Axion couplings to electroweak gauge bosons
Abstract
We determine the modelindependent component of the couplings of axions to electroweak gauge bosons, induced by the minimal coupling to QCD inherent to solving the strong CP problem. The case of the invisible QCD axion is developed first, and the impact on W and Z axion couplings is discussed. The analysis is extended next to the generic framework of heavy true axions and low axion scales, corresponding to scenarios with enlarged confining sector. The mass dependence of the coupling of heavy axions to photons, W and Z bosons is determined. Furthermore, we perform a twocouplingatatime phenomenological study where the gluonic coupling together with individual gauge boson couplings are considered. In this way, the regions excluded by experimental data for the axionWW, axionZZ and axion\(Z\gamma \) couplings are determined and analyzed together with the usual photonic ones. The phenomenological results apply as well to ALPs which have anomalous couplings to both QCD and the electroweak bosons.
1 Introduction
The term “axion” denotes any (pseudo)Nambu–Goldstone boson (pNGB) of a global chiral U(1) symmetry which is exact at the classical level but is explicitly broken only by anomalous couplings to the field strength of QCD and perhaps other confining groups.^{1} Axions are the characteristic byproduct of solutions to the strong CP problem based on an anomalous U(1) axial symmetry, usually called PecceiQuinn (PQ) symmetry [1]. Whatever its mass, an axion necessarily has anomalous gluonic couplings and it only deserves that name if its presence solves the strong CP problem. When the number of axions in a given theory outnumbers the total number of distinct instantoninduced scales other than QCD, one (or more) light axions remain.
Light enough axions (that is, below \(\mathcal {O}(100\,\mathrm {MeV})\)) can participate in astrophysical phenomena [14, 15, 16]. The constraints that follow from their nonobservation in photonic processes lead to very high values for the decay constant, \(f_a\gtrsim 10^8\) GeV. It follows then from Eq. (1.1) that \(m_a \le 10^{2}\) eV for QCD axions.
Nevertheless, in specific QCD axion models the coupling to photons may be suppressed [9, 17] and moreover large uncertainties may hover over the purely hadronic constraints [18]. It is thus important to analyze the axion couplings to other electroweak gauge bosons, as they may become the phenomenologically dominant couplings in certain regions of the parameter space for those models. Couplings of axions and also of axionlike particles (ALPs) to heavy gauge bosons are increasingly explored [19, 20, 21, 22, 23, 24, 25] in view of present and future collider data, and also in view of rare meson decay data. For instance, in addition to LHCrelated signals, recent works [26, 27] consider the oneloop impact of aWW couplings on rare meson decays.
Here we first determine the modelindependent components of the mixing of QCD axions with electroweak gauge bosons, which result from the mixing of the axion with the pseudoscalar mesons of the SM. In other words, we determine the equivalent of the 1.92 factor in the photonic coupling in Eq. (1.3), for the couplings of the axion to W and Z gauge bosons. A chiral Lagrangian formulation will be used for this purpose, determining the leadingorder effects. The heavy electroweak gauge bosons will be introduced in that Lagrangian as external –classical– sources. Our results should impact the analyses for light axions of theories which solve the strong CP problem. They are novel and relevant in particular whenever the axion is lighter than the QCD confining scale and is onshell in either lowenergy or highenergy experiments. They also impact the comparison between the data taken at experiments at low and highenergy. For instance, a null result in NA62 data for \(K\rightarrow \pi a\) does not imply the absence of a signal at high energy in an accelerator such as that from offshell axions at LEP or at a collider. This is because modelindependent contributions are present at the low momenta dominant in rare decays (and cancellations may then take place), while at high energies they are absent.
As a second step, we will extend the analysis to heavy axions which solve the strong CP problem. Axions much heavier than \(\Lambda _{QCD}\) and with low axion scales are possible within dynamical solutions to the strong CP problem, at the expense of enlarging the confining sector of the Standard Model (SM) beyond QCD [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39]. These theories introduce a second and large instantoninduced scale \(\Lambda '\gg \Lambda _{QCD}\) to which the axion also exhibits anomalous couplings, resulting in the bulk of its large mass. Precisely because the axion mass typically lies well above the MeV regime, these heavy axions avoid the stringent astrophysical and laboratory constraints and present and future colliders may discover them. The transition between the light and heavy axion regime will be explored for the coupling of the axion to the photon and to the electroweak gauge bosons.
Finally, the phenomenological part of the analysis will be carried out on a “twocouplingatatime” basis: it will take into account the simultaneous presence of a given electroweak coupling and the axion–gluon–gluon anomalous coupling (essential to solve the strong CP problem). For the analysis of data, for the first time the experimentally excluded areas for the EW couplings \(g_{aWW}\), \(g_{aZZ}\) and \(g_{a\gamma Z}\) will be identified and depicted separately, besides the customary ones for the \(g_{a\gamma \gamma }\) coupling. Furthermore, the relations among the exclusion regions stemming from electroweak gauge invariance will be determined and exploited. Model predictions will be illustrated over the experimental parameter space.
