1 Introduction
###### Abstract

We present a study of a class of effective actions which show typical axion-like interactions, and of their possible effects at the Large Hadron Collider. One important feature of these models is the presence of one pseudoscalar which is a generalization of the Peccei-Quinn axion. This can be very light and very weakly coupled, with a mass which is unrelated to its couplings to the gauge fields, described by Wess Zumino interactions. We discuss two independent realizations of these models, one derived from the theory of intersecting branes and the second one obtained by decoupling one chiral fermion per generation (one right-handed neutrino) from an anomaly-free mother theory. The key features of this second realization are illustrated using a simple example. Charge assignments of intersecting branes can be easily reproduced by the chiral decoupling approach, which remains more general at the level of the solution of its anomaly equations. Using considerations based on its lifetime, we show that in brane models the axion can be dark matter only if its mass is ultralight ( eV), while in the case of fermion decoupling it can reach the GeV region, due to the absence of fermion couplings between the heavy Higgs and the light fermion spectrum. For a GeV axion derived from brane models we present a detailed discussion of its production rates at the LHC.

Axions from Intersecting Branes and Decoupled Chiral Fermions

Claudio Corianò and Marco Guzzi

Dipartimento di Fisica, Università del Salento

and INFN Sezione di Lecce, Via Arnesano 73100 Lecce, Italy

## 1 Introduction

The study of possible signatures of string/brane theory at lower energy has achieved a significant strength with the development, in the last few years, of several extensions of the Standard Model (SM) formulated in scenarios with intersecting branes and large extra dimensions [1, 2, 3, 4, 5, 6], which are characterized by quite distinct features compared to other constructions, such as those based on more traditional anomaly-free supersymmetric formulations. The latter include specific theories like the MSSM but also its further variants such as its next-to-minimal (nMSSM,NMSSM) extensions, eventually with the inclusion of a gauge structure enlarged by an extra anomaly-free gauge symmetry (USSM) [7] (see [8] for an overview).

On the other hand, since anomalous ’s are naturally produced in geometrical compactifications and are an important aspect of brane models, the search for possible signatures of string theory has necessarily to take into consideration the peculiarities of these anomalous extensions, which are characterized by anomalous extra neutral currents, contact interactions of Chern-Simons form at trilinear gauge level [9] [10][11] and several axions of Stückelberg type. Supersymmetric extensions of these classes of models have also been investigated recently [12, 13].

One of the most demanding feature of these formulations, in regard to possible experimental searches, is to clarify the role of gauge anomalies on a substantial sector of collider phenomenology, from precision measurements of leptoproduction to double prompt-photon production [14], just to mention a few processes. These and many more are all affected by the new anomalous trilinear gauge vertices [15] which appear in these models, although their studies are expected to be quite difficult experimentally.

In fact, the limited accuracy of hadron colliders might reduce the expectations in regard to the possible experimental identification of subtle effects due to the mechanism(s) which underline the cancellation of the gauge anomalies. Nevertheless, the presence of an axion-like particle in the spectra of these theories is an important feature of intersecting brane models, which represents a serious departure from the typical anomaly-free formulations - both for supersymmetric and non-supersymmetric models - and provides a natural justification for a light pseudoscalar state.

The higher perturbative order at which these effects start to appear in the perturbative expansion and the limitations of the parton model description seem to indicate that the analysis of anomalous effects are more likely to be the goal of a linear collider rather than that of the LHC, nevertheless the signatures of new physics are manyfold and are not limited to collider physics, but have remarkable implications also in astroparticle physics and cosmology.

Among the aspects that can be addressed within these new formulations are those related to the flavour sector and the connection between these constructions and the traditional solutions of the strong CP-problem, previously addressed with the help of global symmetries, such as in the invisible axion model [16, 17, 18, 19, 20]. We recall that studies of the flavour sector of the SM in the presence of gauged anomalous ’s are not new, having been used in the past in a variety of cases, for example in the construction of realistic scenarios for neutrino mixing [21]. At the same time, the study of axion-like particles is at the center of new important proposals for their detection which are now under an intense investigation at DESY [22, 23]. Other interesting proposals consider the possible implications of axion-like particles in the propagation of gamma rays [24]. We believe that these motivations are sufficient to justify generalized searches of pseudoscalars as a possible solution of the dark matter problem. At the same time anomalous gauge interactions, in combination with quantum gravitational effects, show puzzling features, due to the presence of phantom fields [25, 26] in the local formulation of the trace anomaly [27] which deserve a closer look.

