Pati-Salam Axion

I discuss the implementation of the Peccei-Quinn mechanism in a minimal realization of the Pati-Salam partial unification scheme. The axion mass is shown to be related to the Pati-Salam breaking scale and it is predicted via a two-loop renormalization group analysis to be in the window $m_a \in [10^{-11}, \, 3 \times 10^{-7}]$ eV, as a function of a sliding Left-Right symmetry breaking scale. This parameter space will be fully covered by the late phases of the axion Dark Matter experiments ABRACADABRA and CASPEr-Electric. A Left-Right symmetry breaking scenario as low as 20 TeV is obtained for a Pati-Salam breaking of the order of the reduced Planck mass.


Conclusions 21
A Two-loop running and one-loop matching 22 1 Introduction A central question of physics beyond the Standard Model (SM) is whether there is intermediate-scale physics between the electroweak and the Planck scales and how to possibly test it. It can be reasonably argued that the SM is an effective field theory, valid until some cut-off scale Λ SM ≤ M Pl = 1.2 × 10 19 GeV and that (disregarding the long-pursued naturalness argument of the electroweak scale) the new layer of physical reality might lie much above the TeV scale. This is actually suggested by the inner structure of the SM: flavour and CP violating observables have generically probed scales up to Λ SM 10 6 GeV, while light neutrino masses point to Λ SM 10 14 GeV. At the same time, it is evident that the hypercharge structure of the SM fermions cries out for unification (letting aside the more mysterious origin of flavour). Left-Right symmetric theories [1][2][3][4] provide a most natural route for addressing the origin of hypercharge and neutrino masses, passing through the Pati-Salam partial unification scheme [1] (which also provides a rationale for the quantization of electric charge) and ending up into one SM family plus a right-handed neutrino unified into a spinorial representation of SO(10) [5,6]. Due to the fact that these groups have rank 5, they admit at least an intermediate breaking stage before landing on the SM gauge group, and in the case of Pati-Salam [7] and SO (10) [8][9][10] those are often predicted to lie in between 10 6 GeV and 10 14 GeV by (partial) gauge coupling unification. This picture would gain an additional value if such intermediate-scale physics would be connected to other open issues of the SM, most notably the baryon asymmetry of the Universe and Dark Matter (DM). The former is built-in in the form of thermal leptogenesis [11], which in its simplest realization would suggest Λ SM 10 9 GeV (see e.g. [12]), while DM is often a missing ingredient in minimal realizations of Left-Right symmetric theories (for an exception, see [13]). A natural possibility is then to impose a Peccei-Quinn (PQ) symmetry [14,15] delivering an axion [16,17], which provides at the same time an excellent DM candidate [18][19][20] and solves the strong CP problem. This choice is economical also in the following sense: i) in SO (10) setups the PQ symmetry was often imposed for another reason, namely to enhance the predictivity of the renormalizable (non-supersymmetric) SO(10) Yukawa sector [21] and ii) it is based on a coincidence of scales: the axion decay constant is in fact bounded from astrophysical and cosmological consideration within the range 10 8 GeV f a 10 18 GeV (see [22] for updated limits).
From an experimental point of view, there are now better hopes to catch the axion tail of the story. Axion physics is in fact in a blooming phase with several new detection concepts which promise to open for explorations regions of parameter space which were thought unreachable until few years ago (for updated experimental reviews see [43,44]). In particular, the possibility that the Grand Unified Theory (GUT) axion window could be completely covered by the axion DM experiments CASPEr-Electric [45,46] and ABRA-CADABRA [47] has triggered a revival of studies of GUT×U(1) PQ models, with the axion field residing in a non-singlet representation of the GUT group. In particular, Ref. [37] computed for the first time low-energy axion couplings in SO(10)×U(1) PQ models and considered axion mass predictions in SO(10) models with up to two intermediate breaking stages. Ref. [48] considered instead a minimal non-renormalizable SU(5)×U(1) PQ model based on a PQ extension of [49,50], which due to its minimality allowed to obtain (via the three-loop gauge coupling unification analysis of [51]) a sharp prediction for the axion mass in the neV domain. Subsequently, Refs. [52,53] considered axion mass predictions in other minimal renormalizable SU(5)×U(1) PQ models. Some cosmological consequences of supersymmetric axion GUTs were considered instead in [54].
The study of the Pati-Salam axion considered in the present work has a twofold motivation. On the one hand, the Pati-Salam (partial) unification constraints are genuinely different from SO(10) ones, so their predictions can be in principle discerned from those of SO (10). For instance, while it is notoriously difficult to obtain a low-scale Left-Right symmetry breaking scale in SO(10), we will show that if the Pati-Salam group is broken at the Planck scale, the Left-Right symmetry breaking can be as low as 20 TeV (and even lower in the absence of the PQ). One the other hand, the Pati-Salam gauge group, which is half-way through SO(10), provides a simpler setup and for this reason it can be studied in quite some detail. For instance, although SO(10)×U(1) PQ scalar potentials have been partially classified in [26], they have never been investigated in detail. In the present work we provide also a non-trivial step in that direction, by working out the scalar potential dynamics of a complex scalar adjoint of SU(4) PS , which hosts the axion field as a phase, and whose vacuum expectation value (VEV) is simultaneously responsible for PQ and Pati-Salam breaking down to the Left-Right symmetric gauge group. This allows in turn to constrain the axion mass via a renormalization group (RG) analysis of (partial) gauge coupling unification in Pati-Salam.
The paper is structured as follows. In Sect. 2.1 we describe the logic behind the construction of a minimal renormalizable Pati-Salam×U(1) PQ model. Next, we focus on axion couplings (Sect. 2.2) and on the axion mass prediction from Pati-Salam breaking (Sect. 2.3). This is the main conceptual point of the paper, which relies on the calculable relation between the axion mass and the Pati-Salam breaking scale (cf. Eq. (2.42)). The latter can then be constrained via a RG analysis of (partial) gauge coupling unification within the Pati-Salam model. In Sect. 2.4 we report the results of such RG analysis, which is based both on a one-loop analytical understanding and a more involved twoloop numerical investigation, whose details are deferred to App. A. The main outcome of the RG analysis is that the Pati-Salam breaking scale (the axion mass) becomes a decreasing (increasing) function of a sliding Left-Right symmetry breaking scale. Sect. 3 is devoted to the phenomenology of the model and to the experimental prospects for hunting the Pati-Salam axion. We first collect various cosmological and astrophysical constraints (Sect. 3.1) and then review (Sect. 3.2) the future sensitivity of the axion DM experiments ABRACADABRA and CASPEr-Electric. The main outcome is that the parameter space of the Pati-Salam axion will be fully covered by the late phases of those axion DM experiments, as shown in Fig. 3. We finally consider possible correlated signals due to  and Left-Right (Sect. 3.4) symmetry breaking dynamics. While the former are more difficult to be observable, a sliding Left-Right breaking scale can give rise to more interesting indirect/direct signatures. We conclude in Sect. 4 with a brief recap of the main results, together with a discussion of the critical points of the present setup and an outlook for possible future work.

