Particle Physics Model of Curvaton Inflation in a Stable Universe

PHYSICAL REVIEW D 101, 063533 (2020)

Particle physics model of curvaton inflation in a stable universe

† Zoltán Péli* 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 Péter sétány 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 exponential expansion. In this sense it is not a real multifield ðϕÞ ðϕÞμ ðχÞ 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 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 couplings 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 λ ¼ λϕ þ λχ 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

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TABLE I. Possible signs of the couplings in the scalar potential Vðϕ; χÞ in order to have two nonvanishing real VEVs. Θ is the step function, ΘðxÞ¼1 if x>0 and 0 if x<0.

2 2 2 2 2 2 2 ΘðλÞ ΘðλϕÞ ΘðλχÞ Θð4λϕλχ − λ Þ ΘðμϕÞΘðμχ Þ Θð2λχ μϕ − λμχ ÞΘð2λϕμχ − λμϕÞ 1 1 1 Unconstrained 1 Unconstrained 0 1 1 1 1 Unconstrained 011 1 0 1

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 λvw pffiffiffi 2λχμ − λμχ pffiffiffi 2λϕμχ − λμ ð2θ Þ¼− ð Þ ¼ 2 ϕ ¼ 2 ϕ ð Þ tan S 2 2 : 15 v 2 ;w 2 : 8 λϕv − λχw 4λϕλχ − λ 4λϕλχ − λ h s Using the VEVs, we can express the quadratic couplings as The squares of the masses of the mass eigenstates and (mh and ms) are

2 2 λ 2 2 2 λ 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi μ ¼ λϕv þ w ; μχ ¼ λχw þ v ; ð9Þ ϕ 2 2 2 ¼ λ 2 þ λ 2 ∓ ðλ 2 − λ 2Þ2 þðλ Þ2 ð Þ Mh=H ϕv χw ϕv χw vw 16 so those are both positive if λ > 0.Ifλ < 0, the constraint ≤ (7) ensures that the denominators of the VEVs in Eq. (8) where Mh MH in out convention. At this point either ¼ ¼ are positive, so the VEVs have nonvanishing real values mh=s Mh=H or mh=s MH=h. only if As Mh must be positive, the condition

2 2 2 2 2 2ð4λ λ − λ2Þ 0 ð Þ 2λχμϕ − λμχ > 0 and 2λϕμχ − λμϕ > 0 ð10Þ v w ϕ χ > 17 simultaneously, which can be satisfied if at most one has to be fulfilled. If both VEVs are greater than zero–as of the quadratic couplings is smaller than zero. We needed for two nonvanishing scalar masses–, then this summarize the possible cases for the signs of the couplings condition reduces to the positivity constraint (7), but with in Table I. different meaning. Equation (7) is required to ensure that After spontaneous symmetry breaking, we use the the potential be bounded from below if λ < 0, which has to following convenient parametrization for the scalar fields: be fulfilled at any scale. For λ > 0, the potential is bounded from below even without requiring the constraint (7). The 1 0 0 ϕ ¼ pffiffiffi eiT·ξðxÞ=v ð11Þ inequality in (17) ensures Mh > , which has to be fulfilled 2 v þ h0ðxÞ as long as vw > 0 independently of the sign of λ. The VEV of the BEH field and the mass of the and Higgs boson (at the scale equal to mass of the ð Þ ≃ 262 1 t-quark) are known experimentally, v mt GeV χð Þ¼pffiffiffi iηðxÞ=wð þ 0ð ÞÞ ð Þ ð Þ ≃ 131 55 x e w s x ; 12 and mH mt . GeV [8]. Introducing the abbrevia- 2 λ ¼ 1 2 2 λ ð Þ ≃ 0 126 tion SM 2 mH=v ,wehave SM mt . and we can λ ð Þ with T ¼ðT1;T2;T3Þ denoting the generators of the SU(2) distinguish two cases at the weak scale: (i) ϕ mt > λ ð Þ λ ð Þ λ ð Þ group, ξ ¼ðξ1; ξ2; ξ3Þ and η being real functions. We can SM mt and (ii) SM mt > ϕ mt . Then we can relate use the gauge invariance of the model to choose the unitary the new VEV w to the BEH VEV v and the four couplings λ λ λ λ gauge when SM, ϕ, χ, using Eq. (16) as 1 0 1 ð Þ2ð4ðλ ð Þ − λ ð ÞÞλ ð Þ − λð Þ2Þ ϕ0ð Þ¼pffiffiffi χ0ð Þ¼pffiffiffið þ 0ð ÞÞ w mt ϕ mt SM mt χ mt mt x 0 and x w s x : 2 vþ h ðxÞ 2 ¼ 4 ð Þ2λ ð Þðλ ð Þ − λ ð ÞÞ ð Þ v mt SM mt ϕ mt SM mt : 18 ð13Þ Using Eq. (18), it is convenient to consider w as a We can diagonalize the mass matrix (quadratic terms) of the dependent parameter and scan the parameter space two real scalars (h0 and s0) by the rotation of the remaining three quartic couplings as done below. λ ð Þ¼λ ð Þ We are not interested in the case of ϕ mt SM mt h cos θ − sin θ h0 ¼ S S ð Þ because that prevents the model from interpreting neu- 0 14 s sin θS cos θS s trino masses [24]. λ ð Þ λ ð Þ In case (i) when ϕ mt > SM mt , then MH >mH,so π π where for the scalar mixing angle θS ∈ ð− 4 ; 4Þ we find only h can be the Higgs particle and Mh ¼ mH, while

