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September 2, 2023


For various reasons, some people seem to think that the following modification to Einstein Gravity
(1)S=12dϕ*dϕ+κ 22*+3ϕ192π 2fTr(RR)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 be1. But in the real world, there are nearly massless neutrinos. In the Standard Model, U(1) BLU(1)_{B-L} has a gravitational ABJ anomaly (where, in the real world, the number of generations N f=3N_f=3)
(2)d*J BL=N f192π 2Tr(RR) d * J_{B-L} = \frac{N_f}{192\pi^2} Tr(R\wedge R)
which, by a U(1) BLU(1)_{B-L} rotation, would allow us to entirely remove2 the coupling marked in red in (1). In the real world, the neutrinos are not massless; there’s the Weinberg term
(3)1M(y ij(HL i)(HL j)+h.c.)\frac{1}{M}\left(y^{i j} (H L_i)(H L_j) + \text{h.c.}\right)
which explicitly breaks U(1) BLU(1)_{B-L}. When the Higgs gets a VEV, this term gives a mass m ij=H 2y ijM m^{i j} = \frac{\langle H\rangle^2 y^{i j}}{M} to the neutrinos, So, rather than completely decoupling, ϕ\phi reappears as a (dynamical) contribution to the phase of the neutrino mass matrix
(4)m ijm ije 2iϕ/fm^{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.


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, H=0\langle H \rangle=0 and the effect of a ϕ\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 θ\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 θ\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, 𝒜\mathcal{A}, for U(1) BLU(1)_{B-L} and modify (1) to
(5)S=12(dϕf𝒜)*(dϕf𝒜)+κ 22*+3ϕ24π 2f[18Tr(RR)+d𝒜d𝒜]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) BLU(1)_{B-L} gauge transformations 𝒜 𝒜+dχ ϕ ϕ+fχ Q i e iχ/3Q i u¯ i e iχ/3u¯ i d¯ i e iχ/3d¯ i L i e iχL i e¯ i e iχe¯ i \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 𝒜\mathcal{A} (or would be if we promoted 𝒜\mathcal{A} to a dynamical gauge field). Choosing χ=ϕ/f\chi = -\phi/f eliminates the term in red. Turning back on the Weinberg term (which explicitly breaks U(1) BLU(1)_{B-L}) puts the coupling to ϕ\phi into the neutrino mass matrix (where it belongs).
Posted by distler at September 2, 2023 1:34 PM

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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…)

Posted by: Mitchell Porter on September 3, 2023 2:48 AM | Permalink | Reply to this

Re: 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

In QCD, strong coupling infrared dynamics spontaneously breaks the U(1) AU(1)_A symmetry (which is also explicitly broken by the ABJ anomaly and the quark mass term) leading to the interpretation of the η\eta' as a pseudo-Goldstone boson. (In large-NN QCD, the effect of the ABJ anomaly is suppressed and, for massless quarks, the η\eta' is a Goldstone boson, just like the rest of the pions.)

Gravity is infrared free, and absolutely does nothing like that (create a neutrino condensate analogous to the quark condensate in QCD). They just blithely assume that it does.

That assumption (as far as I can tell) is based on a flawed analogy with QCD. Pure Yang-Mills has a mass-gap. So (for instance) does 1-flavour QCD. Pure gravity does not have a mass gap. Introducing a massless fermion does not suddenly produce a mass gap where there was none before.

As to Putrov-Wang, I should have been clear that the gravitational ABJ anomaly in U(1) BLU(1)_{B-L} is intimately tied to proposals for generating the matter-antimatter asymmetry in the universe (“leptogenesis”). Introducing ϕ\phi as in (1) cancels the anomaly (indeed, this is the 4D version of the famous Green-Schwarz anomaly cancellation mechanism).

Putrov and Wang seem to take more seriously than is warranted the introduction of the right-handed neutrinos that give rise (upon integrating them out) to the Weinberg term (3) in the low-energy effective theory. But that’s the theory which is valid when leptogenesis supposedly takes place. There’s nothing wrong about their analysis (indeed, it is quite pretty), but every useful statement about higher/categorical symmetries associated to U(1) BLU(1)_{B-L} should have a formulation in the effective theory.

Posted by: Jacques Distler on September 3, 2023 4:20 AM | Permalink | PGP Sig | Reply to this

Re: dCS

Hello, Prof. Distler. I just read this post:

What would you say is a good undergrad particle physics book and a graduate particle physics book that you would use for such classes you’d be teaching. I suppose you’d be most likely using your own notes, but I’m curious about what you’d recommend an advanced undergraduate should look at or what a graduate student should look at. Thanks!

Posted by: Tom on October 3, 2023 4:21 PM | Permalink | Reply to this

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