Page 1

arXiv:1107.0630v2 [hep-ph] 6 Nov 2011

B → K1ℓ+ℓ−Decays in a Family Non-universal Z′Model

Ying Li∗, Juan Hua

Department of Physics, Yantai University, Yantai 264-005, China

Kwei-Chou Yang

Department of Physics, Chung Yuan Christian University, Chung-Li, Taiwan 320, Republic of China

(Dated: November 8, 2011)

The implications of the family non-universal Z′model in the B → K1(1270,1400)ℓ+ℓ−(ℓ = e,µ ,τ) decays

are explored, where the mass eigenstates K1(1270,1400) are the mixtures of1P1and3P1states with the mixing

angle θ. In this work, considering the Z′boson and setting the mixing angle θ = (−34±13)◦, we analyze the

branching ratio, the dilepton invariant mass spectrum, the normalized forward-backward asymmetry and lepton

polarization asymmetries of each decay mode. We find that all observables of B→K1(1270)µ+µ−are sensitive

to the Z′contribution. Moreover, the observables of B → K1(1400)µ+µ−are relatively strong θ-dependence;

thus, the Z′contribution will be buried by the uncertainty of the mixing angle θ. Furthermore, the zero crossing

position in the FBA spectrum of B → K1(1270)µ+µ−at low dilepton mass will move to the positive direction

with Z′contribution. For the tau modes, the effects of Z′are not remarkable due to the small phase space. These

results could be tested in the running LHC-b experiment and Super-B factory.

I.INTRODUCTION

The flavor changing neutral currents (FCNC) b → sℓ+ℓ−(ℓ = e,µ,τ), forbidden in the standard model (SM) at the tree level,

are very sensitive to the flavor structure of the SM and to the new physics (NP) beyond the SM. The rare decays B → K1ℓ+ℓ−

involving axial-vector strange mesons, also induced by b → sℓ+ℓ−, have been the subjects of many theoretical studies in the

frame work of the SM [1–4] and some NP models, such as universal extra dimension [5], models involving supersymmetry [6]

and the fourth-generation fermions [7]. Generally, these semileptonic decays provide us with a wealth of information with a

number of physical observables, such as branching ratio, dilepton invariant mass spectrum, the forward backward asymmetry,

lepton polarization asymmetry and other distributions of interest, which play important roles in testing SM and are regarded as

probes of possible NP models.

In the quark model, two lowest nonets of JP= 1+axial-vector mesons are usually expected to be the orbitally excited q¯ q′

states. In the context of the spectroscopic notation n2S+1LJ, where the radial excitation is denoted by the principal number n,

there are two types of lowest p-wave meson, namely,13P1and11P1. The two nonets have distinctiveC quantumnumbers,C =+

orC = −, respectively. Experimentally, the JPC= 1++nonet consists of a1(1260), f1(1285), f1(1420), and K1A, while the 1+−

nonet contains b1(1235), h1(1170), h1(1380) and K1B. The physical mass eigenstates K1(1270) and K1(1400) are mixtures of

K1Aand K1Bstates owing to the mass differenceof the strange and non-strangelight quarks, and the relation could be written as:

?|K1(1270)?

|K1(1400)?

?

= M

?|K1A?

|K1B?

?

, with M =

?sinθ

cosθ

cosθ

−sinθ

?

.

(1)

∗liying@ytu.edu.cn

Page 2

2

In the past few years, many attempts have been made to constrain the mixing angle θ [8–11]. In this study, we will use

θ = −(34±13)◦for numerical calculations, which has been extracted from B → K1(1270)γ and τ → K1(1270)ντby one of us

in [11] , and the minus sign is related to the chosen phase of |K1A? and |K1B?.

To make predictions of these exclusive decays, one requires the additional knowledge about form factors, i.e., the matrix

elements of the effective Hamiltonian between initial and final states. This problem, being a part of the nonperturbative sector

of QCD, lacks a precise solution. To the best of our knowledge, a number of different approaches had been used to calculate

the decay form factors of B → K1decays, such as QCD sum rules [12], light cone sum rules (LCSRs) [13], perturbative QCD

approach [14] and light front quark model [15]. Among them, the results obtained by LCSRs which deal with form factors at

small momentum region, are complementary to the lattice approach and have consistence with perturbative QCD and the heavy

quark limit. On this point, we will use the results of LCSRs [13] in this work.

