Axion-Mediated Forces, CP Violation, and Left-Right Interactions

We compute the CP -violating scalar axion coupling to nucleons in the framework of baryon chiral perturbation theory and we apply the results to the case of left-right symmetry. The correlated constraints with other CP -violating observables show that the predicted axion nucleon coupling is within the reach of present axion-mediated force experiments for M W R up to 1000 TeV.

Introduction.-The axion experimental program has received an impressive boost in the past decade. Novel detection strategies, bridging distant areas of physics, promise to open for exploration the parameter space of the QCD axion in the not-so-far future, possibly addressing the issue of strong CP violation in the standard model (SM) via the Peccei-Quinn (PQ) mechanism [1][2][3][4] and the dark matter (DM) puzzle [5][6][7] (for updated reviews, see Refs. [8][9][10]). Standard axion searches often rely on highly model-dependent axion production mechanisms, as in the case of relic axions (haloscopes) or to a less extent solar axions (helioscopes), while traditional optical setups in which the axion is produced in the lab are still far from probing the standard QCD axion. A different experimental approach, as old as the axion itself [3], consists in searching for axion-mediated macroscopic forces [11]. Given the typical axion Compton wavelength λ a ∼ 2 cm ð10 μeV=m a Þ, an even tiny scalar axion coupling to matter may coherently enhance the force between macroscopic bodies. The sensitivity of these experiments crucially depends on the (pseudo)scalar nature of the axion field, a matter of ultraviolet (UV) physics.
Within QCD the Vafa-Witten theorem [12] ensures that the axion vacuum expectation value (VEV) relaxes on thē θ eff ≡ hai=f a þθ ¼ 0 minimum, whereθ denotes the QCD topological term. However, extra CP violation in the UV invalidate the hypotheses of this theorem, and in general one expects a minimum withθ eff ≠ 0. While the Cabibbo-Kobayashi-Maskawa (CKM) phase in the SM yieldsθ eff ≃ 10 −18 [13], too tiny to be experimentally accessible, CP-violating (CPV) phases from new physics can saturate the neutron electric dipole moment (nEDM) bound jθ eff j ≲ 10 −10 .
Another remarkable consequence of a nonzeroθ eff is the generation of CPV scalar axion couplings to nucleons,ḡ aN , which is probed in axion-mediated force experiments. In particular, given the nEDM bound onθ eff the scalarpseudoscalar combination (also known as monopole-dipole interaction) offers the best chance for detecting the QCD axion. Additionally, the presence of a spin-dependent interaction allows us to use nuclear magnetic resonance (NMR) to enhance the signal. This is the strategy pursued by the ARIADNE experiment [14,15], which aims at probing the monopole-dipole force via a sample of nucleon spins. A similar approach is pursued by QUAX-g p g s [16,17], using instead electron spins. ARIADNE will probe jθ eff j below 10 −10 for axion masses 1 ≲ m a =μeV ≲ 10 4 , a range highly motivated by DM.
In this Letter, we provide a coherent framework for computing the CPV scalar axion coupling to nucleons in terms of new sources of CP violation beyond the SM. This is done in the framework of the baryon chiral Lagrangian that allows us to compute all contributions of meson tadpoles andθ eff at once, as well as isospin-breaking effects. In comparison to previous works [11,[18][19][20], the contributions of the pion tadpole induced by the QCD dipole operator was estimated in Ref. [18] by naive dimensional analysis and in Ref. [19] using current algebra techniques, while isospin breaking was considered in Ref. [20] forθ eff without meson tadpoles. Our result is general and can be systematically applied to any bosonic representation of P-and CP-violating effective operators induced in extensions of the SM.
