The minimal axion minimal linear \(\sigma \) model
Eur. Phys. J. C
The minimal axion minimal linear σ model
L. Merlo 1
F. Pobbe 0
S. Rigolin 0
0 Dipartimento di Fisica e Astronomia, Università di Padova and INFN, Sezione di Padova , via Marzolo 8, 35131 Padua , Italy
1 Departamento de Física Teórica and Instituto de Física Teórica UAM/CSIC, Universidad Autónoma de Madrid , Cantoblanco, 28049 Madrid , Spain
The minimal S O(5)/S O(4) linear σ model is extended including an additional complex scalar field, singlet under the global S O(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 axionlikeparticle. 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 S O(5) → S O(4), electroweak and PQ symmetry breaking. A single minimal scenario is identified and the associated scalar potential is minimised including counterterms needed to ensure oneloop 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 ∼ 105 TeV and therefore falling in the category of the invisible axions; or it is a more massive axionlikeparticle, such as a 1 GeV axion with an associated scale of ∼ 200 TeV, that may show up in collider searches. 1 Introduction . . . . . . . . . . . . . . . . . . . . . 1 2 The axion minimal linear σ model . . . . . . . . . . 2 2.1 The gauge Lagrangian . . . . . . . . . . . . . 3 2.2 The fermionic Lagrangian . . . . . . . . . . . 3 2.3 The scalar Lagrangian . . . . . . . . . . . . . . 5 3 The minimal model . . . . . . . . . . . . . . . . . . 6 4 The scalar potential . . . . . . . . . . . . . . . . . . 8 4.1 Integrating out the heaviest scalar field . . . . . 9 4.2 The case for fs ∼ f ∼ √scm and β, λsφ 1 . 11 4.3 Numerical analysis . . . . . . . . . . . . . . . 12 5 Collider phenomenology and exotic fermions . . . . 15 6 The axion and ALP phenomenology . . . . . . . . . 16 6.1 QCD axion or axionlikeparticle? . . . . . . . 18 7 Concluding remarks . . . . . . . . . . . . . . . . . 19 Generic PQ Transformations . . . . . . . . . . . . . . 20 References . . . . . . . . . . . . . . . . . . . . . . . . 21

Contents
1 Introduction
The last decade experienced a revival of interest for the
socalled Composite Higgs (CH) models: first introduced in the
middle of the 1980s [
1–3
], they have been reconsidered 20
years later with a more economical symmetry content [
4–6
].
The Minimal Composite Higgs Model (MCHM) [
4
] is based
on the nonlinear realisation of the S O(5)/S O(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 wouldbe 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 socalled 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 [
7–10
] 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 S O(5)/S O(4) Linear σ model
(MLσ M) is constructed extending the SM spectrum by the
introduction of an EW singlet scalar field σ and a specific set
of vectorlike fermions in the singlet and in the fundamental
representations of S O(5). In the limit of large σ mass, the
model falls back onto the usual effective nonlinear
description of the MCHM [
4,7,11–13
], that represents a specific
realisation of the socalled Higgs Effective Field Theory [
14–
34
] Lagrangian describing the most general Higgs couplings
to SM gauge bosons and fermions, which preserve the SM
gauge symmetry.
The MLσ 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, S O(5) and EW singlet, the symmetry content
of the model can be supplemented with an extra Peccei–
Quinn (PQ) U (1)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 σ Model
(AMLσ 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 AMLσ 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 treelevel
through the fermion partial compositeness mechanism [
40–
43
], which is the only explicit S O(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 QCDθ parameter
through a shift symmetry transformation, thus solving the
strong CP problem. If instead this requirement is relaxed,
then the PQ GB is dubbed axionlikeparticle (ALP). Both
the possibilities are envisaged in the minimal AMLσ 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 S O(5)/S O(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 S O(5) gets
spontaneously broken down to S O(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
aris1 In Ref. [
38
] the MCHM has been enriched by an additional U (1)
symmetry, that is nonanomalous and therefore does not originate a
QCD axion.
ing at one oneloop 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 nonperturbative QCD effects as in the traditional
PQ models [
35,44–47
]. Its corresponding scale is larger than
∼ 105 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 AMLσ M and to analyse its phenomenological
features. In the next section, the construction of the AMLσ 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 S O(5)/S O(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 σ model
The MLσ M based on the linear S O(5)/S O(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 MLσ M is the following:
1. The four SM gauge bosons associated to the SM gauge
symmetry.
2. A real scalar field φ in the fundamental representation of
S O(5), which includes the three wouldbelongitudinal
components of the SM gauge bosons πi , i = 1, 2, 3, the
Higgs field h and the additional complex scalar field σ ,
singlet under the SM gauge group:
φ = (π1, π2, π3, h, σ )T u.g.
−−→ φ = (0, 0, 0, h, σ )T ,
(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
S O(5) scalar sector through S O(5) invariant
protoYukawa interactions. These fermions transform either in
the fundamental of S O(5), and they will be labelled as ψ ,
or in the singlet representation of S O(5), dubbed χ . 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
SMexotic fermion interactions in the spirit of the fermion
partial compositeness mechanism [
40–43
]. SM fermions
do not come embedded in a complete representation of
S O(5), leading to a soft explicit S O(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 AMLσ M encompasses, in addition to the previous
content,
5. A complex scalar field s, singlet under the global S O(5)×
U (1)X and the SM gauge group. Adopting an exponential
notation,
r
s ≡ √ eia/ fa ,
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 AMLσ M
can be written as the sum of three terms describing
respectively the pure gauge, fermionic and scalar sectors,
L = Lg + Lf + Ls.
The explicit expression for each piece will be detailed in the
following subsections.
(2.2)
(2.3)
αs θ Gaμν Gaμν ,
+ 8π
2.1 The gauge Lagrangian
The first term, Lg, contains the SM gauge kinetic and the
colour anomaly terms,
1 1 1
Lg = − 4 Gaμν Gaμν − 4 W aμν Wμaν − 4 Bμν Bμν
with the indices summed over SU (3)c or SU (2)L , and
1
Gμν ≡ 2 μνρσ Gρσ
(with
The introduction of the axion will provide a natural
explanation for the vanishing of the QCDθ 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
Lf = q L i D/ qL + t Ri D/ tR + bRi D/ bR
(2.4)
(2.5)
+
−
+
+
+ ψ i D/ − M5 ψ + χ i D/ − M1 χ
− y1 ψ L φ χR + y2 ψ R φ χL + h.c.
− z1 χ R χL s + z˜1 χ R χL s∗
+ z5 ψ R ψL s + z˜5 ψ R ψL s∗ + h.c.
1 q L 2×5 ψR +
2 ψ L
× ( 5×1tR ) +
3 χ L tR + h.c.
+ ψ i D/ − M5 ψ
+ χ i D/ − M1 χ
y1 ψ L φ χR + y2 ψ R φ χL + h.c.
− z1 χ R χL s + z˜1 χ R χL s∗
+ z5 ψ R ψL s + z˜5 ψ R ψL s∗ + h.c.
1 q L 2×5 ψR
2 ψ L
5×1bR +
3 χ L bR + h.c. .
