What is the meaning of the mass assigned to an individual quark?

[Dale]

TL;DR Summary

What is the meaning of the mass assigned to an individual quark?

A hydrogen atom is less massive than the sum of an unbound proton and an unbound electron. If I add energy to the atom the system becomes more massive, and when I add enough then I have an unbound electron and proton, each of which has the usual mass. So the mass of an electron is the unbound mass

This doesn’t work for a quark. Because of the strong force potential it would take an infinite amount of energy to unbind a proton into its constituent quarks. So you never have an unbound quark, and the equivalent concept of the quark mass would be infinite.

So since it doesn’t mean that, what does the individual quark mass mean?

 

[PeroK]

Summary:: What is the meaning of the mass assigned to an individual quark?

A hydrogen atom is less massive than the sum of an unbound proton and an unbound electron. If I add energy to the atom the system becomes more massive, and when I add enough then I have an unbound electron and proton, each of which has the usual mass. So the mass of an electron is the unbound mass

This doesn’t work for a quark. Because of the strong force potential it would take an infinite amount of energy to unbind a proton into its constituent quarks. So you never have an unbound quark, and the equivalent concept of the quark mass would be infinite.

So since it doesn’t mean that, what does the individual quark mass mean?

You could try this:

http://pdg.lbl.gov/2017/reviews/rpp2017-rev-quark-masses.pdf

Although one often speaks loosely of quark masses as one would of the mass of the electron or muon, any quantitative statement about the value of a quark mass must make careful reference to the particular theoretical framework that is used to define it.

I don't follow the argument about the infinite potential. If the Coulomb force were stronger, that wouldn't increase the mass of a free electron.

You may also want to read about color confinement:

http://hyperphysics.phy-astr.gsu.edu/hbase/Particles/qbag.html

 

[Dale]

You could try this:

http://pdg.lbl.gov/2017/reviews/rpp2017-rev-quark-masses.pdf

Although one often speaks loosely of quark masses as one would of the mass of the electron or muon, any quantitative statement about the value of a quark mass must make careful reference to the particular theoretical framework that is used to define it.

Thanks for the link. I am going through it.

I don't follow the argument about the infinite potential. If the Coulomb force were stronger, that wouldn't increase the mass of a free electron.

Yes, it would in the following sense. Suppose that the mass of the hydrogen atom were fixed. That mass consists of the unbound mass of the proton plus the unbound mass of the electron minus the mass deficit. If the Coulomb force were stronger then the mass deficit would be larger. To keep the mass of the hydrogen atom fixed if you increase the mass deficit then you would have to correspondingly increase the masses of the unbound proton or electron.

 

[Vanadium 50]

What is the meaning of the mass assigned to an individual quark?

It is a parameter in the theory.

If you want to define a mass with F=ma (or similar), you quickly run into the problem that, because of confinement, you can't move a quark around without also moving gluons around too. So the effective mass is a combination of the quark and the nearby gluons.

Worse, the effect of these gluons isn't even approximately constant - it depends on the experiment. And it's not a small effect, it can be almost a factor of 100.

 

[mfb]

As result of that problem there are multiple mass definitions. What experiments measure is typically not what theorists use in calculations, for example. For the top quark this difference starts to get relevant and people spend time on understand what "top mass" actually means for different people. Here is an overview:

Since pole and $\overline{MS}$ masses are the most popular top-mass schemes [...]
Another mass definition [...]
I wish to remind some other top mass definitions [...]

 

[Dale]

the value of a quark mass must make careful reference to the particular theoretical framework that is used to define it

As result of that problem there are multiple mass definitions.

It is a parameter in the theory.

It seems like all of these are pointing in the same direction.

I am sure this is stretching a bit beyond what I am capable of, but I assume that if you write a typical classical action for a massive particle in a potential and the action for a quark and the strong force that the parameter that we call the mass of the quark shows up in a similar portion of the equations, yes? If so, is there a non-initiate explanation for how the different versions of the mass come about? What is it about the strong force that makes a clear identification challenging?

 

[Vanadium 50]

What is it about the strong force that makes a clear identification challenging?

Confinement. You don't ever have a free quark, so it's always interacting with something else. These alter what you are trying to measure.

 

[anuttarasammyak]

Values of mass come from interaction with Higgs boson or Higgson. We are postulating that Higgsons interact with any particle independently .e.g. same way no matter what the surroundings of the particles are. These surroundings, like gluons and quark-antiquark pair sea are also particles and they too interact with Higgson independently. With this hypothesis of quark mass we try to find the kinetic states and mechanism on generation of the surroundings of quarks. If we fail, change the hypothesis. This is a story I think. So now relying on the above hypothesis, I am not so much bothered if we cannot take out a quark alone.

