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

The ML$\sigma$M based on the linear 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:

  $$\begin{aligned} \phi =\left( \pi _1,\,\pi _2,\,\pi _3,\,h,\,\sigma \right) ^T \xrightarrow {u.g.} \phi =\left( 0,\,0,\,0,\,h,\,\sigma \right) ^T, \end{aligned}$$

  (2.1)

  where the last expression holds when selecting the unitary gauge, which will be used throughout the next sections.

• 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.

The AML$\sigma$M encompasses, in addition to the previous content,

• 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,

  $$\begin{aligned} s \equiv \dfrac{r}{\sqrt{2}}e^{ia/f_a}, \end{aligned}$$

  (2.2)

  the degrees of freedom are defined as the radial component 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.

The complete renormalisable Lagrangian for the AML$\sigma$M can be written as the sum of three terms describing respectively the pure gauge, fermionic and scalar sectors,

$$\begin{aligned} \mathscr {L}=\mathscr {L}_\mathrm{g} + \mathscr {L}_\mathrm{f}+ \mathscr {L}_\mathrm{s}. \end{aligned}$$

(2.3)

The explicit expression for each piece will be detailed in the following subsections.

2.1 The gauge Lagrangian

The first term, $\mathscr {L}_\mathrm{g}$, contains the SM gauge kinetic and the colour anomaly terms,

$$\begin{aligned} \mathscr {L}_\mathrm{g}= & {} -\dfrac{1}{4}G^{a\mu \nu }G^a_{\mu \nu }-\dfrac{1}{4}W^{a\mu \nu }W^a_{\mu \nu }-\dfrac{1}{4}B^{\mu \nu }B_{\mu \nu }\nonumber \\&+\, \dfrac{\alpha _s}{8\pi }\theta G^{a\mu \nu }\widetilde{G}^a_{\mu \nu }, \end{aligned}$$

(2.4)

with the indices summed over $SU(3)_c$ or $SU(2)_L$, and

$$\begin{aligned} \widetilde{G}_{\mu \nu }\equiv \dfrac{1}{2}\epsilon _{\mu \nu \rho \sigma }G^{\rho \sigma } \qquad \qquad (\mathrm{with} \quad \epsilon _{1230}=+1). \end{aligned}$$

(2.5)

The introduction of the axion will provide a natural explanation for the vanishing of the QCD-$\theta$ term.

2.2 The fermionic Lagrangian

According to the spectrum and symmetries described in the previous section, the fermionic part of the renormalisable Lagrangian in agreement with Ref. [39], although with a slightly different notation, reads

(2.6)

The first line contains the kinetic terms for the 3$^{rd}$ generation SM quarks, being $q_L$ the left-handed (LH) $SU(2)_L$ doublet and $t_R$ and $b_R$ the right-handed (RH) singlet counterparts. The second line contains the kinetic and mass terms for the exotic vector fermions, $\psi$ and $\chi$ (with $U(1)_X$ charge 2 / 3). The direct mass terms for the heavy fermions are denoted by $M_{1,5}$ respectively for the fermions in the singlet and fundamental representations. The proto-Yukawa couplings between the heavy fermions and the real scalar quintuplet field $\phi$ are also present in the second line. In the third line, the Yukawa-like couplings of the exotic fermions with the complex scalar singlet 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.

While, the second and third lines of the Lagrangian explicitly preserve 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:

$$\begin{aligned} \psi \sim \left( K,\,Q,\,T_5\right) ,\qquad \qquad \chi \sim T_1, \end{aligned}$$

(2.7)

being 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

$$\begin{aligned} Y=\Sigma _R^{(3)}+X, \end{aligned}$$

(2.8)

with $\Sigma _R^{(3)}$ the third component of the global $SU(2)_R$ (1 / 2 for K and $-1/2$ for Q) and X the $U(1)_X$ charge of the spinor.

The last three lines describe the replicated sector associated to the bottom quark. The exotic vector fermions, $\psi '$ and $\chi '$ have $U(1)_X$ charge $-1/3$ to allow the direct partial compositness coupling with the bottom. Their decomposition in terms of $SU(2)_L$ representations, reads

$$\begin{aligned} \psi '\sim \left( Q',\,K',\,B'_5\right) ,\qquad \qquad \chi '\sim B'_1, \end{aligned}$$

(2.9)

being $Q'$ and $K'$ doublets of $SU(2)_L$ (with $\Sigma _R^{(3)}$ component 1 / 2 and $-1/2$ respectively) and $B'_{1,5}$ singlets of $SU(2)_L$.

The Lagrangian in Eq. (2.6) can be rewritten for later convenience in terms of fermionic vectors regrouping all the spinors components ordered accordingly of their electric charge,

$$\begin{aligned} \Psi =\left( K^u,\,\mathcal {T},\,\mathcal {B},K^{\prime d}\right) , \end{aligned}$$

(2.10)

with

$$\begin{aligned} \mathcal {T}= & {} \left( t,\, Q^u,\,K^d,\,T_5,\, T_1,\,Q^{\prime u}\right) ,\nonumber \\ \mathcal {B}= & {} \left( b,\, Q^{\prime d},\,K^{\prime u},\,B'_5,\, B'_1,\,Q^d\right) . \end{aligned}$$

(2.11)

The fermion mass terms in Eq. (2.6) can then be written as

$$\begin{aligned} \mathscr {L}_\mathcal {M}=-\overline{\Psi }_L\,\mathcal {M}_f(h,\sigma ,r)\,\Psi _R, \end{aligned}$$

(2.12)

where the field dependent fermion mass matrix $\mathcal {M}_f$ is a $14\times 14$ block diagonal matrix,

$$\begin{aligned} \mathcal {M}_f(h,\sigma ,r)= & {} {diag}\Big (M_5(r),\,\mathcal {M}_\mathcal {T}(h,\sigma ,r),\nonumber \\&\,\mathcal {M}_\mathcal {B}(h,\sigma ,r),\,M'_5(r)\Big ). \end{aligned}$$

(2.13)

For the top sector one has explicitly

$$\begin{aligned}&\mathcal {M}_\mathcal {T}(h,\sigma ,r)\nonumber \\&\quad = \left( \begin{array}{cccccc} 0 &{}\quad \Lambda _1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad \Lambda '_1\\ 0 &{}\quad M_5(r) &{}\quad 0 &{}\quad 0 &{}\quad y_1\frac{h}{\sqrt{2}} &{}\quad 0\\ 0 &{}\quad 0 &{}\quad M_5(r) &{}\quad 0 &{}\quad y_1\frac{h}{\sqrt{2}} &{}\quad 0\\ \Lambda _2 &{}\quad 0 &{}\quad &{}\quad M_5(r) &{}\quad y_1 \sigma &{}\quad 0\\ \Lambda _3 &{}\quad y_2\frac{h}{\sqrt{2}} &{}\quad y_2\frac{h}{\sqrt{2}} &{}\quad y_2\sigma &{}\quad M_1(r) &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad M'_5(r) \end{array} \right) ,\nonumber \\ \end{aligned}$$

(2.14)

with

$$\begin{aligned} M_1(r)=M_1+\left( z_1+\tilde{z}_1\right) r, \qquad \qquad M_5(r)=M_5+\left( z_5+\tilde{z}_5\right) r. \end{aligned}$$

(2.15)

The corresponding matrix for the bottom sector, $\mathcal {M}_\mathcal {B}(h,\sigma ,r)$ can be obtained from Eqs. (2.14) and (2.15) by replacing the unprimed couplings with the corresponding primed ones.

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

The scalar part of the Lagrangian introduced in Eq. (2.3) describes scalar-gauge and scalar-scalar interactions:

$$\begin{aligned} \mathscr {L}_\mathrm{s} = \frac{1}{2}(D_\mu \phi )^T (D^\mu \phi ) + (\partial _{\mu }s^*) (\partial ^{\mu }s) - V(\phi ,s), \end{aligned}$$

(2.16)

where the $SU(2)_L\times U(1)_Y$ covariant derivative of the quintuple $\phi$ is given by

$$\begin{aligned} D_\mu \phi =\left( \partial _{\mu }+ i g \Sigma _L^{(i)} W_\mu ^i+ig'\Sigma _R^{(3)}B_\mu \right) \phi , \end{aligned}$$

(2.17)

and $\Sigma _L^i$ and $\Sigma _R^i$ denote respectively the generators of the $SU(2)_L \times SU(2)_R \sim SO(4)'$ subgroup of SO(5), rotated with respect to the SO(4) group preserved from the spontaneous breaking.