### 1.1 An axion with independent gauge couplings and mass

An axion-like particle is characterized by the usual pseudoscalar couplings to the gauge fields (the term, where is an axion) but has a mass which is unrelated to its coupling. Different mass ranges for the axion have quite different implications at a phenomenological level. For instance, for a very light axion ( eV), as for the PQ case, the pseudoscalar can mix with the photon and can generate, in the presence of background galactic magnetic fields, the usual phenomena of birifringence and dichroism for light propagation [28], with important effects at astrophysical level [24, 29] and other experimental signatures [22, 23]. The optical activity of the intergalactic medium due to the presence of background axions, also in this generalized case, is essentially caused by the coupling in the equations of motion of the lagrangean [30][31].

In the invisible axion model astrophysical arguments bound the mass of the axion (and its interaction to the gauge fields) requiring its suppression by a large scale . All the axion couplings and the axion mass

 m≈6⋅10−6eV1012fGeV (1)

are inversely proportional to , where is arbitrary ( GeV experimentally) and makes the axion, indeed, very light. In general, a very light axion, being a quasi-goldstone mode of a global symmetry, is produced copiously at the center of the sun and escapes after its production, with a mean free path which is larger than the radius of the sun. The failure by existing ground-based helioscopes to detect this particle in a detector of Sikivie type [32][33, 34] has been used to bound its mass and its interaction with the gauge fields. The bound can be evaded if the axion has a mass larger than the temperature at the center of the sun, since in this case would not be produced at its center, mass which is not allowed for the invisible axion according to current constraints. For a very light axion interesting effects are allowed, such as its non-relativistic decoupling, since its average momentum at the QCD phase transition is not of the order of the associated temperature, which is in the GeV range, but of the Hubble expansion rate ( eV), and the formation of Bose-Einstein condensates [35].

As we have mentioned, the gauging of the axionic symmetries can lift the typical constraints of the invisible axion model, allowing a wider parameter space, which is the main motivation for our study. In principle, in the extensions that we consider, this pseudoscalar can be very light, while its gauge interactions can be suppressed by a scale which is given by the mass of the lightest extra present in the neutral sector of these models. For this reason, these types of pseudoscalars are naturally associated to the neutral current sector, with new implications at the level of the trilinear gauge interactions.

So far, two models have been developed in which the structure of the effective action allows a physical axion: the MLSOM (the Minimal Low-Scale Orientifold Model) [36] and the USSM-A [37, 13], the first being a non-supersymmetric model, the second a supersymmetric one. In the first model, motivated by a construction based on intersecting branes, the scalar sector involves beside the Stückelberg axions, 2 Higgs doublets. At the same time, the gauge structure of the Standard Model is corrected by the presence of extra neutral currents due to the extra .

To date, a detailed analysis of these models is contained in [38], worked out for a single extra . In the supersymmetric case the presence of a physical axion is guaranteed if the superpotential allows extra superfields which are singlet respect to the Standard Model but are charged under the anomalous ’s. The field content of the superpotential of the nMSSM is sufficient to have a physical axion in the spectrum [37, 13].

### 1.2 Gauged axions

The gauging of axionic symmetries is realized in the low energy effective lagrangeans by introducing (shifting) axions, one for each anomalous present in the gauge structure of a given model. These are accompanied by Wess-Zumino terms in order to restore the gauge invariance of the theory due to the chiral anomalies present in these constructions. These axions (Stückelberg axions) are not all physical fields. In fact, the only physical axion, called the ”axi-Higgs” in [36], is identified in the CP-odd sector of the scalars by a joint analysis of the potential and of the bilinear mixing terms generated by the Stückelberg mass terms which are present for each anomalous . We are going to summarize below the scalar of the scalar potentials which allow a physical axion in the spectrum, either massless or massive. Since the mass of this particle is expected to obtain small non-perturbative corrections due to the instanton vacuum, as for the invisible axion, these small corrections are described by extra terms in the scalar potential which are allowed by the symmetry. These terms make the physical axion part of the scalar potential, but their size remains, in the class of theories that we analyze, essentially unspecified. In supersymmetric models they are expected to correspond to non-holomorphic corrections to the superpotential [37] which involve directly the axion/axino superfield. The size of these corrections depends on the way the fundamental symmetry is broken, and the appearance of the axion in the scalar potential just parameterizes our ignorance of the fundamental mechanism which is responsible for these corrections. For this reason, we focus our analysis on several mass windows for this particle, although the most relevant mass range for collider studies is the GeV region.

