# Axion poperties in GUTs

Anne Ernst,a Luca Di Luzio,b a and Carlos Tamaritc
aDeutsches Elektronen-Synchrotron DESY, Notkestr. 85, D-22607 Hamburg,
University
bDipartimento di Fisica dell’Università di Pisa and INFN, Italy
cPhysik Department T70, Technische Universität München, James Franck Straße 1,
D-85748 Garching, Germany
E-mail

###### Abstract:

We summarize recent studies of realistic nonsupersymmetric Grand Unified
Theories (GUTs) extended with a global $U(1)_{\rm PQ}$ symmetry, so as to
accommodate the axion solution to the strong CP problem. Aside from solving
the CP problem and unifying the gauge structure of the SM, these models can
also yield realistic spectra and mixings, including neutrino masses, and
allowing for a consistent cosmological history that accounts for inflation,
dark matter and baryogenesis. In our studies of $SO(10)$ and $SU(5)$ theories,
we determined the mass and couplings of the axion in terms of the relevant
threshold scales, and assessed how the former are constrained from the
requirements of gauge coupling unification, proton decay searches and collider
bounds. The axion mass ends up being rather constrained for GUT scale axions,
particularly in the case of $SU(5)$, and could be probed by upcoming dark
matter experiments, such as ABRACADABRA and CASPEr-Electric.

##  1 Introduction

The Peccei-Quinn (PQ) solution to the strong CP problem [1, 2] has attracted
much attention in recent times. It involves the postulation of additional
scalar or fermionic degrees of freedom, constrained by a global anomalous
$U(1)_{\rm PQ}$ symmetry known as PQ symmetry. This symmetry becomes
spontaneously broken at an unknown scale and the $\theta$ angle of QCD – which
is replaced by a dynamical field in these models – is set to zero dynamically.
This solves the strong CP problem. The dynamical field that replaces the
$\theta$ angle is the axion, the light pseudo-Goldstone boson of the
spontaneously broken colour anomalous symmetry [3, 4]. The axion couples only
weakly to Standard Model (SM) particles and makes for a good Dark Matter (DM)
candidate [5, 6, 7]. In fact, by the so-called “vacuum realignment” mechanism,
axions can be produced in the early universe in large enough numbers to
explain the observed DM abundance.

It has been pointed out that axion models can also provide a natural candidate
for the inflaton field [8], thereby explaining another unsolved problem of the
SM of particle physics. A recent proposal called SMASH [9, 10] (Standard
Model-Axion-Seesaw-Higgs portal inflation) has shown how a simple axion model
can be combined with the introduction of heavy right-handed neutrinos and
thereby also explain both neutrino masses and the baryon asymmetry of the
universe, as well as give a mechanism for inflation. The SMASH model thus
addresses five fundamental problems of particle physics and cosmology. An
important feature of the model is that the scale of PQ breaking is the only
new fundamental scale of the theory and determines the masses of the heavy
particles up to the effects from dimensionless couplings; for example the
right handed neutrinos acquire masses exclusively from Yukawa couplings to the
PQ breaking scalar.

As in most axion models, the PQ breaking scale that determines the axion mass
is a free parameter. In order to constrain it one can take hints from axion
cosmology and the requirement of obtaining the observed relic abundance from
the realignment mechanism. The latter relies on the fact that after the phase
transition in which the PQ symmetry is broken by a vacuum expectation value
(VEV) $v_{\rm PQ}$ of a complex scalar, the phase of the latter (which is
proportional to the axion field) can in principle assume any initial value or
“misalignment angle” $\theta_{i}$ between 0 and $2\pi$. One can show that
after the axion field gets a mass, its oscillations around the minimum
contribute to the stress-energy tensor of the Universe as pressureless DM –an
axion condensate– and the observed number density of axions today depends on
the value of the initial misalignment angle and on the mass of the axion at
zero temperature. Two different scenarios can then be considered: (1) If the
latest phase transition that breaks the PQ symmetry happens after inflation,
then the initial misalignment angle is expected to be randomly distributed
throughout our observable universe and can be replaced by its average. In this
case, a preferred range for the mass of the axion can be obtained. Including
axion production from other effects beyond the misalignment-induced
oscillations of the axion field, the authors of Ref. [11] predict an axion
mass in the range $50\,\mu{\rm eV}<m_{A}<1.5\,{\rm meV}$. This scenario is
usually referred to as the “post-inflationary” PQ breaking scenario. An
alternative is given by (2), the “pre-inflationary” scenario, in which the
latest state of PQ breaking happens already before or during inflation. In
this case, despite a possible initial random distribution of $\theta_{i}$ in
patches of the size of the Hubble scale during PQ breaking, the exponential
expansion of the Universe implies that our current Hubble patch can be traced
to a region with a uniform value of $\theta_{i}$ . In this scenario – also
named the “anthropic window” – one can tune $\theta_{i}$ to accommodate any
axion mass, and isocurvature perturbations may place a bound on the scale of
inflation. The admissible ranges for the axion mass in both scenarios are
indicated in Fig. 1 in lines 5 to 7.

