANTS: Matching descendants: " + descendants);
return descendants.toString();
} else {
LOGGER.finer("DESCENDANTS: No matching descendants.");
return null;
}
}
/* Used for testing and instrumentation. */
public class Statistics {
public final int nodeCount; // Number of nodes processed by this instance.
public final int queryCount; // Number of queries run by this instance.
public final CacheStatistics includedStats; // Statistics for includedCache.
public final CacheStatistics excludedStats; // Statistics for excludedCache.
public Statistics(int nodeCount, int queryCount,
CacheStatistics includedStats, CacheStatistics excludedStats) {
this.nodeCount = nodeCount;
this.queryCount = queryCount;
this.includedStats = includedCache.statistics();
this.excludedStats = excludedCache.statistics();
}
public String toString() {
return "nodes: " + nodeCount + ", queries: " + queryCount
+ ", includedCache: ( " + includedStats + " ), excludedCache: ( "
+ excludedStats + " )";
}
}
/**
* Returns a structure that contains the accumulated Genealogist and Cache
* statistics.
*
* @return a Genealogist.Statistics structure.
*/
/* Used for testing and instrumentation. */
public synchronized Statistics statistics() {
return new Statistics(nodeCount, queryCount, includedCache.statistics(),
excludedCache.statistics());
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,588 |
Q: How to reference ID property in relationship with NHibernate? How do I map relationship, where child endpoint is exposed via Id property and not via whole Parent object?
Here is the example:
class Parent {
public Guid Id { get; set; }
public List<Child> Chlidren { get; set; }
}
class Child {
public Guid Id { get; set; }
public Guid ParentId { get; set; }
}
Here are the equivalent mappings I'm using:
<?xml version="1.0" encoding="utf-8" ?>
<hibernate-mapping xmlns="urn:nhibernate-mapping-2.2"
assembly="Blabla"
namespace="Blabla"
auto-import="false">
<typedef name="ChildrenList" class="Blabla" />
<class name="Parent" table="Parent" lazy="false">
<id name="Id" column="ID" type="Guid">
<generator class="guid" />
</id>
<bag name="Children" table="Child"
cascade="save-update"
collection-type="ChildrenList"
lazy="false">
<key column="ParentID" not-null="true" />
<one-to-many class="Child" />
</bag>
</class>
<class name="Child" table="Child" lazy="false">
<id name="Id" column="ID" type="Guid">
<generator class="guid" />
</id>
<!-- How to map ParentID here? -->
</class>
</hibernate-mapping>
When I create a parent, add some children to Children collection and then save the parent, everything is fine. But if save a parent object first, then create a child, setting its ParentID property to ID of the parent, then I get
NHibernate.PropertyValueException:
not-null property references a null or transient value Child._Parent.ChildrenBackref
All attempts to map many-to-one relationship resulted in different exceptions while creating NHibernate configuration. Mostly about object type mismatch.
I'm sure NHibernate is capable to handle this scenario. There must something fairly basic that I miss.
EDIT:
I think it make sense to the example test, which fails with above exception:
var child = new Child(Create.Saved<Parent>().Id); // this sets the ParentId property
this.Repository.Save(child); // here I get the exception
My thoughts why NHibernate is raising this: Children property of Parent class mapped in a way that says that a child cannot exist without a parent (<key column="ParentID" not-null="true" />). When I try to persist a child, NHibernate tries to resolve this relationship (to find a parent this child relates to) and fails, since being given no child endpoint (which otherwise would be ParentId property) in the mapping, it check for its own Child._Parent.ChildrenBackref endpoint, whatever it is.
This looks like a desired solution: Mapping ParentId property as child endpoint of the relationship. This would force NHibernate to resolve a parent by using value of ParentId property as parent's primary key.
The thing is I don't know if it's possible.
A: The one-to-many / many-to-one relationships you have in NHibernate always needs to have a dominant side (i.e. the side that manages the "saving").
<bag name="Children" table="Child"
cascade="save-update"
collection-type="ChildrenList"
lazy="false">
<key column="ParentID" not-null="true" />
<one-to-many class="Child" />
</bag>
The above is a one-to-many relationship where the dominant side is the parent. That means, you save the parent ... and that will save the parent first, then, the children (with the ParentId being null), then a subsequent update will be issued to set the child.ParentId.
Note:
The child is inserted first with ParentId=null ... if you have a db or mapping restriction to say ParentId cannot be null, this action will fail.
<bag name="Children" table="Child"
cascade="save-update"
collection-type="ChildrenList"
lazy="false"
inverse=true>
<key column="ParentID" not-null="true" />
<one-to-many class="Child" />
</bag>
Note the inverse=true attribute. This means the child object is dominant in the relationship, meaning the child object is in charge. The parent will be inserted, then the Id will be assiged to the child.ParentId, and then the child will be inserted with the ParentId already set.
In many cases, of course, you want to go either way. The easiest way to do this is to manage the relationship on both ends (unfortunately, you have to do this yourself).
On the Parent, you have a method:
public void AddChild(Child child)
{
Children.Add(child);
child.ParentId = Id;
}
public void RemoveChild(Child child)
{
Children.Remove(child);
child.ParentId = null;
}
On the Child, you have a method:
public void SetParent(Parent parent)
{
ParentId = parent.Id;
parent.Children.Add(this);
}
Using these methods to Add/Remove/Set, both sides are consistent after the action is performed. It, then, wouldn't matter whether you set inverse=true on the bag or not.
see http://www.nhforge.org/doc/nh/en/index.html#collections-example
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,833 |
\subsubsection*{Note added:}
The recent preprint \cite {AGS} discusses the same ${\mathbb Z}_2$ projection
of super-Yang-Mills theory, and examines a number of physical
quantities which can be used as consistency tests of large $N$ equivalence
between parent and daughter theories.
The authors of Ref.~\cite {AGS} argue that all their tests fail,
and therefore assert that
spontaneous breaking of ${\mathbb Z}_2$ symmetry must occur in the daughter theory.
However, the apparent inconsistencies noted in Ref.~\cite {AGS}
are all consequences of incorrect mappings of observables and/or
connected correlators between parent and daughter theories.
When the correct mappings are used, these apparent inconsistencies
disappear.%
\footnote
{
The supposed mis-match in the vacuum structure at $\theta_p = 2\pi$,
discussed in section 2 of Ref.~\cite {AGS},
is a consequence of overlooking the double-valued nature of
the mapping from $\theta_p$ to $\theta_d$.
With a non-zero mass turned on,
it is clear that properties in the unique ground state
of the parent theory at $\theta_p = 2\pi$
correspond to properties in the unique ground state
of the daughter theory at $\theta_d = 0$, not at $\theta_d = \pi$.
The assertion (on pg.~7) that ``the partition functions for
parent and daughter must coincide at large $N$'' is also incorrect.
Both partition functions diverge exponentially as $N \to \infty$.
The correct statement of equivalence for the partition functions is
that the ratio of their free energies must approach two (not one)
as $N \to \infty$, or
$
\lim_{N\to\infty} (2N)^{-2} \ln Z_p
=
\lim_{N\to\infty} (2N^2)^{-1} \ln Z_d
$.
The claimed inconsistency in
gravitational contributions to the axial anomaly,
discussed in section 3 of Ref.~\cite {AGS},
is a result of the use of incorrect mappings between
axial currents, stress-energy tensors,
and topological charge densities.
As discussed in section \ref{sec:anomalies} of this paper,
when the correct relations are used
the chiral anomaly in the parent theory
(including both gauge and gravitational contributions)
is properly mapped to the chiral anomaly in the daughter theory,
as it must be for purely perturbative equivalence to be valid.
The discussion in section 4 of Ref.~\cite {AGS}
of $\langle {\rm tr}(F^1 F^1 + F^2 F^2) \rangle$ as an order parameter
for the massless theory whose non-zero value, if $O(N)$ [not $O(N^2)$],
would signal failure of large $N$ equivalence and hence
imply ${\mathbb Z}_2$ symmetry breaking in the daughter theory is fine ---
except that there is no evidence that this expectation
value, suitably renormalized, is non-vanishing at $O(N)$.
The comparison of domain wall tensions in section 5
asserts that ${\rm tr} \, FF$ maps to ${\rm tr}(F^1 F^1 + F^2 F^2)/\sqrt 2$,
and then asserts that valid large $N$ equivalence requires
that parent and daughter domain wall tensions coincide.
Neither is correct;
${\rm tr} \, FF$ maps to ${\rm tr}(F^1 F^1 + F^2 F^2)$ with no $\sqrt 2$,
and the tension in the parent theory should be compared
with twice the tension of the corresponding wall in the daughter,
as explained in section \ref{sec:walls} above.
Once one corrects the mis-understanding regarding coinciding
vacuum structure in parent and daughter theories, we fail to
see any basis for concluding that domain walls in the
daughter theory can split up into ``fractional domain walls.''
The same mis-understanding regarding the appropriate mappings
of ${\rm tr}(FF)$ and ${\rm tr}(F\tilde F)$, and of vacuum energies
(namely ${\cal E}_{\rm (p)} \to 2 \, {\cal E}_{\rm (d)}$)
are responsible for the apparent inconsistencies presented
in section 6 and the appendix of Ref.~\cite {AGS}.
In summary, the assertions in Ref.~\cite{AGS} that
``Ample evidence ... establishes ... nonperturbative nonequivalence''
and that
``this is the first example of spontaneous breaking of a discrete
symmetry in a strongly coupled gauge theory ever established
analytically in four dimensions'' are unfounded.
Spontaneous breaking of the ${\mathbb Z}_2$ theory space symmetry of
the daughter theory (in infinite volume) produced by a ${\mathbb Z}_2$
projection of ${\cal N}\,{=}\,1$ super-Yang-Mills
remains a logical possibility --- but everything known about
this theory, so far, is consistent with the absence of such
symmetry breaking, and hence with the validity of large $N$
equivalence in this case.
}
\vfill
\begin{acknowledgments}
This work was supported, in part,
by the U.S. Department of Energy under Grant Nos.~DE-FG02-96ER40956
and DE-FG02-91ER40676,
and the National Science Foundation under Grant No.~PHY99-07949.
\end{acknowledgments}
\begin {thebibliography}{99}
\bibitem{Bershadsky-Johansen}
M.~Bershadsky and A.~Johansen,
{\it Large N limit of orbifold field theories,}
\npb{536}{1998}{141},
\hepth{9803249}.
\bibitem{Schmaltz}
M.~Schmaltz,
{\it Duality of non-supersymmetric large N gauge theories,}
\prd{59}{1999}{105018},
\hepth{9805218}.
\bibitem{Strassler}
M.~J.~Strassler,
{\it On methods for extracting exact non-perturbative results in
non-supersymmetric gauge theories,}
\hepth{0104032}.
\bibitem{Dijkgraaf-Neitzke-Vafa}
R.~Dijkgraaf, A.~Neitzke and C.~Vafa,
{\it Large N strong coupling dynamics in
non-supersymmetric orbifold field theories,}
\hepth{0211194}.
\bibitem{KUY1}
P.~Kovtun, M.~\"Unsal and L.~G.~Yaffe,
{\it Non-perturbative equivalences among large $N_c$ gauge theories
with adjoint and bifundamental matter fields,}
\jhep{0312}{2003}{034},
{\tt hep-th/0311098}.
\bibitem{KUY2}
P.~Kovtun, M.~\"Unsal and L.~G.~Yaffe,
{\it Necessary and sufficient conditions for non-perturbative
equivalences of large $N_{\rm c}$ orbifold gauge theories,}
{\tt hep-th/0411177}.
\bibitem{Gorsky-Shifman}
A.~Gorsky and M.~Shifman,
{\it Testing nonperturbative orbifold conjecture,}
\prd{67}{2003}{022003},
\hepth{0208073}.
\bibitem{Tong}
D.~Tong,
{\it Comments on condensates in non-supersymmetric
orbifold field theories,}
\jhep{0303}{2003}{022},
\hepth{0212235}.
\bibitem{ASV}
A.~Armoni, M.~Shifman and G.~Veneziano,
{\it From super-Yang-Mills theory to QCD: planar equivalence and its
implications,}
\hepth{0403071}.
\bibitem{Erlich-Naqvi}
J.~Erlich and A.~Naqvi,
{\it Nonperturbative tests of the parent/orbifold correspondence
in supersymmetric gauge theories,}
\jhep{0212}{2002}{047},
\hepth{9808026}.
\bibitem{Tseytlin:1999ii}
A.~A.~Tseytlin and K.~Zarembo,
{\it Effective potential in non-supersymmetric $SU(N) \times SU(N)$
gauge theory and interactions of type 0 D3-branes,}
\plb {457}{1999}{77},
\hepth{9902095}.
\bibitem{Adams:2001jb}
A.~Adams and E.~Silverstein,
{\it Closed string tachyons, AdS/CFT, and large N QCD,}
\prd{64}{2001}{086001},
\hepth{0103220}.
\bibitem{Klebanov:1999ch}
I.~R.~Klebanov and A.~A.~Tseytlin,
{\it A non-supersymmetric large N CFT from type 0 string theory,}
\jhep {9903}{1999}{015},
\hepth{9901101};
\bibitem{Nekrasov:1999mn}
N.~Nekrasov and S.~L.~Shatashvili,
{\it On non-supersymmetric CFT in four dimensions,}
\prep{320}{1999}{127},
\hepth{9902110}.
\bibitem{Klebanov:1999um}
I.~R.~Klebanov,
{\it Tachyon stabilization in the AdS/CFT correspondence,}
\plb{466}{1999}{166},
\hepth{9906220}.
\bibitem{Dymarsky:2005uh}
A.~Dymarsky, I.~R.~Klebanov and R.~Roiban,
{\it Perturbative search for fixed lines in large N gauge theories,}
\hepth{0505099}.
\bibitem{Witten}
E.~Witten,
{\it Branes and the dynamics of {QCD},}
\npb{507}{1997}{658},
\hepth{9706109}.
\bibitem{Elitzur}
S.~Elitzur,
{\it Impossibility of spontaneously breaking local symmetries,}
\prd{12}{1975}{3978},
\bibitem{F&S}
See, for example,
K.~Fujikawa and H.~Suzuki,
{\it Path integrals and quantum anomalies,}
Oxford, UK: Clarendon (2004).
\bibitem{Dvali}
G.~R.~Dvali and M.~A.~Shifman,
{\it Domain walls in strongly coupled theories,}
\plb{396}{1997}{64};
[Erratum-ibid.\ {\bf B 407} (1997) 452],
\hepth{9612128}.
\bibitem{AGS}
A.~Armoni, A.~Gorsky and M.~Shifman,
{\it Spontaneous Z(2) symmetry breaking in the orbifold daughter of
${\cal N} = 1$
super-Yang-Mills theory, fractional domain walls and vacuum structure,}
\hepth{0505022}.
\end {thebibliography}
\end {document}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,467 |
VxCage
======
VxCage is a WSGI Python application for managing a malware samples repository with a REST API interface.
The successor of VxCage is [Viper](http://viper.li/) and is also available on [github](https://github.com/botherder/viper)
Requirements
------------
In order to install VxCage you need to have Python (2.7) installed. Following are the required libraries.
* [bottle.py](http://www.bottlepy.org/) -- `pip install bottle`
* [sqlalchemy](http://www.sqlalchemy.org) -- `pip install sqlalchemy`
If you want to enable the fuzzy hash, you need to install [ssdeep](http://ssdeep.sourceforge.net/) and the Python bindings, [pydeep](https://github.com/kbandla/pydeep).
VxCage also requires any database engine from the [ones supported](http://docs.sqlalchemy.org/en/rel_0_7/core/engines.html). Depending to which one you pick, you'll need the required Python API. For example, in the case of MySQL you'll also need [MySQLdb](http://mysql-python.sourceforge.net/) (`pip install mysqldb`).
If you plan to run VxCage with Apache, you'll need to have mod_wsgi installed. On Ubuntu/Debian systems ``apt-get install libapache2-mod-wsgi``.
Installation
------------
First thing first, extract VxCage to your selected location and open `api.conf` and configure the path to the local folder you want to use as a storage.
You also need to configure the connection string for your database. For example:
MySQL:
mysql://user:pass@host/database
SQLite:
sqlite:///database.db
PostgreSQL:
postgresql://user:pass@host/database
Refer to [SQLAlchemy](http://docs.sqlalchemy.org/en/latest/core/engines.html)'s documentation for additional details.
Now proceeds installing Apache and required modes:
# apt-get install apache2 libapache2-mod-wsgi
Enable the mod:
# a2enmod wsgi
If you want to enable SSL, you need to generate a certificate with OpenSSL or buy one from a certified authority.
You can also use the `make-ssl-cert` utility as following:
# make-ssl-cert /usr/share/ssl-cert/ssleay.cnf /path/to/apache.pem
Now create a virtual host for the domain you want to host the application on. We'll enable WSGI, SSL and a basic authentication.
A valid template is the following:
<VirtualHost *:443>
ServerName yourwebsite.tld
WSGIDaemonProcess yourapp user=www-data group=www-data processes=1 threads=5
WSGIScriptAlias / /path/to/app.wsgi
<Directory /path/to/app.wsgi>
WSGIProcessGroup yourgroup
WSGIApplicationGroup %{GLOBAL}
Order deny,allow
Allow from all
</Directory>
<Location />
AuthType Basic
AuthName "Authentication Required"
AuthUserFile "/path/to/users"
Require valid-user
</Location>
SSLEngine on
SSLCertificateFile /path/to/apache.pem
ErrorLog /path/to/error.log
LogLevel warn
CustomLog /path/to/access.log combined
ServerSignature Off
</VirtualHost>
Now add your user:
# htpasswd -c /path/to/users username
You should be ready to go. Make sure to restart Apache afterwards:
# /etc/init.d/apache2 restart
For testing purposes, you can also run it with the Bottle.py server just doing:
$ python api.py
Usage
-----
You can interact with your repository with the provided REST API.
Submit a sample:
$ curl -F file=@sample.exe -F tags="tag1 tag2" http://yourdomain.tld/malware/add
Retrieve a sample:
$ curl http://yourdomain.tld/malware/get/<sha256> > sample.exe
Find a sample by MD5:
$ curl -F md5=<md5> http://yourdomain.tld/malware/find
Find a sample by SHA-256:
$ curl -F sha256=<sha256> http://yourdomain.tld/malware/find
Find a sample by Ssdeep (can also search for a substring of the ssdeep hash):
$ curl -F ssdeep=<pattern> http://yourdomain.tld/malware/find
Find a sample by Tag:
$ curl -F tag=<tag> http://yourdomain.tld/malware/find
List existing tags:
$ curl http://yourdomain.tld/tags/list
In case you added a basic authentication, you will need to add `--basic -u "user:pass"`. In case you added SSL support with a generated certificate, you will need to add `--insecure` and obviously make the requests to https://yourdomain.tld.
Console
-------
You can also easily interact with your VxCage server using the provided console interface.
In order to run it, you'll need the following dependencies:
* [requests](http://www.python-requests.org) -- `pip install requests`
* [prettytable](http://code.google.com/p/prettytable/) -- `pip install prettytable`
* [progressbar](http://code.google.com/p/python-progressbar/) -- `pip install progressbar`
This is the help message:
usage: vxcage.py [-h] [-H HOST] [-p PORT] [-s] [-a]
optional arguments:
-h, --help show this help message and exit
-H HOST, --host HOST Host of VxCage server
-p PORT, --port PORT Port of VxCage server
-s, --ssl Enable if the server is running over SSL
-a, --auth Enable if the server is prompting an HTTP
authentication
As you can see, you can specify the `host`, the `port` and enable SSL and HTTP authentication.
For example, you can launch it simply with:
$ python vxcage.py --host yourserver.com --port 443 --ssl --auth
You will be prompted with:
`o O o O .oOo .oOoO' .oOoO .oOo.
O o OoO O O o o O OooO'
o O o o o o O O o O
`o' O O `OoO' `OoO'o `OoOo `OoO'
O
OoO' by nex
Username: nex
Password:
vxcage>
Now you can start typing commands, you can start with:
vxcage> help
Available commands:
help Show this help
tags Retrieve list of tags
find Find a file by md5, sha256, ssdeep, tag or date
get Retrieve a file by sha256
add Upload a file to the server
You can retrieve the list of available tags:
vxcage> tags
+------------------------+
| tag |
+------------------------+
| banker |
| bot |
| carberp |
| citadel |
| zeus |
+------------------------+
Total: 5
You can search for all samples matching a specific tag:
vxcage> find tag carberp
+----------------------------------+------------------------------------------------------------------+--------------+---------------------------------------------------+-----------+
| md5 | sha256 | file_name | file_type | file_size |
+----------------------------------+------------------------------------------------------------------+--------------+---------------------------------------------------+-----------+
| 719354b4b7b182b30e1de8ce7b417d2f | 689a35928f71848fab346b50811c6c0aab95da01b9293c60d74c7be1357dc029 | carberp1.exe | PE32 executable (GUI) Intel 80386, for MS Windows | 132096 |
| 63d8fd55ebe6e2fa6cc9523df942a9a5 | a6d77a5ba2b5b46a0ad85fe7f7f01063fe7267344c0cecec47985cd1e46fa7a4 | carberp2.exe | PE32 executable (GUI) Intel 80386, for MS Windows | 192512 |
| ccf43cdc957d09ea2c60c6f57e4600f0 | b998233b85af152596f5087e64c2cadb1466e4f6da62f416ac3126f87c364276 | carberp3.exe | PE32 executable (GUI) Intel 80386, for MS Windows | 186880 |
+----------------------------------+------------------------------------------------------------------+--------------+---------------------------------------------------+-----------+
Total: 3
You can view details on a specific sample:
vxcage> find md5 719354b4b7b182b30e1de8ce7b417d2f
sha1: 091fcf7378bfc4baec61bc5708e9a64128c5c7e4
tags: banker,carberp
file_type: PE32 executable (GUI) Intel 80386, for MS Windows
file_name: carberp1.exe
created_at: 2012-12-25 00:37:16
file_size: 132096
crc32: 05AF53DC
ssdeep: 3072:fQAsBL+tnecg1OS+x/+SSQSBX8MxaQhJwox:fQAsBoecg1UM3c
sha256: 689a35928f71848fab346b50811c6c0aab95da01b9293c60d74c7be1357dc029
sha512: 844e0010e23571e2bc6a44405a012bca4f01955348db26320d6a95e54e6afc85a81bef574ee65de9d67cdf6e2cf80fd4d1b2c559902596943b1e4ebeb5641650
id: 41
md5: 719354b4b7b182b30e1de8ce7b417d2f
You can download the sample:
vxcage> get 689a35928f71848fab346b50811c6c0aab95da01b9293c60d74c7be1357dc029 /tmp
Download: 100% |:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::| Time: 00:00:00 223.63 K/s
File downloaded at path: /tmp/689a35928f71848fab346b50811c6c0aab95da01b9293c60d74c7be1357dc029
Or upload a new one:
vxcage> add /tmp/malware.exe windows,trojan,something
File uploaded successfully
Copying
-------
VxCage is licensed under [BSD 2-Clause](http://opensource.org/licenses/bsd-license.php) and is copyrighted to Claudio Guarnieri.
Contacts
--------
Twitter: [@botherder](http://twitter.com/botherder)
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,968 |
\section{Introduction\label{sec:intro}}
Core-collapse supernova remnants (SNRs) often interact with large molecular clouds (MCs). This interaction between SNRs and dense MCs is of considerable interest because it provides an opportunity to study the dynamical and chemical processes associated with strong shocks, e.g., how the MCs affect the evolution of SNRs, how the MCs are disrupted by SN shocks, how the shocks change the abundances of MCs, and how molecules are destroyed or reformed. Since the first discovery of the SNR-MC interaction in the SNR IC 443, \citep{cornett77, denoyer79}, 45 SNRs, which are 16\% of the known Galactic SNRs, have been found to show some evidences for the interaction with MCs according to the compilation by \citet{jiang10}.
The SNRs interacting with MCs are of particular interest in relation to the \emph{thermal} \emph{composite} or \emph{mixed} \emph{morphology} (MM-) SNRs, which are the SNRs that appear shell-type in radio continuum but emit bright thermal X-ray in its center. In \citet{rho98}, it was surmised that about $\sim$ 25\% of the whole SNRs detected from the $Einstein$ observation belong to this category.
The center-bright morphology is not consistent with a Sedov-Taylor model \citep{sed46,tay50} where most of the swept-up matter is confined to a dense shell at the boundary. In order to explain the MM-SNRs, several hypotheses have been proposed. Probably the most popular one is the so-called evaporation model of \citet{whi91}. In this model, the ambient medium is clumpy, so there are dense clumps inside the SNR which have survived the passage of the SN shock. Subsequently, these clumps evaporate thereby brightening the X-ray emission. Another hypothesis is based on conduction from a hot interior to cold radiative shell, proposed by \citet{cox99} for the SNR W44.
The other two hypotheses are fossil thermal radiation from a hot interior after the shell of a remnant cools down \citep{sew85} and projection effect of the SNR exploded outside an MC \citep{pet01}.
It is worth noting that a good fraction of these MM-SNRs are the SNRs interacting with MCs \citep{rho98,jiang10}. And among the 45 probable SNRs interacting with MCs in the catalog of \citet{jiang10}, 15 SNRs are classified as the MM-SNRs. It is therefore interesting to investigate if an SNR breaking out of an MC can appear centrally-brightened in X-rays.
Dynamical evolution of SNRs interacting with MCs has been studied numerically by several authors \citep{fal82, ten85, yor89, art91, dohm96, vel01, fer08}. According to their results, the interaction may be divided into three categories.
First, if an SN explosion ($E_{51}$ = 10$^{51}$ erg as thermal energy) occurs deep inside an MC, e.g., 15 pc below the cloud surface \citep{ten85}, its remnant cannot break out of the MC and ends its life inside the cloud. The possible existence of such buried SNRs has been analytically addressed by \citet{wheeler80} and \citet{shu80}, where the SNRs radiate most of the energy in the infrared energy range.
Second, if an SN explosion occurs close to the surface of an MC, the SN blast wave can break out of the MC. This breakout phenomenon is characterized by the acceleration of the blast wave and the ejection of cloud matter across the original cloud surface. If the breakout occurs when an SNR is in the Sedov phase, the accelerated blast wave produces a large half-spherical remnant in the low-density intercloud medium (ICM), whereas the blast wave propagating into the dense cloud matter makes another sphere which is well described by the Sedov-Taylor solution and the relation in \citet{cio88}. \citet{dohm96} investigated the early evolution of SNRs produced very near the cloud surface. If an SNR breaks out the surface of an MC during its snowplow phase, the breakout process is expected to be rather complicated with the radiative shell disrupted by the Rayleigh-Taylor (R-T) instability. The overall morphology of the SNR is considerably elongated along the direction perpendicular to the cloud surface.
Finally, if an SN explodes outside the MC, the cloud is not largely disrupted while the SNR is distorted to a half-sphere. For the SN explosion just outside the cloud, \citet{fer08} presented the results of magneto-hydrodynamic simulations which shows how the reflected wave from the cloud surface moves back to the explosion center. Especially, \citet{ten85} carried out two-dimensional (2-D) simulations with cases belonging to the above three categories and studied the dynamical evolution of SNRs and the cloud disruption efficiency.
In this paper, we explore the dynamical evolution of breakout SNRs (BO-SNRs) with hydrodynamic simulations with the aim of finding the conditions for the SNRs to show center-bright X-ray morphology.
\citet{ten85} suggest that additional matter could be supplied by the radiative shell broken by the R-T instability
so that we may expect the BO-SNRs to show the center-bright X-ray morphology.
Their simulations with low resolution ($\sim$ 100 computational grids in one dimension), however, are limited for investigating the detailed process of breakout such as the disruption of a radiative shell by the R-T instability.
So, by performing three-dimensional (3-D) simulations with higher resolution, we could describe the complex structures of the BO-SNRs such as the R-T unstable structures.
We also synthesize X-ray morphology of BO-SNRs and investigate when they appear centrally-brightened. We apply our result to the SNR 3C 391, which is a prototype of the MM-SNRs.
This paper is organized as follows.
In Section~2, we introduce the methods used in the numerical simulations with cooling and heating processes.
In Section~3, we present the results from several models with different depths and density ratios.
In Section~4.1, we compare the results of numerical simulations to the results of one-dimensional (1-D) spherical experiments, and the semi-analytic solutions of shell and shock by \citet{koo90}.
Also in Section~4.2, we derive X-ray morphology of the simulated SNRs to discuss the origin of thermal emission inside the MM-SNRs. Finally, a simulated SNR model is compared with the prototypical MM-SNR, 3C 391, in the scope of X-ray characteristics.
\section{Numerical Methods}
\subsection{Governing Equations and Numerical Schemes}
To follow up the evolution of SNR interacting with MC, we solve the following Eulerian hydrodynamic equations:
\begin{equation}
\frac{\partial \rho}{\partial t}+ \nabla \cdot (\rho \textbf{v}) = 0,
\end{equation}
\begin{equation}
\frac{\partial }{\partial t} (\rho \textbf{v})+ \nabla \cdot ( \rho \textbf{v} \textbf{v} )+ \nabla P = 0,
\end{equation}
\begin{equation}
\frac{\partial E}{\partial t} + \nabla \cdot [(E+P)\textbf{v}] = \Gamma - \Lambda.
\end{equation}
where the total energy $E$ is defined as $E ={P}/(\gamma-1) + \rho v^2/2 $ and $\mu$ is the mean molecular weight, $\mu = 14 m_{\rm H} /23$ with 10\% helium fraction by number under fully ionized state. Other symbols have their usual meanings.
The energy equation must be integrated with effective cooling effect, $\Lambda_{\rm eff}= \Gamma - \Lambda$ to follow the SNR evolution in the snowplow phase. The radiative cooling rate, $\Lambda = n_{\rm e} n_{\rm H} L(T) $ with the cooling function, $L(T)$, and the hydrogen and electron densities, $n_{\rm H}$ and $n_{\rm e}$. In the fully ionized state and with 10\% helium fraction by number, $n_{\rm e} = 1.2 n_{\rm H}$. $L(T)$ involves different
cooling processes as a function of the temperature.
From 10 K to 10$^4$ K the cooling function of \citet{san02}, which is fitted by a piecewise power-law fit \citep{wol95}, is adopted.
From 10$^{4}$ K to 10$^{8}$ K, the non-equilibrium cooling curve of \citet{sha76} with the solar abundance is adopted.
For higher temperature ($T$ $>$ 10$^8$ K), we include thermal bremsstrahlung process in $L(T)$.
The heating rate, $\Gamma= n_{\rm{H}} G(T)$, comes from a process such as photoelectric heating by starlight, composed of the hydrogen density and the heating function, $G(T)$. The heating function is given by $G(T)=1.2 n_{\rm{i}} L(T_{\rm {i}})$ with initial density and temperature, $n_{\rm {i}}$ and $T_{\rm {i}}$.
Then, the effective cooling effect can be written at the medium as:
%
\begin{eqnarray}
\Lambda_{\rm eff} &=& \Lambda-\Gamma=n_{\rm e}n_{\rm H}L(T)-n_{\rm H}G(T) \nonumber \\
&=& 1.2 n_{\rm H} (n_{\rm H} L(T) - n_{\rm i} L(T_{\rm i})).
\end{eqnarray}
Now the effective radiative cooling can be calculated with the hydrogen number density, $n_{\rm H}$ and the cooling function $L(T)$ with given
initial conditions of $n_{\rm i}$ and $T_{\rm i}$.
Through a model, an MC and the ICM are in thermal equilibrium.
The hydrodynamic equations are solved using the HLL method \citep{har83}, which solves the Riemann problem in an approximate way to obtain intercell fluxes. Since we do not need a full characteristic decomposition of the equations, the HLL Riemann solvers are straightforward to implement and very efficient. The HLL code is tested both for the Sod problem \citep{sod78} and the SNR evolution in the adiabatic state, which are in good agreement with analytic solutions \citep{shu80}.
The coloring method is the scheme to trace a specific component of the multi-fluid using a Lagrangian tracer variable \citep[see][Equation 1]{xu95}. In addition to the usual hydrodynamic equations, we solve the continuity equation for each component. The density is obtained by multiplying the density of the fluid to a Lagrangian tracer variable. In this paper, we trace the MC and the ICM materials, separately (Figure~4), so that we can verify the origin of the complex structure in the evolved stage.
For the cooling process, we first calculate the cooling time scale, $\Delta t_{\rm cool} = {E_{\rm int}^{\rm n}} / { \Lambda _{\rm net} }$, where $E^{\rm n}_{\rm int}$ is the thermal energy and the superscript n represents the n-th time step. We then update the thermal energy using:
\begin{equation}
E_{\rm int}^{\rm n+1} = E_{\rm int}^{\rm n} \exp({- 0.5{\Delta t_{\rm dyn}}/{\Delta t_{\rm cool}}})
\end{equation}
where $\Delta t_{\rm dyn}$ is the dynamical time step set by the Courant condition. Because the above steps are solved before and after the hydrodynamic part, there is 0.5 in Equation (5).
\begin{figure}[t!]
\centering
\includegraphics[trim=45mm 15mm 45mm 25mm, clip, width=80mm]{wankeecho_fig01.eps}
\caption{Three-dimensional schematic overview on our models. Three different regions are drawn in different colors: green for a molecular cloud, orange for the intercloud medium and red for the initial SNR. The $x-$ and $y-$axes on the MC surface are set to be perpendicular to the $z-$axis. $d$ is the depth below the cloud surface where an SN explodes and $\alpha$ is the density ratio of the MC to the ICM.}
\end{figure}
\begin{figure}[t!]
\centering
\includegraphics[width=83mm]{wankeecho_fig02.eps}
\caption{Energy budget of D250 with time. Each solid line represents thermal energy (red), kinetic energy (blue), the total kinetic and thermal energy (black), and energy loss by the radiative cooling (purple), which are normalized by the SN explosion energy of 10$^{51}$ erg. }
\end{figure}
\begin{table}[t!]
\caption{Model parameters with ejecta of 10 $M_\odot$. \label{tbl-1}}
\centering
\begin{tabular}{lrrr}
\toprule
Model & $d$ $^{\rm a}$ & $\alpha$ $^{\rm b}$ & $Res.^{\rm c}$ \\
\midrule
D200 & $2.0$ & $10^3$ & $1/8$ \\
D250 & $2.5$ & $10^3$ & $1/8$ \\
D300 & $3.0$ & $10^3$ & $1/8$ \\
D350 & $3.5$ & $10^3$ & $1/8$ \\
R101 & $2.5$ & $10^1$ & $1/8$ \\
R102 & $2.5$ & $10^2$ & $1/8$ \\
R104 & $2.5$ & $10^4$ & $1/8$ \\
H032 & $2.5$ & $10^3$ & $1/32$ \\
\bottomrule
\end{tabular}
\tabnote{
$^{\rm a}$ The explosion depth $d$ is in the unit of pc.
\\ $^{\rm b}$ The density ratio $\alpha$ is the ratio $n_{\rm MC}/ n_{\rm ICM}$.
\\ $^{\rm c}$ The resolution is in units of pc/grid.
}
\end{table}
\begin{figure*}[t!]
\centering
\includegraphics[width=170mm]{wankeecho_fig03.eps}
\caption{The evolution of model D250. The snapshots of density, pressure, temperature, and speed from left to right columns are drawn at two specific times, i.e., 2.5$\times$10$^{4}$ and 6$\times$10$^{4}$ years after explosion for the upper and lower frames, respectively. The colorbars of density, pressure, and temperature in log scale are shared at each column. To see the overall feature of the remnant, a slice of the quadrant column is copied to the other side with reflective boundary condition assuming symmetry.}
\end{figure*}
\subsection{Model Parameters}
We adopt Cartesian coordinates for our 3-D SNR models. Figure~1 shows the schematic description of our models, an SN (red sphere) explodes at a depth, $d$, below the cloud surface between an MC (filled with green color) and the ICM (orange) with a density ratio, $\alpha$, whose remnant will break out through the MC surface and be ejected into the ICM. Each region of the MC and the ICM has uniform density distribution. We vary the density ratio from 10 to 10$^4$ with a fixed hydrogen number density of 100 cm$^{-3}$ and a fixed temperature of 10 K for the MC. In thermal equilibrium, the resulting density of the ICM ranges from 0.01 cm$^{-3}$ to 10 cm$^{-3}$ and the temperature from 10$^5$ K to 100 K. We set the $z-$axis perpendicular to the cloud surface bearing the $x-$ and the $y-$axes. To save computational cost and time, we find solution in one quadrant column along the $z-$axis. Reflective boundary conditions are adopted in the $xz-$ and the $yz-$planes, while continuous boundary conditions in the rest of planes.
We simulate an SNR from the beginning of the Sedov phase. Because we set the ejecta mass of 10 {$M_\odot$} assuming a core-collapse SNR with a massive progenitor, the total mass of the ejecta and the swept-up MC matter becomes 20 {$M_\odot$}. The remnant matter is distributed uniformly within a sphere in the radius of 0.89 pc with density of 200 cm$^{-3}$. The total energy of SN explosion is assumed to be 10$^{51}$ erg (E$_{51}$) given in thermal. But in few computational time steps, the physical quantities converge to the Sedov-Taylor solution where the kinetic energy is 28\% \citep{sed46,tay50}. In the case of an SN explosion in a uniform medium, a sharp increase of density appears just behind the shock front in the Sedov-Taylor solution; we term it an \emph{adiabatic} \emph{shell} hereafter. In the snowplow phase of the SNR, the dense neutral radiative shell is formed by cooling in the outermost region of the SNR. The semi-analytic solution of \citet{cio88} modified by \citet{koo04} describes the snowplow phase and determined the radiative shell formation radius, $R_{\rm sf}$, to be 2.78 pc and the shell formation time, $t_{\rm sf}$, to be 2.6$\times$10$^{3}$ years
in a medium with $n_{\rm{MC}}$ of 100 cm$^{-3}$. Thus we can divide our models into two groups according to the SN explosion depths: one case being that an SNR already has a radiative shell in an MC before breakout ($d > 2.78$ pc) and the other case that an SNR breaks out of the cloud surface during its Sedov stage ($d < 2.78$ pc).
Models labelled as Dxxx vary in the depths at which the explosion occurs, while those labelled as Rxxx have varying density ratios. The numeral xxx
trailing the character denotes the depth or the density ratio as indicated in Table 1.
The H032 model uses the highest resolution of 32 [grid/pc] and all models use a computational box of size 16$^{2}$ $\times$ 48 [pc$^{3}$] with a larger height in the ICM to track the evolved stages of the breakout SNR.
Figure 2 shows the time variation of energy (normalized by $E_{51}$), upto an age of 2$\times$10$^{4}$ years in the D250 model. Initially, the SN explosion energy is deposited in thermal energy ($E_{the}$/$E_{51}$ = 1.0) and the remnant follows the Sedov-Taylor solution ($E_{the}$/$E_{51}$ $\sim$ 0.7) in a few computational time step, following the red line which denotes the thermal energy variation. Soon the remnant becomes radiative so that the red line drops sharply. At the same time, the purple line shows the energy loss by the radiative cooling starts to rise more rapidly near the shell formation time, $t_{sf}$. The breakout of the remnant from the MC surface causes the thermal energy to decrease more rapidly due to adiabatic expansion of the escaped part of the SNR into the ICM and the kinetic energy (blue line) to decrease very slowly ($E_{kin}$/$E_{51}$ $\sim$ 0.2-0.3) even after $t_{sf}$. The following sections provide more detail on these calculations.
\section{Results}
\subsection{Standard Model: D250}
We set the D250 model as a standard representing SNRs breaking through the surfaces of clouds. Especially, the remnant is produced 2.5 pc below the surface of the MC so that it breaks out of the MC in its Sedov stage. We focus on the separation of the adiabatic shell during breakout in the ICM. Figure~3 shows density, pressure, temperature and speed slices of the remnant at two time epochs of 2.5$\times$10$^{4}$ (upper panels) and 6$\times$10$^{4}$ years (lower panels). In Figure~4, the MC and the ICM matter are traced by the coloring method. We label the key structures of the multi-layered structure as \emph{1st} \emph{layer}, \emph{2nd} \emph{layer}, and \emph{ripples}. The \emph{swept-up} \emph{ICM} and the \emph{R-T} \emph{finger} are also labeled.
\begin{figure}[t]
\centering
\includegraphics[width=83mm]{wankeecho_fig04.eps}
\caption{Prominent features of density structures of the model D250. The left side of each frame denotes the MC matter distribution and the right side the ICM matter. The time epochs are 2.5$\times$10$^{4}$ and 6$\times$10$^{4}$ years for the left and right frames, respectively, and the colorbar is the same to that on the density frames in Figure~3.}
\end{figure}
\begin{figure}[t!]
\includegraphics[width=83mm]{wankeecho_fig05.eps}
\caption{One dimensional profiles along the symmetry axis of density, pressure, temperature and speed from top to bottom. Density, pressure, and temperature are drawn in log scale and speed of the matter is drawn in linear scale as a position function of Z, the distance from the MC surface in the unit of parsec. Each frame has three lines of different colors which means the three time epochs of 5$\times$10$^{3}$ (red), 1.5$\times$10$^{4}$ (blue) and 2.5$\times$10$^{4}$ years (black). Arrows in the density frame indicate the first layer (red arrows) and the separated structures (blue and black arrows) from it.}
\end{figure}
Figure 3 shows the general morphology of break-out SNRs such as the spherical radiative shell in the MC and blown-out morphology in the ICM. Even after break-out, the remnant in the MC evolves like an SNR in uniform medium. We see that the density is highest and the pressure and temperature are lowest at the radiative shell due to radiative cooling. But since the cooling effect is small due to the high temperature at the SN explosion site, the temperature is still high at 2.5$\times$10$^{4}$ years.
But in the ICM, the blown-out part of the remnant makes much more complex structures.
First of all, we see two green layers of enhanced density in the ICM at 10 and 14 pc, far beyond the MC boundary, which we term 1st and 2nd layer, respectively (see Figure~4).
Note that the swept-up matter of the first layer is located quite inside the remnant compared with the Sedov-Taylor solution where most of the shocked matter exists just behind shock front. The ripples appear just below the 1st layer.
The pressure is higher in the 2nd layer and the swept-up ICM matter, while it is much lower around the 1st layer.
The temperature at the swept-up ICM is maintained higher while the temperature near the explosion site of the SNR goes down rapidly due to radiative cooling.
Matter speed of the matter is highest at the lower ends of each layer.
The swept-up ICM matter is colored in bright blue in the density panel, which denotes the shock position propagating into the ICM.
Before describing the dynamical evolution of the late stages of the remnant, we investigate the process of the separation of the adiabatic shell. We can trace the separated layers as peak positions in 1-D density profile along the symmetry axis. From Figure~5, we note that the separation of the shell occurs due to adiabatic expansion in the following steps. First, the blast wave is accelerated and starts to run away from the adiabatic shell as it breaks out of the MC surface. Second, the pressure between the blast wave and the lagged adiabatic shell has decreased due to the adiabatic expansion (the drop zone shown at 2 pc at 5$\times$10$^{3}$ years in Figure~5 and has been expanding with time). Third, along the pressure gradient, matter at the upper side of the adiabatic shell moves to the Contact Discontinuity between the MC and the ICM (hereafter, CD-MI), piles up, and makes the second layer, pointed out with a blue arrow in the top panel of Figure~5 at 1.5$\times$10$^{4}$ years. Moreover, we can see the beginning of the ripple structure marked with purple arrows in Figure~5.
The ripples are formed by matter separating from the adiabatic shell due to the same reasons as the second layer: the MC matter from the lower sides of the first layer moves downward and piles up at the contact-discontinuity between the ejecta of the SN and the MC matter. Such chain reactions cause the piled up matter to form small density peaks around the purple arrow in Figure~5 at 2.5$\times$10$^{4}$ years. The downward motion of the ripples create the speed inversions at 4 pc at 1.5$\times$10$^{4}$ years and also at 7 pc at 2.5$\times$10$^{4}$ years.
Lower frames of Figure 3 show the dynamical evolution at a later, evolved stage of 6$\times$10$^{4}$ years with a single merged shell, Rayleigh-Taylor fingers and ripples. Before 6$\times$10$^{4}$ years, the upper two layers merge into a single shell since the first layer maintains its own speed while the second layer decelerates with outer blast wave. The deceleration results in the R-T instability on the merged layer, shown at 27 pc height in the density frame.
Since, after merging, the merged shell soon enters the snowplow phase and is decelerated more, the Rayleigh-Taylor unstable structures in the shell grow continuously. They will finally stretch in an upward direction and deform the outermost layer of the swept-up ICM, as discussed in detail in the next subsection.
\begin{figure}[t!]
\includegraphics[width=83mm]{wankeecho_fig06.eps}
\caption{The growth of Rayleigh-Taylor fingers. The left frame labeled (a) presents the overview of density distribution of the remnant of H032 model in late stages and the right frames from (b) to (d) focus on the growth of R-T fingers at the top of the remnant. (a): The first and the second layers are merging at the time of 3$\times$10$^{4}$ years. (b)$\sim$(d): R-T fingers deform the layer of swept-up ICM at 6$\times$10$^{4}$, 7$\times$10$^{4}$, and 8$\times$10$^{4}$ years, respectively.}
\end{figure}
\subsection{High Resolution Model: H032}
We need the higher resolution model, H032, to describe the details of the growth of R-T fingers in the late stages of a breakout SNR. The initial conditions of H032 model are the same as those of D250 model but at higher resolution, 32 [grid/pc]. Figure~6 shows the evolution of the R-T fingers and their effects on the environment. The (a) frame captures the moment when the upper two layers are merging.
We see that the second layer exhibits R-T instability as it approaches the thicker upper layer. After merger (frame b), the fingers are seen to be pushing the CD-MI to deform the outermost layer of the swept up ICM (frame c). Finally, the fingers stretch outward and fragment into several blobs as shown in frame (d) of Figure~5.
Tenorio-Tagle et. al. (1985), have already argued that fragments arising from the RT instability may be expected to be seen in the ICM if the breakout occurs in the snowplow phase of the SNR. Here, we see that similar fingers grow and become unstable even if the remnant breaks out of the cloud in its Sedov phase.
\begin{figure*}[t!]
\centering
\includegraphics[width=170mm]{wankeecho_fig07.eps}
\caption{The evolution of model D300. The snapshots of density, pressure, temperature, and speed are drawn from left to right columns. The time epochs are 3$\times$10$^{4}$ (upper) and 8$\times$10$^{4}$ years (lower).}
\end{figure*}
\subsection{Models with Different Depths:\\ D200, D300, and D350}
We simulate a group of models with a range of depths, to investigate the influence of the environment of explosion sites. The SNR of D200 model is produced at 2 pc depth below the MC surface, which locate at shallower depth than the radiative shell formation radius, R$_{sf}$ of 2.78 pc, where $n_{\rm{MC}}$ is 100 cm$^{-3}$. Compared with R$_{sf}$, SNRs of D300 and D350 models are produced deeper at 3 and 3.5 pc depths, respectively.
Figure 7 shows the evolution of the D300 model which does not exhibit the multi-layered structure discussed in the previous section. We can see that the wider radiative shell is elongated in the ICM compared to that inside the MC. The pressure is more uniform compared to the D250 model except in the swept-up ICM layer. And like the D250 model the speed of the matter is fastest at the lower side of CD-EM.
Because the SNR has already entered the snowplow phase before break-out and its shock speed decreases to a few tens of km/s in the MC, the shock is not accelerated enough to run away from the radiative shell after break-out. Hence the second layer does not appear and the shell retains its shape as a single shell, which is the common characteristic of the models with deep explosion sites. In the lower frames, we can see a deformed shell at the top of the remnant under the effect of the R-T instability similar to the standard D250 model and ripples to move downwards from the shell.
Figure 8 shows that the SN explosion depth influences the formation of ML structure as well as the overall shape of the remnant in the ICM. In the ICM, the remnant of D350 model shows a single shell and the outer part of the remnant appears more wedge like shape than D200. For D200 model, the number of the upper layers separated from the adiabatic shell is more than that of D250. And we can see the layer just detached from the adiabatic shell has already undergone the R-T instability and shows vertical structures around 10 pc near the symmetry axis. With time, the separate layers merge into a single shell or fade out with ripples. We can see the shape of the outer part of the remnant of D200 is more spherical compared to the other models.
\subsection{Models with Varying Density Ratios:\\ R101, R102, and R104}
These models are simulated with a fixed explosion depth, 2.5 pc and a fixed number density of the MC, $n_{\rm{MC}}$ = 100 cm$^{-3}$. Only the density ratio $\alpha$ between the MC and the ICM is varied with 10$^1$ (R101), 10$^2$ (R102) and 10$^4$ (R104) compared to 10$^3$ (D250).
Figure 9 shows that the model with higher density ratio shows greater distance between the contact-discontinuities where the multi-layered structure develops. In the upper frames for R101 model, we can see a single shell in the ICM, which is the characteristic density distribution in the ICM of D300 model. Because of the dense ICM, the outer shock cannot run away from the adiabatic shell after breakout. So the adiabatic shell keeps its own shape between the CDs with the multi-layers overlapped. When the shock is decelerated, the shell at the top of the remnant becomes R-T unstable shown in the right frame.
The multi-layers of the R102 model are very close to each other between 4 and 7 pc heights shown in separated form.
But they immediately merge into a single shell. Thus, for small alpha, the ML structure is not seen or survives for a short time. Models with larger alpha (D250 and R104) show the ML structure for a longer time compared with the R102 model since the blast wave proceeds faster into the less dense ICM.
\begin{figure}[t!]
\centering
\includegraphics[width=83mm]{wankeecho_fig08.eps}
\caption{Density snapshots of models D350 (upper) and D200 (lower). The snapshots have different time epochs: D350 at 3$\times$10$^{4}$ and 6$\times$10$^{4}$ years, D200 at 1.5$\times$10$^{4}$ and 3$\times$10$^{4}$ years. All the frames share the same colorbar in log scale.}
\end{figure}
\begin{figure}[t!]
\centering
\includegraphics[width=83mm]{wankeecho_fig09.eps}
\caption{Density snapshots of models R101 (upper), R102 (middle) and R104 (lower) with fixed explosion depths of 2.5 pc. The left frames describes the models of 1.5$\times$10$^{4}$ years and the right frames of 3$\times$10$^{4}$ years.}
\end{figure}
\section{Discussion}
\subsection{One-Dimensional Experiments}
In this section, we will focus on the propagation of the shock from the breakout SNR in the ICM. To this end, we simplify our 3-D models to 1-D models using spherical coordinates. Because the geometry of the MC surface hardly plays a role in the propagation of the shock along the $z-$axis, we expect that the 1-D models will capture the essential features.
Since the 1-D model represents the propagation in spherical coordinates, geometric source terms have to be included in the HLL code which is meant for Cartesian coordinates. The 1-D models are solved at higher resolution, 32 [grid/pc] for a time period of 10$^6$ years (ten times longer than 3-D models) at cheaper computation cost. In Figure~10, we display the propagation of the travelling waves along the radial direction inside MCs with radii 3 pc (left panel) and 2.5 pc (right panel) respectively. The density distributions are plotted in logarithmic scale and the colorbars are different in the two pictures to highlight the difference in the density structures.
In the left frame, the arch is seen to come from the reverse wave which detached from the CD-EM into the MC and rises beyond 1 pc from the explosion center at 4$\times$10$^{3}$ years (upper arrow). As the wave travels in the remnant, it displays the characteristics of a shock such as density increase (seen as a sharp boundary of the arch). When the reverse shock catches up with the outer blast wave in the ICM, the outer shock is pushed out creating yet another reverse shock which again travels towards the center of the remnant. This is seen as a plunging density contrast lasting upto 7$\times$10$^5$ years.
The right frame shows the evolution of an SNR with an MC radius of 2.5 pc which is smaller than R$_{sf}$. Just after breakout, the adiabatic shell expands in the ICM. We can see the adiabatic shell diffused between yellow sharp lines: the upper line for the swept-up ICM and the lower one for the reverse shock. The inclined arch of the reverse shock rises again at 4$\times$10$^{3}$ years, but falls by 3$\times$10$^5$ years, which is much earlier than that in the former case, since the pressure in outermost region is higher.
\begin{figure*}[t!]
\centering
\includegraphics[width=160mm]{wankeecho_fig10.eps}
\caption{Radial density profiles of 1-D SNR simulations as a function of time. The left frame shows the density profiles where the supernova explodes at the center of an MC sphere with a 3-pc radius, while MC with a 2.5-pc radius in the right frame. The arrows are marked at the bottom of each arch to denote the rise and the fall of the reverse shock inside the remnant.}
\end{figure*}
From Figure 11, we might suspect that the shock propagation may be modelled by piecewise straight lines, both in the 1-D (left) and the 3-D case (right).
The shock position is determined by the temperature peak just behind the shock front for 1-D models and along the $z-$axis for 3-D models.
In the left frame, the shock positions are plotted in log scale for four 1-D simulations with different radii from 2 to 3.5 pc. The upper two lines with smaller MCs (2.0 and 2.5 pc radii, smaller than R$_{sf}$) show similar slopes. Inside the MCs, they follow the Sedov-Taylor solution with a slope of $2/5$ \citep{sed46}. The slopes increase suddenly to $3/4$ after breakout and decrease to $3/10$ after the radiative cooling becomes dominant. For the lower two lines, the slopes of remnants with larger MCs (3.0 and 3.5 pc radii, larger than R$_{sf}$) are changed by the arrival of the reverse shock. Inside MCs, the remnants follow the Sedov solution and soon enter the snowplow phase. The slope of the blast wave increases to $3/4$ with breakout and, once the reverse shock catches up with the outer shock, the slope increases to $4/5$. The remnants in the ICM finally enter the snowplow phase again and the slopes follow $3/10$. In the right frame, the shock propagations of 3-D models show similar trends to those of 1-D models, but we can check the shock propagations only at early stages due to the limit of short computational period. Figure~12 shows the schematic descriptions for the shock evolution, and is meant to be a simplified representation of Figure~11. We can see four kinds of slopes at each plot from the Sedov phase ($2/5$), the snowplow phase ($3/10$), the breakout ($3/4$), and the reverse shock ($4/5$).
\begin{figure}[t!]
\centering
\includegraphics[width=83mm]{wankeecho_fig11.eps}
\caption{Shock propagations of 1-D and 3-D simulations for $\alpha$ = 1$\times$10$^{3}$ with different radius of MC: 2.0 pc (solid line), 2.5 pc (dotted line), 3.0 pc (dashed line) and 3.5 pc (dot-dashed line).}
\end{figure}
\begin{figure}[t!]
\centering
\includegraphics[width=83mm]{wankeecho_fig12.eps}
\caption{Schematic plots of shock position with time with different radii of MCs in logarithmic scale from the shock propagations of 1-D cases with the left frame of Figure~11. The broken lines describe the shock propagation of the models that an SN explodes at a shallow depth (left) and that an SN explodes deep inside an MC (right). The numbers over lines denote the powers. The vertical dashed lines mark the points of specific events such as breakout (BO), dominance of radiative cooling (RC) and arrival of the reverse shock (RS).}
\end{figure}
In Figure 13, we compare the density distribution along the $z-$axis of the H032 model with the previous 1-D models and also compare with analytic solutions that describe the reverse shock. With the MC radius set to 2.5 pc, the arch of the reverse shock appears at 4$\times$10$^{3}$ years (this is the same as in the right panel of Figure~10 indicated by the upward arrow).
The dotted line in Figure~13 shows the position of the shock obtained from the 1-D simulation and are seen to be very close to those obtained from the H032 model (the boundary of the blue region). Also shown is the semi-analytic solutions obtained by \citet{koo90} for the shell (dashed line) and shock (dot-dashed line). The semi-analytic solutions differ because they describe the evolution in an exponential medium unlike in the current work which has a jump in the density. However, the position of the shell is seen to reasonably capture the evolution of the first layer. This is because most of the swept-up matter is left in the first layer after breakout.
\begin{figure}[t!]
\centering
\includegraphics[width=83mm]{wankeecho_fig13.eps}
\caption{One-dimensional density profiles of H032 along the $z-$axis with time (background) and shock positions in 1-D simulations (dotted line). The analytic solutions of \citet{koo90} for the shell (dashed line) and the shock (dot-dashed line) are plotted on the profiles. We put the arrow under the time axis to mark the start of arch, which denotes the reverse shock travelling inside the remnant.}
\end{figure}
\begin{figure*}[t!]
\centering
\includegraphics[width=160mm]{wankeecho_fig14.eps}
\caption{Simulated X-ray surface brightnesses in soft (left) and hard (right) X-rays of the models. The colorbar over the left part of frame (a) represents the soft X-ray surface brightness and that over the right part shows the hard one in logarithmic unit of erg/cm$^2$/s/sr. The time epoch for the frames is 1.5$\times$10$^{4}$ years.}
\end{figure*}
\begin{figure*}[t!]
\centering
\includegraphics[width=170mm]{wankeecho_fig15.eps}
\caption{Snapshots of the X-ray surface brightness of model D250. Each frame shows the brightness in soft (left part) and hard (right part) X-ray bands in logarithmic units of erg/cm$^2$/s/sr. The colorbars are set to show the full dynamical range of the brightness at each time epoch of t = 2.5$\times$10$^{3}$, 7.5$\times$10$^{3}$, 1.25$\times$10$^{4}$, and 1.75$\times$10$^{4}$ years (from left to right frames). The bright part at the rim of the remnant inside MC at 1.75$\times$10$^{4}$ years would be artifact since we cannot resolve the contact discontinuity sharply by HLL code.}
\end{figure*}
\subsection{X-ray Morphology}
In this section, we will simulate the surface brightness in X-ray of the MM-SNR to study their morphology since these are diagnosed by showing center-bright
thermal X-ray emission while being shell-bright in radio. The X-ray brightness of our model is calculated using the emissivity tables of \citet{ray77}, which includes recombination, bremsstrahlung, two-photon processes and line emissions. The chemical abundance is taken to be the solar abundance of \citet{and89}.
The X-ray emissivity is multiplied by $n_{\rm{e}}n_{\rm{H}}$ ($n_{\rm{e}} = 1.2 n_{\rm{H}}$) and divided by 4$\pi$ to get the emission per unit solid angle. Using the optically thin approximation, the simulated brightness on the $xz-$plane is obtained by integration along the line of sight.
Figure 14 shows the simulated X-ray brightness of our models in the soft (0.1-3.0 keV, left panel) and the hard (3.0-8.0 keV, right panel) X-ray bands at 1.5$\times$10$^{4}$ years. The upper panels are for varying depths of the explosion, while the lower panels capture the variation with the density ratio. In all cases, the X-ray morphology of the SNRs in the radiative stage resembles a mushroom with a cap in the ICM and a stem through an MC.
Based on Figure 14, we can say that the center-bright X-ray morphology can arise from an SNR which explodes inside a dense medium without invoking any thermal conduction mechanism \citep{til06}. During the early stages of SNR evolution, the soft X-ray brightness tracks the shell-bright morphology, since newly swept-up MC matter ensures that the density and temperature are highest near the shock position (see also Figure~15). However at late stages, the material in shell region cools below 10$^4$K which is too low for X-ray emission, but that in explosion site could still be hot enough to continue emitting bright X-rays inside the MC. We can check the center-bright morphology in soft X-ray band of SNRs inside MCs in Figure~14 and the third and the last frames in Figure~15, while the hard X-ray brightness shows the center-bright morphology from the earlier to the late stages.
In the ICM, otherwise, we can see the shell-bright morphology in SNRs in X-rays in Figure 14. Since the blast wave is re-accelerated by breakout from the MC surface to stay in adiabatic state, the SNR has maximum density and temperature around shock toward the ICM. Especially, the hard X-ray can point out the shocked ICM described a thin layer above CD-MI, while the soft X-ray brightness is shown more diffuse. In the (c) and (f) frames, the cap parts are fading out since the cooling effect is dominated.
The ML structure can also contribute to central-brightening in soft X-rays, as remarked earlier. We notice that soft X-ray emission traces the multi-layered structure in Figure~14. Since the MLs can retain high density inside an SNR after it breaks out through the MC, they can supply enough matter to enhance the surface brightness. In (a), (b), (d), and (e) frames, we can see the bright soft X-ray on the multi-layers. The (a) frame shows several bright X-ray features from 5 pc to 13 pc height along the symmetry axis. In the other frames, only the first and the second layers emit bright soft X-rays unlike (a) frame, because the temperature has been lowered rapidly by adiabatic expansion between the two layers. The soft X-ray of the second layer depends on the ICM density. Moreover, under specific conditions such as higher resolution or larger density ratio, these layers could be distorted or broken apart by R-T instability to form clumpy structures inside the remnants.
There are three prominent points in the simulated X-ray surface brightness of our models. First, the ML structure enhances the soft X-ray brightness, which may reveal the clumpy and complex internal X-ray structures of MM-SNRs. Second, the center-bright morphology in soft X-ray can be formed in the evolved phase of models inside a dense medium. Third, a breakout SNR shows the shell-bright X-ray morphology toward the ICM.
\subsection{3C 391}
3C 391 is a prototype of MM-SNRs \citep{rho98}.
In radio continuum, we can see its blown-out morphology clearly across the surface of an MC.
The remnant is elongated from northwest to southeast and shows a bright rim inside the northwestern MC
indicating the interaction with the remnant \citep{rey93}.
On the other hand, the $Chandra$ images \citep{che01,che04,che05} show
that clumpy X-ray emission of thermal origin is filling the inside of the remnant (see the upper image of Figure~17). We simulate an additional 3-D model to reproduce the X-ray morphology of 3C 391. From the X-ray images of \emph{Einstein} \citep{wan84} and \emph{Chandra} observations of \citet{che04}, we assume that the SN explodes around 2.4 pc depth (1$'$ at 8 kpc distance) under the surface of a dense MC, where we set the hydrogen number density of the MC, 40 cm$^{-3}$, as a upper limit from \citet{che01}. Then the radiative shell formation time and radius become t$_{sf}$ = 4.4$\times$10$^{3}$ years and R$_{sf}$ = 4.1 pc. The age of SNR 3C 391 may be estimated as 8.5$\times$10$^{3}$ years based on the assumed hydrogen density of the MC and the distance from the explosion site to the dense shell of the remnant toward the inside of the cloud, 5.1 pc (2.2$'$), using the semi-analytic solution of \citet{cio88} as modified by \citet{koo04}. For the other initial conditions, we set the density of the ICM of 0.1 cm$^{-3}$ as a typical value of the ICM and the resolution of 8 grids a pc.
Figure 16 shows the density and temperature distributions (left) and the X-ray surface brightness (right) of a new model at 8.5$\times$10$^{3}$ years from SN explosion.
Considering the lower density of the MC of a new model compared to that of the standard one,
we may expect that the part of the remnant inside the MC still keeps its shell-bright morphology in soft X-ray band.
Furthermore, we notice that the evolved ML structure includes R-T fingers and fragments around 5 $\sim$ 9 pc heights around the symmetry axis in the left frame. These fingers and fragments might be the origin of the clumpy structures inside 3C 391, but they are not reflected in the X-ray brightness due to the projection effect and also they cannot enhance the X-ray emission enough to explain the bright clumps of the regions of 6, 7, and 8 in the top image of Figure~17.
\begin{figure*}[t!]
\centering
\includegraphics[width=160mm]{wankeecho_fig16.eps}
\caption{Density and temperature snapshots (left frame) and the soft and hard X-ray surface brightnesses (right frame) of the model for the MM-SNR 3C 391 in the same unit as Figure~14. The time epoch is 8.5$\times$10$^{3}$ years. Each colorbar follows log scale and the unit of the X-ray brightness is erg/cm$^2$/s/sr.}
\end{figure*}
\begin{figure}[t!]
\centering
\includegraphics[trim=0mm 0mm 25mm 0mm, clip, width=83mm]{wankeecho_fig17.eps}
\caption{(top image) $Chandra$ X-ray (0.3-7.0 keV) image \citep{che04} with the FIRST 1.4 GHz radio contours (Becker et al. 1995) of the SNR 3C 391. (middle frame) X-ray brightness profiles along the symmetry axis from our model, where red, green, blue, and black curves are for 0.3-1.5, 1.5-3.0, 3.0-7.0, and 0.3-7.0 keV bands, respectively. The diamond symbols represent the observed brightnesses in
3C 391 which are from \citet{che04}. They are labeled by the region numbers in \citet{che04}. The unit of the X-ray brightness is erg/cm$^2$/s/sr. The red and black lines are almost overlapped, since most of X-rays are radiated in the 0.3-1.5 keV.
(bottom frame) Same as the middle frame but including the attenuation by foreground and MC media (see text for details). }
\end{figure}
We show the image of the $Chandra$ X-ray observation with contours of the VLA FIRST (Very Large Array Faint Images of the Radio Sky at Twenty-cm) 1.4 GHz radio continuum \citep{bec95} of 3C 391 and profiles of the calculated X-ray brightness along the axis of symmetry in Figure~17. In order to compare the observed brightness of the remnant to our simulated X-ray brightness, we select four different regions of 4, 13, 15, and 14 from \citet{che04} in the top image of Figure~17, which represent the regions near the shock front toward the MC and the ICM and the inner regions of the remnant in the MC and the ICM, respectively. We mark the observed X-ray brightness of the regions with diamond symbols on the lower frames of Figure~17. If there is no attenuation of X-rays (middle frame), most of the X-rays are emitted in the soft band (0.3-1.5 keV). However, since the X-ray emitting hot gas is surrounded by the dense MC material, we need to consider the attenuation of X-rays due to the MC and the foreground media (bottom frame). According to \citet{che04}, the column density to the SNR is 2.9$\times$10$^{22}$ cm$^{-2}$ to the ICM area and 3.5$\times$10$^{22}$ cm$^{-2}$ to the MC area. We could expect that extinction will decrease the overall brightness significantly but would not change the morphology considerably since the column density difference between the MC and the ICM areas is only 0.6$\times$10$^{22}$ cm$^{-2}$. We compute the attenuated X-ray emission using the energy-dependent transmission curve of \citet{sew99} with the column densities of \citet{che04}. In detail, we calculate the flux weighted mean transmission with temperature for the column densities in the specific X-ray energy bands. Then we multiply it to the X-ray emission on each grid and integrate the transmitted emission toward the line of sight. The result is shown in the bottom frame of Figure~17.
$Chandra$ X-ray image of 3C 391 shown in the top image of Figure 17
is somewhat different from our simulated X-ray profiles shown in the lower frames.
The SNR has an almost uniform X-ray brightness
while our simulation shows
that the brightness in the ICM is much fainter than that in the MC region.
Toward the SNR bubble within the MC, the emission is from relatively dense ($\sim$ 5 cm$^{-3}$) hot gas near the MC,
while toward the SNR in the ICM,
it is mostly from the gas in the adiabatic shell where the density is much lower ($\sim$ 0.5 cm$^{-3}$) (see Figure~16).
The calculated emission is brighter than the observed for the SNR part in the MC area
while it is fainter for the SNR part in the ICM in the middle frame of Figure~17.
But if we assume extinction, the total brightness is fainter than the observed even toward the SNR part in the MC
(see the bottom frame of Figure~17), and now the dominant emission becomes from 1.5-3.0 keV band (green curve).
As it is easily expected, however, the overall shape of profiles is not changed much.
We suggest that thermal conduction and evaporation of preexisting
cloudlets can explain the difference between the $Chandra$ X-ray observational result
and our results.
For the SNR part embedded in the MC, the interface between the hot gas and the MC might be subject to conduction, which will increase the gas density in the hot bubble \citep{cox99}. A factor of 3 increase in gas density can explain a factor of 10 difference in brightness since the brightness is roughly proportional to the square density.
On the other hand, for the SNR part in the ICM, the difference in the X-ray brightness is more than three orders of magnitude in the bottom frame of Figure~17,
which might be difficult to be explained in a large scale conduction alone.
The $Chandra$ image suggests that the medium is clumpy with dense cloudlets.
The density is, for example, 5-7 cm$^{-3}$ at the regions of 6-8 in the top frame of Figure~17 whereas our characteristic density of those regions is $\sim$0.5 cm$^{-3}$ (Figure~16).
Therefore, there were likely dense clumps in the past, which might have provided additional mass to the hot gas by thermal evaporation \citep{whi91}. The increase of gas density from such evaporation of clumps together with the increase of temperature by thermal conduction from the hotter gas within the bubble might explain the difference. Such scenario may be explored in a future study.
\section{Summary and Conclusions}
We have simulated breakout morphology SNRs with different explosion depths and density ratios to show the evolution of SNRs breaking through molecular clouds (MCs). We have presented a fiducial model where the explosion depth is 2.5 pc below the surface of an MC, which breaks out the surface in its Sedov phase. The outermost shell in the Sedov phase, which we call the adiabatic shell, is separated after the breakout in two thick layers at its upper side and ripples at its lower side, which we call the \emph{multi-layer} structure. It is noticeable that the shocked ambient matter can exist inside a remnant in the form of the ML structure, which cannot be expected in Sedov-Taylor solution.
The environmental effect on the evolution the breakout SNRs is also investigated. When an SNR is produced closer to the MC surface, the number of layers increases at the front of the original adiabatic shell. Also in the more rarefied intercloud medium (ICM), the ML structure survives longer time. If the radiative shell is formed before breakout, we cannot see the ML structure because the outer shock is not fast enough to run away from the radiative shell.
The growth of R-T fingers is another key point in the paper. If the SNR breaks out of the MC surface at its Sedov stage, the R-T fingers grows on the layers in the front side of the original adiabatic shell at the beginning of the ML structure. The simulation with highest resolution shows the evolution of R-T fingers at the top of the remnant after merging of the multi-layers, which are fragmented into several blobs and penetrating the shock front.
We have discussed the shock propagation with simplified one-dimensional (1-D) models. We have noticed that the slope of shock position as a function of time is influenced by the reverse shock, detached from the dense shell in the MC, to rise to $4/5$, while the other slopes are denoted as $3/4$ due to breakout, $2/5$ from Sedov-Taylor solution, and $3/10$ due to radiative cooling. The shock propagations of 3-D models are well described by the simplified 1-D models and the adiabatic shell of the ML structure is fitted well with the semi-analytic solutions of shell in \citet{koo90} for a spherically symmetric blast wave on the ambient medium with density drop.
From the simulated X-ray brightness of our models, three key points can be inferred: (1) The ML structure can enhance the soft X-ray brightness (2) A remnant in an MC can appear centrally-brightened in X-rays around the explosion site in the evolved phase (3) The newly swept-up ICM matter emits hard X-ray at the swept-up ICM behind the outer shock.
Compared with the $Chandra$ images of 3C 391 of \citet{che04} as a fiducial mixed morphology SNR, the simulated surface brightness is consistent in X-ray brightness and the transmitted X-rays of 3C 391 in the MC quantitatively. But we can see difference in X-ray brightness between the model and $Chandra$ observation, which cannot be fully explained by our simplified models without thermal conduction or preexisting cloudlets outside the MC.
\section{Acknowledgements}
This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Science, ICT and future Planning (2014R1A2A2A01002811). Numerical simulations were performed by using a high performance computing cluster in the Korea Astronomy and Space Science Institute. We wish to thank Dr. Chen, Y. for providing $Chandra$ images of 3C 391.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 251 |
Queen Elizabeth II Confirms Royal Family Will 'Support' Prince Harry & Meghan Markle's 'New Lives' In Canada
January 13, 2020 12:38PM EST
Adam Sandler Turns Oscar Snub Into Beautiful Dedication To 'Mama' Kathy Bates
Jenna Lemoncelli
Soon after Adam Sandler's name was not called during the January 13 Oscars nominations, the 'Uncut Gems' star reacted to the snub. He tweeted that 'Sandman got no love from the Academy,' and congratulated his 'Waterboy' co-star, 'mama' Kathy Bates.
The Oscars snubs continue with Adam Sandler being the latest to star speak out. The actor, 53, was noticeably absent from the January 13 nominations, which did not include his wildly popular comedy thriller, Uncut Gems in its Best Actor category. While Adam wasn't feeling the love from the Academy, he had the best response to the snub.
"Bad news: Sandman gets no love from the Academy," he wrote in a tweet on Monday morning. However, he found the silver lining. "Good news: Sandman can stop wearing suits," he continued, possibly hinting that he will not attend the Oscars on February 9. He ended his tweet on a lighter note and praised his good friend, Kathy Bates.
"Congrats to all my friends who got nominated, especially Mama," Adam said in reference to Bates, who played his on-screen mother Helen in 1998's football classic, The Waterboy. He included a photo of Kathy, 71, from the film in his tweet. Kathy is nominated for Best Supporting Actress for playing Barbara Jewell in Richard Jewell. at the 2020 Academy Awards.
Bad news: Sandman gets no love from the Academy.
Good news: Sandman can stop wearing suits.
Congrats to all my friends who got nominated, especially Mama. pic.twitter.com/o1Ep3E7GRB
— Adam Sandler (@AdamSandler) January 13, 2020
Not long after Adam sent out the tweet, Kathy responded with a hilariously cute message.
"I love you my Bobby Boucher!!!," the actress wrote, addressing Adam as his character. "You was robbed!!," she admitted. "But Mama loves you!!! I learned a new urban slang word for you! You da GOAT!! Not the one we eat at home, Son."
I love you my Bobby Boucher!!! You was robbed!! But Mama loves you!!! I learned a new urban slang word for you! You da GOAT!!
Not the one we eat at home, Son. ❤️😎🐐 https://t.co/2KDbfUjIXR
— Kathy Bates (@MsKathyBates) January 13, 2020
Meanwhile, HollywoodLife caught up with Adam Sandler at the start of this awards season at the National Board Of Review in New York, where he won Best Actor for his work in Uncut Gems.
When asked if he'll take on more dramas, after portraying a bit of a darker role as jeweler Howard Ratner, Adam said, "If they come my way and it makes sense, yeah, if it happens." Nonetheless, "I do like doing my comedies, but we'll see what happens," he admitted.
Uncut Gems, where Adam plays a charismatic jeweler and compulsive gambler, has been a hit at the box office following its December 13 debut. The film also received four nominations at the 25th Critics' Choice Awards on Sunday, January 12.
Adam Sandler Kathy Bates
Issa Rae Throws Major Shade After No Female Directors Are Nominated For 2020 Oscars
Eddie Murphy Hoping To Host 2020 Oscars As He Makes Career Comeback: It Would Be 'Magical'
Oscars Highlights 2019 — See The Show's Best Moments | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,937 |
L'épreuve masculine de huit aux Jeux olympiques de 2008 s'est déroulée du 9 au sur le Parc aquatique olympique de Shunyi.
Horaires
Les temps sont donnés en heure standard de la Chine (UTC+8)
Qualifications
Qualifications 1
Qualifications 2
Repêchage
Qualification Rules: 1-4→FA, 5..→FB
Finale B
Finale A
Aviron aux Jeux olympiques d'été de 2008 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 2,807 |
\section{Introduction}
\label{sec:intro}
\IEEEPARstart{H}{igh} speed digital receivers such as those used in
coherent optical
communications~\cite{morero_design_2016,faruk_digital_2017,
fludger_digital_2014,
crivelli_adaptive_2004,crivelli_architecture_2014,
agrell_roadmap_2016, roberts_high_2015,kikuchi_fundamentals_2016,
bosco_advanced_2019} require high bandwidth, high sampling rate
analog to digital converters (ADC). Current coherent receivers
operate at symbol rates around 96 Giga-baud (GBd) and require ADC
bandwidths of about \SI{50}{\giga Hz} and sampling rates close to
\SI{150}{\GHz}. In the near future symbol rates will increase to
\SIrange{128}{150}{\giga\baud} or higher, required bandwidths will be
in the range of \SIrange{65}{75}{\GHz}, and sampling rates in the
\SIrange{200}{250}{\GHz} range. The technique universally applied so
far to achieve these high bandwidths and sampling rates in coherent
transceivers is the time interleaved ADC (TI-ADC)
\cite{laperle_advances_2014,kull_cmos_2016}. Frequency interleaved
ADCs (FI-ADC) may become a promising alternative in the near future
\cite{Passetti20-ISCAS}.
The performance of TI-ADCs is affected by mismatches among the
interleaves, particularly the mismatches of sampling time, gain, and
DC offset\cite{kurosawa_explicit_2001,vogel_impact_2005}. There has
been a large body of literature dedicated to various techniques to
calibrate these mismatches~\cite{reyes_joint_2012,lin_10b_2016,
reyes_design_2017,el-chammas_12-gs/s_2011,wei_8_2014,song_10-b_2017,
haftbaradaran_background_2008, elbornsson_blind_2005,
salib_low-complexity_2017,
matsuno_all-digital_2013,devarajan_12-b_2017,
lee_1_2014,mafi_digital_2017, ali_14-bit_2016,
harpe_oversampled_2014,luna_compensation_2006}. For a thorough
review and discussion of previous TI-ADC calibration techniques,
please see \cite{murmann_digitally_2013,harpe_digitally_2015} and
references therein. In general, existing techniques suffer from one
or more of the following drawbacks: \textit{i)} Dependence on the
properties of the input signal or the oversampling factor for proper
operation\cite{wei_8_2014,song_10-b_2017,elbornsson_blind_2005,salib_low-complexity_2017,haftbaradaran_background_2008,song_10-b_2018};
\textit{ii)} Requiring an extra ADC to provide a
reference\cite{lin_10b_2016,el-chammas_12-gs/s_2011}; \textit{iii)}
Introducing intentional degradations (e.g., \textit{dither}) in the
ADC output in order to find the calibration
parameters\cite{matsuno_all-digital_2013,devarajan_12-b_2017}, and
\textit{iv)} Slow
convergence\cite{reyes_joint_2012,reyes_design_2017}. In this paper
we propose a new background technique that overcomes the above
limitations, and is applicable to receivers for digital communications
such as those used for coherent optical transmission
\cite{morero_design_2016}. The basic idea consists in the use of a low
complexity adaptive equalizer, called \textit{Compensation Equalizer}
(CE), to compensate the mismatches of the TI-ADC. The CE is adapted
using the stochastic gradient descent (SGD) algorithm and a post
processed version of the error available at the slicer of the
receiver, where the post processing is done using the backpropagation
algorithm \cite{rumelhart_learning_1986,goodfellow_deep_2016}.
Next we discuss some of the state-of-the-art TI-ADC calibration
techniques and compare them to the one proposed in this paper. The
approaches developed in~\cite{wei_8_2014, salib_low-complexity_2017,
lin_10b_2016, haftbaradaran_background_2008, elbornsson_blind_2005}
use the autocorrelation of the quantized signal to estimate the timing
mismatches and adjust them in the analog or digital domain. One
serious limitation of this technique is related to the properties of
the input signal~\cite{wei_8_2014, haftbaradaran_background_2008},
which cause the calibration algorithm to diverge for some particular
input frequencies (given by $f_{in}=m\frac{f_s}{2M}$, with
$m\in \left\{0,\cdots,M-1\right\}$ where $f_s$ is the sampling rate
and $M$ is the number of interleaves of the TI-ADC). This makes this
technique not suitable for receivers with a particular oversampling
ratio (OSR).
Other techniques perform the calibration based on statistical
properties of the quantized
signal~\cite{song_10-b_2017,lee_1_2014,mafi_digital_2017,song_10-b_2018}.
Histograms of sub-ADC outputs are created and calculations such as the
variance or cumulative distribution function are made to estimate the
error introduced by the timing mismatch. The effectiveness of these
approaches depends on the input signal characteristics and therefore
their robustness is problematic. In other examples,
\cite{lee_1_2014,song_10-b_2017} an auxiliary channel (working at the
overall sampling rate of the TI-ADC) is required to provide a
reference or to enable the estimation, respectively.
Dither injection techniques are based on the addition of a signal in
either the analog or digital domain to facilitate the estimation of
the calibration parameters~\cite{devarajan_12-b_2017, ali_14-bit_2016,
morie_71db-sndr_2013, harpe_oversampled_2014}. They are often used
in high resolution ADCs ($>10$ \emph{effective number of bits} or
ENOB). Depending on the level of the dither signal, this technique
can limit the swing of the input signal. One of the most common
limitations of existing techniques is their inability to adjust
calibration parameters of \textit{different} nature simultaneously. A
technique where calibration of a given impairment does not depend on
calibration of the other impairments is highly desirable. For example,
the stability of the calibration of the timing mismatch should not
rely on how free from offset, gain or bandwidth mismatches the sampled
signal is.
This work presents a new technique based on adaptive equalization
\cite{solis_background_2020} that runs inherently in background and is
able to overcome the aforementioned limitations. The proposed
technique is intended to be applied in high-speed receivers for
digital communication systems such as those used in coherent optical
transmission \cite{morero_design_2016}.
Adaptive equalization has been shown \cite{luna_compensation_2006,
agazzi_90_2008} to be a powerful technique to compensate errors in
TI-ADCs. However, its application to some types of receivers for
digital communications, particularly coherent optical transceivers,
has been limited by the lack of availability of a suitable error
signal to use in the SGD algorithm. In \cite{luna_compensation_2006,
agazzi_90_2008} the equalizer used to compensate TI-ADC impairments
is the main receiver equalizer, or \textit{Feedforward Equalizer}
(FFE). This is possible in the referenced works because the FFE is
immediately located after the TI-ADC, without any other blocks in
between. Therefore the FFE can access and compensate directly the
impairments of the different interleaves. Also, the slicer error
carries information about the impairments of the individual
interleaves and therefore the FFE adaptation algorithm can drive its
coefficients to a solution that jointly compensates the channel and
the TI-ADC impairments. In the case of coherent optical receivers
there is at least one block, the \textit{Bulk Chromatic Dispersion
Equalizer} (BCD) between the TI-ADC and the FFE. The BCD causes
signal components associated with different interleaves of the TI-ADC
to be mixed in a way that makes the use of the FFE unsuitable to
compensate them. Therefore a \textit{separate} equalizer, immediately
located after the TI-ADC, is necessary. This is the previously
mentioned CE. Although in this way the CE has direct access to the
impairments of the different TI-ADC interleaves, the slicer error is
not directly applicable to adapt it, because error components
associated with different TI-ADC have also been mixed by the BCD (and
possibly other signal processing blocks, depending on the architecture
of the receiver). This paper solves that problem through the use of
the backpropagation algorithm ~\cite{rumelhart_learning_1986}, an
algorithm widely used in machine
learning~\cite{goodfellow_deep_2016}. Its main characteristic is that,
in a multi-stage processing chain where several cascaded blocks have
adaptive parameters, it is able to determine the error generated by
each one of these sets of parameters for all the
stages. Backpropagation is used in combination with the SGD algorithm
to adjust the parameters of the CE in order to minimize the Mean
Squared Error (MSE) at the slicer of the receiver. The use of the CE
in combination with the backpropagation algorithm results in robust,
fast converging background compensation or calibration. As mentioned
before, the compensation is not limited to the impairments of
individual TI-ADCs (which is the case for traditional calibration
techniques), but it extends itself to the entire receiver analog front
end, enabling the compensation of impairments such as time skew
between the in-phase and the quadrature components of the signal in a
receiver based on Quadrature Amplitude Modulation (QAM) or Phase
Modulation (PM).
In the architecture just described, the compensation is achieved by an
all-digital technique. A variant of the technique based on a
mixed-signal calibration is also proposed in this paper. In this
variant, the backpropagation and the SGD algorithms are used to
estimate the TI-ADC mismatch errors, but the equalizer per se is not
built.
Because ultrafast adaptation is usually not necessary, the
backpropagation algorithm can be implemented in a highly subsampled
hardware block which does not require parallel processing. Therefore
the implementation complexity of the proposed technique is low, as
discussed in detail in Section \ref{s:complexity}.
The rest of this paper is organized as follows. Section
\ref{sec:System Model} presents a discrete time model of the TI-ADC
system in a dual polarization optical coherent receiver. The error
backpropagation based adaptive compensation equalizer is introduced in
Section \ref{sec:EBP}. Simulation results are presented and discussed
in Section \ref{s:sim_res}. The hardware complexity of the proposed
compensation scheme is discussed in Section \ref{s:complexity}, and
conclusions are drawn in Section \ref{s:conclusion}.
\section{System Model}
\label{sec:System Model}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{receiver_OFE.eps}
\caption{\label{f:ofe} Optical/analog front-end for a TI-ADC-based
coherent optical receiver. The optical signal is split into four
electrical lanes that are converted by a TI-ADC. \emph{PBS}:
polarization beam splitter; \emph{LO}: local oscillator;
\emph{$90^o$ Hyb}: $90^o$ hybrid coupler.}
\end{figure}
Although the new compensation algorithm proposed in this work can be
applied to any high-speed digital communication receiver, to make the
discussion more concrete we focus the study on dual-polarization (DP)
optical coherent transceivers
\cite{morero_design_2016,faruk_digital_2017, fludger_digital_2014,
crivelli_adaptive_2004,crivelli_architecture_2014}. A block diagram
of an optical front-end (OFE) for a DP coherent receiver is shown in
Fig.~\ref{f:ofe}. The optical input signal is decomposed by the OFE to
obtain four components, the in-phase and quadrature (I/Q) components
of the two polarizations (H/V). The photodetectors convert the
optical signals to photocurrents which are amplified by
trans-impedance amplifiers (TIAs). The analog front-end (AFE) is in
charge of the acquisition and conversion of the electrical signal to
the digital domain. Typically, oversampled digital receivers are used
to compensate the dispersion experienced in optical links (e.g.,
$T_s=\frac{T}{2}$ where $T_s$ and $T$ are the sampling and symbol
periods, respectively)~\cite{crivelli_adaptive_2004}. Next we develop
the model of the optical channel used in the remainder of this paper.
Let
$a_k^{({\mathcal P})}=a_k^{({\mathcal P},{ I})}+ja_k^{({\mathcal
P},{Q})}$ be the $k$-th quadrature amplitude modulation (QAM)
symbol in polarization ${\mathcal P}\in\{H,V\}$. An optical fiber link
with chromatic dispersion (CD) and polarization-mode dispersion (PMD)
can be modeled as a $2 \times 2$ \emph{multiple-input multiple output}
(MIMO) complex-valued channel \cite{crivelli_adaptive_2004}
encompassing four complex filters with impulse responses
${\overline h}_{m,n}(t)$ where $m,n=1,2$. Then, the received
noise-free electrical signals provided by the optical demodulator can
be expressed as \cite{crivelli_adaptive_2004}
\begin{align}
\label{eq:eq0H}
s^{(H)}(t)&=s^{(H,I)}(t)+js^{(H,Q)}(t)\\
\nonumber
&=e^{j\omega_0 t}\left[\sum_k a_k^{(H)}{\overline h}_{1,1}(t-kT)+ a_k^{(V)}{\overline h}_{1,2}(t-kT)\right],\\
\label{eq:eq0V}
s^{(V)}(t)&=s^{(V,I)}(t)+js^{(V,Q)}(t)\\
\nonumber
&=e^{j\omega_0t}\left[\sum_k a_k^{(H)}{\overline h}_{2,1}(t-kT)+ a_k^{(V)}{\overline h}_{2,2}(t-kT)\right],
\end{align}
where $1/T$ is the symbol rate and $\omega_0$ is the optical carrier
frequency offset.
\subsection{Discrete-Time Model of the AFE and the TI-ADC}
In this section we introduce a discrete-time model for the AFE and
TI-ADC system of Fig. \ref{f:ofe} including their impairments. A
simplified model of the analog path for one component
${\mathcal C}\in\{I,Q\}$ in a given polarization
${\mathcal P\in\{H,V\}}$ is shown in Fig.~\ref{f:fig1}. Each lane of
the AFE includes a filter with impulse response
$c^{({\mathcal P},{\mathcal C})}(t)$ that models the response of the
electrical interconnections between the optical demodulator and the
TIA, the TIA response itself, and any other components in the signal
path up to an $M$-parallel TI-ADC system. Mismatches between
$c^{({\mathcal P},I)}(t)$ and $c^{({\mathcal P},Q)}(t)$ may cause time
delay or \emph{skew} between components $I$ and $Q$ of a given
polarization $\mathcal P$, which degrade the receiver performance. As
we shall show here, the proposed background calibration algorithm is
able to compensate not only the imperfections of the TI-ADC, but also
the I/Q skew and any other mismatches among the signal paths.
The independent frequency responses of the $M$ track and hold units in
an $M$-channel TI-ADC system are modeled by blocks
$f^{({\mathcal P},{\mathcal C})}_{m}(t)$ with $m=0,\cdots, M-1$. Each
$M$-way interleaved TI-ADC path is sampled every $M/f_s=MT_s$ seconds
with a proper sampling phase. Parameters
$\delta_m^{({\mathcal P},{\mathcal C})}$ and
$o^{({\mathcal P},{\mathcal C})}_m$ model the sampling time errors and
the DC offsets, respectively. Path gains are modeled by
\begin{equation}
\label{eq:gain_error}
\gamma^{({\mathcal P},{\mathcal C})}_m=1+\Delta_{\gamma^{({\mathcal
P},{\mathcal C})}_m},
\end{equation}
where $\Delta_{\gamma^{({\mathcal P},{\mathcal C})}_m}$ is the gain
error.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{fs_fig1.eps}
\caption{\label{f:fig1} Analog front-end for polarization
$\mathcal P \in\{H,V\}$ and component $\mathcal C \in\{I,Q\}$ in a
TI-ADC-based DP coherent optical receiver.}
\end{figure}
Following \cite{luna_compensation_2006,agazzi_90_2008}, the sampling
phase error $\delta_m^{({\mathcal P},{\mathcal C})}$ and the path gain
$\gamma_m^{({\mathcal P},{\mathcal C})}$ are modeled by an analog
interpolation filter with impulse response
$p_m^{({\mathcal P},{\mathcal C})}(t)$ followed by ideal sampling as
depicted in Fig. \ref{f:fig2}. Assuming that the bit-resolution of the
ADC's is sufficiently high, the quantizer can be modeled as additive
white noise with uniform distribution. Also, at high-frequency (i.e.,
$1/T_s$), the offsets $o^{({\mathcal P},{\mathcal C})}_m$ generate an
$M$-periodic signal denoted as
${\tilde o}^{({\mathcal P},{\mathcal C})}[n]$ such that
${\tilde o}^{({\mathcal P},{\mathcal C})}[n]={\tilde o}^{({\mathcal
P},{\mathcal C})}[n+M]$ with
\begin{equation}
\label{eq:tilde)}
{\tilde o}^{({\mathcal P},{\mathcal C})}[m]={o}^{({\mathcal
P},{\mathcal C})}_m,\quad m=0,\cdots,M-1.
\end{equation}
Then, the digitized high-frequency samples can be expressed as
\begin{equation}
\label{eq:eq1}
y^{({\mathcal P},{\mathcal C})}[n]=r^{({\mathcal P},{\mathcal C})}[n]+{\tilde o}^{({\mathcal P},{\mathcal C})}[n]+q^{({\mathcal P},{\mathcal C})}[n],
\end{equation}
where $r^{({\mathcal P},{\mathcal C})}[n]$ is the signal component
provided by the $M$-channel TI-ADC, and
$q^{({\mathcal P},{\mathcal C})}[n]$ is the quantization noise (see
Fig. \ref{f:fig2}).
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{fs_fig2.eps}
\caption{\label{f:fig2} Modified model of the analog front-end and
TI-ADC for polarization $\mathcal P \in\{H,V\}$ and component
$\mathcal C \in\{I,Q\}$ in a DP coherent optical receiver.}
\end{figure}
We define the total impulse response of a given subchannel as
\begin{equation}
\label{eq:eq2}
h_m^{({\mathcal P},{\mathcal C})}(t)=c^{({\mathcal P},{\mathcal C})}(t)\otimes f_m^{({\mathcal P},{\mathcal C})}(t) \otimes p_m^{({\mathcal P},{\mathcal C})}(t),
\end{equation}
where $m=0,\cdots, M-1$ and $\otimes$ denotes the convolution
operation. Let $H_m^{({\mathcal P},{\mathcal C})}(j\omega)$ and
$S^{({\mathcal P},{\mathcal C})}(j\omega)$ be the Fourier transforms
(FTs) of $h_m^{({\mathcal P},{\mathcal C})}(t)$ and
$s^{({\mathcal P},{\mathcal C})}(t)$, respectively. In digital
communication systems with spectral shaping is
$|S^{({\mathcal P},{\mathcal C})}(j\omega)|\approx 0$ for
$|\omega|\ge \pi/T_s$. Further assuming that
$|H_m^{({\mathcal P},{\mathcal C})}(j\omega)|\approx 0$ for
$|\omega|\ge \pi/T_s$, the analog filtering of Fig. \ref{f:fig2} can
be represented by a real discrete-time model as depicted in
Fig. \ref{f:fig3} by using
\begin{equation}
\label{eq:eq3}
h^{({\mathcal P},{\mathcal C})}_m[n]=T_sh_m^{({\mathcal P},{\mathcal C})}(nT_s),\quad m=0,\cdots,M-1.
\end{equation}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{fs_fig3.eps}
\caption{\label{f:fig3} Equivalent discrete-time model of the analog
front-end and TI-ADC system with impairments for the signal
component given by \eqref{eq:eq4} (i.e., without DC offsets and
quantization noise) for polarization $\mathcal P \in\{H,V\}$ and
component $\mathcal C \in\{I,Q\}$.}
\end{figure}
Therefore it can be shown that the digitized high-frequency signal can
be expressed as:
\begin{equation}
\label{eq:eq4}
r^{({\mathcal P},{\mathcal C})}[n]=\sum_{l} {\tilde h}^{({\mathcal P},{\mathcal C})}_n[l] s^{({\mathcal P},{\mathcal C})}[n-l],
\end{equation}
where
$s^{({\mathcal P},{\mathcal C})}[n]=s^{({\mathcal P},{\mathcal
C})}(nT_s)$ and ${\tilde h}^{({\mathcal P},{\mathcal C})}_n[l]$ is
the impulse response of a time-varying filter, which is an
$M$-periodic sequence such
${\tilde h}^{({\mathcal P},{\mathcal C})}_n[l]={\tilde h}^{({\mathcal
P},{\mathcal C})}_{n+M}[l]$, and defined by
\begin{equation}
\label{eq:eq5}
{\tilde h}^{({\mathcal P},{\mathcal C})}_n[l]={h}^{({\mathcal P},{\mathcal C})}_n[l], \quad n=0,\cdots,M-1,
\forall l,
\end{equation}
with ${h}^{({\mathcal P},{\mathcal C})}_n[l]$ given by
\eqref{eq:eq3}\footnote{See \cite{saleem_adaptive_2010} and references
therein for more details about this formulation.}. We highlight that
\eqref{eq:eq4} includes the impact of both the AFE mismatches and the
$M$-channel TI-ADC impairments. Replacing \eqref{eq:eq4} in
\eqref{eq:eq1}, the digitized high-frequency sequences result
\begin{align}
\nonumber
y^{({\mathcal P},{\mathcal C})}[n]=&\sum_{l} {\tilde h}^{({\mathcal P},{\mathcal C})}_n[l] s^{({\mathcal P},{\mathcal C})}[n-l]+{\tilde o}^{({\mathcal P},{\mathcal C})}[n]+\\
\label{eq:eq1b}
&q^{({\mathcal P},{\mathcal C})}[n].
\end{align}
\subsection{Compensation of AFE Mismatch and TI-ADC Impairments}
Similar to what was done in previous works
\cite{luna_compensation_2006,agazzi_90_2008, saleem_adaptive_2010}, we
propose to use an adaptive digital compensation filter applied after
the mitigation of the offset sequence, i.e.,
\begin{equation}
\label{eq:eq6}
x^{({\mathcal P},{\mathcal C})}[n]=\sum_{l=0}^{L_g-1} {\tilde g}^{({\mathcal P},{\mathcal C})}_n[l]
{w}^{({\mathcal P},{\mathcal C})}[n-l],
\end{equation}
where ${\tilde g}^{({\mathcal P},{\mathcal C})}_n[l]$ is the
$M$-periodic time-varying impulse response of the compensation filter
(i.e.,
${\tilde g}^{({\mathcal P},{\mathcal C})}_n[l]={\tilde g}^{({\mathcal
P},{\mathcal C})}_{n+M}[l]$), $L_g$ is the number of taps of the
compensation filters, and $w^{({\mathcal P},{\mathcal C})}[n]$ is the
offset compensated signal given by
\begin{equation}
\label{eq:w}
w^{({\mathcal P},{\mathcal C})}[n]=y^{({\mathcal P},{\mathcal
C})}[n]-{\hat {\tilde o}}^{({\mathcal P},{\mathcal C})}[n],
\end{equation}
with ${\hat {\tilde o}}^{({\mathcal P},{\mathcal C})}[n]$ being the
estimated $M$-periodic offset sequence. The combination of the offset
compensation blocks and the compensation filters
${\tilde g}^{({\mathcal P},{\mathcal C})}_n[l]$ constitutes the
\emph{Compensation Equalizer} (Fig. \ref{f:f5}).
A proper strategy to estimate the response of the CE is
required. Notice that adaptive calibration techniques based on a
reference ADC such as in \cite{saleem_adaptive_2010} cannot be used to
compensate mismatches between the $I$ and $Q$ signal paths. In the
following we propose the \emph{backpropagation} technique to adapt the
CE.
\section{Error Backpropagation Based Compensation of AFE and TI-ADC
Impairments in DP Optical Coherent Receivers}
\label{sec:EBP}
Based on the previous analysis, Fig. \ref{f:f5} depicts a block
diagram of the AFE+TI-ADC in a dual-polarization optical coherent
receiver with the adaptive calibration block, which includes four
instances of the real filter as defined by \eqref{eq:eq6}. For
simplicity, we modified the notation of the system model of
Fig. \ref{f:fig3}. Note that we use an integer index between 1 and 4
to represent a certain component in a given polarization:
$``(1) "=(H,I), ``(2)"=(H,Q), ``(3)"=(V,I)$, and $``(4)"=(V,Q)$.
\begin{figure}[t]
\centering
\includegraphics[width=\columnwidth]{fig5.eps}
\caption{\label{f:f5} Block diagram of a dual-polarization optical
coherent receiver with the \emph{Compensation Equalizer} (CE) for
mitigating the effects of both AFE mismatches and TI-ADC
impairments.}
\end{figure}
The main receiver functions are included in the digital signal
processing (DSP) block of Fig. \ref{f:f5}, which works with samples
every $T_s$ seconds. In summary, some of the most important DSP
algorithms used in these receivers are the chromatic dispersion
equalizer (or BCD), the MIMO FFE to compensate the polarization-mode
dispersion, \emph{Timing Recovery} (TR) from the received symbols, the
\emph{Fine Carrier Recovery} (FCR) to compensate the carrier phase and
frequency offset, and the \emph{Forward Error Correction} (FEC)
decoder. Readers interested in more details on optical coherent
receivers can see \cite{morero_design_2016, faruk_digital_2017,
fludger_digital_2014} and references therein.
\subsection{All Digital Compensation Architecture}
Let ${g}^{(i)}_m[l]$ with $i=1,\cdots,4$ be the filter impulse
response ${\tilde g}^{(i)}_{n}[l]$ in one period defined as
\begin{equation}
\label{eq:gPC}
{g}^{(i)}_m[l]={\tilde g}^{(i)}_{m+n_0}[l], \quad m=0,\cdots,M-1,
\end{equation}
where $l=0,\cdots,L_g-1$ and $n_0$ is an arbitrary time index multiple
of $M$. The filter taps of the CE ${g}^{(i)}_m[l]$ are adapted by
using the slicer error at the output of the receiver DSP block. Let
$e_k^{(j)}$ be the \emph{slicer error} defined by
\begin{equation}
\label{eq:ePC}
e_k^{(j)}=u_k^{(j)}-{\hat a}_k^{(j)},\quad j=1,\cdots,4,
\end{equation}
where $u_k^{(j)}$ is the input of the slicer and ${\hat a}_k^{(j)}$ is
the $k$-th detected symbol at the slicer output (see
Fig. \ref{f:f5}). Notice that the sampling rate of the slicer inputs
$u_k^{(j)}$ is $1/T$, therefore a subsampling of $T/T_s$ is carried
out after the receiver DSP block. Then, we define the total squared
error at the slicer as
\begin{equation}
\label{eq:eT}
{\mathcal E}_k=\sum_{j=1}^4|e_k^{(j)}|^2.
\end{equation}
Let $E\{{\mathcal E}_k\}$ be the MSE at the slicer with $E\{.\}$
denoting the expectation operator. In this work we use the \emph{least
mean squares} (LMS) algorithm to iteratively adapt the real
coefficients of the CE given by \eqref{eq:gPC}, in order to minimize
the MSE at the slicer:
\begin{equation}
\label{eq:lmsg}
{\bold g}^{(i)}_{m,p+1}={\bold g}^{(i)}_{m,p}- \beta
\nabla_{{\bold g}^{(i)}_{m,p}} E\{{\mathcal
E}_k\},
\end{equation}
where $i=1,\cdots,4,$; $m=0,\cdots,M-1$; $p$ denotes the number of
iteration, ${\bold g}^{(i)}_{m,p}$ is the $L_g$-dimensional
coefficient vector at the $p$-th iteration given by
\begin{equation}
\label{eq:vgPC}
{\bold g}^{(i)}_{m,p}=\left[{g}^{(i)}_{m,p}[0],{g}^{(i)}_{m,p}[1],\cdots,{g}^{(i)}_{m,p}[L_g-1] \right]^T;
\end{equation}
$\beta$ is the adaptation step, and
$\nabla_{{\bold g}^{(i)}_{m,p}} E\{{\mathcal E}_k\}$ is the gradient
of the MSE with respect to the filter vector ${\bold g}^{(i)}_{m,p}$.
We emphasize that the computation of the MSE gradient is not trivial
since ${\mathcal E}_k$ is not the error at the output of the CE
block. To get the proper error samples to adapt the coefficients of
the filters as expressed in \eqref{eq:lmsg}, we propose the
\emph{backpropagation algorithm} widely used in \emph{machine
learning} \cite{rumelhart_learning_1986,
goodfellow_deep_2016}. Towards this end, the slicer errors are
backpropagated as described in~\cite{morero_forward_2018}. Finally,
based on these backpropagated errors we can estimate the gradient
$\nabla_{{\bold g}^{(i)}_{m,p}} E\{{\mathcal E}_k\}$ as usual in the
classical LMS algorithm.
\subsection{Error Backpropagation (EBP)}
\label{s:bp_formulation}
\begin{figure}
\centering \includegraphics[width=\columnwidth]{fs_fig6b.eps}
\caption{\label{f:f6} Block diagram of the proposed \emph{error
backpropagation} (EBP) based adaptation architecture for
AFE+TI-ADC impairment compensation in a dual-polarization optical
coherent receiver with $T/T_s=2$.}
\end{figure}
Without loss of generality, we assume that the receiver DSP block can
be modeled as a real time-varying $4 \times 4$ MIMO $T/2$ fractional
spaced equalizer (i.e., $T_s=T/2$), which is able to compensate CD and
PMD among other optical fiber channel effects. Then, the downsampled
output of the $T/2$ receiver DSP block can be written as (see Fig
\ref{f:f6})
\begin{equation}
\label{eq:u1}
u^{(j)}_k=\sum_{i=1}^4\sum_{l=0}^{L_{\Gamma}-1}
{\Gamma}^{(j,i)}_{2k}[l]{x}^{(i)}[2k-l],\quad j=1,\cdots,4,
\end{equation}
where ${\Gamma}_n^{(j,i)}[l]$ is the time-varying impulse response of
the filter with input $i$ and output $j$, $L_{\Gamma}$ is the number
of taps of the filter, while ${x}^{(i)}[l]$ is the signal at the DSP
block input $i$ given by \eqref{eq:eq6}, i.e.,
\begin{equation}
\label{eq:eq6b}
x^{(i)}[n]=\sum_{l'=0}^{L_g-1} {g}^{(i)}_{\lfloor n\rfloor_M}[l']
{w}^{(i)}[n-l'],\quad i=1,\cdots,4,
\end{equation}
where ${g}^{(i)}_m$ is the impulse response defined by \eqref{eq:gPC},
$\lfloor .\rfloor_M$ denotes the modulo $M$ operation, and
${w}^{(i)}[n]$ is the DC compensated signal \eqref{eq:w}.
As usual with the SGD based adaptation, we replace the gradient of the
MSE, $\nabla_{{\bold g}^{(i)}_{m,p}} E\{{\mathcal E}_k\}$, by a noisy
estimate, $\nabla_{{\bold g}^{(i)}_{m}} {\mathcal E}_k$. In the
Appendix we show that an \emph{instantaneous} gradient of the squared
error \eqref{eq:eT} can be expressed as
\begin{equation}
\label{eq:grad}
\nabla_{{\bold g}^{(i)}_{m}} {\mathcal E}_k=\alpha {\hat e}^{(i)}[m+kM]{\bold w}^{(i)}[m+kM],
\end{equation}
where $\alpha$ is a certain constant, ${\bold w}[n]$ is the
$L_g$-dimensional vector with the samples at the CE input, i.e.,
\begin{equation}
\label{eq:vecw}
{\bold w}^{(i)}[n]=\left[{w}^{(i)}[n],{w}^{(i)}[n-1],\cdots,{w}^{(i)}[n-L_g+1] \right]^T,
\end{equation}
while ${\hat e}^{(i)}[n]$ is the \emph{backpropagated error} given
by
\begin{equation}
\label{eq:bpe}
{\hat e}^{(i)}[n]=\sum_{j=1}^4\sum_{l=0}^{L_{\Gamma}-1}\Gamma^{(j,i)}_{n+l}[l] e^{(j)}[n+l],
\end{equation}
with $e^{(j)}[n]$ being the \emph{oversampled} slicer error obtained
from the \emph{baud-rate} slicer error $e_k^{(j)}$ in \eqref{eq:ePC}
as
\begin{equation}
\label{eq:oe}
e^{(j)}[n] =
\begin{cases}
e_{n/2}^{(j)} & \mbox{if } n= 0,\pm 2,\pm 4,\cdots \\
0 & \mbox{otherwise}
\end{cases}.
\end{equation}
Then, a full digital compensation architecture can be derived by using
an adaptive CE with
\begin{equation}
\label{eq:lmsg2}
{\bold g}^{(i)}_{m,p+1}={\bold g}^{(i)}_{m,p}- \mu \nabla_{{\bold g}^{(i)}_{m,p}} {\mathcal E}_k,
\end{equation}
where $\mu=\alpha \beta$ is the step-size. Furthermore, based on the
backpropagated error \eqref{eq:bpe} it is possible to estimate the DC
offsets in the input samples as follows
\begin{equation}
{\hat {o}}^{(i)}_{p+1}[m]={\hat {o}}^{(i)}_{p}[m]-\mu_o{\hat
e}^{(i)}[n+m], \quad m=0,\cdots, M-1,
\label{eq:tildeo}
\end{equation}
where ${\hat {o}}^{(i)}_{p}[m]$ is the estimate at the $p$-th
iteration of the DC offset sequence in one period (see \eqref{eq:w}),
and $\mu_o$ is the step-size of the DC offset estimator.
Competition between the CE and any adaptive DSP blocks in
$\Gamma_n^{j,i}[l]$ (e.g., the FFE) may generate instability,
therefore an adaptation constraint must be included. For example, one
of the $4M$ sets of the filter coefficients can be limited to only be
a time delay line, for example, ${g}^{(0)}_{0}[l]=\delta_{l,l_d}$
where $l=0,\cdots,L_g-1$ and $l_d=\frac{L_g+1}{2}$ ($L_g$ is assumed
odd).
Since channel impairments change slowly over time, the coefficient
updates given by \eqref{eq:lmsg2} and \eqref{eq:tildeo} do not need to
operate at full rate, and subsampling can be applied. The latter
allows implementation complexity to be significantly
reduced. Additional complexity reduction is enabled by: 1) strobing
the algorithms once they have converged, and/or 2) implementing them
in firmware in an embedded processor, typically available in coherent
optical transceivers. Practical aspects of the hardware implementation
shall be discussed in Section \ref{s:complexity}.
\subsection{Mixed-Signal Compensation Architecture}
\label{sec:mixed}
A mixed-signal based calibration technique can be also derived from
the error backpropagation (EBP) algorithm described in the previous
section. Toward this end, sampling phase, gain, and offsets are
adjusted before the ADC\footnote{Compared
to the full-digital compensation architecture, notice that the
described mixed-signal solution is not able to compensate some
effects such as bandwidth mismatches.} by using the gradient of the backpropagated
slicer error as depicted in Fig. \ref{f:ms_cal_sch}. Similarly to the full
digital solution, the DC offsets in the mixed-signal calibration
approach are compensated by using \eqref{eq:tildeo}. The gain is
iteratively adjusted by using
\begin{equation}
\hat \gamma^{(i)}_{m,p+1}=\hat \gamma^{(i)}_{m,p}- \mu_{\gamma}
{\hat e}^{(i)}[m+kM]w^{(i)}[m+kM],\quad \forall k,
\end{equation}
where $m=0,\cdots,M-1$ and $i=1,2,3,4$. Finally, since the
backpropagated slicer error is available at the ADC outputs, the
sampling phase can be iteratively adjusted by using the \emph{MMSE
timing recovery algorithm}~\cite{lee_digital_2004}, i.e.,
\begin{align}
\hat \tau^{(i)}_{m,p+1}=&\hat \tau^{(i)}_{m,p}-\mu_{\tau}{\hat e}^{(i)}[m+kM]\times\\
\nonumber
&\left(w^{(i)}[m+kM+1]-w^{(i)}[m+kM-1]\right),\quad
\forall k
\end{align}
with $m=0,\cdots,M-1$. The calibration algorithm adjusts analog
elements already present in most implementations of the TI-ADC
\cite{reyes_design_2017,
reyes_energy-efficient_2019,kull_24-72-gs/s_2018}. The clock
sampling phase is adjusted with variable delay lines, gain and offset
can be corrected in the comparator or with programmable gain
amplifiers (PGA), if needed.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{mixed_signal_cal.eps}
\caption{\label{f:ms_cal_sch}Block diagram of the mixed-signal
calibration variant. The calibration with analog elements enables
power consumption reduction. The EBP is the same as the
all-digital variant.}
\end{figure}
\section{Simulation Results}
\label{s:sim_res}
\begin{table}[]
\centering
\caption{Parameters Used in Simulations (UDRVD: Uniformly Distributed
Random Variable. VFS: Full-Scale Voltage).}
\label{t:sim_parameters}
\begin{tabular}{l|c}
\hline
{\textbf{Parameter}} & \textbf{Value}
\\ \hline
Modulation & 16-QAM \\
Symbol Rate ($f_B=1/T$) & 96 GBd \\
Receiver Oversampling Factor ($T/T_s$) & 4/3 \\
Fiber Length & 100 km \\
Differential Group Delay (DGD) & 10 ps \\
Second Order Pol. Mode Disp. (SOPMD) & 1000 ps$^2$ \\
Speed of Rotation of the Pol. at the Tx & 2 kHz \\
Speed of Rotation of the Pol. at the Rx & 20 kHz \\
TI-ADC Resolution & 8 bit \\
TI-ADC Sampling Rate (all interleaves) & 128 GS/s \\
Number of Interleaves of TI-ADC ($M$) & 16 \\
Number of Taps of CE ($L_g$) & 7\\
Rolloff Factor & 0.10\\
Nominal BW of Analog Paths ($B_0$) (see \eqref{eq:B}) & 53 GHz\\
Gain Errors (see \eqref{eq:gain_error}) - UDRV & $\Delta_{\gamma^{(i)}_m}\in [\pm 0.15]$ \\
Sampling Phase Errors - UDRV & $\delta_m^{(i)} \in [\pm 0.10]T$\\
Bandwidth Mismatches (see \eqref{eq:B}) - UDRV & $\Delta_{B_{m}^{(i)}} \in [\pm 0.075]B_0$\\
I/Q Time Skew (see \eqref{eq:tau}) - UDRV & $\tau_H,\tau_V\in [\pm 0.10]T$\\
DC Offsets - UDRV & $o_m^{(i)} \in [\pm 0.025]$VFS\\
\\ \hline
\end{tabular}
\end{table}
\begin{figure}[t]
\centering
\includegraphics[width=\columnwidth]{setup_sim2.eps}
\caption{\label{f:simulation_setup}Block diagram of the system model
used in the simulations.}
\end{figure}
The performance of the proposed backpropagation based adaptive CE is
investigated by running Montecarlo simulations of the setup shown in
Fig.~\ref{f:simulation_setup} and defined in Table
\ref{t:sim_parameters}. Each test consists of 500 cases where the
impairment parameters are obtained by using uniformly distributed
random variables (UDRV). The electrical analog path responses
\eqref{eq:eq2} are simulated with first-order lowpass filters with
3dB-bandwidth defined by
\begin{equation}
\label{eq:B}
B^{(i)}_m=B_0+\Delta_{B^{(i)}_m},\quad i=1,2,3,4;\quad
m=0,\cdots,M-1,
\end{equation}
where $B_0$ is the nominal BW and $\Delta_{B^{(i)}_m}$ is the BW
mismatch. Let $\tau_m^{(i)}$ be the mean group delay of the filter of
the $m$-th channel and $i$-th component. The impact of the I/Q time
skew of polarizations $H$ and $V$ defined as
\begin{equation}
\label{eq:tau}
\tau_H={\overline \tau}^{(1)}-{\overline \tau}^{(2)},\quad \tau_V={\overline \tau}^{(3)}-{\overline \tau}^{(4)}
\end{equation}
with ${\overline \tau}^{(i)}=\frac{1}{M}\sum_{m=0}^{M-1}\tau_m^{(i)}$,
is also investigated. We consider a 16-QAM modulation scheme with a
symbol rate of $1/T=\SI{96}{\giga\baud}$. Raised cosine filters with
rolloff factor $0.10$ for transmit pulse shaping are simulated (i.e.,
the nominal BW of the channel filters is
$B_0=1.1\times \frac{96}{2}\approx 53$ GHz). The optical
signal-to-noise ratio (OSNR) is set to that required to achieve a
bit-error-rate (BER) of $\sim 1.2\times 10^{-3}$ (see
\cite{freude_quality_2012,chan_optical_2010} for the definition of
OSNR). The oversampling factor in the DSP blocks is $T/T_s=4/3$. The
fiber length is \SI{100}{\km} with \SI{10}{\ps} of differential group
delay (DGD) and \SI{1000}{ps^2} of second-order PMD (SOPMD). Rotations
of the state of polarization (SOP) of \SI{2}{\kHz} and \SI{20}{\kHz}
are included at the transmitter and receiver, respectively.
