222 lines
8.1 KiB
TeX
222 lines
8.1 KiB
TeX
% !TeX program = lualatex
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\documentclass[aspectratio=169]{beamer}
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\input{einstellungen.tex}
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% \usepackage[backend=bibtex]{biblatex} %lualatex, biber
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\usepackage{multimedia}
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\usepackage{tcolorbox}
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\usepackage{multicol}
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\usepackage{tikz} % I added this
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\usetikzlibrary{patterns} % I added this
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\usetikzlibrary{calc} % I added this
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\usepackage{mathtools} % I added this
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\usepackage{tcolorbox} % I added this
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\usepackage{vwcol} % I added this
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\usepackage{wrapfig} % I added this
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\usepackage{booktabs} % Top and bottom rules for table,I added this
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\usepackage{animate} % added this for gifs
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\usepackage{dsfont}
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\usepackage{simplewick} %added this
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\usepackage[makeroom]{cancel}
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\definecolor{Darkgreen}{rgb}{0,0.5,0}
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\newcommand{\halflinewidth}{.48\linewidth}
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\newcommand{\fulllinewidth}{.95\linewidth}
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\usepackage{macros} % used commands defined for rm paper
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\newcommand*{\arxivtag}[1]{{\color{pantone315}\texttt{[#1]}}}
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\newcommand{\openlat}{OpenLat}
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\title[$A_\mu^a$ impr. for msl. \& mass. quarks]{Non-singlet axial current improvement for massless and massive sea quarks}
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\author[Justus Kuhlmann]{\textbf{Justus Kuhlmann}\\ Patrick Fritzsch, Jochen Heitger}
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% \institute wird von der Vorlage nicht direkt verwendet
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\institute{Institut für theoretische Physik}
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\date{\today}
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\keywords{Münster}
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\newcommand{\customcite}[1]{{\color{fu-blue}\citename{#1}{author}}, \citefield{#1}{journaltitle}, {\color{pantone315}\citeyear{#1}}}
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\begin{document}
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\begin{frame}[plain]
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\maketitle
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\end{frame}
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% Relevance of the AV-current
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% Relevance in renormalisation adn improvement of other currents
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% so far only in chi lim
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% not exactly given with the ensembles at hand
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% also: we have ensembles close to the symmetric point
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% openLAT so far at sym point
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% differences at sym point?
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% improvement at the symmetric point
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% example
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\begin{frame}
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\frametitle{Relevance for further improvement and physics}
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\begin{itemize}
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\item exp. Wilson-clover fermion framework
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\item massive $\widehat{=}$ at $N_{\rm f}=3$ symmetric point
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\pause
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\vspace{.5cm}
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\item needed for improv. quark current mass
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\item decay constants \& matrix elements
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% \item masses of mesons (e.g. $\chi_\mathrm{c1}$ or $D_\mathrm{1}^\ast$)
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\pause
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\vspace{.5cm}
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\item improvement and renormalisation:
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\begin{itemize}
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\item $\cv$, $c_{\rm T}$, $\za$
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\item no $\ca$ $\Rightarrow$ no improvement of other channels
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\end{itemize}
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\end{itemize}
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\end{frame}
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\begin{frame}
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\frametitle{Determination of $\ca$}
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\begin{itemize}
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\item Schrödinger functional boundary conditions
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\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}
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\item derive from PCAC mass
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\end{itemize}
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\vspace{.5cm}
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$$m_{\rm PCAC} = \frac{\partial_0 f_{\rm A}}{2f_{\rm P}} + \ca~a\frac{\partial^2_0 f_{\rm P}}{2f_{\rm P}} = r + \ca~as$$
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$$m_{\rm PCAC}^{(0)} = m_{\rm PCAC}^{(1)}\quad\Leftrightarrow\quad\ca = - \frac{r^{(1)} - r^{(0)}}{s^{(1)} - s^{(0)}}$$
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\begin{itemize}
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\item states (0) and (1) are the PS ground and first excited state in our setup
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\item PCAC relation holds for both
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\end{itemize}
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\end{frame}
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\begin{frame}
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\frametitle{The wavefunction method}
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\begin{itemize}
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\item construct pseudoscalar states
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\begin{itemize}
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\item H-like basis wavefunctions:
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$\omega_{1} = e^{-r/a_0}\;,\quad\omega_{2} = r~e^{-r/a_0}\;,\quad\omega_{3} = e^{-r/(2a_0)}$
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\item also include $\omega_{4} = {\rm cons.}\;,\quad\omega_{5} = -r^2~e^{-r/a_0}$
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\quad with $r=|\vec{y}-\vec{x}|$
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\end{itemize}
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\pause
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\item diagonalise boundary-to-boundary corr. func. $(F_1)_{i,j} = -\langle O(\omega_{{\rm b}i}) O'(\omega_{{\rm b}j})\rangle$
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\vspace{.5cm}
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\pause
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\item employ eigenvectors of $(F_1)_{i,j}$ to project $f_{\rm A}(x_0)$ and $f_{\rm P}(x_0)$ onto the eigenstates
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% Question: do we include all wavefunctions or just some?
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% How does this interplay with the states that we achieve?
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% Which is the optimal wf combination?
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% \item also: where on the lattice do we define $c_{\rm A}$?
