IPPP2263 TUM-HEP 142122 Dispersion relations for B00form factors Stephan K urten1Marvin Zanke2yBastian Kubis2zand Danny van Dyk1 3x

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IPPP/22/63, TUM-HEP 1421/22

Dispersion relations for B−→`−¯ν``0−`0+form factors

Stephan K¨urten,1, ∗Marvin Zanke,2, †Bastian Kubis,2, ‡and Danny van Dyk1, 3, §

1Physik Department (T31), Technische Universit¨at M¨unchen, 85748 Garching, Germany

2Helmholtz-Institut f¨ur Strahlen- und Kernphysik (Theorie) and

Bethe Center for Theoretical Physics, Universit¨at Bonn, 53115 Bonn, Germany

3Institute for Particle Physics Phenomenology and

Department of Physics, Durham University, Durham DH1 3LE, United Kingdom

Using dispersive methods, we study the B→γ∗form factors underlying the decay B−→

`−¯ν``0− `0+. We discuss the ambiguity that arises from a separation of the full B−→`−¯ν``0− `0+

amplitude into a hadronic tensor and a ﬁnal-state-radiation piece, including eﬀects from nonvan-

ishing lepton masses. For the eligibility of a dispersive treatment, we propose a decomposition of

the hadronic part that leads to four form factors that are free of kinematic singularities. By es-

tablishing a set of dispersion relations, we then relate the B→γ∗form factors to the well-known

B→V,V=ω(782), ρ(770), analogs. Using the combination of a series expansion in a conformal

variable and a vector-meson-dominance ansatz to parameterize the B→γ∗form factors, we infer

the values of the associated unknown parameters from the available input on B→V. The phe-

nomenological application of our formalism includes the determination of the branching ratios and

forward–backward asymmetries of the process B−→`−¯ν``0−`0+.

Keywords: B-meson physics, Nonperturbative eﬀects, Ward identity, Dispersion relations

I. INTRODUCTION

The radiative leptonic decay B−→`−¯ν`γis widely considered to be the best source of information on the leading-

twist B-meson light-cone distribution amplitude (LCDA) by elucidating the inner structure of the Bmeson [1–3].

However, measurements of this decay are likely only possible at the ongoing Belle II experiment and not at the

LHC experiments, primarily the LHCb. This precludes leveraging the upcoming large datasets at the LHC, which

will become available from run 3 onwards. The four-lepton decay of the Bmeson, B−→`−¯ν``0−`0+, with `06=`,

`(0)=e, µ, has been identiﬁed as a suitable candidate for studies at both Belle II and the LHC experiments. This

decay has been studied to some extent in the literature, with a variety of models for the relevant B→γ∗form

factors [4–7]. However, its usefulness to extract B-meson LCDA parameters is hampered by the need for a description

of a virtual photon in the timelike region, which requires careful treatment.

We propose a dispersive approach for B→γ∗, which is based on the fundamental principles of analyticity and

unitarity. Dispersive analyses in the timelike region are commonly done for low-energy processes, such as the pion

vector form factor; see, for instance, Ref. [8] and references therein. Here, we apply methods originally developed for

these processes to hadronic transition form factors of Bmesons. For future analyses, our approach has the potential

to enable the transfer of information from the region of timelike photon momentum to the spacelike region, where

the sensitivity to the LCDA parameters is less aﬀected by soft interactions [3]. We relate the isoscalar and isovector

components of the B→γ∗transition inherent to the hadronic part of the amplitude through B−→`−¯ν`γ∗(→

`0−`0+) to available input on B→ω≡ω(782) and B→ρ≡ρ(770) [9] via a set of dispersion relations in the

photon momentum. Although we use a vector-meson-dominance (VMD) ansatz in this work, our results provide the

groundwork for more sophisticated future analyses. Using dispersion relations requires the form factors to be free of

kinematic singularities. We modify the well-known Bardeen–Tung–Tarrach (BTT) [10, 11] procedure, which has

not been designed for hadronic form factors in weak transitions, to obtain such a set of form factors. At this, we

face a problem: the separation of the amplitude into a hadronic term—containing the nonperturbative dynamics of

the process—and a ﬁnal-state-radiation (FSR) term turns out to be ambiguous; the two terms are not individually

gauge invariant but only their sum is. A further issue is the lack of deﬁnite angular-momentum and parity quantum

numbers of the form factors. Our modiﬁcation to the BTT procedure addresses this issue, and we take special care

not to spoil the singularity-free structure.

