Angular distribution of Lb-pKellell decays comprising Lambda resonances with spin up to 52

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Angular distribution of Λ0

b→pK−`+`−

decays comprising Λresonances

with spin ≤5/2

A. Beck†, T. Blake‡, M. Kreps§.

Department of Physics, University of Warwick, Coventry CV4 7AL, UK

Abstract

This paper describes the angular distribution of

Λ0

b→Λ

(

→pK−

)

`+`−

decays. A full

expression is given for the case of multiple interfering spin-states with spin

≤5

. This

distribution is relevant for future measurements of

Λ0

b→pK−`+`−

decays, where diﬀerent

states cannot easily be separated based on their mass alone. New observables arise when

considering spin-

states as well as interference between states. An exploration of their

behaviour for a variety of beyond the Standard Model scenarios shows that some of these

observables exhibit interesting sensitivity to the Wilson coeﬃcients involved in

b→s`+`−

transitions. Others are insensitive to the Wilson coeﬃcients and can be used to verify the

description of

Λ0

b→Λ

form-factors. A basis of weighting functions that can be used to

determine all of the angular observables described in this paper in a moment analysis of the

experimental data is also provided.

†anja.beck@cern.ch

‡thomas.blake@cern.ch

§michal.kreps@cern.ch

arXiv:2210.09988v3 [hep-ph] 1 Mar 2023

Contents

1 Introduction 1

2 Decomposition of the decay rate 2

2.1 Invariantamplitude................................... 2

2.2 Four-bodyphase-space................................. 3

3 Invariant amplitudes in the helicity formalism 4

3.1 Helicity amplitudes for the Λ0

b→ΛV decay ..................... 6

3.2 Helicity amplitudes for the Λ-resonance decay . . . . . . . . . . . . . . . . . . . . 8

3.3 Helicity amplitudes for the leptonic current . . . . . . . . . . . . . . . . . . . . . 9

4 Angular distribution 9

5 Method of moments 13

6 Explicit expressions for the angular coeﬃcients 13

7 Angular distributions for the decay of unpolarized Λ0

bbaryons to individual

states 17

8 Predictions for the SM and modiﬁed theories 19

8.1 Predictions for the spin-5

2Λ(1820) resonance..................... 19

8.2 Predictions for an ensemble of diﬀerent Λresonances . . . . . . . . . . . . . . . . 22

9 Conclusion 24

A Eﬀective Wilson coeﬃcients 26

B Polarisation vector and Rarita-Schwinger representations 26

C Deﬁnition of the angles 28

D Translation between lattice and quark-model form factors 28

D.1 Translation for 1

+to 1

+transitions ......................... 29

D.2 Translation for 1

+to 3

2−transitions ......................... 31

E Blatt-Weisskopf form factors 33

F Full set of angular coeﬃcients 34

G Translation between the Liand Kjbasis 43

References 44

1 Introduction

In recent years, the

LHCb

collaboration has published several measurements of the rates and

angular distributions of

- to

-quark ﬂavour-changing neutral-current processes [1

–

5]. The

experimental results reveal a pattern of discrepancies with predictions based on the Standard

Model of particle physics (SM). The measurements by the

LHCb

collaboration are reinforced by

compatible observations performed by the BaBar and Belle experiments and the

ATLAS

CMS

and

CDF

collaborations [6

–

18]. Global analyses of

- to

-quark transitions indicate that the

measurements form a coherent picture that could be explained by several proposed extensions of

the SM, see for example Refs. [19

–

22]. Measurements comparing the rates of processes involving

b→sµ+µ−

and

b→se+e−

transitions also show diﬀerences [23, 24], which suggest that the

underlying theory may have non-universal lepton couplings.

Thus far, measurements have mainly focused on analyses of

meson decays. It is important

to conﬁrm the discrepancies in other systems. The most convenient choice for this is through

the decay of the

Λ0

baryon, which is the lightest

-baryon and is produced abundantly at the

LHC

[25]. As of today, the

LHCb

collaboration has measured the branching fraction and angular

distribution of the

Λ0

b→Λ

(1115)

µ+µ−

decay [26, 27], where the label

(1115) is used to refer to

the weakly decaying ground-state baryon. Measurements of the

Λ0

b→Λ

(1115)

µ+µ−

transition

have already been considered in global analyses [19, 28] but larger experimental data sets are

needed to understand the compatibility of the measurements with those in

meson systems.