063533-3 PELI,´ NÁNDORI, and TRÓCSÁNYI PHYS. REV. D 101, 063533 (2020) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 III. RENORMALIZATION GROUP EQUATIONS λϕ − λ 4λϕλχ − λ M ¼ m SM : ð Þ H H λ 4ðλ − λ Þλ − λ2 19 SM ϕ SM χ In order to study the values of the couplings at any energy scale we use the RG equations, 2 The positivity of MH, in addition to the constraint in (17), also requires that dg ¼ aβðgÞð23Þ dt λ2 4ðλ − λ Þλ − λ2 0 λ λ þ ð Þ ϕ SM χ > or ϕ > SM : 20 ¼ 10 ¼ 4λχ where the factor a ln ensures that the RG-time t lnðμ=GeVÞ represents the energy scale μ ¼ 10t GeV rather t 2 2 2 than μ ¼ e GeV and the variable g is a generic notation for In case (ii), m > 2λϕv >M , so only H can be H h any of the five gauge couplings, the four most relevant the Higgs particle and we can express the masses of the Yukawa couplings c , c , cτ, and cν, the two quadratic and scalars as in Eq. (19), with h and H interchanged, or t b three quartic scalar couplings (14 equations in total). In explicitly order to solve this coupled system of differential equations, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi we need to specify the β-functions and the initial conditions λ − λ 4λ λ − λ2 for the couplings. ¼ SM ϕ ϕ χ ð Þ Mh mH λ 2 21 At one-loop in perturbation theory, the β-function SM λ þ 4ðλ − λϕÞλχ SM of a dimensionless coupling g is computed from the formula and MH ¼ mH, which does not require any further constraint to (17). ∂ 1 X In principle, it may happen that one of the VEVs β0ðgÞ¼M −δ þ g δ ð24Þ ∂M g 2 Z;i vanishes at some critical scale tc. In that case, for i t>tc the only scalar particle is the Higgs boson. Thus, beyond t we do not need to assume the c where δg is the one-loop counterterm for a given vertex, validity of the extra constraints beyond the requirements which is proportional to g, while δZ;i are the wave function of stability and the new scalar sector affects only the RG renormalization counterterms for all the i legs of the given equations. vertex. The one-loop β-functions are independent of the renormalization scheme and so is the one-loop equation B. Neutrino Yukawa sector Eq. (24) (see for instance Chapter 12 of Ref. [26]). We Neutrino oscillation experiments prove that neutrinos computed those in perturbation theory at one-loop order for have masses, which in a usual gauge field theoretical the complete model of Ref. [24], and can be found description necessitates the assumption that right-handed in Ref. [27]. neutrinos exist. The existence of the new scalar allows for In order to obtain the running of the scalar couplings, we gauge invariant Majorana-type Yukawa terms of dimension need the β-functions of the scalar sector. According to our four operators for the neutrinos assumption on the smallness of the new gauge couplings, we can set gZ ¼ 0. We also neglect the Yukawa couplings 1 X ν of all charged leptons as well as the quarks, except that of L ¼ − νc ðc Þ ν χ þ h:c: ð22Þ Y 2 i;R R ij j;R the t-quark. With these assumptions the β-functions i;j 2 β0ðgÞ ≡ b0ðgÞ=ð4πÞ of the gauge and Yukawa couplings provided the superscript c denotes the charge conjugate simplify to their forms in the standard model, while those in ð Þ the scalar sector become of the field. The Yukawa coupling matrix cR ij is a real symmetric matrix whose values are not constrained. μ2 3 9 There are other gauge invariant Yukawa terms involving ðμ2 Þ¼μ2 12λ þ 2 χ λ þ 6 2 − 2 − 2 b0 ϕ ϕ ϕ 2 ct gY gL ; the left-handed neutrinos (see Ref. [24] where all possible μϕ 2 2 terms are taken into account for neutrino mass gener- μ2 1 ation), but those must contain small Yukawa couplings, ðμ2Þ¼μ2 8λ þ 4 ϕ λ þ 2 b0 χ χ χ 2 cν with Dirac otherwise the left-handed neutrino masses would violate μχ 2 experimental constraints. In our analysis below we μ2 ¼ μ2 8λ þ 4 ϕ λ þ 2 ð Þ assume that at least one element of the diagonal matrix χ χ 2 cν with Majorana 25 T μχ OcRO , with O being a suitable orthogonal matrix, can take any value in the range (0,2). We denote this element by cν below. neutrino for the quadratic, and

