JHEP12(2019)148

Published for SISSA by Springer

Received: September 27, 2019

Accepted: December 1, 2019

Published: December 20, 2019

Probing effects of new physics in Λ0b → Λ+c µ−νµdecays

Martina Ferrillo, Abhijit Mathad,1 Patrick Owen and Nicola Serra

Physik-Institut, Universität Zürich,

Zürich, Switzerland

E-mail: martina.ferrillo@cern.ch, Abhijit.Mathad@cern.ch,

powen@physik.uzh.ch, Nicola.Serra@cern.ch

Abstract: We present, for the first time, the six-fold differential decay density expression

for Λ0b → Λ+c `−ν`, taking into account the polarisation of the Λ0b baryon and a completebasis of new physics operators. Using the expected yield in the current dataset collected at

the LHCb experiment, we present sensitivity studies to determine the experimental preci-

sion on the Wilson coefficients of the new physics operators with Λ0b → Λ+c µ−νµ decays intwo scenarios. In the first case, unpolarised Λ0b → Λ+c µ−νµ decays with Λ+c → pK+π− areconsidered, whereas polarised Λ0b → Λ+c µ−νµ decays with Λ+c → pK0S are studied in thesecond. For the latter scenario, the experimental precision that can be achieved on the de-

termination of Λ0b polarisation and Λ+c weak decay asymmetry parameter is also presented.

Keywords: Beyond Standard Model, Heavy Quark Physics

ArXiv ePrint: 1909.04608

1Corresponding author.

Open Access, c© The Authors.Article funded by SCOAP3.

https://doi.org/10.1007/JHEP12(2019)148

mailto:martina.ferrillo@cern.chmailto:Abhijit.Mathad@cern.chmailto:powen@physik.uzh.chmailto:Nicola.Serra@cern.chhttps://arxiv.org/abs/1909.04608https://doi.org/10.1007/JHEP12(2019)148

JHEP12(2019)148

Contents

1 Introduction 1

2 Effective Lagrangian for b→ clν 2

3 Decay amplitude 3

4 Decay density 6

5 Experimental sensitivity 7

6 Conclusions 10

A Hadronic amplitudes of Λ0b → Λ+c W ∗− decay 11

B Leptonic amplitudes of W ∗− → `−ν` decay 14

C Hadronic amplitudes of Λ+c → pK0S decay 16

D Phase space 17

E Terms of differential decay density 19

F Decay density for two considered cases 20

G Response matrix 21

1 Introduction

Semileptonic b-hadron decays are highly promising avenues to search for New Physics

(NP) due to their large signal yields and controllable theoretical uncertainties. The hint of

lepton flavour universality violation in B → D(∗)`ν decays [1–8]1 has led to the proposalof various NP scenarios that could affect decays involving b → c`ν transitions [9–11].In addition, numerous studies involving B → D(∗)τν decays have shown the effects ofthese NP contributions on the corresponding angular distributions [12–18]. Global fits to

b → cτν transitions have also been conducted to determine the Wilson coefficients of theNP operators [19–22]. A recent global fit to b → cµν and b → ceν transitions [22] hasproven that a good sensitivity to various different NP operators can be achieved through

studies of b-meson decays involving lighter leptons in the final state.

1As no CP violation is considered in this paper, the inclusion of charge conjugate processes is implied.

– 1 –

JHEP12(2019)148

The baryonic equivalent of these decays, Λ0b → Λ+c `−ν`, is a good candidate to comple-ment the NP sensitivity of the mesonic counterparts, due to the large production cross sec-

tion of Λ0b baryons and the well measured form factors [23–26]. The literature is rich in stud-

ies of the possible effect of NP contributions in unpolarised Λ0b → Λ+c τ−ντ decays [27–31],as well as in subsequent Λc → Λπ decays [32, 33]. More recently, the same investigationhas been extended to Λ0b unpolarised semileptonic decays to lighter leptons [33, 34].

In this study, we present for the first time an expression for the six-fold differential

decay density of Λ0b → Λ+c `−ν` transitions, including the effects of Λ0b polarisation and allthe relevant NP contributions which are encapsulated by Wilson coefficients. These decays

can currently be studied only at the LHCb experiment and present several experimental

challenges. On one side, in the Λ0b → Λ+c τ−ντ case multiple missing neutrinos in the finalstate drastically dilute the resolution on the kinematic variables in addition to contributions

from irreducible backgrounds (such as feed-down Λ0b → Λ+c∗`−ν` and Λ

0b → Λ+c Xc, where

Xc is a charmed meson). Furthermore, Λ0b → Λ+c e−νe decays are challenging to reconstruct

at LHCb due to the poor electron reconstruction efficiency [35]. Therefore, we conduct

sensitivity studies to determine the experimental precision on the Wilson coefficients with

Λ0b → Λ+c µ−νµ channel in two different scenarios, using the expected yield from Run I andII data collected at the LHCb experiment.

In the first scenario, the Λ0b is unpolarised and the Λ+c decay kinematics are integrated

over and is assumed to be reconstructed using the Λ+c →pK−π+ channel. This is anexperimentally favourable signature due to the presence of three charged particles in the

final state and the large branching fraction, which ensure a cleaner reconstruction with

small background contributions at LHCb.

In the second scenario we allow for a non-zero Λ0b polarisation, with a subsequent

Λ+c → pK0S decay accounting for the involved kinematics of the process. The interest inthis case lies in the achievable sensitivity not only to the polarisation of Λ0b (PΛ0b

), but also

to the Λ+c decay asymmetry parameter (αΛ+c ). So far, at the LHC no hint for a non-zero

value of PΛ0bhas been observed [36–38], whereas the Λ+c decay asymmetry has been very

recently measured at the BESIII experiment, but with a large uncertainty [39]. Therefore,

we present an estimate on the experimental precision which could be achieved on PΛ0band

αΛ+c at the LHCb, relying on the large signal yields of semileptonic decays.

The paper is organised in the following way. In section 2, the effective Lagrangian

expression for b→ clν transitions is reported, including all the relevant NP operators. Thedecay amplitude of Λ0b →Λ+c (→p K0S)(`−ν`) is presented in section 3. Section 4 containsthe expression for the six-fold differential decay density for the polarised Λ0b → (pK0S)(`−ν`)decays in the context of NP. In section 5, the results of the sensitivity studies undertaken

on the Wilson coefficients in the two working assumptions are reported. The conclusions

of this work are given in section 6.

2 Effective Lagrangian for b→ clν

The most generic effective Lagrangian of the four-fermion interaction, extending the Stan-

dard Model (SM) within the NP scenario and governing semileptonic b→ clν transitions,

– 2 –

JHEP12(2019)148

is given by:

Heff =4GF√

2Vcb

[(1 + CVL)OCVL + CVROCVR + CSLOCSL + CSROCSR + CTLOCTL

]+ h.c. ,

(2.1)

where the four-fermion operators OCi are defined as:

OCVL = c̄LγµbL ¯̀LγµνL ,

OCVR = c̄RγµbR ¯̀LγµνL ,

OCSL = c̄RbL¯̀RνL ,

OCSR = c̄LbR¯̀RνL ,

OCTL = c̄RσµνbL ¯̀RσµννL .

Here the factors CVL,R , CSL,R , CTL denote the Wilson coefficients of their respective op-

erators, that take a value of zero within the SM. The symbol ` represents the lepton

flavour involved in the interaction. It is noted here that the right-handed tensor operator

OTR = q̄LσµνbR τ̄RσµννL vanishes [13]. As in the case of SM, we assume the absence ofright-handed ν` and left-handed ν`.

2 Since the flavour of the neutrino is not observed,

neutrino mixing effects are also not considered.

3 Decay amplitude

The transition matrix elements for Λ0b → (pK0S)(`−ν`) can be expressed as the product ofamplitudes of unstable particles involved in the decay, i.e. Λ0b , Λ

+c , W

∗−:

TλΛbλl,λν` ,λp,λK0S

=4GFVcb√

2BW (m2pK0S

)∑

i,λΛc ,λW

Ti,λΛbλΛc ,λW

(q2)T i,λWλl,λν`(q2)T

i,λΛcλp,λK0

S

. (3.1)

In eq. (3.1), the term q2 denotes the squared transferred four momentum, defined as

q2 = (PΛb − PΛc)2 = (P`− + Pν`)2; mpK0S is the combined mass of the system pK0S; GF

represents the Fermi constant; the index λ refers to the helicity of the particle involved in

the transition. The propagator term for intermediate Λ+c particle is parametrised as the

relativistic Breit-Wigner and is denoted by BW . Using the narrow-width approximation

for BWΛc , the m2pK0S

dependence is integrated out in the expression of the differential de-

cay density. In eq. (3.1), we have also summed over i, denoting the operator Oi (eq. (2.1))involved in the transition, and the helicities of the intermediate unstable particles. In the

following the helicity index λK0Sis dropped as it is null and λν` is fixed to 1/2 for Λ

0b decays

(or λν` = −1/2 for Λ0b decays). Since the weak decay of Λ+c → pK0S involves the chargedcurrent transition of c→sud, we express the total decay density in terms of the weak decayasymmetry parameter, αΛ+c [39].

2Operators involving right-handed neutrino are considered in refs. [40, 41].

