Heavy-lifting of gauge theories by cosmic inflation

Journal of High Energy Physics, May 2018

Abstract Future measurements of primordial non-Gaussianity can reveal cosmologically produced particles with masses of order the inflationary Hubble scale and their interactions with the inflaton, giving us crucial insights into the structure of fundamental physics at extremely high energies. We study gauge-Higgs theories that may be accessible in this regime, carefully imposing the constraints of gauge symmetry and its (partial) Higgsing. We distinguish two types of Higgs mechanisms: (i) a standard one in which the Higgs scale is constant before and after inflation, where the particles observable in non-Gaussianities are far heavier than can be accessed by laboratory experiments, perhaps associated with gauge unification, and (ii) a “heavy-lifting” mechanism in which couplings to curvature can result in Higgs scales of order the Hubble scale during inflation while reducing to far lower scales in the current era, where they may now be accessible to collider and other laboratory experiments. In the heavy-lifting option, renormalization-group running of terrestrial measurements yield predictions for cosmological non-Gaussianities. If the heavy-lifted gauge theory suffers a hierarchy problem, such as does the Standard Model, confirming such predictions would demonstrate a striking violation of the Naturalness Principle. While observing gauge-Higgs sectors in non-Gaussianities will be challenging given the constraints of cosmic variance, we show that it may be possible with reasonable precision given favorable couplings to the inflationary dynamics.

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Heavy-lifting of gauge theories by cosmic inflation

HJE Heavy-lifting of gauge theories by cosmic in ation Soubhik Kumar 0 1 Raman Sundrum 0 1 0 College Park, MD 20742 , U.S.A 1 Maryland Center for Fundamental Physics, Department of Physics, University of Maryland , USA Future measurements of primordial non-Gaussianity can reveal cosmologically produced particles with masses of order the in ationary Hubble scale and their interactions with the in aton, giving us crucial insights into the structure of fundamental physics at extremely high energies. We study gauge-Higgs theories that may be accessible in this regime, carefully imposing the constraints of gauge symmetry and its (partial) Higgsing. We distinguish two types of Higgs mechanisms: (i) a standard one in which the Higgs scale is constant before and after in ation, where the particles observable in non-Gaussianities are far heavier than can be accessed by laboratory experiments, perhaps associated with gauge uni cation, and (ii) a \heavy-lifting" mechanism in which couplings to curvature can result in Higgs scales of order the Hubble scale during in ation while reducing to far lower scales in the current era, where they may now be accessible to collider and other laboratory experiments. In the heavy-lifting option, renormalization-group running of terrestrial measurements yield predictions for cosmological non-Gaussianities. If the heavy-lifted gauge theory su ers a hierarchy problem, such as does the Standard Model, con rming such predictions would demonstrate a striking violation of the Naturalness Principle. While observing gauge-Higgs sectors in non-Gaussianities will be challenging given the constraints of cosmic variance, we show that it may be possible with reasonable precision given favorable couplings to the in ationary dynamics. Cosmology of Theories beyond the SM; Gauge Symmetry - 3 Squeezed limit of cosmological correlators The in-in formalism for cosmological correlators Useful gauges for general coordinate invariance Observables NG from single eld in ation in the squeezed limit NG from multi eld in ation in the squeezed limit NG from Hubble-scale masses in the squeezed limit 4 Gauge-Higgs theory and cosmological collider physics The central plot and its connections to the literature High energy physics at the Hubble scale Heavy-lifting of gauge-Higgs theory 5 eld slow roll in ation and coupling strengths of e ective theory Visibility of a Higgs scalar Visibility of a massive gauge boson Gauge theory with a heavy Higgs scalar 1 Introduction 2 Preliminaries 8 Concluding remarks and future directions 6.1.1 6.1.2 6.2.1 6.2.2 Visibility of a Higgs scalar Visibility of a massive gauge boson 7 Detailed form of NG mediated by h 6 NG in the E ective Goldstone description of in ationary dynamics 6.1 Minimal Goldstone in ationary dynamics Leading terms in the e ective theory and power spectrum Higher order terms 6.2 Incorporating gauge-Higgs theory into the Goldstone E ective description { i { A.1 Massive elds A.2 In aton mode functions B HJEP05(218) 1 Introduction Cosmic In ation (see [1] for a review), originally invoked to help explain the homogeneity and atness of the universe on large scales, provides an attractive framework for understanding inhomogeneities on smaller scales, such as the spectrum of temperature uctuations in the Cosmic Microwave Background (CMB) radiation. These uctuations are consistent with an almost scale-invariant, adiabatic and Gaussian spectrum of primordial curvature perturbations R [2]. The approximate scale invariance of these uctuations can be naturally modeled as quantum oscillations of the in aton eld in a quasi-de Sitter (dS) spacetime. The adiabaticity property implies that among the elds driving in ation, there is a single \clock", the in aton, which governs the duration of in ation and the subsequent reheating process. Finally, Gaussianity of the present data [3] re ects very weak couplings among in ationary and gravitational elds. While these features point to successes of the in ationary paradigm, few details of the fundamental physics at play during in ation have emerged. Observing small non-Gaussianity (NG) of the uctuations could change this situation radically, giving critical insights not only into the in ationary dynamics itself but also into the particle physics structure of that era. Interactions of the in aton with itself or other elds during or immediately after in ation can lead to a non-Gaussian spectrum of R. However, NG can also be developed after uctuation modes re-enter the horizon at the end of in ation. This can happen for various reasons, including, nonlinear growth of perturbations under gravity during structure formation (see [4, 5] for reviews in the context of CMB and Large-Scale Structure). Therefore { 1 { it is crucial to understand and distinguish this latter type of NG which can \contaminate" the invaluable primordial NG. In this paper we will assume that this separation can be achieved in future experiments involving Large-Scale Structure (LSS) surveys [6] and 21cm cosmology [7, 8], to reach close to a cosmic-variance-only limited precision. With this quali er, a future measurement of NG can reveal important clues as to the underlying in ationary dynamics. For example, for the case single- eld slow-roll in ation, there is a minimal amount of NG mediated by gravitational interactions [9, 10], while lying several orders of magnitude below the current limit on NG, can be achievable in the future. There also exist a variety of models which predict a larger than minimal NG (see [ 11, 12 ] for reviews and references to original papers). A common feature among some of these models is the presence of additional elds beyond the in aton itself. Such non-minimal structure can be motivated by the need to capture in ationary dynamics within a fully theoretically controlled and natural framework. If those additional elds are light with mass, m H (where H denotes the Hubble scale during in ation), they can oscillate and co-evolve along with the in aton during in ation. These elds can generate signi cant NG after in ation, with a functional form approximated by the \local" shape [3, 13, 14]. On the other hand, the additional elds can be heavy with masses m & H. Such elds can be part of \quasi-single- eld in ation" which was introduced in [15] and further developed in [16{22]. In the presence of these massive elds, the three-point correlation function of the curvature perturbation R has a distinctive non-analytic dependence on momenta, hR(~k1)R(~k2)R(~k3)i / k33 k13 in the \squeezed" limit where one of 3-momenta becomes smaller than the other two. In the above, (m) = + i 3 2 r m2 H2 9 4 ; where m is the mass of the new particle. The non-analyticity re ects the fact that the massive particles are not merely virtual within these correlators, but rather are physically present \on-shell" due to cosmological particle production, driven by the in ationary background time-dependence. Such production is naturally suppressed for m H, which is re ected by a \Boltzmann-like suppression" factor in the proportionality constant in (1.1). The only e ect of m H particles is then virtual-mediation of interactions among the remaining light elds [23]. At the other extreme, for m H the distinctive non-analyticity is lost. Hence, we are led to a window of opportunity around H, where the non-analytic dependence of the three-point function is both non-trivial and observable, and can be used to do spectroscopy of masses. Furthermore, if a massive particle has nontrivial spin [21, 24], there will be an angle-dependent prefactor in (1.1), which can enable us to determine the spin as well [25]. These observations point to a program of \Cosmological Collider Physics" [21], which has an unprecedented reach into the structure of fundamental physics measurements is ultimately constrained by cosmic variance, very roughly in the ball park of hRRRi hRRi 2 3 1 pN21-cm where we have assumed the number of modes accessible by a cosmic variance limited 21-cm experiment is N21-cm the full potential of the program. 1016 [7]. Achieving such a precision is very important for realizing In this paper, we couple gauge-Higgs theories with m H to in ationary dynamics and ask to what extent the associated states can be seen via the cosmological collider physics approach. The contributions of massive particle to the three point function hR(~k1)R(~k2)R(~k3)i can be represented via \in-in" diagrams in (quasi-)dS space such as in gure 1. From gure 1(a), we see that since the in aton has to have the internal quantum numbers of the vacuum,1 the same has to be true for the massive particles. The particles must therefore be gauge singlets. Keeping this fact in mind, let us analyze the two scenarios that can arise during in ation. 