Aside from the main focus of the paper on true axions, our analysis applies to and calls for a timely extension of the ALP parameter space. Very interesting bounds on ALPs from LEP and LHC [19, 20, 21, 22, 23, 24, 25, 40, 41] assume often just one electroweak coupling for the axion, and no gluonic coupling. The path to consider any two (or more) couplings at a time will change the experimental perspective on ALPs.
What is the difference between a heavy axion and an ALP with both anomalous electroweak and gluonic couplings? The key distinction is that the former stems from a solution to the strong CP problem while a “gluonic ALP” may not. Both exhibit anomalous couplings to QCD and in both cases there is an external source of mass besides that induced by QCD instantons and mixing. However, for a true heavy axion that extra source of mass does not induce a shift of the \(\theta \) parameter outside the CP conserving minimum (and thus the solution to the strong CP problem is preserved), while for a generic gluonic ALP such a shift may be induced. Nevertheless, this important distinction is not directly relevant for this work, as the novel aspects and phenomenological analysis developed below are valid for both true heavy axions which solve the strong CP problem and for gluonic ALPs. To sum up, all results below for heavy axions apply directly to gluonic ALPs as well. In addition, the conclusions based purely on EW gauge invariance have an even larger reach: they hold for all type of axions and for generic ALPs (that is, ALPs with or without gluonic couplings).
The structure of the paper can be easily inferred from the Table of Contents.
2 The Lagrangian for the QCD axion
2.1 The Lagrangian below the QCD confinement scale
2.1.1 SM light quarks not charged under PQ (\({\mathcal {X}_{\mathbf {u,d}}=\mathbf {0}}\))
The results in Eqs. (2.33)–(2.37) illustrate that the physical lowenergy axion eigenstate acquires \(\pi _3\) and \(\eta _0\) components and thus inherits their couplings to all gauge bosons, weighted down by their mixing with the axion. These results apply to any physical process in which the axion is onshell and the axion mass is lighter than the confinement scale.
We are interested in identifying the modelindependent contributions in the coupling to the electroweak gauge bosons for light axions and for the SM light pseudoscalars. We will first recover in our basis the customary axion–photon couplings, to set the framework.
Axion–photon coupling
2.1.2 SM light quarks charged under PQ (\({\mathcal {X}_{\mathbf {u,d}}\ne \mathbf {0}}\))
Axion–photon coupling
2.2 Axion couplings to EW gauge bosons
The description in terms of the effective chiral Lagrangian is only appropriate for energies/momenta not higher than the cutoff of the effective theory, \(4\pi f_\pi \), that is, the QCD scale as set by the nucleon mass. In this context, the W and Z bosons can be considered as external currents that couple to a QCD axion whose energy/momentum is not higher than \(\Lambda _{QCD}\), for instance a light enough onshell QCD axion. In other words, the W and Z bosons enter the effective chiral Lagrangian as classical sources, alike to the treatment of baryons in the effective chiral Lagrangian.
The modelindependent results obtained here for the coupling of light QCD axions to the SM electroweak bosons may impact axion signals in rare decays in which they participate. For instance, in lowenergy processes the axion could be be photophobic at low energies [17] (or more generally, EWphobic), in models in which the terms in parenthesis cancel approximately, unlike at higher energies at which the modelindependent component disappears and only the modeldependence (encoded in E / N, M / N, Z / N and R / N) is at play.
Gauge invariance
3 Beyond the QCD axion
We discuss in this section the case of a “heavy axion”: an axion whose mass is not given by the QCD axion expression in Eq. (1.1) but receives instead extra contributions. We have in mind a true axion which solves the strong CP problem, for which the source of this extra mass does not spoil the alignment of the CP conserving minimum. This is the case for instance of models in which the confining sector of the SM is enlarged involving a new force with a confining scale much larger than the QCD one [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39]. This avenue is of particular interest as it allows to consider heavy axions and low axion scales (e.g. \(\mathcal {O}\)(TeV)), and still solve the SM strong CP problem. The axion mass can then expand a very large range of values. It can become much larger than the EW scale or, conversely, be in the GeV range or lower. For the purpose of this work, the latter range is to be kept in mind as a general guideline, so as to remain in the range of validity of the effective Lagragian with confined hadrons. The procedure will also serve as a template to show how the mixing effects disappear from the axiongauge couplings as the axion mass is raised.
