Probing CP Violation in Photon Self-Interactions with Cavities

In this paper we study CP violation in photon self-interactions at low energy. These interactions, mediated by the effective operator $FFF\tilde{F}$, where ($\tilde F$) $F$ is the (dual) electromagnetic field strength, have yet to be directly probed experimentally. Possible sources for such interactions are weakly coupled light scalars with both scalar and pseudoscalar couplings to photons (for instance, complex Higgs-portal scalars or the relaxion), or new light fermions coupled to photons via dipole operators. We propose a method to isolate the CP-violating contribution to the photon self-interactions using Superconducting Radio-Frequency cavities and vacuum birefringence experiments. In addition, we consider several theoretical and experimental indirect bounds on the scale of new physics associated with the above effective operator, and present projections for the sensitivity of the proposed experiments to this scale. We also discuss the implications of these bounds on the CP-violating couplings of new light particles coupled to photons.


Introduction
Photon self-interactions are absent in pure Maxwell's theory, but are induced by photon-matter interactions. Therefore, effective photon-photon interactions appear in generic low energy Quantum Electrodynamics (QED) theories. The Euler-Heisenberg (EH) Lagrangian is an example of such an effective theory, where the non-linear term arises at one loop, at energies below the electron mass [1,2], and preserves CP. CP violating (CPV) sources of photon interactions, on the other hand, arise in the Standard Model (SM) only at multiple loop level from the, CKM-suppressed, weak interactions, or are controlled by the tiny strong CP phase, and are thus negligibly small [3]. Given the suppression of the SM contribution, it should in principle be possible to probe (and possibly discover) CPV new physics in this channel, without a significant SM background.
As suggested in [4], the QCD axion, or in general axion-like particles (ALPs), are well motivated new sources of photon self-interactions. Depending on their mass and their coupling to photons, the contributions of such particles to effective photon-photon interactions -which are CP conserving (CPC) -may be comparable to that of the EH term or even exceed it [5,6]. CPV photon self-interactions are instead mediated by degrees of freedom that are not CP-eigenstates. Scalars of this kind appear in theoretically motivated models such as the complex Higgs portal [7] and the relaxion [8,9]. These interactions could also be induced by new fermions with non-vanishing electric and magnetic dipole moments. In the following we will mostly adopt a model-independent approach, where the information of the particular new physics providing photon self-interactions will be encoded in the the Wilson coefficients of the effective photon operators. Effective CPC and CPV photon self-interactions beyond the Standard Model (BSM) can be indirectly constrained by the measured electronic magnetic dipole moment [10] (see [11] for the SM theory calculation), and by the upper limit on the electronic electric dipole moment [12]. As we will show, the corresponding bounds can be easily estimated at energies above the electron mass, and yield relatively strong constraints on the scale of new physics. In contrast, at energies below the electron mass, the reach of direct experimental tests for the presence non-linear photon dynamics is, to date, much more limited. In fact, as reviewed in Section 2, current experiments, looking for vacuum birefringence (e.g. [13]) and the Lamb shift [14,15], have probed only the CPC part of the photon self-couplings. While their sensitivity is still about one order of magnitude above the EH contribution, other experimental proposals such as [16][17][18][19] could be able to measure such a term, and possibly constrain new physics contribution to light-by-light interactions.
In this paper we study the prospects of directly detecting CPV photon self-interactions at energies below the electron mass, described by the effective operator F µν F µν F ρσF ρσ , which has not been directly probed by any current experiment. In particular, we present simple modifications to proposed and currently running experimental setups, such that they can be made sensitive also to CPV phenomena. Crucially, our proposals will be able to disentangle the CPV contribution from the CPC one, providing unique probes of the CPV operator that, as mentioned, are free from the SM background. These experiments could set the first model-independent bound on CPV effective photon interactions at energies below the electron mass.
Our first proposal employs the production and detection of light-by-light interactions in a superconducting radio frequency (SRF) cavity, extending the setup described in [16,17,20]. In this configuration, the self-interactions of background resonance modes pumped into the cavity act as a source, exciting another resonance mode of the cavity. We demonstrate how a particular choice of the pump and signal modes and of the cavity geometry allows singling out CPV photon self-interactions.
Photon nonlinearities are known to induce vacuum birefringence in the presence of an external electromagnetic field. In particular, polarized light acquires a non-vanishing ellipticity and rotation of the polarization plane [21,22]. As a second probe, we discuss an experimental configuration where this phenomenon happens in a ring cavity. While inspired by the linear Fabry-Perot (FP) cavity of the PVLAS experiment [23], which is essentially insensitive to CP-odd effects, we show that a ring cavity geometry is sensitive to CPC and CPV photon interactions simultaneously, which can be distinguished by a temporal analysis of the signal. A similar scheme has been proposed in [24], and applied to CPV dark sectors.
The paper is organized as follows: In Section 2 we define the photon Effective Field Theory (EFT) at low energy and summarize the current direct and indirect bounds on its coefficients. We also discuss the possible contributions to photon self-interactions from simple new physics models. In Section 3 we discuss the prospects of detection of CPV photon interactions using an SRF cavity. In Section 4 we study the detection of vacuum birefringence and dichroism in a ring cavity, and its implication for the CPV operator. We conclude in Section 5.

Agnostic EFT Approach
At energies below the electron mass, interactions among photons are self-consistently described by an effective Lagrangian involving the photon field only. Such a Lagrangian can be conveniently expanded in powers of the two independent gauge invariant CP-even and CP-odd operators, where F µν andF µν are the photon field strength and its dual, and E (B) is the electric (magnetic) field. At leading order, up to dimension-8, it reads (see e.g. [3]) Note that a has no physical effect being F µνF µν a total derivative. The coefficients b, c, d are proportional to four inverse powers of the EFT UV-cutoff, Λ.
