Challenges for a QCD Axion at the 10 MeV Scale

We report on an interesting realization of the QCD axion, with mass in the range $\mathcal{O}(10)$ MeV. It has previously been shown that although this scenario is stringently constrained from multiple sources, the model remains viable for a range of parameters that leads to an explanation of the Atomki experiment anomaly. In this article we study in more detail the additional constraints proceeding from recent low energy experiments and study the compatibility of the allowed parameter space with the one leading to consistency of the most recent measurements of the electron anomalous magnetic moment and the fine structure constant. We further provide an ultraviolet completion of this axion variant and show the conditions under which it may lead to the observed quark masses and CKM mixing angles, and remain consistent with experimental constraints on the extended scalar sector appearing in this Standard Model extension. In particular, the decay of the Standard Model-like Higgs boson into two light axions may be relevant and leads to a novel Higgs boson signature that may be searched for at the LHC in the near future.


I. INTRODUCTION
The Standard Model (SM) provides a consistent description of particle interactions, based on a local and renormalizable gauge theory [1]. The gauge interactions of the theory have been tested with great precision, while the Yukawa interactions of quarks and leptons with the recently discovered Higgs boson are being currently tested at the Large Hadron collider (LHC). In the SM, these Yukawa interactions lead to a rich flavor structure based on the diversity of quark and lepton masses and the CKM and PMNS matrices in the quark and lepton sector, respectively. These matrices include CP violating phases that are sufficient to explain all CP-violating phenomena observed in B and K meson, as well as neutrino experiments.
Although the gluon interactions with quarks preserve the CP symmetry, there is a CP violating interaction in the SM QCD sector that is related to the interactions induced by the θ QCD parameter. Even in the absence of a tree-level value, a non-vanishing value of θ QCD would be generated via the chiral anomaly, by the chiral phase redefinition associated with making the quark masses real. In general, hence, the physical value of θ QCD comes from the addition of the tree-level value and the phase of the determinant of the quark masses, and is naturally of order one. In the presence of an unsuppressed value of θ QCD , however, the neutron electric dipole moment (EDM) would acquire an unobserved sizable value [2]. The experiment measuring neutron EDM has found that such CP phase should be smaller than 10 −10 [3], which leads to a severe fine tuning problem in the SM [1].
One elegant way to solve the neutron EDM problem is to introduce a chiral symmetry, the so-called PQ symmetry, which is spontaneously broken, leading to a Goldstone boson, a, which has been called the axion [4][5][6][7]. The axion interactions lead to a field dependent θ QCD angle, and the strong interactions induce a non-trivial potential which is minimized at vanishing values of θ QCD , therefore leading to the disappearance of the CP interactions in the strong sector in the physical vacuum and the solution of the strong CP problem.
Being associated with a Goldstone boson, the axion field has an approximate shift symmetry, thus it can be naturally light. Its couplings to other particles may be rendered small due to the suppression from large decay constant, f a , proportional to the vacuum expectation value of the field that breaks the chiral PQ symmetry. Since the f a scale is not necessarily related to the weak or QCD scales, the axion mass and couplings to fermions and gauge bosons, which are proportional to 1/f a , can span many orders of magnitude. Thus, there is a rich phenomenology to explore this wide region of parameter space with different experimental setups [1, [8][9][10][11][12]. The original visible axion scenario [6,7], where f a is related to the weak scale, has been severely constrained by laboratory experiments [1, 8,9,12]. There are also invisible axion scenarios, with f a much larger than electroweak scale, where the axion remains hidden due to its ultralight mass and ultra small couplings. A realization of these invisible axions are given in the so-called KSVZ and DFSZ axion scenarios [13][14][15][16].
Regarding the visible axion, there are efforts [17][18][19] to save it from experimental constraints by making it pion-phobic. In this scenario, the mixing with the neutral pion is suppressed by choosing appropriate PQ charges and an accidental cancellation arising from the low energy quark mass and coupling parameters. This visible axion scenario has been combined with the recent experimental anomalies from Atomki collaboration [20,21]. The experiment looks for high excitation states of nuclei from 8 Be and 4 He, which de-excite into ground states and emit a pair of electron and positron. A peak was observed in the angular correlation of e − e + , which can be interpreted as an intermediate pseudo-scalar particle with mass around 17 MeV produced in the de-excitation and later decay into e − e + [18,19,[22][23][24].
