93. Grand Unified Theories

1

93. Grand Unified Theories
Revised August 2023 by A. Hebecker (Heidelberg U.), J. Hisano (Nagoya U.; KMI, Nagoya U.)
and N. Nagata (UTokyo).

93.1 The standard model
The Standard Model (SM) may be defined as the renormalizable field theory with gauge group
GSM = SU (3)C × SU (2)L × U (1)Y , with 3 generations of fermions in the representation(3, 2)1/3 + (3̄, 1)−4/3 + (3̄, 1)2/3 + (1, 2)−1 + (1, 1)2 ,(93.1)

and a scalar Higgs doublet H transforming as (1, 2)1 . Here and below we use boldface numbers
to specify the dimension of representations of non-Abelian groups (in this case fundamental andantifundamental) and lower indices for U (1) charges. The fields of Eq. (93.1) should also be familiar
as [Q, uc , dc , L, ec ], with Q = (u, d) and L = (ν, e) being the quark and lepton SU (2)-doublets and
uc , dc , ec charge conjugate SU (2)-singlets.1 Especially after the discovery of the Higgs, this modelis remarkably complete and consistent with almost all experimental data.
A notable exception are neutrino masses, which are known to be non-zero but are absent in the
SM even after the Higgs acquires its vacuum expectation value (VEV). The minimalist attitude is
to allow for the dimension-five operator (HL)2 [1], which induces (Majorana) neutrino masses. In
the seesaw mechanism [2–4] this operator is generated by integrating out heavy singlet fermions(right-handed (r.h.) neutrinos). Alternatively, neutrinos can have Dirac masses if light singlet
neutrinos are added to the SM spectrum.
Conceptual problems of the SM include the absence of a Dark Matter candidate, of a mechanism
for generating the baryon asymmetry of the Universe, and of any reason for the observed smallnessof the θ parameter of QCD (θQCD ). In addition, the apparently rather complex group-theoretic dataof Eq. (93.1) remains unexplained. Together with the abundance of seemingly arbitrary coupling
constants, this disfavors the SM as a candidate fundamental theory, even before quantum gravity
problems arise at energies near the Planck mass MP .
To be precise, there are 19 SM parameters which have to be fitted to data: Three gauge
couplings2 g3 , g2 and g1 , 13 parameters associated with the Yukawa couplings (9 charged fermion
masses, three mixing angles and one CP phase in the CKM matrix.), the Higgs mass and quartic
coupling, and θQCD . In addition, Majorana neutrinos introduce 3 more masses and 6 mixing angles
and phases. As we will see, the paradigm of grand unification addresses mainly the group theoretic
data of Eq. (93.1) and the values of the three gauge couplings. In many concrete realizations, itthen impacts also the other mentioned issues of the SM, such as the family structure and fermion
mass hierarchy.
More specifically, after precision measurements of the Weinberg angle θW in the LEP experiments, supersymmetric GUTs (SUSY GUTs) have become the leading candidates in the searchfor ‘Physics beyond the SM’. Supersymmetry (SUSY) is a symmetry between bosons and fermions
which requires the addition of superpartners to the SM spectrum. If SUSY is motivated as a solution
to the gauge hierarchy problem (i.e. to the naturalness or fine-tuning problem of the electroweak
scale) [5], superpartners have to be present near the weak scale. Assuming a minimal Higgs sector,
SUSY GUTs [6] then lead to the prediction of θW , in good agreement with subsequent observations [7]. However, the non-discovery of new particles at the LHC puts into question the presence of
new physics at the TeV scale in general and in particular of low-scale supersymmetry. Still, SUSY
may be present just outside the presently explored energy domain.
1

In our convention the electric charge is Q = T3 + Y /2 and all our spinor fields are left-handed(l.h.).
p
Equivalently, the SU (2)L and U (1)Y couplings are denoted as g = g2 and g 0 =3/5 g1 . One also uses
αs = α3 = (g32 /4π), αEM = (e2 /4π) with e = g sin θW and sin2 θW = (g 0 )2 /(g 2 + (g 0 )2 ).
2

S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)
31st May, 2024 10:14am

93. Grand Unified Theories

2

The measured Higgs mass (125 GeV) is in principle consistent with this picture, assuming
superpartners in the region of roughly 10 TeV. Such heavy superpartners then induce radiative
corrections raising the Higgs mass above the Z boson mass mZ [8, 9]. However, from the vantage
point of the hierarchy problem, heavy superpartners are problematic: They also contribute to
SUSY-breaking Higgs mass parameters and thereby to the Higgs potential, tending to raise the Z
mass. As a result, the incarnation of SUSY in terms of the minimal supersymmetric SM (MSSM)
is becoming questionable. Turning the logic around, one may say that compared to expectations
based on the MSSM with superpartner masses below about 1 TeV, the measured Higgs mass
value of 125 GeV is somewhat too high [10]. Independently, the LHC has disfavored light coloredsuperpartners (which does not imply that all superpartners are heavy). These facts represent newhints for future work on SUSY GUTs or on GUTs without TeV-scale supersymmetry.

93.2 Basic group theory and charge quantization
93.2.1 SU (4)C × SU (2)L × SU (2)RHistorically, the first attempt at unification was the Pati-Salam model with gauge group GP S =SU (4)C × SU (2)L × SU (2)R [11]. It unifies SM fermions in the sense that one generation (plus an
extra SM singlet) now comes from the (4, 2, 1) + (4, 1, 2) of GP S . This is easy to verify from thebreaking pattern SU (4)C → SU (3)C × U (1)B−L together with the identification of SM hypercharge
as a linear combination between B − L (baryon minus lepton number) and the T3 generator ofSU (2)R . This model explains charge quantization, that is, why all electric charges are integer
multiples of some smallest charge in the SM. Concretely, the 4 and 4 of SU (4)C identify lepton
number as the 4th colour and the tracelessness of the diagonal generator implies that quark charges
are expressed in terms of 1/Nc fractions of lepton charges. However, GP S is not simple (containing
three simple factors), and thus it does not predict gauge coupling unification.
93.2.2 SU (5)Since GSM has rank four (two for SU (3)C and one for SU (2)L and U (1)Y , respectively), therank-four group SU (5) is the minimal choice for unification in a simple group [12]. The three SM
gauge coupling constants derive from a universal coupling αG at the GUT scale MG . Explicitly
embedding GSM in SU (5) is straightforward, with SU (3)C and SU (2)L corresponding e.g. to theupper-left 3 × 3 and lower-right 2 × 2 blocks, respectively, in traceless 5 × 5 matrices for SU (5)generators of the fundamental representation. The U (1)Y corresponds to matrices generated bydiag(−2/3, −2/3, −2/3, 1, 1) and hence commutes with SU (3)C × SU (2)L ⊂ SU (5). It is theneasy to derive how one SM generation precisely comes from the 10 + 5 of SU (5) (where 10 is theantisymmetric rank-2 tensor):0
ucb −ucg ur dr
−uc
0
ucr
ug dg 
 b

 c

c
0
ub db 10 :  ug −urc−ur −ug −ub0e −dr −dg −db −ec 0




and

 c 
dr
 dc  g 5 :  dcb  .


 e 

(93.2)−νe

In addition to charge quantization this structure explains why the l.h. quark and lepton states fallin SU (2)L doublets while the r.h. states are singlets.
Since SU (5) has 24 generators, SU (5) GUTs have 12 new gauge bosons known as X bosons (orX/Y bosons) in addition to the SM. X bosons form an SU (3)C -triplet and SU (2)L -doublet. Theirinteraction connects quarks and leptons such that baryon and lepton numbers are not conserved
and nucleon decay is predicted. Furthermore, U (1)Y hypercharge is automatically quantized since
it is embedded in SU (5).
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93. Grand Unified Theories

3

In order to break the electroweak symmetry at the weak scale and give mass to quarks andleptons, Higgs doublets are needed. In the minimal SU (5) model, they can sit in either a 5H or
5̄H . The three additional states are referred to as color-triplet Higgs scalars. Their couplings also
violate baryon and lepton numbers, inducing nucleon decay. In order not to violently disagreewith the non-observation of nucleon decay, the triplet mass must be greater than ∼ 1011 GeV [13].
Moreover, in SUSY GUTs [6], in order to cancel anomalies as well as give mass to both up anddown quarks, both Higgs multiplets 5H and 5̄H are required. As we shall discuss later, nucleon
decay now constrains the Higgs triplets to have mass significantly greater than MG in the minimal
SUSY SU (5) GUT since integrating out the Higgs triplets generates dimension-five baryon-numberviolating operators [14]. The mass splitting between doublet and triplet in the 5H (and 5H ) comesfrom their interaction with the SU (5) breaking sector.
93.2.3 SO(10)While SU (5) allows for the minimal GUT models, unification is not complete: Two independent
representations, 10 and 5̄, are required for one SM generation. A further representation, an SU(5)singlet, has to be added to serve as r.h. neutrino in the seesaw mechanism. In this case, ther.h. neutrino masses are not necessarily related to the GUT scale. By contrast, a single 16dimensional spinor representation of SO(10) accommodates a full SM generation together with an
extra singlet, potentially providing a r.h. neutrino [15]. This is most easily understood from thebreaking pattern SO(10) → SU (5) × U (1)X and the associated branching rule3 16 = 10−1 + 5̄3 +1−4 . Here the indices refer to charges under the U (1)X subgroup, which is orthogonal to SU (5) andreflects the fact that SO(10) has rank five. From the above, it is easy to see that U (1)X charges
can be given as 2Y − 5(B − L). Intriguingly, all representations of SO(10) are anomaly free infour dimensions (4d). Thus, the absence of anomalies in an SU (5)-GUT or a SM generation canbe viewed as deriving from this feature.
We now describe in more detail how one family of quarks and leptons appears in the 16. To
understand this, recall that the Γ -matrices of the 10d Clifford algebra give rise to five independent,
anticommuting ‘creation-annihilation’ operators Γ a± = (Γ 2a−1 ± iΓ 2a )/2 with a = 1, ..., 5. These
correspond to five fermionic harmonic oscillators or “spin” 1/2 systems. The 32-dimensional tensor
product of those is reducible since the 10d rotation generators Mmn = −i[Γ m , Γ n ]/4 (m, n =1, ..., 10) always flip an even number of “spins”. This gives rise to the 16 as displayed in Table 93.1.
Next, one also recalls that the natural embedding of SU (5) in SO(10) relies on ‘pairing up’ the
10 real dimensions to produce 5 complex dimensions, R10 ≡ C5 , similarly to the paring up of Γ m sused above. This makes it clear how to associate one |± > system to each complex dimensionof SU (5), which explains the labeling of the “spin” columns in Table 93.1: The first three andlast two “spins” correspond to SU (3)C and SU (2)L , respectively. In fact, an SU (3)C rotation
just raises one color index and lowers another, changing colors {r, g, b}, or changes relative phases
between the three spin states. Similarly, an SU (2)L rotation raises one weak index and lowers
another, thereby flipping the weak isospin from up to down or vice versa, or changes the relative
phase between the two spin states. In this representation U (1)Y hypercharge is simply given byPP
Y = −2/3(
color spins) + (weak spins). SU (5) rotations corresponding to X bosons thenraise (or lower) a color index, while at the same time lowering (or raising) a weak index. It is easy
to see that such rotations can mix the states {Q, uc , ec } and {dc , L} among themselves and ν c
is a singlet. Since SO(10) has 45 generators, additional 21 gauge bosons are introduced including
the U (1)X above. The 20 new SO(10) rotations not in SU (5) are then given by either raising
any two spins or lowering them. With these rotations, 1 and 5 are connected with 10. The lastP
P
SO(10) rotation changes phases of states with weight 2( color spins) + 2( weak spins), which
3