In the SM the coefficients b and c receive leading contribution from the EH effective action [1,2] b EH = 2 45 where m e is the electron mass and α EM is the fine-structure constant. The term F µν F µν F ρσF ρσ violates CP and obtains radiative contributions from the two CPV sources of the SM. First, the contribution from the QCD θ-term has been estimated in chiral perturbation theory in the large N c limit in Ref. [3]. Due to the smallness of θ 10 −10 , it is suppressed by at least 20 orders of magnitudes with respect to the EH CPC self-interactions in Eq. (2). A second contribution to d comes from the CPV phase of the CKM matrix. Although never calculated, to the best of our knowledge it is expected to appear at least at three loops and (due to the GIM mechanism) be extremely suppressed. Given Eq. (2) and the smallness of d, the UV cutoff Λ is therefore of order 10 MeV.
As discussed in the next section, the coefficients of the Lagrangian in Eq. (1) could obtain contributions from BSM physics. On theoretical grounds, they are subject to positivity constraints if L EFT comes from a causal UV theory with an analytic and unitary S-matrix. In particular, b and c must be positive, and d must be bounded [25] |d| < 2 √ bc .
This provides a nontrivial consistency condition on L EFT , that requires the magnitude of the CP-odd term to be bounded by the CP-even ones. In particular, if a violation of this bound is measured, one would need to give up some of the fundamental principles in the UV theory underlying the effective Lagrangian in Eq. (1), such as analyticity, unitarity or causality.
Let us now review the current experimental limits on L EFT at energies below the EH scale. 1 The coefficients b and c are hardly constrained from direct light-by-light scattering [26,27], which provides limits that are more than 10 orders of magnitude above the EH prediction in Eq. (2). The coefficient b induces a correction to the Coulomb potential of the hydrogen atom [14] and a Lamb shift of its 1S and 2S energy levels. For the EH Lagrangian, this correction to the energy levels has been calculated to be 3 × 10 −4 times the leading term [15], while the related measurements still have a precision of 3 × 10 −3 [28]. Since the correction is linear with b, this therefore yields a bound of at 95% confidence level.
As explained in more detail in Section 4, the combination c − 4b induces a non-vanishing ellipticity to polarized light passing through a region permeated by an external magnetic field. This observable has been bounded by BMV [29] and PVLAS [23], and the latest limit [13] where c EH − 4b EH = 3b EH . We obtain a bound on c by combining the 2 σ intervals of Eqs. (4) and (5) − 15 c EH c 19 c EH .
Eqs. (4) and (6) correspond to the strongest direct bounds to date.
Currently, there is no direct experimental constraint on the coefficient d, although there have been theoretical studies [3,30] of its possible detection using vacuum birefringence. On the other hand, from Eqs. (4) and (6) we expect that the EFT is consistent according to Eq. (3) only if which will be denoted as the EFT consistency bound.
At energies above the electron mass, the electron must be included in the EFT, and the measurements of its electric and magnetic dipole moments put an indirect constrain on the contributions of new physics to photon self-interactions. Indeed, in the electron-photon EFT, a single insertion of the 4-photon operators generates a two loop contribution to the magnetic and electric dipole operators a e 4m e eψ e σ µν ψ e F µν , (where ψ e is the electron field and e ≡ √ 4πα EM is the electric charge) which could make the coefficients a e and d e deviate from their SM predictions. Although never evaluated directly, a crude estimate for the deviations induced by new physics via such a diagram is where b BSM , c BSM and d BSM are the BSM contributions to the coefficients, and C 1 and C 2 are order one factors. Assuming C 1 and C 2 of order one, and considering the current bounds on |a exp − a SM e |/a SM e 10 −9 [11] and |d e | < 1.1 × 10 −29 e × cm [12], we obtain a rough estimate for the bounds on the BSM contribution to the EFT coefficients as These are indirect bounds on b BSM , c BSM and d BSM that strongly constrain possible new physics heavier than the electron. In the following, however, we will focus on constraining these operators at energies below the electron mass, as the existing bounds of this type are quite weak (see Eqs. (4) and (6)). In particular, our proposals will be able to set the first model-independent bound on the coefficient d at energies below the electron mass.
In principle, it should be possible to translate the bounds from the electric and magnetic dipole moments in Eq. (10) (valid at energies above m e ) into bounds on the low energy photon EFT in eq. (1), so that they can be applied to new physics lighter than the electron. This however requires matching the photon EFT with the photon-electron Lagrangian, and is beyond the scope of our work.
We finally note that light-by-light scattering has been observed at the LHC in Pb-Pb collisions by the ATLAS and CMS collaborations [31][32][33]. The diphoton invariant mass relevant to this measurement is above 6 GeV, thus well beyond the scale at which the EFT in Eq. (1) is valid. At such energies the EFT coefficients for photon self-interactions, including their CPV part, have been constrained to be b BSM , c BSM , d BSM 10 −10 GeV −4 , see [34].

Contributions from New Physics
As we now show, new particles coupled to photons generically contribute to the effective Lagrangian in Eq. (1), and possibly to the CPV coefficient d.
As a first example, we consider a real scalar singlet under U (1) EM with mass m φ . Even if not coupled to the photon at the renormalizable level, couplings to the photon field strength are present in the dimension-5 Lagrangian: If both g andg are nonzero, φ has no definite transformation properties under CP, and L φ breaks CP explicitly. When φ is integrated out, L φ provides the contributions to b, c, d where d φ is proportional to the CPV combination gg. For ALPs, g = 0 (to leading order) andg provides a contribution to c. This contribution is larger than the EH term ifg/m φ α/m 2 e ≈ 1/(10 MeV) 2 . The couplings g andg arise simultaneously in models in which the scalar is not a CP eigenstate. In particular, in relaxion models [8], the scalar coupling g is nonzero and determined by the relaxion's mixing with the Higgs [9], while the pseudoscalar couplingg is related to the shift-symmetric nature of the axion.
