Accidental Peccei-Quinn Symmetry from Discrete Flavour Symmetry and Pati-Salam

We show how an accidental $U(1)$ Peccei-Quinn (PQ) symmetry can arise from a discrete $A_4$ family symmetry combined with a discrete flavour symmetry $ \mathbb{Z}_3 \times \mathbb{Z}_5^2 $, in a realistic Pati-Salam unified theory of flavour. Imposing only these discrete flavour symmetries, the axion solution to the strong $ CP $ problem is protected from PQ-breaking operators to the required degree. A QCD axion arises from a linear combination of $ A_4 $ triplet flavons, which are also responsible for fermion flavour structures due to their vacuum alignments. We find that the requirement of an accidental PQ symmetry arising from a discrete flavour symmetry constrains the form of the Yukawa matrices, providing a link between flavour and the strong $ CP $ problem. Our model predicts specific flavour-violating couplings of the flavourful axion and thus puts a strong limit on the axion scale from kaon decays.


Introduction
Perhaps the best explanation for why CP violation does not appear in strong interactions is to postulate a Peccei-Quinn (PQ) symmetry: a QCD-anomalous global U (1) symmetry which is broken spontaneously, leading to a pseudo-Goldstone boson called the QCD axion [1]. Typically, the PQ symmetry is realized by introducing heavy vector-like quarks (the KSVZ model) [2] or by extending the Higgs sector (the DFSZ model) [3]. The QCD axion is also a good candidate for dark matter [4] within the allowed window of the axion (or PQ symmetry-breaking) scale f a = 10 9−12 GeV [5].
It has also been realised that the PQ axion need not emerge from an exact global U (1) symmetry, but could result from some discrete symmetry or continuous gauge symmetry leading to an accidental global U (1) symmetry. Considering the observed accuracy of strong-CP invariance, it is enough to protect the PQ symmetry up to some higherdimensional operators [6]. In this regard, it is appealing to consider an approximate PQ symmetry gauranteed by discrete (gauge) symmetries [7]. Alternatively, an attempt to link PQ symmetry protected by continuous gauge symmetries to the flavour problem was made in [8]. 4 Despite being the leading candidate for a resolution to the strong CP problem, the origin of the PQ symmetry and its possible connection with other aspects of physics remains unclear. It is possible that PQ symmetry is related to flavour symmetries, which are a compelling proposal for understanding the origin of the fermion masses and mixing. Indeed it has already been proposed that the PQ symmetry arises from flavour symmetries [10], linking the axion scale to the flavour symmetry-breaking scale, and various attempts have been made to incorporate such a flavourful PQ symmetry as a part of such continuous flavour symmetries [11,12]. The resultant axion is sometimes dubbed a "flaxion" or "axiflavon". We shall refer to it as a "flavourful axion".
In recent years there has been considerable work on discrete flavour symmetries applied to understanding lepton -especially neutrino -masses and mixing parameters [13]. Motivated by this, we wish to put forward a new idea, namely that the PQ symmetry could arise accidentally from such discrete flavour symmetries [13]. 5 In order to include the quarks, as is necessary to resolve the strong CP problem, we shall combine discrete flavour symmetries with unified gauge theories where both quarks and leptons appear on an equal footing [13]. However for a discrete flavoured grand unified theory (GUT) based on SO(10) [16], the flavour symmetry is often partly broken at the GUT scale, and it is hard to accommodate the traditional axion window below 10 12 GeV.
In this work, we explore the possibilities of realizing an approximate or accidental PQ symmetry starting from a discrete flavour symmetry which controls both quarks and leptons via a Pati-Salam unification, which allows a lower flavour breaking scale [17]. The idea is that the discrete flavour symmetry and the resulting accidental PQ symmetry are both spontaneously broken by flavons at around 10 11 GeV, leading to the observed flavour structure as well as the (approximate) QCD axion at the same time. We demonstrate that a QCD axion at the correct scale can be achieved in a variant of the flavoured Pati-Salam (PS) model [17] employing an A 4 × Z 5 flavour symmetry, referred to as the "A to Z" model. We also show such a model is compatible with current quark and lepton mass and mixing data.
