Perturbative Quantum Gravity Induced Scalar Coupling to Electromagnetism

Physicists working on atom interferometers are interested in scalar couplings to electromagnetism of dimensions 5 and 6 which might be induced by quantum gravity. There is a widespread belief that such couplings can only be induced by conjectured non-perturbative effects, resulting in unknown coupling strengths. In this letter we exhibit a completely perturbative mechanism through which quantum gravity induces dimension six couplings with precisely calculable coefficients.


Introduction
There has been much recent interest in searching for exotic processes which might be induced by quantum gravity [1,2]. In particular, it has been suggested [3] that quantum gravitationally induced scalar couplings to electromagnetism might be detected by planned atom interferometers such as MAGIS [4], AION [5,6] and AEDGE [7]. Conventional wisdom has it that perturbative quantum gravity can at best generate couplings of dimension eight, and that couplings of dimensions 5 and 6 could only be induced, with unknown coefficients, by nonperturbative effects [8] such as gravitational instantons [9,10] and wormholes [11].
In this paper we point out that there is a completely perturbative mechanism through which quantum gravity induces a dimension six coupling of a massive scalar with a precisely calculable coefficient. The mechanism is simple: assuming that the scalar is constant in space and time, and that the potential energy from its mass dominates the stress-energy, the background geometry will be de Sitter with a Hubble parameter which depends in a precise way on the scalar. Unlike the graviton propagator in flat space, the coincidence limit of a graviton propagator on de Sitter background goes like the square of the Hubble parameter in any gauge [12,13,14,15,16]. Hence integrating out pairs of graviton fields from the Heisenberg operator Maxwell equation (the Hartree approximation) induces couplings of the desired form with precisely computable coefficients.
This short letter contains only three sections, of which this Introduction is the first. We present the calculation in Section 2. Our conclusions comprise Section 3.

Calculation
Consider a massive, uncharged scalar field which is coupled to electromagnetism and gravity, The corresponding Einstein equation is (2) As discussed in the introduction, we assume that ϕ = ϕ 0 is a constant and also set A µ = 0 to get, The unique, maximally symmetric solution is de Sitter with Hubble constant, where, H is the Hubble constant. This shows that a constant scalar triggers a phase of de Sitter inflation. Because de Sitter is conformally flat, there is no classical effect on electromagnetism in conformal coordinates. However, we will see that the breaking of conformal invariance by gravity does induce a quantum effect.
Consider the Maxwell equation in a general metric g µν , where F µν ≡ ∂ µ A ν − ∂ ν A µ is the field strength tensor and J µ is the current density. We write the quantum metric g µν in terms of the Minkowski metric η µν , where κ 2 ≡ 16πG is the loop counting parameter, a(η) ≡ −1/Hη is the scale factor at conformal time η and h µν is the graviton field. Graviton indices are raised and lowered with the Minkowski metric: h µ ν ≡ η µρ h ρν , h µν ≡ η µρ η νσ h ρσ . The inverse and determinant of the conformally transformed metric are, Here h is the trace of the graviton field h ≡ η µν h µν .
To facilitate dimensional regularization we formulate the theory in D spacetime dimensions. The term inside the square bracket of equation (5) can be expressed in terms of the conformally transformed metric as The terms involvingg µν can be expanded as, Using the Hartree approximation [17,18], we can replace the terms proportional to κ by zero and the terms proportional to κ 2 by the coincidence limit of the graviton propagator The graviton is of course gauge dependent but its coincidence limit on de Sitter is proportional to H D−2 in all gauges [12,13,14,15,16]. In the simplest gauge [12,13] it consists of a sum of three constant tensor factors, each multiplied by a different scalar propagator, The constant tensor factors are, where parenthesized indices are symmetrized andη µν ≡ η µν + δ 0 µ δ 0 ν is the purely spatial part of the Minkowski metric. The three scalar propagators correspond to masses M 2 They obey the propagator equations, where the various kinetic operators are, The coincidence limits of the three scalar propagators are [16], where the constant k is, By employing the relations (11)(12)(13) and (16)(17)(18) which define the coincident graviton propagator in expression (10), and then substituting into the left hand side of Maxwell's equation (5), we obtain the order κ 2 correction, Renormalization is facilitated by expressing the divergent part in terms of the purely spatial componentsF µν ≡η µρηνσ F ρσ of the field strength tensor, The cotangent is divergent as D → 4, The divergences on the final line of (21) can be eliminated by the counterterm, Note the factor of (µ/H) D−4 required to cancel the D-dependence in the factor of H D−2 evident in expression (19) for the constant k. Note also that the need for a noninvariant counterterm arises from the avoidable breaking of de Sitter invariance in the simplest gauge [12,13] and from the unavoidable time-ordering of interactions [19].
Combining the variation of the counterterm (23) with the primitive contribution (21), and then taking the unregulated limit gives, Note the µ-dependence against the scale factor a(η) in the logarithm at the end. This is a vestige of renormalization. Substituting for the Hubble constant from expression (4), and recalling that the loop-counting parameter is κ 2 = 16πG, results in the final dimension six coupling to Maxwell's equation, (25) Note that the scalar could have as easily been placed inside the ∂ ν -as it would have been in varying the counterterm (23) -because the computation was made assuming φ and the induced H were constant. Although a finite renormalization could have eliminated the term proportional to F µν −F µν in (25), the logarithm of φ multiplying the other term is a genuine prediction of the theory, with a specific coefficient which we have just computed.