We observe that for small enough m φ the couplingg is bounded by several probes, including laboratory experiments and astrophysical considerations, see [35][36][37] for reviews. The coupling g is stringently constrained from from fifth force and equivalence principle tests, see [38][39][40] and references therein. Nevertheless d φ in Eq. (12) might easily dominate over the SM contribution (this, for comparison, was estimated to be in its QCD part d QCD 10 −27 MeV −4 where we used θ QCD 10 −10 [3]). For scalars heavier than m e , the couplings g andg are bounded by Eq. (10), see also [41,42] for the full calculation.
Another possibility is to consider a new SM gauge singlet fermion, ψ, which at dimension-5 level will be coupled to photons via dipole operators as where D andD are its magnetic and electric dipole moments respectively. 2 The coefficients b, c, d receive threshold corrections from one loop diagrams with four insertions of D and/orD. In particular, b ψ ∼ 4(D 4 +D 4 )/(16π 2 ) , c ψ ∼ 64D 2D2 /(16π 2 ) and d ψ ∼ 32(D 2 +D 2 )DD/(16π 2 ). As a result, the bounds in Eqs. Contrary to the previous one, this bound applies only for masses m ψ up to few hundreds keV, corresponding to the core plasma temperature of the stars. Moreover, for m ψ m e such operators are (indirectly) constrained by the electric and magnetic dipole moments of the electron in Eq. (8). For instance, a nonvanishingD gives a contribution to d e at two loops, which can be roughly estimated as ∆d e m e m ψD 3 e 2 /(16π 2 ) 2 , and thus the previously mentioned bound on d e yields D 10 −10 eV −1 (m ψ / MeV) 1/3 .

Isolating CP-Violation in an SRF Cavity
The photon self-interactions in Eq. (1) introduce nonlinearities in Maxwell's equations. These nonlinearities act as a source for an electromagnetic field in the presence of background electromagnetic waves. In this section, we discuss the production and the detection of this field in a superconducting radio frequency (SRF) cavity, and point out how the contribution from CPV photon interactions can be singled out by an appropriate choice of the background fields and the cavity dimensions.
We consider a free background field A µ p (satisfying ∂ µ F µν p = 0), and split the total field as F µν p + F µν . At leading order in the photon self-couplings b, c and d, the equations of motion of the Lagrangian in Eq. (1) become a source-equation for the field F µν , i.e.
where the effective current J µ is a function of the background field only, and reads where in the second line we rewrote F µν p in terms of its electric and magnetic fields E p and B p . As a result, the induced field, F µν , is generated proportionally to the cubic power of the background fields. A similar effect occurs if the photon self-interactions are mediated by an off-shell scalar φ with the Lagrangian in Eq. (11). In this case, the effective current reads where, at leading order in g andg, φ is the solution of the Klein-Gordon equation retarded Green's function. The calculation of G R and φ is simplified if m φ is much larger or much Figure 1: Schematic picture of a cylindrical SRF cavity. Two cavity modes with frequency ω 1 and ω 2 are pumped into the cavity and source an additional mode at the frequency ω s > ω 1 , ω 2 as a result of the photon self-interactions of Eq. (1). The cavity geometry is chosen such that ω s is a resonance mode and therefore amplified. In a small filtering region (F) the pump modes are exponentially suppressed and only the signal mode ω s is detected (D). If the modes satisfy Eq. (23), the signal mode, ω s , will be automatically sourced only by the CPV part of the photon interactions (see main text for more details).
smaller than the typical frequency of the background field ω. In these cases where Note that the effective current in Eq. (16) is also cubic in the background fields and reduces to Eq. (15) in the limit m φ ω, with the Wilson coefficients given in Eq. (12).
As suggested in [16], an SRF cavity is a natural setup where the field F µν can be generated and amplified. We consider an SRF cavity, see Fig. 1, that is pumped simultaneously with two cavity modes, with corresponding electric fields E 1 , E 2 and magnetic fields B 1 , B 2 , at frequencies ω 1 and ω 2 respectively, with F µν p = F µν 1 + F µν 2 (we discuss the possibility of pumping the cavity with a single mode at the end of this section). Since the modes of F µν that match resonances of the cavity are amplified by the cavity geometry, the electric field produced in the cavity will mostly be sourced by the projection of J onto these resonance modes. The resonant field E f generated by exciting a cavity eigenmodeÊ s , with a corresponding frequency ω s , can be written as [17] where V is the volume of the cavity, Q s is the quality factor for the frequency ω s andÊ s is dimensionless and normalized as d 3 x|Ê s | 2 = V .
Note that in order to excite the cavity resonance, one of the cavity resonance frequencies must match one of the Fourier components of F µν . Given the cubic dependence of J on the pump fields, and assuming no other background sources, eq. (14) dictates that F µν can only have frequencies ±nω 1 ± mω 2 , with m, n ≥ 0 and m + n = 3. The cavity geometry must be therefore chosen such that there exists a resonance frequency ω s matching one of these combinations (within the cavity's bandwidth), while also verifying that the spatial overlap of the corresponding resonance modeÊ s and the effective current J (i.e. Eq. (18)) is non-zero (and ideally maximal).