TI-ADCs with 8-bit resolution, \SI{128}{\giga\sample\per\s} sampling
rate, and $M=16$ are simulated. The number of taps of the digital
compensation filters is $L_g=7$.
\subsection{Montecarlo Simulations of the Adaptive CE}
\label{sec:montecarlo}
Figs. \ref{f:hist1} and \ref{f:hist2} show the histograms of the BER
for the receiver with and without the CE in the presence of gain
errors, phase errors, I/Q time skew, and BW mismatches. Only one
effect is exercised in each case. Results of 500 random gain and phase
errors uniformly distributed in the interval
$\Delta_{\gamma^{(i)}_m} \in [\pm 0.15]$ (see \eqref{eq:gain_error})
and $\delta_m^{(i)}\in [\pm 0.10]T$, respectively, are depicted in
Fig. \ref{f:hist1}, whereas Fig. \ref{f:hist2} shows results for 500
random BW mismatches (see \eqref{eq:B}) and I/Q time skews (see
\eqref{eq:tau}) uniformly distributed in the interval
$\Delta_{B^{(i)}_m} \in [\pm 0.075]B_0$ and
$\tau_H,\tau_V\in [\pm 0.10]T$, respectively. In all cases, it is
observed that the proposed compensation technique is able to mitigate
the impact of all impairments when they are exercised
separately\footnote{Similar performance has been verified with random
DC offsets \cite{solis_background_2020}.}. In particular, notice
that the proposed CE with $L_g=7$ taps practically eliminates the
serious impact on the receiver performance of the I/Q time skew values
of Table \ref{t:sim_parameters}.
Fig. \ref{f:hist3} shows histograms of the BER for the receiver with
and without the CE in the presence of the combined effects. Results of
500 cases with random gain errors, sampling phase errors, I/Q time
skews, BW mismatches, and DC offsets as defined in Table
\ref{t:sim_parameters}, are presented. Performance of the CE with
$L_g=13$ taps is also depicted. As before, note that the CE is able to
compensate the impact of all combined impairments. Moreover, note that
a slight performance improvement can be achieved when the number of
taps $L_g$ increases from 7 to 13.
\begin{figure}
\centering
\includegraphics[width=.87\columnwidth]{hist_ber_1.eps}
\caption{\label{f:hist1}Histogram of the BER for 500 random cases with
and without CE for a reference BER of $\sim 1.2\times
10^{-3}$. Left: gain errors (only). Right: sampling phase errors
(only). See simulation parameters in Table \ref{t:sim_parameters}.}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=.87\columnwidth]{hist_ber_2.eps}
\caption{\label{f:hist2}Histogram of the BER for 500 random cases with
and without CE for a reference BER of $\sim 1.2\times
10^{-3}$. Left: I/Q time skew (only). Right: BW mismatch (only).
See simulation parameters in Table \ref{t:sim_parameters}.}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=.9\columnwidth]{hist_ber_3.eps}
\caption{\label{f:hist3}Histogram of the BER for 500 random cases with
combined impairments as defined in Table
\ref{t:sim_parameters}. Reference BER of $\sim 1.2\times
10^{-3}$. Top: CE w/$L_g=7$ taps. Middle: CE w/$L_g=13$
taps. Bottom: without CE.}
\end{figure}
As mentioned in Section \ref{s:bp_formulation}, the impairments of the
AFE and TI-ADCs change very slowly over time in multi-gigabit optical
coherent transceivers. Therefore the coefficient updates given by
\eqref{eq:lmsg2} and \eqref{eq:tildeo} do not need to operate at full
rate, and subsampling can be applied. Block processing and frequency
domain equalization based on the Fast Fourier Transform (FFT) are
widely used to implement high-speed coherent optical transceivers
\cite{morero_design_2016}. Then we propose to use block decimation of
the error samples to update the CE. Let $N$ be the block size in
samples to be used for implementing the EBP. Define $D_B$ the block
decimation factor. In this way, only one block of $N$ consecutive
samples of the oversampled slicer error \eqref{eq:oe} every $D_B$
blocks, i.e.,
\begin{equation}
e^{(i)}[k ND_B+n],\quad n=0,1,\cdots,N-1,\forall k
\end{equation}
with $k$ integer, is used to adapt the CE. Fig. \ref{f:ber_conv}
depicts an example of the temporal evolution of the BER in the
presence of combined impairments according to Table
\ref{t:sim_parameters} for different values of the block decimation
factor $D_B$ with $N=8192$. The instantaneous BER is evaluated every
$10^5$ symbols and then processed by a moving average filter of size
$40$. Gear shifting is used to accelerate the convergence of the CE
and reduce the steady-state MSE. In all cases, notice that the use of
block decimation practically does not impact on the resulting
BER. Therefore it can be adopted to drastically reduce the
implementation complexity, as shall be discussed in Section
\ref{s:complexity}.
\begin{figure}[t]
\centering
\includegraphics[width=1.\columnwidth]{ber_convergence_comparative.eps}
\caption{\label{f:ber_conv} Convergence of the CE in the presence
combined impairments for different block decimation factors $D_B$
with $N=8192$.}
\end{figure}
\subsection{Mixed-Signal Compensation of TI-ADC with Highly
Interleaved Architectures}
\label{ss:high_interleaved_sim}
The performance of the mixed-signal scheme of Section \ref{sec:mixed}
is investigated in typical hierarchical ultra high-speed TI-ADCs such
as those used in high speed
receivers~\cite{reyes_energy-efficient_2019, kull_24-72-gs/s_2018,
kim_161-mw_2020}. This hierarchical TI-ADC architecture organizes
the T\&H in two or more ranks with a high number of
sub-ADCs. Fig.~\ref{f:sch_hier_tiadc} depicts an example with two
ranks. Rank 1 includes $M_1$ switches each of which feeds $M_2$ T\&H
stages of Rank 2. Then, $M_1\times M_2$ ADCs are used to digitize the
input signal. Successive approximation register (SAR) ADCs are used
for this application due to their power efficiency at the required
sampling rate and resolution. This approach relaxes the requirements
for the clock generation and synchronization. Furthermore, the impact
on the input bandwidth is reduced in contrast to T\&H with direct
sampling~\cite{greshishchev_40_2010}. As an example of application of
the mixed-signal compensation scheme of Section \ref{sec:mixed}, its
performance in a hierarchical TI-ADC with $M_1=16$ and $M_2=8$ (i.e.,
$M_1\times M_2=128$ individual converters) is evaluated. A clock
jitter of \SI{100}{\fs} RMS is added to this simulation. Notice that
the mixed-signal calibration algorithm adjusts the $M_1$ sampling
phases of the switches in the first rank, and the $M_1\times M_2$
gains and offsets of the individual sub-ADCs.
Fig.~\ref{f:ber_evo_jitter} shows the temporal evolution of both the
BER and the mean \emph{signal-to-noise-and-distortion-ratio} (SNDR)
~\cite{reyes_energy-efficient_2019}. A slower convergence than the
previous simulation is observed as a result of the larger number of
converters (i.e., 128 vs 16). Nevertheless, we verify that the
proposed backpropagation based mixed-signal compensation scheme is
able to mitigate the impact of the impairments in hierarchical
TI-ADC. In particular, note that the SNDR can be improved from
$\sim 20$dB to $\sim 45$dB by using the proposed background
calibration technique.
\begin{figure}
\centering
\includegraphics[width=1.\columnwidth]{tiadc_typ_arch.eps}
\caption{\label{f:sch_hier_tiadc} Example of application of the
proposed backpropagation based mixed-signal calibration in a typical
two-rank hierarchical TI-ADC.}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=.95\columnwidth]{ber_SNDR_conv_128adc.eps}
\caption{\label{f:ber_evo_jitter} BER and SNDR evolution in a
hierarchical TI-ADC based DP optical coherent receiver with the
backpropagation based mixed-signal compensation in the presence
combined impairments. $M_1=16$ and $M_2=8$.}
\end{figure}
\section{Hardware Complexity Analysis}
\label{s:complexity}
This section discusses some practical aspects of the implementation of
the proposed compensation technique. We focus on the two main blocks
of the all digital architecture: the compensation equalizer and the
error backpropagation block.
\subsection{Implementation of the Compensation Equalizer}
\begin{figure}[t]
\centering
\includegraphics[width=1.\columnwidth]{implementacion_V2_ejemplo2.eps}
\caption{\label{f:g_par} Example of a parallel implementation of the
CE with $M=4$, $L_g=3$, and parallelism factor $P=2M=8$. }
\end{figure}
As described in Section~\ref{sec:EBP}, the compensation equalizer in a
DP optical coherent receiver comprises 4 real valued finite impulse
response (FIR) filters $\tilde{g}_n^{(i)}[l]$ with $i=1,2,3,4$, and
$l=0,\cdots, L_g-1$. From computer simulations of
Section~\ref{s:sim_res} it was observed that $L_g=7$ is enough to
properly compensate the AFE and TI-ADC impairments. Therefore a time
domain implementation is preferred for the CE. Each of these filters
has $M$ independent impulse responses $g_n^{(i)}[l]$ which are time
multiplexed as $\tilde{g}_n^{(i)}[l]=g_{\lfloor n\rfloor_M}^{(i)}[l]$
(see \eqref{eq:gPC}). Note that time multiplexing of filters with
independent responses does not translate to additional complexity when
the filter is implemented with a parallel architecture. The use of
parallel implementation is mandatory in high speed optical
communication where parallelism factors on the order of 128 o higher
are typical. In these architectures, the parallelism factor $P$ can be
chosen to be a multiple of the ADC parallelism factor $M$, i.e.,
$P=q\times M$ where $q$ is an integer. Therefore the different time
multiplexed coefficients are used in fixed positions of the
parallelism without incurring in significant additional complexity in
relation to a filter with just one set of coefficients (see
Fig. \ref{f:g_par}). We highlight that the resulting filter is
equivalent in complexity to the I/Q-skew compensation filter already
present in current coherent receivers~\cite{morero_design_2016}. Since
the proposed scheme also corrects skew, the classical skew correction
filter can be replaced by the proposed CE without incurring
significant additional area or power.
\subsection{Implementation of the Error Backpropagation Block}
A straightforward implementation of error backpropagation must include
a processing stage for each DSP block located between the ADCs and the
slicers. Typically these blocks comprise the BCD, FFE, TR
interpolators, and the FCR. All these blocks can be mathematically
modeled as a sub-case of the generic receiver DSP block used in
Section~\ref{s:bp_formulation} and the Appendix. The EBP block is
algorithmically equivalent to its corresponding DSP block with the
only difference that the coefficients are \emph{transposed} (i.e.,
compare \eqref{eq:un} and \eqref{eq:epb}). Therefore, in the worst
case, the EBP complexity would be similar to that of the receiver DSP
block\footnote{Note that the LMS adaptation hardware of the FFE, the
PLL of the FCR and the PLL of the TR do not need to be implemented
in the EBP path, which further reduces the complexity of the
latter.}. Since doubling power and area consumption is not
acceptable for commercial applications, important simplifications must
be provided.
Considering that AFE and TI-ADC impairments change very slowly over
time in multi-gigabit optical coherent transceivers, the coefficient
updates given by \eqref{eq:lmsg2} and \eqref{eq:tildeo} do not need to
operate at full rate, and subsampling can be applied. The latter
allows implementation complexity, and particularly power dissipation,
to be drastically reduced. In Section \ref{sec:montecarlo} we
evaluated the performance with block decimation where one block of $N$
consecutive samples of the oversampled slicer error are used every
$D_B$ blocks. Simulation results not included here have shown a good
performance even with $N=8192$ and $D_B=256$. The block based
decimation approach allows the EBP algorithm to be implemented in the
frequency domain when necessary to reduce complexity (for example in
the EBP of the BCD and FFE). This error decimation reduces the power
dissipation of the EBP to only $1/D_B$ of the power of the
corresponding DSP blocks, equivalent to less than 1\% in the simulated
example. However, the areas of the EBP blocks are still equivalent to
the area of their corresponding DSP blocks. To reduce area, the EBP
blocks could be implemented using a serial
architecture\footnote{Typically, a serial implementation requires that
hardware such as multipliers be reused with variable numerical
values of coefficients, whereas in a parallel implementation
hardware can be optimized for fixed coefficient values. This results
in a somewhat higher power per operation in a serial
implementation. Nevertheless, the drastic power reduction achieved
through decimation greatly outweighs this effect.} or a lower
parallelism factor. If a serial implementation is chosen, an area
reduction proportional to the parallelism factor is expected at the
expense of increasing the latency by a similar amount. The resulting
latency is $2 \times (N_{BCD}+N_{FFE}) \times P$ samples, where
$N_{BCD}$ and $N_{FFE}$ are the block sizes of the FFTs used to
implement the BCD and FFE, respectively (factor 2 includes the FFT /
IFFT pair). The latencies of the EBP blocks for the TR interpolators
and FCR can be neglected. Therefore the CE adaptation speed is not
reduced by a serial implementation of the EBP blocks if
$2 \times (N_{BCD}+N_{FFE}) \times P < N \times D_B$. Details of
efficient architectures for implementing the error backpropagation
block will be addressed in a future work.
\section{Conclusions}
\label{s:conclusion}
A new TI-ADC background calibration algorithm based on the
backpropagation technique has been presented in this paper. Two
implementation variants were presented, one of them all-digital and
the other mixed-signal. Simulation results have shown a fast, robust
and almost ideal compensation/calibration of TI-ADC sampling time,
gain, offset, and bandwidth mismatches as well as I/Q time skew
effects under different test conditions in the example of application
of a DSP-based optical coherent receiver. Hardware complexity is
minimized with serial processing and decimation. As the technique runs
in background, the calibration can track parameter variations caused
by temperature, voltage, aging, etc., without operational
interruptions.
\section*{Acknowledgements}
The authors would like to thank Dr. Ariel Pola for his helpful advice
on various technical issues related to the hardware implementation.
\section*{Appendix}
In this Appendix the stochastic gradient of the squared error defined
by \eqref{eq:grad} is derived. The total squared error \eqref{eq:eT}
is
\begin{equation}
\label{eq:eT2}
{\mathcal E}_k=\sum_{j=1}^4 \left|e_k^{(j)}\right|^2=\sum_{j=1}^4
\left(u_k^{(j)}- \hat a_k^{(j)}\right)^2.
\end{equation}
with $u_k^{(j)}$ given by \eqref{eq:u1}. Define the \emph{average}
squared error as
\begin{equation}
\label{eq:mse}
\overline {\mathcal E}_N=\frac{1}{2N+1}\sum_{k=-N}^{N}\sum_{j=1}^4
\left(u_k^{(j)}-\hat a_k^{(j)}\right)^2.
\end{equation}
The derivative of $\overline {\mathcal E}_N$ with respect to
$g^{(i_0)}_{m_0}[l_0]$ is
\begin{equation}
\label{eq:dEdg}
\frac{\partial { \overline {\mathcal E}_N}}{\partial g^{(i_0)}_{m_0}[l_0]}=\frac{2}{2N+1}\sum_{k=-N}^{N}\sum_{j=1}^4
e_k^{(j)}\frac{\partial u_k^{(j)}}{\partial g^{(i_0)}_{m_0}[l_0]},
\end{equation}
where $l_0\in\{0, 1, \cdots,L_g-1\}$, $m_0\in\{0, 1, \cdots,M-1\}$,
and $i_0\in\{1,2,3,4\}$. From the slicer error $e_{k}^{(j)}$ given by
\eqref{eq:ePC}, define the $T_s=T/2$ oversampled slicer error as
\begin{equation}
e^{(j)}[n] =
\begin{cases}
e_{n/2}^{(j)} & \mbox{if } n= 0,\pm 2,\pm 4,\cdots \\
0 & \mbox{otherwise}
\end{cases}.
\end{equation}
Thus, \eqref{eq:dEdg} can be rewritten as
\begin{equation}
\label{eq:dEdg2}
\frac{\partial { \overline {\mathcal E}_N}}{\partial g^{(i_0)}_{m_0}[l_0]}=\frac{2}{2N+1}\sum_{n=-2N}^{2N}\sum_{j=1}^4
e^{(j)}[n]\frac{\partial u^{(j)}[n]}{\partial g^{(i_0)}_{m_0}[l_0]},
\end{equation}
where $u^{(j)}[n]$ is the oversampled compensation equalizer output
given by
\begin{equation}
\label{eq:un}
u^{(j)}[n]=\sum_{i=1}^4\sum_{l=0}^{L_{\Gamma}-1}
{\Gamma}^{(j,i)}_{n}[l]{x}^{(i)}[n-l],\quad j=1,\cdots,4.
\end{equation}
The time index $n$ can be expressed as
\begin{equation}
\label{eq:n}
n=m+k'M,\quad m=0,1,\cdots,M-1;\quad \forall k',
\end{equation}
with $k'$ integer. Then, omitting the constant factor
$\frac{2}{2N+1}$, the derivative \eqref{eq:dEdg2} can be expressed as
\begin{equation}
\label{eq:dEdg3}
\frac{\partial { \overline {\mathcal E}_N}}{\partial g^{(i_0)}_{m_0}[l_0]}\propto\sum_{k'}\sum_{m=0}^{M-1}\sum_{j=1}^4
e^{(j)}[m+k'M]\frac{\partial u^{(j)}[m+k'M]}{\partial g^{(i_0)}_{m_0}[l_0]}.
\end{equation}
Next we evaluate the derivative
$\frac{\partial u^{(j)}[m+k'M]}{\partial
g^{(i_0)}_{m_0}[l_0]}$. Assuming that the DSP filter coefficients
$\Gamma^{(j,i)}_{n}[l]$ do not depend on $g^{(i_0)}_{m_0}[l_0]$, from
\eqref{eq:un} and \eqref{eq:n} we verify that
\begin{equation}
\label{eq:dundg}
\frac{\partial u^{(j)}[m+k'M]}{\partial g^{(i_0)}_{m_0}[l_0]}=\sum_{i=1}^4\sum_{l=0}^{L_{\Gamma}-1}
{\Gamma}^{(j,i)}_{m+k'M}[l]\frac{\partial {x}^{(i)}[m+k'M-l]}{\partial g^{(i_0)}_{m_0}[l_0]}.
\end{equation}
Based on \eqref{eq:n}, the signal at the DSP block input $i$ given by
\eqref{eq:eq6b} can be rewritten as
\begin{equation}
x^{(i)}[m+k'M]=\sum_{l'=0}^{L_g-1} {g}^{(i)}_{m}[l']
{w}^{(i)}[m+k'M-l']
\end{equation}
Therefore,
\begin{equation}
\label{eq:dxdg}
\frac{\partial x^{(i)}[m+k'M]}{\partial g^{(i_0)}_{m_0}[l_0]}=
{w}^{(i)}[m+k'M-l_0]\delta_{m,m_0}\delta_{i,i_0},
\end{equation}
where $\delta_{n,m}$ is the Kronecker delta function (i.e.,
$\delta_{n,m}=1$ if $n=m$ and $\delta_{n,m}=0$ if $n\ne m$). Replacing
\eqref{eq:dxdg} in \eqref{eq:dundg} we get
\begin{align}
\nonumber
&\frac{\partial u^{(j)}[m+k'M]}{\partial g^{(i_0)}_{m_0}[l_0]}=\\
\label{eq:dudg2}
&\quad\quad\quad \sum_{l=0}^{L_{\Gamma}-1} {\Gamma}^{(j,i_0)}_{m+k'M}[l] {w}^{(i_0)}[m+k'M-l-l_0]\delta_{m,m_0}.
\end{align}
Using \eqref{eq:dudg2} in \eqref{eq:dEdg3}, we obtain
\begin{align}
\frac{\partial { \overline {\mathcal E}_N}}{\partial g^{(i_0)}_{m_0}[l_0]}\propto&\sum_{k'}\sum_{j=1}^4
e^{(j)}[m_0+k'M]\times\\
\nonumber
&\sum_{l=0}^{L_{\Gamma}-1} {\Gamma}^{(j,i_0)}_{m_0+k'M}[l] {w}^{(i_0)}[m_0+k'M-l-l_0].
\end{align}
Finally, we set $kM=k'M-l$ resulting
\begin{equation}
\label{eq:dEdg4}
\frac{\partial { \overline {\mathcal E}_N}}{\partial g^{(i_0)}_{m_0}[l_0]}\propto\sum_{k}{\hat e}^{(i_0)}[m_0+kM] {w}^{(i_0)}[m_0+kM-l_0],
\end{equation}
where
\begin{equation}
\label{eq:epb}
{\hat e}^{(i)}[n]=\sum_{j=1}^4\sum_{l=0}^{L_{\Gamma}-1} {\Gamma}^{(j,i)}_{n+l}[l]e^{(j)}[n+l]
\end{equation}
is the backpropagated error. Notice that \eqref{eq:dEdg4} is the
average of the instantaneous gradient component given by
${\hat e}^{(i_0)}[m_0+kM] {w}^{(i_0)}[m_0+kM-l_0]$. Therefore, an
instantaneous gradient of the square error can be obtained as
\begin{equation}
\nabla_{{\bold g}^{(i)}_{m}} {\mathcal E}_k\propto{\hat e}^{(i)}[m+kM]{\bold w}^{(i)}[m+kM],
\end{equation}
where ${\bold w}[n]$ is the $L_g$-dimensional vector with the samples
at the CE input defined by \eqref{eq:vecw}.
\bibliographystyle{IEEEtran/IEEEtran}
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Unified and efficient Machine Learning since 1999.
Latest release:
[](https://github.com/shogun-toolbox/shogun/releases/latest)
Cite Shogun:
[](https://zenodo.org/badge/latestdoi/1555094)
Develop branch build status:
[](https://dev.azure.com/shogunml/shogun/_build/latest?definitionId=-1)
[](https://codecov.io/gh/shogun-toolbox/shogun)
Donate to Shogun via NumFocus:
[](http://numfocus.org)
Buildbot: https://buildbot.shogun.ml.
* See [doc/readme/ABOUT.md](doc/readme/ABOUT.md) for a project description.
* See [doc/readme/INSTALL.md](doc/readme/INSTALL.md) for installation instructions.
* See [doc/readme/INTERFACES.md](doc/readme/INTERFACES.md) for calling Shogun from its interfaces.
* See [doc/readme/EXAMPLES.md](doc/readme/EXAMPLES.md) for details on creating API examples.
* See [doc/readme/DEVELOPING.md](doc/readme/DEVELOPING.md) for how to hack Shogun.
* See [API examples](http://shogun.ml/examples) for all interfaces.
* See [the wiki](https://github.com/shogun-toolbox/shogun/wiki/) for extended developer information.
## Interfaces
-------------
Shogun is implemented in C++ and offers automatically generated, unified interfaces to Python, Octave, Java / Scala, Ruby, C#, R, Lua. We are currently working on adding more languages including JavaScript, D, and Matlab.
| Interface | Status |
|:----------------:|-----------------------------------------------------------|
|Python | *mature* (no known problems) |
|Octave | *mature* (no known problems) |
|Java/Scala | *stable* (no known problems) |
|Ruby | *stable* (no known problems) |
|C# | *stable* (no known problems) |
|R | Currently disabled due to a [swig bug](https://github.com/swig/swig/issues/1660) |
|Perl | *pre-alpha* (work in progress quality) |
|JS | *pre-alpha* (work in progress quality) |
See [our website](http://shogun.ml/examples) for examples in all languages.
## Platforms
------------
Shogun is supported under GNU/Linux, MacOSX, FreeBSD, and Windows.
## Directory Contents
---------------------
The following directories are found in the source distribution.
Note that some folders are submodules that can be checked out with
`git submodule update --init`.
- *src* - source code, separated into C++ source and interfaces
- *doc* - readmes (doc/readme, submodule), Jupyter notebooks, cookbook (API examples), licenses
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- *tests* - unit tests and continuous integration of interface examples
- *applications* - applications of SHOGUN (outdated)
- *benchmarks* - speed benchmarks
- *cmake* - cmake build scripts
## License
----------
Shogun is distributed under [BSD 3-clause license](doc/license/LICENSE.md), with
optional GPL3 components.
See [doc/licenses](doc/license) for details.
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,522 |
GEO-SLOPE International, makers of the GeoStudio 2007 geotechnical software suite have announced a version 7.16 update for their GeoStudio 2007 software products, which includes SLOPE/W, SIGMA/W and SEEP/W, QUAKE/W and others. The major change is the addition of a Chinese user interface. This is the first I've heard of a geotechnical software package trying to target the Chinese market, very smart.
From GeoStudio Current News. If you are interested, read about the full list of changes in GeoStudio 2007 Version 7.16. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,859 |
Complex design techniques are often time-consuming and, well, complex. Some of theseadvanced effects can add plenty of depth to designs, but when used in the wrong place, they do little more than distract viewers from the project's intended focus. These effects may be precisely what a design needs to have the impact it requires, but even in these cases, they should be complemented by simpler effects.
When you use color sparingly and intelligently in your design, it is so much easier to draw attention to important items. On Interspire's "About Us" page, viewers are quickly drawn in turn to the dash of color in the logo at the top of the page, then straight to the headings and, lastly, to the logo at the right of the page content.
Additionally, using only uppercase letters allows MSNBC to make its incredibly smallbuttons just clear enough to be legible. In this 5-pixel-tall application, lowercase letters such as a, m, x and z would be only 2 to 3 pixels tall and not readable at all.
Sticking with news websites for the moment, CNN misses a lot of opportunities to use case to enhance its pages. To its credit, it uses an all-caps navigation menu, but thevast majority of the page is in traditional case, with only the first letter of each sentence capitalized.
Blurs can also be used to give a sense of depth or layering. Windows Vista's Aero theme blurs anything behind windows for a cool, diffused glass effect. A simpleGaussian Blur tool can create the same effect.
This technique applies not only to text either. Some designers fall back on templates or personal work habits when conceptualizing a design. This can greatly increase the speed with which concepts are turned over to clients; but all too often, it also restricts creativity — especially with regard to alignment.
By trimming all unnecessary components, CSS Remix is left with only the essentials and can display seven premium ads (128 by 96 pixels), 53 favicon ads (16 by 16 pixels) and a whopping 56 websites at one time — all in the top 1000 pixels of the page! Even the website's logo has been trimmed to 53 by 7 pixels. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,118 |
{"url":"http:\/\/daemonmaker.blogspot.ca\/2014\/","text":"## 20141221\n\n### Monitoring Experiments in Pylearn2\n\nIn an earlier post I covered the basics of running experiments in Pylearn2. However I only covered the bare minimum commands required leaving out many details. One fundamental concept to running experiments in Pylearn2 is knowing how to monitor their progress, or \"monitoring\" for short.\n\nIn this tutorial we will look at two forms of monitoring. The basic form which is always done and a new approach for real-time remote monitoring.\n\n#### Basic Monitoring\n\nWe will build upon the bare-bones example from the previous tutorial which means we will be using the MNIST dataset. Most datasets have two or three parts. At a minimum they have a part for training and a part for testing. If a dataset has a third part its purpose is for validation, or measuring the performance of our learner without unduly biasing our learner towards the dataset.\n\nPylearn2 performs monitoring at the end of each epoch and it can monitor any combination of the parts of the dataset. When using Stochastic Gradient Descent (SGD) as the training algorithm one uses the monitoring_dataset parameter to specify which parts of the dataset are to be monitored. For example, if we are only interested in monitoring the training set we would add the following entry to the SGD parameter dictionary:\n\nmonitoring_dataset:\n{\n'train': !obj:pylearn2.datasets.mnist.MNIST { which_set: 'train' }\n}\n\n\nThis will instruct Pylearn2 to calculate statistics about the performance of our learner using the training part of the dataset at the end of each epoch. This will change the default output after each epoch from:\n\nMonitoring step:\nEpochs seen: 1\nBatches seen: 1875\nExamples seen: 60000\nTime this epoch: 2.875921 seconds\n\n\nto:\n\nMonitoring step:\nEpochs seen: 0\nBatches seen: 0\nExamples seen: 0\nlearning_rate: 0.0499996989965\ntotal_seconds_last_epoch: 0.0\ntrain_objective: 2.29713964462\ntrain_y_col_norms_max: 0.164925798774\ntrain_y_col_norms_mean: 0.161361783743\ntrain_y_col_norms_min: 0.158035755157\ntrain_y_max_max_class: 0.118635632098\ntrain_y_mean_max_class: 0.109155222774\ntrain_y_min_max_class: 0.103917405009\ntrain_y_misclass: 0.910533130169\ntrain_y_nll: 2.29713964462\ntrain_y_row_norms_max: 0.0255156457424\ntrain_y_row_norms_mean: 0.018013747409\ntrain_y_row_norms_min: 0.00823106430471\ntraining_seconds_this_epoch: 0.0\nTime this epoch: 2.823628 seconds\n\n\nEach of the entries in the output (e.g. learning_rate, train_objective) are called channels. Channels give one insight into what the learner is doing. The two most frequently used are train_objective and train_y_nll. The channel train_objective reports the cost being optimized by training while train_y_nll monitors the negative log likelihood of the current parameter values. In this particular example these two channels are monitoring the same thing but this will not always be the case.\n\nMonitoring the train part of the dataset is useful for debugging purposes. However it is not enough alone to evaluate the performance of our learner because the learner will likely always improve and at some point it begins to overfit on the training data. In other words it will find parameters that work well on the data used to train it but not on data it has not seen during training. To combat this we use a validation set. MNIST does not explicitly reserve a part of the data for validation but it has become a de facto standard to use the last 10,000 samples from the train part. To specify this one uses the start and stop parameters when instantiating MNIST. If we were only monitoring the validation set our monitoring_dataset parameter to SGD would be:\n\nmonitoring_dataset:\n{\n'valid': !obj:pylearn2.datasets.mnist.MNIST {\nwhich_set: 'train',\nstart: 50000,\nstop: 60000\n}\n}\n\n\nNote that the key to the dictionary, 'valid' in this case, is merely a label. It can be whatever we choose. Each channel monitored for the associated dataset is prepended with this value.\n\nIt's also worth noting that we are not limited to monitoring just one part of the dataset. It is usually helpful to monitor both the train and validation parts of a data set. This is done as follows:\nmonitoring_dataset:\n{\n'train': !obj:pylearn2.datasets.mnist.MNIST {\nwhich_set: 'train',\nstart: 0,\nstop: 50000\n},\n'valid': !obj:pylearn2.datasets.mnist.MNIST {\nwhich_set: 'train',\nstart: 50000,\nstop: 60000\n}\n}\n\n\nNote that here we use the start and stop parameters when loading both the train and valid parts to appropriately partition the dataset. We do not want the learner to validate on the data from the train dataset otherwise we will not be able to identify overfitting.\n\nPutting it all together our our complete YAML now looks like:\n\n!obj:pylearn2.train.Train {\ndataset: &train !obj:pylearn2.datasets.mnist.MNIST {\nwhich_set: 'train'\nstart: 0,\nstop: 50000\n},\nmodel: !obj:pylearn2.models.softmax_regression.SoftmaxRegression {\nbatch_size: 20,\nn_classes: 10,\nnvis: 784,\nirange: 0.01\n},\nalgorithm: !obj:pylearn2.training_algorithms.sgd.SGD {\nlearning_rate: 0.05,\nmonitoring_dataset:\n{\n'train': *train,\n'valid': !obj:pylearn2.datasets.mnist.MNIST {\nwhich_set: 'train',\nstart: 50000,\nstop: 60000\n}\n}\n}\n}\n\nNote that here we have used a YAML trick to reference a previously instantiated object to save ourselves typing. Specifically the dataset has been tagged \"&train\" and when specifying monitor_dataset the reference \"*train\" is used to identify the previously instantiated object.\n\n#### Live Monitoring\n\nThere are two problems with the basic monitoring mechanism in Pylearn2. First the output is raw text. This alone can make it difficult to understand how the values of the various channels are evolving in time. Especially when attempting to track multiple channels simultaneously. Second, due in part to the ability to add channels for monitoring, the amount of output after each epoch can and frequently does grow quickly. Combined these problems make the basic monitoring mechanism difficult to use.\n\nAn alternative approach is to use a new mechanism called live monitoring. To be completely forthright the live monitoring mechanism is something that I developed to combat the aforementioned problems. Furthermore I am interested in feedback regarding its user interface and what additional functionality people would like. Please feel free to send an E-mail to the Pylearn2 users mailing list or leave a comment below with feedback.\n\nThe live monitoring mechanism has two parts. The first part is a training extension, i.e. an optional plug-in that modifies the way training is performed. The second part is a utility class that can query the training extension for data about channels being monitored.\n\nTraining extensions can be selected using the extensions parameter to the train object. In other words add the following to the parameters dictionary for the train object in any YAML:\n\nextensions: [\n!obj:pylearn2.train_extensions.live_monitoring.LiveMonitoring {}\n]\n\nThe full YAML would look like:\n\n!obj:pylearn2.train.Train {\ndataset: &train !obj:pylearn2.datasets.mnist.MNIST {\nwhich_set: 'train'\nstart: 0,\nstop: 50000\n},\nmodel: !obj:pylearn2.models.softmax_regression.SoftmaxRegression {\nbatch_size: 20,\nn_classes: 10,\nnvis: 784,\nirange: 0.01\n},\nalgorithm: !obj:pylearn2.training_algorithms.sgd.SGD {\nlearning_rate: 0.05,\nmonitoring_dataset:\n{\n'train': *train,\n'valid': !obj:pylearn2.datasets.mnist.MNIST {\nwhich_set: 'train',\nstart: 50000,\nstop: 60000\n}\n}\n},\nextensions: [\n!obj:pylearn2.train_extensions.live_monitoring.LiveMonitoring {}\n]\n}\n\nThe LiveMonitoring training extension listens for queries about channels being monitored. To perform queries one need only instantiate LiveMonitor and use it's methods to request data. Currently it has three methods:\n\u2022 list_channels: Returns a list of channels being monitored.\n\u2022 update_channels: Retrieves data about the list of specified channels.\n\u2022 follow_channels: Plots the data for the specified channels. This command blocks other commands from being executed because it repeatedly requests the latest data for the specified channels and redraws the plot as new data arrives.\nTo instantiate LiveMonitor start ipython and execute the following commands:\n\nfrom pylearn2.train_extensions.live_monitoring import LiveMonitor\nlm = LiveMonitor()\n\n\nEach of the methods listed above return a different message object. The data of interest is contained in the data member of that object. As such, given an instance of LiveMonitor, one would view the channels being monitored as follows:\n\nprint lm.list_channels().data\n\nWhich, if we're running the experiment specified by the YAML above, will yield:\n\n['train_objective', 'train_y_col_norms_max',\n'train_y_row_norms_min', 'train_y_nll', 'train_y_col_norms_mean',\n'train_y_max_max_class', 'train_y_min_max_class', 'train_y_row_norms_max',\n'train_y_misclass', 'train_y_col_norms_min', 'train_y_row_norms_mean',\n'train_y_mean_max_class', 'valid_objective', 'valid_y_col_norms_max',\n'valid_y_row_norms_min', 'valid_y_nll', 'valid_y_col_norms_mean',\n'valid_y_max_max_class', 'valid_y_min_max_class', 'valid_y_row_norms_max',\n'valid_y_misclass', 'valid_y_col_norms_min', 'valid_y_row_norms_mean',\n'valid_y_mean_max_class', 'learning_rate', 'training_seconds_this_epoch',\n'total_seconds_last_epoch']\n\nFrom this we can pick channels to plot using follow_channels:\n\nlm.follow_channels(['train_objective', 'valid_objective'])\n\n\nThis command will then display a graph like that in figure 1 and continually updates the plot at the end of each epoch.\n\nFigure 1: Example output from the follow_channels method of the LiveMonitor utility object.\n\nThe live monitoring mechanism is network aware and by default it answers queries on port 5555 of any network interface on the computer wherein the experiment is being executed. It is not necessary for a user to know anything about networking to use live monitoring however. By default the live monitoring mechanism assumes the experiment of interest is being executed on the same computer as the LiveMonitor utility class. If that is not the case and one knows the IP address of the computer on which the experiment is running then one need only specify the address when instantiating LiveMonitor. The live monitoring mechanism will automatically take care of the networking.\n\nLive monitoring is also very efficient. It only ever requests data it does not already have and the underlying networking utility waits for new data without taking unnecessary CPU time.\n\nThe live monitoring mechanism has many benefits including:\n\u2022 The ability to filter the channels being monitored.\n\u2022 The ability to plot data for any given set of channels being monitored.\n\u2022 The ability to retrieve data from an experiment in real-time*\n\u2022 The ability to query for data from an experiment running on a remote machine.\n\u2022 The ability to change which channels are being followed or plotted without restarting an experiment.\n* Updates only occur at the end of each epoch but this is real-time with respect to Pylearn2 experiments.\n\n#### Conclusion\n\nMonitoring the progress of experiments in Pylearn2 is as easy as setting up an experiment. Monitoring is also very flexible and offers output both directly in the terminal as text or graphically via a training extension.\n\n## 20141021\n\n### Thoughts Regarding the Michael Jordan Interview on IEEE Spectrum\n\nFor the few that may not have already seen it Dr. Michael Jordan was interviewed by IEEE Spectrum recently. He offers commentary on a number of topics including computer vision, deep learning, and big data.\n\nOverall I found the article to be an interesting read though it seems to offer little new over what he said on his AMA on Reddit.\n\nUltimately I find my self agreeing with his position on computer vision. Even given the major strides we have made as of late with convnets and the like we are still far from having a system as capable as we are at vision tasks. After all, the state-of-the-art challenge is the classification of just 1,000 classes of objects in high resolution images. This is a hard problem but it is something that we, humans, and many other animals do trivially.\n\nI am a bit torn about his perspective on deep learning. Notably because of the statement \"it\u2019s largely a rebranding of neural networks.\" I have encountered this idea a couple of times now but I argue that it is not accurate. It is true that neural networks are a favored tool amongst those in the deep learning community and that the strides made in the DL community have been seen while using NNs. But as Bengio et al. note in their forth-coming text called Deep Learning, it \"involves learning multiple levels of representation, corresponding to different levels of abstraction.\" Neural networks have been shown to do this but it has not been shown that they are required to perform such a task. On the flip side, they are out performing other methods that could be used.\n\nAnother point that stood out to me were is comments on the singularity. I find myself waffling on this topic and his comments help highlight the reason. Specifically he points out that discussions of the singularity are more philosophical in nature. I rather enjoy philosophy. I often say that if I had another life I would be a mathematician but if I had another one beyond that I would be a philosopher. More so than I am now anyway. I meet so many AI\/ML people that think the singularity folks are just crackpots. And if we are being honest, there do seem to be more than a reasonable proportion of crackpots in the community. However that does not prevent us from approaching the topic with sound and valid argumentation. We just have to be prepared to encounter those that cannot or chose not.\n\nEdit 2014-10-23: It appears Dr. Jordan was a bit displeased with IEEE Spectrum interview as he explains in Big Data, Hype, the Media and Other Provocative Words to Put in a Title. The long and short of it appears to be that he believes his perspective was intentionally distorted for the reason that many of my colleagues have been discussing. Namely the title, and arguably the intro, imply much stronger claims than his subsequent comments in the article seem to allude to. As such he he felt the need to clarify his perspectives.\n\nOn the one hand I though that a careful critical read of the interview allowed one to pick out his perspective fairly well. But in reading his response there appear to be some things that seem to come across just plain wrong. For instance his opinion about whether we should be collecting and exploring these large data sets. In the interview he makes the great point that we must be cognizant of bad correlations that can and will likely arise. But in the context I did get the impression that he was arguing against doing it all, i.e. collecting and analyzing such data sets, whereas in his response he argues that doing it can be a good thing because it can contribute to the development of principals that are currently missing.\n\nAs a side note, I find it interesting that he did not link to the interview but instead gave a link to it. As if to say, let's not lend any more credibility to this article than is absolutely necessary.\n\n## 20141018\n\n### A First Experiment with Pylearn2\n\nVincent Dumoulin recently wrote a great blog post titled Your models in Pylearn2 that shows how to quickly implement a new model idea in Pylearn2. However Pylearn2 has a fair number of models already implemented. This post is meant to compliment his post by explaining how to setup and run a basic experiment using existing components in Pylearn2.\n\nIn this tutorial we will train a very simple single layer softmax regression model on MNIST, a database of handwritten digits. Softmax is a generalization of a binary predictor called logistic regression to the prediction of one of many classes. The task will be to identify which digit was written, i.e. classify the image into the classes 0-9.\n\nThis same task is addressed in the Softmax regression Pylearn2 tutorial. This post will borrow from that tutorial. However Pylearn2 is feature rich allowing one to control everything from which model to train and which dataset to train it on to fine grained control over the training and the ability to monitor and save statistics about an experiment. For the sake of simplicity and understanding we will not be using most of them and as such this tutorial will be simpler.\n\n#### YAML Syntax\n\nA main goal of pylearn2 is to make managing experiments quick and easy. To that end a basic experiment can be executed by writing a description of the experiment in YAML (Yet Another Markup Language) and running the train script (pylearn2\/scripts\/train.py) on it.\n\nYAML is a markup language intended to be very sparse as compared to other markup languages such as XML. A run down of useful features for use with Pylearn2 can be found in the document YAML for Pylearn2 and the full specification can be found on yaml.org in case you need to something particularly out of the ordinary like defining a tuple.\n\nA Pylearn2 YAML configuration file identifies the object that will actually perform the training and the parameters it takes. I believe there is only one type of training object at the moment so it's kind of redundant but it allows for easy incorporation of special training procedures. The existing training object takes a specification of the model to be trained, the dataset on which the model should be trained, and the object representing the algorithm that will actually perform the training.\n\nBasic YAML syntax is extremely straight forward and the only special syntax that is really needed for the simplest of experiments is the !obj: tag. This is a Pylearn2 custom tag that instructs the Pylearn2 to instantiate a python object as specified immediately following the tag. For example the statement:\n!obj:pylearn2.datasets.mnist.MNIST { which: 'train' }\nresults in the instantiation of the MNIST dataset class found amongst the various Pylearn2 datasets in pylearn2.datasets in the file mnist.py specifying a supplies the value 'train' for a parameter called which that identifies the portion (e.g. training, validation, or test) of the dataset that should be loaded via a python dictionary.\n\nNote that the quotes around the value 'train' are required as they indicate that the value is string which is the required data type for the 'which' parameter.\n\nIt's important to note that any parameters required for the instantiation of a class must be provided in the associated dictionary. Check the Pylearn2 documentation for the class you need to instantiation to understand the available parameters and specifically which are required for the task you are attempting to perform.\n\n#### Defining an Experiment\n\nTo define an experiment we need to define a train object and provide it a dataset object, a model object, and an algorithm object via its parameters dictionary.\n\nWe have already seen how to instantiate the MNIST dataset class so lets look next at the algorithm. The Pylearn2 algorithm classes are found in the training_algorithms sub-directory. In this example we are going to use stochastic gradient descent (SGD) because it is arguably the most commonly used algorithm for training neural networks. It requires only one parameter, namely learning_rate, and is instantiated as follows:\n!obj:pylearn2.training_algorithms.sgd.SGD { learning_rate: 0.05 }\nThe final thing we need to do before we can put it all together is to define a model. The Pylearn2 model classes are located in the model sub-directory. The class we want is called SoftmaxRegression and found in softmax_regression. In its most basic form we only need to supply four parameters:\n\u2022 nvis: the number of visible units in the network, i.e. the dimensionality of the input.\n\u2022 n_classes: the number of output units in the network, i.e. the number of classes to be learned.\n\u2022 irange: the range from which the initial weights should be randomly selected. This is a symmetric range about zero and as such it is only necessary to supply the upper bound.\n\u2022 batch_size: the number of samples to be used simultaneously during training. Setting this to 1 results in pure stochastic gradient descent whereas setting it to the size of the training set effectively results in batch gradient descent. Any value in between yields stochastic gradient descent with mini-batches of the size specified.\n\nUsing what we know, we can now construct the train object and in effect the full YAML file as follows:\n!obj:pylearn2.train.Train {\ndataset: !obj:pylearn2.datasets.mnist.MNIST { which_set: 'train' },\nmodel: !obj:pylearn2.models.softmax_regression.SoftmaxRegression {\nbatch_size: 20,\nn_classes: 10,\nnvis: 784,\nirange: 0.01\n},\nalgorithm: !obj:pylearn2.training_algorithms.sgd.SGD { learning_rate: 0.05 }\n}\nNote that a Pylearn2 YAML file can contain definitions for multiple experiments simultaneously. Simply stack them one after the other and they will be executed in order from top to bottom in the file.\n\n#### Executing an Experiment\n\nThe final step is to run the experiment. Assuming the scripts sub-directory is in your path we simply call train.py and supply the YAML file created above. Assuming that file is called basic_example.yaml and your current working directory contains it the command would be:\ntrain.py basic_example.yaml\nPylearn2 will load the YAML, instantiate the specified objects and run the training algorithm on the model using the specified dataset. An example of the output from this YAML looks like:\ndustin@Cortex ~\/pylearn2_tutorials $train.py basic_example.yaml compiling begin_record_entry... compiling begin_record_entry done. Time elapsed: 0.013530 seconds Monitored channels: Compiling accum... Compiling accum done. Time elapsed: 0.000070 seconds Monitoring step: Epochs seen: 0 Batches seen: 0 Examples seen: 0 Time this epoch: 0:02:18.271934 Monitoring step: Epochs seen: 1 Batches seen: 1875 Examples seen: 60000 Time this epoch: 0:02:18.341147 ... Note we have not told the training algorithm under what criteria it should stop so it will run forever! Under the hood the Pylearn2 uses Theano to construct and train many of the models it supports. The first four lines of output, i.e. those related to begin_record_entry and accum, are related to this fact and can be disregarded for our purposes. The rest of the output is related to Pylearn2's monitoring functionality. Since no channels, particular metrics or statics about the training, have been specified the rest of the output is rather sparse. There are no channels listed under the Monitor channels heading and the only things listed under the Monitoring step headings are those things common to all experiments (e.g. epochs seen, batches seen, and examples seen). The only other output is a summary of the time it took to train each epoch. #### Conclusion Pylearn2 to makes specifying and training models easy and fast. This tutorial looked at the most basic of models. However it does not discuss the myriad training and monitoring options provided by Pylearn2. Nor does it show how to build more complicated models like those with multiple layers as in multilayer perceptrons nor those with special connectivity patterns as in convolutional neural networks. My inclination is to continue in the next post by discussing the types of stopping criteria and how to use them. From there I would proceed to discussing the various training options and work my way towards more complicated models. However I'm amenable to the idea of changing this order if there is something of particular interest so let me know what you would like to see next. ## 20141013 ### Harvard Librarians Advise Open Access Publishing Excellent. The Harvard university librarians have written a letter the Harvard faculty and staff encouraging they start publishing in journals that make content free to the public, known as open access journals, as opposed to hidden behind a pay wall. I have been watching this debate for some time as a number of the UofU CS professors have been arguing for exactly this change. I quite like the policy at the Machine Learning Lab here in Montreal which requires us to publish our articles on Arxiv.org, a database for freely publishing and accessing of scholarly works. It\u2019s not without it\u2019s challenges. For instance you never know the quality of a given paper that you find on Arxiv until you have invested time in reading it. Many arguing for the open access model have been actively trying to devise strategies for such problems. Regardless I believe it\u2019s preferable to not having access to a paper that should probably be cited. From a grad student's perspective it is nice because I don\u2019t have to spend time submitting special requests for access to articles and then waiting to receive them. It could end up meaning that I have to pay to have my articles published but I personally prefer this because I want my work available to others to hopefully build upon. ## 20140413 ### Installing RL-Glue and ALE without Root Access In 2012 Bellemare, Naddaf, Veness, and Bowling [1] introduced the Arcade Learning Environment (ALE) for developing and evaluating general AI methods. It interfaces with an Atari 2600 simulator called Stella. They motivate their use of the the Atari 2600 because it permits access to hundreds of \"game environments, each one different, interesting, and designed to be a challenge for human players.\" ALE also interfaces with RL-Glue, a collection of tools for developing and evaluating Reinforcement Learning agents. RL-Glue is a two part system; a language agnostic part, referred to as the core, and a language specific part, referred to as the codec. There are multiple codecs supporting development in C\/C++, Java, Lisp, Matlab, and Python. The following discusses installing RL-Glue and ALE. First I cover installing RL-Glue and then ALE. ### Installing RL-Glue The instructions for installing the RL-Glue core and Python codec are drawn from the technical manuals for versions 3.04 and 2.0 and assume versions 3.04 and 2.02 respectively. However, looking over older versions of the manual it appears these instructions have not changed much, if at all, which means they may also work for future versions with the correct changes in the commands. I leave it as an exercise to the reader to determine what changes are needed. (Don't you love it when an author does that?) Installing RL-Glue in the user space as opposed to system wide requires compiling the core code from source. To do so, execute the following commands. #### RL-Glue Core 1. Download RL-Glue Core $ cd ~ && wget http:\/\/rl-glue-ext.googlecode.com\/files\/rlglue-3.04.tar.gz\n2. Unpack RL-Glue Core\n$tar -xvzf rlglue-3.04.tar.gz 3. Make a directory into which RL-Glue will be installed: $ mkdir ~\/rlglue\n4. Configure RL-Glue Core\nAt this point it is necessary to identify the location wherein you wish to install the core as you must tell this to the configure script. I made a directory called rlglue in my home directory resulting in the following command:\n$cd rlglue-3.04 && .\/configure --prefix=<rlglue> where <rlglue> is the absolute path to the directory into which you want RL-glue installed. 5. Build and install RL-Glue Core $ make && make install\nBecause the core has been installed in a non-standard location it is necessary to inform the system of it's location. This merely entails updating your PATH environment variable to include rlglue\/bin, i.e.:\n$export PATH=~\/rlglue\/bin:$PATH\nNote that you can make this change in your .bashrc to avoid doing it every time you open a new terminal.\n\nAt this point executing rl_glue should result in:\nRL-Glue Version 3.04, Build 909\nRL-Glue is listening for connections on port=4096\nThis indicates that RL-Glue is waiting for programs managing the agent, environment, and experiment to connect. For now you can type ctrl-c to exit the program.\n\n#### Python Codec\n\n$cd ~ && wget http:\/\/rl-glue-ext.googlecode.com\/files\/python-codec-2.02.tar.gz 2. Unpack the codec $ tar -xvzf python-codec-2.02.tar.gz\nThis is another step that is only necessary because we're installing into a non-standard location.\n$export PYTHONPATH=~\/python-codec\/src Note that this is another command that can be placed into your .bashrc to avoid executing it every time you open a new terminal. ### Installing ALE Installing ALE takes a little more effort. This is due to the fact that ALE does not supply a configure script to build a makefile specific for your system. These instruction are for installing ALE 0.4.3 and may not extend to newer versions well so your mileage may vary. As before, execute the following commands. 1. Download ALE $ wget\u00a0http:\/\/www.arcadelearningenvironment.org\/wp-content\/uploads\/2014\/01\/ale_0.4.3.zip\n2. Unpack ALE\n$unzip ale_0.4.3.zip 3. Select Makefile ALE supports Linux, OSX, and Windows and as such a makefile is supplied for each platform. Installing ALE requires making a makefile from one of these. I advise copying the one you want as opposed to renaming it: $ cd ale_0.4.3\/ale_0_4\/ && cp makefile.unix makefile\nUpdate 2014-04-15: Fr\u00e9d\u00e9ric Bastien notes that this step can be avoided by supplying the name of the preferred makefile to make on the command line as follows:\n$make -f makefile.unix Still be sure to change your working directory to ale_0.4.3\/ale_0_4 as the following commands assume that context. 4. Enable RL-Glue support RL-Glue support in ALE is disabled by default. To enable it edit the makefile and change the line: USE_RLGLUE := 0 to USE_RLGLUE := 1 It is also necessary to inform ALE where the RL-Glue headers are located. This can be done by changing the line: INCLUDES := -Isrc\/controllers -Isrc\/os_dependent -I\/usr\/include -Isrc\/environment to INCLUDES :=-Isrc\/controllers -Isrc\/os_dependent -I\/usr\/include -Isrc\/environment -I<rlgluedir>\/include where <rlgluedir> indicates the directory into which rlglue was installed earlier. Similarly it is necessary to inform ALE where the RL-Glue libraries are installed. This is done by changing the line: LIBS_RLGLUE := -lrlutils -lrlgluenetdev to LIBS_RLGLUE := -L<rlgluedir>\/lib -lrlutils -lrlgluenetdev Update 2014-04-15: Fr\u00e9d\u00e9ric Bastien notes that one can override these variables on the command line. For example: $ make USE_RLGLUE=1\nsets USE_RLGLUE to 1. However it is unclear how to append to variables via the command line so this may only work for the\u00a0USE_RLGLUE\u00a0variable without restating the existing variable value.\n\n5. Build ALE\n$make 6. Update LD_LIBRARY_PATH Because the RL-Glue libraries have been installed in a non-standard location it is necessary to tell ALE where to find them. This is done using the LD_LIBRARY_PATH environment variable as follows: $ export LD_LIBRARY_PATH=<rlgluedir>\/lib:$LD_LIBRARY_PATH Note that this is another command you can add to your .bashrc to avoid needing to execute the command every time you open a new terminal. 7. Update your PATH As before it is necessary to update your path to inform the system of the location of the ALE binaries since they are installed in a non-standard location. $ export PATH=~\/ale_0.4.3\/ale_0_4:$PATH Note that this is yet another command you will have to execute every time you open a terminal unless you add this command to your .bashrc. At this point you can execute ale which should result in: A.L.E: Arcade Learning Environment (version 0.4) [Powered by Stella] Use -help for help screen. Warning: couldn't load settings file: .\/stellarc No ROM File specified or the ROM file was not found. Disregard the warnings, they are simply saying that ALE was unable to find your stella configuration and that a game ROM was not specified. This is to be expected since you did not specify the locations for them. ### Conclusion This covers installing RL-Glue and ALE. I have not discussed anything beyond basic testing as such material can be found in the documentation for the tools. In a future post, or possibly more than one, I will outline the processes for making and executing agents, environments, and experiments. ### References [1] Bellemare, Marc G., et al. \"The arcade learning environment: An evaluation platform for general agents.\" arXiv preprint arXiv:1207.4708 (2012). ## 20140404 ### What's Wrong with the Turing Test? If I could answer one question through the course of my research it would be \"what is intelligence?\" This question like no other drives my studies. I wrote this post a while ago but did not post it. I did not post it because I intended to refine it. But the reality is I will always be refining my thoughts on this topic. Tonight I went out to the pub with several of my colleagues at Universit\u00e9 de Montr\u00e9al and this topic came up reminding me that I need to just put this out there. As such I am posting it now, with some small changes. I look forward to your responses. The question \"what is intelligence?\" is non-trivial. We have been seeking an answer for millennia. While many definitions have been offered [1] no single definition has really ever dominated. Even in the sixty or so years that we have been seriously studying how to create an artificial intelligence we have never actually formally defined intelligence. Ask 100 people to define it and you will likely receive 100 different definitions [1], [2]. The de facto standard is the Turing test developed by Alan Turing [3]. History tells us that he was worried about getting mired in a long drawn out philosophical debate which would likely prevent any progress on actually creating an artificially intelligent being. The Turing Test as it has come to be known is a variation on a game known as the imitation game [2] wherein a participant, the interrogator, attempts to identify which of the other two participants, one male and one female, was in fact male. Of course the female's objective was to fool the interrogator. The crux of the game was that the decision had to be made solely from communication patterns. In other words, the interrogator did not meet nor could they see the other participants. Additionally, to avoid cues from writing styles, messages would be passed between them in an anonymous way, such as through a computer terminal. In the Turing Test the objective is to identify the artificial being [4], the computer, as opposed to the male. The hypothesis being that if the interrogator cannot differentiate the artificial being and the human, the artificial being must necessarily be intelligent if we accept that humans are intelligent. This test is clever because it does not require a definition of intelligence nor a measure of intelligence other than agreement that humans are intelligent. However the Turing Test is a behavioral test. Not everyone accepts the test. One of the more well known opponents is John Searle, a professor of philosophy at the University of California, Berkeley. Dr. Searle offers the Chinese Room argument in counter to the Turing Test. According to the Chinese Room argument we can construct a process that appears intelligent but in fact is not. We do so by placing a person in a room, particularly one that does not speak Chinese. The resident will receive messages from outside the room written in Chinese. The resident must then consult a set of instructions that, when followed, dictate the appropriate response. One, that to any outside observer, would have to have been written by someone that knows Chinese. Since the resident does not know Chinese it supposedly follows that intelligence had only been imitated by the process. There are a number of counter arguments to the Chinese Room argument. Some argue that the analogy breaks down as a result of the fact that a human performing the processing would simply take too long. Others argue that the room itself is intelligent. But I digress. While I don't personally accept the Chinese Room argument I do agree there is a flaw in the Turing Test. Specifically, by the nature of its construction, it will permit us to classify a being as intelligent if it behaves like a human. From this we have to conclude that everything else is either not intelligent or at least not classifiable as intelligent without some added criterion. This not only applies to the animals but to all other beings. Consider the scenario wherein we are visited by aliens that can talk to us, that can do incredibly complicated mathematics, even teach us a few things, and possess technologies way beyond our understanding such as a clearly engineered means of interstellar travel which they used to come to Earth. Would we consider these beings intelligent? We can all think of scenarios wherein the answer is \"not necessarily\" but in all likelihood we would agree that they are in fact intelligent. But how likely is it that the Turing test will apply? Of course this problem applies to artificial beings as well, i.e. our computer programs. Have we already created an artificial intelligence? Some might argue we have with Cleverbot garnering 59.3% positive votes from 1,334 participants at the Techniche 2011 festival. Others would likely respond that the real turing test involves physical interaction, i.e. shaking hands with the being, and still not being able to discern a difference. This again highlights the problem. A precise definition of intelligence would address this problem. However it would not only allow us to differentiate between intelligent and not and answer the question of whether we have already created an artificial intelligence. But it could allow for the development of metrics for comparing intelligences and even help us understand why our existing creations are not intelligent, if that is truly the case. [1] Legg & Hutter, 2006, A Collection of Definitions of Intelligence, http:\/\/www.vetta.org\/documents\/A-Collection-of-Definitions-of-Intelligence.pdf [2] Pfeifer, 1999, Understanding Intelligence [3] Turing, 1950, Computing Machinery and Intelligence http:\/\/www.csee.umbc.edu\/courses\/471\/papers\/turing.pdf [4] Russel and Norvig, 2009, Artificial Intelligence: A Modern Approach Third Edition ## 20140308 ### Day One in Canada I crossed the US-Canada border today. The process was fast and simple enough. However many of the \"facts\" on my visa were wrong and the gentleman at the border had to fix them. Evidently my last name was \"Dustin James Webb\" and I had no first or middle names, I was listed as a mechanical engineer not a computer scientist, and I was destined for Calgary, Alberta, not Montreal, Quebec. Oh, and I came to Canada in 2002, not today. I guess all the paper work we submitted to the NY consulate was for show as was the processing time for our visas. And no, my identity was not stolen. I won't go into the details on how I know this though. Shortly after leaving the border I stopped for food. Ironically the first place I saw was a Tim Hortons. Naturally I had soup and a coffee to combat the cold. Even given all the warnings from friends and family about just how cold it is here I was still not prepared. It is insanely cold. And yes, I consider 30-40 degrees + large windchill factor to be insanely cold. I don't look forward to sub-zero temperatures. Unfortunately I lost internet access on my phone at the border. Apparently T-mobile does not offer it to their US customers through their local partner. Worse yet I have had horrible cell reception. Google maps cached the info to get from the border to Toronto. But that took me downtown and I was unable to get direction anywhere else at that point. As such I set out to find a gas station and a hotel which are both very difficult tasks without reliable means of communication. This would not have been a problem if I had actually let my wife book my hotel when she had intended. As it is I am paying nearly twice what I should be for a hotel tonight. C'est la vie! I also had fun trying to fill my tank. For reasons I still don't understand the gas pumps at the station I found would not accept our credit card. So I went into the store to pay in advance. The attendant asked how much I wanted so asked the price. He said it was 1.50 CAD! I was astounded. I have been paying about$3.50 on average throughout this trip. But I shrugged it off and asked for \\$12 because our Prius only has an 8 gallon tank. It turns out they measure gas in litres here. Duh! Place this one in the category \"should have seen that coming.\" Oh well, my tank is currently half full which is better than the gallon or so I had when I found the station.\n\nTomorrow, Montreal!\n\n## 20140116\n\n### Making Learning Fun with Electronics and Robotics\n\nIn my last two posts I discussed my thoughts on making learning fun. In the first post the vehicle was games\u00a0while in second post the vehicle was scientific experimentation.\u00a0Electronics and robotics seem to me another possible vehicle. While I have a fair bit of experience with these topics I have not done much with them as a educational tool.\n\nI'm most excited about using the\u00a0Lego Mindstroms. This is Lego's robotics kits. While it is recommended for kids ages 10 and up it actually allows for construction of real autonomous robots. It comes with several sensors for measuring features of the environment such as light intensity, color, distance to the nearest object, audio, and the rotations of it's own motors. It introduces children to programming using a language called NXT-G which is a visual programming language. Think programming via legos. And of course it's compatible with all things Lego.\n\nI look forward to use the Mindstorms to teach everything from mechanics and programming to how to use sensors and basic motion planning techniques.\n\nMy motivation for using the Mindstorms as an educational tool comes from my experience volunteering for\u00a0First Lego League (FLL). FLL is a program for children between 9 and 15 years of age that promotes science and technology. Each year the kids are given a topic about which they must learn. To aid in the learning process they are given a large game board with lots of challenges built out of Lego pieces. The kids must then build one or more robots to solve the challenges. FLL is part of a larger program called FIRST, but is my favorite because the robots made by the kids are actually truly autonomous. For several years now I have volunteered as a robot design judge and through that effort I have seen just how excited the kids can get about learning and solving real world problems.\n\nMy children are not yet old enough to participate in FLL. However my son and I have started a Jr. First Lego League\u00a0(Jr.FLL) team. Jr.FLL is like FLL but shoots to teach the basics of design, mechanics, research, and team building. In fact he and his team will be showing off they have learned about natural disasters on January 25th (2014) at the University of Utah Student Union building. Please feel free to come talk to them. I must warn you though, the FLL finals will be going on at the same time so the place will be an absolute mad house.\n\nAn alternative to the Mindstorms are Bo & Yana by iPlay. They are meant to teach the basics of programming, sensing, and actuation. We have not received ours yet but they look promising and are even compatible with other systems like the Mindstorms which should allow for a simple transition when the time is right.\n\nStill another alternative are the solutions from\u00a0Modular Robotics\u00a0called\u00a0Cubelets and their latest product called MOSS. These are just cubes with basic sensing and actuation capabilities that snap together using magnets. But in connecting them one is making simple robots. Unfortunately the MOSS Kickstarter came and went before I could get involved. If anyone has these, I would be interested in hearing about your experience with them. Specifically MOSS.\n\nOf course to build a robot it helps to know something about electronics. One need not be an expert by any means but it helps to be able to build simple circuits. A few years ago I encountered a method for teaching children about circuits called Squishy Circuits. The foundation of this idea is to use \"playdough\", i.e. modeling compound, to make circuits. It turns out that if you make a modeling compound using a salt base it conducts electricity. Conversely, if you make a modeling compound with a sugar base it does not conduct electricity. As such you can make small sculptures from the two different types. Circuits can then be formed by connecting conductive portions of the sculpture with discrete components like a battery pack and LEDs.\n\nThere are a lot of other solutions out there for teaching children about electrics and circuits but I have no experience with any of them. Some of the ones I have found are:\n\nAgain, if you have any experience with these I would like to hear your thoughts.\n\n## 20140108\n\n### Making Learning Fun with Experiments\n\nIn my last post I talked about how to take advantage of games to make learning fun. By no means is this an original idea. In fact it's rather obvious. Another approach which is probably again pretty obvious is to do experiments.\n\nTheir are numerous sites describing fun experiments. One that everyone seems to know is the classic baking-soda-vinegar volcano. For anyone that may be unfamiliar you take something like dirt or clay and sculpt a volcanic cone with an extra deep caldera. From there you pour in some baking-powder into the caldera. Finally you pour in some vinegar. The baking soda and vinegar react and discharge carbon dioxide in the form of bubbles that are heavier than air so, when done correctly, the bubbles overflow the volcano and spill down the sides much like a lava flow from a real volcano would.\n\nThis experiment is a great experiment because it opens the door to talking about all kinds of things from the structure of the earth, to how volcanoes form, to how volcanos can lead to other natural disasters like earth quakes and tsunamis, how pyroclastic flow can in a way preserve whatever it hits. For us the volcano devolved to just mixing baking soda and vinegar. After all, who doesn't like watching a vigorous chemical reaction? Even this is great though as it opens the door for talking about things like pressure, surface tension, and chemistry. In a similar vein, the Coke and Mentos experiment is always fun to do. Last I read the reaction wasn't well understood but it's clear that the process is releasing a lot of gas in short order.\n\nAnother experiment we have had fun with is the basic electromagnet. Again for those that are unfamiliar you wrap a length of wire around something like an iron bolt and connect the two ends of the wire to the opposite ends of a battery. It is important that the object around which you wrap the wire is ferrous. We usually call this object the core. The movement of the electrons through the wire then produces an electromagnetic field which is amplified by whatever you use for your core.\u00a0 From there you can use whatever other magnetic objects you have lying around to show the formation and destruction of the magnetic field as you connect and disconnect the battery.\n\nIt is not necessary for the experiment to seem obviously fun though. Take for instance testing soil types. In this experiment you gather a couple of soil samples from different places. Potting soil and dirt from outside are great. You put the potting soil in a jar, the dirt in another, and then mix of both into yet a third. Then fill the jars with water, cap them, and shake. What do you get? Mud! Of course the educational part comes from the discussion that ensues when everything settles and talking about how long it takes to settle. My son and I played with this experiment for a couple of days.\n\nAs I eluded to earlier, the list of possible experiments is endless. They often lead to the same discussions but that doesn't make them any less fun. To close out this post I will leave you with some links to other particularly fun experiments:\n\u2022 Rock Candy: This one takes a bit longer but you get a treat in the end.\n\u2022 Fluorescent Jello: Not so tasty, but it glows in the dark!\n\u2022 Squishy Circuits: This is great for introducing children to both chemistry and electronics.\n\n## 20140101\n\n### Making Learning Fun with Games\n\nIn the last few days I have had a couple of conversations with friends about teaching children. These conversations have inspired me to write a bit about my thoughts on the topic. My experience basically comes from teaching my own son. I strongly believe in the need for parents to supplement their childs education. To the point that I try to work with my kids a little bit every day.\n\nAs anyone with children will likely tell you it can be difficult at times to maintain their interest. Even if you are incredibly passionate about a topic it can be challenging to imbue them with that passion. One thing I have noticed while working with my son is that he gets excited when the solutions come easily. Conversely, failure to immediately understand quickly leads to disinterest. This would seem to imply a need for instant gratification. For that reason I often seek ways of removing that need or replacing that need in some way.\n\nOne obvious but still great method to address this problem while still providing a lesson is through games. Any gamer will tell you as much. Not just because they are attempting to justify their pastime but because any games provide myriad lessons in the guise of entertainment. Two of my favorite games are DragonBox and LightBot.\n\nDragonBox teaches the principles of algebra without focusing on the mathematical foundations. It simply challenges the child with a puzzle that involves isolating an object from a set of others using the rules of algebra. Early in the game it doesn't even use numbers, just pictures, so that the child may focus on the rules. Unfortunately there isn't enough content. My son has beat this game numerous times and basically lost interest.\n\nLightBot teaches the basics of programming. The objective is to get a robot to turn on lights placed throughout an environment. The crux is that the series of commands needed to execute the task must be provided before the robot ever does anything. As with all games it starts off simply and increases in complexity. In this case the complexity comes from restricting the number of commands the player can use, requiring the use of subprocedures, requiring the application of recursion, and the like.\n\nAnother game we have been playing is from MindSnacks. Specifically we've been studying French to help our son prepare for entry into the Montreal educational system where at least a third of the class is taught strictly in French. In total it has nine subgames but only permits the child access to two at the beginning. The child must gain levels to unlock the others. It also focuses the child on certain aspects of the topic of study. In the case of French, and probably the other languages it supports, the topics include numbers, colors, days of the week, and greetings for a total of 50 different topics.\n\nOf course games don't have to be deemed educational to in fact be educational. One of my son's favorite games is Minecraft. He prefers creative mode and will play for hours constructing little houses and zoos for all the animals he hatches. I don't particularly care for the game myself but it has a lot of great educational aspects to it. For instance it is great for talking about Geometry, from the different types of shapes we study to the difference between 1D, 2D, and 3D. Because it is in part focused on crafting it also offers a segue into talking about how real things are made.\n\nAnother \"non-educational\" game that I like is StarMade. This one is inspired by Minecraft but takes place in space. The objective is to make a spacecraft and fly it around collecting materials and fighting space pirates. I particularly like this one because it is more challenging than Minecraft but my son finds it interesting enough to work through the challenges. For instance, he would prefer to play and have to practice his reading to accomplish his goal than not. This is significant because he has not yet found that reading for the sake of reading is fun. It also offers additional lessons to those found in Minecraft. For instance, building a spacecraft requires an understanding of the different parts including the different computers required (e.g. control computer, weapons computers) to engines and shielding.\n\nGames are not the only method for making learning fun. I also like to use experimentation to bring lessons to life but I'll talk more about this in my next entry. I have also been looking for a way to introduce my son to real world electronics and robotics. As part of this we have created a Jr. FLL but this is limited to simple machines and designing solutions. There are a lot of products being made to go beyond this which I will also discuss later.","date":"2018-05-25 18:45:49","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.2855032980442047, \"perplexity\": 1211.3714598119623}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-22\/segments\/1526794867173.31\/warc\/CC-MAIN-20180525180646-20180525200646-00520.warc.gz\"}"} | null | null |
package com.tangyu.component.demo.service.remind;
import android.app.Activity;
import android.content.Intent;
import android.os.Bundle;
import android.os.Handler;
import android.os.Message;
import android.text.format.DateFormat;
import android.view.View;
import android.widget.Button;
import android.widget.TextView;
import com.tangyu.component.R;
import com.tangyu.component.Util;
import com.tangyu.component.service.remind.TYRemindData;
import com.tangyu.component.service.remind.TYRemindService;
import java.util.HashMap;
import java.util.List;
import java.util.Map;
/**
* @author binliu on 12/31/13.
*/
public class ActDemoRemindService extends Activity implements View.OnClickListener {
private TextView mVMsg;
private Button mVBtnStart, mVBtnStop, mVBtnReset;
private DataPersistLayer mDataPersistLayer;
private TYRemindData.RemindDataUtil mRemindDataUtil;
public final int CMD_INIT = -1;
public final int CMD_START = 0;
public final int CMD_STOP = 1;
public final int CMD_RESET = 2;
private final Map<Integer, Integer> mMapIDWithCMD = new HashMap<Integer, Integer>();
private Handler mRefreshRequestHandler = new Handler() {
@Override
public void handleMessage(Message msg) {
super.handleMessage(msg);
if (createDataPersistIfNeed().isInitState()) {
mVMsg.setText("preparing");
} else {
StringBuilder sb = new StringBuilder("current : " + DateFormat.format("kk:mm:ss", System.currentTimeMillis()) + "\n\n") ;
List<RemindData> data = createDataPersistIfNeed().generalTestData();
createDataPersistIfNeed().restoreStatus(data);
if (!Util.isNull(data)) {
for (RemindData item : data) {
sb.append(item.toString() + "\n");
}
}
if (createRemindDataUtilIfNeed().isAllCompleted(data)) {
sb.append("Complete!!!!");
}
mVMsg.setText(sb);
}
mRefreshRequestHandler.removeMessages(0);
mRefreshRequestHandler.sendEmptyMessageDelayed(0, 1000);
}
};
@Override
protected void onCreate(Bundle savedInstanceState) {
super.onCreate(savedInstanceState);
setContentView(R.layout.demo_remind_service);
mVMsg = (TextView) findViewById(R.id.demo_remind_service_show_msg);
mVBtnReset = (Button) findViewById(R.id.demo_remind_service_reset_btn);
mVBtnStart = (Button) findViewById(R.id.demo_remind_service_start);
mVBtnStop = (Button) findViewById(R.id.demo_remind_service_stop);
mVBtnReset.setOnClickListener(this);
mVBtnStart.setOnClickListener(this);
mVBtnStop.setOnClickListener(this);
changeButtonStats(CMD_INIT);
mMapIDWithCMD.put(R.id.demo_remind_service_start, CMD_START);
mMapIDWithCMD.put(R.id.demo_remind_service_stop, CMD_STOP);
mMapIDWithCMD.put(R.id.demo_remind_service_reset_btn, CMD_RESET);
}
@Override
protected void onStart() {
super.onStart();
}
@Override
protected void onStop() {
super.onStop();
}
private DataPersistLayer createDataPersistIfNeed() {
return mDataPersistLayer == null ? mDataPersistLayer = new DataPersistLayer(this) : mDataPersistLayer;
}
private TYRemindData.RemindDataUtil createRemindDataUtilIfNeed() {
return mRemindDataUtil == null ? mRemindDataUtil = new TYRemindData.RemindDataUtil<RemindData>() : mRemindDataUtil;
}
@Override
public void onClick(View v) {
int command = mMapIDWithCMD.get(v.getId());
int commandOfService = changeButtonStats(command);
Intent intent = new Intent(this, TYRemindServiceImpl.class);
intent.putExtra(TYRemindService.INTENT_REMIND_COMMAND, commandOfService);
// intent.putExtra(TYRemindService.INTENT_SERVICE_FOCUSES_STOP, false);
intent.setFlags(Intent.FLAG_ACTIVITY_NEW_TASK);
startService(intent);
}
public int changeButtonStats(int command) {
boolean[] status = null;
int commandOfService = 0;
switch (command) {
case CMD_INIT:
status = new boolean[]{true, false, false};
break;
case CMD_STOP:
status = new boolean[]{true, false, false};
commandOfService = TYRemindService.CMD_REMIND_CANCEL;
break;
case CMD_START:
case CMD_RESET:
commandOfService = TYRemindService.CMD_REMIND_RESCHEDULE;
status = new boolean[]{false, true, true};
createDataPersistIfNeed().reset();
mRefreshRequestHandler.removeMessages(0);
mRefreshRequestHandler.sendEmptyMessage(0);
break;
}
mVBtnStart.setEnabled(status[0]);
mVBtnStop.setEnabled(status[1]);
mVBtnReset.setEnabled(status[2]);
return commandOfService;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,001 |
Warner Bros trying to get a Supergirl movie off the ground: Report
Josh Weiss
@JoshuaHWeiss
Feb 26, 2020, 3:48 PM EST (Updated)
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Tag: Warner Brothers
Tag: DC Extended Universe
Tag: Supergirl 60th
Warner Bros is actively developing a DCEU film based on the character of Supergirl (Kara Zor-El), Deadline reports.
The Cloverfield Paradox scribe Oren Uziel is penning the script, but not much else about the project is known at this point. However, the studio's desire to turn this property into a full-fledged feature film could be indicative that they have more confidence in her than do in Superman, whose Man of Steel sequel was scrapped.
More Supergirl
Why a skirt isn't necessary for Supergirl and Wonder Woman
Right now, Superman's cousin enjoys her own TV show on the CW, where she's played by Melissa Benoist. It was recently renewed for a fourth season that will premiere this October.
Created by Otto Binder and Al Plastino, Kara Zor-El made her comic book debut in 1959's Action Comics #252. She too is from Krypton, but arrived on Earth as a teenager. Supergirl (aka the Girl of Steel) has all the powers of Superman with none of his moral restraints, making her a danger to herself and others. Her chosen human name is Kara Danvers.
The character recently received a new comic book series at DC from writer Marc Andreyko and artist Kevin Maguire. Issue #21 of the goes on sale August 8 and features Kara dealing with the devestation caused by Brian Bendis's new Superman villain, Rogol Zaar.
Video of Supergirl (Seasons 1-2) in 2 Minutes | SYFY WIRE | {
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Q: inconsistent variable when passing it from one method to another I have a problem that I have not been able to solve and it does not occur to me that it could be.
I have a class to which I am passing an InputStream from the main method, the problem is that when transforming the InputString to String with the class IOUtils.toString of AWS, or with the IOUtils of commons-io, they return
an empty String
No matter what the problem may be, since inside the main class, it works correctly and returns the String it should, but when I use it inside the other class (without having done anything), it returns the empty String to me.
these are my classes:
public class Main {
public static void main(String [] args) throws IOException {
InputStream inputStream = new ByteArrayInputStream("{\"name\":\"Camilo\",\"functionName\":\"hello\"}".getBytes());
OutputStream outputStream = new ByteArrayOutputStream();
LambdaExecutor lambdaExecutor = new LambdaExecutor();
String test = IOUtils.toString(inputStream); //this test variable have "{\"name\":\"Camilo\",\"functionName\":\"hello\"}"
lambdaExecutor.handleRequest(inputStream,outputStream);
}
}
and this:
public class LambdaExecutor{
private FrontController frontController;
public LambdaExecutor(){
this.frontController = new FrontController();
}
public void handleRequest(InputStream inputStream, OutputStream outputStream) throws IOException {
//Service service = frontController.findService(inputStream);
String test = IOUtils.toString(inputStream); //this test variable have "" <-empty String
System.exit(0);
//service.execute(inputStream, outputStream, context);
}
}
I used the debug tool, and the InputStream object is the same in both classes
A: The reason you can't is because you can only read from a stream once.
To be able to read twice, you must call the reset() method for it to return to the beginning. After reading, call reset() and you can read it again!
Some sources don't support resetting it so you would actually have to create the stream again. To check if the source supports it, use the markSupported() method of the stream!
A: By the time that you've passed the stream into handleRequest(), you've already consumed the stream:
public static void main(String [] args) throws IOException {
InputStream inputStream = new ByteArrayInputStream("{\"name\":\"Camilo\",\"functionName\":\"hello\"}".getBytes());
OutputStream outputStream = new ByteArrayOutputStream();
LambdaExecutor lambdaExecutor = new LambdaExecutor();
String test = IOUtils.toString(inputStream); //this consumes the stream, and nothing more can be read from it
lambdaExecutor.handleRequest(inputStream,outputStream);
}
When you took that out, the method worked as, as you said in the comments.
If you want the data to be re-useable, you'll have to use the reset() method if you want the same data again, or close and re-open the stream to re-use the object with different data.
// have your data
byte[] data = "{\"name\":\"Camilo\",\"functionName\":\"hello\"}".getBytes();
// open the stream
InputStream inputStream = new ByteArrayInputStream(data);
...
// do something with the inputStream, and reset if you need the same data again
if(inputStream.markSupported()) {
inputStream.reset();
} else {
inputStream.close();
inputStream = new ByteArrayInputStream(data);
}
...
// close the stream after use
inputStream.close();
Always close the stream after you use it, or use a try block to take advantage of AutoCloseable; you can do the same with the output stream:
try (InputStream inputStream = new ByteArrayInputStream(data);
OutputStream outputStream = new ByteArrayOutputStream()) {
lambdaExecutor.handleRequest(inputStream, outputStream);
} // auto-closed the streams
| {
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La nitrendipina è un principio attivo di indicazione specifica contro l'ipertensione, appartenente alla classe dei calcio-antagonisti e simile alla nicardipina.
Indicazioni
È utilizzato come medicinale in cardiologia contro l'insufficienza cardiaca, ipertensione arteriosa, profilassi dell'arteriosclerosi e alcune forme di angina.
Controindicazioni
Controindicata in casi di ipotensione arteriosa, bradicardia ed emorragia cerebrale. Da non utilizzare in caso di gravidanza o allattamento.
Dosaggi
Ipertensione, 20 mg al giorno
Farmacodinamica
I bloccanti dei canali del calcio hanno il ruolo di interferire con il flusso di ioni calcio verso l'interno delle cellule attraverso i canali lenti della membrana plasmatica.
I calcioantagonisti hanno un'azione farmacologica predominante in tessuti dove il calcio regola la coppia eccitazione-contrazione, riducendo la contrattilità miocardica e di riflesso il tono vascolare diminuito e l'impulso elettrico che circola nel cuore può essere depresso, tali luoghi in cui i calcioantagonisti rivestono un ruolo importante sono le cellule miocardiche, cellule del sistema di conduzione del cuore e muscolatura liscia vascolare. Sono dei vasodilatatori arteriosi periferici e coronarici.
Effetti indesiderati
Alcuni degli effetti indesiderati sono bradicardia, cefalea, dolore addominale, edema, vertigini, nausea, vomito, febbre, neutropenia, ipoglicemia, aritmie, affaticamento, vasculiti, ipotensione, broncospasmo, rash, mialgia, dispnea, ansia, dolore toracico, confusione, vampate, sudorazione.
Stereochimica
La nitrendipina contiene uno stereocentro e consiste di due enantiomeri, in particolare due atropisomeri. Si utilizza come racemo, cioè una miscela in rapporto 1:1 delle forme ( R ) e ( S ):
Note
Bibliografia
Altri progetti
Calcioantagonisti
Antiaritmici | {
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{"url":"https:\/\/gateoverflow.in\/tag\/gate2001","text":"# Recent questions tagged gate2001\n\n9 votes\n1 answer\n1\nA sequential circuit takes an input stream of 0's and 1's and produces an output stream of 0's and 1's. Initially it replicates the input on its output until two consecutive 0's are encountered on the input. From then onward, it produces an output stream, which is ... to be used to design the circuit. Give the minimized sum-of-product expression for J and K inputs of one of its state flip-flops\n6 votes\n2 answers\n2\nConsider a relation examinee (regno, name, score), where regno is the primary key to score is a real number. Write an SQL query to list the regno of examinees who have a score greater than the average score.\n1 vote\n1 answer\n3\nConsider a relation $\\text{examinee (regno, name, score)},$ where regno is the primary key to score is a real number. Suppose the relation $\\text{appears (regno, centr_code)}$ specifies the center where an examinee appears. Write an SQL query to list the centr_code having an examinee of score greater than $80.$\n17 votes\n4 answers\n4\nConsider a set of n tasks with known runtimes $r_1, r_2, \\dots r_n$ to be run on a uniprocessor machine. Which of the following processor scheduling algorithms will result in the maximum throughput? Round Robin Shortest job first Highest response ratio next first come first served\n7 votes\n2 answers\n5\nWe wish to construct a $B^+$ tree with fan-out (the number of pointers per node) equal to $3$ for the following set of key values: $80, 50, 10, 70, 30, 100, 90$ Assume that the tree is initially empty and the values are added in the order given. Show ... Intermediate trees need not be shown. The key values $30$ and $10$ are now deleted from the tree in that order show the tree after each deletion.\n15 votes\n2 answers\n6\nConsider a relation examinee (regno, name, score), where regno is the primary key to score is a real number. Write a relational algebra using $( \\Pi, \\sigma, \\rho, \\times)$ to find the list of names which appear more than once in examinee.\n32 votes\n3 answers\n7\nConsider a disk with the $100$ tracks numbered from $0$ to $99$ rotating at $3000$ rpm. The number of sectors per track is $100$ and the time to move the head between two successive tracks is $0.2$ millisecond. Consider a set of disk requests to read ... initially at track $0$ and the elevator algorithm is used to schedule disk requests, what is the worse case time to complete all the requests?\n29 votes\n3 answers\n8\nTwo concurrent processes $P1$ and $P2$ want to use resources $R1$ and $R2$ in a mutually exclusive manner. Initially, $R1$ and $R2$ ... to deadlock. Exchange the statements $Q1$ and $Q3$ and statements $Q2$ and $Q4$. Is mutual exclusion guaranteed now? Can deadlock occur?\n15 votes\n2 answers\n9\nRemove left-recursion from the following grammar: $S \\rightarrow Sa \\mid Sb \\mid a \\mid b$ Consider the following grammar: $S \\rightarrow aSbS\\mid bSaS \\mid \u220a$ Construct all possible parse trees for the string abab. Is the grammar ambiguous?\n6 votes\n1 answer\n10\nThe syntax of the repeat-until statement is given by the following grammar $S \\rightarrow\\text{ repeat }S_1\\text{ until }E$ where E stands for expressions, $S$ and $S_1$ stand for statements. The non-terminals $S$ and $S_1$ have an attribute code that represents ... a statement. Use the operator '\\\\' to concatenate two strings and the function gen(s) to generate a line containing the string s.\n19 votes\n2 answers\n11\nConsider the following grammar with terminal alphabet $\\Sigma =\\{a,(,),+,* \\}$ and start symbol $E$. The production rules of the grammar are: $E \\rightarrow aA$ $E \\rightarrow (E)$ $A \\rightarrow +E$ $A \\rightarrow *E$ $A \\rightarrow \\epsilon$ Compute the FIRST and FOLLOW sets for $E$ and $A$. Complete the LL(1) parse table for the grammar.\n19 votes\n2 answers\n12\nConsider a weighted undirected graph with vertex set $V = \\{ n1, n2, n3, n4, n5, n6 \\}$ and edge set $E = \\{(n1,n2,2), (n1,n3,8), (n1,n6,3), (n2,n4,4), (n2,n5,12), (n3,n4,7), (n4,n5,9), (n4,n6,4)\\}$. The ... tree unique over all possible minimum spanning trees of a graph? Is the maximum among the edge weights of a minimum spanning tree unique over all possible minimum spanning tree of a graph?\n26 votes\n3 answers\n13\nInsert the following keys one by one into a binary search tree in the order specified.$15, 32, 20, 9, 3, 25, 12, 1$Show the final binary search tree after the insertions. Draw the binary search tree after deleting $15$ from it. Complete the statements $S1$, $S2$ and $S3$ in ... = NULL) return 0; x = depth (t -> left); S1: ___________; S2: if (x > y) return __________; S3: else return _______; }\n21 votes\n2 answers\n14\nConsider the following C program: void abc(char*s) { if(s[0]=='\\0')return; abc(s+1); abc(s+1); printf(\"%c\",s[0]); } main() { abc(\"123\"); } What will be the output of the program? If $abc(s)$ is called with a null-terminated string $s$ of length $n$ characters (not counting the null ('\\0') character), how many characters will be printed by $abc(s)$?\n37 votes\n2 answers\n15\nConsider a $5-$stage pipeline - IF (Instruction Fetch), ID (Instruction Decode and register read), EX (Execute), MEM (memory), and WB (Write Back). All (memory or register) reads take place in the second phase of a clock cycle and all writes occur ... Show all data dependencies between the four instructions. Identify the data hazards. Can all hazards be avoided by forwarding in this case.\n18 votes\n1 answer\n16\nA sequential circuit takes an input stream of 0's and 1's and produces an output stream of 0's and 1's. Initially it replicates the input on its output until two consecutive 0's are encountered on the input. From then onward, it produces an output stream, ... The desired output 101100|10110100 0|11 J-K master-slave flip-flops are to be used to design the circuit. Give the state transition diagram\n20 votes\n5 answers\n17\nIs the $3\\text{-variable}$ function $f= \\Sigma(0,1,2,4)$ its self-dual? Justify your answer. Give a minimal product-of-sum form of the $b$ output of the following $\\text{excess-3}$ to $\\text{BCD}$ converter.\n21 votes\n2 answers\n18\nA CPU has $32-bit$ memory address and a $256 \\ KB$ cache memory. The cache is organized as a $4-way$ set associative cache with cache block size of $16$ bytes. What is the number of sets in the cache? What is the size (in bits) of the tag field per ... bits are required to find the byte offset within a cache block? What is the total amount of extra memory (in bytes) required for the tag bits?\n33 votes\n3 answers\n19\nConsider a disk with the following specifications: 20 surfaces, 1000 tracks\/surface, 16 sectors\/track, data density 1 KB\/sector, rotation speed 3000 rpm. The operating system initiates the transfer between the disk and the memory sector-wise. Once the head has been placed on ... transfer? What is the maximum percentage of time the CPU is held up for this disk I\/O for cycle-stealing DMA transfer?\n23 votes\n2 answers\n20\nLet a decision problem $X$ be defined as follows: $X$: Given a Turing machine $M$ over $\\Sigma$ and any word $w \\in \\Sigma$, does $M$ loop forever on $w$? You may assume that the halting problem of Turing machine is undecidable but partially decidable. Show that $X$ is undecidable Show that $X$ is not even partially decidable\n15 votes\n2 answers\n21\nGive a deterministic PDA for the language $L=\\{a^ncb^{2n} \\mid n \\geq 1\\}$ over the alphabet $\\Sigma = \\{a,b,c\\}$. Specify the acceptance state.\n23 votes\n2 answers\n22\nConstruct DFA's for the following languages: $L=\\left\\{w \\mid w \\in \\{a,b\\}^*, \\text{ w has baab as a substring } \\right\\}$ $L=\\left\\{w \\mid w \\in \\{a,b\\}^*, \\text{ w has an odd number of a's and an odd number of b's } \\right\\}$\n2 votes\n3 answers\n23\nConsider the function $h: N \\times N \\rightarrow N$ so that $h(a,b) = (2a +1)2^b - 1$, where $N=\\{0,1,2,3,\\dots\\}$ is the set of natural numbers. Prove that the function $h$ is an injection (one-one). Prove that it is also a Surjection (onto)\n4 votes\n2 answers\n24\nProve that powerset $(A \\cap B) = \\text{powerset}(A) \\cap \\text{powerset}(B)$ Let $sum(n) = 0 + 1 + 2 + ..... + n$ for all natural numbers n. Give an induction proof to show that the following equation is true for all natural numbers $m$ and $n$: $sum(m+n) = sum(m) + sum(n) + mn$\n43 votes\n5 answers\n25\nConsider a relation geq which represents \"greater than or equal to\", that is, $(x,y) \\in$ geq only if $y \\geq x$. create table geq ( ib integer not null, ub integer not null, primary key ib, foreign key (ub) references geq on delete cascade ); Which of the following is possible ... deleted A tuple (z,w) with z > x is deleted A tuple (z,w) with w < x is deleted The deletion of (x,y) is prohibited\n28 votes\n2 answers\n26\nWhich of the rational calculus expression is not safe? $\\left\\{t \\mid \\exists u \\in R_1\\left(t[A] = u[A]\\right) \\land \\neg \\exists s \\in R_2 \\left(t[A] = s[A]\\right)\\right\\}$ ... $\\left\\{t \\mid \\exists u \\in R_1\\left(t[A]=u[A]\\right) \\land \\exists s \\in R_2 \\left(t[A] = s[A]\\right)\\right\\}$\n87 votes\n4 answers\n27\n$R(A,B,C,D)$ is a relation. Which of the following does not have a lossless join, dependency preserving $BCNF$ decomposition? $A \\rightarrow B, B \\rightarrow CD$ $A \\rightarrow B, B \\rightarrow C, C \\rightarrow D$ $AB \\rightarrow C, C \\rightarrow AD$ $A \\rightarrow BCD$\n40 votes\n5 answers\n28\nConsider Peterson's algorithm for mutual exclusion between two concurrent processes i and j. The program executed by process is shown below. repeat flag[i] = true; turn = j; while (P) do no-op; Enter critical section, perform actions, then exit critical section Flag[i] = false; Perform other non- ... turn = i flag[j] = true and turn = j flag[i] = true and turn = j flag[i] = true and turn = i\n26 votes\n3 answers\n29\nConsider a machine with $64$ MB physical memory and a $32$-bit virtual address space. If the page size s $4$ KB, what is the approximate size of the page table? $\\text{16 MB}$ $\\text{8 MB}$ $\\text{2 MB}$ $\\text{24 MB}$\n46 votes\n4 answers\n30\nWhich of the following does not interrupt a running process? 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A 16-year-old Merritt Island teen was arrested after Cocoa police said he made threats over Snapchat warning Space Coast Christian Academy students of a plan to 'blow the place up.' The threat turned out to be a hoax.
"We take these types of cases very seriously," said Cocoa Police Chief Mike Cantaloupe in a statement.
"These kids need to understand the far-reaching consequences of their actions. There are significant resources put into play to deal with these situations; resources that are needed for other potential emergencies."
The teen, who FLORIDA TODAY is not naming because of his age, was arrested Monday night after detectives learned of the threat and tracked the teen to his home. It was not immediately known if he was a student at the same school.
More:Teen charged with threatening to 'shoot up' West Melbourne school
The teen was charged with making a false bomb threat, a second-degree felony that could garner up to 15 years in state prison if convicted in adult court.
The teen, however, will go before a juvenile judge for a hearing instead.
Police were notified about the threat, circulating on social media, by school administrators.
The threat is also the latest to take place at a Brevard campus since Feb. 14 when a 19-year-old man shot and killed 17 people at Marjory Stoneman Douglas High School in Parkland, Florida. Hundreds of other threats have been reported at schools statewide.
The school was shut down briefly as administrators moved students to another location while bomb-sniffing dogs searched the campus.
No devices were located and students were later allowed to return to the campus.
Detectives said that the teen crafted a fake Snapchat account – a popular social media platform – and then sent a screen shot of a threat he placed on the same account to a friend. The teen will go before a juvenile judge Tuesday for a hearing.
Contact Gallop at 321-242-3642, jdgallop@floridatoday and Twitter @JDGallop | {
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'BY ANY NAME' BECOMES FIRST WELSH FILM TO BE RELEASED GLOBALLY ON AMAZON PRIME
An independent action adventure film shot in and around Swansea in just 16 days has gone global after becoming the first ever Welsh film to appear on Amazon Prime. Tanabi Films joined forces with Katherine John (aka Catrin Collier) to produce By Any Name, based on the author's novel of the same name which has…
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Songyun () war ein aus Dunhuang (Gansu) stammender chinesischer buddhistischer Pilger aus der späten Zeit der Fremdherrschaft der Nördlichen Wei-Dynastie, der zusammen mit dem Mönch Huisheng () Zentralasien und Indien bereiste.
Leben
Er war von 518 bis 522 unterwegs und folgte von Dunhuang aus der südlichen Route der Seidenstraße nach Khotan. Er kehrte mit 170 buddhistischen Schriften in seine Heimat zurück. Sein Reisebericht Weiguo yi xi shiyi guo shi () ist heute verloren. Im 5. Kapitel des Werkes Luoyang qielan ji () von Yang Xuanzhi () aus dem 6. Jahrhundert wird ein Werk namens Songyun jia ji () zitiert, manche halten dies für das verlorene Buch.
Songyun bereiste Udyana () und Gandhara. Er lieferte einen wertvollen Bericht über die "weißen Hunnen" oder Hephthaliten (siehe Iranische Hunnen), die damals Baktrien beherrschten; die Gruppe, die Gandhara erobert hatte, war jedoch eher die Alchon (siehe auch Mihirakula).
Literatur
Samuel Beal: Travels of Fah-Hian and Sung-Yun. Asian Educational Service, New Delhi 1993, ISBN 81-206-0824-0
Samuel Beal: Si-Yu-Ki or the Buddhist Records of the Western World. Translated from the Chinese of Hiuen Tsiang AD 629. 2 Bände, India Munshiram Manoharlal Publishers Pvt. Ltd., New Delhi 2004
Edouard Chavannes: Voyage de Song-yun dans l'Udyana et le Gandhara. In: Bulletin de l'Ecole Francaise d'Extreme-Orient. Band 3, 1903, S. 1–63.
Yang Xuanzhi (Autor), Yi-t'ung Wang (Übersetzer): A Record of Buddhist Monasteries in Lo-Yang. Princeton University Press, Princeton, New Jersey 1983
William John Francis Jenner: Memories of Lo-yang: Yang Hsuan-chih and the Lost Capital (493-534). Clarendon Press, Oxford University Press, New York 1981.
Fan Xiangyong (范祥雍): Luoyang qielan ji jiaozhu (洛阳伽蓝记校注). Zhonghua shuju, Peking 1978
Zhou Zumo (周祖谟): Luoyang qielan ji jiaoshi (洛阳伽蓝记校释). Zhonghua shuju, Peking 1963
Siehe auch
Faxian
Xuanzang
Oddiyana
Yang Xuanzhi
Weblinks
Literatur zum Thema
Person des Mahayana-Buddhismus
Buddhistischer Mönch
Buddhismus in China
Buddhismus in Indien
Reiseliteratur
Literatur (6. Jahrhundert)
Buddhistische Literatur
Autor
Geboren im 5. oder 6. Jahrhundert
Gestorben im 6. Jahrhundert
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The community members over the age of 16 are welcome to use the facility as part of the Morning Workout Program during specified hours.
Fitness Center Hours for the Community
The hours for public, community use of the Fitness Center are limited in order to avoid conflict with the hours that are highly used by our students and employees.
Download the Fitness & Wellness Programs Registration Form (pdf)
Community Hours:
Monday - Thursday 6:30 a.m. - 2:00 p.m. & 5:30 - 10:00 p.m.
Friday 6:30 a.m. - 2:00 p.m. & 5:30 – 8:00 p.m.
Saturday* 9:00 a.m. - 4:00 p.m.
Sunday* 12:00 - 4:00 p.m.
*Please Note: Saturday/Sunday hours (subject to change) are in effect only when Cazenovia College academic classes are in session.
Fitness Center is open for public use during community hours only. Participants must be over the age of 16.
Updates to the Fitness Center hours are posted on our Fitness Center Facebook page. Follow us at facebook.com/wildcatworkdown/.
We ask that each person signing up for semester use of the Fitness Center participate in an orientation. To schedule an orientation and set up a workout program, call 315-655-7223. Completed registration, waiver and medical forms must be on file in the Fitness Center prior to a participant's first workout session. Orientations are not given to day-pass patrons. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,978 |
#include <stdint.h>
#include "detector_defines.h"
#include "cl_err.h"
#include "check_utils.h"
#include "overflow_error.h"
#include "util_functions.h"
#include "cpu_check_utils.h"
static uint32_t divide_ceiling(uint32_t a, uint32_t b)
{
uint32_t ret = a / b;
if(a % b)
ret += 1;
return ret;
}
void cpu_parse_canary(cl_command_queue cmd_queue, uint32_t check_len,
uint32_t *map_ptr, kernel_info *kern_info, void *buffer,
uint32_t *dupe)
{
uint32_t max_to_check = divide_ceiling(check_len, sizeof(uint32_t));
for(uint32_t p = 0; p < max_to_check; p++)
{
uint32_t word = *(((uint32_t*)map_ptr) + p);
if (word == poisonFill_32b)
continue;
//find the first byte from this word that mismatched
for(uint32_t q = 0; q < sizeof(uint32_t); q++)
{
uint8_t byte = *(((uint8_t*)map_ptr) + sizeof(uint32_t)*p + q);
if(byte != poisonFill_8b && sizeof(uint32_t)*p + q < check_len)
{
char * backtrace_str = NULL;
if(get_print_backtrace_envvar())
{
//clEnqueueNDRangeKernel->kernelLaunchFunc->verifyBufferInBounds->verify_buffer_on_host->verify_cl_mem->cpu_parse_canary
backtrace_str = get_backtrace_level(5);
}
overflowError(kern_info, buffer, sizeof(uint32_t)*p + q, backtrace_str);
printDupeWarning(kern_info->handle, dupe);
optionalKillOnOverflow(get_exitcode_envvar(), 0);
mendCanaryRegion(cmd_queue, buffer, CL_TRUE, 0, NULL, NULL);
if(backtrace_str)
free(backtrace_str);
break;
}
}
break;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 872 |
Coronavirus pics 65: Bayard Rustin's Down the Line
"In these cases, the police figure prominently in the incidents that triggered the rioting. Sometimes they are not directly involved, but rumors of police brutality flood through the ghetto. Although it may be of some interest to search for a pattern, no very profound purpose is served by concentrating on who struck the match. There are always matches lying around. We must ask why there was also a fuse and why the fuse was connected to a powder keg."
"One cannot argue with the President's position that riots are destructive or that they frighten away allies. Nor can one find fault with his sympathy for the plight of the poor; surely the poor need sympathy. But one can question whether the government has been working seriously enough to eliminate the conditions which lead to frustration-politics and riots. The President's very words, "all this takes time," will be understood by the poor for precisely what they are--an excuse instead of a real program, a cover-up for the failure to establish real priorities, and an indication that the administration has no real commitment to create new jobs, better housing, and integrated schools."
"I am not going to stress the usual argument that the police habitually mistreat Negroes. Every Negro knows this. There is scarcely any black man, woman, or child in the land who at some point or other has not been mistreated by a policeman. (A young man in Watts said, "The riots will continue because I, as a Negro, am immediately considered to be a criminal by the police and, if I have a pretty woman with me, she is a tramp even if she is my wife or mother.")"
"As the civil rights movement progressed, winning victory after victory in public accommodations and voting rights, it became increasingly conscious that these victories would not be secure or far-reaching without a radical improvement in the Negro's socioeconomic position. And so the movement reached out of the South into the urban centers of the North and the West. It moved from public accommodations to employment, welfare, housing, education--to find a host of problems the nation had let fester for a generation.
But these were not problems that affected the Negro alone or that could be solved easily with the movement's traditional protest tactics. These injustices were imbedded not in ancient and obsolete institutional arrangements but in the priorities of powerful vested interests, in the direction of public policy, in the allocation of our national resources. Sit-ins could integrate a lunch counter, but massive social investments and imaginative public policies were required to eliminate the deeper inequalities."
"[The Kerner R]eport does not say that Americans are racist. If it did, the only answer would be to line everybody up, all 200 million of us, then line up 200,000 psychiatrists, and have us all lie on couches for ten years trying to understand the problem and for ten years more learning how to deal with it. All over the country people are beating their breasts crying mea culpa--"I'm so sorry that I am a racist"--which means, really, that they want to cop out because if racism is to be solved on an individual psychological basis, then there is little hope.
What the Kerner Report is really saying is that the institutions of America brutalize not only Negroes but also whites who are not racists but who in many communities have to use racist institutions. When it is put on that basis, we know we cannot solve the fundamental problem by sitting around examining our innards, but by getting out and fighting for institutional change."
"At a street corner meeting in Watts when the riots were over, an unemployed youth of about twenty said to me, "We won." I asked him: "How have you won? Homes have been destroyed, Negroes are lying dead in the streets, the stores from which you buy food and clothes are destroyed, and people are bringing you relief." His reply was significant: "We won because we made the whole world pay attention to us. The police chief never came here before; the mayor always stayed uptown. We made them come." Clearly it was no accident that the riots proceeded along an almost direct path to City Hall.... This is hardly a description of a Negro community that has run amok. The largest number of arrests were for looting—not for arson or shooting. Most of the people involved were not habitual thieves; they were members of a deprived group who seized a chance to possess things that all the dinning affluence of Los Angeles had never given them."
"We are indeed a house divided. But the division between race and race, class and class, will not be dissolved by massive infusions of brotherly sentiment. The division is not the result of bad sentiment, and therefore will not be healed by rhetoric. Rather the division and the bad sentiments are both reflections of vast and growing inequalities in our socioeconomic system--inequalities of wealth, of status, of education, of access to political power. Talk of brotherhood and "tolerance" (are we merely to "tolerate" one another?) might once have had a cooling effect, but increasingly it grates on the nerves. It evokes contempt not because the values of brotherhood are wrong--they are more important than ever--but because it just does not correspond to the reality we see around us. And such talk does nothing to eliminate the inequalities that breed resentment and deep discontent."
"The resort to stereotype is the first refuge and chief strategy of the bigot. Though this is a matter that ought to concern everyone, it should be of particular concern to Negroes. For their lives, as far back as we can remember, have been made nightmares by one kind of bigotry or another.
This urge to stereotype groups and deal wtih them accordingly is an evil urge. Its birthplace is in that sinister back room of the mind where plots and schemes are hatched for the persecution and oppression of other human beings.
It comes out of many things, but chiefly out of a failure or refusal to do the kind of tough, patient thinking that is required of difficult problems of relationship. It comes, as well, out of a desire to establish one's own sense of humanity and worth upon the ruins of someone else's."
The End Times Philosophy Interviews Have Moved From 3:AM
3:16 Paintings on Cardboard
The Scottish Stores Pop Up Cardboard Paintings Exhibition | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 103 |
5G and IoT: Big winners of CTIA Super Mobility 2016
Verifying Location Technologies
Experts on Cyber Security Weigh In
Vehicle Routing Software Survey: Higher expectations drive transformation
What to Expect at CES
Detroit and Silicon Valley Tech Showcased at LA Auto Show's Connected Car Expo
Risk to Automated Vehicles, What if the OEMs take a VW Approach?
CTIA 2015, What Mattered Most, Wireless Location Insights
Post-Mortem on Flight 370 Crowdsource Search
Phones Allow Surreptitious Geo-Tracking
In the two years since the last survey, vehicle routing has begun a transformation that mirrors changes occurring throughout the software industry, pu...
Janice Partyka, Wireless LBS Insider, GPS World Magazine
Not everyone wants to be located. Consumers think they have the ability to turn off the tracking ability of their phones. But can they? More about that later. In other news, there are good reasons why Nokia's HERE mapping is still on the selling block. And blind people are using a no-tech version of a widely used location positioning method that doesn't need canes.
The controls that phone makers have devised to enable consumers to opt out of being located have a big hole. Android-based phones are giving app makers free access to phone data that can be used to surreptitiously geolocate devices. The data comes from an unlikely source: power consumption, and no consent is needed.
The technique, called PowerSpy, was developed by researchers at Stanford and Rafael, Israel's defense research group, and gathers a phone's power usage history. Simplistically, the location of the phone is tracked by using the phone's battery consumption to determine the distance of a phone to a cell tower. The further the distance, or the greater the obstacles blocking the tower, the more power is consumed by the battery. The researchers say they can take into account phone usage battery drain and filter out the noise created by focusing on long-term trends.
At its current level of development, the PowerSpy method requires the snoop to have driven a route (war driving) to identify its power consumption pattern. With tests conducted in San Francisco, the method worked with 90 percent accuracy to identify a correct route from seven choices. The team is working on using the data to detect unknown routes that have not been previewed. How would the hypothetical stalker, crook or unethical mobile advertiser get access to this data? They would entice a person to download an app. The smoke screen app might be a game or a productivity app that is quietly slurping up the power consumption data.
Here Today, Not Gone Tomorrow? Wouldn't you think that Nokia would by now have clinched a deal to sell the mapping division? Given its mapping debacle, Apple was on the top of everyone's list as a buyer, but apparently the company didn't even participate in the bidding, and instead is committed to further development of its self-built mapping database. Contenders — Facebook, Baidu, Tencent and Uber — seem to have dropped out of the competition. Left is a consortium of German automakers — BMW, Daimler and Volkswagen — who feel that they should get a better deal with no other buyers in sight. It is a double-edged sword, as they also worry that if the highly accurate maps are acquired by tech firms, the car makers will lose a competitive advantage in the fight for supremacy of the automated vehicle. High-precision mapping is critical to the success of the auto OEMs.
Who Will Win Connected Vehicles? Follow the Money. Investors who want a piece of the connected vehicle action are placing bets on the tech companies, not the auto OEMs. Many blue chip and small companies are seeing healthy gains in price. Sensor chip makers, car infotainment and telecom companies are some of the winners. With the surge of connectivity required in the Internet of Things, networking technology will also do well.
E911 Innovations. While regulations are in place for eventually requiring technology to automatically identify the location of indoor E911 calls, dispatchers don't yet have that capability. Callers can be inside a large complex, like a dormitory or hotel, and if they are unable to speak or identify their location, response is hampered. Smart911 from Rave Mobile Safety is sending dispatchers floor plans of buildings to help in rescue efforts. The maps are automatically sent with the 911 call and have already been credited with quicker responses.
Quick Business News. Uber acquired Microsoft's geo-imagery team and assets, known at BIT (Bing Imagery Technologies), which is based in Boulder. Microsoft didn't need this technology as it had already outsourced Bing Maps technology to Nokia HERE. Telecommunication Systems (TCS) purchased location-based technology and intellectual property from Loctronix. The purchase will further TCS in developing indoor-location technologies. Denmark has become the first country to use real-time traffic data across a national network. Denmark will use GPS probe data managed by INRIX for congestion management. The Internet of Things relies on multitudes of sensors and a new start-up, Sense360, has built a platform to manage that data.
No-Tech Location Technology. Daniel Kish was a particularly helpful kid who made deliveries for his mom to homes outside of his neighborhood. What is unusual is that Kish is blind and uses echolocation to "see" the space around him. He clicks his tongue to ascertain the unique echoes of his surroundings, starting by identifying areas of high or low density, such as tall buildings, squat houses or open space. And in a version of drive testing, blind users like Kish first walk a neighborhood with a sighted guide and remember the signature echoes. Whether it is solely by ear or with a big computer algorithm like PowerSpy, pattern mapping can be effective.
http://gpsworld.com/phones-allow-surreptitious-geo-tracking/
connect vehicle
Indoor location
IndoorAtlas
Location Accuracy
automated car
automated vehicle
automotive cybersecurity
car hacking
commercial telematics
indoor positioning
location players
mspping
vehicle routing | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,618 |
Q: Deploying Azure Function Swagger API Definition using ARM We are creating an ARM template for our Azure Function that we would like to use for deploying to different environments however we cannot figure out how to deploy the Swagger with it.
As far as we see in the ARM template there is an option to add a link to the Swagger but can the Swagger be deployed as part of the ARM template itself?
