diff --git a/doc/omeas_heavy_mesons.qmd b/doc/omeas_heavy_mesons.qmd index 91774a64b..e5fcb31b1 100644 --- a/doc/omeas_heavy_mesons.qmd +++ b/doc/omeas_heavy_mesons.qmd @@ -20,6 +20,10 @@ csl: acta-ecologica-sinica.csl fontsize: 16pt --- +### Preamble and notation +
+ Show Content + `tmLQCD`can compute the correlators for the heavy mesons $K$ and $D$. The twisted mass formulation of the heavy doublet $(s, c)$ is such that $K$ and $D$ mix in the spectral decomposition @baron2011computing. In fact, the Dirac operator is not diagonal in flavor, @@ -54,8 +58,13 @@ In the following we use the twisted basis $\chi$ for fermion fields: \end{align} +
+ ## Correlators for the $K$ and $D$ +
+ Show Content + The required correlators are given by the following expectation values on the interacting vacuum ${\bra{\Omega} \cdot \ket{\Omega} = \braket{\cdot}}$: @@ -139,8 +148,13 @@ where ${\sigma_3 = 3. The action of $\sigma_1$ swaps the up and down flavor components of a spinor. ::: +
+ ## Wick contractions +
+ Show Content + Using the above remarks, we can write: @@ -208,7 +222,7 @@ Our correlator becomes: Upon a careful calculation for all values $i,j = 0,1$ we find, equivalently (using spacetime translational symmetry): -\begin{equation} +\begin{equation} \label{eq:C.hihj.Gamma1Gamma2} \begin{split} \mathcal{C}^{h_i, h_j}_{\Gamma_1, \Gamma_2}(t, \vec{x}) &= @@ -219,23 +233,52 @@ Upon a careful calculation for all values $i,j = 0,1$ we find, equivalently (usi (S_u)^\dagger (x|0) \gamma_5 \Gamma_1 \right] -\\ -&= -\sum_{y,z} -\operatorname{Tr} -\left[ - (S_h)_{f_i f_j} (x+y|y) - \Gamma_2 \gamma_5 - (S_u)^\dagger (x+z|z) - \gamma_5 \Gamma_1 - \delta_{yz} -\right] -\, . \end{split} \end{equation} This is a generalized case of eq. (A9) of @PhysRevD.59.074503. +
+ + +## Stochastic approximation of the correlators + +### Index dilution + +
+ Show Content + + +We now make the following remark. If we want to invert numerically the system $D_{ij} \psi_{j} = \eta_{i}$ (where the $i,j$ indices include all internal indices), we have: + +\begin{equation} +D_{ij} \psi_{j} = \eta_{i} \, \implies +\psi_{i} = S_{ij} \eta_{j} \, . +\end{equation} + +If $\eta_i \eta_i^{*} = 1$ we have: + +\begin{equation} +S_{ij} = \psi_{i} \eta_{j}^{*} \, . +\end{equation} + +--- + +We can also use **index dilution** in order to select the components we want. In fact, if we define: $\eta_i^{(a)} = \eta \delta_{i}^{a}$, with $\eta^{*} \eta =1$ we have: + +\begin{equation} +S_{ab} = S_{ij} \delta_{i}^{a} \delta_{j}^{b} = +S_{ij} (\eta* \delta_{i}^{a}) (\eta \delta_{j}^{b}) = +(\psi_{i}^{(b)})^{*} \eta_{i}^{a} = \psi^{(b)} \cdot \eta^{(a)} \, . +\end{equation} + +
+ +### Stochastic expression of the correlators + +
