Nuclear 0ν2β decays in B-L symmetric SUSY model and in TeV scale left–right symmetric model

Jin-Lei Yang，Chao-Hsi Chang and Tai-Fu Feng

1 CAS Key Laboratory of Theoretical Physics，Institute of Theoretical Physics，Chinese Academy of Sciences，Beijing 100190，China

2 School of Physical Sciences，University of Chinese Academy of Sciences，Beijing 100049，China

3 Department of Physics，Hebei University，Key Laboratory of High-precision Computation and Application of Quantum Field Theory of Hebei Province，Baoding，071002，China

Abstract In this paper，we take the B-L supersymmetric standard model (B-LSSM) and TeV scale left–right symmetric model (LRSM) as two representations of the two kinds of new physics models to study the nuclear neutrinoless double beta decays(0ν2β)so as to see the senses onto these two kinds of models when the decays are taken into account additionally.Within the parameter spaces allowed by all the existing experimental data，the decay half-life of the nucleus 76Ge and 136Xe，(76Ge，136Xe)，is precisely calculated and the results are presented properly.Based on the numerical results，we conclude that there is greater room for LRSM type models than for B-LSSM type models in foreseeable future experimental observations on the decays.

Keywords: B-LSSM，QCD correction，neutrinoless double beta decay，LRSM

1.lntroduction

Tiny but nonzero neutrino masses explaining neutrino oscillation experiments [1] are unambiguous evidence of new physics (NP) beyond the standard model (SM).It is because in SM there are only left-handed neutrinos so therefore the neutrinos can acquire neither Dirac masses nor Majorana masses.Hence to explore any mechanism inducing the tiny neutrino masses，as well as relevant phenomenology，is an important direction to search for NP.The simple extension of SM is to introduce three right-handed neutrinos in a singlet of the gauge group SU(2) additionally，where the neutrinos acquire Dirac masses，and to fit the neutrino oscillation and nuclear decay experiments as well as astronomy observations，the corresponding Yukawa couplings of Higgs to the neutrinos are requested so tiny as ?10-12，that is quite unnatural.However，neutrino(s) may acquire masses naturally by introducing Majorana mass terms in extended SMs.Once a Majorana mass term is introduced，certain interesting physics arise.One of the consequences is that the lepton-number violation(LNV)processes，e.g.the nuclear neutrinoless double beta decays (0ν2β)，etc may occur.Of them，the 0ν2β decays are especially interesting，because they may tell us sensitively about the nature of the neutrinos，whether Dirac [2] or Majorana [3].When the decays 0ν2β are observed in experiments，most likely，the neutrinos contain Majorana components.Thus studying 0ν2β decays is attracting special attention.

Nowadays there are several experiments running to observe the 0ν2β decays，and the most stringent experimental bounds on the processes are obtained by GERDA [4，5] and KamLAND-Zen [6，7].They adopt suitable approaches and nuclei such as76Ge and136Xe respectively.Now the latest experimental lower bound on the decay half-life given by GERDA experiments is＞1.8 ×1026years (90% C.L.)for nucleus76Ge[8]，and in the near future，the sensitivity can reach up to 1028years [9].For the nucleus136Xe，the most stringent lower bound on the decay half-life is＞1.07 ×1026years (90% C.L.) given by KamLAND-Zen [6]，and the corresponding future sensitivity can reach up to 2.4×1027years [10].Moreover，underground experiments PANDAX，CDEX etc，which are originally designed for searching for WIMP dark matter，are also planning to seek the 0ν2β decays i.e.they may observe the 0ν2β decays with the sensitivity which at least will set a fresh lower bound.

In literature，there are a lot of theoretical analyses on the 0ν2β decays.The analyses are carried out generally by dividing the estimation of the 0ν2β decays into three ‘factors’: one is at quark level to evaluate the amplitude for the‘core’ process d+d →u+u+e+e of the decays; the second one is，from quark level to nucleon level，to involve the quark process into the relevant nucleon one i.e.the ‘initial’quarks d and d involve into the two neutrons in the initial nucleus and the ‘final’ quarks u and u involve into the two protons in the final nucleus; the third one is，from nucleon level to nucleus level，the relevant nucleons involve in the initial nucleus and the final nucleus properly.For the ‘core’process，in [11] a general Lorentz-invariant effective Lagrangian is constructed by dimension-9 operators，and in[12，13] the QCD corrections to all of these dimension-9 operators are calculated.In [14] the short-range effects at quark level are considered，the analyses of the decay rates in the SM effective field theory are presented in [15–17]，the 0ν2β decay rates are derived in [18]，the corresponding nuclear matrix elements (NME) and phase-space factors(PSF) for the second and the third factors of the decays i.e.from quark level to nucleon and nucleus levels，are considered in [19–29]，some theoretical predictions on the 0ν2β for certain models are presented in [30–34]，and the theoretical analyses on the decays are reviewed in [35–37].

