February 10 2023 On the impact of meson mixing on Bseeangular observables at low q2

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February 10, 2023

On the impact of meson mixing on Bs→φee angular

observables at low q2

S´ebastien Descotes-Genon, Ioannis Plakias, Olcyr Sumensari

Universit´e Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France

Abstract

Decays based on the b→sγ transition are expected to yield left-handed photons in the

Standard Model, but could be particularly sensitive to New Physics contributions modifying the

short-distance Wilson coeﬃcients C7and C70deﬁned in the low-energy Eﬀective Field Theory.

These coeﬃcients can be determined by combining observables of several modes, among which

Bs→φee at low q2, in a kinematic range where the photon-pole is dominant. We investigate

the impact of Bs−¯

Bsmixing on the angular observables available for this mode, which induces

a time-dependent modulation governed by interference terms between mixing and decay that are

sensitive to the moduli and phases of C7and C70. These interference terms can be extracted

through a time-dependent analysis but they also aﬀect time-integrated observables. In particular,

we show that the asymmetries A(2)

Tand A(Im);CP

Treceive terms of potentially similar size from

mixing-independent and mixing-induced terms, and we discuss how the constraints coming from

the angular analysis of Bs→φee at low q2should be interpreted in the presence of mixing.

1 Introduction

Processes mediated by ﬂavor-changing neutral currents are loop-suppressed in the Standard Model

(SM), making them useful probes of New Physics. A prominent example is the b→sµµ transition,

for which several discrepancies have been observed in the past several years. Indeed, LHCb data

exhibits deviations close to 3σfrom the SM expectation in the P0

5angular observable of B→K∗µµ

decay [1], and tensions are also seen in branching ratios of b→sµµ exclusive decays [2–6]. Deviations

are also hinted at in Belle data for B→K∗µµ [7]. Global ﬁts [8–12] indicate that these deviations

can be described consistently in the Low-Energy Eﬀective Field Theory (LEFT) at the µ=mbscale,

if we assume that New Physics (NP) modiﬁes the short-distance contributions of the leading SM

operators and/or their chirality-ﬂipped versions. Several NP scenarios can provide an equally good

description of the data, with the interesting possibility of a V−Astructure of the NP contribution,

which would be similar to the SM one.

The closely related b→sγ transition is also known to be a powerful probe of NP eﬀects [13]. This

ﬂavour-changing neutral current is also suppressed in the SM and can be aﬀected by NP eﬀects at

the loop level. Although the number of observables is limited for b→sγ, its theoretical description

is much simpler than its b→s`` counterpart. The V−Astructure of the weak interaction in the

SM means that the photon will be dominantly produced with a left-handed polarisation, and that

the right-handed polarisation is highly suppressed by the ratio of quark masses ms/mb[13]. This is

encoded at the level of the LEFT by the suppression of the Wilson coeﬃcient C70compared to the

leading one C7. However, NP in right-handed currents can alter the SM hierarchy among photon

polarisations, which would provide a very interesting hint on the nature of the NP responsible for

the deviations observed in b→s`` processes. These NP eﬀects could generate additional phases

leading to new forms of CP-violation in b→sγ decays. Finally, in the context of global explanations

of b→s`` and b→cτν anomalies within an EFT framework, an interesting NP scenario consists in

arXiv:2210.11995v2 [hep-ph] 9 Feb 2023

“large” contributions to b→cτ ν and b→sτ τ operators (see e.g. Ref. [14–19]), which could feed into

NP lepton-ﬂavour universal contributions to b→s`` [19,20], but also into possible NP contributions

to b→sγ through radiative corrections.

Many diﬀerent approaches have been proposed to extract information on the photon polarisation

in b→sγ, and more generally on the values of C7and C70[21]. The branching ratio for the inclusive

B→Xsγdecay has reached a high level of precision both theoretically and experimentally [22].

