# Weak nonleptonic decays of vector B-mesons

Mikhail A. Ivanov  Bogoliubov Laboratory of Theoretical Physics, Joint
Institute for Nuclear Research, Dubna, Russia  Zhomart Tyulemissov  Bogoliubov
Laboratory of Theoretical Physics, Joint Institute for Nuclear Research,
Dubna, Russia  The Institute of Nuclear Physics, Ministry of Energy of the
Republic of Kazakhstan, 050032 Almaty, Kazakhstan  Al-Farabi Kazakh National
University, 050040 Almaty, Kazakhstan  Akmaral Tyulemissova  Bogoliubov
Laboratory of Theoretical Physics, Joint Institute for Nuclear Research,
Dubna, Russia  The Institute of Nuclear Physics, Ministry of Energy of the
Republic of Kazakhstan, 050032 Almaty, Kazakhstan  Al-Farabi Kazakh National
University, 050040 Almaty, Kazakhstan

###### Abstract

We study the radiative and weak nonleptonic decays of vector B-mesons within
the covariant confined quark model (CCQM) developed in our previous papers.
First, we calculate the matrix elements and decays widths of the radiative
decays $B^{\ast}\to B\gamma$. The obtained results are compared with those
obtained in other approaches. Then we consider the nonleptonic decays
$B^{\ast}\to D^{\ast}V$ which proceed via tree-level quark diagrams. It is
shown that the analytical expressions for the amplitudes correspond to the
factorization approach. In the framework of our model we calculate the
leptonic decay constants and the form factors of the $B^{\ast}\to D^{\ast}(V)$
transitions in the entire physical region of the momentum transfer squared.
Finally, we calculate the two-body decay widths and compare our results with
other models.

##  I Introduction

In 2009, the Belle Belle:2008ezn  collaboration has reported on the
determination of the masses of the $B_{s}$ and $B_{s}^{\ast}$ mesons
$m_{B_{s}}=5364.4\pm 1.5$ MeV and $m_{B^{\ast}_{s}}=5416.4\pm 0.6$ MeV. The
LHCb LHCb:2012iuq  collaboration announced the discovery of a vector
$B^{\ast}$-meson with the mass $m(B^{\ast})=5324.26(41)$ MeV.

Since the $b(c,s,d,u)$ mesons cannot annihilate into gluons, the excited
states decay to the ground state via the cascade emission of photons or pion
pairs, leading to total widths that are less than a few hundred keV. The
difference between masses $m_{B^{\ast}}-m_{B}=45.21\pm 0.21$ MeV, a
$m_{B^{\ast}_{s}}-m_{B_{s}}=48.5^{+1.8}_{-1.1}$ MeV less than the mass of
lightest meson ($\pi$-meson), ($m_{\pi}=134.98$ MeV), as a result of which
these mesons cannot decay through a strong channel. Consequently, for vector
$B$-mesons, radiative decays will be dominant. As is known, weak nonleptonic
decays of $B^{\ast}$-mesons are always suppressed in comparison with their
electromagnetic decay LHCb:2012iuq . However, due to small cross sections,
these decays have not yet been detected experimentally. The situation can be
improved with the help of the LHC and Belle-II experiments, since the annual
integrated luminosity of Belle-II is expected to reach $\approx 13$ ab-1 and
this makes it possible to detect weak decays with branchings greater than
$\mathcal{O}(10^{-9})$. Moreover, the LHC experiment will also provide new
experimental data for weak decays of $B^{\ast}$-mesons, due to the large
production cross section for $b$-quarks.

A search for excited states of the $B_{c}^{\pm}$-meson has been started at
2014 when the ATLAS collaboration reported on the observation of a new state
ATLAS:2014lga  through its hadronic transition to the ground state
$B_{c}^{\pm}$. The mass of the observed state was found to be $6842(6)$ MeV.
The mass and decay of this state are consistent with expectations for the
second S-wave state of the ground $B_{c}$ state denoted as $B_{c}(2^{1}S_{0})$
or $B_{c}(2S)$.

Search for excited $B_{c}^{+}$ states was started at LHCb LHCb:2017rqe . In
2019, the CMS collaboration observed two states $B_{c}(2S)$ and
$B^{\ast}_{c}(2S)$ CMS:2019uhm  in the $B_{c}\pi^{+}\pi^{-}$ invariant mass
spectrum. They are separated in mass by $\approx 29$ MeV. The mass of the
$B_{c}(2S)$ is measured to be $6871.0\pm 1.6$ MeV.

The LHCb collaboration has reported on the observation of an excited $B_{c}$
state in the the $B_{c}\pi^{+}\pi^{-}$ invariant mass spectrum LHCb:2019bem .
The observed peak has a mass of $6841.2\pm 1.0$ MeV. It is consistent with
expectations of the $B_{c}^{\ast}(2^{3}S_{1})$ state. A second state is seen
with a mass of $6872.1\pm 1.5$ MeV, and is consistent with the
$B_{c}(2^{1}S_{0})$ state.

Radiative and hadronic heavy meson decays of the heavy vector
$B^{\ast}$-mesons have been evaluated using the Heavy Quark Effective Theory
and the Vector Meson Dominance hypothesis Colangelo:1993zq . It was found that
$\Gamma(B^{\ast\,+}\to B^{+}\gamma)=220(90)$ eV and $\Gamma(B^{\ast\,0}\to
B^{0}\gamma)=75(27)$ eV.

The radiative and hadronic decays of vector heavy mesons were analyzed within
the relativistic quark model with confined light quarks Ivanov:1994ji . The
following results were obtained for the value of the constituent bottom quark
mass $m_{b}=5$ GeV, $\Gamma(B^{\ast\,+}\to B^{+}\gamma)=401$ eV and
$\Gamma(B^{\ast\,0}\to B^{0}\gamma)=131$ eV.

The method of QCD sum rules in the presence of the external electromagnetic
field was used to analyze radiative decays of charmed or bottomed mesons
Zhu:1996qy . The calculated values of the decay widths were found to be
$\Gamma(B^{\ast\,+}\to B^{+}\gamma)=380(60)$ eV, $\Gamma(B^{\ast\,0}\to
B^{0}\gamma)=130(30)$ eV and $\Gamma(B_{s}^{\ast\,0}\to
B_{s}^{0}\gamma)=220(40)$ eV.

Radiative transitions in heavy mesons have been considered in a relativistic
quark model Goity:2000dk . The calculated values of the decay widths with the
model parameter $\kappa^{q}=0.55$) were found to be $\Gamma(B^{\ast\,+}\to
B^{+}\gamma)=740(88)$ eV, $\Gamma(B^{\ast\,0}\to B^{0}\gamma)=228(27)$ eV,
$\Gamma(B_{s}^{\ast\,0}\to B_{s}^{0}\gamma)=136(12)$ eV.

Radiative magnetic dipole decays of heavy-light vector mesons into
pseudoscalar mesons have been considered within the relativistic quark model
Ebert:2002xz . The results for the mixture of vector and scalar confining
potentials with the mixing parameter $\varepsilon=-1$ look as
$\Gamma(B^{\ast\,+}\to B^{+}\gamma)=190$ eV, $\Gamma(B^{\ast\,0}\to
B^{0}\gamma)=70$ eV, $\Gamma(B_{s}^{\ast\,0}\to B_{s}^{0}\gamma)=54$ eV.

Decay constants and radiative decays of heavy mesons in light-front quark
model Choi:2007se  were found to be $\Gamma(B^{\ast\,+}\to
B^{+}\gamma)=400(30)$ eV, $\Gamma(B^{\ast\,0}\to B^{0}\gamma)=130(10)$ eV,
$\Gamma(B_{s}^{\ast\,0}\to B_{s}^{0}\gamma)=68(17)$ eV.

In this paper Chang:2019xtj , the light-front quark model (LFQM) was employed
to evaluate the decay widths: $\Gamma(B^{\ast\,+}\to B^{+}\gamma)=349(18)$ eV,
$\Gamma(B^{\ast\,0}\to B^{0}\gamma)=116(6)$ eV, $\Gamma(B_{s}^{\ast\,0}\to
B_{s}^{0}\gamma)=84(10)$ eV.

The vector $B^{\ast}_{c}$ meson physics has attracted theorists from all over
the world since 1994. First attempt to study $B_{c}^{\ast}$ meson was made in
the framework of nonrelativistic quarkonium quantum mechanics by using the
QCD-motivated potential Eichten:1994gt . It was found that
$\Gamma(B_{c}^{\ast\,+}\to B_{c}^{+}\gamma)=135$ eV. Later on the authors
updated this study and published the new work Eichten:2019gig .

In the framework of potential models for heavy quarkonium the mass spectrum
for the system $(\bar{b}c$) was considered Gershtein:1994dxw . Spin-dependent
splittings, taking into account a change of a constant for effective Coulomb
interaction between the quarks, and widths of radiative transitions between
the $(\bar{b}c$) levels were calculated $\Gamma(B_{c}^{\ast\,+}\to
B_{c}^{+}\gamma)=60$ eV.

In the paper Ebert:2002xz , the decay width was found to be
$\Gamma(B_{c}^{\ast\,+}\to B_{c}^{+}\gamma)=33$ eV.

The homogeneous bag model was employed in Ref. Liu:2022bdq  to calculate the
width of the radiative decay $\Gamma(B_{c}^{\ast\,+}\to
B_{c}^{+}\gamma)=53(3)$ eV.

The spectrum of heavy mesons including the excited states was treated in the
framework of the heavy quark effective theory in Ref. Zeng:1994vj .

The $B_{c}$ spectroscopy was investigated in a quantum-chromodynamic potential
model by using a quantum-chromodynamic potential model Gupta:1995ps , within
the lattice Non-Relativistic QCD (NRQCD) Davies:1996gi ,

Phenomenological predictions of the properties of the $B_{c}$ system have been
done in Ref. Fulcher:1998ka  by using Richardson’s potential. In particular,
the width of the radiative decay of $B_{c}^{\ast}$ was found to be
$\Gamma(B_{c}^{\ast\,+}\to B_{c}^{+}\gamma)=59$ eV.

The properties of heavy quarkonia and $B_{c}$ mesons have been studied in the
relativistic quark model Ebert:2002pp .

In Ref. Godfrey:2004ya  the spectrum of the charm-beauty mesons and their
radiative decays were studied by using the relativized quark model
Godfrey:2004ya . It was found that $\Gamma(B_{c}^{\ast\,+}\to
B_{c}^{+}\gamma)=80$ eV.

The observation possibility of $B_{c}$ excitations at LHC was discussed in
Ref. Berezhnoy:2013sla .

So far no experimental measurement of the vector $B^{\ast}_{c}$-meson mass
available. However there are a few reliable calculations made in the Lattice
QCD. In the paper Gregory:2009hq  the prediction was done to be
$m(B^{*}_{c})=6330(7)(2)(6)$ MeV by using the Highly Improved Staggered Quark
formalism to handle charm, strange and light valence quarks in full lattice
QCD, and NRQCD.

The improved results for the $B$ and $D$ meson spectrum from lattice QCD
including the effect of $u/d$, $s$ and $c$ quarks in the sea were presented in
Ref. Dowdall:2012ab . It was found that $m(B^{*}_{c})=6278(9)$ MeV.

Precise predictions of charmed-bottom hadrons from lattice QCD have been
published in Ref. Mathur:2018epb , in particularly,
$m(B_{c}^{\ast})=6331(4)(6)$ MeV.