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,343 |
{"url":"https:\/\/www.nature.com\/articles\/s41467-021-24210-9","text":"Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.\n\n# Shallow slow earthquakes to decipher future catastrophic earthquakes in the Guerrero seismic gap\n\n## Abstract\n\nThe Guerrero seismic gap is presumed to be a major source of seismic and tsunami hazard along the Mexican subduction zone. Until recently, there were limited observations at the shallow portion of the plate interface offshore Guerrero, so we deployed instruments there to better characterize the extent of the seismogenic zone. Here we report the discovery of episodic shallow tremors and potential slow slip events in Guerrero\u00a0offshore. Their distribution, together with that of repeating earthquakes, seismicity, residual gravity and bathymetry, suggest that a portion of the shallow plate interface in the gap undergoes stable slip. This mechanical condition may not only explain the long return period of large earthquakes inside the gap, but also reveals why the rupture from past M\u2009<\u20098 earthquakes on adjacent megathrust segments did not propagate into the gap to result in much larger events. However, dynamic rupture effects could drive one of these nearby earthquakes to break through the entire Guerrero seismic gap.\n\n## Introduction\n\nIn recent years, slow earthquakes have been observed at some subduction zones at shallow depths (0\u201320\u2009km deep) in the updip regions close to the trench1,2,3. Nevertheless, in contrast to downdip slow earthquakes4, shallow slow earthquakes are still not well understood. Some representative observations of shallow slow earthquakes come from the Japan Trench, where episodic tremor and slip (ETS) preceded the great Tohoku-Oki 2011, Mw 9 earthquake, revealing that slow slip and megathrust earthquakes can coexist at shallow depths of the plate interface5. For this reason, the near trench portions of subduction zones should be extensively studied to understand the influence of newly subducted materials on its mechanical properties and assess the actual extent of seismogenic zones, which may produce tsunamigenic and devastating earthquakes.\n\nA proposed earthquake scenario for the Guerrero seismic gap (GG), located along the Pacific coast of Mexico, has been one of a pending rupture of the entire gap capable of generating a large earthquake with Mw\u2009>\u200986. An earthquake of this magnitude in the seismic gap could produce ground accelerations in Mexico City twice as high as those experienced during the disastrous MW 8 Michoac\u00e1n earthquake of 19857, where several hundred buildings collapsed and around 10,000 people died. The associated tsunami could also be catastrophic in large coastal communities such as Acapulco and Zihuatanejo, among many others. Assessing the seismic potential of the GG is therefore one of the most urgent issues worldwide, given the risk to which more than 15 million people in the country\u2019s capital are exposed.\n\nThe north-west segment of the Guerrero seismic gap (NW-GG), with a length of approximately 140\u2009km (Fig.\u00a01A), has not experienced an earthquake with M\u2009>\u20097 since 19118. The largest seismic events in the NW-GG since then have been two near-trench tsunami earthquakes in 2002 (Mw 6.7, Mw\u00a05.9)9,10 and two aftershocks of the 2014 Mw 7.3 Papanoa event with magnitudes Mw 6.5 and Mw\u00a06.111,12. The Guerrero subduction zone is also prone to slow earthquakes, including some of the largest slow slip events (SSE) in the world, with Mw\u2009>\u20097.5, that repeat almost every 4 years8,13. Accordingly, the NW-GG accommodates stress aseismically as revealed by geodetic data, suggesting that coupling at the plate interface is approximately 75% lower than that of the bordering segments14,15. Since these contributions come from onshore observations, our knowledge of the shallow plate interface regions is limited.\n\nOne type of slow earthquake is tectonic tremor, a long duration burst of intermittent seismic signals16. Tremors are a good indicator of shear slip at the plate interface and are typically correlated temporally and spatially to SSE in ETS17. Offshore observations show that shallow tremor (ST) is also accompanied by SSEs in the weakly coupled shallow plate interface5,18. Repeating earthquakes (or repeaters) are the recurrent rupture of small patch-like regions together with surrounding aseismic sliding, making them another useful proxy for aseismic slip19. In Guerrero, tremors have been identified at approximately 200\u2009km downdip and 40\u201350\u2009km depths20,21 along the horizontally subducting slab22 (Fig.\u00a01A). These tremors describe rapid migration and have been a useful monitoring tool to infer the location and occurrence time of slow slip on the plate interface23,24,25, due to the close interconnection they have with SSE and overpressured fluids at the interface21,25,26. All of these observations were carried out using mostly onshore temporary stations and a few permanent stations, limiting tremor detection to selected time periods and regions27. So far, shallow slow earthquakes have not been observed at the\u00a0Mexican segment of the Middle America Trench most likely due to the absence of offshore instrumentation.\n\nHere we analyze the first ever set of offshore observations performed in the GG, corresponding to a one-year period starting from November 2017. An array of seven ocean bottom seismometers (OBS) was deployed inside the NW-GG28 (Fig.\u00a01A) to improve monitoring of seismic activity in search of shallow slow earthquakes. The OBS stations were equipped with three component 1\u2009Hz short period sensors and installed at water depths ranging between 980 and 2350 [m]. Their locations were estimated with a mean uncertainty of 2 metres and the data was corrected to remove time shifts on seismic recordings29. An envelope correlation method30 was used to detect STs as an initial evidence of shallow slow earthquakes in the GG. To explore the marine area with higher spatial resolution, regular earthquakes were also detected using continuous OBS data. Additionally, repeaters were identified using onshore data in a period from 2001 to 2019, and then complemented with offshore data from 2017 to 2018. Based on these results, together with information from residual gravity and residual bathymetry, we seek to answer critical questions such as whether slow earthquakes occur offshore and near the trench, what conditions are required to generate them and what are their characteristics. Most importantly, we seek to improve our understanding of possible future large earthquakes in the Guerrero seismic gap.\n\n## Results\n\n### Seismological observations\n\nWe detected and located over 100 STs (Fig.\u00a01B) within two kilometres uncertainty\u00a0in their location (Supplementary Fig.\u00a01). Most of the STs occurred between 10 and 16\u2009km depth (Supplementary Fig.\u00a01), so we assume they rupture at the plate interface as tremors occurring downdip20,25,26. However, depth is the least constrained parameter in our locations. No clear ST migration was observed (Fig.\u00a02C) as in regions further downdip25, suggesting that ST occur at mechanically isolated locked patches embedded in a weakly coupled plate interface31.\n\nMost of STs are located at short distances from the trench (<30\u2009km), while most detected seismicity lies around 60\u2009km away from the trench close to the coast (Fig.\u00a01B). There is a clear trench-perpendicular separation between STs and earthquakes, with a ~20 km-wide region devoid of seismic activity that we refer to as a \u201csilent zone\u201d hereafter. Repeaters coincide with STs in the east and west regions, indicating aseismic slip in these areas.\n\nA least-squares\u00a0fit with a hypothesis test32 confirms a weak correlation between ST magnitude and distance along dip (Fig.\u00a02B). ST magnitude and duration tend to be larger closer to the coast where seismicity intensifies, and coupling is expected to increase. Further observation is still required to confirm this conjecture.\n\nWe spatially grouped STs into four main clusters, S1 to S4 (Fig.\u00a02A). Each cluster shows characteristic behaviours and source properties in their magnitude and duration (Supplementary Fig.\u00a02). Clusters S3 and S4 are located at the east side of the study area and have fewer number of tremors. However, S4 (close to the coastline) has the largest magnitude tremors with a mean magnitude of 2.0\u2009\u00b1\u20090.14. S1 and S2 are located at the west side of the array of OBS stations. These two clusters contain most of the ST in the region with episodes of increased activity. The increased activity comes as episodic tremor bursts with recurrence periods of three months\u00a0and one\u00a0month for S1 and S2, respectively (Fig.\u00a02C, D). The episodic ST can be employed as preliminary and first near trench observation of where and when SSE occurred offshore Guerrero23,24. Assuming the occurrence of aseismic slip where repeaters and tremors take place, it is most likely that the episodic STs are associated with nearby short-term SSE with one month\u00a0and three\u00a0month recurrence periods. Characteristic source properties and different recurrence intervals of STs in each cluster are an indicator of a heterogeneous plate interface regulating the behaviour of tremors.\n\n### Residual gravity and bathymetry\n\nSeafloor geodetic observations at other subduction zones have revealed that subducting\u00a0relief can increase pore pressure along the megathrust and create a complex system of fractures within the overriding plate33 reducing the interplate coupling and generating heterogeneous stress fields at the plate interface31. This heterogeneity may lead to a mixture of mechanical conditions controlling macroscopic fault slip and thus enhancing the generation of slow earthquakes25,34. Further observations have also shown that the presence of a subducting\u00a0seamount can lead to the development of a creeping region, capable of producing a diversity of slip behaviours including SSE, tremor or even tsunami earthquakes35,36,37. One way to identify subducting\u00a0relief at the plate interface is by interpreting residual gravity and residual bathymetry anomalies (RG&BA) which have also been associated with small earthquakes and creep38.\n\nIn the NW-GG there are some seamounts moving with the oceanic crust towards future subduction (Fig.\u00a01A, Supplementary Fig.\u00a03). Additionally, the 2002 near-trench tsunami earthquake (Mw 6.7) overlaps with a positive RG&BA making it possible for both to be related36 (Fig.\u00a01, Supplementary Fig.\u00a03). The silent zone covers a trench-parallel transition of a negative-positive RG&BAs, while repeaters and most STs are located over positive RG&BAs. The large positive and negative RG&BA may thus be interpreted as an irregular subducting\u00a0relief that increases pore pressure, fracturing and decreases coupling, that then determines the mechanical properties of the plate interface likely to generate tsunami earthquakes, STs, repeaters and\/or the silent zone.\n\n### Plate interface mechanical model\n\nThe spatial distributions of regular earthquakes and STs reveal one of our most interesting observations, which is the silent zone in the centre of the NW-GG (Fig.\u00a01B), referred to above. There are two possibilities to explain the lack\u00a0of seismicity there. Firstly, this area could be completely locked and thus accumulating strain, so that the probability of a large earthquake initiating there is large. Secondly, it could basically be unlocked and sliding freely with low coupling, thus decreasing the possibility of a large earthquake in the GG. Considering that onshore geodetic data suggest a lack of coupling along the NW-GG14,15, together with the inference of aseismic slip from STs and repeaters, it is reasonable to think that the offshore portion of the NW-GG (i.e., the silent zone) better corresponds to the second hypothesis of an almost freely sliding domain. The heterogeneous distribution of STs, repeaters, earthquakes and large variation of RG&BA, however, suggest local changes of mechanical properties at the offshore plate interface that cannot be explained with a simple unlocked model.\n\nSuch heterogeneity in the NW-GG strongly suggests that interplate frictional conditions39 vary significantly in space. Regions prone to velocity-strengthening and velocity-weakening slip should be present and interact with each other, leading to a variety of coexisting sliding behaviours11,40. Therefore, we interpreted our observations to characterize the study area into well-defined regions with similar sliding behaviour and thus frictional conditions.\n\nRegion A extends from the trench up to 30\u2009km downdip (Fig.\u00a03C), it represents the shallowest interface segment\u00a0of the NW-GG. In this region we have episodic STs, repeaters, positive RG&BAs and near-trench tsunami earthquakes (Fig.\u00a04). Given the close connection between tremor and SSE17,21, the episodic ST could be accompanied by slow slip in the area. Such features would correspond to isolated velocity-weakening asperities within a velocity-strengthening matrix which together comprise a frictionally heterogeneous domain where SSEs can occur and generate episodic ST40,41,42.\n\nRegion A\u2019, from 30 to 50\u2009km downdip (Fig.\u00a03C), is formed by the silent zone. It covers an along-strike transition of positive-negative RG&BAs and is surrounded by STs, repeaters, seismicity and rupture areas of M\u2009>\u20097 earthquakes (Fig.\u00a04). The silent zone would correspond to a velocity-strengthening domain where strain is mostly released aseismically, so the possibility of a large earthquake nucleating there is low.\n\nBelow the coast, from 50 to 80\u2009km downdip (Fig.\u00a03C), we find Region B, which also contains STs and repeaters, but experiences a larger concentration of small earthquakes and M 7 class earthquakes in the past (Fig.\u00a04). Additionally, large long-term SSEs originating downdip can penetrate this area. Region B is therefore frictionally heterogeneous too, and together with Region A, should contain significant velocity-weakening patches where rapid slip has produced earthquakes with M\u2009>\u20096, such as the tsunami earthquakes, the aftershocks of the 2014 earthquake, or even larger earthquakes like those in the late 19th and early 20th centuries11 (Fig.\u00a04). This means that locked patches could eventually generate M\u2009>\u20097 earthquakes despite aseismic slip also occurring periodically. We still have limited data to interpret the south-east segment of regions A, A\u2019 and B (Fig. 4).\u00a0At farther distances, in Region C, only long-term SSEs and deep tremors occur13,15,20,21. Region C has similar slip behaviour as Region A, their difference should be in the mechanisms giving rise to slow earthquakes, with possibly subducted relief, increased pore pressure, sediments and a complex system of fractures facilitating slow earthquakes in Region A43.\n\nIn addition to the along-dip variations, there is also a clear along-strike change in the mechanical properties between the NW-GG and the adjacent Petatl\u00e1n segment to the west (Fig.\u00a04). The boundary separating these two segments is outlined by the abrupt end of the silent zone to the west (close to Papanoa) and beyond which STs, repeaters and large earthquakes are more frequent.\n\nIn the Petatl\u00e1n segment, we distinguish three trench-parallel regions (Fig.\u00a03B). From the trench and up to 45\u2009km downdip we find repeaters, ST and possibly SSE, similarly to Region A inside the NW-GG. Slow earthquakes in here may define a transition zone1 between the velocity-strengthening silent zone of the NW-GG and an adjacent velocity-weakening segment offshore Petatl\u00e1n. Further downdip, Region A changes into Region B, a velocity-weakening domain which is absent of STs but seismically active, with repeaters and large earthquakes (M\u2009>\u20097) occurring every ~35 years. At deeper areas we find the same behaviour as in Region C at the NW-GG, where long-term SSEs take place. One of the most important overall differences between the NW-GG and the Petatl\u00e1n segment is that the former seems to be dominated by mechanically unlocked conditions, which makes the initiation of a large earthquake (M\u2009>\u20098) there less likely.\n\n## Discussion\n\nMany large earthquakes (M\u2009>\u20097) have taken place in the Petatl\u00e1n segment (Fig. 4); interestingly, these earthquakes have stopped propagating when entering the NW-GG. It is possible that the silent zone acts as a barrier stopping large earthquakes\u00a0from becoming much larger and rupturing the NW-GG. To do so they will need to continue rupturing across the large velocity-strengthening region in the silent zone. A similar situation is reported at the shallow plate interface of central Peru, where a low coupling and velocity-strengthening region, associated with the subducting Nazca ridge, acts as a barrier for larger earthquakes (M\u2009>\u20098)44.\n\nSubducting relief interpreted from RG&BAs in the region could have created highly heterogeneous frictional conditions that contribute to generate the conditions for slow slip, such as episodic STs, possible short-term SSEs repeaters and\/or a creeping silent zone. All of our observations indicate that the central portion of the NW-GG is dominated by a weakly coupled domain. Weak coupling, offshore and onshore, explains the long recurrence period of large earthquakes in the GG45. This implies that the initiation of a M\u2009>\u20098 earthquake in the NW-GG is less likely to occur as previously expected; however, the risk of a rupture of the GG does not disappear. Stress loading around the creeping silent zone offshore the NW-GG may facilitate a sufficiently large earthquake approaching from an adjacent segment to propagate through the gap driven by dynamic effects, including the cascading rupture of nearby locked patches below the coast, where M\u2009>\u20097 ruptures have occurred in the past. Comparable circumstances where subducted seamounts, seismic and aseismic events coexist in the same interface region has already been described in the Japan Trench46; for these reasons, disaster prevention efforts should not be reduced in Guerrero. New offshore and onshore observations, together with physics-based source modelling, could validate our predictions, which are important for further development of seismic risk mitigation.\n\n## Method\n\n### Tremor detections\n\nTremor detection and location were done following a modified envelope correlation method based on maximum-likelihood30. Envelope waveforms were estimated by (1) band\u2010pass filtering continuous velocity data between 2 and 8\u2009Hz, (2) squaring, (3) low pass filtering at 0.2\u2009Hz and (4) resampling at 1\u2009Hz. Tremor detections were\u00a0done using 300\u2009s time windows with 150\u2009s time steps and a detection threshold for cross correlation coefficient between stations of 0.647.\n\nLocalization of tremors was done considering a normalized envelope waveform $${w}_{i}\\left(t\\right)$$ for all components at all stations30.\n\n$${w}_{i}\\left(t\\right)=\\frac{{w}_{i}^{{\\prime} }(t)-\\bar{{w}_{i}^{{\\prime} }}}{\\sqrt{{\\sum }_{k=1}^{{N}_{t}}{\\left({w}_{i}^{{\\prime} }({t}_{k})-\\bar{{w}_{i}^{{\\prime} }}\\right)}^{2}}},$$\n(1)\n\nwhere the sub index $$i$$ represents the $$i$$-th component, $${w}_{i}^{{\\prime} }(t)$$ is the original envelope waveform, $$\\bar{{w}_{i}^{{\\prime} }}$$ is the temporal mean, $${N}_{t}$$ is the number of time samples and $${t}_{k}$$ is the $$k$$-th time step. $${w}_{i}\\left(t\\right)$$ can also be defined by assuming a source envelope which is a common template waveform $$w(t)$$ at all stations, but shifted in time considering a travel time $${\\triangle t}_{i}(x)$$ between source and position $$x$$, plus a Gaussian error with a distribution $$N(0,{\\sigma }_{i}^{2})$$,\n\n$${w}_{i}\\left(t+{\\triangle t}_{i}(x)\\right)=w\\left(t\\right)+{e}_{i}(t+{\\triangle t}_{i}(x)).$$\n(2)\n\nWith these assumptions we can develop the mathematical expressions to maximize the likelihood of finding the observed envelopes from the combination of a common template waveform $$w\\left(t\\right)$$ and travel times to position $$x$$. The maximum likelihood problem can be solved as a summation of cross-correlations weighted by the error variance. We then maximize the averaged of the weighted cross-correlations (ACC),\n\n$${ACC}\\left(x\\right)=\\frac{{\\sum }_{(i,j)}{\\sum }_{k=1}^{{N}_{t}}\\frac{{w}_{i}\\left({t}_{k}+{\\triangle t}_{i}(x)\\right){w}_{j}\\left({t}_{k}+{\\triangle t}_{j}(x)\\right)}{{\\sigma }_{i}^{2}{\\sigma }_{j}^{2}}}{{\\sum }_{(i,j)}\\frac{1}{{\\sigma }_{i}^{2}{\\sigma }_{j}^{2}}}.$$\n(3)\n\nThe weight, or error variance, is calculated based in the similarity of the template and observed waveforms. This method is looking for the best tremor location $${x}^{{best}}$$, that will maximize ACC, by calculating travel times to any position $$x$$. Travel times are calculated using ray theory and a 1D velocity model for the Guerrero region48.\n\nThe maximization problem is solved using a sequence of two steps; first, a grid search with a fixed 10\u2009km depth considering local maxima, and second, a gradient method search49 (CCSA) to improve hypocentre locations. The grid search is taking into account local maxima because each of these will come as a result of more than one detection inside the same time window; therefore, this method is able to detect more than one event inside the same time window. Finally, the gradient method uses all the local maxima from the grid search as initial values to refine the location.\n\nTo remove outliers, the acquired detections must validate two established conditions. Firstly, cross-correlation coefficients must be larger than 0.6 as initially postulated. However, during the maximization procedure the correlation value could decrease, so this condition must be verified. Secondly, the similarity between envelope and template waveform (Eq.\u00a02) must be good, so if their correlation value is below 0.4, envelopes are rejected. Once outliers are excluded, the localization procedure is repeated starting from the gradient method and continues until no outliers are found. Standard deviation of the final location is estimated using bootstrap, and results with more than two kilometres of error are also excluded.\n\nIt is noteworthy to point out that the velocity structure used in tremor location does not include a slow sedimentary layer located at the ocean floor, where OBS are situated at. This will produce a common bias, in which estimated locations should be deeper than expected. This means that depth is the less constrained parameter in our locations.\n\nSource parameters for tremors are estimated using the original envelope waveform. The average of seismic energy rate can be expressed as\n\n$${\\dot{E}}_{s}\\left(t\\right)=4\\pi \\rho \\beta \\frac{{\\sum }_{(i,j)}\\left(\\frac{{w}_{i}^{{\\prime} 2}\\left(t+{\\triangle t}_{i}({x}^{{best}})\\right){R}_{i}^{2}}{{\\sigma }_{i}^{2}}+\\frac{{w}_{j}^{{\\prime} 2}\\left(t+{\\triangle t}_{j}({x}^{{best}})\\right){R}_{j}^{2}}{{\\sigma }_{j}^{2}}\\right)}{{\\sum }_{(i,j)}\\left(\\frac{1}{{\\sigma }_{i}^{2}}+\\frac{1}{{\\sigma }_{j}^{2}}\\right)}.$$\n(4)\n\nwhere $${R}_{i}$$ is the hypocentral distance to stations, $$\\rho =3000$$ kg\/m3 is density and $$\\beta =2.8$$ km\/s is shear wave velocity. The time when the averaged seismic energy gets to its maximum will be the hypocentral time and duration will be equal to the time required to get one quarter of the maximum energy. Energy magnitude is estimated with $${M}_{e}=\\frac{{\\log }\\left(\\int {\\dot{E}}_{s}(t){dt}\\right)-4.4}{1.5}$$50.\n\n### Earthquake detections\n\nContinuous OBS data was also used to detect small local seismicity and monitor in detail seismic behaviour at the he marine area. Earthquake detections were done using an automatic method of a short time window over a long-time window (STA\/LTA), with a ratio of 2 in at least three stations to trigger a detection. The STA\/LTA threshold was determined empirically by visually detecting 74 earthquakes a priori. LTA and STA windows had 10- and 0.3-seconds length, respectively. A total of 4303 events were detected by the STA\/LTA method. To distinguish earthquakes from noise we used an automatic picking51, to pick P and S waves and discard those detections where no phase was found. A second stage of visual inspection to avoid any false detections was done. Finally, a new manual picking of P and S waves was done to a total of 848 events. Location was done following a maximum likelihood method52 and using a 1D velocity structure for Guerrero49. Density of earthquakes was estimated using a grid covering the complete study area with square elements of 0.1 degrees length. A bivariate histogram was calculated with the number of earthquakes in each element of the grid.\n\nWe compared our OBS earthquake catalogue with an alternative catalogue reported by the National Seismological Service (SSN)53, from inland stations and earthquake detections since 1908 (Supplementary Fig.\u00a04, Supplementary Fig.\u00a05). For the time period of interest (2017\/11\/01- 2018\/12\/01), the SSN catalogue reported a total of 1074 earthquakes in the Guerrero seismic gap region. The OBS and SSN catalogues have 298 common earthquakes that are included in both catalogues. Additionally, the OBS catalogue has 518 new earthquakes not reported by SSN (Supplementary Fig.\u00a05B). Even when the locations of events common to the SSN and OBS catalogues are similar with good correlation values (Supplementary Fig.\u00a04), locations from SSN tend to have a north-west shift with respect to the OBS locations (Supplementary Fig.\u00a05B). These systematic differences must be due to differences in velocity models used for locations, since SSN uses a simple 5 layers 1D velocity model. Moreover, the two networks (offshore and inland) are configured independent of each other and will have difficulty in precisely locating earthquakes occurring outside their range, contributing to a systematic offset.\n\nEarthquake distribution for both catalogues was compared (Supplementary Fig.\u00a05A). The distribution of OBS earthquakes and SSN earthquakes is equivalent. With this we can constrain the earthquake distribution offshore the Guerrero seismic gap with additional certainty. From these extra enquiries, the observations reported in this research are not modified and become better supported. Thus, we can confirm there is no significant artifact that could modify our conclusion and that detectability of reported earthquakes is sufficient to support our conclusions.\n\n### Repeaters\n\nWe searched for repeating earthquakes along the trench by analyzing 440,655 waveforms from 13 permanent stations from the SSN53 corresponding to 75,567 earthquakes recorded between 2001 though 2019. Furthermore, we analyzed the one-year data from the OBS network, using the ad hoc catalogue described in the previous section. To classify events as repeating earthquakes, we computed the correlation coefficient (CC) and spectral coherency (COH) for all pairs of closely located events (<100\u2009km). The CC and COH was estimated in a 25-s time window starting at the onset of the P-wave arrival in a frequency band of 1\u20138\u2009Hz and 4\u201316\u2009Hz for the permanent stations and OBS stations, respectively. We used two different frequency bands to provide a higher signal to noise ratio\u00a0for the OBS stations which are located in a nosier environment. Sequences were formed by linking pairs of events with a COH\u00a0and CC\u00a0higher than 95% for at least two or more stations either permanent or OBS. Clusters of repeaters were formed using hierarchical clustering with \u201csingle\u201d method provided from SciPy library. Hence, we found along the GG, a set of 51 sequences consisting of 2 to 4 repeating earthquakes each; the magnitude of these events ranges from 3.4 to 4.5 using the onshore permanent stations from 2001 through 2019. By means of the OBS network, an additional set of 7 sequences was detected with magnitudes between 2.7 and 3.8 with short burst-type recurrence time (<days, month) during the survey period (2017\u20132018).\n\n### Residual gravity and bathymetry\n\nWe used global compilations of marine gravity anomalies and bathymetry data to generate grids of residual gravity and bathymetry38,54 covering a region of the Middle America subduction zone offshore the Pacific coast of Mexico. Analyzing residual gravity anomalies and residual bathymetry allows us to discern small-scale local features on the forearc because the broader regional signal associated with subduction has been removed. In order to calculate the residual gravity, we used version 29.1 of the Global Gravity grid55 developed and distributed by the Scripps Institution of Oceanography, University of California, San Diego. This global gravity model is provided at 1-arc minute resolution and is constructed using measurements of the sea surface slope collected by several satellite altimeters over separate geodetic missions conducted in the years between 1985 and 2019. We extracted trench-perpendicular profiles every ~25 kilometres along a ~1500-kilometre-long segment of the Middle America subduction zone. The gravity anomaly data (given in units of mGal) along the profiles were then stacked. We then subtracted the trench-perpendicular average gravity anomaly from the original grid to obtain the residual gravity anomalies, which show both positive and negative values in the forearc region above the Guerrero seismic gap. Meanwhile, in order to calculate the residual bathymetry, we requested gridded data from the Global Multi-Resolution Topography (GMRT) Synthesis56 at 7.5-arc second resolution. Offshore, the GMRT grid combines data from multiple sources, including publicly available high-resolution multibeam sonar swath surveys and the General Bathymetric Chart of the Oceans57. We followed a similar procedure to the processing of the residual gravity in extracting profiles of elevation data, averaging these, and then subtracting this trench-perpendicular average to finally obtain the residual bathymetry (for offshore regions) and residual topography (on land).\n\nFrom bathymetry data (Fig.\u00a01A) and residual bathymetry data (Supplementary Fig.\u00a03), seamounts can be found in front of the Guerrero seismic gap, at approximately 100\u2009km away from the coast in between longitudes \u2212100.5\u02da, \u2212100.0 and latitudes 15.7\u02da, 16.5\u02da. On the incoming plate these seamounts have basal widths ranging from 10 to 20 kilometres and rising at least several hundred metres above the surrounding seafloor. Seamounts are moving with the oceanic crust towards future subduction. Areas with positive values in the residual gravity are found landward of seamounts on the incoming Cocos Plate (Supplementary Fig.\u00a03). This suggests that some of the positive residual anomalies that share these features could be associated with subducting\u00a0seamount chains underneath the forearc. Meanwhile, some of the negative residual features are coincident with basins based on interpretation of the seafloor morphology. The gravity data are more directly related to the roughness of subducting topography since it is more sensitive to the structure of the rock basement, whereas the bathymetry measurements are affected by sediment infill or erosional processes occurring on the upper plate.\n\n## Data availability\n\nData set of tremors, earthquakes and repeaters are available in\u00a0Supplementary Information. Codes are available upon request to the correspondence author R. P.-M. (plata.omar.48e.@st.kyoto-u.ac.jp). Ocean bottom seismometers data from SATREPS-UNAM project are subjected to restrictions policies until March 2025. Inland broadband data is available from SSN at: www.ssn.unam.mx.\n\n## References\n\n1. 1.\n\nObana, K. & Kodaira, S. Low-frequency tremors associated with reverse faults in a shallow accretionary prism. Earth Planet. Sci. Lett. 287, 169\u2013174 (2009).\n\n2. 2.\n\nSugioka, H. et al. Tsunamigenic potential of the shallow subduction plate boundary inferred from slow seismic slip. Nat. Geosci. 5, 414\u2013418 (2012).\n\n3. 3.\n\nWallace, L. et al. Slow slip near the trench at the Hikurangi subduction zone, New Zealand. Science 352, 701\u2013704 (2016).\n\n4. 4.\n\nObara, k & Kato, A. Connecting slow earthquakes to huge earthquakes. Science 353, 253\u2013257 (2016).\n\n5. 5.\n\nIto, Y., Hino, R. & Kaneda, Y. Episodic tremor and slip near the Japan Trench prior to the 2011 Tohoku-Oki earthquake. Geophys. Res. Lett. 42, 1725\u20131731 (2015).\n\n6. 6.\n\nSingh, S. & Mortera, F. Source time functions of large Mexican subduction earthquakes, morphology of the Benioff zone, age of the plate, and their tectonic implications. J. Geophys. Res. 96, 21,487\u201321,502 (1991).\n\n7. 7.\n\nKanamori, H., Jennings, P. C., Singh, S. K. & Astiz, L. Estimation of strong ground motions in Mexico City expected for large earthquakes in the Guerrero seismic gap. Bull. Seismol. Soc. Am. 83, 811\u2013829 (1993).\n\n8. 8.\n\nKostoglodov, V. et al. A large silent earthquake in the Guerrero seismic gap, Mexico. Geophys. Res. Lett. 30, 1807 (2003).\n\n9. 9.\n\nIglesias, A. et al. Near-trench Mexican earthquakes have anomalously low peak accelerations. Bull. Seismol. Soc. Am. 93, 953\u2013959 (2003).\n\n10. 10.\n\nFlores-Ibarra, K. \u201cEstudio de la fuente compleja del sismo 18 de abril del 2002 (Mw 6.7), fuera de la costa de Guerrero\u201d, thesis, Universidad Nacional Aut\u00f3noma de M\u00e9xico, Mexico City (2018).\n\n11. 11.\n\nUNAM seismology group. Papanoa, Mexico earthquake of 18 April 2014 (Mw7.3). Geof\u00edsica Internacional 54, 363\u2013386 (2015).\n\n12. 12.\n\nSingh, S. K. et al. Evidence of directivity during the earthquakes of 8 and 10 May 2014 (Mw 6.5, 6.1) in the Guerrero, Mexico seismic gap and some implications. J. Seismol. 23, 683\u2013697 (2019).\n\n13. 13.\n\nRadiguet, M. et al. Slow slip events and strain accumulation in the Guerrero gap, Mexico. J. Geophys. Res. 117, B043305 (2012).\n\n14. 14.\n\nRadiguet, M. et al. Triggering of the 2014 Mw7.3 Papanoa earthquake by a slow slip event in Guerrero, Mexico. Nat. Geosci. 9, 829\u2013833 (2016).\n\n15. 15.\n\nCruz-Atienza, V. M. et al. Short-term interaction between silent and devastating earthquakes in Mexico. Nat. Commun. 12, 2171 (2021).\n\n16. 16.\n\nIde, S., Shelly, D. R. & Beroza, G. C. Mechanism of deep low frequency earthquakes: further evidence that deep non-volcanic tremors is generated by shear slip on the plate interface. Geophys. Res. Lett. 34, L03308 (2007).\n\n17. 17.\n\nRogers, G. & Dragert, H. Episodic tremor and slip on the Cascadia subduction zone: the chatter of silent slip. Science 300, 1942\u20131943 (2003).\n\n18. 18.\n\nNakano, M., Hori, T., Araki, E. & Ide, S. Shallow very low frequency earthquakes accompany slow slip events in the Nankai subduction zone. Nat. Commun. 9, 984 (2018).\n\n19. 19.\n\nUchida, N., Iinuma, T., Nadeau, R. M., B\u00fcrgmann, R. & Hino, R. Periodic slow slip triggers megathrust zone earthquakes in northeastern Japan. Science 351, 488\u2013492 (2016).\n\n20. 20.\n\nHusker, et al. Temporal variations of non-volcanic tremor (NVT) locations in the Mexican subduction zone: finding the NVT sweet spot. Geochem. Geophys. Geosystems, Q03011 (2012).\n\n21. 21.\n\nVillafuerte, C. & Cruz-Atienza, V. M. Insights into the causal relationship between slow slip and tectonic tremor in Guerrero, Mexico. J. Geophys. Res. 122, 6642\u20136656 (2017).\n\n22. 22.\n\nP\u00e9rez-Campos, X. et al. Horizontal subduction and truncation of the Cocos Plate beneath central Mexico. Geophys. Res. Lett. 35, L18303 (2008).\n\n23. 23.\n\nFrank, W. B. et al. Uncovering the geodetic signature of silent slip through repeating earthquakes. Geophys. Res. Lett., 42, 2774\u20132779 (2015).\n\n24. 24.\n\nFrank, W. B. Slow slip hidden in the noise: the intermittence of tectonic release. Geophys. Res. Lett. 43, 10125\u201310133 (2016).\n\n25. 25.\n\nCruz-Atienza, V. M., Villafuerte, C. & Bhat, H. S. Rapid tremor migration and pore-pressure waves in subduction zones. Nat. Commun. 9, 2900 (2018).\n\n26. 26.\n\nKostoglodov, V. et al. The 2006 slow slip event and nonvolcanic tremor in the Mexican subduction zone. Geophys. Res. Lett. 37, L24301 (2010).\n\n27. 27.\n\nMaury, J. et al. Comparative study of tectonic tremor locations: characterization of slow earthquakes in Guerrero, Mexico. J. Geophys. Res. Solid Earth 121, 5136\u20135151 (2016).\n\n28. 28.\n\nCruz-Atienza, V. M. et al. A seismogeodetic amphibious network in the Guerrero seismic gap Mexico. Seismol. Res. Lett. 89, 1435\u20131449 (2018).\n\n29. 29.\n\nShinohara, M. et al. Continuous seismic monitoring by using long-term ocean bottom seismometers. Earth Planets Space 69, 159 (2017).\n\n30. 30.\n\nMizuno, I. & Ide, S. Development of a modified envelope correlation method based on maximum-likelihood method and application to detecting and locating deep tectonic tremors in western Japan. Earth Planet Space 71, 40 (2019).\n\n31. 31.\n\nSaffer, D. & Wallace, L. The frictional, hydrologic, metamorphic and thermal habitat of shallow slow earthquakes. Nat. Geosci. 8, 594\u2013600 (2015).\n\n32. 32.\n\nSanford, W. Applied Linear Regression (John Wiley and Sons, 1985).\n\n33. 33.\n\nDominguez, S., Lallemand, S. E., Malavieille, J. & von Huene, R. Upper plate deformation associated with seamount subduction. Tectonophysics 293, 207\u2013224 (1998).\n\n34. 34.\n\nYokota, Y., Ishikawa, T., Watanabe, S., Tashiro, T. & Asada, A. Seafloor geodetic constraints on interplate coupling of the Nankai Trough megathrust zone. Nature 534, 374\u2013377 (2016).\n\n35. 35.\n\nWang, K. & Bilek, S. Invited review paper: fault creep caused by subduction of rough seafloor relief. Tectonophysics 610, 1\u201324 (2014).\n\n36. 36.\n\nBell, R., Holden, C., Power, W., Wang, X. & Downes, G. Hikurangi margin tsunami earthquake generated by slow seismic rupture over a subducted seamount. Earth Planet. Sci. Lett. 397, 1\u20139 (2014).\n\n37. 37.\n\nTodd, E. et al. Earthquakes and tremor linked to seamount subduction during shallow slow slip at the Hikurangi margin, New Zealand. J. Geophys. Res. 123, 6769\u20136783 (2018).\n\n38. 38.\n\nBassett, D. & Watts, A. Gravity anomalies, crustal structure, and seismicity at subduction zones: 1. Seafloor roughness and subducting relief. Geochem. Geophysics Geosystems 16, 1508\u20131540 (2015).\n\n39. 39.\n\nScholz, C. Earthquakes and friction laws. Nature 391, 37\u201342 (1998).\n\n40. 40.\n\nYabe, S. & Ide, S. Slip-behavior transitions of a heterogeneous linear fault. J. Geophys. Res. 122, 387\u2013410 (2017).\n\n41. 41.\n\nAndo, R., Nakata, R. & Hori, T. A slip pulse model with fault heterogeneity for low\u2010frequency earthquakes and tremor along plate interfaces. Geophys. Res. Lett. 37, L10310 (2010).\n\n42. 42.\n\nPhillips, N. J. et al. Frictional strengths of subduction thrust rocks in the region of shallow slow earthquakes. J. Geophys. Res. 125, e2019JB018888 (2020)\n\n43. 43.\n\nWang, K. & Bilek, S. Do sea mounts generate or stop large earthquakes? Geology 39, 819\u2013822 (2011).\n\n44. 44.\n\nPerfettini, H. et al. Seismic and aseismic slip on the Central Peru megathrust. Nature 465, 78\u201381 (2010).\n\n45. 45.\n\nHusker, A., Ferrari, L., Arango-Galv\u00e1n, C., Corbo-Camargo, F. & Arzate-Flores, J. A. A geologic recipe for transient slip within the seismogenic zone: insight from the Guerrero seismic gap, Mexico. Geol. Soc. Am. 46, 35\u201338 (2018).\n\n46. 46.\n\nMochizuki, K., Yomoaki, Y., Shinohara, M., Yamanaka, Y. & Kanazawa, T. Weak interplate coupling by seamounts and repeating M~7 earthquakes. Science 321, 1194\u20131197 (2008).\n\n47. 47.\n\nIde, S. Striations, duration, migration and tidal response in deep tremor. Nature 446, 356\u2013359 (2010).\n\n48. 48.\n\nSpica, Z. et al. 3-D shear wave velocity model of Mexico and South US: bridging seismic networks with ambient noise cross-correlations (C1) and correlation of coda of correlations (C3). Geophys. J. Int. 206, 1795\u20131813 (2016).\n\n49. 49.\n\nSvanberg, K. A class of globally convergent optimization methods based on conservative convex separable approximations. Soc. Ind. Appl. Math. 12, 555\u2013573 (2002).\n\n50. 50.\n\nChoy, G. L. & Boatwright, J. L. Global patterns of radiated seismic energy and apparent stress. J. Geophys. Res. 100, 18,205\u201318,228 (1995).\n\n51. 51.\n\nYokota, T., Zhou, S., Mizoue, M. & Nakamura, I. An automatic measurement of arrival time of seismic waves and its application to an on-line processing system. Bull. Earth. Res. Inst. 55, 449\u2013484 (1981).\n\n52. 52.\n\nHirata, N. & Matsu\u2019ura, M. Maximun-likelihood estimation of hypocenter with origin time eliminated using nonlineal inversion technique. Phys. Earth Planet. Inter. 47, 50\u201361 (1987).\n\n53. 53.\n\nServicio Sismol\u00f3gico Nacional (SSN), Servicio Sismol\u00f3gico Nacional, Instituto de Geof\u00edsica, Universidad Nacional Aut\u00f3noma de M\u00e9xico (2018).\n\n54. 54.\n\nBassett, D. & Watts, A. Gravity anomalies, crustal structure, and seismicity at subduction zones: 2. Interrelationships between fore-arc structure and seismogenic behavior. Geochem. Geophysics Geosystems 16, 1541\u20131576 (2015).\n\n55. 55.\n\nSandwell, D. T., Harper, H., Tozer, B. & Smith, W. H. F. Gravity field recovery from geodetic altimeter missions. Adv. Space Res. https:\/\/doi.org\/10.1016\/j.asr.2019.09.011 (2019).\n\n56. 56.\n\nRyan, W. B. et al. Global multi\u2010resolution topography synthesis. Geochem. Geophys. Geosystems 10, Q03014 (2009).\n\n57. 57.\n\nWeatherall, P. et al. New digital bathymetric model of the world\u2019s oceans. Earth Space Sci. 2, 331\u2013345 (2015).\n\n58. 58.\n\n59. 59.\n\nFerrari, L., Orozco\u2010Esquivel, T., Manea, T. & Manea, M. The dynamic history of the trans\u2010Mexican volcanic belt and the Mexico subduction zone. Tectonophysics 522\u2010523, 122\u2013149 (2012).\n\n## Acknowledgements\n\nOffshore instrumentation of the NW-GG comes from a collaboration between Japanese and Mexican universities and agencies. This project was funded by Science and Technology Research Partnership for Sustainable Development (SATREPS) and the Japan Science and Technology Agency (JST) with grant number JPMJSA1510, the National Autonomous University of Mexico (UNAM) with PAPIIT grants IN113814, IN120220 and IG100617, the Mexican Council of Science and Technology (CONACyT) with grant PN6471, AMEXCID-SRE and the Ministry of Civil Protection of the State of Guerrero, Mexico. Special thanks to UNAM Coordinaci\u00f3n de la Investigaci\u00f3n Cient\u00edfica for \u201cEl Puma\u201d research vessel time and all the crew. We thank Arturo Ronquillo Arvizu for his technical support and P. Romanet, T. Nishikawa, S. Ohyanagi, T. Chang and C. Villafuerte for valuable discussion.\n\n## Author information\n\nAuthors\n\n### Contributions\n\nR.P.-M. detected tremors and wrote the manuscript, M.S. detected earthquakes, E.G. calculated residual bathymetry, L.A.D. and T.T. detected repeaters and V.M.C.-A. and Y.I. are the PIs of the Mexico-Japan cooperation project. S.I., N.M., Y.Y., A.T., T.Y., J.R. and A.H. contributed with geophysical interpretation and improved the manuscript.\n\n### Corresponding author\n\nCorrespondence to R. Plata-Martinez.\n\n## Ethics declarations\n\n### Competing interests\n\nThe authors declare no competing interests.\n\nPeer review information Nature Communications thanks Hector Gonzalez-Huizar and the other, anonymous, reviewers for their contribution to the peer review of this work.\n\nPublisher\u2019s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.\n\n## Rights and permissions\n\nReprints and Permissions\n\nPlata-Martinez, R., Ide, S., Shinohara, M. et al. Shallow slow earthquakes to decipher future catastrophic earthquakes in the Guerrero seismic gap. Nat Commun 12, 3976 (2021). https:\/\/doi.org\/10.1038\/s41467-021-24210-9\n\n\u2022 Accepted:\n\n\u2022 Published:","date":"2021-09-17 22:05:01","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 2, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.47578686475753784, \"perplexity\": 6021.274653486734}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-39\/segments\/1631780055808.78\/warc\/CC-MAIN-20210917212307-20210918002307-00697.warc.gz\"}"} | null | null |
Zhongxin station (), is a station of Line 21 of the Guangzhou Metro. It started operations on 28 December 2018.
The station has an underground island platform with 4 tracks. Platform 1 is for trains heading east to Zengcheng Square, whilst platform 2 is for trains heading west to Yuancun, along with 2 unused bypass tracks next to the stopping tracks.
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There are 3 exits, lettered A, C and D. Exit D is accessible. All exits are located on Fengguang Road.
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References
Railway stations in China opened in 2018
Guangzhou Metro stations in Zengcheng District | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 576 |
The Cemetery
History Research Group
Nature Group
Monuments Group
History of the cemetery
Map of the cemetery
Chapel Stained-Glass Window
Cemetery Nature Survey
Tree Survey
Charles Norris
Author: Christine Gambles
© Christine Gambles
Charles was born in Newbury on the 15th August and baptised on the 26th August 1825 at St Nicolas Church, he was the son on James & Hannah Norris (nee Edwards) who married on the 24th April 1825 in Inkpen Berkshire.
Other children of James and Hannah were (all baptised at St Nicolas Church):
Sarah 2/3/1828
James 30/5/1830
George 24/2/1833
Mary 12/6/1836
Charles was recorded living with his widowed mother and siblings in Bartholomew Street Newbury in 1841. His father passed away aged 34 in 1836 and was buried on the 16th July at St Nicolas Church
He married Mary Ann Pearce in Newbury in 1848 and they had the following children:
William 1849
Charles c1851
Harry 1852
Mary Jane 1854
Frank c1857
Elizabeth c1860
The family were living in West Mills (1851 and 1861), Wither's Yard Northbrook Street (1871) and Magdala Terrace (1881). Charles worked as a Fellmonger and Glover in 1851, a Leather Dresser in 1861, a Fellmonger & Leather Dresser's Foreman in 1871 and a Fellmonger's Foreman in 1881.
Charles died aged 63 on the 6th May 1888 and was laid to rest in the Newtown Road Cemetery on the 8th May.
His widow Mary Ann died aged 75 on the 27th February 1896 and was laid to rest in the Newtown Road Cemetery on the 3rd March.
Sources:as above
Website designed and maintained by Paul Thompson on behalf of the Friends of Newtown Road Cemetery. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,384 |
Q: A basic question about exponentiation: when is $a^{xy}=(a^x)^y$? This is a silly question but under what conditions is $a^{xy}=(a^x)^y$ true, given all are complex numbers?
A: Let's spell out the definitions. For complex numbers $z$ and $\alpha$, and choice of range of argument, such as $[0,2\pi)$ or $(-\pi, \pi]$, we can define
$$z^{\alpha} = \exp(\alpha \log z) = \exp(\alpha (\log |z| + i\arg z))$$
With this definition, we have
$$a^{xy} = \exp(xy(\log |a| + i\arg a))$$
Now $a^x = \exp(x(\log |a| + i\arg a))$, so $\log |a^x| = \log (\exp(x \log |a|)) = x\log |a|$ and $\arg(a^x) = \arg a + 2n\pi$ for the unique $n_x \in \mathbb{Z}$ making $\arg(a^x)$ live in the desired range. Hence
$$(a^x)^y = \exp(xy(\log |a| + i(\arg a + 2n_x\pi))$$
In other words,
$$(a^x)^y = a^{xy}\exp(2n_xxyi\pi)$$
for some $n \in \mathbb{Z}$. Thus we have $(a^x)^y = a^{xy}$ if and only if $n_x=0$ or $xy \mid n_x$.
If you consider complex exponentiation as a multifunction, the identity $(a^x)^y = a^{xy}$ is true since this issue with the choice of range of argument goes away.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,155 |
Ingo Schmidt ist der Name folgender Personen:
* Ingo Schmidt (Wirtschaftswissenschaftler) (1932–2020), deutscher Wirtschaftswissenschaftler
Ingo Schmidt (Ökonom), deutscher Ökonom und Leiter das Labour Studies Program der Athabasca University in Kanada
Ingo Schmidt-Lucas (* 1972), deutscher Tonmeister
Ingo Schmidt-Wolf (* vor 1961), deutscher Onkologe und Hochschullehrer
Siehe auch:
Ingo Schmitt | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 500 |
Q: C++ macro integers and object definitions I lack of experience with C++ and I am trying to create a Settings file to put all my definitions and global variables there, so my project's classes can access those the values from there. The Settings.h file could look like this:
#ifndef SETTINGS_H_
#define SETTINGS_H_
#define COLOR_BLUE = Vec3b(255, 0, 0);
#define COLOR_GREEN = Vec3b(0, 255, 0);
#define NOT_SET = 0;
#define IN_PROCESS = 1;
#define SET = 2;
#define FGD_PX = 255;
#define BGD_PX = 127;
#include <cv.h>
using namespace cv;
class Settings {
};
#endif /* SETTINGS_H_ */
The idea is to access the variables without instantiate the class but just including it.
Is there any beautiful way to do this?
cheers,
A: #include <cv.h>
using namespace cv;
#ifndef SETTINGS_H_
#define SETTINGS_H_
#define COLOR_BLUE Vec3b(255, 0, 0)
#define COLOR_GREEN Vec3b(0, 255, 0)
#define NOT_SET 0
#define IN_PROCESS 1
#define SET 2
#define FGD_PX 255
#define BGD_PX 127
class Settings {
public:
static int var1;
static float var2;
static short var3;
};
// initialization
int Settings::var1 = SET;
float Settings::var2 = 3.14;
short Settings::var3 = BGD_PX;
#endif /* SETTINGS_H_ */
Usage:
int tmp = Settings::var1;
A: Constants either go in enums, or as static consts. Manifest constants generally are reserved for compiler type options:
#ifndef SETTINGS_H_
#define SETTINGS_H_
class Settings {
public:
static const vec3b color_blue;
static const vec3b color_green;
enum statics {
NOT_SET = 0,
IN_PROCESS = 1,
SET = 2,
FGD_PX = 255,
BGD_PX = 127
};
};
vec3b Settings::color_blue(255, 0, 0);
vec3b Settings::color_green(0, 255, 0);
#include <cv.h>
#endif
A: You shouldn't have equals signs or semicolons with #defines. What do you plan on putting in your Settings class?
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 416 |
Q: Show messages after killing the app I am currently working on android chatting app using io.socket. When the app opened all the thing is working fine but when I kill the app I am not able show messages.
*
*GCM is a bad idea for messaging because I heard it misses some messages.
*service means it will kill the battery.
I want to show messages even if I kill the app(just like watsapp).
Do you have any suggestions, as to, how I can achieve this.
A: When you kill whatsup app its use Gcm and when you Open your Application its uses XMPP protocol to send and recieve Message. Whatsup app using this both feature in it. so It getting message after killing app too. Whats app app also use MessageService after killing app so It get message regarding.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 447 |
{"url":"https:\/\/codedump.io\/share\/SjfYVzQiKo6Q\/1\/how-to-estimate-the-kalman-filter-with-39kfas39-r-package-with-an-ar1-transition-equation","text":"danielkv - 5 months ago 47\nR Question\n\n# How to estimate the Kalman Filter with 'KFAS' R package, with an AR(1) transition equation?\n\nI am using 'KFAS' package from R to estimate a state-space model with the Kalman filter. My measurement and transition equations are:\n\ny_t = Z_t * x_t + \\eps_t (measurement)\n\nx_t = T_t * x_{t-1} + R_t * \\eta_t (transition),\n\nwith \\eps_t ~ N(0,H_t) and \\eta_t ~ N(0,Q_t).\n\nSo, I want to estimate the variances H_t and Q_t, but also T_t, the AR(1) coefficient. My code is as follows:\n\nlibrary(KFAS)\n\nset.seed(100)\n\neps <- rt(200, 4, 1)\nmeas <- as.matrix((arima.sim(n=200, list(ar=0.6), innov = rnorm(200)*sqrt(0.5)) + eps),\nncol=1)\n\nZt <- 1\nHt <- matrix(NA)\nTt <- matrix(NA)\nRt <- 1\nQt <- matrix(NA)\n\nss_model <- SSModel(meas ~ -1 + SSMcustom(Z = Zt, T = Tt, R = Rt,\nQ = Qt), H = Ht)\nfit <- fitSSM(ss_model, inits = c(0,0.6,0), method = 'L-BFGS-B')\n\n\nBut it returns: \"Error in is.SSModel(do.call(updatefn, args = c(list(inits, model), update_args)),: System matrices (excluding Z) contain NA or infinite values, covariance matrices contain values larger than 1e+07\"\n\nThe NA definitions for the variances works well, as documented in the package's paper. However, it seems this cannot be done for the AR coefficients. Does anyone know how can I do this?\n\nNote that I am aware of the SSMarima function, which eases the definition of the transition equation as ARIMA models. Although I am able to estimate the AR(1) coef. and Q_t this way, I still cannot estimate the \\eps_t variance (H_t). Moreover, I am migrating my Kalman filter codes from EViews to R, so I need to learn SSMcustom for other models that are more complicated.\n\nThanks!\n\nIt seems that you are missing something in your example, as your error message comes from the function fitSSM. If you want to use fitSSM for estimating general state space models, you need to provide your own model updating function. The default behaviour can only handle NA's in covariance matrices H and Q. The main goal of fitSSM is just to get started with simple stuff. For complex models and\/or large data, I would recommend using your self-written objective function (with help of logLik method) and your favourite numerical optimization routines manually for maximum performance. Something like this:\n\nlibrary(KFAS)\n\nset.seed(100)\n\neps <- rt(200, 4, 1)\nmeas <- as.matrix((arima.sim(n=200, list(ar=0.6), innov = rnorm(200)*sqrt(0.5)) + eps),\nncol=1)\n\nZt <- 1\nHt <- matrix(NA)\nTt <- matrix(NA)\nRt <- 1\nQt <- matrix(NA)\n\nss_model <- SSModel(meas ~ -1 + SSMcustom(Z = Zt, T = Tt, R = Rt,\nQ = Qt), H = Ht)\n\nobjf <- function(pars, model, estimate = TRUE) {\nmodel$H[1] <- pars[1] model$T[1] <- pars[2]\nmodel$Q[1] <- pars[3] if (estimate) { -logLik(model) } else { model } } opt <- optim(c(1, 0.5, 1), objf, method = \"L-BFGS-B\", lower = c(0, -0.99, 0), upper = c(100, 0.99, 100), model = ss_model) ss_model_opt <- objf(opt$par, ss_model, estimate = FALSE)\n\n\nSame with fitSSM:\n\nupdatefn <- function(pars, model) {\nmodel$H[1] <- pars[1] model$T[1] <- pars[2]\nmodel$Q[1] <- pars[3] model } fit <- fitSSM(ss_model, c(1, 0.5, 1), updatefn, method = \"L-BFGS-B\", lower = c(0, -0.99, 0), upper = c(100, 0.99, 100)) identical(ss_model_opt, fit$model)","date":"2017-03-26 19:57:44","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5826563835144043, \"perplexity\": 6856.317063796618}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-13\/segments\/1490218189245.97\/warc\/CC-MAIN-20170322212949-00237-ip-10-233-31-227.ec2.internal.warc.gz\"}"} | null | null |
It's For You Own Good — второй EP австралийской рок-группы The Living End, вышедший в 1996 году.
Об альбоме
It's For You Own Good записан в Мельбурне в студии Birdland. В первом издании EP на обложке с ошибкой было написано заглавное It's'(I'ts), в последующих переизданиях ошибка была исправлена.
Песня с этого мини-альбома — «From Here on In» попала в ротацию австралийского радио. Это был первый радио-сингл группы.
В 2004 году вышел сборник синглов, который назывался как заглавная песня EP — «From Here on In».
EP был переиздан на двойном компакт-диске (вместе с EP Hellbound) в 2005 году и распространяется компанией EMI.
Список композиций
Музыка и тексты песен Криса Чини, кроме отмеченных.
Участники записи
Крис Чини — гитры и вокал
Скотт Оуэн — контрабас и бэк-вокал
Джоуи Пирипитци — барабаны
Альбомы The Living End | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 856 |
Pan Fusheng (; December 1908 – April 1980) was a Chinese Communist revolutionary and politician. He was the first party secretary of the short-lived Pingyuan Province of the People's Republic of China, and also served as the First Secretary (i.e. party chief) of Henan and Heilongjiang provinces.
During the Great Leap Forward, Pan sympathized with Marshal Peng Dehuai, a critic of Mao Zedong's collectivization policy. As a result, in 1958, he was dismissed as party chief of Henan and subjected to persecution, but was later rehabilitated.
When the Cultural Revolution began, Pan, then party chief of Heilongjiang province, embraced the rebel Red Guards movement and gained the support of Mao. However, he was soon involved in major factional violence, and was dismissed again in 1971 and put under investigation. In 1982, the Chinese Communist Party posthumously criticized him for committing "serious mistakes" during the Cultural Revolution.
Republic of China
Pan Fusheng was born in December 1908 to a poor peasant family in Wendeng, Shandong province. His original name was Liu Kaijun (), and courtesy name Juchuan (). Pan was an excellent student in Wendeng County Senior Elementary School, and was admitted to Wendeng County Normal School with the highest score in the entrance examination; however, he was later forced to drop out owing to poverty, and taught in a rural senior elementary school for five years.
In 1929, he was admitted to the Shandong Number One Rural Normal School, where he was influenced by communist classmates, and joined the Chinese Communist Party in 1931. After the Mukden Incident, which led to the Japanese occupation of Manchuria, Pan organized the Shandong students to join the anti-Japanese and anti-Kuomintang (KMT) protests in the capital Nanjing. In March 1932, he was arrested by the KMT government and held at the Jinan Number One Prison. He was sentenced to ten years for "endangering the Republic", but was released in late 1937, after the eruption of the Second Sino-Japanese War.
After his release from prison, Pan organized and led the Communist guerrillas in Shandong, and later became a leader of the Hebei–Shandong–Henan (Ji–Lu–Yu) Communist base. He participated in the Second Sino-Japanese War and the subsequent Chinese Civil War against the KMT government.
Early PRC
In August 1949, just before the official establishment of the People's Republic of China, Pan Fusheng was appointed Communist Party Secretary of the newly established Pingyuan Province. In March 1950, a number of peasants and cattle froze to death when transporting grain to government storage in Puyang prefecture. Pan took partial responsibility for the "Puyang Incident" and was demoted to deputy party chief.
In November 1952, Pingyuan Province was abolished and most of it was merged into Henan Province, and Pan Fusheng became the party chief (then called First Secretary) of Henan, succeeding Zhang Xi, and the Political Commissar of the Henan Military District. In September 1956, he was elected as an alternate member of the 8th Central Committee of the Chinese Communist Party.
Great Leap Forward
During the Anti-Rightist Movement and the Great Leap Forward, Pan Fusheng sympathized with Marshal Peng Dehuai, who criticized Mao's disastrous collectivization policies. As a result, he became a major target of persecution by the fervent Mao loyalist Wu Zhipu, the Second Secretary of Henan, with the approval of the CPC Central Committee.
During the 9th plenum of the Henan Provincial Party Congress in June 1958, Pan Fusheng and two lower-ranking officials, Yang Jue and Wang Tingdong, were denounced as the "Pan, Yang, Wang right-deviating anti-party clique", and subjected to brutal struggle sessions. In October 1958, the Communist Party calculated that some 1.6 billion big-character posters denouncing them had been written. People who did not put up posters risked being labelled "little Pan Fusheng". Pan was dismissed from office and sent to work as a laborer on a farm. Wu Zhipu replaced Pan as First Secretary of Henan and zealously implemented Mao's collectivization policy. Under Wu's leadership, Henan was one of the worst affected provinces during the Great Chinese Famine, with nearly 3 million people starving to death from 1959 to 1961.
After the end of the Great Leap Forward, Pan was rehabilitated in 1962 and became the minister-level director of the All China Federation of Supply and Marketing Cooperatives.
Cultural Revolution
In October 1965, Pan Fusheng was appointed First Secretary of Heilongjiang Province in Northeast China and Political Commissar of the Heilongjiang Military District. The Cultural Revolution erupted shortly afterwards, and on 31 January 1967, Heilongjiang became the first province to set up a revolutionary committee. By enthusiastically embracing the "power seizure" movement by the rebel Red Guards, Pan was able to gain Mao's support and became a well-known figure throughout China. During the Cultural Revolution, Pan became chairman of the Heilongjiang revolutionary committee in March 1967, one of the only three provincial party chiefs in the country to have transformed from an incumbent provincial First Secretary to head of the revolutionary committee. This was considered a rare feat, as the vast majority of provincial First Secretaries were overthrown through power seizures by radicals and Red Guards, or were removed from power through other means. In April 1968, Pan was elected a member of the Central Committee of the Chinese Communist Party, and in April 1969, he became a member of the Central Military Commission.
Soon after the establishment of the revolutionary committee, however, a faction opposing Pan emerged and sought to remove him from power. Pan became involved in factional violence. He jailed people from the rival faction, including party members and ordinary people. His supporters and opponents fought each other in major armed conflicts, and Pan had serious disagreements with the local military leadership. In June 1971, the central government dismissed him from his posts and put him under investigation.
Death
In April 1980, Pan died of a stroke in Harbin. In 1982, the CPC Central Committee issued a statement criticizing Pan for committing "serious mistakes" during the Cultural Revolution.
References
1908 births
1980 deaths
People's Republic of China politicians from Shandong
Chinese Communist Party politicians from Shandong
Political office-holders in Henan
Political office-holders in Heilongjiang
Politicians from Weihai
People from Wendeng
Members of the 1st Chinese People's Political Consultative Conference
Members of the 4th Chinese People's Political Consultative Conference
Alternate members of the 8th Central Committee of the Chinese Communist Party
CCP committee secretaries of Henan
CPPCC Chairmen of Henan
Governors of Heilongjiang | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,085 |
Air Force badge miniature mirror finish Missile Operator.
Established in 1988 for personnel involved solely in missile operations. Its awarded to officers after graduation from missile school at Vandenberg AFB and upon qualification as a Missile Combat crew member. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,700 |
CREATE OR REPLACE PACKAGE job_history_crud_view
AS
-- --Views for job_history entity
PROCEDURE create_job_history;
PROCEDURE read_job_history;
PROCEDURE update_job_history(
p_employee_id varchar2,
p_start_date varchar2
);
END;
/
--Define the controller package of the application
CREATE OR REPLACE PACKAGE BODY job_history_crud_view
AS
--Return job history
PROCEDURE read_job_history
AS
BEGIN
HTP.HTMLOPEN;
HTP.PRINT('<META HTTP-EQUIV="CACHE-CONTROL" CONTENT="NO-CACHE">');
HTP.HEADOPEN;
hr_crud_view.stylize;
HTP.HEADCLOSE;
HTP.BODYOPEN;
HTP.DIV(cattributes => 'class = "container" style = "width:1280px;"');
hr_crud_view.header;
HTP.HEADER(3, '<span>Job History</span>');
HTP.DIV(cattributes => 'class = "button-wrapper"');
HTP.ANCHOR('job_history_crud_view.create_job_history', 'Add', cattributes => 'class = "button light"');
HTP.PRINT('</DIV>
</BR>');
HTP.CENTEROPEN;
HTP.DIV(cattributes => 'class = "table-wrapper" style = "width:auto;display:inline-block"');
HTP.TABLEOPEN;
HTP.TABLEROWOPEN(cattributes => 'class = "table-row"');
HTP.TABLEHEADER('Id', cattributes => 'class = "table-header light"');
HTP.TABLEHEADER('Start Date', cattributes => 'class = "table-header light"');
HTP.TABLEHEADER('End Date', cattributes => 'class = "table-header light"');
HTP.TABLEHEADER('Job', cattributes => 'class = "table-header light"');
HTP.TABLEHEADER('Department', cattributes => 'class = "table-header light"');
HTP.TABLEHEADER('Remove', cattributes => 'class = "table-header light"');
HTP.TABLEROWCLOSE;
FOR REC IN (SELECT employee_id,
start_date,
end_date,
(SELECT job_title FROM jobs WHERE job_id = job_history.job_id) AS job_id,
(SELECT department_name FROM departments WHERE department_id = job_history.department_id) AS department_id
FROM job_history ORDER BY employee_id) LOOP
HTP.TABLEROWOPEN(cattributes => 'class = "table-row"');
HTP.TABLEDATA(HTF.ANCHOR('job_history_crud_view.update_job_history?p_employee_id=' || REC.EMPLOYEE_ID || chr(38) || 'p_start_date=' || TO_CHAR(REC.START_DATE, 'DD.MM.YYYY'), REC.EMPLOYEE_ID));
HTP.TABLEDATA(TO_CHAR(REC.START_DATE, 'DD.MM.YYYY'));
HTP.TABLEDATA(TO_CHAR(REC.END_DATE, 'DD.MM.YYYY'));
HTP.TABLEDATA(REC.JOB_ID);
HTP.TABLEDATA(REC.DEPARTMENT_ID);
HTP.TABLEDATA('
<DIV class = "button-wrapper" style = "width:70;height:auto;">
' ||
HTF.ANCHOR('job_history_crud_controller.delete_job_history?p_employee_id=' || REC.EMPLOYEE_ID || chr(38) || 'p_start_date=' || TO_CHAR(REC.START_DATE, 'DD.MM.YYYY'), 'Delete', cattributes => 'class = "button strawberry" style = "width:70;height:auto;"') ||
'
</DIV>
');
HTP.TABLEROWCLOSE;
END LOOP;
HTP.TABLECLOSE;
HTP.PRINT('</DIV>');
HTP.CENTERCLOSE;
hr_crud_view.footer;
HTP.PRINT('</DIV>');
HTP.BODYCLOSE;
HTP.HTMLCLOSE;
END read_job_history;
--Create new job_history
PROCEDURE create_job_history
AS
BEGIN
HTP.HTMLOPEN;
HTP.PRINT('<META HTTP-EQUIV="CACHE-CONTROL" CONTENT="NO-CACHE">');
HTP.HEADOPEN;
hr_crud_view.stylize;
HTP.HEADCLOSE;
HTP.BODYOPEN;
HTP.DIV(cattributes => 'class = "container" style = "width:1280px;"');
hr_crud_view.header;
HTP.HEADER(3, '<span>Create new job history</span>');
HTP.CENTEROPEN;
HTP.FORMOPEN('job_history_crud_controller.create_job_history', cattributes => 'id = "form"');
HTP.DIV(cattributes => 'class = "table-wrapper" style = "width:460px;"');
HTP.TABLEOPEN;
HTP.TABLEROWOPEN(cattributes => 'class = "table-row"');
HTP.TABLEDATA('For Employee', cattributes => 'style="vertical-align: middle"');
HTP.PRINT('<td>');
HTP.FORMSELECTOPEN('p_employee_id', cattributes => 'style = "width:255px;height:35px;"');
FOR REC IN (SELECT * FROM employees ORDER BY employee_id) LOOP
HTP.FORMSELECTOPTION(REC.FIRST_NAME || ' ' || REC.LAST_NAME, cattributes => 'class = "dropdown-option" value = "' || REC.EMPLOYEE_ID || '"');
END LOOP;
HTP.FORMSELECTCLOSE;
HTP.PRINT('</td>');
HTP.TABLEROWCLOSE;
HTP.TABLEROWOPEN(cattributes => 'class = "table-row"');
HTP.TABLEDATA('Start Date', cattributes => 'style="vertical-align: middle"');
HTP.TABLEDATA(HTF.FORMTEXT('p_start_date', cattributes => 'class = "search-field"'));
HTP.TABLEROWCLOSE;
HTP.TABLEROWOPEN(cattributes => 'class = "table-row"');
HTP.TABLEDATA('End Date', cattributes => 'style="vertical-align: middle"');
HTP.TABLEDATA(HTF.FORMTEXT('p_end_date', cattributes => 'class = "search-field"'));
HTP.TABLEROWCLOSE;
HTP.TABLEROWOPEN(cattributes => 'class = "table-row"');
HTP.TABLEDATA('Job', cattributes => 'style="vertical-align: middle"');
HTP.PRINT('<td>');
HTP.FORMSELECTOPEN('p_job_id', cattributes => 'style = "width:255px;height:35px;"');
FOR REC IN (SELECT * FROM jobs ORDER BY job_id) LOOP
HTP.FORMSELECTOPTION(REC.JOB_TITLE, cattributes => 'class = "dropdown-option" value = "' || REC.JOB_ID || '"');
END LOOP;
HTP.FORMSELECTCLOSE;
HTP.PRINT('</td>');
HTP.TABLEROWCLOSE;
HTP.TABLEROWOPEN(cattributes => 'class = "table-row"');
HTP.TABLEDATA('Department', cattributes => 'style="vertical-align: middle"');
HTP.PRINT('<td>');
HTP.FORMSELECTOPEN('p_department_id', cattributes => 'style = "width:255px;height:35px;"');
FOR REC IN (SELECT * FROM departments ORDER BY department_id) LOOP
HTP.FORMSELECTOPTION(REC.DEPARTMENT_NAME, cattributes => 'class = "dropdown-option" value = "' || REC.DEPARTMENT_ID || '"');
END LOOP;
HTP.FORMSELECTCLOSE;
HTP.PRINT('</td>');
HTP.TABLEROWCLOSE;
HTP.TABLECLOSE;
HTP.PRINT('</DIV>');
HTP.DIV(cattributes => 'class = "button-wrapper"');
HTP.ANCHOR('#', 'Save', cattributes => 'class = "button light" onclick="document.getElementById(''form'').submit();"');
HTP.PRINT('</DIV>
</BR>');
HTP.FORMCLOSE;
HTP.CENTERCLOSE;
hr_crud_view.footer;
HTP.PRINT('</DIV>');
HTP.BODYCLOSE;
HTP.HTMLCLOSE;
END create_job_history;
--Update details of employee
PROCEDURE update_job_history(
p_employee_id varchar2,
p_start_date varchar2
)
AS
rec job_history%ROWTYPE;
BEGIN
BEGIN
SELECT *
INTO rec
FROM job_history
WHERE employee_id = p_employee_id AND start_date = TO_DATE(p_start_date,'DD.MM.YYYY');
EXCEPTION
WHEN NO_DATA_FOUND THEN
RAISE_APPLICATION_ERROR(-20020, 'No job with such id.');
END;
HTP.HTMLOPEN;
HTP.PRINT('<META HTTP-EQUIV="CACHE-CONTROL" CONTENT="NO-CACHE">');
HTP.HEADOPEN;
hr_crud_view.stylize;
HTP.HEADCLOSE;
HTP.BODYOPEN;
HTP.DIV(cattributes => 'class = "container" style = "width:1280px;"');
hr_crud_view.header;
HTP.HEADER(3, '<span>Update job history details</span>');
HTP.CENTEROPEN;
HTP.FORMOPEN('job_history_crud_controller.update_job_history', cattributes => 'id = "form"');
HTP.FORMHIDDEN('p_employee_id', rec.employee_id);
HTP.DIV(cattributes => 'class = "table-wrapper" style = "width:460px;"');
HTP.TABLEOPEN;
HTP.TABLEROWOPEN(cattributes => 'class = "table-row"');
HTP.TABLEDATA('Start Date', cattributes => 'style="vertical-align: middle"');
HTP.TABLEDATA(HTF.FORMTEXT('p_start_date', cvalue => TO_CHAR(rec.start_date, 'DD.MM.YYYY'), cattributes => 'class = "search-field"'));
HTP.TABLEROWCLOSE;
HTP.TABLEROWOPEN(cattributes => 'class = "table-row"');
HTP.TABLEDATA('End Date', cattributes => 'style="vertical-align: middle"');
HTP.TABLEDATA(HTF.FORMTEXT('p_end_date', cvalue => TO_CHAR(rec.end_date, 'DD.MM.YYYY'), cattributes => 'class = "search-field"'));
HTP.TABLEROWCLOSE;
HTP.TABLEROWOPEN(cattributes => 'class = "table-row"');
HTP.TABLEDATA('Job', cattributes => 'style="vertical-align: middle"');
HTP.PRINT('<td>');
HTP.FORMSELECTOPEN('p_job_id', cattributes => 'style = "width:255px;height:35px;"');
FOR job IN (SELECT * FROM jobs ORDER BY job_id) LOOP
IF job.JOB_ID = rec.JOB_ID THEN
HTP.FORMSELECTOPTION(job.JOB_TITLE, cselected => 'selected', cattributes => 'class = "dropdown-option" value = "' || job.JOB_ID || '"');
ELSE
HTP.FORMSELECTOPTION(job.JOB_TITLE, cattributes => 'class = "dropdown-option" value = "' || job.JOB_ID || '"');
END IF;
END LOOP;
HTP.FORMSELECTCLOSE;
HTP.PRINT('</td>');
HTP.TABLEROWCLOSE;
HTP.TABLEROWOPEN(cattributes => 'class = "table-row"');
HTP.TABLEDATA('Department', cattributes => 'style="vertical-align: middle"');
HTP.PRINT('<td>');
HTP.FORMSELECTOPEN('p_department_id', cattributes => 'style = "width:255px;height:35px;"');
FOR department IN (SELECT * FROM departments ORDER BY department_id) LOOP
IF department.department_id = rec.department_id THEN
HTP.FORMSELECTOPTION(department.DEPARTMENT_NAME, cselected => 'selected', cattributes => 'class = "dropdown-option" value = "' || department.DEPARTMENT_ID || '"');
ELSE
HTP.FORMSELECTOPTION(department.DEPARTMENT_NAME, cattributes => 'class = "dropdown-option" value = "' || department.DEPARTMENT_ID || '"');
END IF;
END LOOP;
HTP.FORMSELECTCLOSE;
HTP.PRINT('</td>');
HTP.TABLEROWCLOSE;
HTP.TABLECLOSE;
HTP.PRINT('</DIV>');
HTP.DIV(cattributes => 'class = "button-wrapper"');
HTP.ANCHOR('#', 'Save', cattributes => 'class = "button light" onclick="document.getElementById(''form'').submit();"');
HTP.PRINT('</DIV>
</BR>');
HTP.FORMCLOSE;
HTP.CENTERCLOSE;
hr_crud_view.footer;
HTP.PRINT('</DIV>');
HTP.BODYCLOSE;
HTP.HTMLCLOSE;
EXCEPTION
WHEN OTHERS THEN
job_history_crud_view.read_job_history;
HTP.PRINT('
<script >
window.location.href="job_history_crud_view.read_job_history";
alert("An error was encountered - ' || REPLACE(SQLERRM, '"', '\"') || '.");
</script>');
END update_job_history;
END;
/
SHOW ERRORS
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,045 |
\section{Introduction}
Recently, many efforts have been made for finding Very Low-Mass (VLM) objects down to the planetary mass regime in young associations. Several reasons can account for this. First, the atmospheres of free-floating planetary mass objects can be easily studied as giant exoplanet analogs, because these objects would not be masked by a bright, primary star. The first such objects that have been found seem to show great diversity of spectral features, even at a fixed age and temperature, which strengthens the possibility those objects are analogs to giant, gaseous exoplanets, as well as demands for more such discoveries. Since young objects are warmer and brighter, they are also easier to study. They even provide good targets for the direct imaging of exoplanets, since such young exoplanets would also be brighter than their old counterparts, and the contrast ratio required to achieve such discoveries would be lower for a fainter primary star. Another great outcome of finding those objects would be the possibility to study the bottom of the Initial Mass Function (IMF) in a coeval environment, not to mention the possibility to improve atmospheric models of young brown dwarfs, which are still imprecise because of the lack of observations, as well as the difficulty of dealing with the large amount of dust contained in their photospheres. We have chosen to focus our search on young very low-mass stars and brown dwarfs in Nearby, Young Associations (NYAs) such as TW Hydrae (TWA; 8 - 12 Myr; \citealp{2004ARA&A..42..685Z}), $\beta$ Pictoris (BPMG; 12 - 22 Myr; \citealp{2001ApJ...562L..87Z}), Tucana-Horologium (THA; 10 - 40 Myr; \citealp{2000AJ....120.1410T}, \citealp{2001ASPC..244..122Z}), Carina (CAR; 10 - 40 Myr; \citealp{2008hsf2.book..757T}), Columba (COL; 10 - 40 Myr; \citealp{2011ApJ...732...61Z}), Argus (ARG; 10 - 40 Myr; \citealp{2011ApJ...732...61Z}) and AB Doradus (50 - 120 Myr ; \citealp{2004ApJ...613L..65Z}). Even though these NYAs are all closer than 100~pc, most of their low-mass members ($>$K5) remain to be identified because their currently known members were uncovered in the Hipparcos mission, which is limited to relatively bright objects. Since those moving groups are close and have ages between 8~\textendash~120 Myr, they present several advantages for the task of identifying young low-mass objects: 1) their members will be even brighter and thus easier to study, 2) they are old enough so that they are no longer embedded in dust, 3) they have not significantly dispersed yet, so their members share similar Galactic position ($XYZ$) and space velocities ($UVW$) and 4) they span a significant age range, which means each NYA will serve as a benchmark for their evolution in time. Furthermore, the 10~\textendash~30 Myr range is well covered, corresponding to the period where gaseous and terrestrial planets form \citep{2003ApJ...599..342S}. However, we must overcome a significant difficulty in order to identify such new members to those NYAs: since they are close-by and have begun dispersing, they cover great portions of the sky as viewed from earth. The fact that we do not have access to parallaxes and radial velocities for most low-mass objects renders this task even more difficult. Even worse, we expect that most of the potential members won't even have been spectroscopically confirmed as brown dwarfs.\\
\begin{figure}[]
\resizebox{\hsize}{!}{\includegraphics[clip=true]{CMD.eps}}
\caption{
\footnotesize
Color-magnitude sequence for field objects (brown line and shaded region), and our candidates with (red dots) and without (blue dots) spectroscopic confirmation of low-gravity. We can see that they are significantly redder than the old sequence, consistent with those being low-gravity objects with more dust in their photosphere.
}
\label{fig:CMD}
\end{figure}
\begin{figure*}[]
\begin{center}
\subfigure{
\includegraphics[width=0.485\textwidth]{dstat.eps}
}
\subfigure{
\includegraphics[width=0.485\textwidth]{vrad.eps}
}
\end{center}
\caption{
\footnotesize
Predicted distances and radial velocities for \emph{Bona Fide} members of NYAs (triangles), synthetic Montecarlo objects drawn from our NYAs models (small dots) and our candidates for which those measurements are available (thick open circles). The dark brown line represents a dependence. Our estimated errors agree well with the observed scatter. True distances (radial velocities) are retrieved within 8.0~\% (1.6~$\mathrm {km s}^{-1}$).
}
\label{fig:dstat}
\end{figure*}
\section{Method}
In order to overcome the precise difficulties previously described, \cite{2013ApJ...762...88M} proposes using bayesian inference in order to identify highly probable candidates to NYAs within possibly large samples of objects for those we have at least a position and proper motion. Here, we use a slightly modified version of this method and apply it to an all-sky sample of 650 000 red objects which we have built from a correlation of the 2MASS \citep{2003yCat.2246....0C} and WISE \citep{2012yCat.2311....0C} surveys. We have limited our search outside of the galactic plane ($\left | b \right | >$ 15\textdegree) in order to avoid crowded fields. Various filters have been applied in order to reject possible contaminants such as red galaxies and giant stars, and to keep only objects whose photometry is consistent with $>$M4 objects. Applying bayesian inference to this sample has yielded more than 300 highly probable candidates to NYAs, which are currently being followed so that we can confirm them as new \emph{Bona Fide} members. Included in the bayesian inference, we have used $J$ - $K_s$, as well as $H$ - $W2$ colors as a function of absolute $W1$ magnitude in order to give a bigger weight to young hypotheses for objects which have very red colors (see Figure~\ref{fig:CMD}). \\
We have estimated the mass distribution of our candidates by using the predicted distance (see Figure~\ref{fig:dstat} for an accuracy assessment of these predictions) to compute 2MASS and WISE absolute magnitudes, as well as the known NYA ages in combination with the AMES-Cond isochrones \cite{2003A&A...402..701B} and the BT-SETTL atmosphere models (\citealp{2013arXiv1302.6559A}, \citealp{Rajpurohit:ta}). The resulting distribution is displayed in Figure~\ref{fig:IMF} and compared with the expected remaining members to be discovered when supposing 1) that the NYAs are approximately complete at masses around 1~M$_\odot$ and 2) that the IMF of NYAs can be approximately described as a fiducial log-normal function with $m_c$ = 0.25~M$_\odot$ and $\sigma$~=~0.5~dex (\citealp{2012EAS....57...45J}, \citealp{2005ASSL..327...41C}). We seem to be finding more planetary mass candidates than predicted with this IMF, which could be a sign that we are finding planetary-mass objects that were not formed as brown dwarfs, but rather as ejected planets. However, we must confirm these candidate members before we can draw any conclusions of this kind. We have estimated our contamination rate to be between 20~\% and 30~\% for M5~\textendash~L0
\begin{figure}[t!]
\resizebox{\hsize}{!}{\includegraphics[clip=true]{hist04_num.eps}}
\caption{\footnotesize
Mass population for (1) \emph{Bona Fide} members of NYAs, (2) a fiducial log-normal IMF (described in text), candidates from this project (3) with and (4) without spectroscopic confirmation of low-gravity. We seem to find too many planetary mass candidates compared to the predictions of the IMF
}
\label{fig:IMF}
\end{figure}
\section{Candidates follow-up}
\begin{figure*}[]
\begin{center}
\subfigure{
\includegraphics[width=0.45\textwidth]{KiJ.eps}
}
\subfigure{
\includegraphics[width=0.45\textwidth]{Hcont.eps}
}
\subfigure{
\includegraphics[width=0.45\textwidth]{FeH.eps}
}
\subfigure{
\includegraphics[width=0.45\textwidth]{NaI2.eps}
}
\end{center}
\caption{
\footnotesize
K~I$_J$, FeH and $H$-cont indices defined by \cite{Allers:2013tq} and Na~I equivalent widths for our intermediate (circles) and very low gravity (triangles) candidates. The black line and beige region delimit the high gravity dwarfs sequence along with its scatter. The dotted grey line delimits the very low gravity region from the one where intermediate gravity objects typically fall. Filled symbols indicate objects with low resolution spectra (R~$\approx$~300) and open symbols indicate those with moderate resolution spectra (R~$\approx$~1200).
}
\label{fig:Allers}
\end{figure*}
We have used SpeX on the 3.0-m IRTF and OSIRIS on the SOAR 4.1-m telescope in order to obtain medium-resolution (R~$\approx$~1200) Near-InfraRed (NIR) spectroscopy of our brightest candidates, in order to confirm their low surface gravity (and thus youth) by using several gravity sensitive indices described in \cite{Allers:2013tq}. As a result, we have yet found 16 $>$M5 objects classified as intermediate (5) or very low (11) gravity displayed in Figure~\ref{fig:Allers}. For two of our faint candidates, we have used FIRES on the MAGELLAN 6.5-m telescope to obtain low-resolution (R~$\approx$~300) NIR spectroscopy, from which we can assess low-gravity in the same way albeit with fewer indices. Both objects turned out as very low-gravity, highly probable candidates to the TWA, which makes them potential 11~\textendash~13 M$_{Jup}$ and 13~\textendash~15 M$_{Jup}$ objects at the boundary of planetary masses. Earlier targets were followed with GMOS-S and GMOS-N on both Gemini telescopes in order to measure the equivalent width of the Na I doublet at $8174~\AA$, which is also gravity sensitive (\citealp{2004MNRAS.355..363L}, \citealp{2011AJ....142..104R}). This has yet yielded 20 M4~\textendash~M8 very strong candidates that show weaker Na I equivalent width, consistent with them being low-gravity and thus young dwarfs (see Figure~\ref{fig:NAI}). Several of our strongest bright candidates are currently being observed with CRIRES at the VLT, in order to obtain high-resolution $K$-band spectroscopy which will allow us to measure their radial velocities at a precision of 1~$\mathrm {km s}^{-1}$, which is sufficient distinguish between field objects and NYAs \emph{Bona Fide} members. More observations for NIR spectroscopy are also already planned for completing the spectroscopic characterization of our whole sample, and more candidates in the $>$L0 regime are also being currently uncovered.
\begin{figure}[]
\resizebox{\hsize}{!}{\includegraphics[clip=true]{NaI.eps}}
\caption{
\footnotesize
NaI index sequence for old dwarfs (thick line and gray region), various NYAs and low-gravity candidates from our survey (open circles), as defined by \cite{2004MNRAS.355..363L}. We can see that their relative gravity corresponds to the 6~\textendash~100 Myr age range, in agreement with the ages of NYAs considered here. Sequences are drawn from \cite{2011AJ....142..104R}.
}
\label{fig:NAI}
\end{figure}
\section{Conclusions}
We have summarized our survey for $>$M4 members to NYAs. We show that bayesian analysis can be used to significantly cut down the initial number of candidates in an all-sky search, and then show first results from this study. It is expected that this project will unveil a significant portion of young, planetary-mass objects in NYAs, which will be easy to study in details due to their short distance and larger intrinsic brightness than old objects.
\begin{acknowledgements}
We would like to address special thanks to Adric Riedel for generously sharing valuable parallax data with our team. This work was supported in part through grants from the Natural Sciences and Engineering Research Council of Canada. This research has made use of the SIMBAD database and VizieR catalogue, operated at Centre de Donn\'ees astronomiques de Strasbourg (CDS), Strasbourg, France.
\end{acknowledgements}
\bibliographystyle{aa}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,246 |
Консольный стілець (, ) — стілець без задніх ніжок, опору якого забезпечують дві передні ніжки, які вигнуті біля підлоги під прямим кутом та часто поєднані на підлозі у вигляді прямокутника. Оскільки навантаження на ніжки такого стільця (дві замість чотирьох, як у звичайного стільця) значно вище, то їх виробляють з більш міцного матеріалу, зазвичай — зі сталі. Крім стільців також є і консольні табурети, лави, крісла та дивани.
Найпершу модель консольного стільця без задніх ніжок «Kragstuhl» спроектував та виготовив нідерландський архітектор та дизайнер Март Стам зі звичайних сталевих дюймових газових труб з фітингами. Вперше ця модель була представлена у вигляді ескізу 22 листопада 1926 року під час наради з підготовки виставки у Вайсенхофі. Пізніше — в готовому вигляді — представлена у 1927 році на цій же виставці. На нараді був присутній Людвіг Міс ван дер Рое, якого дуже надихнула така ідея, тому він розробив свої оригінальні варіанти, які базувались на гнучкості та пружності конструкції (у Стама стілець був жорстким і практично не гнувся).
Незалежно до цієї ідеї прийшов і Марсель Бреєр, який ще раніше експериментував з меблями зі сталевих труб. Його суперечка з Стамом щодо авторського права дійшла до судового розгляду, яке визнало пріоритет Марта Стама.
З дерева консольний стілець вперше спроектував та виготовив Алвар Аалто.
Крім того, дослідження згодом показали, що подібну конструкцію сидінь з 1926 року використовувала в «народному автомобілі» чехословацька компанія Tatra (модель Tatra 12). Тому ідея консольного стільця десь витала у повітрі. Цікаво, що коли Міс ван дер Рое вирішив запатентувати свої стільці та крісла в США, то виявилось, що вже раніше, у 1922 році Гаррі Нолан подав заявку на патент на консольне пружне садове крісло. І отримав у 1924 році патент на цю розробку.
Посилання
Консольный стул
Cars, Furniture, Architecture - How Tatra Car Seating inspired an Iconic Modernist Chair
Стільці
Промисловий дизайн | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 510 |
node::~node()
{
for (int i = 0; i < children.size(); i++)
{
delete children[i];
}
}
node* node::insert(node *child)
{
children.push_back(child);
child->parent = this;
return this;
}
bool node::equal(node *other)
{
if (kind != other->kind ||
text != other->text ||
lo != other->lo ||
hi != other->hi ||
not_ != other->not_)
{
return false;
}
if (children.size() != other->children.size())
{
return false;
}
for (int i = 0; i < children.size(); i++)
{
if (!children[i]->equal(other->children[i]))
{
return false;
}
}
return true;
}
struct state;
typedef void (*parser)(state *s, lexer *lexer);
struct state {
node *tree;
parser parser;
std::string error;
};
static void parser_main(state *s, lexer *lexer);
static void parser_range(state *s, lexer *lexer);
std::string glob_parse(lexer *lexer, node **output)
{
node *root = new node(kind_pattern);
for (struct state s = {root, parser_main}; s.parser; )
{
s.parser(&s, lexer);
if (s.error != "")
{
delete root;
return s.error;
}
}
*output = root;
return "";
}
static void parser_main(state *s, lexer *lexer) {
token token(glob_lexer_token_eof, "", 0);
lexer->next(&token);
switch (token.kind)
{
case glob_lexer_token_eof:
s->parser = NULL;
break;
case glob_lexer_token_error:
s->parser = NULL;
s->error = token.s;
break;
case glob_lexer_token_text:
s->tree->insert(new node(kind_text, token.s));
s->parser = parser_main;
break;
case glob_lexer_token_any:
s->tree->insert(new node(kind_any));
s->parser = parser_main;
break;
case glob_lexer_token_super:
s->tree->insert(new node(kind_super));
s->parser = parser_main;
break;
case glob_lexer_token_single:
s->tree->insert(new node(kind_single));
s->parser = parser_main;
break;
case glob_lexer_token_range_open:
s->parser = parser_range;
break;
case glob_lexer_token_terms_open: {
node *a = new node(kind_any_of);
s->tree->insert(a);
node *p = new node(kind_pattern);
a->insert(p);
s->tree = p;
s->parser = parser_main;
break;
}
case glob_lexer_token_separator: {
node *p = new node(kind_pattern);
s->tree->parent->insert(p);
s->tree = p;
s->parser = parser_main;
break;
}
case glob_lexer_token_terms_close:
s->tree = s->tree->parent->parent;
s->parser = parser_main;
break;
default:
s->parser = NULL;
s->error = "unexpected token";
break;
}
}
static void parser_range(state *s, lexer *lexer)
{
bool not_ = false;
std::string lo, hi;
int lo_cp = 0, hi_cp = 0;
std::string chars;
while (true)
{
token token(glob_lexer_token_eof, "", 0);
lexer->next(&token);
switch (token.kind)
{
case glob_lexer_token_eof:
s->error = "unexpected end";
s->parser = NULL;
return;
case glob_lexer_token_error:
s->parser = NULL;
s->error = token.s;
return;
case glob_lexer_token_not:
not_ = true;
break;
case glob_lexer_token_range_lo: {
int len;
lo_cp = opa_unicode_decode_utf8(token.s.c_str(), 0, token.s.length(), &len);
if (lo_cp < 0 || len != token.s.length())
{
s->parser = NULL;
s->error = "unexpected length of lo character";
return;
}
lo = token.s;
break;
}
case glob_lexer_token_range_between:
break;
case glob_lexer_token_range_hi: {
int len;
hi_cp = opa_unicode_decode_utf8(token.s.c_str(), 0, token.s.length(), &len);
if (hi_cp < 0 || len != token.s.length())
{
s->parser = NULL;
s->error = "unexpected length of hi character";
return;
}
hi = token.s;
if (hi < lo)
{
s->parser = NULL;
s->error = "hi character should be greater than lo character";
return;
}
break;
}
case glob_lexer_token_text:
chars = token.s;
break;
case glob_lexer_token_range_close: {
const bool is_range = lo_cp != 0 && hi_cp != 0;
const bool is_chars = chars != "";
if (is_chars == is_range)
{
s->parser = NULL;
s->error = "could not parse range";
return;
}
if (is_range)
{
s->tree->insert(new node(kind_range, lo, hi, not_));
} else {
s->tree->insert(new node(kind_list, chars, not_));
}
s->parser = parser_main;
return;
}
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,336 |
\section{Introduction}
Construction of high energy electron-electron accelarator is technically
viable [1]. The $e^-e^-$ process is very interesting because it is particularly
suitable for testing a possible lepton violation mechanism. Observation of the
processes such as $e^-e^- \rightarrow \mu^-\mu^-,\;\tau^-\tau^-\;{\rm
or\;}W^-W^-$
will indicate the family lepton number or the total lepton number violation.
Especially interesting is the last process, $e^-e^- \rightarrow W^-W^-$, where
the
total lepton number is violated by two units, $\Delta L=2$. Firstly, it
has been shown
that the standard model's background for such process can be substantially
reduced
by appropriate kinematical
cuts (e.g. below 0.1 fb for a 1 TeV $e^-e^-$
collider [2,3]). Secondly, the occurence of this process will indicate that
there
exist
massive neutrinos (with masses $M_N>M_Z$) of Majorana type. Several papers
have been devoted to such breaking process with $\Delta$L=2 in the last few
years but
with different conclusions. Some are very pesimistic and
indicate that the total cross section for the $e^-e^- \rightarrow W^-W^-$
process
is much below the SM background [4], others [5], are very optimistic and
predict
that $\sigma_{tot}(e^-e^- \rightarrow W^-W^-)$ can be as large as 4 fb
(for $\sqrt{s}=0.5$ TeV) or 64 fb (for $\sqrt{s}=1$ TeV).
We would like to elucidate this point. Is there realy a chance to observe
such process in the future NLC ($\sqrt{s}=0.5$ TeV) or TLC ($\sqrt{s}=1$ TeV)
colliders? The answer depends on the model in which we calculate
the cross section. We take the simplest one -
the Standard Model (SM) with additional heavy right-handed neutrino singlets
(RHS). In that case
the
process takes place by exchange of neutrinos in the t-and u-channels. But even
then the size of the total cross section depends on the way in which light
and heavy neutrino masses are generated. The number of heavy neutrinos,
the magnitudes of their masses and the high energy behaviour of the total
cross section (unitarity) also have strong consequences.
In Section 2 we give the necessary information about the SM with right-handed
neutrinos, the helicity amplitudes for the $e^-e^- \rightarrow W^-W^-$ process
and various limits of the cross section. In the main section (Section 3)
we discuss the numerical results of our calculations and finally we summarize
and conclude in Section 4.
\section{The RHS model and the cross section}
In the RHS model which we consider there are $n_L$ (=3) left-handed and $n_R$
(=1,2,...) right-handed weak
neutrino states transforming under $SU_L(2)$ gauge group
as doublets and singlets,
respectively. The neutrino mass matrix has $n_L+n_R$
dimensions
\begin{equation}
M_{\nu}= { \overbrace{0}^{n_L} \ \overbrace{M_D}^{n_R} \choose M_D^T \ M_R }
{\begin{array}{c}
\} n_L \\ \} n_R.
\end{array}}
\end{equation}
Without Higgs triplet fields the $n_L \times n_L$ dimension part $M_L$ of
$M_{\nu}$ equals zero
\begin{equation}
M_L=0.
\end{equation}
Using ($n_L+n_R$) dimensional unitary matrix $U=\left( \matrix{ K^T \cr
U_R } \right) $
acting on the weak neutrino states we can diagonalize matrix $M_{\nu}$
($U^TM_{\nu}U=M_{diag}$) and get the
physical states. We know from experiments that three of
them are very light ($m_{\nu_e}<5.1\;{\rm eV},\;m_{\nu_{\mu}}<270\;{\rm keV},
\;m_{\nu_{\tau}}<24\;{\rm MeV}$)
and others, if exist, have masses above $M_Z/2$ [6] or even $M_Z$ with
appropriate assumptions about their couplings [7].
Without loosing the
generality we can assume that the charged lepton mass matrix is diagonal, so
then the physical neutrino $N= \left( N_1,...,N_{L+R} \right)^T$
couplings to gauge bosons are defined by
($\hat{l}=(e,\mu,\;\tau)^T,\;P_L=\frac{1}{2}(1-\gamma_5)$)
\begin{eqnarray}
L_{CC}&=&\frac{g}{\sqrt{2}}\bar{N}\gamma^{\mu}KP_L\hat{l}W_{\mu}^+ + h.c., \\
L_{NC}&=&\frac{g}{2\cos{\theta_W}} \left[
\bar{N}\gamma^{\mu}P_L(KK^{\dagger})N \right].
\end{eqnarray}
For $n_R=3$ the matrix K
has the following form
$$
K=
\begin{array}{c}
e\;\; \mu \;\; \tau \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \\
\left. { \begin{array}{ccc}
\cdot & \cdot & \cdot \\
\cdot & \cdot & \cdot \\
\cdot & \cdot & \cdot
\end{array} }
\right\} {\rm light\;\; neutrinos} \\
\left. { \begin{array}{ccc}
\Box & \cdot & \cdot \\
\Box & \cdot & \cdot \\
\Box & \cdot & \cdot
\end{array} }
\right\} {\rm heavy\;\; neutrinos}
\end{array}.
$$
We are specially interested in the relevant for our process `box' couplings
of electrons with heavy neutrinos.
{}From various experimental data we can find the bounds on the mixing matrix
elements $K_{Nl}$ and $(KK^{\dagger})_{NN'}$ [8,9].
Production of two gauge bosons
in two-charged-electrons scattering process in the SM with only
the RHS neutrinos is described by the helicity amplitudes with the same
negative
polarizations of the incoming electrons
$\sigma_1=\sigma_2=-1/2$. The other helicity
polarizations of electrons are connected with right-handed currents which
are absent - Eqs.(3,4)
(the full helicity amplitudes suitable for the L-R models with additional
s-channels and right handed currents
are given e.g. in [4]).
The RHS model's differential cross section is given by
\begin{equation}
\frac{d\sigma(\lambda_1,\lambda_2)}{d\cos{\Theta}}=
\frac{G_F^2\sqrt{1-\gamma^2} }{ 16 \pi }{\mid M(\lambda_1,\lambda_2) \mid }^2
\end{equation}
where $\lambda_1$ and $\lambda_2$ are helicities of
the produced gauge bosons and
$\gamma=\frac{2M_W}{\sqrt{s}}$.
The helicity amplitudes can be written in the form ($\Theta,\;\phi$ are polar
angles of one of the gauge bosons in the CM frame)
\begin{equation}
M( {\lambda}_1,{\lambda}_2 ) =
\left\{ M_t(\lambda_1,\lambda_2)R_t
+M_u(\lambda_1,\lambda_2)R_u \right\}
D^{\mid \lambda_1-\lambda_2 \mid }_{0, \; \lambda_1-\lambda_2} \left(0,
\Theta,\phi \right)
\end{equation}
where $R_t$ and $R_u$ are as follows ($m_a$ - masses of
neutrinos,$\beta=\sqrt{1-
\gamma^2}$)
\begin{eqnarray}
R_{t(u)} &=& - \sum_{a}
K_{ae}^2
\frac{m_a}
{\frac{1+\beta^2}{2} \mp \beta\cos{\Theta}+\frac{m_a^2}{s}}
\end{eqnarray}
and the
reduced helicity amplitudes $M_{t(u)}\left( \lambda_1,
\lambda_2\right) $ for the t- and u-channels are gathered in Table 1.
The sum $\sum_{a}$ is over all light and heavy neutrinos.
Now we can easily find the approximated cross section formulae in some limited
cases.
(i) For very high energy, if ($\sqrt{s} >> M_W,m_a$) only one helicity
amplitude M(0,0) gives non-vanishing contribution and
\begin{equation}
\sigma(s \rightarrow \infty ) = \frac{G_F^2}{4\pi} \mid
\sum_{a} K^2_{ae} m_a \mid ^2,
\end{equation}
In the RHS model, however,
\begin{equation}
\sum_{a} K^2_{ae}m_a=(M_L^{\ast})_{\nu_e \nu_e}=0,
\end{equation}
so unitarity is restored.
(ii) If additional right-handed neutrinos are very heavy and \newline
$m_{heavy(a)} >> \sqrt{s} >> M_W$ we get
\begin{eqnarray*}
\sigma(m_{heavy(a)}>>\sqrt{s}>>M_W)&=&\frac{G_F^2s^2}{4\pi} \mid
\sum_{light(a)}(K_{ae})^2\frac{m_a}{s}+ \sum_{heavy(a)}
(K_{ae})^2\frac{1}{m_a} \mid ^2 \nonumber \\
&=&\frac{G_F^2}{4\pi} \mid
\sum_{heavy(a)} (K_{ae})^2m_a\left[ 1-\frac{s}{m_a^2} \right] \mid ^2,
\end{eqnarray*}
\begin{equation}
\end{equation}
where we used Eq.(9).
Let us note that $s/m_a^2 <<1$ for $m_{\rm heavy(a)}>>\sqrt{s}$ and
the contribution from the light neutrinos dominates.
(iii) If masses of the heavy neutrinos $m_{heavy} \rightarrow \infty,$
then
\begin{eqnarray}
\sigma(m_{heavy} \rightarrow \infty )
&=&\frac{G_F^2}{4 \pi}
\mid \sum_{light(a)} (K_{ae})^2m_a \mid ^2 <
\frac{G_F^2}{4 \pi}
(m_{\nu_e}+m_{\nu_{\mu}}+m_{\nu_{\tau}})^2 \nonumber \\
& \sim & \frac{G_F^2}{4 \pi} (25 MeV)^2
\; or \; \frac{G_F^2}{4 \pi} (30\;eV)^2 \nonumber \\
& \sim & 10^{-2} fb \; or \; 10^{-13} fb.
\end{eqnarray}
The first (second) estimation is connected with limits on the light neutrino
masses
coming from terrestrial experiments (astrophysical and cosmological
observations respectively).
We can see from Eq.(11) that the contribution to $\sigma (e^-e^- \rightarrow
W^-W^-)$
from light neutrino exchange is very small ($<10^{-2}$ fb). It follows from
Eq.(10) that the light neutrino dominates if $m_{\rm heavy(a)} >> \sqrt{s}$,
and from the unitarity $\sigma \rightarrow 0$ if $\sqrt{s} >> m_{\rm heavy(a)}$
(Eq.(9)). So the only region where $\sigma (e^-e^- \rightarrow W^-W^-)$
could be large is for $\sqrt{s} \sim m_{\rm heavy(a)}$. We will see in the
next section that it realy takes place.
\section{The $e^-e^- \rightarrow W^-W^-$ process: numerical results}
All specific features of our lepton-violating process are included in
quantities $R_t$ and $R_u$ given by Eq.(7). Each of them is expressed by a
sum over physical neutrinos with masses $m_a$, where the three are light ones
and the others
are heavy. The magnitudes of the $R_t$ and $R_u$ decide about the size of
the total cross section. First of all, if all $m_a \rightarrow 0,\;R_{t(u)}
\rightarrow 0$ as it should be, because for the Weyl neutrinos there
is no lepton symmetry breaking.
Two factors influence the magnitude
of the $R_{t(u)}$ - the square of the mixing matrix elements $K_{ae}^2$ and
neutrino
masses $m_a$ which are restricted by two constraints:
\begin{itemize}
\item unitarity of the K matrix
\begin{equation}
\sum_{i=light} \mid K_{ie} \mid^2=1-\sum_{j=heavy} \mid K_{je} \mid^2
\end{equation}
and
\item the lack of Higgs triplets from which $M_L=0$ and Eq.(9) follows
\begin{equation}
\Delta_{light} \equiv \sum_{i=light}(K_{ie})^2m_i=-\sum_{j=heavy}(K_{je})^2m_j
\equiv -\Delta_{heavy}.
\end{equation}
\end{itemize}
Just above the threshold for the $W^-W^-$ production $\sqrt{s} \sim 2M_W$
and the cross section is small ($\gamma \rightarrow 1$). As we try to find the
region where the total cross section is largest we take $\sqrt{s} >>M_W$,
so $\gamma \rightarrow 0$ and only one helicity amplitude M(0,0) is important,
the other ones tend to zero like $\gamma^2$ or $\gamma$ (see Table 1). We
consider this limit in our
discussion but numerical results are given without any
approximation.
We can ask now what the contribution to $\sigma$ from light neutrinos is.
This contribution is given by Eq.(11), for $m_{heavy} \rightarrow \infty .$
As the masses are small and $\mid K_{ae} \mid \leq 1,$ $\Delta_{light}$ in
Eq.(13) is very
small too and the cross section is $<10^{-2}$ fb or $10^{-13}$ fb for
laboratory
experiments or
astrophysical observations, respectively. The only possibility to obtain larger
cross section is through heavy neutrino(s) exchange in the t- and u-channels.
But even
if the masses of heavy neutrinos are very large the combination
$\Delta_{heavy}=
-\sum\limits_{j=heavy}K_{je}^2m_j$ must be still very small as $\mid
\Delta_{heavy} \mid=
\mid \Delta_{light} \mid .$
The combination $\Delta_{heavy}$ given by the heavy neutrino
exchange (Eq.(13)) can be small because of two reasons. Firstly, the mixing
matrix element $\mid K_{ie} \mid $ can be small so even for large $m_i$
the combination $\Delta_{\rm heavy}$ is small. As we will see this happens
in the case of the `see-saw' type [10] of the neutrino mass matrix $M_{\nu}$.
There is no chance then to get a reasonably large cross section.
Secondly, $\mid K_{ie} \mid $ are not small but there is destructive
interference
between large
contributions from the different heavy neutrinos. It can happen if the
neutrinos
have opposite CP parities.
The models where this scenario is realized are also considered [11].
There is a possibility to get
`experimentally interesting' value of the cross section in the frame of these
models.
The neutrino propagator
gives the factors $+\frac{m_a^2}{s}$ in the denominators of $R_{t(u)}$
(Eq.(7)).
These factors can disturb the destructive interference in $\Delta_{heavy}$
giving larger
values for $R_{t(u)}$.
If the masses of heavy neutrinos are equal then
we can extract the denominator in $R_{t(u)}$ and the cross section is still
proportional to $\mid \Delta_{heavy} \mid^2$ so it is small.
The same is true if there is only one
heavy neutrino. Then the destructive interference in $\Delta_{\rm heavy}$
is impossible and the cross
section $\sigma(e^-e^- \rightarrow W^-W^-)$ is of the same order of magnitude
as for the light neutrinos. The last observation agrees with the fact that in
our model with one heavy neutrino only the `see-saw' scenario is applicable
[12].
{}From the present limits on the masses of
light neutrinos we can see that at least two conditions must be satisfied to
get experimentally
interesting value of the cross section; (i) there must be two or more heavy
neutrinos and (ii) their masses must be different.
We see also that some heavy neutrino mixing angles $K_{ie}$ must be complex,
which can happen if CP is violated or if CP is conserved and the CP parities
of some heavy neutrinos are opposite.
The spectrum of the neutrino masses and elements of the mixing matrix K are
the
result of the diagonalization of neutrino mass matrix $M_{\nu}$. The elements
of the $M_{\nu}$ are not known and usually some models which guarantee
a reasonable spectrum of neutrino masses are assumed. The popular model
to obtain the light ($\sim$eV) - heavy ($\sim$TeV) spectrum of neutrino masses
is the
`see-saw' model [10]. This means that the $M_R$ and $M_D$ matrices in Eq.(1)
are proportional to different scales of symmetry breaking and
$\mid (M_R)_{ii}\mid >> \mid (M_D)_{lk} \mid $. Then, without any additional
symmetry, the important K matrix elements $K_{ae}$ are proportional to
$<M_D>/m_a$ and are very small for large $m_a$
\begin{equation}
K_{ae} \sim \frac{<M_D>}{m_a}.
\end{equation}
In this case not only $\Delta_{\rm heavy}=-\sum\limits_{a=1}^{n_R}(K_{ae})^2
m_a \simeq \sum\limits_{a=1}^{n_R} \frac{<M_D>}{m_a} $ but also the
quantities $R_{t(u)}$
\begin{equation}
R_{t(u)} \simeq \sum \frac{<M_D>}{m_a \left( \frac{1+\beta^2}{2}\mp
\beta \cos{\Theta}+\frac{m_a^2}{s} \right) }
\end{equation}
are small. The same phenomena of decoupling of the heavy neutrinos in the
`see-saw' type of models at the one-loop level have been also observed (see
e.g. [13]). To find what the size of total cross section is let us take
the neutrino mass matrix in the following `see-saw' forms
\begin{equation}
M_D= \left( \matrix{ 1.0 & 1.0 & 0.9 \cr
1.0 & 1.0 & 0.9 \cr
0.9 & 0.9 & 0.95 } \right)
\;\;{\rm and}\;\;M_R= \left( \matrix{ M & 0.0 & 0.0 \cr
0.0 & AM & 0.0 \cr
0.0 & 0.0 & BM } \right)
\end{equation}
which give a reasonable spectrum of the neutrino masses for M$>$100 GeV,
A,B$>$10
($m_{\rm light} =0 {\rm \;eV},\sim {\rm keV}, \sim {\rm MeV},m_{\rm heavy}
\sim {\rm M,AM,BM}$).
The calculated cross section
$\sigma (e^-e^- \rightarrow W^-W^-)$ for $\sqrt{s}=0.5(1)$ TeV and several
values A and B
as function of mass M and shown in Fig.1.
We can see from Eq.(15) that the cross section is larger for smaller
$m_{\rm heavy}$. The smallest value of $m_{\rm heavy}$ allowed in practice
is $\sim 100$ GeV the `see-saw' type models give the cross sections
$\sigma(e^-e^- \rightarrow W^-W^-)$ which are not experimentally interesting.
However the `see-saw' mechanism is not the only scenario which explains the
small
masses of the known neutrinos. There are models [11,12] where the relations
(14) do
not work and the mixing matrix elements can be large even for large masses of
the heavy neutrinos. In this class of models the smallness of masses of
the known neutrinos is
guaranteed by some special symmetry argument. There are then no simple
relations connecting $m_a$ with $K_{ae}$ and the mixing matrix elements can
be treated as independent parameters, bounded only by experimental data.
{}From existing experimental data only the sum
\begin{equation}
\sum\limits_{a=1}^{n_R} \mid K_{ae} \mid^2 \leq \kappa^2
\end{equation}
can be bounded:
$$\kappa^2=0.015 \;\;\;\; \eqno{ \rm (see\; Ref.[8]) }$$
or with the new LEP results ($m_t=170$ GeV and $m_H=200$ GeV)
$$\kappa^2=0.0054 \;\;\;\; \eqno{ \rm (see\; Ref.[9]).}$$
Let us calculate the total cross section
$\sigma(e^-e^- \rightarrow W^-W^-)$ for different number of right-handed
neutrinos
$n_R$. As was said the case $n_R=1$ is not interesting - the cross section
is very small. \\
$\bullet$ The $n_R=2$ case.
Let us denote the mass of the lightest heavy neutrino by $m_1=M$ and the mass
ratio
$m_2/m_1$ by A. Then, if we assume that $\eta_{CP}(N_1)=+i,\;
\eta_{CP}(N_2)=-i$ and denote $\delta=\Delta_{\rm light}/M <<1$, from Eqs.(13)
and
(17) (assuming the upper bound) we have
\begin{eqnarray}
K_{N_1e}^2&=&\frac{A\kappa^2-\delta}{1+A}, \nonumber \\
K_{N_2e}^2&=&-\frac{\kappa^2-\delta}{1+A}.
\end{eqnarray}
The total cross section for mixing matrix elements (18) as function
of M for different ratios A is given in Fig.2. The largest value of the total
cross section is obtained for $\sqrt{s} \geq M$ (as discussed before)
and for $A \rightarrow \infty$. For very heavy second neutrino
$K_{N_2e} \rightarrow 0$ and the destructive interference in $R_{t(u)}$
functions
(Eq.(7)) between two neutrinos vanishes ($K_{N_1e} \rightarrow \kappa,\;
K_{N_2e} \rightarrow 0$). For $\kappa^2=0.0054$ the maximum of $\sigma_{\rm
tot}$
is obtained for $\sqrt{s}=0.5(1)$ TeV
$$\sigma_{\rm tot}({\rm max}) \simeq 2.3(10) fb\;\;\;{\rm for\;\;M}=400(700)
{\rm \;GeV}.$$
The cross section $\sigma_{\rm tot}$ depends crucially on the value of
$\kappa^2$. If we take for example the older value $\kappa^2=0.015$ we obtain
$$\sigma_{\rm tot}({\rm max}) \simeq 20(90) fb\;\;\;{\rm for\;\;M}=400(700)
{\rm \;GeV}.$$
The similar values of the $\sigma_{\rm tot}(max)$ were obtained in Ref.[5].
The assumption that the $\eta_{CP}$ of the heavy neutrinos is opposite
($\eta_{CP}(N_2)=-\eta_{CP}(N_1)=+i$) is equivalent to changing the sign of
$\delta$
and has no influence on the numerical values of the total cross section.
In the case of CP violation $K_{N_ie}$ are complex but still the bound
(17) is satisfied and $\sigma_{\rm tot}(max)$ is smaller than in the considered
case of the CP conservation. \\
$\bullet$ The $n_R=3$ case.
If we take
$$\eta_{CP}(N_1)=\eta_{CP}(N_2)=-\eta_{CP}(N_3)=+i$$
and parametrize the heavy neutrino masses similarly as in the $n_R=2$ case
$$A=\frac{m_2}{M},\;\;B=\frac{m_3}{M},\;\;K_{N_1e}=x$$
we have from Eqs.(13) and (17)
\begin{eqnarray}
K_{N_2e}^2&=&\frac{B(\kappa^2-x^2)-x^2-\delta}{A+B}, \nonumber \\
K_{N_3e}^2&=&-\frac{A(\kappa^2-x^2)+x^2+\delta}{A+B},
\end{eqnarray}
where
$$0 \leq x^2 \leq \frac{B}{1+B}\kappa^2-\frac{\delta}{1+B}.$$
Now the cross section depends on four parameters M,A,B and x. As before, the
largest value of $\sigma_{\rm total}$ is obtained in the case when the
destructive
interference coming from the neutrino with $\eta_{CP}=-i$ disappears. It
happens for
$B \rightarrow \infty$ ($K_{N_3e} \rightarrow 0$).
The total cross sections
as functions of M for different values A and x ($0 \leq x \leq
\frac{B\kappa^2}{1+B}$) are
given in Fig.(3). We can see that the detailed behaviour of
$\sigma_{\rm tot}$
is now different than in the case $n_R=2$. For small values
of $x^2$, in particular, the $\sigma_{\rm tot}({\rm max})$ is the same as in
the case $n_R=2$
and depends only on the value of $\kappa^2$. The different $\eta_{CP}$
configurations for neutrinos are obtained by the interchange $A \leftrightarrow
B$ and the sign change of $\delta$ and have no influence on $\sigma_{\rm tot}
({\rm max})$.
Finally, we check the influence of the unitary relation (9) on the
$\sigma_{\rm total}$ for the smaller values of $\sqrt{s}$. The unitary
constraints
begin to be important when $\sqrt{s} >> m_{\rm heavy}$ and cause the
$\sigma_{\rm tot} \rightarrow 0$ for $\sqrt{s} \rightarrow \infty$ (Eq.(8)).