Peccei-Quinn extended Pati-Salam
In this Section we propose a simple implementation of the PQ symmetry in a minimal Pati-Salam model, which is inspired by the more studied case of SO(10) × U(1) PQ [23,33,34,37]. A similar Pati-Salam×U(1) PQ construction has been recently considered in Ref. [36]. The main difference compared to [36] is that here the axion field resides dominantly in a non-singlet representation of Pati-Salam. This allows in turn to connect the axion mass and the Pati-Salam breaking scale, with the latter being constrained via a RG analysis of gauge coupling (partial) unification in Pati-Salam.

Minimal model construction
The Pati-Salam gauge group is defined by where P is a discrete symmetry exchanging L ↔ R, which enforces parity restoration in the UV. The color factor is embedded as SU(3) C ×U(1) B−L ⊂ SU(4) PS , thus implementing the idea of lepton number as the fourth color [1]. 1 The embedding of the hypercharge follows the standard one of Left-Right symmetric theories [1][2][3][4] The fermion fields transform under G PS as Q L ∼ (4, 2, 1) and Q R ∼ (4, 1, 2). Explicitly, the embedding of the SM fermions consists in three copies of including also a RH neutrino, ν R . Under P : Q L ↔ Q R , so that P assumes the meaning of space-time parity. 2 The Higgs sector comprises the following representations: S ∼ (15, 1, 1), ∆ L ∼ (10, 3, 1), ∆ R ∼ (10, 1, 3), Φ 1 ∼ (1, 2, 2), Φ 15 ∼ (15, 2, 2). Let us motivate in turn the need for such representations: S is introduced in order to allow for an intermediate breaking stage: We will also consider the case in which the VEV of S breaks spontaneously the symmetry P, so that the unbroken group is G The field ∆ R is needed to provide the final symmetry breaking stage down to the SM gauge group (2.5) 1 Although the original formulation was based on the gauge group SU(4) PS × SU(4) L × SU(4) R , the SU(4) PS × SU(2) L × SU(2) R setup emerged later on as a simpler UV completion of the SM.
2 Instead of P one could consider charge conjugation, C : Q L ↔ Q c R . This latter choice is ultraviolet (UV) motivated by the fact that upon embedding G PS in SO(10), C turns out to be an element of SO(10), also known as D-parity [55,56]. The difference between the two choices mainly regards the structure of CP violation (see e.g. [57][58][59]). In particular, as pointed out in Ref. [57], imposing a PQ symmetry in the case of P allows to non-trivially relax the strong bounds from K on the Left-Right symmetry breaking scale. In this work we will stick for definiteness to the case of P.
The field ∆ L is required in order to make the theory Left-Right symmetric under P : ∆ L ↔ ∆ R . Finally, the (complex) bi-doublets Φ 1 and Φ 15 are needed in order to reproduce SM fermion masses and mixing. The renormalizable Yukawa Lagrangian reads 3 whereΦ 1,15 = Φ * 1,15 (with = iσ 2 ) denote conjugate bi-doublet fields. Further assuming . In fact, without Φ 15 one would end up with a wrong mass relation between downquarks and charged leptons, M d = M T e . This is avoided by introducing Φ 15 which transforms non-trivially under SU(4) PS , so that after SU(4) PS breaking Φ 15 feeds differently into down-quarks and charged leptons [60].
On the other hand, the proliferation of Yukawa matrices in Eq. (2.6) makes the model not predictive for fermions masses and mixings. This provides a rationale for introducing (similarly as was originally proposed for the SO(10) Yukawa sector [21]) a U(1) PQ : which enhances the predictivity of the Pati-Salam Yukawa sector by enforcingỸ 1,15 → 0. After electroweak symmetry breaking, one obtains the following SM fermion mass sum rules (respectively for the up-, down-quarks, charged-leptons, Dirac neutrinos, left-handed and right-handed Majorana neutrinos) 4 where we introduced the VEVs: Light neutrino mass eigenstates follow the standard type-I+II seesaw formula From the above mass relations we qualitatively conclude that since M u and M D are strongly correlated, the top mass eigenvalue prefers an intermediate scale v R v, unless a tuning is invoked into the Dirac neutrino mass term, M D . 