# The minimal axion minimal linear \(\sigma \) model

## Abstract

The minimal *SO*(5) / *SO*(4) linear \(\sigma \) model is extended including an additional complex scalar field, singlet under the global *SO*(5) and the Standard Model gauge symmetries. The presence of this scalar field creates the conditions to generate an axion *à la* KSVZ, providing a solution to the strong CP problem, or an axion-like-particle. Different choices for the PQ charges are possible and lead to physically distinct Lagrangians. The internal consistency of each model necessarily requires the study of the scalar potential describing the \(SO(5)\rightarrow SO(4)\), electroweak and PQ symmetry breaking. A single minimal scenario is identified and the associated scalar potential is minimised including counterterms needed to ensure one-loop renormalizability. In the allowed parameter space, phenomenological features of the scalar degrees of freedom, of the exotic fermions and of the axion are illustrated. Two distinct possibilities for the axion arise: either it is a QCD axion with an associated scale larger than \(\sim 10^{5}\) TeV and therefore falling in the category of the invisible axions; or it is a more massive axion-like-particle, such as a 1 GeV axion with an associated scale of \(\sim 200\) TeV, that may show up in collider searches.

## 1 Introduction

The last decade experienced a revival of interest for the so-called Composite Higgs (CH) models: first introduced in the middle of the 1980s [1, 2, 3], they have been reconsidered 20 years later with a more economical symmetry content [4, 5, 6]. The Minimal Composite Higgs Model (MCHM) [4] is based on the non-linear realisation of the *SO*(5) / *SO*(4) spontaneous breaking, which relies on a not well identified strong dynamics: the four Nambu–Goldstone bosons (GBs), originated from the global symmetry breaking, can be identified with the three would-be longitudinal components of the Standard Model (SM) gauge bosons and the Higgs field. The gauging of the SM symmetry group and the interactions with the SM fermions produce an explicit mass term for the Higgs field, which otherwise would be massless due to the underlying GB shift symmetry. This mechanism provides an elegant solution to the so-called Electroweak (EW) Hierarchy Problem.

A general drawback of these CH constructions is represented by its effective formulation: the generality of the effective approach comes together with its limited energy range of application. References [7, 8, 9, 10] attempted to improve in this respect, providing a renormalisable description of the scalar sector. Following for definiteness the treatment done in Ref. [9], the Minimal *SO*(5) / *SO*(4) Linear \(\sigma \) model (ML\(\sigma \)M) is constructed extending the SM spectrum by the introduction of an EW singlet scalar field \(\sigma \) and a specific set of vector-like fermions in the singlet and in the fundamental representations of *SO*(5). In the limit of large \(\sigma \) mass, the model falls back onto the usual effective non-linear description of the MCHM [4, 7, 11, 12, 13], that represents a specific realisation of the so-called Higgs Effective Field Theory [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34] Lagrangian describing the most general Higgs couplings to SM gauge bosons and fermions, which preserve the SM gauge symmetry.

The ML\(\sigma \)M can also be considered an optimal framework where to look for a solution to the strong CP problem. Indeed, extending the scalar spectrum with an additional complex scalar field *s*, *SO*(5) and EW singlet, the symmetry content of the model can be supplemented with an extra Peccei–Quinn (PQ) \(U(1)_\text {PQ}\) [35], eventually providing a realisation of the KSVZ axion mechanism [36, 37]: the angular component of the extra scalar *s* may indeed represent an axion.^{1} This idea has been firstly developed in Ref. [39] and this class of models will be dubbed Axion Minimal Linear \(\sigma \) Model (AML\(\sigma \)M). Even in this simple setup, the choice of the PQ charge assignment is not unique and different choices lead to physically distinct Lagrangians.

In this paper, a “minimality criterium” in terms of number of parameters will be introduced and only one “minimal scenario”, the minimal AML\(\sigma \)M, is identified among all the constructions presented in Ref. [39]. In order to completely fix the PQ charge assignment the following requirements are imposed: the SM fermion masses are generated at tree-level through the fermion partial compositeness mechanism [40, 41, 42, 43], which is the only explicit *SO*(5) breaking sector; the PQ scalar field *s* couples to (part of) the exotic fermions providing a portal between the axion and the colour interactions. The angular component of *s* can be identified as a QCD axion, requiring in addition that the contributions to the colour anomaly allow to reabsorb the QCD-\(\theta \) parameter through a shift symmetry transformation, thus solving the strong CP problem. If instead this requirement is relaxed, then the PQ GB is dubbed axion-like-particle (ALP). Both the possibilities are envisaged in the minimal AML\(\sigma \)M identified through the conditions aforementioned. Moreover, in this scenario, all the SM fields do not transform under the PQ symmetry and three distinct scales are present, that is the EW scale, the *SO*(5) / *SO*(4) and PQ symmetry breaking scales, the latter being independent from the first two.

A dedicated analysis of the scalar potential and its minima is necessary in order to guarantee that *SO*(5) gets spontaneously broken down to *SO*(4), and that the EW symmetry breaking (EWSB) mechanism occurs providing the correct EW vacuum expectation value (VEV). This analysis requires to take into account contributions to the scalar potential arising at one one-loop from the fermions and the gauge bosons of the model. The renormalisable scalar potential is identified according to the aforementioned requirements. The associated parameter space is studied, both analytically for few limiting cases and numerically, illustrating the main features of this minimal model. The phenomenological analysis reveals that modifications of the Higgs couplings to SM fermions and gauge bosons are present, leading to possibly interesting signals at colliders.

Turning the attention to the PQ GB sector, the axion and the ALP cases are characterised by two distinct phenomenologies. The axion is very light, with a mass generated by non-perturbative QCD effects as in the traditional PQ models [35, 44, 45, 46, 47]. Its corresponding scale is larger than \(\sim 10^5\text { TeV}\) and therefore it enters into the category of the invisible axion models [36, 37, 48, 49]. On the other side, the ALP can be much heavier, but at the price of invoking a soft explicit breaking of the shift symmetry and not necessarily solving the strong CP problem. As its characteristic scale can be much lower, it may give rise to visible effects at colliders.

It is the aim of the present paper to illustrate in details the minimal AML\(\sigma \)M and to analyse its phenomenological features. In the next section, the construction of the AML\(\sigma \)M is described, discussing the fermion content and the main characteristics of the scalar potential, focussing on the renormalisability of the full Lagrangian. In Sect. 3, the minimal scenario is identified, based on a minimality criterium in terms of number of parameters of the whole Lagrangian. Section 4 is devoted to the analytical description of the scalar potential and the *SO*(5) / *SO*(4) spontaneous symmetry breaking mechanism, presenting few relevant limiting cases. The phenomenological features of the model are described in Sects. 4.3 and 6, with the later section dedicated to the analysis of the axion and of the ALP. Finally, conclusions are drawn in Sect. 7, while more technical details are left for the appendix.

## 2 The axion minimal linear \(\sigma \) model

*SO*(5) /

*SO*(4) symmetry breaking realisation has been analysed in Ref. [9]. As usual in this class of minimal models, an additional \(U(1)_X\) is introduced in order to ensure the correct hypercharge assignment. The field content of the original ML\(\sigma \)M is the following:

- 1.
The four SM gauge bosons associated to the SM gauge symmetry.

- 2.A real scalar field \(\phi \) in the fundamental representation of
*SO*(5), which includes the three would-be-longitudinal components of the SM gauge bosons \(\pi _i\), \(i=1,\,2,\,3\), the Higgs field*h*and the additional complex scalar field \(\sigma \), singlet under the SM gauge group:where the last expression holds when selecting the unitary gauge, which will be used throughout the next sections.$$\begin{aligned} \phi =\left( \pi _1,\,\pi _2,\,\pi _3,\,h,\,\sigma \right) ^T \xrightarrow {u.g.} \phi =\left( 0,\,0,\,0,\,h,\,\sigma \right) ^T, \end{aligned}$$(2.1) - 3.
Exotic vector fermions, which couple directly to the

*SO*(5) scalar sector through*SO*(5) invariant proto-Yukawa interactions. These fermions transform either in the fundamental of*SO*(5), and they will be labelled as \(\psi \), or in the singlet representation of*SO*(5), dubbed \(\chi \). For both types of fermions, two distinct \(U(1)_X\) assignments are considered, 2 / 3 and \(-1/3\), as they are necessary to induce mass terms for both the SM up and the down quark sectors. - 4.
SM fermions, which do not couple directly to the Higgs field. SM fermion masses are originated through SM-exotic fermion interactions in the spirit of the fermion partial compositeness mechanism [40, 41, 42, 43]. SM fermions do not come embedded in a complete representation of

*SO*(5), leading to a soft explicit*SO*(5) symmetry breaking. Although the whole SM fermion sector could be considered, only the top and bottom quarks will be retained in what follows. This simplification does not have relevant consequences on the results presented here and the three generation setup can be easily envisaged.

- 5.A complex scalar field
*s*, singlet under the global \(SO(5) \times U(1)_X\) and the SM gauge group. Adopting an exponential notation,the degrees of freedom are defined as the radial component$$\begin{aligned} s \equiv \dfrac{r}{\sqrt{2}}e^{ia/f_a}, \end{aligned}$$(2.2)*r*and the angular one*a*, to be later identified with the physical axion. Following the philosophy adopted in Ref. [9] any direct coupling between the scalar*s*and the SM fermions is not introduced, as it will be discussed in more details in the following.

### 2.1 The gauge Lagrangian

### 2.2 The fermionic Lagrangian

*s*are shown. Two distinct type of couplings,

*z*and \(\tilde{z}\), have been introduced reflecting the freedom in choosing the PQ charges of

*s*and of the fermionic bilinears. The fourth line contains the interactions between the top quark and exotic fermions with \(U(1)_X\) charge equal to 2 / 3.

*SO*(5), the partial compositeness terms in the fourth line, proportional to \(\Lambda _{1,2}\), explicitly break the global

*SO*(5) symmetry. The combinations \(\Lambda _1\Delta _{2\times 5}\) and \(\Lambda _2\Delta _{5\times 1}\) may play the role of spurions [50, 51, 52, 53, 54] that formally ensure the \(SO(5)\times U(1)_X\) invariance of the operators. The exotic fermion spinors can be decomposed under the \(SU(2)_L\) quantum numbers as follows:

*K*and

*Q*doublets while \(T_{1,5}\) singlets of \(SU(2)_L\). The resulting interactions preserve the gauge EW symmetry, with the hypercharge defined as

*K*and \(-1/2\) for

*Q*) and

*X*the \(U(1)_X\) charge of the spinor.

