AA Cosmological Higgs Collider
Shiyun Lu a , Yi Wang a , Zhong-Zhi Xianyu b a Department of Physics, The Hong Kong University of Science and Technology,Clear Water Bay, Kowloon, Hong Kong, P.R.ChinaJockey Club Institute for Advanced Study, The Hong Kong University of Science and Technology,Clear Water Bay, Kowloon, Hong Kong, P.R.China b Department of Physics, Harvard University, 17 Oxford St., Cambridge, MA 02138, USA
Abstract
The quantum fluctuations of the Higgs field during inflation could be a source of primor-dial density perturbations through Higgs-dependent inflaton decay. By measuring primordialnon-Gaussianities, this so-called Higgs-modulated reheating scenario provides us a uniquechance to probe Higgs interactions at extremely high energy scale, which we call the Cosmo-logical Higgs Collider (CHC). We realize CHC in a simple scenario where the inflaton decaysinto Higgs-portal scalars, taking account of the decay of the Higgs fluctuation amplitudeafter inflation. We also calculate the CHC signals of Standard Model particles, namely theirimprints in the squeezed bispectrum, which can be naturally large. The concept of CHCcan be straightforwardly generalized to cosmological isocurvature colliders with other typesof isocurvature perturbations. a r X i v : . [ h e p - t h ] J u l Introduction
The Higgs boson is a central focus of the current study of particle physics. It is the onlyfundamental scalar field experimentally discovered so far. It is also the only scalar particle inthe particle Standard Model (SM), and is responsible for mass generation of SM particles viaspontaneous electroweak symmetry breaking. On the other hand, the Higgs sector of SM suffersfrom a naturalness problem which hints at new physics beyond SM. The Higgs boson is also aunique portal to new physics. A careful study of Higgs properties may eventually lead us to amore fundamental theory beyond SM. (See e.g. [1] for a review.)The current strategy of studying the Higgs boson focuses almost exclusively on particle col-liders. We have learned many properties about the Higgs boson from the LHC. Higgs physics isalso among the most important physical targets of next-generation colliders operating at energiesroughly within O (100GeV ∼ H ,can be as high as O (10 )GeV. Such a huge energy can trigger spontaneous particle productionsthrough quantum fluctuations. The particle production can be effective for particles with massup to m (cid:38) H , this covers essentially all SM particles, including the Higgs boson, and possiblymany new particles beyond SM.The idea of using the huge energy of inflation to probe heavy particles has been put forwardin the context of the cosmological collider physics. See [8–14] for previous works. The essentialidea is that, by couplings to the inflaton, the quantum fluctuations of heavy particles couldleave characteristic imprints on n -point ( n ≥
3) correlations of the inflaton field, known as theprimordial non-Gaussianity. By measuring a particular “channel” — the squeezed limit — of3-point function, one could extract the mass of the heavy particle as resonance peaks [8, 9, 12]and its spin as angular distribution [12–14] of the correlation. Related ideas can turn the veryearly universe physics into classical [15–17] and quantum [18, 19] standard clocks to measure theexpansion history of the very early universe.Previous studies of cosmological collider assumed that the quantum fluctuations of the inflatonfield seeded the large scale inhomogeneity and anisotropy. In this sense, we may say that thecosmological collider is an inflaton collider, which collects long-lived inflaton fluctuations. Westudy indirectly the dynamics of heavy fields by coupling them to the inflaton. This inflatoncollider has been used to consider SM physics [20–23], neutrino physics [24], and other newphysics [25]. In particular, it is shown that the mass of light particles will in general be lifted tothe Hubble scale due to various types of mass corrections. (Light means lighter than the Hubblescale during inflation.) However, such corrections, together with the couplings to the inflaton,introduce lots of free parameters, making it difficult to extract and identify new physics fromthe observables. Worse still, it is in general difficult to get visibly large non-Gaussianities fromheavy particles with generic couplings to the inflaton. Technically, the difficulty comes from thefact that visibly large non-Gaussianity prefers large inflaton-matter coupling, which could easily2enerate a large mass correction to the matter field and render it too heavy to be produced duringinflation .Fortunately, there are alternatives that could be more interesting. In this paper we introducethe idea of a Cosmological Higgs Collider (CHC). The basic idea is that the large scale inhomo-geneities and anisotropies could be seeded by the primordial fluctuations not of the inflaton field,but of the SM Higgs field. This is possible when the rate of the inflaton decay (and thus theefficiency of reheating) depends on the background value of the Higgs. In such cases, the fluc-tuations of the Higgs field will perturb the decay rate of the inflaton, and thus will perturb theexpansion history of local universes. This is in line with the widely studied scenario of modulatedreheating [26–29], and we are simply identifying the Higgs boson as the light field modulating theinflaton decay.To illustrate the difference between the modulated reheating scenario and the conventionalinflation scenario, we show a sketch of the Hubble scale H and the inflaton decay rate Γ asfunctions of space position x and time t after inflation in Fig. 1. In ordinary scenario (left panel),the local Hubble scale H ( t, x ) (orange surface) was perturbed by the inflaton fluctuation duringinflation (shown by the wiggly behavior in x direction). After inflation, H drops as a function of t . When it reaches the decay rate Γ of the inflaton (blue surface), reheating is completed, and thespatial inhomogeneity of H is translated to that of the temperature, which we can observe today.In the modulated reheating scenario (right panel), the inflaton fluctuation is insignificant, sothe Hubble scale H ( t, x ) right after inflation is still quite homogeneous over space. However, whenthe inflaton decay rate is dependent on the background value of some light scalar fields (which wecall the modulating fields), it can develop spatial inhomogeneities due to the primordial fluctuationof modulating fields. Then, during reheating, the spatial inhomogeneities in Γ will be translatedto that of the temperature, and also of the Hubble parameter. ��������� H ( t , x ) Γ ( t , x ) Figure 1: The ordinary inflation scenario versus modulated reheating, illustrating the Hubble scale(orange) and the decay rate of the inflaton (blue) as functions of time and space coordinates.When the SM Higgs boson acts as the modulating scalar, the large scale density fluctuations are There is an exception for this too-heavy-to-produce argument. The coupling between the inflaton and theheavy field may provide the heavy field a chemical potential. In the context of cosmological collider, an exampleof chemical potential is noted in [24]. With a chemical potential, very heavy particles can be produced making useof the inflaton kinetic energy. Just to clarify that we are not identifying the Higgs field as the inflaton. We will assume a typical standardslow-roll inflation sector in this paper, except that we do not ask the inflaton field to generate all the primordialdensity fluctuations. H during inflation, does not give huge mass correction to thematter fields. Therefore, we see that the CHC is not only able to naturally generate sizable non-Gaussian signals, but also free from free-parameter “pollutions” from the unknown inflaton-mattercouplings. h
