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talk-lattice2024/talk.tex
2024-07-24 13:59:26 +02:00

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% !TeX program = lualatex
\documentclass[aspectratio=169]{beamer}
\input{einstellungen.tex}
% \usepackage[backend=bibtex]{biblatex} %lualatex, biber
\usepackage{multimedia}
\usepackage{tcolorbox}
\usepackage{multicol}
\usepackage{tikz} % I added this
\usetikzlibrary{patterns} % I added this
\usetikzlibrary{calc} % I added this
\usepackage{mathtools} % I added this
\usepackage{tcolorbox} % I added this
\usepackage{vwcol} % I added this
\usepackage{wrapfig} % I added this
\usepackage{booktabs} % Top and bottom rules for table,I added this
\usepackage{animate} % added this for gifs
\usepackage{dsfont}
\usepackage{simplewick} %added this
\usepackage[makeroom]{cancel}
\definecolor{Darkgreen}{rgb}{0,0.5,0}
\newcommand{\halflinewidth}{.48\linewidth}
\newcommand{\fulllinewidth}{.95\linewidth}
\usepackage{macros} % used commands defined for rm paper
\newcommand*{\arxivtag}[1]{{\color{pantone315}\texttt{[#1]}}}
\newcommand{\openlat}{OpenLat}
\title[$A_\mu^a$ impr. msl. \& mass. quarks]{Non-singlet axial current improvement for massless and massive sea quarks}
\author[Justus Kuhlmann]{\textbf{Justus Kuhlmann}\\ Patrick Fritzsch, Jochen Heitger}
% \institute wird von der Vorlage nicht direkt verwendet
\institute{Institut für theoretische Physik}
\date{\today}
\keywords{Münster}
\newcommand{\customcite}[1]{{\color{fu-blue}\citename{#1}{author}}, \citefield{#1}{journaltitle}, {\color{pantone315}\citeyear{#1}}}
\addbibresource{"./My Library.bib"}
\begin{document}
\begin{frame}[plain]
\maketitle
\end{frame}
% Relevance of the AV-current
% Relevance in renormalisation adn improvement of other currents
% so far only in chi lim
% not exactly given with the ensembles at hand
% also: we have ensembles close to the symmetric point
% openLAT so far at sym point
% differences at sym point?
% improvement at the symmetric point
% example
\begin{frame}
\frametitle{Relevance for further improvement and physics}
\begin{itemize}
\item needed for improv. determination of the PCAC quark-mass
\item decay constants
\item masses of mesons (e.g. $\chi_\mathrm{c1}$ or $D_\mathrm{1}^\ast$)
\pause
\vspace{.5cm}
\item improvement and renormalisation:
\begin{itemize}
\item $\cv$, $\za$, $c_{\rm T}$
\item no $\ca$ $\Rightarrow$ no improvement of other channels
\end{itemize}
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Determination of $\ca$}
\begin{itemize}
\item Schrödinger functional
\item used already \arxivtag{hep-lat/9609035} for $N_{\rm f} = 2$ \arxivtag{hep-lat/0503003} and std. Wilson-Clover $N_{\rm f} = 3$ \arxivtag{1502.04999}
\item from PCAC mass $m_{\rm PCAC} = \frac{\partial_0 f_{\rm A}}{2f_{\rm P}} + \ca \frac{\partial^2_0 f_{\rm P}}{2f_{\rm P}} = r + \ca s$
\end{itemize}
\vspace{.5cm}
$$m_{\rm PCAC}^{(0)} = m_{\rm PCAC}^{(1)}$$
$$\Leftrightarrow \ca = - \frac{r^{(1)} - r^{(0)}}{s^{(1)} - s^{(0)}}$$
\end{frame}
\begin{frame}
\frametitle{The wavefunction method}
\begin{itemize}
\item mimic pionic sources on boundaries $\pi^{(0)}, \pi^{(1)}$ and require PCAC to hold
\begin{itemize}
\item basis wavefunctions:
$\omega_{\rm b1} = e^{-r/a_0}\;,\quad\omega_{\rm b2} = r~e^{-r/a_0}\;,\quad\omega_{\rm b3} = e^{-r/(2a_0)}$
\item also include $\omega_{\rm b4} = {\rm cons.}\;,\quad\omega_{\rm b5} = -r^2~e^{-r/a_0}$
\end{itemize}
\item eigenvectors of boundary-to-boundary corr. func. $(F_1)_{i,j} = -\langle O(\omega_{{\rm b}i}) O'(\omega_{{\rm b}j})\rangle$ lead to eigenstates $\pi^{(0)}, \pi^{(1)}$
% Question: do we include all wavefunctions or just some?
% How does this interplay with the states that we achieve?
% Which is the optimal wf combination?
% \item also: where on the lattice do we define $c_{\rm A}$?
