Pursuing interpretations of the HERA large-$Q^{2}$ data

We explore interpretations of the anomaly observed by H1 and ZEUS at HERA in deep-inelastic e^+ p scattering at very large Q^2. We discuss the possibilities of new effective interactions and the production of a narrow state of mass 200 GeV with leptoquark couplings. We compare these models with the measured Q^2 distributions: for the contact terms, constraints from LEP2 and the Tevatron allow only a few choices of helicity and flavour structure that could roughly fit the HERA data. The data are instead quite consistent with the Q^2 distribution expected from a leptoquark state. We study the production cross sections of such a particle at the Tevatron and at HERA. The absence of a signal at the Tevatron disfavours the likelihood that any such leptoquark decays only into e^+ q. We then focus on the possibility that the leptoquark is a squark with R-violating couplings. In view of the present experimental limits on such couplings, the most likely production channels are e^+d -> scharm_L or perhaps e^+d->stop, with e^+s->stop a more marginal possibility. Possible tests of our preferred model include the absence both of analogous events in e^- p collisions and of charged current events, and the presence of detectable cascade decays whose kinematical signatures we discuss. We also discuss the possible implications for K->pi nu nubar, neutrinoless double-beta decay, the Tevatron and for e^+ e^- ->q qbar and neutralinos at LEP2.