Equations (2.6), (2.14), and (2.15) contain all the possible couplings invariant under SM gauge group and \(SO(5)\times U(1)_X\) global symmetry that can be constructed following the assumptions described in the previous section. However, it is important to notice that the Lagrangian actually describing the AML\(\sigma \)M can be obtained only after the adoption of a specific choice for the PQ charges: not all the terms are simultaneously allowed. In fact, only one between the \(M_i\), \(z_i\) and \(\tilde{z}_i\) (and corresponding primed) terms is allowed once a specific PQ charge assignment for the fermion chiral components is chosen, assuming obviously a non-vanishing charge for the scalar *s* field. In other words, exotic fermions acquire masses either through the direct mass terms (\(M_i\)) or through the Yukawa-like interactions with *s* (\(z_i\) or \(\tilde{z}_i\)) once the scalar field *s* develops a VEV. In addition, following the assumptions outlined in the previous section, as the scalar quintuplet \(\phi \) does not transform under the PQ symmetry, the presence of the proto-Yukawa interactions (\(y_{i}\)) necessarily depend on the PQ charges of exotic fermions.

Finally, turning the attention to the interactions between exotic and SM fermions, in the fourth and seventh lines of Eq. (2.6), if only the exotic fermions have non-vanishing PQ charges, then these operators are forbidden, unless the \(\Lambda _{i}\) couplings are either promoted to be spurions under the PQ symmetry or substituted by a PQ dynamical field (*s* or \(s^*\)). This would introduce explicit sources for the PQ symmetry breaking or imply that the PQ sector contributes to the dynamics that originate these operators. These issues will be discussed in the next sections, where the conditions that lead to the minimal AML\(\sigma \)M charge assignment are illustrated.

### 2.3 The scalar Lagrangian

*SO*(5), rotated with respect to the

*SO*(4) group preserved from the spontaneous breaking.

*r*, the kinetic term of the axion field

*a*gets canonically normalised, by identifying

*s*, invariant under \(SO(5)\times U(1)_\text {PQ}\) symmetry, broken spontaneously to

*SO*(4):

*SO*(5) and the PQ sectors: if \(\lambda _{s\phi }\sim \mathcal {O}(1)\) then the

*SO*(5) /

*SO*(4) and PQ breaking mechanisms would be linked and they would occur at similar scales; this would represent a possible tension between the naturalness of the AML\(\sigma \)M, which requires

*f*not so much larger than EW scale \(v=246\text { GeV}\), in order to reduce the typical fine-tuning in CH models, and the experimental data on the axion sector, which suggests very high values of \(f_s\) (see Sect. 6). In consequence, values of \(\lambda _{s\phi }\) smaller than 1 are favoured in the AML\(\sigma \)M.

*h*, \(\sigma \) and

*r*take a non trivial VEV, respectively \(v_h\), \(v_\sigma \) and \(v_r\), a spontaneous symmetry breaking for the EW, the global

*SO*(5) and the PQ symmetry, is obtained.

The second term \(V^\mathrm{CW}(\phi , s)\) is the Coleman–Weinberg (CW) one-loop potential that provides an explicit and dynamical breaking of the original symmetries. Its form depends on the explicit structure of the fermionic and bosonic Lagrangians and it will be outlined in the following subsection.

Finally, the term \(V^\mathrm{c.t.}(\phi , s)\), includes all the couplings that need to be introduced at tree-level in order to cancel the divergences potentially arising from the one-loop CW potential, so to make the theory renormalizable.

**The Coleman–Weinberg one-loop potential**

Explicit dynamical breaking of the tree-level symmetries can be introduced at one-loop level through the CW mechanism [55]. Indeed, the presence of *SO*(5) breaking couplings in both the fermionic and the gauge sectors generate *SO*(5) breaking terms at one-loop level. Explicit \(U(1)_\text {PQ}\) breaking contributions may also be generated, depending on the fermion PQ charge assignment.

*SO*(5) breaking terms, while the ones proportional to \(\hat{d}_{2,3}\) are

*SO*(5) preserving. On the other side, \(\hat{d}_{1,2,3}\) terms also explicitly break the PQ symmetry. If in a specific setup these terms were not vanishing, renormalisability of the model would then require the introduction of the corresponding structures in the tree-level potential.

*SO*(5) and/or of the PQ symmetries read:

*SO*(5) symmetry.

The two divergences associated to \(\tilde{a}_1\) and \(\tilde{d}_2\) require the introduction of an \(h^2\) term in the tree-level scalar potential, in order to ensure the renormalisability of the model, while the divergence proportional to the \(\tilde{b}_1\) coefficient requires an additional \(h^4\) term.

## 3 The minimal model

- 1.Mass terms for the SM quarks are originated at tree-level. Generalising the result in Ref. [9], the leading order (LO) contribution to the third generation quark masses is given byand similarly for the bottom mass. In this expression, \(M_{1,5}(v_r)\) refer to the definitions in Eq. (2.14) substituting the dependence on$$\begin{aligned} m_t= & {} \dfrac{y_1\Lambda _1\Lambda _3 v_h}{M_1(v_r)M_5(v_r)-y_1 y_2(v_h^2+v_\sigma ^2)} \nonumber \\&-\, \dfrac{y_1 y_2 \Lambda _1\Lambda _2v_h v_\sigma }{M_1(v_r)M_5^2(v_r)-y_1y_2M_5(v_r)(v_h^2+v_\sigma ^2)}, \end{aligned}$$(3.1)
*r*with its VEV, \(v_r\). In order for this expression not to be vanishing, the conditions \(y_1\ne 0\) and \(\Lambda _1\ne 0\) should hold simultaneously. Then, either \(\Lambda _3\ne 0\) or \(y_2\ne 0 \wedge \Lambda _2\ne 0\) should be verified, depending on whether the leading or sub-leading term in the*v*/*M*expansion is retained. - 2.
The dynamics that generate the partial-composite operators in the fourth line of Eq. (2.6) are associated only to the

*SO*(5) /*SO*(4) breaking sector. This implies that the scales*f*and \(f_s\) are distinct and independent.

On the left-side, the PQ charge assignments where \(n_i\) refers to the *i* field, conventionally fixing the PQ charge of the complex scalar field *s*, \(n_s=1\). On the right-side, the parameters entering the fermionic Lagrangian, together with the information on whether they are compatible (\(\checkmark \)) or not (\(\times \)) with the PQ symmetry. This assignment can be trivially extended to the bottom sector

\(n_{q_L}\) | \(n_{t_R}\) | \(n_{\psi _L}\) | \(n_{\psi _R}\) | \(n_{\chi _L}\) | \(n_{\chi _R}\) | \(y_1\) | \(y_2\) | \(\Lambda _1\) | \(\Lambda _2\) | \(\Lambda _3\) | \(M_5\) | \( M_1\) | \(z_1,\tilde{z}_5\) | \(\tilde{z}_1, z_5\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0 | 0 | \(+1\) | 0 | 0 | \(+1\) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\times \) | \(\checkmark \) | \(\times \) | \(\times \) | \(\checkmark \) | \(\times \) |

0 | 0 | \(-1\) | 0 | 0 | \(-1\) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\times \) | \(\checkmark \) | \(\times \) | \(\times \) | \(\times \) | \(\checkmark \) |

- 3.
No PQ explicit breaking is generated at one-loop from the CW potential.

^{2}This condition is satisfied by imposing \(\hat{d}_i=0\), for \(i=1,\,2,\,3\) (and the equivalent ones for the bottom sector).

*à la*KSVZ a fourth condition is necessary:

- 4.
The complex scalar field

*s*needs to couple to at least one of the exotic fermions (not necessarily to all of them) and the net contribution to the QCD-\(\theta \) term of the colour anomaly needs to be non-vanishing.

The model identified with the PQ charge assignments in Table 1 satisfies to all the previous conditions: using the freedom to fix one of the charges, i.e. the charge of the complex scalar singlet \(n_s=1\), the two cases shown in the table are physically equivalent. This model is contained within the classes of constructions recently presented in Ref. [39].

The model presents a series of interesting features. No PQ charge is assigned to the SM particles and neither to the exotic fermions \(\psi _R\) and \(\chi _L\). The Yukawa-like terms proportional to \(y_{1,2}\) are invariant under \(U(1)_\text {PQ}\), while the term proportional to \(\Lambda _2\) is not and then it cannot be introduced in the Lagrangian. In consequence, the subleading contribution to the SM fermion masses is identically vanishing and the top mass is given only by the leading term in Eq. (3.1) (similarly for the bottom mass). The Dirac mass terms \(M_{1,5}\) are also forbidden and then the exotic fermions \(\psi \) and \(\chi \) receive mass of the order \(z_5 v_r\) (or \(\tilde{z}_5 v_r\) depending on the specific sign of the PQ charge) and \(z_1 v_r\) (or \(\tilde{z}_1 v_r\)), once *r* develops a non-vanishing VEV. As \(v_r\) is typically expected to be of the order of \(f_s\), these fermions decouple from the spectrum when \(f_s \gg f\). Finally, condition 2 implies that the couplings \(\Lambda _{i}\) are neither promoted to spurions nor substituted by a dynamical field (i.e. *s* or \(s^*\)), and this represents a difference with respect to the analysis in Ref. [39].

*SO*(5) breaking terms read

*SO*(5) breaking contribution to the

*h*-mass term arises from the fermionic part of the CW potential, while no \(\sigma \) tadpole contribution is generated. This is different from the analysis performed in Ref. [9], where the only

*SO*(5) symmetry breaking terms considered have been the \(\sigma \) tadpole and the \(h^2\) terms. The minimisation of the scalar potential performed in Ref. [9] does not apply to this model and a new analysis is in order. To obtain a viable

*SO*(5) and EW spontaneous symmetry breaking at least two different

*SO*(5) breaking terms are necessary. Additional unavoidable sources of

*SO*(5) breaking comes from the gauge sector, as shown in Eq. (2.27). The minimal counter-term potential required at tree-level by renormalisability of the theory, once the charge assignment has been chosen, is then given in the unitary gauge by

*SO*(5) /

*SO*(4) phenomenology and the analysis of the scalar potential. The physical dependence on the explicit value of \(n_s\), and then of those of the exotic fermions, can be found in the couplings between the axion and the gauge field strengths, whose coefficients are determined by the chiral anomaly (see Refs. [57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67] for other studies where the axion couplings are modified with respect to those in the original KSVZ model).