3.1 Heavy axion couplings to EW gauge bosons
Gauge invariance
Alike to the discussion after Eq. (2.63), the relations in Eqs. (3.11) and (3.12) apply not only to heavy axions and heavy gluonic ALPs, but also to generic ALPs which only exhibit EW interactions and are much heavier than nucleons. The corollary that at least two EW couplings –if any– must exist for any axion or ALP holds as well.
4 Phenomenological analysis
The impact of the results obtained above on present and future axion searches will be illustrated in this section. Both treelevel and looplevel effects will be taken into account. Indeed the latter are relevant when confronting data on photons, electrons and nucleons, as the experimental constraints on these channels are so strong that they often dominate the bounds on EW axion couplings.
4.1 Loopinduced couplings
The treelevel coupling of the axion to photons can be suppressed in some situations [9, 17] (photophobic ALPs are also possible [25]). Additionally, many models have no treelevel couplings to leptons or suppressed couplings to nucleons [8]. However, all possible effective couplings will mix at the loop level. This affects the renormalization group (RG) evolution, via which all couplings allowed by symmetry will be generated even when assuming only a subset of couplings at some scale.
Before proceeding with the phenomenological analysis, we discuss in this subsection the loopinduced effective interactions arising from the direct coupling to electroweak gauge bosons. Because the experimental and observational limits are usually strongest for photons, electrons, and nucleons, the loopinduced contributions to these channels can give stronger constraints than those stemming from the treelevel impact on other channels.
The combination of Eqs. (4.2) and (4.3) allows to derive the coupling to nucleons induced by the coupling to electroweak bosons.
It is worth noting that chaining the two previous oneloop contributions gives an approximate estimation of the twoloop contribution of a given heavy gauge boson coupling to the axion–photon coupling. As an example, consider \(g_{aWW}\) in Eq. (4.2): it results in an effective fermion coupling \(c_{1\,\mathrm {eff}}^{f}\) which, when subsequently inserted in the second term in Eq. (4.5), results in an effective axion–photon coupling. This can be compared with the third term which gives directly a oneloop contribution of \(g_{aWW}\) to \(g_{a\gamma \gamma }\): for \(m_a\ll m_W\), the second term in Eq. (4.5) may actually be numerically larger than the third term, that is, the twoloop contribution via fermionic couplings may dominate over the oneloop gauge one, as it was pointed out in Refs. [21, 25]. Indeed, this twoloop contribution may be phenomenologically the most relevant one to constrain the axion couplings to heavy electroweak gauge bosons. A caveat is that only a true twoloop computation may settle the dominant pattern, but the analysis discussed is expected to provide an order of magnitude estimate.
4.2 Axion decay channels and lifetime
The plethora of couplings discussed, contributing either at tree or loop level, produces a rich variety of production and decay channels of the axion, depending on its mass and on the relative strength of the couplings. A quantitative evaluation of the lifetime and branching ratios is essential for assessing what experiments or searches are more adequate to test different regions of parameter space.
In the low mass region \(m_a<3m_\pi \), only decays to pairs of electrons, muons or photons are possible. The axion typically becomes long lived enough so as to be stable at collider and flavour experiments. Note that this region is particularly sensitive to a possible cancellation/suppression of the photonic coupling \(g_{a\gamma \gamma }\) [this happens for instance in models of axions in which the modeldependent parameter E / N partially cancels the modelindependent contribution, see Eqs. (2.55) and (3.6)]. This would suppress the decay width to photons and thus enhance the branching fraction to fermions, especially close to the respective mass thresholds.
The hadronic channel plays a central role as soon as it opens. It then dominates the decay of the axion due to the large gluonic coupling. The lightest possible hadronic final state is three neutral pions. At around the GeV scale many other final states become viable, but in this region chiral perturbation theory starts to break down and we refrain from making any quantitative predictions.^{11} At high axion masses above \(3\,\)GeV the inclusive decay to hadrons can be safely estimated within perturbative QCD.^{12}
At much higher energies, treelevel decays to pairs of EW gauge bosons become possible and, though subdominant with respect to the hadronic one, will play a relevant role in collider searches.
4.3 Experimental constraints on the (heavy) axion parameter space
The coloured areas in Fig. 3 show the regions experimentally excluded if taking into account exclusively the axion–gluon coupling \(g_{agg}\) (which in axion models fixes the axion scale \(f_a\)). Although this work focuses on the case where also EW gauge boson couplings are present, this parameter space is also shown for reference.
The resulting greenish regions in Fig. 4a match wellknown exclusion regions for \(g_{a\gamma \gamma }\), although the overlap is not complete because the latter typically only take into account the effective axion–photon coupling; the additional presence in our analysis of the axion–gluon coupling \(g_{agg}\) has a particularly relevant impact in the heavy axion region (see below).
