Minimal Signatures of the Standard Model in Non-Gaussianities

PHYSICAL REVIEW D 101, 023519 (2020) Minimal signatures of the standard model in non-Gaussianities Anson Hook* Maryland Center for Fundamental Physics, University of Maryland, College Park, Maryland 20742, USA † ‡ Junwu Huang and Davide Racco Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada (Received 19 September 2019; published 22 January 2020) We show that the leading coupling between a shift symmetric inflaton and the standard model fermions leads to an induced electroweak symmetry breaking due to particle production during inflation, and as a result, a unique oscillating feature in non-Gaussianities. In this one parameter model, the enhanced production of standard model fermions dynamically generates a new electroweak symmetry breaking minimum, where the Higgs field classically rolls to. The production of fermions stops when the Higgs expectation value and hence the fermion masses become too large, suppressing fermion production. The balance between the above-mentioned effects gives the standard model fermions masses that are uniquely determined by their couplings to the inflaton. In particular, the heaviest standard model fermion, the top quark, can produce a distinct cosmological collider physics signature characterized by a one-to-one relation between amplitude and frequency of the oscillating signal, which is observable at future 21-cm surveys. DOI: 10.1103/PhysRevD.101.023519 I. INTRODUCTION AND SUMMARY induces electroweak symmetry breaking during inflation, whereas a large expectation value of the Higgs field Cosmological collider physics [1–8] provides an oppor- increases the masses of the SM fermions and suppresses tunity to search for new heavy particles that are not their production. This interplay results in a very predictive accessible at particle colliders. These heavy particles, scenario where the strength of the oscillating signature in produced through their interactions with the inflaton field, non-Gaussianities is directly tied to the period of oscillation. can accumulate to large enough densities and affect the We assume that the fermions in the SM couple to the bispectrum of density perturbations, leaving observable inflaton through the lowest dimensional operator respecting signatures in the cosmic microwave background and large the shift symmetry of the inflaton [11,12], scale structure of the Universe [9,10]. Given the exciting potential of this approach, it is worth ∂ ϕ “ L ⊃ μ F†σ¯ μF fc†σ¯ μfc y HFfc; asking What are the minimal signatures of the standard Λ ð þ Þþ f ð1Þ model (SM) in the context of cosmological collider phys- f ics?” Of course the absolutely minimal signature is nothing v h where ϕ is the inflaton, H 0; pþffiffi is the Higgs doublet, if the SM does not couple directly to the inflaton. In this ¼ð 2 Þ and F Q, L and fc uc;dc;ec are left- and right-handed article, we explore the consequences of adding a single ¼ ¼ fermions in the SM in two-component notation.1 When coupling to the SM fermions, the dimension five shift ϕ_ ≠ 0 symmetric coupling between the inflaton and the SM h i , this coupling leads to the production of fermions during inflation whose effective number density is nf ∼ fermions. Somewhat surprisingly, this single coupling by ffiffiffi 2 2 ϕ_ p itself leads to an interplay between the dynamics of the SM mfλ exp −πm =λfH , where λf , mf yfv= 2, H is f ½ f ¼ Λf ¼ fermions and the Higgs: The SM fermion production pffiffiffi the Hubble parameter during inflation, and v= 2 ¼hHi is the vacuum expectation value (vev) of the Higgs field. It is well known that a high density of particles can *[email protected] † [email protected] change the properties of a scalar potential. Thermal effects ‡ [email protected] favor symmetry restoration of the Higgs for temperatures above the electroweak phase transition [13]. On the other Published by the American Physical Society under the terms of hand, chemical potentials favor symmetry breaking and in the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, 1We neglect the typically smaller dimension five anomalous and DOI. Funded by SCOAP3. couplings to the gauge bosons. 