Branching ratios, forward-backward asymmetries and angular distributions of $B\to K_2^*l^+l^-$ in the standard model and new physics scenarios

0.61 MB·English

by  Run-Hui Li

DESY 10-234
Branching ratios, forward-backward asymmetries and angular distributions of
B K∗l+l− in the standard model and two new physics scenarios
→ 2
Run-Hui Lia,b, Cai-Dian Lu¨ a and Wei Wang c
∗ †
a Institute of High Energy Physics, P.O. Box 918(4), Beijing 100049, People’s Republic of China
b Department of Physics & IPAP, Yonsei University, Seoul 120-479, Korea
c Deutsches Elektronen-Synchrotron DESY, Hamburg 22607, Germany
We analyze the B → K∗(→ Kπ)l+l− (with l = e,µ,τ) decay in the standard model and two
2
new physics scenarios: vector-like quark model and family non-universal Z′ model. We derive
1 its differential angular distributions, using the recently calculated form factors in the perturbative
1 QCD approach. Branching ratios, polarizations, forward-backward asymmetries and transversity
0 amplitudesarepredicted,fromwhichwefindapromisingprospectivetoobservethischannelinthe
2 futureexperiment. WeupdatetheconstraintsoneffectiveWilsoncoefficientsand/orfreeparameters
n in these two new physicsscenarios by making useof theB→K∗l+l− and b→sl+l− experimental
a data. Their impact on B → K∗l+l− is subsequently explored and in particular the zero-crossing
2
J
point for theforward-backward asymmetry in these newphysicsscenarios can sizably deviate from

4 thestandardmodel. InadditionwealsogeneralizetheanalysistoasimilarmodeBs →f2′(1525)(→
2 K+K−)l+l−.

]
h PACSnumbers: 13.20.He;12.39.St14.40.Be;
p
-
p
I. INTRODUCTION
e
h
[
Discoveries of new degrees of freedom at TeV energy scale, with contributions to our understanding of the origin

2 of the electroweak symmetry breaking, can proceed in two different ways. One is a direct search of the Higgs boson,
v the last piece to complete the standard model (SM), and particles beyond the SM, to establish new physics (NP)
9
theories. The othereffort,alreadyongoing,is to investigateprocessesinwhichSMis testedwith higherexperimental
2
1 and theoretical precision. Among the latter category, rare B decays are among ideal probes. Besides constraints on
2 the Cabibbo-Kobayashi-Maskawa(CKM) matrix including apexs andangles of the unitary triangle,which have been
2. contributedbysemileptonicb u/candnonleptonicB decaysrespectively,theelectroweakinteractionstructurecan
→
1 also be probed by, for instance, the b sγ and b sl+l modes which are induced by loop effects in the SM and
−
→ →
0 therefore sensitive to the NP interactions.
1
Unlike b sγ and B K γ that has only limited physical observables, b sl+l especially B K l+l ,
: → → ∗ → − → ∗ −
v with a number of observables accessible, provides a wealth of information of weak interactions, ranging from the
i
X forward-backwardasymmetries (FBAs), isospin symmetries, polarizations to a full angular analysis. The last barrier
r to access this mode, the low statistic with a branching faction of the order 10−6, is being cleared by the B factories
a
and the hadron collider [1–3]. The ongoing LHCb experiment can accumulate 6200 events per nominal running year
of 2fb 1 with √s = 14 TeV [4], which allows to probe the short-distance physics at an unprecedented level. For
−
instance the sensitivity to zero-crossingpoint of FBAs can be reduced to 0.5GeV2 and might be further improved as
0.1GeV2 after the upgrade [5]. This provides a good sensitivity to discriminate between the SM and different models
of new physics. There are also a lot of opportunities on the Super B factory [6]. Because of these virtues, theoretical
researchinterests in this mode haveexploded and the precisionis highly improved,see Refs. [7–22] for an incomplete
list.
Towardthedirectiontoelucidatetheelectroweakinteraction,B K l+l anditsSU(3)-relatedmodeB φl+l
∗ − s −
→ →
arenotunique. Inthiswork,weshallpointoutthatB K (1430)l+l andtheB -counterpartB f (1525)l+l 1,
→ 2∗ − s s → 2′ −
which so far have not been investigated in detail [23–26], are also useful in several aspects. Due to the similarities
∗ AlexandervonHumboldtFellow
† Email:[email protected]
1 HereafterwilluseK∗ andf′ toabbreviateK∗(1430) andf′(1525).
2 2 2 2
2
between K and K , all experiment techniques for B K l+l are adjustable to B K l+l . The main decay
∗ 2∗ → ∗ − → 2∗ −
product of K is a pair of charged kaon and pion which are easily detected on the LHCb. Moreover as we will show
2∗
in the following, based on either a direct computation in the perturbative QCD approach [27] or the implication
of experimental data on B K γ process, the branching ratio (BR) of B K l+l is found sizable. Therefore
→ 2∗ → 2∗ −
thousands of signal events can be accumulated on the LHCb per nominal running year.