It will be useful for later convenience to express the scalar Lagrangian in Eq. (2.16) in the unitary gauge, making use of Eqs. (2.1) and (2.2):

$$\begin{aligned} \mathscr {L}_\mathrm{s}= & {} \phantom {+}\frac{1}{2}(\partial _{\mu }h)(\partial ^{\mu }h) + \frac{1}{2}(\partial _{\mu }\sigma )(\partial ^{\mu }\sigma )\nonumber \\&+\, \frac{h^2}{4} \left[ g^2\,W^+_\mu W^{-\mu } + \frac{g^2+{g'}^2}{2} Z_\mu Z^\mu \right] \nonumber \\&+\, \frac{1}{2}(\partial _{\mu }r)(\partial ^{\mu }r) + \frac{r^2}{2 f_a^2}(\partial _{\mu }a)(\partial ^{\mu }a) - V(h,\sigma ,r),\nonumber \\ \end{aligned}$$

(2.18)

Notice that once the $U(1)_\text {PQ}$ gets spontaneously broken through the VEV of r, the kinetic term of the axion field a gets canonically normalised, by identifying

$$\begin{aligned} f_a\equiv v_r. \end{aligned}$$

(2.19)

The scalar potential $V(\phi ,s)$ can then be written as:

$$\begin{aligned} V(\phi ,s) = V^\mathrm{SSB}(\phi , s) + V^\mathrm{CW}(\phi , s) + V^\mathrm{c.t.}(\phi , s). \end{aligned}$$

(2.20)

The first part, $V^\mathrm{SSB}(\phi , s)$, describes the most general potential constructed out of $\phi$ and s, invariant under $SO(5)\times U(1)_\text {PQ}$ symmetry, broken spontaneously to SO(4):

$$\begin{aligned} V^\mathrm{SSB}(\phi ,s)= & {} \lambda (\phi ^T\phi -f^2)^2 +\, \lambda _s (2\,s^*s-f_s^2)^2 \nonumber \\&- 2\lambda _{s\phi }\,(s^*s)\,\left( \phi ^T\phi \right) , \end{aligned}$$

(2.21)

where $\lambda$, $\lambda _s$ and $\lambda _{s\phi }$ are the dimensionless quartic coefficients and the sign in front of $\lambda _{s\phi }$ has been chosen negative for future convenience. Notice that $\lambda _{s\phi }$ plays the role of portal between the 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.

The expression of $V^\mathrm{SSB}$ in the exponential notation will be useful in the following sections:

$$\begin{aligned} V^\mathrm{SSB}(h,\sigma ,r)= & {} \lambda (h^2+\sigma ^2-f^2)^2 +\, \lambda _s (r^2-f_s^2)^2 \nonumber \\&-\, \lambda _{s\phi }\,r^2\, (h^2+\sigma ^2). \end{aligned}$$

(2.22)

When the scalar fields 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.

The one-loop fermionic contributions can be calculated from the field dependent fermion mass matrix $\mathcal {M}_f(h,\,\sigma ,\,r)$ in Eq. (2.13), using the usual CW expression:

$$\begin{aligned} V_f^\mathrm{CW}= & {} -\frac{1}{64\pi ^2} \left( \mathrm{Tr}\left[ \mathcal {M}_f\mathcal {M}_f^\dag \right] \Lambda ^2-\mathrm{Tr}\left[ \left( \mathcal {M}_f\mathcal {M}_f^\dag \right) ^2\right] \right. \nonumber \\&\quad \times \log \left( \frac{\Lambda ^2}{{\mu }^2}\right) + \mathrm{Tr}\left[ \left( \mathcal {M}_f\mathcal {M}_f^\dag \right) ^2 \log \left( \frac{\mathcal {M}_f\mathcal {M}_f^\dag }{\mu ^2} \right) \right] \nonumber \\&\quad \left. -\frac{1}{2} \mathrm{Tr}\left[ \left( \mathcal {M}_f\mathcal {M}_f^\dag \right) ^2\right] \right) , \end{aligned}$$

(2.23)

where $\Lambda$ is the ultraviolet (UV) cutoff scale while $\mu$ is a generic renormalisation scale. The two terms in the first line are divergent, quadratically and logarithmically respectively, while those in the second line are finite. For the model under discussion the possible divergent contributions read

$$\begin{aligned}&\mathrm{Tr}\left[ \mathcal {M}_f\mathcal {M}_f^\dag \right] = c_0 + c_1 (s^*s) + c_2 \, (\phi ^T \phi ), \end{aligned}$$

(2.24)

$$\begin{aligned}&\mathrm{Tr}\left[ \left( \mathcal {M}_f\mathcal {M}_f^\dag \right) ^2\right] = d_0 + d_1\,(s^*s) + d_2\,(\phi ^T\phi )\nonumber \\&+\, d_3\,(s^*s)^2 + d_4\,(\phi ^T \phi )^2 + d_5\,(\phi ^T \phi )(s^*s)\nonumber \\&+\, \tilde{d}_1\sigma + \tilde{d}_2 h^2 + \hat{d}_1\sigma (s+s^*) + \hat{d}_2 (\phi ^T\phi )(s+s^*)\nonumber \\&+\, \hat{d}_3 (\phi ^T\phi ) (s s+s^*s^*). \end{aligned}$$

(2.25)

The terms in Eq. (2.24) are already present in the tree level potential $V^\mathrm{SSB}$ in Eq. (2.22) and therefore the quadratic divergences can be absorbed by a redefinition of the initial Lagrangian parameters. This is not the case for the logarithmic divergent term that contains five new couplings, denoted with $\tilde{d}_{1,2}$ and $\hat{d}_{1,2,3}$ in Eq. (2.25). The ones proportional to $\tilde{d}_{1,2}$ and $\hat{d}_{1}$ are 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.

The expressions for the top sector CW coefficients that provide an explicit breaking of the SO(5) and/or of the PQ symmetries read:

$$\begin{aligned} \tilde{d}_1= & {} 4 (y_1 M_1 + y_2 M_5) \Lambda _2 \Lambda _3 \nonumber \\ \tilde{d}_2= & {} y_2^2 \Lambda _1^2 - 2\,y_1^2 \Lambda _2^2 \nonumber \\ \hat{d}_1= & {} 2 \,y_1 (z_1 + \tilde{z}_1) \Lambda _2 \Lambda _3 + 2 \,y_2 (z_5 + \tilde{z}_5) \Lambda _2 \Lambda _3 \nonumber \\ \hat{d}_2= & {} 2\, y_1 y_2 (z_1 + \tilde{z}_1) M_5 + 2\, y_1 y_2 (z_5 + \tilde{z}_5) M_1\nonumber \\ \hat{d}_3= & {} 2 \, y_1 y_2\,( z_1 z_5 \, +\, \tilde{z}_1 \tilde{z}_5). \end{aligned}$$

(2.26)

Similar contributions for the bottom sector are obtained by substituting the unprimed couplings in Eq. (2.26) with the corresponding primed ones. As stated before, once a specific PQ charge assignment is assumed, some of the couplings in the Lagrangian are forbidden, and consequently the corresponding CW coefficients vanish, as it will be explicitly discussed in the next section.

In a similar way the one-loop gauge boson contributions to the CW potential can be calculated through the CW formula given in Eq. (2.23) just substituting the fermion mass matrix $\mathcal {M}_f$ with the gauge boson one $\mathcal {M}_g$:

$$\begin{aligned}&V_g^\mathrm{CW}= -\frac{1}{64\pi ^2} \nonumber \\&\quad \left( \mathrm{Tr}\left[ \mathcal {M}_g^2\right] \Lambda ^2 - \mathrm{Tr}\left[ \left( \mathcal {M}_g^2\right) ^2 \right] \log \left( \frac{\Lambda ^2}{\mu ^2}\right) \,+ \ldots \right) .\nonumber \\ \end{aligned}$$

(2.27)

The quadratic and logarithmic divergent terms read

$$\begin{aligned} \mathrm{Tr}\left[ \mathcal {M}_g^2\right] = \tilde{a}_1 h^2\quad \mathrm{Tr}\left[ \left( \mathcal {M}_g^2\right) ^2\right] = b_0 + \tilde{b}_1\,h^4, \end{aligned}$$

(2.28)

with

$$\begin{aligned} \tilde{a}_1 = \dfrac{1}{8} \left( g^2 + g^{\prime 2}\right) \quad \tilde{b}_1 = \dfrac{1}{64}\left[ 2\, g^4 + \left( g^2 + g^{\prime 2}\right) ^2\right] , \end{aligned}$$

(2.29)

both explicitly breaking the global 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

There is a large zoology of possible $U(1)_\text {PQ}$ charges that can be assigned to the spectrum discussed in the previous sections (see Ref. [39] for details on more general charge assignments). However, after requiring a few, strong physical conditions, only one single set of charge assignments can be identified, which lead to the identification of the minimal AML$\sigma$M. The requirements are the following:

• 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 by

  $$\begin{aligned} m_t= & {} \dfrac{y_1\Lambda _1\Lambda _3 v_h}{M_1(v_r)M_5(v_r)-y_1 y_2(v_h^2+v_\sigma ^2)} \nonumber \\&-\, \dfrac{y_1 y_2 \Lambda _1\Lambda _2v_h v_\sigma }{M_1(v_r)M_5^2(v_r)-y_1y_2M_5(v_r)(v_h^2+v_\sigma ^2)}, \end{aligned}$$

  (3.1)

  and 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 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.

