aainstitutetext: Physics Department and Center for Education and Research in
Cosmology and Astrophysics (CERCA), Case Western Reserve University,
Cleveland, OH 44106, USAbbinstitutetext: Departamento de Física Teórica, IFIC,
Universitat de Valencia-CSIC, E-46071, Valencia, Spain

# The QCD Axion and Unification

Pavel Fileviez Pérez b Clara Murgui a and Alexis D. Plascencia
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The QCD axion is one of the most appealing candidates for the dark matter in
the Universe. In this article, we discuss the possibility to predict the axion
mass in the context of a simple renormalizable grand unified theory where the
Peccei-Quinn scale is determined by the unification scale. In this framework,
the axion mass is predicted to be in the range $m_{a}\simeq(3-13)\times
10^{-9}\ \rm{eV}$. We study the axion phenomenology and find that the
ABRACADABRA and CASPEr-Electric experiments will be able to fully probe this
mass window.

##  1 Introduction

The need for cold dark matter to explain the features of our Universe is well
established today. The QCD axion Peccei:1977hh ; Wilczek:1977pj ;
Weinberg:1977ma  is one of the most popular candidates for the cold dark
matter in the Universe Preskill:1982cy ; Abbott:1982af ; Dine:1982ah , which
was introduced to solve the so-called strong CP problem. For reviews about
axions see Refs. Raffelt:1990yz ; Dine:2000cj ; Sikivie:2006ni ; Kim:2008hd ;
Jaeckel:2010ni ; Graham:2015ouw ; Irastorza:2018dyq . Using the Peccei-Quinn
(PQ) mechanism Peccei:1977hh  and after QCD confinement the axion mass can be
written as Weinberg:1977ma

$m_{a}^{2}\simeq\frac{m_{u}m_{d}}{(m_{u}+m_{d})^{2}}\frac{m_{\pi}^{2}f_{\pi}^{2}}{f_{a}^{2}},$

where $f_{\pi}$ is the pion constant, $m_{\pi}$ is the pion mass, $m_{u}$ is
the mass of the up quark, $m_{d}$ the mass of the down quark and $f_{a}$ is
the Peccei-Quinn breaking scale. This relation implies that the axion mass can
only be predicted if the scale $f_{a}$ can be computed. See Ref.
diCortona:2015ldu ; Gorghetto:2018ocs  for an improved relation between the
axion mass and $f_{a}$.

There are two simple mechanisms where the interaction between the axion and
the gluons can be understood. Firstly, there is the KSVZ mechanism Kim:1979if
; Shifman:1979if  in which the axion is the imaginary part of the scalar field
that couples to new vector-like quarks, and once these heavy quarks are
integrated out, the coupling between the axion and the gluons $a(x)G\tilde{G}$
is generated. Secondly, there is the DFSZ mechanism Zhitnitsky:1980tq ;
Dine:1981rt  where there are two Higgs doublets which couple to the Standard
Model (SM) fermions and a complex scalar singlet. In this scenario the axion
is mainly the imaginary part of the singlet, it couples to the SM fermions
through scalar mixing, and the $a(x)G\tilde{G}$ coupling is generated after
integrating out the SM quarks. However, these simple models do not predict the
PQ scale, and hence, the axion mass and its interactions with the SM gauge
bosons and fermions cannot be determined.

In this work, we study the idea of predicting the PQ scale in the context of
grand unified theories (GUTs) where the unification of the Standard Model
gauge interactions can be understood. In this context, if the same field
breaking the GUT symmetry contains the axion, the PQ and unification scales
will be related. This idea was first pointed out by M. B. Wise, H. Georgi and
S. Glashow in Ref. Wise:1981ry . In this article, we investigate this idea and
the predictions in the context of a simple renormalizable grand unified theory
where the PQ and seesaw scales are determined by the unification scale. We
focus on the scenario where the PQ symmetry is broken before inflation.

We discuss the implementation of the Peccei-Quinn mechanism in the context of
grand unified theories and show that in a simple renormalizable $\,{\rm
SU}(5)$ theory the KSVZ mechanism can be implemented. We find that in the
context of Adjoint $\,{\rm SU}(5)$ Perez:2007rm , the PQ and seesaw scales are
determined by the unification scale; also, a realistic framework for all
fermion masses including neutrino masses can be achieved. In this theory, the
PQ scale is given by

$f_{a}=\frac{1}{10}\sqrt{\frac{6}{5\pi\alpha_{\rm GUT}}}M_{\rm GUT},$

where $M_{\rm GUT}$ is the unification scale and $\alpha_{\rm GUT}$ is the
value of the gauge couplings at the unification scale. Once we take into
account experimental constraints on the proton lifetime and LHC constraints,
the axion mass is predicted in the range

$m_{a}\simeq(3-13)\times 10^{-9}\ \rm{eV}.$

We discuss in detail the testability of this theory at current and future
axion experiments, and investigate the implications for collider and proton
decay experiments. We find that the ABRACADABRA Kahn:2016aff  and CASPEr-
Electric Budker:2013hfa  axion experiments will be able to fully probe the
predicted mass window.

This article is organized as follows, we discuss the implementation of the PQ
mechanism in grand unified theories in section 2. In section 3, we show how to
implement the PQ mechanism in the context of Adjoint $\,{\rm SU}(5)$, we
discuss the unification of gauge interactions, the constraints from proton
decay experiments, the predictions for the axion mass and the predictions for
different axion experiments. Our main results are summarized in section 4.

##  2 Predicting the PQ Scale and Unification

To investigate the relation between the PQ and unification scales we will
focus on minimal unified theories which are based on the $\,{\rm SU}(5)$ gauge
symmetry. In the minimal $\,{\rm SU}(5)$ theory proposed by H. Georgi and S.
Glashow Georgi:1974sy  the SM matter is unified in ${\bf{\bar{5}}}$ and
${\bf{10}}$, the Higgs sector is composed of ${\bf{5_{H}}}$ and
${\bf{24_{H}}}$, and all the gauge fields live in ${\bf{24_{V}}}$. It is well-
known that this theory is not realistic because: a) the values of the gauge
couplings at the low scale cannot be reproduced, b) it predicts the relation
$Y_{e}=Y_{d}^{T}$, where $Y_{e}$ and $Y_{d}$ are the Yukawa matrices for
charged leptons and down quarks, respectively. Last but not least, c)
neutrinos are massless as in the SM. In order to have a consistent relation
between the fermion masses, $Y_{e}\neq Y_{d}^{T}$, a Higgs in the
${\bf{45_{H}}}$ representation can be introduced Georgi:1979df . In this
scenario, both the ${\bf{5_{H}}}$ and the ${\bf{45_{H}}}$ must couple to
matter in the same manner. This Higgs sector can provide the relevant degrees
of freedom which allow us to have gauge coupling unification in agreement with
the experimental values of the gauge couplings. For a recent study see Ref.
Perez:2016qbo .

In order to predict the PQ scale from unification, the same field that breaks
the grand unifying group must simultaneously break the PQ symmetry. In $\,{\rm
SU}(5)$ the symmetry is broken once the ${\bf{24_{H}}}$ acquires a vacuum
expectation value (VEV); when this field is charged under $\,{\rm U}(1)_{\rm
PQ}$ it becomes a complex field and the axion will be contained in the
imaginary part of the singlet inside the ${\bf{24_{H}}}$. This simple idea is
very appealing because it allows us to write down a simple GUT model with a
dark matter candidate as well. In the previous discussion, we have mentioned
that in renormalizable $\,{\rm SU}(5)$ the ${\bf{5_{H}}}$ and ${\bf{45_{H}}}$
must couple to matter in the same way, then if we implement the PQ mechanism
these Higgses must have the same PQ charge. This implies that the DFSZ
mechanism cannot be implemented in this minimal renormalizable theory.

For the implementation of the KSVZ mechanism we require extra matter fields
coupled to the ${\bf{24_{H}}}$ in order to generate the $a(x)G\tilde{G}$
coupling. We can consider the fermionic ${\bf{24}}$ Ma:1998dn ; Bajc:2006ia ,
which also generates the neutrino masses through the type I Minkowski:1977sc ;
Mohapatra:1979ia ; GellMann:1980vs ; Yanagida:1979as  and type III Foot:1988aq
seesaw mechanisms. Consequently, the KSVZ mechanism can be implemented in the
context of Adjoint $\,{\rm SU}(5)$ Perez:2007rm  and the axion mass can be
predicted in a realistic renormalizable grand unified theory. See e.g. Refs.
Co:2016vsi ; Boucenna:2017fna ; DiLuzio:2018gqe ; Ernst:2018bib  for the
implementation of the PQ mechanism in the context of GUTs.

