An Axion-Like Particle from an SO ( 10 ) Seesaw with U ( 1 ) B − L

We investigate the decoupling of heavy right handed neutrinos in the context of an SO(10) GUT model, where a remnant anomalous symmetry is U(1)B−L. In this model the see-saw mechanism which generates the neutrino masses is intertwined with the Stueckelberg mechanism, which leaves the CP-odd phase of a very heavy Higgs in the low energy spectrum as an axion-like dark matter particle. Such pseudoscalar is predicted to be ultralight, in the 10 eV mass range. In this scenario, the remnant anomalous B−L symmetry of the particles of the Standard Model is interpreted as due to the decoupling of the right-handed neutrino sector which leaves as a signature a direct coupling of the Stueckelberg-like axion to the anomaly. We illustrate this scenario including its realisation in the context of SO(10).


Introduction
Recently there has been considerable interest in the occurrence of axion-like particles [1][2][3] including the appearance in model building af anomalous U (1) symmetries with a Stueckelberg field [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. In this paper we examine the simplest GUT example where this phenomenon is closely related to the see-saw mechanism [20] for generating the neutrino masses and may provide a link between axions and right-handed neutrinos. At the same time our scenario establishes a possible link between leptogenesis and dark matter [21][22][23] in a generalized setting, due to the prediction of an axion in the low energy spectrum. Stueckelberg axions (b(x)) appear in the field theory realization of the Green-Schwarz mechanism of anomaly cancelation of string theory, in the dualization of a 3-form, and correspond to pseudoscalar gauge degrees of freedom (see also the discussion in [18]). As ordinary Nambu-Goldstone modes they undergo under an abelian gauge transformation and are coupled to the anomaly via a dimension-5 operators of the form b(x)/M F ∧ F where F is, generically, the field strength of the gauge fields which share a mixed anomaly with the U (1) symmetry, and M is the Stueckelberg scale. In these scenarios, pseudoscalar gauge degrees of freedom may develop physical components only after the breaking of the shift symmetry by some extra potential. This is expected to occur in the case of phase transitions in a non-abelian gauge theory, when instanton interactions naturally arise and induce a mixing between the Stueckelberg field and the Higgs sector of the theory, with the generation of a periodic potential, after spontaneous symmetry breaking. This scenario in which the CP-odd phases of the scalar sector mix and generate such a potential, has provided the basic template for the emergence of a physical CP-odd state, in a way which is very close to what was conjectured to occur in the case of the electroweak or DFSZ version of the Peccei-Quinn [24] axion (see the review [25]), where the anomalous symmetry is a global rather than a local one. Indeed, we recall that in the DFSZ case one writes down a general potential, function of three scalar fields, which is SU (2) × U (1) invariant. The simplest realization of this scenario is in the two-Higgs doublet model, where the Higgs fields H u and H d are assigned the global symmetry under U (1) P Q and are accompanied by an additional scalar Φ, which is singlet under the Standard with X u + X d = −2X Φ . The potential is given by a combination of terms of the form which is invariant under the Standard Model gauge symmetry and is in addition invariant under the global U (1) P Q . As pointed out in [19] a similar effective theory can be obtained in the case of a gauge symmetry, in a scenario that leaves most of the intermediate steps in the generation of Stueckelberg-like Lagrangian unchanged. In this realization of such Lagrangian, the Stueckelberg pseudoscalar emerges from the phase of the complex scalar field which is responsible for the breaking of the gauged U (1) symmetry. The breaking takes place at the GUT (Grand Unified Theory) scale, which takes the role of the Stueckelberg mass for the low energy effective theory. In our case such abelian symmetry is contained within SO(10) and it is identified with U (1) B−L . This provides the basic observation which motivates our work, which connects the decoupling of a gauge boson corresponding to an U (1) B−L symmetry within SO(10) and of a right-handed neutrino to the appearance of an axion in the spectrum of the low energy theory. Building on a similar analysis by two of us in [4] based on a E 6 × U (1) X , such axion is expected to be ultralight, in the 10 −20 eV mass range.

