A Predictive SO10Model George Lazarides1Rinku Maji2Rishav Roshan3Qaisar Sha4 1School of Electrical and Computer Engineering Faculty of Engineering

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A Predictive SO(10) Model

George Lazarides,1Rinku Maji,2Rishav Roshan,3Qaisar Shaﬁ4

1School of Electrical and Computer Engineering, Faculty of Engineering,

Aristotle University of Thessaloniki, Thessaloniki 54124, Greece

2Theoretical Physics Division, Physical Research Laboratory,

Navarangpura, Ahmedabad 380009, India

3Department of Physics, Kyungpook National University, Daegu 41566, Korea

4Bartol Research Institute, Department of Physics and Astronomy,

University of Delaware, Newark, DE 19716, USA

Abstract

We discuss some testable predictions of a non-supersymmetric SO(10) model supple-

mented by a Peccei-Quinn symmetry. We utilize a symmetry breaking pattern of SO(10)

that yields uniﬁcation of the Standard Model gauge couplings, with the uniﬁcation scale

also linked to inﬂation driven by an SO(10) singlet scalar ﬁeld with a Coleman-Weinberg

potential. Proton decay mediated by the superheavy gauge bosons may be observable at

the proposed Hyper-Kamiokande experiment. Due to an unbroken Z2gauge symmetry from

SO(10), the model predicts the presence of a stable intermediate mass fermion which, to-

gether with the axion, provides the desired relic abundance of dark matter. The model also

predicts the presence of intermediate scale topologically stable monopoles and strings that

survive inﬂation. The monopoles may be present in the Universe at an observable level. We

estimate the stochastic gravitational wave background emitted by the strings and show that

it should be testable in a number of planned and proposed space and land based experi-

ments. Finally, we show how the observed baryon asymmetry in the Universe is realized via

non-thermal leptogenesis.

arXiv:2210.03710v3 [hep-ph] 8 Dec 2022

1 Introduction

A recent paper [1] highlighted some salient features of a non-supersymmetric SO(10) ×U(1)PQ

model [2,3], where U(1)PQ denotes the Peccei-Quinn (PQ) symmetry included to resolve the

strong CP problem [4,5]. The SO(10) symmetry is broken to SU(3)C×U(1)EM by employing

tensor representations such that the Z2subgroup of Z4, the center of SO(10), remains unbroken

[6]. Independent of the symmetry breaking chain, this yields topologically stable cosmic strings

[6,7] with a string tension that is determined by the appropriate symmetry breaking scale. We

focus here on a speciﬁc symmetry breaking pattern of SO(10) which is compatible with the

uniﬁcation of the Standard Model (SM) gauge couplings and also yields topologically stable

intermediate scale monopoles and strings [8,9]. We also take into account primordial inﬂation

driven by an SO(10)×U(1)PQ singlet real scalar ﬁeld with a Coleman-Weinberg potential [10,11].

In light of the most recent measurements [12,13] of nsand r, the scalar spectral index and tensor-

to-scalar ratio respectively, a non-minimal coupling of the inﬂaton to gravity is preferred [14].

Regarding U(1)PQ, we assume that this symmetry is spontaneously broken after inﬂation ends

in which case one should make sure that the axion domain wall problem does not exist. This

is taken care of through the introduction of two fermionic 10-plets whose components acquire

masses from the breaking of U(1)PQ at scale fa[2,15]. An important consequence of these

considerations is the appearance of intermediate scale WIMP-like fermionic dark matter (DM)

whose stability is ensured by the unbroken gauge Z2symmetry that we previously mentioned.

We are therefore led to a scenario in which the observed dark matter in the universe potentially

consists of axions as well as electrically neutral intermediate scale fermions from the SU(2)L

doublet components in the 10-plets. Following Ref. [1] we expand on some of the most important

predictions of this SO(10) ×U(1)PQ model which includes gauge coupling uniﬁcation, inﬂation,

proton decay, axion, and heavy WIMP DM, and non-thermal leptogenesis implemented within

a framework that takes into account the observed fermion masses and mixings.

The paper is organized as follows. In Section 2we summarize the salient features of the model

including the ﬁeld content and the symmetry breaking pattern. The renormalization group

analysis of the SM gauge couplings and proton decay are discussed in Section 3, and the eﬀects

of threshold corrections and dimension-5 operators on the uniﬁcation of the gauge couplings

are discussed in Section 4. In Section 5we sketch the Coleman-Weinberg inﬂationary scenario.