 

[mfb]

Values of mass come from interaction with Higgs boson or Higgson.

It doesn't. It comes from the interaction with the Higgs field. No Higgs bosons involved. But, as discussed in this thread, this isn't necessarily what people consider the mass of quarks.

These surroundings, like gluons and quark-antiquark pair sea are also particles and they too interact with Higgson independently.

Gluons don't get a mass from the Higgs mechanism.

 

[Reggid]

What is it about the strong force that makes a clear identification challenging?

Just to be clear. Each of the different mass schemes that are used have a precicse definition by themselves. It is not that we have problems to define a scheme in QCD. Not at all.

It is just that there is no way to say that this or that is the "physical" mass of a free quark. Obviously because there is no free quark, and even the concept of it does not make sense.
So in principle any of these mass schemes are equally justified (and can only be understood as being just a renormalized parameter in your Lagrangian), and which one to use is just a matter of how well the result convergences in a perturbative expansion (which can be different for different schemes, but also for different observabes and kinematic regions).

Let me say that also for non-colored particles you could define all these different mass schemes, and there is nothing wrong with it. Of course I could also use the MS-scheme for the mass of an electron, and the theory is still fine and well defined. The point is just, that for non-colored particles, that are not subject to confinement, you do have a notion of what the physical mass of that single particle is (the location of the pole of the propagator, that is independet of the renormalization scale). And so it makes a lot of sense to use that definition of the mass scheme. But in principle you don't have to.

The problem of identifying the mass scheme of the top quark that is measeured in experiments, is tied to the fact that they use predictions for the observables that they measeure (and from which they fit the top mass) that are usually made by Monte Carlo event generators. And we do not understand those well enough to say what mass scheme they are effectivley implementing. And this introduces and additional uncertainty when trying to connect to a well-defined scheme, that can be of the same order as the current experimental uncertainties.

But there are also measurements where we do know what scheme they are extracting. Because they use simpler observabes (for example the total ttbar-produciton cross section to measure the pole mass), that we can calculate directly in QFT, which means we have full control over our calculation and which scheme we are using in it. The downside is that this observable is not so sensitive to the top mass, so these measurements are not as precise. But at least we understand what we are measuring ;-)

 

[HomogenousCow]

Isn't this situation quite similar to that of unstable particles, they don't exist as asymptotic states either.

 

[Vanadium 50]

Isn't this situation quite similar to that of unstable particles

I don't think so. I have no trouble weighing a chunk of uranium.

 

[vanhees71]

In a sense yes, but "unstable particles" are usually resonances, and you can measure there "mass" or rather "mass distributions" by looking at the corresponding scattering processes where they occur (in an $s$-channel process). They then occur as peaks with a width in the scattering cross sections as a function of the center-mass energy of the collision.

E.g., a $\rho$ meson appears as a peak at the process $\mathrm{e}^+ + \mathrm{e}^- \rightarrow \text{hadrons}$ as a center-mass energy of about 770 MeV, having a width of about 150 MeV. The width is telling you its mean lifetime, $\tau=\hbar c/\Gamma \simeq 197 \text{MeV} \; \text{fm}/\Gamma$.

With quarks it's way more complicated, because of confinement, as has been described already in this thread. How the current-quark mass are defined is described in the "Review of Particle Physics" (also quoted already above).

The real challenge is to understand the socalled constituent-quark masses or the masses of hadrons. For the light (up and down) quarks within hadrons only about 2% of the mass are due to the Higgs mechanism (leading to current quark masses of a few MeV). The rest is due to the strong interaction. From a group-theoretical point of view there are two sources of mass via the strong interaction: the trace anomaly, related to the gluon field, and spontaneous breaking of the approximate chiral symmetry. It's quite commonly believed that the bulk of the light-hadron mass (including protons and neutrons making up the atomic nuclei in the matter surrounding us) is due to the trace anomaly (i.e., the anomalous breaking of scale invariance of (massless) QCD) and the mass split of chiral-partner hadrons is due to the spontaneous breaking of (approximate) chiral symmetry of QCD in the light-quark sector due to the formation of a quark condensate, $\langle \bar{\psi} \psi \rangle \neq 0$.

