Gravitational Waves, \mu Term & Leptogenesis from B-L Higgs Inflation in Supergravity

# Gravitational Waves, μ Term & Leptogenesis from B−L Higgs Inflation in Supergravity

C. Pallis
Department of Physics, University of Cyprus,
P.O. Box 20537, Nicosia 1678, CYPRUS
cpallis@ucy.ac.cy
###### Abstract

We consider a renormalizable extension of the minimal supersymmetric standard model endowed by an and a gauged symmetry. The model incorporates chaotic inflation driven by a quartic potential, associated with the Higgs field which leads to a spontaneous breaking of , and yields possibly detectable gravitational waves. We employ quadratic Kähler potentials with a prominent shift-symmetric part proportional to and a tiny violation, proportional to , included in a logarithm with prefactor . An explanation of the term of the MSSM is also provided, consistently with the low energy phenomenology, under the condition that one related parameter in the superpotential is somewhat small. Baryogenesis occurs via non-thermal leptogenesis which is realized by the inflaton’s decay to the lightest or next-to-lightest right-handed neutrino with masses lower than . Our scenario can be confronted with the current data on the inflationary observables, the baryon asymmetry of the universe, the gravitino limit on the reheating temperature and the data on the neutrino oscillation parameters, for and gravitino as light as .

Keywords: Cosmology, Inflation, Supersymmetric Models
PACS codes: 98.80.Cq, 12.60.Jv, 95.30.Cq, 95.30.Sf

Published in Universe 4, no. 1, 13 (2018)

Gravitational Waves, Term & Leptogenesis from Higgs Inflation in Supergravity

C. Pallis

Department of Physics, University of Cyprus,

P.O. Box 20537, Nicosia 1678, CYPRUS

## 1 Introduction

One of the primary ideas, followed the introduction of inflation [3] as a solution to longstanding cosmological problems – such as the horizon, flatness and magnetic monopoles problems –, was its connection with a phase transition related to the breakdown of a Grand Unified Theory (GUT). According to this economical and highly appealing scenario – called henceforth Higgs inflation (HI) – the inflaton may be identified with one particle involved in the Higgs sector [4, 7, 6, 5, 8, 9] of a GUT model. In a series of recent papers [10, 11] we established a novel type of GUT-scale, mainly, HI called kinetically modified non-Minimal HI. This term is coined in Ref. [12] due to the fact that, in the non-Supersymmetric (SUSY) set-up, this inflationary model, based on the power-law potential, employs not only a suitably selected non-minimal coupling to gravity but also a kinetic mixing of the form – cf. Ref. [13]. The merits of this construction compared to the original (and certainly more predictive) model [4, 14, 15] of non-minimal inflation (nMI) defined for are basically two:

• For , the observables depend on the ratio and can be done excellently consistent with the the recent data [16, 17] as regards the tensor-to-scalar ratio, . More specifically, all data taken by the Bicep2/Keck Array CMB polarization experiments up to and including the 2014 observing season (BK14) [17] seem to favor ’s of order , since the analysis yields

 r=0.028+0.026−0.025  ⇒  0.003≲r≲0.054at 68\%c.l. (1.0)
• The resulting theory respects the perturbative unitarity [18, 19] up to the Planck scale for and of order unity.

In the SUSY – which means Supergravity (SUGRA) – framework the two ingredients necessary to achieve this kind of nMI, i.e., the non-minimal kinetic mixing and coupling to gravity, originate from the same function, the Kähler potential, and the set-up becomes much more attractive. Actually, the non-minimal kinetic mixing and gravitational coupling of the inflaton can be elegantly realized introducing an approximate shift symmetry [20, 21, 13, 22, 10]. Namely, the constants and introduced above can be interpreted as the coefficients of the principal shift-symmetric term () and its violation () in the Kähler potentials . Allowing also for a variation of the coefficients of the logarithms appearing in the ’s we end up with the most general form of these models analyzed in Ref. [11].

Here, we firstly single out the most promising models from those investigated in Ref. [11], employing as a guiding principle the consistency of the expansion of the ’s in powers of the various fields. Namely, as we mention in Ref. [11], and are the two most natural choices since they require just quadratic terms in some of the ’s considered. From these two choices the one with is privileged since it ensures within Eq. (1) with central value for the spectral index . Armed with the novel stabilization mechanisms for the non-inflaton accompanied field – recently proposed in the context of the Starobinsky-type inflation [23] too –, we concentrate here on ’s including exclusively quadratic terms with . The embedding of the selected models in a complete framework is the second aim of this paper. Indeed, a complete inflationary model should specify the transition to the radiation domination, explain the origin of the observed baryon asymmetry of the universe (BAU) [24] and also, yield the minimal supersymmetric standard model (MSSM) as low energy theory. Although this task was carried out for similar models – see, e.g., Refs. [6, 25, 26] – it would be certainly interesting to try to adapt it to the present set-up. Further restrictions are induced from this procedure.