As a consequence of the unitarity of quark mixing matrix, tree level flavor-changing neutral-current (FCNC) is
forbidden in the SM. When higher order correctionsare taken into account, b sl+l arises from photonic penguin,
−
→
Z penguin and W-box diagram. The large mass scale of virtual states leads to tiny Wilson coefficients in b quark
decays and thus b sl+l would be sensitive to the potential NP effects. In certain NP scenarios, new effective
−
→
operators out of the SM scope can emerge, but in a class of other scenarios, only Wilson coefficients for effective
operators are modified. Among the latter category,vector-like quark model (VQM) [28–36] and family non-universal
Z model [37–42] are simplest and therefore of theoretical interest. In this work we shall also elaborate the impacts
′
of these models on B K l+l .
→ 2∗ −
The rest of the paper is organized as follows. In Sec. II, we collect the necessary hadronic inputs, namely form
factors. Sec.IIcontainstheanalyticformulasfordifferentialdecaydistributionsandintegratedquantities. InSec.IV,
we give a brief overview of two NP models whose effects we will study. Sec. V is our phenomenological analysis: the
predictionsintheSM;update ofthe constraintsonthe VQMandZ modelparameters;the NPeffectonthe physical
′
quantities. We conclude in the last section. In the appendix, we give the effective Hamiltonian in the SM and the
helicity amplitude method.
II. B→K FORM FACTORS
2
B K l+l decay amplitudes contain two separate parts. Short-distance physics, in which contributions at the
→ 2∗ −
weak scale µ is calculated by perturbation theory and the evolution between m and b quark mass scale m is
W W b
organized by the renormalization group. These degrees of freedom are incorporated into Wilson coefficients and the
obtained effective Hamiltonian responsible for b sl+l in the appendix A. The low-energy effect characterizes the
−
→
long-distancephysics andwill be parameterizedby hadronicmatrix elements of effective operators,whichare usually
reduced to heavy-to-lightform factors in semileptonic B decays.
Thespin-2polarizationtensor,whichsatisfiesǫ Pν =0withP beingthe momentum,issymmetricandtraceless.
µν 2 2
It can be constructed via the spin-1 polarization vector ǫ:
1
ǫ ( 2)=ǫ ( )ǫ ( ), ǫ ( 1)= [ǫ ( )ǫ (0)+ǫ ( )ǫ (0)],
µν µ ν µν µ ν ν µ
± ± ± ± √2 ± ±
1 2
ǫ (0)= [ǫ (+)ǫ ( )+ǫ (+)ǫ ( )]+ ǫ (0)ǫ (0). (1)
µν µ ν ν µ µ ν
√6 − − 3
r
In the case ofthe tensormesonmoving onthe z axis,the explicit structures ofǫ in the ordinarycoordinateframe are
chosen as
1 1
ǫµ(0) = (p~K∗ ,0,0,EK∗), ǫµ( )= (0, 1, i,0), (2)
mK∗ | 2| 2 ± √2 ∓ −
2
where EK2∗ and p~K2∗ is the energy and the momentum magnitude of K2∗ in B meson rest frame, respectively. In the
following calculation, it is convenient to introduce a new polarization vector ǫ
T
1
ǫ (h)= ǫ (h)Pν, (3)
Tµ m µν B
B
which satisfies
1 1 1 2
ǫ ( 2)=0, ǫ ( 1)= ǫ(0) P ǫ ( ), ǫ (0)= ǫ(0) P ǫ (0). (4)
Tµ Tµ B µ Tµ B µ
± ± mB √2 · ± mBr3 ·
Thecontractionisevaluatedasǫ(0) PB/mB = p~K∗ /mK∗ andthuswecanseethatthenewvectorǫT playsasimilar
· | 2| 2
role to the ordinary polarization vector ǫ, regardless of the dimensionless constants √12|p~K2∗|/mK2∗ or 23|p~K2∗|/mK2∗.
