Axion mass prediction from minimal grand unification

We propose a minimal realization of the Peccei Quinn mechanism in a realistic SU(5) model, where the axion mass is directly connected to the grand-unification scale. By taking into account constraints from proton decay, collider searches and gauge coupling unification, we predict the axion mass: $m_a \in [4.8, 6.6]$ neV. The upper bound can be relaxed up to $m_a<330$ neV, at the cost of tuning the flavour structure of the proton decay operators. The predicted mass window will be complementarily probed by the axion dark matter experiments ABRACADABRA and CASPER-Electric, which could provide an indirect evidence for the scale of grand unification before the observation of proton decay.

Introduction. It is a widespread belief that the standard model (SM) of particle physics should break down at some intermediate energy between the electroweak and the Planck scale. The quantum numbers of the SM fermions, together with the apparent convergence of the SM gauge couplings at high energies, hint to a unified gauge dynamics around 10 15 GeV. This scale is generically compatible with indirect constraints from the nonobservation of proton decay, the smoking-gun signature of Grand Unified Theories (GUTs). The search for proton decay was vigorously pushed in the past decades, and has slowly reached its limits with the Super-Kamiokande (SK) observatory [1]. Planned large-volume facilities, such as Hyper-Kamiokande (HK) [2], will improve the bound on the proton lifetime by one order of magnitude in the next decade. Though fundamentally important, that translates only into a factor of two on the GUT scale.
Another well-motivated framework which points to energies in between the electroweak and the Planck scale is associated with the Peccei-Quinn (PQ) solution of the strong CP problem [3,4], which predicts the axion as a low-energy remnant [5,6]. The axion needs to be extremely light and decoupled, and in a certain mass range it is also a viable dark matter (DM) candidate [7][8][9]. The experimental program for axion searches is rapidly evolving, with many novel detection techniques and new experiments being proposed recently [10]. It is reasonable to expect that a large portion of the parameter space predicted by the QCD axion will be probed in the next decade. From an experimental point of * luca.di-luzio@durham.ac.uk † andreas.ringwald@desy.de ‡ carlos.tamarit@tum.de view, however, one of the main bottlenecks of axion DM searches (e.g. those exploiting microwave cavities or nuclear magnetic resonance techniques) is the need to perform a fine scan in the axion mass in order to meet a resonance condition. Since the axion mass is not predicted by the PQ mechanism, any extra theoretical information which could pin-down precisely the axion mass would be extremely helpful for experiments. Following recent attempts to revive PQ-GUTs in SO(10) [11] (see also [12][13][14][15][16][17][18]), in this Letter we revisit the more minimal option of SU (5). The simplest implementation of the axion in nonsupersymmetric 1 SU(5) was proposed long ago by Wise, Georgi and Glashow (WGG) [22]. However, similarly to the original SU(5) model of Georgi and Glashow (GG) [23], the WGG model is ruled out in its minimal formulation because of gauge coupling unification and neutrino masses. An elegant and minimal way to fix both these issues in the GG model was put forth some years ago by Bajc and Senjanović [24], which add to the minimal GG field content a single Majorana fermion representation, 24 F , transforming in the adjoint of SU (5). The extra degrees of freedom have the right quantum numbers to generate neutrino masses via a hybrid Type-I+III seesaw mechanism and ensure a proper unification pattern. In particular, the main observable emerging from detailed renormalization group analyses of 1 The reader might wonder why we care for the fine-tuning of θ QCD 10 −10 and not for the electroweak-GUT hierarchy. A possible answer is that the strong CP problem is qualitatively different from the hierarchy problem, and it is conceivable that the solution of the latter does not rely on a stabilizing symmetry (an interesting example is the possibility that a light Higgs might be selected by the cosmological evolution of the universe [19][20][21] [24][25][26]) is a clean correlation between light electroweak triplet states (constrained by the Large Hadron Collider (LHC)) and the unification scale (constrained by SK).
