project_details.tex

\label{sec:proj}

The computation proposed here will use the same ensembles of the previous stage of the project
i.e. Wilson-clover twisted
mass fermions~\cite{Alexandrou:2018egz}. The up/down quarks will be treated
in a fully unitary setting. To avoid flavor mixing lattice artefacts
in the unitary strange/charm sector of this regularisation, we use a
mixed action approach for strange and charm quarks: so-called
Osterwalder-Seiler type~\cite{Frezzotti:2004wz} strange and charm quark doublets
$(s^+ , s^-)^T$ and $(c^+ , c^- )^T$ are added with bare
twisted strange and charm quark mass $\pm \mu_s$ and $\pm \mu_c$
tuned to reproduce the physical mass of the $\phi$ and $J/\Psi$
mesons, respectively, as described in Appendix C of
Ref.\cite{ExtendedTwistedMass:2022jpw}. For more details, we refer to
this reference.


The core of our project is the calculation of inclusive semi-leptonic
decays of the $B$ and $B_s$ mesons, for which we need to determine the
bare $b$-quark mass first.
For the application of what we call the
ratio-method~\cite{ETM:2009sed}, we will generate data at six
different values of the heavy quark mass in the range from 
1 to 3.5 times the charm quark mass. 
The data produced for these six heavy quark mass values can be used at
the same time for the  determination of the $B$ and $B_s$ meson masses
and the total inclusive decay rate for the process $B_{(s)} \to
X\ell\bar\nu$. 



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\subsection{Sub-project 1: total inclusive decay rate $B_s \to X\ell\bar\nu$}

By using the optical theorem, the semi-leptonic decay of a $B$ and $B_s$
to a hadron $X$ with a pair of massless leptons $B_{(s)} \to X\ell\bar\nu$ can be written as
\begin{equation}
  \Gamma = G^2_F\left\{ |V_{bu} |^2 \Gamma_{bu} + |V_{cb} |^2 \Gamma_{bc} + |V_{us} |^2 \Gamma_{su}
  \right\}\,,
\end{equation}
where the different contributions on the right-hand-side correspond at the quark level
to the weak transitions $b \to u$, $b \to c$ and $s \to u$, respectively.
Each contribution can be written as
\begin{equation}\label{eq:Gamma_fg}
  \Gamma_{fg}=\int \frac{d^3p_\nu}{(2\pi)^32E_\nu}\frac{d^3p_\ell}{(2\pi)^32E_\ell}
  L_{\mu\nu}(p_\ell, p_\nu) H^{\mu\nu}_{fg}(p,p-p_\ell-p_\nu)\,,
\end{equation}
where the leptonic tensor is given by
\begin{equation}
  L_{\mu\nu}(p_\ell, p_\nu) =4\left\{p_\ell^\mu p_\nu^\nu +p_\ell^\nu
  p_\nu^\mu - g^{\mu\nu} p_\ell\cdot p_\nu+
  i\epsilon_{\mu\nu\alpha\beta} p_\ell^\alpha p_\nu^\beta\right\}\,,
\end{equation}
while the hadronic tensor reads
\begin{equation}
  H^{\mu\nu}_{fg}(p,p_X)=\frac{1}{2m_{B_{(s)}}}\langle B_{(s)}| J^\mu_{fg}(0)(2\pi)^4
  \delta^4(\mathbb{P}-p_x) J^{\nu\dagger}_{gf} (0)| B_{(s)}\rangle\,.
\end{equation}
$\mathbb{P}$ is the QCD four momentum operator and $J_{fg}$ are the
relevant Minkowski weak currents $J_{fg}^\mu(x)=i\bar
  f(x)\gamma^\mu(1-\gamma_5)g(x)$.
The contribution $\Gamma_{su}$ is zero in the decay of the $B$ for
momentum conservation while  in the decay of the $B_{s}$ it is
suppressed by the integration over the phase space of
\eqref{eq:Gamma_fg}. Moreover, this flavour channel is only inclusive
through $B_s\to B\ell\bar\nu$. 
The calculation of $\Gamma_{bu}$  requires computing disconnected
diagrams, which is computationally more challenging than the connected
part. 
On the other hand the contribution coming from disconnected diagrams
is expected to be small due to the OZI-suppression rule. 
In this project we will focus only on the computation of $\Gamma_{bc}$
and the connected part of $\Gamma_{bu}$ leaving the computation of the
disconnected diagram to a followup project. 
It is important to notice that the decays $B_{(s)} \to X_c\ell\bar\nu$ corresponding to decays into charmed final states $X_c$ are known from the experimental side and thus can be compared directly with our calculations.