This model may be tested with multi-lepton signatures when light mesons decay to multiple axion states [25]. It may also lead to significant dark matter electronic scattering crosssection in the dark matter direct detection [26]. Recently, it has been suggested that higher order SM effects may lead to an effect similar to the one observed by the Atomki experiment [27]. However, further theoretical and experimental analyses are needed to determine whether or not the observed peak may indeed be accounted by SM effects. In this work we shall assume that this is not the case and identify the associated pseudo-scalar particle with a 17 MeV QCD axion. We will, however, comment on how the constraints on the proposed QCD axion at the 10 MeV scale would be modified if its resonant signal were absent. In general, since m a m π f π /f a , an axion mass in the 10 MeV range implies f a 1 GeV. For this scenario to survive the current experimental constraints, the axion must couple only to first generation quarks and charged leptons with suitable couplings determined by their masses, their QCD charges and f a [18,19].
The recent accurate measurement of the fine structure constant [28] revealed that there is a mild discrepancy with the electron magnetic dipole moment measurements [29,30], compared to the SM theoretical calculation [31,32]. The discrepancy, ∆a e , associated with this measurement used to be negative and had a significance of about 2.4 σ. On the other hand, a very recent, more accurate measurement of the fine structure constant [33] suggests a smaller discrepancy for ∆a e and with positive sign. Since the axion has to decay to electronpositron pairs promptly in the Atomki experiment, it has a sizable coupling to electron. As a pseudo-scalar, it can naturally give a negative contribution to electron (g − 2) e at the oneloop level [34]. Although positive correction would be therefore in tension with the one-loop corrections induced by the 17 MeV axion particle, there are relevant two loop corrections that, depending on the axion-η meson mixing, may lead to a positive value of ∆a e .
In addition to experimental constraints from precision measurement of mesons, this scenario also is constrained by low energy electron collider and beam dump experiments. Therefore, it is interesting to ask if this 17 MeV visible axion can evade all current experimental constraints and also be consistent with the current measurements of (g − 2) e and the fine structure constant. It is also important to find a field theoretical realization of this axion.
The most natural realizations are associated with new physics at the weak scale and are therefore subject to further constraints beyond those associated with the effective low en-ergy axion model. In this article, we will present an extension of the SM which leads to a 10 MeV axion and study the associated constraints on this model. This article is organized as follows. In section II, we review the properties of this visible axion model and how the most relevant experimental constraints may be avoided under the pion-phobic limit. In section III, we consider the electron and photon couplings to the axion and show the region of parameters which can lead to a consistency between the measured values of (g − 2) e and α, what leads to a relation between the axion electron and photon couplngs. In section IV, we further show how it can survive the constraints coming from low energy electron colliders and beam dump experiments, while at the same time lead to an explanation of the Atomki experimntal constraints. Furthermore, in section V, we construct a concrete UV model and show how this scenario can couple to only the first generation fermions while successfully generating the CKM matrix. We also discuss the additional constraints implied by the new physics associated with this scenario. We reserve section VI for our conclusions.

II. EFFECTIVE LOW ENERGY MODEL
We consider the effective theory for a variant of the QCD axion which couples mostly to the first generation fermions in the SM. While the low-energy structure of this setup has appeared some time ago [17,35], it has recently been discussed how this scenario remains robust to a variety of constraints [18,19]. From the ultraviolet (UV) perspective, for the axion to couple only to u and d quarks in the infrared (IR), the PQ breaking mechanism must allow for the generation of the appropriate quark and lepton masses as well as the CKM matrix. We discuss how this is achieved in a particular UV model later in section V.
