Abstract
We investigate the direct violation for the decay process of (P,V refer to the pseudoscalar meson and vector meson, resp.) via isospin symmetry breaking effects from the mixing mechanism in PQCD factorization approach. Isospin symmetry breaking arises from the electroweak interaction and the u-d quark mass difference by the strong interaction, which are known to be tiny. However, we find that isospin symmetry breaking at the leading order shifts the violation due to the new strong phases.
1. Introduction
The measurement of violation is an important area in understanding Standard Model (SM) and exploring new physics signals. Cabibbo-Kobayashi-Maskawa (CKM) matrix [1, 2] due to the quark flavour mixing provides us with the weak phases. The weak phase associated with the strong phase is responsible for the source of the CP violation. The strong phase comes from the dynamics of QCD and the other mechanism.
The hadronic matrix elements of the nonleptonic weak decay are known to be associated with the strong phase. We can estimate the power contribution by the factorization method in the limit of ( refers to b quark mass) in B meson decay process. Based on the QCD correction and taking into account transverse momenta, PQCD factorization method safely avoids the infrared divergence by introducing the Sudakov factor which is applied to deal with the decay amplitude related with the hadronic matrix elements. The decay amplitude can be written as the convolution of the meson wave functions and the hard kernel, which show the contributions of the non-perturbative and the perturbative parts, respectively [3–11].
Isospin symmetry plays an important part in the weak decay process of B meson. We can infer sum rule associated with the isospin symmetry to form a triangular shape on a complex plane for the decay amplitude. One can eliminate uncertainty from the penguin diagram by the isospin analysis in B decays [12]. Isospin symmetry breaking via - mixing produces the strong phase to lead to the large CP violation in the three bodies decay process [13, 14]. Isospin symmetry is approximate symmetry due to identical u and d quark masses in Standard Model (SM). The mixing of pseudoscalar mesons -- is from the isospin symmetry breaking within QCD. Isospin symmetry breaking plays a significant role for the decays of , which breaks the triangle relationship in the framework of generalized factorization [15]. -- mixing is discussed by the model-independent way in decay process using flavor SU symmetry [16]. The quark-flavor mixing produces the -- mixing due to the isospin symmetry breaking [17]. Recently, isospin symmetry breaking is discussed by incorporating the Nambu-Jona-Lasinio model in a generalized multiquark interaction scheme [18]. However, one can find that the research rarely pays attention to the CP violation from the effect of isospin symmetry breaking via -- mixing. The strong phase may be introduced to affect the value of CP violation accordingly which is similar to the contribution from the isospin symmetry breaking by the - mixing [13, 14].
The remainder of this paper is organized as follows. In Section 2 we present the form of the effective Hamiltonian. In Section 3 we give the calculating formalism of violation from isospin symmetry breaking in . Input parameters are presented in Section 4. We present the numerical results in Section 5. Summary and discussion are included in Section 6. The related functions defined in the text are given in the Appendix.
2. The Effective Hamiltonian
With the operator product expansion, the effective weak Hamiltonian can be written as [19]where represents Fermi constant, () are the Wilson coefficients, , , and are the CKM matrix elements. The operators have the following forms: 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.
We can obtain numerical values of . When [11],
One can obtain numerical values of . The combinations of Wilson coefficients are defined as [6–8]
3. Violation from Isospin Symmetry Breaking Effects
3.1. Formalism
It is convenient to introduce isospin vector triplet , isospin scalar and isospin scalar which can be distinguished by including strange quark or not. The singlet and octet can be well described by the translation and . The states of , and are identified by , and which are obtained from the quark model, respectively. The physical meson states can be transformed from the , and by unitary matrix [17]: where , and the higher order terms are neglected. In the isospin limit of , we can find that the formula is expressed as the mixing in Eq.(7): where is the mixing angle [20]. The and mixing depends on the quark flavor basises and .
The relevant decay constants can be written as [21, 22] where refers to the momenta of or .
One can understand that isospin symmetry breaking comes from the electroweak interaction and quark mass difference in Standard Model. We can calculate the isospin symmetry breaking correction by chiral perturbative theory which induces the mixing. To the leading order of isospin symmetry breaking, the physical eigenstate , and from Eq.(5)(6) can be written as One can define , . refer to the isospin component in the triplet. We use the values of , , [17].