The layout of the remainder of the paper is as follows. In Section 2 we present the modified A to Z model: its field content, symmetries, and the superpotential responsible for flavour structures. In Section 3 we show how it solves the strong CP problem, taking into account also higher-order corrections. Section 4 details the fermion mass and Yukawa matrices, and we perform a fit of the model to experimental data. Flavour constraints on the PQ-breaking scale are also considered in Section 5. Section 6 concludes.

The A to Z model
In this section we first give an overview of the original model, then propose a modification of it which is suitable for solving the strong CP problem via an accidental global U (1) P Q symmetry emerging from an extended discrete flavour symmetry.

Overview of the original model
We here present the main features of the original A to Z model first introduced in [17], before going on to introduce the modifications necessary to fully realise the automatic U (1) P Q symmetry to the required accuracy. The model is based on the PS gauge group, which unifies left-handed (L) quarks and leptons, F i (4, 2, 1) and charge-conjugated righthanded (R) quarks and leptons F c i (4,1,2), interpreting lepton number as a fourth colour, and where i = 1, 2, 3 is a family index. In order to unify the left-handed families, it postulates an A 4 non-Abelian discrete flavour symmetry. All left-handed SM chiral fields are united in an A 4 triplet F (3,4,2,1), under A 4 × G P S , while right-handed fields are contained in three F c i (1,4,1,2), one for each generation. F couples to A 4 triplet flavons φ u 1,2 (3, 1, 1, 1), φ d 1,2 (3, 1, 1, 1), under A 4 × G P S , whose vacuum expectation values (VEVs) are aligned in particular directions according to constrained sequential dominance (CSD). More precisely, the model realises the CSD(4) alignments along the A 4 triplet directions, first explored in [18]. The fermion Yukawa matrices arise from non-renormalisable terms of the generic form (F ·φ)hF c , where h denotes a Higgs superfield; these terms are realised by a renormalisable superpotential involving messengers, generally denoted X, which are integrated out below the GUT scale.
As discussed in more detail below, CSD(4) leads to up-type quark and neutrino Yukawa matrices (Y u and Y ν , respectively) of the form (in LR convention) while the down-type quark and charged lepton Yukawa matrices (Y d and Y e ) are approximately diagonal, owing to flavons φ d 1,2 whose VEVs are aligned in the (1, 0, 0) and (0, 1, 0) directions. The third family couplings arise from a renormalisable interaction These mass matrix structures yield predictions for quark and lepton mixing, including a natural prediction for the Cabibbo angle θ q 12 ∼ 1/4 and a neutrino reactor angle θ 13 ∼ 9.5 • , subject to small corrections from the off-diagonal elements of Y d,e . The original prediction for θ 13 agreed well with experiment at the time, while more recent global fits (e.g. [19]) prefer smaller θ 13 ≈ 8.5 • . In the modified theory presented in this paper, the precise structure of the Yukawa matrices differ slightly; we show that it can accommodate current experimental data. The Higgs sector is discussed in some detail in [17], with an explicit mechanism given for spontaneous breaking of PS → SM. As all Higgs fields are understood to be neutral under the accidental PQ symmetry, these results remain intact, and will not be discussed further here; we refer interested readers to the original paper.
The original A to Z model also involves a discrete Z 5 flavour symmetry. We will show below that the model very nearly possesses an accidental global U (1) symmetry, suggesting that a PQ solution to the strong CP problem may be realised in this model with only minor modifications. The most significant addition is a discrete Z 3 symmetry under which matter, flavons, and messengers are charged. This ensures certain (previously allowed) operators in the renormalisable superpotential, which explicitly break the accidental U (1) P Q , are forbidden. An additional Z 5 is necessary to forbid higher-order terms up to a required order. Finally, the mechanism originally proposed to drive the flavon VEVs to a particular scale is not suitable since it does not respect the accidental U (1) P Q . Instead we shall discuss an alternative mechanism which preserves U (1) P Q , and discuss its implications for flavour.