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3 9 3 ðλ Þ¼24λ2 þ λ2 − 6 4 þ 4 þ 4 þ 2 2 − λ ð9 2 þ 3 2 Þþ12λ 2 b0 ϕ ϕ ct 8 gY 8 gL 4 gYgL ϕ gL gY ϕct ; 1 ðλ Þ¼20λ2 þ 2λ2 − 4 þ λ 2 b0 χ χ 8 cν χcν with Dirac neutrino; 1 ¼ 20λ2 þ 2λ2 − 4 þ 2λ 2 χ 2 cν χcν with Majorana neutrino; 1 9 3 ðλÞ¼12λλ þ 8λλ þ 4λ2 þ λ 2 þ 6 2 − 2 − 2 b0 ϕ χ 4 cν ct 2 gL 2 gY with Dirac neutrino; 9 3 ¼ 12λλ þ 8λλ þ 4λ2 þ λ 2 þ 6 2 − 2 − 2 ð Þ ϕ χ cν ct 2 gL 2 gY with Majorana neutrino 26

for the quartic couplings where gL and gY are the left- scalar masses [Eq. (20) for case (i) and Eq. (17) for case handed and hypercharge couplings. We solve this system of (ii)], from the initial conditions up to mP, but as long as simplified equations numerically for both types of neu- w>0. A similar analysis was presented in Ref. [32],but trinos. Of course, for cν ¼ 0 the difference between the with Z2 symmetry assumed on the new gauge sector. Our equations for Dirac and Majorana neutrinos disappears. For analysis is based on the simplest, but complete (in the sense cν > 0 the qualitative behavior of the running couplings is of renormalizable quantum field theory) extension of the similar for the two types of neutrinos, but the larger standard model gauge group described in Ref. [24]. coefficients in front of cν for the Majorana neutrino results As seen in Eq. (26), the β-functions are independent of in a stronger effect of the neutrino Yukawa coupling, and both μϕ and μχ, except of course their own β-functions, eventually more constrained parameter space. which decouple from the rest. Thus, in the parameter scan We fix the initial conditions for the standard model we focus on the four-dimensional parameter subspace couplings as done in the two-loop analysis of Ref. [8] of cν, λϕ, λχ, λ by selecting slices at fixed values of cν. (using the two-loop MS scheme). Specifically, we set In addition to the stability conditions, we also require that rffiffiffi the couplings remain in the perturbative region that we 3 defined by ð Þ¼ 0 4626 ð Þ¼0 648 gY mt 5 × . ;gL mt . ; λ ð Þ¼0 126 ð Þ¼262 ð Þ λϕðtÞ < 4π; λχðtÞ < 4π; jλðtÞj < 4π: ð29Þ SM mt . ;vmt GeV; 27 and c ðm Þ¼0.937. Choosing some initial values of the We have restricted the region of the new VEV to t t w<1 TeV because a large value of w is likely to imply quartic couplings λϕðm Þ, λχðm Þ and λðm Þ, we obtain the t t t large kinetic mixing between the two Uð1Þ gauge fields couplings μϕðm Þ and μχðm Þ according to Eq. (9), with t t [24], which is not supported by experiments (see, e.g., sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ref. [33]). This restriction does not influence the allowed λ ðm Þðλϕðm Þ − λ ðm ÞÞ regions for the quartic couplings significantly. ð Þ¼2 ð Þ SM t t SM t w mt v mt 2 ; ð4ðλϕðm Þ − λ ðm ÞÞλχðm Þ − λðm Þ Þ Figures 1 and 2 display our results for the allowed t SM t t t λ ð Þ λ ð Þ regions for the initial conditions of ϕ mt , χ mt and ð Þ λð Þ 28 mt at three selected values of the Dirac neutrino Yukawa coupling as shaded areas where the stability