Warm inflation within a supersymmetric distributed mass model

Warm inflation within a supersymmetric distributed mass model Mar Bastero-Gil,1,* Arjun Berera,2,†Rafael Hernández-Jim´enez,2,‡and João G. Rosa3,§ 1Departamento de Física T´eorica y del Cosmos, Universidad de Granada, Granada-18071, Spain 2School of Physics and Astronomy, University of Edinburgh, Edinburgh, EH9 3FD, United Kingdom 3Departamento de Física da Universidade de Aveiro and CIDMA, Campus de Santiago, 3810-183 Aveiro, Portugal (Received 5 February 2019; published 16 May 2019) We study the dynamics and observational predictions of warm inflation within a supersymmetric distributed mass model. This dissipative mechanism is well described by the interactions between the inflaton and a tower of chiral multiplets with a mass gap, such that different bosonic and fermionic fields become light as the inflaton scans the tower during inflation. We examine inflation for various mass distributions, analyzing in detail the dynamics and observational predictions. We show, in particular, that warm inflation can be consistently realized in this scenario for a broad parametric range and in excellent agreement with the Planck legacy data. Distributed mass models can be viewed as realizations of the landscape property of string theory, with the mass distributions coming from the underlying spectra of the theory, which themselves would be affected by the vacuum of the theory. We discuss the recently proposed swampland criteria for inflation models on the landscape and analyze the conditions under which they can be met within the distributed mass warm inflation scenario. We demonstrate mass distribution models with a range of consistency with the swampland criteria including cases in excellent consistency. DOI: 10.1103/PhysRevD.99.103520 I. INTRODUCTION The most recent cosmological observations once again confirm an expanding universe that is spatially flat, homogeneous, and isotropic on large scales, and where the large scale structure originated from primordial fluctuations with a nearly scale-invariant, adiabatic, and Gaussian spectrum [1]. Inflation [2] remains the dominant paradigm that can consistently explain the observational data. In the standard inflation picture, cold inflation (CI), depicted by a homogeneous scalar “inflaton”field, the short period of quasi–de Sitter accelerated expansion phase quickly dilutes away all traces of any preinflationary matter or radiation density, so that the state of the universe is the vacuum state. However, this generates a supercooled universe and leaves indeterminate a reasonable description of the transition from inflation to the “hot big bang”scenario, required by big bang nucleosynthesis, and the physics of recombination leading to the cosmic microwave background (CMB) that we observe today. This necessarily requires the conversion of inflaton energy density into ordinary matter and radiation and thus to its interactions with other fields. In the conventional inflation picture the inflaton decay can only play a significant role at the end of the slow-roll regime, since particle production is not pictured to occur within the inflationary expansion phase. This leads to cold inflation ending through the standard “(p)reheating”paradigm [3]. The reasoning behind this phase lies in the fact that the perturbative decay width of a particle is generically smaller than its mass, which in turn lies below the Hubble expansion rate for a slowly rolling scalar field. Consequently such interplay between the inflaton and other constituents may perform a negligible role during the slow-roll phase of inflationary models. Nonetheless, it is relevant to note that the perturbative decay width only describes the decay of a field close to the minimum of its potential [4], which is evidently not the case during slow-roll dynamics, and that finite temperature effects can further significantly enhance the rate at which the inflaton dissipates its energy into other degrees of freedom (d.o.f.). Thereby the inflaton field could be coupled to other components and might dissipate its vacuum energy and warm up the universe. This alternative scenario is known as the warm inflation (WI) paradigm [5,6], where dissipative effects and associated particle production can, in fact, sustain a thermal bath concurrently with the accelerated expansion of the universe during inflation. *[email protected] †[email protected] ‡[email protected] §[email protected] Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3. PHYSICAL REVIEW D 99, 103520 (2019) 2470-0010=2019=99(10)=103520(19) 103520-1 Published by the American Physical Society One of the earliest warm inflation models [7] suggested the idea that parameters in an inflation model could be randomly distributed. The distributed-mass model (DM model) [8–11] was subsequently proposed and built on this idea in the context of string theory. It observed [9] that models from string theory have states at many energy levels and the distribution of these levels is ultimately dictated by the string vacuum. In particular, depending on the details of the compactification, the state of Kaluza-Klein modes, and patterns of symmetry breaking, it will induce splittings of string energy levels. Thus, based on the specific properties of a given string realization, the string states will be distributed differently. The idea of such distribution of states can find a more dynamical motivation in the context of the landscape picture, whereby different ground states of string theory will in turn imply changes to the spectrum of string states. In this respect, the DM model was one of the first low-energy realizations following the landscape idea well before the idea was formally stated. The string landscape idea [12,13] claims an unimaginable huge number of possible vacua reaching by some estimates to order 10500. To have any hope of dealing with such a large number of possibilities, one would need to work from the direction of both string theory and phenomenology to identify viable vacua. The string landscape emerges from the