Asymptotic Grand Unication The SO10 case Mohammed Omer Khojaliab1 Alan S. Cornella2 aDepartment of Physics University of Johannesburg PO Box 524 Auckland Park 2006 South

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Asymptotic Grand Uniﬁcation: The SO(10) case

Mohammed Omer Khojalia,b,1, Alan S. Cornella,2

aDepartment of Physics, University of Johannesburg, PO Box 524, Auckland Park 2006, South

Africa

bDepartment of Physics, University of Khartoum, PO Box 321, Khartoum 11115, Sudan

Aldo Deandreac,d,3, Giacomo Cacciapagliac,d,4

cUniversit´e de Lyon, Universit´e Lyon 1, F-69622 Lyon, France

dInstitut de Physique des 2 Inﬁnis (IP2I), UMR5822 CNRS/IN2P3, F-69622 Villeurbanne

Cedex, France

Ammar Abdalgabare,5

eUniversity of Hafr Al Batin, college of Science, department of physics, Hafr Al Batin 39524,

Kingdom of Saudi Arabia

Corentin Cotf,6

fLaboratoire de Physique des 2 Inﬁnis (IJCLab), Universit´e Paris-Saclay, Orsay, France

E-mail: 1khogali11@gmail.com, 2acornell@uj.ac.za , 3deandrea@ipnl.in2p3.fr,

4g.cacciapaglia@ipnl.in2p3.fr, 5aabdalgabar@gmail.com,

6corentin.cot@ijclab.in2p3.fr

Abstract. We explicitly test the asymptotic grand uniﬁcation of a minimal 5-dimensional

model with SO(10) gauge theory compactiﬁed on an S1/Z2×Z0

2orbifold. We consider all

matter ﬁelds as propagating in the bulk and show that the gauge couplings asymptotically run

to a ﬁxed point in the UV. However, the Yukawa couplings will typically hit a Landau pole right

above the compactiﬁcation scale in this class of SO(10) models.

1. Introduction

Theories of grand uniﬁcation continue to play an important role in guiding the searches for

extensions of the Standard Model (SM) [1, 2]. The idea of grand uniﬁcation theories (GUT) is

to reduce all the gauge interactions to one single gauge group and all the fermionic multiplets

into one or two diﬀerent representations for each generation of matter [3, 4, 5]. This single

gauge group corresponds to a uniﬁcation of the three forces described by the SM. Since our

observations are mostly in agreement with a model based on the SM gauge group, we require

that the uniﬁed gauge group has SM gauge group as a subgroup. The SM group is rank 4,

which means that the gauge group Gmust be at least rank 4. In the same way as in the Higgs

mechanism, we demand that the uniﬁed gauge group spontaneously breaks to the SM gauge

group at some higher energy scale. The SO(10) group is a popular candidate for uniﬁcation for

many reasons; it contains both the Pati–Salam group, SU(5) ×U(1) (and hence also SU(5))

arXiv:2210.03596v1 [hep-ph] 7 Oct 2022

as subgroups and is therefore more “uniﬁed” in a sense. It also embeds all SM fermions of a

generation, plus the right-handed neutrino, into one single representation [6, 7] . Since SO(10)

is rank 5, which is one more than SM gauge group, there are several possibilities for symmetry

breaking. On the one hand, this produces a rich variety of phenomenologies, but on the other

hand, it introduces arbitrariness into the model in terms of the choice of scalar sector and

potential. Furthermore, since the scalar and intermediate symmetry breaking steps aﬀect the

renormalization group (RG) running, the chosen breaking procedure can have an eﬀect on the

uniﬁcation scale and hence the related phenomenology.

In this work, we shall study the non-supersymmetric extensions of the SM based on the gauge

group SO(10). In particular, we will study higher-dimensional non-supersymmetric orbifold

models. We will consider a uniﬁcation where the couplings unify asymptotically, as in these

models with a compact extra dimension (which becomes relevant at scales higher than the

electroweak (EW) scale) the gauge symmetry in the bulk is uniﬁed [2]. We study the asymptotic

GUT based on an SO(10) model in a ﬂat S1/Z2×Z0

2orbifold.

The structure of this paper is as follows: In section 2 we outline the model setup, in section

3 we explore the gauge running and asymptotic uniﬁcation, and in section 4 we present the

running of the Yukawa couplings. In section 5 we conclude.

2. Model Setup

We consider here a minimal SO(10) grand uniﬁed model in ﬁve dimensions, where the extra

dimension is compactiﬁed on an S1/Z2×Z0

2orbifold [8, 9]. SO(10) gauge symmetry is broken to

a Pati-Salam model SU (4)C×SU(2)L×SU(2)Rby an Z2×Z0

2orbifold twisting which generates

two inequivalent ﬁxed points. One is the preserved SO(10) symmetric ﬁxed point (we call this

the visible brane) while the other has only a Pati-Salam symmetry (we refer to as the PS hidden

brane) [10, 11].

The breaking is performed with a scalar in the 16 or 126 representations of SO(10), thus

breaking to the SU(5) is done by the ordinary Higgs mechanism on the brane (we will refer to

this as brane breaking):

16 ⇒10 + 5 + 1

126 ⇒50 + 45 + 15 + 10 + 5 + 1.(1)

Both the 16 and 126 representations contain a singlet under SU (5), where we choose the adjoint

scalar in the 16 representation. The minimal content at the SO(10) scale to realise this symmetry

breaking could be either 16 + 16 or 126 + 16. We could also use a 16 + 45 on the SO(10)

symmetric brane to break SO(10) to SU(5). The unbroken gauge group in the overlap of

SU(4)C×SU (2)L×SU(2)Rwith SU(5) is just the SM [12].

2.1. Boundary conditions

All the quarks and leptons, including right-handed neutrinos, in each generation are uniﬁed to

a single 16 and 16 dimensional spinor representation ﬁeld. One family of fermions with an

addition of a right handed neutrino [13, 14] is:

16 = (tL, νL, bL, τL, Bc

L,Tc

L,−Tc

L,−Nc

L, TR, NR, BR,TR, bc

R, τc

R,−tc

R,−νc

R),(2)

and

16 = (TL, NL, BL,TL, bc

L, τc

L,−tc

L,−νc

L, tR, νR, bR, τR, Bc

R,Tc

R,−Tc

R,−Nc

R).(3)

We choose the P0and P1matrices to be:

P0=diag(−1,−1,−1,+1,+1) ⊗diag(+1,+1),

P1=diag(+1,+1,+1,+1,+1) ⊗diag(+1,+1).(4)

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AsymptoticGrandUnication:TheSO(10)caseMohammedOmerKhojalia;b;1,AlanS.Cornella;2aDepartmentofPhysics,UniversityofJohannesburg,POBox524,AucklandPark2006,SouthAfricabDepartmentofPhysics,UniversityofKhartoum,POBox321,Khartoum11115,SudanAldoDeandreac;d;3,GiacomoCacciapagliac;d;4cUniversitedeLyon,Univer...

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Asymptotic Grand Unication The SO10 case Mohammed Omer Khojaliab1 Alan S. Cornella2 aDepartment of Physics University of Johannesburg PO Box 524 Auckland Park 2006 South

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