The Non-triviality of Dynamical Chern-Simons Gravity and the Standard Model

The dCS action is

where $g_{\mu\nu}$ is the spacetime metric with $g$ its determinant, $\phi$ is the dCS pseudoscalar, $R$ is the Ricci scalar, $R^{\mu\nu\rho\sigma}$ is the Riemann tensor with $\tilde{R}_{\mu\nu\rho\sigma}$ its Hodge dual, and $\lambda$ the dCS coupling constant.

Under the global symmetry $\phi(x^{\mu})\to\phi(x^{\mu})+\chi$, Eq. (II.1) changes by a constant shift since the last term can be written as the derivative of the gravitational Chern-Simons (CS) current $K^{\mu}$,

where $\epsilon^{\mu\nu\rho\sigma}$ are the contravariant components of the Levi-Civita tensor, and the trace is taken over the indices of the Lorentz-algebra valued spin connection ${\omega_{\mu}}^{a}_{\;\;b}$. The equations of motion will be invariant under constant scalar shifts. We will call this shift symmetry $\text{U}(1)_{\text{dCS}}$. Its associated Noether current is

which satisfy

upon using $\phi$’s equation of motion $\Box\phi-\lambda\tilde{R}R=0$.³³3We note that the $S_{\text{dCS}}$ is not strictly invariant under global $\phi$ shifts because the dCS Lagrangian changes by a total derivative. Symmetries satisfying this property are often referred to as quasi-symmetries, so a $\text{U}_{\text{dCS}}(1)$ transformation is more properly referred to as a quasi-symmetry of the action, although we will make no such distinction in what follows.

For field theories in a flat spacetime with vanishing boundary conditions, Eq. (4) guarantees the existence of a conserved charge. In curved spacetime, a conserved Noether charge is obtained if we assume a globally hyperbolic and asymptotically flat spacetime. In this case, we then integrate Eq. (4) over a spacetime volume $V$ foliated by spacelike surfaces $\Sigma_{t}$ whose boundaries $\partial\Sigma_{t}$ are 2-spheres with a very large radius (compared to the maximum curvature scale inside $V$), where $t$ is the parameter of the timelike curves perpendicular to $\Sigma_{t}$. We have

Note that $j^{\mu}$ is defined up to a term $\Delta j^{\mu}$ that vanishes at the boundary of $V$. If the last integral in the second line of Eq. (II.1) vanishes, then we have that

is independent of $\Sigma_{t}$ and thus is conserved. Typically it is assumed that $n_{i}j^{i}\to 0$ asymptotically such that the last integral in Eq. (II.1) vanishes and we have charge conservation.

Let us now apply these facts to dCS gravity. Assuming that $\partial^{i}\phi$ vanishes at the boundary, only the gravitational CS current will contribute to the integral of $n_{i}j^{i}$. However, the spin-connection will be pure gauge on the cylinder at infinity, and we can choose a gauge where its time component $\omega_{0}$ vanishes there [31]. In such a case, the CS contribution to $n_{i}j^{i}_{\text{dCS}}$ is null. Then, the conserved charge is

This charge is precisely the momentum of $\phi$, with the contribution $K^{0}$ coming from the fact that the Lagrangian contains a term proportional to $\lambda\dot{\phi}K^{0}$. If $Q_{\text{dCS}}$ were gauge invariant, the expression above would be independent of the gauge choice $\omega_{0}|_{\partial\Sigma_{t}}=0$. As shown below, $Q_{\text{dCS}}$ changes under large Lorentz gauge transformations. Under a Lorentz transformation $\Lambda$, we have that the gravitational CS current transforms as $K^{\mu}\to K^{\prime\mu}$, where

It follows that [32]

and thus

where $\nu(\bf\Lambda)$ is the winding number of the corresponding Lorentz transformation (that might be non-vanishing even with $\Lambda(x^{i})\to\mathbf{1}$ asymptotically at infinity). In particular, we have that the Noether charge is only invariant under transformations such that $\nu(\Lambda)=0$, i.e. for transformations continuously connected to the identity. But the charge transforms non-trivially for large Lorentz transformations. Eq. (10) then shows that the charge $Q_{\text{dCS}}$ associated with global shifts in the dCS scalar transform non-trivially under large gauge transformation while still being conserved.

Some of the above results are analogous to axions coupled to non-Abelian gauge fields [33, 34, 35, 36]. To better understand what can be applied to the gravitational case, we shall review some aspects of axions in the next section.