Figure 4b–d respectively for \(g_{WW}\), \(g_{ZZ}\) and \(g_{\gamma Z}\) are novel. The possibility of measuring four distinct EW observables offers a multiple window approach and a superb crosscheck if a signal is detected, given the fact that for axion masses much smaller (larger) than \(\Lambda _{QCD}\) only three (two) couplings are independent among the set \(\{g_{a\gamma \gamma }\) \(g_{aWW}\), \(g_{aZZ}\), \(g_{a\gamma Z}\}\), see Eq. (2.63) (Eqs. (3.11) and (3.12)).
 A
For LHC searches and \(m_a>3\,m_\pi \), Eqs. (4.8) and (4.9) were used, which for most cases is equivalent to assume \(g_{agg}\gg g_{aXY}\).
 B
For the regions labelled as “photons”, “electrons” and “nucleons” in Fig. 4b–d, the loopinduced bounds have a very mild dependence on the assumption in Eq. (4.10).
Coupling to photons
The combination of astrophysical and terrestrial probes makes this search a particularly powerful tool to test the axion paradigm, especially for low mass axions. Even in the case of relatively large axion masses and/or situations where the coupling to photons can be suppressed, photons still place strong constraints both at treelevel and through loopinduced effects.
The experimental limits on \(g_{a\gamma \gamma }\) are compiled in Fig. 4a. At the lowest axion masses considered here, \(m_a \lesssim 10\mathrm{~meV}\), the most competitive bounds come from the CAST helioscope [51], and will improve in the future with the upgrade to the IAXO experiment [52]. At slightly larger masses up to \(m_a\sim 1\,\mathrm {keV}\), \(g_{a\gamma \gamma }\) is constrained by an energyloss argument applied to Horizontal Branch (HB) stars [14]. A similar argument applies to the supernova SN1987a and constrains larger masses up to the \(100\,\)MeV range, both using extra cooling arguments [53] and by the lack of observation of a photon burst coming from decaying emitted axions [54]. In the same mass range, larger couplings can be constrained using beam dump experiments, with these exclusions led at present by the NuCal [55] experiment together with the 137 [56] and 141 [57] experiments at SLAC. We adapt here the constraints compiled in Ref. [58], noting that these bounds rely solely on the photon coupling.
For yet higher axion masses, where colliders provide the best limits, the gluon coupling plays a relevant role. As long as no hadronic decay channel is open, the LEP constraints based on \(Z\rightarrow \gamma \gamma \) and \(Z\rightarrow \gamma \gamma \gamma \) searches obtained in [19, 20] for ALPs without gluonic couplings are also applicable to heavy axions. However, for masses above \(3m_\pi \), hadronic final states start to dominate and we refrain from claiming any exclusion, as a new dedicated analysis would be required to use these channels. This explains the white gap just left of the grey hatched “no human’s land” region, which should be at least partially covered when the forementioned analysis is performed. It is nevertheless possible to exploit some collider searches, if a relation between the gluonic and the EW couplings is assumed. Assumption A is adopted here. Our reinterpretation of the analysis in Ref. [59], in which the L3 collaboration looked for hadronic final states accompanied by a hard photon, yields the limit labelled “L3” in Fig. 4a, though it is ultimately superseded by LHC exclusions. The region labelled “Flavour” is excluded by data from Babar [60] and LHCb [61], as computed in Ref. [41]. For high axion masses near the TeV scale, the limits from LHC are much stronger than those from LEP, because of the enhanced axion production via gluongluon fusion. We have included the limits obtained in this context in Refs. [62, 63] using run 1 data. These limits are extremely strong and should improve with the addition of run 2 data, especially at higher energies.
Finally, the bounds on \(g_{a\gamma \gamma }\) described above have been translated – using assumption B – into competitive bounds for the other EW axion couplings, by means of their loopmediated impact.
Coupling to fermions
Flavour blind observables involving fermions can be used to constrain gauge boson couplings via the impact of the latter at loop level, see Eq. (4.2). In order to fix the mild logarithmic dependence on the cutoff scale, assumption B will be adopted.
The most relevant constraints on flavourblind axion–fermion interactions are of astrophysical origin and come from either electrons or nucleons. Firstly, a coupling of the axion to electrons allows for efficient extra cooling of some stars, which allows to place a bound on the axionelectron coupling \(g_{aee}\) via the observation of Red Giants (RG) [14]. Secondly, and in a manner similar to the discussion above for photons, a too strong coupling of the axion to nucleons would have shortened the duration of the neutrino burst of the supernova SN1987a. We use the most recent evaluation of this bound calculated in Ref. [15]. These two observations (RG and SN1987a) give the strongest limits on the coupling of axions to gauge bosons for axion masses respectively below \(10\,\mathrm {keV}\) and \(10\,\mathrm {MeV}\), as can be seen in Fig. 4.
In addition, the oneloop induced fermion couplings also play a role in many of the observables considered here. In particular, they open potential axion decay channels into pairs of fermions.
