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\newcommand * { \arxivtag } [1]{ { \color { pantone315} \texttt { [#1]} } }
\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}
% \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}
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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}
\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
\vspace { .5cm}
\item improvement and renormalisation:
\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
\end { itemize}
\end { itemize}
\end { frame}
\begin { frame}
\frametitle { Determination of $ \ca $ }
\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}
\vspace { .5cm}
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$$ m _ { \rm PCAC } = \frac { \partial _ 0 f _ { \rm A } } { 2 f _ { \rm P } } + \ca ~a \frac { \partial ^ 2 _ 0 f _ { \rm P } } { 2 f _ { \rm P } } = r + \ca ~as $$
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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}
\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}
\begin { frame}
\frametitle { The wavefunction method}
\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:
$ \omega _ { 1 } = e ^ { - r / a _ 0 } \; , \quad \omega _ { 2 } = r~e ^ { - r / a _ 0 } \; , \quad \omega _ { 3 } = e ^ { - r / ( 2 a _ 0 ) } $
\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}
\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?
% 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
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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
\end { itemize}
\end { frame}
\begin { frame}
\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}
\begin { tabular} { cc|c|c|c|c}
\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
24& 3.685& 0.1396980& 0.1395500& 0.1394400& 0.120\\
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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\\
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96& 4.37& ---& ---& ---& 0.035\\
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\bottomrule
\end { tabular}
\end { center}
\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}
\end { itemize}
\end { frame}
\begin { frame}
\frametitle { Improvement of the axial-vector current}
\framesubtitle { $ \ca $ estimators}
\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} &
\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}
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\framesubtitle { ... to the symmetric and critical point}
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\begin { itemize}
\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: $$ \Phi ^ { \rm SF } _ 4 = \frac { 3 } { 2 } \, 8 t _ 0 \, |m _ { \rm eff } | \, m _ { \rm eff }
\quad \Rightarrow \quad \Phi ^ { \rm SF} _ 4\bigm \lvert _ { m_ { 0,{ \rm cr} } } = 0\, ,\; \Phi ^ { \rm SF} _ 4\bigm \lvert _ { m_ { 0,{ \rm sym} } } = 1.115$$
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\end { itemize}
\end { frame}
\begin { frame}
\frametitle { Improvement of the axial-vector current}
\framesubtitle { Finding the symmetric and chiral point}
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\begin { tabular} { cc}
\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}
\begin { frame}
\frametitle { Improvement of the axial-vector current}
\framesubtitle { Interpolations in $ c _ { \rm A } $ }
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\begin { tabular} { cc}
\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}
\begin { frame}
\frametitle { Improvement of the axial-vector current}
\framesubtitle { Final interpolations in $ g _ 0 ^ 2 $ }
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\begin { tabular} { cc}
\includegraphics [width=\halflinewidth] { plots/g0sq_ chi.pdf} &
\includegraphics [width=\halflinewidth] { plots/g0sq_ sym.pdf}
\end { tabular}
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\end { frame}
% first study for difference:
\begin { frame}
\frametitle { First scaling test of improvement}
\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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$$ f _ { \rm A } ^ { \rm RI } = Z _ { \rm A } ( 1 + b _ { \rm A } am _ { \rm q } + \bar { b } _ { \rm A } a \Tr [ M _ { \rm q } ] ) \frac { \sqrt { 2 } \mathcal { A } _ { \rm A _ 0 P } } { \sqrt { \mathcal { A } _ { \rm PP } m _ \pi } } $$
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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}
\end { frame}
\begin { frame}
\frametitle { First study of improvement}
\framesubtitle { Results}
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\vspace { -.3cm}
\centering
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\includegraphics [width=.6\linewidth] { plots/test_ f_ part_ ZA_ chi_ imp.pdf}
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\end { frame}
\begin { frame}
\frametitle { Outlook}
\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 } $ )
\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
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\end { itemize}
\end { itemize}
\end { frame}
\end { document}