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)

??

Page 6

6

+

λ

4 ˆ m2

K1

???CK1??2?

?

λ −ˆ u(ˆ s)2

3

?

+??GK1??2?

λ −ˆ u(ˆ s)2

3

λ −ˆ u(ˆ s)2

3

+4 ˆ m2

ℓ(2+2 ˆ m2

K1− ˆ s)

??

λ −ˆ u(ˆ s)2

−

1

2 ˆ m2

K1

Re?BK1CK1∗??

λ?Re?FK1HK1∗?−Re?GK1HK1∗?(1− ˆ m2

with ˆ p = p/mB, ˆ pB= pB/mB, ˆ q = q/mB, ˆ mK1= mK1/mB, and p = pB+ pK1, q = pB− pK1= p++ p−. The auxiliary functions

AK1(ˆ s),···,HK1(ˆ s) are defined in Ref.[11], and we list them in the Appendix for convenience. Here, we also choose ˆ s = ˆ q2

and ˆ u ≡ ( ˆ pB− ˆ p−)2−( ˆ pB− ˆ p+)2as the two independent parameters, which are bounded as 4 ˆ m2

ˆ u ≤ ˆ u(ˆ s), with ˆ u(ˆ s) ≡

The differential forward-backward asymmetry of the B → K1ℓ+ℓ−decay is defined as

dAFB

dˆ s

0

?

(1− ˆ m2

K1− ˆ s)+Re?FK1GK1∗???

K1)?+

3

?

(1− ˆ m2

K1− ˆ s)+4 ˆ m2

ℓλ

??

− 2

ˆ m2

ˆ m2

ℓ

K1

ˆ m2

ˆ m2

ℓ

K1

ˆ sλ??HK1??2.

(23)

l≤ ˆ s ≤ (1− ˆ mK1)2and −ˆ u(ˆ s) ≤

?

λ?1−4 ˆ m2

l/ˆ s?, λ ≡ 1+ ˆ m2

K1+ ˆ s2−2ˆ s−2 ˆ m2

K1(1+ ˆ s).

≡

?ˆ u(ˆ s)

d ˆ ud2Γ

d ˆ udˆ s−

?0

−ˆ u(ˆ s)d ˆ ud2Γ

d ˆ udˆ s.

(24)

Furthermore, the normalized forward-backward asymmetry, which is more useful in the experimental side, can be written in

terms of quantities as

dAFB

dˆ s

≡dAFB

dˆ s

?

dΓ

dˆ s≡ ˆ u(ˆ s)ˆ s

?

Re?BK1EK1∗?+Re?AK1FK1∗??

.

(25)

Here, we do not considerthe hard spectator correctionssince the light-conedistribution amplitudes of K1are not precise enough.

At the end of this section , we pay our attentions on obtaining the lepton polarization asymmetries. In the center mass frame

of dilepton, the three orthogonal unit vectors could be defined as

ˆ eL=? p+,

ˆ eN=

? pK×? p+

|? pK×? p+|,

ˆ eT= ˆ eN× ˆ eL,

(26)

which are related to the spins of leptons by a Lorentz boost. So, the decay width of the B→ K1ℓ+ℓ−decay for any spin direction

ˆ n of the lepton, where ˆ n is a unit vector in the dilepton center mass frame, can be written as:

dΓ(ˆ n)

dˆ s

=1

2

?dΓ

dˆ s

?

0[1+(PLˆ eL+PNˆ eN+PTˆ eT)· ˆ n].

(27)

In the above equation, the subscript ”0” denotes the unpolarized decay width, and PLand PTare the longitudinal and transverse

polarizationasymmetriesin thedecayplane,respectively. PNis the normalpolarizationasymmetryinthe directionperpendicular

to the decay plane. Correspondingly,the lepton polarization asymmetry Pi (i = L,N,T) can be obtained by calculating

Pi(ˆ s) =dΓ(ˆ n = ˆ ei)/dˆ s−dΓ(ˆ n = −ˆ ei)/dˆ s

dΓ(ˆ n = ˆ ei)/dˆ s+dΓ(ˆ n = −ˆ ei)/dˆ s.

(28)

After a straightforward calculation, we obtain:

PL∆ =

?

1−4ˆ m2

l

ˆ s

?

2ˆ sλ

3

Re(A EK1∗)+(λ +12 ˆ m2

K1ˆ s)

3 ˆ m2

K1

Re(BFK1∗)

−λ(1− ˆ m2

−π√ˆ sˆ u(ˆ s)

4 ˆ mK1

K1− ˆ s)

K1

?