of the vacuum determined from Eq. (18). and the constraints set by the positivity requirement on the scalar masses are respected. In order to ease the IV. CONSTRAINTS FROM STABILITY AND interpretation, we show projections of the allowed three- PERTURBATIVITY dimensional region onto two-dimensional subspaces. We also show the running couplings up to the Planck scale at a In order to constrain the parameter space of the new point representing selected values of the initial conditions at couplings, spanned by λϕ, λχ, λ and cν, we require the the electroweak scale. Although the new VEV w is not an validity of the conditions of Table I, i.e., the stability of the independent parameter, we find interesting to present the − vacuum up to the Planck scale mP. Such studies have projections also in the w g subspaces where g denotes one already been presented for various hidden sector (usually of the quartic couplings (see Fig. 3). The foremost con- singlet scalar) extensions of the standard model in clusion is that the parameter space is not empty, but only for λ ð Þ λ Refs. [28–31]. In addition, we also check the validity of case (i), i.e., when ϕ mt > SM. Thus the Higgs particle the constraints set by the positivity requirement on the has the smaller scalar mass always. In fact, we find that the

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λ ð Þ − λð Þ λ ð Þ − λð Þ FIG. 1. Accepted initial conditions (as shaded areas) in the ϕ mt mt plane (left) and χ mt mt plane (right) for the stability of the vacuum and perturbativity preserved up to the Planck mass at different values of the Dirac neutrino Yukawa coupling cν. The star marks the point in the parameter space for which the example of the running couplings up to the Planck scale is presented in Fig. 2.

FIG. 2. Left: same as Fig. 1 in the λϕðmtÞ − λχðmtÞ plane. Right: the running of the couplings up to the Planck scale in a selected point of the parameter space.

λ ð Þ allowed region for ϕ mt is about [0.151, 0.241] (starting to with increasing Yukawa coupling of the right handed ð Þ 1 5 ð Þ≃ ≃ 1 ≃ 1 decrease only for cν mt > . , while min MH mt neutrino up to cν .Abovecν the parameter space 144 GeV. Clearly, the precise values may somewhat change vanishes swiftly. The maximal allowed regions for the in an analysis at the accuracy of higher loops. Even in the parameters are presented for the selected values of cν in allowed region for λϕ, the parameter space for the other Table II. Thus we find that the stability of the vacuum couplings is constrained significantly and decreases slowly requires cν ≲ 1.65 for Dirac neutrinos (cν ≲ 1.15 for

ð Þ − ð Þ ¼ λ ¼ λ λ FIG. 3. Same as Fig. 1 in the w mt g mt planes. Left: g , right: g ϕ and χ.

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TABLE II. Maximal allowed regions of the couplings required by stability of the vacuum and perturbativity of the couplings up to the Planck scale for selected values of the Yukawa coupling of the right-handed neutrino cνðmtÞ and w<1 TeV set explicitly.