Incomplete decoupling of a chiral fermion and global anomalous U(1) B−L
We believe it is useful to scrutinise this within a transparent model where two examples of physics beyond the Standard Model (SM), the non-zero neutrino masses and the Stueckelberg axion are closely related. Since we know from experiment [26] that the first extension exists in Nature, it increases our expectation that the second should be realised. We shall review the group theory of SO(10) including the available irreducible representations for the matter particles and the symmetry breaking. SO(10) naturally provides three right-handed neutrinos which can participate in the see-saw. Because of the decoupling of these additional neutrino states at high masses, the resultant effective theory possesses an anomalous U (1) B−L symmetry. Since we shall discuss neutrino masses it is worth recalling the various possibilities for introducing them into the minimal SM. We shall mention four of these, one being the see-saw mechanism, and reveal why the other three are less attractive. One of them, introduced in [27], once appeared to be compelling when based only on the SuperKamiokande experiment [26] but it predicted maximal solar neutrino mixing which unfortunately was subsequently excluded by the SNO experiment [28]. This left as the most popular possibility the see-saw mechanism which we shall employ in the present model. When neutrino masses were established experimentally in 1998 there was confusion about to whom priority for the see-saw idea belonged and it was temporarily assigned to a number of theory papers published in 1979. Further scholarship revealed, however, that priority belonged to a 1977 paper by Minkowski [20].

SO(10) Grand Unification
The SO(10) model for unifying quarks and leptons was invented over forty years ago in [29,30]. After non-zero masses for neutrinos were discovered, it became the most popular GUT superseding the otherwise more economical SU(5) GUT [31]. A recent discussion of an SO(10) GUT can be found in [32]. In the minimal SM, as in the minimal SU(5) GUT, the neutrinos were assumed to be massless. In the SO(10) GUT, each family in a 16 contains, in addition to the fifteen helicity states of the minimal SM, a right-handed neutrino N . This gives rise to several additional features, beyond the most obvious one that the neutrinos can acquire mass through the see-saw mechanism. An SU(5) GUT subsumes the SM gauge group SU (3) C × SU (2) L × U (1) Y but an SO(10) GUT with one additional rank includes also a U (1) B−L . It is this gauged (B − L) symmetry and its breaking which will play a central role in our present discussion. We note that in the minimal SM and SU(5) models (B-L) is anomalous. In SO(10) it is the N R which renders (B-L) anomaly-free. The group theory underlying the SO(10) GUT is well-known and reviewed in many papers; one reliable such reference is [33]. For the purposes of establishing notation we shall briefly discuss this with special emphasis on the role of (B-L) symmetry which will be treated further in subsequent subsections.

Breaking patterns
The gauge group SO(10) has the dimension 45 of its adjoint. An adjoint of scalars can break the symmetry while preserving rank-5 to We shall need more scalars to give mass to the fermions. Each family is in a 16 irreducible representation. For example the first family is where we have designated the colours as r, g, b (= red, green, blue). The Yukawa couplings which can provide fermion masses require scalar fields which are included in 16 × 16 = 10 s + 120 a + 126 s where the subscripts s, a specify symmetric, antisymmetric. The 10 is the vector representation of SO(10), although the spinor representation 16 is really the defining representation, because one can make 10 from 16, as in Eq. (7), but not vice versa. We first consider the decomposition of SU(5) into The states in the first two lines of Eq.(8) are the familiar ones of one SM family, without a right-handed neutrino, which is why (10 +5) is used in an SU (5) GUT. The scalars in the SU(5) Yukawa couplings must be among5 and we note that the usual Higgs boson, which in this notation is the complex doublet (1, 2) ±1 , appears uniquely in the 5 and 45 of SU(5), as can be seen from Eq. (8). Armed with these preliminaries about SU(5), it is rendered almost trivial to extend the analysis to SO(10), but the (B-L) symmetry means we must tread carefully. We return to Eq.(7) and adopt a new notation in the SO(10) decompositions of (SU (5)) (B−L) . From [33] we are able to decompose the scalar SO(10) irreducible representations into their SU(5) components: All of 10, 120 and 126 necessarily contain a candidate for the SM complex Higgs doublet. From Eq. (8), we can, if needed, translate the SU(5) representations in Eq.(10) into SM representations. This provides all the group theory we shall need in the present article. In the following we shall focus on the breaking of U (1) (B−L) which is intimately related to the mass of the right-handed neutrinos N in Eq.(6) and hence to the see-saw mechanism.