Section 6is devoted to the generation of topological defects (monopoles and cosmic strings)

and the gravitational wave spectrum from the decay of cosmic string loops. In Section 7we

construct the Boltzmann equations for the production of the baryon asymmetry of the Universe

via non-thermal leptogenesis as well as the non-thermal generation of fermionic DM. In Section 8

we analyse the axion contribution to the DM abundance and solve numerically the Boltzmann

equations in two examples. Our conclusions are summarized in Section 9.

2SO(10) ×U(1)PQ Symmetry Breaking

The fermion sector consists of three generations of 16-plets and two generations of 10-plets

denoted as follows:

ψ(i)

16 (1) (i= 1,2,3), ψ(α)

10 (−2) (α= 1,2),(1)

where the numbers within parentheses are the PQ charges of the respective multiplets. The

complex scalar multiplets are

φ10(−2), φ45(4), φ126(2), φ210(0).(2)

For deﬁniteness, we consider the following breaking scheme

SO(10) ×U(1)PQ

h210(0)i

−−−−−→

SU (2)L⊗SU(2)R⊗SU(4)C×U(1)PQ

h(1,1,15)∈210(0)i

−−−−−−−−−−→

SU (2)L⊗SU(2)R⊗SU(3)C⊗U(1)B−L×U(1)PQ h(1,3,1,−2)∈(1,3,10)∈126(−2)i

−−−−−−−−−−−−−−−−−−−→

MII

SU (3)C⊗SU(2)L⊗U(1)Y⊗Z2×U(1)0

h(1,3,1)+(1,1,15)∈45(4)i

−−−−−−−−−−−−−−−→

SU (3)C⊗SU(2)L⊗U(1)Y⊗Z2h(1,2,±1

2)∈10(−2)i

−−−−−−−−−−−−→

SU (3)C⊗U(1)Q⊗Z2.(3)

Here MU,MI, and MII respectively denote the grand uniﬁcation and the two intermediate

gauge symmetry breaking scales and fais the breaking scale of U(1)0

PQ. The representations of

the multiplets that remain massless at diﬀerent stages of gauge symmetry breaking are shown

in Table 1. The vacuum expectation value (VEV) of 210(0) along the (1,1,1) direction breaks

SO(10) to the Pati-Salam (PS) gauge group G2L2R4C[16] at the uniﬁcation scale MU. At this

stage, (1,1,15) ∈210, (1,3,10) and (2,2,15) from 126(−2), (1,3,1) and (1,1,15) from 45(4), and

the bi-doublet from 10(−2) remain massless. The breaking at MIof G2L2R4Cto G2L2R3C1B−Lis

achieved via the VEV of the appropriate component of (1,1,15) ∈210. At MII , a VEV along the

SM-singlet direction in (1,3,1,−2) ∈126(2) breaks G2L2R3C1B−Lto SU(3)C×SU(2)L×U(1)Y,

leaving in addition an unbroken Z2which is the subgroup of the center Z4of Spin(10) [6]. Note

that the U(1)PQ symmetry, so far unbroken, is rotated to another global anomalous U(1)0

symmetry generated by Q0

PQ = 5QPQ −3(B−L) + 4T3

R, where T3

Ris the diagonal generator of

SU (2)R. The VEV of 45(4) ﬁnally breaks the U(1)0

PQ symmetry at the scale fa.

3 Renormalization Group Evolution and Proton Decay

The renormalization group evolution of the gauge couplings gi(i= 1,2, ..., n) in a generic

product gauge group of the form G ≡ G1⊗ G2⊗... ⊗ Gncontaining non-Abelian groups and at

SO(10) ×U(1)P Q G2L2R4C×U(1)P Q G2L2R3C1B−L×U(1)P Q G3C2L1YZ2×U(1)0

P Q

Scalars

210(0) h(1,1,1)i

(1,1,15)h(1,1,1,0)i

(1,1,3,4

3)GB

(1,1,3,−4

3)GB

(1,1,8,0)

(1,3,15)

(3,1,15)

(2,2,6)GB

(2,2,10)

126(−2) (1,3,10) (1,3,1,−2)h(1,1,0)i

(1,1,−1)GB

(1,1,−2)

(1,3,3,−2

(1,3,6,2

(2,2,15) (2,2,1,0) (1,2,±1

(2,2,3,4

(2,2,3,−4

(2,2,8,0)

(1,1,6)

(3,1,10)

45(4) (1,1,15) (1,1,1,0)

(1,1,3,4

(1,1,3,−4

(1,1,8,0)

(1,3,1) (1,3,1,0) (1,1,0)

(1,1,±1)

(3,1,1)

(2,2,6)

10(−2) (2,2,1) (2,2,1,0) (1,2,±1

(1,1,6)

Table 1: Representations of scalar multiplets at diﬀerent stages of gauge symmetry breaking.

