I Are Light Quark Pole Masses Below 1 GeV Meaningful or Non-Physical?

[ohwilleke]

The pole masses of the heavy quarks (c, b and t) are relatively well defined in QCD (i.e. the solution of m²(p²) = p² extrapolated using the beta function and the available data from other values of µ usually obtained based upon model dependent decompositions of hadron masses that include these heavy quarks).

Generally speaking, when we talk about the masses of the light quarks (u, d, and s) we use the masses at µ corresponding to 1 GeV or 2 GeV in an MS renormalization scheme (or some similar alternative) and those are the masses referenced by the particle data group.

Naively extrapolating light quark masses to lower energy scales gives much larger hypothetical pole masses for these particles (approaching constituent quark approximation masses), but it isn't obvious that these pole masses are meaningful because confinement implies that light quarks are always present in hadrons, and there are no hadrons lighter than the pion (ca. 140 MeV) and the protons (a bit under 1 GeV). So, perhaps pole masses for these particles are simply "non-physical".

The discussion of the issue in one paper (http://arxiv.org/pdf/hep-ph/9712201v2.pdf) states:

As we noted already, the values of the light quark masses mq(mq) (q = u, d, s) should not be taken rigidly, because the perturbative calculation below µ ∼ 1 GeV seems to be not reliable. In order to see the reliability of the calculation of αs(µ), in Fig. 4, we illustrate the values of the second and third terms in { } of (B4) in Appendix B separately. The values of the second and third terms exceed one at µ ≃ 0.42 GeV and µ ≃ 0.47 GeV, respectively. Also, in Fig. 5, we illustrate the values of the second and third terms in { } of (4.5) separately. The values of the second and third terms exceed one at µ ≃ 0.58 GeV and µ ≃ 0.53 GeV, respectively. These means that the perturbative calculation is not reliable below µ ≃ 0.6 GeV. Therefore, the values with asterisk in Tables I, II and VI should not be taken strictly. These situations are not improved even if we take the four-loop correction into consideration. For example, for nq = 3, d(αs/π)/d ln µ is given by [22] d(αs/π) d ln µ = − 9 2 αs π 2 " 1 + 1.79 αs π + 4.47 αs π 2 + 21.0 αs π 3 + · · ·# . (5.1) Since the value of αs/π is αs/π ≃ 0.16 at µ ≃ 1 GeV, the numerical values of the right-hand side of (5.1) becomes d(αs/π) d ln µ = − 9 2 αs π 2 [1 + 0.28 + 0.11 + 0.085 + · · ·] , (5.2) so that the fourth term is not negligible compared with the third term. This suggests that the fifth term which is of the order of (αs/π) 6 will also not be negligible below µ ∼ 1 GeV. However, we consider that the evolution of mq(µ) above µ ∼ 1 GeV (from µ ≃ 1 GeV to µ ∼ mZ) is reliable in spite of the large error of αs(µ) at µ ∼ 1 GeV.

Is there a non-perturbative way to determine the light quark pole masses below µ ∼ 1 GeV? Is it simply too hard to calculate but well defined? Or, are these quantities ill defined or truly non-physical?

 

[vanhees71]

The definition of the light-quark masses is indeed a tricky subject. For a review, see

http://pdg.lbl.gov/2015/reviews/rpp2015-rev-quark-masses.pdf

 

[ohwilleke]

Thanks for the reference, although the pdg review dances around the subject rather than really attacking it head on.

 

[vanhees71]

I don't understand what you mean. It explains, how the mass values they quote are determined. As I said, it's a tricky business since quarks are not observable as "asymptotic free states".

 

[ohwilleke]

The crux of the issue is whether determining these values is simply difficult, or whether the concept of quarks have a well defined mass at a scale far less than the smallest hadron masses of hadrons that contain them is itself is in some respect unsound.