complex structure of string theory. One property of this complex structure is the huge number of string states, going into the hundreds of thousands. Inflation models motivated by the landscape are generally built with only a small number of fields and generally do not attempt to utilize the vast spectra of string states. DM models are one of the few that attempt to utilize as part of the dynamical solution this inherent feature that there are many string states available. In the landscape way of thinking therefore, DM models, although phenomenologically motivated, presumably are studying theories emerging from a vast range of vacua that otherwise are missed in inflation models with just a small number of fields. As such, in constructing DM models it is natural to look for distributions that are of observational interest. In this paper we will first look at the generic DM distribution, where the string states in the range relevant to inflation are equally spaced apart, which was the original example studied in the early work [8–11]. This case has many relevant features but we find that it is not consistent with present observational constraints. We will then examine other types of DM distribution that have appealing consequences in comparison with observation. Inflation is assumed to be described by low-energy effective field theory (EFT), although in many models inflation can happen when the inflaton field is superPlanckian, particularly by considering monomial chaotic potentials in cold inflation. Furthermore, an EFT can be ultraviolet (UV) complete if it can successfully be incorporated in a quantum theory of gravity, such as string theory. This paradigm provides a vast landscape of consistent embeddings of the EFT of gravity into a quantum theory, but this does not imply that any EFT coupled to gravity is consequently included in the landscapes. Those EFTs that are, in fact, inconsistent with a quantum theory of gravity lie in the surrounding swamplands [14,15]. Hence, a benchmark is needed to ensure a de Sitter vacuum EFT can live in the desired string landscapes. Recently two swampland criteria relevant for inflationary theories have been proposed [16,17],jΔϕj=MP<Δand MPjVϕj=V > c, provided that V>0, where fΔ;cg∼Oð1Þ. Nonetheless, these criteria have been noted to pose inherent threats to the basic mechanism of slow roll in cold inflation [17]. However, as part of the subsequent analysis we evaluate the aforementioned criteria in the warm inflation scenario. We show models of warm inflation that can be very consistent with the swampland criteria. Inherently the dissipative feature in warm inflation makes it amenable for consistency with swampland criteria, as already noted in the literature [18–22]. In this paper we will examine in detail DM models and explore their consistency with observational data and theoretical viability. This work is organized as follows. In Sec. II we introduce all WI dynamics and the primordial perturbation spectrum. In Sec. III we present in detail the supersymmetric distributed mass model, which is described by the interactions between the inflaton field and other light constituents: fermion (fψi;ψσg) and scalar (fχi;σg) fields. In Sec. IV we calculate all relevant parameters of the dissipation dynamics for the bosonic and fermionic sectors: dissipative coefficients and their corresponding thermal averages of the decay width. In Sec. Vwe develop a form of the mass distribution function. In Sec. VI we analyze in detail various dissipative coefficients for several inflation driven monomial potentials; we apply standard slow-roll methods and identify observationally consistent regions in parameter space. Here we also test the recently proposed swampland criteria. The main conclusions of this work are summarized in Sec. VII. Three Appendixes are also included, where we provide more detailed discussions of some of the results used in our computations. II. WARM INFLATION DYNAMICS AND PRIMORDIAL PERTURBATION SPECTRUM Nonequilibrium effects in the dynamics of a scalar field are generically produced due to interactions with an ambient thermal bath. The leading nonequilibrium effect, for a field evolving slowly compared to the characteristic timescale of the thermal bath, is a dissipative friction term ϒ_ ϕin its equation of motion [23], where ϒ¼ϒðϕ;TÞcan be computed from first principles given the form of the interactions between the scalar field and the thermalized d.o.f. For a homogeneous field, this implies the continuity equation _ ρϕþ3HðρϕþpϕÞ¼−ϒ_ ϕ2, such that overall MAR BASTERO-GIL et al. PHYS. REV. D 99, 103520 (2019) 103520-2 energy-momentum conservation implies the existence of an identical term with the opposite sign in the continuity equation for the thermal fluid. This explicitly proves that dissipative effects in the inflaton’s equation of motion lead to particle production in the thermal bath, so it prevents the exponential dilution of the latter in a quasi–de Sitter background. Therefore the temperature does not drop abruptly, and the universe is able to smoothly cross to the radiation epoch. Taking into account dissipative effects, the evolution equation for the background inflaton field is given by  ϕþð3HþϒÞ_ ϕþVeff;ϕðϕ;TÞ¼0;ð1Þ where Veff;ϕ¼dVeff=dϕ¼VðϕÞ;ϕþVT;ϕincludes the effective thermal potential Veff ¼VðϕÞþVT, where VðϕÞ is the zero temperature potential, and VTis the finite temperature effective potential, including radiative corrections to the potential due to any light component (fermion or scalar fields); ϒis the dissipative coefficient and His the Hubble parameter, given by the Friedmann equation for a flat Friedmann-Roberston-Walker universe: H2¼ρT 3M2 P ;ð2Þ where ρT¼_ ϕ2=2þVeff þTsRis the total energy density of the system, sRis the entropy density, and MPis the reduced Planck mass. One parameter that quantifies dissipation during WI is defined as the dissipative ratio Q¼ϒ=ð3HÞ. Depending on the ratio Qwe can have different regimes: when Q<1, this is called weak dissipative warm inflation; and when Q≳1, we are in strong dissipative warm inflation. Furthermore, the