# Anomalies in Weak Decays of Hadrons Containing a b Quark

###### Abstract

A brief review of the current state of observed deviations of theoretical
predictions from experimental data in semileptonic decays of $B$ and $B_{c}$
mesons is given. A theoretical analysis of these decays is carried out, taking
into account the effects of new physics, which appear due to the introduction
of new four-fermion operators, which are absent in the basis of the Standard
Model (SM) operators. The necessary form factors are calculated within the
framework of the covariant quark model developed in our papers.

a Joint Institute for Nuclear Research, 141980 Dubna, RUSSIA b The Institute
of Nuclear Physics,
Ministry of Energy of the Republic of Kazakhstan, 050032 Almaty, KAZAKHSTAN

## INTRODUCTION

To date, the Standard Model (SM) has been tested and validated by many
experiments. Attention has shifted beyond the SM in search of new particles
and new interactions. It should be noted that so far it has not been possible
to observe new particles in modern accelerators, in particular, at the Large
Hadron Collider at CERN. However, there are indirect indications to the New
Physics (NP) in a number of experiments on the study of weak decays of hadrons
containing a b quark. Among them, semitau decays of b hadrons should be noted
($B$,$B_{s}$, and $B_{c}$ mesons and $\Lambda_{b}$ baryons), which are due to
the transition $b\to c\tau\nu_{\tau}$, as well as rare decays due to the
transition $b\to s\ell^{+}\ell^{-}$. The observed deviations of the
experimental data in a number of observed quantities from their values
obtained as part of the Standard Model have been called “anomalies.” A common
point of view among theorists is that all these deviations can be explained by
the violation of lepton universality [1, 2]. In other words, the possible
manifestations of the New Physics lead to a difference in the interactions of
muons, electrons, and tau leptons with gauge bosons. This unified point of
view suggests a common origin of anomalies from the point of view of physics
beyond the SM, which confirms the arguments in favor of the violation of
lepton universality. In particular, this path opens up new possibilities for
constructing NP models and makes it possible to create a convincing physical
justification for experiments.

## ANOMALIES AND VIOLATION OF LEPTON UNIVERSALITY

There are three generations of leptons in the SM, grouped into lepton–neutrino
doublets:

$\displaystyle\left(\begin{array}[]{c}\nu_{e}\\\\[8.61108pt]
e^{-}\end{array}\right),\qquad\left(\begin{array}[]{c}\nu_{\mu}\\\\[8.61108pt]
\mu^{-}\end{array}\right),\qquad\left(\begin{array}[]{c}\nu_{\tau}\\\\[8.61108pt]
\tau^{-}\end{array}\right).$ (7)

The lepton universality in the SM means that, in the interaction Lagrangian,
the constants characterizing the coupling of weak lepton currents to W bosons
are the same for all three generations of leptons. The violation of lepton
universality implies a difference in the coupling constants for electrons,
muons, and tau leptons.

Semitauonic decays $b\\!\\!\to\\!\\!c\tau\nu$:

These decays are due to charged currents at the tree level in the SM and,
therefore, the decay branchings take values of the order
${\mathcal{O}}(10^{-3})$. Differential decay width, ${\rm d}\Gamma$, for
semileptonic decays involving $D^{(*)}$ mesons in the final state depends on
both $m^{2}_{\ell}$ and $q^{2}$, of the square of the invariant mass of the
lepton pair.

$\displaystyle\frac{{\rm d}\Gamma^{\rm SM}(\bar{B}\to
D^{(\ast)}\ell^{-}\bar{\nu}_{\ell})}{{\rm
d}q^{2}}\,=\underbrace{\frac{G_{F}^{2}\;|V_{cb}|^{2}\;|\boldsymbol{p}^{*}_{D^{(*)}}|\;q^{2}}{96\pi^{3}m_{B}^{2}}\left(1-\frac{m_{\ell}^{2}}{q^{2}}\right)^{2}}_{\text{universal
and phase space factors}}$
$\displaystyle\times\underbrace{\left[(|H_{+}|^{2}+|H_{-}|^{2}+|H_{0}|^{2})\left(1+\frac{m_{\ell}^{2}}{2q^{2}}\right)+\frac{3m_{\ell}^{2}}{2q^{2}}|H_{s}|^{2}\right]}_{\text{hadronic
effects}}~{}.$ (8)

