PhD thesis - CP3-Origins

QCD and Chiral Symmetry Breaking 27Considering the two flavor QCD with equal quark masses m u = m d = m ≠ 0 thevector symmetry in question is the SU(2) V . If this symmetry were broken there wouldbe Goldstone bosons associated with the scalar currents.Let us consider Euclidean correlator〈〉∣C Γ = 0 ∣J ūd (x)J ¯du (y) ∣ 0 , (3.31)where J ūd = ūΓd are quark currents withΓ = 1, γ 5 , iγ µ , γ µ γ 5 , iσ µν . (3.32)Using the results derived in the section 3.2, we can present the correlators asC Γ (x, y) = 1 Z [Π n(m − iλ)] 2 Tr {ΓG(x, y)ΓG(y, x)} , (3.33)where G(x, y) is the Euclidean Green function of the u- and d-quarks in a given gaugefield backgroundG(x, y) = ∑ ku k (x)u † k (y)m − iλ k. (3.34)Note that we have explicitly employed the fact that the masses are degenerate. Inaddition, we have assumed that the common mass is real i.e. the θ angle of the QCDvacuum is zero.Using the symmetry u k → γ 5 u k , λ k → − λ k we can show thatγ 5 G(x, y)γ 5 = ∑ k[γ 5 u k (x)][γ 5 u k (y)] †m − iλ k= ∑ k[u k (x)u † k (y) ∑=m + iλ kk]u k (y)u † k (x)†m − iλ k(3.35)= G † (y, x).The Green function can be expanded over the full basisG(x, y) = s(x, y) + γ 5 p(x, y) + iγ µ v µ (x, y) + γ µ γ 5 a µ a µ (x, y) + 1 2 iσ µνt µν , (3.36)