From the IR perspective, the relevant low-energy interactions are simply written as where Q f denotes the PQ charge of the fermion f , and f a is the axion decay constant, which, as emphasized in the introduction must be f a 1 GeV. For the following discussion, it will be convenient to work with the linear axion couplings to fermions. For instance, the coupling to first generation quark and leptons in this basis is simply given by (2) At energies below the QCD scale, the couplings of the axion to u and d quarks will induce mixing of the axion in particular to the pion as well as other pseudo-scalar mesons. In this regime, the physical axion state and meson mixing angles can be determined by standard chiral perturbation theory (χPT) techniques. For detailed explanations and results of this procedure we refer to Refs. [18,19]. The same mixing angles can be used to parameterize the effective operator controlling the axion coupling to photons where θ aπ is the mixing angle between the axion and neutral pion, and θ aη ud,s are the mixing angles with the eta-mesons in the light-heavy basis. The relationships between these parameters and other bases for the meson octect in leading order of χPT are given also in Ref. [18]. For the purpose of this work, it is important to note that for Q u , Q d O(1), the natural ranges for θ aη ud and θ aηs are in the range of O(10 −3 − 10 −2 ) [18].
Searches for axions in the decay products of pions presents the most stringent constraints for the setup we consider. In particular, the bound on the charged pion decay BR(π + → e + ν e (a → e − e + )) < 0.5 × 10 −10 is quite severe [36]. For m a = O(10) MeV, this translates to a bound on the mixing angle as As we will see the range of couplings needed for (g − 2) e will require the axion width into photons to be subdominant with respect to the one into electrons. Hence, if decays only to SM particles are considered, one expects BR(a → e + e − ) 1. Thus, this variant of the axion is pion-phobic [18,19], in the sense that the mixing of the axion with the pion is below its natural values. Let us emphasize that, in leading order of perturbation theory and where we have chosen an arbitrary normalization of f a with respect to the PQ charges. Since the ratio m u /m d 0.47 ± 0.06 [1], a cancellation of θ aπ 10 −4 demands Q u 2Q d . Even assuming this relation between the quark PQ charges, the value of the mixing θ aπ becomes naturally of the order of a few times 10 −3 and hence an accidental cancellation between the leading order contribution associated with the precise value of the up and down quark masses and the higher order contributions must occur for this scenario to be viable.
One could in principle alleviate this aspect of the model assuming that the axion width is dominated by invisible decays, allowing the reduction of the axion decay branching ratio into electrons. This possibility, however, is strongly constrained by charged Kaon decays constraints, in particular the decay K + → (π + + Missing Energy). Indeed, considering a neutral pion state with a relevant mixing with an axion with dominant invisible decays leads to a bound on the axion-pion mixing θ aπ 10 −4 , and therefore this scenario does not lead to a relaxation of the pion axion mixing bound.
One interesting aspect of a mixing angle θ aπ below but close to 10 −4 is that it leads to a possible resolution of the so-called KTeV anomaly, related to a decay branching ratio of the neutral pion into pairs of electrons larger than the one expected in the SM [18,19]. A pseudo-scalar particle with a mass m a 17 MeV and a coupling to electrons of order m e /f a 5 × 10 −4 for f a 1 GeV contributes in a relevant way to the electron anomalous magnetic moment a e = (g − 2) e /2. The measured deviation of the electron magnetic dipole moment [29,30] from the SM prediction [31,32] extracted from the fine structure constant measured in recent cesium recoil experiments [28] is However, more recently the deviation inferred from a new, independent measurement of the fine-structure constant has been reported as [33] ∆a e = (48 ± 30) × 10 −14 .
In the present case, the largest contribution of the model to ∆a e comes from the tree-level coupling of the axion to electrons, induced by the one-loop diagram in Fig. 1 ∆a 1-loop At the 2-loop level, however, the magnetic moment receives an additional contribution from a Barr-Zee type diagram proportional to the product g e a g γγ a , where the effective coupling to photons is defined by Eq. (3). This contribution is given by where and we have taken an effective cutoff at the scale 4πf π as in [18]. Note that while we have chosen to work in the linear basis of the axion couplings to fermions, a similar calculation for (g − 2) µ has been presented in [37]. There, one trades the linear coupling of the axion to fermions with a derivative coupling and a di-photon coupling. We have checked numerically that the corresponding result for (g − 2) e in the non-linear basis agrees with our results in the linear basis. In fact, using the appendices in [37], we have verified that the chiral lagrangian obtained from the linear couplings matches that obtained in the non-linear basis due to the specific choice of interactions in our model. and red shaded regions are excluded by KLOE and NA64 respectively [34,38]. The cyan region is consistent at the 95% C.L. with the deviation ∆a e obtained from Eq (7). The blue region shows the 95% C.L. region consistent with the value inferred from the recent measurement of the fine structure constant, reported in [33].