(2.6)
The first line contains the kinetic terms for the 3rd generation
SM quarks, being qL the lefthanded (LH) SU (2)L doublet
and tR and bR the righthanded (RH) singlet counterparts.
The second line contains the kinetic and mass terms for the
exotic vector fermions, ψ and χ (with U (1)X charge 2/3).
The direct mass terms for the heavy fermions are denoted
by M1,5 respectively for the fermions in the singlet and
fundamental representations. The protoYukawa couplings
between the heavy fermions and the real scalar quintuplet
field φ are also present in the second line. In the third line,
the Yukawalike couplings of the exotic fermions with the
where the field dependent fermion mass matrix M f is a
14 × 14 block diagonal matrix,
M f (h, σ, r ) = di ag M5(r ), MT (h, σ, r ),
MB(h, σ, r ), M5(r ) .
For the top sector one has explicitly
MT ( h⎛, σ, r )
0
0
⎜⎜⎜⎜⎜
= ⎜⎜⎜
⎝
0
(2.13)
⎞
1 ⎟
00 ⎟⎟⎟
⎟⎟ ,
0 ⎟⎟
0 ⎠
M5(r )
(2.14)
(2.15)
complex scalar singlet s are shown. Two distinct type of
couplings, z and 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 S O(5), the partial compositeness terms in the
fourth line, proportional to 1,2, explicitly break the global
S O(5) symmetry. The combinations 1 2×5 and 2 5×1
may play the role of spurions [
50–54
] that formally ensure
the S O(5) × U (1)X invariance of the operators. The exotic
fermion spinors can be decomposed under the SU (2)L
quantum numbers as follows:
ψ ∼ (K , Q, T5) ,
χ ∼ T1,
being K and Q doublets while T1,5 singlets of SU (2)L . The
resulting interactions preserve the gauge EW symmetry, with
the hypercharge defined as
Y =
(3)
R + X,
with (R3) 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, ψ and
χ 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
ψ ∼
Q , K , B5 ,
χ ∼ B1,
being Q and K doublets of SU (2)L (with (R3) component
1/2 and −1/2 respectively) and B1,5 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,
=
K u , T , B, K d ,
with
B =
T = t, Qu , K d , T5, T1, Q u ,
b, Q d , K u , B5, B1, Qd .
LM = − L M f (h, σ, r ) R ,
The fermion mass terms in Eq. (2.6) can then be written as
(2.7)
(2.8)
(2.9)
(2.10)
(2.11)
(2.12)
M1(r ) = M1+(z1 + z˜1) r,
M5(r ) = M5+(z5 + z˜5) r.
The corresponding matrix for the bottom sector, MB(h, σ, 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 S O(5) ×
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 AMLσ 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 Mi , zi
and z˜i (and corresponding primed) terms is allowed once a
specific PQ charge assignment for the fermion chiral
components is chosen, assuming obviously a nonvanishing charge
for the scalar s field. In other words, exotic fermions acquire
masses either through the direct mass terms (Mi ) or through
the Yukawalike interactions with s (zi or z˜i ) once the scalar
field s develops a VEV. In addition, following the
assumptions outlined in the previous section, as the scalar quintuplet
φ does not transform under the PQ symmetry, the presence
of the protoYukawa interactions (yi ) 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 nonvanishing PQ
charges, then these operators are forbidden, unless the 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 AMLσ M charge assignment are illustrated.
2.3 The scalar Lagrangian The scalar part of the Lagrangian introduced in Eq. (2.3) describes scalargauge and scalarscalar interactions: 1
Ls = 2 (Dμφ)T (Dμφ) + (∂μs∗)(∂μs) − V (φ, s),
where the SU (2)L × U (1)Y covariant derivative of the
quintuple φ is given by
Dμφ =
∂μ + i g L(i)W μi + i g
(R3) Bμ φ,
and Li and iR denote respectively the generators of the
SU (2)L × SU (2)R ∼ S O(4) subgroup of S O(5), rotated
with respect to the S O(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):
Ls =
1 1
2 (∂μh)(∂μh) + 2 (∂μσ )(∂μσ )
+ h42 g2 Wμ+W −μ + g2 +2 g 2
Zμ Z μ
1 r 2
+ 2 (∂μr )(∂μr ) + 2 fa2 (∂μa)(∂μa) − V (h, σ, r ),
Notice that once the U (1)PQ gets spontaneously broken
through the VEV of r , the kinetic term of the axion field
a gets canonically normalised, by identifying
fa ≡ vr .
The scalar potential V (φ, s) can then be written as:
V (φ, s) = V SSB(φ, s) + V CW(φ, s) + V c.t.(φ, s). (2.20)
The first part, V SSB(φ, s), describes the most general
potential constructed out of φ and s, invariant under S O(5) ×
U (1)PQ symmetry, broken spontaneously to S O(4):
V SSB(φ, s) = λ(φT φ − f 2)2 + λs (2 s∗s − fs2)2
−2λsφ (s∗s) φT φ ,
where λ, λs and λsφ are the dimensionless quartic coefficients
and the sign in front of λsφ has been chosen negative for
future convenience. Notice that λsφ plays the role of portal
between the S O(5) and the PQ sectors: if λsφ ∼ O(1) then
(2.16)
(2.17)
(2.18)
(2.19)
(2.21)
the S O(5)/S O(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
AMLσ M, which requires f not so much larger than EW scale
v = 246 GeV, in order to reduce the typical finetuning in CH
models, and the experimental data on the axion sector, which
suggests very high values of fs (see Sect. 6). In consequence,
values of λsφ smaller than 1 are favoured in the AMLσ M.
The expression of V SSB in the exponential notation will
be useful in the following sections:
V SSB(h, σ, r ) = λ(h2 + σ 2 − f 2)2 + λs (r 2 − fs2)2
− λsφ r 2 (h2 + σ 2).
(2.22)
When the scalar fields h, σ and r take a non trivial VEV,
respectively vh , vσ and vr , a spontaneous symmetry
breaking for the EW, the global S O(5) and the PQ symmetry, is
obtained.
The second term V CW(φ, s) is the Coleman–Weinberg
(CW) oneloop 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 c.t.(φ, s), includes all the couplings
that need to be introduced at treelevel in order to cancel
the divergences potentially arising from the oneloop CW
potential, so to make the theory renormalizable.
The Coleman–Weinberg oneloop potential
Explicit dynamical breaking of the treelevel symmetries
can be introduced at oneloop level through the CW
mechanism [
55
]. Indeed, the presence of S O(5) breaking
couplings in both the fermionic and the gauge sectors generate
S O(5) breaking terms at oneloop level. Explicit U (1)PQ
breaking contributions may also be generated, depending on
the fermion PQ charge assignment.
