1

76. Semileptonic b-Hadron Decays, Determination of Vcb , Vub

76. Semileptonic b-Hadron Decays, Determination of Vcb , Vub
Revised August 2023 by A.X. El-Khadra (Physics, Illinois U.) and P. Urquijo (School of Phys. U.
of Melbourne).

76.1 Introduction
Precision determinations of |Vub | and |Vcb | are central to testing the CKM sector of the Standard
Model, and complement the measurements of CP asymmetries in B decays. The length of the sideof the unitarity triangle opposite the well-measured angle β is proportional to the ratio |Vub |/|Vcb |;its precise determination is a high priority of the heavy-flavor physics program.
The transitions b → c`ν̄` and b → u`ν̄` (` = e, µ) each provide two avenues for determining
these CKM matrix elements, namely through measurements of inclusive decay rates, B̄ → X`ν̄`with a sum over all possible hadronic states X or of exclusive rates, where the final state hadron isa specific meson (X = D, D∗ , π, ρ etc.).
Purely leptonic decays, such as Bc− → τ ν̄, B − → τ ν̄, and B − → µν̄, provide a third avenue thatis theoretically very simple (see the RPP mini-review [1]). However, we do not use this information
at present since none of the measurements have reached a competitive level of precision. Hencethe results presented here are solely obtained from exclusive and inclusive semileptonic b-hadron
decays. This article and the values quoted here update the previous review [2].
The theoretical methods underlying the different determinations of |Vqb | are quite mature. The
theoretical approach for inclusive determinations uses the fact that the mass mb of the b quark islarge compared to the scale ΛQCD that determines low-energy hadronic physics. Thus the basis for
precise calculations is a systematic expansion in powers of Λ/mb , where Λ ∼ 500 − 700 MeV is a
hadronic scale of the order of ΛQCD . Such an expansion can be formulated in the framework of aneffective field theory which is described in a separate RPP mini-review [3]. Exclusive determinations
rely on non-perturbatively calculated form factors, that encode the low-energy dynamics of thehadronic transition. Here, lattice QCD provides an, in principle, ab-initio method to calculate thenon-perturbative QCD contributions to the exclusive decay amplitudes. Thanks to a combinationof improved theoretical methods, better algorithms, and large increases in computational power,
precise lattice QCD results are now available for the processes that are used in exclusive |Vub | and|Vcb | determinations. State-of-the-art lattice QCD calculations are based on gauge-field ensembles
that include realistic sea quark effects for degenerate up/down and strange quarks (aka 2 + 1flavor ensembles), and increasingly, also for charm (aka 2 + 1 + 1-flavor ensembles). They employ
ensembles generated at three (or more) lattice spacings, different spatial volumes, among otherparameter variations to allow for quantification of the underlying systematic errors, and include
detailed systematic error analyses. This is described in a separate RPP mini-review [4]. The latticeQCD results discussed in this review are all state-of-the art with fully quantified uncertainties, andtherefore play a central role in exclusive |Vqb | determinations. In the case of exclusive B̄ → D(∗) `ν̄`
decays, heavy quark symmetry (HQS) and heavy quark effective theory (HQET) yield constraints
on the form factors that can be used to improve exclusive |Vcb | determinations, especially whenlattice QCD form factor results are incomplete, as was the case until very recently for B̄ → D∗ `ν̄` .
Light-cone sum rules (LCSR) provide another nonperturbative approach to compute form factors.
However, while the lattice-QCD results employed in this review have well-quantified uncertainties,LCSR results suffer, in general, from hard-to-quantify systematic errors. The two methods typicallyprovide results in opposite regions of phase space. Lattice-QCD calculations work best at low
hadronic recoil or high momentum transfer (q 2 ) to the leptons, while LCSR calculations provide
estimates at high recoil, near q 2 = 0. In general, we don’t use form factor inputs obtained fromS. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)
31st May, 2024 10:21am

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76. Semileptonic b-Hadron Decays, Determination of Vcb , Vub

LCSR in this review.
The measurements discussed in this review are of branching fractions, ratios of branching fractions, and decay kinematic distributions. The determinations of |Vcb | and |Vub | also require ameasurement of the total decay widths of the corresponding b hadrons, determined from lifetimes,
which is the subject of a separate RPP mini-review [5]. The measurements of inclusive semileptonic decays relevant to this review come primarily from e+ e− B factories operating at the Υ (4S)
resonance, while the measurements of exclusive semileptonic decays come from both the e+ e− Bfactories and from the LHCb experiment at CERN.
Semileptonic B-meson decay amplitudes to electrons and muons are well measured and consistent with Standard Model W -boson exchange. As they are expected to be insensitive to the effects
of non-Standard-Model physics, they are used to extract |Vqb |. However, semileptonic decays totau-lepton final states, such as B̄ → D(∗) τ ν̄τ , may be sensitive to effects from beyond the Standard
Model particles due to the large mass of the τ lepton. The currently observed tensions between
Standard Model theory and experiment for these decays indicate that semitauonic decays must
be studied further. For rare decays, tests of lepton-flavor universality violations (LFUV) in ratios
involving µ/e final states provide CKM-free tests of the SM. These and other flavor dependent
angular analyses are also included in this review.
Many of the numerical results quoted in this review have been provided by the Heavy Flavor
Averaging Group (HFLAV) [6].

76.2 Determination of |Vcb |Summary: The determination of |Vcb | from inclusive decays has a relative uncertainty of
about 1.5%; the limitations arise mainly from our ignorance of higher-order perturbative and nonperturbative corrections. Exclusive B̄ → D∗ `ν̄` decays provide a determination of |Vcb | with arelative precision of about 2%, with comparable contributions from theory and experiment to thetotal uncertainty; the value determined from B̄ → D`ν̄` decays is consistent and has an uncertaintyof 2.5%. The values obtained from the inclusive and exclusive B decay determinations discussed
below are:|Vcb | = (42.2 ± 0.5) × 10−3(inclusive)(76.1)−3(exclusive).

(76.2)|Vcb | = (39.8 ± 0.6) × 10

An average of these determinations has p(χ2 ) = 1%, so we scale the error by|Vcb | = (41.1 ± 1.2) × 10−3

(average).

pχ2 /1 = 3.0 to find(76.3)

Given the only marginal consistency, of approximately 3.0 σ, this average should be treated withcaution.
76.2.1 |Vcb | from exclusive decays
Exclusive determinations of |Vcb | make use of semileptonic B decays into the ground statecharmed mesons D and D∗ . The corresponding hadronic matrix elements can be parameterized
in terms of six independent form factors, which depend on the variable w ≡ v · v 0 , where v andv 0 are the four velocities of the initial and final-state hadrons. In the rest frame of the decay
this variable corresponds to the Lorentz factor of the final state D(∗) meson. Determinations of|Vcb | from experimental measurements of B̄ → D(∗) `ν̄` decay rates require precise knowledge ofthese form factors. Fortunately, lattice QCD results for all relevant form factors, including theirrecoil dependence, are now available [7–12]. Heavy Quark Symmetry (HQS) [13, 14] predicts thatin the infinite mass limit the six form factors collapse into a single one, which is normalized atthe “zero recoil point” w = 1, the point of maximum momentum transfer to the leptons. Heavy
31st May, 2024

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76. Semileptonic b-Hadron Decays, Determination of Vcb , Vub

Quark Effective Theory (HQET) provides a framework for obtaining the corrections to the HQS
prediction in a systematic expansion in powers of ΛQCD /mc which is discussed in a separate RPPmini-review [3].
76.2.2 B̄ → D ∗ `ν̄`The decay rate for B̄ → D∗ `ν̄` , assuming massless leptons, is given bydΓ
G2 m5(B̄ → D∗ `ν̄` ) = F 3B |Vcb |2 |ηEW |2 (w2 − 1)1/2 P (w)|F(w)|2 ,dw
48π(76.4)

where P (w) is a phase space factor,
3

2

P (w) = r (1 − r) (w + 1)

2

4w 1 − 2rw + r2
1+
w + 1 (1 − r)2!

,(76.5)

with r = mD∗ /mB . The decay amplitude F(w) can be expressed in terms of the form factors whichparametrize the vector and axial vector current matrix elementshD∗ (v 0 , )|c̄γ µ b|B(v)i= hV (w) εµνρσ vB,ν vD∗ ,ρ ∗σ ,√mB mD∗(76.6)hD∗ (v 0 , )|c̄γ µ γ 5 b|B(v)iµµ∗= ihA1 (w) (1 + w)∗µ − i [hA2 (w)vB+ hA3 (w)vD√∗ ]  · vB ,mB mD∗(76.7)and the ratios
R1 (w) =hV (w),hA1 (w)

R2 (w) =hA3 (w) + r hA2 (w),hA1 (w)(76.8)

asP (w)|F(w)|2 =|hA1 (w)|2
(

×

 )

2
r2 − 2rw + 1
w−1 2
w−1
1
+
R
(w)
+
1
+
(1
−
R
(w))
2
2
(1 − r)2
w+1 1
1−r







.

(76.9)

Note that F at w = 1 is unity by HQS in the infinite-mass limit [15–18]. The effect of assuming
massless leptons is typically very small, but for the muon case can be non-negligible in fits to data
at high hadronic recoil.
The factor ηEW = 1.0066 ± 0.0050 accounts for the leading electroweak corrections to the fourfermion operator mediating the semileptonic decay [19], and includes an estimated uncertainty for
missing long-distance and structure-dependent QED corrections [20]. While some QED radiative
corrections are included in the experimental analysis, the Coulomb correction, which arises when
the final-state hadron is charged, is not taken into account.
The determination of |Vcb | requires knowledge of the normalization, where one commonly uses
a theoretical calculation of the zero recoil input, F(1). Theoretical knowledge of the shapes of theform factors provides crucial compatibility checks between theory and experiment, and improves
the precision of the determinations.
Model-independent shape parametrizations of the form factors make use of analyticity andunitarity constraints and are expressed in terms of the variable√√
w + 1 − 2N√z=√,(76.10)w + 1 + 2N
31st May, 2024

476. Semileptonic b-Hadron Decays, Determination of Vcb , Vuboriginating from a conformal transformation, where N = (t+ − t0 )/(t+ − t− ), t± = (mB ± mD∗ )2 ,and the choice of t0 determines the kinematic point at which z = 0. The form factors (genericallydenoted as F ) may then be written as [21–25]
F (z) =∞
X1
an z nPF (z)φF (z) n=0(76.11)

where the sum is bounded,|an |2 < 1. Furthermore, the function PF (z) takes into account theresonances in the (c̄b) system below the D̄B threshold, and the weighting functions φF (z) are
derived from the unitarity constraint on the corresponding form factor. With the conventional
choice N = 1, z = 0 when w = 1 and kinematic range of the decay corresponds to 0 ≤ z ≤ 0.06. As
a result, only very few terms are needed in the series in z. We refer to Eq. (76.11) as the “BGL”
expansion.
The Caprini-Lellouch-Neubert (CLN) parametrization [26] yields a simple formPhhA1 (w) = hA1 (1) 1 − 8ρ2 z + (53ρ2 − 15)z 2 − (231ρ2 − 91)z 3

i(76.12)

with the slope ρ and normalization hA1 (1) as the only parameters, albeit at the cost of introducing
model dependence. Furthermore, the ratios R1 (w) and R2 (w) are expanded in w − 1. However, thissimple CLN parametrization does not account for higher-order corrections in the 1/mc/b expansion,
which are now relevant [25,27–33]. In addition, the numerical values of the coefficients use outdated
knowledge and should include uncertainties. Thus, this report focuses on recent analyses that
employ different combinations of HQET expansions through O(αs /mb,c , 1/m2b,c ), lattice form factors
at non-zero recoil, as well as the model-independent BGL expansion. Typical fits include up to
three parameters an in Eq. (76.11) for the different form factors.
There now exist three independent lattice QCD calculations by Fermilab/MILC [9], HPQCD[10], and JLQCD [11] of the B̄ → D∗ `ν̄` form factors at non-zero recoil. The three groups each
employ statistically independent gauge-field ensembles and use different lattice actions for the heavy
and light quarks. To obtain results in the continuum and at the physical point, all three analyses
first perform chiral-continuum fits and systematic error studies on the corresponding lattice form
factor data. The resulting continuum form factors are smooth functions (bands) obtained overrecoil ranges which depend on the respective simulation details. In a second step, BGL-expansion
fits are used to obtain the form factors over the entire recoil range. We note that the interplay
between choices for the ChPT-based continuum extrapolations and BGL coefficients needs further
investigation [34]. A comparison of the resulting |F(w)|2 amplitudes is shown in Fig. 76.1. Whilethe lattice results are reasonably consistent in the low-recoil region, there is some variation in theslopes, as is evident from the spread between the bands at large recoil.
Prior lattice calculations obtained results for only the form factor at the zero-recoil point,
F(1), [20, 35] with a total uncertainty at the (1-2)% level. A weighted average of the F(1) resultsfrom Refs. [10, 11, 20] yields
F(1) = 0.903 ± 0.012 .
(76.13)The lattice QCD results in Refs. [9–11] appeared (or were published) after the most recent FLAG
review [36], and will therefore be included in the next FLAG review. Non-lattice estimates basedon zero-recoil sum rules for the form factor tend to yield lower central values for F(1) [37–39]. The
sum rules indicate that F(1) < 0.92 [37, 38, 40, 41], while an explicit estimate that includes excitedstate contributions yields F(1) ≈ 0.86 [39,42]. Lattice-QCD form factors at non-zero recoil provide
valuable shape constraints, and hence enable more refined |Vcb | determinations. However, since
they were not available until recently, some |Vcb | determinations in the literature use the value for31st May, 2024