2HHyH with H The gauge theory may be unbroken during in ation. Gauge singlet 1-particle states then can mediate NG via tree diagrams as shown in gure 1(a). This is also the case that has been analyzed extensively in the literature. On the other hand, gauge charged states can contribute via loops, as shown in gure 1(b), but are expected to be small. Alternatively, the gauge theory may be (partially) Higgsed during in ation. Then the massive particle in gure 1(a) need only be a gauge singlet of a residual gauge symmetry, but may be charged under the full gauge group. This possibility, which has received less attention in the literature (however, see [27, 28] for a related scenario), will be our primary focus. There are two ways in which such a Higgsing can happen, as we discuss now. First, such a breaking can be due to a xed tachyonic mass term for the Higgs H, H. In this case, the gauge-Higgs theory remains Higgsed after in ation ends and its massive states can annihilate away as universe cools giving rise to standard cosmology. Grand uni ed theories are examples of gauge extensions of Standard Model (SM) containing very massive new particles and which are strongly motivated by existing lower energy experimental data. For example, non-supersymmetric uni cation is suggested 1In the context of Higgs in ation [26] however, in aton is the physical charge neutral Higgs eld. { 3 { by the near renormalization-group convergence of SM gauge couplings in the 1013{1014 GeV range, right in the high-scale in ation window [2] of opportunity for cosmological collider physics!2 NG detection of some subset of these massive states could give invaluable clues to the structure and reality of our most ambitious theories. It is also possible that H-mass states revealed in NG are not connected to speci c preconceived theories, but even this might provide us with valuable clues about the far UV. Another very interesting and testable option is a tachyonic \mass" term of the form L cRHyH, where R is the Ricci scalar and c > 0 parametrizes the non-minimal coupling of Higgs to gravity. The e ects of non-tachyonic terms for this form with c < 0 have been considered before (see e.g. [31, 32]). Note, spontaneous breaking triggered by c > 0 is completely negligible at low temperatures, say below 100 GeV. Whereas in the scenario above we needed the gauge-Higgs theory to fortuitously have states with m H, here we naturally get the Higgs particle at H for c O(1). Furthermore, if (gauge coupling Higgs VEV) H, we also get massive gauge bosons at H. In this way such a nonminimal coupling can lift up a gauge theory with a relatively low Higgs scale today, which we can access via collider or other probes, to the window of opportunity of cosmological collider physics during the in ationary era. We will call this the \heavy-lifting" mechanism. To make this idea concrete, we consider the example of heavy lifting the SM. During the in ationary era the SM weak scale v can be lifted to be very high, but we do not know where precisely because of the unknown parameter c (even if we knew H). However, this uncertainty drops out in mass ratios, mh = mZ mh = mW mh = mt p 2 2 h While the top t and W boson can only appear in loops gure 1(b), the physical Higgs h and the Z can appear in gure 1(a) giving us one prediction in this case. However, an important subtlety of the couplings on the r.h.s. of the ratios above is that they are not those measured at the weak scale but rather are the results of running to H. But it is well known that the SM e ective potential develops an instability around 1010{1012 GeV because of the Higgs quartic coupling running negative (see [33] and references therein for older works). Since the in ationary H can be higher, the Higgs eld can sample values in its potential beyond the instability scale. Whether this is potentially dangerous for our universe has been considered before (see e.g. [31{37]). But it is possible that this instability is straightforwardly cured once dark matter (DM) is coupled to the SM. A simple example [38{41] would be if future experiments determine that DM is a SM gauge singlet 2Uni cation at such scales is disfavored in minimal uni cation schemes by proton decay constraints, but viable in non-minimal schemes such as that of refs. [29, 30]. { 4 { scalar S stabilized by a Z2 symmetry, S ! couplings are given by the Higgs portal coupling and scalar self-interaction, S. Then the most general renormalizable new theory up to even high scale in ation energies. Imagine a discovery of such a DM (S) is made in the coming years, along with a measurement of k and its mass mS (and possibly a measurement of or at least a bound on, S). Also, imagine a measurement of H is obtained via detecting the primordial tensor power spectrum. Then we can use the Renormalization Group (RG) to run all the measured couplings to the high scale H. These would then allow us to compute the run-up couplings needed to make a cosmological veri cation of (1.4). Such a veri cation of this Next-to-Minimal SM (NMSM) would give strong evidence that no new physics intervenes between TeV and H. Since this NMSM clearly su ers from a hierarchy problem (worse than the SM), the precision NG measurements would therefore provide us with a test of \un-naturalness " in Nature, perhaps explained by the anthropic principle [42, 43]. Whether the naturalness principle is undercut by the anthropic principle or by other considerations is one of the most burning questions in fundamental physics. Of course, the heavy-lifting mechanism may also apply to non-SM \dark" gauge-Higgs sectors, which we may uncover by lower energy experiments and observations in the coming years, or to gauge-Higgs extensions of the SM which may emerge from collider experiments. In this way, there may be more than one mass ratio of spin-0 and spin-1 particles that might appear in NG which we will be able to predict. As we will show, such new gauge structure may be more easily detectable in NG than the (NM)SM, depending on details of its couplings. It is important to note that di erent gauge theory sectors in the current era, with perhaps very di erent Higgsing scales, can be heavy-lifted to the same rough scale H during in ation, with their contributions to NG being superposed. The heavy-lifting mechanism may not be con ned to unnatural gauge-Higgs theories. For example, if low energy supersymmetry (SUSY) plays a role in stabilizing the electroweak hierarchy, a suitable structure of SUSY breaking may permit the heavy-lifting mechanism to work. Heavy-lifting can then provide us with a new test of naturalness! Possibly non-tachyonic squarks and sleptons in the current era were tachyonic during in ation, higgsing QCD or electro-magnetism back then. We leave a study of the requisite SUSYbreaking structure for future work. Cosmological collider physics studies incorporating SUSY but restricted to gauge singlet elds have appeared in [17, 44]. NG potentially provide us with the boon of an ultra-high energy \cosmological collider", but cosmic variance implies it operates at frustratingly low \luminosity"! We will see that this constrains how much we can hope to measure, even under the best experimental/observational circumstances. For example, a pair of spin-1 particles appearing in the NG will be more di cult to decipher than only one of them appearing, due to { 5 { the more complicated functional form of the pair that must be captured in the limited squeezed regime under cosmic variance. And yet, we would ideally like to see a rich spectrum of particles at H. The key to visibility of new physics under these harsh conditions is then determined by the strength of couplings to the in aton. This is the central technical consideration of this paper, taking into account the signi cant suppressions imposed by (spontaneously broken) gauge invariance. We study this within two e ective eld theory frameworks, one more conservative but less optimistic than the other. Single- eld slow-roll in ation gives the most explicit known construction of in ationary dynamics, but we will see that minimal models under e ective eld theory control give relatively weak NG signals, although still potentially observable. We also consider the more agnostic approach in which the dynamics of in ation itself is parametrized as a given background process [45], but in which the interactions of the gauge-Higgs sector and in aton uctuations are explicitly described. This will allow for larger NG signals, capable in principle of allowing even multiple particles to be discerned. This paper is organized as follows. We start in section 2 by reviewing the in-in formalism and its use in calculation of the relevant non-Gaussian observables. We also include a discussion of di erent gauges and conventions used for characterizing NG. Then in section 3 we review the signi cance of the squeezed limit of cosmological correlators, both in the absence and presence of new elds beyond the in aton. In particular, we review the derivation of (1.1). In section 4 we discuss some general aspects of gauge-Higgs theory dynamics during in ation and elaborate upon the two alternatives for Higgs mechanism discussed above. We then specialize in section 5 to slow-roll in ation where we study the couplings of Higgstype and Z-type bosons to the in aton in an e ective eld theory (EFT) framework. In section 6 we describe parallel considerations in the more agnostic EFT approach mentioned above. The two levels of e ective descriptions are then used in sections 7 and 8 (supplemented by technical appendices A{D) to derive some of the detailed forms of NG due to Higgs-type and Z-type exchanges respectively. We conclude in section 9. Hubble units. In this paper, the Hubble scale during in ation is denoted by H. To reduce clutter, from now on we will set H 1 in most of the numbered equations, with a few exceptions where we explicitly write it for the sake of clarity. Factors of H can be restored via dimensional analysis. However, we will refer explicitly to H in the text throughout, again for ease of reading, and in the unnumbered equations within the text. 2 2.1 Preliminaries The in-in formalism for cosmological correlators Primordial NG induced by in aton uctuations are calculated as \in-in" expectation values [46] of certain gauge-invariant (products of) operators at a xed instant of time towards the end of in ation, denoted by tf . The expectation needs a speci cation of the quantum state. The notion of \vacuum" is ill-de ned because spacetime expansion gives a timedependent Hamiltonian, H(t). However, for very short distance modes/physics at some very early time ti, the expansion is negligible and we can consider the state to be the { 6 { Minkowski vacuum, j i. As such modes redshift to larger wavelengths at tf , the state at tf can then be taken to be given by U (tf ; ti)j i, where t f i R dtH(t) U (tf ; ti) = T e t i In order to capture arbitrarily large wavelengths at tf in this manner, we formally take ti ! 