Charm Quark and Bottom (a.k.a. Beauty) Quark Mass Definitions

Executive Summary

It is possible in principle to convert the MS bar mass of a top quark, bottom quark (a.k.a. beauty quark) or charm quark to the pole mass of that quark, in practice, the current state of the art formula for making that conversion introduces a theoretical uncertainty into the pole mass value of several hundred MeV. This is only about 0.1% of the mass of the top quark, but is about 3% for the bottom quark and about 9% or more for the charm quark. The MS bar mass determinations for the bottom and charm quarks are currently ten times as precise.

But, while the pole mass of a top quark can be determined directly, since it does not hadronize, the pole masses for all other quarks cannot be determined directly because they are always confined in hadrons.

The theoretical error introduced by the conversion factor undermines the immense progress that has been made in recent years in precisely determining the MS bar masses of the heavy quarks by perturbative QCD methods in recent years.

This is a problem because some important particle physics calculations that involve perturbative renormalization physics that involve high energies work best if you use the pole mass rather than the MS bar mass. So, if we are going to calculate the pole masses of bottom quarks and charm quarks, we still need to use the non-perturbative QCD methods of lattice QCD to determine them instead.

Background

There are multiple different definitions of the mass of a quark which are particular relevant in perturbative QCD calculations (i.e. calculations involving the Standard Model equations for interactions involving quarks and gluons at high energy scales). 

Perturbative QCD is less accurate to the point of becoming invalid at low energies, because the more precisely non-perturbative QCD effects that it ignores for mathematical expediency become significant at low energies and in circumstances like conversions from MS bar mass to pole mass. The math is too hard to calculate the full non-perturbative QCD equations analytically (i.e. exactly using algebra and calculus), however, so to make non-perturbative QCD calculations we have to make a discrete approximation to these continuous equations and work out a numerical approximation of the results using methods known as "lattice QCD."

Theoretically, the "pole mass" which is the rest mass of a free quark of a particular type at an energy scale equal to its mass, is the most "natural" definition of the quark mass and is important in some kinds of calculations. The normal definition of the masses of the leptons (electrons, muons, taus, and corresponding neutrinos) is equivalent to the pole mass.

But, pole mass can't be measured directly for any kind of quark other than a top quark, as all other kinds of quarks are found only found in hadrons (e.g. protons, neutrons, pions and kaons), rather than in a free state.

For these quarks, it is easier to use another mass definition, the most popular of which is the MS bar definition.

MS bar masses at an energy scale equal to MS bar mass are of the same order of magnitude as pole masses, and in principle can be converted according to an exact formula (which involves the sum of terms in an infinite series). MS bar masses are defined in a way that is always less than the pole mass for heavy quarks.

Pole Masses v. MS bar Masses for Heavy Quarks

In the case of the top quark, the conversion formula is very precise, converges rapidly, and has an uncertainty of only about one part per thousand.

The conversion formula works less well for the bottom quark (a.k.a. beauty quark), converging slowly and providing less precision, and works even less well for the charm quark where the formula diverges almost immediately. 

The first few terms of the formula are as follows, assuming a strong force coupling constant at the Z boson mass energy scale of the global average value of 0.118:

mt pole = (mt MS bar)(1 + 0.046 + 0.010 + 0.003 + 0.001 + · · ·)
mb pole= (mb MS bar)(1 + 0.096 + 0.048 + 0.035 + 0.033 + · · ·)
mc pole= (mc MS bar)(1 + 0.16 + 0.16 + 0.22 + 0.39 + · · ·)

Basically, the conversion for MS bar mass to pole mass introduces an uncertainty of hundreds of MeVs to the mass of the bottom and charm quarks (whose masses are about 4,200 MeV and 1,275 MeV respectively).

The good news noted in the paper discussed below, is that for processes with only virtual bottom and charm quarks, the more precisely determinable MS bar mass can be used, making the calculation of PDFs (parton distribution functions) which tell you what particles come out of a smashed hadron (often including particles other than the valence quarks of the hadron, for example, causing charm quarks to be emitted from a proton or neutron which has only up or down quarks as valence quarks) more accurate when this is possible.

The bad news is that if the process has an end state that includes bottom and/or charm quarks, there is no escaping the need to use the pole mass and the accuracy of the calculations cannot be improved by using the more precisely known MS bar mass.

The Paper

This is explained in Richard D. Ball, "Charm Production: Pole Mass or Running Mass" (December 12, 2016), and the abstract of the paper is shoddy to the point of being misleading, but the body of the paper tells the story well enough.

The introduction of the paper explains that:

Inclusive processes involving massive quarks are an important ingredient of LHC physics, not least because of their role in determining PDFs. Uncertainties in heavy quark masses can lead to substantial contributions to PDF uncertainties, and thus to uncertainties in predictions for LHC crosssections for a wide range of processes. 

Perturbative coefficient functions can be renormalized to depend on either the pole mass m or the MS running mass m(µ). The perturbative relation between them is now known to four loops, and for top the choice is essentially immaterial. However for charm and beauty nonperturbative corrections are more substantial, and while the MS charm and beauty masses can be determined rather precisely (to a few tens of MeV) through nonperturbative lattice or sum rule calculations, their pole masses are subject to large uncertainties (a few hundreds of MeV). 

Global PDF fits traditionally use pole masses. This is because the experimental observables used in the fits are generally inclusive, to avoid large uncertainties from final state effects. Heavy quarks in the final state are dealt with in the perturbative calculations by putting them on-shell: hadronisation corrections are then of relative order Λ/m, and thus power suppressed. The on-shell condition naturally leads to the mass dependence from heavy quarks in the final state being expressed in terms of the pole mass. In this short note, we will re-examine the possible use of the MS running mass in PDF determinations, and consider other ways in which uncertainties due to charm mass dependence might be reduced at LHC.

The conclusion of the paper explains that:

We have shown that, while for processes with only internal charm quark lines (such as processes with no charm in the final state, or semi-inclusive processes) it is straightforward to calculate using either pole mass or running mass in the hard cross-section, for inclusive processes with charm in the final state there no advantage to using the running mass, since the kinematics produces a nontrivial dependence on the pole mass which cannot be avoided without spoiling the factorized perturbative expansion. It follows that any empirical determination of the charm quark mass from inclusive charm production data has an intrinsic limitation due to nonperturbative corrections of a few hundred MeV. It is easy to see that these considerations generalise straightforwardly to inclusive hadronic processes such as W c or Z c-anti-c  production, and indeed to inclusive beauty production, though here the effect will be less significant. 

A number of recent perturbative determinations of the MS charm mass from inclusive data claim an uncertainty as small as 50 MeV, competitive with the nonperturbative results. The reason for this small uncertainty is probably the use of the theoretical assumption that charm is produced entirely perturbatively, which greatly increases sensitivity to the charm mass. However it is clear from the poor convergence of Equation (4) that perturbation theory close to the charm threshold is unreliable: charm production is subject to large nonperturbative corrections. Relaxing the assumption by fitting a charm PDF, significantly reduces the dependence of the PDFs on the charm mass. This in turn reduces the dependence of high energy cross-sections required at LHC on the charm mass, and thus increases their precision. It will also presumably increase the uncertainty on any empirical determination of the charm mass from inclusive data to a few hundred MeV.