In this work，we are investigating the constrains from the 0ν2β decays for the B-L supersymmetric model (B-LSSM)and for the TeV scale left–right symmetric model (LRSM)comparatively.It is because the two models are typical: both have an LNV source but the mechanisms which give rise to the Majorana mass terms are different [38–51].In the B-LSSM，the tiny neutrino masses are acquired naturally through the so-called type-I seesaw mechanism which is proposed firstly by Weinberg[52].In the LRSM[53，54]，the tiny neutrino masses are acquired by both type-I and type-II seesaw mechanisms，in addition，the new right-handed gauge bosonss introduced in this model，then both left-handed and right-handed currents cause the 0ν2β decays[55–72].As a result，the computations of the decays are much more complicated in the LRSM than those in the B-LSSM.Hence these two models，being representatives of NP models，are typical for the 0ν2β decays，and one may learn the mechanisms in the models well via analyzing the 0ν2β decays comparatively.

In the study here，we will mainly focus on the first‘factor’ about the quark level i.e.the core process，which relates to the applied specific model closely.We will evaluate the Wilson coefficients of the operators relevant to the core process d+d →u+u+e+e etc on the models，whereas the estimation of the other two ‘factors’，i.e.to evaluate ‘NME’and ‘PSF’ etc，we will follow the literature [29–33].With respect to the‘core’process d+d →u+u+e+e，all of the contributions in the B-LSSM can be deduced directly quite well，while the contributions in LRSM cannot be so.As shown in [73]，the calculations in the LRSM are much more complicated and the interference effects are quite hard to be considered well.In this work，a new approximation，i.e.the momenta of the two involved quarks inside the initial or final nuclei is tried to be set equal，is made so as may reduce all contributions in the LRSM quite similar to the case of B-LSSM.Then the calculations in LRSM are simplified quite a lot and under the approximation，the interference effects can be treated comparatively well.Finally，for comparison，we also present the results obtained by the traditional method [65].

The paper is organized as follows: In section 2，for B-LSSM，the seesaw mechanisms which give rise to the tiny neutrino masses，the heavy neutral leptons as well，the relevant interactions etc，the calculations of the 0ν2β decay half-lives of the nuclei are given.Similarly，in section 3，for LRSM，the seesaw mechanisms which give rise to the tiny neutrino masses，as well the heavy neutral leptons，the relevant interactions，and the calculations of the 0ν2β decay half-lives of the nuclei are given.In sections 4.1 and 4.2 the numerical results for B-LSSM and LRSM are presented respectively.Finally，in section 5 brief discussions and conclusions are given.In the appendix，the needed QCD corrections to the effective Lagrangian which contains the dimension-9 operators in the region from the energy scale μ;MWto the energy scale μ1.0 GeV are collected.

2.The B-LSSM for 0ν2β decays

In the B-LSSM，the local gauge group is SU(3)C ?SU(2)L?U(1)Y?U(1)B-L，where B，L denote the baryon number and lepton number respectively，and the details about the gauge fields，their breaking，extra lepton and Higgs fields etc can be found in [42–51].In this model，tiny neutrino masses are acquired by the so-called type-I seesaw mechanism，when the U(1)B-Lsymmetry is broken spontaneously by the two U(1)B-Lsinglet scalars (Higgs).The mass matrix for neutrinos and neutral heavy leptons in the model can be expressed as

and the mass matrix can be diagonalized in terms of a unitary matrix Uνas follows:

where U，S，T，V are matrices of 3×3.

The interactions，being applied later on，in the model are

where ν，N are the four-component fermion fields of the light and heavy neutral leptons respectively.