The oneloop fermionic contributions can be calculated
from the field dependent fermion mass matrix M f (h, σ, r )
in Eq. (2.13), using the usual CW expression:
V fCW
1
= − 64π 2
†
M f M f
2 − Tr
(2.23)
where is the ultraviolet (UV) cutoff scale while μ 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
(2.24)
(2.25)
discussion the possible divergent contributions read
Tr
The terms in Eq. (2.24) are already present in the tree level
potential V 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 d˜1,2 and dˆ1,2,3 in Eq. (2.25). The ones proportional to
d˜1,2 and dˆ1 are S O(5) breaking terms, while the ones
proportional to dˆ2,3 are S O(5) preserving. On the other side,
dˆ1,2,3 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 treelevel potential.
The expressions for the top sector CW coefficients that
provide an explicit breaking of the S O(5) and/or of the PQ
symmetries read:
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 oneloop 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
M f with the gauge boson one Mg:
The quadratic and logarithmic divergent terms read
Tr
Mg = a˜1h2 Tr
2
with
both explicitly breaking the global S O(5) symmetry.
The two divergences associated to a˜1 and d˜2 require the
introduction of an h2 term in the treelevel scalar potential,
in order to ensure the renormalisability of the model, while
the divergence proportional to the b˜1 coefficient requires an
additional h4 term.
3 The minimal model
There is a large zoology of possible U (1)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
AMLσ M. The requirements are the following:
1. Mass terms for the SM quarks are originated at treelevel.
Generalising the result in Ref. [
9
], the leading order (LO)
contribution to the third generation quark masses is given
by
mt =
and similarly for the bottom mass. In this expression,
M1,5(vr ) refer to the definitions in Eq. (2.14) substituting
the dependence on r with its VEV, vr . In order for this
expression not to be vanishing, the conditions y1 = 0 and
1 = 0 should hold simultaneously. Then, either 3 =
0 or y2 = 0 ∧ 2 = 0 should be verified, depending
on whether the leading or subleading term in the v/M
expansion is retained.
2. The dynamics that generate the partialcomposite
operators in the fourth line of Eq. (2.6) are associated only to
the S O(5)/S O(4) breaking sector. This implies that the
scales f and fs are distinct and independent.
In a completely generic model a third condition can be
also considered:
Lagrangian, together with the information on whether they are
compatible ( ) or not (×) with the PQ symmetry. This assignment can be
trivially extended to the bottom sector
nqL
3. No PQ explicit breaking is generated at oneloop from
the CW potential.2 This condition is satisfied by imposing
ˆi = 0, for i = 1, 2, 3 (and the equivalent ones for the
d
bottom sector).
This condition prevents the arising of large contributions to
the axion mass, and it is automatically verified in the class
of AMLσ 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 QCDθ term of the colour
anomaly needs to be nonvanishing.
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
nonperturbative 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 ns = 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 ψR and χL . The Yukawalike terms
proportional to y1,2 are invariant under U (1)PQ, while the term
proportional to 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 M1,5 are
also forbidden and then the exotic fermions ψ and χ receive
mass of the order z5vr (or z˜5vr depending on the specific
sign of the PQ charge) and z1vr (or z˜1vr ), once r develops a
2 The discussion on the consequences of PQ explicit breaking
contributions, on its interest in cosmological studies, and on the case where
the S O(5)/S O(4) and PQ symmetry breaking occur at the same scale
is deferred to Ref. [
56
].
z˜1, z5
×
(3.2)
(3.3)
nonvanishing VEV. As vr is typically expected to be of the
order of fs , these fermions decouple from the spectrum when
fs f . Finally, condition 2 implies that the couplings 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
PQbreaking terms in the fermionic CW potential, dˆi , are
vanishing, while the S O(5) breaking terms read
d˜1 = 0,
d˜2 = y22 21.
In consequence, in this scenario, only a logdivergent S O(5)
breaking contribution to the hmass term arises from the
fermionic part of the CW potential, while no σ tadpole
contribution is generated. This is different from the analysis
performed in Ref. [
9
], where the only S O(5) symmetry
breaking terms considered have been the σ tadpole and the h2
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 S O(5) and EW spontaneous
symmetry breaking at least two different S O(5) breaking terms are
necessary. Additional unavoidable sources of S O(5)
breaking comes from the gauge sector, as shown in Eq. (2.27).
The minimal counterterm potential required at treelevel by
renormalisability of the theory, once the charge assignment
has been chosen, is then given in the unitary gauge by
V c.t.(h, σ ) = −β f 2h2 + γ h4.
Other values for the PQ charges are possible by changing the
explicit value of ns , but they lead to the same physical model
presented above, at least for what concerns the S O(5)/S O(4)
phenomenology and the analysis of the scalar potential. The
physical dependence on the explicit value of ns , 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–
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
caW W Wμ+ν W˜ −μν ,
4 The scalar potential
gauge boson field strengths in the physical basis,
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
αs cagg
gagg ≡ 2π fa
gi ≡ α2eπm cfai ,
where i = {aγ γ , a Z Z , aγ Z , aW W }.
The charge assignment in Table 1 corresponds to the
minimal setup among all the possible AMLσ M constructions,
where the minimality refers to the number of new parameters
that are introduced with respect to the MLσ 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 ( fs , λs and λsφ ), and in
particular only two S O(5) breaking terms are present
(corresponding to β and γ ); the PQ charges also represent degrees
of freedom and the minimal model in Table 1 is univocally
determined by fixing ns . Indeed, conditions 1 and 2 impose
that the difference between the charges of the LH and RH
components of the SM fermions is vanishing, nqL − ntR = 0,
and in consequence it is always possible to redefine the whole
set of PQ charges such that nqL = ntR = 0.
It is worth mentioning that an alternative charge
assignment can be found satisfying to the conditions 14, but this
scenario is not minimal in terms of number of parameters.
In this case, the charges are such that ntR = nχL = nχR =
nψL = nψR ∓ ns = nqL ∓ ns , where the “−” or “+” refer to
the presence of z5 or z˜5 terms in the Lagrangian, respectively.
As discussed in Ref. [
39
], SM fermions transform under
the PQ symmetry, differently from the minimal AMLσ M
3 In the present discussion, only one fermion generation has been
considered. Once extending this study to the realistic case of three
generations [
56
], the values reported in Table 2 will be modified: for example,
assuming that the same charges will be adopted for all the fermion
generations, the numerical values in the table will be multiplied by a factor
3.
in Table 1. Moreover, the Dirac mass term M1 is allowed
in the Lagrangian, while the ψ fermions receive mass from
the Yukawalike term proportional to z5 (or z˜5). Moreover,
the terms proportional to 1,2,3 and y1 are allowed, while
the one with y2 is forbidden. In consequence, the term d˜1 in
Eq. (2.26) is not vanishing and then a σ tadpole needs to be
also added into the counter term potential V c.t.(h, σ ). The
number of S O(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.
As constructed in the previous section, the treelevel
renormalisable scalar potential of the minimal AMLσ M reads
V (h, σ, r ) = λ(h2 + σ 2 − f 2)2 − β f 2h2 + γ h4
+ λs (r 2 − fs2)2 − λsφ r 2 (h2 + σ 2).
When f 2 > 0 and fs2 > 0, the minimum of the potential
allows for the S O(5), U (1)PQ and EW spontaneous
symmetry breaking with nonvanishing VEVs,
(4.1)
where the condition vr ≡ fa 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 hˆ,
σˆ and rˆ the physical fields after SSB breaking. Assuming all
parameters nonvanishing, the following conditions on the
parameters must be imposed:
(i) λ > 0 and λs > 0 in order to lead to a potential bounded
from below.