To measure the power in the excited signal mode, one could use a smaller filtering cavity (of which ω s is still a resonance mode), as is schematically shown in Fig. 1. Assuming ω 1 , ω 2 < ω s , this suppresses the pump fields and isolates E f only [16]. The expected number of signal photons is given by We will now show that, due to the intrinsically different structure of the CPC and CPV couplings, it is possible to select the pump fields E 1 and E 2 , and the signal modeÊ s , to single out CPV (or CPC) phenomena in the generated field of Eq. (18).
First, we observe that the effective current can be written as J µ = F µν ∂ ν f +F µν ∂ νf , where f andf are quadratic functions of the pump fields and can be read off from Eqs. (15) and (16). In particular, the vector current is Let us consider a setup in which the signal modeÊ s is parallel to E 2 , the pump fields are orthogonally polarized, and are chosen such that either B 1 E 2 or (∇f ) B 1 ×E 2 = 0 (with the notation (A) B we refer to the component of A along B). With this choice, the scalar product entering in Eq. (18) simplifies considerably and readŝ The expressions for f andf are also simplified for the above choice of pump fields. From Eq. (15), schematically f ∼ bF 2 p + dF pFp andf ∼ cF pFp + dF 2 p , and the choice of orthogonal pump modes implies F 2 Plugging these expressions into Eq. (20), we see that the CPV terms (proportional to d) contain an odd number of powers of the field '1' and an even number of '2', and vice-versa for the CPC terms (proportional to b or c). This happens because, crucially, only the fields E 1 and B 1 (E 2 and B 2 ) enter in the term operating oñ f (f ) in Eq. (20), thanks to the properties of the pump modes. The same holds for the current in Eq. (16), since only terms linear in F 2 and FF can appear in f and inf .
As a result, for modes that satisfy the conditions above, the CPV part of the signal field E f will only have frequency components i.e. combinations of odd multiplicities of ω 1 , and even multiplicities of ω 2 . The opposite holds for the CPC terms, which only provide the frequency components Therefore, one can distinguish between CPV and CPC photon self-interactions according to the frequency of the component of E f .
As we are interested in observing and constraining CPV photon self-interactions, we would like to amplify only the CP-odd contribution to field in the cavity (proportional to d or gg). This can be done for the choice of pump fields proposed above, by setting the cavity geometry such that there exists a cavity eigenmodeÊ s , parallel to E 2 , with a frequency matching one of the possible CP-odd frequencies in Eq. (21). In particular, since for generic cavity geometries eigenfrequencies are not linear combinations of other eigen-frequencies, it should also be possible to make sure that the CP-even frequencies in Eq. (22) do not match any of the resonant modes of the cavity (therefore preventing contaminations of the signal from the CPC part). Summarizing, the conditions: together with the choice of ω s among ω CPV s , are sufficient for isolating the CPV part of the photon self-interactions from the CPC one, as the signal will be affected only by the CPV part.
Following the proposal in [17], we now estimate the possible reach of the measurement of d, and of the CPV combination √ gg of an off-shell scalar. We choose cavity modes satisfying Eq. (23) and normalize them as , where E 0 is the typical magnitude of the electric field of the pump modes,Ĵ is dimensionless and κ has dimension −3. For the scalar mediator, we define κ = ω s gg/m 2 φ for m φ ω s and κ = gg/ω s for m φ ω s . When working in the EFT limit, κ = 2ω s d. The number of signal photons produced is then where K is an O(1) form factor of the cavity that depends on the pump and signal modes, which will be computed in the following for explicit examples. We will use the notation K ∞ when describing the EFT limit, and K 0 in the m φ ω s case.
We obtain the expected bounds on the appropriate CPV parameters for the experimental phases proposed in [17] we neglect possible backgrounds from impurities of the cavities material, which can induce a noise in the signal mode, see [18]). By following the procedure of Ref. [17], the projected bound on CPV combination of couplings of a scalar field to photons, √ gg, is obtained by comparing the number of thermal photons and signal photons from Eq. (24) and is given by where T is the cavity's temperature (we assume T ω s ), t is the total measurement time, B ω is the signal bandwidth, L is the total length of the cavity (including the filtering region), and SNR is the signal to noise ratio. We set SNR = 2 to obtain the bound at a 95% confidence level. The projected bounds on the coefficient d of the generic EFT in Eq. (1) is As a proof of principle, we consider a cavity with a right cylindrical geometry with radius r = 0.5 m (see Fig. 1). The conditions in Eq. (23) can be satisfied, for example, for the orthogonal pump fields of the form F 1 = TE 0q 1 p 1 and F 2 = TM 0p 2 0 , with the signal mode F s = TM 0ps0 (see e.g. [51] for notation and analytic expressions). With this choice,Ê s is parallel to E 2 and points to the z direction. For these modes, B 1 ×Ê s ∝φ, where ϕ is the azimuthal angle. Since the system (fields and boundary conditions) is azimuthally symmetric, condition (iv) of Eq. (23) is also satisfied. Another class of modes satisfying Eq. (23) is TM 0p 1 0 /TE 0p 2 q 2 /TE 0psqs . As in the previous mode choice, the pump fields are orthogonal andÊ s E 2 . However, for this choice, B 1 E 2 , satisfying condition (iii). In Tables 2 and 3 of Appendix A we list mode combinations of the form TE 0q 1 p 1 /TM 0p 2 0 /TM 0ps0 and TM 0p 1 0 /TE 0p 2 q 2 /TE 0psqs that yield O (0.1 − 1) values for K ∞ and K 0 , for ω s = 2ω 2 − ω 1 .