2470-0010=2020=101(2)=023519(6) 023519-1 Published by the American Physical Society ANSON HOOK, JUNWU HUANG, and DAVIDE RACCO PHYS. REV. D 101, 023519 (2020) the context of the Higgs potential could prevent symmetry restoration even at high temperatures [14]. The coupling in Eq. (1) is very similar to a chemical potential. In fact, if the plus sign were a minus, it would be the familiar chemical potential for fermion number. As such, it is unsurprising to find that the effect of this coupling is to generate a correction to the Higgs potential that favors symmetry breaking of the form y2 πy2h2 δV − f λ2h2 − f : FIG. 2. Contribution to the inflaton bispectrum from a loop of h ¼ 2 f exp ð2Þ f τ 2π 2λfH SM fermions. Two fermions are produced at a time 3 by the interaction with a soft inflaton leg δϕ, and annihilate later at τ1 ∼ τ2 into two hard inflaton legs with k1, k2 ≫ k3. The For λf >H≫ v (v being the electroweak vacuum), EW EW nonanalytic contribution to the bispectrum is due to the time this contribution to the potential induces spontaneous break- propagation of the fermions from τ3 to τ1, τ2. ing of the electroweak symmetry, as shown in Fig. 1. However, as the Higgs vev increases, the fermion masses increase and their particle production is exponentially sup- solely determined by the inflaton coupling λf to that 2 pressed as mf ≳ λfH. Therefore, quite insensitively to any fermion. As a result, the non-Gaussian signal in the other term of the Higgs potential, e.g., the value of the quartic squeezed limit simplifies into λh, the Higgs gets a vev during inflation: pffiffiffi 4 2N P qffiffiffiffiffiffiffiffiffi fðclockÞ ≈ c ζ λ˜13=2; 1 NL 3e f ð4Þ v ∼ λfH: ð3Þ yf depending only on the size of the inflaton fermion coupling _ ˜ λf ϕ in Hubble units λf . This signal oscillates in In the SM, due to the large hierarchy between the top quark ¼ H ¼ ΛfH ˜ and the lighter leptons and quarks, this effect is determined lnðk3=k1Þ with a frequency ∼λf in the squeezed limit. This entirely by the top quark. Incidentally, such a scenario can relation between the amplitude (fðclockÞ) and the frequency only occur for fermions with an Oð1Þ Yukawa coupling, NL ˜ of the oscillating signal offers a simple cross-check of this since λf ¼ λf=H cannot be arbitrarily large. Thus, in what mechanism. Such a signature offers a direct probe of the follows, we focus on the coupling of the top quark with the induced electroweak symmetry breaking during inflation, inflaton. and could help us to shed light on the inflationary sector. The observational signature associated to this coupling is the generation of a large non-Gaussian oscillating pattern in the squeezed limit. The Feynman diagram to be calculated II. PARTICLE PRODUCTION AND is shown in Fig. 2 [11,12]. THE HIGGS POTENTIAL The interesting feature in this case is that the dynamics of In this section, we discuss particle production and how the Higgs potential ensures that the mass of the fermion f the fermion density affects the Higgs potential. To calculate that produces the largest observable non-Gaussianity is the correction to the Higgs mass term, we calculate the diagram shown in Fig. 3 that corrects the energy density of the state. In a companion paper [12] (see in particular Sec. III and Appendix A), we show in detail how to estimate and calculate these diagrams. For a single fermion flavor and color, the diagrams in Fig. 3 contribute as ZZ 0 τ τ 2 αβ α_ β_ d 1 d 2 ⃗ ⃗ Nab ¼ −yfϵ ϵ ab Gaðk1; τ1ÞGbð−k1; τ2Þ −∞ Hτ Hτ Z 1 2 d3q × ðD _ ðp⃗12; τ1; τ2ÞDabβα_ ð−p⃗21; τ1; τ2Þ ð2πÞ3 abαβ þ Dbaαβ_ ð−p⃗12; τ2; τ1ÞDbaβα_ ðp⃗21; τ2; τ1ÞÞ; ð5Þ FIG. 1. Higgs potential at zero temperature and λf ∼ 0 (blue line) and in the presence of a top quark condensate induced by ⃗ where τ denotes conformal time, p⃗12 k1 p⃗21 q⃗ λt >H>v (purple line). The Higgs field sits in the dynami- ¼ þ ¼ EW ⃗ cally generated minimum v during inflation. and jq⃗j ≫ jk1j is the internal momentum. The indices 023519-2 MINIMAL SIGNATURES OF THE STANDARD MODEL IN NON- … PHYS. REV. D 101, 023519 (2020) α; β; α_; β_ ∈ f1; 2g are spinor indices. The antisymmetric ϵ 12 tensors are defined by ϵ ¼−ϵ12 ¼1. The in-in indices a; b take values in fþ1; −1g, denoted respectively by f⊕; ⊖g to distinguishpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi them from spinor indices. The functions ⃗ 2 3 iakτ ⃗ Gaðk;τÞ¼ H =2k ð1−iakτÞe and Dabαβ_ ðk; τ1; τ2Þ are the propagator of the Higgs and the fermion fields respectively (see Sec. A.2 of [12]). By the same logic of FIG. 3. Feynman diagrams for the contribution of the top the electroweak hierarchy problem, the momentum integral fermion condensate to the Higgs potential. For notations, see the in Eq. (5) is quadratically divergent. The leading quadratic main text and [12,15,16]. divergence is absorbed into the definition of the physical Higgs mass that we observe today when λf ¼ 0 (this the Higgs potential to the exponential suppression of the operation automatically removes some of the subleading 3 ˜ fermion density. corrections in the λf expansion). In the massless limit, the There is not a clean analytic formula interpolating integral in Eq. (5) can be simplified (by exploiting proper- between the small and large mass regions and a full result ties of the Whittaker functions appearing in the fermion would need to be obtained numerically.

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