Astrophobic Axions

We propose a class of axion models with generation dependent Peccei-Quinn charges for the known fermions that allow to suppress the axion couplings to nucleons and electrons. Astrophysical limits are thus relaxed, allowing for axion masses up to ${\cal O}(0.1)$ eV. The axion-photon coupling remains instead sizeable, so that next generation helioscopes will be able to probe this scenario. Astrophobia unavoidably implies flavor violating axion couplings, so that experimental limits on flavour-violating processes can provide complementary probes. The astrophobic axion can be a viable dark matter candidate in the heavy mass window, and can also account for anomalous energy loss in stars.

Introduction. One of the main mysteries of the standard model (SM) is the absence of CP violation in strong interactions. The most elegant solution is provided by the Peccei-Quinn (PQ) mechanism [1,2] which predicts the axion as a lowenergy remnant [3,4]. The axion is required to be extremely light and decoupled, and in a certain mass range it can provide a viable dark matter (DM) candidate. The Kim-Shifman-Vainshtein-Zakharov (KSVZ) [5,6] and Dine-Fischler-Srednicki-Zhitnitsky (DFSZ) [7,8] axion models are frequently used as benchmarks to assess experimental sensitivities and to derive astrophysical bounds. However, constraining axion properties solely on the basis of standard benchmarks can be too restrictive, and exploring alternative models whose properties can sizably deviate from those of KSVZ and DFSZ is highly desirable. While it is conceptually easy to build models with suppressed axion-electron couplings g ae [5,6,9] or axion-photon couplings g aγ [10][11][12], it is generally believed that a robust prediction of all axion models is an unsuppressed axion-nucleon coupling g aN . This is particularly important, because g aN is responsible for the often quoted bound on the axion mass m a 20 meV from the neutrino burst duration of the SN1987A [13,14]. In this Letter we argue that a strong suppression of g aN is instead possible in a class of DFSZ-like models with generation-dependent PQ charges. Additional strong bounds on m a are obtained if, as in standard DFSZ, g ae is unsuppressed, since this can affect white-dwarf (WD) cooling rates and red giants (RG) evolution in globular clusters [14]. However, a suppression of g ae can be also arranged in our scenario. Thus, nucleophobia allows to relax the SN bound and electrophobia allows to evade the WD/RG constraints, rendering viable masses up to m a ∼ 0.2 eV. We denote such an axion as astrophobic, although g aγ remains generically sizable, and could still affect the evolution of horizontal branch (HB) stars. Astrophobic axions are interesting in many respects: i) they render viable a parameter space region well beyond the standard DFSZ and KSVZ benchmarks, yet still within the reach of the planned IAXO helioscope [15]. ii) Nucleophobia necessarily implies flavor-violating (FV) axion couplings to the quarks, so that complementary searches can be carried out in flavor experiments. iii) Astrophobic axions can be non-standard DM in the heavy mass window [16][17][18] and iv) can provide an explanation for various hints of anomalous energy loss in stars [19,20].
Axion coupling to nucleons. Let us first recall why g aN cannot be suppressed in KSVZ and DFSZ models. The relevant terms for this discussion are: where N (E) are the QCD (QED) anomaly coefficients, f a = v a /(2N ) with v a = √ 2 φ the vacuum expectation value (VEV) of the PQ symmetry breaking singlet field,G a,µν = 1 2 µνρσ G a ρσ ,F µν = 1 2 µνρσ F ρσ and Q L,R = U L,R , D L,R are vectors containing the left-handed (LH) and right-handed (RH) quarks of the three generations (capital letters denote matrix quantities, q, u, d are used otherwise). In KSVZ the PQ charge matrices X Q L,R vanish, while in DFSZ they are non-zero but generation blind, hence the current in Eq. (1) does not dependent on the quark basis. It is convenient to remove the axion-gluon term via a field dependent chiral rotation of the first generation quarks q = u, d: q L,R → e ∓i a 2fa fq q L,R with f u + f d = 1. Defining z = m u /m d and choosing f u = 1/(1 + z) 2/3 avoids tree level axion-pion mixing. As a result of this rotation the coefficient of the QED term gets shifted as E/N → E/N − f γ (z) with f γ 1.92, while the axion coupling to the first generation quarks becomes The charge dependent part of the couplings is commonly denoted as C 0 q = (X q R − X q L )/(2N ), while the vector couplings vanish upon integration by part because of the equation of motion. Matching Eq. (2) with the non-relativistic axion-nucleon Lagrangian allows to extract the axion couplings to the nucleons N = p, n [21] which are defined in analogy to the couplings to the quarks by ∂ µ a/(2f a )C N N γ µ γ 5 N . It is convenient to recast the results in terms of the two linear combinations where the two numbers in parenthesis correspond to f u +f d = 1 (exact) and f u −f d 1/3 (approximate), while δ s is a correction appearing in DFSZ which is dominated by the s-quark sea contribution. In the models below, using the results from [21] and allowing for the largest possible values of C 0 s,c,b,t , we have |δ s | < ∼ 0.04. Eq. (3) makes clear why it is difficult to decouple the axion from the nucleons. For KSVZ C 0 u = C 0 d = 0 and the model independent contribution survives. For DFSZ we see from Eq. (2) that C 0 u + C 0 d = N l /N with N l the contribution to the QCD anomaly of the first generation (light) quarks. Hence, for generation blind charges C 0 u + C 0 d = 1/3 is an exact result.
