$l^{-}_{\alpha}\to l^{-}_{\beta}\gamma$, $l^{-}_{\alpha}\to
l^{-}_{\beta}l^{-}_{\gamma}l^{+}_{\delta}$ (with $\alpha\neq\beta$) and
$\mu-e$ conversion in nuclei [597]. One-loop contributions to the first two
processes can occur through a dipole and box diagrams as depicted in Figure
16, with a scalar or vector mediator(s) $X(X^{\prime})$ and a SM or exotic
fermion(s) $f(f^{\prime})$ running in the loop. Contributions to $\mu-e$
conversion follows from the penguin (centre) and box (right) diagrams with
$l_{\alpha}=\mu$, $l_{\beta}=e$ and $l_{\gamma,\delta}=q$.

Unified theories often contain a number of exotic states capable of fulfilling
the role of $X$ and $f$ in Fig. 16, violating lepton flavour either by
interactions between the leptons and mediators or by mixing in the leptonic
sector. The latter case is realised in GUT models with heavy neutrinos (c.f.
left-right models in Sec. 4.1), where the mixing between active and sterile
neutrinos enhances the LFV contribution [598, 599, 600]. The contribution to
the branching ratios of the most constraining LFV processes, $\mu\to e\gamma$,
$\mu\to eee$ and $\mu-e$ conversion, in these models, with heavy neutrinos of
mass $M_{N_{I}}$, active-sterile mixing $\Theta_{\alpha I}$, a right-handed
gauge boson $W_{R}$ and left and right-handed scalar triplets $\delta_{L}$ and
$\delta_{R}$, can be written as [601, 602, 174]

$\displaystyle BR(\mu\to e\gamma)$ $\displaystyle\sim 1.5\times
10^{-7}|\Theta_{eI}^{*}\Theta_{\mu
I}|^{2}\left(\frac{g_{R}}{g_{L}}\right)^{4}\left(\frac{m_{N_{I}}}{m_{W_{R}}}\right)^{4}\left(\frac{1\text{
TeV}}{M_{W_{R}}}\right)^{4},$ $\displaystyle BR(\mu\to eee)$
$\displaystyle\sim\frac{1}{2}|\Theta_{eI}^{*}\Theta_{\mu
I}|^{2}|\Theta_{eI}|^{4}\left(\frac{g_{R}}{g_{L}}\right)^{4}\left(\frac{m_{N_{I}}}{m_{W_{R}}}\right)^{4}\left(\frac{M_{W_{R}}^{4}}{M_{\delta_{R}}^{4}}+\frac{M_{W_{R}}^{4}}{M_{\delta_{L}}^{4}}\right),$
(69) $\displaystyle R^{N}(\mu-e)$ $\displaystyle\sim 0.73\times
10^{-9}X_{N}|\Theta_{eI}^{*}\Theta_{\mu
I}|^{2}\left(\frac{g_{R}}{g_{L}}\right)^{4}\left(\frac{m_{N_{I}}}{m_{W_{R}}}\right)^{4}\left(\frac{1\text{
TeV}}{M_{\delta_{R}}}\right)^{4}\left(\log\frac{m_{\delta_{R}}^{2}}{m_{\mu}^{2}}\right)^{2}.$

In supersymmetric GUTs there are many possible sources of lepton flavour
violation, parametrised by the mixing in the slepton sector of the MSSM, which
has contributions to LFV processes of the type [603]

$BR(l_{\alpha}\to l_{\beta}\gamma)\approx\frac{48\pi^{3}\alpha_{\rm
em}}{G_{F}^{2}}\frac{|(m_{\tilde{L}}^{2})_{ij}|^{2}+|(m_{\tilde{e}}^{2})_{ij}|^{2}}{M_{SUSY}^{8}}BR(l_{\alpha}\to
l_{\beta}\nu_{\alpha}\bar{\nu}_{\beta}).$ (70)

Off-diagonal entries in the slepton mass matrices can be the result of non-
minimal flavour violating interactions or non-canonical Yukawa textures at the
GUT scale, where the soft masses are supposed to unify [604, 605]. In
addition, slepton mixing can be induced in minimal flavour violating (MFV)
SUSY via seesaw mechanisms [606, 607, 608, 609, 610, 611, 612] or, for Yukawa-
unified theories (see Sec. 3.3), it can depend on the CKM matrix at the GUT
scale [596]. LFV contributions can also arise in SUSY models where $R$-parity
is violated, explicitly or spontaneously, with interaction terms of the type
$l_{i}l_{j}\tilde{\nu}_{k}$ that induce tree-level contributions to $l\to lll$
decays and $\mu\to e$ conversion, as well as new dipole contributions to $l\to
l\gamma$ [613, 614, 615, 616, 617].

