1

72. Leptonic Decays of Charged Pseudoscalar Mesons

72. Leptonic Decays of Charged Pseudoscalar Mesons
Updated Aug. 2023 by R. A. Briere (Carnegie Mellon U.); written with J. L. Rosner (Chicago U.),
S. L. Stone (Syracuse U.) and R. Van de Water (FNAL)

72.1 Introduction
This review updates one in Ref. [1]. Extensive use is made of results of the Flavor Lattice
Averaging group (FLAG 21 and the 2023 online update), Ref. [2, 3].
Charged mesons formed from a quark and antiquark can decay to a lepton-neutrino pair whenthese objects annihilate via a virtual W boson. Figure 72.1 illustrates this process for the purely
leptonic decay of a D+ meson.

Figure 72.1: The annihilation process for pure D+ leptonic decays in the standard model.
Similar quark-antiquark annihilations via a virtual W + to the `+ ν final states occur for the π + ,K + , Ds+ , and B + mesons. (Whenever pseudoscalar-meson charges are specified in this article, useof the charge-conjugate particles and corresponding decays are also implied.) Let P be any of thesepseudoscalar mesons. To lowest order, the decay width is
Γ

(0)

G2
(P → `ν) = F fP2 m2` MP
8πm2
1 − `2
MP!2|Vq1 q2 |2 .

(72.1)

Here MP is the P mass, m` is the ` mass, Vq1 q2 is the Cabibbo-Kobayashi-Maskawa (CKM) matrix
element between the quarks q1 q̄2 in P , and GF is the Fermi coupling constant. The decay constantfP is proportional to the matrix element of the axial current between the one-P -meson state andthe vacuum:h0|q̄1 γµ γ5 q2 |P (p)i = ipµ fP ,(72.2)and can be thought of as proportional to the “wave function overlap” of the quark and antiquark.
In this article, we use the convention in which fπ ≈ 130 MeV. For brevity, we will often denote thepurely leptonic decay width in Eq. (72.1) by Γ (0) .
The decay of P ± starts with a spin-0 meson, and ends up with a left-handed neutrino or righthanded antineutrino. By angular momentum conservation, the `± must then also be left-handedor right-handed, respectively. In the m` = 0 limit, the decay is forbidden, and can only occur asa result of the finite ` mass. This helicity suppression is the origin of the m2` factor in the decay
width.
S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)
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72. Leptonic Decays of Charged Pseudoscalar Mesons

Experimentally, it is difficult to isolate events in which there are only a lepton and neutrino
in the final state from those with a lepton, neutrino, and soft photon. Thus, radiative contributions must be removed from the experimental measurements a posteriori to obtain Γ (0) . The
radiative contributions can be broken into three pieces: the short-distance contribution to leptonic
and semileptonic decays mediated by a W ± boson that accounts for electroweak corrections notincluded in the definition of GF , the long-distance internal bremsstrahlung (IB) contribution, andthe contribution from photon emission that depends upon the hadron’s structure. The universal
electroweak correction was calculated at O(α) by Sirlin [4], and increases the purely leptonic decayrate by ∼ 1.8–2.2% depending on the decaying meson. The O(α) IB contribution was calculatedby Kinoshita [5], and again is universal for all leptonic decays at this order. Numerically, this universal long-distance contribution lowers the purely leptonic decay rate by ∼ 0.4–2.4%, where thecorrection is smallest for pions and largest for D(s) mesons. The structure-dependent contributions
have been estimated within various effective theories to increase the purely leptonic rate by one toa few percent [6–11].
In this review, we treat the radiative corrections differently for the light, charm, and bottom
meson systems for several reasons. First, the experimental uncertainties on the decay widths
vary substantially Thus, while the inclusion of radiative corrections is essential for the pion, kaon,
and D(s) -meson decay widths, which have been measured to (sub)-percent precision, radiative
corrections can be neglected (for now) for B → τ ν decay. Second, the photons are treated differently
on the experimental side for the different decay processes. For pions and kaons, the experimental
measurements of ΓP `2[γ] are fully inclusive, while for D(s) mesons, the experiments impose cutson the energy of any neutral cluster deposited in the calorimeter, which reduce the soft-photon
background substantially. Some experiments also remove the QED bremsstrahlung in the leadinglogarithmic approximation using the PHOTOS Monte-Carlo generator [12]. Third, the theoreticalknowledge of the structure-dependent corrections varies for each meson system.
Once radiative corrections have been accounted for, measurements of purely leptonic decaybranching fractions and lifetimes allow an experimental determination of the product |Vq1 q2 | fP . If
the decay constant fP is known to sufficient precision from theory, one can obtain the corresponding
CKM element within the standard model. If, on the other hand, one takes the value of |Vq1 q2 |assuming CKM unitarity, one can infer an “experimental measurement” of the decay constant that
can then be compared with theory.
The importance of measuring Γ (P → `ν) depends on the particle being considered. Leptonic
decays of charged pseudoscalar mesons occur at tree level within the standard model. Thus one does
not expect large new-physics contributions to measurements of Γ (P → `ν) for the lighter mesons
P = π + , K + , and these processes, in principle, provide clean standard-model determinations of |Vud |and |Vus |. The situation is different for leptonic decays of charm and bottom mesons. The presenceof new heavy particles such as charged Higgs bosons or leptoquarks could lead to observable effects+
in Γ (P → `ν) for P = D(s), B + [13–17]. Thus, the determination of |Vub | from B + → τ ν decay,
in particular, should be considered a probe of new physics. More generally, the ratio of leptonic
decays to τ ν over µν final states probes lepton universality [13, 18].
The determinations of CKM elements from leptonic decays of charged pseudoscalar mesons
provide complementary information to those from other decay processes. The decay P → `νproceeds in the standard model via the axial-vector current q¯1 γµ γ5 q2 , whereas semileptonic pseudoscalar meson decays P1 → P2 `ν proceed via the vector current q¯1 γµ q2 . Thus the comparison ofdeterminations of |Vq1 q2 | from leptonic and semileptonic decays tests the V − A structure of thestandard-model electroweak charged-current interaction. More generally, a small right-handed admixture to the standard-model weak current would lead to discrepancies between |Vq1 q2 | obtained
from leptonic pseudoscalar-meson decays, exclusive semileptonic pseudoscalar-meson decays, exclu31st May, 2024

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72. Leptonic Decays of Charged Pseudoscalar Mesons

sive semileptonic baryon decays, and inclusive semileptonic decays [19, 20].
Both measurements of the decay rates Γ (P → `ν) and theoretical calculations of the decay+
constants fP for P = π + , K + , D(s)from numerical lattice-QCD calculations are now quite precise.
As a result, the elements of the first row of the CKM matrix |Vud | and |Vus | can be obtained to
sub-percent precision from π + → `ν and K + → `ν, where the limiting error is from continuum
theory radiative corrections. The elements of the second row of the CKM matrix |Vcd(s) | can beobtained from leptonic decays of charmed pseudoscalar mesons to few-percent precision, where here
the limiting error is from experiment. These enable stringent tests of the unitarity of the first andsecond rows of the CKM matrix.
This review is organized as follows. Because the experimental and theoretical issues associatedwith measurements of pions and kaons, charmed mesons, and bottom mesons differ, we discusseach one separately. We begin with the pion and kaon system in Sec. 72.2. First, in Sec. 72.2.1,
we review current measurements of the experimental decay rates. We provide tables of branchingratio measurements and determinations of the product |Vud(s) |fπ+ (K + ) , as well as average values
for these quantities including correlations and other effects needed to combine results. Then,in Sec. 72.2.2, we summarize the status of theoretical calculations of the decay constants. We
provide tables of recent lattice-QCD results for fπ+ , fK + , and their ratio from calculations including
dynamical u, d, s, and (in some cases c) quarks, along with averages including correlations andstrong SU(2)-isospin breaking corrections as needed. We next discuss the charmed meson system
in Sec. 72.3, again reviewing current experimental rate measurements in Sec. 72.3.1 and theoretical
decay-constant calculations in Sec. 72.3.2. Last, we discuss the bottom meson system in Sec. 72.4,
following the same organization as the two previous sections. For almost all of the decay constantspresented in Secs. 72.2.2, 72.3.2, and 72.4.2, we take as our preferred values the four-flavor averages
from the FLAG 21 review [2] as updated online in 2023 [3]. This 2023 web update contains only
one relevant new result since the 2021 version, on fK + /fπ+ from ETM 21 [21].
After having established the status of both experimental measurements and theoretical calculations of leptonic charged pseudoscalar-meson decays, we discuss some implications for phenomenology in Sec. 72.5. For each process discussed in Secs. 72.2–72.4, we combine the average B(P → `ν)with the decay constant fP to infer the associated CKM matrix element. We then compare theseresults with determinations of the same CKM elements from other processes. We also use the
CKM elements obtained from leptonic decays to test the unitarity of the first and second rows of
the CKM matrix. Further, as in previous reviews, we combine the experimental B(P → `ν)s withthe associated CKM elements obtained from CKM unitarity to infer “experimental” values for the
decay constants. The comparison of these values with theory provides a test of lattice and other
QCD approaches, assuming that new-physics contributions to these processes are not significant.