The mixing is generated by the strong interaction from , where we assert , or neglecting higher-order terms for the isospin breaking. The pseudoscalar meson octet in current algebra[23] and the lowest order chiral perturbation theory[24] discuss the - mixing angle which can be written as with , so that the scale is expeted as .
Generally, the mesons and are composed of , , linear combination. One believes that there is gluon in the meson for explaining the large branching ratios for the decay processes and [25, 26]. H.N. Li et al. find the tiny contribution from the gluonic admixture [27, 28]. Z.J. Xiao et al. calculate the branching ratios and CP violations in PQCD for the decay processes and show that the theoretical result is in agreement with the experimental one neglecting the contribution of gluon [29–31]. Thomas shows the contribution from composition of gluon does not make obvious impact on fitted values on the basis of existing experiment data [32].
For the meson function, we use the model [20, 33] where the normalization factor is dependent on the free parameter . is the conjugate variable of the parton transverse momenta . refers to the mass of the meson. For the meson, one can obtain the value of from the light cone sum rule [34].
The wave functions of and are the same in form and can be defined as [26] where and are the momentum and the momentum fraction of (), respectively. Following Ref.[26], we assumed here that the wave function of () is the same as the wave function based on SU flavor symmetry. Depending on the assignment of the momentum fraction , the parameter should be chosen as +1 or -1. In this paper, we will use those distribution amplitudes [20]: where . are the decay constants of scalar (vector) mesons, respectively. The pseudoscalar mesons and have the similar wave functions. The expressions of amplitudes can be obtained by the replacements Gegenbauer polynomials are defined as
The wave function of the meson is applied to describe the formation of hadron from the positive and negative quarks, which provides distribution of the momentum carried by the parton. It is non-perturbative and process independent from partons to hadrons. The wave function of the meson is transverse-momentum-dependent wave fucnction in PQCD. The results show that there is large contribution from transverse-momentum for the heavy meson function. However, the wave function of light meson is less reliant on transverse-momentum. Currently, it is reasonable that the results form Light-cone QCD sum rule and Lattice QCD. The wave function of light pseudoscalar meson is usually obtained by Light-cone QCD sum rule. The branching ratios and CP violations are calculated in the framework of PQCD. The theoretical prediction is in good agreement with the experimental results for the most decay processes of . Hence, the wave function is credible. The transverse-momentum-dependent (TMD) wave fucnction is given with simpler soft subtraction in [35] which presents the more accurate results.
The transverse momentum cannot be neglected for eliminating the divergence in the endpoints. The double logarithms will be obtained when collinear divergent is overlapping with soft divergen considering radiative corrections for the form factor. For the effective perturbative expansion, we need to sum the double logarithmic terms by the resummation technique [36]. The wave function and hard kernel can absorb collinear divergent and infrared divergence, respectively. The other infrared divergences are cancelled out. At the range of large b, we need to sum the double logarithmic terms by renormalization group. Hence, the sudakov factor is introduced. Due to the suppression of sudakov factor, the non-perturbative contribution becomes very small. The amplitude of hard kernal produces the double logarithmic terms from radiative corrections. The threshold resummation introduces the sudakov factor which drops rapidly at the endpoints. Hence, perturbative calculation plays the dominant role when the endpoint singularity is suppressed.
Recently, an evolution for the B meson wave fucntions are constructed in the factorization theorem, whose solutions sum the double logarithms with the light-cone singularities, namely, the rapidity logarithms. The effective heavy-quark field is involved for an energetic light hadron comparing with the Sudakov resummation. The resummation effect maintians the normalization of the B meson wave functions and strengths their convergent behavior at small momentum, which combines the threshold and resummations [37]. Due to the resummation technique with off-light-cone Wilson lines, an evolution for the pion meson wave fucntions is introduced in the factorization theorem, whose solutions sum the mixed logarithm to all orders, with being a parton momentum fraction (transverse momentum). This joint resummation shows strong suppression of the pion wave fucnction in the small and large regions, being the impact parameter conjugate to . The next-to-leading-order contributions are different from those under the conventional resummations at high energy. The scheme-independent formalism can be extended to the factorization of more complicated exclusive processes. [38]. The improvement of the wave function can be applied to a more accurate discussion for the next step.