$Y_{e}\neq Y_{d}^{T}$ | $M_{\nu}\neq 0$ | PQ | Unification
---|---|---|---
${\bf{45_{H}}}$ |  | ${\bf{1_{i}}}$
---
${\bf{10_{H}}}$
${\bf{15_{H}}}$
${\bf{24}}$
| ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\times}$
---
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\times}$
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\times}$
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\checkmark}$
|
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\checkmark}$
---
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\checkmark}$
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\checkmark}$
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\checkmark}$
${\bf{5^{{}^{\prime}}}}$ \+ ${\bf{\bar{5}^{{}^{\prime}}}}$ |  | ${\bf{1_{i}}}$
---
${\bf{15_{H}}}$
${\bf{24}}$
|
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\checkmark}$
---
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\checkmark}$
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\checkmark}$
| ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\times}$
---
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\times}$
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\times}$
Table 1:  Minimal renormalizable gauge theories based on $\,{\rm SU}(5)$. The
third column indicates whether the PQ symmetry leading to the axion can be
implemented. The fourth column indicates whether gauge unification can be
achieved.

Recently, the authors in Ref. DiLuzio:2018gqe  studied the axion mass
predictions in a non-renormalizable $\,{\rm SU}(5)$ theory. The main
differences between the study in Ref. DiLuzio:2018gqe  and our study are the
following: a) Our model is a simple realistic renormalizable model based on
SU(5) where the PQ mechanism can be implemented without requiring Planck scale
suppressed operators, while in Ref. DiLuzio:2018gqe  these operators are
needed in order to achieve realistic masses for the charged fermions, split
the mass of the fermions in the ${\bf{24}}$ and give mass to more than one
neutrino. Generically, this type of operators could break the PQ symmetry. b)
In our model, we have implemented the KSVZ mechanism to generate the axion
mass, while in Ref. DiLuzio:2018gqe  they have two mechanisms, the KSVZ and
the DFSZ.

In Table 1 we present the different renormalizable theories based on $\,{\rm
SU}(5)$ with the minimal number of representations where the Peccei-Quinn
mechanism could be implemented. As we have discussed above, if we use the
${\bf{45_{H}}}$ to correct the charged fermion masses, only the theory with
the fermionic ${\bf{24}}$ can be realistic. In the case where the
${\bf{5^{{}^{\prime}}}}$ \+ ${\bf{\bar{5}^{{}^{\prime}}}}$ vector-like
fermions are used to correct the charged fermion masses and considering the
different possibilities to generate neutrino masses, we need to add more
representations in order to achieve unification. See for example Ref.
FileviezPerez:2018dyf  for the study of $\,{\rm SU}(5)$ theories with extra
${\bf{5^{{}^{\prime}}}}$ \+ ${\bf{\bar{5}^{{}^{\prime}}}}$ vector-like matter.
Notice that when the PQ mechanism is implemented, the splitting in the
${\bf{24}}$ is small and unification cannot be achieved in agreement with
bounds from proton decay, see Ref. FileviezPerez:2018dyf  for details.
Moreover, new Higgses in the fundamental representation could be added, as it
was done in Ref. Wise:1981ry , to implement the DFSZ mechanism but this also
requires extra representations. Then, it can be argued that the theory we will
investigate in this article is the realistic renormalizable theory based on
$\,{\rm SU}(5)$ with the minimal number of representations where the Peccei-
Quinn mechanism can be realized.

##  3 The QCD Axion and Adjoint SU(5)

In Adjoint $\,{\rm SU}(5)$ Perez:2007rm  the KSVZ mechanism can be implemented
if the different representations are charged under the PQ symmetry as follows,

$\displaystyle\bar{5}\to e^{-3i\theta}\bar{5},\hskip 45.52458pt10\to
e^{+i\theta}10,\hskip 34.14322pt5_{H}\to e^{-2i\theta}5_{H},$ $\displaystyle
24_{H}\to e^{-10i\theta}24_{H},\hskip 14.22636pt45_{H}\to
e^{-2i\theta}45_{H},\hskip 14.22636pt24\to e^{+5i\theta}24.$

We note that the relative PQ charges among all representations are fixed by
the consistency of the theory. Then, we can achieve a realistic relation
between the charged fermion masses using the following Yukawa interactions

${\cal{L}}\supset 10\ 10\left(Y_{1}\ 5_{H}\ +\ Y_{2}\ 45_{H}\right)+10\
\bar{5}\left(Y_{3}\ 5_{H}^{*}\ +\ Y_{4}\ 45_{H}^{*}\right)+\rm{h.c.},$ (1)

from which the charged fermion mass matrices read as

$\displaystyle M_{U}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(2\left(Y_{1}+Y_{1}^{T}\right)v_{5}-4\left(Y_{2}-Y_{2}^{T}\right)v_{45}\right),$
(2) $\displaystyle M_{E}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(Y_{3}v_{5}^{*}-6Y_{4}v_{45}^{*}\right),$ (3)
$\displaystyle M_{D}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(Y_{3}^{T}v_{5}^{*}+2Y_{4}^{T}v_{45}^{*}\right),$
(4)

where $\left<5_{H}\right>=v_{5}/\sqrt{2}$,
$\left<45_{H}\right>^{15}_{1}=\left<45_{H}\right>^{25}_{2}=\left<45_{H}\right>^{35}_{3}=v_{45}/\sqrt{2}$
and $\left<45_{H}\right>^{45}_{4}=-3v_{45}/\sqrt{2}$.

In this theory, the neutrino masses are generated through the type I
Minkowski:1977sc ; Mohapatra:1979ia ; GellMann:1980vs ; Yanagida:1979as , type
III Foot:1988aq  and the colored seesaw FileviezPerez:2009ud  mechanisms using
the interactions

${\cal{L}}\supset h_{1}\ \bar{5}\ 24\ 5_{H}\ +\ h_{2}\ \bar{5}\ 24\
45_{H}+\lambda{\rm{Tr}}\left[24^{2}\ 24_{H}\right]+\rm{h.c.}.$ (5)

The neutrino mass matrix is given by

$M_{\nu}=M_{\nu}^{\rm I}+M_{\nu}^{\rm III}+M_{\nu}^{\rm cs},$ (6)

where $M_{\nu}^{\rm I}$, $M_{\nu}^{\rm III}$, and $M_{\nu}^{\rm cs}$ are the
type I, type III and colored seesaw contributions, respectively. See Ref.
Perez:2007rm ; Emmanuel-Costa:2013gia  for the explicit form of the neutrino
mass matrix and further details.

Now, in order to implement the KSVZ mechanism we use the fact that in this
theory the axion lives in the ${\bf{24_{H}}}$, i.e.

${\bf{24_{H}}}\supset\frac{1}{\sqrt{2}}|\Sigma_{0}|\
e^{\frac{ia(x)}{v_{\Sigma}}},$

where $v_{\Sigma}$ is the VEV of $\Sigma_{0}$. This simple idea is appealing
because the Peccei-Quinn scale can be predicted from gauge coupling
unification since the same field breaking the GUT symmetry contains the axion
field. See Appendix A for the explicit form of all fields in the theory and
Refs. Perez:2008ry ; Blanchet:2008cj ; Blanchet:2008ga  for the study of the
phenomenological aspects of Adjoint SU(5).

Figure 1:  Left Panel: Predictions for the GUT scale, $M_{\rm GUT}$, as a
function of $M_{\Phi_{1}}$. We fix the mass of the second Higgs doublet
$H_{2}$ living in the ${\bf 45_{H}}$ to $M_{H_{2}}=1$ TeV. The purple shaded
region shows the current LHC bound on the scalar colored octet
$\Phi_{1}\sim(8,2,1/2)$, $M_{\Phi_{1}}>1$ TeV Hayreter:2017wra ;
Miralles:2019uzg . The region shaded in red shows the current experimental
limit on the proton lifetime from the Super-Kamiokande collaboration,
$\tau(p\to K^{+}\bar{\nu})>5.9\times 10^{33}$ years Abe:2014mwa , and the
green dashed line shows the projected bound for the Hyper-Kamiokande
collaboration, $\tau(p\to K^{+}\bar{\nu})>2.5\times 10^{34}$ years
Yokoyama:2017mnt . When applying the bounds from proton lifetime we take the
most conservative limit which corresponds to taking $M_{\rho_{3}}=10^{15}$
GeV. The mass of the $\Phi_{3}\sim(3,3,-1/3)$ varies between the range
$10^{8.4}-10^{8.63}$ GeV from left to right. Right Panel: Relation between the
GUT scale and $\alpha_{\rm GUT}$. The blue shaded region leads to gauge
unification, the width of this band is determined by varying from
$M_{\rho_{3}}=10^{4.51}$ GeV (left limit determined by the BBN bounds, as
discussed in Appendix C) to $M_{\rho_{3}}=10^{15}$ GeV (right limit determined
by the perturbative bound on the seesaw scale). Here we use $M_{H_{2}}=1$ TeV
as in the left panel.

