Axion Dark Matter, Proton Decay and Unification

We discuss the possibility to predict the QCD axion mass in the context of grand unified theories. We investigate the implementation of the DFSZ mechanism in the context of renormalizable SU(5) theories. In the simplest theory, the axion mass can be predicted with good precision in the range ma = (2 − 16) neV, and there is a strong correlation between the predictions for the axion mass and proton decay rates. In this context, we predict an upper bound for the proton decay channels with antineutrinos, τ(p→ K+ν̄) . 4× 1037 yr and τ(p→ π+ν̄) . 2× 1036 yr. This theory can be considered as the minimal realistic grand unified theory with the DFSZ mechanism and it can be fully tested by proton decay and axion experiments. ar X iv :1 91 1. 05 73 8v 1 [ he pph ] 1 3 N ov 2 01 9

1 Introduction a global PQ symmetry and is realistic, in the sense that reproduces the values of the gauge couplings at the electroweak scale and can explain the masses for the charged fermions. In section 3 we discuss how to implement the DFSZ mechanism in the context of SU (5) GUTs and establish a direct connection between the Peccei-Quinn scale and the GUT scale. We also demonstrate that this theory is able to provide predictions for the axion mass and the proton lifetime in the channel involving anti-neutrinos. In section 4 we discuss the testability of the theory by studying the axion-photon coupling and the axion coupling to the neutron electric dipole moment in the predicted mass window.
allows for the implementation of the DFSZ mechanism once the symmetry is spontaneously broken. In this context, the SU(5) ⊗ U(1) PQ theory has the following Yukawa interactions: L Y = Y u 10 10 5 H + 105 (Y 1 Since the above terms must respect the global PQ symmetry, it follows that the U(1) PQ charges are given by where the PQ charge of 24 H is determined by the potential in Eq. (2.1), which defines the mixing between the axion and the Higgs doublets.
In this theory the mass matrices for the charged fermions are given by 3) Notice that M u = M T u has strong implications for the proton decay channels with antineutrinos [27]. In Appendix B we discuss how to extend this model to explain neutrino masses.
3 The DFSZ mechanism in SU (5) The terms in the scalar potential relevant for the DFSZ mechanism can be written in terms of the elements of the SU(5) representations in the following way V ⊃ (m 2 12 + λH † 1 H 2 )H † 1 H 2 + H † 3 Σ 2 0 (λ a1 H 1 + λ a2 H 2 ) + h.c., (3.1) where, following the notation previously introduced, H 1 ⊂ 5 H , H 2 ⊂ 45 H , H 3 ⊂ 5 H and Σ 0 ⊂ 24 H . After spontaneous symmetry breaking, all neutral fields acquire a vacuum expectation value and one can write their CP-odd component as a function of the two Goldstone bosons arising due to the spontaneous breaking of two global symmetries of the potential: Here,â andâ Z are the phases of the axion and the Goldstone boson that will be eaten by the Z, respectively. The factor q is the contribution related to the electroweak quantum numbers and PQ i parametrizes the presence of the axion in each of the scalar representations. The terms of the scalar potential fix the following conditions for the PQ i charges: Notice that the axion in reality is a pseudo-Goldstone boson since the Peccei-Quinn symmetry, although being a good symmetry classically, it is broken at the quantum level. Linearizing the kinetic terms, Orthogonality of the Goldstone bosons requires the following condition whereas the normalization of the kinetic terms of the axion demands that where n a is the normalization of the CP-odd phaseâ = a/n a . The presence of the axion in each of the scalar representations is given by for the last relation we have taken the limit v Σ v 1 , v 2 , v 3 , which is justified by the fact that M GUT is about 13 orders of magnitude higher than the electroweak scale. Comparing Eqs. (3.2)-(3.5) with Eq. (3.13) we conclude that the axion lives predominantly in the Σ 0 field as expected.
In the broken phase, the Yukawa Lagrangian can be rewritten as where the axion can be rotated away from the Yukawa Lagrangian by performing the following chiral rotations u L/R → e ±i v 2 1 +v 2 2 2v 2 3 a n u L/R , d L/R → e ±i a 2n d L/R , and e L/R → e ±i a 2n e L/R , (3.16) the transformation of the quarks will generate the following aGG term Hence, the Peccei-Quinn scale f a is identified as Then, the connection between the PQ and GUT scales is given by Therefore, if we predict the GUT scale and α GUT we can find the allowed values for the axion mass in this grand unified theory.

