From 259019c2d69b49c7a0fb4f208430a58fb298aca8 Mon Sep 17 00:00:00 2001 From: simone-romiti Date: Tue, 4 Jul 2023 17:15:13 +0200 Subject: [PATCH] is the correlator =0 for i!=j ??? --- doc/omeas_heavy_mesons.qmd | 170 ++++++++++++++++++++++++++++++++----- 1 file changed, 148 insertions(+), 22 deletions(-) diff --git a/doc/omeas_heavy_mesons.qmd b/doc/omeas_heavy_mesons.qmd index 2403b5f0b..0ac8dfa86 100644 --- a/doc/omeas_heavy_mesons.qmd +++ b/doc/omeas_heavy_mesons.qmd @@ -152,7 +152,7 @@ where ${\sigma_3 = Moreover $\sigma_3 = \sigma_3^\dagger$, hence $P_i^\dagger = P_i$. -- Obviously, $P_i$ commutes with the $\gamma$ matrices asthey act on different spaces. +- Obviously, $P_i$ commutes with the $\gamma$ matrices as they act on different spaces. ## Wick contractions @@ -169,8 +169,8 @@ we can write: \\ &= - \braket{ -[\bar{\chi}_{\ell} (\tau_1 P_i) \Gamma_1 \chi_{h}](x) -[\bar{\chi}_{h} (P_j \tau_1) \Gamma_2 \chi_{\ell}](0) +[\bar{\chi}_{\ell} (\sigma_1 P_i) \Gamma_1 \chi_{h}](x) +[\bar{\chi}_{h} (P_j \sigma_1) \Gamma_2 \chi_{\ell}](0) } \, . \end{split} @@ -181,42 +181,71 @@ An implicit summation on flavor indices is understood: \begin{equation} \mathcal{C}^{h_i, h_j}_{\Gamma_1, \Gamma_2}(t, \vec{x}) = - \braket{ -[(\bar{\chi}_{\ell})_{f_1} (\tau_1 P_i)_{f_1 f_2} \Gamma_1 (\chi_{h})_{f_2}](x) -[(\bar{\chi}_{h})_{f_3} (P_j \tau_1)_{f_3 f_4} \Gamma_2 (\chi_{\ell})_{f_4}](0) +[(\bar{\chi}_{\ell})_{f_1} (\sigma_1 P_i)_{f_1 f_2} \Gamma_1 (\chi_{h})_{f_2}](x) +[(\bar{\chi}_{h})_{f_3} (P_j \sigma_1)_{f_3 f_4} \Gamma_2 (\chi_{\ell})_{f_4}](0) } \, . \end{equation} -If we now call $S = D^{-1}$ the inverse of the Dirac operator, +We now call $S = D^{-1}$ the inverse of the Dirac operator and use Wick's theorem: \begin{equation} \braket{\Psi_{a}(x) \bar{\Psi}_{b}(0)} = S_{ab}(x|0) \, , \end{equation} -where $a$ and $b$ are a shortcut for the other indices of the spinor (spin, flavor, color, etc.), -we write: +where $a$ and $b$ are a shortcut for the other indices of the spinor (spin, flavor, color, etc.). +The correlator is: \begin{equation} -\mathcal{C}^{h_i, h_j}_{\Gamma_1, \Gamma_2}(t, \vec{x}) = +\begin{split} +\mathcal{C}^{h_i, h_j}_{\Gamma_1, \Gamma_2}(t, \vec{x}) +&= (S_\ell)_{\substack{f_4 f_1 \\ \alpha_1 \alpha_2}} (0|x) -(\tau_1 P_i)_{f_1 f_2} (\Gamma_1)_{\alpha_2 \alpha_3} +(\sigma_1 P_i)_{f_1 f_2} (\Gamma_1)_{\alpha_2 \alpha_3} (S_h)_{\substack{f_2 f_3 \\ \alpha_3 \alpha_4}} (x|0) -(P_j \tau_1)_{f_3 f_4} (\Gamma_2)_{\alpha_4 \alpha_1} +(P_j \sigma_1)_{f_3 f_4} (\Gamma_2)_{\alpha_4 \alpha_1} +\\ +&= +\operatorname{Tr}_f +\operatorname{Tr} +\left[ + S_h(x|0) (P_j \sigma_1) \Gamma_2 S_\ell(0|x) (\sigma_1 