6. But in general it is some O (1) number. In this paper we will simplytake − / ζ = − H ˙ φ δφ − δ ΓΓ . (7)Correspondingly, the power spectrum of the curvature perturbation is a sum of two parts. (Weassume that the mixing between δφ and δh is small.) P ζ = P ( φ ) ζ + P ( h ) ζ , P ( φ ) ζ = H ˙ φ P δφ , P ( h ) ζ = C h P δh , P δφ = P δh = (cid:16) H π (cid:17) . (8)The factor C h depends on the nature of reheating and the post-inflationary evolution of the Higgsfield, which we shall calculated in Sec. 4. Depending on the relative sizes of P ( h ) ζ and P ( φ ) ζ , we canhave 3 different scenarios: 7. P ( φ ) ζ (cid:29) P ( h ) ζ : The inflaton fluctuations dominates the primordial fluctuations, and we havethe usual inflation scenario.2. P ( φ ) ζ (cid:28) P ( h ) ζ : The Higgs fluctuations dominate, and we have the Higgs-modulated reheating.3. P ( φ ) ζ ∼ P ( h ) ζ : Mixed scenario.We will show in Sec. 5 that the Higgs self-interaction will in general lead to large local non-Gaussianity. In the Higgs-fluctuation-dominated scenario P ( φ ) ζ (cid:28) P ( h ) ζ , this local non-Gaussianityis often too large to be consistent with CMB bound. Therefore, we will consider a general mixedscenario where the inflaton and Higgs fluctuations contribute together to the density perturbation.For later use, we define the fraction of Higgs boson in the “external lines” of CHC diagrams bythe ratio R h , R h ≡ (cid:18) P ( h ) ζ P ζ (cid:19) / . (9)The power spectrum at CMB scale is measured to be P ζ (cid:39) × − . In ordinary inflation (Case1), P ζ is generated from the inflaton, and therefore we have the fixed relation between the rollingspeed of the inflaton ˙ φ and the Hubble scale H inf , namely ˙ φ = (2 πP / ζ ) − H (cid:39) (60 H inf ) . InCase 3 of mixed scenario, the inflaton contribution a fraction of P ζ , namely P ( φ ) ζ = (1 − R h ) P ζ .So, we have ˙ φ = H π [(1 − R h ) P ζ ] / . (10)So ˙ φ will be larger than it is in ordinary scenario (Case 1). This point is important for thestudy of CHC, because when we write down effective operators during inflation, the rolling speed˙ φ / will be the lower bound of the cutoff scale Λ, Λ (cid:38) ˙ φ / , as required by perturbative unitarity.Therefore, we see that the unitarity bound on the cutoff scaler for CHC is higher than the ordinarycosmological collider. In this section we study the evolution of the Higgs field, including its background h and thefluctuation δh . The basic picture is that, during inflation (Sec. 3.1), fluctuations of the Higgsfield can be produced through quantum fluctuation. The longest mode leaving the horizon at thebeginning of observable inflation will look like a uniform background h , while the shorter modeswill be like space-dependent fluctuations. After inflation (Sec. 3.2), the Higgs will start oscillatingin its quartic potential well. The Hubble friction will reduce the amplitude of this oscillation. Itis of great importance to understand how fast this reduction is for the study of Higgs modulatedreheating.Before entering the detailed analysis, we should note that the fluctuations of interest are ofCMB or LSS scales, which are much longer than the Hubble radius of the universe between theend of inflation t end and reheating t reh . Therefore, the fluctuation field can be treated effectivelyas a constant within a Hubble patch during this epoch.8 .1 Higgs Dynamics During Inflation We assume a high scale inflation scenario for the study of this paper. By high scale wemean that the Hubble parameter H inf during inflation is much higher than the electroweak scale H inf (cid:29) O (100GeV). The original negative quadratic term in the SM Higgs potential can thenbe neglected and we are left with a quartic potential with V ( h ) = λh /
4. This corresponds to aclassically massless scalar field h with quartic self-coupling. During inflation when the spacetimeis de Sitter-like, such a scalar field could develop large quantum fluctuations (cid:104) h (cid:105) ∼ H / √ λ .This will introduce a dynamical mass m ∼ λ (cid:104) φ (cid:105) ∼ √ λH to the originally massless scalar h .One can think of this mass as a thermal mass in inflation (though the thermal distribution isnot the conventional Bose-Einstein because of the redshifts from the expansion of the universe),since the Hawking radiation coming from the dS horizon carries a temperature T ∼ H − . To seethis point more quantitatively, we can calculate the 2-point function the scalar field h with loopcorrections. One can do this either in the real-time formalism or by doing Wick rotation into theEuclidean dS. The zero-mode approximation in the Euclidean dS allows us to sum over all loops.The result is [22] m h = (cid:114) λπ H . (11)For small λ , and considering the extra powers of π suppression, the Higgs can be light enough,such that its fluctuation does not depart from scale invariance very much. This contribution,combined with the time dependence of H , and the inflaton fluctuation as indicated in the firstterm in (7), should be possible to fit the observed tilt of the nearly scale-invariant power spectrum.Moreover, to work out a complete story, it is important to consider the post-inflationary evolutionof the Higgs field, which we will do below. We assume that the Higgs field is light during inflation, m h (cid:28) H , which is easily satisfied.Consequently, light Higgs acquires large fluctuations with typical amplitude of O ( H ). Afterinflation, this background will eventually roll back to its origin and also decay to SM particleswhenever it can. Therefore, for the purpose of modulated reheating, it is important to check thatthe Higgs fluctuation at the time of reheating is still large enough.Again we assume H inf (cid:29) v (cid:39) h then follows the equation,¨ h ( t ) + (3 H + Γ h ) ˙ h ( t ) + λh ( t ) = 0 . (12)Here Γ h is the decay width of the Higgs field to SM particles, which is proportional to Higgsbackground value h . The ratio γ h ≡ Γ h /h is almost a constant at different energy scales and It is possible that a non-minimal coupling between the Ricci scalar and the Higgs term ξh R may break theelectroweak symmetry during inflation [22,23]. In this case the Higgs mass and the two-point function are modifiedaccordingly. By fine-tuning ξ , one can make the Higgs light and be used for modulated reheating. Here we willnot consider this possibility in details. h (cid:39) h (cid:39) γ h (cid:39) . × − . Therefore, for Higgs field with background value h ∼ O ( H ), theperturbative decay of the Higgs field can be neglected compared to Hubble friction. RG runningof γ h only brings O (1) corrections and thus is unimportant.We further assume that the inflaton potential at the bottom is nearly quadratic so the universebefore the completion of reheating is effectively dominated by cold matter, and therefore we cantake H = 2 / (3 t ). Then, the equation of Higgs background simplifies to¨ h ( t ) + 2 t ˙ h ( t ) + λh ( t ) = 0 . (13)This equation shows that the Higgs field after inflation will start oscillating in its quartic potentialwell. The oscillation amplitude will decay due to the Hubble friction. The exact solution to (13)should be found numerically, but the qualitative feature of h evolution can be inferred from thefollowing simple argument.It is known that an oscillating classical field in a quartic potential behaves effectively likeradiation. Therefore we expect that the energy density of the Higgs field ρ h = λh will decayaccording to a − . On the other hand, as mentioned above, the energy density right after inflationis dominated by the inflaton potential which we assumed to be quadratic, and therefore we knowthat the universe is effectively dominated by matter, and therefore a ∝ t / . Consequently, we seethat the background value h of the Higgs field decays according to h ∝ ρ / h ∝ a − ∝ t − / .With the above picture in mind, we solve the equation (13) numerically. From the numericalsolution in Fig. 3 we see that the Higgs background does oscillate with decaying amplitude h ∼ t − / .In addition, we are also interested in the time evolution of the Higgs fluctuation δh ( t ). Asnoted previously, all Higgs fluctuations we are interested in are outside the horizon between theend of inflation and the completion of reheating, i.e., they look like constant background within aHubble patch. Therefore, the time evolution of δh ( t ) can be found simply by solving (13) twice,one with initial condition h ( t ini ) = h and the other with h ( t ini ) = h + δh ini . Then the timeevolution of δh is simply given by δh ( t ) = h ( t ) − h ( t ). In all cases we take ˙ h = 0 initially sincewe know that Higgs field is slowly rolling during inflation, and giving Higgs background a smallinitial velocity will not significantly alter its subsequent evolution.We show a numerical solution of δh also in Fig. 3. From the numerical solution to (13) it turnsout that the amplitude of the background oscillation h and of the fluctuation δh ( t ) are well fitby h ( t ) ∼ λ − / H inf ( H inf t ) − / , δh ( t ) ∼ λ − / ( H inf t ) − / δh ini , (14)as long as t is not too large. The behavior h ∼ t − / is just as expected, but the behavior δh ∼ t − / shows that the fluctuation of Higgs field decays a bit slower than the background. Thisis from the nonlinear nature of the equation (13) and certainly it cannot hold for arbitrarily long.If we evolve δh long enough then more complicated behavior will appear. The example in Fig. 3shows that the δh ∼ t / behavior holds well within 10 times of the initial Hubble (time) scale.This is long enough for our following analytical estimate. At first sight, the different scaling in background and fluctuation seems puzzling. Because the splitting between
10 100 1000 10 - t / H inf - | h ( t ) | - - t / H inf - | δ h ( t ) | Figure 3: Left: | h ( t ) | as a function of time t . Right: | δh ( t ) | as a function of time t . In both panelsthe blue solid curves represent numerical solutions of (13), while black dashed curves correspondto analytical fits (14). For illustration we take λ = 0 .
