Transition form factors and angular distributions of the b1520 NKdecay supported by baryon spectroscopy Yu-Shuai Li12Su-Ping Jin3yJing Gao3zand Xiang Liu1245x

2025-05-06 0 0 2.79MB 23 页 10玖币

侵权投诉

Transition form factors and angular distributions of the Λb→Λ(1520)(→N¯

K)`+`−decay

supported by baryon spectroscopy

Yu-Shuai Li1,2,∗Su-Ping Jin3,†Jing Gao3,‡and Xiang Liu1,2,4,5§

1School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, China

2Research Center for Hadron and CSR Physics, Lanzhou University and Institute of Modern Physics of CAS, Lanzhou 730000, China

3School of Physics, Nankai University, Tianjin 300071, China

4Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics of Gansu Province,

and Frontiers Science Center for Rare Isotopes, Lanzhou University, Lanzhou 730000, China

5Key Laboratory of Quantum Theory and Applications of MoE, Lanzhou University, Lanzhou 730000, China

We calculate the weak transition form factors of the Λb→Λ(1520) transition, and further calculate the

angular distributions of the rare decays Λb→Λ(1520)(→N¯

K)`+`−(N¯

K={pK−,n¯

K0}) with unpolarized Λb

and massive leptons. The form factors are calculated by the three-body light-front quark model with the support

of numerical wave functions of Λband Λ(1520) from solving the semirelativistic potential model associated

with the Gaussian expansion method. By ﬁtting the mass spectrum of the observed single bottom and charmed

baryons, the parameters of the potential model are ﬁxed, so this strategy can avoid the uncertainties arising from

the choice of a simple harmonic oscillator wave function of the baryons. With more data accumulated in the

LHCb experiment, our result can help for exploring the Λb→Λ(1520)`+`−decay and deepen our understanding

on the b→s`+`−processes.

I. INTRODUCTION

The ﬂavor-changing neutral-current (FCNC) processes, in-

cluding the high-proﬁle b→s`+`−process, can play a cru-

cial role in indirect searches for physics beyond the Stan-

dard Model (SM). These transitions are forbidden at the tree

level and can only operate through loop diagrams in the SM,

and are therefore highly sensitive to potential new physics

(NP) eﬀects, such as the much-discussed RD(∗)=B(B→

D(∗)τντ)/B(B→D(∗)e(µ)νe(µ)) [1–4]. These processes thus

provided a unique platform to deepen our understanding of

both quantum chromodynamics (QCD) and the dynamics of

weak processes, and to help hunt for NP signs. Therefore,

the rare decays of b→shave attracted the attention of both

theorists and experimentalists [5–11].

For example, the rare decay Λb→Λ`+`−has been the-

oretically studied by various approaches, including lattice

QCD (LQCD) [12,13], QCD sum rules [14], light-cone

sum rule [15–19], covariant quark model [20], nonrelativistic

quark model [21,22], and the Bethe-Salpeter approach [23],

etc., and was ﬁrst measured by the CDF Collaboration [7]

and later by the LHCb Collaboration [8,9]. In addition to

the diﬀerential branching ratio, such abundant phenomenolo-

gies of various angular distributions have also been studied.

Compared with the measured data, the angular distribution of

Λb→Λ`+`−was studied in Refs. [24,25] with unpolarized

Λbbaryon, and with polarized Λbbaryon in Ref. [26]. Fur-

thermore, the authors studied the b→sµ+µ−Wilson coeﬃ-

cients in Ref. [27] using the measured full angular distribution

of the rare decay Λb→Λ(→pπ)µ+µ−by the LHCb Collabo-

ration [9].

∗Electronic address: liysh20@lzu.edu.cn

†Electronic address: jinsuping@nankai.edu.cn

‡Electronic address: 9820210055@nankai.edu.cn

§Electronic address: xiangliu@lzu.edu.cn

With the previous experiences on the decay to the ground

state Λ, it is therefore worth to further testing the b→s`+`−

transition in the baryon sector decaying to the excited hy-

peron with quantum number being JP=3/2−. The form

factors of the weak transition were calculated by the quark

model [21,22], LQCD [28,29], and the heavy quark expan-

sion [30]. The angular analysis was performed in Ref. [31]

and Ref. [32] for massless and massive leptons, respectively.

