Why accurate quark masses Original expression New expression Summary More accurate determination of the quark running masses in QCD 夸克跑动质量的更精确计算 G.X.Peng 彭光雄 Why accurate quark masses Original expression New expression Summary

Why we need accurate running masses 1. Important for hadron physicists who is intend to evaluate observable quantities on the basis of an effective theory. 2. Essential for quark-lepton physicists who want to builde new-physics models and testing their viability at a fundamental energy scale such as the typical scale of grand unified theories for strong and electroweak interactions Possible new physics beyond the standard model appear far above the scale of electroweak symmetry breaking 道可道，非常道，名可名，非常名。无名，天地之始；有名，万物之母。故常无，欲以观其妙；常有，欲以观其徼。此两者，同出而异名，同谓之玄；玄之又玄，众妙之门。

History of the systematic calculations 1. Gasser and Leutwyler, Phys. Rep. 87 (1982) 77. 2. Fusaoka and Y. Koide, Phys. Rev. D 57 (1998) 3986. 3. Z.Z. Xing, H. Zhang, S. Zhou, Phys. Rev. D 77 (2008) 113016. 道可道，非常道，名可名，非常名。无名，天地之始；有名，万物之母。故常无，欲以观其妙；常有，欲以观其徼。此两者，同出而异名，同谓之玄；玄之又玄，众妙之门。

Renormalization-group dependence

Original expression of the running quark masses

Weakpoints of the original expressions 1. 夸克质量直接依赖耦合常数. 这使得耦合常数的不确定性直接传导到夸克质量. 2.耦合常数的表达式不是收敛最快的表达式. 这增加了QCD标度参数的误差. To increse precision, the conventional way is to calculate the beta and gamma functions to higher orders. But this become more and more difficult nowadays.

New expressions for the running quark masses

Expansion coeficients, matching-invariant beta and gamma functions PLB 634 (2006) 413 W.J.Marciano, Phys.Rev.D 29(1984)580

RG dependence of QCD running coupling G.X.Peng, PLB 634 (2006) 413

QCD scale Lambda

QCD mass scales

Comparison of the coupling & quark masses PLB 634 (2006) 413

Summary The renormalization group equations of the QCD running coupling and quark masses are solved in a mathematically strict way. General relations between the standard expansion coeficients and the beta and gamma functions have been accordingly deduced. It is thus found that previous expressions either loses an infinite number of known logarithmic terms, which can be given in a closed form, or includes a large number of incorrect terms. Consequently, a new expansion with the fastest convergence speed is developed. Given the matching-invariant beta and gamma functions to four-loop level, the scale parameters for the coupling and masses are calculated from the new expressions and the latest experimental average of alpha_s(Mz), for the order from 1 to 4 and the number of flavors from 3 to 6. PLB 634 (2006) 413

The end 谢谢 道可道，非常道，名可名，非常名。无名，天地之始；有名，万物之母。故常无，欲以观其妙；常有，欲以观其徼。此两者，同出而异名，同谓之玄；玄之又玄，众妙之门。