1. (For free elds, the state de ned in this way at nite times, is the Bunch-Davies \vacuum".) Then the desired late-time expectation value of a gauge invariant operator Q is given in the Schroedinger picture by, h jU (tf ; ti = 1)yQU (tf ; ti = 1)j i. Now the calculation of the expectation value becomes standard. First, we go over to the interaction picture, and second we employ the standard trick of continuing the early evolution slightly into complex time to project the free vacuum j0i onto the interacting vacuum j i. Thus we arrive at the in-in master formula, h jU (tf ; ti)yQU (tf ; ti)j i = h0jT e +i t R f 1(1+i ) dt2HiInt(t2) QI (tf )T e i t R f 1(1 i ) dt1HiInt(t1) j0i: (2.2) In the above, the subscript I denotes that the corresponding operator is to be evaluated in the interaction picture. Finally, Hint(t) is the interaction part of the Hamiltonian of the uctuations, i.e. H = H0 + Hint with H0 being quadratic in uctuations. We note that the anti-time ordered product also appears in (2.2). The perturbative expansion of cosmological correlators of the above general type is facilitated as usual by expanding in products of Wick contractions, given by in-in propagators. This leads to a diagrammatic form, illustrated in gure 2. 2.2 Useful gauges for general coordinate invariance Metric and in aton uctuations are not gauge invariant under di eomorphisms. Hence we now review two useful gauges and a gauge invariant quantity characterizing the scalar perturbations during in ation. Our discussion will be brief and for more details the reader is referred to [9, 47]. For simplicity, we will specialize here to single- eld slow-roll in ation, but the considerations are more general. The metric of dS space is given by, ds2 = dt2 + a2(t)d~x2; with a(t) = eHt being the scale factor in terms of Hubble scale H. To discuss the gauge choices, it is useful to decompose the spatial metric hij dxidxj in presence of in ationary backreaction as follows [47], hij = a2(t) (1 + A) ij + where, A; B; Ci; ij are two scalars, a divergenceless vector, and a transverse traceless tensor perturbation respectively. The in aton eld can also be decomposed into a classical part 0(t) and a quantum uctuation (t; ~x), (2.3) (2.4) (2.5) (t; ~x) = 0(t) + (t; ~x): { 7 { Using the transformation rules of the metric and scalar eld, it can be shown that the quantity [48], R A 2 1 _ 0 ; is gauge invariant. This is the quantity that is conserved on superhorizon scales for single eld in ation [13, 49{52]. Although R seems to depend on more than one scalar uctuation, there is only one physical scalar uctuation which is captured by it. This is because among the ve scalar uctuations in the metric plus in aton system, two are non-dynamical constraints and two more can be gauged away by appropriate di eomorphisms, leaving only one uctuation. To make this manifest, we can do gauge transformations which set either to zero in (2.6) to go to spatially at and comoving gauge respectively. The rst of these will be most useful for simplifying in-in calculations involving Hubble-scale massive particles external to the in ation dynamics, while the second one is useful for constraining the squeezed limit of the NG due to in ationary dynamics itself. Spatially at gauge [9]. In this gauge the spatial metric (2.4) becomes Gauge invariant answers can be obtained by writing becomes in this gauge, in terms of R using (2.6), which Comoving gauge [9]. In this gauge the spatial metric (2.4) looks like hij = a2(t) ((1 + A) ij + ij ) ; = 0. This means the gauge invariant quantity R evaluated (2.6) (2.7) (2.8) (2.9) (2.10) (2.11) (2.12) (2.13) hij = a2(t) ( ij + ij ) : R = 1 _ 0 : R = A 2 ; Pk hR(~k)R( ~k)i0; { 8 { with quantum in aton eld in the new gauge becomes, which lets us rewrite the spatial metric (2.4) as hij = a2(t) ((1 + 2R) ij + ij ) ; with R being conserved after horizon exit (in single- eld in ation). 2.3 Observables Having discussed the gauge choices, we now move on to discussing the observables. The power spectrum for the density perturbations is given by, where the 0 denotes the notation that momentum conserving delta functions are taken away i.e. hR(~k1) R(~kn)i = (2 )3 3(~k1 + + ~kn)hR(~k1) R(~kn)i0 : The power spectrum can be evaluated to be where the r.h.s. is to be evaluated at the moment of horizon exit k = aH for a given k-mode. Since di erent k-modes exit the horizon at di erent times and H_24 has a slow time dependence, the combination k3Pk is not exactly k-independent, and we can write 0 1 1 The crude estimate of cosmic variance (1.3) translates to F 10 4{10 3. It is often conventional in the literature to typify the size of NG by the value of F at the equilateral point, Since we are mostly interested here in the squeezed limit for future signals, k3 we will explicitly compute F in that limit, referring to fNL only in the context of current NG limits (see subsection 6.1). In terms of the quantum in aton eld , the function F can be rewritten as, data [2] we get, ns power spectrum takes the form, 0:96 and H_24 = 8:7 0 where 1 ns is the tilt of the power spectrum and k is a \pivot" scale. From Planck 10 8 at k = 0:05 Mpc 1. In position space, the hR(~x1)R(~x2)i jx1 1 x2jns 1 : To calculate the bispectrum we will be interested in evaluating hR(~k1)R(~k3)R(~k3)i. By translational invariance the three momenta form a triangle, and by rotational invariance we are only interested in the shape and size of the triangle, not in the orientation of the triangle. Furthermore since we also have approximate scale invariance, we do not care about the overall size of the triangle, so e ectively the momentum dependence of bispectrum is governed only by the ratios kk31 and kk21 . We denote the bispectrum by the function B(k1; k2; k3), B(k1; k2; k3) = hR(~k1)R(~k3)R(~k3)i0: It is convenient to de ne a dimensionless version of this, F (k1; k2; k3) = _ 0 h (~k1) (~k2) (~k3)i0 h (~k1) ( ~k1)i0h (~k3) ( ~k3)i0 jk3 k1;k2 ; and where the r.h.s. is evaluated at the point of horizon exit for each mode. { 9 { (2.15) Squeezed limit of cosmological correlators NG from single eld in ation in the squeezed limit In single eld in ation, NG in the squeezed limit is proportional to the tilt of the in aton power spectrum [9, 53, 54], i.e. F Single Field(k1; k2; k3)jk3 k1;k2 = (1 ns) + O (3.1) k3 k1 2 : Let us go to comoving gauge (2.11) to demonstrate this. We are interested in computing hRh(~k1)Rh(~k2)Rs(~k3)i0, where the subscript h(s) means the associated momentum is hard(soft). We de ne position space coordinates ~xi to be conjugate to momentum ~ki. In the limit k3 j~x1 ~x2j k1; k2 we are interested in an \Operator Product Expansion (OPE)" regime, j~x1 ~x3j. Consider just the leading tree-level structure of the associated diagram in gure 2, and rst focus on just the boxed subdiagram. We see that for this subdiagram the soft line is just a slowly-varying background eld in which we are computing a hard 2-point correlator. Thus, hRh(x1)Rh(x2)Rs(x3)i hhRh(x1)Rh(x2)iRs( x1+2x2 )Rs(x3)i: (3.2) The e ect of the soft mode Rs is just to do the transform ~x ! (1 + Rs)~x of (2.11) within the leading 2-point function of (2.16): hRh(x1)Rh(x2)iRs 1 (jx1 x2j(1 + Rs))ns 1 (jx1 1 x2j)ns 1 (1 ns)Rs x1 + x2 : 2 (3.3) To get the middle expression, we have taken Rs to be approximately constant over distances of order j~x1 ~x2j, a good approximation since k3 ! 0. The last expression follows by expanding in (small) Rs, evaluated at the midpoint (x~1 + x~2)=2. We have also dropped a Rs-independent piece since that goes away when we consider the three point function. Thus the three point function becomes, hRh(x1)Rh(x2)Rs(x3)i hhRh(x1)Rh(x2)iRs( x1 +2x2 )Rs(x3)i (1 ns) (jx1 1 x2j)ns 1 (jx1 1 x3j)ns 1 : (3.4) Fourier transforming to momentum space, hRh(~k1)Rh(~k2)Rs(~k3)i0 (1 (1 1 1 ns) k4 ns k4 ns 3 1 ns)hRh(~k1)Rh( ~k1)i0hRs(~k3)Rs( ~k3)i0; (3.5) leading to (3.1). Subleading corrections proportional to are absent by rotational k3 k1 invariance, so the leading corrections are order The importance of the above expression lies in the fact that in the squeezed limit any value of F Single Field bigger than O(1 ns) will signal the presence of new physics beyond single- eld in ationary dynamics. In particular, next we comment on what can happen to the squeezed limit if we have multiple light (m H) elds (\multi eld in ation") or m 3.2 H elds (\quasi single eld in ation" [16]) during in ation. NG from multi eld in ation in the squeezed limit If we have light elds with m H, other than the in aton, then during in ation those elds can lead to larger NG in the squeezed limit than (3.1), see [55] and references therein. This can be understood again via similar in-in diagrammatics to gure 2. In this case it is again true that we have to evaluate the hard two point function in the background of some soft mode, and correlate the result with a R soft mode. However, since Rs is no longer the only soft mode in the theory, hRh(x1)Rh(x2)isoft mode 6= hRh(x1)Rh(x2)iRs : Thus the derivation in the previous subsection does not go through. Consequently F Multi Field in the squeezed limit is no longer constrained to be order (1 ns), but rather it becomes model dependent. 