The Feynman diagrams which are responsible for the dominated contributions to the 0ν2β decays in the B-LSSM are plotted in figure 1.Note here that the contributions from charged Higgs exchange(s) are ignored safely as they are highly suppressed by the charged Higgs masses and Yukawa couplings，thus in figure 1 the charged Higgs exchange does not appear at all.To evaluate the contributions corresponding to the Feynman diagrams for the 0ν2β decays，and to consider the roles of the neutral leptons (light neutrinos and heavy neutral leptons) in the Feynman diagrams，the useful formulae are collected below so as to deal with the neutrino propagator sandwiched by various chiral project operators

Figure 1.The dominant Feynman diagrams for the 0ν2β decays in the B-LSSM Model.(a)The contributions from the heavy neutral lepton exchanges，(b) The contributions from the light neutrino exchange.

and the propagator

Relating to the exchanges of the heavy neutral leptons(the virtual neutral lepton momentum k has∣k∣ ?0.10 GeV ?MNi) for the decays，from figure 1(a) the effective Lagrangian at the energy scaleμ?MWLcan be read out as

where X，Y，Z=L，R，θCis the Cabibbo angle，mpis proton mass introduced for normalization of the effective Lagrangian，and S1iis the matrix elements in equation (3).

Since the nuclear 0ν2β decays take place at the energy scale of about μ ≈0.10 GeV，obviously we need to consider the QCD corrections for the effective Lagrangian obtained at the energy scaleμ?MWLin equation(7)i.e.to evolute the effective Lagrangian in terms of renormalization group equation (RGE)method from the energy scaleμ?MWLto that μ1.0 GeV first，where the corrections are in perturbative QCD (pQCD)region.Thus，for completeness，the QCD corrections to all of the possible dimension-9 operators which may contribute to the nuclear 0ν2β decays，are calculated by the RGE method，and the details of computations are collected in the appendix.Whereas the QCD corrections in the energy scale region μ1.0 GeV ～μ;0.10 GeV，being in the non-perturbative QCD region，we take them into account by inputting the experimental measurements for the relevant current matrix elements of nucleons，which emerge when calculating the amplitude based on the effective Lagrangian at μ;0.10 GeV.

For the decays when considering contributions from the neutrino (m νi?∣k∣) exchanges as described by the Feynman diagram figure 1(b)，and the interferences between the light neutrinos’and heavy neutral leptons’contributions，to derive the effective Lagrangian for the neutrino contributions is better at the energy scale μ1.0 GeV as heavy neutral leptons too.At this energy scale the effective Lagrangian can be written down according to the Feynman diagram figure 1(b) as below:

Since the light neutrino exchange is of long range，the QCD corrections in the region μ1.0 GeV ～μ;0.10 GeV to the coefficients in equation (8) may be involved like in the above case for the heavy neutral lepton，via inputting the experimental measurements for the relevant current matrix elements of nucleons，which emerge when calculating the matrix elements in the amplitudes at μ ≈0.10 GeV.

To evaluate the half-life≡lnof the 0ν2β decays，the contributions from the heavy neutral leptons(figure 1(a))and those from the neutrinos(figure 1(b))should be summed up for the amplitudes.The half-lifeof the 0ν2β decays can be written as [28]

where G0ν=2.36×10-15(14.56×10-15) yr-1[28] for76Ge(136Xe) is the PSF，M0ν=-6.64±1.06 (-3.60±0.58)[19，28] for76Ge(136Xe)4The NMEs adopted here are obtained within the framework of the microscopic interacting meson model for nuclei [28]，and the uncertainties for NMEs calculations from the various nuclear structure models are quite wild，here we remind only that the NMEs obtained by various approaches are varying by a factor of (2–3) roughly [74].is the NME corresponding to the long range contributions which is defined as

3.The LRSM

For the model LRSM，the gauge fields are SU(3)C?SU(2)L?SU(2)R?U(1)B-L，and the details about the gauge fields and their breaking can be found in [38–41].In this model，the tiny neutrino masses are obtained by both of type-I and type-II seesaw mechanisms due to introducing the righthanded neutral leptons and two triplet Higgs (scalars)accordingly.In the model，the mass matrix for the neutral leptons generally is written as

and the mass matrix equation (13)5The matrix equation(13)with ML=0 indicates the masses are acquired by type-I seesaw mechanism; it with MD=0 indicates the masses are acquired by type-II seesaw mechanism; it in general feature indicates the masses are acquired by type-I+II seesaw mechanism.can be diagonalized in terms of a unitary matrix Uν，whereas the matrix Uνcan be expressed similarly as that in the case of the B-LSSM equation (3).