In the present work we investigate both radiative and weak nonleptonic decays
of $B_{(u,d,s,c)}^{\ast}$ mesons. We use the covariant confined quark model
(CCQM) developed in our previous papers for calculation of the relevant form
factors, branching fractions and decay widths. Our paper is organized as
follows. In Sec. II we give short sketch to the model and discuss its basic
aspects. In Sec. III, we present the detailed calculation of the radiative
decays of vector mesons $B^{\ast}\to B\gamma$ in the framework of our
approach. Then we give the numerical results for calculated decay widths and
compare their values with those obtained in other approaches. In Sec. IV we
study the weak nonleptonic decays $B^{\ast}\to D^{\ast}V$ where
$V=\rho,K^{\ast},D^{\ast},D_{s}^{\ast})$. Since we consider the decays which
proceed via tree-level quark diagrams, the matrix elements of two-body decays
are factorized into the leptonic decays and the weak meson-meson transition.
We calculate the relevant leptonic decay constants and the form factors of
those meson-meson transitions in the framework of the CCQM. Finally, we
compute the two-body decay widths by using the calculated quantities and
compare the obtained results with other approaches. At the end, we make a
brief summary of our main results in Sec. V.

##  II Basic aspects of the covariant confined quark model

The covariant confined quark model is an effective quantum field approach to
hadronic interactions based on an interaction Lagrangian of hadrons
interacting with their constituent quarks Branz:2009cd . The coupling strength
of the hadrons with the constituent quarks is determined by the so-called
compositeness condition $Z_{H}=0$ Salam:1962ap ; Weinberg:1962hj  where
$Z_{H}$ is the wave function renormalization constant of the hadron. Matrix
elements are generated by a set of quark loop diagrams. The ultraviolet
divergences of the quark loops are regularized by including the hadron-quark
vertex functions which, in addition, describe finite size effects due to the
non-pointlike structure of hadrons. By using Schwinger’s
$\alpha$-representation for each local quark propagator and integrating out
the loop momenta, one can write the resulting matrix element expression as an
integral which includes integrations over a simplex of the $\alpha$-parameters
and an integration over a generalized the Fock-Schwinger proper time. By
introducing an infrared cutoff on the upper limit of the proper time one can
avoid the appearance of singularities in any matrix element. The new infrared
cutoff parameter $\lambda$ will be taken to have a common value for all
processes. The CCQM contains only a few model parameters: the light and heavy
constituent quark masses, the size parameters that describe the size of the
distribution of the constituent quarks inside the hadron and unified parameter
$\lambda$. The model parameters are determined by a fit to available
experimental data.

The CCQM was successfully applied for description of both light and heavy
hadron exclusive decays. In particularly, the wide range of the decays of
b-hadrons ($B$, $B_{s}$, $B_{c}$ and $\Lambda_{b}$) have been researched and
described Ivanov:2006ni ; Ivanov:2000aj ; Faessler:2002ut ; Gutsche:2015mxa ;
Ivanov:2005fd ; Ivanov:2016qtw ; Tran:2018kuv ; Ivanov:2002un ;
Issadykov:2018myx ; Dubnicka:2017job ; Gutsche:2018utw ; Ivanov:2019nqd .

In this paper we are going to start with an application of the CCQM to
radiative and nonleptonic decays of the vector excited states $B^{\ast}$. An
interaction Lagrangian of $B$-meson which consists from $\bar{q}b$ quarks
($q=u,d,s,c$) are written as

${\mathcal{L}}_{\rm int}=g_{B}\,B(x)J(x)+H.c.$ (1)

The local interpolating current has a form $J=\bar{q}\Gamma b$ where
$\Gamma=i\gamma_{5}$ for a pseudoscalar meson with spin $S=0$ and
$\Gamma=\gamma^{\mu}$ for a vector meson with spin $S=1$. The Lorentz index
$\mu$ is contracting with the corresponding index of a vector state. In the
CCQM the nonlocal interpolating currents are used for accounting the internal
structure of a hadron. In our case they are written as

$J(x)=\int\\!\\!dx_{1}\\!\\!\int\\!\\!dx_{2}F_{B}(x,x_{1},x_{2})\,\bar{q}_{a}(x_{1})\,\Gamma\,b_{a}(x_{2})$
(2)

where $a=1,2,3$ is the color index. The vertex function $F_{B}(x,x_{1},x_{2})$
characterizes the finite size of the meson. To satisfy translational
invariance the vertex function has to obey the identity
$F_{B}(x+a,x_{1}+a,x_{2}+a)\,=\,F_{B}(x,x_{1},x_{2})$ for any given four-
vector $a\,$. We employ a specific form for the vertex function which
satisfies the translation invariance. One has

$F_{B}(x,x_{1},x_{2})\,=\,\delta^{(4)}\left(x-\sum\limits_{i=1}^{2}w_{i}x_{i}\right)\;\Phi_{B}\biggl{(}(x_{1}-x_{2})^{2}\biggr{)}$
(3)

where $\Phi_{B}$ is the correlation function of the two constituent quarks
with masses $m_{q_{1}}$ and $m_{q_{2}}$. The ratios of the quark masses
$w_{i}$ are defined as

$w_{q_{1}}=\frac{m_{q_{1}}}{m_{q_{1}}+m_{q_{2}}},\quad
w_{q_{2}}=\frac{m_{q_{2}}}{m_{q_{1}}+m_{q_{2}}},\quad w_{1}+w_{2}=1.$ (4)

We choose a simple Gaussian form for the Fourier transform of vertex function
$\widetilde{\Phi}_{B}(-k^{2})=\exp(k^{2}/\Lambda_{B}^{2})$. The parameter
$\Lambda_{B}$ characterizes the size of the meson. Since $k^{2}$ turns into
$-k^{2}_{E}$ in Euclidean space the form $\widetilde{\Phi}_{B}(k^{2}_{E})$ has
the appropriate fall–off behavior in the Euclidean region.

The coupling constant $g_{B}$ in Eq. (1) is determined by the so-called
compositeness condition. The compositeness condition requires that the
renormalization constant $Z_{B}$ of the elementary meson field $B(x)$ is set
to zero, i.e.

$Z_{B}=1-\widetilde{\Pi}^{\prime}_{B}(p^{2})=0,\qquad(p^{2}=m^{2}_{B})$ (5)

where $\Pi^{\prime}_{B}(p^{2})$ is the derivative of the mass function.

S-matrix elements are described by the quark-loop diagrams which are the
convolution of the vertex functions and quark propagators. In the evaluation
of the quark-loop diagrams we use the local Dirac propagator

$S_{q}(k)=\frac{1}{m_{q}-\not\\!k-i\epsilon}=\frac{m_{q}+\not\\!k}{m^{2}_{q}-k^{2}-i\epsilon}$
(6)

with an effective constituent quark mass $m_{q}$.

The meson functions in the case of the pseudoscalar and vector meson are
written as

$\displaystyle\widetilde{\Pi}_{P}(p^{2})$ $\displaystyle=$ $\displaystyle
N_{c}g_{P}^{2}\int\frac{d^{4}k}{(2\pi)^{4}i}\widetilde{\Phi}^{2}_{P}(-k^{2})\mbox{\rm{tr}}\Big{(}\gamma^{5}S_{1}(k+w_{1}p)\gamma^{5}S_{2}(k-w_{2}p)\Big{)},$
(7) $\displaystyle\widetilde{\Pi}^{\mu\nu}_{V}(p^{2})$ $\displaystyle=$
$\displaystyle
N_{c}g_{V}^{2}\int\frac{d^{4}k}{(2\pi)^{4}i}\widetilde{\Phi}^{2}_{V}(-k^{2})\mbox{\rm{tr}}\Big{(}\gamma^{\mu}S_{1}(k+w_{1}p)\gamma^{\nu}S_{2}(k-w_{2}p)\Big{)}$
(8) $\displaystyle=$ $\displaystyle
g^{\mu\nu}\widetilde{\Pi}_{V}(p^{2})+p^{\mu}p^{\nu}\widetilde{\Pi}^{\parallel}_{V}(p^{2}).$

Here $N_{c}=3$ is the number of colors. Since the vector meson is on its mass-
shell $\epsilon_{V}\cdot p=0$ we need to keep the part
$\widetilde{\Pi}_{V}(p^{2})$. Substituting the derivative of the mass
functions into Eq. (5) one can determine the coupling constant $g_{B}$ as a
function of other model parameters. The loop integrations in Eqs. (7) and (8)
proceed by using the Fock-Schwinger representation of quark propagators

$S_{q}(k+wp)=\frac{1}{m_{q}-\not\\!k-w\not\\!p}=(m_{q}+\not\\!k+w\not\\!p)\int\limits_{0}^{\infty}\\!\\!d\alpha\,e^{-\alpha[m_{q}^{2}-(k+wp)^{2}]}.$
(9)

In the obtained integrals over the Fock-Schwinger parameters
$0\leq\alpha_{i}<\infty$ we introduce an additional integration over the
proper time which converts the set of Fock-Schwinger parameters into a
simplex. In general case one has

$\prod\limits_{i=1}^{n}\int\limits_{0}^{\infty}\\!\\!d\alpha_{i}f(\alpha_{1},\ldots,\alpha_{n})=\int\limits_{0}^{\infty}\\!\\!dtt^{n-1}\prod\limits_{i=1}^{n}\int\\!\\!d\alpha_{i}\delta\left(1-\sum\limits_{i=1}^{n}\alpha_{i}\right)f(t\alpha_{1},\ldots,t\alpha_{n}).$
(10)

Finally, we cut the integration over the proper time at the upper limit by
introducing an infrared cutoff $\lambda$.

$\int\limits_{0}^{\infty}dt(\ldots)\to\int\limits_{0}^{1/\lambda^{2}}dt(\ldots),$
(11)

This procedure allows us to remove all possible thresholds present in the
initial quark diagram. Thus the infrared cutoff parameter $\lambda$
effectively guarantees the confinement of quarks within hadrons. This method
is quite general and can be used for diagrams with an arbitrary number of
loops and propagators. In the CCQM the infrared cutoff parameter $\lambda$ is
taken to be universal for all physical processes.

The model parameters are determined by fitting calculated quantities of basic
processes to available experimental data or lattice simulations (for details,
see Ref. Branz:2009cd . The numerical values of the constituent quark masses
and the parameter $\lambda$ are given by (all in GeV)

$\begin{array}[]{cccccc}m_{u}&m_{s}&m_{c}&m_{b}&\lambda\\\ \hline\cr\ \
0.241&\ \ 0.428&\ \ 1.67&\ \ 5.04&\ \ 0.181&\text{GeV}\\\
\hline\cr\end{array}.$ (12)

##  III Radiative decays of vector B-mesons

###  III.1 $V\to P\gamma$: theoretical calculation of matrix elements and
decay widths

As the first step we consider the radiative decays of vector
$B^{\ast}$-mesons. In addition to the strong interaction Lagrangian Eq.(1) we
need the part describing the electromagnetic interactions. The free Lagrangian
of quarks is gauged in the standard manner by using minimal substitution which
gives

$\mathcal{L}^{\rm em}_{\rm int}(x)=e\,A_{\mu}(x)\,J^{\mu}_{\rm em}(x),\qquad
J^{\mu}_{\rm
em}(x)=e_{b}\,\bar{b}(x)\gamma^{\mu}b(x)+e_{q}\,\bar{q}(x)\gamma^{\mu}q(x)$
(13)

where $e_{b}$ and $e_{q}$ are the quark charges in units of the positron
charge. The radiative decays of a vector mesons into a pseudoscalar meson and
photon $V\to P\gamma$ are described by the Feynman diagrams shown in Fig. 1.

Figure 1: Feynman diagrams describing the radiative decays of a vector meson.

One has to note that there is an additional piece in the Lagrangian related to
the gauging nonlocal interactions of hadrons with their constituents
Branz:2009cd . This piece gives the additional contributions to the
electromagnetic processes. However, they are identically zero for the process
$V\to P\gamma$ due to its anomalous nature.