3.3 NG from Hubble-scale masses in the squeezed limit The situation changes quite a lot if we have particles with m H. Such particles can modify the bispectrum in a way that in the squeezed limit F contains a non-analytic part, F nonanalytic / f ( ) 94 and f ( ) is a calculable function of the mass of the particle, which is k3 k1 than H. of the order 1 when 1 but is \Boltzmann suppressed" e for large . We have a proportionality sign in (3.7) because there are model dependent prefactors which can take either large or small values, thus from (3.7) itself we can not get a complete estimate of NG. We will spell out the model dependent prefactors later. The crucial aspect of (3.7) is that F now contains a non-analytic dependence on along with other analytic terms. Importantly this non-analytic behavior can not be captured by any single or multi eld in ation models where all the masses are much smaller (3.6) (3.7) (c) Triple Exchange Diagram. Note that all these diagrams rely on mixing between the in aton uctuation and massive scalar in the (implicit) non-trivial background of rolling 0(t). as a two point function. The 0 background causing mixing is not explicitly shown, as in gure 3. (b) The same \OPE" approximation expressed an in aton-h three point function with one in aton leg set to zero momentum to now explicitly represent the background 0 . This dependence also encodes the information about the mass of the Hubble scale particle, via the exponent , [16{21]. If the massive particle has a nonzero spin(s), then F non-analytic has an additional factor dependent on Legendre polynomials, Ps(cos ), where k1 k^3 = cos . If we can measure this angular dependence precisely enough then we can ^ get the information about spin as well [21, 24]. Furthermore, such angular dependence is absent in purely single- eld and some of the multi eld in ation models. This can in principle help us in distinguishing the \signal" of m H particles from the \background" of m H particles. We see as we go to the region, m H, the leading behavior reverts to being analytic k1 k3 3=2 3=2 , and indistinguishable from purely single- eld or multi- eld in ation. This means it is observationally challenging to reach the region m H and still distinguish and measure m accurately. Also, at the other extreme, for m H cosmological production is strongly Boltzmann suppressed, so observation will again be di cult. Therefore we are led to a window around H for doing spectroscopy of masses and spins. Let us brie y explain the form of (3.7), rst concentrating on just the soft k3dependence. In presence of new particles with m H there are additional contributions to the bispectrum beyond those in gure 2. At tree level we can have three diagrammatic forms, as shown in gure 3. These are called single, double and triple exchange diagram based on the number of massive propagators [24]. In the squeezed limit, we are once again interested in calculating hRh(~k1)Rh(~k2)Rs(~k3)i0 i.e. correlation of two hard modes with a soft mode. In position space, this again corresponds to an \OPE" limit, jx12j jx13j, where the hard subdiagram is given by an e ective local vertex, depicted in gure 4 by the round black blob. The strength of this e ective vertex is then given by the hard two-point function in the background of the massive but k3-soft eld, which is predominantly k3-independent. Tracking only the k3-dependence is then given by the two-point correlator shown in gure 4. The leading k3 dependence can be worked out by the scaling properties of the elds involved, which can be read o from their classical late time asymptotics. For a general scalar eld, 9 . This means, O1(2)(~x) can be thought of as an operator with 4 ! 0, dS isometry generators in 4D acts as generators of the conformal group in 3D space. However, the leading e ect of the in ationary background is to break this conformal invariance of late-time correlators, but only weakly for slow roll. Using the simple scaling symmetry subgroup of the 3D conformal invariance, we nd3;4 hO (~x2)Rs(~x3)iinf / jx23j : should have the factor k323 i However, it is well known that 2-point correlators of di ering scale dimension vanish if conformal invariance is exact, therefore the implicit proportionality \constant" is suppressed by slow-roll parameters here. Fourier transforming and writing . We can now put back the k1 dependence, which again by the above scale invariance can only enter into the expression for F as shown in (3.7). 4 4.1 Gauge-Higgs theory and cosmological collider physics The central plot and its connections to the literature Having commented on NG and the squeezed limit in general, we focus on what kind of signature a gauge theory coupled to the in aton will have on NG. In particular we study signatures of Higgs scalars and gauge bosons. Non-trivial spin of heavy particles in the context of slow-roll in ation was rst considered in ref. [21], primarily for even spin. In ref. [24] both even and odd spin were considered. In both [21] and [24] no assumptions were made on the origins of the heavy masses. Here, we will impose the stringent constraints following from assuming that the heavy masses arise via the Higgs mechanism of weaklycoupled gauge eld theory, in particular for spins 0 and 1. In particular, the relevant non-linear terms coupling the gauge-Higgs sector and the in aton will be more suppressed by requiring gauge invariance than would be the case for massive elds unconnected to a Higgs mechanism. For example, consider the interaction of a pair of massive spin-1 particles, Z , with a pair of in aton elds. Without considering a gauge theoretic origin for Z, a low dimension interaction respecting (approximate) in aton shift symmetry has the form, 1 (3.8) (3.9) (4.1) 3There can be subleading slow-roll (1 ns) corrections to the exponent which are neglected here. 4Note that 4D dimensionful parameters, such as the Planck scale, do not break this 3D conformal or scale invariance. where is of order the cuto of EFT. However, if Z is a Higgsed gauge boson, the analogous interaction must arise from 1 (4.2) Crucially the interaction between Z and in aton has to happen in the presence of the Higgs eld H since we are assuming in aton to be a gauge singlet. Assuming the gauge theory is spontaneously broken, we see that the gauge theory interaction has a suppression of the order m2Z = 2 compared to the non-gauge theoretic case.5 This argument can be generalized to all the gauge boson interaction terms that we consider below. This also makes a general point that it can be harder to see NG due to gauge sector particles compared to non-gauge theoretic states. This is especially true for spin-1 particles as we saw above. In [27, 28] the signature of gauge theory was considered, focusing on unbroken electroweak symmetry during the in ationary phase as well as the scenario of Higgs in ation (in which the in aton is identi ed with the physical Higgs eld). In a general gauge theory with unbroken gauge symmetry, the gauge bosons will be massless up to (small) loop corrections [56, 57]. Non-trivial spectroscopy must then proceed via gauge-charged matter, which can only appear in loops by charge conservation and the singlet nature of the in aton. Such loops are di cult (but depending on speci c models, may not be impossible6) to observe for several reasons. First, at one loop [21], F loop f~( ) kk31 3+2i + g~( ) kk31 3 2i + so the fall-o is faster compared to (3.7) as one goes to smaller k3. Second, for heavier masses the Boltzmann suppression goes as e 2 because there is now a pair of massive particles involved. Thirdly, we will obviously have the loop factor ( 161 2 ) suppression. This gives us motivation to look for bigger tree level e ects which will be present if gauge symmetry is broken spontaneously during in ation. In [27, 28] such a scenario was mentioned although the primary focus was on Higgs-in ation-like scenarios in which the Higgs VEV is very large compared to H and consequently the massive gauge bosons are too heavy to be seen via NG due to Boltzmann suppression. The situation is much better if one keeps the gauge theory and in aton sectors distinct, with gauge symmetry spontaneously broken and Higgs VEV not too much larger than H. This is the case we focus on, and we will see that such scenarios can give rise to observable NG for both spin-0 and spin-1 particles. Since the Hubble scale during in ation can be very high (H . 5 1013GeV), in ation and the study of NG provides an exciting arena to hunt for new particles. In this regard two distinct possibilities arise. We discuss them next. 5The non-gauge theoretic case can be viewed as the limit of the gauge case where mZ . For example, a QCD meson or a spin-one superstring excitation cannot be housed in point-particle EFT, except in the marginal sense where the e ective cuto is mZ, where the constraints of gauge invariance disappear. 