Unification Constraints
The fact that 24 H breaks both SU(5) and U(1) PQ symmetries establishes a connection between the PQ and the GUT scales. Therefore, once M GUT and α GUT are known the axion mass is predicted. In this theory, M GUT is determined from the experimental input on the values of the gauge couplings at the low scale. The following RG equations fix the EW values for the gauge couplings as a function of the GUT parameters: where the subindex i = 1, 2 and 3 refers to the three different SM forces U(1) Y , SU(2) L and SU (3) Table 1 we show their contributions to the beta functions.
In order to obtain the following parameters at the M Z scale: sin 2 θ W (M Z ) = 0.23122, α(M Z ) = 1/127.955, and α s (M Z ) = 0.1181 [29], the following conditions must be satisfied [30] B 23 B 12 = 0.717, and ln where unification at the one-loop level has been assumed.
Fields are only sensitive to the relative splitting between the SU(5) representations. In Tab. 1 we also show their B ij contributions. Among the new scalar sector, the Φ 3 and H 2 from the 45 H , together with H 3 in the 5 H help towards unification since the three of them contribute to enhance the ratio B 23 /B 12 needed to satisfy the first condition from Eq. (3.22) since, as it is well known, in the Georgi and Glashow model this ratio is below the required value. The colored octet Φ 1 in the 45 H indirectly helps to unify since it allows for a larger M GUT range according to the second condition from Eq. (3.22). We will assume the rest of the scalar fields to be at the GUT scale since they do not help to achieve unification.
Unification constraints determine M GUT as a function of M Φ 1 and the doublet masses M H 2 and M H 3 , as shown by the blue region in the left panel in Fig. 1. This figure also shows that the lighter Φ 1 , the larger M GUT can be. However, due to experimental bounds derived from collider physics, Φ 1 cannot be arbitrarily light. According to the recent study in Ref. [31], its mass has to be above 1 TeV, which establishes an upper bound for the GUT scale as the figure reflects. The region shaded in purple shows the parameter space ruled out by the collider bounds on Φ 1 . The mass of the Φ 3 ∼ (3, 3, −1/3) is also obtained from the unification constraints and it is implicitly given in the figure: for GeV, as shown explicitly by the blue dots.
On the right panel of Fig. 1 we show the relation between M GUT and α GUT . The region shaded in blue satisfies the unification constraints in Eq. (3.22), as we vary M H 2 and M H 3 from 1 TeV to the GUT scale. The LHC bound on Φ 1 is shown by a purple line. Experimental constraints for proton decay define a lower bound on the GUT scale. In both of the panels in Fig. 1, we show in red the excluded region by the bound on p → K +ν from the Super-Kamiokande (SK) collaboration, τ (p → K +ν ) > 5.9 × 10 33 years [32], whereas the projected bound on the same decay from the Hyper-Kamiokande (HK) collaboration, τ (p → K +ν ) > 3.2 × 10 34 years [33] and the DUNE collaboration, τ (p → K +ν ) > 5 × 10 34 years [34], are shown with a green and orange dashed lines, respectively. These constraints will be addressed in detail in the next section.
To close this section, we emphasize that with the 45 H alone unification can be achieved, as shown in Fig. 1, and the splitting in the 5 H helps to increase the parameter space where unification occurs. We find that the allowed window for the GUT scale is given by where the upper bound is obtained from the collider bounds on M Φ 1 whereas the lower bound is given by experimental constraints on proton decay. We note that there are two upper (and lower) bounds for the GUT scale, depending on the masses of the doublets H 2 and H 3 . In order to define the GUT scale window, we have taken the conservative approach to consider the larger range possible in the context of this theory. We also note that the consistency with proton decay bounds in this case is ensured by the PQ symmetry, because it forbids the interaction of the scalar leptoquark Φ 3 with the up-quarks. Otherwise, Φ 3 would mediate proton decay interactions through an effective operator suppressed by two powers of its mass. Hence, in this scenario the PQ symmetry allows Φ 3 to be light in agreement with both unification constraints and proton decay.