5 Moreover, v L ∼ v 2 /v R is an induced VEV from the minimization of the scalar potential, which is consistent with the requirement of light neutrino masses. We hence assume the hierarchy of scales In the present setup, where only the fields in Eq. (2.7) are charged under PQ, the VEV new global PQ symmetry (a linear combination of the original PQ and the broken gauge generators), which is eventually broken at the electroweak scale by Φ 1,15 , thus leading to an experimentally untenable Weinberg-Wilczek axion. A natural way to fix this is to complexify the representation S and charge it under the U(1) PQ , thus connecting the 16) and the scalar potential can be written as where V r.i. contains re-phasing invariant terms which are not sensitive to U(1) phases, while V PQ is chosen in such a way to ensure the explicit breaking and reads (see also [36]) where multiple gauge invariant contractions with the same U(1) PQ structure are left understood. 6 The λ 1,15 terms are needed to communicate the PQ breaking from the S to the bi-doublets Φ 1,15 , λ mix and α mix are allowed by the global symmetries left invariant by λ 1,15 , while β 1,15, mix = 0 is required in order to avoid an extra spontaneously broken U(1) global symmetry, with an associated (unwanted) Goldstone boson. Other terms in the third and fourth line of Eq. (2.19) are allowed by gauge invariance and the global symmetries of the system. P invariance (with P : S → S † being a possible definition compatible with the PQ symmetry) implies some restrictions on the scalar potential parameters, e.g., the couplings in the first two rows of Eq. (2.19) need to be real (since the h.c. operator coincides with the P-transformed one). The field transformation properties under G PS × U(1) PQ are collected for convenience in Table 1. 5 Ref. [36] performed the fit to the Pati-Salam fermion mass sum rules fixing v R = 10 14 GeV and assuming the dominance of Type-I seesaw. Similar fermion mass sum rules in the (more constraining) case of SO (10) are also known to yield viable fits (see e.g. [61]). A quantitative assessment of fermion masses and mixing in the minimal Pati-Salam model, leaving the v R scale free, is left for a future study. 6 For instance, S * 2 Φ 2 15 features four linearly-independent invariants: where subscripts denote the type of SU(4) PS contraction. Since in this work we will not address the full minimization of the G PS × U(1) PQ scalar potential, such details are not essential for the following discussion. Table 1: Field content of the minimal G PS × U(1) PQ model.

Axion couplings
In the presence of spontaneously broken gauge symmetries the identification of the canonical axion field and its couplings to SM fields presents some non-trivial steps. The axion field must be properly orthogonalized in order to avoid kinetic mixings with the would-be Goldstone bosons associated with broken Cartan generators [37]. Due to the hypercharge relation in Left-Right symmetric models (cf. Eq. (2.2)), it is indeed enough to require orthogonality with respect to T 3 L and T 3 R [62]. This is particularly relevant for the axion coupling to matter fields (nucleons and electrons) which become functions of vacuum angles, expressed in terms of the gauge symmetry breaking VEVs. Since these couplings will not be phenomenologically relevant for axion mass range discussed in this paper, we will not report here their derivation, but just quote the final results (for a more detailed account, see [62]). On the other hand, the axion coupling to photons depends only on the anomaly coefficients of the PQ current, defined via which can be actually computed in terms of the U(1) PQ charges of Table 1 (generically denoted as X i ), and are found to be (see e.g. [63]) where we used n g = 3 (number of generations), T (3) = 1 2 (Dynkin index of the fundamental of SU(3) C ) and the electric charges Hence, in particular E/N = 8/3, which sets the axion coupling to photons (see below).
The axion effective Lagrangian, including couplings to photons, matter fields (f = p, n, e) and the oscillating neutron Electric Dipole Moment (nEDM), can be written as with the values of the C ax coefficients given by [22,64,65] 29) in terms of model-dependent factors E/N and c 0 u, d, e . In the Pati-Salam axion model they are found to be (up to safely negligible v 2 /v 2 S 1 corrections for the fermion couplings, see [62]) with the index i = 1, 2, 3 denoting generations and (2.31)