Time-dependent analysis of B→K∗γand Bs→φγ can be performed to extract mixing-induced CP-

asymmetries [23–25] containing relevant information on C7and C70[26–29]. The baryonic mode Λb→

Λγalso provides interesting constraints [30–32]. Other approaches such as converted photons [33]

and asymmetries in B→K1(→Kππ)γ[34–38] have also been considered.

Another possibility is to consider semileptonic decays based on the b→see transition at very

low-q2values, in a kinematic range that is not reachable for the muonic channel. In this regime,

the photon pole dominates and the transverse polarisations provide leading contributions to all

observables. Therefore, the angular analysis can provide information on the photon polarisation

through the available observables, namely transverse asymmetries [39, 40]. The main advantage of

these observables is that they are rather theoretically clean, since the relevant form factors cancel

out completely as q2approaches the photon pole. LHCb has performed very accurate measurements

of these asymmetries for B→K∗ee with 9 fb−1, providing signiﬁcant constraints on C7and C70[5].

In particular, these results provide the leading constraints on the real and imaginary parts of C70

as of today. A similar analysis is certainly possible for Bs→φee, given that angular analyses are

available for both B→K∗and Bs→φmodes in the case of the b→sµµ transition [6, 41].

Interestingly, the Bs→φ`` decays are not self-tagging and their ﬁnal states (K+K−or KSKL)

are CP-eigenstates. Therefore, Bs−¯

Bsmixing must be included to fully describe these decays, which

may provide non-negligible corrections given the size of mixing parameters in the Bssystem. The

impact of mixing has been discussed for various B-meson decays [42–45]. For B→V `` [46] and

B→P `` [47] modes, the time-dependence of the branching ratio will involve new observables that

depend on the interference between mixing and decay. This may provide additional information about

the relative moduli and phases of the relevant transversity amplitudes. The modiﬁcations induced by

neutral-meson mixing for the Bs→φ`` mode have already been considered in Refs. [43,46]. However,

the hierarchy of amplitudes change at low-q2values due to the dominance of the photon pole and the

impact of Bs-meson mixing on all the accessible observables is not necessarily intuitive. This article

is thus focused on assessing the impact of Bs-mixing on the low-q2observables for Bs→φee and

demonstrating that additional information can be gathered from the interference between mixing

and decay occurring in this mode.

The remainder of this article is organized as follows. In Sec. 2, we discuss the angular analysis

of Bs→φee, ﬁrstly neglecting neutral-meson mixing, and we determine the observables of interest

close to the photon pole. In Sec. 3, we deal with mixing and consider the time-dependent analysis

of Bs→φee and how time-integrated angular observables will also carry information on the relative

size and phase of C7and C70. In Sec. 4, we perform a thorough numerical analysis, checking the

range of validity of the photon-pole approximation, the constraints obtained from the low-q2angular

observables and their sensitivity to NP eﬀects, showing the importance of taking into account Bs−¯

mixing, before concluding in Sec. 5. Appendices are devoted to conventions regarding kinematics

and helicity amplitudes, to collect lengthy expressions of the angular coeﬃcients in the presence

of mixing, as well as to discuss further transverse asymmetries which prove more diﬃcult to reach

experimentally.

2Bs→φee in the absence of mixing

2.1 Low-energy eﬀective theory Hamiltonian

The b→s`` transitions are described by the usual LEFT Hamiltonian with SM operators, in addition

to the NP ones with a chirality-ﬂipped, scalar or tensor structure [48, 49]:

Heﬀ =−4GF

√2λu[C1(Oc

1− Ou

1) + C2(Oc

2− Ou

2)] + λtX

i∈ICiOi+ h.c. , (1)

where λq=VqbV∗

qs and I∈ {1c, 2c, 3,4,5,6,8,7(0),9(0)`, 10(0)`, S(0)`, P (0)`, T (0)`}. In the following, we

neglect doubly Cabibbo suppressed contributions, of relative size of O(λ2)'4% where λis the usual

parameter of the Wolfenstein parametrisation of the Cabibbo-Kobayashi-Maskawa (CKM) matrix.