In some new physics models, Z′gauge boson could be naturally derived in certain string constructions [16] and E6models

[17] by adding additional U(1)′gauge symmetry [18]. Among many Z′models, the simplest one is the family non-universal

Z′model. It is of interest to note that in such a model the non-universal Z′couplings could lead to FCNCs at tree level as well

as introduce new weak phases, which could explain the CP asymmetries in the current high energy experiments. The effects

of Z′in the B sector have been investigated in the literature, for example see Ref. [19–21]. The recent detailed review is Ref.

[22]. In Ref. [21], Chang et.al obtained the explicit picture of Z′couplings with the data of¯Bs−Bsmixing, B → K(∗)ℓ+ℓ−,

B → µ+µ−, B → Kπ and inclusive decays B → Xsℓ+ℓ−. So, it should be interesting to explore the discrepancy of observables

between predictions of SM and those of the family non-universalZ′model. Motivated by this, we shall address the effects of the

Z′boson in the rare decays B → K1ℓ+ℓ−.

In experiments, B → K1ℓ+ℓ−have not yet been measured, but are expected to be observed at LHC-b [23] and Super-B factory

[24]. In particular, it is estimated that there will be almost 8000 B → K∗µ+µ−events with an integrated luminosity of 2fb−1in

the LHC-b experiment[23, 25]. Althoughthe branchingratio of B→K1(1270)µ+µ−calculated in [1] is one orderof magnitude

smaller than the experimentally measured value of B → K∗µ+µ−[26], we still expect the significant number of events for this

decay.

The remainder of this paper is organized as follows: in section 2, we introduce the effective Hamiltonian responsible for

the b → sℓ+ℓ−transition in both SM and Z′model. Using the effective Hamiltonian and B → K1form factors, we obtain the

branchingratios as well as variousrelated physical observables. In section 3, we numericallyanalyze the consideredobservables

of B → K1ℓ+ℓ−. This section also includes a comparisonof the results obtained in Z′model with those predicted by the SM. We

will summarize this work in the last section.

II. ANALYTIC FORMULAS

A.The Effective Hamiltonian for b → s transition in SM

By integrating out the heavy degrees of freedom including top quark,W±and Z bosons above scale µ = O(mb), the effective

Hamiltonian responsible for the b → sℓ+ℓ−transitions is given as [27, 28]:

Hef f(b → sℓ+ℓ−) = −GF

2√2VtbV∗

ts

10

∑

i=1

Ci(µ)Oi(µ),

(2)

Page 3

3

where we have neglected the terms proportionaltoVubV∗

in [27]. Specifically, the operators O9and O10are given as

uson account of |VubV∗

us/VtbV∗

ts| <0.02. The local operators can be found

O9=e2

g2s

(¯ sγµPLb)(¯ℓγµℓ),

O10=e2

g2s

(¯ sγµPLb)(¯ℓγµγ5ℓ).

(3)

In SM, the Wilson coefficients Ciat scale µ = mbcalculated in the naive dimensional regularization (NDR) scheme [27] are

collected in Table I. It should be stressed that for b → sℓ+ℓ−processes, the quark decay amplitude can also receive additional

TABLE I: The SM Wilson coefficients at the scale µ = mb.

C1(mb)

C2(mb)

C3(mb)

C4(mb)

C5(mb)

C6(mb)

Ceff

7(mb)

−0.302

Ceff

9(mb)−Y(q2)

4.094

C10(mb)

−0.2741.007

−0.0040.0760.0000.001

−4.193

contributions from the matrix element of four-quark operators,

6

∑

i=1?ℓ+ℓ−s|Oi|b?, which are usually absorbed into the effective

7,9in Table I are defined respectively as [29]Wilson coefficients. The effective coefficientsCeff

Ceff

7=4π

αsC7−1

3C3−4

9C4−20

3C5−80

9C6,

Ceff

9=4π

αsC9+YSD(z, ˆ s)+YLD(z, ˆ s),

(4)

with definitions z = mc/mb, ˆ s = q2/m2

from the c¯ c resonances regions, which can be calculated reliably in the perturbative theory. On the contrary, the long-distance

b. YSD(z, ˆ s) represents the short-distance contributions from four-quarkoperators far away

contributions YLD(z, ˆ s) from four-quark operators near the c¯ c resonances cannot be calculated and are usually parameterized

in the form of a phenomenological Breit-Wigner formula. Currently, the light-cone distribution amplitudes of the axial-vector

mesonsactuallyhavenotyet beenwell studied,since contributionsoftwo axial-vectormesonsin thehadronicdispersionrelation

cannot be separated in all cases. Moreover, the width effect of axial-vector meson is so large that the traditional approach like

the sum rules cannot deal with it effectively. The manifest expressions and discussions for YSD(z, ˆ s) and YLD(z, ˆ s), are refereed

to Ref. [30]. Since the contribution of long distance can be vetoed effectively in the experimental side, we will not discuss it in

the current work. Furthermore, for the Ceff

7, we here also ignore the long-distance contribution of the charm quark loop, which

is suppressed heavily by the Breit-Wigner factor.

