PHYSICAL REVIEW D 101, 063533 (2020) Particle physics model of curvaton inflation in a stable universe † Zoltán P´eli* and István Nándori MTA-DE Particle Physics Research Group, H-4010 Debrecen, PO Box 105, Hungary ‡ Zoltán Trócsányi Institute for Theoretical Physics, ELTE Eötvös Loránd University, Pázmány P´eter s´etány 1/A, H-1117 Budapest, Hungary and MTA-DE Particle Physics Research Group, H-4010 Debrecen, PO Box 105, Hungary (Received 26 November 2019; revised manuscript received 17 February 2020; accepted 6 March 2020; published 27 March 2020) We investigate a particle physics model for cosmic inflation based on the following assumptions: (i) there are at least two complex scalar fields; (ii) the scalar potential is bounded from below and remains perturbative up to the Planck scale; (iii) we assume slow-roll inflation with maximally correlated adiabatic and entropy fluctuations 50–60 e-folds before the end of inflation. The energy scale of the inflation is set automatically by the model. Assuming also at least one massive right-handed neutrino, we explore the allowed parameter space of the scalar potential as a function of the Yukawa coupling of this neutrino. DOI: 10.1103/PhysRevD.101.063533 I. INTRODUCTION only open questions in particle physics—such as the origin of neutrino masses—but also those in cosmology. The standard model (SM) of elementary particle inter- One of the vigorously studied questions in our under- actions [1] has been proven experimentally to high pre- standing of the early universe is the physics of cosmic cision at the Large Electron Positrion Collider [2] and also inflation. Today the original idea for inflation [9] is not at the Large Hadron Collider (LHC) [3,4]. At the LHC the favored because it is unclear how to define a proper last missing piece, the Higgs particle has also been mechanism to explain the required reheating of the uni- discovered and its mass has been measured at high verse. A popular solution to this question of reheating is the precision [5,6], which made possible the precise renorm- slow-roll scenario [10,11] in which the ground state starts alization group (RG) flow analysis of the Brout-Englert- from an unstable position and rolls down very slowly to a Higgs potential [7,8]. The perturbative precision of this local or global minimum. The inflation stops when the computation is sufficiently high so that the conclusion potential energy function becomes too steep, which leads to about the instability of the vacuum in the standard model a fast roll. In principle, the slow roll can start from a large cannot be questioned. While the instability may not field value and proceed toward a minimum with a smaller influence the fate of our present Universe if the tunneling field value, or from a small (essentially vanishing) field rate from the false vacuum is sufficiently low (making the value to a larger minimum. These two cases are referred to Universe metastable), one may insist that the vacuum must as large- and small-field slow roll [12]. A problem related to be stable up to the Planck scale. Indeed, if we assume that large-field slow-roll is the initial value problem, namely the characteristic energy scale of particle interactions were one has to explain why the ground state starts from a value close to the Planck scale immediately after the big bang, the much larger than the typical energy scale of inflation. Universe based on the SM was unstable and could not exist, Chaotic inflation [13] was devised to handle this problem, which calls for an extension of the SM. Presently it is but then one has to assume very large—again larger than widely accepted that such an extension should explain not the scale of inflation—fluctuations. The origin of inflation and especially the emergence of the inflationary potential is still an open question in cosmology [14,15]. *[email protected] † [email protected] It is known that scalar fields can mimic the equation of ‡ [email protected] state required for the exponential expansion of the early http://pppheno.elte.hu/ universe [10,11]. As the Higgs boson was discovered, we know that at least one doublet scalar field exists in nature. Published by the American Physical Society under the terms of Hence, it may appear natural to assume that the Brout- the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to Englert-Higgs (BEH) field is the inflaton (see for example the author(s) and the published article’s title, journal citation, Ref. [16]), but such a scenario was criticized, see for and DOI. Funded by SCOAP3. instance Ref. [17]. Many types of scalar potentials have 2470-0010=2020=101(6)=063533(11) 063533-1 Published by the American Physical