3 ˆ m2

Re(BGK1∗+CFK1∗)+

λ2

3 ˆ mK1

Re(CGK1∗)

?

,

(29)

PN∆ =

ˆ ml

ˆ mK1

?Im(FGK1∗)(1+3 ˆ m2

K1− ˆ s)

Page 7

7

TABLE III: Input parameters

mB= 5.279GeV,

τB− = 1.638×10−12sec,

mK1(1400)= 1.403GeV,

mb,pole= 4.8±0.2GeV,

τB0 = 1.530×10−12sec,

mK1A= 1.31GeV ,

αem= 1/129,

mK1(1270)= 1.272GeV ,

|VtbV∗

mK1B= 1.34GeV

ts| = 0.0407 ,

αs(µh) = 0.3,

TABLE IV: Predictions for the non-resonant branching fractions Br(B → K1ℓ+ℓ−)(10−6) in the SM and the non-universal Z′model. The first

errors come from the uncertainty of the θ = (−34±13)◦and the second errors are combination of all uncertainties in the Z′model.

Mode SM S1

24.1+0.2

−3.6

B−→ K−

B−→ K−

B−→ K−

B−→ K−

B−→ K−

B0→ K0

B0→ K0

B0→ K0

B0→ K0

B0→ K0

B0→ K0

S2Extreme Limit

49.6+0.1

44.9+0.2

1.2+0.0

−0.2

1.6−0.4

+4.3

1.3−0.1

+3.5

0.02−0.02

+0.06

46.3+0.1

41.9+0.2

1.1+0.0

−0.1

1.5−0.3

+3.9

1.2−0.1

+3.3

0.02−0.02

+0.06

B−→ K−

1(1270)e+e−

1(1270)µ+µ−

1(1270)τ+τ−

1(1400)e+e−

1(1400)µ+µ−

1(1400)τ+τ−

1(1270)e+e−

1(1270)µ+µ−

1(1270)τ+τ−

1(1400)e+e−

1(1400)µ+µ−

1(1400)τ+τ−

33.7+0.1

29.1+0.1

0.7+0.1

−0.1±0.3

1.2−0.4

0.8−0.0

0.01−0.01

31.5+0.1

−3.3±7.2

27.2+0.1

−2.5±7.2

0.7+0.0

−0.1±0.3

1.1−0.4

0.8−0.1

0.01−0.01

−3.5±7.4

−3.5±7.4

28.8+0.2

24.3+0.0

0.8+0.0

−0.2±0.2

1.0−0.3

0.7−0.1

0.01−0.01

26.9+0.2

−2.8±3.8

22.8+0.1

−2.2±3.8

0.7+0.0

−0.1±0.2

1.0−0.4

0.6−0.0

0.01−0.01

−3.0±3.9

−3.2±3.9

−5.3

−4.2

19.7+0.2

0.8+0.0

−0.2

0.9−0.4

+2.2

0.5−0.0

+1.6

0.01−0.00

+0.04

22.5+0.2

18.4+0.1

0.7+0.0

−0.1

0.8−0.3

+2.2

0.5−0.0

+1.5

0.01−0.01

+0.04

−1.8

+3.0±0.3

+2.4±0.2

+0.04±0.01

+2.7±0.2

+1.9±0.2

+0.05±0.01

−3.4

−1.7

−4.6

−3.9

+2.8±0.2

+2.1±0.2

+0.04±0.01

+2.4±0.2

+1.8±0.2

+0.04±0.01

+Im(FHK1∗)(1− ˆ m2

π√λ ˆ ml

4√ˆ s

K1− ˆ s)−Im(GHK1∗)λ?+2 ˆ mK1ˆ ml[Im(BEK1∗)+Im(A FK1∗)]

4ˆ sRe(A BK1∗)+(1− ˆ m2

ˆ m2

K1

?

,

(30)

PT∆ =

?

K1− ˆ s)

?−Re(BFK1∗)+(1− ˆ m2

K1)Re(BGK1∗)+ ˆ sRe(BHK1∗)?

+

λ

ˆ m2

K1

[Re(CFK1∗)−(1− ˆ m2

K1)Re(CGK1∗)− ˆ sRe(CHK1∗)]

?

.