B.Family Non-universal Z′Model and Parameter Constraint

As stated before, in the family non-universal Z′model, there exists the flavor changing neutral current even at the tree level

due to the non-diagonal chiral coupling matrix. Assuming that the couplings of right-handed quark flavors with Z′boson are

diagonal and ignoring Z−Z′mixing, the Z′part of the effective Hamiltonian for b → sl+l−can be written as [19–21]

ef f(b → sℓ+ℓ−) = −2GF

VtbV∗

ts

HZ′

√2VtbV∗

ts

?

−BL

sbBL

ℓℓ

(¯ sb)V−A(¯ℓℓ)V−A−BL

sbBR

VtbV∗

ℓℓ

ts

(¯ sb)V−A(¯ℓℓ)V+A

?

+h.c.

(5)

To match the effective Hamiltonian in SM, as shown in Eq.(2), the above equation is reformulated as

HZ′

ef f(b → sℓ+ℓ−) = −4GF

√2VtbV∗

ts

?△C′

9O9+△C′

10O10

?+h.c.,

(6)

Page 4

4

with

△C′

9(mW) = −g2

s

e2

BL

sb

VtbV∗

ts

(BL

ℓℓ+BR

ℓℓ), △C′

10(mW) = +g2

s

e2

BL

sb

VtbV∗

ts

(BL

ℓℓ−BR

ℓℓ),

(7)

where BL

sband BL,R

ll

denote the effective chiral Z′couplings to quarks and leptons.

be represented as modifications of the Wilson coefficients of the corresponding semileptonic operators, i.e., C′

CSM

Therefore, the Z′contributions can

9,10(mW) =

9,10(mW)+△C′

the evolution effect from mZ′ to mWhere. Numerically, with the central values of the inputs, we get

9,10(mW). The running from mW scale down to mbis the same as that of SM [31, 32], and we had ignored

C′

9(mb) = 0.0682−28.82BL

sb

VtbV∗

ts

Sℓℓ, C′

10(mb) = −0.0695+28.82BL

sb

VtbV∗

ts

Dℓℓ,

(8)

where Sℓℓ= (BL

ℓℓ+BR

ℓℓ) and Dℓℓ= (BL

ℓℓ−BR

ℓℓ).

C.Form Factor

Following the definitions in Ref. [13], the B(pB) → K1(pK1,λ) form factors could be parameterized as

2

mB+mK1

?

?K1(pK1,λ)|¯ sγµ(1−γ5)b|B(pB)? = −i

εµνρσε∗ν

(λ)pρ

Bpσ

K1AK1(q2)+2mK1

ε∗

(λ)· pB

q2

qµ

?

VK1

3(q2)−VK1

?

0(q2)

?

−(mB+mK1)ε(λ)∗

µ

VK1

1(q2)−(pB+ pK1)µ(ε∗

(λ)· pB)VK1

2(q2)

mB+mK1

,

(9)

?K1(pK1,λ)|¯ sσµνqν(1+γ5)b|B(pB)? = 2TK1

1(q2)εµνρσε∗ν

(λ)pρ

Bpσ

K1−iTK1

3(q2)(ε∗

(λ)·q)

?

qµ−

q2

m2

B−m2

K1

(pK1+ pB)µ

?

− iTK1

2(q2)

?

(m2

B−m2

K1)ε(λ)

∗µ−(ε∗

(λ)·q)(pB+ pK1)µ

?

,

(10)

with q ≡ pB− pK1, γ5≡ iγ0γ1γ2γ3, and ε0123= −1. In the context of equation of motion, the form factors satisfy the following

relations,

VK1

3(0) = VK1

VK1

0(0),

mB+mK1

2mK1

TK1

1(q2)−mB−mK1

1(0) = TK1

VK1

2(0),

3(q2) =

2mK1

VK1

2(q2).

(11)

Because the K1(1270) and K1(1400) are the mixing states of the K1Aand K1B, the B → K1form factors can be parameterized

as:

??K1(1270)|¯ sγµ(1−γ5)b|B?

