Comparisons Of Two Renormalization Scale- Setting Approaches

Quantum chromodynamics(QCD)is a dynamical theory describing the strong interaction between standard particles.When the high-energy physical process is related to the high energy transferring,we find that the strong interaction coupling constant between the standard particles is so small that we can use perturbative quantum chromodynamics(pQCD)theory.When the pQCD is calculated to the higher order,there will be the integral divergence problem.The renormalization theory should be introduced to eliminate the divergence and obtain a reliable pQCD prediction.According to the renormalization group invariance(RGI),the physical quantity should not depend on the renormalization scale and scheme.An inappropriate renormalization scale will lead to the dependence of renormalization scale and scheme between the coupling and the coefficients of pQCD expansions cannot be canceled each other.Then the pQCD predictions at fixed order will depend on the choice of renormalization scale and scheme which constitute one of the uppermost systematic errors in the pQCD theory prediction.More seriously,when we adapt the traditional renormalization scale setting method,i.e.,the typical momentum transfer is set to the renormalization scale,a wrong pQCD prediction may be obtained.Therefore how to obtain a more accurate theoretical prediction and reduce or even eliminate the ambiguities of renormalization scale and scheme by setting the renormalization scale properly is one of the significant problem in pQCD theory.In response to this theoretical problem,Brodsky,Lepage and Mackenzie proposed the BLM mechanism in 1983.The mechanism achieves great success and it was applied to various high-energy physics processes.However BLM mechanism is just a renormalization scale-setting method based on the one-loop QCD calculation.We want to find a method to make an analytic extension of the BLM mechanism from the next leading order to a higher order.In this paper,we will give the complete comparisons between two different methods which can expand the BLM mechanism to higher order,i.e.,the maximum conformal principle(PMC)and sequential extension BLM mechanism(seBLM).The RGI is the basis to discuss the renormalization scale-and scheme-ambiguities.So we introduce the renormalization group(RGE)and the extended renormalization group(eRGE)to discuss RGI,i.e.,the standard RGI and the local RGI.The main idea of PMC mechanism is to absorb all the renormalization group related unconformal terms,i.e.,{?i}-terms into the running coupling constant,based on the standard RGI,then the renormalization scale at each order can be obtained.While seBLM mechanism is designed to improve the pQCD convergence by using the large,?0 approximation to absorb all the {?i}-terms into the running coupling constant without distinguishing,based on the local RGI.Therefore,the main differences between PMC mechanism and seBLM mechanism are given,?)the "conformal coefficients" are different after absorbing the {?i}-terms;?)the purpose of PMC is to eliminate the renormalization scale and scheme ambiguities,where the pQCD convergence is also improved as a good by-product due to the elimination of the divergent renormalon terms;While the purpose of seBLM is just to improve the convergence of pQCD expansion.We also find that seBLM needs to introduce an additional freedom to determine the coefficients of {?i}-terms,which lead to that seBLM can only be applied to QCD calculation at two-loop level.To eliminate this limitation of seBLM,we proposed MseBLM mechanism which determines the {?i}-terms using the degeneracy relation by PMC and absorbs the {?i}-terms using large ?0 approximation.MseBLM can be used to any order of pQCD calculation.As an in-depth comparison,we give the specific implementation algorithms of seBLM,PMC and MseBLM in this paper.Based on the four-loop physical processes,Re+e-and ?(H?bb),the comparisons of properties between seBLM mechanism and PMC mechanism are given.As seBLM can only be applied to next-next leading order,MseBLM will be used at the higher order as the substitute method.It was found that seBLM cannot be able to improve the pQCD convergence,however PMC can obtain a more convergent expansion and a more precise pQCD prediction.At the same time,seBLM leads to the final scale beyond the perturbative limitation due to the wrong absorption of {?i}-terms.Thus the following conclusions can be obtained by the comparisons:seBLM is an effective renormalization scale-setting method with many limitations.It may improve the pQCD convergence for some physical processes;PMC mechanism can be used to eliminate the renormalization scale and scheme ambiguity which is based on the RG and the standard RGI.