^{3}It will be useful for the future discussion to introduce the notation of the effective couplings

The coefficients of the axion couplings to the gauge boson field strengths in the physical basis are reported, where the normalisation is defined in Eq. (3.4)

\(c_{agg}\) | \(c_{a\gamma \gamma }\) | \(c_{aZZ}\) | \(c_{a\gamma Z}\) | \(c_{a W W}\) |
---|---|---|---|---|

8 | 112 / 3 | 49.3 | 17.8 | 108.1 |

The charge assignment in Table 1 corresponds to the minimal setup among all the possible AML\(\sigma \)M constructions, where the minimality refers to the number of new parameters that are introduced with respect to the ML\(\sigma \)M: the number of parameters in the fermionic Lagrangian is the same; in the scalar potential, only three additional parameters are considered, corresponding to the PQ sector (\(f_s\), \(\lambda _s\) and \(\lambda _{s\phi }\)), and in particular only two *SO*(5) breaking terms are present (corresponding to \(\beta \) and \(\gamma \)); the PQ charges also represent degrees of freedom and the minimal model in Table 1 is univocally determined by fixing \(n_s\). Indeed, conditions 1 and 2 impose that the difference between the charges of the LH and RH components of the SM fermions is vanishing, \(n_{q_L}-n_{t_R}=0\), and in consequence it is always possible to redefine the whole set of PQ charges such that \(n_{q_L}=n_{t_R}=0\).

It is worth mentioning that an alternative charge assignment can be found satisfying to the conditions 1-4, but this scenario is not minimal in terms of number of parameters. In this case, the charges are such that \(n_{t_R}=n_{\chi _L}=n_{\chi _R}=n_{\psi _L}=n_{\psi _R}\mp n_s=n_{q_L} \mp n_s\), where the “−” or “\(+\)” refer to the presence of \(z_5\) or \(\tilde{z}_5\) terms in the Lagrangian, respectively. As discussed in Ref. [39], SM fermions transform under the PQ symmetry, differently from the minimal AML\(\sigma \)M in Table 1. Moreover, the Dirac mass term \(M_1\) is allowed in the Lagrangian, while the \(\psi \) fermions receive mass from the Yukawa-like term proportional to \(z_5\) (or \(\tilde{z}_5\)). Moreover, the terms proportional to \(\Lambda _{1,2,3}\) and \(y_1\) are allowed, while the one with \(y_2\) is forbidden. In consequence, the term \(\tilde{d}_1\) in Eq. (2.26) is not vanishing and then a \(\sigma \) tadpole needs to be also added into the counter term potential \(V^\mathrm{c.t.}(h,\sigma )\). The number of *SO*(5) breaking parameters is now increased by one unit with respect to the minimal case discussed above. For this reason, this second scenario is not considered in what follows.

## 4 The scalar potential

*SO*(5), \(U(1)_\text {PQ}\) and EW spontaneous symmetry breaking with non-vanishing VEVs,

- (i)
\(\lambda >0\) and \(\lambda _s>0\) in order to lead to a potential bounded from below.

- (ii)
\(\beta \) and \(\gamma \) should have the same sign in order to guarantee a positive \(v_h^2\) value. Following the sign convention adopted in Eq. (4.1), when both parameters are positive, the explicit symmetry breaking terms sum “constructively” to the quadratic and quartic terms in the potential in the broken phase, and a larger parameter space is allowed. Moreover, the ratio \(\beta /2\gamma <1\) leads to \(v_h<f\), corresponding to the expected ordering in the symmetry breaking scales.

- (iii)\(\lambda _{s\phi }\) should satisfy toin order to enforce positive \(v_\sigma ^2\) and \(v_r^2\) values. For negative \(\lambda _{s\phi }\) values, additional constraints could be enforced depending on the values of the other parameters. The sign convention chosen in Eq. (4.1) guarantees that no cancellation in \(v_\sigma ^2\) and \(v_r^2\) occurs for \(\lambda _{s\phi }>0\).$$\begin{aligned} \lambda _{s\phi }^2 < 4 \lambda \lambda _s \end{aligned}$$(4.3)

- 1.
Integrating out the heaviest scalar dof, whose largest component is the radial scalar field

*r*, and studying the LO terms of the Lagrangian; - 2.
Assuming \(f_s\sim f\), expanding perturbatively in the small \(\beta \) and \(\lambda _{s\phi }\) parameters.

### 4.1 Integrating out the heaviest scalar field

The case for \(\lambda _s\gg 1\), with \(f_s\) of the same order of *f*, corresponds to the \(U(1)_\text {PQ}\) non-linear spontaneous symmetry breaking framework^{4}: this is the traditional axion framework where the only component of *s* in the low-energy spectrum is the axion, while \(\hat{r}\) is integrated out. As the Yukawa-like couplings of the exotic fermions do not depend on \(\lambda _s\), the decoupling of \(\hat{r}\) does not have any impact on the spectrum of the exotic fermions, that depends exclusively of the specific value chosen for \(f_s\). One can then consider in detail the two limiting cases: \(f_s \sim f\) or \(f_s \gg f\). Notice that in the second scenario, when \(f_s\) is much larger than any other mass scale, the exotic fermion sector decouples at the same time as the heavier scalar dof.

*SO*(5) /

*SO*(4) SSB mechanism.

*SO*(2) rotation,

The following two subsections will describe in detail the two limits \(\lambda _s\gg 1\) and \(f_s\gg f \sim \sqrt{s_\mathrm{cm}}\), focusing on the scalar sector.

**The large PQ quartic coupling:** \(\lambda _s \gg 1\) **and** \(f_s \sim f\)

*r*can be expanded in inverse powers of \(\lambda _s\) (see Ref. [10] for a similar analysis): at the NLO, one has

*SO*(5) /

*SO*(4) breaking scale, while the

*SO*(5) quartic coupling \(\lambda =\lambda _R\) remains unchanged. The positivity of \(f_R^2\) translates into a constraint on the couplings \(\lambda _{s\phi }\):

*SO*(5) and the PQ sectors, and therefore once it is vanishing the two sectors are completely decoupled.

**The large PQ SSB scale:**\(f_s \gg f \sim \sqrt{s_\mathrm{cm}}\)

*r*, adopting as new dimensionless expanding parameter \(f/f_s\). Within this setup

*r*at NLO reads

*SO*(5) and PQ sectors are decoupled: in this specific case \(f_R=f\) and the

*SO*(5) SSB sector is not affected by the integration out of the radial dof

*r*.

### 4.2 The case for \(f_s\sim f \sim \sqrt{s_\mathrm{cm}}\) and \(\beta ,\lambda _{s\phi }\ll 1\)

For \(f_s\sim f \sim \sqrt{s_\mathrm{cm}}\), all the three scalar dofs are retained in the low energy spectrum and in general a stronger mixing between the three eigenstate is expected, compared to the previous setups. Complete analytical expression for the masses and mixings cannot be written in particularly elegant and condensed form. Nevertheless, simple analytic results can be obtained under the assumption that \(\beta ,\,\lambda _{s\phi }\ll 1\), which are natural conditions in the AML\(\sigma \)M. The first condition comes from the requirement that \(v_h\) coincides with the EW scale *v*, defined by \(v\equiv 2M_W/g=246\text { GeV}\), and it is much smaller than the *SO*(5) SSB scale, i.e. \(v_h<f\). The smallness of \(\lambda _{s\phi }\) follows, instead, from the assumption that the *SO*(5) and PQ sectors are determined by two distinct dynamics and therefore the two breaking mechanisms occur independently. A large \(\lambda _{s\phi }\) would indicate, instead, a unique origin for the two symmetry breaking mechanisms and would signal the impossibility of disentangling the two sectors.

### 4.3 Numerical analysis

*W*and

*Z*are modified with respect to the SM predictions for the Higgs particle by factor of \(C_1\).

**The scalar potential parameter space**

*f*and \(f_s\). By using the known experimental values of the Higgs VEV, \(v_h=v\equiv 246\text { GeV}\), and mass \(m_1=m_h\equiv 125\text { GeV}\), two of these coefficients can be eliminated in terms of the remaining five. The adopted procedure for the numerical analysis is to express \(\gamma \) as function of \(\beta \) and

*f*, by inverting the \(v_h^2\) expression in Eq. (4.2):

*hZZ*and

*hWW*couplings are obtained by [71], using the so called \(\kappa \)-framework

^{5}:

Figure 1 gives the idea of the interplay between the two scales *f* and \(f_s\) for fixed values of the remaining adimensional parameters. For \(f_s=1\) TeV, LHC can already start to exclude values of \(f \lesssim 0.7\) TeV. However, for the larger value \(f_s=3\) TeV, even values of \(f\approx 0.5\) TeV will lie outside LHC exclusion reach and no precise bound separately on *f* or \(f_s\) can be inferred from the sole measurement of the Higgs couplings to gauge bosons, for most of the parameter space.^{6} Only when \(\lambda , \lambda _s \gg 1\) are taken, the extra scalar dofs are decoupled and the CHM relation of Eq. (4.39) can be exploited. These results are compatible with the ones of Ref. [9], where a detailed study on the allowed range for *f* has been performed in the context of the ML\(\sigma \)M. For completeness in Fig. 1 also the curves corresponding to two values of the mass of the next to lightest scalar, \(m_2 = 1\) TeV and \(m_2=2\) TeV, have been depicted.

In Fig. 2, the masses \(m_2\) and \(m_3\) are shown as a function of \(\lambda _{s\phi }\) (upper left), or \(\lambda =\lambda _s\) (upper right), or \(\lambda \) (lower). The mass \(m_1\) is fixed at \(m_h\), while the scale *f* is taken at \(2\text { TeV}\). Three distinct values for \(f_s\) are considered, \(f_s=1\text { TeV},\,10^3\text { TeV},\,10^6\text { TeV}\), and are shown in the same plot spanning a different region of the parameter space. In the plot in the upper left, the values for \(\lambda \) and \(\lambda _s\) are taken to be equal to 10; in the plot in the upper right, \(\lambda _{s\phi }=0.1\); in the lower plot, \(\lambda _{s\phi }=0.1\) and \(\lambda _s=10\).