Here, $\boldsymbol{p}^{*}_{D^{(*)}}$ is the momentum of the daughter hadron
$B$ in the rest frame of the meson. Four helical amplitudes
$H_{\pm},H_{0},H_{s}$ characterize the effect of the hadronic structure and
depend on the square of the momentum transfer $q^{2}$, changing in the
interval $m^{2}_{\ell}\leq q^{2}\leq(m_{B}-m_{D^{*}})^{2}$. The most optimal
are measurements of the ratios of decay branchings with a tau lepton to
branching with an electron (or muon) in the final state. In this regard, the
dependence on the element of the Cabibbo–Kobayashi–Maskawa (CKM) matrix is
eliminated $|V_{cb}|$, there is a partial reduction in theoretical
uncertainties associated with hadronic effects, and experimental uncertainties
are reduced.

$R(D^{(\ast)})=\frac{\mathcal{B}(B\to
D^{(\ast)}\tau\nu_{\tau})}{\mathcal{B}(B\to D^{(\ast)}\ell\nu_{\ell})},\quad
D^{(\ast)}=D\,\,\text{or}\,\,D^{\ast},\quad\ell=e\,\,\text{or}\,\,\mu$ (9)

The predictions obtained in the SM are as follows:

$R^{\rm SM}(D)=0.299\pm 0.003,\qquad R^{\rm SM}(D^{\ast})=0.258\pm 0.005.$
(10)

Averaging over experimental data obtained up to 2021 gave the following
results:

$R(D)=0.340\pm 0.030,\qquad R(D^{\ast})=0.295\pm 0.014.$ (11)

It can be seen that there are deviations of about 3$\sigma$. Taking into
account reports from LHCb about new measurements performed in 2022, a new
averaging was performed over all available experimental data in [3]. However,
the deviation from the SM practically did not change. The LHCb collaboration
reported measuring a similar branching ratio in semileptonic decay $B\to
J/\psi+\tau\nu_{\tau}$ [4]. The received data is within 2$\sigma$ deviations
from the range of central values predicted by the SM. Thus, the above results
are confirmed by this independent measurement.

Rare decays $b\to s\ell^{+}\ell^{-}$.

Rare decays due to neutral currents with a change in strangeness $b\to
s\ell^{+}\ell^{-}$ appear only at the one-loop SM level and, therefore, are
significantly suppressed. The branchings of such decays are at the level
$10^{-6}$. However, their measurement is available at modern accelerators,
with the exception of the currently unavailable $\tau^{+}\tau^{-}$ mode.

Branching measurements $B\to K^{(\ast)}\ell^{+}\ell^{-}$ ($\ell=e,\mu$), as
well as a number of observables arising in the angular distributions, are
available. The measurement of the ratio of muon-to-electron mode branchings is
of particular interest, because in relation to

$R(K^{(\ast)})=\frac{\mathcal{B}(B\to
K^{(\ast)}\mu^{+}\mu^{-})}{\mathcal{B}(B\to
K^{(\ast)}e^{+}e^{-})}\,,\qquad(K^{(\ast)}=K,K^{\ast})\,,$ (12)

the dependence on model-dependent form factors practically disappears. These
branching ratios are measured by LHCb and Belle. The experimental results
obtained for several bins are consistent with the expectations of the SM at
the level $2,1-2,5~{}\sigma$. However, for the branchings $\mathcal{B}(B\to
K^{\ast}\mu^{+}\mu^{-})$, $\mathcal{B}(B\to K\mu^{+}\mu^{-})$, and
$\mathcal{B}(B_{s}\to\phi\mu^{+}\mu^{-})$ themselves, as well as for some
observables from angular distributions, there are discrepancies between the
predictions of the SM and measurements.

## EFFECTIVE THEORIES

The use of effective theories based on the construction of effective
Hamiltonian is the most convenient way to describe weak decays of b hadrons.
Their construction within the framework of the SM is based on several powerful
methods from the arsenal of quantum field theory. This is an operator
expansion that allows one to separate the contributions of small and large
distances; it is a renormalization group technique that allows one to obtain
numerical values of the Wilson coefficients at energies on the order of the
b-quark mass and, at the final stage, to obtain a set of four-fermionic
operators that make it possible to describe weak decays of hadrons. The main
problem is the calculation of the matrix elements of these operators in the
layers of the physical states of the initial and final particles. Their
calculation requires the use of nonperturbative methods such as lattice
calculations, various sum rules, and quark models. At the final stage, the
amplitude of the weak transition of the initial meson to the final state is
written as

$\displaystyle A(f\to i)=\langle f|\,{\cal H_{\rm
eff}}\,|i\rangle=\frac{G_{F}}{\sqrt{2}}\lambda_{\rm
CKM}\sum\limits_{k}\underbrace{C_{k}(\mu)}_{\rm SD}\,\,\,\underbrace{\langle
f|Q_{k}(\mu)|i\rangle}_{\rm LD}.$ (13)

Here, $C_{k}(\mu)$ is Wilson coefficients and $Q_{k}(\mu)$ is four-fermion
operators.

