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talk-lattice2024/document.tex

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2024-07-24 13:44:10 +02:00
\begin{frame}
\frametitle{How are sigma terms defined? - renormalisation}
\begin{align*}
m_q^\mathrm{ren} = Z_\mathrm{m} \left[m_q \, + \, (r_\mathrm{m} - 1) \frac{\mathrm{Tr} M}{N_\mathrm{f}} \right] + \text{cut-off effects}, \quad \mathrm{Tr} M = \Sigma_q m_q
\end{align*}
{\small $Z_\mathrm{m}$ - renormalisation parameter of the non-singlet scalar density}
%The matrix elements must renormalise in the inverse manner to the masses so that
\begin{align*}
\sigma_{qB}^{\mathrm{ren}} = \left(m_q + \frac{r_\mathrm{m}-1}{3} \mathrm{Tr} M \right) \left(g_{q,S}^B - \frac{r_\mathrm{m}-1}{3r_\mathrm{m}}\mathrm{Tr} g_{S}^B \right) \quad \text{for}\, N_\mathrm{f}= 3,\ \sigma_{\pi B}^\mathrm{ren} = \sigma_{uB}^\mathrm{ren} + \sigma_{dB}^\mathrm{ren}
\end{align*}
The normalisation factor $r_\mathrm{m}$ (ALPHA, RQCD) is\\ the ratio of flavour non-singlet and singlet scalar density renormalisation parameters. \\
$\rightarrow$ accounts for the \textbf{mixing of quark flavours under renormalisation} for Wilson fermions.
\end{frame}