Electroweak Contributions to Quark Masses

Abstract

The divergences in lowest-order electroweak contributions to quark masses are discussed. The example model of U(l) em ®SU(3) co i 0 r is worked out. If the bare mass and color coupling constant are constrained to be independent of the electromagnetic coupling, as is natural, then the lowest-order electromagnetic shifts of the renormalized mass and strong coupling will be infinite. This is the expected result and is true even though we treat the strong interactions to all orders and thereby incorporate the relatively soft ultraviolet behavior of quantum chromodynamics.There has recently been interest in the lowestorder electroweak contribution to hadronic masses in quantum chromodynamics. 1 " 3 Traditionally, such effects were thought to account for the mysterious n-p mass difference. 4 Nowadays, it is believed that symmetry breaking effects within a grand unified scheme are also important. Regardless of the ultimate origins of mass differences, the general problem of gauge boson corrections to hadronic masses must be addressed. Indeed, there is no a priori reason why these corrections need be smaller, say, than those arising from dynamical symmetry-breaking effects. The traditional scheme for performing such calculations has been via the Cottingham formula, 4 originally designed for calculating the purely electromagnetic contribution in terms of a photon loop integral over the trace of the virtual Compton amplitude, T. It can be proven under very general conditions that this contribution is logarithmically divergent. 1 * 2 Loosely speaking, this follows from the observation that the large-# (photon momentum) behavior of T is given by the operator-product expansion (OPE), and since T is a scalar, the leading operator will be the trace of the energy-momentum tensor, 9, which has no anomalous dimension. Thus, the large-# contribution can be calculated as if T, itself, were being treated in perturbation theory. Just as in quantum electrodynamics (QED), this leads to a logarithmically divergent result. The conventional vertex softening due to asymptotic freedom is absent. Because of their different masses and charges, different quark flavors receive different divergent contributions so, in general, even mass differences remain divergent. However, when the bare masses are equal, the gluon term in 6 does not contribute to differences which can be calculated as if the quarks were free "partons" and there were no gluons in the theory. Weinberg 5 has pointed out that in a pure electroweak theory, where the strong interactions are ignored, there is a "natural" possibility for mass differences to be finite. The conventional logarithmic electromagnetic (e.m.) divergence can be cancelled by an analogous contribution from the weak interactions. From our previous remark, we see that such a result would be preserved even in the presence of the strong interactions (QCD), since they have no effect on the leading divergence. Having canceled this divergence, one must still ensure that the next-to-leading term in the OPE converges; th...

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