The authors of Ref. [33] studied the kinematic endpoint rela-

tions for Λb→Λ(1520)`+`−decays and provided the corre-

sponding angular distributions. Amhis et al. [34] used the dis-

persive techniques to provide a model-independent parameter-

ization of the form factors of Λb→Λ(1520) and further inves-

tigated the FCNC decay Λb→Λ(1520)`+`−with the LQCD

data. In addition, Xing et al. also studied the multibody decay

Λb→Λ∗

J(→pK−)J/ψ(→`+`−) [35]. In addition, Amhis et

al. studied the angular distributions of Λb→Λ(1520)`+`−

and talked about the potential to identify NP eﬀects [36]. Ob-

viously, the Λb→Λ(1520) is less studied. Following this

line, we further study the Λb→Λ(1520)(→N¯

K)`+`−with

the N¯

K={pK−,n¯

K0}process and investigate the correspond-

ing angular observables.

From a theoretical point of view, apart from the consider-

ation of new operators beyond the SM, the calculation of the

weak transition form factors is a key issue. In addition, how to

solve the three-body system for the Λbbaryon and Λ∗hyperon

involved is also a challenge. In previous work on baryon weak

decays [37–41], the quark-diquark scheme has been widely

adopted as an approximate treatment. Meanwhile, the spa-

tial wave functions of hadrons are often approximated as sim-

ple harmonic oscillator (SHO) wave functions [37–43], which

makes the results dependent on the relevant parameters. To

avoid the correlative uncertainties of the above approxima-

tions, in this work we calculate the Λb→Λ∗form factors

by the three-body light-front quark model. Moreover, in the

realistic calculation, we take the numerical spatial wave func-

tions as input, where the semirelativistic potential model com-

bined with the Gaussian expansion method (GEM) [44–47] is

arXiv:2210.04640v4 [hep-ph] 1 May 2023

adopted. By ﬁtting the mass spectrum of the observed single

bottom and charmed baryons, the parameters of the semirela-

tivistic potential model can be ﬁxed. Compared with the SHO

wave function approximation, our strategy can avoid the un-

certainties arising from the selection of the spatial wave func-

tions of the baryons.

The structure of this paper is as follows. After the Intro-

duction, we derive the helicity amplitudes of Λb→Λ∗(→

N¯

K)`+`−(N¯

K={pK−,n¯

K0}) processes and deﬁne some an-

gular observables with unpolarized Λbbaryons and massive

leptons in Sec. II. The formulas for the weak transition form

factors are derived in the three-body light-front quark model in

Sec. III. And then, to obtain the spatial wave functions of the

involved baryons, the applied semirelativistic potential model

and GEM are brieﬂy introduced in Sec. IV. In Sec. V, we

present our numerical results, including both the relevant form

factors and the physical observables in Λb→Λ∗(→N¯

K)`+`−

decays. Finally, this paper ends with a short summary in

Sec. VI.

II. THE ANGULAR DISTRIBUTION OF

Λb→Λ∗(→N¯

K)`+`−

In this paper, we use a model-independent approach with

the eﬀective Hamiltonian[48,49]

Heﬀ(b→s`+`−)=−4GF

√2VtbV∗

i=1Ci(µ)Oi(µ) (2.1)

to study the b→s`+`−process, where GF=1.16637 ×

10−5GeV−2is the Fermi coupling constant and |VtbV∗

ts|=

0.04088 [12] is the product of the Cabibbo-Kobayashi-

Maskawa matrix elements. Furthermore, the Wilson coeﬃ-

cients Ci(µ) describe the short-distance physics, while the four

fermion operators Oi(µ) describe the long-distance physics,

where O1,2are the current-current operators, O3−6are the

QCD penguin operators, O7,8denote the electromagnetic and

chromomagnetic penguin operators respectively, and O9,10

stand for the semileptonic operators.