From now on we will drop the indices $fg$
and the procedure we are describing will be equivalent for the $bu$ and $bc$ contributions,
moreover
in the rest frame of the $B_{(s)}$ meson we have $p=(M_{B_{(s)}},\bm{0})$ thus we simplify the notation of
$H^{\mu\nu}(p_X)\equiv H^{\mu\nu}_{fg}(p,p_X)$.
The hadronic tensor $H_{\mu\nu}$ is
the spectral density of the quantity
\begin{equation}
  M_{\mu\nu}(t,{\bf q})= \int_0^{\infty}d \omega H_{\mu\nu} (\omega,{\bf q}) e^{-\omega t}\,,\quad \text{with}\quad p_X=(\omega,\bm{q})\,.
\end{equation}
$H_{\mu\nu}$ can be reconstructed from $M_{\mu\nu}$ using the method presented in
Ref.~\cite{Hansen:2019idp}. $M_{\mu\nu}$ can be directly computed from the
ratio of Euclidean lattice correlators
\begin{gather}
  M_{\mu\nu}(t_2-t_1,{\bf q})=\frac{C_{\mu\nu}(t_{snk},t_2,t_1,t_{src};q)}{e^{-(t_{snk}-t_2)}  C(t_1-t_{src}) }\label{eq:ratio_4pt_2pt}\\
  \label{eq:4pt}
  C_{\mu\nu}(t_{snk},t_2,t_1,t_{src};q)=\int d^3x e^{i{\bf q}\cdot {\bf x}}
  \langle B_{(s)}({\bf 0}, t_{snk}) J^{\dagger}_\mu({\bf x},t_2)  J_\nu(0,t_1)
  B_{(s)}^\dagger({\bf 0}, t_{src})\rangle\,,\\
  \label{eq:2pt}
  C(t_{snk}-t_{src}) =  \langle B_{(s)}({\bf 0}, t_{snk})
  B_{(s)}^\dagger({\bf 0}, t_{src})\rangle\,.
\end{gather}
In general, the inverse problem represented by the extraction of
hadronic spectral densities from Euclidean correlators is notoriously
ill-posed. Recently, a method to cope with these problems has been
proposed in Ref.~\cite{Hansen:2019idp}. It consists of treating the
integrals mentioned above with some $C_\infty$ kernel $K_\sigma$
\begin{equation}
  H^\sigma_{\mu\nu}(\omega^*, {\bf q})= \int d \omega K_\sigma( \omega^*,\omega ) H_{\mu\nu}(\omega, {\bf q})\,,
\end{equation}
making the inverse problem well posed and $H^\sigma_{\mu\nu}$ computable.
In our case the phase space integration in \eqref{eq:Gamma_fg}
provides a sharp smearing kernel function $\theta$. Following
Refs.~\cite{Gambino:2020crt, Gambino:2022dvu}, the phase space integral over
\eqref{eq:Gamma_fg} can be written as
\begin{gather}
  \frac{48 \pi^4}{m_{B_{(s)}}^5}\frac{d\Gamma}{d \bm{ \omega^2} }
  =\sum_{l=0}^2 |\bm{\omega}|^{3-l}\int_0^{\infty}d \omega_0 \Theta^l(\omega_0^{max}-\omega_0) Z^l\,,\quad\quad {\omega}_0^{max}=1-|\bm{\omega}|\,,\quad\Theta^l(x)=x^l\theta(x)\\
  Z^2=Y_3-2Y_1\,,\quad Z^1=2(Y_3-2Y_1-Y_4)\,,\quad Z^0=Y_2+Y_3-2Y_4\,.
\end{gather}
Moreover, the $Y_i$ can be directly computed from the Euclidean
hadronic tensor $H^{\mu\nu}$ as follows
\begin{align}
                                                     & Y^1=-m_{B_{(s)}}\sum_{ij}\hat{n}^i\hat{n}^j H^{ij}(p,q)\,,          &  & Y^2=-m_{B_{(s)}}H^{00}(p,q)\,, \\
                                                     & Y^3=m_{B_{(s)}}\sum_{ij}\hat{\omega}^i\hat{\omega}^j H^{ij}(p,q)\,, &  &
  Y^4=-im_{B_{(s)}}\sum_{i}\hat{\omega}^i H^{0i}(p,q)\,, &
\end{align}
with $\hat{n}$ a unit vector orthogonal to $\bm\omega$.
Thus, the phase space integral provides us a $\theta$-function smearing
kernel. However, the $\theta$-function is not smooth, which is why we
will use a $C_\infty$ kernel that
after taking the infinite volume limit, the
$\theta$-function is recovered as $\sigma\to0$, i.e. $\lim_{\sigma\to 0} K_\sigma^l(x^*,x)=\Theta^l(x^*-x)$.
Therefore,
\begin{gather}
  \frac{48 \pi^4}{m_{B_{(s)}}^5}\frac{d\Gamma}{d \bm{ \omega^2} }
  =\lim_{\sigma\to 0}\sum_{l=0}^2 |\bm{\omega}|^{3-l}\int_0^{\infty}d \omega_0 K_\sigma^l(\omega_0^{max},\omega_0) Z^l\,.
\end{gather}
The spectral reconstruction can be performed at the analysis stage. Thus,
the main cost of this calculation is the production of the
four-point functions \eqref{eq:4pt}.