A GUT based on , where is the gauge group of the standard model and and denote the baryon and lepton number respectively, consists [22, 10, 11] a conveniently simple framework which allows us to exemplify our proposal. Actually, this is a minimal extension of the MSSM which is obtained by promoting the already existing global symmetry to a local one. The Higgs fields which cause the spontaneous breaking of the symmetry to can naturally play the role of inflaton. This breaking provides large Majorana masses to the right-handed neutrinos, , whose the presence is imperative in order to cancel the gauge anomalies and generate the tiny neutrino masses via the seesaw mechanism. Furthermore, the out-of-equilibrium decay of the ’s provides us with an explanation of the observed BAU [27] via non-thermal leptogenesis (nTL) [28] consistently with the gravitino () constraint [29, 30, 31, 32] and the data [33, 34] on the neutrino oscillation parameters. As a consequence, finally, of an adopted global symmetry, the parameter appearing in the mixing term between the two electroweak Higgs fields in the superpotential of MSSM is explained as in Refs. [35, 25] via the vacuum expectation value (v.e.v) of the non-inflaton accompanying field, provided that the relevant coupling constant is rather suppressed.

Below, we present the particle content, the superpotential and the possible Kähler potentials which define our model in Sec. 2. In Sec. 3 we describe the inflationary potential, derive the inflationary observables and confront them with observations. Sec. 4 is devoted to the resolution of the problem of MSSM. In Sec. 5 we analyze the scenario of nTL exhibiting the relevant constraints and restricting further the parameters. Our conclusions are summarized in Sec. 6. Throughout the text, the subscript of type denotes derivation with respect to (w.r.t) the field and charge conjugation is denoted by a star. Unless otherwise stated, we use units where is taken unity.

## 2 Model Description

We focus on an extension of MSSM invariant under the gauge group . Besides the MSSM particle content, the model is augmented by six superfields: a gauge singlet , three ’s, and a pair of Higgs fields and which break . In addition to the local symmetry, the model possesses also the baryon and lepton number symmetries and a nonanomalous symmetry . The charge assignments under these symmetries of the various matter and Higgs superfields are listed in Table 1. We below present the superpotential (Sec. 2.1) and (some of) the Kähler potentials (Sec. 2.2) which give rise to our inflationary scenario.

### 2.1 Superpotential

The superpotential of our model naturally splits into two parts:

 W=WMSSM+WHI,where (2.0)
##### (a)

is the part of which contains the usual terms – except for the term – of MSSM, supplemented by Yukawa interactions among the left-handed leptons () and :

 WMSSM=hijDdciQjHd+hijUuciQjHu+hijEeciLjHd+hijNNciLjHu. (2.\theparentequationa)

Here the th generation doublet left-handed quark and lepton superfields are denoted by and respectively, whereas the singlet antiquark [antilepton] superfields by and [ and ] respectively. The electroweak Higgs superfields which couple to the up [down] quark superfields are denoted by [].

##### (b)

is the part of which is relevant for HI, the generation of the term of MSSM and the Majorana masses for ’s. It takes the form

 WHI=λS(¯ΦΦ−M2/4)+λμSHuHd+λ[Nci]¯ΦNc2i. (2.\theparentequationb)

The imposed symmetry ensures the linearity of w.r.t . This fact allows us to isolate easily via its derivative the contribution of the inflaton into the F-term SUGRA potential, placing at the origin – see Sec. 3.1. It plays also a key role in the resolution of the problem of MSSM via the second term in the right-hand side (r.h.s) of Eq. (2.1) – see Sec. 4.2. The inflaton is contained in the system . We are obliged to restrict ourselves to subplanckian values of since the imposed symmetries do not forbid non-renormalizable terms of the form with – see Sec. 3.3. The third term in the r.h.s of Eq. (2.1) provides the Majorana masses for the ’s – cf. Refs. [6, 25, 26] – and assures the decay [36] of the inflaton to , whose subsequent decay can activate nTL. Here, we work in the so-called -basis, where is diagonal, real and positive. These masses, together with the Dirac neutrino masses in Eq. (2.1), lead to the light neutrino masses via the seesaw mechanism.