In the following we will use Eqs. (25)- (26) to estimate the projected bounds on d and gg achievable using our proposed method. We choose the TE 021 /TM 050 /TM 060 modes configuration, representing the first type of mode combinations mentioned above. As discussed before, (1) as a function of the total measurement time in the SRF cavity (solid lines). The different lines correspond to the typical cavity parameters of the phases in proposed in [17], summarized in Table 1. The disk shows the total measurement time proposed in [17]. The upper dashed line is the consistency bound d ≤ 2 √ bc for the photon EFT from Eq. (3), with the current direct experimental limits on b and c (see Eqs. (4) and (6)). The lower dashed line corresponds to d ≤ 2 √ b EH c EH , which would be the consistency bound on the CPV coefficient assuming the EH effect would be measured, and thus represents the equivalent sensitivity to the EH scale.
by choosing the cavity geometry such that ω s = 2ω 2 − ω 1 , i.e. ω TM 060 = 2ω TM 050 − ω TE 021 , only the CPV part of the Lagrangian will contribute to E f . This is achieved if the cavity length is h = 0.332r = 0.166 m. For this mode choice, we find K ∞ = 0.25 and K 0 = 0.26. We set the total length of the cavity with the filtering region to be L = 2h, and assume the radius of the filtering cavity will be chosen such that E f corresponds to the lowest resonance mode of the filtering cavity. We give our projections for four cases, following the four phases of Ref. [17], where the different operating parameters are given in Table 1. For all cases we assume that the cavity temperature is T = 1.5 K and that E 0 = 45 MV/m. The corresponding expected bound on the EFT coefficient d is plotted in Fig. 2. We observe that such a cavity can easily probe values of d that are within the EFT consistency region given by Eq. (3), providing the first direct limit on this coefficient to date. In particular, the reach could even be a few orders of magnitude below the value 2 √ b EH c EH 0.24α 2 EM /m 4 e , which corresponds to the EFT consistency scale if b and c are measured to be at the values predicted by the EH effect, and thus represents the equivalent effective sensitivity associated with the EH scale. The projected sensitivity for the CPV combination of scalar couplings √ gg, presented in Fig. 3, is  Table 1. In red we show the bounds for the proposal presented in Sec. 4 (ring cavity), for the same experimental parameters as PVLAS. For reference, we also plot the current best bound on √ gg, obtained by combining the constraints on g from fifth force [38,39] and EP tests [46][47][48][49] (summarized in [40]) (5F/EP), and those ong from PVLAS/CAST [13,50] and astrophysics (see e.g. [36]).
still however a few order of magnitudes above the current best bounds on these couplings, but would provide a strong complementary probe. Additionally, when applied to the fermion dipole operators in Eq. (13), our best bound on d would constrain D ,D 5×10 −8 eV −1 , even in the mass range few 100 keV m ψ 10 MeV which is presently not directly constrained by astrophysical observations. We note that it is possible to disentangle the contribution of the CPC and CPV coefficients using just a single pump mode (that self-interacts with itself) with orthogonal electric and magnetic fields, i.e. E p · B p = 0. In this case the scalar product in Eq. (18) iŝ given that F pFp = 4E p · B p = 0. Therefore ifÊ s is chosen to be parallel to E p (B p ) only the term containing f (f ) survives inÊ s · J and the signal will be affected only by b (d). In both cases, the cavity dimensions should be chosen such that the signal resonance mode satisfies ω s = 3ω p in order for the signal be amplified and isolated. This method is however unable to constrain effective CP-even interactions of the form (FF ) 2 (as could be generated by an ALP), since it requires F pFp = 0. For concreteness, we have tested the CPV combination F p = TM 010 and F s = TE 011 , and found K ∞ ∼ 0.09. It is possible that higher form factors are attainable, although they may be harder to optimize, as suggested by the authors of [16].

CP-violation and Vacuum Birefringence
In this section, we discuss the effect of CP-odd photons self-interactions on the birefringence properties of the vacuum. In particular, we will show that a setup where vacuum birefringence takes place in a ring cavity is sensitive to both CPC and CPV phenomena separately.
Nonlinearities in Maxwell's equations are known to introduce a nontrivial response of the vacuum in the presence an external electromagnetic field (see e.g. [3,21,22,52,53]). Consider a light beam with a frequency ω, linearly polarized along thex axis and propagating along theẑ direction through a region of length L permeated by an external static magnetic field B, which lies in the x − y plane (see Fig. 4). The external magnetic field induces two different refractive indices, n 1 and n 2 , along the two orthogonal directionsv 1 andv 2 , shown in Fig. 4. The magnitude of n 1 and n 2 , and the orientation of the axesv 1 andv 2 , are determined by B and the photon self-interactions.
As a result of the anisotropic refractive index, the components of the electric field E i of the beam alongv 1 andv 2 evolve separately, inducing a change in the polarization vector the probe [3,22]. Below we consider two observables that are sensitive to an anisotropic refractive index, both of which, as we will see, can be split into the sum of CP-even and CP-odd components.