The matrix element of the process $V\to P\gamma$ is written down

$\displaystyle
M_{VP\gamma}(p;p^{\prime},q)=eg_{V}g_{P}\epsilon^{V}_{\nu}(p)\epsilon^{\gamma}_{\mu}(q)\int\\!\\!dx\\!\\!\int\\!\\!dy\\!\\!\int\\!\\!dz\,e^{-ipx+ip^{\prime}y+iqz}\langle\,T\\{\bar{J}_{V}^{\nu}(x)J^{\mu}_{\rm
em}(z)J_{P}(y)\\}\rangle_{0}.$ (14)

Using the Fourier transforms of the quark currents, we come to the final
result

$\displaystyle M_{VP\gamma}(p;p^{\prime},q)$ $\displaystyle=$
$\displaystyle(2\pi)^{4}i\,\delta(p-p^{\prime}-q)M(p,p^{\prime}),$
$\displaystyle M(p,p^{\prime})$ $\displaystyle=$
$\displaystyle(-3i)eg_{V}g_{P}\epsilon^{V}_{\nu}(p)\epsilon^{\gamma}_{\mu}(q)\,\left(e_{b}M^{\mu\nu}_{b}+e_{q}M^{\mu\nu}_{q}\right)$
$\displaystyle M^{\mu\nu}_{b}$ $\displaystyle=$
$\displaystyle\int\\!\\!\frac{dk}{(2\pi)^{4}i}\widetilde{\Phi}_{V}(-\ell_{1}^{2})\widetilde{\Phi}_{P}(-\ell_{2}^{2})\mbox{\rm{tr}}\left[S_{q}(k)\gamma^{\nu}S_{b}(k-p)\gamma^{\mu}S_{b}(k-p^{\prime})\gamma^{5}\right]$
$\displaystyle M^{\mu\nu}_{q}$ $\displaystyle=$
$\displaystyle\int\\!\\!\frac{dk}{(2\pi)^{4}i}\widetilde{\Phi}_{V}(-\ell_{3}^{2})\widetilde{\Phi}_{P}(-\ell_{4}^{2})\mbox{\rm{tr}}\left[S_{q}(k+p^{\prime})\gamma^{\mu}S_{q}(k+p)\gamma^{\nu}S_{b}(k)\gamma^{5}\right]$
(15)

where $\ell_{1}=k-w_{2}\,p$, $\ell_{2}=k-w_{2}\,p^{\prime}$ and
$\ell_{3}=k+w_{1}\,p$, $\ell_{2}=k+w_{1}\,p^{\prime}$. The ratios of quark
masses are defined by Eq. (4. Now one has $m_{q_{1}}=m_{b}$ and
$m_{q_{2}}=m_{q}$ with $q=u,d,s$. By using the technique of calculations
outlined in Sec. II and taking into account the transversality conditions
$\epsilon^{\gamma}_{\mu}(q)q^{\mu}=0$ and $\epsilon^{V}_{\nu}(p)p^{\nu}=0$ one
can arrives at the standard form of matrix element

$M(p,p^{\prime})=e\,g_{VP\gamma}\,\varepsilon^{pq\mu\nu}\epsilon^{\gamma}_{\mu}(q)\epsilon^{V}_{\nu}(p),$
(16)

where
$g_{VP\gamma}=e_{b}I_{b}(m^{2}_{V},m^{2}_{P})+e_{q}I_{q}(m^{2}_{V},m^{2}_{P})$
is radiative decay constant. The quantities $I_{b,q}$ are defined by the two-
fold integrals which are calculated numerically. The electromagnetic decay
width is written as

$\Gamma(V\to
P+\gamma)=\frac{\alpha}{24}m_{V}^{3}\left(1-\frac{m_{P}^{2}}{m_{V}^{2}}\right)^{3}g_{VP\gamma}^{2}\,.$
(17)

###  III.2 $V\to P\gamma$: numerical results

The masses of the $B^{\ast}$-meson family is not well established yet. We
collect the experimentally measured masses of B-meson family in Table 1

Table 1: Masses of $B$-meson family taken from PDG ParticleDataGroup:2022pth  (in MeV). $B^{\pm}$ | $B^{0}$ | $B^{0}_{s}$ | $B^{+}_{c}$ | $B^{\ast}$ | $B^{\ast}_{s}$
---|---|---|---|---|---
5279.25(26) |  5279.63(20) |  5366.91(11) |  6274.47(32) |  5324.71(21) |  5415.8(1.5)

Since the $B_{c}^{\ast}$ or $B_{c}(1^{3}S_{1})$ state has not been observed
yet, we will use the value $m(B_{c}^{\ast})~{}=~{}6331(4)(6)$ MeV obtained
from lattice QCD Mathur:2018epb . The values of size parameters are taken from
our previous papers Dubnicka:2017job ; Gutsche:2018utw ; Ivanov:2019nqd .
Their numerical values are given in Table 2.

Table 2: Size parameters of $B$\- meson family (in GeV). $\Lambda_{B}$ | $\Lambda_{B_{s}}$ | $\Lambda_{B_{c}}$ | $\Lambda_{B^{\ast}}$ | $\Lambda_{B^{\ast}_{s}}$ | $\Lambda_{B^{\ast}_{c}}$
---|---|---|---|---|---
1.96 |  2.05 |  2.73 |  1.72 |  1.71 |  2.42

The numerical results for the radiative decay constants are shown in Eq. (18).

$\begin{array}[]{cccccc}\hline\cr
g_{B^{\ast\,+}B^{+}\gamma}&g_{B^{\ast\,0}B^{0}\gamma}&g_{B_{s}^{\ast\,0}B_{s}^{0}\gamma}&g_{B_{c}^{\ast\,+}B_{c}^{+}\gamma}&\\\
\hline\cr\ \ 1.28(13)&\ \ -0.76(8)&\ \ -0.57(6)&\ \ 0.28(3)&{\rm GeV^{-1}}\\\
\hline\cr\end{array}$ (18)

The numerical values for the radiative decay widths are given in Tables 3 and
4. Our findings are compared with the results obtained in other approaches.

Now, we briefly discuss some error estimates within our model. The CCQM
consists of several free parameters: the constituent quark masses $m_{q}$, the
hadron size parameters $\Lambda_{H}$ and the universal infrared cutoff
parameter $\lambda$. These parameters are determined by minimizing the
functional $\chi^{2}=\sum\limits_{i}\frac{(y_{i}^{\rm expt}-y_{i}^{\rm
theor})^{2}}{\sigma^{2}_{i}}$ where $\sigma_{i}$ is the experimental
uncertainty. If $\sigma$ is too small then we take its value of 10$\%$.
Besides, we have observed that the errors of the fitted parameters are of the
order of 10$\%$. Thus, the theoretical error of the CCQM is estimated to be of
the order of 10$\%$ at the level of matrix elements and the order of 15$\%$ at
the level of widths.

Table 3: Widths of $B^{\ast}\to B\gamma$ and $B_{s}^{\ast}\to B_{s}\gamma$ decays (in eV). Mode |  CCQM |  Chang:2019xtj  |  Choi:2007se  |  Ebert:2002xz  |  Goity:2000dk  |  Zhu:1996qy  |  Ivanov:1994ji  |  Colangelo:1993zq
---|---|---|---|---|---|---|---|---
$B^{\ast+}\to B^{+}\gamma$ |  372(56) |  349(18) |  400(30) |  190 |  740(88) |  380(60) |  401 |  220(90)
$B^{\ast\,0}\to B^{0}\gamma$ | 126(19) | 116(6) | 130(10) | 70 | 228(27) | 130(30) | 131 | 75(27)
$B_{s}^{\ast\,0}\to B_{s}^{0}\gamma$ | 90(14) | 84(10) | 68(17) | 54 | 136(12) | 220(40) |  |
Table 4: Widths of $B_{c}^{\ast}\to B_{c}\gamma$ decay (in eV). Mode |  CCQM |  Liu:2022bdq  |  Godfrey:2004ya  |  Fulcher:1998ka  |  Ebert:2002xz  |  Gershtein:1994dxw  |  Eichten:1994gt
---|---|---|---|---|---|---|---
$B_{c}^{\ast\,+}\to B_{c}^{+}\gamma$ |  33(5) |  53(3) |  80 |  59 |  33 |  60 |  135

As can be seen from the Table 3, there is suppression of the neutral mode
$B^{\ast\,0}\to B^{0}\gamma$ compared to the charged mode $B^{\ast+}\to
B^{+}\gamma$. The ratio of those widths is written as

$\frac{\Gamma(B^{\ast\,0}\to B^{0}\gamma)}{\Gamma(B^{\ast+}\to
B^{+}\gamma)}\sim\frac{(I_{b}+I_{d})^{2}}{{(I_{b}-2I_{u})^{2}}}\approx 0.38.$
(19)

Note that in the heavy quark limit $m_{b}\to\infty$ the integral $I_{b}$ is
suppressed as $1/m_{b}$ compare with $I_{q}$ (q=u,d). Therefore our result for
the ratio in Eq. (19) is not far from the the heavy quark limit $0.25$. This
confirms the observation that the heavy quark limit is quite reliable in the
case of b-quark.

##  IV Nonleptonic decays

The effective Hamiltonian describing the nonleptonic decays
$B^{\ast}_{u,d,s}\to D^{\ast}_{u,d,s}\,+\,V$ where
$V=D^{\ast}_{u,d,s},K^{\ast}_{u,d},\rho$ is written down

$\mbox{${\cal H}_{\rm
eff}$}=-\frac{G_{F}}{\sqrt{2}}V^{\ast}_{cb}V_{q_{1}q_{2}}\Big{(}C_{2}\left(\bar{c}_{a}O^{\mu}b_{a}\right)\left(\bar{q}_{1b}O_{\mu}q_{2b}\right)+C_{1}\left(\bar{c}_{a}O^{\mu}b_{b}\right)\left(\bar{q}_{1b}O_{\mu}q_{2a}\right)\Big{)}$
(20)

where $G_{F}$ is the Fermi constant, $V_{cb}$ and $V_{q_{1}q_{2}}$ are the
matrix elements of the CKM-matrix, $C_{1}$ and $C_{2}$ is Wilson coefficients
and $O^{\mu}=\gamma^{\mu}(1-\gamma_{5})$ is the weak matrix with the left
chirality. The Wilson coefficients $C_{2}$ (leading order) and $C_{1}$
(subleading order) are taken at the scale of $b$-quark mass from Ref.
Descotes-Genon:2013vna  (see, Table 3). One has

$C_{1}=-0.2632\qquad\text{and}\qquad C_{2}=1.0111.$ (21)

One has to emphasize that if QCD is neglected, than $C_{1}=0$ and $C_{2}=1$
Buchalla:1995vs ; Buras:1998raa .

We consider the nonleptonic decays $B^{\ast}\to D^{\ast}V$ which proceed via
tree-level quark diagrams shown in Fig. 2. One has to note that the Wilson
coefficients will appear in combinations $a_{1}=C_{2}+\xi C_{1}$ for the
charged emitted meson or $a_{2}=C_{1}+\xi C_{2}$. for the neutral emitted
meson. The color suppression factor $\xi=1/N_{c}$ will be used to be equal to
zero in calculation that corresponds to the large-$N_{c}$ limit $\xi=0$. One
has to remind that an approximation is widely used in the phenomenological
studies of two-body nonleptonic decays because in the case of $N_{c}=3$ the
combination $a_{2}\approx 0.074$, i.e. significantly suppressed.

Figure 2: Tree-level quark diagrams

The Feynman diagram describing the nonleptonic decays $B^{\ast}\to M_{1}M_{2}$
within the CCQM are shown in Fig. 3.