72.2 Pions and kaons
72.2.1 Experimental rate measurements
Experimental rate measurements of pion and kaon leptonic decays are fully radiation inclusive.
Following Refs. [22, 23], and references therein, we combine the O(α) radiative corrections to thepurely leptonic rate as follows:Γ (P → `ν[γ]) = Γ(0)α(P → `ν) 1 + CP ,π

(72.3)

where P = π, K. The full expressions for Cπ and CK are given in Eq. (114) of Ref. [8]. In addition
to the universal short- [4] and long-distance [5] corrections, Cπ and CK include hadronic-structure
dependent contributions [6] through O(αp4 ) in chiral perturbation theory (χPT), where p is the
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72. Leptonic Decays of Charged Pseudoscalar Mesons

4pion or kaon momentum. The inclusion of radiative corrections to the purely leptonic rates isnumerically important given the level of precision achieved on the experimental measurements ofthe π ± → µ± ν and K ± → µ± ν decay widths.
We evaluate δP ≡ (α/π)CP using experimentally-measured meson and lepton masses and coupling constants from the Particle Data Group [24], and taking the low-energy constants (LECs) thatparametrize the hadronic contributions from Refs. [8, 25, 26]. Because the finite non-logarithmic
parts of the LECs were estimated within the large-NC approximation assuming that contributions
from the lowest-lying resonances dominate, we conservatively assign a 100% uncertainty to theLECs, which leads to a ±0.9 error in Cπ,K .1 We obtain the following correction factors to theindividual charged pion and kaon decay widths:δπ = 0.0176(21)andδK = 0.0107(21) .

(72.4)The error on the ratio of kaon-to-pion leptonic decay widths is under better theoretical control
because the hadronic contributions from low-energy constants estimated within the large-Nc framework cancel at lowest order in the chiral expansion. For the ratio, we use the correction factor
δK/π = −0.0069(17) ,(72.5)

where we take the estimated error due to higher-order corrections in the chiral expansion from
Ref. [28].
There have been no new measurements of the pion leptonic decay rate since the review ofRef. [29]. The sum of branching fractions for π − → µ− ν̄ and π − → µ− ν̄γ is 99.98770(4)% [1].
Together with the lifetime 26.033(5) ns [1] this implies Γ (π − → µ− ν̄[γ]) = 3.8408(7) × 107 s−1 .
We then subtract the estimated radiative correction factor δπ in Eq. (72.4) to obtain the purely
leptonic rate Γ (0) (π − → µ− ν̄). Using this rate and the masses from the 2022‘ PDG review [1] in
Eq. (72.1) gives
fπ+ |Vud | = (127.13 ± 0.01 ± 0.13) MeV ,(72.6)
where the errors are from the experimental rate measurement and the radiative correction factor,
respectively.
The uncertainty on fπ− |Vud | is dominated by that from theoretical estimate of the hadronic
structure-dependent radiative corrections. The first direct lattice-QCD calculation of the radiative
corrections to the pion and kaon leptonic decay rates was performed by the RM123-Soton Collaboration [11]. The results for both δπ = 0.0153(19) and δK = 0.0088(9), which are given in theGasser-Rusetsky-Scimemi scheme for separating strong and electromagnetic effects [30], are compatible with our chiral-perturbation-theory estimates above and have smaller quoted uncertainties,
especially for δK . While independent confirmation of these results is needed, they demonstrate apromising approach for reducing the theoretical uncertainties on the pion and kaon leptonic decay
rates in the future.
The world average for the K → µν decay rate was previously obtained from a global fit of several
kaon-decay branching ratios and lifetime measurements by the FlaviaNet Working Group on Kaon
Decays in 2014 [41]; we have now updated to the PDG [1]. The radiation-inclusive branching ratio
B(K + → µ+ ν[γ]) = 63.56(11)% and lifetime τK ± = 12.380(20) ns are now used; this is a verysmall change. These measurements imply Γ (K + → µ+ ν[γ]) = 5.134(12) × 107 s−1 . As before, we
1This uncertainty on Cπ,K is smaller than the error estimated by Marciano and Sirlin in Ref. [27], which predates
the calculations of the hadronic-structure contributions in Refs. [6, 8, 25, 26]. The hadronic LECs incorporate thelarge short-distance electroweak logarithm discussed in Ref. [27], and their dependence on the chiral renormalization
scale cancels the scale-dependence induced by chiral loops, thereby removing the dominant scale uncertainty of theMarciano–Sirlin analysis [27].

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72. Leptonic Decays of Charged Pseudoscalar Mesons

5

Table 72.1: Recent published lattice-QCD results for fπ+ , fK + , andtheir ratio. The upper and lower panels show (2 + 1 + 1)-flavor and (2 + 1)flavor determinations, respectively. When two errors are shown, they arestatistical and systematic, respectively. Results for fπ and fK in the isospin
symmetric limit mu = md are noted with a ‡; they are corrected for isospin
breaking via Eq. (72.13) before computing the averages. The 2+1+1 FLAG
23 averages are dominated by the Fermilab/MILC values.
ReferenceETM 21 [21]CalLat 20 [31]Fermilab/MILC 17 [32] ∗ETM 14 [33] ∗Fermilab/MILC 14 [10] ∗ †HPQCD 13 [34] ∗FLAG 23 average [3]
QCDSF/UKQCD 16 [35]BMW 16 [36]
RBC/UKQCD 14 [37]‡
MILC 10 [38]BMW 10 [39]‡
HPQCD/UKQCD 07 [40]‡FLAG 23 average [3]
∗
†

Nf
2+1+1
2+1+1
2+1+1
2+1+1
2+1+1
2+1+1
2+1+1
2+1
2+1
2+1
2+1
2+1
2+1
2+1

fπ+ (MeV)
–
–
–
–
–
–
–
–
–
130.19(89)129.2(0.4)(1.4)–
132(2)
130.2(8)fK + (MeV)–––
154.4(1.5)(1.3)
155.92(13)(+42−34 )155.37(20)(27)
155.7(3)––155.51(83)156.1(4)(+6−9 )–157(2)
155.7(7)fK + /fπ+
1.1957(44)(7)1.1942(32)(31)
1.1950(15)(+6−18 )1.184(12)(11)† ‡1.1956(10)(+26−18 )1.1916(15)(16)
1.1934(19)1.190(10)(13)
1.178(10)(26)
1.1945(45)1.197(2)(+3−7 )1.192(7)(6)
1.189(2)(7)1.1917(37)PDG 2014 value of fπ+ = 130.41(21) MeV used to set absolute lattice scale.
Superseded by fK + /fπ+ from Fermilab/MILC 17.

