[a,b]Giulia Ricciardi

# Semileptonic $B$ decays and $|V_{xb}|$ update

###### Abstract

We review the status of semileptonic $B$ decays and $|V_{xb}|$ determinations.

##  1 Introduction

Semileptonic $B$ decays are the processes of election when it comes to a
precise determination of the parameters $|V_{xb}|$ of the Cabibbo-Kobayashi-
Maskawa (CKM) matrix. At the lowest order in the SM, they are mediated by tree
level quark decays, and the presence of leptons in the final states simplifies
the QCD analyses, since hadronic and leptonic currents factorize. Precision
studies of semileptonic $B$ meson decays have been made possible by the large
samples of $B$ mesons collected at the $B$-factories and at LHCb, and by a
concomitant progress in theoretical computations, especially in lattice QCD.
Since at least four decades, the parameters $|V_{xb}|$ play a central role in
the analyses of the unitarity triangle and in testing the Standard Model (SM).
A long-standing tension among their values, depending on whether they are
extracted using exclusive or inclusive semi-leptonic $B$ decays, has been an
additional motivation for more and more refined theoretical and experimental
analyses. We briefly review recent progress on the study of semileptonic $B$
decays and on $|V_{xb}|$ determinations 111For brief overviews see for example
[1, 2, 3, 4, 5, 6] and references therein..

##  2 Inclusive decays $B\to X_{c}\,\ell\nu_{\ell}$

In inclusive $B\rightarrow X_{c}\,\ell\,\nu_{\ell}$ decays, the final state
$X_{c}$ is an hadronic state originated by the charm quark. The large
hierarchy between the intrinsic large ’dynamic’ scale of energy release and
the hadronic soft scale leads naturally to $\Lambda_{QCD}/m_{b}$ as an
expansion parameter of non-perturbative effects. The expansion for the total
semileptonic width takes the form

$\Gamma(B\rightarrow
X_{c}l\nu)=\frac{G_{F}^{2}m_{b}^{5}}{192\pi^{3}}|V_{cb}|^{2}\left[c_{3}\langle
O_{3}\rangle+c_{5}\frac{\langle O_{5}\rangle}{m_{b}^{2}}+c_{6}\frac{\langle
O_{6}\rangle}{m_{b}^{3}}+O\left(\frac{\Lambda^{4}_{QCD}}{m_{b}^{4}},\;\frac{\Lambda^{5}_{QCD}}{m_{b}^{3}\,m_{c}^{2}},\dots\right)\right]$
(1)

Here $c_{d}$ ($d=3,5,6\dots$) are short distance coefficients, calculable in
perturbation theory as a series in the strong coupling $\alpha_{s}$, and
$O_{d}$ denote local operators of (scale) dimension $d$. The hadronic
expectation values of the operators are the (normalized) forward matrix
elements, encode the nonperturbative corrections and can be parameterized in
terms of heavy quark expansion (HQE) parameters, whose number grows with
powers of $\Lambda_{QCD}/m_{b}$. Similar expansions hold for sufficiently
inclusive quantities as the moments of distributions of charged-lepton energy,
hadronic invariant mass and hadronic energy.

At order $1/m_{b}^{0}$ in the HQE, the perturbative corrections up to order
$\alpha_{s}^{2}$ to the width and to the moments of the lepton energy and
hadronic mass distributions are known completely [7, 8, 9, 10, 11, 12, 13].
The terms of order $\alpha_{s}^{n+1}\beta_{0}^{n}$, where $\beta_{0}$ is the
first coefficient of the QCD $\beta$ function, $\beta_{0}=(33-2n_{f})/3$, have
also been computed following the Brodsky-Lepage-Mackenzie procedure [10, 14].
Perturbative corrections to the coefficients of the kinetic operator [15, 16]
and the chromomagnetic operator [17, 18, 19] have been evaluated at order
$\alpha_{s}$. Two independent parameters, $\rho^{3}_{D,LS}$, are also needed
to describe matrix elements of operators of dimension six, that is at order
$1/m_{b}^{3}$. Their coefficients have long been known at tree level [20], and
more recently $\alpha_{s}$ corrections to the coefficient of the
$\rho^{3}_{D}$ term have been computed [21]. Starting at order
$\Lambda_{QCD}^{3}/m_{b}^{3}$, terms with an infrared sensitivity to the charm
mass appear, at this order as a $\log m_{c}$ contribution [22, 23, 24].
Presently, the matrix elements have been identified and estimated up to the
order $1/m_{b}^{4}$ and $1/m_{b}^{5}$ [25, 26, 27].