WI paradigm assumes the presence of a thermal bath, at temperature T; hence the evolution of such a dissipative mechanism can be obtained from the evolution equation for the entropy density, given by Tð_ sRþ3HsRÞ¼ϒ_ ϕ2;ð3Þ which in the slow-roll regime reduces to 3HTsR≃ϒ_ ϕ2. Without including the T-dependent corrections in the inflaton potential, we would have the standard relation TsR¼4ρR=3¼4CRT4=3, where CR¼π2geffðTÞ=30, geffðTÞis the effective T-dependent number of relativistic d.o.f., with ρRdenoting the standard energy density of radiation. All relativistic light fields contribute to the effective d.o.f., having geffðTÞ¼90 4π2 sR T3:ð4Þ Since the radiative corrections modify the potential and its derivatives, they alter the standard slow-roll parameters, being from ϵϕ¼M2 PðV;ϕ=VÞ2=2and ηϕ¼M2 PV;ϕϕ=V to ϵeff ¼M2 p 2Veff;ϕ Veff 2 ;ηeff ¼M2 p Veff;ϕϕ Veff ;ð5Þ where Veff;ϕϕ ¼VðϕÞ;ϕϕ þVT;ϕϕ. Recall that inflation happens when _ ϕ2=2≪Veff and so does ðTsRQ−1=2Þ≪Veff, but even if it is small compared to the inflaton effective potential, it can be larger than the expansion rate with ðTsRÞ1=4≳H; by assuming thermalization, this translates roughly into T≳H, so one can consistently obtain a warm inflationary universe with a slow-roll evolution, as long as the radiative corrections are restrained such they do not spoil inflation. Furthermore, one can show that the radiation energy density portrayed by an entropic description can never exceed the inflationary potential in a slow-roll regime, guaranteeing a period of accelerated expansion, TsR Veff ≃2 3 ϵeff 1þQ Q 1þQ;ð6Þ such that consistency of the slow-roll evolution requires ϵeff <1þQ. This in turn also implies that, at the end of the slow-roll regime, when ϵeff ∼1þQ, one may attain TsR∼ Veff if a strong dissipative regime Q≳1can be achieved. In such cases radiation will smoothly become the dominant component at the end of inflation, providing the necessary “graceful exit”into the hot big bang cosmic evolution [24]. Although there may be additional particle production at the end of inflation, no reheating is actually necessary in WI when strong dissipation is reached; otherwise, such a mechanism is needed. In addition to the smooth exit from inflation, WI exhibits several attractive features that have been explored in recent years. For instance, the dissipative friction damps the inflaton’s evolution, making slow roll easier or, equivalently, alleviating the conditions on the flatness of the inflaton potential, expressed now by the slow-roll conditions ϵeff,jηeffj≪1þQ. This may potentially provide a solution to the so-called “eta problem” typically found in string/supergravity inflationary models where generically ηϕ∼Oð1Þ[10,25]. (For other recent reviews of warm inflation, please see [26,27].) Small fluctuations of the inflaton about its homogenous component provide the initial seeds of density perturbation. These density perturbations produced during inflation evolve into the classical inhomogeneities observed in the CMB. For WI scenarios the fluctuations of the inflaton are thermally induced. As such, these initial seeds of density perturbations are already classical upon definition. The general expression for the amplitude of the primordial spectrum is given by [4,28–31] Δ2 R¼H _ ϕ2H 2π21þ2nþ2ffiffiffi 3 pπQ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3þ4πQ pT HGðQÞ; ð7Þ WARM INFLATION WITHIN A SUPERSYMMETRIC DISTRIBUTED …PHYS. REV. D 99, 103520 (2019) 103520-3 where all quantities are evaluated when the relevant CMB modes become superhorizon 50–60 e-folds before inflation ends. In the expression above, ndenotes the inflaton phase space distribution at the horizon crossing. By the strength of the interactions between the inflaton field and other particles in the thermal bath (including, e.g., scattering processes), this might interpolate between the BunchDavies vacuum, n¼0, and the Bose-Einstein distribution at the ambient temperature T,n≃ðeH=T−1Þ−1. We will focus on the latter limiting case in this paper, which we denote as “thermal”inflaton fluctuations. Also the function GðQÞaccounts for the growth of inflaton fluctuations due to the coupling to radiation fluctuations through the temperature dependence of the dissipation coefficient and must be determined numerically. In addition, this function also exhibits a mild dependence on the form of the scalar potential. In general, with thermalized inflation fluctuations 1þ2n¼coth ðH=ð2TÞÞ,wehave Δ2 R≃3H3 ð1þQÞ 2πVeff;ϕ2 ×2ffiffiffi 3 pπQ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3þ4πQ pT HþcothH 2TGðQÞ;ð8Þ then from the amplitude of the curvature power spectrum, we may determine the scalar spectral index ns−1≃dlnΔ2 R=dNe. Since, for T≪MP, gravitational waves are not significantly affected by thermal effects, the primordial tensor spectrum is given by the standard inflationary form Δ2 t¼2H2 =ðπ2M2 PÞ. The tensor-to-scalar ratio r¼Δ2 t=Δ2 Ris nevertheless affected, and, in fact, is typically reduced, by the modifications to the scalar curvature perturbations introduced due to dissipation, which are basically a function of T=Hand Q. We illustrate this fact by using the slow-roll dynamics, where the ratio rcan be written as r≃16ϵeff ð1þQÞ2FðT=H;QÞ; FðT=H;QÞ¼2ffiffiffi 3 pπQ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3þ4πQ pT HþcothH 2TGðQÞ; ð9Þ which is suppressed with respect to the CI prediction by a factor ð1þQÞ2KðT=H;Q Þ>1. Indeed, in [25] it was shown explicitly, even before the BICEP and Planck results, that the presence of radiation and dissipation suppresses the tensor-to-scalar ratio. Authors in [25] computed the tensorto-scalar ratio of the monomial ϕ2and ϕ4models, and it was one of the few analyses at the time that predicted for these inflation driven potentials a low tensor-to-scalar ratio, which now we see is consistent with data. Subsequent work further developed the analysis [32–36], showing that the tensor-to-scalar ratio may attainvalues even below 10−3for a ϕ4potential, and thus potentially distinguishable from, e.g., scenarios with a nonminimal coupling to gravity such