The nucleophobic axion. We take as the defining condition for the nucleophobic axion the (approximate) vanishing of the relations in Eqs. (3), (4). Remarkably, since the axion-pion coupling is proportional to the isospin breaking combination C p − C n [22], nucleophobic axions are also pionphobic. We start by studying Eq. (3). In the approximation in which δ s is neglected, This can only be realized in two ways: (i) either the contributions of the two heavier generations cancel each other (N 2 = −N 3 and N l = N 1 ) or (ii) they vanish identically, in which case it is convenient to assign N l = N 3 and, hoping that no confusion will arise with the usual generation ordering, require for the anomalies of the heavier generations N 1 = N 2 = 0. 1 Clearly both cases require generation dependent PQ charges. A generic matrix of charges for a LH or RH quark q can be written as are proportional to the corresponding SU (3) matrices. In this Letter we are mainly interested in a proof of existence for nucleophobic axions, so we introduce some simplification: we assume just two Higgs doublets H 1,2 (with PQ charges X 1,2 and hypercharge Y = −1/2), and we consider only PQ charge assignments that do not forbid any of the SM Yukawa operators. Under these conditions, it can be shown that two generations must have the same PQ charges [24]. We can then drop the SU (2) breaking λ 3 term so that the matrix X Q = X 0 q I + X 8 q λ 8 respects a SU (2) symmetry acting on the generation indices {1, 2}, and we henceforth refer to such a structure as 2 + 1 . To study which Yukawa structures can enforce the condition N = N l it is then sufficient to consider just one of the generations in 2 together with the generation in 1 carrying index {3}. The relevant Yukawa operators read: whereH = iσ 2 H * , assigning H 1 to the first term is without loss of generality and, according to our assumptions, all the Higgs sub-indices must take values in {1, 2}. It is easy to verify that in each line the charges of the first three quark-bilinears determine the fourth one, e.g.
, while the third term in the second line is obtained by equating X q3 − X q2 as extracted from the second and third terms of both lines. It is now straightforward to classify all the possibilities that yield N l /N = 1. Denoting the Higgs ordering in the two lines of Eq. (5) with their indices ∈ {1, 2}, e.g.
It is easy to verify that in Let us now discuss how the second condition C p −C n ≈ 0 can be realized. We denote by tan β = v 2 /v 1 , the ratio of the H 1,2 VEVs, and we use henceforth the shorthand notation s β = sin β, c β = cos β. The ratio X 1 /X 2 = − tan 2 β is fixed by the requirement that the PQ Goldston boson is orthogonal to the Goldston eaten up by the Z-boson [8], and the charge normalization is given in terms of the light quark anomaly as X 2 − X 1 = ±2N l . Here and below the upper sign holds for (i 1,2 ) and (ii 1 ), and the lower sign for (ii 2 ). From Eq. (6) it follows that in all and (ii 1 ), and for s 2 β = 1/3 in (ii 2 ). We learn that even under some restrictive assumptions, there are four different ways to enforce nucleophobia. More possibilities would become viable by allowing for PQ charges that forbid some Yukawa operator [24]. Note that while C p − C n ≈ 0 requires a specific choice tan β ≈ √ 2, 1/ √ 2, C p +C n ≈ 0 is enforced just by charge assignments. For both values of tan β the top Yukawa coupling remains perturbative up to the Planck scale, however, we stress that these values should be understood as relative to the physical VEVs, rather than resulting from a tree level scalar potential. This is because the large v a would destabilize any lowest order result for v 1,2 . This is of course a naturalness issues common to all invisible axion models.