Conclusion
The usual way a constant scalar background engenders quantum corrections is by giving some field a mass so that the coincidence limit of that field's propagator depends on the scalar. That cannot happen in perturbative quantum gravity because the graviton remains massless to all orders. However, constant scalars can also contribute by changing a field strength [20]. In our case, a constant scalar background changes the cosmological constant, and the graviton propagator in de Sitter background depends upon this cosmological constant [12,13,14,15,16]. We have exploited this mechanism to compute the dimension six coupling (25) to electromagnetism. Similar results could be obtained for couplings to any other low energy field.
Our result comes with several important caveats, both theoretical and phenomenological. On the theoretical side, our computation depended on the scalar being constant throughout spacetime. Although this is not a reasonable assumption, setting the scalar to be constant is the correct way to compute nonderivative couplings, which should remain valid in the resulting low energy effective field theory, even when the assumption of constancy is relaxed. We have also assumed that the scalar potential energy dominates the total stress energy, which is not reasonable for a weak scalar. However, the coupling must still be present in a realistic cosmological background because it must be there in the large field limit. Finally, although the graviton propagator is gauge dependent, dimensional analysis requires its coincidence limit on de Sitter background to go like H D−2 , a fact which is confirmed in all known gauges [12,13,14,15,16]. A recently developed formalism [21,22], based on the S-matrix, can be used to remove gauge dependence from the effective field equations.
It should be possible to verify our beliefs by employing the same techniques which were recently used to compute cosmological Coleman-Weinberg potentials [23,24,25,26]. In this regard, consider the graviton propagator for an evolving cosmological geometry with Hubble parameter H(t) and first slow roll parameter ǫ(t), This geometry is supported by the energy density ρ and pressure p of matter through the Friedmann equations, Our scalar adds to the energy density and pressure, but it need not dominate them, nor need it be constant. Although the graviton propagator is not known for general ǫ(t), it is known for the important special case of constant ǫ [27], which includes well-known epochs of cosmological evolution such as radiation domination (ǫ = 2), matter domination (ǫ = 3 2 ), and vacuum energy domination (ǫ = 0). The full propagator can be found in equation (3.38) of [27], but the part involving dynamical gravitons is a constant tensor factor times the propagator of the massless, minimally coupled scalar. The coincidence limit of this propagator is given by equation (33) of [28], (29) Note the initial factor of [H(t)] D−2 , which can be written in terms of the total energy density ρ using the first Friedmann equation (27). This remains true even if the scalar is only one part of a much larger contribution, and even if the scalar is not constant in time. We believe that a similar result pertains for a scalar which is not constant in space.
An important phenomenological caveat concerns the fact that our coupling (25) is proportional to G 2 m 2 φ 2 , rather than Gφ 2 . Although both couplings correspond to contributions to the action of dimension six, the extra factor of Gm 2 in our result (25) renders it unobservably small for the optimal mass of m ∼ 10 −24 GeV which is relevant to atom interferometers [3]. So our work must be regarded as an illustration of how perturbative quantum gravity can induce couplings to low energy fields rather than as a testable prediction.