Since the limits on the axion mass coming from cosmology are strongly
dependent on the inflationary model and the post-inflationary thermal history,
we turn to theory in the hope of obtaining predictions on the axion mass. We
have considered extensions of axion models in which the PQ solution is
combined with Grand Unified Theories (GUTs) in order to obtain theoretical
constraints on the axion mass. GUTs are attractive as they provide an elegant
way to explain the seemingly ad-hoc SM gauge structure involving a product of
three groups $SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y}$. In unified models,
the SM gauge group is embedded into a larger group which is spontaneously
broken down to the SM at a certain high scale.

The GUT proposal entails the requirement of gauge coupling unification - i.e.
the condition that the running couplings meet in the combinations defined by
the GUT symmetry. This is a rather strict requirement which excludes the most
minimal $SU(5)$ models [12]. In models with additional particles and/or
additional symmetry breaking scales, the requirement of gauge coupling
unification can be implemented successfully. In Refs. [13, 14], we have
considered a class of non-supersymmetric $SO(10)$ and $SU(5)$ models, endowed
with an additional global $U(1)_{\rm PQ}$ symmetry that commutes with the GUT
group111Note that we are not touching two fundamental issues related to such
constructions, namely the gauge hierarchy between the electroweak and the GUT
scale and the origin/quality of the global $U(1)_{\rm PQ}$. It is not
unconceivable that a light Higgs might be dynamically selected during the
evolution of the universe [15, 16, 17], while relaxing the simplifying
hypothesis that the PQ symmetry commutes with the GUT group might offer new
model-building directions to address the second problem [18].. The choices of
$SU(5)$ and $SO(10)$ can be motivated by minimality either of the unified
gauge group –which favours the rank four $SU(5)$ over the rank five $SO(10)$–
or by minimality of the matter representations, which favour instead $SO(10)$,
for which a full generation of SM particles plus a right-handed neutrino can
be embedded into a single spinorial $\bf 16$ representation of $SO(10)$, which
is automatically anomaly free; this is to be contrasted with the $SU(5)$ case,
in which a SM generation fits into 2 representations, $\bar{\bf 5}$ and $\bf
10$, and one needs additional representations to generate neutrino masses.

Although GUT theories extended with a PQ symmetry have been considered before
in the literature, our analysis offers improvements in the systematic
identification of the physical axion field, which must remain orthogonal to
all the heavy gauge bosons, and the determination of its properties such as
couplings to nucleons and domain-wall number. We have also performed detailed
analyses of the constraints coming from unification, with the aim of studying
whether they can have an impact on the allowed values of the scale of PQ
breaking and the axion mass. Further constraints considered are those from
proton decay, which is unavoidable in GUT theories due to the fact that
unifying quarks and leptons into grand-unified representations implies an
explicit breaking of the accidental baryon number symmetry of the Standard
Model222Current experiments limit the lifetime $\tau=1/\Gamma$ of the proton
in the channel $p\rightarrow\pi^{0}e^{+}$ to larger than $1.6\times 10^{34}$
years [19].. Other bounds accounted for are related to axion-induced black
hole superradiance and stellar-cooling effects.

The final goal of our analysis is the identification of a grand unified
version of Refs. [9, 10] – a GUT SMASH. Such a model should preferably also be
able to solve the mentioned problems of the SM and modern cosmology – neutrino
masses and mixings, the strong CP problem, DM, baryogenesis and inflation.

##  2 Axions in $SO(10)\times U(1)_{\rm PQ}$ theories

###  2.1 The role of the PQ symmetry

As explained in the introduction, the inclusion of the SM fermion
representations is quite straightforward in $SO(10)$ GUT models. The next step
in the model building process leaves more room for speculation: the definition
of the scalar sector of the theory. A hint on the possible choices can be
taken from considering the Yukawa interactions. Due to gauge invariance,
fermion mass terms must come from Yukawa couplings to scalars. To identify the
allowed Yukawa interactions, we consider the tensor product of two fermionic
$\bf 16$ representations:

$\displaystyle\bm{16}\times\bm{16}=\bm{10}+\bm{120}+\bm{126}.$ (1)

The most general Yukawa couplings can therefore be constructed using scalar
fields in the representations $\bm{10}$, $\bm{120}$ and $\overline{\bm{126}}$.
For reasons of minimality however we restrict ourselves to only two distinct
Higgs representations333A single Yukawa matrix can always be diagonalized by
rotating the fermion fields, therefore in order to allow for the observed
mixings of the SM fermions we have to employ at least two distinct Higgs
representations.. We adhere to the most studied option and choose to employ
scalar fields in the $\bm{10}_{H}$ and the $\overline{\bm{126}}_{H}$
representations. As pointed out in [20, 21], the $\bm{10}_{H}$ must be taken
to be complex, since otherwise the phenomenologically unacceptable mass
relations $m_{t}\sim m_{b}$ are predicted. With this choice of scalar sector –
i.e. a complex $\bm{10}_{H}$ and a real $\bm{126}_{H}$ the $SO(10)$ symmetric
Yukawa Lagrangian is given by