Figure 3: Diagram describing the nonleptonic decay $B^{\ast}\to M_{1}M_{2}$.

The matrix element of the nonleptonic $B^{\ast}\to M_{1}(p_{1})M_{2}(p_{2})$
decays are given by

$\displaystyle M(B^{\ast}(p)\to M_{1}(p_{1})M_{2}(p_{2}))$ $\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}V_{CKM}a_{W}\,m_{2}f_{M_{2}}\epsilon_{2\,\mu}\,$
(22) $\displaystyle\times$
$\displaystyle\Big{\langle}M_{1}\left(p_{1},\epsilon_{1\,\beta}\right)|\bar{q}_{1}O^{\mu}q_{2}|B^{\ast}(p,\epsilon^{\ast}_{\alpha})\Big{\rangle}$

where $f_{M_{2}}$ is the leptonic decay constant of the vector meson $M_{2}$.
Here $V_{CKM}$ and $a_{W}$ are the relevant CKM-matrix elements and the Wilson
coefficients, respectively. Note that the Eq. (22) is equivalent to the
factorization hypothesis.

The leptonic decay constant of the vector meson $M_{2}$ which consists from
$q_{1}$ and $q_{2}$ quarks is defined by

$N_{c}\,g_{M_{2}}\\!\int\\!\\!\frac{d^{4}k}{(2\pi)^{4}i}\,\widetilde{\Phi}_{M_{2}}(-k^{2})\,\mbox{\rm{tr}}\biggl{[}O^{\,\mu}S_{1}(k+w_{1}p_{2})\not\\!\epsilon_{2}S_{2}(k-w_{2}p_{2})\biggr{]}=m_{2}f_{M_{2}}\epsilon_{2}^{\mu}.$
(23)

The meson is taken on its mass-shell, i.e. $p_{2}^{2}=m^{2}_{2}$ and
$\epsilon_{2}\cdot p_{2}=0$.

The matrix elements of the weak $B^{\ast}-M_{1}$ transition is written as

$\displaystyle\langle
M_{1}(p_{1})|\bar{q}_{1}O^{\mu}q_{2}|B^{\ast}(p)\rangle=$
$\displaystyle=N_{c}\,g_{B^{\ast}}\,g_{M_{1}}\epsilon_{\alpha}(p)\epsilon^{\ast}_{1\,\beta}(p_{1})\\!\\!\int\\!\\!\frac{d^{4}k}{(2\pi)^{4}i}\,\widetilde{\Phi}_{B^{\ast}}\Big{(}-(k+w_{13}p_{1})^{2}\Big{)}\,\widetilde{\Phi}_{M_{1}}\Big{(}-(k+w_{23}p_{2})^{2}\Big{)}$
$\displaystyle\times\mbox{\rm{tr}}\biggl{[}O^{\,\mu}\,S_{1}(k+p_{1})\,\gamma^{\alpha}\,S_{3}(k)\,\gamma^{\beta}\,S_{2}(k+p_{2})\biggr{]}.$
(24)

All particles are on their mass shell: $p^{2}=m^{2}_{B^{\ast}}$,
$\epsilon_{\alpha}\,p^{\alpha}=0$, and $p_{1}^{2}=m^{2}_{1}$,
$\epsilon_{1\,\beta}\,p_{1}^{\beta}=0$. Altogether there are three flavors of
quarks involved in this process. We therefore introduce a notation with two
subscripts $w_{ij}=m_{q_{j}}/(m_{q_{i}}+m_{q_{j}})$ $(i,j=1,2,3)$ such that
$w_{ij}+w_{ji}=1$. The loop integrations in the Eqs. (23) and(24) are
performed by using technique given in Sec. II. We finalize by two- and three-
fold integrals in equations Eqs. (23) and (24), respectively. We calculate
them by using FORTRAN codes with NAG library.

The matrix elements of the weak $B^{\ast}-M_{1}$ transition can be expressed
in terms of six vector form factors $V_{i}$ and four axial form factors
$A_{i}$. One has

$\displaystyle\langle M_{1}(p_{1})|\bar{q}_{1}$ $\displaystyle
O^{\mu}q_{2}|B^{\ast}(p)\rangle=\epsilon_{\alpha}\epsilon^{\ast}_{1\,\beta}\times\Big{[}g^{\alpha\beta}\left(-P^{\mu}V_{1}(q^{2})+q^{\mu}V_{2}(q^{2})\right)$
$\displaystyle+\frac{q^{\alpha}q^{\beta}}{M^{2}-m^{2}_{1}}\left(P^{\mu}V_{3}(q^{2})-q^{\mu}V_{4}(q^{2})\right)-q^{\alpha}g^{\mu\beta}V_{5}(q^{2})+q^{\beta}g^{\mu\alpha}V_{6}(q^{2})\Big{]}$
$\displaystyle+\epsilon_{\alpha}\epsilon^{\ast}_{1\,\beta}\times\Big{[}-i\varepsilon^{\mu\nu\alpha\beta}\left(P^{\nu}A_{1}(q^{2})-q^{\nu}A_{2}(q^{2})\right)$
$\displaystyle+\frac{i\varepsilon^{\mu\nu
qP}}{M^{2}-m^{2}_{1}}\left(g^{\nu\alpha}q^{\beta}A_{3}(q^{2})-g^{\nu\beta}q^{\alpha}A_{4}(q^{2})\right)\Big{]}$
(25)

where $P=p+p_{1}$, $q=p-p_{1}=p_{2}$. The full matrix element of the
nonleptonic decay $B^{\ast}(p)\to M_{1}(p_{1})M_{2}(p_{2})$ is obtained by
contraction of the above two expressions from Eqs.(23) and (25). Keeping in
mind that $\epsilon^{\ast}_{2}\cdot q=(\epsilon^{\ast}_{2}\cdot p_{2})=0$ one
gets that the two vector form factors $V_{2}$ and $V_{4}$ do not contribute to
the full matrix element.

The values of the size parameters of vector mesons are taken from Ref.
Dubnicka:2017job ; Gutsche:2018utw ; Ivanov:2019nqd  and displayed in Table 5.

Table 5: Size parameters of vector mesons (in GeV) $\Lambda_{\rho}$ |  $\Lambda_{K^{\ast}}$ |  $\Lambda_{D^{\ast}}$ |  $\Lambda_{D^{\ast}_{s}}$ |  $\Lambda_{B^{\ast}}$ |  $\Lambda_{B^{\ast}_{s}}$
---|---|---|---|---|---
0.61 | 0.81 | 1.53 | 1.56 | 1.72 | 1.71

The calculated leptonic decay constants $f_{M_{2}}$ are shown in Table 6.

Table 6: Calculated leptonic decay constants $f_{M_{2}}$ (in MeV) |  CCQM |  Expt/Lat
---|---|---
$f_{\rho}$ | 218(22) |  $221(1)\quad$ ParticleDataGroup:2022pth
$f_{K^{\ast}}$ | 227(23) |  $217(7)\quad$ ParticleDataGroup:2022pth
$f_{D^{\ast}}$ | 246(25) |  $223.5(8.4)$ Lubicz:2017asp
$f_{D^{\ast}_{s}}$ | 273(27) |  $268.8(6.6)$ Lubicz:2017asp
$f_{B^{\ast}}$ | 185(19) |  $186.4(7.1)$ Lubicz:2017asp ; ETM:2016nbo
$f_{B^{\ast}_{s}}$ | 260(26) |  $223.1(5.6)$ Lubicz:2017asp ; ETM:2016nbo

We calculate the relevant transition form factors in the full kinematical
region of the transfered momentum squared $q^{2}$. The behavior of the form
factors on the $q^{2}$ are shown in Figs. 4-7.

Figure 4: Form factors of $B^{\ast}-\rho$ transition

Figure 5: Form factors of $B^{\ast}-K^{\ast}$ transition

Figure 6: Form factors of $B^{\ast}-D^{\ast}$ transition

Figure 7: Form factors of $B_{s}^{\ast}-D_{s}^{\ast}$ transition

The numerical results for the form factors are well approximated by a dipole
parametrization

$F(q^{2})=\frac{F(0)}{1-as+bs^{2}},\qquad s=\frac{q^{2}}{M^{2}}$ (26)

where $M$ is the mass of ingoing meson. The dipole approximation is quite
accurate. The error relative to the exact results is less than 1$\%$ over the
entire $q^{2}$ range.

Table 7: Parameters of dipole approximation

$B^{\ast}-\rho$ |  $A_{1}$ |  $A_{2}$ |  $A_{3}$ |  $A_{4}$ |  $V_{1}$ |  $V_{3}$ |  $V_{5}$ |  $V_{6}$
---|---|---|---|---|---|---|---|---
F(0) |  0.57 |  0.51 |  $-0.014$ |  $-0.38$ |  0.22 |  0.027 |  0.45 |  0.075
a | 0.66 | 0.58 | 2.08 | 1.42 | 1.61 | 2.13 | 1.59 | 1.55
b | $-0.26$ | $-0.33$ | 1.13 | 0.39 | 0.58 | 1.17 | 0.57 | 0.52

$B^{\ast}-K^{\ast}$ |  $A_{1}$ |  $A_{2}$ |  $A_{3}$ |  $A_{4}$ |  $V_{1}$ |  $V_{3}$ |  $V_{5}$ |  $V_{6}$
---|---|---|---|---|---|---|---|---
F(0) |  $0.62$ |  $0.51$ |  $-0.027$ |  $-0.35$ |  $0.31$ |  $0.038$ |  $0.62$ | $0.12$
a | $0.66$ | $0.59$ | $1.75$ | $1.34$ | $1.46$ | $1.92$ | $1.44$ | $1.42$
b | $-0.24$ | $-0.31$ | $0.77$ | $0.31$ | $0.44$ | $0.94$ | $0.42$ | $0.40$

$B^{\ast}-D^{\ast}$ |  $A_{1}$ |  $A_{2}$ |  $A_{3}$ |  $A_{4}$ |  $V_{1}$ |  $V_{3}$ |  $V_{5}$ |  $V_{6}\qquad$
---|---|---|---|---|---|---|---|---
F(0) |  0.85 |  0.41 |  $-0.14$ | $-0.41$ |  0.73 |  0.050 |  1.21 |  0.49
a | 0.57 | 0.56 | 1.02 | 1.01 | 0.98 | 1.29 | 0.97 | 0.95
b | $-0.14$ | $-0.15$ | 0.14 | 0.13 | $0.11$ | 0.36 | 0.10 | 0.089

$B_{s}^{\ast}-D_{s}^{\ast}$ |  $A_{1}$ |  $A_{2}$ |  $A_{3}$ |  $A_{4}$ |  $V_{1}$ |  $V_{3}$ |  $V_{5}$ |  $V_{6}\qquad$
---|---|---|---|---|---|---|---|---
F(0) |  0.78 |  0.38 |  $-0.11$ | $-0.34$ |  0.68 |  0.051 |  1.16 |  0.49
a | 0.71 | 0.69 | 1.16 | 1.11 | 1.10 | 1.36 | 1.08 | 1.07
b | $-0.13$ | $-0.16$ | 0.21 | 0.15 | $0.15$ | 0.38 | 0.15 | 0.13

Table 8: Form factors at origin $q^{2}=0$ compared with CLFQM Chang:2019xtj

$B^{\ast}-\rho$ |  CCQM |  CLFQM
---|---|---
$A_{1}$ | 0.57 | 0.27
$A_{2}$ | 0.51 | 0.25
$A_{3}$ | -0.014 | 0.07
$A_{4}$ | -0.38 | 0.06
$V_{1}$ | 0.22 | 0.28
$V_{3}$ | 0.027 | 0.11
$V_{5}$ | 0.45 | 0.60
$V_{6}$ | 0.075 | 0.14