First, the propagation inside the birefringent region will cause the linearly polarized beam to become elliptically polarized, as the projections of the field ontov 1 andv 2 propagate at two different velocities (1/n 1 and 1/n 2 ), and thus acquire a phase difference δϕ which varies along the propagation distance as δϕ = (n 1 − n 2 ) ωz. This can be easily seen by noticing that the evolution of the polarization vector is given by wherek is the propagation direction of the beam, which we assume is either parallel or anti-parallel toẑ. The relative phase between the two components of the field is usually quantified in terms of the ellipticity Ψ, which is defined by the ratio A 1 /A 2 of the axes of the polarization ellipse via tan Ψ ≡ A 2 /A 1 (see Fig. 4). Equivalently, in terms of the electric field in Eq. (28), we may define sin 2Ψ = ik · (E × E * )/|E| 2 [54]. As can be seen from Eq. (28), if the eigen-axisv 1 forms an angle θ with respect to the initial direction of polarizationÊ i , then in the limit ∆n ≡ n 2 − n 1 1 the ellipticity acquired over a distance L is given by (see also e.g. [13]) Second, as we will see shortly, in the presence of photon interactions with a light scalar, the refractive index can acquire a anisotropic imaginary part, expressed in terms of the absorption coefficients κ 1 ≡ Im[n 1 ] and κ 2 ≡ Im[n 2 ]. From Eq. (28), this corresponds to an anisotropic attenuation of the field, which can be interpreted as the decay of one component of the photon field into the on-shell scalar. In particular, if κ 1 = κ 2 the components of the polarization vector alongv 1 andv 2 are depleted differently, and the polarization ellipse rotates by an angle ζ, defined as the angle between the major axis of the ellipse and the initial polarization direction E i (see Fig. 4). In the limit ∆κ ≡ κ 2 − κ 1 1, the acquired rotation over a propagation length L is given by (see also e.g. [13]) In the following, we will specialize the expressions of the ellipticity and rotation (written before for generic refractive indices andv 1 ,v 2 ) to those induced by photon interactions in the background of a magnetic field, mediated either by a light scalar with the Lagrangian in Eq. (11) or via the effective interactions in Eq. (1). We will then show that both the ellipticity and the rotation can be broken into a CP-even and a CP-odd component, denoted by where the superscript e (o) is for the CP even (odd) component.
If the photon self-interactions are mediated by the effective Lagrangian in Eq. (1), the refractive indices n 1 and n 2 are real and read (see Appendix B for the explicit derivation) Note that the difference in refractive indices is proportional to the square of the magnetic field B and to the strength of the photon self-interactions.
As mentioned earlier, the directionsv 1 andv 2 are related to B and to the photon-self interaction coefficients. In the absence of CP violation, i.e. for d = 0, it is easy to show thatv 1 coincides witĥ B. In this case, the orthogonal directionv 2 is given byk i ×B, where k i is the momentum of the initial beam. Note that we define the positive direction of all angles in the polarization plane aŝ B ×k i ×B =k i . Therefore, θ coincides with the angle α between B and E i . Instead, for d = 0, {v 1 ,v 2 } are rotated with respect to {B,k i ×B} by the angle αC P given by (see Appendix B) As a consequence, if CP is broken, θ and α are different and related by θ = α − αC P , as shown in Fig. 4. In particular, according to our convention, the positive direction of αC P is constant and set by k i . As was pointed out in [3], flipping the propagation direction while keeping the polarization vector constant -as is the case upon reflection off of a mirror in a zero incidence angle -is equivalent to flipping the sign of the CPV spurion d and therefore the sign of αC P from Eq. (33), as can be seen by the (k i ·k) term (see also Eq. (56) of Appendix B, which is invariant under k → −k and d → −d). This is a direct consequence of parity violation, which is equivalent to CP violation since charge conjugation is a symmetry of electrodynamics in vacuum.
Plugging Eq. (32) and θ = α − αC P into the general expressions for the ellipticity and the rotation in Eq. (29) and Eq. (30) respectively, we find that the rotation vanishes (i.e. ζ EFT = 0, as was also shown in [21,22]), and the ellipticity has the finite value As anticipated, the ellipticity in Eq. (34) can be broken into the CP-even and CP-odd parts Similarly, if the photon interacts with a light scalar field with the couplings in Eq. (11), the acquired ellipticity Ψ φ and rotation ζ φ are given by (in the small Ψ φ and ζ φ limit, see [22,55,56] for the explicit derivation) where x ≡ m 2 φ L/(2ω). In this case tan αC P = k i ·k g/g, and as before it changes sign under k → −k. The ellipticity and the rotation in Eq. (36) and Eq. (37) can again be broken into the CP-even and CP-odd parts Note that in the limit m φ /ω → ∞ the scalar expression in Eq. (36) consistently reproduces the expression for the EFT in Eq. (34) with the Wilson coefficients in Eq. (12), while ζ φ as expected vanishes.
We note that the CP-even and CP-odd parts of the ellipticity and rotation in Eqs. (35), (38) and (39) have different dependencies on the angle α, and can therefore be studied separately. However, as already noticed also in Refs. [3,22,24,56], while the former does not depend on the relative direction of propagation of the beam, the latter changes signs after switching the direction of propagation. Therefore, if the light beam is reflected within a region permeated by a magnetic field, the total change in ellipticity and rotation after a single round trip is (for perfect mirrors) respectively 2Ψ e and 2ζ e , and the CP-odd part cancels out. As a result, after N trips, the total ellipticity increases by a factor of (2N + 1) only in its CP-even part, while remains unchanged in its CP-odd part. A setup involving multiple reflections can be therefore thought of as an optical path multiplier affecting only the CP-even part of Ψ and ζ. In the PVLAS experiment [23] the CP-even signal is enhanced in this way inside a linear FP cavity. Since there is no amplification of the CP-odd component, the resulting sensitivity to CP-odd photon self-interactions is very weak.
The cancellation of the CP-odd contribution can be avoided by a modification of the optical path such that only part of it will be inside the magnetic field. Therefore we are motivated to consider a ring cavity instead of a linear cavity. If only the lower part of the ring cavity is permeated by the magnetic field, both the CP-even and the CP-odd parts of Ψ and ζ would be accumulated, as the interaction with the magnetic field takes place exclusively for one propagation direction of the beam. A schematic design of the ring cavity proposal in comparison to the PVLAS design is presented in Fig. 5. Note that the essential difference between our ring cavity proposal and the PVLAS setup is the optical path.