576. Semileptonic b-Hadron Decays, Determination of Vcb , Vub

Fermilab/MILC
JLQCD
HPQCD

0.8
0.7
0.6
|F(w)| 2

0.5
0.4
0.3
0.2
0.1
1.0

1.1

1.2

w

1.3

1.4

1.5

Figure 76.1: Comparison of lattice QCD results for |F(w)|2 obtained by the Fermilab Lattice andMILC collaborations (red), HPQCD (purple), and JLQCD (yellow). Plot courtesy of A. Vaquero.

F(1) from Eq. (76.13) as the only quantitative lattice input. We discuss these first, before turning
to results for |Vcb | which employ the lattice-QCD form factors at non-zero recoil from Refs. [9–11].
While Light Cone Sum Rules (LCSR) can provide constraints on the form factors at maximum
recoil, the underlying systematic errors are not fully quantified. A modern LCSR calculation [43]quotes uncertainties in the . 20% range for the B̄ → D(∗) `ν̄` form factors at q 2 = 0.
Dispersive methods [44] can also be used to provide additional constraints on form factors
calculated in lattice QCD. Recent work [45, 46] introduces the dispersive matrix method, which
includes a strategy for computing the susceptibilities that are also needed inputs in z-expansion
fits. They are calculable in lattice QCD from two-point correlation functions, which must be
analyzed with the same care as the form factors themselves. The susceptibilities have also been
computed in perturbative QCD in Ref. [47]. The two sets of results in Refs. [46, 47] differ in somecases outside of the quoted error bars, with a significance of up to 2.5σ. A test of the dispersive
matrix approach for the D → K`ν form factors, which can be computed in lattice-QCD over theentire kinematic range, is presented in Ref. [45]. The implementation of the dispersive matrix in
Ref. [48] imposes unitarity constraints on the input data via a so-called unitarity filter. The filter
has the effect of altering the shapes of the form factor bands while reducing the uncertainties fromextrapolating the form factors over the entire kinematic range. Subsequently, the authors of Ref. [48]determine |Vcb | from a bin-by-bin analysis of the experimental data, which avoids the common shape
constraints in joint z-expansion fits. In follow-up work, this method has also been applied to other
31st May, 2024

676. Semileptonic b-Hadron Decays, Determination of Vcb , Vubsemileptonic decay channels [49–51]. A more recent proposal [52] employs Bayesian inference toimplement unitarity constraints, which allows for a straightforward combination of inputs frommultiple lattice calculations while including experimental data in joint z-expansion fits.
Many experiments [53–65] have measured the differential decay rate as a function of w, employing a variety of methods: using either B + or B 0 decays, with or without B-tagging, and with orwithout explicit reconstruction of the transition pion from D∗ → D decays. In Ref. [6] a subsetof the available experimental measurements were input to a four-dimensional fit for ηEW F(1)|Vcb |,ρ2A1 and the form-factor ratios R1 ∝ A2 /A1 and R2 ∝ V /A1 . The fit has a p-value of 0.9%, so wepscale the uncertainty by a factor χ2 /23 to give ηEW F(1) |Vcb | = (35.00 ± 0.49) × 10−3 (CLN).
The leading sources of uncertainty on ηEW F(1) |Vcb | are due to detection efficiencies and D(∗)
decay branching fractions. Note that the B̄ → D∗ `ν̄` form factor in the fit is parameterized using
the CLN form, which has the drawbacks discussed previously.
Using the value from Eq. (76.13) for F(1) and accounting for the electroweak correction gives|Vcb | = (38.5 ± 0.5 ± 0.6) × 10−3 (B̄ → D∗ `ν̄` , LQCD, CLN).

(76.14)

A safer approach is to use the more general BGL form-factor parameterization [22–25, 29]. The
BABAR and Belle experiments have published analyses with BGL based parametrizations at agiven order in the expansion [62–64]. A preliminary measurement from Belle II [65] has also been
performed with a BGL based parametrization. The main results from these measurements are
summarised in Table 76.1 after multiplying through the value of F(1) quoted in the nominal resultof each respective study. These results have not yet been combined by the Heavy Flavor Averaging
Group.
The Belle analysis [62] and the Belle II analysis [65] are both based on untagged approaches in
the mode B̄ 0 → D∗+ `ν̄` and measure 1-d projections in bins of the hadronic recoil w, and angularvariables cos θ` , cos θV , and χ. The BABAR analysis [63] and the recent Belle analysis [64] arebased on hadronic tagged samples using both charged and neutral B decays. BABAR performs
a full 4-d unbinned analysis of neutral and charged B decay modes, while Belle uses unfolded 1-dprojections of w, cos θ` , cos θV , and χ. Only the BGL form factors are directly determined inthese tagged analysis, not the normalization, which is taken from the world average B̄ → D∗ `ν̄`branching fractions.
Table 76.1: F(1)ηEW |Vcb | results from B → D∗ `ν with BGL expansion fits. Note that the tagged
analyses combined results from B − and B̄ 0 decays, while the untagged analyses study only B̄ 0
decays. The final column listed the non-zero recoil lattice QCD data used in each study.
Measurement
Belle [66]Babar [63]Belle [64]Belle II [65]Tag
Untagged
TaggedTaggedUntagged

F(1)ηEW |Vcb | ×103
35.22 ± 0.95
34.98 ± 0.65
37.00 ± 0.56
36.94 ± 0.91

Fit inputs
FNAL/MILC [20]FNAL/MILC [9], HPQCD [10], JLQCD [11]FNAL/MILC [9]The studies in Refs. [62, 64, 65] publish the fully-differential decay rate data and associatedcovariance matrices. The BGL fit results in the Belle, Belle II and BABAR measurements are allconsistent with corresponding results from fits with the CLN parametrization, Eq. (76.14). Eachstudy reports fit results at low order in the three BGL expansion terms when non-zero lattice data
is not used as an additional constraint, ranging from zero-order to second-order. Studies of the
impact of higher order expansions based on the Belle published decay rate data have been reported
31st May, 2024

76. Semileptonic b-Hadron Decays, Determination of Vcb , Vub

7

in Refs. [30,31], as well as in the recent Belle and Belle II studies [64,65], where it is shown that thefit uncertainty on |Vcb | increases substantially with respect to the results reported at lower order,
while the parameter correlations approach 1. This is due to larger number of degrees of freedom
allowed in the higher order expansions, which are not sufficiently constrained without lattice-QCD
inputs at nonzero recoil.
In Ref. [9], BGL expansion fits to the FNAL/MILC lattice QCD form factors show that they
are insensitive to truncation effects beyond quadratic order. These form factors were subsequently
used in Ref. [9] to extract |Vcb | from combined BGL fits to the Belle [62] and synthetic BABAR [63]B̄ → D∗ `ν̄` data, yielding |Vcb | = (38.74 ± 0.78) × 10−3 . The p-values for the combined (experiment+lattice QCD) BGL fits reported in Ref. [9] are small, reflecting tensions between the data
sets. For a fit combining only the Belle data with the lattice-QCD form factors, Ref. [9] quotes|Vcb | = 38.60(86) × 10−3 , while a fit combining the experimental data with only the lattice QCD
F(1) yields |Vcb | = 39.75(92) × 10−3 . Similarly, Refs. [10, 11] present analyses based on the BGL
expansion, combining the respective lattice form factor results with the new Belle data [64]. The
HPQCD collaboration [10] reports |Vcb | = 39.31(74) × 10−3 , while the JLQCD collaboration [11]reports |Vcb | = 39.19(90) × 10−3 . All these determinations are consistent with the result presented
in Eq. (76.15) as well as with other exclusive determinations, where the HPQCD value needs to be
shifted due to their inclusion of the Coulomb factor in ηEW . As noted in Refs. [9–11,64,65], the form
factor shapes and R2 (w) ratio predicted from FNAL/MILC [9] and HPQCD [10] are in tension withexperimental data yielding small p-values, while those from JLQCD [11] are not. Furthermore, theR2 (w) ratio results from Refs. [9,10] are also in tension with HQET expectations (see, for example,
Refs. [30, 32, 33, 67]). These tensions must be understood better in order to improve the precisionof |Vcb | extractions. Indeed, ongoing experimental analyses as well as theoretical work may resolve
them in the near future.
For this review, we take a naive combination of the results in Table 76.1, assuming minimalcorrelations between measurements, and correct for the value of F(1)ηEW . In the future we expect
HFLAV to perform a combined global fit analysis. The combination has a p-value of 8%. The
nominal result for |Vcb | is therefore|Vcb | = (39.7 ± 0.7) × 10−3 (B̄ → D∗ `ν̄` , LQCD, BGL),(76.15)where the uncertainty contains contributions from experimental, lattice QCD, and ηEW sources. We
note that despite the improvements in the experimental data, theoretical inputs, and |Vcb | extraction
methods, this new exclusive determination of |Vcb |, while higher in central value compared to earlier
ones, is still in tension with the inclusive determinations discussed in Section 76.2.5 and summarizedin Eq. (76.32).
76.2.3 B̄ → D`ν̄`
The differential rate for B̄ → D`ν̄` is given bydΓG2F(B̄ → D`ν̄` ) =|Vcb |2 (mB + mD )2 m3D (w2 − 1)3/2 (ηEW G(w))2 .
dw
48π 3

(76.16)

The form factor is defined in terms ofhD(v 0 )|c̄γ µ b|B(v)i= h+ (w) (vB + vD )µ + h− (w) (vB − vD )µ√mB mDand reads
G(w) = h+ (w) −mB − mDh− (w),
mB + mD

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(76.17)(76.18)876. Semileptonic b-Hadron Decays, Determination of Vcb , Vub

where h+ (1) is normalized to unity due to HQS and h− (1) vanishes in the infinite-mass limit. Thus


G(1) = 1 + O

mB − mD
mB + mD

2

ΛQCD
mc

!

(76.19)and the corrections to the HQS prediction are of order ΛQCD /mc in contrast to the case of F(1).
The lattice-QCD result for the normalization, G(1), obtained in Ref. [7] isG(1) = 1.054 ± 0.004 ± 0.008 .