All these plots present features discussed in the different limiting cases of the previous section. In the three plots, the lines corresponding to \(f_s=10^3\text { TeV}\) and \(f_s=10^6\text { TeV}\) well represent the expressions for the masses in Eq. (4.28). In the upper left plot, the red-dashed line represents the heaviest dof with a constant mass according with Eq. (4.4); the blue-continue line corresponds to the second heaviest dof and it shows an increasing behaviour with a constant slope, corresponding to the expression for \(m_2^2\) that in first approximation is proportional to \(\lambda _{s\phi }\). In the upper right plot, the red area is excluded according to Eq. (4.3): close to this region, the analytic expressions do not closely follow the numerical results, as it appears in the behaviour of the red-dashed line that increases with a constant slope according to Eq. (4.4) only for \(\lambda =\lambda _s\gtrsim 0.1\). The blue-continue line is almost constant, as expected from the expression of \(m_2^2\) in Eq. (4.28), except for the region with small \(\lambda =\lambda _s\). In the lower plot, both the red-dashed and the blue-continue lines are horizontal, as expected having fixed both \(\lambda _s\) and \(\lambda _{s\phi }\).

The mixing coefficients \(C_1\), \(C_2\) and \(C_3\) are shown in Fig. 3: the green-dot-dashed line describes \(C_1^2\), the blue-continue line \(C_2^2\) and the red-dashed line \(C_3^2\). Both plots clearly show that the largest component to \(\hat{h}\) is \(\varphi _1\), that is identified to the physical Higgs particle. The contaminations from \(\varphi _2\) and \(\varphi _3\) are much smaller and at the level of \(\sim 1\%\) at most. This is a typical feature in almost all the parameter space, and in particular for \(f_s\gg f\), whose corresponding plots are very similar to the one in Fig. 3 on the right. The only substantial difference between the two plots shown is the exchange behaviour between \(C_2^2\) and \(C_3^2\): as far as \(f_s>f\) the largest contamination is given by \(\varphi _2\), while for \(f<f_s\) it is given by \(\varphi _3\), as it is confirmed by Eq. (4.34).

The results on the mixing coefficients can be compared to the ones for the equivalent quantities in the ML\(\sigma \)M: in the latter, only two scalar states are present and then only one mixing can be defined, that is between \(\hat{h}\) and \(\hat{\sigma }\); for increasing masses of \(\varphi _2\), which almost coincides with \(\hat{\sigma }\), the sibling of \(C_2^2\) asymptotically approaches the ratio \(v^2/f^2\) and a benchmark value of 0.06 has been taken in the phenomenological analysis. From Fig. 3, the maximal value that \(C_2^2\) (or \(C_3^2\)) can take is of 0.015: this means that some differences are expected between the two models when discussing the EW precision observables (EWPO) and the impact of the exotic fermions.

In a tiny region of the parameter space, \(\varphi _2\) can be lighter than \(\varphi _1\), with \(m_1\) still fixed at the value \(m_h\). This is consistent with the results in Ref. [9]. Although this possibility is experimentally viable, from the theoretical perspective it is not appealing as \(m_2<m_1\) requires \(\lambda _{s\phi }\lesssim 10^{-7}\), corresponding to a highly tuned situation. Similarly, mixing parameters larger than the typical values shown in Fig. 3, for example \(C_2^2\sim 0.1\), can only be achieved for \(\lambda _{s\phi }\lesssim 10^{-4}\), another tuned region of the parameter space. Another possibility for relatively large mixing parameters is for \(f\sim 100\text { GeV}\) and \(f_s\lesssim 1\text { TeV}\), that is very unlikely as it would correspond to the case with the EWSB occurring before the *SO*(5) / *SO*(4) symmetry breaking. In consequence, only the case with \(\varphi _2\) heavier than \(\varphi _1\) and values of \(\lambda _{s\phi }\gtrsim 0.01\) will be considered in the following.

## 5 Collider phenomenology and exotic fermions

Within a specific CH model setup, defined by a coset, the Higgs couplings to fermions depend on the kind of exotic fermions that enrich the spectrum and the chosen symmetry representations. A recent review on the *SO*(5) / *SO*(4) context has been presented in Ref. [12] and the impact at colliders of different realisations has been analysed in Ref. [74]. The ML\(\sigma \)M, and therefore also the AML\(\sigma \)M, seems an interpolation between the so-called \(MCHM_4\) and \(MCHM_5\) scenarios considered in Ref. [74], once only the physical Higgs is retained in the low-energy theory. Typical observables of interest at colliders are the EWPO, the \(Zb\bar{b}\) coupling, couplings of the scalar dofs to gluons and photons [7, 8], and the interactions with fermions. As they have been studied for the ML\(\sigma \)M in Refs. [9, 10], the aim of this section is to extend those results to the AML\(\sigma \)M.

**EWPO**

Deviations to the SM predictions for the *T* and *S* parameters [75] (or equivalently \(\epsilon _1\) and \(\epsilon _3\) [76]) are expected to be relevant. In the ML\(\sigma \)M, the mixing between \(\hat{h}\) and \(\hat{\sigma }\) can reach relatively large values, \(\sim 0.1\), and relevant scalar contributions to *T* and *S* are indeed expected. However, these contributions can always be compensated, in some allowed region of the parameters space, once exotic fermion contributions are included.

In the AML\(\sigma \)M, for the benchmark values chosen in the previous section, the values of the scalar sector mixing parameters result very small, see Fig. 3, and then the contributions to *T* and *S* are expected to be much less relevant. For smaller values of *f* consistent with Fig. 1, the \(\hat{h}\)-\(\hat{\sigma }\) mixing slightly increases, and then larger contributions to *T* and *S* are expected. In addition, relevant contributions to the EWPO from the fermionic sector can also be present. However, exactly as happens in the ML\(\sigma \)M case, it is always possible to evade the *T* and *S* bounds in a non negligible part of the full (fermionic + bosonic) parameter space.

\({\varvec{Zb\bar{b}}}\) **coupling**

The modification of the *Z* couplings to \(b\bar{b}\) is a very good observable to test a model. The most relevant contributions arise from the top-partner fermion, while the ones from the heavier scalar dofs turn out to be negligible. The top-partner induces deviations from the SM prediction of this coupling only at the one-loop level, and the effect of these contributions is soften with respect to those to the EWPO previously discussed. This result holds for both the ML\(\sigma \)M and the AML\(\sigma \)M. As illustrated in Ref. [9], it is easy to accommodate the experimental measure of the \(Zb\bar{b}\) coupling in a large part of the parameter space, and therefore no relevant constraint can be deduced from this observable.

**Couplings with gauge bosons and** \(\sigma \) **production at colliders**

As in the SM, no tree level \(\hat{h}gg\) and \(\hat{h}\gamma \gamma \) couplings are present in the AML\(\sigma \)M. However, effective interactions with gluons and with photons may arise at the one-loop level. In consequence, all the three scalar mass eigenstates, \(\varphi _{1,2,3}\), do couple with gluons and photons, with their interactions weighted by the corresponding mixing coefficients \(C_i^2\), according to Eq. (4.37).

As worked out in details in Ref. [9], the Higgs coupling with two gluons, \(\varphi _1gg\), is mainly due to the top contribution, as the bottom one is negligible and the exotic fermion ones tend to cancel out (due to their vector-like nature). On the other hand, the \(\varphi _2 gg\) and \(\varphi _3 gg\) couplings are suppressed by \(C_2^2\) and \(C_3^2\) respectively, and therefore are typically at least \(10^{-2}\) smaller than \(\varphi _1gg\). Moreover, as the top quark is lighter than \(\varphi _2\) and \(\varphi _3\), its contribution to their couplings are also suppressed, and the dominant terms arise from the exotic fermion sector.

The couplings to photons receive relevant contributions, not only from loops of top quark and of exotic fermions, but also from loops of massive gauge bosons. The latter are the dominant ones in the case of the physical Higgs particle, i.e. for \(\varphi _1\gamma \gamma \), while they are suppressed by \(C_2^2\) and \(C_3^2\) for the heavier scalar dofs and the most relevant contributions to \(\varphi _2\gamma \gamma \) and \(\varphi _3\gamma \gamma \) are those from the exotic fermions.

These results impact on the production mechanisms of the heavier dofs at collider, that are gluon fusion or vector boson fusion. From Fig. 2, the masses for \(\varphi _2\) and \(\varphi _3\) are typically larger than the TeV scale, within the whole range of values for *f* and \(f_s\) shown in Fig. 1. The lowest mass values are then potentially testable at colliders, although it strongly depends on the couplings with gluons and the massive gauge bosons. Ref. [9] concluded that, in the presence of only two scalar dofs, the heaviest one would be constrained only for masses lower than \(0.6\text { TeV}\) and mixing coefficient \(C_2^2>0.1\). Extending this result to the three scalar dofs described in the AML\(\sigma \)M and considering the results presented in Fig. 2, the present LHC data and the future prospects (LHC run-2 with total luminosity of \(3ab^{-1}\)) are not able to put any relevant bound, or in other words the heavier scalar dofs have production cross sections too small to lead to any signal in the present and future run of LHC.

**Impact of the exotic fermions**

The exotic fermion masses partially depend on a distinct set of parameters with respect to those entering the scalar potential. While this is particularly true for the ML\(\sigma \)M, where two arbitrary mass parameters \(M_{1,5}^{(\prime )}\) are introduced in the Lagrangian, in the minimal AML\(\sigma \)M the exotic fermion masses are controlled by \(f_s\), through the parameters \(z_{1,5}^{(\prime )}\) (and/or \(\tilde{z}_{1,5}^{(\prime )}\)). The largeness of \(f_s\) corresponds to large masses for these exotic fermions, consistent with the fermion partial compositeness mechanism. Direct detections would be probably very unlikely, while their effect would manifest in deviations from the SM predictions of SM field couplings. In Ref. [9], the exotic fermions have been integrated out and the induced low-energy operators have been identified. The mayor expected effects consist in decorrelations between observables that are instead correlated in the SM, and the appearance of anomalous couplings: these effects are very much typical of the HEFT setup, where the EWSB is non-linearly realised and the Higgs originates as a GB. For an overview of these analyses see Refs. [21, 22, 23, 29, 32, 77, 78].