Having in mind the possibility of narrowing the axion mass range within a minimal and realistic extension of the WGG model, we extend the latter with a 24 F in analogy to the GG+24 F case. This is actually welcome also from the point of view of the GG+24 F model, which lacks a DM candidate. Within the WGG model (or any realistic extension of it) the axion mass can be put in one-to-one correspondence with the proton decay rate, regardless of the fine details of gauge coupling unification. This allows us to extract a generic upper bound on the axion mass. Including also the detailed information from gauge coupling unification available in the WGG+24 F model, we are also able to set a lower bound on the axion mass from the non-observation of electroweak-triplet states at LHC, thus predicting the following axion mass window: m a ∈ [4.8, 6.6] neV, where the upper bound holds in the absence of tuning of fermionic mixing. Next, we provide the axion coupling to the SM fields and estimate the sensitivity of future axion DM experiments such as ABRACADABRA [27] and CASPEr [28,29] in the relevant mass window.
The WGG model. Let us recall the main features of the WGG model [22]. While the fermion content is that of the original GG SU(5) [23], namely three copies of 5 F and 10 F comprising the chiral SM matter fields, the scalar sector is extended to include a complex 24 H and two fundamentals, 5 H and 5 H . The WGG Lagrangian can be written as L WGG = L kin + L Y − V H , where L kin encodes the (gauge) kinetic terms, the Yukawa Lagrangian is schematically 2 while the scalar potential (which we do not report here entirely) contains two non-trivial invariants which are affected by global re-phasings: Note that the structure of the WGG Lagrangian resembles that of the DFSZ model [30,31]. In fact We have performed the minimization of the full scalar potential in [22] and computed in turn the particle spectrum. In particular, it can be shown that the vacuum expectation value (VEV) configuration breaks SU(5)×U(1) PQ down to the SM gauge group with a single order parameter V . 3 The axion, the (pseudo) Nambu-Goldstone boson of the global U(1) PQ , is dominantly contained in the phase along the SM singlet direction of 24 H , i.e.
A crucial point of the WGG model is that the mass of the heavy vector leptoquark V µ = (3, 2, −5/6) mediating proton decay, (where g 5 denotes the SU(5) gauge coupling) is directly connected to the axion decay constant 4 whereN is the U(1) PQ -SU(3) C -SU(3) C anomaly coefficient, e.g.N = 6 in the WGG model. This implies a generic relation between the axion mass and the proton decay rate. By means of chiral effective field theory techniques, we can recast the master formula for the proton decay mode p → π 0 e + in SU(5) as [33,34]: where we have set unknown fermion mixing rotations to a unit matrix (see [34] for complete expressions). A L = 1.25 encodes the renormalization from the electroweak scale to the proton mass,  [35,36]. Compact expressions for the latter can be found e.g. in Ref. [37]. For instance, running within the SM from 10 15 GeV to the electroweak scale yields A SL = 2.4 and A SR = 2.2.
By using Eqs. (5)-(6) and the relation m a = 5.7 neV (10 15 GeV/f a ) [38,39] we can re-express Eq. (7) in the following parametric form: where we have highlighted in the first parenthesis the current proton decay bound from SK [1]. Remarkably, this translates into an upper bound for the axion mass which, although affected by the model-dependent parameterN , is independent of the fine details of the unification analysis that enter only logarithmically into A SL(R) .
Axion mass prediction in WGG+24 F . The failure of the WGG model in explaining neutrino masses and gauge coupling unification can be readily fixed by adding a single Majorana representation, 24 F , in analogy to the proposal of Ref. [24]. Here, we highlight the main differences due to the presence of the PQ symmetry. The Yukawa Lagrangian is extended by The first term provides a Dirac Yukawa interaction for the fermion triplet and singlet fields contained in 24 F , while the second term generates a Majorana mass for the full multiplet upon SU(5) symmetry breaking. We leave implicit the presence of extra non-renormalizable operators which are needed for two reasons: i) to avoid a rank-one light neutrino mass matrix and ii) to split the mass of the 24 F sub-multiplets (for further details see [24][25][26]). Eq. (9) also fixes the PQ transformation of the new field: 24 F → e −iα/2 24 F ; including the latter the total U(1) PQ -SU(3) C -SU(3) C anomaly yieldsN = 11. The possibility of narrowing down the axion mass range follows directly from unification constraints. The main issue with gauge coupling unification in the SM is the early convergence of the electroweak gauge couplings, α 1 and α 2 , around 10 13 GeV, at odds with proton decay bounds. Hence, the key ingredients for a viable unification pattern are additional particles charged under SU(2) L which can delay the meeting of α 1 and α 2 . Such a role in the WGG+24 F model can be played by the electroweak fermion T F = (1, 3, 0) and scalar T H = (1, 3, 0) triplets contained in the 24 F,H . 5 They are predicted 5 Compared to the GG+24 F case we have in principle extra thresholds due to fact that the 24 H is complex. However, the constraints coming from the minimization of the scalar potential imply that only one real triplet can be light, otherwise a colored octet scalar would be lowered to the triplet mass scale, spoiling nucleosynthesis [24].