For the model LRSM，if the left–right symmetry is not broken manifestly but spontaneously，i.e.gL=gR≡g2and as one consequence，the mass terms of W bosons can be written as

where v1，v2，vL，vR(vL?vR) are the VEVs of new scalars(Higgs) in the LRSM.Then the physical masses of the W bosons can be obtained [72]

The interactions，being applied later on，in the model are

where the definitions for U，S，T，V，ν，N are the same as the ones in the B-LSSM.

In the LRSM，the dominant contributions to the 0ν2β decays are represented by Feynman diagrams figure 2.

Figure 2.The dominant Feynman diagrams for the 0ν2β decays in the LRSM.(a)The contributions from the heavy neutral lepton exchanges，(b) the contributions from the light neutrino exchanges.

The QCD corrections in the energy scale region μ1.0 GeV to0.10 GeV，being of non-perturbative QCD，are taken into account by inputting in the experimental measurements for the relevant current matrix elements of nucleons，which emerge at the effective Lagrangian at μ;0.10 GeV.

In [55–66]，the second term and the third term of equation (19) are defined as η，λ respectively.Extracting the factors

With the ‘frozen approximation’ and ‘the on-shell approximation’ onto the momenta for the out legs as well，all contributions corresponding to figure 2(b) can be well-deduced and the final results can be collected as

where

Then in LRSM，the half-life of 0ν2β decays can be written as[28]

where

where

4.Numerical results

The direct searching for the right-handed gauge boson sets the lower bound for the mass of W2boson asMW2? 4.8 TeV [78–81]，and the WR，WLmixing angle as ζ ?7.7×10-4[65].Finally，we have x ≡v2/v1＞0.02 [75]required by the CP-violation data for the K and B mesons，where v1，v2are the VEVs of the two Higgs particles in the LRSM.Equations(11)and(25)showdepends the NME and phase space factor of the chosen nuclei.However these factors appear in the expression ofin the form of some ratios，and these ratios for76Ge are equal to those for136Xe roughly，hence we can use the same expression offor76Ge，136Xe in the analyses later on.

4.1.The numerical results for the B-LSSM

According to the above analysis，in B-LSSM the contributions from the heavy neutral leptons shown in equation(7)are highly suppressed(S1i≈10-7)for the TeV-scale heavy neutral leptons，hence the dominant contributions to the 0ν2β decays come from the light neutrinos.Then with the neutrino mixing parameters

and the neutrino mass-squared differences as those in equation(29)at 3σ error level，versus mν-lightest(the lightest neutrino mass)for the B-LSSM is plotted in figure 3，where the green (red) points denote the NH (IH) results，the blue(purple)solid line denotes the constraints from the lower 0ν2β decay half-life bound of76Ge (136Xe)，the blue (purple) dashed line denotes the experimental ability of76Ge (136Xe) for the next generation of experiments，the orange solid(dashed)line denotes the constraints from PLANK 2018 for NH(IH)neutrino masses(the meaning for the red points，green points，blue lines，purple lines，orange lines is also adopted accordingly in the figures later on).In figure 3，is well below the experimental upper bounds in the cases of NH and IH，and there is not tighter restriction on the mν-lightestthan that offered by PLANK.Additionally，the blue dashed line shows that there is certainly opportunity to observe the 0ν2β decays in the next generation of experiments，whereas if any of the decays is not observed in the next generation of experiments，then the IH neutrino masses will be excluded completely by the blue dashed line in figure 3.

Figure 3.versus mν-lightest under scanning the neutrino masssquared differences equation(29)at 3σ error level and the parameter regions equation (32).The green (red) points denote the NH (IH)results，the blue (purple) solid line denotes the experimental constraints from the 0ν2β decay half-life of 76Ge (136Xe)，the blue(purple)dashed line denotes the experimental ability of 76Ge(136Xe)for the next generation of experiments，the orange solid(dashed)line denotes the constraints from PLANK 2018 for the case of NH (IH).