Axion Mass and Proton Stability
In this theory the mass matrix for the up-quarks is symmetric and therefore we can predict the decay width for the proton decay channels with antineutrinos as a function of the known mixings at low energy [27]. The decay widths for the p → K +ν and p → π +ν channels in the context of this SU(5) ⊗ U(1) PQ are given by where We note that the C K and C π coefficients are functions of known parameters: the RGE factor A RG , which parametrizes the running between the GUT and the Λ QCD scale [35], and the known values of the CKM matrix. We remark that the calculation of these coefficients in the context of GUTs is only possible if, as in this theory, M u = M T u . For the hadronic matrices we use the results from lattice QCD given in Ref. [36]. Since the proton decay width and the axion mass both depend on the ratio (α GUT /M 2 GUT ), we can relate them by m a 4.5 Therefore, if any of these two decay channels are discovered in proton decay experiments one can automatically predict the other channel and the axion mass, as shown in Fig. 2.
In this figure, we present the prediction of the axion mass from the lifetime of any of the proton decay channels into anti-neutrinos. The red shaded area shows the excluded parameter space from Super-Kamiokande (SK) bounds for both τ (p → K +ν ) > 5.9 × 10 33 years [32] and τ (p → π +ν ) > 3.9 × 10 32 years [37]. The projected bounds on the decay channel p → K +ν from the Hyper-Kamiokande collaboration τ (p → K +ν ) > 3.2 × 10 34 years [33] and the DUNE collaboration τ (p → K +ν ) > 5 × 10 34 years [34] are shown with a green and orange dashed lines, respectively. The purple shaded areas correspond to the parameter space excluded by collider bounds on the colored doublet M Φ 1 > 1 TeV [31], where we have assumed the M H 2,3 = M GUT in order to account for the largest possible range. The white region in Fig. 2 shows the available window for the axion mass in this model, which is predicted to be m a = (1.87 − 16.05) × 10 −9 eV. (3.30) Furthermore, the theory predicts the upper bound on the proton decay lifetime for the channels with antineutrinos, τ (p → K +ν ) 3.5 × 10 37 yr, and τ (p → π +ν ) 1.8 × 10 36 yr, (3.31) which expose the theory to be tested in current or future proton decay experiments. We emphasize that the peculiar feature M u = M T u from this theory allows us to predict the upper bound of the axion mass window.  Figure 2: Correlation between the proton lifetime prediction for the channels p → K +ν and p → π +ν with respect to the axion mass. The regions shaded in red show the parameter space excluded by the proton decay bounds from the Super-Kamiokande collaboration τ (p → K +ν ) > 5.9 × 10 33 years [32] and τ (p → π +ν ) > 3.9 × 10 32 years [37]. The green and orange dashed lines give the projected bounds from the Hyper-Kamiokande collaboration τ (p → K +ν ) > 3.2 × 10 34 years [33] and the DUNE collaboration τ (p → K +ν ) > 5 × 10 34 years [34]. The region shaded in purple is the parameter space ruled out by collider bounds on Φ 1 , i.e. M Φ 1 > 1 TeV [31] assuming M H 2,3 = M GUT . The area in white gives the allowed axion mass and proton lifetimes in the context of this theory.
Unfortunately, the width for the proton decay channel with charged leptons cannot be predicted as a function of known quantities at low energy. The decay width for p → π 0 e + is given by where V e is a flavor factor coming from the combination of some unknown fermion mixing matrices. See Appendix A for further details. As we show in that appendix, although the proton lifetime cannot be predicted in this channel, information about the V e matrix could be inferred from the experimental bounds on proton decay.