It is clear that when g e a and g γγ a have the same sign, the 2-loop contribution can partially cancel the 1-loop result. Thus, both couplings will have a relevant impact to the prediction of ∆a e . In Fig. 2 Relevant constraints on the values of the electron coupling are extracted from the decays of the axion to electron-positron pairs. As these constraints are placed on the couplings to electrons, the constraints will remain the same regardless of which measurement of ∆a e is assumed. Thus, in Fig. 2 we show an upper limit on g e from the KLOE experiment in the purple shaded region [34]. A lower limit can be derived from various beam dump experiments. We show the strongest available bound, from NA64, in the red shaded region [38]. It is important to note that both bounds have been derived assuming BR(a → e + e − ) = 1. We see that without the 2-loop contribution (or g γγ a → 0) the values of ∆a e would be negative and therefore the new determination of the fine structure constant demands sizable two loop corrections for the axion contributions to be consistent with ∆a e . Upper bounds on the couplings of the axion to vector bosons from LEP and LHC have also been explored in various realizations of axions and ALPS [39][40][41][42][43]. However, those bounds are based on the assumption of an axion which mostly decay into photons, contrary to our model in which the dominant axion decay is into electron pairs. Although searches for an axion decaying to electron pairs are difficult at LEP due to large backgrounds, this may be feasible at future runs of the LHC [44].
Let us stress that the values of the axion coupling to electron leads to a constraint on the ratio of the PQ charges of the first generation quark and leptons. For instance, since in this scenario, Q u ∼ 2Q d , from the expression of the axion mass one obtains that Therefore, the coupling of the axion to the electrons, Eq. (2), is given by Taking into account the range of values consistent with (g − 2) e , which from Fig. 2 are between 1.4 × 10 −4 and 1 × 10 −3 , one obtains In particular, if the above ratio of PQ charges takes the values 1/3, 1/2 or 1, the axion is free of experimental constraints from the KLOE and NA64 experiments. This range of values of the ratio of PQ charges will play an important role in the UV model described in section V.

IV. ATOMKI ANOMALY
Recently, it has been shown that the variant of the QCD axion analyzed in this work, with couplings only to the first generation of quarks and leptons, can lead to an explanation of the anomalies reported by the Atomki experiment [19]. The relevant axion-nuclear transition rates were also considered much earlier in [45][46][47][48] with further studies aimed at generic pseudo-scalar couplings to nucleons more recently in [22,49]. In terms of the mixing angles of the axion to mesons, the predicted axion emission rate for 8 Be * (18.15) nuclear transitions can be parameterized as where ∆q parametrizes the proton spin contribution from a quark, q, while for 4 He * (21.01) they obtain We note that since the axion-pion mixing must be small to remain acceptable from pion decay observations, the mixing angles θ aη ud and θ aηs , which are determined by the Atomki anomaly, effectively fix the coupling of the axion to photons. We see in Fig. 2 that for a given value of g γγ a , the combined one-and two-loop contributions to (g − 2) e sets a value for g e which leads to the correct prediction for ∆a e . Thus, combining the regions of mixing angles which favor the Atomki anomalies with the requirement to satisfy (g − 2) e parametrically connects the PQ charges of the u and d quarks to that of the electron, giving complementary information on the viable region of the scenario.
If SM effects are confirmed to be dominant feature of the Atomki anomaly, as in [27], this could imply further constraints on the mixing angles θ aη ud and θ aηs or more specific relations between them. For instance, in order to suppress the resonant signal, one possibility would be that those mixing angles are much smaller than their natural values, leading to a small photon axion coupling. Beyond the fact that such small mixing values are difficult to realize, a small photon axion coupling would be inconsistent with the values of ∆a e associated with the new fine structure constant determination, Eq. (8), but would remain consistent with the old one, Eq. (7). Alternatively, there could be an approximate cancellation between the θ aη ud and θ aηs contributions to Eqs. (15) and (16), what would imply a mixing θ aηs about an order of magnitude larger than θ aη ud .
