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Physics Letters B 771 (2017) 332–338

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Gauge coupling uniﬁcation without leptoquarks

Georgios K. Karananas

∗

, Mikhail Shaposhnikov

Institute of Physics, Laboratory of Particle Physics and Cosmology, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015, Lausanne, Switzerland

a r t i c l e i n f o a b s t r a c t

Article history:

Received

13 March 2017

Received

in revised form 21 April 2017

Accepted

22 May 2017

Available

online 29 May 2017

Editor: A.

Ringwald

We propose an interpretation of the gauge coupling uniﬁcation scale which is not related to any new

particle threshold. We revisit Grand Uniﬁed Theories and show that it is possible to completely eliminate

the scalar as well as vector leptoquarks from the particle physics spectrum. As a consequence, in our

approach the gauge hierarchy problem is put on different grounds, and the proton may be absolutely

stable. In order to achieve that, we employ a number of nonlinear gauge-invariant constraints which only

affect the superheavy degrees of freedom. We illustrate our considerations in a model based on the SU(5)

group, with the generalization to other groups being straightforward. We discuss how scale or conformal

invariance may be added to our proposal.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

1. Introduction and motivation

Arguably, the biggest triumph of utilizing the gauge principle

is the successful description of the strong, weak and electromag-

netic

interactions in the context of a self-consistent theory based

on the groups SU(3) × SU(2) × U(1), the Standard Model (SM) of

particle physics. This framework has been remarkably successful,

for it made it possible to explain a vast number of phenomena at

the subatomic level, and moreover, all the particles that it predicts

have now been discovered.

Nevertheless,

the indications that the SM is not the ﬁnal theory

of Nature are compelling. To start with, it fails to explain the neu-

trino

masses and oscillations, it lacks a candidate for dark matter

and does not incorporate a mechanism for the baryon asymmetry

of the Universe. In addition, the SM is plagued by issues of purely

theoretical nature. These are the strong CP, as well as the hierar-

chy

and cosmological constant problems. The last two are related

to the failure of (naive) dimensional analysis, such that unusually

big ﬁne-tunings are required in order to reconcile the predictions

of the theory with the experimental data. Moreover, the presence

of Landau pole related to the U(1) symmetry and the Higgs sec-

tor,

puts the consistency of the theory in jeopardy. Bear in mind

though that this problem manifests itself at very high energies,

above the Planck scale M

, and might be resolved in the context

of quantum gravity.

Corresponding author.

E-mail

addresses: georgios.karananas@epﬂ.ch (G.K. Karananas),

mikhail.shaposhnikov@epﬂ.ch (M. Shaposhnikov).

Apart from the aforementioned shortcomings, when the SM is

considered from the model-building point of view, it is in a sense

unattractive due to its arbitrariness. Let us be more speciﬁc. First,

the way matter ﬁelds arrange themselves into different group rep-

resentations

appears to be random. Moreover, there are many free

parameters that are unrelated to each other. Finally, the experi-

mental

fact that the electric charge is quantized requires an expla-

nation,

since the U(1) group is Abelian.

Many years ago it was realized that some of the SM prob-

lems

could potentially be resolved once we require that it be the

low energy limit of a gauge theory that enjoys invariance under a

larger group G. This naturally led to Grand Uniﬁcation, i.e. to the

hypothesis that, above a certain energy threshold, the strong and

electroweak interactions are actually one and the same force.

The

models in which this is achieved—the Grand Uniﬁed The-

ories (hereafter

GUTs)—have attracted considerable attention and

have been extensively studied throughout the years. For more de-

tails

on various aspects of these theories, the interested reader

is referred to the classic review by Langacker [5]. Let us note

that the most important GUTs are the Pati–Salam model based on

SU(4) × SU(2) × SU(2) [6], the Georgi–Glashow SU(5) theory [7],

as well as the SO(10) uniﬁcation proposed by Georgi [8] and by

Fritzsch and Minkowski [9].

spectacular aspect of Grand Uniﬁcation is that matter ﬁelds

(quarks and leptons) can be placed neatly into multiplets of the

gauge group G, something that yields nontrivial relations between

It should be noted though that a possible solution may be related to the can-

cellation

of anomalies, see for example [1–4].

http://dx.doi.org/10.1016/j.physletb.2017.05.065

0370-2693/

SCOAP

G.K. Karananas, M. Shaposhnikov / Physics Letters B 771 (2017) 332–338 333

their masses [10,11]. Yet another interesting point about GUTs is

that in their context it is possible to predict the weak mixing angle

sin

[12], or to put in other words, achieve the famous gauge-

coupling

uniﬁcation. Therefore, due to the presence of a larger

symmetry, the various parameters that in the SM appear to be free

or arbitrary, might now be subject to constraints relating them.

Note,

however, that the price to pay for having uniﬁcation

is quite high. The introduction of the extra bosons needed to

gauge the larger symmetry—called leptoquarks and denoted col-

lectively

by X and Y —has a number of consequences. Since they

are charged under color as well as ﬂavor, they couple both to

leptons and quarks. An aftermath of this fact is that the baryon

(and lepton) numbers are not conserved like in the SM. As a re-

sult,

one of the main predictions of these theories is baryon- (and

lepton-) number violating processes, the most signiﬁcant being

the leptoquark-mediated proton decay. To satisfy the experimen-

tal

constraints concerning the lifetime of the proton, it is neces-

sary

that these new vector bosons be superheavy, with masses

X,Y

 10

GeV [5,10,12], i.e. many orders of magnitude heav-

ier

than the SM particle content.