(ii) β and γ should have the same sign in order to
guarantee a positive vh2 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 β/2γ < 1 leads
to vh < f , corresponding to the expected ordering in
the symmetry breaking scales.
(iii) λsφ should satisfy to
2
λsφ < 4λλs
(4.3)
in order to enforce positive vσ2 and vr2 values. For
negative λsφ 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 vσ2 and vr2 occurs for λsφ > 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 fs ∼ f , expanding perturbatively in the small
β and λsφ 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 λs and/or fs : the mass of the
heaviest scalar dof receives a LO contribution proportional
to
m3 ∝
8λs fs .
(4.4)
With increasing values of λs and/or fs , the contaminations
of hˆ and σˆ to the heaviest scalar dof, in this region of the
parameter space, tend to vanish and the only relevant
constituent is the radial component, rˆ. From the expression in
Eq. (4.4), one can envisage two different ways for
integrating out the heaviest dof, either taking the limit λs 1 or
taking fs f ∼ √scm, being √scm the typical centre of
mass energy scale at LHC. These two cases represent two
physically different scenarios that are discussed separately.
The case for λs 1, with fs of the same order of f ,
corresponds to the U (1)PQ nonlinear spontaneous symmetry
breaking framework4: this is the traditional axion framework
where the only component of s in the lowenergy spectrum
is the axion, while rˆ is integrated out. As the Yukawalike
couplings of the exotic fermions do not depend on λs , the
decoupling of rˆ does not have any impact on the spectrum of
4 In the case where an UV strong interacting dynamics is responsible
of the largeness of λs , new resonances are expected at the scale 4π fs
(see the naive dimensional analysis [
68
]).
the exotic fermions, that depends exclusively of the specific
value chosen for fs . One can then consider in detail the two
limiting cases: fs ∼ f or fs f . Notice that in the second
scenario, when fs 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 rˆ
component, leads to an effective scalar potential that, at LO in
the appropriate expansion parameter, reads
VRL O (h, σ ) = λR (h2 + σ 2 − f R2)2 − βR f R2h2 + γ h4 , (4.5)
in terms of conveniently renormalised couplings:
λR = kλλ,
kλ β,
βR = k f
f R2 = kkλf f 2. (4.6)
The finite renormalisation constants kλ 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 hˆ and
σ :
ˆ
vh2 = β2γR f R2,
satisfying to
vh2 + vσ2 = f R2.
vσ2 = f R2 1 − β2γR
The restrictions on the parameters that follow from Eq. (4.2)
hold for the expressions just obtained: βR /γ needs to be
positive in order to guarantee vh2 > 0; f R is required to be
larger than vh to ensure vσ2 > 0. Moreover, if vσ > vh then
the field hˆ is the largest component of the mass eigenstate that
can be interpreted as the physical Higgs particle originated
as a GB of the S O(5)/S O(4) SSB mechanism.
From Eq. (4.5) and using the relations of Eq. (4.7) one
derives the following mass matrix:
2
Ms = 8 λR
(1 + γ /λR )vh2 vh vσ
vh vσ vσ2
whose diagonalisation is obtained by performing an S O(2)
rotation,
di ag m21, m22 = U (ϑ )T Ms2U (ϑ )
with
U (ϑ ) =
cos ϑ sin ϑ
.
− sin ϑ cos ϑ
The expressions for the masses and the mixing obtained from
the LO potential of Eq. (4.5) are given by
(4.7)
(4.8)
(4.9)
(4.10)
m21,2 = 4λR ⎣
2vh vσ
tan 2ϑ = vσ2 − (1 + γ /λR )vh2
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
λR > 0,
γ > 0,
βR > 0,
(4.13)
where the last condition follows from the requirement that
γ and βR should have the same sign in order to guarantee a
positively defined vh2 value, as discussed below Eq. (4.2).
The following two subsections will describe in detail the
two limits λs 1 and fs f ∼ √scm, focusing on the
scalar sector.
The large PQ quartic coupling: λs 1 and fs ∼ f
For λs in the strongly interacting regime, the radial
component r can be expanded in inverse powers of λs (see Ref. [
10
]
for a similar analysis): at the NLO, one has
1
r = fs + λs r1.
Solving the Equations Of Motion (EOMs) perturbatively
allows to determine r1:
λsφ
r1 = 4 fs
h2 + σ 2
Ls = 21 (∂μh)(∂μh) + 21 (∂μσ )(∂μσ ) − h42 Tr VμVμ
+ 21 (∂μa)(∂μa) − λR h2 + σ 2 − f R2 2
+ βR f R h − γ h4 + δLsNLO
2 2
with λR , βR and f R2 defined as in Eq. (4.6) with
kλ = 1,
k f =
1 + 21 λλsφ ffs22 ,
and where the NLO correcting term is given by
δLsNLO = λ4s fs2r12 = 4λλs2φs
(h2 +σ 2)+ 21fs2 ∂μa
In this scenario, f R is the new effective S O(5)/S O(4)
breaking scale, while the S O(5) quartic coupling λ = λR remains
(4.16)
(4.17)
∂μa
2
.
(4.18)
unchanged. The positivity of f R2 translates into a constraint
on the couplings λsφ :
λsφ > −2λ fs2 ,
where λ, f 2 and fs2 are all positive (see the discussion at
the beginning of Sect. 4). The value λsφ = 0 is special:
λsφ represents the portal between the S O(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 λs λR 1 and small β, for
which the expressions in Eqs. (4.11) and (4.12), reduce to
The large PQ SSB scale: fs f ∼ √scm
In the limit fs f ∼ √scm, being λs 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 / fs .
Within this setup r at NLO reads
and λR and f R2 defined in Eq. (4.6), with kλ and k f explicitly
given by
kλ =
,
k f =
An increasing value of fs corresponds to an increasing value
of f R . However, caution is necessary in the case when λsφ is
exactly vanishing, as the S O(5) and PQ sectors are
decoupled: in this specific case f R = f and the S O(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 λR = λ is introduced. To ensure
a potential bounded from below both f R2 and λR need to be
positive, leading to the following constraints on λsφ ,
(4.26)
∧
2
λsφ < 4λλs .
(4.27)
λsφ > −2λ fs2
scale v, defined by v ≡ 2MW /g = 246 GeV, and it is much
smaller than the S O(5) SSB scale, i.e. vh < f . The smallness
of λsφ follows, instead, from the assumption that the S O(5)
and PQ sectors are determined by two distinct dynamics and
therefore the two breaking mechanisms occur independently.
A large λsφ 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 β and λsφ , it leads to the following
simplified expressions:
where in the brackets the dependence on β and λsφ of the
higher order corrections is reported. The scalar squared mass
matrix is given by the following expression
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 λs2φ λλs ,
simplify to
that can be diagonalised performing an orthogonal
transformation,
di ag m21, m22, m23 = U (ϑ12, ϑ23)T Ms2U (ϑ12, ϑ23)
β λ f 2
m21 = 4β f 2 1 − γ λsφ fs2
m22 = 4λsφ fs2 1 + 2
,
λ f 2
λsφ fs2
with the mixing angle given by
tan 2ϑ = 2
β λ f
γ λsφ fs
.
