Unified Origin of Inflation, Baryon Asymmetry, and Neutrino Mass
Ajay Kaladharan Shaikh Saad







[E-mail:]kaladharan.ajay@okstate.edu

Department of Physics, Oklahom State University, Stillwater, OK 74078, USA


[E-mail:]shaikh.saad@unibas.ch

Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland


§ ABSTRACT 
In this work, we present a unified theoretical framework that simultaneously addresses some of the most intriguing puzzles in particle physics and cosmology, namely the origins of neutrino mass, baryon asymmetry, and cosmic inflation. 
In our model, inflation is driven by a combination of the Standard Model Higgs, the type II seesaw Higgs responsible for neutrino mass generation, and the unified symmetry-breaking Higgs field. During inflation, non-zero values of the latter field ensure the absence of the monopole problem. The baryon asymmetry is generated through the Affleck-Dine mechanism, facilitated by the non-zero angular motion in the phase of a complex scalar field, which is part of the inflaton. We find that the successful  parameter region for generating baryon asymmetry through a renormalizable term in the scalar potential requires a rather heavy type II seesaw triplet, with a mass well beyond the TeV scale. Inflationary observables, in particular, the spectral index is in excellent agree with experimental observation, whereas tensor-to scalar ratio is expected to be probed by the future LiteBIRD and CMB-S4 missions. 




Unified Origin of Inflation, Baryon Asymmetry, and Neutrino Mass

    Shaikh Saad
    

==================================================================




§ INTRODUCTION

Although the Standard Model (SM) of particle physics has been very successful in describing the fundamental laws of physics, it fails to incorporate, for example, the observed neutrino oscillation data and the matter-antimatter asymmetry in the universe. Resolving these issues requires going beyond the SM (BSM), and Grand Unified Theories (GUTs) <cit.> are the prime candidates for its ultraviolet (UV) completion. In this work, in particular, we focus on the minimal unified gauge group, namely SU(5) GUT. For neutrino mass generation, within SU(5) GUT, one of the simplest options  is the type II seesaw mechanism <cit.> (numerous efforts have been made to attribute non-zero masses to neutrinos in SU(5) GUT. For instance, in Refs. <cit.>, neutrinos acquire their masses at the tree level, while in Refs. <cit.>, the masses emerge at the loop level).



Note that the generation of neutrino mass in this setup requires only a single generation of 15_H-dimensional Higgs representation. However, achieving a successful thermal leptogenesis demands a second copy of 15_H multiplet. From the point of view of minimality and predictivity, such an extension is not desirable since incorporating correct neutrino oscillation data requires only a single generation of 15_H multiplet. 

As recently pointed out in Ref. <cit.>, one copy of scalar weak triplet with a hypercharge equal to one–employed for type II seesaw–is sufficient to give rise to the correct baryon asymmetry utilizing Affleck-Dine mechanism <cit.>. The prerequisites in realizing this mechanism  are (i) that the triplet carries a charge under a global U_X(1) symmetry (e.g., X=B (baryon number) or L (lepton number)), (ii) the presence of a small term in the Lagrangian that breaks this global symmetry, and (iii) the triplet acquiring a displaced vacuum value in the early universe. 


Interestingly, implementation of type II seesaw to give neutrinos non-zero masses  guarantees that the triplet scalar carries a U_X(1) charge (typically the lepton number) due to its interactions with the SM fermions and the SM Higgs doublet. Moreover, certain mixed terms in the scalar potential breaks this U_X(1) symmetry, which is essential to generate Majorana neutrino masses. Remarkably, this same triplet scalar can also play a role in realizing cosmic inflation <cit.>, as a result, it can naturally get a large displaced vacuum value during inflationary phase <cit.>.   Note that cosmic inflation, which refers to the rapid exponential expansion of the universe, is  another major puzzle that remains unresolved within the SM of cosmology. The theory of inflation elegantly explains several observed features of the universe, such as the large-scale homogeneity and isotropy. GUT models also require inflation to washout superheavy monopoles and other topological defects generated during the subsequent symmetry breaking phases <cit.>. However,  the exact mechanism driving this inflationary phase, as well as its connection to particle physics, remains one of the most significant open questions in cosmology. 


In this work, we consider a minimal SU(5) GUT in which, the  inflationary dynamics is governed by three fields:  the GUT-breaking adjoint Higgs 24_H, the multiplet 15_H  responsible for neutrino mass generation, and the SM Higgs contained in the fundamental representation 5_H. Specifically, we explore inflation with a non-minimal coupling to gravity (in the context of GUT, for earlier works, see, for example, Refs. <cit.>).  Although the global B-L remains preserved in the original Georgi-Glashow <cit.> model of SU(5), in our framework, the generation of neutrino mass breaks this symmetry. This explicit breaking is controlled by a small term in the scalar potential, and the weak triplet within 15_H, being part of the inflaton, attains a displaced vacuum value very early in the universe, ensuring the genesis of baryon asymmetry through the Affleck-Dine mechanism. Intriguingly, the adjoint field that breaks the GUT symmetry also acquires non-zero field values during inflation, thereby evading the monopole problem—a major challenge in GUT models—without the need to introduce any additional fields.  Moreover, inflationary observables are found to be in great agreement with experiments.  Our proposed model, therefore, suggests a common origin for the generation of neutrino mass, matter-antimatter asymmetry, and cosmic inflation within an economical unified theoretical framework.