###  3.1 Unification Constraints and Proton Decay

Figure 2: Predictions for the proton decay lifetime for the channel
$p\to\pi^{0}e^{+}$. The different blue bands show different choices of the
magnitude of the unknown mixing matrix $V_{e}$. The width of the blue bands is
determined by the allowed values of $\alpha_{\rm GUT}$ as we vary
$M_{\rho_{3}}\subset[10^{4.51},10^{15}]$ GeV. The red shaded area shows the
excluded parameter space by the bound on the proton decay lifetime,
$\tau(p\to\pi^{0}e^{+})>1.6\times 10^{34}$ years Miura:2016krn , from the
Super-Kamiokande collaboration. The green dashed line shows the projected
bound on proton decay lifetime from the Hyper-Kamiokande collaboration, i.e.
$\tau(p\to\pi^{0}e^{+})>8\times 10^{34}$ years Yokoyama:2017mnt .

Figure 3: Predictions for the proton decay channels $p\to K^{+}\bar{\nu}$ and
$p\to\pi^{+}\bar{\nu}$. The blue bands show the possible range of variation of
$\alpha_{\rm GUT}$, according to $M_{\rho_{3}}\subset[10^{4.51},10^{15}]$ GeV.
The red shaded areas show the excluded region by the Super-Kamiokande
constraints, $\tau(p\to K^{+}\bar{\nu})>5.9\times 10^{33}$ years Abe:2014mwa
and $\tau(p\to\pi^{+}\bar{\nu})>3.9\times 10^{32}$ years Abe:2013lua . The
green dashed line shows the projected bound by the Hyper-Kamiokande
collaboration on the $p\to K^{+}\bar{\nu}$ channel, $\tau(p\to
K^{+}\bar{\nu})>2.5\times 10^{34}$ years Yokoyama:2017mnt .

In order to predict the PQ scale we need to understand the predictions for the
GUT scale. The renormalization group equations (RGEs) for the gauge couplings
can be written as

$\frac{1}{\alpha_{i}(M_{Z})}-\frac{1}{\alpha_{i}(\mu)}=\frac{1}{2\pi}b_{i}^{\rm
SM}\ {\rm{ln}}\frac{\mu}{M_{Z}}+\frac{1}{2\pi}\sum_{I}b_{iI}\ \Theta(\mu-
M_{I})\,{\rm{ln}}\frac{\mu}{M_{I}},$ (7)

where $i=1,2,3$, $b_{1}^{\rm SM}=41/10$, $b_{2}^{\rm SM}=-19/6$ and
$b_{3}^{\rm SM}=-7$. In Appendix D we list the different contributions to the
beta functions in Adjoint $\,{\rm SU}(5)$. Using the above relations, the
following equations must be satisfied in order to achieve unification at the
one-loop level Giveon:1991zm ,

$\displaystyle\frac{B_{23}}{B_{12}}$ $\displaystyle=$
$\displaystyle\frac{5}{8}\left(\frac{\sin^{2}\theta_{W}(M_{Z})-\alpha(M_{Z})/\alpha_{s}(M_{Z})}{3/8-\sin^{2}\theta_{W}(M_{Z})}\right)=0.717,$
(8) $\displaystyle{\rm{ln}}\frac{M_{\rm GUT}}{M_{Z}}$ $\displaystyle=$
$\displaystyle\frac{16\pi}{5\alpha(M_{Z})}\left(\frac{3/8-\sin^{2}\theta_{W}(M_{Z})}{B_{12}}\right)=\frac{184.95}{B_{12}},$
(9)

where the parameters $B_{ij}$ are defined in Appendix D. Here we have used the
experimental values $\sin^{2}\theta_{W}(M_{Z})=0.23122$,
$\alpha(M_{Z})=1/127.955$, and $\alpha_{s}(M_{Z})=0.1181$ at the $M_{Z}$ scale
Tanabashi:2018oca . It is well-known that in the minimal $\,{\rm SU}(5)$
theory unification cannot be achieved in agreement with the experimental
values of the gauge couplings at the low scale since $B_{23}/B_{12}<0.6$.

In the context of Adjoint $\,{\rm SU}(5)$, in the ${\bf 45_{H}}$
representation there are only three fields that help achieve unification: for
$\Phi_{3}$ and $H_{2}$, the lighter they are the larger the fraction
$B_{23}/B_{12}$ can be. The field $\Phi_{1}$ does not help to increase the
later ratio; however, it helps towards unification indirectly since it gives a
negative contribution to $B_{12}$, thus increasing the GUT scale. This is
necessary, as we will show, to avoid the strong proton decay experimental
constraints. With the help of these three fields, unification can be achieved
in agreement with experimental results. The rest of fields in the ${\bf
45_{H}}$ will be assumed to be, without loss of generality, at the GUT scale
since we are interested in finding the allowed parameter space by proton decay
bounds.

For the case of the $\bf{24}$ representation, the small mass splitting between
the masses of their fields is fixed and given by

$\displaystyle M_{\rho_{0}}=\frac{1}{3}M_{\rho_{3}},\quad
M_{\rho_{8}}=\frac{2}{3}M_{\rho_{3}},\quad
M_{\rho_{(3,2)}}=M_{\rho_{(\bar{3},2)}}=\frac{1}{6}M_{\rho_{3}},$ (10)

which is a direct consequence of ${\bf 24}$ being charged under the PQ-
symmetry. We note that the unification constraints do not depend on the global
mass scale of the fields in the ${\bf 24}$, i.e. $M_{\rho_{3}}$; they are only
sensitive to their mass splitting.

In the left panel in Fig. 1 we show the predictions of the GUT scale for the
different values of $M_{\Phi_{1}}$. We fix the mass of the second Higgs
doublet $H_{2}$ living in the ${\bf 45_{H}}$ to $M_{H_{2}}=1$ TeV. The purple
shaded region shows the current LHC bound on the scalar colored octet
$\Phi_{1}\sim(8,2,1/2)$, $M_{\Phi_{1}}>1$ TeV Hayreter:2017wra ;
Miralles:2019uzg . The region shaded in red shows the current experimental
limit on the proton lifetime from the Super-Kamiokande collaboration,
$\tau(p\to K^{+}\bar{\nu})>5.9\times 10^{33}$ years Abe:2014mwa , and the
green dashed line shows the projected bound for the Hyper-Kamiokande
collaboration, $\tau(p\to K^{+}\bar{\nu})>2.5\times 10^{34}$ years
Yokoyama:2017mnt . When applying the bounds from proton lifetime we take the
most conservative limit which corresponds to taking $M_{\rho_{3}}=10^{15}$
GeV. The mass of the $\Phi_{3}\sim(3,3,-1/3)$ varies between the range
$10^{8.4}-10^{8.63}$ GeV from left to right. This result is quite general
since $H_{2}$ has a small contribution to the running of the gauge couplings.
For the choice $M_{H_{2}}=1$ TeV, the allowed range for the GUT scale is
$M_{\text{GUT}}=\left[10^{15.06}-10^{15.74}\right]$ GeV, the lower limit is
determined by the bound on the proton lifetime from Super-Kamiokande on the
$p\to K^{+}\bar{\nu}$ channel and the upper one by the collider bound on the
mass of $\Phi_{1}$.

The allowed values for $\alpha_{\text{GUT}}$ depend on the values for
$M_{\rho_{3}}$, which define a possible range of values for the GUT gauge
coupling. The fields in the $\bf{24}$ must be at the seesaw scale or below in
order to generate realistic values for the neutrino masses. In the right panel
in Fig. 1, we present the relation between the GUT scale and $\alpha_{\rm
GUT}$. The blue shaded region leads to gauge unification, the width of this
band is determined by varying from $M_{\rho_{3}}=10^{4.51}$ GeV (left limit)
to $M_{\rho_{3}}=10^{15}$ GeV (right limit), and we have used $M_{H_{2}}=1$
TeV. The lower limit $M_{\rho_{3}}=10^{4.51}$ GeV is determined by the bound
from Big Bang Nucleosynthesis (BBN) discussed in Appendix C, while
$M_{\rho_{3}}=10^{15}$ GeV is the perturbative bound using the seesaw relation
for neutrino masses.

In the left panel in Fig. 1 we have demonstrated that, in order to have
unification in agreement with all experimental constraints, the mass of the
$\Phi_{3}$ field must lie in the range $10^{8.4}-10^{8.63}$ GeV. This is
possible only if the coupling of $\Phi_{3}$ to matter, $Y_{2}$, is suppressed
because this field mediates proton decay. It is well-known that if $Y_{2}\sim
Y_{U}$ we need $M_{\Phi_{3}}>10^{12}$ GeV to suppress proton decay Nath:2006ut
. Fortunately, in this theory we can have that $Y_{2}\ll Y_{1}\approx Y_{U}$,
and hence, we can have a consistent scenario for fermion masses where the mass
matrix for up-quarks $M_{U}$ is approximately symmetric. In the case when
$M_{U}$ is symmetric, we can make clean predictions for the proton decay
channels with anti-neutrinos FileviezPerez:2004hn . See Appendix B for more
details.