In Fig. 3 we show the regions of parameters given in [19] favored by the nuclear transition anomalies. We include the correlation between the predicted value of g γγ a in this plane with the values of g e which are consistent with ∆a e , Eq. (8), at the 95% CL. Thus, the red and purple shaded regions are excluded by the NA64 and KLOE experiments, respectively, as shown in Fig. 2. The light gray region is excluded by bounds on the K + → π + (a → e + e − ) branching ratio in the non octet-enhanced regime, see [19]. The regions that would be excluded in the octet enhanced regime, which are therefore not firm bounds, are indicated by gray lines. In the left panel, we show the predictions and bounds for the case when sgn(θ aη ud ) = sgn(θ aηs ), while in the right panel we show the corresponding results when It would be interesting to compare the allowed values of the axion-η mixing parameters with the ones predicted in chiral perturbation theory. However, as emphasized in Ref. [18], this is not a simple task, since these mixing parameters receive relevant contributions at nextto-leading order. The only firm prediction is that these mixing parameters are naturally between 10 −3 and a few 10 −2 , and hence, they are consistent with most of the allowed parameter space shown in Fig. 3. Between the yellow regions the contributions from θ aη ud and θ aηs cancel in Eqs. (15) and (16)  The light and dark blue bands in Fig. 3 also show the regions of parameter space consistent with a ratio of PQ charges Q e /Q d = 1/3 and 1/2, respectively. We see that for opposite signs of the mixing angles, as represented in the right panel of Fig. 3, a ratio Q e /Q d = 1/2 leads to a mixing angle θ aηs that is much larger than its natural values.
The corresponding parameter space consistent with the determination of ∆a e from Eq. (7) is, however, quite different compared to that of Eq. (8). In Fig. 2 we see that, except for 1 When θ aη ud > 0 and θ aηs < 0 we find that there is no mutual parameter space for the (g − 2) e and Atomki anomalies.
values of ∆a e close to or larger than the central value of the measurement, there is no lower bound on g aγγ inferred from NA64. Hence, this constraint on the meson-mixing angles vanishes for couplings consistent with the region [∆a e − 2σ, ∆a e ]. In Fig. 4, we show the corresponding parameter space for the (g − 2) e and Atomki anomalies assuming the correlation of electron and photon couplings consistent with ∆a e , from Eq. 7, at the 95% CL. All color shading follows that of Fig. 3. In this case, natural regions of θ aη ud and θ aηs accounting for both anomalies are completely open. In fact, scenarios motivated by PQ charges Q e /Q d = 1/3 cover almost this whole region. However, this is not the case for scenarios motivated by Q e /Q d = 1/2 which does not differ significantly considering either determination of the electron magnetic moment.

A. The generation of CKM matrix and the couplings to first generation fermions
In this section, we present a possible UV completion of the effective model presented in Eq. (1), where the axion couples exclusively to the first generation fermions in the SM.
Generating the effective interactions to the up-and down-quarks is non-trivial as the PQ breaking mechanism must be carefully intertwined with electroweak symmetry breaking in a way that reproduces the correct flavor structure in the CKM matrix. To this end, we consider an extension of the SM by three additional Higgs doublets and three singlets Higgs fields. The PQ symmetry is realized by assigning charges to the additional Higgs bosons as well as the right-handed SM fermions u R , d R , and e R . The particle content and their PQ charges of the model is summarized in Table I.

The relevant Yukawa interactions allowed by the PQ symmetry of the Higgs doublets,
H f , and those of the SM Higgs, H, are given by where L Yuk PQ gives the couplings of H u,d,e to the first generation fermions, while L Yuk SM gives the couplings of the second and third generation fermions to the SM Higgs. Below the scales of  PQ and electroweak symmetry breaking the up-quark mass matrix is then given by where v and v u are the vacuum expectation values of H and H u respectively. The V uL and V uR matrices given the rotation between the flavor-and mass-eigenstate basis (denoted with superscript m), where u L/R = (u L/R , c L/R , t L/R ) T . For simplicity, we work in a basis where V u R is the identity matrix. The diagonalization of the down-type quarks follows similarly.