To ensure a consistent treatment of lepton-mass eﬀects, we work with nonzero lepton masses throughout our

analysis; taking the limit m`(0)→0 remains possible. While the considerations in this article are mostly restricted to

∗stephan.kuerten91@gmail.com

†zanke@hiskp.uni-bonn.de

‡kubis@hiskp.uni-bonn.de

§danny.van.dyk@gmail.com

arXiv:2210.09832v2 [hep-ph] 9 May 2023

the decay of a negatively charged Bmeson, the decay of a positively charged Bmeson can be calculated in complete

analogy, with some minor adjustments to the formulae given here and completely equivalent numerical results.

The outline of this article is as follows: in Sec. II, we introduce the Lagrangian of the weak eﬀective theory (WET)

that describes semileptonic b→u`¯νtransitions. The amplitude for B−→`−¯ν`γ∗(→`0−`0+) and its decomposition

into a hadronic tensor and an FSR piece is discussed in Sec. III. Using our modiﬁed BTT procedure, the hadronic

tensor is then parameterized in terms of four form factors that are free of kinematic singularities in Sec. IV, where the

ambiguity arising from the separation of the full amplitude is a subject of special attention. In Sec. V, we establish

a set of dispersion relations that relate the B−→γ∗transition inherent to the hadronic part of the amplitude to

available input on B−→Vform factors, V=ω, ρ, and provide predictions for the B−→γ∗form factors. Using

these predictions, we present numerical results for the branching ratios and forward–backward (FB) asymmetries of

the process B−→`−¯ν``0−`0+in Sec. VI. We conclude and give a brief outlook in Sec. VII. Some supplementary

material is outsourced to Apps. A–G.

II. WEAK EFFECTIVE THEORY

At the energy scale of the Bmeson, the standard model’s (SM’s) ﬂavor-changing processes are conveniently described

within an eﬀective ﬁeld theory [12, 13]. The leading terms in this theory arise at mass dimension six, with higher-

dimensional operators being suppressed by at least m2

B/M 2

W≈0.4%. Moreover, such an eﬀective ﬁeld theory allows

us to transparently include potential eﬀects beyond the SM as long as new matter ﬁelds and mediators live above the

scale of electroweak symmetry breaking. For b→u`¯ν`transitions in particular, we use the eﬀective Lagrangian

Lub`ν

WET =4GF

√2Vub X

iCub`ν

iOub`ν

i+ h.c.,(1)

where GFis the Fermi constant as measured in muon decays, Vub is the Cabibbo–Kobayashi–Maskawa (CKM)

matrix element for the b→utransition, and Cub`ν

i≡ Cub`ν

i(µ) are the so-called Wilson coeﬃcients at the scale µ

that multiply the local ﬁeld operators Oub`ν

i≡ Oub`ν

i(x). A convenient basis of operators up to dimension six and

with only left-handed neutrinos is given by

Oub`ν

V,L(R)=¯u(x)γµPL(R)b(x)¯

`(x)γµPLν`(x),Oub`ν

S,L(R)=¯u(x)PL(R)b(x)¯

`(x)PLν`(x),

Oub`ν

T=¯u(x)σµν b(x)¯

`(x)σµν PLν`(x),(2)

where, in the SM, Cub`ν

V,L |SM = 1 + O(αe) and Cub`ν

i|SM = 0 for all other corresponding Wilson coeﬃcients. Here,

PL/R = (1 ∓γ5)/2 are the projection operators onto the left- and right-chiral components and αe=e2/(4π) is

the ﬁne-structure constant. To leading order in the electromagnetic (EM) interaction, matrix elements of the above

operators factorize into matrix elements of a purely hadronic and a purely leptonic current. In this work, we limit

ourselves to the SM operator Oub`ν

V,L and—to a lesser extent—the scalar operator Oub`ν

S,L .