In our calculation, we follow the treatment given in Refs. [6,

23], adding the factorable quark-loop contributions from O1−6

and O8to the eﬀective Wilson coeﬃcients Ceﬀ

7and Ceﬀ

9. The

eﬀective Hamiltonian can be written as

Heﬀ(b→s`+`−)=−4GF

√2VtbV∗

αe

4π(¯s"Ceﬀ

9(µ, q2)γµPL

−2mb

q2Ceﬀ

7(µ)iσµνqνPR#b(¯

`γµ`)

+C10(µ)( ¯sγµPLb)( ¯

`γµγ5`)),

(2.2)

where PR(L)=(1 ±γ5)/2 and σµν =i[γµ, γν]/2. The elec-

tromagnetic coupling constant is αe=1/137. For the lead-

ing logarithmic approximation, we take mb=4.80 GeV [50,

51] and the Wilson coeﬃcients as Ceﬀ

7(mb)=−0.313 and

C10(mb)=−4.669 in the calculation [50–53]. In addition,

the short-distance contributions from the soft-gluon emission

and the one-loop contributions of the four-quark operators O1-

O6, and the long-distance eﬀects due to the charmonium reso-

nances, J/ψ and ψ(2S) are taken into account, where we adopt

the Ceﬀ

9(µ, q2) as [50,54,55]

Ceﬀ

9(µ, q2)=C9(µ)+Ypert( ˆs)+Yres(q2).(2.3)

The Ypert term can be written as

Ypert( ˆs)=g( ˆmc,ˆs)C(µ)

−1

2g(1,ˆs)(4C3(µ)+4C4(µ)+3C5(µ)+C6(µ))

−1

2g(0,ˆs)(C3(µ)+3C4(µ))

9(3C3(µ)+C4(µ)+3C5(µ)+C6(µ)),

(2.4)

where ˆmc=mc/mb, ˆs=q2/m2

b,C(µ)=3C1(µ)+C2(µ)+

3C3(µ)+C4(µ)+3C5(µ)+C6(µ), and [50]

g(z,ˆs)=−8

9ln z+8

27 +4

9x−2

9(2 +x)p|1−x|

×(ln |1+√1−x

1−√1−x| − iπfor x≡4z2/ˆs<1

2 arctan 1

√x−1for x≡4z2/ˆs>1,

g(0,ˆs)=8

27 −8

9ln mb

µ−4

9ln ˆs+4

9iπ.

(2.5)

The Wilson coeﬃcients are used as C1(mb)=−0.248,

C2(mb)=1.107, C3(mb)=0.011, C4(mb)=−0.026,

C5(mb)=0.007, and C6(mb)=−0.031 [50]. Besides,

mc=1.4 GeV [50]. The Yres term can be parametrized by

using the Breit-Wigner ansatz (it is a model-dependent treat-

ment, and one can refer to Refs. [56,57] for more detailed

discussions) as [51]

Yres(q2)=3π

α2

C(0) X

Vi=J/ψ,ψ(2S)

κVi

Γ(Vi→`+`−)mVi

Vi−q2−imViΓVi

,(2.6)

where C(0) =0.362, κJ/ψ =1, and κψ(2S)=2. The masses

and total widths associated with the relevant charmonium res-

onances are taken to be 3.096 GeV and 92.9 keV for J/ψ, and

3.686 GeV and 294 keV for ψ(2S) [58]. The decay widths are

taken as Γ(J/ψ →`+`−)=5.53 keV and Γ(ψ(2S)→`+`−)=

2.33 keV [58].

Since the quarks are conﬁned in hadron, the weak transi-

tion matrix element cannot be calculated in the framework of

perturbative QCD. They are conventionally parametrized in

terms of eight (axial-)vector and six (pseudo-)tensor type di-

mensionless form factors [21,25,31,32,59–61]. In this work,

we adopt the helicity-based form as [31,32]

hΛ∗(k,sΛ∗)|¯sγµb|Λb(p,sΛb)i=¯uα(k,sΛ∗)(pα"fV

t(q2)(mΛb−mΛ∗)qµ

+fV

0(q2)mΛb+mΛ∗

s+ pµ+kµ−(m2

Λb−m2

Λ∗)qµ

q2!