as Higgs inflation [37] or the Starobinsky model [2]. III. SUPERSYMMETRIC DISTRIBUTED MASS MODEL Let us consider the general form of an effective N ¼1 global supersymmetry (SUSY) theory version of the distributed mass model with chiral superfields Φ,Xi, and Yi, described by the superpotential [25,38–40] W¼X ig 2ðΦ−MiÞX2 iþh 2XiY2 i;ð10Þ where gand hare coupling constants and the sum is taken over an arbitrary distribution of supermultiplets Xiand Yi. The chiral superfields Φ,Xi, and Yihave (scalar, fermion) components (ϕ,ψϕ), (χi,ψχi), and (σi,ψσi), respectively. Note that these are complex scalars and Weyl fermions, each with 2 d.o.f. We may use the Majorana representation for the spinors, i.e., use a four-component Majorana spinor built from the same Weyl fermion. In this case, note that a Majorana fermion is its own antiparticle. We have considered different Yifields coupled to each Xifield in the tower to avoid mass mixing at the level of the thermal masses. The scalar interaction terms in the theory are obtained from the superpotential using [38] −LS¼j∂ΦWj2þX ij∂XiWj2þX ij∂YiWj2:ð11Þ On the other hand, the fermion Lagrangian can be computed from the general formula [38] −LF¼1 2X n;m ∂2W ∂ξn∂ξm ¯ ψnPLψmþ1 2X n;m ∂2W† ∂ξ† n∂ξ† m ¯ ψnPRψm; ð12Þ where ξnis a superfield: Φ;Xi;Yi, and PL¼1−PR¼ ð1þγ5Þ=2are the chiral projection operators acting on Majorana four-spinors. Note that ∂W=∂Xi∂Yi¼ ∂W=∂Yi∂Xiand similarly ∂W†=∂X† i∂Y† i¼∂W†=∂Y† i∂X† i. Then we select only the boson field components of the chiral superfields Φ,Xi, and Yi, which are ϕ,χi, and σi, respectively. Nonetheless, see that the inflaton field ϕcan be decomposed into its real and imaginary parts via1 ϕ¼ðϕRþiϕIÞ=ffiffiffi 2 p. Hence, this prescription introduces another decay channel; for instance, the modulus square becomes jϕ−Mij2¼ðϕR=ffiffiffi 2 p−MiÞ2þϕ2 I=2. However, ϕR=ffiffiffi 2 pis the nonzero vacuum expectation value of ϕ, which will be the only term that will contribute to dissipation in the scalar field’s effective equation of motion, 1Similarly we define the complex scalars in terms of the real and imaginary parts for χiand σi. MAR BASTERO-GIL et al. PHYS. REV. D 99, 103520 (2019) 103520-4 so that the imaginary part of the inflaton, ϕI, is not relevant for our subsequent analysis. Moreover, in order to be coherent with further calculations, ϕis going to be considered only as the classical expectation value, without the label Rand the factor 1=ffiffiffi 2 p. Therefore, the relevant Lagrangians that may contribute to dissipation in the scalar field’s effective equation of motion are −LS¼g2X iðϕ−MiÞ2jχij2þgh 2X iðϕ−MiÞ½χiðσ† iÞ2þχ† iσ2 i þh2X ijχij2jσij2þg2 4X ijχij4þh2 4X ijσij4;ð13Þ −LF¼g 2X iðϕ−MiÞ¯ ψχiPLψχiþg 2X iðϕ−MiÞ¯ ψχiPRψχi þh 2X i χi¯ ψσiPLψσiþh 2X i χ† i¯ ψσiPRψσi þhX i σi¯ ψσiPLψχiþhX i σ† i¯ ψσiPRψχi:ð14Þ Note that the bare masses are mψχi¼gðϕ−MiÞ¼mχiat zero temperature for unbroken SUSY.2This is in agreement with [38] upon rescaling the couplings gand hin the superpotential by 1=2factors. At finite temperature, both the χiand the ψχireceive thermal mass corrections. The contributions of the σiand ψσi fields to the latter have been computed in [38] and have been shown to be identical for both χiand ψχi, corresponding to h2T2=8taking into account the coupling normalization differences. The χiscalars also receive thermal corrections from their self-interactions g2jχij4=4. Noting that these interactions give a contribution to their tree-level mass ∂2V=∂χi∂χ† i¼g2jχij2and taking into account the contribution of the χifields to the thermal effective potential ΔVT⊂2×m2 χi 24 T2¼g2 12 T2jχij2þ;ð15Þ where we have taken into account the 2 d.o.f. for complex scalars, this yields a thermal mass correction g2T2=12 to the χifields.3In summary, we obtain Δm2 χi¼g2 12 T2þh2 8T2;Δm2 ψχi¼h2 8T2:ð16Þ We also need to compute the finite temperature decay widths of the χiand ψχifields. However, we only recall such calculations for Dirac fermions [39]; albeit the difference between Majorana and Dirac fermions is only in the overall factors of the decay width. Hence, we can first compute them at zero temperature to set the correct normalization factors in order to identify these global constants. In general, we have for the decay of a particle of mass mat rest into a pair of massless particles Γ¼S 16πmjMj2;ð17Þ where S¼1=2if the particles are identical and S¼1if they are distinct. The χifields may decay via χi→σiσiand χi→ψσiψσi. In the first case, dropping the indices for simplicity, we may decompose the fields into their real and imaginary components via χ¼ðχRþiχIÞ=ffiffiffi 2 pand analogously for σ. This yields the scalar interactions −Lχσ2¼hg 2ffiffiffi 2 pðϕ−MiÞ½χRσ2 R−χRσ2 Iþ2χIσRσI:ð18Þ Hence, the χRscalar may decay into σRor σIpairs, while the χIscalar has only one decay channel χI→σRσI. For each of these decay channels, the vertex factor is −ihgðϕ−MiÞ= ffiffiffi 2 p. Taking into account the S¼1=2factors in the χR→ σRσRand χR→σIσIchannels, we then find that the decay widths are equal for both χRand χI, being given by ΓS χi¼h2g2ðϕ−MiÞ2 32πmχi :ð19Þ The fermionic decay channel comes from the interaction term 1 2hχi¯ ψσiψσi, where the vertex factor is simply −ih. Since the particles are identical in the final state and by computing the matrix element with the usual Feynman rules, this yields ΓF χi¼h2 32πmχi:ð20Þ Noting that, at zero temperature, mχi¼gðϕ−MiÞ, we see that ΓS χi¼ΓF χiin this limit. The fermions ψχican decay as ψχi →σiψσivia the corresponding Yukawa term above, and this yields simply Γψχi¼h2 16πmψχi:ð21Þ Again, note that at zero temperature for unbroken SUSY we have mχi¼mψχi¼gðϕ−MiÞ, which yields identical 2Note that in warm inflation SUSY is broken both by the finite temperature and the inflaton energy density. The latter should arise from an additional Φ-dependent term in the superpotential that we have note included above and that will lead to a small splitting of the mass for the real and imaginary components of the χiscalar fields. 