Table 1 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

In a completely generic model a third condition can be also considered:

• 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).

This condition prevents the arising of large contributions to the axion mass, and it is automatically verified in the class of AML$\sigma$M constructions defined in Eq. (2.6), as it will be explicitly shown in the following.

If one requires additionally to solve the strong CP problem 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.

This last condition, when satisfied, implies condition 3 and therefore for a QCD axion no PQ explicit breaking contributions arise in the scalar potential, besides those due to non-perturbative QCD effects.

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].

Accordingly to the charge assignment in Table 1, the PQ-breaking terms in the fermionic CW potential, $\hat{d}_i$, are vanishing, while the SO(5) breaking terms read

$$\begin{aligned} \tilde{d}_1 = 0,\qquad \qquad \tilde{d}_2 = y_2^2 \Lambda _1^2. \end{aligned}$$

(3.2)

In consequence, in this scenario, only a log-divergent 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

$$\begin{aligned} V^\mathrm{c.t.}(h,\sigma )=-\beta f^2 h^2 + \gamma h^4. \end{aligned}$$

(3.3)

Other values for the PQ charges are possible by changing the explicit value of $n_s$, but they lead to the same physical model presented above, at least for what concerns the 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).

The explicit expression describing the Lagrangian modification under generic PQ transformations are reported in the Appendix 1. The coefficients of the axion couplings with the gauge boson field strengths in the physical basis,

$$\begin{aligned} \delta \mathscr {L}\supset- & {} \dfrac{\alpha _s}{8\pi }\,\frac{a}{f_a} c_{agg} G^a_{\mu \nu }\tilde{G}^{a\mu \nu }-\dfrac{\alpha _{em}}{8\pi }\,\frac{a}{f_a}c_{a\gamma \gamma }F_{\mu \nu }\tilde{F}^{\mu \nu }\nonumber \\- & {} \,\dfrac{\alpha _{em}}{8\pi }\,\frac{a}{f_a}c_{aZZ}Z_{\mu \nu }\tilde{Z}^{\mu \nu }\nonumber \\- & {} \,\dfrac{\alpha _{em}}{8\pi }\,\frac{a}{f_a}c_{a\gamma Z}F_{\mu \nu }\tilde{Z}^{\mu \nu }-\dfrac{\alpha _{em}}{8\pi }\,\frac{a}{f_a}c_{aWW}W^+_{\mu \nu }\tilde{W}^{-\mu \nu },\nonumber \\ \end{aligned}$$

(3.4)

are reported in Table 2 for the PQ scenario under consideration.³ It will be useful for the future discussion to introduce the notation of the effective couplings

$$\begin{aligned} g_{agg}\equiv \dfrac{\alpha _{s}}{2\pi }\,\frac{c_{agg}}{f_a}\qquad \qquad g_{i}\equiv \dfrac{\alpha _\mathrm{em}}{2\pi }\,\frac{c_{i}}{f_a}, \end{aligned}$$

(3.5)

where $i=\{a\gamma \gamma ,\,aZZ,\,a\gamma Z,\,aWW\}$.

Table 2 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)

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

As constructed in the previous section, the tree-level renormalisable scalar potential of the minimal AML$\sigma$M reads

$$\begin{aligned} V(h,\sigma , r)= & {} \lambda (h^2 + \sigma ^2 -f^2)^2 -\beta f^2 h^2 + \gamma h^4\nonumber \\&+\, \lambda _s (r^2 - f_s^2)^2 - \lambda _{s\phi }\,r^2\,(h^2 + \sigma ^2). \end{aligned}$$

(4.1)

When $f^2>0$ and $f_s^2>0$, the minimum of the potential allows for the SO(5), $U(1)_\text {PQ}$ and EW spontaneous symmetry breaking with non-vanishing VEVs,

$$\begin{aligned} v^2_h= & {} \dfrac{\beta }{2 \gamma } f^2 \nonumber \\ v^2_\sigma= & {} \left( 1- \frac{\lambda _{s\phi }^2}{4\lambda \lambda _s}\right) ^{-1}\nonumber \\&\times \left\{ f^2 \left[ \left( 1-\frac{\beta }{2\gamma }\right) + \frac{\beta }{2\gamma }\frac{\lambda _{s\phi }^2}{4 \lambda \lambda _s}\right] + \frac{f_s^2}{2} \dfrac{\lambda _{s\phi }}{\lambda } \right\} \nonumber \\ v^2_r= & {} \left( 1- \frac{\lambda _{s\phi }^2}{4\lambda \lambda _s}\right) ^{-1} \left\{ f_s^2 + \frac{f^2}{2} \frac{\lambda _{s\phi }}{\lambda _s} \right\} \equiv f^2_a, \end{aligned}$$

(4.2)

where the condition $v_r \equiv f_a$ is imposed to have canonically normalised axion kinetic term, see Eqs. (2.18) and (2.19). For sake of definiteness we will indicate in the following with $\hat{h}$, $\hat{\sigma }$ and $\hat{r}$ the physical fields after SSB breaking. Assuming all parameters non-vanishing, the following conditions on the parameters must be imposed:

• (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 to

  $$\begin{aligned} \lambda _{s\phi }^2 < 4 \lambda \lambda _s \end{aligned}$$

  (4.3)

  in 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 $\lambda _{s\phi }>0$.

Once the symmetries are spontaneously broken, mass eigenvalues and eigenstates can be identified. While the general case can be studied only numerically (see Sect. 4.3), simple analytical expressions can be obtained in two specific frameworks:

• 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.

These two cases will be discussed in the following sections.

4.1 Integrating out the heaviest scalar field

A clear hierarchy between the three mass scalar eigenstates is achievable for large values of $\lambda _s$ and/or $f_s$: the mass of the heaviest scalar dof receives a LO contribution proportional to

$$\begin{aligned} m_3\propto \sqrt{8\lambda _s} f_s. \end{aligned}$$

(4.4)

With increasing values of $\lambda _s$ and/or $f_s$, the contaminations of $\hat{h}$ and $\hat{\sigma }$ to the heaviest scalar dof, in this region of the parameter space, tend to vanish and the only relevant constituent is the radial component, $\hat{r}$. From the expression in Eq. (4.4), one can envisage two different ways for integrating out the heaviest dof, either taking the limit $\lambda _s \gg 1$ or taking $f_s \gg f \sim \sqrt{s_\mathrm{cm}}$, being $\sqrt{s_\mathrm{cm}}$ the typical centre of mass energy scale at LHC. These two cases represent two physically different scenarios that are discussed separately.

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 framework⁴: 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.

Considering the scalar sector, integrating out the $\hat{r}$ component, leads to an effective scalar potential that, at LO in the appropriate expansion parameter, reads

$$\begin{aligned} V^{LO}_R (h,\sigma ) = \lambda _R (h^2 + \sigma ^2 - f_R^2)^2 - \beta _R f_R^2 h^2 + \gamma h^4\, , \end{aligned}$$

(4.5)

in terms of conveniently renormalised couplings:

$$\begin{aligned} \lambda _R= k_\lambda \lambda ,\qquad \qquad \beta _R=\dfrac{k_\lambda }{k_f} \beta ,\qquad \qquad f_R^2=\dfrac{k_f}{k_\lambda } f^2. \end{aligned}$$

(4.6)

The finite renormalisation constants $k_\lambda$ and $k_f$ are going to be different in the two limiting cases as it will be detailed in the following subsections.

The minimum of the effective scalar potential in Eq. (4.5) corresponds to the following VEVs for the lighter dofs $\hat{h}$ and $\hat{\sigma }$:

$$\begin{aligned} v^2_h = \dfrac{\beta _R}{2\gamma }f_R^2,\qquad \qquad v^2_\sigma = f^2_R\left( 1-\dfrac{\beta _R}{2\gamma }\right) , \end{aligned}$$

(4.7)

satisfying to

$$\begin{aligned} v^2_h+v^2_\sigma =f^2_R. \end{aligned}$$

(4.8)

The restrictions on the parameters that follow from Eq. (4.2) hold for the expressions just obtained: $\beta _R/\gamma$ needs to be positive in order to guarantee $v^2_h>0$; $f_R$ is required to be larger than $v_h$ to ensure $v^2_\sigma >0$. Moreover, if $v_\sigma >v_h$ then the field $\hat{h}$ is the largest component of the mass eigenstate that can be interpreted as the physical Higgs particle originated as a GB of the SO(5) / SO(4) SSB mechanism.

