# Quark's mass vs an electron's mass

syano
If the mass of an electron is significantly less (at least 3 times less) than the mass of a proton or a neutron, and If every proton or neutron is made up of three quarks. Would it be fair to say that a quark’s mass is greater than an electron's?

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## Answers and Replies

Loren Booda
Good guess, but it eventually depends on the binding energy of the quarks. E=mc2.

Jonathan
Yes, I think by several orders of magnitude.
EDIT: I somehow missed Loren Booda's post, I don't think the energy makes much difference in a comparison of quarks with electrons.

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Zimm
Loren makes a good point, the binding energy of the quarks can give the resulting particle a greater mass "then the sum of its parts"

From Physical Review D by APS the up quark has a mass of 1.5 to 4.5 MeV and the down quark has a mass of 5 to 8.5 MeV (in the mass-independent subtraction scheme as opposed to the higher values assigned in potential models, which is just gobbledygook for, there are more than one way to measure it and this is one way with a certain result) Compare that to an electron's mass of close to 0.511 MeV and you see that the quarks are more massive.

But think of a protons and neutrons mass ~938 MeV each. A proton is uud, up-up-down quarks, which would total 17.5 MeV on the high side, and a neutron is udd, up-down-down quarks, which totals 21.5 MeV. The binding energy of the quarks, caused as I recall by the strong force, is what accounts for the additional "mass" of the proton and neutron. E->m*c^2

Jonathan
Yes, the point is true, but unrelated to the topic, all that was asked was:
qm>em?
Which is true regardless of binding energy, since I'm pretty sure the energy is never negative, esp. to such an extreme degree as to make them anywhere close in mass quantity.

Zimm
Actually, I believe it is related to the topic because no one had yet stated masses for the quarks...you see if the quarks had masses less than the electrons all that would be required is greater binding energy to still make the masses of the proton and neutron. So while you could assume that the quarks must have masses greater than the electron to make the proton, and while that *may* be the case, without knowing the quark masses one could use binding energy arguments to say that quark masses *could* be lower than the electron mass

Jonathan
What are you talking about? The point of stating your point rests on an assumption we know isn't true.

Staff Emeritus
Gold Member
Zimm has a good point. Out of roughly 1000MeV of a proton's mass, at most 20MeV comes from the mass of the constituent quarks, which means that the main ingredient, by far (~98%), is binding energy.

Clearly, this implies that the assertion "me<mq" depends crucially on the magnitude of the binding energy. If you don't know such magnitude, you cannot conclude the inequality, since it could also go the other way.

Taking out the binding energy, one may wrongly conclude (as Jonathan does) that
Yes, I think by several orders of magnitude.
While the difference may be of only a factor 3 or 5.

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Jonathan
The point in my previous post was that I knew that the mass of even the lightest quark is far more than an electron's, I didn't give the number because that's not what he asked for, and I wasn't going to look up the actual number when that's not really relevant to the question. What do you mean 'it depends crucially on the binding energy'? Show me one quark that is lighter than a electron! Come on, do you hear yourselves?! The energy only matters if one wants to know the actual value of a quark's mass. If he wanted that he would have specified what type of quark too. In fact, I just looked up both masses on google. It only took me 30 seconds, and here we are arguing semantics!

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Dearly Missed
Jonathan, you are absolutely right, but just to clinch it here's a http://na47sun05.cern.ch/target/fundamentalparticles.html [Broken] showing that the lightest quarl, the top, has mass .005 GeV/c2, that's 5 MeV/c2 or nearly 10 times the electron's 511 eV/c2.

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Loren Booda
Starting with only the knowledge of proton or neutron mass, its composition by three quarks of two colors and an unknown binding energy, I retain my assertion that such "quarks" could be of any mass less than that of the given nucleon, therefore possibly less than that of an electron.

Staff Emeritus
Gold Member
IMO, the main point of the original question was not if the mass of electrons is smaller than that of quarks. Rather, it asks if the following is a valid reasoning:

IF
mass of an electron is at least 3 times less than the mass of a proton or a neutron
AND
every proton or neutron is made up of three quarks
THEN
a quark’s mass is greater than an electron's

The answer to this question is

NO, it is not a sound argument,
because the total mass of a nucleon has an extra component (the binding energy) that is important enough to possibly affect the result.

It turns out that, once having taken the binding energy into account, the answer is the same, but that does not mean that the original argument was correct, especially since it the binding energy turns out to represent close to 98% of the full mass.

Actually, this can be easily written in a couple of equations. This way things get clearer:

The original question is:

From
1. e < 3n (n: mass of a nucleon)
2. n = q1+q2+q3 (qi: mass of a quark)
can I conclude that q1>e ??

This was probably what he had in mind (i.e., no binding energy in the picture). Even in this case, you can easily find a counterexample.