6For example one can imagine working in an e ective theory of in ation with its cuto & H, however if we have a cuto very close to Hubble then the gauge theory spectrum is no longer separated from the states coming from some UV completion of the eld theory, and measurements of NG cannot be translated robustly into information about the gauge theory alone. Such a scenario, of course, is still interesting, but we do not focus on that in this paper. HJEP05(218) 1 2 c1 + We could imagine a scenario in which there exists some new spontaneously broken gauge theory at H. Then some of the gauge-charged matter and gauge- elds may become singlets under the residual unbroken gauge symmetry. Bosons of this type, spin-0 and spin-1, can therefore have Hubble scale masses, couple to the in aton, and leave their signatures on NG at tree-level. For simplicity here, we focus on spontaneously broken U(1) gauge theory with no residual gauge symmetry, but is straightforward to generalize to the nonabelian case. For example, we can imagine a scalar in the fundamental representation of SU(N ) breaking the symmetry to SU(N 1). Then the gauge boson associated with the broken diagonal generator plays the role of the massive U(1) gauge boson that we consider now. Let us focus on the case of single- eld slow-roll in ation. We write an e ective theory with cuto . Since we are interested in e ects of gauge theory on NG, we will write down higher derivative interaction terms between the gauge sector and in aton. But we will not be explicit about higher derivative terms containing gauge sector elds alone or the in aton alone, although we will ensure that such terms are within EFT control. The lagrangian containing the in aton (with an assumed shift symmetry), the Higgs (H) and gauge bosons (not necessarily the SM Higgs and gauge bosons) has the form 1 2 L = Mp2lR + LGauge Theory 2 V ( ) + Liinntf + Lint inf-gauge; (4.3) where LGauge Theory contains all the terms (including higher derivative terms) containing gauge theory elds alone. V ( ) is a generic slow roll potential. Liinntf contains higher derivative terms containing in aton alone. For our purpose the interesting interaction terms between gauge theory and the in aton are contained in Lint below assuming an UV cuto and a set of dimensionless EFT coe cients ci, inf-gauge, which we write (HyD H) + (HyD H) + : c22 (@ ) H H + 2 y c44 (@ )2Z2 (4.4) In Liinntf, the rst term gives a quadratic mixing between Higgs and Z0. It also couples Higgs, Z and the in aton. But it does not contain any quadratic mixing between the in aton and Z; and also none between the in aton and Higgs. But we do see, from gure 3, that we need one or more quadratic mixings between the in aton and the massive particle of interest. Such quadratic mixing does arise from the second and the fth term, which give quadratic mixing of the in aton with Higgs and Z respectively. The third term gives, among other interactions, the interaction between an in aton and a pair of Zs. We have not written operators coming from the expansion in terms we have already considered. Hy2H since these will be subdominant to the To unpack (4.4) we can go to the unitary gauge for U(1) gauge theory and write down some of the relevant terms, h v ; Lint inf-gauge = 1;Z Z0h + 1;hh h_ + 2v 1;Z Z0h2 + + y H H + 5;h _ h_ + 1 2h v + _ 0 _ 0 Z 1;Z = 5;Z = Im(c1) _0mZ Im(c5) _20mZ v ; 5 1;h = 5;h = Re(c1) _0 2 Re(c5) _20v : 5 ; 2 _h2 + 2v Z h (hp+v) T 2 2 = 2c2 _0v ; 2 where we have expanded the Higgs eld in unitary gauge H = 0 and the in aton eld term via the parameter = c2 2_02 . We also have several quadratic mixing parameters, i, 4.3 Heavy-lifting of gauge-Higgs theory Until now, we have been discussing theories with Higgs physics intrinsically of order H. Now although a future detection of m H particles via NG will be very interesting in its own right, given that H may well be orders of magnitude beyond the energies of foreseeable particle colliders, we would not have valuable complementary access to this physics in the lab. But as discussed in the introduction, the alternative is the \heavy-lifted" scenario, in which m H during in ation and again yields observable NG, and yet m H in the current post-in ationary era and therefore conceivably is accessible to collider and other \low-energy" probes. Given a gauge theory at low energy, we can consider adding a non-minimal coupling of the Higgs to gravity, cRHy H to the lagrangian (4.3), where we will consider c of order one. This gives a Higgs e ective potential of the form, Ve (H) = hjHj 4 2 hjHj 2 cRHyH; While the curvature is negligible in the current era, during in ation we have R so that for c > 0, the symmetry breaking scale setting gauge-Higgs physical masses is naturally of order H. We can also see how this \heavy-lifting" mechanism appears in Einstein frame in which the in aton and Higgs potential get modi ed to (V ( ) + V (H)) ! (V ( ) + V (H))= 4 V ( ) 1 + V (H); (4.8) 4cHyH Mp2l ! (4.5) (4.6) (4.7) 12H2, 2 = 1 + 2cHyH is the Weyl scaling factor used to get to Einstein frame and we have kept the leading correction in cHyH=Mp2l.7 For NG, the discussion in the previous subsection then carries over from this point. As we elaborated in the introduction, one interesting fact about the heavy-lifting mechanism is that it is testable. This requires a knowledge of the couplings of the gauge theory sector in the current era, where they may be accessible at collider energies, and a measurement of H during in ation, as for example via the primordial tensor power spectrum. We can then use the renormalization group to run those couplings up to H, and thereby predict the mass ratios of spin-0 and spin-1 h and Z type particles (bosons charged under the full gauge symmetry which are singlets of the unbroken gauge symmetry) as they were in the in ationary epoch when they contributed to NG. Here the richer the set of h and Z type particles, and hence the larger the set of mass ratios, the less precision we would need to measure each ratio in NG in order to be convinced that we are seeing the same gauge theory in both regimes. 5 eld slow roll in ation We saw in the previous section that the leading interaction between in aton and gauge theory is captured by (4.4) and (4.5). These can be used to estimate the magnitudes of NG induced by h and Z. However, the parameters appearing in those two lagrangians have to satisfy several consistency requirements. We rst discuss such restrictions and then proceed with the estimation of NG. Our discussion in this section will be in the context of slow roll in ation. and coupling strengths of e ective theory We start with the restriction on , which we saw in the previous section sets the most optimistic suppression scale for higher-dimensional interactions relevant to NG. We imagine that roughly represents the mass scale of heavy particles that have been integrated out to give the e ective non-renormalizable couplings we need between the gauge sector and in aton. We can therefore think of them as -mass \mediators" of the requisite e ective interactions. But in general, if such mediators couple substantively to both the in aton and to the gauge sector, they will also mediate in aton (non-renormalizable) self-interactions, roughly powers of . In order for the e ective expansion in these powers to be controlled, we should require to exceed the in ationary kinetic energy [23], 7There may in addition be direct Higgs-in aton couplings even before the Weyl-rescaling to Einstein frame, in which case the Einstein frame couplings may be modi ed from that above. However, even this modi cation would have to share similar features, namely that during in ation the Higgs mass parameter is e ectively raised to the H2-scale and in the current post-in ationary era the Higgs mass parameter is much smaller in order to t the current electroweak data. Therefore, we will not pursue this more general modi ed lagrangian, for simplicity. > q _0: (5.1) In our ensuing discussion of single- eld in ation, we will take this bound to hold. We will assume an approximate in aton shift symmetry during in ation, allowing the 4 to be only as big as the slowly-rolling kinetic energy rather than a larger scale. The potential energy of the in aton eld V ( ) gives rise to an even higher energy scale 1 V 4 , which is bigger than q _0. Approximate shift symmetry during in ation keeps this scale from spoiling the EFT expansion in higher-dimension operators, but after in ation this symmetry may be signi cantly broken and the higher scale can then a ect dynamics signi cantly. In particular, EFT with < V 4 can break down at reheating, signaling that 1 the -scale mediators can be reheated and subsequently decay. However, the NG produced and described by the controlled e ective theory during in ation are already locked in on superhorizon scales and are insensitive to the subsequent post-in ationary breakdown of the EFT. Furthermore, in theories involving large \vacuum" expectation values, non-renormalizable operators in the UV theory can become super-renormalizable (or marginal) in the IR, once some elds are set to their expectation values. There is then the danger of such e ective super-renormalizable couplings becoming strong in the IR, and outside perturbative control, or becoming e ective mass terms which are too large phenomenologically and have to be ne-tuned to be smaller. This general concern is realized in the present context, because of the large classical expectation given by H2, as well as large hHi > H within some of the interesting parameter space. We nd that these issues are avoided for su ciently small ci in (4.4) with, c i O H= q _ 0 ; which we take to hold from now on. We go into more detail on such restrictions in the next subsection. To concretely illustrate the above considerations, consider the following set up. We imagine a theory, with a cuto interact directly. Thus a term like 102 (@ ) H H is absent in the lagrangian. However, we 2 y 0 & V 14 > q _0, in which the in aton and Higgs do not assume the presence of a \mediator" gauge-singlet particle with mass m talks to both the in aton and Higgs separately via the terms, q _0, which will lead to a controlled e ective theory expansion. These parameters are reproduced naturally if we take, H; 0 1 V 4 1=4q _0; m q _0 in i O H= q _ 0 1 0 Then below m , we can integrate out to write an e ective coupling between the in aton Now in the previous paragraphs we have stated that the choice of & q _ 0 and (5.2) (5.3) (5.4) HJEP05(218) e l e l g n isFh-0.005 -0.010 5 5 10 20 50 HJEP05(218) Slow-roll description (B.17) with c2 = p _ m=1.9H m=2.2H m=1.6H Double exchange diagram As derived in appendix B in (B.31), i 2 32 where, A( ) and s( ) are mass dependent coe cients: A( ) = sin( 4 + i 2 ); and s( ) can be represented by the integral, s( ) = R01 dxx2 e ixJ+(x)x3=2+i where, J+(x) is a somewhat complicated function given in (B.23). We exemplify the strength of NG in table 3 for the benchmark values, 2 = 0:2; h = 0:5. We illustrate the momentum dependence of F double in h gure 9 and 10. In the special case of single- eld mass j 0.2 0.0 3 2 1 -1 -2 e l b u o dFh 0 k1 k3 k1 k3 m=1.6H m=1.9H mass mass j j h slow-roll in ation, using lagrangian (5.6), F double takes an identical form to (B.31) except h the coupling constants are now di erent (B.29), k3 k1 3=2 i k3 2k1 + ( ! ) : (7.7) The strength of the NG then, for the same set of benchmark values, c2 = p _ q _0, is shown in table 4. The shape dependence is identical to gures 9 and 10, so H , h = 2H_20 , 0 not shown explicitly. 