Useful references on group theory in the present context include [16] and refs. therein.

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93. Grand Unified Theories

4
corresponds to U (1)X .
Table 93.1:
SO(10).

Quantum numbers of 16-dimensional representation of

state
νc
ec
ur
dr
ug
dg
ub
db
ucr
ucg
ucb
dcr
dcg
dcb
ν
e

Y
0
2
1/3
1/3
1/3
1/3
1/3
1/3
−4/3
−4/3
−4/3
2/3
2/3
2/3
−1
−1

Color
−−−
−−−
+−−
+−−
−+−
−+−
−−+
−−+
−++
+−+
++−
−++
+−+
++−
+++
+++

Weak
−−
++
−+
+−
−+
+−
−+
+−
−−
−−
−−
++
++
++
−+
+−

SU (5)
1

SO(10)10
16

5̄

SO(10) has two inequivalent maximal subgroups and hence breaking patterns, SO(10) →SU (5)×U (1)X and SO(10) → SU (4)C ×SU (2)L ×SU (2)R . In the first case, one can carry on breaking to GSM ⊂ SU (5) precisely as in the minimal SU (5) case above. Alternatively, one can identify
U (1)Y as an appropriate linear combination of U (1)X and the U (1) factor from SU (5), leading to theso-called flipped SU (5) [17] as an intermediate step in breaking SO(10) to GSM . In the second case,
we have an intermediate Pati-Salam model thanks to the branching rule 16 = (4, 2, 1) + (4, 1, 2).
Finally, SO(10) can break directly to the SM at MG . Gauge coupling unification remains intact
in the case of this ‘direct’ breaking and for the breaking pattern SO(10) → SU (5) → GSM (withSU (5) broken at MG ). In the case of intermediate-scale Pati-Salam or flipped SU (5) models, gauge
coupling predictions are modified. The Higgs multiplets in the minimal SO(10) come from the fundamental representation, 10H = 5H + 5̄H . Note, only in SO(10) does the representation type
distinguish SM matter from Higgs fields.
93.2.4 Beyond SO(10)Finally, larger symmetry groups can be considered. For example, the exceptional group E6
has maximal subgroup SO(10) × U (1) [18]. Its fundamental representation branches as 27 =161 + 10−2 + 14 . Another maximal subgroup is SU (3)C × SU (3)L × SU (3)R ⊂ E6 with branchingrule 27 = (3, 3, 1) + (3̄, 1, 3̄) + (1, 3̄, 3). Independently of any underlying E6 , the group [SU (3)]3
with additional permutation symmetry Z3 interchanging the three factors can be considered. This is
known as “trinification” [19]. The E6 → [SU (3)]3 breaking pattern has been used in phenomenological analyses of the heterotic string [20]. However, in larger symmetry groups, such as E6 , SU (6),etc., there are now many more states which have not been observed and must be removed from theeffective low-energy theory.
Intriguingly, the logic by which GSM is a maximal subgroup of SU (5), which together with
U (1)X is a maximal subgroup of SO(10), continues in a very elegant and systematic way up to
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593. Grand Unified Theories

the largest exceptional group. The resulting famous breaking chain E8 →E7 →E6 → SO(10) →SU (5) → GSM together with the special role played by E8 in group and in string theory is atantalizing hint at deeper structures. However, since all representations of E8 and E7 are real andcan not lead to 4d chiral fermions, this is necessarily outside the 4d GUT framework.

93.3 GUT breaking and doublet-triplet splitting
In the standard, 4d field-theoretic approach to GUTs, the unified gauge group is broken spontaneously by an appropriate GUT Higgs sector. Scalar potentials (or superpotentials in SUSY
GUTs) exist whose vacua spontaneously break SU (5) or SO(10). While these potentials are ad hoc(just like the Higgs potential in the SM), the most naive expectation is that all their dimensionful
parameters are O(MG ). In the simplest case of SU (5), the 24 (adjoint) GUT Higgs develops a
VEV along the GSM -singlet direction as hΦi ∝ diag(−2/3, −2/3, −2/3, 1, 1). In order for SO(10)to break to SU (5), the 16 or 126, which have a GSM -singlet with non-zero U (1)X charge, get a
VEV.
The masses of doublet and triplet in the 5H (and 5H ) generically split due to their coupling to
the GUT Higgs. In addition, both the doublet and the triplet masses also get an equal contribution
from an SU (5)-invariant GUT-scale mass term. Without any further structure, an extreme finetuning between two large effects is then necessary to keep the doublet mass at the electroweak scale.
Supersymmetry plays an important role in forbidding large radiative correction to the doublet
mass due to the non-renormalization theorem [5]. However, even in this case we have to fine tune
parameters at tree level. This is the doublet-triplet splitting problem which, in the SUSY context,
is clearly related to the µ-term problem of the MSSM (the smallness of the coefficient of µHu Hd ).
Several mechanisms for natural doublet-triplet splitting have been suggested under the assumption of supersymmetry, such as the sliding singlet [21], missing partner [22], missing VEV [23],and pseudo-Nambu-Goldstone boson mechanisms [24]. Particular examples of the missing partner
mechanism for SU (5) [25], the missing VEV mechanism for SO(10) [26,27] and the pseudo-NambuGoldstone boson mechanism for SU (6) [28] have been shown to be consistent with gauge coupling
unification and nucleon decay. From the GUT-scale perspective, one is satisfied if the triplets arenaturally heavy and the doublets are massless. In other words, one wants to ensure that the fundamental superpotential term µHu Hd is absent (µ ' 0) while the corresponding term for the Higgs
triplets is large. We note however that implementing this by a global symmetry is problematic, seee.g. [29–31].
Once a model with vanishing fundamental µ term has been realized, an effective µ term of order
the SUSY breaking scale can then be induced by, for example, higher-mass-dimension terms in the
Kahler potential [32]. For a review of the µ problem and some suggested solutions in SUSY GUTs
and string theory, see [30, 33–35] and references therein.
In general, GUT-breaking sectors successfully resolving the doublet-triplet splitting problem,
dynamically stabilizing all GUT-scale VEVs and allowing for realistic neutrino masses and Yukawa
couplings (including the GUT-symmetry violation in the latter) require a number of ingredients.
However, for validity of the effective theory, introduction of higher or many representations islimited, otherwise a Landau pole may appear below the Planck scale. In addition, GUTs are only
effective theories below the Planck scale in the 4d field-theoretic approach. Since MG is close tothis scale, the effects of higher-dimension operators are not obviously negligible. In particular,
operators including the GUT-breaking Higgs may affect low-energy predictions, such as quark andlepton masses.
Thus, especially in the context of GUT breaking and doublet-triplet splitting, models beyond
4d field theory appear attractive. While this is mainly the subject of the next section, some advantages can already be noted: In models with extra dimensions, in particular string constructions,
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GUT breaking may occur due to boundary conditions in the compactified dimensions [36–39]. No
complicated GUT breaking sector is then required. Moreover, boundary conditions can give mass
only to the triplet, leaving the doublet massless. This is similar to the ‘missing partner mechanism’
since the effective mass term does not ‘pair up’ the triplets from 5H and 5H but rather each ofthem with further fields which are automatically present in the higher-dimensional theory. Thiscan eliminate dimension-five nucleon decay (cf. Sec. 93.6).