– 3 –

JHEP12(2019)148

The transition amplitude shown in eq. (3.1) can be expanded in terms of the helicity

amplitudes of the involved decay processes as follows:

TλΛbλl,λp

=4GF |Vcb|√

2BW (m2pK0S

)[ ∑λΛc ,λW

ηλW

( ∑j=SM,CVL ,CVR

Hj;λΛbλΛc ,λW

)LSM ;λWλl G

λΛcλp

+∑λΛc

( ∑k=CSL ,CSR

Hk;λΛbλΛc

)LCSLλl

GλΛcλp

+∑

λΛc ,λW ,λ′W

ηλW ηλ′WHCTL ;λΛbλΛc ,λW ,λ

′WLCTL ;λW ,λ

′W

λlGλΛcλp

]. (3.2)

Here H, L and G represent the helicity amplitudes of Λ0b , W∗− and Λ+c decays, respectively,

retaining dependence on all the angular degrees of freedom. In this study the two lowest

allowed spins for W ∗−, i.e. (JW = 0, λW = 0) and (JW = 1, λW = −1, 0, 1), are being con-sidered. To distinguish the former helicity configuration from the latter, we denote λW = t

when JW = 0. The term ηλ denotes a metric factor that originates when we replace the

metric tensor with the polarisation vectors of the virtual W ∗− i.e. gµν =∑

λ ηλ�†µ(λ)�ν(λ)

where η0,±1 = −ηt = −1.The hadronic amplitudes expressed above in eq. (3.2) are related to those involving

vector (V), axial-vector (A), scalar (S), pseudo-scalar (PS), tensor (T) and pseudo-tensor

(PT) currents through the following relations:

HSM ;λΛbλΛc ,λW

=1

2

(HV ;λΛbλΛc ,λW

−HA;λΛbλΛc ,λW), (3.3)

HCVL ;λΛbλΛc ,λW

=CVL

2

(HV ;λΛbλΛc ,λW

−HA;λΛbλΛc ,λW), (3.4)

HCVR ;λΛbλΛc ,λW

=CVR

2

(HV ;λΛbλΛc ,λW

+HA;λΛbλΛc ,λW

), (3.5)

HCSL ;λΛbλΛc

=CSL

2

(HS;λΛbλΛc

−HPS;λΛbλΛc), (3.6)

HCSR ;λΛbλΛc

=CSR

2

(HS;λΛbλΛc

+HPS;λΛbλΛc

), (3.7)

HCTL ;λΛbλΛc ,λW ,λ

′W

=CTL

2

(HT ;λΛbλΛc ,λW ,λ

′W−HPT ;λΛb

λΛc ,λW ,λ′W

). (3.8)

In appendix A the expressions for these amplitudes are provided in the rest frame of Λb,

where Λc momentum has spherical coordinates (p[Λb]Λc

, θ[Λb]Λc

, φ[Λb]Λc

= 0). In further discus-

sions, we drop superscript ‘[Λb ]’ for brevity, specifying that the quantity has been defined

in the Λ0b rest frame.

The polar angle and momentum of Λ+c in this frame are depicted in figure 1. It is

worth noting that these hadronic helicity amplitudes are functions of q2 and θΛc .

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JHEP12(2019)148

x̂ = ŷ × n̂

ŷ = n̂× p̂ [Λb]Λc

ẑ = n̂ = p̂ [lab]Λb × p̂Beam

p̂ [lab]Λb

p̂Beam

p̂[Λb]

ΛcθΛc Λb rest frame

Figure 1. Pictorial representation of the frame (x̂,ŷ,ẑ) in which the hadronic helicity amplitudes

related to Λ0b →Λ+c W ∗− decay are defined. The axis n̂ represents the polarisation axis of Λ0b , chosento be perpendicular to the production plane (p̂

[lab]Λb× p̂Beam).

The leptonic amplitudes, shown in eq. (3.2), are defined as:

LSM ;λWλl =1

2�µ(λW )ūl(λl)γµ(1− γ5)νν̄l , (3.9)

LCSLλl

=1

2ūl(λl)(1− γ5)νν̄l , (3.10)

LCTL ;λW ,λ

′W

λl=−i2�µ(λW )�

ν(λ′W )ūl(λl)σµν(1− γ5)νν̄l . (3.11)

Here ūl, νν̄l and �µ represents the particle helicity spinor of the lepton, the anti-particle

helicity spinor for neutrino and the polarisation vector of W , respectively. In appendix B,

we present the expressions for the leptonic amplitudes defined in the helicity frame of

W ∗−,3 (x̂`,ŷ`,ẑ`), where the `− momentum has spherical coordinates (p

[W ]` , θ

[W ]` , φ

[W ]` ). As

before, the superscript ‘[W ]’ is dropped in further discussions for brevity. The angles

and momentum of the `−, defined in the W ∗− helicity frame, are shown in figure 2. The

leptonic helicity amplitudes expressed above are functions of q2, θl and φl.

The amplitudes corresponding to the weak decay Λ+c → pK0S are given as:

GλΛcλp

= D1/2∗λΛc ,λp

(φp, θp,−φp)gλp , (3.12)

where gλp denotes the rotationally invariant amplitude of the Λc decay, defined in the rest

frame of Λc with the proton moving in positive direction of the z-axis. The Wigner-D

elements, D1/2∗λΛc ,λp

, specify the rotation of the helicity states into the helicity frame of Λc.

In this frame, the proton momentum has spherical coordinates (p[Λc]p , θ

[Λc]p , φ

[Λc]p ). The

superscript ‘[Λc ]’ in further discussions is omitted for brevity. As noted above, after

incoherent sum over λp, we can express the decay density in terms of the weak decay

3In the decay of A→ BC, the helicity frame of A forms the rest frame of A in which the z-axis is in thedirection of its polarisation axis. The latter is chosen to be in direction of the momentum of A in the rest

frame of its parent particle.

– 5 –

JHEP12(2019)148

W ∗ Λc

n̂

θΛc

Λb

Λb rest frameẑp = p̂

[Λb]

Λc

ŷp = n̂× p̂ [Λb]Λc

p̂ [Λc]p

x̂p = ŷp × ẑp

θp

x̂l = −x̂p

p̂ [W∗]

l

ŷl = ẑl × x̂l

ẑl = p̂[Λb]

W∗

θl

x̂p

ŷl = ŷp

x̂l

φl φp

p̂ [Λc] Tpp̂[W∗] Tl

Figure 2. Pictorial representation of W ∗− helicity frame (x̂l,ŷl,ẑl) and Λ+c helicity frame (x̂p,ŷp,ẑp).

(Top) The unit vectors p̂[Λb]W and p̂

[Λb]Λc

denote the direction of propagation of W ∗− and Λ+c in the

Λ0b rest frame, respectively. (Bottom) The unit vector p̂[W ]Tl and p̂

[Λc]Tp denote the direction of the

transverse momentum components of `− and p in W ∗− and Λ+c helicity frames, respectively.

asymmetry parameter αΛ+c , through the substitution:

|g+ 12|2∧

=|g+ 1

2|2∑

λp|gλp |2

=1

2(1 + αΛc) , |g− 1

2|2∧

=|g− 1

2|2∑

λp|gλp |2

=1

2(1− αΛc) . (3.13)

The expressions for the amplitude shown in eq. (3.12), when expanding out the Wigner-D

elements, are given in appendix C.

The transition amplitudes shown in eq. (3.1) apply to the Λ0b → (pK0S)(`−ν`) decaychannel. To obtain the amplitude of the CP conjugate process, we complex conjugate

all the Wilson co-coefficients ({C}) that carry the weak phase, change the sign of all theazimuthal angles involved ({φ}) and change the set of final state particles helicities to thoseof their CP conjugate partner {λ̄} [12], i.e.:

TΛ0b→Λ−c `+ν`

= TΛ0b→Λ+c `−ν`

({λ} → {λ̄}, {φ} → {−φ}, {C} → {C∗}) .

4 Decay density

The full six-fold normalised angular differential decay density considering the Λ0b polarisa-

tion effects is given by:

d6Γ =N

Γ|T |2dΩ′ , (4.1)

– 6 –

JHEP12(2019)148

with

Γ =

∫(d6Γ/dΩ′)dΩ′ , (4.2)

N =G2F |Vcb|2 pl BΛc [pΛc ]

m2pK0

S

=m2Λc

212π5m2Λb

√q2

, dΩ′ = dq2d cos θΛcd cos θpdφpd cos θldφl , (4.3)

and

|T |2 = (1 + PΛb)∑λp,λ`−

|Tλ

Λ0b= 1

2

λ`− ,λp|2 + (1− PΛb)

∑λp,λ`−

|Tλ

Λ0b=− 1

2

λ`− ,λp|2 ,

= K1(1− PΛb cos θΛc) +K2(1 + PΛb cos θΛc) +K3PΛb sin θΛc . (4.4)

More details are provided in appendix D. In eq. (4.3), BΛc denotes the branching fractionof Λ+c → pK0S decay; mΛb and mΛc are the masses of Λ0b and Λ+c ; pΛc , pp and pl denotethe three-momentum magnitudes of Λ+c , proton and lepton respectively, all defined in the

rest frame of their parent particle and expressed in terms of Lorentz invariant quantities

in eq. (D.3). In eq. (4.4), PΛb refers to the polarisation of Λ0b ; the terms Ki depend on all

the phase space observables with the exception of θΛc , which are given in appendix E.

The expression of the decay density intrinsically depends on the assumptions made

on Λ0b polarisation and Λ+c decay kinematics. When Λ

0b is produced unpolarised and the

two-body decay Λ+c → pK0S is considered, the decay density is independent of cos θΛcand φp (eq. (D.8)), and the variable φl can be expressed in terms of χ, defined as the

angle between pK0S and `−ν` decay planes. Conversely, if Λ

0b is produced polarised and

the degrees of freedom related to Λ+c decay are integrated out, the decay density exhibits

dependence on q2, cos θΛc , cos θl and φl (eq. (D.9)), where φl can be expressed in terms of

the angle between the Λ0b polarisation plane (i.e. the one containing n̂ and p̂Λc) and `−ν`

decay plane. In the case that the Λ0b baryon is unpolarised and degrees of freedom related

to the Λ+c decay are integrated out, the decay density depends only on q2 and cos θl.

5 Experimental sensitivity

In this section the sensitivity that can be achieved on the Wilson coefficients, by studying

the differential decay density in different scenarios, is presented.

In the first case, Λ0b baryons are considered to be produced unpolarised. The angular

distribution of the Λ+c decay is integrated over and the q2 and cos θl distributions are

measured. The expression for the employed fit model is shown in eq. (F.1) of appendix F.

The expected signal yield is determined from ref. [26], where the Λ+c → pK+π− decaymode is adopted. When extrapolated to the current LHCb dataset of 9 fb−1, this gives

7.5M expected signal candidates Λ0b → Λ+c µ−νµ. The abundant signal yield suggests thatthe Λ+c → pK+π− decay mode is the most sensitive to NP operators, as favoured by theexperimental signature.

– 7 –

JHEP12(2019)148

For the second scenario the Λ+c → pK0S decay is reconstructed and a non-zero polarisa-tion of Λ0b is foreseen. In this case, the angles cos θp and φp are additionally included in the

differential decay density and thus the four-dimensional distribution is fitted. The expres-

sion for the complete fit model is shown in eq. (F.2) of appendix F. It is worth noting that

the low reconstruction efficiency of long-lived K0S mesons and the smaller Λ+c branching

fraction translates into a signal yield which is reduced by approximately a factor 20 with

respect to the Λ+c → pK+π− three-body decay case.The background level for Λ0b → Λ+c µ−νµ decays is small, as reported in ref. [26].