From Eq. (4.5) and using the relations of Eq. (4.7) one derives the following mass matrix:

$$\begin{aligned} \mathcal {M}^2_s = 8\, \lambda _R \begin{pmatrix} (1+ \gamma / \lambda _R) v_h^2 &{} v_h v_\sigma \\ v_h v_\sigma &{} v_\sigma ^2 \end{pmatrix}, \end{aligned}$$

(4.9)

whose diagonalisation is obtained by performing an SO(2) rotation,

$$\begin{aligned} {diag}\left( m^2_1,\,m^2_2\right) =U(\vartheta )^T\mathcal {M}^2_s U(\vartheta )\nonumber \\ \qquad \mathrm{with}\qquad U(\vartheta )= \begin{pmatrix} \cos \vartheta &{} \sin \vartheta \\ -\sin \vartheta &{} \cos \vartheta \end{pmatrix}. \end{aligned}$$

(4.10)

The expressions for the masses and the mixing obtained from the LO potential of Eq. (4.5) are given by

$$\begin{aligned} m^2_{1,2}= & {} 4\lambda _R \left[ \left( 1+\dfrac{\gamma }{\lambda _R}\right) v_h^2 + v_\sigma ^2 \pm \sqrt{\left( 1+\dfrac{\gamma }{\lambda _R}\right) ^2 v^4_h + 2 \left( 1-\dfrac{\gamma }{\lambda _R}\right) v^2_h v^2_\sigma + v^4_\sigma } \right] \end{aligned}$$

(4.11)

$$\begin{aligned} \tan 2\vartheta= & {} \frac{2 v_h v_\sigma }{v_\sigma ^2-(1+ \gamma /\lambda _R) v_h^2 }. \end{aligned}$$

(4.12)

The positivity of the two mass square eigenvalues is guaranteed imposing that both the trace and the determinant of the mass matrix in Eq. (4.9) are positive: this leads to

$$\begin{aligned} \lambda _R>0,\qquad \qquad \gamma>0,\qquad \qquad \beta _R>0, \end{aligned}$$

(4.13)

where the last condition follows from the requirement that $\gamma$ and $\beta _R$ should have the same sign in order to guarantee a positively defined $v_h^2$ value, as discussed below Eq. (4.2).

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 $f_s \sim f$

For $\lambda _s$ in the strongly interacting regime, the radial component r can be expanded in inverse powers of $\lambda _s$ (see Ref. [10] for a similar analysis): at the NLO, one has

$$\begin{aligned} r = f_s +\dfrac{1}{\lambda _s}r_1. \end{aligned}$$

(4.14)

Solving the Equations Of Motion (EOMs) perturbatively allows to determine $r_1$:

$$\begin{aligned} r_1 = \frac{\lambda _{s\phi }}{4f_s}\left( h^2+\sigma ^2\right) +\frac{1}{8f^3_s}\left( \partial _{\mu }a\right) \left( \partial ^{\mu }a\right) . \end{aligned}$$

(4.15)

The effective Lagrangian at the NLO reads

$$\begin{aligned} \mathscr {L}_s= & {} \frac{1}{2}(\partial _{\mu }h)(\partial ^{\mu }h) + \frac{1}{2}(\partial _{\mu }\sigma )(\partial ^{\mu }\sigma ) - \frac{h^2}{4} \mathrm{Tr}\left( \mathbf {V}^\mu \mathbf {V}_\mu \right) \nonumber \\&+\,\frac{1}{2}(\partial _{\mu }a)(\partial ^{\mu }a)- \lambda _R \left( h^2+\sigma ^2 -f_R^2\right) ^2\nonumber \\&+\, \beta _R f_R^2 h^2 - \gamma h^4 + \delta \mathscr {L}^\mathrm{NLO}_s \end{aligned}$$

(4.16)

with $\lambda _R$, $\beta _R$ and $f_R^2$ defined as in Eq. (4.6) with

$$\begin{aligned} k_\lambda =1,\qquad \qquad k_f = \left( 1 + \frac{1}{2} \frac{\lambda _{s\phi }}{\lambda } \frac{f^2_s}{f^2} \right) , \end{aligned}$$

(4.17)

and where the NLO correcting term is given by

$$\begin{aligned} \delta \mathscr {L}_s^\mathrm{NLO} = \dfrac{4}{\lambda _s} f_s^2 r_1^2 = \frac{\lambda _{s\phi }^2}{4\lambda _s} \left[ (h^2+\sigma ^2) + \frac{1}{2 f_s^2} \left( \partial _{\mu }a\right) \left( \partial ^{\mu }a\right) \right] ^2. \end{aligned}$$

(4.18)

In this scenario, $f_R$ is the new effective 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 $\lambda _{s\phi }$:

$$\begin{aligned} \lambda _{s\phi }> - 2 \lambda \frac{f^2}{f^2_s}, \end{aligned}$$

(4.19)

where $\lambda$, $f^2$ and $f^2_s$ are all positive (see the discussion at the beginning of Sect. 4). The value $\lambda _{s\phi }=0$ is special: $\lambda _{s\phi }$ represents the portal between the SO(5) and the PQ sectors, and therefore once it is vanishing the two sectors are completely decoupled.

A convenient limit that will be used to compare with the numerical analysis, is when $\lambda _s\gg \lambda _R\gtrsim 1$ and small $\beta$, for which the expressions in Eqs. (4.11) and (4.12), reduce to

$$\begin{aligned} m_1^2= & {} 4\beta f^2\left( 1-\dfrac{\beta }{2\gamma }\right) \end{aligned}$$

(4.20)

$$\begin{aligned} m_2^2= & {} 8\lambda f^2\left( 1+\dfrac{\beta ^2}{4\gamma \lambda }\right) +4\lambda _{s\phi }f_s^2 \end{aligned}$$

(4.21)

with the mixing angle defined as

$$\begin{aligned} \tan 2\vartheta = \left( 1-\dfrac{\beta }{\gamma }\right) ^{-1}\sqrt{\dfrac{2\beta }{\gamma }\left( 1-\dfrac{\beta }{2\gamma }\right) }. \end{aligned}$$

(4.22)

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

In the limit $f_s\gg f \sim \sqrt{s_\mathrm{cm}}$, being $\lambda _s$ in either the perturbative or strongly interacting regimes, a similar expansion as in the previous subsection can be performed on the field r, adopting as new dimensionless expanding parameter $f/f_s$. Within this setup r at NLO reads

$$\begin{aligned} r = f_s + \dfrac{f}{f_s}\, r_1. \end{aligned}$$

(4.23)

Solving the EOMs in this case, one gets

$$\begin{aligned} r_1 =\dfrac{\lambda _{s\phi }}{4\lambda _sf}\left( h^2+\sigma ^2\right) . \end{aligned}$$

(4.24)

Once substituting these expressions in Eq. (2.18), the effective Lagrangian in Eq. (4.16) is obtained with

$$\begin{aligned} \delta \mathscr {L}^\mathrm{NLO}_s= & {} \dfrac{\lambda _{s\phi }}{4\lambda _s}\dfrac{\left( h^2+\sigma ^2\right) }{f_s^2}\left[ \left( \partial _\mu a\right) \left( \partial ^\mu a\right) {+} \dfrac{\lambda _{s\phi }^2}{4\lambda _s}\left( h^2+\sigma ^2\right) ^2\right] \nonumber \\&+\,\frac{\lambda _{s\phi }^2}{32\lambda _s^2f_s^2}\partial _\mu \left( h^2+\sigma ^2\right) \partial ^\mu \left( h^2+\sigma ^2\right) , \end{aligned}$$

(4.25)

and $\lambda _R$ and $f_R^2$ defined in Eq. (4.6), with $k_\lambda$ and $k_f$ explicitly given by

$$\begin{aligned} k_\lambda = \left( 1-\dfrac{1}{4}\dfrac{\lambda _{s\phi }^2}{\lambda \lambda _s}\right) ,\qquad \qquad k_f = \left( 1 + \frac{1}{2} \frac{\lambda _{s\phi }}{\lambda } \frac{f^2_s}{f^2} \right) . \end{aligned}$$

(4.26)

An increasing value of $f_s$ corresponds to an increasing value of $f_R$. However, caution is necessary in the case when $\lambda _{s\phi }$ is exactly vanishing, as the 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.