subtract δK in Eq. (72.4) from the radiation-inclusive decay width to obtain Γ (0) (K + → µ+ ν). We
then use Eq. (72.1) to obtainfK + |Vus | = (35.083 ± 0.042 ± 0.037) MeV ,(72.7)

where the errors are from the experimental rate measurement and the radiative correction factor,
respectively.
Short-distance radiative corrections cancel in the ratio of pion-to-kaon decay rates [42]:ΓK`2[γ]Γπ`2[γ]=

2 |f 22
2 2|Vus
K − mK (1 − m` /mK )(1 + δK/π ) ,|Vud |2 fπ2− mπ (1 − m2` /m2π )2(72.8)

where δK/π is given in Eq. (72.5). The left-hand side of Eq. (72.8) is 1.3367(32), which implies|Vus |fK += 0.27599 ± 0.00033 ± 0.00024 ,|Vud |fπ+(72.9)

where the first uncertainty is from the branching fractions and the second is from δK/π . Here, theestimated error on the hadronic structure-dependent radiative corrections is commensurate withthe experimental error.
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72. Leptonic Decays of Charged Pseudoscalar Mesons

6

In summary, the main experimental results pertaining to charged pion and kaon leptonic decays
are|Vud |fπ+ = (127.13 ± 0.01 ± 0.13) MeV ,(72.10)|Vus |fK + = (35.083 ± 0.042 ± 0.037) MeV ,(72.11)|Vus |fK += 0.27599 ± 0.00033 ± 0.00024 ,(72.12)|Vud |fπ+
where the errors are from the experimental uncertainties in the branching fractions and the theoretical uncertainties in the radiative correction factors δP , respectively. These values contain smallchanges from earlier reviews [29, 43, 44].
72.2.2 Theoretical decay-constant calculations
Table 72.1 presents recent published results for the charged pion and kaon decay constants
and their ratio from numerical lattice-QCD calculations with three (Nf = 2 + 1) or four flavors(Nf = 2 + 1 + 1) of dynamical quarks. The uncertainties on both the individual decay constants
and their ratio are at the sub-percent level. The SU(3)-breaking ratio fK + /fπ+ can be obtainedwith especially small errors because statistical errors associated with the Monte Carlo calculations
are correlated between the numerator and denominator, as are some systematics. The resultsin Table 72.1 were obtained using several independent sets of gauge-field configurations, and avariety of lattice fermion actions that are sensitive to different systematic uncertainties.2 Thus, thegood agreement between them indicates that the lattice-QCD uncertainties are controlled and theassociated error estimates are reliable.3
Table 72.1 also shows the three- and four-flavor averages for the pion and kaon decay constants
and their ratio from the FLAG 23 review [2, 3] in the lines labeled “FLAG 23 average.” Thereis no four-flavor average for the pion decay constant in Table 72.1 because all of the four-flavor
calculations use the quantity fπ+ = 130.41(20) MeV [24] as an input to fix the absolute lattice scaleneeded to convert from lattice-spacing units to GeV [10, 33, 34].
All of the results in Table 72.1 were obtained using isospin-symmetric gauge-field configurations,
i.e., the dynamical up and down quarks have the same mass. Fortunately, however, the dominant
effect of strong-isospin breaking is easily included in lattice-QCD calculations as follows. Because
the up-down mass difference ∆mud ≡ (mu − md ) ∼ −2.5 MeV [50, 52] is much less than typical
hadronic scales, the strong-isospin breaking corrections to physical observables can be systematically expanded in the small parameter δmud ≡ ∆mud /ΛQCD . The leading strong-isospin-breaking
corrections to pseudoscalar-meson decay constants arise from the light valence quarks in the initialand final-state hadrons. (See, e.g., the discussion in Ref. [53], for a detailed discussion of isospinbreaking effects in pion and kaon observables.) Thus, to include the effect of non-degenerate upand down-quark masses, most recent lattice-QCD calculations of fπ+ and fK + evaluate the massesof the valence quarks in the pion at the physical mu and md , and the mass of the valence light quarkin the kaon at the physical mu . This procedure yields a correction to the kaon decay constant below
0.5%. Consequently, strong-isospin breaking corrections from the light sea-quark masses — which
are suppressed by an additional power of δmud — can be neglected given present uncertainties.
Some earlier lattice-QCD calculations, however, only provide the decay constants and their ratio
in the SU (2) isospin-symmetric limit [37, 39, 40]. The Flavour Lattice Averaging Group corrects
these results for strong-isospin breaking using chiral perturbation theory before including them in
the averages. The leading strong-isospin-breaking corrections to the pion and kaon decay constants
2

See the PDG mini-review on “Lattice Quantum Chromodynamics” [45–47] for a general review of numericallattice-QCD calculations. Details on the different methods used in modern lattice-QCD calculations are provided inAppendix A of the FLAG “Review[s] of lattice results concerning low energy particle physics" [2, 48–50].
3
See the review by Kronfeld [51] for a summary of the large body of evidence validating the methods employedin modern lattice-QCD calculations.
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72. Leptonic Decays of Charged Pseudoscalar Mesons

7in χPT can be parametrized as [28, 54]
fπ+ = fπ ,fK + = fKq

1 + δSU(2) ,(72.13)where fπ and fK denote the values of the decay constants in the isospin-symmetric limit. The pion
decay constant does not receive corrections linear in mu − md because of the G-parity symmetry4of the pion triplet, so at first order the δmud expansion, strong-isospin breaking corrections arecharacterized by a single parameter, δSU(2) . Next-to-leading order χPT yields numerical valuesfor δSU(2) of approximately −0.004. Recent direct lattice-QCD calculations of δSU(2) give larger
values of around −0.005 to −0.008 [11, 32–34, 53, 55], but further studies are needed. Thus, to beconservative, FLAG includes an uncertainty of 100% on the χPT estimate for δSU(2) when correcting
those decay-constant values that are quoted in the isospin-symmetric limit.
The errors on the decay-constant results in Table 72.1 obtained from (2 + 1)-flavor lattice-QCD
calculations do not include an estimate of the systematic uncertainty from the omission of charmsea quarks in the calculation. Consequently, when the uncertainty on the (2 + 1 + 1)-flavor FLAG
average is comparable to or better than that on the (2 + 1)-flavor FLAG average, we simply use
the four-flavor average as our preferred value. This is not possible, however, for the pion decay
constant. To account for this, we first estimate the systematic uncertainty on pseudoscalar-meson
decay constants associated with the omission of charm sea quarks. We then add this estimate inquadrature to the quoted error on the (2 + 1)-flavor FLAG average for fπ+ to obtain our preferredvalue.
The error introduced by omitting charm sea quarks can be roughly estimated by expanding
the charm-quark determinant in powers of 1/mc [56]; the resulting leading contribution is of order αs (ΛQCD /2mc )2 [57]. Taking the MS values mc (mc ) = 1.275 GeV, ΛQCD ∼ 340 MeV fromFLAG [48], and αs (mc ) ∼ 0.4, leads to an estimate of about 0.7% for the contribution to the decay
constants from charm sea quarks. We can compare this power-counting estimate of charm sea-quark
contributions with the observed differences between the (2+1)- and (2+1+1)-flavor lattice-QCD
averages for kaon, D(s) -meson, and B(s) -decay constants in Tables 72.1, 72.4, and 72.6. Looking
at Table 72.1, the three- and four-flavor averages for fK + agree to much better than our simple
power-counting estimate. Inspection of Tables 72.4 and 72.6 shows, however, that charm sea-quark
effects of this size are still allowed for both D(s) -meson and B(s) -meson decay constants.
Our final preferred theoretical values for the charged pion and kaon decay constants are
fπ+ = 130.2(1.2) MeV , fK + = 155.7(3) MeV ,fK += 1.193(2) ,fπ+(72.14)

where fK + and fK + /fπ+ are simply the four-flavor FLAG 2023 averages [2, 3], and fπ+ is thethree-flavor FLAG 2023 average with the error increased by the estimated 0.7% charm sea-quark
contribution.