III. HADRONIC TENSOR

We study the decay B−(p)→`−(p`)¯ν`(pν)γ∗(q), k=p`+pν, whose amplitude in the SM reads [1]

M(B−→`−¯ν`γ∗) = 4GFVub

√2h`−¯ν`γ∗|Oub`ν

V,L |B−i(3)

up to corrections of O(αe). It is convenient to write the WET operator in terms of the leptonic and hadronic weak

currents Jν

W(x) = ¯

`(x)γν(1 −γ5)ν`(x) and Jν

H(x) = ¯u(x)γν(1 −γ5)b(x) according to

Oub`ν

V,L =1

4JHν(0)Jν

W(0).(4)

At the level of the WET, there are two possible diagrammatic ways for the emission of the (virtual) photon: either

from the constituents of the Bmeson or from the charged ﬁnal-state lepton; the respective diagrams are shown

in Fig. 1.

ℓ−

¯νℓ

B−

pν

pℓ

qγ∗

ℓ−

¯νℓ

B−

pν

pℓ

γ∗

FIG. 1. The diagrams contributing to the decay B−→`−¯ν`γ∗at dimension six in the WET on the hadronic level: pole and cut

contributions of Tµν

H(k, q), e.g., from the intermediate states Bin k2or ππ in q2(left) and emission from the charged ﬁnal-state

lepton in Tµ

FSR(p`, pν, q) (right). The hadronic tensor Tµν

H(k, q) and FSR tensor Tµ

FSR(p`, pν, q) are deﬁned in Eqs. (7) and (8),

respectively. Note that an eﬀective four-particle vertex is discarded here, since it contributes at dimension eight in the WET.

At leading order in the EM coupling, the hadronic matrix element on the right-hand side of Eq. (3) can be written

h`−¯ν`γ∗|JHν(0)Jν

W(0)|B−i=e∗

µhh`−¯ν`|JWν(0)|0iZd4xeiqx h0|T{Jµ

EM(x)Jν

H(0)}|B−i

+h0|JHν(0)|B−iZd4xeiqx h`−¯ν`|T{Jµ

EM(x)Jν

W(0)}|0ii

=e∗

µhQBLνTµν

H(k, q)−ifBpνZd4xeiqx h`−¯ν`|T{Jµ

EM(x)Jν

W(0)}|0ii

=e∗

µQBLνTµν

H(k, q) + Q`Tµ

FSR(p`, pν, q),(5)

where eis the elementary charge and ∗

µ≡∗

µ(q;λ) the polarization vector of the outgoing photon with momentum q

and polarization λ. Furthermore, fBis the decay constant of the B-meson, h0|¯u(0)γνγ5b(0)|B−i= ifBpν, and

Jµ

EM(x) = ¯q(x)Qγµq(x) + X

Q`¯

`(x)γµ`(x) (6)

the EM current, with q(x)=(u(x), d(x), s(x), c(x), b(x))|,Q= diag[2/3,−1/3,−1/3,2/3,−1/3] the quark charge

matrix, and QB=−1 = Q`the charge of the Bmeson and lepton in units of e. With the aim to render the transfer

of our analysis to the positively charged channel more transparent, we will explicitly retain factors of QB=Q`in

our formulae; it is, however, to be kept in mind that further modiﬁcations of the spinor structure apply beyond this

simple alteration. In Eq. (5), we moreover abbreviate the leptonic matrix element Lν= ¯u`γν(1 −γ5)v¯νand introduce

the hadronic tensor Tµν

H(k, q),

QBTµν

H(k, q) = Zd4xeiqx h0|T{Jµ

EM(x)Jν

H(0)}|B−i,(7)

and the FSR tensor Tµ

FSR(p`, pν, q),

Q`Tµ

FSR(p`, pν, q) = −ifBpνZd4xeiqx h`−¯ν`|T{Jµ

EM(x)Jν

W(0)}|0i.(8)

While the hadronic tensor Tµν

H(k, q) describes the genuinely nonperturbative physics of the process, Tµ

FSR(p`, pν, q)

comprises the FSR from the charged lepton and can be reduced to the B-meson decay constant fBand an entirely

perturbative remainder. The former can be decomposed into a set of Lorentz structures and associated scalar-

valued functions, which are commonly referred to as the B→γ∗form factors. The purpose of this work is to study

these form factors within a dispersive framework, which requires knowledge of their singularity structure in the two

independent kinematic variables and of the form factors’ asymptotic behavior, see Sec. IV.