We wish to understand what happens with the shift symmetry and the conserved charge when we quantize the scalar $\phi$ in theory. Because all the results above have analogs for axions and their coupling to gauge fields, we will first consider the system

which includes an axion-like field $a$ coupled to a SU$(2)$ gauge field $A_{\mu}^{a}$ via the term proportional to $\lambda$. We can consider this system in Minkowski space and Cartesian coordinates for simplicity since curvature is unimportant for the axion discussion in this section. Classically, the action above has the global shift symmetry $a\to a+c$ with conserved current

where $C^{\mu}$ is the Chern-Simons current of the gauge field

and the trace is now taken over the adjoint of the gauge group. The Noether charge is

where we chose the gauge $A^{0}=0$ without loss of generality. Under a gauge transformation $A_{\mu}\to g^{-1}A_{\mu}g+g^{-1}\partial_{\mu}g$ where

with $\tau^{a}$ the generators of SU$(2)$ and $f(r)\to 2\pi n$ as $r\to\infty$, the charge $Q_{a}$ transforms as [32, 37]

This gauge dependence does not introduce any issues because once we fix the background gauge field configuration, the system will not be invariant under large gauge transformations with a non-trivial winding number. To see that explicitly, note that a given background configuration $A^{\mu}(t,x^{i})$ should approach a pure gauge one as $r\to\infty$, and we can choose it to be of the form (15) with $f(r)\to 2\pi m$ at infinity. In other words, the homotopy class of the gauge configuration is part of the system’s definition so that large gauge transformations would change the system altogether. Thus, by definition, large gauge transformations cannot be a symmetry of the system, and it is no surprise that such a gauge transformation changes $Q_{a}$.

The conservation of the homotopy-class dependent $Q_{a}$ implies that the dynamical evolution cannot change the winding number of the background configuration $A_{(\nu)}^{\mu}$. In fact, even if we promote $A^{\mu}$ to a dynamical field, there are no solutions to the equations of motion in Lorentzian signature that would allow a change in winding number [31]. This also guarantees that if $A^{\mu}$ is pure gauge asymptotically at $t\to\pm\infty$, the action is invariant under global $a$-shifts: we have

but

since the homotopy class of the gauge field does not change in time.

However, if $A_{\mu}^{a}$ is promoted to a dynamical field that should also be quantized, the argument in the paragraph above cannot be applied. If we also quantize the gauge field, we cannot exclude the possibility of quantum tunneling between gauge configurations of different homotopy classes. This is precisely what happens in quantum chromodynamics (QCD) and general non-Abelian gauge theories, as manifested by instanton solutions. This is the reason why the vacuum structure of gauge theories is non-trivial [37].

With a quantized gauge field, the system cannot be fixed in given homotopy class because there is always a probability for the gauge field to tunnel to configurations of different classes. In other words, the partition function includes a sum over the instantons connecting different homotopy classes,

where we wrote the (homotopically trivial) gauge fixing and Fadeev-Popov contributions in $S_{\text{gauge}}$ and $[d\mu_{(\nu)}]$. Then, the vacuum state of the whole system depends on $\theta$, but all physical observables would be the same regardless of the vacuum $\theta$ we use to compute them. In the quantum theory defined above, the specific value of theta is unphysical because of the shift symmetry at the quantum level: since

a shift in $a$, which should not change any physical observable, induces a shift in $\theta$. Another way of seeing this is by looking at how $Q_{a}$ acts on a given $\theta$ vacuum of the pure gauge sector (in the absence $a$, there is no shift symmetry and $\theta$ is observable). Under a gauge transformation in the homotopy class $n=1$, we have

So, since $Q_{a}$ transforms as Eq. (16), we have [32]

As $Q_{a}$ is conserved (commutes with the Hamiltonian), different theta vacua are degenerated in energy and thus physically indistinguishable.

This degeneracy between the choice of $\theta$ and the zero mode (constant part) of $a$ has a different nature than the usual quantum symmetries. Consider, for instance, the symmetry breaking potential $V(\Phi)=\lambda_{\Phi}\left(|\Phi|^{2}-\mu^{2}_{\Phi}/2\lambda_{\Phi}\right)^{2}$ for the complex scalar field $\Phi(x)=r(x)e^{i\alpha(x)}$. In this case, a vacuum expectation value for $r(x)$ spontaneously breaks the symmetry while $\alpha$ parametrizes the vacuum manifold, with shifts in $\alpha$ being the non-linear realizations of U$(1)$ phase rotation of the complex scalar field. If we think of $a$ as a Goldstone mode after some phase transition (as is the case in the Peccei-Quinn (PQ) proposal), then shifts around the vacuum manifold at constant $\theta$ are generated by the CS-independent part of $Q_{a}$, that we will call $\tilde{Q}_{a}$:

Due to Eq. (22), $\tilde{Q}_{a}$ and $Q_{\rm CS}$ are not individually conserved, and so the conservation of $Q_{a}$ implies that $\tilde{Q}_{a}$ is not conserved: shifts of the vacuum manifold for a given value of $\theta$ are not symmetries of the $\theta$ invariant quantum theory. Instanton effects explicitly break the continuous $\theta$ symmetry transformations. Therefore, there are no Goldstone bosons associated with this instanton induced explitict symmetry breaking. Note that it is a mistake to claim that the axion is a Golsdone boson associated with the anomalous breaking by these instanton effects.