Differently from the previous case, here also a new renormalised quartic couplings $\lambda _R \ne \lambda$ is introduced. To ensure a potential bounded from below both $f_R^2$ and $\lambda _R$ need to be positive, leading to the following constraints on $\lambda _{s\phi }$,

$$\begin{aligned} \lambda _{s\phi }> - 2 \lambda \frac{f^2}{f^2_s} \qquad \wedge \qquad \lambda _{s\phi }^2<4\lambda \lambda _s. \end{aligned}$$

(4.27)

In the limiting case under discussion, the explicit values for the two lightest mass eigenvalues and for their mixing in Eqs. (4.11) and (4.12), assuming for simplicity $\lambda _{s\phi }^2 \ll \lambda \lambda _s$, simplify to

$$\begin{aligned}&m^2_1 = 4\beta f^2\left( 1 - \frac{\beta }{\gamma }\dfrac{\lambda }{\lambda _{s\phi }}\dfrac{f^2}{f_s^2} \right) ,\nonumber \\&\quad m^2_2 = 4 \lambda _{s\phi }f_s^2\left( 1+2\dfrac{\lambda }{\lambda _{s\phi }}\dfrac{f^2}{f_s^2} \right) \end{aligned}$$

(4.28)

with the mixing angle given by

$$\begin{aligned} \tan 2\vartheta&= 2\sqrt{\dfrac{\beta }{\gamma }\dfrac{\lambda }{\lambda _{s\phi }}} \dfrac{f}{f_s}. \end{aligned}$$

(4.29)

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.

Expanding the expressions for the generic VEVs found in Eq. (4.2) for small $\beta$ and $\lambda _{s\phi }$, it leads to the following simplified expressions:

$$\begin{aligned} v^2_h= & {} \dfrac{\beta }{2 \gamma } f^2 \nonumber \\ v^2_\sigma= & {} \left( 1-\frac{\beta }{2\gamma }\right) f^2 + \dfrac{\lambda _{s\phi }}{\lambda }\frac{f_s^2}{2} \,+\, {\mathcal O}\left( \beta ^2,\beta \lambda _{s\phi },\lambda _{s\phi }^2\right) \nonumber \\ v^2_r= & {} f_s^2 + \frac{\lambda _{s\phi }}{\lambda _s}\frac{f^2}{2} \,+\, {\mathcal O}\left( \beta ^2,\beta \lambda _{s\phi },\lambda _{s\phi }^2\right) , \end{aligned}$$

(4.30)

where in the brackets the dependence on $\beta$ and $\lambda _{s\phi }$ of the higher order corrections is reported. The scalar squared mass matrix is given by the following expression

$$\begin{aligned} \mathcal {M}^2_s= 2 \left( \begin{array}{ccc} 4 (\gamma + \lambda ) v_h^2&{} 4\lambda v_h v_\sigma &{} -2\lambda _{s\phi }v_h v_r\\ 4 \lambda v_h v_\sigma &{} 2 \lambda (v_h^2+3 v_\sigma ^2-f^2)-\lambda _{s\phi }v_r^2 &{} -2\lambda _{s\phi }v_\sigma v_r\\ -2 \lambda _{s\phi }v_h v_r &{}-2 \lambda _{s\phi }v_\sigma v_r &{} -\lambda _{s\phi }(v_h^2+v_\sigma ^2)+6\lambda _s v_r^2-2\lambda _s f_s^2 \end{array}\right) \end{aligned}$$

that can be diagonalised performing an orthogonal transformation,

$$\begin{aligned} {diag}\left( m_1^2,\,m_2^2,\,m_3^2\right) =U(\vartheta _{12},\vartheta _{23})^T \mathcal {M}^2_s U(\vartheta _{12},\vartheta _{23}) \end{aligned}$$

(4.31)

with

$$\begin{aligned} U(\vartheta _{12},\vartheta _{23}) = U(\vartheta _{12}) U(\vartheta _{23}), \end{aligned}$$

(4.32)

the product of a rotation in the 12 sector and in the 23 sector respectively, of angles $\vartheta _{12}$ and $\vartheta _{23}$. The resulting mass eigenvalues read

$$\begin{aligned} m_1^2= & {} 4\beta f^2\left( 1-\dfrac{\beta }{2\gamma }\right) + {\mathcal O}\left( \beta ^3,\beta ^2\lambda _{s\phi }\right) \nonumber \\ m_2^2= & {} 8\lambda f^2 \left( 1+\frac{1}{2}\frac{\lambda _{s\phi }}{\lambda }\frac{f_s^2}{f^2}\right) + {\mathcal O}\left( \beta ^2,\beta \lambda _{s\phi },\lambda _{s\phi }^2\right) \nonumber \\ m_3^2= & {} 8\lambda _sf_s^2\left( 1+\frac{1}{2}\frac{\lambda _{s\phi }}{\lambda _s}\frac{f^2}{f_s^2}\right) +{\mathcal O}\left( \beta \lambda _{s\phi },\lambda _{s\phi }^2\right) , \end{aligned}$$

(4.33)

while the mixing angles are given by

$$\begin{aligned}&\tan 2\vartheta _{12}= \sqrt{\dfrac{2\beta }{\gamma } } \left( 1\,+\, {\mathcal O}(\beta , \lambda _{s\phi }) \right) ,\nonumber \\&\quad \tan 2\vartheta _{23}= \dfrac{f f_s}{\lambda _s f_s^2-\lambda f^2} \, \lambda _{s\phi }\left( 1 \,+\, {\mathcal O}(\beta ,\lambda _{s\phi })\right) . \end{aligned}$$

(4.34)

As for Eq. (4.30), only the first two relevant terms in the expansion are reported in the expressions in Eqs. (4.33), while the powers in $\beta$ and $\lambda _{s\phi }$ of the expected next order terms are shown in the brackets. Instead, in the formula for the mixing angles in Eq. (4.34), only the first term is indicated. Notice that, once considering the next order terms in the masses expressions, a rotation in the 13 sector is also necessary to exactly diagonalise the squared mass matrix.

4.3 Numerical analysis

This subsection illustrates the numerical analysis on the parameter space of the scalar potential. The analytic results of the specific cases presented in the previous subsection will be used to discuss the numerical outcome. To this aim, a more general notation with respect to the one previously adopted is introduced. The scalar mass matrix $\mathcal {M}_s$ is real and can be diagonalised by an orthogonal transformation,

$$\begin{aligned}&{diag}(m_1^2,m_2^2,m_3^2)\nonumber \\&\quad = U(\vartheta _{12},\vartheta _{23},\vartheta _{13})^T \mathcal {M}_s^2\, U(\vartheta _{12},\vartheta _{23},\vartheta _{13}), \end{aligned}$$

(4.35)

where $U(\vartheta _{12},\vartheta _{23},\vartheta _{13})\equiv U(\vartheta _{23}) U(\vartheta _{13}) U(\vartheta _{12})$ is the product of three rotations in the 23, 13, and 12 sectors respectively, of angles $\vartheta _{23}$, $\vartheta _{13}$ and $\vartheta _{12}$. The scalar mass eigenstates $\varphi _1$, $\varphi _2$, and $\varphi _3$ are defined by

$$\begin{aligned} \begin{pmatrix} \varphi _1 \\ \varphi _2 \\ \varphi _3 \end{pmatrix} = U(\vartheta _{12},\vartheta _{23},\vartheta _{13})^T \begin{pmatrix} \hat{h} \\ \hat{\sigma } \\ \hat{r} \end{pmatrix} \end{aligned}$$

(4.36)

in terms of the three physical shifts around the vacua. Unless explicitly indicated, in the analysis that follows, $\varphi _1$ will be identified with the Higgs particle and the deviations of its couplings from the SM predicted values are interesting observables at colliders. The $\varphi _1$ couplings to the SM gauge bosons can be deduced from the couplings of $\hat{h}$, as $\hat{\sigma }$ and $\hat{r}$ are singlets under the SM gauge group. The composition of $\hat{h}$ in terms of $\varphi _i$ is explicitly given by

$$\begin{aligned} \hat{h} = c_{12} c_{13} \varphi _1 + c_{13} s_{12} \varphi _2 + s_{13} \varphi _3 \equiv C_1 \varphi _1 + C_2 \varphi _2 + C_3 \varphi _3, \end{aligned}$$

(4.37)

where $c_{ij}$ and $s_{ij}$ stand for $\cos \theta _{ij}$ and $\sin \theta _{ij}$, and the coefficients $C_i$ in the last equality have been introduced for shortness. The couplings with the SM gauge bosons can be written as

$$\begin{aligned}&\frac{g^2}{4} (\hat{h} + v_h)^2 W^+_\mu W^{-\mu }\nonumber \\&\quad = m_W^2 \left( C_1 \frac{\varphi _1}{v_h} + C_2 \frac{\varphi _2}{v_h} + C_3 \frac{\varphi _3}{v_h} +1 \right) ^2 W^+_\mu W^{-\mu },\nonumber \\&\frac{g^2+g^{\prime 2}}{8} (\hat{h} + v_h)^2 Z_\mu Z^\mu \nonumber \\&\quad = \dfrac{m_Z^2}{2} \left( C_1 \frac{\varphi _1}{v_h} + C_2 \frac{\varphi _2}{v_h} + C_3 \frac{\varphi _3}{v_h} +1 \right) ^2 Z_\mu Z^\mu . \end{aligned}$$

(4.38)

Finally, the $\varphi _1$ couplings to the longitudinal components of W and Z are modified with respect to the SM predictions for the Higgs particle by factor of $C_1$.