The

LHCb

experiment also observes large signals of

Λ0

b→pK−µ+µ−

decays, which it has used

to search for

violation in the decay [29] and to test lepton ﬂavour universality by comparing

Λ0

b→pK−e+e−

and

Λ0

b→pK−µ+µ−

decays [30]. A unique feature of the

pK−

spectrum in

these decays is the rich contribution from diﬀerent

resonances, whose states cannot easily be

separated.1

From the theoretical point of view, the semi-leptonic

Λ0

decay to the ground-state

(1115)

baryon has been studied in detail. There are predictions for the form factors for the decay from

light-cone sum-rule techniques [31, 32] and lattice QCD [33

–

35]. Dispersive bounds on the form

factors have also been discussed in Ref. [36]. The angular distribution for the decay is known [37],

even for the case of polarised

Λ0

baryons [38] and the full basis of new physics operators [39, 40].

Much less is known about the decay via other

resonances. The form factors for

Λ0

(1520)

transitions have been determined in lattice QCD [41, 42], in the quark model [43], and studied

in HQET [44]. Dispersive bounds have also been considered in Ref. [45]. For other resonances,

form-factor predictions are only available in the context of the quark model [46, 47]. The full

angular distribution of single spin-

[48, 49] and the spin-

[50, 51] resonances is known but

the distribution of higher-spin resonances and the more general case of overlapping, interfering,

resonances has not been studied. The aim of this paper is to provide a description of the angular

distribution, including up-to spin-

resonances, and to present a method that can be used by

experiments to perform a model-independent analysis of the Λ0

b→pK−`+`−decay.

The following sections begin with a decomposition of the full

Λ0

b→pK−`+`−

decay rate into

subsequent two-body decays. The approach used holds for any decay of a spin-

baryon to a

ﬁnal state involving a spin-

baryon, a spin-0 meson, and two fermions. The amplitudes for the

two-body decays are calculated in the helicity formalism, as described in Section 3. Section 4

provides an expansion for the full angular distribution in terms of a set of basis functions.

Section 5 introduces the method of moments and explains its application to the decay rate

developed in the ﬁrst sections. Section 6 provides explicit expressions for some of the observables

appearing in the angular distribution. Section 7 explores the angular distributions of individual

In order to avoid confusion, the weakly-decaying ground-state will be labelled

(1115) and the strongly decaying

resonance states will be collectively labelled

resonances when referring to the resonances in general and a mass

in parentheses will be used to refer to a speciﬁc state.

resonances. The angular distribution of the spin-

2Λ

(1820) resonance, and a realistic ensemble

states, is explored in Section 8 together with the possibility of observing modiﬁcations of

the angular distribution of the decay in extensions of the SM.

2 Decomposition of the decay rate

The diﬀerential decay rate for the full decay chain can be expressed as

dΓ = |M|2

2mΛb

(2π)4dΦ4,(1)

where

is the invariant amplitude for the decay,

mΛb

is the mass of the

Λ0

baryon and dΦ

the 4-body diﬀerential phase space. The

Λ0

b→pK−`+`−

decay is modelled as three subsequent

two-body decays, where the

Λ0

baryon ﬁrst decays into a

resonance and a virtual vector boson,

labelled below by

. The

resonance then decays strongly into a proton and a kaon, while

the virtual vector boson produces the two leptons. In what follows, the four momenta of the

Λ0

baryon and the

resonance are denoted

and

, respectively. The four momentum of the

`+`−

system is

p−k

. The four momenta of the proton and kaon are

and

and the four

momenta of the

and

`−

are

and

. We use the notation

pµ

= (

p0, ~p

)when referring to the

energy and three momentum of the particles.

2.1 Invariant amplitude

The spin-averaged invariant amplitude-squared,

|M|2

, is obtained by summing over the possible

helicites of the

Λ0

baryon,

λb

±1

, the proton,

λp

±1

, and the two leptons,

λ1

±1

and

λ2=±1

|M|2=X

λb

PλbX

λ1,λ2,λp|Mλb,λp,λ1,λ2|2.(2)

The factor

Pλb

corresponds to the relative amount of the

Λ0

spin state

λb

. The sum of

Pλb

over

the two spin-states is one, i.e. P+1/2+P−1/2= 1.