Rare decays
For axion masses in the MeV–GeV range, \(g_{aWW}\) is best tested by its oneloop impact on rare meson decay experiments, where axions can be produced in flavourchanging neutral current (FCNC) processes. This search was first proposed in Ref. [26] (where ALPs either stable or decaying to photons were considered). Recently, these bounds have been recomputed in Ref. [25] in the context of photophobic ALPs, considering as well the potential decays of the axion to a pair of fermions due to the oneloop induced coupling in Eq. (4.2). We reinterpret these searches, taking into account in addition the effects of the gluonic axion coupling under the assumptions A and B. The main consequence is that, for axion masses \(m_a>3m_\pi \), the sensitivity is drastically reduced because of the opening of hadronic axion decay channels.
At low axion masses below \(2m_\mu \), the axion is longlived enough so that it can be considered stable for experimental purposes. This means that in these regions the axion has to be looked for in invisible searches. The most stringent limits were placed by the E787 and E949 experiments testing the \(K^+\rightarrow \pi ^+X\) channel, with X invisible [49]. Following Ref. [26], we reinterpret this search in terms of axions coupled to W bosons, which yields the constraint shown in Fig. 4b. These bounds will be improved in the near future by the NA62 experiment.
Axions can also be produced from rare meson decays in proton beam dump experiments, where they can be looked for in searches for longlived particles. The current best limits are set by the CHARM experiment [64], where the axion can be produced in Kaon and B meson decays and subsequently decays to a pair of electrons or muons. The framework developed in Ref. [65] has been recast to obtain the limit on \(g_{aWW}\) shown in Fig. 4b.^{14}
Direct couplings to heavy gauge bosons
LEP provides the best environment to directly test the \(g_{a\gamma Z}\) coupling for axion masses below \(m_Z\), as shown in Fig. 4d. The first constraint set assuming only the \(g_{a\gamma Z}\) coupling was placed in Ref. [24] exploiting the limit on the uncertainty of the total Z width [67], \(\Gamma (Z\rightarrow \mathrm {BSM})\lesssim 2\,\mathrm {MeV}\) at \(95\%\) C.L., which allows to set a conservative bound on the process \(Z\rightarrow a\gamma \). Stronger limits can be placed by more specific searches, as studied in Ref. [25]. The best limit at axion masses low enough for the axion to be longlived stem from the \(Z\rightarrow \gamma +\,\mathrm {inv}.\) search. For higher axion masses, the large hadronic branching fraction makes the \(Z\rightarrow \gamma +\,\mathrm {had.}\) search the more fruitful one to look for axions. Under assumption A for the relative strength of the gluonic and EW couplings, we exploit the results of the search performed by the L3 collaboration as presented in Ref. [59] to obtain strong limits for \(m_a\) in the range from \(10\,\)GeV up to the Z mass. Note that, even if the search is the same than that used to constrain the photonic axion coupling, the exclusion for \(g_{a\gamma Z}\) has a larger reach due to the fact that the process is mediated by an onshell Z boson, instead of a very virtual photon.
LHC allows to look for a plethora of processes sensitive to axions. In particular, for heavy axions it provides the best limits on the coupling to heavy EW gauge bosons. The drawback of restricting the analysis to processes that separately involve only one of the EW gauge boson couplings plus the gluon coupling is the reduced number of available searches. Nevertheless, the advantage is that it provides robust and modelindependent constraints.
The authors of Ref. [24] studied the LHC phenomenology of axions that are stable on collider lengths and thus would manifest themselves as missing energy. In particular, monoW and monoZ final states where an axion is radiated as missing energy/momentum can set constraints on the three couplings \(g_{aWW}\), \(g_{aZZ}\) and \(g_{a\gamma Z}\), as shown in Fig. 4. For large axion masses \(m_a>m_Z\), the authors of Ref. [25] suggested that triboson final states place the strongest bounds on ALPs coupling to massive gauge bosons, though the sensitivity of this search is hindered for axions because of the large hadronic branching ratio that we take into account. Adapting their constraints –with assumption A– leads to the exclusion of regions in parameter space near the TeV range, as shown in Fig. 4b, d for \(g_{aWW}\) and \(g_{a\gamma Z}\), respectively. Note that significant exclusions can also be placed through the loopinduced coupling to photons. However, the most promising LHC search is one that –to the best of our knowledge– has not been performed yet. We advocate [68] the use of \(pp\rightarrow a \rightarrow VV^\prime \) processes, which benefit from the large production cross section through the gluonic coupling together with the clean final states that the decay to EW gauge bosons produce. We foresee that this search will have a sensitivity to the couplings of axions to heavy EW gauge bosons similar to the photonic case presented in Fig. 4a. Though potentially very interesting, the detailed analysis that this study requires is beyond the scope of this work and is left for the future [68].
