A Combined Fit on the Annihilation Corrections in Decays Within QCDF
Abstract
Motivated by the possible large annihilation contributions implied by recent CDF and LHCb measurements on nonleptonic annihilation meson decays, and the refined experimental measurements on hadronic meson decays, we study the strength of annihilation contributions within QCD factorization (QCDF) in this paper. With the available measurements of twobody , , decays, a comprehensive fit on the phenomenological parameters (or and ) which are used to parameterize the endpoint singularity in annihilation amplitudes is performed with the statistical approach. It is found that (1) flavor symmetry breaking effects are hardly to be distinguished between and due to the large experimental errors and theoretical uncertainties, where and are related to the nonfactorization annihilation contributions in and decay, respectively. So is a good approximation by now. (2) In principle, parameter which is related to the factorization annihilation contributions and independent of the initial state can be regarded as the same variable for decays. (3) Numerically, two solutions are found, one is and , the other is and . Obviously, nonfactorization annihilation parameter is generally unequal to factorization annihilation parameter , which differ from the traditional treatment. With the fitted parameters, all results for observables of , , decays are in good agreement with experimental data.
pacs:
12.39.St 13.25.Hw 14.40.NdWith the running of the Large Hadron Collider (LHC), many intriguing meson decays are well measured and some interesting phenomena are found by LHCb collaboration in the past years. For example, measurements of branching fractions for the pure annihilation and decays LHCbanni (). Their averaged results given by Heavy Flavor Averaging Group (HFAG) are HFAG ()
(1) 
(2) 
which attract much attention recently xiao1 (); zhu1 (); zhu2 (); chang1 ().
Theoretically, the branching ratios of pure annihilation nonleptonic meson decays are formally power suppressed and expected at level, which roughly agrees with the measurements. In the framework of QCD factorization (QCDF) Beneke1 (), the annihilation amplitudes, together with the chirally enhanced power corrections and possible large strong phase involved in them, play an important role in evaluating the observables of meson decays. However, due to the endpoint singularities, the amplitudes of annihilation topologies are hardly to be exactly calculated. To estimate the endpoint contributions, phenomenological parameter is introduced Beneke2 () as
(3) 
where . The QCDF approach itself cannot give some information/or constraint on parameters and . To simplify the calculation, one usually takes the same parameters and for factorizable and nonfactorizable annihilation topologies. And as a conservative choice, the values of and (named scenario S4) Beneke2 (); Cheng1 (); Cheng2 () are usually adopted in previous studies on decays, which leads to the prediction^{1}^{1}1The second uncertainty comes from parameters and . Cheng1 () and Cheng2 (). Clearly, the QCDF’s prediction on agrees well with the current measurements considering the experimental and theoretical errors, while the QCDF’s prediction on is much smaller than the experimental data Eq.(2) by about , which implies unexpectedly possible large annihilation corrections and possible large flavor symmetry breaking effects between the annihilation amplitudes of and decays zhu2 (); zhu1 (). Motivated by such mismatch, some works have been done for possible solutions and implications.
Within the QCDF framework, using the asymptotic lightcone distribution amplitudes, the building blocks of annihilation amplitudes are simplified as Beneke1 (); Beneke2 ()
(4)  
(5)  
(6) 
where the superscripts “” and “” refer to gluon emission from the initial and final states, respectively; the subscripts “1”, “2” and “3” correspond to three possible Dirac structures, with “1” for , “2” for , and “3” for , respectively; is negligible for light final pseudoscalars due to . The explicit expressions of effective annihilation coefficients could be found in Ref. Beneke1 (); Beneke2 ().
For the annihilation parameters in Eqs. (46), although there are no imperative and a priori reasons for it to be the same in the building blocks ( , , ), the simplification is commonly used in many previous works of nonleptonic decays Beneke1 (); Beneke2 (); Cheng2 (); du1 (); Cheng1 (), independent of mesons involved and topologies. However, the carefully renewed study in Refs. zhu1 (); chang1 () shows that it is hardly to accommodate all available observables of charmless decays simultaneously with the universal and . Recently, a refreshing suggestion was proposed in Refs. zhu2 (); zhu1 () to cope with the parameters . The main points of “new treatment” could be briefly summarized as follow:

As the superscripts of correspond to different topologies, parameters of and should be treated individually.