In Fig. 2 we show the predictions for the proton decay lifetime for the
channel $p\to\pi^{0}e^{+}$. The decay width for this channel is given by

$\displaystyle\Gamma(p\to\pi^{0}e^{+})$ $\displaystyle=$
$\displaystyle\frac{\pi m_{p}}{2}\frac{\alpha_{\rm GUT}^{2}}{M_{\rm
GUT}^{4}}A^{2}\left(\left|(V_{2}^{11}+V_{UD}^{11}(V_{2}V_{UD})^{11})\right|^{2}+\left|V_{3}^{11}\right|^{2}\right)\left|\left<\pi^{0}\vphantom{(ud)_{R}u_{L}p}\right|(ud)_{R}u_{L}\left|p\vphantom{\pi^{0}(ud)_{R}u_{L}}\right>\right|^{2}$
(11) $\displaystyle=$ $\displaystyle\frac{\pi m_{p}}{2}\frac{\alpha_{\rm
GUT}^{2}}{M_{\rm
GUT}^{4}}A^{2}V_{e}^{2}\left|\left<\pi^{0}\vphantom{(ud)_{R}u_{L}p}\right|(ud)_{R}u_{L}\left|p\vphantom{\pi^{0}(ud)_{R}u_{L}}\right>\right|^{2}.$

Here we parametrize the magnitude of the unknown mixings with $V_{e}$, and we
show the predictions in Fig. 2 for two different values: $V_{e}=0.1$ and
$V_{e}=1$. See the Appendix B for more details. The blue bands in Fig. 2
include the possible different values of $\alpha_{\text{GUT}}$ according to
the choice of $M_{\rho_{3}}\subset[10^{4.51},10^{15}]$ GeV. The red shaded
area shows the excluded parameter space by the bound on the proton decay
lifetime, $\tau(p\to\pi^{0}e^{+})>1.6\times 10^{34}$ years Miura:2016krn ,
from the Super-Kamiokande collaboration. The green dashed line shows the
projected bound on proton decay lifetime from the Hyper-Kamiokande
collaboration, i.e. $\tau(p\to\pi^{0}e^{+})>8\times 10^{34}$ years
Yokoyama:2017mnt . Since the lifetime for this channel is a function of an
unknown matrix $V_{e}$, we proceed to investigate the channel with
antineutrinos that can give us a more conservative bound on the GUT scale.

In Fig. 3 we show the predictions for the proton decay channels $p\to
K^{+}\bar{\nu}$ and $p\to\pi^{+}\bar{\nu}$. We have discussed above that we
need to work in the scenario where $Y_{2}\ll Y_{1}$. In this case $M_{U}\simeq
M_{U}^{T}$ and the different proton decay channels can be written as

$\displaystyle\Gamma(p\to K^{+}\bar{\nu})$ $\displaystyle\simeq$
$\displaystyle\frac{\pi m_{p}}{2}\frac{\alpha_{\rm GUT}^{2}}{M_{\rm
GUT}^{4}}\left(1-\frac{m_{K^{+}}^{2}}{m_{p}^{2}}\right)^{2}A^{2}$ (12)
$\displaystyle\times$ $\displaystyle\left(\left|V_{\rm
CKM}^{11}\left<K^{+}\vphantom{(us)_{R}d_{L}p}\right|(us)_{R}d_{L}\left|p\vphantom{K^{+}(us)_{R}d_{L}}\right>\right|^{2}+\left|V_{\rm
CKM}^{12}\left<K^{+}\vphantom{(ud)_{R}s_{L}p}\right|(ud)_{R}s_{L}\left|p\vphantom{K^{+}(ud)_{R}s_{L}}\right>\right|^{2}\right),$
$\displaystyle\Gamma(p\to\pi^{+}\bar{\nu})$ $\displaystyle\simeq$
$\displaystyle\frac{\pi m_{p}}{2}\frac{\alpha_{\rm GUT}^{2}}{M_{\rm
GUT}^{4}}A^{2}\left|V_{\rm
CKM}^{11}\left<\pi^{+}\vphantom{(du)_{R}d_{L}p}\right|(du)_{R}d_{L}\left|p\vphantom{\pi^{+}(du)_{R}d_{L}}\right>\right|^{2},$
(13)

where $V_{\rm CKM}^{ij}$ are the elements of the $V_{\rm CKM}$ matrix. See the
Appendix B for more details. Since in this case the decay lifetime is a
function of known parameters we can make a better prediction for these
channels. The blue bands in Fig. 3 show the possible range of variation of
$\alpha_{\rm GUT}$, according to $M_{\rho_{3}}\subset[10^{4.51},10^{15}]$ GeV.
The red shaded areas show the excluded region by the Super-Kamiokande
constraints, $\tau(p\to K^{+}\bar{\nu})>5.9\times 10^{33}$ years Abe:2014mwa
and $\tau(p\to\pi^{+}\bar{\nu})>3.9\times 10^{32}$ years Abe:2013lua . The
green dashed line shows the projected bound by the Hyper-Kamiokande
collaboration on the $p\to K^{+}\bar{\nu}$ channel, $\tau(p\to
K^{+}\bar{\nu})>2.5\times 10^{34}$ years Yokoyama:2017mnt . Therefore, in the
rest of this article we will use the range of $M_{\rm GUT}$ allowed by the
experimental bounds on proton decay into antineutrinos as shown in Fig. 3,
which are the most conservative allowed values for $M_{\rm GUT}$.

###  3.2 Axion Mass and Couplings

Figure 4: Axion mass as a function of the GUT scale. The blue band shows the
parameter space allowed by unification constraints and proton decay. The red
(purple) shaded area shows the excluded parameter space by the proton decay
bound (LHC bound on $M_{\Phi_{1}}$). The dashed green line gives the projected
sensitivity for Hyper-Kamiokande.

In this theory, the axion is only coupled to the heavy fermions in the
${\bf{24}}$, and hence, we can use the KSVZ mechanism. The interaction comes
from the Yukawa term between ${\bf{24}}$ and ${\bf{24_{H}}}$

${\cal L}\supset\lambda\ \text{Tr}\left[24^{2}24_{H}\right]+{\rm h.c.}.$ (14)

Once the singlet inside the ${\bf{24_{H}}}$ acquires a non-zero vacuum
expectation value, we can replace ${\bf{24_{H}}}$ in the above equation by

$24_{H}\to\frac{v_{\Sigma}}{\sqrt{15}}\
{\rm{diag}}(-1,-1,-1,3/2,3/2)\,e^{ia(x)/v_{\Sigma}},$ (15)

where $a(x)$ corresponds to the axion field, and we find that

$\mathcal{L}\supset\frac{\lambda}{\sqrt{15}}\,v_{\Sigma}\,e^{ia(x)/v_{\Sigma}}\left(-{\rm
Tr}[\rho_{8}\rho_{8}]+\frac{1}{2}{\rm
Tr}[\rho_{(\bar{3},2)}\rho_{(3,2)}]+\frac{3}{2}{\rm
Tr}[\rho_{3}\rho_{3}]+\frac{1}{2}\rho_{0}^{2}\right)+{\rm h.c.}.$ (16)

Following the method proposed by Fujikawa Fujikawa:1980eg , the coupling to
the gluons can be written as

$\mathcal{L}\supset\frac{\alpha_{3}}{8\pi}\frac{a}{v_{\Sigma}}NG_{\mu\nu}\tilde{G}^{\mu\nu}=\frac{\alpha_{3}}{8\pi}\frac{a}{f_{a}}G_{\mu\nu}\tilde{G}^{\mu\nu},$
(17)

where $N$ depends on the chiral fields that have been rotated in order to
absorb the axion field, and in our case $N=5$. The coupling of the axion to
the gluons is generated by the rotations of the colored fields in the ${\bf
24}$ representation. Hence, the relation between $f_{a}$ and the GUT scale is
given by

$f_{a}\equiv\frac{v_{\Sigma}}{N}=\frac{v_{\Sigma}}{5}=\sqrt{\frac{6}{5\pi\alpha_{\rm
GUT}}}\frac{M_{\rm GUT}}{10},$ (18)

where we have used $M_{V}\\!=\\!M_{\rm GUT}\\!=\\!\sqrt{5/6}\,g_{\rm
GUT}\,v_{\Sigma}$. This is one of the most important results of this paper
which allows us to predict the axion mass once we know the allowed values for
$M_{\rm GUT}$ and $\alpha_{\rm GUT}$. In order to show our numerical results
we will use the following relation between $m_{a}$ and $f_{a}$
diCortona:2015ldu

$m_{a}=5.70(7)\,\times 10^{-6}\ \text{eV}\left(\frac{10^{12}\text{
GeV}}{f_{a}}\right).$ (19)

As demonstrated in Sec. 3.1, in Adjoint $\,{\rm SU}(5)$ a small range for the
GUT scale and $\alpha_{\rm GUT}$ can be predicted, this leads to the following
prediction for the axion mass

$m_{a}=\left(2.98-13.4\right)\times 10^{-9}\ {\rm eV}.$ (20)

In Fig. 4 we show the predictions for the axion mass as a function of the GUT
scale. The blue band shows the parameter space allowed by unification
constraints and proton decay. The red (purple) shaded area shows the excluded
parameter space by the proton decay bound (LHC bound on $M_{\Phi_{1}}$). This
prediction is crucial to understand the testability of this theory at axion
experiments because it tells us the specific range for the axion mass where a
signal could be expected. In the allowed mass window, an initial misalignment
angle of $\theta_{i}\approx 10^{-2}$ Preskill:1982cy ; Abbott:1982af ;
Dine:1982ah  must be assumed in order for the axion field to saturate the dark
matter relic abundance $\Omega_{\rm DM}h^{2}=0.1200\pm 0.0012$ Aghanim:2018eyx
.