4.2 The case for fs ∼ f ∼ √scm and β, λsφ
1
For fs ∼ f ∼ √scm, 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 β, λsφ 1, which
are natural conditions in the AMLσ M. The first condition
comes from the requirement that vh coincides with the EW
(4.28)
(4.29)
with
U (ϑ12, ϑ23) = U (ϑ12)U (ϑ23),
the product of a rotation in the 12 sector and in the 23
sector respectively, of angles ϑ12 and ϑ23. The resulting mass
eigenvalues read
f fs
tan 2ϑ23 = λs fs2 − λ f 2 λsφ 1 + O(β, λsφ ) . (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 β and λsφ 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 Ms is real and can be
diagonalised by an orthogonal transformation,
di ag(m12, m22, m32)
= U (ϑ12, ϑ23, ϑ13)T Ms2 U (ϑ12, ϑ23, ϑ13),
(4.35)
where U (ϑ12, ϑ23, ϑ13) ≡ U (ϑ23)U (ϑ13)U (ϑ12) is the
product of three rotations in the 23, 13, and 12 sectors
respectively, of angles ϑ23, ϑ13 and ϑ12. The scalar mass eigenstates
ϕ1, ϕ2, and ϕ3 are defined by
⎛ ⎞ ⎛ ⎞
ϕ1 hˆ
⎝ ϕ2⎠ = U (ϑ12, ϑ23, ϑ13)T ⎝ σˆ ⎠ (4.36)
ϕ3 rˆ
in terms of the three physical shifts around the vacua. Unless
explicitly indicated, in the analysis that follows, ϕ1 will be
identified with the Higgs particle and the deviations of its
couplings from the SM predicted values are interesting
observables at colliders. The ϕ1 couplings to the SM gauge bosons
can be deduced from the couplings of hˆ, as σˆ and rˆ are
singlets under the SM gauge group. The composition of hˆ in
terms of ϕi is explicitly given by
hˆ = c12c13ϕ1 + c13s12ϕ2 + s13ϕ3 ≡ C1ϕ1 + C2ϕ2 + C3ϕ3,
where ci j and si j stand for cos θi j and sin θi j , and the
coefficients Ci in the last equality have been introduced for
shortness. The couplings with the SM gauge bosons can be written
as
(hˆ + vh )2Wμ+W −μ
g2
4
= m2W
g2 + g 2
8
C1
ϕ1
vh
(hˆ + vh )2 Zμ Z μ
+ C2
+ C3
+ 1
ϕ2
vh
ϕ3
vh
2
Wμ+W −μ,
(4.37)
γ =
f
vh
2 β
2
and then extract β, in terms of the remaining five parameters,
by numerically solving the equation m1(β, λ, λs , λsφ , f, fs )
= mh . Consequently, predictions for all the remaining
observables can be obtained by choosing specific values for
(λ, λs , λsφ , f, fs ).
In Fig. 1 the bounds on the C1 parameter in the ( fs , f )
plane for λ = λs = 1 and λsφ = 0.1 are shown. The
dark green region corresponds to C1 < 0.90, while the
light green one to 0.90 < C1 < 0.95. In the white region
m2Z
= 2
C1
ϕ1
vh
+ C2
+ C3
Finally, the ϕ1 couplings to the longitudinal components of
W and Z are modified with respect to the SM predictions for
the Higgs particle by factor of C1.
To have a clear comparison with CHM predictions, one
can write the expression for the C1 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 λR 1, giving
The last term on the righthand side introduces the
parameter ξ , that customary defines the tension between the EW
and the composite scales. This parameter often appears in
CHMs to quantify the level of nonlinearity 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 ξ , as the ones from Refs. [
12,70
],
ξ
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 C1 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 λ, λs , β, γ , λsφ , and two scales f and fs . By using
the known experimental values of the Higgs VEV, vh = v ≡
246 GeV, and mass m1 = mh ≡ 125 GeV, two of these
coefficients can be eliminated in terms of the remaining five. The
adopted procedure for the numerical analysis is to express
γ as function of β and f , by inverting the vh2 expression in
Eq. (4.2):
(4.39)
(4.40)
(4.41)
Fig. 1 C1 contours in the ( fs , f ) plane, for λ = λs = 1 and λsφ = 0.1.
The dark green region corresponds to C1 < 0.90, while the light green
one to 0.90 < C1 < 0.95. In the white region C1 > 0.95. The two
red curves correspond to values for the next to lightest scalar m2 = 1
TeV and m2 = 2 TeV respectively, being the Higgs mass fixed to the
reference value mh = 125 GeV
C1 > 0.95. From this plot one can have an order of
magnitude comparison with present/future experimental bound on
the Higgsgauge boson interaction. The following bounds on
h Z Z and hW W couplings are obtained by [
71
], using the so
called κframework5:
(4.42)
The expressions in Eq. (4.38) enforce the relation κZ =
κW = C1.
Figure 1 gives the idea of the interplay between the two
scales f and fs for fixed values of the remaining
adimensional parameters. For fs = 1 TeV, LHC can already start
to exclude values of f 0.7 TeV. However, for the larger
value fs = 3 TeV, even values of f ≈ 0.5 TeV will lie
outside LHC exclusion reach and no precise bound separately
on f or fs can be inferred from the sole measurement of the
Higgs couplings to gauge bosons, for most of the
parameter space.6 Only when λ, λs 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 MLσ M. For
completeness in Fig. 1 also the curves corresponding to two values
of the mass of the next to lightest scalar, m2 = 1 TeV and
m2 = 2 TeV, have been depicted.
In the following analysis the value f = 2 TeV has been
chosen as benchmark. The parameter space for the
remain5 Notice that in the κframework one assumes that there are no new
particles contributing to the gg H production or H → γ γ decay loops.
6 Limits on the scale f from EWPO will be discussed in the following
section.
ing four variable, λ, λs , λsφ , fs , can be studied, plotting the
behaviour of the scalar mass eigenvalues mi and of the
mixing coefficients squared C 2.
i
In Fig. 2, the masses m2 and m3 are shown as a function
of λsφ (upper left), or λ = λs (upper right), or λ (lower).
The mass m1 is fixed at mh , while the scale f is taken at
2 TeV. Three distinct values for fs are considered, fs =
1 TeV, 103 TeV, 106 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 λ and λs are taken to be equal
to 10; in the plot in the upper right, λsφ = 0.1; in the lower
plot, λsφ = 0.1 and λs = 10.
