What is new with Koide sum rules?

mitchell porter

There is a very phenomenological paper from Koide (and colleague Ishida) today. It seems to be the first paper that talks about adapting Sumino's mechanism to the quarks.

But let's take a step back. Koide found his formula 30 years ago. Koide has proposed a number of field-theoretic models to explain it; so have a few other people (actually, who else has made a proper field-theoretic model, apart from Ernest Ma?). All QFT models of the relation have the problem that there should be deviations from the formula, because of quantum corrections, but empirically it is exact within error.

Yukinari Sumino was the first person to develop a model in which the corrections are cancelled. It's a little complicated, but it involves a family symmetry that is gauged and then spontaneously broken. The heavy family gauge bosons do the cancelling of the corrections coming from QED.

Koide and Yamashita adapted Sumino's mechanism to supersymmetry. The present paper does not mention supersymmetry, but it does assume the modified version of Sumino's mechanism (in which the mass hierarchy of the family gauge bosons is inverted, compared to Sumino's original version).

Koide and Ishida's inspiration is a tiny aberration in the data for B meson decays. I still haven't digested the paper, but they seem to say at the end that, naively, even a Sumino meson shouldn't be able to produce the dip (that may be there, or which may go away with more data). But there could be some enhancement, and, importantly for them, if the dip is due to their family bosons, then a corresponding dip will not appear in another particular measurement.

From my perspective, this paper runs ahead of theory, because we still have no field-theoretic model of any of the generalized Koide relations for quarks, let alone adaptations of the Sumino mechanism for such models. Koide's own recent BSM work generally assumes that there's a nonet of scalars whose VEVs are diag(√m_e,√m_μ,√m_τ), and then he builds mass matrices for all the SM fermions out of couplings to these. It is from within this theoretical context that he will have guessed at the quark couplings with the Sumino mesons.

Since the quarks have their own Koide relations, it seems very unlikely that their masses are produced in the manner of Koide's recent models. Still, it's always useful to have papers that go "too far ahead" - in this case, trying to interpret a known anomaly as a signal of quark-sector Sumino mesons! Thinking about how the ideas in the paper work, may help those of us still struggling to find an approach to "Koide for quarks".