Figure 5: The axion-photon coupling versus the axion mass. The blue line
corresponds to the axion mass prediction in Adjoint $\,{\rm SU}(5)$. The
regions shaded in blue, orange and red correspond to the projected
sensitivities for Phase I, II and III of the ABRACADABRA experiment
Kahn:2016aff  using the broadband approach. The purple lines correspond to the
projected sensitivities for the three different phases in the resonant
approach.

We proceed to study the phenomenology of the axion in the predicted mass
window. The coupling between the axion and photons can be written as

$\mathcal{L}\supset-\frac{g_{a\gamma\gamma}}{4}aF_{\mu\nu}\tilde{F}^{\mu\nu},$
(21)

with the effective coupling $g_{a\gamma\gamma}$ given by

$g_{a\gamma\gamma}=\frac{\alpha_{\tiny{\rm EM}}}{2\pi
f_{a}}\left(\frac{E}{N}-1.92(4)\right)=\frac{\alpha_{\tiny{\rm EM}}}{2\pi
f_{a}}\left(\frac{8}{3}-1.92(4)\right),$ (22)

where second term is the contribution from non-perturbative effects from the
axion coupling to QCD and has been computed at NLO in Ref. diCortona:2015ldu .
The first term in Eq. (22) is given by the anomaly coefficient for the
electromagnetic fields and $E/N\\!=\\!8/3$ since we are working with fermions
in a complete representation of $\,{\rm SU}(5)$.

The ABRACADABRA experiment Kahn:2016aff  is expected to be sensitive to the
axion to photon coupling for very light axions, $m_{a}\lesssim 10^{-6}$ eV. In
Fig. 5 we present the axion to photon coupling together with the projected
reach for ABRACADABRA, as given in Ref. Kahn:2016aff . The regions shaded in
different colors correspond to the projected sensitivity for the broadband
operating mode, while the purple lines show the ones for the resonant
operating mode. The blue solid line corresponds to the prediction in Adjoint
$\,{\rm SU}(5)$. It is of great interest that the third stage of the broadband
mode will be able to probe a portion of this mass range, while the third stage
of the resonant operating mode could fully probe the predicted mass window.

At low energies, the axion field has also a coupling to the electric dipole
moment (EDM) operator for nucleons

$\mathcal{L}\supset-\frac{i}{2}g_{aD}\,a(\bar{\psi}_{N}\,\sigma^{\mu\nu}\gamma^{5}\psi_{N})F_{\mu\nu},$
(23)

where $\sigma^{\mu\nu}\\!=\\!i[\gamma^{\mu},\gamma^{\nu}]/2$, and this term
gives rise to the nucleon electric dipole moment Crewther:1979pi ;
Pospelov:1999mv ; Chupp:2017rkp ; Graham:2013gfa

$d_{n}=g_{aD}\,a\approx 2.4\times 10^{-16}\frac{a}{f_{a}}e\cdot{\rm cm}.$ (24)

In the presence of an oscillating axion background field, $a(t)$, this
interaction generates an oscillating nucleon EDM. Thus, when dark matter
consists of the axion field with $\rho_{\rm DM}\approx
m_{a}^{2}\,a_{0}^{2}/2\approx 0.4$ GeV/cm3 Catena:2009mf  it could lead to an
observable effect on nucleons.

Figure 6: The axion coupling to the nucleon electric dipole moment versus the
axion mass. The blue band is the prediction in Adjoint $\,{\rm SU}(5)$. The
width of the band corresponds to the theoretical error, as given in
Pospelov:1999mv ; Chupp:2017rkp . The region shaded in purple and red
correspond to the projected sensitivities for Phase II and III of the CASPEr-
Electric experiment JacksonKimball:2017elr . The brown dashed line shows the
ultimate limit from magnetization noise.

The CASPEr-Electric experiment Budker:2013hfa  has been proposed to measure
this oscillating EDM by applying nuclear magnetic resonance techniques. This
experiment is expected to probe axions with very low masses, $m_{a}<10^{-9}$
eV, which is the region of interest for theories in which the PQ scale has a
connection to the GUT or the string scale Svrcek:2006yi . In Fig. 6 we present
the predictions for the $g_{aD}$ coupling in the axion mass window predicted
by Adjoint $\,{\rm SU}(5)$. The region shaded in purple (red) corresponds to
the projected sensitivity for Phase II (Phase III) for CASPEr-Electric
JacksonKimball:2017elr , the latter will be able to probe a part of the mass
window predicted by Adjoint $\,{\rm SU}(5)$. The brown dashed line shows the
ultimate limit from magnetization noise. As can be appreciated, these
predictions for the axion mass can be probed in the near future.

In this article, we have showed that combining the signals at proton decay
experiments, the collider signatures of the $\Phi_{1}$ field and the
predictions for axion experiments, it is possible to test a simple grand
unified theory which predicts a small window for the axion mass. In this
theory, we can also have an explanation for the baryon asymmetry in the
Universe using the Baryogenesis via Leptogenesis mechanism. This theory
predicts that $M_{\rho_{0}}=M_{\rho_{3}}/3$, then the lightest field relevant
for leptogenesis is the field responsible for type I seesaw. The leptogenesis
mechanism in the context of Adjoint $\,{\rm SU}(5)$ was investigated in Refs.
Blanchet:2008cj ; Blanchet:2008ga , but the emphasis was in the case where
$\rho_{3}$ is the lightest field. We will investigate this issue in a future
publication.

##  4 Summary

The QCD axion remains one of the most appealing candidates for the dark matter
in the Universe. In this work, we discussed the implementation of the DFSZ and
KSVZ mechanisms in renormalizable grand unified theories based on the gauge
symmetry $\,{\rm SU}(5)$. In order to connect the GUT scale to the Peccei-
Quinn scale we require the same scalar field to break both symmetries. We
showed that in the simplest renormalizable models the ${\bf 5_{H}}$ and ${\bf
45_{H}}$ Higgses must couple in the same way to matter and have the same
Peccei-Quinn charge. Therefore, the DFSZ mechanism cannot be implemented with
the minimal Higgs sector even if we have two Higgs doublets and the singlet
inside the ${\bf 24_{H}}$.

We have shown that in the context of Adjoint $\,{\rm SU}(5)$ the same
fermionic field, ${\bf 24}$, needed to generate neutrino masses can be used to
implement the KSVZ mechanism. The colored and the charged fields in the ${\bf
24}$ generate the couplings of the axion to the gluons and the photons
respectively, while neutrino masses are generated through the seesaw
mechanism. We found that the theory studied in this article can be considered
as the realistic renormalizable theory based on $\,{\rm SU}(5)$ with the
minimal number of representations where the Peccei-Quinn mechanism can be
realized.

We investigated the unification of gauge interactions taking into account
collider and proton decay constraints, and we found that the allowed GUT scale
is in a small window which allows us to predict the axion mass in the range

$m_{a}\simeq(3-13)\times 10^{-9}\ \rm{eV}.$

We focused on the scenario where the PQ symmetry is broken before inflation
and discussed the predictions for the different axion experiments showing that
this theory could be fully tested in the near future by the ABRACADABRA and
the CASPEr-Electric experiments. Additionally, we have shown that this theory
could be probed at proton decay experiments such as the current Super-
Kamiokande and the future Hyper-Kamiokande.

The prediction for the axion mass in this work has some overlap with the one
given in Ref. DiLuzio:2018gqe . In the latter work the authors rely on Planck
suppressed operators that could break the PQ symmetry and used two mechanisms
to generate the axion mass . In our case, we have considered a simple
renormalizable theory that does not rely on Planck suppressed operators and
the axion mass is generated only by the KSVZ mechanism. Our theory is more
predictive due to renormalizability and it has less free parameters.

The theoretical framework presented in this article, based on the $\,{\rm
SU}(5)\times\,{\rm U}(1)_{\rm PQ}$ symmetry, can be considered as an appealing
theory for physics beyond the Standard Model, where it is possible to
understand the unification of the Standard Model forces, the origin of
neutrino masses, dark matter, the strong CP problem and the baryon asymmetry
through leptogenesis. Furthermore, the lifetime of the proton can be predicted
and exotic signatures at colliders are expected.

## Acknowledgments

P. F. P. thanks M. B. Wise for a discussion and the theory group at Caltech
for hospitality. A. D. P. would like to thank C. Tamarit for helpful
discussions. The work of C. M. has been supported in part by Grants No.
FPA2014-53631-C2-1-P, No. FPA2017-84445-P, and No. SEV-2014- 0398 (AEI/ERDF,
EU), and by a La Caixa-Severo Ochoa scholarship.