### 1.3 Organization of this work

The analysis presented in this work concerns the phenomenology of the axi-Higgs in anomalous abelian models with a single anomalous extra and in the non-supersymmetric case. The construction, therefore, is the one typical of the MLSOM, formulated in the context of intersecting branes. A similar analysis can be performed in the supersymmetric case, although it is more complex and will be presented elsewhere. Our analysis, however, is not limited to models of intersecting branes, but to the entire class of effective actions which are characterized by axion-like interactions at low energy, independently from their high energy completion. Typical charge embeddings of brane constructions, as we are going to show, can nevertheless be obtained in our approach starting from an anomaly-free spectrum and decoupling some chiral fermions. Some differences between the two realization remain, at phenomenological level, since the corresponding axion, in the case of decoupled fermions, does not couple to the light fermions which are part of the low-energy spectrum.

Our motivations for working within this more general framework has been motivated by scenarios where a heavy fermion, for instance a right-handed neutrino, decouples from the low energy spectrum leaving one Stückelberg axion (the phase of a Higgs field) in the effective lagrangean. We will come to discuss these points in more detail in one of the sections below. The different completions of these lagrangeans start differing at the level of operators whose mass dimensions is larger than 5, the five dimensional ones being the Wess-Zumino terms.

After reviewing briefly these models in order to make our analysis self-contained, we illustrate how their anomalous content can be obtained by requiring that only some of the anomaly equations are satisfied, taking as a starting point an anomaly-free chiral spectrum and decoupling some chiral fermions. Typical brane models such as the Madrid model [4] are obtained for a particular choice of the free charges allowed by the decoupling of the heavy chiral fermions and are just particular solutions of the anomaly equations. We then move towards a phenomenological analysis of the axi-Higgs in the MLSOM, selecting the GeV mass range for the axion. This region is the most promising one for collider studies of this particle, although in this range, as we are going to show, it is not long-lived. A GeV axion can be long lived, but must have suppressed couplings to the fermions of the low-energy spectrum, and one way of getting this lagrangean is via the mechanism of decoupling of heavy fermions (and of the radial excitations of the associated Higgs field) from the low energy theory. We show, using a simple toy model, how this can occur.

Production and decay rates for this particle are studied for all the mass windows in the MLSOM for typical LHC searches. We give in an appendix a summary of the scalar sector of the lagrangean and the determination of coefficients of the Wess Zumino terms. We have also included a section where we present a discussion and a comparison of the effective action of intersecting brane models versus the analogous one obtained by decoupling a chiral fermion, illustrating briefly the origin of the various operators left in the low energy formulation, with the axion interpreted as the phase of a second Higgs sector, partially decoupled from the 2 Higgs doublets included in the electroweak sector.

## 2 The model: overview of its general structure

We analyze a class of models characterized by a gauge structure of the form , defined in [38], where the gauge symmetry is anomalous and the corresponding gauge boson undergoes mixing with the rest of the gauge bosons of the Standard Model. Details can be found in [38, 38, 39]; here we just summarise the main features of this construction for which we will define rather general charge assignments. As we have already stressed, the reason for keeping our analysis quite general is motivated by the observation that effective actions of intersecting brane models are not uniquely identified. Various completions can generate the same low energy signatures, at least up to operators of dimension 5, which, for anomalous gauge theories, are the Wess-Zumino terms. These points will be illustrated in a section below, where we will solve the basic equations that characterize the charge assignments of the anomalous model, under some assumptions on the fermion spectrum which are essential in order to make our analysis concrete.

### 2.1 The structure of the effective action

The effective action has the structure given by

 S = S0+SYuk+San+SWZ+SCS (2)

where is the classical action which is given in an appendix. It contains the usual gauge degrees of freedom of the Standard Model plus the extra anomalous gauge boson which is already massive, before electroweak symmetry breaking, via a Stückelberg mass term. The scalar potential is the maximal one permitted by the symmetry and allows electroweak symmetry breaking. The structure of the Yukawa sector is very close to that of the Standard Model. In one of the sections below we identify the fundamental physical degrees of freedom of this sector after electroweak symmetry breaking, which, in our analysis, is based on the choice of the largest potential allowed by the symmetry. The model is a canonical gauge theory with dimension-4 operators plus dimension 5 counterterms of Wess-Zumino type.

In Eq. (2) the anomalous contributions coming from the 1-loop triangle diagrams involving abelian and non-abelian gauge interactions are summarized by the expression

 San = 12!⟨TBWWBWW⟩+12!⟨TBGGBGG⟩+13!⟨TBBBBBB⟩ (3) +12!⟨TBYYBYY⟩+12!⟨TYBBYBB⟩,

where the symbols denote integration. For instance, the anomalous contributions in configuration space are given explicitly by

 ⟨TBWWBWW⟩ ≡ ∫dxdydzTλμν,ijBWW(z,x,y)Bλ(z)Wμi(x)Wνj(y) (4)

and so on, where denotes the anomalous triangle diagram with one field and two ’s external gauge lines. The gluons are denoted by .