(31)

III.NUMERICAL RESULTS AND DISCUSSION

In this section, we shall calculate aforementioned observables like the branching ratios (BR), the normalized forward-

backward asymmetries (FBA) and lepton polarization asymmetries, as well as their sensitivities to the new physics due to Z′

boson. The input parameters used in the numerical calculations are listed in Table.III. In discussing the K1(1270) and K1(1400),

we have to draw much attention on the mixing angle θ defined in Eq.(1), although many attempts have been done to constrain it.

The magnitudeof θ was estimated to be |θ|≈34◦∨57◦in Ref. [8], 35◦≤|θ|≤55◦in Ref. [9], and |θ|=37◦∨58◦in Ref. [10].

Nevertheless, the sign of the θ was yet unknown in above studies. From the studies of B → K1(1270)γ and τ → K1(1270)ντ,

one of us recently obtained [11]

θ = −(34±13)◦,

(32)

Page 8

8

?60

?50

?40

?30

?20

?10

0.0

0.1

0.2

0.3

0.4

0.5

0.6

ΘK

dFA?B???K1

??1400?Μ?Μ???dq2

?60

?50

?40

?30

?20

?10

?1.00

?0.95

?0.90

?0.85

?0.80

ΘK

PL?B???K1

??1400?Μ?Μ??

FIG. 1: Normalized differential forward-backward asymmetries (left panel) and longitudinal lepton polarization asymmetry(right panel) of

B → K1(1400)µ+µ−, as a function of θ(in units of degree). The solid, dotted and dashed curves correspond to s = 5GeV2, 7GeV2and

12GeV2, respectively.

05 1015

0

1

2

3

4

q2?GeV2?

dBR?B???K1

??1270?Μ?Μ???dq2?10?7?

13.013.514.014.5 15.015.516.0

0.0

0.1

0.2

0.3

0.4

q2?GeV2?

dBR?B???K1

??1270?Τ?Τ???dq2?10?7?

02468 1012 14

0.00

0.05

0.10

0.15

0.20

0.25

q2?GeV2?

dBR?B???K1

??1400?Μ?Μ???dq2?10?7?

13.0 13.5 14.014.515.0

0.000

0.005

0.010

0.015

0.020

0.025

0.030

q2?GeV2?

dBR?B???K1

??1400?Τ?Τ???dq2?10?7?

FIG. 2: The differential decay rates dBr(B+→ K+

The solid (green), and dotted (blue), dashed (red) and dot-dashed (orange) lines represent results from the standard model, S1, S2 and ELV

1ℓ+ℓ−)/dq2as functions of q2(in units of GeV2). The central values of inputs are used.

parameters, respectively.

Page 9

9

TABLE V: The inputs parameters for the Z′couplings [21].

|BL

1.09±0.22

2.20±0.15

sb|(×10−3)

φL

s[◦]

BL

µµ(×10−2)

−4.75±2.44

−1.83±0.82

BR

µµ(×10−2)

1.97±2.24

0.68±0.85

S1

−72±7

−82±4 S2

0510 15

?0.1

0.0

0.1

0.2

0.3

q2?GeV2?

dAFB?B???K1

??1270?Μ?Μ???dq2

13.013.5 14.0 14.5 15.015.516.0

0.00

0.02

0.04

0.06

0.08

0.10

0.12

q2?GeV2?

dAFB?B???K1

??1270?Τ?Τ???dq2

02468 1012 14

?0.6

?0.4

?0.2

0.0

0.2

0.4

q2?GeV2?

dAFB?B???K1

??1400?Μ?Μ???dq2

13.013.5 14.0 14.515.0

0.00

0.02

0.04

0.06

0.08

0.10

q2?GeV2?

dAFB?B???K1

??1400?Τ?Τ???dq2

FIG. 3: The normalized differential forward-backward asymmetries for the B+→ K+

1ℓ+ℓ−decays as functions of q2(in units of GeV2).

where the minus sign of θ is related to the chosen phase of |K1A? and |K1B?, and we will use this range in the following

discussion.

In Ref. [1], the authors found that in the low s region, where s ≈ 2GeV2, the differential decay rate for B → K1(1400)µ+µ−

with θ = −57◦is enhanced by about 80% compared with that with θ = −34◦, whereas the rate for B → K1(1270)µ+µ−is

not so sensitive to variation of θ. After calculation, we emphasize that all observables of B → K1(1400)µ+µ−are sensitive to

the mixing angle. In Fig.1, for instance, we plot the relations of the normalized forward-backward asymmetry and longitudinal

leptonpolarizationasymmetryof B→K1(1400)µ+µ−withθ varyingfrom−10◦to −60◦, whens=5GeV2(solidline), 7GeV2

(dotted line) and 12GeV2(dashed line). With these figures and data, one can constrain the angle in future, as well as cross check

the bands from other theories and experiments. Additionally, because of small masses of electron and muon, the invariant mass

spectra and branching ratios are almost the same for electron and muon modes. Meanwhile, it is very difficult to measure the

electron polarization, so we only consider B → K1µ+µ−,K1τ+τ−except for numerical results in the following discussions.