The renormalization constants necessary to renormalise the ratio of
correlators \eqref{eq:ratio_4pt_2pt} $Z_A$ and $Z_V$ have already been
computed in previous work of ETMC~\cite{ExtendedTwistedMass:2024myu}.

\begin{figure}
  \centering
  \begin{tikzpicture}
    \begin{feynman}[scale=1.5]
      \vertex[anchor=east,blob, fill=black!0!] (a) at (0,0) {\(B_{(s)}^\dagger\)};
      \vertex[anchor=east,blob, fill=black!0!]  (J1) at (1,0.8) {\(J_W\)};
      \vertex[anchor=east,blob, fill=black!0!] (J2) at (2.4,0.8) {\(J_W\)};
      \vertex[anchor=west,blob, fill=black!0!] (b) at (3,0) {\(B_{(s)}\)};
      \vertex (s1) at (3.4,-0.4) ;
      \vertex (s2) at (3.4, 0.4) ;
      \vertex (s3) at (2.6, 0.7) ;
      \vertex (s4) at (1.8, 1) ;

      \diagram*{
      %				(a) --[fermion,  half right,looseness=0.5,edge label=$s$] (b);
      (b) --[fermion, black, out=220, in=320,edge label=$u(s)$,swap] (a);

      (a) --[fermion, blue, edge label=$b$] (J1) ;
      (J1) --[fermion, sand, edge label=$c$, reversed momentum'=$\bf{q}$] (J2);
      (J2) --[fermion, blue ,edge label=$b$] (b);
      (s1) --[->, half right, edge label=sequential,swap] (s2);
      (s3) --[->, half right, edge label=sequential,swap] (s4);
      };
    \end{feynman}
    \begin{feynman}[scale=1.5]
      \vertex[anchor=east,blob, fill=black!0!] (a) at (6.5,0) {\(B_{(s)}^\dagger\)};
      \vertex[anchor=west,blob, fill=black!0!] (b) at (9.5,0) {\(B_{(s)}\)};