93.4 String-theoretic and higher-dimensional unified models
As noted earlier, the GUT scale is dangerously close to the scale of quantum gravity. It may
hence be necessary to discuss unified models of particle physics in the latter, more ambitious context.
Among the models of quantum gravity, superstring or M-theory stands out as the best-studied andtechnically most developed proposal, possessing in particular a high level of internal, mathematicalconsistency. For our purposes, it is sufficient to know that five 10d and one 11d low-energy effective
supergravity theories arise in this setting (cf. [40] and refs. therein).
Grand unification is realized most naturally in the context of the two ‘heterotic’ theories withgauge groups E8 ×E8 and SO(32), respectively [38,41] (see [42] for some of the more recent results).
Justified in part by the intriguing breaking path E8 → · · · → GSM mentioned above, the focus
has historically largely been on E8 ×E8 . To describe particle physics, solutions of the 10d theory
with geometry R1,3 × M6 are considered, where M6 is a Calabi-Yau (CY) 3-fold (with 6 real dimensions) [38]. The background solution involves expectation values of higher-dimensional componentsof the E8 ×E8 gauge fields. This includes both Wilson lines [36] and non-vanishing field-strengthand leads, in general, to a reduced gauge symmetry and to chirality in the resulting 4d effective
theory. The 4d fermions arise from 10d gauginos.
Given an appropriate embedding4 of GSM in E8 ×E8 , gauge coupling unification is automatic at
leading order. Corrections arise mainly through (string)-loop effects and are similar to the familiar
field-theory thresholds of 4d GUTs5 [43]. Thus, one may say that coupling unification is a genericprediction in spite of the complete absence6 of a 4d GUT at any energy scale. This absence is bothan advantage and a weakness. On the up side, GUT breaking and doublet-triplet splitting [45] aremore naturally realized and dimension-five nucleon decay is relatively easy to avoid. On the downside, there is no reason to expect full GUT representations in the matter sector and flavor modelbuilding is much less tied to the GUT structure than in 4d.
Let us pause to explain the beautiful idea behind the advertised solution of the doublet-triplet
splitting problem: One starts with a simply connected CY X and mods out the action of a discretegroup G (say Z2 ). In the absence of fixed points, X/G is smooth and has a non-contractible 1-cycle.
Furthermore, let G also act on the gauge bundle, according to an embedding G → E8 . Now theparallel transport around the 1-cycle is tied to a gauge rotation (one says a non-trivial Wilson-line
is present). Moreover, this Wilson line can not be continuously turned off since, e.g. in the case of
Z2 , its square is the unit element of the group. The induced ‘Wilson-line breaking’, which comeson top of the breaking by non-zero field strengths, may remove certain sub-representations (e.g.
the triplet of SU (5) → GSM ) while keeping others exactly massless. A simpler and, due to fixed
points, singular version of this will appear below in the context of orbifold GUTs.
One technical problem of heterotic constructions is the dependence on the numerous size andshape parameters of M6 (the so-called moduli), the stabilization of which is poorly understood4

All embeddings of GSM in one E8 factor which are consistent with a breaking-pattern E8 → SU (5) → GSM
are suitable (cf. for example the natural breaking chain from E8 to GSM through maximal subgroups mentioned
at the end of Sect. 93.3). Other embeddings can change the ratios between the three resulting GSM couplings at
the GUT scale by group-theoretic factors. Crucially, due to the single 10d gauge coupling, no continuous tuning ispossible.
5
Field-theory thresholds of 4d GUTs are discussed in 93.5.
6
See however [44].
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93. Grand Unified Theories(see [46] for recent developments). Another is the sheer mathematical complexity of the analysis,
involving in particular the study of (non-Abelian) gauge-bundles on CY spaces [47] (see however[48]).
An interesting aspect of heterotic string constructions is represented by orbifold models [37].
Here the internal space is given by a six-torus, modded out by a discrete symmetry group (e.g.
T 6 /Zn ). More recent progress is reported in [49, 50], including in particular the systematic exploration of the phenomenological advantages of so-called ‘non-prime’ (referring to n) orbifolds. The
symmetry breaking to GSM as well as the survival of Higgs doublets without triplet partners is
ensured by the appropriate embedding of the discrete orbifold group in E8 ×E8 . String theory onsuch spaces, which are locally flat but include singularities, is much more calculable than in the
CY case. The orbifold geometries can be viewed as singular limits of CYs.
An even simpler approach to unified models, which includes many of the advantages of fullfledged string constructions, is provided by Orbifold GUTs [39]. These are (mostly) 5d or 6d
SUSY field theories with unified gauge group (e.g. SU (5) or SO(10)), broken in the process ofcompactifying to 4d. To give a particularly simple example, consider SU (5) on R1,3×S 1 /(Z2 × Z02 ).
Here the compact space is an interval of length πR/2 and the embedding of Z02 in the hypercharge
direction of SU (5) realizes the breaking to GSM . Concretely, 5d X bosons are given Dirichlet BCs at
one endpoint of the interval and thus have no Kaluza-Klein (KK) zero mode. Their lightest modes
have mass ∼ 1/R, making the KK-scale the effective GUT scale. As an implication, the boundary
theory has no SU (5) invariance. Nevertheless, since the SU (5)-symmetric 5d bulk dominates 4dgauge couplings, unification remains a prediction. Many other features but also problems of 4d
GUTs can be circumvented, especially doublet-triplet splitting is easily realized.
With the advent of the string-theory ‘flux landscape’ [51], which is best understood in 10d
type-IIB supergravity, the focus in string model building has shifted to this framework. While
type II string theories have no gauge group in 10d, brane-stacks support gauge dynamics. A
particularly appealing setting (see e.g. [52]) is provided by type IIB models with D7 branes (defining
8d submanifolds). However, in the SO(10) context the 16 is not available and, for SU (5), the topYukawa coupling vanishes at leading order [53]. As a crucial insight, this can be overcome on the
non-perturbative branch of type IIB, also known as F-theory [54, 55]. This setting allows for moregeneral branes, thus avoiding constraints of the Dp-brane framework. GUT breaking can be realizedusing hypercharge flux (the VEV of the U(1)Y field strength), an option not available in heterotic
models. The whole framework combines the advantages of the heterotic or higher-dimensional
unification approach with the more recent progress in understanding moduli stabilization. It thus
represents at this moment the most active and promising branch of theory-driven GUT modelbuilding (see e.g. [56] and refs. therein).
As a result of the flux-breaking, a characteristic ‘type IIB’ or ‘F-theoretic’ tree-level correction
to gauge unification arises [57]. The fact that this correction can be rather significant numericallyis occasionally held against the framework of F-theory GUTs. However, at a parametric level, thiscorrection nevertheless behaves like a 4d threshold, i.e., it provides O(1) additive contributions to
the inverse 4d gauge couplings αi−1 (MG ).
A final important issue in string GUTs is the so-called string-scale/GUT-scale problem [58]. It
arises since, in heterotic compactifications, the Planck scale and the high-scale value of the gauge
coupling unambiguously fix the string-scale to about 1018 GeV. As the compactification radius R
is raised above the string length, the GUT scale (identified with 1/R) goes down and the string
coupling goes up. Within the domain of perturbative string theory, a gap of about a factor ∼ 20
remains between the lowest GUT scale achievable in this way and the phenomenological goal of
2 × 1016 GeV. The situation can be improved by venturing into the non-perturbative regime [58],by considering ‘anisotropic’ geometries with hierarchically different radii R [58, 59] or by including
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93. Grand Unified Theories

GUT scale threshold corrections [60, 61].
In F-theory GUTs, the situation is dramatically improved since the gauge theory lives only infour out of the six compact dimensions. This allows for models with a ‘decoupling limit’, where
the GUT scale is parametrically below the Planck scale [55]. However, moduli stabilization may
not be without problems in such constructions, in part due to a tension between the required large
volume and the desirable low SUSY breaking scale.

93.5 Gauge coupling unification
The quantitative unification of the three SM gauge couplings at the energy scale MG is oneof the cornerstones of the GUT paradigm. It is obviously of direct phenomenological relevance.
Gauge coupling unification is well understood in the framework of effective field theory (EFT) [62].
In the simplest case, the relevant EFT at energies µ  MG has a unified gauge symmetry (saySU (5) for definiteness) and a single running gauge coupling αG (µ). At energies µ  MG , stateswith mass ∼ MG (such as X bosons, GUT Higgs, color-triplet Higgs) have to be integrated out.
The EFT now has three independent couplings and SM (or SUSY SM) matter content. One-loop
renormalization group equations readily allow for an extrapolation to the weak scale,−1
αi−1 (mZ ) = αG(MG ) +

bi
MG
log
2π
mZ




+ δi ,(93.3)(i = 1, 2, 3). Here we defined δi to absorb all sub-leading effects, such as threshold correctionsat or near the weak scale (e.g. from superpartners and the additional Higgs bosons in the caseof the MSSM) and at the GUT scale, and also higher-order corrections. We will discuss them
momentarily.
It is apparent from Eq. (93.3) that the three low-scale couplings can be very different. Thisis due to the large energy range mZ  µ  MG and the non-universal β-function coefficients(bSM= {41/10, −19/6, −7} or bMSSM= {33/5, 1, −3}). Incomplete GUT multiplets, such as gaugeiiand Higgs bosons in the SM and also their superpartners and the additional Higgs bosons in the
MSSM, contribute to the differences between the β functions. Inverting the argument, one expects
that extrapolating the measured couplings to the high scale, we find quantitative unification atµ ∼ MG . While this fails in the SM, it works intriguingly well in the MSSM (cf. Fig. 93.1).
The three equations contained in Eq. (93.3) can be used to determine the three ‘unknowns’
α3 (mZ ), αG (MG ) and MG , assuming that all other parameters entering the equations are given.
−1
Focusing on the SUSY case and using the MS coupling constants αEM(mZ ) and sin2 θW (mZ )from [64],−1αEM(mZ ) = 127.951 ± 0.009 ,(93.4)
sin2 θW (mZ ) = 0.23122 ± 0.00004 ,(93.5)−1
as input, one determines α1,2(mZ ), which then gives−1αG(MG ) ' 24.3

andMG ' 2 × 1016 GeV .