##  Appendix A Adjoint $\,{\rm SU}(5)$ Representations

In the Adjoint $\,{\rm SU}(5)$ Perez:2007rm  theory we have the following
representations:

${\bf{\bar{5}}}=\begin{pmatrix}d^{c}_{1}\\\ d^{c}_{2}\\\ d^{c}_{3}\\\ e\\\
-\nu\end{pmatrix},\\\ \quad\quad\\\
{\bf{10}}=\frac{1}{\sqrt{2}}\begin{pmatrix}0&u^{c}_{3}&-u^{c}_{2}&-u^{1}&-d^{1}\\\
-u^{c}_{3}&0&u_{1}^{c}&-u^{2}&-d^{2}\\\
u^{c}_{2}&-u^{c}_{1}&0&-u^{3}&-d^{3}\\\ u^{1}&u^{2}&u^{3}&0&e^{c}\\\
d^{1}&d^{2}&d^{3}&-e^{c}&0\end{pmatrix},\\\ \quad\quad\\\
{\bf{5}_{H}}=\begin{pmatrix}T_{1}\\\ T_{2}\\\ T_{3}\\\ H^{+}\\\
H^{0}\end{pmatrix},$

${\bf{24_{H}}}=\begin{pmatrix}\Sigma_{8}-\frac{2}{\sqrt{30}}\Sigma_{0}&\Sigma_{(\bar{3},2)}\\\
\Sigma_{({3},2)}&\Sigma_{3}+\frac{3}{\sqrt{30}}\Sigma_{0}\end{pmatrix},\\\
\quad\quad\\\
{\bf{24}}=\begin{pmatrix}\rho_{8}-\frac{1}{\sqrt{15}}\rho_{0}&\rho_{({\bar{3}},2)}\\\
\rho_{({3},2)}&\rho_{3}+\frac{3}{2\sqrt{15}}\rho_{0}\end{pmatrix},$

the components of the ${\bf{24_{H}}}$ have the following quantum numbers,

$\Sigma_{8}\sim(8,1,0),\,\,\Sigma_{3}\sim(1,3,0),\,\,\Sigma_{0}\sim(1,1,0),\,\,\Sigma_{({3},2)}\sim(3,2,-5/6)\
{\rm{and}}\ \Sigma_{(\bar{3},2)}\sim(\bar{3},2,5/6).$

In a similar manner, the components of the ${\bf{24}}$ are given by

$\rho_{8}\sim(8,1,0),\,\,\rho_{3}\sim(1,3,0),\,\,\rho_{0}\sim(1,1,0),\,\,\rho_{({3},2)}\sim(3,2,-5/6)\
{\rm{and}}\ \rho_{(\bar{3},2)}\sim(\bar{3},2,5/6),$

while the gauge bosons are described by

${\bf{A}_{\mu}}=\begin{pmatrix}G_{\mu}-\frac{1}{\sqrt{15}}B_{\mu}&V_{\mu}^{\dagger}\\\
V_{\mu}&W_{\mu}+\frac{3}{2\sqrt{15}}B_{\mu}\end{pmatrix},$

with quantum numbers

$G_{\mu}\sim(8,1,0),\ B_{\mu}\sim(1,1,0),\ V_{\mu}\sim(3,2,-5/6)\ {\rm{and}}\
W_{\mu}\sim(1,3,0).$

The scalar 45 dimensional representation has components

$\displaystyle\displaystyle{\bf{45_{H}}}$ $\displaystyle\sim$
$\displaystyle\underbrace{(8,2,1/2)}_{\Phi_{1}}\oplus\underbrace{(\bar{6},1,-1/3)}_{\Phi_{2}}\oplus\underbrace{(3,3,-1/3)}_{\Phi_{3}}\oplus\underbrace{(\bar{3},2,-7/6)}_{\Phi_{4}}\oplus\underbrace{(3,1,-1/3)}_{\Phi_{5}}\oplus\underbrace{(\bar{3},1,4/3)}_{\Phi_{6}}\oplus\underbrace{(1,2,1/2)}_{H_{2}}.$

##  Appendix B Proton Decay

The proton decay widths for the most relevant channels are given by

$\displaystyle\Gamma(p\to\pi^{0}e^{+})$ $\displaystyle=$
$\displaystyle\frac{m_{p}}{8\pi}A^{2}k_{1}^{4}\left(\left|c(e^{c},d)\left<\pi^{0}\vphantom{(ud)_{L}u_{R}p}\right|(ud)_{L}u_{R}\left|p\vphantom{\pi^{0}(ud)_{L}u_{R}}\right>\right|^{2}+\left|c(e,d^{c})\left<\pi^{0}\vphantom{(ud)_{R}u_{L}p}\right|(ud)_{R}u_{L}\left|p\vphantom{\pi^{0}(ud)_{R}u_{L}}\right>\right|^{2}\right),\hskip
22.76228pt$ (25) $\displaystyle\displaystyle\Gamma(p\to K^{+}\bar{\nu})$
$\displaystyle=$
$\displaystyle\frac{m_{p}}{8\pi}\left(1-\frac{m_{K^{+}}^{2}}{m_{p}^{2}}\right)^{2}A^{2}k_{1}^{4}\sum_{i}\left|c(\nu_{i},d,s^{c})\left<K^{+}\vphantom{(us)_{R}d_{L}p}\right|(us)_{R}d_{L}\left|p\vphantom{K^{+}(us)_{R}d_{L}}\right>\right.$
(26)
$\displaystyle+\left.c(\nu_{i},s,d^{c})\left<K^{+}\vphantom{(ud)_{R}s_{L}p}\right|(ud)_{R}s_{L}\left|p\vphantom{K^{+}(ud)_{R}s_{L}}\right>\right|^{2},$
$\displaystyle\Gamma(p\to\pi^{+}\bar{\nu})$ $\displaystyle=$
$\displaystyle\frac{m_{p}}{8\pi}A^{2}k_{1}^{4}\left|V_{UD}^{11}\left<\pi^{+}\vphantom{(du)_{R}d_{L}p}\right|(du)_{R}d_{L}\left|p\vphantom{\pi^{+}(du)_{R}d_{L}}\right>\right|^{2},$
(27) $\displaystyle\textrm{with}\ A$ $\displaystyle=$ $\displaystyle A_{\rm
QCD}A_{SR}=\left(\frac{\alpha_{3}(m_{b})}{\alpha_{3}(M_{Z})}\right)^{6/23}\left(\frac{\alpha_{3}(Q)}{\alpha_{3}(m_{b})}\right)^{6/25}\left(\frac{\alpha_{3}(M_{Z})}{\alpha_{3}(M_{\rm
GUT})}\right)^{2/7}.$ (28)

In the above equation $k_{1}=g_{\rm GUT}/\sqrt{2}M_{\rm GUT}$ and $A$ defines
the running of the operators. $A_{\rm QCD}\approx 1.2$ corresponds to the
running from the $M_{Z}$ to the $Q\approx 2.3$ GeV scale, while $A_{SR}\approx
1.5$ defines the running from the GUT scale to the electroweak scale. The
c-coefficients FileviezPerez:2004hn  are given by

$\displaystyle
c(e^{c}_{\alpha},d_{\beta})=V_{1}^{11}V_{2}^{\alpha\beta}+(V_{1}V_{UD})^{1\beta}(V_{2}V_{UD}^{\dagger})^{\alpha
1},$ (29) $\displaystyle
c(e_{\alpha},d^{c}_{\beta})=V_{1}^{11}V_{3}^{\beta\alpha},$ (30)
$\displaystyle
c(\nu_{l},d_{\alpha},d_{\beta}^{c})=(V_{1}V_{UD})^{1\alpha}(V_{3}V_{EN})^{\beta
l},$ (31)

where the $V$’s are mixing matrices defined as

$\displaystyle
V_{1}=U_{C}^{\dagger}U,\,\,V_{2}=E_{C}^{\dagger}D,\,\,V_{3}=D_{C}^{\dagger}E,\,\,V_{UD}=U^{\dagger}D\
\textrm{and}\ V_{EN}=E^{\dagger}N.$ (32)

The matrices $U$, $E$, $D$ and $N$ define the Yukawa couplings diagonalization
so that

$\displaystyle
U_{C}^{T}Y_{u}U=Y_{u}^{\text{diag}},\,\,D_{C}^{T}Y_{d}D=Y_{d}^{\text{diag}},\,\,E_{C}^{T}Y_{e}E=Y_{e}^{\text{diag}},\
\textrm{and}\ N^{T}Y_{\nu}N=Y_{\nu}^{\text{diag}}.$ (33)

For the predictions of the proton decay width we have taken the values for the
hadronic matrix elements given in Ref. Aoki:2017puj :