Process | Branch. Frac. | Reference
---|---|---
$\mu^{-}\to e^{-}\gamma$ | $4.2\times 10^{-13}$ | MEG [618]
$\tau^{-}\to e^{-}\gamma$ | $5.4\times 10^{-8}$ | BaBar [619],Belle [620]
$\tau^{-}\to\mu^{-}\gamma$ | $5.0\times 10^{-8}$ | BaBar [619],Belle [620]
$\mu^{-}\to e^{-}e^{-}e^{+}$ | $1.0\times 10^{-12}$ | SINDRUM [621]
$\tau^{-}\to e^{-}e^{-}e^{+}$ | $1.4\times 10^{-8}$ | BaBar [622],Belle [623]
$\tau^{-}\to\mu^{-}\mu^{-}\mu^{+}$ | $1.2\times 10^{-8}$ | ATLAS [624],BaBar [622],Belle [623],LHCb [625]
$\tau^{-}\to\mu^{-}e^{-}e^{+}$ | $1.1\times 10^{-8}$ | BaBar [622],Belle [623]
$\tau^{-}\to e^{-}e^{-}\mu^{+}$ | $0.84\times 10^{-8}$ | BaBar [622],Belle [623]
$\tau^{-}\to e^{-}\mu^{-}\mu^{+}$ | $1.6\times 10^{-8}$ | BaBar [622],Belle [623]
$\tau^{-}\to\mu^{-}\mu^{-}e^{+}$ | $0.98\times 10^{-8}$ | BaBar [622],Belle [623]
$\mu-e$ (Ti) | $1.7\times 10^{-12}$ | SINDRUM II [626]
$\mu-e$ (Pb) | $4.6\times 10^{-11}$ | SINDRUM II [627]
$\mu-e$ (Au) | $8.0\times 10^{-13}$ | SINDRUM II [628]
$d_{e}$ | $1.1\times 10^{-29}~{}e~{}cm$ | ACME II [629]
$d_{\mu}$ | $1.9\times 10^{-19}~{}e~{}cm$ | Muon g-2 [630]
$d_{\tau}$ | $4.5\times 10^{-17}~{}e~{}cm$ | Belle [631]
$a_{e}(10^{-13})$ | $11596521809.1\pm 2.6$ |  [632]
$a_{\mu}(10^{-10})$ | $11659208.9\pm 8.7$ | Muon g-2 [633]
Table 2: _Upper bounds at $90\%$ C.L on LFV processes and EDMs, and
measurements of AMMs, along with the experiments that provided them. The HFLAV
average is quoted for limits by different experiments [549]._

The anomalous electric, $d_{i}$, and magnetic, $a_{i}$, dipole moments of
quarks and leptons follow from processes identical to the diagram on the left
in Fig. 16, where $l_{\alpha}$ and $l_{\beta}$ have the same flavour. Hence
contributions from heavy states running in the loops can have a strong effect
that can be tested experimentally. As with LFV, the SM contribution to
electric dipole moments (EDMs) is tiny, as it is proportional to the CP-
violation phase in the CKM matrix [634, 635]. EDMs have not been observed
experimentally, so deviations from the SM prediction due to CP-violation in
other sectors is strongly constrained [636]. Other sources of CP violation can
appear in neutrino mixing [635], phases in fermion-sfermion couplings [637] or
extended Higgs sectors [638]. Anomalous magnetic moments (AMM), on the other
hand, have been measured with extreme accuracy. In fact, the precision of both
the experimental measurement and theoretical prediction for $a_{\mu}$ has
shown a discrepancy of more than 3 standard deviations [34]. New physics
contributions have been shown to resolve that tension, particularly in the
context of supersymmetry [639].