To have a clear comparison with CHM predictions, one can write the expression for the $C_1$ parameter obtained integrating out all the scalar dofs of our model, but the physical Higgs. The most immediate way to obtain such a result is to start from Eq. (4.12) and expanding it for $\lambda _R\gg 1$, giving

$$\begin{aligned} C_1\simeq 1-\dfrac{1}{2}\dfrac{v_h^2}{v_\sigma ^2}\equiv 1-\dfrac{\xi }{2}, \end{aligned}$$

(4.39)

The last term on the right-hand side introduces the parameter $\xi$, that customary defines the tension between the EW and the composite scales. This parameter often appears in CHMs to quantify the level of non-linearity of the model. The expression in Eq. (4.39) agrees with previous MCHM results present in literature, see for example Ref. [72]. Therefore, the corresponding bounds on $\xi$, as the ones from Refs. [12, 70],

$$\begin{aligned} \xi \lesssim 0.18\quad @\,2\sigma , \end{aligned}$$

(4.40)

strictly apply to the model presented here only in the MCHM limit, i.e. when all the scalar fields, but the Higgs, are extremely massive and can be safely integrated out. If this is not the case, the coefficient $C_1$ is a complicate function of all the scales and parameters effectively present in the model.

The scalar potential parameter space

The parameter space of the scalar sector is spanned by seven independent parameters: five dimensionless coefficients $\lambda$, $\lambda _s$, $\beta$, $\gamma$, $\lambda _{s\phi }$, and two scales 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):

$$\begin{aligned} \gamma = \left( \frac{f}{v_h} \right) ^2 \frac{\beta }{2}. \end{aligned}$$

(4.41)

and then extract $\beta$, in terms of the remaining five parameters, by numerically solving the equation $m_1(\beta ,\lambda ,\lambda _s, \lambda _{s\phi }, f, f_s)=m_h$. Consequently, predictions for all the remaining observables can be obtained by choosing specific values for $(\lambda , \lambda _s, \lambda _{s\phi }, f, f_s)$.

In Fig. 1 the bounds on the $|C_1|$ parameter in the $(f_s,f)$ plane for $\lambda =\lambda _s=1$ and $\lambda _{s\phi }=0.1$ are shown. The dark green region corresponds to $|C_1|<0.90$, while the light green one to $0.90<|C_1|<0.95$. In the white region $|C_1|>0.95$. From this plot one can have an order of magnitude comparison with present/future experimental bound on the Higgs-gauge boson interaction. The following bounds on hZZ and hWW couplings are obtained by [71], using the so called $\kappa$-framework⁵:

$$\begin{aligned} |\kappa _{Z}|= & {} 0.89+0.09-0.08 \quad @ \quad 1\sigma \nonumber \\ |\kappa _{W}|= & {} 1.00+0.00-0.05 \quad @ \quad 1\sigma \end{aligned}$$

(4.42)

The expressions in Eq. (4.38) enforce the relation $\kappa _Z=\kappa _W=C_1$.

Fig. 1

$C_1$ contours in the $(f_s,f)$ plane, for $\lambda = \lambda _s =1$ and $\lambda _{s\phi }= 0.1$. The dark green region corresponds to $|C_1| < 0.90$, while the light green one to $0.90< |C_1| < 0.95$. In the white region $|C_1|>0.95$. The two red curves correspond to values for the next to lightest scalar $m_2 = 1$ TeV and $m_2 = 2$ TeV respectively, being the Higgs mass fixed to the reference value $m_h =125$ GeV

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 the following analysis the value $f = 2$ TeV has been chosen as benchmark. The parameter space for the remaining four variable, $\lambda , \lambda _s, \lambda _{s\phi }, f_s$, can be studied, plotting the behaviour of the scalar mass eigenvalues $m_i$ and of the mixing coefficients squared $C_i^2$.

Fig. 2

The profiles of the scalar masses $m_2$ and $m_3$ as a function of $\lambda _{s\phi }$ (upper left), $\lambda =\lambda _s$ (upper right), and $\lambda$ (lower). The other parameters are chosen at fixed values: $f = 2\text { TeV}$; $f_s = 1\text { TeV},\,10^3\text { TeV},\,10^6 \text { TeV}$; $\lambda = \lambda _s = 10$ (upper left); $\lambda _{s\phi }=0.1$ (upper right); $\lambda _s=10$ and $\lambda _{s\phi }=0.1$ (lower). The red-dashed line represents the heaviest dof with mass $m_3$, while the blue-continue line the next-to-heaviest dof with mass $m_2$. The lightest dof is identified with the Higgs particle with mass $m_1=m_h$. The red area is excluded from the constraint in Eq. (4.3)

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 $\lambda _{s\phi }$.

When $f_s=1\text { TeV}$, the numerical results agree with the analytic expressions in Eqs. (4.21) and (4.33). In the upper left plot, the red-dashed and the blue-continue lines are exchanged with respect to the lines for $f_s=10^3\text { TeV}$ and $f_s=10^6\text { TeV}$: this is in agreement with Eq. (4.33), as indeed for $f>f_s$ the heaviest dof is $\varphi _2$ and the next-to-heaviest is $\varphi _3$. Moreover, the two lines are almost horizontal as the dependence on $\lambda _{s\phi }$ only enters at higher orders. In the upper right plot, both the lines increase with a constant slope, as expected from Eq. (4.33), except for small values of $\lambda =\lambda _s$, that is close to the excluded region. In the lower plot, the red-dashed line is almost horizontal, according to $m_3^2$ in Eq. (4.33), while the blue-continue line increases with $\lambda$, as shown by the expression for $m_2^2$. For $\lambda =2.5$ the two lines cross and $\varphi _2$ becomes the heaviest dof. The same conclusions are expected by analysing the expressions in Eq. (4.21), where $\varphi _3$ is integrated out: the comparison is however more difficult as $m_2^2$ depends explicitly on $\beta$ and $\gamma$, which are only numerically computed in terms of $\lambda , \lambda _s, \lambda _{s\phi }, f_s$. Moreover, when $\lambda >2.5$, $\varphi _2$ should also be integrated out from the low-energy spectrum as its mass reaches the value of the one of $\varphi _3$, and not consistent description is expected for these values of $\lambda$.

Fig. 3

The profiles of the coefficients squared $C_1^2$, $C_2^2$ and $C_3^2$, as a function of $\lambda =\lambda _s$. The other parameters are chosen at fixed values: $f = 2\text { TeV}$; $\lambda _{s\phi }=0.1$; $f_s = 1\text { TeV}$ on the left and $f_s = 3\text { TeV}$ on the right. The green-dot-dashed line describes $C_1^2$, the blue-continue line $C_2^2$ and the red-dashed line $C_3^2$. The red area is excluded from the constraint in Eq. (4.3)

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

The axion coupling to photons is bounded from both astrophysical and low-energy terrestrial data, and they depend on the axion mass. The most recent summary on these constraints can be found in Refs. [117, 118], while the last update for masses below tens of meV is given in Ref. [115]: the upper bounds can be summarised as

$$\begin{aligned} |g_{a\gamma \gamma }|\lesssim & {} 7\times 10^{-11}\text { GeV}^{-1}\qquad \mathrm{for} \qquad m_a\lesssim 10 \text { meV}\nonumber \\ |g_{a\gamma \gamma }|\lesssim & {} 10^{-10}\text { GeV}^{-1}\qquad \mathrm{for}\qquad 10 \text { meV}\lesssim m_a\lesssim 10\text { eV}\nonumber \\ |g_{a\gamma \gamma }|\ll & {} 10^{-12}\text { GeV}^{-1}\qquad \mathrm{for}\qquad 10 \text { eV}\lesssim m_a\lesssim 0.1\text { GeV}\nonumber \\ |g_{a\gamma \gamma }|\lesssim & {} 10^{-3}\text { GeV}^{-1}\qquad \mathrm{for}\qquad 0.1 \text { GeV}\lesssim m_a\lesssim 1\text { TeV}.\nonumber \\ \end{aligned}$$

(6.1)