(93.6)

Here we have set δi = 0 for simplicity. Crucially, one in addition obtains a prediction for the
low-energy observable α3 ,

where

12
5
α3−1 (mZ ) = − α1−1 (mZ ) + α2−1 (mZ ) + ∆3 ,7
7

(93.7)

512∆3 = δ1 − δ2 + δ3 .
7
7

(93.8)

31st May, 2024

93. Grand Unified Theories

9

MSSM: m0=M1/2=2 TeV, A0=0, tanβ=30

SM

60

60

α1
α2
α3

50

50
40
α-1
i (Q)

α-1
i (Q)

40
30

30

20

20

10

10

0

α1
α2
α3

SOFTSUSY 3.6.2

0510
log10(Q/GeV)

0

15

SOFTSUSY 3.6.2

0510
log10(Q/GeV)

15

Figure 93.1: Running couplings in SM and MSSM using two-loop RG evolution. The SUSY
threshold at 2 TeV is clearly visible on the MSSM side. (We thank Ben Allanach for providing theplots created using SOFTSUSY [63].)
Here we followed the elegant formulation in Ref. [65] of the classical analyses of [7]. Of course, it
is a matter of convention which of the three low-energy gauge coupling parameters one ‘predicts’and indeed, early works on the subject discussed the prediction of sin2 θW in terms of αEM andα3 [66, 67].
Remarkably, the leading order result (i.e. Eq. (93.7) with δi = 0) is in excellent agreement withexperiments [64]:α3LO (mZ ) = 0.117

vs.

α3EXP (mZ ) = 0.1179 ± 0.0009 .

(93.9)

However, this near perfection is to some extent accidental. To see this, we now discuss the various
contributions to the δi (and hence to ∆3 ).
(2)
The two-loop running correction from the gauge sector ∆3 and the low-scale threshold correc(l)tion ∆3 from superpartners can be summarized as [65](2)∆3 ' −0.82

and

(l)
∆3 '

19
mSUSY
log
28π
mZ




.

(93.10)

The relevant scale mSUSY can be estimated as [68]
mSUSY

→

3/19 12/19 4/19
mH m e m e ×
H
W

mW
e
meg

!28/19

mel
me
q

!3/19

,

(93.11)

where mH stands for the masses of non-SM Higgs states and superpartner masses are given in
self-evident notation. Detailed analyses including the above effects are best done using appropriate
software packages, such as SOFTSUSY [63] (or alternatively SuSpect [69] or SPheno [70]). See
also [63] for references to the underlying theoretical two-loop analyses.
To get a very rough feeling for these effects, let us assume that all superpartners are de(l)generate at mSUSY = 1 TeV, except for heavier gluinos: mWe /me
g ' 1/3. This gives ∆3 '31st May, 2024

93. Grand Unified Theories

10−0.35 + 0.22 ln(mSUSY /mZ ) ' 0.18. The resulting prediction of α3 (mZ ) ' 0.126 significantly upsets the perfect one-loop agreement found earlier. Before discussing this issue further, it is useful
to introduce yet another important type of correction, the high or GUT scale thresholds.
To discuss high scale thresholds, let us set all other corrections to zero for the moment andwrite down a version of Eq. (93.3) that captures the running near and above the GUT scale morecorrectly. The threshold correction at one-loop level can be evaluated accurately by the simple
step-function approximation for the β functions in the DR scheme7 [74],−1αi−1 (mZ ) = αG
(µ) +

1
µ
µ
µ
µ
bi ln
+ bC
+ bX
+ bΦ
.
i ln
i ln
i ln
2π
mZ
MC
MX
MΦ




(93.12)

Here we started the running at some scale µ  MG , including the contribution of the minimal setof states relevant for the transition from the high-scale SU (5) model to the MSSM. These are thecolor-triplet Higgs multiplets with mass MC , massive vector multiplets of X-bosons with mass MX(including GUT Higgs degrees of freedom), and the remaining GUT-Higgs fields and superpartners
with mass MΦ . The coefficients bC,X,Φ
can be found in Ref. [75]. Crucially, the bi in Eq. (93.12)
iconspire to make the running GUT-universal at high scales, such that the resulting prediction for
α3 does not depend on the value of µ.
To relate this to our previous discussion, we can, for example, define MG ≡ MX and then chooseµ = MG in Eq. (93.12). This gives the high-scale threshold corrections(h)δi

1 C MG
MG
bi ln
+ bΦ
,
i ln
2π
MC
MΦ


=



(93.13)(h)and a corresponding correction ∆3 . To get some intuition for the magnitude, one can furthermore
assume MΦ = MG , finding (with bC
i = {2/5, 0, 1})
(h)
∆3 =
(2)

9
MG
ln
14π
MC




.

(93.14)(l)

To obtain the desired effect of −∆3 − ∆3 ' +0.64, the triplet Higgs would have to be by abouta factor 20 lighter than the GUT scale. While this is ruled out by nucleon decay in the minimalmodel [76] as will be discussed Sec. 93.6, it is also clear that threshold corrections of this order ofmagnitude can, in general, be realized with a certain amount of GUT-scale model building, e.g.
in specific SU (5) [25] or SO(10) [26, 27] constructions. Corrections can also be much larger orof different sign if, as is required in many fully realistic 4d GUT models, many additional (andin particular higher) representations are introduced. Thus, there is considerable model building
freedom. Nevertheless, a significant constraint from getting the right GUT threshold corrections
while keeping the triplet Higgs heavy remains.
The above analysis implicitly assumes universal soft SUSY breaking masses at the GUT scale,
which directly affect the spectrum of SUSY particles at the weak scale. In the simplest case wehave a universal gaugino mass M1/2 , a universal mass for squarks and sleptons m16 and a universal
Higgs mass m10 , as motivated by SO(10). In some cases, threshold corrections to gauge coupling
unification can be exchanged for threshold corrections to soft SUSY parameters (see [77] and refs.
therein). For example, if gaugino masses were not unified at MG and, in particular, gluinos werelighter than winos at the weak scale (cf. Eq. (93.11)), then it is possible that, due to weak scale
threshold corrections, a much smaller or even slightly negative threshold correction at the GUT
scale would be consistent with gauge coupling unification [78].
7
The DR scheme is frequently used in a supersymmetric regularization [71]. The renormalization transformationof the gauge coupling constants from MS to DR scheme is given in Ref. [72]. For an alternative treatment using
holomorphic gauge couplings and NSVZ β-functions see e.g. [73].

31st May, 2024

93. Grand Unified Theories

11

It is also noteworthy that perfect unification can be realized without significant GUT-scalecorrections, simply by slightly raising the (universal) SUSY breaking scale. In this case the dark
matter abundance produced by thermal processes in the early Universe (if the lightest neutralinois the dark matter particle) is too high. However, even if the gaugino mass in the MSSM is about
1 TeV to explain the dark matter abundance, if the Higgsino and the non-SM Higgs boson masses
are about 10-100 TeV, the effective SUSY scale can be raised [79]. This setup is realized in split
SUSY [80] or the pure gravity mediation model [81] based on anomaly mediation [82]. Since thesquarks and sleptons are much heavier than the gaugino masses in those setups, a gauge hierarchy
problem is reintroduced. The facts that no superpartners have so far been seen at the LHC andthat the observed Higgs mass favors heavier stop masses than about 1 TeV force one to accept acertain amount of fine-tuning anyway.
For non-SUSY GUTs or GUTs with a very high SUSY breaking scale to fit the data, new lightstates in incomplete GUT multiplets or multiple GUT breaking scales are required. For example,
non-SUSY models SO(10) → SU (4)C × SU (2)L × SU (2)R → SM, with the second breaking scaleof order an intermediate scale, determined by light neutrino masses using the see-saw mechanism,
can fit the low-energy data for gauge couplings [83] and at the same time survive nucleon decaybounds [84]. Alternatively, one can appeal to string-theoretic corrections discussed in Sec. 93.4 tocompensate for a high SUSY breaking scale. This has, for example, been concretely analyzed in
the context of F-theory GUTs in [85]. Similarly, one may even wonder whether particularly large
GUT threshold corrections could be sufficient to ensure non-SUSY precision unification. Noticehere that the gauge coupling unification predicts just one parameter. When introducing new states,
typically one can fit the data by choice of their masses. This is not the case in SUSY GUTs withlow-scale SUSY breaking scale where the masses are constrained by fine-tuning.
In 5d or 6d orbifold GUTs, certain “GUT scale” threshold corrections come from the KaluzaKlein modes between the compactification scale, Mc ∼ 1/R, and the effective cutoff scale M∗ . In
string theory, this cutoff scale is the string scale. Gauge coupling unification at two loops thenconstrains the values of Mc and M∗ .8 Often, one finds Mc to be lower than the 4d GUT scale.
Since the X-bosons, responsible for nucleon decay, get mass at the compactification scale, this has
significant consequences for nucleon decay.
Finally, it has been shown that non-supersymmetric GUTs in warped 5d orbifolds can be consistent with gauge coupling unification. This assumes (in 4d language) that the r.h. top quark andthe Higgs doublets are composite-like objects with a compositeness scale in the TeV range [87].