The modified model
The field content of the modified A to Z model are given in Table 1, showing their charges under gauge and discrete symmetries, as well as the inferred charges under the accidental U (1) P Q , although we emphasise that this symmetry is not enforced but arises as a consequence of the discrete flavour symmetry. We split the renormalisable superpotential into several parts, The Higgs part W H plays no role in the solution to the strong CP problem since the Higgs fields h i , H c are invariant under U (1) P Q and have zero inferred PQ charges. In fact the only fields whose scalar components get VEVs and which carry PQ charge are the flavons, which are therefore solely responsible for PQ-symmetry breaking. The driving Table 1: The basic Higgs, matter, flavon and messenger content of the model. α = γ = e 2πi/5 , and β = e 2πi/3 . R is a supersymmetric R-symmetry. We emphasise that U (1) P Q is a resulting approximate PQ symmetry which is not imposed directly, but emerges as an accidental result of the discrete flavour symmetry.
superpotential W driving , which sets the scale of flavon VEVs, will be discussed shortly. The "fermion" part W F , containing the couplings of F and F c i to flavons and X messengers as well as messenger couplings to Σ fields, is given by The "Majorana" part W Maj gives the right-handed neutrino mass matrix, and is given by After integrating out X messengers, we obtain the effective superpotentials, which also  preserve U (1) P Q , Figures 1 and 2 show how non-renormalisable terms arise in W eff F and W eff Maj , respectively, from diagrams involving X messengers which receive large masses either from Σ fields which get VEVs, or have direct heavy masses Λ.
It is worth recalling that, in the original model, the superpotential allowed a term which populated the (1,3) and (2,3) elements of Y d and Y e . These terms, which would violate the accidental PQ symmetry, are now both forbidden by the extended flavour symmetry.

Driving sector
Yukawa textures are controlled by vacuum alignments of triplet flavons. In addition, we wish to drive the flavon VEVs to a particular scale. One possible mechanism which achieves this while preserving the strong CP solution is to introduce "conjugate" flavons, labelled φ u,d 1,2 ,ξ, which have charges opposite to φ u,d 1,2 and ξ, respectively, including under U (1) P Q . We are able to drive flavon VEVs to a scale M via a superpotential with each of the five φφ orξξ pairs driven by the F -term equation of the corresponding driving field P , which has R = 2.
It is worth recalling that the original A to Z model permitted bilinear flavon terms φ u 1 φ d 2 and φ u 2 φ d 1 , invariant under all discrete symmetries. These coupled to driving fields P ij , giving rise to driving terms W ⊃ P 12 . However these bilinears have total charge 3 under U (1) P Q , breaking it explicitly. As such, the original mechanism is not compatible with an accidental PQ symmetry. This could have been remedied by removing the driving fields P ij from the theory, but this would have left the question of how flavons acquire non-zero VEVs. 6 In the previous discussion we have doubled the flavon sector by introducing φ u,d 1,2 with opposite effective PQ charges as shown in Table 1 and Eq. 9.
3 Solution to the strong CP problem 3.1 Accidental QCD axion from flavon fields The modified model described in the previous section has the necessary ingredients to realise a PQ symmetry (leading to a QCD axion): a global chiral U (1) symmetry, which is spontaneously broken when A 4 -triplet flavon fields acquire non-zero VEVs, and a colour anomaly, which is ensured by the (standard) left-and right-handed fermions (contained in F and F c i , respectively) transforming differently under the U (1) P Q .