The relation (9) is satisfied for heavy neutrinos if some of them have
opposite CP parities. To find how important the unitarity relation for
$\sqrt{s} \simeq m_{\rm heavy}$ is we assume that the CP parities of all
neutrinos are the same. In Fig.4 we present the behaviour of $\sigma_{\rm tot}$
as function of $\sqrt{s}$ if the unitarity relation is satisfied (dashed lines)
or
not satisfied (solid lines). The lines (h) show the results for three
right-handed
neutrinos with M=1 TeV A=B=2 and $x^2=\frac{\kappa^2}{2}$ (top of the (h) line
from Fig.3). We can see that for
$\sqrt{s}=1$ TeV the cross section where unitarity is not satisfied is
approximately one order of magnitude larger than the cross section which
satisfis the unitarity requirement.
The lines (f) show the results for M=700 GeV A=1 and B=100 ( $\sim$ top of the
(f) line
from Fig.3). The mass
of the third neutrino which causes the right or wrong unitarity behaviour
is large. The contribution of this very massive neutrino to
$\sigma_{\rm tot}$ is small and the difference between the right and wrong
unitarity behaviour is
visible only for very high energy ($\sqrt{s} \sim 10^4$ GeV). At the end we
would like to
stress that the dashed line (f) represents one of the most optimistic results
from
Fig.3.
\section{Conclusions}
We have calculated the total cross section for
the $e^-e^- \rightarrow W^-W^-$ process in the frame of the Standard Model
with additional right-handed neutrino singlets. The cross section resulting
from
the known light neutrino exchange is very small: $\sigma_{\rm tot} < 10^{-2}$
fb
if the laboratory limits for neutrino masses are taken or
$\sigma_{\rm tot} < 10^{-13}$ fb if astrophysical and cosmological bounds
are appropriate. The only chance to get a larger cross section is through
heavy Majorana neutrinos exchange with mass $M \geq M_Z$. If, however,
the small
masses of the existing neutrinos are explained by the `see-saw' mechanism,
the mixing matrix elements between electron and additional neutrinos
are small for large neutrino mass and the cross section is also small,
$\sigma_{\rm tot} < 10^{-3}$ fb (for $M \geq 100$ GeV).
In models where the `see-saw' mechanism is not employed to explain
the small masses of known neutrinos, the mixing matrix elements are
usually not connected with heavy neutrino mass. In such models the mixing
matrix
elements are free parameters which can be bounded from existing experimental
data.
Taking into account the new data from LEP the bounds on the mixing matrix
elements are such that the maximal cross section can be as large as
$\sigma_{\rm tot}({\rm max}) \simeq 2.3(10)$ fb for $\sqrt{s}=0.5(1)$ TeV.
Such large cross sections are possible only if the number of right-handed
neutrinos $n_R$ is greater than one ($n_R>1$) and the masses of heavy neutrinos
are different. The largest value
$\sigma_{\rm tot}({\rm max})$ is obtained for the energy $\sqrt{s}$ not very
different from the mass of the lightest heavy neutrino and only if the
neutrinos
with opposite CP parities are much heavier than the lighter ones. As the
present
experiments give bounds to the sum of the squares of moduli of the
mixing matrix elements
$\sum\limits_{a=1}^{n_R} \mid K_{ae} \mid^2,$ the value
$\sigma_{\rm tot}$
is independent of the number of the right-handed neutrinos. If CP is not
conserved
the $\sigma_{\rm tot}({\rm max})$ is smaller than in the case of CP
conservation.
The unitarity constraints can have a big influence on the value of the
$\sigma_{\rm tot}({\rm max})$ especially in the case where the destructive
interference between neutrinos with the opposite CP parities is large.
\section{Acknowledgement}
This work was supported by Polish Committee for Scientific Researches
under Grant No. PB 136/IF/95.
\section*{References}
\newcounter{bban}
\begin{list}
{$[{\ \arabic {bban}\ }]$}{\usecounter{bban}\setlength{\rightmargin}{
\leftmargin}}
\item See e.g. Proc.of the Workshop on physics and experiments with linear
colliders (Saariselk$\ddot{a}$, Finland, September 1991),edited by
R.Orava,P.Eerola and
M.Nordberg (World Scientific, 1992) and Proc.of the Workshop on physics
and experiments with linear colliders (Waikoloa,Hawaii,April 1993),edited by
F.A.Harris,S.L.Olsen,S.Pakvasa and X.Tata (World Scientific, 1993).
\item J.F.Gunion and A.Tofighi-Niaki,Phys.Rev.D36,2671(1987) and
D38,1433(1988),
F.Cuypers,K.Ko{\l}odziej,O.Korakianitis and
R.R$\ddot{u}$kl,Phys.Lett.B325(1994)243.
\item `Inverse Neutrinoless Double $\beta$-Decay at the NLC?',T.Rizzo,
preprint ANL-HEP-CP-93-24.
\item J.Maalampi,A.Pietil$\ddot{a}$ and J.Vuori, Phys.Lett.B297(1992)327;
J.Gluza and M.Zra{\l}ek,hep-ph/9502284.
\item C.A.Heusch and P.Minkowski,Nucl.Phys.B416(1994)3 and `A strategy
for discovering heavy neutrinos',preprint BUTP-95/11,SCIPP 95/07.
\item Particle Data Group,K.Hikasa et.al.,Phys.Rev.D,S1(1994).
\item O.Adriani et.al.Phys.Lett.B(1992)371.
\item E.Nardi,E.Roulet and D.Tommasini,Nucl.Phys.B386(1992);A.Ilakovic and
A.Pilaftsis Nucl.Phys.B437(1995)491.
\item A.Djoudi,J.Ng and T.G.Rizzo,`New Particles and Interactions',
SLAC-PUB-95-6772.
\item T.Yanagida,Prog.Theor.Phys.B135(1978)66; M.Gell-Mann,
P.Ramond and R.Slansky,in `Supergravity',edsP.Van Nieuwenhuizen
and D.Freedman (North-Holland,Amsterdam,1979)p.315.
\item D.Wyler and L.Wolfenstein,Nucl.Phys.B218(1983)205;R.N.Mohapatra and
J.W.F.Valle,Phys.Rev.D34(1986)1642;\newline
E.Witten,Nucl.Phys.B268(1986)79;
J.Bernabeu et al.,Phys.Lett.B187(1987)303;
J.L.Hewett and T.G.Rizzo,Phys.Rep.183
(1989)193;\newline
P.Langacker and D.London,Phys.Rev.D38(1988)907;E.Nardi,Phys.Rev.D48
(1993)3277;D.Tommasini et al.,`Non decoupling of heavy neutrinos and
leptons flavour violation' (hep-ph/9503228).
\item L.N.Chang,D.Ng and J.N.Ng,Phys.Rev.D50(1994)4589.
\item T.P.Cheng and L.F.Li,Phys.Rev.D44(1991)1502.
\end{list}
\section*{Figure Captions}
\newcounter{bean}
\begin{list}
{\bf Fig.\arabic
{bean}}{\usecounter{bean}\setlength{\rightmargin}{\leftmargin}}
\item The cross section as function of the heavy neutrino mass for `classical'
see-saw models, where the mixing angles between light and heavy neutrinos are
proportional to the inverse of mass of the heavy neutrino. A=10, B=20
(Eq.(16)).
Solid (dashed) line is for the TLC (NLC) collider's energy.
\item The cross section as function of the heavy neutrino masses for the
Standard Model
with two right-handed neutrinos.
Dashed (solid) lines are for $\sqrt{s}=0.5(1)$ TeV colliders.
The curves (a),(b),(c),(d) are for A=2,5,10,$10^5$ respectively (Eq.(18)).
\item The cross section as function of the heavy neutrino masses for the
Standard Model
with three right-handed neutrinos for the TLC collider's energy.
The denotations are as follows (Eq.(19)):
(a):A=4,B=$\infty \;(10^5),x^2=0$;
(b):A=2,B=$\infty ,\;x^2=0$;
(c):A=2,B=$\infty ,\;x^2=0.002$;
(d):A=2,B=$\infty ,\;x^2=0.003$;
(e):A=2,B=$\infty ,\;x^2=0.004$;
(f):A=2,B=$\infty ,\;x^2=\kappa^2$;
(g):A=2,B=2,$\;x^2_{max}=\frac{2}{3}\kappa^2$;
(h):A=2,B=2,$\;x^2=\frac{\kappa^2}{2}$.
\item The influence of the unitarity constraints at the small energy limit for
the
RHS model with three right handed neutrinos (Eq.(19)). Lines denoted
by (f) are for A=1,B=100,M=700 GeV. Lines denoted by (h) are for
A=2,B=2,M=1000 GeV. Solid (dashed) lines are for real (complex)
coupling $K_{N_3e}$ (Eq.(19)).
\end{list}
\begin{table}
\begin{center}
\caption{
The reduced helicity amplitudes for
the $e^-e^- \rightarrow W^-W^-$ process in t- and u-channels.}
\begin{tabular}{||c|c||c||c||} \hline \hline
$\lambda_1$
& $\lambda_2$
& $M_t$
& $M_u$ \\ \hline
1 & 1 &
$ \gamma^2(1-c)$
& $\gamma^2(1+c)$ \\
&&& \\
-1 & -1 & $ \gamma^2(1+c)$
& $ \gamma^2(1-c)$
\\ \cline{1-4}
1 & 0 &
$ -\gamma(1-\beta)$
&
$ \gamma(1-\beta)$ \\
&&& \\
-1 & 0 & $ \gamma(1+\beta)$
& $ -\gamma(1+\beta)$
\\ \cline{1-4}
0 & 1 &
$ -\gamma(1-\beta)$ &
$ \gamma(1-\beta)$ \\
&&& \\
0 & -1 & $ \gamma(1+\beta)$
& $ -\gamma(1+\beta)$
\\ \cline{1-4}
1 & -1 & 0 & 0 \\
-1 & 1 & 0 & 0 \\ \cline{1-4}
&&& \\
0 & 0 & $-\left( 1+\beta^2 - 2c \beta \right) $
&
$- \left( 1+\beta^2 + 2c \beta \right) $
\\ \hline \hline
\end{tabular}
\end{center}
\end{table}
\newpage
\end{document}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,237 |
Toronto - Melbourne Research Training Group
Joint PhD opportunities
Deep analysis of large international clinical trials for bloodstream infections
This joint PhD opportunity is no longer accepting applications until further notice.
There are 50 million cases of sepsis worldwide each year, leading to 11 million sepsis-related deaths.
Bloodstream infections are a key cause and manifestation of sepsis.
Associate Professor Daneman and Associate Professor Tong lead two large international clinical trials for bloodstream infections.
BALANCE, led by Associate Professor Daneman, investigates the optimal duration of antibiotics needed to treat bloodstream infections. SNAP, led by Associate Professor Tong, aims to determine the optimal antibiotic regimens for bloodstream infections caused by Staphylococcus aureus.
These are the two largest clinical trials for bloodstream infections to date and collectively will include more than 10,000 participants, providing an unprecedented opportunity to embed additional mechanistic and epidemiologic sub-studies.
The PhD project will involve multiple nested sub-studies within each of these two trials. These projects are best suited to a clinician aiming to develop expertise in infectious diseases clinical research.
The two different clinical trials each have their distinct project goals. The BALANCE trial involves:
Evaluating the predictive utility of procalcitonin and other novel biomarkers to offer a precision-medicine approach to bloodstream treatment durations.
Exploring the concordance of clinician and patient perceptions of the daily probability of cure during the treatment trajectory, and whether these perceptions are predictive of treatment success and survival.
Examining patient, pathogen, and syndrome predictors of protocol non-adherence (and treatment prolongation) among patients randomized to shorter-duration treatment for bacteremia.
The SNAP trial involves:
Determining predictors of outcomes within specific sub-groups of patients (e.g., those with methicillin-resistant S. aureus [MRSA] infection).
Improving and assessing consent processes, particularly for disadvantaged or marginalised patients.
Microbiology focussed questions such as the impact of the cefazolin inoculum effect on patient outcomes.
Pharmacokinetic / pharmacodynamics (PK/PD) questions such as the impact of β-lactam drug levels on outcomes
Comparison of outcomes for patients enrolled in the trial vs those not enrolled (but who are part of a broader registry of S. aureus bloodstream infections.
Supervision team
The University of Melbourne: Associate Professor Steven Tong and Professor Joshua Davis
The University of Toronto: Associate Professor Nick Daneman and Dr Rob Fowler
*Click on the researcher's name above to learn more about their publication and grant successes.
Who we are looking for
We are seeking a PhD candidate with the following skills:
Demonstrated experience in the field of biomedical sciences.
Demonstrated ability to work independently and as part of a team.
Demonstrated time and project management skills.
Demonstrated ability to write research reports or other publications to a publishable standard (even if not published to date).
Demonstrated organisational skills, time management and ability to work to priorities.
Demonstrated problem-solving abilities.
The ability to work independently and as a member of a team.
The PhD candidate will benefit from the combined expertise of the project supervisors, and the embedding into two research environments.
Associate Professor Steven Tong and Professor Joshua Davis at the University of Melbourne will contribute their expertise in infectious diseases, the conduct of clinical trials, and translational science. Associate Professor Nick Daneman and Dr Rob Fowler at the University of Toronto will contribute expertise in the conduct of clinical trials, critical care medicine, and data-driven implementation science.
This PhD project will be based at the University of Toronto with a minimum 12-month stay at the University of Melbourne.
The candidate will be enrolled in the PhD program at the Department of Institute of Health Policy, Management and Evaluation (IHPME) in the Dalla Lana School of Public Health (DLSPH)at the University of Toronto, and in the PhD program at the Department of Infectious Diseases, Melbourne Medical School at the University of Melbourne.
Apply for a joint PhD with the Toronto-Melbourne Research Training Group.
Discover what researchers from the Toronto-Melbourne Research Training Group are working on right now.
Hear the stories of current and past graduate researchers. Find out about their experiences at the University and where their degrees have taken them.
Find a supervisor at the University of Melbourne | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 36 |
/ Professionals
/ Samuel H. Megerditchian
Samuel H. Megerditchian
Director, Intellectual Property
smegerditchian@gibbonslaw.com
Mr. Megerditchian, a seasoned patent attorney, has over 20 years of experience in chemical, pharmaceutical, biotechnological, electrical, and mechanical matters, with a concentration in the development, management, and enforcement of patent portfolios.
Mr. Megerditchian's experience covers all aspects of intellectual property law, including domestic and international patent preparation and prosecution, counseling, opinion work, patentability studies, due diligences, intellectual property licensing and patent portfolio and life-cycle management.
Mr. Megerditchian is registered to practice before the US Patent and Trademark Office. His scientific background and experience in the areas of medicinal chemistry, organic chemistry, biochemistry and molecular biology, and his work experience in the pharmaceutical and biotech industries, are immeasurable assets to his clients.
Previously, Mr. Megerditchian was Senior Counsel at Hoffmann-La Roche, where he served as U.S. Patent Head for Cardiovascular and Metabolic Diseases and then as U.S. Patent Head for Inflammatory Diseases.
Prior to becoming a lawyer, he was Chief Chemist/Chemistry Laboratory Supervisor at a major multinational corporation.
St. John's University School of Law (J.D.)
Editor, St. John's Law Review
Polytechnic Institute of New York University (M.S.)
Iona College (B.S.)
Vice President, Chemical Society
Professional Admissions
United States District Court for the Southern District of New York
United States Patent & Trademark Office
AllArticleBlog PostClient AlertInterviewQuoteVideo
Biotechnology & Pharmaceutical
Manufacturing & Consumer Products, Including Electronics | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,100 |
using System;
using System.Collections.Generic;
using System.Reflection;
using TruRating.Dto.TruService.V220;
using TruRating.TruModule.ConsoleRunner.Environment;
using TruRating.TruModule.Device;
namespace TruRating.TruModule.ConsoleRunner.Device
{
public class ConsoleDevice : IDevice
{
private readonly string[] _languages;
private readonly IConsoleLogger _consoleLogger;
public ConsoleDevice(IConsoleLogger consoleLogger, string[] languages)
{
_consoleLogger = consoleLogger;
_languages = languages;
}
public void DisplayMessage(string value)
{
_consoleLogger.WriteLine(ConsoleColor.White, "DISPLAY: " + value);
}
public RequestPeripheral GetScreenCapabilities()
{
return new RequestPeripheral
{
Format = Format.TEXT,
Separator = System.Environment.NewLine,
Font = Font.MONOSPACED,
Height = 4,
HeightSpecified = true,
Unit = UnitDimension.LINE,
Width = 16,
WidthSpecified = true
};
}
public SkipInstruction GetSkipInstruction()
{
return SkipInstruction.NONE;
}
public string GetName()
{
return GetType().Name;
}
public string GetFirmware()
{
return Assembly.GetExecutingAssembly().FullName;
}
public void CancelQuestion()
{
KeyPressReader.Cancel();
}
public RequestLanguage[] GetLanguages()
{
var result = new List<RequestLanguage>();
foreach (var language in _languages)
{
result.Add(new RequestLanguage {Rfc1766 = language});
}
return result.ToArray();
}
public string GetCurrentLanguage()
{
return GetLanguages()[0].Rfc1766;
}
public RequestServer GetServer()
{
return null;
}
public void DisplayAcknowledgement(string value, int timeoutMilliseconds, bool hasRated, RatingContext ratingContext)
{
_consoleLogger.WriteLine(ConsoleColor.White, "DISPLAY: " + value);
_consoleLogger.WriteLine(ConsoleColor.Gray, "DISPLAY: waiting {0} ms", timeoutMilliseconds);
try
{
KeyPressReader.ReadKey(timeoutMilliseconds, true);
}
catch (TimeoutException)
{
//Suppress
}
}
public short Display1AQ1KR(string value, int timeoutMilliseconds)
{
try
{
_consoleLogger.WriteLine(ConsoleColor.Cyan, "1AQ1KR : " + value);
_consoleLogger.WriteLine(ConsoleColor.Gray, "1AQ1KR : waiting {0} ms", timeoutMilliseconds);
_consoleLogger.Write(ConsoleColor.Cyan, "1AQ1KR : ");
short result;
if (short.TryParse(KeyPressReader.ReadKey(timeoutMilliseconds, false).KeyChar.ToString(), out result))
{
return result;
}
return -1; //User didn't press a number
}
catch (TimeoutException)
{
return -2; //User timed out
}
catch (Exception)
{
return -4; // Couldn't ask a question or capture the response
}
}
}
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 9,374 |
Syria war: US weighs military action following gas 'attack'
11:39, April 7, 2017
US President Donald Trump says "something should happen" against the Syrian leadership over a suspected chemical weapons attack on a rebel-held town.
His exact intentions remain unclear but the US is reportedly considering a military response.
The BBC's North America editor says a US strike could be imminent.
US Secretary of State Rex Tillerson has said Bashar al-Assad should have no role in a future Syria.
His comments signal an apparent U-turn for the US - only last week the US ambassador to the UN, Nikki Haley, said Washington was no longer prioritising the removal of the Syrian president.
Measures reported to be under discussion between the Pentagon and the White House include the targeting of Syrian radar using cruise missiles launched from US ships.
Dozens of people, including at least 27 children, are reported to have died following an attack on the town of Khan Sheikhoun in Idlib province early on Tuesday.
Syria denies its forces were to blame and is supported by its ally, Russia.
Military action looks certain: Jon Sopel, BBC North America editor
It is a clear and pretty dramatic shift. A month ago, Bashar al-Assad in the eyes of the US was part of the solution, considered useful in the fight against so-called Islamic State.
Then yesterday, President Trump said his position had changed, that the Assad regime had crossed many lines. The implication was that there could be military action.
But given everything that's been said in the past 24 hours, I would say military action looks certain and could be imminent. We could wake up tomorrow morning and find out the Americans have taken action.
Cast your mind back to what President Trump said about Barack Obama, when the then president said a red line had been crossed and he did nothing about it afterwards.
He heaped derision on President Obama. If Mr Trump were not to act now, he would look weak and he wouldn't want that.
Russia's deputy UN envoy warned of "negative consequences" of any US military strike on Syria.
"All the responsibility... will be on shoulders of those who initiated such doubtful and tragic enterprise," Vladimir Safronkov said.
Addressing reporters on his way to Florida to meet Chinese President Xi Jinping, Mr Trump said: "I think what Assad did is terrible. I think what happened in Syria is a disgrace to humanity and he's there, and I guess he's running things, so something should happen."
Also on Thursday, Mr Tillerson said: "Assad's role in the future is uncertain and with the acts that he has taken, it would seem that there would be no role for him to govern the Syrian people.
"We are considering an appropriate response (to the) violations of all previous UN resolutions, violations of international norms."
He said "steps were under way" to organise an international coalition to remove Mr Assad.
UN Secretary-General Antonio Guterres, asked if he was concerned about possible US military action in Syria, said his priority was to ensure accountability for the deadly gas attack.
He said he could not comment on "things that have not yet happened".
Evidence has mounted that the victims of the attack in Khan Sheikhoun were killed with a nerve agent such as Sarin.
Syria has denied dropping chemical weapons from the air while Russia has argued that the mass poisoning was caused by an air strike on a rebel weapons dump where chemical weapons were being stored.
But the claims have been viewed with scepticism from the US and its allies.
Syria says it would only accept a UN investigation into the incident if a list of conditions are met.
Syira Gas Attack
Syria war | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,981 |
\section{Introduction}
\label{intro}
\vspace{-2mm}
Robots are progressively spreading to logistic, social and assistive domains. However, in order to become handy co-workers and helpful assistants, they must be endowed with quite different abilities than their industrial ancestors \citep{Asfour2008,Torras2016}. The ability to deal with articulated objects is relevant for robots operating in domestic environments. For example, robots need to open doors when moving around homes and to open cabinets to pick up objects. The problem of opening doors and drawers with robots has been tackled extensively \citep{Meeussen2010,Ott2005,Jain2009,Kessens2010,Endres2013}. These approaches usually focus either on a particular type of door and handle mechanism or in a certain aspect of the task. Therefore, our contribution is on devising a more general framework that can incorporate different types of door models and that provides adaptive behavior during door operation.
\begin{figure}
\centering
\includegraphics[width=0.72\linewidth]{figures/Fig1.jpg}
\caption{The HSR robot assists a person to enter a room.}
\label{Fig1}
\end{figure}
The paper is organized as follows: in Section \ref{related_work} we review the state-of-the-art in the field; in Section \ref{problem_statement} we state the problem addressed in this work; in Section \ref{detection} we present our door and handle detection model; in Section \ref{grasping} we explain our approach for achieving robust real-time estimation of end-effector grasping poses; in Section \ref{unlatching} we describe a method for unlatching door handles; in Section \ref{learning} we present a Bayesian approach to learn door kinematic models which allows to improve performance by learning from experience as well as from human demonstrations; in Section \ref{door_opening} we discuss the integration of kinematic model inference with a motion planner; in Section \ref{experiments} we experimentally validate our framework; finally, in Section \ref{conclusions} we draw the main conclusions.
\section{Related Work}
\label{related_work}
The detection of doors and handles has been explored based on 2D images, depth data, or both. In \citep{Chen2014}, they presents a deep convolutional neural network for estimating door poses from images. Although doors are accurately located, the identification of handles is not addressed. In \citep{Banerjee2015}, following the requirements from the DARPA Robotics Challenge, the authors develop an algorithm for identifying closed doors and their handles. Doors are detected by finding consecutive pairs of vertical lines at a specific distance from one another in an image of the scene. If a flat surface is found in between, the door is recognized as closed. Handle detection is subsequently carried out by color segmentation. The paper \citep{Llopart2017} addresses the problem of detecting room doors and also cabinet doors. The authors propose the use of a CNN to extract and identify the Region of Interest (ROI) in an RGB-D image. Then, the handle's 3D position is calculated under the assumption that it is the only object contained in the ROI and its color is significantly different from that of the door. Although positive results are obtained in these last two works, they rely on too many assumptions limiting the versatility of the proposed methods.
The door manipulation problem with robotic systems has also been addressed with different approaches. Some works assume substantial previous knowledge about the kinematic model of the door and its parameters, while others are entirely model-free. Among the works that assume an implicit kinematic model, in \citep{Diankov2008} the operation of articulated objects is formulated as a kinematically constrained planning problem. The authors propose to use caging grasps, to relax task constraints, and then use efficient search algorithms to produce motion plans. Another interesting work is \citep{Wieland2009}. The authors combine stereo vision and force feedback for compliant execution of the door opening task. Regarding model-free approaches, in \citep{Lutscher2010} they propose to operate unknown doors based on an impedance control method, which adjusts the guiding speed to achieve two-dimensional planar operation. Another example is the approach presented in \citep{Karayiannidis2013}. Their method relies on force measurements and estimation of the motion direction, rotational axis and distance from the center of rotation. They propose a velocity controller that ensures a desired tangential velocity. Both approaches have their own advantages and disadvantages. By assuming an implicit kinematic model, although in practice a simpler solution is typically achieved, the applicability is limited to a single type of door. On the other hand, model-free approaches release programmers from specifying the motion parameters, but they rely entirely on the compliance of the robot, which requires rich sensory feedback and advanced mechanical and control capabilities.
Alternatively, other works propose probabilistic methods that do not consider interaction forces. In \citep{Nemec2017} the authors combine reinforcement learning with intelligent control algorithms. With their method, the robot is able to learn the door opening policy by trial and error in a simulated environment. Then, the skill is transferred to the real robot. In \citep{Welschehold2017} the authors present an approach to learn door opening action models from human demonstrations. The main limitation of these works is that they do not allow to operate autonomously unknown doors. Finally, the probabilistic approach proposed in \citep{Sturm2013} enables the description of the the geometric relation between object parts to infer the kinematic structure from observations of their motion. We have adopted this approach as a basic reference, but extended its capabilities to improve the performance by using prior information or human demonstrations.
In this paper, we propose a robust and adaptive framework for manipulating general types of door mechanisms. We consider all the stages of the door opening task in a unified framework. The main contributions of our work are: (a) the development of a novel algorithm to estimate the robot\textquoteright s end-effector grasping pose in real-time for multiple handles simultaneously; (b) a versatile framework that provides the robust detection and subsequent door operation for different types of door kinematic models; (c) the analysis of the door kinematic inference process by taking into account door prior information; (d) the testing on real hardware using the Toyota Human Support Robot (HSR) (Figure \ref{Fig1}).
\section{Problem Statement}
\label{problem_statement}
We study the problem of enabling a robot to open doors autonomously, regardless of their form or kinematic model. Thus, the following sub-tasks are to be performed successfully by an autonomous robot:
\begin{itemize}
\item Door and handle detection
\item Grasping of the handle
\item Unlatching the handle
\item Estimating the door kinematic model
\item Planning and executing the door opening motion
\end{itemize}
In this paper, we assume the robot is equipped with a vision system able to capture features in a 3-dimensional space. Additionally, we consider the particular case of a mobile robot and that force/torque feedback is available. Note that these assumptions attempt to be as general as possible, as these requirements are usually met by most service robots nowadays.
\vspace{-3mm}
\section{Door and Handle Detection}
\label{detection}
Doors and handles present a wide variety of geometries, sizes, colours, etc. Thus, a robust detection algorithm is essential. Additionally, in order to achieve real-time estimation, it must operate at speeds of a few frames-per-second (fps). Object detection is the task of simultaneously classifying and localizing multiple objects in an image. In \citep{Redmond2016a} the authors proposed the YOLO algorithm, an open-source state-of-the-art object detector with CNN-based regression. This network uses features from the entire image to predict each bounding box, reasoning globally about the full image and all the objects in the image. It enables end-to-end training and facilitates real-time speeds while maintaining high average precision. For these reasons, we decided to adopt this CNN architecture.
\vspace{-3mm}
\subsection{Model Training}
Training the YOLO network with a custom dataset allows us to build a handle and door detection model. The simplest classification semantics for our objects of interest are ``door'' and ``handle''. However, to increase the detail of the information and also to make our method versatile and extendable to other applications, we propose to split the class door into three classes: ``door'', which refers to a room door, ``cabinet door'', which includes all sorts of small doors such as drawers or a locker door, and ``refrigerator door''. We built a data set using images from the Open Images Dataset \citep{Kuznetsova2018} and annotated a total of $1213$ images containing objects pertaining to our desired object classes; $1013$ of them were used for the training set, and the remaining $200$ for the testing set (the dataset is available in \citep{GitHub2019}). Some examples of the annotated images are shown in Figure \ref{Fig2}. We also applied data augmentation techniques to improve the generalization of our neural network \citep{Taylor2017}.
\begin{figure}
\centering
\includegraphics[width=0.95\linewidth]{figures/Fig2.png}
\caption{Examples of annotated images from the training dataset used for building the door and handle detection model. The bounding boxes enclose the objects, with the corresponding label, that should be identified by the model.}
\label{Fig2}
\end{figure}
\vspace{-3mm}
\subsection{Model Selection}
For selecting the CNN weights and assessing the model quality, we applied cross-validation against the test set. As performance index, we propose to use the mean average precision (mAP). This criterion was defined in the PASCAL VOC 2012 competition and is the standard metric for object detectors \citep{Everingham2015}. Briefly, the mAP computation involves the following steps: (1) Based on the likelihood of the predictions, a precision-recall curve is computed for each class, varying the likelihood threshold. (2) The area under this curve is the average precision. Averaging over the different classes we obtain the mAP. Precision and recall are calculated as:\begin{equation}
\centering
\text{Precision}=\frac{TP}{TP+FP}\quad\quad\text{Recall}=\frac{TP}{TP+FN}
\end{equation} where $TP=$ True Positive, $TN=$ True Negative, $FP=$ False Positive and $FN=$ False Negative. True or false refers to the assigned classification being correct or incorrect, while positive or negative refers to whether the object is assigned or not to a category.
\section{Grasping the Handle}
\label{grasping}
When a robot moves towards an object, it is actually moving towards a pose at which it expects the object to be. For solving the grasping problem, the handle's 6D-pose estimation is essential. The end-effector grasping goal pose can be then easily expressed relative to the handle's pose, and reached by solving the inverse kinematics of the robot. Perception is usually provided by means of an RGB-D sensor, which supplies an RGB image and its corresponding depth map. For estimating the 6-D pose in real-time, we propose to: (1) Identify the region of the RGB image where the door and the handle are located. (2) Filter the RGB-D image to extract the Regions of Interest, clean the noise and down sample. (3) From a set of 3D geometric features of the door and the handle, estimate the grasping pose. We explain in detail these steps in this section. The proposed approach is summarized in the algorithm below:
\begin{algorithm}
\caption{\bf End-Effector Grasping Pose Estimation}
\small
\label{Algorithm1}
\begin{algorithmic}
\renewcommand{\algorithmicrequire}{\textbf{Input:}}
\renewcommand{\algorithmicensure}{\textbf{Output:}}
\Require RGB image $\mathcal{I}$ and point cloud $\mathcal{P}= \left \{ \textbf{p}_j \right \} ^{N_{points}}_{0}$
\Ensure Grasping poses $\mathcal{G}=\left \{ \textbf{g}_k \right \}^{N_{handles}}_1$ with $\textbf{g}_k \in SE(3) $
\State Bounding boxes $\mathcal{B}=\left \{ b_l\right \}^{N_{objects}}_1 \gets \text{Detect\_Objects}(\mathcal{I})$
\ForAll{$b_l\in \mathcal{B}$}
\State $\mathcal{P}^{ROI}_{l} \gets \text{ROI\_Segmentation}(\mathcal{P})$
\State $\mathcal{P}^{denoised}_{l} \gets \text{Remove\_Statistical\_Outliers}(\mathcal{P}^{ROI}_{l})$
\State $\mathcal{P}^{filtered}_{l} \gets \text{Downsample}(\mathcal{P}^{denoised}_{l})$
\If {$\text{class}(b_l)=\text{"handle"}$}
\State $\text{orientation}_l \gets \text{Bounding\_Box\_Dimensions}(b_l)$
\State $\mathcal{P}^{handle}_{l} \gets \text{RANSAC\_Plane\_Outliers}(\mathcal{P}^{ROI}_{l})$
\State $\textbf{O}_l \gets \text{Centroid}(\mathcal{P}^{handle}_l)$
\Else
\State Normal $\textbf{a}_l; \, \mathcal{P}^{door}_{l} \gets \text{RANSAC\_Plane}(\mathcal{P}^{filtered}_{l})$
\State $\textbf{O}_l \gets \text{Centroid}(\mathcal{P}^{door}_l)$
\EndIf
\EndFor
\State $k=1$
\ForAll{$b_l\in \mathcal{B}$ that $\text{class}(b_l)="\text{handle}"$}
\State $\textbf{a}_l \gets \text{Assign\_Closest\_Door}(\textbf{O}_l)$
\State $\textbf{h}_k \in SE(3) \gets \text{Handle\_Transform}(\textbf{a}_l \, ; \, \textbf{O}_l)$
\State $\textbf{g}_k \gets \text{Goal\_Pose}(\textbf{h}_k\, ; \, \text{orientation}_l)$
\State $k \gets k+1$\
\EndFor
\newline
\Return $\mathcal{G}$
\end{algorithmic}
\end{algorithm}
\vspace{-5mm}
\subsection{Point Cloud Filtering}
Raw point clouds contain a large number of point samples, but only a small fraction of them are of interest. Furthermore, they are unavoidably contaminated with noise. Point cloud data needs to be filtered adequately for achieving accurate feature extraction and real-time processing. We propose the following filtering process:
\subsubsection{Regions Of Interest (ROIs) Segmentation}
The points of interest are those that correspond to the doors and the handles in the scene, which can be defined as those contained in the bounding boxes of the proposed object detection CNN. By separating the sets of points that are contained in each ROI, the amount of data to be processed is reduced significantly (Figure \ref{Fig3}). There is a direct correspondence between the pixels in the image and the point cloud indexes, if the latter is indexed according to its spatial distribution. Taking into account that the bounding boxes are usually provided in pixel coordinates, let $\mathcal{P}$ be the raw point cloud. Then, each ROI can be defined as follows:
\begin{equation}
\mathcal{P}^{ROI}=\left\{ \textbf{p}_{j}\in\mathcal{P}\,\vert\,j=\text{width}\cdot y+x\right\}
\end{equation}
where $j$ is the point cloud index; $\text{width}$ is the image width in pixels, $x\in\left[x_{min},\;x_{max}\right]$ and $y\in\left[y_{min},\;y_{max}\right]$, being $\left(x_{min},y_{min}\right)$ and $\left(x_{max},y_{max}\right)$ two opposite corners of the bounding box in pixel coordinates.
\begin{figure}
\centering
\includegraphics[width=1.0\linewidth]{figures/Fig3.png}
\caption{On the left, the Regions of Interest detected by our door and handle detection model. On the right, the subset of the corresponding point cloud enclosed in the ROIs.}
\label{Fig3}
\end{figure}
\subsubsection{Statistical Outlier Filtering}
Measurement errors lead to sparse outliers, which complicate the estimation of local point cloud features such as surface normals. Some of these irregularities can be solved by performing a statistical analysis of each point neighborhood, and trimming those that do not meet a certain criterion. We can carry this analysis at a discrete point level. By assuming that the average distance from every point to all its neighboring points $r_j$, can be described by a Gaussian distribution, the filtered point cloud can be defined as follows:
\begin{equation}
\centering
\mathcal{P}^{denoised}=\left\{ \mathbf{p}_{j}\in\mathcal{P}^{ROI}\mid r_{j}\in\left[\mu_r\pm\alpha\cdot\sigma_r\right]\right\}
\end{equation} where $\alpha$ is a multiplier, and $\mu_{r}$ and $\sigma_{r}$ are the mean distance and the standard deviation, respectively.
\subsubsection{Downsampling}
In order to lighten up the computational load we propose to reduce considerably the amount of data by using a voxelized grid approach (Figure \ref{Fig4}). Unlike other sub-sampling methods, the shape characteristics are maintained. If $s$ is the number of points contained in each voxel $A$, the set of points in each voxel is replaced by:
\begin{equation}
\bar{x}=\frac{1}{s}\sum_{A}x\quad\quad\bar{y}=\frac{1}{s}\sum_{ A}y\quad\quad\bar{z}=\frac{1}{s}\sum_{A}z
\end{equation}
\vspace{-3mm}
\begin{figure}
\centering
\includegraphics[width=1.0\linewidth]{figures/Fig4.png}
\caption{On the left, the raw point cloud. On the right, the downsampled point cloud using a voxelized grid approach.}
\label{Fig4}
\end{figure}
\vspace{-6mm}
\subsection{Grasping Pose Estimation}
We have considered three geometric features of the 3D structure of the door and the handle for the grasping pose estimation: the handle orientation, its position and the door plane normal direction.
\subsubsection{Handle Orientation}
The end-effector orientation for grasping the handle depends on this feature. Since door handles are commonly only oriented vertically or horizontally (for the particular case of a door knob, full orientation is not relevant to grasp it), the binary decision can be made by comparing the dimensions of the sides of the CNN output bounding boxes. If the height is greater than the width, the handle orientation will be vertical and vice versa.
\subsubsection{Door Plane Normal\label{subsec:Door=002019s-Plane-Normal}}
In order to grasp the handle correctly, the normal to the ``palm'' of the robot's end-effector (which we consider similar to the human hand) must be parallel to the door normal. We propose to use the RAndom SAmple Consensus (RANSAC) algorithm \citep{Rusu2013} to compute the normal direction. It is an iterative method to estimate the parameters of an specified mathematical model from a set of observed data that contains outliers, where the outliers do not influence the values of the estimates. Thus, we can fit a planar model to the door point cloud, and calculate the coefficients of its parametric Hessian normal form. In this way, we can determine the door normal direction. In Figure \ref{Fig5} we show some examples of the resulting normal vectors obtained with RANSAC.
\begin{figure}
\centering
\includegraphics[width=0.9\linewidth]{figures/Fig5.png}
\caption{The red arrows show the normal direction of the plane defined by each detected door.}
\label{Fig5}
\end{figure}
\subsubsection{Handle Position}
We make the assumption that the handle position can be represented by its centroid. However, it cannot be directly computed from the sub-point cloud associated to the handle ROI, since the defining bounding box usually may include some points from the door in the background. We can also use the RANSAC algorithm to separate these points \citep{Zuliani2012}. We fit a plane model, and classify the points as inliers and outliers. In this case, the outlier subset corresponds to the handle.
\subsubsection{Goal Pose Generation}
Let $\textbf{O}=\left(O_{x},O_{y},O_{z}\right)$ be the handle centroid and $\mathbf{a}=\left(a_{x},a_{y},a_{z}\right)$ the door plane normal unitary vector, both expressed in an arbitrary reference frame $w$. The handle pose can be defined as the following transform:
\begin{equation}
\small{\mathbf{T}_{w}^{handle}=\left(\begin{array}{cccc}
a_{x} & \frac{a_{y}}{a_x^2+a_y^2} & \frac{a_{x}a_{z}}{a_x^2+a_y^2} & O_{x}\\
a_{y} & -\frac{a_{x}}{a_x^2+a_y^2} & -\frac{a_{y}a_{z}}{a_x^2+a_y^2} & O_{y}\\
a_{z} & 0 & -1 & O_{z}\\
0 & 0 & 0 & 1
\end{array}\right)}
\end{equation}
\begin{figure*}
\centering
\includegraphics[width=0.98\linewidth]{figures/Fig6.png}
\caption{Proposed handle unlatching strategy. First, the wrist is turned anti-clockwise. If torque feedback is above the allowed threshold, the movement is aborted. Then, the wrist is turned clockwise. If torque feedback is also above the threshold, the handle is identified as ``no actuation is required''.}
\label{Fig6}
\end{figure*}
The grasping pose can then be easily specified as a relative transform to the handle reference frame $\mathbf{T}_{\textit{handle}}^{\textit{grasping}}$, taking into account its orientation. Thus, the pose for which the Inverse Kinematics (IK) of the robot must be solved in order to finally grasp the handle, can be computed as:
\begin{equation}
\mathbf{T}_w^{\textit{grasping}}=\mathbf{T}_w^{\textit{handle}}\mathbf{T}_{\textit{handle}}^{\textit{grasping}}
\end{equation}
\section{Unlatching the Handle}
\label{unlatching}
There exists a variety of mechanisms to open a door. Some of them do not require any specific actuation while others generally require a rotation to be applied. A handle usually occupies a small region in the door image. Thus, in order to estimate its kinematic model from visual perception data, the robot camera needs to be placed close to the handle. This operation would increase considerably the time required to perform the task.
Instead, we rely on force feedback for inferring how the handle should be actuated. We propose a simple, trial-and-error strategy for operating different types of handles, illustrated in Figure \ref{Fig6}. The robot tries to turn the handle in both directions and, depending on a torque threshold, either the door is unlatched or no actuation is required.
\vspace{-3mm}
\section{Learning the Door Kinematic Model}
\label{learning}
Opening doors in unstructured environments is challenging for robots because they have to deal with uncertainty, since the kinematic model of the door is not known a priori. What if a robot has no previous knowledge of the door at the time of taking a decision? And, what if previous knowledge is available? To address these questions, we will present a probabilistic framework that allows to infer the kinematic model of the door when no previous knowledge is available and improve the performance based on previous experiences or human demonstrations.
\subsection{Overview of the Probabilistic Framework}
Let $\mathcal{D}=(\mathbf{d}_{1},\dots,\mathbf{d}_{N})$ be the sequence of $N$ relative transformations between an arbitrary fixed reference frame and the door, observed by the robot. We assume that the measurements are affected by Gaussian noise and, also, that some of these observations are outliers but not originated by the noise. Instead, the outliers might be the result of sensor failures. We denote the kinematic link model as $\mathcal{M}$. Its associated parameters are contained in the vector $\boldsymbol{\theta}\in\mathbb{R}^{k}$ (where $k$ is the number of parameters). The model that best represents the data can be formulated in a probabilistic context as \citep{Sturm2010}:
\begin{equation}
(\hat{\mathcal{M}},\hat{\boldsymbol{\theta})}=\argmax_{\mathcal{M},\boldsymbol{\theta}}{p\left(\mathcal{\mathcal{M}},\boldsymbol{\theta}\mid\mathcal{D}\right)}
\end{equation}
This optimization is a two-step process \citep{MacKay2003}. First, a particular model is assumed true and its parameters are estimated from the observations:
\begin{equation}
\hat{\boldsymbol{\theta}}=\argmax_{\boldsymbol{\theta}}p\left(\boldsymbol{\theta}\mid\mathcal{D},\mathcal{M}\right)
\end{equation}
By applying Bayes rule, and assuming that the prior over the parameter space is uniform, this is equivalent to:
\begin{equation}
\hat{\boldsymbol{\theta}}=\argmax_{\boldsymbol{\theta}}{p(\mathcal{D}\mid\boldsymbol{\theta},\mathcal{M}})
\end{equation} which shows that fitting a link model to the observations is equivalent to maximizing the data likelihood. Then, we can compare the probability of different models, and select the one with the highest posterior probability:
\begin{equation}
\hat{\mathcal{M}}=\argmax_{\mathcal{M}}{\int p\left(\mathcal{M},\mathbf{\boldsymbol{\theta}}\mid\mathcal{D}\right)d\boldsymbol{\theta}}
\end{equation}
Summarizing, given a set of observations $\mathcal{D}$, and candidate models $\mathcal{M}$ with parameters $\boldsymbol{\theta}$, the procedure to infer the kinematic model of the door consists in: (1) fitting the parameters of all candidate models; (2) selecting the model that best describes the observed motion.
\subsection{Candidate Models}
When considering the set of doors that can be potentially operated by a service robot, their kinematic models belong to a few generic classes \citep{Sturm2012}. We have considered as candidate kinematic models a prismatic model, and a revolute model, shown in Figure \ref{Fig7}.
\subsubsection{Prismatic model}
Prismatic joints move along a single axis. Their motion describes a translation in the direction of a unitary vector $\mathbf{e}\in\mathbb{R}^{3}$ relative to a fixed origin, $\mathbf{a}\in\mathbb{R}^{3}$. The parameter vector is $\boldsymbol{\theta}=(\mathbf{a};\mathbf{e})$ with $k=6$.
\subsubsection{Revolute model}
Revolute joints rotate around an axis that impose a one-dimensional motion along a circular arc. It can be parametrized by the center of rotation $\mathbf{c}\in\mathbb{R}^{3}$, a radius $\mathbf{r}\in\mathbb{R}$, and the normal vector $\mathbf{n}=\mathbb{R}^{3}$ to the plane where the motion arc is contained. This results in a parameter vector $\boldsymbol{\theta}=(\mathbf{c};\mathbf{n};r)$ with $k=7$.
\begin{figure}
\centering
\includegraphics[width=0.98\linewidth]{figures/Fig7.png}
\caption{Prismatic and revolute candidate kinematic models.}
\label{Fig7}
\end{figure}
\vspace{-5mm}
\subsection{Model Fitting}
In the presence of noise and outliers, finding the parameter vector $\hat{\boldsymbol{\theta}}$ that maximizes the data likelihood is not trivial, as least square estimation is sensitive to outliers. The RANSAC algorithm has proven to be robust in this case, and can be modified in order to maximize the likelihood. This is the approach implemented by the Maximum Likelihood Estimation SAmple Consensus (MLESAC) algorithm \citep{Torr2000}. In this case, the score is defined by the likelihood of the consensus sample. Thus, for estimating the model vector parameter $\boldsymbol{\theta}$, the log-likelihood of a mixture model is maximized \citep{Zuliani2012}:
\begin{equation}
\hat{\boldsymbol{\theta}}=\argmax_{\boldsymbol{\theta}}{\mathcal{L}\left[e(\mathcal{D}\mid\mathcal{M},\boldsymbol{\theta})\right]}
\end{equation}
\vspace{-1.5mm}
\begin{equation}
{\small{}
\begin{split}
\hat{\mathcal{L}}=\sum_{j=1}^{N}\log\bigg(\gamma\cdot p\left[e\mathbf{(d}_{j},\mathcal{M},\boldsymbol{\hat{\theta}})\mid\mathrm{j^{th}element\equiv\:inlier}\right] \\ + \left(1-\gamma\right)p\left[e\mathbf{(d}_{j},\mathcal{M},\boldsymbol{\hat{\theta}})\mid\mathrm{j^{th}element\equiv outlier}\right]\bigg)
\end{split}
}
\end{equation} where $\gamma$ is the mixture coefficient, which is computed with Expectation Maximization. The first and second term correspond to the error distribution $e(\mathbf{d}_j,\mathcal{M},\hat{\boldsymbol{\theta}})$ of the inliers and the outliers, respectively. The error statistics of the inliers is modeled with a Gaussian. On the other hand, the error of the outliers is described with a uniform distribution.
\subsection{Model Selection}
Once all model candidates are fitted to the observations, the model that best explains the data has to be selected \citep{Sturm2011}.
Let $\mathcal{M}_{m}$ ($m=1,...,M$) be the set of candidate models, with vector parameters $\boldsymbol{\theta}_{\mathit{m}}$. Let $p\left(\boldsymbol{\theta}_{m}\vert \mathcal{M}_{m}\right)$ be the prior distribution for the parameters. Then, the posterior probability of a given model is proportional to \citep{Hastie2009}:
\begin{equation}
{\small{}
p\left(\mathcal{M}_{m}\mid\mathcal{D}\right)\propto\int p\left(\mathcal{D}\mid\boldsymbol{\theta}_{m},\mathcal{M}_{m}\right)p\left(\boldsymbol{\theta}_{m}\mid\mathcal{M}_{m}\right)d\mathbf{\boldsymbol{\theta}}_{m}
}
\end{equation}
In general, computing this probability is difficult. Applying the Laplace approximation and assuming a uniform prior for the models, it can be estimated in terms of the Bayesian Information Criterion (BIC):
\begin{equation}
{\small{}
p(\mathcal{M}_{m}\mid \mathcal{D})\approx\frac{\exp\left(-\frac{1}{2}\triangle BIC_{m}\right)}{\sum_{m=1}^{M}\exp\left(-\frac{1}{2}\triangle BIC_{m}\right)}
}
\end{equation} where:{\small{} $\triangle BIC_{m}=BIC_{m}-\min\left\{BIC_{m}\right\}^{M}_1$}, and:
\begin{equation}
BIC_{m}=-2\log\left[\mathcal{L}\left(\mathcal{D}\mid\mathcal{M}_{m},\boldsymbol{\hat{\theta}}_{m}\right)\right]+k\cdot\log N
\end{equation} The first term accounts for the likelihood of the fit, and the second term for the model complexity; smaller $BIC$ are preferred. Thus, model selection can be reduced to selecting the model with the lowest $BIC$:
\begin{equation}
\hat{\mathcal{M}}=\argmin_{\mathcal{M}}BIC(\mathcal{M})
\end{equation}
\subsection{Exploiting Prior Knowledge}
A robot operating in domestic environments can boost its performance by learning priors from previous experiences. A small set of representative models can be used as prior information to improve the model selection and parameter estimation. Suppose that the robot has previously encountered two doors. We have two observation sequences $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$, with $N_{1}$ and $N_{2}$ samples. We must choose then between two distinct models $\mathcal{M}_{1}$ and $\mathcal{M}_{2}$ or a joint model $\mathcal{M}_{1+2}$:
\begin{equation}
p\left(\mathcal{M}_{1+2}\mid\mathcal{D}_{1},\mathcal{D}_{2}\right)>p\left(\mathcal{M}_{1}\mid\mathcal{D}_{1}\right)\cdot p\left(\mathcal{M}_{2}\mid\mathcal{D}_{2}\right)
\end{equation}
Merging the new data with a previous model, the parameter vector is obtained from a larger dataset which leads to a better estimation. If we consider more than two trajectories, this should be repeated for all the possible combinations, becoming hard to compute. Thus, instead, we check if merging the new data with each learned model associated to the door class being opened gives a higher posterior. Finally, we pick the model with the highest posterior and record the new data, which will be used as prior knowledge for future doors. This approach is summarized in algorithm \ref{Algorithm2}.
\vspace{-1mm}
\begin{algorithm}
\small{
\caption{{\bf Model Selection Using Prior Knowledge} \label{Algorithm2}}
\small
\begin{algorithmic}
\renewcommand{\algorithmicrequire}{\textbf{Input:}}
\renewcommand{\algorithmicensure}{\textbf{Output:}}
\Require{New observed trajectory $\mathcal{D}_{\text{new}}=\left \{ \textbf{d}^{\text{new}}_j \right \}^{N}_1$ ;\\door class $c \in \left \{ \text{door},\; \text{cabinet door},\; \text{refrigerator door} \right \}$; \\previously observed trajectories $\mathbb{D}_c= \left \{ \mathcal{D}_s \right \} ^{S}_{1}$}
\Ensure{Best model $\mathbb{M}_{\text{best}}$and prior knowledge updated $\mathbb{D}_c$}
\State $\mathcal{M}_{\text{new}} \gets \text{Kinematic\_Model}\left(\mathcal{D}_{\text{new}}\right)$
\State $\mathbb{M}_{\text{best}}\gets\left \{ \mathcal{M}_{\text{new}} \right \}\:$, $\mathbb{D}_c\gets\mathbb{D}_c\cup\left\{\mathcal{D}_{\text{new}}\right\}$, $\: p_{\text{best}}\gets0$
\ForAll{$\mathcal{D}_s \in \mathbb{D}$}
\State $\mathcal{M}_{s} \gets \text{Kinematic\_Model} \left (\mathcal{D}_s \right )$
\State $\mathcal{M}_{\text{new+s}} \gets \text{Kinematic\_Model} \left ( \mathcal{D}_{\text{new}} \cup \mathcal{D}_{s} \right )$
\small
\If {\scriptsize{}$p\left(\mathcal{M}_{\text{new+s}}\vert\mathcal{D}_{\text{new}},\mathcal{D}_s\right)>p\left(\mathcal{M}_{\text{new}}\vert\mathcal{D}_{\text{new}}\right)p\left(\mathcal{M}_s\vert\mathcal{D}_s\right)\And p\left(\mathcal{M}_{\text{new+s}}\vert\mathcal{D}_{\text{new}},\mathcal{D}_s\right)>p_{\text{best}}$}
\small
\State $\mathbb{M}_{\text{best}} \gets \left\{ \mathcal{M}_{\text{new}},\;\mathcal{M}_s\right\}$
\State $\mathbb{D}_{c} \gets \left\{ \mathcal{D}_{1},\ldots,\mathcal{D}_{\text{new}}\cup\mathcal{D}_s\,\ldots,\mathcal{D}_S\right\}$
\State $p_{\text{best}} \gets p\left( \mathcal{M}_{\text{new+s}}\:\vert\:\mathcal{D}_{\text{new}},\mathcal{D}_s \right)$
\EndIf
\EndFor
\newline
\Return $\mathbb{M}_{\text{best}}$ and $\mathbb{D}_c$
\end{algorithmic}
}
\end{algorithm}
\vspace{-4mm}
\subsection{Learning from Human Demonstrations\label{sec:Learning-for-Human}}
If robots can learn from demonstration, this can boost the scale of the process, since non experts would be able to teach them \citep{Lee2017}. With our probabilistic framework, the only necessary input we need is a set of observations of the door's motion. Using our 6D-pose estimation approach, this tracking behavior can be efficiently achieved. Thus, the robot's prior knowledge can be provided by human demonstrations (Figure \ref{Fig8}).
\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{figures/Fig8.png}
\caption{Observations can be provided by executions of the task by a human teacher. In this case, the robot infers the motion of the cabinet is described by a prismatic model.}
\label{Fig8}
\end{figure}
\section{Execution of the Door Opening Motion}
\label{door_opening}
Computing the motion that enables a mobile manipulator to open a door is challenging because it requires tight coordination between arm and base. This makes the problem high-dimensional and thus hard to plan. In the previous section we have discussed how to learn the door kinematic model from observations of its motion. In order to achieve full autonomy, although observations are not provided before-hand, the robot must also be able to operate the previously unseen door. In this section we will discuss how these issues can be addressed through a suitable motion planning framework and an effective door opening strategy.
\vspace{-3mm}
\subsection{Task Space Region (TSR)}
Task Space Region is a constrained manipulation planning framework presented in \citep{Berenson2011b}. The authors propose a specific constraint representation, that has been developed for planning paths for manipulators with end-effector pose constraints. The framework unifies an efficient constraint representation, constraint satisfaction strategies, and a sampling-based planner, to create a state-of-the-art whole-body manipulation planning algorithm. Once the end-effector pose restrictions are specified in terms of a TSR, the algorithm finds a path that lies in the constraints manifold. To define a TSR, three elements are required:
\begin{itemize}
\item $\textbf{\textit{T}}^{o}_w$: Transform between the origin reference frame $o$ and the TSR frame $w$.
\item$\textbf{\textit{T}}^{w}_e$: End-effector offset transform.
\item$\textbf{\textit{B}}^{w}$: $6\times2$ matrix that defines the end-effector constraints, expressed in the TSR reference frame
\end{itemize}
\begin{equation}
\left(\textbf{\textit{B}}^{w}\right)^T=\left(\begin{array}{cccccc}
x_{\min} & y_{\min} & z_{\min} & \varphi_{\min} & \theta_{\min} & \psi_{\min} \\
x_{\max} & y_{\max} & z_{\max} & \varphi_{\max} & \theta_{\max} & \psi_{\max}
\end{array}\right)
\end{equation} where the first three columns bound the allowable translation along the $x$, $y$ and $z$ axes, and the last three columns bound the allowable translation assuming the Roll-Pitch-Yaw angle convention. Thus, the end-effector constraints for the considered kinematic models can be easily specified as follows.
\subsubsection{TSR Represenation for Prismatic Doors}
By fitting a prismatic model to the observations we can estimate the axis along which the door moves. If this axis is determined, we can specify the TSR reference frame as shown in Figure \ref{Fig9}, and define the end-effector pose constraints as:
\begin{equation}
\left(\textbf{\textit{B}}^{w}\right)^T=\left(\begin{array}{cccccc}
\;0\;\; & 0\;\; & -d\; & 0\;\; & 0\;\; & 0\;\; \\
0\;\; & 0\;\; & \;0\;\; & 0\;\; & 0\;\; & 0\;\;
\end{array}\right)
\end{equation}
\begin{figure}
\centering
\includegraphics[width=1.0\linewidth]{figures/Fig9.png}
\caption{TSR representation for operating prismatic doors. The $x$, $y$ and $z$ axis of each reference frame are red, green and blue respectively.}
\label{Fig9}
\end{figure}
\subsubsection{TSR Representation for Revolute Doors}
In the case of fitting a revolute model to the observations, we can estimate the center of rotation, the radius and the normal axis. With these parameters, we can specify the TSR reference frame as shown in Figure \ref{Fig10}, and define the end-effector pose constraints as:
\begin{equation}
\left(\textbf{\textit{B}}^{w}\right)^T=\left(\begin{array}{cccccc}
\;0\;\; & 0\;\; & 0\;\; & -\varphi\; & 0\;\; & 0\;\; \\
0\;\; & 0\;\; & 0\;\; & \;0\;\; & 0\;\; & 0\;\;
\end{array}\right)
\end{equation}
\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{figures/Fig10.png}
\caption{TSR representation for operating revolute doors. The $x$, $y$ and $z$ axis of each reference frame are red, green and blue respectively.}
\label{Fig10}
\end{figure}
\subsection{Door Opening Procedure}
\begin{figure}
\centering
\includegraphics[width=0.96\linewidth]{figures/Fig11.png}
\caption{Adaptive door opening procedure scheme. The robot opens the door following these steps iteratively.}
\label{Fig11}
\end{figure}
For defining the TSR reference frame, observations of the door motion are also required. Instead of using visual perception, door motion can also be inferred by direct actuation. Once the handle is grasped, the position of the end-effector directly corresponds with the position of the handle. As a result, the robot can make observations of the door motion by solving its forward kinematics. Thus, $\mathcal{D}$ can be obtained by sampling the trajectory. We execute the door opening motion repeating iteratively a series of sequential steps, shown in Figure \ref{Fig11}. After each iteration, we re-estimate the kinematic model of the door and its parameters adding the new observations to $\mathcal{D}$.
To start the opening process, when no observations are available, we make the initial guess that the model is prismatic. Using a compliant controller, the robot's end-effector trajectory is also driven by the forces exerted by the door, adapting its motion to the true model. Thus, a certain error margin is allowed, enabling the robot to operate the door correctly even if the estimation is biased, when only a few observations have been acquired.
\begin{figure*}
\centering
\includegraphics[width=1.0\linewidth]{figures/Fig12.png}
\caption{On top, a series of pictures of the HSR robot grasping different handles, starting from various relative positions. Below, the estimated grasping pose for the handles in the scene, as well as the corresponding detections provided by our CNN. The estimated grasping pose is illustrated through the red, green and blue axis ($x$, $y$ and $z$ respectively). Note the end-effector reference frame is shown at the HSR gripper.}
\label{Fig12}
\vspace{-2mm}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=1.0\linewidth]{figures/Fig13.png}\caption{(a) The posterior of the revolute model vs the number of observations. In the legend, the doors true models are indicated. The means of the executions are displayed as continuous lines. The shaded areas represent a margin of two standard deviations. Next to the plot, the evolution of the posterior along the opening trajectory is shown graphically. (b) Evolution of the revolute posterior mean against the number of observations. The legend indicates the true model of the doors being opened and the predominant prior during the realization.}
\label{Fig13}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=1.0\linewidth]{figures/Fig14.png}\caption{The HSR robot successfully opens different types of doors, without a priori knowledge of their kinematic model.}
\label{Fig14}
\end{figure*}
\vspace{-4mm}
\section{Experimental Evaluation}
\label{experiments}
\vspace{-3mm}
In order to validate experimentally the proposed door operation framework, we implemented it on the Toyota HSR robot, a robot designed to provide assistance. It is equipped with an omni-directional wheeled base, an arm with 4 degrees-of-freedom, a lifting torso and a two-fingered gripper as end-effector. In this work, we take advantage of the sensorial feedback provided by a 6-axis force sensor, located on the wrist, and a RGB-D camera, located on its head. We conducted a series of real-world experiments to test the performance of the presented grasping pose estimator and the proposed kinematic model inference process. We tested the latter in two different scenarios: with and without exploiting prior knowledge. To assess the robustness of our framework, we used different doors such as cabinet, refrigerator, and room doors with their variety of handles. Video demonstrations are available in the following \href{https://www.youtube.com/watch?v=LbDfKPpxEss}{\underline{hyperlink}}.
\subsection{Grasping Pose Estimation}
\vspace{-3mm}
For evaluating the performance of the grasping pose estimation, we focused on accuracy and speed. Regarding the door and handle detection model, due to the limited range of different doors available in the laboratory, its accuracy is best assessed, as discussed in Section \ref{detection}, by computing the mAP on the test set, with a wide variety of doors and handles. The resulting mAP of the selected model, as well as some reference values for comparison are shown in Table \ref{tab:1}. We can see that our model's mAP is just a $10\%$ lower. This performance value is close to that obtained by state-of-the-art object detectors in high-quality image datasets.
Qualitatively, testing the model in the laboratory, the available doors and handles were effectively detected from different viewpoints. Given a successful detection, the algorithm always computed the grasping pose of the handles present in the image correctly. By solving the IK, if the handle was located within the reachable workspace, the robot was always able to grasp the handle. Using an Nvidia Geforce GTX 1080 GPU, we obtained a computation rate of $6$fps. This shows an efficient behavior of the presented real-time grasping pose estimator. In Figure \ref{Fig12} we show a series of pictures that illustrate how the HSR robot reaches the handle in different scenarios, after inferring the grasping pose with our method. An effective grasping is achieved from several starting positions and types of doors. We also show some examples of the estimated goal pose from RGB-D data. We can observe that is accurately located in the observed point cloud for all the handles simultaneously.
\vspace{-3mm}
\renewcommand{\arraystretch}{1.5}
\begin{table}[h]
\centering
\caption{mAP comparison}
\label{tab:1}
\begin{tabular}{||c||c||}
\cline{2-2}
\multicolumn{1}{c||}{} & mAP \\
\hline
YOLO on COCO dataset & 55\% \\
\hline
YOLO on VOC 2012 & 58\% \\
\hline
Our model & 45\% \\
\hline
\end{tabular}
\end{table}
\subsection{Kinematic Model Inference}
\vspace{-3mm}
In order to evaluate the door kinematic model inference process, when no prior knowledge is available, we opened three different types of doors ten times: a drawer, a room and a refrigerator door. The task of the robot was to grasp the handle and open the door while it learned its kinematic model. The robot succeeded $26$ times out of $30$ trials ($87\%$). All four failures were due to the gripper slipping from the door knob, most likely caused by the design of the gripper which is not very suitable to manipulate this kind of objects. No errors were observed during the model learning.
We also studied the convergence of the estimators versus the number of training samples. We considered ten successful openings for each of the two considered kinematic models. Results are shown in Figure \ref{Fig13}(a). During the task, the evolution of the candidate posterior model was evaluated against the number of observations. It can be seen that the posterior probability for both cases converges towards the true model as the number of observations increases. When few observations are acquired, the probability oscillates around $0.5$, which is consistent with considering equal priors. However, they soon diverge from this value, showing an effective behavior regarding the decision criterion. A more convergent behavior is visible in the case of a revolute door. This is due to the difference in complexity between both models. When a prismatic door is opened, the revolute model can fit the data, which does not happen in the opposite case.
Finally, we analyze our approach for exploiting prior knowledge. We reproduced the same experiments when no prior knowledge was available but for three different situations: when the prior is predominantly revolute or prismatic and when both are balanced. Results are shown in Figure \ref{Fig13}(b). It can be observed that the behavior of the posterior depends on the predominant prior. In the case it matches the true model, the posterior converges quickly. If the prior is balanced, the behavior depends on the true model. When few new observations are available, the posterior tends to converge to the simplest model which is prismatic. This is reasonable, since the trajectory is very similar for both models at this point but the complexity is penalized. However, at a relatively low number of observations, the posterior rapidly converges to the true model proving, therefore, an improvement in performance. Finally, in the case the prior does not match the true model, the behavior is symmetric for both doors. At the beginning, the observations converge with the predominant prior model. However, when the number of observations is sufficiently large, they converge towards the true model. In Figure \ref{Fig14} we show a series of pictures that illustrate how the HSR successfully opens different doors. Combining the proposed probabilistic approach, with the TSR constrained manipulation framework, the robot is able to operate doors autonomously in a previously unknown environment.
\section{Conclusion}
\label{conclusions}
In this work, our objective is to push the state-of-the-art towards achieving autonomous door operation. The door opening task involves a series of challenges that have to be addressed. In this regard, we have discussed the detection of doors and handles, the handle grasp, the handle unlatch, the identification of the door kinematic model, and the planning of the constrained opening motion.
The problem of rapidly grasping door handles, leads to the first paper contribution. A novel algorithm to estimate the required end-effector grasping pose for multiple handles simultaneously, in real-time, based on RGB-D has been proposed. We have used a CNN, providing reliable results, and efficient point cloud processing to devise a high-performance algorithm, which proved robust and fast in the conducted experiments. Then, in order to operate the door reliably and independently of its kinematic model, we have devised a probabilistic framework for inferring door models from observations at run time, as well as for learning from robot experiences and from human demonstrations. By combining the grasp and model estimation processes with a TSR robot motion planner, we achieved a reliable operation for various types of doors.
Our desire is to extend this work to include more general and complex kinematic models \citep{Barragan2014,Hoefer2014}. This would enable robots, not only to achieve robust door operations, but would ultimately achieve general articulated object manipulation. Furthermore, the use of non-parametric models, such as Gaussian processes, would allow the representation of even more complex mechanisms. Also, we would like to explore in more depth the possibility of integrating our system in a general Learning from Demonstration (LfD) framework.
\bibliographystyle{abbrvnat}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,458 |
Q: could not find module 'ember-resources' am trying to build an web app with ember and in the process of making a request to server and receiving a response and for that i used resources from ember-resource
yet it always popping the error module not found ember-resources
the js code
import { use, resource } from 'ember-resources';
import { tracked } from '@glimmer/tracking';
class RequestState {
@tracked value;
@tracked error;
get isPending() {
return !this.error && !this.value;
}
}
export default class RoomselectController extends Controller {
@service router;
@use request = resource(({ on }) => {
const mobile = '123123123';
const state = new RequestState();
$.ajax({
url: 'My',
method: 'GET',
dataType: 'json',
data: { mobile },
success: (response) => state.value = response;,
error: (xhr, status, error) => state.error = `${status}: ${xhr.statusText}`,
});
return state;
});
get result() {
return this.request.value || [];
}
}
i installed ember-resource using
ember install ember-resources
also done npm install ember-resources
still showing the same module not found errro what to do?
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,853 |
Q: Google Analytics: How to generate a unique visitor's User ID? AND add it to the Tracker code According to google...I need to generate my own user ID's
And then add ga('set', 'userId', {{USER_ID}}); to my Tracker code...
(function(i,s,o,g,r,a,m){i['GoogleAnalyticsObject']=r;i[r]=i[r]||function(){
(i[r].q=i[r].q||[]).push(arguments)},i[r].l=1*new Date();a=s.createElement(o),
m=s.getElementsByTagName(o)[0];a.async=1;a.src=g;m.parentNode.insertBefore(a,m)
})(window,document,'script','//www.google-analytics.com/analytics.js','ga');
ga('create', 'UA-XXXXXXXX-X', 'auto');
ga('send', 'pageview');
Question 1) How to generate my own user ID?
Right now my code looks like (include ('google-analytics.php')) this:
(function(i,s,o,g,r,a,m){i['GoogleAnalyticsObject']=r;i[r]=i[r]||function(){
(i[r].q=i[r].q||[]).push(arguments)},i[r].l=1*new Date();a=s.createElement(o),
m=s.getElementsByTagName(o)[0];a.async=1;a.src=g;m.parentNode.insertBefore(a,m)
})(window,document,'script','//www.google-analytics.com/analytics.js','ga');
ga('create', '<?=$seo['analytics'];?>', 'auto');
ga('send', 'pageview');
How do I get php (or javascript) to generate a unique id, by site visitor (NOT by clicks)?
In other words, if the same visitor clicks on several pages within the site, I want the ID to remain the same ID and not chage with every click (or refresh).
I tried $seo['userID'] = uniqid(); but it changes with every click or refresh.
Question 2) Where to insert it, within the tracker code?
I have no idea where to place ga('set', 'userId', {{<?=$seo['userID'];?>}}); within all this...
(function(i,s,o,g,r,a,m){i['GoogleAnalyticsObject']=r;i[r]=i[r]||function(){
(i[r].q=i[r].q||[]).push(arguments)},i[r].l=1*new Date();a=s.createElement(o),
m=s.getElementsByTagName(o)[0];a.async=1;a.src=g;m.parentNode.insertBefore(a,m)
})(window,document,'script','//www.google-analytics.com/analytics.js','ga');
ga('create', '<?=$seo['analytics'];?>', 'auto');
ga('send', 'pageview');
...tracker code.
AND if this ga('set', 'userId', {{<?=$seo['userID'];?>}}); is the right way to code it, or this ga('set', 'userId', <?=$seo['userID'];?>});, or this ga('set', 'userId', <?=$seo['userID'];?>);, etc...
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,331 |
{"url":"http:\/\/www.koreascience.or.kr\/article\/JAKO201713059661669.page","text":"# Fundamental Small-signal Modeling of Li-ion Batteries and a Parameter Evaluation Using Levy's Method\n\n\u2022 Zhang, Xiaoqiang (School of Information and Electronics, Beijing Institute of Technology) ;\n\u2022 Zhang, Mao (School of Information and Electronics, Beijing Institute of Technology) ;\n\u2022 Zhang, Weiping (School of Electronic and Information Engineering, North China University of Technology)\n\u2022 Accepted : 2017.01.11\n\u2022 Published : 2017.03.20\n\n#### Abstract\n\nThe fundamental small-signal modeling of lithium-ion (Li-ion) batteries and a parameter evaluation approach are investigated in this study to describe the dynamic behaviors of small signals accurately. The main contributions of the study are as follows. 1) The operational principle of the small signals of Li-ion batteries is revealed to prove that the sinusoidal voltage response of a Li-ion battery is a result of a sinusoidal current stimulation of an AC small signals. 2) Three small-signal measurement conditions, namely stability, causality, and linearity, are proved mathematically proven to ensure the validity of the frequency response of the experimental data. 3) Based on the internal structure and electrochemical operational mechanism of the battery, an AC small-signal model is established to depict its dynamic behaviors. 4) A classical least-squares curve fitting for experimental data, referred as Levy's method, are introduced and developed to identify small-signal model parameters. Experimental and simulation results show that the measured frequency response data fit well within reading accuracy of the simulated results; moreover, the small-signal parameters identified by Levy's method are remarkably close to the measured parameters. Although the fundamental and parameter evaluation approaches are discussed for Li-ion batteries, they are expected to be applicable for other batteries.\n\n#### Acknowledgement\n\nSupported by : Natural Science Foundation of China, Beijing Municipal Natural Science Foundation\n\n#### References\n\n1. J. Jang and J. Yoo, \"Equivalent circuit evaluation method of lithium polymer battery using bode plot and numerical analysis,\" IEEE Trans. Energy Convers., Vol. 26, No. 1, pp. 290-298, Mar. 2011. https:\/\/doi.org\/10.1109\/TEC.2010.2089796\n2. B. Scrosati and J. Garche, \"Lithium batteries: Status, prospects and future,\" Journal of Power Sources, Vol. 195, No. 9, pp. 2419-2430, May 2010. https:\/\/doi.org\/10.1016\/j.jpowsour.2009.11.048\n3. X. Hu, R. Xiong, and B. Egardt, \"Model-based dynamic power assessment of lithium-ion batteries considering different operating conditions,\" IEEE Trans. Ind. Informat., Vol. 10, No. 3, pp. 1948-1959, Aug. 2014. https:\/\/doi.org\/10.1109\/TII.2013.2284713\n4. X. Hu, S. Li, and H. Peng, \"A comparative study of equivalent circuit models for Li-ion batteries,\" Journal of Power Sources, Vol. 198, No. 15, pp. 359-367, Jan. 2012. https:\/\/doi.org\/10.1016\/j.jpowsour.2011.10.013\n5. B. Pattipati, C. Sankavaram, and K. R. Pattipati, \"System identification and estimation framework for pivotal automotive battery management system characteristics,\" IEEE Trans. Syst, Man, Cybern, Part C (Applications and Reviews), Vol. 41, No. 6, pp. 869-884, Nov. 2011. https:\/\/doi.org\/10.1109\/TSMCC.2010.2089979\n6. M. Chen and G. A. Rincon-Mora, \"Accurate electrical battery model capable of predicting runtime and I-V performance, \" IEEE Trans. Energy Convers., Vol. 21, No. 2, pp. 504-511, Jun. 2006. https:\/\/doi.org\/10.1109\/TEC.2006.874229\n7. H. Blanke, O. Bohlen, S. Buller, R. W. De Doncker, B. Fricke, A. Hammouche, D. Linzen, M. Thele, and D. U. Sauer, \"Impedance measurements on lead-acid batteries for state-of-charge, state-of-health and cranking capability prognosis in electric and hybrid electric vehicles,\" Journal of Power Sources, Vol. 144, No. 2, pp. 418-425, Jun. 2005. https:\/\/doi.org\/10.1016\/j.jpowsour.2004.10.028\n8. F. Yusivar, H. Haratsu, M. Sato, S. Wakao, K. Kondo, K. Matsuoka, and T. Kawamatsu, \"The modeling of lead-acid battery by frequency-response characteristics,\" IEEJ Trans. Fundamentals and Material, Vol. 122, No. 8, pp. 715-721, Aug. 2002. https:\/\/doi.org\/10.1541\/ieejfms.122.715\n9. S. Buller, M. Thele, R. W. De Doncker, and E. Karden, \"Impedance based simulation models of super capacitors and Li-ion batteries for power electronic applications,\" in Industry Applications Conference, Oct. 2003.\n10. R. S. Robinson, \"On-line battery testing: a reliable method for determining battery health?\"IEEE Telecommunications Energy Conf., pp. 654-661, Oct. 1996.\n11. J. O'M. Bockris, A. K. Reddy, and M. Gamboa-Aldeco, Modern Electrochemistry 2A:Fundamentals of Electrodics, 2nd edition, Kluwer Academic, Chap. 7, pp. 1138, 2000.\n12. Product catalog of zhicheng champion, http:\/\/www.zhicheng-champion.com\/index.html, 2016.\n13. C. L. Philips, J. M. Parr, and E. Riskin, Signals, Systems, and Transforms, 3th edition, Prentice Hall, Chap. 3, pp. 111-112, 2004.\n14. The definition of the cross-correlation, https:\/\/en.wikipedia.org\/wiki\/Cross-correlationk.\n15. M. Urbain, M. Hinaje, S. Rael, B. Davat, and P. Desprez, \"Energetical modeling of lithium-ion batteries including electrode porosity effects,\" IEEE Trans. Energy Convers., Vol. 25, No. 3, pp. 862-872, Sep. 2010. https:\/\/doi.org\/10.1109\/TEC.2010.2049652\n16. B. E. Conway, Electrochemical Supercapacitors: Scientific Fundamentals and Technological Applications, Kluwer Academic\/Plenum Press, Chap. 2, pp. 91-92, 1999.\n17. J. O'M. Bockris, A. K. Reddy, and M. Gamboa-Aldeco, Modern Electrochemistry:Fundamentals of Electrodics, 2nd edition, Kluwer Academic, Chap. 7, pp. 1133-1134, 2000.\n18. Autolab Application Note EIS03 Electrochemical Impedance Spectroscopy (EIS) Part 3- Data Analysis, ${\\Omega}$ Metrohm Autolab B.V., Jul. 2011.\n19. E. C. Levy, \"Complex-curve fitting,\" IRE Trans. Autom. Control, Vol. AC-4, No. 1, pp. 37-43, May 1959. https:\/\/doi.org\/10.1109\/TAC.1959.6429401\n20. J. Proakis and D. Manolakis, Digital Signal Processing: Principle, Algorithm, and Applications, 3th edition, Prentice-Hall, 1996.\n21. C. Sanathanan and J. Koerner, \"Transfer function synthesis as a ratio of two complex polynomials,\" IEEE Trans. Autom. Control, Vol. 8, No. 1, pp.56-58, Jan. 1963. https:\/\/doi.org\/10.1109\/TAC.1963.1105517","date":"2020-08-05 02:58:50","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6229910850524902, \"perplexity\": 11164.932032829427}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-34\/segments\/1596439735906.77\/warc\/CC-MAIN-20200805010001-20200805040001-00364.warc.gz\"}"} | null | null |
using base::android::ConvertUTF8ToJavaString;
namespace cronet {
static const int kReadBufferSize = 32768;
// Explicitly register static JNI functions.
bool CronetUrlRequestAdapterRegisterJni(JNIEnv* env) {
return RegisterNativesImpl(env);
}
static jlong CreateRequestAdapter(JNIEnv* env,
jobject jurl_request,
jlong jurl_request_context_adapter,
jstring jurl_string,
jint jpriority) {
CronetURLRequestContextAdapter* context_adapter =
reinterpret_cast<CronetURLRequestContextAdapter*>(
jurl_request_context_adapter);
DCHECK(context_adapter);
GURL url(base::android::ConvertJavaStringToUTF8(env, jurl_string));
VLOG(1) << "New chromium network request_adapter: "
<< url.possibly_invalid_spec();
CronetURLRequestAdapter* adapter =
new CronetURLRequestAdapter(context_adapter, env, jurl_request, url,
static_cast<net::RequestPriority>(jpriority));
return reinterpret_cast<jlong>(adapter);
}
CronetURLRequestAdapter::CronetURLRequestAdapter(
CronetURLRequestContextAdapter* context,
JNIEnv* env,
jobject jurl_request,
const GURL& url,
net::RequestPriority priority)
: context_(context),
initial_url_(url),
initial_priority_(priority),
initial_method_("GET"),
load_flags_(context->default_load_flags()) {
DCHECK(!context_->IsOnNetworkThread());
owner_.Reset(env, jurl_request);
}
CronetURLRequestAdapter::~CronetURLRequestAdapter() {
DCHECK(context_->IsOnNetworkThread());
}
jboolean CronetURLRequestAdapter::SetHttpMethod(JNIEnv* env,
jobject jcaller,
jstring jmethod) {
DCHECK(!context_->IsOnNetworkThread());
std::string method(base::android::ConvertJavaStringToUTF8(env, jmethod));
// Http method is a token, just as header name.
if (!net::HttpUtil::IsValidHeaderName(method))
return JNI_FALSE;
initial_method_ = method;
return JNI_TRUE;
}
jboolean CronetURLRequestAdapter::AddRequestHeader(JNIEnv* env,
jobject jcaller,
jstring jname,
jstring jvalue) {
DCHECK(!context_->IsOnNetworkThread());
std::string name(base::android::ConvertJavaStringToUTF8(env, jname));
std::string value(base::android::ConvertJavaStringToUTF8(env, jvalue));
if (!net::HttpUtil::IsValidHeaderName(name) ||
!net::HttpUtil::IsValidHeaderValue(value)) {
return JNI_FALSE;
}
initial_request_headers_.SetHeader(name, value);
return JNI_TRUE;
}
void CronetURLRequestAdapter::DisableCache(JNIEnv* env, jobject jcaller) {
DCHECK(!context_->IsOnNetworkThread());
load_flags_ |= net::LOAD_DISABLE_CACHE;
}
void CronetURLRequestAdapter::SetUpload(
scoped_ptr<net::UploadDataStream> upload) {
DCHECK(!context_->IsOnNetworkThread());
DCHECK(!upload_);
upload_ = upload.Pass();
}
void CronetURLRequestAdapter::Start(JNIEnv* env, jobject jcaller) {
DCHECK(!context_->IsOnNetworkThread());
context_->PostTaskToNetworkThread(
FROM_HERE, base::Bind(&CronetURLRequestAdapter::StartOnNetworkThread,
base::Unretained(this)));
}
void CronetURLRequestAdapter::FollowDeferredRedirect(JNIEnv* env,
jobject jcaller) {
DCHECK(!context_->IsOnNetworkThread());
context_->PostTaskToNetworkThread(
FROM_HERE,
base::Bind(
&CronetURLRequestAdapter::FollowDeferredRedirectOnNetworkThread,
base::Unretained(this)));
}
void CronetURLRequestAdapter::ReadData(JNIEnv* env, jobject jcaller) {
DCHECK(!context_->IsOnNetworkThread());
context_->PostTaskToNetworkThread(
FROM_HERE, base::Bind(&CronetURLRequestAdapter::ReadDataOnNetworkThread,
base::Unretained(this)));
}
void CronetURLRequestAdapter::Destroy(JNIEnv* env, jobject jcaller) {
DCHECK(!context_->IsOnNetworkThread());
context_->PostTaskToNetworkThread(
FROM_HERE, base::Bind(&CronetURLRequestAdapter::DestroyOnNetworkThread,
base::Unretained(this)));
}
void CronetURLRequestAdapter::PopulateResponseHeaders(JNIEnv* env,
jobject jurl_request,
jobject jheaders_list) {
DCHECK(context_->IsOnNetworkThread());
const net::HttpResponseHeaders* headers = url_request_->response_headers();
if (headers == nullptr)
return;
void* iter = nullptr;
std::string header_name;
std::string header_value;
while (headers->EnumerateHeaderLines(&iter, &header_name, &header_value)) {
base::android::ScopedJavaLocalRef<jstring> name =
ConvertUTF8ToJavaString(env, header_name);
base::android::ScopedJavaLocalRef<jstring> value =
ConvertUTF8ToJavaString(env, header_value);
Java_CronetUrlRequest_onAppendResponseHeader(
env, jurl_request, jheaders_list, name.obj(), value.obj());
}
}
base::android::ScopedJavaLocalRef<jstring>
CronetURLRequestAdapter::GetHttpStatusText(JNIEnv* env, jobject jcaller) const {
DCHECK(context_->IsOnNetworkThread());
const net::HttpResponseHeaders* headers = url_request_->response_headers();
return ConvertUTF8ToJavaString(env, headers->GetStatusText());
}
base::android::ScopedJavaLocalRef<jstring>
CronetURLRequestAdapter::GetNegotiatedProtocol(JNIEnv* env,
jobject jcaller) const {
DCHECK(context_->IsOnNetworkThread());
return ConvertUTF8ToJavaString(
env, url_request_->response_info().npn_negotiated_protocol);
}
jboolean CronetURLRequestAdapter::GetWasCached(JNIEnv* env,
jobject jcaller) const {
DCHECK(context_->IsOnNetworkThread());
return url_request_->response_info().was_cached;
}
int64 CronetURLRequestAdapter::GetTotalReceivedBytes(JNIEnv* env,
jobject jcaller) const {
DCHECK(context_->IsOnNetworkThread());
return url_request_->GetTotalReceivedBytes();
}
// net::URLRequest::Delegate overrides (called on network thread).
void CronetURLRequestAdapter::OnReceivedRedirect(
net::URLRequest* request,
const net::RedirectInfo& redirect_info,
bool* defer_redirect) {
DCHECK(context_->IsOnNetworkThread());
DCHECK(request->status().is_success());
JNIEnv* env = base::android::AttachCurrentThread();
cronet::Java_CronetUrlRequest_onRedirect(
env, owner_.obj(),
ConvertUTF8ToJavaString(env, redirect_info.new_url.spec()).obj(),
redirect_info.status_code);
*defer_redirect = true;
}
void CronetURLRequestAdapter::OnResponseStarted(net::URLRequest* request) {
DCHECK(context_->IsOnNetworkThread());
if (MaybeReportError(request))
return;
JNIEnv* env = base::android::AttachCurrentThread();
cronet::Java_CronetUrlRequest_onResponseStarted(env, owner_.obj(),
request->GetResponseCode());
}
void CronetURLRequestAdapter::OnReadCompleted(net::URLRequest* request,
int bytes_read) {
DCHECK(context_->IsOnNetworkThread());
if (MaybeReportError(request))
return;
if (bytes_read != 0) {
JNIEnv* env = base::android::AttachCurrentThread();
base::android::ScopedJavaLocalRef<jobject> java_buffer(
env, env->NewDirectByteBuffer(read_buffer_->data(), bytes_read));
cronet::Java_CronetUrlRequest_onDataReceived(env, owner_.obj(),
java_buffer.obj());
} else {
JNIEnv* env = base::android::AttachCurrentThread();
cronet::Java_CronetUrlRequest_onSucceeded(env, owner_.obj());
}
}
void CronetURLRequestAdapter::StartOnNetworkThread() {
DCHECK(context_->IsOnNetworkThread());
VLOG(1) << "Starting chromium request: "
<< initial_url_.possibly_invalid_spec().c_str()
<< " priority: " << RequestPriorityToString(initial_priority_);
url_request_ = context_->GetURLRequestContext()->CreateRequest(
initial_url_, net::DEFAULT_PRIORITY, this);
url_request_->SetLoadFlags(load_flags_);
url_request_->set_method(initial_method_);
url_request_->SetExtraRequestHeaders(initial_request_headers_);
url_request_->SetPriority(initial_priority_);
if (upload_)
url_request_->set_upload(upload_.Pass());
url_request_->Start();
}
void CronetURLRequestAdapter::FollowDeferredRedirectOnNetworkThread() {
DCHECK(context_->IsOnNetworkThread());
url_request_->FollowDeferredRedirect();
}
void CronetURLRequestAdapter::ReadDataOnNetworkThread() {
DCHECK(context_->IsOnNetworkThread());
if (!read_buffer_.get())
read_buffer_ = new net::IOBufferWithSize(kReadBufferSize);
int bytes_read = 0;
url_request_->Read(read_buffer_.get(), read_buffer_->size(), &bytes_read);
// If IO is pending, wait for the URLRequest to call OnReadCompleted.
if (url_request_->status().is_io_pending())
return;
OnReadCompleted(url_request_.get(), bytes_read);
}
void CronetURLRequestAdapter::DestroyOnNetworkThread() {
DCHECK(context_->IsOnNetworkThread());
delete this;
}
bool CronetURLRequestAdapter::MaybeReportError(net::URLRequest* request) const {
DCHECK_NE(net::URLRequestStatus::IO_PENDING, url_request_->status().status());
DCHECK_EQ(request, url_request_);
if (url_request_->status().is_success())
return false;
int net_error = url_request_->status().error();
VLOG(1) << "Error " << net::ErrorToString(net_error)
<< " on chromium request: " << initial_url_.possibly_invalid_spec();
JNIEnv* env = base::android::AttachCurrentThread();
cronet::Java_CronetUrlRequest_onError(
env, owner_.obj(), net_error,
ConvertUTF8ToJavaString(env, net::ErrorToString(net_error)).obj());
return true;
}
} // namespace cronet
| {
"redpajama_set_name": "RedPajamaGithub"
} | 623 |
Q: Как в codeception для определенной группы тестов использовать свой config? Добрый день.
Вопрос следующий есть множество acceptance тестов.
и есть несколько тестов в acceptance/init которые по плану должны запускаться отдельно и производить первоначальную инициализацию нашего продукта. Вопрос в том как при выполнении тестов acceptance/init подгрузить свой файл конфигурации acceptance.suite.yml? и свой файл фикстур fixture.app.yml ?
A: Если это совсем отдельные тесты требующие отдельных настроек, то как Вам вариант создать новый Suite со своим конфигом? К примеру:
codecept generate:suite init
Далее настроить конфиг сюита и переместить туда все тесты из Init.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,012 |
Size of Presidential Suite is 151 sq.m. and it consists of large studio with sitting area for eight guests, kitchen area with dining table for six, wardrobe, spacious bedroom, large bathroom and sauna. You have also private office and meeting room. There is separate entry as well as toilet for your guests.
In the studio you can light on ethanol fireplace, feel yourself comfortable on the Leather Chaise Lounge Relax Reclining Chair and send your photos to the big 55 inches flat TV.
Presidential Suite maintain our exceptional standard of comfort and feature all equipment, described in other rooms.
In addition, welcome amenities upon arrival, turndown service, complimentary in-room breakfast, use of Sport & Relaxation area and one-way transfer service form Airport or parking. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,859 |
{"url":"https:\/\/www.physicsforums.com\/threads\/triple-integral.329897\/","text":"# Triple Integral\n\nGiven the triple integral $$\\int\\int\\int_{G}$$ xyz dV\nWhere G is the region bounded by x=1, y=x, y=0, z=0, z=2.\nHow do I evaluate it.\n\ndaniel_i_l\nGold Member\nFirst of all, is there any other constraint on x? I'll assume that x>=0.\nDo you know how to change triple integrals into single-variable integrals?\nThe general idea is to first evaluate the integral while pretending that 2 of the variables are constant. Then you use that result to integrate over the other variables.\n\nIn this case it might be easiest to first evaluate the integral in the triangle 0<=x<=1 and\n0<=y<=x while assuming that z is constant. Then integrate that result with z as a variable from 0 to 2. The triangle can be evaluated in a similar way. In other words:\n$$I = \\int^{2}_{0}(\\int^{1}_{0}(\\int^{1-x}_{0} xyz dy)dx)dz$$\n\nCheers. I got an answer of 1\/3","date":"2020-12-05 18:49:38","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9471283555030823, \"perplexity\": 494.51160632887746}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-50\/segments\/1606141748276.94\/warc\/CC-MAIN-20201205165649-20201205195649-00640.warc.gz\"}"} | null | null |
REBELS (2009-2011) #27
REBELS (2009-2011)
Save an additional 10%. Learn How
The behemoth known as Tribulus is unleashed in a last-ditch gambit to save team leader Vril Dox from the mind-slavery of Starro the Conqueror. Will the combined might of Lobo, Adam Strange, Starfire and Captain Comet be enough to stop Starro's latest bid for conquest?
Tony Bedard
Pencils:
Claude St. Aubin
Inks:
Scott Hanna
Colored by:
Richard Horie Tanya Horie
Cover by:
Aaron Lopresti
Superhero Military
The future is now! Brainiac 2 is a hunted man. His cosmic police force safeguarded over 80 worlds until a mysterious adversary suddenly seized control. Now Vril Dox must recruit a new team to win back his command and free countless billions!
R.E.B.E.L.S. rises from the ashes of the L.E.G.I.O.N! Fugitives from the very organization they founded, Vril Dox, Lobo, Stealth now roam as the galaxy on the run from a crime they were forced to commit!
Doom Patrol (2009-2011)
The world's strangest super heroes are back in all new thrilling (and strange) adventures! When it comes to the bizarre, forget the Justice League and call the Doom Patrol!
Outsiders (2007-2011)
Welcome to the Outsiders, an elite group originally brought together by Batman, the Dark Knight Detective, for unique missions away from the public eye. Batman and the Outsiders must fight even dirtier than their enemies...and risk becoming the supervillains they despise!
LEGION (1989-1994)
Vril Dox, ancestor of the Legion of Super-Heroes' Brainiac 5, conceives a plan for a new intergalactic police force to fill the void left by the death of the Green Lantern Corps. Forged in the fires of the INVASION, a rag-tag team of paranormal aliens will grow into a tough, no-nonsense law enforcement organization with a cosmic scope. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,260 |
{"url":"https:\/\/www.sandyman.xyz\/products\/phone-case-for-xiaomi-redmi-7a-matte-black-cover-silicon-tpu-soft-cases-back-cover-for-xiomi-xiaomi-redmi-7a-7-a-a7-redmi7a-case","text":"# Phone Case For Xiaomi Redmi 7A Matte Black Cover Silicon TPU Soft Cases Back Cover For Xiomi Xiaomi Redmi 7A 7 A A7 Redmi7A Case\n\n\\$0.01\nColor:\nDescription\n\nPhone Case For Xiaomi Redmi 7A Matte Black Cover Silicon TPU Soft Cases Back Cover For Xiomi Xiaomi Redmi 7A 7 A A7 Redmi7A Case\n\nProduct Description? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ??\n\n?\n\nReturn & Warranty\n\u2022 100% Secure payment with SSL Encryption.\n\u2022 If you're not 100% satisfied, let us know and we'll make it right.\nShipping Policies\n\u2022 Orders ship within 5 to 10 business days.\n\u2022 Tip: Buying 2 products or more at the same time will save you quite a lot on shipping fees.","date":"2021-04-18 20:44:43","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.969444751739502, \"perplexity\": 703.3676747504561}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-17\/segments\/1618038860318.63\/warc\/CC-MAIN-20210418194009-20210418224009-00498.warc.gz\"}"} | null | null |
\section{Introduction}\label{sec:introduction}
Crowdsourcing, which has become popular in recent years, is a process that requires completion of specific tasks by crowd workers.
In crowdsourced single-answer multiple-choice questions, workers are asked to select one answer out of multiple candidates for each given question.
Such a scheme is used, for instance, in
annotating named entities for microblogs~\cite{Finin2010},
sentiment classification on political blogs~\cite{Hsueh2009},
and image tagging~\cite{Lin2015}.
For the purpose of quality control, crowdsourcing systems usually assign multiple workers to the same questions and then aggregate their answers using some rule or algorithm.
A critical fact here is that the workers often have different levels of expertise, skills, and motivation;
some workers may guess the correct answers with their rich knowledge, while others may answer almost randomly to get a reward without any effort.
Let us call the workers of the former and latter type \emph{experts} and \emph{non-experts}, respectively.
When only a few experts are overwhelmed by a large number of non-experts,
the most intuitive answer aggregation rule, \emph{majority voting}, often fails to acquire the correct answers.
Consider a small example in Table~\ref{table:typical}.
There are six workers $w_1,\dots,w_6$ assigned to eight questions $q_1,\dots,q_8$.
For each question, the candidate answers are A, B, C, D, and E.
Among these workers, $w_1$, $w_2$, and $w_3$ are experts who almost always give the correct answers,
while the others, $w_4$, $w_5$, and $w_6$, are non-experts who give random answers.
Let us apply the majority voting to their answers.
The majority answer for $q_1$ is D, which is the correct answer.
However, the majority answer for $q_8$ is C, which is not the correct answer.
In addition, we need tie-breaking for $q_3$ because B and D get the same number of votes.
It is very likely that as the fraction of non-experts increases, the quality of the majority voting answers deteriorates.
\begin{table}[t]
\caption{A small example of crowdsourced single-answer multiple-choice questions.}\label{table:typical}
\centering
\setlength{\tabcolsep}{4pt}
{\renewcommand{\arraystretch}{1}
\scalebox{1}{
\begin{tabular}{rc||*{8}{c}}
&$\mathcal{W}\backslash\mathcal{Q}$&$q_1$&$q_2$&$q_3$&$q_4$&$q_5$&$q_6$&$q_7$&$q_8$\\\cline{2-10}\addlinespace[1pt]
&\lower3pt\hbox{\scriptsize\shortstack{correct\\answer}}
&D&C&D&E&B&C&A&E\\\addlinespace[1pt]\cline{2-10}
\ldelim\{{3}{27pt}[Experts]&$w_1$&D&C&D&E&B&C&A&E\\
&$w_2$&D&C&B&E&B&C&A&C\\
&$w_3$&D&C&D&D&B&C&A&E\\[2pt]
\ldelim\{{3}{45pt}[Non-experts]&$w_4$&C&B&A&A&E&C&D&B\\
&$w_5$&A&B&E&A&B&E&E&C\\
&$w_6$&C&A&B&E&B&B&A&C
\end{tabular}}}
\end{table}
Various answer aggregation algorithms exist in addition to the majority voting (see Related Work).
However, most of these algorithms implicitly strengthen the majority answers and therefore fail to provide the true answers when the majority answers are incorrect.
To overcome this issue, Li et al.~\cite{LBK2017} recently proposed a sophisticated answer aggregation algorithm for such a hard situation.
More specifically, they introduced the notion of \emph{hyper questions}, each of which is a set of single questions.
Their algorithm applies the majority voting (or other existing answer aggregation algorithms) for the hyper questions
and then decodes the results to votes on individual questions.
Finally, it applies the majority voting again to obtain the final answers.
The results of their experiments demonstrate that their algorithm outperforms existing algorithms.
\subsection{Our Contribution}
In this study, we further investigate the above-mentioned hard situation.
Our contribution can be summarized as follows:
\begin{enumerate}
\leftskip=4pt
\itemsep=1pt
\parsep=1pt
\item We introduce a graph-mining-based efficient algorithm that accurately extracts the set of experts;
\item We propose two types of answer aggregation algorithms based on the above experts extraction algorithm;
\item We provide a theoretical justification of our proposed algorithms;
\item We conduct thorough computational experiments using synthetic and real-world datasets to evaluate the performance of our proposed algorithms.
\end{enumerate}
\paragraph{First result.}
To design a powerful answer aggregation algorithm for the above hard situation, it is crucial to extract reliable experts from the crowd workers.
In the example above, if we recognize that $w_1$, $w_2$, and $w_3$ are experts,
we can obtain the correct answers for all questions by simply applying the majority voting to the answers of the experts.
The fundamental observation we use is as follows: as the experts almost always give the correct answers, each pair of experts frequently gives the same answer to a question.
Let us now consider constructing an edge-weighted complete undirected graph in which each vertex corresponds to a worker
and each edge weight represents the agreement rate of the answers of two workers.
From the above observation, it is very likely that there is a dense component consisting of the experts, which we call the \emph{expert core}.
Note that the formal definition of the expert core will be given in Section~\ref{sec:model}.
Figure~\ref{fig:network} depicts an edge-weighted graph constructed from the example in Table~\ref{table:typical}.
As can be seen, experts $w_1$, $w_2$, and $w_3$ form a dense component.
Although, in this example, we simply set the edge weight to the number of same answers of two workers, we will use more suitable values in our algorithm.
To extract the expert core, we use a well-known dense subgraph extraction algorithm called the \emph{peeling algorithm}~\cite{Asahiro+_00}, which removes the most unreliable worker one by one.
\begin{figure}[htbp]
\centering
\scalebox{1.2}{
\begin{tikzpicture}[thick,xscale=1.6,yscale=1.2,
vertex/.style={circle,fill=blue!10,draw=blue,font=\small,inner sep=1pt},
label/.style={font=\sffamily\small,rectangle,fill=white,opacity=0.8, text opacity=1,inner sep=2pt,rounded corners}]
\draw[fill=red!10,draw=red!50,line width=31pt,rounded corners] (0,0) -- (1,1) -- (1,-1) -- cycle;
\draw[fill=red!10,draw=red!10,line width=30pt,rounded corners] (0,0) -- (1,1) -- (1,-1) -- cycle;
\node[text=red,font=\small] at (0,1) {Experts};
\node[vertex] at (0,0) (1) {$w_1$};
\node[vertex] at (1,1) (2) {$w_2$};
\node[vertex] at (1,-1) (3) {$w_3$};
\node[vertex] at (2,1) (4) {$w_4$};
\node[vertex] at (2,-1) (5) {$w_5$};
\node[vertex] at (3,0) (6) {$w_6$};
\draw[line width=3.0pt] (1) -- (2) node[pos=.7,label] {$6$};
\draw[line width=3.5pt] (1) -- (3) node[pos=.7,label] {$7$};
\draw[line width=0.5pt] (1) -- (4) node[pos=.2,label] {$1$};
\draw[line width=0.5pt] (1) -- (5) node[pos=.2,label] {$1$};
\draw[line width=1.5pt] (1) -- (6) node[pos=.2,label] {$3$};
\draw[line width=2.5pt] (2) -- (3) node[pos=.3,label] {$5$};
\draw[line width=0.5pt] (2) -- (4) node[pos=.5,label] {$1$};
\draw[line width=1.0pt] (2) -- (5) node[pos=.7,label] {$2$};
\draw[line width=2.5pt] (2) -- (6) node[pos=.8,label] {$5$};
\draw[line width=0.5pt] (3) -- (4) node[pos=.7,label] {$1$};
\draw[line width=0.5pt] (3) -- (5) node[pos=.5,label] {$1$};
\draw[line width=1.0pt] (3) -- (6) node[pos=.8,label] {$2$};
\draw[line width=1.0pt] (4) -- (5) node[pos=.3,label] {$2$};
\draw[line width=0.5pt] (4) -- (6) node[pos=.2,label] {$1$};
\draw[line width=1.0pt] (5) -- (6) node[pos=.2,label] {$2$};
\end{tikzpicture}
}
\caption{An edge-weighted graph constructed from the example in Table~\ref{table:typical}.} \label{fig:network}
\end{figure}
\paragraph{Second result.}
Based on the expert core extraction algorithm, we propose two types of answer aggregation algorithms.
The first one incorporates the expert core into existing answer aggregation algorithms such as the majority voting.
Indeed, once we extract the expert core, we can apply existing algorithms to the answers of the workers in the expert core.
The second one utilizes the information pertaining to the reliability of workers provided by the expert core extraction algorithm,
which is quite effective when the task is very hard.
\paragraph{Third result.}
We first demonstrate that the expert core is very unlikely to contain a non-expert if the number of questions is sufficiently large (Theorem~\ref{thm:asymp_excore}).
We then prove that the majority voting is asymptotically correct if there are only experts (Theorem~\ref{thm:mv_good}),
but not reliable if there are a large number of non-experts (Theorem~\ref{thm:mv_bad}).
Theorems~\ref{thm:asymp_excore} and \ref{thm:mv_good} provide a theoretical justification for the first type of our algorithm.
In fact, combining these two theorems, we see that
if the number of questions and the number of workers in the expert core are sufficiently large,
our proposed algorithm (i.e., the majority voting among the workers in the expert core)
gives the correct answer with high probability.
On the other hand, Theorem~\ref{thm:mv_bad} provides the limitation of the majority voting,
which implies that it is quite important to exclude non-experts when we use the majority voting.
\paragraph{Fourth result.}
We conduct thorough computational experiments using synthetic and real-world datasets to evaluate our proposed algorithms.
To simulate the hard situation in this study,
we use six datasets recently collected by Li et al.~\cite{LBK2017} as real-world datasets,
all of which have difficult heterogeneous-answer questions that require specialized knowledge.
We demonstrate that the expert core counterparts of existing answer aggregation algorithms perform much better than their original versions.
Furthermore, we show that our novel algorithm based on the information of the reliability of workers outperforms the other algorithms particularly when the task is quite hard.
\subsection{Related Work}
To date, a large body of work has been devoted to developing algorithms that estimate the quality of workers~\cite{DS1979,Whitehill2009,WBBP2010,RYZ+2010,KOS2011,WJ2011,Bachrach2012,DDC2012,KG2012,LPI2012,LLO+2012,ZPBM2012,AYL+2014,LLG+2014,VGK+2014,FLO+2015,MLL+2015,LBK2017,ZLLSC2017,LBK2018}.
In most existing work, the quality of workers and correct answers are estimated by an iterative approach comprising the following two steps:
(i) infer the correct answers based on the quality of workers estimated and (ii) estimate the quality of workers based on the correct answers inferred.
For example, Whitehill et al.~\cite{Whitehill2009} modeled each worker's ability and each task's difficulty,
and then designed a probabilistic approach, which they called \emph{GLAD} (Generative model of Labels, Abilities, and Difficulties).
Finding a dense component in a graph is a fundamental and well-studied task in graph mining.
A typical application of dense subgraph extraction is to identify components
that have some special role or possess important functions in the underlying system represented by a graph.
Examples include communities or spam link farms extraction in Web graphs~\cite{Dourisboure+_07,Gibson+_05},
identification of molecular complexes in protein--protein interaction graphs~\cite{Bader_Hogue_03},
and expert team formation in collaboration graphs~\cite{Bonchi+_14,Tsourakakis+_13},
The peeling algorithm~\cite{Asahiro+_00} is known to be effective in various optimization problems for dense subgraph extraction
(e.g., the densest subgraph problem and its variations~\cite{Andersen_Chellapilla_09,Charikar_00,Kawase_Miyauchi_17,Miyauchi_Kakimura_18,Khuller_Saha_09} as well as other related problems~\cite{Tsourakakis+_13,Miyauchi_Kawase_15}).
\section{Model}\label{sec:model}
An instance of our problem is a tuple $(\mathcal{W},\mathcal{Q},\mathcal{C},\mathcal{L})$, where each component is defined as follows:
There is a finite set of workers $\mathcal{W}=\{w_1,\ldots,w_n\}$ and a finite set of questions $\mathcal{Q}=\{q_1,\ldots,q_m\}$.
Each question $q$ has a set of candidate answers $\mathcal{C}_q=\{c^q_1,\dots,c^q_s\}$ ($s\geq 2$).
Suppose that, for each question $q$, worker $w$ answers $l_{wq}\in\mathcal{C}_q$ and let $\mathcal{L}=(l_{wq})_{w\in\mathcal{W},\,q\in\mathcal{Q}}$.
Our task is to estimate the unknown correct answers to the questions.
Suppose that, among the workers $\mathcal{W}=\{w_1,\dots,w_n\}$, there are $n_{\mathrm{ex}}$ experts $\mathcal{E}~(\subseteq \mathcal{W})$ who give the correct answer with probability $p_{\mathrm{ex}}~(>1/s)$,
and the other $n_{\mathrm{non}}~(=n-n_{\mathrm{ex}})$ workers are non-experts who give an answer independently and uniformly at random.
If an expert makes a mistake, she selects a wrong answer independently and uniformly at random.
Thus, for a question $q\in\mathcal{Q}$ with a correct answer $a_q\in\mathcal{C}_q$ and an answer $c\in\mathcal{C}_q$, it holds that
\begin{align*}
\Pr[l_{wq}=c]=
\begin{cases}
p_{\mathrm{ex}}&(c=a_q,~w\in\mathcal{E}),\\
\frac{1-p_{\mathrm{ex}}}{s-1}&(c\ne a_q,~w\in\mathcal{E}),\\
1/s&(w\in\mathcal{W}\setminus\mathcal{E}).
\end{cases}
\end{align*}
It should be noted that, by showing the candidate answers in random order for each worker,
we can handle some biases (e.g., some non-expert workers always choose the first candidate of answers) using this model.
Let us consider the probability that a pair of workers $u,v\in\mathcal{W}$ $(u\ne v)$ gives the same answer for a question $q\in\mathcal{Q}$.
If at least one of $u$ and $v$ is a non-expert, then we have \(\Pr[l_{uq}=l_{vq}]=1/s\).
On the other hand, if both workers are experts, then \(\Pr[l_{uq}=l_{vq}]=\frac{(p_{\mathrm{ex}}-1/s)^2}{1-1/s}+\frac{1}{s}\),
which is strictly larger than $1/s$.
For each pair of workers $u,v\in\mathcal{W}$,
let $\tau(u,v)$ be the number of questions such that $u$ and $v$ give the same answer,
that is, $\tau(u,v)=|\{q\in \mathcal{Q}\mid l_{uq}=l_{vq}\}|$.
Here, if $\Pr[l_{uq}=l_{vq}]=p$, then $u$ and $v$ give the same answers for at least $\tau(u,v)$ questions
with probability $\sum_{i=\tau(u,v)}^{m}\binom{m}{i} p^i(1-p)^{m-i}$.
For a given $p\in(0,1)$, a subset of workers $W\subseteq\mathcal{W}$ is called a \emph{$\theta$-expert set} with respect to $p$ if
\begin{align*}
\prod_{v\in W\setminus\{u\}}\sum_{i=\tau(u,v)}^{m}\binom{m}{i} p^i(1-p)^{m-i}\le\theta
\end{align*}
holds for all $u\in W$.
Intuitively, a $\theta$-expert set with small $\theta$ is a set of workers that is very unlikely to contain a non-expert.
Let $\theta(W)$ be the minimum threshold such that $W$ is a $\theta$-expert set, that is,
\begin{align*}
\theta(W)=\max_{u\in W}\prod_{v\in W\setminus\{u\}}\sum_{i=\tau(u,v)}^{m}\binom{m}{i} p^i(1-p)^{m-i}.
\end{align*}
Then, we employ $W\subseteq\mathcal{W}$ that minimizes $\theta(W)$ as the estimated set of experts.
We refer to such a set as the \emph{expert core} (with respect to $p$).
As will be shown in Theorem~\ref{thm:asymp_excore}, the expert core is very unlikely to contain a non-expert if the number of questions is sufficiently large.
\section{Algorithms}\label{sec:algorithm}
In this section, we design an algorithm to compute the expert core, and then propose two types of answer aggregation algorithms.
Our expert core extraction algorithm first constructs an edge-weighted complete undirected graph that represents the similarity of the workers in terms of their answers.
Then, it extracts a dense component in the graph using the peeling algorithm~\cite{Asahiro+_00}.
\subsection{Peeling Algorithm}
We first revisit the peeling algorithm.
The algorithm iteratively removes a vertex with the minimum weighted degree in the current graph until we are left with only one vertex.
Let $G=(V,E,\omega)$ be an edge-weighted graph.
For $S\subseteq V$ and $v\in S$,
let $d_S(v)$ denote the weighted degree of $v$ in the induced subgraph $G[S]$,
that is, $d_S(v)=\sum_{e=\{u,v\}\in E:\,u\in S}\omega(e)$.
Then, the procedure can be summarized in Algorithm~\ref{alg:peeling}.
Note that the algorithm here returns $S_i\in \{S_1,\dots, S_{|V|}\}$ that maximizes $d_{S_i}(v_i)$,
although there are other variants depending on the problem at hand~\cite{Charikar_00,Kawase_Miyauchi_17,Miyauchi_Kawase_15,Tsourakakis+_13}.
Algorithm~\ref{alg:peeling} can be implemented to run in $O(|E|+|V|\log |V|)$ time.
\begin{algorithm2e}[t]
\SetKwInOut{Input}{Input}\Input{ Edge-weighted graph $G=(V,E,\omega)$}
\SetKwInOut{Output}{Output}\Output{ $S\subseteq V$ ($k$-core with maximum $k$)}
\caption{Peeling algorithm}\label{alg:peeling}
$S_{|V|}\leftarrow V$\;
\For{$i\leftarrow |V|,\dots,2$}{
\mbox{Find $v_i\in \mathop{\rm argmin}_{v\in S_i} d_{S_i}(v)$ and $S_{i-1}\leftarrow S_i\setminus\{v_i\}$\;}
}
\Return $S_i\in \{S_1,\dots, S_{|V|}\}$ that maximizes $d_{S_i}(v_i)$\;
\end{algorithm2e}
Algorithm~\ref{alg:peeling} indeed produces the \emph{$k$-core decomposition} of graphs.
For $G=(V,E,\omega)$ and a positive real $k$, a subset $S\subseteq V$ is called a \emph{$k$-core}
if $S$ is a maximal subset in which every vertex $v\in S$ has a weighted degree of at least $k$ in the induced subgraph $G[S]$.
Note that a $k$-core is unique for a fixed $k$.
The $k$-core decomposition reveals the hierarchical structure of $k$-cores in a graph
and is particularly focused on finding the $k$-core with maximum $k$.
Algorithm~\ref{alg:peeling} is suitable for this scenario; in fact, it is evident that the algorithm returns the $k$-core with maximum $k$.
\subsection{Expert Core Extraction Algorithm}
Here, we present an algorithm to compute the expert core.
In particular, we explain the construction of an edge-weighted graph,
which represents the similarity of the workers in terms of their answers.
In our algorithm, we set $p$ to the average agreement probability, that is,
\begin{align*}
p=\frac{1}{m}\sum_{q\in\mathcal{Q}}\textstyle\bigl|\bigl\{\{u,v\}\in \binom{\mathcal{W}}{2}\mid l_{uq}=l_{vq}\bigr\}\bigr|\big/\binom{n}{2},
\end{align*}
where $\binom{\mathcal{W}}{2}=\bigl\{\{u,v\}\mid u,v\in \mathcal{W},\ u\neq v\bigr\}$,
and extract the expert core with respect to this $p$ via the peeling algorithm.
We construct a complete graph $(\mathcal{W},\binom{\mathcal{W}}{2})$ with weight $\gamma(u,v)=-\log \sum_{i=\tau(u,v)}^{m}\binom{m}{i}p^i(1-p)^{m-i}$, where recall that $\tau(u,v)=|\{q\in \mathcal{Q}\mid l_{uq}=l_{vq}\}|$.
Then, we compute the $k$-core with maximum $k$ for the edge-weighted graph $(\mathcal{W},\binom{\mathcal{W}}{2},\gamma)$ using Algorithm~\ref{alg:peeling}.
As a result, we can obtain a set of workers $W\subseteq \mathcal{W}$ such that the following value is maximized:
\begin{align*}
\min_{u\in W}\sum_{v\in W\setminus\{u\}}\gamma(u,v)
&=\min_{u\in W} \left(-\log\prod_{v\in W\setminus\{u\}}\sum_{i=\tau(u,v)}^{m}\binom{m}{i}p^i(1-p)^{m-i}\right)
=-\log\theta(W).
\end{align*}
As $-\log x$ is monotone decreasing, the obtained set minimizes $\theta(W)$ and hence is the expert core.