In the same notations the Wess Zumino (WZ) counterterms are given by

 SWZ = CBBM⟨bFB∧FB⟩+CYYM⟨bFY∧FY⟩+CYBM⟨bFY∧FB⟩ (5) +FM⟨bTr[FW∧FW]⟩+DM⟨bTr[FG∧FG]⟩,

while the gauge dependent CS abelian and non abelian counterterms [10] needed to cancel the mixed anomalies involving a B line with any other gauge interaction of the SM take the form

 SCS = +d1⟨BY∧FY⟩+d2⟨YB∧FB⟩ (6) +c1⟨ϵμνρσBμCSU(2)νρσ⟩+c2⟨ϵμνρσBμCSU(3)νρσ⟩,

with the non-abelian CS forms given by

 CSU(2)μνρ = 16[Wiμ(FWi,νρ+13g2εijkWjνWkρ)+cyclic], (7) CSU(3)μνρ = 16[Gaμ(FGa,νρ+13g3fabcGbνGcρ)+cyclic]. (8)

The only constraint which fixes the coefficients in front of the WZ counterterms is gauge invariance. Specifically, the anomalous variation of is compensated by the variation of . Imposing this condition one discovers that the scale of the WZ counterterms (M) becomes the Stückelberg mass term . This is found in the defining phase of the model, in which the realization of the gauge symmetry is in the Stückelberg form. Obviously, in this phase only the gauge boson is massive (in a Stückelberg phase). The breaking of the electroweak symmetry, triggered by the Higgs potential and the transition to the mass eigenstates determines a rotation of the Stückelberg axion into a physical axion plus some Nambu-Goldstone modes. This rotation brings in a redefinition of the suppression scale , which now coincides with the mass of the extra gauge boson, as shown in an appendix.

### 2.2 The scalar potentials and their axion-dependent phases

In previous studies it has been shown that anomalous abelian models, realized in the case of potentials with 2 Higgs doublets, both in the non-supersymmetric and in the supersymmetric cases, are characterized by the presence of an axion-like particle in the spectrum. In the context of the 2 Higgs doublets model shown in detail in [36, 38] the presence of PQ-breaking terms in the scalar potential allows the axion to become massive. The PQ symmetric contribution is given by

 VPQ(Hu,Hd)=∑a=u,d(μ2aH†aHa+λaa(H†aHa)2)−2λud(H†uHu)(H†dHd)+2λ′ud|HTuτ2Hd|2, (9)

which is a pure Higgs scalar potential, while in the PQ-breaking terms we introduce a dependence on the axion field by means of explicit phases

 VP/Q/(Hu,Hd,b) = b1(H†uHde−iΔqBbM1)+λ1(H†uHde−iΔqBbM1)2 +λ2(H†uHu)(H†uHde−iΔqBbM1)+λ3(H†dHd)(H†uHde−iΔqBbM1)+h.c.

where , has mass squared dimension, while , , are dimensionless couplings. In the scalar potential we can isolate three sectors, namely, two neutral and one charged sector, which are described by the quadratic expansion of the potential around its minimum

 VCP−even(Hu,Hd)+VCP−odd(Hu,Hd,b)+V±(Hu,Hd)= (11) +(ImHu0,ImHd0,a′I)N3⎛⎜⎝ImHu0ImHd0b⎞⎟⎠. (12)
• The Charged Sector

In the charged sector we find a zero eigenvalue of the mass matrix, corresponding to the Goldstone mode and the nonzero eigenvalue

 m2H+ = 4λ′udv2−2(2bv2sin2β+2λ1+tanβλ2+cotβλ3)v2, (13)

corresponding to the charged Higgs mass. The two vevs of the Higgs sector are defined by , with . The rotation matrix into the physical eigenstates is

 (Hu+Hd+)=(sinβ−cosβcosβsinβ)(G+H+). (14)
• The CP-even Sector

In the neutral sector both a CP-even and a CP-odd subsectors are present. The CP-even sector is described by which can be diagonalized by an appropriate rotation matrix in terms of CP-even mass eigenstates as

 (ReHu0ReHd0)=(sinα−cosαcosαsinα)(h0H0), (15)

with

 tanα=N2(1,1)−N2(2,2)−√Δ2N2(1,2) (16)

and

 Δ=(N2(1,1))2−2N2(2,2)N2(1,1)+4(N2(1,2))2+(N2(2,2))2. (17)

The definition of these matrix elements is left to an appendix. The eigenvalues corresponding to the physical neutral Higgs fields are given by

 m2h0 = 12(N2(1,1)+N2(2,2)−√Δ) m2H0 = 12(N2(1,1)+N2(2,2)+√Δ). (18)

We refer to [36] for a more detailed discussion of the scalar sector of the model with more than one extra .