3The mass of the scalar σifields is also corrected by a similar factor (involving only the hcoupling) and, although the resulting masses are below the temperature, they will generically lie above Hin the parametric range relevant to our discussion. This implies that no scalar field other than the inflaton can sustain a slowly rolling background value that drives inflation. WARM INFLATION WITHIN A SUPERSYMMETRIC DISTRIBUTED …PHYS. REV. D 99, 103520 (2019) 103520-5 total decay widths for the scalars and fermionic superpartners, as it should. Once we have computed the decay widths at zero temperature, we identify each vertex factor squared from the decay of a scalar boson to fermions and the decay of a fermion to a scalar boson and a fermion. In the pole approximation (see Appendix A) we take the limit T→0of the corresponding decay widths [41] and then by comparing them with Eqs. (20) and (21), we identify g2 χi¯ ψσiψσi¼h2=4and g2 ψχiσiψσi¼h2. From here wewill use these overall factors for the rest of the computations. IV. DISSIPATIVE INTERACTIONS The particle physics model considered in this work is inspired by string theory exhibiting N¼1global supersymmetry, with the inflaton field coupled to massive modes of the string, as discussed in [42]. Several interactions are identified by the shifted couplings g2ðϕ−MiÞ2χ2 iand gðϕ−MiÞ¯ ψχiψχifor bosons χiand fermions ψχirespectively, with fMigranging over mass scales. This feature yields the name of distributed-mass model (DM model). In the subsequent segments we establish the relevant interplay the inflaton field has with the aforementioned fields. We will restrict to interactions such that the leading contribution will come dominantly from one-loop processes, when the decaying field is light. The key property of the DM model is that for a given temperature T, only the fields with masses g2ðϕ−MiÞ2≲T2will contribute to the dissipation. Henceforth, we will refer to such a configuration of states as thermally excited sites. We will consider separately the dissipative processes associated with the excitation of the scalar χifields, which may decay via χi→σiσior χi→¯ ψσiψσi, and those associated with the excitation of the fermionic ψχifields, which decay via ψχi →σiψσi, with technical details of the computation given in Appendix A. A. Bosonic sector The dissipative coefficient arising from the pattern of interactions among the scalar component and the light states is given by [8,10,11] (see also Appendix A) ϒSðϕ;TÞ¼X t:e: i¼1 32g4 πh2½16m2 χi ˜ m2 χiþ ˜ m2 χi T2 ln2T ˜ mχiðϕ−MiÞ2 ˜ mχi ;ð22Þ where “t.e.”means sum over all thermally excited sites. In the computation of the dissipative coefficient, the total decay rate for all processes is already taken into account, which is given by [8,10,11] (see also Appendix A) Γχi¼Γðχi→σiσiÞþΓðχi→¯ ψσiψσiÞ ¼h2 128π T˜ mχi ωχiðpÞ16 m2 χi ˜ m2 χiþ ˜ m2 χi T2;ð23Þ where ωχiðpÞ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ˜ m2 χiþjpj2 qand ˜ mχiis the full χimass, including the tree-level contribution mχiand the thermal corrections computed above. We need to ensure that all light bosons remain in a nearly thermal state during inflation, so that the above result for the dissipation coefficient is a consistent approximation. For simplicity, we may take the thermal average of the above decay width, given by ¯ Γχi¼1 nBZd3p ð2πÞ3ΓχifB;ð24Þ where fBðp=TÞis the Bose-Einstein distribution and nBð˜ mχi=TÞthe associated number density. This yields in the limit mχi≲Tvia a numerical approximation (see Appendix B) ¯ Γχi≃h2f1=2 128π16 f m2 χi T2þf0.68 ð1þ0.77f1=2ÞT; ð25Þ where f¼fðg; hÞ¼g2=12 þh2=8. Note that the above expression depends on the factor mχi=T ¼gðϕ−MiÞ=T,so that different fields will decay at different rates. However, note that states for which mχi=T ¼0, corresponding to ϕ¼Mi, do not contribute to dissipation according to Eq. (22). Conversely, the heaviest states that can be thermally excited have mχi∼T, and these are the ones that contribute the most to dissipation at any given time. Therefore, in analyzing the consistency of the model, namely whether the χidecay faster than expansion in order to remain close to thermal equilibrium, we will consider the states for which mχi=T ≃1. B. Fermionic sector The calculation of the dissipation coefficient can be done following essentially the same steps as in the “warm little inflaton”model [34–36], yielding ϒFðTÞ¼X t:e: i¼1 CF TT; CF T≃3g2 h2ð1−0.34 logðhÞÞ:ð26Þ This calculation involves computing the finite temperature decay width of all interaction processes, which is given by Γψχi¼Γðψχi →σiψσiÞ ¼h2 16π T2m2 ψi ω2 ψiðpÞjpjFkþ T;ωψiðpÞ T−Fk− T;ωψiðpÞ T; ð27Þ where, neglecting the masses of the decay products σand ψσ, we have ωψiðpÞ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ˜ m2 ψiþjpj2 q,k¼ðωψiðpÞjpjÞ=2, and MAR BASTERO-GIL et al. PHYS. REV. D 99, 103520 (2019) 103520-6 Fðx; yÞ¼xy −x2 2þðy−xÞln1−e−x 1þex−y þLi2ðe−xÞþLi2ð−ex−yÞ;ð28Þ where Li2ðzÞis the dilogarithm function. Note that the mass of the light fields is corrected by a factor of h2T2=8due to their interactions with the thermal bath fields ψσiand σi, which we will take to be dominant over the inflaton contribution; i.e., we take ˜ mψi≃h2T2=8in the above computation of the decay width. We need to ensure that all light fermions remain in a nearly thermal state during inflation, so that the above result for the dissipation coefficient is a consistent approximation. For simplicity, we may take the thermal average of the above decay width in the high-temperature and ultrarelativistic region: ¯ Γψχi¼1 nFZd3p ð2πÞ3ΓψχifF;ð29Þ where fFðp=TÞis the Fermi-Dirac distribution and nFð˜ mχi=TÞthe associated number density. This yields in the limit mψχi≲T ¯ Γψχi ≃10−31−0.875 lnh2 8h4T: ð30Þ V. MASS DISTRIBUTION FUNCTION In the DM model the dissipative mechanism is well described by the interaction between the inflaton ϕand the finite number of fermion and boson fields that are light at any given time as the scalar field scans the tower of states. DM models are distinguished by how the mass sites are distributed. Such an idea has a natural realization with string theory, whereby the inflaton is suggestive of an excited string zero mode, which