Confining Bethe–salpeter Equation in Qcd

We derive a confining qq̄ Bethe–Salpeter equation starting from the same assumptions on the Wilson loop integral already adopted in the derivation of a semirelativistic heavy quark potential. We show that, by standard approximations, an effective meson squared mass operator can be obtained from our BS kernel and that, from this, by 1 m 2 expansion, the corresponding Wilson loop potential is recovered, spin–dependent and velocity–dependent terms included. We also show, that, on the contrary, neglecting spin–dependent terms, relativistic flux tube model is reproduced. In the paper presented by G.M. Prosperi 1 it was shown how the properties of the Wilson loop integral ( we assume Wilson area law and the straight line approximation; see eqs. (2)–(4)) can be used to obtain a confining Bethe–Salpeter equation from first principles. This result was accomplished neglecting the spin of the quarks. In this paper we show that it can be extended to the case of regular QCD with quarks with spin by defining an appropriate operator for the spin dependent part and using a second order formalism. Even in this case the basic object is the quark–antiquark Green function G4(x1, x2, y1, y2) = 1 3 〈0|Tψ 2(x2)U(x2, x1)ψ1(x1)ψ1(y1)U(y1, y2)ψ c 2(y2)|0〉 = = 1 3 Tr〈U(x2, x1)S1(x1, y1;A)U(y1, y2)S̃2(y2, x2;−Ã)〉 (1) where c denotes the charge-conjugate fields, U the path-ordered gauge string U(b, a) = Pba exp (