01 and δh ini = 0 . Now we consider the decay of the inflaton after inflation. The inflaton can decay in manydifferent ways. As a simple example, we will show that the inflaton decay into Higgs-portal scalarparticles can provide a viable realization of Higgs-modulated reheating. We will also considerpossibilities of inflaton decay into SM particles. We will show that it is in general not easy torealize Higgs-modulated reheating if the inflaton only decays into SM particles.Some general remarks are in the following, before we proceed with detailed analysis.As we mentioned earlier, the Higgs-modulation of the inflaton decay happens at the time whenreheating is complete, namely when the decay rate Γ is comparable to the Hubble scale. Then,a perturbed decay rate will either advance or delay reheating and thus generates the observeddensity perturbation. For this purpose, an instantaneous reheating right after inflation (namelythe decay rate is very large and comparable to the Hubble scale at the end of inflation) is certainlynot favorable, because the effect of perturbed decay rate would be too weak.Therefore, it is preferable to have a delayed reheating scenario, where the decay rate Γ issignificantly smaller than the Hubble scale at the end of inflation. In this case, only when theHubble scale drops (typically according to H ∼ t − ) down to the scale of Γ, reheating could thenbe completed.However, we also do not want a too slow reheating for a successful generation of densityperturbation through Higgs modulation. This is because the Higgs background field will decreaseafter inflation, as studied above. If reheating is too slow, then the Higgs background would background and fluctuation is arbitrary and we would have split them in a different way. But note that the scaling h ∼ t − / is an approximate solution and the precise solution is not a power law. The subleading terms can beas large as δh and modify the scaling. To make this point clear, consider a toy function f = t + αt , where α issmall. We may have split f = t − αt and δf = 2 αt . Then the scaling f ∼ t is approximate with subleadingcorrection scales as αt , but clearly there is nothing wrong. The amplification of difference in initial condition isa general feature for a broad class of non-linear systems. As we have mentioned in the text, the approximationcannot hold for arbitrarily long before the peculiar term catches up. But for the period of time we are interestedin, it’s safe to trust this approximation. σ ,and in literature it is usually assumed that the dependence follows a power law, Γ( σ ) ∝ σ n .These two assumptions hold simultaneously only when the background light field does not evolvesignificantly after inflation, which is possible if σ is in a flat direction. But this is certainly notthe case if we identify σ to be the SM Higgs boson, as we have shown previously that the Higgsbackground value decrease according to h ( t ) ∼ t − / after inflation. Therefore, we must takeaccount of the fact that the decay rate Γ is also a function of time.Furthermore, it is also easy to see that a power-law dependence Γ ∝ h n has difficulty ingenerating a successful modulated reheating. This is because, to have the power-law dependenceΓ ∝ h n , one needs to couple the inflaton φ to at least one single power of Higgs field in an operatorlike O ∼ φh · · · with possible spacetime derivatives acting on these fields. Then we see that theinflaton φ can decay to particles represented by “ · · · ” in O with a decay rate Γ ∝ h . That is,the smallest nonzero exponent n in the power-law dependence Γ ∝ h n will be n = 2. Then, from h ( t ) ∝ t − / we see that the decay rate Γ will decrease with time no slower than t − / . Thisdecrease is already faster compared with the Hubble scale H ( t ) ∼ /t , so it is impossible for H ( t )to catch up Γ at a later time to achieve the successful reheating.Two remarks to complete the above argument. First, we may expect that the inflaton couplesto at least a pair of Higgs fields since the Higgs doublet carries gauge charges. But this does notcontradict the above assumption of linear h -dependence in O . Consider for instance the operator O ∼ φ H † H ψ where ψ is arbitrary matter field, one can find a term φh δhψ when evaluating O on the Higgs background h and this term will provide the linear h -dependence. Second, one canalso consider a polynomial dependence in Γ( h ) ∼ Γ + Γ h + · · · which can be easily realized insimple particle models. In this scenario, the inflaton decay rate does not drop quickly to zero dueto the constant piece Γ , while the desired Higgs modulation is achieved through the Γ h term.However, this type of dependence will not generate correct amount of density perturbations, whichcan be seen after going through the analysis similar to what we will do in the next subsection.Given the fact that Γ ∼ h n with power-law dependence decreases faster than Hubble scale H ,we could imagine that we can have a decay rate Γ initially slightly larger than H at the end ofinflation, so we have an instantaneous reheating. The term “instantaneous” is actually misleadinghere because reheating would take at least take a duration of O (Γ − ) to finish. But since Γ isdecreasing fast, we could imagine that reheating is terminated before it is completed. The rest ofinflaton could decay through other much slower channels, e.g., through a gravitational couplingfrom operators suppressed by Planck scale. In this way, a perturbation in the Higgs field h would lead to advanced/delayed termination of reheating, and thus a perturbation in the thermalexpansion of the local universe. While more quantitative study is needed to justify whetherthis scenario is possible or not, here it suffices to mention that a decay rate initially larger thanHubble would typically require a rather large coupling between the inflaton to its decay products,which could be easily constraint from physics during inflation, such as the back reaction to the12nflaton potential or the perturbative unitarity. Also, before the further decay of the inflaton, theoscillatory inflaton background will dilute the existing radiation, which suppresses the effect ofmodulated termination of reheating. Therefore, we will not consider this case any more in thispaper.Then it seems difficult for the SM Higgs boson to modulate reheating. Fortunately, the decayrate Γ can depend on the Higgs background value h through the kinematic factor rather thanthrough a power law. Consider a simple example where the inflaton decays to a pair of massivegauge bosons A µ and the gauge boson gets the mass from the Higgs background. Let the decaybe from the coupling φF (cid:101) F /
Λ where F µν = ∂ µ A ν − ∂ ν A µ and (cid:101) F being the dual of F . The decayrate will have the following form,Γ( φ → AA ) ∝ m φ Λ (cid:18) − m A m φ (cid:19) / . (15)Then we see that Γ depends on the Higgs background h where the gauge boson A get its mass, m A ∝ h . In this way, the Higgs background fluctuation will perturb the decay rate and thusmodulate the reheating process, while the decay rate itself does not decrease significantly, so thata delayed reheating is easily achieved: We just need to have a decay rate Γ initially smaller than H , and wait until H drops to the value of Γ.In the following we will make use of this kinematic dependence on the Higgs background valueto modulate reheating. As mentioned at the beginning of this section, there are many ways tomake the inflaton decay. We will only consider dim-5 operators respecting the shift symmetry φ → φ + const. which frees us from worrying about the back reaction to the inflaton potential.We will begin with the dim-5 couplings to SM particles (fermions, gauge bosons and the Higgsboson) and show why it is quite difficult to modulate reheating through these channels. Then wewill move on to the Higgs-portal scalar channels, where the inflaton decays to Higgs-portal scalarbosons, which turns out to be the easiest way for the CHC to work. The only dim-5 couplings between the inflaton and SM gauge bosons and fermions respectingthe shift symmetry are the following,∆ L = (cid:88) i f,i ¯ f i ( /∂φ ) γ f i + (cid:88) α g,I φF I (cid:101) F I , (16)where f i represents all SM fermions, F I represents the field strengths of SM gauge fields A I , Λ f,i and Λ g,I are corresponding cut-off scale. These are axion-type couplings, and indeed, axion-likeparticles can be perfect inflaton candidate.The two-body decay rates areΓ( φ → f i ¯ f i ) = 12 π Λ f,i m φ m f i (cid:18) − m f i m φ (cid:19) / , (17a)Γ( φ → A I A I ) = 14 π Λ g,I m φ (cid:18) − m A I m φ (cid:19) / . (17b)13e note that the decay rates here are for one species in the final states, and it is understood thatfactors from internal degrees of freedom should be included. For example, an additional factor of3 should be included in (17a) when f i is a quark, and an additional factor of 8 should be includedin (17b) when A I represents the gluon.For fermionic decay channels, the two-body decay is dominated by the heaviest f i satisfying m f i < m φ /
2. But there can be cascade decay of heavy fermions, too. These decays will all becontrolled by the Higgs background value.A first observation here is that the fermionic decay rates depend on the Higgs background h through a power law, Γ ∼ m f ∼ h . The discussion at the beginning of this section showsthat this channel is of no use for modulated reheating if we want a delayed reheating scenario,because the decay rate decreases faster than the Hubble scale. Therefore we will no longer considerthis channel any more in this paper. But before moving on to the gauge boson channel, let usalso mention that the fermion channel cannot be used in the “terminated reheating” scenariomentioned above either, which requires a decay rate initially larger than the Hubble scale. Tosee it, we note that the Higgs background h ∼ H inf initially, and let us consider the top quarkwithout further suppression from Yukawa coupling. Then, the decay rate is,Γ( φ → f f ) ∼ m φ H Λ f . (18)The cutoff scale Λ f cannot be smaller than the unitarity bound, Λ f (cid:38) ˙ φ / which is typicallymuch higher than the Hubble scale (cf. discussion at the end of Sec. 2). Assuming that inflationand reheating are described by the same effective field theory (EFT), we see that Γ( φ → f f ) cannever be larger than the Hubble scale.Now we turn to the gauge boson channel. Apart from the kinematic factor, the decay rate isapproximately Γ( φ → AA ) ∼ m φ Λ g . (19)With the constraint Λ g > m φ and Λ g > ˙ φ / in mind, we see that the decay rate of the gaugeboson channel can be either greater or smaller than the Hubble scale at the end of inflation. Ofcourse, to achieve a delayed reheating, we prefer a smaller Γ and thus a higher cutoff scale.In the gauge boson channel, the fluctuations in h will generate fluctuations in δ Γ through thedependence in the kinematic factor, δ Γ (cid:39) − (cid:88) I g I π m φ ¯ h Λ g,I δh, (20)where we have assumed m A I = gh (cid:28) m φ . To get an idea of how large the curvature fluctuationcan be generated in this channel, let’s further assume that Λ g,I and g I are the same for all species,then, ζ = − δ ΓΓ (cid:12)(cid:12)(cid:12)(cid:12) t = t reh = g h ( t reh ) δh ( t reh )2 m φ , (21)14o see how large curvature fluctuations can be generated from the Higgs-modulated decay, wecompute (cid:104) ζ (cid:105) as follows. (cid:104) ζ k (cid:105) (cid:39) (cid:20) g h ( t reh )2 m φ (cid:21) (cid:10) δh k ( t reh ) (cid:11) (cid:39) (cid:20) g m φ λ / t reh (cid:21) H k . (22)From the relation (cid:104) ζ (cid:105) = (2 π /k ) P ζ and P ζ (cid:39) × − , we can determine the reheating time t reh .Expressing the result in terms of the decay width Γ = 2 / (3 t reh ), we have,Γ = 8 πλ / g (cid:16) m φ H inf (cid:17) (cid:112) P ζ H inf . (23)This can be translated to an expression for Λ g (treating all Λ g,I equal to Λ g ),Λ g = (cid:114) N g π λ / P − / ζ (cid:112) m φ H inf . (24)To get an idea of how large the cutoff is needed to be, we take an example of N = 4, g = 0 . SU (2) L gauge coupling, and thus we are ignoring decays into photons and gluons,since they are massless and not helpful for kinematic modulation of reheating), and λ = 0 .