$B^{\ast}-K^{\ast}$ |  CCQM |  CLFQM
---|---|---
$A_{1}$ | 0.62 | 0.33
$A_{2}$ | 0.51 | 0.27
$A_{3}$ | -0.027 | 0.07
$A_{4}$ | -0.35 | 0.07
$V_{1}$ | 0.31 | 0.33
$V_{3}$ | 0.038 | 0.11
$V_{5}$ | 0.62 | 0.68
$V_{6}$ | 0.12 | 0.16

$B^{\ast}-D^{\ast}$ |  CCQM |  CLFQM
---|---|---
$A_{1}$ | 0.85 | 0.66
$A_{2}$ | 0.41 | 0.35
$A_{3}$ | -0.14 | 0.07
$A_{4}$ | -0.41 | 0.08
$V_{1}$ | 0.73 | 0.67
$V_{3}$ | 0.050 | 0.13
$V_{5}$ | 1.21 | 1.17
$V_{6}$ | 0.49 | 0.48

$B^{\ast}_{s}\to D^{\ast}_{s}$ |  CCQM |  CLFQM
---|---|---
$A_{1}$ | 0.78 | 0.65
$A_{2}$ | 0.38 | 0.38
$A_{3}$ | -0.11 | 0.10
$A_{4}$ | -0.34 | 0.09
$V_{1}$ | 0.68 | 0.66
$V_{3}$ | 0.051 | 0.15
$V_{5}$ | 1.16 | 1.19
$V_{6}$ | 0.49 | 0.53

For convenience of calculation, we further introduce the helicity amplitudes
$H^{M_{1}M_{2}}_{\lambda_{1}\lambda_{2}}$ for the outgoing mesons $M_{1}$ and
$M_{2}$. We imply that the meson $M_{2}$ is the emitted meson whereas $M_{1}$
is the outgoing meson in the transition $B^{\ast}-M_{1}$. One has

$\displaystyle H^{M_{1}M_{2}}_{00}=f_{M_{2}}m_{2}$
$\displaystyle\Big{[}+\frac{|\mathbf{p_{2}}|\left(M^{2}+m_{1}^{2}-m_{2}^{2}\right)}{m_{1}m_{2}}V_{1}+\frac{2M^{2}|\mathbf{p_{2}}|^{3}}{\left(M^{2}-m_{1}^{2}\right)m_{1}m_{2}}V_{3}$
$\displaystyle\,\,-\frac{|\mathbf{p_{2}}|\left(M^{2}-m_{1}^{2}-m_{2}^{2}\right)}{2m_{1}m_{2}}V_{5}+\frac{|\mathbf{p_{2}}|\left(M^{2}-m_{1}^{2}+m_{2}^{2}\right)}{2m_{1}m_{2}}V_{6}\Big{]}\,,$
$\displaystyle H^{M_{1}M_{2}}_{++}=f_{M_{2}}m_{2}$
$\displaystyle\Big{[}+\frac{3M^{2}+m_{1}^{2}-m_{2}^{2}}{2M}A_{1}-\frac{M^{2}-m_{1}^{2}+m_{2}^{2}}{2M}A_{2}+\frac{2|\mathbf{p_{2}}|^{2}M}{M^{2}-m_{1}^{2}}A_{4}-|\mathbf{p_{2}}|V_{5}\Big{]}\,,$
$\displaystyle H^{M_{1}M_{2}}_{--}=f_{M_{2}}m_{2}$
$\displaystyle\Big{[}-\frac{3M^{2}+m_{1}^{2}-m_{2}^{2}}{2M}A_{1}+\frac{M^{2}-m_{1}^{2}+m_{2}^{2}}{2M}A_{2}-\frac{2|\mathbf{p_{2}}|^{2}M}{M^{2}-m_{1}^{2}}A_{4}-|\mathbf{p_{2}}|V_{5}\Big{]}\,,$
$\displaystyle H^{M_{1}M_{2}}_{+0}=f_{M_{2}}m_{2}$
$\displaystyle\Big{[}-\frac{M^{2}-m_{1}^{2}}{m_{2}}A_{1}+m_{2}A_{2}+\frac{2M|\mathbf{p_{2}}|}{m_{2}}V_{1}\Big{]}\,,$
$\displaystyle H^{M_{1}M_{2}}_{-0}=f_{M_{2}}m_{2}$
$\displaystyle\Big{[}+\frac{M^{2}-m_{1}^{2}}{m_{2}}A_{1}-m_{2}A_{2}+\frac{2M|\mathbf{p_{2}}|}{m_{2}}V_{1}\Big{]}\,,$
$\displaystyle H^{M_{1}M_{2}}_{0-}=f_{M_{2}}m_{2}$
$\displaystyle\Big{[}-\frac{M^{2}+3m_{1}^{2}-m_{2}^{2}}{2m_{1}}A_{1}+\frac{M^{2}-m_{1}^{2}-m_{2}^{2}}{2m_{1}}A_{2}-\frac{2|\mathbf{p_{2}}|^{2}M^{2}}{\left(M^{2}-m_{1}^{2}\right)m_{1}}A_{3}$
$\displaystyle\,\,-\frac{M|\mathbf{p_{2}}|}{m_{1}}V_{6}\Big{]}\,,$
$\displaystyle H^{M_{1}M_{2}}_{0+}=f_{M_{2}}m_{2}$
$\displaystyle\Big{[}+\frac{M^{2}+3m_{1}^{2}-m_{2}^{2}}{2m_{1}}A_{1}-\frac{M^{2}-m_{1}^{2}-m_{2}^{2}}{2m_{1}}A_{2}+\frac{2|\mathbf{p_{2}}|^{2}M^{2}}{\left(M^{2}-m_{1}^{2}\right)m_{1}}A_{3}$
$\displaystyle\,\,-\frac{M|\mathbf{p_{2}}|}{m_{1}}V_{6}\Big{]}\,.$

Here $|\mathbf{p_{2}}|=\lambda^{1/2}(M^{2},m_{1}^{2},m_{2}^{2})/2M$ is the
momentum of the daughter meson in the $B^{\ast}$-meson rest frame.

Finally, the amplitudes can be written in terms of helicity amplitudes. The
only two amplitudes contain the contributions from the diagrams with the
charged and neutral emitted mesons. They are written as

$\displaystyle M(B^{\ast-}\to D^{\ast 0}+K^{\ast-})$ $\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}\Big{(}a_{1}V_{cb}V^{\ast}_{us}H_{\lambda_{D^{\ast}}\lambda_{K^{\ast}}}^{D^{\ast}K^{\ast}}+a_{2}V_{cb}V^{\ast}_{us}H_{\lambda_{K^{\ast}}\lambda_{D^{\ast}}}^{K^{\ast}D^{\ast}}\Big{)},$
$\displaystyle M(B^{*-}\to D^{*0}+\rho^{-})$ $\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}\Big{(}a_{1}V_{cb}V^{\ast}_{ud}H_{\lambda_{D^{\ast}}\lambda_{\rho}}^{D^{\ast}\rho}+a_{2}V_{cb}V^{\ast}_{ud}H_{\lambda_{\rho}\lambda_{D^{\ast}}}^{\rho
D^{\ast}}\Big{)}.$ (27)

Other ten amplitudes contain the contributions from the diagrams with the
charged emitted mesons only. One has

Table 9: Amplitudes of $B^{\ast}\to D^{\ast}V_{\rm charge}$ decays.
$M(B^{\ast}\to D^{\ast}V_{\rm
charge})={\bf\frac{G_{F}}{\sqrt{2}}a_{1}V_{cb}}\,\widetilde{M}(B^{\ast}\to
D^{\ast}V_{\rm charge})$
---
Mode | $\widetilde{M}$ | Mode | $\widetilde{M}$
$B^{*-}\to D^{*0}\,D^{*-}$ |  $V^{\ast}_{cd}\,H_{\lambda_{D^{\ast 0}}\lambda_{D^{\ast-}}}^{D^{\ast 0}D^{\ast-}}$ |  $B^{*-}\to D^{*0}\,D^{*-}_{s}$ |  $V^{\ast}_{cs}\,H_{\lambda_{D^{\ast}}\lambda_{D_{s}^{\ast}}}^{D^{\ast}D^{\ast}_{s}}$
$\bar{B}^{*0}\to D^{*+}\,K^{*-}$ | $V^{\ast}_{us}\,H_{\lambda_{D^{\ast}}\lambda_{K^{\ast}}}^{D^{\ast}K^{\ast}}$ | $\bar{B}^{*0}\to D^{*+}\,\rho^{-}$ | $V^{\ast}_{ud}\,H_{\lambda_{D^{\ast}}\lambda_{\rho}}^{D^{\ast}\rho}$
$\bar{B}^{*0}\to D^{*+}\,D^{*-}$ | $V^{\ast}_{cd}\,H_{\lambda_{D^{\ast+}}\lambda_{D^{\ast-}}}^{D^{\ast+}D^{\ast-}}$ | $\bar{B}^{*0}\to D^{*+}\,D_{s}^{*-}$ | $V^{\ast}_{cs}H_{\lambda_{D^{\ast}}\lambda_{D^{\ast}_{s}}}^{D^{\ast}D^{\ast}_{s}}$
$\bar{B}_{s}^{*0}\to D^{*+}_{s}\,K^{*-}$ | $V^{\ast}_{us}\,H_{\lambda_{D_{s}^{\ast}}\lambda_{K^{\ast}}}^{D^{\ast}_{s}K^{\ast}}$ | $\bar{B}_{s}^{*0}\to D^{*+}_{s}\,\rho^{-}$ | $V^{\ast}_{ud}\,H_{\lambda_{D_{s}^{\ast}}\lambda_{\rho}}^{D^{\ast}_{s}\rho}$
$\bar{B}_{s}^{*0}\to D^{*+}_{s}\,D^{\ast-}$ | $V^{\ast}_{cd}\,H_{\lambda_{D_{s}^{\ast}}\lambda_{D^{\ast}}}^{D^{\ast}_{s}D^{\ast}}$ | $\bar{B}_{s}^{*0}\to D^{*+}_{s}\,D^{\ast-}_{s}$ | $V^{\ast}_{cs}\,H_{\lambda_{D_{s}^{\ast+}}\lambda_{D_{s}^{\ast-}}}^{D^{\ast+}_{s}D^{\ast-}_{s}}$

By assuming that the $B^{\ast}$-meson is unpolarized, the decay widths are
calculated by the formulas

$\Gamma\left(B^{\ast\,\lambda}\to
M_{1}^{\lambda_{1}}M_{2}^{\lambda_{2}}\right)=\frac{|\mathbf{p_{2}}|}{24\pi\,M^{2}}\sum\limits_{\lambda_{1}\lambda_{2}}|M(B^{\ast}\to
M_{1}M_{2})|^{2}$ (28)

where $\lambda=-\lambda_{1}+\lambda_{2}$. In Table 10 the branching fractions
of all weak nonleptonic decays considered here are given. As wide accepted, we
used the calculated values of the radiative decay widths as total widths, i.e.
$\Gamma_{\rm tot}(B^{\ast})=\Gamma(B^{\ast}\to B\gamma$). For comparison, we
give the results obtained in Ref. Chang:2019xtj .