The key results of this work are briefly summarized here.  In our setup, inflation is driven by three fields, whereas Ref. <cit.> considered two field scenario. Additionally, due to a unified approach, we have a few significant differences compared to the earlier works. First, unlike Ref. <cit.>, our model does not rely on higher-dimensional operators for generating the baryon asymmetry. Rather, a dimension four-term in the scalar potential (with coefficient λ), which explicitly breaks a global symmetry, plays a major role in producing the matter-antimatter asymmetry.  Second, as a consequence of the renormalizable term, we find a rather narrow range for the mass of the type II seesaw triplet field, specifically, in the window 10^7 GeV≲ m_Δ≲ 10^10 GeV (depending on the value of λ), which is consistent with the observed baryon asymmetry in the universe.  Lastly, as mentioned earlier, since the GUT-breaking Higgs participates in inflation and acquires non-zero field values during this period, our model intriguingly offers a novel resolution to the monopole problem.



This article is organized in the following way. In Sec. <ref>, we introduce the propose model. Fermion mass generation and gauge coupling unification are discussed in Sec. <ref>.  In Sec. <ref>, the inflationary dynamics, along with the generation of baryon asymmetry, are studied. In this same section, by considering relevant washout and perturbativity constraints, we show the parameter space where the correct order of baryon asymmetry can be produced. Finally, we conclude  in Sec. <ref>.







§ MODEL


Proposal:– In this work, we consider a GUT model based on SU(5) gauge group. In this setup, the SM fermions are embedded within the 5_F and 10_F dimensional representations

    5_F=[ d^C_1, d^C_2, d^C_3, e,  -ν_e ]^T,
    
       10_F=1/√(2)[      0  u^C_3 -u^C_2    u_1    d_1; -u^C_3      0  u^C_1    u_2    d_2;  u^C_2 -u^C_1      0    u_3    d_3;   -u_1   -u_2   -u_3      0    e^C;   -d_1   -d_2   -d_3   -e^C      0 ].

In the above, we have suppressed the family index, and mass generation of the charged fermion masses will be discussed later in the text. 

Moreover, GUT symmetry is broken via the vacuum expectation value (VEV) of the adjoint 24_H representations. The SM Higgs doublet, which is embedded within a fundamental representation 5_H, spontaneously breaks the electroweak (EW) symmetry. The decomposition of these Higgs fields, along with their VEV structures, are as follows:

    Φ≡  24_H   =ϕ_8(8,1,0)+ϕ_1(1,3,0)+ϕ_0(1,1,0)
         +ϕ_3(3,2,-5/6)+ϕ_3(3,2,5/6),
    ϕ≡  5_H   =H(1,2,1/2)+T(3,1,-1/3).

and,

    ⟨ 24_H ⟩ =v_24diag( -1, -1, -1, 3/2, 3/2 ), 
       ⟨ 5_H ⟩ = (0    0    0    0    v_5/√(2))^T ,

with v_5=246 GeV. 


As a result of the GUT symmetry breaking, the superheavy vector bosons receive the following masses: 
    M_X,Y=√(25/8) g_GUT v_24,

where, g_GUT is the unified gauge coupling constant.  These gauge fields induce d=6 operators that are responsible for mediating proton decay. Accordingly, the lifetime of the protons can be estimated as <cit.> 

    τ_p∼16π^2 M^4_X/g^4_GUTm^5_p,

where m_p is the proton mass. Then, from the current proton decay bound of τ_p (p→ e^+π^0)> 2.4× 10^34 yrs <cit.>, we obtain a lower bound on the GUT scale M_X∼ M_GUT≳ 6× 10^15 GeV, where we have used g_GUT=0.6. 


Note that with the particle contents given above, neutrinos remain massless. In this work, we adopt one of the simplest ways to give neutrinos a non-zero masses, namely via type-II seesaw <cit.>. This requires the addition of a 15_H dimensional Higgs representation, which has the following decomposition: 

    Δ≡   15_H=Δ_1(1,3,1)+Δ_3(3,2,1/6)+Δ_6(6,1,-2/3).

Once the EW symmetry is broken, the weak triplet, Δ_1, receives an induced VEV and generates neutrino mass (as will be discussed later in the text).  






Scalar potential:–  With the above-mentioned Higgs fields, the complete  renormalizable scalar potential takes the form

    V   =
    -m^2_Φ Tr[Φ^2] + λ^Φ_1 Tr[Φ^2]^2 + λ^Φ_2 Tr[Φ^4] + μ_Φ Tr[Φ^3] 
    
       -m^2_ϕϕ^†ϕ + λ^ϕ_1 (ϕ^†ϕ)^2
    
       +m^2_ΔΔ^†Δ + λ^Δ_1 (Δ^†Δ)^2 + λ^Δ_2 (Δ^†ΔΔ^†Δ)
    
       +μ_1 ϕ^†Φϕ + λ^Φϕ_1 (ϕ^†ϕ) Tr[Φ^2]+ λ^Φϕ_2 (ϕ^†ϕΦΦ) 
    
       +μ_2 Δ^†ΦΔ 
    + λ^ΦΔ_1 (Δ^†Δ) Tr[Φ^2]+ λ^ΦΔ_2 (Δ^†ΔΦΦ)
    
       
    + λ^ΦΔ_3 (Δ^†ΔΦΦ)^' 
    +λ^ϕΔ_1 (ϕ^†ϕ)(Δ^†Δ) +λ^ϕΔ_2 (ϕ^†ϕΔ^†Δ)
    
       
    +{μ_3 ϕϕΔ^† +λ_1 ϕϕΔ^†Φ
    +h.c.}.