The two complex singlet scalars in the effective potential
If we introduce a scalar field Φ, singlet under SU(5) with lepton number L=+2, we can write the Majorana mass M of the right-handed neutrino N i R (i,j =1,2,3) of the three generations as The masses λ ij Φ may be taken to be ∼ 10 10 GeV, far above the weak scale, whereupon we may integrate out the right-handed neutrino N to derive an effective field theory with interesting properties.
In particular, the gauged U (1) (B−L) of the SO (10) GUT has become anomalous, because in the (B−L) 3 triangle diagram N has been removed from the internal states. We note that the 126 of scalars in Eq.(10) contains an SU (5)  Such a statement is obviously not connected to grand unification. Of course, this model is ruled out because neutrinos have non-zero masses so some modification is necessary to the MSM and there is a number of possibilities [34]. The most popular is the addition of right-handed neutrinos which permit the see-saw mechanism for generating neutrino masses. This is achieved most naturally in SO (10) unification. Now we carefully discuss a top-down analysis of SO(10) spontaneous symmetry breaking. At the GUT scale (10 15−16 GeV) the adjoint 45 is used to break the symmetry in a necessarily rank-preserving manner according to

See-Saw Mechanism
In the MSM neutrinos are massless. The minimal standard model involves three chiral neutrino states, but it does not admit renormalizable interactions that can generate neutrino masses. Nevertheless, experimental evidence suggests that both solar and atmospheric neutrinos display flavor oscillations, and hence that neutrinos do have mass. Two very different neutrino squared-mass differences are required to fit the data: where the neutrino masses m i are ordered such that: and the subscripts s and a pertain to solar (s) and atmospheric (a) oscillations respectively. The large uncertainty in ∆ s reflects the several potential explanations of the observed solar neutrino flux: in terms of vacuum oscillations or large-angle or small-angle MSW solutions, but in every case the two independent squared-mass differences must be widely spaced with In a three-family scenario, four neutrino mixing parameters suffice to describe neutrino oscillations, akin to the four Kobayashi-Maskawa parameters in the quark sector. Solar neutrinos may exhibit an energy-independent time-averaged suppression due to ∆ a , as well as energy-dependent oscillations depending on ∆ s /E. Atmospheric neutrinos may exhibit oscillations due to ∆ a , but they are almost entirely unaffected by ∆ s . It is convenient to define neutrino mixing angles as follows: with s i and c i standing for sines and cosines of θ i . For neutrino masses satisfying (13), the vacuum survival probability of solar neutrinos is: whereas the transition probabilities of atmospheric neutrinos are: None of these probabilities depend on δ, the measure of CP violation. Let us turn to the origin of neutrino masses. Among the many renormalizable and gauge-invariant extensions of the standard model that can do the trick are [34] (i) The introduction of a complex triplet of mesons (T ++ , T + , T 0 ) coupled bilinearly to pairs of lepton doublets. They must also couple bilinearly to the Higgs doublet(s) so as to avoid spontaneous (B − L) violation and the appearance of a massless and experimentally excluded majoron. This mechanism can generate an arbitrary complex symmetric Majorana mass matrix for neutrinos. (ii) The introduction of singlet counterparts to the neutrinos with very large Majorana masses. The interplay between these mass terms and those generated by the Higgs boson, the so-called see-saw mechanism, yields an arbitrary but naturally small Majorana neutrino mass matrix. (iii) The introduction of a charged singlet meson f + coupled antisymmetrically to pairs of lepton doublets, and a