### 2.2 Kähler Potentials

HI is feasible if cooperates with one of the following Kähler potentials – cf. Ref. [11]:

 K1 = −Nln(1+c+F++F1X(|X|2))+c−F−, (2.\theparentequationa) K2 = −Nln(1+c+F+)+c−F−+F2X(|X|2), (2.\theparentequationb) K3 = −Nln(1+c+F+)+F3X(F−,|X|2), (2.\theparentequationc)

where , and the complex scalar components of the superfields and are denoted by the same symbol whereas this of by . The functions assist us in the introduction of shift symmetry for the Higgs fields – cf. Ref. [21, 22]. In all ’s, is included in the argument of a logarithm with coefficient whereas is outside it. As regards the non-inflaton fields , we assume that they have identical kinetic terms expressed by the functions with . In Table 2 we expose two possible forms for each following Ref. [23]. These are selected so as to successfully stabilize the scalars at the origin employing only quadratic terms. Recall [37, 23] that the simplest term leads to instabilities for and light excitations of for and . The heaviness of these modes is required so that the observed curvature perturbation is generated wholly by our inflaton in accordance with the lack of any observational hint [27] for large non-Gaussianity in the cosmic microwave background.

As we show in Sec. 3.1, the positivity of the kinetic energy of the inflaton sector requires and . For , our models are completely natural in the ’t Hooft sense because, in the limits and , they enjoy the following enhanced symmetries

 Φ→ Φ+c,¯Φ→ ¯Φ+c∗andXγ→ eiφγXγ, (2.0)

where and are complex and real numbers respectively and no summation is applied over . This enhanced symmetry has a string theoretical origin as shown in Ref. [38]. In this framework, mainly integer ’s are considered which can be reconciled with the observational data. Namely, acceptable inflationary solutions are attained for [] if [ or ] – see Sec. 3.4. However, deviation of the ’s from these integer values is perfectly acceptable [22, 11, 40, 39] and can have a pronounced impact on the inflationary predictions allowing for a covering of the whole plane with quite natural values of the relevant parameters.

## 3 Inflationary Scenario

The salient features of our inflationary scenario are studied at tree level in Sec. 3.1 and at one-loop level in Sec. 3.2. We then present its predictions in Sec. 3.4, calculating a number of observable quantities introduced in Sec. 3.3.

### 3.1 Inflationary Potential

Within SUGRA the Einstein frame (EF) action for the scalar fields and can be written as

 S=∫d4x√−ˆg(−12ˆR+Kα¯βˆgμνDμzαDνz∗¯β−ˆV), (3.\theparentequationa)

where is the Ricci scalar and is the determinant of the background Friedmann-Robertson-Walker metric, with signature . We adopt also the following notation

 Kα¯β=K,zαz∗¯β>0and% Dμzα=∂μzα+igAaμTaαβzβ (3.\theparentequationb)

are the covariant derivatives for the scalar fields . Also, is the unified gauge coupling constant, are the vector gauge fields and are the generators of the gauge transformations of . Also is the EF SUGRA scalar potential which can be found via the formula

where we use the notation

 Kα¯βKα¯γ=δ¯β¯γ,DαWHI=WHI,zα+KαWHI  and  Da=zα(Ta)βαKβ  with  Kα=K,zα. (3.\theparentequationd)

If we express and according to the parametrization

 Φ=ϕeiθ√2cosθΦ,  ¯Φ=ϕei¯θ√2sinθΦ  and  Xγ=xγ+i¯xγ√2, (3.0)

where , we can easily deduce from Eq. (3.1) that a D-flat direction occurs at

 xγ=¯xγ=θ=¯θ=0andθΦ=π/4 (3.0)

along which the only surviving term in Eq. (3.1) can be written universally as

 ˆVHI=eKKSS∗|WHI,S|2=λ2(ϕ2−M2)216f2(1+n)\@fontswitchRwheref\@fontswitchR=1+c+ϕ2 (3.0)

plays the role of a non-minimal coupling to Ricci scalar in the Jordan frame (JF) – see Refs. [37, 22]. Also, we set

 (3.0)