This leads us to neglect the contributions proportional to λuin Heﬀ . The operators O1,..,6and O8

are hadronic operators of the type (¯sΓb)(¯qΓ0q) and (¯sγµν TaPRb)Ga

µν , respectively. These operators

are not likely to receive very large contributions from NP, as they would appear in non-leptonic B

decay amplitudes 1. The main operators of interest O7(0),9(0),10(0)are then given by

O9(0)`=e2

(4π)2[¯sγµPL(R)b][¯

`γµ`],O10(0)`=e2

(4π)2[¯sγµPL(R)b][¯

`γµγ5`],

O7(0)=e

(4π)2mb[¯sσµν PR(L)b]Fµν .

(2)

In the SM, and at a scale µb=O(mb), the Wilson coeﬃcients of interest Eq. (2) are CSM

7(µb)' −0.3,

CSM

70(µb)' −0.006 (suppressed by ms/mbcompared to C7), CSM

9(µb)'4.1 and CSM

10 (µb)' −4.3,

which are identical for `=eand `=µdue to the universality of the SM gauge-couplings to leptons.

Note, in particular, that these Wilson coeﬃcients might be aﬀected by complex NP contributions,

which can also violate LFU, being diﬀerent for `=eand `=µ, see e.g. Ref. [53].

When considering actual matrix elements describing b→s`` or b→sγ transitions, certain com-

binations of these Wilson coeﬃcients naturally arise. One repeatedly encounters the regularisation-

scheme independent combinations of Wilson coeﬃcients Ceﬀ

j=7,8=Cj+P6

i=1 yjiCi, where yiare pure

numbers given by RGE [48]. Moreover, one has to take into account long-distance contributions com-

ing from four-quark operators and corresponding to charm-loop contributions. As we will discuss

below, these contributions can be absorbed into q2- and ﬁnal-state-dependent “eﬀective” Wilson-

coeﬃcients, corresponding to a vector coupling to leptons. This long-distance part is thus absorbed

in C7for the b→sγ transition, while the customary choice for b→s`` transitions is C9(whereas C7

gets only redeﬁned by ultraviolet contributions required by renormalization), see e.g. Ref. [49].

2.2 Bs→φ(→K+K−)`` angular coeﬃcients

In the following, we consider the Bs→φ(→K+K−)`` decays, which has already been considered

by LHCb in the `=µchannel [41] 2. In the absence of mixing, we can give the general expression

for the angular coeﬃcients using the general formalism of Refs. [39,49], with the angular convention

speciﬁed in Appendix A,

d4Γ(Bs→φ(→K+K−)``)

dq2dcos θKdcos θldφ =9

32πJ1ssin2θK+J1ccos2θK+J2ssin2θKcos 2θl

+J2ccos2θKcos 2θl+J3sin2θKsin2θlcos 2φ+J4sin 2θKsin 2θlcos φ

+J5sin 2θKsin θlcos φ+J6ssin2θKcos θl+J6ccos2θKcos θl(3)

+J7sin 2θKsin θlsin φ+J8sin 2θKsin 2θlsin φ+J9sin2θKsin2θlsin 2φ,

1See Refs. [50–52] for a discussion of low-energy constraints on these operators.

2In principle, the Bs→φ(→KSKL)`` mode could also be considered, but it is far more challenging experimentally.

which depend on the invariant mass of the lepton pair (q2), and three angles that we denote θ`,θK

and φ. The angle θ`is deﬁned between the lepton `+direction with respect to the opposite of the

direction of ﬂight of the Bsmeson in the `+`−centre-of-mass frame. Similarly, θKis deﬁned as

the angle of the K−meson with respect to the opposite direction of ﬂight of the Bs-meson in the

φ-meson rest frame, see Fig. 7 in Appendix A. Note that this choice of kinematics diﬀers from the

one considered in the LHCb analysis for self-tagging decays [5, 54–56] (where θ`would be associated

with `−for Bqdecays and `+for ¯

Bqdecays), and from the theory convention [49] (where this angle

would be associated to `−for both Bqand ¯

Bqdecays), but it is the most appropriate when we discuss

non-self-tagging modes such as Bs→φ`` [46].