Furthermore, the acceptance is not expected to be a strong function of the decay variables

as the muon trigger selection at LHCb is efficient [26]. As a result, these effects are neglected

in the following studies. One aspect that cannot be ignored, however, is the dilution of

resolution on q2 and cos θl variables due the unreconstructed neutrino. To take this into

account, we generate the Λ0b → Λ+c µ−νµ signal sample using Pythia [42, 43], requiring thesignal events to be approximately within the LHCb acceptance (i.e. 2 < η < 5). The Λ0bvertex position is then smeared according a spatial resolution inspired in ref. [44]. Data

migration between kinematic bins in q2 and cos θl is included in the fit by convoluting the

decay density with a migration matrix, which is described in more detail in appendix. G.

In order to assess the sensitivity to the Wilson coefficients, pseudo-experiments have

been generated with the expected signal yield and experimental resolution. A binned

maximum likelihood fit, with 20 bins employed in each dimension, is then performed. A

Gaussian constraint is applied to the Λ0b →Λ+c hadronic form factors, using lattice QCD re-sults from [24, 30]. The 95% CL intervals obtained from this study are compared with those

inferred from B → D(∗)`ν decays [22]. No effects of CP violation have been considered,therefore the Wilson coefficients in this study are assumed to be real.

At first, only one Wilson coefficient at a time is varied; the results are shown in

figure 3. As the production fraction of Λ0b decays has a relatively large uncertainty [45],

the normalised differential distribution is fitted which has no sensitivity to the Wilson

coefficient CVL and the CKM matrix element Vcb. The sensitivity to other NP operators is

expected to be significantly better than that of the current constraints, mainly due to the

huge signal yields expected at LHCb.

The interplay between different Wilson coefficients is explored in figure 4 and compared

to ref. [22]. Large non-Gaussian correlations are observed, affecting both the 2D and 4D dif-

ferential widths. When all the Wilson coefficients are fitted at once, then the observed corre-

lations between the tensor and scalar currents become even larger, implying that contribu-

tions from these operators are difficult to disentangle from the explored distributions alone.

The four-dimensional decay density distribution involving Λ+c → pK0S is sensitive toboth the PΛ0b

and αΛ+c . A comparison of the results of this study with existing measurements

from BES III [39] and LHCb [36] is illustrated in figure 5. The expected sensitivity to αΛ+cis currently a world-leading value, whereas the sensitivity to PΛ0b

is slightly less precise than

previous measurements [36–38], but could be improved in the future with a full angular

analysis of Λ0b →Λ+c (→pK0S)µ−νµ decays. A summary of the sensitivity for the variouscases can be found in table 1.

– 8 –

JHEP12(2019)148

-0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20e(CVR)

3.2

3.4

3.6

3.8

4.0

4.2

4.4

4.6

4.8

102

×V c

b

95% confidence intervals

SMB D * l (JHEP01 2019 009)B Dl (JHEP01 2019 009)B Xcl (JHEP01 2019 009)pKS sensitivitypK sensitivity

-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00e(CSR)

3.4

3.6

3.8

4.0

4.2

4.4

4.6

4.8

5.0

102

×V c

b

95% confidence intervals

SMB D * (JHEP01 2019 009)B D (JHEP01 2019 009)pKS sensitivitypK sensitivity

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3e(CTL)

3.00

3.25

3.50

3.75

4.00

4.25

4.50

4.75

5.00

102

×V c

b

95% confidence intervals

SMB D * all (JHEP01 2019 009)BF(B D * ) (JHEP01 2019 009)pKS sensitivitypK sensitivity

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6e(CSL)

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

Prob

abilit

y de

nsity

func

tion

95% confidence intervals

SMpKS sensitivitypK sensitivity

Figure 3. Expected sensitivity to the Wilson coefficients of the NP operators individually fitted

and compared to the constraints obtained from the corresponding mesonic semileptonic decays [22].

As done in ref.([22]), we define here C̃i = Ci/(1 + CVL) and Ṽcb = Vcb/(1 + CVL).

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5e(CSL)

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

e(C T

L)

95% confidence intervals

SMB D * (JHEP01 2019 009)B D (JHEP01 2019 009)pKS sensitivitypK sensitivity

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5e(CSR)

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

e(C S

L)

95% confidence intervals

SMB D * (JHEP01 2019 009)B D (JHEP01 2019 009)pKS sensitivitypK sensitivity

Figure 4. Two-dimensional sensitivity plot between the Wilson coefficients ˜CTL ,˜CSL and

˜CSRwhen compared to the limits obtained from mesonic semileptonic decays [22]. As done in ref.([22]),

we define here C̃i = Ci/(1 + CVL).

– 9 –

JHEP12(2019)148

-0.4 -0.2 0.0 0.2 0.4P b

-1.0

-0.5

0.0

0.5

1.0

c95% confidence intervals

BESIII (arXiv:1905.04707)P b sensitivity (pKS case)LHCb (PLB 724 2013 27)

c sensitivity (pKS case)

Figure 5. Sensitivity on PΛ0b and αΛ+c as obtained from a four-dimensional fit to the Λ0b → Λ+c (→

pK0S)µν differential distribution.

Free parameters pK0S case pK−π+ case

CVR 0.005 0.001

CSR 0.046 0.018

CTL 0.020 0.007

CSL 0.091 0.039

PΛ0b0.13 –

αΛ+c 0.003 –

Table 1. The 68% confidence intervals for the parameters of interest for various cases.

6 Conclusions

In this study a general expression for the effective Lagrangian governing b→ clνl transitionshas been considered, including NP contributions through a complete basis of dimension-six

operators and assuming only left-handed neutrinos. Using this formalism, we presented

for the first time an expression for the six-fold differential decay density for polarised

Λ0b → Λ+c `−ν` decays, with subsequent Λ+c →pK0S decay.In addition, we carried out sensitivity studies to determine the experimental precision

on the Wilson coefficients which can be achieved through the analysis of Λ0b → Λ+c µ−νµdecays at the LHCb experiment. When considering the integrated Run I and Run II data

samples collected at LHCb. The first case considered was the decay channel Λ0b →Λ+c(→pK−π+)µ−νµ, where the 2D distribution in q2 and cos θl was studied. The secondexplored scenario focused on Λ0b →Λ+c (→pK0S)µ−νµ decays, including polarisation effectson the production of Λ0b . At this purpose, the 4D distributions in q

2, cos θl, cos θp and φpvariables were inspected. Since a missing unreconstructed neutrino in the final state spoils

the experimental resolution on q2 and cos θl, in both of the mentioned cases the resolution

effects were folded into the fit model through a migration matrix.

– 10 –

JHEP12(2019)148

The results of the sensitivity studies show that the best precision on the Wil-

son coefficients can be achieved by probing the q2 and cos θl distributions of Λ0b →Λ+c

(→pK−π+)µ−νµ decays collected at LHCb, leading to a good sensitivity to CVR , CTL , CSRand CSL . No sensitivity is expected to overall global factors, such as Vcb and CVL , as the

present study is performed on the normalised differential decay distributions.

Although no enhanced sensitivity to the Wilson coefficients could be achieved through

the investigation of 4D kinematic distributions of Λ0b →Λ+c (→pK0S)µ−νµ decay channel,they do however provide a prospect of measuring αΛ+c and PΛ0b

. The parameter αΛ+c is

particularly promising, with a precision which is two orders of magnitude smaller than

that measured by the BESIII experiment. The precision on PΛ0bcould also be improved by

performing a full six-dimensional angular analysis. That would require large signal yields

expected at the LHCb upgrades and a different treatment of the resolution.

In this paper we have assumed CP conservation and considered the Wilson coefficients

to be real variables. To further distinguish NP models, one could easily extend the present

study allowing Wilson coefficients to be complex to probe CP violation in Λ0b → Λ+c µ−νµdecays. The results of the sensitivity studies have been compared to the model-independent

constraints obtained in global fits to B → D(∗)`−ν` semileptonic decays, presented inref. [22], and have shown a significant improvement in the achievable experimental precision.

As a consequence, it can be concluded that studying Λ0b → Λ+c µ−νµ decays at the LHCbexperiment will not only lead to a more precise measurement on Λ+c decay asymmetry

parameter, but also allows to place stringent world leading constraints on the Wilson

coefficients of the corresponding NP operators.

Acknowledgments

The authors would like to thank their colleagues from the LHCb collaboration (Lucia Grillo,

Michel De Cian, Greg Ciezarek and others) for fruitful discussions. The authors would also

like to extend their gratitude to Matthew Charles for his helpful suggestions to this paper.

This work is supported by the Swiss National Science Foundation (SNF) under Contract

No. 200021 182622.

A Hadronic amplitudes of Λ0b → Λ+c W ∗− decay

In this section, we give expressions of the hadronic helicity amplitudes for Λ0b →Λ+c W ∗−

decay. The hadronic matrix elements shown in eqs. (3.3)–(3.8) are expressed as:

HV ;λΛbλΛc ,λW

= �†µ(λW )〈Λc, λΛc |c̄γµb|Λb, λΛb〉 ,

HA;λΛbλΛc ,λW

= �†µ(λW )〈Λc, λΛc |c̄γµγ5b|Λb, λΛb〉 ,

HS;λΛbλΛc

= 〈Λc, λΛc |c̄b|Λb, λΛb〉 ,

HPS;λΛbλΛc

= 〈Λc, λΛc |c̄γ5b|Λb, λΛb〉 ,

HT ;λΛbλΛc ,λW ,λ

′W

= �†µ(λW )�†ν(λ′W )〈Λc, λΛc |c̄iσµνb|Λb, λΛb〉 ,

HPT ;λΛbλΛc ,λW ,λ

′W

= �†µ(λW )�†ν(λ′W )〈Λc, λΛc |c̄iσµνγ5b|Λb, λΛb〉 .

– 11 –

JHEP12(2019)148

Here ‘�†µ’ denotes the polarisation vector of W ∗−. The definitions of the matrix elements

in terms of the form factors are given in ref. [30].