After diagonalization of both up-and down-type quark sectors, the CKM matrix is given by Since H u is charged under the PQ symmetry while H is not, we expect the axion field to be contained in H u but not H. Therefore, to connect to the effective model, this requires that H u couples only to the first generation up-quark in the mass eigenstate basis, which is equivalent to T , the unitary matrix V u L can be decomposed in terms of the normalized vector (Y j1 ) , and two perpendicular normalized vectors (Y j1 ) ⊥1 and (Y j1 ) ⊥2 where T represents the transposed column vector, and x is an auxiliary rotation angle which does not affect the equality Eq. (23). Similarly for V d L we obtain We choose the auxiliary angles explicitly as and for the Yukawas we take where θ 12 , θ 13 , θ 23 angles are the CKM mixing angles in the SM. Then, the explicit forms of the vectors Y j1 u and Y j1 d are given by, The rest of the Yukawa matrix Y u and Y d can be reconstructed according to Eq. (19). This completes the realization of the CKM matrix in the UV model. We omit details of the procedure of the mixing in the lepton sector. However, this may be accomplished in an analogous way.
Besides the Yukawa interactions, the scalar potential needs to be studied to single out the physical axion field in the model as well as its mixing with other pseudo-scalars. The off-diagonal scalar potential, V PQ , and diagonal scalar potential, V dia , allowed by symmetries where Φ = H, H u , H d , H e , φ u , φ d , φ e . We assume that all the coefficients are real. In Eq. (29), the first three terms generate the couplings of axion to first generation fermions through mixing with H u,d,e . The last two terms reflect the PQ charge assignments of the scalars φ u , φ d and φ e . V dia provides all the diagonal terms which do not provide mixing, but provide masses to CP-even scalars. We omit other operators found by other combinations of the scalar fields allowed by the imposed symmetries. The above potential will be sufficient for a complete description of the physical axion and the interactions relevant for our discussion.
We require that the potential has a minimum, which determines the values of the µ Φ 's.
After applying this condition, we solve the mixing between the seven pseudo-scalars, Φ I ≡ where f = u, d, e, Q f is the associated PQ charge, and we define (−1) f ≡ −1 for f = u and +1 for f = d, e. We have expressed G PQ at leading order in Note that G PQ gives the mixing between the physical axion states among the seven pseudo-scalars, e.g. Φ I ⊃ G PQ a. For example, if we further assume v φ f v f , it is clear that the axion is dominantly composed of φ I f . In the doublet Higgs H f , the axion is contained with the suppressed factor v f /v φ f , while for the SM Higgs H, it is contained with a double After rotating to the physical basis, the couplings of axion to SM fermions are given by where and the second term gives the axion couplings to the 2nd and 3rd generation fermions, generated by mixing with the SM Higgs. Note that this generates an effective PQ charge to heavy fermions, namely The couplings of the QCD axion to the 2nd and 3rd generation fermions are constrained from the exotic decays of heavy mesons. The relevant constraints are J/Ψ → γa leading to |Q PQ c | 0.25(f a /GeV) and Υ → γa leading to |Q PQ b | 0.8 × 10 −2 (f a /GeV) [18]. If we take the hierarchy v f ≈ 100 MeV v and f a ≈ 1 GeV, we have Q PQ F, eff ≈ 5 × 10 −7 which is quite safe from the above constraints.

