Musings

Gravitational Leptogenesis?

I was very excited when I saw a recent paper by Alexander, Peskin and Sheikh-Jabbari (which, I guess, is based on this older paper). They claimed to produce an acceptable lepton-asymmetry during inflation if the inflaton is a (CP-odd) pseudoscalar field, $\phi$, with a coupling

to the curvature, where $F(\phi)$ is an odd function of $\phi$.

Fluctuations generated during inflation drive an expectation-value for $R\tilde{R}$, which generates a lepton-asymmetry via the anomaly,

For reasons, that will become apparent shortly, however, they proceed to say something very unconventional about the anomaly coefficient, $N$:

In general, when heavy right-handed neutrinos are also added to the Standard Model, as is done in the seesaw mechanism for explaining the smallness of the neutrino mass, (2) will be correct in an effective theory valid below a scale $\mu$, of order of the right-handed neutrino mass. More concretely, $N$ can in general be a function of energy. At low energies, below the right-handed neutrino mass scale $N=3$. At higher energies, $N$ could be anywhere between zero to three, depending on the details of the particle physics invoked.

Umh … no. That’s not how anomalies work. The anomaly depends only on the massless spectrum and the anomaly coefficient does not run with energy.

Now, it’s true that, above the scale $\mu$, one can treat the right-handed neutrino, $\Psi$, as massless. In that theory, there’s a new current which is non-anomalous. $$\partial_\mu (j^\mu_L + j^\mu_\Psi) = 0$$ However, once one includes the Majorana mass for $\Psi$, $j^\mu_\Psi$ is not conserved, even classically. The correct, classically-conserved, current in the theory which includes the mass term, $\mu$ is $j^\mu_L$. And the anomaly in that current is the same, whether one calculates it in the high- or low-energy theory.

The motivation for the seemingly bizarre statement about the anomaly (2) becomes apparent a little later on. One would naïvely expect the gravitational contribution to the lepton asymmetry, during inflation, to be tiny, suppressed by at least a factor of ${\left(\tfrac{H}{M_{\text{pl}}}\right)}^2$. They, however, claim that there’s an enhancement over the naïve answer by a factor of ${\left(\tfrac{\mu}{H}\right)}^6$. This enhancement comes about because, in their calculation, $\langle R\tilde{R}\rangle$ is UV-divergent, dominated by short-distance effects near the cutoff, $\mu$.

I don’t really understand their calculation. But, if they are correct that, in the effective theory, $\langle R\tilde{R}\rangle \sim \mu^6$, then what this means is that, in the full theory, where one includes loops of $\Psi$, the contribution of those extra diagrams cancels the erstwhile UV divergence, yielding a finite (but $\mu$-dependent) answer. I don’t see why there should be such a cancellation, here.