$\mathcal{L}_{Y}=\bm{16}_{F}\left(Y_{10}\bm{10}_{H}+\tilde{Y}_{10}\bm{10}_{H}^{\ast}+Y_{126}\overline{\bm{126}}_{H}\right)\bm{16}_{F}+{\rm
h.c.}.$ (2)

After assigning vacuum expectation values (VEVs) to the included scalars in a
meaningful way – i.e. such that the electroweak symmetry breaking can be
reproduced and such that the neutrino masses are generated by the see-saw
mechanism – one can obtain formulae relating the fermion masses and mixings to
the Yukawa matrices and VEVs. The predictive power of the model is weak, since
it still contains three different Yukawa couplings (cf. Eq. (2)). This
motivated the authors of Ref. [21] to impose a PQ symmetry, under which the
fields transform as

$\displaystyle\bm{16}_{F}$ $\displaystyle\rightarrow\bm{16}_{F}e^{i\alpha},$
$\displaystyle\bm{10}_{H}$ $\displaystyle\rightarrow\bm{10}_{H}e^{-2i\alpha},$
(3) $\displaystyle\overline{\bm{126}}_{H}$
$\displaystyle\rightarrow\overline{\bm{126}}_{H}e^{-2i\alpha}\,,$

which forbids the coupling $\tilde{Y}_{10}$ in (2) (see also Ref. [20]). This
is how in many GUT models, a PQ symmetry is invoked without a reference to the
strong CP problem!

###  2.2 Definition of the models

All $SO(10)$ models considered here have in common that they employ fermionic
representations in the $\bm{16}_{F}$ and scalar representations in the
$\bm{10}_{H}$ and the $\overline{\bm{126}}_{H}$, which transform as given in
(3) under a global PQ symmetry. This scalar particle content however is not
sufficient to break the GUT group down to the SM gauge group, and we must
employ at least one additional scalar field. We choose444A more minimal option
would be given by a (real) $\bm{45}_{H}$. This representation was doomed in
the early 80’s [22, 23, 24] due to the emergence of tachyons in the tree-level
scalar spectrum, arising for the phenomenologically relevant non-$SU(5)$-like
breaking patterns required by gauge coupling unification [25, 26]. However, it
was recently shown [27, 28, 29, 30] that the tachyonic instabilities can be
removed at the one-loop level, thus re-opening for exploration the most
minimal $SO(10)$ Higgs sector.  the $\bm{210}_{H}$ [13], which contains a
singlet under the Pati-Salam group $SU(4)_{C}\times SU(2)_{L}\times
SU(2)_{R}$, such that our symmetry breaking chain is

$\displaystyle SO(10)$ $\displaystyle\stackrel{{\scriptstyle
v^{210}-210_{H}}}{{\longrightarrow}}$ $\displaystyle 4_{C}\,2_{L}\,2_{R}\
\stackrel{{\scriptstyle
v_{R}-\overline{126}_{H}}}{{\longrightarrow}}3_{C}\,2_{L}\,1_{Y}\
\stackrel{{\scriptstyle v_{u,d}^{10,126}-10_{H}}}{{\longrightarrow}}\
3_{C}\,1_{\rm em}\,.$ (4)

The symmetry breaking VEVs indicated above the arrows are constrained by the
requirement of gauge coupling unification.

We have not fixed yet the PQ charge of this additional scalar. The most
minimal option might be to leave the $210_{H}$ a PQ singlet. A more careful
analysis of such a model (see also [31]) results however in an axion decay
constant at the electroweak scale – i.e. the resulting model contains a
visible axion which is already excluded by experiments. We must therefore
construct models in which the axion decay constant is lifted from the
electroweak scale. This can be achieved by ensuring that more than one of the
scalar multiplets acquiring large VEVs are charged under PQ, for example by
(1) extending the PQ symmetry to all existing multiplets, (2) postulating an
additional multiplet and extending the PQ symmetry to it or (3) postulating an
additional singlet and extending the PQ symmetry to include it. We consider
models of all three types, where the additional multiplet in models of the
second type transforms as a $\bm{45}_{H}$ representation. The PQ charges of
these models are summarized in table 1. The original versions of these models
are variations of the original DFSZ [7, 32] axion model and have a non-trivial
domain wall number $N_{\rm DW}$ of 3.555In Ref. [13] we clarify how the domain
wall number corresponds to the dimension of the finite group that arises from
the translational symmetry of the axion-Goldstone after the explicit breaking
of the anomaly, and after modding out by unphysical rotations that leave the
scalar fields invariant, as well as by transformations in the center of the
gauge group. We also show how the domain-wall number can be computed entirely
in terms of the charges of the scalars for the physical PQ symmetry that gives
an axion that does not mix with the heavy gauge bosons . The table also
indicates variations of these models in which the domain wall number is
reduced to 1 by including exotic fermions.

