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It is shown that gauge theories give contributions to neutrinoless double-B decay [(B&,] which are not covered by the standard parametrizations. While probably small, their existence raises the question of whether the observation of (PP), implies the existence of a Majorana mass term for the neutrino. For a "natural" gauge theory we argue that this is indeed the case. The associated questions of neutrinoless double-P decay [denoted (PP)&] and neutrino mass have again become of general interest. A comprehensive recent discussion is given by Doi et al. ' and a concise summary by ~o s e n .~ The classical analysis of the (PP)oy process, which of course predated gauge theories by many years, assumed that it arose as a result of neutrino exchange between two effective four-fermion vertices. This is shown in modern language in Fig. l(a), which illustrates the process at the quark level. Thus, the parametrization of the (fib)& process was considered to be given fundamentally by the ---I -2 L 2 Zi (4 ( d ) FIG. 1. Diagrams for neutrinoless double-B decay in an SU(2)XU(l) gauge theory. The standard diagram is Fig. l(a). It is the only one which contains a virtual neitrino (of four-momentum p ) . d and u are the down and up quarks. parametrization of 'the four-fermion single-P-decay interaction.We point out here that the situation is more complicated3 if the weak interactions are described by a gauge theory. For definiteness we consider in the framework of the standard SU(2) x U( 1) gauge theory. Larger gauge groups usually contain SU(2) X U(1) as a subgroup and composite models are usually contrived to also display this symmetry. Thus there is not much loss of generality in doing so.A rather minimal way4 to naturally include lepton-number violation in the theory is to add to the complex Y = 1, I = Higgs doublet a complex Y = 2, I = 1 isotriplet H:The theory need not contain any more than one two-component Weyl neutrino field for each gene r a t i~n .~ In a natural theory (no special adjustment of parameters) both #' and h 0 will develop nonzero vacuum expectation values giving the relations4in standard notation. The vacuum value y is expected to be small compared to h, but the experi-295 1

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We propose an extension of the old, fairly successful, phenomenological prescription for adding spin-1 fields to the chiral Lagrangian of pseudoscalar mesons to the case where the Wess-Zumino term is present. This leads to the possibility of describing within the effective-Lagrangian framework a whole class of "unnatural-parity" hadronic reactions. In particular, we study the wellknown (pure hadronic) w + 3 a and (radiative) w-+@y processes, and find good agreement with experiment, comparable to that for any of the current-algebra "theorems." The construction of the Lagrangian is seen to involve subtle theoretical issues. We are led to examine in more detail Witten's "trial-and-error" gauging of the Wess-Zumino term and note that the Bardeen form of the anomaly seems more suitable than the left-right-symmetric form for constructing an effective model with phenomenological spin-1 fields.