7.3 Triple exchange diagram The triple exchange diagram has been calculated in [61], but we include it here for completeness and comparison to the other diagrams. As derived in appendix B in (B.35), h where, A( ) is the same coe cient as introduced above and t( ) = R01 dxx4 J+(x)2x 23 +i . We exemplify the strength of NG below for the benchmark values, 2 = 0:2 and h = 0:5 in table 5. We illustrate the momentum dependence of F triple in gure 11 and 12. In the special case of single- eld slow-roll in ation, using lagrangian (5.6), F triple takes an identical form except the coupling constants are now di erent (B.34), h The strength of the NG then, for the same set of benchmark values, c2 = p _ q _0, is shown in table 6. The shape dependence is identical to gures 11 and 12, so not shown explicitly. k3 2 3 k1 ) (7.8) h ! ) : (7.9) H , h = 2H_20 , 0 5 10 20 50 lep -2 itr h F 0 -1 -3 -4 -5 k1 k3 h k1 k3 mass 1.6 H 1.9 H 2.2 H j h f triple j mass f single j Detailed form of NG mediated by Z To discuss the form of F for NG mediated by Z, we again rst focus on the Goldstone effective description as before, and specialize to the single- eld slow-roll description following that. Since the triple exchange diagram is too small to make any observable contribution we will restrict ourselves to single and double exchange diagrams. The Goldstone e ective lagrangian needed for this case is given by (6.32) which we rewrite, k^3 and k^1, 2 Im(d1)mZ v _ cZ0 2 f 2 cZ (8.1) In this case in the squeezed limit, F (k1; k2; k3) is a function of kk31 and also the angle between F = f ( ) + f ( ) k1 k1 sin2 = k^3 k^1. We also see that F falls faster with kk31 . The angle dependence, in principle, gives an important handle to determine the spin-1 nature of Z. Recently in [25] it was analyzed to what extent future galaxy surveys can constrain mass and spin. A forecast using 21-cm cosmology would also be important and possibly more constraining. Now we give the expressions for f ( ) for the single exchange diagram, leaving the details for the appendix D. The computation of double exchange diagram will not be performed in this paper, however using the mixed propagator formalism [61] it can be done. Here, we will only give some reasonable estimates. 8.1 Single exchange diagram As derived in appendix D in (D.6), Z v 2 ( 2 2i )(1 i sinh( i cosh( ) k1 +( ! We illustrate the strength of NG, for the parameter choices, = 8H in table 7. We see the strengths are quite weak, hence 21-cm cosmology is critical if we are to see NG due to the single exchange diagram. Note that even an imprecise measurement should be readily distinguishable from scalar-mediated NG and (8.2) (8.3) ! ) ; NG purely due to the in ationary dynamics (analytic in kk31 ), due to the non-trivial angular dependence. We now discuss single- eld slow-roll in ation. The relevant lagrangian for a nonnegligible Z-mediated signal arises when the associated Higgs scalar h is heavy enough that its on-shell propagation is Boltzmann suppressed, but can be integrated out to yield new Z vertices, as in (5.21). It has an identical structure to the Goldstone lagrangian (8.1) above, as shown in D. Hence F can be obtained just by the replacement, vmZ 2 ! 1;Z 2 m2 : We see for 1;Z = 1; 2 = 1; mh = 3H we have roughly the same strength of NG as the e ective Goldstone theory. However, we get parametrically bigger NG in both e ective theories from the double exchange diagram in gure 3, which we now discuss. Double exchange diagram As we mentioned above, in this paper we will give only an estimate of the double exchange diagram. As we have explained in section 3, in the squeezed limit diagrams factorize into contributions from hard and soft processes. This means in gure 3(b), the Z propagator having hard momenta k2 is expected to be a function of O(1) (in Hubble units). In that approximation the diagram then has the same topology as the single exchange diagram. However, as can be seen from the lagrangian (8.1), the parametric strength of the diagram goes like 2 f 2 2 ; (8.5) (8.4) which has the enhancement by f2 . Thus, while we saw that the single-exchange contri2 bution was at best marginally detectable in the future, the double-exchange contribution should be much more promising in magnitude for 5{10H, v 2 3H, with fZ 0:1{1. We leave a precise calculation of this for later work, to hopefully con rm this expectation. Moving to the case of single- eld slow-roll in ation, from (5.21) arising from integrating out the associated heavy h, we see that the double-exchange diagram is parametrically This should yield a weak but detectable signal. enhanced over single-exchange by a factor of Hvm_2h0 , so that fZ 9 Concluding remarks and future directions 0:01 for v q _0; mh 3H. Cosmological Collider Physics builds on the distinctive non-analytic momentum dependence of primordial NG mediated by particles with masses m H, in contrast to the analytic dependence of NG due purely to the in ationary dynamics, driven by elds with m H. In this paper, we focused on the question of whether gauge-theories with such ultra-high H mass scales could be detected by this means, since such theories are obviously very highly motivated. If the gauge symmetry is unbroken during in ation, gaugecharged states can only a ect primordial NG via very small loop-level e ects, di cult to observe. However, we showed that when the gauge-symmetry is (partially) Higgsed, the Z Goldstone EFT Slow-roll Models with 5H with 10H with 60H 1{10 0:1{1 0:1{1 0:01{0:1 0:01{0:1 0:001{0:01 Higgs-type spin-0 and Z-type spin-1 bosons can contribute at tree level to potentially observable NG. The simplest e ective vertices one can write connecting the gauge-Higgs states q _ 0 to the in aton so as to mediate NG are non-renormalizable, suppressed at least by powers of the cuto of the in ationary EFT, , representing the threshold of even heavier physics that has been integrated out. The largest NG will then come by considering the lowest consistent . We studied these NG within two e ective descriptions of the in ationary dynamics: a) generic slow-roll in ation models, and b) the e ective Goldstone description of in aton quantum uctuations. In slow-roll, the minimal cuto is given by the scale of kinetic energy of the rolling in aton eld, 60H. The e ective Goldstone description is more agnostic about in ationary dynamics, treating this as a given classical background process, in which case can be as low as a few H. Of course, the detailed strengths of NG, F , that we get in the two cases are model-parameter dependent, but we can brie y summarize the results in sections 7 and 8 in table 8. The dimensionless bispectrum F (see (2.17), (2.18)) given above is the maximum value taken in the squeezed regime. Based on the above table, several remarks are in order. While the above choices for EFT cuto s lead to an observable strength of NG, we cannot make the cuto s much bigger, since the NG falls rapidly as a function of squeezing and the observable precision is limited by cosmic variance, F 10 4{10 3, (1.3). The scale of Higgsing, v, is also relevant to our theoretical control. Higgsing obviously relaxes the tight constraints of gauge invariance, allowing tree-level NG. But there are non-trivial constraints of the gauge structure following from having to expand observables in powers of v= . In the UV limit v , the constraints of gauge-invariance disappear altogether. To stay in theoretic control, we have chosen v . 31 in our studies. We have used e ective non-renormalizable vertices for this paper, but it is obviously of great interest and importance to seek a more UV-complete level of theoretical description to have greater con dence in the opportunity to detect gauge theory states in NG. We see that the strength of NG is bigger when it is mediated by h's compared to mediation by Z's. Furthermore, if cosmological collider physics turns out to be in a purely gaugetheoretic domain, then we would not see any states with spin > 1, and their associated angular dependences. Spin > 2 mediated NG would signal a breakdown of point-particle eld theories, perhaps signaling the onset of string theoretic structure. On the other hand, observing spins 0; 1 only, with stronger spin-0 signals, would give strong evidence for the structure studied above. While the (NM)SM gives only one h and one Z, extensions of it (for example, even just some colored scalars) or whole new gauge sectors are capable of giving multiple h=Z-type states to observe. We have argued that a strong possibility for mgauge theory H is that they arise via a \heavy-lifting" mechanism from much lower-scale gauge theories in the current era. If these gauge theories are already seen at lower-scale terrestrial experiments, then the renormalization group allows us to predict expected mass ratios in NG. In principle, such corroboration would provide spectacular evidence for the large range of validity of such gauge theories, and the absence of intervening (coupled) states. However, we cannot hope to get a very precise measurement of such mass ratios, given cosmic variance. But if we are ever in the position to predict even a few such ratios, modestly precise measurements in NG would still be compelling. Alternatively, of course, we may discover wholly unexpected gauge-structure within the NG, at least dimly seen. There are multiple future directions which remain to pursue. There is obviously the need for an explicit calculation of the double-exchange diagram involving Z-type particles which would provide a check for our estimates. Cosmological correlations derived from in ationary expansion are famously nearly spatially scale-invariant. But in large regimes of slow-roll in ation or in the Goldstone description, the correlators are actually nearly spatially conformally invariant, that is they are close to the isometries of dS spacetime. In