Finally, to render the axion invisible, H 1,2 need to be coupled via a non Hermitian operator to the scalar singlet φ with PQ charge X φ . This ensures that the PQ symmetry gets spontaneously broken at the scale v a v 1,2 suppressing efficiently all axion couplings. There are two possibilities: H † 2 H 1 φ in which case |X φ | = 2N l = 2N , the axion field has the same periodicity than the θ term and the number of domain walls (DW) is N DW = 1, or H † 2 H 1 φ 2 in which case |X φ | = N l = N and N DW = 2. In contrast, in DFSZ models |X φ | = 2N/3, (2N/6) yield N DW = 3, (6) and a DW problem is always present.
Flavor changing axion couplings. Generation dependent PQ charges imply FV axion couplings. Plugging X Q = X 0 q I + X 8 q λ 8 in Eq. (1) it is readily seen that a misalignment between the Yukawa and the PQ charge matrix becomes physical. Since we are mostly interested in the light quark couplings, we single out X q1 for case (i), and X q3 for (ii): with 3X 8 q = X q1 −X q3 , Λ = 1 3 (I −λ 8 ) = diag(0, 0, 1) and Λ = 1 3 (2I +λ 8 ) = diag(1, 1, 0). In case (i), the matrices of couplings in the Yukawa basis read: where for C 0V Q the equations of motion imply only the vanishing of the diagonal entries, but not of the off-diagonal ones, C 0 Q = C 0 q1 I with C 0 q1 defined below Eq. (2), and denoting by V Q the unitary rotations to the diagonal Yukawa basis, W Q = V † Q ΛV Q . While in the models discussed here W Q R and W Q L are never simultaneously present, this is possible in more general cases [24]. It is now convenient to single out the diagonal (denoted by δ) and off-diagonal (denoted by ω) entries in W Q = δ Q + ω Q : where the condition on δ q follows from Tr(W Q ) = 1, the one on ω Q from the vanishing of the principal minors for the rank one matrix W Q , and δ ij in the first relation is the usual Kronecker symbol. In (ii) the couplings are given by Eqs. (8)-(9) by replacing C 0 q1 → C 0 q3 , (−3) → (+3) and W Q → W Q = V † Q Λ V Q , while the two conditions read i δ qi = 2 and |(ω Q ) ij | 2 = (1 − δ qi )(1 − δ qj ). Information on the LH matrices can be obtained from the CKM matrix: Therefore, to a good approximation we can define a single set of LH parameters δ L = δ u L ≈ δ d L . In contrast, we have no information about the RH matrices so that in general W U R = W D R and δ u R , δ d R are two independent sets. Corrections to the diagonal axial couplings due to quark mixing are listed in Table I. Corrections to the second condition for nucleophobia can be always compensated by changing appropriately the value of tan β to maintain C p − C n ≈ 0. However, this is not so for the first condition, for which large mixing corrections would spoil C p + C n ≈ 0. Actually, only for (ii 1 ) a relatively small correction can improve nucleophobia, and this is because only in this case C 0 s , which determines the sign of δ s in Eq. (3), is negative (C 0 s = −s 2 β ), rendering possible a tuned cancellation −0.50 δ d 3R + 2|δ s | ≈ 0, while for all other cases the value of g aN is increased. Nucleophobia thus generically requires that the quark Yukawa and the PQ charge matrices are aligned to a good approximation (for recent attempts to connect axion physics to flavor dynamics see e.g. [25][26][27]).

Electrophobia.
Electrophobia can be implemented exactly (at the lowest loop order), or approximately (modulo lepton mixing corrections) by introducing an additional Higgs doublet uncharged under the PQ symmetry, and by coupling it respectively to all the leptons, or just to the electron. However, electrophobia can also be implemented without enlarging the Higgs sector at the cost of a fine tuned cancellation between C 0 e and a mixing correction. Of course this requires large lepton mixings and fine tuning. Given that large mixings do characterize the lepton sector, at least the first requirement is not unnatural. It is a bit tedious but straightforward to verify that in all the following cases a cancellation is possible: we can assign the electron: (i l ) to the doublet in 2 + 1, or (ii l ) to the singlet, and in both cases we can consider (12 . .) l or (21 . .) l structures, and then combine these possibilities with the four quark cases. Moreover, for (abab) l type of structures electrophobia is enforced by a cancellation from LH mixing, while for (abba) l from RH mixing. All in all, there are 2 × 2 × 4 × 2 = 32 physically different astrophobic models. However, as regards the axionphoton coupling, there are only four different values of E/N . We have listed them in Table II by picking out four representative models. Phenomenology of the heavy axion window.
• Structure-formation arguments also provide hot DM (HDM) limits on the axion mass: in benchmark models m a 0.8 eV [13,33,34]. However, nucleophobic axions are also pionphobic, and the main thermalization process ππ → πa is then suppressed, relaxing the HDM bound. This also implies that large-volume surveys like EUCLID [35] cannot probe astrophobic axions.