To avoid cluttering, we have suppressed all group indices. The prime here denotes  a different contraction compared to the unprimed one. The SM Higgs quartic coupling is identified as λ^ϕ_1=m^2_h/(2v_5^2).  


Since v_24≫ v_5≫ v_Δ, for all practical purposes, for the analysis of GUT symmetry breaking and computing the mass spectrum, it is sufficient to consider only non-zero v_24. The potential minimum where SU(5) is broken to the SM gauge group has the stationary condition,

    m^2_Φ=(15λ^Φ_1 +7/2λ^Φ_2 )v^2_24 +3/4μ_Φ v_24.



Scalar mass spectrum:–
Utilizing the above relation, the physical masses of states arising from 5_H are given by 

    m^2_H   =-m^2_ϕ+3/2μ_1v_24+(15/2λ^Φϕ_1+9/4λ^Φϕ_2)v_24^2,   
    
    
    m^2_T   =-m^2_ϕ-μ_1v_24+(15/2λ^Φϕ_1+λ^Φϕ_2)v_24^2.

Doublet-triplet splitting is obtained by fixing 

    μ_1=2/3m^2_ϕ/v_24-(5  λ^Φϕ_1-3/2λ^Φϕ_2)v_24  .

The masses of the submultiplets of 24_H are

    m^2_ϕ_0   =3/4μ_Φ v_24+(30λ^Φ_1+7λ^Φ_2)v_24^2,
    
    
    m^2_ϕ_1   =15/4μ_Φ v_24+10λ^Φ_2 v_24^2,
    
    
    m^2_ϕ_8   =-15/4μ_Φ v_24+5/2λ^Φ_2 v_24^2,

whereas, components of 15_H receive the following masses:

    m^2_Δ_1   =m^2_Δ+ 3/2μ_2 v_24+(
    15/2λ^ΦΔ_1
    +9/4λ^ΦΔ_2
    +9/4λ^ΦΔ_3
    )v_24^2,
    
     
    m^2_Δ_3   =m^2_Δ+ 1/4μ_2 v_24+(
    15/2λ^ΦΔ_1
    +13/8λ^ΦΔ_2
    -3/2λ^ΦΔ_3
    )v_24^2,
    
    
    m^2_Δ_6   =m^2_Δ-μ_2 v_24+(
    15/2λ^ΦΔ_1
    + λ^ΦΔ_2
    + λ^ΦΔ_3
    )v_24^2.

In the subsequent sections, we examine the inflationary dynamics and check the consistency of the mass spectrum of the scalar fields as computed above with a successful generation of baryon asymmetry.  



Global symmetry:
Note, in the Georgi-Glashow SU(5) model, although baryon number B and lepton number L are separately broken,  B-L is still a global symmetry; therefore, B-L is conserved (hence, all nucleon decay operators conserved B-L). Consequently, neutrinos are massless. However, in extended scenarios, such as ours with Δ, this global B-L symmetry is broken. Consequently, neutrinos receive non-zero masses. For example, in the scalar potential, both the terms μ_3 and λ_1 violate it (however, Yukawa couplings respect this symmetry). 
More specifically, the unbroken abelian global symmetry, U(1)_X, is identified as <cit.> 

    X=5(B-L)-4Y

such that X[5_F]=-3 and X[10_F]=+1. Therefore,  the scalars carry the following charges under U(1)_X:

    X[ϕ]=-2,  
    X[Δ]=+6, 
    X[Φ]=0.
 

From these charge assignments, one can see that the cubic term μ_3 and the quartic term λ_1 in the scalar potential Eq. (<ref>) explicitly break the global U(1)_X symmetry.   As will be shown below, the latter term is responsible <cit.> for generating the matter-antimatter asymmetry in the universe. The cubic term, on the other hand, must be very small so as not to ruin <cit.> the predictability of the generated asymmetry during the inflation. 







§ FERMION MASS GENERATION


Charged Fermion Sector:–
Among the scalar fields introduced in our model, only the fundamental Higgs can provide masses to the charged fermions. The relevant part of the Yukawa couplings reads

    ℒ_Y⊃ Y_10^ab10_F^a10^b_F5_H+Y_5^ai 10^a_F  5^i_F 5^*_H.

Once the EW symmetry is spontaneously broken, charged fermions acquire the following masses:

    M_U    =√(2)v_5(Y^u+Y^u T),
    
    M_E    = v_5/2Y^d T,
    
    M_D    = v_5/2Y^d.

Since it predicts M_E=M_D^T at the GUT scale, this simplest setup fails to correctly reproduce the observed mass spectrum. Therefore, we introduce a pair of vectorlike fermions  10_F+10_F that allows additional Yukawa interactions of the form

    ℒ_Y⊃  y^'10_F10_F5^*_H + (m_a  + λ_a 24_H) 10_F  10_F^a,

where a=1-4. With these interactions, the modified mass matrices of the down-type quarks and the charged leptons are given by <cit.>

    M_D=
    [  (Y_5)^ijv_5/2 m_i-λ_i v_24/4;  (Y_5)^4jv_5/2 m_4-λ_4 v_24/4 ]_4× 4,
    
       M_E=
    [  (Y_5^T)^ijv_5/2  (Y_5^T)^i4v_5/2; m_j- 3/2λ_j v_24  m_4-3/2λ_4 v_24 ]_4× 4.