We now show in more detail how the A to Z model solves the strong CP problem. The QCD axion a arises as a combination of the phases of the flavons φ u,d 1,2 and ξ, where x ϕ denotes the PQ charge of a flavon ϕ (that is, v ϕ is the VEV of ϕ (i.e. ϕ † ϕ = v 2 ϕ /2), 7 a ϕ is the phase field of ϕ, and v P Q = ϕ x 2 ϕ v 2 ϕ . In order to get the correct scale of the Yukawa couplings for the first and second families of fermions, we require these to acquire VEVs of O(10 11 ) GeV, which is the desired scale of PQ breaking.
The QCD anomaly number N a of the PQ symmetry is determined by the sum of the PQ charges of the fermion fields F, F c i and messengers X. As the X are vector-like, they do not contribute to N a and thus we have N a ≡ |6x F + 2 i x F c i | = 6. 6 It is possible that the flavon VEVs are driven by another mechanism entirely, without introducing new conjugate flavons and driving fields. For instance, in a theory of radiative SUSY breaking, loop corrections may drive the mass parameter to a negative value, leading to non-zero values at the potential minimum [20]. 7 More precisely for A 4 triplets, v ϕ / √ 2 is the constant of proportionality of the alignments described in Eq. 2.
The axion-gluon-gluon coupling is: where f a = v P Q /N a , which leads to the axion mass m a ≈ m π f π /f a . Although our model differs from the DFSZ model in that the standard Higgs doublets are not charged under the PQ symmetry, the QCD anomaly is the same and the axion phenomenology is very similar. A crucial difference comes from flavour-violating axion couplings, as will be discussed in a later section. Since our model has a number of discrete symmetries which are supposed to be broken at around 10 11 GeV, it is vulnerable to a dangerous domain wall problem. This can in principle be evaded if symmetry breaking occurs during inflation. However we shall not discuss inflation here.

Corrections from higher-order operators
Flavour models based on non-Abelian discrete symmetries typically require the inclusion of one or more Abelian "shaping symmetry" groups which ensure the superpotential includes only desirable terms at low order, leading to predictive mass structures [13]. The Z 3 symmetry ensures a U (1) P Q in the renormalisable theory involving the flavons φ u,d 1,2 . However, high-dimensional operators involving powers of the PQ scalar field(s) (in this model, flavons) explicitly violating U (1) P Q may be present to shift the axion potential away from its CP -conserving minimum [6].
We first consider higher-dimensional terms of the form where {φ} n denotes any combination of flavons φ u,d 1,2 (or their φ counterparts) which are allowed by the discrete flavour symmetry and R-symmetry under consideration, but which do not respect the accidental PQ symmetry.
In the context of supergravity, supersymmetry breaking generically leads to the VEV W ∼ m 3/2 M 2 P where m 3/2 is the gravitino mass. The operator in Eq. 12 then generates a PQ-breaking axion mass contribution To preserve the axion solution to the strong CP problem, we require m 2 * /m 2 a < 10 −10 , where the standard axion mass due to QCD instantons is given by m 2 a ≈ m 2 π f 2 π /f 2 a [6]. Taking v P Q ∼ 10 11 GeV and M P = 2 × 10 18 GeV, the above condition is satisfied with n > 7.
We must therefore forbid all flavon combinations up to n = 7, or equivalently superpotential terms with D = 10 (since dim(W ) = 3), to ensure the solution to the strong CP problem protected to sufficient order. This cannot be achieved by Z 5 × Z 3 alone. This requires the additional Z 5 symmetry, which protects the PQ solution, but largely does not alter the renormalisable superpotential or flavour predictions. It is worth noting, however, that if we assume flavon VEVs are driven by the superpotential proposed in Eq. 9, the gauge and A 4 × Z 5 × Z 3 symmetries would also permit renormalisable interactions involving the conjugate flavons φ u,d 1,2 with matter of the form X F φF , which violate U (1) P Q . However the Z 5 ensures these terms are forbidden.