72.3 Charmed mesons
72.3.1 Experimental rate measurements
Measurements have been made of the branching fractions for D+ and Ds+ mesons decaying toboth µ+ ν and τ + ν final states. The CLEO-c and BESIII experiments have made measurements of
D+ decays using e+ e− collisions at the ψ(3770) resonant energy where D− D+ pairs are copiously
produced. They fully reconstruct one of the D mesons; for concreteness, we will take this tobe the D− . Counting the number of these events provides the normalization for the branching
fraction measurement. The experimental analyses then proceed by identifying a candidate µ+ andThis is a combination of charge conjugation, C, and an eiπI2 rotation in isospin space that maps the triplet onto
itself.
4

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8

72. Leptonic Decays of Charged Pseudoscalar Mesons2−−−
forming the missing-mass squared, MM 2 = (ECM − ED− )2 − →p CM − →p D− − →p µ+ , where ECM
and pCM are the center-of-mass energy (which is known) and momentum (which equals zero ine+ e− collisions). A peak at zero MM 2 implies the existence of a missing neutrino, and hence theµ+ ν decay of the D+ . CLEO-c does not explicitly identify the muon, so their data consists of a
combination of µ+ ν and τ + ν, τ + → π + ν events. This permits them to do two fits: in one they fit
for the individual components, and in the other they fix the ratio of τ + ν/µ+ ν events to be thatgiven by the standard-model expectation. Thus, the former measurement should be used for newphysics searches, and the latter for standard-model predictions. Our average uses the fixed-ratio
value; this ratio is implied by Eq. 72.1 which is necessary to extract |Vcd |fD+ .



Table 72.2: Experimental results for B(D+ → µ+ ν[γ]), B(D+ → τ + ν[γ]),and |Vcd |fD+ . The systematic errors on the inferred values of |Vcd |fD+
include those from the D+ lifetime and mass. The error from radiative
corrections is only included in the entries labeled “our average.” In thesecond CLEO line the τ + decays to µ+ νµ ν̄τ .
Experiment
CLEO-c [58, 59]
CLEO-c [58, 59]
BESIII [60]
Our average
CLEO-c [61, 62]
BESIII [63]
Our average

Mode
µ+ ν
+
µ ν+τ + νµ+ νLines 2+3
τ + ν (π + ν)
τ + ν (π + ν)µ+ ν + τ + ν

B(3.93 ± 0.35 ± 0.09) × 10−4
(3.82 ± 0.32 ± 0.09) × 10−4
(3.71 ± 0.19 ± 0.06) × 10−4
(3.74 ± 0.17) × 10−4
< 1.2 × 10−3
(1.20 ± 0.24 ± 0.12) × 10−3

|Vcd |fD+ (MeV)
46.80 ± 2.08 ± 0.55
46.14 ± 1.93 ± 0.56
45.47 ± 1.16 ± 0.38
45.65 ± 1.19
50.08 ± 5.62
45.82 ± 1.10

Table 72.2 shows the available measurements of D+ → µ+ ν, an upper limit on D+ → τ + ν from
CLEO-c, and the first measurement of this decay from BESIII. To extract the values of |Vcd |fD+
via Eq. (72.1), we use [1] the mass mD+ = 1.86966(5) GeV and the lifetime τD+ = 1.033(5) ps,
and apply radiative corrections as described below. For calculating the average µ+ ν number, we
use the CLEO-c result from µ+ ν + τ + ν. Since the last review the D+ lifetime is updated and the
propagated uncertainties on |Vcd |fD+ from the last measurement [63] are corrected.
To obtain the purely leptonic rates Γ (0) (D+ → µ+ (τ + )ν), we subtract the radiative contributions as in Sec. 72.2.1, but use numerical values for the corrections appropriate for D mesons.
First, we reduce both the µ+ ν and τ + ν branching fractions in Table 72.2 by 1.8%, which is theuniversal short-distance electroweak contribution of Sirlin [4] evaluated using the D-meson mass
for the factorization scale. We do not adjust the experimental rates by the universal long-distancecorrection [5]. This is because QED bremsstrahlung contributions have already been subtracted
at leading-log order from the measurements in Table 72.2 using Monte-Carlo estimates computedwith PHOTOS [12]. The µ+ ν rates should also be reduced by the 1% estimate of the structuredependent contributions from Dobrescu and Kronfeld [17]. This correction accounts for tree-level
radiative processes in which the D meson decays into a real photon and an off-shell vector meson, which subsequently decays weakly to a charged lepton and neutrino. It is estimated using
Eq. (12) of Burdman et al. [7] with the CLEO-c cut on the photon energy from Ref. [74], which
is typical of all the measurements. We do not need to apply the structure-dependent correction
to the µ+ ν branching fractions in Table 72.2, however, because the experiments have already included it in their quoted results. Therefore, in summary, we reduce both the D+ → µ+ ν and the
D+ → τ + ν rates by 1.8% to account for radiative corrections. It is worth noting, however, that
31st May, 2024

972. Leptonic Decays of Charged Pseudoscalar Mesons
Table 72.3: Experimental results for B(Ds+ → µ+ ν[γ]), B(Ds+ → τ + ν[γ]),and |Vcs |fDs+ . The systematic errors on the inferred values of |Vcs |fDs+
include those from the Ds+ lifetime and mass. The entries labeled “ouraverage” take into account correlations between systematic errors common
to the experiments, and also include errors from radiative corrections.
Experiment
CLEO-c [61, 62]
BaBar∗ [65]Belle [66]
BESIII [67]
BESIII [68]
Our average
CLEO-c [61, 62]
CLEO-c [69]
CLEO-c [70, 71]
BaBar [65]Belle [66]Belle [66]Belle [66]
BESIII [67]
BESIII [68]
BESIII [72]BESIII [73]
Our average
Our average

Mode
µ+ ν
µ+ ν
µ+ ν
µ+ ν
µ+ ν
µ+ ν
τ + ν (π + ν)
τ + ν (ρ+ ν)
τ + ν (e+ νν)τ + ν (e+ (µ+ )νν)
τ + ν (π + ν)
τ + ν (e+ νν)τ + ν (µ+ νν)τ + ν (π + ν)
τ + ν (π + ν)
τ + ν (e+ νe ν)
τ + ν (π + π 0 ν)τ +νµ+ ν + τ + ν

B(%)
0.565 ± 0.045 ± 0.017
0.602 ± 0.038 ± 0.034
0.531 ± 0.028 ± 0.020
0.517 ± 0.075 ± 0.021
0.535 ± 0.013 ± 0.016
0.543 ± 0.017
6.42 ± 0.81 ± 0.18
5.52 ± 0.57 ± 0.21
5.30 ± 0.47 ± 0.22
5.00 ± 0.35 ± 0.49
6.04 ± 0.43+0.46−0.40
5.37 ± 0.33+0.35−0.31
5.86 ± 0.37+0.34−0.59
3.28 ± 1.83 ± 0.37
5.21 ± 0.25 ± 0.17
5.27 ± 0.10 ± 0.12
5.29 ± 0.25 ± 0.20
5.33 ± 0.13

|Vcs |fDs+ (MeV)
247.6 ± 9.9 ± 4.1
254.3 ± 8.0 ± 7.4
238.8 ± 6.3 ± 4.8
238.9 ± 17.3 ± 4.9
240.9 ± 2.9 ± 4.0
242.6 ± 4.0
267.3 ± 16.9 ± 4.2
247.9 ± 12.8 ± 5.0
242.9 ± 10.8 ± 5.3
236.9 ± 8.3 ± 11.7
260.3 ± 9.3+10.1−8.8
244.5 ± 7.5+8.2−7.4
255.4 ± 8.0+7.6−13.1
193 ± 54 ± 11240.8 ± 5.8 ± 4.3
242.2 ± 2.3 ± 3.2
242.7 ± 5.7 ± 4.0
244.0 ± 3.1
243.5 ± 2.7∗
We do not use a previous unpublished BaBar result from a subsample of data that uses a different technique for obtaining
the branching fraction normalization [64].