Owing to the mixing with the vector-like fermions, the above mass matrices break the wrong mass relation, namely M_E=M_D^T, and a consistent fit to the charged fermion spectrum can be easily obtained. 


Gauge coupling unification:
In the SU(5) setup, gauge coupling unification requires a few states to live below the GUT scale. Interestingly, within our scenario, a pair of vectorlike fermions employed to cure the wrong mass relations can greatly help in achieving coupling unification.  As shown in Ref. <cit.>, the vectorlike quarks Q(3,2,1/6)⊂ 10_F having a mass in the range m_Q∼ 10^3-10^6 GeV can provide gauge coupling unification consistent with current proton decay limits from Super-Kamiokande. In this analysis, the weak triplet ϕ_1 and the color octet ϕ_8 from 24_H are assumed to live in the intermediate mass scale in between m_Q-M_GUT. 





Neutrino Mass:–
In our framework, neutrino mass is generated by the type II seesaw. The corresponding Yukawa interaction is given by

    ℒ_Y⊃1/2 Y^ij_ν5_F^i 5_F^j 15_H
    
       ⊃1/2χ_2 e^iθ_2{1/√(2)(Y_ν)_ij}ν^T_i C ν_j.

From the above Lagrangian, neutrinos receive the following masses:

    m^ν_ij= v_Δ(Y_ν)_ij ,

where the induced VEV of the neutral component of the  15_H multiplet has the expression

    v_Δ   = -v^2_5/2m^2_Δ_1( μ_3+ 3/√(2)λ_1 v_24) _≡μ.

Recall, v_5=246 GeV and v_24∼ 10^16 GeV; therefore, if the second term dominates, the size of λ_1 must be somewhat small to replicate the neutrino mass scale.  As discussed above, to predict the generated baryon asymmetry at the end of the inflation, the cubic term is taken to be rather small. Therefore, within our setup, we have an one-to-one correspondence between λ_1 and v_Δ.   Although these parameters μ_3 and λ_1 do not enter in the expressions of the scalar masses, they are restricted from the EW-precision  (upper bound) and neutrino mass constraints (lower bound due to perturbativity)

    𝒪(1) GeV> v_Δ≳ 5× 10^-11 GeV.









§ INFLATIONARY DYNAMICS

Cosmic inflation addresses the horizon and flatness problems of standard Big Bang cosmology and explains the origin of structure formation in the observable universe. Moreover, the unwanted superheavy  monopoles generated <cit.> during the phase transition

    SU(5)  SU(3)_C × SU(2)_L × U(1)_Y

are also diluted away by inflation. Without inflation, these stable monopoles would overclose the universe. Our proposed scenario is particularly compelling because inflation is achieved using only the minimal set of fields introduced in the previous section.


The inflationary dynamics will be
induced by a combination of the SM Higgs H∈ 5_H, the weak triplet Δ_1∈ 15_H, and the GUT breaking field ϕ_0∈ 24_H. To realize inflation, one must guarantee flat directions in the scalar potential, which we achieve through these fields having  non-minimal couplings to gravity that generically arise in curved spacetime <cit.>. These couplings flatten the scalar potential at large field values, ensuring that the slow-roll parameters are adequately satisfied and that the predicted observational signatures match current CMB measurements <cit.>.  Inflation of this type has been consider, for example, in Refs. <cit.>.



For the consideration of inflation, the only relevant fields are the neutral components that can acquire VEVs, which we denote by

    (1,2,1/2)⊃ϕ^0=1/√(2)ρ_1 e^iθ_1, 
       (1,3,1)⊃Δ^0=1/√(2)ρ_2 e^iθ_2,
       (1,1,0)⊃Φ^0=ρ_3.

Then the Lagrangian, including the non-minimal couplings to gravity, can be written as (in the Jordan (𝒥) frame)

    ℒ^𝒥/√(-g^𝒥)   ⊃ - ( M^2_pl/2+ξ_ϕϕ^†_0ϕ_0+ξ_ΔΔ^†_0Δ_0 +ξ_Φ/2Φ^2_0 )ℛ^𝒥
       
    -g^μν(D_μϕ^0)^†(D_μϕ^0)
    -g^μν(D_μΔ^0)^†(D_μΔ^0)
    
       
    -g^μν(D_μΦ^0)^†(D_μΦ^0)
    -V_inf
    +ℒ_Yukawa.