Let us compare the PVLAS setup to the cavity ring proposal in more detail. In both, a magnetic field is slowly rotating in the plane perpendicular to the wave vector of the incoming light, with an angular frequency ω B ω, such that the approximation of static magnetic field holds. A linearly polarized light (by the polarizer P 1 ) is fed into the cavity. While in the PVLAS setup it bounces between the mirrors M 1 and M 2 , where the optical path is fully under the magnetic filed (left panel of Fig. 5), in the ring cavity setup the light will be bouncing between four mirrors, M 1,2,3,4 , such that only part of the optical path is inside the magnetic field (right panel of Fig. 5). In order to increase the amplitude of the outgoing wave, a time-dependent ellipticity η = η 0 cos(ω η t) is injected via a modulator, G η . In this way, the leading outgoing signal wave will be an interference between Ψ and η, with a linear dependence on Ψ rather than quadratic. The light detector (D) finally collects only the component of the polarization orthogonal to that of the incoming light, selected by P 2 (this component is nonzero thanks a nonvanishing ellipticity and rotation). Thus, the ratio between the incoming (I in ) and outgoing (I out ) wave intensities is [3,23] where α(t) is the angle between the polarization vector E i and the magnetic field. The number of roundtrips inside the cavity is related to the cavity finesse F by N F/π, where for PVLAS F 7 · 10 5 . While the amplification factors for PVLAS are N e = 2N + 1 and N o = 1, for the ring cavity they are N e = N o = N + 1. As in PVLAS, to measure the rotation ζ, one should insert a quarter-wave plate with one of its axes aligned along the initial polarization, such that the rotation is converted to an ellipticity, and interferes with η [13]. Thus, the signal becomes We note that although the polarization plane is not parallel to the mirrors in a ring cavity (unlike in the linear cavity), assuming the mirrors are properly aligned, the polarization vector E i will not be shifted by the mirrors after a full round trip. 5 Therefore, in a ring cavity, the projected sensitivity for the CP-odd ellipticity and rotation could be similar as to their CP-even counterparts. Importantly, since the CP-even and CP-odd signals vary differently with α, see Eqs. (31)- (35), in the case of a measurement the two can be disentangled by a temporal analysis of the outgoing intensity in Eqs. (40) and (41), see [3] for this analysis. A similar idea has been proposed in [24], where instead the magnetic field does not rotate and the CPV part of the photon self-interactions is selected by the choice of the (time-independent) angle α.
By inverting Eq. (35), we can express the reach of the ring cavity in the measurement of d in terms of the minimum measurable ellipticity (Ψ o ) lim (acquired over the full set of round trips) as As a reference, the PVLAS sensitivity for the CP-even part of the ellipticity is (Ψ e ) lim PVLAS = 2σ PVLAS Ψ = 8 · 10 −12 , which is the latest result, reported in [13], obtained over a measurement time of 5 × 10 6 s. This value, using B PVLAS = 2.5 T, L PVLAS = 0.82 m and ω PVLAS = 1.2 eV, provides the bound on c − 4b in Eq. (5).
In Fig. 6 we present the potential reach of the ring cavity for the measurement of d. We show the bound as a function of the minimum measurable ellipticity, normalized for reference to the current bound on this observable by the PVLAS experiment (quoted above). In the plot we assume that the same magnetic field, finesse and cavity length will be employed. If the dominant noise is independent of the finesse, the magnetic field and the length (see [13] for a discussion on the validity of this assumption), the relative bound on d compared to the PVLAS bound on c − 4b scales as where the factor of 2 comes from the fact that the light traveling in the FP cavity crosses the magnetic field twice at each round trip. In this case increasing either B or L could greatly improve the bound.
Since the ring cavity would be sensitive also to the combination c − 4b, the bound on this Figure 6: The expected bound on the coefficient d of the effective operator F F FF from vacuum birefringence in a ring cavity (solid red line), as a function of the experimental sensitivity to the ellipticity Ψ, normalized for convenience to the current sensitivity on this observable in the PVLAS experiment, (Ψ e ) lim PVLAS = 8 · 10 −12 . The upper horizontal dashed line is the EFT consistency bound d ≤ 2 √ bc with the current bounds on b and c, as in Fig. 2. The lower horizontal dashed line corresponds to d ≤ 2 √ b EH c EH , which would be the consistency bound on the CPV coefficient assuming the EH term would be measured. The pink dashed line corresponds to the prospective consistency bound from the improvement in the bound on c considering only the measurement of the CPC signal (using the same cavity). The vertical line corresponds to the sensitivity to Ψ e at which the EH contribution should be observed (analyzing the CP-even part of the signal). In particular, for Ψ e > Ψ EH , the pink dashed line is plotted assuming that EH background can be fully removed in the measurement of the CPC part of the signal. quantity could get stronger, and correspondingly that on c (obtained as in Eq. (6), assuming the same bound on b of Eq. (4)). Therefore, we also present the prospective EFT consistency condition of Eq. (3) with this updated c. As the bound is presented as a function of the improved sensitivity, in Fig. 6 we mark the value of (Ψ e ) lim PVLAS /Ψ at which the EH contribution to the photon self-interactions will be observed, and assume it will be removed completely when deriving the bound. We observe that the left hand side of Eq. (3) is proportional only to √ c. Thus, using a ring cavity, an increase in the sensitivity for vacuum birefringence would improve both the bounds on c and on d, but (assuming the same bound on b of Eq. (4)) still probing values of c and d that are compatible with the unitarity and causality of the UV theory.