In the presence of light leptoquarks anomalous dipole moments of leptons or
quarks are one-loop processes [480]. A special feature of non-chiral
leptoquarks, such as $R_{2}$ with couplings (65), is that both $l_{ij}$ and
$r_{ij}$ are non-zero in the interaction Lagrangian
$\bar{q}^{i}\left[l_{ij}P_{R}+r_{ij}P_{L}\right]\ell^{j}\phi$ which then leads
to the anomalous moment of the muon:

$\begin{split}a_{\mu}=-\frac{3m_{\mu}^{2}}{8\pi^{2}m_{\phi}^{2}}\sum_{q}\Bigg{[}&\left(|l_{q\mu}|^{2}+|r_{q\mu}|^{2}\right)\left(Q_{\phi}/4-1/6\right)-\frac{m_{q}}{m_{\mu}}\log\frac{m_{q}^{2}}{m_{\phi}^{2}}\mathrm{Re}(r_{q\mu}^{*}l_{q\mu})\left(Q_{\phi}-1\right)\Bigg{]},\end{split}$
(71)

where $m_{\phi}$ and $Q_{\phi}$ are the charge and mass of the leptoquark and
$q$ is the flavour of the quark in the loop (see the leftmost diagram in Fig.
16). Shown is the leading order contribution in $m_{q}$. The first term
increases $a_{\mu}$ only when $Q_{\phi}>2/3$ and it is present for all scalar
LQ states that couple to a muon. The second term is relevant for non-chiral
LQs and it is chirally enhanced by $m_{q}/m_{\ell}$, possibly leading to large
effects with moderate couplings to $b$ or $t$ quark. Furthermore, the sign of
the non-chiral term is adjustable. On the other hand, the same mechanism also
enhances dipole LFV transitions, e.g. $\mu\to e\gamma$, $\tau\to\mu\gamma$
[560]. Non-chiral LQs may also generate quark or lepton electric dipole
moments at 1-loop [640, 641].

Whichever the mechanism, it is clear that LFV and anomalous dipole moments are
predicted by many GUT models, with varying strengths for different processes.
The full list of processes and their current experimental upper bounds and
measurements can be seen in Table 2, where the experiments that have studied
the processes are detailed. Furthermore, new experiments are being developed
at this moment that will attempt to improve the limits on processes like
$\mu\to eee$ (Mu3e [642]) and $\mu-e$ conversion (COMET [643], Mu2e [644]),
with projected limits up to four orders of magnitude stronger than previous
studies. Additionally, a new measurement of $a_{\mu}$ has been performed by
the Muon g-2 experiment and it is expected to be released soon [645], which
may confirm the deviation observed before, and thus further motivate the need
of new physics.

###  7.5 Neutrinoless Double Beta Decay

This rare nuclear process corresponding to a simultaneous conversion of two
neutrons to two protons and two electrons within the nucleus [646] is of great
interest for particle physics, as it clearly does not conserve lepton number,
and thus violates the corresponding accidental Abelian global symmetry of the
SM. Consequently, a strong experimental effort is being made to observe this
unique process. Unfortunately, its observation will be very difficult, as
$0\nu\beta\beta$ decay is expected to be extremely rare. There is a number of
experiments, some in operation, other being constructed or planned, attempting
to measure the decay. An overview of the major collaborations is shown in Tab.
3. The current experimental lower limits on its half-life are around $10^{26}$
years [647, 648] and the future searches should reach sensitivities by one or
two orders of magnitude higher.

Experiment | Isotope | Status |  $M$ [kg] |  $T_{1/2}^{0\nu\beta\beta}$ limit [y]
---|---|---|---|---
CUORE |  130Te | running | 200 | ($3.5\times 10^{26}$)
EXO-200 |  136Xe | running | 110 | $1.1\times 10^{25}$
nEXO |  136Xe | R&D | 5000 | ($10^{27}\text{-}10^{28}$)
GERDA |  76Ge | running | 21.6 | $5.3\times 10^{25}$
|  | in progress |  40 | ($\sim 10^{26}$)
KamLAND-Zen |  136Xe | running | 383 | $1.1\times 10^{26}$
|  | in progress | 600 | ($2\times 10^{26}$)
LEGEND |  76Ge | R&D | 200 | ($\sim 10^{27}$)
|  | R&D | 1000 | ($\sim 10^{28}$)
Majorana Dem. |  76sGe | running | 44.1 | $1.9\times 10^{25}$
NEXT |  136Xe | in progress (demo) | 100 | ($5.9\times 10^{25}$)
SNO+ |  130Te | in progress | 1300 | ($2\times 10^{26}$)
SuperNEMO |  82Se (${}^{150}\text{Nd}$) | in progress (demo) | 100 | ($\sim 10^{26}$)
Table 3: _An overview of both current and future major $0\nu\beta\beta$ decay
searches. For each experiment the following information is shown: used
isotope, operational status, the deployed mass $M$ of the isotope in question
and the measured or expected (for experiment in preparation these values are
shown in parentheses) sensitivity $T_{1/2}^{0\nu\beta\beta}$. For some
experiments (GERDA, KamLAND-Zen, LEGEND) characteristics of more stages of
development are given. In case of SuperNEMO, the primary isotope to be tested
is 82Se and in the future the measurement will be repeated with a
${}^{150}\text{Nd}$ source._