Table 10: Branching fractions of the $B^{\ast}\to D^{\ast}V$ deacys ($V=\rho,K^{\ast},D^{\ast},D_{s}^{\ast}$) Decay mode |  CCQM |  CLFQM Chang:2019xtj
---|---|---
$B^{*-}\to D^{*0}+K^{*-}$ | $1.02(15)\times 10^{-9}$ | $1.10^{+0.01+0.19}_{-0.01-0.17}\times 10^{-11}$
$B^{*-}\to D^{*0}+\rho^{-}$ | $1.92(29)\times 10^{-8}$ | $2.23^{+0.04+0.39}_{-0.04-0.35}\times 10^{-10}$
$B^{*-}\to D^{*0}+D^{*-}$ | $1.75(26)\times 10^{-9}$ | $1.44^{+0.11+0.24}_{-0.11-0.22}\times 10^{-11}$
$B^{*-}\to D^{*0}+D^{*-}_{s}$ | $3.85(58)\times 10^{-8}$ | $3.71^{+0.18+0.64}_{-0.18-0.57}\times 10^{-10}$
$\bar{B}^{*0}\to D^{*+}+K^{*-}$ | $4.62(69)\times 10^{-9}$ | $3.40^{+0.24+0.58}_{-0.23-0.52}\times 10^{-11}$
$\bar{B}^{*0}\to D^{*+}+\rho^{-}$ | $8.09(1.21)\times 10^{-8}$ | $6.85^{+0.26+1.17}_{-0.26-1.05}\times 10^{-10}$
$\bar{B}^{*0}\to D^{*+}+D^{*-}$ | $5.12(77)\times 10^{-9}$ | $4.33^{+0.33+0.74}_{-0.32-0.66}\times 10^{-11}$
$\bar{B}^{*0}\to D^{*+}+D^{*-}_{s}$ | $1.17(18)\times 10^{-7}$ | $1.11^{+0.06+0.19}_{-0.05-0.17}\times 10^{-9}$
$\bar{B}^{*0}_{s}\to D^{*+}_{s}+K^{*-}$ | $5.63(84)\times 10^{-9}$ | $4.80^{+0.34+0.83}_{-0.32-0.74}\times 10^{-11}$
$\bar{B}^{*0}_{s}\to D^{*+}_{s}+\rho^{-}$ | $5.21(78)\times 10^{-9}$ | $9.39^{+0.36+1.63}_{-0.35-1.46}\times 10^{-10}$
$\bar{B}^{*0}_{s}\to D^{*+}_{s}+D^{*-}$ | $6.51(98)\times 10^{-9}$ | $6.10^{+0.47+1.03}_{-0.45-0.92}\times 10^{-11}$
$\bar{B}^{*0}_{s}\to D^{*+}_{s}+D^{*-}_{s}$ | $1.48(22)\times 10^{-7}$ | $1.54^{+0.08+0.26}_{-0.07-0.24}\times 10^{-9}$

One can see from Table 10 that our results are almost two order larger in
magnitude than those from Ref. Chang:2019xtj . Unfortunately, the authors of
Chang:2019xtj  did not give the numerical coefficients for the Wilson
coefficients $C_{1}$ and $C_{2}$, there are only the combinations of
$\alpha_{1}=C_{1}+\xi C_{2}$ and $\alpha_{2}=C_{2}+\xi C_{1}$ with
$\xi=1/N_{c}$. However, they refer to the papers by Buras et al.
Buchalla:1995vs ; Buras:1998raa  in which the notation are accepted as
$C_{2}=1$ and $C_{1}=0$ if QCD is neglected. We are using the Wilson
coefficients from Ref. Descotes-Genon:2013vna  (see, Table 3, and references
therein) where their values have been calculated by using the various
corrections and found that $C_{1}=-0.2632$ and $C_{2}=1.0111$. If one uses
these values for $\alpha_{1}$ and $\alpha_{2}$ of Ref. Chang:2019xtj  then one
arrives at $\alpha_{1}\approx 0.074$ for $N_{c}=3$, i.e. strongly suppressed,
whereas the value of $\alpha_{2}=0.923$ is of the leading order. If compare
the analytical expressions for the decay amplitudes from Chang:2019xtj  (see,
Eqs. (13)-(24)) with our results given by Eq. 27 and Table 9, then one finds
that in Ref. Chang:2019xtj  the amplitudes with the charged emitted mesons are
proportinal to $\alpha_{1}$ which is suppressed whereas in our approach those
amplitudes are proportinal to $a_{2}$ which is of leading order. It could
explain two order difference for branching fractions obtained in these two
approaches.

##  V Summary

The radiative and weak nonleptonic decays of vector $B$-mesons have been
studied within the covariant confined quark model (CCQM) developed in our
previous papers. The matrix elements and decays widths of the radiative decays
$B^{\ast}\to B\gamma$ were calculated. The obtained results were compared with
those obtained in other approaches.

The nonleptonic decays $B^{\ast}\to D^{\ast}V$ which proceed via tree-level
quark diagrams have been carefully analized. It was shown that the analytical
expressions for the amplitudes correspond to the factorization approach. In
the framework of our approach we calculated the leptonic decay constants and
the form factors of the $B^{\ast}\to D^{\ast}(V)$ transitions in the entire
physical region of the momentum transfer squared. Finally, we calculated the
two-body decay widths and compared our results with other models.

##  VI Acknowledgements

This work is supported by the JINR grant of young scientists and specialists
No. 22-302-06. Zh.T.’s research has been funded by the Science Committee of
the Ministry of Education and Science of the Republic of Kazakhstan (Grant No.
AP09057862).

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# Operationally independent events can influence each other in quantum theory

Shubhayan Sarkar<EMAIL_ADDRESS>Laboratoire d’Information Quantique,
Université libre de Bruxelles (ULB), Av. F. D. Roosevelt 50, 1050 Bruxelles,
Belgium

###### Abstract

In any known description of nature, two physical systems are considered
independent of each other if any action on one of the systems does not change
the other system. From our classical intuitions about the world, we further
conclude that these two systems are not affecting each other in any possible
way, and thus these two systems are causally disconnected or they do not
influence each other. Building on this idea, we show that in quantum theory
such a notion of “classical independence” is not satisfied, that is, two
quantum systems can still influence each other even if any operation on one of
the systems does not create an observable effect on the other. For our
purpose, we consider the framework of quantum networks and construct a linear
witness utilizing the Clauser-Horne-Shimony-Holt inequality. We also discuss
one of the interesting applications resulting from the maximal violation of
“classical independence” towards device-independent certification of quantum
states and measurements.

Introduction— Nonlocality is one of the most fascinating aspects of quantum
theory, encapsulating the absence of a local description for spatially
separated quantum systems that can not communicate with each other. Discovered
by Bell in 1964 [1, 2] and then experimentally observed in the last decades
[3, 4, 5, 6], the phenomenon of nonlocality clearly establishes the departure
of the quantum world from classical physics. An equivalent way to understand
it is that two quantum systems can be correlated in a stronger way than two
classical systems.

In this work, we pose an even more stringent inquiry: consider two systems
that exhibit no correlation with each other, meaning they are mutually
independent. The fundamental question we address is whether these two
independent systems can influence each other in any manner. Equivalently, can
one system exert an impact on its counterpart when there is no correlation and
no communication between them? Drawing upon our classical understanding of the
natural world, it can be logically deduced that in the absence of
communication and with both systems being mutually independent, they cannot
exert any influence on each other in any conceivable manner. We consider this
viewpoint as a notion of classicality and term it “classical independence”.

Here, we show that the notion of classical independence is violated in quantum
theory, that is, two mutually independent quantum systems might affect each
other if they are individually entangled to some different quantum systems.
For our purpose, we consider the framework of quantum networks, in particular,
the quantum bilocality scenario [7, 8] with weaker constraints on the network.
We then derive a linear inequality inspired by Clasuer-Horne-Shimony-Holt
(CHSH) inequality [9]. Restricting to operationally independent correlations,
we find an upper bound for correlations that can be described in a classically
independent way. We then find a set of quantum states and measurements that
violate this bound. Furthermore, we find the maximum value of the inequality
that can be attained in quantum theory. Interestingly using the methods
presented in [10], also allows us to certify the quantum states and
measurements in a device-independent way from the maximal violation of the
constructed inequality.

Classical independence.— Consider two systems with Alice and Bob such that
measurements $\mathcal{A},\mathcal{B}$ with outcomes $a,b$ can be performed on
them respectively. Now, we define when these two systems can be considered to
be operationally independent of each other.

######  Definition 1 (Operational independence).

Two systems are operationally independent if the probability of obtaining an
outcome when performing a measurement on one system is completely independent
of the other, that is,

$p(a|\mathcal{A},b,\mathcal{B})=p(a|\mathcal{A})\qquad\forall
a,b,\mathcal{A},\mathcal{B}.$ (1)

The resulting joint probability $p(a,b|\mathcal{A},\mathcal{B})$ factors out
using Bayes rule as

$p(a,b|\mathcal{A},\mathcal{B})=p(a|\mathcal{A})p(b|\mathcal{B})\qquad\forall
a,b,\mathcal{A},\mathcal{B}.$ (2)

Inspired by the above definition, we define the principle of no influence as
stated below.

######  Definition 2 (No-influence principle).

Two systems do not influence each other if given additional information
$\lambda$, the probability of obtaining an outcome when performing a
measurement on one system is completely independent of the other, that is,

$p(a|\mathcal{A},b,\mathcal{B},\lambda)=p(a|\mathcal{A},\lambda)\qquad\forall
a,b,\mathcal{A},\mathcal{B},\lambda.$ (3)

It is straightforward to observe that no-influence principle implies
operational independence. This brings us to the definition of classicality,
which we call classical independence.

Figure 1: Weak-bilocality scenario. Alice and Bob each receive a single
particle from their respective sources, which might be correlated to each
other, while Eve receives two particles, one from each of these sources. Alice
and Bob have the freedom to independently select and conduct two dichotomic
measurements on their respective particles. In contrast, Eve’s measurement is
constrained to a single four-outcome measurement. None of the parties can
communicate with each other.

######  Definition 3 (Classical independence).

Two operationally independent events [def. 1] are classically independent of
each other if they do not influence each other [def. 2], or to put it simply
the notion of classical independence means

$\displaystyle\mathrm{Operational\ Independence}\implies\mathrm{No\
Influence}.$ (4)

Equivalently the above definition can be understood as, if the correlations
between two parties are mutually independent for any possible choice of
measurement of both parties, then the correlations must be local.

Let us now construct a scenario where we can observe the violation of
classical independence with quantum states and measurements. A natural
scenario that one could investigate in this regard is the standard Bell
scenario. However, it is quite clear that if the correlations between two
parties are operationally independent (2), then one can never observe any
violation of a Bell inequality. Consequently in this work, we consider the
bilocality scenario [7] with three parties as described below.

The scenario— We consider three parties namely, Alice, Bob and Eve in three
different spatially separated labs. Alice and Bob receive a single particle
from sources $S_{1},S_{2}$ respectively and Eve receives two particles from
both the sources. Unlike the bilocality scenario, the sources here need not be
independent of each other, thus we call it “weak-bilocality scenario”. Now,
Alice and Bob perform two dichotomic measurements on their particles which
they can freely choose. Eve on the other hand can only perform a single four-
outcome measurement. The measurement inputs of Alice and Bob are denoted as
$x,y=0,1$ respectively and their outcomes are denoted as $a,b=0,1$, whereas
the outcomes of Eve are denoted as $e=0,1,2,3$. The scenario is depicted in
Fig. 1.

The experiment is repeated enough times to construct the joint probability
distribution or correlations, $\vec{p}=\\{p(a,b,e|x,y)\\}$ where
$p(a,b,e|x,y)$ denotes the probability of obtaining outcome $a,b,e$ by Alice,
Bob and Eve when they choose the inputs $x,y$ respectively. These
probabilities can be computed in quantum theory using the Born rule as

$\displaystyle p(a,b,e|x,y)=\mathrm{Tr}\left[(N^{A}_{a|x}\otimes
N^{B}_{b|y}\otimes N^{E}_{e})\rho_{ABE}\right]$ (5)

where $N^{A}_{a|x},N^{B}_{b|y},N^{E}_{e}$ denote the measurement elements of
Alice, Bob and Eve corresponding to $x,y$ input and $\rho_{ABE}$ denotes the
joint state generated by the source $S_{1},S_{2}$. The measurement elements
are positive semi-definite and
$\sum_{a}N^{A}_{a|x}=\sum_{b}N^{B}_{b|y}=\sum_{e}N^{E}_{e}=1$ for all $x,y$.
It is important to recall here that Alice and Bob can not communicate with
each other during the experiment.