When considering the transverse polarisation of Λb, the common choice of a Λb rest

frame is the one where the polarisation axis (n̂) is perpendicular to the production plane,

i.e.:

ẑ = n̂ = p̂labΛb × p̂labBeam , ŷ = p̂Λc × ẑ , x̂ = ŷ × ẑ , (A.1)

where ˆ refers to the unit vector, the superscript ‘lab’ indicates the lab frame and p̂Λc is

the unit vector of Λc momentum in Λb rest frame (figure 1). In such a frame, the choice

of azimuthal angle for p̂[Λb]Λc

is arbitrary and is set to zero. Therefore, Λc momentum is

oriented in the direction (θΛc , φΛc = 0) with magnitude pΛc and the virtual W∗− moves in

the opposite direction, which is (π − θΛc , π) with pW ∗− = pΛc .The generic expression for the polarisation vector, helicity spinors and the representa-

tion of gamma matrices used in this work follow refs. [46, 47]. We show below the expression

for helicity spinors of Λ0b (uΛb) with mass mΛb , Λ+c (uΛc) with mass mΛc and polarisation

vector of W ∗− (�µ) in the Λb rest frame.

uΛb

(λΛb =

1

2

)=

√

2mΛb0

0

0

uΛc(λΛc =

1

2

)=

√EΛc +mΛc cos

θΛc2√

EΛc +mΛc sinθΛc2√

EΛc −mΛc cosθΛc2√

EΛc −mΛc sinθΛc2

,

uΛb

(λΛb = −

1

2

)=

0√

2mΛb0

0

, uΛc(λΛc = −

1

2

)=

−√EΛc +mΛc sin

θΛc2√

EΛc +mΛc cosθΛc2√

EΛc −mΛc sinθΛc2

−√EΛc −mΛc cos

θΛc2

,

�µ(λW = t) =

EW√q2

−pΛc sin θΛc√q2

0

−pΛc cos θΛc√q2

, �µ(λW = 1) =

0cos θΛc√

2−i√

2

− sin θΛc√2

,

�µ(λW = 0) =

pΛc√q2

−EW sin θΛc√q2

0

−EW cos θΛc√q2

, �µ(λW = −1) =

0

− cos θΛc√2

− i√2

sin θΛc√2

,

where

EΛc =√p2Λc +m

2Λc, EW =

√p2Λc + q

2 =1

2mΛb( m2Λb − m

2Λc + q

2) ,

pΛc =1

2mΛb

√Q+Q− , Q± = (M

2± − q2) , M± = (mΛb ±mΛc) .

– 12 –

JHEP12(2019)148

The vector (HV ;λΛbλΛc ,λW

) and axial vector (HA;λΛbλΛc ,λW

) amplitudes can then be expressed as:

HV ; 1

212,t

= HV ;− 1

2

− 12,t

= cosθΛc2aV , H

V ; 12

− 12,t

= −HV ;−12

12,t

= − sin θΛc2aV ,

HV ; 1

212,0

= HV ;− 1

2

− 12,0

= cosθΛc2bV , H

V ; 12

− 12,0

= −HV ;−12

12,0

= − sin θΛc2bV ,

HV ; 1

2

− 12,− = H

V ;− 12

12,+

= − cos θΛc2cV , H

V ;− 12

− 12,− = −H

V ; 12

12,+

= − sin θΛc2cV ,

HA; 1

212,t

= −HA;−12

− 12,t

= cosθΛc2aA , H

A; 12

− 12,t

= HA;− 1

212,t

= sinθΛc2aA ,

HA; 1

212,0

= −HA;−12

− 12,0

= cosθΛc2bA , H

A; 12

− 12,0

= HA;− 1

212,0

= sinθΛc2bA ,

HA; 1

2

− 12,− = −H

A;− 12

12,+

= cosθΛc2cA , H

A;− 12

− 12,− = H

A; 12

12,+

= sinθΛc2cA ,

HV (A); 1

212,− = H

V (A);− 12

− 12,+

= 0 , HV (A); 1

2

− 12,+

= HV (A);− 1

212,− = 0 ,

where the q2 dependent quantities are expressed in terms of the form factors F0,+,⊥ and

G0,+,⊥ as:

aV =F0√Q+M−√q2

, bV =F+√Q−M+√q2

, cV =√

2F⊥√Q− ,

aA =G0√Q−M+√q2

, bA =G+√Q+M−√q2

, cA =√

2G⊥√Q+ . (A.2)

Similarly, the scalar (HS;λΛbλΛc

) and pseudo-scalar (HPS;λΛbλΛc

) amplitudes are given by:

HS; 1

212

= HS;− 1

2

− 12

= cosθΛc2aS , H

S; 12

− 12

= −HS;−12

12

= − sin θΛc2aS ,

HPS; 1

212

= −HPS;−12

− 12

= − cos θΛc2aP , H

PS; 12

− 12

= HPS;− 1

212

= − sin θΛc2aP ,

where

aS =F0√Q+M−

mb −mc, aP =

G0√Q−M+

mb +mc. (A.3)

The tensor (HT ;λΛbλΛc ,λW ,λ

′W

) and pseudo-tensor (HPT ;λΛbλΛc ,λW ,λ

′W

) amplitudes can be expressed

as follows:

HT ;± 1

2

± 12,t,0

= HPT ;± 1

2

± 12,1,−1 = cos

θΛc2aT , H

T ;± 12

∓ 12,t,0

= HPT ;± 1

2

∓ 12,1,−1 = ∓ sin

θΛc2aT ,

HT ;∓ 1

2

± 12,t,±1 = H

PT ; 12

− 12,0,−1 = − cos

θΛc2bT , H

T ;− 12

− 12,t,−1 = H

PT ;± 12

± 12,0,±1 = − sin

θΛc2bT ,

HT ;− 1

2

− 12,0,−1 = H

PT ;± 12

± 12,t,±1 = − sin

θΛc2cT , H

T ;∓ 12

± 12,0,±1 = H

PT ; 12

− 12,t,−1 = − cos

θΛc2cT ,

HT ;± 1

2

∓ 12,1,−1 = H

PT ;± 12

∓ 12,t,0

= − sin θΛc2dT , H

T ;± 12

± 12,1,−1 = H

PT ;± 12

± 12,t,0

= ∓ cos θΛc2dT ,

H(P )T ;± 1

2

− 12,t,1

= H(P )T ;± 1

212,t,−1 = 0 , H

(P )T ;± 12

− 12,0,1

= H(P )T ;± 1

212,0,−1 = 0 .

– 13 –

JHEP12(2019)148

The remaining (pseudo-)tensor amplitudes can be obtained through the relations:

HT ; 1

212,t,1

=−HT ;−12

− 12,t,−1 , H

T ; 12

12,0,1

=−HT ;−12

− 12,0,−1 ,

HPT ;− 1

212,t,1

=−HPT ;12

− 12,t,−1 , H

PT ;− 12

12,0,1

=−HPT ;12

− 12,0,−1 ,

H(P )T ;λΛbλΛc ,λW ,λ

′W

=−H(P )T ;λΛbλΛc ,λ

′W ,λW

,H(P )T ;λΛbλΛc ,λW ,λW

= 0 .

In the above expressions, the q2 dependent quantities are given in terms of the tensor form

factors h+, h⊥, h̃+ and h̃⊥ as:

aT = h+√Q− , bT =

√2h⊥M+

√Q−√

q2,

cT =

√2h̃⊥M−

√Q+√

q2, dT = h̃+

√Q+ , (A.4)

It is worth noting that when θΛc = 0 we recover the expressions as quoted in refs. [27, 28, 30].

The helicity amplitudes presented in this section are expressed in terms of the form

factors F0,+,⊥, G0,+,⊥, h+,⊥ and h̃+,⊥, which are defined in such a way that they corre-

spond to time-like (scalar), longitudinal and transverse polarisation with respect to the

momentum-transfer qµ. An alternate parameterisation of form factors (denoted by f1,2,3,

g1,2,3, fT1,2 and g

T1,2) that are based on the large and small projections of massive fermion

spinors, can often be found in the literature [27, 28]. The relation between these two form

factor parameterisation is given in appendix B of ref. [48].

B Leptonic amplitudes of W ∗− → `−ν` decay

In this section, we give the expressions of the leptonic helicity amplitudes for W ∗− →`−ν`decay, shown in eqs. (3.9)–(3.11). The choice of the lepton azimuthal angle in the W ∗ rest

frame is now fixed by n̂, i.e. the Λb polarisation axis (eq. (A.1)):

ẑl = p̂[Λb]W , ŷl = n̂× ẑl , x̂l = ŷl × ẑl . (B.1)

Therefore, the lepton `− momentum is oriented in the direction (θl, φl) with magnitude

pl, whereas the neutrino ν` moves in the opposite direction, which is (π − θl, π + φl) withpν` = pl. This frame of reference is depicted in figure 2.

– 14 –

JHEP12(2019)148

We show below the expressions for helicity spinors of `− (ul) with mass ml, ν` (uν̄l)

and polarisation vector of W ∗− (�µ) in the above defined frame:

�µ(λW = t) =

1

0

0

0

, �µ(λW = 1) =

01√2

− i√2

0

,

�µ(λW = 0) =

0

0

0

−1

, �µ(λW =−1) =

0

− 1√2

− i√2

0

,

ul

(λl =

1

2

)=

√m+El cos

(θl2

)eiφl√m+El sin

(θl2

)√El−mcos

(θl2

)eiφl√El−msin

(θl2

)

, ul(λl =−

1

2

)=

−e−iφl

√m+El sin

θl2√

ml+El cosθl2

e−iφl√El−msin θl2

−√El−ml cos θl2

,

vν`

(λν` =

1

2

)=

e−iφl

√pl cos

θl2√

pl sinθ2

−e−iφl√pl cos θl2−√pl sin θ2

,

where

pl =√q2v2/2 , El = pl +m

2l /√q2 , v =

√1−

m2lq2

. (B.2)

The vector and axial-vector amplitudes (LSM ;λWλl,λν`=

12

) are then given by:

LSM ;t12

= e−iφlal , LSM ;012

= −e−iφl cos θlal ,

LSM ;112

= e−2iφl sin θlal√

2, LSM ;−11

2

= − sin θlal√

2,

LSM ;0− 12

= sin θlbl , LSM ;±1∓ 1

2

= e∓iφl(1± cos θl)bl√2,

LSM ;t− 12

= 0 .