In Table IV, we again summarize the predictions for branching fractions corresponding to θ = −(34±13)◦without consid-

Page 10

10

051015

?1.0

?0.8

?0.6

?0.4

?0.2

0.0

0.2

q2?GeV2?

PL?B???K1

??1270?Μ?Μ??

13.013.5 14.0 14.515.015.5 16.0

?0.5

?0.4

?0.3

?0.2

?0.1

0.0

q2?GeV2?

PL?B???K1

??1270?Τ?Τ??

05 10 15

?1.0

?0.5

0.0

0.5

q2?GeV2?

PL?B???K1

??1400?Μ?Μ??

13.013.5 14.014.5 15.015.5 16.0

?0.7

?0.6

?0.5

?0.4

?0.3

?0.2

?0.1

0.0

q2?GeV2?

PL?B???K1

??1400?Τ?Τ??

FIG. 4: The longitudinal lepton polarization asymmetries for the B+→ K+

1ℓ+ℓ−decays as functions of q2(in units of GeV2).

ering the uncertainties taken by the form factors, which have been discussed in detail in Ref. [1]. The negligible disparities

between our results and those of Ref.[1] are from the difference of Wilson coefficients. From the table, we note that the branch-

ing ratios of B → K1(1270)ℓ+ℓ−are not sensitive to the mixing angle θ, while those of B → K1(1400)ℓ+ℓ−are sensitive to it

seriously. We also find that the branching ratios of B → K1(1270)ℓ+ℓ−are much larger than those of B → K1(1400)ℓ+ℓ−. For

B → K1τ+τ−, the branching ratios are very small due to small phase spaces.

Now, we turn to a discussion of the new physics contribution. Within a family non-universal Z′model, the Z′contribution to

B→ K1ℓ+ℓ−decay involves four new parameters |BL

well measured channels have been done by many groups in the past few years. Combining the constraints from¯Bs−Bsmixing,

B→πK(∗)andρK decays, |BL

of BL,R

ll

from B → Xsµ+µ−, Kµ+µ−and K∗µ+µ−, as well as Bs→ µ+µ−decay. Recently, there have been more data from

Tevatron and LHC on decay processes mentioned above. Many of them might afford stronger upper bounds than before, but the

sb|, φL

s, BL

ℓℓand BR

ℓℓ. The tasks of constrainingthe aboveparameters from the

sb| andφL

shave beenstrictly constrainedby Chang et. al. [21]. Theyalso performedthe constraints

new parameters have not been fitted and we will leave it as our future work. In the current work, we will adopt the parameters

fitted in Ref [21] so as to probe contribution of new physics with the largest possibility. For convenience, we recollect their

numerical results in the Table V, where S1 and S2 correspondto UTfit collaboration’s two fitting results for¯Bs−Bsmixing [33].

Meanwhile, in order to show the maximal strength of Z′, with permitted range in S1, we choose the extreme values

|BL

sb| = 1.31×10−3,φL

s= −79◦,Sll= −6.7×10−2,Dll= −9.3×10−2,

(33)

and name them as extreme limit values (ELV) expediently.

Page 11

11

05 10 15

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

q2?GeV2?

PT?B???K1

??1270?Μ?Μ??

13.013.514.014.515.015.516.0

0.0

0.1

0.2

0.3

0.4

0.5

q2?GeV2?

PT?B???K1

??1270?Τ?Τ??

05 1015

?0.05

0.00

0.05

0.10

q2?GeV2?

PT?B???K1

??1400?Μ?Μ??P

13.013.514.014.515.015.5 16.0

0.0

0.1

0.2

0.3

0.4

0.5

q2?GeV2?

PT?B???K1

??1400?Τ?Τ??

FIG. 5: The transverse lepton polarization asymmetries for the B+→ K+

1ℓ+ℓ−decays as functions of q2(in units of GeV2).

02468 1012 14

0.015

0.020

0.025

0.030

0.035

q2?GeV2?