the universal long-distance electromagnetic contribution estimated for point-like charged mesons
by Kinoshita [5], which we are not including because IB contributions are already subtracted fromthe measurements via PHOTOS, would increase both rates by about 2.5%.
We now discuss the Ds+ decay process; there is no change since our previous review [1]. Measurements of Ds+ leptonic decay branching fractions by several groups are listed in Table 72.3. We
exclude older values obtained by normalizing to Ds+ decay modes that are not well-defined. Manymeasurements, for example, used the φπ + mode. This decay is a subset of the Ds+ → K + K − π +channel which has interferences from other modes populating the K + K − mass region near the φ, themost prominent of which is the f0 (980). Thus, the extraction of the effective φπ + rate is sensitive
to the mass resolution of the experiment and the cuts used to define the φ mass region [75].5
To find Ds decays in the µ+ ν signal channels, the experiments rely on fully reconstructing allof the final state particles except for the neutrino and use a missing-mass technique to infer the
existence of the neutrino. CLEO and BESIII use e+ e− → Ds Ds∗ collisions at 4170 MeV, while
BaBar and Belle use e+ e− → DKnπDs∗ collisions at energies near the Υ (4S). CLEO and BESIII
do a similar analysis as was done for the D+ above. BaBar and Belle do a similar MM 2 calculationby using the reconstructed hadrons, the photon from the Ds∗+ decay and a detected µ+ . To get
the normalization they do a MM 2 fit without the µ+ and use the signal at the Ds+ mass squared
5
We have not included the BaBar result for B(Ds+ → µ+ ν) reported in Ref. [76] because this measurement
determined the ratio of the leptonic decay rate to the hadronic decay rate Γ (Ds+ → `+ ν)/Γ (Ds+ → φπ + ).

31st May, 2024

10

72. Leptonic Decays of Charged Pseudoscalar Mesons

to determine the total Ds+ yield.
When selecting the τ + → π + ν̄ and τ + → ρ+ ν̄ decay modes, CLEO uses both the calculationof the missing-mass and the fact that there should be no extra energy in the event beyond that
deposited by the measured tagged Ds− and the τ + decay products. The τ + → e+ ν ν̄ mode, however,uses only extra energy. BaBar and Belle also use the extra energy to discriminate signal frombackground in their τ + ν measurements. BESIII uses τ + → π + ν decays, where they calculate theMM 2 and discriminate against µ+ from Ds+ → µ+ ν decays, and more recently τ + → e+ νν andτ + → π + π 0 ν decays.
When extracting |Vcs |fDs+ via Eq. (72.1), we first apply the −1.8% universal electroweak correction [4] to all of the µ+ ν and τ + ν branching fractions in Table 72.3; this is the same as forD+ mesons. We also decrease the BaBar and Belle µ+ ν branching fractions by the 1% structuredependent correction [17]. This correction was already included in CLEO and BESIII results forthe µ+ ν branching fractions in Table 72.3. We use the masses and lifetimes mDs+ = 1.96834(7) GeV,mτ + = 1.77686(12) GeV, and τDs+ = 0.504(4) ps [43]. The inferred values for fDs+ |Vcs | are in good
agreement for the µ+ ν and τ + ν decay modes.
It is clear from the discussion of radiative corrections in this section that they are less wellunderstood theoretically for D+ and Ds+ meson decays than for pions and kaons. We therefore
assign a 1.4% systematic uncertainty to the purely leptonic decay rates, which is half the size of theapplied radiative corrections. This translates to a 0.7% error on the products of the decay constanttimes CKM matrix element. Putting everything together, the main experimental results pertaining
to charmed meson leptonic decays are (see the bottom lines of Tables 72.2 and 72.3):|Vcd |fD+ = 45.82 ± 1.05 ± 0.32 = 45.82 ± 1.10 MeV ,(72.15)|Vcs |fDs+ = 243.5 ± 2.1 ± 1.7 = 243.5 ± 2.7 MeV ,(72.16)

where the errors are from the measured branching fractions and the applied radiative corrections,
respectively.
72.3.2 Theoretical decay-constant calculations
Table 72.4 presents recent theoretical calculations of charmed heavy-light meson decay constants
and their ratio in the isospin-symmetric limit mu = md . (As in Sec. 72.2.2, we denote the physicalD+ -meson decay constant by fD+ , and use fD for the isospin-symmetric value.) The upper twopanels show results from lattice-QCD calculations with three (Nf = 2 + 1) or four flavors (Nf =2+1+1) of dynamical quarks. Although there are fewer available results than for the pion and kaon
sector, both fD and fDs have been obtained using multiple sets of gauge-field configurations withdifferent lattice fermion actions, providing independent confirmation. For comparison, the bottom
panel of Table 72.4 shows QCD-model calculations of the D- and Ds -meson decay constants forwhich uncertainty estimates are provided. The lattice and non-lattice results agree, but numericallattice-QCD calculations have now reached significantly greater precision than other approaches.
The lattice-QCD decay-constant results in Table 72.4 were all obtained using isospin-symmetricgauge-field configurations. As discussed in Sec. 72.2.2, however, the leading strong-isospin breaking
corrections to heavy-light pseudoscalar-meson decay constants can be accounted for by using thephysical down (or up) quark in the D (or B) meson. Strong-isospin breaking corrections to heavystrange meson decay constants are roughly an order-of-magnitude smaller because there are nolight valence quarks involved. Recently, the Fermilab Lattice and MILC Collaborations used thisapproach to calculate directly the dominant strong-isospin breaking corrections to both fD and fB ,finding [32]fD+ − fD = +0.58(1)stat (7)syst (1)EM scheme MeV ,31st May, 2024

11

72. Leptonic Decays of Charged Pseudoscalar Mesons
Table 72.4: Recent theoretical determinations of fD , fDs , and their ratio
in the isospin-symmetric limit. The upper panels show results from latticeQCD calculations with (2 + 1 + 1) and (2 + 1) dynamical quark flavors,
respectively. Statistical and systematic errors are quoted separately. The
bottom panel shows estimates from QCD sum rules (QCD SR). These
are not used to obtain our preferred decay-constant values. The 2+1+1
averages are dominated by the Fermilab/MILC values.
Reference
Fermilab/MILC 17 [32]
ETM 14 [33]FLAG 23 average [3]χQCD 20A [77]
RBC/UKQCD 18A [78]†
RBC/UKQCD 17 [79]
χQCD 14 [80]HPQCD 12 [81]Fermilab/MILC 11 [82]HPQCD 10 [83]
FLAG 23 average [3]Pullin 21 [84]Wang 15 [85]‡Gelhausen 13 [86]Narison 12 [87]Lucha 11 [88]Method
LQCD
LQCD
LQCD
LQCD
LQCD
LQCD
LQCD
LQCD
LQCD
LQCD
LQCD
QCD SRQCD SRQCD SRQCD SRQCD SR