The scalar and pseudo-scalar leptonic helicity amplitudes (LSLλl ) become:

LSL12

= e−iφlbl , LSL− 1

2

= 0 . (B.3)

– 15 –

JHEP12(2019)148

The tensor amplitudes (LTL;λW ,λ

′W

λl) are given by:

LTL;t,0− 12

= LTL;1,−1− 12

= sin θlal , LTL;0,−1− 1

2

= LTL;t,−1− 12

= eiφl(1− cos θl)al√

2,

LTL;t,1− 12

= −LTL;0,1− 12

= e−iφl(1 + cos θl)al√

2, LTL;t,01

2

= LTL;1,−112

= −e−iφl cos θlbl ,

LTL;t,112

= −LTL;0,112

= e−2iφl sin θlbl√2, LTL;t,−11

2

= LTL;0,−112

= − sin θlbl√2.

The rest of the tensor amplitudes can be obtained using the relations:

LTL;λW ,λWλl = 0 , LTL;λW ,λ

′W

λl= −LTL;λ

′W ,λW

λl. (B.4)

The q2 dependent terms that appear in the above equations are given by:

al = 2mlv , bl = 2√q2v . (B.5)

We note that the relations obtained here match that of ref. [30] when φl = 0. As

done in ref. [30], in the definition of the polarisation vector the Euler angle has been set to

γ = −φl, contrary to γ = 0 as done in ref. [14]. As a result, the expressions presented inref. [14], differ from ours and those presented in ref. [30] by an unimportant overall phase

factor, e−π.

C Hadronic amplitudes of Λ+c → pK0S decay

In this section, we expand out the Wigner-D elements and provide expressions of the

hadronic amplitudes for Λ+c → pK0S decay, which is shown in eqs. (3.12). These amplitudesare defined in the Λ+c rest frame where the choice of azimuthal angle for the proton is fixed

by n̂ defined in eq. (A.1):

ẑp = p̂[Λb]Λc

, ŷp = n̂× ẑp , x̂p = ŷp × ẑp . (C.1)

Therefore, in this frame, pmomentum is oriented in the direction (θp, φp) with magnitude ppand K0S moves in the opposite direction i.e. (π−θp, π+φp) with pK0S = pp. A transformationfrom this frame to the frame defined in eq. (B.1) can be achieved through rotation by angle

π about yp-axis. This frame of reference is depicted in figure 2.

The hadronic amplitudes (GλΛcλp

), in this frame, are then given by4

G1212

= cosθp2g 1

2, G

12

− 12

= −eiφp sin θp2g− 1

2,

G− 1

212

= e−iφp sinθp2g 1

2, G

− 12

− 12

= cosθp2g− 1

2.

4We employ Wigner sign convention, where Wigner-D elements are defined as DJm′,m(α, β, γ) =

e−im′αdJm′,m(β)e

−imγ with the property DJ∗m′,m(α, β, γ) = (−1)m′−mDJ−m′,−m(γ, β, α), where α, β, γ are

Euler angles and dJm′,m(β) are small-d Wigner elements.

– 16 –

JHEP12(2019)148

D Phase space

The differential decay rate can be written as:

dΓ =|T |2

2mΛbdΦ4(PΛb ;Pp, PK0S

, Pl, Pν`) (D.1)

where T denotes the complex transition amplitude, ‘PA’ is the four-momentum of the

particle ‘A’ in Λb rest frame, mΛb is mass of Λ0b and dΦ4 is the four-body phase space

element that can be written as the product of two-body phase space elements as follows:

dΦ4 = (2π)4δ(4)

(PΛb −

4∑i

Pi

)4∏i=1

d3pi(2π)32Ei

=dm2

pK0S

2π

dq2

2πdΦ2(PΛb ;PΛc , q) dΦ2(PΛc ; P̂p, P̂K0S

) dΦ2(q; P̂l, P̂ν`) .

Here m2pK0S

= (Pp + PK0S)2, q2 = (Pl + Pν`)

2 andˆdenotes that the four-momenta are now

defined in the rest frame of the parent particle. The two-body phase space is given by:

dΦ2(P̂i, P̂j) =1

24π2pimij

d cos θi dφi ,

where pi denotes the magnitude of three-momentum of particle i in the ij rest frame. The

full four-body phase space element then becomes:

dΦ4 =1

214π8dm2pK0S

dq2pΛcmΛb

d cos θΛcdφΛcpp

mpK0Sd cos θpdφp

pl√q2d cos θldφl , (D.2)

where pΛc , pp, pl with their corresponding angles are defined in the Λ0b , Λ

+c and W

∗− rest

frames, respectively. These momenta can be expressed as:

pΛc =

√λ(m2Λb ,m

2pK0S

, q2)

2mΛbpp =

√λ(m2

pK0S,m2p,m

2K0S

)

2mpK0Spl =

q2 −m2l2√q2

, (D.3)

where λ(a, b, c) = a2 + b2 + c2 − 2(ab+ ac+ bc).The differential density shown in eq. (D.1) then becomes:

dΓ4-body =pΛc pp pl

215π8m2ΛbmpK0S

√q2|T |2dΩ , (D.4)

where dΩ = dm2pK0S

dq2d cos θΛcdφΛcd cos θpdφpd cos θldφl.

In the main text, it is highlighted that the choice of φΛc is arbitrary and has been

set to zero, removing its dependence from |T |2. It can also be seen in eq. (3.2) that, fora given helicity of initial and final state, |T |2 would have dependence on m2

pK0Sthrough

the propagation terms BWΛcBW†Λc

. Since the total width of Λ+c is far below its mass

(ΓΛc � mΛc), we can use here the narrow width approximation to give:

BWΛcBW†Λc

=π

mΛcΓΛcδ(m2pK0S

−m2Λc). (D.5)

– 17 –

JHEP12(2019)148

We can also factor out the term (|g+ 12|2 + |g− 1

2|2) from |T |2 and normalise Λ+c decay density

using the relation:

pp26π2m2Λc

(|g+ 1

2|2 + |g− 1

2|2)

=BΛcΓΛc

4π. (D.6)

Substituting all the above three relations in eq. (D.4) and integrating over m2pK0S

and φΛc ,

the differential decay density becomes:

dΓ4-body =pl BΛc [pΛc ]

m2pK0

S

=m2Λc

210π5m2Λb

√q2

|T |2dΩ′ , (D.7)

where dΩ′ = dq2d cos θΛcd cos θpdφpd cos θldφl and BΛc denotes the branching fraction ofΛ+c → pK0S decay.

When the Λ0b is unpolarised (PΛb = 0), it is clear from eq. (4.4), that the dependence

of |T |2 on θΛc is inherently removed. Also the choice of the azimuthal angle φp that waspreviously fixed by the definition of the Λ0b polarisation axis (n̂), now becomes arbitrary.

We therefore set the Λ+c helicity frame (xp, yp, zp) in such a way that angle φp = 0, i.e. the

proton momentum always lies in the xp–zp plane and, as before, x̂l = −x̂p. In this case,the differential decay density becomes:

dΓ4-body =pl BΛc [pΛc ]

m2pK0

S

=m2Λc

28π4m2Λb

√q2

[|T |2

]PΛb=0,φp=0dΩ′′ , (D.8)where dΩ′′ = dq2d cos θpd cos θldφl. Note that with the choice of φp = 0, one can also

express angle φl (figure 2) in terms of the relative angle between the (pK0S) and (`

−ν`)

decay planes, χ. Either of the two relations that are often employed in the literature can

be used, i.e. either φl = π + mod(2π) − χ [49] or φl = χ [14] (However, when chosen thedefinition should be adopted consistently throughout the analysis).

When integrating out the Λ+c dynamics, the three-body phase space element is consid-

ered. This case is very similar to setting Λ+c decay asymmetry to zero (αΛc = 0), where the

dependency of |T |2 on cos θp and φp is inherently removed. The differential decay densitythen takes the form:

dΓ3-body =pΛc pl

29π4m2Λb

√q2

[|T |2

]αΛc=0 dΩ′′′ , (D.9)where dΩ′′′ = dq2d cos θΛcd cos θldφl. The choice of angle φl in the above case is now fixed

through the choice of W ∗− helicity frame, defined with respect to the Λ0b polarisation plane

(i.e. ŷl = n̂ × p̂Λc). Additionally, if Λ0b is unpolarised, the dependence of[|T |2

]αΛc=0 oncos θΛc and φl is inherently removed, leaving the dependence of decay density on only q

2

and cos θl (with an additional factor 4π from integration over the element d cos θΛc dφl).

– 18 –

JHEP12(2019)148

E Terms of differential decay density

We present below explicit expressions for Ki terms defined in the full angular differential

density in eq. (4.4):

K1 =1

8

[cosθp(|g− 1

2|2∧

−|g+ 12|2∧

){−2(1−cos2 θl)(|I1|2−2|I8|2)−2|I4|2(cosθl+1)2+

2I∗5 (I5(2cos2 θl−1)+2I9 cosθl)+4I∗9 (I5 cosθl+I9)

}+4cosθl{

|I4|2+I9I∗5 +I5I∗9}

+2|I5|2{|g− 1

2|2∧

(cosθp+1)−|g+ 12|2∧

cosθp

}+|g− 1

2|2∧{

|I1|2+3|I4|2+

2|I8|2+4|I9|2}

+|g+ 12|2∧{

|I1|2+3|I4|2+2|I5|2+2|I8|2+4|I9|2}

+(2cos2 θl−1){−|I1|2+|I4|2+2|I5|2−2|I8|2

}]+

1

2√

2

[e−i(φl+φp)

√1−cos2 θl

√1−cos2 θp

(|g− 12|2∧

−|g+ 12|2∧

){e2iφlI∗1 (I5 cosθl+I9)−e2iφlI8I∗4 (cosθl+1)+

e2iφp(I1(I∗5 cosθl+I

∗9 )−I4I∗8 (cosθl+1))

}].