+fV

⊥(q2) γµ−2mΛ∗

pµ−2mΛb

kµ!#

+fV

g(q2)"gαµ +mΛ∗

pα

s− γµ−2kµ

mΛ∗

+2(mΛ∗pµ+mΛbkµ)

s+!#)u(p,sΛb),

(2.7)

hΛ∗(k,sΛ∗)|¯sγµγ5b|Λb(p,sΛb)i=−¯uα(k,sΛ∗)γ5(pα"fA

t(q2)(mΛb+mΛ∗)qµ

+fA

0(q2)mΛb−mΛ∗

s− pµ+kµ−(m2

Λb−m2

Λ∗)qµ

q2!

+fA

⊥(q2) γµ+2mΛ∗

s−

pµ−2mΛb

s−

kµ!#

+fA

g(q2)"gαµ −mΛ∗

pα

s+ γµ+2kµ

mΛ∗−2(mΛ∗pµ−mΛbkµ)

s−!#)u(p,sΛb),

(2.8)

hΛ∗(k,sΛ∗)|¯siσµνqνb|Λb(p,sΛb)i=−¯uα(k,sΛ∗)(pα"fT

0(q2)q2

s+ pµ+kµ−(m2

Λb−m2

Λ∗)qµ

q2!

+fT

⊥(q2)(mΛb+mΛ∗) γµ−2mΛ∗

pµ−2mΛb

kµ!#

+fT

g(q2)"gαµ +mΛ∗

pα

s− γµ−2kµ

mΛ∗

+2(mΛ∗pµ+mΛbkµ)

s+!#)u(p,sΛb),

(2.9)

hΛ∗(k,sΛ∗)|¯siσµνqνγ5b|Λb(p,sΛb)i=−¯uα(k,sΛ∗)γ5(pα"fT5

0(q2)q2

s− pµ+kµ−(m2

Λb−m2

Λ∗)qµ

q2!

+fT5

⊥(q2)(mΛb−mΛ∗) γµ+2mΛ∗

s−

pµ−2mΛb

s−

kµ!#

+fT5

g(q2)"gαµ −mΛ∗

pα

s+ γµ+2kµ

mΛ∗−2(mΛ∗pµ−mΛbkµ)

s−!#)u(p,sΛb).

(2.10)

This form deﬁned above is convenient for calculating the corresponding helicity amplitudes, where q2is the transferred momen-

tum square and s±=(mΛb±mΛ∗)2−q2.

A. The helicity amplitudes of the Λb→Λ∗`+`−decay

To calculate the Λb→Λ∗`+`−process, we deﬁne the corresponding helicity amplitudes of the Λb(sΛb)→Λ∗(sΛ∗) transition

H(V,A,T,T5)(sΛb,sΛ∗, λW)=∗

µ(λW)hΛ∗(sΛ∗)|¯sγµ, γµγ5,iσµνqν,iσµνqνγ5b|Λb(sΛb)i,(2.11)

where µ(λW=t,±,0) are the polarization vectors of the virtual gauge boson in the Λbrest frame, sΛband sΛ∗are the polarizations

of Λband Λ∗, respectively. For the vector current, the complete helicity amplitudes HV(sΛb,sΛ∗, λW) read [31]

HV(sΛb,sΛ∗,t)=∗

µ(t)hΛ∗(k,sΛ∗)|¯sγµb|Λb(p,sΛb)i

=fV

t(q2)mΛb−mΛ∗

pq2¯uα(k,sΛ∗)pαu(p,sΛb),(2.12)

HV(sΛb,sΛ∗,0) =∗

µ(0)hΛ∗(k,sΛ∗)|¯sγµb|Λb(p,sΛb)i

=2fV

0(q2)mΛb+mΛ∗

k·∗(0)¯uα(k,sΛ∗)pαu(p,sΛb),(2.13)

HV(sΛb,sΛ∗,±)=∗

µ(±)hΛ∗(k,sΛ∗)|¯sγµb|Λb(p,sΛb)i

=fV

⊥(q2)+fV

g(q2)mΛ∗

s−¯uα(k,sΛ∗)pα/

∗(±)u(p,sΛb)