01 (theHiggs self-coupling at the inflation scale). Then we have,Λ g (cid:39) H inf (cid:18) m φ H inf (cid:19) / . (25)Therefore, with m φ larger than the Hubble scale during inflation, we will have a cutoff quite closeto the unitarity bound Λ g (cid:38) ˙ φ / . (Recall that ˙ φ / (cid:38) − R h ) − / H inf when Higgs fluctuationscontribute R h of the density perturbation in terms of the amplitude.) For models with m φ (cid:29) H inf ,we can have much higher cutoff scale Λ.Some O (1) corrections to the above estimate could arise if we consider more realistic situation.For example, in SM, only W and Z are massive among all gauge bosons. Therefore only φ → W W and ZZ channels contribute nonzero δ Γ in (20). On the other hand, all gauge bosons, including8 colored gluons and the photon, would contribute to Γ. Taking this into account and rederiving(21), it’s easy to see that the net effect is to replace m φ in every formula from (21) to (23) by aneffective (cid:98) m φ , given by, (cid:98) m φ = (cid:18) (cid:88) I =all Λ − g,I (cid:30) (cid:88) I = Z,W ± Λ − g,I (cid:19) m φ . (26)If we again take all Λ g,I ’s equal, then we have (cid:98) m φ = 4 m φ . We should also replace m φ in (24) by m φ /
4, and take N = 12. As a result, the numerical factor in front of (25) should include anotherfactor of √ / µ ∼ max { H inf , m A } where H inf and m A are the Hubble scale and the gauge boson mass during inflation, respectively,and the chemical potential µ ≡ ˙ φ / Λ g . The overproduction can be seen from the exponential15nhancement in the mode function of the gauge field in the presence of a chemical potential. Seethe appendix for details. Given ˙ φ > (60 H inf ) , we see that it is desirable to have Λ g at least O (10 ) larger than the value in (25).There are ways to make Λ g larger. One obvious way to raise the cutoff is to have a largerinflaton mass during reheating as can be seen from (25). Too large inflaton mass at the bottomof the potential well may bring difficulty in the model building of single field inflation. Also, onecannot make the inflaton mass m φ higher than the cutoff Λ g . Multi-field inflation scenarios suchas hybrid inflation [37] may easily achieve a large m φ (in which context m φ is actually the mass ofthe waterfall field). Also, if different stages of inflation are described by different validity regimesof EFT, the constraints on Λ g may get relaxed. To keep the model simple, we will not explorethese multi-field or multi-EFT possibilities here.One can also consider other decay channels in new physics scenario to speed up the decay andthus raise the cutoff further. However, we must keep in mind that presence of additional decaychannels could weaken the dependence on the Higgs background by giving a larger denominatorto (21), unless these new channels themselves are dependence on Higgs background. We leavedetailed studies on new-physics channels to future works.The general conclusion of this subsection is that it is not possible to use SM fermion channel tomodulate reheating. It is possible to make use of the gauge boson channel. However, to generatethe right amount of primordial perturbation would require a rather low cutoff Λ g that is easilyinconsistent with the lower bound from the gauge boson production during inflation. Thereforethe viable parameter space for the gauge boson channel is also quite restricted. Given the difficulty of achieving modulated reheating in the SM gauge boson and fermionchannels, now we move on to the SM Higgs channel. At dim-5 level, we can write down two inde-pendent real couplings between the inflaton and the Higgs boson respecting the shift symmetry,∆ L = 2Λ hr ( ∂ µ φ )Re (cid:0) H † D µ H (cid:1) + 2Λ hi ( ∂ µ φ )Im (cid:0) H † D µ H (cid:1) = 1Λ hr ( ∂ µ φ ) h∂ µ h + g c W Λ hi ( ∂ µ φ ) Z µ h . (27)Here Λ hr and Λ hi are cutoff scales of the two couplings, respectively, and c W = cos θ W with θ W being the Weinberg angle. In the second line we have taken the unitary gauge H = (0 , √ h ) T where all would-be Goldstone bosons are gauged away.For the analysis of inflaton decay, we will only consider the Λ hr -term. The contribution fromthe Λ hi -term is similar, and we will omit it for simplicity. However, for the “cosmological collider”signals to be considered in the next section, the coupling Λ hi will be very interesting since it willintroduce Z -exchanging tree-level diagrams.The Λ hr -term in (27) modifies the dynamical evolution of the Higgs field, both during inflationand after. For now we will proceed as if the previous results on the Higgs evolution are stillapplicable and will justify this treatment later. 16he two-body decay rate of φ → hh from the Λ hr -term isΓ( φ → hh ) = 116 π Λ hr m φ (cid:18) − m h m φ (cid:19) / . (28)The Higgs-dependence in the decay width from this channel is very similar to that in the gaugeboson channel, since the Higgs mass during this period is m h = 6 λh . (29)So we can repeat our analysis for the gauge boson channel in the last subsection. We would expectthat the resulting cutoff scale Λ hr would be rather close to the unitarity bound. One importantdifference, though, is that both operators in (27) do not lead to copious particle production duringinflation, and thus the cutoff scales Λ hr and Λ hi are free from a much stronger bound constrainingthe gauge boson channel.The rest of the analysis is in parallel with the one in the last subsection. First, we write downthe curvature perturbation generated at the time of reheating, ζ = − δ ΓΓ (cid:39) λh δhm φ (cid:12)(cid:12)(cid:12)(cid:12) t = t reh , (30)where we assumed that m φ (cid:29) m h . From this we can express the decay rate in terms of the powerspectrum P ζ as Γ = π √ λ m φ H inf P / ζ . (31)So the required cutoff scale Λ hr isΛ hr = √ λ / π P − / ζ (cid:112) m φ H inf . (32)Taking λ ∼ .