Mass structures and numerical fit
In this section we show that the modified model provides a realistic description of all quark and lepton masses and mixing parameters. From the terms in Eq. 7, when the flavons acquire CSD(4) VEVs, we arrive at the Yukawa matrices where all parameters are in general complex. The parameters a, b, y 0 d , and y 0 s are proportional to the VEVs of flavons φ u 1 , φ u 2 , φ d 1 , and φ d 2 , respectively, while c, y 0 b derive from the renormalisable coupling F h 3 F c 3 . Elements proportional to B arise from the term F φ u 1 h d F c 1 ; B is expected to be an O(1) number. 13,23 parametrise a small admixture of h 3 into h u , which is not specified by the model, while x is related to the mixing between doublets within h d and h d 15 , and is taken to be real. One overall phase in each Yukawa matrix is unphysical; we may choose c and y 0 b to be real without loss of generality. The right-handed neutrino mass matrix M R is given by The neutrino matrix after seesaw is where we have neglected the contributions from 13,23 , as they are small and appear only at O( 2 ). We also neglect contributions from the off-diagonal elements M 13 in M R . The Yukawa matrices differ from those in the original model in two ways. Off-diagonal (1,3) and (2,3) elements of Y d and Y e are now zero, as the effective terms that produced these contributions (Eq. 8) are forbidden by the Z 3 symmetry. Moreover, all phases are now free, unlike in the original model where they were fixed to discrete multiples of 2π/5. This phase fixing derived from the driving superpotential which, as noted above, is incompatible with a PQ solution. In summary, the additional texture zeroes in Y d,e increase overall predictivity, while the additional phase freedom decreases it.
The best fit of the model to quark [21] and lepton [19] data is given in Tables 2 and  3 Table 2: Model predictions in the lepton sector, at the GUT scale. We set tan β = 5, M SUSY = 1 TeV andη b = −0.24. The model interval is a Bayesian 95% credible interval. The bound on m i is taken from [22].
fit to mass and mixing parameters at the GUT scale, and to estimate the predicted ranges of physical parameters. The running of Yukawa couplings and quark mixing angles depends on various SUSY-breaking parameters, including tan β, the scale of SUSY breaking M SUSY , and threshold corrections. These have been studied in [21], which parametrise SUSY threshold effects in terms of several parametersη i . We assume a small correction corresponding to the choiceη b = −0.24, which primarily affects the b quark Yukawa coupling, ensuring y b ≈ y τ at M GUT as predicted by the model. All other threshold parameters are set to zero. We assume no running in the PMNS parameters. The intervals given correspond to so-called 95% credible intervals, which correspond, in standard Bayesian formalism, to the regions in parameter space of highest posterior (probability) density (hpd). 8 The fit gives minimum χ 2 ≈ 7.2. Generically the model predicts a rather large Cabibbo angle θ q 12 ∼ 1/4, while θ q 13,23 are expected to be much smaller. There is sufficient parameter freedom to attain a fit to all quark mass and mixing parameters to within 1σ of their experimental best fits. However in the lepton sector a perfect fit is never attained: the fit consistently predicts an atmospheric angle θ 23 ∼ 38 − 39 • , well below the current experimental value around 42 • . This is the dominant contribution to the χ 2 . This agrees with earlier analytical results: as shown in [18], when the mass parameters m a,b in m ν are fixed to give the correct neutrino mass-squared differences, 9 the mixing angles in CSD (4) are found to obey an approximate sum rule θ 23    θ 23 . This prediction may be tested by increased precision in the measurement of the PMNS matrix. The fit also gives δ ≈ −120 • , in agreement with a previous numerical study of CSD(n) models [23], and encouragingly close to current experimental hints for a normal neutrino hierarchy. However, phase freedom in the mass matrices allows also a fit where δ ≈ +120 • , with other parameters essentially the same. CSD(4) alone cannot predict the sign of δ .