It is usually simpler to express the probabilities in terms of expectation
values as

$\displaystyle\langle\mathcal{A}_{x}\mathcal{B}_{y}N^{E}_{e}\rangle=p(0,0,e|x,y)+p(1,1,e|x,y)$
$\displaystyle-p(0,1,e|x,y)-p(1,0,e|x,y)$ (6)

where $\mathcal{A}_{x},\mathcal{B}_{y}$ denote Alice’s and Bob’s observable
corresponding to the input $x,y$ respectively and can be expressed as
$s_{i}=N^{s}_{0|i}-N^{s}_{1|i}$ for $s=A,B$.

Violation of classical independence.— Let us now restrict that the
correlations $p(a,b|x,y)$ are operationally independent [def. 1], that is,
$p(a,b|x,y)=p(a|x)p(b|y)$.

Now, let us express the joint probability distribution $p(a,b,e|x,y)$ as

$p(a,b,e|x,y)=\sum_{\lambda}p(\lambda)p(a,b,e|x,y,\lambda)$ (7)

Using Bayes rule, the probability $p(a,b,e|x,y,\lambda)$ can be rewritten as

$p(a,b,e|x,y,\lambda)=p(a|x,b,y,e,\lambda)p(b|y,e,\lambda)p(e|\lambda)$ (8)

Assuming no-influence [def. 2] allows us to conclude that
$p(a|x,b,y,e,\lambda)=p(a|x,e,\lambda)$ and thus we get that

$p(a,b,e|x,y)=\sum_{\lambda}p(a|x,e,\lambda)p(b|y,e,\lambda)p(e|\lambda)p(\lambda).$
(9)

Notice that in the bilocality scenario [7], one additionally assumes that
$p(a|x,e,\lambda)=p(a|x,\lambda)$ and $p(b|y,e,\lambda)=p(b|y,\lambda)$.

Inspired by [9], we will now construct a linear functional constructed from
the joint probability distribution $\vec{p}$. In terms of observables, the
functional can be represented as

$\displaystyle\mathcal{I}=\big{\langle}\mathcal{A}_{0}(\mathcal{B}_{0}-\mathcal{B}_{1})\mathcal{E}_{0}+\mathcal{A}_{1}(\mathcal{B}_{0}+\mathcal{B}_{1})\mathcal{E}_{1}\big{\rangle}$
(10)

where $\mathcal{E}_{0}=N_{0}^{E}-N_{1}^{E}-N_{2}^{E}+N_{3}^{E}$ and
$\mathcal{E}_{1}=N_{0}^{E}+N_{1}^{E}-N_{2}^{E}-N_{3}^{E}$. As shown in
Appendix A, the above inequality can be broken up into conditional CHSH
inequalities, up to the presence of Eve’s measurement, which were useful to
prove that every pure entangled state is Bell nonlocal [11] and self-testing
the Bell basis [12].

Let us compute the maximum value of $\mathcal{I}$ (10) achievable using
correlations that satisfy “classical independence". We will further call this
value as the classical bound and denote it as $\beta_{C}$. For this purpose,
let us express the expectation value (Operationally independent events can
influence each other in quantum theory) using (9) as

$\displaystyle\langle\mathcal{A}_{x}\mathcal{B}_{y}N^{E}_{e}\rangle=\sum_{\lambda}p(\lambda)p(e|\lambda)\big{(}p(0|x,e,\lambda)-p(1|x,e,\lambda)\big{)}$
$\displaystyle\big{(}p(0|y,e,\lambda)-p(1|y,e,\lambda)\big{)}\quad$ (11)

which can be simply stated as

$\displaystyle\langle\mathcal{A}_{x}\mathcal{B}_{y}N^{E}_{e}\rangle=\sum_{\lambda}p(\lambda)p(e|\lambda)\langle\mathcal{A}_{x,e,\lambda}\rangle\langle\mathcal{B}_{y,e,\lambda}\rangle.$
(12)

Using the above expression in Appendix A, we calculate the classical bound of
$\mathcal{I}$ (10) which is stated below.

######  Fact 1.

Consider the functional $\mathcal{I}$ (10). The maximum value $\beta_{C}$ that
can be achieved by classically independent correlations is $\beta_{C}=2$.

Consider now the quantum state
$|\psi_{ABE}\rangle=|\phi^{+}_{A\overline{A}}\rangle\otimes|\phi^{+}_{B\overline{B}}\rangle$
such that $E\equiv\overline{AB}$ and $|\phi^{+}\rangle$ is the two-qubit
maximally entangled state. It is easy to check that such a state will always
generate correlations between Alice and Bob that are operationally independent
[def. 1]. Also, consider that Alice’s and Bob’s observables are given by

$\displaystyle\tilde{\mathcal{A}}_{0}$ $\displaystyle=$
$\displaystyle\sigma_{z},\qquad\tilde{\mathcal{A}}_{1}=\sigma_{x}$
$\displaystyle\tilde{\mathcal{B}}_{0}$ $\displaystyle=$
$\displaystyle\frac{\sigma_{z}+\sigma_{x}}{\sqrt{2}},\qquad\tilde{\mathcal{B}}_{1}=\frac{\sigma_{x}-\sigma_{z}}{\sqrt{2}}$
(13)

along with Eve’s measurement given by the Bell-basis as
$\tilde{N}_{i,j}^{E}=|\phi_{i,j}\rangle\\!\langle\phi_{i,j}|$ where
$|\phi_{i,j}\rangle=\frac{1}{\sqrt{2}}(|i\ j\rangle+(-1)^{i}|\overline{i}\
\overline{j}\rangle)$ where $i,j=0,1$ and $N_{i,j}^{E}\equiv N_{2i+j}^{E}$.
Plugging these states and observables in the functional $\mathcal{I}$ (10)
gives us the value $2\sqrt{2}$. Thus, quantum theory violates the notion of
classical independence. Consequently, one can conclude that systems that are
operationally independent can influence each other. We show in Appendix A that
$2\sqrt{2}$ is in fact the maximal value achievable using quantum theory.

We can identify some necessary conditions to violate the notion of classical
independence using quantum states and measurements. The first necessary
condition is that Eve needs to perform an entangled measurement as one can
observe from Eq. (9). This is contrary to the violation of bilocality which
can also happen with separable measurements with Eve. Further on, the sources
generating entangled states between Alice-Eve and Bob-Eve are necessary.
Although the inequality (10) considered in this work requires incompatible
measurements to obtain any violation, one further needs to explore whether
incompatible measurements are a necessity to violate classical independence or
not. Let us now discuss an interesting application arising due to the
violation of classical independence.

Self-testing.— Self-testing is the strongest device-independent scheme that
allows one to certify the quantum states and measurements without making any
assumption on the devices involved apart from the fact that they are governed
by quantum theory [13]. Self-testing in quantum networks has been explored
recently [12, 14, 15, 16, 17, 18, 10], however, all of these schemes also
assume that the sources are independent [see nevertheless [19]]. Here, we do
not need to assume it as we show in Appendix B that the condition of
operational independence [see Eq. (1)] allows one to conclude that the sources
are independent. Let us now state the self-testing result the proof of which
is discussed in Appendix B.

Figure 2: Causality graph of the weak-bilocality scenario. The square boxes
represent the measurement devices and the circles represent the sources. The
grey circle represents a hidden variable that might correlate the sources.

######  Theorem 1.

Assume that the operationally independent correlations $\vec{p}$ attain the
quantum bound of $\mathcal{I}$ (10). Then, (i) The Hilbert spaces of all the
parties decompose as
$\mathcal{H}_{s}=\mathcal{H}_{s^{\prime}}\otimes\mathcal{H}_{s^{\prime\prime}}$
and
$\mathcal{H}_{\overline{s}}=\mathcal{H}_{\overline{s}^{\prime}}\otimes\mathcal{H}_{\overline{s}^{\prime\prime}}$;
(ii) There exist local unitary transformations
$U_{s}:\mathcal{H}_{s}\rightarrow\mathcal{H}_{s}$ and
$U_{\overline{s}}:\mathcal{H}_{\overline{s}}\rightarrow\mathcal{H}_{\overline{s}}$
such that

$(U_{s}\otimes
U_{\overline{s}})|\psi_{s\overline{s}}\rangle=|\phi^{+}_{s^{\prime}\overline{s}^{\prime}}\rangle\otimes|\xi_{s^{\prime\prime}\overline{s}^{\prime\prime}}\rangle$
(14)

for some
$|\xi_{s^{\prime\prime}\overline{s}^{\prime\prime}}\rangle\in\mathcal{H}_{s^{\prime\prime}}\otimes\mathcal{H}_{\overline{s}^{\prime\prime}}$,
and the measurements of all parties are certified as

$\overline{U}\,N_{i,j}^{E}\,\overline{U}^{\dagger}=|\phi_{i,j}\rangle\\!\\!\langle\phi_{i,j}|_{E^{\prime}}\otimes\mathbbm{1}_{E^{\prime\prime}},\quad
U_{s}\,s_{i}\,U_{s}^{\dagger}=\tilde{s}_{i}\otimes\mathbbm{1}_{s^{\prime\prime}}$
(15)

for $i,j=0,1$ where $\overline{U}=\otimes_{s}U_{\overline{s}}$ such that
$s=A,B$ and $E\equiv\overline{AB}$. The observables $\tilde{s}_{i}$ are given
in Eqs. (Operationally independent events can influence each other in quantum
theory).

Discussions— Likewise preparation non-contextuality [20], the above result can
also be considered as a violation of Leibniz principle of indiscernibles [21],
that can be simply stated as principles that hold operationally should also
hold ontologically. In the present context, if two systems are operationally
independent does not imply that they are ontologically independent. Consider
again the no-influence principle which can also be interpreted as the
assumption of local causality in the Bell scenario. However, the striking
difference is the fact that the parties involved in the Bell scenario are not
operationally independent. Consequently, the scenario considered in this work
is weaker when compared to the Bell scenario, that is, we identify non-
classical behavior even in situations where one can not violate a Bell
inequality. Further on, we find that the assumptions considered in this work
are weaker when compared to the bilocality scenario [7] as we allow Eve to
influence Alice’s and Bob’s results. Furthermore, in any quantum network
including the bilocality scenario one needs to further assume that the sources
are statistically independent of each other [see nevertheless [19]]. However,
here we do not put any restrictions on the sources and even allow them to be
entangled. Furthermore, the bilocality scenario has already been
experimentally implemented [22] and thus we believe that the violation of
inequality (10) can be easily tested.

Analyzing the above result from a realist perspective gives an interesting
insight towards understanding whether a measurement on an entangled
counterpart produces a physical change on the other. In the Bell scenario, a
possible explanation of the observed nonlocal correlations is that the
measurement by Alice updates the state with Bob or vice versa. However, such
an explanation in the above-presented scenario is not consistent. First,
consider that Eve performed her entangled measurement before Alice and Bob,
then as the states between Alice and Bob are entangled then one can explain
the violation of classical-independence using a similar realist explanation as
the Bell scenario. However, consider now that Eve has not performed her
entangled measurement, then as the states between Alice and Bob are separable
any measurement by Alice should not alter Bob’s state but can alter Eve’s
state. Consequently, there exist spacelike frames of reference where Alice’s
state update is caused by Bob’s measurement and other frames where it remains
unchanged. Thus, whether the "physical state" of Alice gets updated when Bob
performs a measurement depends on the information about Eve’s result which
again is problematic if one considers that the cause-effect relationship is
not epistemic.