this paper, we have assumed this regime of in ationary dynamics. But it is possible to relax this assumption of approximate conformal invariance, and just keep approximate scale invariance, for example allowing a small speed of in aton uctuations, cs 1, which can give rise to larger NG [45, 63, 64], even allowing us to probe loop e ects of charged states. This remains to be explored. There is also the generic question of how e ciently we can use NG templates to look for simultaneous presence of spin-0 and spin-1 particles, with a \background" of in ationary NG as well as late-time e ects. Recent preliminary studies in these directions appear in [25, 62] which suggest that some of the stronger signals we describe above would be visible with reasonable precision. We can view the heavy-lifting mechanism as leveraging un-naturalness, by noting that the low-dimension Higgs mass term of elementary Higgs elds is very \unstable" to curvature-related corrections. In that sense, con rming heavy-lifting of an unnatural gauge-Higgs theory, such as the (NM)SM, would be a strong sign that naturalness is massively violated in Nature. Of course, the validity of naturalness is one of the burning debates and concerns within fundamental physics. But it is also possible that terrestrial experiments show us a natural theory, such as a supersymmetric gauge theory. One can then consider the possibility of heavy-lifting of such a natural theory. Depending on the nature of supersymmetry-breaking it is possible that the lifted gauge theory exhibits a different pattern of supersymmetry soft breaking and associated Higgsing than the unlifted theory in the current era. We leave an investigation of supersymmetric gauge-Higgs theory for the future. We have seen that invaluable information on the gauge-theoretic structure of the laws of nature can be imprinted on cosmological NG, but we have also seen that these signals are extremely weak given cosmic variance. To have any chance of seeing and deciphering such exciting physics will require pushing experimental precision and understanding of systematic uncertainties to the their limits. Heavy-lifting indeed! 2 2 = 0: This can be solved in terms of Hankel (or equivalently, Bessel) functions. After Fourier transforming to ~k-space, we can write a general classical solution as, ( ; ~k) = c1( ) 23 Hi(1)( k ) + c2( . As usual, to canonically quantize the theory, we elevate the coe cients c1; c2 to linear The quantum eld thereby has the form, combinations of creation and destruction operators, a~y ; a~k, on the Bunch-Davies vacuum. k Acknowledgments The authors would like to thank Nima Arkani-Hamed, Julian Munoz, Marc Kamionkowski, David E. Kaplan for helpful discussions. This research was supported in part by the NSF under Grant No. PHY-1620074 and by the Maryland Center for Fundamental Physics (MCFP). A Scalar elds in dS space The metric for the Poincare patch of dS spacetime in Hubble units can be written as ! 1 lim fk( ) = ( r 1 eik : 2k d 2 + d~x2 2 : A.1 Massive elds We want to get the mode functions for a quantum eld in dS. This can be obtained by rst solving the classical equation of motion and then by canonically quantizing the theory. Let us start by writing the equation of motion, ( ; ~k) = fk( )a~y + fk( )a ~k; k where the mode functions, fk( ) and fk( ) (or equivalently, the linear combinations referred to above), are determined as follows. We rst nd the conjugate momentum demand, [ ( ; ~x); ( ; ~y)] = i 3(~x Wronskian condition on the mode functions at ~y) and [a~k; ay ] = (2 )3 3(~k ~k0 1, ~k0). This gives the fk( )fk0 ( ) fk0 ( )fk( ) = i 2: To impose the Bunch Davies vacuum we demand fk( ) / eik , and using the Wronskian condition (A.6) we can also x the normalization of fk( ) up to a phase. In summary, we demand (A.1) (A.2) (A.3) (A.4) (A.5) (A.6) (A.7) p 2 p 2 fk( ) = (+ie i =4) e =2( ) 23 Hi(1)( k ): This can be satis ed by choosing and Here, we have introduced some phase factors which are just conventions, which drop out HJEP05(218) when we calculate propagators. We note that in (2.2) we have both time and anti-time ordered expressions appearing. Since a propagator involves two mode functions, we can have a total of four types of propagators depending on the mode functions coming from either the time or anti-time ordered part. We denote time(anti-time) ordering by a +( ) sign. As an example, a propagator G+ (k; ; 0) means the mode function with argument ( 0) is coming from time(anti-time) ordering. Similarly, G++(k; ; 0) means both the mode functions are coming from time ordering. Thus we can write G++(k; ; 0) = fk( 0)fk( ) ( 0 ) + fk( )fk( 0) ( G+ (k; ; 0) = fk( 0)fk( ) ; G +(k; ; 0) = fk( )fk( 0) ; G (k; ; 0) = fk( 0)fk( ) ( 0) + fk( )fk( 0) ( 0 0) ; ) : Among these four, G+ ; G++ are conjugates of G +; G respectively, and so we only have two independent propagators. A.2 In aton mode functions Mode functions for massless elds, in particular the in aton, follow by using = 3i=2 in (A.8) and (A.9), which gives fk( ) = fk( ) = (1 ik )eik p2k3 (1 + ik )e ik p2k3 ; : A.3 Some useful relations for diagrammatic calculations For later use we also note a few relations involving Hankel and hypergeometric functions that arise upon evaluating the Feynman diagrams for the NG correlators of interest. We can write the following integral involving Hankel functions in terms of a hypergeometric (A.8) (A.9) (A.10) (A.11) (n+1+i ) (n+1 i )2F1(n+1+i ; n+1 i ; n+3=2; (1 p)=2); (n+1 i ) (n+1+i )2F1(n+1 i ; n+1+i ; n+3=2; (1 p)=2): It is useful later to approximate these expressions for large p by using limiting forms of Hankel and hypergeometric functions, e =2Hi(2)(z) = =2Hi(1)(z) = i (i )(z=2) i e =2 + ( i )(z=2)+i e =2 ; ( i )(z=2)+i e =2 + (+i )(z=2) i e =2 : We also need large negative argument expansion of hypergeometric function, 2F1(a; b; c; z) = (b) (c a) (c) ( z) a + (c) (a (a) (c ( z) b: function (valid for real and i < 12 ), e =2 Z 1 = p = p 0 ( i=2)n 2 (n+3=2) =2 Z 1 0 dxxne ipxHi(2)(x) dxxne+ipxHi(1)(x) B.1 NG due to h exchange Calculation of single exchange diagram We will use the in-in formula (2.2) to calculate NG due to the single exchange diagram which is depicted in gure 3(a). We begin by reviewing this calculation in the context of single- eld slow-roll in ation, as originally performed in [21]. The relevant terms in the lagrangian (5.6) for such a diagram are L 2 _0 : Note in (2.2) we have both time and anti-time ordering. Thus each vertex can contribute either from time or anti-time ordering. So an in-in diagram with n vertices gives rise to 2n subdiagrams. These subdiagrams di er in the type of propagators used for the massive particle and in atons. For example if both the vertices are coming from time ordering, we should use G++ for the massive propagator as de ned in (A.10). We call the subdiagram containing G++ to be I++. Thus for the single exchange diagram we have four subdiagrams which we denote by I++; I+ ; I +; I depending on which kind of massive propagator has been used. However, to compute the entire three point function due to single exchange diagram, we have to consider only two subdiagrams, since the other two are related by complex conjugation. For example, we will calculate only I and I+ which are related to I++ and I + respectively by complex conjugation. We sum all four contributions to (A.12) (A.13) (A.14) (A.15) (A.16) (B.1) get the nal answer. To clarify the above comments, we write the expressions for four subdiagrams schematically, I / ( i)( i) Z d Z d 0 4 0 4 g (k3; 0)g~ (k1; k2; )G (k3; ; 0): (B.2) The prefactors i arise depending on whether we use e i R Hdt for time ordering or e+i R Hdt for anti-time ordering. g ; g~ are in aton bulk-boundary propagators (which we de ne below); and G are bulk-bulk propagators (A.10) for h. B.1.1 Calculation of I+ Let us start with I+ diagram. This diagram factorizes into a product of two integrals with one coming from time ordering and another from anti time ordering. Anti-time ordered contribution. We rst calculate in aton contribution using in aton mode function (A.11), + i 1 2 : g (k; ) h _( ; ~k) ( 0 ! 0; ~k)i = to write the anti time ordered contribution as Anti-time Ordered Contribution = (+i) Using the mode functions (A.9) and relation (A.13) we get Anti-time Ordered Contribution = Time ordered contribution. Let us rst calculate the in aton contribution again. Now we have to do a little more work since based on (B.1) we see that we have to nd the contraction which can be schematically written as h j(@ )2i. Writing g~+(k1; k2; ) as, g~+(k1; k2; ) we get, using in aton mode function (A.11), g~+(k1; k2; ) = 1 2k13 + 2 ( ik1i)( ik2i) 1 2k23 (1 eik12 k k 2 2 4 1 2 ik1 )(1 2k132k23 ik2 ) eik12 ; where we have de ned, k12 = k1 + k2. We can simplify this by removing some -dependent factors by writing the above as an operator D acting on eik12 , where D k k 1 2@k12 + ( ~k1 ~k2)(1 2 2 2 Then, g~+(k1; k2; ) = 2 4k13k23 Deik12 : (B.3) (B.4) (B.5) (B.6) (B.7) (B.8) (B.9) Thus the time ordered contribution looks like Time Ordered Contribution = ( i) 1 4k13k23 D 1 Z 0 d eik12 fk3( ): 2 The integral can be evaluated using (A.12) and (A.8) to get a hypergeometric function. Since we will be interested in squeezed limit, k3 k1; k2, we can expand the answer using (A.16). We then get Time Ordered Contribution 1 = 4p2k13k23pk3 D The action of D simpli es in the squeezed limit, ( 2i ) (1=2+i ) p 1=2 i (2i ) (1=2 i ) p 1=2+i (1=2 i ) 2 (1=2+i ) 2 HJEP05(218) 8 Dk12 = ( 2)k122+ ; (B.10) (B.11) : (B.12) using which, 1 8 2 Thus we have I + I+ 2 = _20 64k12k22k32 1 + ( ! ): Time Ordered Contribution = 4p2k13k23pk3 (3=2+i )(5=2+i ) ( 2i ) (1=2+i ) 2 (1=2 i ) k12 2k3 k12 Now we are ready to put together both the contributions: 1 I+ = _20 64k12k22k32 (1=2 + i )2 ( 2i )(3=2 + i )(5=2 + i ) 1=2 i k1 k3 +( ! ): (B.13) 1=2 i + ( ! ): (B.14) B.1.2 Calculation of I I diagram is in general complicated since it does not factorize into and 0 integrals. But we can still calculate the nonanalytic terms in k3 in the squeezed limit. This is because in the squeezed limit, 0 integral contributes when 0 integral is dominant when O( k112 ). Thus when k3 (A.10) drops out and the integral approximately factorizes. The 0 integral then is identical to what we had for I+ ; whereas the only change for integral is that k12 ! k12. O( k13 ), whereas the contribution of k12, one of the step functions 1 2 + i 2 + i + i k1 k3 12 i 1 ei ( 12 i ) (B.15) h (~k1) (~k2) (~k3)i = _0 16k12k22k32 2 2 1 The full three point function can be written as a sum of I++; I+ ; I +; I . This gives (1+i sinh( This gives F as a function of k3 as de ned in (2.18) to be 1 4 1 2 2 2 2 2 k1 3 2 3 2 1 2 5 2 5 2 3 2 h (1+i sinh( For the case of the Goldstone e ective description of in ation, with f , the relevant terms in the lagrangian are given by (6.28). The calculation of F single follows identical steps as above with the di erence that the operator D = k k 1 2 k12 h 52 + i ! 