##  3 Inclusive $|V_{cb}|$ determination

The shapes of the kinematic distributions in the $B\to X_{c}\ell\nu$ decays
are sensitive to the masses of the $b$ and $c$ quarks and the non-perturbative
HQE parameters, and all these quantities are affected by the particular
theoretical scheme used to define the quark masses. Non perturbative
parameters can be extracted together with $|V_{cb}|$ in a global fit based on
experimentally measured distributions and momenta. Global fit analyses differ
by the data sets they are based onto, the theoretical scheme employed, and the
order of truncation of the HQE expansion. Challenges are experimental
selections applied to the data as well as to properly account for
correlations.

A recent global analysis of the inclusive $B\to X_{c}\ell\nu$ has been done by
HFLAV [28]. In the framework of kinetic scheme, $|V_{cb}|$ is extracted
together with the $b$ and $c$ quark masses and 4 non-perturbative parameters
(namely $\mu^{2}_{\pi}$, $\mu^{2}_{G}$, $\rho^{3}_{D}$ and $\rho^{3}_{LS}$).
Details on the extraction can be find for instance in Ref. [1]. The resulting
value is

$|V_{cb}|=(42.19\pm 0.78)\times 10^{-3}$ (2)

where the quoted uncertainty includes both the experimental and the
theoretical uncertainties. It is worth to mention that the latter ones are
dominating. In this analysis, the excellent fit quality points toward the
validity of the HQE fit, but the small $\chi^{2}$ per degree of freedoms of
$\chi^{2}/ndf=0.32$, could be a signal of some overestimated theoretical
uncertainties, or overestimated correlations between the various moments.

##  4 Exclusive $B\to D^{(\ast)}$ decays into light leptons

One can express the differential ratios for the semi-leptonic CKM favoured
decays $B\to D^{(\ast)}\ell\nu$ in terms of the recoil parameter
$\omega=p_{B}\cdot p_{D^{(\ast)}}/m_{B}\,m_{D^{(\ast)}}$, which corresponds to
the energy transferred to the leptonic pair. For negligible lepton masses
($\ell=e,\mu)$, one has

$\displaystyle\frac{d\Gamma}{d\omega}(B\rightarrow D^{\ast}\,\ell{\nu})$
$\displaystyle\propto G_{F}^{2}(\omega^{2}-1)^{\frac{1}{2}}|V_{cb}|^{2}{\cal
F}(\omega)^{2}$ (3) $\displaystyle\frac{d\Gamma}{d\omega}(B\rightarrow
D\,\ell{\nu})$ $\displaystyle\propto
G_{F}^{2}\,(\omega^{2}-1)^{\frac{3}{2}}\,|V_{cb}|^{2}{\cal G}(\omega)^{2}$

In the heavy quark limit, both form factors are related to a single Isgur-Wise
function, ${\cal F(\omega)}={\cal G(\omega)}={\cal\xi(\omega)}$, which is
normalized to unity at zero recoil, that is ${\cal\xi(\omega}=1)=1$. There are
non-perturbative corrections to this prediction, expressed at the zero-recoil
point by the heavy quark symmetry under the form of powers of
$\Lambda_{QCD}/m$, where $m=m_{c}$ and $m_{b}$. Other corrections are
perturbatively calculable radiative corrections from hard gluons and photons.
Latest estimates for zero recoil form factors come from lattice, and are
reported in Table 1. We have also listed the form factors for
$B_{s}\rightarrow D_{s}^{(\ast)}\ell\nu_{\ell}$ decays, whose lattice
computations are more advantageous, because of the larger mass of the valence
$s$ quark compared to $u$ or $d$ quarks. Since the very recent LHCb
measurement [29], these decays supply a new method for precisely determining
$|V_{cb}|$.