What Loren correctly pointed out is that the assumption implicit in two is incorrect, and that there is an important term B:

2. n = q1 + q2 + q3 + B.

Clearly, this new term makes the original argument even harder to defend, since B can in principle be very close to n (i.e., the binding energy can account for a big fraction of the total mass), in which case you would not be able to say anything about the relation between e and q1, q2, or q3.

Furthermore, as mentioned before, B is actually close to 98% of n. It just so happens that the remaining 2% is big enough to give the result e<qi (i=1,2,3).

In any case, I think what he wanted was some clarification of the relation between masses. I hope the discussion is useful in that sense.

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Dearly Missed
Let's get quantitative. We have

q1 + q2 + q3 + X = 1800m

Where the q's are the quark masses, X is the binding energy and m is the electron mass. The proton is 1800 times as massive as the electron.

We are given that X = 49(q1 + q2+ q3)

Let q1 be the heaviest quark mass, then 3q1 > q1 + q2 + q3

whence 147q1 > X
150q1 > q1 + q2 + q3 + X = 1800m
q1 > (1800/150)m > m

So only knowing the ratio of the electron mass to the proton mass, the fact that the proton is made up of three quarks of unequal mass, and that 98% of the proton mass is binding energy, we easily conclude that at least one quark is more massive than the electron.

But if we didn't know about the binding energy we couldn't have concluded that.

Staff Emeritus
Gold Member
Moreover, it is very tricky to even DEFINE what is meant with "quark mass". It is a parameter that is to a certain extend an arbitrary ingredient in the renormalisation scheme. The reason is the asymptotic freedom, and the counterpart, confinement.
In QED, it is not difficult to define the electron mass, it is the pole of the renormalized propagator, for example, in the very low energy regime (yes, the pole is depending on the scale at which you decide to catch it!). So the mass scale is very natural there. However, in QCD, we know that the perturbative expansions don't make sense below about 1 GeV. So one can't define the quark mass as the pole of the quark propagator in the low energy regime !
The physical reason behind this mathematical problem is confinement: you can't just put a single quark in a static color field and watch how it behaves. However, the mass parameter does start to make sense when considering collisions, say, at 10 GeV. However, in that case, these light quarks are then essentially massless compared to the energy scale. So you better just consider the mass of the light quarks as "fitting parameters" in QCD.

cheers,
Patrick.

Sorry to dredge up such an old post but I was just googling for information on the mass discrepancy between the three constituent quarks and the proton (or neutron) and I came across this thread.

So the three constituent quarks amount to a mass (equiv) of about 15MeV and the neutron a mass of nearly 1000MeV. At first I thought like everyone else here that this must be due to the binding energy, but it’s the wrong way around! The binding energy of parts held together by an attractive force is negative (a negative PE that is) so the mass of the parts should be greater than the mass of the assembly (see ref). You've got to put in work (hence mass) to get them apart right?. So what’s the real story behind this huge mass discrepancy, I’m curious now.

Ref : http://en.wikipedia.org/wiki/Binding_energy
Because a bound system is at a lower energy level than its unbound constituents, its mass must be less than the total mass of its unbound constituents. For systems with low binding energies, this "lost" mass after binding, may be fractionally small. For systems with high binding energies, however, the missing mass may be an easily measurable fraction.

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Homework Helper
Gold Member
Sorry to dredge up such an old post but I was just googling for information on the mass discrepancy between the three constituent quarks and the proton (or neutron) and I came across this thread.

So the three constituent quarks amount to a mass (equiv) of about 15MeV and the neutron a mass of nearly 1000MeV. At first I thought like everyone else here that this must be due to the binding energy, but it’s the wrong way around! The binding energy of parts held together by an attractive force is negative (a negative PE that is) so the mass of the parts should be greater than the mass of the assembly (see ref). You've got to put in work (hence mass) to get them apart right?. So what’s the real story behind this huge mass discrepancy, I’m curious now.

Ref : http://en.wikipedia.org/wiki/Binding_energy

see the excellent post by Vanesch just before yours. The key point is that one really cannot measure the masses of the quarks in the way one measures the masses of other particles, because of confinement. The definition of mass and binding energy you quote supposes that you can take the particle apart and measure the masses of the constituents separately and then compare this to the mass of the bound system. This is not possible for strongly interacting particles. So the way people assign values to the parameters "m" appearing in the QCD lagrangian is very indirect.

Gold Member
Binding energy does not work here as nicely as for the EM field, because the gluon field is charged. In this sense the strong force is more as gravity than as EM.

The problem reflects in the calculation of the mass of the "glueball", it is predicted to be higher than the mass of the proton. But the glueball has 0 quarks, while the proton has 3 quarks. The 3 quarks actually disturb the field, but the proton (nor the neutron) can not get rid of them.