• The CP-odd sector

The symmetric matrix describing the mixing of the CP-odd Higgs sector with the axion field is given by . After the diagonalization we can construct the orthogonal matrix that rotates the Stückelberg field and the CP-odd phases of the two Higgs doublets into the mass eigenstates

 ⎛⎜⎝ImH0uImH0db⎞⎟⎠=Oχ⎛⎜ ⎜⎝χG01G02⎞⎟ ⎟⎠. (19)

The mass matrix of this sector exhibits two zero eigenvalues corresponding to the Goldstone modes and a mass eigenvalue, that corresponds to the physical axion field , with a value

 m2χ=−12cχv2⎡⎣1+(qBu−qBdM1vsin2β2)2⎤⎦=−12cχv2[1+(qBu−qBd)2M21v2uv2dv2], (20)

with the coefficient

 cχ=4(4λ1+λ3cotβ+b1v22sin2β+λ2tanβ). (21)

The mass of this state is positive if . The Goldstone bosons are obtained by orthonormalizing that span a two dimensional space. Notice that, in general, the mass of the axi-Higgs is the result of two effects: the presence of the Higgs vevs and the presence of the Stückelberg mass via the PQ-breaking potential. In the particular case of a charge assignment such that , in the PQ-breaking potential the dependence on the axion field disappears () and the rotation matrix simplifies to

 ⎛⎜⎝ImH0uImH0ub⎞⎟⎠=⎛⎜⎝−cosβsinβ0sinβcosβ0001⎞⎟⎠⎛⎜ ⎜⎝A0G01G02⎞⎟ ⎟⎠. (22)

For this particular assignment of the Higgs charges the and bosons are still massive, as can be seen from eqs. (19, 19). A brief counting of the physical degrees of freedom shows, also in this case, that we expect only one physical particle in the CP-odd sector. Then, in this particular case, it is easily found that the model doesn’t exhibit Higgs-axion mixing because the physical degree of freedom , as identified by the scalar potential, is a combination of the imaginary parts of the two Higgs , while the axion is only part of the Goldstones modes and , identified by an inspection of the derivative couplings.

## 3 Axions from the decoupling of a chiral fermion

Other realizations of these effective models are obtained by studying the decoupling of a chiral fermion from an original anomaly-free theory, due to large Yukawa couplings [40]. The remnant axion, in this particular realization, is the surviving massless phase of a heavy Higgs. We will illustrate briefly this approach sketching the derivation, though in the case of a simple model, in a section below. Obviously, in these types of completions of the anomalous theory, the challenge of the construction would consist in the identification of a pattern of sequential breaking of the underlying anomaly-free theory in order to generate suitable axion-like Wess-Zumino interactions, which are not part of our simple example.

For instance, considerable motivations for this reasoning comes from unified models based on an anomaly-free fermion spectrum assigned to special representations of the gauge symmetry. Specifically, one could consider the 16 of in which find accommodation the fermions of an entire generation of the Standard Model plus a right handed neutrino. The decoupling of a right handed neutrino could leave a remnant pseudoscalar in the spectrum with axion-like couplings. While the explicit realization of this construction and the (sequential) breaking of the original GUT towards the spectrum of the Standard Model is rather complex, the implications of these assumptions can be grasped by a simple model.

To illustrate these points, we introduce a simple toy model and show step by step that a specific form of the decoupling can generate a certain dynamics at low energy which is completely described by an effective action with Stückelberg and a Higgs-Stuckelberg phases, Wess Zumino interactions and higher dimensional operators suppressed by the Stückelberg mass. It should be mentioned that in our example, the low energy gauge boson , which has anomalous effective interactions, would be massive in the Stückelberg form. We recall that the study of the Stückelberg construction has been discussed recently in several works [41, 42] (see also [43]) for non-anomalous theories, with its possible experimental signatures.