4.2.The numerical results for the LRSM

In the LRSM owing to the bosons WL，WRand their mixing，the situation is much more complicated than that in the B-LSSM，and both the light neutrinos and heavy neutral leptons make substantial contributions to the 0ν2β decays.For TeV-scale heavy neutral leptons in the LRSM，the consequences of the type II seesaw dominance are similar to those of the type I seesaw dominance，while the consequences are very different from the ones of type I+II seesaw dominance，because the light-heavy neutral lepton mixing is not much suppressed in this case.Hence we do the numerical computations for the type I and type I+II seesaw dominance cases in our analyses.For simplicity and not losing general features，we assume that there is only one heavy neutral lepton making substantial contributions to the model.It indicates that there are no off-diagonal elements in the matrix MRin equation (13)，i.e.we havediag(MN1，M N2，MN3).Then under the dominance of either the type I seesaw or the type I+II seesaw mass mechanisms，we computenumerically in turn.

4.2.1.The results under type I seesaw dominance.Under the type I seesaw dominance and due to the tiny neutrino masses，the sub-matrix T in equation (3) has T1j?1 (j=1，2，3).It indicates that the contributions from the terms proportional toas shown in equation (23) are highly suppressed.Then scanning the parameter space in equations (29)，(32) and in the parameter space

Figure 4. versus mν-lightest with scanning the parameter space in equations (29)，(32)，(33).

4.2.2.The results under the type I+II seesaw dominance.As pointed out above，the mixing parameters of the light neutrinos and the heavy neutral leptons are not tiny under the type I+II seesaw dominance，hence the Dirac mass matrix MDcan also affect the numerical results via the mixing of the light neutrinos and the heavy neutral leptons.For simplicity and not losing general features，we assume that the mass matrix MDin equation (13) is diagonal as MD=diag (MD11，MD22，MD33).Taking the assumption that only one generation of heavy neutral leptons makes substantial contributions，i.e.MD22and MD33do not affect the results.Scanning the ranges determined by equations (29)，(32) and the parameter space

5.Summary and discussions

In the paper，we take the B-L supersymmetric standard model(B-LSSM) and TeV scale left–right symmetric model(LRSM) as two representations of two kinds of new physics models to study the nuclear neutrinoless double beta decays(0ν2β).As stated in the Introduction，the calculations are those about the factor on the ‘core’ factor of the decays:evaluating the process d+d →u+u+e+e by considering the effective Lagrangian containing the operators with the Wilson coefficients or considering the relevant amplitude etc.Whereas here the estimations of the other necessary ‘factors’for the decays，i.e.to evaluate relevant nuclear matrix element(NME)and phase space factor(PSF)etc，that do not relate to the specific models，are treated by the following literature.

In the B-LSSM，all of the calculations can be well deduced and the results are dominated by the neutrino mass terms.However，in the LRSM，owing to the existence of right-handed gauge boson WR，the calculations are complicated，and the interference effects are hard and easily estimated.In this work，a new approximation，i.e.the momenta of the two involved quarks inside the initial nucleus and inside the final nucleus，are assumed to be equal approximately，is made，then all contributions in the LRSM can be well reduced and summed up all the contributions.Then the calculations in LRSM are simplified and the interference effects can be calculated comparatively easily.To see the consequences of the approximation，we compare the results obtained in this work with the results obtained by the traditional method numerically，and the results coincide with each other well when light-heavy neutral lepton mixing parameteris small orMW2is large.For the effective dimension-9 contact interactions in these two models，the contributions from heavy neutral lepton exchange，the QCD corrections from the energy scaleμ?MWL(orMW1) to the energy scale μ1.0 GeV to all dimension-9 operators in the effective Lagrangian which is responsible for the 0ν2β decays are calculated by the RGE method，and all the QCD corrections including the contributions from light neutrnos in the energy scale region μ1.0 GeV to μ;0.10 GeV，being of non-perturbative QCD，are taken into account alternatively by inputting in the experimental measurements for the relevant current matrix elements of nucleons，which emerge at the effective Lagrangian at μ0.10 GeV.

With necessary input parameters allowed by experimental data，the theoretical predictions on 0ν2β decay halflife76Ge and136Xe are obtained in these two models.In the B-LSSM，the contributions from heavy neutral leptons are highly suppressed by the tiny light-heavy neutral lepton mixing parameters for TeV-scale heavy neutral leptons.Hencedepends on the light neutrino masses mainly，and the numerical results show that the 0ν2β decays may be observed with quite a great opportunity in the near future.Whereas if the decays are not observed in the next generation of 0ν2β experiments，the IH neutrino masses are excluded completely by the lower bound on(as shown in figure 3).