Pati-Salam breaking dynamics
The complex SU(4) PS adjoint representation S is responsible for the initial breaking Since S provides the largest VEV, this dynamics is captured by the S sector of the re-phasing invariant potential (here we restore gauge contractions in SU(4) PS space) where the last invariant reads explicitly and we have set to zero operators of the type Tr S n , with n = 2, 3, 4, consistently with the presence of the PQ symmetry (cf. Eq. (2.16)). We decompose the complex adjoint in terms of canonically normalized real fields as (see also [66]) with SU(4) PS generators (see e.g. Appendix A.10 of [67]), normalized as Tr T α T β = 1 2 δ αβ . We assume S = T 15 v S , with (2.37)

Scalar boson spectrum
The calculation of the scalar spectrum for µ 2 S > 0 yields: • A (perturbatively) massless axion: m 2 a = 0; • A radial mode: m 2 ρ = 1 6 (12(λ • 6 would-be Goldstone modes (eaten by the massive vector leptoquark • Three sets of degenerate scalar modes for a total of 22 massive scalars, with masses: The conditions on the scalar potential parameters leading to a positive mass spectrum are straightforward and they serve to show that the SU (4) configuration can be at least a local minimum. Since they are lengthy and not needed in the following, we do not report them explicitly.

Gauge boson spectrum
The gauge boson spectrum can be determined from the action of the covariant derivative, A α µ , and the kinetic term We find: • 8+1 massless gauge boson associated with the unbroken SU(3) C ×U(1) B−L algebra.

Axion mass from Pati-Salam breaking scale
The axion field resides dominantly in S, hence to a very good approximation (up to  In the next Section we will constrain (M PS , g PS ), and in turn the axion mass, via a RG analysis of (partial) gauge coupling unification in Pati-Salam.

Renormalization group analysis
Let us consider the breaking pattern where M PS , M LR and M Z denote the renormalization scales associated with G PS , G ( / P) LR and G SM , respectively. In order to determine the beta-functions which govern the RG evolution in the two running steps, we assume that the scalar spectrum obeys the socalled extended survival hypothesis (ESH) [69] which requires that at every stage of the symmetry breaking chain only those scalars are present that develop a VEV at the current or the subsequent levels of the spontaneous symmetry breaking. The ESH is equivalent to the requirement of the minimal number of fine-tunings to be imposed onto the scalar potential [70]. The surviving scalars at the M Z and M LR scales are displayed in Table 2   We first focus on a one-loop RG analysis, in order to grasp an analytical understanding of the correlation among mass scales. Starting with the electroweak values of the three SM gauge couplings [71] with the SM beta coefficients given in Eq. (A.4). The tree-level matching of the gauge couplings at M LR is given by where Eq. (2.50) comes from the relation The value of the one-loop beta coefficients are given in Eqs. Reshuffling the RG equations and the matching conditions above one obtains The dependence of Pati-Salam breaking scale from M LR , as well as that of the gauge couplings at M PS is displayed respectively in Fig. 1 .7)). To improve on this point one should take into account the constraints coming form the minimization of the full G PS × U(1) PQ scalar potential, in order to obtain the range of variation of the scalar thresholds allowed by the vacuum manifold. It goes without saying that this is a highly non-trivial task. As a partial justification of the ESH = minimal fine-tuning hypothesis [69,70], we note that strong violations of the latter would be difficult to be reconciled with the idea that gauge hierarchies could arise due to environmental selection/cosmological evolution (see also footnote (10)).
We can now proceed with the main goal of the RG analysis, namely to express the axion mass as a function of M LR , via the relation in Eq. (2.42). This correlation is shown in Fig. 3, where we report directly the two-loop result for the two cases G LR (P) and G / P LR (No P), together with the current bounds from Black Hole Superradiance and the experimental prospects of future axion DM experiments, whose phenomenological implications are described in the next Section.  From the two-loop RG analysis we have inferred the mass windows displayed in Table 3, where the lower bounds take already into account the exclusion limit in Eq. (3.4). In particular, the correlations among the mass scales can be read off Fig. 1 (M PS vs. M LR ) and Fig. 3 (m a vs. M LR ).  In this Section we discuss the phenomenological implications of those mass ranges, both from the point of view of axion physics and the Pati-Salam/Left-Right symmetry breaking scales.