B. Additional Massive Scalar and Pseudo-scalar States
In addition to the two Goldstone bosons, there are five massive pseudo-scalars. For n = 3, e.g. Q PQ e = 1/3, one can only decouple four heavy massive pseudo-scalars leaving one light pseudo-scalar. More specifically, we assume the following parameters where the last term determines the axion decay constant f a ≈ O(1)GeV. In this case, B φ is a dimensionless parameter. There are then three massive pseudo-scalars, composed mostly of linear combinations of the h I f states, with masses given by where f = u, d, e. Additionally, there is one mass of the form which dominantly comes from linear combination of φ I u,d states and may have a mass close to 100 GeV, and one pseudo-scalar of mass The light pseudo-scalar with mass B φ v 2 φ ∼ GeV 2 , which we denote as φ I light , mainly comes from φ I e . The lightness of this state can be understood in the limit B φ = 0. In this case, there is a global U (1) e symmetry and associated charged scalars φ e and H e . As a result, there is one additional exact goldstone boson. With B φ = 0, the U (1) e is explicitly broken in the scalar potential. Therefore, φ I light becomes a pseudo-goldstone boson with mass proportional to B φ v 2 φ f and not related to any other dimensionful trilinear couplings. The associated eigenvector can be easily represented with the simplficiation A φ = A u = 0. In this case, we where we have omitted the normalization factor. Thus, φ I light is dominantly composed of φ I e since v f v φ f . Moreover, it has a mixing of v f /v φ f from H I e and an even smaller mixing from H I d . Therefore, this state mainly couples to the electron with a coupling of ∼ m e /v φ ∼ 5 × 10 −4 . This coupling can be constrained using the Dark photon search at BaBar, e + e − → γA [52]. Refs. [53,54] have converted this constraint to scalars, which requires a coupling smaller than ∼ 5 × 10 −4 for scalar masses 1-10 GeV, and therefore this light pseudo-scalar coupling would be close to the current experimental bounds. The corresponding 1-loop contribution to ∆a e is −1.1 × 10 −14 (taking m φ I light = 1 GeV and coupling 5 × 10 −4 ) making its contribution to ∆a e much smaller than the uncertainty of the experiment.
For n = 2, B φ is dimensionful. In this case, the mass of φ I light is a linear combination of A φ v φ and B φ v φ . One can choose A φ and B φ large enough such that φ I light avoids existing constraints. Moreover, one can also add the term A 1 φ e H † d H e , which can also further raise its mass. As a result, we conclude that for n = 3 the light pseudo-scalar is still viable but is subject to strong constraints, while for n = 2, the five massive pseudo-scalars can be heavy and completely decoupled from the low energy phenomenology.
Having dealt with the pseudo-scalar sector, we briefly discuss the seven massive CPeven scalars. For n = 3, there are seven massive CP-even scalars. Under the assumption find that there is one SM-like Higgs with mass squared 2λ H v 2 . In this limit, there are three CP-even masses squared given by where f = u, d, e, and are all on the order of ∼ (500 GeV) 2 . These states are mainly composed of the neutral CP-even components of doublet Higgs H f . There are two massive scalars with masses squared ∼ A φ v φ f (100 GeV) 2 , which are mostly composed of the CP-even components of singlet Higgs bosons φ f . 2 Similar to the pseudo-scalar sector, there is still one light scalar with mass squared ∼ O(v φ f ) 2 , whose eigenvector is given by Just as φ I light , φ R light couples dominantly to the electron with coupling ∼ m e /v φe and suffers from similar constraints. For n = 2, the light scalar masses become A φ v φ or B φ v φ . Thus, in this case the masses for CP-even scalars are also heavy and can be decoupled from low energy phenomenology.
Finally, we come to the phenomenology related to the SM Higgs decay. Working again where f = u, d, e, and (−1) f ≡ −1 for f = u and +1 for f = d, e, as defined above. We also assume v f v φ f . The SM Higgs decay to aa has a width of which is far smaller than SM Higgs total width of 4 MeV (leading to 0.6% exotic Higgs branching ratio). Assuming the boost factor for the axion is m h /(2m a ) and width of a → e + e − as Γ(a → e + e − ) ≈ 1 8π we obtain the lifetime of axion in the lab frame γcτ a ≈ 1.65 cm for Q e = 1/2 and m a = 17 MeV.
