Radiative Symmetry breaking Cosmic Strings and Observable Gravity Waves in U1Rsymmetric SU5U1

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Radiative Symmetry breaking, Cosmic Strings and

Observable Gravity Waves in U(1)Rsymmetric

SU(5) ×U(1)χ

Waqas Ahmeda1and Umer Zubairb2

aSchool of Mathematics and Physics,

Hubei Polytechnic University, Huangshi 435003, China

bDivision of Science and Engineering,

Pennsylvania State University, Abington, PA 19001, USA

Abstract

We implement shifted hybrid inﬂation in the framework of supersymmetric SU(5)×

U(1)χGUT model which provides a natural solution to the monopole problem ap-

pearing in the spontaneous symmetry breaking of SU(5). The U(1)χsymmetry is

radiatevely broken after the end of inﬂation at an intermediate scale, yielding topo-

logically stable cosmic strings. The Planck’s bound on the gravitational interaction

strength of these strings, characterized by GNµsare easily satisﬁed with the U(1)χ

symmetry breaking scale which depends on the initial boundary conditions at the

GUT scale. The dimension-5 proton lifetime for the decay p→K+¯ν, mediated

by color-triplet Higgsinos is found to satisfy current Super-Kamiokande bounds for

SUSY breaking scale MSUSY &12.5 TeV. We show that with minimal K¨ahler po-

tential, the soft supersymmetry breaking terms play a vital role in bringing the

scalar spectral index nswithin the Planck’s latest bounds, although with small ten-

sor modes r.2.5×10−6and SU(5) gauge symmetry breaking scale in the range

(2 ×1015 .Mα.2×1016) GeV. By employing non-minimal terms in the K¨ahler

potential, the tensor-to-scalar ratio approaches observable values (r.10−3) with

the SU(5) symmetry breaking scale Mα'2×1016 GeV.

1Introduction

SUSY hybrid inﬂation [1, 2, 4, 5] provides fascinating framework to realize inﬂation within

the grand uniﬁed theories (GUTs) of particle physics. Several GUT models such as, SU(5)

[6], Flipped SU(5) [7, 8, 9] and the Pati-Salam symmetry SU(4)C×SU(2)L×SU(2)R[10,

11, 12], have been employed successfully to realize standard, shifted and smooth variants

1E-mail: waqasmit@hbpu.edu.cn

2E-mail: umer@udel.edu

arXiv:2210.13059v1 [hep-ph] 24 Oct 2022

of hybrid inﬂation [13, 14, 15, 16, 17, 18]. The SU(5) ×U(1)χgauge symmetry is another

suitable choice as a GUT model due to its various attractive features [19, 20]. The whole

gauge group of the model is embedded in SU(5) ×U(1)χ⊂SO(10), owing to special

U(1)χcharge assignment [19]. In contrast to the SU(5) [6] model, a discrete Z2symmetry

that avoids rapid proton decay, naturally arises after the breaking of U(1)χfactor. This Z2

symmetry not only serves as the Minimal Supersymmetric Standard Model (MSSM) matter

parity, but also ensures the existence of a stable lightest supersymmetric particle (LSP)

which can be a viable cold dark matter candidate. Furthermore, the right-handed neutrino

mass is naturally generated by the breaking of U(1)χsymmetry after one of the ﬁelds

carrying U(1)χcharge acquires a Vacuum Expectation Value (VEV) at some intermediate

scale. The well-known advantages of U(1)χsymmetry include seesaw physics to explain

neutrino oscillations, and baryogenesis via leptogenesis [21, 22].

The breaking of SU(5) part of the gauge symmetry leads to copious production of mag-

netic monopoles [23] in conﬂict with the cosmological observations whereas, the breaking

of U(1)χfactor yields topologically stable cosmic strings [19, 24, 25]. The cosmic strings

can be made to survive if U(1)χbreaks after the end of inﬂation. In order to avoid the

undesired monopoles, the shifted or smooth variant of hybrid inﬂation [16] can be em-

ployed, where the gauge symmetry is broken during inﬂation and disastrous monopoles

are inﬂated away. In the simplest SUSY hybrid inﬂationary scenario the potential along

the inﬂationary track is completely ﬂat at tree level. The inclusion of radiative corrections

(RC) to the scalar potential provide necessary slope needed to drive inﬂaton towards the

SUSY vacuum and in doing so the gauge symmetry Gbreaks spontaneously to its subgroup