The amplitude for a given set of initial and ﬁnal states corresponds to the sum over all

intermediate resonances, Λ, and their corresponding helicity, λΛ,

Mλb,λp,λ1,λ2=X

ΛX

λΛMΛ

λΛ.(3)

The helicity indices

λb

λp

λ1

, and

λ2

have been suppressed on the right-hand side of the

expression for readability. Equation 3 can be split into two pieces, representing an amplitude for

the decay to Λ`+`−and for the subsequent decay of the Λresonance to pK−,i.e.

MΛ

λΛ=MΛ0

b→Λ`+`−

λΛMΛ→pK−

λΛ.(4)

Using a naive factorisation approach, after integrating out heavy degrees of freedom, the

eﬀective Lagrangian is

Leﬀ =LQCD +LQED +4GF

√2VtbV∗

ts X

iCiOi,(5)

where

Vij

are elements of the Cabibbo-Kobayashi-Maskawa quark-mixing matrix and

is the

Fermi constant. The

and

represent the Wilson coeﬃcients and the corresponding local

operators of the eﬀective theory. The relevant dimension-four operators for

b→s`+`−

transitions

are

O7(0)=e

16π2mb(¯sσµν PR(L)b)Fµν ,

O9(0)=e2

16π2(¯sγµPL(R)b)(¯

`γµ`),

O10(0)=e2

16π2(¯sγµPL(R)b)(¯

`γµγ5`),

(6)

with the left- and right-handed chiral projection operators

PL,R

2(1 ∓γ5)

. Four-quark

current-current and QCD penguin operators, usually denoted

O1−6

, also contribute to

b→s`+`−

transitions. Their impact is discussed in Appendix A and included by using eﬀective Wilson

coeﬃcients

Ceﬀ

. Non-factorisable corrections that can not be expressed in terms of contributions

to Ceﬀ

7,9are neglected in this paper.

Splitting the operators into Λ0

b→Λand dilepton current results in the expression

MΛ0

b→Λ`+`−

λΛ=N1X

iCih``|Oµ

lep,i|0ihΛ|Oν

had,i|Λ0

bigµν (7)

for the

Λ0

b→Λ`+`−

decay amplitude, where the constants have been absorbed into the normali-

sation factor

N1=4GF

√2VtbV∗

ts .(8)

To simplify the calculations, a projection onto an intermediate vector boson is introduced by

expressing the Minkowski metric as

gµν =X

λV

ε∗

ν(λV)εµ(λV)gλVλV.(9)

The virtual vector boson has four polarization states: time-like (

λV

= 0), longitudinal

(

= 1

, λV

= 0), and transverse (

= 1

, λV

1). In the following, these states will be

labelled by

λV

,±

. An explicit form of the polarisation vectors for the diﬀerent polarisation

states is given in Appendix B.

Using Equation 9, the hadronic and leptonic currents can be separated

MΛ0

b→Λ`+`−

λΛ=N1X

λV

gλVλVX

ih``|Oµ

lep,i|0iεµ(λV)

| {z }

MV→`+`−

λV,Oi

CihΛ|Oν

had,i|Λ0

biε∗

ν(λV)

| {z }

MΛ0

b→ΛV

λΛ,λV,Oi

,(10)

and evaluated in independent reference frames. The amplitudes

MV→`+`−

λV,Oi

MΛ0

b→ΛV

λΛ,λV,Oi

, and

MΛ→pK−

λΛare further discussed in Section 3.

2.2 Four-body phase-space

The four-body phase-space for the

Λ0

b→pK−`+`−

decay can be decomposed into three two-body

phase-space elements as

dΦ4=1

4(2π)6|~

mΛb

dΩΛ0

b×1

4(2π)6|~

k1|

√k2dΩpK ·(2π)3dk2×1

4(2π)6|~q1|

pq2dΩ`` ·(2π)3dq2

26(2π)12 |~

mΛb

k1|

√k2|~q1|

pq2dΩΛ0

bdΩpK dΩ``dk2dq2.

(11)

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Angulardistributionof0b!pK`+`decayscomprisingresonanceswithspin5=2A.Beck,T.Blake,M.Kreps.DepartmentofPhysics,UniversityofWarwick,CoventryCV47AL,UKAbstractThispaperdescribestheangulardistributionof0b!(!pK)`+`decays.Afullexpressionisgivenforthecaseofmultipleinterferingspin-stateswithspin52.Th...

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Angular distribution of Lb-pKellell decays comprising Lambda resonances with spin up to 52

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