93.6 Nucleon decay
Quarks and leptons are indistinguishable in any 4d GUT, and both the baryon (B) and lepton
number (L) are not conserved. This leads to baryon-number-violating nucleon decay. In addition
to baryon-number violation, lepton-number violation is also required for nucleon decay since, in
the SM, leptons are the only free fermions which are lighter than nucleons. The lowest-dimension
operators relevant for nucleon decay are (B +L) violating dimension-six four-fermion-terms in theSM, and all baryon-violating operators with dimension less than seven preserve (B −L) [1, 88].
In SU (5) GUTs, the dimension-six operators are induced by X boson exchange. These operators2 ) (Mare suppressed by (1/MX
X is the X boson mass), and the nucleon lifetime is given by τN ∝42
5
MX /(αG mp ) (mp is proton mass). The dominant decay mode of the proton (and the baryonviolating decay mode of the neutron), via X boson exchange, is p → e+ π 0 (n → e+ π − ). In any
simple gauge symmetry, with one universal GUT coupling αG and scale MX , the nucleon lifetime
from gauge boson exchange is calculable. Hence, the GUT scale may be directly observed via theextremely rare decay of the nucleon. The present experimental bounds on the modes come from8

It is interesting to note that a ratio M∗ /Mc ∼ 100, needed for gauge coupling unification to work in orbifold
GUTs, is typically the maximum value for this ratio consistent with perturbativity [86].
31st May, 2024

93. Grand Unified Theories

12Super-Kamiokande. With 450 kton-years of data they find τp /Br(p → e+ π 0 ) > 2.4 × 1034 years
at 90% CL [89]. In addition, Hyper-Kamiokande [90] is planned to reach to τp /Br(p → e+ π 0 ) ∼1035 years. The reader may consult [91] for other planned experiments and some historical context.
The hadronic matrix elements for baryon-number-violating operators are evaluated with lattice
QCD simulations [92]. In SUSY SU (5) GUTs, the lower bound on the X boson mass from null
results in nucleon decay searches is approaching 1016 GeV [93], which is close to the GUT scalesuggested by gauge coupling unification. On the other hand, the prediction for nucleon decay in
non-SUSY GUTs is hard to quantify. The reason is that gauge couplings do not unify with justthe SM particle content. Once extra states or large thresholds are included to ensure precisionunification, a certain range of unification scales is allowed.
In SUSY GUTs there are additional sources for baryon and/or lepton-number violation –dimension-four and five operators [14]. These arise since, in the SUSY SM, quarks and leptons havescalar partners (squarks and sleptons). Although our notation does not change, when discussing
SUSY models our fields are chiral superfields and both fermionic and bosonic matter is implicitly
represented by those. In this language, baryon- and/or lepton-number-violating dimension-fourand five operators are given as so-called F terms of products of chiral superfields, which containtwo fermionic components and the rest scalars or products of scalars. Within the context of SU (5)the dimension-four and five operators have the form
(10 5̄ 5̄) ⊃ (uc dc dc ) + (Q L dc ) + (ec L L),(10 10 10 5̄) ⊃ (Q Q Q L) + (uc uc dc ec ) + B- and L-conserving terms,
respectively.
The dimension-four operators in (10 5̄ 5̄) violate either baryon number or lepton number. The
nucleon lifetime is extremely short if both types of dimension-four operators are present in the
SUSY SM since squark or slepton exchange induces the dangerous dimension-six SM operators.
Even in the case that they violate baryon number or lepton number only but not both, they
are constrained by various phenomena [94]. For example, the primordial baryon number in the
Universe is washed out unless the dimensionless coupling constants are less than 10−7 . Both typesof operators can be eliminated by requiring R parity, which distinguishes Higgs from ordinary
matter multiplets. R parity [95] or its cousin, matter parity [6, 96], act as F → −F, H → H
with F = {10, 5̄}, H = {5̄H , 5H } in SU (5).9 In SU (5), the Higgs multiplet 5̄H and the matter
multiplets 5̄ have identical gauge quantum numbers. In E6 , Higgs and matter multiplets could be
unified within the fundamental 27 representation. Only in SO(10) are Higgs and matter multiplets
distinguished by their gauge quantum numbers. The Z4 center of SO(10) distinguishes 10s from
16s and can be associated with R parity [97].
The baryon-number violating dimension-five operators have a dimensionful coupling. They are
generated by integrating out the color-triplet Higgs with GUT-scale mass in SUSY GUTs suchthat the coefficient is suppressed by 1/MG . Note that both triplet Higgsinos (due to their fermionicnature) and Higgs scalars (due to their mass-enhanced trilinear coupling with matter) contribute tothe operators. The dimension-five operators include squarks and/or sleptons. To allow for nucleon
decay, these must be converted to light quarks or leptons by exchange of a gaugino or Higgsinoin the SUSY SM. The nucleon lifetime is proportional to MG2 m2SUSY /m5p , where mSUSY is the
SUSY breaking scale. Thus, dimension-five operators may predict a shorter nucleon lifetime thandimension-six operators. Unless accidental cancellations are present, the dominant decay modes
from dimension-five operators include a K meson, such as p → K + ν̄ (n → K 0 ν̄). This is due
9This forbids the dimension-four operator (10 5̄ 5̄), but allows the Yukawa couplings for quark and lepton massesof the form (10 5̄ 5̄H ) and (10 10 5H ). It also forbids the dimension-three, lepton-number-violating operator (5̄ 5H )⊃ (L Hu ) as well as the dimension-five, baryon-number-violating operator (10 10 10 5̄H ) ⊃ (Q Q Q Hd ) + · · · .
31st May, 2024

13

93. Grand Unified Theories

to a simple symmetry argument: The operators are given as (Qi Qj Qk Ll ) and (uci ucj dck ecl ),where i, j, k, l (= 1, 2, 3) are family indices and color and weak indices are implicit. They must
be invariant under SU (3)C and SU (2)L so that their color and weak doublet indices must be
anti-symmetrized. Since these operators are given by bosonic superfields, they must be totallysymmetric under interchange of all indices. Thus, the first operator vanishes for i = j = k andthe second vanishes for i = j. Hence a second or third generation member exists in the dominantmodes of nucleon decay unless these modes are accidentally suppressed [96].
The Super-Kamiokande bounds on the proton lifetime severely constrain the dimension-fiveoperators. With 306 kton-years of data they find τp /Br(p → K + ν̄) > 6.61 × 1033 years at 90%
CL [98]. In the minimal SUSY SU (5), τp /Br(p → K + ν̄) is smaller than about 1031 years if thetriplet Higgs mass is 1016 GeV and mSUSY = 1 TeV [99]. The triplet Higgs mass bound from
nucleon decay is then in conflict with gauge coupling unification so that this model is considered
to be ruled out [76].
Since nucleon decay induced by the triplet Higgs is a severe problem in SUSY GUTs, various
proposals for its suppression have been made. First, some accidental symmetry or accidental
structure in non-minimal Higgs sectors in SU (5) or SO(10) theories may suppress the dimensionfive operators [22, 26, 27, 100]. Symmetries to suppress the dimension-five operators are typicallybroken by the VEVs responsible for the color-triplet Higgs masses. Consequently, the dimensionfive operators are generically generated via the triplet Higgs exchange in SUSY SU (5) GUTs,
as mentioned above. In other words, the nucleon decay is suppressed if the Higgs triplets in
5̄H and 5H do not have a common mass term but, instead, their mass terms involve partners
from other SU (5) multiplets. Second, the SUSY breaking scale may be around O(10–100) TeV
in order to explain the observed Higgs boson mass at the LHC. In this case, nucleon decay is
automatically suppressed [80, 101, 102]. Third, accidental cancellations among diagrams due to afine-tuned structure of squark and slepton flavor mixing might suppress nucleon decay [103]. Last,
we have also implicitly assumed a hierarchical structure for Yukawa matrices in the analysis. It
is however possible to fine-tune a hierarchical structure for quarks and leptons which baffles thefamily structure so that the nucleon decay is suppressed [104]. The upper bound on the proton
lifetime from some of these theories is approximately a factor of 10 above the experimental bounds.
Future experiments with larger neutrino detectors, such as JUNO [105], Hyper-Kamiokande [90],DUNE [106] and Theia [107], are planned and will have higher sensitivities to nucleon decay.
Are there ways to avoid the stringent predictions for proton decay discussed above? Orbifold
GUTs and string theories, see Sec. 93.4, contain grand unified symmetries realized in higher dimensions. In the process of compactification and GUT symmetry breaking, the triplet Higgs states
may be removed (projected out of the massless sector of the theory). In such models, the nucleon decay due to dimension-five operators can be severely suppressed or eliminated completely.
However, nucleon decay due to dimension-six operators may be enhanced, since the gauge-bosons
mediating proton decay obtain mass at the compactification scale, Mc , which is typically less than
the 4d GUT scale (cf. Sec. 93.5). Alternatively, the same projections which eliminate the triplet
Higgs may rearrange the quark and lepton states such that the massless states of one family comefrom different higher-dimensional GUT multiplets. This can suppress or completely eliminate evendimension-six proton decay. Thus, enhancement or suppression of dimension-six proton decay ismodel-dependent. In some complete 5d orbifold GUT models [65, 108] the lifetime for the decay
τp /Br(p → e+ π 0 ) can be near the bound of 1 × 1034 years with, however, large model-dependence
and/or theoretical uncertainties. In other cases, the modes p → K + ν̄ and p → K 0 µ+ may be
dominant [65]. Thus, interestingly, the observation of nucleon decay may distinguish string or
higher-dimensional GUTs from 4d ones.
In orbifold GUTs or string theory, new discrete symmetries consistent with SUSY GUTs can
31st May, 2024

93. Grand Unified Theories

14

forbid all dimension-three and four baryon- and lepton-number-violating operators. Even the µ termand dimension-five baryon- and lepton-number-violating operators can be forbidden to all orders
in perturbation theory [35]. The µ term and dimension-five baryon- and lepton-number-violating
operators may then be generated, albeit sufficiently suppressed, via non-perturbative effects. The
simplest example of this is a Z4R symmetry which is the unique discrete R symmetry consistent with
SO(10) [35]. Even though it forbids the dimension-five proton decay operator to the desired level,
it allows the required dimension-five neutrino mass term. In this case, proton decay is dominatedby dimension-six operators, leading to decays such as p → e+ π 0 .