Society PELI,´ NÁNDORI, and TRÓCSÁNYI PHYS. REV. D 101, 063533 (2020) þ already been discussed in the literature as viable scenar- ϕ 1 ϕ1 þ iϕ2 ϕ ¼ ¼ pffiffiffi ð Þ ios for cosmic inflation [17]. There are three major 0 ; 1 ϕ 2 ϕ3 þ iϕ4 categories of scalar inflaton potentials with minimal kinetic terms: (i) the large field, (ii) the small field we assume the existence of a complex scalar χ that and (iii) the hybrid models. In the third case one transforms as a singlet under the SM gauge transforma- introduces more than one field, with one of those being tions. The gauge invariant Lagrangian of the scalar fields is the inflaton and the other field switches off the expo- nential expansion. In this sense it is not a real multifield ðϕÞ Ã ðϕÞμ ðχÞ Ã ðχÞμ Lϕ χ ¼½Dμ ϕ D ϕ þ½Dμ χ D χ − Vðϕ; χÞ: ð2Þ model. The case of hybrid models is excluded by ; experimental observations because those predict a scalar The covariant derivative for the new scalar χ is tilt ns larger than one in contradiction with the observed structure of the thermal fluctuations of the cosmic ðχÞ Dμ ¼ ∂μ − ig Zμ ð3Þ microwave background radiation (CMBR) resulting in Z ns ¼ 0.9677 Æ 0.0060 [18,19]. The tensor and scalar where Zμ is the new gauge field and g is the new gauge power spectra of the CMBR suggest a small value for Z coupling. In the renormalization group analysis below we the tensor-to-scalar ratio r, consistent with zero, which shall concentrate on the phenomenologically relevant case emerges automatically in real multifield models with when the new couplings are superweak g ≪ 1, hence curvaton scenario [20–23]. Z negligible in the scalar sector. Here we consider the simplest possible extension of the In Eq. (2) the potential energy is defined as SM that has the potential to explain the emergence of the inflationary potential. The complete particle physics jϕj2 2 2 2 2 1 2 2 model of this new superweak force was published in Vðϕ; χÞ¼V0 − μ jϕj − μχjχj þ ðjϕj ; jχj ÞC ϕ 2 2 Ref. [24]. We study the renormalization group flow of the jχj scalar couplings of this extension of the SM at one-loop ð4Þ order in perturbation theory. We find that for small values of the new gauge couplings—as suggested by other where jϕj2 ¼jϕþj2 þjϕ0j2 and — phenomenological considerations the only relevant cou- plings are the scalar ones and the largest Yukawa- 2λϕ λ coupling in the neutrino sector if we assume similar C ¼ ð5Þ λ 2λχ hierarchy of the latter as one can observe for u-type quarks in the SM [25]. Hence, the precise formulation of is the coupling matrix. This potential energy function the gauge sector does not influence our conclusions, so in 2 2 contains a coupling term λjϕj jχj of the scalar fields in this article we present only the scalar and relevant addition to the usual quartic terms. The value of the additive Yukawa sector in detail. We also find that the extended constant V0 is irrelevant for particle dynamics, but as we scalar sector can serve as a simple multifield model of shall see, it is relevant for the inflationary model, hence we cosmic inflation with a curvaton scenario. We show that allow a nonvanishing value for it. In order that this potential in a fairly constrained region of the parameter space, the energy be bounded from below, we have to require the model can provide a natural switch on and off mechanism positivity of the self-couplings, λϕ, λχ > 0. The eigenvalues of inflation. of the coupling matrix are II. PARTICLE PHYSICS MODEL qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 2 λÆ ¼ λϕ þ λχ Æ ðλϕ − λχÞ þ λ ; ð6Þ The particle content of the model coincides with that in 2 the standard model of particle interactions, supplemented with λþ > 0 and λ− < λþ. In the physical region the with one complex scalar field and three right-handed potential can be unbounded from below only if λ− < 0 neutrinos. The latter can be either Dirac- or Majorana- and the eigenvector belonging to λ− points into the first type. While we present the relevant renormalization group quadrant, which may occur only when λ < 0. In this case, equations (RGE) for both cases, for the sake of definiteness the potential will be bounded from below if the coupling we present physical predictions only for the case of Dirac matrix is positive definite, i.e., neutrinos. 2 det C ¼ 4λϕλχ − λ > 0: ð7Þ A. Scalar sector Our scalar sector is defined similarly as in the SM, If these conditions are satisfied, we findp theffiffiffi minimum ofp theffiffiffi but in addition to the usual scalar field ϕ that is an potential energy at field values ϕ ¼ v= 2 and χ ¼ w= 2 SUð2Þ-doublet where the vacuum expectation values (VEVs) are 063533-2 PARTICLE PHYSICS MODEL OF CURVATON INFLATION IN A … PHYS.