Cosmological and astrophysical constraints
Let us start by addressing some relevant cosmological and astrophysical constraints. Very light axion DM tends to be overproduced via the misalignment mechanism [18][19][20], and the measured amount of cold DM can only be explained if the PQ symmetry remained broken during inflation and never restored after it, which corresponds to the so-called pre-inflationary PQ breaking scenario (more precisely, the post-inflationary PQ breaking scenario is excluded for m a 30 µeV [72]). A late inflationary phase H I M PS ≈ f a (cf. Eq. (2.40)) is also supported by other cosmological issues of the Pati-Salam axion framework related to the formation of topologically stable defects, which tend to dominate the energy density of the Universe, unless inflated away. These include: i) Magnetic monopoles from Pati-Salam breaking; ii) Domain walls from spontaneous breaking of P; iii) Axion domain walls at the QCD phase transition. While the domain wall problems are not specific of Pati-Salam and have been widely discussed in the literature (for a review see e.g. [73]), we dwell a bit on the less known physics of Pati-Salam monopoles in Sect. 3.1.1. After that we discuss other constraints related to axion DM (Sect. 3.1.1) and Black Hole Superradiance (Sect. 3.1.3).

Pati-Salam monopoles
Pati-Salam monopoles are topologically stable scalar-gauge field configuration arising from G PS → G SM breaking, with magnetic charge Q mag = 4π/e and mass M PS ∼ M PS /α PS . Although they were originally investigated in the context of intermediate symmetry breaking stages of SO(10) [74][75][76], they have some distinct features with respect to GUT monopoles which make them interesting by their own.
A flux of Pati-Salam monopoles hitting an Earth-based detector could actually lead to a spectacular signature, since according to Sen [77,78] they are expected to catalyze ∆(B + L) = 3 violating processes via the weak 't Hooft anomaly with a geometrical cross section, i.e. not suppressed by the Pati-Salam breaking scale or by other nonperturbative factors. The conservative Kibble estimate [79] of one monopole per Hubble horizon provides a lower bound on today's monopole number density (normalized to entropy density) 8 n PS s where T PS ≈ M PS 10 13 GeV is the critical temperature of the Pati-Salam phase transition. Hence, for the lowest value allowed by the RG analysis, M PS = 4.7 × 10 13 GeV (cf. Table 3) one has n PS /s 5.7 × 10 −17 , which overshoots the critical energy density of the Universe 9 and is also well above the indirect Parker's bound [83,84], n PS /s 10 −26 , and the direct detection limits from the MACRO Collaboration [85], n PS /s 2 × 10 −28 .
On the other hand, differently from GUT monopoles, Pati-Salam monopoles are not subject to the much more stringent bounds [86] from the catalysis of proton decay due to the Callan-Rubakov effect [87,88]. Although in the model considered in this paper Pati-Salam monopoles need to be inflated away, we note that the observational window of two orders of magnitude between the Parker's bound and direct detection limits might be populated in models with M PS 10 13 GeV, e.g. in the ballpark of M PS ≈ 10 10 GeV according to the naive Kibble estimate.

Axion relic density and iso-curvature bounds
In the pre-inflationary PQ breaking scenario, the axion's relic abundance depends both on the mass and 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. [72,89]: where we have normalized the axion mass to the upper bound in Table 3 coming from the RG analysis. Thus an axion close to that boundary can reproduce the whole DM, without the need of tuning the initial misalignment angle. In this cosmological scenario, however, iso-curvature quantum fluctuations of a massless axion field during inflation may leave an imprint in the temperature fluctuations of the cosmic microwave background [90,91], whose amplitude is stringently constrained by observations. In the case that the S field hosting the axion stays at the broken minimum of the potential throughout inflation (i.e. for the inflaton field not residing in S), those constraints translate into an upper bound on the Hubble expansion rate during inflation [92][93][94] which is consistent with a late inflationary phase in order to dilute the cosmic density of monopoles and domain walls.

Black Hole Superradiance
Although super-light axions are free from standard astrophysical bounds due to stellar evolution, they are subject to limits from Black Hole Superradiance as long as the axion decay constant approaches the Planck mass. In fact, axions can form gravitational bound states around black holes whenever their Compton wavelength is of the order of the black holes radii. The phenomenon of superradiance [95] then guarantees that the axion occupation numbers grow exponentially, providing a way to extract very efficiently energy and angular momentum from the black hole [96,97]. The rate at which the angular momentum is extracted depends on the black hole mass and so the presence of axions could be inferred by observations of black hole masses and spins. A recent analysis excludes the mass window [98] m a ∈ [6 × 10 −13 , 10 −11 ] eV , (3.4) which is shown as a gray band in Fig. 3. It should be stressed that the Black Hole Superradiance bound does not assume the axion being DM and it just relies on the universal axion coupling to gravity through its mass.