(76.20)We first turn to |Vcb | extractions that rely on only the normalization G(1) from Eq. (76.20).
The most precise measurements of B̄ → D`ν̄` [60, 68, 69] dominate the CLN average [6] value,ηEW G(1)|Vcb | = (41.53 ± 0.98) × 10−3 . Note that this average corresponds to measurements thatare fit to the CLN form factor parameterization; the same concerns expressed above for B̄ → D∗ `ν̄`
apply here. Using the value from Eq. (76.20) for G(1) and accounting for the electroweak correction
as above gives|Vcb | = (39.1 ± 0.9 ± 0.4) × 10−3 (B̄ → D`ν̄` , LQCD, CLN),(76.21)
where the first uncertainty is from experiment, and the second combines the lattice QCD uncertainty
in Eq. (76.20) with the electroweak correction.
For the B̄ → D`ν̄` modes, theoretical input on the shape of the form factor, especially near w ∼1, is especially beneficial to |Vcb | determinations, since experimental measurements of the differential
decays rate near zero recoil are affected by the more severe phase space suppression, compared to
the D∗ case. Hence the best |Vcb | determinations from exclusive B̄ → D`ν̄` decays employ latticeQCD form factors obtained over a range of w ≥ 1 [7, 8], where Ref. [70] provides a lattice averageof the two results. Using the BCL parametrization (z-expansion from Ref. [71]) for the form
factors, they can be combined with binned measurements from Belle [69] and BABAR [68]. Only
Ref. [69] published the full measurement covariance matrix, while Ref. [68] provides the statisticaluncertainty covariance. Nevertheless, Ref. [69] is more precise and dominates the average [70],giving|Vcb | = (40.1 ± 1.0) × 10−3 (B̄ → D`ν̄` , LQCD, BCL).
(76.22)This result is consistent with the value reported in Ref. [47], which is based on the same experimental
and lattice inputs.
The |Vcb | averages from B̄ → D∗ `ν̄` and B̄ → D`ν̄` decays using the BGL (Eq. (76.15)) and BCL(Eq. (76.22)) forms, respectively, are reasonably consistent. The correlations between the latticeuncertainties for B̄ → D∗ `ν̄` and B̄ → D`ν̄` are discussed in Ref. [36], and considered to be 100%for the statistical uncertainty component. We assume an experimental uncertainty correlation oforder 20% and combine the results of Eq. (76.15) and Eq. (76.22), giving|Vcb | = (39.8 ± 0.6) × 10−3

(exclusive).

(76.23)76.2.4 Bs → Ds(∗)− µ+ νµSemileptonic decays of Bs mesons are being studied extensively by the LHCb experiment. On
the theory side, lattice QCD calculations of the corresponding form factors can proceed using thesame methods and gauge-field ensembles as for B-meson decays. In fact, the presence of a strange
spectator quark in the Bs -meson decay amplitudes tends to yield smaller statistical and systematic
errors in the lattice computation. Nevertheless, to-date there are still fewer lattice-QCD resultsfor Bs -meson form factors available than for the B-meson case, but this situation is expected tochange. For the case of the Bs → Ds transition, two lattice QCD calculations of the form factors
31st May, 2024

976. Semileptonic b-Hadron Decays, Determination of Vcb , Vub

over a range of recoil momenta have been performed by the HPQCD collaboration, the first on 2+1flavor gauge field ensembles (generated by the MILC collaboration) using NRQCD b-quarks [72].
The second calculation uses the 2 + 1 + 1-flavor gauge-field ensembles (also generated by MILC),
employing the HISQ action for all valence quarks [73]. The all-HISQ approach avoids the need tocompute renormalization factors separately, at the cost of requiring an extrapolation to the physicalb-quark mass at the lattice spacings used in Ref. [73]. For the Bs → Ds∗− transition, Refs. [35, 74]
provide lattice QCD results for the form factor at zero recoil, where Ref. [74] uses a similar set-up
as Ref. [73] and reads∗
F Bs →Ds (1) = 0.9020(96)stat (90)sys .
(76.24)A lattice QCD calculation of the Bs → Ds∗− form factors at non-zero recoil, employing 2+1+1-flavor
gauge-field ensembles (generated by MILC) in an all-HISQ approach was presented in Ref. [12] andupdated in Ref. [10].
LHCb has extracted |Vcb | from semileptonic Bs0 decays [75]. The measurement uses both Bs0 →Ds− µ+ νµ and Bs0 → Ds∗− µ+ νµ decays. The value of |Vcb | is determined from the observed yields
of Bs0 decays normalized to those of B 0 decays after correcting for the relative reconstruction andselection efficiencies. The normalization channels are B 0 → D− µ+ νµ and B 0 → D∗− µ+ νµ decays,−→ [K + K − ]φ π − ). The
where the D− is reconstructed with the same decay mode of the Ds (D(s)
shape of the form factors are extracted using p⊥ (Ds ), which is the component of the Ds− momentum
perpendicular to the Bs0 flight direction. This variable is correlated with q 2 and the helicity angles
in the Bs0 → Ds∗− µ+ νµ decay. For the Bs → Ds− µ+ νµ decay, for example, |Vcb | is related to themeasured ratio of signal yields, Nsig , and the normalization channel yields, Nref , through therelationNsig
fs B(Ds− → K + K − π − )
1
=ξ
τs
−
+
−
−
0
Nref
fd B(D → K K π ) B(B → D− µνµ )ZdΓ (Bs → Ds µνµ )dw ,dw

where τs is the Bs lifetime, and ξ is the efficiency ratio between the signal and the normalization.
The analysis uses the lattice-QCD form factors obtained for Bs → Ds− over the full kinematic range
in Ref. [73] and the zero-recoil result for the Bs → Ds∗− form factor from Ref. [74]. Fits to both
the CLN parameterization and a 5-parameter version of BGL were performed. The result for |Vcb |,updated with the most recent determination of fs /fd and B(Ds− → K − K + π − ) from Ref. [76], are|Vcb |CLN = (40.8 ± 0.6 ± 0.9 ± 1.1) × 10−3 , and |Vcb |BGL = (41.7 ± 0.8 ± 0.9 ± 1.1) × 10−3 , where thefirst uncertainty is statistical, the second systematic and the third due to the limited knowledge ofthe external inputs, in particular the constant fs /fd × B(Ds− → K + K − π − ). The determination of|Vcb | in Ref. [12], using the lattice form factors obtained therein combined with LHCb data yields
a consistent result, |Vcb | = 42.7(1.5)latt (1.7)exp (0.4)EM × 10−3 . The results obtained are consistent
with the exclusive determinations of |Vcb | using the B 0 and B + decays but are not used in the overall
average in this review at this stage. These channels are important in the measurement of the ratio|Vub |/|Vcb |, which cancels out the Bs normalisation uncertainties, discussed in Section 76.5. LHCb
also perform measurements of the Bs0 → Ds∗− µ+ νµ differential decay rate [77]. Fits are performed
to extract the form factors in CLN and BGL parametrizations.
76.2.5 |Vcb | from inclusive decays
Measurements of the total semileptonic branching decay rate, along with moments of the lepton
energy and hadronic invariant mass spectra in inclusive semileptonic b → c transitions, can beused for a precision determination of |Vcb |. The total semileptonic decay rate can be calculated
quite reliably in terms of non-perturbative parameters that can be extracted from the information
contained in the moments.
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1076. Semileptonic b-Hadron Decays, Determination of Vcb , Vub

76.2.6 Inclusive semileptonic rate
The theoretical foundation for the calculation of the total semileptonic rate is the Operator
Product Expansion (OPE) which yields the Heavy Quark Expansion (HQE) [78, 79]. Details canbe found in the RPP mini-review on Effective Theories [3].
The OPE result for the total rate can be written schematically (details can be found, e.g., in
Ref. [80]) as

G2F m5b (µ)|ηEW |2 ×
3
192π
"


αs (µ) (1)αs (µ) 2 (2)(0)
z0 (r) +z0 (r) + · · ·z0 (r) +ππΓ =|Vcb |2µ2
αs (µ) (1)(0)+ π2 z2 (r) +z2 (r) + · · ·πmbαs (µ) (1)µ2(0)
y2 (r) + · · ·+ G2 y2 (r) +πmbαs (µ) (1)ρ3(0)+ D3 z3 (r) +z3 (r) + · · ·
π
mb



ρ3
αs (µ) (1)(0)+ LS3 y3 (r) +y3 (r) + · · ·
π
mb











#

+ ...

(76.25)where r is the ratio mc /mb and the yi and zi are perturbatively calculable Wilson coefficientsfunctions that appear at different orders of the heavy mass expansion.
The parameters µ2π , µ2G , ρ3D and ρ3LS constitute the non-perturbative input into the heavy quark
expansion; they correspond to certain matrix elements to be discussed below. In the same way the
HQE can be set up for the moments of distributions of charged-lepton energy, hadronic invariant
mass and hadronic energy, e.g.

hEen iEe >Ecut =Z Emax
Ecut

dΓ n
E dEe
dEe e

,Z
Emax
Ecut

dΓ
dEe .
dEe

(76.26)(i)

The coefficients of the HQE are known up to order 1/m5b at tree level [81–84]. The leading term z0
is the parton model, and is known completely through order αs2 [85–87] and now also at αs3 [88]. The
terms of order αsn+1 β0n (where β0 is the first coefficient of the QCD β function, β0 = (33 − 2nf )/3)
have been included following the BLM procedure [80, 89–92]. Corrections of order αs µ2π /m2b havebeen computed in Ref. [93] and Ref. [94], while the αs µ2G /m2b terms have been calculated in Ref. [95]and Ref. [96], and the αs ρ3D /m3b corrections in Refs. [97, 98].
Starting at order 1/m3b contributions with an infrared sensitivity to the charm mass, mc , appear[81, 83, 99, 100]. At order 1/m3b this “intrinsic charm” contribution manifests as a log(mc ) in thecoefficient of the Darwin term ρ3D . At higher orders, terms such as 1/m3b × 1/m2c and αs (mc )1/m3b ×
1/mc appear; numerically these terms may be comparable in size to the contributions of order 1/m4b .
The HQE parameters are given in terms of forward matrix elements of local operators; theµparameters entering the expansion for orders up to 1/m3b are (D⊥= (gµν − vµ vν )Dν , where v =31st May, 2024

1176. Semileptonic b-Hadron Decays, Determination of Vcb , Vub

pB /MB is the four-velocity of the B meson)Λ = MB − mb ,µ2π = −hB|b̄(iD⊥ )2 b|Bi ,µνµ2G = hB|b̄(iD⊥)(iD⊥)σµν b|Bi ,µρ3D = hB|b̄(iD⊥µ )(ivD)(iD⊥)b|Bi ,µ
ν)(ivD)(iD⊥)σµν b|Bi.
ρ3LS = hB|b̄(iD⊥(76.27)

These parameters still depend on the heavy quark mass. Sometimes the infinite mass limits ofthese parameters Λ → ΛHQET , µ2π → −λ1 , µ2G → 3λ2 , ρ3D → ρ1 and ρ3LS → 3ρ2 , are used instead.
Beyond 1/m3 the number of independent HQE parameters starts to proliferate [84,101]. In general,
there are 13 parameters (at tree level) up to order 1/m4 and 31 (at tree level) up to order 1/m5 ,not including Λ . A framework to compute observables at higher order in 1/m [84, 102] has beenused to estimate the HQE parameters of the orders 1/m4b and 1/m5b . Their their impact on the|Vcb | determination has been studied in Ref. [103]. However, as pointed out in Ref. [104] one
may reduce the number of independent parameters in the HQE by exploiting reparametrization
invariance, which is a symmetry of the HQE stemming from Lorentz invariance of QCD. For asubset of observables this allows us to reduce the number of parameters to three up to order 1/m3
(ρ3LS can be absorbed into µ2G by a re-definition) and to 8 up to order 1/m4 [105].
The perturbative QCD expansion for the decay rate does not converge, in general. In addition,
the expansion coefficients in the rates and spectra depend strongly on the scheme used for mb(or equivalently on Λ). It is well known (see eg. [106]) that using the pole mass definition forheavy quark masses leads to particularly badly behaved expansions. This motivates the use of“short-distance” mass definitions, such as the kinetic scheme [37] or the 1S scheme [107–109]. Bothschemes are well suited for the HQE, since they allow the choice of the renormalization scale µ ≤ mb .
Furthermore, in both schemes, the masses can be extracted from other observables with sufficientprecision, enabling a precise determination of |Vcb |, despite of the strong quark-mass dependenceof the total rate.
The 1S scheme eliminates the b quark pole mass by relating it to the perturbative expression
for the mass of the 1S state of the Υ system. The b quark mass in the 1S scheme is half of theperturbatively calculated mass of the 1S state of the Υ system. The best determination of the bquark mass in the 1S scheme is obtained from sum rules for e+ e− → bb̄ [110]. A second alternativeis the so-called “kinetic mass” mkinb (µ), which enters the non-relativistic expression for the kinetic
energy of a heavy quark, and is defined using heavy-quark sum rules [37]. The relation between
S is known through three-loop order [111].
mkin
and mMbb
Finally, we note that the theoretical description of inclusive decays employed here, is based on
the assumption of quark-hadron duality. While there is no evidence of violations of quark-hadron
duality, the theoretical uncertainties due to such violations are not well quantified [3].
76.2.7 Determination of HQE Parameters and |Vcb |Several experiments have measured moments in B̄ → Xc `ν̄` decays [112–120] as a function of theminimum lepton momentum. Recently, moments measurements have been performed as a functionof lepton invariant mass squared, q 2 [121, 122]. The measurements of the moments of the electronenergy spectrum (0th -3rd ) and of the squared hadronic mass spectrum (0th -2nd ) have statisticaluncertainties that are roughly equal to their systematic uncertainties. The 3rd order hadronic mass
spectrum moments have also been measured by some experiments, with relatively large statisticaluncertainty. The sets of moments measured within each experiment have strong correlations; their
31st May, 2024