Nf
2+1+1
2+1+1
2+1+1
2+1
2+1
2+1
2+1
2+1
2+1
2+1
2+1

fD (MeV)
212.1(0.3)(0.5)
207.4(3.7)(0.9)
212.0(0.7)
213(5)–
208.7(2.8)(+2.1−1.8 )–
208.3(1.0)(3.3)
218.9(9.2)(6.6)–
209.0(2.4)
190(15)208(10)
201 +12−13
204(6)
206.2(8.9)

fDs (MeV)
249.9(0.3)(0.3)
247.2(3.9)(1.4)
249.9(0.5)
249(7)–
246.4(1.3)(+1.3−1.9 )
254(2)(4)–260.1(8.9)(6.1)
248.0(1.4)(2.1)
248.0(1.6)
226(17)240(10)
238 +13−23
246(6)
245.3(16.3)

fDs /fD
1.1782(06)(15)∗1.192(19)(11)
1.1783(16)1.16(3)
1.1740(51)(68)
1.1667(77)(+57−43 )–1.187(4)(12)
1.188(14)(21)–
1.174(7)1.19(7)1.15(6)
1.15 +0.04−0.05
1.21(4)
1.193(26)∗
Ref. [32] provides values for fD and fDs in the isopsin-symmetric limit, but not for their ratio. Here we infer the central value
from those of the individual decay constants, and take the statistical and systematic errors to be the same as for the physical ratio
fDs /fD+ .
†Slight difference from preliminary value quoted in 2019 FLAG review.
pole‡
are also given in the paper.
Obtained using mMS
c ; results using mc

fB + − fB = −0.53(5)stat (7)syst (0)EM scheme MeV .

(72.17)

These results agree with independent estimates of the strong-isospin-breaking corrections to heavylight meson decay constants from QCD sum rules [89]. Combined with the determinations of fD
and fB from the same work, Eq. (72.17) implies that the corrections to the SU(3)-flavor breaking
ratios are


fDs
fDs
fBs
fB
=
1 − 0.0027(3) ,= s 1 + 0.0028(5) .
(72.18)
fD+
fD
fB +
fB
These estimated strong-isospin-breaking corrections to fD and fDs /fD above are commensuratewith the uncertainties on the (2+1+1)-flavor FLAG averages in Table 72.4. Consequently, it isimportant to account for isospin-breaking effects before combining the theoretical decay constants
with the corresponding experimental decay rates.
To obtain the charged D+ -meson decay constant, we apply the correction in Eq. (72.17) to the(2+1+1)-flavor 2021 FLAG average for the D-meson decay constant in the isospin-symmetric limit.
Similarly we use Eq. (72.18) to correct the (2+1+1)-flavor 2021 FLAG average for fDs /fD . We
take the four-flavor FLAG 2021 average for fDs directly. Our final preferred theoretical values for
31st May, 2024

12

72. Leptonic Decays of Charged Pseudoscalar Mesons

the charmed pseudoscalar-meson decay constants are
fD+ = 212.0(7) MeV ,fDs = 249.9(5) MeV ,fDs= 1.1783(16) .
fD+(72.19)72.4 Bottom mesons
72.4.1 Experimental rate measurements
The Belle and BaBar collaborations have found evidence for B − → τ − ν decay in e+ e− → B − B +collisions at the Υ (4S) energy. The analysis relies on reconstructing a hadronic or semi-leptonic
B decay tag, finding a τ candidate in the remaining track and photon candidates, and examining
the extra energy in the event which should be close to zero for a real τ − decay to e− ν ν̄ or µ− ν ν̄
opposite a B + tag. The results are listed in Table 72.5. These are unchanged since the previous
review [1].
Table 72.5: Experimental results for B(B − → τ − ν) and |Vub |fB + . To
extract the values of |Vub |fB + via Eq. (72.1), we use the PDG values of
the B + lifetime of 1.638±0.004 ps, and the τ + and B + masses of 1.77686
and 5.27934 GeV, respectively.
Experiment Tag
B (units of 10−4 ) |Vub |fB + (MeV)
Belle [90]Hadronic
0.72+0.27−0.25 ± 0.11
Belle [91]Semileptonic 1.25 ± 0.28 ± 0.27
Belle [91]Average
0.91 ± 0.22
0.72 ± 0.09
+0.53
BaBar [92] Hadronic
1.83 −0.49 ± 0.24
BaBar [93] Semileptonic 1.7 ± 0.8 ± 0.2
BaBar [92] Average
1.79 ± 0.48
1.01 ± 0.14
Our average
1.06 ± 0.20
0.77 ± 0.07
Because there are large backgrounds under the signals for these measurements, as well as substantial systematic errors, the significances of the individual results are still below the 5σ discovery
threshold. Belle quotes 4.6σ for their combined hadronic and semileptonic tags, while BaBar quotes
3.3σ and 2.3σ, for hadronic and semileptonic tags. Greater precision is necessary to determine if
any effects beyond the standard model are present.
We do not correct the measured branching ratios in Table 72.5 for radiative corrections because
the experimental uncertainties are so large. The relative radiative corrections are expected tobe larger, however, for B → µν leptonic decays because the corrections are no longer helicity
suppressed [9], and may be a significant fraction of the purely leptonic rate. More theoreticalwork is needed to understand radiative corrections to leptonic B decays in anticipation of future
measurements with greater precision, and of new decay channels.
72.4.2 Theoretical decay-constant calculations
Table 72.6 presents recent theoretical calculations of bottom heavy-light meson decay constants
and their ratio in the isospin-symmetric limit mu = md . The upper two panels show results fromlattice-QCD calculations with three (Nf = 2 + 1) or four flavors (Nf = 2 + 1 + 1) of dynamicalquarks. For all decay constants, calculations using different gauge-field configurations, light-quark
actions, and b-quark actions provide independent confirmation. For comparison, the bottom panelof Table 72.6 shows QCD-model calculations of the B- and Bs -meson decay constants for which
uncertainty estimates are provided. These are consistent with the lattice values, but with muchlarger uncertainties.
31st May, 2024

72. Leptonic Decays of Charged Pseudoscalar Mesons

13

Table 72.6: Recent theoretical determinations of fB , fBs , and their ratio in the isospin-symmetric limit. The upper panels show results fromlattice-QCD calculations with (2 + 1 + 1) and (2 + 1) dynamical quark
flavors, respectively. When available, statistical and systematic errors arequoted separately. The bottom panel shows estimates from the relativistic
potential model (RPM) and QCD sum rules (QCD SR), which are notused to obtain our preferred decay-constant values. The 2+1+1 FLAG 23
averages are dominated by the Fermilab/MILC values.
Reference
Method Nf
fB (MeV)fBs (MeV)
fBs /fB
Fermilab/MILC 17 [32] LQCD 2+1+1
189.9(1.4)
230.7(1.2)
1.2146(49)
HPQCD 17 [94]∗
LQCD 2+1+1
190(4)
229(5)
1.206(5)
ETM 16 [95]
LQCD 2+1+1
193(6)
229(5)
1.184(25)
HPQCD 13 [96]
LQCD 2+1+1
186(4)
224(5)
1.205(7)FLAG 23 average [3]
LQCD 2+1+1
190.0(1.3)
230.3(1.3)
1.209(5)Aoki 14 [97]†
LQCD 2+1 218.8(6.5)(30.8) 263.5(4.8)(36.7) 1.193(20)(44)
RBC/UKQCD 14 [98]‡ LQCD 2+1 195.6(6.4)(13.3) 235.4(5.2)(11.1) 1.223(14)(70)
HPQCD 12 [99]
LQCD 2+1
191(1)(8)228(3)(10) 1.188(12)(13)
HPQCD 12 [99]
LQCD 2+1
189(3)(3)∗––
HPQCD 11 [100]LQCD 2+1–
225(3)(3)–
Fermilab/MILC 11 [82] LQCD 2+1 196.9(5.5)(7.0) 242.0(5.1)(8.0) 1.229(13)(23)
FLAG 23 average [3]
LQCD 2+1
192.0(4.3)
228.4(3.7)
1.201(16)Pullin 21 [84]Sun 16 [101]§
Wang 15 [85]§Baker 13 [102]
Lucha 13 [103]Gelhausen 13 [86]Narison 12 [87]