K2 =1

16

[2|g− 1

2|2∧{

4(1−cosθp)(I∗6 cosθl(I6 cosθl+I10)+|I7|2(1−cos2 θl)

)−

4I∗10(cosθp−1)(I6 cosθl+I10)+2(cosθp+1)(|I2|2(1−cos2 θl)+|I3|2

(1−cosθl)2)}

+e−iφl{

4√

2√

1−cos2 θl√

1−cos2 θp(|g− 12|2∧

−|g+ 12|2∧

)(−e2iφl−iφp

{I2I∗6 cosθl+I3I

∗7 (cosθl−1)+I2I∗10

}−eiφp

{I∗2 (I6 cosθl+I10)+

I7I∗3 (cosθl−1)

})+4eiφl |g+ 1

2|2∧(

2(cosθp+1)(I6 cosθl+I10)

(I∗6 cosθl+I∗10)+(1−cos2 θl)

{I∗2 (I2−I2 cosθp)+2|I7|2(cosθp+1)

}+

|I3|2(1−cosθl)2(1−cosθp))}]

.

K3 =1

16

[4e−i(2φl+φp)

√1−cos2 θp(|g− 1

2|2∧

−|g+ 12|2∧

){e2iφp

((1−cos2 θl)(I1I∗2−I4I∗3 )−

2e2iφl{

(I6 cosθl+I10)(I∗5 cosθl+I

∗9 )−I7I∗8 (1−cos2 θl)

})+

e2iφl(−2(I5 cosθl+I9)(I∗6 cosθl+I∗10)+(1−cos2 θl)

{2I8I

∗7 +e

2iφl(I2I∗1−I3I∗4 )

})}+

4√

2e−iφl√

1−cos2 θl cosθp(|g− 12|2∧

−|g+ 12|2∧

){e2iφl

(I∗1 (I6 cosθl+I10)+

I2(I∗5 cosθl+I

∗9 )−I3I∗8 (cosθl−1)+I7I∗4 (cosθl+1)

)+cosθl

(I1I∗6 +I5I

∗2−

I8I∗3 +I4I

∗7

)+I1I

∗10+I9I

∗2 +I8I

∗3 +I4I

∗7

}+4√

2e−iφl√

1−cos2 θl{e2iφl

(−I∗1 (I6 cosθl+I10)+I2(I∗5 cosθl+I∗9 )−I3I∗8 (cosθl−1)−I7I∗4 (cosθl+1)

)+

cosθl

(−I1I∗6 +I5I∗2−I8I∗3−I4I∗7

)−I1I∗10+I9I∗2 +I8I∗3−I4I∗7

}].

In the above expression, the normalised helicity amplitudes for Λ+c decay, |g− 12|2∧

and |g+ 12|2∧

,

can be expressed in terms of the Λ+c weak decay asymmetry parameter given by eq. (3.13).

– 19 –

JHEP12(2019)148

In all the above expressions the Ii depend on the complex Wilson coefficients and terms

that are functions of q2, i.e.:

I1 = alcA(1+CVL−CVR)−alcV (1+CVL+CVR)−4bl(bT +cT )CTL ,

I2 = alcA(1+CVL−CVR)+alcV (1+CVL+CVR)+4bl(bT−cT )CTL ,

I3 = blcA(1+CVL−CVR)+blcV (1+CVL+CVR)+4al(bT−cT )CTL ,

I4 = blcA(1+CVL−CVR)−blcV (1+CVL+CVR)−4al(bT +cT )CTL ,

I5 = albA(1+CVL−CVR)+albV (1+CVL+CVR)+4bl(aT−dT )CTL ,

I6 = albA(1+CVL−CVR)−albV (1+CVL+CVR)−4bl(aT +dT )CTL ,

I7 =−bAbl(1+CVL−CVR)+blbV (1+CVL+CVR)+4al(aT +dT )CTL ,

I8 = bAbl(1+CVL−CVR)+blbV (1+CVL+CVR)+4al(aT−dT )CTL ,

I9 = aAal(1+CVL−CVR)+alaV (1+CVL+CVR)−aP bl(CSL−CSR)+aSbl(CSL+CSR) ,

I10 = aAal(1+CVL−CVR)−alaV (1+CVL+CVR)−aP bl(CSL−CSR)−aSbl(CSL+CSR) .(E.1)

Here the q2 dependent terms aV,A,S,P,T,l, bV,A,T,l, cV,A,T and dT are defined in eqs. (A.2),

(A.3), (A.4) and (B.5). Note that in literature [28, 50], the relations (CSL − CSR) = gP ,(CSL + CSR) = gS , (CVR − CVL) = gA and (CVR + CVL) = gV are often used.

F Decay density for two considered cases

For the first case studied in this paper, we present below the decay density as a function

of q2 and cos θl, after integrating eq. (4.1) over all other phase space variables:

d2Γ

dq2 dcosθl=N

Γ

[2π2{

cosθl

(−2cosθl(|I7|2+|I8|2)+2I∗6 (cosθlI6+I10)−cosθlI2I∗2 +

(cosθl−2)I3I∗3 +(cosθl+2)I4I∗4 +2I∗5 (cosθlI5+I9))

+|I2|2+|I3|2+|I4|2+

I∗1 (I1−cosθ2l I1)+2I∗10(cosθlI6+I10)+2I∗9 (cosθlI5+I9)+2I7I∗7 +2I8I∗8}]

,

(F.1)

where the q2 dependent terms Ii are defined eqs. (E.1). It can be seen from the above

equation that the decay density is independent of PΛb and αΛc . The shape of this decay

density is similar to the case when Λ+c is considered stable and Λ0b is unpolarised.

For the second case studied in this paper, we present below the decay density as a

function of q2, cos θl, cos θp and φp, after integrating eq. (4.1) over all other phase space

– 20 –

JHEP12(2019)148

variables

d4Γ

dω=N

Γ

[1

8π

{− αΛc

√1− cos θ2pe−iφpPΛb

(− 2πe2iφp

{(cos θ2l − 1)I7I∗8

+ (cos θlI6 + I10)(cos θlI∗5 + I

∗9 )}− 2π(cos θ2l − 1)I8I∗7

− 2π(cos θlI5 + I9)(cos θlI∗6 + I∗10))− 4(cos θ2l − 1)|I1|2(αΛc cos θp + 1)

+ 4αΛc cos θ2l cos θp|I2|2 − 4αΛc cos θ2l cos θp|I3|2

− 8(αΛc cos θp − 1)(|I8|2(1− cos θ2l ) + cos θlI∗5 (cos θlI5 + I9)

+ I∗9 (cos θlI5 + I9)

)+ 8αΛc cos θ

2l cos θp|I6|2 − 8αΛc cos θ2l cos θp|I7|2

+ 8αΛc cos θl cos θpI6I∗10 + 8αΛc cos θl cos θpI10I

∗6 + 8αΛc cos θl cos θp|I3|2

+ 4(cos θl + 1)2|I4|2(αΛc cos θp + 1) + 8αΛc cos θp|I10|2 − 4αΛc cos θp|I2|2

− 4αΛc cos θp|I3|2 + 8αΛc cos θp|I7|2 − 4 cos θ2l |I2|2 + 4 cos θ2l |I3|2

+ 8 cos θ2l |I6|2 − 8 cos θ2l |I7|2 + 8 cos θlI6I∗10 + 8 cos θlI10I∗6

− 8 cos θl|I3|2 + 8|I10|2 + 4|I2|2 + 4|I3|2 + 8|I7|2}]

, (F.2)

where dω = dq2d cos θld cos θpdφp and the q2 dependent terms Ii are defined eqs. (E.1). In

the above equation, φp dependence is removed when either PΛb or αΛc is zero. The cos θpdependence on the above equation only exists when αΛc is non-zero.

G Response matrix

In this section the migration/response matrix is discussed, which is convolved with the

fit model to account for finite resolution effects on q2 and cos θl variables. It is a four

dimensional object that is a function of the reconstructed and true variables. To obtain

this matrix, we generate a signal sample according to SM using PYTHIA [42, 43] and

require that the signal events lie within the LHCb acceptance of 2 < η < 5. The Λ0b vertex

position is then smeared according the resolution discussed in ref. [44]. To avoid any model

dependence, we bin the reconstructed and true variables very finely. In this study a binning

scheme of 20× 20× 20× 20 is employed.The migration matrix of reconstructed (q2rec, cos(θl)reco) versus true (q

2true,cos(θl)true)

variables is illustrated in figure 6, where the 2D projections are shown. The effect of

migration is pretty uniform for the q2 variable except for the corners. On the contrary, for

cos(θl) the migration effects are prominent at lower cos(θl)reco and cos(θl)true values. In

figure 7, the effects of migration on SM-like Monte Carlo sample are illustrated.

– 21 –

JHEP12(2019)148

2 4 6 8 10q2reco

2

4

6

8

10

q2 tru

eReponse Matrix

0.2

0.4

0.6

0.8

1.01e1

1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00cos( l)reco

1.00

0.75

0.50

0.25

0.00

0.25

0.50

0.75

1.00

cos(

l) tru

e

0.2

0.4

0.6

0.8

1.01e1

Figure 6. 2D projection of the 4D migration matrix of reconstructed versus true variables for (left)

q2 and (right) cos(θl).

2 4 6 8 10q2

1.00

0.75

0.50

0.25

0.00

0.25

0.50

0.75

1.00

cos(

l)

MC True

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

1e 3

2 4 6 8 10q2

1.00

0.75

0.50

0.25

0.00

0.25

0.50

0.75

1.00MC Reco

0.5

1.0

1.5

2.0

2.5

3.0

3.5

1e 3

Figure 7. (Left) The true distribution of q2 and cos(θl) generated according to SM and (right) the

result of the convolution of the true distribution with the migration matrix.

Open Access. This article is distributed under the terms of the Creative Commons

Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in

any medium, provided the original author(s) and source are credited.

References

[1] BaBar collaboration, Evidence for an excess of B̄ → D(∗)τ−ν̄τ decays, Phys. Rev. Lett. 109(2012) 101802 [arXiv:1205.5442] [INSPIRE].

[2] BaBar collaboration, Measurement of an Excess of B̄ → D(∗)τ−ν̄τ Decays and Implicationsfor Charged Higgs Bosons, Phys. Rev. D 88 (2013) 072012 [arXiv:1303.0571] [INSPIRE].

[3] Belle collaboration, Measurement of the branching ratio of B̄ → D(∗)τ−ν̄τ relative toB̄ → D(∗)`−ν̄` decays with hadronic tagging at Belle, Phys. Rev. D 92 (2015) 072014[arXiv:1507.03233] [INSPIRE].