      \diagram*{
      %				(a) --[fermion,  half right,looseness=0.5,edge label=$s$] (b);
      (b) --[fermion, black, out=220, in=320,edge label=$u(s)$,swap] (a);
      (a) --[fermion, blue, out=40, in=140,edge label=$b$,swap] (b);
      % (a) --[fermion, green!40!black, out=40, in=140,edge label=$b$,swap] (b);
      };
    \end{feynman}
  \end{tikzpicture}
  \caption{On the left: Wick contractions required for the calculation
    of the correlator \eqref{eq:4pt} for the contribution $bc$. For
    the contribution $bu$ the $c$-quark propagator must be replaced
    with and $u$-quark propagator. On the right: Wick contractions
    required for the calculation of the mesonic two-pointcorrelator
    \eqref{eq:2pt}.}
  \label{fig:4pt}
\end{figure}

A similar analysis was done in the first stage of the project and the result
are shown in Fig.~\ref{fig:dGammadq_Ds} already extrapolated to the continuum and
to $\sigma\to0$.
The extra difficulties will come from the fact we can not simulate directly at the
physical $b$-quark, but the decay rate will be computed
using a set of heavy quark mass as described in the sub-project 2 of section~\ref{sec:mb}
and then extrapolated to the physical $b$ mass.



\begin{figure}
  \centering
  \includegraphics[scale=0.8]{plots/final_DgammaDq2.pdf}
  \caption{Preliminary values of the differential decay rate of
  $D_s\to X\ell\bar\nu$.
  the contribution form the $cd$ channel (in red) is observed to be smaller than
  the contribution form the $cd$ channel and (in blue) mainly due to $V_{cs}>V_{cd}$.}
  \label{fig:dGammadq_Ds}
\end{figure}