The introduction of allows us to obtain a unique inflationary potential for all the ’s in Eqs. (2.2) – (2.2). For and or or and we get and develops an inflationary plateau as in the original case of non-minimal inflation [4]. Contrary to that case, though, here we have also which dominates the canonical normalization of – see Sec. 3.2 – and allows for distinctively different inflationary outputs as shown in Refs. [12, 10]. Finally, the variation of above and below zero allows for more drastic deviations [22, 11] from the predictions of the original model [4]. In particular, for , remains increasing function of , whereas for , develops a local maximum

 ˆVHI(ϕmax)=λ2n2n16c2+(1+n)2(1+n)  at  ϕmax=1√c+n. (3.0)

In a such case we are forced to assume that hilltop [41] HI occurs with rolling from the region of the maximum down to smaller values. Therefore, a mild tuning of the initial conditions is required which can be quantified somehow defining [42] the quantity

 Δmax⋆=(ϕmax−ϕ⋆)/ϕmax, (3.0)

is the value of when the pivot scale crosses outside the inflationary horizon. The naturalness of the attainment of HI increases with and it is maximized when which result to .

The structure of as a function of is displayed in Fig. 1. We take , and (light gray line), (black line) and (gray line). Imposing the inflationary requirements mentioned in Sec. 3.4 we find the corresponding values of and which are and respectively. The corresponding observable quantities are found numerically to be or and or with in all cases. We see that is a monotonically increasing function of for whereas it develops a maximum at , for , which leads to a mild tuning of the initial conditions of HI since , according to the criterion introduced above. It is also remarkable that increases with the inflationary scale, , which, in all cases, approaches the SUSY GUT scale facilitating the interpretation of the inflaton as a GUT-scale Higgs field.

### 3.2 Stability and one-Loop Radiative Corrections

As deduced from Eq. (3.1) is independent from which dominates, though, the canonical normalization of the inflaton. To specify it together with the normalization of the other fields, we note that, for all ’s in Eqs. (2.2) – (2.2), along the configuration in Eq. (3.1) takes the form

 (Kα¯β)=diag⎛⎜ ⎜ ⎜⎝M±,Kγ¯γ,...,Kγ¯γ8 % \footnotesize elements⎞⎟ ⎟ ⎟⎠ (3.\theparentequationa)

with

 (3.\theparentequationb)

Here and . Upon diagonalization of we find its eigenvalues which are

 κ+=c−(1+Nr±(c+ϕ2−1)/f2\@fontswitchR)≃c−andκ−=c−(1−Nr±/f\@fontswitchR), (3.0)

where the positivity of is assured during and after HI for

 r±

Given that and , Eq. (3.2) implies that the maximal possible is . Given that tends to [] for [ or ], the inequality above discriminates somehow the allowed parameter space for the various choices of ’s in Eqs. (2.2) – (2.2).

Inserting Eqs. (3.1) and (3.2) in the second term of the r.h.s of Eq. (3.1) we can, then, specify the EF canonically normalized fields, which are denoted by hat, as follows

 Kα¯β˙zα˙z∗¯β = (3.\theparentequationa) ≃ 12(˙ˆϕ 2+˙ˆθ 2++˙ˆθ 2−+˙ˆθ 2Φ+˙ˆxγ2+˙ˆ¯x  γ2),

where and the dot denotes derivation w.r.t the cosmic time . The hatted fields of the system can be expressed in terms of the initial (unhatted) ones via the relations

 dˆϕdϕ=J=√κ+,ˆθ+=J√2ϕθ+,ˆθ−=√κ−2ϕθ−,andˆθΦ=√κ−ϕ(θΦ−π4)⋅ (3.\theparentequationb)

As regards the non-inflaton fields, the (approximate) normalization is implemented as follows

 (ˆxγ,ˆ¯xγ)=√Kγ¯γ(xγ,¯xγ). (3.\theparentequationc)

As we show below, the masses of the scalars besides during HI are heavy enough such that the dependence of the hatted fields on does not influence their dynamics – see also Ref. [6].