+fV

g(q2)¯uα(k,sΛ∗)∗α(±)u(p,sΛb).(2.14)

Analogous expressions for the helicity amplitudes of the axial-vector, tensor, and pseudotensor currents are written as

HA(sΛb,sΛ∗,t)=∗

µ(t)hΛ∗(k,sΛ∗)|¯sγµγ5b|Λb(p,sΛb)i

=−fA

t(q2)mΛb+mΛ∗

pq2¯uα(k,sΛ∗)pαγ5u(p,sΛb),(2.15)

HA(sΛb,sΛ∗,0) =∗

µ(0)hΛ∗(k,sΛ∗)|¯sγµγ5b|Λb(p,sΛb)i

=−2fA

0(q2)mΛb−mΛ∗

s−

k·∗(0)¯uα(k,sΛ∗)pαγ5u(p,sΛb),(2.16)

HA(sΛb,sΛ∗,±)=∗

µ(±)hΛ∗(k,sΛ∗)|¯sγµγ5b|Λb(p,sΛb)i

=fA

⊥(q2)−fA

g(q2)mΛ∗

s+¯uα(k,sΛ∗)pα/

∗(±)γ5u(p,sΛb)

−fA

g(q2)¯uα(k,sΛ∗)∗α(±)γ5u(p,sΛb),(2.17)

HT(sΛb,sΛ∗,0) =∗

µ(0)hΛ∗(k,sΛ∗)|¯siσµνqνb|Λb(p,sΛb)i

=−2fT

0(q2)q2

k·ε∗(0)¯uα(k,sΛ∗)pαu(p,sΛb),(2.18)

HT(sΛb,sΛ∗,±)=∗

µ(±)hΛ∗(k,sΛ∗)|¯siσµνqνb|Λb(p,sΛb)i

=−fT

⊥(q2)(mΛb+mΛ∗)+fT

g(q2)mΛ∗

s−¯uα(k,sΛ∗)pα/

∗(±)u(p,sΛb)

−fT

g(q2)¯uα(k,sΛ∗)∗α(±)u(p,sΛb),(2.19)

HT5(sΛb,sΛ∗,0) =∗

µ(0)hΛ∗(k,sΛ∗)|¯siσµνqνγ5b|Λb(p,sΛb)i

=−2fT5

0(q2)q2

s−

k·∗(0)¯uα(k,sΛ∗)pαγ5u(p,sΛb),(2.20)

HT5(sΛb,sΛ∗,±)=∗

µ(±)hΛ∗(k,sΛ∗)|¯siσµνqνγ5b|Λb(p,sΛb)i

=fT5

⊥(q2)(mΛb−mΛ∗)−fT5

g(q2)mΛ∗

s+¯uα(k,sΛ∗)pα/

∗(±)γ5u(p,sΛb)

−fT5

g(q2)¯uα(k,sΛ∗)∗α(±)γ5u(p,sΛb),(2.21)

respectively. Using the kinematic conventions presented in Appendix B 1, the nonzero terms for the above helicity amplitudes

of the vector, axial-vector, tensor, and pseudotensor currents are [31]

HV(+1/2,+1/2,t)=HV(−1/2,−1/2,t)=fV

t(q2)mΛb−mΛ∗

pq2

s+√s−

√6mΛ∗

HV(+1/2,+1/2,0) =HV(−1/2,−1/2,0) =−fV

0(q2)mΛb+mΛ∗

pq2

s−√s+

√6mΛ∗

HV(+1/2,−1/2,+)=HV(−1/2,+1/2,−)=−fV

⊥(q2)s−√s+

√3mΛ∗

HV(−1/2,−3/2,+)=HV(+1/2,+3/2,−)=fV

g(q2)√s+,

(2.22)

HA(+1/2,+1/2,t)=−HA(−1/2,−1/2,t)=fA

t(q2)mΛb+mΛ∗

pq2

s−√s+

√6mΛ∗

HA(+1/2,+1/2,0) =−HA(−1/2,−1/2,0) =−fA

0(q2)mΛb−mΛ∗

pq2

s+√s−

√6mΛ∗

HA(+1/2,−1/2,+)=−HA(−1/2,+1/2,−)=fA

⊥(q2)s+√s−

√3mΛ∗

HA(−1/2,−3/2,+)=−HA(+1/2,+3/2,−)=−fA

g(q2)√s−,

(2.23)