One evidence from this comes from the study of heavy-ion collisions, i.e., the collision of two heavy nuclei at relativistic energies, where for a short time of a few 10th of fm/c a hot and dense fireball of a collectively moving fluid of strongly interacting particles is created. At the highest beam energies at the Relativistic Heavy Ion Collider (RHIC) at BNL and at the Large Hadron Collider (LHC) at CERN, it's pretty sure that even a socalled quark-gluon plasma (QGP), i.e., a fluid in local thermal equilibrium with quark and gluon like effective degrees of freedom is formed.

Now at high enough energies on expects that the chiral symmetry is restored, i.e., the approximate chiral symmetry of hadrons is no longer spontaneously broken. If one could now measure hadron masses in the medium one could study whether or not their masses drop to 0, which would mean that the main mass-generating mechanism is the spontaneous chiral symmetry breaking or whether it's rather the trace anomaly, which survives up to higher temperatures than the quark condensate.

The problem is that it is not easy to measure "in-medium hadron masses", because the hadrons take part in the strong interaction and thus all the time interact with the hot and dense fireball and thus can only be observed when they decouple from the collective motion of the fluid, which happens at quite low temperatures of around 100 MeV ("thermal freezeout").

There's one exception however: The neutral vector bosons ($\rho$, $\omega$, and $\phi$) not only decay to hadrons (mostly pions and kaons) but also very rarely (suppressed by a factor $\alpha_{\text{em}}^2$ from the corresponding QED vertex) to a lepton-antilepton pair ("dileptons", i.e., $\text{e}^+ \text{e}^-$ or $\mu^{+} \mu^{-}$ pairs). These leptons or antileptons do not take part in the strong interaction, and thus for them the hot and dense fireball is dilute, i.e., their mean free path in this medium is much larger than the extent of the fireball of some 10th of a fm. Thus they are produced during the whole time evolution of the fireball and their final-state interaction with the surrounding medium can be neglected. Thus measuring carefully their invariant-mass spectrum they allow for a space-time weighted averaged mass spectrum of the light vector bosons in the medium.

Comparing simulations of these heavy-ion collisions using effective models describing the vector mesons with masses either generated via the spontaneous chiral-symmetry breaking, predicting "dropping masses" in the medium or as keeping their masses more or less as in the vacuum (but their mass spectra becoming much broader due to collisional broadening until becoming so broad that they merge into the continuum when everything is melt to a QGP), leads to the conclusion that the data are most accurately described (over several orders of magnitude in the beam energies of the heavy-ion collision, probing the medium at different temperatures and net-baryon density (or baryo-chemical potentials)) by a model where the light-vector-meson mass stays pretty much the same as in vacuo but with tremendously broadened mass spectra due to the vector-mesons interaction with the medium. This hints at the trace anomaly as the mass-generating formalism rather than the spontaneous breaking of the chiral symmetry.

 

[Vanadium 50]

And we do not understand those well enough to say what mass scheme they are effectivley implementing

I don't think the authors of the generators would agree. They might even feel insulted. What the generators do is well-defined and well-described. It is true that it doesn't match some of the more "analytic" approaches.

However, I think this is unhelpful. In true PF fashion, when asked a question, we immediately glom on to the exceptional case. Dale, I apologize.

Confinement is the answer to your question, but the top quark is so short-lived that it usually decays before that can happen. So it's the exception. However, it's also the case where it makes the least difference. The two masses people are most interested in are the "pole mass", which is the mass parameter in the theory, and the invariant mass of the top quark decay products. These differ by around 1%. Maybe half that.

For light quarks it is different. If I tried to measure the mass of a u-quark by a collision with an "infinitely energetic" electron, I would get a number around 4 MeV. If instead I try and measure it by looking at how much energy it takes to flip the spin in a magnetic field, I get a number more like 300 MeV. Maybe 350.

 

[Reggid]

I don't think the authors of the generators would agree. They might even feel insulted. What the generators do is well-defined and well-described.

Well I think the authors would agree, and they would not be insulted. Because I am not saying that the generators are not well-defined and well-described. I am saying that we don't know what mass scheme they are effectively implementing for the top quark (mostly due to effects in the parton shower related to the shower cutoff). And I am sure they agree with that, otherwise they could just state the scheme and the whole "issue" would be solved.

But anyway, you are right that this is just a technicality related to experimental measurements of the top quark and - though related - not exactly what the OP was asking about.

 

[mfb]

I am saying that we don't know what mass scheme they are effectively implementing for the top quark

Well, that's easy: They implement the MC mass. How that is related to other definitions, however...

The top quark isn't affected by confinement but it's still complicated and we have all these definitions, so it's a good place to start.

 

[Reggid]

Well, that's easy: They implement the MC mass.