Here ξ_ϕ, ξ_Δ, ξ_Φ are real dimensionless (non-minimal) couplings which are taken to be positive, ℛ^𝒥 is the Ricci scalar in the Jordan
frame, and M_pl is the reduced
Planck mass. Moreover, with the fields defined in Eqs. (<ref>)-(<ref>), the scalar potential for inflation V_inf takes the following form:

    V_inf=    
    -1/2m^2_ϕρ_1^2 +λ^ϕ_1/2_≡λ_ϕ/8ρ_1^4
    +1/2m^2_Δρ_2^2 +( λ^Δ_1/4+λ^Δ_2/4)_≡λ_Δ/8ρ_2^4
    
       
    -1/2 m^2_Φρ_3^2 + ( λ^Φ_1/4+7λ^Φ_2/120)_≡λ_Φ/8 ρ_3^4
    +( λ^ϕΔ_1/4+λ^ϕΔ_2/4) _≡λ_ϕΔ/4ρ_1^2ρ_2^2
    
       
    +( λ^Φϕ_1/4+3λ^Φϕ_2/40) _≡λ_Φϕ/4ρ_1^2ρ_3^2
    
       
    +( λ^ΦΔ_1/4+λ^3ΦΔ_2/40 +3λ^ΦΔ_3/40) _≡λ_ΦΔ/4ρ_2^2ρ_3^2
    
       
    +  √(3)μ_1/4√(5)_≡μ̂_1ρ_1^2ρ_3
    +  √(3)μ_2/4√(5)_≡μ̂_2ρ_2^2ρ_3
    +  μ_Φ/4√(15)_≡μ̂_4ρ_3^3
    
       
    +{λ_1 √(3)/4√(10)_≡λ e^i(2θ_1-θ_2)ρ_1^2ρ_2ρ_3+h.c.}
       
    +{μ_3/2√(2)_≡μ̂_3 e^i(2θ_1-θ_2)ρ_1^2ρ_2+h.c.}.

In this work, we will assume the cubic couplings to be sufficiently small (that can be easily arranged) so that they do not affect the inflation dynamics. 



In this setup, the inflation phase starts when one or more of the modulus fields are displaced from their minimum, entering a large-field regime, i.e.,
ξ_ϕρ_1^2+ξ_Δρ_2^2+ξ_Φρ_3^2 ≫ M^2_pl. For the analysis of the inflationary dynamics, it is customary to work in the Einstein frame (ℰ), which is related to the Jordan frame  through a local rescaling of the spacetime metric:

    g^ℰ_μν=Ω^2(ρ_1,ρ_2,ρ_3)g^𝒥_μν,

where, Ω^2(ρ_1,ρ_2,ρ_3)=1+ξ_ϕρ_1^2+ξ_Δρ_2^2+ξ_Φρ_3^2/M_pl^2.

This Weyl transformation serves to both restore the minimal coupling to gravity and flatten the potential for large values of the modulus fields. However, in this Einstein  frame, the metric, g^ℰ_μν,  is no longer trivial due to the  non-canonical form of the kinetic terms <cit.>.    In the large-field regime (Ω^2≫ 1), the Einstein frame scalar potential can be written as

    V^ℰ(ρ_i)=Ω^-4V^𝒥(ρ_i) 
       ≈M_pl^4/8 (ξ_ϕρ_1^2+ξ_Δρ_2^2+ξ_Φρ_3^2)^-2{λ_ϕρ_1^4+λ_Δρ_2^4+λ_Φρ_3^4
    
       
    +2λ_ϕΔρ_1^2ρ_2^2
    +2λ_Φϕρ_1^2ρ_3^2
    +2λ_ΦΔρ_2^2ρ_3^2+16λcos(θ_2)ρ_1^2ρ_2ρ_3}.

In the above potential, the only term that violates the global U(1)_X symmetry is the one with coefficient λ, which needs to be very small so that it does not spoil the inflationary trajectory. This term, however, plays a  crucial role in generating baryon asymmetry.  In this three-field inflationary scenario, hyper-valley condition for realizing inflation successfully demands <cit.>  

    A_ϕΔ>0,   A_ϕΦ>0,   A_ΦΔ>0  ,

where,

    κ_ij   ≡λ_ijξ_i-λ_iξ_j, 
    γ_ij=γ_ji   ≡λ_iλ_j-λ_ij^2>0,
    
    A_ij=A_ji   ≡ξ_kγ_ij+λ_ikκ_ji+λ_jkκ_ij.

As we will show later with an example benchmark point that  all these conditions can be satisfied for 𝒪(1) quartic couplings. For the analysis, we use the following parametrisation for the underlying field space:

    ρ_1=φcosαsinβ,   ρ_2=φsinαsinβ,   ρ_3=φcosβ.

where, α and β are defined as,

    cos^2 α=A_ΔΦ/A_ΔΦ+A_ϕΦ,  cos^2 β=A_ϕΔ/A_ϕΔ+A_ϕΦ+A_ΔΦ.



The above parametrization leads to 

    Ω^2=1+ξφ^2/M_pl^2,

with

    ξ=ξ_ϕcos^2αsin^2β+ξ_Δsin^2 αsin^2β+ξ_Φcos^2 β.