\vspace{.5cm}
\pause
\item evaluate $c_{\rm A}(x_0)$ with these source terms
\item later: choice of $x_0$ and wavefunction basis is part of the improvement condition
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Ensembles}
\framesubtitle{$L\approx 3\,{\rm fm}$ Schrödinger-Functional ensembles, exp. Wilson-Clover fermions}
\begin{center}
\begin{tabular}{cc|c|c|c|c}
\toprule
$L/a$ & $\beta$ & $\kappa_{1}\approx\kappa_{\rm cr}$ & $\kappa_{2}$ & $\kappa_{3}\approx\kappa_{\rm sym}$&$a$\\
\midrule
24&3.685&0.1396980&0.1395500&0.1394400&0.120\\
32&3.80&0.1392500&---&0.1389630&\\
40&3.90&0.1388562&0.1386148&0.1386030&\\
48&4.00&0.1384942&0.1384880&0.1382720&\\
56&4.10&0.1381410&0.1380000&0.1379450&\\
\bottomrule
\end{tabular}
\end{center}
\begin{itemize}
\item interested in 2 LCPs: chiral and at $N_{\rm f}=3$ symmetric point
\item matching sym. point of \openlat~\arxivtag{2201.03874}
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Improvement of the axial-vector current}
\framesubtitle{$\ca$ estimators}
\begin{tabular}{cc}
Close to chiral ensembles&Symmetric ensembles\\
\includegraphics[width=\halflinewidth]{plots/plateaus_chi_0.2_0.3_0124_ee_ee_total_quad.pdf}&
\includegraphics[width=\halflinewidth]{plots/plateaus_sym_0.2_0.3_0124_ee_ee_total_quad.pdf}
\end{tabular}
% systematic errors, to capture "non-plateauness"
\end{frame}
% interpolations
\begin{frame}
\frametitle{Interpolation}
\framesubtitle{Hit symmetric and critical point exactly}
\begin{itemize}
\item ensembles not exactly tuned
\item determine points of interest as in \openlat~ensembles
\item define: $$\Phi^{\rm SF}_4 = \frac{3}{2}\,8t_0\,|m_{\rm eff}|\,m_{\rm eff}
\quad \Rightarrow \quad \Phi^{\rm SF}_4\bigm\lvert_{m_{0,{\rm cr}}} = 0\,,\;\Phi^{\rm SF}_4\bigm\lvert_{m_{0,{\rm sym}}} = 1.115$$
\item interpolate to the desired points
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Improvement of the axial-vector current}
\framesubtitle{Finding the symmetric and chiral point}
\begin{tabular}{cc}
\includegraphics[width=\halflinewidth]{plots/fix_sym_b3.9.pdf}&
\includegraphics[width=\halflinewidth]{plots/fix_sym_b4.pdf}
\end{tabular}
\end{frame}
\begin{frame}
\frametitle{Improvement of the axial-vector current}
\framesubtitle{Interpolations in $c_{\rm A}$}
\begin{tabular}{cc}
\includegraphics[width=\halflinewidth]{plots/ca2sym_b3.9.pdf}&
\includegraphics[width=\halflinewidth]{plots/ca2sym_b4.pdf}
\end{tabular}
\end{frame}
\begin{frame}
\frametitle{Improvement of the axial-vector current}
\framesubtitle{Final interpolations in $g_0^2$}
%\includegraphics[width=\halflinewidth]{plots/plateaus_sym_0.175_0.325_0124.pdf}
\end{frame}
% first study for difference:
\begin{frame}
\frametitle{First scaling test of improvement}
\framesubtitle{Construction}
\begin{itemize}
\item Example: construct a non-trivial observable, for which renormalisation drops out
$$\frac{A^R_{\pm}}{A^R_{ll}} = \frac{Z_{\rm A}}{Z_{\rm A}} \frac{(1+ b_{\rm A} (m_{q+} + m_{q-})/2}{(1+ b_{\rm A} m_{ql})} \frac{(1+\bar{b}_{\rm A}Tr[M_q])}{(1+ \bar{b}_{\rm A}Tr[M_q])} \frac{A_{\pm}}{A_{ll}}$$
with $$(m_{q+} + m_{q-})/2=m_{ql}$$
\item symmetric point \openlat~ensembles
\item unimproved vs improved chi vs improves sym
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{First study of improvement}
\framesubtitle{Results}
\end{frame}
\begin{frame}
\frametitle{Outlook}
\begin{itemize}
\item starting to work on SF determination of
\begin{itemize}
\item vector current improvement ($c_{\rm V}$, $c_{\rm T}$)
\item current quark mass renormalization ($b_{\rm A}-b_{\rm P}$, $b_m$, $Z$)
\end{itemize}
\item determination of $Z_{\rm A}$, $Z_{\rm V}$, $Z_{\rm S}/Z_{\rm P}$ through $\chi$SF
\end{itemize}
\end{frame}
\end{document}