Besides the effects discussed above, it is worth to mention the possibility to investigate the Higgs nature through the physics of the longitudinal components of the SM massive gauge bosons. As the ML\(\sigma \)M and AML\(\sigma \)M deal with the same symmetry of the SM, no additional effects are expected with respect to the analyses carried out in Refs. [79, 80, 81, 82, 83].

## 6 The axion and ALP phenomenology

The axion couplings to SM gauge bosons and fermions have been bounded from several observables [84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116]. Two recent summaries can be found in Refs. [117, 118]. In the following, only the couplings with bosons will be taken into consideration, as in the minimal AML\(\sigma \)M described here no direct interaction is present with SM fermions.^{7} The axion couplings strongly depend on its mass, that moreover determines whether the axion is expected to decay or not inside the collider. On the other side, for the ALP, mass and couplings are not related.

The following constraints hold for both a QCD axion and an ALP.

**Coupling to photons**

**Coupling to gluons**

**Couplings to massive gauge bosons**

*W*gauge bosons (as already discussed, no axion-SM fermion couplings are present at tree-level in the minimal AML\(\sigma \)M). The most relevant observable for axion masses below \(\sim 0.2\text { GeV}\) is \(K^+\rightarrow \pi ^++a\) whose branching ratio has been bounded by the E787 and E949 experiments [92]:

*Z*boson width allows to put a conservative bound on \(Z\rightarrow a\gamma \) interaction:

**The axion mass**

### 6.1 QCD axion or axion-like-particle?

*s*, and in consequence \(f_a\simeq f_s\) in first approximation. The corresponding induced axion mass belongs to the window from tens of meV to the keV, according to Eq. (6.14). For this range of values, the strongest constraints on \(f_a\) come from the axion coupling to two photons \(g_{a\gamma \gamma }\), Eqs. (6.1) and (6.2): specifying the value of \(c_{a\gamma \gamma }\) for the minimal AML\(\sigma \)M charge assignment as reported in Table 2, one gets

For much lighter values of the \(f_s\) scale, instead, the astrophysical bounds on \(g_{a\gamma \gamma }\) coupling can be satisfied only assuming that the axion mass and its characteristic scale \(f_s\) are not correlated. This corresponds to the ALP scenario: differently from the QCD axion, an ALP has a mass that is independent from its characteristic scale \(f_s\), due to additional sources of soft shift symmetry breaking with respect to those in Eqs. (6.12) and (6.13), and does not necessarily solve the strong CP problem.^{8} As an example, a benchmark point that passes all the previous bounds corresponds to a \(1\text { GeV}\) axion with \(f_s\sim 200\text { TeV}\). The most sensitive observables for this particle are its couplings with two *W*’s, two *Z*’s and \(Z\gamma \), see Eq. (6.11), than can be analysed in collider searches. The other class of constraints arising from meson decays are not relevant in this case: the \(K^+\rightarrow \pi ^++a\) decay is kinematically forbidden for this axion mass, while the prediction for the branching ratio of \(B^+\rightarrow K^++a\) is of \(\lesssim 10^{-13}\), much below the future expected sensitivity at Belle II [123].

For this value of \(f_s\), the heaviest scalar dofs, despite being much smaller than in the previous scenario, are expected to have so large masses and so small couplings that will be very unlikely to detect any signal at present or even future LHC runs. Instead, the model can be tested through deviations from the SM predictions of the Higgs couplings or through pure gauge boson observables.

Finally, the difference with respect to the previous scenario is mainly that a massive axion is likely to give signals at colliders, due to the present sensitivity on its couplings with massive gauge bosons. On the other side, no signal at all is expected in the flavour sector, as the expected future improvements in the experimental precision are still very far from the predicted theoretical values.

**The fine-tuning problem**

The presence of different scales in the scalar potential leads to a fine-tuning problem in the model. As already mentioned, the parameter \(\xi \) measures the tension between the EW scale and the *SO*(5) SSB scale, as shown in Eq. (4.39). In models where axions or ALPs are dynamically originated, a new scale \(f_s\) is present and typically much larger than the EW scale. Once the scalar field *s* develops a VEV, the scale *f* receives a contribution proportional to \(\sqrt{\lambda }_{s\phi }f_s\), as can be read in Eq. (2.21). This leads to \(f\approx f_s\gg v\), or \(\lambda _{s\phi }\ll 1\): this represents two sides of the same fine-tuning problem.

In the ALP model presented here \(f_s\sim 200\text { TeV}\) and therefore a value of \(\lambda _{s\phi }\lesssim 10^{-4}\) would be necessary to not modify, excessively, the scale *f*. In generic AML\(\sigma \)M, much larger values for \(f_s\) are typically necessary to pass the different experimental bounds on the axion/ALP couplings and then a much stronger fine-tuning on \(\lambda _{s\phi }\) has to be invoked.^{9}

## 7 Concluding remarks

The AML\(\sigma \)M [39] represents a class of models that extend the ML\(\sigma \)M [9] by the introduction of a complex scalar singlet, that allows to supplement the *SO*(5) and EW symmetries with an extra \(U(1)_\mathrm{PQ}\).

The spectrum of the AML\(\sigma \)M encodes: i) the SM gauge bosons and fermions; ii) three real scalar dofs, one of them, the Higgs particle, being the only uneaten GB of the *SO*(5) / *SO*(4) breaking; iii) two types of vectorial exotic fermions respectively in the fundamental and in the singlet representation of *SO*(5); iv) the PQ GB originated by the spontaneous breaking of the \(U(1)_\mathrm{PQ}\) symmetry. The scale *f* of the *SO*(5) / *SO*(4) breaking is expected to be in the TeV region, in order to solve the Higgs hierarchy problem, while the PQ-breaking scale, \(f_s\), is in principle independent from *f*, spanning over a large range of values.

A detailed analysis of the scalar potential and its minima has been presented for the first time. The appearance of possible *SO*(5) and PQ explicit breaking terms arising from 1-loop fermionic and gauge contributions has been extensively discussed. The type and number of the additional terms required by renormalisability depends on the PQ charges assigned to the fields of the model.

A minimal AML\(\sigma \)M has been identified by introducing few general requirements with the intent to minimize the number of parameters in the whole Lagrangian. In particular, the parameter space of the minimal AML\(\sigma \)M scalar sector is determined by 7 parameters. Two of them can be fixed by identifying one scalar dof with the physical Higgs particle and its VEV with the EW scale. The remaining free parameters correspond to: the quartic couplings \(\lambda \) and \(\lambda _s\) that control the linearity of the EWSB and the PQ symmetry breaking mechanisms, respectively; the scales *f* and \(f_s\) related to the symmetry breaking; the mixed quartic coupling \(\lambda _{s\phi }\) that represents the portal between the EW and PQ sectors. Simplified analytical expressions can be obtained for the scalar sector by integrating out the highest mass dof, either in the strongly interacting regime, \(\lambda _s \gg 1\), keeping free the scales \(f_s\) and *f* either in the perturbative regime, \(\lambda _s \lesssim 1\), but assuming instead a large hierarchy between the scales, \(f_s \gg f\). Interesting analytical expression for the scalar sector in the regime \(f_s \sim f\) can be obtained also in the limit \(\beta , \lambda _{s\phi }\ll 1\).

The analytical and numerical analysis of the parameter space points out that for \(f,\,f_s\gtrsim 1\text { TeV}\) the heavier scalar dofs are unlikely to give signals at the present and future LHC run, while only the non-linearity of the EWSB mechanism would lead to interesting deviations from the SM predictions in Higgs and gauge boson sectors.

The analysis of the PQ GB phenomenology reveals two possible scenarios: a light QCD axion or a heavy ALP. In the first case, the axion mass is expected in the range \([\mathrm{meV},\,\mathrm{keV}]\) and the strong bounds present on the axion coupling to two photons require that its characteristic scale \(f_a\sim f_s\) must be larger than \(10^5\text { TeV}\), strongly suppressing all its interactions. This model represents a minimal invisible axion construction, where the EWSB mechanism is non-linearly realised and the physical Higgs particle arises as a GB. As can be realised from Eqs. (4.6)–(4.26), invisible axion models are, in general, strongly fine-tuned. In fact, the typical *SO*(5) / *SO*(4) breaking scale of the effective theory obtained integrating out the heavy degrees of freedom “naturally runs” to the highest scale, \(f_R \sim f_s\), reintroducing the EW hierarchy problem, \(\xi \ll 1\). Alternatively, the tuning \(\lambda _{s\phi }=0\) can be introduced: this is, however, rather unnatural as no symmetry protects it.

In the second scenario, the ALP typically has a much larger mass, independent from the value of its characteristic scale. The benchmark \(m_a=1\text { GeV}\) and \(f_s=200\text { TeV}\) has been considered for concreteness. Such an ALP would be free from the strong bounds on \(a\gamma \gamma \) and it is likely to be detected at LHC, the best sensitivity being on the *aWW* and *aZZ* couplings, while no signals are expected in flavour observables such as meson decays. Values of \(f_s\) close to \(200\text { TeV}\) introduce a mild fine-tuning on the model, compared to the one that may be encountered in traditional axion models. To obtain more natural ALP models, the minimality conditions stated in this analysis should be, in some way, relaxed, attempting to suppress the *aWW* and *aZZ* couplings (see Ref. [56] for such possibility).

## Footnotes

- 1.
In Ref. [38] the MCHM has been enriched by an additional

*U*(1) symmetry, that is non-anomalous and therefore does not originate a QCD axion. - 2.
The discussion on the consequences of PQ explicit breaking contributions, on its interest in cosmological studies, and on the case where the

*SO*(5) /*SO*(4) and PQ symmetry breaking occur at the same scale is deferred to Ref. [56]. - 3.
In the present discussion, only one fermion generation has been considered. Once extending this study to the realistic case of three generations [56], the values reported in Table 2 will be modified: for example, assuming that the same charges will be adopted for all the fermion generations, the numerical values in the table will be multiplied by a factor 3.

- 4.
In the case where an UV strong interacting dynamics is responsible of the largeness of \(\lambda _s\), new resonances are expected at the scale \(\lesssim 4\pi f_s\) (see the naive dimensional analysis [68]).