Let us briefly summarize the defining properties of each model before moving
on to the computed constraints:

#### Model 1

The $\bm{210}_{H}$ – being the only multiplet that had no charge under
$U(1)_{\rm PQ}$ – is assigned a PQ charge. Its VEV then sets the scale of the
axion decay constant, which is inversely proportional to the axion mass.
Moreover, the VEV of the $\bm{210}_{H}$ is also the one that breaks the GUT
symmetry and is therefore constrained by the requirement of gauge coupling
unification and by proton decay searches. From the above it follows that this
model features an axion decay constant at the unification scale.

#### Models 2.1 and 2.2

Both models feature an additional multiplet in the $\bm{45}_{H}$. This choice
is interesting as it allows for an additional step in the symmetry breaking
chain:

$\displaystyle SO(10)$ $\displaystyle\stackrel{{\scriptstyle M_{\rm
U}-210_{H}}}{{\longrightarrow}}4_{C}\,2_{L}\,2_{R}\,\stackrel{{\scriptstyle
M_{\rm PQ}-4{\bf
5}_{H}}}{{\longrightarrow}}4_{C}\,2_{L}\,1_{R}\,\stackrel{{\scriptstyle M_{\rm
BL}-126_{H}}}{{\longrightarrow}}3_{C}\,2_{L}\,1_{Y}\,\stackrel{{\scriptstyle
M_{Z}-10_{H}}}{{\longrightarrow}}\ 3_{C}\,1_{\rm em}.$ (5)

The scales of symmetry breaking and the responsible multiplets are indicated
above the arrows. This breaking chain occurs only if $M_{\rm PQ}>M_{\rm BL}$,
otherwise the relevant breaking chain is given by (4). The axion decay
constant is given by an intermediate scale $M_{\rm PQ}$ which is constrained
by the gauge coupling unification requirement.

Model 2.2 also features two exotic fermion representations in the
$\bm{10}_{F}$, which ensure a domain-wall number equal to one.

#### Models 3.1 and 3.2

The least constraining extension of our original model is obtained by
including a gauge singlet charged under the PQ symmetry. This singlet is then
not constrained by gauge coupling unification and can take any VEV. In these
models the axion decay constant is not constrained by theoretical
considerations. Model 3.2 is a variation featuring two exotic fermion
representations in the $\bm{10}_{F}$, again giving $N_{DW}=1$.

$\displaystyle\begin{array}[]{ l c c c c c c c c
}\hline\cr\hline\cr&\bm{16}_{F}&\overline{\bm{126}}_{H}&\bm{10}_{H}&\bm{210}_{H}&\bm{45}_{H}&S&\bm{10}_{F}&N_{\rm
DW}\\\ \hline\cr{\bf Model\ 1}&1&-2&-2&4&-&-&-&3\\\ \hline\cr{\bf Model\
2.1}&1&-2&-2&0&4&-&-&3\\\ \hline\cr{\bf Model\ 2.2}&1&-2&-2&0&4&-&-2&1\\\
\hline\cr{\bf Model\ 3.1}&1&-2&-2&0&-&4&-&3\\\ \hline\cr{\bf Model\
3.2}&1&-2&-2&0&-&4&-2&1\\\ \hline\cr\hline\cr\end{array}$ Table 1: Field
content, PQ charge assignments, and resulting domain wall number $N_{\rm DW}$
in the various $SO(10)\times U(1)_{\rm PQ}$ models considered in [13].

###  2.3 Results

We have analyzed each of the above model for the constraints put on the axion
mass by the requirement of gauge coupling unification and experimental limits
[13]. We have made a two-loop analysis, which requires the inclusion of one-
loop threshold corrections. The threshold corrections depend strongly on the
masses of the heavy scalars that need to be integrated out at their
corresponding scales. In lack of a detailed knowledge of the rather involved
scalar sector we have assumed the so-called “extended survival hypothesis”
[33, 34] – yet modified to allow for two light Higgs doublets at low scales –
and scanned over randomized values for the scalar masses. These masses are
assumed to be in the range $[\frac{1}{10\,}M_{T},10\,M_{T}]$, where $M_{T}$ is
a threshold, i.e. a symmetry breaking scale. The possibility of rather large
threshold corrections is the reason for the wide axion mass ranges predicted
in our analysis.

Figure 1:  Possible ranges of the axion mass and decay constant consistent
with gauge coupling unification in our four models. Regions in black are
excluded by constraints from black hole superradiance [35], regions in dark
blue by proton stability constraints, and areas in red by LHC constraints. We
have also included the much more constrained prediction for the axion mass
range in a minimal $SU(5)\times U(1)_{\rm PQ}$ model described in section 3.2
[14], denoting in light blue the region in which the proton decay bounds can
be circumvented with appropriate tunings. Regions in grey are excluded by
stellar cooling constraints from horizontal branch stars in globular clusters
[36]. For comparison, we show also the mass regions preferred by axion DM
(lines 6 to 8), cf. [37]. Here, the dark regions indicate the ranges where the
axion can make up the main part of the observed DM with a tuned misalignment
angle in the pre-inflationary PQ breaking scenario. In the light regions,
axions could still be DM, but not the dominant part. The remaining regions are
not allowed - axions in this mass range would be overabundant. Note that the
region in the $N_{\rm DW}=3$ case has been derived under the assumption that
the PQ symmetry is protected by a discrete symmetry, so that Planck scale
suppressed PQ violating operators are allowed at dimension 10 or higher [38].
In the last two lines the projected sensitivities of various experiments are
indicated [39, 40, 41, 42, 43, 44, 45, 46]. A similar plot was already
published in Ref. [13].