The coeﬃcients of the distribution Ji(q2) contain interference terms of the form Re[AXA∗

Y] and

Im[AXA∗

Y] between the eight transversity amplitudes deﬁned in Appendix B,

AL

0, AR

0, AL

|| , AR

|| , AL

⊥, AR

⊥, At, AS,(4)

which are given by

J1s=(2 + β2

4h|AL

⊥|2+|AL

|| |2+|AR

⊥|2+|AR

|| |2i+4m2

sRe AL

⊥AR∗

⊥+AL

|| AR∗

|| ,

J1c=|AL

0|2+|AR

0|2+4m2

s|At|2+ 2Re(AL

0AR∗

0)+β2

`|AS|2,

J2s=β2

4h|AL

⊥|2+|AL

|| |2+|AR

⊥|2+|AR

|| |2i, J2c=−β2

`|AL

0|2+|AR

0|2,

J3=1

2β2

`h|AL

⊥|2− |AL

|| |2+|AR

⊥|2− |AR

|| |2i, J4=1

√2β2

`hRe(AL

0AL∗

|| +AR

0AR∗

|| )i,

J5=√2β`hRe(AL

0AL∗

⊥−AR

0AR∗

⊥)−m`

√sRe(AL

|| A∗

S+AR∗

|| AS)i,

J6s= 2β`hRe(AL

|| AL∗

⊥−AR

|| AR∗

⊥)i, J6c= 4β`

√sRe(AL

0A∗

S+AR∗

0AS),

J7=√2β`hIm(AL

0AL∗

|| −AR

0AR∗

|| ) + m`

√sIm(AL

⊥A∗

S−AR∗

⊥AS))i,

J8=1

√2β2

`Im(AL

0AL∗

⊥+AR

0AR∗

⊥), J9=β2

`hIm(AL∗

|| AL

⊥+AR∗

|| AR

⊥)i,(5)

where β`=q1−4m2

`/q2. In the massless lepton limit, we have J1s= 3J2sand J1c=−J2cand

J6c= 0. The expression for the various transversity amplitudes and Wilson coeﬃcients can be found

e.g. in Ref. [49].

Similar expressions hold for the CP-conjugate decay ¯

Bs→¯

φ(→K+K−)``, with angular coeﬃ-

cients ¯

Jiinvolving amplitudes denoted by ¯

AX, and obtained from the AXby conjugating all CP-odd

“weak” phases 3. These weak phases appear in the CKM matrix elements involved in the normali-

sation of the weak eﬀective Hamiltonian as well as the short-distance Wilson coeﬃcients Cipresent

in the deﬁnition of the Wilson coeﬃcients. These amplitudes will involve in general [49]

A[VtbV∗

ts,Ci, Fj, hj], A[V∗

tbVts,C∗

i, Fj, hj],(6)

where Fiare local form-factors describing hV|Oi|Biand hjare non-local contributions from charm

loops 4.

3This is opposite to the notation used in Ref. [49] for Band ¯

Bdecays, but in agreement with general discussions

on CP-violation and the discussions of refs. [46, 47].

4We assume that there are no signiﬁcant complex NP contributions to the short-distance four-quark Wilson coeﬃ-

cients C1...6so that these contributions carry no weak phases.