We detail our approach in the case of effective operators from right-handed (RH) currents, and then apply the results in the minimal left-right symmetric model (LRSM) endowed with a PQ symmetry and P parity as LR symmetry. This is an extremely predictive and motivated case for neutrino masses and additional CP violation, with an active collider physics program [21]. We build on the approach detailed in Ref. [22], which presented a study of the kaon CPV observables ε, ε 0 and the nEDM (d n ) in minimal LR scenarios. It was found there that the embedding of a PQ symmetry relaxes the lower bound on the LR scale just at the upper reach of the LHC. In this work we show that the present search for the scalar axion coupling to nucleons provides correlated and complementary constraints, with a sensitivity to the LR scale stronger than other CPV observables. Remarkably, for a nondecoupled LR scale we obtain a lower bound on theḡ aN coupling, thus setting a target for present axion-mediated force experiments.
CPV axion couplings to matter.-Including both CP-conserving and CPV couplings, the axion effective Lagrangian with matter fields (f ¼ p, n, e) reads where the first term can be rewritten in terms of a pseudoscalar density as −g af afiγ 5 f, with g af ¼ C af m f =f a . For protons and neutrons the adimensional axion coupling coefficients are [23] where K a ¼ 0.038ð5Þc s þ 0.012ð5Þc c þ 0.009ð2Þc b þ 0.0035ð4Þc t , and where the (model-dependent) axion couplings to quarks c q are defined via the Lagrangian term c q ð∂ μ a=2f a Þqγ μ γ 5 q. The axion mass and decay constant are related by m a ¼ 5.691ð51Þð10 12 GeV=f a Þ μeV [24,25].
The origin of the CPV scalar couplings to nucleonsḡ aN (N ¼ p, n) can be traced back to sources of either PQ or CP violation. These generically lead to a remnantθ eff ≠ 0 which induces CPV couplings. One finds for the isospin singlet component of the matrix element [11] where we included a 1=2 factor missed in Ref. [11]. A shortcoming of Eq. (4) is that CPV physics can induce not onlyθ eff , but also shifts the chiral vacuum, inducing tadpoles for the π 0 , η 0 , η 8 meson fields. These in turn yield extra contributions toḡ aN , as to other CPV observables such as d n . A derivation of g an;p taking all these effects consistently into account is here obtained in the context of the baryon chiral Lagrangian with axion field, as described below. We findḡ where for clarity we neglected m u;d =m s terms. Here, [26]. The value of b 0 is determined from the pion-nucleon σ-term as b 0 ≃ −σ πN =4m 2 π . From the precise determination in Refs. [27,28], one obtains b 0 ≃ −0.76 AE 0.04 GeV −1 at 90% C.L. Given σ πN ≡ hNjūu þddjNiðm u þ m d Þ=2, the isospin symmetric b 0θeff term reproduces exactly Eq. (4).
Equation (5) represents our general result, including isospin-breaking effects, whereθ eff and the meson VEVs are meant to be computed from a given source of CPV. In generalḡ aN and d n are not proportional, as it would follow from Eq. (4). Exact cancellations among the VEVs can happen for d n [22,29].
Axion coupling and RH currents.-As a paradigmatic application, we explicitly compute the above CPV axionmatter coupling in the case of RH currents, which arise in a wide class of models beyond the SM. Heavy RH currents lead generally to four quark operators that violate P and CP as [29][30][31][32]. Such operators induce meson tadpoles and allow for a nonvanishing correlator with the topological GG term, thus shifting both chiral and axion vacua [19]. At the leading order in momentum expansion the operators O qq 0 1 are represented in the lowenergy meson Lagrangian by combinations of ½U † qq ½U q 0 q 0 terms, where the usual 3 × 3 matrix U represents nonlinearly the meson nonet under Uð3Þ L × Uð3Þ R rotations. By a proper Uð3Þ A field rotation, the axion field is also included in the meson and baryon chiral Lagrangians. Complete notation and details are found in Appendix D of Ref. [22]. Rotating away the axion and meson tadpoles, the new CPV axion-nucleon scalar couplings of Eq. (5) are induced from the baryon Lagrangian.