These arguments are at the core of ’t Hooft’s solution to the U$(1)$ problem in QCD [38, 39]. The discussion about how the U$(1)$ symmetry is affected by non-perturbative effects is very similar to the chiral symmetry of massless fermions coupled to non-Abelian gauge fields. The standard chiral rotation combines with the instanton effects to make physics independent of $\theta$. Equivalently, the chiral rotation alone is explicitly broken by instantons.

Returning to dCS gravity, we can apply the following non-Abelian axion results to the dCS pseudoscalar: ($i$) The dCS action has a continuous constant shift (quasi) symmetry with an associated divergent-free current $j^{\mu}_{\text{dCS}}$; ($ii$) If gravity is quantized in the path integral for the theory, the CS-independent part of the total $\text{U}_{\text{dCS}}(1)$ charge is not conserved due to gravitational instanton effects⁴⁴4See [40] for a discussion on the gravitational theta sector.; ($iii$) The total charge $Q_{\text{dCS}}$ (7) is conserved, but not invariant under homotopically non-trivial gauge transformations.

So, we see that the U$(1)_{\text{dCS}}$ shift symmetry is always present, while the CS-independent part of the charge $\tilde{Q}_{\text{dCS}}$ is only conserved in flat spacetime in the absence of gravitational instantons (which are absent at the semiclassical level). In that regime, $\tilde{Q}_{\text{dCS}}$ is not conserved because the local coupling of $\phi$ with the gravitational Pontryagin density changes the dynamical evolution of the metric and the dCS scalar. For instance, in regions where $\tilde{R}R$ is non-trivial, it will source $\phi$. This does not mean the symmetry under $\phi$ translations is broken: the total charge $Q_{\text{dCS}}$ is conserved (even after quantizing $\phi$). Similarly, one does not worry about non-trivial topological effects when using the axion coupling to $\tilde{F}F$ to compute the axion-photon interactions or birefringence effects [41, 42, 14].

In the next discussion, we shall couple a Dirac fermion with dCS and show that gravitational instantons do not affect a combination of the dCS-shift and fermion-axial currents.

Similar to axions arising from a PQ symmetry breaking, we can realize the dCS pseudoscalar as a Goldstone boson associated with the spontaneous breaking of a U$(1)$ symmetry. The remaining constant scalar shift is then a non-linear realization of the global symmetry below the breaking scale. We review this picture in the following model, recently discussed in Ref. [45] (see also Ref. [46] for the non-Abelian axion case),

where $P_{L,R}=(1\pm\gamma_{5})/2$ are the chiral projectors and $V(|\Phi|)$ is a symmetry-breaking potential for the complex field $\Phi$ whose explicit form is not important in the following. The action in Eq. (IV.1) is invariant under the global transformation

However, the associated path integral is not invariant as the symmetry is anomalous at the quantum level due to the gravitational contribution to the chiral anomaly. This symmetry is spontaneously broken by the vacuum expectation value of $\Phi$. Writing $\Phi(x^{\mu})=(1/\sqrt{2})[v+\rho(x^{\mu})]\exp\left[{i\phi(x^{\mu})/v}\right]$, with $v$ a constant, we obtain

where $\phi$ is massless as expected from a Goldstone mode. In order to obtain dCS gravity, we now integrate out the fermion $\Psi$ and massive scalar $\rho$. In doing so, we obtain an effective field theory (EFT) for the pseudoscalar $\phi$ below the symmetry-breaking scale,

where we simplified the Yukawa coupling by rearranging the phases times chiral projectors into $\exp\left[{i(\phi/v)\gamma_{5}}\right]$. Using the field redefinition

we get

where the last term in the exponential appears due to the non-trivial Jacobian of the path integral measure under the change of variables above and we identified the spinor-covariant derivative $\nabla_{\mu}\Psi=(\partial_{\mu}+\omega_{\mu}^{ab}\gamma_{ab}/4)\Psi$. To leading order, the path integral picks the classical saddle where $\Psi=0=\rho$, so the path integral then becomes