QCD SRRPM
QCD SRQCD SRQCD SRQCD SRQCD SR

192+20−19219(15)194(15)186(14)
192.0(14.6)

207 +17−9
206(7)

225+21−20
266(19)231(16)222 (12)228.0(19.8)
242 +17−12234(5)

1.17(7)1.21(9)1.19(10)
1.19(4)
1.184(24) 
1.17 +3−4
1.14(3)∗

Average of new HPQCD 17 and previous HPQCD 13 results allowing for correlations between the two sets of results.
Obtained with static b quarks (i.e., mb → ∞).
‡Ref. [98] does not provide results in the isospin-symmetric limit, so we show fB + and fBs /fB + for this work.
pole§
are also given in the paper.
Obtained using mMS
b ; results using mb†

The lattice-QCD decay-constant results in Table 72.6 were all obtained using isospin-symmetricgauge-field configurations. Some calculations, however, account for the dominant effect of strongisospin-breaking by using the correct value for the valence light-quark mass in the B meson (mu for
fB + and md for fB 0 ). Early estimates of the strong-isospin-breaking correction obtained fB + −fB ∼−2 MeV [96, 98], which would be significant given the present lattice-QCD uncertainties. It turns
out, however, that these calculations inadvertently introduced a spurious sea-quark contribution,and therefore overestimated the size of the effect. A more recent calculation by the Fermilab/MILC
Collaboration finds very little evidence for isospin violation (fB + − fB ∼ −0.5 MeV) [32], which ismore than two times smaller than the total uncertainties on present lattice-QCD calculations. For
this reason, we quote isospin averages in the current review. Our preferred theoretical values forthe bottom pseudoscalar-meson decay constants are
fB = 190.0(1.3) MeV ,fBs = 230.3(1.3) MeV ,31st May, 2024

fBs
= 1.209(5) ,fB(72.20)14

72. Leptonic Decays of Charged Pseudoscalar Mesons

which are simply the Nf = 2 + 1 + 1 FLAG 2023 averages [3]. Because the uncertainties on
the three-flavor results in Table 72.6 are substantially larger than those on the four-flavor results,
including them in the average leaves the central values unchanged, and decreases the errors only
slightly.

72.5 Phenomenological implications
72.5.1 |Vud |, |Vus |, and status of first-row unitarity
Using the average values for fπ+ |Vud |, fK + |Vus |, and their ratio from Eqs. (72.10)–(72.12) andfor fπ+ , fK + , and their ratio from Eq. (72.14), we obtain the following determinations of the CKM
matrix elements |Vud |, |Vus |, and their ratio from leptonic decays within the standard model:|Vud | = 0.9763(1)(90)(10) ,|Vus | = 0.2253(3)(4)(2),|Vus |= 0.2313(3)(4)(2) ,|Vud |(72.21)

where the errors are from the experimental branching fraction(s), the pseudoscalar decay constant(s), and radiative corrections, respectively. These results enable a test of the unitarity of thefirst row of the CKM matrix from leptonic decays alone (the contribution from |Vub | is negligible).
Using the values of |Vud | and |Vus | from Eq. (72.21), we find
|Vud |2 + |Vus |2 + |Vub |2 − 1 = 0.004(18) ,(72.22)

which is consistent with three-generation unitarity at the few-percent level.
The determinations of |Vud | and |Vus | from leptonic decays in Eq. (72.21) can be comparedto those obtained from other processes. The result above for |Vud | agrees with the determinationfrom super-allowed β-decay (a previous value of |Vud | = 0.97420(21) [104] has been updated to|Vud | = 0.97373(31) [105]) but has a considerably larger error, primarily due to the uncertainty inthe theoretical determination of fπ+ .
The CKM element |Vus | can be determined from semileptonic K + → π 0 `+ ν decay. Here experimental measurements provide a value for the product f+Kπ (0)|Vus |, where f+Kπ (0) is the form
factor at zero four-momentum transfer between the initial state kaon and the final state pion. We
quote a recent analysis by Seng et al. [106] which finds |Vus |f+Kπ (0) = 0.21635(38)(3), with thefirst error denoting experiment and the second reflecting uncertainty in higher order corrections.
Choosing the 2021 2+1+1-flavor FLAG average for f+ (0)Kπ = 0.9698(17) [2] based on the calculations of ETM [107] and Fermilab/MILC [108] gives |Vus | = 0.22309(39)exp (39)LQCD (3)HO fromsemileptonic decay. The determinations of |Vus | from leptonic and semileptonic kaon decays areboth quite precise, with comparable errors, but the central values differ by 2.8σ.
Finally, the combination of the ratio |Vus |/|Vud | from leptonic decays [Eq. (72.21)] with |Vud |from β decay implies an alternative determination of |Vus | = 0.2252(5) which agrees with the value
from leptonic kaon decay, but again disagrees with the semileptonic-decay result at the 2.8σ level.
Given the roughly 3σ tension between |Vus | from leptonic and semileptonic kaon decays, it
is important to scrutinize the uncertainties on the theoretical and experimental inputs to |Vus |and other elements of the first row of the CKM matrix. Seng et al. have introduced a new
approach for calculating radiative corrections to neutron and nuclear beta decays using dispersion
relations [109–112]. These calculations imply a value of |Vud | = 0.97395(23), consistent with theupdated analysis of Ref. [105]. An independent calculation of the radiative corrections by Czarnecki
and Marciano using QCD sum rules yields similar results [113]. Using this value of |Vud | with thedetermination of |Vus | from leptonic kaon decays in Eq. (72.21), we obtain |Vud |2 +|Vus |2 +|Vub |2 −1 =−0.0007(5), again consistent with first-row unitarity.
31st May, 2024

15

72. Leptonic Decays of Charged Pseudoscalar Mesons

Last, we combine the experimental measurement of fπ+ |Vud | in Eq. (72.10) with |Vud | fromsuper-allowed β-decay [105] to infer an “experimental” value for the pion decay constant:fπ“exp”= 130.56(2)(4)(13) MeV ,−(72.23)where the uncertainties are from the errors on Γ , |Vud |, and higher-order corrections, respectively.
Many recent (2+1+1)-flavor lattice-QCD calculations use this quantity to set the overall physicalscale in their simulations, e.g., Refs. [10, 32–34]. Conversely, comparing fπ“exp”
with the 2021−
FLAG (2+1)-flavor average fπ+ = 130.2(8) MeV, which only includes lattice-QCD results that
employ other observables to set the scale [35–40], provides a test of lattice-QCD methods. The
values are in good agreement within present uncertainties. We do not quote an “experimental”
value for the kaon decay constant because the value of |Vus | is less clear given the approximately
3σ tension between the values of |Vus | obtained from leptonic and semileptonic kaon decays.
72.5.2 |Vcd |, |Vcs |, and status of second-row unitarity
Using the average values for |Vcd |fD+ and |Vcs |fDs+ from Eqs. (72.15) and (72.16), and for fD+
and fDs+ from Eq. (72.19), we obtain the following determinations of the CKM matrix elements|Vcd | and |Vcs | from leptonic decays within the standard model:|Vcd | = 0.2161(50)(15)(7)and|Vcs | = 0.9744(84)(16)(20) ,(72.24)