P_i) \Gamma_1 + \right] \, . +\end{split} \end{equation} +Using $\gamma_5$ hermiticity, +${(S_\ell)_{f_1 f_2} = (\sigma_1)_{f_1 g_1} \gamma_5 (S_\ell)_{g_1 g_2}^\dagger \gamma_5 (\sigma_1)_{g_2 f_2}}$, +we find: + +\begin{equation} +\mathcal{C}^{h_i, h_j}_{\Gamma_1, \Gamma_2}(t, \vec{x}) += +\operatorname{Tr}_f +\operatorname{Tr} +\left[ + S_h(x|0) P_j \Gamma_2 S_\ell (x|0) P_i \Gamma_1 +\right] +\end{equation} + +**Check: does this vanish for $i\neq j$ !?!?** + + ## Stochastic sampling Analogously to @foley2005practical, we now use $N_r$ stochastic samples in order to approximate $S$. The difference here is that the dilution is not on time but on Dirac index (spin). -Therefore, our estimator for $S$ is +Therefore, our estimators for $S$ and $S^{\dagger}$ are (cf. eqs. (5) and (7) of @foley2005practical): \begin{equation} +\label{eq:StochasticPropagator} +\begin{split} (S_\phi)_{\substack{f_1 f_2 \\ \alpha_1 \alpha_2}}(x|y) \approx \sum_{\beta=1}^{N_D=4} @@ -225,25 +254,35 @@ Therefore, our estimator for $S$ is (\eta_\phi^{(\beta)})^*_{\substack{f_2 \\ \alpha_2}}(y) } \, , +\\ +(S_\phi^{\dagger})_{\substack{f_1 f_2 \\ \alpha_1 \alpha_2}}(x|y) +\approx +\sum_{\beta=1}^{N_D=4} +\braket{ + (\eta_\phi^{(\beta)})_{\substack{f_1 \\ \alpha_1}}(x) + (\psi_\phi^{(\beta)})^{*}_{\substack{f_2 \\ \alpha_2}}(y) +} +\, , +\end{split} \end{equation} where $\braket{\cdot}$ here denotes the expectation value over the $N_r$ stochastic samples. The sources are such that ${\eta^{(\beta)}_{\alpha} = \delta_{\alpha \beta}}$, -and $\psi^{\beta}$ are the solutions to: +and $\psi^{(\beta)}$ are the solutions to: \begin{equation} \begin{split} &(D_{\phi})_{\substack{f_1 f_2 \\ \alpha_1 \alpha_2}}(x|y) \, -(\psi_\phi)^{\beta}_{\substack{f_2 \\ \alpha_2}}(y) = -(\eta_\phi)^{\beta}_{\substack{f_1 \\ \alpha_1}}(x) +(\psi_\phi^{(\beta)})_{\substack{f_2 \\ \alpha_2}}(y) = +(\eta_\phi^{(\beta)})_{\substack{f_1 \\ \alpha_1}}(x) \\[1em] & \implies \, \, -(\psi_\phi)^{\beta}_{\substack{f_1 \\ \alpha_1}}(x) = +(\psi_\phi^{(\beta)})_{\substack{f_1 \\ \alpha_1}}(x) = (S_{\phi})_{\substack{f_1 f_2 \\ \alpha_1 \alpha_2}}(x|y) \, -(\eta_\phi)^{\beta}_{\substack{f_2 \\ \alpha_1}}(y) +(\eta_\phi^{(\beta)})_{\substack{f_2 \\ \alpha_1}}(y) \end{split} \end{equation} @@ -259,13 +298,13 @@ Therefore, back to our correlators, we can write: & (\psi_{\ell}^{(\beta_1)})_{\substack{f_4 \\ \alpha_1}}(0) (\eta_{\ell}^{(\beta_1)})^*_{\substack{f_1 \\ \alpha_2}}(x) -(\tau_1 P_i)_{f_1 f_2} (\Gamma_1)_{\alpha_2 \alpha_3} +(\sigma_1 P_i)_{f_1 f_2} (\Gamma_1)_{\alpha_2 \alpha_3} \\ \times & (\psi_{h}^{(\beta_2)})_{\substack{f_2 \\ \alpha_3}}(x) (\eta_{h}^{(\beta_2)})^*_{\substack{f_3 \\ \alpha_4}}(0) -(P_j \tau_1)_{f_3 f_4} (\Gamma_2)_{\alpha_4 \alpha_1} +(P_j \sigma_1)_{f_3 f_4} (\Gamma_2)_{\alpha_4 \alpha_1} \, . \end{split} \end{equation} @@ -280,22 +319,109 @@ In terms of dot products (flavor and Dirac indices contracted): \sum_{\beta_1, \beta_2} & \left[ -(\eta_\ell^{(\beta_1)}(x))^{\dagger} (\tau_1 P_i) \Gamma_1 \psi_h^{(\beta_2)}(x) +(\eta_\ell^{(\beta_1)}(x))^{\dagger} (\sigma_1 P_i) \Gamma_1 \psi_h^{(\beta_2)}(x) \right] \\ \times& \left[ -(\eta_h^{(\beta_2)}(0))^{\dagger} (P_j \tau_1) \Gamma_2 \psi_\ell^{(\beta_1)}(0) +(\eta_h^{(\beta_2)}(0))^{\dagger} (P_j \sigma_1) \Gamma_2 \psi_\ell^{(\beta_1)}(0) \right] \end{split} \end{equation} +## Stochastic improvements + +There are some tricks we can use in order to enhance the signal-to-noise ratio. + +- Use the $\gamma_5$ hermiticity of the Dirac operator: + +\begin{equation} +\gamma_5 (S_\ell)_{f_1 f_2} \gamma_5 += (\sigma_1)_{f_1 g_1} (S_\ell)_{g_1 g_2}^\dagger (\sigma_1)_{g_2 f_2} +\, , +\end{equation} + + + + +- We also use the "one-end-trick", namely use the same source for each flavor: + +\begin{equation} +\eta_\phi = \eta \, , +\end{equation} + + + +Therefore (cf. eq. \eqref{eq:StochasticPropagator}): + +\begin{equation} +\begin{split} +&\mathcal{C}^{h_i, h_j}_{\Gamma_1, \Gamma_2}(t, \vec{x}) += +\\[1em] +&\sum_{\beta_1 \beta_2} +(\sigma_1)_{f_1 g_1} +(\gamma_5)_{\alpha_1 \alpha_5} +(\eta^{(\beta_1)})_{\substack{g_1 \\ \alpha_5}} (0) +(\psi_\ell^{(\beta_1)})^{*}_{\substack{g_2 \\ \alpha_6}} (x) +(\gamma_5)_{\alpha_6 \alpha_2} +(\sigma_1)_{g_2 f_4} +(\sigma_1 P_i)_{f_1 f_2} (\Gamma_1)_{\alpha_2 \alpha_3} \\ +&\times +(\psi_h^{(\beta_2)})_{\substack{f_2\\ \alpha_3}} (x) +(\eta^{(\beta_2)})^{*}_{\substack{f_3 \\ \alpha_4}} (0) +(P_j \sigma_1)_{f_3 f_4} (\Gamma_2)_{\alpha_4 \alpha_1} +\\[1em] +=& +\sum_{\beta} +(\sigma_1)_{f_1 g_1} +(\gamma_5)_{\alpha_1 \alpha_4} +(\psi_\ell^{(\beta)})^{*}_{\substack{g_2 \\ \alpha_6}} (x) +(\gamma_5)_{\alpha_6 \alpha_2} +(\sigma_1)_{g_2 f_4} +(\sigma_1 P_i)_{f_1 f_2} (\Gamma_1)_{\alpha_2 \alpha_3} \\ +&\times +(\psi_h^{(\beta)})_{\substack{f_2\\ \alpha_3}} (x) +(P_j \sigma_1)_{g_1 f_4} (\Gamma_2)_{\alpha_4 \alpha_1} +\, . +\end{split} +\end{equation} + + +The stochastic sources $\eta$ are such that: + +\begin{equation} +\braket{ + (\eta^{\beta_1})_{\substack{f_1 \\ \alpha_1}}^{*} (x) + (\eta^{\beta_2})_{\substack{f_2 \\ \alpha_2}}^{*} (y) + }= + \delta_{xy} \delta_{f_1 f_2} \delta_{\alpha_1 \alpha_2} \delta_{\beta_1 \alpha_1} \delta_{\beta_2 \alpha_2} + \, +\end{equation} + + ## Implementation -See comments in the code: `correlators.c`. +See comments in the code: `tmLQCD/meas/correlators.c`.