We have summarized our results in Fig. 1. The allowed ranges of the axion mass
and decay constant are plotted in orange for each of our models. We also
compare our results to the reaches of various axion experiment and also to the
permitted mass ranges in the various inflationary scenarios.

#### Model 1

The most constraining – and also the most minimal – of our $SO(10)$ models is
Model 1. The axion mass is suppressed by the GUT scale, and we predict an
axion in the range

$1.9\times 10^{-11}\,\mathrm{eV}<m_{A}<2.2\times 10^{-9}\,\mathrm{eV}.$ (6)

As indicated in Fig. 1, this model only allows for the pre-inflationary PQ
breaking scenario, since the PQ and GUT breaking scale are tightly connected.
In order for axion cold DM not to become overabundant, the initial value of
the axion field in the causally connected patch which contains the present
universe had to be small,
$10^{-3}\lesssim|\theta_{i}|=|A(t_{i})/f_{A}|\lesssim 10^{-2}$ [11].

Remarkably, the predicted axion mass range of Model 1 will be probed in the
next decade by the CASPEr-Electric experiment [39], cf. Fig. 2, which aims to
probe the axion-induced electric dipole moment (EDM) of the nucleon,
$d_{N}(t)=g_{d}\,\frac{\sqrt{2\rho_{\text{DM}}}}{m_{A}}\cos(m_{A}\,t)$ [47],
where $g_{d}$ is the model-independent coupling of the axion to the EDM
operator
$\mathcal{L}_{A}\supset-\frac{i}{2}g_{d}\,A\,\overline{\Psi}_{N}\sigma_{\mu\nu}\gamma_{5}\Psi_{N}F^{\mu\nu}$
of the nucleon and $\rho_{\rm DM}=0.3\,\text{GeV}/\text{cm}^{3}$ is the local
density of axion DM. If successful and interpreted in terms of Model 1, one
may translate the measurement of the axion mass into an indirect determination
of the mass of the heaviest gauge bosons, i.e. the unification scale,

$\displaystyle M_{\mathrm{U}}\simeq 3g_{U}\sqrt{\chi}/m_{A},$ (7)

where $\chi$ is the topological susceptibility in QCD,
$\chi=[75.6(1.8)(0.9){\rm MeV}]^{4}$ [48, 11] and $g_{U}$ is the unified gauge
coupling at the unification scale. The unification scale can be probed
complementarily by the next generation of experiments looking for signatures
of proton decay, such as Hyper-Kamiokande [49] or DUNE [50].

Figure 2:  Plot adapted from Ref. [39]. Experimental reach of CASPEr-Electric.
The QCD axion is indicated by a band in the parameter space, and the range
predicted by Model 1 is indicated in orange and light green. Black hole
superradiance limits are drawn again in black, and the range excluded by
proton decay limits is indicated in dark green, to avoid confusion with the
ADMX sensitivity range painted in blue. The small light green region labeled
HK indicates the predicted range within the sensitivity of the Hyper-
Kamiokande experiment after 10 years of data collection. The orange band gives
the predicted region outside the reach of Hyper-Kamiokande. The light orange,
red, and maroon regions demonstrate the predicted sensitivity of the CASPEr-
Electric experiment in the phases I-III as explained in Ref. [39]. Phase III
will be able to reach the $SO(10)$ \- GUT axion as predicted by our Model 1.

#### Other models

In Models 2.1 and 2.2, the requirement of gauge coupling unification
constrains the intermediate scale which sets the axion decay constant in these
models if the PQ breaking is correlated with the breaking of a gauge group.
This however is only the case if the PQ breaking scale is larger than the
scale at which the B-L symmetry is broken. Thus, no lower limit on $f_{A}$ can
be derived in these models. An upper bound is given by $f_{A}<1.5\times
10^{13}{\rm GeV}$. In Model 3, no constraints on the axion decay constant can
be derived.

##  3 A predictive $SU(5)\times U(1)_{\rm PQ}$ model

Motivated by the quest for a more predictive axion-GUT framework, we turn to
$SU(5)$ grand unification. The simplest extension of non-supersymmetric
$SU(5)$ to accommodate the axion was proposed by Wise, Georgi and Glashow
(WGG) [51] (see also [52]). Nevertheless, the WGG model is ruled out because
of unsuccessful gauge coupling unification and zero neutrino masses, much like
the original $SU(5)$ model of Georgi and Glashow (GG) [12]. A solution to
these issues in the GG model was proposed in Refs. [53, 54], relying on
extending the field content with one fermion multiplet ${\bf 24}_{F}$ in the
adjoint of $SU(5)$. The latter allows to give rise to neutrino masses through
a combined Type-I+III seesaw mechanism, and can also lead to successful
unification. A thorough renormalization group analysis [53, 54, 55] leads to
the prediction of a sharp correlation between the masses of light electroweak
triplets (which may be probed at the Large Hadron Collider (LHC)) and the
scale of unification, which is constrained by the proton-decay bounds from
Super-Kamiokande (SK)).