In this paper, we implement shifted hybrid inﬂation scenario in the SU(5)×U(1)χGUT

model [20] where the SU(5) symmetry is broken during inﬂation and the U(1)χsymmetry

radiatevely breaks to its Z2subgroup at some intermediate scale. The scalar spectral index

nslies in the observed range of Planck’s results [26] provided the inﬂationary potential

incorporates either the soft supersymmetry (SUSY) breaking terms [27, 28, 32, 30, 31, 32],

or higher-order terms in the K¨ahlar potential [17, 33]. Without these terms, the scalar

spectral index nslies close to 0.98 which is acceptable only if the eﬀective number of light

neutrino species are slightly greater than 3 [34]. We show that, by taking soft SUSY

contribution into account along with the supergravity (SUGRA) corrections in a minimal

K¨ahlar potential setup, the predictions of the model are consistent with the Planck’s latest

bounds on scalar spectral index ns[34], although the values of tensor to scalar ratio remain

small. By employing non-minimal K¨ahler potential, large tensor modes are easily obtained,

approaching observable values potentially measurable by near-future experiments such as,

PRISM [35], LiteBird [36], CORE [37], PIXIE [38], CMB-S4 [39], CMB-HD [40] and PICO

[41]. Moreover, the U(1)χsymmetry radiatively breaks after the end of inﬂation at an

intermediate scale, yielding topologically stable cosmic strings. The Planck’s bound [42, 43]

on the strength of gravitational interaction of the strings, GNµsare easily satisﬁed with

the U(1)χsymmetry breaking scale obtained in the model, which depends on the initial

boundary conditions at the GUT scale. Furthermore, the Super-Kamiokande bounds [44]

on dimension-5 proton decay lifetime are easily satisﬁed for SUSY breaking scale MSUSY &

12.5 TeV.

The rest of the paper is organised as follows. Sec. 2 provides the description of the

SU(5) ×U(1)χmodel. The implementation of shifted hybrid inﬂation including the mass

spectrum of the model, gauge coupling uniﬁcation and dimension-5 proton decay is dis-

cussed in Sec. 3. The results and inﬂationary predictions of the model with minimal K¨ahler

potential are presented in Sec. 4 and with non-minimal K¨ahler potential in Sec. 5. The

radiative breaking of U(1)χsymmetry and cosmic strings is discussed in Sec. 6. Finally

we summarize our results in Sec. 7.

2The U(1)RSymmetric SU(5) ×U(1)χModel

The 10,¯

5and 1dimensional representations of the group SU(5) constitute the 16 (spino-

rial) representation of SO(10) and contains the MSSM matter superﬁelds. Their decom-

position with respect to the MSSM gauge symmetry is

Fi≡(10,−1) = Q(3,2,1/6) + uc(¯

3,1,−2/3) + ec(1,1,1) ,

fi≡(¯

5,+3) = dc(¯

3,1,1/3) + `(1,2,−1/2) ,

νc

i≡(1,−5) = νc(1,1,0) ,(1)

where i= 1,2,3 denotes the generation index. The scalar sector of SU(5)×U(1)χconsists of

the following superﬁelds: a pair of Higgs ﬁveplets, h≡(5,2), ¯

h≡(¯

5,−2), containing the

electroweak Higgs doublets (hd, hu) and color Higgs triplets (Dh,¯

D¯

h); a Higgs superﬁeld Φ

that belongs to the adjoint representation (Φ ≡240) and responsible for breaking SU(5)

gauge symmetry to MSSM gauge group; a pair of superﬁelds (χ, ¯χ) which trigger the

breaking of U(1)χinto a Z2symmetry which is realized as the MSSM matter parity; and

ﬁnally, a gauge singlet superﬁeld Swhose scalar component acts as an inﬂaton. The

decomposition of the above SU(5) representations under the MSSM gauge group is

Φ≡(24,0) = Φ24(1,1,0) + WH(1,3,0) + GH(8,1,0)

+QH(3,2,−5/6) + ¯

QH(3,2,5/6),

h≡(5,2) = Dh(3,1,−1/3) + hu(1,2,1/2) ,

h≡(¯

5,−2) = ¯

D¯

h(¯

3,1,1/3) + hd(1,2,−1/2),

χ≡(1,10),¯χ≡(1,−10), S ≡(1,0),(2)

where the singlets (χ, ¯χ) originate from the decomposition of 126 representation of SO(10)

126 = (1,−10) + (¯

5,−2) + (10,−6) + ( ¯

15,6) + (45,2) + ( ¯

50,−2).(3)

Following [16], the U(1)Rcharge assignment of the superﬁelds is given in Table 1 along with

their transformation properties. The SU(5) ×U(1)χand U(1)R, symmetric superpotential

of the model with the leading-order non-renormalizable terms is given by

W=SκM2−κTr(Φ2)−β

Tr(Φ3) + σχχ¯χ+γ¯

hΦh+δ¯

+y(u)

ij FiFjh+y(d,e)

ij Fi¯

fj¯

h+y(ν)

ij νc

i¯

fjh+λijχνc

iνc

j,(4)