Axion Dark Matter experiments
Axion DM experiments turn out to be the most powerful probes of the Pati-Salam axion (under the assumption that the axion comprises the whole DM), since they can cover the whole mass window m a ∈ [10 −11 , 3.4 × 10 −7 ] eV predicted by the RG analysis (cf. Table 3). In particular, this is possible due to the complementarity of ABRACADABRA [47] and CASPEr-Electric [45,46], which probe the axion mass parameter space region from opposite directions (cf. Fig. 3).

ABRACADABRA
The axion DM experiment ABRACADABRA [47] has very good prospects to probe the axion-photon coupling for masses m a 4 × 10 −7 eV. Such low values are notoriously difficult to be reached for standard cavity experiments due to the need of matching axion wavelengths with extremely large cavity sizes 50 m. ABRACADABRA uses instead a different detection concept based on a toroidal magnet and a pickup loop to detect the variable magnetic flux induced by the oscillating current produced by DM axions in the static (lab) magnetic field. The experiment can operate either in broadband or resonant modus by using an untuned or a tuned magnetometer respectively. A small scale prototype ABRACADABRA-10 cm [99] has already given exclusion limits competitive with astrophysics for axion masses ∈ [3.1×10 −10 , 8.3×10 −9 ] eV. The projected sensitivities (from [47]) on the axion mass are displayed in Fig. 3 via blue bands. They refer to three different phases of the experiment in the resonant approach, which is more sensitive to the standard QCD axion region, and they also assume E/N = 8/3, which applies to the Pati-Salam axion (cf. Eq. (2.30)) and in general to any GUT axion model.

CASPEr
CASPEr-Electric [45,46] employs nuclear magnetic resonance techniques to search for an oscillating nEDM [100] d n (t) = g d √ 2ρ DM m a cos(m a t) , (3.5) where g d = C anγ /(m n f a ) is the model-independent coupling of the axion to the nEDM operator defined in Eq. (2.23) and ρ DM ≈ 0.4 GeV/cm 3 is the local energy density of axion DM. The numerical value of C anγ (cf. Eq. (2.29)) takes over the static nEDM calculation of [101] based on QCD sum rules and amounts to a theoretical error of about 40%. In Fig. 3 we show in red bands the axion mass reach of CASPEr-Electric [46] for phases 2 and 3, including as well on the right side of the plot the size of the QCD error (denoted by ∆ QCD ). Remarkably, the QCD error turns out to be strongly correlated with the projected sensitivities shown in Ref. [46], so that a future reduction of the theoretical error on the static nEDM (e.g. via Lattice QCD techniques [102,103]) might have a non-trivial impact on the sensitivity reach of CASPEr-Electric.
On the other hand, the projected sensitivity of CASPEr-Wind [46], which exploits the axion nucleon (N = p, n) coupling g aN = C aN /(2f a ) (with C aN given in Eqs. (2.25)-(2.26)) to search for an axion DM wind due to the movement of the Earth through the Galactic DM halo [100], misses the preferred coupling vs. mass region by at least two orders of magnitude, even in its phase 2.

Pati-Salam signatures
Given the lower bound on the Pati-Salam breaking scale inferred from the RG analysis, M PS 10 13 , genuine signatures of Pati-Salam dynamics turn out to be difficult to be experimentally accessible, as briefly reviewed in the following.

Rare meson decays
The vector leptoquark X µ mediates tree-level rare meson decays K L , B 0 , B 0 s → i j (for = (e, µ, τ )), which turn out to be loop-and chirally-enhanced with respect to the SM contribution, and hence provide a powerful flavour probe of the Pati-Salam model [104,105]. A recent collection of bounds which takes also into account flavour mixing can be found in Ref. [106]. For instance, for maximal mixing the most sensitive channel is K L → µe, which probes leptoquark masses up to 10 6 GeV, hence still much below the Pati-Salam breaking scale emerging from the RG analysis.