where dots stand for higher-derivative corrections suppressed by the fermionic and $\rho$ masses.⁵⁵5Since the fermionic path integral was done in curved space, there will be also extra curvature contributions, see Refs. [47, 48] and references therein. We see that the dCS gravity term is obtained from integrating-out heavy fermions. This model is thus analogous to the Kim-Shifman-Vainshtein-Zakharov (KSVZ) model for the QCD axion [49, 50]. Again, the shift symmetry of the dCS pseudoscalar is a non-linear realization of the non-anomalous phase transformation of $\Phi$ in Eq. (33). It is worth noting that the theories defined the action in Eq. (34) and Eq. (IV.1) are dual since their path integrals are the same (they differ by a fermionic $\Psi$ field redefinition), with the latter having no $\lambda\phi R\tilde{R}$ coupling. However, once we integrate out $\Psi$, there is no fermionic field redefinition left, and the $\lambda\phi R\tilde{R}$ coupling is a genuine low-energy operator of the effective action in Eq. (38).

In what follows, we wish to make a connection with the Standard Model where the neutrinos get a Majorana mass term at energies below the symmetry-breaking scale, with $\phi$ appearing as a phase in such a mass term. We do so after assuming that the heavy fermion $\Psi$ plays the role of a right-handed sterile neutrino that couples with the lepton and Higgs doublets, $L$ and $H$, respectively,

as

where $L_{\nu}$, $L_{e}$, and $\Psi$ are Majorana spinors:

Note that we need the conjugated $\tilde{H}$ instead of the Higgs doublet $H^{T}=(\phi^{0}\;\;\phi^{+})$ in Eq. (40) to get a hypercharge neutral coupling. We consider a single lepton flavor for simplicity since a generalization for all three generations is straightforward (see the discussion section).

With the new Yukawa coupling (40) and the kinetic terms for $L$ and $H$, the action in Eq. (IV.1) becomes

This action is invariant under Eq. (33) provided $L$ transforms as

Note that since both $L$ and $\psi$ are four-component Majorana spinors, the global chiral transformation acting on those is a phase rotation associated with their fermionic number. We discuss further aspects of this symmetry in the discussion section. After performing the field redefinition in Eq. (36), the path integral is now

We thus see that $\bar{\chi}=-\tilde{y}\bar{L}\tilde{H}e^{-i\frac{\phi}{2v}\gamma_{5}}$ acts like a source for the right-hand part of $\Psi$. Moreover, using $\bar{L}=L^{c}=L^{T}C$, where $C$ is the charge conjugation matrix, we have

and so $\eta=e^{-i\frac{\phi}{2v}\gamma_{5}}\tilde{H}^{\dagger}L$ acts like a source for the left-hand part of $\Psi$. But since $\Psi$ is a Majorana spinor, its left and right components are related, and both $\chi$ and $\eta$ contribute as sources to the path integral of $\Psi$. Hence, the coupling in Eq. (40) will give the contribution

to the effective Lagrangian. Note that the charge conjugation of the chiral sources fixes $\chi$ and $\eta$,

The effective action is given by

where the heavy fermion mass $m_{\Psi}$ can be read from Eq. (35) to be $m_{\Psi}=yv/\sqrt{2}$. The resulting effective action has a modified Weinberg operator [51] that includes a coupling with $\phi$.

Notice that the pseudoscalar arises as a complex phase in the neutrino mass matrix and as a coupling to the gravitational Pontryagin density. A measurement of the mass phase will not constrain its coupling to the gravitational Pontryagin density because strong gravitational effects from compact binary can source a large gravitational Pontryagin density which will source large gradients of $\phi$.

This is analog to QCD, as the axion couples to chromo-electric and magnetic field via the Pontryagin density, astrophysical sources may enhance the axion and a similar story applies to dCS. Here, there are two observable effects to consider. First, an Earth-based measurement of the dynamical field can constrain the value of the pseudoscalar, which is predicted to be small due to the fact that the Pontryagin term is zero in Minkowski space-time. However in compact binary systems strong gravity can give a non-vanishing $R\tilde{R}$ which will in turn source gravitational waves and non-vanishing pseudoscalar amplitude. Also, in models of Higgs-Inflation, Electroweak symmetry is broken during inflation, and the phase can also be non-vanishing in the early universe. In these strong gravity, early universe regimes the mass matrix phase will also be non-vanishing but obviously difficult to measure. Still, it may be possible to study the effects of the phase and its impact on neutrino cosmology. We leave this possibility for future investigation.