The model requires two Higgs fields, here assumed to be two complex scalars, and a potential characterized by a first breaking of the anomaly-free gauge symmetry at a certain scale (), followed by a second breaking at a lower scale (). The heavy Higgs is assumed to decouple (partially) after the first breaking. Specifically, the decoupling involves the radial fluctuations () of the field , and all the interactions which are characterized by operators which are suppressed by a certain power of . We expand the heavy Higgs as

 ϕ∼(vϕ+ρ√2)eiθ (23)

with denoting a massless phase that may be rendered massive during the process of decoupling of the radial excitation by some small tilting, as it occurs for the ordinary Peccei-Quinn axion (PQ). The (almost massless) phase remains in the low energy theory. The Stückelberg axion is identified from in a certain way, that will be specified below. Also we assume, for simplicity, that only one chiral fermion becomes heavy in the course of decoupling of the heavy Higgs, and is integrated out of the low energy spectrum. As we have already stressed, our approach can be made more realistic, but we expect that the crucial steps that bring to its specific effective action at low energy can be part of a more complete theory.

The Yukawa couplings, expanded around the vacuum of the heavy Higgs, show the presence of a complex phase () that we try to remove by a chiral redefinition of the integration measure before we integrate out the heavy fermion. It is this chiral redefinition of the fermionic measure which induces, by Fujikawa’s approach, typical Wess-Zumino terms in the low energy effective theory. This theory, obviously, admits a derivative expansion in terms of the large scale , which can be systematically captured by a derivative expansion in , or equivalently, the Stückelberg mass, since the two scales are related ().

### 3.1 Partial integration

To be specific, we consider a model with 2 fermions and a gauge symmetry of the form , where is vector like and is the anomalous gauge boson. We define the lagrangean

 L = (24) +λ¯ψ(1)Lϕψ(1)R+λ¯ψ(1)Rϕ∗ψ(1)L+|DμH|2+|Dμϕ|2−V(ϕ,H)

where we have neglected the Yukawa coupling of the light fermion(s) , which are proportional to the vev of the light Higgs . For simplicity we may consider a simple scalar potential function of the two Higgs and , such as , that as we have mentioned, admits vacua which are widely separated. While this would induce a hierarchy between the two vevs, and could be the real difficulty in the realization of this scenario, one possible way out would be to consider to be the sum of two separate potentials. Since the phase of the heavy Higgs survives in the low energy theory as a pseudo-goldstone mode, it may acquire a mass if the potential in which it appears is tilted.

We show in Tab. 1 the charge assignments of the model. We define

 DμH = (∂μ+iqHBgBBμ)H Dμϕ = (∂μ+iqϕBgBBμ)ϕ Dμψ(i)L = (∂μ+iq(i)ALgAAμ+iqBLgBBμ)ψ(i)L. (25)

Under a gauge transformation we have

 ψ′(i)L = e−iq(i)LgBθψ(i)L ψ′(i)R = e−iq(i)RgBθψ(i)R (27)

with .

We assume that the charge assignments are such that the model is anomaly-free. Notice also that , in this realization, becomes massive via a first breaking at the large scale and then its mass gets corrected by the second breaking, characterized by the scale .

We parameterize the fluctuations of the field around the first vacuum in the form

 ϕ=vϕ+ρ√2e−iqϕBgBθ (28)

from which we obtain the first contribution to the mass of the gauge boson in the form . As we are going to show next, this mass can be taken to be the Stückelberg mass of a reduced Higgs system if we neglect the radial excitations. In fact we have

 |Dμϕ|2=12(∂μϕ)2+(vϕ+ρ√2)2(qϕBgB)2(−∂μθ+Bμ)2, (29)

and we isolate from the phase of this exact relation a dimensionful field which will be taking the role of a Stückelberg mass term as

 θ=bqϕBgBvϕ. (30)

We can expand (29) in the form

 |Dμϕ|2=12(∂μ−M1Bμ)2+O(ρ/v), (31)

with , defined to be the Stückelberg mass. The decoupling of the radial excitations of the very heavy Higgs from the low energy lagrangean generates a Stückelberg mass term on the rhs of (31), whose phase is at this stage massless. Notice that after the second symmetry breaking, the mass of the gauge boson will acquire an additional contribution proportional to , in analogy to the first breaking, that is

 MB=√M21+(gBqHBvH)2. (32)

Notice also that after the first radial decoupling of the heavy Higgs , the Yukawa mass terms are affected by a phase dependence that can be eliminated from the effective lagrangean via an anomalous transformation. To illustrate this point consider the expansion of the Yukawa term around the vacuum of the heavy Higgs

 λ¯ψ(1)Lϕψ(1)R=λ1√2(vϕ+ρ)¯ψ(1)Lψ(1)Re−iqϕBgBθ (33)

which is affected by a phase that we will try to remove in the course of the elimination of the heavy degrees of freedom of the mother theory. Notice that in this case we do not take a large Yukawa coupling (), as in previous analysis [44, 45], since the large fermion mass of is instead obtained via the large vev of the heavy Higgs, . For this reason, having defined the Stückelberg mass in terms of the same vev, after neglecting the radial contributions we obtain

