Scale-Running Behavior Of The Strong Coupling Constants In The Whole Physical Region And Its Applications

Quantum chromodynamics(QCD)is a standard dynamical theory describing the strong interaction between quarks and gluons.It is a basic component of the Standard Model of particle physics.QCD is a successful application of non-Abelian gauge field theory,which has quark confinement and asymptotic freedom.In 2004,Gross,Politzer,and Wilczek won the Nobel Prize in physics for revealing the asymptotic freedom of QCD.The strong interaction can be determined by the strong coupling constant(αs).On the one hand,the value of as is usually much less than 1 in the high-energy region,so perturbative theory can be used to estimate the value of observable physical quantities;the running behavior of the strong coupling constant with the scale in this region can be described by the renormalized group equation.If the strong coupling constant is determined by the experimental data at a specific energy scale,such as MZ,the exact value of the strong coupling constant in the whole perturbative region can be determined.On the other hand,if the energy scale involved in the high energy process is low,the value of the strong coupling constant is too large,and the perturbative numerical result based on the renormalization group equation is not suitable anymore.People usually use low-energy models to estimate their magnitude,and then the input parameters’values of those low-energy models are obtained by fitting a large number of experimental data in the low-energy region.In general,how to obtain the accurate running behavior of as in the whole physical energy region is an important topic in the field of QCD,and it’s also one of the important studies in our paper.In the process of perturbative theory calculation,there is a non-physical quantity,the renormalization scale,which is artificially introduced,because the high-order calculation requires renormalization to eliminate the ultraviolet radiation.In the finite order,the uncertainty of the renormalization scale not only breaks renormalization group invariance but also directly affects the accuracy of the prediction of perturbative QCD theory.Based on the invariance of the renormalization group equations,only if the QCD calculation is up to infinite order,we can obtain theoretical results independent of the choice of renormalization scheme and scale.However,the current computer’s calculation is limited and the perturbative calculation can only be completed up to a limited low order.In this case,we urgently need a systematic method to get accurate theoretical results at a fixed order and obtain reliable predictions for unknown higher-order terms.Based on the renormalized group equation(RGE),the principle of maximum conformality(PMC)determines the effective strong coupling constant of the high-energy process by using all non-conformal β-terms related to RGE in the perturbative series,and thus the effective momentum flow of this process is also determined(be called the PMC scale).It can be proved that the PMC perturbative series eliminates the ambiguity of the renormalization scheme and scale at the same time.In addition,since the perturbative series no longer has the divergent renormalon terms n!fβinαsn,it also can improve the convergence of the whole series.Compared with the traditional perturbative series with renormalized scale uncertainty,the renormalization scale independent series is more beneficial to obtain accurate predictions for unknown higher-order terms.First,we used the PMC singlet-scale approach(PMCs)to analyze the heavy quarkonium inclusive decays,the QCD contributions to the electroweak parameter p and the properties of Higgs decays to two photons and determined the effective strong coupling constants of those perturbative processes.It’s different from the PMC multi-scales approach,according to the mean value theorem,the PMCs determines a single whole effective PMC scale to replace the multi scales of PMCm.The PMCs not only realized a renormalization scheme and scale independent prediction,but also greatly reduced the residual scale uncertainty from unknown higher-order terms.And we also estimated their unknown higher-order corrections based on the obtained PMC perturbative series.We have found that the theoretical predictions obtained by the PMCs method are consistent with the experimental data within errors.That proved the current method is effective.In order to obtain the exact value of the strong coupling constant in the whole physical energy region,it is necessary not only to obtain accurate perturbative QCD results but also to select an appropriate low energy model to describe the infrared behavior of the strong coupling constant in the non-perturbative region.Based on the Bjorken sum rule(B SR)of the spin structure functions of proton and neutron defines an effective strong coupling constant asg1(Q)in the whole region,which provides a natural platform for us to obtain the values of the strong coupling constant in the whole energy region.In order to solve the problem of inconsistent in previous theoretical studies,we first transform the perturbative expression of the effective strong coupling constant αsg1(Q)expanded by the MS-scheme strong coupling constant into that of asg1(Q)expanded by the V-scheme which defined the strong coupling constant based on the scattering potential energy of the heavy quark and anti-quark pairs.Then,after using the PMCs,we obtain a conformal perturbative series with good perturbative convergence and independent of the initial renormalization scale selection,therefore an accurate behavior of ags1 in the perturbative energy region is obtained.Subsequently,we also use the Pade approximate method to estimate the contribution from the unknown higher-order terms.By requiring that the derivative of the PMC predictions at the point is equal to that of the light-front holography model(LFH)in the non-perturbative region,the appropriate parameter space of the low-energy model isκ=0.64-0.28+0.07 GeV and the critical energy scale of the behavior in the perturbative and non-perturbative energy regions is Q0=1.51-0.62+0.16 GeV.In the end,the results show that the obtained running behavior of the effective strong coupling constant in the whole region matches the measured data of Jefferson laboratory in the United States(JLab)very well,the correlation coefficient reaches 0.96-0.03+0.02,it has been proved that the current method is worked.Besides,this paper also introduces a new way to determine the non-perturbative contributions to BSR.That’s based on the nonperturbative contribution expression of BSR obtained by the operate product expansion(OPE),combining with PMC determined accurate effective strong coupling constant,and after using some general low energy models,e.g.,the model based on the analytic perturbative theory(APT),the Webber model(WEB),the massive pQCD model(MPT)and the model under continuum QCD theory(CON),extracted the non-perturbative correction from Jefferson laboratory measurements.All in all,this paper determines the behavior of the effective strong coupling constant in the whole region by using the PMCs method.In the perturbative energy region,the combination of PMC and Pade approximate method determines the accurate strong couplings of several typical high energy processes and improves the accuracy of perturbative QCD theoretical prediction;in the non-perturbative energy region,using the general low energy models to obtain the infrared behavior of strong coupling constant and determine the non-perturbative correction from experiments.More importantly,our results indicate that the PMC perturbative series can be used to determine the accurate running behavior of strong coupling in the whole physical energy region,we also realized the smooth transition from the small scale to large scale and it ensures the inherent self-consistency of the method.It is not only a successful application of PMC,but also provides an important reference to determine the behavior of strong coupling constants in the whole physical energy region,which is helpful to obtain more accurate perturbative QCD predictions and test the standard model accurately.