q
3
The parametrizationof B K form factors is analogous to the B K ones [25–27, 43]
→ 2∗ → ∗
2V(q2)
hK2∗(P2,ǫ)|s¯γµb|B(PB)i = −mB+mK∗ǫµνρσǫ∗TνPBρP2σ,
2
ǫ q ǫ q
hK2∗(P2,ǫ)|s¯γµγ5b|B(PB)i = 2imK2∗A0(q2) ∗Tq2· qµ+i(mB +mK2∗)A1(q2) ǫ∗Tµ− ∗Tq2· qµ
(cid:20) (cid:21)
iA (q2) ǫ∗T ·q Pµ m2B −m2K2∗qµ ,
− 2 mB+mK∗ " − q2 #
2
K (P ,ǫ)s¯σµνq bB(P ) = 2iT (q2)ǫµνρσǫ P P ,
h 2∗ 2 | ν | B i − 1 ∗Tν Bρ 2σ
q2
hK2∗(P2,ǫ)|s¯σµνγ5qνb|B(PB)i = T2(q2)h(m2B −m2K2∗)ǫ∗Tµ−ǫ∗T ·qPµi+T3(q2)ǫ∗T ·q"qµ− m2B−m2K2∗Pµ#, (5)
where q = PB P2,P = PB +P2. We also have the relation 2mK∗A0(0) = (mB +mK∗)A1(0) (mB mK∗)A2(0)
− 2 2 − − 2
in order to smear the pole at q2 =0.
Using the newly-studied light-cone distribution amplitudes [44], we have computed B K form factors [27] in
→ 2∗
the perturbativeQCDapproach(PQCD)[45]. At the leadingpower,ourpredictions arefound to obey the nontrivial
relations derived from the large energy symmetry. This consistence may imply that the PQCD results for the form
factors are reliable and therefore suitable for the study of the semileptonic B decays. The recent computation in
light-cone QCD sum rules [43] is also consistent with ours. Results in the light-cone sum rules in conjunction with
B-meson wave functions [46], however, are too large and thus not favored by the B K γ data. In our work the
→ 2∗
B K form factors are q2-distributed as [27]
→ 2∗
F(0)
F(q2) = , (6)
(1 q2/m2)(1 a(q2/m2)+b(q2/m2)2)
− B − B B
where F denotes a generic form factor among A ,A ,V,T . Neglecting higher power corrections, A is related to
0 1 1 3 2
−
A and A by
0 1
A2(q2)= mmB2+mqK22∗ (mB +mK2∗)A1(q2)−2mK2∗A0(q2) . (7)
B −
(cid:2) (cid:3)
Numerical results for the B K and B f (1525) form factors at maximally recoil point and the two fitted
→ 2∗ s → 2′
parameters a,b are collected in table I. The two kinds of errors are from: decay constants of B meson and shape
parameter ω ; Λ , the scales ts and the threshold resummation parameter c [27].
b QCD
III. DIFFERENTIAL DECAY DISTRIBUTIONS AND SPIN AMPLITUDES
In this section, we will discuss the kinematics of the quasi four-body decay B K ( Kπ)l+l , define angular
→ 2∗ → −
observables and collect the explicit formulas of helicity amplitudes and/or transversity amplitudes.
A. Differential decay distribution
At the quark level, the decay amplitude for b sl+l is expressed as
−
→
G α C +C C C
M(b→sl+l−) = √F2 πemVtbVt∗s× 9 4 10[s¯b]V−A[¯ll]V+A+ 9−4 10[s¯b]V−A[¯ll]V−A
(cid:18)
qµ qµ
+C m [s¯iσ (1+γ )b] [¯lγνl]+C m [s¯iσ (1 γ )b] [¯lγνl] , (8)
7L b µν 5 q2 × 7R b µν − 5 q2 ×
(cid:19)
where C =C andC = msC in the SM.Sandwiching Eq.(8) betweenthe initial andfinalstates andreplacing
7L 7 7R mb 7L
the spinor product [s¯b] by hadronic matrix elements, one obtains the decay amplitude for hadronic B process. For
the process under scrutiny in this work, the decay observed in experiment is actually B K ( Kπ)l+l which is
→ 2∗ → −
4
TABLE I: B → K2∗ and Bs → f2′(1525) form factors in the PQCD approach. F(0) denotes results at q2 = 0 point while a,b
are the parameters in the parametrization shown in Eq. (6). The two kinds of errors are from: decay constants of B meson
and shape parameter ωb; ΛQCD, factorization scales tsand thethreshold resummation parameter c.