Note that we can construct the edge-weighted graph $(\mathcal{W},\binom{\mathcal{W}}{2},\gamma)$ in $O(n^2m)$ time
and Algorithm~\ref{alg:peeling} for the graph runs in $O(\binom{n}{2}+n\log n)=O(n^2)$ time.
Thus, we have the following theorem.
\begin{theorem}
The expert core can be found in $O(n^2m)$ time.
\end{theorem}
\subsection{Answer Aggregation Algorithms}
Once we extract the expert core, we can apply existing answer aggregation algorithms (e.g., the majority voting)
to the answers of the workers in the expert core,
which implies the expert core counterparts of the existing algorithms.
In addition, we propose a novel answer aggregation algorithm.
The peeling algorithm produces the ordering that represents the reliability of the workers.
With this ordering, we can obtain a majority-voting-based algorithm as follows.
At the beginning, we obtain the ordering of workers using the peeling algorithm for $(\mathcal{W},\binom{\mathcal{W}}{2},\gamma)$.
Then we pick the most reliable pair of workers, that is, the pair left until the second last round of the peeling algorithm.
For each question, if the two workers select the same answer, then we employ this answer.
If the two workers select different answers, then we add the next most reliable worker one by one according to the ordering until two of them give the same answer.
Our algorithm, which we call \textsf{Top-2}, is summarized in Algorithm~\ref{alg:top2}.
Note that the algorithm runs in $O(n^2m)$ time, which is the same as that of the expert core extraction algorithm.
\begin{algorithm2e}[t]
\SetKwInOut{Input}{Input}\Input{ Worker set $\mathcal{W}$; Question set $\mathcal{Q}$; Answer set $\mathcal{L}$}
\SetKwInOut{Output}{Output}\Output{ Estimated true answers $(z_q)_{q\in\mathcal{Q}}$}
\SetInd{0.4em}{0.8em}
\caption{\textsf{Top-2}}\label{alg:top2}
Calculate the average agreement probability $p$\;
Construct the edge-weighted graph $(\mathcal{W},\binom{\mathcal{W}}{2},\gamma)$\;
Let $S_1,\dots,S_n$ be the sets computed in Algorithm~\ref{alg:peeling}\;
\For{$q\in\mathcal{Q}$}{
\For{$i\leftarrow 2,\dots,n$}{
\lIf{$\exists c^*\in\mathcal{C}_q$ s.t. $|\{w\in W_{i}\mid l_{wq}=c^*\}|=2$}{
$z_q\leftarrow c^*$ and \textbf{break}
}
\lElseIf{$i=n$}{
$z_q\leftarrow l_{wq}$, where $\{w\}=S_1$
}
}
}
\Return $(z_q)_{q\in\mathcal{Q}}$\;
\end{algorithm2e}
\section{Theoretical Results}\label{sec:theoretical}
In this section, we provide a theoretical justification for the first type of our proposed algorithm.
The proofs can be found in Appendix~\ref{sec:omitted}.
Suppose that each worker gives answers according to the model described in Section~\ref{sec:model}.
The following theorem states that the expert core does not contain any non-expert with high probability
if the number of questions is sufficiently large.
\begin{theorem}\label{thm:asymp_excore}
Let $W^*\subseteq \mathcal{W}$ be the expert core.
If $n_{\mathrm{ex}}\ge 2$ and $m\ge \frac{2n^4\log\frac{n^2}{\epsilon}}{(p_{\mathrm{ex}}-1/s)^4}$ for $\epsilon>0$, then we have
$\Pr[W^*\subseteq \mathcal{E}]\ge 1-\epsilon$.
\end{theorem}
The next theorem states that the majority voting gives the correct answer with high probability
if the number of workers is sufficiently large and all of them are experts.
Let $\mathtt{MV}_q$ be the output of the majority voting for question $q\in\mathcal{Q}$.
\begin{theorem}\label{thm:mv_good}
If $n=n_{\mathrm{ex}}\ge \frac{2\log\frac{2s}{\epsilon}}{(p_{\mathrm{ex}}-1/s)^2}$ for $\epsilon>0$, then we have
$\Pr\left[\mathtt{MV}_q=a_q\right]\ge 1-\epsilon$ $(\forall q\in\mathcal{Q})$.
\end{theorem}
Finally, the following theorem states that,
the majority voting does not provide the correct answer with high probability if the number of non-experts is much larger than the number of experts.
\begin{theorem}\label{thm:mv_bad}
If $n_{\mathrm{non}}\ge \frac{n_{\mathrm{ex}}^2}{2\pi\epsilon^2}$ for $\epsilon>0$, then we have
$\Pr\left[\mathtt{MV}_q=a_q\right]\le \frac{1}{2}+\epsilon$ $(\forall q\in\mathcal{Q})$.
\end{theorem}
It should be noted that when the number of candidate answers is $2$,
even the random choice gives the correct answer with probability $1/2$.
Thus, the above theorem indicates that the majority voting is no better than the random choice if the number of non-experts is much larger than the number of experts.
\section{Experiments}
In this section, we report the results of computational experiments.
The objective of our experiments is to examine the performance of our proposed algorithms
from various aspects using both synthetic and real-world datasets.
The main component of our experiments comprises comparing our proposed answer aggregation algorithms with the following three existing algorithms:
\textsf{MV} (the majority voting), \textsf{GLAD}~\cite{Whitehill2009},
and \textsf{Hyper-MV}~\cite{LBK2017}.
As our proposed algorithms, we employ the expert core counterparts of the above three algorithms,
which we denote by \textsf{Ex-MV}, \textsf{Ex-GLAD}, and \textsf{Ex-Hyper-MV}, respectively,
in addition to \textsf{Top-2}.
\subsection{Datasets}
As for synthetic datasets, we generate a variety of instances
using the model described in Section~\ref{sec:model}.
Recall the following five parameters:
number of workers $n=|\mathcal{W}|$, number of questions $m=|\mathcal{Q}|$, number of candidate answers (for each question) $s$,
number of ground-truth experts $n_{\mathrm{ex}}$, and the probability $p_{\mathrm{ex}}$ that an expert gives the correct answer.
Throughout the experiments, we set $s=5$.
\begin{table}
\caption{Real datasets used in our experiments. The last column gives the best accuracy rate (i.e., the fraction of the number of correct answers) among the workers. }\label{tab:instance}
\centering
\scalebox{1}{
\renewcommand{\arraystretch}{1.0}
\begin{tabular}{l*{4}{@{\hspace{8mm}}r}}
\toprule
Dataset & $m$ & $n$ & $s$ & \textsc{Best}\\
\midrule
{\em Chinese } & 24 & 50 & 5 & 0.79 \\
{\em English } & 30 & 63 & 5 & 0.70 \\
{\em IT } & 25 & 36 & 4 & 0.84 \\
{\em Medicine } & 36 & 45 & 4 & 0.92 \\
{\em Pok\'{e}mon} & 20 & 55 & 6 & 1.00 \\
{\em Science } & 20 & 111 & 5 & 0.85 \\
\bottomrule
\end{tabular}
}
\end{table}
Table~\ref{tab:instance} summarizes six datasets that we use as real-world datasets.
They were recently collected by Li et al.~\cite{LBK2017}
using Lancers, a commercial crowdsourcing platform in Japan.
Consistent with the hard situation being addressed here,
these datasets have difficult heterogeneous-answer questions that require specialized knowledge.
In fact, as shown later, the majority voting performs poorly on these datasets.
Note that the classical datasets used in other previous work (e.g., \emph{Bluebird} and \emph{Dog} in image tagging~\cite{WBBP2010,ZPBM2012}, \emph{Web} in Web search relevance judging~\cite{ZPBM2012}, \emph{Price} in product price estimation~\cite{LIS2013}, and \emph{RTE} and \emph{Temp} in textual entailment recognition~\cite{Snow+_08})
usually have relatively easy homogeneous-answer questions,
which are not within the scope of this study.
During the collection of the above six datasets, all workers were asked to answer all questions.
This may not be the usual assumption in crowdsourcing but is effective in the current hard situation.
Indeed, if we can identify reliable experts by asking all workers to answer a small number of questions at an early stage,
then it is possible to ask only the identified experts to answer to the remaining questions,
which may reduce the overall cost.
This scenario has been studied in previous work~\cite{LBK2017,LZF2014}.
\subsection{Experts Extraction}
To evaluate the performance of our expert core extraction algorithm in terms of extracting reliable experts,
we conducted simulations using synthetic datasets.
We set the number of workers as $n$ to $20$ and the number of experts as $n_{\mathrm{ex}}$ to $4$.
In these settings, we generated two types of instances to simulate different situations.
In the first type, to investigate the effect of the number of questions,
we vary the number of questions $m$ from 5 to 50 (in increments of 5)
under a fixed value of $p_{\mathrm{ex}}=0.8$.
In the second type, to investigate the effect of the expertise of workers,
we vary the probability $p_{\mathrm{ex}}$ from 0.5 to 1.0 (in increments of 0.05)
under a fixed value of $m=25$.
As the sets of workers extracted by our peeling algorithm,
we employ the Top-2 pair
(i.e., a pair of workers left until the second last round of the peeling algorithm)
and the expert core.
To evaluate the quality of the set of workers at hand, we adopt the following two measures:
the \emph{precision} (number of ground-truth experts in the set divided by the size of the set)
and the \emph{recall} (number of ground-truth experts in the set divided by the number of all ground-truth experts).
The results are shown in Figure~\ref{fig:parameta}.
Each point corresponds to an average over 100 data realizations.
As can be seen, the precision of the expert core becomes better as the number of questions $m$ or the probability $p_{\mathrm{ex}}$ increases.
Moreover, the expert core is robust in terms of the recall;
in fact, even when $p_{\mathrm{ex}}$ is small (i.e., $p_{\mathrm{ex}} \leq 0.7$),
the expert core achieves an average recall of $0.8$.
The Top-2 pair achieves better precision in all parameter settings.
\begin{figure}[t]
\centering
\includegraphics[scale=.82]{extract2.pdf}
\caption{Results of experts extraction for synthetic datasets, where $n=20$ and $n_{\mathrm{ex}}=4$.}\label{fig:parameta}
\end{figure}
\subsection{Ordering Defined by Peeling}
The objective here is to demonstrate that the ordering of workers
defined by the peeling algorithm for $(\mathcal{W},\binom{\mathcal{W}}{2},\gamma)$ reflects the accuracy rate of the workers.
To this end, we perform the peeling algorithm for the real-world datasets.
The results are shown in Figure~\ref{fig:rorder}.
In the subfigures, the horizontal axis represents the ordering of workers defined by the peeling algorithm;
specifically, the peeling algorithm has removed the workers from right to left.
The vertical axis represents the accuracy rate of the workers.
The bars corresponding to the workers contained in the expert core are colored black,
while those corresponding to the other workers are colored gray.
\newcommand{5cm}{5cm}
\begin{figure}[t]
\centering
\subfigure{\label{fig:chinese_order}
\includegraphics[width=5cm]{expert_Chinese_deg-crop.pdf}
}
\subfigure{\label{fig:english_order}
\includegraphics[width=5cm]{expert_English_deg-crop.pdf}
}
\subfigure{\label{fig:it_order}
\includegraphics[width=5cm]{expert_IT_deg-crop.pdf}
}
\subfigure{\label{fig:medicine_order}
\includegraphics[width=5cm]{expert_Medicine_deg-crop.pdf}
}
\subfigure{\label{fig:pokemon_order}
\includegraphics[width=5cm]{expert_Pokemon_deg-crop.pdf}
}
\subfigure{\label{fig:science_order}
\includegraphics[width=5cm]{expert_Science_deg-crop.pdf}
}
\caption{Ordering defined by our peeling algorithm for real-world datasets.}\label{fig:rorder}
\end{figure}
As can be seen, workers with smaller order tend to have higher accuracy rates,
In addition, the expert core contains almost all workers with significantly high accuracy rates.
which demonstrates the reliability of our peeling algorithm.
In particular, for {\em Pok\'{e}mon} and {\em Science},
the ordering defined by the peeling algorithm correctly reproduces the ordering of the top five workers in terms of accuracy rates.
\begin{table*}[t!]
\caption{Results of answer aggregation algorithms for synthetic datasets, where $n=100$ and $m=50$. Each cell gives the average and standard deviation of the accuracy rate over 100 data realizations. For each setting, the best average accuracy rate
is written in bold.}\label{table:synthetic}
\centering
{\renewcommand{\arraystretch}{1.0}
\tabcolsep=3.5mm
\scalebox{1}{
\begin{tabular}{cccccccc}
\toprule
$(p_{\mathrm{ex}},n_{\mathrm{ex}})$ & \textsf{Top-2} & \textsf{Ex-MV} & \textsf{Ex-GLAD} & \textsf{Ex-Hyper-MV} & \textsf{MV} & \textsf{GLAD} & \textsf{Hyper-MV} \\
\midrule
\rule{0pt}{6mm} \raise1mm\hbox{$(0.7, 2)$} & \shortstack{{ \bf 0.519 }\\ {\scriptsize$\pm$0.218}} & \shortstack{0.285\\ {\scriptsize$\pm$0.079}} & \shortstack{0.277\\ {\scriptsize$\pm$0.098}} & \shortstack{0.276\\ {\scriptsize$\pm$0.079}} & \shortstack{0.273\\ {\scriptsize$\pm$0.058}} & \shortstack{0.264\\ {\scriptsize$\pm$0.092}} & \shortstack{0.272\\ {\scriptsize$\pm$0.062}} \\ \Xhline{0.1pt}
\rule{0pt}{6mm} \raise1mm\hbox{$(0.7, 4)$} & \shortstack{{ \bf 0.892 }\\ {\scriptsize$\pm$0.047}} & \shortstack{0.388\\ {\scriptsize$\pm$0.120}} & \shortstack{0.563\\ {\scriptsize$\pm$0.212}} & \shortstack{0.513\\ {\scriptsize$\pm$0.120}} & \shortstack{0.348\\ {\scriptsize$\pm$0.070}} & \shortstack{0.549\\ {\scriptsize$\pm$0.212}} & \shortstack{0.514\\ {\scriptsize$\pm$0.131}} \\ \Xhline{0.1pt}
\rule{0pt}{6mm} \raise1mm\hbox{$(0.7, 6)$} & \shortstack{{ \bf 0.911 }\\ {\scriptsize$\pm$0.040}} & \shortstack{0.624\\ {\scriptsize$\pm$0.247}} & \shortstack{0.886\\ {\scriptsize$\pm$0.073}} & \shortstack{0.782\\ {\scriptsize$\pm$0.247}} & \shortstack{0.426\\ {\scriptsize$\pm$0.066}} & \shortstack{0.865\\ {\scriptsize$\pm$0.083}} & \shortstack{0.755\\ {\scriptsize$\pm$0.106}} \\ \Xhline{0.1pt}
\rule{0pt}{6mm} \raise1mm\hbox{$(0.8, 2)$} & \shortstack{{ \bf 0.750 }\\ {\scriptsize$\pm$0.097}} & \shortstack{0.320\\ {\scriptsize$\pm$0.110}} & \shortstack{0.300\\ {\scriptsize$\pm$0.131}} & \shortstack{0.405\\ {\scriptsize$\pm$0.110}} & \shortstack{0.284\\ {\scriptsize$\pm$0.069}} & \shortstack{0.303\\ {\scriptsize$\pm$0.139}} & \shortstack{0.380\\ {\scriptsize$\pm$0.126}} \\ \Xhline{0.1pt}
\rule{0pt}{6mm} \raise1mm\hbox{$(0.8, 4)$} & \shortstack{{ \bf 0.956 }\\ {\scriptsize$\pm$0.026}} & \shortstack{0.783\\ {\scriptsize$\pm$0.259}} & \shortstack{0.896\\ {\scriptsize$\pm$0.138}} & \shortstack{0.866\\ {\scriptsize$\pm$0.259}} & \shortstack{0.373\\ {\scriptsize$\pm$0.069}} & \shortstack{0.814\\ {\scriptsize$\pm$0.160}} & \shortstack{0.813\\ {\scriptsize$\pm$0.095}} \\ \Xhline{0.1pt}
\rule{0pt}{6mm} \raise1mm\hbox{$(0.8, 6)$} & \shortstack{0.958\\ {\scriptsize$\pm$0.028}} & \shortstack{{ \bf 0.984 }\\ {\scriptsize$\pm$0.018}} & \shortstack{0.982\\ {\scriptsize$\pm$0.020}} & \shortstack{0.969\\ {\scriptsize$\pm$0.018}} & \shortstack{0.484\\ {\scriptsize$\pm$0.065}} & \shortstack{0.950\\ {\scriptsize$\pm$0.031}} & \shortstack{0.962\\ {\scriptsize$\pm$0.030}} \\ \Xhline{0.1pt}
\rule{0pt}{6mm} \raise1mm\hbox{$(0.9, 2)$} & \shortstack{{ \bf 0.879 }\\ {\scriptsize$\pm$0.048}} & \shortstack{0.356\\ {\scriptsize$\pm$0.172}} & \shortstack{0.384\\ {\scriptsize$\pm$0.214}} & \shortstack{0.702\\ {\scriptsize$\pm$0.172}} & \shortstack{0.290\\ {\scriptsize$\pm$0.067}} & \shortstack{0.353\\ {\scriptsize$\pm$0.180}} & \shortstack{0.704\\ {\scriptsize$\pm$0.146}} \\ \Xhline{0.1pt}
\rule{0pt}{6mm} \raise1mm\hbox{$(0.9, 4)$} & \shortstack{{ \bf 0.991 }\\ {\scriptsize$\pm$0.014}} & \shortstack{0.989\\ {\scriptsize$\pm$0.015}} & \shortstack{0.991\\ {\scriptsize$\pm$0.014}} & \shortstack{0.988\\ {\scriptsize$\pm$0.015}} & \shortstack{0.416\\ {\scriptsize$\pm$0.063}} & \shortstack{0.941\\ {\scriptsize$\pm$0.091}} & \shortstack{0.982\\ {\scriptsize$\pm$0.021}} \\ \Xhline{0.1pt}
\rule{0pt}{6mm} \raise1mm\hbox{$(0.9, 6)$} & \shortstack{0.989\\ {\scriptsize$\pm$0.014}} & \shortstack{{ \bf 0.998 }\\ {\scriptsize$\pm$0.005}} & \shortstack{0.998\\ {\scriptsize$\pm$0.006}} & \shortstack{0.996\\ {\scriptsize$\pm$0.005}} & \shortstack{0.550\\ {\scriptsize$\pm$0.073}} & \shortstack{0.984\\ {\scriptsize$\pm$0.016}} & \shortstack{0.997\\ {\scriptsize$\pm$0.007}} \\
\bottomrule
\end{tabular}
}}
\vskip 16pt
\caption{Results of answer aggregation algorithms for real-world datasets. Each cell gives the accuracy rate. The average and standard deviation over 100 runs are listed for the hyper-question-based algorithms. For each dataset, the best accuracy rate is written in bold.}\label{table:real}
\centering
{\renewcommand{\arraystretch}{1.0}
\tabcolsep=3.5mm
\scalebox{1}{
\begin{tabular}{ccccccccc}
\toprule
Dataset &\textsf{Top-2}& \textsf{Ex-MV} & \textsf{Ex-GLAD} & \textsf{Ex-Hyper-MV} & \textsf{MV} & \textsf{GLAD} & \textsf{Hyper-MV} \\
\midrule
Chinese &{\bf 0.750} &0.667 & 0.667 & 0.700 {\footnotesize($\pm$0.026)}& 0.625 & 0.542 & 0.696 {\footnotesize($\pm$0.042)} \\
English&{\bf 0.733}&0.400 & 0.533 & 0.490 {\footnotesize($\pm$0.049)} & 0.467 & 0.567 & 0.542 {\footnotesize($\pm$0.060)} \\
IT&0.800&0.760 & 0.800 & 0.802 {\footnotesize($\pm$0.008)} & 0.760 & 0.720 & {\bf 0.828} {\footnotesize($\pm$0.018)} \\
Medicine& 0.944&{\bf 0.972 }& {\bf 0.972} & 0.951 {\footnotesize($\pm$0.022)} & 0.667 & 0.694 & 0.848 {\footnotesize($\pm$0.019)} \\
Pok\'{e}mon&{\bf 1.000}& {\bf 1.000} & {\bf 1.000} & {\bf 1.000} {\footnotesize($\pm$0.000)} & 0.650 & 0.850 & {\bf 1.000} {\footnotesize($\pm$0.000)} \\
Science& {\bf 0.900} &0.650 & 0.650 & 0.603 {\footnotesize($\pm$0.025)} & 0.550 & 0.550 & 0.606 {\footnotesize($\pm$0.018)} \\
\bottomrule
\end{tabular}
}}
\end{table*}
\subsection{Comparison with Existing Algorithms}
Using both synthetic and real-world datasets,
we compare the performance of our proposed answer aggregation algorithms,
that is, \textsf{Top-2}, \textsf{Ex-MV}, \textsf{Ex-GLAD}, and \textsf{Ex-Hyper-MV},
with existing algorithms, that is, \textsf{MV}, \textsf{GLAD}, and \textsf{Hyper-MV}.
We first explain the details of \textsf{GLAD} and \textsf{Hyper-MV}.
\textsf{GLAD} is a method that takes into account not only the worker expertise, denoted by $\alpha$,
but also the difficulty of each question, denoted by $\beta$,
We set $\alpha \sim \mathcal{N}(1,1)$ and $\beta \sim \mathcal{N}(1,1)$ as in Li et al.~\cite{LBK2017}.
It is known that \textsf{GLAD} runs in $O(nmsT)$ time, where $T$ is the number of iterations to converge.
\textsf{Hyper-MV} is a method that applies the majority voting
to the hyper questions rather than the original individual questions.
Because the number of possible hyper questions may be too large,
Li et al.~\cite{LBK2017} suggested to apply the following random sampling procedure for $r$ times:
(i) shuffle the order of all single questions uniformly at random to generate a permutation and
(ii) from this permutation, pick every $k$ single questions from the beginning of the queue to generate hyper questions as long as they can be picked.
Then, the overall time complexity of \textsf{Hyper-MV} is given by $O(rmn)$.
We performed the sampling procedure with $k=5$ for $r=100$ times, as suggested by Li et al.~\cite{LBK2017}.
Table~\ref{table:synthetic} shows the results for synthetic datasets.
We list the results for nine settings
in which the probability $p_{\mathrm{ex}}$ varies in $\{0.7, 0.8, 0.9\}$ and $n_{\mathrm{ex}}$ varies in $\{2, 4, 6\}$.
We set the number of workers $n$ to 100 and the number of questions $m$ to 50.
As can be seen, each expert core counterpart achieves better performance than the original algorithm.
In particular, \textsf{Ex-MV} significantly improves the performance of \textsf{MV}.
\textsf{Top-2} outperforms the other algorithms particularly when the problem is quite difficult,
although \textsf{Ex-MV} performs better when the problem is relatively easy.
Table~\ref{table:real} summarizes the results for the six real-world datasets.
As can be seen, for all datasets except {\em IT}, our proposed algorithms achieve the best performance.
In fact, for almost all datasets, the performance of the existing algorithms is improved by using the expert core.
Among our proposed algorithms, \textsf{Top-2} provides the best performance;
in particular, for {\em English} and {\em Science}, the accuracy rate of \textsf{Top-2} is even higher than those of the other algorithms.
It should be noted that, for {\em English}, {\em Medicine}, and {\em Science},
the accuracy rate of \textsf{Top-2} is strictly higher than the best accuracy rate among workers (presented in Table~\ref{tab:instance}),
which emphasizes the power of answer aggregation in crowdsourcing.
According to the trend observed in synthetic datasets,
it is very likely that the high performance of \textsf{Top-2} stems from the high fraction of non-experts in real-world datasets.
\section{Conclusion}
In this study, we addressed the answer aggregation task in crowdsourcing.
Specifically, we focused on a hard situation wherein a few experts are overwhelmed by a large number of non-experts.
To design powerful answer aggregation algorithms for such situations,
we introduced the notion of \emph{expert core}, which is a set of workers that is very unlikely to contain a non-expert.
We then designed a graph-mining-based efficient algorithm that exactly computes the expert core.
Based on the expert core extraction algorithm, we proposed two types of answer aggregation algorithms.
The first one incorporates the expert core into existing answer aggregation algorithms.
The second one utilizes the information provided by the expert core extraction algorithm pertaining to the reliability of workers.
In particular, we provided a theoretical justification for the first type of algorithm.
if the number of questions and the number of workers in the expert-core are sufficiently large, our proposed algorithm gives the correct answer with high probability.
Computational experiments using synthetic and real-world datasets
demonstrated that our proposed answer aggregation algorithms outperform state-of-the-art algorithms.
There are several directions for future research.
Our model assumes that all experts give the correct answer with the same probability and all non-experts give an answer independently and uniformly at random.
However, in reality, experts themselves may have different levels of expertise and non-experts may not be completely random.
Although we have already confirmed that our proposed algorithms work well on real-world datasets, it is interesting to extend our model to such a more general scenario.
Another direction is to generalize our model in a higher-level perspective.
This study has focused on crowdsourced \emph{closed-ended} questions, where workers can select an answer from candidates.
On the other hand, there are often cases where we wish to handle crowdsourced \emph{open-ended} questions, where workers have to answer without any candidates.
We believe that our proposed algorithms may be applicable to this more general scenario by introducing a measure of similarity of answers (and thus similarity of workers).
\section*{Acknowledgments}
This work was partially supported by JST ACT-I Grant Number JPMJPR17U79 and JSPS KAKENHI Grant Numbers 16K16005, 17H07357, 18J23034, and 19K20218.
\bibliographystyle{abbrv}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,013 |
Q: Distribute subcommands over head command to avoid writing same words twice? I would like to try to write this command:
while read -r repo; do gh repo delete $repo; done <<< $(gh repo list --no-archived)
which basically cleans up my GitHub and deletes everything that I did not archive because I wanted to keep it on hand as a backup.
I would like there to be some way - without using aliases - to not have to write "gh repo" twice - elegantly. Some way of passing the two subcommands "list" and "delete" into that "head", something like:
gh repo $(list | while read -r repo; do delete $repo; done)
Would there be any possible way to do that? This is Zsh, but if any other shell has that capability, I'd love to know.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,299 |
\section{Introduction}
\label{sec_intro}
The optical colors of galaxies are characterized by a bimodality, with early-type galaxies residing on the red sequence and late-type galaxies populating the blue cloud \citep[e.g.,][]{Str01,Bal04,Bra09}.
The transition from blue to red galaxies is known to be driven by a decrease in star formation \citep[e.g.,][]{Whi12,Tay15}.
Since galaxies require gas to fuel star formation \citep[e.g.,][]{Sai17}, a knowledge of the gain and loss of gas provides a huge leap in the understanding of galaxy evolution.
Many galaxies reside in gravitationally bound groups or clusters, where a high volume density of galaxies is found.
A natural consequence of clustering of galaxies is frequent interactions between galaxies and the environment in which they live \citep[e.g.,][]{Fer14}.
Such interactions can stimulate the gain and loss and heating and cooling of gas in various ways and control the star formation processes in galaxies.
For example, interaction between galaxies can potentially trigger nuclear activity and nuclear starbursts that generate energetic gas outflow/winds to the environments \citep[e.g.,][]{Lar78,Sil98,Spr05,Ell08,Hop08,Fer14}.
Besides, when a galaxy moves at a high speed through a high density region, a complex hydrodynamical interaction of its interstellar medium (ISM) and the surrounding hot intracluster/intergalactic medium takes place.
A large fraction of gas could be removed from a galaxy if the ram-pressure is strong enough to overcome the gravitational force \citep{Gun72}.
These gas removal processes can lead to the interruption of star formation activity.
On the other hand, fresh supplies of gas can sustain star formation in galaxies.
Galaxies can gain gas through accretion of small companions \citep[][]{Bon91,Lac93}.
Moreover, cold gas can form from the cooling of the hot intracluster medium (ICM) or intragroup medium (IGM) and be accreted to galaxies \citep{Ega06,Ode08}.
Such process can rejuvenate early-type and S0 galaxies by supplying gas into their centers.
In our previous paper, \citet[][hereafter \citetalias{Lin17}]{Lin17}, we report the discovery of a giant ionized gaseous (H$\alpha$) blob associated with a dry merger system (i.e., mergers between two early-type galaxies; Figure \ref{fig_Halpha} and \ref{fig_cluster_zoom}) residing in a galaxy group (Figure \ref{fig_cluster}).
The ionized gaseous blob is offset from both of the galactic nuclei.
The gaseous blob might be a result of galaxy interactions, active galactic nucleus (AGN) activity, or a galaxy interacting with or being accreted by the dry merger.
In any of these cases, we may be witnessing an ongoing gain and/or loss of gas in these galaxies.
The H$\alpha$ blob was identified from the first-year MaNGA survey \citep[Mapping Nearby Galaxies at APO;][]{Bun15}, part of SDSS-IV \citep{Bla17}.
The H$\alpha$ blob (nicknamed ``Totoro'') is $\sim$ 3 -- 4 kpc in radius and is $\sim$ 8 kpc away (in projection) from the host galaxy MaNGA target 1-24145 (or MCG+10-24-117; nicknamed ``Satsuki''; 17$^\mathrm{h}$15$^\mathrm{h}$23.26$^\mathrm{s}$, +57$^{\circ}$25$\arcmin$58.36$\arcsec$, stellar mass $M_{\ast}$ $\approx$ 10$^{11}$ M$_{\sun}$).
There is no distinct optical continuum counterpart at the position of Totoro.
The mass of the ionized gas is 8.2 $\times$ 10$^{4}$ M$_{\sun}$ (\citetalias{Lin17}).
The SDSS image indicates that the host galaxy Satsuki has a companion galaxy (nicknamed
``Mei'') located to the south east of Satsuki (Figure \ref{fig_cluster_zoom}).
The companion is also within the hexagon bundle field of view (FoV; $\sim$ 32.5$\arcsec$ in diameter) of MaNGA.
These two galaxies are at similar redshifts ($z$ $\approx$ 0.03), with their line of sight velocity differing by $\sim$ 200 km s$^{-1}$.
Both galaxies are elliptical and thus form a dry (gas-poor) merger, also known as VII Zw 700.
On the large scale, the dry merger (Satsuki and Mei) is located at the overlap region of a group-group merger \citep{Osu19}.
Satsuki and Mei are associated with the less massive northern component, while
the nearby large elliptical galaxy NGC 6338 is the brightest galaxy of the more massive group (Figure \ref{fig_cluster}).
The dry merger (Satsuki and Mei) and NGC 6338 are separated by $\sim$ 42 kpc in projection and by $\sim$ 1400 km s$^{-1}$ in velocity.
The two merging groups are expected to form a galaxy cluster in the future.
The merger velocity of the two groups is as large as 1700 -- 1800 km s$^{-1}$, making this one of the most violent mergers yet observed between galaxy groups \citep{Osu19}.
Our H$\alpha$ blob Totoro has also been observed by \cite{Osu19} using the APO 3.5-m telescope (see their Figure 10), but only the high-luminosity, main blob region was detected.
\cite{Osu19} also show that the H$\alpha$ gas of NGC 6338 consists of three diffuse filaments in the southeast and northwest quadrants, extending out to $\sim$ 9 kpc from the nucleus.
The H$\alpha$ filaments have also been revealed in previous observations by $HST$ and the CALIFA\footnote{Calar Alto Legacy Integral Field Area Survey \citep{San12}.} survey \citep{Mar04,Pan12,Gom16}.
Moreover, \cite{Osu19} show that both NGC 6338 and VII Zw 700 contain potential X-ray cavities, which would indicate past AGN jet activity.
In addition, both systems are observed to contain cool IGM structures correlated with the H$\alpha$ emission \citep[see also][]{Pan12}.
We diagnosed the physical properties of Satsuki and Tororo using MaNGA data in \citetalias{Lin17}:
\begin{itemize}
\item \emph{kinematics}: the ionized (H$\alpha$) gas component reveals that there is a moderate velocity variation ($\leqslant$ 100 km s$^{-1}$) from the position of Satsuki along the connecting arms to Totoro, but there is no velocity gradient across Totoro itself and the velocity and velocity dispersion across the blob are low ($\ll$ 100 km s$^{-1}$; Figure 7 of \citetalias{Lin17}).
\item \emph{excitation state}: the Baldwin-Phillips-Terlevich (BPT) emission line diagnostics \citep{Bal81} indicate LI(N)ER-type excitations for Satsuki and a composite (LI(N)ER-HII mix\footnote{Shocks can also lead to line ratios occupying the composite regions. However, the analysis of shock and photoionization mixing models of \citetalias{Lin17} shows that the shocks are unlikely to be the dominant mechanism that is responsible for ionization of Totoro.}) for Totoro.
\item \emph{gas metallicity}: metallicity\footnote{Most of the metallicity calibrators can only be applied for those regions in which ionization is dominated by star formation, and may not be applicable to regions with ionization parameters or ISM pressure different from typical HII regions. In \citetalias{Lin17}, we adopted the ``N2S2H$\alpha$'' calibrator, which is suggested to be less sensitive to the ionization parameters \citep{Dop16}. However, caution is still needed when interpreting the derived metallicity.} of the gas around Satsuki is close to the solar value; on the other hand, the metallicity of Totoro is higher than that of Satsuki by 0.3 dex.
\end{itemize}
Several possibilities for Totoro were raised in \citetalias{Lin17}, including (1) the gas being ram-pressure stripped from Satsuki, (2) a galaxy interacting with the dry merger (Satsuki and Mei), and (3) gas being ejected or ionized by an AGN associated with Satsuki.
However, the data in \citetalias{Lin17} are not sufficient to provide strong constraints on the nature of Totoro.
In this paper, we present new observations of this system and determine the most plausible origin of Totoro.
Our new observations include:
\begin{enumerate}[label=(\roman*)]
\item \emph{wide-field H$\alpha$ image}: to reveal the distribution of ionized gas at larger scale (i.e., beyond the FoV of the MaNGA bundle),
\item \emph{$u$-band observation}: to constrain the ionizing source, i.e., star formation or not, and to search for the possible continuum counterpart of the H${\alpha}$ blob, and
\item \emph{molecular gas in $^{12}$CO(1--0)}: to constrain the amount and distribution of cold, i.e., potentially star-forming, gas.
\end{enumerate}
Moreover, in addition to the scenarios raised in \citetalias{Lin17}, \cite{Osu19} argue for the ``cooling gas'' hypothesis in Satsuki, therefore, we also use
\begin{itemize}
\item[(iv)] \emph{X-ray data} from \cite{Osu19} to constrain the properties of the surrounding hot medium.
\end{itemize}
The paper is organized as follows.
Section \ref{sec_data} describes the observations and our data reduction.
The results are presented and discussed in Section \ref{sec_results}.
In Section \ref{sec_summary} we summarize our results and list our conclusions.
Throughout this study, we assume a cosmology with $\Omega_\mathrm{m}$ $=$ 0.3, $\Omega_\mathrm{\Lambda}$ $=$ 0.7, and $H_{0}$ $=$ 70 km s$^{-1}$ Mpc$^{-1}$. We use a Salpeter stellar initial mass function.
\begin{figure*}
\begin{center}
\subfigure[]{\label{fig_Halpha}\includegraphics[scale=0.465]{Halpha.pdf}}
\subfigure[]{\label{fig_cluster_zoom}\includegraphics[scale=0.4]{cluster_zoom.pdf}}
\subfigure[]{\label{fig_cluster}\includegraphics[scale=0.4]{cluster.pdf}}
\end{center}
\caption{Optical images of MaNGA 1-24145 (nicknamed ``Satsuki'') and the environment in which it lives. The hexagons show the coverage of MaNGA bundle field of view. Satsuki was observed with the 127 fiber bundle of MaNGA; the hexagon is $\sim$ 32.5$\arcsec$ in diameter. (a) MaNGA H$\alpha$ map of MaNGA 1-24145. An H$\alpha$ blob (nicknamed ``Totoro'') is about 8 kpc northwest of MaNGA 1-24145. The unit of the map is 10$^{-16}$ erg s$^{-1}$ cm$^{-2}$. The data extend to regions just outside the hexagon because of the dithering. (b) The SDSS $gri$ composite image centering on MaNGA 1-24145. The south companion (nicknamed ``Mei'') is also within the MaNGA field of view.
(c) The SDSS $gri$ composite image of MaNGA 1-24145 and the nearby galaxies.
}
\label{fig_clusters}
\end{figure*}
\section{Data}
\label{sec_data}
\subsection{MaNGA}
MaNGA, the largest integral field spectroscopy (IFS) survey of the nearby Universe to date, observed $\sim$ 10,000 galaxies with a median redshift ($z$) of 0.03.
The details of the MaNGA survey, the integral-field-unit (IFU) fiber system, the sample selection, observing strategy, and the data reduction and analysis pipelines are explained in \citet{Dro15}, \citet{Wak17}, \citet{Law15}, \citet{Law16}, and \citet{Wes19}, respectively, and also summarized in \citetalias{Lin17}.
The MaNGA data used in this work were reduced using the MPL-7 version \citep[corresponding to SDSS data release 15,][]{Agu19} of the MaNGA data reduction pipeline.
An earlier version of the pipeline (MPL-4) was used for \citetalias{Lin17}.
The differences between the pipeline products of MPL-4 and MPL-7 are negligible for the current work.
The spectral-line fitting is carried out using the Pipe3D pipeline \citep{San16a,San16b}.
Details of the fitting procedures are described in \cite{San16a,San16b} and summarized in \citetalias{Lin17}.
The method described in \cite{Vog13} is used to compute the reddening using the Balmer decrement at each spaxel.
\subsection{Wide-field H$\alpha$ data}
Deep optical images were taken at the prime focus of the 6-m telescope of the Special Astrophysical Observatory of the Russian Academy of Sciences (SAO RAS) with the SCORPIO-2 multimode focal reducers \citep{Afa11}.
A narrow-band filter AC6775 (the central wavelength CWL $=$ 6769\AA, the bandwidth FWHM $=$ 15\AA) covers the spectral region around the redshifted H$\alpha$ emission line. Two middle-band filters FN655 (CWL $=$ 6559\AA, FWHM $=$ 97\AA) and FN712 (CWL $=$ 7137\AA, FWHM $=$ 209\AA) were used to obtain the blue and red continuum images. We combined the data taken during two nights 05/06 and 06/07 Mar 2017 with seeing of 1.3 -- 1.5$\arcsec$. The total exposure time depended on filter FWHM: 7800, 1300, and 780 sec in the filters AC6775, FN655 and FN712, respectively.
The detector, CCD E2V 42-90 (2K $\times$ 4.5K), operated in the bin 2 $\times$ 2 read-out mode provides 0.35$\arcsec$/px scale in the 6.1$\arcmin$ field of view.
The data reduction was performed in a standard way for SCORPIO-2 direct image processing with IDL-based software \citep[see, for instance,][]{Sit15}.
The underlying stellar continuum from the H$\alpha$ image was subtracted using the linear combination of the images in the filters FN655 and FN712.
The astrometry grid was created using the Astrometry.net project web-interface\footnote{http://nova.astrometry.net/} \citep{Lan10}.
We emphasize that the purpose of the wide-field observation is to reveal the distribution of H$\alpha$ gas outside of the MaNGA FoV, in particular, to look for a potential tidal tail(s) on the other side of Totoro.
To be in line with the measurements and discussions in \citetalias{Lin17}, the MaNGA H$\alpha$ map will be used by default throughout this paper, and the wide-field H$\alpha$ map will be used only where specifically mentioned in the text.
The flux of the new narrow-band image has been calibrated by the MaNGA spectral line data to avoid systematic errors related with narrow-band filter calibration (e.g., only one standard star per night, mismatch between the redshifted lines and a peak of the filter transmission curve, etc.).
This forces the flux of the narrow-band image to be consistent with that of MaNGA spectral line data.
\subsection{$u$-band Data}
The $u$-band observation was taken with the wide-field imaging facility MegaCam with a 1$^{\circ}$ field of view at the Canada–France–Hawaii Telescope (CFHT) from April 27th to June 24th in 2017 (PI: L. Lin; project ID:
17AT008).
The total exposure time is 12,000 seconds.
The MegaCam data were processed and stacked via MegaPipe \citep{Gwy08}.
The final image has a limiting mag of 26.3 mag (1$\arcsec$ aperture in radius)
\subsection{Molecular Gas (CO) Data}
We mapped the $^{12}$CO($J$ $=$ 1 $\rightarrow$ 0; 115.2712 GHz) emission with the NOrthern Extended Millimeter Array (NOEMA) at Plateau de Bure (PI: L. Lin; project ID: S16BE001).
The full width at half power of the primary beam of each NOEMA antenna at the $^{12}$CO(1-0) frequency is $\sim$ 50$\arcsec$, sufficiently large to cover Satsuki, Mei, and Totoro.
The observations were spread across 11 nights from May 25th to July 18th of 2015 with 5 or 7 antennas in D configuration.
The total on-science-source time is $\sim$ 22 hours.
The shortest possible baseline length is $\sim$ 24m, and therefore sources larger than about 15$\arcsec$ might be resolved out.
Nonetheless, in Section \ref{sec_results}, we will see that the distribution of molecular gas matches the H$\alpha$ extremely well.
Data reduction, calibration, and imaging were performed with the CLIC and MAPPING software of GILDAS\footnote{http://www.iram.fr/IRAMFR/GILDAS} using standard procedures.
Images were reconstructed using natural weighting to preserve maximal sensitivity.
The resultant synthesized beam is 3.4$\arcsec$ $\times$ 2.3$\arcsec$ (PA $=$ 73.3$^{\circ}$), with an effective beamsize of 2.8$\arcsec$.
The aim of the observation is to constrain the amount of molecular gas traced by $^{12}$CO(1-0), therefore a relatively low spectral resolution (53.6 km s$^{-1}$) was proposed to maximize the detection probability.
The rms noise in the $^{12}$CO(1-0) cube is 0.31 mJy beam$^{-1}$ per 53.6 km s$^{-1}$.
Figure \ref{fig_CO_spec} shows the $^{12}$CO(1-0) spectrum integrated over Totoro and the connecting arms, outlined by a dashed ellipse with an area of 210 arcsec$^{2}$ on the integrated intensity map in Figure \ref{fig_CO_flux_vel}a.
The $^{12}$CO(1-0) emission is strongly detected in two channels with velocities of $\sim$ 0 -- 100 km s$^{-1}$ and marginally detected in the adjacent channels.
A single Gaussian fit to the line profile is overplotted in Figure \ref{fig_CO_spec}.
The full width half maximum (FWHM) of the line is 86.5$\pm$7.4 km s$^{-1}$ and the line velocity relative to the galaxy velocity is 18.6$\pm$3.3 km s$^{-1}$.
The $^{12}$CO(1-0) (hereafter, CO) line properties are summarized in Table \ref{tab_co}.
\begin{figure}
\centering
\includegraphics[width=0.43\textwidth]{Spec.pdf}
\caption{NOEMA $^{12}$CO(1-0) spectrum integrated over the H$\alpha$ blob (Totoro) and the connecting arms. The area, 210 arcsec$^{2}$, we have used for deriving the spectrum is indicated by the dashed ellipse in Figure \ref{fig_CO_flux_vel}a. A single Gaussian fit to the line profile is overplotted. Significant $^{12}$CO(1-0) emission is detected at Totoro. }
\label{fig_CO_spec}
\end{figure}
\begin{table*}
\begin{center}
\caption{$^{12}$CO (1-0) line properties of the H$\alpha$ blob Totoro.}
\label{tab_co}
\begin{tabular}{ccccc}
\hline
velocity & line flux & line luminosity & line wdith & peak flux \\
(km s$^{-1}$) & (Jy) & (K km s$^{-1}$ pc$^{2}$) & (km s$^{-1}$) & (mJy) \\
\hline
18.6$\pm$3.3 & 1.01$\pm$0.05 &(4.83$\pm$0.23)$\times$10$^{7}$ & 86.5$\pm$7.4& 13.00 $\pm$ 0.74\\
\hline
\end{tabular}
\end{center}
\end{table*}
\section{Results and Discussion}
\label{sec_results}
In this section, we will present and discuss our results following the scenarios mentioned in the Introduction, namely, Section \ref{sec_ram}: the gas being ram-pressure stripped from Satsuki; Section \ref{sec_sep_gal}: an extremely low surface brightness galaxy or ultra-diffuse galaxy; Section \ref{sec_agn}: gas being ejected or ionized by an AGN.
In addition, \citetalias{Lin17} did not discuss the scenario of cooling of the IGM, which we discuss in Section \ref{sec_cooling}.
We will present the analysis and results of data at a specific wavelength described in Section \ref{sec_data} when it is needed to test a specific scenario.
\subsection{Ram-Pressure Stripping}
\label{sec_ram}
Galaxies in dense environments experience ram-pressure \citep{Gun72}.
As pointed out in \citetalias{Lin17}, ram-pressure stripping is not expected to produce a centrally-concentrated blob, but is more likely to form clumpy structures embedded in a jellyfish-like tail \citep[e.g.,][]{Bos16, Pog17,Bel19,Jac19}.
The centrally peaked structure we observe in Totoro in H$\alpha$ is also seen in molecular gas.
The integrated intensity map of CO is shown in Figure \ref{fig_CO_flux_vel}a and the comparison with H$\alpha$ is displayed in Figure \ref{fig_CO_flux_vel}b
The morphology of molecular gas generally agrees well with H$\alpha$ emission.
The CO emission of Totoro is also dominated by a large, centrally-concentrated structure associated with Totoro, but the peak position of CO is offset toward the south of H$\alpha$ peak by $\sim$ 0.3$\arcsec$ ($\sim$ 200 pc).
The multi-wavelength peak coordinates of Totoro, along with other properties that will be derived and discussed in this paper are provided in Table \ref{Tab_totoro}.
The two arm-like structures connecting the extended structure and the central region of the galaxy are also seen in CO.
Moreover, as somewhat expected, an early-type galaxy like Satsuki has a low molecular gas content.
While there is a strong and compact H$\alpha$ emission at the nucleus of Satsuki, no CO detection is found at this position.
Instead, two knots are moderately detected in the northeast and southwest of the nucleus. Although their orientation is consistent with that of the possible past AGN jets indicated by X-ray cavities \citep{Osu19}, deep CO observations are required to confirm the nature of these two knots.
We cannot rule out that the H$\alpha$ and CO morphologies would appear more clumpy if the angular resolution is improved.
The recently reported size of gas clumps in ram-pressure-stripped tails of galaxies range from several hundreds of parsec to several kpc \citep[e.g.,][]{Bel17,Lee18,Jac19,Lop20}.
Although we are not able to resolve individual gas clumps with our $\sim$ 1.7 kpc resolution, if Totoro is intrinsically clumpy as ram-pressured stripped gas, we should see a sign of clumpy sub-structure given that the size of the object is as large as 6 -- 8 kpc in diameter.
Ram-pressure-stripped gas is known to have high velocities (several hundreds of km s$^{-1}$) and velocity dispersions ($>$ 100 km s$^{-1}$) \citep[e.g.,][]{Bel17,Con17}.
Figure \ref{fig_CO_flux_vel}c and \ref{fig_CO_flux_vel}d display the CO and MaNGA H$\alpha$ velocity fields in color scale.
For ease of comparison, contours of CO integrated intensity map are overplotted on both velocity fields.
There is a strong variation in the line of sight H$\alpha$ velocity at the two connecting arms, with
redshifted gas in the left tail and blueshifted gas in the right tail.
Such features are not observed in the CO velocity field.
At the main blob region of Totoro, both CO and H$\alpha$ velocity field show no velocity gradient, little variation, and low velocity (mostly $\leq$ 60 km s$^{-1}$) across the region, suggesting that the system is not in rotation, unless it is perfectly face-on.
The velocity dispersion of H$\alpha$ gas in the region of Totoro is only $\sim$ 50 km s$^{-1}$ (\citetalias{Lin17}); the narrow CO line width also implies a low gas velocity dispersion.
Therefore, the gas kinematics of Totoro is in conflict with that of ram-pressure stripped gas.
We should caution, however, that the velocity resolutions of our CO and H$\alpha$ map are relatively low, $\sim$ 50 and 70 km s$^{-1}$, respectively. Further observations are needed to probe the detailed kinematics of Totoro.
Moreover, if ram-pressure stripping is the main origin of Totoro, the molecular-gas data imply that the cold gas is stripped almost completely from Satsuki.
However, such a scenario is disfavored by simulations of ram-pressure stripping \citep{Ste12,Ste16}.
A galaxy can lose all of its gas only in extreme cases of ram-pressure stripping, e.g., galaxy encounters high ICM densities with very high relative velocity.
In addition, massive galaxies, are less prone to lose their gas due to ram-pressure stripping because the existence of a massive bulge can prevent the stripping of gas and reduce the amount of gas being stripped.
Finally, galaxies that show ram-pressure stripping are mostly gas-rich late-type galaxies \citep[][and series of papers by the GAs Stripping Phenomena in galaxies with MUSE, GASP, team]{Pog17}.
For these reasons, the CO and H$\alpha$ gas are unlikely to be moved from the center of Satsuki to the current position as a result of ram-pressure stripping.
However, we note that this does not mean that the galaxy has not experienced any ram-pressure stripping.
We come back to this discussion in Section \ref{sec_cooling_env}.
\begin{figure*}
\centering
\includegraphics[scale=0.6]{CO_Halpha_natural.pdf}
\includegraphics[scale=0.6]{vel.pdf}
\caption{(a) Map of $^{12}$CO(1-0) integrated intensity map (color and contours) with the MaNGA hexagonal FoV overlaid. The $^{12}$CO contours are in intervals of 2, 3, 5.5, and 7.5$\sigma$, where 1 $\sigma$ corresponds to 40 mJy beam$^{-1}$ km s$^{-1}$. The nucleus of Satsuki is marked with a white cross. The synthesized beam (3.4$\arcsec$ $\times$ 2.34$\arcsec$, PA $=$ 73.3$^{\circ}$) is plotted in the bottom left. The dashed ellipse indicates the area we have used for generating the integrated spectrum in Figure \ref{fig_CO_spec}. (b) Similar to the panel (a), but the color map shows the H$\alpha$ emission from the MaNGA survey. (c) Velocity field of gas traced by $^{12}$CO(1-0) (color scale), with $^{12}$CO(1-0) intensity contours overlaid. (d) Velocity field of gas traced by MaNGA H$\alpha$ (color scale). The contours are the same as in other panels.}
\label{fig_CO_flux_vel}
\end{figure*}
\begin{table}[]
\begin{threeparttable}
\caption{Properties of the offset-cooling gas Totoro.}
\label{Tab_totoro}
\begin{tabular}{ll}
\hline
\multicolumn{2}{c}{General Properties} \\
\hline
\multirow{ 2}{*}{host} & VII Zw 700 \\
& (dry merger: Satsuki and Mei) \\
redshift\tnote{a} & 0.0322 \\
distance to the host & $\sim$ 8 kpc \\
enviroment & merging group \\
\hline
\multicolumn{2}{c}{Peak Position\tnote{b}} \\
\hline
$^{12}$CO(1-0) & 17:15:22.46, $+$57:26:6.90 \\
H$\alpha$\tnote{c} & 17:15:22.40, $+$57:26:7.83 \\
X-ray & 17:15:22.52, $+$57:26:7.89 \\
\hline
\multicolumn{2}{c}{Luminosity} \\
\hline
$^{12}$CO(1-0) & 4.8 $\times$ 10$^{7}$ K km s pc$^{2}$ \\
H$\alpha$\tnote{c} & 5.9 $\times$ 10$^{39}$ erg s$^{-1}$ \\
X-ray & 4.4 $\times$ 10$^{40}$ erg s$^{-1}$ \\
\hline
\multicolumn{2}{c}{Gas Mass} \\
\hline
cold gas (H$_{2}$) & 2.1 $\times$ 10$^{8}$ M$_{\sun}$ ($^{12}$CO (1-0)) \\
& 1.9 $\times$ 10$^{8}$ M$_{\sun}$ ($A_\mathrm{V}$: cloud) \\
& 2.5 $\times$ 10$^{8}$ M$_{\sun}$ ($A_\mathrm{V}$: diffuse) \\
warm gas (H$\alpha$) & 8.2 $\times$ 10$^{4}$ M$_{\sun}$ \\
hot gas (X-ray) & 1.2 $\times$ 10$^{9}$ M$_{\sun}$ \\
\hline
\multicolumn{2}{c}{Other Properties} \\
\hline
SFR\tnote{d} & $<$ 0.047 M$_{\sun}$ yr$^{-1}$ \\
cooling time & 2.2 $\times$ 10$^{8}$ yr \\
\hline
\end{tabular}
\begin{tablenotes}
\item[a] Redshift of VII Zw 700, taken from the NASA-Sloan Atlas (http://nsatlas.org/).
\item[b] Position of intensity peak in images, no centroid fitting is performed.
\item[c] Based on MaNGA H$\alpha$ data.
\item[d] Based on the assumption that all MaNGA-H$\alpha$ fluxes come from star formation.
\end{tablenotes}
\end{threeparttable}
\end{table}
\subsection{Separate Galaxy}
\label{sec_sep_gal}
In \citetalias{Lin17}, we argue that Totoro may be a separate galaxy interacting with the dry merger (Satsuki and Mei).
We can examine this scenario by (1) searching for its underlying stellar component; (2) looking for interaction features; and (3) comparing the molecular gas and star formation properties of this galaxy candidate with other galaxy populations.
\subsubsection{Underlying Stellar Counterpart}
\label{sec_gal_stellar}
The CFHT $u$-, $g$-, $r$, and $i$-band images are presented in Figure \ref{fig_CFHT_all}a -- \ref{fig_CFHT_all}c, respectively.
There are extended stellar halos surrounding the two galaxies, but we find no apparent optical counterpart directly from the images at the position of Totoro.
The limiting magnitude\footnote{The $gri$ limiting surface brightnesses quoted here are different from that given in \citetalias{Lin17}. This is because the former is scaled from the limiting magnitudes using 1$\arcsec$ aperture size while the latter was measured using an aperture size corresponding to one arcsec$^{2}$ area, which however is more affected by the Point Spread Function (PSF) because of the small aperture.} and the surface brightness of Totoro are listed in Table \ref{tab_cfht_mag}. Due to the absence of an optical counterpart at the position of Totoro, only upper limits can be placed on surface brightnesses (i.e., limiting surface brightness of the observations).
In \citetalias{Lin17}, we used a multi-component \texttt{GALFIT} \citep{Pen10} model for searching for a stellar component of Totoro from a $g$-band image and found no sub-structure that is responsible for the blob.
However, the residual of this complex parametric model still showed significant fluctuations that could hinder the detection of small scale sub-structure (see Figure 6 in \citetalias{Lin17}).
The on-going interaction of the main system creates an extended stellar envelope and asymmetric structures that are challenging to the model.
In this paper, we use three different approaches to further search for a potential stellar counterpart to Totoro from multiple-band ( $g$, $r$, and $i$) images.
All three methods are commonly used in literature for background subtraction and for both compact and extended source detection.
\begin{table}[]
\begin{center}
\caption{The 5$\sigma$ limiting magnitudes of CFHT $u$-, $g$-, $r$-, and $i$-band images and the 5$\sigma$ upper limit of the surface brightness of Totoro at each band.}
\label{tab_cfht_mag}
\begin{tabular}{ccc}
\hline
band & limiting magnitude & surface brightness of Totoro \\
& (1$\arcsec$ radius) [mag] & (upper limit) [mag arcsec$^{-2}$] \\
\hline
$u$ & 26.3 & 27.54 \\
$g$ & 25.7 & 26.94 \\
$r$ & 26.2 & 27.44 \\
$i$ & 25.2 & 26.44 \\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{figure*}
\centering
\includegraphics[width=0.9\textwidth]{CFHT-u.pdf}
\caption{CFHT $u$-, $g$-, $r$-, and $i$-band images (from left to right). The first three bands combine archival data downloaded from the CADC server and the data taken in 2015 summer (see \citetalias{Lin17} for the details). The $u$-band data was taken later in 2017. The cyan hexagon and red circle mark the region of MaNGA FoV and the H$\alpha$ blob Totoro.}
\label{fig_CFHT_all}
\end{figure*}
Firstly, we use the Python photometry tool \texttt{sep} \citep{Bar16}\footnote{https://github.com/kbarbary/sep} to generate a ``background'' model of the image with small background box size (5 pixels).
The tool \texttt{sep} uses the same background algorithm as in \texttt{SExtractor} \citep{Ber96}.
We then subtract the ``background'' model from the image to increase the contrast around the main galaxies.
This method has been used to separate galaxy stellar halos and the light from adjacent (background) objects \citep[e.g.,][]{Hua18,Rub18}, and to identify faint, extended emission such as tidal tails \citep[e.g.,][]{Man19}.
As a second approach, we subtract a blurred version of the image from the original one.
The blurred version of the image is created by convolving the image with a circular Gaussian kernel ($\sigma = 4$ pixels).
This procedure is part of the unsharp masking method in digital image processing\footnote{The original unsharp mask technique re-scales the residual and add it back to the original image.} and can also increase the contrast of the image.
Compared to the first approach, the Gaussian convolution makes it more sensitive to a low threshold feature with a sharp edge.
This method has been commonly used to identity HII regions in galaxies (e.g., \citealt{Rah11}; Pan et al. in preparation) and to detect faint embedded spiral and bar features in early-type galaxies \citep[e.g.,][]{Bar02,Kim12}. The method has been applied to IFS data for the later purpose as well by \cite{Gom16}.
For our third method, we perform isophotal fitting using the \texttt{Ellipse} task in \texttt{IRAF} \citep{Tod86,Tod93}.
We first mask out all detected objects other than the main galaxy using \texttt{sep}.
Then we run \texttt{Ellipse} on the masked images, allowing the centroid and the shape of the isophote to vary.
Using the resulting isophotal parameters, we create the corresponding 2-D model using the \texttt{bmodel} task and subtract it from the input image.
Compared to the \texttt{GALFIT} parametric model, the \texttt{Ellipse} one does not depend on the choice of model component and typically leads to smoother residuals.
This approach has been routinely used to determine morphology of elliptical and lenticular galaxies \citep[e.g.,][]{Hao06,Oh17} and to search for low-surface-brightness tidal features in nearby galaxies \citep[e.g.,][]{Tal09,Gu13}.
The residual maps of these methods are shown in Figure \ref{fig_lightfitting}.
From left to right the panels correspond to the analysis of the background, unsharp mask, and isophotal fitting methods; from top to bottom we show the residual maps and annular residual profiles of $u$-, $g$-, $r$- and $i$-band, respectively ($u$-band results will be discussed later in this section).
The annular profiles are centered on the H$\alpha$ peak of Totoro.
We also explore different parameters adopted in these procedures (e.g., the size of the background box or the convolution kernel).
The choice of these parameters within a reasonable range does not affect the results.
Using these methods, we detect a large number of unresolved (point-like) sources on the $g$-, $r$- and $i$-band images, presumably a combination of globular clusters of the main system and background galaxies.
The residual profiles are generally flat, fluctuating around the zero value.
This is true for all $gri$ bands and methods, suggesting that there is no sign of a distinctive stellar counterpart at the position of Totoro.
Therefore, our new analyses confirm the previous result by using \texttt{GALFIT} in \citetalias{Lin17}.
There are two unresolved sources in the Totoro area (red circle in Figure \ref{fig_lightfitting}), but we do not notice any increase or decrease of number density around the Totoro area.
The two sources are associated with neither H$\alpha$ nor CO peak.
We will come back to the nature of these two unresolved sources later.
On the residual maps from the unsharp mask method, we uncover a pair of ``ripple''-like features close to the center of the main galaxy.
These sub-structures remain the same when we vary the convolution kernel used, and they resemble the structure we saw on the \texttt{GALFIT} residual maps in \citetalias{Lin17}.
Such ``ripples'' often relate to recent galaxy interaction or the presence of dust \citep[e.g.,][]{Col01,Kim12,Duc15,Bil16}.
However, it is unclear whether there is any connection between them and Totoro.
\begin{figure
\includegraphics[width=0.45\textwidth]{LightFitting.pdf}
\caption{The residual images and annular residual profiles after subtracting the model images of the dry merger (Satsuki and Mei). The annular profiles are centered on the H$\alpha$ peak of Totoro. Three different approaches (from left to right: background method, unsharp mask, and isophotal fitting) are used to search for distinctive stellar counterpart and star formation of Totoro on the $u$-, $g$-, $r$- and $i$-band images (from top to bottom). The cyan hexagon and the red circle mark the regions of MaNGA FoV and Totoro, respectively. We do not observe a significant, extended stellar component around with Totoro in any residual maps and profiles. There are two point sources around the position of Totoro, they are likely background sources (see text for the details). }
\label{fig_lightfitting}
\end{figure}
Since there is no optical ($g$, $r$, and $i$) continuum counterpart found in Totoro, it is expected to be composed mostly of young stars if it is indeed a galaxy.
In \citetalias{Lin17}, we use the excitation state of optical lines to constrain the presence of young stars. However, the result is method (diagnostic diagram) dependent.
The $u$-band luminosity of a galaxy is dominated by young stars of ages $<$ 1 Gyr, therefore it is more sensitive to any recent star formation than any of the other broad band luminosities available \citep{Mou06,Pre09,Zho17}, and is a more straightforward probe than optical emission line diagnostics.
Similar to the $gri$ bands, the $u$-band emission in the MaNGA hexagonal FoV is dominated by the dry merger (Satsuki and Mei) as shown in Figure \ref{fig_CFHT_all}, but the $u$-band data is less affected by the large stellar halos associated with the dry merger (Satsuki and Mei) than the redder bands, and therefore serves as a better probe for underlying stellar component.
The limiting magnitudes and surface brightness of the $u$-band image are 26.3 mag and 27.54 mag arcsec$^{-2}$, respectively (Table \ref{tab_cfht_mag}).
The point source in the upper-left corner of the dry merger (Satsuki and Mei) is a foreground star according to the MaNGA spectrum.
Although visually there is no distinguishable $u$-band feature (i.e., recent star formation) associated with Totoro, to ensure that the $u$-band counterpart of Totoro is not embedded within the light of the dry merger (Satsuki and Mei), we subtract the photometric models for the merging system from the $u$-band image using the three different methods mentioned above.
The residual maps and profiles are shown in the top row of Figure \ref{fig_lightfitting}, respectively.
The residual maps and profiles are not perfectly smooth, but we find no obvious evidence for a distinctive, extended $u$-band counterpart at Totoro.
The two point sources seen in $g$-, $r$-, and $i$-band residual maps are also seen in the $u$-band residual image of the isophotal fitting (and marginally seen in the background method as well).
This confirms the results in \citetalias{Lin17} that star formation alone can not explain the excitation state of Totoro.
To gain some insight into the nature of the two unresolved objects in the Totoro area, their photometric redshifts are determined using the EAZY \citep{Br08} and P\'{E}GASE 2.0 \citep{fr1997} template fitting the aperture magnitudes measured from the isophotal fitting residual images using GAIA (Graphical Astronomy and Image Analysis Tool).
The default EAZY template is generated from the P\'{E}GASE 2.0 models
using the \citet{br2007} algorithm and then calibrated using
semi-analytic models, plus an additional young and dusty template.
The P\'{E}GASE 2.0 template is a library including
$\sim$3000 models with a variety of star formation histories
and with ages between 1 Myr and 20 Gyr, as described in detail
in \cite{grazian2006}.
Figure \ref{fig_sed} shows the chi-squared of the fit for a given redshift using the EAZY and P\'{E}GASE templates (left column of each panel) and the best-fit SED with observed fluxes at $i$-, $r$-, $g$- and $u$-band overlaid as red circles (right column of each panel).
The results based on the EAZY and P\'{E}GASE templates are presented in the top and bottom rows respectively.
The minimum chi-squared value indicates that the southern source (17$^\mathrm{h}$15$^\mathrm{m}$22.111$^\mathrm{s}$, +57$^{\circ}$26$\arcmin$5.628$\arcsec$) is a background galaxy at $z$ $\sim$ 0.40 and 0.37 using the EAZY and P\'{E}GASE templates, respectively (Figure \ref{fig_PS_S}).
The redshift of our target is marked by a yellow dashed line in the figures. We notice a second minimum at $z$ $<$ 0.03, close to the redshift of our target. However, at such low redshifts, the source would be resolved, not point like.
The most plausible redshift of the northern source (17$^\mathrm{h}$15$^\mathrm{m}$22.573$^\mathrm{s}$, +57$^{\circ}$26$\arcmin$7.659$\arcsec$) is 0.29 according to both templates (Figure \ref{fig_PS_N}), but we cannot rule out other possibilities of $z$ $<$ 0.4, in particular $z$ $\sim$ 0.15, due to the shallower basin-shaped chi-square distribution. Nonetheless, the redshift of Totoro (yellow dashed line) is not associated with any local minimum of the chi-square values. We find poorer fits when using stellar templates, providing further support that the source is not a nearby object.
The best-fitted SEDs from P\'{E}GASE are exported to SED-fitting code \texttt{New-Hyperz}\footnote{\url{http://userpages.irap.omp.eu/~rpello/newhyperz/}} in order to derive the stellar mass ($M_{\ast}$) and star formation rate (SFR) of these two objects.
The results suggest that the southern and northern sources are $\sim$3 $\times$ 10$^{8}$ and $\sim$6 $\times$ 10$^{8}$ M$_{\sun}$ and their specific SFR (sSFR $=$ SFR/$M_{\ast}$) are $\leq$ 10$^{-11}$ yr$^{-1}$.
We also estimate their $M_{\ast}$ assuming they are at the same redshift as Totoro ($z$ $=$ 0.03), and yield $\sim$ 10$^{6}$ and $\sim$ 10$^{7}$ M$_{\sun}$ for the southern and northern sources, respectively.
Given their point-like morphologies ($<$ 1 kpc$^{2}$), we expect to see distinctive stellar mass surface density ($\Sigma_{\ast}$) distributions in the MaNGA data if they are associated with Totoro. However, we find no such features.
It should be noted that the above stellar masses may be subjected to non-negligible uncertainties due to the lack of (near-) infrared measurements.
We use web interface Marvin\footnote{https://dr15.sdss.org/marvin}, a tool to visualise and analyse MaNGA data \citep{Che19}, to search for any signatures in the spectra of these two sources by redshifting strong optical emission lines (e.g., H$\alpha$) based on the derived photometric redshifts.
However, the emission lines are too faint to be seen by MaNGA.
Given their likely high redshift and their positions being offset from the centroid of the H$\alpha$ blob, these two point sources are unlikely to be associated with Totoro.
\begin{figure}[!ht]
\begin{center}
\subfigure[]{\label{fig_PS_S}\includegraphics[scale=0.4]{PS_S.pdf}}
\subfigure[]{\label{fig_PS_N}\includegraphics[scale=0.4]{PS_N.pdf}}
\end{center}
\vspace{-15pt}
\caption{The SED-fitting results of the two unresolved (point-like) sources appearing in the residual images after subtracting the model images of the dry merger (Satsuki and Mei). The results of the southern and northern sources are presented in panel (a) and (b), respectively. For each panel, the sub-panels (1) and (2) show the chi-squared values of the fit for a given redshift using the EAZY template and the best-fit SED with observed fluxes at $i$-, $r$-, $g$- and $u$-band overlaid as red circles, respectively. The corresponding plots using the P\'{E}GASE template are shown in sub-panels (3) and (4). The red solid vertical lines in sub-panels (1) and (3) mark the redshift of minimum chi-squared value. The redshift of our target is indicated by yellow dashed lines.
}
\label{fig_sed}
\end{figure}
\subsubsection{Interaction Features}
\label{sec_gal_ion}
If Totoro is indeed a separate galaxy, it may possess another tidal tail on the other side of the blob (i.e., as opposed to the ones that connect the Totoro and Satsuki), but such tail(s) would not be seen by MaNGA due to the limited FoV.
Figure \ref{fig_Ha_large} shows the new wide-field H$\alpha$ image taken from the SAO RAS 6-m telescope.
The new H$\alpha$ map has a sensitivity comparable to that of MaNGA, but the FoV is $\sim$ 10 times larger.
For comparison, Figure \ref{fig_Ha_large}a zooms in to the region of the MaNGA hexagonal FoV.
The new H$\alpha$ image globally resembles that of the MaNGA H$\alpha$ in Figure \ref{fig_Halpha}, but shows finer structures thanks to higher spatial resolution.
A zoom-out view of the region of interest is displayed in Figure \ref{fig_Ha_large}b.
There are several H$\alpha$ knots beyond the MaNGA FoV towards the east. These are background galaxies or foreground stars.
It is clear that there is no hint of H$\alpha$ emission extending beyond the MaNGA FoV at the opposite side Totoro.
Quantitatively, we can estimate the possible missing flux of MaNGA by comparing the H$\alpha$ flux related with Totoro from the two observations.
The total H$\alpha$ luminosity of Totoro and the surrounding $\sim$15$\arcsec$ ($\sim$ 9 kpc) region outside of the MaNGA FoV (blue circle in Figure \ref{fig_Ha_large}b) is 6.5 $\times$ 10$^{39}$ erg s$^{-1}$.
The total H$\alpha$ luminosity of Totoro measured by MaNGA is 5.9 $\times$ 10$^{39}$ erg s$^{-1}$.
Therefore the possible missing flux of Totoro due to limited MaNGA FoV is no larger than 10\%.
Accordingly, the scenario of a separate galaxy appears less likely unless the tidal tail(s) develop at only one side of a galaxy.
Such cases are relatively rare, though not impossible (e.g., Arp 173, Arp188, and Arp 273), depending on the stage of the interaction and the projected orientation on the sky \citep[e.g.,][]{Mih04,Str12}.
However, the low gas velocity provides additional support against a merger scenario (Figure \ref{fig_CO_flux_vel}c and \ref{fig_CO_flux_vel}d).
On the other hand, it is possible that Totoro is a completely disrupted dwarf galaxy.
The averaged surface brightness of Totoro has an upper limit of $\sim$ 27 mag arcsec$^{-2}$ (Table \ref{tab_cfht_mag}).
It would be classified as a dwarf low-surface-brightness galaxy (LSB; $>$ 23 mag arcsec$^{-2}$) if it is a galaxy.
In the next section, we compare the star formation and cold gas properties of Totoro with other galaxy populations, including LSBs.
As a side note, the nearby galaxy NGC 6338 is encompassed by the wide-field H$\alpha$ observation.
Figure \ref{fig_Ha_large}c shows the H$\alpha$ image of NGC 6338.
There is no strong filaments or bridge linking NGC 6338 and VII Zw 700, presumably because the two are very separate entities (with projected separation of $\sim$ 42 kpc and $\sim$ 1400 km s$^{-1}$ difference in line of sight velocity), but there is a bit more H$\alpha$ emission between galaxies than the rest of the regions in the plotted area.
The H$\alpha$ emission of NGC 6338 is characteristic of three previously-reported filaments in the southeast and northwest quadrants.
The H$\alpha$ intensity contours of 0.035 and 0.09 $\times$ 10$^{-16}$ erg s$^{-1}$ cm$^{-2}$ are overplotted to highlight the asymmetric filaments.
We also refer the reader to \citet{Mar04} and \citet{Osu19} for higher-resolution H$\alpha$ images of NGC 6338 and \citet{Gom16} for optical IFS data analysis of NGC 6338.
\begin{figure*}
\centering
\includegraphics[width=0.99\textwidth]{Ha_large3.pdf}
\caption{New, wide-field narrow-band H$\alpha$ image taken from the SAO RAS 6-m telescope. (a) A zoom-in to the region of the MaNGA observation with the MaNGA hexagonal bundle FoV overlaid. The color scale is the same as in Figure \ref{fig_Halpha}. The nucleus of Satsuki is marked by a cross. (b) A zoom-out view of the region of interest. It is clear that there is no hint of tidal feature extending beyond the MaNGA FoV. Quantitatively, the total H$\alpha$ luminosity in the blue circle (15$\arcsec$ or $\sim$ 9 kpc in radius) is only $\sim$ 10\% higher than the luminosity of Totoro measured by MaNGA, therefore the possible missing flux that is associated with Totoro due to the small MaNGA FoV is at most 10\%. (c) The H$\alpha$ image of NGC 6338 and our target. The two contours correspond to 0.035 and 0.09 $\times$ 10$^{-16}$ erg s$^{-1}$ cm$^{-2}$, respectively.}
\label{fig_Ha_large}
\end{figure*}
\subsubsection{$M_\mathrm{H_{2}}$-SFR Relation of Galaxies}
\label{gas_sfr_mh2}
Another way to constrain the origin of Totoro using data in hand is to look into the question of whether Totoro shares similar gas and star formation properties with nearby galaxies.
This could not be concluded in \citetalias{Lin17} due to the lack of cold gas data.
Figure \ref{fig_CO_ks} compares the star formation rate (SFR) and molecular gas mass ($M_\mathrm{H_{2}}$) of Totoro with other galaxy populations.
The plot resembles the Kennicutt-Schmidt relation \citep{Ken89} assuming that the molecular gas and star-forming regions coexist.
The galaxy data include nearby LSBs \citep{One03,Mat05,Cao17}, nearby star-forming (sSFR $>$ 10$^{-11}$ yr$^{-1}$) and quiescent (sSFR $<$ 10$^{-11}$ yr$^{-1}$) galaxies \citep{Sai17}.
The total H$\alpha$ luminosity of Totoro (including the connecting arms; $\sim$48.1 kpc$^{2}$ area in total) is converted to SFR using the calibration of
\begin{equation}
\frac{\mathrm{SFR}}{[\mathrm{M_{\odot}\, yr^{-1}}]}=7.9\, \times\, 10^{42}\,\frac{L_{\mathrm{H\alpha }}}{[\mathrm{erg\, s^{-1}}]}
\end{equation}
\citep{Ken98}, where $L_\mathrm{H\alpha}$ is H$\alpha$ luminosity.
This yields a SFR of 0.047 M$_{\sun}$ yr$^{-1}$.
Here we assume all of the H$\alpha$ results from star formation, but this is unlikely to be the case as BPT diagnostics indicate a composite nature, LI(N)ER-HII mix excitation, for Totoro (\citetalias{Lin17}).
Therefore the derived SFR is an upper limit.
The total molecular gas mass of Totoro traced by CO is computed using
\begin{multline}
\frac{M\mathrm{_{H_{2}}}}{[\mathrm{M_{\odot}}]}=1.05\, \times\, 10^{4}\, \frac{X_{\mathrm{CO}}}{[2\, \times\, 10^{20}\, \mathrm{cm^{-2}(K\, km\, s^{-1})^{-1}}]}\, \\ \frac{S_{\mathrm{CO}}\Delta v }{[\mathrm{Jy\, km\, s^{-1}}]}\frac{D_\mathrm{L}^{2}}{[\mathrm{Mpc}]}\, (1+z)^{-1}
\end{multline}
,where $X_\mathrm{CO}$, $S\mathrm{_{CO}}\Delta\nu$, $D_{\mathrm{L}}$ are CO-to-H$_{2}$ conversion factor, integrated line flux density, and luminosity distance, respectively \citep{Bol13}.
We adopt a Galactic $X_\mathrm{CO}$ of 2 $\times$ 10$^{20}$ cm$^{-2}$ (K km s$^{-1}$)$^{-1}$ \citep[][]{Bol13}.
The derived $M_\mathrm{H_{2}}$\footnote{We should note that the value of $M_\mathrm{H_{2}}$ depends on the choice of $X_\mathrm{CO}$.
We may overestimate $M_\mathrm{H_{2}}$ by $\sim$ 30\% to a factor of a few if the metallicity of Totoro is indeed higher than the solar metallicity by $\sim$ 0.3 dex as discussed in the Introduction \citep{Bol13}.
\cite{Van17} use the optical thin $^{13}$CO(3-2) emission line to estimate $X_\mathrm{CO}$ of the BCG of the cooling-core cluster RXJ0821$+$0752, finding a $X_\mathrm{CO}$ of a factor of two lower than the Galactic (our adopted) value.
However, this is within the object-to-object scatter from extragalactic sources, and based on several assumptions such as isotopic abundance ratio and excitation line ratios.
Statistical analysis is necessary in order to constrain the $X_\mathrm{CO}$ in BCGs.} of Totoro is (2.1$\pm$0.1) $\times$ 10$^{8}$ M$_{\sun}$.
Note that the uncertainty due to different methodologies to derived the Galactic $X_\mathrm{CO}$ is $\sim$ 30\% \citep{Bol13}.
In addition to CO, the gaseous column density $N_\mathrm{H_{2}}$ (and $M_\mathrm{H_{2}}$) also follows from the amount of extinction $A_\mathrm{V}$, which can be derived from the MaNGA H$\alpha$ and H$\beta$ data.
For reference, the mean $A_\mathrm{V}$ across Totoro is $\sim$ 0.5 mag.
The conversion from extinction to H$_{2}$ column density depends on the medium \citep{Boh78,Eva09}:
\begin{align}
\label{eq_n2av_mc}
\frac{N(\mathrm{H_{2}})}{[\mathrm{cm^{-2}}]}&=6.9\, \times\, 10^{20}\, \frac{A_\mathrm{v}}{[\mathrm{mag}]}\, \mathrm{(molecular\,cloud)} \\
\label{eq_n2av_diffuse}
&= 9.4\, \times\, 10^{20}\, \frac{A_\mathrm{v}}{[\mathrm{mag}]}\, \mathrm{(diffuse\,ISM)}.
\end{align}
These yield a $M_\mathrm{H_{2}}$ of $\sim$ 1.9 $\times$ 10$^{8}$ and $\sim$ 2.5 $\times$ 10$^{8}$ M$_{\odot}$ assuming molecular-cloud- (Equation \ref{eq_n2av_mc}) and diffuse-ISM-type (Equation \ref{eq_n2av_diffuse}) medium, respectively.
The three $M_\mathrm{H_{2}}$ derived using radio and optical measurements agree well with each other.
The H$_{2}$ mass derived from CO is used for the discussion in the rest of the paper.
Although only upper limits could be achieved for many LSB objects, they largely follow the trend established by star forming galaxies towards the lower end in both $M_\mathrm{H_{2}}$ and SFR axes, consistent with the finding of \cite{Mcg17}.
On the other hand, quiescent galaxies, mostly early types, have lower SFR for a given $M_\mathrm{H_{2}}$ and a lower CO detection rate than that of star-forming galaxies \cite[see also][]{Cal18}.
Due to the low SFR, Totoro deviates from the SFR-$M_\mathrm{H_{2}}$ relation formed by LSBs and star-forming galaxies and appears to overlap with quiescent galaxies.
The H$\alpha$ emission in quiescent galaxies is dominated by LI(N)ER excitation \citep{Hsi17,Pan18}.
In the LI(N)ER regions of quiescent galaxies, surface density of H$\alpha$ luminosity ($\Sigma_\mathrm{H\alpha}$) is found to be tightly correlated with underlying $\Sigma_\mathrm{\ast}$ \citep{Hsi17}, in which the H$\alpha$ are primarily powered by the hot, evolved stars rather than recent star formation.
BPT diagnostics suggest that Totoro is powered by a composite (LI(N)ER-HII mix) mechanism (\citetalias{Lin17}).
If Totoro is analogous to a LI(N)ER region in quiescent (early-type) galaxies, the average $\Sigma_\mathrm{H\alpha}$ of Totoro corresponds to a $\Sigma_\mathrm{\ast}$ of $\sim$ 7 $\times$ 10$^{8}$ M$_{\sun}$ kpc$^{-2}$ according to the ``resolved LI(N)ER sequence'' of quiescent galaxies reported by \cite{Hsi17}.
The predicted $\Sigma_{\ast}$ is higher than the mass of Satsuki's stellar halo by a factor of 3 -- 5; however, there is no distinct stellar counterpart at the location of Totoro.
For this reason, in spite of the overlap in Figure \ref{fig_CO_ks}, Totoro is not analogous to the LI(N)ER region in quiescent galaxies.
Lastly, we should note that if the true global SFR of Totoro is considerably lower than the upper limit, Totoro would fall below the main cloud of data points of quiescent galaxies.
As a whole, Totoro is unlikely to be consistent with nearby normal star-forming, early-type, and low-surface-brightness galaxies in terms of the Kennicutt-Schmidt relation and the resolved LI(N)ER sequence. In addition, it is worth noting that the average $A_\mathrm{V}$ of Totoro corresponds to a SFR surface density ($\Sigma_\mathrm{SFR}$) at least 6 times higher than the observed upper limit based on the local $A_\mathrm{V}$-$\Sigma_\mathrm{SFR}$ relation derived from MaNGA galaxies \citep{Li19}. Generally speaking, the dust and star formation relation (if any) of Totoro is also dissimilar to that of star-forming regions in nearby galaxies even when the systematic dependencies on other physical properties (e.g., metallicity) are considered.
While LSBs have been studied for decades, recently, \cite{Van15} have identified a new class of LSBs in the Coma cluster, ultra-diffuse galaxies (UDGs).
These UDGs have a surface brightness as low as $>$ 24.5 mag arcsec$^{-2}$, but their sizes are similar to those of $L^{\ast}$ galaxies.
It is not clear how UDGs were formed.
One possible scenario is that UDGs are failed massive galaxies, which lost their gas at high redshift by ram-pressure stripping or other effects after forming their first generation of stars \citep{Van15,Yoz15}.
Our finding of a large molecular gas reservoir appears in direct conflict with this scenario.
On the other hand, several studies suggest that UDGs are extended dwarf galaxies.
Some simulations predict them to be rapidly rotating \citep{Amo16}.
\cite{Dic17} suggest that the extended sizes of UDGs are the consequence of strong gas outflows driven by starbursts.
However, the SFR of Totoro is extremely low and the system is not rotating.
Altogether, there is no evidence in our data to support Totoro as an UDG.
\begin{figure}
\centering
\includegraphics[width=0.43\textwidth]{KS.pdf}
\caption{Star formation rate versus H$_{2}$ mass. The orange square is Totoro from this work. The gray, green, and magenta symbols are LSBs measurements from \cite{One03}, \cite{Mat05}, and \cite{Cao17}, respectively. Objects with solid detection in CO lines are shown with squares. An arrow indicates that only an upper limit was found. Blue and red squares are nearby star-forming (sSFR $>$ 10$^{-11}$ yr$^{-1}$) and quiescent (sSFR $<$ 10$^{-11}$ yr$^{-1}$) galaxies taken from the xCOLD GASS survey \citep{Sai17}. In all cases a conversion factor of 2.0 $\times$ 10$^{20}$ cm$^{-2}$ (K km s$^{-1}$)$^{-1}$ \citep[][the uncertainty of the conversion factor is $\sim$ 30\%]{Bol13} is used to allow ready comparison between the studies. }
\label{fig_CO_ks}
\end{figure}
In summary, the newly obtained CO, H$\alpha$, and $u$-band data allow us to better constrain the nature of Totoro; however, combining the results in Section \ref{sec_sep_gal}, we argue that Totoro is unlikely to be a separate galaxy interacting with the dry merger (Satsuki and Mei).
The reasons include the lack of stellar counterpart and tidal features and the different star formation, ionized and cold gas, and dust properties (i.e., $M_\mathrm{H_{2}}$-SFR, $\Sigma_{\ast}$-$\Sigma_\mathrm{H\alpha}$, and $A_\mathrm{V}$-$\Sigma_\mathrm{SFR}$ relations and gas kinematics) from that of a variety of nearby galaxy populations (i.e., star-forming and quiescent galaxies, LSBs, and UDGs).
\subsection{AGN-driven Activity}
\label{sec_agn}
\subsubsection{Multi-wavelength Nuclear Characteristics}
{\bf \emph{Radio}}. A possible origin of Totoro is gas that is photoionized by X-ray emission from a misaligned blazar at the core of Satsuki.
A blazar is a sub-class of radio-loud AGN which is characterized by one-sided jet structure \citep{Urr95}.
However, blazars invariably have bright compact radio cores, whereas no detection at 1.4 GHz is found for Satsuki \citep{Wan19,Osu19} and there is only marginal detection at 5 GHz (\citetalias{Lin17}).
For these reasons, a blazar is a very unlikely scenario for Totoro.
{\bf \emph{Optical}}. The H$\alpha$ equivalent width (EW) is $<$ 3\AA\, across the entire H$\alpha$-emitting region and the EW does not present a rise towards the center of Satsuki.
The H$\alpha$ velocity dispersion does not present a central peak either (\citetalias{Lin17}).
Moreover, the H$\alpha$ luminosity of Satsuki follows the continuum emission, i.e., does not present a central peak emission with stronger intensities that follows an $r^{-2}$ or lower decline from the center \citep[][]{Sin13}.
Therefore, Satsuki presents a lack of optical characteristics of a strong AGN.
{\bf \emph{X-ray}}. A comprehensive study of the X-ray emission of the region based on \emph{Chandra} and \emph{XMM-Newton} data has been reported recently by \cite{Osu19}.
Figure \ref{fig_Chandra}a shows the \textit{Chandra} X-ray image of the NGC 6338 group taken from \cite{Osu19}.
X-ray and galaxy velocity studies have shown the group to be a high-velocity near head-on merger \citep{Wan19}, with the two group cores visible as bright clumps of X-ray emission with trailing X-ray tails. The northern and southern clumps are respectively associated with VII Zw 700 and NGC~6338.
Figure \ref{fig_Chandra}b zooms in to the MaNGA observed region.
The northern X-ray clump, around Satsuki, is dominated by a bar of X-ray emission extending roughly southeast-northwest. The clump and bar are centered somewhat to the north of the optical centroid of Satsuki, and the bar is made up of three X-ray knots whose intensities progressively decrease from the western end. The nuclei of Satsuki and Mei are not correlated with the brightest X-ray emission \citep[see also][]{Wan19}.
Nonetheless, after subtracting the overall surface brightness distribution, there is a hint of excess X-ray emission at the position of the nuclei of Satsuki and Mei (see \citealt{Osu19} for the construction of the overall surface brightness distribution).
X-ray luminosities of 4.17 $\times$ 10$^{39}$ erg s$^{-1}$ for the nucleus of Satsuki and $\leqslant$ 2.5 $\times$ 10$^{38}$ erg s$^{-1}$ for Mei are reported \citep{Osu19}.
The values suggest that there are no strong X-ray AGN in Satsuki or Mei or the AGNs are fading.
For reference, the X-ray luminosity of Totoro is (4.4$\pm$0.2) $\times$ 10$^{40}$ erg s$^{-1}$.
\begin{figure*}
\centering
\includegraphics[width=0.9\textwidth]{Chandra_T.pdf}
\caption{\emph{Chandra} 0.5 -- 2 keV maps (a)-(c) and temperature maps (d )-(f) taken from \cite{Osu19}. In each row, from left to right we show the full map, zoom in on the MaNGA FoV with H$\alpha$ contours overlaid, and zoom in on the MaNGA FoV with CO contours overlaid. In panel (a) and (d), the nucleus of NGC 6338 is marked by an open cross. The MaNGA hexagonal FoV is also indicated in the figures. In panel (b), (c), (e), and (f), the red filled crosses indicate the nuclei of Satsuki and Mei. The maps show that the H$\alpha$ and CO emissions are coincident with X-ray structures and low temperature regions.}
\label{fig_Chandra}
\end{figure*}
\subsubsection{Gas Ejected by AGN}
\label{sec_agn_out}
All in all, the multi-wavelength data show no direct evidence for an active ongoing AGN in Satsuki.
However, it is known that the AGN luminosity can vary over timescales as short as 10$^{5}$ years.
Thus, we still cannot rule out the possibility of a recent AGN outflow.
AGN-driven extended outflows are detected in multiple phases, including ionized gas and many molecular lines \citep[][and references therein]{Rob20}.
Even though the observed radial extent of AGN outflows is $\leq$ 1 kpc in most cases, galactic scale outflows, as Totoro would be, given its extreme distance from Satsuki, are also reported \citep[e.g.,][]{Lop19,Leu19,Lop20}.
However, the low gas velocities (Figure \ref{fig_CO_flux_vel}c and \ref{fig_CO_flux_vel}d) do not support the scenario of an energetic outflow.
Nonetheless, we should note that although an AGN does not seem like a plausible mechanism for moving H$\alpha$ or CO out of the galaxy core to the current position of Totoro, the potential cavities identified in Satsuki suggest that in the past ($\sim$ 40 Myr ago) the AGN was indeed active and had non-negligible impact on its surroundings \citep{Osu19}.
\subsubsection{Gas Ionized by AGN}
There have been studies showing that AGNs are able to
ionize gas extending to large distance, such as the well-known ionized cloud Hanny's Voorwerp $\sim$ 20 kpc from its host galaxy IC~2497 \citep[e.g.,][]{Hus08,Lin09,Hus10}.
It is believed that an interaction-triggered, currently fading AGN, illuminated and ionized Hanny's Voorewerp \citep{Lin09,Joz09,Kee12}.
Diffuse ionized gas similar to Hanny's Voorewerp was also found $\sim$ 32 kpc north of the iconic interacting galaxies NGC 5194/5195 or M51 \citep{Wat18}.
The low AGN luminosity and activity and the tidal history provide similarities between Satsuki, IC~2497 and M51.
However, there are also several differences between Hanny's Voorwerp and the M51 cloud and Totoro in terms of excitation state (Seyfert versus LINER-HII, morphology (diffuse for Hanny's Voorwerp and M51's cloud versus centrally-concentrated for Totoro), and the properties of the host galaxy (late type versus early type, see \citetalias{Lin17} for the details).
For these reasons, we argue that this scenario is unlikely.
\subsection{Cooling Gas}
\label{sec_cooling}
\subsubsection{Spatial Comparison of Cold, Warm, and Hot Gas}
\label{sec_cooling_spatial}
A scenario we did not consider in \citetalias{Lin17} is the cooling of the hot IGM or ICM. Observations of the central regions of some galaxy groups and clusters show strong X-ray emission suggesting that the IGM and ICM are undergoing rapid cooling \citep[e.g.,][]{Fab94}, and in some cases ionized and molecular gas which are thought to be the product of that cooling \citep{Bab18,Lak18,Oli19,Rus19}.
\cite{Osu19} shows that the X-ray peak of the southern clump in Figure \ref{fig_Chandra}a is consistent with the optical centroid of NGC~6338. Three X-ray filaments are observed to extend from the galaxy center, following the same branching filamentary structure as the H$\alpha$ gas (Figure \ref{fig_Ha_large}c).
These X-ray filaments are cooler than their surroundings and have very low gas entropies and short cooling times \citep[][]{Osu19}, strongly indicating that they are a locus of cooling from the IGM.
Young X-ray cavities are also found in NGC 6338, suggesting recent AGN outbursts in this galaxy.
As shown in Figure \ref{fig_Chandra}b, the peak of H$\alpha$ emission (contours) at the nucleus of Satsuki is not associated with the bar, but the position of Totoro corresponds to the brightest X-ray peak.
This was first noted by \cite{Osu19} who showed that this was also the coolest part of the X-ray bar, and concluded that the H$\alpha$ was likely material cooled from the IGM, as in the filaments of NGC~6338.
They also suggested that the offset between Satsuki and the center of the X-ray clump is evidence that ram-pressure forces caused by the (supersonic) motion of the galaxy are detaching the gas from the galaxy.
In this scenario, the X-ray bar would once have been centered on Satsuki, but has been pushed back to the north and perhaps along the line of sight, away from the galaxy core.
Figure \ref{fig_Chandra}c shows our CO data overlaid on the \textit{Chandra} image. The molecular gas is well correlated with the western X-ray knot, and (as previously discussed) with the H$\alpha$. Figures \ref{fig_Chandra}d-e show the \textit{Chandra} temperature maps from \cite{Osu19} showing the cool gas around NGC~6338 and Satsuki, and the high-temperature gas between the two, which has been shock-heated by the group-group merger. Overlaid H$\alpha$ and CO contours show that the ionized and molecular gas are located in the coolest part of the IGM, as in NGC~6338 and the centers of other groups and clusters \citep[e.g.,][]{Sal06,Ham12}.
\subsubsection{Cooling Time and Gas Properties}
\label{sec_cooling_time}
Studies have shown that warm and cold gas in the brightest group and cluster galaxies (BCGs) are preferentially observed in systems where the cooling times lie below $\sim$ 1 Gyr \citep{Edg01,Sal03,Cav08,Raf08,Pul18}.
\cite{Osu19} found that the cooling times are as short as $<$1~Gyr in both cores of Satsuki and NGC~6338.
We can also estimate the cooling time around Totoro, i.e., the west end of the X-ray bar as a cylinder of radius 2.8$\arcsec$(1.5 kpc) and length 11.2$\arcsec$ (5.9 kpc, corresponding to the Region 2 in Figure 10b of \citealt{Osu19}).
The temperature, density, and luminosity of the X-ray gas are estimated via spectral fitting of the $Chandra$ data.
Using Equation (1) in \cite{Osu19}, we estimate the cooling time ($t_\mathrm{cool}$) in the Totoro region to be 2.2$^{+0.2}_{-0.1}$ $\times$ 10$^{8}$ yr, well within the regime where warm and cold gas are expected.
If the warm and cold gas have indeed cooled from the hot gas, one would expect the hot gas to be significantly more massive than the cool/warm gas.
The hot gas mass is derived from a radial deprojected profile (Figure 9 in \citealt{Osu19}), centering on the middle of the X-ray bar.
Within a radius of 6.5 kpc, the hot gas mass ($M_\mathrm{X}$) is $1.2_{-0.24}^{+1.07}$ $\times$ 10$^{9}$ M$_{\sun}$.
The derived hot gas mass is indeed significantly higher than the total mass of warm ($\sim$ 10$^{5}$ M$_{\odot}$ from H$\alpha$) and cold gas ($\sim$ 10$^{8}$ M$_{\odot}$ from CO).
The ratio of the cold to hot gas of Totoro is $\sim$ 17\%.
\cite{Pul18} investigate the molecular gas properties of 55 central cluster galaxies.
They found a strong correlation of hot and cold gas mass traced by X-ray and CO, suggesting that the hot and cold gas arise from the same ensemble of clouds.
In other words, the cold gas is unlikely a result of external effects, such as merger or stripping from a plunging galaxy.
The average fraction of cold to hot gas in their sample is $\sim$ 18\%.
The cold to hot gas mass ratio of Totoro is in good agreement with that of these central cluster galaxies \citep[see also][]{Oli19}.
The consistency provides support that a similar process to that in the central cluster galaxies has occurred in Satsuki and Totoro.
In addition, molecular mass has been found to be correlated with the amount of H$\alpha$ gas expressed by $L_\mathrm{H\alpha}$ in the cooling-core galaxies of clusters \citep{Edg01,Sal03}.
In Figure \ref{fig_Cooling} we show the H$_{2}$ gas mass versus $L_\mathrm{H\alpha}$ for data taken from the literatures.
Totoro is overlaid with a orange square, and falls at the low end of the correlation.
The consistency of Totoro with the $M_\mathrm{H_{2}}$- $L_\mathrm{H\alpha}$ relationship of cooling systems again supports a similar process of cooling in the system.
To put it another way, our multi-wavelength data of Totoro show that its position on the mass (or luminosity) relations between cold ($\lesssim$ 100 K; CO), warm ($\sim$ 10$^{4}$ K, H$\alpha$), and hot ($>$ 10$^{7}$ K, X-ray) gas is in line with the gas content of systems with short cooling times.
We summarize the physical properties of Totoro in Table \ref{Tab_totoro}.
As a side note, the cold and warm gas in cooling systems often appear filamentary, but it is worth noting that the multi-phase gas morphologies of our target are strikingly similar to the BCG of cluster Abell 1991 reported by \cite{Ham12} (see their Figure 2).
The H$\alpha$-emitting gas is relatively circular (blob-like) and is spatially coincident with the most rapidly cooling region (X-ray peak) of the ICM.
The peak in the H$\alpha$ and X-ray gas lies roughly 11 kpc to the north of the BCG, and there are connecting arm structures between the peak and the secondary peak at the galactic nucleus.
Moreover, the bulk of the molecular gas with a mass of $\sim$ 8$\times$ 10$^{8}$ M$_{\sun}$ is also found at the location of the cooling region.
However, note that spatial resolution must play an important role in detecting the morphology of cooling gas.
We cannot rule out that the morphologies of Totoro and the cooling gas in Abell 1991 are more filamentary than a single peak.
Finally, the velocity fields of cooling gas in galaxy clusters, traced by CO and H$\alpha$, are characteristic of slow motions (projected velocity $<$ 400 km s$^{-1}$), narrow line widths ($<$ 250 km s$^{-1}$) and a lack of relaxed (e.g., rotating) structures \citep[e.g.,][]{Sal08,Ham12,Oli19}.
The observed velocity structures, despite low velocity resolutions, of Totoro traced by CO and H$\alpha$ (Figures \ref{fig_CO_flux_vel}c and \ref{fig_CO_flux_vel}d) agree with that of other cooling systems.
In addition, the gas velocities of Totoro traced by CO and H$\alpha$ are not exactly identical, but the differences are small, $\sim$ 35 km s$^{-1}$ at the main blob region and 60 -- 90 km s$^{-1}$ at the connecting arms.
This is also consistent with the finding by \cite{Oli19} that the velocity difference between CO and H$\alpha$ gas is well below 100 km s$^{-1}$, providing an additional support for CO and H$\alpha$ gas arising from the same bulk of clouds.
The velocity difference may be related to different velocity resolution of CO and H$\alpha$ observations and line of sight projection effect.
We should note again that our CO and H$\alpha$ observations suffer from low velocity resolutions, therefore the velocity fields must be interpreted with caution. Nonetheless, the maps still provide a guide of the velocity resolution needed for future observation of this system.
Future high velocity-resolution CO and H$\alpha$ observations are required to reveal the detailed gas kinematics of Totoro.
\begin{figure}
\centering
\includegraphics[width=0.43\textwidth]{Cooling.pdf}
\caption{H$\alpha$ luminosity versus molecular gas mass of cooling gas in cluster galaxies. Data points marked with circles, thin diamonds, pentagons, and diamond are taken from \cite{Edg01}, \cite{Sal03}, \cite{Mcd12}, and \cite{Ham12}, respectively. Totoro is shown by an orange square and lies on the relationship of other systems.
}
\label{fig_Cooling}
\end{figure}
\subsubsection{Environments}
\label{sec_cooling_env}
While most of the studies of cooling gas focus on BCGs in rich clusters, cold gas cooling from the hot X-ray medium is also observed at smaller scales in galaxy groups, such as NGC 5044, NGC 4638 and NGC5846 \citep{Dav14,Tem18}.
In fact, the observed cool-core fractions for galaxy groups are slightly higher than those of galaxy clusters \citep{Osu17}.
Therefore, gas cooled from the IGM is not unique to Totoro.
In Section \ref{sec_ram}, we argue that the cold gas in Totoro is unlikely to be the primitive gas of Satsuki being stripped by ram-pressure.
Nonetheless, the X-ray tail (Figure \ref{fig_Chandra}a) and the offset between the center of the X-ray bar and the optical centroid of Satsuki are both evidence that the motion of the dry merger (Satsuki and Mei) is rapid enough to lead to stripping of the hot gas halo \citep{Osu19}.
The question then arises of where and when the ionized and molecular gas we observe was formed; have ram-pressure or other effects changed its location?
We might normally expect to see the most rapid cooling in or near the galaxy center.
However, the offset of the X-ray bar means that the densest, coolest IGM gas is no longer located at the center of Satsuki.
Ram-pressure, by pushing the X-ray halo and bar away from the core of Satsuki, may therefore have caused a reduction in cooling in the galaxy center \citep{Osu19}.
CO emission occurs in dense molecular clouds which are ``self-shielding'' from the ionizing effects of the surrounding environment.
While some of the H$\alpha$ emission likely comes from the outer layers of such molecular cloud complexes, observations show that in some cooling systems the H$\alpha$ emission is considerably more extended than the molecular gas \citep[e.g., in NGC~5044; ][]{Sch20} which may suggest it is associated with a less dense cooled gas component.
Such low-density material would likely move with the surrounding IGM if ram-pressure pushed it back from the galaxy. Dense molecular clouds would not be affected by ram-pressure, and might be expected to fall under gravity toward the center of Satsuki, unless they are connected to the IGM via magnetic fields \citep{Mcc15} or surrounding layers of neutral and ionized gas \citep{Li18}. Even with such connections to the surrounding environment, it seems implausible that the molecular gas could have condensed out of the IGM in the core of Satsuki and then been uplifted. The correlation between the CO, H$\alpha$ and the coolest X-ray gas strongly suggests that the molecular and ionized gas is the product of cooling from the IGM at its current location, i.e., that cooling has occurred (and may be ongoing) in Totoro, well outside the center of Satsuki.
Last but not least, as suggested by \cite{Osu19} and this work, we are witnessing a merger between two groups undergoing rapid radiative cooling.
Further analysis on the multi-wavelength phase of cooling gas in NGC 6338, from cold to hot gas as in this study, will be carried in a separate paper (O'Sullivan et al. in preparation).
\subsubsection{Future of the Gas}
\label{sec_cooling_future}
Cooling gas can potentially serve as the fuel for an AGN and/or central star formation \citep[][]{Ode08,Raf08,Mit09,Hic10,Fog17}.
In X-ray bright groups, like our target, star formation in the central galaxy is generally weak even in systems known to be cooling.
By contrast, as many as 85 -- 90\% of group-dominant galaxies have radio AGN, moreover, dominant galaxies with active or recently active radio jets are relatively common in X-ray bright groups \citep{Kol18,Kol19}.
Galaxy cluster-dominant galaxies, however, seem much more likely to have significant star formation.
Here we consider whether the cooling gas would fuel star formation assuming the gas will fall back into Satsuki.
\cite{Mcd18} compare the cooling rate of the ICM/IGM to the observed SFR in the central galaxy for a sample of isolated ellipticals, groups, and clusters.
They found that the cooling ICM/IGM is not providing the fuel for star formation in systems with cooling rate $<$ 30 M$_{\sun}$ yr$^{-1}$, which are dominated by groups and isolated ellipticals.
On the other hand, SFR increases with increasing cooling rate for the rapidly cooling systems ($>$ 30 M$_{\sun}$ yr$^{-1}$), presumably due to an increase in either the cooling efficiency of the hot gas or the star formation efficiency of the cooled gas \citep[see also][]{Edg01,Sal03,Ode08}.
The cooling rate ($M_\mathrm{X}$/$t_\mathrm{cool}$) of Totoro is $\sim$ $1.2$ $\times$ 10$^{9}$ M$_{\sun}$/2.2 $\times$ 10$^{8}$ yr $\approx$ 5 M$_{\sun}$ yr$^{-1}$.
In this aspect, the cooling gas in our group-dominant, relatively low cooling-rate system is less likely to significantly contribute to star formation.
Moreover, star formation can be suppressed by AGN feedback even in systems with short cooling times.
The summed AGN jet power ($P_\mathrm{cav}$) for both cavities associated with Satsuki from \cite{Osu19} is (0.97 -- 2.67) $\times$ 10$^{41}$ erg s$^{-1}$ (an estimate of heating; the actual value depends on the cavity age used: buoyant rise time, sonic expansion time-scale, or refill time), the energy is comparable to the X-ray luminosity for the whole X-ray emitting gas ($\sim$ 3 $\times$ 10$^{41}$ erg s$^{-1}$; an estimate of cooling).
\cite{Raf08} find a tendency for star-forming systems to have low $P_\mathrm{cav}$/$L_{X}$ ratios ($<$ 1) and quiescent systems to have high $P_\mathrm{cav}$/$L_{X}$ ratios ($>$ 1), supporting the suppression of star formation by AGN feedback.
However, this is not exclusively the case; star-forming cooling systems can have high $P_\mathrm{cav}$/$L_{X}$ ratios, and vice versa.
The estimated heating available from the cavities around Satsuki make it a borderline case, and the ongoing merger and stripping add to the complexities. The fate of Totoro may depend on how effectively energy from the cavities can heat their surroundings.
Finally, it is worth mentioning that Satsuki and Totoro may host a little star-formation activity with an upper limit of $<$ 0.059 and $<$ 0.047 M$_{\sun}$ yr$^{-1}$, respectively, assuming all the H$\alpha$ fluxes come from star formation.
It is unclear if the current star formation (if any) is related to the cooling and gas fueling processes.
\cite{Mcd18} attribute the low-level star formation in low cooling rate systems to recycling of gas lost by evolved stars, namely, the star formation is not related to cooling gas.
\section{Summary}
\label{sec_summary}
In \citetalias{Lin17}, we identified an H$\alpha$ blob Totoro $\sim$ 8 kpc away from a dry merger (Satsuki and Mei) from MaNGA data (Figure \ref{fig_clusters}).
Here we present new optical (wide-field H$\alpha$ and $u$-band), millimeter ($^{12}$CO(1-0)) observations, and published X-ray data \citep{Osu19}, with the aim of providing significant constraints and answers to fundamental questions regarding the nature of Totoro.
The main conclusions of this paper are as follows:
\begin{itemize}
\item The data disfavor the scenario that Totoro is stripped from Satsuki by ram-pressure based on the morphology and kinematics of ionized (H$\alpha$) and molecular gas) and the properties of the host galaxy (Section \ref{sec_ram} and Figure \ref{fig_CO_flux_vel}).
\end{itemize}
We consider whether Totoro is a separate galaxy interacting with the dry merger (Satsuki and Mei) from several aspects:
\begin{itemize}
\item We apply three commonly-used methods to $g$-, $r$-, and $i$-band images (Figure \ref{fig_CFHT_all}) to look for an underlying stellar counterpart of Totoro. However, we find no compact underlying stellar component associated with Totoro (Section \ref{sec_gal_stellar} and Figure \ref{fig_lightfitting}).
\item No tidal tail feature is seen in H$\alpha$ beyond the MaNGA FoV. If Totoro is a galaxy interacting with the dry merger (Satsuki and Mei), it may have a non-typical tidal history and morphology, or it is a completely disrupted low-surface-brightness dwarf galaxy (Section \ref{sec_gal_ion}, Figure \ref{fig_Ha_large} and Table \ref{tab_cfht_mag}).
\item However, Totoro shows different star formation, gas, and dust properties (in terms of $M_\mathrm{H_{2}}$-SFR, $\Sigma_{\ast}$-$\Sigma_\mathrm{H\alpha}$, and $A_\mathrm{V}$-$\Sigma_\mathrm{SFR}$ relations and gas kinematics) from that of a variety of nearby galaxy populations (i.e., star-forming and quiescent galaxies, low-surface-brightness and ultra-diffuse galaxies). Therefore, Totoro is unlikely to be a separate galaxy interacting with the dry merger (Satsuki and Mei) (Section \ref{gas_sfr_mh2} and Figure \ref{fig_CO_ks}).
\item The $u$-band data, which are sensitive to recent star formation, show no strong sign of recent star formation at the position of Totoro. Therefore, the ionized gas of Totoro is unlikely to be powered by star formation, confirming the results of emission line ratios diagnostics in \citetalias{Lin17} and previous bullet-point that Totoro is not an analogue of a star-forming region in nearby galaxy populations (Section \ref{sec_gal_stellar} and Figure \ref{fig_lightfitting}).
\end{itemize}
We consider whether Totoro is a result of AGN activity.
\begin{itemize}
\item However, in spite of possible past AGN outbursts, the multi-wavelength data show no direct evidence for an active ongoing AGN in Satsuki or Mei. Moreover, Totoro is unlikely to be gas being ionized or ejected by an AGN as its physical properties (gas excitation state, morphology, and kinematics, etc.) are distinct from similar objects (Section \ref{sec_agn}).
\end{itemize}
Finally, we consider whether Totoro is formed via cooling of hot IGM as implied by \cite{Osu19}.
\begin{itemize}
\item We compare the spatial distribution of H$\alpha$ and CO with X-ray intensity and temperature maps.
We find that the ionized and molecular gas are related to the most rapidly cooling region of the hot IGM.
The cooling time in the Totoro region is well within the regime where cooling is expected (Section \ref{sec_cooling_spatial} and \ref{sec_cooling_time} and Figure \ref{fig_Chandra}).
\end{itemize}
\begin{itemize}
\item The mass (or luminosity) relations between cold ($<$ 100 K; CO), warm ($\sim$ 10$^{4}$ K, H$\alpha$), and hot ($>$ 10$^{7}$ K, X-ray) gas, as well as gas kinematics are in line with the gas content of cooling systems, supporting again that Totoro originates from the same physical process of cooling gas (Section \ref{sec_cooling_time} and Figure \ref{fig_Cooling}).
\end{itemize}
\begin{itemize}
\item Previous study by \cite{Osu19} suggests that the densest, coolest X-ray gas has been pushed away from the core of the host galaxy Satsuki by ram-pressure. The correlation between the CO, H$\alpha$ and the coolest X-ray gas presented in this work strongly suggests that the molecular and ionized gas is the product of cooling from the hot X-ray gas and is formed at its current location, leading to the observed \emph{offset} cooling and the reduction in cooling in the galaxy core (Section \ref{sec_cooling_env}).
\item The cooling rate of Totoro is considerably lower than that of star-forming cooling systems. The estimated heating available from the
cavities around Satsuki is comparable to the cooling X-ray luminosity. The fate of Totoro may depend on how effectively energy from the cavities can heat their surroundings, but note that the ongoing merger (Satsuki, Mei, and NGC~6338) and stripping add to the complexities (Section \ref{sec_cooling_future}).
\end{itemize}
In the majority of clusters the peaks of optical and X-ray emission are very close to the center of galaxy, so it is difficult to determine whether the appearances of warm or even cold gas are primarily related to the cooling of the IGM/ICM or the host galaxy.
Offset cooling is rare \citep[$<$ 5\%, e.g.,][]{Ham12}, therefore VII Zw 700 provides an exceptional opportunity to constrain the gas cooling process and the interplay between cooling gas and host galaxy.
In future work, we intend to constrain the gas kinematics at the connecting arms with high-spectral-resolution data to quantify the potential of a fueling process for Satsuki. A detailed multi-wavelength analysis, from CO to H$\alpha$, to other optical lines (e.g., H$\beta$, [NII], [OIII], etc.), and to X-ray, will be presented in an upcoming paper (O'Sullivan et al in preperation).
Moreover, the large sample of optical IFU survey MaNGA ($\sim$ 5,000 galaxies in the latest SDSS data releases DR15/16 and $\sim$ 10,000 in the future) along with the \emph{Chandra} archive is a suitable starting point for future multi-wavelength statistical studies of cooling gas properties.
Although MaNGA does not target specific environments, the large sample size of MaNGA ensures observations of numerous galaxies located in groups and clusters.
Given the significant fraction of cooling cores in galaxy clusters/groups \cite[e.g.,][]{Mit09,Osu17}, more cooling gas candidates are expected in the final MaNGA sample.
\acknowledgments
We would like to thank the anonymous referee for constructive comments that helped to improve the manuscript.
H.A.P thanks Eva Schinnerer, Christine Wilson, and Toshiki Saito for useful discussions.
This work is supported by the Academia Sinica under the Career Development Award CDA-107-M03 and the Ministry of Science \& Technology of Taiwan under the grant MOST 108-2628-M-001-001-MY3.
M.J.M.~acknowledges the support of the National Science Centre, Poland
through the SONATA BIS grant 2018/30/E/ST9/00208, the Royal Society of
Edinburgh International Exchange Programme, and the hospitality of the
Academia Sinica Institute of Astronomy and Astrophysics (ASIAA).
EOS gratefully acknowledges the support for this work provided by the National Aeronautics and Space Administration(NASA) through \emph{Chandra} Award Number G07-18162X, issued by the \emph{Chandra} X-ray Center, which is operated by the Smithsonian Astrophysical Observatory for and on behalf of NASA under contract NAS8-03060.
SFS is grateful for the support of a CONACYT grant FC-2016-01-1916, and funding from the PAPIIT-DGAPA-IN100519 (UNAM) project.
J.G.F-T is supported by FONDECYT No. 3180210 and Becas Iberoam\'erica Investigador 2019, Banco Santander Chile.
This project makes use of the MaNGA-Pipe3D dataproducts \citep{San16b,San18}. We thank the IA-UNAM MaNGA team for creating this catalogue, and the Conacyt Project CB-285080 for supporting them.
This work is partly based on observations carried out under project number S16BE001 with the IRAM NOEMA Interferometer. IRAM is supported by INSU/CNRS (France), MPG (Germany) and IGN (Spain).
