An Interesting Fitting Formula of Quark Masses

We show an interesting empirical formula of quark masses here, which is found by implementing a least squares fit. In this formula the measured QCD coupling is almost a ”best fitting coupling”. There are many papers working on the masses of quarks, in theory or in experiment. For theory, the difficulty is, masses of quarks can not been determined by first principle. The masses of known six quarks correspond to six free parameters in standard model (SM). On the other hand, there appears a mystery problem in the SM, that is, comparing to t quarks, the other five quarks can be considered as ones with vanishing mass. However, there are empirical formulae to illuminate this hierarchy problem, such as reference [1]. We show another interesting mass fitting for the known six quarks in this note, which can be compared with the results in [1]. Although it is empirical, this formula is significant agreement with experiment values. We expect this formula can help us to discover the correct theory of flavor. A byproduct of this formula is, that the best fitting coupling is just the experiment one. We first list the MS quark masses in table 1. These data are from reference [2] in different scale. Because pole masses of light quarks are physical meaningless and pole masses of heavy quarks also have evil definitions[3], we do not use the data of pole mass here. ng = 1 ng = 2 ng = 3 ns = 1 3.25 ± 1.75Mev(2Gev) 1.2± 0.2Gev(mc) 174± 5Gev † ns = 2 7.0 ± 2.0Mev(2Gev) 115 ± 35Mev(2Gev) 4.25 ± 0.2Gev(mb) Table 1: Masses of quarks in MS scheme from [2]. ng is the number of generator and ns is the number of ”isospin”: ns = 3(1−|Q|), where Q is the charge of quark. Since reference [2] only gives mass ranges of different quarks, we take its midpoint as our input. For instance, the mass of u quark 3.25± 1.75Mev(2Gev) just corresponds to the range 1.5Mev ≤ mu(2Gev) ≤ 5Mev in reference [2]. The quantities in bracket are scales where we obtained quark masses. : This is the pole mass of top quark, which should be converted into MS mass using Eqn. (1)[2]. We then renormalize different quark masses to the mass of Z particle mZ = 91.2Gev using renormalization group equation. Relevant formulae have been shown in reference [4]. We take here αs(mZ) = 0.117 [2], which corresponds Λ(5) = 197Mev, Λ(4) = 274Mev and