93.7 Yukawa coupling unification
In the SM, masses and mixings for quarks and leptons come from the Yukawa couplings with
the Higgs doublet, but the values of these couplings remain a mystery. GUTs provide at least a
partial understanding since each generation is embedded in unified multiplet(s). Specifically, sincequarks and leptons are two sides of the same coin, the GUT symmetry relates the Yukawa couplings(and hence the masses) of quarks and leptons.
In SU (5), there are two types of independent renormalizable Yukawa interactions given by
λij (10i 10j 5H ) + λ0ij (10i 5̄j 5̄H ). These contain the SM interactions λij (Qi ucj Hu )+ λ0ij (Qi dcj Hd + eci Lj Hd ). Here i, j ( = 1, 2, 3) are, as before, family indices. Hence, at
the GUT scale we have tree-level relations between Yukawa coupling constants for charged leptonand down quark masses, such as λb = λτ in which λb/τ are the bottom quark /τ lepton Yukawa
coupling constants [109, 110]. In SO(10), there is only one type of independent renormalizable
Yukawa interaction given by λij (16i 16j 10H ), leading to relations among all Yukawa coupling
constants and quark and lepton masses within one generation [111, 112] (such as λt = λb = λτ ,with λt the top quark Yukawa coupling constant).
In addition to gauge coupling unification, the ratio of bottom quark and τ lepton mass has
been a central target in the study of GUTs since it was found that this ratio was almost consistent
with observations after including the QCD correction [109]. Today the quark masses and the gauge
coupling constants are known precisely such that, as discussed below, the ratio of bottom quarkand τ lepton mass has become a target of precision analyses in the GUT context.
93.7.1 The third generation, b–τ or t–b–τ unification
Third generation Yukawa couplings are larger than those of the first two generations. Hence,
the fermion mass relations predicted from renormalizable GUT interactions which we introduced
above are expected to be more reliable. In order to compare them with data, we have to include
the radiative correction to these relations from the RG evolution between GUT and fermion massscale, from integrating out heavy particles at the GUT scale, and from weak scale thresholds.
Since testing Yukawa coupling unification is only possible in models with successful gauge
coupling unification, we here focus on SUSY GUTs. In the MSSM, top and bottom quark and τ
lepton masses are related to the Yukawa coupling constants at the scale mZ as
mt (mZ ) = λt (mZ ) vu (1 + δmt /mt ),mb/τ (mZ ) = λb/τ (mZ ) vd (1 + δmb/τ /mb/τ ) ,√√where hHu0 i ≡ vu = sin β v/ 2, hHd0 i ≡ vd = cos β v/ 2, vu /vd ≡ tan β and v ∼ 246 GeV is fixedby the Fermi constant, Gµ . Here, δmf /mf (f = t, b, τ ) represents the threshold correction due tointegrating out SUSY partners. For the bottom quark mass, it is found [113] that the dominantcorrections come from the gluino-sbottom and from the Higgsino-stop loops,


δmb
mb



g 2 meg µ∼ 32 2tan β
6π mSUSY
g3



and

δmb
mb

31st May, 2024



∼
λt

λ2t
At µ
tan β ,
16π 2 m2SUSY(93.16)

15

93. Grand Unified Theories

where meg , µ, and At stand for gluino and Higgsino masses and trilinear stop coupling, respectively.
Note that Eq. (93.16) only illustrates the structure of the corrections – non-trivial functional dependences on several soft parameters ∼ mSUSY have been suppressed. For the full one-loop correction
to the bottom quark mass see, for example, Ref. [114].
Note also that the corrections do not go to zero as SUSY particles become much heavier thanmZ . They may change the bottom quark mass at the O(10)% level for tan β = O(10). The total
effect is sensitive to the relative phase between gluino and Higgsino masses since At ∼ −meg due tothe infrared fixed point nature of the RG equation for At [115] in settings where SUSY breaking
terms come from Planck scale dynamics, such as gravity mediation. The τ lepton mass also receives
a similar correction, though only at the few % level. The top quark mass correction, not being
proportional to tan β, is at most 10% [116].
Including one loop threshold corrections at mZ and additional RG running, one finds the top,bottom and τ pole masses. In SUSY GUTs, b–τ unification has two possible solutions with tan β ∼ 1or O(10). The small tan β solution may be realized in the MSSM if superpartner masses are
O(10) TeV, as suggested by the observed Higgs mass [101]. The large tan β limit such as tan β ∼ 40–
50 overlaps the SO(10) symmetry relation [116, 117]. When tan β is large, there are significant
threshold corrections to down quark masses as mentioned above, and Yukawa unification is only
consistent with low-energy data in a restricted region of SUSY parameter space, with importantconsequences for SUSY searches [116, 118]. More recent analyses of Yukawa unification after LHC
Run-I are found in Ref. [119].
Gauge coupling unification is also successful in the scenario of split supersymmetry [80], inwhich squarks and sleptons have mass at a scale m̃  mZ , while gauginos and/or Higgsinos havemasses of order the weak scale. Unification of b–τ Yukawa couplings requires tan β to be fine-tuned
close to 1 [101]. If by contrast, tan β & 1.5, b–τ Yukawa unification only works for m̃ . 104 GeV.
This is because the effective theory between the gaugino mass scale and m̃ includes only one Higgs
doublet, as in the standard model. As a result, the large top quark Yukawa coupling tends toincrease the ratio λb /λτ due to the vertex correction, which is absent in supersymmetric theories,
as one runs down in energy below m̃. This is opposite to what happens in the MSSM where thelarge top quark Yukawa coupling lowers the ratio λb /λτ [110].
93.7.2 Beyond leading order: three-family models
Simple Yukawa unification is not possible for the first two generations. Indeed, the simplest
implementation of SU (5) implies λs = λµ , λd = λe and hence λs /λd = λµ /λe . This is an RGinvariant relation which extrapolates to ms /md = mµ /me at the weak scale, in serious disagreementwith data (ms /md ∼ 20 and mµ /me ∼ 200). An elegant solution to this problem was given by
Georgi and Jarlskog [120] (for a recent analysis in the SUSY context see [121]).
More generally, we have to recall that in all of the previous discussion of Yukawa couplings, we
assumed renormalizable interactions as well as the minimal matter and Higgs content. Since the
GUT scale is close to the Planck scale, higher-dimension operators involving the GUT-breaking
Higgs may modify the predictions, especially for lower generations. An example is provided by theoperators 10 5̄ 5̄H 24H with 24H the GUT-breaking Higgs of SU (5). We can fit parameters to
the observed fermion masses with these operators, though some fine-tuning is introduced in doing
so. The SM Higgs doublet may come in part from higher representations of the GUT group. For
example, the 45 of SU (5) includes an SU (2)L doublet with appropriate U (1)Y charge [120]. This
45 can, in turn, come from the 120 or 126 of SO(10) after its breaking to SU (5) [122, 123]. These
fields may also have renormalizable couplings with quarks and leptons. The relations among the
Yukawa coupling constants in the SM are modified if the SM Higgs doublet is a linear combinationof several such doublets from different SU (5) multiplets. Finally, the SM fermions may not be
31st May, 2024

16

93. Grand Unified Theories

embedded in GUT multiplets in the minimal way. Indeed, if all quarks and leptons are embedded
in 16s of SO(10), the renormalizable interactions with 10H cannot explain the observed CKM
mixing angles. This situation improves when extra matter multiplets, such as 10, are introduced:
After U (1)X , which distinguishes the 5s coming from the 16 and the 10 of SO(10), is broken(e.g. by a VEV of 16H or 126H ), the r.h. down quarks and l.h. leptons in the SM can be linear
combinations of components in 16s and 10s. As a result, λ 6= λ0 in SU (5) [124].
To construct realistic three-family models, some or all of the above effects can be used. Evenso, to achieve significant predictions for fermion masses and mixing angles grand unification alone
is not sufficient. Other ingredients, for example additional global family symmetries are needed(in particular, non-Abelian symmetries can strongly reduce the number of free parameters). Thesefamily symmetries constrain the set of effective higher-dimensional fermion mass operators discussed
above. In addition, sequential breaking of the family symmetry can be correlated with the hierarchyof fermion masses [125]. One simple, widely known idea in this context is to ensure that each 10i
enters Yukawa interactions together with a suppression factor 3−i ( being a small parameter). Thisway one automatically generates a stronger hierarchy in up-type quark Yukawas as compared todown-type quark and lepton Yukawas and no hierarchy for neutrinos, which agrees with observations
at the O(1)-level. Three-family models exist which fit all the data, including neutrino masses andmixing [27, 126].
Finally, a particularly ambitious variant of unification is to require that the fermions of all
three generations come from a single representation of a large gauge group. A somewhat weakerassumption is that the flavor group (e.g. SU (3)) unifies with the SM gauge group in a simple gauge
group at some energy scale M ≥ MG . Early work on such ‘flavor-unified GUTs’, see e.g. [124, 127],
has been reviewed in [128, 129]. For a selection of more recent papers see [130]. In such settings,
Yukawa couplings are generally determined by gauge couplings together with symmetry breaking
VEVs. This is reminiscent of heterotic string GUTs, where all couplings come from the 10d gauge
coupling. However, while the E8 → SU (3)×E6 branching rule 248 = (8, 1) + (1, 78) + (3, 27) +(3, 27) looks very suggestive in this context, the way in which most modern heterotic models arriveat three generations is actually more complicated.