The main results for astrophobic axions are summarized in Fig. 1 and compared to the KSVZ and DFSZ benchmarks. The lines are broken at: • marks, which indicate the upper bounds on m a from SN1987A, and marks, corresponding to the combined SN/WD constraints for DFSZ models. As anticipated, for KSVZ and DFSZ, axion masses above m a ∼ 10 −2 eV are precluded by the SN/WD limits (dark brown bullet for KSVZ and green stars for DFSZ). For astrophobic axions the SN/WD bounds get significantly relaxed (they cannot evaporate completely because of the contribution δ s in Eq. (3) to g aN ). We obtain m a < 0.20 eV for M1/M2 (blue bullets), m a < 0.25 eV for M3 and m a < 0.12 eV for M4 (red bullets).
Searches with helioscopes: Helioscopes are sensitive to g aγ which is not particularly suppressed in astrophobic models. The solid black line in Fig. 1 shows the present limits from CAST [36], while the dotted black lines show the projected sensitivities • �� of next generation helioscopes. While the improvement in mass reach will be limited for TASTE [37] and BabyIAXO [38], we see that IAXO [15,39] and its upgrade IAXO+ [20] will be able to cover the whole interesting region up to m a ∼ 0.2 eV.
Flavor Violation Experiments: The strongest limits on FV axion couplings to quarks come from K + → π + a [40]. Comparing the model prediction with the current limit [41] gives where ω 2 l2 = |ω 12 | 2 = δ 1L δ 2L , δ d 1R δ d 2R for (i 1,2 ), ω 2 l2 = |ω 32 | 2 for (ii 1 ), while in (ii 2 ) the branching ratio vanishes (see Table I). For models (i 1,2 ), by taking CKM-like entries in V d L,R , ω 2 12 > ∼ 10 −8 and the limit would be saturated. This implies that NA62, which is expected to improve by a factor of ∼70 the limit on K + → π + a [42,43] can probe these models (the explorable "flavor window" corresponds to the vertical magenta line in Fig. 1). In contrast, in case (ii 1 ) for CKM-like mixings the limit Eq. (11) would be exceeded by six orders of magnitude, which renders this case rather unrealistic. If we allow for only two Higgs doublets, electrophobic models necessarily have FV axion couplings to leptons. The strongest limits come from searches for µ → eγa [44,45]. They yield f a /ω eµ > ∼ 2 · 10 9 GeV which implies m a 2.7 · 10 −3 /ω eµ eV. Recalling that ω eµ = δ e δ µ and that δ e ∼ O(1) is needed to cancel C 0 e , we need to impose δ µ 10 −4 to avoid overconstraining the large mass window. This implies δ τ ∼ O(1) from which we can predict B τ →ea 7 · 10 −6 ma 0.2 eV 2 , about three orders of magnitude below the present bound [46]. Axion DM in the heavy mass window: For m a ∼ 0.2 eV the misalignment mechanism cannot fulfill Ω a Ω DM . In post-inflationary scenarios, if N DW > 1 [16,17] an additional contribution from topological defects can concur to saturate Ω DM . This requires an explicit PQ breaking to trigger DW decays, and a fine tuning not to spoil the solution to the strong CP problem. For N DW = 1 a contribution to the relic abundance can come from axion production via a parametric resonance in the oscillations of the axion field radial mode [18], in which case f a < ∼ 10 18 GeV is needed. In both cases, the lower values of f a allowed by the astrophobic models can help to match the required conditions. Stellar cooling anomalies: Hints of anomalous energy loss in stars [19,20] can be more easily accommodated in astrophobic axion models. Ref. [20] finds, as the best-fit point for extra axion cooling, g aγ ∼ 0.14 · 10 −10 GeV −1 and g ae ∼ 1.5 · 10 −13 . While in the DFSZ model this point is in tension with the SN bound, it is comfortably within the allowed parameter space of the astrophobic axion.
Conclusions. We have discussed a class of DFSZlike axion models with generation dependent PQ charges that allows to relax the SN1987A bound on g aN and the WD/RG limit on g ae , and to extend the viable axion mass window up to m a ∼ 0.2 eV. This scenario is characterized by compelling connections with flavor physics. Complementary informations to direct axion searches can be provided by experimental searches for FV meson/lepton decays and, conversely, the discovery of this type of astrophobic axions would provide evidences that the quark Yukawa matrices are approximately diagonal in the interaction basis, conveying valuable information on the SM flavor structure. While we have restricted our analysis to PQ charge assignments which do not forbid any of the SM Yukawa operators, it would be interesting to relax this condition, and explore to which extent the PQ symmetry could play a role as a flavor symmetry in determining specific textures for the SM Yukawa matrices.