2 2 2 cν λðtÞ λχðtÞ ðμϕðtÞ=GeVÞ ðμχðtÞ=GeVÞ wðtÞ=GeV MHðtÞ=GeV 0.0 ½−0.26; 0.23 [0.011,0.191] ½−4362; 4962½1122; 5802 [213,1000] [144,558] 1.0 ½−0.24; 0.22 [0.031,0.181] ½−4362; 4362½1142; 5482 [220,1000] [144,557] 1.5 ½−0.26; 0.22 [0.141,0.211] ½−3742; 4962½1112; 6032 [203,994] [144,598]

Majorana neutrinos). It is also interesting to remark that the The single-field inflationary models predict purely cur- allowed regions are also very sensitive to the value of the vature perturbations, resulting from energy density fluctu- ð Þ ≃ 1 1 Yukawa coupling of the t quark. For instance, at ct mt . ations. Having multiple fields allows for multiple types of the allowed parameter space vanishes completely. fluctuations, hence several observable quantities, such as the tilts corresponding to curvature, isocurvature (emerging V. COSMOLOGICAL INFLATION due to fluctuations in the relative number density of particles) and a correlation angle Δ [34]. We now explore the cosmic inflation of the two-field Following Ref. [35], we introduce a local rotation of model with potential energy defined in Eq. (4). We consider ðϕ; χÞ into ðσ;sÞ, with σ and s referring to the adiabatic and slow-roll inflation when the potential energy has a large, entropy field. The adiabatic field is the path length along almost flat area for small field values and a global minimum the classical trajectory, while s remains a constant. Using at large values of the VEVs. Such a potential energy allows the power spectra PRðkÞ and PSðkÞ of these comoving for slow roll of the fields from small values toward the curvature and isocurvature fluctuations [36], as well as the global minimum, resulting in cosmic inflation. The spectrum of the cross-correlation CRSðkÞ of the two fields, required form of the potential energy function appears the correlation angle is defined as the dimensionless ratio naturally at some high energy scale, for certain values of the scalar couplings at the mass of the t-quark m . As Eq. (8) C ð Þ t pffiffiffiffiffiffiffiffiffiffi Δ ¼ RS k ð Þ cos 1 2 1 2 : 30 shows, the VEVs are inversely proportional to det C. P = ðkÞP = ðkÞ Fig. 4 shows the running of det C together with that R S of the couplings from initial values at mt chosen from the As the fluctuations can communicate in super-horizon — stability region. We see a narrow wedge like an inverse scales, in order to acquire the observable primordial — resonance where det C becomes very small, implying fluctuations one has to evolve in time the curvature 105 VEVs at around field values of GeV. The figure shows R and isocurvature fluctuations S calculated at horizon an example with vanishing Yukawa coupling cν of the right- exit. This is achieved by applying the transfer handed neutrino, but below we show that the value of cν matrix [34]: influences only the size of the parameter space of the scalar couplings where this phenomenon leads to such potential R 1 TRS R energy function that can support cosmic inflation in accor- ¼ : ð31Þ S 0 T S dance with current values of relevant observables. SS

The TRS and TSS transfer coefficients are related to the correlation angle. At the first order in the slow-roll expansion, we have

T 1 cos Δ ≃ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRS and sin Δ ≃ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð32Þ 1 þ 2 1 þ 2 TRS TRS Also, at the leading order of the slow-roll expansion the curvature power spectrum is

2 P ¼ P ð1 þ Þ ð Þ R R TRS ; 33

2 2 so PR ¼ PR sin Δ. As a result, a factor of sin Δ appears in the tensor-to-scalar ratio, as compared to the single-field case: r ¼ 16ϵðsin ΔÞ2. FIG. 4. Running of the scalar couplings and of det C. g means The number of slow-roll parameters also increase. In the 3 p any of the couplings. single-field case, in addition to the parameter ϵ ¼ 2 ðρ þ 1Þ,

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FIG. 5. Left: A possible trajectory of the rolling of the scalar fields. The black dots denote the extrema of the potential. Right: Projection of the trajectory onto the ϕ − χ plane. The red star denotes the end of the inflation on the trajectory, marked with a red dot on the three-dimensional picture.