"A" MC mass, not "the" MC mass. It is not even clear whether they are the same for different parton showers, hadronization models, observables, ...

 

[Vanadium 50]

Well, that's easy: They implement the MC mass. How that is related to other definitions, however...

The fact that what's in the MC doesn't exactly match what's in, say, the MSbar scheme could equally be the problem of the mass scheme authors. (Who did MSbar originally? John Collins maybe?)

If I were in a grumpy mood, I'd even say that the only important numbers are the pole mass, which is what comes into precision electroweak, and the experimental mass. MSbar, DR, DRbar etc. are just distractions. I'm not in a grumpy mood at the moment, so I will concede some value with these schemes, but not enough to blame the MC authors.

In any event, for the top, this is around a GeV out of 175 GeV. That's 0.6%, but we could always argue about whether 1 GeV is really 1.2, 1.5, 0.9, 0.6, etc. It's peanuts compared to light quarks.

 

[Vanadium 50]

That mass consists of the unbound mass of the proton plus the unbound mass of the electron minus the mass deficit.

This picture will lead you astray in QCD. The pion weighs about 140 MeV, and is a bound state of u and d quarks, each weighing a few MeV> (4 and 7 are reasonable estimates). So, what will happen if I raise both masses to 10 MeV? You'd think the pion mass will move up by a few MeV- in fact, it will likely go down.

The reason is that the dynamics is quite complicated. including in this case an approximate symmetry driving the pion mass low. The details don't matter all that much: it's just that interactions between quarks are complicated and simple pictures that don't capture this complexity tend not to work well.

 

[Reggid]

If I were in a grumpy mood, I'd even say that the only important numbers are the pole mass,...

Then I guess you will be really grumpy once we have an $e^+ e^-$ collider with sufficient energy for $t\bar{t}$ produciton ;-) Because then the pole mass will be obsolete for doing precision measurements...

The pole mass of the top quark has an intrinsic renormalon ambiguity of ca. 200 MeV. When you want to be more precise than that, then you have to use a suitable renormalon-free mass scheme. That is not a distraction, that is a necessitiy if you want to reach a precision of around 50 MeV, as projected for top mass measurements at lepton-colliders (even at the HL-LHC some projections go down to 200 MeV).

And people are putting a lot effort in being very careful about which scheme to use for which situation, because it does have an impact on the quality of your results and the convergence of your perturbation series, so they will be grateful that you concede some value to their work ;-)

...but not enough to blame the MC authors.

Who is blaming the MC authors? I'm certainly not. Nobody is blaming anyone.
It is just a well accepted statement that we do not know how the mass scheme in a MC generator (and therefore the mass that is extracted in measurements relying on the reconstruction of the top decay products) is related to a renormalized mass in QFT. That is not saying that some MC authors were doing something wrong. Just that we have to investigate furhter, if we want to understand our measurments at the LHC better.

I also don't see how "conceding" more value to other mass schemes than the pole mass has anything to do with the issue of the MC mass at all (or how else should I read your "but not enough to bame the MC authors"?) Because the MC mass will not be equivalent to the pole mass either, so this issue is independet of your preference for one particular mass scheme.

In any event, for the top, this is around a GeV out of 175 GeV. That's 0.6%, but we could always argue about whether 1 GeV is really 1.2, 1.5, 0.9, 0.6, etc. It's peanuts compared to light quarks.

But it is not peanuts compared to the experimental uncertainties, and that is what matters when you have to decide whether it is a relevant effect that you should address in your theory predictions or not.

Now we went again astray discussing about top quark mass definitions and measurements, and I hope it's no problem for Dale since it is his thread, but it's hard to leave some statements uncommented.

Anyway, related to the OP questions one has to say that due to confinement the mass of any quark in any renormalization scheme is not more than a Lagrangian parameter, much like any other coupling, and not to be interpreted as the "physical mass" of a free quark (as you would do with the pole mass for non-colored particles). I think we all agree on that, independent of whether you like the pole mass much more than other schemes.

 

[Vanadium 50]

Then I guess you will be really grumpy once we have an e+e- collider with sufficient energy for ttbar produciton

Naw...I'll be pushing up daisies by that point. (Or would that be pushing up DESYs?)

 

[vanhees71]

If I were in a grumpy mood, I'd even say that the only important numbers are the pole mass, which is what comes into precision electroweak, and the experimental mass. MSbar, DR, DRbar etc. are just distractions. I'm not in a grumpy mood at the moment, so I will concede some value with these schemes, but not enough to blame the MC authors.

The pole masses are also gauge invariant!