The form of the angular distribution for the CP-conjugated decay depends on the way the kine-

matical variables are deﬁned. In the case in which the same conventions are used for the lepton angle

irrespective of whether the decaying meson is a Bsor a ¯

Bs, we have [46, 49]

dΓ[Bs→φ(→K+K−)`+`−]

dq2dcos θ`dcos θMdφ =X

Ji(q2)fi(θ`, θM, φ),(7)

dΓ[ ¯

Bs→φ(→K+K−)`+`−]

dq2dcos θ`dcos θMdφ =X

ζi¯

Ji(q2)fi(θ`, θM, φ) = X

Ji(q2)fi(θ`, θM, φ),(8)

where fi(θ`, θM, φ) are deﬁned by Eq. (3). These expressions feature two diﬀerent angular coeﬃcients

Jiand ¯

Jiwhich are CP conjugates of Ji:

•The angular coeﬃcients e

Jiformed by replacing AXby e

AX≡AX(¯

Bs→fCP ) (without CP-

conjugation applied on fCP ), which will appear in the study of time evolution due to mixing,

where both Bsand ¯

Bsdecay into the same ﬁnal state fCP .

•The angular coeﬃcients ¯

Ji, obtained by considering ¯

AX≡AX(¯

Bs→fCP ) (with CP-conjugation

applied to fCP ), which can be obtained from AXby changing the sign of all weak phases, and

arise naturally when discussing CP violation (and CP conjugation) from the theoretical point

of view.

As discussed in Ref. [46, 47], we have e

AX=ηX¯

AX, with ηXgiven by

ηX=ηfor X=L0, L||, R0, R||, t ;ηX=−ηfor X=L⊥, R⊥, S , (9)

with η= 1 in the Bs→φ(→K+K−)`` case. Therefore ¯

Jican be obtained from Jiby changing the

sign of all weak phases, and ˜

Ji=ζi¯

Jiwith

ζi= 1 for i= 1s, 1c, 2s, 2c, 3,4,7 ; ζi=−1 for i= 5,6s, 6c, 8,9.(10)

Since the ﬁnal state is not self-tagging, an untagged measurement of the diﬀerential decay rate

(e.g. at LHCb, where the production asymmetry is tiny) yields

dΓ(Bs→fCP ) + dΓ( ¯

Bs→fCP )

dq2dcos θ`dcos θKdφ =X

[Ji+e

Ji]fi(θ`, θK, φ) = X

[Ji+ζi¯

Ji]fi(θ`, θK, φ),(11)

which involves CP-averages for some of the observables, but CP-asymmetries for the others. The

diﬀerence between the two decay rates can only be measured through ﬂavour-tagging, and it involves

Ji−e

Ji=Ji−ζi¯

Ji, providing access to other averages and asymmetries,

dΓ(Bs→fCP )−dΓ( ¯

Bs→fCP )

dq2dcos θ`dcos θKdφ =X

[Ji−e

Ji]fi(θ`, θK, φ) = X

[Ji−ζi¯

Ji]fi(θ`, θK, φ).(12)

We emphasize that we have neglected any eﬀect coming from mixing at this stage and that we

assumed that there was no asymmetry in the production of Bsand ¯

Bsmeson.

2.3 Bs→φ(→K+K−)ee angular observables

We focus now on the Bs→φee observables that can be extracted experimentally. To this purpose,

we deﬁne [49]

Si= (Ji+¯

Ji).d(Γ + ¯

Γ)

dq2, Ai= (Ji−¯

Ji).d(Γ + ¯

Γ)

dq2,(13)

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February10,2023OntheimpactofmesonmixingonBs!eeangularobservablesatlowq2SebastienDescotes-Genon,IoannisPlakias,OlcyrSumensariUniversiteParis-Saclay,CNRS/IN2P3,IJCLab,91405Orsay,FranceAbstractDecaysbasedontheb!stransitionareexpectedtoyieldleft-handedphotonsintheStandardModel,butcouldbeparticularlys...

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February 10 2023 On the impact of meson mixing on Bseeangular observables at low q2

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