Collaboration | Refs. | ${\cal F}(1)$ | Refs. | ${\cal G}(1)$
---|---|---|---|---
FNAL/MILC | [30] | $0.906\pm 0.004\pm 0.012$ | [31] | $1.054\pm 0.004\pm 0.008$
HPQCD | [32] | $0.895\pm 0.010\pm 0.024$ | [33] | $1.035\pm 0.040$
HPQCD | [34] | $0.914\pm 0.024$ |  |
|  |  ${\cal F}^{B_{s}\to D^{\ast}_{s}}$(1) |  |  ${\cal G}^{B_{s}\to D_{s}}$(1)
HPQCD | [34] | $0.9020\pm 0.0096\pm 0.0090$ | [35] | $1.068\pm 0.004$
Atoui et al. |  |  | [36] | $1.052\pm 0.046$
Table 1: Latest lattice form factor estimates at zero recoil (From Ref. [1]).

In the computation of the form factors, the advantage provided by the heavy
quark symmetries has the hindrance that the differential rates in (3) vanish
at zero recoil. Thus one needs to extrapolate the experimental points taken at
$\omega\neq 1$ to the zero recoil point $\omega=1$, using a parameterization
of the dependence on $\omega$ of the form factors, which introduces additional
uncertainties. Commonly used parameterizations are the CLN (Caprini-Lellouch-
Neubert) [37], the BGL (Boyd-Grinstein-Lebed) [38] and the BCL (Bourrely-
Caprini-Lellouch) [39] parameterizations. In all of them, $\omega$ is mapped
onto a complex variable $z$ via a conformal transformation; form factors are
written in the form of an expansion in $z$, which converges rapidly in the
kinematical region of heavy hadron decays.

Form factor estimates via zero recoil sum rules [40, 41, 42] give, in general,
relatively higher values of $|V_{cb}|$ and a theoretical error more than twice
the error in the lattice determinations. Recent progress includes form factors
determinations from $B$ meson light-cone sum rules (LCSR) beyond leading twist
in the $B\to D^{\ast}\ell\nu$ channel [43].

##  5 Exclusive $|V_{cb}|$ determination

In 2017, for the first time, the unfolded fully-differential decay rate and
associated covariance matrix have been published, by the Belle collaboration
[44], prompting independent determinations of $|V_{cb}|$ [45, 46, 47, 48, 49,
32]. Based on the Belle measurement [44], the accuracy of the CLN
parameterization was questioned in both $B\to D\,l\nu$ [50] and $B\to
D^{\ast}\,l\nu$ [46, 45] channels, in favour of the BGL one [45, 46, 47].
Higher central values (closer to the inclusive values) were found in the
latter approach, and the possibility to have solved the long standing
inclusive/exclusive tension was aired. However, one year later, more data were
provided by Belle [51] and in 2019 by BaBar [52]. Both these analyses showed
no sign of discrepancy on $|V_{cb}|$ between the BGL and CLN
parameterizations, within the uncertainties. Belle also in this case released
the data in a format that allows them to be fitted by outside groups,
prompting a new analysis by some among the authors of the 2017 fits [53],
which gave consistent results with both CLN and BGL parameterizations, in
different configurations. The situation described above is summarized in Fig.
1 222For details refer to Ref. [1].. The exclusive values are compared with
the inclusive HFLAV average of Eq. (2). A recent estimate from QCD LCSR [54]
is reported as well, together with the recent $|V_{cb}|$ determination by
LHCb, the first one at a hadron collider and the first one to use $B^{0}_{s}$
decays [29].

Figure 1: Recent $|V_{cb}|$ determinations (see text).

The ultimate way out of parametrization dependence is to compute the form
factors at nonzero recoil values. In the case of $B\to D\ell\nu$ decays, the
form factors in the unquenched lattice-QCD approximation have been made
available for a range of non-zero recoil momenta since 2015 by the FNAL/MILC
[31] and the HPQCD Collaboration [33]. Both estimates are in good agreement.
Recently, the HPQCD Collaboration has presented a lattice QCD determination of
the $B_{s}\rightarrow D_{s}\ell\nu_{\ell}$ scalar and vector form factors over
the full physical range of momentum transfer [55].