In Ref. [14] we have extended the WGG model with a ${\bf 24}_{F}$, in analogy
to the GG+${\bf 24}_{F}$ case666From the point of view of the GG+${\bf
24}_{F}$ model, the PQ extension is also motivated by the absence of a DM
candidate.. Next we will give a succinct review of the WWG model and its
+${\bf 24}_{F}$ extension, focusing in particular on the connection of the
axion mass with the GUT scale. Combining constraints from proton stability,
LHC and unification we obtain the axion mass prediction:
$m_{A}\in[4.8,6.6(330)]$ neV, where the number in parenthesis corresponds to
the case of a tuned flavour structure in the operators that mediate proton
decay.

###  3.1 The Wise-Georgi-Glashow model

Here we review the main characteristics of the WGG model [51]. As in the
original GG model [12], the SM fermions fit into three copies of $\bar{\bf
5}_{F}$ and ${\bf 10}_{F}$, and the scalar sector includes a complex ${\bf
24}_{H}$ and two fundamentals, ${\bf 5}_{H}$ and ${\bf 5}^{\prime}_{H}$. The
Yukawa Lagrangian is777It should be noted that obtaining the correct ratio
between the masses of down quarks and charged leptons requires the addition of
nonrenormalizable operators or new scalar representations.

$\mathcal{L}_{Y}=\bar{\bf 5}_{F}{\bf 10}_{F}5^{\prime*}_{H}+{\bf 10}_{F}{\bf
10}_{F}{\bf 5}_{H}\,,$ (8)

whereas the scalar potential contains the following interactions transforming
nontrivially under global rephasings,

$V_{H}\supset{\bf 5}^{\prime{\dagger}}_{H}{\bf 24}^{2}_{H}{\bf 5}_{H}+{\bf
5}^{\prime{\dagger}}_{H}{\bf 5}_{H}\mbox{Tr}\,({\bf 24}^{2}_{H})\,,$ (9)

which are similar to those in DFSZ models [32, 56]. It turns out that the
terms in the previous equation, and the entire Lagrangian, are invariant under
the following $U(1)_{\rm PQ}$ symmetry: $\bar{\bf 5}_{F}\to
e^{-i\alpha/2}\bar{\bf 5}_{F}$, ${\bf 10}_{F}\to e^{-i\alpha/2}{\bf 10}_{F}$,
${\bf 5}_{H}\to e^{i\alpha}{\bf 5}_{H}$, ${\bf 5}^{\prime}_{H}\to
e^{-i\alpha}{\bf 5}^{\prime}_{H}$ and ${\bf 24}_{H}\to e^{-i\alpha}{\bf
24}_{H}$.

The GUT gauge group is broken down to the SM group by means of a VEV in the
${\bf 24}_{H}$. Representing the adjoint field by a $5\times 5$ matrix, the
appropriate VEV, the associated axion excitation and the axion decay constant
$f_{A}$ are given by

$\left\langle{\bf
24}_{H}\right\rangle=V\tfrac{1}{\sqrt{30}}\,\text{diag}(2,2,2,-3,-3),\quad{\bf
24}_{H}\supset\left\langle{\bf
24}_{H}\right\rangle\tfrac{1}{\sqrt{2}}e^{i\,A/V},\quad
f_{A}=\frac{V}{\hat{N}},$ (10)

with $\hat{N}$ the anomaly coefficient of the $U(1)_{\rm PQ}$ symmetry under
$SU(3)_{C}$ – equal to 6 in the WGG+${\bf 24}_{F}$ model. $V$ fixes the mass
of the heavy leptoquark vector bosons, which are also proportional to the
$SU(5)$ gauge coupling $g_{5}$:

$m_{V}=\sqrt{\tfrac{5}{6}}g_{5}V=\sqrt{\tfrac{5}{6}}\hat{N}g_{5}f_{A}\,.$ (11)

Crucially, the heavy gauge boson mass, which is constrained by proton decay
experiments, is connected to the axion decay constant, which is delimited by
axion searches because it fixes the axion mass. Both types of experiments are
complementary probes of the model and there is a direct connection between the
axion mass and the proton decay rate. To see this connection explicitly we may
start from the following formula of the proton decay rate in the
$p\to\pi^{0}e^{+}$ channel, obtained within chiral perturbation theory and
recasted for $SU(5)$ [57, 58]:

$\displaystyle\Gamma_{p\to\pi^{0}e^{+}}=\,\frac{m_{p}}{16\pi
f_{\pi}^{2}}A^{2}_{L}\left|\alpha\right|^{2}(1+D+F)^{2}\left(\frac{g_{5}^{2}}{2m^{2}_{V}}\right)^{2}\left[4A^{2}_{SL}+A^{2}_{SR}\right],$
(12) $\displaystyle\begin{array}[]{c c c}m_{p}=938.3\text{
MeV},&f_{\pi}=139\text{ MeV},&\alpha=-0.011\text{ GeV}^{3},\\\
A_{L}=1.25,&D=0.81,&F=0.44.\end{array}$