Baryon number violation
A standard signature of Pati-Salam dynamics are ∆B = 2 processes, in particular n-n oscillations [107], which are described by d = 9 SM operators of the type (3.6) Present bounds on nuclear instability and direct reactor oscillations experiments yield bounds at the level of Λ ∆B=2 100 TeV [108]. In the present Pati-Salam×U(1) PQ model (see also [36]) only operators of the type q L q L q L q L d R d R are generated, which are mediated by color sextet scalar di-quark fields ∆ qq contained in the Pati-Salam representations ∆ R ∼ (10, 1, 3) and ∆ L ∼ (10, 3, 1), with strength (schematically) where η and y denote respectively the scalar coupling of ∆ 2 R ∆ 2 L (cf. Eq. (2.19)) and the Yukawa coupling of ∆ qq to quarks (cf. Eq. (2.6)). Hence, n-n oscillations could be visible only if those color sextets were unnaturally light m ∆ qq M PS .
Nucleon decay from the scalar sector of Pati-Salam is possible as well, as originally observed in [60,109]. In the presence of the Higgs multiplet Φ 15 ∼ (15, 2, 2), required by a realistic fit to SM fermion masses, the spontaneous breaking of U(1) B−L can lead to B + L preserving nucleon decay modes of the type N → + meson or even threelepton decay modes N → + + [110], associated respectively to d = 9 and d = 10 SM effective operators, whose short-distance origin can be traced back into the scalar potential couplings γ 15 and δ 15 in Eq. (2.19) (see [36] for a more detailed account). As in the case of n-n oscillations, for these exotic nucleon decay modes to be observable, one or more scalar fragments of Φ 15 and ∆ R,L mediating those nucleon decay operators need to be 100 TeV, unnaturally light compared to the Pati-Salam breaking scale.

Left-Right signatures
A sliding Left-Right symmetry breaking scale M LR ∈ [2.0×10 4 , 4.7×10 13 ] GeV (e.g. in the case of P broken at the M LR scale, cf. Table 3) offers potentially observables signatures related to the Left-Right breaking dynamics, together with the possibility of observing a correlated signal with axion physics. In the following, we consider two opposite scenarios in which the Left-Right symmetry is broken either at high or low scales.

High-scale Left-Right breaking
The high-scale Left-Right breaking scenario, corresponding to a single step breaking with M LR ≈ M PS ≈ 4.7 × 10 13 GeV is motivated by naturalness arguments. In fact, it simultaneously minimizes the parameter space tuning in three sectors of the theory: i) it avoids the extra 10 (M LR /M PS ) 2 tuning of triplet fields (cf. Table 2); ii) it reduces the tuning of the initial axion misalignment angle in order not to overshoot the axion DM relic density (cf. Eq. (3.2)); iii) it mitigates the tuning in the Dirac neutrino mass matrix, which turns out to be strongly correlated with the up-quark mass matrix (cf. Eq. (2.8) and (2.11)), and hence it prefers high values of M LR in order not to overshoot light neutrino masses (cf. Eq. (2.15)). Moreover, we note that the single-step breaking corresponds to the lower end of the axion mass window m a ≈ 3 × 10 −7 GeV, which will be one of the first region to be tested, already in Phase 1 of ABRACADABRA (cf. Fig. 3).
In fact, a detailed fit of SM fermion masses and mixings within the minimal renormalizable Yukawa sector (cf. the mass sum rules in Eqs. TeV. An independent argument for high-scale Left-Right breaking is given by thermal leptogenesis [11], which in its simplest realization would suggest M LR 10 9 GeV (see e.g. [12]).
Finally, in the presence of a strong first-order phase transition the Left-Right symmetry breaking can lead to the production of a stochastic gravitational wave background, that might leave its imprint on the gravitational wave spectrum of forthcoming spacebased interferometers [113]. This can happen in some parameter space regions of the Left-Right symmetric scalar potential, resembling an approximate scale invariance [114]. The latter work focussed on Left-Right breaking scales close to the TeV scale, but in principle detectable gravitational wave signals might arise also for M LR TeV.

Low-scale Left-Right breaking
The Left-Right symmetry breaking scale can be as low as 20 TeV (230 TeV) in the case where P is broken at the M LR (M PS ) scale (cf. Table 3). 11 In particular, the lower bound is saturated in both cases for M PS = 1.4 × 10 18 GeV, or equivalently (cf. Eq. (2.40)) for a Pati-Salam/Peccei-Quinn breaking order parameter v S = 3.4 × 10 18 GeV, that is of the order of the reduced Planck mass M Pl / √ 8π. Although this numerical coincidence should not be taken too seriously, since it might be spoiled by scalar threshold effects, it is suggestive of a possible connection with gravity and it could be seen as a mild theoretical motivation for a low-scale Left-Right symmetry breaking scenario. 12 In particular, the former case of P broken at M LR , corresponds to the most constrained version of the Left-Right symmetric model [3,4], whose phenomenology has beed studied in great detail in the recent years (see e.g. [115][116][117]). Although a W R mass of the order of 20 TeV is well beyond the direct/indirect ≈ 6 TeV reach of the LHC [118][119][120][121], flavour [117] and CP [57,58] violating observables offer sensitivities up to hundreds of TeV. A future 100 TeV hadron collider would be able instead to directly probe the lower end of the M LR range, possibly in correlation with an axion signal at CASPEr-Electric for m a 10 −11 eV.