 λ¯ψ(1)Lϕψ(1)R=κM1¯ψ(1)Lψ(1)Re−iqϕBgBθ,κ=λ√2qϕBgB. (34)

Before performing the partial integration on the heavy fermion , it is convenient to define a change of variables in the functional integral, in order to remove the phase-dependence on present in the Yukawa couplings. For this reason, let’s consider the part of the partition function directly related to the heavy fermion , which is involved in the procedure of partial integration. This is given by

 Z(1)(A,B)=∫Dψ(1)LD¯ψ(1)LDψ(1)RD¯ψ(1)Rei∫d4xL(1) (35)

where

 L(1)=ψ(1)LD/ψ(1)L+¯ψ(1)RD/ψ(1)R+κM1¯ψ(1)Lψ(1)Re−iqϕBgBθ+h.c. (36)

and we have neglected the contributions proportional to the radial excitation of the heavy Higgs. At this point we try to remove the phase from the Yukawa couplings by performing a field redefinition in the functional integral of the heavy fermion. We set

 ψ(1)BL=e−iq(1)BLgBθψ′(1)BL ψ(1)BR=e−iq(1)BRgBθψ′(1)BR, (37)

where from gauge invariance we have

 q(1)BR+qϕB−q(1)BL=0. (38)

The field redefinition induces in the integration measures two jacobeans

 Dψ(1)LD¯ψ(1)L = JLDψ′(1)LD¯ψ′(1)L Dψ(1)RD¯ψ(1)R = JRDψ′(1)LD¯ψ′(1)R (39)

which are computed using Fujikawa’s approach (see for instance [46]). We obtain

 JL = e−iq(1)BL132π2⟨θF∧F⟩L JR = e−iq(1)BR132π2⟨θF∧F⟩R. (40)

In this case contains both gauge fields and the corresponding gauge charges of the heavy fermions such as, for instance,

 FμνL,R=iq(1)AL,RFAμν+iq(1)BL,RFBμν. (41)

The structure of the effective action after the field redefinition takes the form

 Z(1)(A,B)=∫Dψ′(1)LD¯ψ′(1)LDψ′(1)RD¯ψ′(1)Rei∫d4xL′(1)+LWZ (42)

where

 L′(1)=ψ′(1)L(D/−iq(1)BL∂/θ)ψ′(1)L+¯ψ′(1)R(D/−iq(1)BL∂/θ)ψ′(1)R+κM1¯ψ′(1)Lψ′(1)R+h.c. (43)

with the Wess-Zumino (WZ) lagrangean obtained from the expansion of the terms. These are suppressed by the Stückelberg mass term .

At this point we can perform the Grassmann integration over the heavy fermion, which trivially gives the functional determinant of an operator, , explicitly given by

 P=v′ϕ⎛⎜ ⎜ ⎜ ⎜⎝D/−iq(1)BLgB∂/θv′ϕ11D/−iq(1)BRgB∂/θv′ϕ⎞⎟ ⎟ ⎟ ⎟⎠, (44)

where . The remaining terms in the total partition function of the model can be obtained from the functional integral

 Zeff∼∫Dψ(2)LD¯ψ(2)LDψ(2)RD¯ψ(2)RDHDbDθei∫d4xLeff (45)

where

 Leff=L′(2)+LWZ+TrlogP+12(∂μ−M1Bμ)2+|DμH|2−V(H,θ) (46)

with

 L′(2)=−14F2A−14F2B+ψ(2)LD/ψ(2)L+¯ψ(2)RD/ψ(2)R. (47)

The derivative expansion of the effective action can be organized in terms of corrections in the Stückelberg mass. Obviously, a similar approach can be followed for the integration of a Majorana fermion, which is slightly more involved. The basic physical principle, however, remains the same also in this second variant. In this case the functional determinant can be organized as in [47].

There are some implications concerning the two realizations of this class of effective actions, especially in regard to the possible mass of the axion as a dark matter candidate in the various models that share the effective actions that we have presented. The first observation concerns the absence of a direct Yukawa coupling between the heavy Higgs and the light fermion spectrum, which is part of the effective action after partial integration on the heavy fermion modes. This feature is absent in the MLSOM, and turns out to be rather important since it affects drastically the lifetime of the axion, as we are going to elaborate in the following sections. We will find that a GeV axion is favoured by the mechanism of partial decoupling but is not allowed in the MLSOM. In this second case a very light axion is necessary in order to have a state which is long lived and that can be a good dark matter candidate.