+ Show Content + We now approximate the propagator using stochastic sources. Additionally, we use: @@ -267,99 +310,36 @@ Additionally, we use: \end{equation} +Therefore, we can approximate the correlators of eq. \eqref{eq:C.hihj.Gamma1Gamma2} above with: -Therefore, we can use spin dilutions to rephrase the correlator in a form which will turn out to be convenient later -($c$ is the color index): \begin{equation} +\begin{split} \mathcal{C}^{h_i, h_j}_{\Gamma_1, \Gamma_2}(t, \vec{x}) = -[(S_h)_{f_i f_j}]_{\alpha_1 \beta_1} (x|0) -[\eta^{(\alpha_2)}_{\beta_1}]_c -(\Gamma_2 \gamma_5)_{\alpha_2 \alpha_3} -[{(\eta^\dagger)}^{(\alpha_3)}_{\beta_2}]_c -[(S_u)^\dagger]_{\beta_2 \alpha_4} (x|0) +& +\left( + (\psi_h)^{(f_j, \alpha_2)}(x) + \cdot + \eta^{(f_i, \alpha_1)}(0) +\right) +(\Gamma_2 \gamma_5)_{\alpha_2 \alpha_3} +\\ +& +\left( + (\psi_u)^{(0, \alpha_4)}(x) + \cdot + \eta^{(0, \alpha_3)}(0) +\right) (\gamma_5 \Gamma_1)_{\alpha_4 \alpha_1} -\, . +\end{split} \end{equation} -We now define our spinor propagators. -If $\eta^{(\beta, \phi)}$ is the diluted source: - -\begin{align} -& (D_{\ell/h})_{\alpha_1 \alpha_2} (x|y) ({\psi}_{\ell/h}^{(\beta, \phi)})_{\alpha_2} (y) -= (\eta^{(\beta, \phi)})_{\alpha_1} (x) -\\ -& \, \implies \, -(\psi_{\ell/h}^{(\beta, \phi)})_{\alpha_1} (x) -= -(S_{\ell/h})_{\alpha_2 \alpha_1} (x | y) -\eta^{(\beta, \phi)}_{\alpha_2} (y) -= -(S_{\ell/h})_{\alpha_2 \alpha_1} (x | 0) -\eta^{(\beta, \phi)}_{\alpha_2} (0) -\\ -& \, \implies \, -(\psi_{\ell/h}^{(\beta, \phi)})^{*}_{\alpha_1} (x) -= -(\eta^{(\beta, \phi)})^{*}_{\alpha_2} (y) -(S_{\ell/h}^\dagger)_{\alpha_2 \alpha_1} (x | y) -= -(\eta^{(\beta, \phi)})^\dagger_{\alpha_2} (0) -(S_{\ell/h}^\dagger)_{\alpha_2 \alpha_1} (x | 0) -\end{align} - -This means that for our matrix of correlators we have to do ${4_D \times 2_f \times 2_{h,\ell}} = 16$ inversions. +
-::: {.remark} -1. For the light doublet `tmLQCD` computes only $S_u$, which is obtained with $(\psi_h^{(\beta, f_0)})_{f_0}$. This is the only propagator we need. -2. For the heavy propagator, we can access the $(i,j)$ component of $S_h$ with $(\psi_h^{(\beta, f_j)})_{f_i}$. -::: -Our correlator is given by the following expectation value -(no summation on flavor indices): - -\begin{equation} -\begin{split} -\mathcal{C}^{h_i, h_j}_{\Gamma_1, \Gamma_2}(t, \vec{x}) -&= -\langle -[(\psi_h^{(\alpha_2, f_j)})_{f_i}]_{\alpha_1}(x) -(\Gamma_2 \gamma_5)_{\alpha_2 \alpha_3} -[(\psi_\ell^{(\alpha_3, f_0)})_{f_0}^\dagger]_{\alpha_4} (x) -(\gamma_5 \Gamma_1)_{\alpha_4 \alpha_1} -\rangle -\\ -&= -\braket{ -(\psi_\ell^{(\alpha_3, f_0)})_{f_0}^\dagger (x) -\cdot -(\gamma_5 \Gamma_1) -\cdot -(\psi_h^{(\alpha_2, f_j)})_{f_i}(x) -} -\, -(\Gamma_2 \gamma_5)_{\alpha_2 \alpha_3} -\\ -&= -\mathcal{R}^{\alpha_3 \alpha_2} (\Gamma_2 \gamma_5)_{\alpha_2 \alpha_3} -\end{split} -\end{equation} - -More explicitly: - -\begin{equation} -\begin{split} -\Gamma_2=1 &\implies \mathcal{C}^{h_i, h_j}_{\Gamma_1, \Gamma_2}(t, \vec{x}) -= \mathcal{R}^{00}+\mathcal{R}^{11}-\mathcal{R}^{22}-\mathcal{R}^{33} -\\ -\Gamma_2=\gamma_5 &\implies \mathcal{C}^{h_i, h_j}_{\Gamma_1, \Gamma_2}(t, \vec{x}) -= \mathcal{R}^{00}+\mathcal{R}^{11}+\mathcal{R}^{22}+\mathcal{R}^{33} -\end{split} -\end{equation} -