# Are There Reasonable Definitions Of The Pole Masses Of Light Quarks?

For the heavy quarks and charged leptons, the "pole mass" of a fundamental particle is its mass (which varies as a function of energy scale in the Standard Model) when the energy scale it is evaluated at and its mass at the energy scale are the same. A "pole mass" is an interesting energy scale at which to evaluate the mass of a fundamental particle, among other reasons, because it is independent of any arbitrarily chosen energy scale and instead has an energy scale with an inherently physical basis.

Usually, the masses of the up, down and strange quarks are determined at a 1 GeV energy scale, corresponding approximately to the mass of a nucleon, or at a 2 GeV energy scale, corresponding approximately to the mass of two nucleons (the value at 1 GeV is 35% larger than the value at 2 GeV per PDG), in a renormalization scheme such as the $\overline{\mathrm{MS}}$ renormalization scheme. See, e.g., the Particle Data Group entry for the up quark.

Extrapolating blindly to low energies using perturbative QCD formulas to determine the pole masses of the light quarks is problematic because those formulas are beyond their domain of applicability in the infrared. If you do so, you get masses much larger than the $\overline{\mathrm{MS}}$ renormalization scheme values (even in excess of 50% of the measured neutral pion mass of 134.977 MeV per PDG for up and down quarks).

Koide (1994) calculates the running of the three light quark masses down to their pole masses, even though these values have little practical application and they are acknowledged to be problematic, in both a five quark and three quark flavor model. In the five quark flavor model he comes up with pole masses for the up quark of 346.3 MeV, for the down quark of 352.4 MeV and for the strange quark of 489 MeV. In a three quark flavor model he comes up with pole masses of 163.1 MeV for the up quark, 169 MeV for the down quark, and 338 MeV for the strange quark. One could get somewhat lower light quark pole mass values still in a two quark flavor model.

Koide updated these calculations in 1997 and concluded that the pole mass of the up quark was 501 MeV, the pole mass of the down quark was 517 MeV and the pole mass of the strange quark was 687 MeV (based on their measured values at other energy scales), although all sub-1 GeV values were noted with an "*" mark.

A more recent update of the calculations can be found at Xing (2008) (which does not consider masses running to very low energy scales for light quarks, explaining that "The pole masses of three light quarks are not listed, simply because the perturbative QCD calculation is not reliable in that energy region.").

Is there any other sensible way to determine the pole masses of the light quarks?

Or, is pole mass an inherently unsound concept as applied to light quarks, and, if so, why?

I have tried to discern the answer from Manohar, et al. (2018), but if there is an answer there I don't understand it properly. Perhaps it is not a coincidence that this issue is discussed in Section 66.6 of that article. ;) The key language in that discussion states:

The pole mass cannot be used to arbitrarily high accuracy because of nonperturbative infrared effects in QCD. The full quark propagator has no pole because the quarks are confined, so that the pole mass cannot be defined outside of perturbation theory. The relation between the pole mass mQ and the MS mass mQ is known to three loops.

But, this discussion does not explain why sensible pole masses (e.g. masses that don't violate the pion paradox) can be determined for charm and bottom quarks (which are also confined) but not for the up, down and strange quarks.

• Indeed, the pole mass is an inherently unsound concept as applied to light quarks. Your last, PDG, link reminds you their light masses, 2 and 5 MeV , eqn (66.6) is far beyond perturbation theory and RG methods. Instead, people use lattice methods and chiral perturbation theory, the heart of low energy QCD! to get them, eqns (99.9-10). You are really barking up the wrong tree. Read up on chiral perturbation theory. – Cosmas Zachos Jul 30 '18 at 22:02
• I don't see equations 99.9-10, so I'm confused about that reference. Did you mean 66.9 and 66.10? I understand that lattice and chiral perturbation theory are used in low energy QCD. Those formulas seem to provide scale independent mass ratios, but not absolute masses. What I don't get is whether there is a concept in those theories that is analogous to pole mass in perturbation theory at which there is a way to find a mass that is "natural" without the arbitrarily assumed energy scale of MS mass. – ohwilleke Jul 30 '18 at 22:12
• yes... i'm upside down... indeed, 66.6, 66.9, 66.10..... – Cosmas Zachos Jul 30 '18 at 22:18
• The yukawa coupling is the fundamental concept. So a unified perspective would understand how the concepts of quark mass in lattice QCD, chiral perturbation theory, and perturbative QCD all arise from the yukawas. – Mitchell Porter Jul 31 '18 at 8:33
• @MitchellPorter Do yukawas generalize nicely into the non-perturbative QCD energy scale? – ohwilleke Aug 13 '18 at 20:23