The previous formula ignores fermion mixing effects. $A_{L}$ includes effects
from renormalization between the proton mass and the electroweak scales, while
$A_{SL(R)}$ are renormalization factors from the electroweak to the GUT scale
[59, 60, 29]. These depend on intermediate thresholds, and for SM running up
to $10^{15}$ GeV one has $A_{SL(R)}=2.4\,(2.2)$. The rest of the constants in
(12) are phenomenological parameters of the chiral Lagrangian. Substituting
the gauge boson mass (11) and using the relation
$m_{A}=5.7\,\text{neV}\,(10^{15}\,\text{GeV}/f_{A})$ [48, 11] one obtains:

$\Gamma_{p\to\pi^{0}e^{+}}\simeq\left(1.6\times 10^{34}\
\text{yr}\right)^{-1}\left(\frac{m_{A}}{3.7\
\text{neV}}\right)^{4}\left(\frac{6}{\hat{N}}\right)^{4}\left[0.83\left(\frac{A_{SL}}{2.4}\right)^{2}+0.17\left(\frac{A_{SR}}{2.2}\right)^{2}\right]\,,$
(13)

where the first factor is the current proton decay bound from SK [19].

###  3.2 Axion mass prediction in Wise-Georgi-Glashow$+{\bf 24}_{F}$ model

In the presence of the additional fermion representation in the ${\bf
24}_{F}$, the Yukawa Lagrangian is augmented by

$\Delta\mathcal{L}_{Y}=\bar{\bf 5}_{F}{\bf 24}_{F}{\bf 5}_{H}+\mbox{Tr}\,{\bf
24}^{2}_{F}{\bf 24}^{*}_{H}\,.$ (14)

The first term generates Yukawa interactions for the SM triplet and singlets
within the ${\bf 24}_{F}$, and the second interaction gives a Majorana mass
for the full multiplet after the breaking of $SU(5)$. One also needs
nonrenormalizable operators to get at least two distinct light neutrino
masses, and to split the masses of the ${\bf 24}_{F}$ sub-multiplets [53, 54,
55], but for our purposes it suffices to note that (14) fixes the PQ charge of
the ${\bf 24}_{F}$ at $-1/2$, which in turn gives an anomaly coefficient of
$\hat{N}=11$ for the PQ symmetry under $SU(3)_{C}$.

Unification constraints are rather strict, and allow to narrow down the axion
mass range due to the connection between the axion mass and the unification
scale. In the SM, unification fails because the electroweak couplings
$\alpha_{1}$ and $\alpha_{2}$ meet at a scale around $10^{13}$ GeV, which is
ruled out by proton decay bounds. Thus, viable unification requires matter
multiplets that can delay the meeting of $\alpha_{1}$ and $\alpha_{2}$, mainly
by making $\alpha_{2}$ increase faster for a growing scale (additional
multiplets with nonzero hypercharge will make $\alpha_{1}$ increase faster and
worsen the problem). In the WGG+${\bf 24}_{F}$ model, each $\bf 24$
representation (fermion or scalar) contains the following matter multiplets of
the SM group $SU(3)\times SU(2)_{L}\times U(1)_{Y}$:

$\displaystyle{\bf 24}=({\bf 8},{\bf 1},0)+({\bf 3},{\bf
2},-5/6)+({\bf{\bar{3}}},{\bf 2},5/6)+({\bf 1},{\bf 3},0)+({\bf 1},{\bf
1},0).$ (15)

Then the appropriate saviour role can be played by the $SU(2)$ triplet
$T_{F}=({\bf 1},{\bf 3},0)\subset{\bf 24}_{F}$, and similarly for the triplet
scalar $T_{H}=({\bf 1},{\bf 3},0)$ within the ${\bf 24}_{H}$. Both triplets
are predicted to be near the weak scale, so as to maximize the effect on the
running of $\alpha_{2}$, and could potentially be probed at the LHC. The
fermion triplet can give rise to lepton-number violation, same sign di-lepton
events [61]; current CMS bounds for $m_{T_{F}}$ are near $840$ GeV [62], while
High Luminosity LHC (HL-LHC) projections can reach up to 2 TeV [63, 64]. In
turn, scalar triplets can impact the di-photon Higgs signal, yet the effect is
model-dependent and smaller to that of the fermionic triplet [65].

Finally, requiring also the convergence of $\alpha_{3}$ with $\alpha_{1}$ and
$\alpha_{2}$ demands heavier colored particles, which can be found in the the
$({\bf 8},{\bf 1},0)$ color-octet scalars and fermions within the ${\bf
24}_{F,H}$. Their masses are required to be around $10^{8}$ GeV, far from the
LHC reach.