$\displaystyle\left<\pi^{0}\vphantom{(ud)_{R}u_{L}p}\right|(ud)_{R}u_{L}\left|p\vphantom{\pi^{0}(ud)_{R}u_{L}}\right>=-0.131\text{
GeV}^{2},\quad$
$\displaystyle\left<\pi^{+}\vphantom{(du)_{R}d_{L}p}\right|(du)_{R}d_{L}\left|p\vphantom{\pi^{+}(du)_{R}d_{L}}\right>=-0.186\text{
GeV}^{2},$
$\displaystyle\left<K^{+}\vphantom{(us)_{R}d_{L}p}\right|(us)_{R}d_{L}\left|p\vphantom{K^{+}(us)_{R}d_{L}}\right>=-0.049\text{
GeV}^{2},\quad$
$\displaystyle\left<K^{+}\vphantom{(ud)_{R}s_{L}p}\right|(ud)_{R}s_{L}\left|p\vphantom{K^{+}(ud)_{R}s_{L}}\right>=-0.134\text{
GeV}^{2}.$

##  Appendix C BBN bounds on the $\rho_{8}$ lifetime

We have discussed above the unification constraints and mentioned that the
allowed value of the gauge coupling at the GUT scale is sensitive to the mass
of the fields in the ${\bf{24}}$ representation. The fields $\rho_{0}$ and
$\rho_{3}$ have two body decays and can decay before Big Bang Nucleosynthesis
(BBN), while the colored fields $\rho_{8}$, $\rho_{(3,2)}$ and
$\rho_{(\bar{3},2)}$ have only three body decays. Here we discuss only the
bounds on the mass of $\rho_{8}$ by imposing that the decay lifetime must be
smaller than one second to avoid issues with BBN Fields:2014uja . Since the
masses of all fields in ${\bf{24}}$ are related it is enough to understand the
limit on the mass of $\rho_{8}$. Neglecting all SM fermion masses, the decay
width for $\rho_{8}$ decaying into three quarks is given by

$\Gamma(\rho_{8})\simeq\frac{Y_{t}^{2}h_{1}^{2}}{48(2\pi)^{3}}\frac{M_{\rho_{8}}^{5}}{M_{T}^{4}},$
(34)

where $M_{T}$ corresponds to the mass of the scalar triplet inside the ${\bf
5_{H}}$. Here we used the main decay channel defined by the coupling of $T$ to
the third generation quarks. Naively, the $\rho_{3}$ contribution to neutrino
masses can be written as

$M_{\nu}\simeq\frac{h_{1}^{2}v_{0}^{2}}{M_{\rho_{3}}},$ (35)

and using this result we find that

$\Gamma(\rho_{8})\simeq\frac{Y_{t}^{2}}{32(2\pi)^{3}}\frac{M_{\nu}}{v_{0}^{2}}\frac{M_{\rho_{8}}^{6}}{M_{T}^{4}}.$
(36)

In Fig. 7 we present the predictions for the $\rho_{8}$ lifetime using
different $M_{T}$ masses ($M_{T}>10^{12}$ GeV from bounds on the proton
lifetime) and $M_{\nu}\sim 0.1$ eV. The region shaded in blue shows the region
that leads to $\tau_{\rho_{8}}>1\ {\rm{sec}}$ and it is excluded by a naive
bound coming from Big Bang Nucleosynthesis. Therefore, the mass of $\rho_{3}$
should lie in the window

$10^{4.51}\,\,{\rm GeV}<M_{\rho_{3}}<10^{15}\,\,{\rm GeV},$ (37)

where the latter bound comes from the perturbativity of the Yukawa coupling in
Eq. (35).

Figure 7: Predictions of the $\rho_{8}$ lifetime for different $M_{T}$ masses,
we have $M_{T}>10^{12}$ GeV from bounds on the proton lifetime. The blue
shaded regions shows the predictions for $\tau_{\rho_{8}}\\!<\\!1\ {\rm{sec}}$
and excluded by a naive bound coming from Big Bang Nucleosynthesis.

##  Appendix D Beta Functions

The extra contributions to the running of the gauge couplings in Adjoint
$\,{\rm SU}(5)$ are listed in Table 2.

Fields | $b_{1}$ | $b_{2}$ | $b_{3}$
---|---|---|---
$T$ | 1/15 | 0 | 1/6
$\Phi_{1}$ | 4/5 | 4/3 | 2
$\Phi_{2}$ | 2/15 | 0 | 5/6
$\Phi_{3}$ | 1/5 | 2 | 1/2
$\Phi_{4}$ | 49/30 | 1/2 | 1/3
$\Phi_{5}$ | 1/15 | 0 | 1/6
$\Phi_{6}$ | 16/15 | 0 | 1/6
$H_{2}$ | 1/10 | 1/6 | 0
$\rho_{8}$ | 0 | 0 | 2
$\rho_{3}$ | 0 | 4/3 | 0
$\rho_{(3,2)}$ | 5/3 | 1 | 2/3
$\rho_{(\bar{3},2)}$ | 5/3 | 1 | 2/3
Table 2:  Contributions from each field to the renormalization group equations
for the gauge couplings.

The $B_{ij}$ parameters that enter in Eqs. (8)-(9) correspond to
$B_{ij}=B_{i}-B_{j}$ with

$B_{i}=b_{i}^{\rm SM}+\sum_{I}b_{iI}\,\frac{\ln\left(M_{\rm
GUT}/M_{I}\right)}{\ln\left(M_{\rm GUT}/M_{Z}\right)},$ (38)

where $M_{I}$ is the mass of the new particle living between the electroweak
and the GUT scales.