Flavour constraints on the flavourful axion scale
Contrary to the usual KSVZ or DFSZ axion model, a flavourful axion model allows general flavour-violating couplings of the axion which may constrain the axion scale more strongly. As noted in [12], a severe limit is obtained from the kaon decay K + → π + a. The Yukawa structure in Eq. 14 leads to the mass matrix mis-aligned with the axion coupling matrix due to the flavour-dependent PQ charges. As a result, our model predicts a specific flavour-violating coupling to down (d) and strange (s) quarks where m d 21 = B|y 0 d |v d / √ 2. To understand the above equation, note that a flavon field ϕ is effectively expressed by ϕ = v ϕ e iaϕ/vϕ where the phase field contains the axion component as a ϕ = x ϕ v ϕ a/v P Q + · · · (see Eq. 10). Thus we get ϕ = v ϕ (1 + ix ϕ a/v P Q + · · · ) inducing the axion coupling matrix to down-type quarks Therefore, one finds the a-s-d coupling of Eq. 17 in the mass basis after diagonalising Y d in Eq. 14.

Conclusion
We have investigated the possibility that an accidental PQ symmetry could arise from discrete flavour symmetry, which represents the first study of its kind. The ingredients of the model are a discrete flavour symmetry which encompasses both leptons and quarks, where the PQ symmetry is not imposed by hand but emerges accidentally, and is spontaneously broken by flavons, resulting in a flavourful axion.
To be concrete, we have presented a solution to the strong CP problem in a supersymmetric unified model of flavour, which is a modification of the A to Z of flavour Pati-Salam model, based on Pati-Salam and A 4 symmetry, together with an Abelian discrete flavour symmetry. With some modifications to the original model, an accidental Peccei-Quinn symmetry is realised at the renormalisable level, and spontaneously broken by the VEVs of A 4 triplet flavons, which are also responsible for explaining the flavour structures of quarks and leptons via the CSD(4) vacuum alignments. For the first time, we have shown how a PQ symmetry can arise purely from discrete flavour symmetries, where we have ensured that the accidental U (1) P Q is protected to sufficiently high order, with all higher-order operators suppressed by the Planck mass are forbidden up to dimension 10.
To achieve this, in addition to the original A 4 × Z 5 discrete flavour symmetry, we also introduced a Z 3 × Z 5 symmetry, under which flavons as well as right-handed SM matter fields (contained in PS multiplets F c i ) are charged. However, no new scalar fields were necessary to realise the U (1) P Q symmetry: the same flavons already introduced to explain flavour structures also give rise to a QCD axion. The Z 3 symmetry is sufficient to ensure the PQ symmetry at the renormalisable level. It also necessarily forbids several terms in the Yukawa superpotential allowed in the original model, modifying the predictions for flavour. The Z 5 is primarily responsible for protecting the PQ solution against higherorder terms suppressed by powers of M P , up to D = 10.
The originally proposed superpotential which drives the flavons to have non-zero VEVs, commonly used to drive VEVs in models of this kind, turned out to be generally incompatible with a PQ symmetry. In order to overcome this we have suggested an alternative mechanism, which respects the PQ solution, wherein flavons φ couple to "conjugate" flavons φ which have opposite charges under all symmetries. The Z 5 symmetry is then essential also in forbidding dangerous renormalisable couplings of conjugate flavons to matter, which spoils both the PQ symmetry and the predictive flavour structures.
The accidental QCD axion arising from the flavourful PQ symmetry in our model shares many phenomenological properties with the conventional DFSZ axion. A crucial difference comes from its flavour-violating couplings determined by the predicted Yukawa structure and flavour-dependent PQ charges. The specific prediction for the a-s-d coupling allows us to probe the axion scale f a up to 3 × 10 10 GeV in the NA62 experiment. We look forward to a new era in flavourful axion model building and phenomenology, where the discrete symmetries responsible for flavour may also accidentally yield a global PQ symmetry and resolve the strong CP problem.