RΜ

13.013.514.0 14.5 15.015.516.0

0.010

0.015

0.020

0.025

q2?GeV2?

RΤ

FIG. 6: The ratio of the decay distributions, Rµ(left panel) and Rτ(right panel), as a function of the dilepton invariant mass q2(in units of

GeV2). The legends are the same as in Fig. 2

Considering the Z′contribution with two sets of parameters, we calculate the non-resonant branching ratios of concerned

decay modes and tabulate them in the third and fourth columns of the Table.IV, where the first errors come from the mixing

angle θ and the second errors are from all uncertainties of Z′model by adding all the theoretical errors in quadrature. With the

ELV parameters, the extreme results are listed in the last column of Table.IV, and the errors are only from mixing angle.

In Fig.2-5, we plot the differential branching ratios, forward-backward asymmetries and polarization asymmetries of the

Page 12

12

leptons of B+→ K1(1270)+ℓ+ℓ−and B+→ K1(1400)+ℓ+ℓ−, respectively. In all figures, the bands with solid (green) lines are

results from the standard model with θ = −(34±13)◦, and dotted (blue), dashed (red) and dot-dashed (orange) lines represent

the results with the S1, S2 and ELV parameters by fixing θ = −34◦, respectively. Some discussions of the above results are in

order.

• From the Table. IV, we find that the effect of S1 is more significant than that of S2. For the central values, compared

with predictions of the SM, the branching ratio of B−→ K1(1270)−µ+µ−can be enhanced about by 48% in S1, and by

23% in S2. If we choose the extreme limit of S1, the branching ratio can be enhanced one time at most by new physics

contribution of Z′. As concerns B → K1(1400)ℓ+ℓ−, their branching ratios are more sensitive to the mixing angle than to

a new physics contribution, and then it is very hard to differentiate the Z′effects.

• For the dilepton invariant mass spectrum of B−→ K1(1270)−µ+µ−, the effects of the Z′boson are quite distinctive from

that of the SM, as shown in Fig.2. The reason for the enlargement is the relative change of the absolute values of Ceff

and C10, though the latter is q2independent. For B−→ K1(1400)−µ+µ−, Z′boson could change the shape effectively,

however this contribution would be clouded by the uncertainties from the mixing angle. For the tauon modes, with large

9

tauon mass and small phase space, it is very difficult to disentangle the new physics contribution from the predictions of

SM, unless choosing the extreme limit case.

• We plot the normalized forward-backwardasymmetries in Fig.3. For B → K1µ+µ−, there exist zero crossing positions in

SM, S1 and S2. We would like to emphasize that the hadronic uncertainties and mixing angle almost have no influence

on zero crossing positions, as shown in figures. Specifically, for B → K1(1270)µ+µ−, the zero crossing s0positions are

2.3GeV, 3.3GeVand2.9GeVinSM,S1andS2. Accordingly,forB→K1(1400)µ+µ−,s0=2.8GeV,3.5GeV,3.2GeV.

It is obvious that s0moves to the positive direction with the Z′boson effects. And in the limit values, the zero crossing

positions disappear. Thus, the measurement of zero position is very important for searching for new physics contribution

in the experiments. For B → K1(1400)τ+τ−, with the central value of S1 and S2, the forward-backward asymmetries

are almost the same as the predictions from SM. However, these asymmetries become almost zero in both low and large

momentum regions in the ELV case.

• Just like the BR and AFB, the polarization asymmetries of leptons are also good observables for probing the new physics

signals. In order to show the effects due to the Z′, we figure out the longitudinal PLand transverse PT polarization

asymmetries as functions of q2in Fig.4 and Fig.5, respectively. The PNparts are too tiny to be measured experimentally

even in the designed Super-B factory, so we will not discuss them in this work. In the case of B → K1(1270)µ+µ−, the

longitudinal (transverse) polarization asymmetry of lepton is enhanced (decreased) remarkably by new physics effects.

In SM, the longitudinal polarization asymmetry for muon is around −1 in the large momentum region, while it changes

to −0.6(−0.7) in S1 (S2). In the Z′model, a large value of differential decay rate will suppress the absolute value of

longitudinal polarization asymmetry in the large q2part. With the extreme values, PLflips the sign in the low q2region

and approaches to zero in the large q2region. If there exist large Z′−b−s and Z′−l−l couplings, we can check them

by measuring the above observables. Similar effects can be found in tau modes, but the deviations are too small to be

measured experimentally.