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\vspace{.5cm}
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\pause
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\item evaluate $c_{\rm A}(x_0)$ with projected correlation functions
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\item later: choice of $x_0$ and wavefunction basis is part of the improvement condition
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\end{itemize}
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\end{frame}
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\begin{frame}
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\frametitle{Ensembles}
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\framesubtitle{$T=L\approx 3\,{\rm fm}$ Schrödinger-Functional ensembles, exp. Wilson-Clover fermions}
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\begin{center}
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\begin{tabular}{cc|c|c|c|c}
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\toprule
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$L/a$ & $\beta$ & $\kappa_{1}\approx\kappa_{\rm cr}$ & $\kappa_{2}$ & $\kappa_{3}\approx\kappa_{\rm sym}$&$\approx a\;{\rm [fm]}$\\
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\midrule
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24&3.685&0.1396980&0.1395500&0.1394400&0.120\\
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32&3.80&0.1392500&---&0.1389630&0.095\\
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40&3.90&0.1388562&0.1386148&0.1386030&0.080\\
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48&4.00&0.1384942&0.1384880&0.1382720&0.064\\
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56&4.10&0.1381410&0.1380000&0.1379450&0.055\\
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96&4.37&---&---&---&0.035\\
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\bottomrule
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\end{tabular}
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\end{center}
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\begin{itemize}
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\item interested in 2 LCPs: chiral and $N_{\rm f}=3$ sym. point
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\item matching sym. point of \openlat~\arxivtag{2201.03874}
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\end{itemize}
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\end{frame}
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\begin{frame}
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\frametitle{Improvement of the axial-vector current}
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\framesubtitle{$\ca$ estimators}
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\begin{tabular}{cc}
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Critical point ensembles&Symmetric point ensembles\\
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\includegraphics[width=\halflinewidth]{plots/plateaus_chi_0.2_0.3_0124_ee_ee_total_quad.pdf}&
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\includegraphics[width=\halflinewidth]{plots/plateaus_sym_0.2_0.3_0124_ee_ee_total_quad.pdf}
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\end{tabular}
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% systematic errors, to capture "non-plateauness"
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\end{frame}
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% interpolations
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\begin{frame}
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\frametitle{Interpolation}
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\framesubtitle{... to the symmetric and critical point}
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\begin{itemize}
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\item ensembles not exactly tuned
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\item able to interpolate to the desired points due to 2 or 3 ensembles per $\beta$
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\item determine points of interest as in \openlat~ensembles \arxivtag{2201.03874}
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\item define: $$\Phi^{\rm SF}_4 = \frac{3}{2}\,8t_0\,|m_{\rm eff}|\,m_{\rm eff}
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\quad \Rightarrow \quad \Phi^{\rm SF}_4\bigm\lvert_{m_{0,{\rm cr}}} = 0\,,\;\Phi^{\rm SF}_4\bigm\lvert_{m_{0,{\rm sym}}} = 1.115$$
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\end{itemize}
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\end{frame}
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\begin{frame}
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\frametitle{Improvement of the axial-vector current}
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\framesubtitle{Finding the symmetric and chiral point}
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\begin{tabular}{cc}
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\includegraphics[width=\halflinewidth]{plots/fix_sym_b3.9.pdf}&
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\includegraphics[width=\halflinewidth]{plots/fix_sym_b4.pdf}
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\end{tabular}
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\end{frame}
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\begin{frame}
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\frametitle{Improvement of the axial-vector current}
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\framesubtitle{Interpolations in $c_{\rm A}$}
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\begin{tabular}{cc}
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\includegraphics[width=\halflinewidth]{plots/ca2sym_b3.9.pdf}&
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\includegraphics[width=\halflinewidth]{plots/ca2sym_b4.pdf}
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\end{tabular}
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\end{frame}
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\begin{frame}
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\frametitle{Improvement of the axial-vector current}
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\framesubtitle{Final interpolations in $g_0^2$}
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\begin{tabular}{cc}
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\includegraphics[width=\halflinewidth]{plots/g0sq_chi.pdf}&
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\includegraphics[width=\halflinewidth]{plots/g0sq_sym.pdf}
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\end{tabular}
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\end{frame}
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% first study for difference:
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\begin{frame}
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\frametitle{First scaling test of improvement}
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\begin{itemize}
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\item Example: Calculate $f_\pi/K$ with stabilised Wilson fermions
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\item symmetric point \openlat~ensembles
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\item improve with $\ca = 0$ vs $\ca(g_0^2)|_{\rm chi}$ vs $\ca(g_0^2)|_{\rm sym}$
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$$f_{\rm A}^{\rm RI} = Z_{\rm A} (1+b_{\rm A} am_{\rm q}+\bar{b}_{\rm A} a\Tr[M_{\rm q}])\frac{\sqrt{2} \mathcal{A}_{\rm A_0P}}{\sqrt{\mathcal{A}_{\rm PP} m_\pi}}$$
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\item renormalisation: $Z_{\rm A}$ preliminary, $b_{\rm A}$ from pert. theory, $\bar{b}_{\rm A}$ neglected
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\end{itemize}
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\end{frame}
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\begin{frame}
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\frametitle{First study of improvement}
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\framesubtitle{Results}
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\includegraphics{plots/test_f_part_ZA_chi_imp.pdf}
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\end{frame}
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\begin{frame}
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\frametitle{Outlook}
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\begin{itemize}
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\item finish $\zv$, $\bV$ and $\bar{b}_{\rm V}$ through SF
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\item further improvement and renormalisation currently in the works:
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\begin{itemize}
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\item vector and tensor current improvement ($c_{\rm V}$, $c_{\rm T}$)
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\item current quark mass renormalisation ($b_{\rm A}-b_{\rm P}$, $b_m$, $Z$)
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\item determination of $Z_{\rm A}$, $Z_{\rm V}$, $Z_{\rm S}/Z_{\rm P}$ through $\chi$SF
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\end{itemize}
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\end{itemize}
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\end{frame}
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\end{document}
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