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China (NNSFC) under Grants No.11 821 505 No.12 047 503，No.12 075 301 and No.11 705 045.It was also supported in part by the Key Research Program of Frontier Sciences，CAS，Grant No.QYZDY-SSW-SYS006.The authors(J-L Yang and C-H Chang) would like to thank Prof J-H Yu(ITP，CAS) for helpful discussions and suggestions.

Appendix.QCD corrections to the relevant dimension-9 operators

The essential process d+d →u+u+e+e for the 0ν2β decays involves six ‘fermion legs’，so the effective Lagrangian for the decays is composed generally of a set of independent dimension-9 operators as follows [14]:

In the region fromMWLto 1.0 GeV for the energy scale μ，the pQCD is applicable，so the QCD corrections to the effective Lagrangian for the 0ν2β decays can be carried out by the renormalization group equation method [82，83].The corresponding QCD corrections for the 0ν2β decays were also calculated in [12]，so here we describe how the corrections are calculated briefly.In the RGE method the renormalized operator matrix elements＜(Oiis defined in equation (A2)) for pQCD relate to their bare ones up to oneloop level generally as the following form:

Finally，according to the numerical results of the present comparative studies on the 0ν2β decays for the two typical models LRSM and B-LSSM，it may be concluded that the room for LRSM type models for the foreseeable future decay experiments is greater than that of B-LSSM type models，and having the right-handed gauge bosons，the feature of the LRSM type models is more complicated than that of the B-LSSM type models.〈 Oj〉 are given by

where

and CF=(N2-1)/(2N) is the SU(N) color factor (N=3).The singularities in equation (A4) are required to be cancelled，then we have

Then Zijcan be read out and written as

Considering the 4-quark-leg operatorsO(qbare) in the effective Lagrangian equation (A1) are constructed by the bare quark fieldqbare，and the corresponding coefficientsCbareare also bare.Thenqbare，Cbarerelate to the renormalized ones as

Hence we have

and the matrix elements for the QCD corrections are read as

Combining equation (A11)and equation(A5)，we can obtain

Due to the fact that the bare quantitiesCibaredo not depend on the renormalization energy scale μ，we have

which can be rewritten as

Equation (A14) is the RGE accordingly for the Wilson coefficients，whereis written as a vector form.Then the anomalous dimension matrixγ?can be written as

Combining with the one-loop expression in the-scheme

wheres the coefficient matrix of 1/ε in equation(A8)，the‘a(chǎn)nomalous dimension matrix’ to the leading order can be written as

Then solving equation (A14)，the evolution of Wilson coefficients→(μ) can be expressed by(Λ ) in terms of the μevolution matrixμ，Λ):

where precisely

and

withis γijin matrix form.The running coupling constant to one-loop level of QCD can be written as

with β0=(33-2f)/3，and f is the number of the active quark flavors which is varied with the energy scale μ，and only the quark qfwith mass mfsmaller than the upper bound of the considered energy scale region are‘a(chǎn)ctive’.Thus in the region from ～1.0 GeV to mc～1.3 GeV we have f=3，in the region from mcto mb～4.6 GeV we have f=4 and in the region from mbto MWwe have f=5.As a result，the required matrix to describe the QCD RGE evolution from μ=MWto μ1.0 GeV is

To the leading order，the QCD corrections of the operators in equation (A2) correspond to figure 9 as follows.

Calculating the diagrams respectively，the operator matrix elements corresponding to those in equation(A4)have the following structures

Figure 9.One-loop QCD corrections to the dimension-9 operators for the 0ν2β decays in the effective Lagrangian.

where Γiare the Lorentz structures of the operators in equation (A2)，and Taare the generators of SU(N).Since the lepton sector is irrelevant with the QCD corrections，in the calculation the leptonic factor in the operators is irrelevant.According to equations (A4)–(A17)，the anomalous dimension matrix elements can be extracted from equations (A23)–(A28).

Summarizing the obtained anomalous dimension matrix elements forand，we have

It is easy to realize that our results on the anomalous dimensions for the matrix elementscoincide with those in [13]，but do not coincide with those in [12].Hence the calculational details about the anomalous dimensions ofare given below:

The Fierz transformation formalisms used in the above calculation read

and for the generators Taof SU(N) we have

Combining equations (A20)–(A22) with equation (A29)，we can compute the numerical QCD RGE evolution matrices from Λ =MWLto μ1.0 GeV as

ORClD iDs