In terms of φ, the Lagrangian in the Jordan Frame is given by,

    ℒ^𝒥/√(-g^𝒥)   =-M_pl^2/2ℛ^𝒥 -ξ/2φ^2ℛ^𝒥 -1/2 g^μν∂_μφ∂_νφ
       -1/2φ^2 sin^2 αsin^2 β  g^μν∂_μθ_2 ∂_νθ_2-V(φ,θ_2),

where the potential is given by 

    V(φ,θ_2)=-1/2 m^2φ^2+(μ̃+2μ̂cosθ_2)φ^3+1/4λ̃φ^4+2λ^'cosθ_2 φ^4 ,

and we have defined the following quantities:

    m^2=m^2_ϕcos^2αsin^2 β+m^2_Δsin^2 αsin^2 β+ m^2_Φcos^2 β, 
       μ̃=μ̂_1 cos^2αsin^2 βcosβ+μ̂_2sin^2 αsin^2 βcosβ+μ̂_4cos^3β,
       μ̂=μ̂_3 cos^2 αsinαsin^3 β,
       λ̃=λ_ϕ/2cos^4αsin^4 β+λ_Δ/2sin^4 αsin^4 β+λ_Φ/2cos^4 β
       +λ_ϕΔcos^2αsin^2αsin^4 β + λ_ϕΦcos^2 αsin^2 βcos^2 β
       +λ_ΔΦsin^2 αsin^2 βcos^2 β, 
       λ^'=λcos^2 αsinαsin^3 βcosβ .



Now reparametrizing φ in terms of canonically normalized field, which we denote by χ,

    dχ/dφ=√(6ξ^2φ^2/M^2_pl+Ω^2)/Ω^2,

it allows us to write χ in terms of φ,

    χ(φ)/M_pl=1/ξ    ( √(1+6ξ)sinh^-1 ( √(ξ+6ξ^2)φ/M_pl ) . 
       . -√(6ξ)sinh^-1 ( √(6ξ^2)φ/M_pl/√(1+ξφ^2/M^2_pl) )   ).

Therefore, the Lagrangian in the Einstein frame takes the following form:  

    ℒ^ℰ/√(-g)=   -M^2_pl/2R-1/2 g^μν∂_μχ∂_νχ
        -1/2 f(χ)g^μν∂_μθ_2 ∂_νθ_2-U(χ,θ_2),

where 

    f(χ)=φ(χ)^2sin^2αsin^2 β/Ω^2,   
    U(χ,θ_2)=V(φ(χ),θ_2)/Ω^4.



From the above Lagrangian, the equations of motion become,

    χ̈-1/2 f_,χθ̇_̇2̇^2+3Hχ̇+U_,χ=0,
       θ̈_̈2̈+f_,χ/f(χ)θ̇_̇2̇χ̇+3Hθ̇_2+1/f(χ)U_,θ_2=0,

and the  Hubble parameter is given by

    H^2=1/3M^2_p ( 1/2 f(χ)θ̇_2^2+1/2χ̇^2+U(χ,θ_2)  ).

Using slow-roll approximation, one can straightforwardly finds,

    χ̇≈ -M_p U_,χ/√(3U)  ,    θ̇_2 ≈ -M_p U_,θ_2/f(χ)√(3U).



We reparametrizing τ=t H_0, where H_0=m_s/2  (m_s=3× 10^13 GeV is the Starobinsky mass scale <cit.>) and denote a derivative with respect to τ by a prime. We, therefore, get

    χ^''-1/2f_,χθ_2^'^2+3H̃/M_plχ^'+U_,χ/H_0^2=0,
       θ_2^''+f_,χ/f(χ)θ_2^'χ^'+3H̃/M_plθ_2^'+1/f(χ)H_0^2U_,θ_2=0,

where we have defined the reduced Hubble parameter as

    H̃^2=1/3 ( 1/2χ^'^2+1/2 f(χ)θ_2^'^2+U/H_0^2 ).







 §.§ Inflationary Observables




Since Eq. (<ref>) describes an approximate single-field inflationary setup, we can utilize it to analyze the evolution of the inflationary phase and associated predictions. 
We choose the relevant parameters such that the inflationary trajectory is preserved and the dynamics of θ_2 negligibly affect inflation.
As discussed above, non-minimal couplings are responsible for flattening the potential in the large field limit, and the Affleck-Dine mechanism is realized via the motion of the dynamical field θ_2. The amount of the asymmetry generated is determined by the size of the non-trivial motion induced in θ_2 sourced by inflation. Since during inflationary phase, m, μ̃, μ̂≪ϕ, and λ^' is small, the quartic coupling λ dominates the inflation dynmaics.  Then the scalar potential Eq. (<ref>), in this regime, can be written as 

    U(χ)≃3/4 m^2_s M^2_pl( 1-e^-√(2/3)χ/M_pl)^2,

which mirrors the Starobinsky form <cit.>, and therefore, inflationary observables are expected to be in full agreement with inflationary observations.  

From the scalar potential Eq. <ref>, the slow-roll parameter is given by

    ϵ≃3/4N^-2_*,

where N_* represents the  e-foldings number from the  horizon
exit to the end of the inflation, which is required to be in the range 50< N_*< 60. N_* is computed using 

    N_*≃ M^-2_pl∫^χ_*_χ_end dχV/dV/dχ.
 
Two of the observables, spectral index and the tensor-to-scalar ratio, are given by

    n_s≃ 1-2N^-1_*,    
    r≃ 12 N^-2_*,

which must satisfy the Planck observations <cit.>:

    n_s = 0.9649 ± 0.042    (68%C.L.) , 
    
       
    r < 0.036    (95%C.L.).
  