93.8 Neutrino masses
We see from atmospheric and solar neutrino oscillation observations, along with long baseline
accelerator and reactor experiments, that neutrinos have finite masses. By adding three “sterile”
neutrinos νic with Yukawa couplings λν,ij (νic Lj Hu ) (i, j = 1, 2, 3), one easily obtains three massive
Dirac neutrinos with mass mν = λν vu , analogously to quark and charged lepton masses. However,in order to obtain a τ neutrino with mass of order 0.1 eV, one requires the exceedingly small
coupling ratio λντ /λτ . 10−10 . By contrast, in GUTs the seesaw mechanism naturally explains
such tiny neutrino masses as follows [2–4]: The sterile neutrinos have no SM gauge quantum
numbers so that there is no symmetry other than global lepton number which forbids the Majorana
mass term 21 Mij νic νjc . Note also that sterile neutrinos can be identified with the r.h. neutrinos
necessarily contained in complete families of SO(10) or Pati-Salam models. Since the Majorana
mass term violates U (1)X in SO(10), one might expect Mij ∼ MG . The heavy sterile neutrinos
can be integrated out, defining an effective low-energy theory with only three light active Majorana
neutrinos with the effective dimension-five operator
1− Lef f = cij (Li Hu ) (Lj Hu ) ,2(93.17)

where c = λTν M −1 λν . This then leads to a 3×3 Majorana neutrino mass matrix m = mTν M −1 mν .
The seesaw mechanism implemented by r.h. neutrinos is sometimes called the type-I seesaw
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93. Grand Unified Theories

model. There are variant models in which the dimension-five operator for neutrino masses is
induced in different ways: In the type-II model, an SU (2)L triplet Higgs boson Σ is introduced
to have couplings ΣL2 and also ΣHu2 [122, 131]. In the type-III model, an SU (2)L triplet offermions Σ̃ with a Yukawa coupling Σ̃LHu is introduced [132]. In these models, the dimension-fiveoperator is induced by integrating out the triplet Higgs boson or fermions. Such models can also
be implemented in GUTs by introducing Higgs bosons in the 15 or fermions in the 24 in SU (5)GUTs or the 126 in SO(10) GUTs. Notice that the gauge non-singlet fields in the type-II and III
models have masses at the intermediate scale. Thus, gauge coupling unification is not automatic ifthese variant mechanisms are implemented in SUSY GUTs.
Atmospheric neutrino oscillations discovered by Super-Kamiokande [133] require neutrino masses
with ∆m2ν ∼ 2.5 × 10−3 eV2 with maximal mixing [64], in the simplest
scenario of two neutrinopdominance. With hierarchical neutrino masses this implies mντ = ∆m2ν ∼ 0.05 eV. Next, we cantry to relate the neutrino Yukawa coupling to the top quark Yukawa coupling, λντ = λt at the
GUT scale, as in SO(10) or SU (4) × SU (2)L × SU (2)R models. This gives M ∼ 1014 GeV, whichis remarkably close to the GUT scale.
Neutrinos pose a special problem for GUTs. The question is why the quark mixing angles in the
CKM matrix are small while there are two large lepton mixing angles in the PMNS matrix. Global
fits of neutrino masses and mixing angles can, for example, be found in Refs. [134] and [135]. For
SUSY GUT models which fit quark and lepton masses, see Ref. [136, 137] for reviews. Finally, fora compilation of the range of SUSY GUT predictions for neutrino mixing, see [138].
We should also note that, both within and independently of the GUT framework, much work
has been devoted to neutrino-mass and more generally flavor model building on the basis of discrete
symmetries (see e.g. [137,139]). A more recent interesting twist in this context is the use of the (nonlinearly realized) modular group SL(2, Z) as a discrete flavor symmetry [140]. This group plays a
key role in string theory as the group of large diffeomorphisms of the 2-torus, the latter appearing
as the string worldsheet and, more relevant in the present context, as part of the compactification
space of torus orbifold models (cf. also Sect. 93.4). For a selection from the recent significant
activity in these directions see e.g. [141].

93.9 Selected topics
93.9.1 Global symmetries
As we discussed, global symmetries are frequently introduced to control higher-dimension operators in GUT models. This is particularly important in the context of nucleon decay but also plays
a role in GUT-based flavor model building and cosmological applications, such as baryogenesis
and inflation. However, we should note that appealing to global symmetries to suppress specificinteractions may not be as straightforward as it naively seems. Indeed, there are two possibilities:
On the one hand, the relevant symmetry might be gauged at a higher scale. Effects of the VEVs
responsible for the spontaneous breaking are then in principle dangerous and need to be quantified. On the other hand, the symmetry might be truly only global. This must e.g. be the casefor anomalous symmetries, which are then also violated by field-theoretic non-perturbative effects.
The latter can in principle be exponentially small. It is, however, widely believed that global symmetries are always broken in quantum gravity (see e.g. [142]). One then needs to understand whichpower or functional form the Planck scale suppression of the relevant interaction has. For example,
dimension-five baryon number violating operators suppressed by just one unit of the Planck or
string scale are completely excluded in the supersymmetric models.
In view of the above, it is also useful to recall that in string models 4d global symmetries
generally originate in higher-dimensional gauge symmetries [40, 143]. Here ‘global’ implies that thegauge boson has acquired a Stückelberg-mass. This is a necessity in the anomalous case (Green31st May, 2024

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93. Grand Unified Theories

Schwarz mechanism [144]) but can also happen to non-anomalous symmetries. One expects nosymmetry violation beyond the well-understood non-perturbative effects. Discrete symmetries arise
as subgroups of continuous gauge symmetries, such as ZN ⊂ U (1). In particular, non-anomalous
subgroups of Stückelberg-massive U (1)s represent unbroken discrete gauge symmetries and as suchare non-perturbatively exact (see e.g. [145]). Of course, such discrete gauge symmetries may also
arise as remnants of continuous gauge symmetries after conventional 4d spontaneous breaking.
93.9.2 Anomaly constraints vs. GUT paradigm
As emphasized at the very beginning, the fact that the SM fermions of one generation fill out
the 10 + 5 of SU (5) appears to provide overwhelming evidence for some form of GUT embedding.
However, one should be aware that a counterargument can be made which is related to the issueof ‘charge quantization by anomaly cancellation’ (see [146, 147] for some early papers and [148] fora more detailed reference list): Imagine we only knew that the low-energy gauge group were GSM
and the matter content included the (3, 2)Y , i.e. a ‘quark doublet’ with U (1)-charge Y . One canthen ask which possibilities exist of adding further matter to ensure the cancellation of all triangle
anomalies. It turns out that this problem has only three different, minimal10 solutions [147]. Oneof those is precisely a single SM generation, with the apparent ‘SU (5)-ness’ emerging accidentally.
Thus, if one randomly picks models from the set of consistent gauge theories, preconditioning on
GSM and (3, 2)Y , one may easily end up with ‘10 + 5’ of an SU (5) that is in no way dynamicallypresent. This is precisely what happens in the context of non-GUT string model building [149].
93.9.3 Topological defects and phase transitions
Of the various topological defects which may exist in the broken phase of GUT models, themost famous ones are magnetic monopoles. These are typically localized classical solutions carrying
magnetic charge under an unbroken U (1) symmetry [150]. These magnetic monopoles with massof order MG /αG can be produced during a possible GUT phase transition in the early Universe.
The flux of magnetic monopoles is experimentally found to be less than ∼ 10−16 cm−2 s−1 sr−1[151]. Many more are however predicted, hence the GUT monopole problem. In fact, one of theoriginal motivations for inflation was to solve the monopole problem by exponential expansion
after the GUT phase transition [152] and hence dilution of the monopole density. Other possible
solutions to the monopole problem include: sweeping them away by domain walls [153], U (1)electromagnetic symmetry breaking at high temperature [154] or GUT symmetry non-restoration[155]. Parenthetically, it was also shown that GUT monopoles can catalyze nucleon decay [156].
A significantly stronger bound on the monopole flux can then be obtained by considering X-ray
emission from radio pulsars due to monopole capture and the subsequent nucleon decay catalysis[157].
Note that the present upper bound on the inflationary vacuum energy density is very close to
1/4
the GUT scale, Vinf = (1.88 × 1016 GeV) × (r/0.10)1/4 , with the scalar-to-tensor ratio constrained
to r < 0.036 [158], This guarantees that reheating does not lead to temperatures above MG andhence the monopole problem is solved by inflation (unless MG is unexpectedly low).
As already noted, topological defects other than monopoles are also common, see e.g. [159].
They can be cosmologically relevant in GUT models based on a gauge group larger than SU(5),say, SO(10), where an intermediate gauge theory may appear. The breaking of this intermediate
gauge symmetry can a priori occur anywhere between the GUT and the electroweak scale. Righthanded neutrino masses favor a value near O(1014 ) GeV and non-SUSY coupling unification may
profit from such a two-step GUT breaking scheme (cf. Sect. 93.5). Crucially for the present context,
the last breaking step may occur after inflation, generating topological defects (like cosmic strings)
10
Adding extra vector-like sets of fields, e.g. two fermions which only transform under U (1) and have charges Y
and −Y , is considered to violate minimality.

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which are not diluted. Their signature can be probed through gravitational waves [160], with themetastable string case being of particular phenomenological interest [161,162]. Metastability arises
because the nucleation of a monopole-anti-monopole pair, each of which serves as an endpoint
for the string, allow the cosmic string to break. The lifetime becomes long and hence the string
stable as the ratio between monopole mass and the square root of the string tension grows. Thisis related to the ratio of scales of the two symmetry breaking steps. It is of phenomenologicalinterest that metastability allows the string network to evade the recent bound of Gµ . 10−10 (or√µ . 1014 GeV) [162]. Moreover the spectrum is blue-tilted compared to the stable string case.
Finally, we note that if the intermediate gauge symmetry breaking proceeds with a (strong) firstorder phase transition, this could also generate gravitational waves [162, 163]. However, due to therelatively high energy scales involved the peak frequency tends to be high, with signals hopefully
to be explored in future gravitational wave experiments.
93.9.4 Flavor violation
Yukawa interactions of GUT-scale particles with quarks and leptons may leave imprints on theflavor violation induced by SUSY breaking parameters [164]. To understand this, focus first on the
MSSM with universal Planck-scale boundary conditions (as e.g. in gravity mediation). Working ina basis where up-quark and lepton Yukawas are diagonal, one finds that the large top-quark Yukawa
coupling reduces the l.h. squark mass squareds in the third generation radiatively. It turns out that
only the l.h. down-type squark mass matrix has sizable off-diagonal terms in the flavor basis after
CKM-rotation. However, in GUTs the color-triplet Higgs has flavor violating interactions from the
Yukawa coupling λij (10i 10j 5H ), such that flavor-violating r.h. slepton mass terms are radiatively
generated in addition [165]. In SU (5) extension of the type-I seesaw model, where r.h. neutrinos
are introduced as SU (5) singlets with interactions λ00ij (1i 5̄j 5H ), the doublet and color-triplet
Higgses acquire another type of Yukawa coupling, respectively. They then radiatively generate
flavor-violating l.h. slepton [166] and r.h. down squark masses [167]. These flavor-violating SUSY
breaking terms induce new contributions to FCNC processes in quark and lepton sectors, such asµ → eγ and K 0 –K̄ 0 and B 0 –B̄ 0 mixing. Note that even if the flavor-universal SUSY breaking
terms are generated at MG , the r.h. neutrino Yukawa coupling may induce sizable flavor violation
in l.h. slepton masses due to the running between MG and the right-handed-neutrino mass scale.
EDMs are also induced when both l.h. and r.h. squarks/sleptons have flavor-violating mass termswith relative phases, as discussed for SO(10) in [168] or for SU (5) with r.h. neutrinos in [169]. Thus,
such low-energy observables constrain GUT-scale interactions.
93.9.5 From GUT baryogenesis to leptogenesis and B/L-violating transitions
During inflation, any conserved quantum number is extremely diluted. Thus, one expects
the observed baryon asymmetry of the Universe to originate at reheating or in the subsequentcosmological evolution. In detail, the situation is slightly more involved: Both baryon number B
and lepton number L are global symmetries of the SM. However, (B+L) is anomalous and violatedby thermal fluctuations in the early Universe, via so-called sphaleron processes. Moreover, it isviolated in GUT models, as is most apparent in proton decay. By contrast, (B−L) is anomaly freeand preserved by both the SM as well as SU (5) or SO(10) gauge interactions.
Now, the old idea of GUT baryogenesis [170, 171] is to generate a (B +L) and hence a baryon
asymmetry by the out-of-equilibrium decay of the color-triplet Higgs. However such an asymmetry,generated at GUT temperatures, is washed out by sphalerons11 . This can be overcome [173] using
lepton-number violating interaction of neutrinos to create a (B −L) from the (B +L) asymmetry,
before sphaleron processes become sufficiently fast at T < 1012 GeV. This (B − L) asymmetry
11