to be at the TeV scale, so that a large enough unification scale can be achieved. Both types of triplets, if light enough, can give interesting signatures at the LHC. The fermionic component leads to same sign di-lepton events which violate lepton number [40]. A recent CMS analysis [41] sets a 95% CL exclusion at 840 GeV, while projected limits at the High Luminosity LHC (HL-LHC) [42,43] give m T F 2 TeV. Bosonic triplets can affect the di-photon Higgs signal strength, but the bound is milder compared to the fermionic triplet and model-dependent [44]. Here we assume a conservative m T H 200 GeV.
The complete unification pattern including also the convergence of α 3 with α 1 and α 2 requires heavier colored particles. These are the color-octet fermions and scalars contained in the 24 F,H , whose masses are required to be around 10 8 GeV, well beyond the LHC energy range.
The main prediction of gauge coupling unification is hence a clean correlation between a triplet mass parameter (whose analytical form is a consequence of the α 2 beta function), and the unification scale. The latter is operatively defined as the energy scale where α 1 and α 2 meet up to GUT-scale thresholds [45,46], and it can be identified with m V , the mass of the heavy vector leptoquark V µ mediating proton decay. Thanks to Eqs. (5)-(6), we can trade m V for the axion mass, which allows us to present the unification constraints in the (m a , m 3 ) plane. Following Ref. [26], we have performed a gauge coupling unification analysis including the leading NNLO corrections coming from the 2-loop matching coefficients and the 3-loop beta functions due to the fermion and scalar triplets. The extra thresholds affecting the evolution of α 1 and α 2 are fixed in such a way that the value of m 3 is maximized (cf. [26] for more details), which defines the parameter m max Future projections at HL-LHC (where we represent only the sensitivity to the fermion triplet mass) and HK (10 years data taking [2]) can complementary test this scenario. We remark that the SK bound was imposed via Eq. (8), which does not account for possible cancellations in the flavour structure of the proton decay operators. By considering different proton decay channels and accounting for flavour rotations, one can still extract a model-independent bound on the unification scale which is about an order of magnitude smaller [47,48]. The absolute upper bound on the axion mass is obtained by tuning to zero all the main proton decay channels, except those involving strange mesons. Using the results of Ref. [47] for the case of heavy Majorana neutrinos and updated with the latest experimental limit τ /B(p → K 0 µ + ) > 1.3 × 10 33 yr [49], we obtain m a < 330 neV. Similarly, from the projections at HK (10 years data taking [2]) in the p → K + ν channel we estimate m a < 160 neV.
Sensitivity of future axion DM searches. An axion in this mass range is extremely weakly coupled to SM particles, since its couplings to e.g. photons (γ), electrons (e), protons (p), and neutrons (n) are inversely proportional to the axion decay constant, (12) while the coefficients C ax are of order unity. In the WGG+24 F model, we find: where we introduced the ratio of the electroweak VEVs, tan β = 5 H / 5 H . This makes the GUT axion clearly invisible for purely laboratory based experiments. However, axions in this mass range are known to be excellent DM candidates [7][8][9] which can be searched for in axion DM direct detection experiments. In fact, very light axion DM even tends to be overproduced and can only be reconciled with the measured amount of cold DM if the PQ symmetry remained broken during and after inflation in the early universe. 6 In this case, the relative contribution of axion DM to the energy density of the universe depends not only on the mass, but also on the initial value of the axion field a i in units of the decay constant, θ i = a i /f a , inside the causally connected region which is inflated into our visible universe, cf. [39,50]: Thus an axion in the neV mass range can make 100 % of DM, if the initial field value θ i is of order 10 −2 . 7 In this cosmological scenario, however, quantum fluctuations of a massless axion field during inflation may lead to isocurvature density fluctuations that get imprinted in the temperature fluctuations of the cosmic microwave background (CMB) [52,53], whose amplitude is stringently constrained by observations. In the case that the 24 H stays at a broken minimum of the potential throughout inflation (e.g. for a SM-singlet inflaton), those constraints translate in an upper bound on the Hubble expansion rate during inflation [54][55][56]: Intriguingly, these isocurvature constraints can disappear completely in the case of non-minimal chaotic inflation [57][58][59] along one of the components of the 24 H . In this case, during inflation the 24 H is not at a minimum, Goldstone's theorem does not apply, and the lightest fluctuations orthogonal to the inflaton can have masses above H I as long as the parameter ξ 24 H , describing the non-minimal coupling to the Ricci scalar, S ⊃ − d 4 x √ −g ξ 24 H Tr (24 2 H )R, is larger than ∼ 0.01. For ξ 24 H above this value, the power spectra of the isocurvature fluctuations become exponentially suppressed and the CMB bounds can be avoided. In such scenarios, one still needs to ensure that the PQ symmetry is never restored after inflation; we expect that this might be possible for small enough quartic and Yukawa couplings of the 24 H , but a dedicated analysis generalizing the non-pertubative and perturbative reheating calculations in Ref. [50] is needed.