– 22 –

https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1103/PhysRevLett.109.101802https://doi.org/10.1103/PhysRevLett.109.101802https://arxiv.org/abs/1205.5442https://inspirehep.net/search?p=find+EPRINT+arXiv:1205.5442https://doi.org/10.1103/PhysRevD.88.072012https://arxiv.org/abs/1303.0571https://inspirehep.net/search?p=find+EPRINT+arXiv:1303.0571https://doi.org/10.1103/PhysRevD.92.072014https://arxiv.org/abs/1507.03233https://inspirehep.net/search?p=find+EPRINT+arXiv:1507.03233

JHEP12(2019)148

[4] Belle collaboration, Measurement of the branching ratio of B̄0 → D∗+τ−ν̄τ relative toB̄0 → D∗+`−ν̄` decays with a semileptonic tagging method, Phys. Rev. D 94 (2016) 072007[arXiv:1607.07923] [INSPIRE].

[5] Belle collaboration, Measurement of the τ lepton polarization and R(D∗) in the decay

B̄ → D∗τ−ν̄τ , Phys. Rev. Lett. 118 (2017) 211801 [arXiv:1612.00529] [INSPIRE].

[6] LHCb collaboration, Measurement of the ratio of branching fractions

B(B̄0 → D∗+τ−ν̄τ )/B(B̄0 → D∗+µ−ν̄µ), Phys. Rev. Lett. 115 (2015) 111803 [Erratum ibid.115 (2015) 159901] [arXiv:1506.08614] [INSPIRE].

[7] LHCb collaboration, Measurement of the ratio of the B0 → D∗−τ+ντ and B0 → D∗−µ+νµbranching fractions using three-prong τ -lepton decays, Phys. Rev. Lett. 120 (2018) 171802

[arXiv:1708.08856] [INSPIRE].

[8] Belle collaboration, Measurement of R(D) and R(D∗) with a semileptonic tagging method,arXiv:1904.08794 [INSPIRE].

[9] M. Tanaka, Charged Higgs effects on exclusive semitauonic B decays, Z. Phys. C 67 (1995)

321 [hep-ph/9411405] [INSPIRE].

[10] S. Davidson, D.C. Bailey and B.A. Campbell, Model independent constraints on leptoquarks

from rare processes, Z. Phys. C 61 (1994) 613 [hep-ph/9309310] [INSPIRE].

[11] W. Buchmüller, R. Ruckl and D. Wyler, Leptoquarks in Lepton-Quark Collisions, Phys. Lett.

B 191 (1987) 442 [Erratum ibid. B 448 (1999) 320] [INSPIRE].

[12] Z. Ligeti, M. Papucci and D.J. Robinson, New Physics in the Visible Final States of

B → D(∗)τν, JHEP 01 (2017) 083 [arXiv:1610.02045] [INSPIRE].

[13] M. Tanaka and R. Watanabe, New physics in the weak interaction of B̄ → D(∗)τ ν̄, Phys.Rev. D 87 (2013) 034028 [arXiv:1212.1878] [INSPIRE].

[14] D. Bečirević, M. Fedele, I. Nǐsandžić and A. Tayduganov, Lepton Flavor Universality tests

through angular observables of B → D(∗)`ν decay modes, arXiv:1907.02257 [INSPIRE].

[15] M. Duraisamy and A. Datta, The Full B → D∗τ−ν̄τ Angular Distribution and CP-violatingTriple Products, JHEP 09 (2013) 059 [arXiv:1302.7031] [INSPIRE].

[16] D. Hill, M. John, W. Ke and A. Poluektov, Model-independent method for measuring the

angular coefficients of B0 → D∗−τ+ντ decays, JHEP 11 (2019) 133 [arXiv:1908.04643][INSPIRE].

[17] A. Ray, S. Sahoo and R. Mohanta, Model independent analysis of B∗ → P`ν̄` decayprocesses, Eur. Phys. J. C 79 (2019) 670 [arXiv:1907.13586] [INSPIRE].

[18] A.K. Alok, D. Kumar, S. Kumbhakar and S. Uma Sankar, Resolution of RD/RD∗ puzzle,

Phys. Lett. B 784 (2018) 16 [arXiv:1804.08078] [INSPIRE].

[19] C. Murgui, A. Peñuelas, M. Jung and A. Pich, Global fit to b→ cτν transitions, JHEP 09(2019) 103 [arXiv:1904.09311] [INSPIRE].

[20] A.K. Alok, D. Kumar, J. Kumar, S. Kumbhakar and S.U. Sankar, New physics solutions for

RD and RD∗ , JHEP 09 (2018) 152 [arXiv:1710.04127] [INSPIRE].

[21] A.K. Alok, D. Kumar, S. Kumbhakar and S. Uma Sankar, New Physics solutions for b→ cτ ν̄anomalies before and after Moriond 2019, arXiv:1903.10486 [INSPIRE].

[22] M. Jung and D.M. Straub, Constraining new physics in b→ c`ν transitions, JHEP 01(2019) 009 [arXiv:1801.01112] [INSPIRE].

– 23 –

https://doi.org/10.1103/PhysRevD.94.072007https://arxiv.org/abs/1607.07923https://inspirehep.net/search?p=find+EPRINT+arXiv:1607.07923https://doi.org/10.1103/PhysRevLett.118.211801https://arxiv.org/abs/1612.00529https://inspirehep.net/search?p=find+EPRINT+arXiv:1612.00529https://doi.org/10.1103/PhysRevLett.115.159901https://arxiv.org/abs/1506.08614https://inspirehep.net/search?p=find+EPRINT+arXiv:1506.08614https://doi.org/10.1103/PhysRevLett.120.171802https://arxiv.org/abs/1708.08856https://inspirehep.net/search?p=find+EPRINT+arXiv:1708.08856https://arxiv.org/abs/1904.08794https://inspirehep.net/search?p=find+EPRINT+arXiv:1904.08794https://doi.org/10.1007/BF01571294https://doi.org/10.1007/BF01571294https://arxiv.org/abs/hep-ph/9411405https://inspirehep.net/search?p=find+EPRINT+hep-ph/9411405https://doi.org/10.1007/BF01552629https://arxiv.org/abs/hep-ph/9309310https://inspirehep.net/search?p=find+EPRINT+hep-ph/9309310https://doi.org/10.1016/S0370-2693(99)00014-3https://doi.org/10.1016/S0370-2693(99)00014-3https://inspirehep.net/search?p=find+J+%22Phys.Lett.,B191,442%22https://doi.org/10.1007/JHEP01(2017)083https://arxiv.org/abs/1610.02045https://inspirehep.net/search?p=find+EPRINT+arXiv:1610.02045https://doi.org/10.1103/PhysRevD.87.034028https://doi.org/10.1103/PhysRevD.87.034028https://arxiv.org/abs/1212.1878https://inspirehep.net/search?p=find+EPRINT+arXiv:1212.1878https://arxiv.org/abs/1907.02257https://inspirehep.net/search?p=find+EPRINT+arXiv:1907.02257https://doi.org/10.1007/JHEP09(2013)059https://arxiv.org/abs/1302.7031https://inspirehep.net/search?p=find+EPRINT+arXiv:1302.7031https://doi.org/10.1007/JHEP11(2019)133https://arxiv.org/abs/1908.04643https://inspirehep.net/search?p=find+EPRINT+arXiv:1908.04643https://doi.org/10.1140/epjc/s10052-019-7183-8https://arxiv.org/abs/1907.13586https://inspirehep.net/search?p=find+EPRINT+arXiv:1907.13586https://doi.org/10.1016/j.physletb.2018.07.001https://arxiv.org/abs/1804.08078https://inspirehep.net/search?p=find+EPRINT+arXiv:1804.08078https://doi.org/10.1007/JHEP09(2019)103https://doi.org/10.1007/JHEP09(2019)103https://arxiv.org/abs/1904.09311https://inspirehep.net/search?p=find+EPRINT+arXiv:1904.09311https://doi.org/10.1007/JHEP09(2018)152https://arxiv.org/abs/1710.04127https://inspirehep.net/search?p=find+EPRINT+arXiv:1710.04127https://arxiv.org/abs/1903.10486https://inspirehep.net/search?p=find+EPRINT+arXiv:1903.10486https://doi.org/10.1007/JHEP01(2019)009https://doi.org/10.1007/JHEP01(2019)009https://arxiv.org/abs/1801.01112https://inspirehep.net/search?p=find+EPRINT+arXiv:1801.01112

JHEP12(2019)148

[23] F.U. Bernlochner, Z. Ligeti, D.J. Robinson and W.L. Sutcliffe, New predictions for Λb → Λcsemileptonic decays and tests of heavy quark symmetry, Phys. Rev. Lett. 121 (2018) 202001

[arXiv:1808.09464] [INSPIRE].

[24] W. Detmold, C. Lehner and S. Meinel, Λb → p`−ν̄` and Λb → Λc`−ν̄` form factors fromlattice QCD with relativistic heavy quarks, Phys. Rev. D 92 (2015) 034503

[arXiv:1503.01421] [INSPIRE].

[25] F.U. Bernlochner, Z. Ligeti, D.J. Robinson and W.L. Sutcliffe, Precise predictions for

Λb → Λc semileptonic decays, Phys. Rev. D 99 (2019) 055008 [arXiv:1812.07593] [INSPIRE].

[26] LHCb collaboration, Measurement of the shape of the Λ0b → Λ+c µ−νµ differential decay rate,Phys. Rev. D 96 (2017) 112005 [arXiv:1709.01920] [INSPIRE].

[27] A. Ray, S. Sahoo and R. Mohanta, Probing new physics in semileptonic Λb decays, Phys.

Rev. D 99 (2019) 015015 [arXiv:1812.08314] [INSPIRE].

[28] X.-Q. Li, Y.-D. Yang and X. Zhang, Λb → Λcτντ decay in scalar and vector leptoquarkscenarios, JHEP 02 (2017) 068 [arXiv:1611.01635] [INSPIRE].

[29] E. Di Salvo, F. Fontanelli and Z.J. Ajaltouni, Detailed Study of the Decay Λb → Λcτ ν̄τ , Int.J. Mod. Phys. A 33 (2018) 1850169 [arXiv:1804.05592] [INSPIRE].

[30] A. Datta, S. Kamali, S. Meinel and A. Rashed, Phenomenology of Λb → Λcτντ using latticeQCD calculations, JHEP 08 (2017) 131 [arXiv:1702.02243] [INSPIRE].

