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Justus Kuhlmann 2024-07-30 20:36:38 +02:00
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talk.tex
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@ -75,29 +75,32 @@
\frametitle{Determination of $\ca$} \frametitle{Determination of $\ca$}
\begin{itemize} \begin{itemize}
\item Schrödinger functional boundary conditions \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 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 \item derive from PCAC mass
\end{itemize} \end{itemize}
\vspace{.5cm} \vspace{.5cm}
$$m_{\rm PCAC} = \frac{\partial_0 f_{\rm A}}{2f_{\rm P}} + \ca~a\frac{\partial^2_0 f_{\rm P}}{2f_{\rm P}} = r + \ca~as$$ $$m_{\rm PCAC} = \frac{\partial_0 f_{\rm A}}{2f_{\rm P}} + \ca~a\frac{\partial^2_0 f_{\rm P}}{2f_{\rm P}} = r + \ca~as$$
$$m_{\rm PCAC}^{(0)} = m_{\rm PCAC}^{(1)}\quad\Leftrightarrow\quad\ca = - \frac{r^{(1)} - r^{(0)}}{s^{(1)} - s^{(0)}}$$ $$m_{\rm PCAC}^{(0)} = m_{\rm PCAC}^{(1)}\quad\Leftrightarrow\quad\ca = - \frac{r^{(1)} - r^{(0)}}{s^{(1)} - s^{(0)}}$$
\begin{itemize}
\item states (0) and (1) are the PS ground and first excited state in our setup
\end{itemize}
\end{frame} \end{frame}
\begin{frame} \begin{frame}
\frametitle{The wavefunction method} \frametitle{The wavefunction method}
\begin{itemize} \begin{itemize}
\item mimic pionic sources on boundaries $\pi^{(0)}, \pi^{(1)}$ and require PCAC relation to hold for both \item construct pseudoscalar states
\begin{itemize} \begin{itemize}
\item basis wavefunctions: \item H-like basis wavefunctions:
$\omega_{\rm b1} = e^{-r/a_0}\;,\quad\omega_{\rm b2} = r~e^{-r/a_0}\;,\quad\omega_{\rm b3} = e^{-r/(2a_0)}$ $\omega_{1} = e^{-r/a_0}\;,\quad\omega_{2} = r~e^{-r/a_0}\;,\quad\omega_{3} = e^{-r/(2a_0)}$
\item also include $\omega_{\rm b4} = {\rm cons.}\;,\quad\omega_{\rm b5} = -r^2~e^{-r/a_0}$ \item also include $\omega_{4} = {\rm cons.}\;,\quad\omega_{5} = -r^2~e^{-r/a_0}$
\quad with $r=|\vec{y}-\vec{x}|$ \quad with $r=|\vec{y}-\vec{x}|$
\end{itemize} \end{itemize}
\pause \pause
\item eigenvectors of boundary-to-boundary corr. func. $(F_1)_{i,j} = -\langle O(\omega_{{\rm b}i}) O'(\omega_{{\rm b}j})\rangle$ \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} \vspace{.5cm}
\pause \pause
\item diagonalise $(F_1)_{i,j}$ \& project $f_{\rm A}(x_0)$ and $f_{\rm P}(x_0)$ onto the eigenstates \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? % Question: do we include all wavefunctions or just some?
% How does this interplay with the states that we achieve? % How does this interplay with the states that we achieve?
% Which is the optimal wf combination? % Which is the optimal wf combination?
@ -192,7 +195,7 @@
\item Example: Calculate $f_\pi/K$ with stabilised Wilson fermions \item Example: Calculate $f_\pi/K$ with stabilised Wilson fermions
\item symmetric point \openlat~ensembles \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 improve with $\ca = 0$ vs $\ca(g_0^2)|_{\rm chi}$ vs $\ca(g_0^2)|_{\rm sym}$
$$f_{\rm A}^{RI} = Z_{\rm A} (1+b_{\rm A} m_{\rm q})(1+\bar{b}_{\rm A} \Tr[M_{\rm q}])f\frac{\sqrt{2} \mathcal{A}_{\rm A_0P}}{\sqrt{\mathcal{A}_{\rm PP} m_\pi}}$$ $$f_{\rm A}^{RI} = Z_{\rm A} (1+b_{\rm A} m_{\rm q}+\bar{b}_{\rm A} \Tr[M_{\rm q}])\frac{\sqrt{2} \mathcal{A}_{\rm A_0P}}{\sqrt{\mathcal{A}_{\rm PP} m_\pi}}$$
\item renormalisation: $Z_{\rm A}$ preliminary, $b_{\rm A}$ from pert. theory, $\bar{b}_{\rm A}$ neglected \item renormalisation: $Z_{\rm A}$ preliminary, $b_{\rm A}$ from pert. theory, $\bar{b}_{\rm A}$ neglected
\end{itemize} \end{itemize}
\end{frame} \end{frame}