All these plots present features discussed in the different
limiting cases of the previous section. In the three plots, the
lines corresponding to fs = 103 TeV and fs = 106 TeV well
represent the expressions for the masses in Eq. (4.28). In the
upper left plot, the reddashed line represents the heaviest
dof with a constant mass according with Eq. (4.4); the
bluecontinue line corresponds to the second heaviest dof and it
shows an increasing behaviour with a constant slope,
corresponding to the expression for m22 that in first approximation
is proportional to λsφ . 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 reddashed line that
increases with a constant slope according to Eq. (4.4) only
for λ = λs 0.1. The bluecontinue line is almost constant,
as expected from the expression of m22 in Eq. (4.28), except
for the region with small λ = λs . In the lower plot, both
the reddashed and the bluecontinue lines are horizontal, as
expected having fixed both λs and λsφ .
When fs = 1 TeV, the numerical results agree with
the analytic expressions in Eqs. (4.21) and (4.33). In the
upper left plot, the reddashed and the bluecontinue lines
are exchanged with respect to the lines for fs = 103 TeV
and fs = 106 TeV: this is in agreement with Eq. (4.33), as
indeed for f > fs the heaviest dof is ϕ2 and the
nexttoheaviest is ϕ3. Moreover, the two lines are almost horizontal
as the dependence on λsφ 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
λ = λs , that is close to the excluded region. In the lower plot,
the reddashed line is almost horizontal, according to m23 in
Eq. (4.33), while the bluecontinue line increases with λ, as
shown by the expression for m22. For λ = 2.5 the two lines
cross and ϕ2 becomes the heaviest dof. The same
conclusions are expected by analysing the expressions in Eq. (4.21),
where ϕ3 is integrated out: the comparison is however more
difficult as m22 depends explicitly on β and γ , which are only
numerically computed in terms of λ, λs , λsφ , fs . Moreover,
when λ > 2.5, ϕ2 should also be integrated out from the
lowenergy spectrum as its mass reaches the value of the one
of ϕ3, and not consistent description is expected for these
values of λ.
The mixing coefficients C1, C2 and C3 are shown in Fig. 3:
the greendotdashed line describes C12, the bluecontinue
line C22 and the reddashed line C32. Both plots clearly show
that the largest component to hˆ is ϕ1, that is identified to the
physical Higgs particle. The contaminations from ϕ2 and ϕ3
are much smaller and at the level of ∼ 1% at most. This is
a typical feature in almost all the parameter space, and in
particular for fs 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 C22 and C32: as far as fs > f the largest
the heaviest dof with mass m3, while the bluecontinue line the
nexttoheaviest dof with mass m2. The lightest dof is identified with the
Higgs particle with mass m1 = mh . The red area is excluded from the
constraint in Eq. (4.3)
contamination is given by ϕ2, while for f < fs it is given by
ϕ3, 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 MLσ M: in the
latter, only two scalar states are present and then only one
mixing can be defined, that is between hˆ and σˆ ; for
increasing masses of ϕ2, which almost coincides with σˆ , the sibling
of C22 asymptotically approaches the ratio v2/ f 2 and a
benchmark value of 0.06 has been taken in the phenomenological
analysis. From Fig. 3, the maximal value that C22 (or C 2)
3
can take is of 0.015: this means that some differences are
expected between the two models when discussing the EW
Fig. 3 The profiles of the coefficients squared C2, C22 and C2, as a
1 3
function of λ = λs . The other parameters are chosen at fixed values:
f = 2 TeV; λsφ = 0.1; fs = 1 TeV on the left and fs = 3 TeV on
the right. The greendotdashed line describes C12, the bluecontinue
line C22 and the reddashed line C32. The red area is excluded from the
constraint in Eq. (4.3)
precision observables (EWPO) and the impact of the exotic
fermions.
In a tiny region of the parameter space, ϕ2 can be lighter
than ϕ1, with m1 still fixed at the value mh . This is
consistent with the results in Ref. [
9
]. Although this possibility is
experimentally viable, from the theoretical perspective it is
not appealing as m2 < m1 requires λsφ 10−7,
corresponding to a highly tuned situation. Similarly, mixing parameters
larger than the typical values shown in Fig. 3, for example
C22 ∼ 0.1, can only be achieved for λsφ 10−4, another
tuned region of the parameter space. Another possibility for
relatively large mixing parameters is for f ∼ 100 GeV and
fs 1 TeV, that is very unlikely as it would correspond to
the case with the EWSB occurring before the S O(5)/S O(4)
symmetry breaking. In consequence, only the case with ϕ2
heavier than ϕ1 and values of λsφ 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 S O(5)/S O(4)
context has been presented in Ref. [
12
] and the impact at colliders
of different realisations has been analysed in Ref. [
74
]. The
MLσ M, and therefore also the AMLσ M, seems an
interpolation between the socalled MC H M4 and MC H M5
scenarios considered in Ref. [
74
], once only the physical Higgs
is retained in the lowenergy theory. Typical observables of
interest at colliders are the EWPO, the Z bb¯ coupling,
couplings of the scalar dofs to gluons and photons [
7,8
], and the
interactions with fermions. As they have been studied for the
MLσ M in Refs. [
9,10
], the aim of this section is to extend
those results to the AMLσ M.
EWPO
Deviations to the SM predictions for the T and S
parameters [
75
] (or equivalently 1 and 3 [
76
]) are expected to
be relevant. In the MLσ M, the mixing between hˆ and σˆ can
reach relatively large values, ∼ 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 AMLσ 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
hˆσˆ 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 MLσ 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.
Zbb¯ coupling
The modification of the Z couplings to bb¯ is a very good
observable to test a model. The most relevant contributions
arise from the toppartner fermion, while the ones from the
heavier scalar dofs turn out to be negligible. The toppartner
induces deviations from the SM prediction of this coupling
only at the oneloop level, and the effect of these
contributions is soften with respect to those to the EWPO
previously discussed. This result holds for both the MLσ M and
the AMLσ M. As illustrated in Ref. [
9
], it is easy to
accommodate the experimental measure of the Z bb¯ 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 σ production at collid
ers
As in the SM, no tree level hˆ gg and hˆ γ γ couplings
are present in the AMLσ M. However, effective interactions
with gluons and with photons may arise at the oneloop
level. In consequence, all the three scalar mass eigenstates,
ϕ1,2,3, do couple with gluons and photons, with their
interactions weighted by the corresponding mixing coefficients C 2,
i
according to Eq. (4.37).
As worked out in details in Ref. [
9
], the Higgs coupling
with two gluons, ϕ1gg, 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 vectorlike nature). On the
other hand, the ϕ2 gg and ϕ3gg couplings are suppressed by
C22 and C32 respectively, and therefore are typically at least
10−2 smaller than ϕ1gg. Moreover, as the top quark is lighter
than ϕ2 and ϕ3, 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 ϕ1γ γ , while they are suppressed by C22 and C32 for
the heavier scalar dofs and the most relevant contributions to
ϕ2γ γ and ϕ3γ γ 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 ϕ2 and ϕ3 are
typically larger than the TeV scale, within the whole range
of values for f and fs 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 TeV and mixing coefficient C22 > 0.1.