This work is partly based on observations obtained with MegaPrime/MegaCam, a joint project of CFHT and CEA/DAPNIA, at the Canada-France-Hawaii Telescope (CFHT) which is operated by the National Research Council (NRC) of Canada, the Institut National des Sciences de l'Univers of the Centre National de la Recherche Scientifique of France, and the University of Hawaii.
The authors also wish to recognize and acknowledge the very
significant cultural role and reverence that the summit of Maunakea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain.
This work is partly based on observations obtained with the 6-m telescope of the Special Astrophysical Observatory of the Russian Academy of Sciences carried out with the financial support of the Ministry of Science and Higher Education of the Russian Federation (including agreement No. 05.619.21.0016, project ID RFMEFI61919X0016).
The analysis of the ionized gas distribution according SCORPIO-2 data was supported by the grant of Russian Science Foundation project 17-12-01335 ``Ionized gas in galaxy discs and beyond the optical radius''.
This work use GAIA to derive aperture photometry.
GAIA is a derivative of the SKYCAT catalogue and image display tool, developed as part of the VLT project at ESO. SKYCAT and GAIA are free software under the terms of the GNU copyright. The 3D facilities in GAIA use the VTK library.
This research made use of APLpy, an open-source plotting package for Python \citep{Rob12}.
Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions. SDSS-IV acknowledges
support and resources from the Center for High-Performance Computing at
the University of Utah. The SDSS web site is www.sdss.org.
SDSS-IV is managed by the Astrophysical Research Consortium for the
Participating Institutions of the SDSS Collaboration including the
Brazilian Participation Group, the Carnegie Institution for Science,
Carnegie Mellon University, the Chilean Participation Group, the French Participation Group, Harvard-Smithsonian Center for Astrophysics,
Instituto de Astrof\'isica de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and Mathematics of the Universe (IPMU) /
University of Tokyo, the Korean Participation Group, Lawrence Berkeley National Laboratory,
Leibniz Institut f\"ur Astrophysik Potsdam (AIP),
Max-Planck-Institut f\"ur Astronomie (MPIA Heidelberg),
Max-Planck-Institut f\"ur Astrophysik (MPA Garching),
Max-Planck-Institut f\"ur Extraterrestrische Physik (MPE),
National Astronomical Observatories of China, New Mexico State University,
New York University, University of Notre Dame,
Observat\'ario Nacional / MCTI, The Ohio State University,
Pennsylvania State University, Shanghai Astronomical Observatory,
United Kingdom Participation Group,
Universidad Nacional Aut\'onoma de M\'exico, University of Arizona,
University of Colorado Boulder, University of Oxford, University of Portsmouth,
University of Utah, University of Virginia, University of Washington, University of Wisconsin,
Vanderbilt University, and Yale University.
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} | 9,944 |
{"url":"https:\/\/tex.stackexchange.com\/questions\/346286\/a-nice-collection-of-exercises-for-students","text":"# A nice collection of exercises for students\n\nI would to build, for my students, a creative collection of exercises and problems. I am not very able, with my source in attachment, to create the same figure shown in attachment. Someone, please, could you help me?\n\n \\documentclass[italian]{book}\n\\usepackage{amssymb,latexsym, mathtools}\n\\usepackage[utf8]{inputenc}\n\\usepackage{babel}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\usepackage{stackengine,xcolor}\n\\let\\svitem\\item\n\\newcommand\\difbox[1]{\\stackengine{0pt}{\\color{gray!30}\\rule{5ex}{1.15ex}}{%\n\\color{red}$\\mkern1mu\\makeballs{#1}$}{O}{c}{F}{F}{L}}\n\\def\\makeballs#1{\\ifnum#1>0\\relax{\\bullet}%\n\\expandafter\\makeballs\\the\\numexpr#1-1\\relax\\fi}\n\\newenvironment{benumerate}\n{\\renewcommand\\item[1][1]{\\def\\difficulty{##1}\\svitem}%\n\\def\\labelenumi{\\smash{\\stackunder[1pt]{\\color{teal}%\n\\bfseries\\sffamily\\large\\theenumi}{\\difbox{\\difficulty}}}}%\n\\enumerate}{\\endenumerate}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\usepackage{siunitx}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\usepackage{titlesec}\n\\usepackage{multicol}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\usepackage[left=2.5cm, right=2.5cm, top=3cm, bottom=2cm]{geometry}\n% \\usepackage[a4paper,left=2.5cm,right=2.5cm,top=1.5cm,bottom=1.5cm,\n% marginparsep=3mm,marginparwidth=18mm,\n%]{geometry}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\usepackage{fancyhdr}\n\\usepackage{tikz}\n\\definecolor{gray}{cmyk}{0,0,0,0.4}\n\\definecolor{darkgrey}{cmyk}{0,0,0,0.6}\n\n\\pagestyle{fancy}\n\\renewcommand{\\chaptermark}[1]{\\markboth{#1}{}}\n\\fancyfoot{} % Remove foot fields\n\\renewcommand{\\footrulewidth}{0pt}\n\\vskip-\\baselineskip\\vskip4pt\n\\ifodd\\count0\\hfill\\begin{tikzpicture}\n\\end{tikzpicture}\\else\\begin{tikzpicture}\n\\end{tikzpicture}\\fi}\n\\parindent0pt\n\\parskip6pt\n\\makeatletter\n\\makeatother\n\\newcounter{myExercise}[section]\n\\setcounter{myExercise}{1}\n\\newcommand\\exercise{\\textbf{Esercizi \\thesection.\\stepcounter{myExercise}\\themyExercise.\\,}}\n\\newcommand\\mySol[1]{\\textcolor{cyan!20!blue}{[$#1$]}}\n\\setlength{\\columnsep}{12pt}\n\\setlength\\columnseprule{0pt}\n\\begin{document}\n\n\\section*{Velocit\u00e0}\n\n\\begin{multicols}{2}\n\\begin{benumerate}\n\n\\item[1]\n\nUn'automobile transita al km 25 di un'autostrada alle ore 8:25 e transita al km 29 alle ore 8:27. Qual \u00e8 la sua velocit\u00e0 media in km\/h? \\hfill\\mySol{120\\,\\, \\text{km\/h}}\n\n\\item[2] In autostrada ogni kilometro \u00e8 contrassegnato da un numero. Guardando fuori dal finestrino, ti accorgi che passano 36 s tra un cartello e l'altro. A quale velocit\u00e0 stai procedendo? \\hfill\\mySol{100\\,\\, \\text{km\/h}}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n%\\item\n\n\\hfill\\mySol{\\ldots}\n\\end{benumerate}\n\n\\end{multicols}\n\\end{document}\n\n\u2022 A few suggestions that have nothing to do with your question: to typeset quantities, use the package siunitx, it will give you much more consistency; since you are using the utf8 input encoding, if you have an Italian keyboard, you can write directly the accented letters \u00e0\u00e8\u00e9\u00ec\u00f2\u00f9, without the need of using the escape sequences \\'e, etc.; for setting margins, you can use the geometry package. \u2013\u00a0Massimo Ortolano Dec 29 '16 at 21:17\n\u2022 Note that questions about graphic design are off topic for this site (there is a separate stack exchange site for that) once you have chosen your design, asking how to implement it in TeX is of course on topic. (The topic boundaries are of course very vague and lots of overlap is allowed but I would say this question is pretty clearly off topic) \u2013\u00a0David Carlisle Dec 29 '16 at 21:40\n\u2022 Why \\setlength{\\topskip}{0in} ? (It will lead to uneven settings with the position of the first line of each page varying depending on the content of the line.) \u2013\u00a0David Carlisle Dec 29 '16 at 21:49\n\u2022 As I said before, this is not a good way to go about learning LaTeX. You need to break down This Complicated Thing into individual steps you can tackle one at a time. Then go about learning how to do each step. If you get stuck, ask a question about that particular step. By the way, in addition to the other comments, your font configuration could do with some attention. times is deprecated for a start. You are worrying far too much about all the twiddly bits, in my opinion, and not taking the time you need to learn the basics. \u2013\u00a0cfr Dec 30 '16 at 2:24\n\u2022 Don't change the layout dimensions manually if using geometry. If you say \\setlength{\\headheight}{24pt}, geometry doesn't know about this and things are bound to go wrong. For the new picture, look at tcolorbox and just work on that element. It would not be my priority if my page layout wasn't consistent yet, which is far more basic, but if you are determined to focus on bling, at least try to deal with one variety of bling at a time. tcolorbox will take some time and work on your part: it is a powerful package. \u2013\u00a0cfr Dec 31 '16 at 3:38\n\nI tried to reproduce the graphics and look, but more is to do and is left to the O.P, since there are some hard coded parts not really configurable.\n\nThe easiest positioning of the grade\/skill level box can be achieved by using a tcolorbox and a TikZ node.\n\nThe \\mylib code is 'stolen' from the tcolorbox documentation.\n\n\\documentclass[italian]{book}\n\\usepackage{amssymb}\n\\usepackage{latexsym}\n\\usepackage{mathtools}\n\n\\usepackage[utf8]{inputenc}\n\\usepackage[italian]{babel}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\usepackage{stackengine}\n\\usepackage{xcolor}\n\n\\newcommand\\difbox[1]{\\stackengine{0pt}{\\color{gray!30}\\rule{5ex}{1.15ex}}{%\n\\color{red}$\\mkern1mu\\makeballs{#1}$}{O}{c}{F}{F}{L}}\n\\def\\makeballs#1{\\ifnum#1>0\\relax{\\bullet}%\n\\expandafter\\makeballs\\the\\numexpr#1-1\\relax\\fi}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\usepackage[copy-decimal-marker,color=blue]{siunitx}\n\\usepackage{multicol}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\usepackage[left=2.5cm, right=2.5cm, top=3cm, bottom=2cm]{geometry}\n% \\usepackage[a4paper,left=2.5cm,right=2.5cm,top=1.5cm,bottom=1.5cm,\n% marginparsep=3mm,marginparwidth=18mm,\n%]{geometry}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\usepackage{fancyhdr} % Load fancyhdr after geometry!\n\n\\usepackage{tikz}\n\n\\definecolor{gray}{cmyk}{0,0,0,0.4}\n\\definecolor{darkgrey}{cmyk}{0,0,0,0.6}\n\n\\pagestyle{fancy}\n\\renewcommand{\\chaptermark}[1]{\\markboth{#1}{}}\n\\fancyfoot{} % Remove foot fields\n\n\\renewcommand{\\footrulewidth}{0pt}\n\n\\vskip-\\baselineskip\\vskip4pt\n\\ifodd\\count0\\hfill\\begin{tikzpicture}\n\\end{tikzpicture}\\else\\begin{tikzpicture}\n\\end{tikzpicture}\\fi}\n\n\\parindent0em\n\\parskip6pt\n\n\\setlength{\\columnsep}{12pt}\n\\setlength{\\columnseprule}{0pt}\n\n\\usepackage[most]{tcolorbox}\n\n\\newtcbox{\\mylib}{enhanced,nobeforeafter,tcbox raise base,boxrule=0.4pt,top=0mm,bottom=0mm,\nright=0mm,left=4mm,arc=1pt,boxsep=2pt,before upper={\\vphantom{dlg}},\ncolframe=green!50!black,coltext=green!25!black,colback=green!10!white,\noverlay={\\begin{tcbclipinterior}\\fill[green!75!blue!50!white] (frame.south west)\nrectangle node[text=white,font=\\sffamily\\bfseries\\tiny,rotate=90] {Tema} ([xshift=4mm]frame.north west);\\end{tcbclipinterior}}}\n\n\\makeatletter\n\n\\newcommand{\\exercisesidebox}[1]{%\n\\def\\difficulty{#1}%\n\\stackunder[1pt]{\\color{teal}\\bfseries\\sffamily\\large\\the\\c@tcb@cnt@exercise}{\\difbox{\\difficulty}}%\n}\n\\makeatother\n\n\\newcommand\\mySol[1]{\\textcolor{cyan!20!blue}{[$#1$]}}\n\n\\newtcolorbox[auto counter, number within=section]{exercise}[2][]{%\nright skip={20pt},\nenhanced jigsaw,\nsharp corners,\nframe hidden,\ncolback=white,\noverlay={\\node[xshift=0.3em,yshift=-1.5em] (A) at (frame.north west) {\\exercisesidebox{#2}};} %Change the shift values to place the box more appropiately.\n#1\n}\n\n\\begin{document}\n\n\\section*{\\mylib{Velocit\u00e0}}\n\n\\begin{multicols}{2}\n\n\\begin{exercise}{1}\nUn'automobile transita al \\SI{25}{\\kilo\\meter} di un'autostrada alle ore 8:25 e transita al \\SI{29}{\\kilo\\meter} alle ore 8:27. Qual \u00e8 la sua velocit\u00e0 media in \\si{\\kilo\\meter\/\\hour}? \\hfill\\mySol{\\SI{120}{\\kilo\\meter\/\\hour}}\n\\end{exercise}\n\\begin{exercise}{2}\nIn autostrada ogni kilometro \u00e8 contrassegnato da un numero. Guardando fuori dal finestrino, ti accorgi che passano \\SI{36}{\\second} tra un cartello e l'altro. A quale velocit\u00e0 stai procedendo? \\hfill\\mySol{\\SI{100}{\\kilo\\meter\/\\hour}}\n\\end{exercise}\n\\begin{exercise}{3}\nProof that there is a typesetting system better than \\LaTeXe.\n\\end{exercise}\n\\end{multicols}\n\\end{document}\n\n\n\u2022 Please note that this is a starter only... I don't claim to fulfill all requirements here \u2013\u00a0user31729 Jan 1 '17 at 19:19\n\u2022 Christian're a big. Casper :-) compared to you was an amateur: -? Allow me to joke with you. I certainly do not would never have done. When you have a bit more time doing others aesthetic retouch with \\tcolorbox will very nice. At least I have a memory of your work. The size of the exercises numbers can be increased with a bold font? \u2013\u00a0Sebastiano Jan 1 '17 at 20:18\n\u2022 @Sebastiano: Actually, I've provided much of what you wanted. A bigger number is not really advantageous, in my point of view \u2013\u00a0user31729 Jan 1 '17 at 20:51\n\u2022 You are a friend. Thank you very much. I have checked your answer. \u2013\u00a0Sebastiano Jan 1 '17 at 21:37\n\u2022 @Sebastiano: Unaccepting answers won't give you a badge ;-) You must not vote for answers but also for questions. You should vote for 10 questions and another 30 votes may distributed at will amongst questions and answers. See meta.stackexchange.com\/a\/188732\/261926 \u2013\u00a0user31729 Feb 8 '17 at 8:57","date":"2020-02-27 03:01:40","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5574601888656616, \"perplexity\": 3673.7719175389484}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-10\/segments\/1581875146643.49\/warc\/CC-MAIN-20200227002351-20200227032351-00383.warc.gz\"}"} | null | null |
The AlarmSense manual call point has a red LED to indicate when it has been activated. When installed as part of a CFP AlarmSense 2-Wire conventional fire alarm panel system, the unit can be detected as an operated call point as opposed to a detector that has changed in to the alarm state.
This is the technical data for the AlarmSense 2-Wire Surface Mounted Manual Call Point. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,983 |
Q: Send parameter to request on every action for composite component I have a composite component, which has an id I would like to send as a parameter when executing one of many posiible actions inside the composite component. I know I can use something like;
<h:form id="testForm">
<p:commandButton value="#{testReqBean.label}"
actionListener="#{testReqBean.perform()}"
process="@this or @form" update="@form" ajax="true" >
<f:param value="#{cc.attrs.id}" name="CC-Id" />
</p:commandButton>
</h:form>
now, imagine I have many forms or buttons with specific actions inside the composite component... is there a way to define the parameter I want to send in the request just once ? I mean not adding an f:param inside each form/button (depending on the process @form or @this) but one for the whole composite component?
Thanks in advance!
A: Maybe one solution would be to use viewparam but this only works if you can add a request parameter.
<f:metadata>
<f:viewParam value="#{your_bean.your_property_name}" name="request_param"/>
</f:metadata>
The only problem here is that whoever implements your composite component would have to set the above when needed, but it still an abstraction to this problem of having to set the same property for all components in same page.
A: I gather that the <h:form> is enclosed in the composite component itself.
Just use a plain HTML hidden input field.
<h:form>
<input type="hidden" name="CC-Id" value="#{cc.attrs.id}" />
...
<p:commandButton />
<p:commandButton />
<p:commandButton />
...
</h:form>
Unrelated to the concrete problem, having an entire form in a composite is kind of strange. This is then food for read: When to use <ui:include>, tag files, composite components and/or custom components?
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,711 |
{"url":"https:\/\/www.physicsforums.com\/threads\/gravitational-attraction-problem.197841\/","text":"# Gravitational attraction problem\n\n1. Nov 12, 2007\n\n### steven10137\n\n1. The problem statement, all variables and given\/known data\nAn astronaut working in the cargo bay of a space shuttle accidentally released the lifeline when picking up a toolbox. The mission control computers quickly calculated the astronaut was drifting away from the space shuttle at a constant speed of 0.350m\/s. The astronaut, toolbox and space suit had a combined mass of 187.5kg and the space shuttle had a total mass of 6.55x10^3 kg\n\na) What is the gravitational attraction between the astronaut with equipment and the space shuttle when their centres of mass are 8.00m apart?\n\nb) Explain how the astronaut could use the toolbox of mass 17.5 kg to get back to the cargo bay within about 20 seconds when she is 10m away.\n\n2. Relevant equations\n$$\\begin{array}{l} F = \\frac{{Gm_1 m_2 }}{{r^2 }} \\\\ v = \\frac{s}{t} \\\\ \\end{array}$$\n\n3. The attempt at a solution\na)\n$$\\;F = \\frac{{Gm_1 m_2 }}{{r^2 }} = \\frac{{\\left( {6.67 \\times 10^{ - 11} } \\right)\\left( {187.5} \\right)\\left( {6.55 \\times 10^3 } \\right)}}{{\\left( {8.00} \\right)^2 }} = 1.28 \\times 10^{ - 6} \\;N$$\nRight?\n\nb) I'm a little confused here.\nThe only thing I could think of involving the toolbox is to drop it, but how would that make her drop back to the cargo bay??\n\nThanks\nSteven\n\n2. Nov 13, 2007\n\n### catkin\n\nHave you considered conservation of momentum?\n\n3. Nov 13, 2007\n\n### steven10137\n\nhmm ok\n\nso we have the equation;\n$$\\Delta {\\bf{p}}_1 = - \\Delta {\\bf{p}}_2$$\n\nBut I'm still a little confused as to how this would help me in solving part b of the question.\n\n$$\\begin{array}{l} m_1 v_1 = - m_2 v_2 \\\\ \\left( {187.5} \\right)\\left( {0.350} \\right) = - \\left( {17.5} \\right)\\left( {v_2 } \\right) \\\\ v_2 = 3.75\\;ms^{ - 1} \\\\ \\end{array}$$\n\nAny further help would be appreciated.\n\nThanks\nSteven.\n\n4. Nov 13, 2007\n\n### catkin\n\nWhat you have just calculated is the velocity she (so PC!) would have to throw the toolbox to stop her drift away from the space shuttle. What if she threw it harder?\n\n5. Nov 13, 2007\n\n### steven10137\n\nRight so if 3.75 m\/s is the velocity she would have to throw the toolbox to stop her from drifting away, she needs to throw it harder.\n\nWe are told that she needs to drift back within 20 seconds, when she is 10m away.\n\nThis requires a velocity of 10\/20 = 0.5 m\/s\n\nSo she must throw it with a velocity;\nV - 3.75 = 0.5\ntherefore\nV=4.25m\/s?\n\n6. Nov 13, 2007\n\n### catkin\n\nNot quite. You need to take account of the relative masses of astronut and toolbox\n\n7. Nov 13, 2007\n\n### steven10137\n\nargh i'm still having trouble.\n\nThe weight of the astronaut is 170kg and the weight of the toolbox is 17.5 kg.\nA minimum velocity of 3.75 m\/s is required to stop the astronaut drifting away.\n\nI don't know what to do to form some kind of equation ... really quite stuck.\nAny further help would be appreciated.\n\n8. Nov 13, 2007\n\n### catkin\n\nUsing subscript A for astronut and T for toolbox\n$$\\Delta p_A = - \\Delta p_T$$\nMomentum is mass x velocity\n$$\\Delta(m_{A}v_{A}) = - \\Delta(m_{T}v_{T})$$\nMass does not change\n$$m_{A}\\Delta v_{A} = - m_{T} \\Delta v_{T}$$\nThe question is what $\\Delta v_{T}$ is required to change the astronut's velocity from 0.350 away from the shuttle to 0.5 toward the shuttle ...\n\n9. Nov 13, 2007\n\n### steven10137\n\nAhh ok.\n\nSo it should be;\n$$m_{A}\\Delta v_{A} = - m_{T} \\Delta v_{T}$$\n\n$$\\begin{array}{l} \\left( {170} \\right)\\left( { - 0.35 - 0.5} \\right) = - \\left( {17.5} \\right)\\left( {\\Delta v_T } \\right) \\\\ \\Delta v_T = \\frac{{170 \\times - 0.85}}{{ - 17.5}} = 8.26\\;ms^{ - 1} \\\\ \\end{array}$$\n\n10. Nov 14, 2007\n\n### catkin\n\nLooks good\n\n11. Nov 14, 2007\n\n### steven10137\n\nexcellent\n\n12. Nov 15, 2007\n\n### JenDM\n\nHi,\n\nI have a question about escape velocities, is this a good place to post it?\n(I'm new, so sorry if I'm asking a stupid question, or if this is the wrong place to post it)\n\nThanks\n\n13. Nov 16, 2007\n\n### steven10137\n\nWelcome to PF!\n\nYou are in the right place, but just create a new post in the 'introductory physics' area using the 'new topic' button\n\nSteven\n\n14. Nov 18, 2007\n\n### JenDM\n\nOkay, Thanks!\n\nJen","date":"2016-12-08 00:16:32","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.20757430791854858, \"perplexity\": 2004.092213497001}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2016-50\/segments\/1480698542288.7\/warc\/CC-MAIN-20161202170902-00028-ip-10-31-129-80.ec2.internal.warc.gz\"}"} | null | null |
\section{Introduction}
Statistical methods that make use of algebraic topological
ideas to summarize and visualize complex data
are called
topological data analysis (TDA). In particular persistent homology and a method called the persistence diagram are used to measure the persistence of topological features.
As we expect many readers may not be familiar with these concepts, Section~\ref{s:simple-ex}
discusses two examples without going into technical details though a few times it is unavoidable to refer to the terminology used in persistent homology. Section~\ref{s:1:background} discusses the use of the persistence diagram and related summary statistics
and motivates why in Section~\ref{s:1:APF}
a new
functional summary statistic called the accumulative persistence function (APF) is introduced. The remainder of the paper demonstrates the use of the APF in various statistical settings concerned with point clouds and brain artery trees.
\subsection{Examples of TDA}\label{s:simple-ex}
The mathematics underlying TDA
uses technical definitions
and results from persistent homology, see \cite{fasy:etal:14} and the references therein. This theory will not be needed for the present paper. Instead we provide an understanding through examples of the notion of persistence of
$k$-dimensional topological features for a sequence of compact subsets $C_t$ of the $d$-dimensional Euclidean space $\mathbb R^d$, where $t\ge0$, $k=0,1,\ldots,d-1$, and either $d=2$ (Section~\ref{s:toy}) or $d=3$ (Section~\ref{s:brain}).
Recalling that a set $A\subseteq \mathbb R^d$ is path-connected if any two points in $A$ are connected by a curve in $A$,
a $0$-dimensional topological feature of a compact set $C\subset\mathbb R^d$ is a maximal path-connected subset of $C$, also called a {\it connected component} of $C$. The meaning of a $1$-dimensional topological feature is simply understood when $d=2$, and we appeal to this in a remark at the end of Section~\ref{s:toy} when $d=3$.
\subsubsection{A toy example}\label{s:toy}
Let $C\subset \mathbb R^2$ be the union of the three circles depicted in the
top-left panel of Figure~\ref{fig:TDA_tuto}.
The three circles are the $0$-dimensional topological features (the connected components) of $C$, as any curve that goes from a circle to another will be outside $C$. The complement $\mathbb R^2\setminus S$ has four connected components, one of which is unbounded, whilst the three others are the $1$-dimensional topological features of $C$, also called the
{\it loops} of $C$ (the boundary of each bounded connected component is a closed curve with no crossings; in this example the closed curve is just a circle).
\begin{figure}
\centering
\begin{tabular}{ccc}
\includegraphics[scale=0.15]{TDA_tuto1.eps} & \includegraphics[scale=0.15]{TDA_tuto2.eps} &
\includegraphics[scale=0.15]{TDA_tuto3.eps} \\
\includegraphics[scale=0.15]{TDA_tuto4.eps} & \includegraphics[scale=0.15]{TDA_tuto5.eps} &
\includegraphics[scale=0.15]{TDA_tuto6.eps}
\end{tabular}
\caption{The four first panels show a simple example of spherically growing circles centred at (-1,-1), (1,-1), and (0,1) and all with radii 0.5 when time is $0$. The fifth panel shows
the persistence diagram for the connected components ($k=0$) and the loops ($k=1$). The
final panel shows the corresponding accumulated persistence functions.}
\label{fig:TDA_tuto}
\end{figure}
For $t\ge0$, let $C_{t}$ be the subset of points in $\mathbb R^2$ within distance $t$ from $C$.
Thinking of $t$ as time, $C_{t}$ results when each point on $C$ grows as a disc with constant speed one. The algebraic topology (technically speaking the Betti numbers) changes exactly at the times $t=0,0.5,0.62,0.75$, see the first four panels of Figure~\ref{fig:TDA_tuto}: For each topological dimension $k=0,1$, let $t_i^{(k)}$ denote the time of the $i$th change. First, $C_{0}=C$ has three connected components and three loops as
given above; we say that they are {\it born} at time $t_1^{(0)}=t_1^{(1)}=0$ (imaging there was nothing before time 0).
Second, the loops disappear and two connected components merge into one connected component; we say that the loops and one of the connected components {\it die} at time $t_2^{(0)}=t_2^{(1)}=0.5$;
since the two merging connected components were born at the same time, it is decided uniformly at random
which one should die respectively survive; the one which survives then represent the new merged connected component.
Third, at time $t_3^{(0)}=t_3^{(1)}=0.62$, a new loop is born and the two connected component merge into one which is represented by the oldest (first born) connected component whilst the other connected component (the one which was retained when the two merged at time $t_2^{(0)}=0.5$)
dies; the remaining connected component "lives forever" after time $0.62$ and it will be discarded in our analysis.
Finally, at time $t_4^{(1)}=0.75$, the loop dies.
Hence, for each $k=0,1$, there is a multiset of points specifying the appearance and disappearance of each $k$-dimensional topological feature as $t$ grows. A scatter plot of these points is called a {\it persistence diagram}, see Figure~\ref{fig:TDA_tuto} (bottom-middle panel): For $k=0$ (the connected components), the points are $(0,0.5)$ and $(0,0.62)$ (as $(0,\infty)$ is discarded from the diagram) with multiplicities 1 and 1, respectively; and for $k=1$ (the loops), the points are $(0,0.5)$ and $(0.62,0.75)$ with multiplicities 3 and 1, respectively. The term "persistence" refers to that
distant connected components and large
loops are present for a long time; which here of course just corresponds to the three circles/connected components of $C$ and their loops persist for long whilst the last appearing loop has a short lifetime and hence is considered as "noise".
Usually in practice $C$ is not known but a finite subset of points $\{x_1,\ldots,x_N\}$ has been collected as a
sample on $C$, possibly
with noise. Then we redefine $C_{t}$ as the union of closed discs
of radius $t$ and with centres given by the point cloud.
Hence the connected components of $C_0$ are just the points $x_1,\ldots,x_N$, and $C_0$ has no loops.
For $t>0$, it is in general difficult to directly compute the connected components and loops of $C_t$, but
a graph in $\mathbb R^m$ (with $m\ge2$) can be constructed
so that its connected components correspond to those of $C_t$ and moreover
the triangles of the graph may be filled or not in a way so that the loops of the obtained triangulation correspond to those of $C_t$.
\noindent {\it Remark:} Such a construction can also be created in the case where $C_{t}$ is the union of $d$-dimensional closed balls
of radius $t$ and with centres given by a finite point pattern $\{x_1,\ldots,x_N\}\subset\mathbb R^d$. The construction is a so-called simplicial complex such as the \v{C}ech-complex, where $m$ may be much larger than $d$, or the Delaunay-complex (or alpha-complex), where $m=d$, and a technical result (the Nerve Theorem) establishes that it is possible to identify the topological features of $C_t$ by the \v{C}ech or Delaunay-complex, see e.g.\ \cite{Edelsbrunner:Harer:10}. It is unnecessary for this paper to understand the precise definition of these notions, but as $d=2$ or $d=3$ is small in our examples, it is computationally convenient to use the Delaunay-complex.
When $d=3$, we may still think of a 1-dimensional topological feature as a loop, i.e.\ a closed curve with no crossings; again the simplicial complex is used for the "book keeping" when determining the persistence of
a loop. For example, a 2-dimensional sphere has no loops, and a torus in $\mathbb R^3$ has two.
Finally, when $d\ge3$, a $k$-dimensional topological feature is a $k$-dimensional manifold (a closed surface if $k=2$) that cannot "be filled in", but for this paper we omit the precise definition since it is technical and not needed.
\subsubsection{Persistent homology for brain artery trees}\label{s:brain}
The left panel of Figure~\ref{fig:brain_artery_tree} shows an example of one of the 98 brain artery trees analysed in
\cite{steve:16}. The data for each tree specifies a graph in $\mathbb R^3$ consisting of a dense cloud of about $10^5$ vertices (points) together with the edges (line segments) connecting the neighbouring vertices; further details about the data are
given in Section~\ref{s:brains}.
As in \cite{steve:16}, for each tree
we only consider the $k$-dimensional topological features when $k=0$ or $k=1$, using different types of data and sets $C_t$ as described below. Below we consider the tree in Figure~\ref{fig:brain_artery_tree} and let $B\subset\mathbb R^3$ denote
the union of its edges.
Following~\cite{steve:16}, if $k=0$, let
$C_t=\{(x,y,z)\in B: z\le t\}$ be the sub-level set of the height function for the tree at level $t\ge0$ (assuming $C_t$ is empty for $t<0$).
Thus the 0-dimensional topological features at "time/level" $t$ are the connected components of $C_t$.
As illustrated in the left panel of Figure~\ref{fig:brain_artery_tree}, instead of time we may think of $t$ as "water level": As the water level increases,
connected components of the part of $B$ surrounded by water (the part in blue) may be born or die; we refer to this as sub-level persistence.
As in Section~\ref{s:toy}, we represent the births and deaths of the connected component in a persistence diagram which is shown in Figure~\ref{fig:brain_artery_tree} (middle panel).
The persistence of the connected components for all brain artery trees will be studied in several examples later on.
As in \cite{steve:16}, if $k=1$, we let $B$ be represented by a point pattern $C$ of 3000 points subsampled from $B$, and redefine $C_t$ to be the union of balls of radii $t\ge0$ and centres given by $C$ (as considered in the remark at the end of Section~\ref{s:toy}). The loops of $C_t$ are then determined by the corresponding Delaunay-complex.
The right panel of Figure~\ref{fig:brain_artery_tree} shows the corresponding persistence diagram.
The persistence of the loops for all trees will be studied in the examples to follow.
\begin{figure}
\centering \setlength{\tabcolsep}{0.03\textwidth}
\begin{tabular}{ccc}
\includegraphics[scale=0.23]{brain_artery_tree.eps}& \includegraphics[scale=0.15]{brain_artery_tree_PD0.eps} & \includegraphics[scale=0.15]{brain_artery_tree_PD1.eps}
\end{tabular}
\caption{A brain artery tree with the "water level" indicated (left panel) and the persistence diagrams of connected components (middle panel) and loops (right pane).}
\label{fig:brain_artery_tree}
\end{figure}
\subsection{Further background and objective}\label{s:1:background}
The persistence diagram is a popular graphical representation of the persistence of the topological features of a sequence of compact sets $C_t\subset\mathbb R^d$, $t\ge0$.
As exemplified above it consists
for each topological dimension $k=0,\ldots,d-1$
of a multiset $\mathrm{PD}_k$ of points $(b_i,d_i)$ with multiplicities $c_i$, where $b_i$ and $d_i$ is a pair of birth-death times for a $k$-dimensional topological feature obtained as time $t$ grows.
In the majority of literature on TDA, including the analysis in \cite{steve:16} of brain artery trees, long lifetimes
are of main interest whereas short lifetimes are considered as topological noise.
Short lifetimes are of interest in the study of complex structures
such as branch polymers and fractals, see \cite{macpherson:12}; and for brain artery trees \cite{steve:16} noticed in one case that "not-particularly-high persistence have the most distinguishing power in our specific application".
In our examples we demonstrate that short lifetimes will also be of key interest in many situations, including when analysing the brain artery trees dataset from \cite{steve:16}.
\cite{chazal:etal:13}~and \cite{chen:etal:15} note that it is
difficult to apply statistical methodology to persistent diagrams.
Alternative functional summary statistics have been
suggested: \cite{bubenik:12:publish} introduces a sequence of
one-dimensional functions called the persistent landscape,
where his first function is denoted $\lambda_1$ and is considered to be of main interest,
since it provides a measure of the dominant topological
features, i.e.\ the longest lifetimes; therefore we call $\lambda_1$
the dominant function.
\cite{chazal:etal:13} introduce
the silhouette which is
a weighted average of the functions of the persistent landscape,
where the weights control whether the focus is on topological features with long or short lifetimes.
Moreover, \cite{chen:etal:15} consider a kernel estimate of
the intensity function for the persistent diagram viewed as a point pattern.
The dominant function, the silhouette, and the intensity estimate are one-dimensional functions and hence easier to handle than the persistence diagram, however, they provide selected and not
full information about the persistence diagram.
In Section~\ref{s:1:APF}, we introduce another one-dimensional
functional summary statistic called the accumulative persistence function
and discuss its advantages and how it differs from the existing functional summary statistics.
\subsection{The accumulated persistence function}\label{s:1:APF}
For simplicity and specificity, for each topological dimension $k=0,1,\ldots,d-1$, we always assume that the persistence diagram $\mathrm{PD}_k=\{(b_1,d_1,c_1),\ldots,(b_n,d_n,c_n)\}$ is such that $n<\infty$ and $0\le b_i<d_i<\infty$ for $i=1,\ldots,n$.
This assumption will
be satisfied in our examples (at least with probability one).
Often in the TDA literature, $\mathrm{PD}_k$ is transformed to the \textit{rotated and rescaled persistence diagram} (RRPD) given by
$\mathrm{RRPD}_k= \lbrace
(m_1,l_1,c_1), \ldots, (m_n,l_n,c_n) \rbrace$, where $m_i= (b_i+d_i)/2$ is the meanage and $l_i=d_i-b_i$ is the lifetime.
This transformation is useful when
defining our \textbf{\textit{accumulative persistence function} (APF)} by
\begin{equation}\label{e:(a)}
\mathrm{APF}_k(m)=\sum_{i=1}^n c_i l_i 1(m_i\le m),\quad m\geq 0,
\end{equation}
where $1(\cdot)$ is the indicator function and we suppress in the notation
that $\mathrm{APF}_k$ is a function of $\mathrm{RRPD}_k$. The remainder of this section comments on this definition.
Formally speaking, when $\mathrm{RRPD}_k$ is considered to be random, it is viewed as
a finite point process with multiplicities, see e.g.\ \cite{daley:vere-jones:03}.
It what follows it will always be clear from the context whether $\mathrm{PD}_k$ and $\mathrm{RRPD}_k$
are considered as being random or observed, and hence whether $\mathrm{APF}_k$ is a deterministic or random function.
In the latter case, because $\mathrm{APF}_k(m)$ is an accumulative function, its random fluctuations typically increase as $m$ increases.
Depending on the application, the jumps and/or the shape of $\mathrm{APF}_k$ may be of interest as demonstrated later in our examples.
A large jump of $\mathrm{APF}_k$ corresponds to a large lifetime (long persistence). In the simple example shown in Figure~\ref{fig:TDA_tuto}, both jumps of $\mathrm{APF}_0$ are large and indicate the three connected components (circles), whilst only the first jump of $\mathrm{APF}_1$ is large and indicates the three original loops. For the more complicated examples considered in the following it may be hard to recognize the individual jumps.
In particular, as in the remark at the end of Section~\ref{s:toy}, suppose $C_t$ is the union of $d$-dimensional balls of radius $t$ and with centres given by a finite point pattern $\{x_1,\ldots,x_N\}\subset\mathbb R^d$. Roughly speaking we may then have the following features as illustrated later in Example~1.
For small meanages $m$, jumps of $\mathrm{APF}_0(m)$ correspond to balls that merge together for small values of $t$.
Thus, if the point pattern is aggregated (e.g.\ because of clustering), we expect that
$\mathrm{APF}_0(m)$ has jumps and is hence large for small meanages $m$, whilst
if the point pattern is regular (typically because of inhibition between the points), we expect the jumps of $\mathrm{APF}_0(m)$ to happen and to be large for modest values of $m$ (as illustrated later in the middle panel of Figure~\ref{fig:simEx2} considering curves for the Mat{\'e}rn cluster process and the determinantal point process).
For large meanages, jumps of $\mathrm{APF}_0(m)$ are most likely to happen in the case of aggregation. Accordingly, the shape of $\mathrm{APF}_0$ can be very different for these two cases (as illustrated in the first panel of Figure~\ref{fig:simEx2}).
Similar considerations lead us to expect different shapes of $\mathrm{APF}_1$ for different types of point patterns; we expect that
$\mathrm{APF}_1(m)$ is large respective small for the case of aggregation respective regularity when $m$ is small, and the opposite happens when $m$ is large (as illustrated in the last panel of Figure~\ref{fig:simEx2}).
Clearly, $\mathrm{RRPD}_k$ is in one-to-one correspondence to $\mathrm{PD}_k$. In turn,
if all $c_i=1$ and the $m_i$ are pairwise distinct,
then
there is a one-to-one correspondence
between $\mathrm{RRPD}_k$ and its corresponding $\mathrm{APF}_k$. For $k=0$,
this one-to-one correspondence would easily be lost if we had used $b_i$ in place of $m_i$ in \eqref{e:(a)}.
We need to be careful with not over-stating this possible one-to-one correspondence.
For example, imagine we want to compare two APFs with respect to $L^q$-norm ($1\le q\le\infty$) and let
$\mathrm{PD}_k^{(1)}$ and $\mathrm{PD}_k^{(2)}$ be the underlying persistence diagrams. However, when points $(b,d)$ close to the diagonal are considered as topological noise (see Section~\ref{s:toy}), usually the
so-called bottleneck distance $W_\infty(\mathrm{PD}_k^{(1)},\mathrm{PD}_k^{(2)})$ is used, see e.g.\ \cite{fasy:etal:14}. Briefly, for $\epsilon>0$, let
$\mathcal N=\{(b,d):\,b \le d,\,l\le 2 \epsilon\}$ be the set of points at distance $\sqrt{2}\epsilon$ of the diagonal in the persistence diagram, and let $S(b,d)=\{(x,y):\,|x-b|\le \epsilon,\,|y-d|\le \epsilon\}$ be the square with center $(b,d)$, sides parallel to the $b$- and $d$-axes, and of side length $2\epsilon$. Then
$W_\infty(\mathrm{PD}_k^{(1)},\mathrm{PD}_k^{(2)})\le \epsilon$ if $\mathrm{PD}_k^{(2)}$ has exactly one point in each square $S(b_i,d_i)$, with $(b_i,d_i)$ a point of $\mathrm{PD}_k^{(1)}$ (repeating this condition $c_i$ times).
However, small values
of $W_\infty(\mathrm{PD}_k^{(1)},\mathrm{PD}_k^{(2)})$ does not correspond to
closeness of the two
corresponding APFs with respect to $L^q$-norm.
Note that
the dominant function, the silhouette,
and the intensity estimate (see Section~\ref{s:1:background})
are in general not in a one-to-one correspondence with $\mathrm{RRPD}_k$.
Like these functions, $\mathrm{APF}_k$ is a one-dimensional function,
and so it is easier to handle than the sequence of functions
for the persistent landscape in~\cite{bubenik:12:publish}
and the intensity estimate in~\cite{chen:etal:15}
--- e.g.\ confidence regions become easier to plot.
Contrary to the dominant function and the silhouette,
the APF provides information about topological features
without distinguishing between long and short lifetimes.
\subsection{Outline}\label{s:1:outlines}
Our paper discusses various methods based on APFs in different contexts and illustrated
by simulation studies related to spatial point process applications
and
by re-analysing the brain artery trees dataset previously analysed in~\cite{steve:16}.
Section~\ref{s:1:dataset} specifies the setting for these examples.
Sections~\ref{s:conf for APFS}, \ref{s:one sample}, and \ref{s:two sample problem} consider the case of a single APF, a sample of APFs, and two samples of APFs, respectively. Further examples and details appear in Appendix~A-F.
\section{Datasets}\label{s:1:dataset}
\subsection{Simulated data}\label{s:1:simulated dataset}
In our simulation studies
we consider a planar point cloud, i.e.\ a finite point pattern $\{x_1,\ldots,x_N\} \subset \mathbb R^2$,
and study as at the end of Section~\ref{s:toy} how the topological features of $C_t$, the union of closed discs of radii $t$ and centred at $x_1,\ldots,x_N$, change as $t$ grows.
Here
$\lbrace x_1,\ldots,x_N \rbrace$ will be a realisation of a point process
${\bf X}\subset \mathbb R^2$, where the count $N$ is finite.
Thus $\mathrm{PD}_k$ and $\mathrm{RRPD}_k$ can be viewed as finite planar point processes (with multiplicities) and
$\mathrm{APF}_k$ as a random function. Note that $N$ may be random, and conditional on $N$, the points in ${\bf X}$ are not necessarily
independent and identically distributed (IID). This is a common situation in spatial statistics, e.g.\ if the focus is on the point process ${\bf X}$ and the purpose
is to assess the goodness of fit for a specified point process model of ${\bf X}$ when $\lbrace x_1,\ldots,x_N \rbrace$ is observed.
\subsection{Brain artery trees dataset}\label{s:brains}
The dataset in~\cite{steve:16} comes from 98 brain artery trees which can be included
within a cube of side length at most 206 mm; one tree is excluded
"as the java/matlab function crashed" (e-mail correspondence with Sean Skwerer).
They want to capture how the arteries bend through space and to detect age and gender effects.
For $k=0$, sub-level persistence of the connected components of each tree represented by a union of line segments is considered,
cf.\ Section~\ref{s:brain}; then for all meanages, $m_i\le 137$; and the number of connected components is always below 3200.
For $k=1$,
persistence of the loops for the union of growing balls with centres at a point cloud representing the tree is considered,
cf.\ Section~\ref{s:brain};
the loops have a finite death time but
some of them do not die during the allocated time $T=25$
(that is, \cite{steve:16} stop the growth of balls when $t>25$). Thus we shall only consider meanages $m_i\le 25$; then the number of loops is always below 2700.
For each tree and $k=0,1$, most $c_i=1$ and sometimes $c_i>1$.
For each tree and $k=0,1$,
\cite{steve:16} use only the 100 largest lifetimes in their analysis.
Whereas their principal component analysis clearly reveal age effects,
their permutation test based on the mean lifetimes for the male and females subjects
only shows a clear difference when considering $\mathrm{PD}_1$.
Accordingly, when demonstrating the usefulness of $\mathrm{APF}_0$ and $\mathrm{APF}_1$, we will focus on the gender effect and consider the same 95 trees as in \cite{steve:16} (two transsexual subjects are excluded) obtained from $46$ female
subjects and $49$ male subjects; in contrast to \cite{steve:16}, we consider all observed meanages and lifetimes.
In accordance to the allocated time $T=25$, we need to redefine $\mathrm{APF}_1$ by
\begin{equation}\label{e:(b)}
\mathrm{APF}_1(m)=\sum_{i=1}^n c_i l_i 1(m_i\le m,\,m_i+l_i/2\le T),\quad m\geq 0.
\end{equation}
For simplicity we use the same notation $\mathrm{APF}_1$ in \eqref{e:(a)} and \eqref{e:(b)}; although all methods and results in this paper will be presented with the definition \eqref{e:(a)} in mind, they apply as well when considering \eqref{e:(b)}.
Finally, we write
$\mathrm{APF}^F_k$ and $\mathrm{APF}^M_k$ to distinguish between APF's for females and males, respectively.
\section{A single accumulated persistence function} \label{s:conf for APFS}
There exists several constructions and
results on confidence sets for persistence diagrams when the aim is to separate topological signal from noise, see~\cite{fasy:etal:14}, \cite{chazal:fasy:14}, and the references therein.
Appendix~\ref{s:sep} and its accompanying Example~5 discuss
the obvious idea of transforming such a confidence region into one for an accumulate persistence function, where the potential problem is that the bottleneck metric is used for persistence diagrams and this is not corresponding to closeness of APFs, cf.\ Section~\ref{s:1:APF}.
In this section we focus instead on spatial point process model assessment using APFs or more traditional tools.
Suppose a realization of a finite spatial point process ${\bf X}_0$ has been observed and copies ${\bf X}_1,\ldots,{\bf X}_r$ have been simulated under a claimed model for ${\bf X}_0$
so that the joint distribution of ${\bf X}_0,{\bf X}_1,\ldots, {\bf X}_r$ should be exchangeable. That is, for any permutation
$(\sigma_0,\ldots,\sigma_r)$ of $(0,\ldots,r)$, $({\bf X}_{\sigma_0},\ldots,
{\bf X}_{\sigma_r})$ is claimed to be distributed as $({\bf X}_0,\ldots, {\bf X}_r)$; e.g.\ this is the case if ${\bf X}_0,{\bf X}_1,\ldots, {\bf X}_r$ are IID. This is a
common situation for model assessment in spatial point process analysis when a distribution for ${\bf X}_0$ has been specified (or estimated),
see
e.g.\
\cite{Baddeley:Rubak:Wolf:15} and \cite{moeller:waagepetersen:16}.
Denote the $\mathrm{APF}_k$s for ${\bf X}_0,\ldots, {\bf X}_r$ by
$A_0,\ldots,A_r$, respectively, and the null hypothesis that the joint distribution of
$A_0,\ldots,A_r$ is exchangeable by $ \mathcal{H} _0$. Adapting ideas from \cite{myllymaki:etal:16},
we will discuss how
to construct a goodness-of-fit test for $ \mathcal{H} _0$ based on
a so-called global rank envelope
for
$A_0$; their usefulness will be demonstrated in Example~1.
In functional data analysis, to measure how extreme $A_0$ is in comparison
to $A_1,\ldots,A_r$, a so-called depth function is used for ranking
$A_0,\ldots,A_r$, see e.g.\ \cite{lopez:romo:09}.
We suggest using a depth ordering called extreme rank in \cite{myllymaki:etal:16}:
Let $T>0$ be a
user-specified parameter chosen such that it is the behaviour of $A_0(m)$ for $0\le m\le T$
which is of interest.
For $l=1,2,\ldots$, define the $l$-th bounding curves of $A_0,\ldots,A_r$
by
\[ A^{l}_{\mathrm{low} } ( m) = \min_{i = 0,\ldots, r} {} \hspace{-1.5mm} ^{l} A_i(m) \quad \mathrm{and} \quad A^{l}_{\mathrm{upp} } ( m) =\max_{i = 0,\ldots, r} {} \hspace{-1.5mm} ^{l} A_i(m), \quad 0\leq m \leq T, \]
where $ \min {} \hspace{-0.5mm} ^{l}$ and $ \max {} \hspace{-0.5mm} ^{l}$
denote the $l$-th smallest and largest values, respectively, and where $l\le r/2$. Then, for $i=0,\ldots,r$, the extreme rank of $A_i$
with respect to $A_0,\ldots,A_r$ is
\begin{equation*}
R_i = \max \left \lbrace l \ : \ A^{l}_{\mathrm{low} } ( m) \leq A_i(m) \leq A^{l}_{\mathrm{upp} } (m) \quad \mbox{for all } m \in [0,T] \right \rbrace.
\end{equation*}
The larger $R_i$ is, the deeper or more central $A_i$ is among $A_0,\ldots,A_r$.
Now, for a given $\alpha \in (0,1)$, the extreme rank ordering is used to define the $100(1-\alpha)\%$-global rank envelope as the band delimited by the curves $A^{l_\alpha}_{\mathrm{low} }$ and $A^{l_\alpha}_{\mathrm{upp} }$ where
\begin{equation*}
l_\alpha = \max \left\lbrace l \ : \ \frac{1}{r+1} \sum_{i=0}^{r} 1 ( R_i <l ) \leq \alpha \right\rbrace.
\end{equation*}
Under $ \mathcal{H} _0$, with probability at least $ 1-\alpha$,
\begin{equation}\label{e:APFenvelope}
A^{l_\alpha}_{\mathrm{low} } (m) \leq A_0(m) \leq A^{l_\alpha}_{\mathrm{upp} } (m) \quad \mbox{for all } m \in [0,T],
\end{equation}
see~\cite{myllymaki:etal:16}. Therefore, the $100(1-\alpha)\%$-global rank envelope is specifying a
conservative statistical test called the extreme rank envelope test and which
accepts $ \mathcal{H} _0$ at level $100\alpha\%$ if \eqref{e:APFenvelope} is satisfied or equivalently if
\begin{align}\label{e:APFtest}
\frac{1}{r+1} \sum_{i=0}^{r} 1 ( R_i <R_0 ) >\alpha,
\end{align}
cf.\ \cite{myllymaki:etal:16}.
A plot of the extreme rank envelope allows a graphical interpretation of the
extreme rank envelope test and may in case of rejection suggest an alternative model for ${\bf X}_0$.
There exist alternatives to the extreme rank envelope test,
in particular a liberal extreme rank envelope test and a so-called global scaled maximum absolute difference envelope, see \cite{myllymaki:etal:16}.
It is also possible to combine several extreme rank envelopes, for instance by combining $\mathrm{APF}_0$ and $\mathrm{APF}_1$, see \cite{myllymaki:etal:16b}.
In the following example we focus on \eqref{e:APFenvelope}-\eqref{e:APFtest} and briefly remark on results obtained by combining $\mathrm{APF}_0$ and $\mathrm{APF}_1$.
\paragraph*{Example 1 (simulation study).} Recall that a homogeneous Poisson process is a model for
complete spatial randomness (CSR), see e.g.\ \cite{moeller:waagepetersen:00} and the simulation in the
first panel of Figure~\ref{fig:simEx2}.
Consider APFs $A_0, A_1, \ldots, A_r$ corresponding
to independent point processes ${\bf X}_0, {\bf X}_1, \ldots, {\bf X}_r$ defined on a unit square and where ${\bf X}_i$ for $i>0$ is CSR with a given intensity $\rho$ (the mean number of points). Suppose ${\bf X}_0$
is claimed to be CSR with intensity $\rho$, however, the model for ${\bf X}_0$ is given by one of the following four point process models, which we refer to as the true model:
\begin{enumerate}
\item[(a)] CSR; hence the true model agrees with the claimed model.
\item[(b)] A Baddeley-Silverman cell process; this has the same
second-order moment properties as under CSR, see \cite{baddely:silverman:84}. Though from a
mathematical point of view, it is a cluster process, simulated realisations
will exhibit both aggregation and regularity at different
scales, see the second panel of Figure~\ref{fig:simEx2}.
\item[(c)] A Matérn cluster process; this is a model for clustering where each cluster is a homogenous Poisson process within a disc and the centers of the discs are not observed and constitute a stationary Poisson process, see \cite{matern:86}, \\ \cite{moeller:waagepetersen:00}, and the third panel of Figure~\ref{fig:simEx2}.
\item[(d)] A most repulsive Bessel-type determinantal point process (DPP);
this is a model for regularity, see \cite{LMR15}, \cite{biscio:lavancier:16}, and the fourth panel of Figure~\ref{fig:simEx2}.
\end{enumerate}
We let $\rho=100$ or $400$. This specifies completely the
models in (a) and (d), whereas the remaining parameters in the cases (b)-(c) are defined to be the same as those
used in~\cite{robins:turner:16}. In all cases of
Figure~\ref{fig:simEx2}, $\rho=400$. Finally, following the recommendation in~\cite{myllymaki:etal:16}, we let $r=2499$.
\begin{figure
\begin{center}
\resizebox{\columnwidth}{!}{
\begin{tabular}{cccc}
\includegraphics[scale=0.2]{ex_PP_poisson.eps}& \includegraphics[scale=0.2]{ex_PP_cell.eps} &
\includegraphics[scale=0.2]{ex_PP_clust.eps} & \includegraphics[scale=0.2]{ex_PP_dpp.eps}
\end{tabular} }
\caption{Simulated point patterns for a homogeneous Poisson process (first panel),
a Baddeley-Silverman cell process (second panel),
a Matérn cluster process (third panel),
and a most repulsive Bessel-type DPP (fourth panel).}\label{fig:simEx2}
\end{center}
\end{figure}
For each value of $\rho=100$ or $400$, we simulate each point process in (a)-(d)
with the {\sf \bf R}-package
{\tt spatstat}. Then, for each dimension $k=0$ or $1$, we compute the extreme rank envelopes
and extreme rank envelope tests with the {\sf \bf R}-package {\tt spptest}. We repeat all this $500$ times.
Table~\ref{table:reject_envelope_intensite} shows for each case (a)-(d) the percentage of rejection of the hypothesis that ${\bf X}_0$
is a homogeneous Poisson process with known intensity $\rho$.
In case of CSR,
the type one error of the test is small except when $k=0$ and $\rho=100$. As expected in case of (b)-(d), the power of the test is increased when $\rho$ is increased.
For both the Baddeley-Silverman process and the DPP, when $k=0$ and/or $\rho=400$, the power is high and even 100\% in two cases.
For the Matérn cluster process, the power is 100\% when both $\rho=100$ and 400;
this is also the case when instead the radius of a cluster becomes 10 times larger and hence it is not so easy to distinguish the clusters as in the third panel of Figure~\ref{fig:simEx2}.
When we combine the extreme rank envelopes for $\mathrm{APF}_0$ and $\mathrm{APF}_1$, the results are better or close to the best results obtained when considering only one extreme rank envelope.
\begin{table}
\newcommand{\mc}[3]{\multicolumn{#1}{#2}{#3}}
\def1{0.9}
\resizebox{\columnwidth}{!}{
\begin{tabular}[c]{l|c c c c c c c c}
\mc{1}{c|}{} & \mc{2}{c|}{CSR} & \mc{2}{c|}{DPP} & \mc{2}{c|}{Matérn cluster} & \mc{2}{c|}{Baddeley-Silverman} \\
\mc{1}{c|}{} & $\rho=100$ & \mc{1}{c|}{$\rho=400$} & $\rho=100$ & \mc{1}{c|}{$\rho=400$} & $\rho=100$ & \mc{1}{c|}{$\rho=400$} & $\rho=100$ & \mc{1}{c|}{$\rho=400$} \\ \cline{1-9}
\mc{1}{l|}{ $\mathrm{APF}_0$ } & 3.6 & \mc{1}{c|}{4} & 77.4 & \mc{1}{c|}{100} & 100 & \mc{1}{c|}{100} & 45.6 & \mc{1}{c|}{99.6} \\
\mc{1}{l|}{ $\mathrm{APF}_1$ } & 3.8 & \mc{1}{c|}{4.6} & 28.2 & \mc{1}{c|}{57.8} & 100 & \mc{1}{c|}{100} & 65.8 & \mc{1}{c|}{100} \\
\mc{1}{l|}{ $\mathrm{APF}_0$, $\mathrm{APF}_1$ } & 4.8 & \mc{1}{c|}{3.6} & 82.4 & \mc{1}{c|}{100} & 100 & \mc{1}{c|}{100} & 60.8 & \mc{1}{c|}{100}
\end{tabular}
}
\caption{Percentage of point patterns for which the $95\%$-extreme rank envelope test rejects the hypothesis of CSR (a homogeneous Poisson process on the unit square with intensity $\rho=100$ or $\rho=400$)
when the true model is either CSR or one of three alternative point process models.}\label{table:reject_envelope_intensite}
\end{table}
Figure~\ref{fig:graph:extreme_rank_test} illustrates for one of the $500$ repetitions and for each dimension $k=0$ and $k=1$
the deviation of $\mathrm{APF}_k$ from the extreme rank envelope obtained when the true model is not CSR.
For each of the three non-CSR models, $\mathrm{APF}_k$ is outside the extreme rank envelope, in particular when $k=0$ and
both the meanage and lifetime are small, cf.\ the middle panel. This means that
small lifetimes are not noise but of particular importance, cf.\ the discussion
in Section~\ref{s:1:background}.
Using an obvious notation, for small $m$, we may expect that
$\mathrm{APF}^{\mathrm{DPP}}_0(m)<\mathrm{APF}^{\mathrm{CSR}}_0(m)<\mathrm{APF}^{\mathrm{MC}}_0(m)$ which is
in agreement with the middle panel.
For large $m$, we may expect that
$\mathrm{APF}^{\mathrm{DPP}}_0(m)>\mathrm{APF}^{\mathrm{CSR}}_0(m)$ and $\mathrm{APF}^{\mathrm{CSR}}_0(m)>\mathrm{APF}^{\mathrm{MC}}_0(m)$, but only the last relation is detected by the extreme rank envelope in the left panel.
Similarly, we may expect $\mathrm{APF}^{\mathrm{MC}}_1(m)> \mathrm{APF}^{\mathrm{CSR}}_1(m)$ for small $m$, whereas $\mathrm{APF}^{\mathrm{MC}}_1(m)< \mathrm{APF}^{\mathrm{CSR}}_1(m)$
for large $m$, and both cases
are detected in the right panel. Note that
for the Baddeley-Silverman cell process and $k=0,1$, $\mathrm{APF}^{\mathrm{BS}}_k$ has a rather similar behaviour as $\mathrm{APF}^{\mathrm{DPP}}_k$,
i.e.\ like a regular point process and probably because clustering is a rare phenomena.
\begin{figure}
\centering
\begin{tabular}{ccc}
\includegraphics[scale=0.22]{graph_ext_rank_test_H0_800_800.eps}& \includegraphics[scale=0.22]{graph_ext_rank_test_H0_800_800_zoom.eps} & \includegraphics[scale=0.22]{graph_ext_rank_test_H1_800_800.eps} \end{tabular}
\caption{ $95\%$-extreme rank envelope for $\mathrm{APF}_i$ when $i=0$ (left panel and the enlargement shown in the middle panel) or $i=1$ (right panel) together with the curves for the three non-CSR models (Baddeley-Silverman cell process, Matérn cluster process, and Bessel-type DPP). The envelope is
obtained from $2499$ realisations of a CSR model on the unit square and with intensity $100$.}\label{fig:graph:extreme_rank_test}
\end{figure}
A similar simulation study is discussed in~\cite{robins:turner:16} for the models in (a)-(c), but notice that they fix the number of points to be $100$ and
they
use a testing procedure based on the persistent homology rank function, which
in contrast to our one-dimensional APF is a two-dimensional
function and is not summarizing all the topological features represented in a persistent diagram.
\cite{robins:turner:16} show that a test for CSR based on the persistent homology rank function is
useful as compared to various tests implemented in {\tt spatstat}
and which only concern first and second-order moment properties.
Their method is in particular useful, when the true model is a Baddeley-Silverman cell process
with the same first and second-order moment properties as under CSR.
Comparing Figure~4 in~\cite{robins:turner:16} with the results in Table~\ref{table:reject_envelope_intensite} when the true model is a Baddeley-Silverman cell process and $\rho=100$,
the extreme rank envelope test seems less powerful than the
test they suggest.
On the other hand,~\cite{robins:turner:16} observe that
the latter test performs poorly when the true model is a Strauss process (a model for inhibition) or a Matérn cluster process;
as noticed for the Matérn cluster process, we obtain a perfect power when using the extreme rang envelope test.
\section{A single sample of accumulated persistence functions}\label{s:one sample}
\subsection{Functional boxplot} \label{s:2:functional boxplot}
This section discusses the use of a functional boxplot \citep{sun:genton:11} for a sample $A_1,\ldots,A_r$ of $\mathrm{APF}_k$s those joint distribution is exchangeable. The plot
provides a representation of the variation of the curves given by $A_1,\ldots,A_r$ around the most central curve, and it can be used for outlier detection, i.e.\ detection of curves that are too extreme with respect to the others in the sample.
This is illustrated in Example~2 for the brain artery trees dataset and in Appendix~\ref{s:appendix functional boxplot} and its accompanying Example~6 concerning a simulation study.
The functional boxplot is based on an
ordering of the $\mathrm{APF}_k$s obtained using a so-called depth function. For specificity we make the standard choice called the modified band depth function (MBD), cf.\ \cite{lopez:romo:09} and \cite{sun:genton:11}:
For a user-specified parameter $T>0$ and $h,i,j=1,\ldots,r$ with $i<j$, define
\[B_{h,i,j}= \left\lbrace m \in [0,T]: \ \min \left\lbrace A_i\left( m
\right), A_j\left( m \right) \right\rbrace \leq A_h(m) \leq \max
\left\lbrace A_i\left( m \right), A_j\left( m \right) \right\rbrace
\right\rbrace, \]
and denote the Lebesgue measure on $[0,T]$ by $\left| \cdot \right|$. Then the MBD of
$A_h$ with respect to $A_1,\ldots,A_r$ is
\begin{align}\label{e:definition MBD}
\MBD_r(A_h ) = \frac{2}{r(r-1)} \sum_{ 1 \leq i < j \leq r } \left| B_{h,i,j} \right| .
\end{align}
This is the average proportion of $A_h$ on $[0,T]$ between all possible pairs of $A_1,\ldots,A_r$.
Thus, the larger the value of the MBD of a curve is, the more central or deeper it is in the sample. We call the region delimited by the $50\%$
most central curves the central envelope. It is often assumed that a curve outside the central envelope inflated by $1.5$ times
the range of the central envelope is an outlier or abnormal curve ---
this is just a generalisation of a similar criterion for the boxplot of a sample of real numbers ---
and the range may be changed if it is more suitable for the application at hand, see the discussion in~\cite{sun:genton:11} and Example~2 below.
\paragraph*{Example 2 (brain artery trees).}
For the brain artery trees dataset (Section~\ref{s:brains}), Figure~\ref{fig:functional boxplot brain} shows
the functional boxplots of $\text{APF}_k$s for females (first and third panels) respective males (second and fourth panels) when $k=0$ (first and second panels) and $k=1$ (third and fourth panels): The most central curve is plotted in black, the central envelope in purple, and the upper and lower bounds obtained from all the curves except the outliers in dark blue.
Comparing the two left panels (concerned with connected components), the shape of the central envelope is clearly different for females and males, in particular on the interval $[40,60]$, and
the upper and lower bounds of the non-outlier are closer to the central region for females, in particular on the interval $[0,50]$. For the two right panels (concerned with loops), the main difference
is observed on the interval $[15,25]$ where the central envelope is larger for females than for males.
\begin{figure}
\centering
\begin{tabular}{cccc}
\includegraphics[scale=0.17]{fbplot_brain_H0_sex1.eps} & \includegraphics[scale=0.17]{fbplot_brain_H0_sex2.eps}
\includegraphics[scale=0.17]{fbplot_brain_H1_sex1.eps} & \includegraphics[scale=0.17]{fbplot_brain_H1_sex2.eps}
\end{tabular}
\caption{Functional boxplots of APFs for females and males obtained from the brain
artery trees dataset: $\mathrm{APF}^F_0$ (first panel), $\mathrm{APF}^M_0$ (second panel),
$\mathrm{APF}^F_1$ (third panel), $\mathrm{APF}^M_1$ (fourth panel).
The dashed lines show the outliers detected by the 1.5 criterion.}\label{fig:functional boxplot brain}
\end{figure}
The dashed lines in Figure~\ref{fig:functional boxplot brain} show the APFs detected as outliers by the 1.5 criterion, that is $6$ $\mathrm{APF}^F_0$s (first panel), $3$ $\mathrm{APF}^F_1$s (third panel),
$6$ $\mathrm{APF}^M_0$s (second panel), and $4$ $\mathrm{APF}^M_1$s (fourth panel).
For the females, only for one point pattern both $\mathrm{APF}^F_0$ and $\mathrm{APF}^F_1$ are outliers, where
$\mathrm{APF}^F_1$ is the steep dashed line in the bottom-left panel; and
for the males, only for two point patterns
both $\mathrm{APF}^M_0$ and $\mathrm{APF}^M_1$ are outliers, where in one case
$\mathrm{APF}^M_1$ is the steep dashed line in the bottom-right panel. For this case, Figure~\ref{fig:functional boxplot brain - male outlier} reveals an obvious issue: A large part on the right
of the corresponding tree
is missing!
Examples 3 and 4 discuss to what extent our analysis of the brain artery trees will
be sensitive to whether we include or exclude the detected outliers.
\begin{figure}
\begin{center}
\vspace*{0.8cm}
\includegraphics[scale=0.23]{brain_male_outlier.eps}
\caption{Brain artery tree of a male subject with $\mathrm{APF}^M_0$ and $\mathrm{APF}^M_1$
detected as outliers by the 1.5 criterion.}\label{fig:functional boxplot brain - male outlier}
\end{center}
\end{figure}
\subsection{Confidence region for the mean function}\label{s:confidence region mean}\label{s:crmean}
This section considers an asymptotic confidence region for the mean function
of a sample $A_1,\ldots,A_r$ of IID $\mathrm{APF}_k$s. We assume that $D_1,\ldots,D_r$ are the underlying IID $\mathrm{RRPD}_k$s
for the sample so that with probability one, there exists an upper bound $T<\infty$ on the death times
and there exists an upper bound $n_{\mathrm{max}}<\infty$ on the
number of $k$-dimensional topological features. Note that
the state space for such $\mathrm{RRPD}_k$s is
\begin{align*}
\mathcal{D}_{k,T,n_{\mathrm{max}}}=\{\lbrace (m_1,l_1,c_1),\ldots,(m_n,l_n,c_n) \rbrace: \sum_{i=1}^n c_i \leq n_{\mathrm{max}},\,
m_i + l_i/2 \leq T,\, i=1,\ldots,n\}
\end{align*}
and
only the existence and not the actual values of
$n_{\mathrm{max}}$ and $T$ play a role when applying our method below.
For example, in the settings (i)-(ii) of
Section~\ref{s:1:simulated dataset}
it suffices to assume that $\mathbf X$ is included in a bounded region of $\mathbb R^2$ and that the number of points $N$ is bounded by a constant;
this follows from the two versions of the Nerve Theorem presented in~\cite{fasy:etal:14}
and~\cite{Edelsbrunner:Harer:10}, respectively.
We adapt an empirical bootstrap procedure (see e.g.\ \cite{Vaart:Wellner:96}) which in
\cite{chazal:etal:13} is used for a confidence region for the mean of the dominant function
of the persistent landscape and which in our case works as follows.
For $0\le m\le T$, the mean function is given by
$\mu(m)=\mathrm E\left\{A_1(m)\right\}$ and estimated by the empirical
mean function $\overline{A}_r(m) =\frac{1}{r} \sum_{i=1}^r A_{i}(m)$.
Let $A^*_1,\ldots,A^*_r$ be independent uniform draws with replacement from the set $\{A_1,\ldots, A_r\}$
and set $\overline{A^*_r}=\frac{1}{r} \sum_{i=1}^r A^*_{i}$
and $\theta^*=\sup_{m\in[0,T]} \sqrt{r} \left|\overline{A_r}(m)- \overline{A^*_r}(m)\right|$.
For a given integer $B>0$, independently repeat this procedure $B$ times to obtain $\theta_1^*,\ldots,\theta_B^*$.
Then, for $0<\alpha<1$, the $100(1-\alpha)\%$-quantile in the distribution of $\theta^*$ is estimated
by
\begin{align*}
\hat{q}^B_\alpha = \inf \lbrace q\geq 0: \, \frac{1}{B} \sum_{i=1}^B 1( \theta_i^*>q) \leq \alpha \rbrace.
\end{align*}
The following theorem is verified in Appendix~\ref{s:proof of theorem confidence region}.
\vspace{0.5cm}
\begin{theorem}\label{t:1}
Let the situation be as described above.
For large values of $r$ and $B$, the functions $\overline{A_r} \pm \hat{q}^B_\alpha/\sqrt r$
provide the bounds for an asymptotic conservative $100(1-\alpha)\%$-confidence region for the mean APF, that is
\begin{equation*}
\lim_{r \rightarrow \infty} \lim_{B\rightarrow \infty} \mathrm P\left(
\mu(m) \in [\overline{A_r}(m) - \hat{q}_\alpha^B / \sqrt{r}, \overline{A_r}(m) + \hat{q}_\alpha^B / \sqrt{r}] \mbox{ for all } m \in [0,T] \right) \geq 1-\alpha.
\end{equation*}
\end{theorem}
\paragraph*{Example 3 (brain artery trees).}
The brain artery trees are all contained in a bounded region
and presented by a bounded number of points,
so it is obvious that $T$ and $n_{\mathrm{max}}$ exist for $k=0,1$.
To establish confidence regions for the mean of the
$\mathrm{APF}^M_k$s respective $\mathrm{APF}^F_k$s, we apply the bootstrap procedure with $B=1000$.
The result is shown in Figure~\ref{fig:confidence region brain} when all 95 trees
are considered:
In the left panel, $k=0$ and approximatively half of each confidence region overlap with the other confidence region; it is not clear if there is a difference between genders. In the right panel, $k=1$ and the difference is more pronounced, in particular on the interval $[15,25]$.
Similar results and conclusions are obtained
if we exclude the APFs detected as outliers in Example~2.
Of course we should supply with a statistical test to assess the gender effect and such a test is established in Section~\ref{s:two sample problem} and applied in Example~4.
Appendix~\ref{s:appendix crmean} provides the additional Example~7 for a simulated dataset along with a discussion on the geometrical interpretation of the confidence region obtained.
\begin{figure}
\centering
\begin{tabular}{cc}
\includegraphics[scale=0.2]{confidence_region_brain_H0.eps} & \includegraphics[scale=0.2]{confidence_region_brain_H1.eps}
\end{tabular}
\caption{
Bootstrap confidence regions for the mean $\mathrm{APF}^M_k$ and the mean $\mathrm{APF}^F_k$
when $k=0$ (left panel) and $k=1$ (right panel).}\label{fig:confidence region brain}
\end{figure}
\section{Two samples of accumulated persistence functions}\label{s:two sample problem}
This section concerns a two-sample test for comparison of two samples of APFs.
Appendix~\ref{s:group of APFs} presents both a clustering method (Appendix~\ref{s:clusteringappendix}, including Example~9) and a unsupervised classification method (Appendix~\ref{s:supervised classification}, including Example~10) for two or more samples.
Consider two samples of independent $\mathrm{RRPD}_k$s $D_1,\ldots,D_{r_1}$ and $E_1,\ldots,E_{r_2}$,
where each $D_i$ ($i=1,\ldots,r_1$) has distribution $\mathrm{P}_D$
and each $E_j$ has distribution $\mathrm{P}_E$ ($j=1,\ldots,r_2$), and suppose we
want to test the null hypothesis $\mathcal{H}_0$: $\mathrm{P}_D = \mathrm{P}_E=\mathrm{P}$.
Here, the common distribution $\mathrm P$ is unknown and as in Section~\ref{s:confidence region mean} we assume it is concentrated
on $\mathcal{D}_{k,T,n_{\mathrm{max}}}$ for some integer $n_{\mathrm{max}}>0$ and number $T>0$.
Below, we adapt a two-sample test statistic studied in~\cite{praestgaard:95} and \cite{Vaart:Wellner:96}.
Let $r = r_1+r_2$. Let $A_1,\ldots,A_r$ be the $\mathrm{APF}_k$ corresponding to $( D_1, \ldots,D_{r_1},E_1,\ldots,E_{r_2} )$, and denote by $\overline{ A_{r_1}}$ and $\overline{ A_{r_2}}$
the empirical means of $A_1,\ldots, A_{r_1}$ and $A_{r_1+1},\ldots, A_{r_1+r_2}$, respectively.
Let $I=[T_1,T_2]$ be a user-specified interval with $0\le T_1<T_2\le T$ and used for
defining a two-sample test statistic by
\begin{align}\label{e:KSfirst}
KS_{r_1,r_2} &= \sqrt{\frac{r_1 r_2}{r}} \sup_{ m \in I} \left|\overline{ A_{r_1}} (m) - \overline{ A_{r_2}} (m) \right|,
\end{align}
where large values are critical for $\mathcal{H}_0$.
This may be rewritten as
\begin{align} \label{e:stat KS alternative}
KS_{r_1,r_2} = \sup_{ m \in I} \left|\sqrt{\frac{r_2}{r}} G^{r_1}_D(m) - \sqrt{\frac{r_1}{r}}G^{r_2}_E(m) + \sqrt{\frac{r_1 r_2}{r}} \mathrm E \left\{ A_D-A_E \right\}(m) \right|,
\end{align}
where $G_D^{r_1} = \sqrt{r_1} \left( \overline{ A_{r_1}} - \mathrm E\left\{A_D\right\} \right)$
and $G_E^{r_2} = \sqrt{r_2} \left( \overline{ A_{r_2}} - \mathrm E\left\{A_E\right\}\right)$.
By Lemma~\ref{lemma APF donsker}
in Appendix~\ref{s:proof of theorem confidence region} and by the independence of the samples, $G_D^{r_1}$ and $G_E^{r_2}$
converge in distribution to two independent zero-mean Gaussian processes on $ I$, denoted $G_D$ and $G_E$, respectively.
Assume that $ r_1/r \rightarrow \lambda\in (0,1)$ as $r \rightarrow \infty$.
Under
$\mathcal{H}_0$, in the sense of convergence in distribution,
\begin{align}\label{e:stat KS convergence}
\lim_{r \rightarrow \infty} KS_{r_1,r_2} = \sup_{ m \in I} \left|\sqrt{1-\lambda} G_D(m) - \sqrt{\lambda}G_E(m) \right|,
\end{align}
where $\sqrt{1-\lambda} G_D - \sqrt{\lambda}G_E $ follows the same distribution as $G_D$.
If $\mathcal{H}_0$ is not true and $\sup_{ m \in I} \left| \mathrm E \left\{ A_1-A_{r_1+1}\right\}(m)\right| >0$, then
$KS_{r_1,r_2}\rightarrow\infty$ as $r \rightarrow \infty$, see~\cite{Vaart:Wellner:96}.
Therefore, for $0<\alpha<1$ and letting $q_\alpha = \inf \lbrace q: \, \mathrm P( \sup_{ m \in I} \left| G_D(m) \right| > q ) \leq \alpha \rbrace$,
the asymptotic test that rejects $\mathcal{H}_0$ if $KS_{r_1,r_2} \le q_\alpha$
is of level $100\alpha\%$ and of power $100\%$.
As $q_\alpha $ depends on the unknown distribution $\mathrm P$, we
estimate $q_\alpha $ by a bootstrap method:
Let $A^*_1,\ldots, A^*_r$ be independent uniform draws with replacement from $\lbrace A_1,\ldots, A_r \rbrace$.
For $0\le m\le T$, define the empirical mean functions
$ \overline{ A^*_{r_1}} (m) = \frac{1}{r_1} \sum_{i=1}^{r_1} A^*_{i} (m)$ and
$ \overline{ A^*_{r_2}} (m) = \frac{1}{r_2} \sum_{i=r_1+1}^{r_1+r_2} A^*_{i} (m)$,
and compute
\begin{align}
\theta^* &= \sqrt{\frac{r_1 r_2}{r}} \sup_{ m \in I} \left|\overline{ A^*_{r_1}} (m) - \overline{ A^*_{r_2}} (m) \right|. \label{critical value KS}
\end{align}
For a given integer $B>0$, independently repeat this procedure $B$ times to obtain $\theta_1^*,\ldots,\theta_B^*$.
Then we estimate $q_\alpha $ by the $100(1-\alpha)\%$-quantile of the empirical distribution of $\theta_1^*,\ldots,\theta_B^*$, that is
\begin{align*}
\hat{q}^B_\alpha = \inf \lbrace q\geq 0: \, \frac{1}{B} \sum_{i=1}^B 1( \theta_i^*>q) \leq \alpha \rbrace.
\end{align*}
The next theorem is a direct application of Theorem 3.7.7 in~\cite{Vaart:Wellner:96} noticing that the $\mathrm{APF}_k$s are uniformly bounded by $Tn_{\mathrm{max}}$ and they
form a so-called Donsker class, see Lemma~\ref{lemma APF donsker}
and its proof in Appendix~\ref{s:proof of theorem confidence region}.
\vspace{0.5cm}
\begin{theorem}\label{th convergence KS}
Let the situation be as described above.
If $r \rightarrow\infty$ such that $ r_1/r \rightarrow \lambda$ with $\lambda\in (0,1)$,
then under $\mathcal{H}_0$
\begin{align*}
\lim_{r \rightarrow \infty} \lim_{B\rightarrow \infty} \mathrm P\left( KS_{r_1,r_2}> {\hat{q}^B_\alpha} \right) = \alpha,
\end{align*}
whilst if $\mathcal{H}_0$ is not true and $\sup_{ m \in I} \left| \mathrm E \left\{ A_{1}-A_{r_1+1}\right\}(m)\right| >0$, then
\begin{align*}
\lim_{r \rightarrow \infty} \lim_{B\rightarrow \infty} \mathrm P\left( KS_{r_1,r_2}> {\hat{q}^B_\alpha} \right) = 1.
\end{align*}
\end{theorem}
Therefore, the test that rejects $\mathcal{H}_0$
if $KS_{r_1,r_2} > \hat{q}^B_\alpha $ is of asymptotic level $100\alpha\%$ and power $100\%$.
As remarked in~\cite{Vaart:Wellner:96}, by their Theorem~3.7.2 it is possible to present a permutation two-sample test so that the critical value $\hat q^B_\alpha$ for the bootstrap two-sample test has the same asymptotic properties as the critical value for the permutation test.
Other two-sample test statistics than \eqref{e:KSfirst} can be constructed by considering
other measurable functions of $\overline{ A_{r_1}}-\overline{ A_{r_2}}$,
e.g.\ we may consider the two-sample test statistic
\begin{align}
M_{r_1,r_2} &= \int_I \left| \overline{ A_{r_1}} (m) - \overline{ A_{r_2}} (m) \right| \,\mathrm dm \label{critical value L1}.
\end{align}
Then by similar arguments as above but redefining $\theta^*$ in \eqref{critical value KS} by
\begin{align*}
\theta^* &= \sqrt{\frac{r_1 r_2}{r}} \int_{ m \in I} \left|\overline{ A^*_{r_1}} (m) - \overline{ A^*_{r_2}} (m) \right|\,\mathrm dm,
\end{align*}
the test that rejects $\mathcal{H}_0$
if $ M_{r_1,r_2}>\hat{q}^B_\alpha$ is of asymptotic level $100\alpha\%$ and power $100\%$.
\paragraph*{Example 4 (brain artery trees).}
To distinguish between male and female subjects of the brain artery trees dataset, we use the two-sample test statistic $KS_{r_1,r_2}$
under three different settings:
\begin{enumerate}
\item[(A)] For $k=0,1$, we
let $\mathrm{PD}_k'$ be the subset of $\mathrm{PD}_k$ corresponding to the $100$ largest lifetimes.
Then $D_1,\ldots,D_{46}$ and $E_1,\ldots,E_{49}$ are the $\mathrm{RRPD}_k$s obtained from the
$\mathrm{PD}_k'$s associated to female and male subjects, respectively. This is the setting used in \cite{steve:16}.
\item[(B)] For $k=0,1$, we consider all lifetimes and let $D_1,\ldots,D_{46}$ and $E_1,\ldots,E_{49}$ be the $\mathrm{RRPD}_k$s
associated to female and male subjects, respectively.
\item[(C)] The samples are as in setting (B) except that we exclude the $\mathrm{RRPD}_k$s
where the corresponding $\mathrm{APF}_k$ was detected as an outlier in Example~2.
Hence, $r_1=40$ and $r_2=43$ if $k=0$, and $r_1=43$ and $r_2=45$ if $k=1$.
\end{enumerate}
\cite{steve:16} perform a permutation test based on the mean lifetimes for the male and female subjects and
conclude that gender effect is recognized when considering $\mathrm{PD}_1$ ($p$-value $=3\%$) but not $\mathrm{PD}_0$ ($p$-value $=10\%$). For comparison,
under each setting (A)-(C), we perform the two-sample test for $k=0,1$, different intervals $I$, and $B=10000$. In each case, we estimate the $p$-value,
i.e.\ the smallest $\alpha$ such that the two-sample test with significance level $100\alpha\%$ does not reject $\mathcal{H}_0$,
by $\hat p=\frac{1}{B}\sum_{i=1}^B 1( \theta_i^* > KS_{r_1,r_2})$.
Table~\ref{table:gender two sample test} shows the results. Under each setting (A)-(C), using $\mathrm{APF}_0$ we have a smaller $p$-value than in \cite{steve:16} if $I=[0,137]$ and an even larger $p$-value if $I=[0,60]$; and for $k=1$ under setting~(B), our $p$-value is about seven times larger than the $p$-value in \cite{steve:16} if $I=[0,25]$, and else it is similar or smaller. For $k=1$ and $I=[0,25]$, the large difference between our $p$-values under settings (B) and (C) indicates
that the presence of outliers violates the result of Theorem~\ref{th convergence KS} and care should hence be taken. In our opinion we can better trust the results without outliers, where in contrast to \cite{steve:16} we see a clear gender effect when considering the connected components. Notice also that in agreement with the discussion of Figure~\ref{fig:functional boxplot brain} in Example~2, for each setting A, B, and C and each dimension $k=0,1$, the $p$-values in Table~\ref{table:gender two sample test} are smallest when considering the smaller interval $I=[0,60]$ or $I=[15,25]$.
Appendix~\ref{s:appendix two sample test} provides an additional Example~8 illustrating the use of two-sample test in a simulation study.
\begin{table}
\newcommand{\mc}[3]{\multicolumn{#1}{#2}{#3}}
\def1{1}
\begin{center}
\begin{tabular}[c]{l|c c c c}
\mc{1}{c|}{} & \mc{2}{c|}{$\mathrm{APF}_0$} & \mc{2}{c|}{$\mathrm{APF}_1$} \\
\mc{1}{c|}{} & $I=[0,137]$ & \mc{1}{c|}{$I=[0,60]$} & $I=[0,25]$ & \mc{1}{c|}{$I=[15,25]$} \\ \cline{1-5}
\mc{1}{l|}{Setting (A)} & 5.26 & \mc{1}{c|}{ 3.26} & 3.18 & \mc{1}{c|}{ 2.72 } \\
\mc{1}{l|}{Setting (B)} & 7.67 & \mc{1}{c|}{ 3.64} & 20.06 & \mc{1}{c|}{ 1.83 } \\
\mc{1}{l|}{Setting (C)} & 4.55 & \mc{1}{c|}{ 2.61} & 0.92 & \mc{1}{c|}{0.85}
\end{tabular}
\end{center}
\caption{Estimated $p$-values given in percentage of the two-sample test based on $KS_{r_1,r_2}$
used with $\mathrm{APF}_0$ and $\mathrm{APF}_1$ on different intervals $I$ to distinguish
between male and female subjects under settings (A), (B), and (C) described in Example 4.}\label{table:gender two sample test}
\end{table}
\subsubsection*{Acknowledgements}
Supported
by The Danish Council for Independent Research | Natural Sciences, grant 7014-00074B, "Statistics for point processes in space and beyond", and
by the "Centre for Stochastic Geometry and Advanced Bioimaging",
funded by grant 8721 from the Villum Foundation. Helpful discussions with Lisbeth Fajstrup on persistence homology is acknowledged. In connection to the brain artery trees dataset we thank James Stephen Marron and Sean Skwerer for helpful discussions and
the CASILab at The University of North Carolina at Chapel Hill for providing the data
distributed by the MIDAS Data Server at Kitware, Inc. We are grateful to the editors and the referees for useful comments.
| {
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} | 9,598 |
from djinni.support import MultiSet # default imported in all files
from djinni.exception import CPyException # default imported in all files
from djinni.pycffi_marshal import CPyObject, CPyObjectProxy, CPyString
from PyCFFIlib_cffi import ffi, lib
from djinni import exception # this forces run of __init__.py which gives cpp option to call back into py to create exception
class SetStringHelper:
c_data_set = MultiSet()
@staticmethod
def check_c_data_set_empty():
assert len(SetStringHelper.c_data_set) == 0
@ffi.callback("size_t(struct DjinniObjectHandle *)")
def __get_size(cself):
return len(CPyObjectProxy.toPyObj(None, cself))
@ffi.callback("struct DjinniObjectHandle *()")
def __python_create():
c_ptr = ffi.new_handle(SetStringProxy(set()))
SetStringHelper.c_data_set.add(c_ptr)
return ffi.cast("struct DjinniObjectHandle *", c_ptr)
@ffi.callback("void(struct DjinniObjectHandle *, struct DjinniString *)")
def __python_add(cself, el):
CPyObjectProxy.toPyObj(None, cself).add(CPyString.toPy(el))
@ffi.callback("void(struct DjinniObjectHandle * )")
def __delete(c_ptr):
assert c_ptr in SetStringHelper.c_data_set
SetStringHelper.c_data_set.remove(c_ptr)
@ffi.callback("struct DjinniString *(struct DjinniObjectHandle *)")
def __python_next(cself):
try:
with CPyString.fromPy(next(CPyObjectProxy.toPyIter(cself))) as py_obj:
_ret = py_obj.release_djinni_string()
assert _ret != ffi.NULL
return _ret
except Exception as _djinni_py_e:
CPyException.setExceptionFromPy(_djinni_py_e)
return ffi.NULL
@staticmethod
def _add_callbacks():
lib.set_string_add_callback___delete(SetStringHelper.__delete)
lib.set_string_add_callback__get_size(SetStringHelper.__get_size)
lib.set_string_add_callback__python_create(SetStringHelper.__python_create)
lib.set_string_add_callback__python_add(SetStringHelper.__python_add)
lib.set_string_add_callback__python_next(SetStringHelper.__python_next)
SetStringHelper._add_callbacks()
class SetStringProxy:
def iter(d):
for k in d:
yield k
def __init__(self, py_obj):
self._py_obj = py_obj
if py_obj is not None:
self._py_iter = iter(py_obj)
else:
self._py_iter = None
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,927 |
Home Blog Summerlin Energy Solar Company Executive Charged with Theft
Summerlin Energy Solar Company Executive Charged with Theft
Although Summerlin Energy Solar is now bankrupt, the drama surrounding the native Southern Nevada company continues.
Just this month, a high-level executive, Drew Dean Levy, who owned 30 percent of Summerlin Energy Solar, was indicted on theft-related charges after more than 100 homeowners in Las Vegas lost a total of over 1 million dollars.
Energy Inefficient
Levy's responsibilities within the company included day-to-day operations as well as serving as director of sustainable construction and an executive vice president. The indictment reveals that Levy and a business partner, Henry Bankey, instructed their staff members to issue new contracts for solar panel installation—and to continue collecting fees from customers—even though executives knew the company was deep in debt and could not afford to complete the installations.
As if it wasn't bad enough to leave customers with unfinished projects, customers were never refunded their money. Instead, the money that had been collected by Summerlin Energy Solar Company was spent on executive salaries and a variety of other superfluous expenses.
A Series of Missteps
In February of 2016, after a slew of upset customers led the Nevada State Contractors Board to suspend Summerlin Energy Solar's licenses, the company was forced to file for bankruptcy in Las Vegas. As of this report, a warrant has been issued for Drew Dean Levy's arrest.
In 2007, before Levy's business partner, Bankey, served as Summerlin Energy's president and CEO, he had faced criminal charges linked to a company in Utah. He was one of several people federally indicted on 34 counts including bank fraud and money laundering. Bankey will not face charges in the case regarding Summerlin Energy Solar because, sadly, he was killed in November of 2015 when his ex-wife shot him during an argument. Levy will need an experienced criminal defense attorney to help him mount a vigorous defense to the alleged crimes.
By Half Price Lawyers on Friday, September 22, 2017 | {
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Donald Trump promised to resign from his companies, but there's no record he's done so and the existing documents to this subject "are not public at this time." When they will be?
Update, Jan. 23, 2017: The Trump Organization is now filing paperwork on President Trump's resignation from his companies. As of 5:30 p.m., it has filed paperwork for at least 14 companies in Florida.
The documents, which we've posted, are dated Monday, Jan. 23.
CNN also reported on a Trump Organization document, dated Jan. 19, in which Trump states his resignation from more than 400 companies. Trump can resign from his businesses with a private letter. But in order to complete the process, he needs to file with states, each of which has its own deadline.
None of these moves include divesting from his companies, which ethics experts say Trump should do.
At a news conference last week, now-President Donald Trump said he and his daughter, Ivanka, had signed paperwork relinquishing control of all Trump-branded companies. Next to him were stacks of papers in manila envelopes – documents he said transferred "complete and total control" of his businesses to his two sons and another longtime employee.
To transfer ownership of his biggest companies, Trump has to file a long list of documents in Florida, Delaware and New York. We asked officials in each of those states whether they have received the paperwork. As of 3:15 p.m. today, the officials said they have not.
ProPublica's questions to the transition team were referred to an outside public relations firm, Hiltzik Strategies, which declined to comment. The president's team did not allow reporters to view documents, which they said were legal records separating Trump from his eponymous business empire. Dillon's law firm, Morgan Lewis, has not released the records and they declined further comment, saying it doesn't comment on client issues.
ProPublica looked at more than a dozen of Trump's largest companies, which are registered or incorporated in three states. Officials in New York and Delaware said documents are logged as soon as they are received. In Florida, officials told us there is typically a day or two before documents are logged into the system.
Business filings for Trump Organization LLC, Trump's primary holding company, had not been changed, according to New York's Department of State. Wollman Rink Operations LLC, which runs the Wollman Rink in Central Park through an agreement with New York City, hasn't been updated either. Trump is listed as the sole authorized representative of the company.
Ivanka Trump is still listed as the authorized officer on records for two entities related to the Old Post Office in Washington, D.C., which the Trump family bought and turned into a hotel. No changes have been filed for either of the companies, which are registered in Delaware.
In Delaware, where the majority of Trump's businesses are registered, state officials told ProPublica that no amendments have been filed for four businesses tied to the Old Post Office and that the most recent filings for two businesses related to the Trump National Golf Club in Washington, D.C., were made more than a year ago.
In Florida, no changes have been made for years to three key Trump businesses operating there: the Trump International Golf Club in Palm Beach, the Mar-A-Lago Club, and DJT Holdings, which has controlling interest in most of Trump's golf courses in the U.S. and abroad, according to the state's Division of Corporations.
Even if Trump hands over his companies to a new trust, the plan fails to solve many of his bigger business conflicts, experts say. Terms of the trust that would insulate the president from the Trump Organization haven't been made public. Trump's decision not to divest his assets has also been heavily criticized by several former White House attorneys and ethics chiefs. | {
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Nice newer 1 bedroom in Shelton. Attached Carport & 0ff-street parking, 704 sq ft, electric heat, dishwasher & W/D hookups. Great South side of town location. Close to all of town amenities. Water, sewer/septic and garbage included. Credit score of 600 or above required. Minimum of a one year lease. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,392 |
Chicago Med is een Amerikaanse ziekenhuisserie gemaakt door Dick Wolf en Matt Olmstead. Deze televisieserie is een spin-off van Chicago Fire en de eerste aflevering werd in Amerika op 17 november 2015 uitgezonden. De televisieserie laat het wel en wee zien van de spoedeisende hulp van het Gaffney Chicago Medical Center. De televisieserie wordt uitgezonden door NBC.
En in Nederland door Fox en in Vlaanderen door VTM.
Rolverdeling
Hoofdrollen
|-
| Nick Gehlfuss || Will Halstead || arts op de spoedeisende hulp || 1-heden
|-
| Torrey DeVitto || Natalie Manning || (kinder)arts op de spoedeisende hulp || 1-7
|-
| Colin Donnell || Connor Rhodes || trauma/hartchirurg || 1-5
|-
| Brian Tee || Ethan Choi || arts op de spoedeisende hulp || 1-heden
|-
| Oliver Platt || Daniel Charles || arts/psychiater || 1-heden
|-
| Rachel DiPillo || Sarah Reese || geneeskundestudente / psychiater|| 1-4
|-
| S. Epatha Merkerson || Sharon Goodwin || hoofd ziekenhuis || 1-heden
|-
| Yaya DaCosta || April Sexton || verpleegster || 1-6
|-
| Marlyne Barrett || Maggie Lockwood || verpleegster || 1-heden
|-
| Ato Essandoh || Isidore Latham || hartchirurg || 2-heden
|-
| Norma Kuhling || Ava Bekker || chirurg || 2-5
|-
| Dominic Rains || Crockett Marcel || arts || 5-heden
|}
Terugkerende rollen
|-
| Peter Mark Kendall || Joey Thomas || labmedewerker en vriend van Sarah Reese || 1-4
|-
| Roland Buck III || Noah Sexton || geneeskundestudent en broer van April Sexton || 1-5
|-
| Brennan Brown || Sam Abrams || neuroloog || 1-heden
|-
| Jeremy Shouldis || Marty Peterson || arts || 1-heden
|-
| Courtney Rioux || Courtney || ambulancemedewerkster || 1-heden
|-
| Cesar Jaime || Cesar || ambulancemedewerker || 1-heden
|-
| Desmond Gray || Desmond || ambulancemedewerker || 1-heden
|-
| Kara Killmer || Sylvie Brett || ambulancemedewerkster || 1-heden
|-
| Marc Grapey || Peter Kalmick || advocaat van ziekenhuis || 1-heden
|-
| Mia Park || Beth || verpleegster chirurgie || 1-heden
|-
| Julie Marie Berman || Samantha "Sam" Zanetti || chirurg || 1
|-
| Gregg Henry || David Downey || hartchirurg en mentor van dr. Rhodes || 1
|-
| Deron J. Powell || Tate Jenkins || vriend/verloofde van April Sexton || 1-2
|-
| Jeff Hephner || Jeff Clarke || geneeskundestudent || 1-2
|-
| Patti Murin || Nina Shore || patholoog en vriendin van dr. Halstead || 1-2
|-
| Mekia Cox || Robyn Charles || epidemioloog en dochter van dr. Charles || 2-5
|-
| Eddie Jemison || Stanley "de Trol" Stohl || arts en hoofd eerste hulp || 2-4
|-
| Shay Rose Aljadeff || Leah Bardovi || arts || 2-3
|-
| Elena Marisa Flores || Rosado || agente || 3-heden
|-
| Ian Harding || Phillip Davis || 'ex-verloofde' Natalie Manning || 5
|-
| Marie Tredway || Trini || verpleegster || 5-heden
|-
| Jesse Lee Soffer || Jay Halstead || politieman en broer van Will || 1-heden
|}
Afleveringen
Cross-overs
Chicago Med heeft ook regelmatig een cross-over met de televisieseries Chicago Fire en Chicago P.D..
Amerikaanse ziekenhuisserie
Amerikaanse dramaserie
Programma van NBC | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,793 |
Doris H. Sherwood (nee Harkenrider), 97, died Wednesday (Aug. 18, 2010).
Beloved wife of the late John F. Sherwood; dearest mother of Richard J. (Cindy), Gregory (Marcia), Jeffrey W. (Lisa) Sherwood and the late Janice L. Sherwood; devoted grandmother of Neal, Erin, Jessica, Kelsey, Patrick, Andrew and Ian; sister-in-law of Margaret Ann "Muggs" Fitzpatrick and Joyce Sherwood; also survived by many loving nieces, grand-nieces, nephews, grand-nephews and cousins.
Predeceased by her brothers and sisters. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,360 |
{"url":"https:\/\/thanhlongmaybaobi.vn\/puppis-constellation-buvaqut\/43ca01-prime-factorization-definition","text":"Shame And Scandal Ukulele Chords, Deskriptibong Pananaliksik Pdf, Say Cow 10 Times, Jojo Siwa Vs Maddie Ziegler Net Worth, Harrison Maine Real Estate, How Many Climate Zones Are There In China?, \" \/>Shame And Scandal Ukulele Chords, Deskriptibong Pananaliksik Pdf, Say Cow 10 Times, Jojo Siwa Vs Maddie Ziegler Net Worth, Harrison Maine Real Estate, How Many Climate Zones Are There In China?, \" \/>\n\nThe process of finding the Prime Factors of 3045 is called Prime Factorization of 3045. See more ideas about Prime factorization, Math, Teaching math. See: Prime Number. - Definition & Overview, What is Perimeter? Does 2 * 5 * 7 really equal 70? $$2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47$$ Prime Factorization Using the Factor Tree Method . Remember that exponents are used to indicate the number of times a number is multiplied by itself. imaginable degree, area of Suppose f(n) and g(n) are multiplicative and that f(p^r) = g(p^r) for each r and each prime p. Prove that f(n) = g(n) for all n. Factor the following: [{MathJax fullWidth=?false? Let's check our work. 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You get to choose an expert you'd like to work with. - Definition & Examples, What is a Fraction? Remember that your answer should only contain prime numbers, or those numbers that are only divisible by themselves and one. Study.com has thousands of articles about every All other trademarks and copyrights are the property of their respective owners. As we break it down, we create branches, and when we get to the smallest factors, we see the leaves. To get the Prime Factors of 3045, you divide 3045 by the smallest prime number possible. Here's a hint: we can always check our work. If, when multiplied, the product is the original number you began with, you have successfully completed prime factorization. You can test out of the Log in or sign up to add this lesson to a Custom Course. In order to check your work, all you need to do is multiply together the factors you found for your answer. First, we could consider the definition of time complexity class $$\\small P$$ to include weakly-polynomial algorithms from above, but this isn't well accepted. x^2+4x-12 }] and polynomial, [{MathJax fullWidth=?false? \u00a9 copyright 2003-2020 Study.com. Get access risk-free for 30 days, Example: The prime factors of 15 are 3 and 5 (because 3\u00d75=15, and 3 and 5 are prime numbers). However, we aren't done yet because the question asked us to write our answer in exponential form. By the time we're done, you should be skilled at breaking down numbers into their smallest parts! credit-by-exam regardless of age or education level. - Definition & Examples, Free Online Accounting Courses with a Certificate, Tech and Engineering - Questions & Answers, Health and Medicine - Questions & Answers. D\u00e9finitions de Prime_factorization, synonymes, antonymes, d\u00e9riv\u00e9s de Prime_factorization, dictionnaire analogique de Prime_factorization (anglais) We've got the best prices, check out yourself! It was probably deleted, or it never existed here. A helpful method for doing this is factor trees, where the ''leaves'' at the ends of the ''branches'' are the prime factors of the number you started off with. They vary quite a bit in sophistication and complexity. It would be pretty difficult to perform prime factorization if we didn't first refresh our memory on prime numbers. Yes, the number 46! Visit the High School Algebra II: Help and Review page to learn more. HCF and LCM are two such concepts that find importance not only for school-level Mathematics but also in various other exams, like CAT, MAT, recruitment exams for government jobs, etc. Earn Transferable Credit & Get your Degree, What is a Prime Factor? breaking up a number, like 75, into a product of prime numbers. - Definition & Example, How to Find the Prime Factorization of a Number, What are Variables in Math? Select a subject to preview related courses: Since both of these numbers are prime, we've completed our prime factorization. How to Find Prime Factorization. What is the Difference Between Blended Learning & Distance Learning? The tree is the given number. Repeat this process until you end up with 1. - Definition & Examples, What are Equivalent Fractions? Since 2 and 5 are both prime numbers, they cannot be broken down any further and we're now finished. Prime Factorization The prime number factors that multiply to get a composite number. Competitors' price is calculated using statistical data on writers' offers on Studybay, We've gathered and analyzed the data on average prices offered by competing websites. All rights reserved. - Definition & Examples, What is a Multiple in Math? A prime number is a number which has only two factors, i.e. Make sure you leave a few more days if you need the paper revised. | {{course.flashcardSetCount}} Prime factorisation is a method to find the prime factors of a given number, say a composite number. Prime definition is - the second of the canonical hours. Definition of prime factorization in the Definitions.net dictionary. Is there a number we can multiply times 2 to give us 92? Each factor is called a primary. Let's look at an example using the number 70. First, let's do a factor tree for 92. Since 46 is divisible by numbers other than itself and 1, we need to keep going. 3x^2-7x-6. }] Example: The prime factors of 21 are 3 and 7 (because 3\u00d77=21, where 3 and 7 are prime \u2026 - Definition, Methods & Examples, Introduction to Statistics: Help and Review, NC EOC Assessment - Math I: Test Prep & Practice, NY Regents Exam - Algebra I: Test Prep & Practice, Cambridge Pre-U Mathematics: Practice & Study Guide, Introduction to Statistics: Tutoring Solution, ORELA Middle Grades Mathematics: Practice & Study Guide, High School Algebra II: Tutoring Solution, TExES Mathematics 7-12 (235): Practice & Study Guide, GED Math: Quantitative, Arithmetic & Algebraic Problem Solving, High School Algebra I: Homeschool Curriculum. How to use prime in a sentence. The leaves tell you that the prime factorization of 70 is 2 * 5 * 7, so we would still get the same answer! Simply multiply your factors to be sure that they result in your original value. Prime Factorization : Prime Factorization is a method to depict a number as the product of the prime numbers. {{courseNav.course.mDynamicIntFields.lessonCount}} lessons Advantages of Self-Paced Distance Learning, Texas Native American Tribes: History & Culture, The Ransom of Red Chief: Theme, Conflict & Climax, Real Estate Agent & Broker Conduct in New Hampshire, Captain Beatty in Fahrenheit 451: Character Analysis & Quotes, Quiz & Worksheet - Irony in Orwell's 1984, Quiz & Worksheet - Impact of Density & Buoyancy on Plate Tectonics, Quiz & Worksheet - The Iliad Meaning & Purpose, Quiz & Worksheet - Homer's Portrayal of the Gods in The Iliad, Flashcards - Real Estate Marketing Basics, Flashcards - Promotional Marketing in Real Estate, What is Project-Based Learning? Prime factorization: writing a composite number as a product of its prime factors. Improvisation: Test all integers less than \u221an A large enough number will still mean a great deal of work. What is the Prime Factorization of 18. Provide a step-by-step solution. When the numbers are sufficiently large, no efficient, non-quantum integer factorization algorithm is known. So, yes! Biology Lesson Plans: Physiology, Mitosis, Metric System Video Lessons, Lesson Plan Design Courses and Classes Overview, Online Typing Class, Lesson and Course Overviews, Diary of an OCW Music Student, Week 4: Circular Pitch Systems and the Triad, Personality Disorder Crime Force: Study.com Academy Sneak Peek. How Do I Use Study.com's Assign Lesson Feature? Find the prime factorization of 180 and express it in exponential form. Specify when you would like to receive the paper from your writer. Prime Factor. 's' : ''}}. Plus, get practice tests, quizzes, and personalized coaching to help you To learn more, visit our Earning Credit Page. Can every knot be factored into a finite number of prime knots? This time, let's try a factor tree using two different factors that give us 70. What is the prime factorization of 92 expressed in exponential form? (A prime number is a whole number... (A prime number is a whole number... Show Ads If you are sure that the error is due to our fault, please, contact us , and do not forget to specify the page from which you get here. Prime Factorization Algorithms Many algorithms have been devised for determining the prime factors of a given number (a process called prime factorization). succeed. In math, we often use factor trees as a method to perform prime factorization. For example, 3 \u00d7 5 is a factorization of the integer 15, and is a factorization of the polynomial x2 \u2013 4. - Definition & Example, Fractions & Decimals: Real World Applications, What is Ascending Order in Math? The factorization of a number into its constituent primes, also called prime decomposition. What should the definition of a prime knot be? {{courseNav.course.topics.length}} chapters | You can always rely on prime numbers to simplify things! Did you know\u2026 We have over 220 college Here are a few prime numbers to get you started: They might seem like a random bunch of numbers, but they do have that one very important thing in common: they're only divisible by one and themselves. The numbers we are left with cannot be broken down any further. In this lesson, learn about prime numbers and how to find the prime factorization of a number. Let's take a look! credit by exam that is accepted by over 1,500 colleges and universities. - Definition & Examples, What are Whole Numbers? The prime factorization of a number, then, is all of the prime numbers that multiply to create the original number. courses that prepare you to earn K Prime Factorization of Knots. It is also referred as Integer factorization. In mathematics, factors are the numbers that multiply to create another number. Create an account to start this course today. Illustrated definition of Prime Factorization: Finding which prime numbers multiply together to make the original number. Now that we only have prime numbers left, we can find the prime factorization of 70 by looking at the leaves of our tree. Create your account. This will become a handy part of your math skills as you move into higher-level algebra or when you're working with numbers that you're not very familiar with. One way to think about solving for the prime factorization of a number is to picture leaves on a tree. all the prime numbers that when multiplied together equal 4096. Provide a step-by-step solution with an explanation on how each number is obtained. What does prime factorization mean? Or make slid show with them. Hello, BodhaGuru Learning proudly presents an animated video in English which explains how to do prime factorization. At this point, we know that the prime factorization of 92 is 2 * 2 * 23. What about 10? - Definition and Types, What is Subtraction in Math? Finding the prime factorization of a number is breaking it down into the smallest numbers possible. Unlike with other companies, you'll be working directly with your project expert without agents or intermediaries, which results in lower prices. - Definition & Overview, What Is The Order of Operations in Math? If any of the prime factors appear more than once, like 2 in the prime factorization of 92 (2 * 2 * 23), then you can write out the prime factorization in exponential form so that you only have to write the recurring prime factor once, using an exponent to show how many times it recurs. All the prime numbers that \u2026 - Definition & Formula, What Are Like Fractions? When we multiply 7 * 10, we still get 70. Then you take the result from that and divide that by the smallest prime number. Let's do another example. more ... A factor that is a prime number. In mathematics, factorization or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind. Factorization is not usually considered meaningful within number systems possessing division, such \u2026 A prime number is a number that can only be divided by 1 and itself. Example: 24 = 2 * 2 * 2 * 3 Note: all these factors are prime numbers. Today, we will examine the prime number factors of those composite numbers.CFU (include connection to LO) 3. If these factors are further restricted to prime numbers, the process is called prime factorization. What if you had the fraction of 70\/92, and you were asked to simplify it? You'll get 20 more warranty days to request any revisions, for free. | PBL Ideas & Lesson Plans, High School Physics: Homeschool Curriculum, ASSET Geometry Test: Practice & Study Guide, Introduction to Business: Certificate Program, Praxis Physical Education (5091): Practice & Study Guide, The Risk of Eating Disorders to Nutrition, Glencoe Math Connects Chapter 12: Area and Volume, Quiz & Worksheet - Converting Float to Int in Java, Quiz & Worksheet - Function of Graduated Cylinders, Quiz & Worksheet - Theories of Life Science, Quiz & Worksheet - Cross-Bridge Formation, What is Ontology? traduction factorization dans le dictionnaire Anglais - Espagnol de Reverso, voir aussi 'factor into',factor in',factorial',factor', conjugaison, expressions idiomatiques In other words: any of the prime numbers that can be multiplied to give the original number. Hence we can list the prime factors from the list of factors of 48. prime factorization translation in English - French Reverso dictionary, see also 'prime minister',prime minister',prime mover',prime number', examples, definition, conjugation Would we still get the same answer?' Already registered? In this case, instead of having a 2 * 2, we have to express that as 2 to the power of 2, or 2 squared. Information and translations of prime factorization in the most comprehensive dictionary definitions resource on the web. For example, 3\u20222\u20222\u20222 is the prime factorization of 24, since the numbers multiply to 24, and are all \u2026 Prime Factorization of 48. Anyone can earn So how do we show Prime Factorization is in polynomial time? Every number has its own unique set of prime factors, regardless of the path we take to find them. Not sure what college you want to attend yet? Mar 11, 2017 - Explore Scourge's board \"Math - Prime Factorization\", followed by 467 people on Pinterest. You can be confident that the answer is fully simplified because you did the prime factorization of 70 and 92, and there are no other common factors. Think about factors that will give us a product of 70. (Compare definitions and questions for the surface sum.) 7 is a prime number because it is only divisible by itself and 1. first two years of college and save thousands off your degree. According to the prime factor definition, we know that the prime factor of a number is the product of all the factors that are prime that is a number that divides by itself and only one. We do not mention 4 *6 because these factors are not prime numbers. Services. 5 * 7 = 35, so let's break that down once more. 2 * 5 gives us 10, so let's add it to our tree. For example, 2 is a prime number which has two factors, 2 \u00d7 1. The factorization of a number into its prime factors and expression of the number as a product of its prime factors is known as the prime factorization of that number. We already know that 2 is a prime number, so let's try it first. As a member, you'll also get unlimited access to over 83,000 But what if we chose two different factors for 70 during the first step? and career path that can help you find the school that's right for you. The prime factorization of 70 is 2 * 5 * 7. The prime factorization of a number includes ONLY the prime factors, not any products of those prime factors. Enrolling in a course lets you earn progress by passing quizzes and exams. Given a positive integer, the prime factorization is written where the s are the prime factors, each of order. The knot sum operation suggests a definition and some questions. study You may want to refer to the following list of prime numbers less than $$50$$ as you work through this section. But how do we know this is correct? The prime factorization of a number, then, is all of the prime numbers that multiply to create the original number. In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. To unlock this lesson you must be a Study.com Member. What are some examples of prime knots? In 2019, Fabrice Boudot, Pierrick Gaudry, Aurore Guillevic, Nadia Heninger, \u2026 The connection to trees isn't an accident. 1 and the number itself. How to Multiply and Divide Rational Expressions, Over 83,000 lessons in all major subjects, {{courseNav.course.mDynamicIntFields.lessonCount}}, Using Tables and Graphs in the Real World, Scatterplots and Line Graphs: Definitions and Uses, Parabolas in Standard, Intercept, and Vertex Form, Multiplying Binomials Using FOIL and the Area Method, Multiplying Binomials Using FOIL & the Area Method: Practice Problems, How to Factor Quadratic Equations: FOIL in Reverse, Factoring Quadratic Equations: Polynomial Problems with a Non-1 Leading Coefficient, Solving Quadratic Trinomials by Factoring, How to Solve a Quadratic Equation by Factoring, Parabola Intercept Form: Definition & Explanation, Biological and Biomedical Well, I'm sure that fractions are one of your favorite topics, so let's touch on those for a moment. If you hate working with large numbers like 5,733, learn how to turn it into 3 x 3 x 7 x 7 x 13 instead. 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Lancelot Thomas Hogben (Portsmouth, 9 de diciembre de 1895 – Wrexham, 22 de agosto de 1975) fue un científico multidisciplinar de origen británico, miembro de la Royal Society. Fue especialmente conocido en el campo de la fisiología experimental, introduciendo el Xenopus laevis (un anfibio muy parecido al sapo) como modelo para la investigación biológica; en el campo de la estadística médica, oponiéndose activamente al movimiento pro-eugenesia; y en el campo de la divulgación científica. Hogben escribió su autobiografía poco antes de morir, sin llegar a entregar una versión definitiva, por lo cual sus hijos Adrian y Annie Hogben la publicaron como autobiografía no autorizada.
Primeros años
Hogben nació en Portsmouth en 1895, hijo de padres metodistas. Tras mudarse a Londres, asistió a la escuela secundaria de Tottenham, en cuya biblioteca pública descubrió las obras darwinistas y obtuvo una formación extra en zoología de manera autodidacta. Mediante una beca, estudió medicina en el Trinity College de Cambridge, graduándose en fisiología en 1915. Adquirió convicciones socialistas, y participó en la Sociedad Fabiana de Cambridge. Adoptó una visión crítica del dogmatismo marxista, y acabó siendo un miembro muy activo del Partido Laborista. Finalmente, acabaría definiéndose como humanista. Durante la Primera Guerra Mundial, la conciencia pacifista de Hogben le llevó a negarse a realizar el servicio militar obligatorio, por lo cual fue encarcelado durante tres meses. En 1918 Hogben se casó con la economista y sindicalista Enid Charles, a la cual había conocido un año antes en Cambridge.