The LHCb collaboration has measured the ratio of the branching fractions
$\Lambda^{0}_{b}\to p\mu^{-}\bar{\nu}_{\mu}$ and
$\Lambda^{0}_{b}\to\Lambda_{c}^{+}\mu^{-}{\bar{\nu}}_{\mu}$ [56], from which
they have determined the first direct measurement of the ratio
$|V_{ub}|/|V_{cb}|$. Even if this is not a direct measurement of $|V_{cb}|$,
by taking $|V_{ub}|$ from external inputs it is possible to determine
$|V_{cb}|$. The LHCb analysis, besides being the first one to use $B$-baryon
decays, has paved the way to extract $|V_{ub}|/|V_{cb}|$ from the ratio
${\cal{B}}(B_{s}^{0}\to K^{-}\mu^{+}\nu_{\mu})/{\cal{B}}(B_{s}^{0}\to
D_{s}^{-}\mu^{+}\nu_{\mu})$[57].

##  6 Inclusive $|V_{ub}|$ determination

In order to extract $|V_{ub}|$ from semileptonic $B\to X_{u}\ell\nu$ decays
one has to reduce the $b\to c$ semileptonic background through experimental
cuts. Such cuts enhance the relevance of the so-called threshold region in the
phase space, jeopardizing the use of HQE. In order to face this problem, that
is absent in the inclusive determination of $|V_{cb}|$, different theoretical
schemes have been devised, which are tailored to analyze data in the threshold
region, but differ in their treatment of perturbative corrections and the
parametrization of non-perturbative effects. In Table 2 we present the results
of four theoretical different approaches, that is ADFR [58, 59, 60], BLNP [61,
62, 63], DGE[64] and GGOU [65], as analyzed by BaBar [66, 67], Belle [68, 69]
and HFLAV [70] collaborations. The most recent (2016) averaged HFLAV
determinations [70] are the ones used in the latest (2019) update of PDG [71].
The results are consistent within the uncertainties.

Inclusive decays |  | $|V_{ub}|\times 10^{3}$ |  |
---|---|---|---|---
|  ADFR |  BNLP |  DGE |  GGOU
Belle 2020 [69] | $4.01^{+0.08,+0.15,+0.18}_{-0.08,-0.16,-0.18}$ | $4.01^{+0.08,+0.15,+0.18}_{-0.08,-0.16,-0.19}$ | $4.12^{+0.08,+0.16,+0.11}_{-0.09,-0.16,-0.12}$ | $4.11^{+0.08,+0,16,+0.08}_{-0.09,-0,16,-0.09}$
HFLAV 2016 [70] | $4.08\pm 0.13^{+0.18}_{-0.12}$ | $4.44\pm 0.15^{+0.21}_{-0.22}$ | $4.52\pm 0.16^{+0.15}_{-0.16}$ | $4.52\pm 0.15^{+0.11}_{-0.14}$
BaBar 2011 [66] | $4.29\pm 0.24^{+0.18}_{-0.19}$ | $4.28\pm 0.24^{+0.18}_{-0.20}$ | $4.40\pm 0.24^{+0.12}_{-0.13}$ | $4.35\pm 0.24^{+0.09}_{-0.10}$
Belle 2009 [68] | $4.48\pm 0.30^{+0.19}_{-0.19}$ | $4.47\pm 0.27^{+0.19}_{-0.21}$ | $4.60\pm 0.27^{+0.11}_{-0.13}$ | $4.54\pm 0.27^{+0.10}_{-0.11}$
Table 2: Status of inclusive $|V_{ub}|$ determinations.

The most recent estimates (2020, still preliminary), provided by Belle [69],
are also included in Table 2: the uncertainties are statistical, systematic
and from the theory calculation, respectively. Their arithmetic average of the
results obtained from the four different QCD predictions gives [69]
$|V_{ub}|=(4.06\pm 0.09\pm 0.16\pm 0.15)\times 10^{-3}$. This value is smaller
than the previous inclusive measurements, reducing the discrepancy with the
exclusive measurement of about 2-3 to 1.4 standard deviations.

##  7 Exclusive $|V_{ub}|$ determination

The CKM-suppressed decay $B\to\pi\ell\nu$ with light final leptons is the
typical exclusive channel used to extract $|V_{ub}|$. It is well-controlled
experimentally and several measurements have been performed by both BaBar and
Belle collaborations [72, 73, 74, 75, 76, 77, 78]. Since the $u$-quark is not
heavy, heavy quark symmetries are not as binding as in $b\to c$ decays.
Lattice determinations of the form factors in this channel have been obtained
by the HPQCD [79, 80], the Fermilab/MILC [81, 82] and the RBC/UKQCD [83]
collaborations. The HFLAV $|V_{ub}|$ determination comes from a combined fit
of a $B\to\pi$ form factor parameterization to theory predictions and the
average $q^{2}$ spectrum in data. The theory input included in the fit are the
results from the FLAG lattice average [84] and the LCSR result at $q^{2}$ = 0
GeV2 [85]. The results give [28] $|V_{ub}|=(3.67\pm 0.09\pm 0.12)\times
10^{-3}$, where the first error comes from the experiment and the second one
from the theory.