Extending this result to the three scalar dofs described in the
AMLσ M and considering the results presented in Fig. 2,
the present LHC data and the future prospects (LHC
run2 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 MLσ M, where
two arbitrary mass parameters M1(,)5 are introduced in the
Lagrangian, in the minimal AMLσ M the exotic fermion
masses are controlled by fs , through the parameters z1(,)5
(and/or z˜1(,)5). The largeness of fs 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 lowenergy 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
nonlinearly realised and the Higgs originates as a GB. For an
overview of these analyses see Refs. [
21–23, 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 MLσ M and AMLσ M deal with the
same symmetry of the SM, no additional effects are expected
with respect to the analyses carried out in Refs. [
79–83
].
6 The axion and ALP phenomenology
The axion couplings to SM gauge bosons and fermions have
been bounded from several observables [
84–116
]. 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 AMLσ 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 lowenergy 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
7 Indirect couplings arise from the same mechanism that generate SM
fermion masses. However, experimental constraints are present on axion
couplings with only light SM fermions, the strongest being on axion
couplings with two electrons. As in the minimal AMLσ M only the
third generation fermions are considered, no relevant bound can be
deduced considering these constraints. This analysis is postponed to
further investigation [
56
].
bounds can be summarised as
7 × 10−11 GeV−1
10−10 GeV−1
10−12 GeV−1
10−3 GeV−1
for
for
for
for
ma
10 meV
For masses between 10 eV and 0.1 GeV, and in
particular for the socalled MeV window, the coupling gaγ γ is
constrained by (model dependent) cosmological data [
107
].
These bounds can be translated in terms of fa /caγ γ  through
Eq. (3.5): taking αem = 1/137.036,
2 × 107 GeV
107 GeV
109 GeV
for
for
1 GeV
for
0.1 GeV
for
ma
10 meV
10 meV
ma
10 eV
10 eV
ma
ma
0.1 GeV
In Ref. [
39
] a dedicated analysis of the axion coupling to
photons within the AMLσ M is presented, including constraints
and prospects from current experiments.
Coupling to gluons
The axion coupling to gluons has been constrained by
axionpion mixing effects [
84,87
] and monojet searches at
colliders [
104,105,109,113
]. The bounds can be summarised
as follows:
1.1 × 10−5 GeV−1
10−4 GeV−1
for
for
60 MeV
ma
ma
60 MeV
0.1 GeV
(6.3)
gaγ γ 
gaγ γ 
gaγ γ 
gaγ γ 
fa
caγ γ 
fa
caγ γ 
fa
caγ γ 
fa
caγ γ 
gagg
gagg
fa
cagg
fa
cagg
(6.5)
(6.6)
(6.8)
(6.9)
(6.10)
ing ratio has been bounded by the E787 and E949
experiments [
92
]:
B R(K + → π + + a) < 7.3 × 10−11.
For larger masses up to a few GeV’s, the B+ → K + + a
decay provides the most stringent bound: BaBar experiment
has proven that [
96
]
B R(B+ → K + + a)
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 AMLσ 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 gaW W effective coupling read
[
112
]:
3 × 10−6 GeV−1
10−4 GeV−1 for 0.2 GeV
for
ma
ma
0.2 GeV
5 GeV
(6.7)
gaW W 
gaW W 
fa
caW W 
fa
caW W 
that can be translated in terms of fa /cagg as
2×103 GeV
2×102 GeV
for
for
ma
60 MeV
60 MeV
ma
0.1 GeV
taking αs (MZ2 ) = 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
axionSM fermion couplings are present at treelevel in the
minimal AMLσ M). The most relevant observable for axion
masses below ∼ 0.2 GeV is K + → π + + a whose
branch(6.2)
that can be translated in terms of fa /caW W  as
4 × 102 GeV
for
ma
0.2 GeV
10 GeV
for0.2 GeV
ma
5 GeV.
(6.4)
ga Zγ 
1.8 × 10−3 GeV−1.
Collider searches are able to put independent constraints on
gaW W as well as on couplings with other gauge bosons.
Following Ref. [
113
], considering LHC data with √scm =
13 TeV and for axion masses ma 1 GeV, the monoW,
pp → aW (W → μνμ), and monoZ, pp → a Z (Z → ee),
signals put the following constraints:
gaW W 
5 × 10−7 GeV−1, ga Z Z 
3 × 10−7 GeV−1.
The Z boson width allows to put a conservative bound on
Z → aγ interaction:
The corresponding bounds on fa /ci  are given by:
fa
caW W 
fa
ca Z Z 
fa
ca Zγ 
2 × 103 GeV,
4 × 103 GeV,
0.6 GeV.
(6.11)
(6.12)
(6.13)
(6.14)
The axion mass
There are two distinct contributions to the axion mass
(gravitational and/or Planckscale sources [
119–122
] 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
]
ma ∼ 6μeV
1012 GeV
fa /cagg
,
for values of fa typically taken to be larger than 106 GeV.
The second is due to the extra fermions that couple to the
axion, such as in the KSVZ invisible axion model [
36,37
]:
√
Z αs2 fπ mπ ln
ma = 1 + Z π 2 fa
m2
ψ
mu md
,
where Z mu /md and fπ ∼ 94 MeV is the pion decay
constant and mψ is the generic mass of the exotic fermions.
This contribution is a decreasing function with fa for values
of fa > 10 MeV: considering similar values of fa and mψ ,
it follows that
ma ∼ 100 keV for fa ∼ 1 GeV
ma ∼ 0.2 keV for fa ∼ 103 GeV
ma ∼ 0.3 eV for fa ∼ 106 GeV
ma ∼ 0.004 eV for fa ∼ 108 GeV.
Notice that, for the last two cases, the QCD mass in Eq. (6.12)
is relevant and provides the dominant contributions of 60 eV
and 0.6 eV respectively. These benchmarks are interesting
for the discussion that follows.
6.1 QCD axion or axionlikeparticle?
In Sect. 4.3, three values for fs have been considered: fs =
1 TeV, fs = 103 TeV and fs = 106 TeV. Eq. (2.19) links
the axion scale fa to the VEV of the radial component of
s, and in consequence fa fs 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 fa come
from the axion coupling to two photons gaγ γ , Eqs. (6.1) and
(6.2): specifying the value of caγ γ for the minimal AMLσ M
charge assignment as reported in Table 2, one gets
fs
3.7 × 108 GeV.
(6.15)
It follows that a QCD axion, consistent with all the present
data, can only be generated in the minimal AMLσ M if the
scale fs , associated to the PQ breaking, is of the order of
108 GeV or larger. As discussed in Ref. [
39
], the
resulting axion falls into the category of the socalled
invisible axions [
36,37,48,49
], as such a large fs scale strongly
suppresses all the couplings with SM fermions and gauge
bosons, preventing any possible detection at colliders or at
lowenergy (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 MLσ M narrows the
range of possible values that the ratio caγ γ /cagg may take:
the minimal AMLσ M presented here provides a very sharp
prediction for this ratio,
(6.16)
caγ γ
cagg
Moreover, in the minimal AMLσ M with fs 108 GeV the
lowenergy theory is not exactly the SM, but the EWSB
mechanism is nonlinearly realised and the Higgs
particle originates as a GB. This model may be confirmed, or
excluded, by a precise measure of caγ γ /cagg 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 lowenergy 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 AMLσ M.