For masses between $10\text { eV}$ and $0.1\text { GeV}$, and in particular for the so-called MeV window, the coupling $g_{a\gamma \gamma }$ is constrained by (model dependent) cosmological data [107]. These bounds can be translated in terms of $f_a/|c_{a\gamma \gamma }|$ through Eq. (3.5): taking $\alpha _\mathrm{em}=1/137.036$,

$$\begin{aligned} \dfrac{f_a}{|c_{a\gamma \gamma }|}\gtrsim & {} \,2\times 10^{7}\text { GeV}\qquad \mathrm{for}\qquad m_a\,\,\lesssim 10 \text { meV}\nonumber \\ \dfrac{f_a}{|c_{a\gamma \gamma }|}\gtrsim & {} \,10^{7}\text { GeV}\qquad \mathrm{for}\qquad 10 \text { meV}\lesssim m_a\,\,\lesssim 10\text { eV}\nonumber \\ \dfrac{f_a}{|c_{a\gamma \gamma }|}\gg & {} \,10^{9}\text { GeV}\qquad \mathrm{for}\qquad 10 \text { eV}\lesssim m_a\,\,\lesssim 0.1\text { GeV}\nonumber \\ \dfrac{f_a}{|c_{a\gamma \gamma }|}\gtrsim & {} \, 1\text { GeV}\qquad \mathrm{for}\qquad 0.1 \text { GeV}\lesssim m_a\,\,\lesssim 1\text { TeV}.\nonumber \\ \end{aligned}$$

(6.2)

In Ref. [39] a dedicated analysis of the axion coupling to photons within the AML$\sigma$M is presented, including constraints and prospects from current experiments.

Coupling to gluons

The axion coupling to gluons has been constrained by axion-pion mixing effects [84, 87] and mono-jet searches at colliders [104, 105, 109, 113]. The bounds can be summarised as follows:

$$\begin{aligned} |g_{agg}|\lesssim & {} 1.1\times 10^{-5}\text { GeV}^{-1}\qquad \mathrm{for}\qquad m_a\,\lesssim 60 \text { MeV}\nonumber \\ |g_{agg}|\lesssim & {} 10^{-4}\text { GeV}^{-1}\qquad \mathrm{for}\qquad 60 \text { MeV}\lesssim m_a\,\lesssim 0.1\text { GeV}\nonumber \\ \end{aligned}$$

(6.3)

that can be translated in terms of $f_a/|c_{agg}|$ as

$$\begin{aligned} \dfrac{f_a}{|c_{agg}|}\gtrsim & {} 2\times 10^{3}\text { GeV}\qquad \mathrm{for}\qquad m_a\,\lesssim 60 \text { MeV}\nonumber \\ \dfrac{f_a}{|c_{agg}|}\gtrsim & {} 2\times 10^{2}\text { GeV}\qquad \mathrm{for}\qquad 60 \text { MeV}\lesssim m_a\,\lesssim 0.1\text { GeV}\nonumber \\ \end{aligned}$$

(6.4)

taking $\alpha _s(M_Z^2)=0.1184$.

Couplings to massive gauge bosons

Rare meson decays provide strong constraints of axion couplings to two 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]:

$$\begin{aligned} BR(K^+\rightarrow \pi ^++a)<7.3\times 10^{-11}. \end{aligned}$$

(6.5)

For larger masses up to a few GeV’s, the $B^+\rightarrow K^++a$ decay provides the most stringent bound: BaBar experiment has proven that [96]

$$\begin{aligned} BR(B^+\rightarrow K^++a)\lesssim 3.2\times 10^{-5}. \end{aligned}$$

(6.6)

In Refs. [112, 113, 118], meson decays, with the axion subsequently decaying into photons, have also been considered: these observables are not relevant in the minimal AML$\sigma$M, being the axion–photon coupling so strongly bounded that no signals for these observables are expected in present or future experiments.

The induced bounds on $g_{aWW}$ effective coupling read [112]:

$$\begin{aligned} |g_{aWW}|\lesssim & {} \,3\times 10^{-6}\text { GeV}^{-1}\qquad \mathrm{for}\qquad m_a\,\,\lesssim 0.2 \text { GeV}\nonumber \\ |g_{aWW}|\lesssim & {} \, 10^{-4}\text { GeV}^{-1}\quad \mathrm{for}\quad 0.2 \text { GeV}\lesssim m_a\,\,\lesssim 5\text { GeV}\nonumber \\ \end{aligned}$$

(6.7)

that can be translated in terms of $f_a/|c_{aWW}|$ as

$$\begin{aligned} \dfrac{f_a}{|c_{aWW}|}\gtrsim & {} \,4\times 10^{2}\text { GeV}\qquad \mathrm{for}\qquad m_a\,\,\lesssim 0.2 \text { GeV}\nonumber \\ \dfrac{f_a}{|c_{aWW}|}\gtrsim & {} \,10\text { GeV}\qquad \mathrm{for} 0.2 \text { GeV}\lesssim m_a\,\,\lesssim 5\text { GeV}.\nonumber \\ \end{aligned}$$

(6.8)

Collider searches are able to put independent constraints on $g_{aWW}$ as well as on couplings with other gauge bosons. Following Ref. [113], considering LHC data with $\sqrt{s_\mathrm{cm}}=13\text { TeV}$ and for axion masses $m_a\lesssim 1\text { GeV}$, the mono-W, $pp\rightarrow a W(W\rightarrow \mu \nu _\mu )$, and mono-Z, $pp\rightarrow aZ(Z\rightarrow ee)$, signals put the following constraints:

$$\begin{aligned} |g_{aWW}|\lesssim 5\times 10^{-7}\text { GeV}^{-1},\quad |g_{aZZ}|\lesssim 3\times 10^{-7}\text { GeV}^{-1}. \end{aligned}$$

(6.9)

The Z boson width allows to put a conservative bound on $Z\rightarrow a\gamma$ interaction:

$$\begin{aligned} |g_{aZ\gamma }|\lesssim 1.8\times 10^{-3}\text { GeV}^{-1}. \end{aligned}$$

(6.10)

The corresponding bounds on $f_a/|c_i|$ are given by:

$$\begin{aligned}&\dfrac{f_a}{|c_{aWW}|}\gtrsim 2\times 10^{3}\text { GeV},\nonumber \\&\quad \dfrac{f_a}{|c_{aZZ}|}\gtrsim 4\times 10^{3}\text { GeV},\nonumber \\&\quad \dfrac{f_a}{|c_{aZ\gamma }|}\gtrsim 0.6\text { GeV}. \end{aligned}$$

(6.11)

The axion mass

There are two distinct contributions to the axion mass (gravitational and/or Planck-scale sources [119,120,121,122] will not be discussed here). The first is due to purely QCD effects (axion mixing with neutral pions), which is estimated to be [37, 46, 47]

$$\begin{aligned} m_a\sim 6 {\mu }eV\left( \dfrac{10^{12}\text { GeV}}{f_a/c_{agg}}\right) , \end{aligned}$$

(6.12)

for values of $f_a$ typically taken to be larger than $10^6\text { GeV}$. The second is due to the extra fermions that couple to the axion, such as in the KSVZ invisible axion model [36, 37]:

$$\begin{aligned} m_a=\dfrac{\sqrt{Z}}{1+Z}\dfrac{\alpha _s^2}{\pi ^2}\dfrac{f_\pi }{f_a}m_\pi \ln \left( \dfrac{m_\psi ^2}{m_u m_d}\right) , \end{aligned}$$

(6.13)

where $Z\simeq m_u/m_d$ and $f_\pi \sim 94\text { MeV}$ is the pion decay constant and $m_\psi$ is the generic mass of the exotic fermions. This contribution is a decreasing function with $f_a$ for values of $f_a>10\text { MeV}$: considering similar values of $f_a$ and $m_\psi$, it follows that

$$\begin{aligned} m_a\sim & {} 100 \text { keV}\quad \mathrm{for}\quad f_a\sim 1\text { GeV}\nonumber \\ m_a\sim & {} 0.2 \text { keV}\quad \mathrm{for}\quad f_a\sim 10^3\text { GeV}\nonumber \\ m_a\sim & {} 0.3 \text { eV}\quad \mathrm{for}\quad f_a\sim 10^6\text { GeV}\nonumber \\ m_a\sim & {} 0.004 \text { eV}\quad \mathrm{for}\quad f_a\sim 10^8\text { GeV}. \end{aligned}$$

(6.14)

Notice that, for the last two cases, the QCD mass in Eq. (6.12) is relevant and provides the dominant contributions of $60\text { eV}$ and $0.6\text { eV}$ respectively. These benchmarks are interesting for the discussion that follows.