12 12 + i 32 + i . Hence the nal answer into, the replacement reads (taking d2 = 1), h (1+i sinh( Calculation of double exchange diagram We see from the gure 3(a) that there is a quadratic mixing between in aton and Higgs eld h. For numerical simpli cation one can de ne a mixed propagator which captures this mixing [61]. Using this mixed propagator, we then calculate double exchange diagram numerically. We rst focus on single eld slow roll in ation. B.2.1 Mixed propagators A mixed propagator is characterized by the 3 momentum (say k) owing through the line and bulk time coordinate . We have to sum over all time instants 0 where the mixing occurs. The bulk time coordinate can be part of either time or anti-time ordering. Let us rst consider the case where comes from time ordering, and denote that mixed propagator by, D+( ; k). Then we have two possibilities, (a) 0 comes also from time ordering, in which case the we can write the contribution of the mixing part of the entire diagram as, k3 12 +i k1 k3 32 +i k1 k3 32 +i k1 (B.16) (B.17) (B.18) ) : I+ = Z 0 d 0 1 0 4 g+(k; 0) ( 0)f ( )f ( 0) + ( 0 )f ( 0)f ( ) ; (B.19) and, (b) 0 comes from anti-time ordering, for which we get, I = + g (k; 0)f ( 0)f ( ): (B.20) The overall signs are due to again e i R dtH and we have omitted a factor of +i for simplicity which we will restore in the nal expression for the mixed propagator. We can then rewrite I+ as f ( ) 0 4 g+(k; 0)f ( 0): (B.21) So the whole contribution of the mixed propagator when comes from time ordering is D+( ; k) = (+i)( 2)(I+ +I ) = (+i)( 2) 2iIm(I )+f ( ) 0 4 g+(k; 0)f ( 0) f ( ) 0 4 g+(k; 0)f ( 0) : We have restored the factor of +i and also put the mixing vertex Using the mode functions for in atons (A.11) and massive scalars (A.8) and (A.9), and also the relations, (A.13), (A.12), we can evaluate the mixed propagator analytically. For convenience, we de ne the function J+( k) = 8k3 D+( ; k): J+(x) = x 2 2 sech( 3 hp )e =2 Hi(2)(x)ei =4 + H(1i) (x)e i =4 2pxHi(1)(x) (csch( 2pxHi(2)(x) (csch( )F( ; x) + (1 coth( )F( ; x) (1 + coth( ; x)) where x = k , and F( ; x) is given in terms of hypergeometric function 2F2, F( ; x) = 1 1 x i ) 2 + i 2 2F2 1 2 1 2 3 2 i ; i ; i ; 1 2i ; 2ix : (B.24) We will be using the small argument limit of J+(x), 3 J+(x) = A( )x 2 (x=2)i + A( 3 )x 2 (x=2) i ; where A( ) = ) ( i ) sin( 4 + i 2 ). B.2.2 Three-point function After we incorporate appropriate internal lines into mixed propagators, the double exchange diagram e ectively contains a single vertex. Thus we only have two diagrams to calculate, with one of them being the conjugate of the other. Let us start with the time ordered contribution. Since the lagrangian (5.6) contains a term of the form _ 0 _h2 then we get + i 0 Z d 4 k 2 2 1 2k13 eik1 D+( ; k2)D+( ; k3): (B.26) (B.22) (B.23) (B.25) Once again we will be interested in the limit k3 k1, so using the small argument Z d2 eik1 D+( ; k1)D+( ; k3) expansion of J+, +i _ 0 1 2k1 2 2 2 2 2 64k13k33 2 _0 2 2 16 k3 k1 k3 k1 3=2 k3 2k1 3=2 k3 2k1 k1 2 2 _0 64k13k33 A( )s( ) k3 k1 3=2 k3 2k1 k1 3=2 k3 2k1 i ! ; (B.27) where we have de ned s( ) = R01 dxx2 e ixJ+(x)x3=2+i . The full three point function then becomes, after adding the anti-time ordered contribution, )s ( + ( ! ): (B.28) This gives F as a function of kk31 after summing over permutation k1 $ k2, A ( )s ( + ( ! ): (B.29) terms of the function, s( ) = R01 dxx2 e ixI+(x)x3=2 we have, For m < 32H we get an appropriately modi ed version of the above expression. In 2 2 2 8 k1 k3 3=2+ . F double = h B( )Im(s( )) (B.30) where B( ) 2 +1 ( )p2= sec( kept a contribution of the form For the Goldstone description we see from (6.28) that the functional form for F double h is identical to above. For the overall coe cient we change, 22 ! 21 2( 2v)2f 2, and hence, ) sin( =4 =2). In deriving the above we have not i 2 32 F double = 2( 2vf )2 h A ( )s ( k3 k1 3=2 k3 2k1 + ( ! ): (B.31) B.3 Calculation of triple exchange diagram Using the mixed propagator, the triple exchange diagram can also be calculated in an identical manner [61]. Using the cubic Higgs vertex, h2v h3 we can write the time ordered diagram as, ( i) hv Z d 2 4 D+( ; k1)D+( ; k2)D+( ; k3): (B.32) Again in the limit k3 k1 we can use the small argument expansion of J+(x), ( i) hv Z d 2 = ( i) hv 2 4 D+( ; k1)D+( ; k2)D+( ; k3) 3 3 2 83k13k33 A( )t( ) 3 k3 2 k3 k1 2k1 3 k3 2 k3 k1 2k1 i ! ; (B.33) ) ( i ) sin( 4 + i 2 ) and t( ) = R01 dxx4 J+(x)2x 32 +i . After adding the anti-time ordered contribution and permutation k1 $ k2 we get h A ( )t ( k3 2k1 + ( For the Goldstone description we see from (6.28) that the functional form for F triple is identical to above. For the overall coe cient we change, 32 ! ( 2v)3, and hence, h A ( )t ( k1 k3 2k1 + ( 128 d x 4 p g 4 1 2 m2Z2 ; r F = m2Z : r Z = 0: C Massive vector elds in dS space Here we will derive mode functions for massive spin-1 elds [24], which will be useful in the next appendix in computing NG mediated by Z-type particles. C.1 Mode functions in momentum space We start with the lagrangian, where F = r Z r Z . Variation of the action yields the equation of motion, Taking the divergence of both sides and also using the fact that r r F / R F = 0, we get Mode functions for Z are then obtained by solving (C.2) and (C.3). The NG correlators involve mixing the in aton with the Z, which is constrained by the spatial rotation and translation invariance. Therefore only the longitudinal Z polarization, which is a spatial scalar, can appear since the in aton is obviously scalar. We are interested in the mode functions for this degree of freedom. It is shared between the timelike component Z and the longitudinal spatial component Zlong = Z~ k^ with k^ being a unit vector pointing in the direction of propagation. Fourier transforming from ( ; ~x) to ( ; ~k) coordinates, the constraint equation reads, 2 From (C.2) and (C.4) we get the equation of motion for the component Z , 2 (m2 + 2) 2 Z = 0: i ): (B.34) h ): (B.35) (C.1) (C.2) (C.3) (C.4) (C.5) This is almost identical to the equation of motion for the scalar (A.3). The solutions are again given in terms of Hankel functions, but with After we obtain the mode function for Z , that for Zlong is simply obtained from the constraint equation (C.4). In parallel with the case of scalars, the quantum eld is obtained by elevating the free superposition coe cients in the general classical solution to linear combinations of creation and destruction operators on the Bunch-Davies vacuum, 2 = m2 + 2 94 = m2 1 4 Z ( ; ~k) = hk;0( )b~y + hk;0( )b ~k; k (C.6) (D.1) where the mode functions are hk;0( ) = N e 2 ( hk;l( ) = N e 2 2k ) 23 Hi(1)( k ) 1 ) 2 For single- eld slow-roll in ation, the lagrangian involving the in aton and spin-1 Z relevant for single-exchange (diagram (a) in gure 3) is given by (5.21) to be Hi(1)( k )+k Hi(1+)1( k ) k H(1) ( k ) ; (C.8) i 1 i p k . In the above b and by are annihilation and creation operators for the ik1i 2k13 3 2 longitudinal degree of freedom for spin-1. D NG due to Z exchange L = _Z + _ 0 where contribution to future work, although we have estimated its strength and it seems readily = 1m;Z2h 2 . As discussed earlier, we defer the computation of the double-exchange detectable in future measurements. We have also shown earlier that the triple-exchange contribution is suppressed and can be neglected. We now parallel the steps taken in the calculation of NG for the case of single-exchange of a scalar h. We start with I + diagram. In this case the time ordered and anti-time ordered contribution factorize, and we can evaluate them separately. 3 2 f1 ; 2p +ipf1 ; 2p +f2 ; 2p ipf2 ; 2p f3 ; 2p +ipf3 ; 2p 2 2 Time ordered contribution. (+i ) k32 02 2k33 Anti-time ordered contribution. i 0 = _0 16k2k12k323 mZ 4 p 2 k 2 2 2 2k23 e i4 (k^1 k^3) Z 0 d 1 1 1 1 2 1 2 eik3 0 hk;0( ) = 1 p 3 4 2 mZ k32 De ning p = kk13 , 3 2 + i 3 2 : (D.2) e ik2 e ik1 (1+ik1 )k^3ihk;l( ) (D.3) We multiply the above two contributions and also sum over I++; I ; I+ diagrams. Finally we sum over permutations ~k1 $ ~k2 to get, Z 2 3 2 2 cosh( ( 2 2i )(1 i sinh( k3 52 +i k1 = k^1 k^3 and we have used large negative argument expansion of hypergeometric For the Goldstone description, we see from (6.32) that the functional form of F single is identical to the above. The overall coe cient is changed to ! 2 vmZ (taking Im(d1) = 1). where function. Hence we get Z (D.4) (D.5) (D.6) HJEP05(218) where di erent integrals involving Hankel functions have been evaluated using (A.13), f1(n; p) = f2(n; p) = (+i) f3(n; p) = ( i) (n + 3=2) p 1 (+i=2)n (n + 3=2) p (+i=2)n (n + 3=2) p 1 1 (n + 1 i ) (n + 1 + i ) (n + 2 i ) (n + i ) (n + 2 + i ) (n i ) 2F1(n + i ; n + 2 i ; n + 3=2; (1 p)=2) ; 2F1(n + 2 + i ; n i ; n + 3=2; (1 p)=2) : 2F1(n + 1 + i ; n + 1 i ; n + 3=2; (1 p)=2) ; v 2 2 1 2 3 2 2 2 cosh( ( 2 2i )(1 i sinh( k1 This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. [1] D. Baumann, In ation, in Physics of the large and the small, proceedings of The Theoretical Advanced Study Institute in Elementary Particle Physics (TASI 09), Boulder, Colorado, [arXiv:0907.5424] [INSPIRE]. [2] Planck collaboration, P.A.R. Ade et al., Planck 2015 results. XX. 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Soubhik Kumar, Raman Sundrum. Heavy-lifting of gauge theories by cosmic inflation, Journal of High Energy Physics, 2018, 11, DOI: 10.1007/JHEP05(2018)011