F F(0) a b
VBK2∗ 0.21+0.04+0.05 1.73+0.02+0.05 0.66+0.04+0.07
−0.04−0.03 −0.02−0.03 −0.05−0.01
ABK2∗ 0.18+0.04+0.04 1.70+0.00+0.05 0.64+0.00+0.04
0 −0.03−0.03 −0.02−0.07 −0.06−0.10
ABK2∗ 0.13+0.03+0.03 0.78+0.01+0.05 −0.11+0.02+0.04
1 −0.02−0.02 −0.01−0.04 −0.03−0.02
ABK2∗ 0.08+0.02+0.02 −− −−
2 −0.02−0.01
TBK2∗ 0.17+0.04+0.04 1.73+0.00+0.05 0.69+0.00+0.05
1 −0.03−0.03 −0.03−0.07 −0.08−0.11
TBK2∗ 0.17+0.03+0.04 0.79+0.00+0.02 −0.06+0.00+0.00
2 −0.03−0.03 −0.04−0.09 −0.10−0.16
TBK2∗ 0.14+0.03+0.03 1.61+0.01+0.09 0.52+0.05+0.15
3 −0.03−0.02 −0.00−0.04 −0.01−0.01
VBsf2′ 0.20+0.04+0.05 1.75+0.02+0.05 0.69+0.05+0.08
−0.03−0.03 −0.00−0.03 −0.01−0.01
ABsf2′ 0.16+0.03+0.03 1.69+0.00+0.04 0.64+0.00+0.01
0 −0.02−0.02 −0.01−0.03 −0.04−0.02
ABsf2′ 0.12+0.02+0.03 0.80+0.02+0.07 −0.11+0.05+0.09
1 −0.02−0.02 −0.00−0.03 −0.00−0.00
ABsf2′ 0.09+0.02+0.02 −− −−
2 −0.01−0.01
TBsf2′ 0.16+0.03+0.04 1.75+0.01+0.05 0.71+0.03+0.06
1 −0.03−0.02 −0.00−0.05 −0.01−0.08
TBsf2′ 0.16+0.03+0.04 0.82+0.00+0.04 −0.08+0.00+0.03
2 −0.03−0.02 −0.04−0.06 −0.09−0.08
TBsf2′ 0.13+0.03+0.03 1.64+0.02+0.06 0.57+0.04+0.05
3 −0.02−0.02 −0.00−0.06 −0.01−0.09
K−
l− φ
θ
θ K
l B
FIG. 1: Kinematics variables in the B → K¯∗(→ K−π+)l+l− process. The moving direction of K∗ in B rest frame is chosen
2 2
asthez axis. ThepolarangleθK (θl) isdefinedastheangle betweentheflightdirection ofK− (µ−)andthez axisin theK2∗
(lepton pair) rest frame. The convention also applies to Bs →f2′(→K+K−)l+l− transition.
a quasi four-body decay. The convention on the kinematics is illustrated in Fig. 1. The moving direction of K in B
2∗
meson rest frame is chosen as z axis. The polar angle θ (θ ) is defined as the angle between the flight direction of
K l
K (µ ) and the z axis in K (lepton pair) rest frame. φ is the angle defined by decay planes of K and the lepton
− − 2∗ 2∗
pair.
Using the technique of helicity amplitudes described in the appendix B, we obtain the partial decay width
d4Γ 3
= 2, (9)
dq2dcosθ dcosθ dφ 8|MB|
K l
5
with the mass correction factor β = 1 4m2/q2. The function 2 is decomposed into 11 terms
l − l |MB|
p
2 = IcC2+2IsS2+(IcC2+2IsS2)cos(2θ )+2I S2sin2θ cos(2φ)+2√2I CSsin(2θ )cosφ
|MB| 1 1 2 2 l 3 l 4 l
h+2√2I CSsin(θ )cosφ+2I S2cosθ +2√2I CSsin(θ )sinφ
5 l 6 l 7 l
+2√2I CSsin(2θ )sinφ+2I S2sin2θ sin(2φ) , (10)
8 l 9 l
i
with the angular coefficients
m2 m2
Ic = (A 2+ A 2)+8 l Re[A A ]+4 l A 2,
1 | L0| | R0| q2 L0 ∗R0 q2 | t|
3 4m2 4m2
Is = [A 2+ A 2+ A 2+ A 2] 1 l + l Re[A A +A A ],
1 4 | L⊥| | L||| | R⊥| | R||| − 3q2 q2 L⊥ ∗R⊥ L|| ∗R||
(cid:18) (cid:19)
Ic = β2(A 2+ A 2),
2 − l | L0| | R0|
1
Is = β2(A 2+ A 2+ A 2+ A 2),
2 4 l | L⊥| | L||| | R⊥| | R|||
1
I = β2(A 2 A 2+ A 2 A 2),
3 2 l | L⊥| −| L||| | R⊥| −| R|||
1
I = β2[Re(A A )+Re(A A ], I =√2β [Re(A A ) Re(A A )],
4 √2 l L0 ∗L|| R0 ∗R|| 5 l L0 ∗L⊥ − R0 ∗R⊥
I = 2β [Re(A A ) Re(A A )], I =√2β [Im(A A ) Im(A A )],
6 l L|| ∗L⊥ − R|| ∗R⊥ 7 l L0 ∗L|| − R0 ∗R||
1
I = β2[Im(A A )+Im(A A )], I =β2[Im(A A )+Im(A A )]. (11)
8 √2 l L0 ∗L⊥ R0 ∗R⊥ 9 l L|| ∗L⊥ R|| ∗R⊥
C = C(K ) and S = S(K ) for B K l+l . Without higher order QCD corrections, I is zero and I ,I are tiny
2∗ 2∗ → 2∗ − 7 8 9
in the SM and the reason is that only C has an imaginary part. In this sense these coefficients can be chosen as an
9
ideal window to probe new physics signals.
The amplitudes A are generated from the hadronic B K V amplitudes through A =
i → 2∗ Hi i
√λq2βl (K Kπ)
r3·32m3Bπ3B 2∗ → Hi
√λ 1 λ
AL0 = NK2∗√6mBmK2∗ 2mK2∗ q2 (cid:20)(C9−C10)[(m2B −m2K2∗ −q2)(mB +mK2∗)A1− mB+mK2∗A2]
p λ
+2mb(C7L−C7R)[(m2B +3m2K2∗ −q2)T2− m2B −m2K2∗T3]#,
√λ √λ
AL = NK∗ (C9 C10)[(mB +mK∗)A1 V]
± 2√8mBmK2∗ " − 2 ∓ mB+mK2∗
2m (C +C ) 2m (C C )
− b 7qL2 7R (±√λT1)+ b 7qL2− 7R (m2B −m2K2∗)T2 ,
(cid:21)
√λ √λ
ALt = NK∗ (C9 C10) A0, (12)
2√6mBmK∗ − q2
2
with NK2∗ =[3·2G12F0πα52emm3B|VtbVt∗s|2q2λ1/2 1− 4pqm22l 1/2B(K2∗ →Kπ)]1/2. For convenience,we have introduced transver-
sity amplitudes as (cid:16) (cid:17)
1
A = (A A ),
L⊥/|| √2 L+∓ L−
√λ √λV 2m (C +C )
AL⊥ = −√2√8mBmK2∗NK2∗"(C9−C10)mB +mK2∗ + b 7qL2 7R √λT1#,
√λ 2m (C C )
AL|| = √2√8mBmK2∗NK2∗(cid:20)(C9−C10)(mB +mK2∗)A1+ b 7qL2− 7R (m2B−m2K2∗)T2(cid:21), (13)
6
and the right-handed decay amplitudes are similar
A = A . (14)
Ri Li|C10→−C10
The combination of the timelike decay amplitude is used in the differential distribution
√λ √λ
At =ARt ALt =2NK∗ C10 A0. (15)
− 2√6mBmK∗ q2
2
p
B. Dilepton spectrum distribution
Integrating out the angles θ ,θ and φ, we obtain the dilepton mass spectrum
l K
dΓ 1
= (3Ic+6Is Ic 2Is), (16)
dq2 4 1 1 − 2 − 2
and its expression in the massless limit
dΓ
i = (A 2+ A 2), (17)
dq2 | Li| | Ri|
with i = 0, 1 or i = 0, , . After some manipulations in the appendix, the correspondence of the above equations
± ⊥ ||
and Eq. (20) with results in Ref. [25] can be shown.
C. Polarization distribution
The longitudinal polarization distribution for B →K∗2l+l− is defined as
df dΓ dΓ 3Ic Ic
L 0 = 1 − 2 , (18)
dq2 ≡ dq2 dq2 3Ic+6Is Ic 2Is
. 1 1 − 2 − 2
in which dΓ0 can be reduced into Ic in the case of m = 0 since Ic = Ic. The integrated polarization fraction is
dq2 1 l 1 − 2
given as
f Γ0 = dq2ddΓq20. (19)
L ≡ Γ R dq2ddqΓ2
R
D. Forward-backward asymmetry
The differential forward-backwardasymmetry of B →K∗2l+l− is defined by
dA 1 0 d2Γ 3
FB
= dcosθ = I , (20)
dq2 − ldq2dcosθ 4 6
(cid:20)Z0 Z−1(cid:21) l
while the normalized differential FBA is given by
dAFB dAdqF2B 3I6
= = . (21)
dq2 dΓ 3Ic+6Is Ic 2Is
dq2 1 1 − 2 − 2
In the massless limit, we have
dAdqF2B = 8m2Bλm2K∗ λ5q122Gπ2F5mα2e3Bm |VtbVt∗s|2Re"C9C10A1V +C10(C7L+C7R)mb(mBq+2 mK2∗)A1T1
2
mb(mB mK∗)
+C (C C ) − 2 T V . (22)
10 7L− 7R q2 2 #
In the SM where C is small, the zero-crossingpoint s of FBAs is determined by the equation
7R 0
mb(mB +mK∗) mb(mB mK∗)
C A (s )V(s )+C 2 A (s )T (s )+C − 2 T (s )V(s )=0. (23)
9 1 0 0 7L 1 0 1 0 7L 2 0 0
s s
0 0
7
E. Spin amplitudes and transverse asymmetries
Using the above helicity/spin amplitudes, it is also possible to construct several useful quantities which are ratios
of different amplitudes. The following ones, widely studied in the B K case, are stable against the uncertainties
∗
→
from hadronic form factors
A(T1) = ΓΓ−−+ΓΓ+ = −A2R2e(+A||AA∗⊥2),
+
− | ⊥| | |||
A 2 A 2
A(2) = | ⊥| −| ||| ,
T A 2+ A 2
| ⊥| | |||
A A +A A
A(3) = | L0 ∗L|| R0 ∗R|||,
T A 2 A 2
0
| | | ⊥|
A A A A
A(T4) = |AL0pA∗L⊥−+AR0A∗R⊥|, (24)
| L0 ∗L R0 ∗R |
|| ||
with the notation
AiA∗j =ALiA∗Lj +ARiA∗Rj. (25)
Due to thehierarchyinthe SMΓ Γ , A(1) iscloseto 1andthereforeits deviationfrom1ismoreusefultoreflect
− ≫ + T
the size of the NP effects.
IV. TWO NP MODELS
Theb sl+l hasasmallbranchingfractionsincetheSMislackoftreelevelFCNC.Itisnotnecessarilythesame
−
→
inextensions. InthissectionwewillbrieflygiveanoverviewoftwoNPmodels,whichallowtree-levelFCNC.Bothof
these two models, vector-like quark model and family non-universal Z model, do not introduce new type operators
′
but instead modify the Wilson coefficients C ,C . To achieve this goal, they introduce an SU(2) singlet down-type
9 10
quark or a new gauge boson Z .
′
A. Vector-like quark model: Z-mediated FCNCs
In the vector-like quark model, the new SU(2) singlet down quarks D and D modify the Yukawa interaction
L L R
sector
= Q¯ Y Hd +h Q¯ HD +m D¯ D +h.c., (26)
Y L D R D L R D L R
L
where the flavor indices have been suppressed. Q (H) is the SU(2) quark (Higgs) doublet, Y and h are the
L D D
Yuakwa couplings and m is the mass of exotic quark before electroweak symmetry breaking. When the Higgs field
D
acquires the vacuum expectation value (VEV), the mass matrix of down type quark becomes
Yij hi
D | D
m = , (27)
d  
− − −
0 m
D
 | 
 
which can be diagonalized by two unitary matrices
mddia = VDLmdVDR†. (28)
The SM coupling of Z-boson to fermions is flavor blind, and the flavor in the process with exchange of Z-boson
is conserved at tree level. Unlikely although the right-handed sector in the VQM is the same as the SM, the new
8
left-handed quark is SU(2) singlet, which carries the same hypercharge as right-handed particles. Therefore the
L
gauge interactions of left-handed down-type quarks with Z-boson are given by
g g
= Q¯ (I sin2θ Q)Z/Q +D¯ ( sin2θ Q)Z/D , (29)
Z L 3 W L L W L
L cosθ − cosθ −
W W
where g is the coupling constant of SU(2) , θ is the Weinberg’s angle, P =(1 γ )/2. I and Q are operators
L W R(L) 5 3
±
for the third component of the weak isospin and the electric charge, respectively.
Since the ratio ξ of the coupling constants deviates from unity: ξ = sin2θ Q /(IF sin2θ Q ), tree level
D D − W D 3 − W F
FCNC can be induced after the diagonalization of the down-type quarks. For instance, the interaction for b-s-Z in
the VQM is given by
gcsλ
= L sbs¯γµP bZ +h.c., (30)
b s L µ
L → cosθW
where λ is introduced as the new free parameter:
sb
λsb =(ξD−1)(VDL)sD(VDL)∗bD ≡|λsb|exp(iθs) .
Using Eq. (30), the effective Hamiltonian for b sl+l mediated by Z-boson is found by
−
→
2G
Z = Fλ cs(s¯b) cℓ(ℓ¯ℓ) +cℓ (ℓ¯ℓ) . (31)
Hb→sl+l− √2 sb L V−A L V−A R V+A
(cid:2) (cid:3)
The Wilson coefficients C are modified accordingly
9,10
4π λ cs(cℓ +cℓ ) 4π λ cs(cℓ cℓ )
CVLQ = CSM sb L L R , CVLQ =CSM+ sb L L− R . (32)
9 9 − α V V 10 10 α V V
em t∗s tb em t∗s tb
Makinguseoftheexperimentaldataofb sl+l ,ourpreviouswork[47]hasplacedaconstraintonthenewcoupling
−
→
constant
λ <1 10 3, (33)
sb −
| | ×
but its phase θ is less constrained. In the following, we shall see that the constraint can be improved by taking into
s
account the experimental data of the exclusive process B K l+l .
∗ −
→
B. Family non-universal Z′ model
The SM can be extended by including an additional U(1) symmetry, and the currents canbe givenas following in
′
a proper gauge basis
JZµ′ =g′ ψ¯iγµ[ǫψiLPL+ǫψiRPR]ψi, (34)
i
X
where i is the family index and ψ labels the fermions (up- or down-type quarks, or charged or neutral leptons).
According to some string construction or GUT models such as E , it is possible to have family non-universal Z
6 ′
couplings, namely, even though ǫL,R are diagonal the gauge couplings are not family universal. After rotating to the
i
physical basis, FCNCs generally appear at tree level in both LH and RH sectors. Explicitly,
BψL =VψLǫψLVψ†L, BψR =VψRǫψRVψ†R. (35)
Moreover,these couplings may contain CP-violating phases beyond that of the SM.
The Lagrangianof Z ¯bs couplings is given as
′
LZFC′NC =−g′(BsLbs¯LγµbL+BsRbs¯RγµbR)Z′µ+h.c.. (36)
It contributes to the b sℓ+ℓ decay at tree level with the effective Hamiltonian
−
→
Z′ = 8GF(ρLs¯ γ b +ρRs¯ γ b )(ρLℓ¯ γµℓ +ρRℓ¯ γµℓ ) , (37)
Heff √2 sb L µ L sb R µ R ll L L ll R R
9
where
g M
ρL,R ′ ZBL,R (38)
ff′ ≡ gMZ′ ff′
with the coupling g associatedwith the SU(2) groupin the SM. In this paper we shallnot take the renormalization
L
group running effects due to these new contributions into consideration because they are expected to be small. For
the couplings are all unknown, one can see from Eq. (37) that there are many free parameters here. For the purpose
of illustration and to avoid too many free parameters, we put the constraint that the FCNC couplings of the Z and
′
quarks only occur in the left-handed sector. Therefore, ρR = 0, and the effects of the Z FCNC currents simply
sb ′
modify the Wilsoncoefficients C andC in Eq.(A1). We denote these twomodified Wilsoncoefficients by CZ′ and
9 10 9
CZ′, respectively. More explicitly,
10
CZ′ = C 4π ρLsb(ρLll +ρRll), CZ′ =C + 4π ρLsb(ρLll −ρRll). (39)
9 9− α V V 10 10 α V V
em tb t∗s em tb t∗s
Comparedwith the Wilson coefficients in the vector-likequark model in Eq.(32), we can see that the Z contribu-
′
tions in Eq. (39) have similar forms and the correspondence lies in the coupling constants
λ cs ρL, cl ρL,R. (40)
sb L → sb L,R → ll
However the number of free parameters is increased from 2 to 4 since cl in the VQM is the same as the SM.
L,R
V. PHENOMENOLOGICAL ANALYSIS
Inthis section,we will presentourtheoreticalresults in the SM, giveanupdate ofthe constraintsinthe abovetwo
NP models and investigate their effects on B K µ+µ and B f µ+µ . For convenience, branching ratios of
→ 2∗ − s → 2′ −
K and f decays into Kπ and KK¯ will not be taken into account in the numerical analysis.
2∗ 2′
A. SM predictions
With the B K form factors computed in the PQCD approach [27], the BR, zero-crossing point of FBAs and
→ 2∗
polarization fractions are predicted as
(B K µ+µ ) = (2.5+1.6) 10 7,
B → 2∗ − −1.1 × −
fL(B →K2∗µ+µ−) = (66.6±0.4)%,
s0(B →K2∗µ+µ−) = (3.49±0.04)GeV2,
B(B →K2∗τ+τ−) = (9.6+−64..25)×10−10,
fL(B →K2∗τ+τ−) = (57.2±0.7)%. (41)
Theerrorsarefromtheformfactors,namely,fromtheB mesonwavefunctionsandthePQCDsystematicparameters.
Most of the uncertainties from form factors will cancel in the polarization fractions and the zero-crossing point s .
0
Similarly results for B f l+l are given as
s 2 −
→
(B f µ+µ ) = (1.8+1.1) 10 7,
B s → 2′ − −0.7 × −
f (B f µ+µ ) = (63.2 0.7)%,
L s → 2′ − ±
s (B f µ+µ ) = (3.53 0.03)GeV2,
0 s → 2′ − ±
(B f τ+τ ) = (5.8+3.7) 10 10,
B s → 2′ − −2.1 × −
f (B f τ+τ ) = (53.9 0.4)%. (42)
L s → 2′ − ±
We also show the q2-dependence of their differential branching ratios (in units of 10 7) in Fig. 2.
−
Charm-loop effects, due to the large Wilson coefficient and the large CKM matrix element, might introduce im-
portant effects. In a very recent work [20], the authors have adopted QCD sum rules to investigate both factorizable
10
0.5
0.012
0.4
0.010
®0LK2 0.3 ®0LK2 0.008
dB0HB2dq 0.2 dB0HB2dq 0.006
0.004
0.1
0.002
0.0 0.000
0 2 4 6 8 10 12 14 13.0 13.5 14.0 14.5
q2HGeV2L q2HGeV2L
0.4
0.008
0.3
®HLLf15252 0.2 ®HLLf15252 00..000046
dB0HBs2dq 0.1 dB0HBs2dq 0.002
0.0 0.000
0 2 4 6 8 10 12 14 13.0 13.5 14.0 14.5
q2HGeV2L q2HGeV2L
FIG. 2: Differential branching ratios of B →K2∗l+l− (upper) and Bs →f2′l+l− (lower) ( in units of 10−7): the left panel for
l=µand theright panel for l=τ.
diagrams and nonfactorizable diagrams. Their results up to the region q2 =m2 are parameterizedin the following
J/ψ
form,
r(i) 1 q¯2 +∆C(i)(q¯2)q¯2
∆C(i)B→K∗(q2)= 1 − q2 9 q2, (43)
9 (cid:16) 1+(cid:17)r2(i)q¯m2−2q2
J/ψ
where the three results correspond to different Lorentz structures: i = 1,2,3 for terms containing V, A and A
1 2
respectively. The numerical results are quoted as follows
∆C(1)(q¯2)=0.72+0.57, r(1) =0.10, r(1) =1.13,
9 0.37 1 2
−
∆C(2)(q¯2)=0.76+0.70, r(2) =0.09, r(2) =1.12,
9 0.41 1 2
−
∆C(3)(q¯2)=1.11+1.14, r(3) =0.06, r(3) =1.05. (44)
9 0.70 1 2
−
It should be pointed out that not all charm-loop effects in B K l+l are the same as the ones in B K l+l .
→ 2∗ − → ∗ −
Among various diagramsthe factorizable contributions, which can be simply incorporatedinto C given in Eq. (A3),
9
arethe same. Thenonfactorizableonesaremoresubtle. Inparticularthe light-conesumrules(LCSR)withB-meson
distributionamplitudesareadoptedinRef.[20],inwhichintermediatestateslikeK arepickedupasthegroundstate.
∗
ThegeneralizationisnotstraightforwardtothecaseofK sinceinthisapproachstatesbelowK maycontributeina
2∗ 2∗
substantialmanner. Howeverinanotherviewpoint,i.e. the conventionalLCSR,they maybe related. Inourprevious
work we have shown that the light-cone distribution amplitudes of K is similar with K in the dominant region of
2∗ ∗
the PQCD approach. If it were also the same in the conventionalLCSR, one may expect that the charm-loopeffects
in the processes under scrutiny have similar behaviors with the ones in B K l+l . Therefore as the first step to
∗ −
→
proceed, we will use their results to estimate the sensitivity in our following analysis and to be conservative, we use
∆C(i)B→K2∗(q¯2)=(1 1)∆C(i)B→K∗(q2) (45)
9 ± 9
intheregionof1GeV2 <q2 <6GeV2. The centralvaluesforq2-dependentparameterswillbe usedforsimplicityand
in this procedure, the factorizable corrections to C given in Eq. (A3) should be set to 0 to avoid double counting.
9