01 and m φ (cid:39) H inf , we have Λ hr ∼ H inf , which is already lower than the unitaritybound. Here it is even impossible to raise the cutoff by giving larger mass to m φ because we donot expect m φ to be larger than the cutoff scale anyway in single field inflation described by asingle EFT. Inclusion of the Λ hi -term will not change the result significantly. Therefore, it is notlikely to realize Higgs-modulated reheating through this channel.Although we have reached a no-go result, we will still comment on the effect of the operatorsin (27) on the Higgs evolution, only to make sure that the above analysis is valid.It is clear that the Λ hi -term is irrelevant during inflation. On the other hand, the Λ hr -termcontribute a friction term ( ˙ φ / Λ hr ) ˙ h to h ’s equation of motion. The coefficient ˙ φ / Λ hr is muchgreater than the Hubble scale. This makes the Higgs field roll slower than in the usual case.But the Higgs field does not dominate the energy budget during inflation, and thus this slow-down has no consequence on the overall amplitude of the Higgs fluctuation, unlike the case of theordinary inflation. However, decreasing ˙ h would bring corrections to the slow-roll parametersand thus corrections to the scale dependence of the power spectrum. As before, we assume thatthis change can always be compensated by modifying the inflaton potential, so that the overallscale dependence agrees with the CMB measurements.17fter inflation, the term ( ˙ φ / Λ hr ) ˙ h is still present in the equation of motion. But it is nolonger a friction term since φ is fast oscillating. We assume that the inflaton mass at the bottomof its potential well is much greater than the Hubble scale at the end of inflation. Then, the aboveterm provides a fast oscillating external force to h . Just like in ordinary forced oscillation, a fastoscillating external force with frequency ∼ m φ has essentially no effect on the oscillator, whoseintrinsic frequency is m h ∼ √ λh (cid:28) m φ . Therefore, we can safely neglect this term even it issuperficially much greater than the Hubble friction term. One can also understand this by simplytaking the average of this over a time scale longer than m − φ . Then this oscillating term will beaveraged to zero. As we see from the above analysis, the main difficulty of realizing Higgs-modulated reheatingfor SM-only channels is from the fact that the Higgs background decreases quite fast after inflation,and therefore we need a quick decay of the inflaton, which in turn requires a rather low cutoff scalethat is easily inconsistent with various constraints. Now we move on to a simple BSM channel,where the inflaton decays to N heavy real scalars S i that couple to SM through a Higgs portal.Since the Higgs-portal coupling is not constrained, we can assume a large enough coupling toincrease the effect of δh , and we can also make use of the number of scalars N > Z parity to thescalar fields S i → − S i . Then the relevant Lagrangian is∆ L = −
12 ( ∂ µ S i ) − m S S i − αS i | H | + 1Λ S ( ∂ µ φ ) S i ∂ µ S i . (33)Here α -term is the usual Higgs-portal coupling and the last term with dim-5 operator is thecoupling to the inflaton through which we allow the inflaton to decay. We assume that m S is nottoo smaller than the inflation scale H inf so that the background value of S i sits at the minimum S i = 0 during inflation, and thus does not modify the inflaton or Higgs evolution. However,we note that this assumption on m S is only meant to simplify the analysis, and a scenario with m S (cid:28) H inf can well be viable since S i can also receive mass from other background contributionduring inflation. Interestingly, for α ∼ O (1), putting m S ∼ O (10)TeV will make S i consistentwith a thermally produced dark matter candidate.After inflation, the inflaton will gradually decay into S i . At this stage, the mass of the S i willbe dependent on the Higgs background value h as, m S ( h ) = m S + αh . (34)At the same time, the Higgs background h will also decay. We assume that m S > λh so thatthe Higgs does not decay into S i . Then, as discussed above, the Higgs background h decaysmostly due to the Hubble friction, and the result (14) will apply in this case. Consequently, wecan perform an analysis similar to the Higgs channel. The curvature perturbation generated fromthe Higgs-modulated decay φ → S i S i is, ζ = − δ ΓΓ (cid:39) αh δh m φ (cid:12)(cid:12)(cid:12)(cid:12) t = t reh , (35)18here we again assume m φ > m S to simplify the expression. Then we see that to generate thecorrect amount of primordial fluctuations requires the cutoff scale Λ S to beΛ S (cid:39) (cid:114) N α π λ / P − / ζ (cid:112) m φ H inf . (36)Assuming the dimensionless coupling α = 1, N = 10, m φ = 10 H inf , and again λ (cid:39) .
01, we seethat the cutoff scale will be Λ S ≥ H inf and this can be well above the unitairty bound. Wecan further raise the cutoff in this case by giving a larger mass to m φ . Again, the dim-5 operator( ∂ µ φ ) S∂ µ S does not lead to copious production of S particles even when the cutoff scale Λ S is aslow as the unitarity bound, so we see that the Higgs-portal channel is a viable way to realize theHiggs-modulated reheating.After the completion of reheating, the thermal collisions of S i particles will further produce theHiggs boson and other SM particles. The good thing about this scenario is that the inflaton doesnot couple directly to SM fields, and thus the SM mass spectrum during inflation only receivescorrections from the thermal background, which is calculable and quite definite, and this allowsus to find rather predictable and clean signals in the squeezed bispectrum. Several assumptions have been made in the study of the inflaton decay in this section, including1) single-field slow-roll inflation, 2) perturbative inflaton decay, 3) EFT couplings of lowest orderbetween the inflaton and matter fields, respecting the shift symmetry of the inflaton. Morepossibilities are available without making these assumptions. Below we will mention several ofthem as interesting alternatives, and leave detailed analysis to future studies.As we see from the previous discussions, one major difficulty in realizing Higgs-modulatedinflaton decay is the too-fast decreasing of the Higgs background, which requires a large decayrate of the inflaton. The large decay rate is usually in tension with various bounds on the cutoffscale Λ during inflation. This tension can be released in several ways if we go beyond the threeassumptions made above.First, if we go beyond the single-field slow-roll paradigm and consider a two-field inflationmodel such as hybrid inflation [37], then the slow-rolling field φ will be different from the field φ that reheats the universe at the end of inflation. In this case there will be no direct relation betweenthe φ -SM couplings and φ -SM couplings. This will provide us more freedom in realizing theHiggs modulated reheating. Also, even in the framework of single field inflation, during inflationthe inflaton may have rolled a longer distance than one EFT can describe, such that inflation andreheating might be described by different EFTs with different cutoffs.Second, we can imagine that the inflaton decays nonperturbatively through preheating [38, 39]or tachyonic instabilities [40]. This will greatly enhance the decay rate of the inflaton even whenthe perturbative decay is forbidden.Third, we can go beyond the EFT couplings and consider specific inflaton-matter couplingsvalid at large φ . For instance, if the inflaton-matter coupling has the following form, e − φ/ Λ (cid:48) Λ ( ∂ µ φ ) O µ , (37)19here O µ is some vector operator formed by matter fields, and Λ, Λ (cid:48) are two independent cutoffscales. This operator can lead to quick inflaton decay when φ oscillates around φ = 0 provided asmall cutoff Λ. On the other hand, this operator will be suppressed during inflation by the large φ/ Λ (cid:48) so a low cutoff Λ is free from the unitarity constraint during inflation. In the previous section we analyzed possible realizations of Higgs-modulated reheating, and wesee that the easiest way to modulate the inflaton decay through the SM Higgs field is to introducea couple of Higgs-portal scalar fields S i . We assume that the inflaton couples only to S i but notdirectly to SM particles. In this section, we are going to explore the CHC signals mainly in thisscenario. At the same time, we will also mention possible new signals when the inflaton is allowedto couple directly to SM.As in the ordinary case of the cosmological collider, by CHC signal we mean the bispectrum ofthe scalar perturbation in the squeezed limit, namely the three-point correlation of the curvaturefluctuation (cid:104) ζ ( k ) ζ ( k ) ζ ( k ) (cid:105) where one external momentum, say k , is much smaller than theother two k , . The CHC then means the oscillations in the bispectrum as a function of the“squeezeness” parameter k /k . This oscillation is a result of the intermediate massive fieldattached to the soft external leg and going on-shell. And as usual, by measuring the frequency ofthe oscillation, we will be able to read the mass of the intermediate massive particle in the unitof Hubble scale H inf . The special feature of the CHC is that the external legs can be viewed aslong-lived Higgs mode, and thus the intermediate massive particles couples to the external linesthrough Higgs coupling.The advantage of the Higgs-portal scenario for CHC is clear: the SM fields do not coupledirectly to the inflaton background, and thus the SM particles’ mass will receive contributiononly from the Higgs background and the thermal loops, which are predictable. There will befurther corrections from the inflaton background if we integrate out the S i fields and introduceeffective couplings between the inflaton and the SM fields. But the contribution from theseeffective operators is expected to be small. In addition, the couplings between the SM fieldsand the primordial scalar fluctuations are nothing but the Higgs couplings which are known.Therefore, we expect quite definite predictions of SM signals at CHC. This will be very helpfulto tell