then interacts with massive string levels. Such a construction of DM models from string theory was shown in [9]. String levels can be highly degenerate, and the distribution of mass states was suggested in [9] to emerge as a fine structure splitting of such levels. Thus the pattern of splitting a level will depend on the properties of the string state, ultimately governed by the underlying string vacuum. In this respect different string vacua should imply different distributions of mass states. A first principles determination of such distributions is difficult and minimally requires a detailed study of string theory case by case for each different possible vacua. However, we can adopt a phenomenological approach and look at various types of mass distributions to see what type of inflation they can lead to. This would be a minimum first step to test the viability of such models. In this section we develop some basic properties of the mass distribution function. In general, the mass of states labeled by iis as discussed in Sec. III,m2 χi¼m2 ψχi¼g2ðϕ−MiÞ2, and so is governed by the parameter Mi. It is the distribution of these mass sites Mithat the underlying theory should determine, but here we treat them as phenomenological parameters and examine different types of distributions of Miover mass sites labeled by i. To understand the meaning of such mass sites, suppose the inflaton field ϕsat in the middle of some mass sites. Thus for all mass sites ϕ−T=g < Mi<ϕþT=g, the corresponding fields χiand ψχiwould be thermally excited. As some terminology, we will call the region Δϕ, surrounding the inflaton ϕthat has thermally excited fields, a thermal interval. Some examples of this are shown in Fig. 1. The idea of warm inflation in such models is that the inflaton rolls through a region with many such mass sites, thus thermally exciting for some time a given field; then once ϕis far enough away, that field again is no longer thermally excited. This implies, as the inflaton slow rolls during inflation, a thermal interval surrounding it moves with it. With no underlying theory to dictate such mass distributions, we will simply consider one type of construction to make the idea more tangible. We will assume the mass sites can be written as Mi¼ϕþiΔMðϕ;T;m;gÞ;ð31Þ where ΔMis the mass gap in the tower, which may in general be a dynamical function of the inflaton’s expectation value and the ambient temperature, as well as of some intrinsic mass scale mand couplings such as g.For instance, in a theory with compact extra dimensions, as is the case of string/M-theory, the mass gap in the KaluzaKlein tower is inversely proportional to the size of the extra dimensions, generically set by the expectation value of one or more moduli fields. If such moduli interact with the inflaton and/or the thermal bath, their expectation value may become dynamical (although slowly evolving) during FIG. 1. Schematic representation of the time evolution in the model assuming that the temperature decreases. WARM INFLATION WITHIN A SUPERSYMMETRIC DISTRIBUTED …PHYS. REV. D 99, 103520 (2019) 103520-7 inflation, exhibiting a ϕand/or Tdependence.4With this motivation in mind, we will take an effective field theory approach and consider functional forms of the mass gap that lead to simple forms for the dissipative coefficient, analyzing their dynamical and observational consequences. In the expression above iis an integer ranging from mass sites closest to ϕat i¼1up to a maximum mass site that is thermally excited determined by imaxΔM¼T=g. During the course of the observable range of inflation (around 50–60 e-folds), the inflaton will traverse some distance covering many thermal intervals Δϕ, and in general, one needs mass distributions where ΔM≪Δϕ, so that the inflaton crosses many mass sites in each thermal interval. One expects at least tens of mass sites per thermal interval and hundreds to thousands of thermal intervals crossed during the observable period of inflation. In other words, in each e-fold of inflation one expects tens or a few hundred mass sites to be crossed. Thus thousands of string states will be thermally excited for a brief period of time in the course of such a warm inflation period. With this form of mass distribution function, one can then obtain expressions for the dissipative coefficient and the effective potential. For the dissipative coefficient the sum that is required is X imax i¼1ðϕ−MiÞ2¼X imax i¼1 i2ΔM2≈i3 maxΔM2≃T3=ðg3ΔMÞ: ð32Þ For example, if we wanted ϒ∼ϕ, it means considering ΔM∼T=ðg3ϕÞand similarly for other forms of the dissipation function. In the landscape context, the idea is that there are a huge number of different vacua, where some subset can produce a distributed mass model and some subset of those could have a distribution of mass sites that gives a dissipative coefficient that leads to observably consistent warm inflation. Since the possible vacua is so huge, it is simply seen as a statistical possibility that somewhere in this landscape a mass distribution of a particular type can be found. One would need to see this in the anthropic way that there could be inflation of many different forms occurring over the landscape, but some will occur in a way favorable to create a universe like ours. In this respect choosing a function ΔM boils down to finding the type of functions that can lead to a universe that looks like ours. If that could successfully be realized, as we will examine in the next section, then this model would have phenomenological viability. One could then explore as a much bigger step whether it is possible to find such particular types of DM models from string theory, although we will not be exploring that question here. We can follow the above prescription in order to evaluate the effective finite temperature potential, where for the sake of simplicity we evaluate only the bare masses of the χ’s and ψχ’s fields. Hence both bosonic and fermionic sectors can be estimated using [43–46] [see Eqs. (C7) and (C1)] Vχi T≃T5 gΔM−π2 45 þ1 12 þ1 6π−1 32π2lnμ2 T2−cb; ð33Þ Vψχi T≃T5 gΔM−7π2 360 þ1 24 þ1 