For the FSR tensor in the case of a massless charged lepton, one ﬁnds the remarkably simple result [1, 4, 5, 14, 15]

Tµ

FSR,0(p`, pν, q) = fBLµ.(9)

The case of nonzero mass leads to the more intricate formula [16, 17]

Tµ

FSR,m`(p`, pν, q) = fBLµ+m`¯u`

2pµ

`+γµ/

(p`+q)2−m2

(1 −γ5)v¯ν.(10)

For our purpose, it proves convenient to bring the FSR contribution into such a form that it shares a common factor

of Lνwith its hadronic counterpiece, i.e.,

h`−¯ν`γ∗|Jν

W(0)JHν(0)|B−i=eQB∗

µTµν

H(k, q) + Tµν

FSR(p`, pν, q)Lν.(11)

It is straightforward to achieve such a description for the massless case, m`= 0, Eq. (9). For the massive case, m`6= 0,

we make use of the Chisholm identity [18]

iµνρσγσγ5=γµγνγρ−gµν γρ+gµργν−gνργµ,(12)

with the convention 0123 = +1. From this, we obtain

Tµν

FSR(p`, pν, q) = fBgµν +2pµ

`pν

`+pµ

`qν+qµpν

`−(p`·q)gµν + iµνρσ(p`)ρqσ

(p`+q)2−m2

`,(13)

which is valid only when contracted with the leptonic matrix element Lν.1

Because of gauge invariance, the full amplitude complies with the Ward identity

qµTµν

H(k, q) + Tµν

FSR(p`, pν, q)Lν= 0.(14)

However, the hadronic and FSR tensor are not individually gauge invariant but satisfy [1, 4, 5]

qµTµν

H(k, q) = −fB(k+q)ν,

qµTµν

FSR(p`, pν, q) = fB(k+q)ν,(15)

so that gauge invariance only holds for the sum of both contributions. Based on Eq. (15), we split the hadronic tensor

into a homogeneous part and an inhomogeneous part by means of Tµν

H(k, q) = Tµν

H,hom.(k, q) + Tµν

H,inhom.(k, q), which

obey

qµTµν

H,hom.(k, q)=0,

qµTµν

H,inhom.(k, q) = −fB(k+q)ν.(16)

We have not yet made any choice of Lorentz decomposition for Tµν

H(k, q) or its (in)homogeneous part. In App. A,

we demonstrate that any choice for the decomposition of the hadronic tensor leads to the relation

kνTµν

H,hom.(k, q) = Tµ

P(k, q) + fB(k+q)µ−kνTµν

H,inhom.(k, q),(17)

where the pseudoscalar tensor Tµ

P(k, q) is deﬁned in terms of the pseudoscalar weak current JP(x) = ¯u(x)γ5b(x) via

QBTµ

P(k, q)=(mb+mu)Zd4xeiqx h0|T{Jµ

EM(x)JP(0)}|B−i,(18)

with mband muthe MS masses of the b- and u-quarks. As also shown in App. A, this tensor is not gauge invariant

but, similar to Eq. (15), fulﬁlls

qµTµ

P(k, q) = −fBm2

B.(19)

For this reason, we proceed in analogy to Eq. (16) and split Tµ

P(k, q) = Tµ

P,hom.(k, q) + Tµ

P,inhom.(k, q), where

qµTµ

P,hom.(k, q)=0,

qµTµ

P,inhom.(k, q) = −fBm2

B.(20)

In this work, we additionally impose that the homogeneous part of the hadronic tensor fulﬁlls

kνTµν

H,hom.(k, q)!

=Tµ

P,hom.(k, q),(21)

1Note that one can, in principle, further make the replacement pν

`→kνin Eq. (13) by virtue of the Dirac equation for the neutrino.

which, using Eq. (17), leads to the condition

Tµ

P,inhom.(k, q) + fB(k+q)µ−kνTµν

H,inhom.(k, q)=0.(22)

This choice is natural because it relates one of the hadronic form factors of the axial-vector current with that of the

pseudoscalar current, as is the case for hadronic form factors in other weak transitions, too.