[31] R. Dutta, Λb → (Λc, p) τ ν decays within standard model and beyond, Phys. Rev. D 93(2016) 054003 [arXiv:1512.04034] [INSPIRE].

[32] S. Shivashankara, W. Wu and A. Datta, Λb → Λcτ ν̄τ Decay in the Standard Model and withNew Physics, Phys. Rev. D 91 (2015) 115003 [arXiv:1502.07230] [INSPIRE].

[33] P. Böer, A. Kokulu, J.-N. Toelstede and D. van Dyk, Angular Analysis of

Λb → Λc(→ Λπ)`ν̄, arXiv:1907.12554 [INSPIRE].

[34] N. Penalva, E. Hernández and J. Nieves, Further tests of lepton flavour universality from the

charged lepton energy distribution in b→ c semileptonic decays: The case of Λb → Λc`ν̄`,arXiv:1908.02328 [INSPIRE].

[35] LHCb collaboration, Angular analysis of the B0 → K∗0e+e− decay in the low-q2 region,JHEP 04 (2015) 064 [arXiv:1501.03038] [INSPIRE].

[36] ATLAS collaboration, Measurement of the parity-violating asymmetry parameter αb and the

helicity amplitudes for the decay Λ0b → J/ψ + Λ0 with the ATLAS detector, Phys. Rev. D 89(2014) 092009 [arXiv:1404.1071] [INSPIRE].

[37] LHCb collaboration, Differential branching fraction and angular analysis of Λ0b → Λµ+µ−decays, JHEP 06 (2015) 115 [Erratum ibid. 09 (2018) 145] [arXiv:1503.07138] [INSPIRE].

[38] CMS collaboration, Measurement of the Λb polarization and angular parameters in

Λb → J/ψΛ decays from pp collisions at√s = 7 and 8 TeV, Phys. Rev. D 97 (2018) 072010

[arXiv:1802.04867] [INSPIRE].

[39] BESIII collaboration, Measurements of Weak Decay Asymmetries of Λ+c → pK0S, Λπ+,Σ+π0 and Σ0π+, Phys. Rev. D 100 (2019) 072004 [arXiv:1905.04707] [INSPIRE].

[40] S. Fajfer, J.F. Kamenik, I. Nisandzic and J. Zupan, Implications of Lepton Flavor

Universality Violations in B Decays, Phys. Rev. Lett. 109 (2012) 161801 [arXiv:1206.1872]

[INSPIRE].

– 24 –

https://doi.org/10.1103/PhysRevLett.121.202001https://arxiv.org/abs/1808.09464https://inspirehep.net/search?p=find+EPRINT+arXiv:1808.09464https://doi.org/10.1103/PhysRevD.92.034503https://arxiv.org/abs/1503.01421https://inspirehep.net/search?p=find+EPRINT+arXiv:1503.01421https://doi.org/10.1103/PhysRevD.99.055008https://arxiv.org/abs/1812.07593https://inspirehep.net/search?p=find+EPRINT+arXiv:1812.07593https://doi.org/10.1103/PhysRevD.96.112005https://arxiv.org/abs/1709.01920https://inspirehep.net/search?p=find+EPRINT+arXiv:1709.01920https://doi.org/10.1103/PhysRevD.99.015015https://doi.org/10.1103/PhysRevD.99.015015https://arxiv.org/abs/1812.08314https://inspirehep.net/search?p=find+EPRINT+arXiv:1812.08314https://doi.org/10.1007/JHEP02(2017)068https://arxiv.org/abs/1611.01635https://inspirehep.net/search?p=find+EPRINT+arXiv:1611.01635https://doi.org/10.1142/S0217751X18501695https://doi.org/10.1142/S0217751X18501695https://arxiv.org/abs/1804.05592https://inspirehep.net/search?p=find+EPRINT+arXiv:1804.05592https://doi.org/10.1007/JHEP08(2017)131https://arxiv.org/abs/1702.02243https://inspirehep.net/search?p=find+EPRINT+arXiv:1702.02243https://doi.org/10.1103/PhysRevD.93.054003https://doi.org/10.1103/PhysRevD.93.054003https://arxiv.org/abs/1512.04034https://inspirehep.net/search?p=find+EPRINT+arXiv:1512.04034https://doi.org/10.1103/PhysRevD.91.115003https://arxiv.org/abs/1502.07230https://inspirehep.net/search?p=find+EPRINT+arXiv:1502.07230https://arxiv.org/abs/1907.12554https://inspirehep.net/search?p=find+EPRINT+arXiv:1907.12554https://arxiv.org/abs/1908.02328https://inspirehep.net/search?p=find+EPRINT+arXiv:1908.02328https://doi.org/10.1007/JHEP04(2015)064https://arxiv.org/abs/1501.03038https://inspirehep.net/search?p=find+EPRINT+arXiv:1501.03038https://doi.org/10.1103/PhysRevD.89.092009https://doi.org/10.1103/PhysRevD.89.092009https://arxiv.org/abs/1404.1071https://inspirehep.net/search?p=find+EPRINT+arXiv:1404.1071https://doi.org/10.1007/JHEP06(2015)115https://arxiv.org/abs/1503.07138https://inspirehep.net/search?p=find+EPRINT+arXiv:1503.07138https://doi.org/10.1103/PhysRevD.97.072010https://arxiv.org/abs/1802.04867https://inspirehep.net/search?p=find+EPRINT+arXiv:1802.04867https://doi.org/10.1103/PhysRevD.100.072004https://arxiv.org/abs/1905.04707https://inspirehep.net/search?p=find+EPRINT+arXiv:1905.04707https://doi.org/10.1103/PhysRevLett.109.161801https://arxiv.org/abs/1206.1872https://inspirehep.net/search?p=find+EPRINT+arXiv:1206.1872

JHEP12(2019)148

[41] X.-G. He and G. Valencia, B decays with τ leptons in nonuniversal left-right models, Phys.

Rev. D 87 (2013) 014014 [arXiv:1211.0348] [INSPIRE].

[42] T. Sjöstrand, S. Mrenna and P.Z. Skands, PYTHIA 6.4 Physics and Manual, JHEP 05

(2006) 026 [hep-ph/0603175] [INSPIRE].

[43] T. Sjöstrand, S. Mrenna and P.Z. Skands, A Brief Introduction to PYTHIA 8.1, Comput.

Phys. Commun. 178 (2008) 852 [arXiv:0710.3820] [INSPIRE].

[44] LHCb collaboration, LHCb VELO (VErtex LOcator): Technical Design Report,

CERN-LHCC-2001-011 (2001).

[45] LHCb collaboration, Measurement of b-hadron fractions in 13 TeV pp collisions, Phys. Rev.

D 100 (2019) 031102 [arXiv:1902.06794] [INSPIRE].

[46] H.E. Haber, Spin formalism and applications to new physics searches, in Spin structure in

high-energy processes: Proceedings, 21st SLAC Summer Institute on Particle Physics,

26 July–6 August 1993, Stanford, CA, pp. 231–272 (1994) [hep-ph/9405376] [INSPIRE].

[47] P.R. Auvil and J.J. Brehm, Wave Functions for Particles of Higher Spin, Phys. Rev. 145

(1966) 1152 [INSPIRE].

[48] T. Feldmann and M.W.Y. Yip, Form Factors for Λb → Λ Transitions in SCET, Phys. Rev.D 85 (2012) 014035 [Erratum ibid. D 86 (2012) 079901] [arXiv:1111.1844] [INSPIRE].

[49] A. Kadeer, J.G. Korner and U. Moosbrugger, Helicity analysis of semileptonic hyperon decays

including lepton mass effects, Eur. Phys. J. C 59 (2009) 27 [hep-ph/0511019] [INSPIRE].

[50] T. Bhattacharya et al., Probing Novel Scalar and Tensor Interactions from (Ultra)Cold

Neutrons to the LHC, Phys. Rev. D 85 (2012) 054512 [arXiv:1110.6448] [INSPIRE].

– 25 –

https://doi.org/10.1103/PhysRevD.87.014014https://doi.org/10.1103/PhysRevD.87.014014https://arxiv.org/abs/1211.0348https://inspirehep.net/search?p=find+EPRINT+arXiv:1211.0348https://doi.org/10.1088/1126-6708/2006/05/026https://doi.org/10.1088/1126-6708/2006/05/026https://arxiv.org/abs/hep-ph/0603175https://inspirehep.net/search?p=find+EPRINT+hep-ph/0603175https://doi.org/10.1016/j.cpc.2008.01.036https://doi.org/10.1016/j.cpc.2008.01.036https://arxiv.org/abs/0710.3820https://inspirehep.net/search?p=find+EPRINT+arXiv:0710.3820http://cds.cern.ch/record/504321https://doi.org/10.1103/PhysRevD.100.031102https://doi.org/10.1103/PhysRevD.100.031102https://arxiv.org/abs/1902.06794https://inspirehep.net/search?p=find+EPRINT+arXiv:1902.06794https://arxiv.org/abs/hep-ph/9405376https://inspirehep.net/search?p=find+EPRINT+hep-ph/9405376https://doi.org/10.1103/PhysRev.145.1152https://doi.org/10.1103/PhysRev.145.1152https://inspirehep.net/search?p=find+J+%22Phys.Rev.,145,1152%22https://doi.org/10.1103/PhysRevD.85.014035https://doi.org/10.1103/PhysRevD.85.014035https://arxiv.org/abs/1111.1844https://inspirehep.net/search?p=find+EPRINT+arXiv:1111.1844https://doi.org/10.1140/epjc/s10052-008-0801-5https://arxiv.org/abs/hep-ph/0511019https://inspirehep.net/search?p=find+EPRINT+hep-ph/0511019https://doi.org/10.1103/PhysRevD.85.054512https://arxiv.org/abs/1110.6448https://inspirehep.net/search?p=find+EPRINT+arXiv:1110.6448

IntroductionEffective Lagrangian for b –> c l nuDecay amplitudeDecay densityExperimental sensitivityConclusionsHadronic amplitudes of Lambda**(0) (b) –> Lambda**(+) (c) W**(* - ) decayLeptonic amplitudes of W**(* - ) –> l**(-) overlinenu(l) decayHadronic amplitudes of lambda**(+) (c) –> p K**(0) (S) decayPhase spaceTerms of differential decay densityDecay density for two considered casesResponse matrix