32π2lnμ2 T2−cf; ð34Þ hence if we select ΔM∼T=ðg3ϕÞ, the total effective finite temperature potential becomes Vχi TþVψχi T≃g2T3 24 ϕ−π2þ3þ4 πþ3 4π2ðcb−cfÞ:ð35Þ Since the thermal corrections to the inflaton potential are, for this particular case, linear in the field, they contribute to the ϵeff parameter and need to be taken into account. However, in the scenarios with a quartic chaotic potential VðϕÞ¼λϕ4that we analyze in more detail below, we typically find T=ϕ∼10−5and λ∼10−15 −10−14, such that these thermal corrections give a negligible contribution to both the effective potential and its first derivative for any value of the coupling gin the perturbative regime. In addition, in the scenarios where the mass gap is field independent, thermal corrections to the inflaton potential do not modify the slow-roll parameters. VI. RESULTS Let us now analyze the inflationary dynamics for a quartic scalar potential, VðϕÞ¼λϕ4, taking into account both scalar and fermionic contributions to the dissipation coefficient. In addition, different types of mass distribution functions will be used to lead to dissipative coefficients of various forms, namely ϒ∝T2,ϒ∝T, and ϒ∝ϕ. Finite temperature corrections to the effective potential are computed for each case. These effects can be controlled, thus preventing thermal effects from generating large contributions to the inflaton mass that could reintroduce the ηproblem. A. ϒ∝T2 For a homogeneous distribution of states in the tower, with constant mass splitting ΔM, as proposed by [8,10,11], the resulting scalar coefficients grow with temperature as ϒ∝T2, with fermions contributing only about 20% to the 4Note that the bare masses of the fields in the tower relevant for the dissipative process, gMi∼gϕ, are typically very large and that, at any given time, only a finite number of states become light due to the inflaton field reducing their mass. Therefore, this scenario does not require a low Kaluza-Klein/string scale to be implemented in the context of string/M-theory. MAR BASTERO-GIL et al. PHYS. REV. D 99, 103520 (2019) 103520-8 total dissipation as first estimated in [8,10,11]. Because of the adiabatic condition ¯ ΓS χ=H > 1, one requires large values of T=H; quantitatively we have that at least T=H≳150, which in turn yields values of Q≳10. Our results here are consistent with those reported in [8,10,11]. However, in the strong dissipation regime it is now understood that the primordial perturbations have a growing mode [29,30,47,48]. The consequences of this growing mode are to tilt the spectrum of perturbations toward the blue-tilted region, ns>1, and so are inconsistent with the CMB data. Thus, the DM model with a homogeneous mass distribution resides outside the observational window provided by Planck data [1]. Although this particular mass distribution is not consistent with observations, the basic idea of the model shows appealing features, and one can explore other types of mass distributions as we will now do. B. ϒ= ˜ Cϕϕand VðϕÞ=λϕ4 We want to study this model within the strong dissipative regime, since we expect that this model fits perfectly in it. This facilitates all calculations, so we will use standard analytical tools; i.e., we may implement the standard slowroll parameters ϵϕ¼M2 PðV;ϕ=VÞ2=2and ηϕ¼M2 PV;ϕϕ=V. Also we introduce another slow-roll parameter to take into account the variation of ϒ, β¼M2 P ϒ;ϕV;ϕ ϒV;ð36Þ and the slow-roll conditions are now given by ϵϕ<Q, jηϕj<Q,βϕ<Q. We use the slow-roll equations, at Q≫1, in order to find a direct relation between Qand ϕ, yielding Q¼ ˜ Cϕ ffiffiffiffiffi 3λ pMP ϕ;ð37Þ and the slow-roll parameters are ϵϕ¼8M2 P ϕ2;ηϕ¼12M2 P ϕ2;βϕ¼4M2 P ϕ2;ð38Þ so in the strong dissipative regime inflation ends at ϵϕ≃Q, and this yields the inflaton value ϕend ¼8ffiffiffiffiffi 3λ p ˜ Cϕ MP:ð39Þ The number of e-folds during inflation in the aforementioned regime is Ne¼−1 M2 PZϕend ϕ dϕQV Vϕ¼− ˜ Cϕ 4ffiffiffiffiffi 3λ pMPZϕend ϕ dϕ ¼ ˜ Cϕ 4ffiffiffiffiffi 3λ pϕ MP−ϕend MP:ð40Þ Hence the inflaton value at the horizon crossing is ϕ¼4ffiffiffiffiffi 3λ pðNeþ2Þ ˜ Cϕ MP¼2ffiffiffiffiffiffiffiffiffiffiffiffiffiffi Neþ2 Q sMP:ð41Þ Once we have determined ϕ, we can evaluate the observables at 50–60 e-folds before inflation ends. Before we proceed, one should note that in the strong dissipative regime many parameters and formulas simplify; in fact, one of them is an analytic approximation to the density perturbation amplitude. Recall that for T>H, the dominant contributions to the primordial perturbation spectrum are thermal fluctuations of the inflaton field, as opposed to the conventional quantum fluctuations in cold inflation models. Upon exiting the horizon these thermal fluctuations freeze out as classical perturbations, and during slow roll at Q≫1the amplitude of the curvature perturbation power spectrum is given by [28,49] Δ2 R≃9 4π2 H5 TQ5=2  V2 ϕ ;ð42Þ where all quantities are evaluated at horizon crossing. Hence the scalar spectral index ns−1≃dln Δ2 R=dNeis modified as well, becoming [28] ns¼1þ3 2ηV Q−3 2 ϵV Q−3 2 βV Q¼1−9 42ffiffiffiffiffi 3λ p ˜ Cϕ MP ϕ ¼1−9 41 Neþ2:ð43Þ Remarkably, note that nsdepend only on the number of e-folds: nsðNe¼50Þ¼0.9567 and nsðNe¼60Þ¼0.9637, where at 60 e-folds this scalar spectral index agrees outstandingly with Planck data [1]. The tensor-to-scalar ratio r¼Δ2 t=Δ2 R, as mentioned above, is typically reduced by the modifications to the scalar curvature perturbations introduced due to dissipation, which is basically a function of Q. We illustrate this fact by using the slow-roll dynamics, where the ratio rcan be written as r¼2H2  π2M2 PΔ2 R¼2λ 3π2Δ2 R ϕ4  M4 P¼32λ 3π2Δ2 R ðNeþ2Þ2 Q2:ð44Þ The value of λis fixed by using the normalization of the amplitude of the primordial spectrum Δ2 R≃2.2×10−9[1]. Using the slow-roll equations in Eq. (42), we have λ¼ðΔ2 RÞ4=3ð12π4C1=2 RÞ2=3 ðNeþ2Þ3;ð45Þ where CR¼π2geff=30, and geff ¼1þ15NM=4,NMbeing the no of bosonic χi(fermionic ψi) light d.o.f. at horizon crossing. The tensor-to-scalar ratio is then just a function of Qand Ne, WARM INFLATION WITHIN A SUPERSYMMETRIC DISTRIBUTED …PHYS. REV. D 99, 103520 (2019) 103520-9 Then mψσiis the fermionic ψσimass and ωψσi ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðkψσi Þ2þm2 ψσi q; kψσi ¼1 2 pp01− 4m2 ψσi p2 0−p21=2 :ðA8Þ An analysis of Eq. (A1) indicates that the behavior of the dissipation coefficient in the different temperature and interaction regimes is determined by an interplay between the spectral function and the thermal occupation numbers. When the temperature is large, the occupation numbers are also large, so in this regime, the poles of the spectral functions will dominate the integral, and this will be referred to as the pole approximation. As such, the pole approximation works well in the high-Tregion. The dissipation coefficient about its pole at p0≃ωpis ϒ¼X NM i¼1 2 Tg2 22 ðϕ−MiÞ2Zd3p ð2πÞ3 nBð1þnBÞ Γχiω2 p :ðA9Þ For on-shell χimodes, decays into light scalars and fermions are equally probable, with partial decay widths given by Eqs. (A3) and (A6). By treating the scalar σiand the fermion ψσias light fields, we evaluate both decay rates in the limit mσi≪T(mψσi ≪T), so ωσi =T≃kσi =T (ωψσi =T ≃ kψσi =T); this estimation also yields that kσi ¼kψσi ¼ jp0pj=2. Finally, in the limit mχi≪T, we have that both partial decay widths are given by Γχiðχi→σiσiÞ¼h2 8π m2 χi ˜ mχi T ωp ;ðA10Þ Γχiðχi→¯ ψσiψσiÞ¼ h2 128π ˜ m3 χi ωpT;ðA11Þ yielding a total decay rate, Γχi¼h2 128π T˜ mχi ωχiðpÞ16 m2 χi ˜ m2 χiþ ˜ m2 χi T2:ðA12Þ Finally, we compute the integral in Eq. (A9) by following the same procedure as in [8], obtaining the total scalar dissipative coefficient, ϒSðϕ;TÞ¼X NM i¼1 32g4 πh2½16 m2 χi ˜m2 χiþ ˜ m2 χi T2 ln2T ˜ mχiðϕ−MiÞ2 ˜ mχi : ðA13Þ APPENDIX B: SCALAR DECAY WIDTH AVERAGE The thermal average of the scalar decay width is given by ¯ Γχi¼1 nBZd3p ð2πÞ3ΓχifB;ðB1Þ where fBðp=TÞis the Bose-Einstein distribution and nBð˜ mχi=TÞthe associated number density. Hence we have ¯ Γχi¼h2T˜ mχi 128π16 m2 χi ˜ m2 χiþ ˜ m2 χi T2Ið˜ mχi=TÞ; Ið˜ mχi=TÞ¼Rd3p ð2πÞ3 fB ωχiðpÞ Rd3p ð2πÞ3fB :ðB2Þ The integral factor Ið˜ mχi=TÞcan be obtained numerically and is well approximated by Ið˜ mχi=TÞ¼1 TRdxx2 ffiffiffiffiffiffiffiffiffi x2þa2 pðeffiffiffiffiffiffiffiffiffi x2þa2 p−1Þ−1 Rdxx2ðeffiffiffiffiffiffiffiffiffi x2þa2 p−1Þ−1 ≃1 T 0.68 ð1þ0.77aÞ; ðB3Þ where we have defined x¼p=T and a¼˜ mχi=T. Hence the thermal average of the scalar decay width is ¯ Γχi≃h2˜ mχi 128π16 m2 χi ˜ m2 χiþ ˜ m2 χi T20.68 ð1þ0.77f1=2Þ:ðB4Þ Moreover, in the limit mχi=T ≲1we have that the effective finite thermal mass ˜ mχican fittingly be taken as ˜ mχi≃f1=2T, where f¼fðg; hÞ¼g2=12 þh2=8. Therefore the average decay width becomes ¯ Γχi≃h2f1=2 128π16 f m2 χi T2þf0.68 ð1þ0.77f1=2ÞT: ðB5Þ APPENDIX C: EFFECTIVE POTENTIAL AT FINITE TEMPERATURE We start by considering the contribution of the light fermions in the tower to the finite temperature effective potential, which is given by [43–46] Vψχi T≃1 2X i−7π2 180T4þ ˜ m2 ψχiT2 12 þ ˜ m4 ψχi 16π2lnμ2 T2−cf; ðC1Þ where μis the MS renormalization scale, cf¼2.635, and the effective thermal fermion masses ˜ m2 ψi¼g2ðϕ−MiÞ2þ h2T2=8. Note that the overall factor of 1=2in front of the sum is related to the Majorana nature of the fermions in the SUSY model. In the continuum limit we may write this in the form MAR BASTERO-GIL et al. PHYS. REV. D 99, 103520 (2019) 103520-16 Vψi T≃T4 2X i−7π2 180þg2ðϕ−MiÞ2 T2þh2 81 12þg4ðϕ−MiÞ4 T4þh2 4 g2ðϕ−MiÞ2 T2þh4 641 16π2lnμ2 T2−cf ≃T4 2−7π2 180þh2 96þh4 1024π2lnμ2 T2−cfZMþ M− dMnðMÞþg2 T21 12þh2 64π2lnμ2 T2−cfZMþ M− dMnðMÞðϕ−MÞ2 þ1 16π2 g4 T4lnμ2 T2−cfZMþ M− dMnðMÞðϕ−MÞ4 ≃T4 2−7π2 180þh2 96þh4 1024π2lnμ2 T2−cf2T gnðϕÞþþg2 T21 12þh2 64π2lnμ2 T2−cf2 3 T3 g3nðϕÞþ þ1 16π2 g4 T4lnμ2 T2−cf2 5 T5 g5nðϕÞþ ≃T5 2gnðϕÞ−7π2 90 þ1 18þh2 48þ1 16π22 5þh2 6þh4 32lnμ2 T2−cfþ;ðC2Þ where the integrals are in general ZMþ M− dMnðMÞðϕ−MÞn¼ZT=g −T=g dxnðϕÞ−xn0ðϕÞþx2 2! n00ðϕÞþxn ¼nðϕÞxnþ1 nþ1−n0ðϕÞxnþ2 nþ2þn00ðϕÞxnþ3 nþ3þ T=g −T=g ¼8 > > > < > > > : 2 nþ1T gnþ1 nðϕÞþ;n¼0;2;4;… −2 nþ2T gnþ2 n0ðϕÞþ;n¼1;3;5;… :ðC3Þ We have considered the local approximation, i.e., that only states in the vicinity of M¼ϕðtÞare light at any given time. Note that to obtain the derivatives of this potential correction with respect to the field, one needs to take into account that both the integrand and the integration limits are ϕdependent, the end result corresponding to differentiating Eq. (C2), such that Vψi T;ϕ≃T5 2gn0ðϕÞ−7π2 90 þ1 18 þh2 48 þ1 16π22 5þh2 6þh4 32lnμ2 T2−cfþ;ðC4Þ Vψi T;ϕϕ ≃T5 2gn00ðϕÞ−7π2 90 þ1 18 þh2 48 þ1 16π22 5þh2 6þh4 32lnμ2 T2−cfþ:ðC5Þ Nevertheless, in the local approximation the density of states could be taken only as temperature-dependent n∼T−1, and independent of the field; therefore n0ðϕÞ¼n00ðϕÞ¼0. Thus we have NM≃2Tn=g, and hence n≃gNM=ð2TÞ; but most importantly Vψχi T;ϕ¼Vψi T;ϕϕ ≃0. Finally, the finite temperature corrections to the effective potential due to the fermionic tower, in the local approximation at leading order, can be written as Vψi T≃NM 4T4−7π2 90 þ1 18 þh2 48 þ1 16π22 5þh2 6þh4 32lnμ2 T2−cf:ðC6Þ Next we examine the contribution of the light bosons in the tower to the finite temperature effective potential given by [43–46] Vχi T≃2X i−π2 90 T4þ ˜ m2 χiT2 24 − ˜ m3 χiT 12π− ˜ m4 χi 64π2lnμ2 T2−cb;ðC7Þ WARM INFLATION WITHIN A SUPERSYMMETRIC DISTRIBUTED …PHYS. REV. D 99, 103520 (2019) 103520-17 where again μdenotes the MS renormalization scale, cb¼5.41, and the effective thermal boson masses ˜ m2 χi¼g2ðϕ−MiÞ2þg2T2=12 þh2T2=8. Note the overall factor of 2 in front of the sum, which represents the fact that the χi’s scalars are complex fields. Following the same procedure as for the fermionic tower, we have for the bosonic sector Vχi T≃2T5 gnðϕÞ−π2 45 þf2ðg; hÞ 16πlnffiffiffiffiffiffiffiffiffiffiffiffiffiffi fðg; hÞ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þfðg; hÞ pþ1þfðg; hÞ 12 −ð2þ5fðg; hÞÞ 48πffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þfðg; hÞ p −1 16π21 10 þfðg; hÞ 3þf2ðg; hÞ 2lnμ2 T2−cbþ;ðC8Þ where fðg; hÞ¼g2=12 þh2=8. In the local approximation nðϕÞ∼T−1, hence nðϕÞ≃gNχ=ð2TÞ, and the inflaton ϕ derivatives are zero: Vχi T;ϕ¼Vχi T;ϕϕ ≃0. 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