We denote the gauge symmetry by the subscripts of the caligraphy G. The numbers represent

the dimension of the multiplets under the non-Abelian gauge groups along with the charges

under the Abelian gauge groups. The multiplets in bold fonts are those that remain massless

and contribute to the Renormalization Group Equations (RGEs) of gauge couplings whereas

the ones with the subscript GB are Goldstone bosons eaten by the gauge ﬁelds. The rest of the

multiplets are integrated out at the breaking scale of the parent gauge symmetry. We keep one

linear combination of the four SM doublets light after the breaking of the left-right symmetry

at MII .

most a single Abelian group is governed by the equations [17–25]:

µdgi

dµ =1

16π2big3

i+1

(16π2)2

j=1

bijg3

ig2

j,(4)

where µis the renormalization scale parameter and

bi=4

3κT (Fi)DFi+1

3ηT (Si)DSi−11

3C2(Gi),

bij =20

3C2(Gi)δij + 4C2(Fj)κT (Fi)DFi

+2

3C2(Gi)δij + 4C2(Sj)ηT (Si)DSi−34

3(C2(Gi))2δij (5)

are the one- and two-loop β-coeﬃcients respectively. The representation of a ﬁeld multiplet

is denoted as R= (R1, R2, ..., Rn) where R≡Ffor fermions and R≡Sfor scalars. Here,

κ= 1 (1/2) for Dirac (Weyl) fermions, η= 1 (1/2) for complex (real) scalars, T(Ri) is the

normalization of the representation Ri,C2(Gi) is the quadratic Casimir operator for the group Gi,

and C2(Ri) is the quadratic Casimir operator for the representation Ri. Also, DRi=Qj6=iD(Rj)

with D(Ri) being the dimension of the ith representation in the multiplet. For an Abelian group

Gi=U(1)iand a representation Riwith charge qi, we set T(Ri) = C2(Ri) = q2

iand C2(Gi) = 0.

The extended survival hypothesis (ESH) [26] states that at the level of unbroken gauge sym-

metry, the only scalars that remain light are the ones required to provide the VEVs for breaking

this and the subsequent gauge symmetries. This hypothesis provides a prescription for choosing

a minimal scalar sector at any stage of gauge symmetries. We use the ESH to choose the scalar

sector of the model as given in Table 1. The multiplets in bold fonts are those that remain mass-

less at each stage of symmetry breaking and contribute to the RGEs of the gauge couplings. The

rest of the multiplets are heavy and decoupled at the parent gauge symmetry breaking scale.

We keep one linear combination of the four SM doublets to be light after the breaking of the

G2L2R3C1B−Lsymmetry at MII . Table 2shows the one- and two-loop beta coeﬃcients for the

renormalization group evolution of the gauge couplings at diﬀerent stages of gauge symmetry

starting from the scale MUto the mass mDM of the fermionic DM particles.

G2L2R4C×U(1)P Q G2L2R3C1B−L×U(1)P Q G3C2L1YZ2×U(1)0

P Q











,





268

351 525

51 884

1245

105

249

1109













−4

−17







,





39 12 3

9 66 12 27

2−2

187













−17

−11

163





,





−2

12 133

164

667

150







Table 2: One- and two-loop beta coeﬃcients for the renormalization group evolution of the gauge

couplings at diﬀerent stages of gauge symmetry. The light scalar multiplets that contribute to

the RGEs are listed in bold fonts in Table 1.

The dimension-6 operators that mediate the decay p→π0e+are given in the physical basis

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APredictiveSO(10)ModelGeorgeLazarides,1RinkuMaji,2RishavRoshan,3QaisarSha41SchoolofElectricalandComputerEngineering,FacultyofEngineering,AristotleUniversityofThessaloniki,Thessaloniki54124,Greece2TheoreticalPhysicsDivision,PhysicalResearchLaboratory,Navarangpura,Ahmedabad380009,India3DepartmentofPh...

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A Predictive SO10Model George Lazarides1Rinku Maji2Rishav Roshan3Qaisar Sha4 1School of Electrical and Computer Engineering Faculty of Engineering

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