Genética y fisiología
En 1917 Hogben ya era profesor en el departamento de zoología de la escuela universitaria de Birkbeck, y en 1919 pasó al Imperial College London de ciencia, tecnología y medicina, que contaba con un buen laboratorio de fisiología comparada. Allí Hogben dirigió un estudio sobre el proceso que siguen los pares de cromosomas justo antes de dividirse. Contra la teoría dominante de la tele-sinapsis (conexión a distancia entre cromosomas), Hogben planteó la posibilidad de para-sinapsis (conexión lateral). Esta sería una aportación importante a la teoría del cromosoma.
A partir de 1922, Hogben prosiguió su actividad en Edimburgo, estudiando la fisiología de la glándula pituitaria en colaboración con otros biólogos como Julian Huxley y J. B. S. Haldane. Observaron la capacidad de ciertos vertebrados arcaicos, especialmente ranas, para oscurecer o aclarar el color de piel según el tono cromático de su entorno, lo cual les llevó a plantear un proceso de control endocrino sobre la mutación cromática. Así, aislaron una nueva hormona, la hormona estimulante melanocito, o MSH (del inglés melanocyte stimulating hormone).
En 1923 Hogben co-fundó la Sociedad de Biología Experimental y su revista, junto con Julian Huxley y J. B. S. Haldane, con los cuales compartía una postura anti-eugenesia. Contra la separación estricta entre herencia y ambiente, Hogben planteaba la interdependencia entre la herencia natural y el apoyo ambiental (en inglés, nature & nurture).
En 1927 Hogben y su familia van a vivir a Ciudad del Cabo, Sudáfrica, en una zona con gran diversidad biológica. Allí Hogben fue pionero en el estudio de una especie de sapo, el Xenopus laevis, que posteriormente ha sido muy estudiado en diversos campos de la biología. Hogben demostró que la función ovaria del Xenopus laevis depende de una hormona pituitaria, la gonadotropina. Este hallazgo dio lugar a un test de embarazo relativamente fiable (libre de falsos positivos, si bien con posibilidad de falsos negativos), que fue de gran importancia clínica y social durante algunas décadas, hasta la introducción de nuevos tests inmunológicos en la década de 1960.
A nivel social, en Sudáfrica Hogben y su esposa se opusieron activamente a las políticas de discriminación racial.
Hogben retornó a Inglaterra para asumir la nueva cátedra de Biología Social en la London School of Economics, donde coincidió con figuras importantes de la genética y la demografía, como Ronald Fisher (figura destacada del movimiento pro-eugenesia), J. B. S. Haldane, Lionel Penrose, y Sewall Wright. En 1931 Hogben publicó Genetic Principles in Medicine and Social Science (Principios Genéticos en Medicina y Ciencia Social), tras tratar la materia en una serie de conferencias exitosas en la universidad de Birmingham. Adquirió una casa de campo en Devon, que es evocada por su hijo Adrian Hogben en un comentario que sugiere un enfoque limitado de la ciencia genética:
La Royal Society de Edinburgo concedió a Hogben un premio por su contribución a la genética. En 1946, Hogben publicó An Introduction to Mathematical Genetics (Una introducción a la genética matemática).
Divulgación científica
Ya en 1937 Hogben publicó el manual divulgativo Mathematics for the million: a popular self-educator (Matemáticas para la multitud: un auto-educador popular), cuyo título denotaba la voluntad de Hogben por divulgar el conocimiento científico a una gran multitud de gente. Si bien el libro no consiguió vender un millón de copias, como algunos criticaron con sarcasmo, de hecho llegó a vender medio millón ya en 1978. Gran parte del éxito de este libro se debió a los dibujos y gráficos imaginativos que concibió Hogben, influenciado por el tratamiento gráfico que había observado en el libro Outline of History de H. G. Wells (1922).
En 1938 Hogben publicó el manual divulgativo Science for the Citizen: a self-educator (Ciencia para el ciudadano: un auto-educador), que obtuvo menos éxito que el anterior.
Tras editar The loom of language (El surgimiento del lenguaje) de F. Bodmer, Hogben publicó en 1943 Interglossa: A draft of an auxiliary for a democratic world order (Interglossa: Un esbozo de una lengua auxiliar para un orden mundial democrático). Hogben había observado la dificultad de sus estudiantes para memorizar términos biológicos, debido a su desconocimiento de la etimología, y se acostumbró a aclararles las principales raíces greco-latinas mediante ejemplos que resultaban de valor mnemotécnico. Así fue como empezó a recopilar un vocabulario universal formado por raíces latinas y griegas. Ya durante la Segunda Guerra Mundial, en Birmingham Hogben desarrolló unas pautas de sintaxis y completó el esbozo de una nueva lengua auxiliar internacional basada en el léxico de la ciencia moderna:
Estadística médica
Hogben se encontraba en Oslo dando unas conferencias sobre la falsedad de las teorías raciales nazis, cuando estos ocuparon Noruega, y tuvo que escapar a través de Suecia, Siberia y Japón, emigrando temporalmente a Canadá con su familia. En 1942 volvió a Gran Bretaña, y en 1946 la universidad de Birmingham creó una nueva cátedra en Estadística Médica para él. En 1957, cuatro años antes de su retiro, escribió un análisis crítico de los errores o puntos débiles de la metodología estadística, Statistical Theory: an examination of the contemporary crisis in statistical theory from a behaviourist viewpoint (Teoría estadística: un examen de la crisis contemporánea en teoría estadística desde un punto de vista conductista). Hogben trataba de clarificar el misticismo que rodea el concepto de significación estadística. Esta obra fue reeditada en 1970 como primera parte del libro The Significance Test Controversy (La controversia del test de significación). Así, Hogben cuestionaba las bases de la estadística fisheriana, enfoque predominante sobre análisis de datos en medicina y ciencias sociales.
Humanismo científico
En Dangerous Thoughts (Pensamientos peligrosos), de 1939, Hogben se autodefinía como humanista científico:
Algunas obras
Lancelot T. Hogben (1926). Comparative physiology. London: Sedgwick & Jackson; New York: MacMillan.
— (1927). Principles of evolutionary biology. Cape Town and Johannesburg: Juta.
— (1930). The nature of living matter. London: Kegan Paul.
— Traducción por Miguel López Atocha. Qué es la materia viva? Madrid : [s.n.], 1935 (Tall. Espasa-Calpe).
— (1933). Nature and nurture. London: Williams & Norgate.
— (1936). Mathematics for the million: a popular self-educator. London: Allen & Unwin; New York: Norton 1937.
— Traducción por Eduardo Condeminas Abós. La matemática en la vida del hombre. Barcelona : Joaquín Gil, [1941].
— (1938). Science for the citizen: a self-educator based on the social background of scientific discovery. London: Allen & Unwin; New York: Norton.
— (1939). Dangerous thoughts. London: Allen & Unwin.
— (1943). Interglossa. A draft of an auxiliary for a democratic world order, being an attempt to apply semantic principles to language design. Harmondsworth and New York: Penguin Books.
— (1949). From cave painting to comic strip: a kaleidoscope of human communication. London: Max Parrish.
— Traducción por Luis Jordá. De la pintura rupestre a la historieta gráfica : Un valeidoscopio de los medios humanos de expresión gráfica. Barcelona : Omega, [1953].
— (1955). Man must measure: The wonderful world of mathematics. London: Rathbone.
— Traducción por Ernesto Mascaró. 25000 años de Matemáticas : Historia ilustrada de los números, el cálculo y las medidas. Barcelona : Daimon; Bilbao : Grijelmo, [1959].
— (1957). Statistical theory: An examination of the contemporary crisis in statistical theory from a behaviorist viewpoint. London: Allen & Unwin.
— (1960). Mathematics in the making. London: Macdonald.
— Traducción por Oriol Prats Comes. El universo de los números : Historia y evolución de las matemáticas. Barcelona : Destino, [1966].
— (1963). Essential World English. London: Michael Joseph.
— (1964). The mother tongue. London: Secker & Warburg.
— (1967). Mathematics for the million (4a ed., extensivamente revisasada con material adicional y completamente re-ilustrada). London: George Allen & Unwin; New York: Norton 1937.
— (1968). The wonderful world of mathematics. London: Macdonald.
— Traducción por Gonzalo Medina. El maravilloso mundo de las matemáticas. [Madrid] : Aguilar, [1972].
— (1968). The wonderful world of energy. London: Macdonald.
— Traducción por María Luisa Bravo García. El maravilloso mundo de la energía. [Madrid] : Aguilar, [1972].
— (1969). The wonderful world of communication. London: Macdonald.
— Traducción por Javier Sánchez Cuenca. El maravilloso mundo de la comunicación. [Madrid] : Aguilar, [1972].
Referencias
Enlaces externos
Keynes, Milo. "Lancelot Hogben, FRS (1895-1975)". The Galton Institute, 2001. .
Sahotra Sarkar. "Anecdotal, Historical And Critical Commentaries on Genetics. Lancelot Hogben, 1895—1975". Genetics Society of America, 1996.
Científicos del Reino Unido
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\section*{Introduction}
It is a historic irony that while Einstein devised the EPR experiment in order to show via special relativity that quantum mechanics is incomplete, many current physicists think the EPR experiment exhibits instantaneous effects that contradict strict Lorentz invariance. To be sure, these effects cannot be used to send superluminal signals, so most physicists feel free to ignore this glaring contradiction to special relativity.
This attitude seems incongruous to me, for although it is certainly a consequence of relativity that superluminal signals cannot be sent, this is far from its only significance. Lorentz invariance describes fundamental global symmetries of the universe. As Emmy Noether first showed, such symmetries imply the existence of basic observables. For instance, Galilean invariance leads to the existence of observables such as energy and momentum, whereas Lorentz invariance implies that energy and momentum are no longer observables; rather it is the four-vector consisting of a combination of energy and momentum that is now an observable.
Moreover, if the EPR experiment exhibits superluminal effects, then causality is violated. For in the Born form of EPR, if the two particles are space-like separated, and the measurement of the spin of the first particle has the instantaneous effect of giving a value to the spin of the second, then in a Lorentz frame in which the first is in the future of the second, we have backwards causation. Causality is not a consequence of the formalism in either classical or quantum physics, but is so imbedded in our thinking that it is generally accepted as a universal principle. In fact, causality has real consequences in physics, such as the Kramers-Kronig relations [1].
Not all physicists are willing to hide this apparent contradiction between quantum theory and relativity under the rug. Roger Penrose writes in his popular book ([2] p.446):
``The process does not seem to make sense at all when described in ordinary space-time terms. Consider a quantum state for a pair of particles. Such a state would normally be a correlated state. … Then an observation on one of the particles will affect the other in a non-local way which cannot be described in ordinary space-time terms consistently with special relativity (EPR: the Einstein-Podolsky-Rosen effect)."
I agree with Penrose that if the EPR experiment yields a superluminal effect, then it becomes necessary to re-think space-time geometry. However, I shall argue below that a careful application of Born's Rule to EPR shows no such instantaneous effects, and that EPR is consistent with full Lorentz invariance. I then discuss how it is possible to understand the EPR correlations for non-commuting spin components in different directions without recourse to non-local effects. Finally, I use these results to discuss the Free Will Theorem.
The arguments below are not the only ones against non-local effects of EPR. For instance, the paper [3] gives an argument based on a quantum erasure experiment to show the effects are local. Given the continuing controversy over non-locality, I may perhaps be forgiven for the careful exposition of well-known elementary topics.
\section*{Born's Rule}
It will suffice to state the rule for a discrete observable $A$ of a system which is in a pure state given by the vector $\Omega$. Let $A = \sum \lambda_i P_i$ be the spectral decomposition of $A$. Then Born's rule says that the probability that a measurement of $A$ will give the value $\lambda_k$ is $<\Omega, P_k \Omega>$, and the system after the measurement is in the (unnormalized) state $P_k\Omega$ by the projection postulate.
In particular, if $\Omega$ is an eigenstate of $A$, so that $A \Omega = \lambda_k \Omega$ , say, then the observable $A$ has probability 1 of giving the value $\lambda_k$ and being in the state $P_k\Omega$ after the measurement of $A$. Thus, the observable $A$ is certain to give the value $\lambda_k$ if it is measured. Nothing is said about the value of $A$ before the measurement, and in particular that $A$ already has the value $\lambda_k$ \footnote{The same point is made in various text books. For instance, in [4] we see: ``Thus we can conclude that an eigenstate $\psi_j$ of an operator $P$ is a state in which the system yields with certainty the value $p_j$ [the eigenvalue belonging to $\psi_j$] \textit{when the observable corresponding to $P$ is measured}." [italics added]}. If an eclipse is certain to happen tomorrow, we do not say that it has already happened.
Of course, if the state $\Omega$ has been prepared by a prior measurement of $A$, then $A$ already has the value $\lambda_k$, and the second measurement of $A$ would merely confirm that value. More generally, a prior measurement of an observable $A^\prime = \sum \lambda^\prime_i P_i$ with distinct $\lambda^\prime_i$, suffices. What is important is not the eigenvalues but the set of spectral projections $P_i$ , and the Boolean $\sigma$-algebra $B$ they generate.
It is this Boolean algebra $B$, \textit{the interaction algebra}, that acquires truth values as a result of the measurement, namely, if the measurement gives the value $\lambda_k$, then $P_k$ is true and the other $P_i$'s are false, and this determines the truth values of the generated elements of $B$.
That a measurement gives rise to a Boolean algebra of properties with truth values is not simply a consequence of quantum theory, but is, as noted by Kolmogorov in his classic book on probability [5], the case for any theory. Consider a classical experiment of throwing a die. There are six possible outcomes, the elementary events. Compound events, such as ``an even number of dots" or ``not two dots", are then among the 64 $(= 2^6)$ elements of the Boolean algebra generated by the elementary events. Similarly, the quantum Stern-Gerlach experiment to measure the $z$ component of spin of a spin 1 particle has three possible outcomes on the detector screen, which generate the eight element Boolean algebra of compound events. The outcomes correspond to properties of the measured system, in this case the spin properties $S_z = 0,+1,-1,$ given by the spectral decomposition of $S_z$. These three properties generate a Boolean algebra, the interaction algebra, which is isomorphic to the Boolean algebra of events. The actual spot on the detector determines the truth values of the compound events, and hence of all the elements of the interaction algebra of properties. In the case of an infinite number of properties the algebra generated forms a Boolean $\sigma$-algebra.(See [6] for details).
\section*{EPR}
We will discuss the EPR experiment in the Bohm form of two spin $\frac{1}{2}$ particles in the singlet state $\Gamma$ of total spin 0. Suppose that in that state the two particles are separated and the spin component $s_z$ of particle 1 is measured in some direction $z$. That means that the observable $s_z\otimes I$ of the combined system is being measured.
Let $P_z^\pm = \frac{1}{2} I \pm s_z.$ We have the spectral decomposition $s_z\otimes I = (\frac{1}{2})P^+_z\otimes I + (-\frac{1}{2})P_z^-\otimes I,$ so the interaction algebra is the Boolean algebra $B_1 = \{ P_z^+\otimes I, P_z^-\otimes I,1,0\}.$ We may write the singlet state $\Gamma = \sqrt{\frac{1}{2}}(\psi^+_z\otimes \psi^-_z - \psi_z^- \otimes \psi_z^+),$ where $P_z^\pm \psi_z^\pm = \psi_z^\pm.$
Then \\ $P_z^+\otimes I(\Gamma) = \sqrt{\frac{1}{2}} (\psi_z^+\otimes \psi_z^-)\\ P_z^-\otimes I(\Gamma) = \sqrt{\frac{1}{2}} (\psi_z^- \otimes \psi_z^+).$
By Born's Rule, a measurement of $s_z\otimes I,$ the $z$-component of spin of particle 1, will yield the a value of $\frac{1}{2}$ with probability $\frac{1}{2}$ and a new state $\psi_z^+\otimes \psi_z^-,$ or a value of $-\frac{1}{2}$ with probability $\frac{1}{2}$ and new state $\psi_z^-\otimes \psi_z^+.$
Now, $\psi_z^+\otimes \psi_z^-$ and $\psi_z^-\otimes \psi_z^+$ are eigenstates of $I\otimes s_z$ with respective eigenvalues $-\frac{1}{2}$ and $\frac{1}{2}$. It follows again from Born's Rule that if the $z$-spin component of particle 2 is measured, then it is certain to have the opposite $z$-component of spin.
The operator $I\otimes s_z$ has the spectral decomposition, $I\otimes s_z = (\frac{1}{2})I\otimes P_z^+ + (-\frac{1}{2})I\otimes P_z^-,$ so the interaction algebra of a measurement of particle 2 is the Boolean algebra $B_2 = \{ I\otimes P_z^+, I\otimes P_z^-, 1, 0\}.$\footnote{ The interaction algebra of a measurement of both $s_z\otimes I$ and $I\otimes s_z$ (in any temporal order) is the Boolean algebra $B_1\oplus B_2,$ the 16 element Boolean algebra generated by $P_z^+\otimes I,$ and $I\otimes P_z^-.$ }
Our conclusion is that if particle 1 is measured and its spin is, say, $\frac{1}{2}$ then if particle 2 is measured, it is certain to have the value $-\frac{1}{2}$. It does not mean that particle 2 has acquired the opposite spin before it is measured, as a result of the measurement on particle 1. Those who claim this are reverting to the classical notion of intrinsic properties. The interaction algebra $B_1$ does not contain the properties $I\otimes P_z^+$ or $I\otimes P_z^-$ . The spin components do not have values unless and until the appropriate interaction happens.
As we see, Born's Rule gives us the correct temporal order of events. \emph{A measurement of particle 1 giving a value, say, of $+\frac{1}{2}$ does not imply that particle 2 has acquired a spin value of $-\frac{1}{2}$. What it implies is that if and when it is measured, particle 2 will be certain to yield a spin value of $-\frac{1}{2}$. }
We give two further arguments against non-locality. The first argument is by continuity. If after particle 1 is measured to have spin $\frac{1}{2}$ in the $z$ direction particle 2 is measured in some other direction, at an angle $\theta$ to the $z$-axis, it is easily seen that it will give a value $-\frac{1}{2}$ with probability $\cos^2{(\frac{\theta}{2})}$. It cannot possibly mean that the second particle already has spin in that direction since there is a non-zero probability of $\sin^2{(\frac{\theta}{2})}$ that the spin is $+\frac{1}{2}$. Now, if particle 2's spin does not have a value as $\theta$ tends to 0, it should, by continuity, not have a value at $\theta = 0$, in the $z$ direction.
Here is a second, related argument. A Hermitean operator decomposes Hilbert space into a direct sum of its eigenspaces. The operator $s_z\otimes s_z$ decomposes the four dimensional Hilbert space of the two particles as a direct sum $V_1\oplus V_2$ of the two 2 dimensional eigenspaces with the eigenvalues $\frac{1}{2}$ and $-\frac{1}{2}$. These subspaces have respective bases $\{\psi_z^+\otimes \psi_z^+, \psi_z^-\otimes \psi_z^-\}$ and $\{\psi_z^-\otimes\psi_z^+, \psi_z^+\otimes \psi_z^-\}$. Now consider the direct sum operator $s_d\oplus s_z$ , where $s_d$ is the spin operator in an arbitrary direction $d$. Note that $I\otimes s_z = s_z\oplus s_z$, and that $s_d\oplus s_z$ has eigenstates $\psi_z^-\otimes \psi_z^+$ and $\psi_z^+\otimes\psi_z^-$ with eigenvalues $+\frac{1}{2}$ and $-\frac{1}{2}$. As in EPR, with the two particles in the singleton state $\Gamma$, measure $s_z\otimes I$. Then the pair of particles will enter an eigenstate of $s_d\oplus s_z$ with opposite eigenvalue to $s_z\otimes I.$ Born's Rule tells us that a measurement of $s_d\oplus s_z$ is certain to give this opposite value to $s_z\otimes I.$ If however one maintains that $I\otimes s_z$ acquires an opposite value even before its measurement, it seems equally reasonable that for all $d$ each of the $s_d\oplus s_z$ acquires such an opposite value. However, the operators $s_d\oplus s_z$ do not pairwise commute for different values of $d$ and so do not form a Boolean algebra, as we expect of events that have happened.
A final point: Born's Rule tells us that a given state assigns probabilities to the interaction algebra of every observable. Is it possible that Born's Rule does not exhaust the meaning of a state as a complex vector? After all, the rule gives the probability value $<\Gamma, P \Gamma > = || P\Gamma ||^2$ to the new state. Could the complex vector contain more information than its absolute value squared, which possibly instantaneously effects the other particle? The answer is no, because of the following theorem. An assignment of a probability $p(P)$ to every projection which is a probability measure on every Boolean $\sigma$-algebra of projections determines a unique density operator w such that $p(P) = tr(wP)$. For pure states, which are extreme points of the convex set of such probabilities, $w$ is a rank 1 projection $P = | \Gamma ><\Gamma |,$ and $p(P) = < \Gamma, P \Gamma >.$ This is a consequence of Gleason's Theorem. (See [6] for details). So an assignment of probabilities to the eigenvalues of all observables, as is done by Born's Rule, is sufficient to determine the complex vector, up to a phase factor, defining the state.
\section*{How EPR correlations are possible in different directions}
An EPR correlation in a single direction $z$ is no more difficult to understand than a correlation of two classical spinning particles of total angular momentum zero. For the fixed direction $z$ the EPR correlation may exist simply because the spin components in the $z$ direction have opposite values, and they were set up to be so correlated by preparing the total spin component $S_z$ of both particles to be zero.
Thus, if we prepare the two particles to be in the state given by $S =1$ and $ S_z = 0,$ i.e. $\Gamma = \sqrt{\frac{1}{2}}(\psi_z^+\otimes \psi_z^- + \psi_z^-\otimes \psi_z^+)$, then the above calculation of the EPR correlation applies here to show the same correlation of opposite spin $z$-components. No one can claim for this case that we have superluminal effects, for they only exhibit the correlations which were set up by the initial state that $S_z = 0$ in a single direction $z$.
The new feature of EPR that is so puzzling is that the singleton state $S = 0$ yields spin correlations in different directions, say $z$ and $x$, where the spins $s_z$ and $s_x$ in those directions do not commute, and so cannot both have values simultaneously. How can correlations between spin components of two particles subsist when these spin components do not have values?
To understand how this can happen it is necessary to distinguish between events that have already happened and future contingent events. Thus, for instance, if $a \vee b$ \footnote{$a^\prime$ means ``not $a$", $a\ .\ b$ means ``$a$ and $b$", $a \vee b$ means ``$a$ or $b$", and $a\leftrightarrow b$ means ``$a$ if and only if $b$".} is currently true, then either $a$ is true or $b$ is true. When future events are considered, this no longer the case: if $a \vee b$ is certain to happen, it is not the case that $a$ is certain to happen or $b$ is certain to take place. A simple example of this is a single spin $\frac{1}{2}$ particle. For the spin projections $P_z^\pm$ given by $s_z= \pm\frac{1}{2},$ we have $P_z^+\vee P_z^- = 1,$ so $s_z = \frac{1}{2} \vee s_z = -\frac{1}{2}$ is always certain, for every direction $z$, but neither $s_z = \frac{1}{2}$ nor $s_z = -\frac{1}{2}$ is certain.
Born's Rule deals with precisely this distinction. We have seen that a given state gives the probabilities of the interaction Boolean algebra of projections defined by any observable which may be measured. If a particular observable is measured, the interaction algebra, which gives the events that have actually happened, now takes truth values, and the system enters a projected new state, with the corresponding new probabilities for contingent future properties.
This feature is not restricted to quantum mechanics, but is common to phenomena with uncertain outcomes. Events $a$ and $b$ may each have probabilities other than 1 of occurring, while $a \vee b$ has probability 1. A famous example is Aristotle's sea battle. In [7],Chapter 9, he wrote
``A sea-fight must either take place to-morrow or not, but it is not necessary that it should take place to-morrow, neither is it necessary that it should not take place, yet it is necessary that it either should or should not take place tomorrow. Since propositions correspond with facts, it is evident that when in future events there is a real alternative, and a potentiality in contrary directions, the corresponding affirmation and denial have the same character."
For future contingencies Aristotle retains the Law of Excluded Middle, $a \vee a^\prime$, but drops the Principle of Bivalence, that a proposition is either true or false when it comes to necessary future truth. So for Aristotle the disjunction $a\vee b$ may be certain and so have a truth value, although neither $a$ nor $b$, being uncertain, have a truth value. This extends to other logical connectives (which are definable from $\vee$ and $^\prime$). Thus, logical equivalence $a\leftrightarrow b$ (which may be defined as $(a\ .\ b) \vee (a^\prime \ . \ b^\prime))$ may be certain, while $a$ and $b$ have no truth value. For example, the correlation, given by the equivalence ``My cat stays indoors $\leftrightarrow$ it rains in Princeton" has a truth value (e.g. it may be false, if she stayed out in the rain last Tuesday) even though ``My cat stays indoors" and ``it rains in Princeton" are only assigned probabilities, depending on the weather and my cat's temperament.
Now let us consider the correlations of spin components in the $z$ and $x$ directions in the EPR experiment. These can be written as
\begin{equation}
s_z\otimes I = \frac{1}{2} \leftrightarrow I\otimes s_x = -\frac{1}{2} \hspace{1em} (\mbox{ i.e. } P_z^+\otimes I \leftrightarrow I\otimes P_z^-)
\end{equation}
\begin{equation}
s_x\otimes I = \frac{1}{2} \leftrightarrow I\otimes s_x = -\frac{1}{2} \hspace{1em}(\mbox{ i.e. } P_z^+ \otimes I \leftrightarrow I \otimes P_x^-)
\end{equation}
Now, the Boolean functions (1) $P_z^+\otimes I\leftrightarrow I\otimes P_z^-$ and (2) $P_x^+\otimes I \leftrightarrow I\otimes P_x^-$ are projections, and a calculation shows that these projections commute. One way to measure (1) is to measure $s_z\otimes I$ and $I\otimes s_z$ and check that they have opposite spins. However, this precludes one checking (2) by measuring $s_x\otimes I$ and $I\otimes s_x$, since $s_z\otimes I$ and $s_x\otimes I$ do not commute. A calculation shows that projection (1) is identical to the projection $1-S_z^2$ (i.e.the property $S_z = 0$) and (2) is the projection $1- S_x^2$ (i.e. $S_x = 0$). Here $S_z$ and $S_x$ refer to the combined components of spin of the two particles in the $z$ and $x$ directions. Since $S_z^2$ and $S_x^2$ commute we can measure both and check whether (1) and (2) are simultaneously true. In fact, a calculation shows that the conjunction $S_z = 0\ . \ S_x = 0$ is equal to the projection $S = 0.$ So a measurement of the total spin $S$ suffices to check whether (1) and (2) are simultaneously certain. In particular, if we prepare the two particles to have $S=0$, and separate them carefully so that no outside torque is applied, then the combined system will continue to have $S=0$, by the conservation of angular momentum. In this way we could check that both correlations (1) and (2) are true. Of course, we would not determine the values of the spin components of the individual particles in either the $z$ or $x$ directions this way, but that is the whole point of this discussion: that a compound statement such as a correlation, can have a truth value without the component statements having a value. Just as $a \vee b$ may be certain even though $a$ and $b$ are not, correlations (1) and (2) may be certain even when $s_z\otimes I$ and $s_x\otimes I$ are not.
This way of understanding the EPR correlations for non-commutative spins is not an ad hoc approach to this particular experiment. In [6] we show that any correlation between two systems can be understood in a like manner.
\section*{Discussion}
This is not simply a controversy about semantics, so that in the final analysis, there is no real difference between non-local effects and correlations. We have already mentioned that an eminent relativist, Penrose, was willing to countenance non-causal effects and a change in space-time geometry because of his belief in the non-local effects of EPR. Spin correlations are defined in spin space, and have nothing to do with spatial separation or non-locality. If two classical particles have total spin zero, then they will have opposite components of spin in any direction, whether they are separated or not. They have opposite spins because they were correlated by the condition of total spin zero. Two quantum spin particles of total spin 0 will equally have opposite spins when measured in any direction because they were set up to be correlated by having total spin zero, just as with classical correlations. The new quantum aspect is rooted in correlations such as (1) and (2) above, which are true even though the spins in those directions cannot both have truth values; but as we have argued above, any theory of future contingencies must allow for the truth of compound properties such as $a\leftrightarrow b$ without their constituents having truth values.
How did the idea that EPR exhibits non-locality arise? I believe it is based on a conflation of the two concepts of state and property. If a spin $\frac{1}{2}$ particle is in the state $\psi_z^+$, we say that it is in a state of ``spin $\frac{1}{2}$ in the $z$ direction." This an innocent conflation of the state $\psi_z^+$ and the property $P_z^+.$ The reason that it is harmless is that $s_z$ is a rank 1 operator, so that $P_z^+(= |\psi_z^+ >< \psi_z^+|)$ has a one-dimensional image, viz. the ray containing $\psi_z^+$, and preparing the state $\psi_z^+$ also affirms the truth of the property $P_z^+$ in the interaction algebra $\{ P_z^+, P_z^-, 1, 0\}$.
When we consider a pair of particles in EPR, the situation is different. The observable $s_z\otimes I$ is a rank 2 operator, and $P_z^+\otimes I$ has a 2-dimensional image. The resulting state $\psi_z^+\otimes\psi_z^-$ is a vector lying in this image plane. It is tempting to similarly conflate this state with the conjunction of the properties $s_z\otimes I = +\frac{1}{2}$ and $I\otimes s_z = -\frac{1}{2}$, i.e. a state of spin $+\frac{1}{2}$ for particle 1 and spin $-\frac{1}{2}$ for particle 2. However, only the first particle has been measured, to give truth values to the interaction algebra $\{ P_z^+\otimes I, P_z^-\otimes I, 1, 0\}.$ All we can say of particle 2 is that it is certain to have the property $I\otimes P_z^-$ of spin $-\frac{1}{2}$ if and when it is measured.
\section*{EPR and the Free Will Theorem}
In [8] we proved the Free Will Theorem (FWT) on the basis of three axioms. One of these, the TWIN Axiom, was the EPR experiment adapted to the spin 1 case. We used Lorentz invariance via the third axiom, MIN, to give a contradiction to the assumption that the particles' responses are determined by the past. It is clearly important to the proof that, as we have argued above, there are no superluminal effects in the application of the TWIN axiom. We shall now give the relevant part of the proof of the FWT that uses the EPR axiom TWIN.
At the same time, we recast the FWT by replacing MIN by a new axiom, LIN (Lorentz Invariance). Axiom LIN has the advantage over MIN in separating out the free will of the experimenters, and in dealing with a single experimenter rather than both. LIN is experimentally verifiable in principle, and is, I believe, universally accepted as true.
The axiom LIN says that the result of an experiment is Lorentz covariant: a change of Lorentz frames does not change the results of the experiment. This principle is called Lorentz Symmetry, and has been tested to a high degree of accuracy. (See [9].) To see what this means in the particular setting of the FWT, consider a Stern-Gerlach measurement of a spin $\frac{1}{2}$ particle. To see the covariance in a simple geometric way, we think of the detector screen in the form of diamond, i.e. a square rotated 45 degrees, with a horizontal diagonal that separates the spin up from the spin down spots on the detector. Now a change of Lorentz frame will change the square to a parallelogram, but the diagonal will remain a diagonal, and the spin up and down spots will remain above and below the diagonal in the new frame. It is in this sense that the result of the Stern-Gerlach experiment is Lorentz covariant. A similar covariance for measurement of squared spin components in the axiom SPIN is the content of LIN.
We now state our new axioms.
{\bf SPIN Axiom:} Measurements of the squared components of spin of a spin 1 particle in three orthogonal directions give the answers 1, 0, 1 in some order.
{\bf TWIN Axiom:} For twinned spin 1 particles of total spin $0$, suppose experimenter $A$ performs a triple experiment of measuring the squared spin components of particle $a$ in three orthogonal directions $x, y, z$, while experimenter $B$ measures the squared spin component of particle $b$ in one direction $w$. Then if $w$ happens to be in the same direction as one of $x, y, z,$ experimenter $B$'s measurement will necessarily yield the same answer as the corresponding measurement of $A$.
{\bf LIN Axiom:} For each of the 40 orthogonal triple experiments of $A$, particle $a$'s response as recorded on the detector screen does not depend on the inertial frame with respect to which the response is recorded. Similarly, for each of the 33 single experiments of $B$, particle $b$'s response on the detector is independent of the inertial frame.
(The 33 experiments of $B$ refer to the measurement of $S_z^2$ in the 33 directions $z$ specified in [8], and the 40 experiments of $A$ refer to the triple experiments measuring $(S_x^2, S_y^2, S_z^2)$ in the 40 orthogonal frames $(x,y,z)$ formed out of the 33 directions.)
{\bf The Free Will Theorem:} The axioms SPIN, TWIN and LIN imply that if the experimenters $A$ and $B$ are free to choose from the respective 40 and 33 experiments, the response of each of the spin 1 particles is also free, in the sense that the response is not a function of properties outside its future light cone, i.e. of that part of the universe that is earlier than this response with respect to any given inertial frame.
{\bf Proof:} We divide the proof into two parts: in the first part, the reduction of domain of the two functions $\Theta_a^F$ and $\Theta_b^G$ defined below, we copy the proof in [8], which did not use MIN. In the second part we use LIN in place of MIN to show that these two functions do not exist.
(1) We suppose to the contrary that particle $a$'s response $(i, j, k)$ to the triple experiment with directions $x, y, z$ is given by a function of properties $\alpha, \ldots$ that are earlier than this response with respect to some inertial frame $F$. We write this as \[\Theta_a^F(\alpha) = \mbox{ one of } (0,1,1),(1,0,1),(1,1,0)\] (in which only a typical one of the properties $\alpha$ is indicated). Similarly we suppose that $b$'s response $0$ or $1$ for the direction $w$ is given by a function \[\Theta^G_b(\beta)= \mbox{ one of } 0 \mbox{ or } 1\] of properties $\beta, \ldots$ that are earlier with respect to a possibly different inertial frame $G$ \footnote{Both in the EPR experiment and here $a$'s response may well be a function of future events, such as $b$'s response. In EPR, $b$'s future response of, say, $ -\frac{1}{2}$ implies that $a$'s present response is $+\frac{1}{2}$. Thus $a$'s response is conditioned on $b$'s future response. The conditional probability $p(x|y)$ of $x$ given $y$ is by its definition as $p(x\ .\ y)/p(y)$ independent of the time order of $x$ and $y$, and, in particular, so is the case when this probability is 1, so that $y$ implies $x$. We are here interested not in conditionality, but rather in not violating causality, and so we rule out future events by fiat in the definition of $\theta_a^F(\alpha)$ and $\theta_b^G(\beta).$ }.
(i) If either one of these functions, say $\Theta_a^F,$ is influenced by some information that is free in the above sense (i.e. not a function of $A$'s choice of directions and events $F$-earlier than that choice), then there must be an earliest (``infimum") $F$-time $t_0$ after which all such information is available to $a$. Since the non-free information is also available at $t_0$, all these information bits, free and non-free, must have a value $0$ or $1$ to enter as arguments in the function $\Theta_a^F.$ So we regard $a$'s response as having started at $t_0$.
If indeed, there is any free bit that influences $a$, the universe has by definition taken a free decision near $a$ by time $t_0$, and we remove the pedantry by ascribing this decision to particle $a$.
(ii) From now on we can suppose that no such new information bits influence the particles' responses, and therefore that $\alpha$ and $\beta$ are functions of the respective experimenters' choices and of events earlier than those choices.
Now an $\alpha$ can be expected to vary with $x, y, z$ and may or may not vary with $w$. However, whether the function varies with them or not, we can introduce all of $x, y, z, w$ as new arguments and rewrite $\Theta_a^F$ as a new function (which for convenience we give the same name)
\begin{equation}\tag{*}
\Theta_a^F(x, y, z, w; \alpha^\prime)
\end{equation}
of $x,y,z,w$ and properties $\alpha^\prime$ independent of $x,y,z,w.$
To see this, replace any $\alpha$ that does depend on $x,y,z,w$ by the constant values $\alpha_1, \ldots, \alpha_{1320}$ it takes for the $40\times 33 = 1320$ particular quadruples $x,y,z,w$ we shall use. Alternatively, if each $\alpha$ is some function $\alpha(x,y,z,w)$ of $x,y,z,w,$ we may substitute these functions in (*) to obtain information bits independent of $x,y,z,w.$
Similarly, we can rewrite $\Theta_b^G$ as a function \[\Theta_b^G(x,y,z,w; \beta^\prime)\] of $x,y,z,w$ and properties $\beta^\prime$ independent of $x,y,z,w$.
Now for the particle choice of $w$ that $B$ will make, there is a value $\beta_0$ for $\beta^\prime$ for which \[\Theta_b^G(x,y,z,w; \beta_0)\] is defined. By the above independence of $\beta^\prime$ from $x,y,z,w$, the function $\Theta_b^G(x,y,z,w; \beta_0)$ is defined with the same value $\beta_0$ for all $33$ values of $w$, so we may write it as \[\Theta^G_b(x,y,z,w).\]
Similarly there is a value $\alpha_0$ of $\alpha^\prime$ for which the function \[\Theta^F_a(x,y,z,w) = \Theta^F_a(x,y,z,w;\alpha_0)\] is defined for all 40 triples $x,y,z$ and it is also independent of $\alpha_0$, which argument we have therefore omitted.
(2) Now assume that $A$ and $B$ are space-like separated. For an inertial frame $H$ in which $B$'s choice of the direction $w$ and $b$'s response occurs later than $A$'s choice of $x,y,z$ and $a$'s response, it follows from the definition of $\Theta_a^H$ as a function of properties that are earlier than $a$'s response in the frame $H$ that $w$ is not in the domain of $\Theta_a^H.$ Hence, by LIN, the function $\Theta_a^F$ is also independent of $w$, and we may write it as $\Theta_a^F(x,y,z).$ By similar reasoning, the function $\Theta_b^G$ is independent of $x,y,z$ and we may write it as $\Theta_b^G(w).$
By TWIN, \[\theta_a^F(x,y,z) = (\theta_b^G(x), \theta_b^G(y), \theta_b^G(z)).\]
Since $\theta_a^F(x,y,z)$ takes the value $(1,0,1), (1,1,0),$ or $(0,1,1),$ it follows that $\theta_b^G$ is a function of the given 33 directions which takes the values 1,0,1 in some order on every one of the 40 orthogonal triples $(x,y,z).$ This contradicts the Kochen-Specker Theorem as proved in [8], and shows that our assumption that the response of the particles is determined is false, and proves the theorem.
\newpage
\input{BORNrefs.tex}
\end{document}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,030 |
OM3-T
The Olympus OM system and a camera to rediscover: the OM-2s
Take any line of manual focus 35mm reflex camera from the eighties and mid-nineties, Leica R included. Comparable models will be worth less, on the second hand market, than an Olympus OM-4T, not to mention the OM-3 and its ultra-rare and ultra-expensive offspring, the OM-3T. Why, in spite of their very serious limitations, are the single digit OM cameras so sought after? In this test of the OM-2s, the little brother of the OM-4, we'll try and find out why.
The OM system
Olympus OM-1n next to a 35mm film cartridge. The competition needed almost 10 years to introduce more compact SLRs, but they were designed for beginners. In the enthusiast-amateur and pro categories, the OM family remains unchallenged to this day.
Launched in the early 70s, the Olympus OM-1 and its system of lenses and accessories were incredibly compact, very well designed, and at the same time solid enough to please the pros and the very serious amateurs. The competition (Nikon in particular) needed years to develop models approaching the size of the OM-1, which sold by the millions.
The OM-2, introduced in 1974 with the same ergonomics and a similar external appearance, was the automatic exposure version of the OM-1. It pioneered the use of direct exposure metering in the film chamber, and was the first camera with Through The Lens Flash metering. The competitors followed Olympus' example, and almost every SLR cameras introduced after 1985 measures the exposure in the film chamber and offers TTL flash metering.
The OM-2s, OM-3 and OM-4 which followed in the eighties were relatively minor updates of the previous models. They shared a new body and had much more elaborate metering options, but they retained the relatively slow shutter of the OM-1 and OM-2. Their viewfinders were not as great as the ones of the OM-1 and OM-2, and the first models had some reliability issues. The OM-3T and OM-4T (with titanium top and bottom plates and more reliable electronics) raised the level of quality of the OM line, and soldiered on until Olympus finally stopped the production of film cameras, in 2002.
The limitations of OM-x cameras
The incredible prices reached by the OM-3T and to a lesser extent by the OM-4T on the second hand market could lead us to believe that those cameras are perfect. They have some unique qualities (more about them in the next section), but they also suffer from serious limitations.
The textile focal plane shutter of the OM-2S is virtually identical to the mechanism mounted in the other automatic OM cameras. In this version it is limited to 1/1000sec. The OM-3 and OM-4 reach 1/2000sec, but the synchro flash speed remains the same for all models: 1/60sec.
The textile focal plane shutter.
The shutter of the OM-1 and OM-2 cameras was in line with what the competition proposed in 1975 (1/1000 Sec, Synchro Flash at 1/60sec), but ten years later, the OM-3 and OM-4 were confronted to the Nikon FM2n, FE2 and FA or the Canon T90, all with metallic shutters reaching 1/4000sec and offering 1/250sec synchro flash. A slow shutter is a serious limitation now that 400 ISO films are the de facto standard (it means that in sunlight you can only work with an aperture of f:16 or f:11), and it reduces the opportunities to use the fill-in flash technique in the open air.
The reliability and the power consumption of the electronics
Olympus experienced some difficulties when the engineering of cameras became almost exclusively focused on electronics. All models suffered from glitches, some models more than others. The first OM-2S cameras, in particular, were plagued with electronics related issues. The bad apples have most probably been eliminated in the past 25 years, and reliability should not be too much of a concern for a photographer buying one of those cameras now. Excessive power consumption affected all the models launched in the first half of the eighties (when using a flash or at rest), and was only brought back under control with the OM-3T and OM-4T models.
Olympus missed the auto-focus revolution
Olympus' first auto-focus SLR was really horrible (and the lenses were not better – with no focusing ring at all). The auto-focus system was rapidly withdrawn from the market. Olympus did not even try to improve on it. They just placed the manual focus OM system on life support without any significant upgrade for the subsequent 15 years. Contrarily to Nikon's or Pentax's film SLRs, which can still share some lenses and accessories with modern digital SLRs, the OM bodies and lenses have very little in common with today's digital cameras. If brought along modern Olympus digital cameras on a photo-shoot , they will need their own lenses, which at least doubles the weight and the volume of the equipment to be carried around.
Unique qualities of the OM series
The cameras of the OM series also have unique qualities, which gained them the unconditional support of a small group of passionate photographers.
The size and the fit and finish:
When Olympus launched the OM-1, it was the smallest 35mm Single Lens Reflex camera ever made. The competition needed almost 10 years to catch up, but if their cameras were marginally smaller, they were entry level models made of plastic, and could not be compared to the high end "single digit" OM cameras available at the same time. The fit and finish of the OM-1 was impressive, and the black or grey ""T' and "ti" models remain among the nicest cameras ever made.
The Viewfinder:
Bring an OM-1 to your eye, and you will still be impressed by the viewfinder. With 97% coverage and 92% magnification, it presents a very large image of the subject. They eye relief is short (at 14mm approximately), but since the viewfinder does not provide any information at the periphery of the image, it is not an issue, and even photographers wearing glasses can see all of the scene easily. Only much bulkier and high-end cameras (such as the Nikon F3) offer a better user experience. The subsequent models were still very good, but not as impressive: over the years, Olympus had to shoe-horn LCD displays at the periphery of the viewfinder (in support of the increasingly elaborate metering system of the cameras), and introduced a diopter adjustment mechanism, which led to a progressive reduction of the magnification. But the user experience was still far better than what a Nikon FE2 or FE had to offer.
(More about the viewfinder of SLRs in this Post of CamerAgx)
The ergonomics and the level of control offered to the photographer
Until the end, Olympus remained attached to the ergonomics defined with the OM-1: the shutter speed selection ring was positioned around the lens mount, and the aperture ring was pushed at the front of the lens. It made a lot of sense in 1970 – when the command of the mechanical shutter was stiff and the ring needed to be as large as possible, and when the settings on a lens were limited to the focus and the aperture. The generalization of (very soft) electronic shutter commands and of zoom lenses made the Olympus ergonomics a bit of anachronistic at the end of the seventies.
Olympus OM-2S. The shutter speed command ring is positioned around the lens mount, and the aperture ring is at the front of the lens. 1/60 and B shutter speeds are mechanical (marked in red). Setting the camera at 1/60 when at rest addresses the battery leak issues this camera is famous for.
More important is the conscious decision made by Olympus to put the photographer in total control of metering. When most of the cameras manufacturers were trying to make photography less intimidating (using databases and analysis algorithms to make the process of exposure determination transparent to the photographer), Olympus decided to offer spot metering (OM-2s/OM2-SP) and multi-spot metering (OM-3 and OM-4 bodies). On the OM-3 and OM-4 cameras, the photographer could make up to eight successive spot measurements, whose result were presented in the viewfinder on an analog bar scale showing each individual result and the average, letting the photographer determine the exposure of the picture following the principles of the zone system. The cameras also had a "shadow" and a "highlight" push button, for the times when you have to shoot a Scottish terrier on a pile of coal and a white cat on the white sofa (more about metering in this post of Cameragx)
This approach did not get a lot of traction on the marketplace (matrix metering has become the standard) but the small group of photographers who really wanted to be in control of the exposure of their pictures still use OM-3 and OM-4 cameras today. Their unique metering capabilities also explain why they kept their value so well.
Olympus used the same body for the OM-2S and for the OM-3 and OM-4. They are marginally larger than the OM-1 and OM-2 models, but have a built-in flash mount.
The control of metering and exposure modes of the OM2S. The OM-2S operates in aperture priority and in program mode with a center weighted average metering, and in semi-auto mode with spot metering. Those combinations make a lot of sense.
Olympus OM-2S was named OM-2 spot/program (or SP) on some markets.
Using the OM-2S
The OM-2S is not as sought after as its OM-4 or OM-3 brothers. It is true that the latter gained their status of "classics" when the "T" and "ti" versions were launched. The OM-2S never benefited from the titanium parts of the later OM bodies, and remained an entry-level model during its short commercial career. The OM-2S also gained a bad reputation because of power leak issues (the integrated circuit was not properly designed, and when a flash was mounted on the body, it tended to drain the battery of the camera rapidly).The OM2-s is a very pleasant camera to use, though. Compact, with a large viewfinder, it leaves total control of the exposure to the photographer in the semi-auto mode, which is logically combined with the Spot metering system. The exposure parameters are presented on bar graph at the left of the viewfinder. They're visual and very easy to read. When determining the right exposure is not too tricky, the photographer can rely on the automatic (aperture priority) mode, which uses a more conventional center weighted average metering. The Program mode is almost useless, because the photographer has no way of knowing the aperture selected by the camera (the selected shutter speed is displayed in the viewfinder, but aperture is not).
The OM2-s' biggest limitation is the shutter (its fastest speed is 1/1000 sec). The situation is made worse by lenses which can not be set to an aperture smaller than F:16. When taking pictures on a bright sunny day with 400 ISO film (like Kodak's CN400 that I use a lot because it's still easy to have it processed), the photographer can only use a narrow combination of speed and aperture, and can not play with the depth of field as much as he would like.
Not too common but not eagerly sought after by the Olympus fanatics, the OM-2S is a good pick for a photographer looking for a compact SLR with a big viewfinder and exceptional control of the metering. With unconventional but well designed commands, the OM-2S is simpler to use than the OM-3 and OM-4 with their complex multi-spot and zone system metering functions. The OM-2S can be found for less than $150 (stores specialized in second hand cameras) and even cheaper ($50 to $80) on eBay. Small aperture Zuiko (Olympus) lenses are dirt cheap (a 28mm f:3.5 or a 135mm f:3.5 can be had for less than $30), but wide aperture Zuikos are very rare and very expensive.
More about the Olympus OM-2S
Ken Norton's Zone-10 web site has interesting reviews of Olympus cameras (film and digital) and wrote a few pages about the OM2-S.
Photography in Malaysia is primarily focused on Nikon cameras, but they have very good pages about the OM-1, OM-2 and OM-2S cameras.
Historic center of Powder Springs (GA). Olympus OM-2s with OM-Zuiko 28mm f:3.5. Kodak CN400
June 23, 2010 August 15, 2018 xtalfu
Olynpusom-1om-2OM-3OM-4OM-4TOM3-TOM4-TI | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,871 |
\section{Introduction}
Discussions of searches for intermediate mass Higgs bosons usually
concentrate on the decay $H\rightarrow \gamma\gamma$ as the discovery
mode \cite{hhg}. Other modes considered
include $H\rightarrow Z\gamma$ and $H\rightarrow b\bar{b}$. The importance of
radiative processes led us to consider the class of decays $H\rightarrow f
\bar{f} \gamma$, where $f$ is a light fermion. The dominant contributions to
these decays occur at the one loop level, and their calculation is related to
that of the process $e\bar{e} \rightarrow H\gamma$, which we recently
completed \cite{ab-cdr,ddhr}.
Typical results for Higgs boson $f\bar{f}\gamma$ decay appear in the
calculation of $\Gamma(H\rightarrow e\bar{e}\,\gamma)$,
which, for $m_H\gtrsim 100\,$ GeV, receives a large contribution from the $Z$
pole. Additionally, our calculations show that the photon pole makes a
substantial
correction to the estimate obtained by simply multiplying the width for $H
\rightarrow Z\,\gamma$ by the branching ratio $B(Z\rightarrow e\,\bar{e})$.
This feature is common to all light fermions, and we present results for all
decays of the type $H \rightarrow f\bar{f}\gamma$.
In the next section, we present expressions for the decay amplitudes, the decay
matrix element and results for the fermion invariant mass distributions and for
the widths. This is followed by a discussion. Complete expressions for the
various amplitudes are given in the appendices.
\section{Calculation of Higgs decay widths}
Contributions to the decay amplitudes arise from the diagrams illustrated
\cite{tri} in
Fig\,(\ref{fig1}). They are of two basic types: ({\it a}) pole diagrams in which
the $f\bar{f}$ emerge from a virtual gauge boson; and ({\it b}) box diagrams
containing virtual gauge bosons and fermions in the loop. The diagrams are
evaluated using the non--linear gauges discussed in Ref.\cite{ab-cdr}. In these
gauges, the collection of diagrams consists of four separately gauge invariant
contributions, the photon pole, the $Z$ pole, $Z$ boxes and $W$ boxes. The
amplitudes for these contributions are
\begin{eqnarray}
{\cal M}^{\gamma}_{\rm pole} & = & -\frac{\alpha^2m_W}{\sin\!\theta_W}\,
\bar{u}(p_1)\gamma_{\mu}v(p_2)\Bigl(\frac{
\delta_{\mu\nu}k\!\cdot\!(p_1 + p_2) - k_{\mu}(p_1 + p_2)_{\nu}}{m_{f\bar{f}}^2
}\Bigr)\hat{\epsilon}_{\nu}(k){\cal A}_{\gamma}(m_{f\bar{f}}^2)\,, \\ [4pt]
{\cal M}^{Z}_{\rm pole} & = & -\frac{\alpha^2m_W}{\sin^3\!\theta_W}\,\bar{u}
(p_1)\gamma_{\mu}(v_f + \gamma_5)v(p_2)\Bigl(\frac{\delta_{\mu\nu}k\!\cdot
\!(p_1 + p_2) - k_{\mu}(p_1 + p_2)_{\nu}}{(m_{f\bar{f}}^2 - m_Z^2) +
im_Z\Gamma_Z}\Bigr)\hat{\epsilon}_{\nu}(k){\cal A}_{Z}(m_{f\bar{f}}^2)\,,
\\ [4pt]
{\cal M}_{\rm box}^Z & = &\,\frac{\alpha^2m_Z}{4\sin^3\!\theta_W\cos^3\!
\theta_W}\,\bar{u}(p_1)\gamma_{\mu}(v_f + \gamma_5)^2v(p_2)
\Bigl\{\Bigl[\delta_{\mu\nu}k\!\cdot\!p_1 - k_{\mu}(p_1)_{\nu}
\Bigr]{\cal B}_Z(m_{f\bar{f}}^2,m_{f\gamma}^2,m_{\bar{f}\gamma}^2)
\nonumber \\
& &+ \Bigl[\delta_{\mu\nu}k\!\cdot\!p_2 - k_{\mu}(p_2)_{\nu}\Bigr]
{\cal B}_Z(m_{f\bar{f}}^2,m_{\bar{f}\gamma}^2,m_{f\gamma}^2)\Bigr\}
\hat{\epsilon}_{\nu}(k)\;, \\ [4pt]
{\cal M}_{\rm box}^{W} & = &-\frac{\alpha^2m_W}{2\sin^3\!\theta_W}\,
\bar{u}(p_1)\gamma_{\mu}(1 + \gamma_5)^2v(p_2)
\Bigl\{\Bigl[\delta_{\mu\nu}k\!\cdot\!p_1 - k_{\mu}(p_1)_{\nu}
\Bigr]{\cal B}_W(m_{f\bar{f}}^2,m_{f\gamma}^2,m_{\bar{f}\gamma}^2)\nonumber \\
& &+ \Bigl[\delta_{\mu\nu}k\!\cdot\!p_2 - k_{\mu}(p_2)_{\nu}\Bigr]
{\cal B}_W(m_{f\bar{f}}^2,m_{\bar{f}\gamma}^2,m_{f\gamma}^2)\Bigr\}
\hat{\epsilon}_{\nu}(k)\;,
\end{eqnarray}
where $m_{f\bar{f}}^2 = -(p_1 + p_2)^2,\,m_{f\gamma}^2 = -(k + p_1)^2\,$ and
$m_{\bar{f}\gamma}^2 = -(k + p_2)^2$. Here,
$v_f$ denotes the $f\bar{f}Z$ vector coupling constant, $v_f = 1 -
4|e_f|\sin^2\!\theta_W$, and $e_f$ is the fermion charge in units of the proton
charge. To calculate the
invariant amplitudes ${\cal B}_Z(m_{f\bar{f}}^2,m_{f\gamma}^2,m_{\bar{f}\gamma}
^2)$ and ${\cal B}_W(m_{f\bar{f}}^2,m_{f\gamma}^2,m_{\bar{f}\gamma}^2)$, we
use the approach of Ref.\,\cite{ab-cdr}. Here, too, we find a logarithmic
dependence on the fermion mass at intermediate stages of the calculation, but
this dependence cancels enabling us to take the limit of zero fermion mass.
Explicit expressions for the invariant amplitudes ${\cal A}_{\gamma}$, ${\cal
A}_Z$, ${\cal B}_Z$ and ${\cal B}_W$ are given in the Appendices.
The invariant mass distribution $d\Gamma/dm_{f\bar{f}}^2$ is given by
\begin{equation}\label{dgam}
\frac{d\Gamma(H\rightarrow f\bar{f}\gamma)}{dm_{f\bar{f}}^2} =
\frac{1}{256\pi^3}\frac{1}{m_H^3}
\int_{(m_{\bar{f}\gamma}^2)_{\rm min}}^{(m_{\bar{f}\gamma}^2)_{\rm max}}
dm_{\bar{f}\gamma}^2\,\sum_{\rm spin}|{\cal M}|^2\,,
\end{equation}
with the amplitude ${\cal M}$ given by the sum of Eqs.\,(1)--(4). For an
$f\bar{f}\gamma$ final state, the limits on the $dm_{\bar{f}\gamma}^2$
integration are
\begin{eqnarray}
(m_{\bar{f}\gamma}^2)_{\rm min} & = &\;m_f^2 +
\case{1}{2}(m_H^2 - m_{f\bar{f}}^2)\left(1 - \sqrt{1 - \frac{\displaystyle
4m_f^2}{\displaystyle m_{f\bar{f}}^2}}\;\;\right)\,, \\ [4pt]
(m_{\bar{f}\gamma}^2)_{\rm max} & = &\;m_f^2 +
\case{1}{2}(m_H^2 - m_{f\bar{f}}^2)\left(1 + \sqrt{1 -
\frac{\displaystyle 4m_f^2}{\displaystyle m_{f\bar{f}}^2}}\;\;\right)\,.
\end{eqnarray}
The fermion mass is retained in the phase space integration since, as shown
below, there is a $(m_{f\bar{f}}^2)^{-1}$ factor associated with the photon
pole. Explicitly, $\sum_{\rm spin}|{\cal M}|^2$ is
\begin{eqnarray}
\sum_{\rm spin}|{\cal M}|^2 & = &\;\frac{\alpha^4\,m_W^2\,m_{f\bar{f}}^2}
{16\sin^6\!\theta_W\,\cos^8\!\theta_W}
\biggl\{((m_{f\gamma}^2)^2 + (m_{\bar{f}\gamma}^2)^2)\Bigl[|\tilde{\cal A}_
{\gamma}|^2 +
2v_f{\rm Re}(\tilde{\cal A}_{\gamma}\tilde{\cal A}_Z^*)
\nonumber \\
& &\;+ (1 + v_f^2)|\tilde{\cal A}_Z|^2\Bigr]
+ (m_{f\gamma}^2)^2\biggl[2(1 + v_f^2){\rm Re}(\tilde{\cal A}_{\gamma}
\tilde{\cal B}_Z^*) + 4{\rm Re}(\tilde{\cal A}_{\gamma}\tilde{\cal B}_W^*)
\nonumber \\ [4pt]
& &\;+ 2(v_f^3 + 3v_f){\rm Re}(\tilde{\cal A}_Z\tilde{\cal B}_Z^*)
+ 4(1 + v_e){\rm Re}(\tilde{\cal A}_Z\tilde{\cal B}_W^*)
+ (1 + 6v_f^2 + v_f^4)|\tilde{\cal B}_Z|^2 \nonumber \\ [4pt]
& &\;+ 4(1 + v_f)^2{\rm Re}(\tilde{\cal B}_Z\tilde{\cal B}_W^*)
+ 8|\tilde{\cal B}_W|^2\biggr] +
(m_{\bar{f}\gamma}^2)^2\biggl[m_{f\gamma}^2\leftrightarrow m_{\bar{f}\gamma}^2
\biggr]\biggr\}\;.
\end{eqnarray}
Here, $\tilde{\cal A}_{\gamma}$, $\tilde{\cal A}_Z$, $\tilde{\cal B}_Z$ and
$\tilde{B}_W$ are
\begin{eqnarray}
\tilde{\cal A}_{\gamma} & = &\,4\sin^2\!\theta_W\!\cos^4\theta_W\frac{{\cal A}_
{\gamma}(m_{f\bar{f}}^2)}{m_{f\bar{f}}^2}\,, \\ [4pt]
\tilde{\cal A}_Z & = &\,4\cos^4\!\theta_W\frac{{\cal A}_Z(m_{f\bar{f}}^2)}
{(m_{f\bar{f}}^2 - m_Z^2) +im_Z\Gamma_Z}\,, \\ [4pt]
\tilde{\cal B}_Z & = &\,-{\cal B}_Z(m_{f\bar{f}}^2,m_{f\gamma}^2,
m_{\bar{f}\gamma}^2)\,, \\ [4pt]
\tilde{\cal B}_W & = &\,2\cos^4\!\theta_W\,{\cal B}_W(m_{f\bar{f}}^2,
m_{f\gamma}^2,m_{\bar{f}\gamma}^2)\,.
\end{eqnarray}
Using our results for the invariant amplitudes, the $dm_{\bar{f}\gamma}^2$
integration can be performed numerically to obtain the invariant mass
distribution.
The invariant mass distribution $d\Gamma(H\rightarrow e\bar{e}\gamma)
/dm_{e\bar{e}}$ is illustrated in Fig.\,(\ref{fig2}). The striking feature of
these distributions is the large peak at small $m_{e\bar{e}}^2$ due to the
photon pole. There is no singularity in the physical region since
$m_{e\bar{e}}^2\geq 4m_e^2$. In fact, as can be seen from Eqs.\,(6) and (7), the
$dm_{\bar{e}\gamma}^2$ integral in Eq.\,(\ref{dgam}) vanishes when
$m_{e\bar{e}}^2 =
4m_e^2$. Nevertheless, the residual effect of the photon pole is sufficient to
contribute $\sim$ 10-20\% of the events in the distribution. It is also evident
that the box diagrams make only a small contribution. Curiously, the main effect
of the box diagrams is to smooth the distribution by cancelling the kinks in the
pole contributions at the $WW$ threshold. The invariant mass distribution for
the remaining lepton channel $H\rightarrow \nu\bar{\nu}\gamma$, which has no
contribution from the photon pole, is illustrated in Fig.\,(\ref{fig3}).
The various partial widths can be obtained by integrating Eq.\,(\ref{dgam}).
This results in the contributions illustrated in Fig.\,(\ref{fig4}). Also shown
in the lepton panel of Fig.\,(\ref{fig4}) is the contribution from the $Z$ pole.
The figure clearly shows that the widths are enhanced significantly in the
complete calculation, even in the case of neutrino decays. For
$m_H\gtrsim 160$ GeV, the up--type quark contributions are basically the same.
This is also true for the down--type quarks.
\section{Discussion}
As can be seen from Fig.\,(\ref{fig2}), the invariant mass distributions are
basically determined by the photon and $Z$ pole contributions. The box diagrams
make corrections to the high mass side of the distribution where they are of the
same order as the pole terms.
This being the case, it is possible to obtain a simplified expression for
$d\Gamma/dm_{f\bar{f}}^2$ by retaining only the ${\cal A}_{\gamma}$ and ${\cal
A}_Z$ terms in Eq.\,(\ref{dgam}). After performing the $dm_{\bar{f}\gamma}^2$
integration, one finds
\begin{eqnarray}
\frac{d\Gamma}{dm_{f\bar{f}}^2} & = &
\frac{\alpha^4\,m_W^2}{(8\pi)^3\sin^6\!\theta_Wm_H^3}
\Biggl[\sin^4\!\theta_W\frac{|{\cal A}_{\gamma}(m_{f\bar{f}}^2)|^2}
{m_{f\bar{f}}^2} + 2\sin^2\!\theta_W\,v_f{\rm Re}\biggl(\frac{{\cal A}_{\gamma}
(m_{f\bar{f}}^2){\cal A}_{Z}^*(m_{f\bar{f}}^2)}{(m_{f\bar{f}}^2 - m_Z^2) -
im_Z\Gamma_Z}\biggr) \nonumber \\
& &+ \frac{(1 + v_f^2)\,m_{f\bar{f}}^2|{\cal A}_{Z}(m_{f\bar{f}}^2)|^2}
{(m_{f\bar{f}}^2 - m_Z^2)^2 + m_Z^2\Gamma_Z^2}\Biggr](m_H^2 - m_{f\bar{f}}^2)
\sqrt{1 - \frac{\displaystyle 4m_f^2}{\displaystyle m_{f\bar{f}}^2}}\Biggl
[\biggl(m_H^2 + 2m_f^2 - m_{f\bar{f}}^2\biggr)^2 \nonumber \\
& &+\;\case{1}{3}
\biggl(m_H^2 - m_{f\bar{f}}^2\biggr)^2\biggl(1 - \frac{4m_f^2}{m_{f\bar{f}}^2}
\biggr)\Biggr]\,.
\end{eqnarray}
This expression reproduces the dashed lines in Fig.\,(\ref{fig2}).
The total width for $H\rightarrow f\bar{f}\gamma$ is shown in Fig.\,
(\ref{fig5}), where the neutrino, electron, muon, up quark, down quark and
strange quark contributions are added. The dashed line in this figure
corresponds to $H\rightarrow\gamma\gamma$, and it can be seen that the
$f\bar{f}\gamma$ width exceeds the $\gamma\gamma$ width for $m_H\gtrsim 140$
GeV.
\acknowledgements
This research was supported in part by the National Science Foundation under
grant PHY-93-07980 and by the United States Department of Energy under Contract
No. DE-FG013-93ER40757.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,362 |
\section{Introduction}
\label{sec:intro}
Moore's Law is dead; the future will bring improved efficiency and speed to the computational sciences, but at a slower rate than before \citep[e.g.][]{wang15,wang16,bonetti20}. Consequently, the demand for alternative models independent of computational limitations (e.g., analytic calculations) are becoming increasingly urgent for the first time in over half a century.
Galactic and extragalactic globular clusters (GCs) are an example of an astrophysical system predominantly modeled today via complex numerical simulations. One particular challenge in studying GCs is related to their collisional nature; central stellar densities are so high ($>$ 10$^5$ M$_{\odot}$ pc$^{-3}$) that direct encounters between single and binary stars occur frequently. The exact rate depends on the host cluster properties but, for the densest GCs, the time-scale for direct single-binary and binary-binary encounters to occur is of order 1-10 Myr \citep[e.g.][]{leigh11,geller15}. These two timescales are roughly equal for binary fractions $f_b$ $\sim$ 10\% \citep{sigurdsson93,leigh11}, such that the rate of single-binary interactions dominates over that of binary-binary interactions in clusters with $f_b \lesssim 10\%$. These multibody interactions are expensive to resolve in direct N-body simulations, but play a key role in (i) determining the overall cluster evolution, due to the release of gravitational potential energy via ``binary burning'' (as first suggested by \citealt{henon61}), and (ii) producing exotic sources of electromagnetic \citep{leigh11b, ivanova06, ivanova08} or gravitational wave \citep{portegies00, rodriguez16a, askar17, samsing18} radiation.
At present, a number of highly sophisticated computational tools have been developed to simulate the time evolution of dense stellar systems, including direct interactions involving binary stars. Among the most successful of these are $N$-body simulations \citep[e.g.][]{aarseth75,aarseth85,aarseth99,aarseth03}, which calculate directly the gravitational acceleration exerted on every particle in the system, summed over all other particles. The code then propagates the system forward through time using an appropriately chosen time-step, and step-by-step the cluster is evolved. $N$-body simulations originated over half a century ago \citep[e.g.][]{vonhoerner60,vonhoerner63,aarseth63}. They have been evolving ever since, growing increasingly sophisticated over time. Many new software and hardware techniques have been introduced and incorporated, including three-body regularization \citep{aarseth74}, chain regularization \citep{mikkola93,mikkola98}, the Hermite scheme \citep{makino92}, the GRAPE hardware \citep{sugimoto90,makino96,nitadori12} and, more recently, the introduction of GPUs for accelerated computing. Due to the increased computational expense, $N$-body simulations struggle to simulate self-gravitating systems with large particle numbers (this includes the largest globular clusters, but also most nuclear star clusters) and high binary fractions. For this general category of initial conditions (i.e., massive clusters with high binary fractions), long simulation run times, of order a year, can be needed to complete even a single simulation \citep[e.g.][]{wang15,wang16}.
Monte Carlo (MC) simulations for GC evolution have also proven highly successful, covering a range of parameter space inaccessible to more computationally-expensive $N$-body simulations. MC models rely on statistical approximations to calculate the time-dependent diffusion of energy throughout cluster due to two-body relaxation. Consequently, MC models are able to handle much larger particle numbers, including larger numbers of binaries, and hence are able to simulate more massive and denser GCs than are $N$-body models, at a significantly reduced computational expense. MC models are incredibly fast; they are able to evolve a million cluster stars for a Hubble time in a matter of days (roughly two orders of magnitude faster than state-of-the-art N-body simulations). The MC method goes back to the pioneering works of \citet{henon71} and \citet{spitzer71}, with many other researchers later building upon these earliest ideas \citep[e.g.][]{shapiro78,stodolkiewicz82,stodolkiewicz86,giersz98,giersz01}. Modern MC codes supplement their diffusive treatment of bulk cluster evolution with embedded few-body integrators, most notably FEWBODY \citep{fregeau04}. These are typically used to treat direct three- (i.e., single-binary) and four-body (i.e., binary-binary) interactions in MC simulations \citep{giersz08,downing10,hypki13,leigh15}. As a result, MC models are useful for simulations of massive star clusters, capturing the time evolution of temporarily bound and highly chaotic three- and four-body systems. These are thought to be the main source of stellar collisions and/or mergers in dense star clusters and an important formation channel for many exotic stellar systems \citep{leonard89,pooley06,sills97,sills01,leigh11,leigh12,leigh18b}.
In spite of these successes, MC models are still limited, in that they cannot simulate small particle numbers (i.e., $N \lesssim 10^4$) \citep[e.g.][]{giersz98,giersz01,kremer20}. Thus, we are still lacking a single tool capable of covering the entire range of parameter space relevant to real star clusters, ranging from open to globular and even nuclear clusters.
While $N$-body and MC simulations often find good agreement for bulk cluster properties \citep{giersz13}, detailed comparisons find areas of disagreement, such as phases of deep core collapse \citep{rodriguez16b}. The assumptions of existing MC codes also prevent simulation of some clusters of interest, for example those with net rotation\footnote{Although see \citet{fiestas06, kim08} for an example of an MC code which can simulate rotating clusters, albeit at the cost of a special type of assumed isotropy.} (which can accelerate core collapse, possibly increasing the rates of exotica production, \citealt{ernst07}) or those where vector resonant relaxation has created strong deviations from spherical symmetry \citep{meiron19, szolgyen19}. Such clusters nonetheless contain collisionally evolving binaries, and it would be useful to have a computationally efficient tool with which to model them.
Another limitation of MC codes is that three- and four-body interactions, along with stellar and binary evolution, are assumed to occur in isolation, and so do not allow for the interruption of ongoing interactions and other perturbative effects that occur in a live star cluster environment \citep[e.g.][]{geller15,leigh16c}. For example, binaries are continually perturbed by the distant flybys of other stars in the cluster. These perturbations happen at every time-step in an $N$-body code, often leading to random walks in the orbital eccentricity that drive mergers. MC codes do not capture this aspect of the N-body dynamics \citep{giersz98,giersz08,kremer20}, again suggesting the need for an additional efficient computational tool.
Beyond the successes and limitations
of the aforementioned computational tools for simulating GC evolution, we are still lacking a robust and
transparent
\textit{physical} model that can be used to understand the dominant physical processes dynamically re-shaping the properties of binary populations over cosmic time. In other words, we can give a set of initial conditions to a computer simulation and it will compute for us the final result. But how do we understand, and even quantitatively characterize, what this output is telling us about everything that happened in between providing the input and reading the output?
In this paper, we directly address these practical and conceptual issues, by formulating a self-consistent statistical mechanics model
to describe the time evolution of the properties of a population of binaries, based on a master-type equation. Our semi-analytic model quantifies the dominant gravitational physics driving the time evolution of the binary orbital parameter distributions in dense stellar environments, namely direct three-body interactions between single and binary stars. A practical solution of our master equation can be obtained in the Fokker-Planck limit, which we
complete using the
formulation for the outcomes of chaotic three-body interactions introduced in \citet{stoneleigh19}.
We compare the results of our analytic calculations to a suite of N-body simulations, and show that the agreement is excellent for the range of initial conditions considered here.
In an Appendix, we go on to discuss how to adapt our base model to also include four-body or binary-binary interactions. Although four-body interactions are not always dominant over three-body interactions, they are always occurring for non-zero binary fractions, and contribute to the underlying dynamical evolution of the binary orbital parameter distributions. Hence, a complete model must include this contribution which, as described in the Appendix, will be the focus of future work. In principle, this will allow us to address the key thermodynamic issue of whether or not (and on what timescale) stellar multiplicity in dense cluster environments ever reaches a steady- or equilibrium-state is discussed, as quantified by the ratio of single, binary and triple stars. In steady-state, their relative fractions should remain approximately constant.
\section{The Model} \label{model}
In this section, we present our model for dynamically evolving in time the distribution functions for an entire population of binary star systems in dense stellar environments, accounting for the effects of repeated binary-single encounters. In other words, we model the time evolution of a population of binaries embedded in a ``heat bath'' of single stars. This formalism is similar in spirit to the pioneering early work of \citet{goodman93}, but it is both (i) more general, in that we present a multi-dimensional version of their underlying master equation, and (ii) also more accurate, in that we take advantage of recent advances in the statistical theory of chaotic multi-body encounters (\citealt{stoneleigh19}, but see also \citealt{ginat20, kol20, manwadkar21}) and the secular theory of weak encounters \citep{hamers19a, hamers19b}. We begin with a simple, 1D, second-order (Fokker-Planck) method to quantify the time evolution of the binary orbital energy distribution. We then present the more complete multi-dimensional master equation that describes the full time evolution of all binary variables of interest.
This equation is complex enough that we do not solve it fully in this paper, but we show how it can reduce to the simplified Fokker-Planck limit presented earlier, and analyze various 1D solutions.
We also present a simple model to quantify the backreaction effects on the properties of the host star cluster (e.g., the time evolution of the core radius, the time at which core-collapse occurs, and so on).
\subsection{A Fokker-Planck equation in energy space} \label{boltzmann}
We begin by reviewing the features of a 1D Fokker-Planck equation valid to second-order (in the fractional changes of the variable of interest). In the next subsections, we show that this formalism is the relevant limit for some aspects of a more general master equation for binary evolution. Since this is the primary limit of binary evolution we will investigate in this paper, we begin with a brief review, focusing on distributions of binary energy $E_{\rm B}$ for specificity.
Assuming diffusive time evolution of the binary orbital energy probability distribution function $\mathcal{N}(E_{\rm B}) = {\rm d} N_{\rm B} / {\rm d}E_{\rm B}$, we can apply a standard Fokker-Planck equation:
\begin{align}
\frac{\partial \mathcal{N}}{\partial t} = & -\frac{\partial }{\partial E_{\rm B}}\Big[ \mathcal{N}(E_{\rm B})\langle {\Delta}E_{\rm B}\rangle \Big] \label{eq:FP1} \\
&+ \frac{1}{2}\frac{\partial ^2}{\partial E_{\rm B}^2}\Big[\mathcal{N}(E_{\rm B})\langle {\Delta}E_{\rm B}^2\rangle \Big], \notag
\end{align}
where $\langle {\Delta}E_{\rm B} \rangle$ and $\langle {\Delta}E_{\rm B}^2 \rangle$ are the first- and second-order diffusion coefficients. The boundary conditions of the equation above are set by the cluster hard-soft boundary (i.e., $|E_{\rm B}| = |E_{\rm HS}| = \frac{1}{2}\bar{M}\sigma^2$, where $\bar{M}$ is the mean stellar mass in the cluster) at the loosely bound end of the distribution, and by the criterion for a contact binary (i.e., $|E_{\rm B}| = |E_{\rm coll}| \approx \frac{GM^2}{R}$, where $M$ is the mass and $R$ is the radius of the test star species) at the compact end.
This equation arises from the assumption that the binary orbital energy distribution function evolves diffusively in its host cluster. In this way, changes to this distribution can be described as a local process, with binaries flowing mostly in to and out of adjacent energy bins.
We can understand the origin of the Fokker-Planck equation as follows. If we consider a general process of binary evolution through energy space, then the binary energy distribution at a time $t+\Delta t$ will be
\begin{equation}
\label{eqn:eqboltzmann2}
\mathcal{N}(E_{\rm B},t+{\Delta}t) = \int \mathcal{N}(E_{\rm B}-{\Delta}E_{\rm B},t)P(E_{\rm B}-{\Delta}E_{\rm B},{\Delta}E_{\rm B})d{\Delta}E_{\rm B},
\end{equation}
where the transfer function,
\begin{equation}
\label{eqn:transferfunc}
P(E_{\rm B}-{\Delta}E_{\rm B},{\Delta}E_{\rm B}) = \Gamma(E_{\rm B},E_{\rm 0})\mathcal{F}(E_{\rm B},E_{\rm 0}) \Delta t,
\end{equation}
represents the probability that in a differential time interval $\Delta t$, a binary will reach a final energy $E_{\rm B}$ from an initial energy $E_{\rm B} - \Delta E_{\rm B}$.
Here the function $\mathcal{F}(E_{\rm B},E_{\rm 0})$ is the differential probability of a single ``encounter'' transitioning a test binary from initial energy $E_0 = E_{\rm B}-\Delta E_{\rm B}$ to a final energy $E_{\rm B}$, and $\Gamma$ is the rate of all such encounters (here our language is general, but our eventual application will be to consider ``encounter'' arising from single-binary encounters).
Following \citet{spitzer87} and Taylor expanding Equation~\ref{eqn:eqboltzmann2}, we obtain:
\begin{align}
&\mathcal{N}(E_{\rm B},t) + {\Delta}t\frac{\partial \mathcal{N}}{\partial t} = \\ &\int \Big[\mathcal{N}(E_{\rm B})P(E_{\rm B}-{\Delta}E_{\rm B},{\Delta}E_{\rm B})\notag \\
& - \frac{\partial }{\partial E_{\rm B}}\Big(\mathcal{N}(E_{\rm B},t)P(E_{\rm B}-{\Delta}E_{\rm B},{\Delta}E_{\rm B})\Big){\Delta}E_{\rm B}\notag \\
& + \frac{1}{2}\frac{\partial ^2}{\partial E_{\rm B}^2}\Big(\mathcal{N}(E_{\rm B},t)P(E_{\rm B}-{\Delta}E_{\rm B},{\Delta}E_{\rm B})\Big){\Delta}E_{\rm B}^2\Big]d{\Delta}E_{\rm B}, \notag
\end{align}
which can, after eliminating a single factor of $\mathcal{N}(E_{\rm B}, t)$ from both sides, be simplified to Eq. \ref{eq:FP1}.
Here we now see the origin of the diffusion coefficients in our assumption that the relevant ``encounters'' are perturbative in nature, or in other words that our Taylor expansion was validly truncated at second order:
\begin{equation}
\label{eqn:diffusion1}
\langle{\Delta}E_{\rm B}\rangle = \int \Gamma(E_{\rm B},E_{\rm 0})\mathcal{F}(E_{\rm B},E_{\rm 0}){\Delta}E_{\rm B}d{\Delta}E_{\rm B},
\end{equation}
and
\begin{equation}
\label{eqn:diffusion2}
\langle{\Delta}E_{\rm B}^2\rangle = \int \Gamma(E_{\rm B},E_{\rm 0})\mathcal{F}(E_{\rm B},E_{\rm 0}){\Delta}E_{\rm B}^2d{\Delta}E_{\rm B}.
\end{equation}
Equation~\ref{eq:FP1} is a partial differential equation that can be solved numerically. To do this, we must close the system by writing the per-binary encounter rate. Focusing on strong single-binary scatterings for specificity, the gravitationally focused scattering rate is:
\begin{equation}
\label{eqn:mfp2}
\Gamma = \frac{3{\pi}n_{\rm s}G^2(m_{\rm B}+m_{\rm s})m_{\rm B}}{{\sigma}|E_{\rm B}-{\Delta}E_{\rm B}|},
\end{equation}
where $\sigma$ is the cluster velocity dispersion, $m_{\rm B}$ and $m_{\rm s}$ are the binary and single star masses, respectively, and $n_{\rm s}$ is the density of scatterers.
The total encounter energy can then be written:
\begin{equation}
\label{eqn:entot}
E_{\rm 0} = E_{\rm B} - {\Delta}E_{\rm B} + \frac{1}{2}\frac{m_{\rm B}m_{\rm s}}{m_{\rm B}+m_{\rm s}}\sigma^2
\end{equation}
The distribution of binary orbital energies for hard binaries left over after single-binary interactions is often approximated as a power law: \citep{monaghan76a,monaghan76b,valtonen06}:
\begin{equation}
\label{eqn:outcome}
\mathcal{F}_{\rm VK}(|E_{\rm B}|)d|E_{\rm B}| = (n-1)|E_{\rm 0}|^{n-1}|E_{\rm B}|^{-n}d|E_{\rm B}|,
\end{equation}
The parameter $n$ depends on the total interaction angular momentum $L$, and has been fit to numerical simulations as \citep{valtonen06}:
\begin{equation}
\label{eqn:n}
n = 3 + 18\tilde{L}^2,
\end{equation}
where $\tilde{L}$ is a normalized version of the total encounter angular momentum (see \citet{valtonen06} for more details).
Alternatively, one may use first-principles estimates for $\mathcal{F}$ from the ergodic formalism of \citet{stoneleigh19}, which will be our approach later in this paper. With $\mathcal{F}$ specified,
Equations~\ref{eqn:mfp2} and~\ref{eqn:entot} can be plugged in to Equation~\ref{eq:FP1} such that it becomes straight-forward to solve numerically.
\begin{comment}
Formally, the limits of integration should be $|E_{\rm HS}|$ and $|E_{\rm coll}|$ to include only hard binaries which denote, respectively, the binary orbital energies corresponding to the hard-soft boundary and a contact state. Alternatively, if only soft binaries are to be included, then the limits become 0 and $|E_{\rm HS}|$.
An analogous formalism as presented in this section for the orbital energy distribution function $f_{\rm B}(|E_{\rm B}|)$ can be carried out trivially for the orbital eccentricity distribution function $g_{\rm B}(e)$. In the next section, however, we do so more robustly via a Master Equation and its Fokker-Planck limit.
\end{comment}
\subsection{The Master Equation and its Fokker-Planck Limit} \label{master}
In this section, we further generalize our formulation to include angular momentum, moving away from the quasi-empirical outcome distribution provided in \citet{valtonen06} and toward the more robust formulation provided in \citet{stoneleigh19}. The latter self-consistently accounts for angular momentum in the density-of-states formulation, whereas the former uses numerical scattering experiments to obtain approximate outcome distributions as a function of the total angular momentum. We will also present the evolution of the binary distribution from a first-principles kinetic perspective, by working in the Fokker-Planck limit of a master equation.
We consider the evolution of a population of binaries inside a dense, collisional, spherically symmetric star cluster. The cluster has an ambient density of single stars $n(r)$ and a velocity dispersion $\sigma(r)$ that are both assumed to vary with radius $r$. We assume for now that the cluster is relaxed and isotropic, so that the stellar velocity distribution is Maxwellian,
\begin{equation}
f(v) = \sqrt{ \frac{2}{\pi} } \frac{v^2}{\sigma^3} \exp(-v^2 / 2\sigma^2). \label{eq:maxwellian}
\end{equation}
The binaries in this cluster can be characterized by six binary orbital elements, $\vec{B}$. For practical purposes, we focus on their internal energy $E_{\rm B}$, internal angular momentum $L_{\rm B}$ (or equivalently their semimajor axis $a_{\rm B}$ and eccentricity $e_{\rm B}$), and orbital orientation $C_{\rm B}$; other, angular orbital elements can generally be assumed to be distributed isotropically. We define $C_{\rm B} = \cos I_{\rm B}$, where $I_{\rm B}$ is the inclination angle between the binary angular momentum vector and an arbitrary reference direction.
The number of binaries with energy $E_{\rm B}$, angular momentum $L_{\rm B}$, and orientation $C_{\rm B}$ within an infinitesimal range ${\rm d}E_{\rm B}{\rm d}L_{\rm B}{\rm d}C_{\rm B}$ is $\mathcal{N}(E_{\rm B}, L_{\rm B}, C_{\rm B}){\rm d}E_{\rm B}{\rm d}L_{\rm B}{\rm d}C_{\rm B}$. The binary probability distribution\footnote{Note that throughout this paper, we use the symbol $\mathcal{N}$ to denote distributions of binary orbital elements over varied numbers of dimensions. However, the dimensionality of the distribution should always be clear from either context or explicit labeling of its arguments.} $\mathcal{N}(E_{\rm B}, L_{\rm B}, C_{\rm B})$ will evolve due to scatterings with single stars and with each other. These scatterings may be weak, distant encounters, in which case the tools of secular theory can be used to understand the exchange of energy and angular momentum \citep{HeggieRasio96}. Alternatively, some scatterings may form temporarily bound (``resonant'') triple systems, which will eventually disintegrate into a survivor binary and an escaping single star; the outcomes of these strong scatterings can be understood through the ergodic hypothesis \citep{stoneleigh19}.
The long-term evolution of a single-mass binary distribution function will be governed by a master equation:
\begin{align}
\frac{\partial \mathcal{N}}{\partial t} = & \idotsint {\rm d}^6{\Delta \vec{B}}\Big[\Psi(\vec{B}-\Delta \vec{B}, \Delta\vec{B})\mathcal{N}(\vec{B}-\Delta\vec{B})\\
& - \Psi(\vec{B}, \Delta\vec{B})\mathcal{N}(\vec{B}) \Big] - \left(\frac{\partial \mathcal{N}}{\partial t} \right)_{\rm sink}. \label{eq:masterEq}
\end{align}
Here $\Psi(\vec{B}, \Delta\vec{B})$ is a transition probability describing the differential rate that stars are scattered into the phase space region $\vec{B}+\Delta\vec{B}$ from the original phase space coordinates $\vec{B}$, and the final term is a catchall ``sink'' representing ways binaries may be destroyed. Equation \ref{eq:masterEq} is the most general kinetic formulation for the local evolution of a population of single-mass binaries, but if we assume isotropy, we need only to consider a two-dimensional integral and a two-dimensional distribution $\mathcal{N}(E_{\rm B}, L_{\rm B})$.
In principle, one may Taylor-expand Equation \ref{eq:masterEq} in the small $\Delta\vec{B}$ limit, and obtain a Fokker-Planck equation generalizing Eq. \ref{eq:FP1}:
\begin{align}
\frac{\partial \mathcal{N}}{\partial t} = & -\frac{\partial}{\partial E_{\rm B}}(\langle \Delta E_{\rm B} \rangle \mathcal{N}) + \frac{1}{2}\frac{\partial^2}{\partial^2 E_{\rm B}}(\langle (\Delta E_{\rm B})^2 \rangle \mathcal{N})\notag \\
& -\frac{\partial}{\partial L}(\langle \Delta L_{\rm B} \rangle \mathcal{N}) + \frac{1}{2}\frac{\partial^2}{\partial^2 L_{\rm B}}(\langle (\Delta L_{\rm B})^2 \rangle \mathcal{N}) \label{eq:FP} \\
& + \frac{\partial^2}{\partial E_{\rm B}\partial L_{\rm B}}(\langle \Delta E_{\rm B}\Delta L_{\rm B} \rangle \mathcal{N}) - \left(\frac{\partial \mathcal{N}}{\partial t} \right)_{\rm sink}. \notag
\end{align}
Here terms such as $\langle \Delta E \rangle$ are effective diffusion coefficients. These diffusion coefficients (and the transition probability $\Psi$ from which they originate) represent the cumulative effect of interactions between a population of binaries and their perturbers. This includes both self-interactions (i.e. binary-binary scatterings) and external interactions (e.g. binary-single scatterings). In this work, we will focus only on binary-single scatterings, which dominate the rate of binary evolution so long as the binary fraction is sufficiently low $\lesssim$ 10\% \citep[e.g.][]{sigurdsson93,leigh11}.
Qualitatively different types of binary-single scatterings are possible, which we can break down into four categories: (i) ionizations, (ii) flybys, (iii) prompt exchanges, and (iv) resonant encounters. A closer look at some of these categories reveals the inconsistency of the (full) Fokker-Planck limit of the master equation: while energy ($E_{\rm B}$) evolution is indeed diffusive, angular momentum ($L_{\rm B}$) evolves in a strongly non-diffusive way during resonances and prompt exchanges. We will postpone a full solution of the master equation for future work. In this paper, we work with a hybrid, ``two-timescale'' approach to understand the evolution of $E_{\rm B}$ and $L_{\rm B}$ separately. For simplicity, we will treat ionizations by assuming that all binaries with energy below the hard-soft boundary are promptly ionized.
\subsection{Binary energies}
Here we will neglect flybys and prompt exchanges, since we are concerned with hard binaries (soft binaries are short-lived and quickly ionized). In hard binaries, resonant and non-resonant encounters contribute roughly equally to total energy evolution \citep{hut84}, although resonant encounters do dominate the largest energy shifts. We will thus only consider energy evolution through resonant encounters, an approximation that should be valid at a factor $\approx 2$ level, and which is motivated primarily by the lack of an analytic formalism for energy exchange in strong but non-resonant encounters \citep{heggie93} \footnote{Such a formalism exists for weak non-resonant encounters \citep{heggie75, heggie93}, but these are always sub-dominant in energy evolution.}. For simplicity, we treat ionizations by assuming that all binaries with energy below the hard-soft boundary are promptly ionized.
The mean change in binary energy in a given resonant encounter, with conserved energy $E_0$, conserved angular momentum $L_0$, and a mass triplet $\vec{m} = \{m_1, m_2, m_3\}$ is
\begin{equation}
\Delta E_{\rm B} = \iiint E_{\rm B} \mathcal{T}(E_0, L_0, \vec{m}){\rm d}E_{\rm B}{\rm d}L_{\rm B}{\rm d}C_{\rm B}. \label{eq:situationalDC}
\end{equation}
Here we have used the ergodic outcome distribution, $\mathcal{T}(E_0, L_0, \vec{m}) = {\rm d}V / {\rm d}E_{\rm B}{\rm d}L_{\rm B}{\rm d}C_{\rm B}$, which partitions outcomes of non-hierarchical triple disintegration uniformly across a high-dimensional phase volume ($V$), ultimately giving non-trivial outcome distributions in the survivor binary's $E_{\rm B}$, $L_{\rm B}$, and cosine-inclination $C_{\rm B}$ \citep{stoneleigh19}.
While Equation \ref{eq:situationalDC} gives a moment of the outcome distribution for a particular combination of $E_0$, $L_0$, and $\vec{m}$ (and can easily be generalized to produce $\Delta L_{\rm B}$, $(\Delta E_{\rm B})^2$, etc.) we are interested in computing rate-averaged diffusion coefficients. If resonant encounters happen at a differential rate ${\rm d}\Gamma / {\rm d}E_0{\rm d}L_0{\rm d}C_0$, the rate-averaged $k$th diffusion coefficient in energy is
\begin{equation}
\langle \Delta E_{\rm B}^k \rangle = \iiint \Delta E_{\rm B}^k(E_0, L_0, \vec{m}) \frac{{\rm d}\Gamma}{{\rm d}E_0{\rm d}L_0{\rm d}C_0}{\rm d}E_0{\rm d}L_0{\rm d}C_0.
\end{equation}
Here we have introduced an additional variable absent from $\mathcal{T}$, $C_0$, which represents the cosine of the inclination between the pre-encounter binary's orbital plane, and the mutual orbital plane between the binary and the single star it is encountering.
Differential encounter rates are more easily expressed in terms of the impact parameter $b$ and relative velocity at infinity, $v_\infty$. Specifically, the differential encounter rate is:
\begin{equation}
\label{eqn:gammanick}
\frac{{\rm d}\Gamma}{{\rm d}v_\infty {\rm d}b {\rm d}C_0} = 2\pi n b.
\end{equation}
Here $n$ is the local number density of single stars. These variables are related to $E_0$ and $L_0$ through the following equations:
\begin{align}
E_0 = & E_{\rm b} + \frac{1}{2}\mu v_\infty^2 \label{eq:energydef}\\
L_0 = & (L_{\rm b}^2 + \mu^2b^2v_\infty^2 + 2\mu b v_\infty C_0 L_{\rm b})^{1/2},
\end{align}
where $\mu = m_{\rm b} m_3 / (m_{\rm b}+m_3)$ is the reduced mass of the encounter, and we have denoted the variables of the pre-scattering binary with lower-case ``b'' subscripts. Performing a change of variables, we find that
\begin{equation}
d\Gamma = \frac{\pi n L_0 f(E_0, E_{\rm b})}{(2\mu (E_0 - E_{\rm b}) )^{3/2}} \left( 1 -\frac{C_0 L_{\rm b} }{\sqrt{L_0^2 - L_{\rm b}^2(1-C_0^2)}} \right) {\rm d}E_0 {\rm d}L_0 {\rm d}C_0.
\end{equation}
Here we have assumed that $v_\infty$ is drawn from a velocity distribution $f(v)$ (which can be written as a function of $E_0$ and $E_{\rm b}$ through Eq. \ref{eq:energydef}), although we have left this general for the moment (rather than specifying a Maxwellian). Integrating this differential encounter rate ${\rm d}C_{\rm 0}$ (from $-1$ to $1$, i.e. under the assumption of velocity isotropy) eliminates the $L_{\rm b}$ dependence:
\begin{equation}
\label{eqn:FP}
{\rm d}^2\Gamma = \frac{2\pi n L_0 f(E_0, E_{\rm b})}{(2\mu (E_0 - E_{\rm b}) )^{3/2}}{\rm d}E_0 {\rm d}L_0.
\end{equation}
Eq. \ref{eq:FP} is, unfortunately, not valid across all timescales. While binary energy evolution is a fundamentally diffusive process (even strongly resonant encounters rarely change individual binary energies by more than a factor $\approx 2$; \citealt{hut84, stoneleigh19}), resonances lead to highly non-diffusive evolution of binary angular momentum\footnote{Unless the binary is much higher mass than the population of field stars it scatters against; however, this more extreme mass ratio limit greatly reduces the importance of resonant scatterings.}. We therefore analyze two different limits of Eq. \ref{eq:FP}: first, a relatively simple, 1D equation in energy space, where we assume a steady-state eccentricity distribution. In this limit, we have
\begin{equation}
\frac{\partial \mathcal{N}}{\partial t} = -\frac{\partial}{\partial E_{\rm B}}(\langle \Delta E_{\rm B} \rangle \mathcal{N}) + \frac{1}{2}\frac{\partial^2}{\partial^2 E_{\rm B}}(\langle \Delta E_{\rm B}^2 \rangle \mathcal{N}) - \left(\frac{\partial \mathcal{N}}{\partial t} \right)_{\rm sink}. \label{eq:FP2}
\end{equation}
We allow $E_{\rm B}$ to range from $E_{\rm HS}$ to $E_{\rm coll}$.
At these boundaries, we impose a Dirichlet-type boundary condition, with $N(E_{\rm B})=0$. Alternatively, one could allow soft binaries to be included, in which case the limits would become 0 and $E_{\rm coll}$. We neglect soft binaries because of their rapid rate of ionization, but this could in the future be modeled with an appropriate choice of volumetric sink function.
Equation~\ref{eq:FP2} is a partial differential equation that can be solved numerically. To do this, it is necessary to write explicit diffusion coefficients. We find remarkably simple rates of diffusion across the space of orbital elements using $\mathcal{T}$ taken from \citet{stoneleigh19}, assuming a Maxwellian distribution of relative velocities, and considering particles all of equal mass, we find that
\begin{equation}
\langle \Delta E_{\rm B}\rangle = \frac{3}{64} \sqrt{\frac{\pi}{2}} \frac{G^2 n m^3}{\sigma |E_{\rm B}|} \Big(E_{\rm B} + m\sigma^2(1-\exp(E_{\rm B}/m\sigma^2)) \Big)
\end{equation}
and
\begin{align}
\langle \Delta E_{\rm B}^2\rangle =& \frac{1}{32}\sqrt{\frac{\pi}{2}} \frac{G^2 n m^3}{\sigma |E_{\rm B}|} \Big(2m\sigma^2 E_{\rm B} \notag \\ & +E_{\rm B}^2 + 2m^2\sigma^4(1-\exp(E_{\rm B}/m\sigma^2)) \Big).
\end{align}
The one approximation needed to derive these diffusion coefficients is to approximate $\iint \mathcal{T} {\rm d}L_{\rm B} {\rm d}C_{\rm B} = {\rm d}V/{\rm d}E_{\rm B} \propto E_{\rm B}^{-4}$, an approximate scaling that is quite accurate for equal-mass resonant scatterings \citep{stoneleigh19}.
Numerical results are shown in Fig. \ref{fig:FokkerPlanckSolution}. We see that the binary population steadily depletes over time, initially at low energies but eventually at higher energies as well. Eventually, a quasi-steady state energy distribution is reached, with a constant shape but continuously decaying normalization.
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{RealDiffusion_Res128_log.pdf}
\end{center}
\caption{Numerical results for the solution of Equation \ref{eq:FP2}. Different colors show snapshots of the 1D binary distribution function $\mathcal{N}$($E_{\rm B}$) at different dimensionless times $\tau$, which is physical time normalized by units of the local (core) relaxation time. Initial conditions (taken from \"Opik's Law, modified by exponential cutoffs near boundaries) are shown as a solid black line, while later steps in the evolution of the binary distribution function have been shown as colored dashed lines. The absorbing boundary conditions are dimensionless (i.e., normalized by the hard-soft boundary energy $E_{\rm HS}$) where the binary energies $E_{\rm B}=1$ and $E_{\rm B}=50$ represent the hard-soft boundary and the point of direct binary collisions, respectively.}
\label{fig:FokkerPlanckSolution}
\end{figure}
\subsection{Binary angular momenta}
\label{sec:angmom}
Strong encounters (resonances and prompt exchanges) essentially randomize the angular momentum of a given binary, making a Fokker-Planck approach ill-equipped to deal with this aspect of the problem. However, in the time between strong encounters, the cumulative effect of many weak flybys {\it will} lead to diffusive evolution of an individual binary's angular momentum, an effect largely neglected in existing approaches to binary evolution in dense clusters. We therefore treat the problem of binary angular momentum in a two-timescale form. An individual binary will undergo a random walk in angular momentum space {\it until} it suffers a strong binary-single scattering, at which point its new angular momentum can be drawn from the relevant distribution. We will thus arrive at a steady state angular momentum distribution by solving a 1D Fokker-Planck equation in $e_{\rm B}$-space. This {\it population-level} approach uses the diffusion coefficients computed in \citet[][their Eqs. 25a/25b]{hamers19b} to account for weak, distant encounters, and the mildly superthermal outcome distribution of \citet{stoneleigh19}, also found by \citet{ginat22}:
\begin{equation}
\mathcal{N}_{\rm res}(e_{\rm B}) = \frac{{\rm d}\sigma}{{\rm d}e_{\rm B}} = \frac{6}{5}e_{\rm B}(1+e_{\rm B}) \label{eq:superthermal}
\end{equation}
as the ``initial conditions'' in a source term $S^+(e_{\rm B}) = \Gamma \mathcal{N}_{\rm res}(e_{\rm B})$ that accounts for the generation of new binaries after resonant encounters (a sink term $S^-(e_{\rm B}) = -\Gamma \mathcal{N}(e_{\rm B})$ is likewise used to remove existing binaries in resonant encounters). Here we take a gravitationally focused strong scattering rate
\begin{equation}
\Gamma = \frac{2\pi G m_{\rm tot} n a_{\rm B}}{\sigma}, \label{eq:gammaEvaluated}
\end{equation}
i.e. the integral of Eq. \ref{eqn:gammanick} over an isotropic Maxwellian velocity distribution.
Note that in all cases we remain in the equal-mass limit, so that $m_{\rm tot}=3m$. For practical calculations, it is somewhat easier to use the variable $\mathcal{R} = 1-e_{\rm B}^2$, which can be viewed as a dimensionless angular momentum. Combining the resonant source/sink terms with the perturbative time evolution terms, we have
\begin{align}
\frac{\partial \mathcal{N}}{\partial t} = & -\frac{\partial}{\partial \mathcal{R}}(\langle \Delta \mathcal{R} \rangle \mathcal{N}) + \frac{1}{2}\frac{\partial^2}{\partial^2 \mathcal{R}}(\langle (\Delta \mathcal{R})^2 \rangle \mathcal{N}) - S^-_e + S^+_e. \label{eq:FP3}
\end{align}
Unlike Eq. \ref{eq:FP2}, the diffusion coefficients in this Fokker-Planck equation reflect the cumulative effect of many weak, perturbative flybys rather than that of repeated strong scatterings. The diffusion coefficients $\langle\Delta \mathcal{R} \rangle$ and $\langle(\Delta \mathcal{R})^2 \rangle$ are derived in Appendix \ref{app:DCs} using the results of \citep{hamers19b} as a starting point.
We use a Dirichlet-type $\mathcal{N}=0$ boundary condition at high eccentricity $e_{\rm B} = e_{\rm coll}$, i.e. $\mathcal{R}=1-e_{\rm coll}^2$ (corresponding to collisions, or, in the case of black hole binaries, gravitational wave inspirals), and a zero-flux boundary condition at $e_{\rm B}=0$ ($\mathcal{R}=1$).
By evolving this PDE forward in time, we can find a population-level steady state solution for arbitrary initial conditions.
Alternatively, we can take initial conditions corresponding to the {\it outcomes} of the strong scatterings (Eq. \ref{eq:superthermal}), and evolve the eccentricity distribution forward under the influence of weak scatterings. This {\it snapshot-level} approach describes the time evolution of an ensemble of binaries since their last strong scattering. While the population-level approach provides astrophysically realistic eccentricity distributions, the snapshot-level approach is more useful for building physical understanding, specifically by disaggregating the effects of strong and weak scatterings.
Snapshot-level results are shown in Fig. \ref{fig:eccEvolution}. We see that the initially super-thermal distribution (describing an ensemble of binaries shortly after their last resonant encounter) is initially eroded primarily by the Dirichlet boundary condition at high eccentricity. After a time $t\sim \Gamma^{-1}$, however, the effect of many weak encounters serves to further redistribute binary orbits to the low-$e_{\rm B}$ side of the spectrum. If we had considered weak scatterings only (i.e. zero-flux boundary conditions at both ends), we would have found a steady-state eccentricity distribution $\mathcal{N}(e_{\rm B}) \propto e_{\rm B}^{-4/25}$ \citep{hamers19b}, i.e. a moderately sub-thermal distribution biased towards circular orbits. This analytic curve is shown for comparison in Fig. \ref{fig:eccEvolution}, but it is not achieved even for the rare subset of the binary population that survives for a time $10\Gamma^{-1}$ without a resonant encounter, attesting to the importance of the collisional boundary condition at high $e_{\rm B}$.
The combined effects of weak flybys, resonances, and a collisional boundary condition are fully visible in the population-level solutions in Fig. \ref{fig:eccDepletion}, which shows three different solutions for three different values of $e_{\rm coll}$. Here we see that the steady-state $N(e_{\rm B})$ distributions are neither thermal, strictly super-thermal (as in \citealt{stoneleigh19}, nor strictly sub-thermal (as in \citealt{hamers19b}). The distributions are peaked at an intermediate eccentricity $\sim 0.1-0.5$, but highly depleted at large $e_{\rm B}$ (due to the collisional boundary condition) and at nearly circular $e_{\rm B}$ (due to resonances). We caution that our results do depend on the value of tertiary pericenter $Q_{\rm min}$ that transitions between resonant and non-resonant encounters. Appendix \ref{app:DCs} contains a fuller discussion of this, but Figs. \ref{fig:eccEvolution} and \ref{fig:eccDepletion} demonstrate that this dependence is modest.
We also note here that a similar approach could be taken to the diffusive (weak scattering) and non-diffusive (resonant encounter) evolution of binary angular momentum orientations in a {\it non-isotropic cluster}. Since resonant encounters preferentially produce survivor binaries with angular momentum vectors $L_{\rm B}$ aligned with the total angular momentum $\vec{L}_0$ of the three-body encounter, a non-isotropic velocity field will produce a preferential plane of binary rotation. For example, a rotating star cluster with net angular momentum $\vec{L}_{\rm cl}$ would be expected to have binaries with orbital planes biased towards prograde alignment with $\vec{L}_{\rm cl}$. In this first paper, however, we remain focused on the isotropic limit.
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{eccForwardInTimeAlt.pdf}
\end{center}
\caption{Time evolution of the 1D Fokker-Planck equation in the space of dimensionless square angular momentum $\mathcal{R} = 1-e_{\rm B}^2$ ($\mathcal{R}=1$ is a circular orbit; $\mathcal{R}=0$ a radial one). Here we show the number of stars per unit square angular momentum $\mathcal{N}(\mathcal{R}; a_{\rm B})$ at fixed semimajor axis $a_{\rm B}$. We have assumed that the stellar radius $R_{\rm coll} = 10^{-2}a_{\rm B}$ (i.e. there is an absorbing boundary condition at $1-e_{\rm B}=10^{-2}$). Different colors are labeled according to the fraction of the time between a strong (resonant) binary-single scattering event, which resets the binary eccentricity distribution to the mildly superthermal result discussed in the text. However, the cumulative (diffusive) effect of many weak flybys will quickly evolve this superthermal input distribution into a more complicated one, with high-$e_{\rm B}$ (low-$\mathcal{R}$) orbits strongly depleted by direct collisions. The subset of uncommon binaries that survive for more than a few resonant encounter times will achieve a distribution that is significantly more sub-thermal than the ${\rm d}N/{\rm d}e_{\rm B} \propto e_{\rm B}^{-4/25}$ distribution predicted in the absence of collisions \citep[][shown as a dot-dashed black line]{hamers19b}. For comparison, the usual thermal eccentricity distribution is shown as a dotted black line. Our results depend modestly on $Q_{\rm min}$, the critical tertiary pericenter that is assumed to separate resonant from non-resonant encounters. Solid colored lines show our fiducial value $Q_{\rm min}=3.4 a_{\rm B}$, while dashed colored lines show a more extreme choice of $Q_{\rm min}=2.4 a_{\rm B}$. The net effect is modest.}
\label{fig:eccEvolution}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{eccSteadyStatesAlt.pdf}
\end{center}
\caption{Steady-state solutions of the 1D Fokker-Planck equation in the space of dimensionless square angular momentum $\mathcal{R}$. Here we show the steady-state distributions achieved when combining (i) weak perturbations from secular flybys with (ii) less common resonant scatterings, whose outcomes are determined ergodically. The solid green, blue and purple curves show steady-state distributions for populations of binaries that will merge when $\mathcal{R}$ equals $10^{-3}$, $10^{-2}$, and $10^{-1}$, respectively. For comparison, the black dotted line is a thermal eccentricity distribution, the black dashed line is the superthermal ergodic outcome distribution for an ensemble of resonant scatterings, and the black dot-dashed line is the ${\rm d}N/{\rm d}e_{\rm B} \propto e_{\rm B}^{-4/25}$ distribution from \citet{hamers19b}. In all cases the steady-state eccentricity (or $\mathcal{R}$) distribution is neither super- nor sub-thermal; instead it is peaked at intermediate eccentricity and depleted of both highly radial and highly circular orbits. As in Fig. \ref{fig:eccEvolution}, solid colored lines represent $Q_{\rm min}=3.4 a_{\rm B}$, and dashed colored lines $Q_{\rm min}=2.4 a_{\rm B}$. }
\label{fig:eccDepletion}
\end{figure}
\subsection{Energy backreaction to the host cluster} \label{backreaction}
In this section, we discuss and develop a toy model to quantify the effects of binary hardening via single-binary interactions for the energy content of the host star cluster. Energy is systematically imparted to single stars via single-binary interactions, providing a heat source to the cluster core and subsequently further hardening those interacting hard binaries. Integrating over the binary distribution, we can compute the flow of energy from this binary reservoir into the core, and its subsequent diffusion throughout the entire stellar system via two-body relaxation: indirect (predominantly long-range) single-single interactions.
In principle, one could model the interactions between the binary population and the host cluster by adding extra dimensions to a Fokker-Planck approach. The global evolution of star clusters is often treated in a Fokker-Planck way: assuming spherical symmetry, a population of stars (single- or multi-species) can be evolved forward in time in the space of orbital energy $\mathcal{E}$ and orbital angular momentum $\mathcal{L}$, with populations slowly diffusing due to two-body scatterings (indeed, solving this problem by probabilistically sampling stellar distributions is the basis of the Monte Carlo codes mentioned in \S \ref{sec:intro}). While we hope to more rigorously explore this integrated approach in future work, for now we content ourselves to develop a ``two-zone'' model, where a star cluster with total mass $M_{\rm tot}$ is divided into a core of radius $r_{\rm c}$ and a halo extending further out, with a half-mass radius $r_{\rm h}$. The cluster is collisionally relaxed and therefore is quasi-isothermal with a velocity dispersion $\sigma$ that is a weak function of radius. For the sake of concreteness, we will use the analytic potential-density pair of \citet{stone15}. In this three-parameter model ($M_{\rm tot}$, $r_{\rm h}$, $r_{\rm c}$), the mass density profile of the cluster is
\begin{equation}
\rho(r) = \frac{\rho_{\rm c}}{(1+r^2/r_{\rm c}^2) (1+r^2/r_{\rm h}^2)},
\end{equation}
with a central core density of
\begin{equation}
\rho_{\rm c} = \frac{M_{\rm tot}(r_{\rm c} + r_{\rm h})}{2\pi^2 r_{\rm c}^2 r_{\rm h}^2}
\end{equation}
and a core velocity dispersion
\begin{equation}
\sigma_{\rm c}^2 \approx \frac{6}{\pi} (\pi^2/8 -1) \frac{G M_{\rm tot}}{r_{\rm h}}.
\end{equation}
The approximate equality in the last expression indicates that this formula for the core velocity dispersion is taken in the $r_{\rm c} \ll r_{\rm h}$ limit (though it is correct to within $\sim 10\%$ for $r_{\rm c} \sim r_{\rm h}$).
We use this simple model to estimate the rate of orbital energy flow into the core (from binary-single scatterings) and out of the core (from two-body relaxation, primarily between single stars): $\dot{E}_{\rm c}={\rm d}E_{\rm c}/{\rm d}t$.
The net flow is then
\begin{equation}
\label{eqn:energyflow}
\dot{E}_{\rm c} = \frac{{\rm d}E_{\rm c}}{{\rm d}r_{\rm c}}\frac{{\rm d}r_{\rm c}}{{\rm d}t} = \dot{E}_{\rm rel} + \dot{E}_{\rm bin},
\end{equation}
where $E_{\rm c} \approx$ -$M_{\rm c}\sigma_{\rm c}^2$/2 is the binding energy of the core, $r_{\rm c}$ is the core radius and $\sigma_{\rm c}$ is the 3D velocity dispersion in the core. Here the mass of the core is roughly (again, in the limit of $r_{\rm c} \ll r_{\rm h}$)
\begin{equation}
M_{\rm c} \approx \frac{2(1-\pi /4)}{\pi} \frac{r_{\rm c}}{r_{\rm h}} M_{\rm tot}.
\end{equation}
The diffusive energy flow due to two-body relaxation is set by the radial gradient in ``temperature'' (i.e. velocity dispersion $\sigma$) across the cluster core and halo out to the half-mass radius. In the limit of a single-mass cluster, it is roughly
\begin{equation}
\dot{E}_{\rm rel} = -A_{\rm cond} \frac{M_{\rm c}\sigma_{\rm c}^2}{2 t_{\rm r,c}},
\end{equation}
where $t_{\rm r, c}$ is the core relaxation time and $A_{\rm cond}$ is a dimensionless conductivity constant (i.e. an encapsulation of the very small gradient ${\rm d}\sigma / {\rm d}r$) that can be measured from Fokker-Planck (e.g. in single-mass clusters, $A_{\rm cond} \sim 10^{-3}$; \citealt{cohn80}) and N-body simulations. In a multi-mass cluster, $A_{\rm cond}$ is higher by one to two orders of magnitude depending on the exact mass spectrum. $\dot{E}_{\rm rel}$ is negative-definite because two-body relaxation conducts heat outwards, allowing the core to collapse and become more tightly bound.
The rate of energy injection from binary burning can be computed by calculating the time evolution of the total energy in all binaries in the core,
\begin{equation}
E_{\rm bin} = \int E_{\rm B} \mathcal{N}(E_{\rm B}) {\rm d}E_{\rm B}.
\end{equation}
As the binary burning rate is an integral over the distribution function, it turns Eq. \ref{eqn:energyflow} into an integro-differential equation.
One particularly interesting limit is when $\frac{dr_{\rm c}}{dt} \rightarrow 0$, since this defines the time of core "bounce", or the moment when core collapse halts and is reversed by single-binary interactions. $N$-body and other approaches show that steady state solutions do not generally exist near this limit, which is the turning point in gravothermal oscillations.
We now make a ``two-zone'' approximation for the binary populations, in which the binaries are divided into a portion in the cluster core (with distribution $\mathcal{N}_{\rm c}$) and a portion in the halo (with distribution $\mathcal{N}_{\rm h}$). Note that since encounter rates scale as the local relaxation time, there are different diffusion coefficients for both the core (e.g. $\langle\Delta_{\rm c} E_{\rm B} \rangle$) and the halo (e.g. $\langle\Delta_{\rm h} E_{\rm B} \rangle$). Binaries may move back and forth between these two zones via ejection from the core and dynamical friction on halo binaries. These produce interchange rates that are
\begin{equation}
\frac{\partial N_{\rm ej}}{\partial t} = A_{\rm ej} \frac{1}{t_{\rm r,c}}
\end{equation}
and
\begin{equation}
\frac{\partial N_{\rm sink}}{\partial t} = \frac{N_{\rm h}}{t_{\rm r,c}} \frac{r_{\rm c}^2}{r_{\rm h}^2} \sqrt{\frac{m_{\rm B}}{\langle m_\star \rangle}},
\end{equation}
respectively. In both of these equations, $t_{\rm r,c}$ is the core relaxation time (which increases by a factor $r_{\rm h}^2 / r_{\rm c}^2$ when one considers the relaxation time at the half-mass radius $r_{\rm h}$), and in the first, $A_{\rm ej}\sim 0.1-1$ is a dimensionless number computed by integrating over the Maxwellian velocity dispersion of the core stars.
