15 lines
1 KiB
TeX
15 lines
1 KiB
TeX
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\begin{frame}
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\frametitle{How are sigma terms defined? - renormalisation}
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\begin{align*}
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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
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\end{align*}
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{\small $Z_\mathrm{m}$ - renormalisation parameter of the non-singlet scalar density}
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%The matrix elements must renormalise in the inverse manner to the masses so that
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\begin{align*}
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\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}
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\end{align*}
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The normalisation factor $r_\mathrm{m}$ (ALPHA, RQCD) is\\ the ratio of flavour non-singlet and singlet scalar density renormalisation parameters. \\
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$\rightarrow$ accounts for the \textbf{mixing of quark flavours under renormalisation} for Wilson fermions.
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\end{frame}
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