Since the axion a is pretty light, the positron and electron pair (we will refer to them as electrons collectively) in the a → e + e − decay will be highly collimated with a separation of θ ∼ m a /E a ≈ 3 × 10 −4 radian at the decay vertex. With the 3.8 Tesla magnetic field in the tracker, the two tracks will separate with the curvature given by R = |p T /(q × B)| = (p T /GeV) × 0.88 meter. The separation angle between two tracks at the EM calorimeter cells will be θ sep ≈ 2 arcsin(r/(2R)), where r is the radius for the EM calorimeter cells located in the transverse plane. Assuming the electrons have p T ∼ 10 GeV, this angle will be as large as 0.11 radian and the two tracks can in principle be recognized as electron and positron with the correct charge assignment. However, the separation is still smaller than the cone size of the typical EM cluster ∆R = 0.2 [42]. Therefore, it is likely to be constructed as an EM cluster. The displaced vertex of a can be reconstructed, which is displaced to the primary vertex by O(cm). This is similar to the signature of a converted photon in SM events, where the photon passing through the material in the detector has been converted into an electron-positron pair. In that case, one can sum the momentum of e + and e − and obtain the momentum for the converted photon. Extrapolating the direction of the momentum from the vertex, it will go back to the primary vertex 3 . Therefore, the converted photon has zero impact parameter. In the case of the axion, since it will be highly boosted, we can assume the electrons in its decay are parallel. The electron tracks can have non-zero impact parameters, where d a is the distance between the displaced vertex (where a decay) and primary vertex in the transverse plane. Since d a ∼ O(cm), we see d 0 ∼ 10 −3 cm which is too close to measure. Therefore, a highly boosted a decaying into e + e − indeed looks similar to the converted photon in SM event. But there is one critical difference in that the conversion vertex for the signal depends only on the lifetime of a, while in SM events it has to be in the detector material.
Refs. [56][57][58] have considered the case of a light and highly boosted A , which decays into electron pairs, A → e + e − . Such A can be reconstructed as converted photon, as we previously discussed for light a → e + e − decay. Ref. [58] further pointed out that whether it is reconstructed as a converted photon highly depends on the A decay location. If it decays before the first layer of the pixel tracker (about 3.4 cm away from the central axis of the detector in the barrel region), the electrons (including positrons) will leave hits in the first layer of pixel detector. In this case, since the conversion occurs outside the material the event cannot be reconstructed as a converted photon. In the flowchart for reconstruction of electrons or photons [59], such event is also classified as "ambiguous". Thus, as the electrons cannot pass the isolation criteria, this will be identified as a lepton-jet signature. If A decays after the first layer of the pixel detector but before the 3rd-to-the -last layer of silicon-strip detectors (SCT), it can be identified as a converted photon.
In our scenario, the lifetime of a → e + e − in the lab is about ∼ 1.65 cm for the benchmark.
Therefore, it has the probability 1 − e −3.4cm/(1.65cm) ≈ 87% to decay before the first layer of the pixel detector. As a result, it will dominantly look like a lepton-jet event. There is a ∼ 13% probability to decay after the first layer of the pixel detector. From Ref. [59], the two Si tracks (SCT tracks) have an efficiency of 0.35 to be reconstructed as converted photon for E true T ∼ 20 GeV. Therefore, with two a both reconstructed as converted photons, the probability is as small as 0.002. As a result, we conclude that the signal h → aa with a → e + e − will be negligible in the contribution to the h → γγ branching ratio. It also does not fit to the h → e + e − searches because the electrons in the signal is not isolated.
The lepton-jet signature has been generally discussed and also specifically for two leptonjets from the SM Higgs decay in Refs. [60][61][62][63]. The ATLAS and CMS collaboration have searched for lepton jets from a light boson [64][65][66][67][68], where the lepton jets come from the decay topology h → χ 2 χ 2 → χ 1 χ 1 A A , where A is the dark photon, and χ 2 is an excited fermion DM, which decay promtly to χ 1 , and χ 1 is the DM, recognized as missing energy at the collider. The A decay is displaced but Refs. [

VI. CONCLUSIONS
The strong CP problem is one of the most intriguing aspects of the Standard Model of particle physics. It can be naturally solved by the introduction of an axion field, implying the presence of a new pseudo-scalar particle in the spectrum. Generically, in order to avoid experimental constraints, the axion decay constant is assumed to be much larger than the weak scale, implying a very light axion particle. In this article, we have investigated the suggestion that in spite of these constraints, the axion may be as heavy as 17 MeV, opening the possibility that this axion particle may lead to an explanation of the di-lepton resonance