In LR effective setups the operator O ud 1 generates typically the leading contribution to d n . We show in this work that it also generates the dominant contribution toḡ ap;n . We denote its low scale Wilson coefficient as C ud 1 , and similarly for other flavors. When O ud 1 is considered, we find [22,30,32] PHYSICAL REVIEW LETTERS 126, 081801 (2021) The axion VEV no longer cancels the originalθ term, leaving a calculableθ eff . As expected, the pion VEV is isospin odd (u ↔ d), while the other VEVs are even. The low-energy constant c 3 is estimated in the large N limit as c 3 ∼ F 4 π B 2 0 =4. Another estimate, based on SUð3Þ chiral symmetry, is given in Ref. [29]. Analogously, for O us 1 we find One notices in both Eqs. (6) and (7) the m s =m d enhancement of hπ 0 i over the other meson VEV.
As observed in Refs. [22,29], the CPV couplingḡ npπ computed using the VEVs (6) vanishes identically. On the other hand, when O us 1 is considered,ḡ nΣ − K þ cancels in turn. In either case the meson VEVs cancel exactly againstθ eff , a result which is made transparent in the basis of Ref. [26].
Such a cancellation is not present for the CPV axionnucleon couplingsḡ an;p , obtained via Eq. (5) using Eqs. (6) and (7), so that the typically unsuppressed O ud 1 operator dominates. In the large m s limit the complete result can be written as A few comments on Eqs. (5) and (8) are in order. The chiral approach allows us to consistently derive and account for the meson and axion tadpole contributions, thus properly addressing interference and comparison among the various contributions. It further includes LO isospin-breaking effects that enter through the pion VEV (via the b D;F couplings) and from theθ eff term. Within the range of hadronic parameters here considered, it leads to aḡ ap coupling about 60% larger thanḡ an . Finally, the results in Eqs. (5)- (8) are general enough to apply to any axion model with effective RH currents, since the model-dependent derivative axion couplings do not enter the scalar coupling.
Experimental probes forḡ an;p .-At present, the best sensitivity on the QCD axion exploiting axion-mediated forces is obtained by combining limits on monopolemonopole interactions with astrophysical limits of pseudoscalar couplings [33]. On the other hand, monopole-dipole forces will become the best constraining combination in laboratory experiments. In fact, monopole-monopole interactions are doubly suppressed inθ eff while dipoledipole forces have large backgrounds from ordinary magnetic forces. State-of-the-art limits on monopole-dipole forces can be found in Ref. [34]: the resulting lower bounds are at most at the level of f a ≳ ffiffiffiffiffiffi f θ eff p 10 13 GeV. A new detection concept by Arvanitaki and Geraci [14], exploited by the ARIADNE Collaboration [15], plans to use NMR techniques to probe the axion field sourced by unpolarized tungsten 184 W and detected by laser-polarized 3 He. In its current version, the experiment is sensitive tō g a 184 W g a 3 He . The CPV coupling axion coupling to tungsten is approximated byḡ a 184 W ≃ 74ðḡ ap þḡ ae Þ þ 110ḡ an [10], where for the QCD axionḡ ae ¼ 0 at tree level. It is convenient to define an average coupling to nucleons (weighting isospin breaking) as The CP-conserving term, g a 3 He ¼ g an , is only sensitive to neutrons because protons and electrons are paired in the detection sample. Thanks to NMR, ARIADNE can improve the sensitivity of previous searches and astrophysical limits by up to 2 orders of magnitude in ðḡ aN g an Þ 1=2 (for m a ∈ ½1; 10 4 μeV depending on the spin relaxation time), before passing to a scaled-up version with a larger 3 He cell reaching liquid density.
To provide an example of the testing power of these future experiments, as a definite model of RH currents we consider the paradigmatic case of the LR symmetric model, with a PQ symmetry.
Application to left-right models.-In the minimal LRSM [35][36][37][38][39], the gauge group SUð3Þ C × SUð2Þ L × SUð2Þ R × Uð1Þ B−L is spontaneously broken by a scalar triplet VEV hΔ 0 R i ¼ v R and eventually by the VEVs of a bidoublet field R sets the electroweak scale and tan β ≡ t β ¼ v 2 =v 1 . The single phase α is the source of the new CP violation. An important phenomenological parameter is the mixing between left and right gauge bosons, ζ ≃ −e iα sin 2βM 2 W L =M 2 W R , bound to jζj < 4 × 10 −4 from direct search limits on W R .