For the factorizable annihilation topologies, the information of initial state has been included in the decay constant of meson and taken outside from the building blocks of . Only the wave functions of final states are involved in the convolution integral of subamplitudes. Additionally, the same asymptotic light cone distribution amplitude is commonly applied to the final pseudoscalar and vector mesons. So, the parameter should be universal for factorizable annihilation amplitudes of both and nonleptonic decays.

For the nonfactorizable annihilation topologies, the initial meson entangles with the final states via gluon exchange. The wave functions of all participating hadrons, including the initial meson, are involved in the convolution integral of subamplitudes. Hence, the parameter might be different from the parameter generally. Moreover, due to the mass relationship resulting in the flavor symmetry breaking, it is usually assumed that the momentum fraction of the valence quark in meson should be larger than that of the spectator , quark in meson. The flavor symmetry breaking effects might be embodied in parameter , i.e. two parameters, and , should be introduced for nonfactorizable annihilation topologies of and meson decay, respectively, while the isospin symmetry holds approximately. Generally, it is not required that must be equal or unequal to , i.e., and are independent variables.
With this assumption, authors of Refs.zhu2 (); zhu1 () reanalyzed , , decays without considering theoretical uncertainties and found that the experimental data on in Eq.(2) could be explained with large . Compared with in Beneke2 (); Cheng1 () for , it seems to imply unexpectedly large flavor symmetry breaking effects, then the predictive power of QCD will be rather limited. Thanks to the large experimental errors, can be fitted within a large range of including zhu1 (); chang2 (). Therefore, flavor symmetry might be restored as both aforementioned decays could be accommodated by a common set of . It is interesting and essential to systematically evaluate the exact strength of annihilation contribution and further test the aforementioned points, especially the flavor asymmetry effects.
As it is well known, additional phenomenological parameters (or and ), like to Eq.(3), were introduced to regulate the endpoint singularity in the hard spectator scattering (HSS) corrections involving the twist3 light cone distribution amplitudes of light final states Beneke1 (); Beneke2 (); Cheng2 (); du1 (); Cheng1 (). The phenomenological importance of HSS corrections to the colorsuppressed tree contributions which are enhanced by the large Wilson coefficient has already been recognized by Refs.Cheng1 (); pipipuz (); chang2 () in explicating the current experimental measurements on and . Because the wave functions are also involved in the HSS convolution integral, the flavor symmetry breaking effects might be also embodied in parameter .
Following the ansatz in Ref. zhu2 (); zhu1 (), we preform a global fit on the annihilation parameters combining available experimental data on , , decays with a statistical analysis. Based on our previous analysis chang2 (), the approximation, (, ) (, ), is acceptable by current measurements on decays (see scenario III in Ref. chang2 () for details), which lessens effectively the unknown variables. Hence, the approximation is assumed for decays in the following analysis. The detailed explanation on the fitting approach could be found in the Appendix C of Ref. chang2 (). The values of input parameters used in our evaluations are summarized in Table 1.
, , , CKMfitter () 
GeV, GeV, GeV, 
, MeV, GeV PDG14 () 
MeV, MeV, DecayCon () 
MeV, MeV PDG14 () 
, , , BallZwicky () 
, , , BallG () 
Firstly, to clarify the flavor symmetry breaking effects on parameters , we perform a fit on the annihilation parameters for and decays, respectively. For parameters of , the constraints come from observables of the , , decays. The fitted results are shown in Fig.1. For parameters of , they have been fitted with the constraints from , , decays, especially, focusing on the socalled “” and “” puzzles (see Ref. chang2 () for the details). Their allowed regions (green points) at 68% C.L. are also shown in Fig.1 for a comparison with .
From Fig.1(a), it is seen clearly that (1) the region of cannot be seriously constrained by now, because the current measurements on , , decays are not accurate enough and the theoretical uncertainties are also still large. Moreover, a relatively large with in system suggested by recent studies zhu2 (); zhu1 (); chang1 () is allowed. (2) The conventional choice of and Beneke2 (); Cheng1 (); Cheng2 () is ruled out, because the assumption is used in our study to enhance the magnitude and the strong phase of the colorsuppressed tree amplitude via spectator interactions and to solve both “” and “” puzzles chang2 (). Besides, a relatively large with is allowed by which has large experimental error and theoretical uncertainties until now, and is also consistent with Fig.7(a) of Ref.1409.3252 () for decays using the similar statistical fit approach with parameters . (3) The allowed regions of can still overlap with the ones of in part, around (), which implies that the treatment from flavor symmetry breaking effects in Ref. zhu2 (); zhu1 () is not absolutely sure, at least not necessary with current experimental and theoretical precision.
From Fig.1(b), it is seen clearly that (1) there are two allowed solutions for parameters of both and . Besides the commonly used value Beneke1 (); Beneke2 (); Cheng2 (); du1 (); Cheng1 (), there is another bestfit value . (2) It is interesting that the allowed regions for overlap entirely with those for , which confirms the suggestion zhu2 () that (or , ) is universal for and system.
Moreover, comparing Fig.1 (a) with (b), it is seen that (1) generally, the allowed region of is different from that of , and is not always required to be equal to . So, the “new treatment” on parameters according to either factorizable or nonfactorizable annihilation topologies may be reasonable and appropriate for decays. (2) The flavor symmetry breaking effects on parameters could be very small even negligible under the existing circumstances with less available experimental constraints from decays.
[GeV]  