1276. Semileptonic b-Hadron Decays, Determination of Vcb , Vub

use in a global fit requires fully specified statistical and systematic covariance matrices. Measurements of photon energy moments (0th -2nd ) in B → Xs γ decays [123–127] as a function of theminimum accepted photon energy are also used in some fits; the dominant uncertainties on thesemeasurements are statistical.
Global fits [120, 124, 128–134] to the moments of the electron energy spectrum and the squared
hadronic mass spectrum have been performed in the 1S and kinetic schemes. The semileptonic moments alone determine a linear combination of mb and mc very accurately but leave the orthogonal
combination poorly determined (See e.g. [135]); additional input is required to allow a precise determination of mb . This additional information can come from the radiative B → Xs γ moments (withthe caveat that the OPE for b → sγ breaks down beyond leading order in ΛQCD /mb ), which provide
complementary information on mb and µ2π , or from precise determinations of the charm and beauty
quark masses [36, 70, 136]. The values obtained in the kinetic scheme fits [6, 132, 133] with thesetwo constraints are consistent. Based on the charm and beauty quark mass constraints [36, 134]
MS = 4.198 ± 0.012)GeV a fit in the kinetic scheme [134]mMSc (3 GeV) = 0.988 ± 0.007 GeV, and mb
obtains|Vcb | = (42.16 ± 0.51) × 10−3

(76.28)where the error include experimental and theoretical uncertainties. This analysis [134] includes
calculations of the third-order corrections to the semi-leptonic b → c`ν decay width [88] and mass
relations [111,137] Theoretical uncertainties from higher orders in 1/m as well as in αs are estimatedand included in performing the fits. Similar values for the parameters are obtained with a varietyof assumptions about the theoretical uncertainties and their correlations. The χ2 /dof is well below
unity, which could suggest that the theoretical uncertainties may be overestimated. However, while
one could obtain a satisfactory fit with smaller uncertainties, this would result in unrealisticallysmall uncertainties on the extracted HQE parameters, which are used as input to other calculations(e.g. the determination of |Vub |).
A fit to the measured moments in the 1S scheme [6, 124, 131] gives|Vcb | = (41.98 ± 0.45) × 10−3 ,(76.29)m1S
b = (4.691 ± 0.037) GeV.

(76.30)This fit uses moments measurements from semileptonic and radiative decays and constrains thechromomagnetic operator using the B ∗ -B and D∗ -D mass differences, but does not include theconstraint on mc nor all known higher order corrections.
Measurements of higher order moments and moments of additional variables, such as q 2 , canfurther improve the sensitivity of the fits to higher-order terms in the HQE [105]. The q 2 moments
depend on a reduced set of operators, thereby possibly making it easier to extract higher-order terms
from data. A fit to the q 2 moments measured by the Belle [121] and Belle II [122] experiments
combined with the world average inclusive branching fractions, finds [138]:|Vcb | = (41.69 ± 0.63) × 10−3 ,(76.31)

which is compatible with the fit to the other moments. Note there is a strong correlation with theother determinations due to the use of common inputs for the branching fractions.
These fits give consistent results for |Vcb | and, after translation to a common renormalization
scheme, for mb . We take the fit in the kinetic scheme [134], which includes higher-order perturbative
corrections, as the inclusive determination of |Vcb |:|Vcb | = (42.2 ± 0.5) × 10−3 (inclusive).
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(76.32)

13

76. Semileptonic b-Hadron Decays, Determination of Vcb , Vub

76.3 Determination of |Vub |Summary: Currently the best determinations of |Vub | are from B̄ → π`ν̄` decays, where combined fits to theory and experimental data as a function of q 2 provide a precision of about 4%;
the uncertainties from experiment and theory are comparable in size. Determinations based on
inclusive semileptonic B decays are based on different observables and use different strategies tosuppress the b → c background. Most of the determinations are consistent and provide a precisionof about 7%, with comparable contributions to the uncertainty from experiment and theory.
The values obtained from inclusive and exclusive B decay determinations are
0.13−3|Vub | = (4.13 ± 0.12 +− 0.14 ± 0.18) × 10
−3

|Vub | = (3.70 ± 0.10 ± 0.12) × 10

(inclusive),(76.33)(exclusive),(76.34)

where the last uncertainty on the inclusive result was added by the authors of this review and isdiscussed below. The exclusive and inclusive determinations are independent, and the dominant
uncertainties are on multiplicative factors. The results from the two determinations are compatible
to within two standard deviations.
This can be compared to the values derived from |Vub |/|Vcb | ratio measurements, described in
the next section. Taking the value for this ratio from Eq. (76.54), and |Vcb | from Eq. (76.3) we
obtain.
|Vub | = (3.43 ± 0.32) × 10−3

(Bs , Λb ),(76.35)We choose to combine only the direct determinations of |Vub |, based on the inclusive and exclusive B measurements. In the combination they are weighted by their relative errors, where theuncertainties are treatedas normally distributed. The resulting average has p(χ2 ) = 15%, so wepscale the error by χ2 /1 = 1.4 to find|Vub | = (3.82 ± 0.20) × 10−3

(average).

(76.36)

In a future update we may consider the constraints in the |Vub |-|Vcb | plane from the direct determinations together with the LHCb |Vub /|Vcb | results, although the agreement among all inputs issomewhat marginal.
76.3.1 |Vub | from inclusive decays
The theoretical description of inclusive B̄ → Xu `ν̄` decays is based on the Heavy Quark Expansion and leads to a predicted total decay rate with uncertainties below 5% [108, 139]. However,
the total decay rate is hard to measure due to the large background from CKM-favored B̄ → Xc `ν̄`transitions, and hence the theoretical methods differ from the B̄ → Xc `ν̄` case. For a calculation of
the partial decay rate in regions of phase space where B̄ → Xc `ν̄` decays are suppressed one cannot
use the HQE as for b → c, rather one needs to introduce non-perturbative distribution functions,the “shape functions” (SF) [140, 141]. Their exact form is not known, but its moments can be
related to the HQE parameters known e.g from the b → c case.
The shape functions become important when the light-cone momentum component P+ ≡ EX −|PX | is not large compared to ΛQCD , as is the case near the endpoint of the B̄ → Xu `ν̄` lepton
spectrum. Partial rates for B̄ → Xu `ν̄` are predicted and measured in a variety of kinematic regionsthat differ in their sensitivity to shape-function effects.
At leading order in 1/mb only a single shape function (SF) appears, which is universal for all
heavy-to-light transitions [140, 141] and can be extracted in B̄ → Xs γ decays. At subleading orderin 1/mb , several shape functions appear [142], along with small “resolved photon contributions”
31st May, 2024

14

76. Semileptonic b-Hadron Decays, Determination of Vcb , Vubspecific for B̄ → Xs γ [143–145], and thus the prescriptions that relate directly the partial rates forB̄ → Xs γ and B̄ → Xu `ν̄` decays [146–154] are limited to leading order in 1/mb .
Existing approaches use parametrizations of the leading SF that respect constraints on thenormalization and on the first and second moments, which are given in terms of the HQE parametersΛ = MB − mb and µ2π , respectively. The relations between SF moments and the HQE parameters
are known to second order in αs [155]; as a result, measurements of HQE parameters from global
fits to B̄ → Xc `ν̄` and B̄ → Xs γ moments can be used to constrain the SF moments, as wellas to provide accurate values of mb and other parameters for use in determining |Vub |. Flexible
parametrizations of the SF using orthogonal basis functions [156] or artificial neural networks [157]allow global fits to inclusive B meson decay data [145] that incorporate the known short-distancecontributions and renormalization properties of the SF.
HFLAV performs fits on the basis of several approaches, with varying degrees of model dependence. We will consider here the approaches documented in Ref. [158] (BLNP), Ref. [159] (GGOU)
and Ref. [160] (DGE).
The triple differential rate in the variablesP` = MB − 2E` ,P− = EX + |P~X |,P+ = EX − |P~X |(76.37)isd3 Γ
G2 |Vub |2= F 2 (MB − P+ )dP+ dP− dP`
16π(76.38)n(P− − P` )(MB − P− + P` − P+ )F1o+ (MB − P− )(P− − P+ )F2 + (P− − P` )(P` − P+ )F3 .
The “structure functions”, Fi , can be calculated using factorization theorems that have been provento subleading order in the 1/mb expansion [161].
The BLNP [158] calculation uses these factorization theorems to write the Fi terms as functions of perturbatively calculable hard coefficients and jet functions, which are convolved with the(soft) light-cone distribution functions.The calculation of O(αs2 ) contributions [162, 163] is not yet
complete and is not included in the |Vub | determination given below.
The leading order term in the 1/mb expansion of the Fi terms contains a single non-perturbative
function and is calculated to subleading order in αs , while at subleading order in the 1/mb expansion
there are several independent non-perturbative functions that have been calculated only at tree levelin the αs expansion.
A distinct approach (GGOU) [159] uses a hard, Wilsonian cut-off that matches the definitionof the kinetic mass. The non-perturbative input is similar to what is used in BLNP, but the shapefunctions are defined differently. In particular, they are defined at finite mb and depend on the lightcone component k+ of the b quark momentum and on the momentum transfer q 2 to the leptons.
These functions include subleading effects to all orders; as a result they are non-universal, with one
shape function corresponding to each structure function in Eq. (76.38). Their k+ moments can becomputed in the OPE.
Going to subleading order in αs requires the definition of a renormalization scheme for the HQEparameters and for the SF. The relation between the moments of the SF and the forward matrix
elements of local operators appearing the HQE is plagued by ultraviolet problems and requires
additional renormalization. A scheme for improving this behavior was suggested in Refs. [158,164],
which introduce a definition of the quark mass (the so-called shape-function scheme) based on the
31st May, 2024