doubly-charged singlet meson g ++ coupled bilinearly both to pairs of lepton singlets and to pairs of f-mesons. An arbitrary Majorana neutrino mass matrix is generated in two loops. (iv) The introduction of a charged singlet meson f + coupled antisymmetrically to pairs of lepton doublets and (also antisymmetrically) to a pair of Higgs doublets. This simple mechanism was first proposed in [27] and results at one loop in a Majorana mass matrix in the flavor basis (e, µ, τ ) of a special form: This Zee model is attractive as an simple extension of the SM. It predicts maximal solar neutrino mixing, θ 12 = π 4 , a value which was strongly disfavoured by SNO data [28,35]. Of all the models preserving only the three chiral left-handed neutrinos of the SM -models (i), (iii) and (iv) abovemodel (iv) is surely the most appealing and it fails. Therefore one is led to additional neutrino states, typically two or more massive right-handed neutrinos which we denote N I (i = 1, 2, . . . , p). In the model we shall discuss p is necessarily p = 3 because each of the three quark-lepton families is in a 16 of SO(10) and each contains one N state. There has been considerable interest in more minimal models with p = 2 as introduced in the so-called FGY model of [36]. This choice has the property of reducing the number of free parameters such that the CP-violating phase in N i mixing matrix is simply related to the CP-violating phase, δ, in Eq. (16). This means that the measurement of δ in long-baseline neutrino oscillation experiment would shine light on the origin of matter-antimatter asymmetry arising from leptogenesis [37] where it arises from N i decay. In general, this connection does not exist so that an optimistic logic could argue that the FGY model, sometimes called the minimal see-saw, is possibly correct. For the present case of p = 3 we introduce a mass basis Coincidentally, and suggestively, such a mass fits well with the mass required for successful leptogenesis [37]. This discussion exhibits the great advantage of the see-saw mechanism compared to the alternative models discussed above: the smallness of the neutrino masses relative to those of the quarks and leptons occurs naturally. That being said, the other side of the coin is that experimental observation of the very massive N i is challenging. The crucial observation for our present purposes is to consider the U (1) B−L triangle anomalies. If we keep all the states in Eq.(6) for one family 45 ⊃ 1 + 10 +10 + 24 (28) so that the 24 can provide the rank-preserving SU (5) → SU (3) × SU (2) × U (1). We recall that in SO (10), 45 decomposes as in Eq. 10 within which the 24 can provide the rankpreserving SU (5) → SU (3) × SU (2) × U (1) Y symmetry breaking. The fermion masses arise from the Yukawa couplings which may be understood to contain the coupling of Eq.(11). We adopt the convention that Latin indices a, b, c, . . . run from 1 to 10 and Greek indices α, β, γ, . . . run from 1 to 16. The vector field V is V a and the adjoint A is A ab = −A ba so that all the V and A couplings up to quartic in the Higgs potential can be written, bearing in mind that 10 × 10 ⊃ 1 + 45 + 50 10 × 45 ⊃ 10 + 120 + 320 45 × 45 ⊃ 1 + 45 + 54 + 210 + 770 + 945.
in the form among other terms.
To deal with the 126 it is essential to introduce the Γ matrices Γ a αβ (32) which are ten 16 × 16 matrices which roughly generalise the four 4 × 4 Dirac matrices γ µ pertinent to O(4), and likewise satisfy a Clifford algebra. The Φ field of the 126 is a symmetric scalar field satisfying the trace condition Γ a ij Φ ji = T r(Γ a Φ) = 0 to write the Higgs potential terms involving Φ such as among other terms including mixed Φ − A terms possible under SO(10) symmetry, as can be seen from Eqs. (30) and (34). We take note of the cubic scalar coupling 16.16.126 which may be written and which we shall use in the next section.