For much lighter values of the fs scale, instead, the
astrophysical bounds on gaγ γ coupling can be satisfied only
assuming that the axion mass and its characteristic scale fs
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 fs , 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 GeV axion
with fs ∼ 200 TeV. The most sensitive observables for this
particle are its couplings with two W ’s, two Z ’s and Z γ , 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 + → π + + a decay is
kinematically forbidden for this axion mass, while the prediction for
the branching ratio of B+ → K + + a is of 10−13, much
below the future expected sensitivity at Belle II [
123
].
8 In the ALP scenario, a solution to the Strong CP problem is not
guaranteed and therefore the condition 4 is not required. An additional
scenario satisfying conditions 1, 2, and 3, can be considered: in this case,
nqL = nψL = nψR = nχR = ntR ± ns = nχL ± ns (with the “+” or “−”
are associated to the presence of the z1 or z˜1 terms in the Lagrangian,
respectively), and the induced renormalisable scalar potential turns out
to be the same as in Eq. (4.1).
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
],
d ≈
104
ci2
MeV
ma
4
fs
GeV
2
 pa 
GeV
m,
(6.17)
where ci are the couplings in Table 2 and the typical
momentum considered is 100 GeV. For the selected benchmark
considered, ma ∼ 1 GeV and fs ∼ 200 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 fs , 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 finetuning problem
The presence of different scales in the scalar potential
leads to a finetuning problem in the model. As already
mentioned, the parameter ξ measures the tension between the
EW scale and the S O(5) SSB scale, as shown in Eq. (4.39).
In models where axions or ALPs are dynamically originated,
a new scale fs 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 √λsφ fs , as can be
read in Eq. (2.21). This leads to f ≈ fs v, or λsφ 1:
this represents two sides of the same finetuning problem.
In the ALP model presented here fs ∼ 200 TeV and
therefore a value of λsφ 10−4 would be necessary to not modify,
excessively, the scale f . In generic AMLσ M, much larger
values for fs are typically necessary to pass the different
experimental bounds on the axion/ALP couplings and then a
much stronger finetuning on λsφ has to be invoked.9
9 In Ref. [
56
], an ALP model in the MLσ M will be presented where
the finetuning problem is solved, but at the price of renouncing to one
of the assumptions listed in Sect. 2.
7 Concluding remarks
The AMLσ M [
39
] represents a class of models that extend
the MLσ M [
9
] by the introduction of a complex scalar
singlet, that allows to supplement the S O(5) and EW
symmetries with an extra U (1)PQ.
The spectrum of the AMLσ 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
S O(5)/S O(4) breaking; iii) two types of vectorial exotic
fermions respectively in the fundamental and in the singlet
representation of S O(5); iv) the PQ GB originated by the
spontaneous breaking of the U (1)PQ symmetry. The scale f
of the S O(5)/S O(4) breaking is expected to be in the TeV
region, in order to solve the Higgs hierarchy problem, while
the PQbreaking scale, fs , 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 S O(5) and PQ explicit breaking terms arising from 1loop
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 AMLσ 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 AMLσ 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 λ and λs
that control the linearity of the EWSB and the PQ
symmetry breaking mechanisms, respectively; the scales f and fs
related to the symmetry breaking; the mixed quartic coupling
λsφ 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, λs 1, keeping free the
scales fs and f either in the perturbative regime, λs 1,
but assuming instead a large hierarchy between the scales,
fs f . Interesting analytical expression for the scalar
sector in the regime fs ∼ f can be obtained also in the limit
β, λsφ 1.
The analytical and numerical analysis of the parameter
space points out that for f, fs 1 TeV the heavier scalar
dofs are unlikely to give signals at the present and future LHC
run, while only the nonlinearity 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 [meV, keV]
and the strong bounds present on the axion coupling to two
photons require that its characteristic scale fa ∼ fs must
be larger than 105 TeV, strongly suppressing all its
interactions. This model represents a minimal invisible axion
construction, where the EWSB mechanism is nonlinearly
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 finetuned. In fact, the typical
S O (5)/S O (4) breaking scale of the effective theory obtained
integrating out the heavy degrees of freedom “naturally runs”
to the highest scale, f R ∼ fs , reintroducing the EW
hierarchy problem, ξ 1. Alternatively, the tuning λsφ = 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 ma = 1 GeV and fs = 200 TeV has been
considered for concreteness. Such an ALP would be free
from the strong bounds on aγ γ and it is likely to be detected
at LHC, the best sensitivity being on the aW W and a Z Z
couplings, while no signals are expected in flavour
observables such as meson decays. Values of fs close to 200 TeV
introduce a mild finetuning 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 aW W and a Z Z couplings (see
Ref. [
56
] for such possibility).
Acknowledgements The authors thank R. Alonso, F. Feruglio, P.
Machado, and A. Nelson for useful discussions, and I. Brivio and B.
Gavela for comments and suggestions on the preliminary version of
the paper. L.M. thanks the department of Physics and Astronomy of
the Università degli Studi di Padova and the Fermilab Theory Division
for hospitality during the writing up of the paper. F.P. and S.R. thank
the University of Washington for hospitality during the writing up of
the paper. The authors acknowledge partial financial support by the
European Union’s Horizon 2020 research and innovation programme
under the Marie SklodowskaCurie Grant agreements No 690575 and
No 674896. L.M. acknowledges partial financial support by the
Spanish MINECO through the “Ramón y Cajal” programme
(RYC201517173), and by the Spanish “Agencia Estatal de Investigación” (AEI)
and the EU “Fondo Europeo de Desarrollo Regional” (FEDER) through
the project FPA201678645P, and through the Centro de excelencia
Severo Ochoa Program under Grant SEV20160597.
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
Generic PQ Transformations
The Lagrangian containing the axion couplings, in the basis
where fermionic terms are shiftsymmetry preserving, can
be written as
La =
∂μa∂ μa +
∂μa
ψ 2 fa
ψ γ μγ 5ψ +
ψ
αs
∂μa
ψ +
ψ
+
χ +
χ
ψ +
ψ
Wμaν W˜ aμν +
6 ψ
2YX2 +2YQ2 +YT25 +6
χ YT21 +
2Y 2 Q + Y 2
X + 2Y 2
B5
Bμν B˜ μν ,
(A.1)
1
2
+
− 8π fa
α2 a
Gaμν G˜ aμν +
#
where Yi are the Hypercharges of the components of ψ and
χ (see Eq. (2.9)) and f ≡ n fL − 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:
ψ +
ψ
+
χ +
χ
Gaμν G˜ aμν +
6 ψ 1 + 2YX2 + 2YQ2 + YT25 + 6 χ YT21 +
6 ψ
1
tan θ W2 + tan2 θW 2YX2 + 2YQ2 + YT25
+ tan2 θW
αem a
− 8π fa
+ 6 ψ
αem a
− 8π fa
#
#
+ 6 χ tan2 θW YT21 +
1
tan2 θW
− 12 χ tan θW YT21 +
+ 6 χ tan2 θW YB21
Zμν Z˜ μν ,
12 ψ
1
αem a
ψ +
ψ
Wμ+ν W˜ −μν
(A.2)
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