μ2 2 ∼ ð1–3 104Þ describing the deviation from the equation of state of the de i =GeV O × . For such values we have ρ ¼ − ρ Sitter space-time ( p, with denoting the energy found that the ηij parameters increase much faster than density and p is the pressure), there is only one other slow- ϵ, reaching 1, while ϵ remaining small, about Oð10−30Þ. roll parameter η, which essentially measures the acceler- Hence, we set the end of inflation by the condition η ¼ 1. η ij ation of the fields. In our example we have three In practice the parameter ηχχ increases the fastest. We show parameters that can be expressed approximately from the an example of such a trajectory in Fig. 5. This trajectory potential as induces N>200 e-folds. The value of ηss refers to the value of η at 60 e-folds before the end of inflation. ∂ V ss η ≃ 2 ij ¼ ϕϕ χχ ϕχ σσ σ ð Þ The smallness of ϵ in our model contradicts to the ij mP ;ij ; ; ; ss; ;s 34 V swampland conjecture that states that it has to be O(1) during inflation [37]. If the conjecture finds a proof, then (note that ηϕϕ þ ηχχ ¼ ησσ þ η ), while ss our model will be excluded, but at present it is not yet the 2 case. Strictly speaking, this conjecture, if it were true, 1 ∂σV ϵ ≃ 2 ð Þ would forbid the slow-roll models of inflation, which 2 mP : 35 V stimulated a lot of activity aiming at refining the conjecture (see, e.g., [38,39]). In this context, we have not studied if ϵ η In principle, inflation is possible only until both and ij our phenomenological model remains a consistent theory at are small, resulting in the slow roll. the Planck scale, and thus it has similar status as many other To set the exact conditions of slow roll, we numerically studies of inflationary models. The model of slow-roll solved the equations of motion, namely inflation we propose here is based on a scalar potential that is not connected to an exact de Sitter vacuum, but have an ∂V ϕ̈þ 3Hϕ_ þ ¼ 0; equation of state that resembles that of the de Sitter ∂ϕ vacuum, with ϵ ¼ 0 meaning exact coincidence. The very ∂ ϵ χ̈þ 3 χ_ þ V ¼ 0 ð Þ small value of implies exponential inflation, whose end is H ∂χ ; 36 marked by the increasing acceleration of the new scalar field χ, giving a distinct role (or necessity) for χ in the with the integration variable transformed to the number particle physics model (in addition to stabilizing the of e-folds N, and terminated the process, when any vacuum up to the Planck scale). of the slow-roll parameters reached unity. We set the The observables are constructed from the slow-roll starting point of the trajectory at vanishing field values. parameters taken 50–60 e-folds before the end of inflation. For the parameter values of the potential energy we used This corresponds to an even smaller ϵ, which reduces the values allowed by the perturbativity and stability conditions tensor-to-scalar ratio, r ¼ 16ϵðsin ΔÞ2 to essentially zero. −1 −2 mentioned in Sec. II, namely jλj, λi ∼ Oð10 –10 Þ and Such a small r is not excluded by current cosmological

063533-8 PARTICLE PHYSICS MODEL OF CURVATON INFLATION IN A … PHYS. REV. D 101, 063533 (2020) measurements. The smallness of r however is in conflict In the curvaton scenario we have to find a new condition 1=4 with the traditional cosmological normalization to set the scale of inflation ðV0Þ . Neglecting ϵ we have:

16 4 − 1 ¼ 2η ð Þ V0 ≃ rð1.6 × 10 GeVÞ : ð37Þ ns ss 38

η This conflict may be resolved by assuming that the Considering ss as a function of V0 (see Eq. (34) with V0 in adiabatic and entropy fluctuations were maximally corre- V in the denominator), we normalize it to produce the scalar – tilt in agreement with the most recent data, ns ≃ 0.966, lated at 50 60 e-folds before the end of inflation, implying 42 4 cos Δ ¼ 1, and hence predicting zero for the tensor-to- yielding V0 ≃ 5 × 10 GeV , which corresponds to the a 1=4 10 scalar ratio. The case of Δ ¼ 0, i.e., maximally correlated scale of inflation V0 ≃ 5 × 10 GeV. fluctuations are referred to as the curvaton scenario. In this Having fixed the value of V0, we propose the following case, the various tilts coincide. inflationary scenario. The scalar potential energy is given

FIG. 6. Projections of the allowed regions of the parameter space of scalar couplings onto the λ − λχ plane (left) and λ − λϕ plane (right) at three different ranges of the scale of inflation and at three selected values of the cν Yukawa coupling.