This requirement is the core of the infamous gauge hierarchy

problem, ﬁrst pointed out by Gildener [13]. The Higgs mass is

extremely sensitive to radiative corrections, so if such ﬁelds are

present, we would expect that its mass be of the order of M

X,Y

. It

is then clear that in order to reconcile the theoretical value with

the observed one, the relevant parameters have to be adjusted

enormously, and in many orders of perturbation theory.

this point we should mention that there have been numer-

ous

attempts to address the hierarchy problem, many of which

require the existence of new dynamics above the electroweak

scale. Among the most interesting ones are low-energy supersym-

metry [14–18],

composite Higgs models [19–21] and large extra

dimensions [22,23]. However, so far there is no sign of New Physics

up to the energies which are accessible to particle physics experi-

ments.

the other hand, there is a plethora of arguments why an

intermediate particle physics scale at energies between the elec-

troweak

and Planck scales might not be really needed, see [24].

In fact, the observational puzzles of the SM can potentially be

resolved with the presence of new physics only at the aforemen-

tioned

energies. Since M

, being related to gravitational interac-

tions

which are mediated by the massless graviton, is qualitatively

different from the particle physics scales, it may not necessarily

be associated with supermassive degrees of freedom. This way,

the Higgs mass may be stable against radiative corrections due

to the absence of diagrams with heavy particles running in the

loops [24–27].

Our

purpose in this Letter is to argue that there might be an

alternative way for a GUT symmetry to be realized, which is differ-

ent

from spontaneous symmetry breaking. Even though the latter

mechanism plays the central role in the SM, it might not neces-

sarily

be the case that Nature repeats itself at the gauge coupling

uniﬁcation scale. We will illustrate that by employing a number of

appropriate nonlinear gauge-invariant constraints, it is indeed possi-

ble

to achieve uniﬁcation and simultaneously eliminate completely

the heavy particles.

Let

us note that in the conventional treatment of GUTs, there

exists a built-in mechanism for suppressing, but not eliminating,

the supermassive states: as long as they are suﬃciently massive

and we are working at energies well below their masses (equiva-

lently

the uniﬁcation scale), they can be integrated out by virtue

Actually, for the minimal SU(5) GUT, it is the longevity of the proton that prac-

tically

ruled out the theory.

of their equations of motion. From the low-energy perspective, the

theory looks the same as the “constrained” one constructed in this

work. However the implications for the hierarchy problem can be

drastically different. The former is an effective ﬁeld theory whose

cutoff is a physical scale where the particles that UV-complete the

theory exist; thus, the power-like divergencies cannot be discarded

and will eventually give sizeable contributions to the Higgs mass.

On the contrary, the cutoff scale of the latter theory is not related

to the presence of any new states, so the Higgs mass does not re-

ceive

important corrections and can be naturally small.

Meanwhile,

all the ﬁelds associated with the SM are not af-

fected

and in addition, all the elegant characteristics of GUTs, such

as the group structure of the fermionic multiplets, the prediction

of sin

, and the charge quantization, will be retained. For sim-

plicity,

we develop our idea by considering the most economic GUT

based on SU(5), since the generalization to other groups is quite

straightforward.

also speculate about the potential implications of global

scale (or conformal) invariance in this setup. It is quite attrac-

tive

to assume that the scales we observe in Nature are all related

to each other and are generated via the spontaneous breaking of

these symmetries [25,28–30]. In practice, this is achieved by letting

the different scales be sourced by the vacuum expectation value of

a scalar degree of freedom, the dilaton. This approach allows us to

keep the quantum corrections to the Higgs mass under control and

without ﬁne-tuning, as long as two requirements are met. First,

there should not be any contribution from superheavy particles,

something that resonates nicely with our approach here. Second,

the regularization prescription must preserve the symmetry of the

system [30–32]. However, for this to be possible, the requirement

of renormalizability has to be abandoned.

The outline of this article is as follows. In Sec. 2, we brieﬂy

review some basics about GUTs. In Sec. 3, we discuss what are

the appropriate constraints in order for the heavy particles to be

absent from the spectrum of the theory. In Sec. 4, we conjecture

about the potential role of scale or conformal symmetry in our

setup. Our conclusions are presented in Sec. 5.

2. GUTs: a reminder

The main idea behind the canonical SU(5) model [7], is as fol-

lows.

First, the 15 SM fermions of each generation are placed in

the 5

∗

and 10 representations of the group. Then, two scalar ﬁelds

are introduced, one belonging to the adjoint (24) and the other to

the fundamental (5). In what follows we denote them by  and

H, respectively. Their role is to effectuate the chain of spontaneous

symmetry breaking

SU(5) −→

SU(3) × SU(2) × U(1) −→

SU(3) × U(1). (1)

The scalar ﬁeld in the adjoint representation of the group is ex-

pressed

as (a = 1, ..., 24)

 = 

, (2)

with T

the generators, and summation over repeated indices is

understood. For the ﬁrst part of the breaking pattern to take place,

this ﬁeld should acquire a vacuum expectation value proportional

to the hypercharge generator T

, i.e.

=v

GU T

diag(1, 1, 1, −3/2, −3/2), (3)

where v

GU T

is a parameter with mass dimension.

This should not be considered to be a major drawback, since realistic theories

contain gravity which in any case is nonrenormalizable.

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