- 5.
Notice that in the \(\kappa \)-framework one assumes that there are no new particles contributing to the

*ggH*production or \(H \rightarrow \gamma \gamma \) decay loops. - 6.
Limits on the scale

*f*from EWPO will be discussed in the following section. - 7.
Indirect couplings arise from the same mechanism that generate SM fermion masses. However, experimental constraints are present on axion couplings with only light SM fermions, the strongest being on axion couplings with two electrons. As in the minimal AML\(\sigma \)M only the third generation fermions are considered, no relevant bound can be deduced considering these constraints. This analysis is postponed to further investigation [56].

- 8.
In the ALP scenario, a solution to the Strong CP problem is not guaranteed and therefore the condition 4 is not required. An additional scenario satisfying conditions 1, 2, and 3, can be considered: in this case, \(n_{q_L}=n_{\psi _L}=n_{\psi _R}=n_{\chi _R} =n_{t_R}\pm n_s=n_{\chi _L}\pm n_s\) (with the “\(+\)” or “−” are associated to the presence of the \(z_1\) or \(\tilde{z}_1\) terms in the Lagrangian, respectively), and the induced renormalisable scalar potential turns out to be the same as in Eq. (4.1).

- 9.

## Notes

### Acknowledgements

The authors thank R. Alonso, F. Feruglio, P. Machado, and A. Nelson for useful discussions, and I. Brivio and B. Gavela for comments and suggestions on the preliminary version of the paper. L.M. thanks the department of Physics and Astronomy of the Università degli Studi di Padova and the Fermilab Theory Division for hospitality during the writing up of the paper. F.P. and S.R. thank the University of Washington for hospitality during the writing up of the paper. The authors acknowledge partial financial support by the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant agreements No 690575 and No 674896. L.M. acknowledges partial financial support by the Spanish MINECO through the “Ramón y Cajal” programme (RYC-2015-17173), and by the Spanish “Agencia Estatal de Investigación” (AEI) and the EU “Fondo Europeo de Desarrollo Regional” (FEDER) through the project FPA2016-78645-P, and through the Centro de excelencia Severo Ochoa Program under Grant SEV-2016-0597.

