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\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}
		\item derive from PCAC mass 
	\end{itemize}
	\vspace{.5cm}
	$$m_{\rm PCAC} = \frac{\partial_0 f_{\rm A}}{2f_{\rm P}} + \ca~a\frac{\partial^2_0 f_{\rm P}}{2f_{\rm P}} = r + \ca~as$$
	$$m_{\rm PCAC}^{(0)} = m_{\rm PCAC}^{(1)}\quad\Leftrightarrow\quad\ca =  - \frac{r^{(1)} - r^{(0)}}{s^{(1)} - s^{(0)}}$$
\end{frame}

\begin{frame}
	\frametitle{The wavefunction method}
	\begin{itemize}
		\item mimic pionic sources on boundaries $\pi^{(0)}, \pi^{(1)}$ and require PCAC relation to hold for both
		\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 $\omega_{\rm b4} = {\rm cons.}\;,\quad\omega_{\rm b5} = -r^2~e^{-r/a_0}$
			\quad with $r=|\vec{y}-\vec{x}|$
		\end{itemize}
		\pause
		\item eigenvectors of 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 diagonalise $(F_1)_{i,j}$ \& 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 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{$T=L\approx 3\,{\rm fm}$ Schrödinger-Functional ensembles, exp. Wilson-Clover fermions}
	\begin{center}
		\begin{tabular}{cc|c|c|c|c}
			\toprule
			$L/a$ & $\beta$ & $\kappa_{1}\approx\kappa_{\rm cr}$ & $\kappa_{2}$ & $\kappa_{3}\approx\kappa_{\rm sym}$&$a$\\
			\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{Final 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 $\ca(g_0^2)|_{\rm sym}$
		\item $f_{\rm A}^{RI} = Z_{\rm A} (1+b_{\rm A} m_{\rm q})(1+\bar{b}_{\rm A} \Tr[M_{\rm q}])f\frac{\sqrt{2} A}{\sqrt{A m_\pi}}$
		\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 study of improvement}
	\framesubtitle{Results}
	%\includegraphics{plots/test_f.pdf}
\end{frame}

\begin{frame}
	\frametitle{Outlook}
	\begin{itemize}
		\item finish $\zv$, $\bV$ 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}
\end{document}