6.1 QCD axion or axion-like-particle?

In Sect. 4.3, three values for $f_s$ have been considered: $f_s=1\text { TeV}$, $f_s=10^3\text { TeV}$ and $f_s=10^6\text { TeV}$. Eq. (2.19) links the axion scale $f_a$ to the VEV of the radial component of 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

$$\begin{aligned} f_s\gtrsim 3.7\times 10^8\text { GeV}. \end{aligned}$$

(6.15)

It follows that a QCD axion, consistent with all the present data, can only be generated in the minimal AML$\sigma$M if the scale $f_s$, associated to the PQ breaking, is of the order of $10^8\text { GeV}$ or larger. As discussed in Ref. [39], the resulting axion falls into the category of the so-called invisible axions [36, 37, 48, 49], as such a large $f_s$ scale strongly suppresses all the couplings with SM fermions and gauge bosons, preventing any possible detection at colliders or at low-energy (flavour) experiments.

The difference with respect to the traditional invisible axion models resides partly in the axion couplings to photons and gluons, and in the EWSB sector. As underlined in Ref. [39], adding a KSVZ axion to the ML$\sigma$M narrows the range of possible values that the ratio $c_{a\gamma \gamma }/c_{agg}$ may take: the minimal AML$\sigma$M presented here provides a very sharp prediction for this ratio,

$$\begin{aligned} \dfrac{c_{a\gamma \gamma }}{c_{agg}}=\dfrac{14}{3}. \end{aligned}$$

(6.16)

Moreover, in the minimal AML$\sigma$M with $f_s\gtrsim 10^8\text { GeV}$ the low-energy theory is not exactly the SM, but the EWSB mechanism is non-linearly realised and the Higgs particle originates as a GB. This model may be confirmed, or excluded, by a precise measure of $c_{a\gamma \gamma }/c_{agg}$ and by a dedicated analysis of the EW sector. In particular, this case corresponds to the scenario where only the physical Higgs remains in the low-energy spectrum, while the other two scalar dofs are very massive. In consequence, only indirect searches on Higgs couplings or the physics associated to the longitudinal components of the SM gauge bosons may have the potential to constrain the minimal AML$\sigma$M.

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].

By increasing the axion mass, its decay length decreases and this may open up to another class of observables: if the axion decays inside the detector, then it would not show up as missing energy, but as a couple of gauge bosons, as discussed in Refs. [112, 113, 118]. The distance travelled by the axion after being produced may be casted as follows [113],

$$\begin{aligned} d\approx \dfrac{10^4}{c_i^2}\left( \dfrac{\text { MeV}}{m_a}\right) ^4\left( \dfrac{f_s}{\text { GeV}}\right) ^2\left( \dfrac{|p_a|}{\text { GeV}}\right) \text { m}, \end{aligned}$$

(6.17)

where $c_{i}$ are the couplings in Table 2 and the typical momentum considered is $\gtrsim 100\text { GeV}$. For the selected benchmark considered, $m_a\sim 1\text { GeV}$ and $f_s\sim 200\text { TeV}$, the decay length is of tens of meters for decays into two photons. This axion can therefore evade detection at colliders, although for a slightly larger masses this is not guaranteed.

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).

Generic PQ Transformations

The Lagrangian containing the axion couplings, in the basis where fermionic terms are shift-symmetry preserving, can be written as

$$\begin{aligned} \mathscr {L}_a=&\dfrac{1}{2}\partial _\mu a\partial ^\mu a+\Delta _\psi \dfrac{\partial _\mu a}{2f_a}\overline{\psi }\gamma ^\mu \gamma ^5\psi +\Delta _\chi \dfrac{\partial _\mu a}{2f_a}\overline{\chi }\gamma ^\mu \gamma ^5\chi \nonumber \\&+\Delta _{\psi '}\dfrac{\partial _\mu a}{2f_a}\overline{\psi '}\gamma ^\mu \gamma ^5\psi '+ \Delta _{\chi '}\dfrac{\partial _\mu a}{2f_a}\overline{\chi '}\gamma ^\mu \gamma ^5\chi '+\nonumber \\&-\dfrac{\alpha _s}{8\pi }\,\frac{a}{f_a}\sum \Big [5\left( \Delta _{\psi }+\Delta _{\psi '}\right) +\left( \Delta _{\chi }+\Delta _{\chi '}\right) \Big ]\nonumber \\&\quad G^a_{\mu \nu }\tilde{G}^{a\mu \nu }+\nonumber \\&-\dfrac{\alpha _2}{8\pi }\,\frac{a}{f_a}\sum 6\left( \Delta _{\psi }+\Delta _{\psi '}\right) W_{\mu \nu }^a\tilde{W}^{a\mu \nu }+\nonumber \\&-\dfrac{\alpha _1}{8\pi }\,\frac{a}{f_a}\sum \Big [6\Delta _{\psi }\left( 2Y_{X}^2+2Y_{Q}^2+Y_{T_5}^2\right) +6\Delta _{\chi }Y_{T_1}^2+\nonumber \\&\quad +6\Delta _{\psi '}\left( 2Y_{X'}^2+2Y_{Q'}^2+Y_{B_5}^2\right) \nonumber \\&\quad +6\Delta _{\chi '}Y_{B_1}^2\Big ]B^{\mu \nu }\tilde{B}^{\mu \nu }\,, \end{aligned}$$

(A.1)

where $Y_i$ are the Hypercharges of the components of $\psi$ and $\chi$ (see Eq. (2.7)) and $\Delta _f\equiv n_{f_L}-n_{f_R}$. The sum is meant over the different generations: in the specific setup considered here, it reduces to the third family only.

Moving to the gauge boson physical basis, the axion couplings to the gauge field strengths are given by:

$$\begin{aligned}&-\dfrac{\alpha _s}{8\pi }\,\frac{a}{f_a}\sum \Big [5\left( \Delta _{\psi }+\Delta _{\psi '}\right) +\left( \Delta _{\chi }+\Delta _{\chi '}\right) \Big ] G^a_{\mu \nu }\tilde{G}^{a\mu \nu }+\nonumber \\&-\dfrac{\alpha _{em}}{8\pi }\,\frac{a}{f_a}\sum \Big [6\Delta _{\psi }\left( 1+ 2Y_{X}^2+2Y_{Q}^2+Y_{T_5}^2\right) +6\Delta _{\chi }Y_{T_1}^2+\nonumber \\&\quad +6\Delta _{\psi '}\left( 1+ 2Y_{X'}^2+2Y_{Q'}^2+Y_{B_5}^2\right) \nonumber \\&\quad +6\Delta _{\chi _R}Y_{B_1}^2\Big ]F_{\mu \nu }\tilde{F}^{\mu \nu }\,,\nonumber \\&-\dfrac{\alpha _{em}}{8\pi }\,\frac{a}{f_a}\sum \Bigg \{6\Delta _{\psi }\left[ \dfrac{1}{\tan \theta _W^2}+ \tan ^2\theta _W\left( 2Y_{X}^2+2Y_{Q}^2+Y_{T_5}^2\right) \right] \nonumber \\&\quad +6\Delta _{\chi }\tan ^2\theta _W\, Y_{T_1}^2+\nonumber \\&\quad +6\Delta _{\psi '}\left[ \dfrac{1}{\tan ^2\theta _W}+ \tan ^2\theta _W\left( 2Y_{X'}^2+2Y_{Q'}^2+Y_{B_5}^2\right) \right] \nonumber \\&\quad +6\Delta _{\chi '}\tan ^2\theta _W\,Y_{B_1}^2\Bigg \}Z_{\mu \nu }\tilde{Z}^{\mu \nu }\,,\nonumber \\&-\dfrac{\alpha _{em}}{8\pi }\,\frac{a}{f_a}\sum \Bigg \{12\Delta _{\psi }\left[ \dfrac{1}{\tan \theta _W}- \tan \theta _W\left( 2Y_{X}^2+2Y_{Q}^2+Y_{T_5}^2\right) \right] \nonumber \\&\quad -12\Delta _{\chi }\tan \theta _W\, Y_{T_1}^2+\nonumber \\&\quad +12\Delta _{\psi '}\left[ \dfrac{1}{\tan \theta _W}- \tan \theta _W\left( 2Y_{X'}^2+2Y_{Q'}^2+Y_{B_5}^2\right) \right] \nonumber \\&\quad -12\Delta _{\chi '}\tan \theta _W\,Y_{B_1}^2\Bigg \}F_{\mu \nu }\tilde{Z}^{\mu \nu }\,,\nonumber \\&-\dfrac{\alpha _{em}}{8\pi }\,\frac{a}{f_a}\dfrac{12}{\sin ^2\theta _W}\left( \Delta _{\psi }+\Delta _{\psi '}\right) W^+_{\mu \nu }\tilde{W}^{-\mu \nu } \end{aligned}$$

(A.2)

where $\theta _W$ is the Weinberg angle.