Solution A  
Solution B 
Based on the above analyses and discussions, we present the most simplified (flaour conserving) scenario for the annihilation parameters that both and are universal for both and decay modes to lessen phenomenological parameters, where and are independent variables. To get their exact values, we perform a global fit by combining available experimental data for , , decays, which involve 16 decay modes and 42 observables. In our fit, besides of (), the inverse moment , which is used to parameterize integral of the meson distribution amplitude and a hot topic by now (see Ref.Beneke5 () for details), is also treated as a free parameter and taken into account. We present the allowed parameter spaces in Fig.2 and the corresponding numerical results in Table 2.
Decay mode  Exp. data  This work  Cheng Cheng2 () 

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Decay mode  Exp. data  This work  Cheng Cheng2 () 

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—  
—  
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Decay mode  Exp. data  This work  Cheng Cheng2 () 

—  
—  
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— 
As Fig. 2 shows, the allowed spaces of and are strongly restricted by combined constraints from , , decays, especially for . There are two solutions (named solution A and B, respectively). It is easily found that the allowed regions and the bestfit point of are so alike that one can hardly distinguish one from these two solutions. For each solution, there is no common overlap at 68% C.L. between the allowed regions of and , i.e., the nonfactorizable and factorizable annihilation parameters and should be treated as independent parameters, which confirms the suggestion of Ref.zhu2 (); zhu1 (). Numerically, as listed in Table 2, the fitted result is similar to, but with smaller uncertainties, the results in Ref.chang2 () where the decay modes are not considered. In fact, the two sets of parameters values give the same annihilation contributions. From Table 2, it can be seen that a relatively small value of 0.2 GeV which has been found by, for instance, Refs. Beneke2 (); Cheng2 (); chang2 (); Beneke5 () and a relatively large value of with are favored in the phenomenological aspect of nonleptonic decays. They will enable the HSS corrections to play an important role in evaluating observables of penguin dominated decays, and have significant enhancement, assisted with the large Wilson coefficient , to the colorsuppressed tree amplitude with a large strong phase. As noticed and discussed in Refs. Beneke1 (); Beneke2 (); Beneke6 (), the vertex corrections, including NLO and NNLO contributions, to the color suppressed tree coefficient exhibit a serious cancellation of the real part of (for example, see the first line of Eq.(54) in Ref. Beneke6 ()), but the HSS mechanism can compensate for the destructive interference and enhance the with a large magnitude. The value of including NNLO vertex and HSS corrections Beneke6 () obtained with and 0.35 GeV still cannot accommodate the experimental data on branching ratio decay. So a relatively large HSS corrections arising from might be a crucial key for the “ puzzle”. The branching rate of decay and the asymmetry of decay, they both are sensitive to the choice of coefficient , and can provide substantial constraints on parameter . With the bestfit values of both and in this analysis, one can get , which provides a possible solution to the socalled “ and ” puzzles simultaneously. Of course, one can have