Figure 17: _The standard mass mechanism of $0\nu\beta\beta$ decay (left) and a
non-standard contribution that can be constructed in the left-right symmetric
models using vector currents of opposite chiralities (right)._

As can be shown, the existence of $0\nu\beta\beta$ decay implies the Majorana
nature of neutrinos (and vice versa) [649, 650, 651] and as such it represents
one of the best probes of this BSM hypothesis. Therefore, in GUT models
allowing for Majorana neutrino mass generation $0\nu\beta\beta$ decay can be
in principle always triggered. This, however, does not say anything about the
underlying mechanism and the resulting $0\nu\beta\beta$ decay rate. The
prominent _standard (mass) mechanism_ of $0\nu\beta\beta$ decay assumes a
light Majorana neutrino exchange between the two beta-decaying neutrons. In
the SM with light massive neutrinos this process can be depicted as shown in
Fig. 17 (left). Besides the standard scenario a number of _non-standard
mechanisms_ triggering $0\nu\beta\beta$ decay can be constructed. The
effective treatment of these exotic mechanisms can be conveniently employed,
see e.g. Refs. [652, 653, 654, 655, 656].

As for the UV-complete $0\nu\beta\beta$ decay mechanisms, a variety of
interesting ones can be constructed within GUTs. For instance, in the left-
right symmetric models (where, of course, the standard light neutrino exchange
is available) one can think of several exotic mechanisms involving exchange of
heavy neutrino as well as light and heavy $W$ vector bosons [657]. In the
simplest exotic case the light neutrino exchange is substituted by a heavy
right-handed neutrino exchange, which means that the involved vector currents
and emitted electrons must be also right-handed. Due to the large mass of the
propagating neutrino, the interaction can be considered to be contact and we
refer to this contribution as to short-range mechanism. Since the right-handed
currents are present in left-right symmetric models, it is also possible to
draw $0\nu\beta\beta$ decay mechanisms, in which the neutrino exchange does
not violate chirality. This means that the contribution is not proportional to
the neutrino mass and the two outgoing electrons are of opposite chiralities.
A possible mechanism of this type is depicted in Fig. 17 (right). As apparent,
the diagram involves one right-handed and one left-handed vector current and
since the light neutrino propagator is present, one refers to this
contribution as to a long-range $0\nu\beta\beta$ decay mechanism.

Figure 18: _The exotic $0\nu\beta\beta$ decay mechanisms involving scalar or
vector leptoquarks $S$, $V^{\mu}$._

Leptoquarks, particles appearing prominently in GUTs, can also trigger non-
standard $0\nu\beta\beta$ decay contributions. It has been described that this
is the case, when different leptoquark multiplets mix via a possible
leptoquark-Higgs coupling violating lepton number [658, 659]. Diagrams of this
type of contributions to $0\nu\beta\beta$ decay are shown in Fig. 18. The
specific helicity structure of the effective four-fermion interaction leads to
the fact that this contribution can dominate over the standard mass mechanism.
The current lower limits on $0\nu\beta\beta$ decay half-life then allow to
derive the bounds on corresponding leptoquark parameters.

Neutrinoless double beta decay can be triggered also in supersymmetric
theories aspiring for grand unification. In the simplest case, if the MSSM
with broken $R$ parity is considered, $0\nu\beta\beta$ decay diagrams
involving $R$-parity-violating couplings and supersymmetric mediators can be
drawn [660, 661, 662, 663, 664]. An example of a supersymmetric
$0\nu\beta\beta$ decay mechanism is depicted in Fig. 19. Again, from non-
observation of $0\nu\beta\beta$ decay it is possible to derive limits on the
unknown model parameters.

Figure 19: _An example of a $0\nu\beta\beta$ decay mechanism that can be
triggered in R-parity-violating MSSM._

##  8 Outlook and future prospects for GUTs

Among the vast landscape of theories beyond the Standard Model, Grand Unified
Theories stand out as appealing candidates. As we have seen, GUTs are a
collection of ideas from group theory, supersymmetry, neutrino physics,
flavour physics and more, which positions them as some of the most complete
and attractive theories in the literature. Indeed they are among the few BSM
theories capable of simultaneously affecting the highest energy scales,
influencing the cosmology of the early Universe, and the low energies, within
reach of colliders and terrestrial experiments.