The tensors Tµν

H(k, q) and Tµν

FSR(p`, pν, q) emerge in predictions for the decay B−(p)→`−(p`)¯ν`(pν)`0−(q1)`0+(q2),

with `06=`,q=q1+q2,

M(B−→`−¯ν``0−`0+) = 4GFVub

√2h`−¯ν``0−`0+|Oub`ν

V,L |B−i

=GFVub

√2

q2QBTµν

H(k, q) + Tµν

FSR(p`, pν, q)lµLν,(23)

where we abbreviate the leptonic matrix element lµ= ¯u`0γµv¯

`0. The discussion of the decay with identical lep-

ton ﬂavors, `0=`, is more involved [4, 19], since an additional diagram has to be taken into account due to the

interchangeability of two ﬁnal-state fermions, which is beyond the scope of this article.

IV. B→γ∗FORM FACTORS

We develop a method that closely resembles the BTT procedure [10, 11] to parameterize the homogeneous part of

the hadronic tensor, see App. B. Compared to the BTT procedure, our method has the advantage that the emerging

form factors have deﬁnite angular-momentum and parity quantum numbers. Our result reads

Tµν

H,hom.(k, q) = 1

[(k·q)gµν −kµqν]F1(k2, q2) + 1

mBhq2

k2kµkν−k·q

k2qµkν+qµqν−q2gµν iF2(k2, q2)

mBhk·q

k2qµkν−q2

k2kµkνiF3(k2, q2) + i

µνρσkρqσF4(k2, q2),(24)

where the form factors F1(k2, q2) and F2(k2, q2) have axial-vector, F3(k2, q2) has pseudoscalar, and F4(k2, q2) vector

quantum numbers with respect to the weak current.2Assuming no modiﬁcation due to the inhomogeneous part

Tµν

H,inhom.(k, q), our form factors are free of kinematic singularities in k2and q2as well as kinematic zeroes in q2.

However, to ensure a ﬁnite amplitude at k2= 0, the relation F2(0, q2) = F3(0, q2) must hold for all q2. The factors

of mBand the imaginary unit in Eq. (24) render the form factors dimensionless and—with the phase of the Bmeson

chosen appropriately—real-valued below the onset of the ﬁrst branch cut.

The relations given in Eq. (16) constrain the inhomogeneous part of the hadronic tensor to the generic form

Tµν

H,inhom.(k, q) = −fBagµν +bkµkν

k·q+ckµqν

k·q+ (1 −b)qµkν

q2+ (1 −a−c)qµqν

q2,(25)

where a≡a(k2, q2), b≡b(k2, q2), and c≡c(k2, q2) are arbitrary real-valued coeﬃcients. The Levi-Civita tensor is

absent in this expression because it carries the wrong quantum numbers in light of the fact that the inhomogeneity is

entirely due to the axial-vector part of Eq. (7). On account of Eq. (20), the inhomogeneous part of the pseudoscalar

tensor furthermore takes the generic form

Tµ

P,inhom.(k, q) = −fBm2

Bdkµ

k·q+ (1 −d)qµ

q2,(26)

where d≡d(k2, q2) is an arbitrary real-valued coeﬃcient. Adopting the condition imposed in Eq. (22), we ﬁnd that

d=(1 + a+c)(k·q) + bk2

,(27)

which ﬁxes Tµ

P,inhom.(k, q) once Tµν

H,inhom.(k, q) is speciﬁed. We collect four diﬀerent choices for the coeﬃcients, labeled

Athrough D, in Table I. With regard to the dispersive treatment of the form factors in this article, i.e., the

2Note that for on-shell photons, only the form factors F1(k2, q2) and F4(k2, q2) contribute, which correspond to transverse polarizations.

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IPPP/22/63,TUM-HEP1421/22DispersionrelationsforB!```0`0+formfactorsStephanKurten,1,MarvinZanke,2,yBastianKubis,2,zandDannyvanDyk1,3,x1PhysikDepartment(T31),TechnischeUniversitatMunchen,85748Garching,Germany2Helmholtz-InstitutfurStrahlen-undKernphysik(Theorie)andBetheCenterforTheoreticalPhysic...

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IPPP2263 TUM-HEP 142122 Dispersion relations for B00form factors Stephan K urten1Marvin Zanke2yBastian Kubis2zand Danny van Dyk1 3x

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