this scenario of CHC from the ordinary inflation.In the following we will study the CHC signals of SM particles. We will first consider thesimplest Higgs-portal scenario, where inflaton couples only to the Higgs-portal scalars S i . Thenwe will consider an example of BSM couplings between the inflaton and the SM fields. Throughoutthe section we will use H to denote the Hubble scale during inflation, namely, H = H inf .Some general discussions before entering the detail. First, from (8) and (9), the 3-pointfunction from the Higgs external legs can be written as (cid:104) ζ ( k ) ζ ( k ) ζ ( k ) (cid:105) (cid:48) h = (cid:18) πH P / ζ R h (cid:19) (cid:104) δh ( k ) δh ( k ) δh ( k ) (cid:105) (cid:48) . (38)Here and everywhere a prime on the correlator (cid:104)· · ·(cid:105) (cid:48) means that the δ -function of momentumconservation is removed. The subscript on the left hand side (cid:104)· · ·(cid:105) h means the contribution from20he Higgs. On the other hand, the shape function S ( k , k , k ) of the bispectrum is conventionallydefined through (cid:104) ζ ( k ) ζ ( k ) ζ ( k ) (cid:105) (cid:48) = (2 π ) S ( k , k , k ) 1( k k k ) P ζ . (39)Comparing the above two equations, we have S ( k , k , k ) = 12 πH R h P − / ζ ( k k k ) (cid:104) δh ( k ) δh ( k ) δh ( k ) (cid:105) (cid:48) . (40)The correlator (cid:104) δh (cid:105) can be calculated using the Schwinger-Keldysh formalism, which is verysimilar to calculating Feynman diagrams [41]. The overall size of the shape function S can berepresented by a dimensionless number which is conventionally called f NL . We also note thatthere are O (1) differences in different definitions of f NL . Very often it is useful just to estimate f NL without doing detailed calculation. The estimate goes as f NL ∼ R h P − / ζ π × loop factors × vertices × propagators . (41)Every dimensionful parameter in (41) is measured in the unit of Hubble H . The loop factoris the usual one from the momentum integral. For example, we can estimate the 1-loop factoras 1 / (16 π ). The vertices refer to the coupling coefficients from usual Feynman rules, and thepropagators can be estimated as O (1) for fields with mass close to or smaller than Hubble, m (cid:46) H .For heavier mass, each propagator contributing the non-analytical oscillations will be suppressedby a Boltzmann factor e − π ( m − µ ) /H with µ being the corresponding chemical potential. If there isadditional source of particle production for the internal line other than the vacuum fluctuation,one should also include corresponding enhancing factors.Here we use the formula (41) to estimate the local non-Gaussianity. This does not belong tothe CHC we are interested in, but it is still important to check that the local non-Gaussianityproduced in this scenario is consistent with current constraints from CMB.Generally, we expect two sources contributing to local non-Gaussianity. One is from thesuper-horizon evolution of the Higgs fluctuation due to its self-interaction. The induced cubicinteraction with coupling ∼ λh will introduce a secular growth of local non-Gaussianity. Forsmall λ , we expect that the resulting f NL will be proportional to the number of e -folds. Thesecond contribution from the non-linear modulation rate, which means that we have additionalnon-linear terms ζ = z δh + z δh + · · · when converting δh to ζ . In the Higgs-portal scalarchannel (35) we have z (cid:39) αN/ (3 m φ ). Combining these two contributions, we can estimate thelocal non-Gaussianity as, f NL (local) ∼ −O (1) R h πP / ζ λN e + O (1) R h (2 π ) P ζ αN ( m φ /H inf ) , (42)We see that both terms in (42) can be quite large due to the inverse powers of P ζ . The secondterm can be suppressed if we have a larger inflaton mass at the end of inflation. The first termgives a more stringent constraint on the parameter space, especially on R h . Given the currentCMB constraints | f NL (local) | (cid:46)
5, we have, R h (cid:46) min (cid:26) . (cid:18) λ . (cid:19) − / (cid:18) N e (cid:19) − / , (cid:27) . (43)21his means that the Higgs fluctuation cannot be the only source of the primordial density fluc-tuation unless with a very tiny self-coupling λ ∼ O (10 − ). However, we will see below that theCHC signals can still be observably large given the R h suppression derived here. In this subsection we consider SM signals in the simplest and cleanest scenario, namely theinflaton decay through the Higgs-portal channel.In this scenario all particles appear at 1-loop level. From the estimate (41) we see that thelargest signal should come from particles with strongest couplings with the Higgs, namely
W/Z boson and the top quark. We may also see the Higgs-portal scalars S i in this way because thescenario under consideration prefers a large coupling between S i and H . However, the mass of S i is a free parameter, and a mass slightly larger than Hubble by several times could easily render thesignal invisible. We actually prefer a large mass for S i because heavier S i can help to suppress thecoupling between the inflaton and SM fields. Therefore, we will not consider S i signals further. Gauge boson signals.
The signals in the squeezed bispectrum from massive SM gauge bosonshave been calculated in [22]. The corresponding signals at CHC is almost the same as the “inflatoncollider” signals in [22]. Here we will outline the main steps of the calculation and refer readersto [22] for more details. We use the diagrammatic representation for each process followingSchwinger-Keldysh formalism. See [41] for a review.The Higgs-gauge couplings are from the following Lagrangian,12 | D µ H | ⊃ g ( h + δh ) (cid:16) W + µ W − µ + 12 c W Z µ Z µ (cid:17) (44)Apart from the tree-level gauge boson mass from the h background, the W/Z boson also receivemasses from infrared-enhanced Higgs loop [22]. The loop contribution dominates when Higgs islight, which is true in our case. Therefore we will directly quote the loop mass calculated in [22]as following, m W = 3 g H π m H , m Z = m W /c W , (45)where c W = cos θ W and θ W is the Weinberg angle. From the couplings in (44) we see that thefollowing two 1-loop diagrams contribute to the 3-point function of the Higgs fluctuation. (a)4 + (b)2 (46)Here we have used the diagrammatic representation introduced in [41]. We first consider theDiagram (46a). Following the diagrammatic rules in [41], the amplitude corresponding to this22iagram reads,Diag. (46a) = 12 · i g · i g h (cid:88) a,b = ± ab (cid:90) d τ | Hτ | d τ | Hτ | G a ( k ; τ ) G a ( k ; τ ) G b ( k ; τ ) × (cid:90) d q (2 π ) (cid:104) D µν ( q, τ , τ ) (cid:105) ab (cid:104) D µν ( | k − q | , τ , τ ) (cid:105) ab , (47)where a and b are SK contour indices, G and D µν represent the propagators of the Higgs fluctuation δh and the W boson. The Higgs is light and we can use approximately the massless propagatorfor G , G ± ( k ; τ ) = H k (1 ± i kτ ) e ∓ i kτ . (48)The explicit expression for D is collected in the Appendix but we will not use it at this point.Here we have written down the expression for one W boson. To take account of two chargeeigenstates W ± , we need to multiply the expression by 2, and for the Z boson, we need to replacethe couplings as g → g /c W and also replace the mass implicit in the propagator D .The loop integral can be evaluated either directly or using the real-space representation. Thelatter was elaborated in [22]. Here we shall quote the main steps and refer readers to [22] for moredetails. The basic idea is that the loop integral can be written as I A ( k ; τ , τ ) = (cid:90) d X e − i k · X (cid:104) A µ ( x ) A µ ( x ) A ν ( x (cid:48) ) A ν ( x (cid:48) ) (cid:105) , (49)where x = ( τ , x ), x (cid:48) = ( τ , x (cid:48) ) and X = x − x (cid:48) . The oscillatory signals in the squeezed bispectrumappear when x and x (cid:48) are far apart, where the loop integral I A develops non-analytic dependenceon the spacetime coordinates as non-integer power of τ τ /X . We will call this part of theintegral the nonlocal part, and it is free of UV divergence. The nonlocal part of the correlator (cid:104) A ( x ) A ( x (cid:48) ) (cid:105) was calculated in [22]. The result in [22] contains an incorrect factor whose effectis subdominant. After correcting this, we find the result as (cid:104) A ( x ) A ( x (cid:48) ) (cid:105) = 3 H π m A (cid:20) (1 + 2 µ ) Γ ( µ )Γ ( 52 − µ ) (cid:16) τ τ (cid:48) X (cid:17) − µ + c.c. (cid:21) , (50)where m A is the mass of the gauge boson and µ ≡ (cid:112) / − ( m A /H ) . The above expressionassumes that µ is complex, namely m A > H/
2. In the opposite case where m A < H/ µ isreal, one should drop the “c.c.” term. All expressions below should be understood in the sameway.To calculate the loop integral I A from the above correlator, we use the integral, (cid:90) d Xe − i k · X X − γ = 4 πk − γ Γ(2 − γ ) sin( πγ ) . (51)Then it is straightforward to get I A ( k ; τ , τ ) = 3 H π m A k (1 + 2 µ ) Γ ( µ )Γ ( 52 − µ )Γ( − µ ) sin(2 πµ )( k τ τ ) − µ . (52)23herefore we finally haveDiag. (46a) = 3 g H h π m A k (cid:20) µ − (1 + 2 µ ) (2 − µ ) sin ( πµ ) cos( πµ ) × Γ (2 − µ )Γ ( 52 − µ )Γ ( µ )Γ( − µ ) (cid:16) k k (cid:17) − µ + c.c. (cid:21) . (53)The right diagram of (46) is more complicated. We shall make some simplifying assumptions.The hard internal line contributes mostly at the resonant point | kτ | (cid:39) m/H and therefore weshall evaluate the propagator at this point without keeping all momentum and time dependence.Since the propagator at sub-horizon scales is O ( H ), and the additional time integral over τ wouldalso contribute a factor of H − , we see that the net effect will be multiplying the result of thefirst diagram by a factor of 2 g h /H , where the factor of 2 is because (46a) contains a symmetryfactor 1 / S A ( k , k , k ; g, m A ) = R h g H h π m A (cid:16) g h H (cid:17)(cid:20) µ − (1 + 2 µ ) (2 − µ ) sin ( πµ ) cos( πµ ) × Γ (2 − µ )Γ ( 52 − µ )Γ ( µ )Γ( − µ ) (cid:16) k k (cid:17) − µ + c.c. (cid:21) . (54)Applying the above result to W and Z bosons, we get the corresponding shape functions, S W = 2 S A ( k , k , k ; g, m W ) , S Z = S A ( k , k , k ; g/c W , m Z ) . (55) Top quark signal.
