% !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. for 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}}} \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 exp. Wilson-clover fermion framework \item massive $\widehat{=}$ at $N_{\rm f}=3$ symmetric point \pause \vspace{.5cm} \item needed for improv. quark current mass \item decay constants \& matrix elements % \item masses of mesons (e.g. $\chi_\mathrm{c1}$ or $D_\mathrm{1}^\ast$) \pause \vspace{.5cm} \item improvement and renormalisation: \begin{itemize} \item $\cv$, $c_{\rm T}$, $\za$ \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 boundary conditions \item similar to quenched \arxivtag{hep-lat/9609035}, $N_{\rm f} = 2$ \arxivtag{hep-lat/0503003} and std. Wilson-Clover $N_{\rm f} = 3$ \arxivtag{1502.04999, hep-lat/0703006} \item derive from PCAC mass \end{itemize} \vspace{.5cm} $$m_{\rm PCAC} = \frac{\partial_0 f_{\rm A}}{2f_{\rm P}} + \ca~a\frac{\partial^2_0 f_{\rm P}}{2f_{\rm P}} = r + \ca~as$$ $$m_{\rm PCAC}^{(0)} = m_{\rm PCAC}^{(1)}\quad\Leftrightarrow\quad\ca = - \frac{r^{(1)} - r^{(0)}}{s^{(1)} - s^{(0)}}$$ \begin{itemize} \item states (0) and (1) are the PS ground and first excited state in our setup \item PCAC relation holds for both \end{itemize} \end{frame} \begin{frame} \frametitle{The wavefunction method} \begin{itemize} \item construct pseudoscalar states \begin{itemize} \item H-like basis wavefunctions: $\omega_{1} = e^{-r/a_0}\;,\quad\omega_{2} = r~e^{-r/a_0}\;,\quad\omega_{3} = e^{-r/(2a_0)}$ \item also include $\omega_{4} = {\rm cons.}\;,\quad\omega_{5} = -r^2~e^{-r/a_0}$ \quad with $r=|\vec{y}-\vec{x}|$ \end{itemize} \pause \item diagonalise boundary-to-boundary corr. func. $(F_1)_{i,j} = -\langle O(\omega_{{\rm b}i}) O'(\omega_{{\rm b}j})\rangle$ \vspace{.5cm} \pause \item employ eigenvectors of $(F_1)_{i,j}$ to project $f_{\rm A}(x_0)$ and $f_{\rm P}(x_0)$ onto the eigenstates % 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 projected correlation functions \item later: choice of $x_0$ and wavefunction basis is part of the improvement condition \end{itemize} \end{frame} \begin{frame} \frametitle{Ensembles} \framesubtitle{$T=L\approx 3\,{\rm fm}$ Schrödinger-Functional ensembles, exp. Wilson-Clover fermions} \begin{center} \begin{tabular}{cc|c|c|c|c} \toprule $L/a$ & $\beta$ & $\kappa_{1}\approx\kappa_{\rm cr}$ & $\kappa_{2}$ & $\kappa_{3}\approx\kappa_{\rm sym}$&$\approx a\;{\rm [fm]}$\\ \midrule 24&3.685&0.1396980&0.1395500&0.1394400&0.120\\ 32&3.80&0.1392500&---&0.1389630&0.095\\ 40&3.90&0.1388562&0.1386148&0.1386030&0.080\\ 48&4.00&0.1384942&0.1384880&0.1382720&0.064\\ 56&4.10&0.1381410&0.1380000&0.1379450&0.055\\ 96&4.37&---&---&---&0.035\\ \bottomrule \end{tabular} \end{center} \begin{itemize} \item interested in 2 LCPs: chiral and $N_{\rm f}=3$ sym. 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} Critical point ensembles&Symmetric point 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{... to the symmetric and critical point} \begin{itemize} \item ensembles not exactly tuned \item able to interpolate to the desired points due to 2 or 3 ensembles per $\beta$ \item determine points of interest as in \openlat~ensembles \arxivtag{2201.03874} \item define: $$\Phi^{\rm SF}_4 = \frac{3}{2}\,8t_0\,|m_{\rm eff}|\,m_{\rm eff} \quad \Rightarrow \quad \Phi^{\rm SF}_4\bigm\lvert_{m_{0,{\rm cr}}} = 0\,,\;\Phi^{\rm SF}_4\bigm\lvert_{m_{0,{\rm sym}}} = 1.115$$ \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{Interpolations in $g_0^2$} \begin{tabular}{cc} \includegraphics[width=\halflinewidth]{plots/g0sq_chi.pdf}& \includegraphics[width=\halflinewidth]{plots/g0sq_sym.pdf} \end{tabular} \end{frame} % first study for difference: \begin{frame} \frametitle{First scaling test of improvement} \begin{itemize} \item Example: Calculate $f_{\pi K}$ with stabilised Wilson fermions \item symmetric point \openlat~ensembles \item improve with $\ca = 0$ vs $\ca(g_0^2)|_{\rm chi}$ vs $\ca(g_0^2)|_{\rm sym}$ $$f_{\rm A}^{\rm RI} = Z_{\rm A} (1+b_{\rm A} am_{\rm q}+\bar{b}_{\rm A} a\Tr[M_{\rm q}])\frac{\sqrt{2} \mathcal{A}_{\rm A_0P}}{\sqrt{\mathcal{A}_{\rm PP} m_\pi}}$$ with $C_{XX} \propto \mathcal{A}_{XX}{\rm e}^{-mx_0}$ \item renormalisation: $Z_{\rm A}$ preliminary, $b_{\rm A}$ from pert. theory, $\bar{b}_{\rm A}$ neglected \end{itemize} \end{frame} \begin{frame} \frametitle{First scaling test of improvement} \framesubtitle{Results} \vspace{-.3cm} \centering \includegraphics[width=.6\linewidth]{plots/test_f_part_ZA_chi_imp.pdf} \end{frame} \begin{frame} \frametitle{Outlook} \begin{itemize} \item finish $\za$, $\zv$, $\bV$ and $\bar{b}_{\rm V}$ through SF \item further improvement and renormalisation currently in the works: \begin{itemize} \item vector and tensor current improvement ($c_{\rm V}$, $c_{\rm T}$) \item current quark mass renormalisation ($b_{\rm A}-b_{\rm P}$, $b_m$, $Z$) \item determination of $Z_{\rm A}$, $Z_{\rm V}$, $Z_{\rm S}/Z_{\rm P}$ through $\chi$SF \end{itemize} \end{itemize} \end{frame} \begin{frame} \frametitle{Backup} \framesubtitle{$\Delta r$ and $\Delta s$ behaviour} \begin{tabular}{cc} \includegraphics*[width=\halflinewidth]{plots/T48L48_b4.00_k0.1384942/plot_dr_ds10-0124projection-wgcorr.pdf}& \includegraphics*[width=\halflinewidth]{plots/T56L56_b4.10_k0.1381410/plot_dr_ds10-0124projection-wgcorr.pdf} \end{tabular} \end{frame} \begin{frame} \frametitle{Backup} \framesubtitle{Behaviour of $\fp$ projected} \begin{tabular}{cc} \includegraphics*[width=\halflinewidth]{plots/T48L48_b4.00_k0.1384942/plot_m_eff_P-0124projection-wgcorr.pdf}& \includegraphics*[width=\halflinewidth]{plots/T56L56_b4.10_k0.1381410/plot_m_eff_P-0124projection-wgcorr.pdf} \end{tabular} \end{frame} \end{document}