We have carried out a gauge coupling unification analysis following [55],
accounting for the 3-loop beta functions induced by the scalar and fermion
triplets, and the leading NNLO corrections from the 2-loop threshold effects.
Including the constraints from the present LHC bounds and SK the resulting
axion mass window is

$m_{A}\in[4.8,6.6]\ \text{neV}\,.$ (16)

The upper bound can be relaxed when allowing for cancellations in the flavour
structure of proton decay operators [66, 67], which can be used to tune to
zero the proton decay rate in a number of channels. Unitarity of the mixing
matrices however implies that some channels will always survive, and the
weakest constraint is obtained when only decays to strange mesons are allowed
[66]. This gives $\tau/\mathcal{B}(p\to K^{0}\mu^{+})>1.3\times 10^{33}$ yr
[68], which implies an absolute upper bound for the axion mass of $m_{A}<330$
neV. Concerning future experiments, the projected sensitivity of HK [69] in
the $p\to K^{+}\bar{\nu}$ channel could rule out $m_{A}>160$ neV.

Aside from studying unification constraints and their effect on axion masses,
we calculated the axion couplings to SM particles and studied the reach of
axion experiments probing the coupling to photons or nucleons. The ABRACADABRA
experiment [42] is posed to reach enough sensitivity to probe the axion photon
coupling $g_{A\gamma}$, defined by ${\cal
L}_{A}\supset\frac{1}{4}g_{A\gamma}\,a\,F_{\mu\nu}{\tilde{F}}^{\mu\nu}$, in
the relevant mass region. We show this in the left panel of Fig. 3, from which
we conclude that the late stages of the experiment can test the whole
parameter space of the WGG+${\bf 24}_{F}$ model, with the tuned region
included. In the right panel of Fig. 3 we display the projected sensitivity of
CASPEr-Electric [39, 70]. We took the non–perturbative estimate of the axion
coupling $g_{AD}$ to the EDM operator of the nucleon from Ref. [71] and the
QCD axion band in the figure shows the theoretical error. Phase III of CASPEr-
Electric could probe the preferred axion mass window (16). A dedicated
measurement time focused on the preferred mass region could give a factor of
three improvement in the reach, since the sensitivity in $g_{AD}$ improves
with the scanning time as $t^{1/4}$. We denote this optimal reach with a
short, full blue line in the right panel of Fig. 3.

Figure 3:  Left panel: Axion coupling to photons, $g_{A\gamma}$, against the
axion mass $m_{A}$. The blue areas give the estimated sensitivities of the
broadband (“Broad”) and resonant (“Res.”) search modes of ABRACADABRA, taken
from Ref. [42]. Right panel: Axion coupling $g_{AD}$ to the EDM operator of
the nucleon against the axion mass $m_{A}$. The blue areas show the estimated
sensitivities of CASPEr-Electric, taken from Ref. [70]. The short, full blue
line corresponds to a factor of three enhancement of the sensitivity for an
optimized search focused on the favoured mass region. Both figures have been
published in Ref. [14].

##  4 Conclusions

The axion is a well-motivated hypothetical particle that is not only part of a
mechanism that solves the strong CP problem, but is also able to explain the
DM abundance of the universe. As a consequence, a comprehensive experimental
program has developed that could probe large swaths of the parameter space
predicted by the QCD axion in the coming decade. Nevertheless, this is a vast
parameter space, as the PQ breaking scale that determines the axion mass and
its couplings to SM particles is typically a free parameter. For these reasons
it becomes interesting to look for possible theoretical constraints that may
motivate specific choices of parameters for which experiments or scanning
strategies could be optimized.

In our recent work [13, 14], we investigated whether embedding the axion into
Grand Unified Theories may provide such theoretical constraints. With GUTs
themselves offering a compelling –and constrained– unified framework for the
gauge structure of the SM, while at the same time allowing for consistent
cosmological histories, the study of the interplay between axion physics and
the requirements from unification is well motivated. In our research we
studied the properties of the axion in $SO(10)$ and $SU(5)$ theories extended
with an anomalous global $U(1)_{\rm PQ}$ symmetry. We calculated the axion
mass in terms of the threshold scales, and obtained the axion couplings to the
SM particles. A detailed renormalization group analysis allowed us to assess
the impact of the constraints for unification in the axion mass, for which we
also accounted from bounds from proton decay and collider searches.

In those $SO(10)$ and $SU(5)$ models, in which all the Higgs multiplets
involved in the breaking towards the SM carried PQ charges, the PQ breaking
scale and thus the axion mass is rather constrained by gauge coupling
unification and can be probed by axion DM experiments, such as ABRACADABRA and
CASPEr-Electric. Moreover, the relation between the GUT and PQ scale implies
that the axion can be constrained as well by proton decay searches. The
$SU(5)$ WGG+${\bf 24}_{F}$ model is particularly predictive and gives an
allowed window of $m_{a}\in[4.8,6.6(330)]$ neV, with the upper limit depending
on whether one allows for tuning in the fermion mixing matrices.
Interestingly, in this case unification demands light electroweak triplets and
the axion mass can also be probed indirectly through collider searches, so
that the narrow mass window could be fully explored by both HL-LHC and HK,
aside from the direct DM searches.

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