By F. J. Ynduráin (auth.)

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1) 24 QeD as a Field Theory where FI'P = al'A P- aPA 1', DI' = al'ieA 1', is not gauge invariant because of the gauge-fixing term - (A/2)(aA )2. Gauge invariance may, however, be restored by means of the following trick. 2) to J g with w a massless field without interactions. 3). The restoration of gauge invariance was comparatively easy here; because A has no charge, and thus no self-interactions, we could take W to be real and free. However, the simplicity of J w does not mean that it has no deep consequences.

6) ~=I-A -I. It is very easy to see what v dependence we need . We recall that the Vo entered in the combination dDk = dDk V4 - D (27T)D 0 ' so the only dependence on Vo is in the divergent part: r(2/ €)(47T), /2(vJ)' /2. (I )(V) -g- + . . 7 b) 300ur version of the MS scheme is slightly different (although equivalent) to the standa rd one . The Callan-Symanzik Equation 49 cP) the coefficients of the logv 2 terms are the same we have already calculated up to a sign. 2 in the p. scheme, to lowest order .

6b) then, fR(ApI" ' " ApN-I ; g(v) ,m(v),~(v) ;p) = APrfR(PI' . . ,PN-I ; g(A),m(A),ii(A) -I ;V) X exp{ - L'OgAdlogA'Yr(g(N),m(N) ,ii(N) -')} . 7». P This realization is nontrivial due to the infinite character of renorrnalization, which introduces an extraneous mass scale. 32The dimension of a field is easily deduced, noting that the action ,if = Jd 4x J' ( x ) must be dimensionle ss. Hence, [q) = [M)3 / 2, [w) = [M)', [B) = [M]' . The dimension of r is obtained from those of the fields that it embodie s: for example , Ps = -I (3/2 + 3/2 for the fields, and - 4 from d 4x ), for the fermion propagator S.

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