Born in order to feature the spontaneous origin of the SM parity breaking, the model is endowed with the discrete parity P, assumed exact at high scale and broken spontaneously by v R . P exchanges the gauge groups, the fermion representations Q L ↔ Q R , and conjugates the bidoublet Φ ↔ Φ † . As a result, the Yukawa Lagrangian L Y 1 Q L ðYΦ þỸΦÞQ R þ H:c: requires Hermitian Y,Ỹ. The diagonalization of quark masses gives rise to a new CKM matrix V R in the W R charged currents. Only for nonzero α the masses are non-Hermitian and V R departs from the standard V L . An analytical form for V R is found perturbatively in the small parameter y ¼ js α t 2β j ≲ 2m b =m t ≃ 0.05 [40,41]. While the left and right mixing angles can be considered equal for our aims, V R has new external CP phases. For later convenience we denote them as θ q , with PHYSICAL REVIEW LETTERS 126, 081801 (2021) V R ¼ diagfe iθ u ; e iθ c ; e iθ t gV L diagfe iθ d ; e iθ s ; e iθ b g. All θ q are small deviations of OðyÞ around 0 or π, corresponding to 32 physically different sign combinations of the quark mass eigenvalues [22,41]. For details on the relevant features of the minimal LR model, we refer to Refs. [21,22] and references therein.
On the other hand, the construction of a LR DFSZ model, with SM quarks carrying PQ charges, turns out to be less trivial. This is due mainly to the fact that chiral PQ charges X Q L ≠ X Q R forbid one of the Yukawa terms in L Y , implying unphysical mass matrices. Hence, either the LR field content must be extended [46,47] (e.g., with a second bidoublet) or effective operators must be invoked in the Yukawa sector [48,49]. Finally, a complex singlet S to decouple the PQ scale from v R and v is needed. A complete ultraviolet LR DFSZ model description is not needed here [50]; it is enough to report the axion couplings to quarks and charged-leptons: While the minimal LR model with P is a predictive theory even in the strong CP sector [51,52], the axion hypothesis can relax predictivity in the fermion as well as in the strong CP sector, if other fields as a second bidoublet are introduced. Below we stick to the LR KSVZ or the LR DFSZ case with a single bidoublet and a nonrenormalizable Yukawa term. The axion washes outθ (and renormalizations [51,53]), and observables such as, e.g., d n andḡ an;p , are tightly predicted.
With this choice, quark masses set as usual a perturbativity limit on t β , mainly due to m t =m b : one finds t β ≲ 0.5 [54] or ≳2. The two ranges are equivalent in the minimal model (swapping Y andỸ), but they become physically different when the PQ symmetry acts on Φ. Within this perturbative domain the pseudoscalar axion coupling to nucleons Eqs. (2) and (3) can never vanish.
Axion and CPV probes of LR scale.-The RH currents in the LRSM induce the axion couplings described above. For details on the LRSM short-distance and the extended chiral Lagrangian, we refer to Ref. [22]. We just recall that the short-distance coefficients C qq 0 i depend on the relevant CKM entries, carrying the additional CP phases of V R , and on the LR gauge mixing ζ. The C qq 0 i are renormalized at the 1 GeV hadronic scale and matched with the chiral lowenergy constants.