Using the approximated analytical formula given in Eq. (<ref>), one sees that within the viable range of e-folding number, namely 50< N_*< 60, the spectrum index is predicted to be 0.96≲ n_s≲ 0.9667, which is fully consistent with current measurements.  On the other hand, from Eq. (<ref>), the tensor-to-scalar ratio is predicted to be in the range 3.3× 10^-3≲ r ≲ 4.8× 10^-3. Interestingly, r in this range can be probed by the  upcoming LiteBIRD
telescope <cit.> and CMB-S4 experiment <cit.>. By solving the corresponding equations numerically, the predicted spectral index and the tensor-to-scalar ratio as a function of the e-folding number are depicted in Fig. <ref>. 


Furthermore, to match the measured scalar power spectrum amplitude at the horizon exit, we demand <cit.>

    A_s≃1/12π^2V^3/M^6_P|dV/dχ|^2 = 2.1× 10^-9

Fulfilling this condition requires

    λ/ξ^2∼ 4.4× 10^-10,

which restricts the size of the λ coupling. 




 §.§ Generation of Baryon Asymmetry

The observed baryon asymmetry of the universe is measured to be <cit.> 

    η_N^obs =n_B/s≃ 8.5× 10^-11,

where  s=(2π^2/45)g_*T^3 is the entropy density    and 
n_B=-(28/79)n_X/s is the  baryon number density of the universe. As aforementioned, in our scenario, the asymmetry n_X is generated through the Affleck-Dine mechanism <cit.>. This mechanism is based on inducing non-zero angular motion in the phase of a complex scalar field, which is charged under a global Abelian symmetry, such as U(1)_X within our framework. In the early universe, this complex scalar field attains a large initial field value and begins to oscillate once the Hubble parameter drops below its mass. If the scalar potential includes an explicit U(1)_X breaking term, this motion generates a net X charge asymmetry. Therefore, if this symmetry is associated with the global X=B or X=L symmetries, a baryon asymmetry can be established before the electroweak phase transition. 


From the analysis performed above, the U(1)_X charge asymmetry is generated during inflation (Fig. <ref>), and its value at the end of inflation is determined as

    n_X=   Q_B-Lφ_end^2sin^2 αsin^2 βθ̇_2, 
    ≈    -Q_L4φ_end^2M_plλ^'sinθ_end/√(3λ̃),

where slow roll approximation is used. 
This net asymmetry will be transferred to the baryonic sector through
equilibrium sphaleron processes prior to the electroweak phase transition. 


The field value at the end of inflation corresponds to χ≈ 0.67 M_pl, as can be seen from Fig. <ref>, which shows the evolution of the field χ during the inflation.  
As mentioned above, we assume that among the terms (m, μ, μ̂, λ̃, λ^') in the potential Eq. (<ref>), the quartic term with λ̃  dominates the inflationary dynamics, whereas the asymmetry is generated due to the presence of the  λ^' coupling (which has negligible effect on the inflation).  The values of quartic coupling and non-minimal couplings to gravity that lead to this inflationary trajectory are provided in Tab. <ref>.




To find the final value of the asymmetry, one needs to evolve the asymmetry after the end of inflation. The spontaneous symmetry breaking of the GUT field after the end of inflation will disrupt the inflationary trajectory described by Eq. (<ref>).  After the ending of the inflationary phase, the inflation starts oscillating and eventually decays. During this time, we assume that the universe goes through a matter-like epoch.  Therefore, the total baryon asymmetry generated during inflation, which is transferred to the SM thermal plasma during reheating, is given by:

    n_X≃    n_X^end×( a_end/a_rh)^3= n_X^end×( H_rh/H_end)^2 .

For a typical parameter space, H_rh/H_end∼ 10^-2 <cit.>, which leads to

    η_B≡. n_B/s |_T=T_reh=η_B^obs ( n^end_X   /10^-16 M^3_pl ) ( g_⋆/112.75 )^-1/4.

Here, to obtain an approximated good estimation,  following Higgs inflation, we have used the reheating temperature T_rh≃ 2.2× 10^14 GeV <cit.> (see also, Refs. <cit.>). A dedicated analysis of reheating may
be different in our scenario due to the presence of additional scalars on top of the SM Higgs, which, however, is beyond the scope of this work. 
Moreover, in our analysis, not to violate unitarity,  we consider ξ < 350 <cit.>.



Solution to the monopole problem: Typically, GUT models suffer from the overproduction of monopoles and require inflation to dilute their density. In SU(5) GUT, the adjoint Higgs breaks the GUT symmetry to the SM, and stable superheavy monopoles are formed at this stage. Typically, GUT singlets are introduced to realize inflation <cit.>. Our model, on the other hand, has the special feature that the monopole problem is naturally solved since the adjoint Higgs participates in inflation and already has non-zero field values during inflation. Therefore, the associated topological defects are inflated away. The model proposed in this work is highly attractive since no additional scalar fields (such as singlets) are required beyond the minimal set for the implementation of inflation.  


Collider constraints:
Since the VEV of the triplet is expected to be somewhat small within our scenario, its primary decay is to SM leptons. Current LHC searches of doubly-charged scalars utilizing pp→Δ^++_1Δ^–_1→ℓ^+ℓ^+ℓ^-ℓ^- provides a lower limit on their masses ≳ 800 GeV <cit.>. Future colliders can probe this mass up to about 4 TeV <cit.>. 
Therefore, neutrino mass, when combined with collider bounds and suppressing the washout  effects, leaves up with the valid range of 0.05 eV≲ v_Δ≲ 10^-5 GeV. As we will show below, a stronger bound on this VEV is obtained from the condition of preventing the generated asymmetry from being washed out.