Several interesting problems follow up from our work. The most interesting
among them would be to explore in detail whether a cause-effect relationship
between two events is consistent in quantum theory or not. A simpler problem
will be to extend the weak-bilocality scenario to the multipartite regime with
arbitrary number of sources or higher number of outcomes. Furthemore, it will
be extremely interesting if one can use the above-presented self-testing
result to construct a device-independent key distribution scheme or for
randomness certification.

###### Acknowledgements.

We thank Stefano Pironio for insightful comments. This project was funded
within the QuantERA II Programme (VERIqTAS project) that has received funding
from the European Union’s Horizon 2020 research and innovation programme under
Grant Agreement No 101017733.

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##  Appendix A Classical bound

######  Fact 1.

Consider the functional $\mathcal{I}$ (10). The maximum value $\beta_{C}$ that
can be achieved by classically independent correlations is $\beta_{C}=2$.

###### Proof.

For our purpose, we rewrite the functional $\mathcal{I}$ (10) as

$\displaystyle\mathcal{I}=\sum_{i,j=0,1}\langle\mathcal{I}_{CHSH}^{i,j}N_{i,j}^{E}\rangle$
(16)

such that $N_{i,j}^{E}\equiv N_{2i+j}^{E}$ and

$\displaystyle\mathcal{I}_{CHSH}^{i,j}=(-1)^{i}\big{\langle}(-1)^{j}\mathcal{A}_{0}(\mathcal{B}_{0}-\mathcal{B}_{1})+\mathcal{A}_{1}(\mathcal{B}_{0}+\mathcal{B}_{1})\big{\rangle}$

Now, taking the absolute value of Eq. (16) and using triangle inequality gives
us

$\displaystyle|\mathcal{I}|\leq\sum_{i,j=0,1}\left|\langle\mathcal{I}_{CHSH}^{i,j}N_{i,j}^{E}\rangle\right|.$
(18)

Expanding the terms on the right-hand side individually and using Eq. (12)
gives us

$\displaystyle\left|\langle\mathcal{I}_{CHSH}^{i,j}N_{i,j}^{E}\rangle\right|\leq\sum_{\lambda}p(\lambda)p(i,j|\lambda)\Big{(}\Big{|}\langle\mathcal{A}_{0,i,j,\lambda}\rangle(\langle\mathcal{B}_{0,i,j,\lambda}\rangle-\langle\mathcal{B}_{1,i,j,\lambda}\rangle)\Big{|}+\Big{|}(-1)^{j}\langle\mathcal{A}_{1,i,j,\lambda}\rangle(\langle\mathcal{B}_{0,i,j,\lambda}\rangle+\langle\mathcal{B}_{1,i,j,\lambda}\rangle)\big{\rangle}\Big{|}\Big{)}.$
(19)

Now, using the fact that $|\langle\mathcal{A}_{x,e,\lambda}\rangle|\leq 1$ for
any $x,e,\lambda$ we obtain that

$\displaystyle\left|\langle\mathcal{I}_{CHSH}^{i,j}N_{i,j}^{E}\rangle\right|\leq\sum_{\lambda}p(\lambda)p(i,j|\lambda)\Big{(}\Big{|}\langle\mathcal{B}_{0,i,j,\lambda}\rangle-\langle\mathcal{B}_{1,i,j,\lambda}\rangle\Big{|}+\Big{|}\langle\mathcal{B}_{0,i,j,\lambda}\rangle+\langle\mathcal{B}_{1,i,j,\lambda}\big{\rangle}\Big{|}\Big{)}.\quad$
(20)

Furthermore,

$\displaystyle\Big{|}\langle\mathcal{B}_{0,i,j,\lambda}\rangle-\langle\mathcal{B}_{1,i,j,\lambda}\rangle\Big{|}+\Big{|}\langle\mathcal{B}_{0,i,j,\lambda}\rangle+\langle\mathcal{B}_{1,i,j,\lambda}\big{\rangle}\Big{|}=2\max(|\langle\mathcal{B}_{0,i,j,\lambda}\rangle|,|\langle\mathcal{B}_{1,i,j,\lambda}\rangle|)\leq
2.$ (21)

Consequently, we obtain from Eq. (20) that

$\displaystyle\left|\langle\mathcal{I}_{CHSH}^{i,j}N_{i,j}^{E}\rangle\right|\leq
2p(i,j)$ (22)

which summing over $i,j$ finally gives us $|\mathcal{I}|\leq 2$. This
completes the proof. ∎

##  Appendix B Device-independent certification

We first show here that in the weak-bilocality scenario [see Fig. 1],
operational independence implies that the sources $S_{1},S_{2}$ are
independent.

######  Lemma 1.

Consider the weak-bilocality scenario such that the state shared between
Alice, Bob and Eve is given by $\rho_{ABE}$. If Alice and Bob are
operationally independent [def. 1], then
$\rho_{ABE}=\rho_{AE}\otimes\rho_{BE}$.

###### Proof.

If Alice and Bob are operationally independent, then as discussed in the
manuscript

$\displaystyle
p(a,b|\mathcal{A},\mathcal{B})=p(a|\mathcal{A})p(b|\mathcal{B})\qquad\forall
a,b,\mathcal{A},\mathcal{B}.$ (23)

Putting quantum states and restricting to rank-one projective quantum
measurements gives us

$\displaystyle\langle\psi^{a}|\langle\psi^{b}|\rho_{AB}|\psi^{a}\rangle|\psi^{b}\rangle=\langle\psi^{a}|\rho_{A}|\psi^{a}\rangle\langle\psi^{b}|\rho_{B}|\psi^{b}\rangle$
(24)

for all $|\psi^{a}\rangle,|\psi^{b}\rangle$. Thus, we can conclude that

$\displaystyle\langle\psi^{a}|\langle\psi^{b}|\rho_{AB}|\psi^{a}\rangle|\psi^{b}\rangle=\langle\psi^{a}|\langle\psi^{b}|\rho_{A}\otimes\rho_{B}|\psi^{a}\rangle|\psi^{b}\rangle$
(25)

and consequently we have that $\rho_{AB}=\rho_{A}\otimes\rho_{B}$. This
completes the proof.

∎

######  Theorem 1.

Assume that the operationally independent correlations $\vec{p}$ attain the
quantum bound of $\mathcal{I}$ (10). Then, (i) The Hilbert spaces of all the
parties decompose as
$\mathcal{H}_{s}=\mathcal{H}_{s^{\prime}}\otimes\mathcal{H}_{s^{\prime\prime}}$
and
$\mathcal{H}_{\overline{s}}=\mathcal{H}_{\overline{s}^{\prime}}\otimes\mathcal{H}_{\overline{s}^{\prime\prime}}$;
(ii) There exist local unitary transformations
$U_{s}:\mathcal{H}_{s}\rightarrow\mathcal{H}_{s}$ and
$U_{\overline{s}}:\mathcal{H}_{\overline{s}}\rightarrow\mathcal{H}_{\overline{s}}$
such that

$(U_{s}\otimes
U_{\overline{s}})|\psi_{s\overline{s}}\rangle=|\phi^{+}_{s^{\prime}\overline{s}^{\prime}}\rangle\otimes|\xi_{s^{\prime\prime}\overline{s}^{\prime\prime}}\rangle$
(26)

for some
$|\xi_{s^{\prime\prime}\overline{s}^{\prime\prime}}\rangle\in\mathcal{H}_{s^{\prime\prime}}\otimes\mathcal{H}_{\overline{s}^{\prime\prime}}$,
and the measurements of all parties are certified as

$\overline{U}\,N_{i,j}^{E}\,\overline{U}^{\dagger}=|\phi_{i,j}\rangle\\!\\!\langle\phi_{i,j}|_{E^{\prime}}\otimes\mathbbm{1}_{E^{\prime\prime}},\quad
U_{s}\,s_{i}\,U_{s}^{\dagger}=\tilde{s}_{i}\otimes\mathbbm{1}_{s^{\prime\prime}}$
(27)

for $i,j=0,1$ where $\overline{U}=\otimes_{s}U_{\overline{s}}$ such that
$s=A,B$ and $E\equiv\overline{AB}$. The observables $\tilde{s}_{i}$ are given
in Eqs. (Operationally independent events can influence each other in quantum
theory).

###### Proof.

Let us consider the functional $\mathcal{I}$ written in the form in Eq. (16).
As shown in [23], one can use the sum-of-squares decomposition to obtain that
$\mathcal{I}_{CHSH}^{0,0}\leq 2\sqrt{2}\mathbbm{1}$ and similarly one can show
that $\mathcal{I}_{CHSH}^{i,j}\leq 2\sqrt{2}\mathbbm{1}$. As $N_{i,j}^{E}$ are
positive, we can conclude from Eq. (16) that

$\displaystyle\mathcal{I}\leq 2\sqrt{2}\sum_{i,j=0,1}\langle\mathbbm{1}\otimes
N_{i,j}^{E}\rangle\leq 2\sqrt{2}.$ (28)

Consequently, one can conclude that the quantum bound of $\mathcal{I}$ is
$2\sqrt{2}$.

Now, if one attains the value $2\sqrt{2}$ of the functional $\mathcal{I}$,
then we have that

$\displaystyle\mathcal{I}=\sum_{i,j=0,1}\langle\mathcal{I}_{CHSH}^{i,j}N_{i,j}^{E}\rangle=2\sqrt{2}$
(29)

which can be rewritten as

$\displaystyle\sum_{i,j=0,1}\langle(2\sqrt{2}\mathbbm{1}-\mathcal{I}_{CHSH}^{i,j})N_{i,j}^{E}\rangle=0.$
(30)

Consider now the states
$p(i,j|E)\rho_{AB}^{i,j}=\mathrm{Tr}_{E}(N_{i,j}^{E}\rho_{ABE})$ where
$p(i,j|E)=\mathrm{Tr}(N_{i,j}^{E}\rho_{ABE})$ using which the above expression
can be written as

$\displaystyle\sum_{i,j=0,1}p(i,j|E)\mathrm{Tr}[(2\sqrt{2}\mathbbm{1}-\mathcal{I}_{CHSH}^{i,j})\rho_{AB}^{i,j}]=0.$
(31)

As we assume that the local state of Eve is full-rank and thus every outcome
of Eve occurs due to which $p(i,j|E)\neq 0$ for any $i,j$. Thus, we have that

$\displaystyle\mathrm{Tr}(\mathcal{I}_{CHSH}^{i,j}\rho_{AB}^{i,j})=2\sqrt{2}\qquad\forall
i,j.$ (32)

Now, one can straightway adapt the proof presented in [10] to first self-test
the states $\rho_{AB}^{i,j}$ to be

$\displaystyle U_{A}\otimes U_{B}\rho_{AB}^{i,j}U_{A}^{\dagger}\otimes
U_{B}^{\dagger}=|\phi_{i,j}\rangle\\!\\!\langle\phi_{i,j}|\otimes\rho_{A^{\prime\prime}B^{\prime\prime}}^{i,j}$
(33)

where $|\phi_{i,j}\rangle$ are given in the manuscript. The measurements of
Alice and Bob are also certified as given in Eq. (27). Now, using Lemma 1 and
then the above-certified states (33) one can then self-test the states
generated by the sources to be the maximally entangled state (26) which is
then used to certify Eve’s measurement (27). It is important to notice that in
Ref. [10], one additionally assumes that the probabilities $p(i,j|E)$ are
equal for all $i,j$. However, one can straightaway adapt the proof without
such an assumption and get exactly the same conclusions. This completes the
proof.

∎