Superﬁelds Representations under

SU(5) ×U(1)χ

Global

U(1)R

Matter sector

Fi(10,−1) 3/10

fi(¯

5,3) 1/10

νc

i(1,−5) 1/2

Scalar sector

Φ (24,0) 0

h(5,2) 2/5

h(¯

5,−2) 3/5

χ(1,10) 0

¯χ(1,−10) 0

S(1,0) 1

Table 1: The representations of matter and scalar superﬁelds under SU(5) ×U(1)χgauge

symmetry and global U(1)Rsymmetry in shifted hybrid inﬂation model.

where Mis a superheavy mass and mP= 2.43×1018 GeV is the reduced Planck mass. The

terms in square bracket in the ﬁrst line are relevant for shifted hybrid inﬂation while, the

last two terms are involved in the solution of doublet-triplet splitting problem, as discussed

in section 3.2. The Yukawa couplings y(u)

ij ,y(d,e)

ij ,y(ν)

ij in the second line of (4) generate

Dirac masses for quarks and leptons after the electroweak symmetry breaking, whereas

mνij =λijhχiis the right-handed neutrino mass matrix, generated after χacquires a VEV

through radiative breaking of U(1)χsymmetry, as discussed in Sec. 6.

The superpotential Wexhibits a number of interesting features as a consequence of

global U(1)Rsymmetry. First, it allows only linear terms in Sin the superpotential,

omitting the higher order ones, such as S2which could generate an inﬂaton mass of Hubble

size, invalidating the inﬂationary scenario. Second, the U(1)Rsymmetry naturally avoids

the so called ηproblem [3], that appears when SUGRA corrections are included. Finally,

several dangerous dimension-5 proton decay operators are highly suppressed.

3Shifted Hybrid SU(5) ×U(1)χInﬂation

In this section, the eﬀective scalar potential is computed considering contributions from

the F- and D-term sectors. The superpotential terms relevant for shifted hybrid inﬂation

are

W⊃SκM2−κTr(Φ2)−β

Tr(Φ3)+γ¯

hΦh+δ¯

+σχSχ¯χ+λijχνc

iνc

j.(5)

In component form, the above superpotential is expanded as follows,

W⊃S"κM2−κ1

φ2

i−β

4mP

dijkφiφjφk#+δ¯

haha+γT i

abφi¯

hahb

+σχSχ¯χ+λijχνc

iνc

j,(6)

where Φ = φiTiwith Tr[TiTj] = 1

2δij and dijk = 2Tr[Ti{Tj, Tk}] in the SU(5) adjoint basis.

The F-term scalar potential obtained from the above superpotential is given by

VF=

κM2−κ1

φ2

i−β

4mP

dijkφiφjφk+σχχ¯χ

i

κSφi+3β

4mP

dijkSφjφk−γT i

ab ¯

hahb

bγT i

abφi¯

ha+δ¯

hb

2+X

bγT i

abφiha+δhb

+σχS¯χ+λijνc

iνc

j

2+|σχSχ |2+|2λijχνc

i|2,(7)

where the scalar components of the superﬁelds are denoted by the same symbols as the

corresponding superﬁelds. The VEV’s of the ﬁelds at the global SUSY minimum of the

above potential are given by,

S0=h0

a=¯

a=νc0

i= 0, χ0= ¯χ0= 0 (8)

with φ0

isatisfying the following equation:

i=1

(φ0

i)2+β

2κmP

dijkφ0

iφ0

jφ0

k= 2M2.(9)

The superscript ‘0’ denotes the ﬁeld value at its global minimum. The superﬁeld pair (χ, ¯χ)

break U(1)χto Z2, the matter parity. This symmetry ensures the existence of a lightest

supersymmetric particle (LSP) which could play the role of cold dark matter. Further, as

discussed in [20], this Z2symmetry yields topologically stable cosmic strings.

Using SU(5) symmetry transformation the VEV matrix Φ0=φ0

iTican be aligned in

the 24-direction,

Φ0

24 =φ0

√15 (1,1,1,−3/2,−3/2) .(10)

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RadiativeSymmetrybreaking,CosmicStringsandObservableGravityWavesinU(1)RsymmetricSU(5)U(1)WaqasAhmeda1andUmerZubairb2aSchoolofMathematicsandPhysics,HubeiPolytechnicUniversity,Huangshi435003,ChinabDivisionofScienceandEngineering,PennsylvaniaStateUniversity,Abington,PA19001,USAAbstractWeimplementshif...

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Radiative Symmetry breaking Cosmic Strings and Observable Gravity Waves in U1Rsymmetric SU5U1

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