Throughout this review article we have provided a rough sketch of the status
of GUTs and some of the associated research in the field. We have described
the basic principles behind them and their first appearance in the world of
particle physics in the late 70s. A lot of effort was spent in the years after
and many of the greatest models were designed at that time. Research in GUTs
has continued since, focusing either on specific topics within and alongside
them, or on particular models that compiled a few advances in the field. We
have thus summarized a non-exhaustive selection of topics and models that are
at the forefront of research in unified theories, aiming to provide an
overview of the current state of the art.

We are fortunate enough to live in a time where experimental searches are
abundant and they cover a rather vast range of fronts. The most cutting-edge
technologies have been and are being developed to push the boundaries of our
current understanding of particle physics and cosmology. Grand unified
theories are and will be put under the microscope by many of these
experimental advances, which will confirm, constrain or outright exclude some
of the existing models.

The recent observation of gravitational wave signatures opens a new window
into the history of the Universe, where events and phenomena that ocurred in
the early Universe can be observed with gravitational wave detectors. Cosmic
phase transitions associated with patterns of symmetry breaking in unified
theories are such events, as they can be the source of stochastic
gravitational waves that can be observed today. Transition temperatures above
the EW scale, typically associated with the breaking of some intermediate step
in a GUT model, can be studied by future gravitational wave experiments such
as a LISA, the Einstein telescope, Kagra, the Cosmic Explorer, BBO and DEIGO.
Also in the cosmological frontier, GUTs can have a serious impact on the
inflationary epoch of the Universe, testable in measurements of the CMB, and
can contribute to the baryon asymmetry of the Universe, via baryo and
leptogenesis.

At the time of writing we have reached the end of the second run of the LHC,
with an outstanding recorded integrated luminosity of about 150 ${\rm
fb}^{-1}$. Analyses of the accumulated data, however, are still under way and
they will probably spill well into the start run 3 in 2021. Many of the
analyses already published have strong consequences for the predictions of
unified theories, as are direct searches for supersymmetry, leptoquarks or
other exotics at ATLAS and CMS. Upcoming results from ongoing and future
analyses of the results from the LHC experiments may strengthen the bounds on
light states as predicted by SUSY GUTs and other models, or they might show
hints of the existence of new particles, whose relevance for GUTs would need
to be determined. Upgraded versions of the LHC (HL-LHC, VLHC or FCC) or other
future colliders (ILC, CLIC) will certainly boost this programme with
increased accuracy and higher energies, which will further probe the low-
hanging states predicted by GUTs.

Where colliders search for the low scale predictions of GUTs, precision
experiments can explore the intermediate and high scales associated with
unification. Nucleon decay limits are often among the strongest probes of
fully unified theories and future experiments such as Hyper-Kamiokande and
DUNE may set even stronger exclusion limits or perhaps measure signs of proton
decay, which would be a smoking gun for GUTs. Furthermore, GUTs can provide
contributions to a number of flavour and precision observables, such as LFU,
LFV, EDM, AMM or $0\nu\beta\beta$, some of which are in tension with the SM.
Confirmation of these flavour anomalies with more collected data by LHCb and
other experiments would be undeniable evidence of the need for new physics
models and GUTs are very well suited for that purpose.

To conclude, Grand Unified Theories are still at the vanguard of research in
BSM models. They can explain many of the issues of the SM and can accommodate
the recent results from the cosmological, precision and collider frontiers
with relative ease. Contrary to “simplified” models, GUTs are complete
theories that can simultaneously make a large number of testable predictions
on the different fronts. Fortunately, these predictions can be explored by
upcoming analyses and future experiments, which can set strong exclusion
limits in a subset of GUT models. On a more optimistic note, any observation
in, for instance, SUSY searches at colliders, gravitational waves signatures
or proton decay will stack the odds in favour of some GUT models and will
significantly shape the future of the research in particle physics.

## Acknowledgements

The authors would like to thank P. Athron, F. Deppisch, A. Kvellestad and Y.
Zhang for helpful discussions. T.E.G. was partly funded by the Research
Council of Norway under FRIPRO project number 230546/F20 and partly supported
by the ARC Centre of Excellence for Particle Physics at the Tera-scale, grant
CE110001004. N.K. was supported by the Slovenian Research Agency under the
research core funding grant P1-0035 and No. J1-8137. TRIUMF receives federal
funding via a contribution agreement with the National Research Council of
Canada and the Natural Science and Engineering Research Council of Canada.

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