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If you model the quarks as massless, there should be no flavor mixing, because flavor mixing is achieved through the CKM matrix, which is a mass matrix.

However, if quarks are massless, there ought to be an axial flavor symmetry, but there isn't.

So to reconcile this, we must spontaneously break the axial flavor symmetry. This is done by introducing a "fermion condensate" flavor mixing field: [tex]\bar{\Psi}^{\alpha j}P_L \Psi_{\alpha i} [/tex] where [tex]\alpha[/tex] is the color index and 'i' the flavor index (refers to the flavor of quark) and 'j' is also the flavor index (refers to flavor of the antiquark).

How do you spontaneously break this field? You need to add some terms to the Lagrangian involving this composite field, to spontaneously break it, right? And how come when you spontaneously break it, you don't set it equal to some constant VEV (as you do with the Higgs), but allow the VEV to vary in spacetime, and call this variation the pion? Then you treat the vacuum expectation value as separate, independent field, and give the pion field its own Lagrangian?

The flavor symmetry in the book is called [tex] SU(2)_LxSU(2)_R [/tex] because the book is assuming only two flavors of quarks (up and down) for simplicity. So since there are 6 quarks, is the real flavor symmetry [tex] SU(6)_LxSU(6)_R [/tex]? The problem I have with this is some of the arguments of the book rely on the fact that SO(4)=SU(2)xSU(2), which is no longer true if 2=6.

However, if quarks are massless, there ought to be an axial flavor symmetry, but there isn't.

So to reconcile this, we must spontaneously break the axial flavor symmetry. This is done by introducing a "fermion condensate" flavor mixing field: [tex]\bar{\Psi}^{\alpha j}P_L \Psi_{\alpha i} [/tex] where [tex]\alpha[/tex] is the color index and 'i' the flavor index (refers to the flavor of quark) and 'j' is also the flavor index (refers to flavor of the antiquark).

How do you spontaneously break this field? You need to add some terms to the Lagrangian involving this composite field, to spontaneously break it, right? And how come when you spontaneously break it, you don't set it equal to some constant VEV (as you do with the Higgs), but allow the VEV to vary in spacetime, and call this variation the pion? Then you treat the vacuum expectation value as separate, independent field, and give the pion field its own Lagrangian?

The flavor symmetry in the book is called [tex] SU(2)_LxSU(2)_R [/tex] because the book is assuming only two flavors of quarks (up and down) for simplicity. So since there are 6 quarks, is the real flavor symmetry [tex] SU(6)_LxSU(6)_R [/tex]? The problem I have with this is some of the arguments of the book rely on the fact that SO(4)=SU(2)xSU(2), which is no longer true if 2=6.

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