The DM experiment ABRACADABRA [27], has very good prospects to probe the axion photon coupling, g aγ = α C aγ /(2πf a ), in the relevant mass region. This is shown in Fig. 2, from which we infer Axion coupling to the nucleon EDM operator, gaD, versus axion mass ma. The blue regions give the projected sensitivities of CASPEr-Electric from Ref. [29]. The short, full blue line reflects a factor of three improvement in sensitivity for a search just concentrated on the preferred mass region. that the whole parameter space of the WGG+24 F model (including the tuned region) can be tested in the third phase of the broadband and resonant search modes of ABRACADABRA. In Fig. 3, we confront our axion mass prediction with the projected sensitivity of the experiment CASPEr-Electric [28,29], which aims to search for oscillating nucleon electric dipole moments (EDM) d n (t) = g aD √ 2ρ DM ma cos(m a t) [60], where g aD is the model-independent coupling of the axion to the nucleon EDM operator, L a ⊃ − i 2 g aD a Ψ N σ µν γ 5 Ψ N F µν , and ρ DM = 0.3 GeV/cm 3 is the local energy density of axion DM. The QCD axion band in Fig. 3 indicates the theoretical uncertainty of the non-perturbative estimates of g aD . We used the result in [61], obtained with QCD sum rules; for other evaluations see e.g. [62,63]. 8 We infer from Fig. 3, that the preferred axion mass window (11) could definitely be probed in phase III of CASPEr-Electric. 9 On the other hand, the projected sensitivity of CASPEr-Wind [29], which exploits the axion nucleon coupling g aN = C aN /(2f a ) (N = p, n) to search for the axion DM wind due to the movement of the Earth through the Galactic DM halo [60], misses the preferred coupling vs. mass region by two orders of magnitude or more, even in its phase II. We show this in Fig. 4, where the theoretical uncertainty of the axion band is obtained from the errors in the coefficients of Eq. (13), and from varying tan β ∈ [0. 28,140] in the perturbative unitarity domain [65].
Conclusions. In this Letter we have proposed a minimal implementation of the PQ mechanism in a realistic SU(5) model, which predicts a narrow axion mass window (cf. Eq. (11)) which can be directly tested at future axion DM experiments and indirectly probed by collider and proton decay experiments. In principle, a precise determination of m a (via ABRACADABRA and/or CASPEr-Electric) would lead to a direct determination of the GUT scale, possibly discriminating among GUT models, and setting a target for proton decay measurements. Although we exemplified our predictions in the case of the WGG+24 F model, it would be interesting to compare axion properties in other minimal extensions of the WGG model which can simultaneously address neutrino masses and gauge coupling unification (see e.g. [66,67]), or in realistic SO(10) models [11].
Finally, the intriguing possibility that the 24 H field could also be responsible for inflation would make the WGG+24 F model a potential candidate for a minimal and predictive GUT-SMASH [50,68] variant aiming at a self-contained description of particle physics, from the electroweak scale to the Planck scale, and of cosmology, from inflation until today. We leave a detailed investigation of this scenario for future studies.