This gives a set of coupled differential equations for the time evolution of the cluster:
\begin{align}
\dot{E}_{\rm c} =& \dot{E}_{\rm rel} + \dot{E}_{\rm bin} + \frac{1}{2}\sigma_{\rm c}^2\frac{\partial N_{\rm ej}}{\partial t} - \frac{1}{2}\sigma_{\rm c}^2\frac{\partial N_{\rm sink}}{\partial t} \label{eq:twozone} \\
\frac{\partial N_{\rm c}}{\partial t} =& -\frac{\partial}{\partial E_{\rm B}} \left(\langle \Delta_{\rm c} E_{\rm B} \rangle N_{\rm c} \right) + \frac{1}{2} \frac{\partial^2}{\partial E_{\rm B}^2} \left(\langle \Delta_{\rm c} E_{\rm B}^2 \rangle N_{\rm c} \right)\notag \\
+& \frac{\partial N_{\rm sink}}{\partial t} - \frac{\partial N_{\rm ej}}{\partial t} \notag \\
\frac{\partial N_{\rm h}}{\partial t} =& -\frac{\partial}{\partial E_{\rm B}} \left(\langle \Delta_{\rm h} E_{\rm B} \rangle N_{\rm h} \right) + \frac{1}{2} \frac{\partial^2}{\partial E_{\rm B}^2} \left(\langle \Delta_{\rm h} E_{\rm B}^2 \rangle N_{\rm h} \right) \notag \\
-& \frac{\partial N_{\rm sink}}{\partial t} + \frac{\partial N_{\rm ej}}{\partial t} \notag
\end{align}
Note that this is a system of two diffusive-type PDEs and one integro-differential ODE.
\section{Results} \label{results}
In this section, we apply our model to dynamically evolve a population of binary star systems in a dense cluster environment and compare the results to N-body simulations. We choose \"Opik's Law for our fiducial initial conditions, which gives the initial distribution of binary orbital energies. We begin with a one-zone model, before presenting the results for our two-zone (core/halo) model, using Eqs. \ref{eq:twozone}.
\begin{comment}
\subsection{First-order formulation} \label{simple2}
In this section, we evolve an initial distribution of binary orbital energies set by \"Opik's Law, as well as a thermal initial distribution of orbital eccentricities, forward through time using our simple analytic first-order formalism presented in Section~\ref{simple}.
\subsubsection{The orbital energy distribution}
We solve numerically Equation~\ref{eqn:ratef} in order to dynamically evolve our chosen initial binary orbital energy distribution forward through time. To first order, this assumes conservation of the total number of binaries. This assumption should remain approximately valid until the fraction of binaries destroyed (i.e., pushed to harder orbital energies beyond our chosen limit or cut-off at the hard end, $|E_{\rm B,max}|$) becomes of order unity, which only occurs very late in the overall cluster evolution.
At the soft end of the binary orbital energy distribution, binaries start to flow out of our lower cut-off bin(s) to harder orbital energies, depleting bins at the soft-end. Once depleted, a given bin at the soft end will remain depleted; binaries can only flow to harder orbital energies in our first-order formulation. Consequently, this region of parameter space tends to become (rapidly) depleted.
In Figure~\ref{fig:fig1} we show the results of this exercise for a total integration time of 10 Gyr. We adopt an initial number of binaries of $N_{\rm b} =$ 10$^5$. To compute the rate of collisions we assume $\rho_{\rm 0} =$ 10$^6$ M$_{\odot}$ pc$^{-3}$ and $\sigma_{\rm 0} =$ 5 km s$^{-1}$, along with a mean stellar mass of $m =$ 0.35 M$_{\odot}$. Assuming core and half-mass radii of 0.3 and 5 pc, respectively, Equation 15 in \citet{stone15} yields a core relaxation time of 2.5 Myr.
As is clear from Figure~\ref{fig:fig1}, binaries harder than the hard-soft boundary get harder over time, at a rate that broadly increases with decreasing absolute binary orbital energy. This explains the shift along the orbital energy axis to larger absolute orbital energies, which initially occurs the fastest at the soft end of the distribution; the widest hard binaries undergo encounters at the highest rate, and so get hardened to bins corresponding to larger absolute orbital energies the fastest.
Soft binaries have even shorter interaction times due to their larger orbital separations and hence collisional cross-sections. So they tend to experience many direct 1+2 interactions at a high rate, and are consequently assumed to be immediately destroyed in our model. Meanwhile, those hard binaries closest to the hard-soft boundary, relative to other even harder binaries, have the shortest timescales for undergoing 1+2 interactions. Again, their wider orbital separations lead to a reduced interaction time, and they are much more numerous than the hardest binaries. In our model, the rate of depletion at the soft end outweighs the probability being higher for more tightly bound binaries to get diffused to higher orbital energies in a single encounter, as dictated by Liouville's Theorem. This effect should also contribute to depleting binaries at the hardest end of the distribution. But, in our model, the diffusion-based depletion observed at the hard end of the distribution is driven by the fact that it is much more sparsely populated by the most compact binaries, relative to the soft end. Thus, it is the softest hard binaries that are the first to show evidence of the underlying dynamical evolution that is occurring, pushing these hard binaries to become even harder and depleting these orbital energy bins of binaries. This effect is counter-balanced by the depletion of binaries at the hard end. If finite particle radii or dissipative effects were to be considered, these hardest binaries would eventually be driven to collision or merger. In the end, the competing rates of diffusion at the soft and hard ends of the distribution drive it toward a log-normal form, the peak of which is then diffused over time to higher absolute energies while otherwise preserving its functional form.
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{fig1.eps}
\end{center}
\caption[The results of integrating over Equation~\ref{eqn:ratef} to evolve an initial distribution of binary orbital energies forward through time.]{The results of integrating over Equation~\ref{eqn:ratef} to evolve an initial distribution of binary orbital energies (given by \"Opik's Law) forward through time. The (logarithm of the) probability density function is shown on the y-axis, and the (logarithm of the) absolute value of the binary orbital energy is shown on the x-axis. The red line shows the initial distribution of binary orbital energies. In order of increasing line width, the black lines show the distributions of binary orbital energies evaluated at 100 Myr, 3 Gyr, 9 Gyr and 10 Gyr.}
\label{fig:fig1}
\end{figure}
\subsubsection{The orbital eccentricity distribution}
In Figure~\ref{fig:fig5} we show the time evolution of the orbital eccentricity distribution for a total integration time of 10 Gyr. For this exercise, we assume a thermal initial eccentricity distribution, and take the binary apocentre distance as the geometric size of a binary, with a typical orbital separation of 0.01 AU. Otherwise, our assumptions are the same as in the preceding section.
As is clear from Figure~\ref{fig:fig5}, the distribution of orbital eccentricities evolves toward a sub-thermal distribution. This is because the tendency is for binaries to be diffused to higher eccentricities at a rate that increases with increasing orbital eccentricity, as found by \citet{stoneleigh19}. Hence, many binaries in our initial population are rapidly diffused to e $\sim$ 1 and then removed from our sample, since these would collide or merge if finite particle radii or dissipative effects were considered.
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{fig5.eps}
\end{center}
\caption[The results of evolving an initial distribution of binary orbital eccentricities forward through time.]{The results of evolving an initial distribution of binary orbital eccentricities (given by a thermal distribution) forward through time. The axes and lines are defined the same as in Figure~\ref{fig:fig1}.}
\label{fig:fig5}
\end{figure}
\subsection{Second-order Boltzmann equation} \label{boltzmann2}
Next, we dynamically evolve a population of binary star systems in a dense cluster environment using the Boltzmann Equation derived in Section~\ref{master}. Specifically, we numerically solve Equation~\ref{eq:FP2}, a second-order partial differential equation. The initial distribution of binary orbital energies is again chosen (according to \"Opik's Law) to be $\mathcal{N}(E_{\rm B}) = k|E_{\rm B}|^{-1}$, where $k$ is a normalization constant given by Equation~\ref{eqn:knorm2}. The results have al in Figure~\ref{fig:FokkerPlanckSolution} at several different core relaxation times $\tau$, indicated by the different coloured lines. We begin by assuming a one-zone model as before, where all interactions are isolated to the cluster core.
\end{comment}
\subsection{One-Zone Model}
We begin by dynamically evolving a population of binary star systems through energy space in a dense cluster environment, by numerically solving the Fokker-Planck equation derived in Section~\ref{master}.
The initial distribution of binary orbital energies is again chosen (according to \"Opik's Law) to be $f_{\rm B}(E_{\rm B}) = k|E_{\rm B}|^{-1}$, where $k$ is a normalization constant. The results have already been shown in Figure~\ref{fig:FokkerPlanckSolution} at several different core relaxation times $t_{\rm r,c}$, indicated by the different coloured lines, but here we will discuss them in more detail.
In Figure~\ref{fig:FokkerPlanckSolution}, we see slow evolution of the hardest binaries, and relatively quick evolution of ones near the hard-soft boundary. This is unsurprising, given the small (large) scattering cross-sections of the former (latter). Over time, a quasi-steady state is reached at low binary energies, which slowly propagates to higher energies. Once this quasi-steady state solution reaches the collisional (high-$|E_{\rm B}|$) boundary, the shape of the distribution freezes in, and further evolution only decreases its normalization. At all times, the drift coefficient $\langle E_{\rm B} \rangle$ is creating a net flow towards softer energies, though the diffusion term $\langle (E_{\rm B})^2 \rangle$ is responsible for the turnover at the hardest energies, once a quasi-steady state has been established everywhere.
\subsection{Two-Zone Model}
Next, we add many of the extra pieces of the two-zone model, i.e. Eq. \ref{eq:twozone}. Because our goal is to eventually compare to $N$-body simulations with limited run-time (and limited evolution of the cluster density profile, i.e. limited global energy transfer), here we solve only the two coupled PDEs that exchange binary populations between the cluster core and halo. Simultaneous solution of the energy equation would require more numerical development which we defer to future work.
As we will show in the subsequent sections, we initially observe a quick depletion of binaries in the core, due to the recoil imparted by single-binary interactions, and a corresponding increase in the halo population. At later times, these binaries mass segregate back into the core, increasing the population of binaries in the core and decreasing that in the halo. Eventually, an approximate steady-state balance is achieved between the rate of binaries being ejected from the core due to single-binary interactions and binaries re-entering the core by leaving the halo due to mass segregation. This is because we assume all equal-mass particles in our model, such that binaries are the heaviest objects in the cluster.
\subsection{Comparisons to N-body simulations} \label{comparisons}
In this section, we present the results of our preliminary N-body simulations, and compare them to the predictions of our second-order analytic models described in the previous sections, with a focus on the improved two-zone model.
\subsubsection{Initial Conditions and Assumptions}
For each simulation, we adopt a Plummer density profile initially, with 10$^4$ stars and 200 binaries. The initial cluster has a core radius of 0.3 pc and a half-mass radius is 0.8 pc. We assume identical point-particles with masses of 1 M$_{\odot}$. As in our analytic calculations, we assume \"Opik's Law initially for the distribution of binary orbital energies and a thermal eccentricity distribution. At the hard end, we truncate our initial energy distribution at a minimum value of 4 times the radius of the Sun (i.e., corresponding to slightly wider than a contact state for 1 M$_{\odot}$ stars). At the soft end, we truncate at twice the hard-soft boundary, calculated as:
\begin{equation}
\label{eqn:HS}
a_{\rm HS} = \frac{Gm}{\sigma^2},
\end{equation}
where $\sigma$ is the core velocity dispersion, which is initially 2.0 km s$^{-1}$.
The initial core and half-mass relaxation times for our simulated clusters are 7.6 Myr and 25 Myr, respectively.
We perform 10 simulations all with the same initial conditions each perturbed slightly using a different random seed. These simulations are then stacked together, to increase our sample size for the number of binaries evolving dynamically due to single-binary interactions, bringing the total sample size up to 2000. This stacking is done to increase the statistical significance of our results, and verify the robustness of our N-body simulations by quantifying the stochastic contribution of chaos to the observed differences in each simulation.
The time evolution of several core properties in each of the 10 simulations, namely the core density, velocity dispersion, radius, and binary fraction are illustrated in Figure \ref{fig:binplot}, with the average values illustrated in black. Time is normalized by the cluster's initial core relaxation time $t_{r,c}^0$ and the error bars represent the standard deviation about the mean. For $t/t_{r,c}^0 < 20$, the core evolution of all 10 models is very similar, with core density and velocity dispersion staying nearly constant while the core radius and binary fraction slowly decrease with time. However, near $t/t_{r,c}^0 = 20$, the clusters undergo a mild core collapse. In the post-core collapse stage, the core density and radius of individual simulations slowly start to diverge with significant fluctuations between timesteps. The core velocity dispersion, however, remains close to its original value with a few brief fluctuations and the core binary fraction slowly decreases beyond $t/t_{r,c}^0 = 20$ due to binary destruction, after most binaries outside of the core have had enough time to mass segregate into the core. We emphasize that the comparison is most reliable before core-collapse occurs, due to the increased stochasticity in the time evolution of the cluster properties beyond this point. In other words, the simulations begin to diverge significantly beyond core collapse, yielding increasingly different cluster properties between the simulations over time. Also, our neglect of energy backreaction from the binary population becomes worse at this time.
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{binplot.pdf}
\end{center}
\caption{The time evolution, normalized by the initial core relaxation time, of the core density, core velocity dispersion, core radius, and core binary fraction for all 10 simulations with the mean value illustrated as a solid black line. The error bars mark the standard deviation about the mean.}
\label{fig:binplot}
\end{figure}
\subsubsection{One-Zone Model}
In this section, we discuss the time evolution of the binary orbital energy distribution for our one-zone model, focusing on the cluster core. The binary orbital energy distributions of our N-body simulations are shown in Figure~\ref{fig:onezone} at several different core relaxation times. The black solid line show the initial distribution given by \:Opik's Law.
The time evolution of the binary orbital energy distribution behaves as expected, slowly pushing hard binaries to become more compact and soft binaries to become disrupted. The evolution of the one-zone model is initially faster than the simulations as $N(E_B)$ quickly decreases after just 5 core relaxation times. It is not until 40 core relaxation times does $N(E_B)$ for the simulations fall below the one-zone model. This discrepancy is likely because binaries flow into the core from the halo due to two-body relaxation in our simulations, and out of the core due to the recoil imparted post-single-binary interaction. But, in our analytic model, we assume that anything happening in the core stays in the core. Hence, binaries are depleted in the core at a faster rate, both due to mergers at the hard end of the distribution and the destruction of wide binaries at the soft end of the distribution.
After a given number of core relaxation times, Figure~\ref{fig:onezone} illustrates a strong agreement between the one-zone model and the simulations at the hard- and soft-ends of the distribution. However, the model over-predicts the number of binaries in the intermediate regime relative to the simulations by a factor $\sim$ 3 (when error bars are included this typically means that the N-body simulation results disagree with our analytic model at the level of $\gtrsim$ 3$\sigma$). We attribute this disagreement to the overall faster core evolution in our one-zone model, depleting binaries in the intermediate regime by pushing hard binaries to become harder, and wide binaries to become wider. In the subsequent section, we will move on to our two-zone model, and show that this does indeed correct this disagreement.
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{Figure_5.pdf}
\end{center}
\caption{The distribution of binary energies in the core of the cluster, shown at nine different snapshots in time. Black indicates the initial distribution corresponding to $t = 0t_{\rm r,c}^0$, where $t_{\rm r,c}^0$ is the initial core relaxation time. We then evolve our simulation for in total 45 core relaxation times, in steps of 5 $t_{\rm r,c}^0$. We plot the distributions for each time-step using different colours, as indicated in the upper-right inset in the figure. Data points are binned-up binary energies from the N-body simulations, while curves are predictions from the one-zone Fokker-Planck approach.}
\label{fig:onezone}
\end{figure}
\subsubsection{Two-Zone Model}
In this section, we discuss the time evolution of the binary orbital energy distribution for our two-zone model, considering now both the core and the halo. Binaries are now allowed to flow out of the core in our analytic model due to the recoil imparted by linear momentum conservation post-single-binary interaction, and later flow back into the core due to mass segregation.
As is clear from Figures~\ref{fig:binCompareHalo} and~\ref{fig:binCompareCore}, the results from our N-body simulations, showed by the coloured data points, now agree with our analytic two-zone model typically to within 1$\sigma$, with only a few outliers within 2$\sigma$ from our analytic model. We conclude that our improved two-zone model does indeed improve the agreement between the simulations and our analytic theory. We emphasize that this is but one of the many changes that can be accommodated by our model, to even further improve the agreement between the simulated and calculated results. Possible improvements and how to implement them will be discussed in more detail in the subsequent section.
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{twoZoneHalo.pdf}
\end{center}
\caption{The distribution of binary energies in the halo of the cluster, shown at three different snapshots in time. Black, blue, and green correspond to $t=0 t_{\rm r,c}^0$, $t=10 t_{\rm r,c}^0$, and $t=40 t_{\rm r,c}^0$, respectively. Data points are binned-up binary energies from the N-body simulations (error bars show asymmetric $1\sigma$ Poissonian error range; \citealt{Gehrels86}), while curves are predictions from the two-zone Fokker-Planck approach.}
\label{fig:binCompareHalo}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{twoZoneCore.pdf}
\end{center}
\caption{The same as Figure~\ref{fig:binCompareHalo}, but for the cluster core, where statistics are poorer.}
\label{fig:binCompareCore}
\end{figure}
\section{Discussion} \label{discussion}
In this section, we discuss the key features and assumptions going into our model, their short-comings and future efforts that should be made for constructing an improved analytic model for dynamically evolving forward through time populations of binary star systems in dense stellar environments due to single-binary interactions. We further discuss the significance of our analytic models and their results to pertinent problems characteristic of modern astrophysics.
\subsection{A new tool independent of computational methods for dynamically evolving populations of binaries} \label{newtool}
The Boltzmann-type equation derived in this paper presents an alternative and complementary tool to computational $N$-body or Monte Carlo simulations. These methods compute the time evolution of their binaries using a combination of stellar dynamics, stellar and binary evolution, etc. The method presented here isolates the dynamical effects of single-binary scatterings on the time evolution of binary orbital parameter distribution functions. Thus, our method is meant to be directly complementary to analogous computer simulations. In principle, comparing the results of the model presented here with more complicated $N$-body and Monte Carlo simulations should allow us to better isolate the effects of dynamics in evolving a binary population in a given environment, thereby isolating the effects of stellar and binary evolution, etc. (i.e., usually included in $N$-body and MC models in an approximate way, assuming the evolution proceeds in isolation), while also comparing the relevant timescales for each of these critical effects to operate. Thus, our model allows us to quantify and characterize the dominant physics evolving a given binary population in a given star cluster at any time, by rigorously quantifying and isolating the effects of single-binary interactions.
Recently, \citet{geller19} took a significant step forward in this direction by developing a semi-analytic Monte Carlo-based model for the dynamical evolution of binaries that is analogous to a first-order diffusion-based model.
They compared the results to direct $N$-body and Monte Carlo models for GC evolution with very similar initial conditions, and found excellent agreement over several Gyr of cluster evolution. This model was subsequently applied in \citet{trani21} to quantify the impact of binary-binary scatterings on the results, since the authors considered only single-binary interactions. As illustrated in this paper, adopting the higher-order diffusion-based model presented herein only serves to improve the agreement between the analytic theory and computational simulations.
\subsection{Improving the model} \label{future}
Next, we discuss the key features and assumptions going into our model, and how they can be improved upon in future work.
First, we begin with a one zone model for our calculations, including only those binaries and singles inside the cluster core. However, in reality, objects are free to migrate in and out of the core. In particular, we expect the core binary fraction to initially increase over time due to binary stars mass segregating into the core due to two-body relaxation. We also expect that the probability of a given binary being ejected from the core due to the recoil from linear momentum conservation during a single-binary interaction will be higher for more compact binaries. These are likely to eject singles at higher velocities, imparting a larger recoil velocity to the binary and increasing the probability of ejecting it from the core, or even the cluster.
All of these aspects of our model can easily be improved upon by adopting more realistic boundary conditions. One way to go about this is to increase the number of zones, and expanding the model to include regions of the cluster outside of the core (e.g., adopting a shell-like structure for the cluster). As illustrated in Figures~\ref{fig:binCompareHalo} and~\ref{fig:binCompareCore}, we have shown that adapting our model to a two-zone model does indeed improve the agreement, treating the core and halo independently. This allows for binaries to flow into and out of the cluster core and halo, due to being ejected from the core via single-binary interactions and subsequent mass segregation back into the core from the halo.
We also do not allow for binaries to be kicked out of the cluster, since in our simulations these events are very rare. However, for those clusters where such events are more likely, additional sink/source terms could be included in our existing Boltzmann equation, improving the accuracy of our existing two-zone model even further.
Second, we do not consider binary formation via three-body interactions involving initially all isolated single stars, since these events are exceedingly rare in our simulations (or do not occur at all). This effect should become important primarily at very high densities, such as during core collapse or in very massive, dense star clusters and galactic nuclei.
In future work, this aspect of our model can be improved upon by including an additional density-dependent source term in our Boltzmann equation.
Third, we have assumed that the properties of the core are constant (i.e., a constant density and velocity dispersion throughout the core at a given time) in our analytic model since this is approximately true for the initial cluster conditions considered in this paper. However, we manually update the averaged core properties at regular intervals upon performing our comparisons to the $N$-body simulations. The inclusion in our analytic model of a more realistic gradient in the core gravitational potential would introduce a dependence of the encounter rate on the distance from the cluster centre $r$. In future work, this aspect of our model can be improved upon by including a radial $r$ dependence into Equation~\ref{eqn:gammanick}. We do not expect this to have significantly affected the comparison between our analytic model and the results of our $N$-body simulations, since we terminate the comparison before the point at which most simulations reach core collapse and begin to diverge in their time evolution.
Fourth, we have neglected binary-binary interactions, which should also always be occurring for non-zero binary fractions, and will hence contribute to the dynamical evolution of the binary orbital properties. This should not have significantly affected the results of our comparison between our analytic model and the simulations. This is because single-binary interactions occur more frequently than binary-binary interactions for binary fractions $\lesssim$ 10\% \citep{leigh11}, and we intentionally adopt very low initial binary fractions in our simulations to minimize the contribution of binary-binary interactions relative to single-binary interactions. We compensate by performing additional simulations in order to generate the required statistical significance (i.e., the total number of simulated binaries) for the comparisons between our analytic model and the simulations, by stacking the results of these simulations all having nearly identical initial conditions. This aspect of our work can be improved upon in future studies, using the methodology described in the Appendix.
Finally, we assume identical mass particles throughout this paper. However, our model can easily be improved to include a mass-dependence using, for example, the method described in Section 3.1 of \citet{leigh20}.
\subsection{Astrophysical implications: A focus on black hole binaries in globular clusters} \label{BHs}
It turns out that the assumptions needed for the methods and modeling techniques presented in this paper to be valid are particularly well-suited to treat the dynamics of stellar-mass black hole (BH) binaries in globular clusters. This is because the assumptions underlying the application of our analytic formalism for single-binary scatterings require low virial ratios - i.e., the total kinetic energy must be a small fraction of the total binding energy of the interaction, to help maximize the fraction of long-lived, chaotic, resonant interactions, for which our formalism is most directly applicable (i.e., the assumption of ergodicity is upheld). Due to energy equipartition, BH binaries can end up with the lowest velocities in the cores of clusters, and also tend to have very large absolute orbital energies due to the larger component masses. Both effects push us toward preferentially low virial ratio interactions, ideal for applying the methods introduced in this paper to study the dynamical evolution of BH-BH binaries in dense star clusters \citep{leigh16b}. We intend to explore this potentially interesting connection and application for our model in forthcoming work.
Finally, it is also in principle possible to use our model to constrain the primordial properties of star clusters. In particular, if a given cluster is observed to be in core-collapse and the observer can measure the binary population properties (i.e., the present-day number of binaries and their orbital parameter distributions), then one can constrain the initial conditions using our model. This is because our model assumes a given set of initial conditions, and these become a good candidate for the true set of initial conditions if the model reaches core-collapse on the correct timescale. Degeneracies in the initial conditions yielding approximately the same time of core collapse are likely to exist, but these can easily be identified and quantified using our analytic model in future work, which indicates one of its main strengths and utility for astrophysical research: a quick and fast exploration of the relevant parameter space to help constrain the underlying sets of initial cluster conditions that could have evolved over a Hubble time to reproduce what we observe today (i.e., central density, total mass, surface brightness profile, etc.).
\section{Summary} \label{summary}
Moore's Law is dead.
It follows that the demand for alternative models independent of computational limitations is increasing, and the field of gravitational dynamics is no exception. With this in mind, we recently derived, using the ergodic hypothesis \citep{monaghan76a,monaghan76b}, analytic outcome distributions for the products of single-binary \citep{stoneleigh19} scatterings (similar techniques may also apply to binary-binary scatterings, \citealt{leigh16b}). With these outcome distributions in hand, it becomes possible to construct a diffusion-based approach to dynamically evolve an entire population of binaries due to single-binary (and eventually including binary-binary) scatterings.
We present in this paper a self-consistent statistical mechanics-based analytic model formulated in terms of a master equation (in the spirit of \citealt{goodman93}), and eventually evolved in its Fokker-Planck limit. Our model evolves the binary orbital parameter distributions in dense stellar environments forward through time due to
strong single-binary interactions, using the analytic outcome distribution functions found in \citet{stoneleigh19}. The effects of weaker, perturbative scatterings are incorporated using the secular theory of \citet{hamers19a, hamers19b}.
We have applied our formalism in various simplified limits, working for now in the equal-mass case. In the space of binary eccentricity $e_{\rm B}$, we find that the combined effect of strong (resonant) scatterings and more distant, perturbative flybys is to create steady-state $e_{\rm B}$ distributions that are strongly depleted at both the lowest and highest eccentricities. The resulting binary populations do not match the strongly subthermal $e_{\rm B}$ distributions arising from weak scatterings only \citep{hamers19b}, nor the mildly superthermal distributions coming from strong scatterings only \citep{stoneleigh19}, but rather are peaked at an intermediate eccentricity $e_{\rm B} \sim 0.1-0.5$.
In the space of binary energy, we compare the predictions of our semi-analytic model to the results of numerical $N$-body simulations performed using the NBODY6 code. We find good agreement between the simulations and our analytic model for the initial conditions considered here, and the adopted time intervals over which the results are compared (i.e., the first 20 core relaxation times, and roughly up until the time at which core collapse occurs).
The semi-analytic model presented in this paper represents a first step toward the development of more sophisticated models, which will be the focus of future work. For example, as shown in Section~\ref{backreaction}, it is in principle possible to
couple binary evolution equations to a few-zone model to produce physically transparent models of cluster core collapse and gravothermal evolution. More speculatively, the binary evolution formalism presented here could be applied to study the collisional evolution of binaries in (simplified models of) dense star systems with fewer degrees of symmetry, where $N$-body simulations can be prohibitively expensive and Monte Carlo techniques do not currently work. Rotating star clusters (though see \citealt{fiestas06}) and inclination-segregated stellar disks \citep{meiron19} are two examples of such systems.
In future work, we intend to further expand upon the base model presented in this paper to include binary-binary interactions. These can occur more frequently than single-binary interactions in clusters with high binary fractions (i.e., binary-binary interactions dominate over single-binary interactions for binary fractions $\gtrsim$ 10\% \citep{sigurdsson93,leigh11}), and hence cannot be neglected in this cluster regime (e.g., open clusters and low-mass globular clusters). In an Appendix, we consider this added complication to our base model, touching upon some basic predictions that motivate the need for further development in this direction. Specifically, we identify a potential steady-state balance between the binary and triple fractions in star clusters, based on simple thermodynamics-based considerations, and show via a proof-of-concept calculation that this could in principle be used to directly constrain the initial primordial binary and triple fractions, in addition to the primordial properties of the multiple star populations in dense stellar environments.
\section{Acknowledgments}
NWCL gratefully acknowledges the generous support of a Fondecyt Iniciaci\'on grant 11180005, as well as support from Millenium Nucleus NCN19-058 (TITANs) and funding via the BASAL Centro de Excelencia en Astrofisica y Tecnologias Afines (CATA) grant PFB-06/2007. NWCL also thanks support from ANID BASAL project ACE210002 and ANID BASAL projects ACE210002 and FB210003. NCS received financial support from the Israel Science Foundation (Individual Research grant 2565/19), and the BSF portion of a NSF-BSF joint research grant (NSF grant No. AST-2009255 / BSF grant No. 2019772). WL acknowledges support from NASA via grant 20-TCAN20-001 and NSF via grant AST-2007422.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,308 |
\section{Introduction}
With particle colliders extending the energy frontier, the need to understand QCD in the high energy limit becomes essential. The new kinematic regime explored at the LHC and potentially in the future deep inelastic scattering (DIS) machines, EIC \cite{Deshpande:2005wd,Boer:2011fh} and LHeC \cite{Dainton:2006wd,Klein:2009zz}, requires detailed analyses of the high energy and density limit in QCD.
At these high energies it is expected that the parton densities will become very large. In particular, as the Bjorken $x$ becomes very small one needs to take into account the effects of parton saturation \cite{Gribov:1984tu,Mueller:1985wy}. Evolution into the small $x$ region, which incorporates parton saturation phenomenon, is governed by the nonlinear Balitsky-Kovchegov (BK) \cite{Kovchegov:1999yj,Kovchegov:1999ua,Balitsky:1995ub,Balitsky:1998ya,Balitsky:2001re,Balitsky:1998kc} equation. A general framework which systematically incorporates the effects of the large gluon density is the
Color Glass Condensate (CGC) \cite{McLerran:1993ka,McLerran:1993ni} model.
Within the CGC model the evolution into the small $x$ region is given by the
renormalization group - type equation, the JIMWLK equation \cite{JalilianMarian:1997gr,JalilianMarian:1997dw,Weigert:2000gi,Iancu:2000hn,Iancu:2001ad,Ferreiro:2001qy,Mueller:2001uk}. The CGC framework contains the BK evolution equation as well as the evolution of the higher point gluon correlators, for a recent study see \cite{Dumitru:2011vk}. The BK equation is an extension
of the linear BFKL evolution equation for small $x$ \cite{Fadin:1975cb,Balitsky:1978ic,Lipatov:1985uk} as it takes into account parton recombination effects. These effects are included through the additional nonlinear term in parton density. As a result the solution to this equation cannot exceed unity, which is the unitarity bound for the dipole (i.e. quark-antiquark pair)-target scattering amplitude.
Despite the fact that the BK equation is closed and a relatively simple non-linear equation, no exact analytical solution yet exists. Nevertheless, there have been numerous analytical studies \cite{Levin:1999mw,Levin:2000mv,Mueller:2002zm,Munier:2003sj} as well as extensive numerical analyses \cite{Braun:2001kh,Lublinsky:2001yi,Armesto:2001fa,Lublinsky:2001bc,GolecBiernat:2001if,Rummukainen:2003ns,Albacete:2007yr}, and the properties of the solution are currently very well known. The solution has also been used to successfully describe the experimental data on the structure function $F_2$ \cite{Lublinsky:2001yi,Albacete:2009fh,Albacete:2010sy}. Furthermore, it has been used in the prediction of a large number of processes in hadron and heavy ion collisions, such as multiplicites and single inclusive spectra \cite{Albacete:2007sm,Albacete:2010bs,Albacete:2010fs}.
In most of these analyses one utilizes an assumption of the impact parameter independence of the solution. To be precise, the dipole target scattering amplitude in this approximation depends only on the dipole size and not on the position in impact parameter space. This leads to a significant simplification of the problem and drastically reduces the CPU time needed to numerically solve the equation. On the other hand, the impact parameter is an important ingredient for many of the phenomenological predictions. For example, the total multiplicities in heavy ion collisions depend strongly on the centrality and hence knowledge of the impact parameter distribution of the partons is essential. Usually this problem is circumvented by assuming an average parton density distribution for each of the centralities, (for a more refined approach which includes nucleon configurations in the nucleus in the initial condition see \cite{ALbacete:2010ad}). In the context of the saturation physics it is expected that the saturation scale $Q_s$, which characterizes the dense system, is impact parameter dependent (i.e. $Q_s(b)$). The saturation scale will thus have a larger value close to the dense center of the interaction region and smaller value at the periphery of the interaction region where the partons form a dilute system. The knowledge of the parton density distribution in impact parameter is essential not only in the context of heavy ion collisions but also for hadronic collisions (for example in the problem of the underlying event) and in particular for diffractive processes. For example, in exclusive diffractive production of vector mesons in DIS the momentum transfer dependence crucially depends on the impact parameter profile of the dipole scattering amplitude \cite{Munier:2001nr,Kowalski:2003hm,Kowalski:2006hc}
The impact parameter dependence has been taken into account in Monte Carlo simulations based on dipole evolution and scattering \cite{Salam:1995zd,Salam:1995uy,Avsar:2005iz,Avsar:2006gw,Avsar:2006jy,Avsar:2007xh} as well as in numerical solutions to the BK evolution equation,
see \cite{GolecBiernat:2003ym,Berger:2010sh} and \cite{Gotsman:2004ra}. It has been also discussed in the context of the conformal properties of the equation \cite{Gubser:2011qva}.
The results so far indicate that there are important modifications to the solution when the impact parameter is taken into consideration. In particular, it has been found that the parton (or more precisely dipole) density distribution in impact parameter possesses long range Coulomb-like power tails \cite{Salam:1995uy,Salam:1995zd,GolecBiernat:2003ym,Berger:2010sh}, which are the direct consequence of the form of the perturbative branching kernel. These tails need to be regularized by the appropriate cuts on the large dipole sizes which mimic confinement effects \cite{Gotsman:2004ra,Avsar:2005iz,Avsar:2006jy,Avsar:2006gw,Avsar:2007xh}. The other important observation was that the dipole scattering amplitude decreases with increasing dipole size at large dipole sizes (when it is evaluated at fixed value of impact parameter). This has to be contrasted with the impact parameter independent solutions for which the amplitude always saturates to unity for arbitrarily large values of the dipole size.
In the earlier work \cite{Berger:2010sh} we explored the dynamics of the BK equation with impact parameter by taking into account subleading effects such as kinematical cuts and the running of the strong coupling.
The goal of this paper is to use the solutions to the BK equation with impact parameter dependence to compute the cross sections and structure functions for the deep inelastic process.
In order to perform this analysis we introduce the initial condition with physical scales and also modify the branching kernel to account for non-perturbative confinement effects.
This is done by introducing a mass parameter $m$ into the kernel which modifies the long distance behavior of the dipole-target scattering amplitude. This parameter restricts the splitting of dipoles into daughter dipoles which are larger than $r_{\rm max}=\frac{1}{m}$ in order to account for confinement. With such a setup we then compute
$F_2$ and $F_L$ structure functions using the resulting solutions and compare them with experimental data from HERA \cite{Collaboration:2010ry,:2009wt}.
The resulting dipole scattering amplitude was then compared with the parametrizations
available in the literature \cite{Kowalski:2006hc} which include impact parameter.
In particular, we find that although the
dynamically generated amplitude from the BK equation is similar for small values of the dipole size to the Glauber-Mueller like parametrization, for larger dipole sizes one observes notable differences. To be precise, we find that the BK equation generates solutions which possess specific correlations between the dipole size and impact parameter, an effect which is totally absent in the Glauber-Mueller type parametrizations.
The outline of the paper is the following: in Sec.~\ref{sec:CSdipole} we state the basic formulae for the inclusive cross section within the framework of the dipole model. In Sec.~\ref{sec:Evol} we briefly discuss the most substantial features of the solution with the impact parameter dependence and discuss the differences with respect to the impact parameter independent scenario. We then introduce modifications to the evolution kernel by including the mass which mimics confinement. We discuss the properties of the solutions which result from these modifications. Both fixed coupling and running coupling cases are considered as well as the various methods that we used to implement the mass parameter which regulates the large dipole sizes. Later, in Sec.~\ref{sec:results} we show the results for the dipole amplitude and we make the first comparison to the data for $F_2$ and $F_L$. Finally, in Sec.\ref{sec:conclusion} we state the conclusions.
\section{DIS inclusive cross section within the dipole model}
\label{sec:CSdipole}
The dipole model \cite{Nikolaev:1990ja,Nikolaev:1991et} is a very useful tool in evaluating many processes at small values of $x$. One of the advantages of this approach is the possibility of including multiple parton scattering effects. It has been originally formulated for the description of deep inelastic lepton-proton (or nucleus) scattering at small $x$. In this picture, utilizing the leading logarithmic approximation in $x$, the incoming electron emits a virtual photon which fluctuates into a quark-antiquark pair, a dipole. The color dipole then subsequently interacts with the parton constituents of the nucleon, as is illustrated in Fig.~\ref{fig:dipolemodel1}. The interaction of the dipole pair with the target is given by the scattering amplitude $N$. The $q\bar{q}$ pair is characterized by a dipole size which is defined as a separation distance of the color charges $\xb_{01} = \xb_0 - \xb_1$ (where $\xb_0$ and $\xb_1$ are the positions of the $q$ and $\bar{q}$ in transverse space).\footnote{ In this paper we shall denote vector quantities in bold, otherwise they should be read as magnitudes of the associated vector. Also, alternatively we will be also using here the notation for the dipole size to be $r=x_{01}$ and impact parameter $b=\frac{|\xb_0 + \xb_1|}{2}$. } The transverse momentum of the quarks in the dipole is of the order of $\sim \frac{1}{x_{01}}$ where large dipoles correspond to the infra-red region and need to be regulated, as will be discussed in detail in later sections. The interaction of the dipole with the target is described by the scattering amplitude $N(\rb,\bb;Y)$ which contains all the information about the dynamics of the strong interaction. In the following analysis the full dependence of the scattering amplitude on the impact parameter $\bb$ will be taken into account. The evolution of the amplitude in rapidity $Y$ can be represented as emissions of daughter dipoles from the original parent dipole. When the original dipole ${01}$ splits into two dipoles $02$ and $12$ a new coordinate appears, $\xb_2$. These two daughter dipoles are produced with sizes $\xb_{12}$ and $\xb_{02}$ at impact parameters $\bb_{12}$ and $\bb_{02}$ as illustrated in Fig.~\ref{fig:dipolemodel2}.
\begin{figure}
\centering
\subfigure[]{\label{fig:dipolemodel1}\includegraphics[angle=0,width=0.4\textwidth]{DipolePicture.eps}}
\subfigure[]{\label{fig:dipolemodel2}\includegraphics[angle=0,width=0.4\textwidth]{DipoleCascade.eps}}
\caption{ Plot (a): schematic representation of the dipole picture in DIS. The incoming virtual photon $\gamma^*$ splits into a color dipole (quark-antiquark pair) of size $r$ which subsequently interacts with a target, where $N$ is the dipole-target scattering amplitude. Plot (b): depiction in transverse space of single branching of the parent dipole ${01}$ into two daughter dipoles. The sizes of dipoles are denoted by $x_{ij}$. The impact parameter variables $b_{ij}$ of all the dipoles are shown relative to the target.}
\label{fig:dipolemodel}
\end{figure}
The dipole-target amplitude $N(\rb,\bb;Y)$ at high values of rapidity $Y$ (or small $x$) is found from the solution to the BK evolution equation which can be represented in the following form:
\begin{equation}
\frac{\partial N_{\xb_0\xb_1}}{\partial Y} = \int\frac{d^2\xb_2}{2\pi}\, {\cal K}(x_{01},x_{12},x_{02}; \alpha_s,m)\left[N_{\xb_0\xb_2}+N_{\xb_2\xb_1}-N_{\xb_0\xb_1}-N_{\xb_0\xb_2}N_{\xb_2\xb_1}\right] \; .
\label{eq:BK}
\end{equation}
In the above equation we used the shorthand notation for the arguments of the amplitude $N_{\xb_i\xb_j}\equiv N({\rb_{ij}=\xb_i-\xb_j},\bb_{ij}=\frac{1}{2}(\xb_i+\xb_j);Y)$
which depends on the two transverse positions $\xb_i$ and $\xb_j$ and on the rapidity $Y$. The branching kernel ${\cal K}(x_{01},x_{12},x_{02}; \alpha_s,m)$ depends on the dipole sizes involved and contains all information about the splitting of the dipoles. In addition it depends on the running coupling $\alpha_s$. We have also indicated that it depends on the infra-red cutoff $m$ imposed on large dipoles.
The solution to Eq.~(\ref{eq:BK}) is the dipole-target scattering amplitude for arbitrarily small $x$. In order to compute the
structure functions $F_2$ and $F_L$ for the proton we use the following standard formulae in the dipole picture in the transverse coordinate representation
\begin{equation}
F_2(Q^2,x) = \frac{Q^2}{4 \pi^2 \alpha_{em}}\int{d^2 {\bf r} \int_0^1 dz \left(|\Psi_T(r,z,Q^2)|^2+|\Psi_L(r,z,Q^2)|^2\right) \sigma_{\rm dip}({\bf r},x)} \; ,
\label{eq:F2}
\end{equation}
and
\begin{equation}
F_L(Q^2,x) = \frac{Q^2}{4 \pi^2 \alpha_{em}}\int{d^2 {\bf r} \int_0^1 dz |\Psi_L(r,z,Q^2)|^2 \sigma_{\rm dip} (\rb,x)} \; .
\label{eq:FL}
\end{equation}
Here, $\sigma_{dip}$ is the (dimensionful) dipole cross section obtained from the (dimensionless) scattering amplitude by integrating over the impact parameter
\begin{equation}
\sigma_{\rm dip}(\rb,x) = 2 \int{d^2\bb \, N(\rb,\bb;Y)} \; , \; \; \; \; \; \;Y=\ln 1/x \; .
\label{eq:sigmadip}
\end{equation}
The $\Psi(\rb,Q^2,Y)_{T/L}$ functions are the photon wave functions. They describe the dissociation of a photon into a $q$$\bar{q}$ pair and can be calculated from perturbation theory. The photon wave function has the following form for the case of transverse photon polarization
\begin{equation}
|\Psi_T(r,z,Q^2)|^2 = \frac{3 \alpha_{em}}{2 \pi^2} \sum_f e_f^2\left(\left[z^2 + (1-z)^2\right]\bar{Q}^2_f K_1^2\left(\bar{Q}_f r\right) + m_f^2 K_0^2\left(\bar{Q}_f r\right)\right) \; ,
\label{eq:PhotonT}
\end{equation}
and for longitudinal polarization
\begin{equation}
|\Psi_L(r,z,Q^2)|^2 = \frac{3 \alpha_{em}}{2 \pi^2} \sum_f e_f^2\left(4Q^2z^2(1-z)^2K_0^2\left(\bar{Q}_f r\right)\right) \; .
\label{eq:PhotonL}
\end{equation}
In the above equations $\bar{Q}^2_f = z(1-z)Q^2 + m_f^2$, where $-Q^2$ is the photon virtuality and $z,(1-z)$ are the fractions of the longitudinal momentum of the photon carried by the quarks. In addition $K_{0,1}$ are modified Bessel functions of the second kind. The summations are over the active quark flavors $f$ of charge $e_f$ and mass $m_f$.
\section{Dipole evolution with impact parameter dependence}
\label{sec:Evol}
The solution to the BK equation with impact parameter dependence is found numerically, the description of the procedure was outlined in \cite{GolecBiernat:2003ym,Berger:2010sh}.
The technical complication when trying to solve BK with impact parameter is the increased number of arguments in the dipole amplitude. In the $b$-independent scenario the amplitude depends only on two variables: rapidity and dipole size.
In the $b$-dependent case there
are 5 variables: rapidity, dipole size (vector, 2-dim.) and impact parameter (vector, also 2-dim.). Alternatively, one can choose the coordinate variables to be parametrized by the dipole size, impact parameter and two angles: one parametrizing the absolute orientation of the dipole-target system in the coordinate space and the second describing the relative orientation of the dipole with respect to the target.
With the assumption of the global rotational symmetry (i.e. independence of the first angle) the number of variables reduces to 4: rapidity, dipole size, impact parameter and the angle between $\rb$ and $\bb$. Such large number of variables requires working with a very large multidimensional grid and it leads to a significant increase of computation time per each step of evolution in rapidity.
The BK equation was solved numerically by discretizing the scattering amplitude in terms of variables $(\log_{10}r,\log_{10}b,\cos \theta)$, where $\theta$ is the angle between impact parameter $\bb$ and dipole size $\rb$. The amplitude $N(r,b,\cos\theta)$ was placed on a grid with dimensions $200_r\times200_b\times20_\theta$. More details can be found in Refs.~\cite{GolecBiernat:2003ym,Berger:2010sh}.
Let us briefly summarize the most important results of \cite{GolecBiernat:2003ym,Berger:2010sh} which pertain to the properties of the solution with the impact parameter and the differences with respect to the impact parameter independent approximation. We note that for the solutions presented in Fig.~\ref{fig:OldData} we used the same initial condition at $Y=0$ as in Ref.~\cite{Berger:2010sh}, which was taken to be
\begin{equation}
N^0=1-\exp(-c_1 r^2 \exp(-c_2 b^2)) \; ,
\label{eq:n0simp}
\end{equation}
with $c_1=10, \, c_2=0.5$.
The most distinctive feature of the solution with impact parameter dependence is that the amplitude for large dipole sizes goes to zero. This is clearly illustrated in Fig.~\ref{fig:OldData1} where the solution for fixed value of impact parameter $b$ is shown as a function of the dipole size $r$. This property of the amplitude has to be contrasted with the impact parameter independent solution, in which case the amplitude is always equal to unity for sufficiently large dipole sizes. The fact that the $b$-dependent amplitude drops for large dipole sizes has a rather simple physical interpretation: the interaction region possesses finite extension in impact parameter space. The size of this region is set at low rapidity by the initial conditions (in our case it is parameter $c_2$ in Eq.~\ref{eq:n0simp}) and is later increased in the course of the evolution by the diffusion of the dipoles in transverse coordinate space. The probability of the scattering for dipoles which have sizes larger than the extension of the interaction region is very small and therefore the amplitude will tend to go to zero for such configurations.
The dipole-target amplitude therefore is largest for the scattering of dipoles with sizes comparable with the typical size of the target.
As a result, the amplitude decreases either for small or for very large dipole sizes, which in each case are very different than the extension of the target, yielding a maximum contribution for some intermediate sizes of dipoles. The configurations for which the amplitude is small are schematically illustrated in Fig.~\ref{fig:DipoleTarget}, and they correspond to the tails of the distribution depicted in Fig.~\ref{fig:OldData1}. In the course of the evolution in rapidity this distribution in the dipole size broadens due to diffusion. As a consequence, the amplitude in the impact parameter dependent scenario has two fronts as is evident from Fig.~\ref{fig:OldData1}. The first front (for small dipoles) is similar to the front in the impact parameter independent case, and one can define the saturation scale which divides the dense (i.e. where $N\sim 1$ ) and dilute ( $N \ll 1$ ) regime for small dipoles. This saturation scale can be parametrized in the form $Q_{sL}^2(Y,b)=Q_{0,sL}^{2} \exp( \lambda_{sL} Y)$ with $\lambda_{sL}\simeq 4.4$ which is consistent with the analytical predictions and with the solutions for the impact parameter independent case.
The second front expands towards larger dipoles with increasing rapidity.
One can also define the corresponding 'saturation scale' for large dipoles which can be similarly parametrized as $Q_{sR}^2(Y,b)= Q_{0,sR}^{2}\exp(- \lambda_{sR} Y)$ with $\lambda_{sR}\simeq5.8$. Note the '-' sign in the exponent, which originates from the fact that the second front expands towards larger dipoles with increasing rapidity, and therefore this second 'saturation scale' decreases with rapidity. One has to stress however that the region of large dipoles is going to be heavily modified by the non-perturbative effects (see the discussion later in this section) and therefore this second saturation scale is most likely only an artefact of the perturbative expansion in the leading logarithmic (LL) approximation.
\begin{figure}
\centering
\subfigure[ \hspace*{0.1cm} Small impact parameter: $b=1$. ]{\label{fig:OldData1}\includegraphics[angle=270,width=0.49\textwidth]{UcorRdepa2.ps}}
\subfigure[ \hspace*{0.1cm} Large impact parameter: $b=100$.]{\label{fig:OldData2}\includegraphics[angle=270,width=0.49\textwidth]{RdepUcor2b.ps}}
\caption{The dipole scattering amplitude $N(\rb,\bb;Y)$ as a function of the dipole size $r$ from the solution to the LL BK equation with impact parameter dependence. The strong coupling is fixed $\bar{\alpha}_s=0.2$. The consecutive curves shown in plots are for rapidities $Y=10,20,30,40,50$ on the plot (a) and in rapidity intervals of $5$, until $Y=50$ on plot (b). The dotted-dashed line in the left plot is the initial condition at $Y=0$ given by Eq.~\ref{eq:n0simp}. The initial condition is not visible on the right plot as it is very close to zero. The orientation of the dipole with respect to the target is such that ${\rb} \perp {\bb}$.}
\label{fig:OldData}
\end{figure}
This novel feature of the solution in impact parameter dependent case, namely the decrease of the amplitude at large dipole sizes is directly related to the profile in the impact parameter space. It was found in the LL case \cite{GolecBiernat:2003ym,Berger:2010sh} that the solution has a power-like tail in impact parameter. This is related to the fact that there are no mass scales in the perturbative evolution and hence the interaction is long range \cite{Kovner:2001bh,Kovner:2002yt}. As we will discuss later in this section, the branching kernel in the equation needs to be modified by including the effective gluon mass in order to regulate the power-like behavior of the amplitude for large dipole sizes and include the effects of confinement.
Another distinctive feature of the solutions with impact parameter dependence is the presence of the strong correlations between the dipole size and the impact parameter.
It has been observed that the amplitude is largest for specific configurations and orientations of the dipole size ${\rb}$ and impact parameter ${\bb}$ vectors. In particular, the amplitude has a peak
when the dipole size is equal twice the impact parameter.
This is clearly illustrated in
Fig.~\ref{fig:OldData2} where the peak in the amplitude at $r=2b$ emerges.
In this case there is also a non-negligible dependence on the angle between the dipole size vector and the impact parameter.
It turns out that the amplitude is largest when the angle between the dipole size vector and the impact parameter vector is equal to $0$ or $\pi$, that is when the vectors ${\rb}$ and ${\bb}$ are parallel or anti-parallel. In this configuration one of the color charges scatters off the center of the target. This enhancement can also be seen analytically from the conformal eigenfunction representation as discussed in \cite{Berger:2010sh}.
By integrating the resulting amplitude $N(\rb,\bb;Y)$ over the impact parameter $\bb$, as in Eq.~\ref{eq:sigmadip}, one obtains the dipole cross section as a function of the dipole size and rapidity. Even though the amplitude $N$ never exceeds unity, the resulting dipole cross section increases very strongly with the rapidity \cite{Kovner:2001bh,Kovner:2002yt}, due to the rapid diffusion of dipoles in the impact parameter space. In the leading logarithmic approximation this increase is exponential in rapidity, $\sigma_{\rm dip}\sim \exp(\lambda_B Y) $, with $\lambda_B\simeq 2.6$ ~\cite{GolecBiernat:2003ym,Berger:2010sh}. This behavior is very different from the features observed in the $b$-independent solution.
In the latter case one assumes that the dipole amplitude does not depend on the position in coordinate space, but only on the absolute value of the dipole size and rapidity, $N(r;Y)$. The solution still saturates to unity which is the fixed point of the equation in this approximation as well. To obtain the dipole cross section one needs then to multiply the amplitude by a dimensionful coefficient i.e.
$$
\sigma_{\rm dip} \; = \; {\cal S}_0 \; N(r;Y) \; ,
$$
where ${\cal S}_0$ can be interpreted as the integral
$$
{\cal S}_0=\int_{\cal R} d^2 \bb
$$
over the interaction region ${\cal R}$ in the impact parameter and is entirely introduced by hand. The behavior of ${\cal S}_0$ on rapidity (whether it is constant or increasing) is thus not determined by the $b$-independent BK evolution equation.
We see therefore, that the $b$-dependent solution to the BK equation, when integrated over the impact parameter does not reduce to the solution in the previously studied approximation when the impact parameter is neglected. This is an important qualitative and quantitative difference between these two solutions. In fact, the solution with impact parameter dependence is more physically motivated as it gives the increase of the dipole cross section with rapidity. It will also naturally lead to the increase with the energy of the diffractive slope for the vector meson production. We note however, that the results in the LL approximation, even with the running coupling, are not compatible with the experimental data. This is due to the fact that the diffusion in impact parameter is very strong (since it is driven by the purely perturbative physics) and results in a fast increase of the interaction radius which is not supported by the data. In order to tame this growth further corrections are necessary, in particular the inclusion of subleading corrections and a mass parameter into the evolution as will be discussed later in this section.
\begin{figure}
\centering
\subfigure[ \hspace*{0.1cm}Small dipole scattering.]{\label{fig:DipoleTarget1}\includegraphics[width=0.17\textwidth]{smalldiptar.eps}}
\hspace*{4cm}
\subfigure[ \hspace*{0.1cm}Large dipole scattering.]{\label{fig:DipoleTarget2}\includegraphics[width=0.24\textwidth]{largediptar.eps}}
\caption{Two different dipole-target configurations for which the dipole scattering amplitude is small (away from unitarity limit). }
\label{fig:DipoleTarget}
\end{figure}
\subsection{Initial condition for the evolution towards small $x$}
\label{sec:initial}
For the rest of the numerical simulations presented in this paper we will use the initial conditions of the similar form as in (\ref{eq:n0simp}) but with parameters which were adjusted to obtain the predictions consistent with experimental data. We will use as our initial condition the parametrization from Ref.~\cite{Kowalski:2003hm}, where the parametrization in the Glauber-Mueller form was used
\begin{equation}
N_{\rm GM}(r,b;Y=\ln 1/x) \, = \, 1 - \exp{\left(-\frac{\pi^2}{2N_c}r ^2 x g(x,\eta^2) T(b)\right)} \; ,
\label{eq:glaubermueller}
\end{equation}
with
\begin{equation}
T(b) \, = \, \frac{1}{8 \pi} e^{\frac{-b^2}{2B_G}} \; .
\label{eq:profile}
\end{equation}
The formula (\ref{eq:glaubermueller}) is used as an initial condition for the BK evolution for dipoles with sizes smaller than the cutoff $1/m$. For dipoles larger than the cutoff the initial condition is set to zero ( see the discussion in the next subsection).
In formula (\ref{eq:glaubermueller}) the function $xg(x,\eta^2)$ is the integrated gluon density function and $T(b)$ is the density profile of the target in transverse space with $B_G=4 \; {\rm GeV^{-2}}$. This parameter was set to fit the $t$ slope of the diffractive $J/\Psi$ production in \cite{Kowalski:2006hc}. Also, the scale $\eta$ was set according to \cite{Kowalski:2006hc} to be equal to $\eta^2=\frac{C}{r^2}+\eta_0^2$ with parameters $C=4$ and $\eta_0^2=1.16 \; {\rm GeV}^2$. The integrated gluon density in (\ref{eq:glaubermueller}) was also taken from fits performed \cite{Kowalski:2006hc}. We use (\ref{eq:glaubermueller}) as the initial condition at $Y_0=\ln 1/x_0$, $x_0=10^{-2}$ and evolve the amplitude with the BK equation to obtain the solution at lower values of $x<x_0$.
We also note that the initial condition (\ref{eq:glaubermueller}) depends only on the absolute values of the dipole size and impact parameter. The nontrivial dependence on the angle between vectors $\rb$ and $\bb$ is not present in the initial condition, instead being dynamically generated when the initial condition is evolved with the BK equation.
\subsection{Including the effective gluon mass into the evolution kernel}
Currently it is unknown how to introduce a massive cutoff on a fundamental level into the small $x$ evolution as it is an entirely non-perturbative problem. We have tested various prescriptions and found that there is a rather large sensitivity of the resulting solutions to the details of confinement implementation. In addition, there is a strong dependence of the solutions on the way the running coupling is regularized.
This sensitivity stems from the behavior of the $b$-dependent solution at large dipole sizes, as discussed previously. One important difference to note between the solution with and without the impact parameter dependence is that in the latter case the running of the strong coupling is naturally regularized by the saturation scale, provided the latter is in the semi-hard regime. For example, it was observed in Ref.~\cite{GolecBiernat:2001if} that different prescriptions of the regularizations for the running coupling gave similar results in the case of the non-linear evolution. The emergence of the semi-hard saturation scale $Q_s$ and its role as an infrared cutoff is one of the most prominent and useful features of the nonlinear evolution. In the case when the impact parameter is taken into account, the saturation scale strongly varies with $b$. In the dense region this scale is large, and is providing a natural cutoff for the running coupling in the same way as in the impact parameter independent solutions. There is however always a peripheral interaction region in impact parameter where the scattering amplitude is small and the system is dilute. Consequently the coupling is not regularized in this region by the saturation scale which is very small for large $b$. As a result, the solution in the dilute peripheral regime is governed by the linear evolution and becomes extremely sensitive to the region of large dipole sizes.
The specific value of the massive cutoff which is implemented into the evolution kernel should correspond to the non-perturbative scale which is related to confinement. Lattice simulations \cite{Dudal:2010tf,Oliveira:2010xc} suggest the presence of an effective gluon mass which would regulate the large-dipole size regime. In the case of BK equation an important ingredient that one has to take into account is the fact that the evolution in impact parameter is strongly correlated with the evolution of the dipole sizes.
Therefore the cutoff on the latter will crucially influence the size of the interaction region in impact parameter and its variation with the collision energy.
In the Monte Carlo analysis of dipole evolution in \cite{Flensburg:2008ag,Avsar:2006jy} the parameter $r_{\max}$ was set to be around $3 \; {\rm GeV^{-1}}$ (to be precise it was set to $2.9 \; {\rm GeV^{-1}}$ in \cite{Flensburg:2008ag} and $3.1 \; {\rm GeV^{-1}}$ in \cite{Avsar:2006jy}). We set the value of the cutoff to be of the same order, i.e.$m=\frac{1}{r_{\rm max}}=0.35 \; {\rm GeV}$ which corresponds to $r_{\rm max}\simeq 2.86 \; {\rm GeV^{-1}}$.
Let us finally note here
that the impact parameter profile can be accessed through the measurement of the diffractive production of the vector mesons \cite{Kowalski:2006hc,Kowalski:2003hm}.
From the experimental data \cite{Aktas:2005xu} at $|t|<1.2 \; {\rm GeV}^2$ it is known that the diffractive slope $B_D$ of the $J/\Psi$ production ($\frac{d\sigma}{dt}\sim e^{B_D t}$) is of the order of $\sim 4.57 \; {\rm GeV}^{-2}$ for $Q^2 \lesssim 1 \; {\rm GeV}^2$ and $\sim 3.5 \; {\rm GeV}^{-2}$ for $Q^2>5 \; {\rm GeV}^2$ in the energy range $40 < W_{\gamma p}<160 \; {\rm GeV}$. It is also slowly growing with the increasing energy $W$ of the $\gamma^* p$ system.
The value of $r_{\rm max}$ we set in the calculation will strongly influence the variation of the width of the impact parameter profile with the energy. Therefore in principle $r_{\rm max}$ can be related to the dependence of the $B_D$ with the energy, \cite{BerStaDiff}.
\subsubsection{ Regularization of the kernel at leading logarithmic accuracy with fixed coupling}
\label{sec:LOKernel}
The regularization of the large dipole sizes in the dipole kernel is a purely non-perturbative effect. One possible implementation, which is physically motivated, is to introduce the effective gluon mass $m$ into the propagators. This mass will set a correlation length $r_{\rm max}\equiv1/m$ which will limit the propagation of the strong color force. One can then derive the dipole kernel by computing the emission of the gluon from the initial $q\bar{q}$ dipole \cite{Nikolaev:1993th,Nikolaev:1993ke}. The Fourier transform of the momentum space expression of the $q\bar{q}g$ lightcone density into the coordinate space results in the expression which contains the modified Bessel functions instead of powers as in the LL expression. This means that instead of the long range Coulomb-type interaction present in the perturbative evolution there is now screened Yukawa force with finite range given by $r_{\rm max}$. The modified branching kernel for dipoles with effective gluon mass $m$ has the following form \cite{Nikolaev:1993th,Nikolaev:1993ke}
\begin{equation}
{\cal K}_{LL}^{(1)} \, =\, \frac{{\alpha}_s N_c}{\pi} \,m^2 \left[ K_1^2(m x_{02})+K_1^2(m x_{12}) - 2 K_1(m x_{02}) K_1(m x_{12})\frac{\xb_{02} \cdot \xb_{12}}{x_{02} \, x_{12}}\right] \;.
\label{eq:KernLOBessMass}
\end{equation}
In the limit when the dipole sizes are small compared to the cutoff, $x_{ij} \ll \frac{1}{m}$, the Bessel functions are approximated as $K_1(m x_{ij})\simeq \frac{1}{mx_{ij}}$. In this limit the kernel (\ref{eq:KernLOBessMass}) obviously reduces to the well known expression in the LL approximation \cite{Mueller:1993rr,Nikolaev:1993th,Nikolaev:1993ke}. On the other hand, the production of large dipoles $x_{ij} \gtrsim \frac{1}{m}$ is exponentially suppressed.
This form of the modification of the kernel (\ref{eq:KernLOBessMass}) was also used in the later version of the Monte Carlo simulation for the dipole splitting and evolution \cite{Flensburg:2008ag}.
By inspecting expression (\ref{eq:KernLOBessMass}) it is clear that this kernel does not vanish completely when one of the dipole sizes is larger than $1/m$ but the other is smaller than $1/m$. In other words even for very large parent dipole size the above kernel permits the splitting, provided one of the daughter dipoles sizes is below the cutoff. It means that despite the presence of the cutoff $1/m$ there is still a diffusion into the region of arbitrarily large dipole sizes.
As an alternative to the above scenario we have thus used a second prescription where the splitting is suppressed whenever any of the daughter dipole sizes is larger than the cutoff. To be precise, we have tried the second ansatz of the form
\begin{equation}
{\cal K}_{LL}^{(2)} \,=\, \frac{{\alpha}_s N_c}{\pi} \frac{x_{01}^2}{x_{02}^2x_{12}^2} \; \Theta(\frac{1}{m^2} - x_{02}^2)\Theta(\frac{1}{m^2} - x_{12}^2) \; .
\label{eq:KernLOTheta}
\end{equation}
The kernel is thus set to zero whenever any of the dipoles is larger than $1/m$.
This form gives more suppression of the dipole splitting than the first modification (\ref{eq:KernLOBessMass}) and results in an overall slower evolution in rapidity.
\subsubsection{Regularization in the presence of the running coupling }
\label{sec:rc}
The implementations of the cutoff presented above can be used with a fixed coupling. The running coupling correction brings in an additional complication as the form of the kernel changes rather significantly and the coupling is no longer a simple multiplicative factor. The running coupling corrections to the BK evolution have been computed in two independent calculations \cite{Balitsky:2006wa} and \cite{Kovchegov:2006vj}. The two schemes differ by the non-trivial subtraction term as was demonstrated in detail in \cite{Albacete:2007yr} that we will not consider in this work.
For the purpose of the current work we use the scheme of \cite{Balitsky:2006wa}
\begin{equation}
{\cal K}_{rcLL}^{\rm Bal} = \frac{\alpha_s(x_{01}^2) N_c}{\pi} \left[\frac{1}{x_{02}^2}\left(\frac{\alpha_s(x_{02}^2)}{\alpha_s(x_{12}^2)} - 1\right) + \frac{1}{x_{12}^2}\left(\frac{\alpha_s(x_{12}^2)}{\alpha_s(x_{02}^2)} - 1\right) + \frac{x_{01}^2}{x^2_{12} x_{02}^2}\right]
\label{eq:KernLOBal} \;.
\end{equation}
The other scheme derived \cite{Kovchegov:2006vj} tends to be more time consuming per one evaluation which in the case of the impact parameter dependent simulations with large grids greatly extends the time necessary for the evolution. Therefore for purely practical reasons we utilized only Balitsky prescription (\ref{eq:KernLOBal}).
Kernel (\ref{eq:KernLOBal}) reduces to the LL kernel with the strong coupling evaluated at the smallest value of the dipole size, for configurations of dipole sizes which are strongly ordered. For example, in the case when $x_{01} \sim x_{02} \gg x_{12}$ we have
\begin{equation}
{\cal K}_{rcLL}^{\rm Bal} \simeq
\frac{ \alpha_s(x_{01}^2) N_c}{\pi} \left[ \frac{1}{x_{12}^2}\left(\frac{\alpha_s(x_{12}^2)}{\alpha_s(x_{02}^2)} - 1\right) + \frac{1}{x^2_{12} }\right]\simeq \frac{N_c \alpha_s(x_{12}^2)}{\pi} \frac{1}{x^2_{12} } \; ,
\end{equation}
and for $x_{01} \ll x_{02}\sim x_{12}$ we
can take $\frac{\alpha_s(x_{02}^2)}{\alpha_s(x_{12}^2)} \sim 1$ and obtain
$$
{\cal K}_{rcLL}^{\rm Bal} \simeq \frac{\alpha_s(x_{01}^2) N_c}{\pi}\frac{x_{01}^2}{x^2_{12} x_{02}^2} \; .
$$
Thus for the reference we have also used an alternative prescription which is the minimal dipole scenario defined as
\begin{equation}
{\cal K}_{rcLL}^{\rm min} \simeq\alpha_s(\min(x_{01}^2,x_{12}^2,x_{02}^2))\frac{N_c}{\pi}\frac{x_{01}^2}{x^2_{12} x_{02}^2} \;.
\label{eq:dipmin}
\end{equation}
As we shall see later, even though formally kernel (\ref{eq:KernLOBal}) reduces to (\ref{eq:dipmin}), at least in the cases considered above, there are notable numerical differences between the two prescriptions (\ref{eq:KernLOBal},\ref{eq:dipmin}). In particular, we found that the evolution with kernel (\ref{eq:KernLOBal}) is significantly slower than with (\ref{eq:dipmin}) which is consistent with previous numerical results \cite{Albacete:2007yr} obtained without impact parameter.
In the case of the kernel with running coupling (\ref{eq:KernLOBal}) the implementation of the cutoff in the form analogous to
${\cal K}_{LL}^{(1)}$ is not entirely trivial due to the rather complicated form of (\ref{eq:KernLOBal}). We have thus used the simplest
scenario, imposing the cuts on the daughter dipoles as in (\ref{eq:KernLOTheta}). To be precise, for the case of the running coupling with scenario \cite{Balitsky:2006wa} we used
\begin{eqnarray}
{\cal K}_{rcLL,m}^{\rm Bal} = \frac{ \alpha_s(x_{01}^2) N_c}{\pi}\left[\frac{1}{x_{02}^2}\left(\frac{\alpha_s(x_{02}^2)}{\alpha_s(x_{12}^2)} - 1\right) + \frac{1}{x_{12}^2}\left(\frac{\alpha_s(x_{12}^2)}{\alpha_s(x_{02}^2)} - 1\right) + \frac{x_{01}^2}{x^2_{12} x_{02}^2}\right]\Theta(\frac{1}{m^2} - x_{02}^2)\Theta(\frac{1}{m^2} - x_{12}^2) \; .
\label{eq:KernLOBalTheta}
\end{eqnarray}
For comparison,
we have also used the minimal dipole prescription for the running coupling (\ref{eq:dipmin}) where all of the scenarios with masses (\ref{eq:KernLOBessMass},\ref{eq:KernLOTheta}) were extended in this case. This is possible as the minimal dipole prescription gives the multiplicative factor in the LL kernel, see (\ref{eq:dipmin}).
In this paper we use the expression of the QCD running coupling with a mass parameter $\mu$ to regulate the strong coupling at large dipoles, which is of the following form\footnote{Note that, this is the expression for the running coupling in coordinate space. In the literature one finds various forms of the running of the coupling in coordinate space which include different normalizations for the argument, i.e. $\ln \frac{C}{\Lambda^2 x^2}$ with differing values of the constant $C$. We are using here the convention from Ref.~\cite{Balitsky:2008zza} where $C=1$.}
\begin{equation}
\alpha_s(x^2) = \frac{1}{b \, \ln\left[\Lambda_{\rm QCD}^{-2}\left(\frac{1}{x^2} + \mu^2\right)\right]} \; ,
\label{eq:coupling}
\end{equation}
Here $b = \frac{33 - 2n_f}{12\pi}$, $n_f$ is the number of active flavors. The $\mu$ parameter effectively freezes the coupling at large dipole sizes at $\alpha_{s,{\rm fr }} = \frac{1}{b \ln\left[\Lambda^{-2}\mu^2\right]}$. In our simulations we used $\mu=0.52 \; {\rm GeV}$ as an infra-red regulator for the strong coupling, and $\Lambda_{\rm QCD}=0.25 \; {\rm GeV}$.
\begin{figure}
\centering
\subfigure[\hspace*{0.1cm}Kernel given by Eq.~(\ref{eq:KernLOBessMass}) with running coupling $\alpha_s(\min(x_{01}^2,x_{02}^2,x_{12}^2))$.]{\label{fig:mar05-2}\includegraphics[angle=270,width=0.49\textwidth]{mar05CUT-0smallb.ps}}
\subfigure[\hspace*{0.1cm}Kernel given by Eq.~(\ref{eq:KernLOTheta}) with running coupling $\alpha_s(\min(x_{01}^2,x_{02}^2,x_{12}^2))$.]{\label{fig:mar05-0}\includegraphics[angle=270,width=0.49\textwidth]{mar05CUT-2smallb.ps}}
\caption{Dipole amplitude as a function of the dipole size from the evolution to the BK equation with mass $m=0.35 \; {\rm GeV}$. The dotted-dashed line is the initial condition (\ref{eq:glaubermueller}) at $Y=0$ and each solid line represents a progression in two units of rapidity until $Y=8$.
The solutions are evaluated at fixed impact parameter $b=0.0001 \; {\rm GeV}^{-1}$ and for
the orientation of the dipole defined as ${\bf r} \perp {\bf b}$. }
\label{fig:mar05}
\end{figure}
\section{Results}
\label{sec:results}
\subsection{Properties of the dipole amplitude and comparison with the Glauber-Mueller model}
As we saw explicitly in Sec.~\ref{sec:Evol}, the solutions to the BK equation with impact parameter dependence possess very interesting and novel properties as compared to the case without the impact parameter. In this section we investigate in detail the features of the solutions when the mass $m$ is included in the kernel. We study the BK solutions using two different prescriptions for the infrared regulators which were introduced in the previous section, see Eqs.~(\ref{eq:KernLOBessMass}) and (\ref{eq:KernLOTheta}). We also investigate in detail the differences between the two running coupling scenarios, i.e. (\ref{eq:dipmin})
and (\ref{eq:KernLOBalTheta}). It turns out that the differences between the simulations have rather significant impact on the phenomenology.
First, we used kernels (\ref{eq:KernLOBessMass},\ref{eq:KernLOTheta}), where the running coupling has been implemented using the minimal dipole size as the scale (see Eq.~\ref{eq:dipmin}). This was done in order to consistently trace the differences between the two implementations of the massive cutoff.
In Fig.~\ref{fig:mar05-2} we present the simulations using kernel (\ref{eq:KernLOBessMass}).
We observe that despite the fact that the effective gluon mass $m$ is incorporated into the branching kernel, the scattering amplitude is non-vanishing at arbitrary large dipole sizes. For any rapidity there is still a significant diffusion into the large dipole size region. This feature is simple to understand by inspecting the form of (\ref{eq:KernLOBessMass}). One observes that,
for any value of $x_{01}$ there are configurations where one daughter dipole
is very large and above the cutoff but the second daughter dipole
can be still below the cutoff $\frac{1}{m}$. These configurations lead to the non-vanishing contributions of the kernel even at large values of $x_{01}$. Because of this effect the evolution with modified kernel (\ref{eq:KernLOBessMass}) still proceeds into the large dipole regime as indicated by the results in Fig.~\ref{fig:mar05-2}.
In Fig.~\ref{fig:mar05-0} we show the solution using the kernel with the theta functions imposed as in (\ref{eq:KernLOTheta}).
We see from this figure that the prescription (\ref{eq:KernLOTheta}) leads to the solution which completely vanishes for large dipoles in stark contrast with the simulation shown in Fig.~\ref{fig:mar05-2}.
The interesting feature about this scenario is that even though the initial condition (shown by dotted-dashed line) was cut at $x_{01} = \frac{1}{m}$, the evolution does not respect this cut and moves it to a larger value equal to $x_{01} = \frac{2}{m}$. This happens because the kernel (\ref{eq:KernLOTheta}) has cuts only on daughter dipoles and not on the parent dipole. It means that
the amplitude for sizes of dipoles which are larger than $\frac{1}{m}$ is non-vanishing.
Until a point at which the parent dipole is twice the cutoff size there exist configurations where neither emitted daughter dipoles are above the cutoff size. These symmetric states exist until $x_{01} = \frac{2}{m}$ at which point all of the configurations are cut, since at least one daughter dipole is larger than the cutoff.
From this part of the analysis we conclude that
when the massive cutoff is imposed in the form of (\ref{eq:KernLOBessMass}), the evolution still proceeds into the region of large dipole sizes, which is dominated by the large value of the strong coupling constant. This has rather important consequences and we found that this scenario is actually disfavoured by the data.
As a next step, we have performed the comparison of the two solutions using the minimal dipole prescription (\ref{eq:dipmin}) and the Balitsky prescription (\ref{eq:KernLOBalTheta}) for the running coupling. The results of this comparison are shown in Fig.~\ref{fig:MinBalComp}. In both of these cases we have implemented the mass as in (\ref{eq:KernLOTheta}), i.e. setting the kernel to zero whenever any of the daughter dipoles are larger than the cutoff. It is evident that the kernel with the running coupling given by (\ref{eq:KernLOBalTheta})
leads to a much slower evolution than the kernel with the running coupling as in (\ref{eq:dipmin}). For example at rapidity $Y=4$, the evolution front in the minimal dipole scenario is almost at the same position (for small dipoles) as for the scenario (\ref{eq:KernLOBalTheta}) at rapidity $Y=8$. As we will see later, this will translate into large differences for the observable structure functions depending on the prescription used. It has a large effect, in particular on the $x$ slope of $F_2$. This feature has been found also in earlier calculation which did not include the impact parameter dependence and masses, see Ref.~\cite{Albacete:2007yr}.
\begin{figure}
\includegraphics[angle=270,width=0.6\textwidth]{MinBalComp.ps}
\caption{Dipole amplitude as a function of the dipole size for the fixed value of impact parameter. Solid line corresponds to the simulation with the running coupling using the minimum dipole prescription Eq.~(\ref{eq:dipmin}). Dashed line corresponds to the running coupling using Eq.~(\ref{eq:KernLOBalTheta}). In both cases the mass $m=0.35 \, {\rm GeV}$ is included in the kernel, in the form of two theta functions, like in Eq.~(\ref{eq:KernLOTheta}). Two sets of curves correspond to rapidities $Y=4$ and $Y=8$.
The dashed - dotted curve is the initial condition at $Y=0$ for both calculations.}
\label{fig:MinBalComp}
\end{figure}
We have also compared
the solutions to the BK equation with the Glauber-Mueller model, Eq.~\ref{eq:glaubermueller}. In the latter model, the parameters were obtained from a fit to the HERA data ~\cite{Kowalski:2006hc}.
The initial condition for the solution to BK was also taken to be of the form of Eq.~(\ref{eq:glaubermueller})
at $Y=0$ which we choose to correspond to $x_0=0.01$ in this calculation. For consistency the initial condition for the BK equation is set to zero for dipoles which exceed the cutoff.
The comparison between the solution to BK and Glauber-Mueller model is presented in Fig.~\ref{fig:BalKow1}.
We see that the solution to the BK equation agrees quite well with the parametrization (\ref{eq:glaubermueller}) for small values of dipole sizes. On the other hand there are sizeable differences in the larger dipole regime, where one sees that (\ref{eq:glaubermueller}) extends indefinitely whereas the BK solution is cut off by the massive regulator. This has a non-negligible impact on the phenomenology of $F_2$ at low $Q^2$
as will be illustrated in the next subsection.
The difference in the solutions was examined, when the initial condition is not cut at $\frac{1}{m}$ but at $\frac{2}{m}$ and still the kernel (\ref{eq:KernLOBalTheta}) is used with the cutoff $\frac{1}{m}$ in the evolution. The result is shown in Fig.~\ref{fig:BalKow2}. In this case it is evident that the cutoff is not moved due to the reasons described earlier in this section. However, there is a peculiar structure of the solution, where a second peak of the amplitude emerges for dipoles somewhat smaller than the cutoff. The solution in Fig.~\ref{fig:BalKow2} is nevertheless very close to the solution shown in Fig.~\ref{fig:BalKow1} for small values of dipole sizes, i.e. smaller than $\frac{1}{m}$.
We also note that for the values of the cutoff $m$ used here, which are motivated by the profile in impact parameter, both the initial condition and the BK solutions do not completely saturate, although the amplitude is close to $1$ for large rapidities.
\begin{figure}
\centering
\subfigure[\hspace*{0.1cm} { Initial condition regulated at $\frac{1}{m}$.}]{\label{fig:BalKow1}\includegraphics[angle=270,width=0.44\textwidth]{mar07-0.ps}}
\subfigure[\hspace*{0.1cm} { Initial condition regulated at $\frac{2}{m}$.}]{\label{fig:BalKow2}\includegraphics[angle=270,width=0.49\textwidth]{BalKowalCompare.ps}}
\caption{ Dipole scattering amplitude as a function of the dipole size for the fixed value of impact parameter $b$. Solid line corresponds to the model (\ref{eq:glaubermueller}) with parameters from \cite{Kowalski:2006hc}.
The dashed line is the solution to the BK equation with the kernel (\ref{eq:KernLOBalTheta}). The dashed-dotted line denotes the model (\ref{eq:glaubermueller}) at $x_0=0.01$. The cutoff at $r_{\rm max}=\frac{1}{m}$ (plot (a)) and $r_{\rm max}=\frac{2}{m}$ (plot (b)) is also indicated. Impact parameter is fixed to $b= 1 \; {\rm GeV}^{-1}$.}
\label{fig:BalKow}
\end{figure}
It should be also noted that the solutions to the BK equation with the kernel (\ref{eq:KernLOBal}) possesses an interesting dependence on the regularization procedure of the running coupling. This is related to the fact that this kernel is a complicated, non-linear function of $\alpha_s$. In particular, the factors of the coupling in the denominators in expression(\ref{eq:KernLOBal}) lead to a non-trivial dependence on the regularization parameter of the strong coupling.
It was found that by increasing the value of $\mu$, see Eq.(\ref{eq:coupling}), and thus decreasing the maximal value at which the coupling freezes, there was a region of dipole sizes where the solution obtained from evolution with the kernel (\ref{eq:KernLOBal}) was actually increased, contrary to what could be naively expected. We stress that this behavior was observed for some range of dipole sizes only. With the minimal dipole size prescription the amplitude was of course always decreasing with increasing value of $\mu$ as expected.
In general, it was found that the solution with the running coupling in the form (\ref{eq:KernLOBal}) possessed a larger sensitivity to the way the coupling is regulated than the evolution with the minimal dipole prescription (\ref{eq:dipmin}).
This sensitivity persists even
at small dipole sizes which are far away from the scale $\frac{1}{\mu}$.
It suggests that the terms with inverse coupling in kernel (\ref{eq:KernLOBal}) increase the sensitivity to scales which are different than the scale set by the parent dipole $x_{01}$. It is also important to note that this behavior was found both in the evolution with and without impact parameter dependence. The solution using the minimum dipole size prescription does not exhibit such a large sensitivity. It would be interesting to investigate these features further in order to determine whether this is a physical behavior or it is an artefact of the truncation of the resummation of perturbative series which lead to this result \cite{Balitsky:2006wa}.
Finally, the dependence of the amplitude on impact parameter for fixed value of the dipole size was analyzed.
The diffusion property of the solution in impact parameter space is illustrated in Fig.~\ref{fig:bdep} for the running coupling case in scenario (\ref{eq:KernLOBalTheta}).
Plots in Fig.\ref{fig:bdeplog} and Fig.~\ref{fig:bdeplin} differ only by the choice of horizontal scale.
The solution to the BK equation is compared with the profile in impact parameter using the model (\ref{eq:glaubermueller}) with the parameters from \cite{Kowalski:2006hc}. We observe that for the small values of $b$ in general the BK solution is fairly close numerically to model (\ref{eq:glaubermueller}). There is however a significant difference in the shape of the amplitudes between the BK calculation and the Glauber-Mueller model (\ref{eq:glaubermueller}). This is especially manifest at large values of impact parameter where the BK solution has a more extended tail in $b$.
In the BK solution there is a clear increase of the width of the distribution in impact parameter with increasing rapidity.
\begin{figure}
\centering
\subfigure[\hspace*{0.2cm} Dipole size $r=1.0 \; {\rm GeV}^{-1}$. Logarithmic horizontal axis.]{\label{fig:bdeplog}\includegraphics[angle=270,width=0.49\textwidth]{Blog1.ps}}
\subfigure[\hspace*{0.2cm} Dipole size $r=1.0 \; {\rm GeV}^{-1}$. Linear horizontal axis.]{\label{fig:bdeplin}\includegraphics[angle=270,width=0.49\textwidth]{Blinear1.ps}}
\caption{Dipole scattering amplitude as a function of the impact parameter for fixed dipole size and dipole orientation $\theta=\pi/2$. The solid lines represent the model (\ref{eq:glaubermueller}) used in \cite{Kowalski:2006hc}. The dashed lines correspond to the solution of the BK equation with the kernel (\ref{eq:KernLOBalTheta}), $m = 0.35\;{\rm GeV}$. The dashed - dotted line represents the initial conditions at $Y=0 \, (x_0=0.01)$ also taken from model in \cite{Kowalski:2006hc}.}
\label{fig:bdep}
\end{figure}
The diffusion property in impact parameter is best illustrated in Fig.~\ref{fig:baverage}
where we show the average width squared, defined as
\begin{equation}
\langle b^2 \rangle \; = \; \frac{\int d^2 \bb \, b^2 \, N(\rb,\bb;Y)}{\int d^2 \bb \, N(\rb,\bb;Y)} \; ,
\label{eq:bvariance}
\end{equation}
as a function of rapidity for fixed value of the dipole size $r$.
\begin{figure}
\centering
\includegraphics[angle=270,width=0.6\textwidth]{Bave.ps}
\caption{The value of the average squared width $\langle b^2 \rangle$, defined in Eq.~(\ref{eq:bvariance}), as a function of rapidity for fixed value of the dipole size $r$. The solid line is the model (\ref{eq:glaubermueller}) with parameters taken from \cite{Kowalski:2006hc} and the dashed line is obtained from solution to the BK equation with the kernel (\ref{eq:KernLOBalTheta}).}
\label{fig:baverage}
\end{figure}
We compared the value of $\langle b^2 \rangle$ extracted from the solution to the BK equation with the value obtained from model (\ref{eq:glaubermueller}). The model (\ref{eq:glaubermueller})
gives almost constant width, independent of rapidity, which is to be expected. On the contrary, in the case of the BK equation the width clearly increases with rapidity. For the rapidities considered here, we observe that it is almost a linear growth, with slightly faster increase at the highest values of rapidity $\sim 6-8$ along with mild dependence of the slope on the value of the dipole size.
\subsection{Description of the $F_2$ and $F_L$ structure function data}
In our calculation of the $F_2$ and $F_L$ structure functions we used the solution to the BK equation with impact parameter dependence and a gluon mass $m$ as implemented in scenario (\ref{eq:KernLOBalTheta}). The initial condition for the calculation was set as in Sec.~\ref{sec:initial}. In addition, we imposed a cutoff on the dipole sizes to be equal to $x_{01} = \frac{2}{m}$ in the initial condition. As discussed earlier, this is necessary in order to be consistent with the cutoff placed in the evolution kernel.
The data on the structure function correspond to the combined H1 and ZEUS data \cite{:2009wt}, and only the points below $x=0.01$ are used here.
In this calculation there are the following parameters: $C$, $\eta_0$, $B_G$ in the initial condition (\ref{eq:glaubermueller}), mass $m$ in the kernel, strong coupling regulator $\mu$ and $\Lambda_{QCD}$. In principle all of these parameters can be varied to obtain the fit to the data. In practice however, due to very long time needed to find solution for a given set of parameters (of the order of a day on 32 cores), variation of all parameters is prohibitive. We have instead chosen to fix all of the parameters with the exception of $\mu$ and $m$ which we varied in a very limited range.
Since we use an initial condition which is cut at large dipole sizes, the data at values of $x$ around $0.01$ are underestimated by our model. This is because the initial model (\ref{eq:glaubermueller}) from \cite{Kowalski:2006hc} was fitted to the data without any cuts.
This indicates a rather large sensitivity of the $F_2$ obtained from (\ref{eq:F2}) to the region of large dipole sizes even for moderate values of $Q^2$. This effect is well known and corresponds to the presence of the aligned jet configurations in the transverse structure function \cite{Golec-Biernat:1998js,Ewerz:2011ph}. This can be seen by inspecting Eqs.~(\ref{eq:F2},\ref{eq:PhotonT}) as this expression receives large contributions from the endpoints $z\sim 0,1$. As a result, at a given value of $Q^2$ the dipoles which contribute to $F_2$ form a rather wide distribution in dipole size. We have verified that for the model given by (\ref{eq:glaubermueller}) with a cut of the order of $\frac{1}{m}$ and $\frac{2}{m}$, with $m=0.35 \; {\rm GeV}$ the contribution to $F_2$ at $Q^2= 4.5 \; {\rm GeV}^2$ from dipoles larger than the cut is about $30\%$ and $10\%$ respectively.
Hence, the $F_2$ structure function contains significant non-perturbative contributions even at the moderate values of $Q^2$. In order to compensate for this non-perturbative off-set one could of course move the cutoff $m$ towards smaller masses. However, since the cut on the dipole sizes is strongly correlated with the profile in impact parameter it would result in the much larger width of the impact parameter profile, which would be inconsistent with the data on diffractive $J/\Psi$ production. Therefore, we choose to work with the value of the cut which is more consistent with the number obtained from the $J/\Psi$ diffractive slope.
As a result, in order to compensate for the offset, a separate non-perturbative contribution is added which is important at low values of $Q^2$, of the order of $< 15 \; {\rm GeV}^2$.
We stress that this property stems from the fact that in the BK evolution equation the impact parameter and dipole size are strongly correlated. This has to be contrasted with the Glauber-Mueller like parametrization (\ref{eq:glaubermueller}) where the dipole sizes and impact parameter are decoupled.
The non-perturbative part originating from large dipole sizes was parametrized in the following form
\begin{equation}
F_{2}^{\rm soft} = \frac{Q^2}{2 \pi \alpha_{em}} \sigma_0 \int_{\frac{1}{m}} r \,dr\int_0^1 dz \left(|\Psi_L|^2 + |\Psi_T|^2\right) \; .
\label{eq:F2soft}
\end{equation}
The total structure function is then taken to be of the form
\begin{equation}
F_{2}^{\rm tot} = F_2^{\rm BK} + F_2^{\rm soft} \; ,
\label{eq:F2tots}
\end{equation}
where $F_2^{\rm BK}$ denotes the contribution obtained by using the solution to the BK equation (which is the perturbative part).
In the formula (\ref{eq:F2soft}) we assumed that the dominant part of the integral is where the dipole - proton amplitude is almost flat and therefore replaced the dipole cross section with the constant $\sigma_0$. Note that the integral over dipole size in (\ref{eq:F2soft}) is now cut from below by $1/m$, and the integral extends into the large dipole regime.
$\sigma_0$ is a constant that is used to fit the data at $x=0.01$ and lowest bin in $Q^2$.
The $F_2^{\rm soft}$ contribution is slowly varying with $Q^2$ and it accounts well for the non-perturbative dipoles at low $Q^2$. This procedure of adding separate contributions from small (perturbative) and large (non-perturbative) dipoles is similar to the one employed in Refs.~\cite{Kwiecinski:1987tb,Martin:1998dr}.
One could argue that the non-perturbative contribution could be accounted for by including the contribution from the vector meson dominance model \cite{Brodsky:1969iz,Gribov:1968gs,Ritson:1970yu,Sakurai:1972wk}. The vector meson dominance (VMD) contribution can be written as
\begin{equation}
F_{2}^{(\rm VMD)} = \frac{Q^2}{4\pi}\Sigma_{v=\rho,\omega,\phi} \left(\frac{m_v^4 \sigma_v(W^2)}{\gamma^2_v (Q^2 + m_v^2)^2}+\frac{Q^2 m_v^2 \sigma_v(W^2)}{\gamma_v^2 (Q^2 + m_v^2)^2}\xi_0\left(\frac{m_v^2}{Q^2 + m_v^2}\right) ^2\right) \; ,
\label{eq:F2vmd}
\end{equation}
see for example \cite{Kwiecinski:1987tb}.
Here $m_v$ denotes the vector meson mass and $\sigma_v$ is the vector meson-proton cross section which is a function of the energy. These cross sections can be taken
to be
\begin{eqnarray}
\sigma_\rho &=& \sigma_\omega = \frac{1}{2} \left(\sigma(\pi^+p)+\sigma(\pi^-p)\right) \; ,
\\\sigma_\phi &=& \sigma(K^+p) + \sigma(K^-p) -\frac{1}{2} \left(\sigma(\pi^+p)+\sigma(\pi^-p)\right) \; ,
\end{eqnarray}
where the parameterization for the energy dependence of $\sigma(\pi^{\pm}p),\sigma(K^{\pm}p)$ was taken using the soft pomeron model from \cite{Donnachie:1992ny}. The $\gamma_v$ terms relate to the leptonic width of the vector meson \cite{Kwiecinski:1987tb} in question and are defined by $\gamma_v^2 = \frac{\pi \alpha_{em}^2 m_v}{3 \Gamma_{v\rightarrow e^+e^-}}$ where the leptonic decay width is taken from \cite{Nakamura:2010zzi} and $\xi_0$ is a constant taken to be $0.7$ \cite{Kwiecinski:1987tb}.
The computations for $F_2$ using the non-perturbative contribution of either (\ref{eq:F2soft}) or (\ref{eq:F2vmd}) are presented in Figs.~\ref{fig:mar21-0F2NT1},\ref{fig:mar21-0F2NT2}.
As we see the curves with the VMD term undershoot the data whereas the curves with the soft term are systematically closer. The VMD contribution is almost negligible with the exception of the lowest $Q^2$ bin. We see that the two calculations, with soft term and VMD are close for values of $Q^2>15 \, {\rm GeV}^2$ which means in this region the data are only described by the perturbative component. We also conclude that the VMD model is not sufficient as the only soft contribution for the description of the data in the present setup. VMD only contributes to the very low values of $Q^2$ (less than $4 \; {\rm GeV}^2$) and an additional soft component is needed which has a flatter $Q^2$ dependence. This result is consistent with the results of the previous analyses \cite{Martin:1998dr}.
We observe that the slope of the calculations is too steep in $x$ in all bins of $Q^2$ which implies that the LL evolution with the running coupling leads to a faster slope in $\ln 1/x$ than the data indicate. This can be remedied by lowering the scale $\Lambda_{QCD}$ from the value which we used, i.e. $ 0.25\, {\rm GeV}$, and taking it as a fitting parameter.
This was effectively done in the fits presented in \cite{Albacete:2009fh,Albacete:2010sy}. We estimated that it would require $\Lambda_{QCD}$ to be well below the pion mass, of the order of tens of $\rm MeV$ or so to fit the data.
The fact that the LL evolution with running coupling in our scenario has a steeper slope than the data is not unexpected as one needs to take into account the next-to-leading corrections to BK equation. These have been computed in \cite{Balitsky:2008zza}, but a detailed analysis of the BK equation which includes them still needs to be performed. Preliminary analysis in the momentum space was recently performed \cite{Avsar:2011ds} using the method of the saturation boundary\cite{Mueller:2002zm}. The results from this analysis indicate that next-to-leading corrections to the BK equation, which are not due to the running coupling, are indeed substantial and can lead to the instabilities of the evolution despite the presence of saturation \cite{Beuf:2010aw}. This strongly suggests that a resummation of subleading corrections in $\ln 1/x$ is needed in addition to the saturation corrections \cite{Motyka:2009gi,Beuf:2011}.
Finally, we also compared the calculation to the experimental data on the structure function $F_L$ \cite{Collaboration:2010ry}, this is illustrated in Fig.~\ref{fig:mar07-0FL}.
We see that the calculation is consistent with the experimental data in all bins of $Q^2$. However, the data on $F_L$ have very large errors. Note that, in the figures presented, the range in $x$ in each of the bins is very small, and therefore the $F_L$ structure function is very flat in each of these bins.
\section{Conclusions}
\label{sec:conclusion}
In this paper we have analyzed the nonlinear BK equation with impact parameter dependence and running coupling in the presence of a mass scale, which regulates dipole splitting in the infrared regime. This effective gluon mass is responsible for the non-perturbative effect of confinement. Using the resulting solution for the dipole scattering amplitude we have performed a comparison with experimental DIS data on the structure functions $F_2$ and $F_L$. Let us summarize the main points of this investigation :
\begin{enumerate}
\item The details of the evolution in rapidity strongly depend on the way the large dipoles are regularized. In particular, the speed of the evolution with rapidity is affected by the choice of regularization. Two different scenarios have been tested:
modified Bessel functions (\ref{eq:KernLOBessMass}), and
a more stringent cutoff with theta functions (\ref{eq:KernLOTheta}).
The first scenario possesses
a physical motivation and can be derived from the computation of the branching kernel for dipoles in the presence of the effective gluon mass in the propagators.
\item The scenario (\ref{eq:KernLOBessMass})
does not entirely tame the evolution into the large dipole size region and in the presence of the running coupling it results in a rather fast evolution in rapidity.
The resulting amplitude is also much larger than in scenario (\ref{eq:KernLOTheta}), not only in the large dipole regime but also in the small dipole region as well. We found that the scenario with the cutoff on all the large dipoles in the form (\ref{eq:KernLOTheta}) results in solutions which are preferred by the experimental data.
\item The running coupling prescription (\ref{eq:KernLOBal}) gives, in general, much slower evolution than the prescription (\ref{eq:dipmin}) with the minimal dipole as the scale of the running coupling. This happens despite the fact that the two kernels are formally equivalent in the limits when the dipole sizes are strongly ordered. We also found that the evolution with kernel (\ref{eq:KernLOBal}) is more dependent on the details of the regularization of the running coupling. Most likely this is caused by the highly nonlinear form of (\ref{eq:KernLOBal}), i.e. by the fact that it contains the inverse powers of the strong coupling. We found that the scenario which gives results closest to the data is the kernel with Balitsky prescription for the running coupling and the massive regulator taken in the form (\ref{eq:KernLOBalTheta}).
\item Comparison with the data on structure function $F_2$ shows that the slope in $x$ of the calculation is too steep for the data in the case of the LL evolution with running coupling. This could be cured by changing the value of $\Lambda_{QCD}$, but it would require unrealistically small values of this scale in order to match the data. We stress that the full fit which would include the variation of all parameters, in the case of the BK equation with impact parameter, would be extremely demanding as far as computing resources are concerned (using similar techniques as presented here). The fact that the scale in the running coupling has to be adjusted quite a bit to fit the data is consistent with previous calculations (which were done without impact parameter dependence) and calls for the inclusion of the remaining corrections beyond the leading - logarithmic order.
\item An important feature of the solution to the BK equation is the fact that the impact parameter and dipole size are strongly correlated. This property stems from the basic form of the dipole evolution kernel. This has to be contrasted with the models previously used in the literature, for example of the form (\ref{eq:glaubermueller}). This correlation introduces novel effects and leads to more constraints on the calculations. For example, incorporating the effective gluon mass in the kernel introduces a scale in impact parameter. Strong variation of this scale is not possible if one requires consistency with the observed slope in diffractive data. This scale then results in the truncation of the large dipole size region, which in turn causes the offset in the $F_2$ calculation, particularly at small values of $Q^2$. Therefore one needs to include an additional (soft) component to $F_2$ which is entirely non-perturbative. The BK solution also exhibits the diffusion in impact parameter, a feature that is completely absent in the models of the form (\ref{eq:glaubermueller}). We found that in the region of large impact parameters and high rapidities the differences between the BK solution and the model (\ref{eq:glaubermueller}) were substantial.
\end{enumerate}
\section*{Acknowledgments}
We would like to thank Emil Avsar, Dionysis Triantafyllopoulos and Yuri Kovchegov for many useful discussions. We also thank Henri Kowalski for discussions as well as his assistance by allowing us usage of parts of his fortran code for the evaluation of the initial conditions. This work was supported by the MNiSW grant No. N202 249235 and the DOE OJI grant No. DE - SC0002145. A.M.S. is supported by the Sloan Foundation.
\bibliographystyle{h-physrev4}
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{"url":"http:\/\/www.lastfm.com.br\/user\/benpearce0309\/library\/music\/Blondes\/_\/You+Mean+So+Much+To+Me?setlang=pt","text":"# Biblioteca\n\nM\u00fasica \u00bb Blondes \u00bb\n\n## You Mean So Much To Me\n\n10 execu\u00e7\u00f5es | Ir para p\u00e1gina da faixa\n\nFaixas (10)\nFaixa \u00c1lbum Dura\u00e7\u00e3o Data\nYou Mean So Much To Me 9:02 Out 3 2012, 17h20\nYou Mean So Much To Me 9:02 Set 6 2012, 9h04\nYou Mean So Much To Me 9:02 Ago 21 2012, 17h31\nYou Mean So Much To Me 9:02 Ago 14 2012, 12h55\nYou Mean So Much To Me 9:02 Jul 12 2012, 14h19\nYou Mean So Much To Me 9:02 Mai 9 2012, 12h11\nYou Mean So Much To Me 9:02 Mar 30 2012, 7h56\nYou Mean So Much To Me 9:02 Mar 16 2012, 15h39\nYou Mean So Much To Me 9:02 Fev 14 2012, 18h09\nYou Mean So Much To Me 9:02 Fev 13 2012, 10h38","date":"2015-05-24 14:34:02","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.913463294506073, \"perplexity\": 8617.688367410514}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-22\/segments\/1432207928019.31\/warc\/CC-MAIN-20150521113208-00337-ip-10-180-206-219.ec2.internal.warc.gz\"}"} | null | null |
Переволок — деревня в Осьминском сельском поселении Лужского района Ленинградской области.
История
Деревня Переволок упоминается на карте Санкт-Петербургской губернии 1792 года, А. М. Вильбрехта.
Деревня Переволок обозначена на карте Санкт-Петербургской губернии Ф. Ф. Шуберта 1834 года.
ПЕРЕВОЛОКА — деревня принадлежит её величеству, число жителей по ревизии: 38 м. п., 42 ж. п. (1838 год)
Как деревня 00000 из 65 дворов она отмечена на карте профессора С. С. Куторги 1852 года.
ПЕРЕВОЛОКА — деревня господина Ханыкова, по просёлочной дороге, число дворов — 15, число душ — 41 м. п. (1856 год)
ПЕРЕВОЛОКА — деревня удельная при реке Сабе, число дворов — 17, число жителей: 58 м. п., 69 ж. п.(1862 год)
В XIX — начале XX века деревня административно относилась к Осьминской волости 2-го земского участка 1-го стана Гдовского уезда Санкт-Петербургской губернии.
По данным «Памятной книжки Санкт-Петербургской губернии» за 1905 год деревня называлась Переволока и входила в Переволокское сельское общество.
С 1917 по 1919 год деревня Переволока входила в состав Осьминской волости Гдовского уезда.
С 1920 года, в составе Переволокского сельсовета Осьминской волости Кингисеппского уезда.
С 1924 года, в составе Лужницкого сельсовета.
С 1925 года, в составе Николаевского сельсовета.
Согласно топографической карте 1926 года деревня насчитывала 31 крестьянский двор. В деревне находилось братское кладбище, а к северу от неё — водяная мельница.
С 1927 года, в составе Осьминского района.
В 1928 году население деревни Переволока составляло 219 человек.
По данным 1933 года деревня Переволок входила в состав Николаевского сельсовета Осьминского района.
C 1 августа 1941 года по 31 января 1944 года деревня находилась в оккупации.
С 1961 года, в составе Сланцевского района.
С 1963 года, в составе Лужского района.
В 1965 году население деревни Переволок составляло 25 человек.
По данным 1966 года деревня Переволок входила в состав Николаевского сельсовета Лужского района.
По данным 1973 и 1990 годов деревня Переволок входила в состав Рельского сельсовета.
По данным 1997 года в деревне Переволок Рельской волости проживали 5 человек, в 2002 году — 14 человек (все русские).
В 2007 году в деревне Переволок Осьминского СП проживали 2 человека.
География
Деревня расположена в северо-западной части района близ автодороги (Рель — Николаевское).
Расстояние до административного центра поселения — 23 км.
Расстояние до ближайшей железнодорожной станции Молосковицы — 82 км.
Деревня находится на правом берегу реки Саба.
Демография
Улицы
Ольховая.
Примечания
Населённые пункты Лужского района | {
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{"url":"https:\/\/www.bartleby.com\/questions-and-answers\/find-parametric-equations-for-the-lines-in-the-line-through-the-point-p3-4-1-parallel-to-the-vector-\/202d13c6-b470-42bf-9d67-d53ead19c3c5","text":"# Find parametric equations for the lines in The line through the point P(3, -4, -1) parallel to the vector i + j + k\n\nQuestion\n\nFind parametric equations for the lines in The line through the point P(3, -4, -1) parallel to the vector\ni + j + k","date":"2021-07-27 22:44:04","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9020174741744995, \"perplexity\": 218.78274028326786}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-31\/segments\/1627046153491.18\/warc\/CC-MAIN-20210727202227-20210727232227-00261.warc.gz\"}"} | null | null |
I was thinking abaout this.
whem we use ajp conector for apache cluster jboss don't have a engine that's only permit ip's known by jboss.
Would be interesting to set up a way to block access of other ips.
And for denied access, jboss returns a message like "Access denied for this address".
I think that if you're already using apache httpd, then you should allow it to manage access control.
It was not exactly what I meant.
Example, i have my production apache server running very well.
Another person start a new apache server whithout authorization, so the jboss will accept requisitions from both.
The idea is, configure the ajp connector to accpet connections only from known apache servers.
In that case, you should consider some firewall configuration (iptables on linux) to control network traffic access to your JBoss instance.
The AJP authentication was implemented on wildfly 8.1 Final. | {
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State Police investigating death at prison in Hudson
Anchors aweigh as Navy's 50,000th sailor graduates at Kesselring nuclear site
Brian Nearing
1of23Admiral Kirkland Donald, Director, Naval Nuclear Propulsion during the U.S. Naval Nuclear Propulsion Program's official ceremony commemorating the 50,000th nuclear trained sailor to graduate from the Naval Nuclear Propulsion Training Unit at the Knolls Atomic Power Laboratory in West Milton Tuesday May 29, 2012. (John Carl D'Annibale / Times Union)John Carl D'AnnibaleShow MoreShow Less
2of23Naval and civilian personel exit the U.S. Naval Nuclear Propulsion Program's official ceremony commemorating the 50,000th nuclear trained sailor to graduate from the Naval Nuclear Propulsion Training Unit at the Knolls Atomic Power Laboratory in West Milton Tuesday May 29, 2012. At right is the D1G prototype for the former guided missile cruiser BAINBRIDGE, now in the process of being dismantled. (John Carl D'Annibale / Times Union)John Carl D'AnnibaleShow MoreShow Less
4of23Admiral Kirkland H. Donald, left, Director, Naval Nuclear Propulsion presents an award commemorating the 50,000th Graduate to MM3 Jenna Swindt during the U.S. Naval Nuclear Propulsion Program's official ceremony commemorating the 50,000th nuclear trained sailor to graduate from the Naval Nuclear Propulsion Training Unit at the Knolls Atomic Power Laboratory in West Milton Tuesday May 29, 2012. (John Carl D'Annibale / Times Union)John Carl D'AnnibaleShow MoreShow Less
5of23Under Secretary for Nuclear Security& Administrator, National Nuclear Security Administration Thomas D'Agostino, left, and Congressman Paul Tonko during the U.S. Naval Nuclear Propulsion Program's official ceremony commemorating the 50,000th nuclear trained sailor to graduate from the Naval Nuclear Propulsion Training Unit at the Knolls Atomic Power Laboratory in West Milton Tuesday May 29, 2012. (John Carl D'Annibale / Times Union)John Carl D'AnnibaleShow MoreShow Less
7of23Students listen during the U.S. Naval Nuclear Propulsion Program's official ceremony commemorating the 50,000th nuclear trained sailor to graduate from the Naval Nuclear Propulsion Training Unit at the Knolls Atomic Power Laboratory in West Milton Tuesday May 29, 2012. (John Carl D'Annibale / Times Union)John Carl D'AnnibaleShow MoreShow Less
8of23Captain Brian Fort, Commanding Officer, NPTU-Ballston Spa speaks during the U.S. Naval Nuclear Propulsion Program's official ceremony commemorating the 50,000th nuclear trained sailor to graduate from the Naval Nuclear Propulsion Training Unit at the Knolls Atomic Power Laboratory in West Milton Tuesday May 29, 2012. (John Carl D'Annibale / Times Union)John Carl D'AnnibaleShow MoreShow Less
10of23Students listen during the U.S. Naval Nuclear Propulsion Program's official ceremony commemorating the 50,000th nuclear trained sailor to graduate from the Naval Nuclear Propulsion Training Unit at the Knolls Atomic Power Laboratory in West Milton Tuesday May 29, 2012. (John Carl D'Annibale / Times Union)John Carl D'AnnibaleShow MoreShow Less
11of23Instructor ET 1st class Robert Weber of Saratoga Springs stands in formation during the U.S. Naval Nuclear Propulsion Program's official ceremony commemorating the 50,000th nuclear trained sailor to graduate from the Naval Nuclear Propulsion Training Unit at the Knolls Atomic Power Laboratory in West Milton Tuesday May 29, 2012. At right is the D1G prototypefor the former guided missile cruiser BAINBRIDGE, now in the process of being dismantled. (John Carl D'Annibale / Times Union)John Carl D'AnnibaleShow MoreShow Less
13of23Ensign Bryce Taft of Corning, NY, during the U.S. Naval Nuclear Propulsion Program's official ceremony commemorating the 50,000th nuclear trained sailor to graduate from the Naval Nuclear Propulsion Training Unit at the Knolls Atomic Power Laboratory in West Milton Tuesday May 29, 2012. (John Carl D'Annibale / Times Union)John Carl D'AnnibaleShow MoreShow Less
14of23Admiral Kirkland Donald, at podium, right, Director, Naval Nuclear Propulsion speaks during the U.S. Naval Nuclear Propulsion Program's official ceremony commemorating the 50,000th nuclear trained sailor to graduate from the Naval Nuclear Propulsion Training Unit at the Knolls Atomic Power Laboratory in West Milton Tuesday May 29, 2012. (John Carl D'Annibale / Times Union)John Carl D'AnnibaleShow MoreShow Less
16of23Captain Brian Fort, Commanding Officer, NPTU-Ballston Spa following the U.S. Naval Nuclear Propulsion Program's official ceremony commemorating the 50,000th nuclear trained sailor to graduate from the Naval Nuclear Propulsion Training Unit at the Knolls Atomic Power Laboratory in West Milton Tuesday May 29, 2012. The D1G prototype for the former guided missile cruiser BAINBRIDGE ,at top, is now in the process of being dismantled. (John Carl D'Annibale / Times Union)John Carl D'AnnibaleShow MoreShow Less
17of23Buildings at the Knolls Atomic Power Laboratory in West Milton Tuesday May 29, 2012. (John Carl D'Annibale / Times Union)John Carl D'AnnibaleShow MoreShow Less
19of23The D1G "ball", a prototype for the former guided missile cruiser BAINBRIDGE, now in the process of being dismantled at the Knolls Atomic Power Laboratory in West Milton Tuesday May 29, 2012. (John Carl D'Annibale / Times Union)John Carl D'AnnibaleShow MoreShow Less
WEST MILTON — Dwight Eisenhower was president when the first U.S. Navy sailor came to the Kenneth A. Kesselring Site to learn to operate reactors to power the nation's nuclear submarines. On Friday, a former emergency technician from northern California became the site's 50,000th graduate.
Hundreds of people gathered inside a flag-bedecked tent to watch 26-year-old Jenna Swindt graduate with 245 other sailors to become the latest generation to guide the Navy's nuclear fleet.
"It came as quite a surprise, that I was the 50,000th graduate," said Swindt, a native of Santa Rosa and the first member of her family to serve in the military. She will stay at Kesselring for the next two years as an instructor.
The head of the Navy's nuclear propulsion unit gave the graduates a sense of what they had accomplished in the rigorous two-year training program to operate nuclear reactors that power submarines and aircraft carriers. Nearly half the sailors who operate the nuclear fleet are Kesselring graduates, a list that includes former U.S. President Jimmy Carter.
Grade inflation may have devalued some academic achievements in the civilian world, but that has not been allowed to happen at Kesselring.
"During the history of the program, no one here has ever had a 4.0 grade point average ... The sailors here are graded against a standard that has been the same since I came through here in 1976," said Adm. Kirkland H. Donald, director of the U.S. Naval Nuclear Propulsion Program.
Among those watching the ceremony was 24-year-old Bryce Taft, of Corning, Steuben County, who is due to graduate in September. His average day? In class by 7 a.m. for 12 hours of instruction. And on the weeks when he is "on crew" — working with instructors who operate one of Kesselring's two prototype nuclear reactors — he logs about 80 hours. "A lot of coffee makes this place run," he said.
Still to come for Taft are the oral exams, in which trainees are quizzed by instructors about reactor operations. After that, Taft, who graduated from Pennsylvania State University with a degree in nuclear engineering, expects to be posted to a nuclear submarine. "Not sure where, though," he said.
Capt. Brian Fort, commanding officer at Kesselring, said that New York has a special relationship with the Navy, given that its birthplace is in Whitehall, Washington County. During the Revolutionary War, when the village was known as Skenesborough, American troops there built vessels from the forests at the south end of Lake Champlain that sailed north to battle the British fleet coming down from Canada.
bnearing@timesunion.com • 518-454-5094 • @Bnearing10
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A ) Our annual Influenza vaccination is now available. We will run evenings flu clinics and on Saturday mornings. We prefer that the children attend our practice nurses during the day time clinics rather than in the evening clinics. Please contact our receptionist for your flu vaccination. They will keep you right and book an appointment for one of the clinics.
Who is eligible for a influenza vaccination ?
All under 65 with risk factors (respiratory, heart, liver, neurological dis (Stroke, TIA, polio syndrome sufferers, MS, Cerebral palsy, neurological disease of the muscles and learning disability), Diabetic patients and immunosuppressed patients (ie Cancer patients, HIV patients etc), patients without a spleen, all nursing and care home patients), carers.
B ) We also continue with the Shingles immunisation as mentioned in Newsletter 2. If you are 70, 78 or 79 years old on 01.09.14 you are eligible to receive the immunisation. We know that shingles can leave you with significant pain for prolonged time. The immunisation is supposed to reduce the severity and time you experience the pain caused by shingles. Phone and book in with the practice nurse to receive the immunisation.
C ) Over the next few months we will send invitations to patients with chronic disease, like Asthma, COPD, Diabetes mellitus, Mental Health problems etc.to see one of our nursing team for an annual review. This is a good chance of discussing prevention of long term problems and medication issues. Please use this appointment and come.
D ) We already started phoning patients with longer than normal appointments to remind them of the appointment time. If necessary we can reappoint the patients or use the appointments for other patients. However, unfortunately we still experience a considerable number of patients who do not attend in advance booked return / review appointment. These appointments are deemed to be necessary by the doctors to review a medical condition after a previous consultation. Not attending these appointments can put patients at risk and the appointment is wasted.
Please remember it is very important to notify us if you can not come to your appointment. We always have patients waiting for appointments who could be seen by doctors and / or nurses instead.
E ) We are still trying to install a phone queuing system to make it easier to contact us in particular when you phone for an appointment in the morning. If you phone for other reasons than an appointment please phone after 9 am to ease our phone lines until we have the queuing system. Thank you for your help. We will keep you informed on the progress. | {
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