where the errors are from the measured branching fractions, radiative corrections, and decay constants, respectively. These results enable a test of the unitarity of the second row of the CKM
matrix. Taking |Vcb | = 40.8(1.4) × 10−3 [1], we obtain|Vcd |2 + |Vcs |2 + |Vcb |2 − 1 = −0.002(22) ,(72.25)in agreement with three-generation unitarity.
The uncertainty on |Vcd | in Eq. (72.24) is limited by the measurement error on the D+ → µ+ ν
decay rate. For |Vcs |, however, the experimental and radiative-correction errors are commensurate.
The value of |Vcs | from leptonic Ds decays decreased substantially from the value of 1.007(17) in an
earlier version of this review [29, 114], and is now below unity as expected in the three-generation
CKM framework. This change is due to our more consistent treatment of the radiative corrections,which lower the purely leptonic decay rates for Ds+ → µ+ ν and Ds+ → τ + ν. We emphasize, however,that we have taken a generous uncertainty on these estimates, and that more theoretical work isneeded to really pin down the sizes of the radiative corrections to D(s) -meson leptonic decays.
The CKM matrix elements |Vcd | and |Vcs | can also be obtained from semileptonic D+ → π 0 `+ ν
and Ds+ → K 0 `+ ν decays, respectively. Here experimental measurements determine the product of
the form factor times the CKM element, and theory provides the value for the form factor at zerofour-momentum transfer between the initial D(s) meson and the final pion or kaon.
The latest experimental averages (2023, with data as of 2021) from the Heavy Flavor Averaging
Group (HFLAV) are f+Dπ (0)|Vcd | = 0.1426(18) and f+DK (0)|Vcs | = 0.7180(33) [115]. There were
not enough published lattice-QCD calculations of the zero-momentum D-meson semileptonic form
factors with Nf ≥ 3 to permit an average by the FLAG Collaboration; instead, we summarize
the 2+1+1 flavor results here. The newest calculation, from the Fermilab Lattice and MILC
Collaborations [116] appeared after the FLAG 2023 online update [3]. They report f+Dπ (0) =0.6300(51) and f+DK (0) = 0.7452(31); these results are the most precise and include both decay
modes. For the decay D → K`ν, the HPQCD Collaboration [117] calculates f+DK (0) = 0.7380(44),about 1.3σ lower. An update from HPQCD [118] gives f+DK (0) = 0.7441(40), in agreement with
Fermilab-MILC. Note that the two HPQCD results and Fermilab-MILC all share MILC gauge
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72. Leptonic Decays of Charged Pseudoscalar Mesons

configurations and are hence not independent. The ETM Collaboration [119], has results for both
decays, obtaining f+Dπ (0) = 0.612(35) and f+DK (0) = 0.765(31), which are consistent but lessprecise. A naive combination of the HFLAV f (0)|V | products and the new Fermilab-MILC results
gives results for |Vcd | and |Vcs | consistent with the leptonic decays in Eq. (72.24). However, as withthe leptonic case, one must exercise some care with corrections; this is beyond the scope of thepresent review. We note that Ref. [116] presents a detailed discussion of CKM element extractions
using their LQCD results, including consideration of QED, electroweak and isospin corrections; acomparison of leptonic and semileptonic results and unitarity tests are also included.
We can combine the experimental measurements of fD+ |Vcd | and fDs+ |Vcs | from Tables 72.2
and 72.3 with |Vcd | = 0.22486(67) and |Vcs | = 0.97349(16) from the PDG 2022 global unitaritytriangle analysis [1] to infer “experimental” values for the decay constants within the standard
model. We take the CKM elements from the global fit because they are based on many input
quantities, thereby reducing the sensitivity to any one outlying measurement or calculation. We
obtain for the decay constants= 203.8(4.7)(0.6)(1.4) MeV ,fD“exp”+“exp”
fD+ = 250.1(2.2)(0.04)(1.8) MeV ,s(fDs+ /fD+ ) “exp” = 1.228(30)(4)(9) .

(72.26)

where the uncertainties are from the errors on Γ (0) , CKM matrix elements, and radiative corrections, respectively. Here, the third uncertainty on the ratio is taken as 0.7%, the same as on the firsttwo results, due to presumed partial cancellation. Possible further cancellation is not very relevant
given the current statistical uncertainty. The “experimental” values fD+ (fDs+ /fD+ ) are consistent
within 1σ with the (2+1+1)-flavor lattice-QCD averages in Eq. (72.19). The CKM matrix element|Vcd | is approximately equal to |Vus |, as first shown by Wolfenstein [120, 121]. Thus, resolving theinconsistencies between determinations of |Vus | from leptonic and semileptonic decays discussed inSec. 72.5.1 may benefit from the constraints observed here.
Last, we can test lepton-flavor universality in charm meson decays by checking the following
relationship derived from Eq. (72.1):
2

2
m2τ 1 − m2τ /MDΓ (Ds+ → τ + ν)s=
2 = 9.75 ,Γ (Ds+ → µ+ ν)22
2
m 1 − m /Mµµ

(72.27)

Ds

where the uncertainties from the masses are negligible to the number of digits quoted. The measured
ratio of τ + ν to µ+ ν rates is 9.82 ± 0.40, consistent with this standard-model expectation.
72.5.3 |Vub | and other applications
Using the average value for |Vub |fB + from Table 72.5, and for fB + from Eq. (72.20), we obtainthe following determination of the CKM matrix element |Vub | from leptonic decays within thestandard model:|Vub | = 4.05(37)(3) × 10−3 ,(72.28)where the errors are from experiment and theory, respectively. One should bear in mind wheninterpreting Eq. (72.28) that none of the experimental measurements of the branching fractions
that enter the average for |Vub |fB + have individually reached the 5σ discovery level (see Sec. 72.4.1).
Further, decays involving the third generation of quarks and leptons may be particularly sensitive
to new physics associated with electroweak symmetry breaking due to their larger masses [13,15], so
Eq. (72.28) is more likely to be influenced by new physics than the determinations of the elementsof the first and second rows of the CKM matrix in the previous sections.
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72. Leptonic Decays of Charged Pseudoscalar Mesons

The CKM element |Vub | can also be obtained from semileptonic B-meson decays. A longstanding 2-3σ tension between the determinations of |Vub | from exclusive B → π`ν decay andfrom inclusive B → Xu `ν decay has diminished somewhat; here, Xu denotes all hadronic states
populated by the b → u weak transition [24,122–126]. The currently most precise determination of|Vub |excl = 3.73(14) × 10−3 is obtained from a joint z-fit by FLAG [2] of the vector and scalar form
factors f+Bπ (q 2 ) and f0Bπ (q 2 ) calculated in (2+1)-flavor lattice QCD [127–129] and experimental
measurements of the differential decay rate from BaBar [130, 131] and Belle [132, 133]. This isconsistent with the value of |Vub |excl = 3.70(16) × 10−3 obtained in the 2022 PDG review on
semileptonic determinations of Vcb , Vub [134]. On the other hand, the inclusive determination in thatsame review, obtained using the theoretical frameworks in Refs. [135–137], is |Vub |incl = 4.13(26) ×
10−3 [1,134]. This value is significantly lower than the 4.49(28)×10−3 value from the corresponding
PDG 2018 review [134], reducing the exclusive-inclusive tension to less than 2σ. The value of |Vub |from leptonic B → τ ν decay in Eq. (72.28) lies between these inclusive and exclusive determinations,and is compatible (within large uncertainties) with both.
Given the large uncertainties on the experimental measurements of B(B − → τ − ν), and thelingering tension between |Vub | obtained from inclusive and exclusive semileptonic B decays, we do
not present an “experimental” value of the decay constant fB + .

72.6 Concluding remarks
The increasing precision of measurements of leptonic decays of charged pseudoscalar mesons
has begun to require control over radiative corrections at a level superior to that presented here.
Improvements will be called for both in theory and in experiment (for example, by BESIII andBelle II).
Unitarity of the CKM matrix is in reasonable shape. However, there is a lingering almost 3σ
tension between the values of Vus determined by semileptonic and leptonic kaon decays. It will beinteresting to follow experimental and theoretical developments on this quantity.

Acknowledgements
I thank J. L. Rosner for useful discussions on this review. Previous assistance from C. Davies, S.
Gottlieb, S. Hashimoto, A. Kronfeld, H.-L. Ma, W. Marciano, and S. Stone is also acknowledged.
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