15

76. Semileptonic b-Hadron Decays, Determination of Vcb , Vub

first moment of the measured B̄ → Xs γ photon energy spectrum. Likewise, the HQE parameters
can be defined from measured moments of spectra, corresponding to moments of the SF.
There are various ideas to model the SF, but this requires additional assumptions. One approach (DGE) is the so-called “dressed gluon exponentiation” [160], where the perturbative result
is continued into the infrared regime using the renormalon structure obtained in the large β0 limit,
where β0 has been defined following Eq. (76.25). Other approaches make even stronger assumptions, such as in Ref. [165], which assumes an analytic behavior for the strong coupling in theinfrared to perform an extrapolation of perturbation theory.
In order to reduce sensitivity to SF uncertainties, measurements that use a combination ofcuts on the leptonic momentum transfer q 2 and the hadronic invariant mass mX , as suggested inRef. [166,167], have been made. In general, experimental measurements of B̄ → Xu `ν̄` into charmdominated regions (in order to reduce SF uncertainties) are sensitive to the modeling of B̄ → Xu `ν̄`and B̄ → Xc `ν̄` decays. The measurements quoted below have used a variety of functional forms
to parametrize the leading SF; a specific error budget for one determination is quoted in the next
section. In no case is the parametrization uncertainty estimated to be more than a 2% on |Vub |.
Weak Annihilation [159, 168, 169] (WA) can in principle contribute significantly in the high-q 2
region of B̄ → Xu `ν̄` decays. Estimates based on semileptonic Ds decays [100, 166, 167, 169] leadto a ∼ 2% uncertainty on the total B̄ → Xu `ν̄` rate from the Υ (4S). The q 2 spectrum of the
WA contribution is not well known, but from the OPE it is expected to contribute predominantly
at high q 2 . Other theoretical investigations [100, 170, 171], a direct search [172], and B(B 0 →Xu `ν̄)/B(B + → Xu `ν̄) ratio measurements [173–176] indicate that WA is a small effect, but may
become a significant source of uncertainty for |Vub | measurements that accept only a small fractionof the full B̄ → Xu `ν̄` phase space.
76.3.2 Measurements
We summarize the measurements used in the determination of |Vub | below. Given the improved
precision and more rigorous theoretical interpretation of more recent measurements, determinations [177–180] done with LEP data are not considered in this review.
Inclusive electron momentum measurements [181, 182] reconstruct a single charged electron todetermine a partial decay rate for B̄ → Xu `ν̄` near the kinematic endpoint. This results in a
selection efficiency of order 50% and only modest sensitivity to the modeling of detector response.
The inclusive electron momentum spectrum from B B̄ events, after subtraction of the e+ e− → q q̄
continuum background, is fitted to a model B̄ → Xu `ν̄` spectrum and several components (D`ν̄` ,D∗ `ν̄` , ...) of the B̄ → Xc `ν̄` background; the dominant uncertainties are related to this subtraction
and modelling. The decay rate can be cleanly extracted for Ee > 2.3 GeV, but this is deep in the
SF region, where theoretical uncertainties are large. More recent measurements have increased theaccessed phase phase. The resulting |Vub | values for various Ee cuts are given in Table 76.2.
The most recent BABAR measurement [183] is based on the inclusive electron spectrum anddetermines the partial branching fraction and |Vub | for Ee > 0.8 GeV. The analysis shows that thepartial branching fraction measurements can have signal model dependence when the kinematic
acceptance includes regions dominated by B̄ → Xc `ν̄` background. The model dependence enters primarily through the partial branching fractions, and arises because the signal yield fit has
sensitivity to B̄ → Xu `ν̄` decays only in regions with good signal to noise.
An untagged “neutrino reconstruction” measurement [184] from BABAR uses a combination [185]of a high-energy electron with a measurement of the missing momentum vector. This allows
S/B ∼ 0.7 for Ee > 2.0 GeV and a ≈ 5% selection efficiency, but at the cost of a smaller accepted phase space for B̄ → Xu `ν̄` decays and uncertainties associated with the determination of
the missing momentum.
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76. Semileptonic b-Hadron Decays, Determination of Vcb , Vub

The large samples accumulated at the B factories allow studies in which one B meson is fullyreconstructed and the recoiling B decays semileptonically [186–189]. The experiments can fullyreconstruct a “tag” B candidate in about 0.5% (0.3%) of B + B − (B 0 B̄ 0 ) events. An electron ormuon with center-of-mass momentum above 1.0 GeV is required amongst the charged tracks not
assigned to the tag B and the remaining particles are assigned to the Xu system. The full set ofkinematic properties (E` , mX , q 2 , etc.) are available for studying the semileptonically decaying B,
making possible selections that accept up to 90% of the full B̄ → Xu `ν̄` rate; however, the sensitivity
to B̄ → Xu `ν̄` decays is still driven by the regions where B̄ → Xc `ν̄` decays are suppressed. Despite
requirements (e.g. on the square of the missing mass) aimed at rejecting events with additional
missing particles, undetected or mis-measured particles from B̄ → Xc `ν̄` decay (e.g., KL0 andadditional neutrinos) remain an important source of uncertainty.
A recent recoil method measurement of partial branching fractions in three phase-space regions,covering about 31% to 86% of the accessible phase space, was performed by Belle [173], wheremachine learning techniques were used to reduce background levels. The measurement contains
substantial decay model updates and supersedes previous measurements from Belle [187,188]. The
measurement of the partial branching fraction obtained in the E`B > 1GeV region, the most precise
one, is used to obtain |Vub | [173]. Belle also reports measurements of the background subtracted,
and unfolded differential decay spectra in bins of mX , mX and q 2 , P+ and E` [190] for use in global
fit analyses.
Earlier measurements by BABAR [186] were also performed with cuts on mX , mX and q 2 ,P+ and E` using the recoil method. In each case the experimental systematic uncertainties havesignificant contributions from the modeling of B̄ → Xu `ν̄` and B̄ → Xc `ν̄` decays and from thedetector response to charged particles, photons and neutral hadrons.
To reduce modelling uncertainties associated to B̄ → Xc `ν̄` decays a recent Belle tagged measurement [176] uses a novel data driven approach to correct the kinematics and normalisation ofthis background component, while using less restrictive event selection with higher efficiency thanthe other recent Belle result. In the analysis the ratio |Vub |/|Vcb | is directly extracted from inclusive
decays for the first time. While a value for |Vub | is provided here, it is not yet included in the
combination.
The corresponding |Vub | values are given in Table 76.2.
76.3.3 |Vub | from inclusive partial rates
The measured partial rates and theoretical calculations from BLNP, GGOU and DGE described
previously are used to determine |Vub | from all measured partial B̄ → Xu `ν̄` rates [6]; selected values are given in Table 76.2. The correlations amongst the multiple BABAR recoil-based measurements [186] are fully accounted for in the average. The statistical correlations amongst the othermeasurements used in the average are small (due to small overlaps among signal events and large
differences in S/B ratios) and have been ignored. Correlated systematic and theoretical errors aretaken into account, both within an experiment and between experiments. As an illustration of therelative sizes of the uncertainties entering |Vub | we give the error breakdown for the GGOU average: statistical—1.3%; experimental—1.6%; B̄ → Xc `ν̄` modeling—0.9%; B̄ → Xu `ν̄` modeling—
1.7%; HQE parameters (mb ) —1.8%; higher-order corrections—1.5%; q 2 modeling—1.3%; Weak
Annihilation—+0.0−1.1 %; SF parametrization—0.1%.
The averages quoted here are based on the following mb values: mSF= 4.582±0.023±0.018 GeVb
M S = 4.188 ± 0.043 GeV for DGE. The mkinfor BLNP, mkin
=
4.554
±
0.018
GeV
for
GGOU,
and
m
b
b
b
value is determined in a global fit to moments in the kinetic scheme [6]; this value is translated into
S at fixed order in α . The second uncertainty quoted on m arises from the scheme
mSF
and mMsbbbtranslation.
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17

76. Semileptonic b-Hadron Decays, Determination of Vcb , Vub

Table 76.2: |Vub | (in units of 10−5 ) from inclusive B̄ → Xu `ν̄` measurements. The first uncertainty on |Vub | is experimental, while the second includes both theoretical and HQE parameter
uncertainties. The values are generally listed in order of increasing kinematic acceptance, fu (0.19
to 0.90), except for the BABAR Ee >0.8 GeV measurement; those below the horizontal bar are
based on recoil methods. The Belle 2023 measurement is not yet included in the combination.
Ref.
CLEO [181]
BABAR [184]Belle [182]
BABAR [183]cut (GeV)
Ee > 2.1
Ee – q 2
Ee > 1.9
Ee > 0.8BLNP
29422 ± 49 +
− 34
+ 33
471 ± 32 − 3826493 ± 46 +
− 29
+ 27
441 ± 12 − 27GGOU
22423 ± 49 +− 31
not available
16495 ± 46 +
− 21
+ 17
396 ± 10 − 17DGE
25386 ± 45 +
− 27
+ 28
435 ± 29 − 3023482 ± 45 +− 23
+ 8385 ± 11 − 7

q 2 >8mX <1.724433 ± 23 +
− 27
+ 26
425 ± 26 − 2716
410 ± 23 +− 17
+ 9444 ± 24 − 108415 ± 24 +− 9
+ 8425 ± 25 − 918424 ± 22 +
− 21
+ 28
417 ± 25 − 3721422 ± 23 +
− 27
+ 12
445 ± 24 − 1311416 ± 24 +
− 12
+ 11
426 ± 25 − 1211419 ± 12 +− 12

9392 ± 10 +− 10

BABAR [186]
BABAR [186]
BABAR [186]
BABAR [186]Belle [173]Belle [176]

P+ < 0.66
mX < 1.7
E` > 1.3
E` > 1E` > 126432 ± 23 +
− 28
+ 25
409 ± 25 − 2522403 ± 22 +
− 22
+ 19
433 ± 24 − 2118405 ± 23 +
− 20
+ 18
415 ± 24 − 20

HFLAV [6]Combination

20428 ± 13 +− 21

Hadronization uncertainties also impact the |Vub | determination. The theoretical expressions
are valid at the parton level and do not incorporate any resonant structure (e.g. B̄ → π`ν̄` ); thismust be added to the simulated B̄ → Xu `ν̄` event samples, since the detailed final state multiplicity
and structure impacts the estimates of experimental acceptance and efficiency. The experiments
have adopted procedures to input resonant structure while preserving the appropriate behavior in
the kinematic variables (q 2 , E` , mX ) averaged over the sample, but these prescriptions are ad hoc
and ultimately require in situ calibration. The resulting uncertainties have been estimated to be∼ 1-2% on |Vub |. A recent inclusive |Vub | Belle measurement is the first to perform a fit to final
state pion multiplicity for in situ calibration of the hadronization models used. The measurement
simultaneously extracts inclusive and exclusive |Vub |, with compatible results [191].
All calculations yield compatible |Vub | values and similar error estimates. The arithmetic meanof the values and errors of the HFLAV combinations listed in Table 76.2 is |Vub | = (4.13 ±−3
0.12exp +0.13−0.14 theo ) × 10 , although there is a spread of approximately 10% in the evaluations withthe three theoretical models. For reasons discussed below, we assign an additional uncertainty
due to model dependence that is not reflected in the HFLAV averages. As highlighted in the
BABAR analysis [173, 183], model dependence entering measurement procedures can be sizeable,
and is not consistently treated across analyses. Many of the analyses shown in Table 76.2 werebased on partial branching fraction measurements determined in a single model (i.e. the one usedby that analysis when simulating B̄ → Xu `ν̄` decays), although in some cases simulated eventswere weighted to match the expected spectra in other models and the differences introduced assystematic uncertainties, e.g. Ref. [188]. The |Vub | value quoted by HFLAV for each model are,typically, derived from this unique partial branching fraction combined with another model-specificpartial rate calculation. The model dependence in the partial branching fraction is sensitive to how
the model predictions compare in the restricted region with good signal-to-noise, not by how they
compare when integrated over the full kinematic range used in the fit. Ideally this effect needs tobe accounted for by the experiments; the published information is insufficient to determine it. To
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76. Semileptonic b-Hadron Decays, Determination of Vcb , Vub

account for the range in results using the different theoretical models, we take half of the spreadof the averages as an additional systematic uncertainty, denoted ∆model. With this addition, theinclusive |Vub | average is−3|Vub | = (4.13 ± 0.12exp +0.13(inclusive).
−0.14 theo ± 0.18∆model ) × 10

(76.39)76.3.4 |Vub | from exclusive decays
Exclusive charmless semileptonic decays offer a complementary means of determining |Vub |. For
the experiments, the specification of the final state provides better background rejection, but thebranching fraction to a specific final state is typically only a few percent of that for inclusive decays.
For theory, the calculation of the form factors for decays of B-mesons into exclusive final states ischallenging, but brings in a different set of uncertainties from those encountered in inclusive decays.
In this review we focus on B̄ → π`ν̄` , as it is the most promising decay mode for both experimentand theory. However, we also discuss B̄s → K`ν̄` decays, for which there are recent interesting
results. Measurements of other exclusive B̄ → Xu `ν̄` decays can be found in Refs. [192–205].
76.3.5 B̄ → π`ν̄` form factor calculations
The relevant form factors for the decay B̄ → π`ν̄` are usually defined ashπ(pπ )|V µ |B(pB )i ="2

f+ (q )

m2 − m2
pµB + pµπ − B 2 π q µq#+ f0 (q 2 )m2B − m2π µqq2(76.40)in terms of which the rate becomes (in the limit m` → 0)dΓG2F |Vub |2=|pπ |3 |f+ (q 2 )|2 ,
dq 2
24π 3