3.2. Calculation Details
In the framework of PQCD, we can calculate the violation for the decay process via mixing. Firstly, we calculate the amplitudes and , which can be decomposed in terms of tree and penguin contributions depending on the CKM matrix elements and . Next, we take the decay process and as examples for the study of the mixing mechanism.
3.2.1. The Violation for the Decay Modes of Except
We take the decay process of as example to introduce the violation via mixing. The decay amplitude A of in PQCD can be written as where and are the amplitudes form tree and penguin contributions, respectively. The tree level amplitude can be given as and the penguin level amplitude can be written as where the refers to the decay constant. The individual decay amplitudes in the above equations, such as , , , , , and arise from the , and operators, respectively, and will be given in Appendix.
Based on the CKM matrix elements of and , we can express the decay amplitudes as follows: The contributions of and for the decay amplitudes can be written as for the formula of .
The amplitudes T and P from the decay process of with mixing can be written as
One can see that the Eq.(23) without mixing is reduced to which are expressed in Eq.(15) and Eq.(16).
The relevant weak phase and strong phase are obtained as follows: where the parameter represents the absolute value of the ratio of penguin and tree amplitudes: The strong phase associated with can be given as where where , , are the Wolfenstein parameters.
The violation, , can be written as
3.2.2. The Violation of via Mixing
Due to the interference between and , the effect of the isospin symmetry breaking is more significant for the decay process of . Hence, the mixing, which including or meson, may shift the phase larger so as to have a bigger impact on violation. The decay amplitudes of with isospin symmetry are defined as
Taking into account of mixing, the decay amplitudes for in Eq.(30) can be written as We can define where and we have ignored the higher order term of . One can express .
In the same way, we can present the decay amplitudes for with mixing in Eq.(31)
Hence, depending on the CKM matrix elements and , we can express the decay amplitudes as follows: where and refer to the tree and penguin contributions from and in Eq.(32),(37), respectively. The relevant amplitudes can be obtained from the decay processes of , , , and . Combined with Eq.(32), (33), (34), (35), (37), (38) we can also obtain violation from the Eqs.(26), (27), (28) and (29).
4. Input Parameters
The CKM matrix, the elements of which are determined from experiments, can be expressed in terms of the Wolfenstein parameters , , and [39, 40]:where corrections are neglected. The latest values for the parameters in the CKM matrix are [41] where From Eqs. (40) (41) we have The other parameters are given as follows [39–41]:
5. Numerical Results
The CP violation depends on the weak phase differences from the CKM matrix elements and the strong phase differences associated with QCD. The CKM matrix elements are determined by the parameters of , , and . We find that the results for the violation are less reliant on and in the course of calculations. Hence, we present the violation from the weak phases associated with the and in the CKM matrix elements while the and are assigned for the central values. In Table 1, we show the values of violation of decay modes from isospin symmetry and isospin symmetry breaking via mixing. From Table 1, it can be seen that the increasing rate of the violation, which is defined % (where , represent the violation values from isospin symmetry and isospin symmetry breaking, respectively.), is larger in decay process compared with the other decay channels we are considering. It is intelligible that the final states for the decay process include the or meson. Due to the isospin symmetry breaking, the interference between the and mesons is stronger than other decay channels whose final states do not contain or meson. Hence, these decay channels including or meson make the strong phase larger resulting in a great impact on violation. We can find that the violation of the decay mode has not changed much in Table 1. The violation of the decay mode is changed from % to %. From Table 1, one can also see that the isospin symmetry breaking changes the sign of the violation, for example, from % to % for the decay channel of , from % to % for the decay channel of . The increasing rate of violation for the decay mode is % for the central value.
From Table 1, we can find great changes between the values of violation from isospin symmetry and isospin symmetry breaking via mixing. In order to study the influence of the weak phase on violation and understand the mixing mechanism, we present the violation as a function of and in Figure 1 while taking the mixing parameters and as central values. We vary from the limiting values to , respectively, in Figure 1. Due to the effect of weak phases from CKM matrix elements, the value of violation for the decay process of changes from % to % taking into account of isospin symmetry breaking.