• From the Table.IV, we obtain Br(B → K1(1270)ℓ+ℓ−) ≫ Br(B → K1(1400)ℓ+ℓ−). It should be helpful to define the ratio

Page 13

13

Rℓ, as mentioned in Ref.[1],

Rℓ≡dΓ(B−→ K−

dΓ(B−→ K−

1(1400)ℓ+ℓ−)/ds

1(1270)ℓ+ℓ−)/ds.

(34)

To cross check this conclusion that the ratios are insensitive to new physics contribution, we presented the Rµand Rτas

functions of q2in Fig. 6, where the solid (green), and dotted (blue), dashed (red) and dot-dashed (orange) lines repre-

sent results from the standard model, S1, S2 and ELV parameters, respectively. We show that Rℓ(ℓ = µ,τ) are almost

unchanged, so that they are not suitable for searching for Z′effects. These results confirm the conclusion in Ref.[1].

IV.SUMMARY

A new family non-universal Z′boson could be naturally derived in many extensions of SM. One of the possible way to get

such non-universal Z′boson is to include an additionU′(1) gauge symmetry, which has been studied by many groups. With the

data, people had fitted two sets of coupling constants, S1 and S2 namely. In this work, we have considered the contributions of

family non-universalZ′model at the tree level in semi-leptonic B decays involvingaxial-vector meson K1in the final states. The

strange axial-vector mesons, K1(1270) and K1(1400), are the mixtures of the K1Aand K1B, which are the 13P1and 11P1states,

respectively. We show that the mixing angle could be constrained by measuring some observables of B → K1(1400)ℓ+ℓ−, such

as the normalized differential forward-backward asymmetry and longitudinal lepton polarization asymmetry. With θ = −34◦,

the branching ratio of B → K1(1270)µ+µ−is enhanced about by 50%(30%) with respect to the corresponding SM values by

Z′in S1 (S2). We also found FBA and lepton polarization asymmetries show quite significant discrepancies with respect to the

SM values. The zero crossing position in the FBA spectrum at low dilepton mass will move to the positive direction with Z′

boson contribution. We also note that B → K1(1400)µ+µ−is not suitable to probe new physics, which will be buried by the

uncertainty from the mixing angle. While for the tauon modes, the new physics contributions are not remarkable due to small

phase spaces except in the extreme limit. These results could be tested in the running LHC-b experiment and designed Super-B

factory.

Acknowledgement

The work of Y. Li is supported in part by the NSFC ((Nos.10805037 and 11175151)) and the Natural Science Foundation

of Shandong Province (ZR2010AM036). K. C. Y. is supported in part by the National Center for Theoretical Sciences and the

National Science Council of R.O.C. under Grant No. NSC99-2112-M-003-005-MY3.

Appendix

AK1(ˆ s) =

2

1+ ˆ mK1

Ceff

9(ˆ s)AK1(ˆ s)+4 ˆ mb

ˆ s

Ceff

7TK1

1(ˆ s),

(35)

BK1(ˆ s) = (1+ ˆ mK1)

?

(1− ˆ mK1)Ceff

Ceff

9(ˆ s)VK1

1(ˆ s)+2 ˆ mb

ˆ s

(1− ˆ mK1)Ceff

7TK1

2(ˆ s)

?

,

(36)

CK1(ˆ s) =

1

1− ˆ m2

K1

?

9(ˆ s)VK1

2(ˆ s)+2 ˆ mbCeff

7

?

TK1

3(ˆ s)+1− ˆ m2

K1

ˆ s

TK1

2(ˆ s)

??

,

Page 14

14

(37)

DK1(ˆ s) =

1

ˆ s

?

Ceff

9(ˆ s)

?

(1+ ˆ mK1)VK1

1(ˆ s)−(1− ˆ mK1)VK1

2(ˆ s)−2 ˆ mK1VK1

0(ˆ s)

?

−2 ˆ mbCeff

7TK1

3(ˆ s)

?

,

(38)

EK1(ˆ s) =

2

1+ ˆ mK1

C10AK1(ˆ s),

(39)

FK1(ˆ s) = (1+ ˆ mK1)C10VK1

1(ˆ s),

(40)

GK1(ˆ s) =

1

1+ ˆ mK1

1

ˆ sC10

C10VK1

2(ˆ s),

(41)

HK1(ˆ s) =

?

(1+ ˆ mK1)VK1

1(ˆ s)−(1− ˆ mK1)VK1

2(ˆ s)−2 ˆ mK1VK1

0(ˆ s)

?

.

(42)

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