4.4 Impact on (heavy) axion models and gluonic ALPs
The black oblique line in Fig. 3 corresponds to the linear relation between \(1/f_a\) and \(m_a\) for the QCD axion, Eq. (1.1). The horizontal blue branch is one example of how that relation changes after Eq. (3.3) for an illustrative example of a true heavy axion.
Maximum and minimum values of the modeldependent coefficients for the benchmark KSVZ models with only one exotic fermion representation depicted in Fig. 4
\(SU(3)_c\times SU(2)_L\times U(1)_Y \)  

\(\left( E/N\right) _{\mathrm{max}} = 44/3\)  \((3,3,4/3)\) 
\(\left( E/N\right) _{\mathrm{min}} = 5/3\)  \((3,2,+1/6)\) 
\(\left( L/N\right) _{\mathrm{max}} = 4\)  \((3,3,\,\,Y\,)\) 
\(\left( L/N\right) _{\mathrm{min}} = 2/3\)  \((8,2,1/2)\) 
\(\left( Z/N\right) _{\mathrm{max}} = 2.9\)  \((3,3,4/3)\) 
\(\left( Z/N\right) _{\mathrm{min}} = 0.4\)  \((8,2,1/2)\) 
\(\left( 2R/N\right) _{\mathrm{max}} = 5.9\)  \((3,3,1/3)\) 
\(\left( 2R/N\right) _{\mathrm{min}} = 0.7\)  \((8,2,1/2)\) 

The expectation for the pure QCD axion is depicted by grey and black lines. The bands in Fig. 4 delimited by grey lines correspond to just one exotic KSVZ fermion representation. The values of the modeldependent parameters delimiting these benchmark bands [17] are summarized in Table 1. The black line is instead an illustrative case with two fermion representations such that the coupling to photons \(g_{a\gamma \gamma } \) cancels up to theoretical uncertainties [17]. The upward bending of the lines obeys the expected change of the prediction for axion masses larger than the \(\eta '\) mass, a regime in which the last term in the parentheses in Eqs. (2.55)–(2.58) is absent.

The expectations for heavy axions are illustrated with blue lines. The big dots which fall on the QCD axion lines correspond to \(M=0\) in Eqs. (3.1)–(3.3). The heavy axion trajectories start on those points and the prediction moves on each blue line towards the right as M grows. As the value of the axion mass gets near the pion and the \(\eta '\) masses, the prediction reflects the “resonances” found in the pseudoscalar mixing angles and the physical couplings to gauge bosons. For larger values of M the mixing effects progressively vanish, as physically expected and reflected in Eqs. (3.5)–(3.8), and the predictions stabilize again. The asymptotic value of the couplings is then induced only by the modeldependent parameters (E, L, Z, R), and it is often higher than for heavy axions lighter than the pion, for which the partial cancellation between the modeldependent and modelindependent mixing effects may operate.
The parameter space for heavy axion models spans in fact most of the region to the right of the oblique band for the QCD axion: parallel sets of horizontal lines above and below the blue ones depicted are possible and expected for other values of the heavy axion parameters. For a given \(f_a\), varying M (that is, varying \(m_a\)) is tantamount to move right or left on a horizontal blue line, while varying \(f_a\) displaces up or down the set of horizontal blue lines. Finally, all these considerations for heavy axion models apply as well to gluonic ALPs, as argued earlier.
Gauge invariance
For heavy axions or any type of ALP with masses \(m_a\gg \Lambda _{QCD}\), couplings to EW gauge bosons are directly tested and gauge invariance imposes the two relations in Eqs. (3.11) and (3.12). Therefore, the combination of the experimental constrains on two of the operators in the set \(\{ g_{a\gamma \gamma },\, g_{aWW}, g_{aZZ},\, g_{a\gamma Z}\}\) translates in model independent bounds on the other two couplings. For light masses \(m_a\le \Lambda _{QCD}\), only Eq. (2.63) applies instead. These bounds based solely on EW gauge invariance have been depicted by black curves on the upper right corner of Fig. 4b–d. They susbtantially reduce the latter parameter space, especially in the cases of \(g_{aWW}\) and \(g_{aZZ}\), whose current direct constraints are less powerful. They are to be taken with caution, though, since in each of the exclusion plots only one EW coupling was taken into account, while the relations deduced from gauge invariance involve several nonvanishing EW couplings. Furthermore, in a future multiparameter analysis where treelevel axion–fermion couplings are included, those relations could be corrected via oneloop effects.
4.5 Implications for heavy axion models

\(m_a\gg m_\eta '\). The modelindependent effects due to the mixings with SM pseudoscalars have become negligible, and \(g_{a\gamma \gamma }\) is a direct measure of the product \((1/f_a) E/N\).

