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Merge pull request #614 from etmc/heavy_mesons_onlinemeas_gamma5herm
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Heavy mesons onlinemeas gamma5herm
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@simone-romiti
simone-romiti authored Sep 25, 2024
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192 changes: 100 additions & 92 deletions doc/omeas_heavy_mesons.qmd
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Expand Up @@ -20,6 +20,10 @@ csl: acta-ecologica-sinica.csl
fontsize: 16pt
---

### Preamble and notation
<details>
<summary>Show Content</summary>

`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,
Expand Down Expand Up @@ -54,8 +58,13 @@ In the following we use the twisted basis $\chi$ for fermion fields:
\end{align}
<!-- -->

</details>

## Correlators for the $K$ and $D$

<details>
<summary>Show Content</summary>

The required correlators are given by the following
expectation values on the interacting vacuum
${\bra{\Omega} \cdot \ket{\Omega} = \braket{\cdot}}$:
Expand Down Expand Up @@ -139,8 +148,13 @@ where ${\sigma_3 =
3. The action of $\sigma_1$ swaps the up and down flavor components of a spinor.
:::

</details>

## Wick contractions

<details>
<summary>Show Content</summary>

Using the above remarks,
we can write:
<!-- -->
Expand Down Expand Up @@ -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})
&=
Expand All @@ -219,23 +233,53 @@ 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.

</details>


## Stochastic approximation of the correlators

### Index dilution

<details>
<summary>Show Content</summary>


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 $\eta_i \eta_j^{*} = \delta_{ij}$ 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})^{*}
= \eta^{(a)} \cdot \psi^{(b)} \, .
\end{equation}

</details>

### Stochastic expression of the correlators

<details>
<summary>Show Content</summary>

We now approximate the propagator using stochastic sources.
Additionally, we use:

Expand All @@ -246,7 +290,7 @@ Additionally, we use:
\begin{equation}
\eta^{(\alpha)}_{\beta, c} =
\eta_c \, \delta^\alpha_\beta \,\, , \, \,
\eta^\dagger_c \eta_c = 1 \, .
\eta_c \otimes \eta^\dagger_c = \mathbb{1}_{N_c \times N_c} \, .
\end{equation}
<!-- -->
- Flavor dilution: sources have an additional flavor index $\phi$, such that their flavor component different from the index vanish. The value of the flavor component however is the same, it changes only its position in the doublet:
Expand All @@ -267,99 +311,63 @@ 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}
\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)
(\gamma_5 \Gamma_1)_{\alpha_4 \alpha_1}
\, .
\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)
\left(
\eta^{(f_i, \alpha_1)}(0)
\cdot
\psi_h^{(f_j, \alpha_2)}(x)
\right)
(\Gamma_2 \gamma_5)_{\alpha_2 \alpha_3}
[(\psi_\ell^{(\alpha_3, f_0)})_{f_0}^\dagger]_{\alpha_4} (x)
\left(
\eta^{(0, \alpha_3)}(0)
\cdot
\psi_u^{(0, \alpha_4)}(x)
\right)^{*}
(\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}
&=
(\psi_h)_{f_i, \alpha_1}^{(f_j, \alpha_2)}(x)
(\Gamma_2 \gamma_5)_{\alpha_2 \alpha_3}
(\psi_u)_{\alpha_3}^{(0, \alpha_4)}(x)^{*}
(\gamma_5 \Gamma_1)_{\alpha_4 \alpha_1}
\end{split}
\end{equation}
<!-- -->

More explicitly:
<!-- -->
In the end we have:

\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}
\mathcal{C}^{h_i, h_j}_{\Gamma_1, \Gamma_2}(t, \vec{x})
=
[\Gamma_2 \gamma_5 (\psi_u)^{(0, \alpha_2)}(x)^{*}]_{\alpha_1}
[\gamma_5 \Gamma_1 (\psi_h)_{f_i}^{(f_j, \alpha_1)}(x) ]_{\alpha_2}
\end{equation}
<!-- -->

In our case $\Gamma_{1,2} = 1,\gamma_5$. Since we use the chiral basis ${\Gamma_{1,2}^{*} = (\Gamma_{1,2}^{T})^\dagger = \Gamma_{1,2}}$.
Therefore we can equivalently write the correlator as:

\begin{equation}
\mathcal{C}^{h_i, h_j}_{\Gamma_1, \Gamma_2}(t, \vec{x})
=
[\Gamma_2 \gamma_5 (\psi_u)^{(0, \alpha_2)}(x)]^{*}_{\alpha_1}
\cdot
[\gamma_5 \Gamma_1 (\psi_h)_{f_i}^{(f_j, \alpha_1)}(x) ]_{\alpha_2} \,
\end{equation}

where $\cdot$ is the dot product in color space (NOTE: it complex-conjugates the 1st vector).



</details>






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