## References

- 1.D.B. Kaplan, H. Georgi, \(SU(2) \times U(1)\) breaking by vacuum misalignment. Phys. Lett. B
**136**, 183–186 (1984)ADSCrossRefGoogle Scholar - 2.D.B. Kaplan, H. Georgi, S. Dimopoulos, Composite Higgs Scalars. Phys. Lett. B
**136**, 187–190 (1984)ADSCrossRefGoogle Scholar - 3.T. Banks, Constraints on \(SU(2) \times U(1)\) breaking by vacuum misalignment. Nucl. Phys. B
**243**, 125–130 (1984)ADSCrossRefGoogle Scholar - 4.K. Agashe, R. Contino, A. Pomarol, The minimal composite Higgs model. Nucl. Phys. B
**719**, 165–187 (2005). arXiv:hep-ph/0412089 ADSCrossRefGoogle Scholar - 5.B. Gripaios, A. Pomarol, F. Riva, J. Serra, Beyond the minimal composite Higgs model. JHEP
**04**, 070 (2009). arXiv: 0902.1483 ADSCrossRefGoogle Scholar - 6.J. Mrazek, A. Pomarol, R. Rattazzi, M. Redi, J. Serra, A. Wulzer, The other natural two Higgs doublet model. Nucl. Phys. B
**853**, 1–48 (2011). arXiv:1105.5403 ADSCrossRefzbMATHGoogle Scholar - 7.R. Barbieri, B. Bellazzini, V.S. Rychkov, A. Varagnolo, The Higgs Boson from an extended symmetry. Phys. Rev. D
**76**, 115008 (2007). arXiv:0706.0432 ADSCrossRefGoogle Scholar - 8.H. Gertov, A. Meroni, E. Molinaro, and F. Sannino, Theory and phenomenology of the elementary goldstone Higgs Boson. Phys. Rev. D
**92**(9), 095003 (2015). arXiv: 1507.06666 - 9.F. Feruglio, B. Gavela, K. Kanshin, P.A.N. Machado, S. Rigolin, S. Saa, The minimal linear sigma model for the Goldstone Higgs. JHEP
**06**, 038 (2016). arXiv:1603.05668 ADSCrossRefGoogle Scholar - 10.M.B. Gavela, K. Kanshin, P.A.N. Machado, S. Saa, The linear–non-linear frontier for the Goldstone Higgs. Eur. Phys. J. C
**76**(12), 690 (2016). arXiv: 1610.08083 - 11.R. Alonso, I. Brivio, B. Gavela, L. Merlo, S. Rigolin, Sigma decomposition. JHEP
**12**, 034 (2014). arXiv:1409.1589 ADSCrossRefGoogle Scholar - 12.G. Panico, A. Wulzer, The composite Nambu-Goldstone Higgs. Lect. Notes Phys.
**913**, 1–316 (2016). arXiv:1506.01961 - 13.I.M. Hierro, L. Merlo, S. Rigolin, Sigma decomposition: the CP-Odd Lagrangian. JHEP
**04**, 016 (2016). arXiv:1510.07899 ADSMathSciNetzbMATHGoogle Scholar - 14.F. Feruglio, The chiral approach to the electroweak interactions. Int. J. Mod. Phys. A
**8**, 4937–4972 (1993). arXiv:hep-ph/9301281 ADSCrossRefGoogle Scholar - 15.B. Grinstein, M. Trott, A Higgs–Higgs bound state due to new physics at a TeV. Phys. Rev. D
**76**, 073002 (2007). arXiv:0704.1505 ADSCrossRefGoogle Scholar - 16.R. Contino, C. Grojean, M. Moretti, F. Piccinini, R. Rattazzi, Strong double Higgs production at the Lhc. JHEP
**05**, 089 (2010). arXiv:1002.1011 ADSCrossRefGoogle Scholar - 17.R. Alonso, M. B. Gavela, L. Merlo, S. Rigolin, J. Yepes, The effective chiral lagrangian for a light dynamical “Higgs particle”. Phys. Lett. B
**722**, 330–335 (2013). arXiv: 1212.3305 [Erratum: Phys. Lett. B**726**, 926 (2013)] - 18.R. Alonso, M.B. Gavela, L. Merlo, S. Rigolin, J. Yepes, Minimal flavour violation with strong Higgs dynamics. JHEP
**06**, 076 (2012). arXiv:1201.1511 ADSCrossRefGoogle Scholar - 19.R. Alonso, M.B. Gavela, L. Merlo, S. Rigolin, J. Yepes, Flavor with a light dynamical “Higgs particle”. Phys. Rev. D
**87**(5), 055019 (2013). arXiv: 1212.3307 - 20.G. Buchalla, O. Catà, C. Krause, Complete Electroweak Chiral Lagrangian with a Light Higgs at NLO. Nucl. Phys. B
**880**, 552–573 (2014). arXiv:1307.5017 [Erratum: Nucl. Phys. B**913**, 475 (2016)] - 21.I. Brivio, T. Corbett, O.J.P. Éboli, M.B. Gavela, J. Gonzalez-Fraile, M.C. Gonzalez-Garcia, L. Merlo, S. Rigolin, Disentangling a dynamical Higgs. JHEP
**03**, 024 (2014). arXiv:1311.1823 ADSCrossRefGoogle Scholar - 22.I. Brivio, O.J.P. Éboli, M.B. Gavela, M.C. Gonzalez-Garcia, L. Merlo, S. Rigolin, Higgs ultraviolet softening. JHEP
**12**, 004 (2014). arXiv:1405.5412 ADSCrossRefGoogle Scholar - 23.M.B. Gavela, J. Gonzalez-Fraile, M.C. Gonzalez-Garcia, L. Merlo, S. Rigolin, J. Yepes, CP violation with a dynamical Higgs. JHEP
**10**, 044 (2014). arXiv:1406.6367 ADSCrossRefGoogle Scholar - 24.M.B. Gavela, K. Kanshin, P.A.N. Machado, S. Saa, On the renormalization of the electroweak chiral lagrangian with a Higgs. JHEP
**03**, 043 (2015). arXiv:1409.1571 ADSCrossRefGoogle Scholar - 25.R. Alonso, E.E. Jenkins, A.V. Manohar, A geometric formulation of higgs effective field theory: measuring the curvature of scalar field space. Phys. Lett. B
**754**, 335–342 (2016). arXiv: 1511.00724 - 26.B.M. Gavela, E.E. Jenkins, A.V. Manohar, L. Merlo, Analysis of general power counting rules in effective field theory. Eur. Phys. J. C
**76**(9), 485 (2016). arXiv:1601.07551 - 27.R. Alonso, E.E. Jenkins, A.V. Manohar, Sigma models with negative curvature. Phys. Lett. B
**756**, 358–364 (2016). arXiv:1602.00706 - 28.O.J.P. Éboli, M.C. Gonzalez–Garcia, Classifying the bosonic quartic couplings. Phys. Rev. D
**93**(9), 093013 (2016). arXiv:1604.03555 - 29.I. Brivio, J. Gonzalez-Fraile, M.C. Gonzalez-Garcia, L. Merlo, The complete HEFT Lagrangian after the LHC run I. Eur. Phys. J. C
**76**(7), 416 (2016). arXiv:1604.06801 - 30.R. Alonso, E.E. Jenkins, A.V. Manohar, Geometry of the scalar sector. JHEP
**08**, 101 (2016). arXiv:1605.03602 ADSMathSciNetCrossRefGoogle Scholar - 31.LHC Higgs Cross Section Working Group Collaboration, D. de Florian et. al.,
*Handbook of Lhc Higgs Cross Sections: 4. Deciphering the Nature of the Higgs Sector*. arXiv: 1610.07922 - 32.L. Merlo, S. Saa, M. Sacristán-Barbero, Baryon non-invariant couplings in Higgs effective field theory. Eur. Phys. J. C
**77**(3), 185 (2017). arXiv:1612.04832 - 33.R. Alonso, K. Kanshin, S. Saa, Renormalization group evolution of Higgs effective field theory. arXiv:1710.06848
- 34.G. Buchalla, O. Cata, A. Celis, M. Knecht, C. Krause, Complete one-loop renormalization of the Higgs-electroweak chiral lagrangian. arXiv:1710.06412
- 35.R.D. Peccei, H.R. Quinn, CP conservation in the presence of instantons. Phys. Rev. Lett.
**38**, 1440–1443 (1977)ADSCrossRefGoogle Scholar - 36.J.E. Kim, Weak interaction singlet and strong CP invariance. Phys. Rev. Lett.
**43**, 103 (1979)ADSCrossRefGoogle Scholar - 37.M.A. Shifman, A.I. Vainshtein, V.I. Zakharov, Can confinement ensure natural CP invariance of strong interactions? Nucl. Phys. B
**166**, 493–506 (1980)ADSMathSciNetCrossRefGoogle Scholar - 38.B. Gripaios, M. Nardecchia, T. You, On the structure of anomalous composite Higgs models. Eur. Phys. J. C
**77**(1), 28 (2017). arXiv:1605.09647 - 39.I. Brivio, B. Gavela, S. Pascoli, R. del Rey, S. Saa, The axion and the Goldstone Higgs. arXiv:1710.07715
- 40.D.B. Kaplan, Flavor at Ssc energies: a new mechanism for dynamically generated fermion masses. Nucl. Phys. B
**365**, 259–278 (1991)ADSCrossRefGoogle Scholar - 41.R. Contino, A. Pomarol, Holography for fermions. JHEP
**11**, 058 (2004). arXiv:hep-th/0406257 ADSMathSciNetCrossRefGoogle Scholar - 42.M.J. Dugan, H. Georgi, D.B. Kaplan, Anatomy of a composite Higgs model. Nucl. Phys. B
**254**, 299–326 (1985)ADSCrossRefGoogle Scholar - 43.J. Galloway, J.A. Evans, M.A. Luty, R.A. Tacchi, Minimal conformal technicolor and precision electroweak tests. JHEP
**10**, 086 (2010). arXiv:1001.1361 zbMATHGoogle Scholar - 44.F. Wilczek, Problem of strong P and T invariance in the presence of instantons. Phys. Rev. Lett.
**40**, 279–282 (1978)ADSCrossRefGoogle Scholar - 45.S. Weinberg, A new light boson? Phys. Rev. Lett.
**40**, 223–226 (1978)ADSCrossRefGoogle Scholar - 46.W.A. Bardeen, S.H.H. Tye, J.A.M. Vermaseren, Phenomenology of the new light Higgs Boson search. Phys. Lett.
**76B**, 580–584 (1978)ADSCrossRefGoogle Scholar - 47.P. Di Vecchia, G. Veneziano, Chiral dynamics in the large N limit. Nucl. Phys. B
**171**, 253–272 (1980)ADSMathSciNetCrossRefGoogle Scholar - 48.M. Dine, W. FisCHLer, M. Srednicki, A simple solution to the strong CP problem with a harmless axion. Phys. Lett. B
**104**, 199–202 (1981)ADSCrossRefGoogle Scholar - 49.A.R. Zhitnitsky, On possible suppression of the axion hadron interactions (in Russian). Sov. J. Nucl. Phys.
**31**, 260 (1980). [Yad. Fiz. 31, 497 (1980)]Google Scholar - 50.G. D’Ambrosio, G.F. Giudice, G. Isidori, A. Strumia, Minimal flavor violation: an effective field theory approach. Nucl. Phys. B
**645**, 155–187 (2002). arXiv:hep-ph/0207036 ADSCrossRefGoogle Scholar - 51.V. Cirigliano, B. Grinstein, G. Isidori, M.B. Wise, Minimal flavor violation in the lepton sector. Nucl. Phys. B
**728**, 121–134 (2005). arXiv:hep-ph/0507001 ADSCrossRefGoogle Scholar - 52.S. Davidson, F. Palorini, Various definitions of minimal flavour violation for leptons. Phys. Lett. B
**642**, 72–80 (2006). arXiv:hep-ph/0607329 ADSCrossRefGoogle Scholar - 53.R. Alonso, G. Isidori, L. Merlo, L.A. Munoz, E. Nardi, Minimal flavour violation extensions of the seesaw. JHEP
**06**, 037 (2011). arXiv:1103.5461 ADSCrossRefzbMATHGoogle Scholar - 54.D.N. Dinh, L. Merlo, S.T. Petcov, R. Vega-Álvarez, Revisiting minimal lepton flavour violation in the light of leptonic CP violation. JHEP
**07**, 089 (2017). arXiv:1705.09284 ADSCrossRefGoogle Scholar - 55.S.R. Coleman, E.J. Weinberg, Radiative corrections as the origin of spontaneous symmetry breaking. Phys. Rev. D
**7**, 1888–1910 (1973)ADSCrossRefGoogle Scholar - 56.J. Alonso-González, L. Merlo, F. Pobbe, S. Rigolin, in ProgressGoogle Scholar
- 57.S. Dimopoulos, L. Susskind, Mass without scalars. Nucl. Phys. B
**155**, 237 (1979)ADSCrossRefGoogle Scholar - 58.G.F. Giudice, R. Rattazzi, A. Strumia, Unificaxion. Phys. Lett. B
**715**, 142–148 (2012). arXiv:1204.5465 ADSCrossRefGoogle Scholar - 59.M. Redi, A. Strumia, Axion-Higgs unification. JHEP
**11**, 103 (2012). arXiv:1208.6013 ADSMathSciNetCrossRefGoogle Scholar - 60.M. Redi, R. Sato, Composite accidental axions. JHEP
**05**, 104 (2016). arXiv:1602.05427 ADSMathSciNetCrossRefzbMATHGoogle Scholar - 61.L. Di Luzio, F. Mescia, E. Nardi, Redefining the axion window. Phys. Rev. Lett.
**118**(3), 031801 (2017). arXiv:1610.07593 - 62.M. Farina, D. Pappadopulo, F. Rompineve, A. Tesi, The photo-philic QCD axion. JHEP
**01**, 095 (2017). arXiv:1611.09855 ADSCrossRefGoogle Scholar - 63.Y. Ema, K. Hamaguchi, T. Moroi, K. Nakayama, Flaxion: a minimal extension to solve puzzles in the standard model. JHEP
**01**, 096 (2017). arXiv:1612.05492 ADSMathSciNetCrossRefzbMATHGoogle Scholar - 64.L. Calibbi, F. Goertz, D. Redigolo, R. Ziegler, J. Zupan, Minimal axion model from flavor. Phys. Rev. D
**95**(9), 095009 (2017). arXiv:1612.08040 - 65.L. Di Luzio, F. Mescia, E. Nardi, The window for preferred axion models. arXiv:1705.05370
- 66.R. Coy, M. Frigerio, M. Ibe, Dynamical clockwork axions. arXiv:1706.04529 [hep-ph]