To analyze the predicted ðg anḡaN Þ 1=2 as a function of M W R , we study together the four CPV observables (ε; ε 0 ; d n ;ḡ aN ), while marginalizing on tan β, the CP phase α, and the 32 signs. As in Refs. [22,55], we introduce a parameter h i for each observable, normalizing the LR contributions to the experimental central value (ε, ε 0 ) or upper bound (d n ). For the latter we take the updated 90% C.L. result d n < 1.8 × 10 −26 e cm [56]. The LR contributions to the indirect CPV parameter ε in kaon mixing was thoroughly analyzed in Ref. [55], to which we refer the reader for details. For the direct CPV parameter ε 0 the latest lattice result [57] for the K → ππ matrix element of the leading QCD penguin operator supports the early chiral quark model prediction [58,59], confirmed by the resummation of the pion rescattering [60], as well as more recent chiral Lagrangian reassessments [61,62], including a detailed analysis of isospin breaking. All of the above point to a SM prediction in the ballpark of the experimental value, albeit with a large error [63]. We consider below two benchmark cases: 50% and 15% of ε 0 induced by LR physics [64,65].
The average CPV nucleon coupling in Eq. (9) is computed using Eq. (8). With the updated d n bound and including the strange quark contributions, we obtain where α qq 0 ¼ α − θ q − θ q 0 . We recall that all phases θ q depend on a single parameter. Also, α ud ≃ α us modulo π for M W R ≲ 30 TeV from the h ε constraint [55], which plays an important role in enforcing a tight correlation between the above observables. The subleading role of the Cabibbo suppressed us Wilson coefficient inḡ aN is clear, unlike the case of d n where the leading ud contribution is canceled as mentioned above [22]. The model-dependent pseudoscalar coupling g an in the monopole-dipole interaction is taken for the LR DFSZ case via Eq. (10). Similar results are obtained for LR KSVZ, for which, however, g an is compatible with zero; see Eq. (3).
In Fig. 1 we show the allowed regions of ðg anḡaN Þ 1=2 as a function of M W R , together with the reach of three phases of ARIADNE (1 s, 1000 s, projected) [14,15] and the SQUID sensitivity limit. We scale the coupling combination by f a ∝ 1=m a , making the prediction independent from it. With this normalization the experiment sensitivities vary mildly with m a , and we show their best reach, attained for m a ∼ 10 2−3 μeV. Present limits from astrophysics [33] and monopole-dipole experiments [34] lie above the plot and are hence ineffective to probe the LR scale.
The predicted regions depend on the constraints on h ε , h ε 0 , and h d n . In the colored area the LR contribution to ε 0 is allowed up to 15%, while in light gray we relax it to 50%, given the present theoretical uncertainties. In either case, a lower bound onḡ aN arises, for M W R ≲ 20 or 13 TeV, respectively. The origin of this lower bound is traced to the fact that, in the LRSM with P, for a few TeV M W R the CPV effects cannot be eliminated by taking α → 0: an exceedingly large contribution to h ε would remain from the CKM phase in V R ; thus a destructive interference from additional CP phases is required [55]. Thus, for instance, a positive detection from ARIADNE below 2 × 10 −18 with m a ≈ 100 μeV would falsify such a TeV-scale LR DFSZ scenario. Instead, a measurement above 10 −17 would result in a rejection of the LR DFSZ model or a sharp upper bound on M W R , at the reach of a future collider.
Given the square root in ðg anḡaN Þ 1=2 , the probed observable depends mildly on the new physics scale. Indeed, the upper boundary of the shaded region decreases as 1=M W R , and we find that within the ARIADNE sensitivity the model provides possible signals up to M W R ∼ 1000 TeV. Standard flavor observables, decoupling as 1=M 2 W R , have a more limited reach.
The effect of the present and future constraints on d n are shown with increasingly darker shadings, from a most conservative h d n < 2 (accounting for hadronic uncertainties), to a most stringent future bound of h d n < 0.01. The bounds on d n limit from above the predicted axionmediated force. For instance, h d n < 0.1 implies a prediction at the level of the ARIADNE 1000 s sensitivity.
To conclude, we provided a complete and consistent calculation of the CPV axion couplings to matter and applied it to the case RH currents, showing that axion-mediated forces provide a powerful probe of the CPV structure and scale of minimal LR PQ scenarios. It is amusing that the first hints of high-energy parity restoration may possibly be revealed in a condensed matter lab.