A similar discussion applies to the ellipticity and rotation in Eqs. (36) and (37) generated by a light scalar, both of which, as mentioned, can be split into a CP-even and CP-odd part. In a ring cavity the bound on √ gg is obtained both from the measurement of the ellipticity and from the measurement of the rotation, and we show the strongest of the two, i.e.
in Fig. 3 (red line). In plotting the bound we assumed that the ring cavity will be able to get to the same sensitivity as PVLAS, i.e.
, with the same magnetic field and cavity length. Notice that bound on √ gg from the rotation dominates at small masses and becomes m φ -independent for m φ /ω 1 (see Fig. 3), while the one from the ellipticity dominates at higher masses, for which |gg| lim ∝ m 2 φ (see Eq. (44)). As expected, these limits are of the same order as other laboratory bounds and not yet competitive with other current bounds. For the fermionic dipole operators, our bound (assuming Ψ ≈ (Ψ e ) lim PVLAS and similar finesse) yields D ,D 10 −7 eV −1 .

Conclusions
In this paper, we considered the possibility that the photon is subject to CP violating (CPV) selfinteractions, encoded in the low-energy effective operator F µν F µν F ρσF ρσ , with (F µν ) F µν being the (dual) electromagnetic field strength, which are highly suppressed in the Standard Model (SM) but could receive contributions from new physics.
We discussed two simple experimental approaches to isolate such interactions at energies below the electron mass. In particular, we estimated the corresponding sensitivities of two table-top cavity experiments to the above non-linear operator -one using a superconducting radio frequency (SRF) cavity and one using a ring Fabry-Perot (FP) cavity, see Figs. 2 and 5. These are expected to give the first direct limit on CP violation in non-linear Electromagnetism (EM) in vacuum.
In passing, we also qualitatively estimated the indirect bounds on the above CPV operator at energies above the electron mass. We found that at these energies the indirect bounds from the experimental limits on the electric and magnetic dipole moments of the electron are stronger, and therefore provide more stringent constraints on possible generic new physics if heavier than the electron.
In addition to considering an Effective Field Theory (EFT) approach towards the CPV effects discussed above, we derived the corresponding bounds for the case where CPV photon interactions are mediated by light scalars and fermions. We found that the constraints on the CPV combination of the couplings of a scalar to photons are not competitive with fifth-force searches and searches for violation of the equivalence principle. We further found that our experimental setup would be ω 1  able to reach world-record sensitivity to the presence of new fermions with electric and magnetic dipole moments, provided their mass lies between few hundreds keV to 10 MeV (for masses smaller than few hundreds keV, astrophysical bounds become stronger). However, one can indirectly set a stronger bound on the corresponding dipole interactions (at least for fermions heavier than the electron) using the constraints associated with the electric dipole moment of the electron.
Finally, we note that in the two SRF-cavities setup proposed in Ref. [18,19], it might be possible to constrain the CPV part of the non-linear EM interactions, in a manner similar to this work.  Table 3: TM/TE/TE mode combinations with K ∞ , K 0 0.1 for CPV interactions.

B Derivation of The Refractive Indices
In this Appendix we derive the refractive indices and the eigen-axes of propagation for a linearly polarized light beam (with electric field E and magnetic field B) traveling through a region permeated by a constant and homogeneous background magnetic field B 0 (orthogonal to the beam), in the presence of the photon self-interactions with the Lagrangian in Eq. (1). We will assume that the light beam field is negligible with respect to the background field, i.e. that E B 0 and B B 0 . The derivation below mostly follows Refs. [3,57].
The Euler-Lagrange equations of the Lagrangian in Eq. (1) can be equivalently written in terms of the electric and magnetic fields E i ≡ F 0i and B i ≡ − 1 2 ijk F jk as ∇ · B = 0 , ∇ · D = 0 , where D and H are defined by In the setup under consideration, the fields are E = E and B = B + B 0 . From Eqs. (1) and (47) it follows that where terms with higher powers of E or B were omitted since they are negligible with respect to the last three terms of Eqs. (48) and (49), given the assumption E B 0 and B B 0 . Notice that Eqs. (48) and (49) have the form where ij , µ ij , ψ DB ij and ψ HE ij are constant matrices that depend on B 0 and on the photon selfinteraction coefficients ( ij and µ ij are the equivalent of the electric permittivity and magnetization tensors). These matrices can be read off of Eqs. (48) and (49).
We assume the light beam to travel perpendicularly to B 0 , i.e. B 0 ·k = 0, and that the beam's momentum while propagating in the magnetic field isk = ±k i , wherek i is the propagation direction of the initial beam. Therefore, {B 0 ,k i ×B 0 } forms a constant orthonormal basis spanning the polarization plane of the beam, where E lies. In this convenient basis, Eq. (54) can be brought into the matrix form In the limit bB 2 0 , cB 2 0 , dB 2 0 1, and expanding at leading order in n − 1, the previous equation is simplified as The eigenvectors of the matrix in Eq. (56) are the propagation eigenstates, denoted in the main text asv 1 andv 2 , in the basis {B 0 ,k i ×B 0 }. Any vector proportional to one of the propagation eigenstate solves Eq. (56) (and therefore the the initial equations of motion with ansatz in Eq. (52)) provided the refractive index coincides with the eigenvalues of the matrix above, which are n 1,2 in Eq. (32).
As mentioned in the main text, for d = 0 the eigenvectors {v 1 ,v 2 } coincide with {B 0 ,k i ×B 0 } and are otherwise rotated with respect to {B 0 ,k i ×B 0 } by the angle αC P in Eq. (33), which positive direction isB 0 ×k i ×B 0 =k i . Moreover, flipping the sign of k is equivalent to d → −d as Eq. (56) remains invariant, and this corresponds to changing the sign of αC P , given Eq. (33).