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Mass of the proton with massless quarks? 
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#1
Jan2313, 05:48 AM

PF Gold
P: 2,938

A usual lore from chiral perturbation theory is that the mass of the pion is proportional to the sum of the up and down masses, and then it is going to be zero when such masses are zero.
Now, for the proton, I notice the following remark from Chris Quigg 


#2
Jan2313, 06:25 AM

P: 328

I am not really sure, but I think the point is that the pion is a quasiGoldstone boson for the chiral symmetry. It means that in the limit of exactly broken chiral symmetry (i.e. the mass quark vanishes) it should be completely massless. On the other hand, in this situation the nucleons (and so the protons) acquire a mass.
Again, I am not really sure. 


#3
Jan2313, 08:11 AM

PF Gold
P: 2,938

Perhaps the question is, why the chiral expansion of the nucleon has a constant term while the pion hasn't?
It is a little bit as the expansions of cos(x) and sin(x), but in this later case we know that one of the expansions must be even and the other must be odd, so it is crystalclear. 


#4
Jan2313, 12:04 PM

P: 883

Mass of the proton with massless quarks?
There's no reason for the nucleon to be massless at zero quark mass. In general, we should expect hadrons to have masses of order the characteristic scale of QCD; call it ~1 GeV.
The thing that needs explaining is why the pion is massless at zero quark mass. That happens because of chiral symmetry and Goldstone's theorem, as Einj said. 


#5
Jan2413, 07:54 PM

PF Gold
P: 2,938

Of course chiral symmetry is also explicitly broken because of the quark masses, but I fail to see how this mechanism compete with the condensation. 


#6
Jan2513, 11:26 AM

P: 883




#7
Jan2513, 02:16 PM

PF Gold
P: 2,938

 First, the up and down are not massless. But they are light respect to the QCD chiral scale, which is about 100 MeV.  Second, the up and down have not the same mass. So the SU(2)_V symmetri is approximate too. I was thinking which could be the relative contribution of each source to the mass of the pion, and wondering if the second one could be relevant too, or even more relevant. For instance, imagine the up is massless. Then, should we have an exact chiral U(1)L x U(1)R and a massless neutral pion, with massive charged pions due to the breaking of SU(2)_V? 


#8
Jan2613, 02:30 AM

Sci Advisor
P: 5,464

As said the mass scale of the nucleon is rather natural (~ 1GeV) whereas the nearly massless pions are explained via the Goldstone mechanism. It is interesting to see what happens w/o spontaneous chiral symmetry breaking. So let's look at the eta prime meson (η') meson which is the flavorsingulet of the SU(3) generated by Isospin and Strangeness.
The eta meson is a Goldstone boson with mass 548 MeV (rather large compared to pions Due to the mass of the strange quark) whereas the eta prime is NOT a Goldstone boson b/c the singulet U(1) symmetry is not broken via the Goldstone mechanism but via the axial anomaly. Therefore the eta prime has a mass of 958 MeV which is rather close to the nucleon mass. 


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