Stueckelberg Axion
In order to illustrate how the mixing of the CP-odd phases takes place in the breaking of SO(10) → SU (5) × U (1) we consider specific terms in the potential, describing the conditions which need to be satisfied in order to generate a periodic potential function of a single gauge invariant field. The latter takes the role of a physical axion and will be denoted by χ.
The periodic potential is assumed to be generated at the scale at which the SO(10) symmetry is broken to SU (5) × U (1) while, always at this scale, instanton effects are present. In order to illustrate how this may happen, one starts by considering a typical SO(10) invariant term in the original theory such as 16 × 16 × 126 (37) which is built out of the spinorial (16) where the two contributions in brackets refer respectively to the 126 s and to the 10 s of SO (10).
It is quite clear that a term of this form in the potential allows to induce a mixing of the CP-odd phases of the two SU(5) singlet representations in such a way that one linear combination of these will correspond to a physical axion while the second one will be part of the Nambu-Goldstone mode generated by the breaking of U (1) B−L . We will be denoting with σ and φ the two fields corresponding to the 1 −10 and 1 10 respectively, denoting their vevs with v σ and v φ respectively. We will assume that v φ will be large in such a way to provide a mass term for the right-handed neutrino, as specified in (11) using the Majorana operator N R N R φ.
In order to characterize the structure of the Stueckelberg Lagrangian at classical level we focus our attention on the extra (periodic) potential related to σ and φ The coupling λ, assumed to be instanton generated at the scale M I at which the N R decouples, as we are going to discuss next, provides a drastic suppression in V p . We parameterize both fields around their vevs as and we will allow v φ to reach the large scale M I ∼ 10 10 GeV where the decoupling of the N R takes place. The parameterization of V p in a broken phase is made possible by the remaining -non periodic -general scalar potential, which is expected to assume a typical mexican-hat shape as for an ordinary U (1) symmetry. Both σ and φ are charged under U (1) B−L and therefore their vevs break the gauged (B − L) which as we have discussed survives as an anomalous U (1) in the effective theory at low energies. We denote with g B the gauge coupling of the U (1) B−L gauge boson (B µ ), while ±q B will denote the corresponding B − L charges of the scalars. Their normalization, equal to ±10 in the normalization of [33], is indeed arbitrary. The role of the Stueckelberg field is taken by b(x) in the polar parameterization of φ, which is normalized to 1 in mass dimension, while F σ is massless. The two covariant derivatives of the scalars take the form with the typical Stueckelberg kinetic term generated from the decoupling of the radial fluctuations of the φ field with M = q B g B v φ ∼ M I takes the role of the Stueckelberg scale. In general it is natural to assume that both v φ and v σ are of the same order, and the mass of B µ , the B-L gauge boson, will be given as a mean of both vevs The quadratic action, neglecting the contribution of the radial excitations of σ and φ, can be easily written down for such σ − φ combination from which, after diagonalization of the mass terms we obtain where we are neglecting all the other terms generated from the decomposition which will not contribute to the breaking. We can identify the linear combinations corresponding to the physical axion χ B , and to a massless Nambu-Goldstone mode G B . The rotation matrix that allows the change of variables (σ 2 , b) → (χ, G B ) is given by with The potential, as shown in similar analysis [5], is periodic in χ/f χ where f χ ∼ M I takes the role of the axion decay constant. As already stressed before, the origin of this potential is nonperturbative and linked to the presence of instantons at the SO(10) GUT phase transition. For such reason, the size of the constants λ in such potential are exponentially suppressed with λ i ∼ e −2π/α GU T , with the value of the coupling α GU T fixed at the scale M GU T when the SO(10) instantons are exact. The value of α GU T here is in the range 1/33 ≤ α I ≤ 1/32, giving 10 −91 ≤ λ ij ≤ 10 −88 , determining an axion mass given by m 2 χ ∼ λM 2 I in the range 10 −22 eV < m χ < 10 −20 eV (53) corresponding to an ultralight axion, which has been invoked for the resolution of several astrophysical constraints [42].

Conclusions
We believe that we have merely identified the general tracts of this mechanism to which we hope to return in the near future in a more extensive analysis.