?K1(1400)|¯ sγµ(1−γ5)b|B?

??K1(1270)|¯ sσµνqν(1+γ5)b|B?

?

?

= M

??K1A|¯ sγµ(1−γ5)b|B?

?K1B|¯ sγµ(1−γ5)b|B?

??K1A|¯ sσµνqν(1+γ5)b|B?

?

,

(12)

?K1(1400)|¯ sσµνqν(1+γ5)b|B?

= M

?K1B|¯ sσµνqν(1+γ5)b|B?

0,1,2and TK1

?

,

(13)

with the mixing matrix M being given in Eq. (1). Thus the form factors AK1,VK1

1,2,3satisfy following relations:

?AK1(1270)/(mB+mK1(1270))

AK1(1400)/(mB+mK1(1400))

?(mB+mK1(1270))VK1(1270)

?

?

= M

?AK1A/(mB+mK1A)

AK1B/(mB+mK1B)

?(mB+mK1A)VK1A

?

,

(14)

1

(mB+mK1(1400))VK1(1400)

1

= M

1

(mB+mK1B)VK1B

1

?

,

(15)

Page 5

5

TABLE II: Form factors for B → K1A,K1Btransitions obtained in the LCSRs calculation [13] are fitted to the 3-parameter form in Eq. (21).

FF(0) ab

VBK1A

1

0.34 0.64 0.21

1

VBK1A

2

0.41 1.51 1.18

2

VBK1A

0

0.22 2.40 1.78

0

ABK1A0.45 1.60 0.97

ABK1B−0.37 1.72

TBK1A

1

0.31 2.01 1.50

1

TBK1A

2

0.31 0.63 0.39

2

TBK1A

3

0.28 1.36 0.72

3

FF(0)

ab

VBK1B

VBK1B

VBK1B

−0.29 0.73

−0.17 0.92

−0.45 1.34

0.07

0.86

0.69

0.91

TBK1B

TBK1B

TBK1B

−0.25 1.59

−0.25 0.38

−0.11 −1.61 10.2

0.79

−0.76

?VK1(1270)

2

2

/(mB+mK1(1270))

/(mB+mK1(1400))

?mK1(1270)VK1(1270)

?TK1(1270)

B−m2

(m2

VK1(1400)

?

?

?

?

?

= M

?VK1A

?mK1AVK1A

?TK1A

1

?(m2

(m2

?TK1A

3

2

/(mB+mK1A)

/(mB+mK1B)

?

?

B−m2

B−m2

?

VK1B

2

?

,

(16)

0

mK1(1400)VK1(1400)

0

= M

0

mK1BVK1B

0

,

(17)

1

TK1(1400)

1

= M

1

TK1B

,

(18)

?(m2

K1(1270))TK1(1270)

B−m2

2

K1(1400))TK1(1400)

?TK1(1270)

2

= M

K1A)TK1A

K1B)TK1B

2

2

?

,

(19)

3

TK1(1400)

3

= M

3

TK1B

,

(20)

where we have assumed pµ

K1(1270),K1(1400)≃ pµ

K1A≃ pµ

K1Bfor simplicity. For the form factors, we will use results calculated with

LCSRs [13], which are exhibitedin Table II. In the whole kinematical region, the dependenceof each form factor on momentum

transfer q2is parameterized in the double-pole form:

F(q2) =

F(0)

B)+b(q2/m2

1−a(q2/m2

B)2.

(21)

And the nonperturbativeparameters aiand bican be fitted by the magnitudesof form factors correspondingto the small momen-

tum transfer calculated in the LCSRs approach.

D. Formulas of Observables

With the same convention and notation as [1], the dilepton invariant mass spectrum of the lepton pair for the B → K1ℓ+ℓ−

decay is given as

dΓ(B → K1ℓ+ℓ−)

dˆ s

=

G2

Fα2

210π5

emm5

B

|VtbV∗

ts|2ˆ u(ˆ s)∆

(22)

and

∆ =

??AK1??2

1

4 ˆ m2

K1

3

ˆ sλ

?

1+2ˆ m2

ℓ

ˆ s

?

+??EK1??2ˆ sˆ u(ˆ s)2

λ −ˆ u(ˆ s)2

3

3

+

???BK1??2?

+8 ˆ m2

K1(ˆ s+2 ˆ m2

ℓ)

?

+??FK1??2?

λ −ˆ u(ˆ s)2

3

+8 ˆ m2

K1(ˆ s−4 ˆ m2

ℓ)

??