### 3.2 Parametric solutions of the anomaly equations

It is clear that the typical effective action isolated by the decoupling of (one or more) chiral fermions can be organized in terms of the defining lagrangean plus the WZ counterterms, which restore the gauge invariance of the model. Therefore, up to operators of mass dimension 5, the two lagrangeans are quite overlapping at operatorial level. For this reason, we will construct a complete charge assignments for these models, starting from an anomaly-free theory, with a spectrum that we deliberately choose to include one right-handed neutrino per generation, and which we will decouple from the low energy dynamics according to the procedure described above. Of course, other choices are also possible. As we have already stressed, the motivations for selecting this approach are not just of practical nature, although it allows to generate effective anomalous models with ease. For instance, one could envision a scenario, inspired by leptogenesys, which could offer a realization of this decoupling mechanism, although its details remain, at the moment, rather general. We will not pursue the analysis of this point any further, and leave it as an interesting possibility for future studies. However, we will discover, by using the decoupling approach, that a significant class of charge assignments of intersecting brane models can be easily reproduced by the free gauge charges which parametrize the violation of the conditions of cancellation of the anomaly equations. We should also mention that the dependence of our results on the various charge assignments is truly small, showing that the relevant parameters of the models are the Stückelberg mass, the anomalous coupling and the parameters of the potential, which control the axion mass in each realization.

To proceed, we impose first the conditions of cancellation of the gauge and of the mixed gravitational- anomalies, thereby fixing the charges, followed by the conditions of invariance of the Yukawa couplings, in order to determine the charges of the two Higgs [48]. We take the fermion charges to be family-independent in order to avoid possible constraints from flavor-changing neutral current processes. We label the generic fermion charges under the additional group as shown in Table 2.

For every anomalous triangle we allow, in general, a WZ counterterm whose coefficient has to be tuned in order to satisfy the conditions for anomaly cancellation. For the fermion charges we find the following constraints

 BSU(2)SU(2): qBL+3qBQL−CBWW=0, BSU(3)SU(3): qBdR+qBuR−2qBQL−CBgg=0, BYY: 3qBeR+6qBQL+3qBuR−32CBWW−CBYY−CBgg, (48)

where the coefficients appearing in front of the WZ counterterms are proportional to the charge asymmetries

 CBWW∝∑fθfL, CBgg∝∑QθBQ, CBYY∝∑fθBYYf, CBBB∝∑fθBBBf, (49)

which are detailed in an appendix, and with the hypercharges of given in Tab. (3).

If we consider the charges as free parameters of the model, can be in principle expressed in terms of these parameters. The other three conditions coming from the gauge invariance give the following further constraints

 YBB: −3(qBdR)2−3(qBeR)2+3(qBL)2−3(qBQL)2+6(qBuR)2−CYBB=0 BBB: 9(qBdR)3+3(qBeR)3−6(qBL)3−18(qBQL)3+9(qBuR)3−CBBB=0 BRR: 9qBdR+9qBuR+3qBeR−6qBL−18qBQL−3CBGG=0, (50)

where the condition on the BRR triangle comes from the mixed gravitational- anomaly cancellation. From the gauge invariance of the Yukawa couplings (see Lagrangian (73)), we obtain

 qBQL−qBd2−qBdR = 0, qBQL+qBu2−qBuR = 0, qBL−qBd2−qBeR = 0, qBL+qBu2 = 0, (51)

which can be used to constrain the charges of the two Higgs doublets and the counterterms . Collecting the constraints in eqs. (3.2), (48) and (50) we obtain a set of ten equations whose solution allows us to identify a class of charge assignments that we call

 f(qBQL,qBL,ΔqB)=(qBQL,qBuR;qBdR,qBL,qBeR,qBu,qBd). (52)

These depend only upon the three free parameters , , where . The explicit dependences are shown in Table 3, while the related WZ counterterms take the form

 CBYY=−32(qBL−5qBQL)+2ΔqB, (53) CYBB=3(qBL)2−32[18(qBQL)2+8qBQLΔqB+(ΔqB)2], (54) CBBB=−6(qBL)3+78(qBQL)3+72(qBQL)2ΔqB+18qBQL(ΔqB)2+32(ΔqB)3, (55) CBgg=12ΔqB, (56) CBWW=qBL+3qBQL, (57)

where in particular, from the charge assignment shown in Table 3, we identify the counterterm for the mixed gravitational- anomaly with

 CBGG=2(−qBL+qBQL+ΔqB). (58)

Then the WZ counterterms, as defined in general in eqs. (194), can now be specialized in terms of the different charge assignments , just by substituting the corresponding chiral asymmetries. This function will appear in several of our plots.

Finally, since in the case the matrix would become trivial, we require the following relation between the Higgs charges

 qBu