\(m_a \ll m_\pi \). In this case the measured \(g_{a\gamma \gamma }\) value is undistinguishable from that for the QCD axion, with the E / N and \(f_a\) dependence given by Eq. (2.55). In other words, it would indicate either a heavy axion or a QCD axion with some degree of photophobia, as in that region they become undistinguishable. This is so at least for the lowest axion masses and/or without the help of other measurements involving heavy EW gauge bosons.
5 Conclusions
Among the novel results of this work, we have first determined at leading order in the chiral expansion the modelindependent components of the coupling of the QCD axion to heavy EW gauge bosons: \(g_{a\gamma Z}\), \(g_{aZZ}\) and \(g_{aWW}\). They stem from the axion\(\eta '\)pion mixing induced by the anomalous QCD couplings of these three pseudoscalars. Our results extend to heavy EW gauge bosons the well known result for the photonic coupling of the axion \(g_{a\gamma \gamma }\). They must be taken into account whenever an axion lighter than \(\Lambda _{QCD}\) is onshell and/or the energy and momenta involved in a physical process are of the order of the QCD confining scale or lower. As a previous step, we rederived pedagogically the leading contributions to \(g_{a\gamma \gamma }\) for the case of the most general axion couplings (“App. C”), and then proceeded to the determination of the couplings to the SM heavy gauge bosons.
This analysis of the EW couplings of the QCD axion may have rich consequences when comparing the presence/absence of signals at two different energy regimes. For instance, the axion could be photophobic at low energies [17] or even EWphobic (e.g. in rare meson decays) because of cancellations between the modelindependent and modeldependent components, while an axion signal may appear at accelerators or other experiments at higher energies at which the modelindependent component disappears.
We have next extended those results to the case of heavy axions which solve the strong CP problem. This has allowed to explore how the mixing of the axion with the pion and \(\eta '\) evolves with rising axion mass, and in consequence how the modelindependent contributions to all four EW axion couplings vanish as the axion mass increases above the QCD confinement scale. We have determined the modified expression for \(g_{a\gamma \gamma }\) relevant for heavy axions, which may have rich consequences: an hypothetical measurement of that coupling outside the QCD axion band could point to either a heavy axion or a photophobic QCD axion. The analogous expressions for \(g_{a\gamma Z}\), \(g_{aZZ}\) and \(g_{aWW}\) have been also worked out.
On the purely phenomenological analysis, we developed a “two simultaneous coupling” approach in order to determine the regions experimentally excluded by present data for \(g_{a\gamma \gamma }\), \(g_{a\gamma Z}\), \(g_{aZZ}\) and \(g_{aWW}\) versus the axion mass. Each EW coupling has been considered simultaneously with the anomalous gluonic coupling essential to solve the strong CP problem. The allowed/excluded experimental areas have been depicted for each of those couplings as a function of the axion mass. This is the first such reinterpretation for \(g_{a\gamma Z}\), \(g_{aZZ}\) and \(g_{aWW}\). Even for \(g_{a\gamma \gamma }\), the results of previous studies often did not apply and must be reanalysed: for instance the present bounds extracted from LEP and LHC data tend to focus on ALPs which would not have gluonic couplings, with very few exceptions [20, 41, 62, 63]. Furthermore, we have included an estimation of the oneloop induced bounds for each EW axion coupling, which leads to supplementary constraints.
The expectations from KSVZtype of theories have been then projected and illustrated over the obtained experimental regions, both for the QCD axion and for heavy axions. The compatibility of hypothetical a priori contradictory signals in high and low energy experiments in terms of a given axion has been pointed out. Furthermore, we discussed how to interpret an eventual signal (or null result) outside the QCD axion band in terms of the value of the new high confining scale generically present in heavy axion theories.
A simple point with far reaching consequences results from EW gauge invariance. In all generality, not all couplings of axions to EW gauge bosons are independent among the four physical ones in the set \(\{g_{a\gamma \gamma }\) \(g_{a\gamma Z}\), \(g_{aZZ}\) and \(g_{aWW}\}\). In particular, the relations obtained imply that at least two EW gauge couplings –if any– must be nonvanishing for any axion or ALP. These facts have been used to project the exclusion limits for the presently best constrained couplings onto the parameter space for the less constrained ones. In particular, for axions/ALPs much heavier than \(\Lambda _{QCD}\), those relations have been projected on the parameter space for \(g_{aWW}\), \(g_{aZZ}\) and \(g_{a\gamma Z}\), reinforcing their constraints. A future multiparameter analysis may correct them via loop corrections. Nevertheless, the results obtained here clear up the uncharted experimental regions and may be of use in setting a search strategy. More in general, the existence of four physical axion couplings to EW bosons at experimental reach constitutes a phenomenal tool to overconstrain the axion parameter space and to check the origin of an eventual axion signal.