different mechanism for enhancement of the in QCDF, for example, the finalstate rescattering effect ^{2}^{2}2 Considering the final state interaction effects, the coefficients and Cheng1 (). Notice that (1) the above coefficient has similar magnitude module to ours, and the large module of is helpful to accommodate the “” puzzle. (2) The coefficient has similar magnitude imaginary to ours, and the large imaginary part of results in a large strong phase difference to solve the “” puzzle. advocated in Ref.Cheng1 () and the Principle of Maximum Conformality proposed recently in Ref.Qiao (), where the allowed regions for parameters and might be different.
With the inputs in Table 1 and the bestfit values of parameters listed in Table 2, we present our theoretical results for observables of , , decays in the third column of Tables 3, 4 and 5. The results Cheng2 () with the traditional treatment including flavor symmetry breaking effects are also listed in the last column for comparison. The results for , , decays are not listed here, because they are similar to those given in Ref.chang2 (). From these results, it could be found that (1) all QCDF results of , , decays could be accommodated to the experimental data within errors. (2) Our results of branching ratios for decays are twice as large as those with the traditional treatment Cheng2 (). And is in good agreement with the data within one experimental error. Meanwhile, our result is twice as large as the traditional result . Moreover, there are also some other differences between the two sets of theoretical results more or less. So, the future accurate measurements on the nonleptonic meson decays would be helpful to probe the annihilation contributions and to explore the underlying dynamical mechanism.
In summary, we studied the nonfactorizable and factorizable annihilation contributions to , , decays with QCDF approach. To clarify the independence of annihilation parameters and and the possible flavor symmetry breaking effects therein, a statistical analysis is performed for nonleptonic and decays. It is found that (1) and are independent parameters, which differs from the traditional treatment with annihilation parameters and verifies the proposal of Ref.zhu2 (). (2) The flavor symmetry breaking effects might be small for nonleptonic and decays by now due to the large experimental errors and theoretical uncertainties. With the simplifications and , a comprehensive global fit on the annihilation parameters and the wave function parameter is done based on the current available measurements on , , decays. Two allowed solutions are found. With the bestfit parameters summarized in Table 2, the QCDF results for , , decays are consistent with the present experimental data within errors. It is expected that the measuremental precision of nonleptonic decays could be much improved by LHCb and superB experiments in the following years, so more information about annihilation contributions could be revealed.
Acknowledgments
The work is supported by the National Natural Science Foundation of China (Grant Nos. 11105043, 11147008, 11275057, 11475055 and U1232101). Q. Chang is also supported by the Foundation for the Author of National Excellent Doctoral Dissertation of P. R. China (Grant No. 201317) and the Program for Science and Technology Innovation Talents in Universities of Henan Province (Grant No. 14HASTIT036). Thanks Referees for their helpful comments and Xinqiang Li for helpful discussions.
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