To be precise, if lepton flavor numbers Li (i = 1–3) are nonzero and (Li − Lj ) 6= 0 (i 6= j), one may obtain
nonzero values for B and L even if (B −L) = 0 [172].

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can then survive the subsequent sphaleron dominated phase. Note that this does not work in theminimal SUSY GUT setting, with the triplet Higgs above the GUT scale. The reason is that acorrespondingly high reheating temperature would be required which, as explained above, is ruledout by Planck data.
However, the most widely accepted simple way out of the dilemma is to directly generate anet (B − L) asymmetry dynamically in the early Universe, also using r.h. neutrinos. Indeed, we
have seen that neutrino oscillations suggest a new scale of physics of order 1014 GeV. This scale isassociated with heavy Majorana neutrinos in the seesaw mechanism. If in the early Universe, thedecay of the heavy neutrinos is out of equilibrium and violates both lepton number and CP , thena net lepton number may be generated. This lepton number will then be partially converted into
baryon number via electroweak processes [174]. This mechanism is called leptogenesis.
If the three heavy Majorana neutrino masses are hierarchical, the net lepton number is produced
by decay of the lightest one, and it is proportional to the CP asymmetry in the decay. The
CP asymmetry is bounded from above, and the lightest neutrino mass is required to be larger
than 109 GeV in order to explain the observed baryon asymmetry [175]. This implies that thereheating temperature after inflation should be larger than 109 GeV so that the heavy neutrinos
are thermally produced12 . In supersymmetric models, there is a tension between leptogenesis and
Big Bang Nucleosynthesis (BBN) if gravitinos decay in the BBN era. The gravitino problem gives
a constraint on the reheating temperature . 106−10 GeV though the precise value depends on the
SUSY breaking parameters [177]. Recent reviews of leptogenesis can be found in Ref. [178].
One of the important tests of leptogenesis are searches for neutrinoless double-β (0νββ) decays13 . In a 0νββ decay, only two electrons but no (anti-)neutrinos are emitted by the decaying
nucleus. This is in contrast to ordinary double-β decay. Thus, 0νββ decays are lepton-numberviolating with ∆L = 2. At the nucleon level, this is described by dimension-nine effective operatorsfor nn → ppee. These operators may in turn come from SM operators of with dimension less thannine, in combination with SM weak interactions. The lowest one is the dimension-five operator
generating the Majorana neutrino mass terms (Eq. (93.17)). Thus, if the lepton-number violating
effective interactions come from physics at energies much above the weak scale, the 0νββ decay
rates are proportional to the Majorana neutrino masses. The latest experimental results are reviewed in [64]. For recent studies of the 0νββ decay including SM operators up to mass dimension
nine, see [179] and refs. therein.
In addition to L-violation, one can consider (B − L) and B violating phenomena. They areinteresting in their own right and may also be relevant to baryogenesis. The relevant operators havehigher mass dimension than the familiar dimension-six (B + L)-violating operators (cf. Sec. 93.6).
They may be predicted in SO(10) GUTs with an intermediate scale, at which baryogenesis isrealized, such as in [180]. First, one may have nucleon decays with ∆(B−L) = 2, such as n → e− π + .
This is induced by dimension-seven effective operators in the SM, which are suppressed by the SM
Higgs VEV or derivatives. Second, there are neutron-antineutron (n–n̄) oscillations, which are
induced by ∆B = 2 dimension-nine effective operators in the SM. The upper bound on the meantime for n–n̄ transitions is directly derived using free neutrons [181]. It is also constrained from thelower limit on the lifetime for neutrons bound in 16 O, derived by Super-Kamiokande [182]. Theirresults are very similar. Super-Kamiokande also searches for dinucleon decays with ∆B = 2, suchas pp → π + π + and nn → π ± π ∓ [183].
12

This constraint may be avoided in resonant leptogenesis [176], in which the right-handed neutrinos are required
to be almost degenerate in mass.
13
Another important test of leptogenesis would be the observation of CP violation in neutrino oscillations.
Strictly speaking, the CP phase in the PMNS matrix does not contribute to 1 in the seesaw model. Nevertheless,the observation of CP violation in neutrino oscillations would suggest that the seesaw mechanism is associated witha large CP violation, similarly to the quark sector.
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93.9.6 The QCD Axion
The axion is a hypothetical pseudoscalar particle which is capable of resolving the strong CP
problem [184] and may potentially be related to GUT physics. In particular, the strength of thedirect coupling of the QCD axion to photons, relative to its coupling to gluons, may be seen
as a phenomenological signature of grand unification. It is quantified by the predicted anomaly
ratio E/N = 8/3 (see [185–187]). This ratio follows from the GUT normalization of the relevant
generators together with the completeness of GUT multiplets contributing to the anomaly.14 It
hence arises in axion models of both DFSZ [188] and KSVZ [189] type as long as the Peccei-Quinn
symmetry commutes with the GUT group (see e.g. [190]). Specifically in the case of a DFSZ axion,
extra SU (2) multiplets contribute to the running of gauge couplings between Peccei-Quinn and
GUT scale, potentially affecting unification. By contrast, the KSVZ axion can be implemented
without such effects.
We also note that, in bottom-up approaches and in the presence of split GUT multiplets, otheranomaly ratios can appear (cf. e.g. [191]). Their embedding in a UV GUT structure, possibly with
explicit GUT symmetry breaking, requires case-by-case study. The ratio E/N = 8/3 also arises instring-theoretic GUTs [186,192]. In this case, the axion does not come from the phase of a complex
scalar but is a fundamental shift-symmetric real field, coupling through a higher-dimension operator
directly to the product of the GUT field-strength and its dual.

93.10 Conclusion
Most conservatively, grand unification means that (some of) the SM gauge interactions of U (1)Y ,SU (2)L and SU (3)C become part of a larger, unifying gauge symmetry at a high energy scale.
In most models, especially in the simplest and most appealing variants of SU (5) and SO(10)unification, the statement is much stronger: One expects the three gauge couplings to unify (up tosmall threshold corrections) at a unique scale, MG , and the proton to be unstable due to exchangeof gauge bosons of the larger symmetry group. Supersymmetric grand unified theories provide,
by far, the most predictive and economical framework allowing for perturbative unification. Many
more details than could be discussed in the present article can be found in some of the classic
reviews [128, 193] and the four books [194] (see also [195] for two recent overviews).
Thus, the three classical pillars of GUTs are gauge coupling unification at MG ∼ 2 × 1016 GeV,
low-energy supersymmetry (with a large SUSY desert), and nucleon decay. The first of these may
be viewed as predicting the value of the strong coupling – a prediction which has already beenverified (see Fig. 93.1). Numerically, this prediction remains intact even if SUSY partner masses
are somewhat above the weak scale. However, at the conceptual level a continuously increasing
lower bound on the SUSY scale is nevertheless problematic for the GUT paradigm: Indeed, if
the independent, gauge-hierarchy-based motivation for SUSY is completely abandoned, the SUSY
scale and hence α3 become simply free parameters and the first two pillars crumble. Thus, it isimportant to keep pushing bounds on proton decay which, although again not completely universal
in all GUT constructions, is arguably a more generic part of the GUT paradigm than low-energy
SUSY.
Whether or not Yukawa couplings unify is more model dependent. However, irrespective of
possible (partial) Yukawa unification, there certainly exists a very interesting and potentially fruitfulinterplay between flavor model building and grand unification. Especially in the neutrino sector
this is strongly influenced by the developing experimental situation.
It is probably fair to say that, due to limitations of the 4d approach, including especially remaining ambiguities (free parameters or ad hoc assumptions) in models of flavor and GUT breaking,the string theoretic approach has become more important in GUT model building. In this frame14

See e.g. eq. (312) of [187]31st May, 2024

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work, challenges include learning how to deal with the many vacua of the ‘landscape’ as well as,
for each vacuum, developing the tools for reliably calculating detailed, phenomenological observables. Finally, due to limitations of space, the present article has barely touched on the interestingcosmological implications of GUTs. They may become more important in the future, especially inthe case that a high inflationary energy scale is established observationally.
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