(76.41)

where pπ is the pion momentum in the B meson rest frame.
Lattice-QCD calculations of the form factors for the B → π`ν̄ and Bs → K`ν̄ transitions areavailable from the Fermilab/MILC [206,207], HPQCD [208], RBC/UKQCD [209,210], and JLQCD
2[211] collaborations. The lattice form factors are obtained in the large q 2 region, qmax
> q2 &
15 GeV2 and the calculations differ in actions employed for the b quark. While HPQCD [208] is using
lattice-nonrelativistic QCD, the results from Fermilab/MILC [206, 207]and RBC/UKQCD [209]are obtained with relativistic b quark actions based on the Fermilab approach [212], albeit using
independently generated gauge-field configurations. The JLQCD collaboration is using all-domainwall-fermion set-up, The results agree within the quoted errors. For the B → π form factor
f+ (q 2 = 20 GeV2 ), Ref. [206] quotes uncertainty of 3.4%, where the leading contribution is dueto the chiral-continuum extrapolation fit, which includes statistical and heavy-quark discretization
errors.
The extrapolation to small values of q 2 can be performed using guidance from analyticity andunitarity together with a conformal mapping of the kinematical variables onto the complex unit
disc, which yields series in the variablep√t+ − q 2 − t+ − t0z=p,(76.42)√t+ − q 2 + t+ − t0
where t± = (MB ± mπ )2 and the choice of t0 determines the range of z that corresponds to the
2 . The choicekinematic range of the decay. Setting t0 = t− results in z = 0 when q 2 = qmax√√2
t0 = (mB + mπ )/( mB − mπ ) centers the kinematic range around z = 0, so that |z| < 0.3.
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76. Semileptonic b-Hadron Decays, Determination of Vcb , Vub

We note that the B̄s → K`ν̄` decay is an example were the implementation of unitarity bounds
is complicated by the fact that the location of the cut in the q 2 plane is different from the Bs K
threshold, because the cut starts at Bπ production threshold. Modifications of the unitarity constraints in this and similar examples are discussed in Refs. [210, 213–216].
The ”BCL” parameterization of Ref. [71] is commonly used for these decays, as it is consistent
with the asymptotic scaling (q 2 → ∞) as well as near threshold, q 2 = t+ . As pointed out in Ref. [217]the unitarity sum of the z-expansion coefficients for f+ is expected to be of order (Λ/m)3 , hence
yielding a rapidly converging series. We note that relatively small differences in different lattice
QCD form factor results can be amplified by a long extrapolation to the low-q 2 region. However,such extrapolations are not needed, when lattice form factors are combined with experimental
measurements of the differential decay rate. The joint fits provide constraints on the shape ofthe form factor over the entire kinematic range, in addition to enabling precise determinations of|Vub | [218].
LCSR calculations provide estimates for the product fB f+ (q 2 ), valid in the region 0 < q 2 .
12 GeV. The determination of f+ (q 2 ) itself requires knowledge of the decay constant fB , which isusually obtained by replacing fB by its two-point QCD (SVZ) sum rule [219] in terms of perturbativeand condensate contributions. The advantage of this procedure is the approximate cancellation ofvarious theoretical uncertainties in the ratio (fB f+ )/(fB ). The LCSR for fB f+ is based on thelight-cone OPE of the relevant vacuum-to-pion correlation function, calculated in full QCD at
finite b-quark mass. The resulting expressions comprise a triple expansion: in the twist, t, of theoperators near the light-cone, in αs , and in the deviation of the pion distribution amplitudes fromtheir asymptotic form, which is fixed from conformal symmetry. The state-of-the-art calculations
include the leading twists two, three and four with full one-loop αs corrections [220,221] and partial
two-loop corrections [222]. Higher-twist contributions have been investigated in Ref. [223] and twoparticle higher twist contributions are studied in Ref. [43]. Nevertheless, estimates based on LCSR
generally suffer from difficult to quantify systematic uncertainties.
A detailed statistical analysis including the various correlations has been performed in Ref. [224],
also including unitarity bounds on the form factor. The results obtained are numerically compatible
with the lattice QCD calculations of the form factor. A more recent analysis [225] employs a
modified BCL expansion to extrapolate the LCSR form factors obtained at low q 2 to the high q 2
region, where good agreement with lattice QCD form factor predictions is found. The LCSR form
factors can be combined with experimental data in joint z-expansion fits to determine |Vub |.
76.3.6 B̄ → π`ν̄` measurements
The B̄ → π`ν̄` measurements fall into two broad classes: untagged, in which case the reconstruction of the missing momentum of the event serves as an estimator for the unseen neutrino,
and tagged, in which the second B meson in the event is fully reconstructed in either a hadronic orsemileptonic decay mode. The tagged measurements have better q 2 resolution, high and uniform
acceptance and S/B as high as 10, but lower statistical power. The untagged measurements havehigher background (S/B < 1) and make slightly more restrictive kinematic cuts, but still provide
statistical power precision on the q 2 dependence of the form factor.
CLEO has analyzed B̄ → π`ν̄` and B̄ → ρ`ν̄` using an untagged analysis [199–201]. Similar
analyses have been done at BABAR [202–205] and Belle [226]. The leading systematic uncertainties
in the untagged B̄ → π`ν̄` analyses are associated with modeling the missing momentum reconstruction, with background from B̄ → Xu `ν̄` decays and e+ e− → q q̄ continuum events, and withvarying the form factor used to model B̄ → ρ`ν̄` decays.
Analyses [194, 227] based on reconstructing a B in the D̄(∗) `+ ν` decay mode and looking for a
B̄ → π`ν̄` or B̄ → ρ`ν̄` decay amongst the remaining particles in the event make use of the fact
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76. Semileptonic b-Hadron Decays, Determination of Vcb , Vubthat the B and B̄ are back-to-back in the Υ (4S) frame to construct a discriminant variable thatprovides a signal-to-noise ratio above unity for all q 2 bins. A related technique was discussed inRef. [228]. BABAR [227] and Belle [192] have used their samples of B mesons reconstructed in
hadronic decay modes to measure exclusive charmless semileptonic decays, resulting in very clean
but smaller samples. The dominant systematic uncertainties in the tagged analyses arise from tag
calibration.
|Vub | can be obtained from the average B̄ → π`ν̄` branching fraction and the measured q 2
spectrum. Fits to the q 2 spectrum using a theoretically motivated parametrization (e.g. “BCL”
from Ref. [71]) remove most of the model dependence from theoretical uncertainties in the shapeof the spectrum. The most sensitive method for determining |Vub | from B̄ → π`ν̄` decays employs
a simultaneous fit [6, 206, 217, 218, 229, 230] to measured experimental partial rates and lattice
points versus q 2 (or z) to determine |Vub | and the first few coefficients of the expansion of theform factor in z. We quote the result from Ref. [6], which uses as experimental input an averageof the measurements in Refs. [192, 202, 205, 226] and an average [231] of the LQCD input from
Refs. [206] and [209]. The p-value of the q 2 measurement average is 6%. The average for the total
B 0 → π − `+ ν` branching fraction is obtained by summing up the partial branching fractions:
B(B 0 → π − `+ ν` ) = (1.50 ± 0.02stat ± 0.06syst ) × 10−4

(76.43)The corresponding value of |Vub | with this approach is found to be|Vub | = (3.70 ± 0.10 ± 0.12 ) × 10−3

(exclusive),(76.44)

where the first uncertainty is experimental and the second is from theory. A consistent result
for |Vub | was reported in Ref. [36], which uses the same experimental and lattice-QCD inputs.
Adding additional constraints using input from LCSR [222] gives [6] |Vub | = (3.67 ± 0.09 ± 0.12 ) ×
10−3 (exclusive, LQCD+LCSR). Other recent LCSR estimates [225, 232] have been used to obtainconsistent results for |Vub |, albeit with slightly higher central values. Ref. [232] also presents resultsfor |Vub | from joint fits that exclude some of the experimental data in order to bring the exclusive
values into agreement with the inclusive ones.
76.3.7 B̄s → K`ν̄`
The LHCb experiment have conducted the first observation of the decay Bs0 → K − µ+ νµ andthe measurements of its branching fraction normalised to the Bs0 → Ds− µ+ νµ decays [233]. The
measurement has been performed in two bins of q 2 , derived using the Bs0 flight direction and the
known Bs0 mass. The analysis uses a BDT classifier to suppress semileptonic b-hadron background.
The results of the partial branching fractions have been translated into measurements of |Vub |/|Vcb |using form factor calculations from LCSR for q 2 < 7 GeV2 [234], and a recent LQCD calculation
for q 2 > 7 GeV2 [207]. The results are|Vub |/|Vcb | = 0.0607 ± 0.0021 ± 0.0030, q 2 < 7 GeV2 ,22|Vub |/|Vcb | = 0.0946 ± 0.0041 ± 0.0068, q > 7 GeV ,(76.45)(76.46)

where the first uncertainties include also experimental and external input sources, and the latterare due to the form factor calculations. The sizeable discrepancy between the values of |Vub |/|Vcb |for the low and high q 2 , requires further theoretical and experimental investigation. On the theoryside, FLAG [36] perform a joint z-fit to the lattice QCD results in Refs. [207–209], with error
inflation to account for the tensions between lattice results for f0 . They then use the form factors
for B̄s → K`ν̄` and B̄s → Ds∗ `ν̄` to determine the ratio |Vub |/|Vcb | in combination with the LHCb
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76. Semileptonic b-Hadron Decays, Determination of Vcb , Vub

measurements in the high- and low-q 2 regions, finding good agreement between the two:

|Vub |/|Vcb | = 0.0819 ± 0.0029 ± 0.0072, q 2 < 7 GeV2 ,2(76.47)

2|Vub |/|Vcb | = 0.0860 ± 0.0038 ± 0.0037, q > 7 GeV .

(76.48)

While the experimental measurement has higher purity in the low q 2 region, the lattice QCD
form factors are constrained more reliably in high-q 2 region. RBC/UKQCD recently presented inRef. [210] an update of their previous calculation. They studied the effect of different strategies for
obtaining the form factors f+,0 in the continuum limit, finding sizeable differences for f0 , but not f+ .
This issue needs to be studied further. In particular, such differences were not seen in a similar study,
albeit for D-meson decay form factors [235]. However, the resulting effect on |Vub |/|Vcb | is relatively
small, as the f0 form factor enters the determination only via the kinematic constraint at q 2 = 0.
A very recent analysis in Ref. [236] combines the lattice QCD form factors from Refs. [208, 210]
with the LHCb data in |Vub |/|Vcb | determinations that are compatible with Eq. (76.48). Whenthey combine their updated LCSR results with the low-q 2 LHCb data, they find |Vub |/|Vcb | =0.057±0.005, compatible with Eq. (76.45), and hence also in tension with Eq. (76.48). Experimental
measurements of the differential rates in smaller bins would provide valuable shape constraints andadditional compatibility tests, improving the reliability of the |Vub |/|Vcb | determinations by reducing
the need for extrapolations of form factors.

76.4 Semileptonic b-baryon decays
Summary: A significant sample of Λ0b baryons is available at the LHCb experiment, and methods
have been developed to study their semileptonic decays. Both Λ0b → pµν̄ and Λ0b → Λ+c µν̄ decays
have been measured at LHCb, and the ratio of branching fractions to these two decay modes isused to determine the ratio |Vub /Vcb |.
076.4.1 Λ0b → Λ+c µν̄ and Λb → pµν̄0The Λ0b → Λ+c and Λb → p semileptonic transitions are described in terms of six form factors
each. The three form factors corresponding to the vector current can be defined as [237]00hF (p , s )|q̄ γµ b|Λ0b (p, s)i = ūF (p0 , s0 )

f0 (q 2 ) (MΛ0 − mF )bMΛ 0 + m F qµbpµ + p0µ − 2 (MΛ2 0 − m2F )
b
s+
q
!)
2MΛ0
2mF
b
+f⊥ (q 2 ) γµ −pµ −p0uΛ0 (p, s) ,
b
s+
s+ µ

qµ
q2

+f+ (q 2 )(76.49)22
where F = p or Λ+c and where we define s± = (MΛ0b ± mF ) − q . At vanishing momentum transfer,
q 2 → 0, the kinematic constraint f0 (0) = f+ (0) holds. The form factors are defined in such a waythat they correspond to time-like (scalar), longitudinal and transverse polarization with respect
to the momentum-transfer q µ for f0 , f+ and f⊥ , respectively. Likewise, the expression for the
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76. Semileptonic b-Hadron Decays, Determination of Vcb , Vub

axial-vector current ishF (p0 , s0 )|q̄ γµ γ5 b|Λ0b (p, s)i =00− ūF (p , s )γ5 g0 (q 2 ) (MΛ0 + mF )b

2+g+ (q )MΛ0 − mF bs−+g⊥ (q 2 ) γµ +

qµ
q2

qµ
pµ + p0µ − 2 (MΛ2 0 − m2F )
q

2MΛ0
2mF
b
pµ −
p0
s−
s− µ



b

!)