## References

  * (1) R. D. Peccei and H. R. Quinn, _CP Conservation in the Presence of Instantons_ , _Phys. Rev. Lett._ 38 (1977) 1440–1443.
  * (2) F. Wilczek, _Problem of Strong $P$ and $T$ Invariance in the Presence of Instantons_, _Phys. Rev. Lett._ 40 (1978) 279–282.
  * (3) S. Weinberg, _A New Light Boson?_ , _Phys. Rev. Lett._ 40 (1978) 223–226.
  * (4) J. Preskill, M. B. Wise and F. Wilczek, _Cosmology of the Invisible Axion_ , _Phys. Lett._ B120 (1983) 127–132.
  * (5) L. F. Abbott and P. Sikivie, _A Cosmological Bound on the Invisible Axion_ , _Phys. Lett._ B120 (1983) 133–136.
  * (6) M. Dine and W. Fischler, _The Not So Harmless Axion_ , _Phys. Lett._ B120 (1983) 137–141.
  * (7) G. G. Raffelt, _Astrophysical methods to constrain axions and other novel particle phenomena_ , _Phys. Rept._ 198 (1990) 1–113.
  * (8) M. Dine, _TASI lectures on the strong CP problem_ , in _Flavor physics for the millennium. Proceedings, Theoretical Advanced Study Institute in elementary particle physics, TASI 2000, Boulder, USA, June 4-30, 2000_ , pp. 349–369, 2000, hep-ph/0011376.
  * (9) P. Sikivie, _Axion Cosmology_ , _Lect. Notes Phys._ 741 (2008) 19–50, [astro-ph/0610440].
  * (10) J. E. Kim and G. Carosi, _Axions and the Strong CP Problem_ , _Rev. Mod. Phys._ 82 (2010) 557–602, [0807.3125].
  * (11) J. Jaeckel and A. Ringwald, _The Low-Energy Frontier of Particle Physics_ , _Ann. Rev. Nucl. Part. Sci._ 60 (2010) 405–437, [1002.0329].
  * (12) P. W. Graham, I. G. Irastorza, S. K. Lamoreaux, A. Lindner and K. A. van Bibber, _Experimental Searches for the Axion and Axion-Like Particles_ , _Ann. Rev. Nucl. Part. Sci._ 65 (2015) 485–514, [1602.00039].
  * (13) I. G. Irastorza and J. Redondo, _New experimental approaches in the search for axion-like particles_ , _Prog. Part. Nucl. Phys._ 102 (2018) 89–159, [1801.08127].
  * (14) G. Grilli di Cortona, E. Hardy, J. Pardo Vega and G. Villadoro, _The QCD axion, precisely_ , _JHEP_ 01 (2016) 034, [1511.02867].
  * (15) M. Gorghetto and G. Villadoro, _Topological Susceptibility and QCD Axion Mass: QED and NNLO corrections_ , _JHEP_ 03 (2019) 033, [1812.01008].
  * (16) J. E. Kim, _Weak Interaction Singlet and Strong CP Invariance_ , _Phys. Rev. Lett._ 43 (1979) 103.
  * (17) M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, _Can Confinement Ensure Natural CP Invariance of Strong Interactions?_ , _Nucl. Phys._ B166 (1980) 493–506.
  * (18) A. R. Zhitnitsky, _On Possible Suppression of the Axion Hadron Interactions. (In Russian)_ , _Sov. J. Nucl. Phys._ 31 (1980) 260.
  * (19) M. Dine, W. Fischler and M. Srednicki, _A Simple Solution to the Strong CP Problem with a Harmless Axion_ , _Phys. Lett._ 104B (1981) 199–202.
  * (20) M. B. Wise, H. Georgi and S. L. Glashow, _SU(5) and the Invisible Axion_ , _Phys. Rev. Lett._ 47 (1981) 402.
  * (21) P. Fileviez Perez, _Renormalizable adjoint SU(5)_ , _Phys. Lett._ B654 (2007) 189–193, [hep-ph/0702287].
  * (22) Y. Kahn, B. R. Safdi and J. Thaler, _Broadband and Resonant Approaches to Axion Dark Matter Detection_ , _Phys. Rev. Lett._ 117 (2016) 141801, [1602.01086].
  * (23) D. Budker, P. W. Graham, M. Ledbetter, S. Rajendran and A. Sushkov, _Proposal for a Cosmic Axion Spin Precession Experiment (CASPEr)_ , _Phys. Rev._ X4 (2014) 021030, [1306.6089].
  * (24) H. Georgi and S. L. Glashow, _Unity of All Elementary Particle Forces_ , _Phys. Rev. Lett._ 32 (1974) 438–441.
  * (25) H. Georgi and C. Jarlskog, _A New Lepton - Quark Mass Relation in a Unified Theory_ , _Phys. Lett._ 86B (1979) 297–300.
  * (26) P. Fileviez Perez and C. Murgui, _Renormalizable SU(5) Unification_ , _Phys. Rev._ D94 (2016) 075014, [1604.03377].
  * (27) E. Ma, _Pathways to naturally small neutrino masses_ , _Phys. Rev. Lett._ 81 (1998) 1171–1174, [hep-ph/9805219].
  * (28) B. Bajc and G. Senjanovic, _Seesaw at LHC_ , _JHEP_ 08 (2007) 014, [hep-ph/0612029].
  * (29) P. Minkowski, _$\mu\to e\gamma$ at a Rate of One Out of $10^{9}$ Muon Decays?_, _Phys. Lett._ 67B (1977) 421–428.
  * (30) R. N. Mohapatra and G. Senjanovic, _Neutrino Mass and Spontaneous Parity Nonconservation_ , _Phys. Rev. Lett._ 44 (1980) 912.
  * (31) M. Gell-Mann, P. Ramond and R. Slansky, _Complex Spinors and Unified Theories_ , _Conf. Proc._ C790927 (1979) 315–321, [1306.4669].
  * (32) T. Yanagida, _Horizontal gauge symmetry and masses of neutrinos_ , _Conf. Proc._ C7902131 (1979) 95–99.
  * (33) R. Foot, H. Lew, X. G. He and G. C. Joshi, _Seesaw Neutrino Masses Induced by a Triplet of Leptons_ , _Z. Phys._ C44 (1989) 441.
  * (34) R. T. Co, F. D’Eramo and L. J. Hall, _Supersymmetric axion grand unified theories and their predictions_ , _Phys. Rev._ D94 (2016) 075001, [1603.04439].
  * (35) S. M. Boucenna and Q. Shafi, _Axion inflation, proton decay, and leptogenesis in $SU(5)\times U(1)_{PQ}$_, _Phys. Rev._ D97 (2018) 075012, [1712.06526].
  * (36) L. Di Luzio, A. Ringwald and C. Tamarit, _Axion mass prediction from minimal grand unification_ , _Phys. Rev._ D98 (2018) 095011, [1807.09769].
  * (37) A. Ernst, A. Ringwald and C. Tamarit, _Axion Predictions in $SO(10)\times U(1)_{\rm PQ}$ Models_, _JHEP_ 02 (2018) 103, [1801.04906].
  * (38) P. Fileviez Perez, A. Gross and C. Murgui, _Seesaw scale, unification, and proton decay_ , _Phys. Rev._ D98 (2018) 035032, [1804.07831].
  * (39) P. Fileviez Perez and M. B. Wise, _On the Origin of Neutrino Masses_ , _Phys. Rev._ D80 (2009) 053006, [0906.2950].
  * (40) D. Emmanuel-Costa, C. Simoes and M. Tortola, _The minimal adjoint-SU(5) x $Z_{4}$ GUT model_, _JHEP_ 10 (2013) 054, [1303.5699].
  * (41) P. Fileviez Perez, H. Iminniyaz and G. Rodrigo, _Proton Stability, Dark Matter and Light Color Octet Scalars in Adjoint SU(5) Unification_ , _Phys. Rev._ D78 (2008) 015013, [0803.4156].
  * (42) S. Blanchet and P. Fileviez Perez, _Baryogenesis via Leptogenesis in Adjoint SU(5)_ , _JCAP_ 0808 (2008) 037, [0807.3740].
  * (43) S. Blanchet and P. Fileviez Perez, _On the Role of Low-Energy CP Violation in Leptogenesis_ , _Mod. Phys. Lett._ A24 (2009) 1399–1409, [0810.1301].
  * (44) A. Hayreter and G. Valencia, _LHC constraints on color octet scalars_ , _Phys. Rev._ D96 (2017) 035004, [1703.04164].
  * (45) V. Miralles and A. Pich, _LHC bounds on coloured scalars_ , 1910.07947.
  * (46) Super-Kamiokande collaboration, K. Abe et al., _Search for proton decay via $p\to\nu\nu K^{+}$ using 260 kiloton·year data of Super-Kamiokande_, _Phys. Rev._ D90 (2014) 072005, [1408.1195].
  * (47) Hyper-Kamiokande Proto collaboration, M. Yokoyama, _The Hyper-Kamiokande Experiment_ , in _Proceedings, Prospects in Neutrino Physics (NuPhys2016): London, UK, December 12-14, 2016_ , 2017, 1705.00306.
  * (48) Super-Kamiokande collaboration, K. Abe et al., _Search for proton decay via $p\to e^{+}\pi^{0}$ and $p\to\mu^{+}\pi^{0}$ in 0.31 megaton·years exposure of the Super-Kamiokande water Cherenkov detector_, _Phys. Rev._ D95 (2017) 012004, [1610.03597].
  * (49) Super-Kamiokande collaboration, K. Abe et al., _Search for Nucleon Decay via $n\to\bar{\nu}\pi^{0}$ and $p\to\bar{\nu}\pi^{+}$ in Super-Kamiokande_, _Phys. Rev. Lett._ 113 (2014) 121802, [1305.4391].
  * (50) A. Giveon, L. J. Hall and U. Sarid, _SU(5) unification revisited_ , _Phys. Lett._ B271 (1991) 138–144.
  * (51) Particle Data Group collaboration, M. Tanabashi et al., _Review of Particle Physics_ , _Phys. Rev._ D98 (2018) 030001.
  * (52) P. Nath and P. Fileviez Perez, _Proton stability in grand unified theories, in strings and in branes_ , _Phys. Rept._ 441 (2007) 191–317, [hep-ph/0601023].
  * (53) P. Fileviez Perez, _Fermion mixings versus d = 6 proton decay_ , _Phys. Lett._ B595 (2004) 476–483, [hep-ph/0403286].
  * (54) K. Fujikawa, _Path Integral for Gauge Theories with Fermions_ , _Phys. Rev._ D21 (1980) 2848.
  * (55) Planck collaboration, N. Aghanim et al., _Planck 2018 results. VI. Cosmological parameters_ , 1807.06209.
  * (56) R. J. Crewther, P. Di Vecchia, G. Veneziano and E. Witten, _Chiral Estimate of the Electric Dipole Moment of the Neutron in Quantum Chromodynamics_ , _Phys. Lett._ 88B (1979) 123.
  * (57) M. Pospelov and A. Ritz, _Theta vacua, QCD sum rules, and the neutron electric dipole moment_ , _Nucl. Phys._ B573 (2000) 177–200, [hep-ph/9908508].
  * (58) T. Chupp, P. Fierlinger, M. Ramsey-Musolf and J. Singh, _Electric dipole moments of atoms, molecules, nuclei, and particles_ , _Rev. Mod. Phys._ 91 (2019) 015001, [1710.02504].
  * (59) P. W. Graham and S. Rajendran, _New Observables for Direct Detection of Axion Dark Matter_ , _Phys. Rev._ D88 (2013) 035023, [1306.6088].
  * (60) R. Catena and P. Ullio, _A novel determination of the local dark matter density_ , _JCAP_ 1008 (2010) 004, [0907.0018].
  * (61) D. F. Jackson Kimball et al., _Overview of the Cosmic Axion Spin Precession Experiment (CASPEr)_ , 1711.08999.
  * (62) P. Svrcek and E. Witten, _Axions In String Theory_ , _JHEP_ 06 (2006) 051, [hep-th/0605206].
  * (63) Y. Aoki, T. Izubuchi, E. Shintani and A. Soni, _Improved lattice computation of proton decay matrix elements_ , _Phys. Rev._ D96 (2017) 014506, [1705.01338].
  * (64) B. D. Fields, P. Molaro and S. Sarkar, _Big-Bang Nucleosynthesis_ , _Chin. Phys._ C38 (2014) 339–344, [1412.1408].