Covariant quark model:

To calculate the matrix elements of four-fermionic operators, we will use the
covariant quark model, originally formulated in [5] and then developed in
subsequent works. It is based on nonlocal interpolation currents with the
corresponding quantum numbers of hadrons. Therefore, for example, the currents
describing mesons, baryons, and tetraquarks have the following form:

$\displaystyle J_{M}(x)$ $\displaystyle=$
$\displaystyle\int\\!\\!dx_{1}\\!\\!\int\\!\\!dx_{2}\,F_{M}(x;x_{1},x_{2})\cdot\bar{q}^{a}_{f_{1}}(x_{1})\,\Gamma_{M}\,q^{a}_{f_{2}}(x_{2})$
$\displaystyle J_{B}(x)$ $\displaystyle=$
$\displaystyle\int\\!\\!dx_{1}\\!\\!\int\\!\\!dx_{2}\\!\\!\int\\!\\!dx_{3}\,F_{B}(x;x_{1},x_{2},x_{3})$
$\displaystyle\times\,\Gamma_{1}\,q^{a_{1}}_{f_{1}}(x_{1})\,\Big{[}\varepsilon^{a_{1}a_{2}a_{3}}q^{T\,a_{2}}_{f_{2}}(x_{2})C\,\Gamma_{2}\,q^{a_{3}}_{f_{3}}(x_{3})\Big{]}$
$\displaystyle J_{T}(x)$ $\displaystyle=$
$\displaystyle\int\\!\\!dx_{1}\ldots\int\\!\\!dx_{4}\,F_{T}(x;x_{1},\ldots,x_{4})$
$\displaystyle\times\,\Big{[}\varepsilon^{a_{1}a_{2}c}q_{f_{1}}^{T\,a_{1}}(x_{1})\,C\Gamma_{1}\,q_{f_{2}}^{a_{2}}(x_{2})\Big{]}\cdot\Big{[}\varepsilon^{a_{3}a_{4}c}\bar{q}_{f_{3}}^{T\,a_{3}}(x_{3})\,\Gamma_{2}C\,\bar{q}_{f_{4}}^{a_{4}}(x_{4})\Big{]}$

Vertex function $F_{H}(x;x_{1},\ldots,x_{n})$ is chosen in the translation-
invariant form

$F_{H}(x;x_{1},\ldots,x_{n})=\delta\left(x-\sum\limits_{i=1}^{n}w_{i}x_{i}\right)\Phi_{H}\left(\sum\limits_{i<j}(x_{i}-x_{j})^{2}\right),\quad
w_{i}=m_{i}/\sum\limits_{j=1}^{n}m_{j},$

where $m_{i}$ is the mass of the quark described by the field $q(x_{i})$. The
Fourier transform of a function $\Phi_{H}$ is chosen as a Gaussian exponent
decreasing at infinity in the Euclidean direction. As an example, we present
the explicit form of the matrix element that arises when calculating the weak
lepton decay of a pseudoscalar meson:

$N_{c}\,g_{P}\\!\int\\!\\!\frac{d^{4}k}{(2\pi)^{4}i}\,\widetilde{\Phi}_{P}(-k^{2})\,{\rm
tr}\biggl{[}O^{\,\mu}S_{1}(k+w_{1}p)\gamma^{5}S_{2}(k-w_{2}p)\biggr{]}=f_{P}p^{\mu}.$

Here, $S_{i}(k)=1/(m_{i}-\not k_{i})$ is the Dirac quark propagator,
$O^{\,\mu}$ is a weak Dirac matrix with left helicity, and $f_{P}$ is the
calculated constant of weak lepton decay.