uΛ0 (p, s) ,b

(76.50)with the kinematic constraint g0 (0) = g+ (0) at q 2 → 0.
In the heavy-quark limit, where both the b and c quarks are treated as heavy, all the form
factors reduce to the Isgur Wise function ξB for baryons [237]:
f0 = f+ = f⊥ = g0 = g+ = g⊥ = ξB

(76.51)

With a light baryon in the final state, the form factors are related through the heavy quark symmetries of the Λ0b , reducing the the number of independent form factors to two. It should be noted2
that the differential Λ0b → (p/Λ+c )µν̄ decay rates peak at high q , so that the kinematic regions
where both lattice QCD calculations and experimental measurements are precise are well matched.
The form factors for Λ0b decays have been studied with lattice QCD [238]. Based on these results0
the differential rates for both Λ0b → Λ+c µν̄ as well as for Λb → pµν̄ can be predicted in the fullphase space. In particular, for the experimentally interesting region they find the ratio of decay
rates to be [238]B(Λ0b → pµν̄)q2 >15 GeV2
B(Λ0b → Λ+c µν̄)q 2 >7 GeV2

= (1.471 ± 0.095 ± 0.109)

Vub 2
Vcb

(76.52)where the first uncertainty is statistical and the second, systematic.
76.4.2 Measurements at LHCb
The LHCb experiment has measured the branching fractions of the semileptonic decays Λ0b →+Λc µν̄ and Λ0b → pµν̄, from which they determine |Vub |/|Vcb |. This is the first such determination
at a hadron collider, the first to use a b baryon decay, and the first observation of Λ0b → pµν̄.
Excellent vertex resolution allows the pµ and production vertices to be separated, which permits0
the calculation of the transverseq momentum p⊥ of the pµ pair relative to the Λb flight direction.
The corrected mass, mcorr = p2⊥ + m2pµ + p⊥ , peaks at the Λ0b mass for signal decays and provides
good discrimination against background combinations. The topologically similar decay Λ0b → Λ+c µν̄is also measured, which eliminates the need to know the production cross-section or absolute efficiencies. Using vertex and Λ0b mass constraints, q 2 can be determined up to a two-fold ambiguity.
The LHCb analysis requires both solutions to be in the high q 2 region to minimise contaminationfrom the low q 2 region. Their result [239], rescaled [6] to take into account the recent branching− +fraction measurement [240] B(Λ+c → pK π ) = (6.28 ± 0.32)%, is
B(Λ0b → pµν̄)q2 >15GeV2
B(Λ0b → Λ+c µν̄)q 2 >7GeV2= (0.92 ± 0.04 ± 0.07) × 10−2.

(76.53)− +The largest systematic uncertainty is from the measured B(Λ+c → pK π ); uncertainties due to+trigger, tracking and the Λc selection efficiency are each about 3%.
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76. Semileptonic b-Hadron Decays, Determination of Vcb , Vub

A LHCb analysis [241] measures the normalized q 2 spectrum and finds good agreement withthe shape calculated with lattice QCD [238].
The decay rate for Λ0b → pµν̄ peaks at high q 2 where the calculation of the associated form
factors using lattice QCD is under good control. Using the measured ratio from Eq. (76.53) alongwith the calculated ratio from Ref. [238] (see Eq. (76.52)) results in [6]|Vub |/|Vcb | = 0.079 ± 0.004 ± 0.004(Λb ).

(76.54)

where the first uncertainty is experimental and the second is from the LQCD calculation.

76.5 The ratio |Vub |/|Vcb |The ratio of matrix elements, |Vub |/|Vcb |, is often required when testing the compatibility of aset of measurements with theoretical predictions. It can be determined from the ratio of branching
fractions and has been directly measured by the LHCb (quoted in the previous sections) and Belleexperiments. It can also be calculated based on the |Vub | and |Vcb | values quoted earlier in thisreview.
The average of the LHCb results extracted at high q 2 from Bs and Λb decays, taken to be
uncorrelated, is|Vub |/|Vcb | = 0.083 ± 0.004(Λb , Bs ).

(76.55)

The first direct extraction of this ratio from inclusive B decays was recently performed byBelle [176]. Belle measures the ratio of B̄ → Xu `ν̄` and B̄ → Xc `ν̄` branching fractions with alepton energy threshold of 1.0 GeV. The partial decay widths, omitting the CKM factors, are takenfrom BLNP and GGOU for |Vub | and the kinetic scheme for |Vcb | to extract the ratio of CKM
factors. The results are found to be|Vub |/|Vcb | = 0.097 ± 0.008(BLNP),(76.56)|Vub |/|Vcb | = 0.100 ± 0.006(GGOU).

(76.57)

These values are in excellent agreement with |Vub |/|Vcb | from independent inclusive determinations,
given below, but involve non-zero correlations in both the theoretical and experimental uncertainties. We will therefore leave it out of the combination for this edition.
Given the similarities in the theoretical frameworks used for charmed and charmless decays, wechoose to also quote separate ratios of |Vub |/|Vcb | for inclusive and exclusive B decays, as discussed
earlier:|Vub |/|Vcb | = 0.098 ± 0.006(inclusive),(76.58)|Vub |/|Vcb | = 0.093 ± 0.004(exclusive).

(76.59)The respective determinations of |Vub | and |Vcb | are taken to be uncorrelated in the ratio, although
there could be some small cancellations of the uncertainties in both the experimental and the theoretical input. We average the B decay values, along with the Bs and baryonic result in Eq. (76.55),
weighting by relative errors to find|Vub |/|Vcb | = 0.089 ± 0.005
where the uncertainty has been scaled by a factorp(average).

χ2 /2 = 1.6.

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76. Semileptonic b-Hadron Decays, Determination of Vcb , Vub

76.6 Lepton flavour universality violation
76.6.1 Semitauonic decays
Summary: Semileptonic decays to third-generation leptons provide sensitivity to non-Standard
Model amplitudes, such as from a charged Higgs boson [242–245] and from leptoquarks [246–252].
The ratios of branching fractions of semileptonic decays involving tau leptons to those involving` = e/µ, R(D(∗) ) ≡ B(B̄ → D(∗) τ ν̄τ )/B(B̄ → D(∗) `ν̄` ), are predicted with good precision andconsistency in the Standard Model [7–11, 27, 28, 30, 47, 48], using a variety of different strategies.
Because this ratio is independent of |Vcb |, it can, in principle, be computed entirely from within
the Standard Model, without using experimental decay rate data. For example, in Refs. [7, 9] thelattice-QCD only ratios are obtained as:
R(D)LQCD,FNAL−MILC = 0.284 ± 0.014 ,R(D∗ )LQCD,FNAL−MILC = 0.265 ± 0.013 .

(76.61)

Without constraints from experimental data at large recoil, such SM-theory only evaluations tend
to be less precise than those obtained from joint fits that employ experimental data as well asStandard-Model theory inputs (lattice-QCD form factors, HQET constraints, etc.). Other recent
lattice-QCD only results are found in Refs. [10,11]. Here we use an average of Refs. [27,28,47,48,253]
for R(D) and Refs. [27, 28, 30, 63, 253] for R(D∗ ).
R(D)SM+Exp = 0.298 ± 0.004 ,R(D∗ )LQCD+Exp = 0.254 ± 0.005 .

(76.62)

These use various combinations of experimental data, lattice-QCD form factor inputs, HQET constraints, and LCSR estimates and are consistent with the values quoted above.
Measurements [254–261], of these ratios yield higher values; averaging B-tagged measurementsof R(D) and R(D∗ ) at the Υ (4S) and at LHCb yields [6]R(D)meas = 0.357 ± 0.029 ,R(D∗ )meas = 0.284 ± 0.012 ,(76.63)

with a linear correlation of −0.37. These values exceed the Standard Model predictions of Eq. (76.62)
by 2.0σ and 2.2σ, respectively. A variety of new physics models have been proposed to explain thisexcess, see eg. Refs. [242–249] as well as reviews in Refs. [262–264].
76.6.2 Sensitivity of B̄ → D (∗) τ ν̄τ to additional amplitudes
In addition to the helicity amplitudes present for decays to eν̄e and µν̄µ , decays proceeding
through τ ν̄τ include a scalar amplitude Hs . The differential decay rate is given by [265]G2F |Vcb |2 |p∗D(∗) |q 2
dΓ
=
dq 2
96π 3 m2B

m2
1 − 2τq!2 "m2(|H+ | + |H− | + |H0 | ) 1 + τ2
2q
2

2

2

!

#

3m2τ
+
|Hs |2 ,2q 2(76.64)

where |p∗D(∗) | is the 3-momentum of the D(∗) in the B̄ rest frame and the helicity amplitudes
H depend on the four-momentum transfer q 2 . All four helicity amplitudes contribute to B̄ →D∗ τ ν̄τ , while only H0 and Hs contribute to B̄ → Dτ ν̄τ ; as a result, new physics contributions
can produce larger effects in the latter mode. Semi-leptonic B decays into a τ lepton provide a
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76. Semileptonic b-Hadron Decays, Determination of Vcb , Vub

stringent test of the two-Higgs doublet model of type II (2HDMII), i.e. where the two Higgs doublets
couple separately to up- and down-type quarks. The distinct feature of the 2HDMII is that thecontributions of the charged scalars scale as m2τ /m2H + , since the couplings to the charged Higgs
particles are proportional to the mass of the lepton. As a consequence, one may expect visible
effects in decays into a τ , but only small effects for decays into e and µ. The present data disfavors
the 2HDMII, see below.
76.6.3 Measurement of R(D (∗) )B̄ → D(∗) τ ν̄τ decays have been studied at the Υ (4S) resonance and in pp collisions. At theΥ (4S), the majority of experimental measurements are based on signatures that consist of a D or
D∗ meson, an electron or muon (denoted here by `) from the decay τ → `ντ ν̄` , a fully-reconstructeddecay of the second B meson in the event, and multiple missing neutrinos. One analysis reconstructs
the τ in a hadronic mode. The analyses that use hadronic B tags separate signal decays fromB̄ → D(∗) `ν̄` decays using the lepton momentum and the measured missing mass squared; decays
with only a single missing neutrino peak sharply at zero in this variable, while the signal is spreadout to positive values. When a semileptonic B tag is used, the discrimination between signal andB̄ → D(∗) `ν̄` decays comes from the calorimeter energy that is not associated with any particle
used in the reconstruction of the B meson candidates, the measured missing mass squared and thecosine of the angle between the D∗ ` system and its parent B meson, which is calculated under theassumption that only one particle (a neutrino) is missing. In both these approaches, backgroundfrom B̄ → D∗∗ `ν̄` decays with one or more unreconstructed particles is challenging to separate fromsignal, as is background from B̄ → D(∗) Hc̄ X (where Hc̄ is a hadron containing a c̄ quark) decays.
The leading sources of systematic uncertainty are due to the limited size of simulation samples usedin constructing the PDFs, the composition of the D∗∗ states, efficiency corrections, and cross-feed(swapping soft particles between the signal and tag B).
The most precise measurement from Belle [258] uses semileptonic B tags and leptonic τ decays
to simultaneously measure R(D∗ ) and R(D). The measurement combines results from charged andneutral B decays for both R(D∗ ) and R(D), and is compatible with the Standard Model expectation
to approximately 1σ. The first result from Belle II [259] uses a hadronic tag to measure R(D∗ )in B̄ → D∗ τ − ν̄τ decays with a 189 fb−1 dataset. The statistical power of the Belle II dataset has
been found to outperform that of Belle in this channel.
In addition to the ratio measurements, the Belle experiment has performed polarization measurements of the τ [257] and D∗ [266] respectively. The τ polarization measurement uses hadronic
B tags and τ − decays to π − ντ or ρ− ντ . The main discriminant variables are the measured missing
mass squared and the unassociated calorimeter energy. This measurement provides the first determination of the τ polarization in the B̄ → D∗ τ ν̄τ decay, Pτ (D∗ ) = −0.38 ± 0.51 +0.21−0.16 , compatible
with the Standard Model expectation [30], Pτ (D∗ ) = −0.476+0.037.
−0.034The main uncertainties on the R(D∗ ) measurement come from the composition of the hadronic
B background and from modeling of semileptonic B decays and mis-reconstructed D∗ mesons.
The D∗ polarization measurement uses an inclusive tag approach based on Refs. [267, 268], andreconstructs the τ decays in `ντ ν̄` and π + ν̄τ channels. The main discriminant variables are Xmiss , aquantity that approximates missing mass but does not depend on tag B reconstruction, the visibleenergy of the event, and the beam-energy constrained mass, Mbc , of the inclusively reconstructed
tag side B. This measurement provides the first determination of the D∗ longitudinal polarization
fraction in the B̄ → D∗ τ ν̄τ decay, FL (D∗ ) = −0.38 ± 0.60 +0.08−0.04 , compatible with the Standard
Model expectation [269] within 1.7σ.
The LHCb experiment has performed a simultaneous analysis of R(D∗ ) and R(D) where τ →µντ ν̄µ . The reconstructed decay modes are B̄ 0 → D∗+ τ − ν̄τ with D∗+ → D0 π + , B − → D∗0 τ − ν̄τ
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76. Semileptonic b-Hadron Decays, Determination of Vcb , Vub

