diff --git a/talk.tex b/talk.tex index 350e3ec..c5afee4 100644 --- a/talk.tex +++ b/talk.tex @@ -75,29 +75,32 @@ \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 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 + \end{itemize} \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 + \item construct pseudoscalar states \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 $\omega_{\rm b4} = {\rm cons.}\;,\quad\omega_{\rm b5} = -r^2~e^{-r/a_0}$ + \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 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} \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? % How does this interplay with the states that we achieve? % Which is the optimal wf combination? @@ -192,7 +195,7 @@ \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 $\ca(g_0^2)|_{\rm sym}$ - $$f_{\rm A}^{RI} = Z_{\rm A} (1+b_{\rm A} m_{\rm q})(1+\bar{b}_{\rm A} \Tr[M_{\rm q}])f\frac{\sqrt{2} \mathcal{A}_{\rm A_0P}}{\sqrt{\mathcal{A}_{\rm PP} m_\pi}}$$ + $$f_{\rm A}^{RI} = Z_{\rm A} (1+b_{\rm A} m_{\rm q}+\bar{b}_{\rm A} \Tr[M_{\rm q}])\frac{\sqrt{2} \mathcal{A}_{\rm A_0P}}{\sqrt{\mathcal{A}_{\rm PP} m_\pi}}$$ \item renormalisation: $Z_{\rm A}$ preliminary, $b_{\rm A}$ from pert. theory, $\bar{b}_{\rm A}$ neglected \end{itemize} \end{frame}