We can verify that the inflationary direction in Eq. (3.1) is stable w.r.t the fluctuations of the non-inflaton fields. To this end, we construct the mass-squared spectrum of the scalars taking into account the canonical normalization of the various fields in Eq. (3.\theparentequationa) – for details see Ref. [22]. In the limit , we find the expressions of the masses squared (with and ) arranged in Table 3. These results approach rather well the quite lengthy, exact expressions taken into account in our numerical computation. The various unspecified there eigenvalues are defined as follows

 h±=(hu±hd)/√2,¯h±=(¯hu±¯hd)/√2andˆψ±=(ˆψΦ+±ˆψS)/√2, (3.\theparentequationa)

where the (unhatted) spinors and associated with the superfields and are related to the normalized (hatted) ones in Table 3 as follows

 ˆψΦ±=√κ±ψΦ±withψΦ±=(ψΦ±ψ¯Φ)/√2. (3.\theparentequationb)

From Table 3 it is evident that assists us to achieve – in accordance with the results of Ref. [23] – and also enhances the ratios for w.r.t the values that we would have obtained, if we had used just canonical terms in the ’s. On the other hand, requires

 λμ<λ(1+c+ϕ2/N)/4(1/ϕ2+c+) for  K=K1; (3.\theparentequationa) λμ<λϕ2(1+1/NX)/4 for  K=K2  and  K3. (3.\theparentequationb)

In both cases, the quantity in the r.h.s of the inequality takes its minimal value at and numerically equals to . Similar numbers are obtained in Ref. [25] although that higher order terms in the Kähler potential are invoked there. We do not consider such a condition on as unnatural, given that in Eq. (2.1) is of the same order of magnitude too – cf. Ref. [43]. Note that the due hierarchy in Eqs. (3.\theparentequationa) and (3.\theparentequationb) between and differs from that imposed in the models [35] of F-term hybrid inflation, where plays the role of inflaton and and are confined at zero. Indeed, in that case we demand [35] so that the tachyonic instability in the direction occurs first, and the system start evolving towards its v.e.v, whereas and continue to be confined to zero. In our case, though, the inflaton is included in the system while and the system are safely stabilized at the origin both during and after HI. Therefore, is led at its vacuum whereas , and take their non-vanishing electroweak scale v.e.vs afterwards.

In Table 3 we display also the mass of the gauge boson which becomes massive having ‘eaten’ the Goldstone boson . This signals the fact that is broken during HI. Shown are also the masses of the corresponding fermions – note that the fermions and , associated with and remain massless. The derived mass spectrum can be employed in order to find the one-loop radiative corrections, to . Considering SUGRA as an effective theory with cutoff scale equal to , the well-known Coleman-Weinberg formula [44] can be employed self-consistently taking into account the masses which lie well below , i.e., all the masses arranged in Table 3 besides and . Therefore, the one-loop correction to reads

 ΔˆVHI = 164π2(ˆm4θ+lnˆm2θ+Λ2+2ˆm4slnˆm2sΛ2+4ˆm4h+lnˆm2h+Λ2+4ˆm4h−lnˆm2h−Λ2 (3.0) + 23∑i=1(ˆm4i~νclnˆm2i~νcΛ2−ˆm4iNclnˆm2iNcΛ2)−4ˆm4ψ±lnˆm2ψ±Λ2⎞⎠,

where is a renormalization group (RG) mass scale. The resulting lets intact our inflationary outputs, provided that is determined by requiring or . These conditions yield and render our results practically independent of since these can be derived exclusively by using in Eq. (3.1) with the various quantities evaluated at – cf. Ref. [22]. Note that their renormalization-group running is expected to be negligible because is close to the inflationary scale – see Fig. 1.

### 3.3 Inflationary Observables

A period of slow-roll HI is determined by the condition – see e.g. Ref. [45]

 \footnotesizemax{ˆϵ(ϕ),|ˆη(ϕ)|}≤1, (3.0)

where

 ˆϵ=12⎛⎜⎝ˆVHI,ˆϕˆVHI⎞⎟⎠2=12J2(ˆVHI,ϕˆVHI)2≃8(1−nc+ϕ2)2c−ϕ2f2\@fontswitchR (3.\theparentequationa)

and

 ˆη=ˆVHI,ˆϕˆϕˆVHI=1J2(ˆVHI,ϕϕˆVHI−ˆVHI,ϕˆVHIJ,ϕJ)=43−3(1+3n)c+ϕ2+n(1+4n)c2+ϕ4c−ϕ2f2\@fontswitchR⋅ (3.\theparentequationb)

Expanding and for we can find from Eq. (3.0) that HI terminates for such that

 ϕf≃\footnotesize max{2√2/c−√1+16(1+n)r±,2√3/c−√1+36(1+n)r±}. (3.0)

The number of e-foldings, , that the pivot scale suffers during HI can be calculated through the relation

 ˆN⋆=∫ˆϕ⋆ˆϕfdˆϕˆVHIˆVHI,ˆϕ≃{((1+c+ϕ2⋆)2−1)/16r±for