### dCS

For various reasons, some people seem to think that the following modification to Einstein Gravity

(1)$S= \int \tfrac{1}{2} d\phi\wedge *d\phi + \tfrac{\kappa^2}{2} *\mathcal{R} + {\color{red} \tfrac{3 \phi}{192\pi^2 f}Tr(R\wedge R)}$

is

interesting to consider. In some toy world, it might be

^{1}. But in the real world, there are nearly massless neutrinos. In the Standard Model,

$U(1)_{B-L}$ has a gravitational

ABJ anomaly (where, in the real world, the number of generations

$N_f=3$)

(2)$d * J_{B-L} = \frac{N_f}{192\pi^2} Tr(R\wedge R)$

which, by a

$U(1)_{B-L}$ rotation, would allow us to

*entirely remove*^{2} the coupling marked in red in (

1).
In the real world, the neutrinos are not massless; there’s the Weinberg term

(3)$\frac{1}{M}\left(y^{i j} (H L_i)(H L_j) + \text{h.c.}\right)$

which explicitly breaks

$U(1)_{B-L}$. When the Higgs gets a VEV, this term gives a mass

$m^{i j} = \frac{\langle H\rangle^2 y^{i j}}{M}$
to the neutrinos, So, rather than completely decoupling,

$\phi$ reappears as a (dynamical) contribution to the

*phase* of the neutrino mass matrix

(4)$m^{i j} \to m^{i j}e^{2i\phi/f}$

Of course there

*is* a CP-violating phase in the neutrino mass matrix. But its effects are so tiny that its (presumably nonzero) value is

still unknown. Since (

4) is rigourously equivalent to (

1), the effects of the term in red in (

1) are similarly unobservably small. Assertions that it could have dramatic consequences — whether for LIGO or large-scale structure — are …

*bizarre*.

#### Update:

The claim that (

1) has some observable effect is even more bizarre if you are seeking to find one (say) during inflation. Before the electroweak phase transition,

$\langle H \rangle=0$ and the effect of a

$\phi$-dependent phase in the Weinberg term (

3) is

*even more* suppressed.

^{1} An analogy with Yang Mills might be helpful. In pure Yang-Mills, the $\theta$-parameter is physical; observable quantities depend on it. But, if you introduce a massless quark, it becomes unphysical and all dependence on it drops out. For massive quarks, only the *sum* of $\theta$ and phase of the determinant of the quark mass matrix is physical.

^{2} The easiest way to see this is to introduce a background gauge field,

$\mathcal{A}$, for

$U(1)_{B-L}$ and modify (

1) to

(5)$S= \int \tfrac{1}{2} (d\phi-f\mathcal{A})\wedge *(d\phi-f\mathcal{A}) + \tfrac{\kappa^2}{2} *\mathcal{R} + {\color{red} \tfrac{3 \phi}{24\pi^2 f}\left[\tfrac{1}{8}Tr(R\wedge R)+d\mathcal{A}\wedge d\mathcal{A}\right]}$

Turning off the Weinberg term, the theory is invariant under

$U(1)_{B-L}$ gauge transformations

$\begin{split}
\mathcal{A}&\to \mathcal{A}+d\chi\\
\phi&\to \phi+ f \chi\\
Q_i&\to e^{i\chi/3}Q_i\\
\overline{u}_i&\to e^{-i\chi/3}\overline{u}_i\\
\overline{d}_i&\to e^{-i\chi/3}\overline{d}_i\\
L_i&\to e^{-i\chi}L_i\\
\overline{e}_i&\to e^{i\chi}\overline{e}_i\\
\end{split}$
where the anomalous variation of the fermions cancels the variation of the term in red. Note that the first term in (

5) is a gauge-invariant mass term for

$\mathcal{A}$ (or would be if we promoted

$\mathcal{A}$ to a dynamical gauge field). Choosing

$\chi = -\phi/f$ eliminates the term in red. Turning back on the Weinberg term (which explicitly breaks

$U(1)_{B-L}$) puts the coupling to

$\phi$ into the neutrino mass matrix (where it belongs).

Posted by distler at September 2, 2023 1:34 PM
TrackBack URL for this Entry: https://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/3490

## Generalized symmetries

Pardon the slight digression… I have nothing to say about “dynamical Chern-Simons gravity”, but this certainly brings to mind work of Dvali and Funcke that has perplexed me for a long time, e.g. this which claims that a gravitational axial anomaly of neutrinos has all kinds of phenomenological consequences. I also note that Putrov and Wang, in their conclusion (part 3), speculate on similar matters, but they frame it all in terms of the newly popular noninvertible, categorical, etc., symmetries.

It would also be nice to see a review of which such phenomenological proposals do and don’t make sense, from a string theory perspective. (I’m not saying you should write it! Maybe such a review already exists in the swampland literature…)