Conclusions
In this work we have discussed the implementation of the PQ mechanism in a minimal realization of the Pati-Salam (partial) unification scheme, where the axion mass is related to the Pati-Salam breaking scale. The latter was shown to be constrained by a RG analysis of (partial) gauge coupling unification. The main physics result is displayed in Fig. 3, which shows that the whole parameter space of the Pati-Salam axion will be probed in the late phases of the axion DM experiments ABRACADABRA and CASPEr-Electric. Possible correlated signatures connected with the breaking of the Left-Right symmetry group include future collider/flavour probes of a low-scale Left-Right breaking (as low as 20 TeV for a Pati-Salam breaking of the size of the reduced Planck mass -cf. Fig. 1) and, less generically, the imprint on the gravitational wave spectrum of the Left-Right phase transition. Other indirect constraints on the scale of Left-Right symmetry breaking might arise from a detailed fit of SM fermion masses and mixings within the minimal renormalizable Pati-Salam Yukawa sector or from a successful implementation of thermal leptogenesis. Both of them could be worth a future investigation, in order to further narrow down an axion mass range. Finally, some of the ingredients discussed in the present paper might serve as building blocks for a detailed investigation of what could arguably be considered the minimal SO(10)×U(1) PQ model, based on a 45 H +126 H +10 H reducible Higgs representation [122,123], with a complex adjoint hosting the SO(10) axion.
While in the Introduction we have praised the nice aspects of the whole setup, here we would like to conclude with a more critical note. In the present formulation (as well as in all GUT×U(1) PQ models known to the author and despite some earlier attempts of obtaining an automatic U(1) PQ from SU(9) [124]) the PQ symmetry is imposed by hand, while it would be more satisfactory for it to arise as an accidental symmetry, possibly due to some underlying gauge dynamics. 13 Moreover, since global symmetries need not to be exact, it is unclear why the PQ symmetry should be an extremely good symmetry of UV physics, and in particular of quantum gravity, not to spoil the solution of the strong CP problem [125][126][127]. This is particularly problematic for axion GUTs, since the issue gets worse in the f a → M Pl limit. While we have nothing to say on this important problem, it would be desirable to have some fresh new ideas on how to tackle it (especially in GUTs). For the time being, we can pragmatically postpone the question until the discovery of the axion.
Acknowledgments I wish to thank Enrico Nardi for a careful reading of the manuscript and for insightful observations. I also thank Stefano Bertolini, Maurizio Giannotti, Lukas Graf, Ramona Gröber, Alberto Mariotti, Fabrizio Nesti, Miha Nemevšek, Andreas Ringwald, Shaikh Saad, Goran Senjanović and Carlos Tamarit for helpful and interesting discussions. This work is supported by the Marie Sk lodowska-Curie Individual Fellowship grant AXION-RUSH (GA 840791).

A Two-loop running and one-loop matching
In this Appendix we collect the two-loop beta functions and the one-loop matching coefficient employed in the RG analysis.

Two-loop beta functions
Let us denote the product of gauge factors G = G 1 ×. . .×G N . The two-loop RG equations for the corresponding gauge couplings g i (i = 1, . . . , N ) can be written as where α i = g 2 i /(4π). The one-and two-loop beta coefficients in the MS scheme are [128] (no summation over i) where G i denotes the i-th gauge factor, S 2 (R i ) = T (R i )d(R)/d(R i ) in terms of the Dynkin index (with normalization 1 2 for the fundamental) of the representation R i , T (R i ), and the multiplicity factor d(R)/d(R i ), with d(R i ) (d(R)) denoting the dimension of the representation under G i (G). The latter are related to the Casimir invariant, C 2 (R i ), via C 2 (R i )d(R i ) = T (R i )d(G i ), where d(G i ) is the dimension of G i . κ = 1, 1 2 for Dirac, Weyl fermions (F ) and η = 1, 1 2 for complex, real scalar (S) fields, respectively. The Yukawa contribution in the two-loop beta coefficient has been neglected. 14 Specifically, for the two running stages considered in this work we have (using the ESH intermediate-scale scalars in Table 2

One-loop matching coefficients
The general form of the one-loop matching condition between effective theories in the framework of dimensional regularization was derived in [129,130] (see also [10] for the inclusion of U(1) mixing). Considering for definiteness the case of a simple group G spontaneously broken into subgroups G i , the one-loop matching (at the matching scale µ) for the gauge couplings can be written as [131] α −1 i (µ) = α −1 (µ) − 4πλ i (µ) , (A.7) where λ i (µ) = 1 12π (C 2 (G) − C 2 (G i )) with V , F and S denoting the massive vectors, fermions and scalars that are integrated out at the matching scale µ. Note that differently from [129,130] the (Feynman gauge) Goldstone bosons have been conveniently included in the scalar part of the expression, so that the matching coefficients resembles the structure of the one-loop beta coefficients in Eq. (A.2). Specifically, for the two matching scales considered in this work we have: • M LR (G SM ↔ G