with D∗0 → D0 π 0 and D∗0 → D0 π 0 , and B − → D0 τ − ν̄τ . Their analysis [261] takes advantage ofthe measurable flight lengths of b and c hadrons and τ leptons. A multivariate discriminant is used
to select decays where no additional charged particles are consistent with coming from the signal
decay vertices. The separation between the primary and B decay vertices is used to calculate
the momentum of the B decay products transverse to the B flight direction. The longitudinalcomponent of the B momentum can be estimated based on the visible decay products; this allows
a determination of the B rest frame, with modest resolution, and enables the calculation of similar
discrimination variables to those available at the e+ e− B factories. The (rest frame) muon energy,
missing mass-squared and q 2 are used in a 3-d fit.
A complementary LHCb result [260] on R(D∗ ) uses three-prong τ decays that take advantageof their excellent vertex resolution to isolate the τ decay from hadronic background. A 3-d fit isperformed to determine the signal yield, based on the τ -ντ pair q 2 , the τ lifetime, as well as aboosted decision tree classifier based on isolation, invariant mass and flight distance information.
The leading sources of systematic uncertainty are due to the size of the simulation sample usedin constructing the fit templates, uncertainties in modelling the background from hadronic B̄ →D(∗) Hc̄ X decays, as well as reconstruction and trigger effects. The result is normalized to B 0 →D∗− π + π − π + and found to be 1σ from the Standard Model expectation (using the expectation
value quoted here). An analogous measurement of Bc → J/ψτ ν̄τ was performed by the LHCb
measurement [270], in leptonic τ decays. The result, R(J/ψ) = 0.71 ± 0.17 ± 0.18, while relatively
high is compatible to within 2σ of a recent Standard Model evaluation [271] based on a lattticeQCD calculation [272] of the form factors. Systematic uncertainties are dominated by form factors,
as Bc decays are relatively unexplored.
Measurements from BABAR [254, 255], Belle [256–258], Belle II [259] and LHCb [260, 261]
result in values for R(D) and R(D∗ ) that exceed Standard Model predictions. Table 76.3 lists
these values and their average. The simultaneous measurements of R(D) and R(D∗ ) have linear
correlation coefficients of −0.27 (BABAR [254,255]), −0.49 (Belle hadronic tag [256]), −0.51 (Bellesemileptonic tag [258]), and 0.43 (LHCb τ − → µ− ντ ν̄µ ); the R(D) and R(D∗ ) averages havea correlation of −0.40. Two early untagged Belle measurements [267, 268] are subject to larger
systematic uncertainties, with a breakdown of the respective contributions that is inconsistent
with the more recent determinations, hence they cannot be reliably combined in the average. All
three experiments assume the Standard Model kinematic distributions for B̄ → D(∗) τ ν̄τ in theirdeterminations of the branching fraction ratios.
Table 76.3: Measurements of R(D) and R(D∗ ), their correlations, ρ, and the combined averages[6].

B0, B+BABAR [254, 255]
Belle [256]Belle [257, 273]
Belle [258]
LHCb [261]
LHCb [260]Belle II [259]

B0, B+
B0, B+
B0, B+
B + (R(D)), B 0
B0
B0, B+

Average

B0, B+

R(D) × 102
44.0 ± 5.8 ± 4.2
37.5 ± 6.4 ± 2.6
30.7 ± 3.7 ± 1.6
44.1 ± 6.0 ± 6.6

35.7 ± 2.9

R(D∗ ) × 102
33.2 ± 2.4 ± 1.8
29.3 ± 3.8 ± 1.5+ 2.8
27.0 ± 3.5 −2.5
28.3 ± 1.8 ± 1.4
28.1 ± 1.8 ± 2.4
25.7 ± 1.2 ± 1.8+4.1 +3.5
26.2−3.9−3.2
28.4 ± 1.2

ρ
−0.27
−0.49
−0.51
−0.43

−0.40

The measurement combination in the R(D) − R(D∗ ) plane is shown in Fig. 76.2, compared
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76. Semileptonic b-Hadron Decays, Determination of Vcb , Vub

R(D*)

27

0.4

HFLAV
Preliminary 2023

68% CL contours
BelleaBaBar

0.35
BelleII
Belleb

0.3

Average

0.25

LHCbb

0.2

HFLAV SM Prediction

LHCba

BellecWorld Average

R(D) = 0.357 ± 0.029total
R(D*) = 0.284 ± 0.012total
ρ = -0.37
P(χ2) = 33%

R(D) = 0.298 ± 0.004
R(D*) = 0.254 ± 0.005

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

R(D)
Figure 76.2: Measurements of R(D) and R(D∗ ) and their two-dimensional average comparedwith the average predictions for R(D) and R(D∗ ) given in Eq. 76.62. Contours correspond to 68%
CL. The experiment-lattice QCD-HQET predictions and the experimental average deviate fromeach other by 3.3σ.

with the predictions in Eq. (76.61) (LQCD) and Eq. (76.62) (Experiment+LQCD+HQET). The
measurement combination is based on Ref. [6] and online updates with the inclusion of recent
results from LHCb and Belle II. The tension between the Standard Model prediction in Eq. (76.62)
and the measurements is at the level of 2.0σ (R(D)) and 2.2σ (R(D∗ )); if one considers these
deviations together the significance is 3.3σ. This motivates speculation on possible new physics
contributions. The measurements reported in Refs. [259, 260, 273] are all in reasonable agreementwith the Standard-Model predictions. There is some tension in the combination coming from the
BABAR measurement and the most recent LHCb measurement [261]. The BABAR measurement is
the only one to claim a deviation from the Standard Model of more than 3σ, although the p−valueof the full combination is an acceptable 33%.
The current discussion of R(D) and R(D∗ ) may be embedded in the theoretical analysis of
the other anomalies that have been observed in semileptonic FCNC (b → s``) transitions. Moresophisticated approaches fit the data to a general effective Hamiltonian. Matching this effective
Hamiltonian to simplified models, the current situation of the anomalies seems to be compatible
with scenarios with an additional Z 0 or a leptoquark scenario, see eg. [246–252].
76.6.4 Measurements of R(X)
We note that in addition to exclusive modes, inclusive modes also offer an opportunity to test
the presence of non-Standard Model contributions, although typically with lower precision. The
first measurement of R(X) from a B-factory was recently performed by Belle II [274], R(X) =0.228 ± 0.039, and found to be compatible with an average of Standard Model predictions R(X) =0.223 ± 0.005 [275–277]. Prior measurements of the branching fractions of b → Xτ ν were performed
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76. Semileptonic b-Hadron Decays, Determination of Vcb , Vub

by LEP experiments and also found to be compatible with Standard Model predictions.
76.6.5 Ratios of semileptonic decay rates with e to µ final states
Several tests of lepton flavor universality in semileptonic B decays have recently been undertaken
at Belle and Belle II. This includes ratios of rates to electron and muon modes (i.e. R(e/µ))
performed as part of CKM matrix element extraction analyses (Table 76.4), as well as dedicatedflavour dependent angular analyses [278]. Refs. [62, 64, 65] provide measurements of LFUV in
B̄ → D∗ `ν̄ decays at Belle (untagged, and hadron tagged) and Belle II (hadron tagged). The
corresponding theoretical prediction is B(B → D∗ eν)/B(B → D∗ µν) = 1.0041 ± 0.0001 [33].
Refs. [176, 279] provide measurements of LFUV in B → Xc `ν̄ in hadron tagged events at Belle andBelle II, where the theoretical expectation is B(B → Xc eν)/B(B → Xc µν) = 1.006 ± 0.001 [276].
Finally, Ref. [173] provides a result for LFUV in B → Xu `ν decays where the ratio is expected to beclose to 1, as in the B → Xc `ν case. With the exception of the charmless decay measurement, theexperimental uncertainties are dominated by lepton identification uncertainties that do not cancel
in the ratios. All results are found to be highly compatible with SM expectations.
Table 76.4: Measurements of R(e/µ) = B(B → Xeν)/B(B → Xµν) where the specific decay
process is listed.
Ref.
Belle [64]Belle [62]Belle II [65]Belle II [279]Belle [176]Belle [173]Decay
B̄ → D∗ `ν̄
B 0 → D∗+ `ν
B 0 → D∗+ `ν
B → Xc `ν
B → Xc `ν
B → Xu `νR(e/µ) × 100
99.2 ± 2.3 ± 2.3
101 ± 1 ± 3
99.8 ± 0.9 ± 2.0
100.7 ± 0.9 ± 1.9
100.3 ± 0.5 ± 2.4
97 ± 9 ± 4

76.7 Conclusion
The study of semileptonic B meson decays continues to be an active area for both theory andexperiment. The application of HQE calculations to inclusive decays is well established, and fits tomoments of B̄ → Xc `ν̄` decays provide precise values for |Vcb | and, in conjunction with input on
mc or from B → Xs γ decays, provide precise and consistent values for mb . Recent developments
will make use of more types of moments and better theoretical inputs to constrain higher ordereffects. Further into the future, new nonperturbative methods to compute inclusive decay rates,for example, in lattice QCD [280–282], may provide interesting new insights.
The determination of |Vub | from inclusive B̄ → Xu `ν̄` decays is based on multiple calculational
approaches and independent measurements over a variety of kinematic regions, all of which provide
consistent results. Further progress in this area is possible, but will require better theoreticalcontrol over higher-order terms, improved experimental knowledge of the B̄ → Xc `ν̄` backgroundand improvements to the modeling of the B̄ → Xu `ν̄` signal distributions.
In both b → u and b → c exclusive channels there has been significant recent progress in latticeQCD calculations, resulting in improved precision on both |Vub | and |Vcb |. These calculations now
provide information on the form factors well away from the high q 2 region, allowing better useof experimental data. For |Vcb | recent measurements have provided binned data enabling modelindependent fits.
The values from the inclusive and exclusive determinations of |Vcb | and |Vub | are only marginallyconsistent. This is a long-standing puzzle that the recent improvements have, unfortunately, notyet resolved.
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76. Semileptonic b-Hadron Decays, Determination of Vcb , Vub

Both |Vcb | and |Vub | are indispensable inputs into unitarity triangle fits. In particular, knowing|Vub | with good precision allows a test of CKM unitarity in a most direct way, by comparing thelength of the |Vub | side of the unitarity triangle with the measurement of sin(2β). This comparisonof a “tree” process (b → u) with a “loop-induced” process (B 0 − B̄ 0 mixing) provides sensitivity topossible contributions from new physics.
The observation of semileptonic decays into τ leptons has opened a new window to the physics
of the third generation. The measurements indicate a tension between the data and StandardModel predictions, which could be a hint for new physics, manifesting itself as a violation of lepton
universality beyond the Standard-Model couplings to the Higgs. It should be noted that noneof the most recent measurements alone claim discovery of a deviation from the Standard Model.
Combining the data of the semitauonic decays with the anomalies observed in the FCNC b → s``transitions allows an interpretation in terms of additional Z 0 or in terms of additional leptoquarks,
but the current data does not allow us to draw a definite conclusion.

Acknowledgements
We thank J. Flynn, E. Kou, C. Schwanda, and K. Vos for useful discussions and commentson this review, and A. Vaquero for creating Fig. 76.1. Previous assistance from D. Ferlewicz,
P. Gambino, S. Hashimoto, Z. Ligeti, and M. Rotondo is also acknowledged.
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