It can be seen from the Eq.(29) that the value of direct violation is also dependent on and . We take the decay channel of as an example. When and are taken as the central value for the CKM matrix elements, we present the direct violation as a function of , in Figure 2(a) from the isospin symmetry and in Figure 2(b) from isospin symmetry breaking via mixing. Comparing the Figure 2(a) with the Figure 2(b), the violation value has a great change. Only considering the central value, the value of violation changes from % in Figure 2(a) to % in Figure 2(b) and shifts the sign. In Figures 3 and 4, we give the numerical result of and for the decay process of . Comparing Figure 3(a) with Figure 3(b), we can find that the value of changes the sign from in Figure 3(a) to the central value in Figure 3(b). In Figure 4, the central value of changes large comparing the result of isospin symmetry breaking in Figure 4(b) to the value from isospin symmetry in Figure 4(a). Based on the changes of and , large violation is obtained from isospin symmetry breaking via mixing.
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6. Summary and Conclusion
In this paper, we study the violation for the decay process of in Perturbative QCD. It is found that the violation can be shifted via mixing from the isospin symmetry breaking. The violation arises from the weak phase difference in CKM matrix and the strong phase difference. The violation changes small for the decay mode via mixing and the central value of the increasing rate is %. The rate of increase of the violation is larger for the decay process of than other decay channels. Due to the breaking of isospin symmetry, the interference between the and mesons is stronger than other decay channels which in the final state does not contain or meson. For the decay process and , the isospin symmetry breaking changes the sign of the violation.
In order to achieve the required energy and luminosity requirements, the Large Hadron Collider (LHC), which has currently started at CERN, has been upgraded many times. The LHC Run I data started in 2010. The peak instantaneous luminosity documentary during Run I was . The center-of mass energy was primarily = 7 TeV and was raised to 8 TeV in 2012 [42]. This was followed by the first long shutdown period (LS1), which was devoted to upgrades essential for increasing beam energy to = 13 TeV centre of mass energy and peak instantaneous luminosity [43, 44]. In the following years, there are two primary detector (CMS and ATLAS) upgrades happening after Run II and Run III. Phase-I and II upgrade prepares for an instantaneous luminosity of and [45], respectively. With a series of modifications and upgrades, the LHC gives access to high energy frontier at TeV scale and an occasion to further improve the consistency test for the CKM matrix. The production rates for heavy quark flavors will be great at the LHC, and the production cross section will be of the order 0.5 mb, providing as many as bottom events per year [42, 46]. The heavy quark physics is one of the major topics of LHC experiments. Especially, the LHCb experiment exploits amounts of mesons, produced in proton-proton collisions at the LHC to search for violation. Recently, LHCb Collaboration presents observation of the decay meson. Obtaining more data from LHC, it is possible to make further analysis for violation of decays [47]. We expect that our results is valuable for measurement of violation of decays in the following LHCb experiments.
Appendix
Related Functions Defined in the Text
The functions related with the tree and penguin contributions are presented for the factorization and non-factorization amplitudes with PQCD approach [9–11, 20].
The hard scales are chosen as
The function comprises the jet function arising from the threshold re-summation[48] and the propagator of virtual quark and gluon [9–11, 20]. They are defined by where .
The re-sums the threshold logarithms appearing in the hard kernels to all orders and it has been parameterized as with . In the nonfactorizable contributions, gives a very small numerical effect on the amplitude [49]. Therefore, we drop in and .
The evolution factors and are given by [9–11, 20] in which the Sudakov exponents are defined as with the quark anomalous dimension . Replacing the kinematic variables of to in , we can get the expression for . The explicit form for the function is [9–11, 20]: where the variables are defined by and the coefficients and are is the number of the quark flavors and is the Euler constant. We will use the one-loop running coupling constant; i.e., we pick up the four terms in the first line of the expression for the function [9–11, 20].
The , and refer to the contributions from operators, operators and operators, respectively. The form factor of can be given [9–11, 20] as(i) operators: (ii) operators: (iii) operators:
where the color factor and represents the corresponding Wilson coefficients from differen decay channels. , where refers to the chiral scale parameter.(i) operators: (ii) operators: (iii) operators:
The functions are related with the annihilation type process, whose contributions are:(i) operators: (ii) operators: (iii) operators: (i) operators: (ii) operators: (iii) operators:
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by National Natural Science Foundation of China (Project Numbers 11605041), and the Research Foundation of the young core teacher from Henan province.