HT(+1/2,+1/2,0) =HT(−1/2,−1/2,0) =fT

0(q2)qq2s−√s+

√6mΛ∗

HT(+1/2,−1/2,+)=HT(−1/2,+1/2,−)=fT

⊥(q2)(mΛb+mΛ∗)s−√s+

√3mΛ∗

HT(−1/2,−3/2,+)=HT(+1/2,+3/2,−)=−fT

g(q2)√s+,

(2.24)

HT5(+1/2,+1/2,0) =−HT5(−1/2,−1/2,0) =−fT5

0(q2)qq2s+√s−

√6mΛ∗

HT5(+1/2,−1/2,+)=−HT5(−1/2,+1/2,−)=fT5

⊥(q2)(mΛb−mΛ∗)s+√s−

√3mΛ∗

HT5(−1/2,−3/2,+)=−HT5(+1/2,+3/2,−)=−fT5

g(q2)√s−,

(2.25)

respectively.

Similarly, we deﬁne the leptonic helicity amplitudes as

L(V,A)(s`−,s`+, λW)=¯µ(λW)h`−`+|¯

`−γµ, γµγ5`+|0i

=¯µ(λW)¯u(p`,s`−)γµ, γµγ5v(−p`,s`+),(2.26)

where ¯µ(λW=t,±,0) are the polarization vectors of the virtual gauge boson in the dilepton rest frame. Using the kinematic

conventions presented in Appendix B 2, the nonzero terms are obtained as [32]

LV(±1/2,±1/2,0) =±2m`cos θ`,LV(±1/2,∓1/2,0) =−qq2sin θ`,

LV(+1/2,+1/2,±)=∓√2m`sin θ`,LV(−1/2,−1/2,∓)=∓√2m`sin θ`,

LV(±1/2,∓1/2,±)=∓pq2

√2(1 +cos θ`),LV(±1/2,∓1/2,∓)=∓pq2

√2(1 −cos θ`),

LA(±1/2,±1/2,t)=−2m`,LA(±1/2,∓1/2,0) =∓sin θ`qq2β`,

LA(±1/2,∓1/2,±)=−pq2

√2(1 +cos θ`)β`,LA(±1/2,∓1/2,∓)=−pq2

√2(1 −cos θ`)β`,

(2.27)

where β`≡q1−4m2

`/q2.

B. The helicity amplitudes of the Λ∗→N¯

K decay

We use the eﬀective Lagrangian approach to describe the strong decay process Λ∗→N¯

K. The concerned eﬀective Lagrangian

LΛ∗KN =gΛ∗KN ¯

Nγ5Λ∗

α∂αK,(2.28)

where gΛ∗KN is the coupling constant. So the decay amplitude for the Λ∗→N¯

Kprocess can be expressed as

MΛ∗→N¯

K(sΛ∗,sN)=gΛ∗KN ¯uN(sN)γ5uΛ∗,α(sΛ∗)kα

2,(2.29)

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

Transitionformfactorsandangulardistributionsoftheb!(1520)(!NK)`+`decaysupportedbybaryonspectroscopyYu-ShuaiLi1;2,Su-PingJin3,yJingGao3,zandXiangLiu1;2;4;5x1SchoolofPhysicalScienceandTechnology,LanzhouUniversity,Lanzhou730000,China2ResearchCenterforHadronandCSRPhysics,LanzhouUniversityandInstitut...

展开>> 收起<<

Transition form factors and angular distributions of the b1520 NKdecay supported by baryon spectroscopy Yu-Shuai Li12Su-Ping Jin3yJing Gao3zand Xiang Liu1245x.pdf

共23页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Transition form factors and angular distributions of the b1520 NKdecay supported by baryon spectroscopy Yu-Shuai Li12Su-Ping Jin3yJing Gao3zand Xiang Liu1245x

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: