CP violation induced by the double resonance for pure annihilation decay process in Perturbative QCD

# Cp violation induced by the double resonance for pure annihilation decay process in Perturbative QCD

Gang Lü111Email: ganglv66@sina.com, Ye Lu222Email: luye189@163.com, Sheng-Tao Li and Yu-Ting Wang College of Science, Henan University of Technology, Zhengzhou 450001, China
Department of Physics, Guangxi Normal University, Guilin 541004, China
###### Abstract

In Perturbative QCD (PQCD) approach we study the direct violation in the pure annihilation decay process of induced by the and double resonance effect. Generally, the violation is small in the pure annihilation type decay process. However, we find that the violation can be enhanced by double interference when the invariant masses of the pairs are in the vicinity of the resonance. For the decay process of , the maximum violation can reach 28.64%.

###### pacs:
11.30.Er, 12.39.-x, 13.20.He, 12.15.Hh

## I Introduction

violation is an important area in searching new physics signals beyond the standard model(SM). It is generally believed that the meson system provides rich information about violation. The theoretical work has been done in this direction in the past few years. violation arises from the weak phase in the Cabibbo-Kobayasgi-Maskawa (CKM) matrix cab (); kob () in SM. Meanwhile, it is remarkable that violation can still be produced by the interference effects between the tree and penguin amplitudes. Since the kinematic suppression, the strong phase associated with long distance rescattering is generally neglected during the past decades. Recently, the LHCb Collaboration found the large violation in the three-body decay channels of and J.M. (); R.A.1 (); R.A.2 (). Hence, the nonleptonic meson decay from the three-body and four-body decay channels has been become an important area in searching for violation.

A mixing between the and flavor leads to the breaking of isospin symmetry for the system. The chiral dynamics has been shown restore the isospin symmetry chi-Nu (). The mixing matrix element gives rise to isospin violation, where is the Mandelstam variable. The magnitude has been extracted by the pion form factor through the cross section of . We can separate the into two contribution of the direct coupling of and the mixing of . The emergence of arises from the inclusion of a nonresonant contribution to . The appearance of the and resonance is associated with complex strong phase from relatively broad resonance region. Especially, there is perhaps larger strong phase from double and interference. The violation origins from the weak phase difference and the strong phase difference. Hence, the decay process of is a great candidate for studying the origin of the violation.

Meanwhile, it is known that the violation is extremely tiny from the pure annihilation decay process in experiment. There is relatively large error in dealing with the decay amplitudes from the QCD factorization approach qcdf (). The perturbative QCD (PQCD) factorization approach Ali:1997nh (); 9804363 (); pqcd (); pqcd1 () is based on factorization. The amplitude can be divided into the convolution of the Wilson coefficients, the light cone wave function, and hard kernels by the low energy effective Hamiltonian. The endpoint singularity can be eliminated by introducing the transverse momentum. However, The transverse momentum integration leads to the double logarithm term which is resummed into the Sudakov form factor. The nonperturbative dynamics are included in the meson wave function which can be extracted from experiment. The hard one can be calculated by perturbation theory.

The remainder of this paper is organized as follows. In Sec. II we present the form of the effective Hamiltonian. In Sec. III we give the calculating formalism and calculation details of violation from mixing in the decay. In Sec. IV we show input parameters. We present the numerical results in Sec. V. Summary and discussion are included in Sec. VI. The related function defined in the text are given in the Appendix.

## Ii The effective hamiltonian

With the operator product expansion, the effective weak Hamiltonian can be written as buch ()

 Heff = GF√2{VubV∗uq[C1(μ)Qu1(μ)+C2(μ)Qu2(μ)]−VtbV∗tq[10∑i=3Ci(μ)Qi(μ)]}+% H.c., (1)

where , represents Fermi constant, (i=1,…,10) are the Wilson coefficients, ( and represent quarks) is the CKM matrix element, and is the four quark operator. The operators have the following forms:

 Ou1 = ¯dαγμ(1−γ5)uβ¯uβγμ(1−γ5)bα, Ou2 = ¯dγμ(1−γ5)u¯uγμ(1−γ5)b, O3 = ¯dγμ(1−γ5)b∑q′¯q′γμ(1−γ5)q′, O4 = ¯dαγμ(1−γ5)bβ∑q′¯q′βγμ(1−γ5)q′α, O5 = ¯dγμ(1−γ5)b∑q′¯q′γμ(1+γ5)q′, O6 = ¯dαγμ(1−γ5)bβ∑q′¯q′βγμ(1+γ5)q′α, O7 = 32¯dγμ(1−γ5)b∑q′eq′¯q′γμ(1+γ5)q′, O8 = 32¯dαγμ(1−γ5)bβ∑q′eq′¯q′βγμ(1+γ5)q′α, O9 = 32¯dγμ(1−γ5)b∑q′eq′¯q′γμ(1−γ5)q′, O10 = 32¯dαγμ(1−γ5)bβ∑q′eq′¯q′βγμ(1−γ5)q′α, (2)

where and are color indices, and or quarks. In Eq.(2) and are tree operators, are QCD penguin operators and are the operators associated with electroweak penguin diagrams. can be written pqcd1 (),

 C1 = −0.2703,C2=1.1188, C3 = 0.0126,C4=−0.0270, C5 = 0.0085,C6=−0.0326, C7 = 0.0011,C8=0.0004, C9 = −0.0090,C10=0.0022. (3)

So, we can obtain numerical values of . The combinations of Wilson coefficients are defined as usual 9804363 ():

 a1 = C2+C1/3,a2=C1+C2/3, a3 = C3+C4/3,a4=C4+C3/3, a5 = C5+C6/3,a6=C6+C5/3, a7 = C7+C8/3,a8=C8+C7/3, a9 = C9+C10/3,a10=C10+C9/3. (4)

## Iii Cp violation in ¯B0s→ρ0(ω)ρ0(ω)→π+π−π+π−

### iii.1 Formalism

The amplitudes of the process can be written Kram1991 ()

 Aσ=ϵ∗1μ(σ)ϵ∗2ν(σ)(agμν+bm1m2pμpν+icm1m2ϵμναβp1αp2β) (5)

where is the helicity of the vector meson. () and () are the polarization vectors (momenta) of and , respectively. and refer to the masses of the vector mesons and . The invariant amplitudes a, b, c are associated with the amplitude ( i refer to the three kind of polarizations, longitudinal (L), normal (N) and transverse (T)). Then we have

 Aσ=M2BsAL+M2BsANϵ∗1μ(σ=T)⋅ϵ∗2μ(σ=T)+iATϵαβγρϵ∗1α(σ)ϵ∗2α(σ)p1γp2ρ (6)

The longitudinal , transverse of helicity amplitudes can be expressed , . The decay width is written

 Γ=Pc8πM2BsA(σ)+A(σ)=Pc8πM2Bs|H0|2+|H+|2+|H−|2. (7)

The interaction of the photon and the hadronic matter can be described by the vector meson dominance model (VMD) Sakurai1969 (). The photon can couple to the hadronic field through a meson. The mixing matrix element is extracted from the data of the cross section for Connell1997 (); Connell1997-1 (). The nonresonant contribution of has been effectively absorbed into which leads to the explicit dependence of oco (). We can make the expansion . However, one can neglect the dependence of in practice. The mixing parameters were determined in the fit of Gardner and O’Connell gard ():

 Re˜Πρω(m2ω) = −3500±300MeV2, Im˜Πρω(m2ω) = −300±300MeV2, ˜Π′ρω(m2ω) = 0.03±0.04. (8)

The formalism of the violation is presented for the meson decay process in the following. The amplitude () for the decay process () can be written as:

 A=<π+π−π+π−|HT|¯B0s>+<π+π−π+π−|HP|¯B0s>, (9)
 ¯A=<π+π−π+π−|HT|B0s>+<π+π−π+π−|HP|B0s>, (10)

where and refer to the tree and penguin operators in the Hamiltonian, respectively. We define the relative magnitudes and phases between the tree and penguin operator contributions as follows:

 (11) (12)

where and are strong and weak phases, respectively. The weak phase difference can be expressed as a combination of the CKM matrix elements: . The parameter is the absolute value of the ratio of tree and penguin amplitudes:

 r≡∣∣∣⟨π+π−π+π−|HP|¯B0s⟩⟨π+π−π+π−|HT|¯B0s⟩∣∣∣. (13)

The parameter of violating asymmetry, , can be written as

 ACP=|A|2−|¯A|2|A|2+|¯A|2=−2(T20r0sinδ0+T2+r+sinδ++T2−r−sinδ−)sinϕ∑i=0+−T2i(1+r2i+2ricosδicosϕ), (14)

where

 |A|2=∑σA(σ)+A(σ)=|H0|2+|H+|2+|H−|2 (15)

and represent the tree-level helicity amplitudes. We can see explicitly from Eq. (14) that both weak and strong phase differences are responsible for violation. mixing introduces the strong phase difference and well known in the three body decay processes of the bottom hadron guo1 (); guo11 (); lei (); guo2 (); gang1 (); gang2 (); gang3 (). Due to interference from the u and d quark mixing, we can write the following formalism in an approximate from the first order of isospin violation:

 ⟨π+π−π+π−|HT|¯B0s⟩=2g2ρs2ρsω˜Πρωtρω+g2ρs2ρtρρ, (16) ⟨π+π−π+π−|HP|¯B0s⟩=2g2ρs2ρsω˜Πρωpρω+g2ρs2ρpρρ, (17)

where and are the tree (penguin) amplitudes for and , respectively, is the coupling for , is the effective mixing amplitude which also effectively includes the direct coupling . , and (= or ) is the inverse propagator, mass and decay rate of the vector meson , respectively.

 sV=s−m2V+imVΓV, (18)

with being the invariant masses of the pairs. There are double interference in the decay process of . Hence, a factor of 2 appears in Eqs. (16), (17) compared with the case of single interference eno (); gar (); guo1 (); guo2 (); guo11 (); lei (); gang1 (); gang2 (); gang3 (). From Eqs. (9)(11)(16)(17) one has

 reiδeiϕ=2˜Πρωpρω+sωpρρ2˜Πρωtρω+sωtρρ, (19)

Defining

 pρωtρρ≡r′ei(δq+ϕ),tρωtρρ≡αeiδα,pρρpρω≡βeiδβ, (20)

where , and are strong phases, one finds the following expression from Eqs. (19)(20):

 reiδ=r′eiδq2˜Πρω+βeiδβsω2˜Πρωαeiδα+sω. (21)

In order to obtain the violating asymmetry in Eq. (14), sin and cos are needed, where is determined by the CKM matrix elements. In the Wolfenstein parametrization wol (), one has

 sinϕ = −η√ρ2+η2, cosϕ = −ρ√ρ2+η2. (22)

### iii.2 Calculation details

We can decompose the decay amplitude for the decay process in terms of tree-level and penguin-level contributions depending on the CKM matrix elements of and . Due to the equations (14)(19)(20), we calculate the amplitudes , , and in perturbative QCD approach. The and function associated with the decay amplitudes can be found in the appendix from the perturbative QCD approach.

There are four types of Feynman diagrams contributing to (,= or ) annihilation decay mode at leading order. The pure annihilation type process can be classified into factorizable diagrams and non-factorizable diagrams kphi (); krho (). Through calculating these diagrams, we can get the amplitudes , where standing for the longitudinal and two transverse polarizations. Because these diagrams are the same as those of and decays kphi (); krho (), the formulas of or are similar to those of and . We just need to replace some corresponding wave functions, Wilson coefficients and corresponding parameters.

With the Hamiltonian (1), depending on CKM matrix elements of and , the decay amplitudes for in PQCD can be written as

 √2A(i)(¯B0s→ρ0ρ0)=VubV∗ustiρρ−VtbV∗tspiρρ, (23)

The tree level amplitude can written as

 tiρρ = GF√2{fBsFLL,iann[a2]+MLL,iann[C2]}, (24)

where refers to the decay constant of meson.

The penguin level amplitude are expressed in the following

 piρρ = GF√2{fBsFLL,iann[2a3+12a9]+fBsFLR,iann[2a5+12a7] (25) +MLL,iann[2C4+12C10]+MSP,iann[2C6+12C8]}.

The decay amplitude for can be written as

 2A(i)(¯B0s→ρ0ω) = VubV∗ustiρω−VtbV∗tspiρω. (26)

We can give the tree level the contribution in the following

 tiρω = GF√2{fBsFLL,iann[a2]+MLL,iann[C2]}, (27)

and the penguin level contribution are given as following

 piρω = GF√2VtbV∗ts{fBsFLL,iann[32a9]+fBsFLR,iann[32a7] (28)

Based on the definition of (20), we can get

 αeiδα = tρωtρρ, (29) βeiδβ = pρρpρω, (30) r′eiδq = pρωtρρ×∣∣∣VtbV∗tsVubV∗us∣∣∣, (31)

where

 ∣∣∣VtbV∗tsVubV∗us∣∣∣=√ρ2+η2λ2(ρ2+η2).\vspace2mm (32)

## Iv Input parameters

The CKM matrix, which elements are determined from experiments, can be expressed in terms of the Wolfenstein parameters , , and wol ():

 ⎛⎜ ⎜ ⎜⎝1−12λ2λAλ3(ρ−iη)−λ1−12λ2Aλ2Aλ3(1−ρ−iη)−Aλ21⎞⎟ ⎟ ⎟⎠, (33)

where corrections are neglected. The latest values for the parameters in the CKM matrix are ganglvpqcdbc ():

 λ=0.22537±0.00061,A=0.814+0.023−0.024, ¯ρ=0.117±0.21,¯η=0.353±+0.013. (34)

where

 ¯ρ=ρ(1−λ22),¯η=η(1−λ22). (35)

From Eqs. (34) ( 35) we have

 0.121<ρ<0.158,0.336<η<0.363. (36)

The other parameters and the corresponding references are listed in Table.1.

## V The numerical results of Cp violation in ¯B0s→ρ0(ω)ρ0(ω)→π+π−π+π−

In the numerical results, we find that the violation can be enhanced via double mixing for the pure annihilation type decay channel when the invariant mass of is in the vicinity of the resonance within perturbative QCD scheme. The violation depends on the weak phase difference from CKM matrix elements and the strong phase difference which is difficult to control. The CKM matrix elements, which relate to , , and , are given in Eq.(34). The uncertainties due to the CKM matrix elements come from , , and . In our numerical calculations, we let , , and vary among the limiting values. The numerical results are shown from Fig. 1 to Fig. 3 with the different parameter values of CKM matrix elements. The dash line, dot line and solid line corresponds to the maximum, middle, and minimum CKM matrix element for the decay channel of , respectively. We find the results are not sensitive to the values of , , and . In Fig. 1, we give the plot of violating asymmetry as a function of . From the Fig. 1, one can see the violation parameter is dependent on and changes rapidly due to mixing when the invariant mass of is in the vicinity of the resonance. From the numerical results, it is found that the maximum violating parameter reaches in the case of (, ).

From Eq.(14), one can see that the violating parameter depend on both sin and . The plots of and as a function of are shown in Fig. 2, and Fig. 3, respectively. It can be seen that ( and ) vary sharply at the range of the resonance in Fig. 2. One can see that change largely in the vicinity of the resonance.

## Vi Summary and conclusion

In this paper, we study the violation for the pure annihilation type decay process of in perturbative QCD. It has been found that the violation can be enhanced greatly at the area of resonance. The maximum violation value can reach due to double and resonance.

The theoretical errors are large which follows to the uncertainties of results. Generally, power corrections beyond the heavy quark limit give the major theoretical uncertainties. This implies the necessity of introducing power corrections. Unfortunately, there are many possible power suppressed effects and they are generally nonperturbative in nature and hence not calculable by the perturbative method. There are more uncertainties in this scheme. The first error refers to the variation of the CKM parameters, which are given in Eq.(34). The second error comes from the hadronic parameters: the shape parameters, form factors, decay constants, and the wave function of the meson. The third error corresponds to the choice of the hard scales, which vary from 0.75t to 1.25t, which characterizing the size of next-to-leading order QCD contributions. Therefore, the results for violating asymmetrie of the decay process is given as following:

 ACP(¯B0s→π+π−π+π−)=28.43+0.21+0.25+5.62−0.25−0.16−3.98%, (37)

where the first uncertainty is corresponding to the CKM parameters, the second comes from the hadronic parameters, and the third is associated with the hard scales. The LHC experiment may detect the large violation for the decay process in the region of the resonance.

## Vii APPENDIX: Related functions defined in the text

In this appendix we present explicit expressions of the factorizable and non-factorizable amplitudes with Perturbative QCD in Eq.(23) and Eq.(26) pqcd (); pqcd1 (); Lcd:the two-body (); prd81014022 (). The factorizable amplitudes , and (i=L,N,T) are written as

 fBsFLL,Nann(ai) = fBsFLR,Nann(ai) (38)
 fBsFLL,Nann(ai) = −8πCFM4BsfBsr2r3∫10dx2dx3∫∞0b2db2b3db3{Ea(tc)ai(tc)ha(x2,1−x3,b2,b3)) (39) [(2−x3)(ϕv2(x2)ϕv3(x3)+ϕa2(x2)ϕa3(x3))+x3(ϕv2(x2)ϕa3(x3)+ϕa2(x2)ϕv3(x3))] −ha(1−x3,x2,b3,b2)[(1+x2)(ϕv2(x2)ϕv3(x3)+ϕa2(x2)ϕa3(x3)) −(1−x2)(ϕv2(x2)ϕa3(x3)+ϕa2(x2)ϕv3(x3))]Ea(t′c)ai(t′c)}.
 fBsFLL,Tann(ai) = −fBsFLR,Tann(ai) (40)
 fBsFLL,Tann(ai) = −16πCFM4BsfBsr2r3∫10dx2dx3∫∞0b2db2b3db3{[x3(ϕv2(x2)ϕv3(x3)+ϕa2(x2)ϕa3(x3)) (41) +(2−x3)(ϕv2(x2)ϕa3(x3)+ϕa2(x2)ϕv3(x3))]Ea(tc)ai(tc)ha(x2,1−x3