Lepton Flavor Violation: In this setup, both the singly-charged (Δ^±_1) and doubly-charged (Δ^±±_1) scalars within the weak triplet lead to lepton flavor violating processes <cit.>. The most stringent constraint comes from the tree-level decay of Δ^±±_1 leading to μ→ 3e. On the other hand, μ→ eγ is generated at one-loop order via both  Δ^±_1 and Δ^±±_1 fields. Low energy experiments probing such lepton flavor violating rare decays can put a limit as high as about 10^3-10^4 TeV on the triplet mass. The exact bounds on these masses largely depend on the size of the relevant Yukawa couplings. However, within our framework, washout constraints on the parameter space provide even stronger bounds than the lepton flavor violation. 


Washout:
Owing to  large reheating temperature,  the triplet Δ_1 will rapidly
thermalize at the beginning of the radiation era, and subsequently washout the generated asymmetry. Therefore, we require that the LL↔ HH is never in thermal equilibrium, which, when combined with the requirement of reproducing the correct neutrino mass scale, leads to m_Δ≲ 5× 10^11    GeV <cit.>. Moreover, to ensure that processes like LL ↔Δ_1 and  HH ↔Δ_1 do not co-exist demands v_Δ m^1/2_Δ≲ 3.6× 10^-4 GeV^2/3 <cit.>. 


We illustrate the constraints in Figure <ref>, by plotting the quartic coupling λ versus the mass of the triplet, m_Δ, assuming μ̂ is negligible and considering λ^' = 10^-1λ which holds true for a typical parameter space. In this scenario, the lower bound for λ≳𝒪(10^−14) comes from producing the observed baryon asymmetry. The λ^' term should be sufficiently small that it does not significantly alter the inflationary trajectory. Therefore, we restrict ourselves with   λ≲ 10^−9.   Moreover, as aforementioned, the approximate collider bound on the weak triplet mass, m_Δ≳ 1 TeV, translates into v_Δ≲ 10^-5 GeV. However, we find that a more stringent constraint arises from preventing washout and generating appropriate baryon asymmetry that requires μ>10^-2 GeV, which translates into m_Δ≳ 10^7 GeV and thereby providing an upper bound on the VEV v_Δ≲ 10^-7 GeV. For a fixed value of λ^', the upper limit of the m_Δ is provided by the perturbativity constraints of Yukawa couplings. After taking into account all these theoretical and experimental constraints, the remaining white space in Figure <ref> corresponds to a viable parameter space for generating the correct baryon asymmetry of the universe.    As can be seen from Figure <ref>, a viable range for the mass of the type II seesaw triplet field  is 10^7 GeV≲ m_Δ≲ 10^10 GeV, which is rather heavy and a very different window as obtained in Ref. <cit.>. This difference is owing to the fact that unlike Ref. <cit.>, in this work, we considered a renormalizable scalar potential. In our setup, the baryon asymmetry is generated by utilizing a dimension four term in the Lagrangian.  












§ CONCLUSIONS

The origins of neutrino mass and baryon asymmetry represent two of the most profound and unresolved questions in particle physics. Simultaneously, cosmic inflation—an essential mechanism for explaining several fundamental issues in the standard Big Bang model—remains a major enigma in cosmology. In this work, we have proposed a unified theoretical framework, in particular, a minimal SU(5) Grand Unified Theory, that simultaneously addresses the origins of cosmic inflation, baryon asymmetry, and neutrino mass generation. Our model integrates these phenomena into a coherent structure where inflation is governed by three essential fields: the GUT-breaking adjoint Higgs 24_H, the Standard Model Higgs residing in the fundamental representation 5_H, and the multiplet 15_H, which is crucial for neutrino mass generation. Specifically, we have adopted inflation with a non-minimal coupling to gravity. The weak triplet within 15_H, part of the inflaton, achieves a displaced vacuum value early in the universe, leading to baryon asymmetry via the Affleck-Dine mechanism.  In this setup, we have found that the viable mass range for the type II seesaw triplet field is 10^7 GeV≲ m_Δ≲ 10^10 GeV, which depends on the value of the quartic coupling (λ_1)–the coefficient of a renormalizable term in the scalar potential responsible for explicitly breaking a global symmetry and for the genesis of baryon asymmetry.   Additionally, the adjoint Higgs responsible for breaking the GUT symmetry acquires non-zero field values during inflation, which helps evade the monopole problem.   Within our model, the inflationary observables, especially the spectral index, show excellent agreement with experimental data, while the tensor-to-scalar ratio is expected to be probed by the future LiteBIRD and CMB-S4 experiments.  In summary, our unified approach addresses key challenges in particle physics and cosmology, providing a novel resolution to these issues.    






§ ACKNOWLEDGEMENT

We thank Rabindra N. Mohapatra for discussion. The work of A.K. is supported by the U.S. Department of Energy under grant number DE-SC 0016013.  S.S. acknowledges the Center for Theoretical Underground Physics and Related Areas (CETUP* 2024) and the Institute for Underground Science at Sanford Underground Research Facility (SURF) for providing a conducive environment for the finalization of this work.  Some computing for this project was performed at the High Performance Computing Center at Oklahoma State University, supported in part through the National Science Foundation grant OAC-1531128.




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