Analysis of new physics in decays$B\to D^{(*)}\tau\nu_{\tau}$:

The possible influence of NP effects in decays $B\to D^{(*)}\tau\nu_{\tau}$
was investigated in our work [6, 7, 8] within the covariant quark model. The
effective Hamiltonian for the quark transition $b\to
c\tau^{-}\bar{\nu}_{\tau}$ taking into account the effects of NP is written in
the form

${\mathcal{H}}_{eff}\propto
G_{F}\,V_{cb}\,[(1+V_{L})\mathcal{O}_{V_{L}}+V_{R}\mathcal{O}_{V_{R}}+S_{L}\mathcal{O}_{S_{L}}+S_{R}\mathcal{O}_{S_{R}}+T_{L}\mathcal{O}_{T_{L}}]$
(14)

where the four-fermionic NP operators have the form

$\displaystyle\mathcal{O}_{V_{L}}$ $\displaystyle=$
$\displaystyle\left(\bar{c}\gamma^{\mu}P_{L}b\right)\left(\bar{\tau}\gamma_{\mu}P_{L}\nu_{\tau}\right)\qquad\mathcal{O}_{V_{R}}=\left(\bar{c}\gamma^{\mu}P_{R}b\right)\left(\bar{\tau}\gamma_{\mu}P_{L}\nu_{\tau}\right)$
$\displaystyle\mathcal{O}_{S_{L}}$ $\displaystyle=$
$\displaystyle\left(\bar{c}P_{L}b\right)\left(\bar{\tau}P_{L}\nu_{\tau}\right)\qquad\qquad\,\mathcal{O}_{S_{R}}=\left(\bar{c}P_{R}b\right)\left(\bar{\tau}P_{L}\nu_{\tau}\right)$
$\displaystyle\mathcal{O}_{T_{L}}$ $\displaystyle=$
$\displaystyle\left(\bar{c}\sigma^{\mu\nu}P_{L}b\right)\left(\bar{\tau}\sigma_{\mu\nu}P_{L}\nu_{\tau}\right)$
(15)

Here $\sigma_{\mu\nu}=i\left[\gamma_{\mu},\gamma_{\nu}\right]/2$,
$P_{L,R}=(1\mp\gamma_{5})/2$, $V_{L,R}$, $S_{L,R}$, and $T_{L}$ are the
complex Wilson coefficients governing the NP. In the SM we have
$V_{L,R}=S_{L,R}=T_{L}=0$. It is assumed that all neutrinos have left-handed
helicity. It is also assumed that NP affects only the third generation of
leptons. If we assume that, in addition to the SM contribution, only one of
the NP operators is turned on simultaneously and the NP affects only the tau
modes, we can describe the experimental data on $R(D^{(\ast)})$ with the
following values of the new Wilson coefficients:

$\displaystyle V_{L}$ $\displaystyle=$ $\displaystyle-1.33+i\,1.11,\qquad
V_{R}=0.03-i\,0.60,$ $\displaystyle S_{L}$ $\displaystyle=$
$\displaystyle-1.79-i\,0.22,\qquad T_{L}=0.38-i\,0.06.$ (16)

Note that the operator $\mathcal{O}_{S_{R}}$ is excluded at the level of
deviations $2\sigma$.

## DECAYS $B_{c}\to J/\psi+\bar{\ell}\nu_{\ell}$ AND $B_{c}\to J/\psi+K(\pi)$

The LHCb collaboration reported relationship measurements for the following
branchings [9, 10, 4]:

$\displaystyle\mathcal{R}_{K^{+}/\pi^{+}}$ $\displaystyle=$
$\displaystyle\frac{\mathcal{B}(B_{c}^{+}\rightarrow J/\psi
K^{+})}{\mathcal{B}(B_{c}^{+}\rightarrow
J/\psi\pi^{+})}=\left\\{\begin{array}[]{l}0.069\pm 0.019({\rm stat})\pm
0.005({\rm syst})\\\ 0.079\pm 0.007({\rm stat})\pm 0.003({\rm
syst})\end{array}\right.$ (19) $\displaystyle\mathcal{R}_{J/\psi}$
$\displaystyle=$ $\displaystyle\frac{\mathcal{B}(B_{c}^{+}\rightarrow
J/\psi\tau^{+}\nu_{\tau})}{\mathcal{B}(B_{c}^{+}\rightarrow
J/\psi\mu^{+}\nu_{\mu})}=0.71\pm 0.17(\rm stat)\pm 0.18(\rm syst)$ (20)

|
---|---
Figure 1:  Graphical representation of semileptonic and nonleptonic decays of a $B_{c}$ meson. |
---|---
Figure 2: Left panel: comparison of theoretical predictions for the ratio
$\mathcal{R_{J/\psi}}$ with LHCb data [12]. The solid line is the central
experimental value; the dotted lines are the experimental error bar. Right
panel: comparison of theoretical predictions for the ratio
$\mathcal{R_{K^{+}/\pi^{+}}}$ with LHCb data [9, 10]. Two solid lines are
central experimental values, dashed-and-dotted lines are the experimental
error bar from [9], and dotted lines are the experimental error bar from [10].