- 67.F. Arias-Aragon, L. Merlo, The minimal flavour violating axion. JHEP
**10**, 168 (2017). arXiv:1709.07039 ADSCrossRefGoogle Scholar - 68.A. Manohar, H. Georgi, Chiral quarks and the nonrelativistic quark model. Nucl. Phys. B
**234**, 189–212 (1984)ADSCrossRefGoogle Scholar - 69.S. Heinemeyer et al., LHC Higgs Cross Section Working Group,
*Handbook of LHC Higgs Cross Sections: 3. Higgs Properties*. arXiv:1307.1347 - 70.ATLAS, CMS Collaboration, G. Aad et al., Measurements of the Higgs Boson production and decay rates and constraints on its couplings from a combined Atlas and Cms analysis of the Lhc PP collision data at \( \sqrt{s}=7 \) and 8 TeV. JHEP
**08**, 045 (2016). arXiv:1606.02266 - 71.CMS Collaboration, Combined measurements of the Higgs boson’s coupling at \(\sqrt{s} = 13 TeV\), CMS PAS HIG-17-031Google Scholar
- 72.B. Bellazzini, C. Csáki, J. Serra, Composite Higgses. Eur. Phys. J. C
**74**(5), 2766 (2014). arXiv:1401.2457 - 73.C. Grojean, O. Matsedonskyi, G. Panico, Light top partners and precision physics. JHEP
**1310**, 160 (2013). arXiv:1306.4655 ADSCrossRefGoogle Scholar - 74.M. Carena, L. Da Rold, E. Pontón, Minimal composite Higgs models at the LHC. JHEP
**06**, 159 (2014). arXiv:1402.2987 ADSCrossRefGoogle Scholar - 75.M.E. Peskin, T. Takeuchi, Estimation of oblique electroweak corrections. Phys. Rev. D
**46**, 381–409 (1992)ADSCrossRefGoogle Scholar - 76.G. Altarelli, R. Barbieri, Vacuum polarization effects of new physics on electroweak processes. Phys. Lett. B
**253**, 161–167 (1991)ADSCrossRefGoogle Scholar - 77.I. Brivio, M.B. Gavela, L. Merlo, K. Mimasu, J.M. No, R. del Rey, V. Sanz, Non-linear Higgs portal to dark matter. JHEP
**04**, 141 (2016). arXiv:1511.01099 ADSGoogle Scholar - 78.P. Hernandez-Leon, L. Merlo, The complete bosonic basis for a Higgs-like dilaton. Phys. Rev. D
**96**(7), 075008 (2017). arXiv:1703.02064 - 79.D. Espriu, B. Yencho, Longitudinal WW scattering in light of the Higgs Boson discovery. Phys. Rev. D
**87**(5), 055017 (2013). arXiv:1212.4158 - 80.D. Espriu, F. Mescia, B. Yencho, Radiative corrections to \(W\_L W\_L\) scattering in composite Higgs models. Phys. Rev. D
**88**, 055002 (2013). arXiv:1307.2400 ADSCrossRefGoogle Scholar - 81.R.L. Delgado, A. Dobado, F.J. Llanes-Estrada, One-loop \(W\_LW\_L and Z\_LZ\_L\) scattering from the electroweak chiral lagrangian with a light Higgs-like scalar. JHEP
**02**, 121 (2014). arXiv:1311.5993 ADSCrossRefGoogle Scholar - 82.R.L. Delgado, A. Dobado, M.J. Herrero, J.J. Sanz-Cillero, One-loop \(\gamma \gamma \rightarrow W\_L^+ W\_L^- and \gamma \gamma \rightarrow Z\_L Z\_L\) from the electroweak chiral lagrangian with a light Higgs-like scalar. JHEP
**07**, 149 (2014). arXiv:1404.2866 ADSCrossRefGoogle Scholar - 83.R.L. Delgado, A. Dobado, D. Espriu, C. García-Garcia, M.J. Herrero, X. Marcano, J.J. Sanz-Cillero, Production of vector resonances at the LHC via WZ-scattering: a unitarized ECHL analysis. arXiv:1707.04580
- 84.K. Choi, K. Kang, J.E. Kim, Effects of \(\eta ^\prime \) in low-energy axion physics. Phys. Lett. B
**181**, 145–149 (1986)ADSCrossRefGoogle Scholar - 85.S. De Panfilis, A.C. Melissinos, B.E. Moskowitz, J.T. Rogers, Y.K. Semertzidis, W. Wuensch, H.J. Halama, A.G. Prodell, W.B. Fowler, F.A. Nezrick, Limits on the abundance and coupling of cosmic axions at \(4.5{\mu } eV < M(A) < 5.0 {\mu } eV\). Phys. Rev. Lett.
**59**, 839 (1987)ADSCrossRefGoogle Scholar - 86.J.D. Bjorken, S. Ecklund, W.R. Nelson, A. Abashian, C. Church, B. Lu, L.W. Mo, T.A. Nunamaker, P. Rassmann, Search for neutral metastable penetrating particles produced in the SLAC beam dump. Phys. Rev. D
**38**, 3375 (1988)ADSCrossRefGoogle Scholar - 87.M. Carena, R.D. Peccei, The effective lagrangian for axion emission from SN1987A. Phys. Rev. D
**40**, 652 (1989)ADSCrossRefGoogle Scholar - 88.W. Wuensch, S. De Panfilis-Wuensch, Y.K. Semertzidis, J.T. Rogers, A.C. Melissinos, H.J. Halama, B.E. Moskowitz, A.G. Prodell, W.B. Fowler, F.A. Nezrick, Results of a laboratory search for cosmic axions and other weakly coupled light particles. Phys. Rev. D
**40**, 3153 (1989)ADSCrossRefGoogle Scholar - 89.C. Hagmann, P. Sikivie, N.S. Sullivan, D.B. Tanner, Results from a search for cosmic axions. Phys. Rev. D
**42**, 1297–1300 (1990)ADSCrossRefGoogle Scholar - 90.A.D.M.X. Collaboration, S.J. Asztalos et al., An improved RF cavity search for halo axions. Phys. Rev. D
**69**, 011101 (2004). arXiv:astro-ph/0310042 CrossRefGoogle Scholar - 91.G.G. Raffelt, Astrophysical axion bounds. Lect. Notes Phys.
**741**, 51–71 (2008). arXiv:hep-ph/0611350 - 92.E787, E949 Collaboration, S. Adler et. al., Measurement of the \(K^+\rightarrow \pi ^+ \nu \nu \) branching ratio. Phys. Rev. D
**77**, 052003 (2008). arXiv:0709.1000 - 93.A.D.M.X. Collaboration, S.J. Asztalos et al., A squid-based microwave cavity search for dark-matter axions. Phys. Rev. Lett.
**104**, 041301 (2010). arXiv:0910.5914 ADSCrossRefGoogle Scholar - 94.G. Borexino Collaboration, Bellini et al., Search for solar axions produced in \(p(d,{}^3He)A\) reaction with Borexino detector. Phys. Rev. D
**85**, 092003 (2012). arXiv:1203.6258 - 95.A. Friedland, M. Giannotti, M. Wise, Constraining the axion–photon coupling with massive stars. Phys. Rev. Lett.
**110**(6), 061101 (2013). arXiv:1210.1271 - 96.BaBar Collaboration, J.P. Lees et al., Search for \(B \rightarrow K^{(*)} \nu \overline{\nu }\) and invisible quarkonium decays. Phys. Rev. D
**87**(11), 112005 (2013). arXiv:1303.7465 - 97.E. Armengaud et al., Axion searches with the Edelweiss-II experiment. JCAP
**1311**, 067 (2013). arXiv:1307.1488 ADSCrossRefGoogle Scholar - 98.G. Carosi, A. Friedland, M. Giannotti, M.J. Pivovaroff, J. Ruz, J.K. Vogel, Probing the axion–photon coupling: phenomenological and experimental perspectives. a snowmass white paper, in
*Proceedings, 2013 Community Summer Study on the Future of U.S. Particle Physics: Snowmass on the Mississippi*(Cs\(S^2\)013):*Minneapolis, MN, USA, July 29–August 6, 2013*(2013). arXiv:1309.7035 - 99.A. Salvio, A. Strumia, W. Xue, Thermal axion production. JCAP
**1401**, 011 (2014). arXiv:1310.6982 ADSCrossRefGoogle Scholar - 100.J.D. Clarke, R. Foot, R.R. Volkas, Phenomenology of a very light scalar \((100 MeV < m\_h < 10 GeV)\) mixing with the Sm Higgs. JHEP
**02**, 123 (2014). arXiv:1310.8042 ADSCrossRefGoogle Scholar - 101.N. Viaux, M. Catelan, P.B. Stetson, G. Raffelt, J. Redondo, A.A.R. Valcarce, A. Weiss, Neutrino and axion bounds from the globular ClusterM 5 (Ngc 5904). Phys. Rev. Lett.
**111**, 231301 (2013). arXiv:1311.1669 ADSCrossRefGoogle Scholar - 102.XENON100 Collaboration, E. Aprile et al., First axion results from the Xenon100 experiment. Phys. Rev. D
**90**(6), 062009 (2014). arXiv:1404.1455 [Erratum: Phys. Rev. D**95**(2), 029904 (2017)] - 103.A. Ayala, I. Domínguez, M. Giannotti, A. Mirizzi, O. Straniero, Revisiting the bound on axion–photon coupling from globular clusters. Phys. Rev. Lett.
**113**(19), 191302 (2014). arXiv:1406.6053 - 104.CMS Collaboration, V. Khachatryan et. al., Search for dark matter, extra dimensions, and unparticles in monojet events in proton–proton collisions at \(\sqrt{s} = 8 TeV\). Eur. Phys. J. C
**75**(5), 235 (2015). arXiv:1408.3583 - 105.K. Mimasu, V. Sanz, Alps at colliders. JHEP
**06**, 173 (2015). arXiv:1409.4792 ADSCrossRefGoogle Scholar - 106.M.J. Dolan, F. Kahlhoefer, C. McCabe, K. Schmidt-Hoberg, A taste of dark matter: flavour constraints on pseudoscalar mediators. JHEP
**03**, 171 (2015). arXiv:1412.5174. Erratum: JHEP**07**, 103 (2015)] - 107.M. Millea, L. Knox, B. Fields, New bounds for axions and axion-like particles with KeV–GeV masses. Phys. Rev. D
**92**(2), 023010 (2015). arXiv:1501.04097 - 108.N. Vinyoles, A. Serenelli, F.L. Villante, S. Basu, J. Redondo, J. Isern, New axion and hidden photon constraints from a solar data global fit. JCAP
**1510**(10), 015 (2015). arXiv:1501.01639 - 109.
**ATLAS**Collaboration, G. Aad et al., Search for new phenomena in final states with an energetic jet and large missing transverse momentum in PP collisions at \(\sqrt{s}=\)8*TeV with the atlas detector*. Eur. Phys. J. C**75**(7), 299 (2015). arXiv:1502.01518 [Erratum: Eur. Phys. J. C**75**(9), 408 (2015)] - 110.G. Krnjaic, Probing light thermal dark-matter with a Higgs portal mediator. Phys. Rev. D
**94**(7), 073009 (2016). arXiv:1512.04119 - 111.W.J. Marciano, A. Masiero, P. Paradisi, M. Passera, Contributions of axionlike particles to lepton dipole moments. Phys. Rev. D
**94**(11), 115033 (2016). arXiv:1607.01022 - 112.E. Izaguirre, T. Lin, B. Shuve, A new flavor of searches for axion-like particles. Phys. Rev. Lett.
**118**(11), 111802 (2017). arXiv:1611.09355 - 113.I. Brivio, M.B. Gavela, L. Merlo, K. Mimasu, J.M. No, R. del Rey, V. Sanz, ALPs effective field theory and collider signatures. Eur. Phys. J. C
**77**, 572 (2017). arXiv:1701.05379 ADSCrossRefGoogle Scholar - 114.M. Bauer, M. Neubert, A. Thamm, LHC as an axion factory: probing an axion explanation for \((g-2)\_{\mu }\) with exotic Higgs decays. Phys. Rev. Lett.
**119**(3), 031802 (2017). arXiv:1704.08207 - 115.
**CAST**Collaboration, V. Anastassopoulos et al., New cast limit on the axion–photon interaction. Nat. Phys.**13**, 584–590 (2017). arXiv:1705.02290 - 116.M.J. Dolan, T. Ferber, C. Hearty, F. Kahlhoefer, K. Schmidt-Hoberg, Revised constraints and Belle II sensitivity for visible and invisible axion-like particles. arXiv:1709.00009
- 117.J. Jaeckel, M. Spannowsky, Probing MeV to 90 GeV axion-like particles with LEP and LHC. Phys. Lett. B
**753**, 482–487 (2016). arXiv:1509.00476 - 118.M. Bauer, M. Neubert, A. Thamm, Collider probes of axion-like particles. arXiv:1708.00443
- 119.S.M. Barr, D. Seckel, Planck scale corrections to axion models. Phys. Rev. D
**46**, 539–549 (1992)ADSCrossRefGoogle Scholar - 120.M. Kamionkowski, J. March-Russell, Planck scale physics and the Peccei–Quinn mechanism. Phys. Lett. B
**282**, 137–141 (1992). arXiv:hep-th/9202003 ADSCrossRefGoogle Scholar - 121.R. Holman, S.D.H. Hsu, T.W. Kephart, E.W. Kolb, R. Watkins, L.M. Widrow, Solutions to the strong CP problem in a world with gravity. Phys. Lett. B
**282**, 132–136 (1992). arXiv:hep-ph/9203206 ADSCrossRefGoogle Scholar - 122.R. Alonso, A. Urbano, Wormholes and masses for Goldstone bosons. arXiv:1706.07415
- 123.S. Cunliffe, Prospects for Rare B Decays at Belle II, in
*Meeting of the Aps Division of Particles and Fields (Dpf 2017) Batavia, Illinois, USA, July 31–August 4, 2017*(2017). arXiv:1708.09423

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