\subsection{Sub-project 2: Bottom quark mass $m_b$}
\label{sec:mb}

In this sub-project we describe the determination of the $b$-quark
mass. We can extract the mass of the $B$ and $B_s$ pseudo-scalar mesons ($M_{B}$ and $M_{B_s}$)
from the right diagram of Figure~\ref{fig:4pt} representing the two-point function
\begin{equation}
  \langle B_{(s)}(\bm{0},t) B_{(s)}^\dagger({\bf 0},0)\rangle\xrightarrow{t>>a, (T-t)>>a}
  \frac{1}{2M_{B_{(s)}}}|\langle 0 | B_{(s)} | B_{(s)}\rangle|^2\left(e^{-M_{B_{(s)}} t}+e^{-M_{B_{(s)}}(T-t)}\right)
  \,
  \label{eq:Mb}
\end{equation}
with the operators
\[
B({\bf 0},t)=\frac{1}{L^3}\sum_{\bf x}\bar b\gamma_5 u({\bf x},t)\,,\qquad
B_s({\bf 0},t)=\frac{1}{L^3}\sum_{\bf x}\bar b\gamma_5 s({\bf x},t)\,.
\]
In quantities involving a (valence) $b$-quark, we expect lattice
artefacts of the order $(am_b)^2$, which can be significant even with
the lattice spacing values employed in this project. Therefore, it is
advisable not to work directly at the $b$-quark mass, but to apply the
ratio method: The $b$-quark mass is obtained as an interpolation
between quark masses in the charm region and the static limit.
Thus, we compute Eq.~(\ref{eq:Mb}) replacing the $b$ quark with six heavy quarks $h$
in the range of $m_h\in [m_c,3.5m_c]$
and for each we compute the ratio
\begin{equation}
  Q_m = \frac{M_{hs}}{M_{h\ell}^\gamma M_{cs}^{(1-\gamma)}}\,,
  \label{eq:ratio_Q}
\end{equation}
where $M_{hs}$ and $M_{hl}$ are the heavy-strange and heavy-light
pseudo-scalar masses, respectively, while $M_{cs}$ denotes
the mass of the pseudo-scalar meson made out of a charm
and a strange quark. The parameter $\gamma$ is a free parameter in the range $[0, 1)$.
The asymptotic behaviour of $Q_m$ can be computed using HQET reading
\begin{equation}
  \lim_{ m^{pole}_h\to \infty}
  \frac{M_{hs}}{( m^{pole}_h)^{(1-\gamma)} M_{h\ell}^\gamma}=\mbox{const.}\,,
  \label{eq:yHQFTlim}
\end{equation}
where $ m^{pole}_h$ is the pole mass of the heavy quark.
We then consider a sequence of heavy quark masses such that any two
successive masses have a common and fixed ratio i.e.
$ m_h^{(n)}=\lambda m_h^{(n-1)}, n=2,3,...$ and we construct the
following ratios at given lattice spacing $a$
\begin{equation}
  \begin{split}
    y_Q( m^{(n)}_h,a)&=\frac{Q_m( m_h^{(n)},a)}{Q_m( m_h^{(n-1)},a)}\cdot
    \left(\frac{ m_{h}^{(n)} \rho( m_{h}^{(n)})}{ m_{h}^{(n-1)}\rho( m_{h}^{(n-1)})}\right)^{(\gamma-1)}\\
    &=\lambda^{(\gamma-1)}\frac{Q_m( m_h^{(n)},a)}{Q_m( m_h^{(n-1)},a)}\cdot
    \left(\frac{ \rho( m_{h}^{(n)})}{\rho(
      m_{h}^{(n-1)})}\right)^{(\gamma-1)}\,,
  \end{split}
\end{equation}
where we have used the relation  $ m^{pole}_h= m_{h}^{} \rho( m_{h}^{})$
between the $\overline{\mbox{MS}}$ renormalised mass and the pole
mass, which is known perturbatively up to N$^3$LO~\cite{Chetyrkin:1999pq}.
For each pair of heavy quark masses, we carry out a continuum
extrapolation separately
\footnote{We do not need a chiral extrapolation since we are already at physical point.}
obtaining $y_Q( m^{(n)}_h)=y_Q( m^{(n)}_h,a=0)$.
In the continuum limit, the ratios $y_Q( m^{(n)}_h)$ can be described
in terms of the heavy mass $m_h$ as~\cite{ETM:2011zey}
\begin{equation}
  y_Q( m^{(n)}_h) = 1 + \frac{\eta_1}{ m_h}+ \frac{\eta_2}{ m_h^2}\,,
  \label{eq:fity}
\end{equation}
where $\eta_{1,2}$ are parameters to be fitted to the data and the limit in
Eq.~(\ref{eq:yHQFTlim}) has already been taken into account.
Finally, the $b$-quark mass can be computed with
\begin{equation}
  y_Q( m^{(2)}_h)y_Q( m^{(3)}_h)...y_Q( m^{(K+1)}_h)\frac{Q_m( m_h^{(1)},a)}{\lambda^{K(\gamma-1)}}
  \left(\frac{ \rho( m_{h}^{(K+1)})}{\rho( m_{h}^{(1)})}\right)^{(1-\gamma)}=Q_m( m_h^{(K+1)},a)
  \,,
\end{equation}
where the ratios $y_Q$ on the left-hand-side are evaluated using the
fit function Eq.~(\ref{eq:fity}) and the parameters $\lambda$, $K$ and
$ m_h^1$ chosen such that $Q_m( m_h^{(K+1)},a)$ matches 
the experimental value of the ratio $M_{B_s}/(M_{B}^\gamma M_{D_s}^{(1-\gamma)})$.


\endinput