063533-9 PELI,´ NÁNDORI, and TRÓCSÁNYI PHYS. REV. D 101, 063533 (2020) by Eq. (4). After the big bang the characteristic energy scale In Fig. 6 we present the results of such scan of the of particle interactions is near the Planck scale, hence the parameter space. These plots show different planar pro- scalar fields are fluctuating around zero. As the universe jections of the three dimensional parameter space, spanned λ ð Þ λ ð Þ λð Þ expands, the characteristic energy scale decreases and the by ϕ mt ; χ mt , and mt . The shape and size of the ð Þ scalar couplings run according to their renormalization supported regions is affected by the choice of cν mt ,as group equations, exhibiting the wedge for det C at an seen in the titles of the figures. We find that the parameter μ 1016 energy scale inf (around GeV) that we call the space of the scalar couplings is not empty, but constrained characteristic energy scale of inflation [40]. At this scale strongly if we assume that cosmic inflation took place as the global minimum of the potential energy function described above. This assumption constrains the smallest 5 ð Þ increases to about 10 GeV and the fields start to roll value of w mt to around 265 GeV. slowly toward this minimum, resulting in cosmic inflation. This accelerated expansion continues until the acceleration VII. CONCLUSIONS (second time derivative of the fields) remain negligible in the equation of the motion, determined by max ηij ¼ 1. We proposed a particle physics model of cosmic infla- After this the universe starts its Hubble expansion, decreas- tion. It requires at least two scalar fields. We explored the ing the characteristic energy scale, and the global minimum parameter space where the scalar potential remains of the scalar potential quickly returns to small field values bounded from below up to the Planck scale. We found as observed today, preventing any further inflationary that in a small region of the parameter space of the scalar periods. couplings allowed by the requirement of stability, the determinant of the scalar quartic coupling matrix becomes VI. PREDICTING THE SCALAR COUPLINGS very small at an energy scale around 1016 GeV. As a result the field values at the global minimum of the scalar The cosmological inflation as described in the previous potential increase significantly, allowing for an accelerated section occurs only in a restricted region of the parameter expansion of the universe by a slow-roll model of inflation. space of the scalar couplings, which we define at the We assume the curvaton scenario of inflation, i.e., max- electroweak scale (mt). imally correlated adiabatic and entropy fluctuations at The wedge in the running of det C appears only for 50–60 e-folds before the end of inflation, which implies λð Þ 0 mt < . We have scanned this side of the parameter vanishing tensor-to-scalar ratio. To set the scale of inflation space by selecting fixed initial values for the scalar λ ð Þ λ ð Þ (the normalization of the potential at vanishing field couplings ϕ mt , χ mt , and scanning the allowed initial values), we required that the model reproduces the mea- values of the third one to find those points where the wedge sured value of the scalar tilt. The inflation stops when the ≲ ð10−4Þ in the running of det C appears with a minimum O . parameter that measures the acceleration of the fields starts During this scanning, we need to search only for those to increase quickly. After this the position of the global ð Þ— points where w mt given by Eq. (28), with minimum of the potential decreases preventing the appear- λ ð Þ¼1 ð Þ2 ð Þ2 ≃ 0 126— SM mt 2 mH mt =v mt . and the masses ance of another period of inflation. The model is consistent of the scalars given by Eq. (16) remain positive. We find with observations as long as the tensor-to-scalar ratio is that the parameter space is constrained to a shell on the measured to be very small, r ≪ 1. surface of the region allowed by the conditions of stability and perturbativity of V. The width of the shell is affected by ACKNOWLEDGMENTS the allowed depth of the minimum of det C: the smaller det C, the thinner the shell. Furthermore, we have also This work was supported by Grant No. K 125105 of the found that the minimum value of the location of the wedge National Research, Development and Innovation Fund in μ 1014 ð Þ inf is around GeV, depending slightly on cν mt . Hungary.

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