Decay modes with charmless meson states heavier than the charged pion have
been studied to a lesser extent; in general they present more challenges in
the experimental reconstruction due to possibly higher backgrounds, more
complex form-factor dependencies and wider decay widths. No lattice unquenched
QCD calculation of their form factors is available yet, exception made for
preliminary results from the SPQcdR Collaboration [86]. The LCSR computation
of form factors for $B\to\rho/\omega\,\ell\bar{\nu}_{\ell}$ decays has been
exploited to estimate $|V_{ub}|$ [87, 88]. The $B\to\eta^{\prime}$ form
factors have also been computed in the LCSR framework [94]. Measurements of
the branching fractions for $B^{+}\to\eta/\eta^{\prime}\ell\nu_{\ell}$, where
$\ell$ stands for either an electron or a muon, have been reported by CLEO
[89, 90], BaBar [74, 91, 92, 77] and Belle [93],

##  8 The $|V_{cb}|$ and $|V_{ub}|$ puzzles

The discrepancy between inclusive and exclusive values of $|V_{cb}|$
($|V_{ub}|$) is generally referred to as the $|V_{cb}|$ ($|V_{ub}|$) puzzle.
In Fig. 2 HFLAV [28] exclusive and inclusive determinations of $|V_{cb}|$ are
summarized and compared with the analogous determinations of $|V_{ub}|$. The
magenta and green vertical bands represent the two different exclusive
determinations of $|V_{cb}|$: both show a discrepancy with the striped
vertical band, which is relative to the $|V_{cb}|$ inclusive determination in
the kinetic scheme. The bands relative to the exclusive $B\to D$ and $B\to
D^{\ast}$ decays are the HFLAV averages done with the CLN parameterizations.
Also considering the slightly larger uncertainty associated with the BGL fit,
the discrepancy with the inclusive determination remains significant,
amounting to about 3$\sigma$. The grey vertical band corresponds to the LHCb
result with $B_{s}\to D_{s}^{(*)}\mu\nu_{\mu}$ decays: although affected by
large uncertainties, it is compatible with both inclusive and exclusive
determinations of $|V_{cb}|$. The oblique bands represent the constraints on
the $|V_{ub}|/|V_{cb}|$ ratio determined by LHCb in baryon decays [56] and
more recently in $B_{s}^{0}\to K^{-}\mu^{+}\nu_{\mu}$ decays [57].

In Fig. 2, the blu horizontal band gives the exclusive $|V_{ub}|$
determination from semileptonic $B\to\pi$ decay. The values for inclusive
$|V_{ub}|$ with different QCD calculations are given by the four points with
vertical error bars. The shift along the x-axis of these four points is just
arbitrary and has no meaning. By considering their arithmetic average, and the
latest Belle results [69], the observed $|V_{ub}|$ discrepancy reduces from
2-3 to 1.4 standard deviations, as discussed in Sect.6.

It is also possible to determine $|V_{cb}|$ and $|V_{ub}|$ indirectly, using
the CKM unitarity relations together with CP violation and flavour data,
excluding direct information on decays, as done by the CKMfitter [95] and by
the UTfit Collaborations [96].

Figure 2: $V_{ub}-V_{cb}$ HFLAV averages.

Future prospects seems to be promising. Belle II is expected to reduce the
experimental uncertainty on inclusive and exclusive $|V_{cb}|$ to 1% and 1.5%,
and on inclusive and exclusive $|V_{ub}|$ to 3% and 2%, respectively [97]. A
synergy between theoretical advances, Belle II and the upgraded LHCb may well
say the final word on the present puzzles.

## Acknowledgements

The author thanks M. Rotondo for useful discussions. This work received
partial financial support from MIUR under Project No. 2015P5SBHT and from the
INFN research initiative ENP.

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