Finally, all results obtained for heavy axions apply as well to ALPs which have both EW and gluonic anomalous couplings. The constraints stemming from EW gauge invariance extend even to generic ALPs which do not couple to gluons. In consequence, this work also extends automatically the usual parameter space for ALPs that do not intend to solve the strong CP problem, adding to the incipient efforts to go towards a multiparameter strategy.
Footnotes
 1.
The SM \(\eta '\) is excluded from this definition since the associated \(U(1)_A\) symmetry is also explicitely broken by the nonzero quark masses.
 2.
 3.
Obviously, only lefthanded quarks may contribute to \(L_0\); in any case, it is always possible to work in a convention in which only lefthanded quarks are PQ charged.
 4.
In “App. C” it will be explicitly shown that they do not have physical impact on mixing. Note that possible flavour nondiagonal couplings are not considered.
 5.
This expression for the axion–pion mixing agrees with the result of Ref. [18] for the case where the only PQ charged fermions are the up and down quarks, i.e. \(N_0=0\).
 6.
As expected from the triangle diagram, all fermions (including the up and down quarks) run in the loop and contribute to \(N=N_0+\sum _{u,\,d} 2\, \mathcal {X}\,T(R)=N_0+\mathcal {X}_d+\mathcal {X}_u\).
 7.
Obviously, the milder constrain in Eq. (2.63) also applies here.
 8.
For energies below the QCD scale, radiative corrections involving light quarks have to be evaluated using chiral Lagrangian methods. This was achieved in Ref. [21], the conclusion being that the results remain qualitatively right once the quarks masses are substituted by an appropriate hadronic scale, \(m_\pi \) for u and d and \(m_\eta \) for s. However, in the presence of gluonic couplings this contribution is subdominant to the one computed in the previous sections and will thus not be considered here.
 9.
The value for \(g_{a\gamma \gamma }\) used in this figure corresponds approximately to \(f_a= 10\) TeV and \(E=1\) in axion models. Rescaling for other values of the couplings can be achieved by taking into account that \(\Gamma _i\propto g_{aXY}^2\).
 10.
The widths used in determining the coloured regions in Fig. 4 induced at treelevel derive directly from the effective \(g_{aXY}\) value for each point of the parameter space.
 11.
After this paper was completed, Ref. [48] appeared which discusses the hadronic decays of axions in this region \(1\,\mathrm {GeV}<m_a<3\,\mathrm {GeV}\).
 12.
Note that heavyflavour tagging can allow to distinguish final states involving heavy quarks, but this separation will not be taken into account here.
 13.
The axion mass \(m_a\) is a combination of M and the instanton contribution related to the first term, as previously explained (see Eq. 3.3).
 14.
After this work was completed, Ref. [66] appeared which revisits the CHARM exclusion contour and provides projections of the expected NA62 and SHiP sensitivities.
 15.
For the reader interested in generic gluonic ALPs rather than heavy axions, we provide the exclusion plots without any superimposed lines as auxiliary files.
 16.
The \(m_u m_d/(m_u+m_d)^2\) factor in the QCD contribution is not shown for simplicity.
 17.
Note that this reparametrization invariance differs from that in Ref. [44].
 18.
The same Lagrangian can be obtained by making use of the reparametrization invariance in Eq. (C.2) and choosing \(\alpha _u=c_1^u\) and \(\alpha _d=c_1^d\).
Notes
Acknowledgements
We acknowledge J. Fernandez de Troconiz, R. Houtz, J. Jaeckel, J.M. No, R. del Rey, O. Sumensari and V. Sanz for very interesting conversations and comments. We also thank J. Jaeckel for his useful remarks on the manuscript. M.B.G and P. Q. acknowledge IPMU at Tokyo University, where the last part of this work was developed. G. A. thanks the IFT at the Universidad Autónoma de Madrid and the University of Washington for the hospitality during the first and last stages of this work, respectively. This project has received support from the European Union’s Horizon 2020 research and innovation programme under the Marie SklodowskaCurie Grant Agreements No. 690575 (RISE InvisiblesPlus) and No. 674896 (ITN ELUSIVES). M.B.G and P. Q. also acknowledge support from the the Spanish Research Agency (Agencia Estatal de Investigación) through the grant IFT Centro de Excelencia Severo Ochoa SEV20160597, as well as from the “Spanish Agencia Estatal de Investigación” (AEI) and the EU “Fondo Europeo de Desarrollo Regional” (FEDER) through the Project FPA201678645P. The work of G. A. was supported through an ESR contract of the H2020 ITN Elusives (H2020MSCAITN2015//674896ELUSIVES) and through a “La Caixa” predoctoral Grant of Fundación La Caixa. The work of P.Q. was supported through a “La CaixaSevero Ochoa” predoctoral grant of Fundación La Caixa.
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