Next we consider the top quark’s contribution to (cid:104) δh (cid:105) at 1-loop level, fromthe following diagram. (56)Again, there are only two soft internal lines contributing to the nonlocal part of the diagram, andwe will approximate the hard internal line by a factor of 1 /H . Then the diagram can be writtenasDiag. (56) = 2 · · i y t H · i y t √ (cid:88) a,b = ± ab (cid:90) d τ | Hτ | d τ | Hτ | G a ( k ; τ ) G b ( k ; τ ) G c ( k ; τ ) I ψ ( k ; τ , τ ) . (57)Similar to our treatment for the gauge boson loop, here we have written the loop integral as theexpectation value constructed from a 2-component Weyl spinor ψ , I ψ ( k ; τ , τ ) = (cid:90) d X e − i k · X (cid:104) ¯ ψψ ( x ) ¯ ψψ ( x (cid:48) ) (cid:105) , (58)24he factor 2 in front of (57) takes account of the fact that a top quark can be written as twoindependent Weyl spinors, and the factor 3 counts 3 colors. The nonlocal part of the correlatorhas been calculated in [24] and the result is, (cid:104) ¯ ψψ ( x ) ¯ ψψ ( x (cid:48) ) (cid:105) = − H π (1 − (cid:101) m t )Γ (2 − i (cid:101) m t )Γ ( −
12 + i (cid:101) m t ) (cid:16) τ τ (cid:48) X (cid:17) − (cid:101) m + c.c. . (59)From this we can find the following result for the nonlocal part of Diag. (56),Diag. (56) = 9i y t H √ π k (cid:20) − (cid:101) m t (1 − (cid:101) m t ) cosh ( π (cid:101) m t ) sinh( π (cid:101) m t ) × Γ (2 − i (cid:101) m t )Γ ( 12 + i (cid:101) m t )Γ (3 − (cid:101) m t )Γ( − (cid:101) m t ) (cid:16) k k (cid:17) − (cid:101) m t + c.c. (cid:21) , (60)where (cid:101) m t = m t /H . Then the shape function of the top signal is S top = R h √ y t H π (cid:20) − (cid:101) m t (1 − (cid:101) m t ) cosh ( π (cid:101) m t ) sinh( π (cid:101) m t ) × Γ (2 − i (cid:101) m t )Γ ( 12 + i (cid:101) m t )Γ (3 − (cid:101) m t )Γ( − (cid:101) m t ) (cid:16) k k (cid:17) − (cid:101) m t + c.c. (cid:21) . (61) Summary.
Above we have calculated the oscillatory signals in the squeezed bispectrum from
W/Z boson and the top quark. In all cases the shape function S exhibits non-integer powerdependence on the momentum ratio k /k , with the following form, S = (cid:20) C ( m ) (cid:16) k k (cid:17) α +i β + c.c. (cid:21) = 2 (cid:12)(cid:12) C ( m ) (cid:12)(cid:12)(cid:16) k k (cid:17) α cos (cid:16) β log k k + δ (cid:17) . (62)Here C ( m ) is a complex coefficient dependent on the loop particle’s mass m and also its couplingto the Higgs field. α and β are real numbers, and they also depend on the mass m . In the secondequality, we spell out the oscillation explicitly, with the frequency β and the phase δ . We definethe “clock amplitude” f NL,clock by f NL,clock = 2 |C| which provides a dimensionless measure of thesignal strength. Compare this number with the observational constraint/limit can provides us arough idea of the visibility of the signals. The name “clock” is from the fact that such signalscan also be used as “standard clocks” measuring the expansion history of the primordial universe,although the signals is truly oscillatory only when m is large enough ( m > H, , H for spin0, ,1 fields, respectively).In the case of a detection of clock signals, in principle we can measure both the amplitude f NL,clock and the frequency β . The frequency is directly related to the mass of the intermediateparticle, while the amplitude depends on both the mass and the coupling to the external lines,namely the Higgs field in our case. Both of these quantities are calculable as elaborated above, andin Fig. 4 we show the signals from W/Z and top loops calculated above in the ( f NL,clock , m ) plane.For illustration we take R h = 0 .
14, and for SM parameters we take approximately λ = 0 . g = 0 . y t = 1, c W = 0 .
88. Of course these parameters also undergo renormalization grouprunning from the electroweak scale where they are measured to the inflation scale. In generalRG running introduces small O (1) corrections. But in the special case of Higgs self-coupling, RG25unning could also introduce order-of-magnitude change. In fact, our choice of λ = 0 .
01 in Fig. 4is partially motivated by the fact that λ can be much smaller during inflation than its value atthe electroweak scale. Of course if one sticks to the result of SM RG running, then λ will turnnegative for high scale inflation. For our purpose of realizing CHC, we will assume either thatthe inflation scale is lower than the scale of Higgs instability or that new physics will change theRG running of Higgs self-coupling at high scales.Given the fact that the clock amplitude f NL,clock is exponentially sensitive to the mass of theloop particle, and that our knowledge of particles’ mass during inflation is not as precise, we alsovary the masses of the internal particles in Fig. 4 to illustrate the variations in the signal strength.The two peaks at m/H = (cid:112) /
16 and m/H = 1 / W and Z curves come from thefactor Γ( − µ ) in the loop integral (52), which signals the failure of convergence of the loopintegral. More careful treatment of the loop integral should remove this divergence. So, even wemay expect enhancement of the signals near these two values of m/H , the two peaks themselvesin Fig. 4 should be taken with a grain of salt. - m / H f N L , c l o c k W Z top
Figure 4: SM signals at CHC. The blue, purple, and black curves show the signals strength f NL,clock as functions of particles’ mass for W , Z , and top quark, respectively. The choice of parametersare R h = 0 . λ = 0 . g = 0 . y t = 1 and c W = 0 .
88. Even larger non-Gaussianities arepossible if smaller λ is chosen. Above we showed that the SM fields appear in the CHC signals at 1-loop if we assume thatthey couple to the Higgs field only through SM couplings. However, beyond-SM couplings can wellbe present during inflation. Some of these new couplings will lead to interesting new signals thatdo not exist within SM. In this subsection we consider one such example from Sec. 4.2, namelythe Λ hi -term in (27),∆ L = g c W Λ hi ( ∂ µ φ ) Z µ h ⇒ − g ˙ φ c W Λ hi a − ( τ ) Z ( h + δh ) . (63)26e can have a tree-level diagram propagating Z boson in the squeezed spectrum, (64)We can simply estimate the magnitude of the oscillatory signals from the above diagram as, f NL of Diag. (64) ∼ R h P − / ζ π (cid:18) ˙ φ Λ hi (cid:19) h . (65)As before, we have taken H = 1 in this estimate. Given m Z ∼ h ∼ H , the unitarity boundΛ hi > ˙ φ / , and the relation ˙ φ / > H , we see that the magnitude of (64) can be huge. Withhigh scale inflation H ∼ GeV, even Λ hi ∼ M Pl will give f NL ∼ O (1) R h . A lower cutoff scalecan give rise to clock signals much larger than that of the SM. In this manner the CHC may besensitive to BSM physics. Now let us evaluate the diagram (64) using diagrammatic rules,Diag. (64) = (cid:16) − i g ˙ φ c W Λ hi (cid:17) h (cid:88) a,b = ± ab (cid:90) d τ | Hτ | d τ | Hτ | G a ( k ; τ ) G a ( k ; τ ) G b ( k ; τ ) (cid:2) D ( k ; τ , τ ) (cid:3) ab . (66)Here D is the 00-component of the propagator for Z . The 0 component of polarization tensoris nonvanishing only for the longitudinal mode. Its polarization vector is e ( L ) µ ( k ) = (cid:18) − km Z a , (cid:114) (cid:16) km Z a (cid:17) (cid:98) k (cid:19) . (67)Therefore we have (see the appendix), D ( k ; τ , τ ) = 2 k H m Z (cid:20) Γ ( − µ ) π ( k τ τ / µ +3 / + ( µ → − µ ) (cid:21) . (68)The result is S = R h g ˙ φ h P − / ζ / π c W Λ hi Hm Z (cid:20) (7 − µ )(1 + sin πµ )Γ ( µ )Γ ( − µ )(1 + 2 µ ) (cid:18) k k (cid:19) / − µ + c.c. (cid:21) . (69)We note that the shape function does not depend on the angle between k and k as one maynaively expect from an internal spin-1 line. This is because the Z boson couple to the externalHiggs background through its 0-component. So it behaves effectively like a scalar field. The modulated reheating provides a simple way to generate primordial density fluctuations.In the context of the cosmological collider physics, we can say that the modulated reheating27cenario turns inflation into a light scalar collider, where the light scalar is the field modulatingthe inflaton decay.In this paper we have shown that the SM Higgs boson can be a perfect modulating field. ThisHiggs-modulated reheating scenario thus provides us a Higgs collider working at the inflationscale and opens up new possibilities of studying Higgs physics at energies potentially far aboveany ground-based colliders.We have provided a simple realization of Higgs-modulated reheating where the inflaton decaysinto Higgs-portal scalars, which can also be a dark matter candidate. We have calculated theSM signals at the CHC in this scenario. We have seen that the oscillatory/non-analytic scalingsignals in the squeezed bispectrum can be naturally large.As a demonstration of using CHC, we further studied signals of SM fields in the squeezedbispectrum, including the massive gauge boson loop and the top loop, which could be the leadingsignals of SM. In preferable scenarios one could observe them simultaneously.We see that the non-Gaussianities produced in the Higgs-modulated reheating scenario aregenerally quite large. In particular, the large local-shape non-Gaussianity limits the parameterrange of CHC. As a result, the density perturbation cannot be generated all by the Higgs (unlessthe Higgs self-coupling is very small). Consequently, the CHC signals are suppressed by a factorof R h . But as we see from Sec. 5, the size of CHC signals are exponentially sensitive to the massof intermediate particles and they can be enhanced a lot by changing the mass by a small amount.So, the CHC signal in this case can still be observably large.A far more interesting possibility is to study the signals of new physics. The CHC providesan unmatched energy for Higgs collisions, and one can imagine to study any new physics scenarioassociated with the Higgs boson at high energies up to H inf ∼ GeV. Some simple examplesinclude a Higgs-portal scalar, and BSM Higgs- Z couplings, which we briefly considered. We leavemore studies in this respect for future works.In a broader sense, CHC can be considered as an example of a class of cosmological isocur-vature colliders. The isocurvature perturbation may have converted to curvature perturbationthrough curvaton [42–44], modulated reheating or multi-brid [45, 46] mechanisms. Alternatively,the isocurvature perturbation may survive and be directly observed for very light fields such asaxions. These isocurvature fluctuations may carry the information of mass and spin of the heavyparticles during inflation. We hope to study these possibilities in the future. Acknowledgment
We thank Xingang Chen for discussions. YW thanks Dong-Gang Wang fordiscussions. ZZX thanks Prateek Agrawal, Junwu Huang, Hayden Lee, Davide Racco, and LiantaoWang for discussions. SL and YW are supported in part by ECS Grant 26300316 and GRF Grant16301917 and 16304418 from the Research Grants Council of Hong Kong. YW thanks Universityof Warsaw for hospitality where part of this work was done. ZZX thanks TD Lee Institute fortheir hospitality when this work was in progress.28
SK Propagators of a Spin-1 Field
In this appendix we work out the mode function and the late-time limit of the propagator fora spin-1 massive gauge field. We will include a chemical potential for the gauge boson arisingfrom the coupling φF (cid:101) F where φ is the inflaton.We begin with the following Lagrangian, L = √− g (cid:20) − F µν F µν − m A µ A µ (cid:21) − λφF µν (cid:101) F µν , (70)where (cid:101) F µν = (cid:15) µνρσ F ρσ , (cid:15) = 1. The equation of motion of A µ = ( A , A ) can be derived, afterimposing the gauge condition ∂ µ ( √− gA µ ) = 0, to be A (cid:48)(cid:48) − ∇ A + a m A − λa ˙ φ ∇ × A = 0 (71)Or, in the momentum space, A (cid:48)(cid:48) + k A + a m A − i aλ ˙ φ k × A = 0 . (72)Let k = (0 , , k ) be in the 3-direction. Then we can redefine A ± = √ ( A ± i A ) and A L = A .Then the equation of motion becomes A (cid:48)(cid:48)± + ( k + a m ± aµk ) A ± = 0 , A (cid:48)(cid:48) L + ( k + a m ) A L = 0 , (73)where µ ≡ λ ˙ φ . The solutions with positive-frequency initial condition are A ± = e ∓ πµ/ H √ k W ± κ,ν (2i kτ ) , A L = √ π e i πν/ √− τ H (1) ν ( − kτ ) , (74)where W κ,ν is one of the Whittaker functions and H (1) ν is H¨ankel function of first kind. Theindices are κ ≡ i µ/H , ν ≡ (cid:112) / − ( m/H ) . The normalization is determined by the canonicalcommutation relation [ A i ( x ) , A (cid:48) j ( y )] = i δ ij δ (3) ( x − y ). The late-time ( τ →
0) behavior of thesemodes are A ± (cid:39) e − i π (1 / ν/ √ k e ∓ πµ/ Γ( − ν )Γ( ∓ i µ − ν ) ( − kτ ) ν +1 / + ( ν → − ν ) , (75) A L (cid:39) e − i π (1 / − ν/ √ k Γ( − ν ) √ π ( − kτ / ν +1 / + ( ν → − ν ) . (76)The non-local parts of the propagators are D µν ( k ; τ , τ ) = (cid:88) α e αµ ( k ) e αν ( k ) D ( α ) ( k ; τ , τ ) , (77) D ( ± ) ( k ; τ , τ ) = e ∓ πµ k (cid:20) Γ ( − ν )Γ( + i µ − ν )Γ( − i µ − ν ) (4 k τ τ ) ν +1 / + ( ν → − ν ) (cid:21) ,D ( L ) ( k ; τ , τ ) = 12 k (cid:20) Γ ( − ν ) π ( k τ τ / ν +1 / + ( ν → − ν ) (cid:21) . (78)We see that only the transverse parts are affected by the chemical potential. The chemicalpotential will bring exponential enhancement/suppression to the two transverse polarizations,and not-too-large back reaction will put stringent bound on the cutoff scale Λ as discussed inSec. 4.1. 29 eferences [1] Particle Data Group
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