Pole mass and non stationary mass of a particle

  • #1
Is the pole mass of the particle it's characteristic mass? When particle physicists calculate the mass of elementary particles, like top quark, do they mean the pole mass? If not what makes the apparent mass of a particle different from pole mass?

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  • #2
I don't think that the term "characteristic mass" has a defined meaning in this context. But for non-colored particles the pole mass represents - by definition - the pole of the free particle propagator. So yes, that is what you might call the physical observable mass of that particle.

For quarks the situation is in fact different, because they carry color charge and are subject to confinement. So the concept of a free particle propagator does not make sense, because there are no free quarks to beginn with. And indeed it turns out that the propagator of a colored particle does not have a pole, due to non-perturbative effects. You can still define a particular mass scheme that we call the "pole mass" also for quarks, by just doing the equivalent subtractions in the renormalization procdure as you would for the pole mass for non-colored particles, but in this case it is not more physical than any other scheme you might choose, because that interpretation of the mass as the pole of the free particle propagator is anyways lost.

For the top quark in some measurements the pole mass is used, in other measurments other mass schemes are used. For the other quarks you will usually not see results in terms of the pole mass (it just does not make a lot of sense for them, due to an intrinsic renormalon ambiguity in the definition of the pole mass for quarks that becomes more and more relevant the lighter the quark is). Results for the masses of the other quarks are usually qutoted in the MSbar scheme.
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  • #3
For the top quark in some measurements the pole mass is used, in other measurments other mass schemes are used.
The agreed top mass for now is 173 GEV/c2 so that is the pole mass? And this pole mass is derived from some calculation using the measured mass?
  • #4
The PDG ( https://pdg.lbl.gov/2021/tables/rpp2021-sum-quarks.pdf ) quotes as their average of the results for top quark pole mass measurements

$$ m_t = 172.5 \pm 0.7\, \rm{GeV} $$

How this works is that you calculate the ## t \bar{t} ## production cross section to high precision as a function of the pole mass, then measure this cross section at a collider and from that fit for the mass.

For direct top quark mass measurements (from taking the invariant mass of all the top's reconstructed decay products and comparing the resulting distribution to a Monte Carlo template for different values of the mass parameter) they give as their average

$$ m_t = 172.76 \pm 0.30 \, \rm{GeV} $$

This is not exactly the pole mass but will include an additional, not yet fully understood shift with respect to the pole mass of up to ## \mathcal{O}(1 \,\rm{GeV}) ##, but you see that you are in the same range as for the pole mass measurments above.

So yeah, with your 173 GeV that you wrote above you are compatible within the current uncertainties with the pole mass.
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  • #5
In the Standard Model of Particle Physics (SM), due to the running of all of the experimentally measured SM physical constants with energy scale, no particle or coupling constant has one true value, but the beta function of the mass of each Standard Model particle governs how the mass of the particle changed with energy scale and can be determined exactly from first principles without resort to experimental measurement (although the experimental measurements, of course, confirm that accuracy of the calculated beta functions to the full extent permitted by experimental accuracy). This running is due to the role of renormalization in SM physics but it isn't just a theoretical artifact, it is reflected in real, observable experimental measurements.

When available, people quote the mass of a fundamental particle as its pole mass because that is the most "natural" distinctive energy scale (a.k.a. momentum transfer scale) to use as a reference point.

Crudely speaking, pole mass is the mass of a particle at an energy scale identical to its rest mass. The usually quoted masses of the charged leptons (i.e. electrons, muons, and tau leptons) are all pole masses. The top quark is the only quark for which something close to a direct measurement of its pole mass is possible since it doesn't hadronize before it decays.

Pole mass is estimated, usually using lattice QCD, in the case of charm quarks and bottom quarks, from the masses of hadrons that have charm quark and bottom quark components respectively, usually based upon a determination made in the MS bar scheme.

Due to technical definitional issues one has to make a conversion from MS bar mass to pole mass, however, which is somewhat significant. The first few terms of the formula to convert from one to the other for the top, bottom and charm quarks 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 + · · ·)

A more technical discussion of the definition of pole mass in the case of a charm quark (which is the source for the formulas above) can be found here. 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.
Pole mass is ill defined for the up quark, down quark and strange quark, because due to the mass of the lightest hadrons in which they are valence quarks, and the reality of quark confinement, these quarks are never observed at energy scales remotely close to their inferred MS mass from QCD. Instead, the masses of these three light quarks is customarily quoted at the 1 GeV or 2 GeV energy scale, usually using the MS bar scheme.

If you are interested in the running masses of quarks and charged leptons at energy scales other than pole mass, they are calculated at selected energy scales in this 2008 paper.
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