Axion phenomenology
In this section, we study the axion couplings to SM particles in the predicted mass window. We focus on the axion to photon coupling and the interaction between the axion and the electric dipole moment of the neutron (nEDM) for which there exists experiments that  The axion coupling to the neutron EDM as a function of the axion mass. The blue band corresponds to the theoretical error on the g aD ocupling [38]. The region shaded in red gives the projected sensitivity to Phase III of the CASPEr-Electric [39] experiment. could probe this scenario. In this work, we consider the case in which the PQ symmetry is broken before inflation and in order to achieve the correct dark matter relic abundance we assume an initial misalignment angle of θ i ≈ 10 −2 [4][5][6].
The interaction between the axion and photons can be obtained by rotating the axion field from the Yukawa terms of the charged fermions with the effective coupling g aγγ given by where second term is the contribution from non-perturbative effects from the axion coupling to QCD and has been computed at NLO in Ref. [40]. We present our results in Fig. 3. The solid blue line corresponds to the g aγγ coupling in the predicted mass window and we show the projected sensitivites of the ABRACADABRA experiment [25]. The broadband approach in its Phase III (B max = 5 T and V = 100 m 3 ) will be sensitive to a portion of this mass window, while Phase III of the resonant approach will be able to cover most of the predicted mass window.
The dark matter axion background field induces an oscillating electric dipole moment for the neutron, given by 3) The CASPEr-Electric [26,39] experiment aims to measure this oscillating nEDM and Phase III of this experiment will probe the lower portion of the predicted mass window as shown in Fig. 3. When the projected sensitivities for ABRACADABRA and CASPEr-Electric are combined, these experiments will be able to fully probe the mass window in Eq. (3.30).
One important difference between this scenario and the one studied in Ref. [19] is that here the axion has a tree-level coupling to electrons. However, all the experimental constraints on this coupling are well above the prediction for the QCD axion with mass around 10 −9 eV. It is important to emphasize that this theory can be fully tested using the predictions for the axion mass and upper bounds on the proton decay lifetimes.

Summary
We discussed the implementation of the DFSZ mechanism for the QCD axion mass in the context of grand unified theories. We have shown that using the idea of Wise, Georgi and Glashow the axion mass can be predicted in a realistic renormalizable grand unified theory. The PQ scale is determined by the GUT scale which allows us to predict the axion mass: m a (2 − 16) × 10 −9 eV.
We have shown that the predictions for the axion couplings can be tested at ABRA-CADABRA and CASPEr-Electric experiments.
The fact that the mass matrix for up-quarks is symmetric allows us to make definitive predictions for the proton decay channels with a meson and antineutrinos. In this theory, the upper bounds on the proton decay lifetimes with antineutrinos are given by τ (p → K +ν ) 4 × 10 37 yr, and τ (p → π +ν ) 2 × 10 36 yr.
This theory is unique due to the fact that it can be fully probed by proton decay experiments such as DUNE and axion experiments such as ABRACADABRA.
A Proton Decay: p → π 0 e + The decay rate for the proton decay channel p → π 0 e + is given by In Fig. 4 we show the predictions for p → π 0 e + , together with the Super-Kamiokande constraints (red area), τ (p → π 0 e + ) > 1.6 × 10 34 years [41], and the Hyper-Kamiokande projected bound (green line), τ (p → π 0 e + ) > 8 × 10 34 years [42]. To illustrate the effect of the unkown matrix V e , we show the numerical predictions for two possible values of this matrix V e = 0.1 and V e = 1. Notice that the matrix V e can be constrained using the proton decay experimental bounds for the decay into charged leptons, but in general the lifetime for this channel cannot be predicted.  Figure 4: Predictions for the proton lifetime from the decay channel p → π 0 e + . The red area shows the Super-Kamiokande constraints, τ (p → π 0 e + ) > 1.6 × 10 34 years [41], while the green dashed line shows the Hyper-Kamiokande collaboration projected bound, τ (p → π 0 e + ) > 8 × 10 34 years [42]. The region shaded in purple is ruled out by the collider constraints on the Φ 1 field, M Φ 1 > 1 TeV [31].

B Neutrino masses
The theory we have discussed so far does not have a mechanism to generate neutrino masses. However, this can be addressed by adding three neutrinos singlets, ν C i , with a Majorana mass term and the interaction Y ν5 5 H ν C , we can then implement the type I seesaw mechanism [43][44][45][46]. This will fix the PQ charges to β = −2α, or β = α/2 if we use 5 H instead of 5 H . The baryon asymmetry of the Universe can then be explained through thermal leptogenesis [47] or through out-of-equilibrium decays of the heavy colored Higgs [48].
An alternative is to introduce a 10 H [49] and implement the Zee mechanism [50], in which neutrinos acquire mass at the radiative level. In this scenario, the relevant Lagrangian for neutrino masses is given by We note that the Higgs in the 45 H plays a twofold role: it corrects the mass relation between charged leptons and down-type quarks and contributes to the generation of neutrino masses at the quantum level. We also note that the implementation of this mechanism in the renormalizable SU (5)  These are basically the simplest possibilities to generate neutrino masses when implementing only the DFSZ mechanism for the axion mass. See our recent study in Ref. [19] for the discussion of other possibilities.