In work [11], these ratios were calculated in of the SM using the necessary
form factors calculated in the covariant quark model. The results were
compared with other theoretical approaches. It turned out that the theoretical
predictions of the ratio $\mathcal{R}_{J/\psi}$ were more than 2$\sigma$ less
experimental data. This may indicate the possibility of NP effects in this
decay, by analogy with the relation $\mathcal{R}_{D^{(\ast)}}$. At the same
time, predictions for the ratio $\mathcal{R}_{K/\pi}$ are in good agreement
with the experimental data. This may be a fairly convincing indication that
the possible effects of NP manifest themselves in the lepton sector, leading
to a violation of lepton universality, rather than in the hadron sector.

Manifestation of new physics in decays $B_{c}\to(J/\psi,\eta_{c})\tau\nu$:

Work [13] provided a detailed analysis of the decays
$B_{c}\to(J/\psi,\eta_{c})\tau\nu$ taking into account the NP operators.
Constraints on the Wilson coefficients in the effective Hamiltonian equation
(14) and taking into account the NP effects in the tauon sector can be
obtained with the simultaneous use of experimental data for the branching
ratios $R_{D}=0,407\pm 0,046$ and $R_{D^{\ast}}=0,304\pm 0,015$ [14] and
$R_{J/\psi}=0,71\pm 0,25$ [12]. It should be noted that, in the SM, our
calculation gives $R_{D}=0,267$, $R_{D^{\ast}}=0,238$, and $R_{J/\psi}=0,24$.
We take into account the theoretical error in $10\%$ for our ratios. In
addition, we assume the dominance of only one NP operator in addition to the
SM contribution, which means that only one NP Wilson coefficient is considered
at any time.

|  |
---|---|---
|  |
Figure 3: Restrictions on Wilson coefficients $V_{R}$, $V_{L}$ and $T_{L}$
from ratio dimensions $R_{J/\psi}$, $R_{D}$ and $R_{D^{\ast}}$ within
$1\sigma$ (top panels) and $2\sigma$ (bottom panels), as well as from the
restrictions on branching $\mathcal{B}(B_{c}\to\tau\nu)$ (dashed curve).

The top panels of Fig. 3 present the restrictions on the vector $V_{L,R}$ and
tensor $T_{L}$ Wilson coefficients. Within $1\sigma$ there is no place for
these coefficients. Moreover, they are excluded mainly due to an additional
restriction from $R_{J/\psi}$, not from $\mathcal{B}(B_{c}\to\tau\nu)$. This
is true in case $T_{L}$, since the operator $\mathcal{O}_{T_{L}}$ does not
affect $\mathcal{B}(B_{c}\to\tau\nu)$. The bottom panels in Fig. 3 show the
allowed areas for $V_{L,R}$ and $T_{L}$ within $2\sigma$. In each allowed
area, $2\sigma$ we find the best value for each NP connection. The best fit is
achieved with the following values $V_{L}=-1,05+i1,15$, $V_{R}=0,04+i0,60$,
$T_{L}=0,38-i0,06$. In the figure, these values are marked with an asterisk.

|  $<R_{\eta_{c}}>$ |  $<R_{J/\psi}>$
---|---|---
SM |  $0.26$ |  $0.24$
$V_{L}$ |  $(0.28,0.39)$ |  $(0.26,0.37)$
$V_{R}$ |  $(0.28,0.51)$ |  $(0.26,0.37)$
$T_{L}$ |  $(0.28,0.38)$ |  $(0.24,0.36)$
Table 1: Ratios averaged over $q^{2}$ branchings in the SM and in the presence
of NP.

Ratios averaged over $q^{2}$ branchings $R_{J/\psi}$ and $R_{\eta_{c}}$ are
shown in Table 1. The row labeled SM contains our predictions obtained within
the SM using the form factors computed in our model. Predicted ranges of
ratios in the presence of NP are given according to allowed areas $2\sigma$ NP
coefficients shown in Fig. 3. It can be seen that the most noticeable effect
is given by the operator $\mathcal{O}_{V_{R}}$, which can increase the average
ratio $<R_{\eta_{c}}>$ twofold.

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